\\beta ){\\Vert }_F^2 +\\frac{{\\Vert }\\mathbf {P}^i\\cdot \\mathbf {P}_0^i{\\Vert }_F^2 }{{\\Vert }\\mathbf {P}_0^i{\\Vert }_F^2}\\Big ].$ Trace regularization: Based on Eqn.","REF with $\\lambda =1$ , we regularize the intensity of additive traces $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ and inpainting region $\\mathbf {P}$ .","The regularizer loss is denoted as: $L_{R} = \\mathbb {E}_{\\mathbf {i}\\sim \\mathcal {L}\\cup \\mathcal {S}}\\Big [{\\Vert }\\mathbf {B}{\\Vert }_F^2 +{\\Vert }\\mathbf {C}{\\Vert }_F^2 +{\\Vert }\\mathbf {T}{\\Vert }_F^2 +{\\Vert }\\mathbf {P}{\\Vert }_F^2\\Big ].$ Synthesized spoof loss: Synthesized spoof data come with ground truth spoof traces.","As a result, we are able to define a supervised pixel loss for the generator to disentangle the exact spoof traces that were added: $ L_{S} = \\mathbb {E}_{\\mathbf {i}\\sim \\mathcal {L}, \\mathbf {j}\\sim \\mathcal {S}}\\Big [{\\Vert }G(\\lceil G^{-j\\rightarrow i} \\rceil ) -\\lceil G^{j\\rightarrow i}\\rceil {\\Vert }_F^1\\Big ],$ where $G^{j\\rightarrow i}$ is the overall effect of $\\lbrace \\mathbf {P}^j,\\mathbf {I}_P^j,\\mathbf {B}^j,\\mathbf {C}^j,\\mathbf {T}^j\\rbrace $ after warping to subject $i$ , and $\\lceil \\cdot \\rceil $ is the stop_gradient operation.","Without stopping the gradient, $G^{j\\rightarrow i}$ may collapse to 0.","Depth map loss for “harder” samples: We send the “harder” synthesized spoof data to depth estimation network to improve the data diversity, and hope to increase the FAS model's generalization: $L_{\\textit {H}} = \\frac{1}{K^2}\\mathbb {E}_{i\\sim \\hat{\\mathcal {S}}}\\Big [{\\Vert }\\mathbf {M}^i-\\mathbf {M}_0^i{\\Vert }_F\\Big ],$ where $\\hat{\\mathcal {S}}$ denotes the domain of synthesized spoof faces.","Training steps and total loss: Each training iteration has 3 training steps.","In the training step 1, live faces $\\mathbf {I}_{\\textit {live}}$ and spoof faces $\\mathbf {I}_{\\textit {spoof}}$ are fed into generator $G(\\cdot )$ to disentangle the spoof traces.","The spoof traces are used to reconstruct the live counterpart $\\hat{\\mathbf {I}}_{\\textit {live}}$ and synthesize new spoof $\\hat{\\mathbf {I}}_{\\textit {spoof}}$ .","The generator is updated with respect to the depth map loss $L_{\\textit {depth}}$ , adversarial loss $L_G$ , inpainting mask loss $L_P$ , and regularizer loss $L_R$ : $L = \\alpha _1L_{\\textit {depth}} + \\alpha _2 L_{G} + \\alpha _3 L_P + \\alpha _4 L_R.$ In the training step 2, the discriminators are supervised with the adversarial loss $L_D$ to compete with the generator.","In the training step 3, $\\mathbf {I}_{\\textit {live}}$ and $\\hat{\\mathbf {I}}_{\\textit {hard}}$ are fed into the generator with the ground truth label and trace to minimize the synthesized spoof loss $L_S$ and depth map loss $L_H$ : $L = \\alpha _5 L_{S} + \\alpha _6L_H,$ where $\\alpha _1$ -$\\alpha _6$ are the weights to balance the multitask training.","To note that, we send the original live faces $\\mathbf {I}_{\\textit {live}}$ with $\\hat{\\mathbf {I}}_{\\textit {hard}}$ for a balanced mini-batch, which is important when computing the moving average in the batch normalization layer.","We execute all 3 steps in each minibatch iteration, but reduce the learning rate for discriminator step by half.","The whole training process is depicted in Alg.","REF ."],["Experiments"," In this section, we first introduce the experimental setup, and then present the results in the known, unknown, and open-set spoof scenarios, with comparisons to respective baselines.","Next, we quantitatively evaluate the spoof traces by performing a spoof medium classification, and conduct an ablation study on each design in the proposed method.","Finally, we provide visualization results on the spoof trace disentanglement, new spoof synthesis and t-SNE visualization.","Table: The evaluation on four protocols in OULU-NPU.","Bold indicates the best score in each protocol."],["Experimental Setup","Databases We conduct experiments on three major databases: Oulu-NPU [67], SiW [18], and SiW-M [19].","Oulu-NPU and SiW include print/replay attacks, while SiW-M includes 13 spoof types.","We follow all the existing testing protocols and compare with SOTA methods.","Similar to most prior works, we only use the face region for training and testing.","Evaluation metrics Two common metrics are used in this work for comparison: EER and APCER/BPCER/ACER.","EER describes the theoretical performance and predetermines the threshold for making decisions.","APCER/BPCER/ACER[68] describe the practical performance given a predetermined threshold.","For both evaluation metrics, lower value means better performance.","The threshold for APCER/BPCER/ACER is computed from either training set or validation set.","In addition, we also report the True Detection Rate (TDR) at a given False Detection Rate (FDR).","This metric describes the spoof detection rate at a strict tolerance to live errors, which is widely used to evaluate real-world systems [69].","In this work, we report TDR at FDR$=0.5\\%$ .","For TDR, the higher the better.","Parameter setting PhySTD is implemented in Tensorflow with an initial learning rate of $5e$ -5.","We train in total $150,000$ iterations with a batch size of 8, and decrease the learning rate by a ratio of 10 every $45,000$ iterations.","We initialize the weights with $[0,0.02]$ normal distribution.","$\\lbrace \\alpha _1,\\alpha _2,\\alpha _3,\\alpha _4,\\alpha _5,\\alpha _6\\rbrace $ are set to be $\\lbrace 100,5,1,1e$ -$4,10,1\\rbrace $ , and $\\beta =0.1$ .","$\\alpha _0$ is empirically determined from the training or validation set.","We use the open-source face alignment [70] and 3DMM fitting [60] to crop the face and provide 140 landmarks.","Table: The evaluation on three protocols in SiW Dataset.","We compare with the top 7 performances."],["Anti-Spoofing for Known Spoof Types","Oulu-NPU Oulu-NPU[67] is a commonly used face anti-spoofing benchmark due to its high-quality data and challenging testing protocols.","Tab.","REF shows our anti-spoofing performance on Oulu-NPU, compared with SOTA algorithms.","Our method achieves the best overall performance on this database.","Compared with our preliminary version [31], we demonstrate improvements in all 4 protocols, with significant improvement on protocol 1 and protocol 3, i.e., reducing the ACER by $63.6\\%$ and $32.1\\%$ respectively.","Compared with the SOTA, our approach achieves similar best performances on the first three protocols and outperforms the SOTA on the fourth protocol, which is the most challenging one.","To note that, in protocol 3 and protocol 4, the performances of testing camera 6 are much lower than those of cameras 1-5: the ACER for camera 6 are $6.4\\%$ and $10.2\\%$ , while the average ACER for the other cameras are $1.0\\%$ and $2.0\\%$ respectively.","Compared with other cameras, we notice that camera 6 has stronger sensor noises and our model recognizes them as unknown spoof traces, which leads to an increased false negative rate (i.e., BPCER).","How to separate sensor noises from spoof traces can be an important future research topic.","Table: The evaluation and ablation study on SiW-M Protocol I: known spoof detection.Table: The evaluation on SiW-M Protocol II: unknown spoof detection.SiW SiW[18] is another recent high-quality database.","It includes fewer capture cameras but more spoof mediums and environment variations, such as pose, illumination, and expression.","The comparison on three protocols is shown in Tab.","REF .","We outperform the previous works on the first two protocols and rank in the middle on protocol 3.","Protocol 3 aims to test the performance of unknown spoof detection, where the model is trained on one spoof attack (print or replay) and tested on the other.","As we can see from Fig.REF -REF , the traces of print and replay are significantly different, where the replay traces are more on the high-frequency part (i.e., trace component $\\textbf {T}$ ) and the print traces are more on the low-frequency part (i.e., trace component $\\textbf {S}$ ).","These pattern divergence leads to the adaption gap of our method while training on one attack and testing on the other.","SiW-M SiW-M[19] contains a large diversity of spoof types, including print, replay, 3D mask, makeup, and partial attacks.","This allows us to have a comprehensive evaluation of the proposed approach with different spoof attacks.","To use SiW-M for known spoof detection, we randomly split the data of all types into train/test set with a ratio of $60\\%$ vs. $40\\%$ , and the results are shown in Tab.","REF .","Compared to the preliminary version [31], our method outperforms on most spoof types as well as the overall EER performance by $47.9\\%$ relatively, which demonstrates the superiority of our anti-spoofing on known spoof attacks.","For experiments on SiW-M (protocol I, II, and III), we additionally report the TPR at FNR equal to $0.5\\%$ .","While EER and ACER provide the theoretical evaluation, the users in real-world applications care more about the true spoof detection rate under a given live detection error rate, and hence TPR can better reflect how well the model can detect one or a few spoof attacks in practices.","As shown in Tab.","REF , we improve the overall TDR of our preliminary version [31] by $29.5\\%$ .","Table: The evaluation on SiW-M Protocol III: openset spoof detection.Figure: Examples of each spoof trace components.","(a) the input sample faces.","(b) 𝐁\\textbf {B}.","(c) 𝐂\\textbf {C}.","(d) 𝐓\\textbf {T}.","(e) 𝐏\\textbf {P}.","(f) the final live counterpart reconstruction and zoom-in details.","(g) results from .","(h) results from Step1+Step2 with a single trace representation."],["Anti-Spoofing for Unknown and Open-set Spoofs"," Another important aspect is to test the anti-spoofing performance on unknown spoof.","To use SiW-M for unknown spoof detection, The work [19] defines the leave-one-out testing protocols, termed as SiW-M Protocol II.","In this protocol, each model (i.e., one column in Tab.","REF ) is trained with 12 types of spoof attacks (as known attacks) plus the $80\\%$ of the live faces, and tested on the remaining 1 attack (as unknown attack) plus the $20\\%$ of live faces.","As shown in Tab.","REF , our PhySTD achieves significant improvement over our preliminary version, with relatively $11.7\\%$ on the overall EER, $19.0\\%$ on the overall ACER, $50.4\\%$ on the overall TPR.","Specifically, we reduce the EERs of half mask, paper glasses, transparent mask, replay attack, and partial paper relatively by $47.6\\%$ , $40.4\\%$ , $37.7\\%$ , $31.6\\%$ , $56.3\\%$ , respectively.","Overall, compared with the top 7 performances, we outperform the SOTA performance of EER/TPR and achieve comparable ACER.","Among all, the detection of silicone mask, paper-crafted mask, mannequin head, impersonation makeup, and partial paper attacks are relatively good, with the detection accuracy (i.e., TPR@FNR=$0.5\\%$ ) above $65\\%$ .","Obfuscation makeup is the most challenging one with TPR of $0\\%$ , where we predict all the spoof samples as live.","This is due to the fact that the makeup looks very similar to the live faces, while being dissimilar to any other spoof types.","However, once we obtain a few samples, our model can quickly recognize the spoof traces on the eyebrow and cheek, synthesize new spoof samples, and successfully detect the attack (TPR=$41.1 \\%$ in Tab.","REF ).","Figure: Examples of spoof trace disentanglement on SiW (a-h) and SiW-M (i-x).","(a)-(d) items are print attacks and (e)-(h) items are replay attacks.","(i)-(x) items are live, print, replay, half mask, silicone mask, paper mask, transparent mask, obfuscation makeup, impersonation makeup, cosmetic makeup, paper glasses, partial paper, funny eye glasses, and mannequin head.","The first column is the input face, the second column is the overall spoof trace (𝐈-𝐈 ^\\textbf {I}-\\hat{\\textbf {I}}), the third column is the reconstructed live.Moreover, in the real-world scenario, the testing samples can be either a known spoof attack or an unknown one.","Thus, we propose SiW-M Protocol III to evaluate this open-set testing situation.","In Protocol III, we first follow the train/test split from protocol I, and then further remove one spoof type as the unknown attack.","During the testing, we test on the entire unknown spoof samples as well the test split set of the know spoof samples.","The results are reported in Tab.","REF .","Compared to the SOTA face anti-spoofing method [18], our approach substantially outperforms it in all three metrics.","Table: Confusion matrices of spoof mediums classification based on spoof traces.","The results are compared with the previous method .","Green represents improvement over .","Red represents performance drop.Table: Confusion matrices of 6-class spoof traces classification on SiW-M database.Figure: Examples of the spoof data synthesis.","The first row are the source spoof faces, the first column are the target live faces, and the remaining are the synthesized spoof faces from the live face with the corresponding spoof traces."],["Spoof Traces Classification"," To quantitatively evaluate the spoof trace disentanglement, we perform a spoof medium classification on the disentangled spoof traces and report the classification accuracy.","The spoof traces should contain spoof medium-specific information, so that they can be used for clustering without seeing the face.","To make a fair comparison with [24], we remove the additional spoof type information from the preliminary mask $\\mathbf {P}_0$ .","That is, for this specific experiment, we only use the additive traces $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ to learn the trace classification.","After $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ finish training with only binary labels, we fix PhySTD and apply a simple CNN (i.e., AlexNet) on the estimated additive traces to do a supervised spoof medium classification.","We follow the same 5-class testing protocol in [24] in Oulu-NPU Protocol 1.","We report the classification accuracy as the ratio between correctly predicted samples from all classes and all testing samples.","Shown in Tab.","REF .","Our model can achieve a 5-class classification accuracy of $83.3\\%$ .","If we treat two print attacks as the same class and two replay as the same class, our model can achieve a 3-class classification accuracy of $92.0\\%$ .","Compared with the prior method [24], we show an improvement of $29\\%$ on the 5-class model.","In addition, we train the same CNN on the original images instead of the estimated spoof traces for the same spoof medium classification task, and the classification accuracy can only reach $80.6\\%$ .","This further demonstrates that the estimated traces do contain significant information to distinguish different spoof mediums.","We also execute the spoof traces classification task on more spoof types in SiW-M database.","We leverage the train/test split on SiW-M Protocol 1.","We first train the PhySTD till convergence, and use the estimated traces from the training set to train the trace classification network.","We explore the 6-class scenario, shown in Tab.","REF .","Our 6-class model can achieve the classification accuracy of $92.0\\%$ .","Since the traces are more distinct among different spoof types, this performance is even better than 5-class classification on print/replay scenario in Oulu-NPU Protocol 1.","This further demonstrates that PhySTD can estimate spoof traces that contain significant information of spoof mediums and can be applied to multiple spoof types."],["Ablation Study"," In this section, we show the importance of each design of our proposed approach on the SiW-M Protocol I, in Tab.REF .","Our baseline is the auxiliary FAS [18], without the temporal module.","It consists of the backbone encoder and depth estimation network.","When including the image decomposition, the baseline becomes the training step 1 in Alg.","REF , as the traces are not activated without the training step 2.","To validate the effectiveness of GAN training, we report the results from the baseline model with our GAN design, denoted as Step1+Step2.","We also provide the control experiment where the traces are represented by a single component to demonstrate the effectiveness of the proposed 5-element trace representation.","This model is denoted as Step1+Step2 with single trace.","In addition, we evaluate the effect of training with more synthesized data via enabling the training step 3 as Step1+Step2+Step3, which is our final approach.","As shown in Tab.","REF , the baseline model (Auxiliary) can achieve a decent performance of EER $6.7\\%$ .","Adding image decomposition to the baseline (Step 1) can improve the EER from $6.7\\%$ to $4.3\\%$ , but more live samples are predicted with higher scores, causing a worse ACER.","Adding simple GAN design (Step1+Step2 with single trace) may lead to a similar EER performance of $5.8\\%$ , but based on the TPR ($59.3\\%\\rightarrow 74.8\\%$ ) its practical performance may be improved.","With the proper physics-guided trace disentanglement, we can improve the EER to $2.8\\%$ and TPR to $89.7\\%$ .","And our final design can achieve the performance of HTER $2.8\\%$ , EER $2.5\\%$ , and TPR $91.2\\%$ .","Compared with our preliminary version, the EER is improved by $47.9\\%$ , HTER is improved by $31.7\\%$ and TPR is improved by $29.5\\%$ .","Figure: NO_CAPTIONFigure: NO_CAPTIONSpoof trace components In Fig.REF , we provide illustration of each spoof trace component.","Strong color distortion (low-frequency trace) shows up in the print attacks.","Moiré patterns in the replay attack are well detected in the high-frequency trace.","The local specular highlights in transparent mask are well presented in the low- and mid-frequency components, and the inpainting process further fine-tunes the most highlighted area.","For the two glasses attacks, the color discrepancy is corrected in the low-frequency trace, and the sharp edges are corrected in the mid- and high-frequency traces.","Each component shows a consistent semantic meaning on different spoof samples, and this successful trace disentanglement can lead to better final visual results.","As shown on the right side of Fig.","REF , we compare with our preliminary version [31] and the ablated GAN design with a single trace representation.","The result of single trace representation shows strong artifacts on most of the live reconstruction.","The multi-scale from our preliminary version has already shown a large visual quality improvement, but still have some spoof traces (e.g., glass edges) remained in the live reconstruction.","In contrast, our approach can further handle the missing traces and achieve better visualization.","Live reconstruction In Fig.","REF , we show more examples from different spoof types in SiW and SiW-M databases.","The overall trace is the exact difference between the input face and its live reconstruction.","For the live faces, the trace is zero, and for the spoof faces, our method removes spoof traces without unnecessary changes, such as identity shift, and make them look like live faces.","For example, strong color distortion shows up in print/replay attacks (Fig.","REF a-h) and some 3D mask attacks (Fig.","REF l-o).","For makeup attacks (Fig.","REF q-s), the fake eyebrows, lipstick, artificial wax, and cheek shade are clearly detected.","The folds and edges (Fig.","REF t-w) are well detected and removed in paper-crafted masks, paper glasses, and partial paper attacks.","Spoof synthesis Additionally, we show examples of new spoof synthesis using the disentangled spoof traces, which is an important contribution of this work.","As shown in Fig.","REF , the spoof traces can be precisely transferred to a new face without changing the identity of the target face.","Due to the additional inpainting process, spoof attacks such as transparent mask and partial attacks can be better attached to the new live face.","Thanks to the proposed 3D warping layer, the geometric discrepancy between the source spoof trace and the target face can be corrected during the synthesis.","Especially on the second source spoof, the right part of the traces is successfully transferred to the new live face while the left side remains to be still live.","It demonstrates that our trace regularization can suppress unnecessary artifacts generated by the network.","Both the live reconstruction results in Fig.","REF and the spoof synthesis results in Fig.","REF demonstrate that our approach disentangles visually convincing spoof traces that help face anti-spoofing.","Spoof trace removing process As shown in Fig.","REF , we illustrate the effects of trace components by progressively removing them one by one.","For the replay attack, the spoof sample comes with strong over-exposure as well as clear Moiré pattern.","Removing the low-frequency trace can effectively correct the over-exposure and color distortion caused by the digital screen.","And removing the texture pattern in the high-frequency trace can peel off the high-frequency grid effect and reconstruct the live counterpart.","For the makeup attack, since there is no strong color range bias, removing estimated low-frequency trace would mainly remove the lip-stick color and fake eyebrow, but in the meantime bring a few artifacts at the edges.","Next, while removing the content pattern, the shadow on the cheek and the fake eyebrows are adequately lightened.","Finally, removing the texture pattern would significantly correct the spoof traces from artificial wax, eyeliner, and shadow on the cheek.","Similarly, in mask and partial attacks, the reconstruction will be gradually refined as we removing components one by one.","t-SNE visualization We use t-SNE [71] to visualize the encoder features $\\mathbf {F}_1$ ,$\\mathbf {F}_2$ ,$\\mathbf {F}_3$ , and the features that produce $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ and $\\lbrace \\mathbf {P},\\mathbf {I}_P\\rbrace $ .","The t-SNE is able to project the output of features from different scales and layers to 2D by preserving the KL divergence distance.","As shown in Fig.","REF , among the three feature scales in the encoder, $F_3$ is the most separable feature space, the next is $F_1$ , and the worst is $F_2$ .","The features for additive traces $\\lbrace \\textbf {B},\\textbf {C},\\textbf {T}\\rbrace $ are well-clustered as semantic sub-groups of live, makeup, mask, and partial attacks.","As we know the inpainting masks for live samples are close to zero, the feature for inpainting traces $\\lbrace \\textbf {P},\\textbf {I}_P\\rbrace $ shows the inpainting process mostly update the partial attacks, and then some makeup attacks and mask attacks, i.e., the green dots being further away from the black dots means they have greater magnitude.","This validates our prior knowledge of the inpainting process.","This work proposes a physics-guided spoof traces disentanglement network (PhySTD) to tackle the challenging problem of disentangling spoof traces from the input faces.","With the spoof traces, we reconstruct the live faces as well as synthesize new spoofs.","To correct the geometric discrepancy in synthesis, we propose a 3D warping layer to deform the traces.","The disentanglement not only improves the SOTA of face anti-spoofing in known, unknown, and open-set spoof settings, but also provides visual evidence to support the model's decision."],["Acknowledgment","This research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R$\\&$ D Contract No.","2017-17020200004.","The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government.","The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.","[Figure: NO_CAPTION [Figure: NO_CAPTION"]],"string":"[\n [\n \"Physics-Guided Spoof Trace Disentanglement for Generic Face\\n Anti-Spoofing\"\n ],\n [\n \"Abstract Prior studies show that the key to face anti-spoofing lies in the subtle image pattern, termed \\\"spoof trace\\\", e.g., color distortion, 3D mask edge, Moire pattern, and many others.\",\n \"Designing a generic face anti-spoofing model to estimate those spoof traces can improve not only the generalization of the spoof detection, but also the interpretability of the model's decision.\",\n \"Yet, this is a challenging task due to the diversity of spoof types and the lack of ground truth in spoof traces.\",\n \"In this work, we design a novel adversarial learning framework to disentangle spoof faces into the spoof traces and the live counterparts.\",\n \"Guided by physical properties, the spoof generation is represented as a combination of additive process and inpainting process.\",\n \"Additive process describes spoofing as spoof material introducing extra patterns (e.g., moire pattern), where the live counterpart can be recovered by removing those patterns.\",\n \"Inpainting process describes spoofing as spoof material fully covering certain regions, where the live counterpart of those regions has to be \\\"guessed\\\".\",\n \"We use 3 additive components and 1 inpainting component to represent traces at different frequency bands.\",\n \"The disentangled spoof traces can be utilized to synthesize realistic new spoof faces after proper geometric correction, and the synthesized spoof can be used for training and improve the generalization of spoof detection.\",\n \"Our approach demonstrates superior spoof detection performance on 3 testing scenarios: known attacks, unknown attacks, and open-set attacks.\",\n \"Meanwhile, it provides a visually-convincing estimation of the spoof traces.\",\n \"Source code and pre-trained models will be publicly available upon publication.\"\n ],\n [\n \"Introduction\",\n \" In recent years, the vulnerability of face biometric systems has been widely recognized and increasingly brought attention to the computer vision community.\",\n \"The attacks to the face biometric systems attempt to deceive the systems to make wrong identity recognition: either recognize the attackers as a target person (i.e., impersonation), or cover up the original identity (i.e., obfuscation).\",\n \"Figure: The proposed approach can detect spoof faces, disentangle the spoof traces, and reconstruct the live counterparts.\",\n \"It can be applied to diverse spoof types and estimate distinct traces (e.g., Moiré pattern in replay attack, artificial eyebrow and wax in makeup attack, color distortion in print attack, and specular highlights in 3D mask attack).\",\n \"Zoom in for details.There are various types of digital and physical attacks, including face morphing [1], [2], [3], face adversarial attacks [4], [5], [6], face manipulation attacks (e.g., deepfake, face swap) [7], [8], and face spoofing [9], [10], [11].\",\n \"Among the above-mentioned attacks, face spoofing is the only physical attack to deceive the systems, where attackers present faces from spoof mediums, such as photograph, screen, mask and makeup, instead of a live human.\",\n \"These spoof mediums can be easily manufactured by ordinary people, and hence they pose huge threats to face biometric applications such as mobile face unlock, building access control, and transportation security.\",\n \"Therefore, face biometric systems need to be secured with face anti-spoofing (FAS) techniques to distinguish the source of the face before performing the face recognition task.\",\n \"Figure: The comparison of different deep-learning based face anti-spoofing.\",\n \"(a) direct FAS only provides a binary decision of spoofness; (b) auxiliary FAS can provide simple interpretation of spoofness.\",\n \"𝐌\\\\mathbf {M} denotes the auxiliary task, such as depth map estimation; (c) generative FAS can provide more intuitive interpretation of spoofness, but only for a limited number of spoof attacks; (d) the proposed method can provide spoof trace estimation for generic face spoof attacks.As most face recognition systems are based on a monocular RGB camera, monocular RGB based face anti-spoofing has been studied for over a decade, and one of the most common approaches is based on texture analysis [12], [13], [14].\",\n \"Researchers noticed that presenting faces from spoof mediums introduces special texture differences, such as color distortions, unnatural specular highlights, Moiré patterns, etc.\",\n \"Those texture differences are inherent within spoof mediums and thus hard to remove or camouflage.\",\n \"Conventional approaches build a feature extractor plus classifier pipeline, such as LBP+SVM and HOG+SVM [15], [16], and show good performance on several small databases with constraint environments.\",\n \"In recent years, many works leverage deep learning techniques and show great progress in face anti-spoofing performance [17], [18], [19], [20], [21].\",\n \"Deep learning based methods can be generally grouped into 3 categories: direct FAS, auxiliary FAS, and generative FAS, as illustrated in Fig.\",\n \"REF .\",\n \"Early works [22], [23] build vanilla CNN with binary output to directly predict the spoofness of an input face (Fig.REF a).\",\n \"Methods [18], [21] propose to learn an intermediate representation, e.g., depth, rPPG, reflection, instead of binary classes, which can lead to better generalization and performance (Fig.REF b).\",\n \"[24], [25], [26] additionally attempt to generate the visual patterns existing in the spoof samples (Fig.REF c), providing a more intuitive interpretation of the sample's spoofness.\",\n \"Despite the success, there are still at least three unsolved problems in the topic of deep learning-based face anti-spoofing.\",\n \"First, most prior works are designed to tackle limited spoof types, either print/replay or 3D mask solely, while a real-world anti-spoofing system may encounter a wide variety of spoof types including print, replay, various 3D masks, facial makeup, and even unseen attack types.\",\n \"Therefore, to better reflect real-world performance, we need a benchmark to evaluate face anti-spoofing under known attacks, unknown attacks, and their combination (termed open-set setting).\",\n \"Second, many approaches formulate face anti-spoofing as a classification/regression problem, with a single score as the output.\",\n \"Although auxiliary FAS and generative FAS attempt to offer some extent of interpretation by fixation, saliency, or noise analysis, there is little understanding on what the exact differences are between live and spoof, and what patterns the classifier's decision is based upon.\",\n \"A better interpretation can be estimating the exact patterns differentiating a spoof face and its live counterpart, termed spoof trace.\",\n \"Thirdly, compared with other face analysis tasks such as recognition or alignment, the data for face anti-spoofing has several limitations.\",\n \"Most FAS databases are captured in the constraint indoor environment, which has limited intra-subject variation and environment variation.\",\n \"For some special spoof types such as makeup and customized silicone mask, they require highly skilled experts to apply or create, with high cost, which results in very limited samples (i.e., long-tail data).\",\n \"Thus, how to learn from data with limited variations or samples is a challenge for FAS.\",\n \"In this work, we aim to design a face anti-spoofing model that is applicable to a wide variety of spoof types, termed generic face anti-spoofing.\",\n \"We equip this model with the ability to explicitly disentangle the spoof traces from the input faces.\",\n \"Some examples of spoof trace disentanglement are shown in Fig.\",\n \"REF .\",\n \"This is a challenging objective due to the diversity of spoof traces and the lack of ground truth during model learning.\",\n \"However, we believe that fulfilling this objective can bring several benefits: Binary classification for face anti-spoofing would harvest any cue that helps classification, which might include spoof-irrelevant cues such as lighting, and thus hinder generalization.\",\n \"In contrast, spoof trace disentanglement explicitly tackles the most fundamental cue in spoofing, upon which the classification can be more grounded and witness better generalization.\",\n \"With the trend of pursuing explainable AI [27], [28], it is desirable for the face anti-spoofing model to generate the spoof patterns that support its binary decision, since spoof trace serves as a good visual explanation of the model's decision.\",\n \"Certain properties (e.g., severity, methodology) of spoof attacks might potentially be revealed from the traces.\",\n \"Disentangled spoof traces can enable the synthesis of realistic spoof samples, which addresses the issue of limited training data for the minority spoof types, such as special 3D masks and makeup.\",\n \"As shown in Fig.\",\n \"REF d, we propose a Physics-guided Spoof Trace Disentanglement (PhySTD) to explore the spoof traces for generic face anti-spoofing.\",\n \"To model all types of spoofs, we formulate the spoof trace disentanglement as a combination of additive process and inpainting process.\",\n \"Additive process describes spoofing as spoof material introducing extra patterns (e.g., moire pattern), where the live counterpart can be recovered by removing those patterns.\",\n \"Inpainting process describes spoofing as spoof material fully covering certain regions of the original face, where the live counterpart of those regions has to be “guessed\\\" [29], [30].\",\n \"We further decompose the spoof traces into frequency-dependent components, so that traces with different frequency properties can be equally handled.\",\n \"For the network architecture, we extend a backbone network for auxiliary FAS with a decoder to perform the disentanglement.\",\n \"With no ground truth of spoof traces, we adopt an overall GAN-based training strategy.\",\n \"The generator takes an input face, estimates its spoofness, and disentangles the spoof trace.\",\n \"After obtaining the spoof trace, we can reconstruct the live counterpart from the spoof and synthesize new spoof from the live.\",\n \"The synthesized samples are then sent to multiple discriminators with real samples for adversarial training.\",\n \"The synthesized spoof samples are further utilized to train the generator in a fully supervised fashion, thanks to disentangled spoof traces as ground truth for the synthesized samples.\",\n \"To correct possible geometric discrepancy during spoof synthesis, we propose a novel 3D warping layer to deform spoof traces toward the target live face.\",\n \"A preliminary version of this work was published in the Proceedings European Conference on Computer Vision (ECCV) 2020 [31].\",\n \"We extend the work from three aspects.\",\n \"1) Guided by the physics of how a spoof is generated, we introduce a spoof generation function (SGF) to model the spoof trace disentanglement as a combination of additive and inpainting processes.\",\n \"SGF has a better and more natural modeling of generic spoof attacks, such as paper glass.\",\n \"2) Previous trace components $\\\\lbrace \\\\mathbf {S},\\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ are not supervised hierarchically so that there exists semantic ambiguity.\",\n \"In this work, we introduce several hierarchical designs in the GAN framework to remedy such ambiguity.\",\n \"3) We propose an open-set testing scenario to further evaluate the real-world performance for face anti-spoofing models.\",\n \"Both known and unknown attacks are included in the open-set testing.\",\n \"We perform a side-by-side comparison between the proposed approach and the state-of-the-art (SOTA) face anti-spoofing solutions on multiple datasets and protocols.\",\n \"In summary, the main contributions of this work are as follows: $\\\\bullet $ We for the first time study spoof trace for generic face anti-spoofing, where a wide variety of spoof types are tackled with one unified framework; $\\\\bullet $ We propose a novel physics-guided model to disentangle spoof traces, and utilize the spoof traces to synthesize new data samples for enhanced training; $\\\\bullet $ We propose novel protocols for a generic open-set face anti-spoofing; $\\\\bullet $ We achieve SOTA anti-spoofing performance and provide convincing visualization for a wide variety of spoof types.\"\n ],\n [\n \"Related Work\",\n \"Face Anti-Spoofing Face anti-spoofing has been studied for more than a decade and its development can be roughly divided into three stages.\",\n \"In the early years, researchers leverage spontaneous human movement, such as eye blinking and head motion, to detect simple print photograph or static replay attacks [32], [33].\",\n \"However, when facing counter attacks, such as print face with eye region cut, and replaying a face video, those methods would fail.\",\n \"In the second stage, researchers pay more attention to texture differences between live and spoof, which are inherent to spoof mediums.\",\n \"Researchers mainly extract hand-crafted features from the faces, e.g., LBP [12], [15], [34], [35], HoG [16], [36], SIFT [14] and SURF [13], and train a classifier to split the live vs. spoof, e.g., SVM and LDA.\",\n \"Recently, face anti-spoofing solutions equipped with deep learning techniques have demonstrated significant improvements over the conventional methods.\",\n \"Methods in [37], [38], [39], [22] train a deep neural network to learn a binary classification between live and spoof.\",\n \"In [17], [18], [19], [20], [21], additional supervisions, such as face depth map and rPPG signal, are utilized to help the network to learn more generalizable features.\",\n \"As the latest approaches achieving saturated performance on several benchmarks, researchers start to explore more challenging cases, such as few-shot/zero-shot face anti-spoofing [19], [40], [41] and domain adaptation in face anti-spoofing [20], [42].\",\n \"In this work, we aim to solve an interesting yet very challenging problem: disentangling and visualizing the spoof traces from an input face.\",\n \"A related work [24] also adopts GAN seeking to estimate the spoof traces.\",\n \"However, they formulate the traces as low-intensity noises, which is limited to print and replay attacks only and cannot provide convincing visual results.\",\n \"In contrast, we explore spoof traces for a much wider range of spoof attacks, visualize them with novel disentanglement, and also evaluate the proposed method on the challenging cases, e.g., zero-shot face anti-spoofing.\",\n \"Disentanglement Learning Disentanglement learning is often adopted to better represent complex data and features.\",\n \"DR-GAN [43] disentangles a face into identity and pose vectors for pose-invariant face recognition and view synthesis.\",\n \"Similarly in gait recognition, [44] disentangles the representations of appearance, canonical, and pose features from an input gait video.\",\n \"3D reconstruction works [45], [46] also disentangle the representation of a 3D face into identity, expressions, poses, albedo, and illuminations.\",\n \"For image synthesis, [47] disentangles an image into appearance and shape with U-Net and Variational Auto Encoder (VAE).\",\n \"Different from [45], [43], [44], we intend to disentangle features that have different scales and contain geometric information.\",\n \"We leverage the multiple outputs to represent features at different scales, and adopt multiple-scale discriminators to properly learn them.\",\n \"Moreover, we propose a novel warping layer to tackle the geometric discrepancy during the disentanglement and reconstruction.\",\n \"Image Trace Modeling Image traces are certain signals existing in the image that can reveal information about the capturing camera, imaging setting, environment, and so on.\",\n \"Those signals often have much lower energy compared to the image content, which needs proper modeling to explore them.\",\n \"[48], [49], [50] observe the difference of image noises, and use them to recognize the capture cameras.\",\n \"From the frequency domain, [25] shows the image noises from different cameras obey different noise distributions.\",\n \"Such techniques are applied to the field of image forensics, and later [51], [52] propose methods to remove such traces for image anti-forensics.\",\n \"Recently, image trace modeling is widely used in image forgery detection and image adversarial attack detection [53], [54].\",\n \"In this work, we attempt to explore the traces of spoof face presentation.\",\n \"Due to different spoof mediums, spoof traces show large variations in content, intensity, and frequency distribution.\",\n \"We propose to disentangle the traces as additive traces and inpainting trace.\",\n \"And for additive traces, we further decompose them based on different frequency bands.\",\n \"Figure: Overview of the proposed Physics-guided Spoof Trace Disentanglement (PhySTD).\"\n ],\n [\n \"Problem Formulation\",\n \" Let the domain of live faces be denoted as $\\\\mathcal {L}\\\\!\",\n \"\\\\subset \\\\!\",\n \"\\\\mathbb {R}^{N\\\\!\\\\times \\\\!N \\\\!\\\\times \\\\!3}$ and spoof faces as $\\\\mathcal {S}\\\\!\",\n \"\\\\subset \\\\!\\\\mathbb {R}^{N\\\\!\\\\times \\\\!N\\\\!\",\n \"\\\\times \\\\!3}$ , where $N$ is the image size.\",\n \"We intend to obtain not only the correct prediction (live vs. spoof) of the input face, but also a convincing estimation of the spoof trace and live face reconstruction.\",\n \"To represent the spoof trace, our preliminary version assumes an additive relation between live and spoof, and uses 4 trace components $\\\\lbrace \\\\mathbf {S},\\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ at different frequency bands as: $\\\\mathbf {I}_{\\\\textit {spoof}} = (1+\\\\lfloor \\\\mathbf {S}\\\\rfloor _{n_1})\\\\mathbf {I}_{\\\\textit {live}} + \\\\lfloor \\\\mathbf {B}\\\\rfloor _{n_1} + \\\\lfloor \\\\mathbf {C}\\\\rfloor _{n_2} + \\\\mathbf {T},$ where $\\\\mathbf {S},\\\\mathbf {B}$ represent low-frequency traces, $\\\\mathbf {C}$ represents mid-frequency ones, and $\\\\mathbf {T}$ represents high-frequency ones.\",\n \"$\\\\lfloor \\\\cdot \\\\rfloor $ is the low bandpass filtering operation, and in practice, we achieve this by downsampling the original image and upsampling it back.\",\n \"In the previous setting, $n_1\\\\!=\\\\!1$ and $n_2\\\\!=\\\\!64$ .\",\n \"Compared to the simple representation with only a single component [24], this multi-scale representation of $\\\\lbrace \\\\mathbf {S},\\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ can largely improve disentanglement quality and suppress undesired artifacts due to its coarse-to-fine process.\",\n \"The model is designed to provide a valid estimation of spoof traces $\\\\lbrace \\\\mathbf {S},\\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ without respective ground truth.\",\n \"Our preliminary version [31] aims to find a minimum intensity change that transfers an input face to the live domain: $\\\\operatornamewithlimits{arg\\\\,min}_{\\\\hat{\\\\mathbf {I}}} \\\\Vert \\\\mathbf {I} - \\\\hat{\\\\mathbf {I}}\\\\Vert _F \\\\; s.t.\",\n \"\\\\; \\\\mathbf {I} \\\\in (\\\\mathcal {S}\\\\cup \\\\mathcal {L}) \\\\; \\\\text{and} \\\\; \\\\hat{\\\\mathbf {I}} \\\\in \\\\mathcal {L},$ where $\\\\mathbf {I}$ is the source face, $\\\\hat{\\\\mathbf {I}}$ is the target face to be optimized, and $\\\\mathbf {I}-\\\\hat{\\\\mathbf {I}}$ is defined as the spoof trace.\",\n \"When the source face is live $\\\\mathbf {I}_{\\\\text{live}}$ , $\\\\mathbf {I}-\\\\hat{\\\\mathbf {I}}$ should be 0 as $\\\\mathbf {I}$ is already in $\\\\mathcal {L}$ .\",\n \"When the source face is spoof $\\\\mathbf {I}_{\\\\text{spoof}}$ , $\\\\mathbf {I}-\\\\hat{\\\\mathbf {I}}$ should be regularized to prevent unnecessary changes such as identity shift.\",\n \"Despite the effectiveness of this representation, there are still two drawbacks: First, the spoof trace disentanglement is mainly formulated as an additive processing.\",\n \"The optimization of Eqn.\",\n \"REF limits the trace intensity, and the reconstruction for spoof regions with large appearance divergence might be sub-optimal, such as spoof glasses or mask.\",\n \"For those spoof regions, the physical relationship between the live and the spoof is better described as replacement rather than addition; Second, while our preliminary version representing the traces with hierarchical components, these components are learned with losses on their summation.\",\n \"Without careful supervision, the learned components can be ambiguous in their semantic meanings, e.g., the high-frequency component may include low-frequency information.\",\n \"To address the first drawback, we introduce a spoof generation function (SGF) as an additive process followed by an inpainting process: $\\\\mathbf {I}_{\\\\textit {spoof}} = (1-\\\\mathbf {P})(\\\\mathbf {I}_{\\\\textit {live}} +\\\\mathbf {T}_A) + \\\\mathbf {P}\\\\cdot \\\\mathbf {T}_P,$ where $\\\\mathbf {T}_A\\\\!\\\\in \\\\mathbb {R}^{N\\\\!\\\\times \\\\!N\\\\!\\\\times \\\\!3}$ indicates the traces from additive process, $\\\\mathbf {T}_P$ indicates the traces from inpainting process, and $\\\\mathbf {P} \\\\in \\\\mathbb {R}^{N\\\\!\\\\times \\\\!N\\\\!\\\\times \\\\!1}$ denotes the inpainting region.\",\n \"Given a spoof face, one may reconstruct the live counterpart by inversing Eqn.\",\n \"REF : $ \\\\hat{\\\\mathbf {I}}_{\\\\textit {live}} =(1-\\\\mathbf {P})(\\\\mathbf {I}_{\\\\textit {spoof}} - \\\\mathbf {T}_\\\\text{A}) + \\\\mathbf {P}\\\\cdot (\\\\hat{\\\\mathbf {I}}_{\\\\textit {live}} + \\\\mathbf {I}_{\\\\textit {spoof}} - \\\\mathbf {T}_P),$ As the inpainting physically replaces content, the spoof trace $\\\\mathbf {T}_P$ in the inpainting region $\\\\mathbf {P}$ is identical to the spoof image $\\\\mathbf {I}_{\\\\textit {spoof}}$ in the same region, and thus both cancel out in the second term of Eqn.\",\n \"REF .\",\n \"We further rename the $\\\\hat{\\\\mathbf {I}}_{\\\\textit {live}}$ in the second term as $\\\\mathbf {I}_P$ to indicate the inpainting content within the inpainting region that should be estimated from the model.\",\n \"Therefore, the reconstruction of the live image becomes: $ \\\\hat{\\\\mathbf {I}}_{\\\\textit {live}} =(1-\\\\mathbf {P})(\\\\mathbf {I}_{\\\\textit {spoof}} - \\\\mathbf {T}_\\\\text{A}) + \\\\mathbf {P}\\\\cdot \\\\mathbf {I}_P,$ where $\\\\mathbf {T}_A \\\\!=\\\\!\",\n \"\\\\lfloor \\\\mathbf {B}\\\\rfloor _{n_1} \\\\!+\\\\!\",\n \"\\\\lfloor \\\\mathbf {C}\\\\rfloor _{n_2}\\\\!+\\\\!\",\n \"\\\\mathbf {T}$ denotes the additive trace represented by three hierarchical components.\",\n \"$n_1$ and $n_2$ are set to be 32 and 128 respectively.\",\n \"With a larger $n_1$ , the effect of component $\\\\mathbf {S}$ in the preliminary version can be incorporated into $\\\\mathbf {B}$ , and hence we remove $\\\\mathbf {S}$ for simplicity.\",\n \"Besides the additive traces, the model is further required to estimate the inpainting region $\\\\mathbf {P}$ and inpainting live content $\\\\mathbf {I}_P$ .\",\n \"$\\\\mathbf {I}_P$ is estimated based on the rest of the live facial region without intensity constraint.\",\n \"We use a function $G(\\\\cdot )$ to represent the reconstruction process of Eqn.\",\n \"REF .\",\n \"Accordingly, the optimization of Eqn.\",\n \"REF is re-formulated by replacing $\\\\hat{\\\\mathbf {I}}$ with Eqn.\",\n \"REF as: ${\\\\begin{array}{c}\\\\operatornamewithlimits{arg\\\\,min}_{\\\\mathbf {T}_A,\\\\mathbf {P},\\\\mathbf {I}_P} \\\\Vert \\\\mathbf {I}-(1-\\\\mathbf {P})(\\\\mathbf {I} -\\\\mathbf {T}_A) - \\\\mathbf {P}\\\\cdot \\\\mathbf {I}_P\\\\Vert _F \\\\; \\\\\\\\\\\\rightarrow \\\\operatornamewithlimits{arg\\\\,min}_{\\\\mathbf {T}_A,\\\\mathbf {P},\\\\mathbf {I}_P} \\\\Vert (1-\\\\mathbf {P})\\\\mathbf {T}_A\\\\Vert _F + \\\\Vert \\\\mathbf {P}\\\\cdot (\\\\mathbf {I}-\\\\mathbf {I}_P)\\\\Vert _F.\",\n \"\\\\; \\\\\\\\\\\\end{array}}$ As we do not wish to impose any intensity constraint on $\\\\mathbf {I}_P$ , the final objective is formulated as: ${\\\\begin{array}{c}\\\\operatornamewithlimits{arg\\\\,min}_{\\\\mathbf {T}_A,\\\\mathbf {P}} \\\\Vert (1\\\\!-\\\\!\\\\mathbf {P})\\\\mathbf {T}_A\\\\Vert _F+\\\\lambda \\\\Vert \\\\mathbf {P}\\\\Vert _F \\\\;s.t.\",\n \"\\\\; \\\\mathbf {I} \\\\!\",\n \"\\\\in \\\\!\",\n \"\\\\mathcal {S}\\\\cup \\\\mathcal {L}, \\\\hat{\\\\mathbf {I}} \\\\in \\\\mathcal {L},\\\\end{array}}$ where $\\\\lambda $ is a weight to balance two terms.\",\n \"In addition, based on Eqn.\",\n \"REF , we can define another function $G^-(\\\\cdot )$ to synthesize new spoof faces, by transferring the spoof traces from $\\\\mathbf {I}^i$ to $\\\\mathbf {I}^j$ : $\\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}^{i\\\\rightarrow j} = G^-(\\\\mathbf {I}^j|\\\\mathbf {I}^i) = (1-\\\\mathbf {P}^i)(\\\\mathbf {I}^j+ \\\\mathbf {T}_\\\\text{A}^i) + \\\\mathbf {P}^i\\\\cdot \\\\mathbf {I}^i.$ Note that $\\\\mathbf {T}_P$ in Eqn.\",\n \"REF has been replaced with $\\\\mathbf {I}^i$ since the spoof image $\\\\mathbf {I}^i$ contains the spoof trace for the inpainting region.\",\n \"Figure: The proposed PhySTD network architecture.\",\n \"Except the last layer, each conv and transposed conv is concatenated with a batch normalizstion layer and a leaky ReLU layer.\",\n \"‘k3c64s2’ indicates the kernel size of 3×33\\\\times 3, the convolution channel of 64 and the stride of 2.Estimating $\\\\lbrace \\\\mathbf {T}_\\\\text{A},\\\\mathbf {P},\\\\mathbf {I}_P\\\\rbrace $ from an input face $\\\\mathbf {I}$ is termed as spoof trace disentanglement.\",\n \"Given that no ground truth of traces is available, this disentanglement can be achieved via generative adversarial based training.\",\n \"As shown in Fig.\",\n \"REF , the proposed Physics-guided Spoof Trace Disentanglement (PhySTD) consists of a generator and discriminator.\",\n \"Given an input image, the generator is designed to predict the spoofness (represented by the pseudo depth map) as well as estimate the additive traces $\\\\lbrace \\\\mathbf {B}, \\\\mathbf {C}, \\\\mathbf {T}\\\\rbrace $ and the inpainting components $\\\\lbrace \\\\mathbf {P},\\\\mathbf {I}_P\\\\rbrace $ .\",\n \"With the traces, we can apply function $G(\\\\cdot )$ to reconstruct the live counterpart and function $G^-(\\\\cdot )$ to synthesize new spoof faces.\",\n \"We adopt a set of discriminators at multiple image resolutions to distinguish the real faces $\\\\lbrace \\\\mathbf {I}_{\\\\textit {live}},\\\\mathbf {I}_{\\\\textit {spoof}}\\\\rbrace $ with the synthetic faces $\\\\lbrace \\\\hat{\\\\mathbf {I}}_{\\\\textit {live}},\\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}\\\\rbrace $ .\",\n \"To remedy the semantic ambiguity during $\\\\lbrace \\\\mathbf {B}, \\\\mathbf {C}, \\\\mathbf {T}\\\\rbrace $ learning, three trace component combinations, $\\\\lbrace \\\\mathbf {B}\\\\rbrace $ , $\\\\lbrace \\\\mathbf {B}, \\\\mathbf {C}\\\\rbrace $ , and $\\\\lbrace \\\\mathbf {B}, \\\\mathbf {C}, \\\\mathbf {T}\\\\rbrace $ , will contribute to the synthesis of live reconstruction at one particular resolution, which is then supervised by a respective discriminator (details in Sec.REF ).\",\n \"To learn a proper inpainting region $\\\\mathbf {P}$ , we leverage both the prior knowledge and the information from the additive traces.\",\n \"In the rest of this section, we present the details of the generator, the discriminators, the details of face reconstruction and synthesis, and the losses and training steps used in PhySTD.\"\n ],\n [\n \"Disentanglement Generator\",\n \" As shown in Fig.\",\n \"REF , the disentanglement generator consists of a backbone encoder, a spoof trace decoder and a depth estimation network.\",\n \"The backbone encoder aims to extract multi-scale features, the depth estimation network leverages the features to estimate the facial depth map, and a spoof trace decoder to estimate the additive trace components $\\\\lbrace \\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ and the inpainting components $\\\\lbrace \\\\mathbf {P},\\\\mathbf {I}_P\\\\rbrace $ .\",\n \"The depth map and the spoof traces will be used to compute the final spoofness score.\",\n \"Figure: The visualization of image decomposition for different input faces: (a) live face (b) 3D mask attack (c) replay attack (d) print attack.Backbone encoder Backbone encoder extracts features from the input images for both depth map estimation and spoof trace disentanglement.\",\n \"As shown in our preliminary work [31], the spoof traces consists of components from different frequency bands: low-frequency traces includes color distortion, mid-frequency traces includes makeup strikes, and high-frequency traces includes Moiré patterns and mask edges.\",\n \"However, a vanilla CNN model might overlook high-frequency traces since the energy of high-frequency traces is often much weaker than that of low-frequency traces.\",\n \"In order to encourage the network to equally regard traces with different physical properties, we explicitly decompose the image into three elements $\\\\lbrace \\\\mathbf {I}_\\\\mathbf {B},\\\\mathbf {I}_\\\\mathbf {C},\\\\mathbf {I}_\\\\mathbf {T}\\\\rbrace $ as: $\\\\begin{split}\\\\mathbf {I}_\\\\mathbf {B}=&\\\\lfloor \\\\mathbf {I}\\\\rfloor _{n_1},\\\\\\\\\\\\mathbf {I}_\\\\mathbf {C}=&\\\\lfloor \\\\mathbf {I}\\\\rfloor _{n_2}-\\\\lfloor \\\\mathbf {I}\\\\rfloor _{n_1},\\\\\\\\\\\\mathbf {I}_\\\\mathbf {T}=& \\\\mathbf {I} - \\\\lfloor \\\\mathbf {I}\\\\rfloor _{n2}, \\\\\\\\\\\\end{split}$ where $n_1\\\\!=\\\\!32$ , $n_2\\\\!=\\\\!128$ and the image size $N\\\\!=\\\\!256$ .\",\n \"In addition, we amplify the value in $\\\\mathbf {I}_\\\\mathbf {C}, \\\\mathbf {I}_\\\\mathbf {T}$ by two constants 15 and 25, and then feed the concatenation of three elements to the backbone network.\",\n \"Fig.\",\n \"REF provides the visualization of image decomposition.\",\n \"We observe that the traces that are less distinct in the original images become more highlighted in the $\\\\mathbf {I}_\\\\mathbf {T}$ component: 3D mask and replay attack bring unique patterns different with the live face pattern, while print attack is lacking of necessary high frequency details.\",\n \"Semantically, $\\\\mathbf {I}_\\\\mathbf {B},\\\\mathbf {I}_\\\\mathbf {C},\\\\mathbf {I}_\\\\mathbf {T}$ share the same frequency domains with $\\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}$ respectively, and thus the decomposition potentially eases the learning of $\\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}$ .\",\n \"After that, the encoder progressively downsamples the decomposed image components 3 times to obtain features $\\\\mathbf {F}_1\\\\!\",\n \"\\\\in \\\\!\\\\mathbb {R}^{128\\\\!\\\\times \\\\!128\\\\!\\\\times \\\\!64}$ , $\\\\mathbf {F}_2\\\\!\",\n \"\\\\in \\\\!\\\\mathbb {R}^{64\\\\!\\\\times \\\\!64\\\\!\\\\times \\\\!96}$ , $\\\\mathbf {F}_3\\\\!\",\n \"\\\\in \\\\!\\\\mathbb {R}^{32\\\\!\\\\times \\\\!32\\\\!\\\\times \\\\!128}$ via conv layers.\",\n \"Spoof trace decoder The decoder upsamples the feature $\\\\mathbf {F}_3$ with transpose conv layers back to the input face size 256.\",\n \"The last layer outputs both additive traces $\\\\lbrace \\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ and inpainting components $\\\\lbrace \\\\mathbf {P},\\\\mathbf {I}_P\\\\rbrace $ .\",\n \"Similar to U-Net [55], we apply the short-cut connection between the backbone encoder and decoder to bypass the multiple scale details for a high-quality trace estimation.\",\n \"Depth estimation network We still recognize the importance of the discriminative supervision used in auxiliary FAS, and thus introduce a depth estimation network to perform the pseudo-depth estimation for face anti-spoofing, as proposed in [18].\",\n \"The depth estimation network takes the concatenated features of $\\\\mathbf {F}_1$ , $\\\\mathbf {F}_2$ , $\\\\mathbf {F}_3$ from the backbone encoder and $\\\\mathbf {U}_3$ from the decoder as input.\",\n \"The features are put through a spatial attention mechanism from [56] and resize to the same size of $K=32$ .\",\n \"It outputs a face depth map $\\\\mathbf {M}\\\\!\\\\in \\\\!\",\n \"\\\\mathbb {R}^{32\\\\!\\\\times \\\\!32}$ , where the depth values are normalized within $[0,1]$ .\",\n \"Regarding the number of parameters, both spoof trace decoder and depth estimation network are light weighed, while the backbone network is much heavier.\",\n \"With more network layers being shared to tackle both depth estimation and spoof trace disentanglement, the knowledge learnt from spoof trace disentanglement can be better shared with depth estimation task, which can lead to a better anti-spoofing performance.\",\n \"Final scoring In the testing phase, we use the norm of the depth map and the intensity of spoof traces for real vs. spoof classification: $\\\\text{score} = \\\\frac{1}{2K^2}\\\\Vert \\\\mathbf {M}\\\\Vert _1 \\\\!+ \\\\!\\\\frac{\\\\alpha _0}{2N^2}(\\\\Vert \\\\mathbf {B}\\\\Vert _1\\\\!+\\\\!\\\\Vert \\\\mathbf {C}\\\\Vert _1\\\\!+\\\\!\\\\Vert \\\\mathbf {T}\\\\Vert _1\\\\!+\\\\!\\\\Vert \\\\mathbf {P}\\\\Vert _1),$ where $\\\\alpha _0$ is the weight for the spoof trace.\"\n ],\n [\n \"Reconstruction and Synthesis\",\n \" There are multiple options to use the disentangled spoof traces: 1) live reconstruction, 2) spoof synthesis, and 3) “harder\\\" sample synthesis, which will be described below respectively.\",\n \"Live reconstruction: Based on Eqn.\",\n \"REF , we propose a hierarchical reconstruction of the live face counterpart from the input images.\",\n \"To reconstruct faces at a certain resolution, each additive trace is included only if its frequency domain is lower than the target resolution.\",\n \"We apply $\\\\lbrace \\\\textit {hi},\\\\textit {mid},\\\\textit {low}\\\\rbrace $ three resolution settings as: $\\\\begin{split}\\\\hat{\\\\mathbf {I}}_{\\\\textit {hi}}=&(1-\\\\mathbf {P})(\\\\mathbf {I} - \\\\lfloor \\\\mathbf {B}\\\\rfloor _{n_1} - \\\\lfloor \\\\mathbf {C}\\\\rfloor _{n_2} - \\\\mathbf {T}) + \\\\mathbf {P}\\\\cdot \\\\mathbf {I}_P,\\\\\\\\\\\\hat{\\\\mathbf {I}}_{\\\\textit {mid}}=&(1-\\\\mathbf {P})(\\\\lfloor \\\\mathbf {I}\\\\rfloor _{n_2} - \\\\lfloor \\\\mathbf {B}\\\\rfloor _{n_1} - \\\\lfloor \\\\mathbf {C}\\\\rfloor _{n_2}) + \\\\mathbf {P}\\\\cdot \\\\mathbf {I}_P,\\\\\\\\\\\\hat{\\\\mathbf {I}}_{\\\\textit {low}}=&(1-\\\\mathbf {P})(\\\\lfloor \\\\mathbf {I}\\\\rfloor _{n_1} - \\\\lfloor \\\\mathbf {B}\\\\rfloor _{n_1}) + \\\\mathbf {P}\\\\cdot \\\\mathbf {I}_P.\\\\\\\\\\\\end{split}$ Spoof synthesis: Based on Eqn.\",\n \"REF , we can obtain a new spoof face via applying the spoof traces disentangled from a spoof face $\\\\mathbf {I}_i$ to a live face $\\\\mathbf {I}_j$ .\",\n \"However, spoof traces may contain face-dependent content associated with the original spoof subject.\",\n \"Directly applying them to a new face with different shapes or poses may result in mis-alignment and strong visual implausibility.\",\n \"Therefore, the spoof trace should go through a geometry correction before performing this synthesis.\",\n \"We propose an online 3D warping layer and will introduce it in the following subsection.\",\n \"“Harder\\\" sample synthesis: The disentangled spoof traces can not only reconstruct live and synthesize new spoof, but also synthesize “harder\\\" spoof samples by removing or amplifying part of the spoof traces.\",\n \"We can tune one or some of the trace elements $\\\\lbrace \\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T},\\\\mathbf {P}\\\\rbrace $ to make the spoof sample to become “less spoofed\\\", which is thus closer to a live face since the spoof traces are weakened.\",\n \"Such spoof data can be regarded as harder samples and may benefit the generalization of the disentanglement generator.\",\n \"For instance, while removing the low frequency element $\\\\mathbf {B}$ from a replay spoof trace, the generator may be forced to rely on other elements such as high-level texture patterns.\",\n \"To synthesize the “harder\\\" sample $\\\\hat{\\\\mathbf {I}}_{\\\\textit {hard}}$ , we follow Eqn.\",\n \"REF with two minor changes: 1) generate 3 random weights between $[0,1]$ and multiple each with one component of $\\\\lbrace \\\\mathbf {B}, \\\\mathbf {C}, \\\\mathbf {T}\\\\rbrace $ ; 2) randomly remove the inpainting process (i.e., set $\\\\mathbf {P}=0$ ) with a probability of $0.5$ .\",\n \"Compared with other methods, such as brightness and contrast change [57], reflection and blurriness effect [21], or 3D distortion [58], our approach can introduce more realistic and effective data samples, as shown in Sec.\",\n \".\"\n ],\n [\n \"Online 3D Warping Layer\",\n \" We propose an online 3D warping layer to correct the shape discrepancy.\",\n \"To obtain the warping, previous methods in [59], [18] use offline face swapping and pre-computed dense offset respectively, where both methods are non-differentiable as well as memory intensive.\",\n \"In contrast, our warping layer is designed to be both differentiable and computationally efficient, which is necessary for online synthesis during the training.\",\n \"Figure: The online 3D warping layer.\",\n \"(a) Given the corresponding dense offset, we warp the spoof trace and add them to the target live face to create a new spoof.\",\n \"E.g.\",\n \"pixel (x,y)(x,y) with offset (3,5)(3,5) is warped to pixel(x+3,y+5)(x+3,y+5) in the new image.\",\n \"(b) To obtain a dense offsets from the spare offsets of the selected face shape vertices, Delaunay triangulation interpolation is adopted.First, the live reconstruction of a spoof face $\\\\mathbf {I}^i$ can be expressed as: $G^i = G(\\\\mathbf {I}^i)[\\\\mathbf {p}_0],$ where $\\\\mathbf {p}^0=\\\\lbrace (0,0),(0,1),...,(255,255)\\\\rbrace \\\\in \\\\mathbb {R}^{256\\\\times 256\\\\times 2}$ enumerates pixel locations in $\\\\mathbf {I}^i$ .\",\n \"To align the spoof traces while synthesizing a new spoof face, a dense offset $\\\\Delta \\\\mathbf {p}^{i\\\\rightarrow j}\\\\in \\\\mathbb {R}^{256\\\\times 256\\\\times 2}$ is required to indicate the deformation between face $\\\\mathbf {I}^i$ and face $\\\\mathbf {I}^j$ .\",\n \"A discrete deformation can be acquired from the distances of the corresponding facial landmarks between two faces.\",\n \"During the data preparation, we use [60] to fit a 3DMM model and extract the 2D locations of $Q$ facial vertices for each face: $\\\\mathbf {s}=\\\\lbrace (x_0,y_0),(x_1,y_1),...,(x_N,y_N)\\\\rbrace \\\\in \\\\mathbb {R}^{Q\\\\times 2}.$ A sparse offset on the corresponding vertices can then be computed two faces as $\\\\Delta \\\\mathbf {s}^{i\\\\rightarrow j} = \\\\mathbf {s}^j - \\\\mathbf {s}^i$ .\",\n \"To convert the sparse offset $\\\\Delta \\\\mathbf {s}^{i\\\\rightarrow j}$ to the dense offset $\\\\Delta \\\\mathbf {p}^{i\\\\rightarrow j}$ , we apply a triangulation interpolation: $\\\\Delta \\\\mathbf {p}^{i\\\\rightarrow j}=\\\\text{Tri}(\\\\mathbf {p}^0, \\\\mathbf {s}^i,\\\\Delta \\\\mathbf {s}^{i\\\\rightarrow j}),$ where $\\\\text{Tri}(\\\\cdot )$ is the interpolation, $\\\\mathbf {s}^i$ denotes the vertex locations, $\\\\Delta \\\\mathbf {s}^{i\\\\rightarrow j}$ are the vertex values, and we adopt Delaunay triangulation.\",\n \"The warping operation can be denoted as: $G^{-i\\\\rightarrow j} = G^-(\\\\mathbf {I}^j|\\\\mathbf {I}^i)[\\\\mathbf {p}^0+\\\\Delta \\\\mathbf {p}^{i\\\\rightarrow j}],$ where the offset $\\\\Delta \\\\mathbf {p}^{i\\\\rightarrow j}$ applies to all subject $i$ related elements $\\\\lbrace \\\\mathbf {T}_A^i, \\\\mathbf {I}^i, \\\\mathbf {P}^i\\\\rbrace $ .\",\n \"Since the offset $\\\\Delta \\\\mathbf {p}^{i\\\\rightarrow j}$ is typically composed of fractional numbers, we implement the bilinear interpolation to sample the fractional pixel locations.\",\n \"We select $Q=140$ vertices to cover the face region so that they can represent non-rigid deformation, due to pose and expression.\",\n \"As the pixel values in the warped face are a linear combination of pixel values of the triangulation vertices, this entire process is differentiable.\",\n \"This process is illustrated in Fig.\",\n \"REF .\",\n \"[t] Input: live faces $\\\\mathbf {I}_{\\\\textit {live}}$ and facial landmarks $\\\\mathbf {s}_{\\\\textit {live}}$ , spoof faces $\\\\mathbf {I}_{\\\\textit {spoof}}$ and facial landmarks $\\\\mathbf {s}_{\\\\textit {spoof}}$ , ground truth depth map $\\\\mathbf {M}_0$ , preliminary mask $\\\\mathbf {P}_0$ ; Output: reconstructed live $\\\\hat{\\\\mathbf {I}}_{\\\\textit {live}}$ , synthesized spoof $\\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}$ , spoof traces $\\\\lbrace \\\\mathbf {T}_A^{\\\\textit {l}}$ , $\\\\mathbf {P}^{\\\\textit {l}}$ , $\\\\mathbf {I}_P^{\\\\textit {l}}, \\\\mathbf {T}_A^{\\\\textit {s}}$ , $\\\\mathbf {P}^{\\\\textit {s}}$ , $\\\\mathbf {I}_P^{\\\\textit {s}}\\\\rbrace $ , depth maps $\\\\lbrace \\\\mathbf {M}^{\\\\textit {l}}$ , $\\\\mathbf {M}^{\\\\textit {s}}\\\\rbrace $ ; iteration $<$ max_iteration training step 1 $\\\\textbf {1}$ : compute $\\\\mathbf {T}_A^{\\\\textit {l}}$ , $\\\\mathbf {P}^{\\\\textit {l}}$ , $\\\\mathbf {I}_P^{\\\\textit {l}}$ $\\\\leftarrow $ $G(\\\\mathbf {I}_{\\\\textit {live}})$ and compute $\\\\mathbf {T}_A^{\\\\textit {s}}$ , $\\\\mathbf {P}^{\\\\textit {s}}$ , $\\\\mathbf {I}_P^{\\\\textit {s}}$ $\\\\leftarrow $ $G(\\\\mathbf {I}_{\\\\textit {spoof}})$ $\\\\textbf {2}$ : estimate the depth map $\\\\mathbf {M}^{\\\\textit {l}}$ , $\\\\mathbf {M}^{\\\\textit {s}}$ $\\\\textbf {3}$ : compute losses $L_{\\\\textit {depth}}$ , $L_P$ , $L_R$ training step 2 $\\\\textbf {4}$ : compute $\\\\hat{\\\\mathbf {I}}_{\\\\textit {low}}$ , $\\\\hat{\\\\mathbf {I}}_{\\\\textit {mid}}$ , $\\\\hat{\\\\mathbf {I}}_{\\\\textit {hi}}$ from $\\\\mathbf {T}_A^s$ , $\\\\mathbf {P}^s$ , $\\\\mathbf {I}_P^s$ and $\\\\mathbf {I}_{\\\\textit {spoof}}$ (Eqn.\",\n \"REF ) $\\\\textbf {5}$ : compute warping offset $\\\\Delta \\\\mathbf {p}^{s\\\\rightarrow l}$ from $\\\\mathbf {s}_{\\\\textit {live}}$ , $\\\\mathbf {s}_{\\\\textit {spoof}}$ (Eqn.\",\n \"REF ) $\\\\textbf {6}$ : compute $\\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}$ from warped $\\\\mathbf {T}_A^{s\\\\rightarrow l}$ , $\\\\mathbf {P}^{s\\\\rightarrow l}$ and $\\\\mathbf {I}_{\\\\textit {live}}$ (Eqn.\",\n \"REF ) $\\\\textbf {7}$ : send $\\\\mathbf {I}_{\\\\textit {live}}$ , $\\\\mathbf {I}_{\\\\textit {spoof}}$ , $\\\\hat{\\\\mathbf {I}}_{\\\\textit {low}}$ , $\\\\hat{\\\\mathbf {I}}_{\\\\textit {mid}}$ , $\\\\hat{\\\\mathbf {I}}_{\\\\textit {hi}}$ , $\\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}$ to discriminators $\\\\textbf {8}$ : compute the adversarial loss for generator $L_{\\\\textit {G}}$ and for discriminators $L_{\\\\textit {D}}$ training step 3 $\\\\textbf {9}$ : create harder samples $\\\\mathbf {I}_{\\\\textit {hard}}$ from $\\\\mathbf {T}_A^{s\\\\rightarrow l}$ , $\\\\mathbf {P}^{s\\\\rightarrow l}$ and $\\\\mathbf {I}_{\\\\textit {live}}$ with random perturbation on traces $\\\\textbf {10}$ : compute $\\\\mathbf {T}_A^h$ , $\\\\mathbf {P}^h$ , $\\\\mathbf {I}_P^h$ $\\\\leftarrow $ $G(\\\\mathbf {I}_{\\\\textit {hard}})$ $\\\\textbf {11}$ : compute depth map $\\\\mathbf {M}^h$ for $\\\\mathbf {I}_{\\\\textit {hard}}$ $\\\\textbf {12}$ : compute losses $L_S$ , $L_H$ back propagation $\\\\textbf {13}$ : back-propagate the losses from step $3,8,12$ to corresponding parts and update the network; PhySTD Training Iteration.\"\n ],\n [\n \"Multi-scale Discriminators\",\n \" Motivated by [61], we adopt multiple discriminators at different resolutions (e.g., 32, 96, and 256) in our GAN architecture.\",\n \"We follow the design of PatchGAN [62], which essentially is a fully convolutional network.\",\n \"Fully convolutional networks are shown to be effective to not only synthesize high-quality images [62], [61], but also tackle face anti-spoofing problems [18].\",\n \"For each discriminator, we adopt the same structure but do not share the weights.\",\n \"As shown in Fig.\",\n \"REF , we use in total 4 discriminators in our work: $D_1$ , working in the lowest resolution of 32, focuses on low frequency elements since the higher-frequency traces are erased by downsampling.\",\n \"$D_2$ , working at the resolution of 96, focuses on the middle level content pattern.\",\n \"$D_3$ and $D_4$ , working on the highest resolution of 256, focus on the fine texture details.\",\n \"Our preliminary version resizes real and synthetic samples $\\\\lbrace \\\\mathbf {I}, \\\\hat{\\\\mathbf {I}}\\\\rbrace $ to different resolutions and assign to each discriminator.\",\n \"To remove semantic ambiguity and provide correspondence to the trace components, we instead assign the hierarchical reconstruction from Eqn.\",\n \"REF to the discriminators: we send low frequency pairs $\\\\lbrace \\\\mathbf {I}_{\\\\textit {live}}, \\\\hat{\\\\mathbf {I}}_{\\\\textit {low}}\\\\rbrace $ to $D_1$ , middle frequency pairs $\\\\lbrace \\\\mathbf {I}_{\\\\textit {live}}, \\\\hat{\\\\mathbf {I}}_{\\\\textit {mid}}\\\\rbrace $ to $D_2$ , high frequency pairs $\\\\lbrace \\\\mathbf {I}_{\\\\textit {live}}, \\\\hat{\\\\mathbf {I}}_{\\\\textit {hi}}\\\\rbrace $ to $D_3$ , and real/synthetic spoof $\\\\lbrace \\\\mathbf {I}_{\\\\textit {spoof}}, \\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}\\\\rbrace $ to $D_4$ .\",\n \"Each discriminator outputs a 1-channel map in the range of $[0,1]$ , where 0 denotes fake and 1 denotes real.\"\n ],\n [\n \"Loss Functions and Training Steps\",\n \" We utilize multiple loss functions to supervise the learning of depth maps and spoof traces.\",\n \"Each training iteration consists of three training steps.\",\n \"We first introduce the loss function, followed by how they are used in the training steps.\",\n \"Depth map loss: We follow the auxiliary FAS [18] to estimate an auxiliary depth map $\\\\mathbf {M}$ , where the depth ground truth $\\\\mathbf {M}_0$ for a live face contains face-like shape and the depth for spoof should be zero.\",\n \"We apply the $\\\\mathcal {L}$ -1 norm on this loss as: $L_{\\\\textit {depth}} = \\\\frac{1}{K^2}\\\\mathbb {E}_{i\\\\sim \\\\mathcal {L}\\\\cup \\\\mathcal {S}}{\\\\Vert }\\\\mathbf {M}^i-\\\\mathbf {M}^i_0{\\\\Vert }_F, $ where $K\\\\!=\\\\!32$ is the size of $\\\\mathbf {M}$ .\",\n \"We apply the dense face alignment [60] to estimate the 3D shape and render the depth ground truth $\\\\mathbf {M}_0$ .\",\n \"Figure: Preliminary mask 𝐏 0 \\\\mathbf {P}_0 for the negative term in inpainting mask loss.\",\n \"White pixels denote 1 and black pixels denote 0.\",\n \"White indicates the area should not be inpainted.\",\n \"𝐏 0 \\\\mathbf {P}_0 for: (a) print, replay; (b) 3D mask and makeup; (c) partial attacks that cover the eye portion; (d) partial attacks that cover the mouth portion.Adversarial loss for $G$ : We employ the LSGANs [63] on reconstructed live faces and synthesized spoof faces.\",\n \"It encourages the reconstructed live to look similar to real live from domain $\\\\mathcal {L}$ , and the synthesized spoof faces to look similar to faces from domain $\\\\mathcal {S}$ : $\\\\begin{aligned}L_{G} = \\\\mathbb {E}_{i\\\\sim \\\\mathcal {L}, j\\\\sim \\\\mathcal {S}}\\\\Big [{\\\\Vert }D_1(\\\\hat{\\\\mathbf {I}}_{\\\\textit {low}}^j)\\\\!- \\\\!\\\\mathbf {1}{\\\\Vert }_F^2 + {\\\\Vert }D_2(\\\\hat{\\\\mathbf {I}}_{\\\\textit {mid}}^j)\\\\!- \\\\!\\\\mathbf {1}{\\\\Vert }_F^2 + \\\\\\\\{\\\\Vert }D_3(\\\\hat{\\\\mathbf {I}}_{\\\\textit {hi}}^j)\\\\!- \\\\!\\\\mathbf {1}{\\\\Vert }_F^2 +{\\\\Vert }D_4(\\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}^{j\\\\rightarrow i})\\\\!\",\n \"-\\\\mathbf {1}{\\\\Vert }_F^2\\\\Big ].\\\\end{aligned}$ Adversarial loss for $D$ : The adversarial loss for discriminators encourages $D(\\\\cdot )$ to distinguish between real live vs. reconstructed live, and real spoof vs. synthesized spoof: $\\\\begin{aligned}L_{D} = \\\\mathbb {E}_{i\\\\sim \\\\mathcal {L}, j\\\\sim \\\\mathcal {S}}\\\\Big [{\\\\Vert }D_1(\\\\mathbf {I}^i)\\\\!- \\\\!\\\\mathbf {1}{\\\\Vert }_F^2 +{\\\\Vert }D_2(\\\\mathbf {I}^i)\\\\!- \\\\!\\\\mathbf {1}{\\\\Vert }_F^2 + \\\\\\\\{\\\\Vert }D_3(\\\\mathbf {I}^i)\\\\!- \\\\!\\\\mathbf {1}{\\\\Vert }_F^2 +{\\\\Vert }D_4(\\\\mathbf {I}^j)\\\\!\",\n \"-\\\\mathbf {1}{\\\\Vert }_F^2 +{\\\\Vert }D_1(\\\\hat{\\\\mathbf {I}}_{\\\\textit {low}}^j){\\\\Vert }_F^2 + \\\\\\\\{\\\\Vert }D_2(\\\\hat{\\\\mathbf {I}}_{\\\\textit {mid}}^j){\\\\Vert }_F^2 +{\\\\Vert }D_3(\\\\hat{\\\\mathbf {I}}_{\\\\textit {hi}}^j){\\\\Vert }_F^2 + {\\\\Vert }D_4(\\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}^{j\\\\rightarrow i}){\\\\Vert }_F^2\\\\Big ].\\\\end{aligned}$ Inpainting mask loss: The ground truth inpainting region for all spoof attacks is barely possible to obtain, hence a fully supervised training [64] for inpainting mask is out of the question.\",\n \"However, we may still leverage the prior knowledge of spoof attacks to facilitate the estimation of inpainting masks.\",\n \"The inpainting mask loss consists of a positive term and a negative term.\",\n \"First, the positive term encourages certain region to be inpainted.\",\n \"As the goal of inpainting process is to allow certain region to change without intensity constraint, the region with larger magnitude of additive traces would have a higher probability to be inpainted.\",\n \"Hence, the positive term adopts a $\\\\mathcal {L}$ -2 norm between the inpainting region $\\\\mathbf {P}$ and the region where the additive trace is larger than a threshold $\\\\beta $ .\",\n \"Second, the negative term discourages certain region to be inpainted.\",\n \"While the ground truth inpainting mask is unknown, it's straightforward to mark a large portion of region that should not be inpainted.\",\n \"For instance, the inpainting region for funny eye glasses should not appear in the lower part of a face.\",\n \"Hence, we provide a preliminary mask $\\\\mathbf {P}_0$ to indicate the not-to-be-inpainted region, and adopt a normalized $\\\\mathcal {L}$ -2 norm on the masked inpainting region $\\\\mathbf {P}\\\\cdot \\\\mathbf {P}_0$ as the negative term.\",\n \"The preliminary mask $\\\\mathbf {P}_0$ is illustrated in Fig.\",\n \"REF .\",\n \"Overall, the inpainting mask loss is formed as: $L_{P} = \\\\mathbb {E}_{\\\\mathbf {i}\\\\sim \\\\mathcal {S}}\\\\Big [{\\\\Vert }\\\\mathbf {P}^i-(\\\\mathbf {T}_A^i>\\\\beta ){\\\\Vert }_F^2 +\\\\frac{{\\\\Vert }\\\\mathbf {P}^i\\\\cdot \\\\mathbf {P}_0^i{\\\\Vert }_F^2 }{{\\\\Vert }\\\\mathbf {P}_0^i{\\\\Vert }_F^2}\\\\Big ].$ Trace regularization: Based on Eqn.\",\n \"REF with $\\\\lambda =1$ , we regularize the intensity of additive traces $\\\\lbrace \\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ and inpainting region $\\\\mathbf {P}$ .\",\n \"The regularizer loss is denoted as: $L_{R} = \\\\mathbb {E}_{\\\\mathbf {i}\\\\sim \\\\mathcal {L}\\\\cup \\\\mathcal {S}}\\\\Big [{\\\\Vert }\\\\mathbf {B}{\\\\Vert }_F^2 +{\\\\Vert }\\\\mathbf {C}{\\\\Vert }_F^2 +{\\\\Vert }\\\\mathbf {T}{\\\\Vert }_F^2 +{\\\\Vert }\\\\mathbf {P}{\\\\Vert }_F^2\\\\Big ].$ Synthesized spoof loss: Synthesized spoof data come with ground truth spoof traces.\",\n \"As a result, we are able to define a supervised pixel loss for the generator to disentangle the exact spoof traces that were added: $ L_{S} = \\\\mathbb {E}_{\\\\mathbf {i}\\\\sim \\\\mathcal {L}, \\\\mathbf {j}\\\\sim \\\\mathcal {S}}\\\\Big [{\\\\Vert }G(\\\\lceil G^{-j\\\\rightarrow i} \\\\rceil ) -\\\\lceil G^{j\\\\rightarrow i}\\\\rceil {\\\\Vert }_F^1\\\\Big ],$ where $G^{j\\\\rightarrow i}$ is the overall effect of $\\\\lbrace \\\\mathbf {P}^j,\\\\mathbf {I}_P^j,\\\\mathbf {B}^j,\\\\mathbf {C}^j,\\\\mathbf {T}^j\\\\rbrace $ after warping to subject $i$ , and $\\\\lceil \\\\cdot \\\\rceil $ is the stop_gradient operation.\",\n \"Without stopping the gradient, $G^{j\\\\rightarrow i}$ may collapse to 0.\",\n \"Depth map loss for “harder” samples: We send the “harder” synthesized spoof data to depth estimation network to improve the data diversity, and hope to increase the FAS model's generalization: $L_{\\\\textit {H}} = \\\\frac{1}{K^2}\\\\mathbb {E}_{i\\\\sim \\\\hat{\\\\mathcal {S}}}\\\\Big [{\\\\Vert }\\\\mathbf {M}^i-\\\\mathbf {M}_0^i{\\\\Vert }_F\\\\Big ],$ where $\\\\hat{\\\\mathcal {S}}$ denotes the domain of synthesized spoof faces.\",\n \"Training steps and total loss: Each training iteration has 3 training steps.\",\n \"In the training step 1, live faces $\\\\mathbf {I}_{\\\\textit {live}}$ and spoof faces $\\\\mathbf {I}_{\\\\textit {spoof}}$ are fed into generator $G(\\\\cdot )$ to disentangle the spoof traces.\",\n \"The spoof traces are used to reconstruct the live counterpart $\\\\hat{\\\\mathbf {I}}_{\\\\textit {live}}$ and synthesize new spoof $\\\\hat{\\\\mathbf {I}}_{\\\\textit {spoof}}$ .\",\n \"The generator is updated with respect to the depth map loss $L_{\\\\textit {depth}}$ , adversarial loss $L_G$ , inpainting mask loss $L_P$ , and regularizer loss $L_R$ : $L = \\\\alpha _1L_{\\\\textit {depth}} + \\\\alpha _2 L_{G} + \\\\alpha _3 L_P + \\\\alpha _4 L_R.$ In the training step 2, the discriminators are supervised with the adversarial loss $L_D$ to compete with the generator.\",\n \"In the training step 3, $\\\\mathbf {I}_{\\\\textit {live}}$ and $\\\\hat{\\\\mathbf {I}}_{\\\\textit {hard}}$ are fed into the generator with the ground truth label and trace to minimize the synthesized spoof loss $L_S$ and depth map loss $L_H$ : $L = \\\\alpha _5 L_{S} + \\\\alpha _6L_H,$ where $\\\\alpha _1$ -$\\\\alpha _6$ are the weights to balance the multitask training.\",\n \"To note that, we send the original live faces $\\\\mathbf {I}_{\\\\textit {live}}$ with $\\\\hat{\\\\mathbf {I}}_{\\\\textit {hard}}$ for a balanced mini-batch, which is important when computing the moving average in the batch normalization layer.\",\n \"We execute all 3 steps in each minibatch iteration, but reduce the learning rate for discriminator step by half.\",\n \"The whole training process is depicted in Alg.\",\n \"REF .\"\n ],\n [\n \"Experiments\",\n \" In this section, we first introduce the experimental setup, and then present the results in the known, unknown, and open-set spoof scenarios, with comparisons to respective baselines.\",\n \"Next, we quantitatively evaluate the spoof traces by performing a spoof medium classification, and conduct an ablation study on each design in the proposed method.\",\n \"Finally, we provide visualization results on the spoof trace disentanglement, new spoof synthesis and t-SNE visualization.\",\n \"Table: The evaluation on four protocols in OULU-NPU.\",\n \"Bold indicates the best score in each protocol.\"\n ],\n [\n \"Experimental Setup\",\n \"Databases We conduct experiments on three major databases: Oulu-NPU [67], SiW [18], and SiW-M [19].\",\n \"Oulu-NPU and SiW include print/replay attacks, while SiW-M includes 13 spoof types.\",\n \"We follow all the existing testing protocols and compare with SOTA methods.\",\n \"Similar to most prior works, we only use the face region for training and testing.\",\n \"Evaluation metrics Two common metrics are used in this work for comparison: EER and APCER/BPCER/ACER.\",\n \"EER describes the theoretical performance and predetermines the threshold for making decisions.\",\n \"APCER/BPCER/ACER[68] describe the practical performance given a predetermined threshold.\",\n \"For both evaluation metrics, lower value means better performance.\",\n \"The threshold for APCER/BPCER/ACER is computed from either training set or validation set.\",\n \"In addition, we also report the True Detection Rate (TDR) at a given False Detection Rate (FDR).\",\n \"This metric describes the spoof detection rate at a strict tolerance to live errors, which is widely used to evaluate real-world systems [69].\",\n \"In this work, we report TDR at FDR$=0.5\\\\%$ .\",\n \"For TDR, the higher the better.\",\n \"Parameter setting PhySTD is implemented in Tensorflow with an initial learning rate of $5e$ -5.\",\n \"We train in total $150,000$ iterations with a batch size of 8, and decrease the learning rate by a ratio of 10 every $45,000$ iterations.\",\n \"We initialize the weights with $[0,0.02]$ normal distribution.\",\n \"$\\\\lbrace \\\\alpha _1,\\\\alpha _2,\\\\alpha _3,\\\\alpha _4,\\\\alpha _5,\\\\alpha _6\\\\rbrace $ are set to be $\\\\lbrace 100,5,1,1e$ -$4,10,1\\\\rbrace $ , and $\\\\beta =0.1$ .\",\n \"$\\\\alpha _0$ is empirically determined from the training or validation set.\",\n \"We use the open-source face alignment [70] and 3DMM fitting [60] to crop the face and provide 140 landmarks.\",\n \"Table: The evaluation on three protocols in SiW Dataset.\",\n \"We compare with the top 7 performances.\"\n ],\n [\n \"Anti-Spoofing for Known Spoof Types\",\n \"Oulu-NPU Oulu-NPU[67] is a commonly used face anti-spoofing benchmark due to its high-quality data and challenging testing protocols.\",\n \"Tab.\",\n \"REF shows our anti-spoofing performance on Oulu-NPU, compared with SOTA algorithms.\",\n \"Our method achieves the best overall performance on this database.\",\n \"Compared with our preliminary version [31], we demonstrate improvements in all 4 protocols, with significant improvement on protocol 1 and protocol 3, i.e., reducing the ACER by $63.6\\\\%$ and $32.1\\\\%$ respectively.\",\n \"Compared with the SOTA, our approach achieves similar best performances on the first three protocols and outperforms the SOTA on the fourth protocol, which is the most challenging one.\",\n \"To note that, in protocol 3 and protocol 4, the performances of testing camera 6 are much lower than those of cameras 1-5: the ACER for camera 6 are $6.4\\\\%$ and $10.2\\\\%$ , while the average ACER for the other cameras are $1.0\\\\%$ and $2.0\\\\%$ respectively.\",\n \"Compared with other cameras, we notice that camera 6 has stronger sensor noises and our model recognizes them as unknown spoof traces, which leads to an increased false negative rate (i.e., BPCER).\",\n \"How to separate sensor noises from spoof traces can be an important future research topic.\",\n \"Table: The evaluation and ablation study on SiW-M Protocol I: known spoof detection.Table: The evaluation on SiW-M Protocol II: unknown spoof detection.SiW SiW[18] is another recent high-quality database.\",\n \"It includes fewer capture cameras but more spoof mediums and environment variations, such as pose, illumination, and expression.\",\n \"The comparison on three protocols is shown in Tab.\",\n \"REF .\",\n \"We outperform the previous works on the first two protocols and rank in the middle on protocol 3.\",\n \"Protocol 3 aims to test the performance of unknown spoof detection, where the model is trained on one spoof attack (print or replay) and tested on the other.\",\n \"As we can see from Fig.REF -REF , the traces of print and replay are significantly different, where the replay traces are more on the high-frequency part (i.e., trace component $\\\\textbf {T}$ ) and the print traces are more on the low-frequency part (i.e., trace component $\\\\textbf {S}$ ).\",\n \"These pattern divergence leads to the adaption gap of our method while training on one attack and testing on the other.\",\n \"SiW-M SiW-M[19] contains a large diversity of spoof types, including print, replay, 3D mask, makeup, and partial attacks.\",\n \"This allows us to have a comprehensive evaluation of the proposed approach with different spoof attacks.\",\n \"To use SiW-M for known spoof detection, we randomly split the data of all types into train/test set with a ratio of $60\\\\%$ vs. $40\\\\%$ , and the results are shown in Tab.\",\n \"REF .\",\n \"Compared to the preliminary version [31], our method outperforms on most spoof types as well as the overall EER performance by $47.9\\\\%$ relatively, which demonstrates the superiority of our anti-spoofing on known spoof attacks.\",\n \"For experiments on SiW-M (protocol I, II, and III), we additionally report the TPR at FNR equal to $0.5\\\\%$ .\",\n \"While EER and ACER provide the theoretical evaluation, the users in real-world applications care more about the true spoof detection rate under a given live detection error rate, and hence TPR can better reflect how well the model can detect one or a few spoof attacks in practices.\",\n \"As shown in Tab.\",\n \"REF , we improve the overall TDR of our preliminary version [31] by $29.5\\\\%$ .\",\n \"Table: The evaluation on SiW-M Protocol III: openset spoof detection.Figure: Examples of each spoof trace components.\",\n \"(a) the input sample faces.\",\n \"(b) 𝐁\\\\textbf {B}.\",\n \"(c) 𝐂\\\\textbf {C}.\",\n \"(d) 𝐓\\\\textbf {T}.\",\n \"(e) 𝐏\\\\textbf {P}.\",\n \"(f) the final live counterpart reconstruction and zoom-in details.\",\n \"(g) results from .\",\n \"(h) results from Step1+Step2 with a single trace representation.\"\n ],\n [\n \"Anti-Spoofing for Unknown and Open-set Spoofs\",\n \" Another important aspect is to test the anti-spoofing performance on unknown spoof.\",\n \"To use SiW-M for unknown spoof detection, The work [19] defines the leave-one-out testing protocols, termed as SiW-M Protocol II.\",\n \"In this protocol, each model (i.e., one column in Tab.\",\n \"REF ) is trained with 12 types of spoof attacks (as known attacks) plus the $80\\\\%$ of the live faces, and tested on the remaining 1 attack (as unknown attack) plus the $20\\\\%$ of live faces.\",\n \"As shown in Tab.\",\n \"REF , our PhySTD achieves significant improvement over our preliminary version, with relatively $11.7\\\\%$ on the overall EER, $19.0\\\\%$ on the overall ACER, $50.4\\\\%$ on the overall TPR.\",\n \"Specifically, we reduce the EERs of half mask, paper glasses, transparent mask, replay attack, and partial paper relatively by $47.6\\\\%$ , $40.4\\\\%$ , $37.7\\\\%$ , $31.6\\\\%$ , $56.3\\\\%$ , respectively.\",\n \"Overall, compared with the top 7 performances, we outperform the SOTA performance of EER/TPR and achieve comparable ACER.\",\n \"Among all, the detection of silicone mask, paper-crafted mask, mannequin head, impersonation makeup, and partial paper attacks are relatively good, with the detection accuracy (i.e., TPR@FNR=$0.5\\\\%$ ) above $65\\\\%$ .\",\n \"Obfuscation makeup is the most challenging one with TPR of $0\\\\%$ , where we predict all the spoof samples as live.\",\n \"This is due to the fact that the makeup looks very similar to the live faces, while being dissimilar to any other spoof types.\",\n \"However, once we obtain a few samples, our model can quickly recognize the spoof traces on the eyebrow and cheek, synthesize new spoof samples, and successfully detect the attack (TPR=$41.1 \\\\%$ in Tab.\",\n \"REF ).\",\n \"Figure: Examples of spoof trace disentanglement on SiW (a-h) and SiW-M (i-x).\",\n \"(a)-(d) items are print attacks and (e)-(h) items are replay attacks.\",\n \"(i)-(x) items are live, print, replay, half mask, silicone mask, paper mask, transparent mask, obfuscation makeup, impersonation makeup, cosmetic makeup, paper glasses, partial paper, funny eye glasses, and mannequin head.\",\n \"The first column is the input face, the second column is the overall spoof trace (𝐈-𝐈 ^\\\\textbf {I}-\\\\hat{\\\\textbf {I}}), the third column is the reconstructed live.Moreover, in the real-world scenario, the testing samples can be either a known spoof attack or an unknown one.\",\n \"Thus, we propose SiW-M Protocol III to evaluate this open-set testing situation.\",\n \"In Protocol III, we first follow the train/test split from protocol I, and then further remove one spoof type as the unknown attack.\",\n \"During the testing, we test on the entire unknown spoof samples as well the test split set of the know spoof samples.\",\n \"The results are reported in Tab.\",\n \"REF .\",\n \"Compared to the SOTA face anti-spoofing method [18], our approach substantially outperforms it in all three metrics.\",\n \"Table: Confusion matrices of spoof mediums classification based on spoof traces.\",\n \"The results are compared with the previous method .\",\n \"Green represents improvement over .\",\n \"Red represents performance drop.Table: Confusion matrices of 6-class spoof traces classification on SiW-M database.Figure: Examples of the spoof data synthesis.\",\n \"The first row are the source spoof faces, the first column are the target live faces, and the remaining are the synthesized spoof faces from the live face with the corresponding spoof traces.\"\n ],\n [\n \"Spoof Traces Classification\",\n \" To quantitatively evaluate the spoof trace disentanglement, we perform a spoof medium classification on the disentangled spoof traces and report the classification accuracy.\",\n \"The spoof traces should contain spoof medium-specific information, so that they can be used for clustering without seeing the face.\",\n \"To make a fair comparison with [24], we remove the additional spoof type information from the preliminary mask $\\\\mathbf {P}_0$ .\",\n \"That is, for this specific experiment, we only use the additive traces $\\\\lbrace \\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ to learn the trace classification.\",\n \"After $\\\\lbrace \\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ finish training with only binary labels, we fix PhySTD and apply a simple CNN (i.e., AlexNet) on the estimated additive traces to do a supervised spoof medium classification.\",\n \"We follow the same 5-class testing protocol in [24] in Oulu-NPU Protocol 1.\",\n \"We report the classification accuracy as the ratio between correctly predicted samples from all classes and all testing samples.\",\n \"Shown in Tab.\",\n \"REF .\",\n \"Our model can achieve a 5-class classification accuracy of $83.3\\\\%$ .\",\n \"If we treat two print attacks as the same class and two replay as the same class, our model can achieve a 3-class classification accuracy of $92.0\\\\%$ .\",\n \"Compared with the prior method [24], we show an improvement of $29\\\\%$ on the 5-class model.\",\n \"In addition, we train the same CNN on the original images instead of the estimated spoof traces for the same spoof medium classification task, and the classification accuracy can only reach $80.6\\\\%$ .\",\n \"This further demonstrates that the estimated traces do contain significant information to distinguish different spoof mediums.\",\n \"We also execute the spoof traces classification task on more spoof types in SiW-M database.\",\n \"We leverage the train/test split on SiW-M Protocol 1.\",\n \"We first train the PhySTD till convergence, and use the estimated traces from the training set to train the trace classification network.\",\n \"We explore the 6-class scenario, shown in Tab.\",\n \"REF .\",\n \"Our 6-class model can achieve the classification accuracy of $92.0\\\\%$ .\",\n \"Since the traces are more distinct among different spoof types, this performance is even better than 5-class classification on print/replay scenario in Oulu-NPU Protocol 1.\",\n \"This further demonstrates that PhySTD can estimate spoof traces that contain significant information of spoof mediums and can be applied to multiple spoof types.\"\n ],\n [\n \"Ablation Study\",\n \" In this section, we show the importance of each design of our proposed approach on the SiW-M Protocol I, in Tab.REF .\",\n \"Our baseline is the auxiliary FAS [18], without the temporal module.\",\n \"It consists of the backbone encoder and depth estimation network.\",\n \"When including the image decomposition, the baseline becomes the training step 1 in Alg.\",\n \"REF , as the traces are not activated without the training step 2.\",\n \"To validate the effectiveness of GAN training, we report the results from the baseline model with our GAN design, denoted as Step1+Step2.\",\n \"We also provide the control experiment where the traces are represented by a single component to demonstrate the effectiveness of the proposed 5-element trace representation.\",\n \"This model is denoted as Step1+Step2 with single trace.\",\n \"In addition, we evaluate the effect of training with more synthesized data via enabling the training step 3 as Step1+Step2+Step3, which is our final approach.\",\n \"As shown in Tab.\",\n \"REF , the baseline model (Auxiliary) can achieve a decent performance of EER $6.7\\\\%$ .\",\n \"Adding image decomposition to the baseline (Step 1) can improve the EER from $6.7\\\\%$ to $4.3\\\\%$ , but more live samples are predicted with higher scores, causing a worse ACER.\",\n \"Adding simple GAN design (Step1+Step2 with single trace) may lead to a similar EER performance of $5.8\\\\%$ , but based on the TPR ($59.3\\\\%\\\\rightarrow 74.8\\\\%$ ) its practical performance may be improved.\",\n \"With the proper physics-guided trace disentanglement, we can improve the EER to $2.8\\\\%$ and TPR to $89.7\\\\%$ .\",\n \"And our final design can achieve the performance of HTER $2.8\\\\%$ , EER $2.5\\\\%$ , and TPR $91.2\\\\%$ .\",\n \"Compared with our preliminary version, the EER is improved by $47.9\\\\%$ , HTER is improved by $31.7\\\\%$ and TPR is improved by $29.5\\\\%$ .\",\n \"Figure: NO_CAPTIONFigure: NO_CAPTIONSpoof trace components In Fig.REF , we provide illustration of each spoof trace component.\",\n \"Strong color distortion (low-frequency trace) shows up in the print attacks.\",\n \"Moiré patterns in the replay attack are well detected in the high-frequency trace.\",\n \"The local specular highlights in transparent mask are well presented in the low- and mid-frequency components, and the inpainting process further fine-tunes the most highlighted area.\",\n \"For the two glasses attacks, the color discrepancy is corrected in the low-frequency trace, and the sharp edges are corrected in the mid- and high-frequency traces.\",\n \"Each component shows a consistent semantic meaning on different spoof samples, and this successful trace disentanglement can lead to better final visual results.\",\n \"As shown on the right side of Fig.\",\n \"REF , we compare with our preliminary version [31] and the ablated GAN design with a single trace representation.\",\n \"The result of single trace representation shows strong artifacts on most of the live reconstruction.\",\n \"The multi-scale from our preliminary version has already shown a large visual quality improvement, but still have some spoof traces (e.g., glass edges) remained in the live reconstruction.\",\n \"In contrast, our approach can further handle the missing traces and achieve better visualization.\",\n \"Live reconstruction In Fig.\",\n \"REF , we show more examples from different spoof types in SiW and SiW-M databases.\",\n \"The overall trace is the exact difference between the input face and its live reconstruction.\",\n \"For the live faces, the trace is zero, and for the spoof faces, our method removes spoof traces without unnecessary changes, such as identity shift, and make them look like live faces.\",\n \"For example, strong color distortion shows up in print/replay attacks (Fig.\",\n \"REF a-h) and some 3D mask attacks (Fig.\",\n \"REF l-o).\",\n \"For makeup attacks (Fig.\",\n \"REF q-s), the fake eyebrows, lipstick, artificial wax, and cheek shade are clearly detected.\",\n \"The folds and edges (Fig.\",\n \"REF t-w) are well detected and removed in paper-crafted masks, paper glasses, and partial paper attacks.\",\n \"Spoof synthesis Additionally, we show examples of new spoof synthesis using the disentangled spoof traces, which is an important contribution of this work.\",\n \"As shown in Fig.\",\n \"REF , the spoof traces can be precisely transferred to a new face without changing the identity of the target face.\",\n \"Due to the additional inpainting process, spoof attacks such as transparent mask and partial attacks can be better attached to the new live face.\",\n \"Thanks to the proposed 3D warping layer, the geometric discrepancy between the source spoof trace and the target face can be corrected during the synthesis.\",\n \"Especially on the second source spoof, the right part of the traces is successfully transferred to the new live face while the left side remains to be still live.\",\n \"It demonstrates that our trace regularization can suppress unnecessary artifacts generated by the network.\",\n \"Both the live reconstruction results in Fig.\",\n \"REF and the spoof synthesis results in Fig.\",\n \"REF demonstrate that our approach disentangles visually convincing spoof traces that help face anti-spoofing.\",\n \"Spoof trace removing process As shown in Fig.\",\n \"REF , we illustrate the effects of trace components by progressively removing them one by one.\",\n \"For the replay attack, the spoof sample comes with strong over-exposure as well as clear Moiré pattern.\",\n \"Removing the low-frequency trace can effectively correct the over-exposure and color distortion caused by the digital screen.\",\n \"And removing the texture pattern in the high-frequency trace can peel off the high-frequency grid effect and reconstruct the live counterpart.\",\n \"For the makeup attack, since there is no strong color range bias, removing estimated low-frequency trace would mainly remove the lip-stick color and fake eyebrow, but in the meantime bring a few artifacts at the edges.\",\n \"Next, while removing the content pattern, the shadow on the cheek and the fake eyebrows are adequately lightened.\",\n \"Finally, removing the texture pattern would significantly correct the spoof traces from artificial wax, eyeliner, and shadow on the cheek.\",\n \"Similarly, in mask and partial attacks, the reconstruction will be gradually refined as we removing components one by one.\",\n \"t-SNE visualization We use t-SNE [71] to visualize the encoder features $\\\\mathbf {F}_1$ ,$\\\\mathbf {F}_2$ ,$\\\\mathbf {F}_3$ , and the features that produce $\\\\lbrace \\\\mathbf {B},\\\\mathbf {C},\\\\mathbf {T}\\\\rbrace $ and $\\\\lbrace \\\\mathbf {P},\\\\mathbf {I}_P\\\\rbrace $ .\",\n \"The t-SNE is able to project the output of features from different scales and layers to 2D by preserving the KL divergence distance.\",\n \"As shown in Fig.\",\n \"REF , among the three feature scales in the encoder, $F_3$ is the most separable feature space, the next is $F_1$ , and the worst is $F_2$ .\",\n \"The features for additive traces $\\\\lbrace \\\\textbf {B},\\\\textbf {C},\\\\textbf {T}\\\\rbrace $ are well-clustered as semantic sub-groups of live, makeup, mask, and partial attacks.\",\n \"As we know the inpainting masks for live samples are close to zero, the feature for inpainting traces $\\\\lbrace \\\\textbf {P},\\\\textbf {I}_P\\\\rbrace $ shows the inpainting process mostly update the partial attacks, and then some makeup attacks and mask attacks, i.e., the green dots being further away from the black dots means they have greater magnitude.\",\n \"This validates our prior knowledge of the inpainting process.\",\n \"This work proposes a physics-guided spoof traces disentanglement network (PhySTD) to tackle the challenging problem of disentangling spoof traces from the input faces.\",\n \"With the spoof traces, we reconstruct the live faces as well as synthesize new spoofs.\",\n \"To correct the geometric discrepancy in synthesis, we propose a 3D warping layer to deform the traces.\",\n \"The disentanglement not only improves the SOTA of face anti-spoofing in known, unknown, and open-set spoof settings, but also provides visual evidence to support the model's decision.\"\n ],\n [\n \"Acknowledgment\",\n \"This research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R$\\\\&$ D Contract No.\",\n \"2017-17020200004.\",\n \"The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government.\",\n \"The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.\",\n \"[Figure: NO_CAPTION [Figure: NO_CAPTION\"\n ]\n]"},"id":{"kind":"string","value":"2012.05185"}}},{"rowIdx":3681,"cells":{"text":{"kind":"list like","value":[["Direct measurement of ferroelectric polarization in a tunable semimetal"],["Abstract Ferroelectricity, the electrostatic counterpart to ferromagnetism, has long been thought to be incompatible with metallicity due to screening of electric dipoles and external electric fields by itinerant charges.","Recent measurements, however, demonstrated signatures of ferroelectric switching in the electrical conductance of bilayers and trilayers of WTe$_2$, a semimetallic transition metal dichalcogenide with broken inversion symmetry.","An especially promising aspect of this system is that the density of electrons and holes can be continuously tuned by an external gate voltage.","This degree of freedom enables investigation of the interplay between ferroelectricity and free carriers, a previously unexplored regime.","Here, we employ capacitive sensing in dual-gated mesoscopic devices of bilayer WTe$_2$ to directly measure the spontaneous polarization in the metallic state and quantify the effect of free carriers on the polarization in the conduction and valence bands, separately.","We compare our results to a low-energy model for the electronic bands and identify the layer-polarized states that contribute to transport and polarization simultaneously.","Bilayer WTe$_2$ is thus shown to be a canonical example of a ferroelectric metal and an ideal platform for exploring polar ordering, ferroelectric transitions, and applications in the presence of free carriers."],["Direct measurement of ferroelectric polarization in a tunable semimetal Sergio C. de la Barrera These authors contributed equally.","Qingrui Cao These authors contributed equally.","Yang Gao Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Yuan Gao Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China Vineetha S. Bheemarasetty Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Jiaqiang Yan Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA David G. Mandrus Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Wenguang Zhu International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China Di Xiao Benjamin M. Hunt bmhunt@andrew.cmu.edu Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Ferroelectricity, the electrostatic counterpart to ferromagnetism, has long been thought to be incompatible with metallicity due to screening of electric dipoles and external electric fields by itinerant charges.","Recent measurements, however, demonstrated signatures of ferroelectric switching in the electrical conductance of bilayers and trilayers of WTe2, a semimetallic transition metal dichalcogenide with broken inversion symmetry .","An especially promising aspect of this system is that the density of electrons and holes can be continuously tuned by an external gate voltage.","This degree of freedom enables investigation of the interplay between ferroelectricity and free carriers, a previously unexplored regime.","Here, we employ capacitive sensing in dual-gated mesoscopic devices of bilayer WTe2 to directly measure the spontaneous polarization in the metallic state and quantify the effect of free carriers on the polarization in the conduction and valence bands, separately.","We compare our results to a low-energy model for the electronic bands and identify the layer-polarized states that contribute to transport and polarization simultaneously.","Bilayer WTe2 is thus shown to be a canonical example of a ferroelectric metal and an ideal platform for exploring polar ordering, ferroelectric transitions, and applications in the presence of free carriers.","Polar materials exhibit charge separation in the absence of an applied electric field, an effect of broken inversion symmetry and a unique polar axis in the crystal , .","In certain polar systems, the charge polarization can be switched by an external electric field, an effect known as ferroelectricity.","In principle, the presence or absence of ferroelectric effects depends only on the crystal class and not the details of the electronic structure.","Despite this, nearly all known conventional ferroelectrics are electrically insulating.","Since the first theoretical proposals for ferroelectric metals in 1965 , only a handful of experimental claims of ferroelectric-like phases in metallic systems have been reported , , , , , and no clear case for a canonical ferroelectric metal has emerged.","Many such claims fail to demonstrate two key signatures of ferroelectric behavior, direct evidence of the polarization and ferroelectric switching, due to bulk screening effects.","Here, we focus on the polar, semimetallic van der Waals crystal, T$_\\text{d}$ -WTe2, in the limit of two atomic layers, thin enough to admit an external electric field (Fig.","REFd-e).","Few-layer crystals of WTe2 have drawn recent interest for exhibiting a wide variety of low-temperature phases , , , , .","Recent transport measurements showed that bilayer (2L) and trilayer (3L) WTe2 exhibit intrinsic, switchable electrical polarization in the conducting state , and separately, surfaces of bulk WTe2 crystals display hysteresis in piezoresponse force microscopy .","Subsequent first-principles calculations indicated that the net polarization points only in the out-of-plane direction, and that the underlying mechanism results from a subtle interlayer sliding between the layers in two stable configurations , .","These findings are exciting given the semimetallic and tunable nature of bilayer WTe2, which enables reaching both electron and hole bands by electrostatic gating in the ferroelectric state.","While the hysteretic behavior observed in previous experiments is promising, a direct measurement of the metallic polarization is still missing.","Due to methodological limitations it was previously not possible to measure the polarization while varying a pure electric field.","More importantly, these limitations also prevented observing the effect of free carriers on the polarization and its dependence on carrier density, a fundamental open question for ferroelectric metals.","In this work, we directly measure the charge polarization and electronic compressibility as a function of density for electrons and holes with independent control of the electric field.","We study the simplest polar WTe2 system, a bilayer, via capacitive sensing in a dual-gated, planar capacitance device (Fig.","REFb).","Capacitance measures the electronic compressibility (and thus metallicity) of a 2D system.","In a bilayer 2D system the top-gate and bottom-gate capacitances provide a direct measurement of the layer-specific charge distribution , and thus the out-of-plane polarization.","Furthermore, the parallel-plate geometry enables this charge sensing with simultaneous and independent control of the vertical electric field and the carrier density in the bilayer by electrostatic gating.","Our devices each consist of a bilayer WTe2 crystal encapsulated by two hexagonal boron nitride (hBN) dielectric layers, with metallic top and bottom gates, and contacts integrated into the top hBN layer (Fig REFa).","Figure: Parameters employed in 𝐤·𝐩\\mathbf {k \\cdot p} calculation, following Eqs.","–.These parameters are obtained by fitting the polarization from the low-energy model to Fig.","a, and hence are different from those in Ref.","du2018band."]],"string":"[\n [\n \"Direct measurement of ferroelectric polarization in a tunable semimetal\"\n ],\n [\n \"Abstract Ferroelectricity, the electrostatic counterpart to ferromagnetism, has long been thought to be incompatible with metallicity due to screening of electric dipoles and external electric fields by itinerant charges.\",\n \"Recent measurements, however, demonstrated signatures of ferroelectric switching in the electrical conductance of bilayers and trilayers of WTe$_2$, a semimetallic transition metal dichalcogenide with broken inversion symmetry.\",\n \"An especially promising aspect of this system is that the density of electrons and holes can be continuously tuned by an external gate voltage.\",\n \"This degree of freedom enables investigation of the interplay between ferroelectricity and free carriers, a previously unexplored regime.\",\n \"Here, we employ capacitive sensing in dual-gated mesoscopic devices of bilayer WTe$_2$ to directly measure the spontaneous polarization in the metallic state and quantify the effect of free carriers on the polarization in the conduction and valence bands, separately.\",\n \"We compare our results to a low-energy model for the electronic bands and identify the layer-polarized states that contribute to transport and polarization simultaneously.\",\n \"Bilayer WTe$_2$ is thus shown to be a canonical example of a ferroelectric metal and an ideal platform for exploring polar ordering, ferroelectric transitions, and applications in the presence of free carriers.\"\n ],\n [\n \"Direct measurement of ferroelectric polarization in a tunable semimetal Sergio C. de la Barrera These authors contributed equally.\",\n \"Qingrui Cao These authors contributed equally.\",\n \"Yang Gao Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Yuan Gao Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China Vineetha S. Bheemarasetty Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Jiaqiang Yan Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA David G. Mandrus Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Wenguang Zhu International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China Di Xiao Benjamin M. Hunt bmhunt@andrew.cmu.edu Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Ferroelectricity, the electrostatic counterpart to ferromagnetism, has long been thought to be incompatible with metallicity due to screening of electric dipoles and external electric fields by itinerant charges.\",\n \"Recent measurements, however, demonstrated signatures of ferroelectric switching in the electrical conductance of bilayers and trilayers of WTe2, a semimetallic transition metal dichalcogenide with broken inversion symmetry .\",\n \"An especially promising aspect of this system is that the density of electrons and holes can be continuously tuned by an external gate voltage.\",\n \"This degree of freedom enables investigation of the interplay between ferroelectricity and free carriers, a previously unexplored regime.\",\n \"Here, we employ capacitive sensing in dual-gated mesoscopic devices of bilayer WTe2 to directly measure the spontaneous polarization in the metallic state and quantify the effect of free carriers on the polarization in the conduction and valence bands, separately.\",\n \"We compare our results to a low-energy model for the electronic bands and identify the layer-polarized states that contribute to transport and polarization simultaneously.\",\n \"Bilayer WTe2 is thus shown to be a canonical example of a ferroelectric metal and an ideal platform for exploring polar ordering, ferroelectric transitions, and applications in the presence of free carriers.\",\n \"Polar materials exhibit charge separation in the absence of an applied electric field, an effect of broken inversion symmetry and a unique polar axis in the crystal , .\",\n \"In certain polar systems, the charge polarization can be switched by an external electric field, an effect known as ferroelectricity.\",\n \"In principle, the presence or absence of ferroelectric effects depends only on the crystal class and not the details of the electronic structure.\",\n \"Despite this, nearly all known conventional ferroelectrics are electrically insulating.\",\n \"Since the first theoretical proposals for ferroelectric metals in 1965 , only a handful of experimental claims of ferroelectric-like phases in metallic systems have been reported , , , , , and no clear case for a canonical ferroelectric metal has emerged.\",\n \"Many such claims fail to demonstrate two key signatures of ferroelectric behavior, direct evidence of the polarization and ferroelectric switching, due to bulk screening effects.\",\n \"Here, we focus on the polar, semimetallic van der Waals crystal, T$_\\\\text{d}$ -WTe2, in the limit of two atomic layers, thin enough to admit an external electric field (Fig.\",\n \"REFd-e).\",\n \"Few-layer crystals of WTe2 have drawn recent interest for exhibiting a wide variety of low-temperature phases , , , , .\",\n \"Recent transport measurements showed that bilayer (2L) and trilayer (3L) WTe2 exhibit intrinsic, switchable electrical polarization in the conducting state , and separately, surfaces of bulk WTe2 crystals display hysteresis in piezoresponse force microscopy .\",\n \"Subsequent first-principles calculations indicated that the net polarization points only in the out-of-plane direction, and that the underlying mechanism results from a subtle interlayer sliding between the layers in two stable configurations , .\",\n \"These findings are exciting given the semimetallic and tunable nature of bilayer WTe2, which enables reaching both electron and hole bands by electrostatic gating in the ferroelectric state.\",\n \"While the hysteretic behavior observed in previous experiments is promising, a direct measurement of the metallic polarization is still missing.\",\n \"Due to methodological limitations it was previously not possible to measure the polarization while varying a pure electric field.\",\n \"More importantly, these limitations also prevented observing the effect of free carriers on the polarization and its dependence on carrier density, a fundamental open question for ferroelectric metals.\",\n \"In this work, we directly measure the charge polarization and electronic compressibility as a function of density for electrons and holes with independent control of the electric field.\",\n \"We study the simplest polar WTe2 system, a bilayer, via capacitive sensing in a dual-gated, planar capacitance device (Fig.\",\n \"REFb).\",\n \"Capacitance measures the electronic compressibility (and thus metallicity) of a 2D system.\",\n \"In a bilayer 2D system the top-gate and bottom-gate capacitances provide a direct measurement of the layer-specific charge distribution , and thus the out-of-plane polarization.\",\n \"Furthermore, the parallel-plate geometry enables this charge sensing with simultaneous and independent control of the vertical electric field and the carrier density in the bilayer by electrostatic gating.\",\n \"Our devices each consist of a bilayer WTe2 crystal encapsulated by two hexagonal boron nitride (hBN) dielectric layers, with metallic top and bottom gates, and contacts integrated into the top hBN layer (Fig REFa).\",\n \"Figure: Parameters employed in 𝐤·𝐩\\\\mathbf {k \\\\cdot p} calculation, following Eqs.\",\n \"–.These parameters are obtained by fitting the polarization from the low-energy model to Fig.\",\n \"a, and hence are different from those in Ref.\",\n \"du2018band.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05261"}}},{"rowIdx":3682,"cells":{"text":{"kind":"list like","value":[["Decomposition of $(2k+1)$-regular graphs containing special spanning\n $2k$-regular Cayley graphs into paths of length $2k+1$"],["Abstract A $P_\\ell$-decomposition of a graph $G$ is a set of paths with $\\ell$ edges in $G$ that cover the edge set of $G$.","Favaron, Genest, and Kouider (2010) conjectured that every $(2k+1)$-regular graph that contains a perfect matching admits a $P_{2k+1}$-decomposition.","They also verified this conjecture for $5$-regular graphs without cycles of length $4$.","In 2015, Botler, Mota, and Wakabayashi verified this conjecture for $5$-regular graphs without triangles.","In this paper, we verify it for $(2k+1)$-regular graphs that contain the $k$th power of a spanning cycle; and for $5$-regular graphs that contain special spanning $4$-regular Cayley graphs."],["Introduction","All graphs in this paper are simple, i.e., have no loops nor multiple edges.","A decomposition of a graph $G$ is a set $of edge-disjoint subgraphs of $ G$ that cover its edge set.If every element of $ is isomorphic to a fixed graph $H$ , then we say that $ is an \\emph {$ H$-decomposition}.In this paper, we focus on the case $ H$is the simple path with $ 2k+1$ edges, which we denote by $ P2k+1$.Note that this notation is not standard.In 1957, Kotzig~\\cite {Ko57} (see also~\\cite {BoFo83})proved that a $ 3$-regular graph $ G$admits a $ P3$-decomposition if and only if $ G$ contains a perfect matching.In 2010, Favaron, Genest, and Kouider~\\cite {FaGeKo10} extended this result by provingthat every $ 5$-regular graph thatcontains a perfect matching and no cycles of length $ 4$admits a $ P5$-decomposition;and proposed the following conjecture toextend Kotzig^{\\prime }s result.\\begin{conjecture}[Favaron--Genest--Kouider, 2010]If G is a (2k+1)-regular graph that contains a perfect matching,then G admits a P_{2k+1}-decomposition.\\end{conjecture}$ In 2015, Botler, Mota, and Wakabayashi [3] verified Conjecture for triangle-free 5-regular, and, more recently, Botler, Mota, Oshiro, and Wakabayashi [2] generalized this result for $(2k+1)$ -regular graphs with girth at least $2k$ .","It is clear that a 5-regular graph contains a perfect matching if and only if it contains a spanning 4-regular graph.","In fact, by using a theorem of Petersen [10], one can prove that a $(2k+1)$ -regular graph contains a perfect matching if and only if it contains a spanning $2k^{\\prime }$ -regular graph for every $k^{\\prime }\\le k$ .","Theorem 1 (Petersen, 1891) If $G$ is a $2k$ -regular graph, then $G$ admits a decomposition into spanning 2-regular graphs.","In this paper, we explore Conjecture for $(2k+1)$ -regular graphs that contain special spanning $2k$ -regular graphs as follows.","Throughout the text, $\\Gamma $ denotes a finite group of order $n$ ; $+$ denotes the group operation of $\\Gamma $ ; and 0 denotes the identity of $\\Gamma $ .","As usual, for each $x\\in \\Gamma $ , we denote by $-x$ the inverse of $x$ , i.e., the element $y\\in \\Gamma $ for which $x+y=0$ , and the operation $-$ denotes the default binary operation $(x,y)\\mapsto x+(-y)$ .","Let $S\\subseteq \\Gamma $ be a set not containing 0, and such that $-x\\in S$ for every $x\\in S$ (i.e., $S$ is closed under taking inverses).","The Cayley graph $X(\\Gamma ,S)$ is the graph $H$ with $V(H)=\\Gamma $ , and $E(H) = \\big \\lbrace xy\\colon y-x\\in S\\big \\rbrace $ (see [7]).","In this paper, we allow $S$ to be a set not generating $\\Gamma $ , and hence $X(\\Gamma ,S)$ is not necessarily connected.","We say that $H$ is simply commutative if (i) $x + y = y + x$ for every $x,y\\in S$ , and (ii) $-x\\ne x$ for every $x\\in S$ .","Condition (ii) implies that $H$ has no multiple edges, and, since $0\\notin S$ , $H$ is simple.","It is not hard to check that, in such a graph, the neighborhood of a vertex $v\\in \\Gamma $ is $N(v) = \\lbrace v+x\\colon x\\in S\\rbrace $ .","Although the definition of Cayley graphs can be extended to multigraphs and directed graphs, Conjecture considers only simple graphs.","In fact, we explore some structure of the colored directed Cayley graph (see [5]) in which the edge set consists of the pairs $(x,x+s)$ with color $s\\in S$ .","We present two results regarding Conjecture .","We verify it for $(2k+1)$ -regular graphs that contain the $k$ th power of a spanning cycle (see Section ); and for 5-regular graphs that contain spanning simply commutative 4-regular Cayley graphs (see Section ).","Since the graphs in these classes may contain cycles of lengths 3 and 4, these results extend the family of graphs for which Conjecture is known to hold.","We believe that, due to the underlying group structure, the techniques developed here can be extended for dealing with $(2k+1)$ -regular graphs that contain more general spanning Cayley graphs, and also $(2k+1)$ -regular graphs that contain special spanning Schreier graphs, which could give us significant insight with respect to the general case of Conjecture (see Section ).","Notation.","A graph $T$ is a trail if there is a sequence $x_0,\\ldots , x_{\\ell }$ of its vertices for which $E(T)=\\lbrace x_{i}x_{i+1}\\colon 0\\le i \\le \\ell -1\\rbrace $ and $x_{i}x_{i+1}\\ne x_{j}x_{j+1}$ , for every $i\\ne j$ .","Moreover, if $x_i\\ne x_j$ for every $i \\ne j$ , we say that $T$ is a path.","A subgraph $F$ of a graph $G$ is a factor of $G$ or a spanning subgraph of $G$ , if $V(F)=V(G)$ .","If, additionally, $F$ is $r$ -regular, then we say that $F$ is an $r$ -factor.","In particular, a perfect matching is the edge set of a 1-factor.","Moreover, we say that $F$ is an $H$ -factor if $F$ is a factor that consists of vertex-disjoints copies of $H$ .","The reader may refer to [1] for standard definitions of graph theory."],["Regular graphs that contain powers of cycles","Given a perfect matching $M$ in a graph $G$ , we say that a $P_{\\ell }$ -decomposition $ of a graph $ G$ is$ M$-\\emph {centered} if for every$ P=a0a1a-1a , we have $a_{i}a_{i+1} \\in M$ for $i=(\\ell -1)/2$ , i.e., if the middle edges of the paths in $ are precisely the edges of $ M$.The next results are examples of $ M$-centered decompositionthat are used in the proof of Theorems~\\ref {theorem:no-g2r2} and~\\ref {theorem:2g2r!=0}.$ Proposition 2 If $G$ is a 5-regular graph that contains a spanning copy $K$ of $K_{4,4}$ , and ${M = E(G)\\setminus E(K)}$ , then $G$ admits an $M$ -centered $P_5$ -decomposition.","Let $G$ , $K$ , and $M$ be as in the statement.","Let $(R,L)$ be the bipartition of $K$ , where $R=\\lbrace r_1,r_2,r_3,r_4\\rbrace $ and $L=\\lbrace l_1,l_2,l_3,l_4\\rbrace $ .","Since $K$ is a complete bipartite graph, if $xy\\in M$ , then either $x,y \\in R$ or $x,y \\in L$ .","Thus, we may assume, without loss of generality, that $M=\\lbrace r_1r_2,r_3r_4,l_1l_2,l_3l_4\\rbrace $ , and hence, ${\\lbrace l_1r_1l_3l_4r_2l_2,~l_3r_3l_1l_2r_4l_4,~r_1l_2r_3r_4l_1r_2,~r_3l_4r_1r_2l_3r_4\\rbrace }$ is an $M$ -centered decomposition of $G$ as desired (see Figure REF ).","Figure: P 5 P_5-decomposition of a 5-regular graph that contains a spanning copy of a K 4,4 K_{4,4}.Given positive integers $k$ and $n$ , the $k$ th power of the cycle on $n$ vertices, denoted by $C_n^k$ , is the graph on the vertex set $\\lbrace 0,\\ldots ,n-1\\rbrace $ and such that, for every vertex $v$ , we have $x\\in N(v)$ if and only if $x = v+r\\pmod {n}$ , where $r\\in \\lbrace -k,\\ldots ,-1\\rbrace \\cup \\lbrace 1,\\ldots ,k\\rbrace $ .","Proposition 3 Let $n$ and $k$ be positive integers for which $k < n/2$ .","If $G$ is a simple $(2k+1)$ -regular graph on $n$ vertices that contains a copy $C$ of $C_n^k$ , and $M = E(G) \\setminus E(C)$ , then $G$ admits an $M$ -centered $P_{2k+1}$ -decomposition.","Let $G$ , $C$ , and $M$ be as in the statement, and let $V(C) = \\lbrace 0,\\ldots , n-1\\rbrace $ as above.","The operations on the vertices of $C$ are taken modulo $n$ .","Since $C$ is a $2k$ -regular graph, $M$ is a perfect matching of $G$ .","Given $i\\in V(C)$ , let $Q_i$ be the path $v_0v_1\\cdots v_k$ in which $v_0=i$ ; and, for $j=1,\\ldots ,k$ , we have $v_j = v_{j-1}+j$ if $j$ is odd; and $v_j = v_{j-1}-j$ if $j$ is even (see Figure REF ).","Note that for every $j=1,\\ldots ,k$ , the path $Q_i$ contains an edge $xy$ such that $|x-y|=j$ .","Also, we have $V(Q_i) = \\big \\lbrace i+r\\colon r\\in \\lbrace -\\lfloor k/2\\rfloor ,-\\lfloor k/2\\rfloor +1,\\ldots , \\lceil k/2\\rceil \\rbrace \\big \\rbrace $ .","It is not hard to check that the set $\\mathcal {Q}=\\big \\lbrace Q_i\\colon i\\in V(C)\\big \\rbrace $ is a $P_k$ -decomposition of $C$ .","Given an edge $e=ij\\in M$ , let $P_e = Q_i \\cup \\lbrace ij\\rbrace \\cup Q_j$ .","Since $Q_i$ and $Q_j$ have, respectively, $i$ and $j$ as end vertices, and $E(Q_i)\\cap E(Q_j)=\\emptyset $ , the graph $P_e$ is a trail of length $2k+1$ .","Thus, since $\\mathcal {Q}$ is a $P_k$ -decomposition of $C$ , and $M$ is a perfect matching of $G$ , the set $\\lbrace P_e\\colon e\\in M\\rbrace $ is a decomposition of $G$ into trails of length $2k+1$ .","We claim that, in fact, $is a $ P2k+1$-decomposition of $ G$.For that, we prove that if $ ijM$,then $ V(Qi)V(Qj)=$.Indeed, note that for every $ e=ijM$,we have $ |i-j|>k$, otherwise we have $ ijE(C)$.Now, suppose that there is a vertex $ v$ in $ V(Qi)V(Qj)$.Then, there are $ r1,r2$ with$ -k/2r1,r2k/2$,and such that $ i+r1 = v = j+r2$.Suppose, without loss of generality, that $ i>j$.Then, we have $ r2-r1 = i-j > k$,but $ r2-r1 k/2+k/2= k$,a contradiction.$ Figure: The path P uv P_{uv}, with u=2u=2 and v=8,v=8, in the proof of Proposition for a 7-regular graph that contains a spanning copy of a C 10 3 C^3_{10}.Note that, from the proof of Proposition REF we also obtain a construction for the Hamilton path decomposition of complete graphs of even order.","Corollary 3.1 Let $\\ell $ be odd.","The complete graph $K_{\\ell +1}$ admits a $P_\\ell $ -decomposition.","A slight variation of the proof of Proposition REF also provides the following result.","Proposition 4 Let $\\ell $ be odd, and let $G$ be an $\\ell $ -regular graph with a perfect matching $M$ .","If each component of $G\\setminus M$ is the $\\frac{\\ell -1}{2}$ -th power of a cycle, then $G$ admits an $M$ -centered $P_\\ell $ -decomposition.","Let $G_1$ and $G_2$ be disjoint graphs with perfect matchings $M_1$ and $M_2$ , respectively.","Let $a_1b_1,\\ldots ,a_kb_k \\in M_1$ and $x_1y_1,\\ldots , x_ky_k\\in M_2$ be distinct edges, and let $G$ be the graph obtained from the disjoint union $G_1\\cup G_2$ by removing $a_1b_1,\\ldots ,a_kb_k,x_1y_1,\\ldots , x_ky_k$ and adding the edges $a_1x_1,b_1y_1,\\ldots ,a_kx_k,b_ky_k$ .","We say that $G$ is a collage of $G_1$ and $G_2$ over edges of $M_1$ and $M_2$ , and denote by $M_G$ the perfect matching ${\\big (M_1\\cup M_2 \\cup \\lbrace a_1x_1,b_1y_1,\\ldots ,a_kx_k,b_ky_k\\rbrace \\big ) \\setminus \\lbrace a_1b_1,\\ldots ,a_kb_k,x_1y_1,\\ldots , x_ky_k\\rbrace }$ .","When $M_1$ and $M_2$ are clear from the context, we say simply that $G$ is a collage of $G_1$ and $G_2$ .","Note that $G$ is $\\ell $ -regular if and only if $G_1$ and $G_2$ are $\\ell $ -regular.","Let $G$ be an $\\ell $ -regular graph, where $\\ell $ is an odd positive integer, and let $M$ be a perfect matching of $G$ .","We say that $G$ is $M$ -constructable if either $G$ admits an $M$ -centered $P_\\ell $ -decomposition, or $G$ is the collage of an $M_1$ -constructable graph and an $M_2$ -constructable graph over edges of $M_1$ and $M_2$ and $M=M_G$ .","The next straightforward result is a useful tool in the proof of Theorem REF .","Lemma 5 Let $\\ell $ be odd, and $G$ be an $\\ell $ -regular graph.","If $G$ is $M$ -constructable, then $G$ admits an $M$ -centered $P_\\ell $ -decomposition.","The proof follows by induction on $|V(G)|$ .","By the definition of $M$ -constructable, we may assume that $G$ is the collage of an $M_1$ -constructable graph $G_1$ and an $M_2$ -constructable graph $G_2$ over edges of $M_1$ and $M_2$ .","By the induction hypothesis, $G_i$ admits an $M_i$ -centered $P_\\ell $ -decomposition $i$ , for $i=1,2$ .","Let $a_i,b_i,x_i,y_i$ , for $i=1,\\ldots ,k$ be such that $G$ is the graph obtained from $G_1\\cup G_2$ by removing $a_1b_1,\\ldots ,a_kb_k,x_1y_1,\\ldots , x_ky_k$ and adding $a_1x_1,b_1y_1,\\ldots ,a_kx_k,b_ky_k$ as above.","For ${i=1,\\ldots ,k}$ , let $P_i\\in 1$ and $Q_i\\in 2$ be the paths containing the edges $a_ib_i$ and $x_iy_i$ , respectively.","By the definition of $M_1$ - and $M_2$ -centered $P_\\ell $ -decomposition, for $i=1,\\ldots ,k$ , we may write $P_i = P_{i,1}a_ib_i P_{i,2}$ and $Q_i = Q_{i,1}x_iy_iQ_{i,2}$ , where $P_{i,1},P_{i,2},Q_{i,1}$ and $Q_{i,2}$ are paths of length $(\\ell -1)/2$ .","Since $G_1$ and $G_2$ are disjoint, $V(P_{i,j})\\cap V(Q_{i,j}) = \\emptyset $ for $i=1,\\ldots ,k$ and $j=1,2$ .","Let $R_{i,1} = P_{i,1}\\cup \\lbrace a_ix_i\\rbrace \\cup Q_{i,1}$ and $R_{i,2} = P_{i,2}\\cup \\lbrace b_iy_i\\rbrace \\cup Q_{i,2}$ , and note that $ \\big (1\\setminus \\lbrace P_1,\\ldots ,P_k\\rbrace \\big )\\cup \\big (2\\setminus \\lbrace Q_1,\\ldots ,Q_k\\rbrace \\big )\\cup \\lbrace R_{i,j}\\colon i=1,\\ldots ,k\\text{ and }j=1,2\\rbrace $ is an $M_G$ -centered $P_\\ell $ -decomposition of $G$ as desired.","By Proposition REF , if $G$ contains a spanning copy $K$ of $K_{4,4}$ and $M=E(G)\\setminus E(K)$ , then $G$ is $M$ -constructable.","Therefore, Lemma REF yields the following result.","Corollary 5.1 If $G$ is a 5-regular graph that contains a $K_{4,4}$ -factor $K$ and ${M=E(G)\\setminus E(K)}$ , then $G$ admits an $M$ -centered $P_5$ -decomposition."],["5-regular graphs that contain Cayley graphs","In this section, we explore 5-regular graphs that contain spanning simply commutative 4-regular Cayley graphs.","Botler, Mota, and Wakabayashi [3] showed that every triangle-free 5-regular graph $G$ that has a perfect matching admits a $P_5$ -decomposition.","For that, they applied the following strategy: i) to find an initial decomposition of $G$ into paths and trails; and ii) to perform exchanges of edges between the elements of $, preserving a special invariant, while minimizing the number of trails that are not paths.$ The proof of our main result (Theorem REF ) consists of four steps.","First, we deal with a somehow degenerate case (Theorem REF ).","After that, we follow the framework used by Botler, Mota, Wakabayashi [3], i.e, from the structure of Cayley graphs, we find an initial decomposition into trails, not necessarily paths (Proposition REF ), and then we exchange edges between the elements of the decomposition in order to reduce the number of bad elements (the trails that are not paths).","For that, we first show that the bad elements are distributed in a circular fashion (Lemma REF ), and then we show how to deal with these “cycles of bad elements” (Theorem REF ).","The invariants preserved by the operations in the proofs of Lemma REF and Theorem REF are presented, respectively, in Definitions REF and REF .","The following lemma is used often throughout the text.","Lemma 6 Let $\\ell $ be odd, and $G$ be an $\\ell $ -regular graph.","If $ is a decomposition of~$ G$ into trails of length $$,then each vertex of $ G$ is the end vertex of precisely one element of~$ .","Let $k$ , $G$ and $ be as in the statement.Let $ n=|V(G)|$.Given an element $ T, we denote by $o(T)$ the number of vertices $v$ in $T$ for which $d_T(v)$ is odd, and given a vertex $v\\in V(G)$ , we denote by $v)$ the number of trails in $ for which $ dT(v)$ is odd.Clearly, $ T o(T) = vV(G)v)$.Moreover, for every trail $ T$, we have $ o(T)2$.Also, since every element of $ has $\\ell $ edges, we have $| = \\frac{1}{\\ell }|E(G)|= \\frac{1}{\\ell }\\frac{1}{2}\\ell n = \\frac{1}{2}n$ .","Thus, we have $\\sum _{T\\in o(T) \\le 2| = n.Now, since v\\in V(G) has odd degree (in G),v must have odd degree in at least one element of ,and hence v) \\ge 1.Thus, we have \\sum _{v\\in V(G)}v)\\ge n,and hence n \\le \\sum _{v\\in V(G)}D(v) = \\sum _{T\\in o(T) \\le n.This implies that v) = 1 for every v\\in V(G), as desired.","}}Recall that $$ is a finite group of order $ n$ and operation $ +$.Fix two elements $ g,r$ of~$$,we say that $ {g,r}$ is a \\emph {simple commutative generator} (SCG)if (a) $ 0{g, r, 2g, 2r}$;(b) $ g{r,-r}$;and (c) $ g+r = r+g$.Let $ {g,r}$ be an SCG, put $ S = {g, -g, r, -r}$, and consider the Cayley graph $ X=X(,S)$.By construction, $ X$ is a simply commutative Cayley graph (see Section~\\ref {sec:introduction}).Conditions (a) and (b) guarantee that $ X$ is a simple graph,while condition (c) introduces the main restriction explored in this paper.In this case, we say that~$ X$ is the graph \\emph {generated} by $ {g,r}$,and that $ {g,r}$ is the \\emph {generator} of $ X$.Finally, we say that a simple $ 5$-regular graph $ G$ with vertex set $$is a \\emph {$ {g,r}$-graph}if $ G$ contains a spanning Cayley graph $ X$ generated by $ {g,r}$.We say that $ G$ is a \\emph {simply commutative generated graph} or, for short, \\emph {SCG-graph} if~$ G$is a $ {g,r}$-graph for some SCG $ {g,r}$.In this section, we verify Conjecture~\\ref {conj:favaron} for SCG-graphs.","In particular,Proposition~\\ref {lemma:cycle-factorization} implies that every $ {g, r}$-graph for which $ 2g = r$ admits an $ Mg,r$-centered decomposition;and as a consequence of Corollary~\\ref {corollary:K44}, we obtain the following result,which is also a special case of our main result.$ Theorem 7 Every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r=0$ and $2g-2r=0$ admits an $M_{g,r}$ -centered decomposition.","Let $G$ be a $\\lbrace g,r\\rbrace $ -graph for which $2g+2r=0$ and $2g-2r=0$ and put ${M=M_{g,r}}$ .","Note that we also have $4g=4r=0$ .","Let $u$ be a vertex of $G$ , and let $H$ be the component of $G\\setminus E(M)$ that contains $u$ .","In what follows, we prove that $H$ is a copy of $K_{4,4}$ .","Since $g$ and $r$ commute, if $v\\in V(H)$ , we have $v = u + ig + jr$ , where $i,j\\in \\mathbb {N}$ .","Since $4g=4r=0$ , we may assume $i,j\\in \\lbrace 0,1,2,3\\rbrace $ .","Moreover, since $2g - 2r = 0$ (and hence $2g=2r$ ), we may assume $j\\in \\lbrace 0,1\\rbrace $ .","Therefore, there are at most eight vertices in $H$ , namely, $V(H) = \\lbrace u, u+g,u+2g,u+3g, u+r,u+2g+r,u+3g+r\\rbrace $ .","We claim that $H$ is bipartite.","Indeed, suppose that there is an odd cycle $C$ in $H$ .","Then, there is an element $x \\in V(C)$ such that $x+ig+jr=x$ , where $i,j \\in \\mathbb {N}$ .","Note that $i+j$ can be obtained from the length of $C$ by ignoring pairs of edges with the same color and different directions.","Since $C$ is odd, precisely one between $i$ and $j$ is odd.","Suppose, without loss of generality, that $i$ is odd, and hence $j$ is even.","Note that, since $2g=2r$ , we have $jr = jg$ .","Therefore, $(i+j)g = ig + jr = 0$ .","Let $s\\in \\lbrace 1,3\\rbrace $ be such that $i+j = 4q + s$ for some $q\\in \\mathbb {N}$ .","Then we have $0 = (i+j)g = 4qg + sg$ , which implies $sg = 0$ .","Thus, if $s = 1$ , then $g=0$ ; and if $s = 3$ , then $g = 4g-sg = 0$ , a contradiction to the definition of SCG.","Thus, since $H$ is 4-regular, $H$ is a copy of $K_{4,4}$ .","Now, since every component of $G\\setminus E(M)$ is isomorphic to $K_{4,4}$ .","Therefore, $G$ is a 5-regular graph that contains a $K_{4,4}$ -factor, and hence by Corollary REF , $G$ admits an $M$ -centered decomposition as desired.","If $X$ is the graph generated by an SCG $\\lbrace g,r\\rbrace $ , and $x\\in \\lbrace g,r\\rbrace $ , then we denote by $F_x$ the 2-factor of $X$ with edge set $E(F_x) = \\lbrace v+x\\colon v\\in \\Gamma \\rbrace $ .","If $G$ is a $\\lbrace g,r\\rbrace $ -graph, then we denote by $M_{g,r}$ the perfect matching $G\\setminus E(F_g \\cup F_r)$ , and the triple $\\lbrace M_{g,r},F_g,F_r\\rbrace $ is called the base factorization of $G$ .","Although $G$ is a simple graph, for ease of notation, we refer to an edge $uv\\in F_x$ , with $x\\in \\lbrace g,r\\rbrace $ , as a green (resp.","red) out edge of $u$ and in edge of $v$ if $v = u+x$ and $x=g$ (resp.","$x=r$ ).","In the figures throughout the text, the edges in $F_g$ , $F_r$ , $M_{g,r}$ are illustrated, respectively, in dotted green, dashed red, and double black patterns, while edges without specific affiliation are illustrated in straight gray pattern.","Moreover, if such an edge has a specific direction (i.e., in edge or out edge), it is illustrated accordingly.","Note that each vertex $u$ has precisely one edge of each type (green in edge, green out edge, red in edge, red out edge), and is incident to precisely one edge of $M_{g,r}$ .","In particular, the group structure overcomes Theorem REF by giving a decomposition of $X$ into 2-factors in terms of the elements $g$ and $r$ .","In the rest of the paper we deal with the case $2g+2r\\ne 0$ .","For that, we characterize the elements of the forthcoming decompositions.","Definition 1 We say that a trail $T$ in a $\\lbrace g,r\\rbrace $ -graph is of type A, B, C, or D if $T$ can be written as $a_0a_1a_2a_3a_4a_5$ , where $a_0,a_1,a_2,a_3,a_4$ are distinct vertices, as follows.","$a_2 = a_5$ , $a_2a_3\\in M_{g,r}$ , $a_2a_1,a_3a_4\\in F_g$ , $a_4a_5 \\in F_r$ , and $a_1a_0\\in F_g\\cup F_r\\cup M_{g,r}$ , i.e., $a_1a_0$ is an out edge of $a_1$ , or $a_1a_0\\in M_{g,r}$ (see Figure REF fig:typeA).","In this case, we say that $a_3$ is the primary connection vertex of $T$ , $a_2$ is the secondary connection vertex of $T$ ; $a_1$ is the auxiliary vertex of $T$ ; and $a_4$ is the tricky vertex of $T$ .","We denote these vertices, respectively, by ${\\rm {cv}}_1(T)$ , ${\\rm {cv}}_2(T)$ , ${\\rm {aux}}(T)$ , and ${\\rm {tr}}(T)$ ; $a_5\\notin \\lbrace a_0,a_1,a_2,a_3,a_4\\rbrace $ , $a_2a_3\\in M_{g,r}$ $a_2a_1,a_3a_4\\in F_g$ , $a_1a_0,a_4a_5\\in F_g\\cup F_r\\cup M_{g,r}$ (see Figure REF fig:typeB); $a_5\\notin \\lbrace a_0,a_1,a_2,a_3,a_4\\rbrace $ , $a_2a_1,a_4a_3\\in F_g$ , $a_3a_2,a_4a_5\\in F_r$ , $a_1a_0\\in F_g\\cup F_r\\cup M_{g,r}$ , and, moreover, we have $a_2a_4\\in E(G)$ and $a_2a_4 \\in M_{g,r}$ (see Figure REF fig:typeC); $a_5\\notin \\lbrace a_0,a_1,a_2,a_3,a_4\\rbrace $ , $a_1a_0,a_4a_5\\in F_r$ , $a_1a_2,a_3a_4\\in M_{g,r}$ , and $a_3a_2\\in F_g$ (see Figure REF fig:typeD).","Figure: The main types of trails.We remark that elements of type A are not paths, while elements of type B, C, and D are paths.","Moreover, the connection vertices are defined only for elements of type A, and the connection vertices of an element $T$ are always incident to an edge of $M_{g,r}$ in $T$ , and hence, no vertex of a $\\lbrace g,r\\rbrace $ -graph is a connection vertex of two edge-disjoint elements of type A in a graph.","Given a trail (not necessarily a path) $T= a_0a_1a_2a_3a_4a_5$ in a decomposition $ of a $ {g,r}$-graph $ G$,we say that the edge $ a1a0$ (resp.","$ a4a5$) is a \\emph {hanging edge}at $ a1$ (resp.","$ a4$) if $ a1a0Mg,rFgFr$ (resp.","$ a4a5Mg,rFgFr$),i.e., the hanging edges of $ T$ are the end edges of $ T$ that are in $ Mg,r$ or that are in edges of its end vertices.By Definition~\\ref {def:types}, all end edges of elements of type~A, B, C, or D are hanging edges.Note that if $ T$ is an element of type~A where $ a5 = a2$, then $ a1a0$, $ a2a3$ and $ a4a2$ are hanging edgesof $ T$ at, respectively, $ a1$, $ a3$, and $ a4$.Given a trail decomposition $ of a graph $G$ and a vertex $u\\in V(G)$ , we denote by ${\\rm {hang}}_u)$ the number of edges of $G$ that are hanging edges at $u$ .","The next lemma presents a consequence of the exchange of hanging edges at primary connection vertices.","Lemma 8 If $T= a_0a_1a_2a_3a_4a_5$ is an element of type A in a decomposition of a $\\lbrace g,r\\rbrace $ -graph $G$ into trails of length 5, where $a_5=a_2$ and $a_2a_3\\in M_{g,r}$ , and $u\\in V(G)$ is such that $a_3u$ is a hanging edge at $a_3 = {\\rm {cv}}_1(T)$ , then $T^{\\prime }= a_0a_1a_2a_4a_3u$ is of type C. Let $T$ , $u$ , and $T^{\\prime }$ be as in the statement.","Since $a_3a_4$ is a green out edge of $a_3$ and $a_2a_3$ is an edge of $M_{g,r}$ incident to $a_3$ , we conclude that $a_3u$ is a red out edge of $a_3$ , and hence $u = a_3 + r$ .","Now, since $G$ is simple, we have $u\\notin \\lbrace a_2,a_3,a_4\\rbrace $ ; if $u=a_1$ , then we have $a_3 + r= u = a_1 = a_3 + g + r + g$ , which implies $2g=0$ , a contradiction to the definition of SCG.","Finally, by Lemma REF we have $u\\ne a_0$ .","Thus, $T^{\\prime }$ is a path.","Since $a_3u\\in F_r$ , $T^{\\prime }$ is of type C. Figure: Exchange of edges performed in the proof of Lemma .The following lemma shows how two elements of type A may be connected.","Lemma 9 If $T_1$ and $T_2$ are two edge-disjoint elements of type A in a $\\lbrace g,r\\rbrace $ -graph $G$ for which ${\\rm {tr}}(T_2) = {\\rm {cv}}_1(T_1)$ , then ${\\rm {aux}}(T_2) = {\\rm {cv}}_2(T_1)$ .","Let $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ , where $a_5 = a_2$ and $b_5 = b_2$ and $a_2a_3,b_2b_3\\in M_{g,r}$ .","If ${\\rm {cv}}_1(T_1)={\\rm {tr}}(T_2)$ , then $a_3=b_4$ .","Since $b_1 = b_4 + r+g$ and $a_2 = a_3 + g + r$ .","Thus, ${\\rm {aux}}(T_2) = b_1 = b_4 + r + g = a_3 + r + g = a_2 = {\\rm {cv}}_2(T_1)$ , as desired (see Figure REF ).","Figure: Identities given by Lemma when b 3 =0b_3 = 0."],["Complete decompositions","The following definition consists of two properties that are invariant under a series of operations performed throughout the proof of Lemma REF .","Definition 2 A decomposition $ of a $ {g,r}$-graph $ G$into trails of length $ 5$is \\emph {complete} if the following hold for every $ T. $T$ is of type A, B, C or D; If $T$ is of type A, then ${\\rm {hang}}_({\\rm {cv}}_1(T)\\big )\\ge 2$ and ${\\rm {hang}}_({\\rm {cv}}_2(T)\\big )\\ge 1$ .","The first step of our proof is given by the next proposition, which presents an initial decomposition for the graphs studied.","Proposition 10 If $G$ is a $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ , then $G$ admits a complete decomposition.","Let $\\lbrace M_{g,r},F_g,F_r\\rbrace $ be the base factorization of $G$ .","For each $e=xy\\in M_{g,r}$ , let $P_e=a_0a_1a_2a_3a_4a_5$ , where $a_1a_0,a_4a_5\\in F_r$ , $a_2a_1,a_3a_4\\in F_g$ , $a_2=x$ , and $a_3 = y$ .","We claim that $ \\lbrace P_e\\colon e\\in M_{g,r}\\rbrace $ is complete.","Clearly, $P_e$ is an element of type A or B, for every $e\\in M_{g,r}$ , and hence $ satisfiesDefinition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-types}).Moreover, note that $ a0a1$ (resp.","$ a4a5$) is a hanging edgeat $ a1$ (resp.","$ a4$).Thus, given $ zV(G)$, let $ e' = xyMg,r$ be such that $ x= z-g$,then $ Pe'$ contains a hanging edge at $ z$,namely, the red out edge of $ z$,and hence there is a hanging edge at every vertex of $ G$.Moreover, if $ z=cv1(T)$ for some element $ T of type A, and $e\\in M_{g,r}\\cap E(T)$ , then $e$ is a second hanging edge at $z$ .","This proves Definition REF (REF ).","We say that an element $T$ of type A in a decomposition $ is \\emph {free}if $ tr(T)cvi(T')$ for every element $ T' of type A and $i\\in \\lbrace 1,2\\rbrace $ .","An A-chain is a sequence $T_0, T_1, \\ldots , T_{s-1}$ of elements of type A such that for each $j\\in \\lbrace 0,\\ldots ,s-1\\rbrace $ , we have ${\\rm {tr}}(T_j) = {\\rm {cv}}_i(T_{j-1})$ , for some $i \\in \\lbrace 1,2\\rbrace $ (subtraction on the indexes are taken modulo $s$ ).","Note that A-chains do not consider the auxiliary vertex when allowing two elements to be consecutive.","Thus, elements, say $T$ and $T^{\\prime }$ , of type A that are not consecutive in an A-chain, or that are in different A-chains, may still share a vertex $u$ for which ${\\rm {cv}}_i(T) = u = {\\rm {aux}}(T^{\\prime })$ .","Given a decomposition $ of a graph $ G$ into trails of length $ 5$, denote by$ ($ the number of elements that are not paths.By exchanging edges between the elements of a decompositiongiven by Proposition~\\ref {proposition:initial-decomposition},we can show that a complete decomposition that minimizes $ ($has no free element, and hence its elements of type~A are partitionedinto A-chains.$ Lemma 11 Every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ admits a complete decomposition in which the elements of type A are partitioned into A-chains.","Let $g$ and $r$ be as in the statement, let $G$ be a $\\lbrace g,r\\rbrace $ -graph, and put ${M=M_{g,r}}$ .","By Proposition REF , $G$ admits a complete decomposition.","Let $ be a complete decomposition of~$ G$ that minimizes $ ($.In what follows, we prove that $ contains no free element of type A.","For that, we prove three claims regarding the relation between some elements of $.In the proof of each such claim, we exchange edges between some elements of $ and obtain a complete decomposition $$ into trails of length 5 such that $\\tau () <\\tau ($ , which is a contradiction to the minimality of $.To check that $$ is a complete decomposition, we observe the two following items:(i) the vertices $ u$ for which $ hangD(u) >hang(u)$ are verticesthat are not connection vertices of $$,for example, tricky vertices of free elements of type~A,or connection vertices of elements of type~A in $ that become paths in $$ .","Hence, Definition REF (REF ) holds for $$ ; and (ii) every element of $$ that is not an element of $, i.e.,the elements involved in the exchange of edges, are of type~A, B, C, or~D,and hence \\ref {def:complete-commutative}(\\ref {def:complete-commutative-types})holds for $$.$ Claim 11.1 No element of type B or C has a hanging edge at the primary connection vertex of a free element of type A.","Let $T_1 = a_0a_1a_2a_3a_4a_5\\in , where $ a5=a2$ and $ a2a3M$, be a free element of type~A,and let $ T2= b0b1b2b3b4b5 be an element of type B or C that contains a hanging edge at ${\\rm {cv}}_1(T_1)$ .","We divide the proof depending on whether $T_2$ is of type B or C. $\\mathbf {T_2}$ is of type B.","Suppose, for a contradiction, that $b_4={\\rm {cv}}_1(T_1)=a_3$ .","Put ${T_1^{\\prime } = a_0a_1a_2a_4a_3b_5}$ , $T_2^{\\prime } = b_0b_1b_2b_3b_4a_2$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime }\\rbrace $ .","Note that $$ is a decomposition of $G$ into trails of length 5.","By Lemma REF , $T_1^{\\prime }$ is an element of Type C. In what follows, we prove that $T_2^{\\prime }$ is of type B, i.e., $a_2\\notin \\lbrace b_0,b_1,b_2,b_3,b_4\\rbrace $ .","Indeed, since $G$ has no loops or multiple edges, $a_2\\notin \\lbrace b_3,b_4\\rbrace $ .","Since $M$ is a matching, $a_2\\ne b_2$ .","If $a_2 = b_1$ , then $b_2 = b_1-g = a_2 -g = a_3 + g +r -g = b_5$ , and hence $T_2$ is of type A, a contradiction.","Finally, by Lemma REF , $a_2 \\ne b_0$ .","Thus, $T^{\\prime }_2$ is an element of type B, and hence Definition REF (REF ) holds for $$ .","Note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_4\\rbrace .Since a_4 is not a connection vertex of ,and by Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}),we have {\\rm {hang}}_{\\big ({\\rm {cv}}_1(T)\\big )\\ge 2 and {\\rm {hang}}_{\\big ({\\rm {cv}}_2(T)\\big )\\ge 1for every T\\in ,we have{\\rm {hang}}_{}\\big ({\\rm {cv}}_1(T)\\big )\\ge 2 and {\\rm {hang}}_{}\\big ({\\rm {cv}}_2(T)\\big )\\ge 1for every T\\in ,Thus Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}) holds for .Therefore, is a complete decomposition such that\\tau ()=\\tau (-1<\\tau (, a contradictionto the minimality of .","}\\begin{figure}[H]\\centering \\begin{subfigure}{.45}\\centering \\scalebox {.8}{\\begin{tikzpicture}[scale=1.5]\\end{tikzpicture}[fatpath,backcolor1] (120:1)-- ((-1,0)+(120:1)) -- (120:1) -- (0,0) -- (0:1) -- (60:1) -- (0,0) -- (60:1);}\\node (0) [black vertex] at ((-1,0)+(120:1)) {};\\node (1) [black vertex] at (120:1) {};\\node (2) [black vertex] at (0,0) {};\\node (3) [black vertex] at (0:1) {};\\node (4) [black vertex] at (60:1) {};\\node () [] at ((2,0)+(60:1)) {};\\end{subfigure}\\end{figure}\\node (4^{\\prime }) [black vertex] at (1,0) {};\\node (3^{\\prime }) [black vertex] at ((4^{\\prime })-(60:1)) {}; \\node (2^{\\prime }) [black vertex] at ((3^{\\prime })-(1,0)) {};\\node (1^{\\prime }) [black vertex] at ((2^{\\prime })+(120:1)) {};\\node (0^{\\prime }) [black vertex] at ((2^{\\prime })+(-1,0)+(120:1)) {};\\node (5^{\\prime }) [black vertex] at ((4^{\\prime })+(1,0)) {};}}\\node () [] at ($ (0)+(90:.2)$) {a_0};\\node () [] at ($ (1)+(90:.2)$) {a_1};\\node () [] at ($ (2)+(-90:.2)$) {a_2};\\node () [] at ($ (3)+(45:.3)$) {a_3};\\node () [] at ($ (4)+(90:.2)$) {a_4};$ ) [] at ($(0^{\\prime })+(90:.2)$ ) $b_0$ ; ) [] at ($(1^{\\prime })+(45:.3)$ ) $b_1$ ; ) [] at ($(2^{\\prime })+(-90:.2)$ ) $b_2$ ; ) [] at ($(3^{\\prime })+(-90:.2)$ ) $b_3$ ; ) [] at ($(4^{\\prime })+(-45:.3)$ ) $b_4$ ; ) [] at ($(5^{\\prime })+(90:.2)$ ) $b_5$ ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.3pt,color=gray,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (5'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); .45 [scale=1.5] [fatpath,backcolor1] (120:1) – ($(-1,0)+(120:1)$ ) – (120:1) – (0,0) (60:1) – (0,0) – (60:1) – (0:1) – ($(2,0)$ ) – (0:1); 0) [black vertex] at ($(-1,0)+(120:1)$ ) ; 1) [black vertex] at (120:1) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (0:1) ; 4) [black vertex] at (60:1) ; ) [] at ($(2,0)+(60:1)$ ) ; 4') [black vertex] at (1,0) ; 3') [black vertex] at ($(4^{\\prime })-(60:1)$ ) ; 2') [black vertex] at ($(3^{\\prime })-(1,0)$ ) ; 1') [black vertex] at ($(2^{\\prime })+(120:1)$ ) ; 0') [black vertex] at ($(2^{\\prime })+(-1,0)+(120:1)$ ) ; 5') [black vertex] at ($(4^{\\prime })+(1,0)$ ) ; ) [] at ($(0)+(90:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(-90:.2)$ ) $a_2$ ; ) [] at ($(3)+(45:.3)$ ) $a_3$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(90:.2)$ ) $b_0$ ; ) [] at ($(1^{\\prime })+(45:.3)$ ) $b_1$ ; ) [] at ($(2^{\\prime })+(-90:.2)$ ) $b_2$ ; ) [] at ($(3^{\\prime })+(-90:.2)$ ) $b_3$ ; ) [] at ($(4^{\\prime })+(-45:.3)$ ) $b_4$ ; ) [] at ($(5^{\\prime })+(90:.2)$ ) $b_5$ ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.3pt,color=gray,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (5'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); Exchange of edges between elements of type A and B in the proof of Claim REF .","$\\mathbf {T_2}$ is of type C. We may assume $b_3b_2\\in F_r$ .","In this case we have $b_4b_3\\in F_g$ .","Since $T_2$ contains a hanging edge at ${\\rm {cv}}_1(T_1)$ , we have $a_3={\\rm {cv}}_1(T_1)\\in \\lbrace b_1,b_4\\rbrace $ .","If $b_4=a_3$ , then there are two green out edges at $a_3$ , namely $a_3a_4, b_4a_3$ , a contradiction.","Thus, we may assume that $a_3 = b_1$ .","Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_0$ , $T_2^{\\prime } = a_2b_1b_2b_4b_3b_5$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime }\\rbrace $ .","Note that $$ is a decomposition of $G$ into trails of length 5.","By Lemma REF , $T_1^{\\prime }$ is an element of Type C. In what follows we prove that $T_2^{\\prime }$ is a path.","For that, we prove that $a_2\\notin \\lbrace b_0,b_1,b_2,b_3,b_4\\rbrace $ .","Indeed, since $G$ is simple, ${a_2\\notin \\lbrace b_1,b_2\\rbrace }$ .","If $a_2 = b_4$ , then $a_2a_1$ and $b_4b_3$ are two green out edges at $a_2$ , a contradiction.","By Lemma REF , $a_2\\ne b_5$ .","Finally, $a_3 + g + r = a_2$ and ${b_1 = b_3 + r + g}$ , if $a_2 = b_3$ , then we have $a_3 + g + r = a_2 = b_3 = b_1 - g - r = a_3 - g - r$ , which implies $2g+2r = 0$ , a contradiction.","Thus, $T^{\\prime }_2$ is an element of type C, and hence Definition REF (REF ) holds for $$ .","Analogously to the case above $$ is a complete decomposition of $G$ such that $\\tau ()=\\tau (-1<\\tau ($ , a contradiction to the minimality of $.","$ Figure: Exchange of edges between elements of type Aand C in the proof of Claim .Claim 11.2 Let $T_1$ and $T_2$ be two elements of type A in $.If $ T1$ is free and $ T2$ contains a hangingedge on $ cv1(T1)$, then no element of type~A, B, or C in $ {T1,T2}$contains a hanging edge at $ cv2(T2)$.","$ Let $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ be two elements of $,where $ a5=a2$ and $ b5=b2$ and $ a2a3,b2b3M$.First, we prove that $ cv1(T1)=tr(T2)$,and hence, by Lemma~\\ref {lemma:type12}, we have $ cv2(T1) = aux(T2)$.Suppose, for contradiction, that $ cv1(T1)tr(T2)$.Since $ b2b3M$, we must have $ b1=cv1(T1)=a3$.Now, put $ T1' = a0a1a2a4a3b0$,$ T2' = a2b1b2b3b4b2$ (see Figure~\\ref {fig:case3}) and let$ = ({T1,T2}){T1',T2'}$.By Lemma~\\ref {lemma:T^{\\prime }1-is-path}, $ T1'$ is an element of type~C.We claim that $ T2'$ is an element of type~A.For that we prove that $ a2 {b1,b2,b3,b4}$.Again, since $ G$ is a simple graph, we have $ a2{b1,b2}$.Since every vertex is incident to precisely one edge of $ M$, we have $ a2b3$,and if $ a2 = b4$, then we have $ a3 + g + r = a2 = b4 = a3 - g - r$, which implies $ 2g+2r=0$, a contradiction.Thus, Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-types}) holds for $$.Analogously to the cases above,$$ is a complete decomposition such that$ ()=(-1<($, a contradictionto the minimality of $ .","Finally, by Lemma REF , we have ${\\rm {cv}}_2(T_1) = {\\rm {aux}}(T_2)$ .","Figure: Exchange of edges between two elements of type Ain the proof of Claim .Now, let $T_3\\in \\lbrace T_1,T_2\\rbrace $ be an element of type A, B, or C, and suppose, for a contradiction, that $T_3$ contains a hanging edge at ${\\rm {cv}}_2(T_2)$ .","Since ${\\rm {cv}}_1(T_1)={\\rm {tr}}(T_2)$ and ${\\rm {cv}}_2(T_1) = {\\rm {aux}}(T_2)$ , we have $a_3=b_4$ , $a_5=a_2=b_1$ and $b_5=b_2$ .","In what follows, we divide the proof according to the type of $T_3$ .","$\\mathbf {T_3}$ is of type A.","Let $T_3 = c_0c_1c_2c_3c_4c_5$ , where $c_2=c_5$ and $c_2c_3\\in M$ .","Since each vertex is incident to precisely one edge of $M$ we have $c_3\\ne b_2 = {\\rm {cv}}_2(T_2)$ .","Therefore, we have ${\\rm {cv}}_2(T_2)\\in \\lbrace c_1,c_4\\rbrace $ .","Suppose that ${\\rm {cv}}_2(T_2)=c_4$ .","Thus, we have $b_5 = b_2 = c_4$ .","Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_5$ , $T_2^{\\prime } = b_0b_1b_4b_3b_2c_2$ , $T_3^{\\prime } = b_1c_4c_3c_2c_1c_0$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .","In what follows, we prove that $T_1^{\\prime }$ , $T_2^{\\prime }$ and $T_3^{\\prime }$ are paths.","By Lemma REF , $T_1^{\\prime }$ is an element of type C. Since $G$ is simple, we have $c_2\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_2,c_3,c_4\\rbrace $ .","By Lemma REF , $c_2 \\ne b_0,~b_1 \\ne c_0$ .","Therefore, $T_2^{\\prime }$ is an element of type D. If $b_1 = c_1$ , then $b_2b_1$ and $c_2c_1$ are two green in edges at $c_1$ , a contradiction.","Thus, $T^{\\prime }_3$ is an element of type B, and hence Definition REF (REF ) holds for $$ .","Analogously to the cases above, we have ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)\\ge 0for every v\\in V(G)\\setminus \\lbrace a_3,a_4,b_3,c_3\\rbrace .Since a_3,a_4,b_3,c_3 are not connection vertices in ,Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}) holds for .Thus, is a complete decomposition such that\\tau ()=\\tau (-3<\\tau (, a contradiction to the minimality of~.","}\\begin{figure}\\centering \\begin{subfigure}{.45}\\centering \\scalebox {.8}{\\begin{tikzpicture}[scale=1.5]\\end{tikzpicture}[fatpath,backcolor1] (-0.5,0.8) -- (-1.5,0.8) -- (-0.5,0.8) -- (0,0) -- (0,0) -- (1,0) -- (0.5,0.8) -- (0,0);[fatpath,backcolor2] (-0.5,-0.8) -- (-1.5,-0.8)-- (-0.5,-0.8) --(0,-1.6) -- (1,-1.6) -- (0.5,-0.8) -- (0,-1.6);}\\node (0) [black vertex] at (-1.5,0.8) {};\\node (1) [black vertex] at (-0.5,0.8) {};\\node (2) [black vertex] at (0,0) {};\\node (3) [black vertex] at (1,0) {};\\node (4) [black vertex] at (0.5,0.8) {};\\node () [] at ((0,0)+(60:1)) {};\\end{subfigure}\\end{figure}\\node (4^{\\prime }) [black vertex] at ($ (1,0)$) {};\\node (3^{\\prime }) [black vertex] at ($ (3)+(0.5,-0.8)$) {}; \\node (2^{\\prime }) [black vertex] at ($ (2)+(0.5,-0.8)$) {};\\node (1^{\\prime }) [black vertex] at ($ (0,0)$) {};\\node (0^{\\prime }) [black vertex] at ($ (2')-(1,0)$) {};\\node (5^{\\prime }) [] at ($ (2,0)$) {};$ 4”) [black vertex] at ($(2^{\\prime })$ ) ; 3”) [black vertex] at ($(3)-(0,1.6)$ ) ; 2”) [black vertex] at ($(2)-(0,1.6)$ ) ; 1”) [black vertex] at ($(4^{\\prime \\prime })-(1,0)$ ) ; 0”) [black vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(2,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.6)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(-90:.3)-(0.35,0)$ ) $b_0=c_1$ ; ) [] at ($(1^{\\prime })+(180:.4)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\prime })+(-90:.25)+(0.7,0)$ ) $b_2=c_4$ ; ) [] at ($(3^{\\prime })+(0:.3)$ ) $b_3$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.4)$ ) ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.3)$ ) $c_3$ ; ) [] at ($(4^{\\prime \\prime })+(90:.3)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.3pt,color=gray,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (0.5,0.8) – (1,0) – (0.5,-0.8); [fatpath,backcolor2] (-0.5,-0.8) – (-1.5,-0.8)– (-0.5,-0.8) –(0,-1.6) – (1,-1.6) – (0.5,-0.8) – (0,0); 0) [black vertex] at (-1.5,0.8) ; 1) [black vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [black vertex] at (0.5,0.8) ; ) [] at ($(0,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(2^{\\prime })-(1,0)$ ) ; 5') [] at ($(2,0)$ ) ; 4”) [black vertex] at ($(2^{\\prime })$ ) ; 3”) [black vertex] at ($(3)-(0,1.6)$ ) ; 2”) [black vertex] at ($(2)-(0,1.6)$ ) ; 1”) [black vertex] at ($(4^{\\prime \\prime })-(1,0)$ ) ; 0”) [black vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(2,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.6)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(-90:.3)-(0.35,0)$ ) $b_0=c_1$ ; ) [] at ($(1^{\\prime })+(180:.4)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\prime })+(-90:.25)+(0.7,0)$ ) $b_2=c_4$ ; ) [] at ($(3^{\\prime })+(0:.3)$ ) $b_3$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.4)$ ) ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.3)$ ) $c_3$ ; ) [] at ($(4^{\\prime \\prime })+(90:.3)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.3pt,color=gray,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange of edges between three elements of type A in the proof of Claim REF .","Thus, we may assume ${\\rm {cv}}_2(T_2) = c_1$ .","This implies that $b_5=b_2=c_1$ , and hence we have $b_3 = b_5 - r - g = c_1 - g-r = c_4$ .","Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\prime } = b_1c_1c_2c_3c_4c_2$ (see Figure REF ) and $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .","Again, by Lemma REF , $T_1^{\\prime }$ is an element of type C. We claim that $T_2^{\\prime }$ , $T_3^{\\prime }$ are, respectively, of type D and A.","Since $G$ is simple, $c_0\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_1,c_2,c_4\\rbrace $ .","By Lemma REF we have $c_0 \\ne b_0$ .","Therefore, $T_2^{\\prime }$ is of type D. Finally, if $b_1 = c_3$ , then $d(b_1)\\ge 7>5$ , a contradiction.","Thus, $T^{\\prime }_3$ is an element of type A, and hence Definition REF (REF ) holds for $$ .","Analogously to the case above, we have ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)\\ge 0 for every v \\in V(G)\\setminus \\lbrace a_4,a_3,b_3\\rbrace .Since a_4,a_3,b_3 are not connection vertices in ,Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}) holds for .Thus, is a complete decomposition such that {\\tau () = \\tau ( -2<\\tau (}, a contradiction to the minimality of~.","}$ Figure: Exchange of edges between three elements of type Ain the proof of Claim .$\\mathbf {T_3}$ is of type B.","Let $T_3 = c_0c_1c_2c_3c_4c_5$ be an element of type B.","Since $T_3$ contains a hanging edge on ${\\rm {cv}}_2(T_2)=b_2$ , we have $b_2\\in \\lbrace c_1,c_4\\rbrace $ .","By symmetry we may assume $b_2=c_1$ .","Thus, put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\prime } = b_1c_1c_2c_3c_4c_5$ (see Figure REF ) and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .","Again, by Lemma REF , $T_1^{\\prime }$ is an element of type C. We prove that $T_2^{\\prime }$ and $T_3^{\\prime }$ are, respectively, of type D and B.","Since $G$ is simple, we have $c_0\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_1,c_2\\rbrace $ .","By Lemma REF , we have $c_0 \\ne b_0$ and $b_1 \\ne c_5$ .","Therefore, $T_2^{\\prime }$ is an element of type D. Since $c_4=c_3+g$ and $b_1=c_1+g$ , if $c_4=b_1$ , then $c_3=c_1$ , a contradiction.","If $b_1 \\in \\lbrace c_3,c_4\\rbrace $ , then $d(b_1)\\ge 7>5$ , a contradiction.","Therefore, $T_3^{\\prime }$ is an element of type B. Analogously to the case above, $$ is a complete decomposition such that $\\tau () = \\tau ( -2<\\tau ($ , a contradiction to the minimality of $.$ Figure: Exchange of edges between two elements of type Aand an element of type Bin the proof of Claim .$\\mathbf {T_3}$ is of type C. Let $T_3 = c_0c_1c_2c_3c_4c_5$ be an element of type C, where $c_3c_2\\in F_r$ .","This implies that $c_4c_3\\in F_g$ .","Since $T_3$ contains a hanging edge on ${\\rm {cv}}_2(T_2)=b_2$ , we have $b_2\\in \\lbrace c_1,c_4\\rbrace $ .","If $b_2 = c_4$ , then $c_4c_3$ and $b_2b_1$ are two green out edges of $b_2$ , a contradiction.","Thus, we may assume $b_2 = c_1$ .","Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\prime } = b_1c_1c_2c_3c_4c_5$ (see Figure REF ) and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .","Again, by Lemma REF , $T_1^{\\prime }$ is an element of type C. We prove that $T_2^{\\prime }$ and $T_3^{\\prime }$ are, respectively, of type D and C. Since $G$ is simple, $c_0\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_1,c_2\\rbrace $ .","By Lemma REF , we have $c_0 \\ne b_0$ and $b_1 \\ne c_5$ .","Therefore, $T_2^{\\prime }$ is an element of type D. Analogously to the case above, If $b_1 \\in \\lbrace c_3,c_4\\rbrace $ , then $d(b_1)\\ge 7>5$ , a contradiction.","Therefore, $T_3^{\\prime }$ is an element of type C. Once more, analogously to the cases above, $$ is a complete decomposition such that $\\tau () = \\tau ( -2<\\tau ($ , a contradiction to the minimality of $.$ Figure: Exchange of edges between two elements of type Aand an element of type Cin the proof of Claim .Claim 11.3 There is no free element of type A.","Suppose, for a contradiction, that $contains a free element, say $ T1$, of type~A.By Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}),there are two hanging edges $ e2$ and $ e2'$ at $ cv1(T1)$.We may assume $ e2 E(T1)$.Let $ T2$ be the element of $ that contains $e_2$ .","By Claim REF , $T_2$ is not of type B or C, and since $M$ is a matching, $T_2$ is not of type D. Thus, $T_2$ is of type A.","By Definition REF (REF ), there is a hanging edge $e_3$ on ${\\rm {cv}}_2(T_2)$ .","Note that $e_3\\notin E(T_2)$ .","Let $T_3$ be the element of $ that contains $ e3$.By Claim~\\ref {claim:AAB,AAC},$ T3$ is of type~D,which implies that there are two edges of $ M$incident to $ cv2(T2)$, a contradiction.$ Now, consider the auxiliary directed graph $D_ in which $ V(D = and $(T_1,T_2)$ is an arc of $D_ if and only if $ tr(T2) = cvi(T1)$ for some $ i{1,2}$.It is clear that the elements of type~A in $ are partitioned into A-chains if and only if $D_ consists of vertex-disjoint directed cycles and isolated vertices.","Since every vertex of $ G$ is a connection vertex of at most one element of $ , by Claim REF , every vertex of $D_ has in degree precisely $ 1$.$ Note also that given two elements $T_1$ and $T_2$ we have ${\\rm {tr}}(T_1) \\ne {\\rm {tr}}(T_2)$ , otherwise there would be a vertex with two green in edges.","This implies that every vertex of $D_ has out degree at most $ 2$.Now, if $ T1$ and $ T2$ are two elements of type~A in $ such that ${\\rm {cv}}_1(T_1) = {\\rm {tr}}(T_2) = u_1$ , by Lemma REF , we have ${\\rm {aux}}(T_2) = {\\rm {cv}}_2(T_1) = u_2$ , which means that $E(T_1)\\cup E(T_2)$ contains the four edges in $E(G)$ incident to $u_1$ and five edges incident to $u_2$ , and hence, no other element of $ contains $ u2$,and no other element of $ has $u_1$ as its tricky vertex.","This implies that every vertex of $D_ has out degree at most $ 1$,and hence $ D consists of vertex-disjoint directed cycles and isolated vertices as desired."],["Admissible decompositions","In this section, we present a new decomposition invariant, which we call admissibility, and conclude our proof.","For that, we introduce an important object, the exceptional pair.","Let $G$ be a $\\lbrace g,r\\rbrace $ -graph, and let $ be a decomposition of $ G$ into trails of length $ 5$.We say that a pair $ (T1,T2)$ of elements of $ is an exceptional pair if $T_1$ and $T_2$ are elements of type A and C, respectively, and can be written as $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ such that $a_2a_3\\in M_{g,r}$ , $a_2=a_5 = b_3$ , and $a_2a_1,a_3a_4,b_2b_1,b_4b_3\\in F_g$ , $a_4a_5,b_3b_2,b_4b_5\\in F_r$ , $a_1a_0,b_1b_0\\in M_{g,r}\\cup F_g\\cup F_r$ (see Figure REF ).","Note that since $G$ is a simple graph, we have $b_4\\ne a_3$ .","Also, if $2g + 2r \\ne 0$ , then we have $b_1 \\ne a_3$ .","This yields the following remark.","Remark 1 If $G$ is a $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ and $(T_1,T_2)$ is an exceptional pair, then $T_2$ does not contain a hanging edge at ${\\rm {cv}}_1(T_1)$ .","Figure: An exceptional pair.An open chain is a sequence $T_0, T_1,\\ldots , T_{s-1}$ of $s\\ge 2$ elements of $with the following properties.","(i) $ T0$ is a free element of type~A;(ii) $ Tj$ is an element of type~A and $ tr(Tj) = cvi(Tj-1)$,for every $ j {0,...,s-2}$ and some $ i{1,2}$; and(iii) $ Ts-1$ is an element of type~C for which $ (Ts-2,Ts-1)$ is an exceptional pair.We remark that open chains are not A-chains.The next definition describes the invariant studied in this section.$ Definition 3 We say that a decomposition $ of a $ {g,r}$-graph $ G$into trails of length $ 5$ is \\emph {admissible} if the following hold.\\begin{enumerate}[(a)]\\item Every element in is either a path or an element of type~A;\\item For every element T\\in of type~A, we have {\\rm {hang}}\\big ({\\rm {cv}}_1(T)\\big )\\ge 2,and there is at most one element T\\in of type~A for which{\\rm {hang}}\\big ({\\rm {cv}}_2(T)\\big ) = 0,and, in this case, there is an open chain {S = T_0,\\ldots , T_{s-2},T_{s-1}} in ,for which T_{s-2} = T;\\item The elements of type~A in are partitioned into A-chains and at most one open chain.\\end{enumerate}$ It is not hard to check that the decomposition given by Lemma REF is an admissible decomposition.","Therefore, every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ admits an admissible decomposition.","By performing a few more exchanges of edges between the elements of a same A-chain of an admissible decomposition, we can show that an admissible decomposition that minimizes its number of elements of type A is in fact a $P_5$ -decomposition.","Theorem 12 Every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ admits a $P_5$ -decomposition.","Let $g$ and $r$ be as in the statement, let $G$ be a $\\lbrace g,r\\rbrace $ -graph, and put ${M=M_{g,r}}$ .","By Lemma REF , $G$ admits an admissible decomposition.","Let $ be an admissible decomposition of $ G$ that minimizes $ ($.In what follows, we prove that $ ( = 0$.Suppose, for a contradiction, that $ (>0$.We divide A-chains into three types, according to the connections between its elements.Given $ i{1,2}$,we say that an A-chain $ S=T0, T1,..., Ts-1$ is of type~$ i$ if $ tr(Tj)=cvi(Tj-1)$for every $ j {0,...,s-1}$;and we say that $ S$ is a \\emph {mixed} A-chain if $ S$ is not of type~$ 1$ or $ 2$.$ Similarly to the proof of Lemma REF , in each step, we exchange edges between some elements of $and obtain an admissible decomposition $$ into trails of length~5such that $ () <($, which is a contradiction to the minimality of $ .","To check that $$ is an admissible decomposition, we observe the three following items: (i) The only connection vertex that has fewer hanging edges in $$ than in $is the secondary connection vertex of an element $ T1$ of type~A,and in this case there is an element $ T2$ of type~Csuch that $ (T1,T2)$ is an exceptional pair,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$;(ii) every element of $$ that is not an element of $ , i.e., the elements involved in the exchange of edges, is a path or an element of type A, and hence REF () holds for $$ ; (iii) either an open chain is shortened by at least one element, an A-chain is converted into an open chain, or all the elements of an A-chain are replaced by paths of length 5, and hence REF () holds for $$ .","Claim 12.1 Every A-chain in $ is mixed.","$ Suppose, for a contradiction, that there is an A-chain $S = T_0,T_1,\\ldots , T_{s-1}$ of type 1 or 2.","Let $T_j=a_{0,j}a_{1,j}a_{2,j}a_{3,j}a_{4,j}a_{5,j}$ , where $a_{5,j} = a_{2,j}$ , $a_{2,j}a_{3,j}\\in M$ , ${a_{2,j}a_{1,j}, a_{3,j}a_{4,j}\\in F_g}$ , $a_{4,j}a_{2,j}\\in F_r$ and $a_{1,j}a_{0,j}\\in M\\cup F_g \\cup F_r$ .","For $i\\in \\lbrace 1,2,3,4,5\\rbrace $ , the edge $a_{i-1,j}a_{i,j}$ is called the $i$ th edge of $T_j$.","In what follows, we divide the proof according to the type of S. $\\mathbf {S}$ is of type 1.","In this case, we have ${a_{3,j} = {\\rm {cv}}_1(T_{j}) = {\\rm {tr}}(T_{j+1}) = a_{4,j+1}}$ for each ${j\\in \\lbrace 0,\\ldots ,s-1}\\rbrace $ , and hence, by Lemma REF , we have $a_{2,j} = {\\rm {cv}}_2(T_{j}) = {\\rm {aux}}(T_{j+1})=a_{1,j+1}$ .","Now, for each $j=0,\\ldots ,s-1$ , let $T^{\\prime }_j = a_{2,j+1}a_{3,j}a_{4,j}a_{1,j}a_{2,j}a_{0,j+1}$ (see Figure REF ).","Note that $T^{\\prime }_j = T_j- a_{1,j}a_{0,j} + a_{1,j+1}a_{0,j+1}- a_{2,j}a_{3,j} + a_{2,j-1}a_{3,j-1}- a_{4,j}a_{2,j} + a_{4,j+1}a_{2,j+1}$ .","More specifically, $a_{2,j+1}a_{3,j} = a_{4,j+1}a_{5,j+1}$ is the 5th edge of $T_{j+1}$ ; $a_{3,j}a_{4,j}$ is the 4th edge of $T_{j}$ ; $a_{4,j}a_{1,j} = a_{2,j-1}a_{3,j-1}$ is the 3rd edge of $T_{j-1}$ ; $a_{1,j}a_{2,j}$ is the 2nd edge of $T_{j}$ ; $a_{2,j}a_{0,j+1} = a_{1,j+1}a_{0,j+1}$ is the 1st edge of $T_{j+1}$ .","Clearly, $T^{\\prime }_j$ is a trail of length 5.","Moreover, since, for each $i\\in \\lbrace 1,2,3,4,5\\rbrace $ , the element $T^{\\prime }_j$ contains the $i$ th edge of an element of $S$ , and, if $j\\ne j^{\\prime }$ , the elements $T^{\\prime }_j$ and $T^{\\prime }_{j^{\\prime }}$ contain the $i$ th edge of different elements of $S$ , the set $=\\big (\\lbrace T_j\\colon j=0,\\ldots ,s-1\\rbrace \\big )\\cup \\lbrace T^{\\prime }_j\\colon j=0,\\ldots ,s-1\\rbrace $ is a decomposition of $G$ into trails of length 5.","We may regard $$ as the decomposition obtained by reversing the direction of two components of $F_g$ , namely, the green edges in $S$ , and applying the same strategy used in Proposition REF .","Figure: Exchange of edges between the elements of an A-chain of type 1 withfive elements in the proof of Claim .In order to prove that $T^{\\prime }_j$ is a path, we show that $a_{2,j+1},a_{0,j+1}\\notin \\lbrace a_{3,j},a_{4,j},a_{1,j},a_{2,j}\\rbrace $ .","Note that, since for each $j\\in \\lbrace 0,\\ldots ,s-1\\rbrace $ , $T_j$ is a path, we have $a_{i,j} \\ne a_{i^{\\prime },j}$ for every $i\\ne i^{\\prime }$ .","Since $G$ is a simple graph, we have $a_{2,j+1}\\notin \\lbrace a_{3,j},a_{4,j},a_{2,j}\\rbrace $ and $a_{0,j+1}\\notin \\lbrace a_{3,j},a_{4,j},a_{1,j},a_{2,j}\\rbrace $ ; and if $a_{2,j+1} = a_{1,j}$ , then $a_{4,j+1}a_{2,j+1}$ and $a_{4,j-1,}a_{2,j-1}$ are two distinct red in edges of $a_{1,j}$ , a contradiction.","We claim that $$ is an admissible decomposition.","Indeed, the only vertices of the elements of $S$ that can be connection vertices of elements in $$ are the vertices $a_{0,j}$ , for $j=0,\\ldots ,s-1$ .","But a hanging edge at a vertex $a_{0,j}$ is in $T_{j^{\\prime }}\\in if and only if $ a0,j = a3,j'$ for some $ j'j$,and, in this case $ a3,j'$ is not a connection vertex in $$ becauseall edges incident to it are in elements of $ {T'jj=0,...,s-1}$.Therefore, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$.Moreover, since $ T'j$ is a path, for $ j=0,...,s-1$, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:complete-commutative-types}) holds for $$.Finally, $ and $$ have the same number of open chains, and hence Definition REF () holds for $$ .","Therefore, $$ is an admissible decomposition of $G$ such that ${\\tau () = \\tau ( - s}$ , a contradiction to the minimality of $.$ $\\mathbf {S}$ is of type 2.","In this case, for each $j=0,\\ldots ,s-1$ , we have $a_{2,j} = {\\rm {cv}}_2(T_{j}) = {\\rm {tr}}(T_{j+1}) = a_{4,j+1}$ .","Now, for each $j=0,\\ldots ,s-1$ , let $T^{\\prime }_j = a_{0,j}a_{1,j}a_{2,j}a_{3,j}a_{4,j}a_{4,j-1}$ (see Figure REF ).","Clearly, $T^{\\prime }_j$ is a trail of length 5.","Note that $T^{\\prime }_j = T_j - a_{4,j}a_{2,j} + a_{4,j-1}a_{2,j-1}$ , i.e., $T^{\\prime }_j$ is the element obtained from $T_j$ by exchanging its 5th edge by the 5th edge of $T_{j-1}$ .","Thus, the set $=\\big (\\lbrace T_j\\colon j=0,\\ldots ,s-1\\rbrace \\big )\\cup \\lbrace T^{\\prime }_j\\colon j=0,\\ldots ,s-1\\rbrace $ is a decomposition of $G$ into trails of length 5.","We may regard $$ as the decomposition obtained by reversing the direction of one component of $F_r$ and applying the same strategy used in Proposition REF .","In order to prove that $T^{\\prime }_j$ is a path, we show that $a_{4,j-1}\\notin \\lbrace a_{0,j},a_{1,j},a_{2,j},a_{3,j},a_{4,j}\\rbrace $ .","Note that, since for each $j\\in \\lbrace 0,\\ldots ,s-1\\rbrace $ , $T_j$ is a path, we have $a_{i,j} \\ne a_{i^{\\prime },j}$ for every $i\\ne i^{\\prime }$ .","Since $G$ is a simple graph, we have $a_{4,j-1}\\notin \\lbrace a_{2,j},a_{3,j},a_{4,j}\\rbrace $ ; also, by Lemma REF , we have $a_{4,j-1}\\ne a_{0,j}$ ; and if $a_{4,j-1} = a_{1,j}$ , then $a_{2,j}a_{1,j}$ and $a_{3,j-1}a_{4,j-1}$ are two distinct green in edges of $a_{4,j-1}$ , a contradiction.","Figure: Exchange of edges between the elements of an A-chain of type 2 withfive elements in the proof of Claim .We claim that $$ is an admissible decomposition.","Indeed, the only vertices that have hanging edges in $ and may not have hanging edges in $$are the vertices $ a3,j$ and $ a4,j = a2,j-1$, for $ j=0,...,s-1$,but these vertices are connection vertices of the elements in $ S$,and hence can^{\\prime }t be connection vertices of elements in $$.Therefore, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$.Moreover, since $ T'j$ is a path, for $ j=0,...,s-1$, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:complete-commutative-types}) holds for $$.Finally, $ and $$ have the same number of open chains, and hence Definition REF () holds for $$ .","Therefore, $$ is an admissible decomposition of $G$ such that $\\tau () = \\tau ( - s$ , a contradiction to the minimality of $.$ Claim 12.2 Every A-chain contains at least four elements First, note that if an A-chain consists of two elements, then $G$ contains a parallel edge, which is a contradiction.","Thus, let $S$ be an A-chain in $ with precisely three elements, say $ T1$, $ T2$, and $ T3$.By Claim~\\ref {claim:mixed-A-chain}, we may assume $ tr(T1)=cv2(T3)$,$ tr(T2)=cv1(T1)$ and $ tr(T3)=cvi(T2)$, for $ i {1,2}$.","In what follows, we divide the proof depending on whether $ i=1$ or $ i=2$.$ Let $T_1=a_0a_1a_2a_3a_4a_5$ , $T_2=b_0b_1b_2b_3b_4b_5$ and $T_3 = c_0c_1c_2c_3c_4c_5$ be the elements of $S$ where $a_4={\\rm {tr}}(T_1)={\\rm {cv}}_2(T_3)=c_2$ , $b_4={\\rm {tr}}(T_2)={\\rm {cv}}_1(T_1)=a_3$ and ${c_4={\\rm {tr}}(T_3)={\\rm {cv}}_i(T_2)}$ .","Case $\\mathbf {i=1}$ .","In this case, $c_4={\\rm {tr}}(T_3)={\\rm {cv}}_1(T_2)=b_3$ .","Since $b_3=c_4$ , we have $b_2=c_1$ and $c_0=a_1$ .","Put $T_1^{\\prime } = a_0a_1a_2a_3a_4c_1$ , $T_2^{\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_3^{\\prime } = c_0c_1c_4c_3c_2a_2$ (see Figure REF ) and ${ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace }$ .","We claim that $T_1^{\\prime }$ , $T_2^{\\prime }$ and $T_3^{\\prime }$ are paths.","By Lemma REF , $T_2^{\\prime }$ is a path.","Since $G$ is simple, $c_1\\notin \\lbrace a_2,a_3,a_4\\rbrace $ and $a_2\\notin \\lbrace c_1,c_2,c_3,c_4\\rbrace $ .","By Lemma REF , we have $c_1 \\ne a_0$ and $a_2\\ne c_0$ .","Therefore, $T_3^{\\prime }$ is a path.","Finally, if $c_1 = a_1$ , then $c_2c_1$ and $a_2a_1$ are two green out edges at $a_1$ , a contradiction.","Therefore, $T_1^{\\prime }$ is a path, and hence definition REF () holds.","Also, ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_3,b_3,c_3\\rbrace .Thus, definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) holds for .Since is admissible and the new elements are paths, the elements of type~Aare still partitioned into A-chains and at most one open chain,and hence \\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free})holds for~.Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 3, a contradiction to the minimality of~.","}\\begin{figure}[H]\\centering \\begin{subfigure}{.45}\\centering \\scalebox {.8}{\\begin{tikzpicture}[scale=1.5]\\end{tikzpicture}[fatpath,backcolor1] (-0.5,0.8) -- (-1.5,0.8) -- (-0.5,0.8) -- (0,0) -- (0,0) -- (1,0) -- (0.5,0.8) -- (0,0);[fatpath,backcolor2] (0.5,-0.8) -- (-0.5,-0.8)-- (0.5,-0.8) -- (1,-1.6) -- (2,-1.6) -- (1.5,-0.8) -- (1,-1.6);}\\end{subfigure}\\node (0) [black vertex] at (-1.5,0.8) {};\\node (1) [blue vertex] at (-0.5,0.8) {};\\node (2) [black vertex] at (0,0) {};\\node (3) [black vertex] at (1,0) {};\\node (4) [red vertex] at (0.5,0.8) {};\\node () [] at ((1,0)+(60:1)) {};\\end{figure}$4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [red vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\prime })$ ) ; 0”) [blue vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(180:.2)$ ) $b_0$ ; ) [] at ($(1^{\\prime })+(-90:.3)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\prime })+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.25)$ ) $c_3$ ; ) [] at ($(4^{\\prime \\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (1,0) – (0.5,0.8); [fatpath,backcolor1] (1,-1.6) – (0.5,-0.8) –(1,-1.6) – (0.5,-0.8); [fatpath,backcolor2] (0.5,-0.8) – (-0.5,-0.8) – (0.5,-0.8) – (-0.5,-0.8); [fatpath,backcolor2] (2,-1.6) – (1,-1.6) – (2,-1.6) – (1.5,-0.8) – (0.5,-0.8) – (1.5,-0.8); [fatpath,backcolor2] (0,0) – (0.5,0.8) – (0,0) – (0.5,0.8); 0) [black vertex] at (-1.5,0.8) ; 1) [blue vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [red vertex] at (0.5,0.8) ; ) [] at ($(1,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [red vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\prime })$ ) ; 0”) [blue vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(180:.2)$ ) $b_0$ ; ) [] at ($(1^{\\prime })+(-90:.3)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\prime })+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.25)$ ) $c_3$ ; ) [] at ($(4^{\\prime \\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange performed in the proof of Claim REF in the case ${\\rm {tr}}(T_3) = {\\rm {cv}}_1(T_2)$ .","The red (resp.","blue) circles illustrate the same vertex, i.e., $a_4=c_2$ (resp.","$a_1=c_0$ ).","Case $\\mathbf {i=2}$ .","In this case, $c_4={\\rm {tr}}(T_3)={\\rm {cv}}_2(T_2)=b_2$ .","Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_5$ , ${T_2^{\\prime } = b_0b_1b_4b_3b_2c_2}$ , $T_3^{\\prime } = c_0c_1c_2c_3c_4b_1$ (see Figure REF ) and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .","By Lemma REF , $T_1^{\\prime }$ is a path.","Since $G$ is simple, we have $c_2\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_2,c_3,c_4\\rbrace $ .","By Lemma REF , $b_2 \\ne a_0, ~c_2 \\ne b_0,~b_1 \\ne c_0$ .","Therefore, $T_2^{\\prime }$ is a path.","If $b_1 = c_1$ , then $b_2b_1$ and $c_2c_1$ are two green in edges at $c_1$ , a contradiction.","Therefore, $T_3^{\\prime }$ is a path.","Analogously to the case above, $$ is an admissible decomposition of $G$ such that $\\tau () = \\tau ( - 3$ , a contradiction to the minimality of $.$ Figure: Exchange performed in the proof of Claim in the case tr (T 3 )= cv 2 (T 2 ){\\rm {tr}}(T_3) = {\\rm {cv}}_2(T_2).","The red circles illustrate the same vertex.Claim 12.3 $ contains an open chain.$ Suppose, for a contradiction, that there is no open chain in $.Since $ (>0$, {~contains} an A-chain $ S = T0,T1,..., Ts-1$.By Claim~\\ref {claim:mixed-A-chain}, $ S$ is a mixed A-chain.Then we can find three consecutive elements in $ S$, say $ Tj, Tj+1,Tj+2$,such that $ cv2(Tj)= tr(Tj+1)$ and $ cv1(Tj+1) = tr(Tj+2)$.By the cyclic structure of $ S$, we may assume, without loss of generality, that $ j=0$.By Claim~\\ref {claim:no-short-A-chain}, we have $ s4$,and hence there is an element $ T3 such that ${\\rm {tr}}(T_3) = {\\rm {cv}}_i(T_2)$ , for some $i\\in \\lbrace 1,2\\rbrace $ .","In what follows, the proof is divided according to $i$ .","Let $T_0 = a_0a_1a_2a_3a_4a_5$ , $T_1 = b_0b_1b_2b_3b_4b_5$ , $T_2 = c_0c_1c_2c_3c_4c_5$ , and $T_3 = d_0d_1d_2d_3d_4d_5$ , where $a_5 = a_2$ , $b_5 = b_2$ , $c_5 = c_2$ , $d_5 = d_2$ , and $a_2a_3, b_2b_3, c_2c_3, d_2d_3 \\in M$ .","By the choice of $T_0$ , $T_1$ , and $T_2$ , we have ${b_4 = {\\rm {tr}}(T_1) = {\\rm {cv}}_2(T_0) = a_2}$ , ${c_4 = {\\rm {tr}}(T_2) = {\\rm {cv}}_1(T_1) = b_3}$ .","The exchanges of edges performed here are analogous to the exchanges performed on the proof of Claim REF of Lemma REF for elements of type A.","Case $\\mathbf {{\\mathbf {\\rm }{tr}}(T_3) = {\\mathbf {\\rm }{cv}}_1(T_2)}$.","In this case, we have $d_4 = c_3$ and, by Lemma REF , $c_2 = d_1$ .","Put $T_1^{\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_2^{\\prime } = c_0c_1c_4c_3c_2d_0$ , $T_3^{\\prime } = c_1d_1d_2d_3d_4d_2$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .","By Lemma REF , $T^{\\prime }_1$ is an element of type C. In what follows, we prove that $T_2^{\\prime }$ is a path and $T_3^{\\prime }$ is an element of type A.","Since $G$ is simple, we have $d_0\\notin \\lbrace c_1,c_2,c_3,c_4\\rbrace $ , and $c_1\\notin \\lbrace d_1,d_2,d_4\\rbrace $ .","By Lemma REF , $d_0 \\ne c_0$ , and hence, $T_2^{\\prime }$ is a path.","If $c_1 = d_3$ , then $b_2b_3$ and $d_2d_3$ are two edges of $M$ incident to $c_1$ , a contradiction.","Therefore, $T_3^{\\prime }$ is an element of type A, and hence definition REF () holds.","Also, ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_2=b_4,b_3,c_3\\rbrace .Note also that b_4b_2 is a hanging edge at a_2 = {\\rm {cv}}_2(T_0) in ,but not in .However, (T_0,T_1^{\\prime }) is an exceptional pair.Also, b_3 and c_3 are not connection vertices of .Since c_3 is not a connection vertex in ,the element T^{\\prime }_3 is free.Therefore, S^{\\prime }=T^{\\prime }_3,\\ldots , T_{s-1},T_1,T^{\\prime }_2 is an open chain,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for .Finally, note that an element T of type~A in \\lbrace T_1,T_2,T_3\\rbrace is either in an A-chain of different from S,which implies that T is in an A-chain of ,or is in S, which implies that T is in S^{\\prime }.Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 2, a contradiction to the minimality of .","}\\begin{figure}[h]\\centering \\begin{subfigure}{.45}\\centering \\scalebox {.8}{\\begin{tikzpicture}[scale=1.5]\\end{tikzpicture}[fatpath,backcolor1] (-0.5,0.8) -- (-1.5,0.8) -- (-0.5,0.8) -- (0,0) -- (0,0) -- (1,0) -- (0.5,0.8) -- (0,0);[fatpath,backcolor2] (0.5,-0.8) -- (-0.5,-0.8)-- (0.5,-0.8) -- (1,-1.6) -- (2,-1.6) -- (1.5,-0.8) -- (1,-1.6);}\\end{subfigure}\\node (0) [black vertex] at (-1.5,0.8) {};\\node (1) [black vertex] at (-0.5,0.8) {};\\node (2) [black vertex] at (0,0) {};\\node (3) [black vertex] at (1,0) {};\\node (4) [black vertex] at (0.5,0.8) {};\\node () [] at ((1,0)+(60:1)) {};\\end{figure}$4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [black vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\prime })$ ) ; 0”) [black vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $b_0$ ; ) [] at ($(1)+(90:.2)$ ) $b_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4)+(90:.2)$ ) $b_4=a_2$ ; ) [] at ($(0^{\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime })+(-90:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\prime })+(0:.55)$ ) $c_3=d_4$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $d_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.3)-(0.3,0)$ ) $d_1=c_2$ ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $d_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.25)$ ) $d_3$ ; ) [] at ($(4^{\\prime \\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (0.5,0.8) – (1,0) – (0.5,-0.8); [fatpath,backcolor2] (0,0)– (0.5,-0.8) –(1,-1.6) – (2,-1.6) – (1.5,-0.8) – (1,-1.6); 0) [black vertex] at (-1.5,0.8) ; 1) [black vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [black vertex] at (0.5,0.8) ; ) [] at ($(1,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [black vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\prime })$ ) ; 0”) [black vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $b_0$ ; ) [] at ($(1)+(90:.2)$ ) $b_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4)+(90:.2)$ ) $b_4=a_2$ ; ) [] at ($(0^{\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime })+(-90:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\prime })+(0:.55)$ ) $c_3=d_4$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $d_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.3)-(0.3,0)$ ) $d_1=c_2$ ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $d_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.25)$ ) $d_3$ ; ) [] at ($(4^{\\prime \\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange performed in the proof of Claim REF in the case ${\\rm {tr}}(T_3) = {\\rm {cv}}_1(T_2)$ .","Case $\\mathbf {{\\mathbf {\\rm }{tr}}(T_3) = {\\mathbf {\\rm }{cv}}_2(T_2)}$.","Put $T_1^{\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_2^{\\prime } = c_0c_1c_4c_3c_2d_2$ , $T_3^{\\prime } = d_0d_1d_2d_3d_4c_1$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .","By Lemma REF , $T^{\\prime }_1$ is an element of type C. In what follows, we prove that $T_2^{\\prime }$ and $T_3^{\\prime }$ are paths.","Since $G$ is simple, we have $d_2\\notin \\lbrace c_1,c_2,c_3,c_4\\rbrace $ , and $c_1\\notin \\lbrace d_2,d_3,d_4\\rbrace $ .","By Lemma REF , $d_2 \\ne c_0$ , and $c_1 \\ne d_0$ .","Therefore, $T_2^{\\prime }$ is a path.","If $c_1 = d_1$ , then $d_2d_1$ and $c_2c_1$ are two green in edges of $c_1$ .","Therefore, $T_3^{\\prime }$ is a path, and hence definition REF () holds.","Also, ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_2=b_4,b_3,c_3,d_3\\rbrace .Note also that b_4b_2 is a hanging edge at a_2 in but not in .However, (T_0,T_1^{\\prime }) is an exceptional pair.Also, b_3, c_3 and d_3 are not connection vertices of .Thus, since d_2 and d_3 are not connection vertices in ,the element T_4 (or T_1, if s=4) is free.Therefore, S^{\\prime }=T_4,\\ldots , T_{s-1},T_1,T^{\\prime }_2 is an open chain,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for .","Finally, note that an element T of type~A in \\lbrace T_1,T_2,T_3\\rbrace is either in an A-chain of different from S,which implies that T is in an A-chain of ,or is in S, which implies that T is in S^{\\prime }.Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 3, a contradiction to the minimality of .","}$ Figure: Exchange performed in the proof of Claim in the case tr (T 3 )= cv 2 (T 2 ){\\rm {tr}}(T_3) = {\\rm {cv}}_2(T_2).Now, let $S = T_0, T_1,\\ldots , T_{s-1}$ be an open chain in $.Let $ Tj = a0,ja1,ja2,ja3,ja4,ja5,j$, for $ j{0,...,s-1}$, where $ a2,ja3,jM$ and $ a5,j=a2,j$ for $ j{0,...,s-2}$.$ Claim 12.4 $T_1$ is an element of type A and ${\\rm {tr}}(T_1) = {\\rm {cv}}_1(T_0)$ .","Suppose, for a contradiction, that $T_1$ is not an element of type A or ${{\\rm {tr}}(T_1) = {\\rm {cv}}_2(T_0)}$ .","We claim that $T_1$ does not contain a hanging edge at ${\\rm {cv}}_1(T_0)$ .","Indeed, if $T_1$ is not an element of type A, then, by the definition of open chain, $T_1$ is an element of type C, and hence, by Remark REF , $T_1$ does not have a hanging edge at ${\\rm {cv}}_1(T_0)$ ; and if $T_1$ is an element of type A for which ${\\rm {tr}}(T_1) = {\\rm {cv}}_2(T_0)$ , then we have $a_{4,1} = {\\rm {tr}}(T_1) = {\\rm {cv}}_2(T_0) = a_{2,0}$ , and hence, if $a_{1,1} = a_{3,0}$ , then we have $a_{4,1} + r + g = a_{1,1} = a_{3,0} = a_{2,0} - r - g$ , which implies that $2g + 2r = 0$ , a contradiction.","Therefore, $T_1$ does not contain a hanging edge at ${\\rm {cv}}_1(T_0)$ .","By Definition REF (), there are two hanging edges at ${\\rm {cv}}_1(T_0)$ .","Thus, there is an element $T = u_0u_1u_2u_3u_4u_5$ in $\\lbrace T_0,T_1\\rbrace $ that contains a hanging edge, say $u_1u_0$ , at ${\\rm {cv}}_1(T_0)$ .","Note that all the edges incident to $a_{2,0}$ are in $E(T_0)\\cup E(T_1)$ .","Let $T_0^{\\prime } = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}u_0$ and $T^{\\prime } = a_{2,0}u_1u_2u_3u_4u_5$ and put $ = \\big (\\lbrace T_0,T\\rbrace \\big )\\cup \\lbrace T_0^{\\prime },T^{\\prime }\\rbrace $ .","By Lemma REF , $T_0^{\\prime }$ is a path; and since all the edges incident to $a_{2,0}$ are in $E(T_0)\\cup E(T_1)$ , we have $a_{2,0}\\notin \\lbrace u_1,u_2,u_3,u_4,u_5\\rbrace $ , and hence if $T$ is a path (resp.","an element of type A), then $T^{\\prime }$ is a path (resp.","an element of type A).","Thus Definition REF (REF ) holds for $$ .","To check that $$ is an admissible decomposition first note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_{4,0}\\rbrace .Thus, since T_0 is a free element, a_{4,0} is not a connection vertex in ,and hence a_{4,0} is not a connection vertex in .Note also that T_1 is either an element of type~C or a free element of type~A,and hence S^{\\prime } = T_1,\\ldots , T_{s-1} is an open chain.Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for .Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 1, a contradiction to the minimality of .","}$ By Claim REF , we have $s\\ge 3$ , and hence, there is an element $T_2$ in $S$ .","Note that, by Lemma REF , since ${\\rm {tr}}(T_1) = {\\rm {cv}}_1(T_0)$ , we have ${\\rm {aux}}(T_1) = {\\rm {cv}}_2(T_0)$ .","This implies that $a_{1,1}a_{0,1}\\in F_r$ because all the edges incident to $a_{1,1} = {\\rm {cv}}_2(T_0)$ are in $E(T_0)\\cup E(T_1)$ .","Claim 12.5 $T_2$ is of type A.","Suppose, for a contradiction, that $T_2$ is not of type A, then $T_2$ is an element of type C and $(T_1,T_2)$ is an exceptional pair.","Thus, we can write $T_2 = a_{0,2}a_{1,2}a_{2,2}a_{3,2}a_{4,2}a_{5,2}$ such that $a_{2,1}=a_{5,1} = a_{3,2}$ , and $a_{2,2}a_{1,2},a_{4,2}a_{3,2}\\in F_g$ , $a_{3,2}a_{2,2},a_{4,2}a_{5,2}\\in F_r$ , $a_{1,2}a_{0,2}\\in M_{g,r}\\cup F_g\\cup F_r$ .","We claim that $a_{0,1} = a_{1,2}$ .","Indeed, since ${a_{1,1}a_{0,1}\\in F_r}$ , we have ${a_{0,1} = a_{1,1} + r = a_{2,1} + g + r}$ , but by the definition of type C, we have ${a_{1,2} = a_{2,2} + g = a_{3,2} + r + g}$ .","Thus, since $a_{3,2} = a_{2,1}$ , we obtain $a_{0,1} = a_{1,2}$ .","Now, put $T^{\\prime }_0 = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}a_{2,1}$ , $T^{\\prime }_1 = a_{1,1}a_{4,1}a_{3,1}a_{2,1}a_{2,2}a_{0,1}$ , and $T^{\\prime }_2 = a_{0,2}a_{1,2}a_{1,1}a_{3,2}a_{4,2}a_{5,2}$ (see Figure REF ), and put $ = \\big (\\lbrace T_0,T_1,T_2\\rbrace \\big )\\cup \\lbrace T^{\\prime }_0,T^{\\prime }_1,T^{\\prime }_2\\rbrace $ .","By Lemma REF , $T_0^{\\prime }$ is a path; since $G$ is a simple graph, $a_{2,2}\\notin \\lbrace a_{1,1},a_{4,1},a_{3,1},a_{2,1},a_{0,1}\\rbrace $ , and hence $T^{\\prime }_1$ is a path; and since all edges incident to $a_{1,1}$ are in $E(T_0)\\cup E(T_1)$ , we have $a_{1,1} \\notin V(T_2)$ , which implies that $T^{\\prime }_2$ is a path.","Thus Definition REF (REF ) holds for $$ .","Figure: Exchange of edges between two elements of type Aand the elements of an exceptional pairin the proof of Claim .To check that $$ is an admissible decomposition first note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_{4,0},a_{1,1},a_{3,1}\\rbrace .Thus, since T_0 is a free element, a_{4,0} is not a connection vertex in ,and hence a_{4,0} is not a connection vertex in ;and since the edges of M incident to a_{1,1} and a_{3,1} are in T^{\\prime }_1,the vertices a_{1,1} and a_{3,1} are not connection vertices in .Note also that no element of S is in , and hence there are no open chains in .Thus, Definitions~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) and~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free})hold for~.Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 2, a contradiction to the minimality of~.","}$ Now, by Claim REF , we have $s\\ge 4$ .","In what follows, we divide the proof depending on whether ${\\rm {tr}}(T_2) = {\\rm {cv}}_1(T_1)$ or ${\\rm {tr}}(T_2) = {\\rm {cv}}_2(T_1)$ .","Case $\\mathbf {{\\mathbf {\\rm }{tr}}(T_2) = {\\mathbf {\\rm }{cv}}_1(T_1)}$.","By Lemma REF , we have $a_{1,2}={\\rm {aux}}(T_2) = {\\rm {cv}}_2(T_1)=a_{2,1}$ .","Put $T_0^{\\prime } = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}a_{2,1}$ , $T_1^{\\prime } = a_{0,1}a_{1,1}a_{4,1}a_{3,1}a_{2,1}a_{0,2}$ , $T_2^{\\prime } = a_{1,1}a_{1,2}a_{2,2}a_{3,2}a_{4,2}a_{2,2}$ (see Figure REF ), let $ = \\big (\\lbrace T_0,T_1,T_2\\rbrace \\big )\\cup \\lbrace T_0^{\\prime },T_1^{\\prime },T_2^{\\prime }\\rbrace $ , and let $S^{\\prime } = T^{\\prime }_2, T_3, \\dots , T_{s-1}$ .","By Lemma REF , $T^{\\prime }_0$ is an element of type C. In what follows, we prove that $T_1^{\\prime }$ is a path and $T_2^{\\prime }$ is an element of type A.","Since $G$ is simple, we have $a_{0,2}\\notin \\lbrace a_{1,1},a_{2,1},a_{3,1},a_{4,1}\\rbrace $ , and $a_{1,1}\\notin \\lbrace a_{1,2},a_{2,2},a_{4,2}\\rbrace $ .","By Lemma REF , $a_{0,2} \\ne a_{0,1}$ , and hence, $T_1^{\\prime }$ is a path.","If $a_{1,1} = a_{3,2}$ , then $a_{2,0}a_{3,0}$ and $a_{2,2}a_{3,2}$ are two edges of $M$ incident to $a_{1,1}$ , a contradiction.","Therefore, $T_2^{\\prime }$ is an element of type A.","Figure: Exchange of edges between the three first elements in an open A-chain with at least four elements, in the case tr (T 3 )= cv 1 (T 2 ){\\rm {tr}}(T_3) = {\\rm {cv}}_1(T_2).To check that $$ is an admissible decomposition first note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_{3,0}, a_{3,1}, a_{4,0}\\rbrace ,but since T_0 is free, a_{4,0} is not a connection vertex in ,and hence a_{3,0}, a_{3,1},a_{4,0} are not connection vertices in .Thus, since a_{3,1} is not a connection vertex in ,the element T^{\\prime }_2 is free.Therefore, S^{\\prime } is an open chain,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for .Since, T^{\\prime }_0 and T^{\\prime }_1 are paths and T^{\\prime }_2 is an element of type~A,Definition~\\ref {def:semi-complete-commutative}(\\ref {def:complete-commutative-types}) holds for .","Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 2, a contradiction to the minimality of .","}\\smallskip \\textbf {Case} $ tr(T2) = cv2(T1)$\\textbf {.","}Put $ T0' = a0,0a1,0a2,0a4,0a3,0a2,1$,$ T1' = a0,1a1,1a4,1a3,1a2,1a2,2$,$ T2' = a0,2a1,2a2,2a3,2a4,2a1,1$ (see Figure~\\ref {fig:case5-similar2}),let $ = ({T0,T1,T2}){T0',T1',T2'}$,and let $ S' = T3, ..., Ts-1$.By Lemma~\\ref {lemma:T^{\\prime }1-is-path}, $ T'1$ is an element of type~C.","In what follows, we prove that $ T1'$ and $ T2'$ are paths.Since $ G$ is simple, we have $ a2,2{a1,1,a2,1,a3,1,a4,1}$,and $ a1,1{a2,2,a3,2,a4,2}$.By Lemma~\\ref {lemma:no-cycles}, $ a2,2 a0,1$, and $ a1,1 a0,2$.Therefore, $ T2'$ is a path.If $ a1,1 = a1,2$,then $ a2,2a1,2$ and $ a2,1a1,1$ are two green in edges of $ a1,1$.Therefore, $ T2'$ is a path.$ Figure: Exchange of edges between the three first elements in an open A-chain with at least four elements, in the case tr (T 3 )= cv 2 (T 2 ){\\rm {tr}}(T_3) = {\\rm {cv}}_2(T_2).To check that $$ is an admissible decomposition first note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_{3,0}, a_{3,1}, a_{3,2},a_{4,0}\\rbrace ,but since T_0 is free, a_{3,0}, a_{3,1}, a_{3,2},a_{4,0} are not connection vertices in ,and hence a_{4,0} is not a connection vertex in .Thus, since a_{2,2} and a_{3,2} are not connection vertices in~,the element T_3 is free.Therefore, S^{\\prime } is either an open chain or contains only one element,which is of type~C,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for~.Since, T^{\\prime }_0, T^{\\prime }_1 and T^{\\prime }_2 are paths,Definition~\\ref {def:semi-complete-commutative}(\\ref {def:complete-commutative-types}) holds for .","Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 3, a contradiction to the minimality of .This concludes the proof.","}$ Recall that a $\\lbrace g,r\\rbrace $ -graph $G$ is a 5-regular graph that contains the Cayley graph $X(\\Gamma ,S)$ , where $S = \\lbrace g,-g,r,-r\\rbrace $ .","Thus, since $S$ is closed under taking inverses, $G$ is also a $\\lbrace g,-r\\rbrace $ -, $\\lbrace -g,r\\rbrace $ -, $\\lbrace -g,-r\\rbrace $ -graph, which yields the following corollary of Theorem REF .","Corollary 12.1 Every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r \\ne 0$ or $2g-2r\\ne 0$ admits a $P_5$ -decomposition.","The main result of this paper is a straightforward consequence of Corollary REF and Theorem REF .","Theorem 13 Every $\\lbrace g,r\\rbrace $ -graph admits a $P_5$ -decomposition."],["Conclusion and future works","In this paper, we verified Conjecture for (i) $(2k +1)$ -regular graphs containing a spanning $2k$ -regular power of a cycle, and (ii) 5-regular graphs containing special spanning 4-regular Cayley graphs.","We believe that the techniques developed here can be extended for a more general class of graphs, such as Schreier Coset Graphs (see [8]).","Let $G$ be a group and let $H$ be a subgroup of $G$ .","For $s \\in G$ , the right coset of $H$ corresponding to $s$ is the set $Hs=\\lbrace hs\\colon h \\in H\\rbrace $ .","Left cosets can be defined analogously.","Let $g_1,\\dots ,g_r$ be a sequence in $G$ whose members generate $G$ , the Schreier Right Coset Graph (SRCG) is defined as follows.","Its vertex set is the set of right cosets of $H$ in $G$ , for each coset $H^{\\prime }$ and each generator $g_i$ there is an edge from $H^{\\prime }$ to the right coset $H^{\\prime }g_i$ .","In particular, a Cayley graph is an SRCG where $H=\\lbrace 0\\rbrace $ .","Schreier coset graphs are generalization of Cayley “color” graphs using cosets of some specified subgroup as vertices instead of group elements.","In 1977, Gross [8] showed that every connected regular graph of even degree is an SRCG.","This implies that, if we extend our result for 5-regular graphs that contain any spanning 4-regular SRCG, then we verify the conjecture for $k=2$ .","Finally, we can also explore others graphs containing special spanning Cayley graphs.","For instance, a natural step is to examine 7-regular graphs containing a spanning 4- or 6-regular Cayley graph.","Also, note that the definitions of simple commutative generator and $\\lbrace g,r\\rbrace $ -graph are equivalent to Cayley graphs under the restriction of the equation $g+r = r+g$ for every pair of generators.","Therefore, we plan to explore other restrictions, such as $g+r \\ne r+g$ , which would extend our result for 5-regular graphs containing every spanning 4-regular Cayley graph."],["Acknowlegments","This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.","F. Botler is partially supported by CNPq (Grant 423395/2018-1) and by FAPERJ (Grant 211.305/2019)."],["Conclusion and future works","In this paper, we verified Conjecture for (i) $(2k +1)$ -regular graphs containing a spanning $2k$ -regular power of a cycle, and (ii) 5-regular graphs containing special spanning 4-regular Cayley graphs.","We believe that the techniques developed here can be extended for a more general class of graphs, such as Schreier Coset Graphs (see [8]).","Let $G$ be a group and let $H$ be a subgroup of $G$ .","For $s \\in G$ , the right coset of $H$ corresponding to $s$ is the set $Hs=\\lbrace hs\\colon h \\in H\\rbrace $ .","Left cosets can be defined analogously.","Let $g_1,\\dots ,g_r$ be a sequence in $G$ whose members generate $G$ , the Schreier Right Coset Graph (SRCG) is defined as follows.","Its vertex set is the set of right cosets of $H$ in $G$ , for each coset $H^{\\prime }$ and each generator $g_i$ there is an edge from $H^{\\prime }$ to the right coset $H^{\\prime }g_i$ .","In particular, a Cayley graph is an SRCG where $H=\\lbrace 0\\rbrace $ .","Schreier coset graphs are generalization of Cayley “color” graphs using cosets of some specified subgroup as vertices instead of group elements.","In 1977, Gross [8] showed that every connected regular graph of even degree is an SRCG.","This implies that, if we extend our result for 5-regular graphs that contain any spanning 4-regular SRCG, then we verify the conjecture for $k=2$ .","Finally, we can also explore others graphs containing special spanning Cayley graphs.","For instance, a natural step is to examine 7-regular graphs containing a spanning 4- or 6-regular Cayley graph.","Also, note that the definitions of simple commutative generator and $\\lbrace g,r\\rbrace $ -graph are equivalent to Cayley graphs under the restriction of the equation $g+r = r+g$ for every pair of generators.","Therefore, we plan to explore other restrictions, such as $g+r \\ne r+g$ , which would extend our result for 5-regular graphs containing every spanning 4-regular Cayley graph."],["Acknowlegments","This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.","F. Botler is partially supported by CNPq (Grant 423395/2018-1) and by FAPERJ (Grant 211.305/2019)."]],"string":"[\n [\n \"Decomposition of $(2k+1)$-regular graphs containing special spanning\\n $2k$-regular Cayley graphs into paths of length $2k+1$\"\n ],\n [\n \"Abstract A $P_\\\\ell$-decomposition of a graph $G$ is a set of paths with $\\\\ell$ edges in $G$ that cover the edge set of $G$.\",\n \"Favaron, Genest, and Kouider (2010) conjectured that every $(2k+1)$-regular graph that contains a perfect matching admits a $P_{2k+1}$-decomposition.\",\n \"They also verified this conjecture for $5$-regular graphs without cycles of length $4$.\",\n \"In 2015, Botler, Mota, and Wakabayashi verified this conjecture for $5$-regular graphs without triangles.\",\n \"In this paper, we verify it for $(2k+1)$-regular graphs that contain the $k$th power of a spanning cycle; and for $5$-regular graphs that contain special spanning $4$-regular Cayley graphs.\"\n ],\n [\n \"Introduction\",\n \"All graphs in this paper are simple, i.e., have no loops nor multiple edges.\",\n \"A decomposition of a graph $G$ is a set $of edge-disjoint subgraphs of $ G$ that cover its edge set.If every element of $ is isomorphic to a fixed graph $H$ , then we say that $ is an \\\\emph {$ H$-decomposition}.In this paper, we focus on the case $ H$is the simple path with $ 2k+1$ edges, which we denote by $ P2k+1$.Note that this notation is not standard.In 1957, Kotzig~\\\\cite {Ko57} (see also~\\\\cite {BoFo83})proved that a $ 3$-regular graph $ G$admits a $ P3$-decomposition if and only if $ G$ contains a perfect matching.In 2010, Favaron, Genest, and Kouider~\\\\cite {FaGeKo10} extended this result by provingthat every $ 5$-regular graph thatcontains a perfect matching and no cycles of length $ 4$admits a $ P5$-decomposition;and proposed the following conjecture toextend Kotzig^{\\\\prime }s result.\\\\begin{conjecture}[Favaron--Genest--Kouider, 2010]If G is a (2k+1)-regular graph that contains a perfect matching,then G admits a P_{2k+1}-decomposition.\\\\end{conjecture}$ In 2015, Botler, Mota, and Wakabayashi [3] verified Conjecture for triangle-free 5-regular, and, more recently, Botler, Mota, Oshiro, and Wakabayashi [2] generalized this result for $(2k+1)$ -regular graphs with girth at least $2k$ .\",\n \"It is clear that a 5-regular graph contains a perfect matching if and only if it contains a spanning 4-regular graph.\",\n \"In fact, by using a theorem of Petersen [10], one can prove that a $(2k+1)$ -regular graph contains a perfect matching if and only if it contains a spanning $2k^{\\\\prime }$ -regular graph for every $k^{\\\\prime }\\\\le k$ .\",\n \"Theorem 1 (Petersen, 1891) If $G$ is a $2k$ -regular graph, then $G$ admits a decomposition into spanning 2-regular graphs.\",\n \"In this paper, we explore Conjecture for $(2k+1)$ -regular graphs that contain special spanning $2k$ -regular graphs as follows.\",\n \"Throughout the text, $\\\\Gamma $ denotes a finite group of order $n$ ; $+$ denotes the group operation of $\\\\Gamma $ ; and 0 denotes the identity of $\\\\Gamma $ .\",\n \"As usual, for each $x\\\\in \\\\Gamma $ , we denote by $-x$ the inverse of $x$ , i.e., the element $y\\\\in \\\\Gamma $ for which $x+y=0$ , and the operation $-$ denotes the default binary operation $(x,y)\\\\mapsto x+(-y)$ .\",\n \"Let $S\\\\subseteq \\\\Gamma $ be a set not containing 0, and such that $-x\\\\in S$ for every $x\\\\in S$ (i.e., $S$ is closed under taking inverses).\",\n \"The Cayley graph $X(\\\\Gamma ,S)$ is the graph $H$ with $V(H)=\\\\Gamma $ , and $E(H) = \\\\big \\\\lbrace xy\\\\colon y-x\\\\in S\\\\big \\\\rbrace $ (see [7]).\",\n \"In this paper, we allow $S$ to be a set not generating $\\\\Gamma $ , and hence $X(\\\\Gamma ,S)$ is not necessarily connected.\",\n \"We say that $H$ is simply commutative if (i) $x + y = y + x$ for every $x,y\\\\in S$ , and (ii) $-x\\\\ne x$ for every $x\\\\in S$ .\",\n \"Condition (ii) implies that $H$ has no multiple edges, and, since $0\\\\notin S$ , $H$ is simple.\",\n \"It is not hard to check that, in such a graph, the neighborhood of a vertex $v\\\\in \\\\Gamma $ is $N(v) = \\\\lbrace v+x\\\\colon x\\\\in S\\\\rbrace $ .\",\n \"Although the definition of Cayley graphs can be extended to multigraphs and directed graphs, Conjecture considers only simple graphs.\",\n \"In fact, we explore some structure of the colored directed Cayley graph (see [5]) in which the edge set consists of the pairs $(x,x+s)$ with color $s\\\\in S$ .\",\n \"We present two results regarding Conjecture .\",\n \"We verify it for $(2k+1)$ -regular graphs that contain the $k$ th power of a spanning cycle (see Section ); and for 5-regular graphs that contain spanning simply commutative 4-regular Cayley graphs (see Section ).\",\n \"Since the graphs in these classes may contain cycles of lengths 3 and 4, these results extend the family of graphs for which Conjecture is known to hold.\",\n \"We believe that, due to the underlying group structure, the techniques developed here can be extended for dealing with $(2k+1)$ -regular graphs that contain more general spanning Cayley graphs, and also $(2k+1)$ -regular graphs that contain special spanning Schreier graphs, which could give us significant insight with respect to the general case of Conjecture (see Section ).\",\n \"Notation.\",\n \"A graph $T$ is a trail if there is a sequence $x_0,\\\\ldots , x_{\\\\ell }$ of its vertices for which $E(T)=\\\\lbrace x_{i}x_{i+1}\\\\colon 0\\\\le i \\\\le \\\\ell -1\\\\rbrace $ and $x_{i}x_{i+1}\\\\ne x_{j}x_{j+1}$ , for every $i\\\\ne j$ .\",\n \"Moreover, if $x_i\\\\ne x_j$ for every $i \\\\ne j$ , we say that $T$ is a path.\",\n \"A subgraph $F$ of a graph $G$ is a factor of $G$ or a spanning subgraph of $G$ , if $V(F)=V(G)$ .\",\n \"If, additionally, $F$ is $r$ -regular, then we say that $F$ is an $r$ -factor.\",\n \"In particular, a perfect matching is the edge set of a 1-factor.\",\n \"Moreover, we say that $F$ is an $H$ -factor if $F$ is a factor that consists of vertex-disjoints copies of $H$ .\",\n \"The reader may refer to [1] for standard definitions of graph theory.\"\n ],\n [\n \"Regular graphs that contain powers of cycles\",\n \"Given a perfect matching $M$ in a graph $G$ , we say that a $P_{\\\\ell }$ -decomposition $ of a graph $ G$ is$ M$-\\\\emph {centered} if for every$ P=a0a1a-1a , we have $a_{i}a_{i+1} \\\\in M$ for $i=(\\\\ell -1)/2$ , i.e., if the middle edges of the paths in $ are precisely the edges of $ M$.The next results are examples of $ M$-centered decompositionthat are used in the proof of Theorems~\\\\ref {theorem:no-g2r2} and~\\\\ref {theorem:2g2r!=0}.$ Proposition 2 If $G$ is a 5-regular graph that contains a spanning copy $K$ of $K_{4,4}$ , and ${M = E(G)\\\\setminus E(K)}$ , then $G$ admits an $M$ -centered $P_5$ -decomposition.\",\n \"Let $G$ , $K$ , and $M$ be as in the statement.\",\n \"Let $(R,L)$ be the bipartition of $K$ , where $R=\\\\lbrace r_1,r_2,r_3,r_4\\\\rbrace $ and $L=\\\\lbrace l_1,l_2,l_3,l_4\\\\rbrace $ .\",\n \"Since $K$ is a complete bipartite graph, if $xy\\\\in M$ , then either $x,y \\\\in R$ or $x,y \\\\in L$ .\",\n \"Thus, we may assume, without loss of generality, that $M=\\\\lbrace r_1r_2,r_3r_4,l_1l_2,l_3l_4\\\\rbrace $ , and hence, ${\\\\lbrace l_1r_1l_3l_4r_2l_2,~l_3r_3l_1l_2r_4l_4,~r_1l_2r_3r_4l_1r_2,~r_3l_4r_1r_2l_3r_4\\\\rbrace }$ is an $M$ -centered decomposition of $G$ as desired (see Figure REF ).\",\n \"Figure: P 5 P_5-decomposition of a 5-regular graph that contains a spanning copy of a K 4,4 K_{4,4}.Given positive integers $k$ and $n$ , the $k$ th power of the cycle on $n$ vertices, denoted by $C_n^k$ , is the graph on the vertex set $\\\\lbrace 0,\\\\ldots ,n-1\\\\rbrace $ and such that, for every vertex $v$ , we have $x\\\\in N(v)$ if and only if $x = v+r\\\\pmod {n}$ , where $r\\\\in \\\\lbrace -k,\\\\ldots ,-1\\\\rbrace \\\\cup \\\\lbrace 1,\\\\ldots ,k\\\\rbrace $ .\",\n \"Proposition 3 Let $n$ and $k$ be positive integers for which $k < n/2$ .\",\n \"If $G$ is a simple $(2k+1)$ -regular graph on $n$ vertices that contains a copy $C$ of $C_n^k$ , and $M = E(G) \\\\setminus E(C)$ , then $G$ admits an $M$ -centered $P_{2k+1}$ -decomposition.\",\n \"Let $G$ , $C$ , and $M$ be as in the statement, and let $V(C) = \\\\lbrace 0,\\\\ldots , n-1\\\\rbrace $ as above.\",\n \"The operations on the vertices of $C$ are taken modulo $n$ .\",\n \"Since $C$ is a $2k$ -regular graph, $M$ is a perfect matching of $G$ .\",\n \"Given $i\\\\in V(C)$ , let $Q_i$ be the path $v_0v_1\\\\cdots v_k$ in which $v_0=i$ ; and, for $j=1,\\\\ldots ,k$ , we have $v_j = v_{j-1}+j$ if $j$ is odd; and $v_j = v_{j-1}-j$ if $j$ is even (see Figure REF ).\",\n \"Note that for every $j=1,\\\\ldots ,k$ , the path $Q_i$ contains an edge $xy$ such that $|x-y|=j$ .\",\n \"Also, we have $V(Q_i) = \\\\big \\\\lbrace i+r\\\\colon r\\\\in \\\\lbrace -\\\\lfloor k/2\\\\rfloor ,-\\\\lfloor k/2\\\\rfloor +1,\\\\ldots , \\\\lceil k/2\\\\rceil \\\\rbrace \\\\big \\\\rbrace $ .\",\n \"It is not hard to check that the set $\\\\mathcal {Q}=\\\\big \\\\lbrace Q_i\\\\colon i\\\\in V(C)\\\\big \\\\rbrace $ is a $P_k$ -decomposition of $C$ .\",\n \"Given an edge $e=ij\\\\in M$ , let $P_e = Q_i \\\\cup \\\\lbrace ij\\\\rbrace \\\\cup Q_j$ .\",\n \"Since $Q_i$ and $Q_j$ have, respectively, $i$ and $j$ as end vertices, and $E(Q_i)\\\\cap E(Q_j)=\\\\emptyset $ , the graph $P_e$ is a trail of length $2k+1$ .\",\n \"Thus, since $\\\\mathcal {Q}$ is a $P_k$ -decomposition of $C$ , and $M$ is a perfect matching of $G$ , the set $\\\\lbrace P_e\\\\colon e\\\\in M\\\\rbrace $ is a decomposition of $G$ into trails of length $2k+1$ .\",\n \"We claim that, in fact, $is a $ P2k+1$-decomposition of $ G$.For that, we prove that if $ ijM$,then $ V(Qi)V(Qj)=$.Indeed, note that for every $ e=ijM$,we have $ |i-j|>k$, otherwise we have $ ijE(C)$.Now, suppose that there is a vertex $ v$ in $ V(Qi)V(Qj)$.Then, there are $ r1,r2$ with$ -k/2r1,r2k/2$,and such that $ i+r1 = v = j+r2$.Suppose, without loss of generality, that $ i>j$.Then, we have $ r2-r1 = i-j > k$,but $ r2-r1 k/2+k/2= k$,a contradiction.$ Figure: The path P uv P_{uv}, with u=2u=2 and v=8,v=8, in the proof of Proposition for a 7-regular graph that contains a spanning copy of a C 10 3 C^3_{10}.Note that, from the proof of Proposition REF we also obtain a construction for the Hamilton path decomposition of complete graphs of even order.\",\n \"Corollary 3.1 Let $\\\\ell $ be odd.\",\n \"The complete graph $K_{\\\\ell +1}$ admits a $P_\\\\ell $ -decomposition.\",\n \"A slight variation of the proof of Proposition REF also provides the following result.\",\n \"Proposition 4 Let $\\\\ell $ be odd, and let $G$ be an $\\\\ell $ -regular graph with a perfect matching $M$ .\",\n \"If each component of $G\\\\setminus M$ is the $\\\\frac{\\\\ell -1}{2}$ -th power of a cycle, then $G$ admits an $M$ -centered $P_\\\\ell $ -decomposition.\",\n \"Let $G_1$ and $G_2$ be disjoint graphs with perfect matchings $M_1$ and $M_2$ , respectively.\",\n \"Let $a_1b_1,\\\\ldots ,a_kb_k \\\\in M_1$ and $x_1y_1,\\\\ldots , x_ky_k\\\\in M_2$ be distinct edges, and let $G$ be the graph obtained from the disjoint union $G_1\\\\cup G_2$ by removing $a_1b_1,\\\\ldots ,a_kb_k,x_1y_1,\\\\ldots , x_ky_k$ and adding the edges $a_1x_1,b_1y_1,\\\\ldots ,a_kx_k,b_ky_k$ .\",\n \"We say that $G$ is a collage of $G_1$ and $G_2$ over edges of $M_1$ and $M_2$ , and denote by $M_G$ the perfect matching ${\\\\big (M_1\\\\cup M_2 \\\\cup \\\\lbrace a_1x_1,b_1y_1,\\\\ldots ,a_kx_k,b_ky_k\\\\rbrace \\\\big ) \\\\setminus \\\\lbrace a_1b_1,\\\\ldots ,a_kb_k,x_1y_1,\\\\ldots , x_ky_k\\\\rbrace }$ .\",\n \"When $M_1$ and $M_2$ are clear from the context, we say simply that $G$ is a collage of $G_1$ and $G_2$ .\",\n \"Note that $G$ is $\\\\ell $ -regular if and only if $G_1$ and $G_2$ are $\\\\ell $ -regular.\",\n \"Let $G$ be an $\\\\ell $ -regular graph, where $\\\\ell $ is an odd positive integer, and let $M$ be a perfect matching of $G$ .\",\n \"We say that $G$ is $M$ -constructable if either $G$ admits an $M$ -centered $P_\\\\ell $ -decomposition, or $G$ is the collage of an $M_1$ -constructable graph and an $M_2$ -constructable graph over edges of $M_1$ and $M_2$ and $M=M_G$ .\",\n \"The next straightforward result is a useful tool in the proof of Theorem REF .\",\n \"Lemma 5 Let $\\\\ell $ be odd, and $G$ be an $\\\\ell $ -regular graph.\",\n \"If $G$ is $M$ -constructable, then $G$ admits an $M$ -centered $P_\\\\ell $ -decomposition.\",\n \"The proof follows by induction on $|V(G)|$ .\",\n \"By the definition of $M$ -constructable, we may assume that $G$ is the collage of an $M_1$ -constructable graph $G_1$ and an $M_2$ -constructable graph $G_2$ over edges of $M_1$ and $M_2$ .\",\n \"By the induction hypothesis, $G_i$ admits an $M_i$ -centered $P_\\\\ell $ -decomposition $i$ , for $i=1,2$ .\",\n \"Let $a_i,b_i,x_i,y_i$ , for $i=1,\\\\ldots ,k$ be such that $G$ is the graph obtained from $G_1\\\\cup G_2$ by removing $a_1b_1,\\\\ldots ,a_kb_k,x_1y_1,\\\\ldots , x_ky_k$ and adding $a_1x_1,b_1y_1,\\\\ldots ,a_kx_k,b_ky_k$ as above.\",\n \"For ${i=1,\\\\ldots ,k}$ , let $P_i\\\\in 1$ and $Q_i\\\\in 2$ be the paths containing the edges $a_ib_i$ and $x_iy_i$ , respectively.\",\n \"By the definition of $M_1$ - and $M_2$ -centered $P_\\\\ell $ -decomposition, for $i=1,\\\\ldots ,k$ , we may write $P_i = P_{i,1}a_ib_i P_{i,2}$ and $Q_i = Q_{i,1}x_iy_iQ_{i,2}$ , where $P_{i,1},P_{i,2},Q_{i,1}$ and $Q_{i,2}$ are paths of length $(\\\\ell -1)/2$ .\",\n \"Since $G_1$ and $G_2$ are disjoint, $V(P_{i,j})\\\\cap V(Q_{i,j}) = \\\\emptyset $ for $i=1,\\\\ldots ,k$ and $j=1,2$ .\",\n \"Let $R_{i,1} = P_{i,1}\\\\cup \\\\lbrace a_ix_i\\\\rbrace \\\\cup Q_{i,1}$ and $R_{i,2} = P_{i,2}\\\\cup \\\\lbrace b_iy_i\\\\rbrace \\\\cup Q_{i,2}$ , and note that $ \\\\big (1\\\\setminus \\\\lbrace P_1,\\\\ldots ,P_k\\\\rbrace \\\\big )\\\\cup \\\\big (2\\\\setminus \\\\lbrace Q_1,\\\\ldots ,Q_k\\\\rbrace \\\\big )\\\\cup \\\\lbrace R_{i,j}\\\\colon i=1,\\\\ldots ,k\\\\text{ and }j=1,2\\\\rbrace $ is an $M_G$ -centered $P_\\\\ell $ -decomposition of $G$ as desired.\",\n \"By Proposition REF , if $G$ contains a spanning copy $K$ of $K_{4,4}$ and $M=E(G)\\\\setminus E(K)$ , then $G$ is $M$ -constructable.\",\n \"Therefore, Lemma REF yields the following result.\",\n \"Corollary 5.1 If $G$ is a 5-regular graph that contains a $K_{4,4}$ -factor $K$ and ${M=E(G)\\\\setminus E(K)}$ , then $G$ admits an $M$ -centered $P_5$ -decomposition.\"\n ],\n [\n \"5-regular graphs that contain Cayley graphs\",\n \"In this section, we explore 5-regular graphs that contain spanning simply commutative 4-regular Cayley graphs.\",\n \"Botler, Mota, and Wakabayashi [3] showed that every triangle-free 5-regular graph $G$ that has a perfect matching admits a $P_5$ -decomposition.\",\n \"For that, they applied the following strategy: i) to find an initial decomposition of $G$ into paths and trails; and ii) to perform exchanges of edges between the elements of $, preserving a special invariant, while minimizing the number of trails that are not paths.$ The proof of our main result (Theorem REF ) consists of four steps.\",\n \"First, we deal with a somehow degenerate case (Theorem REF ).\",\n \"After that, we follow the framework used by Botler, Mota, Wakabayashi [3], i.e, from the structure of Cayley graphs, we find an initial decomposition into trails, not necessarily paths (Proposition REF ), and then we exchange edges between the elements of the decomposition in order to reduce the number of bad elements (the trails that are not paths).\",\n \"For that, we first show that the bad elements are distributed in a circular fashion (Lemma REF ), and then we show how to deal with these “cycles of bad elements” (Theorem REF ).\",\n \"The invariants preserved by the operations in the proofs of Lemma REF and Theorem REF are presented, respectively, in Definitions REF and REF .\",\n \"The following lemma is used often throughout the text.\",\n \"Lemma 6 Let $\\\\ell $ be odd, and $G$ be an $\\\\ell $ -regular graph.\",\n \"If $ is a decomposition of~$ G$ into trails of length $$,then each vertex of $ G$ is the end vertex of precisely one element of~$ .\",\n \"Let $k$ , $G$ and $ be as in the statement.Let $ n=|V(G)|$.Given an element $ T, we denote by $o(T)$ the number of vertices $v$ in $T$ for which $d_T(v)$ is odd, and given a vertex $v\\\\in V(G)$ , we denote by $v)$ the number of trails in $ for which $ dT(v)$ is odd.Clearly, $ T o(T) = vV(G)v)$.Moreover, for every trail $ T$, we have $ o(T)2$.Also, since every element of $ has $\\\\ell $ edges, we have $| = \\\\frac{1}{\\\\ell }|E(G)|= \\\\frac{1}{\\\\ell }\\\\frac{1}{2}\\\\ell n = \\\\frac{1}{2}n$ .\",\n \"Thus, we have $\\\\sum _{T\\\\in o(T) \\\\le 2| = n.Now, since v\\\\in V(G) has odd degree (in G),v must have odd degree in at least one element of ,and hence v) \\\\ge 1.Thus, we have \\\\sum _{v\\\\in V(G)}v)\\\\ge n,and hence n \\\\le \\\\sum _{v\\\\in V(G)}D(v) = \\\\sum _{T\\\\in o(T) \\\\le n.This implies that v) = 1 for every v\\\\in V(G), as desired.\",\n \"}}Recall that $$ is a finite group of order $ n$ and operation $ +$.Fix two elements $ g,r$ of~$$,we say that $ {g,r}$ is a \\\\emph {simple commutative generator} (SCG)if (a) $ 0{g, r, 2g, 2r}$;(b) $ g{r,-r}$;and (c) $ g+r = r+g$.Let $ {g,r}$ be an SCG, put $ S = {g, -g, r, -r}$, and consider the Cayley graph $ X=X(,S)$.By construction, $ X$ is a simply commutative Cayley graph (see Section~\\\\ref {sec:introduction}).Conditions (a) and (b) guarantee that $ X$ is a simple graph,while condition (c) introduces the main restriction explored in this paper.In this case, we say that~$ X$ is the graph \\\\emph {generated} by $ {g,r}$,and that $ {g,r}$ is the \\\\emph {generator} of $ X$.Finally, we say that a simple $ 5$-regular graph $ G$ with vertex set $$is a \\\\emph {$ {g,r}$-graph}if $ G$ contains a spanning Cayley graph $ X$ generated by $ {g,r}$.We say that $ G$ is a \\\\emph {simply commutative generated graph} or, for short, \\\\emph {SCG-graph} if~$ G$is a $ {g,r}$-graph for some SCG $ {g,r}$.In this section, we verify Conjecture~\\\\ref {conj:favaron} for SCG-graphs.\",\n \"In particular,Proposition~\\\\ref {lemma:cycle-factorization} implies that every $ {g, r}$-graph for which $ 2g = r$ admits an $ Mg,r$-centered decomposition;and as a consequence of Corollary~\\\\ref {corollary:K44}, we obtain the following result,which is also a special case of our main result.$ Theorem 7 Every $\\\\lbrace g,r\\\\rbrace $ -graph for which $2g+2r=0$ and $2g-2r=0$ admits an $M_{g,r}$ -centered decomposition.\",\n \"Let $G$ be a $\\\\lbrace g,r\\\\rbrace $ -graph for which $2g+2r=0$ and $2g-2r=0$ and put ${M=M_{g,r}}$ .\",\n \"Note that we also have $4g=4r=0$ .\",\n \"Let $u$ be a vertex of $G$ , and let $H$ be the component of $G\\\\setminus E(M)$ that contains $u$ .\",\n \"In what follows, we prove that $H$ is a copy of $K_{4,4}$ .\",\n \"Since $g$ and $r$ commute, if $v\\\\in V(H)$ , we have $v = u + ig + jr$ , where $i,j\\\\in \\\\mathbb {N}$ .\",\n \"Since $4g=4r=0$ , we may assume $i,j\\\\in \\\\lbrace 0,1,2,3\\\\rbrace $ .\",\n \"Moreover, since $2g - 2r = 0$ (and hence $2g=2r$ ), we may assume $j\\\\in \\\\lbrace 0,1\\\\rbrace $ .\",\n \"Therefore, there are at most eight vertices in $H$ , namely, $V(H) = \\\\lbrace u, u+g,u+2g,u+3g, u+r,u+2g+r,u+3g+r\\\\rbrace $ .\",\n \"We claim that $H$ is bipartite.\",\n \"Indeed, suppose that there is an odd cycle $C$ in $H$ .\",\n \"Then, there is an element $x \\\\in V(C)$ such that $x+ig+jr=x$ , where $i,j \\\\in \\\\mathbb {N}$ .\",\n \"Note that $i+j$ can be obtained from the length of $C$ by ignoring pairs of edges with the same color and different directions.\",\n \"Since $C$ is odd, precisely one between $i$ and $j$ is odd.\",\n \"Suppose, without loss of generality, that $i$ is odd, and hence $j$ is even.\",\n \"Note that, since $2g=2r$ , we have $jr = jg$ .\",\n \"Therefore, $(i+j)g = ig + jr = 0$ .\",\n \"Let $s\\\\in \\\\lbrace 1,3\\\\rbrace $ be such that $i+j = 4q + s$ for some $q\\\\in \\\\mathbb {N}$ .\",\n \"Then we have $0 = (i+j)g = 4qg + sg$ , which implies $sg = 0$ .\",\n \"Thus, if $s = 1$ , then $g=0$ ; and if $s = 3$ , then $g = 4g-sg = 0$ , a contradiction to the definition of SCG.\",\n \"Thus, since $H$ is 4-regular, $H$ is a copy of $K_{4,4}$ .\",\n \"Now, since every component of $G\\\\setminus E(M)$ is isomorphic to $K_{4,4}$ .\",\n \"Therefore, $G$ is a 5-regular graph that contains a $K_{4,4}$ -factor, and hence by Corollary REF , $G$ admits an $M$ -centered decomposition as desired.\",\n \"If $X$ is the graph generated by an SCG $\\\\lbrace g,r\\\\rbrace $ , and $x\\\\in \\\\lbrace g,r\\\\rbrace $ , then we denote by $F_x$ the 2-factor of $X$ with edge set $E(F_x) = \\\\lbrace v+x\\\\colon v\\\\in \\\\Gamma \\\\rbrace $ .\",\n \"If $G$ is a $\\\\lbrace g,r\\\\rbrace $ -graph, then we denote by $M_{g,r}$ the perfect matching $G\\\\setminus E(F_g \\\\cup F_r)$ , and the triple $\\\\lbrace M_{g,r},F_g,F_r\\\\rbrace $ is called the base factorization of $G$ .\",\n \"Although $G$ is a simple graph, for ease of notation, we refer to an edge $uv\\\\in F_x$ , with $x\\\\in \\\\lbrace g,r\\\\rbrace $ , as a green (resp.\",\n \"red) out edge of $u$ and in edge of $v$ if $v = u+x$ and $x=g$ (resp.\",\n \"$x=r$ ).\",\n \"In the figures throughout the text, the edges in $F_g$ , $F_r$ , $M_{g,r}$ are illustrated, respectively, in dotted green, dashed red, and double black patterns, while edges without specific affiliation are illustrated in straight gray pattern.\",\n \"Moreover, if such an edge has a specific direction (i.e., in edge or out edge), it is illustrated accordingly.\",\n \"Note that each vertex $u$ has precisely one edge of each type (green in edge, green out edge, red in edge, red out edge), and is incident to precisely one edge of $M_{g,r}$ .\",\n \"In particular, the group structure overcomes Theorem REF by giving a decomposition of $X$ into 2-factors in terms of the elements $g$ and $r$ .\",\n \"In the rest of the paper we deal with the case $2g+2r\\\\ne 0$ .\",\n \"For that, we characterize the elements of the forthcoming decompositions.\",\n \"Definition 1 We say that a trail $T$ in a $\\\\lbrace g,r\\\\rbrace $ -graph is of type A, B, C, or D if $T$ can be written as $a_0a_1a_2a_3a_4a_5$ , where $a_0,a_1,a_2,a_3,a_4$ are distinct vertices, as follows.\",\n \"$a_2 = a_5$ , $a_2a_3\\\\in M_{g,r}$ , $a_2a_1,a_3a_4\\\\in F_g$ , $a_4a_5 \\\\in F_r$ , and $a_1a_0\\\\in F_g\\\\cup F_r\\\\cup M_{g,r}$ , i.e., $a_1a_0$ is an out edge of $a_1$ , or $a_1a_0\\\\in M_{g,r}$ (see Figure REF fig:typeA).\",\n \"In this case, we say that $a_3$ is the primary connection vertex of $T$ , $a_2$ is the secondary connection vertex of $T$ ; $a_1$ is the auxiliary vertex of $T$ ; and $a_4$ is the tricky vertex of $T$ .\",\n \"We denote these vertices, respectively, by ${\\\\rm {cv}}_1(T)$ , ${\\\\rm {cv}}_2(T)$ , ${\\\\rm {aux}}(T)$ , and ${\\\\rm {tr}}(T)$ ; $a_5\\\\notin \\\\lbrace a_0,a_1,a_2,a_3,a_4\\\\rbrace $ , $a_2a_3\\\\in M_{g,r}$ $a_2a_1,a_3a_4\\\\in F_g$ , $a_1a_0,a_4a_5\\\\in F_g\\\\cup F_r\\\\cup M_{g,r}$ (see Figure REF fig:typeB); $a_5\\\\notin \\\\lbrace a_0,a_1,a_2,a_3,a_4\\\\rbrace $ , $a_2a_1,a_4a_3\\\\in F_g$ , $a_3a_2,a_4a_5\\\\in F_r$ , $a_1a_0\\\\in F_g\\\\cup F_r\\\\cup M_{g,r}$ , and, moreover, we have $a_2a_4\\\\in E(G)$ and $a_2a_4 \\\\in M_{g,r}$ (see Figure REF fig:typeC); $a_5\\\\notin \\\\lbrace a_0,a_1,a_2,a_3,a_4\\\\rbrace $ , $a_1a_0,a_4a_5\\\\in F_r$ , $a_1a_2,a_3a_4\\\\in M_{g,r}$ , and $a_3a_2\\\\in F_g$ (see Figure REF fig:typeD).\",\n \"Figure: The main types of trails.We remark that elements of type A are not paths, while elements of type B, C, and D are paths.\",\n \"Moreover, the connection vertices are defined only for elements of type A, and the connection vertices of an element $T$ are always incident to an edge of $M_{g,r}$ in $T$ , and hence, no vertex of a $\\\\lbrace g,r\\\\rbrace $ -graph is a connection vertex of two edge-disjoint elements of type A in a graph.\",\n \"Given a trail (not necessarily a path) $T= a_0a_1a_2a_3a_4a_5$ in a decomposition $ of a $ {g,r}$-graph $ G$,we say that the edge $ a1a0$ (resp.\",\n \"$ a4a5$) is a \\\\emph {hanging edge}at $ a1$ (resp.\",\n \"$ a4$) if $ a1a0Mg,rFgFr$ (resp.\",\n \"$ a4a5Mg,rFgFr$),i.e., the hanging edges of $ T$ are the end edges of $ T$ that are in $ Mg,r$ or that are in edges of its end vertices.By Definition~\\\\ref {def:types}, all end edges of elements of type~A, B, C, or D are hanging edges.Note that if $ T$ is an element of type~A where $ a5 = a2$, then $ a1a0$, $ a2a3$ and $ a4a2$ are hanging edgesof $ T$ at, respectively, $ a1$, $ a3$, and $ a4$.Given a trail decomposition $ of a graph $G$ and a vertex $u\\\\in V(G)$ , we denote by ${\\\\rm {hang}}_u)$ the number of edges of $G$ that are hanging edges at $u$ .\",\n \"The next lemma presents a consequence of the exchange of hanging edges at primary connection vertices.\",\n \"Lemma 8 If $T= a_0a_1a_2a_3a_4a_5$ is an element of type A in a decomposition of a $\\\\lbrace g,r\\\\rbrace $ -graph $G$ into trails of length 5, where $a_5=a_2$ and $a_2a_3\\\\in M_{g,r}$ , and $u\\\\in V(G)$ is such that $a_3u$ is a hanging edge at $a_3 = {\\\\rm {cv}}_1(T)$ , then $T^{\\\\prime }= a_0a_1a_2a_4a_3u$ is of type C. Let $T$ , $u$ , and $T^{\\\\prime }$ be as in the statement.\",\n \"Since $a_3a_4$ is a green out edge of $a_3$ and $a_2a_3$ is an edge of $M_{g,r}$ incident to $a_3$ , we conclude that $a_3u$ is a red out edge of $a_3$ , and hence $u = a_3 + r$ .\",\n \"Now, since $G$ is simple, we have $u\\\\notin \\\\lbrace a_2,a_3,a_4\\\\rbrace $ ; if $u=a_1$ , then we have $a_3 + r= u = a_1 = a_3 + g + r + g$ , which implies $2g=0$ , a contradiction to the definition of SCG.\",\n \"Finally, by Lemma REF we have $u\\\\ne a_0$ .\",\n \"Thus, $T^{\\\\prime }$ is a path.\",\n \"Since $a_3u\\\\in F_r$ , $T^{\\\\prime }$ is of type C. Figure: Exchange of edges performed in the proof of Lemma .The following lemma shows how two elements of type A may be connected.\",\n \"Lemma 9 If $T_1$ and $T_2$ are two edge-disjoint elements of type A in a $\\\\lbrace g,r\\\\rbrace $ -graph $G$ for which ${\\\\rm {tr}}(T_2) = {\\\\rm {cv}}_1(T_1)$ , then ${\\\\rm {aux}}(T_2) = {\\\\rm {cv}}_2(T_1)$ .\",\n \"Let $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ , where $a_5 = a_2$ and $b_5 = b_2$ and $a_2a_3,b_2b_3\\\\in M_{g,r}$ .\",\n \"If ${\\\\rm {cv}}_1(T_1)={\\\\rm {tr}}(T_2)$ , then $a_3=b_4$ .\",\n \"Since $b_1 = b_4 + r+g$ and $a_2 = a_3 + g + r$ .\",\n \"Thus, ${\\\\rm {aux}}(T_2) = b_1 = b_4 + r + g = a_3 + r + g = a_2 = {\\\\rm {cv}}_2(T_1)$ , as desired (see Figure REF ).\",\n \"Figure: Identities given by Lemma when b 3 =0b_3 = 0.\"\n ],\n [\n \"Complete decompositions\",\n \"The following definition consists of two properties that are invariant under a series of operations performed throughout the proof of Lemma REF .\",\n \"Definition 2 A decomposition $ of a $ {g,r}$-graph $ G$into trails of length $ 5$is \\\\emph {complete} if the following hold for every $ T. $T$ is of type A, B, C or D; If $T$ is of type A, then ${\\\\rm {hang}}_({\\\\rm {cv}}_1(T)\\\\big )\\\\ge 2$ and ${\\\\rm {hang}}_({\\\\rm {cv}}_2(T)\\\\big )\\\\ge 1$ .\",\n \"The first step of our proof is given by the next proposition, which presents an initial decomposition for the graphs studied.\",\n \"Proposition 10 If $G$ is a $\\\\lbrace g,r\\\\rbrace $ -graph for which $2g+2r\\\\ne 0$ , then $G$ admits a complete decomposition.\",\n \"Let $\\\\lbrace M_{g,r},F_g,F_r\\\\rbrace $ be the base factorization of $G$ .\",\n \"For each $e=xy\\\\in M_{g,r}$ , let $P_e=a_0a_1a_2a_3a_4a_5$ , where $a_1a_0,a_4a_5\\\\in F_r$ , $a_2a_1,a_3a_4\\\\in F_g$ , $a_2=x$ , and $a_3 = y$ .\",\n \"We claim that $ \\\\lbrace P_e\\\\colon e\\\\in M_{g,r}\\\\rbrace $ is complete.\",\n \"Clearly, $P_e$ is an element of type A or B, for every $e\\\\in M_{g,r}$ , and hence $ satisfiesDefinition~\\\\ref {def:complete-commutative}(\\\\ref {def:complete-commutative-types}).Moreover, note that $ a0a1$ (resp.\",\n \"$ a4a5$) is a hanging edgeat $ a1$ (resp.\",\n \"$ a4$).Thus, given $ zV(G)$, let $ e' = xyMg,r$ be such that $ x= z-g$,then $ Pe'$ contains a hanging edge at $ z$,namely, the red out edge of $ z$,and hence there is a hanging edge at every vertex of $ G$.Moreover, if $ z=cv1(T)$ for some element $ T of type A, and $e\\\\in M_{g,r}\\\\cap E(T)$ , then $e$ is a second hanging edge at $z$ .\",\n \"This proves Definition REF (REF ).\",\n \"We say that an element $T$ of type A in a decomposition $ is \\\\emph {free}if $ tr(T)cvi(T')$ for every element $ T' of type A and $i\\\\in \\\\lbrace 1,2\\\\rbrace $ .\",\n \"An A-chain is a sequence $T_0, T_1, \\\\ldots , T_{s-1}$ of elements of type A such that for each $j\\\\in \\\\lbrace 0,\\\\ldots ,s-1\\\\rbrace $ , we have ${\\\\rm {tr}}(T_j) = {\\\\rm {cv}}_i(T_{j-1})$ , for some $i \\\\in \\\\lbrace 1,2\\\\rbrace $ (subtraction on the indexes are taken modulo $s$ ).\",\n \"Note that A-chains do not consider the auxiliary vertex when allowing two elements to be consecutive.\",\n \"Thus, elements, say $T$ and $T^{\\\\prime }$ , of type A that are not consecutive in an A-chain, or that are in different A-chains, may still share a vertex $u$ for which ${\\\\rm {cv}}_i(T) = u = {\\\\rm {aux}}(T^{\\\\prime })$ .\",\n \"Given a decomposition $ of a graph $ G$ into trails of length $ 5$, denote by$ ($ the number of elements that are not paths.By exchanging edges between the elements of a decompositiongiven by Proposition~\\\\ref {proposition:initial-decomposition},we can show that a complete decomposition that minimizes $ ($has no free element, and hence its elements of type~A are partitionedinto A-chains.$ Lemma 11 Every $\\\\lbrace g,r\\\\rbrace $ -graph for which $2g+2r\\\\ne 0$ admits a complete decomposition in which the elements of type A are partitioned into A-chains.\",\n \"Let $g$ and $r$ be as in the statement, let $G$ be a $\\\\lbrace g,r\\\\rbrace $ -graph, and put ${M=M_{g,r}}$ .\",\n \"By Proposition REF , $G$ admits a complete decomposition.\",\n \"Let $ be a complete decomposition of~$ G$ that minimizes $ ($.In what follows, we prove that $ contains no free element of type A.\",\n \"For that, we prove three claims regarding the relation between some elements of $.In the proof of each such claim, we exchange edges between some elements of $ and obtain a complete decomposition $$ into trails of length 5 such that $\\\\tau () <\\\\tau ($ , which is a contradiction to the minimality of $.To check that $$ is a complete decomposition, we observe the two following items:(i) the vertices $ u$ for which $ hangD(u) >hang(u)$ are verticesthat are not connection vertices of $$,for example, tricky vertices of free elements of type~A,or connection vertices of elements of type~A in $ that become paths in $$ .\",\n \"Hence, Definition REF (REF ) holds for $$ ; and (ii) every element of $$ that is not an element of $, i.e.,the elements involved in the exchange of edges, are of type~A, B, C, or~D,and hence \\\\ref {def:complete-commutative}(\\\\ref {def:complete-commutative-types})holds for $$.$ Claim 11.1 No element of type B or C has a hanging edge at the primary connection vertex of a free element of type A.\",\n \"Let $T_1 = a_0a_1a_2a_3a_4a_5\\\\in , where $ a5=a2$ and $ a2a3M$, be a free element of type~A,and let $ T2= b0b1b2b3b4b5 be an element of type B or C that contains a hanging edge at ${\\\\rm {cv}}_1(T_1)$ .\",\n \"We divide the proof depending on whether $T_2$ is of type B or C. $\\\\mathbf {T_2}$ is of type B.\",\n \"Suppose, for a contradiction, that $b_4={\\\\rm {cv}}_1(T_1)=a_3$ .\",\n \"Put ${T_1^{\\\\prime } = a_0a_1a_2a_4a_3b_5}$ , $T_2^{\\\\prime } = b_0b_1b_2b_3b_4a_2$ (see Figure REF ), and let $ = \\\\big (\\\\lbrace T_1,T_2\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime }\\\\rbrace $ .\",\n \"Note that $$ is a decomposition of $G$ into trails of length 5.\",\n \"By Lemma REF , $T_1^{\\\\prime }$ is an element of Type C. In what follows, we prove that $T_2^{\\\\prime }$ is of type B, i.e., $a_2\\\\notin \\\\lbrace b_0,b_1,b_2,b_3,b_4\\\\rbrace $ .\",\n \"Indeed, since $G$ has no loops or multiple edges, $a_2\\\\notin \\\\lbrace b_3,b_4\\\\rbrace $ .\",\n \"Since $M$ is a matching, $a_2\\\\ne b_2$ .\",\n \"If $a_2 = b_1$ , then $b_2 = b_1-g = a_2 -g = a_3 + g +r -g = b_5$ , and hence $T_2$ is of type A, a contradiction.\",\n \"Finally, by Lemma REF , $a_2 \\\\ne b_0$ .\",\n \"Thus, $T^{\\\\prime }_2$ is an element of type B, and hence Definition REF (REF ) holds for $$ .\",\n \"Note that ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)for every v\\\\in V(G)\\\\setminus \\\\lbrace a_4\\\\rbrace .Since a_4 is not a connection vertex of ,and by Definition~\\\\ref {def:complete-commutative}(\\\\ref {def:complete-commutative-hanging-edge}),we have {\\\\rm {hang}}_{\\\\big ({\\\\rm {cv}}_1(T)\\\\big )\\\\ge 2 and {\\\\rm {hang}}_{\\\\big ({\\\\rm {cv}}_2(T)\\\\big )\\\\ge 1for every T\\\\in ,we have{\\\\rm {hang}}_{}\\\\big ({\\\\rm {cv}}_1(T)\\\\big )\\\\ge 2 and {\\\\rm {hang}}_{}\\\\big ({\\\\rm {cv}}_2(T)\\\\big )\\\\ge 1for every T\\\\in ,Thus Definition~\\\\ref {def:complete-commutative}(\\\\ref {def:complete-commutative-hanging-edge}) holds for .Therefore, is a complete decomposition such that\\\\tau ()=\\\\tau (-1<\\\\tau (, a contradictionto the minimality of .\",\n \"}\\\\begin{figure}[H]\\\\centering \\\\begin{subfigure}{.45}\\\\centering \\\\scalebox {.8}{\\\\begin{tikzpicture}[scale=1.5]\\\\end{tikzpicture}[fatpath,backcolor1] (120:1)-- ((-1,0)+(120:1)) -- (120:1) -- (0,0) -- (0:1) -- (60:1) -- (0,0) -- (60:1);}\\\\node (0) [black vertex] at ((-1,0)+(120:1)) {};\\\\node (1) [black vertex] at (120:1) {};\\\\node (2) [black vertex] at (0,0) {};\\\\node (3) [black vertex] at (0:1) {};\\\\node (4) [black vertex] at (60:1) {};\\\\node () [] at ((2,0)+(60:1)) {};\\\\end{subfigure}\\\\end{figure}\\\\node (4^{\\\\prime }) [black vertex] at (1,0) {};\\\\node (3^{\\\\prime }) [black vertex] at ((4^{\\\\prime })-(60:1)) {}; \\\\node (2^{\\\\prime }) [black vertex] at ((3^{\\\\prime })-(1,0)) {};\\\\node (1^{\\\\prime }) [black vertex] at ((2^{\\\\prime })+(120:1)) {};\\\\node (0^{\\\\prime }) [black vertex] at ((2^{\\\\prime })+(-1,0)+(120:1)) {};\\\\node (5^{\\\\prime }) [black vertex] at ((4^{\\\\prime })+(1,0)) {};}}\\\\node () [] at ($ (0)+(90:.2)$) {a_0};\\\\node () [] at ($ (1)+(90:.2)$) {a_1};\\\\node () [] at ($ (2)+(-90:.2)$) {a_2};\\\\node () [] at ($ (3)+(45:.3)$) {a_3};\\\\node () [] at ($ (4)+(90:.2)$) {a_4};$ ) [] at ($(0^{\\\\prime })+(90:.2)$ ) $b_0$ ; ) [] at ($(1^{\\\\prime })+(45:.3)$ ) $b_1$ ; ) [] at ($(2^{\\\\prime })+(-90:.2)$ ) $b_2$ ; ) [] at ($(3^{\\\\prime })+(-90:.2)$ ) $b_3$ ; ) [] at ($(4^{\\\\prime })+(-45:.3)$ ) $b_4$ ; ) [] at ($(5^{\\\\prime })+(90:.2)$ ) $b_5$ ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.3pt,color=gray,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (5'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); .45 [scale=1.5] [fatpath,backcolor1] (120:1) – ($(-1,0)+(120:1)$ ) – (120:1) – (0,0) (60:1) – (0,0) – (60:1) – (0:1) – ($(2,0)$ ) – (0:1); 0) [black vertex] at ($(-1,0)+(120:1)$ ) ; 1) [black vertex] at (120:1) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (0:1) ; 4) [black vertex] at (60:1) ; ) [] at ($(2,0)+(60:1)$ ) ; 4') [black vertex] at (1,0) ; 3') [black vertex] at ($(4^{\\\\prime })-(60:1)$ ) ; 2') [black vertex] at ($(3^{\\\\prime })-(1,0)$ ) ; 1') [black vertex] at ($(2^{\\\\prime })+(120:1)$ ) ; 0') [black vertex] at ($(2^{\\\\prime })+(-1,0)+(120:1)$ ) ; 5') [black vertex] at ($(4^{\\\\prime })+(1,0)$ ) ; ) [] at ($(0)+(90:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(-90:.2)$ ) $a_2$ ; ) [] at ($(3)+(45:.3)$ ) $a_3$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\\\prime })+(90:.2)$ ) $b_0$ ; ) [] at ($(1^{\\\\prime })+(45:.3)$ ) $b_1$ ; ) [] at ($(2^{\\\\prime })+(-90:.2)$ ) $b_2$ ; ) [] at ($(3^{\\\\prime })+(-90:.2)$ ) $b_3$ ; ) [] at ($(4^{\\\\prime })+(-45:.3)$ ) $b_4$ ; ) [] at ($(5^{\\\\prime })+(90:.2)$ ) $b_5$ ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.3pt,color=gray,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (5'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); Exchange of edges between elements of type A and B in the proof of Claim REF .\",\n \"$\\\\mathbf {T_2}$ is of type C. We may assume $b_3b_2\\\\in F_r$ .\",\n \"In this case we have $b_4b_3\\\\in F_g$ .\",\n \"Since $T_2$ contains a hanging edge at ${\\\\rm {cv}}_1(T_1)$ , we have $a_3={\\\\rm {cv}}_1(T_1)\\\\in \\\\lbrace b_1,b_4\\\\rbrace $ .\",\n \"If $b_4=a_3$ , then there are two green out edges at $a_3$ , namely $a_3a_4, b_4a_3$ , a contradiction.\",\n \"Thus, we may assume that $a_3 = b_1$ .\",\n \"Put $T_1^{\\\\prime } = a_0a_1a_2a_4a_3b_0$ , $T_2^{\\\\prime } = a_2b_1b_2b_4b_3b_5$ (see Figure REF ), and let $ = \\\\big (\\\\lbrace T_1,T_2\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime }\\\\rbrace $ .\",\n \"Note that $$ is a decomposition of $G$ into trails of length 5.\",\n \"By Lemma REF , $T_1^{\\\\prime }$ is an element of Type C. In what follows we prove that $T_2^{\\\\prime }$ is a path.\",\n \"For that, we prove that $a_2\\\\notin \\\\lbrace b_0,b_1,b_2,b_3,b_4\\\\rbrace $ .\",\n \"Indeed, since $G$ is simple, ${a_2\\\\notin \\\\lbrace b_1,b_2\\\\rbrace }$ .\",\n \"If $a_2 = b_4$ , then $a_2a_1$ and $b_4b_3$ are two green out edges at $a_2$ , a contradiction.\",\n \"By Lemma REF , $a_2\\\\ne b_5$ .\",\n \"Finally, $a_3 + g + r = a_2$ and ${b_1 = b_3 + r + g}$ , if $a_2 = b_3$ , then we have $a_3 + g + r = a_2 = b_3 = b_1 - g - r = a_3 - g - r$ , which implies $2g+2r = 0$ , a contradiction.\",\n \"Thus, $T^{\\\\prime }_2$ is an element of type C, and hence Definition REF (REF ) holds for $$ .\",\n \"Analogously to the case above $$ is a complete decomposition of $G$ such that $\\\\tau ()=\\\\tau (-1<\\\\tau ($ , a contradiction to the minimality of $.\",\n \"$ Figure: Exchange of edges between elements of type Aand C in the proof of Claim .Claim 11.2 Let $T_1$ and $T_2$ be two elements of type A in $.If $ T1$ is free and $ T2$ contains a hangingedge on $ cv1(T1)$, then no element of type~A, B, or C in $ {T1,T2}$contains a hanging edge at $ cv2(T2)$.\",\n \"$ Let $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ be two elements of $,where $ a5=a2$ and $ b5=b2$ and $ a2a3,b2b3M$.First, we prove that $ cv1(T1)=tr(T2)$,and hence, by Lemma~\\\\ref {lemma:type12}, we have $ cv2(T1) = aux(T2)$.Suppose, for contradiction, that $ cv1(T1)tr(T2)$.Since $ b2b3M$, we must have $ b1=cv1(T1)=a3$.Now, put $ T1' = a0a1a2a4a3b0$,$ T2' = a2b1b2b3b4b2$ (see Figure~\\\\ref {fig:case3}) and let$ = ({T1,T2}){T1',T2'}$.By Lemma~\\\\ref {lemma:T^{\\\\prime }1-is-path}, $ T1'$ is an element of type~C.We claim that $ T2'$ is an element of type~A.For that we prove that $ a2 {b1,b2,b3,b4}$.Again, since $ G$ is a simple graph, we have $ a2{b1,b2}$.Since every vertex is incident to precisely one edge of $ M$, we have $ a2b3$,and if $ a2 = b4$, then we have $ a3 + g + r = a2 = b4 = a3 - g - r$, which implies $ 2g+2r=0$, a contradiction.Thus, Definition~\\\\ref {def:complete-commutative}(\\\\ref {def:complete-commutative-types}) holds for $$.Analogously to the cases above,$$ is a complete decomposition such that$ ()=(-1<($, a contradictionto the minimality of $ .\",\n \"Finally, by Lemma REF , we have ${\\\\rm {cv}}_2(T_1) = {\\\\rm {aux}}(T_2)$ .\",\n \"Figure: Exchange of edges between two elements of type Ain the proof of Claim .Now, let $T_3\\\\in \\\\lbrace T_1,T_2\\\\rbrace $ be an element of type A, B, or C, and suppose, for a contradiction, that $T_3$ contains a hanging edge at ${\\\\rm {cv}}_2(T_2)$ .\",\n \"Since ${\\\\rm {cv}}_1(T_1)={\\\\rm {tr}}(T_2)$ and ${\\\\rm {cv}}_2(T_1) = {\\\\rm {aux}}(T_2)$ , we have $a_3=b_4$ , $a_5=a_2=b_1$ and $b_5=b_2$ .\",\n \"In what follows, we divide the proof according to the type of $T_3$ .\",\n \"$\\\\mathbf {T_3}$ is of type A.\",\n \"Let $T_3 = c_0c_1c_2c_3c_4c_5$ , where $c_2=c_5$ and $c_2c_3\\\\in M$ .\",\n \"Since each vertex is incident to precisely one edge of $M$ we have $c_3\\\\ne b_2 = {\\\\rm {cv}}_2(T_2)$ .\",\n \"Therefore, we have ${\\\\rm {cv}}_2(T_2)\\\\in \\\\lbrace c_1,c_4\\\\rbrace $ .\",\n \"Suppose that ${\\\\rm {cv}}_2(T_2)=c_4$ .\",\n \"Thus, we have $b_5 = b_2 = c_4$ .\",\n \"Put $T_1^{\\\\prime } = a_0a_1a_2a_4a_3b_5$ , $T_2^{\\\\prime } = b_0b_1b_4b_3b_2c_2$ , $T_3^{\\\\prime } = b_1c_4c_3c_2c_1c_0$ (see Figure REF ), and let $ = \\\\big (\\\\lbrace T_1,T_2,T_3\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime },T_3^{\\\\prime }\\\\rbrace $ .\",\n \"In what follows, we prove that $T_1^{\\\\prime }$ , $T_2^{\\\\prime }$ and $T_3^{\\\\prime }$ are paths.\",\n \"By Lemma REF , $T_1^{\\\\prime }$ is an element of type C. Since $G$ is simple, we have $c_2\\\\notin \\\\lbrace b_1,b_2,b_3,b_4\\\\rbrace $ and $b_1\\\\notin \\\\lbrace c_2,c_3,c_4\\\\rbrace $ .\",\n \"By Lemma REF , $c_2 \\\\ne b_0,~b_1 \\\\ne c_0$ .\",\n \"Therefore, $T_2^{\\\\prime }$ is an element of type D. If $b_1 = c_1$ , then $b_2b_1$ and $c_2c_1$ are two green in edges at $c_1$ , a contradiction.\",\n \"Thus, $T^{\\\\prime }_3$ is an element of type B, and hence Definition REF (REF ) holds for $$ .\",\n \"Analogously to the cases above, we have ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)\\\\ge 0for every v\\\\in V(G)\\\\setminus \\\\lbrace a_3,a_4,b_3,c_3\\\\rbrace .Since a_3,a_4,b_3,c_3 are not connection vertices in ,Definition~\\\\ref {def:complete-commutative}(\\\\ref {def:complete-commutative-hanging-edge}) holds for .Thus, is a complete decomposition such that\\\\tau ()=\\\\tau (-3<\\\\tau (, a contradiction to the minimality of~.\",\n \"}\\\\begin{figure}\\\\centering \\\\begin{subfigure}{.45}\\\\centering \\\\scalebox {.8}{\\\\begin{tikzpicture}[scale=1.5]\\\\end{tikzpicture}[fatpath,backcolor1] (-0.5,0.8) -- (-1.5,0.8) -- (-0.5,0.8) -- (0,0) -- (0,0) -- (1,0) -- (0.5,0.8) -- (0,0);[fatpath,backcolor2] (-0.5,-0.8) -- (-1.5,-0.8)-- (-0.5,-0.8) --(0,-1.6) -- (1,-1.6) -- (0.5,-0.8) -- (0,-1.6);}\\\\node (0) [black vertex] at (-1.5,0.8) {};\\\\node (1) [black vertex] at (-0.5,0.8) {};\\\\node (2) [black vertex] at (0,0) {};\\\\node (3) [black vertex] at (1,0) {};\\\\node (4) [black vertex] at (0.5,0.8) {};\\\\node () [] at ((0,0)+(60:1)) {};\\\\end{subfigure}\\\\end{figure}\\\\node (4^{\\\\prime }) [black vertex] at ($ (1,0)$) {};\\\\node (3^{\\\\prime }) [black vertex] at ($ (3)+(0.5,-0.8)$) {}; \\\\node (2^{\\\\prime }) [black vertex] at ($ (2)+(0.5,-0.8)$) {};\\\\node (1^{\\\\prime }) [black vertex] at ($ (0,0)$) {};\\\\node (0^{\\\\prime }) [black vertex] at ($ (2')-(1,0)$) {};\\\\node (5^{\\\\prime }) [] at ($ (2,0)$) {};$ 4”) [black vertex] at ($(2^{\\\\prime })$ ) ; 3”) [black vertex] at ($(3)-(0,1.6)$ ) ; 2”) [black vertex] at ($(2)-(0,1.6)$ ) ; 1”) [black vertex] at ($(4^{\\\\prime \\\\prime })-(1,0)$ ) ; 0”) [black vertex] at ($(1^{\\\\prime \\\\prime })+(-1,0)$ ) ; 5”) [] at ($(2,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.6)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\\\prime })+(-90:.3)-(0.35,0)$ ) $b_0=c_1$ ; ) [] at ($(1^{\\\\prime })+(180:.4)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\\\prime })+(-90:.25)+(0.7,0)$ ) $b_2=c_4$ ; ) [] at ($(3^{\\\\prime })+(0:.3)$ ) $b_3$ ; ) [] at ($(4^{\\\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\\\prime \\\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\\\prime \\\\prime })+(225:.4)$ ) ; ) [] at ($(2^{\\\\prime \\\\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\\\prime \\\\prime })+(0:.3)$ ) $c_3$ ; ) [] at ($(4^{\\\\prime \\\\prime })+(90:.3)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.3pt,color=gray,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (0.5,0.8) – (1,0) – (0.5,-0.8); [fatpath,backcolor2] (-0.5,-0.8) – (-1.5,-0.8)– (-0.5,-0.8) –(0,-1.6) – (1,-1.6) – (0.5,-0.8) – (0,0); 0) [black vertex] at (-1.5,0.8) ; 1) [black vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [black vertex] at (0.5,0.8) ; ) [] at ($(0,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(2^{\\\\prime })-(1,0)$ ) ; 5') [] at ($(2,0)$ ) ; 4”) [black vertex] at ($(2^{\\\\prime })$ ) ; 3”) [black vertex] at ($(3)-(0,1.6)$ ) ; 2”) [black vertex] at ($(2)-(0,1.6)$ ) ; 1”) [black vertex] at ($(4^{\\\\prime \\\\prime })-(1,0)$ ) ; 0”) [black vertex] at ($(1^{\\\\prime \\\\prime })+(-1,0)$ ) ; 5”) [] at ($(2,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.6)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\\\prime })+(-90:.3)-(0.35,0)$ ) $b_0=c_1$ ; ) [] at ($(1^{\\\\prime })+(180:.4)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\\\prime })+(-90:.25)+(0.7,0)$ ) $b_2=c_4$ ; ) [] at ($(3^{\\\\prime })+(0:.3)$ ) $b_3$ ; ) [] at ($(4^{\\\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\\\prime \\\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\\\prime \\\\prime })+(225:.4)$ ) ; ) [] at ($(2^{\\\\prime \\\\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\\\prime \\\\prime })+(0:.3)$ ) $c_3$ ; ) [] at ($(4^{\\\\prime \\\\prime })+(90:.3)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.3pt,color=gray,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange of edges between three elements of type A in the proof of Claim REF .\",\n \"Thus, we may assume ${\\\\rm {cv}}_2(T_2) = c_1$ .\",\n \"This implies that $b_5=b_2=c_1$ , and hence we have $b_3 = b_5 - r - g = c_1 - g-r = c_4$ .\",\n \"Put $T_1^{\\\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\\\prime } = b_1c_1c_2c_3c_4c_2$ (see Figure REF ) and $ = \\\\big (\\\\lbrace T_1,T_2,T_3\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime },T_3^{\\\\prime }\\\\rbrace $ .\",\n \"Again, by Lemma REF , $T_1^{\\\\prime }$ is an element of type C. We claim that $T_2^{\\\\prime }$ , $T_3^{\\\\prime }$ are, respectively, of type D and A.\",\n \"Since $G$ is simple, $c_0\\\\notin \\\\lbrace b_1,b_2,b_3,b_4\\\\rbrace $ and $b_1\\\\notin \\\\lbrace c_1,c_2,c_4\\\\rbrace $ .\",\n \"By Lemma REF we have $c_0 \\\\ne b_0$ .\",\n \"Therefore, $T_2^{\\\\prime }$ is of type D. Finally, if $b_1 = c_3$ , then $d(b_1)\\\\ge 7>5$ , a contradiction.\",\n \"Thus, $T^{\\\\prime }_3$ is an element of type A, and hence Definition REF (REF ) holds for $$ .\",\n \"Analogously to the case above, we have ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)\\\\ge 0 for every v \\\\in V(G)\\\\setminus \\\\lbrace a_4,a_3,b_3\\\\rbrace .Since a_4,a_3,b_3 are not connection vertices in ,Definition~\\\\ref {def:complete-commutative}(\\\\ref {def:complete-commutative-hanging-edge}) holds for .Thus, is a complete decomposition such that {\\\\tau () = \\\\tau ( -2<\\\\tau (}, a contradiction to the minimality of~.\",\n \"}$ Figure: Exchange of edges between three elements of type Ain the proof of Claim .$\\\\mathbf {T_3}$ is of type B.\",\n \"Let $T_3 = c_0c_1c_2c_3c_4c_5$ be an element of type B.\",\n \"Since $T_3$ contains a hanging edge on ${\\\\rm {cv}}_2(T_2)=b_2$ , we have $b_2\\\\in \\\\lbrace c_1,c_4\\\\rbrace $ .\",\n \"By symmetry we may assume $b_2=c_1$ .\",\n \"Thus, put $T_1^{\\\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\\\prime } = b_1c_1c_2c_3c_4c_5$ (see Figure REF ) and let $ = \\\\big (\\\\lbrace T_1,T_2,T_3\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime },T_3^{\\\\prime }\\\\rbrace $ .\",\n \"Again, by Lemma REF , $T_1^{\\\\prime }$ is an element of type C. We prove that $T_2^{\\\\prime }$ and $T_3^{\\\\prime }$ are, respectively, of type D and B.\",\n \"Since $G$ is simple, we have $c_0\\\\notin \\\\lbrace b_1,b_2,b_3,b_4\\\\rbrace $ and $b_1\\\\notin \\\\lbrace c_1,c_2\\\\rbrace $ .\",\n \"By Lemma REF , we have $c_0 \\\\ne b_0$ and $b_1 \\\\ne c_5$ .\",\n \"Therefore, $T_2^{\\\\prime }$ is an element of type D. Since $c_4=c_3+g$ and $b_1=c_1+g$ , if $c_4=b_1$ , then $c_3=c_1$ , a contradiction.\",\n \"If $b_1 \\\\in \\\\lbrace c_3,c_4\\\\rbrace $ , then $d(b_1)\\\\ge 7>5$ , a contradiction.\",\n \"Therefore, $T_3^{\\\\prime }$ is an element of type B. Analogously to the case above, $$ is a complete decomposition such that $\\\\tau () = \\\\tau ( -2<\\\\tau ($ , a contradiction to the minimality of $.$ Figure: Exchange of edges between two elements of type Aand an element of type Bin the proof of Claim .$\\\\mathbf {T_3}$ is of type C. Let $T_3 = c_0c_1c_2c_3c_4c_5$ be an element of type C, where $c_3c_2\\\\in F_r$ .\",\n \"This implies that $c_4c_3\\\\in F_g$ .\",\n \"Since $T_3$ contains a hanging edge on ${\\\\rm {cv}}_2(T_2)=b_2$ , we have $b_2\\\\in \\\\lbrace c_1,c_4\\\\rbrace $ .\",\n \"If $b_2 = c_4$ , then $c_4c_3$ and $b_2b_1$ are two green out edges of $b_2$ , a contradiction.\",\n \"Thus, we may assume $b_2 = c_1$ .\",\n \"Put $T_1^{\\\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\\\prime } = b_1c_1c_2c_3c_4c_5$ (see Figure REF ) and let $ = \\\\big (\\\\lbrace T_1,T_2,T_3\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime },T_3^{\\\\prime }\\\\rbrace $ .\",\n \"Again, by Lemma REF , $T_1^{\\\\prime }$ is an element of type C. We prove that $T_2^{\\\\prime }$ and $T_3^{\\\\prime }$ are, respectively, of type D and C. Since $G$ is simple, $c_0\\\\notin \\\\lbrace b_1,b_2,b_3,b_4\\\\rbrace $ and $b_1\\\\notin \\\\lbrace c_1,c_2\\\\rbrace $ .\",\n \"By Lemma REF , we have $c_0 \\\\ne b_0$ and $b_1 \\\\ne c_5$ .\",\n \"Therefore, $T_2^{\\\\prime }$ is an element of type D. Analogously to the case above, If $b_1 \\\\in \\\\lbrace c_3,c_4\\\\rbrace $ , then $d(b_1)\\\\ge 7>5$ , a contradiction.\",\n \"Therefore, $T_3^{\\\\prime }$ is an element of type C. Once more, analogously to the cases above, $$ is a complete decomposition such that $\\\\tau () = \\\\tau ( -2<\\\\tau ($ , a contradiction to the minimality of $.$ Figure: Exchange of edges between two elements of type Aand an element of type Cin the proof of Claim .Claim 11.3 There is no free element of type A.\",\n \"Suppose, for a contradiction, that $contains a free element, say $ T1$, of type~A.By Definition~\\\\ref {def:complete-commutative}(\\\\ref {def:complete-commutative-hanging-edge}),there are two hanging edges $ e2$ and $ e2'$ at $ cv1(T1)$.We may assume $ e2 E(T1)$.Let $ T2$ be the element of $ that contains $e_2$ .\",\n \"By Claim REF , $T_2$ is not of type B or C, and since $M$ is a matching, $T_2$ is not of type D. Thus, $T_2$ is of type A.\",\n \"By Definition REF (REF ), there is a hanging edge $e_3$ on ${\\\\rm {cv}}_2(T_2)$ .\",\n \"Note that $e_3\\\\notin E(T_2)$ .\",\n \"Let $T_3$ be the element of $ that contains $ e3$.By Claim~\\\\ref {claim:AAB,AAC},$ T3$ is of type~D,which implies that there are two edges of $ M$incident to $ cv2(T2)$, a contradiction.$ Now, consider the auxiliary directed graph $D_ in which $ V(D = and $(T_1,T_2)$ is an arc of $D_ if and only if $ tr(T2) = cvi(T1)$ for some $ i{1,2}$.It is clear that the elements of type~A in $ are partitioned into A-chains if and only if $D_ consists of vertex-disjoint directed cycles and isolated vertices.\",\n \"Since every vertex of $ G$ is a connection vertex of at most one element of $ , by Claim REF , every vertex of $D_ has in degree precisely $ 1$.$ Note also that given two elements $T_1$ and $T_2$ we have ${\\\\rm {tr}}(T_1) \\\\ne {\\\\rm {tr}}(T_2)$ , otherwise there would be a vertex with two green in edges.\",\n \"This implies that every vertex of $D_ has out degree at most $ 2$.Now, if $ T1$ and $ T2$ are two elements of type~A in $ such that ${\\\\rm {cv}}_1(T_1) = {\\\\rm {tr}}(T_2) = u_1$ , by Lemma REF , we have ${\\\\rm {aux}}(T_2) = {\\\\rm {cv}}_2(T_1) = u_2$ , which means that $E(T_1)\\\\cup E(T_2)$ contains the four edges in $E(G)$ incident to $u_1$ and five edges incident to $u_2$ , and hence, no other element of $ contains $ u2$,and no other element of $ has $u_1$ as its tricky vertex.\",\n \"This implies that every vertex of $D_ has out degree at most $ 1$,and hence $ D consists of vertex-disjoint directed cycles and isolated vertices as desired.\"\n ],\n [\n \"Admissible decompositions\",\n \"In this section, we present a new decomposition invariant, which we call admissibility, and conclude our proof.\",\n \"For that, we introduce an important object, the exceptional pair.\",\n \"Let $G$ be a $\\\\lbrace g,r\\\\rbrace $ -graph, and let $ be a decomposition of $ G$ into trails of length $ 5$.We say that a pair $ (T1,T2)$ of elements of $ is an exceptional pair if $T_1$ and $T_2$ are elements of type A and C, respectively, and can be written as $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ such that $a_2a_3\\\\in M_{g,r}$ , $a_2=a_5 = b_3$ , and $a_2a_1,a_3a_4,b_2b_1,b_4b_3\\\\in F_g$ , $a_4a_5,b_3b_2,b_4b_5\\\\in F_r$ , $a_1a_0,b_1b_0\\\\in M_{g,r}\\\\cup F_g\\\\cup F_r$ (see Figure REF ).\",\n \"Note that since $G$ is a simple graph, we have $b_4\\\\ne a_3$ .\",\n \"Also, if $2g + 2r \\\\ne 0$ , then we have $b_1 \\\\ne a_3$ .\",\n \"This yields the following remark.\",\n \"Remark 1 If $G$ is a $\\\\lbrace g,r\\\\rbrace $ -graph for which $2g+2r\\\\ne 0$ and $(T_1,T_2)$ is an exceptional pair, then $T_2$ does not contain a hanging edge at ${\\\\rm {cv}}_1(T_1)$ .\",\n \"Figure: An exceptional pair.An open chain is a sequence $T_0, T_1,\\\\ldots , T_{s-1}$ of $s\\\\ge 2$ elements of $with the following properties.\",\n \"(i) $ T0$ is a free element of type~A;(ii) $ Tj$ is an element of type~A and $ tr(Tj) = cvi(Tj-1)$,for every $ j {0,...,s-2}$ and some $ i{1,2}$; and(iii) $ Ts-1$ is an element of type~C for which $ (Ts-2,Ts-1)$ is an exceptional pair.We remark that open chains are not A-chains.The next definition describes the invariant studied in this section.$ Definition 3 We say that a decomposition $ of a $ {g,r}$-graph $ G$into trails of length $ 5$ is \\\\emph {admissible} if the following hold.\\\\begin{enumerate}[(a)]\\\\item Every element in is either a path or an element of type~A;\\\\item For every element T\\\\in of type~A, we have {\\\\rm {hang}}\\\\big ({\\\\rm {cv}}_1(T)\\\\big )\\\\ge 2,and there is at most one element T\\\\in of type~A for which{\\\\rm {hang}}\\\\big ({\\\\rm {cv}}_2(T)\\\\big ) = 0,and, in this case, there is an open chain {S = T_0,\\\\ldots , T_{s-2},T_{s-1}} in ,for which T_{s-2} = T;\\\\item The elements of type~A in are partitioned into A-chains and at most one open chain.\\\\end{enumerate}$ It is not hard to check that the decomposition given by Lemma REF is an admissible decomposition.\",\n \"Therefore, every $\\\\lbrace g,r\\\\rbrace $ -graph for which $2g+2r\\\\ne 0$ admits an admissible decomposition.\",\n \"By performing a few more exchanges of edges between the elements of a same A-chain of an admissible decomposition, we can show that an admissible decomposition that minimizes its number of elements of type A is in fact a $P_5$ -decomposition.\",\n \"Theorem 12 Every $\\\\lbrace g,r\\\\rbrace $ -graph for which $2g+2r\\\\ne 0$ admits a $P_5$ -decomposition.\",\n \"Let $g$ and $r$ be as in the statement, let $G$ be a $\\\\lbrace g,r\\\\rbrace $ -graph, and put ${M=M_{g,r}}$ .\",\n \"By Lemma REF , $G$ admits an admissible decomposition.\",\n \"Let $ be an admissible decomposition of $ G$ that minimizes $ ($.In what follows, we prove that $ ( = 0$.Suppose, for a contradiction, that $ (>0$.We divide A-chains into three types, according to the connections between its elements.Given $ i{1,2}$,we say that an A-chain $ S=T0, T1,..., Ts-1$ is of type~$ i$ if $ tr(Tj)=cvi(Tj-1)$for every $ j {0,...,s-1}$;and we say that $ S$ is a \\\\emph {mixed} A-chain if $ S$ is not of type~$ 1$ or $ 2$.$ Similarly to the proof of Lemma REF , in each step, we exchange edges between some elements of $and obtain an admissible decomposition $$ into trails of length~5such that $ () <($, which is a contradiction to the minimality of $ .\",\n \"To check that $$ is an admissible decomposition, we observe the three following items: (i) The only connection vertex that has fewer hanging edges in $$ than in $is the secondary connection vertex of an element $ T1$ of type~A,and in this case there is an element $ T2$ of type~Csuch that $ (T1,T2)$ is an exceptional pair,and hence Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$;(ii) every element of $$ that is not an element of $ , i.e., the elements involved in the exchange of edges, is a path or an element of type A, and hence REF () holds for $$ ; (iii) either an open chain is shortened by at least one element, an A-chain is converted into an open chain, or all the elements of an A-chain are replaced by paths of length 5, and hence REF () holds for $$ .\",\n \"Claim 12.1 Every A-chain in $ is mixed.\",\n \"$ Suppose, for a contradiction, that there is an A-chain $S = T_0,T_1,\\\\ldots , T_{s-1}$ of type 1 or 2.\",\n \"Let $T_j=a_{0,j}a_{1,j}a_{2,j}a_{3,j}a_{4,j}a_{5,j}$ , where $a_{5,j} = a_{2,j}$ , $a_{2,j}a_{3,j}\\\\in M$ , ${a_{2,j}a_{1,j}, a_{3,j}a_{4,j}\\\\in F_g}$ , $a_{4,j}a_{2,j}\\\\in F_r$ and $a_{1,j}a_{0,j}\\\\in M\\\\cup F_g \\\\cup F_r$ .\",\n \"For $i\\\\in \\\\lbrace 1,2,3,4,5\\\\rbrace $ , the edge $a_{i-1,j}a_{i,j}$ is called the $i$ th edge of $T_j$.\",\n \"In what follows, we divide the proof according to the type of S. $\\\\mathbf {S}$ is of type 1.\",\n \"In this case, we have ${a_{3,j} = {\\\\rm {cv}}_1(T_{j}) = {\\\\rm {tr}}(T_{j+1}) = a_{4,j+1}}$ for each ${j\\\\in \\\\lbrace 0,\\\\ldots ,s-1}\\\\rbrace $ , and hence, by Lemma REF , we have $a_{2,j} = {\\\\rm {cv}}_2(T_{j}) = {\\\\rm {aux}}(T_{j+1})=a_{1,j+1}$ .\",\n \"Now, for each $j=0,\\\\ldots ,s-1$ , let $T^{\\\\prime }_j = a_{2,j+1}a_{3,j}a_{4,j}a_{1,j}a_{2,j}a_{0,j+1}$ (see Figure REF ).\",\n \"Note that $T^{\\\\prime }_j = T_j- a_{1,j}a_{0,j} + a_{1,j+1}a_{0,j+1}- a_{2,j}a_{3,j} + a_{2,j-1}a_{3,j-1}- a_{4,j}a_{2,j} + a_{4,j+1}a_{2,j+1}$ .\",\n \"More specifically, $a_{2,j+1}a_{3,j} = a_{4,j+1}a_{5,j+1}$ is the 5th edge of $T_{j+1}$ ; $a_{3,j}a_{4,j}$ is the 4th edge of $T_{j}$ ; $a_{4,j}a_{1,j} = a_{2,j-1}a_{3,j-1}$ is the 3rd edge of $T_{j-1}$ ; $a_{1,j}a_{2,j}$ is the 2nd edge of $T_{j}$ ; $a_{2,j}a_{0,j+1} = a_{1,j+1}a_{0,j+1}$ is the 1st edge of $T_{j+1}$ .\",\n \"Clearly, $T^{\\\\prime }_j$ is a trail of length 5.\",\n \"Moreover, since, for each $i\\\\in \\\\lbrace 1,2,3,4,5\\\\rbrace $ , the element $T^{\\\\prime }_j$ contains the $i$ th edge of an element of $S$ , and, if $j\\\\ne j^{\\\\prime }$ , the elements $T^{\\\\prime }_j$ and $T^{\\\\prime }_{j^{\\\\prime }}$ contain the $i$ th edge of different elements of $S$ , the set $=\\\\big (\\\\lbrace T_j\\\\colon j=0,\\\\ldots ,s-1\\\\rbrace \\\\big )\\\\cup \\\\lbrace T^{\\\\prime }_j\\\\colon j=0,\\\\ldots ,s-1\\\\rbrace $ is a decomposition of $G$ into trails of length 5.\",\n \"We may regard $$ as the decomposition obtained by reversing the direction of two components of $F_g$ , namely, the green edges in $S$ , and applying the same strategy used in Proposition REF .\",\n \"Figure: Exchange of edges between the elements of an A-chain of type 1 withfive elements in the proof of Claim .In order to prove that $T^{\\\\prime }_j$ is a path, we show that $a_{2,j+1},a_{0,j+1}\\\\notin \\\\lbrace a_{3,j},a_{4,j},a_{1,j},a_{2,j}\\\\rbrace $ .\",\n \"Note that, since for each $j\\\\in \\\\lbrace 0,\\\\ldots ,s-1\\\\rbrace $ , $T_j$ is a path, we have $a_{i,j} \\\\ne a_{i^{\\\\prime },j}$ for every $i\\\\ne i^{\\\\prime }$ .\",\n \"Since $G$ is a simple graph, we have $a_{2,j+1}\\\\notin \\\\lbrace a_{3,j},a_{4,j},a_{2,j}\\\\rbrace $ and $a_{0,j+1}\\\\notin \\\\lbrace a_{3,j},a_{4,j},a_{1,j},a_{2,j}\\\\rbrace $ ; and if $a_{2,j+1} = a_{1,j}$ , then $a_{4,j+1}a_{2,j+1}$ and $a_{4,j-1,}a_{2,j-1}$ are two distinct red in edges of $a_{1,j}$ , a contradiction.\",\n \"We claim that $$ is an admissible decomposition.\",\n \"Indeed, the only vertices of the elements of $S$ that can be connection vertices of elements in $$ are the vertices $a_{0,j}$ , for $j=0,\\\\ldots ,s-1$ .\",\n \"But a hanging edge at a vertex $a_{0,j}$ is in $T_{j^{\\\\prime }}\\\\in if and only if $ a0,j = a3,j'$ for some $ j'j$,and, in this case $ a3,j'$ is not a connection vertex in $$ becauseall edges incident to it are in elements of $ {T'jj=0,...,s-1}$.Therefore, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$.Moreover, since $ T'j$ is a path, for $ j=0,...,s-1$, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:complete-commutative-types}) holds for $$.Finally, $ and $$ have the same number of open chains, and hence Definition REF () holds for $$ .\",\n \"Therefore, $$ is an admissible decomposition of $G$ such that ${\\\\tau () = \\\\tau ( - s}$ , a contradiction to the minimality of $.$ $\\\\mathbf {S}$ is of type 2.\",\n \"In this case, for each $j=0,\\\\ldots ,s-1$ , we have $a_{2,j} = {\\\\rm {cv}}_2(T_{j}) = {\\\\rm {tr}}(T_{j+1}) = a_{4,j+1}$ .\",\n \"Now, for each $j=0,\\\\ldots ,s-1$ , let $T^{\\\\prime }_j = a_{0,j}a_{1,j}a_{2,j}a_{3,j}a_{4,j}a_{4,j-1}$ (see Figure REF ).\",\n \"Clearly, $T^{\\\\prime }_j$ is a trail of length 5.\",\n \"Note that $T^{\\\\prime }_j = T_j - a_{4,j}a_{2,j} + a_{4,j-1}a_{2,j-1}$ , i.e., $T^{\\\\prime }_j$ is the element obtained from $T_j$ by exchanging its 5th edge by the 5th edge of $T_{j-1}$ .\",\n \"Thus, the set $=\\\\big (\\\\lbrace T_j\\\\colon j=0,\\\\ldots ,s-1\\\\rbrace \\\\big )\\\\cup \\\\lbrace T^{\\\\prime }_j\\\\colon j=0,\\\\ldots ,s-1\\\\rbrace $ is a decomposition of $G$ into trails of length 5.\",\n \"We may regard $$ as the decomposition obtained by reversing the direction of one component of $F_r$ and applying the same strategy used in Proposition REF .\",\n \"In order to prove that $T^{\\\\prime }_j$ is a path, we show that $a_{4,j-1}\\\\notin \\\\lbrace a_{0,j},a_{1,j},a_{2,j},a_{3,j},a_{4,j}\\\\rbrace $ .\",\n \"Note that, since for each $j\\\\in \\\\lbrace 0,\\\\ldots ,s-1\\\\rbrace $ , $T_j$ is a path, we have $a_{i,j} \\\\ne a_{i^{\\\\prime },j}$ for every $i\\\\ne i^{\\\\prime }$ .\",\n \"Since $G$ is a simple graph, we have $a_{4,j-1}\\\\notin \\\\lbrace a_{2,j},a_{3,j},a_{4,j}\\\\rbrace $ ; also, by Lemma REF , we have $a_{4,j-1}\\\\ne a_{0,j}$ ; and if $a_{4,j-1} = a_{1,j}$ , then $a_{2,j}a_{1,j}$ and $a_{3,j-1}a_{4,j-1}$ are two distinct green in edges of $a_{4,j-1}$ , a contradiction.\",\n \"Figure: Exchange of edges between the elements of an A-chain of type 2 withfive elements in the proof of Claim .We claim that $$ is an admissible decomposition.\",\n \"Indeed, the only vertices that have hanging edges in $ and may not have hanging edges in $$are the vertices $ a3,j$ and $ a4,j = a2,j-1$, for $ j=0,...,s-1$,but these vertices are connection vertices of the elements in $ S$,and hence can^{\\\\prime }t be connection vertices of elements in $$.Therefore, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$.Moreover, since $ T'j$ is a path, for $ j=0,...,s-1$, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:complete-commutative-types}) holds for $$.Finally, $ and $$ have the same number of open chains, and hence Definition REF () holds for $$ .\",\n \"Therefore, $$ is an admissible decomposition of $G$ such that $\\\\tau () = \\\\tau ( - s$ , a contradiction to the minimality of $.$ Claim 12.2 Every A-chain contains at least four elements First, note that if an A-chain consists of two elements, then $G$ contains a parallel edge, which is a contradiction.\",\n \"Thus, let $S$ be an A-chain in $ with precisely three elements, say $ T1$, $ T2$, and $ T3$.By Claim~\\\\ref {claim:mixed-A-chain}, we may assume $ tr(T1)=cv2(T3)$,$ tr(T2)=cv1(T1)$ and $ tr(T3)=cvi(T2)$, for $ i {1,2}$.\",\n \"In what follows, we divide the proof depending on whether $ i=1$ or $ i=2$.$ Let $T_1=a_0a_1a_2a_3a_4a_5$ , $T_2=b_0b_1b_2b_3b_4b_5$ and $T_3 = c_0c_1c_2c_3c_4c_5$ be the elements of $S$ where $a_4={\\\\rm {tr}}(T_1)={\\\\rm {cv}}_2(T_3)=c_2$ , $b_4={\\\\rm {tr}}(T_2)={\\\\rm {cv}}_1(T_1)=a_3$ and ${c_4={\\\\rm {tr}}(T_3)={\\\\rm {cv}}_i(T_2)}$ .\",\n \"Case $\\\\mathbf {i=1}$ .\",\n \"In this case, $c_4={\\\\rm {tr}}(T_3)={\\\\rm {cv}}_1(T_2)=b_3$ .\",\n \"Since $b_3=c_4$ , we have $b_2=c_1$ and $c_0=a_1$ .\",\n \"Put $T_1^{\\\\prime } = a_0a_1a_2a_3a_4c_1$ , $T_2^{\\\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_3^{\\\\prime } = c_0c_1c_4c_3c_2a_2$ (see Figure REF ) and ${ = \\\\big (\\\\lbrace T_1,T_2,T_3\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime },T_3^{\\\\prime }\\\\rbrace }$ .\",\n \"We claim that $T_1^{\\\\prime }$ , $T_2^{\\\\prime }$ and $T_3^{\\\\prime }$ are paths.\",\n \"By Lemma REF , $T_2^{\\\\prime }$ is a path.\",\n \"Since $G$ is simple, $c_1\\\\notin \\\\lbrace a_2,a_3,a_4\\\\rbrace $ and $a_2\\\\notin \\\\lbrace c_1,c_2,c_3,c_4\\\\rbrace $ .\",\n \"By Lemma REF , we have $c_1 \\\\ne a_0$ and $a_2\\\\ne c_0$ .\",\n \"Therefore, $T_3^{\\\\prime }$ is a path.\",\n \"Finally, if $c_1 = a_1$ , then $c_2c_1$ and $a_2a_1$ are two green out edges at $a_1$ , a contradiction.\",\n \"Therefore, $T_1^{\\\\prime }$ is a path, and hence definition REF () holds.\",\n \"Also, ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)for every v\\\\in V(G)\\\\setminus \\\\lbrace a_3,b_3,c_3\\\\rbrace .Thus, definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge}) holds for .Since is admissible and the new elements are paths, the elements of type~Aare still partitioned into A-chains and at most one open chain,and hence \\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-free})holds for~.Therefore, is an admissible decomposition of Gsuch that \\\\tau () = \\\\tau ( - 3, a contradiction to the minimality of~.\",\n \"}\\\\begin{figure}[H]\\\\centering \\\\begin{subfigure}{.45}\\\\centering \\\\scalebox {.8}{\\\\begin{tikzpicture}[scale=1.5]\\\\end{tikzpicture}[fatpath,backcolor1] (-0.5,0.8) -- (-1.5,0.8) -- (-0.5,0.8) -- (0,0) -- (0,0) -- (1,0) -- (0.5,0.8) -- (0,0);[fatpath,backcolor2] (0.5,-0.8) -- (-0.5,-0.8)-- (0.5,-0.8) -- (1,-1.6) -- (2,-1.6) -- (1.5,-0.8) -- (1,-1.6);}\\\\end{subfigure}\\\\node (0) [black vertex] at (-1.5,0.8) {};\\\\node (1) [blue vertex] at (-0.5,0.8) {};\\\\node (2) [black vertex] at (0,0) {};\\\\node (3) [black vertex] at (1,0) {};\\\\node (4) [red vertex] at (0.5,0.8) {};\\\\node () [] at ((1,0)+(60:1)) {};\\\\end{figure}$4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [red vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\\\prime })$ ) ; 0”) [blue vertex] at ($(1^{\\\\prime \\\\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\\\prime })+(180:.2)$ ) $b_0$ ; ) [] at ($(1^{\\\\prime })+(-90:.3)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\\\prime })+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4^{\\\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\\\prime \\\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\\\prime \\\\prime })+(225:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\\\prime \\\\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\\\prime \\\\prime })+(0:.25)$ ) $c_3$ ; ) [] at ($(4^{\\\\prime \\\\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (1,0) – (0.5,0.8); [fatpath,backcolor1] (1,-1.6) – (0.5,-0.8) –(1,-1.6) – (0.5,-0.8); [fatpath,backcolor2] (0.5,-0.8) – (-0.5,-0.8) – (0.5,-0.8) – (-0.5,-0.8); [fatpath,backcolor2] (2,-1.6) – (1,-1.6) – (2,-1.6) – (1.5,-0.8) – (0.5,-0.8) – (1.5,-0.8); [fatpath,backcolor2] (0,0) – (0.5,0.8) – (0,0) – (0.5,0.8); 0) [black vertex] at (-1.5,0.8) ; 1) [blue vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [red vertex] at (0.5,0.8) ; ) [] at ($(1,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [red vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\\\prime })$ ) ; 0”) [blue vertex] at ($(1^{\\\\prime \\\\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\\\prime })+(180:.2)$ ) $b_0$ ; ) [] at ($(1^{\\\\prime })+(-90:.3)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\\\prime })+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4^{\\\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\\\prime \\\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\\\prime \\\\prime })+(225:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\\\prime \\\\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\\\prime \\\\prime })+(0:.25)$ ) $c_3$ ; ) [] at ($(4^{\\\\prime \\\\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange performed in the proof of Claim REF in the case ${\\\\rm {tr}}(T_3) = {\\\\rm {cv}}_1(T_2)$ .\",\n \"The red (resp.\",\n \"blue) circles illustrate the same vertex, i.e., $a_4=c_2$ (resp.\",\n \"$a_1=c_0$ ).\",\n \"Case $\\\\mathbf {i=2}$ .\",\n \"In this case, $c_4={\\\\rm {tr}}(T_3)={\\\\rm {cv}}_2(T_2)=b_2$ .\",\n \"Put $T_1^{\\\\prime } = a_0a_1a_2a_4a_3b_5$ , ${T_2^{\\\\prime } = b_0b_1b_4b_3b_2c_2}$ , $T_3^{\\\\prime } = c_0c_1c_2c_3c_4b_1$ (see Figure REF ) and let $ = \\\\big (\\\\lbrace T_1,T_2,T_3\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime },T_3^{\\\\prime }\\\\rbrace $ .\",\n \"By Lemma REF , $T_1^{\\\\prime }$ is a path.\",\n \"Since $G$ is simple, we have $c_2\\\\notin \\\\lbrace b_1,b_2,b_3,b_4\\\\rbrace $ and $b_1\\\\notin \\\\lbrace c_2,c_3,c_4\\\\rbrace $ .\",\n \"By Lemma REF , $b_2 \\\\ne a_0, ~c_2 \\\\ne b_0,~b_1 \\\\ne c_0$ .\",\n \"Therefore, $T_2^{\\\\prime }$ is a path.\",\n \"If $b_1 = c_1$ , then $b_2b_1$ and $c_2c_1$ are two green in edges at $c_1$ , a contradiction.\",\n \"Therefore, $T_3^{\\\\prime }$ is a path.\",\n \"Analogously to the case above, $$ is an admissible decomposition of $G$ such that $\\\\tau () = \\\\tau ( - 3$ , a contradiction to the minimality of $.$ Figure: Exchange performed in the proof of Claim in the case tr (T 3 )= cv 2 (T 2 ){\\\\rm {tr}}(T_3) = {\\\\rm {cv}}_2(T_2).\",\n \"The red circles illustrate the same vertex.Claim 12.3 $ contains an open chain.$ Suppose, for a contradiction, that there is no open chain in $.Since $ (>0$, {~contains} an A-chain $ S = T0,T1,..., Ts-1$.By Claim~\\\\ref {claim:mixed-A-chain}, $ S$ is a mixed A-chain.Then we can find three consecutive elements in $ S$, say $ Tj, Tj+1,Tj+2$,such that $ cv2(Tj)= tr(Tj+1)$ and $ cv1(Tj+1) = tr(Tj+2)$.By the cyclic structure of $ S$, we may assume, without loss of generality, that $ j=0$.By Claim~\\\\ref {claim:no-short-A-chain}, we have $ s4$,and hence there is an element $ T3 such that ${\\\\rm {tr}}(T_3) = {\\\\rm {cv}}_i(T_2)$ , for some $i\\\\in \\\\lbrace 1,2\\\\rbrace $ .\",\n \"In what follows, the proof is divided according to $i$ .\",\n \"Let $T_0 = a_0a_1a_2a_3a_4a_5$ , $T_1 = b_0b_1b_2b_3b_4b_5$ , $T_2 = c_0c_1c_2c_3c_4c_5$ , and $T_3 = d_0d_1d_2d_3d_4d_5$ , where $a_5 = a_2$ , $b_5 = b_2$ , $c_5 = c_2$ , $d_5 = d_2$ , and $a_2a_3, b_2b_3, c_2c_3, d_2d_3 \\\\in M$ .\",\n \"By the choice of $T_0$ , $T_1$ , and $T_2$ , we have ${b_4 = {\\\\rm {tr}}(T_1) = {\\\\rm {cv}}_2(T_0) = a_2}$ , ${c_4 = {\\\\rm {tr}}(T_2) = {\\\\rm {cv}}_1(T_1) = b_3}$ .\",\n \"The exchanges of edges performed here are analogous to the exchanges performed on the proof of Claim REF of Lemma REF for elements of type A.\",\n \"Case $\\\\mathbf {{\\\\mathbf {\\\\rm }{tr}}(T_3) = {\\\\mathbf {\\\\rm }{cv}}_1(T_2)}$.\",\n \"In this case, we have $d_4 = c_3$ and, by Lemma REF , $c_2 = d_1$ .\",\n \"Put $T_1^{\\\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_2^{\\\\prime } = c_0c_1c_4c_3c_2d_0$ , $T_3^{\\\\prime } = c_1d_1d_2d_3d_4d_2$ (see Figure REF ), and let $ = \\\\big (\\\\lbrace T_1,T_2,T_3\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime },T_3^{\\\\prime }\\\\rbrace $ .\",\n \"By Lemma REF , $T^{\\\\prime }_1$ is an element of type C. In what follows, we prove that $T_2^{\\\\prime }$ is a path and $T_3^{\\\\prime }$ is an element of type A.\",\n \"Since $G$ is simple, we have $d_0\\\\notin \\\\lbrace c_1,c_2,c_3,c_4\\\\rbrace $ , and $c_1\\\\notin \\\\lbrace d_1,d_2,d_4\\\\rbrace $ .\",\n \"By Lemma REF , $d_0 \\\\ne c_0$ , and hence, $T_2^{\\\\prime }$ is a path.\",\n \"If $c_1 = d_3$ , then $b_2b_3$ and $d_2d_3$ are two edges of $M$ incident to $c_1$ , a contradiction.\",\n \"Therefore, $T_3^{\\\\prime }$ is an element of type A, and hence definition REF () holds.\",\n \"Also, ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)for every v\\\\in V(G)\\\\setminus \\\\lbrace a_2=b_4,b_3,c_3\\\\rbrace .Note also that b_4b_2 is a hanging edge at a_2 = {\\\\rm {cv}}_2(T_0) in ,but not in .However, (T_0,T_1^{\\\\prime }) is an exceptional pair.Also, b_3 and c_3 are not connection vertices of .Since c_3 is not a connection vertex in ,the element T^{\\\\prime }_3 is free.Therefore, S^{\\\\prime }=T^{\\\\prime }_3,\\\\ldots , T_{s-1},T_1,T^{\\\\prime }_2 is an open chain,and hence Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge})holds for .Finally, note that an element T of type~A in \\\\lbrace T_1,T_2,T_3\\\\rbrace is either in an A-chain of different from S,which implies that T is in an A-chain of ,or is in S, which implies that T is in S^{\\\\prime }.Thus, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\\\tau () = \\\\tau ( - 2, a contradiction to the minimality of .\",\n \"}\\\\begin{figure}[h]\\\\centering \\\\begin{subfigure}{.45}\\\\centering \\\\scalebox {.8}{\\\\begin{tikzpicture}[scale=1.5]\\\\end{tikzpicture}[fatpath,backcolor1] (-0.5,0.8) -- (-1.5,0.8) -- (-0.5,0.8) -- (0,0) -- (0,0) -- (1,0) -- (0.5,0.8) -- (0,0);[fatpath,backcolor2] (0.5,-0.8) -- (-0.5,-0.8)-- (0.5,-0.8) -- (1,-1.6) -- (2,-1.6) -- (1.5,-0.8) -- (1,-1.6);}\\\\end{subfigure}\\\\node (0) [black vertex] at (-1.5,0.8) {};\\\\node (1) [black vertex] at (-0.5,0.8) {};\\\\node (2) [black vertex] at (0,0) {};\\\\node (3) [black vertex] at (1,0) {};\\\\node (4) [black vertex] at (0.5,0.8) {};\\\\node () [] at ((1,0)+(60:1)) {};\\\\end{figure}$4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [black vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\\\prime })$ ) ; 0”) [black vertex] at ($(1^{\\\\prime \\\\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $b_0$ ; ) [] at ($(1)+(90:.2)$ ) $b_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4)+(90:.2)$ ) $b_4=a_2$ ; ) [] at ($(0^{\\\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\\\prime })+(-90:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\\\prime })+(0:.55)$ ) $c_3=d_4$ ; ) [] at ($(4^{\\\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\\\prime \\\\prime })+(180:.2)$ ) $d_0$ ; ) [] at ($(1^{\\\\prime \\\\prime })+(225:.3)-(0.3,0)$ ) $d_1=c_2$ ; ) [] at ($(2^{\\\\prime \\\\prime })+(180:.2)$ ) $d_2$ ; ) [] at ($(3^{\\\\prime \\\\prime })+(0:.25)$ ) $d_3$ ; ) [] at ($(4^{\\\\prime \\\\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (0.5,0.8) – (1,0) – (0.5,-0.8); [fatpath,backcolor2] (0,0)– (0.5,-0.8) –(1,-1.6) – (2,-1.6) – (1.5,-0.8) – (1,-1.6); 0) [black vertex] at (-1.5,0.8) ; 1) [black vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [black vertex] at (0.5,0.8) ; ) [] at ($(1,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [black vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\\\prime })$ ) ; 0”) [black vertex] at ($(1^{\\\\prime \\\\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $b_0$ ; ) [] at ($(1)+(90:.2)$ ) $b_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4)+(90:.2)$ ) $b_4=a_2$ ; ) [] at ($(0^{\\\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\\\prime })+(-90:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\\\prime })+(0:.55)$ ) $c_3=d_4$ ; ) [] at ($(4^{\\\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\\\prime \\\\prime })+(180:.2)$ ) $d_0$ ; ) [] at ($(1^{\\\\prime \\\\prime })+(225:.3)-(0.3,0)$ ) $d_1=c_2$ ; ) [] at ($(2^{\\\\prime \\\\prime })+(180:.2)$ ) $d_2$ ; ) [] at ($(3^{\\\\prime \\\\prime })+(0:.25)$ ) $d_3$ ; ) [] at ($(4^{\\\\prime \\\\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange performed in the proof of Claim REF in the case ${\\\\rm {tr}}(T_3) = {\\\\rm {cv}}_1(T_2)$ .\",\n \"Case $\\\\mathbf {{\\\\mathbf {\\\\rm }{tr}}(T_3) = {\\\\mathbf {\\\\rm }{cv}}_2(T_2)}$.\",\n \"Put $T_1^{\\\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_2^{\\\\prime } = c_0c_1c_4c_3c_2d_2$ , $T_3^{\\\\prime } = d_0d_1d_2d_3d_4c_1$ (see Figure REF ), and let $ = \\\\big (\\\\lbrace T_1,T_2,T_3\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_1^{\\\\prime },T_2^{\\\\prime },T_3^{\\\\prime }\\\\rbrace $ .\",\n \"By Lemma REF , $T^{\\\\prime }_1$ is an element of type C. In what follows, we prove that $T_2^{\\\\prime }$ and $T_3^{\\\\prime }$ are paths.\",\n \"Since $G$ is simple, we have $d_2\\\\notin \\\\lbrace c_1,c_2,c_3,c_4\\\\rbrace $ , and $c_1\\\\notin \\\\lbrace d_2,d_3,d_4\\\\rbrace $ .\",\n \"By Lemma REF , $d_2 \\\\ne c_0$ , and $c_1 \\\\ne d_0$ .\",\n \"Therefore, $T_2^{\\\\prime }$ is a path.\",\n \"If $c_1 = d_1$ , then $d_2d_1$ and $c_2c_1$ are two green in edges of $c_1$ .\",\n \"Therefore, $T_3^{\\\\prime }$ is a path, and hence definition REF () holds.\",\n \"Also, ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)for every v\\\\in V(G)\\\\setminus \\\\lbrace a_2=b_4,b_3,c_3,d_3\\\\rbrace .Note also that b_4b_2 is a hanging edge at a_2 in but not in .However, (T_0,T_1^{\\\\prime }) is an exceptional pair.Also, b_3, c_3 and d_3 are not connection vertices of .Thus, since d_2 and d_3 are not connection vertices in ,the element T_4 (or T_1, if s=4) is free.Therefore, S^{\\\\prime }=T_4,\\\\ldots , T_{s-1},T_1,T^{\\\\prime }_2 is an open chain,and hence Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge})holds for .\",\n \"Finally, note that an element T of type~A in \\\\lbrace T_1,T_2,T_3\\\\rbrace is either in an A-chain of different from S,which implies that T is in an A-chain of ,or is in S, which implies that T is in S^{\\\\prime }.Thus, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\\\tau () = \\\\tau ( - 3, a contradiction to the minimality of .\",\n \"}$ Figure: Exchange performed in the proof of Claim in the case tr (T 3 )= cv 2 (T 2 ){\\\\rm {tr}}(T_3) = {\\\\rm {cv}}_2(T_2).Now, let $S = T_0, T_1,\\\\ldots , T_{s-1}$ be an open chain in $.Let $ Tj = a0,ja1,ja2,ja3,ja4,ja5,j$, for $ j{0,...,s-1}$, where $ a2,ja3,jM$ and $ a5,j=a2,j$ for $ j{0,...,s-2}$.$ Claim 12.4 $T_1$ is an element of type A and ${\\\\rm {tr}}(T_1) = {\\\\rm {cv}}_1(T_0)$ .\",\n \"Suppose, for a contradiction, that $T_1$ is not an element of type A or ${{\\\\rm {tr}}(T_1) = {\\\\rm {cv}}_2(T_0)}$ .\",\n \"We claim that $T_1$ does not contain a hanging edge at ${\\\\rm {cv}}_1(T_0)$ .\",\n \"Indeed, if $T_1$ is not an element of type A, then, by the definition of open chain, $T_1$ is an element of type C, and hence, by Remark REF , $T_1$ does not have a hanging edge at ${\\\\rm {cv}}_1(T_0)$ ; and if $T_1$ is an element of type A for which ${\\\\rm {tr}}(T_1) = {\\\\rm {cv}}_2(T_0)$ , then we have $a_{4,1} = {\\\\rm {tr}}(T_1) = {\\\\rm {cv}}_2(T_0) = a_{2,0}$ , and hence, if $a_{1,1} = a_{3,0}$ , then we have $a_{4,1} + r + g = a_{1,1} = a_{3,0} = a_{2,0} - r - g$ , which implies that $2g + 2r = 0$ , a contradiction.\",\n \"Therefore, $T_1$ does not contain a hanging edge at ${\\\\rm {cv}}_1(T_0)$ .\",\n \"By Definition REF (), there are two hanging edges at ${\\\\rm {cv}}_1(T_0)$ .\",\n \"Thus, there is an element $T = u_0u_1u_2u_3u_4u_5$ in $\\\\lbrace T_0,T_1\\\\rbrace $ that contains a hanging edge, say $u_1u_0$ , at ${\\\\rm {cv}}_1(T_0)$ .\",\n \"Note that all the edges incident to $a_{2,0}$ are in $E(T_0)\\\\cup E(T_1)$ .\",\n \"Let $T_0^{\\\\prime } = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}u_0$ and $T^{\\\\prime } = a_{2,0}u_1u_2u_3u_4u_5$ and put $ = \\\\big (\\\\lbrace T_0,T\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_0^{\\\\prime },T^{\\\\prime }\\\\rbrace $ .\",\n \"By Lemma REF , $T_0^{\\\\prime }$ is a path; and since all the edges incident to $a_{2,0}$ are in $E(T_0)\\\\cup E(T_1)$ , we have $a_{2,0}\\\\notin \\\\lbrace u_1,u_2,u_3,u_4,u_5\\\\rbrace $ , and hence if $T$ is a path (resp.\",\n \"an element of type A), then $T^{\\\\prime }$ is a path (resp.\",\n \"an element of type A).\",\n \"Thus Definition REF (REF ) holds for $$ .\",\n \"To check that $$ is an admissible decomposition first note that ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)for every v\\\\in V(G)\\\\setminus \\\\lbrace a_{4,0}\\\\rbrace .Thus, since T_0 is a free element, a_{4,0} is not a connection vertex in ,and hence a_{4,0} is not a connection vertex in .Note also that T_1 is either an element of type~C or a free element of type~A,and hence S^{\\\\prime } = T_1,\\\\ldots , T_{s-1} is an open chain.Thus, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge})holds for .Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\\\tau () = \\\\tau ( - 1, a contradiction to the minimality of .\",\n \"}$ By Claim REF , we have $s\\\\ge 3$ , and hence, there is an element $T_2$ in $S$ .\",\n \"Note that, by Lemma REF , since ${\\\\rm {tr}}(T_1) = {\\\\rm {cv}}_1(T_0)$ , we have ${\\\\rm {aux}}(T_1) = {\\\\rm {cv}}_2(T_0)$ .\",\n \"This implies that $a_{1,1}a_{0,1}\\\\in F_r$ because all the edges incident to $a_{1,1} = {\\\\rm {cv}}_2(T_0)$ are in $E(T_0)\\\\cup E(T_1)$ .\",\n \"Claim 12.5 $T_2$ is of type A.\",\n \"Suppose, for a contradiction, that $T_2$ is not of type A, then $T_2$ is an element of type C and $(T_1,T_2)$ is an exceptional pair.\",\n \"Thus, we can write $T_2 = a_{0,2}a_{1,2}a_{2,2}a_{3,2}a_{4,2}a_{5,2}$ such that $a_{2,1}=a_{5,1} = a_{3,2}$ , and $a_{2,2}a_{1,2},a_{4,2}a_{3,2}\\\\in F_g$ , $a_{3,2}a_{2,2},a_{4,2}a_{5,2}\\\\in F_r$ , $a_{1,2}a_{0,2}\\\\in M_{g,r}\\\\cup F_g\\\\cup F_r$ .\",\n \"We claim that $a_{0,1} = a_{1,2}$ .\",\n \"Indeed, since ${a_{1,1}a_{0,1}\\\\in F_r}$ , we have ${a_{0,1} = a_{1,1} + r = a_{2,1} + g + r}$ , but by the definition of type C, we have ${a_{1,2} = a_{2,2} + g = a_{3,2} + r + g}$ .\",\n \"Thus, since $a_{3,2} = a_{2,1}$ , we obtain $a_{0,1} = a_{1,2}$ .\",\n \"Now, put $T^{\\\\prime }_0 = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}a_{2,1}$ , $T^{\\\\prime }_1 = a_{1,1}a_{4,1}a_{3,1}a_{2,1}a_{2,2}a_{0,1}$ , and $T^{\\\\prime }_2 = a_{0,2}a_{1,2}a_{1,1}a_{3,2}a_{4,2}a_{5,2}$ (see Figure REF ), and put $ = \\\\big (\\\\lbrace T_0,T_1,T_2\\\\rbrace \\\\big )\\\\cup \\\\lbrace T^{\\\\prime }_0,T^{\\\\prime }_1,T^{\\\\prime }_2\\\\rbrace $ .\",\n \"By Lemma REF , $T_0^{\\\\prime }$ is a path; since $G$ is a simple graph, $a_{2,2}\\\\notin \\\\lbrace a_{1,1},a_{4,1},a_{3,1},a_{2,1},a_{0,1}\\\\rbrace $ , and hence $T^{\\\\prime }_1$ is a path; and since all edges incident to $a_{1,1}$ are in $E(T_0)\\\\cup E(T_1)$ , we have $a_{1,1} \\\\notin V(T_2)$ , which implies that $T^{\\\\prime }_2$ is a path.\",\n \"Thus Definition REF (REF ) holds for $$ .\",\n \"Figure: Exchange of edges between two elements of type Aand the elements of an exceptional pairin the proof of Claim .To check that $$ is an admissible decomposition first note that ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)for every v\\\\in V(G)\\\\setminus \\\\lbrace a_{4,0},a_{1,1},a_{3,1}\\\\rbrace .Thus, since T_0 is a free element, a_{4,0} is not a connection vertex in ,and hence a_{4,0} is not a connection vertex in ;and since the edges of M incident to a_{1,1} and a_{3,1} are in T^{\\\\prime }_1,the vertices a_{1,1} and a_{3,1} are not connection vertices in .Note also that no element of S is in , and hence there are no open chains in .Thus, Definitions~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge}) and~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-free})hold for~.Therefore, is an admissible decomposition of Gsuch that \\\\tau () = \\\\tau ( - 2, a contradiction to the minimality of~.\",\n \"}$ Now, by Claim REF , we have $s\\\\ge 4$ .\",\n \"In what follows, we divide the proof depending on whether ${\\\\rm {tr}}(T_2) = {\\\\rm {cv}}_1(T_1)$ or ${\\\\rm {tr}}(T_2) = {\\\\rm {cv}}_2(T_1)$ .\",\n \"Case $\\\\mathbf {{\\\\mathbf {\\\\rm }{tr}}(T_2) = {\\\\mathbf {\\\\rm }{cv}}_1(T_1)}$.\",\n \"By Lemma REF , we have $a_{1,2}={\\\\rm {aux}}(T_2) = {\\\\rm {cv}}_2(T_1)=a_{2,1}$ .\",\n \"Put $T_0^{\\\\prime } = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}a_{2,1}$ , $T_1^{\\\\prime } = a_{0,1}a_{1,1}a_{4,1}a_{3,1}a_{2,1}a_{0,2}$ , $T_2^{\\\\prime } = a_{1,1}a_{1,2}a_{2,2}a_{3,2}a_{4,2}a_{2,2}$ (see Figure REF ), let $ = \\\\big (\\\\lbrace T_0,T_1,T_2\\\\rbrace \\\\big )\\\\cup \\\\lbrace T_0^{\\\\prime },T_1^{\\\\prime },T_2^{\\\\prime }\\\\rbrace $ , and let $S^{\\\\prime } = T^{\\\\prime }_2, T_3, \\\\dots , T_{s-1}$ .\",\n \"By Lemma REF , $T^{\\\\prime }_0$ is an element of type C. In what follows, we prove that $T_1^{\\\\prime }$ is a path and $T_2^{\\\\prime }$ is an element of type A.\",\n \"Since $G$ is simple, we have $a_{0,2}\\\\notin \\\\lbrace a_{1,1},a_{2,1},a_{3,1},a_{4,1}\\\\rbrace $ , and $a_{1,1}\\\\notin \\\\lbrace a_{1,2},a_{2,2},a_{4,2}\\\\rbrace $ .\",\n \"By Lemma REF , $a_{0,2} \\\\ne a_{0,1}$ , and hence, $T_1^{\\\\prime }$ is a path.\",\n \"If $a_{1,1} = a_{3,2}$ , then $a_{2,0}a_{3,0}$ and $a_{2,2}a_{3,2}$ are two edges of $M$ incident to $a_{1,1}$ , a contradiction.\",\n \"Therefore, $T_2^{\\\\prime }$ is an element of type A.\",\n \"Figure: Exchange of edges between the three first elements in an open A-chain with at least four elements, in the case tr (T 3 )= cv 1 (T 2 ){\\\\rm {tr}}(T_3) = {\\\\rm {cv}}_1(T_2).To check that $$ is an admissible decomposition first note that ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)for every v\\\\in V(G)\\\\setminus \\\\lbrace a_{3,0}, a_{3,1}, a_{4,0}\\\\rbrace ,but since T_0 is free, a_{4,0} is not a connection vertex in ,and hence a_{3,0}, a_{3,1},a_{4,0} are not connection vertices in .Thus, since a_{3,1} is not a connection vertex in ,the element T^{\\\\prime }_2 is free.Therefore, S^{\\\\prime } is an open chain,and hence Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge})holds for .Since, T^{\\\\prime }_0 and T^{\\\\prime }_1 are paths and T^{\\\\prime }_2 is an element of type~A,Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:complete-commutative-types}) holds for .\",\n \"Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\\\tau () = \\\\tau ( - 2, a contradiction to the minimality of .\",\n \"}\\\\smallskip \\\\textbf {Case} $ tr(T2) = cv2(T1)$\\\\textbf {.\",\n \"}Put $ T0' = a0,0a1,0a2,0a4,0a3,0a2,1$,$ T1' = a0,1a1,1a4,1a3,1a2,1a2,2$,$ T2' = a0,2a1,2a2,2a3,2a4,2a1,1$ (see Figure~\\\\ref {fig:case5-similar2}),let $ = ({T0,T1,T2}){T0',T1',T2'}$,and let $ S' = T3, ..., Ts-1$.By Lemma~\\\\ref {lemma:T^{\\\\prime }1-is-path}, $ T'1$ is an element of type~C.\",\n \"In what follows, we prove that $ T1'$ and $ T2'$ are paths.Since $ G$ is simple, we have $ a2,2{a1,1,a2,1,a3,1,a4,1}$,and $ a1,1{a2,2,a3,2,a4,2}$.By Lemma~\\\\ref {lemma:no-cycles}, $ a2,2 a0,1$, and $ a1,1 a0,2$.Therefore, $ T2'$ is a path.If $ a1,1 = a1,2$,then $ a2,2a1,2$ and $ a2,1a1,1$ are two green in edges of $ a1,1$.Therefore, $ T2'$ is a path.$ Figure: Exchange of edges between the three first elements in an open A-chain with at least four elements, in the case tr (T 3 )= cv 2 (T 2 ){\\\\rm {tr}}(T_3) = {\\\\rm {cv}}_2(T_2).To check that $$ is an admissible decomposition first note that ${\\\\rm {hang}}_{}(v) \\\\ge {\\\\rm {hang}}_{(v)for every v\\\\in V(G)\\\\setminus \\\\lbrace a_{3,0}, a_{3,1}, a_{3,2},a_{4,0}\\\\rbrace ,but since T_0 is free, a_{3,0}, a_{3,1}, a_{3,2},a_{4,0} are not connection vertices in ,and hence a_{4,0} is not a connection vertex in .Thus, since a_{2,2} and a_{3,2} are not connection vertices in~,the element T_3 is free.Therefore, S^{\\\\prime } is either an open chain or contains only one element,which is of type~C,and hence Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-commutative-hanging-edge})holds for~.Since, T^{\\\\prime }_0, T^{\\\\prime }_1 and T^{\\\\prime }_2 are paths,Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:complete-commutative-types}) holds for .\",\n \"Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\\\ref {def:semi-complete-commutative}(\\\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\\\tau () = \\\\tau ( - 3, a contradiction to the minimality of .This concludes the proof.\",\n \"}$ Recall that a $\\\\lbrace g,r\\\\rbrace $ -graph $G$ is a 5-regular graph that contains the Cayley graph $X(\\\\Gamma ,S)$ , where $S = \\\\lbrace g,-g,r,-r\\\\rbrace $ .\",\n \"Thus, since $S$ is closed under taking inverses, $G$ is also a $\\\\lbrace g,-r\\\\rbrace $ -, $\\\\lbrace -g,r\\\\rbrace $ -, $\\\\lbrace -g,-r\\\\rbrace $ -graph, which yields the following corollary of Theorem REF .\",\n \"Corollary 12.1 Every $\\\\lbrace g,r\\\\rbrace $ -graph for which $2g+2r \\\\ne 0$ or $2g-2r\\\\ne 0$ admits a $P_5$ -decomposition.\",\n \"The main result of this paper is a straightforward consequence of Corollary REF and Theorem REF .\",\n \"Theorem 13 Every $\\\\lbrace g,r\\\\rbrace $ -graph admits a $P_5$ -decomposition.\"\n ],\n [\n \"Conclusion and future works\",\n \"In this paper, we verified Conjecture for (i) $(2k +1)$ -regular graphs containing a spanning $2k$ -regular power of a cycle, and (ii) 5-regular graphs containing special spanning 4-regular Cayley graphs.\",\n \"We believe that the techniques developed here can be extended for a more general class of graphs, such as Schreier Coset Graphs (see [8]).\",\n \"Let $G$ be a group and let $H$ be a subgroup of $G$ .\",\n \"For $s \\\\in G$ , the right coset of $H$ corresponding to $s$ is the set $Hs=\\\\lbrace hs\\\\colon h \\\\in H\\\\rbrace $ .\",\n \"Left cosets can be defined analogously.\",\n \"Let $g_1,\\\\dots ,g_r$ be a sequence in $G$ whose members generate $G$ , the Schreier Right Coset Graph (SRCG) is defined as follows.\",\n \"Its vertex set is the set of right cosets of $H$ in $G$ , for each coset $H^{\\\\prime }$ and each generator $g_i$ there is an edge from $H^{\\\\prime }$ to the right coset $H^{\\\\prime }g_i$ .\",\n \"In particular, a Cayley graph is an SRCG where $H=\\\\lbrace 0\\\\rbrace $ .\",\n \"Schreier coset graphs are generalization of Cayley “color” graphs using cosets of some specified subgroup as vertices instead of group elements.\",\n \"In 1977, Gross [8] showed that every connected regular graph of even degree is an SRCG.\",\n \"This implies that, if we extend our result for 5-regular graphs that contain any spanning 4-regular SRCG, then we verify the conjecture for $k=2$ .\",\n \"Finally, we can also explore others graphs containing special spanning Cayley graphs.\",\n \"For instance, a natural step is to examine 7-regular graphs containing a spanning 4- or 6-regular Cayley graph.\",\n \"Also, note that the definitions of simple commutative generator and $\\\\lbrace g,r\\\\rbrace $ -graph are equivalent to Cayley graphs under the restriction of the equation $g+r = r+g$ for every pair of generators.\",\n \"Therefore, we plan to explore other restrictions, such as $g+r \\\\ne r+g$ , which would extend our result for 5-regular graphs containing every spanning 4-regular Cayley graph.\"\n ],\n [\n \"Acknowlegments\",\n \"This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.\",\n \"F. Botler is partially supported by CNPq (Grant 423395/2018-1) and by FAPERJ (Grant 211.305/2019).\"\n ],\n [\n \"Conclusion and future works\",\n \"In this paper, we verified Conjecture for (i) $(2k +1)$ -regular graphs containing a spanning $2k$ -regular power of a cycle, and (ii) 5-regular graphs containing special spanning 4-regular Cayley graphs.\",\n \"We believe that the techniques developed here can be extended for a more general class of graphs, such as Schreier Coset Graphs (see [8]).\",\n \"Let $G$ be a group and let $H$ be a subgroup of $G$ .\",\n \"For $s \\\\in G$ , the right coset of $H$ corresponding to $s$ is the set $Hs=\\\\lbrace hs\\\\colon h \\\\in H\\\\rbrace $ .\",\n \"Left cosets can be defined analogously.\",\n \"Let $g_1,\\\\dots ,g_r$ be a sequence in $G$ whose members generate $G$ , the Schreier Right Coset Graph (SRCG) is defined as follows.\",\n \"Its vertex set is the set of right cosets of $H$ in $G$ , for each coset $H^{\\\\prime }$ and each generator $g_i$ there is an edge from $H^{\\\\prime }$ to the right coset $H^{\\\\prime }g_i$ .\",\n \"In particular, a Cayley graph is an SRCG where $H=\\\\lbrace 0\\\\rbrace $ .\",\n \"Schreier coset graphs are generalization of Cayley “color” graphs using cosets of some specified subgroup as vertices instead of group elements.\",\n \"In 1977, Gross [8] showed that every connected regular graph of even degree is an SRCG.\",\n \"This implies that, if we extend our result for 5-regular graphs that contain any spanning 4-regular SRCG, then we verify the conjecture for $k=2$ .\",\n \"Finally, we can also explore others graphs containing special spanning Cayley graphs.\",\n \"For instance, a natural step is to examine 7-regular graphs containing a spanning 4- or 6-regular Cayley graph.\",\n \"Also, note that the definitions of simple commutative generator and $\\\\lbrace g,r\\\\rbrace $ -graph are equivalent to Cayley graphs under the restriction of the equation $g+r = r+g$ for every pair of generators.\",\n \"Therefore, we plan to explore other restrictions, such as $g+r \\\\ne r+g$ , which would extend our result for 5-regular graphs containing every spanning 4-regular Cayley graph.\"\n ],\n [\n \"Acknowlegments\",\n \"This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.\",\n \"F. Botler is partially supported by CNPq (Grant 423395/2018-1) and by FAPERJ (Grant 211.305/2019).\"\n ]\n]"},"id":{"kind":"string","value":"2012.05145"}}},{"rowIdx":3683,"cells":{"text":{"kind":"list like","value":[["The ACTIONFINDER: An unsupervised deep learning algorithm for\n calculating actions and the acceleration field from a set of orbit segments"],["Abstract We introduce the \"ACTIONFINDER\", a deep learning algorithm designed to transform a sample of phase-space measurements along orbits in a static potential into action and angle coordinates.","The algorithm finds the mapping from positions and velocities to actions and angles in an unsupervised way, by using the fact that points along the same orbit have identical actions.","Here we present the workings of the method, and test it on simple axisymmetric models, comparing the derived actions to those generated with the Torus Mapping technique.","We show that it recovers the Torus actions for halo-type orbits in a realistic model of the Milky Way to $\\sim 0.6$% accuracy with as few as 1024 input phase-space measurements.","These actions are much better conserved along orbits than those estimated with the St\\\"ackel fudge.","In our case, the reciprocal mapping from actions and angles to positions and velocities can also be learned.","One of the advantages of the ACTIONFINDER is that it does not require the underlying potential to be known in advance, indeed it is designed to return the acceleration field.","We expect the algorithm to be useful for analysing the properties of dynamical systems in numerical simulations.","However, our ultimate goal with this effort will be to apply it to real stellar streams to recover the Galactic acceleration field in a way that is relatively agnostic about the underlying dark matter properties or the behavior of gravity."],["Introduction","Actions and angles are the natural phase-space coordinates to represent the state of the constituents of an integrable, or close to integrable dynamical system, such as typical Galactic potentials [3].","In these coordinates, motion along an orbit is very simple, as the actions ${J}$ are preserved, while their canonically conjugate angles ${\\theta }$ advance through a cycle at a uniform rate in time (with corresponding fixed frequencies ${\\Omega }$ ).","These are also the natural coordinates for perturbation theory [15].","Furthermore, the actions are adiabatic invariants, so slow changes in the evolution of the parent system will preserve these quantities.","With the superb astrometry being gathered by the Gaia mission [10], it has now become possible to study the real six-dimensional phase space structure of our Galaxy in unprecedented detail.","Action-angle variables are being used extensively in this exploration [5], [27], [22], [29], [6], [23], notably because actions can be used to construct equilibrium distribution functions from the Jeans theorem, because they provide (in principle) the best “archaeological” information on the dynamics of the Galaxy, and because they are convenient for exploring perturbations from, for example, the Galactic bar [19], [20], [17], [26].","While actions and angles are thus undoubtedly useful, they are not always easy to calculate.","Indeed, the only potentials in which the actions are expressed analytically are those of from the family of isochrone models of which the Kepler and spherical harmonic potentials are special cases.","Although this model is very useful, it does not provide a good approximation to interesting stellar systems (other than those dominated by a central object).","Substantial theoretical efforts were therefore undertaken over the past decades to calculate approximations for the transformation from position $x$ and velocity $v$ to $J$ and $\\theta $ , and conversely, for more general and realistic galactic potentials.","[18] developed a method known as the Torus Mapping [2], starting from action-angle coordinates in an isochrone potential, close enough but different from the true potential.","Their insight was to search for a generating function in order to transform from the true actions and angles to those of the isochrone.","This generating function is expressed as a Fourier series expansion on the isochrone angles, the fitted Fourier coefficients being such that the true Hamiltonian remains, after the associated canonical transformation, constant for a set of true actions.","Once this generating function is found, the transformation from actions and angles to positions and velocities is known.","For the reciprocal transformation, one usually relies on separable potentials, denoted Stäckel potentials [7], [9], for which three exact integrals of the motion exist.","These potentials are best expressed in spheroidal coordinates, associated to a focal distance directly related to the first and second derivatives of the Stäckel potential.","This focal distance can be computed for the true potential at any configuration space point as if the potential was a Stäckel one, and the corresponding integrals of the motion and respective actions can be evaluated.","This method introduced by [1] is known as the “Stäckel fudge\".","Unfortunately, this transformation is not the exact inverse of the one going from actions and angles to positions and velocities as obtained from the Torus Mapping.","All these methods are reviewed in [25], and have all been implemented in the AGAMA dynamics package [28], [30].","Our aim in the present work is to provide an innovative calculation of action-angle coordinates, a method which would be able to jointly determine, from a sample of segments of orbits in positions and velocities, (i) the corresponding actions and angles, as well as (ii) the true acceleration field in which these orbits reside.","Moreover, (iii) the inverse transformation from actions and angles to positions and velocities should be determined easily.","To this end we will build a deep neural network that will try to learn on its own, and in an unsupervised way, the coordinate transformation from observables into $J$ and $\\theta $ for the particular dataset under study.","Figure: Sketch of the method.","We consider a dynamical system for which we have measured the kinematics of NN particles or “stars” along each of a sample of SS different orbits which we loosely refer to as “streams”.","In panel (a) we have sketched two such “streams”, each with four “stars”.","The observable sky positions, parallaxes (or distances) as well as the radial velocities and proper motions are first converted by the algorithm into the Cartesian system ( i n x, i n y)({}_{i}^{n}{x}, {}_{i}^{n}{y}), where the subscript denotes the stream number and the superscript denotes the star number in that stream.","The Cartesian coordinates are converted into action-angle coordinates of an analytic isochrone model (b), which serves as a “toy” first approximation to the real actions.","The central ingredient of the algorithm is the network that proposes trial generating functions G(θ,J ' )G({\\theta }, {J^{\\prime }}) of the canonical transformation between the toy coordinates (θ,J)({\\theta }, {J}) and the refined coordinates (θ ' ,J ' )({\\theta ^{\\prime }}, {J^{\\prime }}).","Note that because J ' {J^{\\prime }} is not known in advance, we need to iterate at this stage (c) to find the J ' {J^{\\prime }} values that are consistent with the isochrone's values of J{J}.","Finally (d), we also allow for a simpler canonical point-transformation with the generating function P(θ ' )P({\\theta ^{\\prime }}) to calculate the target coordinates (θ '' ,J '' )({\\theta ^{\\prime \\prime }}, {J^{\\prime \\prime }}).","The two networks (green arrows) are refined over the course of the training procedure by attempting to find the network weights that minimize the spread in the J '' {J^{\\prime \\prime }} values in each stream."],["The Algorithm","We designed the algorithm around the fact that different phase space points along an orbit have identical $J$ (and frequencies $\\Omega $ ).","For this to be useful, we imagine having access to groups of such phase space points along the same orbit.","This could be the case if we had the output of a numerical N-body simulation at a series of different time steps, or alternatively we might have such information from observations of a real dynamical system.","In the code we refer to a group of such phase space points along the same orbit as a “stream”, while each individual phase space point is referred to as a “star”.","Our use of the terms “streams” and “stars” in the software obviously betrays our ultimate motive for building it, but we note here that star streams do not precisely follow orbits, and that a significant amount of further work is required to adapt the present algorithm to properly model such structures.","It is relatively straightforward to devise an algorithm to implement the coordinate mapping from Cartesian $\\eta \\equiv (x, y, z, v_x, v_y, v_z)$ to $\\xi \\equiv (\\theta _1, \\theta _2, \\theta _3, J_1, J_2, J_3)$ with a deep neural network for a system for which we have a set of known $\\eta $ and $\\xi $ values.","(This is what we construct for the inverse transformations in Section ).","In this case of so-called “supervised learning”, the network calculates the derivatives of the loss function with respect to the system parameters (the millions of weights in the neural net), and adapts these parameters using a stochastic gradient descent algorithm (in our work we use the “Adam” optimization algorithm, [16]) so as to mimimize the difference between the predicted actions and angles and the corresponding “ground truth” values.","The learned map can subsequently be used to attribute action and angle values to data not previously seen by the network.","However, for the forward transformation in the present work we do not wish to rely on external software to provide us with a training dataset of action-angle coordinates in galaxy models of interest.","The fact that nature does not provide us with ground-truth action and angle coordinate labels provides the incentive to be able to calculate these quantities without reference to a training set.","The potentials of galaxies probably also do not follow simple mathematical forms, so we have a further motivation to be able to calculate the canonical transformation in a model-independent way.","How then can the stochastic gradient algorithm be steered towards the right solution automatically without labeled data?","As stated above, we know that points along the same orbit should have identical actions and frequencies.","In addition, the map from $\\eta $ to $\\xi $ obeys the symplectic condition for the Jacobian matrix: $M^T \\mathbb {J} M = \\mathbb {J} ,$ where $M$ is the symplectic Jacobian matrix $M_{ij} = {{\\partial \\eta _i}/{\\partial \\xi _j}}$ , and $\\mathbb {J}$ is the antisymmetric block matrix $\\mathbb {J} =\\begin{pmatrix}0 & \\mathbb {I}_3\\\\-\\mathbb {I}_3 & 0\\end{pmatrix} \\, ,$ with $\\mathbb {I}_3$ being the $3\\times 3$ identity matrix.","This makes it tempting to simply build a deep network that directly maps $\\eta $ to $\\xi $ , while including the above conditions as terms in the loss function (together with additional conditions that will be presented in Section ).","We were disappointed to find that this did not work very well.","In order to approximate the coordinate transformation to $\\sim 1$ % accuracy or better, one requires a deep neural net with tens of millions of parameters.","Yet the symplectic condition creates a very complex loss function landscape in this parameter space, which the stochastic gradient descent optimizer explores in a very inefficient way.","To overcome this problem, we decided to make use of a generating function for the canonical transformation, to guarantee that the coordinate transformation will be symplectic.","As pioneered by [18], the transformation can be made much simpler by using the action and angle coordinates of a “toy” model as a stepping stone.","Our procedure is sketched in Fig.","REF .","We begin by converting the observed astrometric data into Cartesian coordinates $({x},{v})$ .","It is worth noting that thanks to the automatic differentiation module in the pytorch [21] machine learning library, the gradients of the output quantities with respect to the astrometric observables can be calculated easily; we will make use of this feature in future work to account for observational uncertainties.","For the toy model, we use an isochrone potential: $\\Phi (r) = - {{G M}\\over {b + \\sqrt{b^2+r^2}}} \\, .$ Here $r$ is the (spherical) radius coordinate, $G$ is the universal gravitational constant, and the two model parameters are the mass $M$ and scale radius $b$ .","One may then choose the three actions in the isochrone model to be: $\\begin{split}J_{1, \\rm iso} &= L_z\\\\J_{2, \\rm iso} &= L - |L_z|\\\\J_{3, \\rm iso} &= {{G M}\\over {\\sqrt{-2 E}}} - {{1}\\over {2}} \\Bigg ( L + {{1}\\over {2}} \\sqrt{L^2 - 4 G M b} \\Bigg ) \\, ,\\end{split}$ where $L$ is the angular momentum of the particle, $L_z$ is the $z$ -component of angular momentum, and $E$ is the particle's energy.","The procedure to convert from $({x},{v})$ to the isochrone model's $({\\theta },{J})$ is detailed in [18], and our pytorch version is heavily inspired by the galpy isochrone implementation [4].","[18] found that their algorithm converged irrespective of the initial chosen values of the two isochrone parameters $M$ and $b$ , although values closer to the real system produced faster convergence.","Our algorithm does not appear to be very sensitive to this initial choice either, as long as $M$ is set high enough that all orbits are bound.","In our algorithm both the $M$ and $b$ parameters can be fitted by pytorch in the stochastic gradient descent procedure, or if desired, they can be held fixed at their initial values.","We now aim to find new coordinates $({\\theta ^{\\prime }},{J^{\\prime }})$ that are closer to those of the real system.","To this end we define an indirect type 2 generating function $G=G({\\theta },{J^{\\prime }})$ , whose derivatives give the implicit transformation: ${J} = \\, {J^{\\prime }} + {{\\partial G}\\over {\\partial {\\theta }}} \\\\{\\theta ^{\\prime }} = \\, {\\theta } + {{\\partial G}\\over {\\partial {J^{\\prime }}}} \\, .","$ A deep learning network will be used to propose trial variations on $G$ , and again thanks to the automatic differentiation in pytorch it is straightforward to find the ${J}$ and ${\\theta ^{\\prime }}$ values generated by $G$ .","Since we are actually interested in ${J^{\\prime }}$ , we iteratively find the ${J^{\\prime }}$ value that yields values of ${J}$ from Eqn.","REF that are the same as those of the toy isochrone.","We now finally update the $({\\theta ^{\\prime }},{J^{\\prime }})$ coordinates using a further transformation ${\\theta ^{\\prime \\prime }}=P({\\theta ^{\\prime }})$ that is only dependent on the updated angles.","This point-transformation is much simpler than $G$ , as we can obtain the updated angles and actions directly with no need for an iterative procedure, ${\\theta ^{\\prime \\prime }} = \\, P({\\theta ^{\\prime }}) \\\\{J^{\\prime \\prime }} = \\, {{\\partial {\\theta ^{\\prime }}}\\over {\\partial {\\theta ^{\\prime \\prime }}}} {J^{\\prime }} \\, ,$ and is therefore easy for the algorithm to fit.","As demonstrated by [14], such a transformation allows for deformations of the toy model that may be better adapted to the geometry of the real system's orbits.","In our experiments we found that this additional network improved the quality of the predicted actions by up to $\\sim 50$ %.","Python style pseudocode for the central function of the algorithm is shown in the Appendix in Listing .","Figure: Sketch of the (shorter) net P {\\rm net}_P neural network.","The ResNet-like unit blocks (left, and Eqn.","), are incorporated as a series of layers (right) that progressively increase in complexity up to a chosen maximum width, and then decrease symmetrically up to the layer immediately before the output.","The number of features nf {\\rm nf} in each unit block is indicated."],["The Action-Angle Network","All of the neural networks used here have the same basic architecture.","Thus, the generating function $G$ is given by: $G = {\\rm net}_G(\\cos ({\\theta }),\\sin ({\\theta }), {J^{\\prime }}) \\, , $ while the canonical point-transformation $P$ has the form: $P = {\\rm net}_P(\\cos ({\\theta }),\\sin ({\\theta })) \\, .","$ After extensive experimentation, we found that the angle coordinates were best learned by first introducing the pair of auxiliary variables $t_{x}\\equiv \\cos (\\theta )$ and $t_{y}\\equiv \\sin (\\theta )$ , as used in Eqns.","REF and REF .","This naturally takes care of the cyclic property of the angle variables.","The ${\\rm net}_G$ network consists of a series of 15 blocks similar to a Residual neural network (ResNet, [12]), adapted for the case of Dense (Linear) layersA linear layer $W$ with bias $b$ simply transforms an input vector $x$ to $y=x W^T+b$ .","The parameters $W$ and $b$ are learned by the algorithm.. An initial fully-connected linear layer takes the input features (9 in the case of ${\\rm net}_G$ and 6 in the case of ${\\rm net}_P$ ) and passes them onto a layer of 64 features.","Deeper layers increase in width, becoming a factor of 2 wider in the number of features per layer, up to a maximum of 1024 features per layer.","After the chosen maximum depth is reached (at layer 6), the layers decrease in size in a symmetric way.","A final fully-connected linear layer is applied to give the chosen output features, which in the case of the generating functions $G$ and $P$ is a scalar value.","This architecture is sketched in Fig.","REF .","The ${\\rm net}_P$ network is identical to ${\\rm net}_G$ , but consists of only 11 blocks.","The unit blocks were constructed as follows: $\\begin{split}{\\rm out} = W_2(& D_2( {\\rm ReLU}( \\\\W_1(& D_1( {\\rm ReLU}( {\\rm input} ))) \\,\\, ))) + {\\rm input}\\end{split}$ where $D_1$ and $D_2$ are fully connected linear layers, ${\\rm ReLU}(x)=\\max (0,x)$ is the Rectified Linear Unit activation function, and $W_1$ and $W_2$ are Weight Normalisation layers [24].","This setup is very similar to the canonical ResNet, designed so that the blocks learn successive corrections to the input information.","We show the pseudo-code of this module in Listing in the Appendix.","Each unit block is connected to the following block with a Dense layer, without applying an activation function.","Our use of Weight Normalisation layers in Eqn.","REF is noteworthy.","Most modern networks in computer vision make use of the “Batch Normalization” (BN, [13]) procedure to decouple as much as possible the fitting of the parameters in the different layers of a network.","This mitigates against the co-variance between layers, allowing the parameters of a deep layer to be refined even though the parameters of the higher layers are also being adjusted at the same time by the optimization algorithm.","However, after much experimentation, we found that the BN layers we initially used were limiting the accuracy of our algorithm.","This is due to the fact that this procedure operates (during training) on the data presented to it in each “batch” (i.e.","in small sub-samples which are chosen so that the data may fill the graphics card memory), from which it normalizes the data using the mean and standard deviation of the sub-sample.","Since the data at different positions within the batch belong to different streams, the unavoidable shot noise then creates substantial variation between batches, resulting in unacceptably large errors for a study such as ours.","We devised a work-around by training in the normal way with the BN procedure until it reached an equilibrium state, and then restarting with the BN layers frozen using parameter values calculated from the whole dataset.","We later realised that the “Weight Normalization” scheme gives similar accuracy to our BN “hack”, and have adopted it for the present work as it is an accepted machine learning method.","Between them, with the chosen depth of 15 and 11 layers, the two generating function networks have a total of $\\sim 44$ M free parameters.","The chosen loss function is very simple: $L = L_{J^{\\prime \\prime }, \\, \\rm spread} + L_{J_{2,3}^{\\prime \\prime }>0} + \\alpha _1 L_{{\\theta _0}^{\\prime \\prime }} \\, .$ $L_{J^{\\prime \\prime }, \\, \\rm spread}$ is the mean absolute deviation of the difference between the predicted action $J^{\\prime \\prime }$ of the stars in a stream and the mean action of that stream $\\langle J^{\\prime \\prime }\\rangle $ .","The term $L_{J_{2,3}^{\\prime \\prime }>0}$ penalises unphysical negative values of ${J_2}^{\\prime \\prime }$ and ${J_3}^{\\prime \\prime }$ , as follows: $L_{J_{2,3}^{\\prime \\prime }>0} = \\langle | J_2-|J_2| | \\rangle + \\langle | J_3-|J_3| | \\rangle \\, .$ Finally, the loss term $\\alpha _1 L_{{\\theta _0}^{\\prime \\prime }}$ is the mean absolute deviation of the target angles ${\\theta ^{\\prime \\prime }}$ when $\\theta =0$ is fed into the network (i.e.","it encourages the zero-point of the target angles to coincide with those of the isochrone toy model).","Following common machine learning practise, we normalised the position and velocity variables (to $20{\\rm \\,kpc}$ and $200{\\rm \\,km\\,s^{-1}}$ , respectively); thus the actions are normalized to $4000 {\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ .","However, for convenience, we left the angle variables in radians.","We chose to set the hyper-parameter to $\\alpha _1=0.1$ in Eqn.","REF , so as compensate for this difference in the range of the angle and action variables.","The algorithm proceeds by iterating over discrete “epochs” when the network derivatives are calculated and the parameters are consequently updated to obtain improved estimates for $J^{\\prime \\prime }$ and $\\theta ^{\\prime \\prime }$ ."],["The acceleration network","After the algorithm has converged on values of ${J^{\\prime \\prime }}$ and ${\\theta ^{\\prime \\prime }}$ that minimize the spread of ${J^{\\prime \\prime }}$ in the streams, we can use these transformed coordinates to study the system they inhabit.","To this end, we re-run the network once more with an updated loss function, and with an additional network (the acceleration network) turned on.","The structure of the acceleration network is set up as an additive correction to the acceleration of the toy isochrone model ${a_{\\rm iso}}$ : $a = {\\rm net}_{a}(x, y, z, {a_{\\rm iso}}) + a_{\\rm iso} \\, .$ Note that here we only feed in the position part of the $\\eta $ phase space vector.","We use a depth of 15 blocks for this network, which contains $\\sim 17.5$ M free parameters.","The structure of ${\\rm net}_{a}$ is essentially identical to that of ${\\rm net}_{G}$ , except that it outputs a vector.","If ${J^{\\prime \\prime }}$ and ${\\theta ^{\\prime \\prime }}$ are only functions of position ${x}$ and velocity ${v}$ , the chain rule implies that the mapping is constrained by the following relations: ${{d {J_i}^{\\prime \\prime }}\\over {d t}} &= {{\\partial {J_i}^{\\prime \\prime }}\\over {\\partial x_j}} \\dot{x}_j + {{\\partial {J_i}^{\\prime \\prime }}\\over {\\partial \\dot{x}_j}} \\ddot{x}_j = 0\\\\{{d {\\theta _i}^{\\prime \\prime }}\\over {d t}} &= {{\\partial {\\theta _i}^{\\prime \\prime }}\\over {\\partial x_j}} \\dot{x}_j + {{\\partial {\\theta _i}^{\\prime \\prime }}\\over {\\partial \\dot{x}_j}} \\ddot{x}_j = {\\Omega _i}^{\\prime \\prime } \\, ,$ where the $i$ index here is associated to each of the (usually three) integrals of motion $J_i$ , and $j$ is a dummy index denoting Einstein summation.","A brief inspection of Eqn.","REF reveals that once we have access to a network that delivers predictions for the position and velocity derivatives of $J^{\\prime \\prime }$ , we can solve the simultaneous equations to calculate the acceleration terms $\\ddot{x}$ .","While this suggests that the acceleration network (Eqn.","REF ) is superfluous, we nevertheless opted to employ a network to calculate $a$ for two main reasons.","First, it allows us to train a network that can be subsequently applied to make predictions for the acceleration on new unseen data (for instance, for “stars” without full 6-dimensional information).","Second, we found that solving Eqn.","REF for $\\ddot{x}$ using linear algebra can occasionally be problematic, due to ill-conditioned matrices.","Creating a separate acceleration network eliminates these problems.","In this second run through the algorithm, we choose the following loss function: $\\begin{split}L =& L_{dJ^{\\prime \\prime }/dt} + \\alpha _2 L_{\\rm symmetry} + \\alpha _3 L_{\\Omega ^{\\prime \\prime }, \\, \\rm spread} \\, ,\\end{split}$ where $L_{dJ^{\\prime \\prime }/dt}$ is the mean absolute value of $dJ^{\\prime \\prime }_i/dt$ , as calculated from Eqn.","REF .","$L_{\\rm symmetry}$ is an optional term to enforce a desired symmetry on the solution of the acceleration network.","For instance for axisymmetric potential models, we experimented with a cosine anti-similarity criterion $L_{\\rm symmetry} = 1 + \\vec{a}_R \\cdot \\vec{R}/(||\\vec{a}_R|| \\, ||\\vec{R}||)$ between the cylindrical-$R$ component of the acceleration field $\\vec{a}_R$ and the same component of the (Galactocentric) position vector $\\vec{R}$ .","Although this loss term helps to constrain the acceleration, for the purpose of the tests presented in Section we decided to suppress this symmetry constraint on $a$ by setting $\\alpha _2=0$ .","Finally, $L_{\\Omega ^{\\prime \\prime }, \\, \\rm spread}$ is another optional loss term to ensure that the frequencies calculated from Eqn.","for each “stream” have minimum spread.","This is done by calculating the mean absolute difference of the frequency of each “star” from the mean of the frequencies in the “stream”.","However, we found in our tests that including this term introduced slight additional scatter in the solutions (most likely because angles are much less accurately constrained than actions in our algorithm), so it too was suppressed by setting $\\alpha _3=0$ in the present work."],["Inverse Transformation","Deep learning can also be used to calculate the inverse transformation from actions and angles back to positions and velocities.","However, there are many different situations that one can envisage being confronted with.","For instance, one may or may not have frequencies in addition to the actions and angles, or one may already know the potential.","Alternatively, it is possible that the potential and frequencies are not known, but one may have groups of particles along orbits (i.e.","our “stars??","in “streams??).","Each of these cases would require a different network to be constructed in order to learn the inverse transformation in an unsupervised way.","The approach we take here is slightly different, but perhaps more realistic given the forward transformation we have presented previously.","We imagine having derived the forward transformation from $\\eta $ to $\\xi $ for a number of orbits.","This information can now be used to train a supervised network to predict $\\eta $ from $\\xi $ .","Since this is a supervised learning task, compared to the previous problem, it is a much easier for the network to find the correct direction for the stochastic gradient descent to improve the loss function.","We therefore choose a simple network architecture: $(t_x,t_y,J) = {\\rm net}_{\\rm inverse}(t_x^{\\prime \\prime },t_y^{\\prime \\prime },J^{\\prime \\prime }) \\, ,$ from which the toy model angles are calculated with $\\theta =\\operatorname{atan2}(t_y,t_x) + \\pi /2$ .","Just like the forward network, the inverse transformation thus employs the isochrone model as a stepping stone, with the final step being the analytic inversion of the toy model's $(\\theta ,J)$ coordinates to the 6-dimensional vector $\\eta $ .","Since this inverse network is not constrained by differential equations (and hence does not require any Jacobians to be calculated) it is feasible to implement it as a deeper network of 23 (or more if necessary) layers of blocks (Eqn.","REF ), using a total of $\\sim 34$ M free parameters."],["Results","The algorithm was constructed to accept as input data a set of $S$ “streams”, each with $N$ “stars”.","The method works with $N\\ge 2$ “stars” per “stream”, but for these initial tests we have chosen a more conservative $N=8$ , simply to make it easier for the algorithm to verify that the actions and frequencies calculated for each “stream” are constant.","Pytorch achieves its speed by processing the data in parallel, and so it is much more efficient to pass pytorch “tensors” of the same size onto the graphics processing unit (GPU).","Because of this, we expect that when the method is applied to real data, it will be convenient to break up streams with a large number of known members into smaller sub-groups (of the same size $N$ ).","In all the experiments below, we split the input sample into two sets, a training set with 50% and a test set with 50% of the sample.","Clearly, with real data we would be much more sparing with the fraction allocated to the test set!","Typically, we run the training process with a learning rate of $10^{-4}$ .","In all the tests reported below we iterate for 1024 “epochs” even though the training losses often stabilize much earlier.","We found that re-running the network with smaller learning rates (which is the standard procedure in machine learning to improve accuracy), resulted in only very marginal improvements to the loss function values.","Test samples are used in machine learning primarily to verify that the training procedure is not overfitting the data.","When this occurs, the loss of the training set continues to improve, while the loss values in the test set (which the algorithm does not see during training) become worse.","We simply ignore all further epochs once the algorithm begins to overfit.","Both the data and network are expressed internally as double precision floating-point numbers, as we found that the Jacobian matrices were not always calculated to sufficient precision with single precision numbers, resulting in a network that would not update correctly due to vanishing gradients.","Because of the fact that we find the actions iteratively starting from an initial guess provided by the toy model, the speed of the algorithm depends on how close the toy model is to the target system.","But in typical cases using $S=1024$ and $N=8$ , the algorithm takes $\\sim 3$ hours to complete a training run of 1024 epochs on an NVIDIA GV100 GPU with 32 GB of card memory.","Running times on larger datasets should scale approximately linearly with the number of data."],["Fitting isochrone models","Although we have built the ACTIONFINDER as a series of transformations from the analytic isochrone, it is still worth checking whether the software can fit a sample of orbital points drawn from different models of this family.","To this end we generated orbits within an isochrone potential with $M=3.334\\times 10^{11}{\\rm \\,M_\\odot }$ and $b=5{\\rm \\,kpc}$ (i.e.","$M=1.5\\times 10^6$ in N-body units where $G=1$ and distances are in ${\\rm \\,kpc}$ and time is in ${\\rm \\,Gyr}$ ), which gives a circular velocity at the Solar neighborhood ($R_\\odot =8.122{\\rm \\,kpc}$ , [11]) of $v_c(R_\\odot )=217{\\rm \\,km\\,s^{-1}}$ .","One of the advantages of the ACTIONFINDER is that we do not need to provide it with a fair sample of orbits within the system, almost any sample that covers the region of interest will do.","In the following, we imagine having access to the orbits of some objects in the “Galaxy” halo.","To model this situation we select a random 3-dimensional initial radius drawn uniformly in Galactocentric distance between $r=[6$ –$16]{\\rm \\,kpc}$ , together with a velocity vector that is oriented randomly (i.e.","isotropic) with magnitude drawn from a Gaussian of dispersion $150{\\rm \\,km\\,s^{-1}}$ .","Using a symplectic Leapfrog scheme, we integrate from these initial phase space locations for $0.1{\\rm \\,Gyr}$ , ensuring energy conservation to 1 part in $10^7$ , and randomly select $N=8$ phase space points along each path.","Since our aim is to eventually work with real astrometric data, we convert the positions and velocities of the set of orbit locations to the observable quantities: $d, \\ell , b, v_h, \\mu _\\ell , \\mu _b$ , as they would be measured from our vantage point in the Galaxy, where $d$ is Heliocentric distance, ($\\ell , b$ ) are Galactic coordinates, $v_h$ is the Heliocentric radial velocity, and $\\mu _\\ell $ and $\\mu _b$ are the proper motions along the Galactic coordinate directions.","With the correct input values of $M$ and $b$ , the network quickly finds the correct $J^{\\prime \\prime }$ and $\\theta ^{\\prime \\prime }$ to better than 0.01% (with a training set size of $S=1024$ ), which is reassuring given that the algorithm effectively just has to learn (in an unsupervised way) the identity operation.","A more interesting case occurs when we try to fit the isochrone “stream” sample above using fixed and incorrect reference $M$ and $b$ values.","With $M$ and $b$ both 10% lower (higher) in the toy model compared to the simulated data, using $S=1024$ we obtain action errors of $\\delta J=1.2 (2.0) {\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ and angle errors of $0.7^\\circ (1.2^\\circ )$ .","When $M$ and $b$ are 20% higher, the action error is $\\delta J=3.0 {\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ and the angle error is $1.4^\\circ $ .","The algorithm fails with a fixed 20% lower value of $M$ , because some orbits are unbound in the toy model, and thus do not have valid $(\\theta ,J)$ coordinate values.","Using $S=1024$ with fixed 20% higher values of $M$ and $b$ in the toy model, the inverse transformation from $(\\theta ^{\\prime },J^{\\prime })$ to $(x,v)$ is recovered to an accuracy of $\\delta x=0.3{\\rm \\,kpc}$ and $\\delta v=2.6{\\rm \\,km\\,s^{-1}}$ (again, mean absolute deviation errors)."],["Fitting a realistic Galaxy model","[8] fitted plausible axisymmetric density models of the main Galactic components to kinematic observations of the Milky Way to derive the potential of the Galaxy.","These useful models have been incorporated into the AGAMA package [30], which now allows one to transform the actions and angles of test particles in these potentials to positions and velocities with the Torus Mapping.","We created a set of random orbits (each with fixed triplet of actions) in the [8] model `1' in a similar way to that described in Section REF , and as before we selected $N=8$ random points along the orbits, recording their input actions and angles values, their $(x, v)$ values generated with the Torus Mapping, as well as their output $(\\theta ^{\\prime \\prime }, J^{\\prime \\prime })$ values found by ACTIONFINDER.","For these tests the toy isochrone $M$ and $b$ parameters are left free.","Using $S=1024$ , the ACTIONFINDER recovers the Torus Mapping input action-angle coordinates with an error of $\\delta J^{\\prime \\prime }=8.0{\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ in action (i.e.","$\\sim 0.4$ % of the Sun's action) and $\\delta \\theta ^{\\prime \\prime }=7.4^\\circ $ in angle.","The model accelerations are also recovered to $\\delta a=2.6$ %.","These errors are calculated as the mean absolute deviation between the Torus Mapping input values and the ACTIONFINDER predictions.","With these noise-less data (possessing only shot-noise), the derived $(\\theta ^{\\prime \\prime },J^{\\prime \\prime })$ are not very sensitive to the sample size.","If instead we set $S=128 (16)$ , we obtain an action error of $\\delta J^{\\prime \\prime }=11.2 (18.1){\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ and the angle error is $\\delta \\theta ^{\\prime \\prime }=8.0^\\circ $ ($9.1^\\circ $ ).","In Fig.","REF we show the evolution of the loss function in these three tests.","As a comparison, we also transformed back the positions and velocities of each particle into action-space with the Stäckel-fudge in AGAMA.","In this case, the mean absolute deviation between the Torus Mapping input actions and the Stäckel-fudge estimate is $44 {\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ , 5 times less accurate than with the ACTIONFINDER.","Figure: Convergence of the loss function for different sample sizes.","All three tests use simulated orbits in the potential model `1', each with N=8N=8 phase-space points.","The sample sizes are S=1024S=1024, S=128S=128 and S=16S=16 for the top, middle and bottom panels, respectively.","In each case, the blue line shows the training loss while the orange line shows the validation loss.","The red dot shows the position of the best validation loss, which defines the epoch at which the (θ '' ,J '' )(\\theta ^{\\prime \\prime },J^{\\prime \\prime }) coordinates are extracted.With $S=1024$ the inverse transformation algorithm (which is basically the same transformation as the one done with the Torus Mapping to generate positions and velocities) recovers the $(x,v)$ originally generated with the Torus Mapping to an accuracy of $\\delta x=0.24{\\rm \\,kpc}$ and $\\delta v=4.4{\\rm \\,km\\,s^{-1}}$ ."],["Discussion and Conclusions","Using modern deep learning tools we have constructed a method to transform observable astrometric coordinates into action and angle coordinates with a scheme that is effectively a back-to-front version of that developed by [18], together with the canonical point-transformation improvement by [14].","The deep neural nets used here are able to generate non-linear functions that are more flexible than a linear decomposition into Fourier coefficients, and this flexibility may make it easier to fit more general dynamical systems.","A further advantage of our method is that it simultaneously uses all the data at its disposal to fit the generating functions of the canonical transformations.","This contrasts to earlier methods where each orbit is fitted independently to derive the best set of Fourier coefficients for that orbit, bearing in mind that each such orbit requires many constraining data points due to the large number of Fourier coefficients that have to be fit.","Moreover, once the transformation from positions and velocities is found, it is easy to inverse it in a supervised way.","The main advantage of the method, however, is that we do not need to know in advance the Hamiltonian or potential of the system under investigation.","The algorithm finds it for us.","The data used in the tests presented here are noise free (apart from shot noise), and as such they do not portray a realistic picture of the limitations of the method as applied to real data.","However, it is beyond the scope of the present contribution to attempt an exploration of the consequences of the observational limitations.","With this caveat in mind, we are able to train the network to predict the actions from positions and velocities in a realistic axisymmetric Milky Way model, with an uncertainty of $\\sim 0.6$ % of the action value of the Sun (compared to the input actions transformed into positions and velocities with the Torus Mapping), by training on only $N=8$ phase-space points along $S=128$ orbit segments, i.e.","with only 1024 phase-space points.","With $S=1024$ such orbit segments, an action uncertainty of $\\sim 0.4$ % of the action value of the Sun can be attained.","This is 5 times more accurate than the estimate of actions from positions and velocities with the Stäckel fudge in AGAMA.","In principle, the present algorithm is not limited to axisymmetric systems, and given that the canonical coordinate transformation encoded in the generating function networks can be made arbitrarily complex by increasing the depth of the networks, one may be able to learn and model complex mass distributions given sufficient data and computational resources.","However, the presence of chaotic dynamics in complex potentials gives cause for concern, since our method relies on the actions being conserved, which is not the case for chaotic orbits.","It is thus likely that in such systems the method will only work for a subset of the orbits, while returning nonsensical results for the rest.","We suspect that this could be turned from a bug into a feature of the method, as it may allow the present algorithm to be adapted into an automatic orbit classifier.","We have attempted some very preliminary tests of the method on a non-axisymmetric system by applying the algorithm to orbits integrated in a triaxial logarithmic halo model.","We found that the present algorithm is not able to adapt automatically to fit all orbital families in a triaxial model simultaneously.","The reason for this may be both because of the presence of chaotic orbits, and because the isochrone model is not a good starting point for the canonical transformation in a triaxial potential.","Nevertheless, we were encouraged to find that we were able to fit a generating function that gave small scatter in $J^{\\prime \\prime }$ between points on the same orbit, by selecting the input sample of “streams” from high angular momentum tube orbits.","In future work we will attempt to replace the isochrone model with a triaxial Stäckel model [7] as the “toy” starting point; it is plausible that this may provide the key to unlock unsupervised fitting in triaxial and more general systems.","But we will also need to be able to identify chaotic orbits in the system.","The method currently relies on the fact that the potential is static.","However, with the same caveats as above regarding chaos, it may be possible to generalize the algorithm so that motion is analysed in a rotating frame in which, for instance, a barred structure would appear static.","This might allow the method to be used for analysing non-axisymmetric rotating systems as well.","The algorithm was deliberately built to accept astrometric data as inputs.","Because pytorch processes data in parallel, this architecture makes it very simple to supply the network with multiple inputs for the same star, where the different instances could, for example, sample over the uncertainties in the astrometry, or scan over missing information in some input dimensions.","Thus, if a star has a missing radial velocity measurement, one may attempt to find the radial velocity value that makes the derived actions for the star agree with those of the group.","It is thus plausible that the present software can be adapted into a new method for detecting stellar streams, especially structures that do not possess any obvious spatial correlation.","The unsupervised learning technique developed here of building a network based on potentially complex corrections to simple analytic models and applying the physical constraints as loss function terms may have substantially wider applications.","The pytorch tensor structure is particularly powerful for this purpose, with its ability to implement automatically the derivatives of the analytic model and the correction function.","As we have seen, this provides an easy means to incorporate differential equations into a network.","A simplified demonstration version of the code, along with a sample of test data from the [8] potential model `1' will be made available on github upon publication of this manuscript.","RI, BF, and GM acknowledge funding from the Agence Nationale de la Recherche (ANR project ANR-18-CE31-0006, ANR-18-CE31-0017 and ANR-19-CE31-0017), from CNRS/INSU through the Programme National Galaxies et Cosmologie, and from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No.","834148)."],["Appendix","[emphstyle=, caption=Pytorch pseudocode for the ActionFinder module., emph=forward, ActionFinder,label=list:DECODE] class ActionFinder(torch.nn.Module): \"\"\" Inputs: d,l,b,vh,mul,mub astrometric phase space coordinates Output: Jddvals - the J” values Tddvals - the theta” values lossJddspread - loss due to the spread of the J” values \"\"\" def init(self,verbose=False): super().init() self.convinput2xv = convInput2xv(verbose=verbose) self.isochroneanalytic = IsochroneAnalytic(verbose=verbose) self.encoderGFG = EncoderGFG(verbose=verbose) self.encoderGFP = EncoderGFP(verbose=verbose) def forward(self,inputs): inputsxv = self.convinput2xv(inputs) convert d,l,b,vh,mul,mub to xv (J,Theta) in isochrone toy model Jiso, Tiso = self.isochroneanalytic(inputsxv, M, b) We now iterate to find J'.","Start with J' = mean(Jiso for each stream) Jdmean = torch.mean( Jiso, dim=1, keepdim=True) for iter in range(itermax): Jdtrial = Jdmean.clone() trial J' Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) expand mean value to fill NStars dimension TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) (T,J') GFG = self.encoderGFG(TJd) generating function G J-J' = d(G)/d(T): JmJd = jacobian(Tiso,GFG)[:,:,0,:].data remove from computational graph GFG.detach() remove from computational graph update Jdmean = torch.mean( Jiso - JmJd, dim=1, keepdim=True) if ( torch.max( torch.abs(Jdtrial - Jdmean)) < 5.e-5 and test for acceptable convergence torch.mean(torch.abs(Jdtrial - Jdmean)) < 1.e-6): break Find Jdmean once more, but now retain the computational graph Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) GFG = self.encoderGFG(TJd) JmJd = jacobian(Tiso,GFG)[:,:,0,:] Jdvals = Jiso - JmJd J' given by generating function G Jdmean = torch.mean( Jdvals, dim=1, keepdim=True) Calculate T' from generating function derivative Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) GFG = self.encoderGFG(TJd) T' = T + d(G)/d(J'): Tdvals = (Tiso + jacobian(Jdmeanfill,GFG)[:,:,0,:]) Add in the extra freedom of a point-transformation Tddvals = self.encoderGFP( Tdvals ) final T” from generating function P dTdddTd = jacobian(Tdvals,Tddvals) d(T”)/d(T') dTddTdd = torch.inverse( dTdddTd ) d(T')/d(T”) Ji” = dT'/dTi” .","J' Jddvals = torch.einsum('bsji,bsj->bsi',dTddTdd,Jdvals) final J” values Jddmean = torch.mean( Jddvals, dim=1, keepdim=True) Jddspread = Jddmean - Jddvals lossJddspread = torch.mean( torch.abs( Jddspread ) ) return Jddvals, Tddvals, lossJddspread [emphstyle=, caption=Pytorch pseudocode for the unit ResDense block., emph=forward, ResDenseblock,label=list:UNITBLOCK] class ResDenseblock(torch.nn.Module): \"\"\" The ResDense unit block.","This layer does not change the number of features. \"\"\"","def init(self, nunits): super().init() \"\"\" Declaration of the layers that will be used: nunits is the number of features. \"\"\"","self.nunits = nunits self.dense1 = torch.nn.Linear(infeatures=self.nunits, outfeatures= self.nunits, bias=False) self.WN1 = torch.nn.utils.weightnorm(self.dense1) self.dense2 = torch.nn.Linear(infeatures=self.nunits, outfeatures= self.nunits) self.WN2 = torch.nn.utils.weightnorm(self.dense2) def forward(self, input): \"\"\" The computational graph (how the above layers are used): \"\"\" xx = torch.relu(input) xx = self.dense1(xx) xx = self.WN1(xx) xx = torch.relu(xx) xx = self.dense2(xx) xx = self.WN2(xx) output = xx + input return output"]],"string":"[\n [\n \"The ACTIONFINDER: An unsupervised deep learning algorithm for\\n calculating actions and the acceleration field from a set of orbit segments\"\n ],\n [\n \"Abstract We introduce the \\\"ACTIONFINDER\\\", a deep learning algorithm designed to transform a sample of phase-space measurements along orbits in a static potential into action and angle coordinates.\",\n \"The algorithm finds the mapping from positions and velocities to actions and angles in an unsupervised way, by using the fact that points along the same orbit have identical actions.\",\n \"Here we present the workings of the method, and test it on simple axisymmetric models, comparing the derived actions to those generated with the Torus Mapping technique.\",\n \"We show that it recovers the Torus actions for halo-type orbits in a realistic model of the Milky Way to $\\\\sim 0.6$% accuracy with as few as 1024 input phase-space measurements.\",\n \"These actions are much better conserved along orbits than those estimated with the St\\\\\\\"ackel fudge.\",\n \"In our case, the reciprocal mapping from actions and angles to positions and velocities can also be learned.\",\n \"One of the advantages of the ACTIONFINDER is that it does not require the underlying potential to be known in advance, indeed it is designed to return the acceleration field.\",\n \"We expect the algorithm to be useful for analysing the properties of dynamical systems in numerical simulations.\",\n \"However, our ultimate goal with this effort will be to apply it to real stellar streams to recover the Galactic acceleration field in a way that is relatively agnostic about the underlying dark matter properties or the behavior of gravity.\"\n ],\n [\n \"Introduction\",\n \"Actions and angles are the natural phase-space coordinates to represent the state of the constituents of an integrable, or close to integrable dynamical system, such as typical Galactic potentials [3].\",\n \"In these coordinates, motion along an orbit is very simple, as the actions ${J}$ are preserved, while their canonically conjugate angles ${\\\\theta }$ advance through a cycle at a uniform rate in time (with corresponding fixed frequencies ${\\\\Omega }$ ).\",\n \"These are also the natural coordinates for perturbation theory [15].\",\n \"Furthermore, the actions are adiabatic invariants, so slow changes in the evolution of the parent system will preserve these quantities.\",\n \"With the superb astrometry being gathered by the Gaia mission [10], it has now become possible to study the real six-dimensional phase space structure of our Galaxy in unprecedented detail.\",\n \"Action-angle variables are being used extensively in this exploration [5], [27], [22], [29], [6], [23], notably because actions can be used to construct equilibrium distribution functions from the Jeans theorem, because they provide (in principle) the best “archaeological” information on the dynamics of the Galaxy, and because they are convenient for exploring perturbations from, for example, the Galactic bar [19], [20], [17], [26].\",\n \"While actions and angles are thus undoubtedly useful, they are not always easy to calculate.\",\n \"Indeed, the only potentials in which the actions are expressed analytically are those of from the family of isochrone models of which the Kepler and spherical harmonic potentials are special cases.\",\n \"Although this model is very useful, it does not provide a good approximation to interesting stellar systems (other than those dominated by a central object).\",\n \"Substantial theoretical efforts were therefore undertaken over the past decades to calculate approximations for the transformation from position $x$ and velocity $v$ to $J$ and $\\\\theta $ , and conversely, for more general and realistic galactic potentials.\",\n \"[18] developed a method known as the Torus Mapping [2], starting from action-angle coordinates in an isochrone potential, close enough but different from the true potential.\",\n \"Their insight was to search for a generating function in order to transform from the true actions and angles to those of the isochrone.\",\n \"This generating function is expressed as a Fourier series expansion on the isochrone angles, the fitted Fourier coefficients being such that the true Hamiltonian remains, after the associated canonical transformation, constant for a set of true actions.\",\n \"Once this generating function is found, the transformation from actions and angles to positions and velocities is known.\",\n \"For the reciprocal transformation, one usually relies on separable potentials, denoted Stäckel potentials [7], [9], for which three exact integrals of the motion exist.\",\n \"These potentials are best expressed in spheroidal coordinates, associated to a focal distance directly related to the first and second derivatives of the Stäckel potential.\",\n \"This focal distance can be computed for the true potential at any configuration space point as if the potential was a Stäckel one, and the corresponding integrals of the motion and respective actions can be evaluated.\",\n \"This method introduced by [1] is known as the “Stäckel fudge\\\".\",\n \"Unfortunately, this transformation is not the exact inverse of the one going from actions and angles to positions and velocities as obtained from the Torus Mapping.\",\n \"All these methods are reviewed in [25], and have all been implemented in the AGAMA dynamics package [28], [30].\",\n \"Our aim in the present work is to provide an innovative calculation of action-angle coordinates, a method which would be able to jointly determine, from a sample of segments of orbits in positions and velocities, (i) the corresponding actions and angles, as well as (ii) the true acceleration field in which these orbits reside.\",\n \"Moreover, (iii) the inverse transformation from actions and angles to positions and velocities should be determined easily.\",\n \"To this end we will build a deep neural network that will try to learn on its own, and in an unsupervised way, the coordinate transformation from observables into $J$ and $\\\\theta $ for the particular dataset under study.\",\n \"Figure: Sketch of the method.\",\n \"We consider a dynamical system for which we have measured the kinematics of NN particles or “stars” along each of a sample of SS different orbits which we loosely refer to as “streams”.\",\n \"In panel (a) we have sketched two such “streams”, each with four “stars”.\",\n \"The observable sky positions, parallaxes (or distances) as well as the radial velocities and proper motions are first converted by the algorithm into the Cartesian system ( i n x, i n y)({}_{i}^{n}{x}, {}_{i}^{n}{y}), where the subscript denotes the stream number and the superscript denotes the star number in that stream.\",\n \"The Cartesian coordinates are converted into action-angle coordinates of an analytic isochrone model (b), which serves as a “toy” first approximation to the real actions.\",\n \"The central ingredient of the algorithm is the network that proposes trial generating functions G(θ,J ' )G({\\\\theta }, {J^{\\\\prime }}) of the canonical transformation between the toy coordinates (θ,J)({\\\\theta }, {J}) and the refined coordinates (θ ' ,J ' )({\\\\theta ^{\\\\prime }}, {J^{\\\\prime }}).\",\n \"Note that because J ' {J^{\\\\prime }} is not known in advance, we need to iterate at this stage (c) to find the J ' {J^{\\\\prime }} values that are consistent with the isochrone's values of J{J}.\",\n \"Finally (d), we also allow for a simpler canonical point-transformation with the generating function P(θ ' )P({\\\\theta ^{\\\\prime }}) to calculate the target coordinates (θ '' ,J '' )({\\\\theta ^{\\\\prime \\\\prime }}, {J^{\\\\prime \\\\prime }}).\",\n \"The two networks (green arrows) are refined over the course of the training procedure by attempting to find the network weights that minimize the spread in the J '' {J^{\\\\prime \\\\prime }} values in each stream.\"\n ],\n [\n \"The Algorithm\",\n \"We designed the algorithm around the fact that different phase space points along an orbit have identical $J$ (and frequencies $\\\\Omega $ ).\",\n \"For this to be useful, we imagine having access to groups of such phase space points along the same orbit.\",\n \"This could be the case if we had the output of a numerical N-body simulation at a series of different time steps, or alternatively we might have such information from observations of a real dynamical system.\",\n \"In the code we refer to a group of such phase space points along the same orbit as a “stream”, while each individual phase space point is referred to as a “star”.\",\n \"Our use of the terms “streams” and “stars” in the software obviously betrays our ultimate motive for building it, but we note here that star streams do not precisely follow orbits, and that a significant amount of further work is required to adapt the present algorithm to properly model such structures.\",\n \"It is relatively straightforward to devise an algorithm to implement the coordinate mapping from Cartesian $\\\\eta \\\\equiv (x, y, z, v_x, v_y, v_z)$ to $\\\\xi \\\\equiv (\\\\theta _1, \\\\theta _2, \\\\theta _3, J_1, J_2, J_3)$ with a deep neural network for a system for which we have a set of known $\\\\eta $ and $\\\\xi $ values.\",\n \"(This is what we construct for the inverse transformations in Section ).\",\n \"In this case of so-called “supervised learning”, the network calculates the derivatives of the loss function with respect to the system parameters (the millions of weights in the neural net), and adapts these parameters using a stochastic gradient descent algorithm (in our work we use the “Adam” optimization algorithm, [16]) so as to mimimize the difference between the predicted actions and angles and the corresponding “ground truth” values.\",\n \"The learned map can subsequently be used to attribute action and angle values to data not previously seen by the network.\",\n \"However, for the forward transformation in the present work we do not wish to rely on external software to provide us with a training dataset of action-angle coordinates in galaxy models of interest.\",\n \"The fact that nature does not provide us with ground-truth action and angle coordinate labels provides the incentive to be able to calculate these quantities without reference to a training set.\",\n \"The potentials of galaxies probably also do not follow simple mathematical forms, so we have a further motivation to be able to calculate the canonical transformation in a model-independent way.\",\n \"How then can the stochastic gradient algorithm be steered towards the right solution automatically without labeled data?\",\n \"As stated above, we know that points along the same orbit should have identical actions and frequencies.\",\n \"In addition, the map from $\\\\eta $ to $\\\\xi $ obeys the symplectic condition for the Jacobian matrix: $M^T \\\\mathbb {J} M = \\\\mathbb {J} ,$ where $M$ is the symplectic Jacobian matrix $M_{ij} = {{\\\\partial \\\\eta _i}/{\\\\partial \\\\xi _j}}$ , and $\\\\mathbb {J}$ is the antisymmetric block matrix $\\\\mathbb {J} =\\\\begin{pmatrix}0 & \\\\mathbb {I}_3\\\\\\\\-\\\\mathbb {I}_3 & 0\\\\end{pmatrix} \\\\, ,$ with $\\\\mathbb {I}_3$ being the $3\\\\times 3$ identity matrix.\",\n \"This makes it tempting to simply build a deep network that directly maps $\\\\eta $ to $\\\\xi $ , while including the above conditions as terms in the loss function (together with additional conditions that will be presented in Section ).\",\n \"We were disappointed to find that this did not work very well.\",\n \"In order to approximate the coordinate transformation to $\\\\sim 1$ % accuracy or better, one requires a deep neural net with tens of millions of parameters.\",\n \"Yet the symplectic condition creates a very complex loss function landscape in this parameter space, which the stochastic gradient descent optimizer explores in a very inefficient way.\",\n \"To overcome this problem, we decided to make use of a generating function for the canonical transformation, to guarantee that the coordinate transformation will be symplectic.\",\n \"As pioneered by [18], the transformation can be made much simpler by using the action and angle coordinates of a “toy” model as a stepping stone.\",\n \"Our procedure is sketched in Fig.\",\n \"REF .\",\n \"We begin by converting the observed astrometric data into Cartesian coordinates $({x},{v})$ .\",\n \"It is worth noting that thanks to the automatic differentiation module in the pytorch [21] machine learning library, the gradients of the output quantities with respect to the astrometric observables can be calculated easily; we will make use of this feature in future work to account for observational uncertainties.\",\n \"For the toy model, we use an isochrone potential: $\\\\Phi (r) = - {{G M}\\\\over {b + \\\\sqrt{b^2+r^2}}} \\\\, .$ Here $r$ is the (spherical) radius coordinate, $G$ is the universal gravitational constant, and the two model parameters are the mass $M$ and scale radius $b$ .\",\n \"One may then choose the three actions in the isochrone model to be: $\\\\begin{split}J_{1, \\\\rm iso} &= L_z\\\\\\\\J_{2, \\\\rm iso} &= L - |L_z|\\\\\\\\J_{3, \\\\rm iso} &= {{G M}\\\\over {\\\\sqrt{-2 E}}} - {{1}\\\\over {2}} \\\\Bigg ( L + {{1}\\\\over {2}} \\\\sqrt{L^2 - 4 G M b} \\\\Bigg ) \\\\, ,\\\\end{split}$ where $L$ is the angular momentum of the particle, $L_z$ is the $z$ -component of angular momentum, and $E$ is the particle's energy.\",\n \"The procedure to convert from $({x},{v})$ to the isochrone model's $({\\\\theta },{J})$ is detailed in [18], and our pytorch version is heavily inspired by the galpy isochrone implementation [4].\",\n \"[18] found that their algorithm converged irrespective of the initial chosen values of the two isochrone parameters $M$ and $b$ , although values closer to the real system produced faster convergence.\",\n \"Our algorithm does not appear to be very sensitive to this initial choice either, as long as $M$ is set high enough that all orbits are bound.\",\n \"In our algorithm both the $M$ and $b$ parameters can be fitted by pytorch in the stochastic gradient descent procedure, or if desired, they can be held fixed at their initial values.\",\n \"We now aim to find new coordinates $({\\\\theta ^{\\\\prime }},{J^{\\\\prime }})$ that are closer to those of the real system.\",\n \"To this end we define an indirect type 2 generating function $G=G({\\\\theta },{J^{\\\\prime }})$ , whose derivatives give the implicit transformation: ${J} = \\\\, {J^{\\\\prime }} + {{\\\\partial G}\\\\over {\\\\partial {\\\\theta }}} \\\\\\\\{\\\\theta ^{\\\\prime }} = \\\\, {\\\\theta } + {{\\\\partial G}\\\\over {\\\\partial {J^{\\\\prime }}}} \\\\, .\",\n \"$ A deep learning network will be used to propose trial variations on $G$ , and again thanks to the automatic differentiation in pytorch it is straightforward to find the ${J}$ and ${\\\\theta ^{\\\\prime }}$ values generated by $G$ .\",\n \"Since we are actually interested in ${J^{\\\\prime }}$ , we iteratively find the ${J^{\\\\prime }}$ value that yields values of ${J}$ from Eqn.\",\n \"REF that are the same as those of the toy isochrone.\",\n \"We now finally update the $({\\\\theta ^{\\\\prime }},{J^{\\\\prime }})$ coordinates using a further transformation ${\\\\theta ^{\\\\prime \\\\prime }}=P({\\\\theta ^{\\\\prime }})$ that is only dependent on the updated angles.\",\n \"This point-transformation is much simpler than $G$ , as we can obtain the updated angles and actions directly with no need for an iterative procedure, ${\\\\theta ^{\\\\prime \\\\prime }} = \\\\, P({\\\\theta ^{\\\\prime }}) \\\\\\\\{J^{\\\\prime \\\\prime }} = \\\\, {{\\\\partial {\\\\theta ^{\\\\prime }}}\\\\over {\\\\partial {\\\\theta ^{\\\\prime \\\\prime }}}} {J^{\\\\prime }} \\\\, ,$ and is therefore easy for the algorithm to fit.\",\n \"As demonstrated by [14], such a transformation allows for deformations of the toy model that may be better adapted to the geometry of the real system's orbits.\",\n \"In our experiments we found that this additional network improved the quality of the predicted actions by up to $\\\\sim 50$ %.\",\n \"Python style pseudocode for the central function of the algorithm is shown in the Appendix in Listing .\",\n \"Figure: Sketch of the (shorter) net P {\\\\rm net}_P neural network.\",\n \"The ResNet-like unit blocks (left, and Eqn.\",\n \"), are incorporated as a series of layers (right) that progressively increase in complexity up to a chosen maximum width, and then decrease symmetrically up to the layer immediately before the output.\",\n \"The number of features nf {\\\\rm nf} in each unit block is indicated.\"\n ],\n [\n \"The Action-Angle Network\",\n \"All of the neural networks used here have the same basic architecture.\",\n \"Thus, the generating function $G$ is given by: $G = {\\\\rm net}_G(\\\\cos ({\\\\theta }),\\\\sin ({\\\\theta }), {J^{\\\\prime }}) \\\\, , $ while the canonical point-transformation $P$ has the form: $P = {\\\\rm net}_P(\\\\cos ({\\\\theta }),\\\\sin ({\\\\theta })) \\\\, .\",\n \"$ After extensive experimentation, we found that the angle coordinates were best learned by first introducing the pair of auxiliary variables $t_{x}\\\\equiv \\\\cos (\\\\theta )$ and $t_{y}\\\\equiv \\\\sin (\\\\theta )$ , as used in Eqns.\",\n \"REF and REF .\",\n \"This naturally takes care of the cyclic property of the angle variables.\",\n \"The ${\\\\rm net}_G$ network consists of a series of 15 blocks similar to a Residual neural network (ResNet, [12]), adapted for the case of Dense (Linear) layersA linear layer $W$ with bias $b$ simply transforms an input vector $x$ to $y=x W^T+b$ .\",\n \"The parameters $W$ and $b$ are learned by the algorithm.. An initial fully-connected linear layer takes the input features (9 in the case of ${\\\\rm net}_G$ and 6 in the case of ${\\\\rm net}_P$ ) and passes them onto a layer of 64 features.\",\n \"Deeper layers increase in width, becoming a factor of 2 wider in the number of features per layer, up to a maximum of 1024 features per layer.\",\n \"After the chosen maximum depth is reached (at layer 6), the layers decrease in size in a symmetric way.\",\n \"A final fully-connected linear layer is applied to give the chosen output features, which in the case of the generating functions $G$ and $P$ is a scalar value.\",\n \"This architecture is sketched in Fig.\",\n \"REF .\",\n \"The ${\\\\rm net}_P$ network is identical to ${\\\\rm net}_G$ , but consists of only 11 blocks.\",\n \"The unit blocks were constructed as follows: $\\\\begin{split}{\\\\rm out} = W_2(& D_2( {\\\\rm ReLU}( \\\\\\\\W_1(& D_1( {\\\\rm ReLU}( {\\\\rm input} ))) \\\\,\\\\, ))) + {\\\\rm input}\\\\end{split}$ where $D_1$ and $D_2$ are fully connected linear layers, ${\\\\rm ReLU}(x)=\\\\max (0,x)$ is the Rectified Linear Unit activation function, and $W_1$ and $W_2$ are Weight Normalisation layers [24].\",\n \"This setup is very similar to the canonical ResNet, designed so that the blocks learn successive corrections to the input information.\",\n \"We show the pseudo-code of this module in Listing in the Appendix.\",\n \"Each unit block is connected to the following block with a Dense layer, without applying an activation function.\",\n \"Our use of Weight Normalisation layers in Eqn.\",\n \"REF is noteworthy.\",\n \"Most modern networks in computer vision make use of the “Batch Normalization” (BN, [13]) procedure to decouple as much as possible the fitting of the parameters in the different layers of a network.\",\n \"This mitigates against the co-variance between layers, allowing the parameters of a deep layer to be refined even though the parameters of the higher layers are also being adjusted at the same time by the optimization algorithm.\",\n \"However, after much experimentation, we found that the BN layers we initially used were limiting the accuracy of our algorithm.\",\n \"This is due to the fact that this procedure operates (during training) on the data presented to it in each “batch” (i.e.\",\n \"in small sub-samples which are chosen so that the data may fill the graphics card memory), from which it normalizes the data using the mean and standard deviation of the sub-sample.\",\n \"Since the data at different positions within the batch belong to different streams, the unavoidable shot noise then creates substantial variation between batches, resulting in unacceptably large errors for a study such as ours.\",\n \"We devised a work-around by training in the normal way with the BN procedure until it reached an equilibrium state, and then restarting with the BN layers frozen using parameter values calculated from the whole dataset.\",\n \"We later realised that the “Weight Normalization” scheme gives similar accuracy to our BN “hack”, and have adopted it for the present work as it is an accepted machine learning method.\",\n \"Between them, with the chosen depth of 15 and 11 layers, the two generating function networks have a total of $\\\\sim 44$ M free parameters.\",\n \"The chosen loss function is very simple: $L = L_{J^{\\\\prime \\\\prime }, \\\\, \\\\rm spread} + L_{J_{2,3}^{\\\\prime \\\\prime }>0} + \\\\alpha _1 L_{{\\\\theta _0}^{\\\\prime \\\\prime }} \\\\, .$ $L_{J^{\\\\prime \\\\prime }, \\\\, \\\\rm spread}$ is the mean absolute deviation of the difference between the predicted action $J^{\\\\prime \\\\prime }$ of the stars in a stream and the mean action of that stream $\\\\langle J^{\\\\prime \\\\prime }\\\\rangle $ .\",\n \"The term $L_{J_{2,3}^{\\\\prime \\\\prime }>0}$ penalises unphysical negative values of ${J_2}^{\\\\prime \\\\prime }$ and ${J_3}^{\\\\prime \\\\prime }$ , as follows: $L_{J_{2,3}^{\\\\prime \\\\prime }>0} = \\\\langle | J_2-|J_2| | \\\\rangle + \\\\langle | J_3-|J_3| | \\\\rangle \\\\, .$ Finally, the loss term $\\\\alpha _1 L_{{\\\\theta _0}^{\\\\prime \\\\prime }}$ is the mean absolute deviation of the target angles ${\\\\theta ^{\\\\prime \\\\prime }}$ when $\\\\theta =0$ is fed into the network (i.e.\",\n \"it encourages the zero-point of the target angles to coincide with those of the isochrone toy model).\",\n \"Following common machine learning practise, we normalised the position and velocity variables (to $20{\\\\rm \\\\,kpc}$ and $200{\\\\rm \\\\,km\\\\,s^{-1}}$ , respectively); thus the actions are normalized to $4000 {\\\\rm \\\\,km\\\\,s^{-1}}{\\\\rm \\\\,kpc}$ .\",\n \"However, for convenience, we left the angle variables in radians.\",\n \"We chose to set the hyper-parameter to $\\\\alpha _1=0.1$ in Eqn.\",\n \"REF , so as compensate for this difference in the range of the angle and action variables.\",\n \"The algorithm proceeds by iterating over discrete “epochs” when the network derivatives are calculated and the parameters are consequently updated to obtain improved estimates for $J^{\\\\prime \\\\prime }$ and $\\\\theta ^{\\\\prime \\\\prime }$ .\"\n ],\n [\n \"The acceleration network\",\n \"After the algorithm has converged on values of ${J^{\\\\prime \\\\prime }}$ and ${\\\\theta ^{\\\\prime \\\\prime }}$ that minimize the spread of ${J^{\\\\prime \\\\prime }}$ in the streams, we can use these transformed coordinates to study the system they inhabit.\",\n \"To this end, we re-run the network once more with an updated loss function, and with an additional network (the acceleration network) turned on.\",\n \"The structure of the acceleration network is set up as an additive correction to the acceleration of the toy isochrone model ${a_{\\\\rm iso}}$ : $a = {\\\\rm net}_{a}(x, y, z, {a_{\\\\rm iso}}) + a_{\\\\rm iso} \\\\, .$ Note that here we only feed in the position part of the $\\\\eta $ phase space vector.\",\n \"We use a depth of 15 blocks for this network, which contains $\\\\sim 17.5$ M free parameters.\",\n \"The structure of ${\\\\rm net}_{a}$ is essentially identical to that of ${\\\\rm net}_{G}$ , except that it outputs a vector.\",\n \"If ${J^{\\\\prime \\\\prime }}$ and ${\\\\theta ^{\\\\prime \\\\prime }}$ are only functions of position ${x}$ and velocity ${v}$ , the chain rule implies that the mapping is constrained by the following relations: ${{d {J_i}^{\\\\prime \\\\prime }}\\\\over {d t}} &= {{\\\\partial {J_i}^{\\\\prime \\\\prime }}\\\\over {\\\\partial x_j}} \\\\dot{x}_j + {{\\\\partial {J_i}^{\\\\prime \\\\prime }}\\\\over {\\\\partial \\\\dot{x}_j}} \\\\ddot{x}_j = 0\\\\\\\\{{d {\\\\theta _i}^{\\\\prime \\\\prime }}\\\\over {d t}} &= {{\\\\partial {\\\\theta _i}^{\\\\prime \\\\prime }}\\\\over {\\\\partial x_j}} \\\\dot{x}_j + {{\\\\partial {\\\\theta _i}^{\\\\prime \\\\prime }}\\\\over {\\\\partial \\\\dot{x}_j}} \\\\ddot{x}_j = {\\\\Omega _i}^{\\\\prime \\\\prime } \\\\, ,$ where the $i$ index here is associated to each of the (usually three) integrals of motion $J_i$ , and $j$ is a dummy index denoting Einstein summation.\",\n \"A brief inspection of Eqn.\",\n \"REF reveals that once we have access to a network that delivers predictions for the position and velocity derivatives of $J^{\\\\prime \\\\prime }$ , we can solve the simultaneous equations to calculate the acceleration terms $\\\\ddot{x}$ .\",\n \"While this suggests that the acceleration network (Eqn.\",\n \"REF ) is superfluous, we nevertheless opted to employ a network to calculate $a$ for two main reasons.\",\n \"First, it allows us to train a network that can be subsequently applied to make predictions for the acceleration on new unseen data (for instance, for “stars” without full 6-dimensional information).\",\n \"Second, we found that solving Eqn.\",\n \"REF for $\\\\ddot{x}$ using linear algebra can occasionally be problematic, due to ill-conditioned matrices.\",\n \"Creating a separate acceleration network eliminates these problems.\",\n \"In this second run through the algorithm, we choose the following loss function: $\\\\begin{split}L =& L_{dJ^{\\\\prime \\\\prime }/dt} + \\\\alpha _2 L_{\\\\rm symmetry} + \\\\alpha _3 L_{\\\\Omega ^{\\\\prime \\\\prime }, \\\\, \\\\rm spread} \\\\, ,\\\\end{split}$ where $L_{dJ^{\\\\prime \\\\prime }/dt}$ is the mean absolute value of $dJ^{\\\\prime \\\\prime }_i/dt$ , as calculated from Eqn.\",\n \"REF .\",\n \"$L_{\\\\rm symmetry}$ is an optional term to enforce a desired symmetry on the solution of the acceleration network.\",\n \"For instance for axisymmetric potential models, we experimented with a cosine anti-similarity criterion $L_{\\\\rm symmetry} = 1 + \\\\vec{a}_R \\\\cdot \\\\vec{R}/(||\\\\vec{a}_R|| \\\\, ||\\\\vec{R}||)$ between the cylindrical-$R$ component of the acceleration field $\\\\vec{a}_R$ and the same component of the (Galactocentric) position vector $\\\\vec{R}$ .\",\n \"Although this loss term helps to constrain the acceleration, for the purpose of the tests presented in Section we decided to suppress this symmetry constraint on $a$ by setting $\\\\alpha _2=0$ .\",\n \"Finally, $L_{\\\\Omega ^{\\\\prime \\\\prime }, \\\\, \\\\rm spread}$ is another optional loss term to ensure that the frequencies calculated from Eqn.\",\n \"for each “stream” have minimum spread.\",\n \"This is done by calculating the mean absolute difference of the frequency of each “star” from the mean of the frequencies in the “stream”.\",\n \"However, we found in our tests that including this term introduced slight additional scatter in the solutions (most likely because angles are much less accurately constrained than actions in our algorithm), so it too was suppressed by setting $\\\\alpha _3=0$ in the present work.\"\n ],\n [\n \"Inverse Transformation\",\n \"Deep learning can also be used to calculate the inverse transformation from actions and angles back to positions and velocities.\",\n \"However, there are many different situations that one can envisage being confronted with.\",\n \"For instance, one may or may not have frequencies in addition to the actions and angles, or one may already know the potential.\",\n \"Alternatively, it is possible that the potential and frequencies are not known, but one may have groups of particles along orbits (i.e.\",\n \"our “stars??\",\n \"in “streams??).\",\n \"Each of these cases would require a different network to be constructed in order to learn the inverse transformation in an unsupervised way.\",\n \"The approach we take here is slightly different, but perhaps more realistic given the forward transformation we have presented previously.\",\n \"We imagine having derived the forward transformation from $\\\\eta $ to $\\\\xi $ for a number of orbits.\",\n \"This information can now be used to train a supervised network to predict $\\\\eta $ from $\\\\xi $ .\",\n \"Since this is a supervised learning task, compared to the previous problem, it is a much easier for the network to find the correct direction for the stochastic gradient descent to improve the loss function.\",\n \"We therefore choose a simple network architecture: $(t_x,t_y,J) = {\\\\rm net}_{\\\\rm inverse}(t_x^{\\\\prime \\\\prime },t_y^{\\\\prime \\\\prime },J^{\\\\prime \\\\prime }) \\\\, ,$ from which the toy model angles are calculated with $\\\\theta =\\\\operatorname{atan2}(t_y,t_x) + \\\\pi /2$ .\",\n \"Just like the forward network, the inverse transformation thus employs the isochrone model as a stepping stone, with the final step being the analytic inversion of the toy model's $(\\\\theta ,J)$ coordinates to the 6-dimensional vector $\\\\eta $ .\",\n \"Since this inverse network is not constrained by differential equations (and hence does not require any Jacobians to be calculated) it is feasible to implement it as a deeper network of 23 (or more if necessary) layers of blocks (Eqn.\",\n \"REF ), using a total of $\\\\sim 34$ M free parameters.\"\n ],\n [\n \"Results\",\n \"The algorithm was constructed to accept as input data a set of $S$ “streams”, each with $N$ “stars”.\",\n \"The method works with $N\\\\ge 2$ “stars” per “stream”, but for these initial tests we have chosen a more conservative $N=8$ , simply to make it easier for the algorithm to verify that the actions and frequencies calculated for each “stream” are constant.\",\n \"Pytorch achieves its speed by processing the data in parallel, and so it is much more efficient to pass pytorch “tensors” of the same size onto the graphics processing unit (GPU).\",\n \"Because of this, we expect that when the method is applied to real data, it will be convenient to break up streams with a large number of known members into smaller sub-groups (of the same size $N$ ).\",\n \"In all the experiments below, we split the input sample into two sets, a training set with 50% and a test set with 50% of the sample.\",\n \"Clearly, with real data we would be much more sparing with the fraction allocated to the test set!\",\n \"Typically, we run the training process with a learning rate of $10^{-4}$ .\",\n \"In all the tests reported below we iterate for 1024 “epochs” even though the training losses often stabilize much earlier.\",\n \"We found that re-running the network with smaller learning rates (which is the standard procedure in machine learning to improve accuracy), resulted in only very marginal improvements to the loss function values.\",\n \"Test samples are used in machine learning primarily to verify that the training procedure is not overfitting the data.\",\n \"When this occurs, the loss of the training set continues to improve, while the loss values in the test set (which the algorithm does not see during training) become worse.\",\n \"We simply ignore all further epochs once the algorithm begins to overfit.\",\n \"Both the data and network are expressed internally as double precision floating-point numbers, as we found that the Jacobian matrices were not always calculated to sufficient precision with single precision numbers, resulting in a network that would not update correctly due to vanishing gradients.\",\n \"Because of the fact that we find the actions iteratively starting from an initial guess provided by the toy model, the speed of the algorithm depends on how close the toy model is to the target system.\",\n \"But in typical cases using $S=1024$ and $N=8$ , the algorithm takes $\\\\sim 3$ hours to complete a training run of 1024 epochs on an NVIDIA GV100 GPU with 32 GB of card memory.\",\n \"Running times on larger datasets should scale approximately linearly with the number of data.\"\n ],\n [\n \"Fitting isochrone models\",\n \"Although we have built the ACTIONFINDER as a series of transformations from the analytic isochrone, it is still worth checking whether the software can fit a sample of orbital points drawn from different models of this family.\",\n \"To this end we generated orbits within an isochrone potential with $M=3.334\\\\times 10^{11}{\\\\rm \\\\,M_\\\\odot }$ and $b=5{\\\\rm \\\\,kpc}$ (i.e.\",\n \"$M=1.5\\\\times 10^6$ in N-body units where $G=1$ and distances are in ${\\\\rm \\\\,kpc}$ and time is in ${\\\\rm \\\\,Gyr}$ ), which gives a circular velocity at the Solar neighborhood ($R_\\\\odot =8.122{\\\\rm \\\\,kpc}$ , [11]) of $v_c(R_\\\\odot )=217{\\\\rm \\\\,km\\\\,s^{-1}}$ .\",\n \"One of the advantages of the ACTIONFINDER is that we do not need to provide it with a fair sample of orbits within the system, almost any sample that covers the region of interest will do.\",\n \"In the following, we imagine having access to the orbits of some objects in the “Galaxy” halo.\",\n \"To model this situation we select a random 3-dimensional initial radius drawn uniformly in Galactocentric distance between $r=[6$ –$16]{\\\\rm \\\\,kpc}$ , together with a velocity vector that is oriented randomly (i.e.\",\n \"isotropic) with magnitude drawn from a Gaussian of dispersion $150{\\\\rm \\\\,km\\\\,s^{-1}}$ .\",\n \"Using a symplectic Leapfrog scheme, we integrate from these initial phase space locations for $0.1{\\\\rm \\\\,Gyr}$ , ensuring energy conservation to 1 part in $10^7$ , and randomly select $N=8$ phase space points along each path.\",\n \"Since our aim is to eventually work with real astrometric data, we convert the positions and velocities of the set of orbit locations to the observable quantities: $d, \\\\ell , b, v_h, \\\\mu _\\\\ell , \\\\mu _b$ , as they would be measured from our vantage point in the Galaxy, where $d$ is Heliocentric distance, ($\\\\ell , b$ ) are Galactic coordinates, $v_h$ is the Heliocentric radial velocity, and $\\\\mu _\\\\ell $ and $\\\\mu _b$ are the proper motions along the Galactic coordinate directions.\",\n \"With the correct input values of $M$ and $b$ , the network quickly finds the correct $J^{\\\\prime \\\\prime }$ and $\\\\theta ^{\\\\prime \\\\prime }$ to better than 0.01% (with a training set size of $S=1024$ ), which is reassuring given that the algorithm effectively just has to learn (in an unsupervised way) the identity operation.\",\n \"A more interesting case occurs when we try to fit the isochrone “stream” sample above using fixed and incorrect reference $M$ and $b$ values.\",\n \"With $M$ and $b$ both 10% lower (higher) in the toy model compared to the simulated data, using $S=1024$ we obtain action errors of $\\\\delta J=1.2 (2.0) {\\\\rm \\\\,km\\\\,s^{-1}}{\\\\rm \\\\,kpc}$ and angle errors of $0.7^\\\\circ (1.2^\\\\circ )$ .\",\n \"When $M$ and $b$ are 20% higher, the action error is $\\\\delta J=3.0 {\\\\rm \\\\,km\\\\,s^{-1}}{\\\\rm \\\\,kpc}$ and the angle error is $1.4^\\\\circ $ .\",\n \"The algorithm fails with a fixed 20% lower value of $M$ , because some orbits are unbound in the toy model, and thus do not have valid $(\\\\theta ,J)$ coordinate values.\",\n \"Using $S=1024$ with fixed 20% higher values of $M$ and $b$ in the toy model, the inverse transformation from $(\\\\theta ^{\\\\prime },J^{\\\\prime })$ to $(x,v)$ is recovered to an accuracy of $\\\\delta x=0.3{\\\\rm \\\\,kpc}$ and $\\\\delta v=2.6{\\\\rm \\\\,km\\\\,s^{-1}}$ (again, mean absolute deviation errors).\"\n ],\n [\n \"Fitting a realistic Galaxy model\",\n \"[8] fitted plausible axisymmetric density models of the main Galactic components to kinematic observations of the Milky Way to derive the potential of the Galaxy.\",\n \"These useful models have been incorporated into the AGAMA package [30], which now allows one to transform the actions and angles of test particles in these potentials to positions and velocities with the Torus Mapping.\",\n \"We created a set of random orbits (each with fixed triplet of actions) in the [8] model `1' in a similar way to that described in Section REF , and as before we selected $N=8$ random points along the orbits, recording their input actions and angles values, their $(x, v)$ values generated with the Torus Mapping, as well as their output $(\\\\theta ^{\\\\prime \\\\prime }, J^{\\\\prime \\\\prime })$ values found by ACTIONFINDER.\",\n \"For these tests the toy isochrone $M$ and $b$ parameters are left free.\",\n \"Using $S=1024$ , the ACTIONFINDER recovers the Torus Mapping input action-angle coordinates with an error of $\\\\delta J^{\\\\prime \\\\prime }=8.0{\\\\rm \\\\,km\\\\,s^{-1}}{\\\\rm \\\\,kpc}$ in action (i.e.\",\n \"$\\\\sim 0.4$ % of the Sun's action) and $\\\\delta \\\\theta ^{\\\\prime \\\\prime }=7.4^\\\\circ $ in angle.\",\n \"The model accelerations are also recovered to $\\\\delta a=2.6$ %.\",\n \"These errors are calculated as the mean absolute deviation between the Torus Mapping input values and the ACTIONFINDER predictions.\",\n \"With these noise-less data (possessing only shot-noise), the derived $(\\\\theta ^{\\\\prime \\\\prime },J^{\\\\prime \\\\prime })$ are not very sensitive to the sample size.\",\n \"If instead we set $S=128 (16)$ , we obtain an action error of $\\\\delta J^{\\\\prime \\\\prime }=11.2 (18.1){\\\\rm \\\\,km\\\\,s^{-1}}{\\\\rm \\\\,kpc}$ and the angle error is $\\\\delta \\\\theta ^{\\\\prime \\\\prime }=8.0^\\\\circ $ ($9.1^\\\\circ $ ).\",\n \"In Fig.\",\n \"REF we show the evolution of the loss function in these three tests.\",\n \"As a comparison, we also transformed back the positions and velocities of each particle into action-space with the Stäckel-fudge in AGAMA.\",\n \"In this case, the mean absolute deviation between the Torus Mapping input actions and the Stäckel-fudge estimate is $44 {\\\\rm \\\\,km\\\\,s^{-1}}{\\\\rm \\\\,kpc}$ , 5 times less accurate than with the ACTIONFINDER.\",\n \"Figure: Convergence of the loss function for different sample sizes.\",\n \"All three tests use simulated orbits in the potential model `1', each with N=8N=8 phase-space points.\",\n \"The sample sizes are S=1024S=1024, S=128S=128 and S=16S=16 for the top, middle and bottom panels, respectively.\",\n \"In each case, the blue line shows the training loss while the orange line shows the validation loss.\",\n \"The red dot shows the position of the best validation loss, which defines the epoch at which the (θ '' ,J '' )(\\\\theta ^{\\\\prime \\\\prime },J^{\\\\prime \\\\prime }) coordinates are extracted.With $S=1024$ the inverse transformation algorithm (which is basically the same transformation as the one done with the Torus Mapping to generate positions and velocities) recovers the $(x,v)$ originally generated with the Torus Mapping to an accuracy of $\\\\delta x=0.24{\\\\rm \\\\,kpc}$ and $\\\\delta v=4.4{\\\\rm \\\\,km\\\\,s^{-1}}$ .\"\n ],\n [\n \"Discussion and Conclusions\",\n \"Using modern deep learning tools we have constructed a method to transform observable astrometric coordinates into action and angle coordinates with a scheme that is effectively a back-to-front version of that developed by [18], together with the canonical point-transformation improvement by [14].\",\n \"The deep neural nets used here are able to generate non-linear functions that are more flexible than a linear decomposition into Fourier coefficients, and this flexibility may make it easier to fit more general dynamical systems.\",\n \"A further advantage of our method is that it simultaneously uses all the data at its disposal to fit the generating functions of the canonical transformations.\",\n \"This contrasts to earlier methods where each orbit is fitted independently to derive the best set of Fourier coefficients for that orbit, bearing in mind that each such orbit requires many constraining data points due to the large number of Fourier coefficients that have to be fit.\",\n \"Moreover, once the transformation from positions and velocities is found, it is easy to inverse it in a supervised way.\",\n \"The main advantage of the method, however, is that we do not need to know in advance the Hamiltonian or potential of the system under investigation.\",\n \"The algorithm finds it for us.\",\n \"The data used in the tests presented here are noise free (apart from shot noise), and as such they do not portray a realistic picture of the limitations of the method as applied to real data.\",\n \"However, it is beyond the scope of the present contribution to attempt an exploration of the consequences of the observational limitations.\",\n \"With this caveat in mind, we are able to train the network to predict the actions from positions and velocities in a realistic axisymmetric Milky Way model, with an uncertainty of $\\\\sim 0.6$ % of the action value of the Sun (compared to the input actions transformed into positions and velocities with the Torus Mapping), by training on only $N=8$ phase-space points along $S=128$ orbit segments, i.e.\",\n \"with only 1024 phase-space points.\",\n \"With $S=1024$ such orbit segments, an action uncertainty of $\\\\sim 0.4$ % of the action value of the Sun can be attained.\",\n \"This is 5 times more accurate than the estimate of actions from positions and velocities with the Stäckel fudge in AGAMA.\",\n \"In principle, the present algorithm is not limited to axisymmetric systems, and given that the canonical coordinate transformation encoded in the generating function networks can be made arbitrarily complex by increasing the depth of the networks, one may be able to learn and model complex mass distributions given sufficient data and computational resources.\",\n \"However, the presence of chaotic dynamics in complex potentials gives cause for concern, since our method relies on the actions being conserved, which is not the case for chaotic orbits.\",\n \"It is thus likely that in such systems the method will only work for a subset of the orbits, while returning nonsensical results for the rest.\",\n \"We suspect that this could be turned from a bug into a feature of the method, as it may allow the present algorithm to be adapted into an automatic orbit classifier.\",\n \"We have attempted some very preliminary tests of the method on a non-axisymmetric system by applying the algorithm to orbits integrated in a triaxial logarithmic halo model.\",\n \"We found that the present algorithm is not able to adapt automatically to fit all orbital families in a triaxial model simultaneously.\",\n \"The reason for this may be both because of the presence of chaotic orbits, and because the isochrone model is not a good starting point for the canonical transformation in a triaxial potential.\",\n \"Nevertheless, we were encouraged to find that we were able to fit a generating function that gave small scatter in $J^{\\\\prime \\\\prime }$ between points on the same orbit, by selecting the input sample of “streams” from high angular momentum tube orbits.\",\n \"In future work we will attempt to replace the isochrone model with a triaxial Stäckel model [7] as the “toy” starting point; it is plausible that this may provide the key to unlock unsupervised fitting in triaxial and more general systems.\",\n \"But we will also need to be able to identify chaotic orbits in the system.\",\n \"The method currently relies on the fact that the potential is static.\",\n \"However, with the same caveats as above regarding chaos, it may be possible to generalize the algorithm so that motion is analysed in a rotating frame in which, for instance, a barred structure would appear static.\",\n \"This might allow the method to be used for analysing non-axisymmetric rotating systems as well.\",\n \"The algorithm was deliberately built to accept astrometric data as inputs.\",\n \"Because pytorch processes data in parallel, this architecture makes it very simple to supply the network with multiple inputs for the same star, where the different instances could, for example, sample over the uncertainties in the astrometry, or scan over missing information in some input dimensions.\",\n \"Thus, if a star has a missing radial velocity measurement, one may attempt to find the radial velocity value that makes the derived actions for the star agree with those of the group.\",\n \"It is thus plausible that the present software can be adapted into a new method for detecting stellar streams, especially structures that do not possess any obvious spatial correlation.\",\n \"The unsupervised learning technique developed here of building a network based on potentially complex corrections to simple analytic models and applying the physical constraints as loss function terms may have substantially wider applications.\",\n \"The pytorch tensor structure is particularly powerful for this purpose, with its ability to implement automatically the derivatives of the analytic model and the correction function.\",\n \"As we have seen, this provides an easy means to incorporate differential equations into a network.\",\n \"A simplified demonstration version of the code, along with a sample of test data from the [8] potential model `1' will be made available on github upon publication of this manuscript.\",\n \"RI, BF, and GM acknowledge funding from the Agence Nationale de la Recherche (ANR project ANR-18-CE31-0006, ANR-18-CE31-0017 and ANR-19-CE31-0017), from CNRS/INSU through the Programme National Galaxies et Cosmologie, and from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No.\",\n \"834148).\"\n ],\n [\n \"Appendix\",\n \"[emphstyle=, caption=Pytorch pseudocode for the ActionFinder module., emph=forward, ActionFinder,label=list:DECODE] class ActionFinder(torch.nn.Module): \\\"\\\"\\\" Inputs: d,l,b,vh,mul,mub astrometric phase space coordinates Output: Jddvals - the J” values Tddvals - the theta” values lossJddspread - loss due to the spread of the J” values \\\"\\\"\\\" def init(self,verbose=False): super().init() self.convinput2xv = convInput2xv(verbose=verbose) self.isochroneanalytic = IsochroneAnalytic(verbose=verbose) self.encoderGFG = EncoderGFG(verbose=verbose) self.encoderGFP = EncoderGFP(verbose=verbose) def forward(self,inputs): inputsxv = self.convinput2xv(inputs) convert d,l,b,vh,mul,mub to xv (J,Theta) in isochrone toy model Jiso, Tiso = self.isochroneanalytic(inputsxv, M, b) We now iterate to find J'.\",\n \"Start with J' = mean(Jiso for each stream) Jdmean = torch.mean( Jiso, dim=1, keepdim=True) for iter in range(itermax): Jdtrial = Jdmean.clone() trial J' Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) expand mean value to fill NStars dimension TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) (T,J') GFG = self.encoderGFG(TJd) generating function G J-J' = d(G)/d(T): JmJd = jacobian(Tiso,GFG)[:,:,0,:].data remove from computational graph GFG.detach() remove from computational graph update Jdmean = torch.mean( Jiso - JmJd, dim=1, keepdim=True) if ( torch.max( torch.abs(Jdtrial - Jdmean)) < 5.e-5 and test for acceptable convergence torch.mean(torch.abs(Jdtrial - Jdmean)) < 1.e-6): break Find Jdmean once more, but now retain the computational graph Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) GFG = self.encoderGFG(TJd) JmJd = jacobian(Tiso,GFG)[:,:,0,:] Jdvals = Jiso - JmJd J' given by generating function G Jdmean = torch.mean( Jdvals, dim=1, keepdim=True) Calculate T' from generating function derivative Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) GFG = self.encoderGFG(TJd) T' = T + d(G)/d(J'): Tdvals = (Tiso + jacobian(Jdmeanfill,GFG)[:,:,0,:]) Add in the extra freedom of a point-transformation Tddvals = self.encoderGFP( Tdvals ) final T” from generating function P dTdddTd = jacobian(Tdvals,Tddvals) d(T”)/d(T') dTddTdd = torch.inverse( dTdddTd ) d(T')/d(T”) Ji” = dT'/dTi” .\",\n \"J' Jddvals = torch.einsum('bsji,bsj->bsi',dTddTdd,Jdvals) final J” values Jddmean = torch.mean( Jddvals, dim=1, keepdim=True) Jddspread = Jddmean - Jddvals lossJddspread = torch.mean( torch.abs( Jddspread ) ) return Jddvals, Tddvals, lossJddspread [emphstyle=, caption=Pytorch pseudocode for the unit ResDense block., emph=forward, ResDenseblock,label=list:UNITBLOCK] class ResDenseblock(torch.nn.Module): \\\"\\\"\\\" The ResDense unit block.\",\n \"This layer does not change the number of features. \\\"\\\"\\\"\",\n \"def init(self, nunits): super().init() \\\"\\\"\\\" Declaration of the layers that will be used: nunits is the number of features. \\\"\\\"\\\"\",\n \"self.nunits = nunits self.dense1 = torch.nn.Linear(infeatures=self.nunits, outfeatures= self.nunits, bias=False) self.WN1 = torch.nn.utils.weightnorm(self.dense1) self.dense2 = torch.nn.Linear(infeatures=self.nunits, outfeatures= self.nunits) self.WN2 = torch.nn.utils.weightnorm(self.dense2) def forward(self, input): \\\"\\\"\\\" The computational graph (how the above layers are used): \\\"\\\"\\\" xx = torch.relu(input) xx = self.dense1(xx) xx = self.WN1(xx) xx = torch.relu(xx) xx = self.dense2(xx) xx = self.WN2(xx) output = xx + input return output\"\n ]\n]"},"id":{"kind":"string","value":"2012.05250"}}},{"rowIdx":3684,"cells":{"text":{"kind":"list like","value":[["A Deep Learning Approach to Anomaly Sequence Detection for\n High-Resolution Monitoring of Power Systems"],["Abstract A deep learning approach is proposed to detect data and system anomalies using high-resolution continuous point-on-wave (CPOW) or phasor measurements.","Both the anomaly and anomaly-free measurement models are assumed to have unknown temporal dependencies and probability distributions.","Historical training samples are assumed for the anomaly-free model, while no training samples are available for the anomaly measurements.","By transforming the anomaly-free observations into uniform independent and identically distributed sequences via a generative adversarial network, the proposed approach deploys a uniformity test for anomaly detection at the sensor level.","A distributed detection scheme that combines sensor level detections at the control center is also proposed that combines local detections to form more reliable detections.","Numerical results demonstrate significant improvement over the state-of-the-art solutions for various bad-data cases using real and synthetic CPOW and PMU data sets."],["Introduction","Bad-data detection is an essential and challenging part of a power grid monitoring system.","Bad data arise in many ways, from natural data anomaly caused by malfunctioning sensors to data attacks that are carefully designed to evade detection.","Whereas anomaly-free data share common characteristics, bad data may have vastly different types; in practice, rarely there are adequate historical anomaly data that can be used as training samples for bad-data detection and classification.","A less discussed and more challenging problem, especially for power system monitoring, is bad-sequence detection (BSD).","By BSD we mean, in particular, the testing of a hypothesis on whether a part of the measurement time series or a block of measurements is corrupted.","Instead of detecting a single anomaly data sample that may be an isolated incidence of little significance, BSD aims to capture bad sequences that are indicative of persistent anomalies.","A direct application of BSD is anomaly detection in the phasor-measurement unit (PMU) data stream where there are strong dependencies among data samples, and such data anomalies tend to happen not in isolation but in blocks of samples.","More significantly, BSD is essential in the detection of dynamic data attacks that aim to affect dispatch sequences from the control center [2], [3], [4].","Because such attacks are strategically designed to alter measurement sequences in specific patterns, a sample-by-sample bad-data detection is not effective.","Bad-data detection, identification, and removal in power system monitoring have been studied extensively [5], although the topic of BSD has not been treated.","Here we highlight aspects of bad-data detection most relevant to BSD and the use of machine learning techniques.","Some methods reviewed here are not designed specifically for BSD but the underlying ideas apply.","Originally proposed by Schweppe and Wildes [6], [7], a classic technique of bad-data detection is the residue error test (a.k.a $J(x)$ -test) in which the system state is first estimated using the weighted least-squares method as if there were no bad data.","Data anomaly is declared when the residue error (computed using the estimated state) is greater than a certain threshold.","The conventional residue error test and its many variations belong to the category of post-estimation bad-data detections where state estimation must be performed first.","As a result, inaccuracy of state estimation caused by bad data circulates back to affect the detection of bad data.","An alternative to post-estimation detection is the class of pre-estimation bad-data detection techniques that detect data anomaly prior to state estimation, thus breaking the path of estimation error propagation.","A key idea is to replace the estimated state used in the post-estimation scheme by the predicted state using the past measurements and apply residue test on the predicted measurement [8], [9].","A more direct pre-estimation approach is to detect anomalies directly based on features of anomaly-free data.","One of the earliest such techniques is the use of a neural network classifier trained by anomaly-free data [10].","A separate line of approaches is to extract features from the anomaly-free data.","Examples include the use of principal component analysis to characterize the signal subspace of the anomaly-free data [11] and the formulation of the problem as the detection of a change in measurement probability distribution [12].","The pre-estimation techniques are particularly relevant here because many of these techniques involve time-series data, and the idea of extracting innovation features of data for anomaly detection is particularly powerful and amenable to machine learning techniques.","A significant line of existing work on bad-data detection focuses on detecting the so-called false data attack by an adversary who can inject, remove, and substitute data to affect system and market operations [13], [14], [2], [15], [16], [17], [18].","When such an attacker is able to manipulate data from a sufficiently large number of sensors, the bad data introduced by the attacker become unobservable.","Specifically, the attacker can create a fake sequence of system states such that the manipulated measurements and the fake state sequence satisfy the underlying power flow equation.","Therefore, such a type of unobservable attack cannot be detected by techniques that rely only on the power flow model.","Consequently, post-estimation bad-data detection techniques tend to be ineffective.","There is a growing literature on the use of machine learning for bad-data detection in power systems since the mid-1990s [10].","A natural classification of these techniques can be based on how data are used in learning.","Supervised learning requires the labeled training data in the anomaly-free and anomaly cases [19], [20], [21], semi-supervised learning requires training samples for the anomaly-free data [4], [1], [22], and unsupervised learning requires no training data [19], [12].","An ensemble learning techniques is proposed in [23] that combines a collection of bad-data detectors.","Because it is difficult to obtain labeled anomaly data for training, the semi-supervised and unsupervised learning paradigms are of special significance.","Although not designed for power system state estimation, two types of semi-supervised anomaly detectors that use only training samples under the anomaly-free model can be applied for bad-data detection in power systems.","One is the one-class support vector machine (OC-SVM) [24] that separates anomaly and anomaly-free data deterministically.","The other is based on the idea of auto-encoder in deep neural network [25].","In both cases, there is an implicit assumption that anomaly data share non-overlapping domains with that of the anomaly-free data.","Such assumptions rarely hold in power system measurement models.","Statistical learning approaches to anomaly detection start from the premise that anomaly and anomaly-free data come from different probability distributions.","To this end, a recent work of particular relevance is [4] that focuses on dynamic data attacks of power system state estimation.","Although the attack models in [4] clearly suggests a BSD problem, the proposed mitigation strategy is a sample-by-sample detection scheme based on anomaly-free probability distributions from historical samples.","The idea of universal bad-data detection methods developed in [1], [22] is a semi-supervised learning technique that learns the inverse generative model of the anomaly-free data using a Wasserstein GAN approach [26], followed by a coincidence test.","The approach developed in [1], [22] is perhaps the only one that offers a detection consistency guarantee.","Specifically, under the assumption that measurement data samples are independent and identically distributed (i.i.d.","), the detection by an ideally trained deep learning network has both the false-positive and false-negative probabilities go to zero as the number of samples used in detection increases.","The assumption on i.i.d.","observations, however, does not hold for sequence detection.","For PMU measurements, in particular, measurements are strongly correlated, and simple decorrelation techniques do not produce independent samples essential to guarantee detection consistency."],["Summary of approach and contributions","We consider the problem of detecting BSD as part of the power system state estimation when neither the distribution of anomaly-free nor that of the anomaly measurements is known.","Also assumed unknown are the temporal dependencies of the measurements.","The main contribution of this work is a deep learning technique based on training samples from anomaly-free data.","The proposed approach builds upon and extends two classic ideas in the literature.","One is on the classical problem of uniformity test based on occupancy statistics [27], [28], [29] It was shown in [28] that, when the anomaly-free data are uniformly distributed i.i.d.","samples, the coincidence test achieves diminishing false positive and false negative error probabilities.","Such consistency is lost when the samples are correlated.","The power system measurements, however, are neither uniformly distributed nor i.i.d.","Combining generative adversarial network (GAN) learning and independent component analysis (ICA), we propose an ICA-GAN neural network trained to transform highly dependent time series measurements with unknown distributions to (approximately) uniformly distributed and i.i.d samples.","The output sequence of the ICA-GAN generator is used in a uniformity test that combines coincidence statistics to form an asymptotic most powerful test.","The proposed technique is applied to the measurements from the EPFL PMU network [30] and a larger synthetic North Texas network with PMU measurements [31]."],["System and Measurement model","The power system state $s_t$ at time $t$ is a column vector consisting of real and imaginary parts of the voltage phasors.","The measurement vector $z_{t}$ may include standard types of measurements such as branch power flows, power injections, and current magnitudes.","The measurement vector $z_{t}$ and the system state $s_{t}$ satisfy $ z_{t}=h(s_{t})+w_{t},$ where $h(\\cdot )$ is defined by the network topology, locations of measurement devices, and network parameters such as the network admittance matrix.","Measurement noise is denoted by $w_{t}$ .","Without loss of generality, we can treat the measurement variables as real by taking either the rectangular or the polar form of the complex variables and modifying and $h(\\cdot )$ in (REF ) accordingly."],["Models of data anomaly","We consider two types of data anomalies: natural data anomaly and adversarial data anomaly.","In both cases, the measurement under anomaly is given by $z_t=h(s_t) + w_t + a_t,$ where $a_t$ is the additional measurement error beyond the standard ambient noise.","Under natural anomaly, $a_t$ is likely non-Gaussian and independent across sensor locations.","Under the adversarial data anomaly, very little can be assumed about the statistical properties of $a_t$ , because an attacker with access to a substantial number of measurements and detailed system-level information can design $a_t$ to evade detection.","An extreme caseNote that the BSD model considered here do not make assumptions on the underlying network model and do not require information on the network topology and network parameters.","is when the attacker injects bad data $a_t=h(s_t+\\Delta _t)-h(s_t)$ into the measurements such that the system control center observes $z_{t}=h(s_{t}) + w_{t} + a_{t}=h(s_{t} + \\Delta _{t}) + w_{t},$ which means that the control center is deceived to believe that the actual state is $s^{\\prime }_{t}=s_{t}+ \\Delta _{t}$ using only $z_{t}$ for state estimation, .","Despite that the extreme adversarial attack is unobservable by any algebraic means, the attack vector $a_{t}$ does change the underlying distribution of $z$ , which is where the proposed BSD scheme can be effective in detecting such an unobservable attack."],["BSD as non-parametric hypothesis testing","We modeled the bad data in the most general form.","For simplicity, consider a stationary random sequence $(z_{t})$ .","Let the null hypothesis ${\\cal H}_0$ model the anomaly-free data and the alternative ${\\cal H}_1$ for the anomaly data.","We assume a simple hypothesis ${\\cal H}_0$ with (anomaly-free) distribution $f_0$ and a composite hypothesis ${\\cal H}_1$ consisting of all distributions some distance away from $f_0$ .","Specifically, $\\begin{array}{ll}{\\cal H}_0: z_{t} \\sim f_0~~{\\rm vs.}~~{\\cal H}_1: z_{t}\\sim f_1 \\in {F}_\\epsilon \\\\{F}_\\epsilon :=\\lbrace f, ||f-f_0|| \\ge \\epsilon \\rbrace \\end{array}$ where ${F}_\\epsilon $ is the set of probability distributions that are $\\epsilon $ distance (divergence) away from $f_0$ .","We assume that neither $f_0$ nor $f_1$ is known.","Under ${\\cal H}_0$ , however, we have some training samples distributed in $f_0$ and none under ${\\cal H}_1$ .","In the context of modern statistical machine learning, the model assumed here is a semi-supervised non-parametric inference.","Fig.","REF shows a schematic of the proposed technique.","At time $t$ , a measurement vector $Z_{t}=(z_{t},z_{t-1}, \\cdots ,z_{t-(M-1)})$ is passed through the ICA-GAN, which is a deep neural network trained to produce a block of uniformly-distributed independent components $V_{t}=(v_{t,1},v_{t,2},\\cdots , v_{t,N})$ .","The training of ICA-GAN is discussed in Sec REF , and it requires some historical measurements under the anomaly-free model.","A stratification (binning) layer, discussed in Sec.","REF , maps independent component vector $v$ to a $K$ -dimensional occupancy vector $x_t=(x_{t,1}, \\cdots , x_{t,K})$ for detection.","The output layer is linear classifier that produces a binary classification label $y_{t} \\in \\lbrace 0, 1\\rbrace $ , “1” for anomaly and “0” for anomaly-free.","The uniformity test is implemented as a coincidence test that passes a linear combination of occupancy statistics through a threshold designed to control the false positive rate of the test.","The threshold depends on the size of the test, the quantization level $K$ , and the number of samples for the coincidence test $N$ .","Figure: A schematics of ICA-GAN for BSD."],["Uniformity Test","The key idea that allows us to distinguish the null hypothesis under $f_0$ from the alternative distributions in ${F}_\\epsilon $ is rooted in the classical birthday problem [32]: given $N$ people in a room, what is the coincidence probability $P_c$ that there are at least two people having the same birthday?","It turns out that this probability is the lowest when the underlying birthday distribution is uniform [32].","This suggests that a test on some measure of coincidence can serve as a way to distinguish the uniform distribution from all other distributions.","Such a test was proposed earlier by David in [27] and more recently by Paninski [28].","The coincidence test by Paninski [28] considers the following binary hypotheses using conditionally i.i.d.","samples $(X_i \\in \\lbrace 1,\\cdots , K\\rbrace )$ from $K$ -alphabet discrete distributions $&{\\cal H}_0^{\\prime }:& X_i \\sim P_0=(\\frac{1}{K},\\cdots ,\\frac{1}{K}),\\nonumber \\\\&{\\cal H}_1^{\\prime }:& X_i \\sim P_1 \\in \\lbrace p=(p_1,\\cdots , p_K)|~ ||p-P_0|| \\ge \\epsilon \\rbrace .\\nonumber $ The intuition of the coincidence test is that, when $X_i$ are from the uniform distribution, the probability of coincidence is the lowest.","Specifically, let $K_1(x)$ be the number of alphabets that are realized by a single sample.","When $(x_i)$ s are realizations of i.i.d.","uniformly distributed random variables, $K_1(x)$ is statistically the highest.","$K_1(x)\\begin{array}{c}{\\small {\\cal H}_0^{\\prime }}\\\\\\gtrless \\\\{\\small {\\cal H}_1^{\\prime }}\\\\\\end{array}T_\\alpha $ where the threshold $T_\\alpha $ is a function of false positive level $\\alpha $ as well as the alphabet size (quantization level) $K$ , and the sample size $N$ .","Paninski showed that the coincidence test is consistent so long as $N$ grows faster than $\\sqrt{K}$ as $N = o(\\frac{1}{\\epsilon ^4} \\sqrt{K})$ .","Remarkably, the sample complexity can be significantly less than the size of the alphabet.","Huan and Meyn [33] later showed that the coincidence-based test also attains the best achievable probability of error decay rate with $P_e = exp(-(N^2/K)J(1 + o(1)))$ for some $J>0.2$ ."],["Stratification","The coincidence test is efficient to test uniformity on discrete distributions.","However, testing the uniformity of continuously distributed random samples without any assumptions on the density function is nontrivial [34].","Here we apply the $K$ -level uniform quantization to $v_t$ , which gives us $K$ -ary discrete random samples $x_t$ .","The quantization level $K$ and the threshold $T_\\alpha $ are the hyperparameters of the test.","Selecting a number of bins $K$ smaller than the number of samples would make the $K_{1}$ number not the highest under the uniform distribution and it would perform poorly.","However, a very big $K$ can also underperform as the alternative distribution also would have a large $K_{1}$ .","Guided by the asymptotic properties of $K_{1}(x)$ shown in [28], [33], we choose $K$ using a simple rule of the square of the block size of the test vector.","The threshold of the test statistics affects the true and false-positive probabilities of the detection.","The threshold $T_\\alpha $ of the $K_1$ coincidence test with the constraint on the false-positive probability to no greater than $\\alpha $ is given by $T_\\alpha = \\min \\lbrace k: \\Pr (K_1\\le k; {\\cal H}_0) \\le \\alpha \\rbrace .$ The computation of $T_\\alpha $ amounts to evaluating $P_k := \\Pr (K_1 = k; {\\cal H}_0)$ , which was given by Von Mises in [32]: $P_k=\\sum _{j=k}^{K}(-1)^{j+k} \\binom{j}{k} \\binom{K}{j}\\frac{ N!}{(N-j)!","}\\frac{(K-j)^{N-j}}{K^{N}}.$"],["Asymptotically most powerful test.","Extending the definition of coincidence we can define $K_{i}$ -coincidence as the number of bins which exactly have $i$ samples.","The original test proposed by David used $K_{0}$ coincidence to test.","Paninski used the $K_{1}$ coincidence.","The higher coincidence statistics $K_{i}, i\\ge 2$ , can also be used as each of these statistics provides additional information.","The importance of these statistics depend on the number of bins $K$ , and the number of samples $N$ .","Viktorova proposed to use a linear combination of statistics $(K_{i})$ with the optimal coefficients as a function of $K$ and $N$ to achieve the asymptotically most powerful test, $\\sum \\limits _{i=0}^{r}c_{i}K_{i}(x)\\begin{array}{c}{\\tiny {\\cal H}_0}\\\\\\gtrless \\\\{\\tiny {\\cal H}_1}\\end{array} T_{\\alpha ^{\\prime }}$ where the coefficient $c_{i}$ given in [29].","In this paper, in addition to using only $K_0(x)$ , only $K_1(x)$ and the linear combination of all higher coincidence $(K_i(x))$ , we also considered the use of a nonlinear combination of $(K_i(x))$ via an OC-SVM.","We trained OC-SVM with the $K_{0}, K_{1}, \\cdots K_{r}$ as input.","We trained the model with the anomaly-free samples and analyzed the results in the simulation section."],["ICA-GAN for Time Series","The coincidence test is developed to test anomalies on independent samples from distributions.","However, in power systems the measurements are highly correlated over time.","To this end, we propose ICA-GAN, a neural network that produces, approximately, i.i.d.","uniformly distributed samples under ${\\cal H}_0$ .","Independent Component Analysis (ICA) is a statistical technique originally proposed in [35] to find a linear transformation to represent a set of random variables as statistically independent random variables.","Nonlinear ICA was later proposed in [35] where a measurement vector $Z_t$ is represented as a nonlinear transform of latent vector $V_t$ with statistical independent components: $Z_{t} = f(V_{t})$ The goal is then to recover the inverse function $f^{-1}$ as well as the independent components $V_{t}$ based on observations of $Z_{t}$ alone.","ICA can be achieved by various machine learning models [36], [37].","ICA-GAN transforms temporally dependent data into independent components.","Fig.","REF shows a learning architecture of ICA-GAN that consists of two simultaneously trained neural networks: (i) an ICA generator and (ii) an a discriminator.","The training data passes through the ICA generator and the output is tested against synthetic uniformly distributed data by a discriminator.","Ideally, the ICA generator converges to a function that transforms the distribution of the data to the uniform distribution.","Figure: Learning of an ICA generator.","The measurement vector Z t Z_{t} is used as input to the generator.","The discriminator is fed by the output of the generator (converges to V t V_t) and i.i.d.","uniformly distributed 𝒰 t {\\cal U}_t.ICA-GAN.","The experiments in the paper used the values $\\alpha = 0.0001$ , $\\lambda = 0.1$ , $b = 100$ , $c = 10$ , $M = 80$ , $N = 50$ .","[1] : $\\alpha $ , the learning rate.","$\\lambda $ , the gradient penalty coefficient.","$b$ , the batch size.","$c$ , the number of iterations of the discriminator per generator iteration.","$M$ , the block size for the data sequence.","$N$ , the number of the independent components for ICA.","Number of training iterations $k = 0,1,\\cdots ,c$ $l = 1,\\cdots ,b$ Sample the uniform distribution samples $U_t = (u_{t,1},u_{t,2}\\cdots , u_{t,N}) $ , $u_{t,i} \\stackrel{{\\normalfont \\mbox{\\tiny \\rm i.i.d}}}{\\sim }{\\cal U}(0,1)$ .","Choose a random time $t$ for the time sequence.","Get $Z_{t}=(z_{t},z_{t-1}, \\cdots ,z_{t-(M-1)})$ measurements sequence from real data sequence.","$\\tilde{U} \\leftarrow g_{\\theta }(Z_t)$ $\\hat{U} \\leftarrow \\epsilon U+ (1-\\epsilon ) \\tilde{U}$ $L_{l} \\leftarrow {f_{\\omega }(\\tilde{U})}-{f_{\\omega }(U)}+ \\lambda (\\Vert \\nabla _{\\hat{U}} f_{\\omega (\\hat{U})}\\Vert _{2}-1)^{2}$ Update the discriminator parameters $\\omega $ by descending its stochastic gradient: $\\nabla _{\\omega } \\big [ \\frac{1}{b}\\sum \\limits _{l=1}^{b}{L_{l}}\\big ]$ $l = 1,\\cdots ,b$ Choose a random time $t$ for the time sequence.","Get $Z^{l}_{t}=(z_{t},z_{t-1}, \\cdots ,z_{t-(M-1)})$ measurements sequence from real data sequence.","Update the ICA-GAN generator parameters $\\theta $ by descending its stochastic gradient: $\\nabla _{\\theta } \\big [ \\frac{1}{b}\\sum \\limits _{i=1}^{b}{-f_{\\omega }(g_{\\theta }(Z^{l}_{\\lbrace t\\rbrace })} \\big ]$"],["Training ICA-GAN","The pseudo-code for the training of ICA-GAN is given in Algorithm REF .","The weights in both networks $\\theta $ , and $\\omega $ are initialized randomly and updated with the learning rate of $\\alpha $ .","Then ICA is trained to transform the observed measurement sequence $Z$ to $N$ independent components.","We use the 1-Wasserstein distance to measure the similarity between probability distributions.","As shown in [26], the Wasserstein distance measure tends to have improved stability of the training process.","To enforce the Lipschitz constraint of the 1-Wasserstein distance, we used gradient penalty on the discriminator's loss function as it is proposed in [38].","The discriminator $f_{\\omega }$ is updated more frequently than the ICA generator $g_{\\theta }$ .","We used Adam algorithm [39] for the weight updates."],["Systems simulated","Two sets of simulations were performed.","One was a four-bus system used in the EPFL Smart Grid Project[30].","We used the 50 Hz PMU measurements collected on April 1st, 2016 from 5 PM to 6 PM.","The second was based on the 133-bus synthetic North Texas transmission system [31] where one hour of 30 Hz PMU measurements were used.","We simulated non-interacting bad-sequences and unobservable attack sequences for each system.","For both systems, we used the one-phase equivalent of the three-phase systems."],["Performance measure","For BSD, the test was based on a block of N samples.","The detection performance of the algorithms was evaluated using receiver operating characteristic (ROC) curves.","We varied the thresholds of the algorithms to obtain a set of different true positive rates (TPR) and false positive rates (FPR) the tests.","We plotted the ROC curves with these values.","In power system applications, low false positive rates are preferred.","To this end, we paid special attention to the standard setting of 0.05 FPR in our performance evaluation.","We presented the true positive rate for the 0.05 false positive rate for each of the algorithms."],["ICA-GAN Implementation. ","We trained the generator with three hidden layers and 100 neurons at each hidden layer.","Rectified Linear Units (ReLU) at hidden layers and a linear activation function at the final layer were used as the activation functions.","For the discriminative network, we also used three hidden layers with 100 neurons.","ReLU at hidden layers and a linear activation function at the final layer were used as the activation functions.","Adam optimization [39] algorithm with mini-batches of 100 was used as the optimizer.","We separated the available data into training, validation and test sets.","Using EPFL data sequence we created a training set that has 600 batches of 80 consecutive anomaly-free samples, a validation set that has 400 batches of 80 consecutive anomaly-free samples and a test set that has 500 batches of 80 anomaly sequences and 500 batches of 80 anomaly-free sequences for each measurement.","Each test sequence consisted of 1.6 seconds of PMU measurement.","Using the North Texas data similarly we created a training, validation and test sets with 600, 300, 225 and another 225 batches of 80 consecutive anomaly-free samples and anomaly samples.","Each test sequence consisted of 2.66 seconds of PMU measurement.","The Wasserstein ICA-GAN was trained to obtain the transformation function from the measurements to the independent components.","$b = 80$ consecutive measurements were used as inputs for the generator and $N = 50$ -dimensional output of the generator was transferred to the discriminator.","We fed another 50-dimensional i.i.d.","uniform samples to the discriminator.","As a preprocessing step before applying ICA-GAN, we used a linear least-squares prediction to decorrelate the measurement samples.","The input layer of the ICA-GAN neural network was a linear least-squares predictor that whitens the input sequence.","After the ICA-GAN generator was used to convert the samples to i.i.d.","sequence samples, we used an additional step to convert the distribution of the ICA sequence to uniform distribution.","We used the empirical CDF of anomaly-free training samples to achieve this transformation.","After these steps, with the trained ICA-GAN we constructed the BSD algorithm.","We used the samples to apply the $K_{1}$ -coincidence test as defined in (REF ).","We used this approach for each measurement sequence individually.","If at least one anomaly measurement sequence was detected we assumed it was a successful detection.","The histogram of anomaly-free and anomaly test samples for one measurement are shown in Fig.","REF anomaly-free test data followed a uniform distribution, where the anomaly data distribution was far from that.","Figure: Histograms of anomaly-free test samples and anomaly test samples after ICA-GAN for simulation Case 1."],["Benchmark Techniques and Implementations","We compared the performance of ICA-GAN with $J(x)$ test [5], OC-SVM [24] and F-AnoGAN [25].","The $J(x)$ -test is the post-estimation detection method that requires no historical data for training.","When dealing with bad data sequence, we used the standard recursive techniques that apply the J(x)-test to isolate the measurement with the largest total residue error that was calculated over the sequence.","If it failed the $J(x)$ -test, the data will be declared bad and removed from the system until either the measurement data pass the test or the system becomes unobservable.","OC-SVM is a machine learning method that uses only anomaly-free historical data.","To deal with multiple bad data we implemented F-AnoGAN and OC-SVM for each individual measurement sequence individually.","Then, if at least one anomaly measurement sequence was detected we assumed it was a successful detection.","We used models that takes 80 consecutive measurement samples from a single measurement unit and gives an anomaly score as output.","We used the radial basis function as a nonlinear kernel.","We evaluated the results on the test sequences using the anomaly score function we achieved.","We varied the parameter $\\nu $ of SVM to get different points on the ROC curve.","F-AnoGAN is an auto-encoder based technique trained based on anomaly-free data.","We used a generator and a discriminator with 3 hidden layers in Wasserstein GAN and a deep neural network with 2 hidden layers and 100 neurons in each layer in the auto-encoder.","We used the 80 consecutive measurement samples as input and reduced the dimension to 50.","We evaluated the results considering the reconstruction error of the auto-encoder.","We thresholded the reconstruction error with different values to get the different points on the ROC curve."],["Case 1: Natural Anomalies","We considered natural anomalies involving multiple non-interacting noises.","We used a Gaussian Mixture noise on these multiple measurements.","We used bad data sequence on 4 of the measurements out of 10 measurements on EPFL system data, and on 6 measurements out of 266 measurements on North Texas synthetic system data.","Figure: ROC curves for anomaly case 1.","Left: The EPFL System.","Right:The Synthetic North Texas system.In Fig.","REF we presented the ROC curve for case 1.","It was seen that the ICA-GAN achieved a higher TPR than the compared methods for each level of FPR.","We also presented the TPR for $\\alpha = 0.05$ significance level in the in the legend of Fig.","REF .","ICA-GAN achieved the highest true positive rate on both systems.","The $J(x)$ -test did not perform well on the EPFL system simulation compared to Texas system simulation possibly because bad sequences are used on a larger ratio of measurements.","OC-SVM had close to ICA-GAN's performance on EPFL system, however it was less successful on the Texas system simulation.","F-AnoGAN had a worse performance than ICA-GAN and OC-SVM in both cases.","ICA-GAN had the best performance on both systems and had higher than $90\\%$ TPR even for small FPRs."],["Case 2: Adversarial Anomalies","We simulated the unobservable attacks to test the performance of the algorithms.","An unobservable data attack is constructed on the EPFL system and the North Texas system.","We used the method in [13] to obtain an attack vector $a_{t}$ .","We assumed the measurements in the system makes the system barely observable.","Then it was possible to design an unobservable data attack by manipulating some of the measurements.","The attack vector can be added to the measurements and it was physically not possible to detect.","$z_{t}^{\\prime }=z_{t} + w_{t} a_{t}.$ where we constructed $w_{t}$ from independent samples from a Gaussian Mixture Model.","In Fig.","REF we presented the ROC curve for case 2.","It was also seen in this case ICA-GAN had better performance than all compared methods with each FPR.It also had the highest TPR for 0.05 FPR level.","The $J(x)$ -test performed as if it was a random selection without using measurements in both cases.","OC-SVM performance was the closest to ICA-GAN, but there was still a significant difference.","F-AnoGAN had a worse performance than ICA-GAN and OC-SVM on both systems.","ICA-GAN had the best performance in both cases because; i) the independent component analysis approach transformed the correlated measurements to an independent sequence of samples that is easier to work on.","ii) it detected the changes in the probability distribution rather than trying to find outlier samples.","Figure: ROC curves for anomaly case 2.","Left: The EPFL System.","Right:The Synthetic North Texas system."],["Variants of the coincidence test","We compared four variants of the coincidence test: David's test [27] based on $K_0$ , Paniski's test [28] based on $K_1$ , Victorova-Chistyakov (VC) test [29] based on a linear combination of $\\lbrace K_i(x), i=1,\\cdots 15\\rbrace $ , and OC-SVM that linearly combines $\\lbrace K_i(x)\\rbrace $ using labeled samples under anomaly-free historical data.","Table REF shows the TPR under FPR=0.05.","We observed that under this setting VC test was the most powerful, i.e.","that minimizes the type 2 error.","$K_{0}$ and One-Class SVM was close to each other and $K_{1}$ test was the worst.","The asymptotically most powerful coincidence test also was the most powerful among the tests we tried in our simulation.","Table: TPR of different coincidence tests for 0.05 FPR constraint.","Case 1.a and Case 1.b are for natural anomalies for the EPFL and the North Texas systems, respectively.","Case 2.a and Case 2.b are for adversarial anomalies for the EPFL and the North Texas systems, respectively."],["Conclusion","We developed a deep learning architecture for BSD and cleansing for state estimation.","A major contribution of this work is to extend the capacities of the universal bad-data detection to time-sequential data using independent component analysis approaches.","We improved the universal anomaly detection algorithm by proposing to use a combination of coincidences instead of only $K_{1}$ .","We simulated bad-data scenarios including the challenging unobservable data attack.","Numerical tests show a considerable gain for the detection performance and state estimation performances over the state-of-the-art benchmark solutions."]],"string":"[\n [\n \"A Deep Learning Approach to Anomaly Sequence Detection for\\n High-Resolution Monitoring of Power Systems\"\n ],\n [\n \"Abstract A deep learning approach is proposed to detect data and system anomalies using high-resolution continuous point-on-wave (CPOW) or phasor measurements.\",\n \"Both the anomaly and anomaly-free measurement models are assumed to have unknown temporal dependencies and probability distributions.\",\n \"Historical training samples are assumed for the anomaly-free model, while no training samples are available for the anomaly measurements.\",\n \"By transforming the anomaly-free observations into uniform independent and identically distributed sequences via a generative adversarial network, the proposed approach deploys a uniformity test for anomaly detection at the sensor level.\",\n \"A distributed detection scheme that combines sensor level detections at the control center is also proposed that combines local detections to form more reliable detections.\",\n \"Numerical results demonstrate significant improvement over the state-of-the-art solutions for various bad-data cases using real and synthetic CPOW and PMU data sets.\"\n ],\n [\n \"Introduction\",\n \"Bad-data detection is an essential and challenging part of a power grid monitoring system.\",\n \"Bad data arise in many ways, from natural data anomaly caused by malfunctioning sensors to data attacks that are carefully designed to evade detection.\",\n \"Whereas anomaly-free data share common characteristics, bad data may have vastly different types; in practice, rarely there are adequate historical anomaly data that can be used as training samples for bad-data detection and classification.\",\n \"A less discussed and more challenging problem, especially for power system monitoring, is bad-sequence detection (BSD).\",\n \"By BSD we mean, in particular, the testing of a hypothesis on whether a part of the measurement time series or a block of measurements is corrupted.\",\n \"Instead of detecting a single anomaly data sample that may be an isolated incidence of little significance, BSD aims to capture bad sequences that are indicative of persistent anomalies.\",\n \"A direct application of BSD is anomaly detection in the phasor-measurement unit (PMU) data stream where there are strong dependencies among data samples, and such data anomalies tend to happen not in isolation but in blocks of samples.\",\n \"More significantly, BSD is essential in the detection of dynamic data attacks that aim to affect dispatch sequences from the control center [2], [3], [4].\",\n \"Because such attacks are strategically designed to alter measurement sequences in specific patterns, a sample-by-sample bad-data detection is not effective.\",\n \"Bad-data detection, identification, and removal in power system monitoring have been studied extensively [5], although the topic of BSD has not been treated.\",\n \"Here we highlight aspects of bad-data detection most relevant to BSD and the use of machine learning techniques.\",\n \"Some methods reviewed here are not designed specifically for BSD but the underlying ideas apply.\",\n \"Originally proposed by Schweppe and Wildes [6], [7], a classic technique of bad-data detection is the residue error test (a.k.a $J(x)$ -test) in which the system state is first estimated using the weighted least-squares method as if there were no bad data.\",\n \"Data anomaly is declared when the residue error (computed using the estimated state) is greater than a certain threshold.\",\n \"The conventional residue error test and its many variations belong to the category of post-estimation bad-data detections where state estimation must be performed first.\",\n \"As a result, inaccuracy of state estimation caused by bad data circulates back to affect the detection of bad data.\",\n \"An alternative to post-estimation detection is the class of pre-estimation bad-data detection techniques that detect data anomaly prior to state estimation, thus breaking the path of estimation error propagation.\",\n \"A key idea is to replace the estimated state used in the post-estimation scheme by the predicted state using the past measurements and apply residue test on the predicted measurement [8], [9].\",\n \"A more direct pre-estimation approach is to detect anomalies directly based on features of anomaly-free data.\",\n \"One of the earliest such techniques is the use of a neural network classifier trained by anomaly-free data [10].\",\n \"A separate line of approaches is to extract features from the anomaly-free data.\",\n \"Examples include the use of principal component analysis to characterize the signal subspace of the anomaly-free data [11] and the formulation of the problem as the detection of a change in measurement probability distribution [12].\",\n \"The pre-estimation techniques are particularly relevant here because many of these techniques involve time-series data, and the idea of extracting innovation features of data for anomaly detection is particularly powerful and amenable to machine learning techniques.\",\n \"A significant line of existing work on bad-data detection focuses on detecting the so-called false data attack by an adversary who can inject, remove, and substitute data to affect system and market operations [13], [14], [2], [15], [16], [17], [18].\",\n \"When such an attacker is able to manipulate data from a sufficiently large number of sensors, the bad data introduced by the attacker become unobservable.\",\n \"Specifically, the attacker can create a fake sequence of system states such that the manipulated measurements and the fake state sequence satisfy the underlying power flow equation.\",\n \"Therefore, such a type of unobservable attack cannot be detected by techniques that rely only on the power flow model.\",\n \"Consequently, post-estimation bad-data detection techniques tend to be ineffective.\",\n \"There is a growing literature on the use of machine learning for bad-data detection in power systems since the mid-1990s [10].\",\n \"A natural classification of these techniques can be based on how data are used in learning.\",\n \"Supervised learning requires the labeled training data in the anomaly-free and anomaly cases [19], [20], [21], semi-supervised learning requires training samples for the anomaly-free data [4], [1], [22], and unsupervised learning requires no training data [19], [12].\",\n \"An ensemble learning techniques is proposed in [23] that combines a collection of bad-data detectors.\",\n \"Because it is difficult to obtain labeled anomaly data for training, the semi-supervised and unsupervised learning paradigms are of special significance.\",\n \"Although not designed for power system state estimation, two types of semi-supervised anomaly detectors that use only training samples under the anomaly-free model can be applied for bad-data detection in power systems.\",\n \"One is the one-class support vector machine (OC-SVM) [24] that separates anomaly and anomaly-free data deterministically.\",\n \"The other is based on the idea of auto-encoder in deep neural network [25].\",\n \"In both cases, there is an implicit assumption that anomaly data share non-overlapping domains with that of the anomaly-free data.\",\n \"Such assumptions rarely hold in power system measurement models.\",\n \"Statistical learning approaches to anomaly detection start from the premise that anomaly and anomaly-free data come from different probability distributions.\",\n \"To this end, a recent work of particular relevance is [4] that focuses on dynamic data attacks of power system state estimation.\",\n \"Although the attack models in [4] clearly suggests a BSD problem, the proposed mitigation strategy is a sample-by-sample detection scheme based on anomaly-free probability distributions from historical samples.\",\n \"The idea of universal bad-data detection methods developed in [1], [22] is a semi-supervised learning technique that learns the inverse generative model of the anomaly-free data using a Wasserstein GAN approach [26], followed by a coincidence test.\",\n \"The approach developed in [1], [22] is perhaps the only one that offers a detection consistency guarantee.\",\n \"Specifically, under the assumption that measurement data samples are independent and identically distributed (i.i.d.\",\n \"), the detection by an ideally trained deep learning network has both the false-positive and false-negative probabilities go to zero as the number of samples used in detection increases.\",\n \"The assumption on i.i.d.\",\n \"observations, however, does not hold for sequence detection.\",\n \"For PMU measurements, in particular, measurements are strongly correlated, and simple decorrelation techniques do not produce independent samples essential to guarantee detection consistency.\"\n ],\n [\n \"Summary of approach and contributions\",\n \"We consider the problem of detecting BSD as part of the power system state estimation when neither the distribution of anomaly-free nor that of the anomaly measurements is known.\",\n \"Also assumed unknown are the temporal dependencies of the measurements.\",\n \"The main contribution of this work is a deep learning technique based on training samples from anomaly-free data.\",\n \"The proposed approach builds upon and extends two classic ideas in the literature.\",\n \"One is on the classical problem of uniformity test based on occupancy statistics [27], [28], [29] It was shown in [28] that, when the anomaly-free data are uniformly distributed i.i.d.\",\n \"samples, the coincidence test achieves diminishing false positive and false negative error probabilities.\",\n \"Such consistency is lost when the samples are correlated.\",\n \"The power system measurements, however, are neither uniformly distributed nor i.i.d.\",\n \"Combining generative adversarial network (GAN) learning and independent component analysis (ICA), we propose an ICA-GAN neural network trained to transform highly dependent time series measurements with unknown distributions to (approximately) uniformly distributed and i.i.d samples.\",\n \"The output sequence of the ICA-GAN generator is used in a uniformity test that combines coincidence statistics to form an asymptotic most powerful test.\",\n \"The proposed technique is applied to the measurements from the EPFL PMU network [30] and a larger synthetic North Texas network with PMU measurements [31].\"\n ],\n [\n \"System and Measurement model\",\n \"The power system state $s_t$ at time $t$ is a column vector consisting of real and imaginary parts of the voltage phasors.\",\n \"The measurement vector $z_{t}$ may include standard types of measurements such as branch power flows, power injections, and current magnitudes.\",\n \"The measurement vector $z_{t}$ and the system state $s_{t}$ satisfy $ z_{t}=h(s_{t})+w_{t},$ where $h(\\\\cdot )$ is defined by the network topology, locations of measurement devices, and network parameters such as the network admittance matrix.\",\n \"Measurement noise is denoted by $w_{t}$ .\",\n \"Without loss of generality, we can treat the measurement variables as real by taking either the rectangular or the polar form of the complex variables and modifying and $h(\\\\cdot )$ in (REF ) accordingly.\"\n ],\n [\n \"Models of data anomaly\",\n \"We consider two types of data anomalies: natural data anomaly and adversarial data anomaly.\",\n \"In both cases, the measurement under anomaly is given by $z_t=h(s_t) + w_t + a_t,$ where $a_t$ is the additional measurement error beyond the standard ambient noise.\",\n \"Under natural anomaly, $a_t$ is likely non-Gaussian and independent across sensor locations.\",\n \"Under the adversarial data anomaly, very little can be assumed about the statistical properties of $a_t$ , because an attacker with access to a substantial number of measurements and detailed system-level information can design $a_t$ to evade detection.\",\n \"An extreme caseNote that the BSD model considered here do not make assumptions on the underlying network model and do not require information on the network topology and network parameters.\",\n \"is when the attacker injects bad data $a_t=h(s_t+\\\\Delta _t)-h(s_t)$ into the measurements such that the system control center observes $z_{t}=h(s_{t}) + w_{t} + a_{t}=h(s_{t} + \\\\Delta _{t}) + w_{t},$ which means that the control center is deceived to believe that the actual state is $s^{\\\\prime }_{t}=s_{t}+ \\\\Delta _{t}$ using only $z_{t}$ for state estimation, .\",\n \"Despite that the extreme adversarial attack is unobservable by any algebraic means, the attack vector $a_{t}$ does change the underlying distribution of $z$ , which is where the proposed BSD scheme can be effective in detecting such an unobservable attack.\"\n ],\n [\n \"BSD as non-parametric hypothesis testing\",\n \"We modeled the bad data in the most general form.\",\n \"For simplicity, consider a stationary random sequence $(z_{t})$ .\",\n \"Let the null hypothesis ${\\\\cal H}_0$ model the anomaly-free data and the alternative ${\\\\cal H}_1$ for the anomaly data.\",\n \"We assume a simple hypothesis ${\\\\cal H}_0$ with (anomaly-free) distribution $f_0$ and a composite hypothesis ${\\\\cal H}_1$ consisting of all distributions some distance away from $f_0$ .\",\n \"Specifically, $\\\\begin{array}{ll}{\\\\cal H}_0: z_{t} \\\\sim f_0~~{\\\\rm vs.}~~{\\\\cal H}_1: z_{t}\\\\sim f_1 \\\\in {F}_\\\\epsilon \\\\\\\\{F}_\\\\epsilon :=\\\\lbrace f, ||f-f_0|| \\\\ge \\\\epsilon \\\\rbrace \\\\end{array}$ where ${F}_\\\\epsilon $ is the set of probability distributions that are $\\\\epsilon $ distance (divergence) away from $f_0$ .\",\n \"We assume that neither $f_0$ nor $f_1$ is known.\",\n \"Under ${\\\\cal H}_0$ , however, we have some training samples distributed in $f_0$ and none under ${\\\\cal H}_1$ .\",\n \"In the context of modern statistical machine learning, the model assumed here is a semi-supervised non-parametric inference.\",\n \"Fig.\",\n \"REF shows a schematic of the proposed technique.\",\n \"At time $t$ , a measurement vector $Z_{t}=(z_{t},z_{t-1}, \\\\cdots ,z_{t-(M-1)})$ is passed through the ICA-GAN, which is a deep neural network trained to produce a block of uniformly-distributed independent components $V_{t}=(v_{t,1},v_{t,2},\\\\cdots , v_{t,N})$ .\",\n \"The training of ICA-GAN is discussed in Sec REF , and it requires some historical measurements under the anomaly-free model.\",\n \"A stratification (binning) layer, discussed in Sec.\",\n \"REF , maps independent component vector $v$ to a $K$ -dimensional occupancy vector $x_t=(x_{t,1}, \\\\cdots , x_{t,K})$ for detection.\",\n \"The output layer is linear classifier that produces a binary classification label $y_{t} \\\\in \\\\lbrace 0, 1\\\\rbrace $ , “1” for anomaly and “0” for anomaly-free.\",\n \"The uniformity test is implemented as a coincidence test that passes a linear combination of occupancy statistics through a threshold designed to control the false positive rate of the test.\",\n \"The threshold depends on the size of the test, the quantization level $K$ , and the number of samples for the coincidence test $N$ .\",\n \"Figure: A schematics of ICA-GAN for BSD.\"\n ],\n [\n \"Uniformity Test\",\n \"The key idea that allows us to distinguish the null hypothesis under $f_0$ from the alternative distributions in ${F}_\\\\epsilon $ is rooted in the classical birthday problem [32]: given $N$ people in a room, what is the coincidence probability $P_c$ that there are at least two people having the same birthday?\",\n \"It turns out that this probability is the lowest when the underlying birthday distribution is uniform [32].\",\n \"This suggests that a test on some measure of coincidence can serve as a way to distinguish the uniform distribution from all other distributions.\",\n \"Such a test was proposed earlier by David in [27] and more recently by Paninski [28].\",\n \"The coincidence test by Paninski [28] considers the following binary hypotheses using conditionally i.i.d.\",\n \"samples $(X_i \\\\in \\\\lbrace 1,\\\\cdots , K\\\\rbrace )$ from $K$ -alphabet discrete distributions $&{\\\\cal H}_0^{\\\\prime }:& X_i \\\\sim P_0=(\\\\frac{1}{K},\\\\cdots ,\\\\frac{1}{K}),\\\\nonumber \\\\\\\\&{\\\\cal H}_1^{\\\\prime }:& X_i \\\\sim P_1 \\\\in \\\\lbrace p=(p_1,\\\\cdots , p_K)|~ ||p-P_0|| \\\\ge \\\\epsilon \\\\rbrace .\\\\nonumber $ The intuition of the coincidence test is that, when $X_i$ are from the uniform distribution, the probability of coincidence is the lowest.\",\n \"Specifically, let $K_1(x)$ be the number of alphabets that are realized by a single sample.\",\n \"When $(x_i)$ s are realizations of i.i.d.\",\n \"uniformly distributed random variables, $K_1(x)$ is statistically the highest.\",\n \"$K_1(x)\\\\begin{array}{c}{\\\\small {\\\\cal H}_0^{\\\\prime }}\\\\\\\\\\\\gtrless \\\\\\\\{\\\\small {\\\\cal H}_1^{\\\\prime }}\\\\\\\\\\\\end{array}T_\\\\alpha $ where the threshold $T_\\\\alpha $ is a function of false positive level $\\\\alpha $ as well as the alphabet size (quantization level) $K$ , and the sample size $N$ .\",\n \"Paninski showed that the coincidence test is consistent so long as $N$ grows faster than $\\\\sqrt{K}$ as $N = o(\\\\frac{1}{\\\\epsilon ^4} \\\\sqrt{K})$ .\",\n \"Remarkably, the sample complexity can be significantly less than the size of the alphabet.\",\n \"Huan and Meyn [33] later showed that the coincidence-based test also attains the best achievable probability of error decay rate with $P_e = exp(-(N^2/K)J(1 + o(1)))$ for some $J>0.2$ .\"\n ],\n [\n \"Stratification\",\n \"The coincidence test is efficient to test uniformity on discrete distributions.\",\n \"However, testing the uniformity of continuously distributed random samples without any assumptions on the density function is nontrivial [34].\",\n \"Here we apply the $K$ -level uniform quantization to $v_t$ , which gives us $K$ -ary discrete random samples $x_t$ .\",\n \"The quantization level $K$ and the threshold $T_\\\\alpha $ are the hyperparameters of the test.\",\n \"Selecting a number of bins $K$ smaller than the number of samples would make the $K_{1}$ number not the highest under the uniform distribution and it would perform poorly.\",\n \"However, a very big $K$ can also underperform as the alternative distribution also would have a large $K_{1}$ .\",\n \"Guided by the asymptotic properties of $K_{1}(x)$ shown in [28], [33], we choose $K$ using a simple rule of the square of the block size of the test vector.\",\n \"The threshold of the test statistics affects the true and false-positive probabilities of the detection.\",\n \"The threshold $T_\\\\alpha $ of the $K_1$ coincidence test with the constraint on the false-positive probability to no greater than $\\\\alpha $ is given by $T_\\\\alpha = \\\\min \\\\lbrace k: \\\\Pr (K_1\\\\le k; {\\\\cal H}_0) \\\\le \\\\alpha \\\\rbrace .$ The computation of $T_\\\\alpha $ amounts to evaluating $P_k := \\\\Pr (K_1 = k; {\\\\cal H}_0)$ , which was given by Von Mises in [32]: $P_k=\\\\sum _{j=k}^{K}(-1)^{j+k} \\\\binom{j}{k} \\\\binom{K}{j}\\\\frac{ N!}{(N-j)!\",\n \"}\\\\frac{(K-j)^{N-j}}{K^{N}}.$\"\n ],\n [\n \"Asymptotically most powerful test.\",\n \"Extending the definition of coincidence we can define $K_{i}$ -coincidence as the number of bins which exactly have $i$ samples.\",\n \"The original test proposed by David used $K_{0}$ coincidence to test.\",\n \"Paninski used the $K_{1}$ coincidence.\",\n \"The higher coincidence statistics $K_{i}, i\\\\ge 2$ , can also be used as each of these statistics provides additional information.\",\n \"The importance of these statistics depend on the number of bins $K$ , and the number of samples $N$ .\",\n \"Viktorova proposed to use a linear combination of statistics $(K_{i})$ with the optimal coefficients as a function of $K$ and $N$ to achieve the asymptotically most powerful test, $\\\\sum \\\\limits _{i=0}^{r}c_{i}K_{i}(x)\\\\begin{array}{c}{\\\\tiny {\\\\cal H}_0}\\\\\\\\\\\\gtrless \\\\\\\\{\\\\tiny {\\\\cal H}_1}\\\\end{array} T_{\\\\alpha ^{\\\\prime }}$ where the coefficient $c_{i}$ given in [29].\",\n \"In this paper, in addition to using only $K_0(x)$ , only $K_1(x)$ and the linear combination of all higher coincidence $(K_i(x))$ , we also considered the use of a nonlinear combination of $(K_i(x))$ via an OC-SVM.\",\n \"We trained OC-SVM with the $K_{0}, K_{1}, \\\\cdots K_{r}$ as input.\",\n \"We trained the model with the anomaly-free samples and analyzed the results in the simulation section.\"\n ],\n [\n \"ICA-GAN for Time Series\",\n \"The coincidence test is developed to test anomalies on independent samples from distributions.\",\n \"However, in power systems the measurements are highly correlated over time.\",\n \"To this end, we propose ICA-GAN, a neural network that produces, approximately, i.i.d.\",\n \"uniformly distributed samples under ${\\\\cal H}_0$ .\",\n \"Independent Component Analysis (ICA) is a statistical technique originally proposed in [35] to find a linear transformation to represent a set of random variables as statistically independent random variables.\",\n \"Nonlinear ICA was later proposed in [35] where a measurement vector $Z_t$ is represented as a nonlinear transform of latent vector $V_t$ with statistical independent components: $Z_{t} = f(V_{t})$ The goal is then to recover the inverse function $f^{-1}$ as well as the independent components $V_{t}$ based on observations of $Z_{t}$ alone.\",\n \"ICA can be achieved by various machine learning models [36], [37].\",\n \"ICA-GAN transforms temporally dependent data into independent components.\",\n \"Fig.\",\n \"REF shows a learning architecture of ICA-GAN that consists of two simultaneously trained neural networks: (i) an ICA generator and (ii) an a discriminator.\",\n \"The training data passes through the ICA generator and the output is tested against synthetic uniformly distributed data by a discriminator.\",\n \"Ideally, the ICA generator converges to a function that transforms the distribution of the data to the uniform distribution.\",\n \"Figure: Learning of an ICA generator.\",\n \"The measurement vector Z t Z_{t} is used as input to the generator.\",\n \"The discriminator is fed by the output of the generator (converges to V t V_t) and i.i.d.\",\n \"uniformly distributed 𝒰 t {\\\\cal U}_t.ICA-GAN.\",\n \"The experiments in the paper used the values $\\\\alpha = 0.0001$ , $\\\\lambda = 0.1$ , $b = 100$ , $c = 10$ , $M = 80$ , $N = 50$ .\",\n \"[1] : $\\\\alpha $ , the learning rate.\",\n \"$\\\\lambda $ , the gradient penalty coefficient.\",\n \"$b$ , the batch size.\",\n \"$c$ , the number of iterations of the discriminator per generator iteration.\",\n \"$M$ , the block size for the data sequence.\",\n \"$N$ , the number of the independent components for ICA.\",\n \"Number of training iterations $k = 0,1,\\\\cdots ,c$ $l = 1,\\\\cdots ,b$ Sample the uniform distribution samples $U_t = (u_{t,1},u_{t,2}\\\\cdots , u_{t,N}) $ , $u_{t,i} \\\\stackrel{{\\\\normalfont \\\\mbox{\\\\tiny \\\\rm i.i.d}}}{\\\\sim }{\\\\cal U}(0,1)$ .\",\n \"Choose a random time $t$ for the time sequence.\",\n \"Get $Z_{t}=(z_{t},z_{t-1}, \\\\cdots ,z_{t-(M-1)})$ measurements sequence from real data sequence.\",\n \"$\\\\tilde{U} \\\\leftarrow g_{\\\\theta }(Z_t)$ $\\\\hat{U} \\\\leftarrow \\\\epsilon U+ (1-\\\\epsilon ) \\\\tilde{U}$ $L_{l} \\\\leftarrow {f_{\\\\omega }(\\\\tilde{U})}-{f_{\\\\omega }(U)}+ \\\\lambda (\\\\Vert \\\\nabla _{\\\\hat{U}} f_{\\\\omega (\\\\hat{U})}\\\\Vert _{2}-1)^{2}$ Update the discriminator parameters $\\\\omega $ by descending its stochastic gradient: $\\\\nabla _{\\\\omega } \\\\big [ \\\\frac{1}{b}\\\\sum \\\\limits _{l=1}^{b}{L_{l}}\\\\big ]$ $l = 1,\\\\cdots ,b$ Choose a random time $t$ for the time sequence.\",\n \"Get $Z^{l}_{t}=(z_{t},z_{t-1}, \\\\cdots ,z_{t-(M-1)})$ measurements sequence from real data sequence.\",\n \"Update the ICA-GAN generator parameters $\\\\theta $ by descending its stochastic gradient: $\\\\nabla _{\\\\theta } \\\\big [ \\\\frac{1}{b}\\\\sum \\\\limits _{i=1}^{b}{-f_{\\\\omega }(g_{\\\\theta }(Z^{l}_{\\\\lbrace t\\\\rbrace })} \\\\big ]$\"\n ],\n [\n \"Training ICA-GAN\",\n \"The pseudo-code for the training of ICA-GAN is given in Algorithm REF .\",\n \"The weights in both networks $\\\\theta $ , and $\\\\omega $ are initialized randomly and updated with the learning rate of $\\\\alpha $ .\",\n \"Then ICA is trained to transform the observed measurement sequence $Z$ to $N$ independent components.\",\n \"We use the 1-Wasserstein distance to measure the similarity between probability distributions.\",\n \"As shown in [26], the Wasserstein distance measure tends to have improved stability of the training process.\",\n \"To enforce the Lipschitz constraint of the 1-Wasserstein distance, we used gradient penalty on the discriminator's loss function as it is proposed in [38].\",\n \"The discriminator $f_{\\\\omega }$ is updated more frequently than the ICA generator $g_{\\\\theta }$ .\",\n \"We used Adam algorithm [39] for the weight updates.\"\n ],\n [\n \"Systems simulated\",\n \"Two sets of simulations were performed.\",\n \"One was a four-bus system used in the EPFL Smart Grid Project[30].\",\n \"We used the 50 Hz PMU measurements collected on April 1st, 2016 from 5 PM to 6 PM.\",\n \"The second was based on the 133-bus synthetic North Texas transmission system [31] where one hour of 30 Hz PMU measurements were used.\",\n \"We simulated non-interacting bad-sequences and unobservable attack sequences for each system.\",\n \"For both systems, we used the one-phase equivalent of the three-phase systems.\"\n ],\n [\n \"Performance measure\",\n \"For BSD, the test was based on a block of N samples.\",\n \"The detection performance of the algorithms was evaluated using receiver operating characteristic (ROC) curves.\",\n \"We varied the thresholds of the algorithms to obtain a set of different true positive rates (TPR) and false positive rates (FPR) the tests.\",\n \"We plotted the ROC curves with these values.\",\n \"In power system applications, low false positive rates are preferred.\",\n \"To this end, we paid special attention to the standard setting of 0.05 FPR in our performance evaluation.\",\n \"We presented the true positive rate for the 0.05 false positive rate for each of the algorithms.\"\n ],\n [\n \"ICA-GAN Implementation. \",\n \"We trained the generator with three hidden layers and 100 neurons at each hidden layer.\",\n \"Rectified Linear Units (ReLU) at hidden layers and a linear activation function at the final layer were used as the activation functions.\",\n \"For the discriminative network, we also used three hidden layers with 100 neurons.\",\n \"ReLU at hidden layers and a linear activation function at the final layer were used as the activation functions.\",\n \"Adam optimization [39] algorithm with mini-batches of 100 was used as the optimizer.\",\n \"We separated the available data into training, validation and test sets.\",\n \"Using EPFL data sequence we created a training set that has 600 batches of 80 consecutive anomaly-free samples, a validation set that has 400 batches of 80 consecutive anomaly-free samples and a test set that has 500 batches of 80 anomaly sequences and 500 batches of 80 anomaly-free sequences for each measurement.\",\n \"Each test sequence consisted of 1.6 seconds of PMU measurement.\",\n \"Using the North Texas data similarly we created a training, validation and test sets with 600, 300, 225 and another 225 batches of 80 consecutive anomaly-free samples and anomaly samples.\",\n \"Each test sequence consisted of 2.66 seconds of PMU measurement.\",\n \"The Wasserstein ICA-GAN was trained to obtain the transformation function from the measurements to the independent components.\",\n \"$b = 80$ consecutive measurements were used as inputs for the generator and $N = 50$ -dimensional output of the generator was transferred to the discriminator.\",\n \"We fed another 50-dimensional i.i.d.\",\n \"uniform samples to the discriminator.\",\n \"As a preprocessing step before applying ICA-GAN, we used a linear least-squares prediction to decorrelate the measurement samples.\",\n \"The input layer of the ICA-GAN neural network was a linear least-squares predictor that whitens the input sequence.\",\n \"After the ICA-GAN generator was used to convert the samples to i.i.d.\",\n \"sequence samples, we used an additional step to convert the distribution of the ICA sequence to uniform distribution.\",\n \"We used the empirical CDF of anomaly-free training samples to achieve this transformation.\",\n \"After these steps, with the trained ICA-GAN we constructed the BSD algorithm.\",\n \"We used the samples to apply the $K_{1}$ -coincidence test as defined in (REF ).\",\n \"We used this approach for each measurement sequence individually.\",\n \"If at least one anomaly measurement sequence was detected we assumed it was a successful detection.\",\n \"The histogram of anomaly-free and anomaly test samples for one measurement are shown in Fig.\",\n \"REF anomaly-free test data followed a uniform distribution, where the anomaly data distribution was far from that.\",\n \"Figure: Histograms of anomaly-free test samples and anomaly test samples after ICA-GAN for simulation Case 1.\"\n ],\n [\n \"Benchmark Techniques and Implementations\",\n \"We compared the performance of ICA-GAN with $J(x)$ test [5], OC-SVM [24] and F-AnoGAN [25].\",\n \"The $J(x)$ -test is the post-estimation detection method that requires no historical data for training.\",\n \"When dealing with bad data sequence, we used the standard recursive techniques that apply the J(x)-test to isolate the measurement with the largest total residue error that was calculated over the sequence.\",\n \"If it failed the $J(x)$ -test, the data will be declared bad and removed from the system until either the measurement data pass the test or the system becomes unobservable.\",\n \"OC-SVM is a machine learning method that uses only anomaly-free historical data.\",\n \"To deal with multiple bad data we implemented F-AnoGAN and OC-SVM for each individual measurement sequence individually.\",\n \"Then, if at least one anomaly measurement sequence was detected we assumed it was a successful detection.\",\n \"We used models that takes 80 consecutive measurement samples from a single measurement unit and gives an anomaly score as output.\",\n \"We used the radial basis function as a nonlinear kernel.\",\n \"We evaluated the results on the test sequences using the anomaly score function we achieved.\",\n \"We varied the parameter $\\\\nu $ of SVM to get different points on the ROC curve.\",\n \"F-AnoGAN is an auto-encoder based technique trained based on anomaly-free data.\",\n \"We used a generator and a discriminator with 3 hidden layers in Wasserstein GAN and a deep neural network with 2 hidden layers and 100 neurons in each layer in the auto-encoder.\",\n \"We used the 80 consecutive measurement samples as input and reduced the dimension to 50.\",\n \"We evaluated the results considering the reconstruction error of the auto-encoder.\",\n \"We thresholded the reconstruction error with different values to get the different points on the ROC curve.\"\n ],\n [\n \"Case 1: Natural Anomalies\",\n \"We considered natural anomalies involving multiple non-interacting noises.\",\n \"We used a Gaussian Mixture noise on these multiple measurements.\",\n \"We used bad data sequence on 4 of the measurements out of 10 measurements on EPFL system data, and on 6 measurements out of 266 measurements on North Texas synthetic system data.\",\n \"Figure: ROC curves for anomaly case 1.\",\n \"Left: The EPFL System.\",\n \"Right:The Synthetic North Texas system.In Fig.\",\n \"REF we presented the ROC curve for case 1.\",\n \"It was seen that the ICA-GAN achieved a higher TPR than the compared methods for each level of FPR.\",\n \"We also presented the TPR for $\\\\alpha = 0.05$ significance level in the in the legend of Fig.\",\n \"REF .\",\n \"ICA-GAN achieved the highest true positive rate on both systems.\",\n \"The $J(x)$ -test did not perform well on the EPFL system simulation compared to Texas system simulation possibly because bad sequences are used on a larger ratio of measurements.\",\n \"OC-SVM had close to ICA-GAN's performance on EPFL system, however it was less successful on the Texas system simulation.\",\n \"F-AnoGAN had a worse performance than ICA-GAN and OC-SVM in both cases.\",\n \"ICA-GAN had the best performance on both systems and had higher than $90\\\\%$ TPR even for small FPRs.\"\n ],\n [\n \"Case 2: Adversarial Anomalies\",\n \"We simulated the unobservable attacks to test the performance of the algorithms.\",\n \"An unobservable data attack is constructed on the EPFL system and the North Texas system.\",\n \"We used the method in [13] to obtain an attack vector $a_{t}$ .\",\n \"We assumed the measurements in the system makes the system barely observable.\",\n \"Then it was possible to design an unobservable data attack by manipulating some of the measurements.\",\n \"The attack vector can be added to the measurements and it was physically not possible to detect.\",\n \"$z_{t}^{\\\\prime }=z_{t} + w_{t} a_{t}.$ where we constructed $w_{t}$ from independent samples from a Gaussian Mixture Model.\",\n \"In Fig.\",\n \"REF we presented the ROC curve for case 2.\",\n \"It was also seen in this case ICA-GAN had better performance than all compared methods with each FPR.It also had the highest TPR for 0.05 FPR level.\",\n \"The $J(x)$ -test performed as if it was a random selection without using measurements in both cases.\",\n \"OC-SVM performance was the closest to ICA-GAN, but there was still a significant difference.\",\n \"F-AnoGAN had a worse performance than ICA-GAN and OC-SVM on both systems.\",\n \"ICA-GAN had the best performance in both cases because; i) the independent component analysis approach transformed the correlated measurements to an independent sequence of samples that is easier to work on.\",\n \"ii) it detected the changes in the probability distribution rather than trying to find outlier samples.\",\n \"Figure: ROC curves for anomaly case 2.\",\n \"Left: The EPFL System.\",\n \"Right:The Synthetic North Texas system.\"\n ],\n [\n \"Variants of the coincidence test\",\n \"We compared four variants of the coincidence test: David's test [27] based on $K_0$ , Paniski's test [28] based on $K_1$ , Victorova-Chistyakov (VC) test [29] based on a linear combination of $\\\\lbrace K_i(x), i=1,\\\\cdots 15\\\\rbrace $ , and OC-SVM that linearly combines $\\\\lbrace K_i(x)\\\\rbrace $ using labeled samples under anomaly-free historical data.\",\n \"Table REF shows the TPR under FPR=0.05.\",\n \"We observed that under this setting VC test was the most powerful, i.e.\",\n \"that minimizes the type 2 error.\",\n \"$K_{0}$ and One-Class SVM was close to each other and $K_{1}$ test was the worst.\",\n \"The asymptotically most powerful coincidence test also was the most powerful among the tests we tried in our simulation.\",\n \"Table: TPR of different coincidence tests for 0.05 FPR constraint.\",\n \"Case 1.a and Case 1.b are for natural anomalies for the EPFL and the North Texas systems, respectively.\",\n \"Case 2.a and Case 2.b are for adversarial anomalies for the EPFL and the North Texas systems, respectively.\"\n ],\n [\n \"Conclusion\",\n \"We developed a deep learning architecture for BSD and cleansing for state estimation.\",\n \"A major contribution of this work is to extend the capacities of the universal bad-data detection to time-sequential data using independent component analysis approaches.\",\n \"We improved the universal anomaly detection algorithm by proposing to use a combination of coincidences instead of only $K_{1}$ .\",\n \"We simulated bad-data scenarios including the challenging unobservable data attack.\",\n \"Numerical tests show a considerable gain for the detection performance and state estimation performances over the state-of-the-art benchmark solutions.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05163"}}},{"rowIdx":3685,"cells":{"text":{"kind":"list like","value":[["Iterative Collision Resolution for Slotted ALOHA with NOMA for\n Heterogeneous Devices"],["Abstract In this paper, the problem of using uncoordinated multiple access (UMA) to serve a massive amount of heterogeneous users is investigated.","Leveraging the heterogeneity, we propose a novel UMA protocol, called iterative collision resolution for slotted ALOHA (IRSA) with non-orthogonal multiple access (NOMA), to improve the conventional IRSA.","In addition to the inter-slot successive interference cancellation (SIC) technique used in existing IRSA-based schemes, the proposed protocol further employs the intra-slot SIC technique that enables collision resolution for certain configurations of collided packets.","A novel multi-dimensional density evolution is then proposed to analyze and to optimize the proposed protocol.","Simulation results show that the proposed IRSA with NOMA protocol can efficiently exploit the heterogeneity among users and the multi-dimensional density evolution can accurately predict the throughput performance.","Last, an extension of the proposed IRSA with NOMA protocol to the frame-asynchronous setting is investigated, where a boundary effect similar to that in spatially-coupled low-density parity check codes can be observed to bootstrap the decoding process."],["Introduction","To realize the overarching ambition of internet of things (IoT) [1], [2], the current 3GPP Specification of the fifth generation cellular network technology has defined a use scenario, termed massive machine-type communication (mMTC) [3], [4].","This use scenario is expected to accommodate a massive number of users with sporadic activities.","Moreover, in such applications, the payload is assumed to be rather small.","In this circumstance, traditional philosophy of first coordinating then communicating becomes extremely inefficient, as there are so many users to coordinate while so small amount of data for each user to transmit.","Hence, uncoordinated multiple access (UMA) techniques that can communicate without first establishing coordination are highly desirable.","Among many UMA techniques, slotted ALOHA (SA) [5] has been adopted in many practical communication systems.","In an SA protocol, each user transmits a packet at the time slot next to that in which the data packet arrived.","When multiple users send packets at the same slot, i.e., the packets collide, these collided packets are discarded black (i.e., collision channel model) and retransmissions are scheduled according to some back-off mechanism.","Recently, this concept has even been adopted in modern cellular systems such as SigFoX and LoRaWAN [6].","Notwithstanding the success and popularity, it is well known that the conventional SA achieves an efficiency (to be defined in Section ) of at most $1/e\\approx 36.8 \\%$ [5].","To improve the efficiency, Casini et al.","in [7] proposed contention resolution diversity slotted ALOHA (CRDSA) that improves the efficiency to be at most $55\\%$ blackasymptotically as the frame size goes to infinity.","In CRDSA, each active user (those having data to send) sends replicated packets on twoblackHigher efficiencies can be achieved by sending more replicas.","randomly selected slots .","The collided packets are not discarded in CRDSA; instead, they are stored with the hope that inter-slotHere, we particularly highlight the term “inter-slot\" for distinguishing it from the intra-slot SIC discussed in Section .","successive interference cancellation (SIC) can resolve collisions by their replicas.","Due to its improved efficiency, a version of CRDSA has been included into the digital video broadcasting (DVB) standardization [8].","After the paradigm-shifting idea [7], much effort has been devoted to improve CRDSA by making connections to error-correction codes [9].","To the best of our knowledge, the connection was first made in [9] where a CRDSA system was shown to have a left-regular bipartite graph representation and inter-slot SIC is interpreted as belief propagation (BP) decoding [10].","After this connection, Liva in [9] then proposed irregular repetition slotted ALOHA (IRSA), in which each active user randomly chooses the number of repetitions and randomly selects time slots to send those replicas.","It can be observed that an IRSA scheme corresponds to an irregular bipartite graph.","It was shown in [9] that by optimizing the left degree distribution with the help of density evolution [10], one can achieve a significantly improved efficiency of $96.5\\%$ blackasymptotically.","It was then recognized in [11] that the coding problem induced by IRSA is similar to that of LT codes [12] with the peeling decoder [13].","Remarkably, with this observation, Narayanan and Pfister showed that applying the robust Soliton distribution as the left degree distribution of IRSA achieves the efficiency of $100\\%$ asymptotically [11], which is optimal among all IRSA schemes.","blackAlso, in [14], Liva et al.","proposed grouping many frames together to form a super frame and employing spatially-coupled IRSA on the super frame to achieve the efficiency of $100\\%$ asymptotically.","The performance of these UMA schemes were then assessed in a more realistic model in [15].","Several variants of IRSA have also been proposed and analyzed.","In [16], the frame-asynchronous version of IRSA was investigated and a boundary effect similar to that exhibited in spatially-coupled low-density parity-check (LDPC) codes [17] is observed.","Thanks to this boundary effect, it was shown that regular left degree distributions suffice to attain the optimal efficiency for frame-asynchronous IRSA.","In [18], a coded slotted ALOHA protocol was proposed where maximum distance separable (MDS) codes replace repetition codes when sending replicas.","It was shown that this class of schemes can achieve rates higher than that obtained by IRSA.","In [19], a combination of IRSA and physical-layer network coding [20] (a.k.a.","compute-and-forward [21]) was proposed, where the receiver decodes the received signal at a slot to an integer linear combination of collided packets.","These combinations are later used to recover individual packets that would not be decodable in conventional IRSA.","In contrast to the aforementioned works focusing on the asymptotic analysis, another series of work in the literature investigated non-asymptotic analysis of IRSA or coded slotted ALOHA.","For example, using packet loss rate and frame error rate as the metrics, the performance of IRSA in the waterfall region was analyzed in [22].","Moreover, the performance of coded slotted ALOHA in the error floor region and that in the waterfall region were separately investigated in [23] and [24], respectively.","Another highly related line of research named unsourced multiple access was initiated by Polyanskiy in [25], where a large number of homogeneous users wish to communicate with a receiver who is only interested in recovering as many users' data as possible without concern for their identities.","Using random Gaussian codebooks together with maximum likelihood decoding, [25] presented finite block-length achievability bounds.","A low-complexity coding scheme which employs the concatenation of an inner compute-and-forward code [21] and a outer binary adder channel code was then introduced and analyzed for unsourced multiple access in [26].","The current state of the art is [27] where a coded compress sensing algorithm followed by a tree-like code was proposed and analyzed.","Most of the works in the UMA literature, including those discussed above, considered homogeneous users.","However, in many applications in the mMTC use scenarios, several heterogeneous types of users coexist [4], [28], [29].","Therefore, it is imperative to understand how the heterogeneity would influence the system performance and it is crucial to design UMA schemes that can exploit such heterogeneity among users.","To address the heterogeneity among users in UMA, in this work, we incorporate the power-domain non-orthogonal multiple access (NOMA) [30], [31], [32], [33] into IRSA and proposed the IRSA with NOMA protocol.","In the proposed scheme, leveraging the power-domain NOMA in the physical layer, when packets coming from heterogenous types of users collide at a slot, it is possible that all the packets can be resolved by intra-slot SIC.","These decoded packets are then used to resolve collisions in other slots by inter-slot SIC.","We must emphasize that this is in sharp contrast to the idea of multi-packet reception [34].","In SA with multi-packet reception [35], [36], as long as the number of collided packets at a slot is smaller than some predefined parameter, all the packets can be decoded.","Evidently, those users are still homogeneous and the benefit of having heterogeneous users remains unexploited.","However, in our proposed IRSA with NOMA, not only the number of collided packets but also the types of users who transmit the packets will affect the decodabilityA more detailed comparison between SA with multi-packet reception and that with NOMA is relegated to Section ..","The main contributions of this paper are provided as follows.","A novel IRSA protocol that incorporates the power-domain NOMA, namely the IRSA with NOMA scheme, is proposed.","In such a protocol, depending on the types of users, multiple collided packets at a time slot may be decodable by intra-slot SIC.","The decoded packets are then used to resolve collisions in other slots by inter-slot SIC.","The connection between the proposed IRSA with NOMA and a bipartite graph with heterogeneous variable nodes and special check nodes is drawn.","This connection is then leveraged to propose a novel multi-dimensional density evolution, which assists us in analyzing the proposed IRSA with NOMA protocol.","According to the proposed multi-dimensional density evolution, a constrained optimization problem for maximizing the efficiency is formulated.","Through solving this constrained optimization problem, the best degree distribution of each type of users for the proposed IRSA with NOMA protocol can be found.","Extensive simulations are provided.","For the considered simulation environment with a practical frame size, it is shown that when there are two and three heterogeneous types, the proposed scheme achieves the sum (over types) efficiencies of up to $132\\%$ ($143\\%$ asymptotically) and $170\\%$ ($185\\%$ asymptotically), respectively.","Moreover, it also indicates that the analysis based on the proposed multi-dimensional density evolution can accurately predict the actual performance of the proposed scheme.","The extension of the proposed protocol to the frame-asynchronous IRSA is presented and the corresponding multi-dimensional density evolution is analyzed.","It is observed that under the frame-asynchronous circumstance, the proposed IRSA with NOMA also exhibits a boundary effect similar to that in [16].","Hence, left-regular degree distributions again suffice to achieve performance that is as good as that achieved by an optimal degree distribution.","blackDuring the revision of this paper, we became aware of a pioneering work [37], in which CRDSA with power unbalance is investigated from a joint physical and MAC layers perspective.","By allowing each user randomly selects its power according to a log-normal distribution, Mengali et al.","show that significant gains can be obtained by intra-slot SIC.","However, how to exploit heterogeneity inherent in the network, how to analyze the asymptotic performance, and how to obtain optimal degree distributions when there are multiple types in the network are left undiscussed.","In Section , we formally state the problem of UMA with heterogeneous types of users and propose the IRSA with NOMA protocol.","In Section , we present a graph representation of the proposed protocol and translate the decoding process as a modified peeling decoder for the corresponding graph.","In Section , to analyze the asymptotic performance of the proposed protocol, we propose a multi-dimensional density evolution technique, with which we formulate an optimization problem for obtaining optimal policies for our proposed protocol.","Optimal policies are then obtained by the differential evolution technique.","Simulation results are presented in Section , followed by blackthe extension of the proposed IRSA with NOMA scheme to the frame asynchronous setting in Sections .","Finally, some conclusion remarks are given in Section ."],["Notational Convention","We use $\\mathcal {F}[x]$ to denote a polynomial with variable $x$ and we denote by $\\mathcal {F}^{\\prime }[x]=\\mathrm {d}\\mathcal {F}[x]/\\mathrm {d}x$ and $\\mathcal {F}^{\\lbrace k\\rbrace }[x]=\\mathrm {d^k\\mathcal {F}[x]}/\\mathrm {d} x^k$ the first and $k$ -th derivative of $\\mathcal {F}[x]$ , respectively.","We denote by $\\mathbb {N}$ the set of natural numbers.","For two natural numbers $N_1, N_2\\in \\mathbb {N}$ with $N_1 < N_2$ , we denote by $[N_1]=\\lbrace 1, 2, \\ldots , N_1\\rbrace $ and $[N_1:N_2]=\\lbrace N_1, N_1+1, \\ldots , N_2\\rbrace $ .","We consider an UMA setting where a massive number of users coming from $T\\in \\mathbb {N}$ different types of applications wish to communicate with the base station as shown in Fig.","REF .","Suppose time is slotted and synchronized.","Furthermore, every $N\\in \\mathbb {N}$ successive slots are organized into a frame; frames are also perfectly synchronous among users, as in other conventional IRSA schemes [7], [9], [11].","A frame-asynchronous version of this problem is relegated to Section .","A user who has data to send within a frame is said to be active in this frame.","Potentially, there could be an arbitrary number of users for each type; Suppose $K^{(t)}\\in \\mathbb {N}$ of type $t\\in [T]$ users are active.","Each active user of type $t\\in [T]$ constructs $L$ replicas of its packet, where the value $L$ is sampled from a degree distribution $\\mathcal {L}^{(t)}$ ; then, the user sends the $L$ replicas in $L$ slots out of $N$ slots chosen uniformly at random in the frame.","The degree distribution is our design issue for maximizing efficiency (that will be discussed soon).","Figure: UMA with T=3T=3 heterogeneous types of users.blackLeveraging the natural heterogeneity and the maturity of SIC technique [38], we propose the decoder that employs power-domain NOMA [30], [31], [33], [32] at each slot.","Specifically, without loss of generality, we assume the decoding order of power-domain NOMA in the physical layer is given by $1\\rightarrow 2\\rightarrow \\ldots \\rightarrow T$ and we define the intra-SIC decodable patterns as follows.","Definition 1 (Type $t$ decodable pattern) For a slot $n\\in [N]$ , we define the transmission pattern $\\mathbf {c}_n=(c_1,\\ldots ,c_T)$ to be a $T$ -dimensional vector where the entry $c_t$ represents the total number of unresolved type $t$ packets at this slot.","Such a vector $\\mathbf {c}_n$ is type $t$ decodable if and only if the following conditions hold: $&\\text{(i) $c_t=1$},\\quad \\quad \\quad \\text{(ii) $c_{t^{\\prime }}\\le 1$ for $t^{\\prime }\\in [t-1]$}\\\\&\\text{(iii) $c_{\\tilde{t}}\\le \\tilde{t}-t$ for $\\tilde{t}\\in [t+1:T]$,} \\quad \\text{(iv) $\\sum _{\\tilde{t}=t+1}^Tc_{\\tilde{t}}\\le T-t$.","}$ The type $t$ decodable pattern is defined based on the principle of power-domain NOMA.","In power-domain NOMA, it is assumed that a packet can be successfully decoded by intra-slot SIC as long as each interfered type has at most one packet collided at the same slot.","This justifies the above conditions (i) and (ii).","Moreover, types having later decoding order means that they will cause smaller interference.","Therefore, trading a type $\\tilde{t}\\in [t+1:T]$ packet for a type $\\hat{t}>\\tilde{t}$ packet would not affect the decodability, justifying the conditions (iii) and (iv).","Definition 2 (Type $t$ decodable set) The type $t$ decodable set $\\mathcal {D}^{(t)}$ consists of all the type $t$ decodable patterns.","i.e., $\\mathcal {D}^{(t)}=\\lbrace \\mathbf {c}:\\mathbf {c}~\\text{is type $t$ decodable}\\rbrace $ .","Example 3 When $T=2$ , it is easy to check that $\\mathcal {D}^{(1)}=\\lbrace (1,0),(1,1)\\rbrace $ and $\\mathcal {D}^{(2)}=\\lbrace (0,1),(1,1)\\rbrace .$ When $T=3$ , we have $\\mathcal {D}^{(1)}&=\\lbrace (1,0,0), (1,0,1), (1,0,2),(1,1,0), (1,1,1)\\rbrace , \\\\\\mathcal {D}^{(2)}&=\\lbrace (0,1,0), (0,1,1), (1,1,0), (1,1,1)\\rbrace , \\\\\\mathcal {D}^{(3)}&=\\lbrace (0,0,1), (0,1,1), (1,0,1), (1,1,1)\\rbrace .$ The decoding process performed at the end of a frame then proceeds as follows.","For every slot $n\\in [N]$ during a frame, the corresponding transmission pattern $\\mathbf {c}_n$ is computedThe transmission patterns $\\mathbf {c}_n$ can evolve with iteration; however, we ignore the index for iterations to avoid heavy notation..","The intra-slot SIC is then employed to resolve a type $t$ packet if there is a $\\mathbf {c}_n$ for $n\\in [N]$ belonging to the type $t$ decodable set $\\mathcal {D}^{(t)}$ .","We assume that in the burst payload of each packet, there is information about the (other) slots containing copies of this packet.","Hence, the decoded packets are then used to cancel other replicas via inter-slot SIC.","This completes an iteration and the decoding process continues iteratively until no more packets are decodable for all $t\\in [T]$ and all $n\\in [N]$ or all the packets in this frame are decoded.","This process is done, independently, for every frame.","An example of the proposed IRSA with NOMA for $T=2$ can be found in Fig.","REF .","In this figure, the replicas of packets transmitted from the type 1 users, namely users 1 and 2, are colored in blue and that from the type 2 users, namely users 3 and 4, are colored in red.","In this example, user 1's packet can be easily decoded from slot 1.","Subtracting user 1's packet at slot 3 from the decoded packet, there's no more degree-1 slot and traditional IRSA would stop the decoding process.","For the proposed IRSA with NOMA, at slot 2, we receive the collision of a packet from type 1 (user 2) and another one from type 2 (user 4), both packets can be decoded via intra-slot SIC.","Hence, our proposed IRSA with NOMA would continue the decoding process by subtracting users 2 and 4's packets in slot 4 and decode user 4's packet.","Figure: An example of SA with heterogeneous users transmitting in a frame of 5 slots.","Here, we have T=2T=2 types where the first type consists of users 1 and 2 and the second type consists of users 3 and 4.At the end of the decoding process, suppose the base station can successfully decode $\\hat{K}^{(t)}$ packets of type $t\\in [T]$ , we define the (actual) efficiency of type $t\\in [T]$ achieved by the scheme as $\\hat{\\eta }^{(t)} = \\frac{\\hat{K}^{(t)}}{N},\\quad t\\in [T],$ and the (actual) sum efficiency of the scheme as $ \\hat{\\eta } = \\sum _{i=1}^T \\hat{\\eta }^{(t)}.$ It is clear that choosing different $\\mathcal {L}^{(t)}$ would result in different efficiency.","Throughout the paper, we call a collection of $T$ degree distributions $\\lbrace \\mathcal {L}^{(1)}, \\mathcal {L}^{(2)}, \\ldots , \\mathcal {L}^{(T)}\\rbrace $ , one for each type $t\\in [T]$ , a policy.","The goal of this paper is then to design the policy, such that the achieved efficiency is maximized."],["Comparison with existing models","In this subsection, we discuss differences between the model considered in this paper and existing models.","Remark 4 It is not difficult to see that when $T=1$ , i.e., there is only one type of homogeneous users, the proposed protocol reduces to the IRSA protocol in [9].","In light of this, the proposed protocol can be regarded as a generalization of the IRSA protocol to accommodate heterogeneous types of users.","Remark 5 We note that both multi-packet reception [34] and power-domain NOMA [30], [31], [32], [33] are based on the same physical-layer model, namely the multiple access channel [39].","However, the multi-packet reception technique treats all the users equally and assumes collided packets are decodable as long as the number of users collided at a slot is smaller than a predefined number.","Hence, in this homogeneous setting, the problem in the physical layer becomes to establish reliable multiple access with homogeneous users that have the same codebook and code rate.","To the best of our knowledge, this is not an easy task that is tackled with some success in [40] and [41], which involves either extremely large blocklength due to spatial coupling [40] or very complex operations of coding over prime fieldsThis is due to the adoption of Construction A lattices [42].. On the contrary, the power-domain NOMA lets the users transmit with different power levels and/or rates so that low-complex SIC can be employed to achieve a corner point of the capacity regionblackHere, the standard information-theoretic sense of achievability [39] is considered.. An illustration can be found in Fig.","REF .","In this paper, the considered problem exhibits natural heterogeneity among users; hence, it is more natural and suitable to work with power-domain NOMA so that the heterogeneity can be exploited by intra-SIC.","The main theme of this paper then centers around analyzing policies for IRSA with power-domain NOMA and designing policies that optimally exploit the heterogeneity inherent in the considered problem.","Figure: The capacity region of a two-user multiple access channel.Remark 6 The combination of NOMA and SA has been investigated before in [43], [44] by Choi.","However, in these works, it is assumed that there is only one type of homogeneous users.","The focus of [43] is mainly to analyze the performance of SA when users randomly choose their power levels and the receiver employs intra-slot SIC.","Moreover, in both works [43], [44], the inter-slot SIC technique is not considered.","Remark 7 We also note that our model is fundamentally different from IRSA with capture effect discussed in [9] and [45].","With capture effect, it is assumed that the receiver may be able to decode more than one packet with some probability.","This opportunity usually comes from the effect of random channel fading.","On the other hand, in the considered model, the receiver is able to decode more than one packet deterministically for certain configurations of collided users, due to the natural heterogeneity inherent in the problem."],["Implementation Issues","We now address potential implementation issues of the proposed IRSA with NOMA.","We would like to note that many of these issues have been encountered and addressed by the conventional IRSA schemes [9].","However, we still present them here for the sake of self-containedness.","Slot synchronization: This issue is inherent in all slotted multiple access schemes.","One easy solution to provide slot synchronization is to rely on stable clocks and a small amount of guard time between packets [5].","blackMoreover, for applications with cheap user devices, stable clocks may be unaffordable.","For such scenarios, [46] provides a technique to achieve slot synchronization, where each node constantly re-synchronizes its clock with the base station by exploiting the ACK signals fed back from the base station.","Another source of slot synchronization errors which is less seen in traditional communication systems is from significantly different travel distances experienced by different devices' signals.","This type of source of slot synchronization errors has become important in non-terrestrial networks (NTN) [47].","Fortunately, in NTN, each device is required to be GNSSGNSS stands for Global Navigation Satellite System.-capable for acquiring its own position and together with satellite ephemeris.","Each user can calculate and then pre-compensate its timing advance (TA) (see Section 6.3 in [47]).","To reduce devices' computation loading, another candidate technique currently under discussion in 3GPP is that the base station estimates and broadcasts TA to a group of devices who have similar TA (see Section 6.3 in [47]).","Frame synchronization: This issue is also not new and has to be cope with in every frame-synchronous systems.","This issue can be addressed by letting the base station periodically broadcasting a beacon signal at the beginning of each frame.","In Section , extension to frame asynchronous setting will be discussed.","Inter-slot SIC: Similar to CRDSA [7] and IRSA schemes [9], to enable inter-slot SIC, the proposed protocol requires the knowledge of the locations of other replicas once a packet is successfully decoded.","blackThis can be achieved by including pointers to the locations of other replicas into the burst payload or by some pseudo-random mechanism known by both the transmit and receive ends.","For example, the DVB system [8] adopts the latter one, where each active user reports its MAC address and a pseudo-random number to the base station.","Both the base station and the user can then compute the slots used for sending replicas by a common deterministic function of the MAC address and the pseudo-random number.","Intra-slot SIC: blackFor the proposed scheme, in order to enable intra-slot SIC at a slot, it appears that the receiver has to know which types of packets are collided in this slot.","This issue is new in our proposed protocol.","blackWe provide two ways to address this issue.","We note that in DVB-RCS2 [8], each active user selects the number of transmissions and a random seed for uniquely determining a pseudo random pattern.","This information is then stored in the burst payload and sent to the base station.","Given the existing data structure, to enable intra-slot SIC, each user can include additional $\\log (T)$ bits to the payload for revealing its type.","The second method described in the sequel eliminates the need for sending extra $\\log (T)$ bits by increasing the decoding burden.","Let $R^{(t)}$ be the rate of the physical-layer channel code adopted by users of type $t\\in [T]$ .","Let us assume the We assume that the rate tuple $(R^{(1)}, R^{(2)}, \\ldots , R^{(T)})$ can be achieved by intra-slot SIC with ascending decoding order $1\\rightarrow 2\\rightarrow \\ldots \\rightarrow T$ , without loss of generality.","Following the NOMA principle, a means to address this issue is as follows: Starting from $t=1$ , the base station tries to decode a packet of type $t$ ; if it succeedsThe success of decoding can be checked by a cyclic redundancy check mechanism., the decoded packet is subtracted from the received signal and we set $t=t+1$ ; otherwise, directly set $t=t+1$ and look for decoding opportunity of the next type.","This procedure would continue until $t=T$ ."],["Graph Representation","In this section, we introduce a graph representation of the proposed IRSA with NOMA protocol for determining an optimal policy in Section ."],["Bipartite Graph and Degree Distributions","We first note that in UMA with $T$ types of heterogeneous users, each type has its own transmission policy and number of users that may be different from other types.","We construct a bipartite graph representation of the proposed IRSA with NOMA protocol.","The graph consists of $\\sum _{t=1}^T K^{(t)}$ variable nodes and $N$ super check nodes, as shown in Fig.","REF .","Each variable node $\\mathsf {v}^{(t)}_j$ represents a user $j\\in [K^{(t)}]$ of type $t\\in [T]$ .","Each super check node consists of $T$ check nodes $\\mathsf {c}^{(t)}_n$ , one for each type $t\\in [T]$ .","A user $j$ of type $t\\in [T]$ transmits a packet in slot $n\\in [N]$ if and only if there is an edge connecting $\\mathsf {v}^{(t)}_j$ with $\\mathsf {c}^{(t)}_n$ .","For simplicity, we use $\\mathsf {v}^{(t)}$ and $\\mathsf {c}^{(t)}$ to denote a generic variable and check nodes of type $t\\in [T]$ .","Figure: A bipartite graph representation of the proposed IRSA with NOMA.With this bipartite graph representation, a user of type $t$ who transmits $d$ replicas corresponds to a degree $d$ variable node $\\mathsf {v}^{(t)}$ .","Moreover, a time slot in which $d$ type $t$ variable nodes transmit corresponds to a degree $d$ check node $\\mathsf {c}^{(t)}$ .","We define the node perspective left and right degree distributions as $\\mathcal {L}^{(t)}[x] = \\sum _{d} L^{(t)}_d x^d \\quad \\text{and}\\quad \\mathcal {R}^{(t)}[x] = \\sum _d R^{(t)}_d x^d,$ respectively, where $L^{(t)}_d$ is the probability that a variable node $\\mathsf {v}^{(t)}$ has degree $d$ and $R^{(t)}_d$ is the probability that a check node $\\mathsf {c}^{(t)}$ has degree $d$ .","Through the node perspective degree distributions, the edge perspective left and right degree distributions can be given by $\\lambda ^{(t)}[x] = \\sum _d \\lambda ^{(t)}_d x^{d-1}=\\frac{\\mathcal {L}^{^{\\prime }(t)}[x]}{\\mathcal {L}^{^{\\prime }(t)}[1]},$ and $\\rho ^{(t)}[x] &= \\sum _d\\rho ^{(t)}_d x^{d-1} =\\frac{\\mathcal {R}^{^{\\prime }(t)}[x]}{\\mathcal {R}^{^{\\prime }(t)}[1]},$ respectively, where $\\lambda ^{(t)}_d$ stands for the probability that an edge connects to a type $t$ variable node of degree $d$ and $\\rho ^{(t)}_d$ is the probability that an edge connects to a type $t$ check node of degree $d$ .","The target efficiency of type $t\\in [T]$ is then defined as $\\eta ^{(t)}= \\frac{K^{(t)}}{N} = \\frac{\\mathcal {R}^{^{\\prime }(t)}[1]}{\\mathcal {L}^{^{\\prime }(t)}[1]},$ and the target sum efficiency of the scheme is defined as $ \\eta = \\sum _{t=1}^T \\eta ^{(t)}.$ Note that the target efficiency $\\eta ^{(t)}$ is different from the actual efficiency $\\hat{\\eta }^{(t)}$ in (REF ).","In general, $\\hat{\\eta }^{(t)}\\le \\eta ^{(t)}$ with equality if and only if all packets are successfully decoded.","Example 8 Consider the scheme described in Fig.","REF where there is one type 1 variable node of degree 2 and one type 1 variable node of degree 3.","Hence, $\\mathcal {L}^{(1)}[x]=0.5x^2 +0.5x^3$ .","One can obtain $\\mathcal {L}^{(2)}[x]=x^3$ in a similar fashion.","Also, there are three type 1 check nodes of degree 1 and one type 1 check node of degree 2.","Therefore, $\\mathcal {R}^{(1)}[x]=0.75x+0.25x^2$ .","Similarly, $\\mathcal {R}^{(2)}[x]=0.5x+0.5x^2.$ Moreover, one can easily verify that the edge perspective degree distributions are given by $\\lambda ^{(1)}[x]= \\frac{2}{5}x + \\frac{3}{5}x^2, \\quad \\lambda ^{(2)}[x]=x^2, \\quad \\rho ^{(1)}[x]=\\frac{3}{5}+\\frac{2}{5}x,\\quad \\rho ^{(2)}[x]=\\frac{1}{3}+\\frac{2}{3}x.$ The graph representation of this example can be found in Fig.","REF -(a).","Figure: The graph representation of the scheme in Fig.","and its decoding output at each iteration of the modified peeling decoder."],["Modified Peeling Decoder","Here, we propose a modified peeling decoder for the bipartite graph that corresponds to the decoding procedure of the proposed IRSA with NOMA.","The decoder operates in an iterative fashion.","For each iteration, the decoder looks at every super check node $n\\in [N]$ and computes the remaining degrees of check nodes $\\mathsf {c}^{(t)}_1, \\mathsf {c}^{(t)}_2, \\ldots , \\mathsf {c}^{(t)}_T$ , which are stored in the transmission pattern $\\mathbf {c}_n$ .","We then look for an $n$ such that $\\mathbf {c}_n$ belongs to the type $t$ decodable set $\\mathcal {D}^{(t)}$ for some $t\\in [T]$ .","If such an $n$ exists, we use intra-slot SIC to decode all the type $t$ packets satisfying $\\mathbf {c}_n\\in \\mathcal {D}^{(t)}$ in this super check node $n$ .","For a variable node $\\mathsf {v}^{(t)}$ that is decoded, all the edges connected with it are removed from the graph.","This concludes an iteration.","The modified peeling decoder operates iteratively until no more decoding opportunity is found.","Example 8 (continued) In this example, we illustrate how to use the modified peeling decoder to decode the scheme in Example REF .","As shown in Fig.","REF -(b), in iteration 1, $\\mathsf {c}^{(1)}_1$ has degree 1 and $\\mathsf {c}^{(2)}_1$ has degree 0, i.e., $\\mathbf {c}_1=(1,0)\\in \\mathcal {D}^{(1)}$ ; hence, $\\mathsf {v}^{(1)}_1$ is decoded and all its edges are removed.","Then, in iteration 2 shown in Fig.","REF -(c), $\\mathsf {c}^{(1)}_2$ has degree 1 and $\\mathsf {c}^{(2)}_2$ has degree 1, i.e., $\\mathbf {c}_2=(1,1)$ belonging to both $\\mathcal {D}^{(1)}$ and $\\mathcal {D}^{(2)}$ ; thus, both $\\mathsf {v}^{(1)}_2$ and $\\mathsf {v}^{(2)}_2$ can be decoded by intra-slot SIC.","All the edges connected to $\\mathsf {v}^{(1)}_2$ and $\\mathsf {v}^{(2)}_2$ are subsequently removed.","Last, in iteration 3 shown in Fig.","REF -(d), one can easily see that $\\mathsf {v}^{(2)}_1$ can be decoded."],["Multi-Dimensional Density Evolution and Convergence Analysis","In this section, to analyze the IRSA with NOMA, we propose a novel multi-dimensional density evolution based on the graph representation described in Section .","With this multi-dimensional density evolution, we then provide the convergence analysis of the IRSA with NOMA under modified peeling decoding.","With an assist from the convergence analysis, we formulate and numerically solve an optimization problem that finds best left degree distributions, leading to the highest asymptotic efficiency.","For clearly explaining the ideas behind our scheme and analysis, we focus on $T=2$ .","The general $T$ case can be similarly analyzed."],["Proposed Multi-Dimensional Density Evolution","It is well known that for LDPC codes over a binary erasure channel, BP and peeling decoders have the same performanceIt essentially means that the order of the limit of average residue erasure probability as the number of iterations tends to $\\infty $ and the limit of that as the blocklength tends to $\\infty $ is exchangeable.","The interested reader is referred to [10]The reason that we propose using the peeling decoder instead of BP is because the peeling decoder potentially leads to a smaller decoding latency in practice as it does not have to wait until the end of the frame in order to start decoding; it can start decoding right away once a decodable slot shows up..","Moreover, density evolution is an outstanding tool for analyzing the performance of BP decoding over a sparse graph [10].","Thus, this section proposes a novel multi-dimensional density evolution to analyze the performance of the proposed scheme under BP decoding.","blackSimilar to that introduced in [10], the BP decoding is an iterative algorithm where in each iteration, each variable node computes the belief of its own message and passes an extrinsic version of it to each connected check node, while each check node computes its belief of the message of each connected variable node and passes an extrinsic version to that variable node.","After a predefined number of iterations, the decoding algorithm halts and outputs hard decisions according to the last beliefs held by the variable nodes.","blackConsider BP decoding described above.","For $t\\in [2]$ , we denote by $x_\\ell ^{(t)}$ and $y_\\ell ^{(t)}$ the average erasure probability of the message passed along an edge from $\\mathsf {v}^{(t)}$ to $\\mathsf {c}^{(t)}$ and that of the message passed along an edge from $\\mathsf {c}^{(t)}$ to $\\mathsf {v}^{(t)}$ in iteration $\\ell $ , respectively.","At iteration $\\ell =0$ , according to the decodable set $\\mathcal {D}^{(t)}$ specified in Example REF , for an edge connected to a type $t$ check node $\\mathsf {c}^{(t)}$ , the outgoing message is not in erasure if and only if a) $\\mathsf {c}^{(t)}$ has degree 1; b) $\\mathsf {c}^{(\\bar{t})}$ has a degree either 0 or 1 for every $\\bar{t}\\in [2],~\\bar{t}\\ne t$ .","Hence, we initialize $y_0^{(t)}$ for initial iteration $\\ell =0$ to be $y_0^{(t)}= 1-\\rho ^{(t)}[0]\\cdot \\left( \\mathcal {R}^{(\\bar{t})}[0] + \\mathcal {R}^{^{\\prime }(\\bar{t})}[0] \\right),$ where $\\mathcal {R}^{(\\bar{t})}[0]=R^{(\\bar{t})}_0$ and $\\mathcal {R}^{^{\\prime }(\\bar{t})}[0]=R^{(\\bar{t})}_1$ are the fractions of type $\\bar{t}$ check nodes having degree 0 and degree 1, respectively.","Moreover, the term $\\rho ^{(t)}[0]=\\rho _1^{(t)}$ is the probability that an edge connects to a type $t$ check node of degree 1.","Suppose we have obtained $x_\\ell ^{(t)}$ and $y_\\ell ^{(t)}$ for all $t\\in [T]$ from iteration $\\ell $ .","In iteration $\\ell +1$ , for an edge incident to a variable node $\\mathsf {v}^{(t)}$ with degree $d$ , the only possibility that the message along this edge to a $\\mathsf {c}^{(t)}$ is in erasure is that all the other $d-1$ edges are in erasure.","Therefore, the probability that the message passed along this edge is in erasure is $(y_\\ell ^{(t)})^{d-1}$ .","Now, averaging over all the edges results in the average erasure probability $x_\\ell ^{(t)}= \\sum _d \\lambda ^{(t)}_d (y_\\ell ^{(t)})^{d-1} = \\lambda ^{(t)}[y_\\ell ^{(t)}].$ For an edge incident to a check node $\\mathsf {c}^{(t)}$ with degree $d$ , the message passed along this edge to a $\\mathsf {v}^{(t)}$ is not in erasure if and only if a) all the other $d-1$ edges incident to $\\mathsf {c}^{(t)}$ are not in erasure in the previous iteration; and b) all but at most one of the edges incident to $\\mathsf {c}^{(\\bar{t})}$ are not in erasure in the previous iteration for every $\\bar{t}\\in [2],~\\bar{t}\\ne t$ .","Suppose in the same super check node $n$ , $\\mathsf {c}^{(\\bar{t})}_n$ has degree $d_{\\bar{t}}$ .","Then the above event has probability $(1-x_\\ell ^{(t)})^{d-1}\\cdot \\left( (1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}} + d_{\\bar{t}} x_\\ell ^{(\\bar{t})}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-1}\\right),$ where $(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}}$ is the probability that all $d_{\\bar{t}}$ edges are not erased and $d_{\\bar{t}} x_\\ell ^{(\\bar{t})}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-1}$ is the probability that all but one edges are not erased.","Now, averaging over all the edges and over all the type $\\bar{t}$ check nodes for $\\bar{t}\\in [2],~\\bar{t}\\ne t$ shows that the average probability of correct decoding is given by $&\\sum _d \\lambda ^{(t)}_d(1-x_\\ell ^{(t)})^{d-1}\\cdot \\nonumber \\sum _{d_{\\bar{t}}}R^{(\\bar{t})}_{d_{\\bar{t}}}\\left( (1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}} + d_{\\bar{t}} x_\\ell ^{(\\bar{t})}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-1}\\right) \\nonumber \\\\&=\\rho ^{(t)}[1-x_\\ell ^{(t)}] \\left( \\mathcal {R}^{(\\bar{t})}[1-x_\\ell ^{(\\bar{t})}] + x_\\ell ^{(\\bar{t})}\\mathcal {R}^{^{\\prime }(\\bar{t})}[1-x_\\ell ^{(\\bar{t})}]\\right).$ Therefore, the average erasure probability becomes $y_{\\ell +1}^{(t)}= 1-\\rho ^{(t)}[1-x_\\ell ^{(t)}]\\cdot \\left( \\mathcal {R}^{(\\bar{t})}[1-x_\\ell ^{(\\bar{t})}] + x_\\ell ^{(\\bar{t})}\\mathcal {R}^{^{\\prime }(\\bar{t})}[1-x_\\ell ^{(\\bar{t})}]\\right).$ Plugging (REF ) into (REF ) leads to the evolution of average erasure probability of a type $t$ check node as shown in (REF ) in the bottom of next page.","Figure: NO_CAPTION"],["Convergence and Stability Condition","After obtaining the density evolution in (REF ), for any given degree distributions $\\mathcal {L}^{(t)}$ (or $\\lambda ^{(t)}$ ) and $\\mathcal {R}^{(t)}$ (or $\\rho ^{(t)}$ ), one can now analyze whether the average erasure probability converges to 0 by checking whether $y_\\ell ^{(t)}> y_{\\ell +1}^{(t)}$ for every $\\ell $ , starting from $y_0^{(t)}=1$ for $t\\in [T]$ .","Moreover, to make sure that the average erasure probability indeed vanishes instead of hovering around 0, we derive the following stability conditionblackNote that the term “stability\" used here is referred to the stability of fixed points rather than that in the stability analysis of slotted ALOHA [48], [5].. We enforce $\\lambda _1^{(t)}=0$ for all $t\\in [2]$ because we certainly do not want degree 1 variable nodes.","Then, assuming $y^{(t)}$ is very small for all $t\\in [2]$ , we expand the degree distributions and approximate them by keeping only the linear terms as follows, $\\lambda ^{(t)}[y^{(t)}] &\\approx \\lambda _2^{(t)}y^{(t)}, \\\\\\rho ^{(t)}[1-\\lambda ^{(t)}[y^{(t)}]] &\\approx 1-\\rho ^{^{\\prime }{(t)}}[1]\\lambda _2^{(t)}y^{(t)}, \\\\R^{(\\bar{t})}[1-\\lambda ^{(\\bar{t})}[y^{(\\bar{t})}]] &\\approx 1-R^{^{\\prime }(\\bar{t})}[1] \\lambda _2^{(\\bar{t})} y^{(\\bar{t})}, \\\\R^{^{\\prime }(\\bar{t})}[1-\\lambda ^{(\\bar{t})}[y^{(\\bar{t})}]] &\\approx R^{^{\\prime }(\\bar{t})}[1] - R^{^{\\prime \\prime }(\\bar{t})}[1]\\lambda _2^{(\\bar{t})}y^{(\\bar{t})}.$ We can now linearize the recursion around 0 by plugging (REF )-() into (REF ) to get $y^{(t)}> 1- (1-\\rho ^{^{\\prime }{(t)}}[1]\\lambda _2^{(t)}y^{(t)}) \\left(1-R^{^{\\prime \\prime }(\\bar{t})}[1](\\lambda _2^{(t)}y^{(\\bar{t})})^2\\right) \\approx \\rho ^{^{\\prime }{(t)}}[1]\\lambda _2^{(t)}y^{(t)},$ which leads to the following stability condition $\\lambda _2^{(t)}< \\frac{1}{\\rho ^{^{\\prime }{(t)}}[1]}\\quad \\text{for $t\\in [2]$}.$"],["Optimization Problem","Before we formulate the optimization problem, we note that there are two sets of degree distributions $\\lbrace \\mathcal {L}^{(t)}[x]\\rbrace $ and $\\lbrace \\mathcal {R}^{(t)}[x]\\rbrace $ in the proposed problem, but we only have control over $\\lbrace \\mathcal {L}^{(t)}[x]\\rbrace $ .","The behavior of $\\lbrace \\mathcal {R}^{(t)}[x]\\rbrace $ ; however, are completely determined by how the users behave.","Specifically, according to the protocol, a type $t$ user $\\mathsf {v}^{(t)}$ having degree $L$ will send a replica in slot $\\mathsf {c}^{(t)}$ with probability $L/N$ .","Hence the average probability that a user $\\mathsf {v}^{(t)}$ sending a replica in slot $\\mathsf {c}^{(t)}$ is given by $ \\frac{\\mathcal {L}^{^{\\prime }(t)}[1]}{N} = \\frac{\\mathcal {R}^{^{\\prime }(t)}[1]}{K^{(t)}}.$ Thus, the degrees of $\\mathsf {c}^{(t)}$ follows the Binomial distribution with parameter $\\frac{\\mathcal {L}^{^{\\prime }(t)}[1]}{N}$ .","Similar to [9], by Poisson approximation [49], we have $\\mathcal {R}^{(t)}[x] \\approx \\exp \\left( -\\mathcal {R}^{^{\\prime }(t)}[1](1-x) \\right) = \\exp \\left( -\\eta ^{(t)}\\mathcal {L}^{^{\\prime }(t)}[1](1-x) \\right).$ Moreover, from (REF ), we have $\\rho ^{(t)}[x]= \\exp \\left( -\\eta ^{(t)}\\mathcal {L}^{^{\\prime }(t)}[1](1-x) \\right).$ blackNow, the convergence condition becomes (REF ) in the bottom of the this page by plugging (REF ) into (REF ) and noting that it is sufficient that $y^{(t)}_{\\ell +1} y_{\\ell +1}^{(t)}$ for every $\\ell $ and every $y_\\ell ^{(t)}>0$ .","In what follows, we again derive the stability condition.","Similar to the $T=2$ case, we enforce $\\lambda _1^{(t)}=0$ for all $t\\in [T]$ .","Assuming $y^{(t)}$ is very small for all $t\\in [T]$ , we expand the degree distributions and approximate them by keeping only the linear terms as follows, black $\\lambda ^{(t)}[y^{(t)}] &\\approx \\lambda _2^{(t)}y^{(t)}, \\\\\\rho ^{(t)}[1-\\lambda ^{(t)}[y^{(t)}]] &\\approx 1-\\rho ^{^{\\prime }{(t)}}[1]\\lambda _2^{(t)}y^{(t)}, \\\\R^{(\\bar{t})}[1-\\lambda ^{(\\bar{t})}[y^{(\\bar{t})}]] &\\approx 1-R^{^{\\prime }(\\bar{t})}[1] \\lambda _2^{(\\bar{t})} y^{(\\bar{t})}, \\\\(R^{(\\bar{t})})^{\\lbrace k\\rbrace }[1-\\lambda ^{(\\bar{t})}[y^{(\\bar{t})}]] &\\approx (R^{(\\bar{t})})^{\\lbrace k\\rbrace }[1] \\nonumber \\\\&- (R^{(\\bar{t})}[1])^{\\lbrace k+1\\rbrace }\\lambda _2^{(\\bar{t})}y^{(\\bar{t})}.$ We can now linearize the recursion around 0 by plugging (REF )-() into (REF ) to get the same stability condition $\\lambda _2^{(t)}< \\frac{1}{\\rho ^{^{\\prime }{(t)}}[1]}\\quad \\text{for $t\\in [T]$}.$"],["Optimization Problem and Optimized Degree Distributions","Similar to the $T=2$ case, we again apply the Poisson approximation and rewrite the convergence condition in (REF ) as (REF ) in the bottom of the next page.","Figure: NO_CAPTIONFinally, we are able to formulate the optimization problem that maximizes the target efficiency subject to derived conditions: $&\\!\\max _{\\lbrace \\mathcal {L}^{(t)}[x]\\rbrace } &\\qquad & \\eta =\\sum _{t=1}^T \\eta ^{(t)}\\\\&\\text{subject to} & & \\text{convergence condition~}(\\ref {eqn:DE_full_poisson_T}),\\\\& & & \\lambda _1^{(t)}=0~\\text{for all $t\\in [T]$},\\\\& & & L_d^{(t)}\\ge 0, ~\\text{for all $d\\in [d_{\\mathrm {max}}^{(t)}]$ and $t\\in [T]$}, \\\\& & & \\mathcal {L}^{(t)}[1]=1~\\text{for all $t\\in [T]$}.$ Similar to $T=2$ , one may include the stability condition (REF ) into the above optimization problem to make sure that the packet loss rate indeed vanishes as $N\\rightarrow \\infty $ ."]],"string":"[\n [\n \"Iterative Collision Resolution for Slotted ALOHA with NOMA for\\n Heterogeneous Devices\"\n ],\n [\n \"Abstract In this paper, the problem of using uncoordinated multiple access (UMA) to serve a massive amount of heterogeneous users is investigated.\",\n \"Leveraging the heterogeneity, we propose a novel UMA protocol, called iterative collision resolution for slotted ALOHA (IRSA) with non-orthogonal multiple access (NOMA), to improve the conventional IRSA.\",\n \"In addition to the inter-slot successive interference cancellation (SIC) technique used in existing IRSA-based schemes, the proposed protocol further employs the intra-slot SIC technique that enables collision resolution for certain configurations of collided packets.\",\n \"A novel multi-dimensional density evolution is then proposed to analyze and to optimize the proposed protocol.\",\n \"Simulation results show that the proposed IRSA with NOMA protocol can efficiently exploit the heterogeneity among users and the multi-dimensional density evolution can accurately predict the throughput performance.\",\n \"Last, an extension of the proposed IRSA with NOMA protocol to the frame-asynchronous setting is investigated, where a boundary effect similar to that in spatially-coupled low-density parity check codes can be observed to bootstrap the decoding process.\"\n ],\n [\n \"Introduction\",\n \"To realize the overarching ambition of internet of things (IoT) [1], [2], the current 3GPP Specification of the fifth generation cellular network technology has defined a use scenario, termed massive machine-type communication (mMTC) [3], [4].\",\n \"This use scenario is expected to accommodate a massive number of users with sporadic activities.\",\n \"Moreover, in such applications, the payload is assumed to be rather small.\",\n \"In this circumstance, traditional philosophy of first coordinating then communicating becomes extremely inefficient, as there are so many users to coordinate while so small amount of data for each user to transmit.\",\n \"Hence, uncoordinated multiple access (UMA) techniques that can communicate without first establishing coordination are highly desirable.\",\n \"Among many UMA techniques, slotted ALOHA (SA) [5] has been adopted in many practical communication systems.\",\n \"In an SA protocol, each user transmits a packet at the time slot next to that in which the data packet arrived.\",\n \"When multiple users send packets at the same slot, i.e., the packets collide, these collided packets are discarded black (i.e., collision channel model) and retransmissions are scheduled according to some back-off mechanism.\",\n \"Recently, this concept has even been adopted in modern cellular systems such as SigFoX and LoRaWAN [6].\",\n \"Notwithstanding the success and popularity, it is well known that the conventional SA achieves an efficiency (to be defined in Section ) of at most $1/e\\\\approx 36.8 \\\\%$ [5].\",\n \"To improve the efficiency, Casini et al.\",\n \"in [7] proposed contention resolution diversity slotted ALOHA (CRDSA) that improves the efficiency to be at most $55\\\\%$ blackasymptotically as the frame size goes to infinity.\",\n \"In CRDSA, each active user (those having data to send) sends replicated packets on twoblackHigher efficiencies can be achieved by sending more replicas.\",\n \"randomly selected slots .\",\n \"The collided packets are not discarded in CRDSA; instead, they are stored with the hope that inter-slotHere, we particularly highlight the term “inter-slot\\\" for distinguishing it from the intra-slot SIC discussed in Section .\",\n \"successive interference cancellation (SIC) can resolve collisions by their replicas.\",\n \"Due to its improved efficiency, a version of CRDSA has been included into the digital video broadcasting (DVB) standardization [8].\",\n \"After the paradigm-shifting idea [7], much effort has been devoted to improve CRDSA by making connections to error-correction codes [9].\",\n \"To the best of our knowledge, the connection was first made in [9] where a CRDSA system was shown to have a left-regular bipartite graph representation and inter-slot SIC is interpreted as belief propagation (BP) decoding [10].\",\n \"After this connection, Liva in [9] then proposed irregular repetition slotted ALOHA (IRSA), in which each active user randomly chooses the number of repetitions and randomly selects time slots to send those replicas.\",\n \"It can be observed that an IRSA scheme corresponds to an irregular bipartite graph.\",\n \"It was shown in [9] that by optimizing the left degree distribution with the help of density evolution [10], one can achieve a significantly improved efficiency of $96.5\\\\%$ blackasymptotically.\",\n \"It was then recognized in [11] that the coding problem induced by IRSA is similar to that of LT codes [12] with the peeling decoder [13].\",\n \"Remarkably, with this observation, Narayanan and Pfister showed that applying the robust Soliton distribution as the left degree distribution of IRSA achieves the efficiency of $100\\\\%$ asymptotically [11], which is optimal among all IRSA schemes.\",\n \"blackAlso, in [14], Liva et al.\",\n \"proposed grouping many frames together to form a super frame and employing spatially-coupled IRSA on the super frame to achieve the efficiency of $100\\\\%$ asymptotically.\",\n \"The performance of these UMA schemes were then assessed in a more realistic model in [15].\",\n \"Several variants of IRSA have also been proposed and analyzed.\",\n \"In [16], the frame-asynchronous version of IRSA was investigated and a boundary effect similar to that exhibited in spatially-coupled low-density parity-check (LDPC) codes [17] is observed.\",\n \"Thanks to this boundary effect, it was shown that regular left degree distributions suffice to attain the optimal efficiency for frame-asynchronous IRSA.\",\n \"In [18], a coded slotted ALOHA protocol was proposed where maximum distance separable (MDS) codes replace repetition codes when sending replicas.\",\n \"It was shown that this class of schemes can achieve rates higher than that obtained by IRSA.\",\n \"In [19], a combination of IRSA and physical-layer network coding [20] (a.k.a.\",\n \"compute-and-forward [21]) was proposed, where the receiver decodes the received signal at a slot to an integer linear combination of collided packets.\",\n \"These combinations are later used to recover individual packets that would not be decodable in conventional IRSA.\",\n \"In contrast to the aforementioned works focusing on the asymptotic analysis, another series of work in the literature investigated non-asymptotic analysis of IRSA or coded slotted ALOHA.\",\n \"For example, using packet loss rate and frame error rate as the metrics, the performance of IRSA in the waterfall region was analyzed in [22].\",\n \"Moreover, the performance of coded slotted ALOHA in the error floor region and that in the waterfall region were separately investigated in [23] and [24], respectively.\",\n \"Another highly related line of research named unsourced multiple access was initiated by Polyanskiy in [25], where a large number of homogeneous users wish to communicate with a receiver who is only interested in recovering as many users' data as possible without concern for their identities.\",\n \"Using random Gaussian codebooks together with maximum likelihood decoding, [25] presented finite block-length achievability bounds.\",\n \"A low-complexity coding scheme which employs the concatenation of an inner compute-and-forward code [21] and a outer binary adder channel code was then introduced and analyzed for unsourced multiple access in [26].\",\n \"The current state of the art is [27] where a coded compress sensing algorithm followed by a tree-like code was proposed and analyzed.\",\n \"Most of the works in the UMA literature, including those discussed above, considered homogeneous users.\",\n \"However, in many applications in the mMTC use scenarios, several heterogeneous types of users coexist [4], [28], [29].\",\n \"Therefore, it is imperative to understand how the heterogeneity would influence the system performance and it is crucial to design UMA schemes that can exploit such heterogeneity among users.\",\n \"To address the heterogeneity among users in UMA, in this work, we incorporate the power-domain non-orthogonal multiple access (NOMA) [30], [31], [32], [33] into IRSA and proposed the IRSA with NOMA protocol.\",\n \"In the proposed scheme, leveraging the power-domain NOMA in the physical layer, when packets coming from heterogenous types of users collide at a slot, it is possible that all the packets can be resolved by intra-slot SIC.\",\n \"These decoded packets are then used to resolve collisions in other slots by inter-slot SIC.\",\n \"We must emphasize that this is in sharp contrast to the idea of multi-packet reception [34].\",\n \"In SA with multi-packet reception [35], [36], as long as the number of collided packets at a slot is smaller than some predefined parameter, all the packets can be decoded.\",\n \"Evidently, those users are still homogeneous and the benefit of having heterogeneous users remains unexploited.\",\n \"However, in our proposed IRSA with NOMA, not only the number of collided packets but also the types of users who transmit the packets will affect the decodabilityA more detailed comparison between SA with multi-packet reception and that with NOMA is relegated to Section ..\",\n \"The main contributions of this paper are provided as follows.\",\n \"A novel IRSA protocol that incorporates the power-domain NOMA, namely the IRSA with NOMA scheme, is proposed.\",\n \"In such a protocol, depending on the types of users, multiple collided packets at a time slot may be decodable by intra-slot SIC.\",\n \"The decoded packets are then used to resolve collisions in other slots by inter-slot SIC.\",\n \"The connection between the proposed IRSA with NOMA and a bipartite graph with heterogeneous variable nodes and special check nodes is drawn.\",\n \"This connection is then leveraged to propose a novel multi-dimensional density evolution, which assists us in analyzing the proposed IRSA with NOMA protocol.\",\n \"According to the proposed multi-dimensional density evolution, a constrained optimization problem for maximizing the efficiency is formulated.\",\n \"Through solving this constrained optimization problem, the best degree distribution of each type of users for the proposed IRSA with NOMA protocol can be found.\",\n \"Extensive simulations are provided.\",\n \"For the considered simulation environment with a practical frame size, it is shown that when there are two and three heterogeneous types, the proposed scheme achieves the sum (over types) efficiencies of up to $132\\\\%$ ($143\\\\%$ asymptotically) and $170\\\\%$ ($185\\\\%$ asymptotically), respectively.\",\n \"Moreover, it also indicates that the analysis based on the proposed multi-dimensional density evolution can accurately predict the actual performance of the proposed scheme.\",\n \"The extension of the proposed protocol to the frame-asynchronous IRSA is presented and the corresponding multi-dimensional density evolution is analyzed.\",\n \"It is observed that under the frame-asynchronous circumstance, the proposed IRSA with NOMA also exhibits a boundary effect similar to that in [16].\",\n \"Hence, left-regular degree distributions again suffice to achieve performance that is as good as that achieved by an optimal degree distribution.\",\n \"blackDuring the revision of this paper, we became aware of a pioneering work [37], in which CRDSA with power unbalance is investigated from a joint physical and MAC layers perspective.\",\n \"By allowing each user randomly selects its power according to a log-normal distribution, Mengali et al.\",\n \"show that significant gains can be obtained by intra-slot SIC.\",\n \"However, how to exploit heterogeneity inherent in the network, how to analyze the asymptotic performance, and how to obtain optimal degree distributions when there are multiple types in the network are left undiscussed.\",\n \"In Section , we formally state the problem of UMA with heterogeneous types of users and propose the IRSA with NOMA protocol.\",\n \"In Section , we present a graph representation of the proposed protocol and translate the decoding process as a modified peeling decoder for the corresponding graph.\",\n \"In Section , to analyze the asymptotic performance of the proposed protocol, we propose a multi-dimensional density evolution technique, with which we formulate an optimization problem for obtaining optimal policies for our proposed protocol.\",\n \"Optimal policies are then obtained by the differential evolution technique.\",\n \"Simulation results are presented in Section , followed by blackthe extension of the proposed IRSA with NOMA scheme to the frame asynchronous setting in Sections .\",\n \"Finally, some conclusion remarks are given in Section .\"\n ],\n [\n \"Notational Convention\",\n \"We use $\\\\mathcal {F}[x]$ to denote a polynomial with variable $x$ and we denote by $\\\\mathcal {F}^{\\\\prime }[x]=\\\\mathrm {d}\\\\mathcal {F}[x]/\\\\mathrm {d}x$ and $\\\\mathcal {F}^{\\\\lbrace k\\\\rbrace }[x]=\\\\mathrm {d^k\\\\mathcal {F}[x]}/\\\\mathrm {d} x^k$ the first and $k$ -th derivative of $\\\\mathcal {F}[x]$ , respectively.\",\n \"We denote by $\\\\mathbb {N}$ the set of natural numbers.\",\n \"For two natural numbers $N_1, N_2\\\\in \\\\mathbb {N}$ with $N_1 < N_2$ , we denote by $[N_1]=\\\\lbrace 1, 2, \\\\ldots , N_1\\\\rbrace $ and $[N_1:N_2]=\\\\lbrace N_1, N_1+1, \\\\ldots , N_2\\\\rbrace $ .\",\n \"We consider an UMA setting where a massive number of users coming from $T\\\\in \\\\mathbb {N}$ different types of applications wish to communicate with the base station as shown in Fig.\",\n \"REF .\",\n \"Suppose time is slotted and synchronized.\",\n \"Furthermore, every $N\\\\in \\\\mathbb {N}$ successive slots are organized into a frame; frames are also perfectly synchronous among users, as in other conventional IRSA schemes [7], [9], [11].\",\n \"A frame-asynchronous version of this problem is relegated to Section .\",\n \"A user who has data to send within a frame is said to be active in this frame.\",\n \"Potentially, there could be an arbitrary number of users for each type; Suppose $K^{(t)}\\\\in \\\\mathbb {N}$ of type $t\\\\in [T]$ users are active.\",\n \"Each active user of type $t\\\\in [T]$ constructs $L$ replicas of its packet, where the value $L$ is sampled from a degree distribution $\\\\mathcal {L}^{(t)}$ ; then, the user sends the $L$ replicas in $L$ slots out of $N$ slots chosen uniformly at random in the frame.\",\n \"The degree distribution is our design issue for maximizing efficiency (that will be discussed soon).\",\n \"Figure: UMA with T=3T=3 heterogeneous types of users.blackLeveraging the natural heterogeneity and the maturity of SIC technique [38], we propose the decoder that employs power-domain NOMA [30], [31], [33], [32] at each slot.\",\n \"Specifically, without loss of generality, we assume the decoding order of power-domain NOMA in the physical layer is given by $1\\\\rightarrow 2\\\\rightarrow \\\\ldots \\\\rightarrow T$ and we define the intra-SIC decodable patterns as follows.\",\n \"Definition 1 (Type $t$ decodable pattern) For a slot $n\\\\in [N]$ , we define the transmission pattern $\\\\mathbf {c}_n=(c_1,\\\\ldots ,c_T)$ to be a $T$ -dimensional vector where the entry $c_t$ represents the total number of unresolved type $t$ packets at this slot.\",\n \"Such a vector $\\\\mathbf {c}_n$ is type $t$ decodable if and only if the following conditions hold: $&\\\\text{(i) $c_t=1$},\\\\quad \\\\quad \\\\quad \\\\text{(ii) $c_{t^{\\\\prime }}\\\\le 1$ for $t^{\\\\prime }\\\\in [t-1]$}\\\\\\\\&\\\\text{(iii) $c_{\\\\tilde{t}}\\\\le \\\\tilde{t}-t$ for $\\\\tilde{t}\\\\in [t+1:T]$,} \\\\quad \\\\text{(iv) $\\\\sum _{\\\\tilde{t}=t+1}^Tc_{\\\\tilde{t}}\\\\le T-t$.\",\n \"}$ The type $t$ decodable pattern is defined based on the principle of power-domain NOMA.\",\n \"In power-domain NOMA, it is assumed that a packet can be successfully decoded by intra-slot SIC as long as each interfered type has at most one packet collided at the same slot.\",\n \"This justifies the above conditions (i) and (ii).\",\n \"Moreover, types having later decoding order means that they will cause smaller interference.\",\n \"Therefore, trading a type $\\\\tilde{t}\\\\in [t+1:T]$ packet for a type $\\\\hat{t}>\\\\tilde{t}$ packet would not affect the decodability, justifying the conditions (iii) and (iv).\",\n \"Definition 2 (Type $t$ decodable set) The type $t$ decodable set $\\\\mathcal {D}^{(t)}$ consists of all the type $t$ decodable patterns.\",\n \"i.e., $\\\\mathcal {D}^{(t)}=\\\\lbrace \\\\mathbf {c}:\\\\mathbf {c}~\\\\text{is type $t$ decodable}\\\\rbrace $ .\",\n \"Example 3 When $T=2$ , it is easy to check that $\\\\mathcal {D}^{(1)}=\\\\lbrace (1,0),(1,1)\\\\rbrace $ and $\\\\mathcal {D}^{(2)}=\\\\lbrace (0,1),(1,1)\\\\rbrace .$ When $T=3$ , we have $\\\\mathcal {D}^{(1)}&=\\\\lbrace (1,0,0), (1,0,1), (1,0,2),(1,1,0), (1,1,1)\\\\rbrace , \\\\\\\\\\\\mathcal {D}^{(2)}&=\\\\lbrace (0,1,0), (0,1,1), (1,1,0), (1,1,1)\\\\rbrace , \\\\\\\\\\\\mathcal {D}^{(3)}&=\\\\lbrace (0,0,1), (0,1,1), (1,0,1), (1,1,1)\\\\rbrace .$ The decoding process performed at the end of a frame then proceeds as follows.\",\n \"For every slot $n\\\\in [N]$ during a frame, the corresponding transmission pattern $\\\\mathbf {c}_n$ is computedThe transmission patterns $\\\\mathbf {c}_n$ can evolve with iteration; however, we ignore the index for iterations to avoid heavy notation..\",\n \"The intra-slot SIC is then employed to resolve a type $t$ packet if there is a $\\\\mathbf {c}_n$ for $n\\\\in [N]$ belonging to the type $t$ decodable set $\\\\mathcal {D}^{(t)}$ .\",\n \"We assume that in the burst payload of each packet, there is information about the (other) slots containing copies of this packet.\",\n \"Hence, the decoded packets are then used to cancel other replicas via inter-slot SIC.\",\n \"This completes an iteration and the decoding process continues iteratively until no more packets are decodable for all $t\\\\in [T]$ and all $n\\\\in [N]$ or all the packets in this frame are decoded.\",\n \"This process is done, independently, for every frame.\",\n \"An example of the proposed IRSA with NOMA for $T=2$ can be found in Fig.\",\n \"REF .\",\n \"In this figure, the replicas of packets transmitted from the type 1 users, namely users 1 and 2, are colored in blue and that from the type 2 users, namely users 3 and 4, are colored in red.\",\n \"In this example, user 1's packet can be easily decoded from slot 1.\",\n \"Subtracting user 1's packet at slot 3 from the decoded packet, there's no more degree-1 slot and traditional IRSA would stop the decoding process.\",\n \"For the proposed IRSA with NOMA, at slot 2, we receive the collision of a packet from type 1 (user 2) and another one from type 2 (user 4), both packets can be decoded via intra-slot SIC.\",\n \"Hence, our proposed IRSA with NOMA would continue the decoding process by subtracting users 2 and 4's packets in slot 4 and decode user 4's packet.\",\n \"Figure: An example of SA with heterogeneous users transmitting in a frame of 5 slots.\",\n \"Here, we have T=2T=2 types where the first type consists of users 1 and 2 and the second type consists of users 3 and 4.At the end of the decoding process, suppose the base station can successfully decode $\\\\hat{K}^{(t)}$ packets of type $t\\\\in [T]$ , we define the (actual) efficiency of type $t\\\\in [T]$ achieved by the scheme as $\\\\hat{\\\\eta }^{(t)} = \\\\frac{\\\\hat{K}^{(t)}}{N},\\\\quad t\\\\in [T],$ and the (actual) sum efficiency of the scheme as $ \\\\hat{\\\\eta } = \\\\sum _{i=1}^T \\\\hat{\\\\eta }^{(t)}.$ It is clear that choosing different $\\\\mathcal {L}^{(t)}$ would result in different efficiency.\",\n \"Throughout the paper, we call a collection of $T$ degree distributions $\\\\lbrace \\\\mathcal {L}^{(1)}, \\\\mathcal {L}^{(2)}, \\\\ldots , \\\\mathcal {L}^{(T)}\\\\rbrace $ , one for each type $t\\\\in [T]$ , a policy.\",\n \"The goal of this paper is then to design the policy, such that the achieved efficiency is maximized.\"\n ],\n [\n \"Comparison with existing models\",\n \"In this subsection, we discuss differences between the model considered in this paper and existing models.\",\n \"Remark 4 It is not difficult to see that when $T=1$ , i.e., there is only one type of homogeneous users, the proposed protocol reduces to the IRSA protocol in [9].\",\n \"In light of this, the proposed protocol can be regarded as a generalization of the IRSA protocol to accommodate heterogeneous types of users.\",\n \"Remark 5 We note that both multi-packet reception [34] and power-domain NOMA [30], [31], [32], [33] are based on the same physical-layer model, namely the multiple access channel [39].\",\n \"However, the multi-packet reception technique treats all the users equally and assumes collided packets are decodable as long as the number of users collided at a slot is smaller than a predefined number.\",\n \"Hence, in this homogeneous setting, the problem in the physical layer becomes to establish reliable multiple access with homogeneous users that have the same codebook and code rate.\",\n \"To the best of our knowledge, this is not an easy task that is tackled with some success in [40] and [41], which involves either extremely large blocklength due to spatial coupling [40] or very complex operations of coding over prime fieldsThis is due to the adoption of Construction A lattices [42].. On the contrary, the power-domain NOMA lets the users transmit with different power levels and/or rates so that low-complex SIC can be employed to achieve a corner point of the capacity regionblackHere, the standard information-theoretic sense of achievability [39] is considered.. An illustration can be found in Fig.\",\n \"REF .\",\n \"In this paper, the considered problem exhibits natural heterogeneity among users; hence, it is more natural and suitable to work with power-domain NOMA so that the heterogeneity can be exploited by intra-SIC.\",\n \"The main theme of this paper then centers around analyzing policies for IRSA with power-domain NOMA and designing policies that optimally exploit the heterogeneity inherent in the considered problem.\",\n \"Figure: The capacity region of a two-user multiple access channel.Remark 6 The combination of NOMA and SA has been investigated before in [43], [44] by Choi.\",\n \"However, in these works, it is assumed that there is only one type of homogeneous users.\",\n \"The focus of [43] is mainly to analyze the performance of SA when users randomly choose their power levels and the receiver employs intra-slot SIC.\",\n \"Moreover, in both works [43], [44], the inter-slot SIC technique is not considered.\",\n \"Remark 7 We also note that our model is fundamentally different from IRSA with capture effect discussed in [9] and [45].\",\n \"With capture effect, it is assumed that the receiver may be able to decode more than one packet with some probability.\",\n \"This opportunity usually comes from the effect of random channel fading.\",\n \"On the other hand, in the considered model, the receiver is able to decode more than one packet deterministically for certain configurations of collided users, due to the natural heterogeneity inherent in the problem.\"\n ],\n [\n \"Implementation Issues\",\n \"We now address potential implementation issues of the proposed IRSA with NOMA.\",\n \"We would like to note that many of these issues have been encountered and addressed by the conventional IRSA schemes [9].\",\n \"However, we still present them here for the sake of self-containedness.\",\n \"Slot synchronization: This issue is inherent in all slotted multiple access schemes.\",\n \"One easy solution to provide slot synchronization is to rely on stable clocks and a small amount of guard time between packets [5].\",\n \"blackMoreover, for applications with cheap user devices, stable clocks may be unaffordable.\",\n \"For such scenarios, [46] provides a technique to achieve slot synchronization, where each node constantly re-synchronizes its clock with the base station by exploiting the ACK signals fed back from the base station.\",\n \"Another source of slot synchronization errors which is less seen in traditional communication systems is from significantly different travel distances experienced by different devices' signals.\",\n \"This type of source of slot synchronization errors has become important in non-terrestrial networks (NTN) [47].\",\n \"Fortunately, in NTN, each device is required to be GNSSGNSS stands for Global Navigation Satellite System.-capable for acquiring its own position and together with satellite ephemeris.\",\n \"Each user can calculate and then pre-compensate its timing advance (TA) (see Section 6.3 in [47]).\",\n \"To reduce devices' computation loading, another candidate technique currently under discussion in 3GPP is that the base station estimates and broadcasts TA to a group of devices who have similar TA (see Section 6.3 in [47]).\",\n \"Frame synchronization: This issue is also not new and has to be cope with in every frame-synchronous systems.\",\n \"This issue can be addressed by letting the base station periodically broadcasting a beacon signal at the beginning of each frame.\",\n \"In Section , extension to frame asynchronous setting will be discussed.\",\n \"Inter-slot SIC: Similar to CRDSA [7] and IRSA schemes [9], to enable inter-slot SIC, the proposed protocol requires the knowledge of the locations of other replicas once a packet is successfully decoded.\",\n \"blackThis can be achieved by including pointers to the locations of other replicas into the burst payload or by some pseudo-random mechanism known by both the transmit and receive ends.\",\n \"For example, the DVB system [8] adopts the latter one, where each active user reports its MAC address and a pseudo-random number to the base station.\",\n \"Both the base station and the user can then compute the slots used for sending replicas by a common deterministic function of the MAC address and the pseudo-random number.\",\n \"Intra-slot SIC: blackFor the proposed scheme, in order to enable intra-slot SIC at a slot, it appears that the receiver has to know which types of packets are collided in this slot.\",\n \"This issue is new in our proposed protocol.\",\n \"blackWe provide two ways to address this issue.\",\n \"We note that in DVB-RCS2 [8], each active user selects the number of transmissions and a random seed for uniquely determining a pseudo random pattern.\",\n \"This information is then stored in the burst payload and sent to the base station.\",\n \"Given the existing data structure, to enable intra-slot SIC, each user can include additional $\\\\log (T)$ bits to the payload for revealing its type.\",\n \"The second method described in the sequel eliminates the need for sending extra $\\\\log (T)$ bits by increasing the decoding burden.\",\n \"Let $R^{(t)}$ be the rate of the physical-layer channel code adopted by users of type $t\\\\in [T]$ .\",\n \"Let us assume the We assume that the rate tuple $(R^{(1)}, R^{(2)}, \\\\ldots , R^{(T)})$ can be achieved by intra-slot SIC with ascending decoding order $1\\\\rightarrow 2\\\\rightarrow \\\\ldots \\\\rightarrow T$ , without loss of generality.\",\n \"Following the NOMA principle, a means to address this issue is as follows: Starting from $t=1$ , the base station tries to decode a packet of type $t$ ; if it succeedsThe success of decoding can be checked by a cyclic redundancy check mechanism., the decoded packet is subtracted from the received signal and we set $t=t+1$ ; otherwise, directly set $t=t+1$ and look for decoding opportunity of the next type.\",\n \"This procedure would continue until $t=T$ .\"\n ],\n [\n \"Graph Representation\",\n \"In this section, we introduce a graph representation of the proposed IRSA with NOMA protocol for determining an optimal policy in Section .\"\n ],\n [\n \"Bipartite Graph and Degree Distributions\",\n \"We first note that in UMA with $T$ types of heterogeneous users, each type has its own transmission policy and number of users that may be different from other types.\",\n \"We construct a bipartite graph representation of the proposed IRSA with NOMA protocol.\",\n \"The graph consists of $\\\\sum _{t=1}^T K^{(t)}$ variable nodes and $N$ super check nodes, as shown in Fig.\",\n \"REF .\",\n \"Each variable node $\\\\mathsf {v}^{(t)}_j$ represents a user $j\\\\in [K^{(t)}]$ of type $t\\\\in [T]$ .\",\n \"Each super check node consists of $T$ check nodes $\\\\mathsf {c}^{(t)}_n$ , one for each type $t\\\\in [T]$ .\",\n \"A user $j$ of type $t\\\\in [T]$ transmits a packet in slot $n\\\\in [N]$ if and only if there is an edge connecting $\\\\mathsf {v}^{(t)}_j$ with $\\\\mathsf {c}^{(t)}_n$ .\",\n \"For simplicity, we use $\\\\mathsf {v}^{(t)}$ and $\\\\mathsf {c}^{(t)}$ to denote a generic variable and check nodes of type $t\\\\in [T]$ .\",\n \"Figure: A bipartite graph representation of the proposed IRSA with NOMA.With this bipartite graph representation, a user of type $t$ who transmits $d$ replicas corresponds to a degree $d$ variable node $\\\\mathsf {v}^{(t)}$ .\",\n \"Moreover, a time slot in which $d$ type $t$ variable nodes transmit corresponds to a degree $d$ check node $\\\\mathsf {c}^{(t)}$ .\",\n \"We define the node perspective left and right degree distributions as $\\\\mathcal {L}^{(t)}[x] = \\\\sum _{d} L^{(t)}_d x^d \\\\quad \\\\text{and}\\\\quad \\\\mathcal {R}^{(t)}[x] = \\\\sum _d R^{(t)}_d x^d,$ respectively, where $L^{(t)}_d$ is the probability that a variable node $\\\\mathsf {v}^{(t)}$ has degree $d$ and $R^{(t)}_d$ is the probability that a check node $\\\\mathsf {c}^{(t)}$ has degree $d$ .\",\n \"Through the node perspective degree distributions, the edge perspective left and right degree distributions can be given by $\\\\lambda ^{(t)}[x] = \\\\sum _d \\\\lambda ^{(t)}_d x^{d-1}=\\\\frac{\\\\mathcal {L}^{^{\\\\prime }(t)}[x]}{\\\\mathcal {L}^{^{\\\\prime }(t)}[1]},$ and $\\\\rho ^{(t)}[x] &= \\\\sum _d\\\\rho ^{(t)}_d x^{d-1} =\\\\frac{\\\\mathcal {R}^{^{\\\\prime }(t)}[x]}{\\\\mathcal {R}^{^{\\\\prime }(t)}[1]},$ respectively, where $\\\\lambda ^{(t)}_d$ stands for the probability that an edge connects to a type $t$ variable node of degree $d$ and $\\\\rho ^{(t)}_d$ is the probability that an edge connects to a type $t$ check node of degree $d$ .\",\n \"The target efficiency of type $t\\\\in [T]$ is then defined as $\\\\eta ^{(t)}= \\\\frac{K^{(t)}}{N} = \\\\frac{\\\\mathcal {R}^{^{\\\\prime }(t)}[1]}{\\\\mathcal {L}^{^{\\\\prime }(t)}[1]},$ and the target sum efficiency of the scheme is defined as $ \\\\eta = \\\\sum _{t=1}^T \\\\eta ^{(t)}.$ Note that the target efficiency $\\\\eta ^{(t)}$ is different from the actual efficiency $\\\\hat{\\\\eta }^{(t)}$ in (REF ).\",\n \"In general, $\\\\hat{\\\\eta }^{(t)}\\\\le \\\\eta ^{(t)}$ with equality if and only if all packets are successfully decoded.\",\n \"Example 8 Consider the scheme described in Fig.\",\n \"REF where there is one type 1 variable node of degree 2 and one type 1 variable node of degree 3.\",\n \"Hence, $\\\\mathcal {L}^{(1)}[x]=0.5x^2 +0.5x^3$ .\",\n \"One can obtain $\\\\mathcal {L}^{(2)}[x]=x^3$ in a similar fashion.\",\n \"Also, there are three type 1 check nodes of degree 1 and one type 1 check node of degree 2.\",\n \"Therefore, $\\\\mathcal {R}^{(1)}[x]=0.75x+0.25x^2$ .\",\n \"Similarly, $\\\\mathcal {R}^{(2)}[x]=0.5x+0.5x^2.$ Moreover, one can easily verify that the edge perspective degree distributions are given by $\\\\lambda ^{(1)}[x]= \\\\frac{2}{5}x + \\\\frac{3}{5}x^2, \\\\quad \\\\lambda ^{(2)}[x]=x^2, \\\\quad \\\\rho ^{(1)}[x]=\\\\frac{3}{5}+\\\\frac{2}{5}x,\\\\quad \\\\rho ^{(2)}[x]=\\\\frac{1}{3}+\\\\frac{2}{3}x.$ The graph representation of this example can be found in Fig.\",\n \"REF -(a).\",\n \"Figure: The graph representation of the scheme in Fig.\",\n \"and its decoding output at each iteration of the modified peeling decoder.\"\n ],\n [\n \"Modified Peeling Decoder\",\n \"Here, we propose a modified peeling decoder for the bipartite graph that corresponds to the decoding procedure of the proposed IRSA with NOMA.\",\n \"The decoder operates in an iterative fashion.\",\n \"For each iteration, the decoder looks at every super check node $n\\\\in [N]$ and computes the remaining degrees of check nodes $\\\\mathsf {c}^{(t)}_1, \\\\mathsf {c}^{(t)}_2, \\\\ldots , \\\\mathsf {c}^{(t)}_T$ , which are stored in the transmission pattern $\\\\mathbf {c}_n$ .\",\n \"We then look for an $n$ such that $\\\\mathbf {c}_n$ belongs to the type $t$ decodable set $\\\\mathcal {D}^{(t)}$ for some $t\\\\in [T]$ .\",\n \"If such an $n$ exists, we use intra-slot SIC to decode all the type $t$ packets satisfying $\\\\mathbf {c}_n\\\\in \\\\mathcal {D}^{(t)}$ in this super check node $n$ .\",\n \"For a variable node $\\\\mathsf {v}^{(t)}$ that is decoded, all the edges connected with it are removed from the graph.\",\n \"This concludes an iteration.\",\n \"The modified peeling decoder operates iteratively until no more decoding opportunity is found.\",\n \"Example 8 (continued) In this example, we illustrate how to use the modified peeling decoder to decode the scheme in Example REF .\",\n \"As shown in Fig.\",\n \"REF -(b), in iteration 1, $\\\\mathsf {c}^{(1)}_1$ has degree 1 and $\\\\mathsf {c}^{(2)}_1$ has degree 0, i.e., $\\\\mathbf {c}_1=(1,0)\\\\in \\\\mathcal {D}^{(1)}$ ; hence, $\\\\mathsf {v}^{(1)}_1$ is decoded and all its edges are removed.\",\n \"Then, in iteration 2 shown in Fig.\",\n \"REF -(c), $\\\\mathsf {c}^{(1)}_2$ has degree 1 and $\\\\mathsf {c}^{(2)}_2$ has degree 1, i.e., $\\\\mathbf {c}_2=(1,1)$ belonging to both $\\\\mathcal {D}^{(1)}$ and $\\\\mathcal {D}^{(2)}$ ; thus, both $\\\\mathsf {v}^{(1)}_2$ and $\\\\mathsf {v}^{(2)}_2$ can be decoded by intra-slot SIC.\",\n \"All the edges connected to $\\\\mathsf {v}^{(1)}_2$ and $\\\\mathsf {v}^{(2)}_2$ are subsequently removed.\",\n \"Last, in iteration 3 shown in Fig.\",\n \"REF -(d), one can easily see that $\\\\mathsf {v}^{(2)}_1$ can be decoded.\"\n ],\n [\n \"Multi-Dimensional Density Evolution and Convergence Analysis\",\n \"In this section, to analyze the IRSA with NOMA, we propose a novel multi-dimensional density evolution based on the graph representation described in Section .\",\n \"With this multi-dimensional density evolution, we then provide the convergence analysis of the IRSA with NOMA under modified peeling decoding.\",\n \"With an assist from the convergence analysis, we formulate and numerically solve an optimization problem that finds best left degree distributions, leading to the highest asymptotic efficiency.\",\n \"For clearly explaining the ideas behind our scheme and analysis, we focus on $T=2$ .\",\n \"The general $T$ case can be similarly analyzed.\"\n ],\n [\n \"Proposed Multi-Dimensional Density Evolution\",\n \"It is well known that for LDPC codes over a binary erasure channel, BP and peeling decoders have the same performanceIt essentially means that the order of the limit of average residue erasure probability as the number of iterations tends to $\\\\infty $ and the limit of that as the blocklength tends to $\\\\infty $ is exchangeable.\",\n \"The interested reader is referred to [10]The reason that we propose using the peeling decoder instead of BP is because the peeling decoder potentially leads to a smaller decoding latency in practice as it does not have to wait until the end of the frame in order to start decoding; it can start decoding right away once a decodable slot shows up..\",\n \"Moreover, density evolution is an outstanding tool for analyzing the performance of BP decoding over a sparse graph [10].\",\n \"Thus, this section proposes a novel multi-dimensional density evolution to analyze the performance of the proposed scheme under BP decoding.\",\n \"blackSimilar to that introduced in [10], the BP decoding is an iterative algorithm where in each iteration, each variable node computes the belief of its own message and passes an extrinsic version of it to each connected check node, while each check node computes its belief of the message of each connected variable node and passes an extrinsic version to that variable node.\",\n \"After a predefined number of iterations, the decoding algorithm halts and outputs hard decisions according to the last beliefs held by the variable nodes.\",\n \"blackConsider BP decoding described above.\",\n \"For $t\\\\in [2]$ , we denote by $x_\\\\ell ^{(t)}$ and $y_\\\\ell ^{(t)}$ the average erasure probability of the message passed along an edge from $\\\\mathsf {v}^{(t)}$ to $\\\\mathsf {c}^{(t)}$ and that of the message passed along an edge from $\\\\mathsf {c}^{(t)}$ to $\\\\mathsf {v}^{(t)}$ in iteration $\\\\ell $ , respectively.\",\n \"At iteration $\\\\ell =0$ , according to the decodable set $\\\\mathcal {D}^{(t)}$ specified in Example REF , for an edge connected to a type $t$ check node $\\\\mathsf {c}^{(t)}$ , the outgoing message is not in erasure if and only if a) $\\\\mathsf {c}^{(t)}$ has degree 1; b) $\\\\mathsf {c}^{(\\\\bar{t})}$ has a degree either 0 or 1 for every $\\\\bar{t}\\\\in [2],~\\\\bar{t}\\\\ne t$ .\",\n \"Hence, we initialize $y_0^{(t)}$ for initial iteration $\\\\ell =0$ to be $y_0^{(t)}= 1-\\\\rho ^{(t)}[0]\\\\cdot \\\\left( \\\\mathcal {R}^{(\\\\bar{t})}[0] + \\\\mathcal {R}^{^{\\\\prime }(\\\\bar{t})}[0] \\\\right),$ where $\\\\mathcal {R}^{(\\\\bar{t})}[0]=R^{(\\\\bar{t})}_0$ and $\\\\mathcal {R}^{^{\\\\prime }(\\\\bar{t})}[0]=R^{(\\\\bar{t})}_1$ are the fractions of type $\\\\bar{t}$ check nodes having degree 0 and degree 1, respectively.\",\n \"Moreover, the term $\\\\rho ^{(t)}[0]=\\\\rho _1^{(t)}$ is the probability that an edge connects to a type $t$ check node of degree 1.\",\n \"Suppose we have obtained $x_\\\\ell ^{(t)}$ and $y_\\\\ell ^{(t)}$ for all $t\\\\in [T]$ from iteration $\\\\ell $ .\",\n \"In iteration $\\\\ell +1$ , for an edge incident to a variable node $\\\\mathsf {v}^{(t)}$ with degree $d$ , the only possibility that the message along this edge to a $\\\\mathsf {c}^{(t)}$ is in erasure is that all the other $d-1$ edges are in erasure.\",\n \"Therefore, the probability that the message passed along this edge is in erasure is $(y_\\\\ell ^{(t)})^{d-1}$ .\",\n \"Now, averaging over all the edges results in the average erasure probability $x_\\\\ell ^{(t)}= \\\\sum _d \\\\lambda ^{(t)}_d (y_\\\\ell ^{(t)})^{d-1} = \\\\lambda ^{(t)}[y_\\\\ell ^{(t)}].$ For an edge incident to a check node $\\\\mathsf {c}^{(t)}$ with degree $d$ , the message passed along this edge to a $\\\\mathsf {v}^{(t)}$ is not in erasure if and only if a) all the other $d-1$ edges incident to $\\\\mathsf {c}^{(t)}$ are not in erasure in the previous iteration; and b) all but at most one of the edges incident to $\\\\mathsf {c}^{(\\\\bar{t})}$ are not in erasure in the previous iteration for every $\\\\bar{t}\\\\in [2],~\\\\bar{t}\\\\ne t$ .\",\n \"Suppose in the same super check node $n$ , $\\\\mathsf {c}^{(\\\\bar{t})}_n$ has degree $d_{\\\\bar{t}}$ .\",\n \"Then the above event has probability $(1-x_\\\\ell ^{(t)})^{d-1}\\\\cdot \\\\left( (1-x_\\\\ell ^{(\\\\bar{t})})^{d_{\\\\bar{t}}} + d_{\\\\bar{t}} x_\\\\ell ^{(\\\\bar{t})}(1-x_\\\\ell ^{(\\\\bar{t})})^{d_{\\\\bar{t}}-1}\\\\right),$ where $(1-x_\\\\ell ^{(\\\\bar{t})})^{d_{\\\\bar{t}}}$ is the probability that all $d_{\\\\bar{t}}$ edges are not erased and $d_{\\\\bar{t}} x_\\\\ell ^{(\\\\bar{t})}(1-x_\\\\ell ^{(\\\\bar{t})})^{d_{\\\\bar{t}}-1}$ is the probability that all but one edges are not erased.\",\n \"Now, averaging over all the edges and over all the type $\\\\bar{t}$ check nodes for $\\\\bar{t}\\\\in [2],~\\\\bar{t}\\\\ne t$ shows that the average probability of correct decoding is given by $&\\\\sum _d \\\\lambda ^{(t)}_d(1-x_\\\\ell ^{(t)})^{d-1}\\\\cdot \\\\nonumber \\\\sum _{d_{\\\\bar{t}}}R^{(\\\\bar{t})}_{d_{\\\\bar{t}}}\\\\left( (1-x_\\\\ell ^{(\\\\bar{t})})^{d_{\\\\bar{t}}} + d_{\\\\bar{t}} x_\\\\ell ^{(\\\\bar{t})}(1-x_\\\\ell ^{(\\\\bar{t})})^{d_{\\\\bar{t}}-1}\\\\right) \\\\nonumber \\\\\\\\&=\\\\rho ^{(t)}[1-x_\\\\ell ^{(t)}] \\\\left( \\\\mathcal {R}^{(\\\\bar{t})}[1-x_\\\\ell ^{(\\\\bar{t})}] + x_\\\\ell ^{(\\\\bar{t})}\\\\mathcal {R}^{^{\\\\prime }(\\\\bar{t})}[1-x_\\\\ell ^{(\\\\bar{t})}]\\\\right).$ Therefore, the average erasure probability becomes $y_{\\\\ell +1}^{(t)}= 1-\\\\rho ^{(t)}[1-x_\\\\ell ^{(t)}]\\\\cdot \\\\left( \\\\mathcal {R}^{(\\\\bar{t})}[1-x_\\\\ell ^{(\\\\bar{t})}] + x_\\\\ell ^{(\\\\bar{t})}\\\\mathcal {R}^{^{\\\\prime }(\\\\bar{t})}[1-x_\\\\ell ^{(\\\\bar{t})}]\\\\right).$ Plugging (REF ) into (REF ) leads to the evolution of average erasure probability of a type $t$ check node as shown in (REF ) in the bottom of next page.\",\n \"Figure: NO_CAPTION\"\n ],\n [\n \"Convergence and Stability Condition\",\n \"After obtaining the density evolution in (REF ), for any given degree distributions $\\\\mathcal {L}^{(t)}$ (or $\\\\lambda ^{(t)}$ ) and $\\\\mathcal {R}^{(t)}$ (or $\\\\rho ^{(t)}$ ), one can now analyze whether the average erasure probability converges to 0 by checking whether $y_\\\\ell ^{(t)}> y_{\\\\ell +1}^{(t)}$ for every $\\\\ell $ , starting from $y_0^{(t)}=1$ for $t\\\\in [T]$ .\",\n \"Moreover, to make sure that the average erasure probability indeed vanishes instead of hovering around 0, we derive the following stability conditionblackNote that the term “stability\\\" used here is referred to the stability of fixed points rather than that in the stability analysis of slotted ALOHA [48], [5].. We enforce $\\\\lambda _1^{(t)}=0$ for all $t\\\\in [2]$ because we certainly do not want degree 1 variable nodes.\",\n \"Then, assuming $y^{(t)}$ is very small for all $t\\\\in [2]$ , we expand the degree distributions and approximate them by keeping only the linear terms as follows, $\\\\lambda ^{(t)}[y^{(t)}] &\\\\approx \\\\lambda _2^{(t)}y^{(t)}, \\\\\\\\\\\\rho ^{(t)}[1-\\\\lambda ^{(t)}[y^{(t)}]] &\\\\approx 1-\\\\rho ^{^{\\\\prime }{(t)}}[1]\\\\lambda _2^{(t)}y^{(t)}, \\\\\\\\R^{(\\\\bar{t})}[1-\\\\lambda ^{(\\\\bar{t})}[y^{(\\\\bar{t})}]] &\\\\approx 1-R^{^{\\\\prime }(\\\\bar{t})}[1] \\\\lambda _2^{(\\\\bar{t})} y^{(\\\\bar{t})}, \\\\\\\\R^{^{\\\\prime }(\\\\bar{t})}[1-\\\\lambda ^{(\\\\bar{t})}[y^{(\\\\bar{t})}]] &\\\\approx R^{^{\\\\prime }(\\\\bar{t})}[1] - R^{^{\\\\prime \\\\prime }(\\\\bar{t})}[1]\\\\lambda _2^{(\\\\bar{t})}y^{(\\\\bar{t})}.$ We can now linearize the recursion around 0 by plugging (REF )-() into (REF ) to get $y^{(t)}> 1- (1-\\\\rho ^{^{\\\\prime }{(t)}}[1]\\\\lambda _2^{(t)}y^{(t)}) \\\\left(1-R^{^{\\\\prime \\\\prime }(\\\\bar{t})}[1](\\\\lambda _2^{(t)}y^{(\\\\bar{t})})^2\\\\right) \\\\approx \\\\rho ^{^{\\\\prime }{(t)}}[1]\\\\lambda _2^{(t)}y^{(t)},$ which leads to the following stability condition $\\\\lambda _2^{(t)}< \\\\frac{1}{\\\\rho ^{^{\\\\prime }{(t)}}[1]}\\\\quad \\\\text{for $t\\\\in [2]$}.$\"\n ],\n [\n \"Optimization Problem\",\n \"Before we formulate the optimization problem, we note that there are two sets of degree distributions $\\\\lbrace \\\\mathcal {L}^{(t)}[x]\\\\rbrace $ and $\\\\lbrace \\\\mathcal {R}^{(t)}[x]\\\\rbrace $ in the proposed problem, but we only have control over $\\\\lbrace \\\\mathcal {L}^{(t)}[x]\\\\rbrace $ .\",\n \"The behavior of $\\\\lbrace \\\\mathcal {R}^{(t)}[x]\\\\rbrace $ ; however, are completely determined by how the users behave.\",\n \"Specifically, according to the protocol, a type $t$ user $\\\\mathsf {v}^{(t)}$ having degree $L$ will send a replica in slot $\\\\mathsf {c}^{(t)}$ with probability $L/N$ .\",\n \"Hence the average probability that a user $\\\\mathsf {v}^{(t)}$ sending a replica in slot $\\\\mathsf {c}^{(t)}$ is given by $ \\\\frac{\\\\mathcal {L}^{^{\\\\prime }(t)}[1]}{N} = \\\\frac{\\\\mathcal {R}^{^{\\\\prime }(t)}[1]}{K^{(t)}}.$ Thus, the degrees of $\\\\mathsf {c}^{(t)}$ follows the Binomial distribution with parameter $\\\\frac{\\\\mathcal {L}^{^{\\\\prime }(t)}[1]}{N}$ .\",\n \"Similar to [9], by Poisson approximation [49], we have $\\\\mathcal {R}^{(t)}[x] \\\\approx \\\\exp \\\\left( -\\\\mathcal {R}^{^{\\\\prime }(t)}[1](1-x) \\\\right) = \\\\exp \\\\left( -\\\\eta ^{(t)}\\\\mathcal {L}^{^{\\\\prime }(t)}[1](1-x) \\\\right).$ Moreover, from (REF ), we have $\\\\rho ^{(t)}[x]= \\\\exp \\\\left( -\\\\eta ^{(t)}\\\\mathcal {L}^{^{\\\\prime }(t)}[1](1-x) \\\\right).$ blackNow, the convergence condition becomes (REF ) in the bottom of the this page by plugging (REF ) into (REF ) and noting that it is sufficient that $y^{(t)}_{\\\\ell +1} y_{\\\\ell +1}^{(t)}$ for every $\\\\ell $ and every $y_\\\\ell ^{(t)}>0$ .\",\n \"In what follows, we again derive the stability condition.\",\n \"Similar to the $T=2$ case, we enforce $\\\\lambda _1^{(t)}=0$ for all $t\\\\in [T]$ .\",\n \"Assuming $y^{(t)}$ is very small for all $t\\\\in [T]$ , we expand the degree distributions and approximate them by keeping only the linear terms as follows, black $\\\\lambda ^{(t)}[y^{(t)}] &\\\\approx \\\\lambda _2^{(t)}y^{(t)}, \\\\\\\\\\\\rho ^{(t)}[1-\\\\lambda ^{(t)}[y^{(t)}]] &\\\\approx 1-\\\\rho ^{^{\\\\prime }{(t)}}[1]\\\\lambda _2^{(t)}y^{(t)}, \\\\\\\\R^{(\\\\bar{t})}[1-\\\\lambda ^{(\\\\bar{t})}[y^{(\\\\bar{t})}]] &\\\\approx 1-R^{^{\\\\prime }(\\\\bar{t})}[1] \\\\lambda _2^{(\\\\bar{t})} y^{(\\\\bar{t})}, \\\\\\\\(R^{(\\\\bar{t})})^{\\\\lbrace k\\\\rbrace }[1-\\\\lambda ^{(\\\\bar{t})}[y^{(\\\\bar{t})}]] &\\\\approx (R^{(\\\\bar{t})})^{\\\\lbrace k\\\\rbrace }[1] \\\\nonumber \\\\\\\\&- (R^{(\\\\bar{t})}[1])^{\\\\lbrace k+1\\\\rbrace }\\\\lambda _2^{(\\\\bar{t})}y^{(\\\\bar{t})}.$ We can now linearize the recursion around 0 by plugging (REF )-() into (REF ) to get the same stability condition $\\\\lambda _2^{(t)}< \\\\frac{1}{\\\\rho ^{^{\\\\prime }{(t)}}[1]}\\\\quad \\\\text{for $t\\\\in [T]$}.$\"\n ],\n [\n \"Optimization Problem and Optimized Degree Distributions\",\n \"Similar to the $T=2$ case, we again apply the Poisson approximation and rewrite the convergence condition in (REF ) as (REF ) in the bottom of the next page.\",\n \"Figure: NO_CAPTIONFinally, we are able to formulate the optimization problem that maximizes the target efficiency subject to derived conditions: $&\\\\!\\\\max _{\\\\lbrace \\\\mathcal {L}^{(t)}[x]\\\\rbrace } &\\\\qquad & \\\\eta =\\\\sum _{t=1}^T \\\\eta ^{(t)}\\\\\\\\&\\\\text{subject to} & & \\\\text{convergence condition~}(\\\\ref {eqn:DE_full_poisson_T}),\\\\\\\\& & & \\\\lambda _1^{(t)}=0~\\\\text{for all $t\\\\in [T]$},\\\\\\\\& & & L_d^{(t)}\\\\ge 0, ~\\\\text{for all $d\\\\in [d_{\\\\mathrm {max}}^{(t)}]$ and $t\\\\in [T]$}, \\\\\\\\& & & \\\\mathcal {L}^{(t)}[1]=1~\\\\text{for all $t\\\\in [T]$}.$ Similar to $T=2$ , one may include the stability condition (REF ) into the above optimization problem to make sure that the packet loss rate indeed vanishes as $N\\\\rightarrow \\\\infty $ .\"\n ]\n]"},"id":{"kind":"string","value":"2012.05159"}}},{"rowIdx":3686,"cells":{"text":{"kind":"list like","value":[["Smoothed Quantile Regression with Large-Scale Inference"],["Abstract Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects.","In this paper, we consider statistical inference for quantile regression with large-scale data in the \"increasing dimension\" regime.","We provide a comprehensive and in-depth analysis of a convolution-type smoothing approach that achieves adequate approximation to computation and inference for quantile regression.","This method, which we refer to as {\\it{conquer}}, turns the non-differentiable quantile loss function into a twice-differentiable, convex and locally strongly convex surrogate, which admits a fast and scalable Barzilai-Borwein gradient-based algorithm to perform optimization, and multiplier bootstrap for statistical inference.","Theoretically, we establish explicit non-asymptotic bounds on both estimation and Bahadur-Kiefer linearization errors, from which we show that the asymptotic normality of the conquer estimator holds under a weaker requirement on the number of the regressors than needed for conventional quantile regression.","Moreover, we prove the validity of multiplier bootstrap confidence constructions.","Our numerical studies confirm the conquer estimator as a practical and reliable approach to large-scale inference for quantile regression.","Software implementing the methodology is available in the \\texttt{R} package \\texttt{conquer}."],["Introduction","Quantile regression (QR) is a useful statistical tool for modeling and inferring the relationship between a scalar response $y$ and a $p$ -dimensional predictor $$ [35].","Compared to the least squares regression that focuses on modeling the conditional mean of $y$ given $$ , QR allows modeling of the entire conditional distribution of $y$ given $$ , and thus provides valuable insights into heterogeneity in the relationship between $$ and $y$ .","Moreover, quantile regression is robust against outliers and can be performed for skewed or heavy-tailed response distributions without a correct specification of the likelihood.","These advantages make QR an appealing method to explore data features that are invisible to the least squares regression.","We refer the reader to [33] and [36] for an extensive overview of QR in terms of methods, theory, computation, and various extensions under complex data structures.","Quantile regression involves a convex optimization problem with a piecewise linear loss function, also known as the check function and pinball loss.","One can formulate QR as a linear programming problem, solvable by the interior point method with a computational complexity of $(n^{1+\\alpha } p^3 \\log n)$ with $\\alpha \\in (0, 1/2)$ , where $n$ is the sample size and $p$ is the parametric dimension.","When $n$ is large relative to $p$ , an efficient algorithm based on pre-processing has an improved complexity of $\\lbrace (np)^{2(1+\\alpha )/3} p^3 \\log n + np\\rbrace $ , and thus can be more efficient than solving a least squares problem when $p$ is small [48].","However, primarily due to the non-differentiability and lack of strong convexity of the loss function, QR remains computationally expensive for large-scale data when both $n$ and $p$ are large.","We refer to Chapter 5 of [36] for an overview of prevailing computational methods for quantile regression, such as simplex-based algorithms [4], [37], interior point methods [48], and alternating direction method of multipliers among other first-order proximal methods [44].","We consider conducting large-scale inference for quantile regression under the setting in which $p$ is large and $n$ is even larger.","Two general principles have been widely used to suit this purpose.","The first uses a nonparametric estimate of the asymptotic variance [23] that involves the conditional density of the response given the covariates, yet such an estimate can be fairly unstable.","Even if the asymptotic variance is well estimated, its approximation accuracy to the finite-sample variance depends on the design matrix and the quantile level.","Resampling methods, on the other hand, provide a more reliable approach to inference for QR under a wide variety of settings [45], [25], [31], [19].","Inevitably, the resampling approach requires repeatedly computing QR estimates up to thousands of times, and therefore is unduly expensive for large-scale data.","Theoretically, valid statistical inference is often justified by asymptotic normal approximations to QR estimators.","The Bahadur-Kiefer representation of the nonlinear QR estimators are essential to this end, as shown in [2] and [24].","In large-$p$ (non)asymptotic settings in which the parametric dimension $p$ may tend to infinity with the sample size, we refer to [54], [26], and [43] for normal approximation results of the QR estimators under fixed and random designs.","The question of how large $p$ can be relative to $n$ to ensure asymptotic normality has been addressed by those authors.","It is now recognized that we may have to pay a price here as compared to $M$ -estimators with smooth loss functions that are at least continuously twice differentiable.","To circumvent the non-differentiability of the QR loss function, [27] proposed to smooth the indicator part of the check function via the survival function of a kernel.","This smoothing method, which we refer to as Horowitz's smoothing throughout, has been widely used for various QR-related problems [55], [30], [21], [52], [56], [14], [10].","However, Horowitz's smoothing gains smoothness at the cost of convexity, which inevitably raises optimization issues.","In general, computing a global minimum of a non-convex function is intractable: finding an $\\epsilon $ -suboptimal point for a $k$ -times continuously differentiable function $f:\\mathbb {R}^p \\rightarrow $ requires at least as many as $(1/\\epsilon )^{p/k}$ evaluations of the function and its first $k$ derivatives [42].","As we shall see from the numerical studies in Section , the convergence of gradient-based algorithms can be relatively slow for high and low quantile levels.","To address the aforementioned issue, [20] proposed a convolution-type smoothing method that yields a convex and twice differentiable loss function, and studied the asymptotic properties of the smoothed estimator when $p$ is fixed.","To distinguish this approach from Horowitz's smoothing, we adopt the term conquer for convolution-type smoothed quantile regression.","In this paper, we first provide an in-depth statistical analysis of conquer under various array (non)asymptotic settings in which $p$ increases with $n$ .","Our results reveal a key feature of the smoothing parameter, often referred to as the bandwidth: the bandwidth adapts to both the sample size $n$ and dimensionality $p$ , so as to achieve a tradeoff between statistical accuracy and computational stability.","Since the convolution smoothed loss function is globally convex and locally strongly convex, we propose an efficient gradient descent algorithm with the Barzilai-Borwein stepsize and a Huber-type initialization.","The proposed algorithm is implemented via RcppArmadillo [17] in the R package conquer.","We next focus on large-scale statistical inference (hypothesis testing and confidence estimation) with large $p$ and large $n$ .","We propose a bootstrapped conquer method that has reduced computational complexity when the conquer estimator is used as initialization.","Under appropriate restrictions on dimension, we establish the consistency (or concentration), Bahadur representation, asymptotic normality of the conquer estimator as well as the validity of the bootstrap approximation.","In the following, we provide more details on the computational and statistical contributions of this paper.","Theoretically, by allowing $p$ to grow with $n$ , the `complexity' of the function classes that we come across in the analysis also increases with $n$ .","Conventional asymptotic tools for proving the bootstrap validity are based on weak convergence arguments [50], which are not directly applicable in the finite-sample setting.","In this paper we turn to a more refined and self-contained analysis, and prove a new local restricted strong convexity (RSC) property for the empirical smoothed quantile loss.","This validates the key merit of convolution-type smoothing, i.e., local strong convexity.","The smoothing method involves a bandwidth, denoted by $h$ .","Theoretically, we show that with sub-exponential random covariates (relaxing the bounded covariates assumption in [20]), conquer exhibits an $\\ell _2$ -error in the order of $\\sqrt{(p+t)/n}+h^2$ with probability at least $1-2e^{-t}$ .","When $h$ is of order $\\lbrace (p+t)/n \\rbrace ^{\\gamma }$ for any $\\gamma \\in [1/4, 1/2]$ , the conquer estimation is first-order equivalent to QR.","Under slightly more stringent sub-Gaussian condition on the covariates, we show that the Bahadur-Kiefer linearization error of conquer is of order $(p+t)/(nh^{1/2})+ h \\sqrt{(p+t)/n}+h^3$ with probability at least $1-3e^{-t}$ .","Based on such a representation, we establish a Berry-Esseen bound for linear functionals of conquer, which lays the theoretical foundation for testing general linear hypotheses, encompassing covariate-effect analysis, analysis of variance, and model comparisons, to name a few.","It is worth noting that with a properly chosen $h$ , the linear functional of conquer is asymptotically normal as long as $p^{8/3}/n\\rightarrow 0$ , which improves the best known growth condition on $p$ for standard QR [54], [26], [43].","We attribute this gain to the effect of smoothing.","Under similar conditions, we further establish upper bounds on both estimation and Bahadur-Kiefer linearization errors for the bootstrapped conquer estimator.","In the context of nonparametric density or regression estimation, it is known that when higher-order kernels are used (and if the density or regression function has enough derivatives), the bias is proportional to $h^\\nu $ for some $\\nu \\ge 4$ which is of better order than $h^2$ .","While a higher-order kernel has negative parts, the resulting smoothed loss is non-convex and thus brings the computational issue once again.","Motivated by the two-stage procedure proposed by [7] whose original idea is to improve an initial estimator that is already consistent but not efficient, we further propose a one-step conquer estimator using higher-order kernels but without the need for solving a large-scale non-convex optimization.","With increasing degrees of smoothness, the one-step conquer is asymptotically normal under a milder dimension constraint of roughly $p^2/n \\rightarrow 0$ .","To better appreciate the computational feasibility of conquer for large-scale problems, we compare it with standard QR on large synthetic datasets, where the latter is implemented by the R package quantreg [34] using the Frisch-Newton approach after preprocessing “pfn”.","We generate independent data vectors $\\lbrace y_i, _i\\rbrace _{i = 1}^n$ from a linear model $y_i = \\beta ^*_0 + \\langle _i , \\beta ^* \\rangle + \\varepsilon _i$ , where $(\\beta ^*_0, {\\beta ^*}^^ (1, \\ldots , 1)^^{p+1}$ , $_i \\sim \\mathcal {N}_p(0, )$ , and the independent errors $\\varepsilon _i \\sim t_2$ , for $i = 1, 2, \\dots , n$ .","We report the estimation error and elapsed time for increasing sample sizes $n\\in \\lbrace 1000,5000,10000,\\ldots ,100000\\rbrace $ and dimension $p = \\lfloor n^{1/2} \\rfloor $ , the largest integer that is less than or equal to $n^{1/2}$ .","Figure REF displays the average estimation error and average elapsed time based on 20 Monte Carlo samples.","This experiment shows promise of conquer as a practically useful tool for large-scale quantile regression analysis.","More empirical evidence will be given in the latter section.","Figure: RuntimeThe rest of the paper is organized as follows.","We start with a brief review of linear quantile regression and the convolution-type smoothing method in Section .","Explicit forms of the smoothed check functions are provided for several representative kernel functions in nonparametric statistics.","We introduce the multiplier bootstrap for statistical inference in Section REF .","In Section , we provide a comprehensive theoretical study of conquer from a nonasymptotic viewpoint, which directly leads to array asymptotic results.","Specifically, the bias incurred by smoothing the quantile loss is characterized in Section REF .","In Section REF , we establish the rate of convergence, Bahadur-Kiefer representation, and Berry-Esseen bound for conquer in a large-$p$ and larger-$n$ regime.","Results for its bootstrap counterpart are provided in Section REF .","A Barzilai-Borwein gradient-based algorithm with a Huber-type warm start is detailed in Section .","We conclude the paper with an extensive numerical study in Section to illustrate the finite-sample performance of conquer in large-scale quantile regression analysis.","We defer the proofs of all theoretical results as well as the full details of the one-step conquer to online supplementary materials.","Notation: For every integer $k\\ge 1$ , we use $^k$ to denote the the $k$ -dimensional Euclidean space.","The inner product of any two vectors $=(u_1, \\ldots , u_k)^ =(v_1, \\ldots ,v_k)^^k$ is defined by $^= \\langle , \\rangle = \\sum _{i=1}^k u_i v_i$ .","We use $\\Vert \\cdot \\Vert _p$ $(1\\le p \\le \\infty )$ to denote the $\\ell _p$ -norm in $^k$ : $\\Vert \\Vert _p = ( \\sum _{i=1}^k | u_i |^p )^{1/p}$ and $\\Vert \\Vert _\\infty = \\max _{1\\le i\\le k} |u_i|$ .","Throughout this paper, we use bold capital letters to represent matrices.","For $k\\ge 2$ , $_k$ represents the identity matrix of size $k$ .","For any $k\\times k$ symmetric matrix $\\in ^{k\\times k}$ , $\\Vert \\Vert _2$ denotes the operator norm of $$ .","If $$ is positive semidefinite, we use $\\Vert \\cdot \\Vert _{}$ to denote the vector norm linked to $$ given by $\\Vert \\Vert _{} = \\Vert ^{1/2} \\Vert _2$ , $\\in ^k$ .","For $r \\ge 0$ , define the Euclidean ball and sphere in $^k$ as $\\mathbb {B}^k(r) = \\lbrace \\in ^k : \\Vert \\Vert _2 \\le r\\rbrace $ and $\\mathbb {S}^{k-1}(r) = \\partial \\mathbb {B}^k(r) = \\lbrace \\in ^k: \\Vert \\Vert _2 =r \\rbrace $ , respectively.","For two sequences of non-negative numbers $\\lbrace a_n \\rbrace _{n\\ge 1}$ and $\\lbrace b_n \\rbrace _{n\\ge 1}$ , $a_n \\lesssim b_n$ indicates that there exists a constant $C>0$ independent of $n$ such that $a_n \\le Cb_n$ ; $a_n \\gtrsim b_n$ is equivalent to $b_n \\lesssim a_n$ ; $a_n \\asymp b_n$ is equivalent to $a_n \\lesssim b_n$ and $b_n \\lesssim a_n$ ."],["The linear quantile regression model","Given a univariate response variable $y\\in $ and a $p$ -dimensional covariate vector $= (x_1, \\ldots , x_p)^^p$ with $x_1 \\equiv 1$ , the primary goal here is to learn the effect of $$ on the distribution of $y$ .","Let $F_{y|}(\\cdot )$ be the conditional distribution function of $y$ given $$ .","The dependence between $y$ and $$ is then fully characterized by the conditional quantile functions of $y$ given $$ , denoted as $F^{-1}_{y|}(\\tau )$ , for $0<\\tau <1$ .","We consider a linear quantile regression model at a given $\\tau \\in (0,1)$ , that is, the $\\tau $ -th conditional quantile function is $F^{-1}_{y|} (\\tau ) = \\langle , \\beta ^*(\\tau ) \\rangle ,$ where $\\beta ^*(\\tau ) = (\\beta ^*_{1}(\\tau ) , \\ldots , \\beta ^*_{ p}(\\tau ) )^^p$ is the true quantile regression coefficient.","Let $\\lbrace (y_i, _i) \\rbrace _{i=1}^n$ be a random sample from $(y,)$ .","The standard quantile regression estimator [35] is then given as $ \\hat{\\beta }(\\tau ) \\in \\min _{\\beta \\in ^p } \\hat{Q} (\\beta ) = \\min _{\\beta \\in ^p } \\frac{1}{n} \\sum _{i=1}^n\\rho _\\tau ( y_i - \\langle _i , \\beta \\rangle ) ,$ where $ \\rho _\\tau (u) = u \\lbrace \\tau - \\mathbb {1}(u<0)\\rbrace $ is the $\\tau $ -quantile loss function, also referred to as the pinball loss and check function.","Statistical properties of $\\hat{\\beta }(\\tau )$ have been extensively studied.","We refer the reader to [33] and [36] for more details."],["Convolution-type smoothing","We consider the convolution-type smoothing approach proposed by [20].","For every $\\beta \\in ^p$ , let $\\hat{F}(\\cdot ;\\beta )$ be the empirical distribution function of the residuals $\\lbrace r_i(\\beta ) := y_i - \\langle _i,\\beta \\rangle \\rbrace _{i=1}^n$ , i.e., $\\hat{F}(t ;\\beta ) = (1/n) \\sum _{i=1}^n\\mathbb {1}\\lbrace r_i(\\beta ) \\le t \\rbrace $ .","Given a fixed quantile level $\\tau $ , the empirical quantile loss $\\hat{Q}(\\cdot )$ in (REF ) can be represented as $\\hat{Q}(\\beta ) = \\int _{-\\infty }^\\infty \\rho _\\tau (u) \\,{\\rm d} \\hat{F}(u;\\beta ).","$ Given a bandwidth $h=h_n >0$ , let $\\hat{F}_h(\\cdot ;\\beta )$ be the distribution function of the classical Rosenblatt–Parzen kernel density estimator: $\\hat{F}_h(u;\\beta ) = \\int _{-\\infty }^u \\hat{f}_h(t;\\beta ) \\,{\\rm d} t~~\\mbox{ with }~~ \\hat{f}_h(t;\\beta ) = \\frac{1}{nh} \\sum _{i=1}^nK\\bigg ( \\frac{t-r_i(\\beta )}{h} \\bigg ) , $ where $K:\\mathbb {R} \\rightarrow ^+ := \\mathbb {[}0,\\infty )$ is a symmetric kernel that integrates to one.","Replacing the empirical distribution function in (REF ) with $\\hat{F}_h $ yields the following smoothed empirical quantile loss $\\hat{Q}_h(\\beta ) := \\int _{-\\infty }^\\infty \\rho _\\tau (u) \\,{\\rm d} \\hat{F}_h(u;\\beta ) = \\frac{1}{nh} \\sum _{i=1}^n\\int _{-\\infty }^\\infty \\rho _\\tau (u) K \\bigg ( \\frac{u + \\langle _i, \\beta \\rangle - y_i }{h} \\bigg ) \\,{\\rm d} u .","$ Accordingly, the smoothed QR estimator is given by $ \\hat{\\beta }_h = \\hat{\\beta }_h(\\tau ) \\in _{\\beta \\in ^p} \\hat{Q}_h(\\beta ) .$ As we shall see later, the ideal choice of bandwidth should adapt to the sample size $n$ and dimension $p$ , while the quantile level $\\tau $ is prespecified and fixed.","Thus, the dependence of $\\hat{\\beta }_h$ and $\\hat{Q}_h(\\cdot )$ on $\\tau $ will be assumed without display.","Given a kernel function $K(\\cdot )$ and bandwidth $h>0$ , the smoothed quantile loss $\\hat{Q}_h(\\cdot )$ defined in (REF ) can be equivalently written as $\\hat{Q}_h(\\beta ) = (1/n) \\sum _{i=1}^n\\ell _h(y_i - \\langle _i, \\beta \\rangle )$ , where $\\ell _h(u) = (\\rho _\\tau * K_h )(u) = \\int _{-\\infty }^{\\infty } \\rho _\\tau (v) K_h(v- u ) \\, {\\rm d} v $ with $*$ denoting the convolution operator and $K_h(u)=(1/h)K(u/h)$ .","Therefore, as stated in the Introduction, we refer to the aforementioned smoothing method as conquer throughout the paper.","Commonly used kernel functions include: (a) uniform kernel $K(u) = (1/2) \\mathbb {1} (|u|\\le 1)$ , (b) Gaussian kernel $K(u) = \\phi (u):= (2\\pi )^{-1/2} e^{-u^2/2}$ , (c) logistic kernel $K(u) = e^{-u}/(1+ e^{-u})^2 $ , (d) Epanechnikov kernel $K(u) = (3 / 4) (1 - u^2) \\mathbb {1}(|u|\\le 1)$ , and (e) triangular kernel $K(u) = (1 - |u|) \\mathbb {1}(|u|\\le 1)$ .","Explicit expressions of the corresponding smoothed loss function $\\rho _\\tau * K_h$ will be given in Section .","The convolution-type kernel smoothing yields an objective function $\\beta \\mapsto \\hat{Q}_{h}(\\beta )$ that is twice continuously differentiable with gradient and hessian matrix $\\nabla \\hat{Q}_{h}(\\beta ) = \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace {K}\\bigl ( -r_i(\\beta )/h \\bigr ) - \\tau \\bigr \\rbrace _i ~~\\mbox{ and }~~\\nabla ^2 \\hat{Q}_{h}(\\beta ) = \\frac{1}{n} \\sum _{i=1}^nK_h( r_i(\\beta )) _i _i^ $ respectively, where $K(u) := \\int _{-\\infty }^u K(t) \\,{\\rm d} t$ .","Provided that $K$ is non-negative, $\\hat{Q}_{h}(\\cdot )$ is a convex function for any $h>0$ , and $\\hat{\\beta }_h = \\hat{\\beta }_h(\\tau ) $ satisfies the first-order condition $\\nabla \\hat{Q}_{ h}(\\hat{\\beta }_h ) = \\frac{1}{n} \\sum _{i=1}^n\\Biggl \\lbrace {K}\\Biggl ( \\frac{\\langle _i , \\hat{\\beta }_h \\rangle - y_i }{h} \\Biggr ) - \\tau \\Biggr \\rbrace _i = \\textbf {0} .\\nonumber $ The smoothness and convexity of $\\hat{Q}_{h}$ warrant the superior computation efficiency of first-order gradient based algorithms for solving large-scale smoothed quantile regressions.","Theoretically, with $p$ fixed and $n \\rightarrow \\infty $ , [20] showed that the convolution-type smoothed QR estimator has a lower asymptotic mean squared error than Horowitz's smoothed estimator, and also has a smaller Bahadur linearization error than the standard QR in the almost sure sense.","The optimal order of the bandwidth based on the asymptotic mean squared error is unveiled as a function of $n$ .","Our theoretical results complement the results in [20] by providing a comprehensive analysis of conquer in the regime in which $p$ is allowed to increase with $n$ ."],["Connection to related existing work","The idea of smoothing the piecewise linear loss, to the best of our knowledge, dates back to [1] in the context of median regression.","Specifically, [1] proposed a smoothed approximation of the absolute value function that has the form $u \\mapsto u + 2 h \\log (1 + e^{-u / h}) , \\ \\ u\\in ,$ where $h= n^{-\\gamma }$ with $ \\gamma \\in (1/3, 1/2)$ .","This smoothing device, however, is mainly used therein to simplify the analysis of the asymptotic behavior of a two-stage median regression estimator.","[27] proposed a kernel smoothing method of similar type for bootstrapping the median regression estimator.","The idea is to replace the indicator function in $\\rho _\\tau (u) = u \\lbrace \\tau - \\mathbb {1}(u<0) \\rbrace $ with a smoothed counterpart, leading to $ \\ell ^{\\mathrm {Horo}}_{ h}(u) = u \\bigl \\lbrace \\tau - {K}(-u / h) \\bigr \\rbrace ,$ where $K(u) = \\int _{-\\infty }^u K(t) \\,{\\rm d} t$ , and $K(\\cdot )$ is a symmetric kernel function.","The key difference between the conquer loss (REF ) and Horowitz's loss (REF ) is that the former is globally convex, while Horowitz's loss is not.","This is illustrated in Figure REF .","We refer to [20] for an in-depth comparison between convolution smoothing and Horowitz's smoothing in terms of asymptotic mean squared errors of the smoothed estimators.","A closer inspection reveals that the smoothed function considered by [1] is a convolution-type smoothed loss with a logistic kernel.","More recently, [57] considered a smoothing approximation for the quantile loss, which they refer to as the Huber approximation.","Their goal was to compute the regularized QR estimator but via a smoothed optimization method.","The corresponding loss function also falls into the general framework of convolution-type smoothing, with $K$ taken as the uniform kernel.","The general statistical theory developed in this paper applies to those special cases.","Figure: Uniform kernel under τ=0.7\\tau = 0.7."],["Multiplier bootstrap inference","In this section, we consider a multiplier bootstrap procedure to construct confidence intervals for conquer.","Independent of the observed sample $\\mathcal {X}_n = \\lbrace (y_i,_i) \\rbrace _{i=1}^n$ , let $\\lbrace w_i\\rbrace _{i = 1}^n$ be independent and identically distributed random variables with $(w_i)=1$ and $\\textnormal {var}(w_i)=1$ .","Recall that $\\hat{\\beta }_h = \\hat{\\beta }_h(\\tau ) = \\min _{\\beta \\in ^p} \\hat{Q}_{h}(\\beta )$ is the conquer estimator.","If the minimizer is not unique, we take any of the minima as $\\hat{\\beta }_h = (\\hat{\\beta }_{h,1},\\ldots , \\hat{\\beta }_{h,p})^.$ The proposed bootstrap method, which dates back to [16] and [3], is based on reweighting the summands of $\\hat{Q}_{h}(\\cdot )$ with random weights $w_i$ .","The resulting weighted quantile loss $\\hat{Q}^{\\flat }_h :^p \\rightarrow $ is $\\hat{Q}_{ h}^\\flat (\\beta ) = \\frac{1}{n} \\sum _{i=1}^nw_i \\ell _{h}(y_i - \\langle _i, \\beta \\rangle ) , $ where $\\ell _{ h}(u) = (\\rho _\\tau * K_h)(u)$ is as in (REF ).","We refer to [9] for a general asymptotic theory for weighted bootstrap for estimating equations, where a class of bootstrap weights is considered.","Extensions to semiparametric $M$ -estimation can be found in [41] and [12].","Let $\\mathbb {E}^*$ and $^*$ be the conditional expectation and probability given the observed data $_n$ , respectively.","Observe that $\\mathbb {E}^*\\lbrace \\hat{Q}_{h}^{\\flat }(\\beta ) \\rbrace = \\hat{Q}_{h}(\\beta )$ for any $\\beta \\in ^{p}$ .","Consequently, we have $_{\\beta \\in ^p} ^* \\lbrace \\hat{Q}_{h}^{\\flat }(\\beta ) \\rbrace = _{\\beta \\in ^p} \\hat{Q}_{h}(\\beta ) = \\hat{\\beta }_h .$ This simple and yet important observation motivates the following multiplier bootstrap statistic: $\\hat{\\beta }_h^\\flat = \\hat{\\beta }_h^\\flat (\\tau ) \\in _{\\beta \\in ^p } \\hat{Q}_{h}^{\\flat }(\\beta ) .","$ To retain convexity of the loss function, non-negative random weights are preferred, such as $w_i \\sim {\\rm Exp}(1)$ , i.e., exponential distribution with rate 1, and $w_i =1 + e_i$ , where $e_i$ are independent Rademacher random variables.","We can construct confidence intervals based on the bootstrap estimates using one of the three classical methods, the percentile method, the pivotal method, and the normal-based method.","To be specific, for each $q\\in (0,1)$ and $1\\le j\\le p$ , define the (conditional) $q$ -quantile of $\\hat{\\beta }^\\flat _{h,j}$ —the $j^{{\\rm th}}$ coordinate of $\\hat{\\beta }^\\flat _h \\in ^p$ —given the observed data as ${\\rm c}^\\flat _j(q) = \\inf \\lbrace t \\in : ^* ( \\hat{\\beta }^\\flat _{h,j} \\le t ) \\ge q \\rbrace $ .","Then, for a prespecified nominal level $\\alpha \\in (0,1)$ , the corresponding $1-\\alpha $ bootstrap percentile and pivotal confidence intervals (CIs) for $\\beta ^*_j$ ($j=1,\\ldots ,p$ ) are, respectively, $\\left[ {\\rm c}^\\flat _j(\\alpha /2 ) , \\, {\\rm c}^\\flat _j( 1 -\\alpha /2) \\right] ~\\mbox{ and }~ \\left[ 2\\hat{\\beta }_{h,j} - {\\rm c}^\\flat _j( 1 -\\alpha /2) , \\, 2 \\hat{\\beta }_{h,j} - {\\rm c}^\\flat _j(\\alpha /2 ) \\right] .$ Numerically, ${\\rm c}^\\flat _j(q)$ ($q\\in \\lbrace \\alpha , 1-\\alpha /2\\rbrace $ ) can be calculated with any specified precision by the simulation.","In the R package conquer, the default number of bootstrap replications is set as $B=1000$ .","In the next section, we will present a finite-sample theoretical framework for convolution-type smoothed quantile regression, including the concentration inequality and nonasymptotic Bahadur representation for both the conquer estimator (REF ) and its bootstrap counterpart (REF ) using Rademacher multipliers.","As a by-product, a Berry-Esseen-type inequality (see Theorem REF ) states that, under certain constraints on the (growing) dimensionality and bandwidth, the distribution of any linear projection of $\\hat{\\beta }_h$ converges to a normal distribution as the sample size increases to infinity.","Informally, for any given deterministic vector $\\in ^p$ , the scaled statistic $n^{1/2} \\langle , \\hat{\\beta }^*_h - \\beta ^* \\rangle $ is asymptotically normally distributed with asymptotic variance $\\sigma _\\tau ^2 := \\tau (1-\\tau ) \\,^^{-1} ^{-1} $ .","To make inference based on such asymptotic results, we need to consistently estimate the asymptotic variance.","[20] suggested the following estimators $\\hat{}_h := \\nabla ^2 \\hat{}_h(\\hat{\\beta }_h) = \\frac{1}{n} \\sum _{i=1}^nK(\\hat{\\varepsilon }_i /h) \\cdot _i^~\\mbox{ and }~~ \\hat{}_h := \\frac{1}{n h} \\sum _{i=1}^n\\bigl \\lbrace K(-\\hat{\\varepsilon }_i/ h) - \\tau \\bigr \\rbrace ^2 ^{align*}of and \\tau (1-\\tau ) \\, , respectively, where \\hat{\\varepsilon }_i = y_i - \\langle _i, \\hat{\\beta }_h\\rangle are fitted residuals.","The ensuing 1-\\alpha normal-based CIs are given by \\hat{\\beta }_{h,j} \\pm \\Phi ^{-1}(1-\\alpha /2) \\cdot n^{-1/2} (\\hat{}_h^{-1} \\hat{}_h \\hat{}_h^{-1} )_{jj}, j=1,\\ldots ,p.However, the normal approximations to the CI suffer from the sensitivity to the smoothing needed to estimated the conditional densities, namely, the matrix = \\lbrace f_{\\varepsilon | }(0) ^.","This is especially true when \\tau is in the upper or lower tail.","See Section~\\ref {sec:normal-boot} for a numerical comparison between normal approximation and bootstrap calibration for confidence construction at high and low quantile levels.","As we shall see, the normal-based CIs can be exceedingly wide and thus inaccurate under these situations.$"],["Statistical Analysis","Under the linear quantile regression model in (REF ), we write, for convenience, the generic data vector $(y,)$ in a linear model form: $y = \\langle , \\beta ^*(\\tau ) \\rangle + \\varepsilon (\\tau ), $ where the random variable $\\varepsilon (\\tau )$ satisfies $\\lbrace \\varepsilon (\\tau ) \\le 0 | \\rbrace = \\tau $ .","Let $ f_{\\varepsilon |}(\\cdot )$ be the conditional density function of the regression error $\\varepsilon = \\varepsilon (\\tau )$ given $$ .","We first derive upper bounds for the smoothing bias under mild regularity conditions on the conditional density $ f_{\\varepsilon |}$ and the kernel function."],["Smoothing bias","[Kernel function] Let $K(\\cdot )$ be a symmetric and non-negative function that integrates to one, that is, $K(u)= K(-u)$ , $K(u) \\ge 0$ for all $u \\in $ and $\\int _{-\\infty }^\\infty K(u) \\, {\\rm d}u = 1$ .","Moreover, $\\int _{-\\infty }^\\infty u^2 K(u) \\, {\\rm d} u <\\infty $ .","We will use the notation $\\kappa _k = \\int _{-\\infty }^\\infty |u |^k K(u) \\,{\\rm d} u$ for $k \\ge 1$ .","Furthermore, we define the population smoothed loss function $Q_{ h} (\\beta ) = \\lbrace \\hat{Q}_{ h}(\\beta )\\rbrace $ , $\\beta \\in ^p$ and the pseudo parameter $\\beta ^*_h (\\tau ) \\in _{\\beta \\in ^p } Q_{ h}(\\beta ), $ which is the population minimizer under the smoothed quantile loss.","For simplicity, we write $\\beta ^* = \\beta ^*(\\tau )$ and $\\beta ^*_h = \\beta _h^*(\\tau )$ hereinafter.","In general, $\\beta ^*_h$ differs from $\\beta ^*$ , and we refer to $\\Vert \\beta ^*_h - \\beta ^* \\Vert _2$ as the approximation error or smoothing bias.","[Conditional density] There exist $\\bar{f} \\ge {f} >0$ such that $ {f} \\le f_{\\varepsilon |}(0 ) \\le \\bar{f}$ almost surely (for all $$ ).","Moreover, there exists a constant $l_0 >0$ such that $|f_{\\varepsilon |}(u)- f_{\\varepsilon |}(0)| \\le l_0 |u |$ for all $u \\in $ almost surely.","[Random design: moments] The (random) vector $\\in \\mathbb {R}^p$ of covariates satisfies $\\mu _3 := \\sup _{\\in \\mathbb {S}^{p-1}} \\bigl ( | \\langle ^{-1/2} , \\rangle |^3 \\bigr )<\\infty $ , where $= (^$ is positive definite.","If $$ is Gaussian, then Condition REF holds trivially.","Heavier-tailed distributions of $$ is excluded here so that we can expect standard rates of convergence for the quantile regression estimates.","The following result characterizes the smoothing bias from a nonasymptotic viewpoint.","Assume Conditions REF –REF hold.","There exist constants $c_1, c_2>0$ depending only on $({f} , l_0, \\mu _3, \\kappa _1, \\kappa _2)$ such that for any $00$ .","Then $ \\biggl \\Vert \\beta ^*_h - \\beta ^* + \\frac{1}{2} \\kappa _2 h^2 \\cdot ^{-1} \\bigl \\lbrace f_{\\varepsilon |}^{\\prime }( 0 )\\bigr \\rbrace \\biggr \\Vert _{}\\le c_3 h^3,$ where $c_3 >0$ is a constant depending only on $({f} , l_0 , l_1 , \\mu _3 )$ and the kernel $K$ .","To better understand inequality (REF ), consider for simplicity a special case where $ f_{\\varepsilon |} ( 0 ) $ is a constant (independent of $$ ).","Hence, $= f_{\\varepsilon |} ( 0 ) $ and (REF ) becomes $( \\langle , \\beta ^*_h -\\beta ^* \\rangle ^2 )^{1/2} = \\Vert \\beta ^*_h -\\beta ^* \\Vert _{} \\le f^{-1/2}_{\\varepsilon |} ( 0 ) c_2 h^2$ .","On the left-hand side, $\\Vert \\beta ^*_h -\\beta ^* \\Vert _{}$ is the average prediction smoothing error.","Interestingly, the upper bound on the right-hand side could be dimension-free given $h$ .","Another interesting implication is that, when both $f_{\\varepsilon |}(0)$ and $f^{\\prime }_{\\varepsilon |}(0)$ are independent of $$ , i.e., $f_{\\varepsilon |}(0)=f_{\\varepsilon }(0)$ and $f^{\\prime }_{\\varepsilon |}(0)=f^{\\prime }_{\\varepsilon }(0)$ , the leading term in the bias simplifies to $\\frac{1}{2} \\kappa _2 h^2 \\cdot ^{-1} \\bigl \\lbrace f_{\\varepsilon |}^{\\prime }(0)\\bigr \\rbrace = \\frac{f^{\\prime }_{\\varepsilon }(0)}{2f_{ \\varepsilon }(0)} \\kappa _2 h^2 \\cdot ^{-1} () = \\frac{f^{\\prime }_{\\varepsilon }(0)}{2f_{\\varepsilon }(0)} \\kappa _2 h^2 \\cdot \\begin{bmatrix}1 \\\\\\textbf {0}_{p-1}\\end{bmatrix} .","\\nonumber $ In other words, the smoothing bias is concentrated primarily on the intercept.","In the asymptotic setting where $p$ is fixed, and $h=o(1)$ as $n\\rightarrow \\infty $ , we refer to Theorem 1 in [20] for the expression of asymptotic bias."],["Finite sample theory","In this section, we provide two nonasymptotic results, the concentration inequality and the Bahadur-Kiefer representation, for the conquer estimator under random design.","[Random design: sub-exponential case] The predictor $\\in ^p$ is sub-exponential: there exists $ \\upsilon _0>0$ such that $\\lbrace |\\langle , \\rangle | \\ge \\upsilon _0 \\Vert \\Vert _2 t \\rbrace \\le 2e^{-t }$ for all $\\in ^p$ and $t \\ge 0$ , where $= ^{-1/2} $ with $= (^$ positive-definite.","Condition REF assumes a sub-exponential condition on the random covariates, which encompasses the bounded case considered by [20].","For the standardized predictor $= ^{-1/2}$ , we define the uniform moment parameters $\\mu _k = \\sup _{\\in \\mathbb {S}^{p-1} } | \\langle , \\rangle |^k , \\ \\ k = 1,2 ,\\ldots , $ with $\\mu _2=1$ .","Under Condition REF , a straightforward calculation shows that $\\mu _k \\le 2 \\upsilon _0^k k!$ , valid for all $k\\ge 1$ , where $\\upsilon _0$ is defined in Condition REF .","In general, $\\upsilon _0$ may increase with $p$ , but in this paper we are primarily interested in the cases where $\\upsilon _0$ is a dimension-free constant.","This is indeed the case if $$ is multivariate Gaussian, multivariate Bernoulli, uniform on $[-1,1]^p$ , or uniform on the sphere/ball with radius $p^{1/2}$ ; see Section 3.4 in [51] for more examples.","Assume that Conditions REF –REF hold with $\\kappa _l = \\min _{|u|\\le 1} K(u) >0$ .","For any $t > 0$ , the smoothed quantile regression estimator $\\hat{\\beta }_{h}$ with $\\sqrt{(p+t)/n} \\lesssim h \\lesssim 1$ satisfies $\\Vert \\hat{\\beta }_h - \\beta ^* \\Vert _{} \\le C (\\kappa _l {f} )^{-1} \\Biggl ( \\sqrt{\\frac{p+t}{n}} + h^2 \\Biggr ), $ with probability at least $1-2e^{-t}$ , where $C>0$ is a constant depending only on $(\\upsilon _0, \\kappa _2,l_0)$ .","The estimation error in (REF ) is upper bounded by two terms, $h^2$ and $\\sqrt{(p+t)/n}$ , and can be interpreted as the upper bounds for bias and variance, respectively.","The condition $\\min _{|u|\\le 1} K(u) >0$ can be relaxed to $ \\min _{|u|\\le c} K(u) >0$ for some $c\\in (0,1)$ , which will only change the constants encountered in the proof.","In particular, for kernels that are compactly supported on $[-1,1]$ , we may choose $c=1/2$ in Theorem REF .","Next, we establish a Bahadur representation for conquer.","To this end, we impose a slightly more stringent sub-Gaussian condition on the covariates.","[Random design: sub-Gaussian case] The predictor $\\in ^p$ is sub-Gaussian: there exists $ \\upsilon _1 >0$ such that $\\lbrace |\\langle , \\rangle | \\ge \\upsilon _1 \\Vert \\Vert _2 t \\rbrace \\le 2 e^{-t^2/2}$ for all $\\in ^p$ and $t\\ge 0$ , where $= ^{-1/2} $ .","We are primarily concerned with the cases where $\\upsilon _1$ is a dimension-free constant that does not depend on $p$ ; see the remarks made earlier about $\\upsilon _0$ .","Assume that the data $\\lbrace (y_i, _i) \\rbrace _{i=1}^n$ are generated from the conditional quantile model (REF ), and write $\\varepsilon _i = y_i - \\langle _i, \\beta ^* \\rangle $ which satisfy $(\\varepsilon _i\\le 0 | _i) =\\tau $ .","Assume Conditions REF , REF , and REF hold with $\\kappa _l = \\min _{|u|\\le 1} K(u) >0$ and $\\kappa _u = \\sup _{u \\in } K(u)$ .","Moreover, assume $\\sup _{u \\in } f_{\\varepsilon | } (u) \\le \\bar{f} $ almost surely.","For any $t > 0$ , let the bandwidth $h$ satisfy $\\sqrt{(p+t)/n} \\lesssim h \\lesssim 1$ .","Then, $\\Biggl \\Vert \\hat{\\beta }_h - \\beta ^* - ^{-1} \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace \\tau - K(- \\varepsilon _i /h) \\bigr \\rbrace _i \\Biggr \\Vert _2 \\le C \\Biggl ( \\frac{p+ t}{ h^{1/2} n } + h\\sqrt{\\frac{p+t}{n}} + h^3\\Biggr ) , $ with probability at least $1-3e^{-t}$ , where $= \\bigl \\lbrace f_{\\varepsilon |}( 0 ) ^\\rbrace $ , and $C>0$ is a constant depending only on $(\\upsilon _1, \\kappa _2, \\kappa _u, \\kappa _l, l_0,\\bar{f}, {f})$ .","The Bahadur representation can be used to establish the limiting distribution of the estimator or its functionals.","Here we consider a fundamental statistical inference problem for testing the linear hypothesis $H_0: \\langle , \\beta ^* \\rangle = 0$ , where $\\in ^p$ is a deterministic vector that defines a linear functional of interest.","It is then natural to consider a test statistic that depends on $n^{1/2} \\langle , \\hat{\\beta }_h \\rangle $ .","Based on the nonasymptotic result in Theorem REF , we establish a Berry-Esseen bound for the linear projection of conquer.","Assume that the conditions in Theorem REF hold, and $\\sqrt{(p+\\log n)/n}\\lesssim h \\lesssim 1$ .","Then, $\\Delta _{n,p}( h) :=\\sup _{x\\in , \\, \\in ^p } \\left| \\left\\lbrace \\frac{n^{1/2} \\langle , \\hat{\\beta }_h - \\beta ^* \\rangle }{\\sqrt{\\tau (1-\\tau ) \\, ^^{-1} ^{-1} }} \\le x \\right\\rbrace - \\Phi (x ) \\right| \\lesssim \\frac{p + \\log n}{(n h)^{1/2}} + n^{1/2} h^2 , $ where $\\Phi (\\cdot )$ denotes the standard normal distribution function.","In particular, with the choice of bandwidth $h \\asymp \\lbrace (p+\\log n)/n \\rbrace ^{2/5}$ , the normal approximation error $ \\Delta _{n,p}( h) $ is of order $(p+\\log n)^{4/5} n^{-3/10}$ .","It is worth noticing that although the bound in (REF ) holds uniformly over all $$ , the asymptotic variance $\\sigma _\\tau ^2 := \\tau (1-\\tau ) \\, ^^{-1} ^{-1} $ depends on $$ .","[Large-$p$ asymptotics] A broader view of classical asymptotics recognizes that the parametric dimension of appropriate model sequences may tend to infinity with the sample size; that is $p=p_n \\rightarrow \\infty $ as $n\\rightarrow \\infty $ .","Results with increasing $p$ are available in the context of regularized quantile regression [6], [53], [36].","In the large-$p$ and larger-$n$ setting with quantile regression estimation without regularization, [54] shows that $p^3(\\log n)^2 / n \\rightarrow 0$ suffices for a normal approximation, which provides some support to the viability of observed rates of parametric growth in the applied literature.","[32] considers a sample of 733 wage models appeared in the econometric literature, and finds that $p_n = (n^{1/4})$ is roughly consistent with empirical practice.","For the (convolution-type) smoothed quantile regression, the Berry-Esseen bound (REF ) in Theorem REF immediately yields a large-$p$ asymptotic result.","Taking $h=h_n = \\lbrace (p+\\log n) /n\\rbrace ^{2/5}$ therein, we find that $n^{1/2} \\langle , \\hat{\\beta }_h - \\beta ^* \\rangle $ , for any given $\\in ^p$ , is asymptotically normally distributed as long as $p^{8/3} /n \\rightarrow 0$ , which improves the best known growth condition on $p$ for quantile regression [54].","For smooth robust regression estimators, asymptotic normality can be proven under less restrictive conditions on $p$ .","[28] showed that if the loss is twice differentiable, the asymptotic normality for $\\langle , \\hat{\\beta }\\rangle $ , where $\\in ^p$ , holds if $p^3/n \\rightarrow 0$ as $n$ increases.","[46] weakened the condition to $(p \\log n)^{3/2} /n \\rightarrow 0$ if the loss function is four times differentiable and the error distribution is symmetric.","For Huber loss that has a Lipschitz continuous derivative, [26] obtained the scaling $p^2 \\log p = o(n)$ that ensures the asymptotic normality of arbitrary linear combinations of $\\hat{\\beta }$ .","Table 1 summarizes our discussion here and show that the smoothing for conquer helps ensure asymptotic normality of the estimator under weaker conditions on $p$ than what we need for the usual quantile regression estimator.","Table: Summary of scaling conditions required for normal approximation under various loss functions.In this paper, we show that the accuracy of conquer-based inference via the Bahadur representation (and normal approximations) has an error of rate faster than $n^{-1/4}$ yet slower than $n^{-1/2}$ ; see Theorems REF and REF .","For standard regression quantiles, [47] proposed an alternative expansion for the quantile process using the “Hungarian\" construction of Komlós, Major and Tusnády.","This stochastic approximation yields an error of order $n^{-1/2}$ (up to a factor of $\\log n$ ), and hence provides a theoretical justification for accurate approximations for inference in regression quantile models."],["Theoretical guarantees for bootstrap inference","We next investigate the statistical properties of the bootstrapped estimator defined in (REF ), with a particular focus on the Rademacher multiplier bootstrap.","To be specific, we use, in this section and the rest of the paper, the random weights $w_i=1+e_i$ for $i=1,\\ldots , n$ , where $e_1,\\ldots , e_n$ are independent Rademacher random variables, that is, $(e_i=1) = (e_i=-1)=1/2$ .","As before, we consider array (non)asymptotics, so that the bootstrap approximation errors depend explicitly on $(n,p)$ .","Assume Conditions REF –REF , and REF hold with $\\kappa _l = \\min _{|u|\\le 1}K(u)>0$ .","For any $t\\ge 0$ , there exists some event $(t)$ with $\\lbrace (t) \\rbrace \\ge 1-5e^{-t}$ such that, with $^*$ -probability at least $1-2e^{-t}$ conditioned on $(t)$ , the bootstrap estimator $\\hat{\\beta }^\\flat _h$ given in (REF ) satisfies $\\Vert \\hat{\\beta }^\\flat _h - \\beta ^* \\Vert _{} \\le C \\Biggl ( \\sqrt{\\frac{p+t}{n} } + h^2 \\Biggr )$ as long as $\\max \\lbrace (p+\\log n + t )/n, \\sqrt{(p+t)/n } \\rbrace \\lesssim h \\lesssim 1$ , where $C>0$ is a constant depending only on $(\\upsilon _1,\\kappa _2, \\kappa _l, l_0, {f})$ .","Analogously to Theorem REF , we further provide a Bahadur representation result for the bootstrap counterpart $\\hat{\\beta }^\\flat _h $ , which lays the theoretical foundation for the validity of Rademacher multiplier bootstrap for conquer.","Assume Conditions REF –REF and REF hold with $\\kappa _l = \\min _{|u|\\le 1}K(u)>0$ and $\\kappa _u = \\sup _{u\\in } K(u)$ .","Moreover, assume $\\sup _{u\\in } f_{\\varepsilon | } (u) \\le \\bar{f}$ almost surely in $$ .","Let the bandwidth satisfy $h \\asymp \\lbrace (p+\\log n)/n\\rbrace ^{\\delta }$ for some $\\delta \\in [1/3, 1/2)$ .","Provided that $n\\gtrsim p + \\log n$ , there exists a sequence of events $\\lbrace _n\\rbrace $ with $(_n) \\ge 1-6n^{-1}$ such that, with $^*$ -probability at least $1-2n^{-1}$ conditioned on $_n$ , $\\hat{\\beta }^\\flat _h - \\hat{\\beta }_h = ^{-1 } \\frac{1}{n} \\sum _{i=1}^ne_i \\cdot _i \\bigl \\lbrace \\tau - K(- \\varepsilon _i /h) \\bigr \\rbrace + ^\\flat _n, $ where $^{\\flat }_n$ depends on $\\lbrace (e_i, y_i, _i) \\rbrace _1^n$ and satisfies $\\Vert ^\\flat _n \\Vert _2 = _{^*}(\\chi _n)$ , where $\\chi _n = \\chi _n(\\lbrace y_i, _i \\rbrace _{i=1}^n) = _{}\\lbrace \\sqrt{(p+\\log n) p \\log n} / (h^{1/2} n) + (p+\\log n)^{1/2} p \\log (n) /n \\rbrace $ .","As suggested by Theorem REF and Remark REF , if we take $h$ to be $\\lbrace (p+\\log n)/n \\rbrace ^{2/5}$ , the normal approximation to the conquer estimator is asymptotically accurate provided that $p^{8/3} = o(n)$ as $n\\rightarrow \\infty $ .","For the same $h$ , the remainder $^\\flat _n$ in (REF ) satisfies $\\Vert ^\\flat _n \\Vert _2 = o_{^*}(n^{-1/2})$ as long as $p^3 (\\log n)^2 = o(n)$ .","Taken these two parts together, we have the following asymptotic bootstrap approximation result.","If the dimension $p=p_n$ satisfies $p^3 (\\log n)^2 = o(n)$ , then, as $n\\rightarrow \\infty $ , $\\sup _{x\\in } \\, \\bigl | \\bigl ( n^{1/2} \\langle , \\hat{\\beta }_h - \\beta ^* \\rangle \\le x \\bigr ) - ^*\\bigl ( n^{1/2} \\langle , \\hat{\\beta }^\\flat _h - \\hat{\\beta }_h \\rangle \\le x \\bigr ) \\bigr | \\stackrel{}{\\rightarrow } 0 .","$ The proof of (REF ) follows the same argument as that in the proof of Theorem REF , and therefore is omitted.","The additional logarithmic factor in the scaling may be an artifact of the proof technique.","For standard quantile regression, [19] established a fixed-$p$ asymptotic bootstrap approximation result for wild bootstrap under fixed design."],["Gradient Descent Methods for Conquer","To solve optimization problems (REF ) and (REF ) with non-negative weights, arguably the simplest algorithm is a vanilla gradient descent algorithm (GD).","For a prespecified $\\tau \\in (0,1)$ and bandwidth $h>0$ , recall that $ \\hat{Q}_h (\\beta ) = (1/n) \\sum _{i=1}^n\\ell _h(y_i - \\langle _i, \\beta \\rangle )$ .","Starting with an initial value $\\hat{\\beta }^0 \\in ^p$ , at iteration $t=0, 1,2,\\ldots $ , GD computes $\\hat{\\beta }^{t+1} = \\hat{\\beta }^t - \\eta _t \\cdot \\nabla \\hat{Q}_h (\\hat{\\beta }^t) ,$ where $\\eta _t>0$ is the stepsize and $\\nabla \\hat{Q}_h (\\beta ) = \\frac{1}{n} \\sum _{i=1}^n\\biggl \\lbrace {K}\\biggl ( - \\frac{ y_i - \\langle _i, \\beta \\rangle }{h} \\biggr ) - \\tau \\biggr \\rbrace _i ~~\\mbox{ with }~~{K}(u ) = \\int _{-\\infty }^u K(t) \\,{\\rm d} t .","\\nonumber $ In the classical GD method, the stepsize is usually obtained by employing line search techniques.","However, line search is computationally intensive for large-scale settings.","One of the most important issues in GD is to determine a proper update step $\\eta _t$ decay schedule.","A common practice in the literature is to use a diminishing stepsize or a best-tuned fixed stepsize.","Neither of these two approaches can be efficient, at least compared to the Newton-Frisch algorithm with preprocessing [48].","Recall that the smoothed loss $\\hat{Q}_h(\\cdot )$ is twice differentiable with Hessian $\\nabla ^2 \\hat{Q}_h (\\beta ) = \\frac{1}{n h } \\sum _{i=1}^nK \\biggl ( \\frac{y_i - \\langle _i, \\beta \\rangle }{h} \\biggr ) _i _i^ \\nonumber $ It is therefore intriguing to employ the Newton-Raphson method, which at iteration $t$ would read $\\hat{\\beta }^{t+1} = \\hat{\\beta }^t - \\bigl \\lbrace \\nabla ^2 \\hat{Q}_h (\\hat{\\beta }^t) \\bigr \\rbrace ^{-1} \\nabla \\hat{Q}_h (\\hat{\\beta }^t) .$ Computing the inverse of the Hessian for large problems can be an expensive operation.","Moreover, when $h$ is small, $\\nabla ^2 \\hat{Q}_h (\\cdot )$ can have a very large condition number, which leads to slow convergence.","Below we list the explicit expressions of the convolution smoothed check functions using several commonly used kernels.","Recall that the check function can be written as $\\rho _\\tau (u) =|u|/2+ (\\tau -1/2)u$ , which, after convolution smoothing, becomes $\\ell _h (u) = \\int _{-\\infty }^\\infty \\frac{1}{2}|u + hv | K (v) \\, {\\rm d} v + (\\tau -1/2)u .$ (Uniform kernel $K(u) = (1/2) \\mathbb {1} (|u|\\le 1)$ ): $\\ell _h(u) = (h/2) \\ell ^{{\\rm U}}(u/h) + (\\tau - 1/2) u$ , where $ \\ell ^{{\\rm U}}(u) := (u^2/2 + 1/2)\\mathbb {1}( |u|\\le 1) + |u| \\mathbb {1}(|u|>1)$ is a Huber-type loss [28].","(Gaussian kernel $K(u)=(2\\pi )^{-1/2} e^{-u^2/2}$ ): $\\ell _{h}(u) = (h/2)\\ell ^{{\\rm G}}(u/h) + (\\tau - 1/2 ) u$ , where $ \\ell ^{{\\rm G}}(u) := (2/\\pi )^{1/2} e^{-u^2/2 } + u \\lbrace 1- 2\\Phi (-u) \\rbrace $ .","(Logistic kernel $K(u) = e^{-u}/(1+ e^{-u})^2 $ ): $\\ell _h (u) = (h/2) \\ell ^{{\\rm L}}(u/h) + (\\tau -1/2) u$ , where $ \\ell ^{{\\rm L}}(u) := u + 2\\log (1+e^{-u})$ .","(Epanechnikov kernel $K(u) = (3 / 4) (1 - u^2) \\mathbb {1}(|u|\\le 1)$ ): $\\ell _{h}(u) = (h/2) \\ell ^{{\\rm E}}(u/h) +(\\tau - 1/2)u$ , where $ \\ell ^{{\\rm E}}(u) := (3u^2/4 - u^4/8+ 3/8) \\mathbb {1}(|u|\\le 1) + |u|\\mathbb {1}(|u|>1)$ .","(Triangular kernel $K(u) = (1 - |u|) \\mathbb {1}(|u|\\le 1)$ ): $ \\ell _{h}(u) = (h/2) \\ell ^{{\\rm T}}(u/h) + ( \\tau - 1/2) u$ , where $ \\ell ^{{\\rm T}}(u ) := ( u^2 - |u|^3/3 + 1/3 ) \\mathbb {1}(|u|\\le 1) + |u|\\mathbb {1}(|u| >1) $ ."],["The Barzilai-Borwein stepsize rule","In this paper, we propose to solve conquer by means of the gradient descent with a Barzilai-Borwein update step [5], which we refer to as the GD-BB algorithm.","The BB method, which is motivated by quasi-Newton methods, has been proven to be very successful in solving nonlinear optimization problems.","Recall the Newton-Raphson method (REF ), which can be computationally expensive and needs modifications if the Hessian is ill-conditioned especially when $h$ is too small.","For this reason, many quasi-Newton methods seek a simple approximation of the inverse Hessian matrix, say $(^t)^{-1}$ , which satisfies the secant equation $^t \\delta ^t = ^t$ , where for $t=1, 2, \\ldots $ , $\\delta ^t = \\hat{\\beta }^t - \\hat{\\beta }^{t-1} ~~\\mbox{ and }~~ ^t = \\nabla \\hat{Q}_h (\\hat{\\beta }^t) - \\nabla \\hat{Q}_h (\\hat{\\beta }^{t-1}).","$ To relieve the computational bottleneck of inverting a large matrix, the BB method chooses $\\eta $ so that $\\eta \\nabla \\hat{Q}_h (\\hat{\\beta }^t) = (\\eta ^{-1} _p)^{-1}\\nabla \\hat{Q}_h (\\hat{\\beta }^t)$ “approximates\" $(^t)^{-1} \\nabla \\hat{Q}_h (\\hat{\\beta }^t)$ .","Since $^t$ satisfies $^t \\delta ^t = ^t$ , it is more practical to choose $\\eta $ such that $ (1/\\eta ) \\delta ^t \\approx ^t$ or $ \\delta ^t \\approx \\eta ^t$ .","Via least squares approximations, one may use $\\eta _{1,t}^{-1} = _{ \\alpha } \\Vert \\alpha \\delta ^t - ^t \\Vert _2^2$ or $\\eta _{2,t} = _{ \\eta } \\Vert \\delta ^t - \\eta ^t \\Vert _2^2$ .","The BB stepsizes are then defined as $\\eta _{1,t} = \\frac{\\langle \\delta ^t , \\delta ^t\\rangle }{\\langle \\delta ^t, ^t \\rangle } ~~\\mbox{ and }~~ \\eta _{2,t} = \\frac{\\langle \\delta ^t , ^t \\rangle }{ \\langle ^t, ^t \\rangle } .","$ Consequently, the BB iteration takes the form $\\hat{\\beta }^{t+1} = \\hat{\\beta }^t - \\eta _{\\ell ,t} \\nabla \\hat{Q}_h (\\hat{\\beta }^t) , \\quad \\ell = 1 {\\rm ~or~} 2 .","$ Note that the BB step starts at iteration 1, while at iteration 0, we compute $\\hat{\\beta }^1$ using the standard gradient descent with an initial estimate $\\hat{\\beta }^0$ .","The procedure is summarized in Algorithm REF .","Based on extensive numerical studies, we find that at a fixed $\\tau $ , the number of iterations is insensitive to varying $(n,p)$ combinations.","Moreover, as $h$ increases, the number of iterations declines because the loss function is “more convex\" for larger $h$ .","In Algorithm REF , the quantity $\\delta >0$ is called the gradient tolerance, ensuring that the obtained estimate, say $\\hat{\\beta }$ , satisfies $\\Vert \\nabla \\hat{Q}_h(\\hat{\\beta }) \\Vert _2 \\le \\delta $ .","Provided that $\\delta \\lesssim \\sqrt{p/n}$ , the statistical theory developed in Section prevails.","In our R package conquer, we set $\\delta = 10^{-4}$ as the default value; this value can also be specified by the user.","As $\\tau $ approaches 0 or 1, the Hessian matrix becomes ill-conditioned.","As a result, the stepsizes computed in GD-BB may sometimes vibrate drastically, causing instability of the algorithm.","Therefore, in practice, we set a lower bound for the stepsizes by taking $\\eta _t = \\min \\lbrace \\eta _{1,t}, \\eta _{2,t} , 100\\rbrace $ , for $t=1, 2,\\ldots $ .","Another cause of an ill-conditioned Hessian arises when we have covariates with very different scales.","In this case, the stepsize should be different for each covariate, and a constant stepsize will be either too small or too large for one or more covariates, which correspond to slow convergence.","To address this issue, we scale the covariate inputs to have zero mean and unit variance before applying gradient descent.","[!t] Gradient descent with Barzilai-Borwin stepsize (GD-BB) for solving conquer.","Input: data vectors $\\lbrace (y_i, _i)\\rbrace _{i=1}^n$ , $\\tau \\in (0,1)$ , bandwidth $h\\in (0,1)$ , initialization $\\hat{\\beta }^{(0)}$ , and gradient tolerance $\\delta $ .","[1] Compute $\\hat{\\beta }^1 \\leftarrow \\hat{\\beta }^0 - \\nabla \\hat{Q}_h( \\hat{\\beta }^0)$ $t=1,2 \\ldots $ $\\delta ^t \\leftarrow \\hat{\\beta }^t - \\hat{\\beta }^{t-1}$ , $^t \\leftarrow \\nabla \\hat{Q}_h (\\hat{\\beta }^{t})-\\nabla \\hat{Q}_h (\\hat{\\beta }^{t-1})$ $\\eta _{1,t} \\leftarrow \\langle \\delta ^t, \\delta ^t \\rangle / \\langle \\delta ^t, ^t \\rangle $ , $\\eta _{2,t} \\leftarrow \\langle \\delta ^t, ^t\\rangle / \\langle ^t, ^t \\rangle $ $\\eta _t \\leftarrow \\min \\lbrace \\eta _{1,t},\\eta _{2,t}, 100\\rbrace $ if $\\eta _{1,t} >0$ and $\\eta _t \\leftarrow 1$ otherwise $ \\hat{\\beta }^{t+1} \\leftarrow \\hat{\\beta }^t - \\eta _t \\nabla \\hat{Q}_h (\\hat{\\beta }^{t})$ when $\\Vert \\nabla \\hat{Q}_h (\\hat{\\beta }^{t}) \\Vert _2\\le \\delta $"],["Warm start via Huber regression","A good initialization helps reduce the number of iterations for GD, and hence facilitates fast convergence.","Recall from Remark that with a uniform kernel, the smoothed check function is proximal to a Huber loss [28].","Motivated by this subtle proximity, we propose using the Huber $M$ -estimator as an initial estimate, and then proceed by iteratively applying gradient descent with BB update step.","Let $H_\\gamma (u) = (u^2/2) \\mathbb {1}(|u|\\le \\gamma ) + \\gamma (|u| - \\gamma /2) \\mathbb {1}(|u|> \\gamma )$ be the Huber loss parametrized by $ \\gamma >0$ .","The Huber $M$ -estimator is then defined as $\\widetilde{\\beta }_\\gamma \\in _{\\beta \\in ^p} \\hat{}_\\gamma (\\beta ), ~~\\mbox{where}~~ \\hat{}_\\gamma (\\beta ) = \\frac{1}{n} \\sum _{i=1}^nH_\\gamma (y_i - \\langle _i, \\beta \\rangle ).","$ Note that $H_1(u) = \\ell ^{{\\rm U}}(u)-1/2$ for $\\ell ^{{\\rm U}}(\\cdot )$ defined in Remark , we have $ \\hat{Q}_{1/2, \\gamma }(\\beta ) = \\hat{}_ \\gamma (\\beta )$ .","The quantity $ \\gamma $ is a shape parameter that controls the amount of robustness.","The main reason for choosing a fixed (neither diminishing nor diverging) tuning parameter $ \\gamma $ in [29] is to guarantee robustness towards arbitrary contamination in a neighborhood of the model.","This is at the core of the robust statistics idiosyncrasy.","In particular, [29] proposed $ \\gamma =1.35 \\sigma $ to gain as much robustness as possible while retaining 95% asymptotic efficiency for normally distributed data, where $\\sigma >0$ is the standard deviation of the random noise.","We estimate $\\sigma $ using the median absolute deviation on the residuals at each iteration, i.e., ${\\rm MAD}(\\lbrace r_i^t \\rbrace _{i=1}^n)= \\text{median}(|r_i^t - r^t_{\\textrm {median}}|)$ , where $r^t_{\\textrm {median}}$ is the median of $r_1^t,\\ldots ,r_n^t$ .","Noting that the Huber loss is continuously differentiable, convex, and locally strongly convex, we use the GD-BB method described in the previous section to solve the optimization problem (REF ).","Starting at iteration 0 with $\\widetilde{\\beta }^0 = \\textbf {0}$ , at iteration $t=0, 1,2,\\ldots $ , we compute $\\widetilde{\\beta }^{t+1} = \\widetilde{\\beta }^t - \\eta _t \\nabla \\hat{}_\\gamma (\\widetilde{\\beta }^t) = \\widetilde{\\beta }^t + \\frac{\\eta _t}{n} \\sum _{i=1}^n\\psi _\\gamma (y_i-\\langle _i, \\widetilde{\\beta }^t \\rangle ) _i$ with $\\eta _t>0$ automatically obtained by the BB method, where $\\psi _\\gamma (u) = H_\\gamma ^{\\prime }(u) =\\min \\lbrace \\max (- \\gamma , u), \\gamma \\rbrace $ .","We summarize the details in Algorithm REF .","[!t] GD-BB method for solving (REF ).","Input: $\\lbrace (y_i, _i)\\rbrace _{i=1}^n$ and convergence criterion $\\delta $ .","[1] Initialize $\\tilde{\\beta }^{(0)} = \\textbf {0}$ Compute $ \\gamma ^0 = 1.35 \\cdot {\\rm MAD}(\\lbrace r_i^0 \\rbrace _{i=1}^n)$ , where $r^0_i \\leftarrow y_i - \\langle _i, \\widetilde{\\beta }^{0} \\rangle $ , $i=1,\\ldots , n$ , where $\\mathrm {MAD}(\\cdot )$ is the median absolute deviation $\\tilde{\\beta }^1 \\leftarrow \\tilde{\\beta }^0 - \\nabla \\hat{L}_{\\gamma ^0}( \\tilde{\\beta }^0)$ $t=1,2 \\ldots $ $\\gamma ^t = 1.35 \\cdot {\\rm MAD}(\\lbrace r^t_i\\rbrace _{i=1}^n)$ , where $r^t_i \\leftarrow y_i - \\langle _i, \\widetilde{\\beta }^{t} \\rangle $ , $i=1,\\ldots , n$ $\\delta ^t \\leftarrow \\tilde{\\beta }^t - \\tilde{\\beta }^{t-1}$ , $^t \\leftarrow \\nabla \\hat{}_{\\gamma ^t}(\\tilde{\\beta }^{t})-\\nabla \\hat{}_{\\gamma ^t}(\\tilde{\\beta }^{t-1})$ $\\eta _{1,t} \\leftarrow \\langle \\delta ^t, \\delta ^t \\rangle / \\langle \\delta ^t, ^t \\rangle $ , $\\eta _{2,t} \\leftarrow \\langle \\delta ^t, ^t\\rangle / \\langle ^t, ^t \\rangle $ .","$\\eta _t \\leftarrow \\min \\lbrace \\eta _{1,t},\\eta _{2,t}, 100\\rbrace $ if $\\eta _{1,t} >0$ and $\\eta _t \\leftarrow 1$ otherwise $ \\widetilde{\\beta }^{t+1} \\leftarrow \\tilde{\\beta }^t - \\eta _t \\nabla \\hat{}_{\\gamma ^t}(\\tilde{\\beta }^{t})$ when $\\Vert \\nabla \\hat{}_{\\gamma ^t}(\\tilde{\\beta }^{t}) \\Vert _2\\le \\delta $ The Huber loss $H_\\gamma (\\cdot )$ approximates the quantile loss function with $\\tau =1/2$ as $\\gamma \\rightarrow 0$ .","Therefore, an alternative method for QR computing is to solve the Huber regression via gradient descent with a shrinking gamma.","To evaluate its performance, we implement the above idea by setting $\\gamma ^{t} = c \\cdot \\gamma ^{t-1}$ for some $c\\in (0,1)$ at the $t$ -th iteration.","We found that the aforementioned idea is not stable numerically across several simulated data sets, unless one is very careful in controlling the minimal magnitude of $\\gamma $ .","In addition, the solution obtained has a higher estimation error than that of standard QR and conquer."],["Numerical Studies","In this section, we assess the finite-sample performance of conquer via extensive numerical studies.","We compare conquer to standard QR [35] and Horowitz's smoothed QR [27].","Both the convolution-type and Horowitz's smoothed methods involve a smoothing parameter $h$ .","In view of Theorem REF , we take $h = \\lbrace (p+\\log n) /n\\rbrace ^{2/5}$ in all of the numerical experiments.","As we will see from Figure REF , the proposed method is insensitive to the choice of $h$ .","Therefore, we leave the fine tuning of $h$ as an optional rather than imperative choice.","In all the numerical experiments, the convergence criterion in Algorithms REF and REF is taken as $\\delta =10^{-4}$ .","We first generate the covariates $_i = (x_{i , 1}, \\dots , x_{i , p})^ from a multivariate uniform distribution on the cube $ 31/2 [-1, 1]p$ with covariance matrix $ =(0.7|j - k|)1j,kp$ using the \\texttt {R} package \\texttt {MultiRNG} \\cite {F1999}.The random noise $ i$ are generated from two different distributions: (i) Gaussian distribution, $ N(0, 4)$; and (ii) $ t$ distribution with degrees of freedom 2, $ t2$.Let $ i- = (xi, 1, ..., xi, p - 1), $\\beta ^* = (1, \\dots , 1)^{p - 1}$ , and $\\beta ^*_0 = 1$ .","Given $\\tau \\in (0, 1)$ , we then generate the response $y_i$ from the following homogeneous and heterogeneous models, all of which satisfy the Assumption REF : Homogeneous model: $y_i = \\beta ^*_0 + \\langle _i^- , \\beta ^* \\rangle + \\lbrace \\varepsilon _i - F^{-1}_{\\varepsilon _i}(\\tau ) \\rbrace , ~~~~i = 1, \\dots , n; $ Linear heterogeneous model: $y_i = \\beta ^*_0 + \\langle _i^- , \\beta ^* \\rangle + (0.5 x_{i, p} + 1) \\lbrace \\varepsilon _i - F^{-1}_{\\varepsilon _i}(\\tau ) \\rbrace , ~~~~i = 1, \\dots , n; $ Quadratic heterogeneous model: $y_i = \\beta ^*_0 + \\langle _i^- , \\beta ^* \\rangle + 0.5\\lbrace 1+ ( x_{i, p}-1)^2 \\rbrace \\lbrace \\varepsilon _i - F^{-1}_{\\varepsilon _i}(\\tau )\\rbrace , ~~~~i = 1, \\dots , n. $ To evaluate the performance of different methods, we calculate the estimation error under the $\\ell _2$ -norm, i.e., $\\Vert \\hat{\\beta }-\\beta ^*\\Vert _2$ , and record the elapsed time.","The details are in Section REF .","In Section REF , we examine the finite-sample performance of the multiplier bootstrap method for constructing confidence intervals in terms of coverage probability, width of the interval, and computing time."],["Estimation","For all the numerical studies in this section, we consider a wide range of the sample size $n$ , with the size-dimension ratio fixed at $n / p = 20$ .","That is, we allow the dimension $p$ to increase as a function of $n$ .","We implement conquer with four different kernel functions as described in Remark : (i) Gaussian; (ii) uniform; (iii) Epanechnikov; and (iv) triangular.","The classical quantile regression is implemented via a modified version of the Barrodale and Roberts algorithm [37], [38] by setting method= “br\" in the R package quantreg, which is recommended for problems up to several thousands of observations in [34].","For very large problems, the Frisch-Newton approach after preprocessing “pfn\" is preferred.","Since the same size taken to be at most 5000 throughout this section, the two methods, “br” and “pfn\", have nearly identical runtime behaviors.","In some applications where there are a lot of discrete covariates, it is advantageous to use method “sfn\", a sparse version of Frisch-Newton algorithm that exploits sparse algebra to compute iterates [39].","Moreover, we implement Horowitz's smoothed quantile regression using the Gaussian kernel, and solve the resulting non-convex optimization via gradient descent with random initialization and stepsize calibrated by backtracking line search (Section 9.3 of [8]).","The results, averaged over 100 replications, are reported.","Figure REF depicts estimation error of the different methods under the simulation settings described in Section with $\\tau = 0.9$ .","We see that conquer has a lower estimation error than the classical QR across all scenarios, indicating that smoothing can improve estimation accuracy under the finite-sample setting.","Moreover, compared to Horowitz's smoothing, conquer has a lower estimation error in most settings.","Estimation error under various quantile levels $\\tau \\in \\lbrace 0.1, 0.3, 0.5, 0.7\\rbrace $ under the $t_2$ random noise are also examined.","The results are reported in Figure REF in Appendix , from which we observe evident advantages of conquer, especially at low and high quantile levels.","Figure: Model () with t 2 t_2 error.To assess the computational efficiency, we compute the elapsed time for fitting the different methods.","Figures REF and REF in Appendix report the runtime for the different methods with growing sample size and dimension under the same settings as in Figures REF and REF , respectively.","We observe that conquer is computationally efficient and stable across all scenarios, and the runtime is insensitive to the choice of kernel functions.","In contrast, the runtime for classical quantile regression grows rapidly as the sample size and dimension increase.","Figures REF and REF in Appendix show that the runtime of Horowitz's smoothing method increases significantly at extreme quantile levels $\\tau \\in \\lbrace 0.1,0.9\\rbrace $ , possibly due to the combination of its non-convex nature and flatter gradient.","In summary, we conclude that conquer significantly improves computational efficiency while retaining high statistical accuracy for fitting large-scale linear quantile regression models.","Next, we conduct a sensitivity analysis for conquer regarding the smoothing bandwidth $h$ .","We set $(n, p) = (2000, 100)$ and consider the simulation settings (REF )–(REF ) with $\\mathcal {N}(0, 4)$ and $t_2$ noise.","We perform conquer with $h \\in \\lbrace 0.2, 0.22,\\dots , 0.5\\rbrace $ , including the default value $h_{{\\rm de}} = \\lbrace (p+\\log n) /n\\rbrace ^{2/5} = 0.3107$ , and compare the estimation error with that of QR in Figure REF .","We see that the estimation error of conquer is uniformly lower than that of QR over a range of $h$ , suggesting that conquer is insensitive to the choice of bandwidth $h$ .","Figure: Model () with t 2 t_2 error.Figure: Model () with t 2 t_2 error."],["Inference","In this section, we assess the performance of the multiplier bootstrap procedure for constructing confidence interval for each of the regression coefficients obtained from conquer.","We implement conquer using the Gaussian kernel, and construct three types of confidence intervals: (i) the percentile mb-per; (ii) pivotal mb-piv; (iii) and regular mb-norm confidence intervals, as described in Section REF .","We also refer to the proposed multiplier bootstrap procedure as mb-conquer for simplicity.","We compare the proposed method to several widely used inference methods for QR.","In particular, we consider confidence intervals by inverting a rank score test, rank ([23]; Section 3.5 of [33]); a bootstrap variant based on pivotal estimating functions, pwy [45]; and wild bootstrap with Rademacher weights, wild [19].","The three methods rank, pwy, and wild are implemented using the R package quantreg.","Note that rank is a non-resampling based procedure that relies on prior knowledge on the random noise, i.e., a user needs to specify whether the random noise are independent and identically distributed.","In our simulation studies, we provide rank an unfair advantage by specifying the correct random noise structure.","We set $(n, p) = (800, 20)$ , $\\tau \\in \\lbrace 0.5, 0.9\\rbrace $ , and significance level $\\alpha = 0.05$ .","All of the resampling methods are implemented with $B = 1000$ bootstrap samples.","To measure the reliability, accuracy, and computational efficiency of different methods for constructing confidence intervals, we calculate the average empirical coverage probability, average width of confidence interval, and the average runtime.","The average is taken over all regression coefficients without the intercept.","Results based on 200 replications are reported in Figure REF , and Figures REF –REF in Appendix .","Figure: Runtime under model ().In Figure REF , and Figures REF –REF in Appendix , we use the rank-inversion method, rank, as a benchmark since we implement rank using information about the true underlying random noise, which is practically infeasible.","In the case of $\\tau =0.9$ , pwy is most conservative as it produces the widest confidence intervals with slightly inflated coverage probability, and wild gives the narrowest confidence intervals but at the cost of coverage probability.","The proposed methods mb-per, mb-piv, and mb-norm achieve a good balance between reliability (high coverage probability) and accuracy (narrow CI width), and moreover, has the lowest runtime.","To further highlight the computational gain of the proposed method, we now perform numerical studies with larger $n$ and $p$ .","In this case, the rank inversion method rank is computationally infeasible.","For example, when $(n,p)= (5000,250)$ , rank inversion takes approximately 80 minutes while conquer with multiplier bootstrap takes 41 seconds for constructing confidence intervals.","We therefore omit rank from the following comparison.","We consider the quadratic heterogeneous model (REF ) with $(n, p) = (4000, 100)$ and $t_2$ noise.","The results are reported in Figure REF .","We see that pwy and wild take approximately 300 seconds while mb-conquer takes approximately 15 seconds.","In summary, mb-conquer leads to a huge computational gain without sacrificing statistical efficiency.","Figure: Runtime"],["Comparison between normal approximation and bootstrap calibration","Finally, we complement the above studies with a comparison between the normal approximation and bootstrap calibration methods for confidence estimation.","We consider model (REF ) with $(n,p) = (2000,10)$ .","We use the same $\\beta ^*\\in ^{p-1}$ and $\\beta ^*_0$ as before, and generate random covariates and noise from a multivariate uniform distribution and $t_{1.5}$ -distribution, respectively.","For each of the $p-1$ regression coefficients, we apply the proposed bootstrap percentile method and the normal-based method [20] to construct pointwise confidence intervals at quantile indices close to 0 and 1, that is, $\\tau \\in \\lbrace 0.05, 0.1, 0.9, 0.95\\rbrace $ .","Boxplots of the empirical coverage and CI width for the two methods are reported in Figure REF .","Considering that extreme quantile regressions are notoriously hard to estimate, the bootstrap method can produce much more reliable (high coverage) and accurate (narrow width) confidence intervals than the normal-based counterpart.","Therefore, for applications in which extreme quantiles are of particular interest, such as the problem of forecasting the conditional value-at-risk of a financial institution [13], the bootstrap provides a more reliable approach for quantifying the uncertainty of the estimates.","Figure: Boxplots of CI width"],["Discussion","In this paper, we provide a comprehensive study on the statistical properties of conquer—namely, convolution-type smoothed quantile regression, under the array nonasymptotic setting in which $p$ is allowed to increase as a function of $n$ while $p/n$ being small.","An efficient gradient-based algorithm is proposed to compute conquer, which proves to be scalable to large dimensions and even larger sample sizes.","Recently, there has been a growing interest in studying the asymptotic behavior of regression estimates under the regime in which $p$ grows proportionally with the sample size $n$ [15], [40].","The current studies are mainly focused on $M$ -estimation with strongly convex loss functions.","New theoretical tools are needed to establish results for conquer under such a regime, and we leave it for future work.","In the high-dimensional setting in which $p \\gg n$ , various authors have studied the regularized quantile regression under the sparsity assumption that most of the regression coefficients are zero [6], [53].","Using $\\ell _1$ penalties, the computation of regularized QR is based on either a linear programming reformulation or alternating direction method of multiplier algorithms [22].","Since the conquer loss is convex and twice differentiable, we expect that gradient-based algorithms, such as coordinate gradient descent or proximal gradient descent, will enjoy superior computational efficiency for solving regularized conquer without sacrificing statistical accuracy.","One future work is to establish the statistical theory and computational complexity of the regularized conquer estimator."],["One-step Conquer with Higher-order Kernels","As noted in Section REF , the smoothing bias is of order $h^2$ when a non-negative kernel is used.","The ensuing empirical loss $\\beta \\mapsto (1/n) \\sum _{i=1}^n(\\rho _\\tau * K_h)(y_i - \\langle _i , \\beta \\rangle )$ is not only twice-differentiable and convex, but also (provably) strongly convex in a local vicinity of $\\beta ^*$ with high probability.","Kernel smoothing is ubiquitous in nonparametric statistics.","The order of a kernel, $\\nu $ , is defined as the order of the first non-zero moment.","The order of a symmetric kernel is always even.","A kernel is called high-order if $\\nu >2$ , which inevitably has negative parts and thus is no longer a probability density.","Thus far we have focused on conquer with second-order kernels, and the resulting estimator achieves an $\\ell _2$ -error of the order $\\sqrt{p/n} + h^2$ .","Let $G(\\cdot )$ be a higher-order symmetric kernel with order $\\nu \\ge 4$ , and $b >0$ be a bandwidth.","Again, via convolution smoothing, we may consider a bias-reduced estimator that minimizes the empirical loss $\\beta \\mapsto \\hat{Q}_b^G(\\beta ):= (1/n) \\sum _{i=1}^n(\\rho _\\tau * G_b )(y_i - \\langle _i , \\beta \\rangle )$ .","This, however, leads to a non-convex optimization.","Without further assumptions, finding a global minimum is computationally intractable: finding an $\\epsilon $ -suboptimal point for a $k$ -times continuously differentiable loss function requires at least $\\Omega \\lbrace (1/\\epsilon )^{p/k}\\rbrace $ evaluations of the function and its first $k$ derivatives, ignoring problem-dependent constants; see Section 1.6 in [42].","Instead, various gradient-based methods have been developed for computing stationary points, which are points $\\beta $ with sufficiently small gradient $\\Vert \\nabla \\hat{Q}_b^G(\\beta ) \\Vert _2 \\le \\epsilon $ , where $\\epsilon \\ge 0$ is optimization error.","However, the equation $\\nabla \\hat{Q}_b^G(\\beta ) = \\textbf {0}$ does not necessarily have a unique solution, whose statistical guarantees remain unknown.","Motivated by the classical one-step estimator [7], we further propose a one-step conquer estimator using high-order kernels, which bypasses solving a large-scale non-convex optimization.","To begin with, we choose two symmetric kernel functions, $K: \\rightarrow [0,\\infty )$ with order two and $G(\\cdot )$ with order $\\nu \\ge 4$ , and let $h, b >0$ be two bandwidths.","First, compute an initial conquer estimator $\\beta \\in _{\\beta \\in ^p} \\hat{Q}_{h}^K(\\beta )$ , where $\\hat{Q}_{h}^K(\\beta ) = (1/n) \\sum _{i=1}^n(\\rho _\\tau * K_h) (y_i - \\langle _i , \\beta \\rangle )$ .","Denote by $\\bar{r}_i = y_i - \\langle _i, \\beta \\rangle $ for $i=1,\\ldots , n$ the fitted residuals.","Next, with slight abuse of notation, we define the one-step conquer estimator $\\hat{\\beta }$ as a solution to the equation $ \\nabla ^2 \\hat{Q}_b^G (\\beta ) ( \\hat{\\beta }- \\beta ) = - \\nabla \\hat{Q}_b^G(\\beta )$ , or equivalently, $\\Biggl \\lbrace \\frac{1}{n } \\sum _{i=1}^nG_b ( \\bar{r}_i ) _i _i^\\rbrace ( \\hat{\\beta }- \\beta ) = \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace G ( \\bar{r}_i / b ) + \\tau - 1 \\bigr \\rbrace _i .","$ where $\\hat{Q}_b^G(\\beta ) = (1/n) \\sum _{i=1}^n(\\rho _\\tau * G_b ) (y_i - \\langle _i , \\beta \\rangle )$ .","Provided that $\\nabla ^2 \\hat{Q}_b^G (\\beta )$ is positive definite, the one-step conquer estimate $\\hat{\\beta }$ essentially performs a Newton-type step based on $\\beta $ : $\\hat{\\beta }= \\beta - \\bigl \\lbrace \\nabla ^2 \\hat{Q}_b^G (\\beta ) \\bigr \\rbrace ^{-1} \\nabla \\hat{Q}_b^G(\\beta ).","$ In this case, $\\hat{\\beta }$ can be computed by the conjugate gradient method [62].","Theoretical properties of the one-step estimator $\\hat{\\beta }$ defined in (REF ), including the Bahadur representation and asymptotic normality with explicit Berry-Esseen bound, will be provided in on-line supplementary materials.","For practical implementation, we consider higher-order Gaussian-based kernels.","For $r=1,2,\\ldots ,$ the $(2r)$ -th order Gaussian kernels are $G_{2r}( u) = \\frac{(-1)^r \\phi ^{(2r-1)}(u)}{2^{r-1} (r-1)!","u} = \\sum _{\\ell =0}^{r-1} \\frac{(-1)^\\ell }{2^\\ell \\ell !}","\\phi ^{(2\\ell )} (u) ;$ see Section 2 of [70].","Integrating $G_{2r}(\\cdot )$ yields ${G}_{2r}(v) = \\int _{-\\infty }^v G_{2r}( u) \\, {\\rm d}u = \\sum _{\\ell =0}^{r-1} \\frac{(-1)^\\ell }{2^\\ell \\ell !}","\\phi ^{(2\\ell -1)}(v).$ In fact, both $G_{2r}$ and ${G}_{2r}$ have simpler forms $G_{2r}(u) = p_r(u) \\phi (u)$ and ${G}_{2r}(u) = \\Phi (u)+ P_r(u) \\phi (u)$ , where $p_r(\\cdot )$ and $P_r(\\cdot )$ are polynomials in $u$ .","For example, $p_1(u) = 1$ , $P_1(u)=0$ , $p_2(u) =(-u^2+3)/2$ , $P_2(u)=u/2$ , $p_3(u)= (u^4 - 10 u^2 + 15)/8$ , and $P_3(u)=(-u^3 +7u)/8$ .","We refer to [65] for more details when $r$ is large."],["Proof of Proposition ","For every $r>0$ , define the ellipse $\\Theta (r) = \\lbrace \\in ^p: \\Vert \\Vert _{} \\le r \\rbrace $ and the local vicinity $\\Theta ^* = \\lbrace \\beta \\in ^p: \\beta - \\beta ^* \\in \\Theta ( \\kappa _2^{1/2} h ) \\rbrace $ .","Let $\\eta = \\sup \\lbrace u \\in [0,1] : u (\\beta ^*_h -\\beta ^*) \\in \\Theta (\\kappa _2^{1/2} h) \\rbrace $ and $\\widetilde{\\beta }^* = \\beta ^* + \\eta (\\beta ^*_h -\\beta ^*)$ .","By definition, $\\eta = 1$ if $\\beta ^*_h \\in \\Theta ^*$ and $\\eta <1$ if $\\beta ^*_h \\notin \\Theta ^*$ .","In the latter case, $\\widetilde{\\beta }^* \\in \\partial \\Theta ^*$ .","By the convexity of $\\beta \\mapsto Q_h(\\beta )$ and Lemma C.1 in the supplementary material of [68], $0 & \\le \\langle \\nabla Q_h( \\widetilde{\\beta }^* ) - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle \\nonumber \\\\& \\le \\eta \\cdot \\langle \\nabla Q_h(\\beta ^*_h ) - \\nabla Q_h(\\beta ^*) , \\beta ^*_h - \\beta ^* \\rangle = \\langle - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle .","$ It follows from the mean value theorem for vector-valued functions that $\\nabla Q_h( \\widetilde{\\beta }^* ) - \\nabla Q_h(\\beta ^*) = \\int _0^1 \\nabla ^2 Q_h( (1-t) \\beta ^* + t \\widetilde{\\beta }^* ) \\, {\\rm d} t \\, \\big ( \\widetilde{\\beta }^* - \\beta ^* \\bigr ) , $ where $\\nabla ^2 Q_h( \\beta ) = \\bigl \\lbrace K_h( y - \\langle , \\beta \\rangle ) ^\\rbrace $ for $\\beta \\in ^p$ .","With $\\delta = \\beta - \\beta ^*$ , note that $\\bigl \\lbrace K_h( y - \\langle , \\beta \\rangle ) | \\bigr \\rbrace = \\frac{1}{h} \\int _{-\\infty }^\\infty K\\bigg ( \\frac{u- \\langle , \\delta \\rangle }{h} \\bigg ) f_{\\varepsilon |} (u) \\, {\\rm d} u = \\int _{-\\infty }^\\infty K (v ) f_{\\varepsilon |} ( \\langle , \\delta \\rangle + h v ) \\, {\\rm d} v .","\\nonumber $ By the Lipschitz continuity of $f_{ \\varepsilon |}(\\cdot )$ , $\\bigl \\lbrace K_h( y - \\langle , \\beta \\rangle ) | \\bigr \\rbrace = f_{ \\varepsilon |}( 0 ) + R_h(\\delta ) $ with $R_h(\\delta )$ satisfying $|R_h(\\delta )| \\le l_0 \\bigl ( |\\langle , \\delta \\rangle | + \\kappa _1 h \\bigr )$ .","Substituting (REF ) into (REF ) and (REF ) yields $& \\langle \\nabla Q_h( \\widetilde{\\beta }^* ) - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle \\nonumber \\\\& \\ge \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 - \\frac{l_0}{2} | \\langle , \\widetilde{\\beta }^* - \\beta ^* \\rangle |^3 - l_0 \\kappa _1 h \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 \\nonumber \\\\& \\ge \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 - \\frac{l_0}{2} \\mu _3 \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert ^3_{} - l_0 \\kappa _1 h \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 .","$ On the other hand, under model (REF ) we have $& \\langle - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle \\le \\Vert ^{-1/2} \\nabla Q_h(\\beta ^*) \\Vert _2 \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} , \\nonumber $ where $\\nabla Q_h(\\beta ^*) = \\lbrace K(-\\varepsilon /h) - \\tau \\rbrace $ .","By integration by parts and a Taylor series expansion, $\\bigl \\lbrace K(-\\varepsilon /h) | \\bigr \\rbrace & = \\int _{-\\infty }^\\infty K(-t/h) \\, {\\rm d} F_{ \\varepsilon |}( t) \\nonumber \\\\& = -\\frac{1}{h}\\int _{-\\infty }^\\infty K(-t/h) F_{ \\varepsilon |}( t) \\, {\\rm d}t = \\int _{-\\infty }^\\infty K(u) F_{\\varepsilon |}( -h u) \\, {\\rm d}u \\nonumber \\\\& = \\tau + \\int _{-\\infty }^\\infty K(u) \\int _{0}^{-hu} \\bigl \\lbrace f_{ \\varepsilon |} ( t) - f_{\\varepsilon |} (0) \\bigr \\rbrace \\, {\\rm d} t \\, {\\rm d}u ,\\nonumber $ from which it follows that $| \\lbrace K(-\\varepsilon /h) | \\rbrace - \\tau | \\le \\frac{l_0}{2} \\kappa _2 h^2$ .","Consequently, $& \\Vert ^{-1/2} \\nabla Q_h(\\beta ^*) \\Vert _2 = \\sup _{\\in \\mathbb {S}^{p-1}} \\bigl \\lbrace K(-\\varepsilon /h) -\\tau \\bigr \\rbrace \\langle ^{-1/2} , \\rangle \\le \\frac{l_0}{2} \\kappa _2 h^2 .","$ Putting together the pieces, we conclude that $\\langle - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle \\le \\frac{l_0}{2} \\kappa _2 h^2 \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} .","$ Recall that $f_{\\varepsilon | }(0) \\ge {f} >0$ almost surely.","Combining (REF ) and (REF ) with (REF ) yields ${f} \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 & \\le {f} ^{1/2} \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} \\nonumber \\\\& \\le \\bigg (\\frac{\\mu _3 +1 }{2} \\kappa _2 + \\kappa _1 \\kappa _2^{1/2} \\bigg ) l_0 h^2 \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} .","\\nonumber $ Canceling $\\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}$ on both sides we obtain $\\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} \\le \\underbrace{ \\bigg (\\frac{ \\mu _3 +1 }{2} \\kappa _2^{1/2} + \\kappa _1 \\bigg ) }_{= c_K} \\frac{l_0 h }{{f} } \\kappa _2^{1/2} h = \\frac{c_K l_0 h}{{f}} \\kappa _2^{1/2} h .","\\nonumber $ Provided that $h < {f}/(c_K l_0)$ , $\\widetilde{\\beta }^*$ falls in the interior of $\\Theta ^*$ , i.e., $ \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}< \\kappa _2^{1/2} h$ , enforcing $\\eta =1$ (otherwise, by construction $\\widetilde{\\beta }^*$ must lie on the boundary which leads to contradiction) and hence $\\widetilde{\\beta }^* = \\beta ^*_h$ .","In addition, by (REF ), $Q_h(\\cdot )$ is strictly convex in a neighborhood of $\\beta ^*_h$ so that $\\beta ^*_h$ is the unique minimizer and satisfies the stated bound (REF ).","Next, to investigate the leading term in the bias, define $\\Delta = ^{-1/2} \\bigl \\lbrace \\nabla Q_h(\\beta ^*_h) - \\nabla Q_h(\\beta ^*) \\bigr \\rbrace - ^{1/2} ( \\beta ^*_h -\\beta ^* ).","\\nonumber $ Again, by the mean value theorem for vector-valued functions, $\\Delta & = \\Bigg \\lbrace ^{-1/2} \\int _0^1 \\nabla ^2 Q_h((1-t) \\beta ^* + t \\beta ^*_h) \\, {\\rm d}t \\, ^{-1/2} - _{p} \\Bigg \\rbrace ^{1/2} ( \\beta ^*_h - \\beta ^* ) .","$ Write $= ^{-1/2}$ , we have $& \\Bigg \\Vert ^{-1/2} \\int _0^1 \\nabla ^2 Q_h((1-t) \\beta ^* + t \\beta ^*_h) \\, {\\rm d}t \\, ^{-1/2} - _p \\Bigg \\Vert _2 \\nonumber \\\\&=\\Bigg \\Vert \\int _0^1 \\int _{-\\infty }^\\infty K(u) \\bigl \\lbrace f_{\\varepsilon |}( t \\langle , \\beta ^*_h -\\beta ^* \\rangle - hu ) - f_{ \\varepsilon |}( 0) \\bigr \\rbrace \\, {\\rm d} u \\, {\\rm d} t \\, ^\\Vert _2 \\nonumber \\\\& \\le l_0 \\sup _{\\in \\mathbb {S}^{p-1}} \\int _0^1 \\int _{-\\infty }^\\infty K(u) \\bigl ( | t \\langle , \\beta ^*_h - \\beta ^* \\rangle | + h |u | \\bigr ) \\, {\\rm d}u \\, {\\rm d} t \\, \\langle , \\rangle ^2 \\nonumber \\\\& \\le \\frac{l_0}{2 } \\sup _{\\in \\mathbb {S}^{p-1}} \\bigl ( |\\langle , \\beta ^*_h - \\beta ^* \\rangle | \\langle , \\rangle ^2 \\bigr ) + \\frac{l_0}{{f}} \\kappa _1 h \\nonumber \\\\& \\le \\frac{l_0}{2{f}} \\mu _3 \\Vert \\beta _h^* - \\beta ^* \\Vert _{} + \\frac{l_0}{{f}} \\kappa _1 h. \\nonumber $ This bound, together with (REF ), implies $\\Vert \\Delta \\Vert _2 \\le \\frac{l_0 }{{f}} \\bigl ( 0.5 \\mu _3 \\Vert \\beta _h^* - \\beta ^* \\Vert _{} + \\kappa _1 h \\bigr ) \\Vert \\beta _h^* - \\beta ^* \\Vert _{} .","$ Moreover, applying a second-order Taylor series expansion to $f_{\\varepsilon |}$ yields $& \\bigl \\lbrace K(-\\varepsilon /h) | \\bigr \\rbrace - \\tau \\nonumber \\\\& = \\int _{-\\infty }^\\infty K(u) \\int _{0}^{-h u} \\bigl \\lbrace f_{ \\varepsilon |} ( t) - f_{\\varepsilon |} ( 0) \\bigr \\rbrace \\, {\\rm d} t \\, {\\rm d}u \\nonumber \\\\& = \\frac{1 }{2} \\kappa _2 h^2 \\cdot f_{ \\varepsilon |}^{\\prime }(0 ) + \\int _{-\\infty }^\\infty \\int _{0}^{-hu} \\int _0^t K(u) \\bigl \\lbrace f_{ \\varepsilon |}^{\\prime }(v) - f_{ \\varepsilon |}^{\\prime }( 0 ) \\bigr \\rbrace \\, {\\rm d} v \\, {\\rm d} t \\, {\\rm d}u .","\\nonumber $ Recalling that $\\nabla Q_h(\\beta ^*) = \\lbrace K(-\\varepsilon /h) - \\tau \\rbrace $ , we get $\\bigg \\Vert ^{-1} \\nabla Q_h(\\beta ^*) - \\frac{1}{2} \\kappa _2 h^2 \\cdot ^{-1} \\bigl \\lbrace f_{\\varepsilon |}^{\\prime }( 0)\\bigr \\rbrace \\bigg \\Vert _{} \\le \\frac{l_1}{6 {f}^{1/2}} \\kappa _3 h^3 .","$ Finally, combining (REF ) and (REF ) proves (REF ).","$\\Box $"],["Proof of Theorem ","Recall from (REF ) that $\\hat{\\beta }_h $ is the smoothed quantile regression estimator obtained by minimizing $\\hat{Q}_{ h} (\\cdot )$ .","For any given $\\tau \\in (0,1)$ and $h>0$ , it follows from the optimality of $\\hat{\\beta }_h $ that $\\nabla \\hat{Q}_h ( \\hat{\\beta }_h ) = \\mathbf {0} $ and by convexity, $ \\langle \\nabla \\hat{Q}_h ( \\hat{\\beta }_h ) - \\nabla \\hat{Q}_h ( \\beta ^*) , \\hat{\\beta }_h - \\beta ^* \\rangle \\ge 0$ .","Recall from the proof of Proposition REF that $\\Theta (t) = \\lbrace \\in ^p : \\Vert \\Vert _{} \\le t \\rbrace $ for $t\\ge 0$ .","For some $r>0$ to be determined, let $\\eta = \\sup \\lbrace u \\in [0,1] : u (\\hat{\\beta }_h -\\beta ^*) \\in \\Theta (r) \\rbrace $ and $\\widetilde{\\beta }= \\beta ^* + \\eta (\\hat{\\beta }_h -\\beta ^*)$ .","Thus, by definition, $\\eta =1$ if $\\hat{\\beta }_h \\in \\beta ^* + \\Theta (r)$ , and $\\eta <1$ if $\\hat{\\beta }_h \\notin \\beta ^* + \\Theta (r)$ .","In the latter case, $\\widetilde{\\beta }\\in \\beta ^* + \\partial \\Theta (r)$ , where $\\partial \\Theta (r)$ is the boundary of $\\Theta (r)$ .","The symmetrized Bregman divergence associated with $\\hat{Q}_h(\\cdot )$ for points $\\beta _1, \\beta _2$ is given by $ D (\\beta _1, \\beta _2) = \\langle \\nabla \\hat{Q}_h (\\beta _1)- \\nabla \\hat{Q} _h (\\beta _2), \\beta _1 - \\beta _2 \\rangle .$ By Lemma C.1 in [68], the three points $\\hat{\\beta }_h$ , $\\widetilde{\\beta }$ , and $\\beta ^*$ satisfy $D( \\widetilde{\\beta }, \\beta ^* ) \\le \\eta D ( \\hat{\\beta }, \\beta ^*)$ .","Together with the properties that $ \\widetilde{\\beta }- \\beta ^* = \\eta (\\hat{\\beta }_h - \\beta ^* ) \\in \\Theta (r)$ and $\\nabla \\hat{Q}_h (\\hat{\\beta }_h) = \\textbf {0}$ , we obtain $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{}^2 \\cdot \\frac{ D( \\widetilde{\\beta }, \\beta ^* ) }{\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{}^2} \\le - \\eta \\langle \\nabla \\hat{Q}_h (\\beta ^* ), \\hat{\\beta }_h - \\beta ^* \\rangle \\le \\Vert ^{-1/2} \\nabla \\hat{Q}_h (\\beta ^* ) \\Vert _2 \\cdot \\Vert \\widetilde{\\beta }- \\beta ^* \\Vert _{} .","\\nonumber $ Canceling $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{}$ on both sides, we obtain $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{} \\le \\frac{ \\Vert ^{-1/2} \\nabla \\hat{Q}_h (\\beta ^* ) \\Vert _2 }{\\inf _{\\beta \\in \\beta ^* + \\Theta (r)} \\lbrace D( \\beta , \\beta ^* ) / \\Vert \\beta -\\beta ^* \\Vert _{}^2 \\rbrace } .","$ The following two lemmas provide, respectively, upper and lower bounds on $\\Vert ^{-1/2} \\nabla \\hat{Q}_h (\\beta ^* ) \\Vert _2$ and $\\inf _{\\beta \\in \\beta ^* + \\Theta (r)} \\lbrace D( \\beta , \\beta ^* ) / \\Vert \\beta -\\beta ^* \\Vert _{}^2\\rbrace $ .","Assume that Conditions REF –REF hold.","For any $t\\ge 0$ , $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2 \\le 1.46 \\upsilon _0 \\Biggl \\lbrace C_\\tau ^{1/2 }\\sqrt{ \\frac{ 4p +2 t }{n}} + 2\\max (1-\\tau , \\tau ) \\frac{ 2p+t }{n} \\Biggr \\rbrace + \\frac{1}{2} l_0\\kappa _2 h^2 $ with probability at least $1-e^{-t}$ , where $C_\\tau = \\tau (1-\\tau ) + (1+\\tau ) l_0 \\kappa _2 h^2$ .","Under Condition REF , there exist constants $\\bar{f}_{ h} \\ge {f}_{ h}$ such that ${f}_{ h} \\le \\inf _{|u|\\le h/2} f_{\\varepsilon | } (u) \\le \\sup _{|u|\\le h/2} f_{\\varepsilon | } (u) \\le \\bar{f}_{ h} $ almost surely.","In fact, by the Lipschitz continuity, we can take $\\bar{f}_{ h} = \\bar{f} + l_0 h/2$ and ${f}_{ h} = {f} - l_0 h/2$ .","Throughout the following, we assume (REF ) holds.","For every $\\delta \\in (0,1]$ , we define $\\eta _\\delta \\ge 0$ as $\\eta _\\delta = \\inf \\bigl \\lbrace \\eta >0 : \\bigl \\lbrace \\langle \\delta , \\rangle ^2 \\mathbb {1}\\big ( |\\langle \\delta , \\rangle | > \\eta _\\delta \\big ) \\bigr \\rbrace \\le \\delta ~\\mbox{ for all } \\delta \\in \\mathbb {S}^{p-1} \\bigr \\rbrace , $ where $= ^{-1/2} $ is the standardized predictor that satisfies $(^ = _p$ .","It can be shown that $\\eta _\\delta $ depends only on $\\delta $ and $\\upsilon _0$ in Condition REF , and the map $\\delta \\mapsto \\eta _\\delta $ is non-increasing with $\\eta _\\delta \\downarrow 0$ as $\\delta \\uparrow 1$ .","For any $t>0$ , $0 < h < 2 {f} / l_0$ and $0< r \\le h/(4 \\eta _{1/4})$ with $\\eta _{1/4}$ defined in (REF ), we have $\\inf _{\\beta \\in \\beta ^* + \\Theta (r) } \\frac{ D( \\beta , \\beta ^* ) }{ \\kappa _l \\Vert \\beta - \\beta ^* \\Vert _{}^2 } \\ge \\frac{3}{4} {f}_{h } - \\bar{f}_{ h}^{1/2} \\left( \\frac{5}{4}\\sqrt{\\frac{h p}{r^2 n}} + \\sqrt{\\frac{h t}{8r^2 n}} \\, \\right) - \\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .","In view of (REF ), (REF ), and (REF ), we take $r = h/(4 \\eta _{1/4})$ so that as long as $(p+t)/n \\lesssim h\\lesssim 1$ , $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{} < \\frac{3 \\upsilon _0 }{ \\kappa _l {f} } \\Biggl \\lbrace C_\\tau ^{1/2} \\sqrt{ \\frac{ 4p + 2t }{n}} + 2\\max (1-\\tau , \\tau ) \\frac{2 p+t }{n} \\Biggr \\rbrace + \\frac{l_0 \\kappa _2 }{ \\kappa _l {f} } h^2 $ with probability at least $1-2e^{-t}$ .","With this choice of $r$ , we see that under the constraint $\\sqrt{(p+t)/n} \\lesssim h \\lesssim 1$ , $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{} 0$ , define the function $\\varphi _R(u)=u^2 \\mathbb {1}(|u|\\le R/2) + \\lbrace u(u) -R \\rbrace ^2 \\mathbb {1}( R/2 < |u| \\le R)$ , which is $R$ -Lipschitz continuous and satisfies $u^2 \\mathbb {1}(|u| \\le R / 2) \\le \\varphi _R(u) \\le u^2 \\mathbb {1}(|u| \\le R ) .","$ Moreover, note that $\\varphi _{cR}(cu) = c^2 \\varphi _{R}(u)$ for any $c > 0$ and $\\varphi _0(u) = 0$ .","For $\\beta \\in \\beta ^* + \\Theta (r)$ , consider a change of variable $\\delta = ^{1/2} (\\beta -\\beta ^*)/\\Vert \\beta -\\beta ^* \\Vert _{} $ so that $\\delta \\in \\mathbb {S}^{p-1}$ .","Together, () and (REF ) imply $\\frac{ D(\\beta , \\beta ^* ) }{\\Vert \\beta -\\beta ^* \\Vert _{}^2} \\ge \\kappa _l \\cdot \\underbrace{ \\frac{1}{n h } \\sum _{i=1}^n\\omega _i \\cdot \\varphi _{ h/(2r) } (\\langle _i, \\delta \\rangle ) }_{=: D_0( \\delta )} , $ where $\\omega _i := \\mathbb {1}( | \\varepsilon _i | \\le h/2 )$ with $\\varepsilon _i = y_i - \\langle _i, \\beta ^* \\rangle $ , and $_i = ^{-1/2} _i$ .","Next, we bound the expectation $\\lbrace D_0(\\delta ) \\rbrace $ and the random fluctuation $ D_0(\\delta ) - \\lbrace D_0(\\delta )\\rbrace $ , separately, starting with the former.","By (REF ), ${f}_{ h} h \\le ( \\omega _i | _i ) = \\int _{-h/2}^{h/2} f_{\\varepsilon _i | _i} (u) \\, {\\rm d}u \\le \\bar{f}_{ h} h. $ Moreover, define $\\xi _{\\delta } = \\langle , \\delta \\rangle $ such that $(\\xi _{\\delta }^2) =1$ .","By (REF ) and (REF ), $& \\bigl \\lbrace \\omega _i \\cdot \\varphi _{ h/(2r)} (\\langle _i , \\delta \\rangle ) \\bigr \\rbrace \\ge {f}_{ h} h \\cdot \\varphi _{ h/(2r) } (\\langle _i , \\delta \\rangle )\\ge {f}_{ h} h \\cdot \\bigl [ 1 - \\bigl \\lbrace \\xi _{\\delta }^2 \\mathbb {1}_{ | \\xi _{\\delta } | > h/(4r) } \\bigr \\rbrace \\bigr ] , \\nonumber $ from which it follows that $\\inf _{ \\delta \\in \\mathbb {S}^{p-1} } \\lbrace D_0(\\delta ) \\rbrace \\ge {f}_h \\cdot \\left( 1 - \\sup _{\\in \\mathbb {S}^{p-1}} \\bigl \\lbrace \\langle , \\rangle ^2 \\mathbb {1}_{ |\\langle , \\rangle | >h/(4r)} \\bigr \\rbrace \\right) .","$ By the definition of $\\eta _\\delta $ in (REF ), we see that as long as $00$ , $\\Delta & \\le (\\Delta ) + \\lbrace ( \\Delta )\\rbrace ^{1/2} \\sqrt{\\frac{h t}{4 r^2 n}} + \\sqrt{ \\bar{f}_h h \\frac{ t }{ 8 r^2 n }} + \\frac{h t}{48 r^2 n} \\le \\frac{5}{4} (\\Delta ) + \\sqrt{ \\bar{f}_h h \\frac{ t }{ 8 r^2 n }} + \\frac{h t}{3 r^2 n} $ with probability at least $1-e^{-t}$ , where the second step follows from the inequality that $ab\\le a^2/4 + b^2$ for all $a,b\\in $ .","It then remains to bound the expectation $(\\Delta ) $ .","Define $(\\delta ; _i) &= \\frac{1}{h } \\omega _i\\varphi _{ h/(2r) } (\\langle _i, \\delta \\rangle )= \\frac{1}{h} \\varphi _{ \\omega _i h/(2r) } (\\langle \\omega _i _i , \\delta \\rangle ) , \\ \\ \\delta \\in \\mathbb {S}^{p-1} ,\\nonumber $ where $_i = (_i , \\varepsilon _i )$ and $\\omega _i = \\mathbb {1} ( | \\varepsilon _i | \\le h/2 ) \\in \\lbrace 0, 1\\rbrace $ .","By Rademacher symmetrization, $(\\Delta ) \\le 2 \\Bigg \\lbrace \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^ne_i (\\delta ; _i) \\Bigg \\rbrace , \\nonumber $ where $e_1,\\ldots , e_n$ are independent Rademacher random variables.","Since $\\varphi _R(\\cdot )$ is $R$ -Lipschitz, $(\\delta ; _i)$ is a $(1/2r)$ -Lipschitz function in $\\langle \\omega _i _i , \\delta \\rangle $ , i.e., for any sample $_i = (_i, \\varepsilon _i)$ and parameters $\\delta , \\delta ^{\\prime }\\in \\mathbb {S}^{p-1}$ , $\\bigl | (\\delta ; _i) - (\\delta ^{\\prime }; _i) \\bigr | \\le \\frac{1}{2r} \\bigl | \\langle \\omega _i _i , \\delta \\rangle - \\langle \\omega _i _i , \\delta ^{\\prime } \\rangle \\bigr | .$ Moreover, observe that $(\\delta ; _i) = 0$ for any $\\delta $ such that $ \\langle \\omega _i _i , \\delta \\rangle = 0$ .","With the above preparations, we are ready to use Talagrand's contraction principle to bound $( \\Delta )$ .","Define the subset $T\\subseteq ^n$ as $T = \\bigl \\lbrace = (t_1, \\ldots , t_n)^ t_i = \\langle \\omega _i_i , \\delta \\rangle , i=1,\\ldots , n, \\, \\delta \\in \\mathbb {S}^{p-1} \\bigr \\rbrace ,$ and contractions $\\phi _i:\\rightarrow $ as $\\phi _i(t) = (2r/h) \\cdot \\varphi _{ h \\omega _i/(2r)} (t)$ .","By (REF ), $|\\phi (t)-\\phi (s)|\\le |t-s|$ for all $t, s \\in $ .","Applying Talagrand's contraction principle (see, e.g., Theorem 4.12 and (4.20) in [64]), we have $(\\Delta ) & \\le 2 \\Bigg \\lbrace \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^ne_i (\\delta ; _i) \\Bigg \\rbrace = \\frac{1}{r} \\Bigg \\lbrace \\sup _{\\in T} \\frac{1}{n} \\sum _{i=1}^ne_i \\phi _i(t_i) \\Bigg \\rbrace \\le \\frac{1}{r} \\Bigg ( \\sup _{\\in T} \\frac{1}{n} \\sum _{i=1}^ne_i t_i \\Bigg ) \\nonumber \\\\&= \\frac{1}{r}\\Bigg \\lbrace \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^ne_i \\langle \\omega _i _i , \\delta \\rangle \\Bigg \\rbrace \\le \\frac{1}{r} \\bigg \\Vert \\frac{1}{n} \\sum _{i=1}^ne_i \\omega _i _i \\bigg \\Vert _2 \\le \\frac{1}{r} \\sqrt{ \\bar{f}_h h \\frac{p}{n} }.","\\nonumber $ This, together with (REF ) and (REF ), yields $\\Delta \\le \\bar{f}_h^{1/2} \\Biggl ( \\frac{5}{4}\\sqrt{\\frac{h p}{r^2 n}} + \\sqrt{\\frac{h t}{8r^2 n}} \\Biggr ) + \\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .","Finally, combining (), (REF ), (REF ), and (REF ) proves (REF ).","$\\Box $"],["Proof of Theorem ","We keep the notation used in the proof of Theorem REF , and for any $t\\ge 0$ , let $r=r(n,p,t) \\asymp \\sqrt{(p+t)/n} + h^2 >0$ be such that $\\lbrace \\hat{\\beta }_h \\in \\beta ^* + \\Theta (r) \\rbrace \\ge 1-2e^{-t}$ , provided $\\sqrt{(p+t)/n} \\lesssim h \\lesssim 1$ .","Define the vector-valued random process $\\Delta (\\beta ) = ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}_h(\\beta ) - \\nabla \\hat{Q}_h(\\beta ^*) - ( \\beta - \\beta ^* ) \\bigr \\rbrace , $ where $= \\lbrace f_{\\varepsilon | } (0) ^$ .","Since $\\hat{\\beta }_h$ falls in a local neighborhood of $\\beta ^*$ with high probability, it suffices to bound the local fluctuation $\\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta (\\beta ) \\Vert _2$ .","By the triangle inequality, $\\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta (\\beta ) \\Vert _2 \\le \\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta (\\beta ) \\Vert _2+ \\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta (\\beta ) - \\Delta (\\beta ) \\Vert _2 :=I_1+I_2.$ We now provide upper bounds for $I_1$ and $I_2$ , respectively.","Upper bound for $I_1$ : By the mean value theorem for vector-valued functions, $\\Delta (\\beta ) & = ^{-1/2} \\biggl \\langle \\int _0^1 \\nabla ^2 Q_h\\bigl ( (1-t) \\beta ^* + t \\beta \\bigr ) {\\rm d}t , \\beta - \\beta ^* \\biggr \\rangle - ^{-1/2} (\\beta - \\beta ^* ) \\nonumber \\\\& =\\biggl \\langle ^{-1/2} \\int _0^1 \\nabla ^2 Q_h\\bigl ( (1-t) \\beta ^* + t \\beta \\bigr ) {\\rm d}t \\, ^{-1/2} - _0 , ^{1/2} ( \\beta - \\beta ^* ) \\biggr \\rangle , \\nonumber $ where $_0 := ^{-1/2} ^{-1/2} = \\lbrace f_{\\varepsilon | }(0) ^$ .","By law of iterative expectation and by a change of variable, $^{-1/2} \\nabla ^2 Q_h(\\beta ) ^{-1/2} & = \\bigl \\lbrace K_h( \\langle , \\beta \\rangle - y ) ^\\rbrace = \\Biggl \\lbrace \\int _{-\\infty }^\\infty K(u) f_{\\varepsilon | }(\\langle , \\beta - \\beta ^* \\rangle - h u ) \\, {\\rm d} u \\cdot ^\\rbrace .","\\nonumber $ For notational convenience, let $= ^{-1/2} ( \\beta -\\beta ^*)$ with $\\beta \\in \\beta ^* + \\Theta (r)$ , so that $\\Vert \\Vert _2 \\le r$ and $\\nabla ^2 Q_h \\bigl ( (1-t) \\beta ^* + t \\beta \\bigr ) = \\Biggl \\lbrace \\int _{-\\infty }^\\infty K(u) f_{\\varepsilon | }( t \\langle ,\\rangle - h u ) \\, {\\rm d} u \\cdot ^\\rbrace .$ By the Lipschitz continuity of $f_{\\varepsilon |}(\\cdot )$ , i.e.","$|f_{\\varepsilon |}(u)- f_{\\varepsilon |}(0)| \\le l_0 |u |$ for all $u \\in $ almost surely for $$ , we have $& \\bigl \\Vert ^{-1/2} \\nabla ^2 Q_h\\bigl ( (1-t) \\beta ^* + t \\beta \\bigr ) ^{-1/2} - _0 \\bigr \\Vert _2 \\nonumber \\\\& = \\Biggl \\Vert \\int K(u) \\bigl \\lbrace f_{\\varepsilon |} ( t \\langle , \\rangle - h u) - f_{\\varepsilon |} ( 0 ) \\bigr \\rbrace \\, {\\rm d} u \\, \\cdot ^\\Vert _2 \\nonumber \\\\& \\le l_0 t \\sup _{\\Vert \\Vert _2 =1} \\bigl ( \\langle , \\rangle ^2 |\\langle , \\rangle | \\bigr )+ l_0 \\kappa _1 h \\sup _{\\Vert \\Vert _2 =1} \\langle , \\rangle ^2 \\nonumber \\\\& \\le l_0 t \\,\\Biggl ( \\sup _{\\Vert \\Vert _2 =1} | \\langle , \\rangle |^3 \\Biggr )^{2/3} \\bigl ( |\\langle , \\rangle |^3 \\bigr )^{1/3} + l_0 \\kappa _1 h \\le l_0 \\big ( \\mu _3 r t + \\kappa _1 h \\big ),$ where the third inequality holds by the Cauchy-Schwarz inequality.","Consequently, $\\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta ( \\beta ) \\Vert _2 \\le l_0 \\bigl (\\mu _3 r /2 + \\kappa _1 h \\bigr ) \\cdot r .","$ Upper bound for $I_2$ : Next, we provide an upper bound for $\\Delta (\\beta ) - \\Delta (\\beta )$ .","Define the centered gradient process $G(\\beta ) = ^{-1/2} \\lbrace \\nabla \\hat{Q}_h(\\beta ) - \\nabla Q_h(\\beta ) \\rbrace $ , so that $\\Delta (\\beta ) - \\Delta (\\beta ) = G(\\beta ) - G(\\beta ^*)$ .","Again, by a change of variable $= ^{ 1/2} ( \\beta - \\beta ^*)$ , we have $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta (\\beta ) - \\Delta (\\beta ) \\Vert _2& \\le \\sup _{ \\beta \\in \\beta ^* + \\Theta (r) } \\Vert G(\\beta ) - G(\\beta ^*) \\Vert _2 \\nonumber \\\\& = \\sup _{ \\Vert \\Vert _2 \\le r } \\Vert \\underbrace{ G(\\beta ^* + ^{-1/2} ) - G(\\beta ^*) }_{ =: \\Delta _0() } \\Vert _2 .\\nonumber $ We will employ Theorem A.3 in [67] to bound the supremum $\\sup _{ \\Vert \\Vert _2 \\le r } \\Vert \\Delta _0() \\Vert _2$ , where $\\Delta _0(\\cdot )$ defined above satisfies $\\Delta _0(\\textbf {0}) = \\textbf {0}$ and $\\lbrace \\Delta _0()\\rbrace =\\textbf {0}$ .","Taking the gradient with respect to $$ yields $\\nabla \\Delta _0 () = \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace K_{i,} _i _i^ \\bigl ( K_{i,} _i _i^) \\bigr \\rbrace , \\nonumber $ where $K_{i,} : = K_h(\\langle _i, \\rangle - \\varepsilon _i )$ satisfies $0\\le K_{i,} \\le \\kappa _u h^{-1}$ .","For any $, \\in \\mathbb {S}^{p-1}$ and $\\lambda \\in $ , using the elementary inequality $|e^u- 1 - u| \\le u^2 e^{|u|}/2$ , we obtain $& \\exp \\bigl \\lbrace \\lambda n^{1/2} \\langle , \\nabla \\Delta _0 () \\rangle \\bigr \\rbrace \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl \\lbrace 1 + \\frac{\\lambda ^2 }{2n} e^{ \\frac{ \\bar{f} |\\lambda | }{ \\sqrt{n}} | \\langle _i, \\rangle \\langle _i, \\rangle |}\\bigl \\lbrace K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle - ( K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle ) \\bigr \\rbrace ^2 e^{\\frac{ \\kappa _u|\\lambda | }{ h \\sqrt{n}} | \\langle _i, \\rangle \\langle _i, \\rangle | } \\Biggr \\rbrace \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl \\lbrace 1 + \\frac{\\lambda ^2 }{2n} e^{ \\frac{ \\bar{f} |\\lambda | }{ \\sqrt{n}} }\\bigl \\lbrace K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle - ( K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle ) \\bigr \\rbrace ^2 e^{\\frac{ \\kappa _u|\\lambda | }{ h \\sqrt{n}} | \\langle _i, \\rangle \\langle _i, \\rangle | } \\Biggr \\rbrace , $ where we use the bound $| \\langle _i, \\rangle \\langle _i, \\rangle | \\le (\\langle _i, \\rangle ^2 )^{1/2} (\\langle _i, \\rangle ^2 )^{1/2} = 1$ in the second inequality.","Moreover, the first and second conditional moments of $K_{i,}$ can be rewritten as follows: $ \\,\\bigl ( K_{i,} | _i \\bigr ) = \\frac{1}{h} \\int _{-\\infty }^\\infty K \\biggl ( \\frac{\\langle _i, \\rangle - t}{h} \\biggr ) f_{\\varepsilon _i(\\tau ) | _i } (t) \\, {\\rm d} t = \\int _{-\\infty }^\\infty K(u) f_{\\varepsilon _i (\\tau ) | _i } (\\langle _i, \\rangle - hu ) \\, {\\rm d} u; \\nonumber \\\\\\bigl ( K^2_{i,} | _i \\bigr ) = \\frac{1}{h^2} \\int _{-\\infty }^\\infty K^2 \\biggl ( \\frac{\\langle _i, \\rangle - t}{h} \\biggr ) f_{\\varepsilon _i(\\tau ) | _i } (t) \\, {\\rm d} t = \\frac{1}{h} \\int _{-\\infty }^\\infty K^2(u) f_{\\varepsilon _i(\\tau ) | _i } (\\langle _i, \\rangle - hu ) \\, {\\rm d} u, \\nonumber $ from which it follows that $|( K_{i,} | _i ) | \\le \\bar{f} $ and $( K_{i,}^2 | _i )\\le \\kappa _u \\bar{f} h^{-1}$ almost surely.","By the Cauchy-Schwarz inequality and the inequality $ab\\le a^2/2+b^2/2$ , $a,b\\in $ , we have $& ( \\langle _i, \\rangle \\langle _i, \\rangle )^2 e^{t | \\langle _i, \\rangle \\langle _i, \\rangle | } \\nonumber \\\\& \\le ( \\langle _i, \\rangle \\langle _i, \\rangle )^2 e^{ \\frac{ t }{ 2 } \\langle _i, \\rangle ^2 +\\frac{ t }{ 2 } \\langle _i, \\rangle ^2 } \\nonumber \\\\& \\le \\Bigl ( \\langle _i, \\rangle ^4 e^{ t \\langle _i, \\rangle ^2 } \\Bigr )^{1/2} \\Bigl ( \\langle _i, \\rangle ^4 e^{ t \\langle _i, \\rangle ^2 } \\Bigr )^{1/2} , ~\\mbox{ valid for any } t>0 .\\nonumber $ Given a unit vector $$ , let $\\chi = \\langle , \\rangle ^2/(2\\upsilon _1)^2$ so that under Condition REF , $( \\chi \\ge u) \\le 2 e^{-2u}$ for any $u\\ge 0$ .","It follows that $(e^\\chi ) = 1 + \\int _0^\\infty e^u (\\chi \\ge u) {\\rm d}u \\le 1 + 2 \\int _0^\\infty e^{-u} {\\rm d}u = 3$ , and $(\\chi ^2 e^{\\chi } ) = \\int _0^\\infty (u^2 + 2u) e^{u} (\\chi \\ge u) {\\rm d} u \\le 2 \\int _0^\\infty (u^2 + 2u) e^{-u} {\\rm d} u = 8.","\\nonumber $ Taking the supremum over $\\in \\mathbb {S}^{p-1}$ , we have $\\sup _{\\in \\mathbb {S}^{p-1} } e^{\\langle , \\rangle ^2 / (2\\upsilon _1)^2 } \\le 3~\\mbox{ and } ~ \\sup _{\\in \\mathbb {S}^{p-1} } \\langle , \\rangle ^4 e^{\\langle , \\rangle ^2 / (2\\upsilon _1)^2 } \\le 8(2\\upsilon _1)^4 .$ Substituting the above bounds into (REF ) yields that, for any $|\\lambda | \\le \\min \\lbrace n^{1/2} h/( 4 \\kappa _u \\upsilon _1^2 ), n^{1/2} / \\bar{f} \\rbrace $ , $& \\exp \\bigl \\lbrace \\lambda n^{1/2} \\langle , \\nabla \\Delta _0 () \\rangle \\bigr \\rbrace \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl [ 1 + \\frac{ e\\lambda ^2 }{2n}\\bigl \\lbrace K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle - (K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle ) \\bigr \\rbrace ^2 e^{ | \\langle _i, \\rangle \\langle _i, \\rangle | /(4\\upsilon _1^2)} \\Biggr ] \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl [ 1 + \\frac{ e\\lambda ^2 }{ n}\\bigl ( K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle \\bigr )^2 e^{ | \\langle _i, \\rangle \\langle _i, \\rangle | / (4\\upsilon _1^2) } \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \\frac{e \\lambda ^2}{n} \\bigl \\lbrace \\bigl ( K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle \\bigr )\\bigr \\rbrace ^2 e^{ | \\langle _i, \\rangle \\langle _i, \\rangle |/ (4\\upsilon _1^2) } \\Biggr ] \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl ( 1 + C_0^2 \\frac{\\lambda ^2}{2 n h} \\Biggr ) \\le \\exp \\bigl \\lbrace C_0^2 \\lambda ^2 /(2h) \\bigr \\rbrace ,\\nonumber $ where $C_0>0$ depends only on $(\\upsilon _1, \\kappa _u, \\bar{f} )$ .","We have thus verified condition (A.4) in [67] with ${\\rm g}= \\min \\lbrace h /( 4\\kappa _u \\upsilon _1^2 ), 1 / \\bar{f} \\rbrace (n/2)^{1/2}$ and $\\nu _0= C_0h^{-1/2}$ .","Applying Theorem A.3 therein, we obtain that with probability at least $1-e^{-t}$ , $\\sup _{ \\Vert \\Vert _2 \\le r } \\Vert \\Delta _0() \\Vert _2 \\le 6 C_0 r \\sqrt{\\frac{4 p + 2 t}{n h }} \\nonumber $ as long as $h \\ge 8 \\kappa _u \\upsilon _1^2 \\sqrt{(2p+t)/n}$ and $n\\ge 4 \\bar{f}^2 (2p+t)$ .","Together with (REF ) and (REF ), this implies that with probability at least $1-e^{-t}$ , $\\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta ( \\beta ) \\Vert _2 \\le 6 C_0 r \\sqrt{\\frac{4 p + 2 t}{n h }} + l_0 \\bigl ( \\mu _3 r /2 + \\kappa _1 h \\bigr ) r .$ Recall from the beginning of the proof that $\\hat{\\beta }_h \\in \\beta ^* + \\Theta (r)$ with probability at least $1-2e^{-t}$ with $r=r(n,p,t) \\asymp \\sqrt{(p+t)/n} +h^2$ .","Combined with (REF ), we conclude that with probability at least $1-3e^{-t}$ , $\\Vert \\Delta (\\hat{\\beta }_h ) \\Vert _2 \\lesssim (p+t)/(h^{1/2}n) + h \\sqrt{(p+t)/n} +h^3$ , as claimed.","$\\Box $"],["Proof of Theorem ","Let $\\in ^p$ be an arbitrary vector defining a linear functional of interest.","Given $h=h_n>0$ , define $S_n = n^{-1/2} \\sum _{i=1}^n\\gamma _i \\xi _i$ and its centered version $S_n^0= S_n - (S_n)$ , where $\\xi _i = \\tau - K(-\\varepsilon _i /h)$ and $\\gamma _i = \\langle ^{-1} , _i \\rangle $ .","Moreover, write $\\delta _n = (p+\\log n)/n$ .","By the Lipschitz continuity of $f_{\\varepsilon | }$ around 0 and the fundamental theorem of calculus, it can be shown that $| (\\xi _i | _i) |\\le 0.5 l_0 \\kappa _2 h^2$ , from which it follows by the law of iterated expectation that $|(\\gamma _i \\xi _i) | \\le 0.5 l_0 \\kappa _2 \\Vert ^{-1} \\Vert _{} \\cdot h^2$ .","Applying (REF ) and (REF ) with $t=\\log n$ and the triangle inequality, we obtain that under the constraint $\\delta _n^{1/2} \\lesssim h \\lesssim 1$ , $& \\bigl | n^{1/2} \\langle , \\hat{\\beta }_h - \\beta ^* \\rangle - S_n^0 \\bigr | \\nonumber \\\\&= n^{1/2} \\biggl | \\bigg \\langle ^{ 1/2} ^{-1} , ^{-1/2} (\\hat{\\beta }_h - \\beta ^* ) - ^{-1/2} \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace \\tau - K(-\\varepsilon _i/h) \\bigr \\rbrace _i \\bigg \\rangle \\biggr | + | (S_n) | \\nonumber \\\\& \\le c_1 \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr ) $ with probability at least $1 - 3 n^{-1}$ for some constant $c_1>0$ .","For the centered partial sum $S_n^0 = S_n - (S_n) = n^{-1/2} \\sum _{i=1}^n(1-) \\gamma _i \\xi _i$ , we have $\\textnormal {var}(S_n^0) = \\textnormal {var}(S_n) = (\\gamma \\xi )^2 - \\lbrace (\\gamma \\xi )\\rbrace ^2$ , where $\\gamma = \\langle ^{-1} , \\rangle $ and $\\xi = \\tau - K(-\\varepsilon /h)$ .","By the Berry-Esseen inequality (see, e.g., [69]), $\\sup _{x\\in } \\bigl | \\bigl \\lbrace S_n^0 \\le \\textnormal {var}(S_n)^{1/2} x \\bigr \\rbrace - \\Phi (x) \\bigr | \\le \\frac{\\lbrace |\\gamma \\xi - (\\gamma \\xi ) |^3\\rbrace }{2 [( \\gamma \\xi )^2 - \\lbrace ( \\gamma \\xi )\\rbrace ^2 ]^{3/2} \\sqrt{n} } .","$ We have shown that $|(\\gamma \\xi ) | \\lesssim \\Vert ^{-1} \\Vert _{} \\cdot h^2$ .","Following a similar argument in the proof of Lemma REF , we have $(\\xi ^2| )\\lesssim \\tau (1-\\tau )+h^2$ .","For $h$ sufficiently small, $\\textnormal {var}(S_n)= \\lbrace \\tau (1-\\tau ) + O(h)\\rbrace \\Vert ^{-1} \\Vert _{}^2$ and $(|\\gamma \\xi |^3) \\le \\max (\\tau , 1-\\tau ) (\\xi ^2 |\\gamma |^3) \\le \\mu _3 \\lbrace \\tau (1-\\tau ) + O(h^2) \\rbrace \\Vert ^{-1} \\Vert _{}^3$ .","Substituting these bounds into (REF ) yields $\\sup _{x\\in } \\bigl | \\bigl \\lbrace S_n^0 \\le \\textnormal {var}(S_n)^{1/2} x \\bigr \\rbrace - \\Phi (x) \\bigr | \\le c_2 n^{-1/2} $ for some constant $c_2>0$ .","Set $\\sigma _\\tau ^2 = \\tau (1-\\tau )\\, \\Vert ^{-1} \\Vert _{}^2$ .","By an application of Lemma A.7 in the supplement of [66], for sufficiently small $h$ , we have $\\sup _{x\\in } \\bigl | \\Phi ( x/\\textnormal {var}(S_n)^{1/2} ) -\\Phi (x/\\sigma _\\tau ) \\bigr | \\le c_3 h. $ Before proceeding, we note that the constants $c_1$ –$c_3$ appeared above are all independent of $$ .","Let $G\\sim N(0,1)$ .","Putting together the above derivations, for any $x\\in $ and $\\in ^p$ , we obtain $& \\bigl ( n^{1/2} \\langle , \\hat{\\beta }_h -\\beta ^* \\rangle \\le x \\bigr ) \\nonumber \\\\& \\le \\bigl \\lbrace S_n^0 \\le x + c_1 \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr ) \\bigr \\rbrace + 3 n^{-1} \\nonumber \\\\& \\le \\bigl \\lbrace \\textnormal {var}(S_n)^{1/2} G \\le x+ c_1 \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr ) \\bigr \\rbrace +c_2n^{-1/2} + 3 n^{-1} \\nonumber \\\\&\\le \\bigl \\lbrace \\sigma _\\tau G \\le x+ c_1 \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr ) \\bigr \\rbrace + c_2 n^{-1/2} + c_3 h + 3 n^{-1} \\nonumber \\\\& \\le \\bigl ( \\sigma _\\tau G \\le x \\bigr ) + c_1(2\\pi )^{-1/2} \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr )/\\sigma _{\\tau } + c_2 n^{-1/2} + c_3 h^2 + 3 n^{-1} , \\nonumber $ where the first, second, and third inequalities holds by (REF ), (REF ), and (REF ), respectively, and the last inequality holds by the fact that for any $a\\le b$ , $\\Phi (b/\\sigma _\\tau ) - \\Phi (a/\\sigma _\\tau ) \\le (2\\pi )^{-1/2}(b-a)/\\sigma _\\tau $ .","A similar argument leads to a series of reverse inequalities.","Note the above bounds are independent of $x$ and $$ , and therefore hold uniformly over all $x$ and $$ .","Putting together the pieces, we conclude that under the bandwidth requirement $\\delta _n^{1/2} \\lesssim h \\lesssim 1$ , $\\sup _{x\\in , \\, \\in ^p } \\bigl | \\bigl ( n^{1/2} \\langle , \\hat{\\beta }_h -\\beta ^* \\rangle \\le \\sigma _\\tau x \\bigr ) - \\Phi (x ) \\bigr | \\lesssim \\frac{p+ \\log n}{ (n h)^{1/2}} + n^{1/2} h^2 , \\nonumber $ as claimed.","$\\Box $"],["Proof of Theorem ","Keep the notation used in the proof of Theorem REF .","With nonnegative weights $w_i$ , the corresponding weighted loss $\\hat{Q}_h^\\flat $ in (REF ) is convex, and thus the first-order condition $\\nabla \\hat{Q}_h^\\flat (\\hat{\\beta }_h^\\flat ) = \\textbf {0}$ holds.","We use the same localized argument as in the proof of Theorem REF .","For some $r>0$ to be determined, define $\\widetilde{\\beta }_h^\\flat = \\beta ^* + \\eta (\\hat{\\beta }_h^\\flat - \\beta ^*)$ , where $\\eta = \\sup \\lbrace u\\in [0,1] : u (\\hat{\\beta }_h^\\flat - \\beta ^*) \\in \\Theta (r) \\rbrace $ .","Similar to (REF ), we have $\\Vert \\widetilde{\\beta }_h^\\flat - \\beta ^* \\Vert _{} \\le \\frac{\\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^*) \\Vert _2 }{\\inf _{\\beta \\in \\beta ^* + \\Theta (r)} D^\\flat (\\beta , \\beta ^*) / \\Vert \\beta - \\beta ^* \\Vert _{}^2 } , $ where $D^\\flat : ^p\\times ^p \\rightarrow ^+$ denotes the symmetrized Bregman divergence associated with $\\hat{Q}_h^\\flat $ .","Next, we present two lemmas on the upper and lower bounds of the numerator and denominator of (REF ), respectively.","Assume that Conditions REF , REF , and REF hold.","For any $t\\ge 0$ , there exists some event $_1(t)$ with $\\lbrace _1(t) \\rbrace \\ge 1- 2e^{-t}$ such that, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_1(t)$ , $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) \\bigr \\Vert _2 \\le C_1 \\Biggl ( \\sqrt{\\frac{p+t}{n}} + h^2 \\Biggr )$ as long as $n\\gtrsim p+t$ , where $C_1>0$ is a constant depending only on $(\\upsilon _1, \\kappa _2, l_0)$ and $\\tau $ .","Let $r= h/(4\\eta _{1/4})$ with $\\eta _{1/4}$ defined in (REF ).","For any $t\\ge 0$ , there exists some event $_2(t)$ with $\\lbrace _2(t) \\rbrace \\ge 1- 3e^{-t}$ such that, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_2(t)$ , $^* \\Biggl \\lbrace \\inf _{\\beta \\in \\beta ^* + \\Theta (r)}\\frac{D^\\flat (\\beta , \\beta ^* )}{\\kappa _l \\Vert \\beta -\\beta ^* \\Vert _{}^2} \\ge \\frac{1}{2} {f} \\Biggr \\rbrace \\ge 1-e^{-t} .","$ as long as $ (p+\\log n + t )/n \\lesssim h \\lesssim 1$ .","Recall from the proof of Theorem REF that the upper bound on $\\Vert \\hat{\\beta }_h - \\beta ^* \\Vert _{}$ depends crucially on $\\Vert ^{-1/2} \\nabla \\hat{Q}_h(\\beta ^* ) \\Vert _2$ and $\\inf _{\\beta \\in \\beta ^* + \\Theta (r)} \\lbrace D(\\beta , \\beta ^*)/\\Vert \\beta - \\beta ^* \\Vert _{}^2\\rbrace $ .","In view of Lemmas REF and REF , we take $r=h/( 4 \\eta _{1/4})$ and for $t\\ge 0$ , set $(t) = _1(t) \\cup _2(t)$ so that $\\lbrace (t) \\rbrace \\ge 1-5 e^{-t}$ and $\\Vert \\hat{\\beta }_h - \\beta ^* \\Vert _{} \\le C_0 \\Biggl ( \\sqrt{\\frac{p+t}{n}} + h^2 \\Biggr ) ~\\mbox{ on $(t)$ provided that } \\sqrt{\\frac{p+t}{n}} \\lesssim h \\lesssim 1,$ where $C_0>0$ depends only on $(\\upsilon _1, \\kappa _2, \\kappa _l, l_0, {f} )$ and $\\tau $ .","Moreover, by (REF ), with $^*$ -probability at least $1-2e^{-t}$ conditioned on $(t)$ , $\\Vert \\widetilde{\\beta }_h^\\flat - \\beta ^* \\Vert _{} \\le \\frac{2C_1}{\\kappa _l {f} }\\Biggl ( \\sqrt{\\frac{p+t}{n}} + h^2 \\Biggr ) < r \\nonumber $ as long as $\\max \\lbrace (p+\\log n + t )/n, \\sqrt{(p+t)/n } \\rbrace \\lesssim h \\lesssim 1$ .","Consequently, $\\widetilde{\\beta }_h^\\flat $ falls in the interior of $\\beta ^* + \\Theta (r)$ , thereby implying $\\eta =1$ and $\\hat{\\beta }_h^\\flat = \\widetilde{\\beta }_h^\\flat $ .","This completes the proof.","$\\Box $"],["Proof of Lemma ","By the triangle inequality, we have $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) \\bigr \\Vert _2 \\le \\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) - ^{-1/2}\\nabla \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2 + \\bigl \\Vert ^{-1/2}\\nabla \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2:= I_1+I_2.$ The term $I_2$ can be upper bounded by Lemma REF , and thus it suffices to bound $I_1$ .","To this end, we will employ Hoeffding's and Bernstein's inequalities.","Recall from Section REF that $w_i$ is such that $(w_i)=1$ and $\\textnormal {var}(w_i)=1$ .","Thus, $^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) = (1/n) \\sum _{i=1}^nw_i \\xi _i _i$ and $^* \\lbrace ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) \\rbrace = ^{-1/2} \\hat{Q}_h (\\beta ^* ) $ , where $_i = ^{-1/2} _i$ and $\\xi _i = K(-\\varepsilon _i /h) - \\tau $ with $\\varepsilon _i = y_i - \\langle _i, \\beta ^* \\rangle $ .","Using a similar covering argument as in the proof of Lemma REF , for any $\\epsilon \\in ( 0, 1)$ , there exists an $\\epsilon $ -net $_\\epsilon $ of the unit sphere $\\mathbb {S}^{p-1}$ with $|_\\epsilon |\\le ( 1+2/\\epsilon )^p$ such that $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) - ^{-1/2}\\nabla \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2 \\le \\frac{1}{1-\\epsilon } \\max _{\\in _\\epsilon } \\frac{1}{n} \\sum _{i=1}^ne_i \\xi _i \\langle , _i \\rangle , \\nonumber $ where $e_i = w_i-1$ are i.i.d.","Rademacher random variables.","By Hoeffding's inequality, for any $u\\ge 0$ , $^* \\left\\lbrace \\frac{1}{n} \\sum _{i=1}^ne_i \\xi _i \\langle , _i \\rangle \\ge \\Biggl ( \\frac{1}{n} \\sum _{i=1}^n\\xi _i^2 \\langle , _i \\rangle ^2 \\Biggr )^{1/2} \\sqrt{\\frac{2 u }{n}} \\right\\rbrace \\le e^{-u}.$ For the data-dependent quantity $(1/n) \\sum _{i=1}^n\\xi _i^2 \\langle , _i \\rangle ^2 $ , as in the proof of Lemma REF we have $| \\xi _i | \\le B_\\tau := \\max ( \\tau , 1-\\tau )$ , $(\\xi _i^2 | _i ) \\le C_\\tau := \\tau (1-\\tau ) + (1+\\tau ) l_0 \\kappa _2 h^2$ almost surely and note that $\\langle , _i \\rangle ^2 =1$ .","Moreover, for $k=2,3,\\ldots $ , $\\bigl ( \\xi _i^2 \\langle , _i \\rangle ^2 \\bigr )^k & \\le C_\\tau B_\\tau ^{2(k-1)} \\upsilon _1^{2k} \\cdot 2k \\int _0^\\infty \\bigl ( | \\langle , _i \\rangle | \\ge \\upsilon _1 u \\bigr ) u^{2k-1} \\, {\\rm d} u \\nonumber \\\\& \\le C_\\tau B_\\tau ^{2(k-1)} \\upsilon _1^{2k} \\cdot 4k \\int _0^\\infty u^{2k-1} e^{-u^2/2 } \\, {\\rm d} u \\nonumber \\\\& = C_\\tau B_\\tau ^{2(k-1)} 2^k \\upsilon _1^{2k} \\cdot 2k \\int _0^\\infty v^{ k-1} e^{-v } \\, {\\rm d}v = C_\\tau B_\\tau ^{2(k-1)} 2^{k+1} \\upsilon _1^{2k} k!",".","\\nonumber $ In particular, $(\\xi _i^4 \\langle , _i \\rangle ^4)\\le B_\\tau ^2 C_\\tau 16 \\upsilon _1^4$ and $( \\xi _i^2 \\langle , _i \\rangle ^2 )^k \\le \\frac{k!","}{2} \\cdot B_\\tau ^2 C_\\tau 16 \\upsilon _1^4 \\cdot (2 B_\\tau ^2 \\upsilon _1^2 )^{k-2}$ for $k\\ge 3$ .","Since $(\\xi _i^2 | _i ) \\le C_\\tau $ , it then follows from Bernstein's inequality that, for any $v\\ge 0$ , $\\Biggl ( \\frac{1}{n} \\sum _{i=1}^n\\xi _i^2 \\langle , _i \\rangle ^2 \\ge C_\\tau + 4 B_\\tau C_\\tau ^{1/2} \\upsilon _1^2 \\sqrt{\\frac{2 v}{n}} + B_\\tau ^2 \\upsilon _1^2 \\frac{2v}{n} \\Biggr ) \\le e^{-v} .$ In the above analysis, we set $\\epsilon = 2/(e^2-1)$ so that $(1+2/\\epsilon )^p = e^{2p}$ , and take $u=v=2p +t$ .","Applying the union bound, we conclude that $\\max _{\\in _\\epsilon } \\frac{1}{n} \\sum _{i=1}^n\\xi _i^2 \\langle , _i \\rangle ^2 \\le C_\\tau + 4 B_\\tau C_\\tau ^{1/2} \\upsilon _1^2 \\sqrt{\\frac{4p+2t}{n}} + 2 B_\\tau ^2 \\upsilon _1^2 \\frac{2p+t}{n} $ with probability at least $1-e^{-t}$ .","Let $_1(t)$ be the event that (REF ) holds.","Then, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_1(t)$ , and by an application of union bound, we obtain $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) - ^{-1/2} \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2 \\le C \\sqrt{\\frac{p+t}{n}}$ as long as $n\\gtrsim p+t$ , where $C>0$ depends only on $(\\upsilon _1,\\kappa _2, l_0)$ and $\\tau $ .","This, together with (REF ) and Lemma REF , proves the claimed bound.","$\\Box $"],["Proof of Lemma ","Using arguments similar to (), (), and (REF ) in the proof of Lemma REF , we obtain $D^\\flat (\\beta , \\beta ^*) &= \\frac{1}{n} \\sum _{i=1}^n\\left\\lbrace {K} \\left( \\frac{\\langle _i, \\beta \\rangle - y_i}{h}\\right) - {K}\\left( \\frac{ -\\varepsilon _i}{h} \\right) \\right\\rbrace w_i \\langle _i, \\beta - \\beta ^* \\rangle \\nonumber \\\\&\\ge \\kappa _l \\Vert \\beta - \\beta ^* \\Vert _{}^2 \\cdot \\underbrace{ \\frac{1}{n h } \\sum _{i=1}^nw_i \\omega _i \\varphi _{h/(2r)} ( \\langle _i , \\delta \\rangle ) }_{ =: D_0^\\flat (\\delta ) } , \\nonumber $ where $\\delta = ^{1/2}( \\beta - \\beta ^*) /\\Vert \\beta - \\beta ^* \\Vert _{}$ , $\\omega _i = \\mathbb {1}( |\\varepsilon _i | \\le h/2)$ , and $\\varphi (\\cdot )$ is as defined in (REF ).","Recall from (REF ) the definition of $D_0 (\\delta )$ .","We have $\\inf _{\\beta \\in \\beta ^* + \\Theta (r)}\\frac{D^\\flat (\\beta , \\beta ^* )}{\\kappa _l \\Vert \\beta -\\beta ^* \\Vert _{}^2} \\ge \\inf _{\\delta \\in \\mathbb {S}^{p-1} } D_0(\\delta ) - \\sup _{\\delta \\in \\mathbb {S}^{p-1} } \\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\rbrace .$ We now obtain lower and upper bounds for $ D_0(\\delta ) $ and $\\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\rbrace $ , respectively.","Recall that $w_i$ is a random variable that is independent of $_i$ with $(w_i)=1$ .","Let $e_i = w_i -1 \\in \\lbrace -1,1\\rbrace $ .","Then, we have $D_0^\\flat (\\delta ) - D_0(\\delta ) = \\frac{1}{n h } \\sum _{i=1}^ne_i \\omega _i \\varphi _{h/(2r)} ( \\langle _i , \\delta \\rangle ) .$ Define $\\Gamma _n = \\gamma (e_1,\\ldots ,e_n) = \\sup _{ \\delta \\in \\mathbb {S}^{p-1}} \\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\rbrace $ .","Recall that $\\varphi _{R}(u) \\le (R/2)^2 $ , we have $^* \\lbrace e_i \\omega _i \\varphi _{h/(2r)} ( \\langle _i , \\delta \\rangle ) \\rbrace ^2 \\le (h/4r)^4 \\omega _i $ and $| e_i \\omega _i \\varphi _{h/(2r)} ( \\langle _i , \\delta \\rangle ) |\\le (h/4r)^2$ .","Then, by the Talagrand's inequality (see Theorem 7.3 in [60]), for every $t\\ge 0$ , $\\Gamma _n & \\le ^*( \\Gamma _n ) + \\sqrt{ \\frac{h^2}{(4r)^4}\\frac{ \\sum _{i=1}^n\\omega _i}{ n} \\frac{2t}{n} + 4^*( \\Gamma _n ) \\frac{h}{(4r)^2}\\frac{t}{n} } + \\frac{h}{(4r)^2} \\frac{t}{3 n} \\nonumber \\\\& \\le 2^*( \\Gamma _n ) + \\frac{h}{(4r)^2} \\Biggl \\lbrace \\Biggl ( \\frac{1}{n}\\sum _{i=1}^n\\omega _i \\Biggr )^{1/2} \\sqrt{ \\frac{2t}{n} } + \\frac{4t}{3 n} \\Biggr \\rbrace \\nonumber $ with probability at least $1-e^{-t}$ .","Further, by the Lipschitz continuity of $u\\rightarrow \\varphi _R(u)$ and Talagrand's contraction principle, $^*( \\Gamma _n ) & \\le \\frac{1}{2 r} ^* \\Biggl \\lbrace \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^ne_i \\langle \\omega _i _i, \\delta \\rangle \\Biggr \\rbrace \\nonumber \\\\& \\le \\frac{1}{2 r} ^* \\Biggl \\Vert \\frac{1}{n} \\sum _{i=1}^ne_i \\omega _i _i \\Biggr \\Vert _2 \\nonumber \\\\& \\le \\frac{1}{2 r n } \\Biggl ( \\sum _{i=1}^n\\omega _i \\Vert _i \\Vert _2^2 \\Biggr )^{1/2} \\nonumber \\\\& \\le \\frac{\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 }{2 r \\sqrt{n}} \\Biggl ( \\frac{1}{n}\\sum _{i=1}^n\\omega _i \\Biggr )^{1/2} .","\\nonumber $ Together, the last two displays imply $\\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\bigl \\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\bigr \\rbrace \\le \\Biggl ( \\frac{1}{n}\\sum _{i=1}^n\\omega _i \\Biggr )^{1/2} \\Biggl \\lbrace \\frac{\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 }{r \\sqrt{n}} +\\frac{h}{(4r)^2} \\sqrt{\\frac{2t}{n} } \\Biggr \\rbrace + \\frac{h}{(4r)^2} \\frac{4t}{3 n} $ with $^*$ -probability at least $1-e^{-t}$ .","Next we provide upper bounds for the data-dependent terms $\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 $ and $(1/n) \\sum _{i=1}^n\\omega _i$ .","As in the proof of Lemma REF , for any $\\epsilon \\in (0,1)$ , there exits $_\\epsilon \\subseteq \\mathbb {S}^{p-1}$ with $|_\\epsilon | \\le (1+2/\\epsilon )^p$ such that $\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 \\le (1-\\epsilon )^{-1} \\max _{1\\le i\\le n} \\max _{\\in _\\epsilon } \\langle , _i \\rangle $ .","Given $1\\le i\\le n$ and $\\in _\\epsilon $ , Condition REF indicates $( |\\langle , _i \\rangle | \\ge \\upsilon _1 u) \\le 2 e^{-u^2/2}$ for any $u \\in $ .","Taking the union bound over $i$ and $$ , and setting $u=\\sqrt{2t + 2\\log (2n)+2p\\log (1+2/\\epsilon )}$ ($t>0$ ), we obtain that with probability at least $1-2 n (1+2/\\epsilon )^p e^{-u^2/2}= 1-e^{-t}$ , $\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 \\le (1-\\epsilon )^{-1} \\upsilon _1 \\sqrt{2t + 2\\log (2n)+2p\\log (1+2/\\epsilon )}$ .","Minimizing this upper bound with respect to $\\epsilon \\in (0,1)$ , we conclude that for any $t>0$ , $\\biggl [ \\max _{1\\le i\\le n} \\Vert _i \\Vert _2^2 \\ge 2 \\upsilon _1^2 \\bigl \\lbrace 3.7p + \\log (2n) + t \\bigr \\rbrace \\biggr ] \\le e^{-t}.","$ Moreover, applying Bernstein's inequality to $(1/n) \\sum _{i=1}^n\\omega _i = (1/n) \\sum _{i=1}^n\\mathbb {1}( |\\varepsilon _i | \\le h/2)$ yields $\\frac{1}{n} \\sum _{i=1}^n\\omega _i \\le (\\omega _i) + \\sqrt{ 2(\\omega _i) \\frac{ t}{n}} + \\frac{t}{3n} \\le \\Biggl ( \\sqrt{(\\omega _i)} + \\sqrt{\\frac{t}{2n}} \\Biggr )^2$ with probability greater than $1-e^{-t}$ .","Note that $(\\omega _i) \\le \\bar{f}_{ h} h$ .","Putting together the pieces, we conclude that conditioned on some event that occurs with probability at least $1-2e^{-t}$ , $& \\sup _{\\delta \\in \\mathbb {S}^{p-1} } \\bigl \\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\bigr \\rbrace \\nonumber \\\\& \\le \\bigl \\lbrace ( \\bar{f}_h h )^{1/2} + (0.5 t/ n)^{1/2} \\bigr \\rbrace \\left\\lbrace \\frac{\\upsilon _1 \\bigl ( 7.4 p + 2\\log (2n) + 2 t \\bigr )^{1/2} }{r \\sqrt{n}} + \\frac{h}{(4r)^2} \\sqrt{\\frac{2t}{n} } \\right\\rbrace + \\frac{h}{(4r)^2}\\sqrt{\\frac{4 t}{3 n}}$ with $^*$ -probability greater than $1-e^{-t}$ .","Turning to $D_0(\\delta )$ , applying Lemma REF yields that as long as $0< r\\le h/(4\\eta _{1/4} )$ , $\\inf _{\\delta \\in \\mathbb {S}^{p-1}} D_0(\\delta ) \\ge \\frac{3}{4} {f}_h - \\bar{f}_h^{1/2} \\left( \\frac{5}{4}\\sqrt{\\frac{h p}{r^2 n}} + \\sqrt{\\frac{h t}{8r^2 n}} \\right) - \\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .","Substituting (REF ) and (REF ) into (REF ), and taking $r=h/(4\\eta _{1/4} )$ with $h\\gtrsim t/n$ , we conclude that conditioned on some event that occurs with probability at least $1-3 e^{-t}$ , $\\inf _{\\beta \\in \\beta ^* + \\Theta (r)}\\frac{D^\\flat (\\beta , \\beta ^* )}{\\kappa _l \\Vert \\beta -\\beta ^* \\Vert _{}^2} \\ge \\frac{3}{4} {f}_h - C \\sqrt{\\frac{p + \\log n + t}{ n h }} \\nonumber $ with $^*$ -probability greater than $1-e^{-t}$ , where $C>0$ depends only on $(\\upsilon _1, l_0 ,\\bar{f} )$ .","This completes the proof of (REF ).","$\\Box $"],["Proof of Theorem ","The proof is based on an argument similar to that used in the proof of Theorem REF .","To begin with, define the random process $\\Delta ^\\flat (\\beta ) = ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}_h^\\flat (\\beta ) - \\nabla \\hat{Q}_h^\\flat (\\beta ^*) - (\\beta -\\beta ^* )\\bigr \\rbrace , ~~ \\beta \\in ^p.$ For a prespecified $r>0$ , a key step is to bound the local fluctuation $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta ^\\flat (\\beta ) \\Vert _2$ .","Since $(w_i)=1$ , we have $^* \\lbrace \\nabla \\hat{Q}_h^\\flat (\\beta ) \\rbrace = \\nabla \\hat{Q}_h (\\beta )$ .","Define the (conditionally) centered process $G^\\flat (\\beta ) = ^{-1/2} \\big \\lbrace \\nabla \\hat{Q}_h^\\flat (\\beta ) - \\nabla \\hat{Q}_h (\\beta ) \\big \\rbrace = \\frac{1}{n} \\sum _{i=1}^ne_i \\lbrace K(-(y_i - \\langle _i, \\beta \\rangle )/h) - \\tau \\rbrace _i,$ so that $\\Delta ^\\flat (\\beta )$ be be written as $\\Delta ^\\flat (\\beta ) = \\bigl \\lbrace G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\bigr \\rbrace + \\Delta (\\beta ) , \\nonumber $ where $\\Delta (\\beta )$ is defined in (REF ).","By the triangle inequality, $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta ^\\flat (\\beta ) \\Vert _2 \\le \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\Vert _2 + \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta (\\beta ) \\Vert _2 .","$ It suffices to bound the first term on the right-hand side of (REF ).","By a change of variable $= ^{1/2} ( \\beta -\\beta ^*)$ , we have $y_i -\\langle _i, \\beta \\rangle = \\varepsilon _i - \\langle _i, \\rangle $ and $& \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\Vert _2 \\nonumber \\\\ & \\le \\sup _{ \\in \\mathbb {B}^p(r), \\in \\mathbb {B}^p(1) } \\langle G^\\flat ( \\beta ^* + ^{-1/2} ) - G^\\flat ( \\beta ^* ) , \\rangle \\nonumber \\\\& = r^{-1} n^{-1/2} \\sup _{ , \\in \\mathbb {B}^p(r) } \\underbrace{ n^{1/2} \\bigl \\langle G^\\flat ( \\beta ^* + ^{-1/2} ) - G^\\flat ( \\beta ^* ) , \\bigr \\rangle }_{ =: \\Psi ( , ) } , $ where $\\Psi (, ) = n^{-1/2} \\sum _{i=1}^ne_i\\langle _i, \\rangle \\lbrace K( \\langle _i, \\rangle / h - \\varepsilon _i /h ) - K(-\\varepsilon _i/h) \\rbrace $ .","To characterize the magnitude of $\\sup _{, \\in \\mathbb {B}^p(r)} \\Psi (, )$ , define the following classes of measurable functions $_1 =\\bigl \\lbrace ( , \\varepsilon ) \\mapsto \\langle , \\rangle : \\in \\mathbb {B}^p(r) \\bigr \\rbrace ~\\mbox{ and }~ _2 = \\bigl \\lbrace ( , \\varepsilon ) \\mapsto \\langle , \\rangle - \\varepsilon : \\in \\mathbb {B}^p(r) \\bigr \\rbrace .$ Also, define functions $f_0 : ( , \\varepsilon ) \\rightarrow K(-\\varepsilon /h)$ and $\\phi : u \\rightarrow K(u/h)$ .","With this notation, we can write $\\sup _{, \\in \\mathbb {B}^p(r)}\\Psi (, ) = \\sup _{g\\in } \\frac{1}{\\sqrt{n}} \\sum _{i=1}^ne_i g(_i), $ where $_i = (_i , \\varepsilon _i ) \\in ^p\\times $ and $= _1 \\cdot (\\phi \\circ _2 - f_0)$ denotes the pointwise product between $_1$ and $\\phi \\circ _2 - f_0$ .","Let $Z = ^* \\lbrace \\sup _{g\\in } (1/n)\\sum _{i=1}^ne_i g(_i) \\rbrace $ be the conditional Rademacher average.","By Theorem 13 in [59] and the bound $\\sup _{1\\le i\\le n, g\\in } g(_i )\\le r \\max _{1\\le i\\le n} \\Vert _i \\Vert _2$ , we have $\\bigl [ \\lbrace Z - (Z)\\rbrace ^{2k}_+ \\bigr ]^{1/(2k)} \\le 2 \\sqrt{(Z) \\cdot k \\kappa r \\frac{M_{n,k}}{n}} + 2 k \\kappa r \\frac{M_{n,k}}{n} , ~\\mbox{ valid for any } k\\ge 1, $ where $\\kappa = \\sqrt{e} / (2\\sqrt{e}-2)$ and $M_{n,k} = \\lbrace (\\max _{1\\le i\\le n} \\Vert _i \\Vert _2^{2k})\\rbrace ^{1/(2k)}$ .","Next we bound $(Z)$ via a maximal inequality specialized to VC type classes.","Note that the function class $_1$ is a $p$ -dimensional vector space, and $_2$ is included in a $(p + 1)$ -dimensional vector space.","By Lemma 2.6.15 of [50], both $_1$ and $_2$ have VC-subgraph of index smaller than or equal to $p+2$ .","Moreover, since $\\phi (\\cdot ) = K(\\cdot /h)$ is an non-decreasing function, it follows from Lemma 9.9 of [63] that $\\phi \\circ _2$ is VC with index less than or equal $ p+2$ , and so is $\\phi \\circ _2 - f_0$ .","Let $F_1(, \\varepsilon ) = r \\Vert \\Vert _2$ and $F_2 (, \\varepsilon ) \\equiv 1$ be the envelops of $_1$ and $\\phi \\circ _2 - f_0$ , respectively.","Hence, by Theorem 9.3 in [63] (or Theorem 2.6.7 in [50]), we obtain the following bounds on the covering numbers of function classes $_1$ and $\\phi \\circ _2 - f_0$ : there exists a universal constant $A_0>0$ such that, for any probability measure $Q$ with $\\Vert F_1 \\Vert _{Q,2} >0$ and any $\\epsilon \\in (0,1)$ , $N( _1, L_2(Q) , \\epsilon \\Vert F_1 \\Vert _{Q,2} ) \\le A_0 (p+2)(4e)^{p+2} \\biggl ( \\frac{2}{\\epsilon } \\biggr )^{2(p+1)}; \\nonumber \\\\N( \\phi \\circ _2 - f_0 , L_2(Q) , \\epsilon \\Vert F_2 \\Vert _{Q,2} ) \\le A_0 (p+2)(4e)^{p+2} \\biggl ( \\frac{2}{\\epsilon } \\biggr )^{2(p+1)} .\\nonumber $ For the product class $$ with envelop $G(, \\varepsilon )= F_1(, \\varepsilon ) F_2(, \\varepsilon ) = r \\Vert \\Vert _2$ , applying Corollary A.1 in the supplement of [61] yields that, for any $\\epsilon \\in (0 ,1)$ , $& N( , L_2(Q) , \\epsilon \\Vert G \\Vert _{Q,2} ) \\nonumber \\\\& \\le N( _1, L_2(Q) , 2^{-1/2} \\epsilon \\Vert F_1 \\Vert _{Q,2} ) N( \\phi \\circ _2 - f_0 , L_2(Q) , 2^{-1/2} \\epsilon \\Vert F_2 \\Vert _{Q,2} ) \\nonumber \\\\& \\le A_0^2 (p+2)^2 (4 e)^{2(p+2)} \\biggl ( \\frac{2^{3/2}}{\\epsilon } \\biggr )^{4(p+1)} \\le \\biggl ( \\frac{A_1}{\\epsilon } \\biggr )^{4(p+2)} \\nonumber $ for some $A_1 > e$ .","Another key quantity is the variance proxy $\\sup _{g\\in } \\lbrace g^2(, \\varepsilon )\\rbrace $ .","Given some $g = g_{, } \\in $ with $, \\in \\mathbb {B}^p(r)$ , we have $\\lbrace g^2(, \\varepsilon )\\rbrace & = ( \\langle , \\rangle ^2 [K\\lbrace ( \\langle , \\rangle - \\varepsilon ) /h\\rbrace - K(-\\varepsilon /h) ]^2) \\nonumber \\\\& = \\biggl (\\langle , \\rangle ^2 \\int _{-\\infty }^\\infty \\bigl \\lbrace K\\bigl ( ( \\langle , \\rangle -u ) /h \\bigr ) - K(-u/h) \\bigr \\rbrace ^2 f_{\\varepsilon | } (u) {\\rm d} u\\biggr ) \\nonumber \\\\& = h \\, \\biggl (\\langle , \\rangle ^2 \\int _{-\\infty }^\\infty \\bigl \\lbrace K \\bigl ( \\langle , \\rangle /h + v \\bigr ) - K(v) \\bigr \\rbrace ^2 f_{\\varepsilon | } (-vh) {\\rm d} v\\nonumber \\biggr ) \\\\& \\le \\bar{f} h^{-1} \\biggl (\\langle , \\rangle ^2 \\langle , \\rangle ^2 \\int _{-\\infty }^\\infty \\biggl \\lbrace \\int _0^1 K(v+ s \\langle ,\\rangle /h ) {\\rm d} s \\biggr \\rbrace ^2 {\\rm d} v\\biggr ) \\nonumber \\\\& \\le \\bar{f} h^{-1} \\biggl (\\langle , \\rangle ^2 \\langle , \\rangle ^2\\Biggl [ \\int _0^1 \\biggl \\lbrace \\int _{-\\infty }^\\infty K^2(v+ s \\langle ,\\rangle /h ) {\\rm d} v \\biggr \\rbrace ^{1/2} {\\rm d}s \\Biggr ]^2 \\nonumber \\biggr ) \\\\& \\le \\kappa _u \\bar{f} h^{-1} (\\langle , \\rangle ^2 \\langle , \\rangle ^2 ), \\nonumber $ where the second inequality follows from an application of generalized Minkowski's integral inequality.","Hence, $\\sup _{g\\in } \\lbrace g^2(, \\varepsilon )\\rbrace \\le \\kappa _u \\bar{f} \\mu _4 r^4h^{-1}$ .","For the envelop $G$ for $$ , it can be shown that $\\lbrace G^2(, \\varepsilon )\\rbrace = r^2 p$ , $\\max _{1\\le i\\le n}\\sup _{g\\in } | g(_i) | \\le r\\max _{1\\le i\\le n} \\Vert _i \\Vert _2$ , and $\\lbrace \\max _{1\\le i\\le n}G^2(_i, \\varepsilon _i)\\rbrace = r^2 M_{n,1}^2$ with $M_{n,1}$ given in (REF ).","Equipped with the above bounds, we apply Corollary 5.1 in [61] to obtain $(Z) \\lesssim r^2 \\sqrt{\\frac{p}{nh} \\log ( A_2 p h /r^2 ) } + r M_{n,1} \\log ( A_2 p h /r^2 ) \\frac{ p }{n} , $ where $A_2>0$ depends on $(\\bar{f} , \\kappa _u, \\mu _4)$ and $A_1$ .","It remains to bound $M_{n,1}$ , which appears in both (REF ) and (REF ).","In fact, the exponential tail probability (REF ) can be directly used to bound the expectation.","Given a non-negative random variable $X$ and constants $A, a>0$ , we have $(X) & = \\int _0^\\infty (X\\ge t) {\\rm d} t \\le A + \\int _A^\\infty (X\\ge t) {\\rm d} t \\nonumber \\\\& = A + \\int _0^\\infty (X\\ge A+ t) {\\rm d} t = A + a \\int _0^\\infty (X\\ge A + a s ) {\\rm d} s .","\\nonumber $ Taking $X= \\max _{1\\le i\\le n} \\Vert _i \\Vert _2^2$ , $A= 2 \\upsilon _1^2 \\lbrace 3.7 p + \\log (2n) \\rbrace $ and $a=2 \\upsilon _1^2 $ , we obtain $M_{n,1}^2 = \\Biggl ( \\max _{1\\le i\\le n} \\Vert _i \\Vert _2^2 \\Biggr ) \\le 2 \\upsilon _1^2 \\bigl \\lbrace 3.7 p + \\log (2e n) \\bigr \\rbrace .","$ Back to the supremum $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\Vert _2$ , first by (REF ), (REF ), and the definition of $Z$ below (REF ), we obtain $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\Vert _2 = O_{^*} (r^{-1} Z ) .","$ For the conditional Rademacher average $Z$ , it follows from Markov's inequality that $\\lbrace Z \\ge (Z) + u \\rbrace \\le u^{-2} \\lbrace Z- (Z)\\rbrace _+^2$ , valid for any $u >0$ .","Applying the bound (REF ) with $k=1$ , (REF ) and (REF ) yields $& \\bigl [ \\lbrace Z- (Z)\\rbrace _+^2 \\bigr ]^{1/2} \\lesssim \\sqrt{ (Z) \\cdot (p +\\log n)^{1/2} \\frac{r}{n} } + (p+\\log n)^{1/2} \\frac{r}{n}; \\nonumber \\\\&(Z) \\lesssim r^2 \\sqrt{ \\frac{p}{n h} \\log (A_2ph / r^2 ) } + r \\log (A_2ph / r^2 ) (p+\\log n)^{1/2} \\frac{p}{n} , \\nonumber $ from which we conclude that $r^{-1 }Z = O_{} \\Biggl ( r \\sqrt{ \\frac{p}{n h} \\log (A_2ph / r^2 ) } + \\log (A_2ph / r^2 ) (p+\\log n)^{1/2} \\frac{p}{n} \\Biggr ).$ Turning to the second term on the right-hand side of (REF ), the earlier bound (REF ) implies that there exists some event $_1(t)$ with $\\lbrace _1(t) \\rbrace \\ge 1-e^{-t}$ such that $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta (\\beta ) \\Vert _2 \\le C_1 \\Biggl ( r \\sqrt{\\frac{p+t}{n h}} + h r + r^2 \\Biggr ) ~\\mbox{ on } _1 (t), $ where $C_1>0$ depends on $( \\upsilon _1, \\kappa _1, \\kappa _u, l_0, \\bar{f} , {f})$ .","With the above preparations, we are ready to prove the claim.","Assume that the triplet $(n,p,h)$ satisfies the scaling $ \\sqrt{(p+\\log n)/n} \\lesssim h \\lesssim 1$ .","By Theorems REF and REF , there exist some event $_n$ with $(_n ) \\ge 1-5n^{-1}$ and some $r=r(n,p,h)\\asymp \\sqrt{(p+\\log n)/n} + h^2>0$ such that, $\\Vert \\hat{\\beta }_h - \\beta ^* \\Vert _{} \\le r$ on the event $_n$ , $ \\Vert \\hat{\\beta }_h^\\flat - \\beta ^* \\Vert _{} \\le r$ with $^*$ -probability at least $1-2 n^{-1}$ conditioned on $_n$ .","Taking $t=\\log n$ and $_{1n} = _1(\\log n)$ in (REF ), we have on the event $_{1n} \\cap _n$ that $\\chi _{1n} := \\Vert \\Delta (\\hat{\\beta }_h ) \\Vert _2 \\lesssim \\frac{p + \\log n}{ h^{1/2} n } + h \\sqrt{\\frac{p + \\log n}{n}} + h^3.","\\nonumber $ Moreover, with $^*$ -probability at least $1-2n^{-1}$ conditioned on $_n$ , $\\Biggl \\Vert ^{-1/2} \\Biggl [ ( \\hat{\\beta }_h^\\flat - \\beta ^* ) - \\frac{1}{n} \\sum _{i=1}^nw_i \\bigl \\lbrace \\tau - K(- \\varepsilon _i /h) \\bigr \\rbrace _i \\Biggr ] \\Biggr \\Vert _2 \\le \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta ^\\flat (\\beta ) \\Vert _2 .","\\nonumber $ Let $\\chi _{2n} = ^* \\lbrace \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta ^\\flat (\\beta ) \\Vert _2 \\rbrace $ , for which it follows from (REF ), (REF ), and (REF ) with $t=\\log n$ that $\\chi _{2n} = O_{} \\Biggl \\lbrace \\frac{\\sqrt{(p+\\log n)p \\log n}}{h^{1/2} n} + h^{3/2} \\sqrt{\\frac{p \\log n }{n}} + h \\sqrt{\\frac{p + \\log n}{n}} + h^3 + (p+\\log n)^{1/2} \\frac{p \\log n}{n} \\Biggr \\rbrace .$ With $h=h_n\\asymp \\lbrace (p+\\log n)/n \\rbrace ^{\\delta }$ for some $\\delta \\in (1/3 , 1/2]$ , the first term on the right-hand side dominates the rest except the last one.","Then, putting together the pieces establishes the claim.","$\\Box $"],["Theoretical Properties of One-step Conquer","In this section, we provide theoretical properties of the one-step conquer estimator $\\hat{\\beta }$ , defined in Section REF .","The key message is that, when higher-order kernels are used (and if the conditional density $f_{\\varepsilon |}(\\cdot )$ has enough derivatives), the asymptotic normality of the one-step estimator holds under weaker growth conditions on $p$ .","For example, the scaling condition $p=o(n^{3/8})$ that is required for the conquer estimator can be reduced to roughly $p=o(n^{7/16})$ for the one-step conquer estimator using a kernel of order 4.","Let $G(\\cdot )$ be a symmetric kernel of order $\\nu >2$ , that is, $\\int _{-\\infty }^\\infty u^k G(u) \\, {\\rm d}u=0$ for $k=1,\\ldots , \\nu -1$ and $\\int _{-\\infty }^\\infty u^\\nu G(u) \\, {\\rm d} u \\ne 0$ .","Moreover, $\\kappa ^G_k = \\int _{-\\infty }^\\infty |u^k G(u) | \\, {\\rm d}u <\\infty $ for $1\\le k\\le \\nu $ , $G$ is uniformly bounded with $\\kappa ^G_u = \\sup _{u\\in } |G(u) | <\\infty $ and is $L^G$ -Lipschitz continuous for some $L^G>0$ .","The use of a higher-order kernel does not necessarily reduce bias unless the conditional density $ f_{\\varepsilon | }(\\cdot )$ of $\\varepsilon $ given $$ is sufficiently smooth.","Therefore, we further impose the following smoothness conditions on $f_{\\varepsilon | }(\\cdot )$ .","Let $\\nu \\ge 4$ be the integer in Condition .","The conditional density $ f_{\\varepsilon | }(\\cdot )$ is $(\\nu -1)$ -times differentiable, and satisfies $| f_{\\varepsilon | }^{(\\nu -2)} (u) - f_{\\varepsilon | }^{(\\nu -2)} (0) | \\le L_{\\nu -2} |u|$ for all $u\\in $ almost surely (over the random vector $$ ), where $L_{\\nu -2}>0$ is a constant.","Also, there exists some constant $C_G>0$ such that $ \\int _{-\\infty }^\\infty | u^{\\nu -1} G(u)| \\cdot \\sup _{|t| \\le |u|} | f_{\\varepsilon | }^{(\\nu -1)} (t)- f_{\\varepsilon | }^{(\\nu -1)} (0) | \\, {\\rm d}u \\le C_G$ almost surely.","Notably, we have $\\nabla Q^G_b( \\beta ) = \\bigl \\lbrace G\\big ( ( \\langle , \\beta \\rangle - y) /b \\big ) - \\tau \\bigr \\rbrace ~~\\mbox{ and }~~\\nabla ^2 Q^G_b (\\beta ) = \\big \\lbrace G_b( y- \\langle , \\beta \\rangle ) ^\\rbrace ,$ representing the population score and Hessian of $Q^G_b (\\cdot ) = \\hat{Q}^G_b(\\cdot )$ .","As $b \\rightarrow 0$ , we expect $\\nabla Q^G_b( \\beta ^*)$ and $\\nabla ^2 Q^G_b (\\beta ^*)$ to converge to $\\textbf {0}$ (zero vector in $^p$ ) and $= \\lbrace f_{\\varepsilon | } (0) ^$ , respectively.","The following proposition validates this claim by providing explicit error bounds.","Let $b\\in (0, 1)$ be a bandwidth.","Under Conditions and , we have $\\big \\Vert ^{-1/2} \\nabla Q^G_b(\\beta ^*) \\big \\Vert _2 & \\le L_{\\nu -2} \\kappa ^G_\\nu \\, b^\\nu / \\nu !","\\nonumber \\\\~~\\mbox{ and }~~\\big \\Vert ^{-1/2} \\nabla ^2 Q^G_b(\\beta ^*) ^{-1/2} - _0 \\big \\Vert _2 & \\le C_G \\, b^{\\nu -1} / (\\nu -1)!",", \\nonumber $ where $_0 = ^{-1/2} ^{-1/2} = \\lbrace f_{\\varepsilon |} (0) ^$ with $= ^{-1/2} $ .","Proposition shows that when a higher-order kernel is used, the bias is significantly reduced in the sense that $\\Vert \\nabla Q^G_b(\\beta ^*) \\Vert _2 = ( b^\\nu )$ and $\\Vert \\nabla ^2 Q^G_b(\\beta ^*) - \\Vert _2 = (b^{\\nu -1})$ , where $\\nu \\ge 4$ is an even integer.","Notably, even if the kernel $H$ has negative parts, the population Hessian $\\nabla ^2 Q^G_b(\\beta ^*) $ preserves the positive definiteness of $$ as long as the bandwidth $g$ is sufficient small, To construct the one-step conquer estimator, two key quantities are the sample Hessian $\\nabla ^2 \\hat{Q}^G_b( \\cdot ) $ and sample gradient $\\nabla \\hat{Q}^G_b(\\cdot )$ , both evaluated at $\\beta $ , a consistent initial estimate.","In the next two propositions, we establish uniform convergence results of the Hessian and gradient of the empirical smoothed loss to their population counterparts.","As a direct consequence, $\\nabla ^2 \\hat{Q}^G_b( \\beta )$ is positive definite with high probability, provided that $\\beta $ is consistent (i.e., in a local vicinity of $\\beta ^*$ ).","To be more specific, for $r>0$ , we define the local neighborhood $\\Theta ^*(r) = \\bigl \\lbrace \\beta \\in ^p : \\Vert \\beta - \\beta ^* \\Vert _{} \\le r \\bigr \\rbrace .","$ Conditions , and REF ensure that, with probability at least $1-e^{-t}$ , $& \\sup _{\\beta \\in \\Theta ^*(r)}\\big \\Vert ^{-1/2} \\big \\lbrace \\nabla ^2 \\hat{Q}^G_b(\\beta ) - \\nabla ^2 Q^G_b(\\beta ) \\big \\rbrace \\nonumber ^{-1/2} \\big \\Vert _2 \\lesssim \\sqrt{\\frac{p \\log n + t}{n b }} + \\frac{p\\log n+t}{n b } + \\frac{(p + t)^{1/2} r }{n b^2} \\nonumber $ as long as $n\\gtrsim p+t$ .","Conditions , and REF ensure that, with probability at least $1- e^{-t}$ , $& \\sup _{\\beta \\in \\Theta ^*(r)} \\big \\Vert ^{-1/2} \\big \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) - ( \\beta - \\beta ^*) \\big \\rbrace \\big \\Vert _2 \\lesssim r \\,\\Biggl ( \\sqrt{\\frac{p+ t}{n b }} + r +b^{\\nu -1} \\Bigg ) $ as long as $\\sqrt{(p+t) /n } \\lesssim b \\lesssim 1$ .","With the above preparations, we are ready to present the Bahadur representation for the one-step conquer estimator $\\hat{\\beta }$ .","Assume Conditions REF , REF and REF in the main text and Conditions and hold.","For any $t>0$ , let the sample size $n$ , dimension $p$ and the bandwidths $h , b>0$ satisfy $n\\gtrsim p(\\log n)^2 +t$ , $\\sqrt{(p+t)/n} \\lesssim h \\lesssim \\lbrace (p+t)/n \\rbrace ^{1/4}$ and $\\sqrt{(p+t)/n } \\lesssim b \\lesssim \\lbrace (p+t)/n \\rbrace ^{1/(2\\nu )}$ .","Then, the one-step conquer estimator $\\hat{\\beta }$ satisfies the bound $ \\Bigg \\Vert \\, \\hat{\\beta }- \\beta ^* - ^{-1} \\frac{ 1}{n} \\sum _{i=1}^n\\bigl \\lbrace \\tau - G(-\\varepsilon _i/b ) \\bigr \\rbrace _i \\Bigg \\Vert _2 \\lesssim \\Biggl \\lbrace \\underbrace{ \\sqrt{ (p\\log n + t)/(n b ) } }_{{\\rm variance~ term}}+ \\underbrace{ b^{\\nu -1} }_{{\\rm bias~term}} \\Biggr \\rbrace \\sqrt{\\frac{p+t}{n}}$ with probability at least $1-5e^{-t}$ , where $G(u) = \\int _{-\\infty }^u G(v)\\,{\\rm d}v$ .","Theorem shows that using a higher-order kernel ($\\nu \\ge 4$ ) allows one to choose larger bandwidth, thereby reducing the “variance\" and the total Bahadur linearization error.","Similarly to Theorem REF in the main text, the following asymptotic normal approximation result for linear projections of one-step conquer is a direct consequence of Theorem .","Assume Conditions REF , REF and REF in the main text and Conditions and hold.","Let the bandwidths satisfy $( \\frac{p+\\log n}{n} )^{1/2} \\lesssim h \\lesssim ( \\frac{p+\\log n}{n} )^{1/4}$ and $( \\frac{p+\\log n}{n} )^{1/2} \\lesssim b \\lesssim (\\frac{p+\\log n}{n})^{1/(2\\nu )}$ .","Then, $\\sup _{x\\in , \\, \\in ^p } \\bigl | \\bigl ( n^{1/2} \\langle , \\hat{\\beta }- \\beta ^* \\rangle \\le \\sigma _\\tau x \\bigr ) - \\Phi (x ) \\bigr | \\lesssim \\sqrt{\\frac{ (p+\\log n)p \\log n}{n b}} + n^{1/2} b^{\\nu } , $ where $\\sigma _\\tau ^2 = \\tau (1-\\tau ) \\Vert ^{-1} \\Vert _{ }^2$ .","In particular, with a choice of bandwidth $b \\asymp (\\frac{p+\\log n}{n})^{2/(2\\nu +1)}$ , $\\sup _{x\\in , \\, \\in ^p } \\bigl | \\bigl ( n^{1/2} \\langle , \\hat{\\beta }- \\beta ^* \\rangle \\le \\sigma _\\tau x \\bigr ) - \\Phi (x) \\bigr | \\rightarrow 0 \\nonumber $ as $n, p \\rightarrow \\infty $ under the scaling $p^{4\\nu /(2\\nu -1)} (\\log n)^{(2\\nu +1 )/(2\\nu -1)} =o(n)$ .","Let $G(\\cdot )$ be a kernel of order $\\nu =4$ .","In view of Theorem , we take $h \\asymp \\lbrace (p+\\log n)/n\\rbrace ^{2/5}$ as in the main text and $b = \\lbrace (p+\\log n)/n\\rbrace ^{2/9}$ , thereby obtaining that $n^{1/2}\\langle , \\hat{\\beta }-\\beta \\rangle $ , for an arbitrary $\\in ^p$ , is asymptotically normally distributed as long as $p^{16/7} (\\log n)^{9/7} = o(n)$ as $n\\rightarrow \\infty $ ."],["Proof of Proposition ","We start from the gradient $^{-1/2}\\nabla Q^G_b(\\beta ^*)=\\lbrace G ( - \\varepsilon /b ) - \\tau \\rbrace $ with $=^{-1/2} $ .","Let $_{}$ be the conditional expectation given $$ .","By integration by parts, $_{}G ( -\\varepsilon /b ) & = \\int _{-\\infty }^\\infty G(-t/b ) \\, {\\rm d} F_{\\varepsilon | } (t) = \\int _{-\\infty }^\\infty G(u) F_{\\varepsilon | } (- b u ) \\, {\\rm d} u.","$ Applying a Taylor expansion with integral remainder on $F_{\\varepsilon | } (- b u )$ yields $F_{\\varepsilon | } (- b u ) &= F_{\\varepsilon | }(0) + \\sum _{\\ell =1}^{\\nu -1} F_{\\varepsilon | }^{(\\ell )} (0) \\frac{(- b u)^\\ell }{\\ell !}","+ \\frac{(-b u)^{\\nu -1}}{(\\nu -2)!}","\\int _0^1 (1-w)^{\\nu -2} \\big \\lbrace F_{\\varepsilon | }^{(\\nu -1)} (- b u w) - F_{\\varepsilon | }^{(\\nu -1)} (0) \\bigr \\rbrace {\\rm d} w \\\\&= \\tau + \\sum _{\\ell =0}^{\\nu -2} f_{\\varepsilon | }^{(\\ell )} (0) \\frac{(- b u)^{\\ell +1}}{(\\ell +1)!}","+ \\frac{(- b u)^{\\nu -1}}{(\\nu -2)!}","\\int _0^1 (1-w)^{\\nu -2} \\big \\lbrace f_{\\varepsilon | }^{(\\nu -2)} (- b u w) - f_{\\varepsilon | }^{(\\nu -2)} (0) \\bigr \\rbrace {\\rm d} w .$ Recall that $G$ is a kernel of order $\\nu \\ge 4$ (an even integer) and $\\kappa ^G_\\nu = \\int _{-\\infty }^\\infty |u^\\nu G (u) |\\, {\\rm d}u <\\infty $ .","Substituting the above expansion into (REF ), we obtain $_{}G ( -\\varepsilon /b ) = \\tau - \\frac{b^{\\nu -1}}{(\\nu -2)!}","\\int _{-\\infty }^\\infty \\int _0^1 u^{\\nu -1} G(u) (1-w)^{\\nu -2} \\big \\lbrace f_{\\varepsilon | }^{(\\nu -2)} (- b u w) - f_{\\varepsilon | }^{(\\nu -2)} (0) \\bigr \\rbrace {\\rm d} w {\\rm d} u .\\nonumber $ Furthermore, by the Lipschitz continuity of $f_{\\varepsilon | }^{(\\nu -2)}(\\cdot )$ around 0, $| _{}G ( -\\varepsilon /b ) - \\tau | & \\le \\frac{L_{\\nu -2} b^{\\nu } }{(\\nu -2)!}","\\int _{-\\infty }^\\infty \\int _0^1 | u^{\\nu } G(u)| (1-w)^{\\nu -2} w \\,{\\rm d} w {\\rm d} u \\nonumber \\\\& = B(2,\\nu -1) L_{\\nu -2} \\kappa ^G_\\nu \\, b^{\\nu } / (\\nu -2)!",", \\nonumber $ where $B(x,y) := \\int _0^1 t^{x-1} (1-t)^{y-1} {\\rm d}t$ denotes the beta function.","In particular, $B(2,\\nu -1)= \\Gamma (2)\\Gamma (\\nu -1)/\\Gamma (\\nu +1)= (\\nu -2)!/\\nu !$ .","Putting together the pieces yields $\\Vert ^{-1/2} \\nabla Q^G_b(\\beta ^*) \\Vert _2 & = \\sup _{\\in \\mathbb {S}^{p-1}} \\, _{} \\big \\lbrace G(-\\varepsilon /b ) - \\tau \\big \\rbrace ^\\le L_{\\nu -2} \\kappa ^G_\\nu \\, b^\\nu / \\nu !",".","\\nonumber $ Turning to the Hessian, note that $& \\big \\Vert ^{-1/2} \\nabla ^2 Q^G_b(\\beta ^*) ^{-1/2} - _0 \\big \\Vert _2 = \\Bigg \\Vert \\int _{-\\infty }^\\infty G(u) \\big \\lbrace f_{\\varepsilon | } (- b u) - f_{\\varepsilon | }(0) \\big \\rbrace {\\rm d}u \\, ^\\Vert _2 .\\nonumber $ Applying a similar Taylor expansion as above, we have $f_{\\varepsilon | } (t) &= f_{\\varepsilon | }(0) + \\sum _{\\ell =1}^{\\nu -1} f_{\\varepsilon | }^{(\\ell )} (0) \\frac{t^\\ell }{\\ell !}","+ \\frac{t^{\\nu -1}}{(\\nu -2)!}","\\int _0^1 (1-w)^{\\nu -2} \\big \\lbrace f_{\\varepsilon | }^{(\\nu -1)} (t w ) - f_{\\varepsilon | }^{(\\nu -1)} (0) \\bigr \\rbrace {\\rm d} w .","$ Under Conditions and , it follows that $& \\big \\Vert ^{-1/2} \\nabla ^2 Q^G_b(\\beta ^*) ^{-1/2} - _0 \\big \\Vert _2 \\\\& \\le \\frac{b^{\\nu -1}}{(\\nu -2)!}","\\sup _{, \\delta \\in \\mathbb {S}^{p-1}} \\int _{-\\infty }^\\infty \\int _0^1 u^{\\nu -1} G(u) (1-w)^{\\nu -2}\\big \\lbrace f_{\\varepsilon | }^{(\\nu -1)} (- b u w) - f_{\\varepsilon | }^{(\\nu -1)} (0) \\bigr \\rbrace {\\rm d} w\\, {\\rm d}u \\, \\big \\langle , \\big \\rangle \\big \\langle , \\delta \\big \\rangle \\\\& \\le \\frac{b^{\\nu -1}}{(\\nu -1)!}","\\sup _{, \\delta \\in \\mathbb {S}^{p-1}} \\int _{-\\infty }^\\infty | u^{\\nu -1} G(u) | \\sup _{|t| \\le b |u|} \\big | f_{\\varepsilon | }^{(\\nu -1)} (t) - f_{\\varepsilon | }^{(\\nu -1)} (0) \\bigr | \\, {\\rm d}u \\cdot | \\langle , \\rangle \\langle , \\delta \\big \\rangle | \\\\& \\le \\frac{C_G b^{\\nu -1 }}{(\\nu -1)!}","\\sup _{ \\in \\mathbb {S}^{p-1}} \\big \\langle , \\big \\rangle ^2 = \\frac{C_G}{(\\nu -1)!}","b^{\\nu -1 } .$ This completes the proof.","$\\Box $"],["Proof of Proposition ","Consider the change of variable $\\delta = ^{1/2} (\\beta - \\beta ^*)$ , so that $\\beta \\in \\Theta ^*(r)$ is equivalent to $\\delta \\in \\mathbb {B}^p(r)$ .","Write $_i = ^{-1/2} _i \\in ^p$ , which are isotropic random vectors, and define $_n(\\delta ) = ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} = \\frac{1}{n} \\sum _{i=1}^nG_b(\\varepsilon _i - _i^) _i _i^ \\quad (\\delta ) = \\bigl \\lbrace _n(\\delta ) \\bigr \\rbrace .$ For any $\\epsilon \\in (0,r)$ , there exists an $\\epsilon $ -net $\\lbrace \\delta _1, \\ldots , \\delta _{d_\\epsilon }\\rbrace $ with $d_\\epsilon \\le (1+ 2r/\\epsilon )^p$ satisfying that, for each $\\delta \\in \\mathbb {B}^p(r)$ , there exists some $1\\le j\\le d_\\epsilon $ such that $\\Vert \\delta - \\delta _j \\Vert _2 \\le \\epsilon $ .","Hence, $& \\Vert _n(\\delta ) - (\\delta ) \\Vert _2 \\nonumber \\\\& \\le \\Vert _n(\\delta ) - _n(\\delta _j) \\Vert _2 + \\Vert _n(\\delta _j) - (\\delta _j) \\Vert _2 + \\Vert (\\delta _j) - (\\delta ) \\Vert _2 \\nonumber \\\\& =: I_1(\\delta ) + I_2 (\\delta _j)+ I_3(\\delta ).","\\nonumber $ For $I_1(\\delta )$ , note that $G_b(u) = (1/b)G(u/b)$ is Lipschitz continuous, i.e.","$|G_b(u) -G_b(v)| \\le L^G b^{-2} |u-v|$ for all $u,v \\in $ .","It follows that $I_1(\\delta ) & \\le \\sup _{, \\in \\mathbb {S}^{p-1}} \\frac{1}{n} \\sum _{i=1}^n| G_b(\\varepsilon _i - _i^) - G_b(\\varepsilon _i -_i^_j ) | \\cdot | _i^\\cdot _i^| \\nonumber \\\\& \\le L^G b^{-2}\\sup _{, \\in \\mathbb {S}^{p-1}} \\frac{1}{n} \\sum _{i=1}^n| _i^ \\delta -\\delta _j ) \\cdot _i^\\cdot _i^| \\nonumber \\\\& \\le L^G b^{-2} \\epsilon \\underbrace{ \\sup _{\\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^n| _i^|^3 }_{=: M_{n,3}} .","$ Next, we use the standard covering argument to bound $M_{n,3}$ .","Given $\\epsilon _1 \\in (0,1)$ , let $_1$ be an $\\epsilon _1$ -net of the unit sphere $\\mathbb {S}^{p-1}$ with $d_{\\epsilon _1} := | _1 | \\le (1+2/\\epsilon _1)^p$ such that for every $\\in \\mathbb {S}^{p-1}$ , there exists some $\\in _1$ satisfying $\\Vert - \\Vert _2 \\le \\epsilon _1$ .","Define the (standardized) design matrix $_n = n^{-1/3}(_1,\\ldots , _n)^^{n\\times p}$ , so that $M_{n,3} = \\sup _{\\in \\mathbb {S}^{p-1}} \\Vert _n \\Vert _3^3$ .","By the triangle inequality, $\\Vert _n \\Vert _3 & \\le \\Vert _n \\Vert _3 + \\Vert _n ( - ) \\Vert _3 \\nonumber \\\\& = \\Vert _n \\delta \\Vert _3 +\\bigg ( \\frac{1}{n} \\sum _{i=1}^n| _i^ - ) |^3 \\bigg )^{1/3} \\le \\Vert _n \\Vert _3 + \\epsilon _1 M_{n,3}^{1/3} .","\\nonumber $ Taking the maximum over $\\in _1$ , and then taking the supremum over $\\in \\mathbb {S}^{p-1}$ , we arrive at $M_{n,3} \\le (1-\\epsilon _1)^{-3} \\widetilde{M}_{n, 3} , $ where $ \\widetilde{M}_{n, 3} = \\max _{\\in _1 } (1/n) \\sum _{i=1}^n| _i^|^3$ .","For every $\\in _1$ , note that $( |_i^|^3 \\ge y\\rbrace \\le 2e^{-y^{2/3}/(2\\upsilon _1^2)}$ for any $y>0$ .","Hence, by inequality (3.6) in [58] with $s=2/3$ , we obtain that for any $z\\ge 3$ , $\\frac{1}{n} \\sum _{i=1}^n|_i^|^3 \\le |^|^3 + C_1 \\Biggl ( \\sqrt{\\frac{z }{n}} + \\frac{z^{3/2}}{n} \\Bigg )$ with probability at least $1-e^{3-z}$ , where $C_1>0$ is a constant depending only on $\\upsilon _1$ .","Taking the union bound over all vectors $$ in $_1$ yields that, with probability at least $1-d_{\\epsilon _1} e^{3-z} \\ge 1- e^{3+p\\log (1+2/\\epsilon _1) - z}$ , $\\widetilde{M}_{n,3} \\le \\mu _3 + C_1\\Biggl ( \\sqrt{\\frac{z }{n}} + \\frac{z^{3/2}}{n} \\Bigg )$ where $\\mu _3 = \\sup _{\\in \\mathbb {S}^{p-1} } |^|^3$ .","Reorganizing the terms, we have $\\widetilde{M}_{n,3} \\le \\mu _3 + C_1 \\Bigg [ \\sqrt{\\frac{p\\log (1+2/\\epsilon _1) +4+t }{n}} + \\frac{\\lbrace p\\log (1+2/\\epsilon _1) + 4 + t \\rbrace ^{3/2}}{n} \\Bigg ] $ with probability at least $1- e^{-t}/2$ .","Hence, taking $\\epsilon =1/8$ in (REF ) and (REF ) implies $M_{n,3} \\le 1.5\\mu _3 + 1.5 C_1 \\Bigg \\lbrace \\sqrt{\\frac{3p+ 4 +t }{n}} + \\frac{ ( 3p + 4+ t )^{3/2}}{n} \\Bigg \\rbrace .","\\nonumber $ Plugging the above bound into (REF ) yields that as long as $n \\gtrsim p+t$ , $\\sup _{\\delta \\in \\mathbb {B}^p(r)} I_1(\\delta ) \\lesssim (p + t)^{1/2} b^{-2} \\epsilon $ with probability at least $1- e^{-t}/2$ .","For $I_3(\\delta )$ , it can be similarly obtained that $I_3(\\delta ) \\le L^G b^{-2} \\sup _{, \\in \\mathbb {S}^{p-1}} | ^ \\delta - \\delta _j ) \\cdot ^\\cdot ^| \\le L^G \\mu _3 \\, b^{-2}\\epsilon $ uniformly over all $\\delta \\in \\mathbb {B}^p(r)$ .","Turning to $I_2(\\delta _j)$ , note that $_n(\\delta _j) - (\\delta _j ) = (1/n) \\sum _{i=1}^n(1-) \\phi _{ij} _i _i^, where $ ij = Gb(i - ij )$ satisfy $ |ij| Gu b-1$ and{\\begin{@align}{1}{-1}\\bigl ( \\phi _{ij}^2 | _i \\bigr ) = \\frac{1}{b^2} \\int _{-\\infty }^\\infty G^2 \\biggl ( \\frac{ \\langle _i , \\delta \\rangle - t}{b} \\biggr ) f_{\\varepsilon _i | _i } (t) \\, {\\rm d} t = \\frac{1}{b} \\int _{-\\infty }^\\infty G^2(u) f_{\\varepsilon _i | _i } (_i^- b u ) \\, {\\rm d} u \\le \\frac{ \\bar{f} m_{G^2 } }{b} \\nonumber \\end{@align}}almost surely, where $ mG2 := -G2(u) du <$.Given $ 2(0,1/2)$, there exits an $ 2$-net $$ of the sphere $ Sp-1$ with $ ||(1+2/2)p$ such that $ n($\\delta $ j) - ($\\delta $ j )2 (1-22)-1 | n($\\delta $ j) - ($\\delta $ j ) }|$.","Given $$ and $ k=2,3,...$, we bound the higher order moments of $ ij (i)2$ by{\\begin{@align}{1}{-1}|\\phi _{ij} (_i^)^2 |^k & \\le \\bar{f} m_{G^2 } b^{-1} \\cdot (\\kappa ^G_u b^{-1})^{k-2} \\upsilon _1^{2k} \\cdot 2k \\int _0^\\infty \\bigl ( |_i^| \\ge \\upsilon _1 u \\bigr ) u^{2k-1} {\\rm d} u \\nonumber \\\\& \\le \\bar{f} m_{G^2 } b^{-1} \\cdot ( \\kappa ^G_u b^{-1})^{k-2} \\upsilon _1^{2k} \\cdot 4k \\int _0^\\infty u^{2k-1} e^{-u^2/2} {\\rm d} u \\nonumber \\\\& \\le \\bar{f} m_{G^2} b^{-1} \\cdot ( \\kappa ^G_u b^{-1})^{k-2} \\upsilon _1^{2k} \\cdot 2^{k+1} k!",".\\nonumber \\end{@align}}In particular, $ ij2 (i)4 16 14 f mG2 b-1 $, and $ |ij (i)2 |kk!2 16 14 f mG2 b-1 (212Gu b-1 )k-2$ for $ k3$.","Applying Bernstein^{\\prime }s inequality and the union bound, we find that for any $ u0$,{\\begin{@align}{1}{-1}& \\Vert _n(\\delta _j) - (\\delta _j ) \\Vert _2 \\nonumber \\\\& \\le \\frac{1}{1-2\\epsilon _2} \\max _{\\in } \\Biggl | \\frac{1}{n} \\sum _{i=1}^n(1-) \\phi _{ij} (_i^)^2 \\Biggr | \\le \\frac{ 2\\upsilon _1^2}{1- 2\\epsilon _2} \\Biggl ( 2 \\sqrt{2 \\bar{f} m_{G^2 } \\frac{u}{n b }} +\\kappa ^G_u \\frac{u}{n b} \\Biggr ) \\nonumber \\end{@align}}with probability at least $ 1-2 (1+2/2)p e-u = 1 - (1/2) e(4) + p(1+2/2) - u$.","Setting $ 2=2/(e3-1)$ and $ u= (4) + 3p + v$, it follows that with probability at least $ 1 -e-v/2$,{\\begin{@align}{1}{-1}I_2(\\delta _j) \\lesssim \\sqrt{ \\frac{ p + v }{n b } } + \\frac{ p + v }{n b} .","\\nonumber \\end{@align}}Once again, taking the union bound over $ j=1,..., d$ and setting $ v= p (1 + 2r/)+ t$, we obtain that with probability at least $ 1-de-v 1-e-t/2$,{\\begin{@align}{1}{-1}\\max _{1\\le j\\le N_{\\epsilon }} I_2(\\delta _j) \\lesssim \\sqrt{\\frac{p\\log (3 e r/\\epsilon )+t}{n b }} + \\frac{p\\log (3e r/\\epsilon )+t}{n b} .","\\end{@align}}$ Finally, combining (REF ), (REF ) and (), and taking $\\epsilon = r/n \\in (0, r)$ in the beginning of the proof, we conclude that with probability at least $1- e^{-t}$ , $& \\sup _{\\beta \\in \\Theta ^*(r)}\\big \\Vert ^{-1/2} \\big \\lbrace \\nabla ^2 \\hat{Q}^G_b(\\beta ) - \\nabla ^2 Q^G_b(\\beta ) \\big \\rbrace \\nonumber ^{-1/2} \\big \\Vert _2 \\lesssim \\sqrt{\\frac{p \\log n + t}{n b }} + \\frac{p\\log n+t}{n b } + \\frac{(p + t)^{1/2} r }{n b^2} \\nonumber $ as long as $n\\gtrsim p+t$ .","This completes the proof.","$\\Box $"],["Proof of Proposition ","Define the stochastic process $\\Delta _b(\\beta ) = ^{-1/2} [ \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\rbrace - ( \\beta - \\beta ^*) ]$ .","By the triangle inequality, $\\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) \\Vert _2 \\le \\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) \\Vert _2 + \\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) - \\Delta _b(\\beta ) \\Vert _2$ Recall that $_0 = ^{-1/2} ^{-1/2} = \\lbrace f_{\\varepsilon |}(0) ^$ .","For the first term on the right-hand side, using the mean value theorem for vector-valued functions yields $\\Delta _b(\\beta ) = \\Biggl \\lbrace ^{-1/2} \\int _0^1 \\nabla ^2 Q^G_b\\bigl ( (1-s) \\beta ^* + s \\beta \\bigr ) \\, {\\rm d}s \\, ^{-1/2} - _0 \\Biggr \\rbrace ^{1/2} ( \\beta -\\beta ^* ) .$ By a change of variable $\\delta = ^{1/2}(\\beta - \\beta ^*)$ , $\\nabla ^2 Q^G_b\\bigl ( (1-s) \\beta ^* + s \\beta \\bigr ) = \\int _{-\\infty }^\\infty G(u) f_{\\varepsilon | } (s ^- b u ) \\, {\\rm d}u \\cdot ^$ For every $s \\in [0,1]$ and $u\\in $ , it ensures from $f_{\\varepsilon | }(\\cdot )$ being Lipschitz that $| f_{\\varepsilon | } (s ^- b u ) - f_{\\varepsilon | } ( - b u ) | \\le l_0 s \\cdot | ^| .$ Moreover, by the Taylor expansion (REF ), $f_{\\varepsilon | } ( - b u ) & = f_{\\varepsilon | }(0) + \\sum _{\\ell =1}^{\\nu -1} f_{\\varepsilon | }^{(\\ell )}(0) \\frac{( - b u )^\\ell }{\\ell !}","+ \\frac{( - b u)^{\\nu -1}}{(\\nu -2)!}","\\int _0^1 (1-w)^{\\nu -2} \\bigl \\lbrace f_{\\varepsilon | }^{(\\nu -1)}( - b u w ) -f_{\\varepsilon | }^{(\\nu -1)}(0)\\bigr \\rbrace \\, {\\rm d}w.$ Consequently, $& \\Bigg \\Vert ^{-1/2} \\int _0^1 \\nabla ^2 Q^G_b\\bigl ( (1-s) \\beta ^* + s\\beta \\bigr )\\, {\\rm d}s \\, ^{-1/2} - _0 \\Bigg \\Vert _2 \\\\&\\le \\Bigg \\Vert \\frac{b^{\\nu -1}}{(\\nu -2)!}","\\int _{-\\infty }^\\infty \\int _0^1 u^{\\nu -1} G(u) (1-w)^{\\nu -2} \\bigl \\lbrace f_{\\varepsilon | }^{(\\nu -1)}( - b u w ) -f_{\\varepsilon | }^{(\\nu -1)}(0)\\bigr \\rbrace \\, {\\rm d}w {\\rm d}u \\cdot ^\\Vert _2 \\\\&~~~~ + \\frac{l_0}{2} \\cdot \\big \\Vert \\, |^| \\cdot ^\\Vert _2 \\\\& \\le \\frac{C_G b^{\\nu -1}}{(\\nu -1)!}","\\sup _{\\in \\mathbb {S}^{p-1}} \\big \\langle , \\big \\rangle ^2 + \\frac{l_0}{2} \\sup _{\\in \\mathbb {S}^{p-1}} \\big ( | ^| \\langle , \\rangle ^2 \\big ) \\\\& \\le \\Bigg ( \\frac{ C_G }{(\\nu -1)! }","b^{\\nu -1} + 0.5 l_0 \\mu _3\\Vert \\delta \\Vert _2 \\Bigg ) .$ Taking the supremum over $\\beta \\in \\Theta ^*(r)$ , or equivalently $\\delta \\in \\mathbb {B}^p(r)$ , yields $\\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) \\Vert \\le \\Bigg ( \\frac{ C_G }{(\\nu -1)! }","b^{\\nu -1} + 0.5 l_0 \\mu _3 r \\Bigg ) r \\lesssim (b^{\\nu -1} + r ) r .","\\nonumber $ For the stochastic term $ \\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) - \\Delta _b(\\beta ) \\Vert _2$ , following the proof of Theorem REF , it can be similarly shown that with probability at least $1-e^{-t}$ , $\\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) - \\Delta _b(\\beta ) \\Vert _2 \\lesssim r \\sqrt{\\frac{p+t}{n b}} \\nonumber $ as long as $\\sqrt{(p+t)/n} \\lesssim b\\lesssim 1$ .","Combining the last two displays completes the proof of the claim (REF ).","$\\Box $"],["Proof of Theorem ","Step 1 (Consistency of the initial estimate).","First, note that the consistency of the initial estimator $\\beta $ —namely, $\\beta $ lies in a local neighborhood of $\\beta ^*$ with high probability, is a direct consequence of Theorem REF .","Given a non-negative kernel $K(\\cdot )$ and for any $t>0$ , the ensuing estimator $\\beta $ satisfies $\\Vert \\beta - \\beta ^* \\Vert _{} \\le r_n \\asymp \\sqrt{\\frac{p+t}{n}} $ with probability at least $1-2e^{-t}$ as long as $ (\\frac{p+t}{n})^{1/2} \\lesssim h \\lesssim (\\frac{p+t}{n})^{1/4}$ .","Provided that the sample Hessian $\\nabla ^2 \\hat{Q}^G_b(\\beta )$ is invertible, by the definition of $\\hat{\\beta }$ we obtain $^{1/2} ( \\hat{\\beta }- \\beta ) & = - \\bigl \\lbrace ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} \\bigr \\rbrace ^{-1} ^{-1/2}\\nabla \\hat{Q}^G_b(\\beta ) \\nonumber \\\\& = - \\bigl \\lbrace ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} \\bigr \\rbrace ^{-1} \\bigl [ ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\bigr \\rbrace - ^{-1/2} (\\beta - \\beta ^* ) \\bigr ] \\nonumber \\\\& ~~~~~ - \\bigl \\lbrace ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} \\bigr \\rbrace ^{-1} \\bigl \\lbrace ^{-1/2} (\\beta - \\beta ^* ) + ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* )\\bigr \\rbrace ,\\nonumber $ or equivalently, $^{1/2} (\\hat{\\beta }- \\beta ^*) & = ^{1/2} (\\beta -\\beta ^* ) - _0^{-1} _0 ^{1/2} (\\beta - \\beta ^* ) - _0^{-1} ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* ) \\nonumber \\\\& ~~~~ - _0^{-1} ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) - (\\beta - \\beta ^* ) \\bigr \\rbrace , \\nonumber $ where $_0 = \\lbrace f_{\\varepsilon |}(0) ^ = ^{-1/2} ^{-1/2}$ and $_0 := ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} $ .","It follows that $& \\bigl \\Vert ^{1/2} ( \\hat{\\beta }- \\beta ^* ) + ^{1/2} ^{-1} \\nabla \\hat{}_b^G(\\beta ^* ) \\bigr \\Vert _2= \\bigl \\Vert ^{1/2} (\\hat{\\beta }- \\beta ^*) + _0^{-1} ^{-1/2}\\nabla \\hat{}_b^G(\\beta ^* ) \\bigr \\Vert _2 \\nonumber \\\\& \\le \\underbrace{ \\Vert _p - _0^{-1} _0 \\Vert _2 }_{ {\\rm see~Proposition~\\ref {prop:Hessian.uniform}} }\\cdot \\bigl \\lbrace \\Vert \\beta -\\beta ^* \\Vert _{} + \\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* ) \\Vert _2 \\bigr \\rbrace \\\\&~~~~~ + \\Vert _0^{-1} \\Vert _2 \\cdot \\underbrace{ \\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) - (\\beta - \\beta ^* ) \\bigr \\rbrace \\big \\Vert _2 }_{ {\\rm see~Proposition~\\ref {prop:gradient.uniform}}}.","\\nonumber $ According to the above bound, we need to further control the following three terms: $\\Vert _p - _0^{-1} _0 \\Vert _2 , ~~\\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* ) \\Vert _2~\\mbox{ and }~\\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) -(\\beta - \\beta ^* ) \\bigr \\rbrace \\big \\Vert _2.$ Step 2 (Invertibility of the sample Hessian $\\nabla ^2 \\hat{Q}^G_b(\\beta )$ ).","Recall that $_0 = ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} $ and $_0 = ^{-1/2} ^{-1/2}$ .","By the triangle inequality, $& \\Vert _0 - _0 \\Vert _2 \\le \\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla ^2 \\hat{Q}^G_b(\\beta ) - \\nabla ^2 Q^G_b( \\beta ^*) \\bigr \\rbrace ^{-1/2} \\big \\Vert _2 + \\big \\Vert ^{-1/2} \\nabla ^2 Q^G_b( \\beta ^*) ^{-1/2} - _0 \\big \\Vert _2 .","\\nonumber $ Let the bandwidth $b$ satisfy $ \\max \\lbrace \\frac{p\\log n + t}{n} , n^{-1/2}\\rbrace \\lesssim b \\lesssim 1$ .","Then, applying Propositions and with $r=r_n$ yields that with probability $1-3e^{-t}$ , $& \\Vert _0 - _0 \\Vert _2 \\le \\delta _n \\asymp \\sqrt{\\frac{p\\log n + t}{ nb }} + b^{\\nu -1} .","\\nonumber $ Note that, under Condition REF , $0< {f} \\le \\lambda _{\\min }(_0) \\le \\lambda _{\\max }(_0) \\le \\bar{f}$ .","For sufficiently large $n$ , this implies $\\Vert _0^{-1} _0 - _p \\Vert _2 \\le {f}^{-1} \\delta _n < 1$ , and hence $\\Vert _0^{-1} _0 - _p \\Vert _2 \\le \\frac{\\delta _n}{ {f} - \\delta _n } ~~\\mbox{ and }~~ \\Vert _0^{-1} \\Vert _2 \\le \\frac{1}{{f} - \\delta _n} .$ Step 3 (Controlling the score).","In view of (REF ), it remains to control $\\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* )\\Vert _2$ and $\\Vert ^{-1/2} \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\rbrace - ^{-1/2} (\\beta - \\beta ^* ) \\Vert _2$ .","For the latter, applying the concentration bound (REF ) and Proposition yields that, with probability at least $1-3e^{-t}$ , $\\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\bigr \\rbrace - ^{-1/2} (\\beta - \\beta ^* ) \\big \\Vert _2 \\lesssim \\frac{p+ t}{n b^{1/2}} + b^{\\nu -1} \\sqrt{\\frac{p+t}{n}} $ as long as $ (\\frac{p+t}{n})^{1/2} \\lesssim b \\lesssim 1$ .","Turning to $\\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* )\\Vert _2$ , it follows from Lemma REF and Proposition that with probability at least $1-e^{-t}$ , $\\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* ) \\Vert _2 \\lesssim \\sqrt{\\frac{p+ t}{n}} + b^\\nu .","$ Finally, combining the bounds (REF )–(REF ), we conclude that with probability at least $1-5e^{-t}$ , $\\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\bigr \\rbrace - ^{-1/2} (\\beta - \\beta ^* ) \\big \\Vert _2\\lesssim \\Biggl ( \\sqrt{\\frac{p\\log n+ t}{n b}}+ b^{\\nu -1} \\Biggr ) \\Bigg ( \\sqrt{\\frac{p+t}{n}} + b^\\nu \\Bigg ) , \\nonumber $ provided that $ \\max \\bigl \\lbrace \\frac{p\\log n + t}{n} , (\\frac{p + t}{n})^{1/2} \\bigr \\rbrace \\lesssim b \\lesssim 1$ .","Under the sample size requirement $n\\gtrsim p \\,(\\log n)^2+t$ and additional constraint $b \\lesssim (\\frac{p+t}{n})^{1/(2\\nu )}$ , this leads to the claimed bound (REF ).","$\\Box $"],["Additional Simulation Results","In this section, we present additional results for the numerical studies as described in Section in Figures REF –REF ."]],"string":"[\n [\n \"Smoothed Quantile Regression with Large-Scale Inference\"\n ],\n [\n \"Abstract Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects.\",\n \"In this paper, we consider statistical inference for quantile regression with large-scale data in the \\\"increasing dimension\\\" regime.\",\n \"We provide a comprehensive and in-depth analysis of a convolution-type smoothing approach that achieves adequate approximation to computation and inference for quantile regression.\",\n \"This method, which we refer to as {\\\\it{conquer}}, turns the non-differentiable quantile loss function into a twice-differentiable, convex and locally strongly convex surrogate, which admits a fast and scalable Barzilai-Borwein gradient-based algorithm to perform optimization, and multiplier bootstrap for statistical inference.\",\n \"Theoretically, we establish explicit non-asymptotic bounds on both estimation and Bahadur-Kiefer linearization errors, from which we show that the asymptotic normality of the conquer estimator holds under a weaker requirement on the number of the regressors than needed for conventional quantile regression.\",\n \"Moreover, we prove the validity of multiplier bootstrap confidence constructions.\",\n \"Our numerical studies confirm the conquer estimator as a practical and reliable approach to large-scale inference for quantile regression.\",\n \"Software implementing the methodology is available in the \\\\texttt{R} package \\\\texttt{conquer}.\"\n ],\n [\n \"Introduction\",\n \"Quantile regression (QR) is a useful statistical tool for modeling and inferring the relationship between a scalar response $y$ and a $p$ -dimensional predictor $$ [35].\",\n \"Compared to the least squares regression that focuses on modeling the conditional mean of $y$ given $$ , QR allows modeling of the entire conditional distribution of $y$ given $$ , and thus provides valuable insights into heterogeneity in the relationship between $$ and $y$ .\",\n \"Moreover, quantile regression is robust against outliers and can be performed for skewed or heavy-tailed response distributions without a correct specification of the likelihood.\",\n \"These advantages make QR an appealing method to explore data features that are invisible to the least squares regression.\",\n \"We refer the reader to [33] and [36] for an extensive overview of QR in terms of methods, theory, computation, and various extensions under complex data structures.\",\n \"Quantile regression involves a convex optimization problem with a piecewise linear loss function, also known as the check function and pinball loss.\",\n \"One can formulate QR as a linear programming problem, solvable by the interior point method with a computational complexity of $(n^{1+\\\\alpha } p^3 \\\\log n)$ with $\\\\alpha \\\\in (0, 1/2)$ , where $n$ is the sample size and $p$ is the parametric dimension.\",\n \"When $n$ is large relative to $p$ , an efficient algorithm based on pre-processing has an improved complexity of $\\\\lbrace (np)^{2(1+\\\\alpha )/3} p^3 \\\\log n + np\\\\rbrace $ , and thus can be more efficient than solving a least squares problem when $p$ is small [48].\",\n \"However, primarily due to the non-differentiability and lack of strong convexity of the loss function, QR remains computationally expensive for large-scale data when both $n$ and $p$ are large.\",\n \"We refer to Chapter 5 of [36] for an overview of prevailing computational methods for quantile regression, such as simplex-based algorithms [4], [37], interior point methods [48], and alternating direction method of multipliers among other first-order proximal methods [44].\",\n \"We consider conducting large-scale inference for quantile regression under the setting in which $p$ is large and $n$ is even larger.\",\n \"Two general principles have been widely used to suit this purpose.\",\n \"The first uses a nonparametric estimate of the asymptotic variance [23] that involves the conditional density of the response given the covariates, yet such an estimate can be fairly unstable.\",\n \"Even if the asymptotic variance is well estimated, its approximation accuracy to the finite-sample variance depends on the design matrix and the quantile level.\",\n \"Resampling methods, on the other hand, provide a more reliable approach to inference for QR under a wide variety of settings [45], [25], [31], [19].\",\n \"Inevitably, the resampling approach requires repeatedly computing QR estimates up to thousands of times, and therefore is unduly expensive for large-scale data.\",\n \"Theoretically, valid statistical inference is often justified by asymptotic normal approximations to QR estimators.\",\n \"The Bahadur-Kiefer representation of the nonlinear QR estimators are essential to this end, as shown in [2] and [24].\",\n \"In large-$p$ (non)asymptotic settings in which the parametric dimension $p$ may tend to infinity with the sample size, we refer to [54], [26], and [43] for normal approximation results of the QR estimators under fixed and random designs.\",\n \"The question of how large $p$ can be relative to $n$ to ensure asymptotic normality has been addressed by those authors.\",\n \"It is now recognized that we may have to pay a price here as compared to $M$ -estimators with smooth loss functions that are at least continuously twice differentiable.\",\n \"To circumvent the non-differentiability of the QR loss function, [27] proposed to smooth the indicator part of the check function via the survival function of a kernel.\",\n \"This smoothing method, which we refer to as Horowitz's smoothing throughout, has been widely used for various QR-related problems [55], [30], [21], [52], [56], [14], [10].\",\n \"However, Horowitz's smoothing gains smoothness at the cost of convexity, which inevitably raises optimization issues.\",\n \"In general, computing a global minimum of a non-convex function is intractable: finding an $\\\\epsilon $ -suboptimal point for a $k$ -times continuously differentiable function $f:\\\\mathbb {R}^p \\\\rightarrow $ requires at least as many as $(1/\\\\epsilon )^{p/k}$ evaluations of the function and its first $k$ derivatives [42].\",\n \"As we shall see from the numerical studies in Section , the convergence of gradient-based algorithms can be relatively slow for high and low quantile levels.\",\n \"To address the aforementioned issue, [20] proposed a convolution-type smoothing method that yields a convex and twice differentiable loss function, and studied the asymptotic properties of the smoothed estimator when $p$ is fixed.\",\n \"To distinguish this approach from Horowitz's smoothing, we adopt the term conquer for convolution-type smoothed quantile regression.\",\n \"In this paper, we first provide an in-depth statistical analysis of conquer under various array (non)asymptotic settings in which $p$ increases with $n$ .\",\n \"Our results reveal a key feature of the smoothing parameter, often referred to as the bandwidth: the bandwidth adapts to both the sample size $n$ and dimensionality $p$ , so as to achieve a tradeoff between statistical accuracy and computational stability.\",\n \"Since the convolution smoothed loss function is globally convex and locally strongly convex, we propose an efficient gradient descent algorithm with the Barzilai-Borwein stepsize and a Huber-type initialization.\",\n \"The proposed algorithm is implemented via RcppArmadillo [17] in the R package conquer.\",\n \"We next focus on large-scale statistical inference (hypothesis testing and confidence estimation) with large $p$ and large $n$ .\",\n \"We propose a bootstrapped conquer method that has reduced computational complexity when the conquer estimator is used as initialization.\",\n \"Under appropriate restrictions on dimension, we establish the consistency (or concentration), Bahadur representation, asymptotic normality of the conquer estimator as well as the validity of the bootstrap approximation.\",\n \"In the following, we provide more details on the computational and statistical contributions of this paper.\",\n \"Theoretically, by allowing $p$ to grow with $n$ , the `complexity' of the function classes that we come across in the analysis also increases with $n$ .\",\n \"Conventional asymptotic tools for proving the bootstrap validity are based on weak convergence arguments [50], which are not directly applicable in the finite-sample setting.\",\n \"In this paper we turn to a more refined and self-contained analysis, and prove a new local restricted strong convexity (RSC) property for the empirical smoothed quantile loss.\",\n \"This validates the key merit of convolution-type smoothing, i.e., local strong convexity.\",\n \"The smoothing method involves a bandwidth, denoted by $h$ .\",\n \"Theoretically, we show that with sub-exponential random covariates (relaxing the bounded covariates assumption in [20]), conquer exhibits an $\\\\ell _2$ -error in the order of $\\\\sqrt{(p+t)/n}+h^2$ with probability at least $1-2e^{-t}$ .\",\n \"When $h$ is of order $\\\\lbrace (p+t)/n \\\\rbrace ^{\\\\gamma }$ for any $\\\\gamma \\\\in [1/4, 1/2]$ , the conquer estimation is first-order equivalent to QR.\",\n \"Under slightly more stringent sub-Gaussian condition on the covariates, we show that the Bahadur-Kiefer linearization error of conquer is of order $(p+t)/(nh^{1/2})+ h \\\\sqrt{(p+t)/n}+h^3$ with probability at least $1-3e^{-t}$ .\",\n \"Based on such a representation, we establish a Berry-Esseen bound for linear functionals of conquer, which lays the theoretical foundation for testing general linear hypotheses, encompassing covariate-effect analysis, analysis of variance, and model comparisons, to name a few.\",\n \"It is worth noting that with a properly chosen $h$ , the linear functional of conquer is asymptotically normal as long as $p^{8/3}/n\\\\rightarrow 0$ , which improves the best known growth condition on $p$ for standard QR [54], [26], [43].\",\n \"We attribute this gain to the effect of smoothing.\",\n \"Under similar conditions, we further establish upper bounds on both estimation and Bahadur-Kiefer linearization errors for the bootstrapped conquer estimator.\",\n \"In the context of nonparametric density or regression estimation, it is known that when higher-order kernels are used (and if the density or regression function has enough derivatives), the bias is proportional to $h^\\\\nu $ for some $\\\\nu \\\\ge 4$ which is of better order than $h^2$ .\",\n \"While a higher-order kernel has negative parts, the resulting smoothed loss is non-convex and thus brings the computational issue once again.\",\n \"Motivated by the two-stage procedure proposed by [7] whose original idea is to improve an initial estimator that is already consistent but not efficient, we further propose a one-step conquer estimator using higher-order kernels but without the need for solving a large-scale non-convex optimization.\",\n \"With increasing degrees of smoothness, the one-step conquer is asymptotically normal under a milder dimension constraint of roughly $p^2/n \\\\rightarrow 0$ .\",\n \"To better appreciate the computational feasibility of conquer for large-scale problems, we compare it with standard QR on large synthetic datasets, where the latter is implemented by the R package quantreg [34] using the Frisch-Newton approach after preprocessing “pfn”.\",\n \"We generate independent data vectors $\\\\lbrace y_i, _i\\\\rbrace _{i = 1}^n$ from a linear model $y_i = \\\\beta ^*_0 + \\\\langle _i , \\\\beta ^* \\\\rangle + \\\\varepsilon _i$ , where $(\\\\beta ^*_0, {\\\\beta ^*}^^ (1, \\\\ldots , 1)^^{p+1}$ , $_i \\\\sim \\\\mathcal {N}_p(0, )$ , and the independent errors $\\\\varepsilon _i \\\\sim t_2$ , for $i = 1, 2, \\\\dots , n$ .\",\n \"We report the estimation error and elapsed time for increasing sample sizes $n\\\\in \\\\lbrace 1000,5000,10000,\\\\ldots ,100000\\\\rbrace $ and dimension $p = \\\\lfloor n^{1/2} \\\\rfloor $ , the largest integer that is less than or equal to $n^{1/2}$ .\",\n \"Figure REF displays the average estimation error and average elapsed time based on 20 Monte Carlo samples.\",\n \"This experiment shows promise of conquer as a practically useful tool for large-scale quantile regression analysis.\",\n \"More empirical evidence will be given in the latter section.\",\n \"Figure: RuntimeThe rest of the paper is organized as follows.\",\n \"We start with a brief review of linear quantile regression and the convolution-type smoothing method in Section .\",\n \"Explicit forms of the smoothed check functions are provided for several representative kernel functions in nonparametric statistics.\",\n \"We introduce the multiplier bootstrap for statistical inference in Section REF .\",\n \"In Section , we provide a comprehensive theoretical study of conquer from a nonasymptotic viewpoint, which directly leads to array asymptotic results.\",\n \"Specifically, the bias incurred by smoothing the quantile loss is characterized in Section REF .\",\n \"In Section REF , we establish the rate of convergence, Bahadur-Kiefer representation, and Berry-Esseen bound for conquer in a large-$p$ and larger-$n$ regime.\",\n \"Results for its bootstrap counterpart are provided in Section REF .\",\n \"A Barzilai-Borwein gradient-based algorithm with a Huber-type warm start is detailed in Section .\",\n \"We conclude the paper with an extensive numerical study in Section to illustrate the finite-sample performance of conquer in large-scale quantile regression analysis.\",\n \"We defer the proofs of all theoretical results as well as the full details of the one-step conquer to online supplementary materials.\",\n \"Notation: For every integer $k\\\\ge 1$ , we use $^k$ to denote the the $k$ -dimensional Euclidean space.\",\n \"The inner product of any two vectors $=(u_1, \\\\ldots , u_k)^ =(v_1, \\\\ldots ,v_k)^^k$ is defined by $^= \\\\langle , \\\\rangle = \\\\sum _{i=1}^k u_i v_i$ .\",\n \"We use $\\\\Vert \\\\cdot \\\\Vert _p$ $(1\\\\le p \\\\le \\\\infty )$ to denote the $\\\\ell _p$ -norm in $^k$ : $\\\\Vert \\\\Vert _p = ( \\\\sum _{i=1}^k | u_i |^p )^{1/p}$ and $\\\\Vert \\\\Vert _\\\\infty = \\\\max _{1\\\\le i\\\\le k} |u_i|$ .\",\n \"Throughout this paper, we use bold capital letters to represent matrices.\",\n \"For $k\\\\ge 2$ , $_k$ represents the identity matrix of size $k$ .\",\n \"For any $k\\\\times k$ symmetric matrix $\\\\in ^{k\\\\times k}$ , $\\\\Vert \\\\Vert _2$ denotes the operator norm of $$ .\",\n \"If $$ is positive semidefinite, we use $\\\\Vert \\\\cdot \\\\Vert _{}$ to denote the vector norm linked to $$ given by $\\\\Vert \\\\Vert _{} = \\\\Vert ^{1/2} \\\\Vert _2$ , $\\\\in ^k$ .\",\n \"For $r \\\\ge 0$ , define the Euclidean ball and sphere in $^k$ as $\\\\mathbb {B}^k(r) = \\\\lbrace \\\\in ^k : \\\\Vert \\\\Vert _2 \\\\le r\\\\rbrace $ and $\\\\mathbb {S}^{k-1}(r) = \\\\partial \\\\mathbb {B}^k(r) = \\\\lbrace \\\\in ^k: \\\\Vert \\\\Vert _2 =r \\\\rbrace $ , respectively.\",\n \"For two sequences of non-negative numbers $\\\\lbrace a_n \\\\rbrace _{n\\\\ge 1}$ and $\\\\lbrace b_n \\\\rbrace _{n\\\\ge 1}$ , $a_n \\\\lesssim b_n$ indicates that there exists a constant $C>0$ independent of $n$ such that $a_n \\\\le Cb_n$ ; $a_n \\\\gtrsim b_n$ is equivalent to $b_n \\\\lesssim a_n$ ; $a_n \\\\asymp b_n$ is equivalent to $a_n \\\\lesssim b_n$ and $b_n \\\\lesssim a_n$ .\"\n ],\n [\n \"The linear quantile regression model\",\n \"Given a univariate response variable $y\\\\in $ and a $p$ -dimensional covariate vector $= (x_1, \\\\ldots , x_p)^^p$ with $x_1 \\\\equiv 1$ , the primary goal here is to learn the effect of $$ on the distribution of $y$ .\",\n \"Let $F_{y|}(\\\\cdot )$ be the conditional distribution function of $y$ given $$ .\",\n \"The dependence between $y$ and $$ is then fully characterized by the conditional quantile functions of $y$ given $$ , denoted as $F^{-1}_{y|}(\\\\tau )$ , for $0<\\\\tau <1$ .\",\n \"We consider a linear quantile regression model at a given $\\\\tau \\\\in (0,1)$ , that is, the $\\\\tau $ -th conditional quantile function is $F^{-1}_{y|} (\\\\tau ) = \\\\langle , \\\\beta ^*(\\\\tau ) \\\\rangle ,$ where $\\\\beta ^*(\\\\tau ) = (\\\\beta ^*_{1}(\\\\tau ) , \\\\ldots , \\\\beta ^*_{ p}(\\\\tau ) )^^p$ is the true quantile regression coefficient.\",\n \"Let $\\\\lbrace (y_i, _i) \\\\rbrace _{i=1}^n$ be a random sample from $(y,)$ .\",\n \"The standard quantile regression estimator [35] is then given as $ \\\\hat{\\\\beta }(\\\\tau ) \\\\in \\\\min _{\\\\beta \\\\in ^p } \\\\hat{Q} (\\\\beta ) = \\\\min _{\\\\beta \\\\in ^p } \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\rho _\\\\tau ( y_i - \\\\langle _i , \\\\beta \\\\rangle ) ,$ where $ \\\\rho _\\\\tau (u) = u \\\\lbrace \\\\tau - \\\\mathbb {1}(u<0)\\\\rbrace $ is the $\\\\tau $ -quantile loss function, also referred to as the pinball loss and check function.\",\n \"Statistical properties of $\\\\hat{\\\\beta }(\\\\tau )$ have been extensively studied.\",\n \"We refer the reader to [33] and [36] for more details.\"\n ],\n [\n \"Convolution-type smoothing\",\n \"We consider the convolution-type smoothing approach proposed by [20].\",\n \"For every $\\\\beta \\\\in ^p$ , let $\\\\hat{F}(\\\\cdot ;\\\\beta )$ be the empirical distribution function of the residuals $\\\\lbrace r_i(\\\\beta ) := y_i - \\\\langle _i,\\\\beta \\\\rangle \\\\rbrace _{i=1}^n$ , i.e., $\\\\hat{F}(t ;\\\\beta ) = (1/n) \\\\sum _{i=1}^n\\\\mathbb {1}\\\\lbrace r_i(\\\\beta ) \\\\le t \\\\rbrace $ .\",\n \"Given a fixed quantile level $\\\\tau $ , the empirical quantile loss $\\\\hat{Q}(\\\\cdot )$ in (REF ) can be represented as $\\\\hat{Q}(\\\\beta ) = \\\\int _{-\\\\infty }^\\\\infty \\\\rho _\\\\tau (u) \\\\,{\\\\rm d} \\\\hat{F}(u;\\\\beta ).\",\n \"$ Given a bandwidth $h=h_n >0$ , let $\\\\hat{F}_h(\\\\cdot ;\\\\beta )$ be the distribution function of the classical Rosenblatt–Parzen kernel density estimator: $\\\\hat{F}_h(u;\\\\beta ) = \\\\int _{-\\\\infty }^u \\\\hat{f}_h(t;\\\\beta ) \\\\,{\\\\rm d} t~~\\\\mbox{ with }~~ \\\\hat{f}_h(t;\\\\beta ) = \\\\frac{1}{nh} \\\\sum _{i=1}^nK\\\\bigg ( \\\\frac{t-r_i(\\\\beta )}{h} \\\\bigg ) , $ where $K:\\\\mathbb {R} \\\\rightarrow ^+ := \\\\mathbb {[}0,\\\\infty )$ is a symmetric kernel that integrates to one.\",\n \"Replacing the empirical distribution function in (REF ) with $\\\\hat{F}_h $ yields the following smoothed empirical quantile loss $\\\\hat{Q}_h(\\\\beta ) := \\\\int _{-\\\\infty }^\\\\infty \\\\rho _\\\\tau (u) \\\\,{\\\\rm d} \\\\hat{F}_h(u;\\\\beta ) = \\\\frac{1}{nh} \\\\sum _{i=1}^n\\\\int _{-\\\\infty }^\\\\infty \\\\rho _\\\\tau (u) K \\\\bigg ( \\\\frac{u + \\\\langle _i, \\\\beta \\\\rangle - y_i }{h} \\\\bigg ) \\\\,{\\\\rm d} u .\",\n \"$ Accordingly, the smoothed QR estimator is given by $ \\\\hat{\\\\beta }_h = \\\\hat{\\\\beta }_h(\\\\tau ) \\\\in _{\\\\beta \\\\in ^p} \\\\hat{Q}_h(\\\\beta ) .$ As we shall see later, the ideal choice of bandwidth should adapt to the sample size $n$ and dimension $p$ , while the quantile level $\\\\tau $ is prespecified and fixed.\",\n \"Thus, the dependence of $\\\\hat{\\\\beta }_h$ and $\\\\hat{Q}_h(\\\\cdot )$ on $\\\\tau $ will be assumed without display.\",\n \"Given a kernel function $K(\\\\cdot )$ and bandwidth $h>0$ , the smoothed quantile loss $\\\\hat{Q}_h(\\\\cdot )$ defined in (REF ) can be equivalently written as $\\\\hat{Q}_h(\\\\beta ) = (1/n) \\\\sum _{i=1}^n\\\\ell _h(y_i - \\\\langle _i, \\\\beta \\\\rangle )$ , where $\\\\ell _h(u) = (\\\\rho _\\\\tau * K_h )(u) = \\\\int _{-\\\\infty }^{\\\\infty } \\\\rho _\\\\tau (v) K_h(v- u ) \\\\, {\\\\rm d} v $ with $*$ denoting the convolution operator and $K_h(u)=(1/h)K(u/h)$ .\",\n \"Therefore, as stated in the Introduction, we refer to the aforementioned smoothing method as conquer throughout the paper.\",\n \"Commonly used kernel functions include: (a) uniform kernel $K(u) = (1/2) \\\\mathbb {1} (|u|\\\\le 1)$ , (b) Gaussian kernel $K(u) = \\\\phi (u):= (2\\\\pi )^{-1/2} e^{-u^2/2}$ , (c) logistic kernel $K(u) = e^{-u}/(1+ e^{-u})^2 $ , (d) Epanechnikov kernel $K(u) = (3 / 4) (1 - u^2) \\\\mathbb {1}(|u|\\\\le 1)$ , and (e) triangular kernel $K(u) = (1 - |u|) \\\\mathbb {1}(|u|\\\\le 1)$ .\",\n \"Explicit expressions of the corresponding smoothed loss function $\\\\rho _\\\\tau * K_h$ will be given in Section .\",\n \"The convolution-type kernel smoothing yields an objective function $\\\\beta \\\\mapsto \\\\hat{Q}_{h}(\\\\beta )$ that is twice continuously differentiable with gradient and hessian matrix $\\\\nabla \\\\hat{Q}_{h}(\\\\beta ) = \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\bigl \\\\lbrace {K}\\\\bigl ( -r_i(\\\\beta )/h \\\\bigr ) - \\\\tau \\\\bigr \\\\rbrace _i ~~\\\\mbox{ and }~~\\\\nabla ^2 \\\\hat{Q}_{h}(\\\\beta ) = \\\\frac{1}{n} \\\\sum _{i=1}^nK_h( r_i(\\\\beta )) _i _i^ $ respectively, where $K(u) := \\\\int _{-\\\\infty }^u K(t) \\\\,{\\\\rm d} t$ .\",\n \"Provided that $K$ is non-negative, $\\\\hat{Q}_{h}(\\\\cdot )$ is a convex function for any $h>0$ , and $\\\\hat{\\\\beta }_h = \\\\hat{\\\\beta }_h(\\\\tau ) $ satisfies the first-order condition $\\\\nabla \\\\hat{Q}_{ h}(\\\\hat{\\\\beta }_h ) = \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\Biggl \\\\lbrace {K}\\\\Biggl ( \\\\frac{\\\\langle _i , \\\\hat{\\\\beta }_h \\\\rangle - y_i }{h} \\\\Biggr ) - \\\\tau \\\\Biggr \\\\rbrace _i = \\\\textbf {0} .\\\\nonumber $ The smoothness and convexity of $\\\\hat{Q}_{h}$ warrant the superior computation efficiency of first-order gradient based algorithms for solving large-scale smoothed quantile regressions.\",\n \"Theoretically, with $p$ fixed and $n \\\\rightarrow \\\\infty $ , [20] showed that the convolution-type smoothed QR estimator has a lower asymptotic mean squared error than Horowitz's smoothed estimator, and also has a smaller Bahadur linearization error than the standard QR in the almost sure sense.\",\n \"The optimal order of the bandwidth based on the asymptotic mean squared error is unveiled as a function of $n$ .\",\n \"Our theoretical results complement the results in [20] by providing a comprehensive analysis of conquer in the regime in which $p$ is allowed to increase with $n$ .\"\n ],\n [\n \"Connection to related existing work\",\n \"The idea of smoothing the piecewise linear loss, to the best of our knowledge, dates back to [1] in the context of median regression.\",\n \"Specifically, [1] proposed a smoothed approximation of the absolute value function that has the form $u \\\\mapsto u + 2 h \\\\log (1 + e^{-u / h}) , \\\\ \\\\ u\\\\in ,$ where $h= n^{-\\\\gamma }$ with $ \\\\gamma \\\\in (1/3, 1/2)$ .\",\n \"This smoothing device, however, is mainly used therein to simplify the analysis of the asymptotic behavior of a two-stage median regression estimator.\",\n \"[27] proposed a kernel smoothing method of similar type for bootstrapping the median regression estimator.\",\n \"The idea is to replace the indicator function in $\\\\rho _\\\\tau (u) = u \\\\lbrace \\\\tau - \\\\mathbb {1}(u<0) \\\\rbrace $ with a smoothed counterpart, leading to $ \\\\ell ^{\\\\mathrm {Horo}}_{ h}(u) = u \\\\bigl \\\\lbrace \\\\tau - {K}(-u / h) \\\\bigr \\\\rbrace ,$ where $K(u) = \\\\int _{-\\\\infty }^u K(t) \\\\,{\\\\rm d} t$ , and $K(\\\\cdot )$ is a symmetric kernel function.\",\n \"The key difference between the conquer loss (REF ) and Horowitz's loss (REF ) is that the former is globally convex, while Horowitz's loss is not.\",\n \"This is illustrated in Figure REF .\",\n \"We refer to [20] for an in-depth comparison between convolution smoothing and Horowitz's smoothing in terms of asymptotic mean squared errors of the smoothed estimators.\",\n \"A closer inspection reveals that the smoothed function considered by [1] is a convolution-type smoothed loss with a logistic kernel.\",\n \"More recently, [57] considered a smoothing approximation for the quantile loss, which they refer to as the Huber approximation.\",\n \"Their goal was to compute the regularized QR estimator but via a smoothed optimization method.\",\n \"The corresponding loss function also falls into the general framework of convolution-type smoothing, with $K$ taken as the uniform kernel.\",\n \"The general statistical theory developed in this paper applies to those special cases.\",\n \"Figure: Uniform kernel under τ=0.7\\\\tau = 0.7.\"\n ],\n [\n \"Multiplier bootstrap inference\",\n \"In this section, we consider a multiplier bootstrap procedure to construct confidence intervals for conquer.\",\n \"Independent of the observed sample $\\\\mathcal {X}_n = \\\\lbrace (y_i,_i) \\\\rbrace _{i=1}^n$ , let $\\\\lbrace w_i\\\\rbrace _{i = 1}^n$ be independent and identically distributed random variables with $(w_i)=1$ and $\\\\textnormal {var}(w_i)=1$ .\",\n \"Recall that $\\\\hat{\\\\beta }_h = \\\\hat{\\\\beta }_h(\\\\tau ) = \\\\min _{\\\\beta \\\\in ^p} \\\\hat{Q}_{h}(\\\\beta )$ is the conquer estimator.\",\n \"If the minimizer is not unique, we take any of the minima as $\\\\hat{\\\\beta }_h = (\\\\hat{\\\\beta }_{h,1},\\\\ldots , \\\\hat{\\\\beta }_{h,p})^.$ The proposed bootstrap method, which dates back to [16] and [3], is based on reweighting the summands of $\\\\hat{Q}_{h}(\\\\cdot )$ with random weights $w_i$ .\",\n \"The resulting weighted quantile loss $\\\\hat{Q}^{\\\\flat }_h :^p \\\\rightarrow $ is $\\\\hat{Q}_{ h}^\\\\flat (\\\\beta ) = \\\\frac{1}{n} \\\\sum _{i=1}^nw_i \\\\ell _{h}(y_i - \\\\langle _i, \\\\beta \\\\rangle ) , $ where $\\\\ell _{ h}(u) = (\\\\rho _\\\\tau * K_h)(u)$ is as in (REF ).\",\n \"We refer to [9] for a general asymptotic theory for weighted bootstrap for estimating equations, where a class of bootstrap weights is considered.\",\n \"Extensions to semiparametric $M$ -estimation can be found in [41] and [12].\",\n \"Let $\\\\mathbb {E}^*$ and $^*$ be the conditional expectation and probability given the observed data $_n$ , respectively.\",\n \"Observe that $\\\\mathbb {E}^*\\\\lbrace \\\\hat{Q}_{h}^{\\\\flat }(\\\\beta ) \\\\rbrace = \\\\hat{Q}_{h}(\\\\beta )$ for any $\\\\beta \\\\in ^{p}$ .\",\n \"Consequently, we have $_{\\\\beta \\\\in ^p} ^* \\\\lbrace \\\\hat{Q}_{h}^{\\\\flat }(\\\\beta ) \\\\rbrace = _{\\\\beta \\\\in ^p} \\\\hat{Q}_{h}(\\\\beta ) = \\\\hat{\\\\beta }_h .$ This simple and yet important observation motivates the following multiplier bootstrap statistic: $\\\\hat{\\\\beta }_h^\\\\flat = \\\\hat{\\\\beta }_h^\\\\flat (\\\\tau ) \\\\in _{\\\\beta \\\\in ^p } \\\\hat{Q}_{h}^{\\\\flat }(\\\\beta ) .\",\n \"$ To retain convexity of the loss function, non-negative random weights are preferred, such as $w_i \\\\sim {\\\\rm Exp}(1)$ , i.e., exponential distribution with rate 1, and $w_i =1 + e_i$ , where $e_i$ are independent Rademacher random variables.\",\n \"We can construct confidence intervals based on the bootstrap estimates using one of the three classical methods, the percentile method, the pivotal method, and the normal-based method.\",\n \"To be specific, for each $q\\\\in (0,1)$ and $1\\\\le j\\\\le p$ , define the (conditional) $q$ -quantile of $\\\\hat{\\\\beta }^\\\\flat _{h,j}$ —the $j^{{\\\\rm th}}$ coordinate of $\\\\hat{\\\\beta }^\\\\flat _h \\\\in ^p$ —given the observed data as ${\\\\rm c}^\\\\flat _j(q) = \\\\inf \\\\lbrace t \\\\in : ^* ( \\\\hat{\\\\beta }^\\\\flat _{h,j} \\\\le t ) \\\\ge q \\\\rbrace $ .\",\n \"Then, for a prespecified nominal level $\\\\alpha \\\\in (0,1)$ , the corresponding $1-\\\\alpha $ bootstrap percentile and pivotal confidence intervals (CIs) for $\\\\beta ^*_j$ ($j=1,\\\\ldots ,p$ ) are, respectively, $\\\\left[ {\\\\rm c}^\\\\flat _j(\\\\alpha /2 ) , \\\\, {\\\\rm c}^\\\\flat _j( 1 -\\\\alpha /2) \\\\right] ~\\\\mbox{ and }~ \\\\left[ 2\\\\hat{\\\\beta }_{h,j} - {\\\\rm c}^\\\\flat _j( 1 -\\\\alpha /2) , \\\\, 2 \\\\hat{\\\\beta }_{h,j} - {\\\\rm c}^\\\\flat _j(\\\\alpha /2 ) \\\\right] .$ Numerically, ${\\\\rm c}^\\\\flat _j(q)$ ($q\\\\in \\\\lbrace \\\\alpha , 1-\\\\alpha /2\\\\rbrace $ ) can be calculated with any specified precision by the simulation.\",\n \"In the R package conquer, the default number of bootstrap replications is set as $B=1000$ .\",\n \"In the next section, we will present a finite-sample theoretical framework for convolution-type smoothed quantile regression, including the concentration inequality and nonasymptotic Bahadur representation for both the conquer estimator (REF ) and its bootstrap counterpart (REF ) using Rademacher multipliers.\",\n \"As a by-product, a Berry-Esseen-type inequality (see Theorem REF ) states that, under certain constraints on the (growing) dimensionality and bandwidth, the distribution of any linear projection of $\\\\hat{\\\\beta }_h$ converges to a normal distribution as the sample size increases to infinity.\",\n \"Informally, for any given deterministic vector $\\\\in ^p$ , the scaled statistic $n^{1/2} \\\\langle , \\\\hat{\\\\beta }^*_h - \\\\beta ^* \\\\rangle $ is asymptotically normally distributed with asymptotic variance $\\\\sigma _\\\\tau ^2 := \\\\tau (1-\\\\tau ) \\\\,^^{-1} ^{-1} $ .\",\n \"To make inference based on such asymptotic results, we need to consistently estimate the asymptotic variance.\",\n \"[20] suggested the following estimators $\\\\hat{}_h := \\\\nabla ^2 \\\\hat{}_h(\\\\hat{\\\\beta }_h) = \\\\frac{1}{n} \\\\sum _{i=1}^nK(\\\\hat{\\\\varepsilon }_i /h) \\\\cdot _i^~\\\\mbox{ and }~~ \\\\hat{}_h := \\\\frac{1}{n h} \\\\sum _{i=1}^n\\\\bigl \\\\lbrace K(-\\\\hat{\\\\varepsilon }_i/ h) - \\\\tau \\\\bigr \\\\rbrace ^2 ^{align*}of and \\\\tau (1-\\\\tau ) \\\\, , respectively, where \\\\hat{\\\\varepsilon }_i = y_i - \\\\langle _i, \\\\hat{\\\\beta }_h\\\\rangle are fitted residuals.\",\n \"The ensuing 1-\\\\alpha normal-based CIs are given by \\\\hat{\\\\beta }_{h,j} \\\\pm \\\\Phi ^{-1}(1-\\\\alpha /2) \\\\cdot n^{-1/2} (\\\\hat{}_h^{-1} \\\\hat{}_h \\\\hat{}_h^{-1} )_{jj}, j=1,\\\\ldots ,p.However, the normal approximations to the CI suffer from the sensitivity to the smoothing needed to estimated the conditional densities, namely, the matrix = \\\\lbrace f_{\\\\varepsilon | }(0) ^.\",\n \"This is especially true when \\\\tau is in the upper or lower tail.\",\n \"See Section~\\\\ref {sec:normal-boot} for a numerical comparison between normal approximation and bootstrap calibration for confidence construction at high and low quantile levels.\",\n \"As we shall see, the normal-based CIs can be exceedingly wide and thus inaccurate under these situations.$\"\n ],\n [\n \"Statistical Analysis\",\n \"Under the linear quantile regression model in (REF ), we write, for convenience, the generic data vector $(y,)$ in a linear model form: $y = \\\\langle , \\\\beta ^*(\\\\tau ) \\\\rangle + \\\\varepsilon (\\\\tau ), $ where the random variable $\\\\varepsilon (\\\\tau )$ satisfies $\\\\lbrace \\\\varepsilon (\\\\tau ) \\\\le 0 | \\\\rbrace = \\\\tau $ .\",\n \"Let $ f_{\\\\varepsilon |}(\\\\cdot )$ be the conditional density function of the regression error $\\\\varepsilon = \\\\varepsilon (\\\\tau )$ given $$ .\",\n \"We first derive upper bounds for the smoothing bias under mild regularity conditions on the conditional density $ f_{\\\\varepsilon |}$ and the kernel function.\"\n ],\n [\n \"Smoothing bias\",\n \"[Kernel function] Let $K(\\\\cdot )$ be a symmetric and non-negative function that integrates to one, that is, $K(u)= K(-u)$ , $K(u) \\\\ge 0$ for all $u \\\\in $ and $\\\\int _{-\\\\infty }^\\\\infty K(u) \\\\, {\\\\rm d}u = 1$ .\",\n \"Moreover, $\\\\int _{-\\\\infty }^\\\\infty u^2 K(u) \\\\, {\\\\rm d} u <\\\\infty $ .\",\n \"We will use the notation $\\\\kappa _k = \\\\int _{-\\\\infty }^\\\\infty |u |^k K(u) \\\\,{\\\\rm d} u$ for $k \\\\ge 1$ .\",\n \"Furthermore, we define the population smoothed loss function $Q_{ h} (\\\\beta ) = \\\\lbrace \\\\hat{Q}_{ h}(\\\\beta )\\\\rbrace $ , $\\\\beta \\\\in ^p$ and the pseudo parameter $\\\\beta ^*_h (\\\\tau ) \\\\in _{\\\\beta \\\\in ^p } Q_{ h}(\\\\beta ), $ which is the population minimizer under the smoothed quantile loss.\",\n \"For simplicity, we write $\\\\beta ^* = \\\\beta ^*(\\\\tau )$ and $\\\\beta ^*_h = \\\\beta _h^*(\\\\tau )$ hereinafter.\",\n \"In general, $\\\\beta ^*_h$ differs from $\\\\beta ^*$ , and we refer to $\\\\Vert \\\\beta ^*_h - \\\\beta ^* \\\\Vert _2$ as the approximation error or smoothing bias.\",\n \"[Conditional density] There exist $\\\\bar{f} \\\\ge {f} >0$ such that $ {f} \\\\le f_{\\\\varepsilon |}(0 ) \\\\le \\\\bar{f}$ almost surely (for all $$ ).\",\n \"Moreover, there exists a constant $l_0 >0$ such that $|f_{\\\\varepsilon |}(u)- f_{\\\\varepsilon |}(0)| \\\\le l_0 |u |$ for all $u \\\\in $ almost surely.\",\n \"[Random design: moments] The (random) vector $\\\\in \\\\mathbb {R}^p$ of covariates satisfies $\\\\mu _3 := \\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\bigl ( | \\\\langle ^{-1/2} , \\\\rangle |^3 \\\\bigr )<\\\\infty $ , where $= (^$ is positive definite.\",\n \"If $$ is Gaussian, then Condition REF holds trivially.\",\n \"Heavier-tailed distributions of $$ is excluded here so that we can expect standard rates of convergence for the quantile regression estimates.\",\n \"The following result characterizes the smoothing bias from a nonasymptotic viewpoint.\",\n \"Assume Conditions REF –REF hold.\",\n \"There exist constants $c_1, c_2>0$ depending only on $({f} , l_0, \\\\mu _3, \\\\kappa _1, \\\\kappa _2)$ such that for any $00$ .\",\n \"Then $ \\\\biggl \\\\Vert \\\\beta ^*_h - \\\\beta ^* + \\\\frac{1}{2} \\\\kappa _2 h^2 \\\\cdot ^{-1} \\\\bigl \\\\lbrace f_{\\\\varepsilon |}^{\\\\prime }( 0 )\\\\bigr \\\\rbrace \\\\biggr \\\\Vert _{}\\\\le c_3 h^3,$ where $c_3 >0$ is a constant depending only on $({f} , l_0 , l_1 , \\\\mu _3 )$ and the kernel $K$ .\",\n \"To better understand inequality (REF ), consider for simplicity a special case where $ f_{\\\\varepsilon |} ( 0 ) $ is a constant (independent of $$ ).\",\n \"Hence, $= f_{\\\\varepsilon |} ( 0 ) $ and (REF ) becomes $( \\\\langle , \\\\beta ^*_h -\\\\beta ^* \\\\rangle ^2 )^{1/2} = \\\\Vert \\\\beta ^*_h -\\\\beta ^* \\\\Vert _{} \\\\le f^{-1/2}_{\\\\varepsilon |} ( 0 ) c_2 h^2$ .\",\n \"On the left-hand side, $\\\\Vert \\\\beta ^*_h -\\\\beta ^* \\\\Vert _{}$ is the average prediction smoothing error.\",\n \"Interestingly, the upper bound on the right-hand side could be dimension-free given $h$ .\",\n \"Another interesting implication is that, when both $f_{\\\\varepsilon |}(0)$ and $f^{\\\\prime }_{\\\\varepsilon |}(0)$ are independent of $$ , i.e., $f_{\\\\varepsilon |}(0)=f_{\\\\varepsilon }(0)$ and $f^{\\\\prime }_{\\\\varepsilon |}(0)=f^{\\\\prime }_{\\\\varepsilon }(0)$ , the leading term in the bias simplifies to $\\\\frac{1}{2} \\\\kappa _2 h^2 \\\\cdot ^{-1} \\\\bigl \\\\lbrace f_{\\\\varepsilon |}^{\\\\prime }(0)\\\\bigr \\\\rbrace = \\\\frac{f^{\\\\prime }_{\\\\varepsilon }(0)}{2f_{ \\\\varepsilon }(0)} \\\\kappa _2 h^2 \\\\cdot ^{-1} () = \\\\frac{f^{\\\\prime }_{\\\\varepsilon }(0)}{2f_{\\\\varepsilon }(0)} \\\\kappa _2 h^2 \\\\cdot \\\\begin{bmatrix}1 \\\\\\\\\\\\textbf {0}_{p-1}\\\\end{bmatrix} .\",\n \"\\\\nonumber $ In other words, the smoothing bias is concentrated primarily on the intercept.\",\n \"In the asymptotic setting where $p$ is fixed, and $h=o(1)$ as $n\\\\rightarrow \\\\infty $ , we refer to Theorem 1 in [20] for the expression of asymptotic bias.\"\n ],\n [\n \"Finite sample theory\",\n \"In this section, we provide two nonasymptotic results, the concentration inequality and the Bahadur-Kiefer representation, for the conquer estimator under random design.\",\n \"[Random design: sub-exponential case] The predictor $\\\\in ^p$ is sub-exponential: there exists $ \\\\upsilon _0>0$ such that $\\\\lbrace |\\\\langle , \\\\rangle | \\\\ge \\\\upsilon _0 \\\\Vert \\\\Vert _2 t \\\\rbrace \\\\le 2e^{-t }$ for all $\\\\in ^p$ and $t \\\\ge 0$ , where $= ^{-1/2} $ with $= (^$ positive-definite.\",\n \"Condition REF assumes a sub-exponential condition on the random covariates, which encompasses the bounded case considered by [20].\",\n \"For the standardized predictor $= ^{-1/2}$ , we define the uniform moment parameters $\\\\mu _k = \\\\sup _{\\\\in \\\\mathbb {S}^{p-1} } | \\\\langle , \\\\rangle |^k , \\\\ \\\\ k = 1,2 ,\\\\ldots , $ with $\\\\mu _2=1$ .\",\n \"Under Condition REF , a straightforward calculation shows that $\\\\mu _k \\\\le 2 \\\\upsilon _0^k k!$ , valid for all $k\\\\ge 1$ , where $\\\\upsilon _0$ is defined in Condition REF .\",\n \"In general, $\\\\upsilon _0$ may increase with $p$ , but in this paper we are primarily interested in the cases where $\\\\upsilon _0$ is a dimension-free constant.\",\n \"This is indeed the case if $$ is multivariate Gaussian, multivariate Bernoulli, uniform on $[-1,1]^p$ , or uniform on the sphere/ball with radius $p^{1/2}$ ; see Section 3.4 in [51] for more examples.\",\n \"Assume that Conditions REF –REF hold with $\\\\kappa _l = \\\\min _{|u|\\\\le 1} K(u) >0$ .\",\n \"For any $t > 0$ , the smoothed quantile regression estimator $\\\\hat{\\\\beta }_{h}$ with $\\\\sqrt{(p+t)/n} \\\\lesssim h \\\\lesssim 1$ satisfies $\\\\Vert \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\Vert _{} \\\\le C (\\\\kappa _l {f} )^{-1} \\\\Biggl ( \\\\sqrt{\\\\frac{p+t}{n}} + h^2 \\\\Biggr ), $ with probability at least $1-2e^{-t}$ , where $C>0$ is a constant depending only on $(\\\\upsilon _0, \\\\kappa _2,l_0)$ .\",\n \"The estimation error in (REF ) is upper bounded by two terms, $h^2$ and $\\\\sqrt{(p+t)/n}$ , and can be interpreted as the upper bounds for bias and variance, respectively.\",\n \"The condition $\\\\min _{|u|\\\\le 1} K(u) >0$ can be relaxed to $ \\\\min _{|u|\\\\le c} K(u) >0$ for some $c\\\\in (0,1)$ , which will only change the constants encountered in the proof.\",\n \"In particular, for kernels that are compactly supported on $[-1,1]$ , we may choose $c=1/2$ in Theorem REF .\",\n \"Next, we establish a Bahadur representation for conquer.\",\n \"To this end, we impose a slightly more stringent sub-Gaussian condition on the covariates.\",\n \"[Random design: sub-Gaussian case] The predictor $\\\\in ^p$ is sub-Gaussian: there exists $ \\\\upsilon _1 >0$ such that $\\\\lbrace |\\\\langle , \\\\rangle | \\\\ge \\\\upsilon _1 \\\\Vert \\\\Vert _2 t \\\\rbrace \\\\le 2 e^{-t^2/2}$ for all $\\\\in ^p$ and $t\\\\ge 0$ , where $= ^{-1/2} $ .\",\n \"We are primarily concerned with the cases where $\\\\upsilon _1$ is a dimension-free constant that does not depend on $p$ ; see the remarks made earlier about $\\\\upsilon _0$ .\",\n \"Assume that the data $\\\\lbrace (y_i, _i) \\\\rbrace _{i=1}^n$ are generated from the conditional quantile model (REF ), and write $\\\\varepsilon _i = y_i - \\\\langle _i, \\\\beta ^* \\\\rangle $ which satisfy $(\\\\varepsilon _i\\\\le 0 | _i) =\\\\tau $ .\",\n \"Assume Conditions REF , REF , and REF hold with $\\\\kappa _l = \\\\min _{|u|\\\\le 1} K(u) >0$ and $\\\\kappa _u = \\\\sup _{u \\\\in } K(u)$ .\",\n \"Moreover, assume $\\\\sup _{u \\\\in } f_{\\\\varepsilon | } (u) \\\\le \\\\bar{f} $ almost surely.\",\n \"For any $t > 0$ , let the bandwidth $h$ satisfy $\\\\sqrt{(p+t)/n} \\\\lesssim h \\\\lesssim 1$ .\",\n \"Then, $\\\\Biggl \\\\Vert \\\\hat{\\\\beta }_h - \\\\beta ^* - ^{-1} \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\bigl \\\\lbrace \\\\tau - K(- \\\\varepsilon _i /h) \\\\bigr \\\\rbrace _i \\\\Biggr \\\\Vert _2 \\\\le C \\\\Biggl ( \\\\frac{p+ t}{ h^{1/2} n } + h\\\\sqrt{\\\\frac{p+t}{n}} + h^3\\\\Biggr ) , $ with probability at least $1-3e^{-t}$ , where $= \\\\bigl \\\\lbrace f_{\\\\varepsilon |}( 0 ) ^\\\\rbrace $ , and $C>0$ is a constant depending only on $(\\\\upsilon _1, \\\\kappa _2, \\\\kappa _u, \\\\kappa _l, l_0,\\\\bar{f}, {f})$ .\",\n \"The Bahadur representation can be used to establish the limiting distribution of the estimator or its functionals.\",\n \"Here we consider a fundamental statistical inference problem for testing the linear hypothesis $H_0: \\\\langle , \\\\beta ^* \\\\rangle = 0$ , where $\\\\in ^p$ is a deterministic vector that defines a linear functional of interest.\",\n \"It is then natural to consider a test statistic that depends on $n^{1/2} \\\\langle , \\\\hat{\\\\beta }_h \\\\rangle $ .\",\n \"Based on the nonasymptotic result in Theorem REF , we establish a Berry-Esseen bound for the linear projection of conquer.\",\n \"Assume that the conditions in Theorem REF hold, and $\\\\sqrt{(p+\\\\log n)/n}\\\\lesssim h \\\\lesssim 1$ .\",\n \"Then, $\\\\Delta _{n,p}( h) :=\\\\sup _{x\\\\in , \\\\, \\\\in ^p } \\\\left| \\\\left\\\\lbrace \\\\frac{n^{1/2} \\\\langle , \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\rangle }{\\\\sqrt{\\\\tau (1-\\\\tau ) \\\\, ^^{-1} ^{-1} }} \\\\le x \\\\right\\\\rbrace - \\\\Phi (x ) \\\\right| \\\\lesssim \\\\frac{p + \\\\log n}{(n h)^{1/2}} + n^{1/2} h^2 , $ where $\\\\Phi (\\\\cdot )$ denotes the standard normal distribution function.\",\n \"In particular, with the choice of bandwidth $h \\\\asymp \\\\lbrace (p+\\\\log n)/n \\\\rbrace ^{2/5}$ , the normal approximation error $ \\\\Delta _{n,p}( h) $ is of order $(p+\\\\log n)^{4/5} n^{-3/10}$ .\",\n \"It is worth noticing that although the bound in (REF ) holds uniformly over all $$ , the asymptotic variance $\\\\sigma _\\\\tau ^2 := \\\\tau (1-\\\\tau ) \\\\, ^^{-1} ^{-1} $ depends on $$ .\",\n \"[Large-$p$ asymptotics] A broader view of classical asymptotics recognizes that the parametric dimension of appropriate model sequences may tend to infinity with the sample size; that is $p=p_n \\\\rightarrow \\\\infty $ as $n\\\\rightarrow \\\\infty $ .\",\n \"Results with increasing $p$ are available in the context of regularized quantile regression [6], [53], [36].\",\n \"In the large-$p$ and larger-$n$ setting with quantile regression estimation without regularization, [54] shows that $p^3(\\\\log n)^2 / n \\\\rightarrow 0$ suffices for a normal approximation, which provides some support to the viability of observed rates of parametric growth in the applied literature.\",\n \"[32] considers a sample of 733 wage models appeared in the econometric literature, and finds that $p_n = (n^{1/4})$ is roughly consistent with empirical practice.\",\n \"For the (convolution-type) smoothed quantile regression, the Berry-Esseen bound (REF ) in Theorem REF immediately yields a large-$p$ asymptotic result.\",\n \"Taking $h=h_n = \\\\lbrace (p+\\\\log n) /n\\\\rbrace ^{2/5}$ therein, we find that $n^{1/2} \\\\langle , \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\rangle $ , for any given $\\\\in ^p$ , is asymptotically normally distributed as long as $p^{8/3} /n \\\\rightarrow 0$ , which improves the best known growth condition on $p$ for quantile regression [54].\",\n \"For smooth robust regression estimators, asymptotic normality can be proven under less restrictive conditions on $p$ .\",\n \"[28] showed that if the loss is twice differentiable, the asymptotic normality for $\\\\langle , \\\\hat{\\\\beta }\\\\rangle $ , where $\\\\in ^p$ , holds if $p^3/n \\\\rightarrow 0$ as $n$ increases.\",\n \"[46] weakened the condition to $(p \\\\log n)^{3/2} /n \\\\rightarrow 0$ if the loss function is four times differentiable and the error distribution is symmetric.\",\n \"For Huber loss that has a Lipschitz continuous derivative, [26] obtained the scaling $p^2 \\\\log p = o(n)$ that ensures the asymptotic normality of arbitrary linear combinations of $\\\\hat{\\\\beta }$ .\",\n \"Table 1 summarizes our discussion here and show that the smoothing for conquer helps ensure asymptotic normality of the estimator under weaker conditions on $p$ than what we need for the usual quantile regression estimator.\",\n \"Table: Summary of scaling conditions required for normal approximation under various loss functions.In this paper, we show that the accuracy of conquer-based inference via the Bahadur representation (and normal approximations) has an error of rate faster than $n^{-1/4}$ yet slower than $n^{-1/2}$ ; see Theorems REF and REF .\",\n \"For standard regression quantiles, [47] proposed an alternative expansion for the quantile process using the “Hungarian\\\" construction of Komlós, Major and Tusnády.\",\n \"This stochastic approximation yields an error of order $n^{-1/2}$ (up to a factor of $\\\\log n$ ), and hence provides a theoretical justification for accurate approximations for inference in regression quantile models.\"\n ],\n [\n \"Theoretical guarantees for bootstrap inference\",\n \"We next investigate the statistical properties of the bootstrapped estimator defined in (REF ), with a particular focus on the Rademacher multiplier bootstrap.\",\n \"To be specific, we use, in this section and the rest of the paper, the random weights $w_i=1+e_i$ for $i=1,\\\\ldots , n$ , where $e_1,\\\\ldots , e_n$ are independent Rademacher random variables, that is, $(e_i=1) = (e_i=-1)=1/2$ .\",\n \"As before, we consider array (non)asymptotics, so that the bootstrap approximation errors depend explicitly on $(n,p)$ .\",\n \"Assume Conditions REF –REF , and REF hold with $\\\\kappa _l = \\\\min _{|u|\\\\le 1}K(u)>0$ .\",\n \"For any $t\\\\ge 0$ , there exists some event $(t)$ with $\\\\lbrace (t) \\\\rbrace \\\\ge 1-5e^{-t}$ such that, with $^*$ -probability at least $1-2e^{-t}$ conditioned on $(t)$ , the bootstrap estimator $\\\\hat{\\\\beta }^\\\\flat _h$ given in (REF ) satisfies $\\\\Vert \\\\hat{\\\\beta }^\\\\flat _h - \\\\beta ^* \\\\Vert _{} \\\\le C \\\\Biggl ( \\\\sqrt{\\\\frac{p+t}{n} } + h^2 \\\\Biggr )$ as long as $\\\\max \\\\lbrace (p+\\\\log n + t )/n, \\\\sqrt{(p+t)/n } \\\\rbrace \\\\lesssim h \\\\lesssim 1$ , where $C>0$ is a constant depending only on $(\\\\upsilon _1,\\\\kappa _2, \\\\kappa _l, l_0, {f})$ .\",\n \"Analogously to Theorem REF , we further provide a Bahadur representation result for the bootstrap counterpart $\\\\hat{\\\\beta }^\\\\flat _h $ , which lays the theoretical foundation for the validity of Rademacher multiplier bootstrap for conquer.\",\n \"Assume Conditions REF –REF and REF hold with $\\\\kappa _l = \\\\min _{|u|\\\\le 1}K(u)>0$ and $\\\\kappa _u = \\\\sup _{u\\\\in } K(u)$ .\",\n \"Moreover, assume $\\\\sup _{u\\\\in } f_{\\\\varepsilon | } (u) \\\\le \\\\bar{f}$ almost surely in $$ .\",\n \"Let the bandwidth satisfy $h \\\\asymp \\\\lbrace (p+\\\\log n)/n\\\\rbrace ^{\\\\delta }$ for some $\\\\delta \\\\in [1/3, 1/2)$ .\",\n \"Provided that $n\\\\gtrsim p + \\\\log n$ , there exists a sequence of events $\\\\lbrace _n\\\\rbrace $ with $(_n) \\\\ge 1-6n^{-1}$ such that, with $^*$ -probability at least $1-2n^{-1}$ conditioned on $_n$ , $\\\\hat{\\\\beta }^\\\\flat _h - \\\\hat{\\\\beta }_h = ^{-1 } \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\cdot _i \\\\bigl \\\\lbrace \\\\tau - K(- \\\\varepsilon _i /h) \\\\bigr \\\\rbrace + ^\\\\flat _n, $ where $^{\\\\flat }_n$ depends on $\\\\lbrace (e_i, y_i, _i) \\\\rbrace _1^n$ and satisfies $\\\\Vert ^\\\\flat _n \\\\Vert _2 = _{^*}(\\\\chi _n)$ , where $\\\\chi _n = \\\\chi _n(\\\\lbrace y_i, _i \\\\rbrace _{i=1}^n) = _{}\\\\lbrace \\\\sqrt{(p+\\\\log n) p \\\\log n} / (h^{1/2} n) + (p+\\\\log n)^{1/2} p \\\\log (n) /n \\\\rbrace $ .\",\n \"As suggested by Theorem REF and Remark REF , if we take $h$ to be $\\\\lbrace (p+\\\\log n)/n \\\\rbrace ^{2/5}$ , the normal approximation to the conquer estimator is asymptotically accurate provided that $p^{8/3} = o(n)$ as $n\\\\rightarrow \\\\infty $ .\",\n \"For the same $h$ , the remainder $^\\\\flat _n$ in (REF ) satisfies $\\\\Vert ^\\\\flat _n \\\\Vert _2 = o_{^*}(n^{-1/2})$ as long as $p^3 (\\\\log n)^2 = o(n)$ .\",\n \"Taken these two parts together, we have the following asymptotic bootstrap approximation result.\",\n \"If the dimension $p=p_n$ satisfies $p^3 (\\\\log n)^2 = o(n)$ , then, as $n\\\\rightarrow \\\\infty $ , $\\\\sup _{x\\\\in } \\\\, \\\\bigl | \\\\bigl ( n^{1/2} \\\\langle , \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\rangle \\\\le x \\\\bigr ) - ^*\\\\bigl ( n^{1/2} \\\\langle , \\\\hat{\\\\beta }^\\\\flat _h - \\\\hat{\\\\beta }_h \\\\rangle \\\\le x \\\\bigr ) \\\\bigr | \\\\stackrel{}{\\\\rightarrow } 0 .\",\n \"$ The proof of (REF ) follows the same argument as that in the proof of Theorem REF , and therefore is omitted.\",\n \"The additional logarithmic factor in the scaling may be an artifact of the proof technique.\",\n \"For standard quantile regression, [19] established a fixed-$p$ asymptotic bootstrap approximation result for wild bootstrap under fixed design.\"\n ],\n [\n \"Gradient Descent Methods for Conquer\",\n \"To solve optimization problems (REF ) and (REF ) with non-negative weights, arguably the simplest algorithm is a vanilla gradient descent algorithm (GD).\",\n \"For a prespecified $\\\\tau \\\\in (0,1)$ and bandwidth $h>0$ , recall that $ \\\\hat{Q}_h (\\\\beta ) = (1/n) \\\\sum _{i=1}^n\\\\ell _h(y_i - \\\\langle _i, \\\\beta \\\\rangle )$ .\",\n \"Starting with an initial value $\\\\hat{\\\\beta }^0 \\\\in ^p$ , at iteration $t=0, 1,2,\\\\ldots $ , GD computes $\\\\hat{\\\\beta }^{t+1} = \\\\hat{\\\\beta }^t - \\\\eta _t \\\\cdot \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^t) ,$ where $\\\\eta _t>0$ is the stepsize and $\\\\nabla \\\\hat{Q}_h (\\\\beta ) = \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\biggl \\\\lbrace {K}\\\\biggl ( - \\\\frac{ y_i - \\\\langle _i, \\\\beta \\\\rangle }{h} \\\\biggr ) - \\\\tau \\\\biggr \\\\rbrace _i ~~\\\\mbox{ with }~~{K}(u ) = \\\\int _{-\\\\infty }^u K(t) \\\\,{\\\\rm d} t .\",\n \"\\\\nonumber $ In the classical GD method, the stepsize is usually obtained by employing line search techniques.\",\n \"However, line search is computationally intensive for large-scale settings.\",\n \"One of the most important issues in GD is to determine a proper update step $\\\\eta _t$ decay schedule.\",\n \"A common practice in the literature is to use a diminishing stepsize or a best-tuned fixed stepsize.\",\n \"Neither of these two approaches can be efficient, at least compared to the Newton-Frisch algorithm with preprocessing [48].\",\n \"Recall that the smoothed loss $\\\\hat{Q}_h(\\\\cdot )$ is twice differentiable with Hessian $\\\\nabla ^2 \\\\hat{Q}_h (\\\\beta ) = \\\\frac{1}{n h } \\\\sum _{i=1}^nK \\\\biggl ( \\\\frac{y_i - \\\\langle _i, \\\\beta \\\\rangle }{h} \\\\biggr ) _i _i^ \\\\nonumber $ It is therefore intriguing to employ the Newton-Raphson method, which at iteration $t$ would read $\\\\hat{\\\\beta }^{t+1} = \\\\hat{\\\\beta }^t - \\\\bigl \\\\lbrace \\\\nabla ^2 \\\\hat{Q}_h (\\\\hat{\\\\beta }^t) \\\\bigr \\\\rbrace ^{-1} \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^t) .$ Computing the inverse of the Hessian for large problems can be an expensive operation.\",\n \"Moreover, when $h$ is small, $\\\\nabla ^2 \\\\hat{Q}_h (\\\\cdot )$ can have a very large condition number, which leads to slow convergence.\",\n \"Below we list the explicit expressions of the convolution smoothed check functions using several commonly used kernels.\",\n \"Recall that the check function can be written as $\\\\rho _\\\\tau (u) =|u|/2+ (\\\\tau -1/2)u$ , which, after convolution smoothing, becomes $\\\\ell _h (u) = \\\\int _{-\\\\infty }^\\\\infty \\\\frac{1}{2}|u + hv | K (v) \\\\, {\\\\rm d} v + (\\\\tau -1/2)u .$ (Uniform kernel $K(u) = (1/2) \\\\mathbb {1} (|u|\\\\le 1)$ ): $\\\\ell _h(u) = (h/2) \\\\ell ^{{\\\\rm U}}(u/h) + (\\\\tau - 1/2) u$ , where $ \\\\ell ^{{\\\\rm U}}(u) := (u^2/2 + 1/2)\\\\mathbb {1}( |u|\\\\le 1) + |u| \\\\mathbb {1}(|u|>1)$ is a Huber-type loss [28].\",\n \"(Gaussian kernel $K(u)=(2\\\\pi )^{-1/2} e^{-u^2/2}$ ): $\\\\ell _{h}(u) = (h/2)\\\\ell ^{{\\\\rm G}}(u/h) + (\\\\tau - 1/2 ) u$ , where $ \\\\ell ^{{\\\\rm G}}(u) := (2/\\\\pi )^{1/2} e^{-u^2/2 } + u \\\\lbrace 1- 2\\\\Phi (-u) \\\\rbrace $ .\",\n \"(Logistic kernel $K(u) = e^{-u}/(1+ e^{-u})^2 $ ): $\\\\ell _h (u) = (h/2) \\\\ell ^{{\\\\rm L}}(u/h) + (\\\\tau -1/2) u$ , where $ \\\\ell ^{{\\\\rm L}}(u) := u + 2\\\\log (1+e^{-u})$ .\",\n \"(Epanechnikov kernel $K(u) = (3 / 4) (1 - u^2) \\\\mathbb {1}(|u|\\\\le 1)$ ): $\\\\ell _{h}(u) = (h/2) \\\\ell ^{{\\\\rm E}}(u/h) +(\\\\tau - 1/2)u$ , where $ \\\\ell ^{{\\\\rm E}}(u) := (3u^2/4 - u^4/8+ 3/8) \\\\mathbb {1}(|u|\\\\le 1) + |u|\\\\mathbb {1}(|u|>1)$ .\",\n \"(Triangular kernel $K(u) = (1 - |u|) \\\\mathbb {1}(|u|\\\\le 1)$ ): $ \\\\ell _{h}(u) = (h/2) \\\\ell ^{{\\\\rm T}}(u/h) + ( \\\\tau - 1/2) u$ , where $ \\\\ell ^{{\\\\rm T}}(u ) := ( u^2 - |u|^3/3 + 1/3 ) \\\\mathbb {1}(|u|\\\\le 1) + |u|\\\\mathbb {1}(|u| >1) $ .\"\n ],\n [\n \"The Barzilai-Borwein stepsize rule\",\n \"In this paper, we propose to solve conquer by means of the gradient descent with a Barzilai-Borwein update step [5], which we refer to as the GD-BB algorithm.\",\n \"The BB method, which is motivated by quasi-Newton methods, has been proven to be very successful in solving nonlinear optimization problems.\",\n \"Recall the Newton-Raphson method (REF ), which can be computationally expensive and needs modifications if the Hessian is ill-conditioned especially when $h$ is too small.\",\n \"For this reason, many quasi-Newton methods seek a simple approximation of the inverse Hessian matrix, say $(^t)^{-1}$ , which satisfies the secant equation $^t \\\\delta ^t = ^t$ , where for $t=1, 2, \\\\ldots $ , $\\\\delta ^t = \\\\hat{\\\\beta }^t - \\\\hat{\\\\beta }^{t-1} ~~\\\\mbox{ and }~~ ^t = \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^t) - \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^{t-1}).\",\n \"$ To relieve the computational bottleneck of inverting a large matrix, the BB method chooses $\\\\eta $ so that $\\\\eta \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^t) = (\\\\eta ^{-1} _p)^{-1}\\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^t)$ “approximates\\\" $(^t)^{-1} \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^t)$ .\",\n \"Since $^t$ satisfies $^t \\\\delta ^t = ^t$ , it is more practical to choose $\\\\eta $ such that $ (1/\\\\eta ) \\\\delta ^t \\\\approx ^t$ or $ \\\\delta ^t \\\\approx \\\\eta ^t$ .\",\n \"Via least squares approximations, one may use $\\\\eta _{1,t}^{-1} = _{ \\\\alpha } \\\\Vert \\\\alpha \\\\delta ^t - ^t \\\\Vert _2^2$ or $\\\\eta _{2,t} = _{ \\\\eta } \\\\Vert \\\\delta ^t - \\\\eta ^t \\\\Vert _2^2$ .\",\n \"The BB stepsizes are then defined as $\\\\eta _{1,t} = \\\\frac{\\\\langle \\\\delta ^t , \\\\delta ^t\\\\rangle }{\\\\langle \\\\delta ^t, ^t \\\\rangle } ~~\\\\mbox{ and }~~ \\\\eta _{2,t} = \\\\frac{\\\\langle \\\\delta ^t , ^t \\\\rangle }{ \\\\langle ^t, ^t \\\\rangle } .\",\n \"$ Consequently, the BB iteration takes the form $\\\\hat{\\\\beta }^{t+1} = \\\\hat{\\\\beta }^t - \\\\eta _{\\\\ell ,t} \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^t) , \\\\quad \\\\ell = 1 {\\\\rm ~or~} 2 .\",\n \"$ Note that the BB step starts at iteration 1, while at iteration 0, we compute $\\\\hat{\\\\beta }^1$ using the standard gradient descent with an initial estimate $\\\\hat{\\\\beta }^0$ .\",\n \"The procedure is summarized in Algorithm REF .\",\n \"Based on extensive numerical studies, we find that at a fixed $\\\\tau $ , the number of iterations is insensitive to varying $(n,p)$ combinations.\",\n \"Moreover, as $h$ increases, the number of iterations declines because the loss function is “more convex\\\" for larger $h$ .\",\n \"In Algorithm REF , the quantity $\\\\delta >0$ is called the gradient tolerance, ensuring that the obtained estimate, say $\\\\hat{\\\\beta }$ , satisfies $\\\\Vert \\\\nabla \\\\hat{Q}_h(\\\\hat{\\\\beta }) \\\\Vert _2 \\\\le \\\\delta $ .\",\n \"Provided that $\\\\delta \\\\lesssim \\\\sqrt{p/n}$ , the statistical theory developed in Section prevails.\",\n \"In our R package conquer, we set $\\\\delta = 10^{-4}$ as the default value; this value can also be specified by the user.\",\n \"As $\\\\tau $ approaches 0 or 1, the Hessian matrix becomes ill-conditioned.\",\n \"As a result, the stepsizes computed in GD-BB may sometimes vibrate drastically, causing instability of the algorithm.\",\n \"Therefore, in practice, we set a lower bound for the stepsizes by taking $\\\\eta _t = \\\\min \\\\lbrace \\\\eta _{1,t}, \\\\eta _{2,t} , 100\\\\rbrace $ , for $t=1, 2,\\\\ldots $ .\",\n \"Another cause of an ill-conditioned Hessian arises when we have covariates with very different scales.\",\n \"In this case, the stepsize should be different for each covariate, and a constant stepsize will be either too small or too large for one or more covariates, which correspond to slow convergence.\",\n \"To address this issue, we scale the covariate inputs to have zero mean and unit variance before applying gradient descent.\",\n \"[!t] Gradient descent with Barzilai-Borwin stepsize (GD-BB) for solving conquer.\",\n \"Input: data vectors $\\\\lbrace (y_i, _i)\\\\rbrace _{i=1}^n$ , $\\\\tau \\\\in (0,1)$ , bandwidth $h\\\\in (0,1)$ , initialization $\\\\hat{\\\\beta }^{(0)}$ , and gradient tolerance $\\\\delta $ .\",\n \"[1] Compute $\\\\hat{\\\\beta }^1 \\\\leftarrow \\\\hat{\\\\beta }^0 - \\\\nabla \\\\hat{Q}_h( \\\\hat{\\\\beta }^0)$ $t=1,2 \\\\ldots $ $\\\\delta ^t \\\\leftarrow \\\\hat{\\\\beta }^t - \\\\hat{\\\\beta }^{t-1}$ , $^t \\\\leftarrow \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^{t})-\\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^{t-1})$ $\\\\eta _{1,t} \\\\leftarrow \\\\langle \\\\delta ^t, \\\\delta ^t \\\\rangle / \\\\langle \\\\delta ^t, ^t \\\\rangle $ , $\\\\eta _{2,t} \\\\leftarrow \\\\langle \\\\delta ^t, ^t\\\\rangle / \\\\langle ^t, ^t \\\\rangle $ $\\\\eta _t \\\\leftarrow \\\\min \\\\lbrace \\\\eta _{1,t},\\\\eta _{2,t}, 100\\\\rbrace $ if $\\\\eta _{1,t} >0$ and $\\\\eta _t \\\\leftarrow 1$ otherwise $ \\\\hat{\\\\beta }^{t+1} \\\\leftarrow \\\\hat{\\\\beta }^t - \\\\eta _t \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^{t})$ when $\\\\Vert \\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }^{t}) \\\\Vert _2\\\\le \\\\delta $\"\n ],\n [\n \"Warm start via Huber regression\",\n \"A good initialization helps reduce the number of iterations for GD, and hence facilitates fast convergence.\",\n \"Recall from Remark that with a uniform kernel, the smoothed check function is proximal to a Huber loss [28].\",\n \"Motivated by this subtle proximity, we propose using the Huber $M$ -estimator as an initial estimate, and then proceed by iteratively applying gradient descent with BB update step.\",\n \"Let $H_\\\\gamma (u) = (u^2/2) \\\\mathbb {1}(|u|\\\\le \\\\gamma ) + \\\\gamma (|u| - \\\\gamma /2) \\\\mathbb {1}(|u|> \\\\gamma )$ be the Huber loss parametrized by $ \\\\gamma >0$ .\",\n \"The Huber $M$ -estimator is then defined as $\\\\widetilde{\\\\beta }_\\\\gamma \\\\in _{\\\\beta \\\\in ^p} \\\\hat{}_\\\\gamma (\\\\beta ), ~~\\\\mbox{where}~~ \\\\hat{}_\\\\gamma (\\\\beta ) = \\\\frac{1}{n} \\\\sum _{i=1}^nH_\\\\gamma (y_i - \\\\langle _i, \\\\beta \\\\rangle ).\",\n \"$ Note that $H_1(u) = \\\\ell ^{{\\\\rm U}}(u)-1/2$ for $\\\\ell ^{{\\\\rm U}}(\\\\cdot )$ defined in Remark , we have $ \\\\hat{Q}_{1/2, \\\\gamma }(\\\\beta ) = \\\\hat{}_ \\\\gamma (\\\\beta )$ .\",\n \"The quantity $ \\\\gamma $ is a shape parameter that controls the amount of robustness.\",\n \"The main reason for choosing a fixed (neither diminishing nor diverging) tuning parameter $ \\\\gamma $ in [29] is to guarantee robustness towards arbitrary contamination in a neighborhood of the model.\",\n \"This is at the core of the robust statistics idiosyncrasy.\",\n \"In particular, [29] proposed $ \\\\gamma =1.35 \\\\sigma $ to gain as much robustness as possible while retaining 95% asymptotic efficiency for normally distributed data, where $\\\\sigma >0$ is the standard deviation of the random noise.\",\n \"We estimate $\\\\sigma $ using the median absolute deviation on the residuals at each iteration, i.e., ${\\\\rm MAD}(\\\\lbrace r_i^t \\\\rbrace _{i=1}^n)= \\\\text{median}(|r_i^t - r^t_{\\\\textrm {median}}|)$ , where $r^t_{\\\\textrm {median}}$ is the median of $r_1^t,\\\\ldots ,r_n^t$ .\",\n \"Noting that the Huber loss is continuously differentiable, convex, and locally strongly convex, we use the GD-BB method described in the previous section to solve the optimization problem (REF ).\",\n \"Starting at iteration 0 with $\\\\widetilde{\\\\beta }^0 = \\\\textbf {0}$ , at iteration $t=0, 1,2,\\\\ldots $ , we compute $\\\\widetilde{\\\\beta }^{t+1} = \\\\widetilde{\\\\beta }^t - \\\\eta _t \\\\nabla \\\\hat{}_\\\\gamma (\\\\widetilde{\\\\beta }^t) = \\\\widetilde{\\\\beta }^t + \\\\frac{\\\\eta _t}{n} \\\\sum _{i=1}^n\\\\psi _\\\\gamma (y_i-\\\\langle _i, \\\\widetilde{\\\\beta }^t \\\\rangle ) _i$ with $\\\\eta _t>0$ automatically obtained by the BB method, where $\\\\psi _\\\\gamma (u) = H_\\\\gamma ^{\\\\prime }(u) =\\\\min \\\\lbrace \\\\max (- \\\\gamma , u), \\\\gamma \\\\rbrace $ .\",\n \"We summarize the details in Algorithm REF .\",\n \"[!t] GD-BB method for solving (REF ).\",\n \"Input: $\\\\lbrace (y_i, _i)\\\\rbrace _{i=1}^n$ and convergence criterion $\\\\delta $ .\",\n \"[1] Initialize $\\\\tilde{\\\\beta }^{(0)} = \\\\textbf {0}$ Compute $ \\\\gamma ^0 = 1.35 \\\\cdot {\\\\rm MAD}(\\\\lbrace r_i^0 \\\\rbrace _{i=1}^n)$ , where $r^0_i \\\\leftarrow y_i - \\\\langle _i, \\\\widetilde{\\\\beta }^{0} \\\\rangle $ , $i=1,\\\\ldots , n$ , where $\\\\mathrm {MAD}(\\\\cdot )$ is the median absolute deviation $\\\\tilde{\\\\beta }^1 \\\\leftarrow \\\\tilde{\\\\beta }^0 - \\\\nabla \\\\hat{L}_{\\\\gamma ^0}( \\\\tilde{\\\\beta }^0)$ $t=1,2 \\\\ldots $ $\\\\gamma ^t = 1.35 \\\\cdot {\\\\rm MAD}(\\\\lbrace r^t_i\\\\rbrace _{i=1}^n)$ , where $r^t_i \\\\leftarrow y_i - \\\\langle _i, \\\\widetilde{\\\\beta }^{t} \\\\rangle $ , $i=1,\\\\ldots , n$ $\\\\delta ^t \\\\leftarrow \\\\tilde{\\\\beta }^t - \\\\tilde{\\\\beta }^{t-1}$ , $^t \\\\leftarrow \\\\nabla \\\\hat{}_{\\\\gamma ^t}(\\\\tilde{\\\\beta }^{t})-\\\\nabla \\\\hat{}_{\\\\gamma ^t}(\\\\tilde{\\\\beta }^{t-1})$ $\\\\eta _{1,t} \\\\leftarrow \\\\langle \\\\delta ^t, \\\\delta ^t \\\\rangle / \\\\langle \\\\delta ^t, ^t \\\\rangle $ , $\\\\eta _{2,t} \\\\leftarrow \\\\langle \\\\delta ^t, ^t\\\\rangle / \\\\langle ^t, ^t \\\\rangle $ .\",\n \"$\\\\eta _t \\\\leftarrow \\\\min \\\\lbrace \\\\eta _{1,t},\\\\eta _{2,t}, 100\\\\rbrace $ if $\\\\eta _{1,t} >0$ and $\\\\eta _t \\\\leftarrow 1$ otherwise $ \\\\widetilde{\\\\beta }^{t+1} \\\\leftarrow \\\\tilde{\\\\beta }^t - \\\\eta _t \\\\nabla \\\\hat{}_{\\\\gamma ^t}(\\\\tilde{\\\\beta }^{t})$ when $\\\\Vert \\\\nabla \\\\hat{}_{\\\\gamma ^t}(\\\\tilde{\\\\beta }^{t}) \\\\Vert _2\\\\le \\\\delta $ The Huber loss $H_\\\\gamma (\\\\cdot )$ approximates the quantile loss function with $\\\\tau =1/2$ as $\\\\gamma \\\\rightarrow 0$ .\",\n \"Therefore, an alternative method for QR computing is to solve the Huber regression via gradient descent with a shrinking gamma.\",\n \"To evaluate its performance, we implement the above idea by setting $\\\\gamma ^{t} = c \\\\cdot \\\\gamma ^{t-1}$ for some $c\\\\in (0,1)$ at the $t$ -th iteration.\",\n \"We found that the aforementioned idea is not stable numerically across several simulated data sets, unless one is very careful in controlling the minimal magnitude of $\\\\gamma $ .\",\n \"In addition, the solution obtained has a higher estimation error than that of standard QR and conquer.\"\n ],\n [\n \"Numerical Studies\",\n \"In this section, we assess the finite-sample performance of conquer via extensive numerical studies.\",\n \"We compare conquer to standard QR [35] and Horowitz's smoothed QR [27].\",\n \"Both the convolution-type and Horowitz's smoothed methods involve a smoothing parameter $h$ .\",\n \"In view of Theorem REF , we take $h = \\\\lbrace (p+\\\\log n) /n\\\\rbrace ^{2/5}$ in all of the numerical experiments.\",\n \"As we will see from Figure REF , the proposed method is insensitive to the choice of $h$ .\",\n \"Therefore, we leave the fine tuning of $h$ as an optional rather than imperative choice.\",\n \"In all the numerical experiments, the convergence criterion in Algorithms REF and REF is taken as $\\\\delta =10^{-4}$ .\",\n \"We first generate the covariates $_i = (x_{i , 1}, \\\\dots , x_{i , p})^ from a multivariate uniform distribution on the cube $ 31/2 [-1, 1]p$ with covariance matrix $ =(0.7|j - k|)1j,kp$ using the \\\\texttt {R} package \\\\texttt {MultiRNG} \\\\cite {F1999}.The random noise $ i$ are generated from two different distributions: (i) Gaussian distribution, $ N(0, 4)$; and (ii) $ t$ distribution with degrees of freedom 2, $ t2$.Let $ i- = (xi, 1, ..., xi, p - 1), $\\\\beta ^* = (1, \\\\dots , 1)^{p - 1}$ , and $\\\\beta ^*_0 = 1$ .\",\n \"Given $\\\\tau \\\\in (0, 1)$ , we then generate the response $y_i$ from the following homogeneous and heterogeneous models, all of which satisfy the Assumption REF : Homogeneous model: $y_i = \\\\beta ^*_0 + \\\\langle _i^- , \\\\beta ^* \\\\rangle + \\\\lbrace \\\\varepsilon _i - F^{-1}_{\\\\varepsilon _i}(\\\\tau ) \\\\rbrace , ~~~~i = 1, \\\\dots , n; $ Linear heterogeneous model: $y_i = \\\\beta ^*_0 + \\\\langle _i^- , \\\\beta ^* \\\\rangle + (0.5 x_{i, p} + 1) \\\\lbrace \\\\varepsilon _i - F^{-1}_{\\\\varepsilon _i}(\\\\tau ) \\\\rbrace , ~~~~i = 1, \\\\dots , n; $ Quadratic heterogeneous model: $y_i = \\\\beta ^*_0 + \\\\langle _i^- , \\\\beta ^* \\\\rangle + 0.5\\\\lbrace 1+ ( x_{i, p}-1)^2 \\\\rbrace \\\\lbrace \\\\varepsilon _i - F^{-1}_{\\\\varepsilon _i}(\\\\tau )\\\\rbrace , ~~~~i = 1, \\\\dots , n. $ To evaluate the performance of different methods, we calculate the estimation error under the $\\\\ell _2$ -norm, i.e., $\\\\Vert \\\\hat{\\\\beta }-\\\\beta ^*\\\\Vert _2$ , and record the elapsed time.\",\n \"The details are in Section REF .\",\n \"In Section REF , we examine the finite-sample performance of the multiplier bootstrap method for constructing confidence intervals in terms of coverage probability, width of the interval, and computing time.\"\n ],\n [\n \"Estimation\",\n \"For all the numerical studies in this section, we consider a wide range of the sample size $n$ , with the size-dimension ratio fixed at $n / p = 20$ .\",\n \"That is, we allow the dimension $p$ to increase as a function of $n$ .\",\n \"We implement conquer with four different kernel functions as described in Remark : (i) Gaussian; (ii) uniform; (iii) Epanechnikov; and (iv) triangular.\",\n \"The classical quantile regression is implemented via a modified version of the Barrodale and Roberts algorithm [37], [38] by setting method= “br\\\" in the R package quantreg, which is recommended for problems up to several thousands of observations in [34].\",\n \"For very large problems, the Frisch-Newton approach after preprocessing “pfn\\\" is preferred.\",\n \"Since the same size taken to be at most 5000 throughout this section, the two methods, “br” and “pfn\\\", have nearly identical runtime behaviors.\",\n \"In some applications where there are a lot of discrete covariates, it is advantageous to use method “sfn\\\", a sparse version of Frisch-Newton algorithm that exploits sparse algebra to compute iterates [39].\",\n \"Moreover, we implement Horowitz's smoothed quantile regression using the Gaussian kernel, and solve the resulting non-convex optimization via gradient descent with random initialization and stepsize calibrated by backtracking line search (Section 9.3 of [8]).\",\n \"The results, averaged over 100 replications, are reported.\",\n \"Figure REF depicts estimation error of the different methods under the simulation settings described in Section with $\\\\tau = 0.9$ .\",\n \"We see that conquer has a lower estimation error than the classical QR across all scenarios, indicating that smoothing can improve estimation accuracy under the finite-sample setting.\",\n \"Moreover, compared to Horowitz's smoothing, conquer has a lower estimation error in most settings.\",\n \"Estimation error under various quantile levels $\\\\tau \\\\in \\\\lbrace 0.1, 0.3, 0.5, 0.7\\\\rbrace $ under the $t_2$ random noise are also examined.\",\n \"The results are reported in Figure REF in Appendix , from which we observe evident advantages of conquer, especially at low and high quantile levels.\",\n \"Figure: Model () with t 2 t_2 error.To assess the computational efficiency, we compute the elapsed time for fitting the different methods.\",\n \"Figures REF and REF in Appendix report the runtime for the different methods with growing sample size and dimension under the same settings as in Figures REF and REF , respectively.\",\n \"We observe that conquer is computationally efficient and stable across all scenarios, and the runtime is insensitive to the choice of kernel functions.\",\n \"In contrast, the runtime for classical quantile regression grows rapidly as the sample size and dimension increase.\",\n \"Figures REF and REF in Appendix show that the runtime of Horowitz's smoothing method increases significantly at extreme quantile levels $\\\\tau \\\\in \\\\lbrace 0.1,0.9\\\\rbrace $ , possibly due to the combination of its non-convex nature and flatter gradient.\",\n \"In summary, we conclude that conquer significantly improves computational efficiency while retaining high statistical accuracy for fitting large-scale linear quantile regression models.\",\n \"Next, we conduct a sensitivity analysis for conquer regarding the smoothing bandwidth $h$ .\",\n \"We set $(n, p) = (2000, 100)$ and consider the simulation settings (REF )–(REF ) with $\\\\mathcal {N}(0, 4)$ and $t_2$ noise.\",\n \"We perform conquer with $h \\\\in \\\\lbrace 0.2, 0.22,\\\\dots , 0.5\\\\rbrace $ , including the default value $h_{{\\\\rm de}} = \\\\lbrace (p+\\\\log n) /n\\\\rbrace ^{2/5} = 0.3107$ , and compare the estimation error with that of QR in Figure REF .\",\n \"We see that the estimation error of conquer is uniformly lower than that of QR over a range of $h$ , suggesting that conquer is insensitive to the choice of bandwidth $h$ .\",\n \"Figure: Model () with t 2 t_2 error.Figure: Model () with t 2 t_2 error.\"\n ],\n [\n \"Inference\",\n \"In this section, we assess the performance of the multiplier bootstrap procedure for constructing confidence interval for each of the regression coefficients obtained from conquer.\",\n \"We implement conquer using the Gaussian kernel, and construct three types of confidence intervals: (i) the percentile mb-per; (ii) pivotal mb-piv; (iii) and regular mb-norm confidence intervals, as described in Section REF .\",\n \"We also refer to the proposed multiplier bootstrap procedure as mb-conquer for simplicity.\",\n \"We compare the proposed method to several widely used inference methods for QR.\",\n \"In particular, we consider confidence intervals by inverting a rank score test, rank ([23]; Section 3.5 of [33]); a bootstrap variant based on pivotal estimating functions, pwy [45]; and wild bootstrap with Rademacher weights, wild [19].\",\n \"The three methods rank, pwy, and wild are implemented using the R package quantreg.\",\n \"Note that rank is a non-resampling based procedure that relies on prior knowledge on the random noise, i.e., a user needs to specify whether the random noise are independent and identically distributed.\",\n \"In our simulation studies, we provide rank an unfair advantage by specifying the correct random noise structure.\",\n \"We set $(n, p) = (800, 20)$ , $\\\\tau \\\\in \\\\lbrace 0.5, 0.9\\\\rbrace $ , and significance level $\\\\alpha = 0.05$ .\",\n \"All of the resampling methods are implemented with $B = 1000$ bootstrap samples.\",\n \"To measure the reliability, accuracy, and computational efficiency of different methods for constructing confidence intervals, we calculate the average empirical coverage probability, average width of confidence interval, and the average runtime.\",\n \"The average is taken over all regression coefficients without the intercept.\",\n \"Results based on 200 replications are reported in Figure REF , and Figures REF –REF in Appendix .\",\n \"Figure: Runtime under model ().In Figure REF , and Figures REF –REF in Appendix , we use the rank-inversion method, rank, as a benchmark since we implement rank using information about the true underlying random noise, which is practically infeasible.\",\n \"In the case of $\\\\tau =0.9$ , pwy is most conservative as it produces the widest confidence intervals with slightly inflated coverage probability, and wild gives the narrowest confidence intervals but at the cost of coverage probability.\",\n \"The proposed methods mb-per, mb-piv, and mb-norm achieve a good balance between reliability (high coverage probability) and accuracy (narrow CI width), and moreover, has the lowest runtime.\",\n \"To further highlight the computational gain of the proposed method, we now perform numerical studies with larger $n$ and $p$ .\",\n \"In this case, the rank inversion method rank is computationally infeasible.\",\n \"For example, when $(n,p)= (5000,250)$ , rank inversion takes approximately 80 minutes while conquer with multiplier bootstrap takes 41 seconds for constructing confidence intervals.\",\n \"We therefore omit rank from the following comparison.\",\n \"We consider the quadratic heterogeneous model (REF ) with $(n, p) = (4000, 100)$ and $t_2$ noise.\",\n \"The results are reported in Figure REF .\",\n \"We see that pwy and wild take approximately 300 seconds while mb-conquer takes approximately 15 seconds.\",\n \"In summary, mb-conquer leads to a huge computational gain without sacrificing statistical efficiency.\",\n \"Figure: Runtime\"\n ],\n [\n \"Comparison between normal approximation and bootstrap calibration\",\n \"Finally, we complement the above studies with a comparison between the normal approximation and bootstrap calibration methods for confidence estimation.\",\n \"We consider model (REF ) with $(n,p) = (2000,10)$ .\",\n \"We use the same $\\\\beta ^*\\\\in ^{p-1}$ and $\\\\beta ^*_0$ as before, and generate random covariates and noise from a multivariate uniform distribution and $t_{1.5}$ -distribution, respectively.\",\n \"For each of the $p-1$ regression coefficients, we apply the proposed bootstrap percentile method and the normal-based method [20] to construct pointwise confidence intervals at quantile indices close to 0 and 1, that is, $\\\\tau \\\\in \\\\lbrace 0.05, 0.1, 0.9, 0.95\\\\rbrace $ .\",\n \"Boxplots of the empirical coverage and CI width for the two methods are reported in Figure REF .\",\n \"Considering that extreme quantile regressions are notoriously hard to estimate, the bootstrap method can produce much more reliable (high coverage) and accurate (narrow width) confidence intervals than the normal-based counterpart.\",\n \"Therefore, for applications in which extreme quantiles are of particular interest, such as the problem of forecasting the conditional value-at-risk of a financial institution [13], the bootstrap provides a more reliable approach for quantifying the uncertainty of the estimates.\",\n \"Figure: Boxplots of CI width\"\n ],\n [\n \"Discussion\",\n \"In this paper, we provide a comprehensive study on the statistical properties of conquer—namely, convolution-type smoothed quantile regression, under the array nonasymptotic setting in which $p$ is allowed to increase as a function of $n$ while $p/n$ being small.\",\n \"An efficient gradient-based algorithm is proposed to compute conquer, which proves to be scalable to large dimensions and even larger sample sizes.\",\n \"Recently, there has been a growing interest in studying the asymptotic behavior of regression estimates under the regime in which $p$ grows proportionally with the sample size $n$ [15], [40].\",\n \"The current studies are mainly focused on $M$ -estimation with strongly convex loss functions.\",\n \"New theoretical tools are needed to establish results for conquer under such a regime, and we leave it for future work.\",\n \"In the high-dimensional setting in which $p \\\\gg n$ , various authors have studied the regularized quantile regression under the sparsity assumption that most of the regression coefficients are zero [6], [53].\",\n \"Using $\\\\ell _1$ penalties, the computation of regularized QR is based on either a linear programming reformulation or alternating direction method of multiplier algorithms [22].\",\n \"Since the conquer loss is convex and twice differentiable, we expect that gradient-based algorithms, such as coordinate gradient descent or proximal gradient descent, will enjoy superior computational efficiency for solving regularized conquer without sacrificing statistical accuracy.\",\n \"One future work is to establish the statistical theory and computational complexity of the regularized conquer estimator.\"\n ],\n [\n \"One-step Conquer with Higher-order Kernels\",\n \"As noted in Section REF , the smoothing bias is of order $h^2$ when a non-negative kernel is used.\",\n \"The ensuing empirical loss $\\\\beta \\\\mapsto (1/n) \\\\sum _{i=1}^n(\\\\rho _\\\\tau * K_h)(y_i - \\\\langle _i , \\\\beta \\\\rangle )$ is not only twice-differentiable and convex, but also (provably) strongly convex in a local vicinity of $\\\\beta ^*$ with high probability.\",\n \"Kernel smoothing is ubiquitous in nonparametric statistics.\",\n \"The order of a kernel, $\\\\nu $ , is defined as the order of the first non-zero moment.\",\n \"The order of a symmetric kernel is always even.\",\n \"A kernel is called high-order if $\\\\nu >2$ , which inevitably has negative parts and thus is no longer a probability density.\",\n \"Thus far we have focused on conquer with second-order kernels, and the resulting estimator achieves an $\\\\ell _2$ -error of the order $\\\\sqrt{p/n} + h^2$ .\",\n \"Let $G(\\\\cdot )$ be a higher-order symmetric kernel with order $\\\\nu \\\\ge 4$ , and $b >0$ be a bandwidth.\",\n \"Again, via convolution smoothing, we may consider a bias-reduced estimator that minimizes the empirical loss $\\\\beta \\\\mapsto \\\\hat{Q}_b^G(\\\\beta ):= (1/n) \\\\sum _{i=1}^n(\\\\rho _\\\\tau * G_b )(y_i - \\\\langle _i , \\\\beta \\\\rangle )$ .\",\n \"This, however, leads to a non-convex optimization.\",\n \"Without further assumptions, finding a global minimum is computationally intractable: finding an $\\\\epsilon $ -suboptimal point for a $k$ -times continuously differentiable loss function requires at least $\\\\Omega \\\\lbrace (1/\\\\epsilon )^{p/k}\\\\rbrace $ evaluations of the function and its first $k$ derivatives, ignoring problem-dependent constants; see Section 1.6 in [42].\",\n \"Instead, various gradient-based methods have been developed for computing stationary points, which are points $\\\\beta $ with sufficiently small gradient $\\\\Vert \\\\nabla \\\\hat{Q}_b^G(\\\\beta ) \\\\Vert _2 \\\\le \\\\epsilon $ , where $\\\\epsilon \\\\ge 0$ is optimization error.\",\n \"However, the equation $\\\\nabla \\\\hat{Q}_b^G(\\\\beta ) = \\\\textbf {0}$ does not necessarily have a unique solution, whose statistical guarantees remain unknown.\",\n \"Motivated by the classical one-step estimator [7], we further propose a one-step conquer estimator using high-order kernels, which bypasses solving a large-scale non-convex optimization.\",\n \"To begin with, we choose two symmetric kernel functions, $K: \\\\rightarrow [0,\\\\infty )$ with order two and $G(\\\\cdot )$ with order $\\\\nu \\\\ge 4$ , and let $h, b >0$ be two bandwidths.\",\n \"First, compute an initial conquer estimator $\\\\beta \\\\in _{\\\\beta \\\\in ^p} \\\\hat{Q}_{h}^K(\\\\beta )$ , where $\\\\hat{Q}_{h}^K(\\\\beta ) = (1/n) \\\\sum _{i=1}^n(\\\\rho _\\\\tau * K_h) (y_i - \\\\langle _i , \\\\beta \\\\rangle )$ .\",\n \"Denote by $\\\\bar{r}_i = y_i - \\\\langle _i, \\\\beta \\\\rangle $ for $i=1,\\\\ldots , n$ the fitted residuals.\",\n \"Next, with slight abuse of notation, we define the one-step conquer estimator $\\\\hat{\\\\beta }$ as a solution to the equation $ \\\\nabla ^2 \\\\hat{Q}_b^G (\\\\beta ) ( \\\\hat{\\\\beta }- \\\\beta ) = - \\\\nabla \\\\hat{Q}_b^G(\\\\beta )$ , or equivalently, $\\\\Biggl \\\\lbrace \\\\frac{1}{n } \\\\sum _{i=1}^nG_b ( \\\\bar{r}_i ) _i _i^\\\\rbrace ( \\\\hat{\\\\beta }- \\\\beta ) = \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\bigl \\\\lbrace G ( \\\\bar{r}_i / b ) + \\\\tau - 1 \\\\bigr \\\\rbrace _i .\",\n \"$ where $\\\\hat{Q}_b^G(\\\\beta ) = (1/n) \\\\sum _{i=1}^n(\\\\rho _\\\\tau * G_b ) (y_i - \\\\langle _i , \\\\beta \\\\rangle )$ .\",\n \"Provided that $\\\\nabla ^2 \\\\hat{Q}_b^G (\\\\beta )$ is positive definite, the one-step conquer estimate $\\\\hat{\\\\beta }$ essentially performs a Newton-type step based on $\\\\beta $ : $\\\\hat{\\\\beta }= \\\\beta - \\\\bigl \\\\lbrace \\\\nabla ^2 \\\\hat{Q}_b^G (\\\\beta ) \\\\bigr \\\\rbrace ^{-1} \\\\nabla \\\\hat{Q}_b^G(\\\\beta ).\",\n \"$ In this case, $\\\\hat{\\\\beta }$ can be computed by the conjugate gradient method [62].\",\n \"Theoretical properties of the one-step estimator $\\\\hat{\\\\beta }$ defined in (REF ), including the Bahadur representation and asymptotic normality with explicit Berry-Esseen bound, will be provided in on-line supplementary materials.\",\n \"For practical implementation, we consider higher-order Gaussian-based kernels.\",\n \"For $r=1,2,\\\\ldots ,$ the $(2r)$ -th order Gaussian kernels are $G_{2r}( u) = \\\\frac{(-1)^r \\\\phi ^{(2r-1)}(u)}{2^{r-1} (r-1)!\",\n \"u} = \\\\sum _{\\\\ell =0}^{r-1} \\\\frac{(-1)^\\\\ell }{2^\\\\ell \\\\ell !}\",\n \"\\\\phi ^{(2\\\\ell )} (u) ;$ see Section 2 of [70].\",\n \"Integrating $G_{2r}(\\\\cdot )$ yields ${G}_{2r}(v) = \\\\int _{-\\\\infty }^v G_{2r}( u) \\\\, {\\\\rm d}u = \\\\sum _{\\\\ell =0}^{r-1} \\\\frac{(-1)^\\\\ell }{2^\\\\ell \\\\ell !}\",\n \"\\\\phi ^{(2\\\\ell -1)}(v).$ In fact, both $G_{2r}$ and ${G}_{2r}$ have simpler forms $G_{2r}(u) = p_r(u) \\\\phi (u)$ and ${G}_{2r}(u) = \\\\Phi (u)+ P_r(u) \\\\phi (u)$ , where $p_r(\\\\cdot )$ and $P_r(\\\\cdot )$ are polynomials in $u$ .\",\n \"For example, $p_1(u) = 1$ , $P_1(u)=0$ , $p_2(u) =(-u^2+3)/2$ , $P_2(u)=u/2$ , $p_3(u)= (u^4 - 10 u^2 + 15)/8$ , and $P_3(u)=(-u^3 +7u)/8$ .\",\n \"We refer to [65] for more details when $r$ is large.\"\n ],\n [\n \"Proof of Proposition \",\n \"For every $r>0$ , define the ellipse $\\\\Theta (r) = \\\\lbrace \\\\in ^p: \\\\Vert \\\\Vert _{} \\\\le r \\\\rbrace $ and the local vicinity $\\\\Theta ^* = \\\\lbrace \\\\beta \\\\in ^p: \\\\beta - \\\\beta ^* \\\\in \\\\Theta ( \\\\kappa _2^{1/2} h ) \\\\rbrace $ .\",\n \"Let $\\\\eta = \\\\sup \\\\lbrace u \\\\in [0,1] : u (\\\\beta ^*_h -\\\\beta ^*) \\\\in \\\\Theta (\\\\kappa _2^{1/2} h) \\\\rbrace $ and $\\\\widetilde{\\\\beta }^* = \\\\beta ^* + \\\\eta (\\\\beta ^*_h -\\\\beta ^*)$ .\",\n \"By definition, $\\\\eta = 1$ if $\\\\beta ^*_h \\\\in \\\\Theta ^*$ and $\\\\eta <1$ if $\\\\beta ^*_h \\\\notin \\\\Theta ^*$ .\",\n \"In the latter case, $\\\\widetilde{\\\\beta }^* \\\\in \\\\partial \\\\Theta ^*$ .\",\n \"By the convexity of $\\\\beta \\\\mapsto Q_h(\\\\beta )$ and Lemma C.1 in the supplementary material of [68], $0 & \\\\le \\\\langle \\\\nabla Q_h( \\\\widetilde{\\\\beta }^* ) - \\\\nabla Q_h(\\\\beta ^*) , \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\rangle \\\\nonumber \\\\\\\\& \\\\le \\\\eta \\\\cdot \\\\langle \\\\nabla Q_h(\\\\beta ^*_h ) - \\\\nabla Q_h(\\\\beta ^*) , \\\\beta ^*_h - \\\\beta ^* \\\\rangle = \\\\langle - \\\\nabla Q_h(\\\\beta ^*) , \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\rangle .\",\n \"$ It follows from the mean value theorem for vector-valued functions that $\\\\nabla Q_h( \\\\widetilde{\\\\beta }^* ) - \\\\nabla Q_h(\\\\beta ^*) = \\\\int _0^1 \\\\nabla ^2 Q_h( (1-t) \\\\beta ^* + t \\\\widetilde{\\\\beta }^* ) \\\\, {\\\\rm d} t \\\\, \\\\big ( \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\bigr ) , $ where $\\\\nabla ^2 Q_h( \\\\beta ) = \\\\bigl \\\\lbrace K_h( y - \\\\langle , \\\\beta \\\\rangle ) ^\\\\rbrace $ for $\\\\beta \\\\in ^p$ .\",\n \"With $\\\\delta = \\\\beta - \\\\beta ^*$ , note that $\\\\bigl \\\\lbrace K_h( y - \\\\langle , \\\\beta \\\\rangle ) | \\\\bigr \\\\rbrace = \\\\frac{1}{h} \\\\int _{-\\\\infty }^\\\\infty K\\\\bigg ( \\\\frac{u- \\\\langle , \\\\delta \\\\rangle }{h} \\\\bigg ) f_{\\\\varepsilon |} (u) \\\\, {\\\\rm d} u = \\\\int _{-\\\\infty }^\\\\infty K (v ) f_{\\\\varepsilon |} ( \\\\langle , \\\\delta \\\\rangle + h v ) \\\\, {\\\\rm d} v .\",\n \"\\\\nonumber $ By the Lipschitz continuity of $f_{ \\\\varepsilon |}(\\\\cdot )$ , $\\\\bigl \\\\lbrace K_h( y - \\\\langle , \\\\beta \\\\rangle ) | \\\\bigr \\\\rbrace = f_{ \\\\varepsilon |}( 0 ) + R_h(\\\\delta ) $ with $R_h(\\\\delta )$ satisfying $|R_h(\\\\delta )| \\\\le l_0 \\\\bigl ( |\\\\langle , \\\\delta \\\\rangle | + \\\\kappa _1 h \\\\bigr )$ .\",\n \"Substituting (REF ) into (REF ) and (REF ) yields $& \\\\langle \\\\nabla Q_h( \\\\widetilde{\\\\beta }^* ) - \\\\nabla Q_h(\\\\beta ^*) , \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\rangle \\\\nonumber \\\\\\\\& \\\\ge \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{}^2 - \\\\frac{l_0}{2} | \\\\langle , \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\rangle |^3 - l_0 \\\\kappa _1 h \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{}^2 \\\\nonumber \\\\\\\\& \\\\ge \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{}^2 - \\\\frac{l_0}{2} \\\\mu _3 \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert ^3_{} - l_0 \\\\kappa _1 h \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{}^2 .\",\n \"$ On the other hand, under model (REF ) we have $& \\\\langle - \\\\nabla Q_h(\\\\beta ^*) , \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\rangle \\\\le \\\\Vert ^{-1/2} \\\\nabla Q_h(\\\\beta ^*) \\\\Vert _2 \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{} , \\\\nonumber $ where $\\\\nabla Q_h(\\\\beta ^*) = \\\\lbrace K(-\\\\varepsilon /h) - \\\\tau \\\\rbrace $ .\",\n \"By integration by parts and a Taylor series expansion, $\\\\bigl \\\\lbrace K(-\\\\varepsilon /h) | \\\\bigr \\\\rbrace & = \\\\int _{-\\\\infty }^\\\\infty K(-t/h) \\\\, {\\\\rm d} F_{ \\\\varepsilon |}( t) \\\\nonumber \\\\\\\\& = -\\\\frac{1}{h}\\\\int _{-\\\\infty }^\\\\infty K(-t/h) F_{ \\\\varepsilon |}( t) \\\\, {\\\\rm d}t = \\\\int _{-\\\\infty }^\\\\infty K(u) F_{\\\\varepsilon |}( -h u) \\\\, {\\\\rm d}u \\\\nonumber \\\\\\\\& = \\\\tau + \\\\int _{-\\\\infty }^\\\\infty K(u) \\\\int _{0}^{-hu} \\\\bigl \\\\lbrace f_{ \\\\varepsilon |} ( t) - f_{\\\\varepsilon |} (0) \\\\bigr \\\\rbrace \\\\, {\\\\rm d} t \\\\, {\\\\rm d}u ,\\\\nonumber $ from which it follows that $| \\\\lbrace K(-\\\\varepsilon /h) | \\\\rbrace - \\\\tau | \\\\le \\\\frac{l_0}{2} \\\\kappa _2 h^2$ .\",\n \"Consequently, $& \\\\Vert ^{-1/2} \\\\nabla Q_h(\\\\beta ^*) \\\\Vert _2 = \\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\bigl \\\\lbrace K(-\\\\varepsilon /h) -\\\\tau \\\\bigr \\\\rbrace \\\\langle ^{-1/2} , \\\\rangle \\\\le \\\\frac{l_0}{2} \\\\kappa _2 h^2 .\",\n \"$ Putting together the pieces, we conclude that $\\\\langle - \\\\nabla Q_h(\\\\beta ^*) , \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\rangle \\\\le \\\\frac{l_0}{2} \\\\kappa _2 h^2 \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{} .\",\n \"$ Recall that $f_{\\\\varepsilon | }(0) \\\\ge {f} >0$ almost surely.\",\n \"Combining (REF ) and (REF ) with (REF ) yields ${f} \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{}^2 & \\\\le {f} ^{1/2} \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{} \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{} \\\\nonumber \\\\\\\\& \\\\le \\\\bigg (\\\\frac{\\\\mu _3 +1 }{2} \\\\kappa _2 + \\\\kappa _1 \\\\kappa _2^{1/2} \\\\bigg ) l_0 h^2 \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{} .\",\n \"\\\\nonumber $ Canceling $\\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{}$ on both sides we obtain $\\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{} \\\\le \\\\underbrace{ \\\\bigg (\\\\frac{ \\\\mu _3 +1 }{2} \\\\kappa _2^{1/2} + \\\\kappa _1 \\\\bigg ) }_{= c_K} \\\\frac{l_0 h }{{f} } \\\\kappa _2^{1/2} h = \\\\frac{c_K l_0 h}{{f}} \\\\kappa _2^{1/2} h .\",\n \"\\\\nonumber $ Provided that $h < {f}/(c_K l_0)$ , $\\\\widetilde{\\\\beta }^*$ falls in the interior of $\\\\Theta ^*$ , i.e., $ \\\\Vert \\\\widetilde{\\\\beta }^* - \\\\beta ^* \\\\Vert _{}< \\\\kappa _2^{1/2} h$ , enforcing $\\\\eta =1$ (otherwise, by construction $\\\\widetilde{\\\\beta }^*$ must lie on the boundary which leads to contradiction) and hence $\\\\widetilde{\\\\beta }^* = \\\\beta ^*_h$ .\",\n \"In addition, by (REF ), $Q_h(\\\\cdot )$ is strictly convex in a neighborhood of $\\\\beta ^*_h$ so that $\\\\beta ^*_h$ is the unique minimizer and satisfies the stated bound (REF ).\",\n \"Next, to investigate the leading term in the bias, define $\\\\Delta = ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla Q_h(\\\\beta ^*_h) - \\\\nabla Q_h(\\\\beta ^*) \\\\bigr \\\\rbrace - ^{1/2} ( \\\\beta ^*_h -\\\\beta ^* ).\",\n \"\\\\nonumber $ Again, by the mean value theorem for vector-valued functions, $\\\\Delta & = \\\\Bigg \\\\lbrace ^{-1/2} \\\\int _0^1 \\\\nabla ^2 Q_h((1-t) \\\\beta ^* + t \\\\beta ^*_h) \\\\, {\\\\rm d}t \\\\, ^{-1/2} - _{p} \\\\Bigg \\\\rbrace ^{1/2} ( \\\\beta ^*_h - \\\\beta ^* ) .\",\n \"$ Write $= ^{-1/2}$ , we have $& \\\\Bigg \\\\Vert ^{-1/2} \\\\int _0^1 \\\\nabla ^2 Q_h((1-t) \\\\beta ^* + t \\\\beta ^*_h) \\\\, {\\\\rm d}t \\\\, ^{-1/2} - _p \\\\Bigg \\\\Vert _2 \\\\nonumber \\\\\\\\&=\\\\Bigg \\\\Vert \\\\int _0^1 \\\\int _{-\\\\infty }^\\\\infty K(u) \\\\bigl \\\\lbrace f_{\\\\varepsilon |}( t \\\\langle , \\\\beta ^*_h -\\\\beta ^* \\\\rangle - hu ) - f_{ \\\\varepsilon |}( 0) \\\\bigr \\\\rbrace \\\\, {\\\\rm d} u \\\\, {\\\\rm d} t \\\\, ^\\\\Vert _2 \\\\nonumber \\\\\\\\& \\\\le l_0 \\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\int _0^1 \\\\int _{-\\\\infty }^\\\\infty K(u) \\\\bigl ( | t \\\\langle , \\\\beta ^*_h - \\\\beta ^* \\\\rangle | + h |u | \\\\bigr ) \\\\, {\\\\rm d}u \\\\, {\\\\rm d} t \\\\, \\\\langle , \\\\rangle ^2 \\\\nonumber \\\\\\\\& \\\\le \\\\frac{l_0}{2 } \\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\bigl ( |\\\\langle , \\\\beta ^*_h - \\\\beta ^* \\\\rangle | \\\\langle , \\\\rangle ^2 \\\\bigr ) + \\\\frac{l_0}{{f}} \\\\kappa _1 h \\\\nonumber \\\\\\\\& \\\\le \\\\frac{l_0}{2{f}} \\\\mu _3 \\\\Vert \\\\beta _h^* - \\\\beta ^* \\\\Vert _{} + \\\\frac{l_0}{{f}} \\\\kappa _1 h. \\\\nonumber $ This bound, together with (REF ), implies $\\\\Vert \\\\Delta \\\\Vert _2 \\\\le \\\\frac{l_0 }{{f}} \\\\bigl ( 0.5 \\\\mu _3 \\\\Vert \\\\beta _h^* - \\\\beta ^* \\\\Vert _{} + \\\\kappa _1 h \\\\bigr ) \\\\Vert \\\\beta _h^* - \\\\beta ^* \\\\Vert _{} .\",\n \"$ Moreover, applying a second-order Taylor series expansion to $f_{\\\\varepsilon |}$ yields $& \\\\bigl \\\\lbrace K(-\\\\varepsilon /h) | \\\\bigr \\\\rbrace - \\\\tau \\\\nonumber \\\\\\\\& = \\\\int _{-\\\\infty }^\\\\infty K(u) \\\\int _{0}^{-h u} \\\\bigl \\\\lbrace f_{ \\\\varepsilon |} ( t) - f_{\\\\varepsilon |} ( 0) \\\\bigr \\\\rbrace \\\\, {\\\\rm d} t \\\\, {\\\\rm d}u \\\\nonumber \\\\\\\\& = \\\\frac{1 }{2} \\\\kappa _2 h^2 \\\\cdot f_{ \\\\varepsilon |}^{\\\\prime }(0 ) + \\\\int _{-\\\\infty }^\\\\infty \\\\int _{0}^{-hu} \\\\int _0^t K(u) \\\\bigl \\\\lbrace f_{ \\\\varepsilon |}^{\\\\prime }(v) - f_{ \\\\varepsilon |}^{\\\\prime }( 0 ) \\\\bigr \\\\rbrace \\\\, {\\\\rm d} v \\\\, {\\\\rm d} t \\\\, {\\\\rm d}u .\",\n \"\\\\nonumber $ Recalling that $\\\\nabla Q_h(\\\\beta ^*) = \\\\lbrace K(-\\\\varepsilon /h) - \\\\tau \\\\rbrace $ , we get $\\\\bigg \\\\Vert ^{-1} \\\\nabla Q_h(\\\\beta ^*) - \\\\frac{1}{2} \\\\kappa _2 h^2 \\\\cdot ^{-1} \\\\bigl \\\\lbrace f_{\\\\varepsilon |}^{\\\\prime }( 0)\\\\bigr \\\\rbrace \\\\bigg \\\\Vert _{} \\\\le \\\\frac{l_1}{6 {f}^{1/2}} \\\\kappa _3 h^3 .\",\n \"$ Finally, combining (REF ) and (REF ) proves (REF ).\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"Recall from (REF ) that $\\\\hat{\\\\beta }_h $ is the smoothed quantile regression estimator obtained by minimizing $\\\\hat{Q}_{ h} (\\\\cdot )$ .\",\n \"For any given $\\\\tau \\\\in (0,1)$ and $h>0$ , it follows from the optimality of $\\\\hat{\\\\beta }_h $ that $\\\\nabla \\\\hat{Q}_h ( \\\\hat{\\\\beta }_h ) = \\\\mathbf {0} $ and by convexity, $ \\\\langle \\\\nabla \\\\hat{Q}_h ( \\\\hat{\\\\beta }_h ) - \\\\nabla \\\\hat{Q}_h ( \\\\beta ^*) , \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\rangle \\\\ge 0$ .\",\n \"Recall from the proof of Proposition REF that $\\\\Theta (t) = \\\\lbrace \\\\in ^p : \\\\Vert \\\\Vert _{} \\\\le t \\\\rbrace $ for $t\\\\ge 0$ .\",\n \"For some $r>0$ to be determined, let $\\\\eta = \\\\sup \\\\lbrace u \\\\in [0,1] : u (\\\\hat{\\\\beta }_h -\\\\beta ^*) \\\\in \\\\Theta (r) \\\\rbrace $ and $\\\\widetilde{\\\\beta }= \\\\beta ^* + \\\\eta (\\\\hat{\\\\beta }_h -\\\\beta ^*)$ .\",\n \"Thus, by definition, $\\\\eta =1$ if $\\\\hat{\\\\beta }_h \\\\in \\\\beta ^* + \\\\Theta (r)$ , and $\\\\eta <1$ if $\\\\hat{\\\\beta }_h \\\\notin \\\\beta ^* + \\\\Theta (r)$ .\",\n \"In the latter case, $\\\\widetilde{\\\\beta }\\\\in \\\\beta ^* + \\\\partial \\\\Theta (r)$ , where $\\\\partial \\\\Theta (r)$ is the boundary of $\\\\Theta (r)$ .\",\n \"The symmetrized Bregman divergence associated with $\\\\hat{Q}_h(\\\\cdot )$ for points $\\\\beta _1, \\\\beta _2$ is given by $ D (\\\\beta _1, \\\\beta _2) = \\\\langle \\\\nabla \\\\hat{Q}_h (\\\\beta _1)- \\\\nabla \\\\hat{Q} _h (\\\\beta _2), \\\\beta _1 - \\\\beta _2 \\\\rangle .$ By Lemma C.1 in [68], the three points $\\\\hat{\\\\beta }_h$ , $\\\\widetilde{\\\\beta }$ , and $\\\\beta ^*$ satisfy $D( \\\\widetilde{\\\\beta }, \\\\beta ^* ) \\\\le \\\\eta D ( \\\\hat{\\\\beta }, \\\\beta ^*)$ .\",\n \"Together with the properties that $ \\\\widetilde{\\\\beta }- \\\\beta ^* = \\\\eta (\\\\hat{\\\\beta }_h - \\\\beta ^* ) \\\\in \\\\Theta (r)$ and $\\\\nabla \\\\hat{Q}_h (\\\\hat{\\\\beta }_h) = \\\\textbf {0}$ , we obtain $\\\\Vert \\\\widetilde{\\\\beta }-\\\\beta ^* \\\\Vert _{}^2 \\\\cdot \\\\frac{ D( \\\\widetilde{\\\\beta }, \\\\beta ^* ) }{\\\\Vert \\\\widetilde{\\\\beta }-\\\\beta ^* \\\\Vert _{}^2} \\\\le - \\\\eta \\\\langle \\\\nabla \\\\hat{Q}_h (\\\\beta ^* ), \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\rangle \\\\le \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h (\\\\beta ^* ) \\\\Vert _2 \\\\cdot \\\\Vert \\\\widetilde{\\\\beta }- \\\\beta ^* \\\\Vert _{} .\",\n \"\\\\nonumber $ Canceling $\\\\Vert \\\\widetilde{\\\\beta }-\\\\beta ^* \\\\Vert _{}$ on both sides, we obtain $\\\\Vert \\\\widetilde{\\\\beta }-\\\\beta ^* \\\\Vert _{} \\\\le \\\\frac{ \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h (\\\\beta ^* ) \\\\Vert _2 }{\\\\inf _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\lbrace D( \\\\beta , \\\\beta ^* ) / \\\\Vert \\\\beta -\\\\beta ^* \\\\Vert _{}^2 \\\\rbrace } .\",\n \"$ The following two lemmas provide, respectively, upper and lower bounds on $\\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h (\\\\beta ^* ) \\\\Vert _2$ and $\\\\inf _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\lbrace D( \\\\beta , \\\\beta ^* ) / \\\\Vert \\\\beta -\\\\beta ^* \\\\Vert _{}^2\\\\rbrace $ .\",\n \"Assume that Conditions REF –REF hold.\",\n \"For any $t\\\\ge 0$ , $\\\\bigl \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h (\\\\beta ^* ) \\\\bigr \\\\Vert _2 \\\\le 1.46 \\\\upsilon _0 \\\\Biggl \\\\lbrace C_\\\\tau ^{1/2 }\\\\sqrt{ \\\\frac{ 4p +2 t }{n}} + 2\\\\max (1-\\\\tau , \\\\tau ) \\\\frac{ 2p+t }{n} \\\\Biggr \\\\rbrace + \\\\frac{1}{2} l_0\\\\kappa _2 h^2 $ with probability at least $1-e^{-t}$ , where $C_\\\\tau = \\\\tau (1-\\\\tau ) + (1+\\\\tau ) l_0 \\\\kappa _2 h^2$ .\",\n \"Under Condition REF , there exist constants $\\\\bar{f}_{ h} \\\\ge {f}_{ h}$ such that ${f}_{ h} \\\\le \\\\inf _{|u|\\\\le h/2} f_{\\\\varepsilon | } (u) \\\\le \\\\sup _{|u|\\\\le h/2} f_{\\\\varepsilon | } (u) \\\\le \\\\bar{f}_{ h} $ almost surely.\",\n \"In fact, by the Lipschitz continuity, we can take $\\\\bar{f}_{ h} = \\\\bar{f} + l_0 h/2$ and ${f}_{ h} = {f} - l_0 h/2$ .\",\n \"Throughout the following, we assume (REF ) holds.\",\n \"For every $\\\\delta \\\\in (0,1]$ , we define $\\\\eta _\\\\delta \\\\ge 0$ as $\\\\eta _\\\\delta = \\\\inf \\\\bigl \\\\lbrace \\\\eta >0 : \\\\bigl \\\\lbrace \\\\langle \\\\delta , \\\\rangle ^2 \\\\mathbb {1}\\\\big ( |\\\\langle \\\\delta , \\\\rangle | > \\\\eta _\\\\delta \\\\big ) \\\\bigr \\\\rbrace \\\\le \\\\delta ~\\\\mbox{ for all } \\\\delta \\\\in \\\\mathbb {S}^{p-1} \\\\bigr \\\\rbrace , $ where $= ^{-1/2} $ is the standardized predictor that satisfies $(^ = _p$ .\",\n \"It can be shown that $\\\\eta _\\\\delta $ depends only on $\\\\delta $ and $\\\\upsilon _0$ in Condition REF , and the map $\\\\delta \\\\mapsto \\\\eta _\\\\delta $ is non-increasing with $\\\\eta _\\\\delta \\\\downarrow 0$ as $\\\\delta \\\\uparrow 1$ .\",\n \"For any $t>0$ , $0 < h < 2 {f} / l_0$ and $0< r \\\\le h/(4 \\\\eta _{1/4})$ with $\\\\eta _{1/4}$ defined in (REF ), we have $\\\\inf _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r) } \\\\frac{ D( \\\\beta , \\\\beta ^* ) }{ \\\\kappa _l \\\\Vert \\\\beta - \\\\beta ^* \\\\Vert _{}^2 } \\\\ge \\\\frac{3}{4} {f}_{h } - \\\\bar{f}_{ h}^{1/2} \\\\left( \\\\frac{5}{4}\\\\sqrt{\\\\frac{h p}{r^2 n}} + \\\\sqrt{\\\\frac{h t}{8r^2 n}} \\\\, \\\\right) - \\\\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .\",\n \"In view of (REF ), (REF ), and (REF ), we take $r = h/(4 \\\\eta _{1/4})$ so that as long as $(p+t)/n \\\\lesssim h\\\\lesssim 1$ , $\\\\Vert \\\\widetilde{\\\\beta }-\\\\beta ^* \\\\Vert _{} < \\\\frac{3 \\\\upsilon _0 }{ \\\\kappa _l {f} } \\\\Biggl \\\\lbrace C_\\\\tau ^{1/2} \\\\sqrt{ \\\\frac{ 4p + 2t }{n}} + 2\\\\max (1-\\\\tau , \\\\tau ) \\\\frac{2 p+t }{n} \\\\Biggr \\\\rbrace + \\\\frac{l_0 \\\\kappa _2 }{ \\\\kappa _l {f} } h^2 $ with probability at least $1-2e^{-t}$ .\",\n \"With this choice of $r$ , we see that under the constraint $\\\\sqrt{(p+t)/n} \\\\lesssim h \\\\lesssim 1$ , $\\\\Vert \\\\widetilde{\\\\beta }-\\\\beta ^* \\\\Vert _{} 0$ , define the function $\\\\varphi _R(u)=u^2 \\\\mathbb {1}(|u|\\\\le R/2) + \\\\lbrace u(u) -R \\\\rbrace ^2 \\\\mathbb {1}( R/2 < |u| \\\\le R)$ , which is $R$ -Lipschitz continuous and satisfies $u^2 \\\\mathbb {1}(|u| \\\\le R / 2) \\\\le \\\\varphi _R(u) \\\\le u^2 \\\\mathbb {1}(|u| \\\\le R ) .\",\n \"$ Moreover, note that $\\\\varphi _{cR}(cu) = c^2 \\\\varphi _{R}(u)$ for any $c > 0$ and $\\\\varphi _0(u) = 0$ .\",\n \"For $\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)$ , consider a change of variable $\\\\delta = ^{1/2} (\\\\beta -\\\\beta ^*)/\\\\Vert \\\\beta -\\\\beta ^* \\\\Vert _{} $ so that $\\\\delta \\\\in \\\\mathbb {S}^{p-1}$ .\",\n \"Together, () and (REF ) imply $\\\\frac{ D(\\\\beta , \\\\beta ^* ) }{\\\\Vert \\\\beta -\\\\beta ^* \\\\Vert _{}^2} \\\\ge \\\\kappa _l \\\\cdot \\\\underbrace{ \\\\frac{1}{n h } \\\\sum _{i=1}^n\\\\omega _i \\\\cdot \\\\varphi _{ h/(2r) } (\\\\langle _i, \\\\delta \\\\rangle ) }_{=: D_0( \\\\delta )} , $ where $\\\\omega _i := \\\\mathbb {1}( | \\\\varepsilon _i | \\\\le h/2 )$ with $\\\\varepsilon _i = y_i - \\\\langle _i, \\\\beta ^* \\\\rangle $ , and $_i = ^{-1/2} _i$ .\",\n \"Next, we bound the expectation $\\\\lbrace D_0(\\\\delta ) \\\\rbrace $ and the random fluctuation $ D_0(\\\\delta ) - \\\\lbrace D_0(\\\\delta )\\\\rbrace $ , separately, starting with the former.\",\n \"By (REF ), ${f}_{ h} h \\\\le ( \\\\omega _i | _i ) = \\\\int _{-h/2}^{h/2} f_{\\\\varepsilon _i | _i} (u) \\\\, {\\\\rm d}u \\\\le \\\\bar{f}_{ h} h. $ Moreover, define $\\\\xi _{\\\\delta } = \\\\langle , \\\\delta \\\\rangle $ such that $(\\\\xi _{\\\\delta }^2) =1$ .\",\n \"By (REF ) and (REF ), $& \\\\bigl \\\\lbrace \\\\omega _i \\\\cdot \\\\varphi _{ h/(2r)} (\\\\langle _i , \\\\delta \\\\rangle ) \\\\bigr \\\\rbrace \\\\ge {f}_{ h} h \\\\cdot \\\\varphi _{ h/(2r) } (\\\\langle _i , \\\\delta \\\\rangle )\\\\ge {f}_{ h} h \\\\cdot \\\\bigl [ 1 - \\\\bigl \\\\lbrace \\\\xi _{\\\\delta }^2 \\\\mathbb {1}_{ | \\\\xi _{\\\\delta } | > h/(4r) } \\\\bigr \\\\rbrace \\\\bigr ] , \\\\nonumber $ from which it follows that $\\\\inf _{ \\\\delta \\\\in \\\\mathbb {S}^{p-1} } \\\\lbrace D_0(\\\\delta ) \\\\rbrace \\\\ge {f}_h \\\\cdot \\\\left( 1 - \\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\bigl \\\\lbrace \\\\langle , \\\\rangle ^2 \\\\mathbb {1}_{ |\\\\langle , \\\\rangle | >h/(4r)} \\\\bigr \\\\rbrace \\\\right) .\",\n \"$ By the definition of $\\\\eta _\\\\delta $ in (REF ), we see that as long as $00$ , $\\\\Delta & \\\\le (\\\\Delta ) + \\\\lbrace ( \\\\Delta )\\\\rbrace ^{1/2} \\\\sqrt{\\\\frac{h t}{4 r^2 n}} + \\\\sqrt{ \\\\bar{f}_h h \\\\frac{ t }{ 8 r^2 n }} + \\\\frac{h t}{48 r^2 n} \\\\le \\\\frac{5}{4} (\\\\Delta ) + \\\\sqrt{ \\\\bar{f}_h h \\\\frac{ t }{ 8 r^2 n }} + \\\\frac{h t}{3 r^2 n} $ with probability at least $1-e^{-t}$ , where the second step follows from the inequality that $ab\\\\le a^2/4 + b^2$ for all $a,b\\\\in $ .\",\n \"It then remains to bound the expectation $(\\\\Delta ) $ .\",\n \"Define $(\\\\delta ; _i) &= \\\\frac{1}{h } \\\\omega _i\\\\varphi _{ h/(2r) } (\\\\langle _i, \\\\delta \\\\rangle )= \\\\frac{1}{h} \\\\varphi _{ \\\\omega _i h/(2r) } (\\\\langle \\\\omega _i _i , \\\\delta \\\\rangle ) , \\\\ \\\\ \\\\delta \\\\in \\\\mathbb {S}^{p-1} ,\\\\nonumber $ where $_i = (_i , \\\\varepsilon _i )$ and $\\\\omega _i = \\\\mathbb {1} ( | \\\\varepsilon _i | \\\\le h/2 ) \\\\in \\\\lbrace 0, 1\\\\rbrace $ .\",\n \"By Rademacher symmetrization, $(\\\\Delta ) \\\\le 2 \\\\Bigg \\\\lbrace \\\\sup _{ \\\\delta \\\\in \\\\mathbb {S}^{p-1} } \\\\frac{1}{n} \\\\sum _{i=1}^ne_i (\\\\delta ; _i) \\\\Bigg \\\\rbrace , \\\\nonumber $ where $e_1,\\\\ldots , e_n$ are independent Rademacher random variables.\",\n \"Since $\\\\varphi _R(\\\\cdot )$ is $R$ -Lipschitz, $(\\\\delta ; _i)$ is a $(1/2r)$ -Lipschitz function in $\\\\langle \\\\omega _i _i , \\\\delta \\\\rangle $ , i.e., for any sample $_i = (_i, \\\\varepsilon _i)$ and parameters $\\\\delta , \\\\delta ^{\\\\prime }\\\\in \\\\mathbb {S}^{p-1}$ , $\\\\bigl | (\\\\delta ; _i) - (\\\\delta ^{\\\\prime }; _i) \\\\bigr | \\\\le \\\\frac{1}{2r} \\\\bigl | \\\\langle \\\\omega _i _i , \\\\delta \\\\rangle - \\\\langle \\\\omega _i _i , \\\\delta ^{\\\\prime } \\\\rangle \\\\bigr | .$ Moreover, observe that $(\\\\delta ; _i) = 0$ for any $\\\\delta $ such that $ \\\\langle \\\\omega _i _i , \\\\delta \\\\rangle = 0$ .\",\n \"With the above preparations, we are ready to use Talagrand's contraction principle to bound $( \\\\Delta )$ .\",\n \"Define the subset $T\\\\subseteq ^n$ as $T = \\\\bigl \\\\lbrace = (t_1, \\\\ldots , t_n)^ t_i = \\\\langle \\\\omega _i_i , \\\\delta \\\\rangle , i=1,\\\\ldots , n, \\\\, \\\\delta \\\\in \\\\mathbb {S}^{p-1} \\\\bigr \\\\rbrace ,$ and contractions $\\\\phi _i:\\\\rightarrow $ as $\\\\phi _i(t) = (2r/h) \\\\cdot \\\\varphi _{ h \\\\omega _i/(2r)} (t)$ .\",\n \"By (REF ), $|\\\\phi (t)-\\\\phi (s)|\\\\le |t-s|$ for all $t, s \\\\in $ .\",\n \"Applying Talagrand's contraction principle (see, e.g., Theorem 4.12 and (4.20) in [64]), we have $(\\\\Delta ) & \\\\le 2 \\\\Bigg \\\\lbrace \\\\sup _{ \\\\delta \\\\in \\\\mathbb {S}^{p-1} } \\\\frac{1}{n} \\\\sum _{i=1}^ne_i (\\\\delta ; _i) \\\\Bigg \\\\rbrace = \\\\frac{1}{r} \\\\Bigg \\\\lbrace \\\\sup _{\\\\in T} \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\phi _i(t_i) \\\\Bigg \\\\rbrace \\\\le \\\\frac{1}{r} \\\\Bigg ( \\\\sup _{\\\\in T} \\\\frac{1}{n} \\\\sum _{i=1}^ne_i t_i \\\\Bigg ) \\\\nonumber \\\\\\\\&= \\\\frac{1}{r}\\\\Bigg \\\\lbrace \\\\sup _{ \\\\delta \\\\in \\\\mathbb {S}^{p-1} } \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\langle \\\\omega _i _i , \\\\delta \\\\rangle \\\\Bigg \\\\rbrace \\\\le \\\\frac{1}{r} \\\\bigg \\\\Vert \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\omega _i _i \\\\bigg \\\\Vert _2 \\\\le \\\\frac{1}{r} \\\\sqrt{ \\\\bar{f}_h h \\\\frac{p}{n} }.\",\n \"\\\\nonumber $ This, together with (REF ) and (REF ), yields $\\\\Delta \\\\le \\\\bar{f}_h^{1/2} \\\\Biggl ( \\\\frac{5}{4}\\\\sqrt{\\\\frac{h p}{r^2 n}} + \\\\sqrt{\\\\frac{h t}{8r^2 n}} \\\\Biggr ) + \\\\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .\",\n \"Finally, combining (), (REF ), (REF ), and (REF ) proves (REF ).\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"We keep the notation used in the proof of Theorem REF , and for any $t\\\\ge 0$ , let $r=r(n,p,t) \\\\asymp \\\\sqrt{(p+t)/n} + h^2 >0$ be such that $\\\\lbrace \\\\hat{\\\\beta }_h \\\\in \\\\beta ^* + \\\\Theta (r) \\\\rbrace \\\\ge 1-2e^{-t}$ , provided $\\\\sqrt{(p+t)/n} \\\\lesssim h \\\\lesssim 1$ .\",\n \"Define the vector-valued random process $\\\\Delta (\\\\beta ) = ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla \\\\hat{Q}_h(\\\\beta ) - \\\\nabla \\\\hat{Q}_h(\\\\beta ^*) - ( \\\\beta - \\\\beta ^* ) \\\\bigr \\\\rbrace , $ where $= \\\\lbrace f_{\\\\varepsilon | } (0) ^$ .\",\n \"Since $\\\\hat{\\\\beta }_h$ falls in a local neighborhood of $\\\\beta ^*$ with high probability, it suffices to bound the local fluctuation $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r) } \\\\Vert \\\\Delta (\\\\beta ) \\\\Vert _2$ .\",\n \"By the triangle inequality, $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r) } \\\\Vert \\\\Delta (\\\\beta ) \\\\Vert _2 \\\\le \\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r) } \\\\Vert \\\\Delta (\\\\beta ) \\\\Vert _2+ \\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r) } \\\\Vert \\\\Delta (\\\\beta ) - \\\\Delta (\\\\beta ) \\\\Vert _2 :=I_1+I_2.$ We now provide upper bounds for $I_1$ and $I_2$ , respectively.\",\n \"Upper bound for $I_1$ : By the mean value theorem for vector-valued functions, $\\\\Delta (\\\\beta ) & = ^{-1/2} \\\\biggl \\\\langle \\\\int _0^1 \\\\nabla ^2 Q_h\\\\bigl ( (1-t) \\\\beta ^* + t \\\\beta \\\\bigr ) {\\\\rm d}t , \\\\beta - \\\\beta ^* \\\\biggr \\\\rangle - ^{-1/2} (\\\\beta - \\\\beta ^* ) \\\\nonumber \\\\\\\\& =\\\\biggl \\\\langle ^{-1/2} \\\\int _0^1 \\\\nabla ^2 Q_h\\\\bigl ( (1-t) \\\\beta ^* + t \\\\beta \\\\bigr ) {\\\\rm d}t \\\\, ^{-1/2} - _0 , ^{1/2} ( \\\\beta - \\\\beta ^* ) \\\\biggr \\\\rangle , \\\\nonumber $ where $_0 := ^{-1/2} ^{-1/2} = \\\\lbrace f_{\\\\varepsilon | }(0) ^$ .\",\n \"By law of iterative expectation and by a change of variable, $^{-1/2} \\\\nabla ^2 Q_h(\\\\beta ) ^{-1/2} & = \\\\bigl \\\\lbrace K_h( \\\\langle , \\\\beta \\\\rangle - y ) ^\\\\rbrace = \\\\Biggl \\\\lbrace \\\\int _{-\\\\infty }^\\\\infty K(u) f_{\\\\varepsilon | }(\\\\langle , \\\\beta - \\\\beta ^* \\\\rangle - h u ) \\\\, {\\\\rm d} u \\\\cdot ^\\\\rbrace .\",\n \"\\\\nonumber $ For notational convenience, let $= ^{-1/2} ( \\\\beta -\\\\beta ^*)$ with $\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)$ , so that $\\\\Vert \\\\Vert _2 \\\\le r$ and $\\\\nabla ^2 Q_h \\\\bigl ( (1-t) \\\\beta ^* + t \\\\beta \\\\bigr ) = \\\\Biggl \\\\lbrace \\\\int _{-\\\\infty }^\\\\infty K(u) f_{\\\\varepsilon | }( t \\\\langle ,\\\\rangle - h u ) \\\\, {\\\\rm d} u \\\\cdot ^\\\\rbrace .$ By the Lipschitz continuity of $f_{\\\\varepsilon |}(\\\\cdot )$ , i.e.\",\n \"$|f_{\\\\varepsilon |}(u)- f_{\\\\varepsilon |}(0)| \\\\le l_0 |u |$ for all $u \\\\in $ almost surely for $$ , we have $& \\\\bigl \\\\Vert ^{-1/2} \\\\nabla ^2 Q_h\\\\bigl ( (1-t) \\\\beta ^* + t \\\\beta \\\\bigr ) ^{-1/2} - _0 \\\\bigr \\\\Vert _2 \\\\nonumber \\\\\\\\& = \\\\Biggl \\\\Vert \\\\int K(u) \\\\bigl \\\\lbrace f_{\\\\varepsilon |} ( t \\\\langle , \\\\rangle - h u) - f_{\\\\varepsilon |} ( 0 ) \\\\bigr \\\\rbrace \\\\, {\\\\rm d} u \\\\, \\\\cdot ^\\\\Vert _2 \\\\nonumber \\\\\\\\& \\\\le l_0 t \\\\sup _{\\\\Vert \\\\Vert _2 =1} \\\\bigl ( \\\\langle , \\\\rangle ^2 |\\\\langle , \\\\rangle | \\\\bigr )+ l_0 \\\\kappa _1 h \\\\sup _{\\\\Vert \\\\Vert _2 =1} \\\\langle , \\\\rangle ^2 \\\\nonumber \\\\\\\\& \\\\le l_0 t \\\\,\\\\Biggl ( \\\\sup _{\\\\Vert \\\\Vert _2 =1} | \\\\langle , \\\\rangle |^3 \\\\Biggr )^{2/3} \\\\bigl ( |\\\\langle , \\\\rangle |^3 \\\\bigr )^{1/3} + l_0 \\\\kappa _1 h \\\\le l_0 \\\\big ( \\\\mu _3 r t + \\\\kappa _1 h \\\\big ),$ where the third inequality holds by the Cauchy-Schwarz inequality.\",\n \"Consequently, $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r) } \\\\Vert \\\\Delta ( \\\\beta ) \\\\Vert _2 \\\\le l_0 \\\\bigl (\\\\mu _3 r /2 + \\\\kappa _1 h \\\\bigr ) \\\\cdot r .\",\n \"$ Upper bound for $I_2$ : Next, we provide an upper bound for $\\\\Delta (\\\\beta ) - \\\\Delta (\\\\beta )$ .\",\n \"Define the centered gradient process $G(\\\\beta ) = ^{-1/2} \\\\lbrace \\\\nabla \\\\hat{Q}_h(\\\\beta ) - \\\\nabla Q_h(\\\\beta ) \\\\rbrace $ , so that $\\\\Delta (\\\\beta ) - \\\\Delta (\\\\beta ) = G(\\\\beta ) - G(\\\\beta ^*)$ .\",\n \"Again, by a change of variable $= ^{ 1/2} ( \\\\beta - \\\\beta ^*)$ , we have $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert \\\\Delta (\\\\beta ) - \\\\Delta (\\\\beta ) \\\\Vert _2& \\\\le \\\\sup _{ \\\\beta \\\\in \\\\beta ^* + \\\\Theta (r) } \\\\Vert G(\\\\beta ) - G(\\\\beta ^*) \\\\Vert _2 \\\\nonumber \\\\\\\\& = \\\\sup _{ \\\\Vert \\\\Vert _2 \\\\le r } \\\\Vert \\\\underbrace{ G(\\\\beta ^* + ^{-1/2} ) - G(\\\\beta ^*) }_{ =: \\\\Delta _0() } \\\\Vert _2 .\\\\nonumber $ We will employ Theorem A.3 in [67] to bound the supremum $\\\\sup _{ \\\\Vert \\\\Vert _2 \\\\le r } \\\\Vert \\\\Delta _0() \\\\Vert _2$ , where $\\\\Delta _0(\\\\cdot )$ defined above satisfies $\\\\Delta _0(\\\\textbf {0}) = \\\\textbf {0}$ and $\\\\lbrace \\\\Delta _0()\\\\rbrace =\\\\textbf {0}$ .\",\n \"Taking the gradient with respect to $$ yields $\\\\nabla \\\\Delta _0 () = \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\bigl \\\\lbrace K_{i,} _i _i^ \\\\bigl ( K_{i,} _i _i^) \\\\bigr \\\\rbrace , \\\\nonumber $ where $K_{i,} : = K_h(\\\\langle _i, \\\\rangle - \\\\varepsilon _i )$ satisfies $0\\\\le K_{i,} \\\\le \\\\kappa _u h^{-1}$ .\",\n \"For any $, \\\\in \\\\mathbb {S}^{p-1}$ and $\\\\lambda \\\\in $ , using the elementary inequality $|e^u- 1 - u| \\\\le u^2 e^{|u|}/2$ , we obtain $& \\\\exp \\\\bigl \\\\lbrace \\\\lambda n^{1/2} \\\\langle , \\\\nabla \\\\Delta _0 () \\\\rangle \\\\bigr \\\\rbrace \\\\nonumber \\\\\\\\& \\\\le \\\\prod _{i=1}^n \\\\Biggl \\\\lbrace 1 + \\\\frac{\\\\lambda ^2 }{2n} e^{ \\\\frac{ \\\\bar{f} |\\\\lambda | }{ \\\\sqrt{n}} | \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle |}\\\\bigl \\\\lbrace K_{i,} \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle - ( K_{i,} \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle ) \\\\bigr \\\\rbrace ^2 e^{\\\\frac{ \\\\kappa _u|\\\\lambda | }{ h \\\\sqrt{n}} | \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle | } \\\\Biggr \\\\rbrace \\\\nonumber \\\\\\\\& \\\\le \\\\prod _{i=1}^n \\\\Biggl \\\\lbrace 1 + \\\\frac{\\\\lambda ^2 }{2n} e^{ \\\\frac{ \\\\bar{f} |\\\\lambda | }{ \\\\sqrt{n}} }\\\\bigl \\\\lbrace K_{i,} \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle - ( K_{i,} \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle ) \\\\bigr \\\\rbrace ^2 e^{\\\\frac{ \\\\kappa _u|\\\\lambda | }{ h \\\\sqrt{n}} | \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle | } \\\\Biggr \\\\rbrace , $ where we use the bound $| \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle | \\\\le (\\\\langle _i, \\\\rangle ^2 )^{1/2} (\\\\langle _i, \\\\rangle ^2 )^{1/2} = 1$ in the second inequality.\",\n \"Moreover, the first and second conditional moments of $K_{i,}$ can be rewritten as follows: $ \\\\,\\\\bigl ( K_{i,} | _i \\\\bigr ) = \\\\frac{1}{h} \\\\int _{-\\\\infty }^\\\\infty K \\\\biggl ( \\\\frac{\\\\langle _i, \\\\rangle - t}{h} \\\\biggr ) f_{\\\\varepsilon _i(\\\\tau ) | _i } (t) \\\\, {\\\\rm d} t = \\\\int _{-\\\\infty }^\\\\infty K(u) f_{\\\\varepsilon _i (\\\\tau ) | _i } (\\\\langle _i, \\\\rangle - hu ) \\\\, {\\\\rm d} u; \\\\nonumber \\\\\\\\\\\\bigl ( K^2_{i,} | _i \\\\bigr ) = \\\\frac{1}{h^2} \\\\int _{-\\\\infty }^\\\\infty K^2 \\\\biggl ( \\\\frac{\\\\langle _i, \\\\rangle - t}{h} \\\\biggr ) f_{\\\\varepsilon _i(\\\\tau ) | _i } (t) \\\\, {\\\\rm d} t = \\\\frac{1}{h} \\\\int _{-\\\\infty }^\\\\infty K^2(u) f_{\\\\varepsilon _i(\\\\tau ) | _i } (\\\\langle _i, \\\\rangle - hu ) \\\\, {\\\\rm d} u, \\\\nonumber $ from which it follows that $|( K_{i,} | _i ) | \\\\le \\\\bar{f} $ and $( K_{i,}^2 | _i )\\\\le \\\\kappa _u \\\\bar{f} h^{-1}$ almost surely.\",\n \"By the Cauchy-Schwarz inequality and the inequality $ab\\\\le a^2/2+b^2/2$ , $a,b\\\\in $ , we have $& ( \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle )^2 e^{t | \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle | } \\\\nonumber \\\\\\\\& \\\\le ( \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle )^2 e^{ \\\\frac{ t }{ 2 } \\\\langle _i, \\\\rangle ^2 +\\\\frac{ t }{ 2 } \\\\langle _i, \\\\rangle ^2 } \\\\nonumber \\\\\\\\& \\\\le \\\\Bigl ( \\\\langle _i, \\\\rangle ^4 e^{ t \\\\langle _i, \\\\rangle ^2 } \\\\Bigr )^{1/2} \\\\Bigl ( \\\\langle _i, \\\\rangle ^4 e^{ t \\\\langle _i, \\\\rangle ^2 } \\\\Bigr )^{1/2} , ~\\\\mbox{ valid for any } t>0 .\\\\nonumber $ Given a unit vector $$ , let $\\\\chi = \\\\langle , \\\\rangle ^2/(2\\\\upsilon _1)^2$ so that under Condition REF , $( \\\\chi \\\\ge u) \\\\le 2 e^{-2u}$ for any $u\\\\ge 0$ .\",\n \"It follows that $(e^\\\\chi ) = 1 + \\\\int _0^\\\\infty e^u (\\\\chi \\\\ge u) {\\\\rm d}u \\\\le 1 + 2 \\\\int _0^\\\\infty e^{-u} {\\\\rm d}u = 3$ , and $(\\\\chi ^2 e^{\\\\chi } ) = \\\\int _0^\\\\infty (u^2 + 2u) e^{u} (\\\\chi \\\\ge u) {\\\\rm d} u \\\\le 2 \\\\int _0^\\\\infty (u^2 + 2u) e^{-u} {\\\\rm d} u = 8.\",\n \"\\\\nonumber $ Taking the supremum over $\\\\in \\\\mathbb {S}^{p-1}$ , we have $\\\\sup _{\\\\in \\\\mathbb {S}^{p-1} } e^{\\\\langle , \\\\rangle ^2 / (2\\\\upsilon _1)^2 } \\\\le 3~\\\\mbox{ and } ~ \\\\sup _{\\\\in \\\\mathbb {S}^{p-1} } \\\\langle , \\\\rangle ^4 e^{\\\\langle , \\\\rangle ^2 / (2\\\\upsilon _1)^2 } \\\\le 8(2\\\\upsilon _1)^4 .$ Substituting the above bounds into (REF ) yields that, for any $|\\\\lambda | \\\\le \\\\min \\\\lbrace n^{1/2} h/( 4 \\\\kappa _u \\\\upsilon _1^2 ), n^{1/2} / \\\\bar{f} \\\\rbrace $ , $& \\\\exp \\\\bigl \\\\lbrace \\\\lambda n^{1/2} \\\\langle , \\\\nabla \\\\Delta _0 () \\\\rangle \\\\bigr \\\\rbrace \\\\nonumber \\\\\\\\& \\\\le \\\\prod _{i=1}^n \\\\Biggl [ 1 + \\\\frac{ e\\\\lambda ^2 }{2n}\\\\bigl \\\\lbrace K_{i,} \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle - (K_{i,} \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle ) \\\\bigr \\\\rbrace ^2 e^{ | \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle | /(4\\\\upsilon _1^2)} \\\\Biggr ] \\\\nonumber \\\\\\\\& \\\\le \\\\prod _{i=1}^n \\\\Biggl [ 1 + \\\\frac{ e\\\\lambda ^2 }{ n}\\\\bigl ( K_{i,} \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle \\\\bigr )^2 e^{ | \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle | / (4\\\\upsilon _1^2) } \\\\nonumber \\\\\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \\\\frac{e \\\\lambda ^2}{n} \\\\bigl \\\\lbrace \\\\bigl ( K_{i,} \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle \\\\bigr )\\\\bigr \\\\rbrace ^2 e^{ | \\\\langle _i, \\\\rangle \\\\langle _i, \\\\rangle |/ (4\\\\upsilon _1^2) } \\\\Biggr ] \\\\nonumber \\\\\\\\& \\\\le \\\\prod _{i=1}^n \\\\Biggl ( 1 + C_0^2 \\\\frac{\\\\lambda ^2}{2 n h} \\\\Biggr ) \\\\le \\\\exp \\\\bigl \\\\lbrace C_0^2 \\\\lambda ^2 /(2h) \\\\bigr \\\\rbrace ,\\\\nonumber $ where $C_0>0$ depends only on $(\\\\upsilon _1, \\\\kappa _u, \\\\bar{f} )$ .\",\n \"We have thus verified condition (A.4) in [67] with ${\\\\rm g}= \\\\min \\\\lbrace h /( 4\\\\kappa _u \\\\upsilon _1^2 ), 1 / \\\\bar{f} \\\\rbrace (n/2)^{1/2}$ and $\\\\nu _0= C_0h^{-1/2}$ .\",\n \"Applying Theorem A.3 therein, we obtain that with probability at least $1-e^{-t}$ , $\\\\sup _{ \\\\Vert \\\\Vert _2 \\\\le r } \\\\Vert \\\\Delta _0() \\\\Vert _2 \\\\le 6 C_0 r \\\\sqrt{\\\\frac{4 p + 2 t}{n h }} \\\\nonumber $ as long as $h \\\\ge 8 \\\\kappa _u \\\\upsilon _1^2 \\\\sqrt{(2p+t)/n}$ and $n\\\\ge 4 \\\\bar{f}^2 (2p+t)$ .\",\n \"Together with (REF ) and (REF ), this implies that with probability at least $1-e^{-t}$ , $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r) } \\\\Vert \\\\Delta ( \\\\beta ) \\\\Vert _2 \\\\le 6 C_0 r \\\\sqrt{\\\\frac{4 p + 2 t}{n h }} + l_0 \\\\bigl ( \\\\mu _3 r /2 + \\\\kappa _1 h \\\\bigr ) r .$ Recall from the beginning of the proof that $\\\\hat{\\\\beta }_h \\\\in \\\\beta ^* + \\\\Theta (r)$ with probability at least $1-2e^{-t}$ with $r=r(n,p,t) \\\\asymp \\\\sqrt{(p+t)/n} +h^2$ .\",\n \"Combined with (REF ), we conclude that with probability at least $1-3e^{-t}$ , $\\\\Vert \\\\Delta (\\\\hat{\\\\beta }_h ) \\\\Vert _2 \\\\lesssim (p+t)/(h^{1/2}n) + h \\\\sqrt{(p+t)/n} +h^3$ , as claimed.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"Let $\\\\in ^p$ be an arbitrary vector defining a linear functional of interest.\",\n \"Given $h=h_n>0$ , define $S_n = n^{-1/2} \\\\sum _{i=1}^n\\\\gamma _i \\\\xi _i$ and its centered version $S_n^0= S_n - (S_n)$ , where $\\\\xi _i = \\\\tau - K(-\\\\varepsilon _i /h)$ and $\\\\gamma _i = \\\\langle ^{-1} , _i \\\\rangle $ .\",\n \"Moreover, write $\\\\delta _n = (p+\\\\log n)/n$ .\",\n \"By the Lipschitz continuity of $f_{\\\\varepsilon | }$ around 0 and the fundamental theorem of calculus, it can be shown that $| (\\\\xi _i | _i) |\\\\le 0.5 l_0 \\\\kappa _2 h^2$ , from which it follows by the law of iterated expectation that $|(\\\\gamma _i \\\\xi _i) | \\\\le 0.5 l_0 \\\\kappa _2 \\\\Vert ^{-1} \\\\Vert _{} \\\\cdot h^2$ .\",\n \"Applying (REF ) and (REF ) with $t=\\\\log n$ and the triangle inequality, we obtain that under the constraint $\\\\delta _n^{1/2} \\\\lesssim h \\\\lesssim 1$ , $& \\\\bigl | n^{1/2} \\\\langle , \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\rangle - S_n^0 \\\\bigr | \\\\nonumber \\\\\\\\&= n^{1/2} \\\\biggl | \\\\bigg \\\\langle ^{ 1/2} ^{-1} , ^{-1/2} (\\\\hat{\\\\beta }_h - \\\\beta ^* ) - ^{-1/2} \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\bigl \\\\lbrace \\\\tau - K(-\\\\varepsilon _i/h) \\\\bigr \\\\rbrace _i \\\\bigg \\\\rangle \\\\biggr | + | (S_n) | \\\\nonumber \\\\\\\\& \\\\le c_1 \\\\Vert ^{-1} \\\\Vert _{} \\\\cdot n^{1/2} \\\\bigl ( h^{-1/2} \\\\delta _n + h^2 \\\\bigr ) $ with probability at least $1 - 3 n^{-1}$ for some constant $c_1>0$ .\",\n \"For the centered partial sum $S_n^0 = S_n - (S_n) = n^{-1/2} \\\\sum _{i=1}^n(1-) \\\\gamma _i \\\\xi _i$ , we have $\\\\textnormal {var}(S_n^0) = \\\\textnormal {var}(S_n) = (\\\\gamma \\\\xi )^2 - \\\\lbrace (\\\\gamma \\\\xi )\\\\rbrace ^2$ , where $\\\\gamma = \\\\langle ^{-1} , \\\\rangle $ and $\\\\xi = \\\\tau - K(-\\\\varepsilon /h)$ .\",\n \"By the Berry-Esseen inequality (see, e.g., [69]), $\\\\sup _{x\\\\in } \\\\bigl | \\\\bigl \\\\lbrace S_n^0 \\\\le \\\\textnormal {var}(S_n)^{1/2} x \\\\bigr \\\\rbrace - \\\\Phi (x) \\\\bigr | \\\\le \\\\frac{\\\\lbrace |\\\\gamma \\\\xi - (\\\\gamma \\\\xi ) |^3\\\\rbrace }{2 [( \\\\gamma \\\\xi )^2 - \\\\lbrace ( \\\\gamma \\\\xi )\\\\rbrace ^2 ]^{3/2} \\\\sqrt{n} } .\",\n \"$ We have shown that $|(\\\\gamma \\\\xi ) | \\\\lesssim \\\\Vert ^{-1} \\\\Vert _{} \\\\cdot h^2$ .\",\n \"Following a similar argument in the proof of Lemma REF , we have $(\\\\xi ^2| )\\\\lesssim \\\\tau (1-\\\\tau )+h^2$ .\",\n \"For $h$ sufficiently small, $\\\\textnormal {var}(S_n)= \\\\lbrace \\\\tau (1-\\\\tau ) + O(h)\\\\rbrace \\\\Vert ^{-1} \\\\Vert _{}^2$ and $(|\\\\gamma \\\\xi |^3) \\\\le \\\\max (\\\\tau , 1-\\\\tau ) (\\\\xi ^2 |\\\\gamma |^3) \\\\le \\\\mu _3 \\\\lbrace \\\\tau (1-\\\\tau ) + O(h^2) \\\\rbrace \\\\Vert ^{-1} \\\\Vert _{}^3$ .\",\n \"Substituting these bounds into (REF ) yields $\\\\sup _{x\\\\in } \\\\bigl | \\\\bigl \\\\lbrace S_n^0 \\\\le \\\\textnormal {var}(S_n)^{1/2} x \\\\bigr \\\\rbrace - \\\\Phi (x) \\\\bigr | \\\\le c_2 n^{-1/2} $ for some constant $c_2>0$ .\",\n \"Set $\\\\sigma _\\\\tau ^2 = \\\\tau (1-\\\\tau )\\\\, \\\\Vert ^{-1} \\\\Vert _{}^2$ .\",\n \"By an application of Lemma A.7 in the supplement of [66], for sufficiently small $h$ , we have $\\\\sup _{x\\\\in } \\\\bigl | \\\\Phi ( x/\\\\textnormal {var}(S_n)^{1/2} ) -\\\\Phi (x/\\\\sigma _\\\\tau ) \\\\bigr | \\\\le c_3 h. $ Before proceeding, we note that the constants $c_1$ –$c_3$ appeared above are all independent of $$ .\",\n \"Let $G\\\\sim N(0,1)$ .\",\n \"Putting together the above derivations, for any $x\\\\in $ and $\\\\in ^p$ , we obtain $& \\\\bigl ( n^{1/2} \\\\langle , \\\\hat{\\\\beta }_h -\\\\beta ^* \\\\rangle \\\\le x \\\\bigr ) \\\\nonumber \\\\\\\\& \\\\le \\\\bigl \\\\lbrace S_n^0 \\\\le x + c_1 \\\\Vert ^{-1} \\\\Vert _{} \\\\cdot n^{1/2} \\\\bigl ( h^{-1/2} \\\\delta _n + h^2 \\\\bigr ) \\\\bigr \\\\rbrace + 3 n^{-1} \\\\nonumber \\\\\\\\& \\\\le \\\\bigl \\\\lbrace \\\\textnormal {var}(S_n)^{1/2} G \\\\le x+ c_1 \\\\Vert ^{-1} \\\\Vert _{} \\\\cdot n^{1/2} \\\\bigl ( h^{-1/2} \\\\delta _n + h^2 \\\\bigr ) \\\\bigr \\\\rbrace +c_2n^{-1/2} + 3 n^{-1} \\\\nonumber \\\\\\\\&\\\\le \\\\bigl \\\\lbrace \\\\sigma _\\\\tau G \\\\le x+ c_1 \\\\Vert ^{-1} \\\\Vert _{} \\\\cdot n^{1/2} \\\\bigl ( h^{-1/2} \\\\delta _n + h^2 \\\\bigr ) \\\\bigr \\\\rbrace + c_2 n^{-1/2} + c_3 h + 3 n^{-1} \\\\nonumber \\\\\\\\& \\\\le \\\\bigl ( \\\\sigma _\\\\tau G \\\\le x \\\\bigr ) + c_1(2\\\\pi )^{-1/2} \\\\Vert ^{-1} \\\\Vert _{} \\\\cdot n^{1/2} \\\\bigl ( h^{-1/2} \\\\delta _n + h^2 \\\\bigr )/\\\\sigma _{\\\\tau } + c_2 n^{-1/2} + c_3 h^2 + 3 n^{-1} , \\\\nonumber $ where the first, second, and third inequalities holds by (REF ), (REF ), and (REF ), respectively, and the last inequality holds by the fact that for any $a\\\\le b$ , $\\\\Phi (b/\\\\sigma _\\\\tau ) - \\\\Phi (a/\\\\sigma _\\\\tau ) \\\\le (2\\\\pi )^{-1/2}(b-a)/\\\\sigma _\\\\tau $ .\",\n \"A similar argument leads to a series of reverse inequalities.\",\n \"Note the above bounds are independent of $x$ and $$ , and therefore hold uniformly over all $x$ and $$ .\",\n \"Putting together the pieces, we conclude that under the bandwidth requirement $\\\\delta _n^{1/2} \\\\lesssim h \\\\lesssim 1$ , $\\\\sup _{x\\\\in , \\\\, \\\\in ^p } \\\\bigl | \\\\bigl ( n^{1/2} \\\\langle , \\\\hat{\\\\beta }_h -\\\\beta ^* \\\\rangle \\\\le \\\\sigma _\\\\tau x \\\\bigr ) - \\\\Phi (x ) \\\\bigr | \\\\lesssim \\\\frac{p+ \\\\log n}{ (n h)^{1/2}} + n^{1/2} h^2 , \\\\nonumber $ as claimed.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"Keep the notation used in the proof of Theorem REF .\",\n \"With nonnegative weights $w_i$ , the corresponding weighted loss $\\\\hat{Q}_h^\\\\flat $ in (REF ) is convex, and thus the first-order condition $\\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\hat{\\\\beta }_h^\\\\flat ) = \\\\textbf {0}$ holds.\",\n \"We use the same localized argument as in the proof of Theorem REF .\",\n \"For some $r>0$ to be determined, define $\\\\widetilde{\\\\beta }_h^\\\\flat = \\\\beta ^* + \\\\eta (\\\\hat{\\\\beta }_h^\\\\flat - \\\\beta ^*)$ , where $\\\\eta = \\\\sup \\\\lbrace u\\\\in [0,1] : u (\\\\hat{\\\\beta }_h^\\\\flat - \\\\beta ^*) \\\\in \\\\Theta (r) \\\\rbrace $ .\",\n \"Similar to (REF ), we have $\\\\Vert \\\\widetilde{\\\\beta }_h^\\\\flat - \\\\beta ^* \\\\Vert _{} \\\\le \\\\frac{\\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^*) \\\\Vert _2 }{\\\\inf _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} D^\\\\flat (\\\\beta , \\\\beta ^*) / \\\\Vert \\\\beta - \\\\beta ^* \\\\Vert _{}^2 } , $ where $D^\\\\flat : ^p\\\\times ^p \\\\rightarrow ^+$ denotes the symmetrized Bregman divergence associated with $\\\\hat{Q}_h^\\\\flat $ .\",\n \"Next, we present two lemmas on the upper and lower bounds of the numerator and denominator of (REF ), respectively.\",\n \"Assume that Conditions REF , REF , and REF hold.\",\n \"For any $t\\\\ge 0$ , there exists some event $_1(t)$ with $\\\\lbrace _1(t) \\\\rbrace \\\\ge 1- 2e^{-t}$ such that, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_1(t)$ , $\\\\bigl \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^* ) \\\\bigr \\\\Vert _2 \\\\le C_1 \\\\Biggl ( \\\\sqrt{\\\\frac{p+t}{n}} + h^2 \\\\Biggr )$ as long as $n\\\\gtrsim p+t$ , where $C_1>0$ is a constant depending only on $(\\\\upsilon _1, \\\\kappa _2, l_0)$ and $\\\\tau $ .\",\n \"Let $r= h/(4\\\\eta _{1/4})$ with $\\\\eta _{1/4}$ defined in (REF ).\",\n \"For any $t\\\\ge 0$ , there exists some event $_2(t)$ with $\\\\lbrace _2(t) \\\\rbrace \\\\ge 1- 3e^{-t}$ such that, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_2(t)$ , $^* \\\\Biggl \\\\lbrace \\\\inf _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)}\\\\frac{D^\\\\flat (\\\\beta , \\\\beta ^* )}{\\\\kappa _l \\\\Vert \\\\beta -\\\\beta ^* \\\\Vert _{}^2} \\\\ge \\\\frac{1}{2} {f} \\\\Biggr \\\\rbrace \\\\ge 1-e^{-t} .\",\n \"$ as long as $ (p+\\\\log n + t )/n \\\\lesssim h \\\\lesssim 1$ .\",\n \"Recall from the proof of Theorem REF that the upper bound on $\\\\Vert \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\Vert _{}$ depends crucially on $\\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h(\\\\beta ^* ) \\\\Vert _2$ and $\\\\inf _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\lbrace D(\\\\beta , \\\\beta ^*)/\\\\Vert \\\\beta - \\\\beta ^* \\\\Vert _{}^2\\\\rbrace $ .\",\n \"In view of Lemmas REF and REF , we take $r=h/( 4 \\\\eta _{1/4})$ and for $t\\\\ge 0$ , set $(t) = _1(t) \\\\cup _2(t)$ so that $\\\\lbrace (t) \\\\rbrace \\\\ge 1-5 e^{-t}$ and $\\\\Vert \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\Vert _{} \\\\le C_0 \\\\Biggl ( \\\\sqrt{\\\\frac{p+t}{n}} + h^2 \\\\Biggr ) ~\\\\mbox{ on $(t)$ provided that } \\\\sqrt{\\\\frac{p+t}{n}} \\\\lesssim h \\\\lesssim 1,$ where $C_0>0$ depends only on $(\\\\upsilon _1, \\\\kappa _2, \\\\kappa _l, l_0, {f} )$ and $\\\\tau $ .\",\n \"Moreover, by (REF ), with $^*$ -probability at least $1-2e^{-t}$ conditioned on $(t)$ , $\\\\Vert \\\\widetilde{\\\\beta }_h^\\\\flat - \\\\beta ^* \\\\Vert _{} \\\\le \\\\frac{2C_1}{\\\\kappa _l {f} }\\\\Biggl ( \\\\sqrt{\\\\frac{p+t}{n}} + h^2 \\\\Biggr ) < r \\\\nonumber $ as long as $\\\\max \\\\lbrace (p+\\\\log n + t )/n, \\\\sqrt{(p+t)/n } \\\\rbrace \\\\lesssim h \\\\lesssim 1$ .\",\n \"Consequently, $\\\\widetilde{\\\\beta }_h^\\\\flat $ falls in the interior of $\\\\beta ^* + \\\\Theta (r)$ , thereby implying $\\\\eta =1$ and $\\\\hat{\\\\beta }_h^\\\\flat = \\\\widetilde{\\\\beta }_h^\\\\flat $ .\",\n \"This completes the proof.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Lemma \",\n \"By the triangle inequality, we have $\\\\bigl \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^* ) \\\\bigr \\\\Vert _2 \\\\le \\\\bigl \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^* ) - ^{-1/2}\\\\nabla \\\\hat{Q}_h (\\\\beta ^* ) \\\\bigr \\\\Vert _2 + \\\\bigl \\\\Vert ^{-1/2}\\\\nabla \\\\hat{Q}_h (\\\\beta ^* ) \\\\bigr \\\\Vert _2:= I_1+I_2.$ The term $I_2$ can be upper bounded by Lemma REF , and thus it suffices to bound $I_1$ .\",\n \"To this end, we will employ Hoeffding's and Bernstein's inequalities.\",\n \"Recall from Section REF that $w_i$ is such that $(w_i)=1$ and $\\\\textnormal {var}(w_i)=1$ .\",\n \"Thus, $^{-1/2} \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^* ) = (1/n) \\\\sum _{i=1}^nw_i \\\\xi _i _i$ and $^* \\\\lbrace ^{-1/2} \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^* ) \\\\rbrace = ^{-1/2} \\\\hat{Q}_h (\\\\beta ^* ) $ , where $_i = ^{-1/2} _i$ and $\\\\xi _i = K(-\\\\varepsilon _i /h) - \\\\tau $ with $\\\\varepsilon _i = y_i - \\\\langle _i, \\\\beta ^* \\\\rangle $ .\",\n \"Using a similar covering argument as in the proof of Lemma REF , for any $\\\\epsilon \\\\in ( 0, 1)$ , there exists an $\\\\epsilon $ -net $_\\\\epsilon $ of the unit sphere $\\\\mathbb {S}^{p-1}$ with $|_\\\\epsilon |\\\\le ( 1+2/\\\\epsilon )^p$ such that $\\\\bigl \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^* ) - ^{-1/2}\\\\nabla \\\\hat{Q}_h (\\\\beta ^* ) \\\\bigr \\\\Vert _2 \\\\le \\\\frac{1}{1-\\\\epsilon } \\\\max _{\\\\in _\\\\epsilon } \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\xi _i \\\\langle , _i \\\\rangle , \\\\nonumber $ where $e_i = w_i-1$ are i.i.d.\",\n \"Rademacher random variables.\",\n \"By Hoeffding's inequality, for any $u\\\\ge 0$ , $^* \\\\left\\\\lbrace \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\xi _i \\\\langle , _i \\\\rangle \\\\ge \\\\Biggl ( \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\xi _i^2 \\\\langle , _i \\\\rangle ^2 \\\\Biggr )^{1/2} \\\\sqrt{\\\\frac{2 u }{n}} \\\\right\\\\rbrace \\\\le e^{-u}.$ For the data-dependent quantity $(1/n) \\\\sum _{i=1}^n\\\\xi _i^2 \\\\langle , _i \\\\rangle ^2 $ , as in the proof of Lemma REF we have $| \\\\xi _i | \\\\le B_\\\\tau := \\\\max ( \\\\tau , 1-\\\\tau )$ , $(\\\\xi _i^2 | _i ) \\\\le C_\\\\tau := \\\\tau (1-\\\\tau ) + (1+\\\\tau ) l_0 \\\\kappa _2 h^2$ almost surely and note that $\\\\langle , _i \\\\rangle ^2 =1$ .\",\n \"Moreover, for $k=2,3,\\\\ldots $ , $\\\\bigl ( \\\\xi _i^2 \\\\langle , _i \\\\rangle ^2 \\\\bigr )^k & \\\\le C_\\\\tau B_\\\\tau ^{2(k-1)} \\\\upsilon _1^{2k} \\\\cdot 2k \\\\int _0^\\\\infty \\\\bigl ( | \\\\langle , _i \\\\rangle | \\\\ge \\\\upsilon _1 u \\\\bigr ) u^{2k-1} \\\\, {\\\\rm d} u \\\\nonumber \\\\\\\\& \\\\le C_\\\\tau B_\\\\tau ^{2(k-1)} \\\\upsilon _1^{2k} \\\\cdot 4k \\\\int _0^\\\\infty u^{2k-1} e^{-u^2/2 } \\\\, {\\\\rm d} u \\\\nonumber \\\\\\\\& = C_\\\\tau B_\\\\tau ^{2(k-1)} 2^k \\\\upsilon _1^{2k} \\\\cdot 2k \\\\int _0^\\\\infty v^{ k-1} e^{-v } \\\\, {\\\\rm d}v = C_\\\\tau B_\\\\tau ^{2(k-1)} 2^{k+1} \\\\upsilon _1^{2k} k!\",\n \".\",\n \"\\\\nonumber $ In particular, $(\\\\xi _i^4 \\\\langle , _i \\\\rangle ^4)\\\\le B_\\\\tau ^2 C_\\\\tau 16 \\\\upsilon _1^4$ and $( \\\\xi _i^2 \\\\langle , _i \\\\rangle ^2 )^k \\\\le \\\\frac{k!\",\n \"}{2} \\\\cdot B_\\\\tau ^2 C_\\\\tau 16 \\\\upsilon _1^4 \\\\cdot (2 B_\\\\tau ^2 \\\\upsilon _1^2 )^{k-2}$ for $k\\\\ge 3$ .\",\n \"Since $(\\\\xi _i^2 | _i ) \\\\le C_\\\\tau $ , it then follows from Bernstein's inequality that, for any $v\\\\ge 0$ , $\\\\Biggl ( \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\xi _i^2 \\\\langle , _i \\\\rangle ^2 \\\\ge C_\\\\tau + 4 B_\\\\tau C_\\\\tau ^{1/2} \\\\upsilon _1^2 \\\\sqrt{\\\\frac{2 v}{n}} + B_\\\\tau ^2 \\\\upsilon _1^2 \\\\frac{2v}{n} \\\\Biggr ) \\\\le e^{-v} .$ In the above analysis, we set $\\\\epsilon = 2/(e^2-1)$ so that $(1+2/\\\\epsilon )^p = e^{2p}$ , and take $u=v=2p +t$ .\",\n \"Applying the union bound, we conclude that $\\\\max _{\\\\in _\\\\epsilon } \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\xi _i^2 \\\\langle , _i \\\\rangle ^2 \\\\le C_\\\\tau + 4 B_\\\\tau C_\\\\tau ^{1/2} \\\\upsilon _1^2 \\\\sqrt{\\\\frac{4p+2t}{n}} + 2 B_\\\\tau ^2 \\\\upsilon _1^2 \\\\frac{2p+t}{n} $ with probability at least $1-e^{-t}$ .\",\n \"Let $_1(t)$ be the event that (REF ) holds.\",\n \"Then, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_1(t)$ , and by an application of union bound, we obtain $\\\\bigl \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^* ) - ^{-1/2} \\\\hat{Q}_h (\\\\beta ^* ) \\\\bigr \\\\Vert _2 \\\\le C \\\\sqrt{\\\\frac{p+t}{n}}$ as long as $n\\\\gtrsim p+t$ , where $C>0$ depends only on $(\\\\upsilon _1,\\\\kappa _2, l_0)$ and $\\\\tau $ .\",\n \"This, together with (REF ) and Lemma REF , proves the claimed bound.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Lemma \",\n \"Using arguments similar to (), (), and (REF ) in the proof of Lemma REF , we obtain $D^\\\\flat (\\\\beta , \\\\beta ^*) &= \\\\frac{1}{n} \\\\sum _{i=1}^n\\\\left\\\\lbrace {K} \\\\left( \\\\frac{\\\\langle _i, \\\\beta \\\\rangle - y_i}{h}\\\\right) - {K}\\\\left( \\\\frac{ -\\\\varepsilon _i}{h} \\\\right) \\\\right\\\\rbrace w_i \\\\langle _i, \\\\beta - \\\\beta ^* \\\\rangle \\\\nonumber \\\\\\\\&\\\\ge \\\\kappa _l \\\\Vert \\\\beta - \\\\beta ^* \\\\Vert _{}^2 \\\\cdot \\\\underbrace{ \\\\frac{1}{n h } \\\\sum _{i=1}^nw_i \\\\omega _i \\\\varphi _{h/(2r)} ( \\\\langle _i , \\\\delta \\\\rangle ) }_{ =: D_0^\\\\flat (\\\\delta ) } , \\\\nonumber $ where $\\\\delta = ^{1/2}( \\\\beta - \\\\beta ^*) /\\\\Vert \\\\beta - \\\\beta ^* \\\\Vert _{}$ , $\\\\omega _i = \\\\mathbb {1}( |\\\\varepsilon _i | \\\\le h/2)$ , and $\\\\varphi (\\\\cdot )$ is as defined in (REF ).\",\n \"Recall from (REF ) the definition of $D_0 (\\\\delta )$ .\",\n \"We have $\\\\inf _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)}\\\\frac{D^\\\\flat (\\\\beta , \\\\beta ^* )}{\\\\kappa _l \\\\Vert \\\\beta -\\\\beta ^* \\\\Vert _{}^2} \\\\ge \\\\inf _{\\\\delta \\\\in \\\\mathbb {S}^{p-1} } D_0(\\\\delta ) - \\\\sup _{\\\\delta \\\\in \\\\mathbb {S}^{p-1} } \\\\lbrace D_0(\\\\delta ) -D_0^\\\\flat (\\\\delta ) \\\\rbrace .$ We now obtain lower and upper bounds for $ D_0(\\\\delta ) $ and $\\\\lbrace D_0(\\\\delta ) -D_0^\\\\flat (\\\\delta ) \\\\rbrace $ , respectively.\",\n \"Recall that $w_i$ is a random variable that is independent of $_i$ with $(w_i)=1$ .\",\n \"Let $e_i = w_i -1 \\\\in \\\\lbrace -1,1\\\\rbrace $ .\",\n \"Then, we have $D_0^\\\\flat (\\\\delta ) - D_0(\\\\delta ) = \\\\frac{1}{n h } \\\\sum _{i=1}^ne_i \\\\omega _i \\\\varphi _{h/(2r)} ( \\\\langle _i , \\\\delta \\\\rangle ) .$ Define $\\\\Gamma _n = \\\\gamma (e_1,\\\\ldots ,e_n) = \\\\sup _{ \\\\delta \\\\in \\\\mathbb {S}^{p-1}} \\\\lbrace D_0(\\\\delta ) -D_0^\\\\flat (\\\\delta ) \\\\rbrace $ .\",\n \"Recall that $\\\\varphi _{R}(u) \\\\le (R/2)^2 $ , we have $^* \\\\lbrace e_i \\\\omega _i \\\\varphi _{h/(2r)} ( \\\\langle _i , \\\\delta \\\\rangle ) \\\\rbrace ^2 \\\\le (h/4r)^4 \\\\omega _i $ and $| e_i \\\\omega _i \\\\varphi _{h/(2r)} ( \\\\langle _i , \\\\delta \\\\rangle ) |\\\\le (h/4r)^2$ .\",\n \"Then, by the Talagrand's inequality (see Theorem 7.3 in [60]), for every $t\\\\ge 0$ , $\\\\Gamma _n & \\\\le ^*( \\\\Gamma _n ) + \\\\sqrt{ \\\\frac{h^2}{(4r)^4}\\\\frac{ \\\\sum _{i=1}^n\\\\omega _i}{ n} \\\\frac{2t}{n} + 4^*( \\\\Gamma _n ) \\\\frac{h}{(4r)^2}\\\\frac{t}{n} } + \\\\frac{h}{(4r)^2} \\\\frac{t}{3 n} \\\\nonumber \\\\\\\\& \\\\le 2^*( \\\\Gamma _n ) + \\\\frac{h}{(4r)^2} \\\\Biggl \\\\lbrace \\\\Biggl ( \\\\frac{1}{n}\\\\sum _{i=1}^n\\\\omega _i \\\\Biggr )^{1/2} \\\\sqrt{ \\\\frac{2t}{n} } + \\\\frac{4t}{3 n} \\\\Biggr \\\\rbrace \\\\nonumber $ with probability at least $1-e^{-t}$ .\",\n \"Further, by the Lipschitz continuity of $u\\\\rightarrow \\\\varphi _R(u)$ and Talagrand's contraction principle, $^*( \\\\Gamma _n ) & \\\\le \\\\frac{1}{2 r} ^* \\\\Biggl \\\\lbrace \\\\sup _{ \\\\delta \\\\in \\\\mathbb {S}^{p-1} } \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\langle \\\\omega _i _i, \\\\delta \\\\rangle \\\\Biggr \\\\rbrace \\\\nonumber \\\\\\\\& \\\\le \\\\frac{1}{2 r} ^* \\\\Biggl \\\\Vert \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\omega _i _i \\\\Biggr \\\\Vert _2 \\\\nonumber \\\\\\\\& \\\\le \\\\frac{1}{2 r n } \\\\Biggl ( \\\\sum _{i=1}^n\\\\omega _i \\\\Vert _i \\\\Vert _2^2 \\\\Biggr )^{1/2} \\\\nonumber \\\\\\\\& \\\\le \\\\frac{\\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2 }{2 r \\\\sqrt{n}} \\\\Biggl ( \\\\frac{1}{n}\\\\sum _{i=1}^n\\\\omega _i \\\\Biggr )^{1/2} .\",\n \"\\\\nonumber $ Together, the last two displays imply $\\\\sup _{ \\\\delta \\\\in \\\\mathbb {S}^{p-1} } \\\\bigl \\\\lbrace D_0(\\\\delta ) -D_0^\\\\flat (\\\\delta ) \\\\bigr \\\\rbrace \\\\le \\\\Biggl ( \\\\frac{1}{n}\\\\sum _{i=1}^n\\\\omega _i \\\\Biggr )^{1/2} \\\\Biggl \\\\lbrace \\\\frac{\\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2 }{r \\\\sqrt{n}} +\\\\frac{h}{(4r)^2} \\\\sqrt{\\\\frac{2t}{n} } \\\\Biggr \\\\rbrace + \\\\frac{h}{(4r)^2} \\\\frac{4t}{3 n} $ with $^*$ -probability at least $1-e^{-t}$ .\",\n \"Next we provide upper bounds for the data-dependent terms $\\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2 $ and $(1/n) \\\\sum _{i=1}^n\\\\omega _i$ .\",\n \"As in the proof of Lemma REF , for any $\\\\epsilon \\\\in (0,1)$ , there exits $_\\\\epsilon \\\\subseteq \\\\mathbb {S}^{p-1}$ with $|_\\\\epsilon | \\\\le (1+2/\\\\epsilon )^p$ such that $\\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2 \\\\le (1-\\\\epsilon )^{-1} \\\\max _{1\\\\le i\\\\le n} \\\\max _{\\\\in _\\\\epsilon } \\\\langle , _i \\\\rangle $ .\",\n \"Given $1\\\\le i\\\\le n$ and $\\\\in _\\\\epsilon $ , Condition REF indicates $( |\\\\langle , _i \\\\rangle | \\\\ge \\\\upsilon _1 u) \\\\le 2 e^{-u^2/2}$ for any $u \\\\in $ .\",\n \"Taking the union bound over $i$ and $$ , and setting $u=\\\\sqrt{2t + 2\\\\log (2n)+2p\\\\log (1+2/\\\\epsilon )}$ ($t>0$ ), we obtain that with probability at least $1-2 n (1+2/\\\\epsilon )^p e^{-u^2/2}= 1-e^{-t}$ , $\\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2 \\\\le (1-\\\\epsilon )^{-1} \\\\upsilon _1 \\\\sqrt{2t + 2\\\\log (2n)+2p\\\\log (1+2/\\\\epsilon )}$ .\",\n \"Minimizing this upper bound with respect to $\\\\epsilon \\\\in (0,1)$ , we conclude that for any $t>0$ , $\\\\biggl [ \\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2^2 \\\\ge 2 \\\\upsilon _1^2 \\\\bigl \\\\lbrace 3.7p + \\\\log (2n) + t \\\\bigr \\\\rbrace \\\\biggr ] \\\\le e^{-t}.\",\n \"$ Moreover, applying Bernstein's inequality to $(1/n) \\\\sum _{i=1}^n\\\\omega _i = (1/n) \\\\sum _{i=1}^n\\\\mathbb {1}( |\\\\varepsilon _i | \\\\le h/2)$ yields $\\\\frac{1}{n} \\\\sum _{i=1}^n\\\\omega _i \\\\le (\\\\omega _i) + \\\\sqrt{ 2(\\\\omega _i) \\\\frac{ t}{n}} + \\\\frac{t}{3n} \\\\le \\\\Biggl ( \\\\sqrt{(\\\\omega _i)} + \\\\sqrt{\\\\frac{t}{2n}} \\\\Biggr )^2$ with probability greater than $1-e^{-t}$ .\",\n \"Note that $(\\\\omega _i) \\\\le \\\\bar{f}_{ h} h$ .\",\n \"Putting together the pieces, we conclude that conditioned on some event that occurs with probability at least $1-2e^{-t}$ , $& \\\\sup _{\\\\delta \\\\in \\\\mathbb {S}^{p-1} } \\\\bigl \\\\lbrace D_0(\\\\delta ) -D_0^\\\\flat (\\\\delta ) \\\\bigr \\\\rbrace \\\\nonumber \\\\\\\\& \\\\le \\\\bigl \\\\lbrace ( \\\\bar{f}_h h )^{1/2} + (0.5 t/ n)^{1/2} \\\\bigr \\\\rbrace \\\\left\\\\lbrace \\\\frac{\\\\upsilon _1 \\\\bigl ( 7.4 p + 2\\\\log (2n) + 2 t \\\\bigr )^{1/2} }{r \\\\sqrt{n}} + \\\\frac{h}{(4r)^2} \\\\sqrt{\\\\frac{2t}{n} } \\\\right\\\\rbrace + \\\\frac{h}{(4r)^2}\\\\sqrt{\\\\frac{4 t}{3 n}}$ with $^*$ -probability greater than $1-e^{-t}$ .\",\n \"Turning to $D_0(\\\\delta )$ , applying Lemma REF yields that as long as $0< r\\\\le h/(4\\\\eta _{1/4} )$ , $\\\\inf _{\\\\delta \\\\in \\\\mathbb {S}^{p-1}} D_0(\\\\delta ) \\\\ge \\\\frac{3}{4} {f}_h - \\\\bar{f}_h^{1/2} \\\\left( \\\\frac{5}{4}\\\\sqrt{\\\\frac{h p}{r^2 n}} + \\\\sqrt{\\\\frac{h t}{8r^2 n}} \\\\right) - \\\\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .\",\n \"Substituting (REF ) and (REF ) into (REF ), and taking $r=h/(4\\\\eta _{1/4} )$ with $h\\\\gtrsim t/n$ , we conclude that conditioned on some event that occurs with probability at least $1-3 e^{-t}$ , $\\\\inf _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)}\\\\frac{D^\\\\flat (\\\\beta , \\\\beta ^* )}{\\\\kappa _l \\\\Vert \\\\beta -\\\\beta ^* \\\\Vert _{}^2} \\\\ge \\\\frac{3}{4} {f}_h - C \\\\sqrt{\\\\frac{p + \\\\log n + t}{ n h }} \\\\nonumber $ with $^*$ -probability greater than $1-e^{-t}$ , where $C>0$ depends only on $(\\\\upsilon _1, l_0 ,\\\\bar{f} )$ .\",\n \"This completes the proof of (REF ).\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"The proof is based on an argument similar to that used in the proof of Theorem REF .\",\n \"To begin with, define the random process $\\\\Delta ^\\\\flat (\\\\beta ) = ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ) - \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ^*) - (\\\\beta -\\\\beta ^* )\\\\bigr \\\\rbrace , ~~ \\\\beta \\\\in ^p.$ For a prespecified $r>0$ , a key step is to bound the local fluctuation $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert \\\\Delta ^\\\\flat (\\\\beta ) \\\\Vert _2$ .\",\n \"Since $(w_i)=1$ , we have $^* \\\\lbrace \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ) \\\\rbrace = \\\\nabla \\\\hat{Q}_h (\\\\beta )$ .\",\n \"Define the (conditionally) centered process $G^\\\\flat (\\\\beta ) = ^{-1/2} \\\\big \\\\lbrace \\\\nabla \\\\hat{Q}_h^\\\\flat (\\\\beta ) - \\\\nabla \\\\hat{Q}_h (\\\\beta ) \\\\big \\\\rbrace = \\\\frac{1}{n} \\\\sum _{i=1}^ne_i \\\\lbrace K(-(y_i - \\\\langle _i, \\\\beta \\\\rangle )/h) - \\\\tau \\\\rbrace _i,$ so that $\\\\Delta ^\\\\flat (\\\\beta )$ be be written as $\\\\Delta ^\\\\flat (\\\\beta ) = \\\\bigl \\\\lbrace G^\\\\flat (\\\\beta ) - G^\\\\flat (\\\\beta ^*) \\\\bigr \\\\rbrace + \\\\Delta (\\\\beta ) , \\\\nonumber $ where $\\\\Delta (\\\\beta )$ is defined in (REF ).\",\n \"By the triangle inequality, $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert \\\\Delta ^\\\\flat (\\\\beta ) \\\\Vert _2 \\\\le \\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert G^\\\\flat (\\\\beta ) - G^\\\\flat (\\\\beta ^*) \\\\Vert _2 + \\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert \\\\Delta (\\\\beta ) \\\\Vert _2 .\",\n \"$ It suffices to bound the first term on the right-hand side of (REF ).\",\n \"By a change of variable $= ^{1/2} ( \\\\beta -\\\\beta ^*)$ , we have $y_i -\\\\langle _i, \\\\beta \\\\rangle = \\\\varepsilon _i - \\\\langle _i, \\\\rangle $ and $& \\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert G^\\\\flat (\\\\beta ) - G^\\\\flat (\\\\beta ^*) \\\\Vert _2 \\\\nonumber \\\\\\\\ & \\\\le \\\\sup _{ \\\\in \\\\mathbb {B}^p(r), \\\\in \\\\mathbb {B}^p(1) } \\\\langle G^\\\\flat ( \\\\beta ^* + ^{-1/2} ) - G^\\\\flat ( \\\\beta ^* ) , \\\\rangle \\\\nonumber \\\\\\\\& = r^{-1} n^{-1/2} \\\\sup _{ , \\\\in \\\\mathbb {B}^p(r) } \\\\underbrace{ n^{1/2} \\\\bigl \\\\langle G^\\\\flat ( \\\\beta ^* + ^{-1/2} ) - G^\\\\flat ( \\\\beta ^* ) , \\\\bigr \\\\rangle }_{ =: \\\\Psi ( , ) } , $ where $\\\\Psi (, ) = n^{-1/2} \\\\sum _{i=1}^ne_i\\\\langle _i, \\\\rangle \\\\lbrace K( \\\\langle _i, \\\\rangle / h - \\\\varepsilon _i /h ) - K(-\\\\varepsilon _i/h) \\\\rbrace $ .\",\n \"To characterize the magnitude of $\\\\sup _{, \\\\in \\\\mathbb {B}^p(r)} \\\\Psi (, )$ , define the following classes of measurable functions $_1 =\\\\bigl \\\\lbrace ( , \\\\varepsilon ) \\\\mapsto \\\\langle , \\\\rangle : \\\\in \\\\mathbb {B}^p(r) \\\\bigr \\\\rbrace ~\\\\mbox{ and }~ _2 = \\\\bigl \\\\lbrace ( , \\\\varepsilon ) \\\\mapsto \\\\langle , \\\\rangle - \\\\varepsilon : \\\\in \\\\mathbb {B}^p(r) \\\\bigr \\\\rbrace .$ Also, define functions $f_0 : ( , \\\\varepsilon ) \\\\rightarrow K(-\\\\varepsilon /h)$ and $\\\\phi : u \\\\rightarrow K(u/h)$ .\",\n \"With this notation, we can write $\\\\sup _{, \\\\in \\\\mathbb {B}^p(r)}\\\\Psi (, ) = \\\\sup _{g\\\\in } \\\\frac{1}{\\\\sqrt{n}} \\\\sum _{i=1}^ne_i g(_i), $ where $_i = (_i , \\\\varepsilon _i ) \\\\in ^p\\\\times $ and $= _1 \\\\cdot (\\\\phi \\\\circ _2 - f_0)$ denotes the pointwise product between $_1$ and $\\\\phi \\\\circ _2 - f_0$ .\",\n \"Let $Z = ^* \\\\lbrace \\\\sup _{g\\\\in } (1/n)\\\\sum _{i=1}^ne_i g(_i) \\\\rbrace $ be the conditional Rademacher average.\",\n \"By Theorem 13 in [59] and the bound $\\\\sup _{1\\\\le i\\\\le n, g\\\\in } g(_i )\\\\le r \\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2$ , we have $\\\\bigl [ \\\\lbrace Z - (Z)\\\\rbrace ^{2k}_+ \\\\bigr ]^{1/(2k)} \\\\le 2 \\\\sqrt{(Z) \\\\cdot k \\\\kappa r \\\\frac{M_{n,k}}{n}} + 2 k \\\\kappa r \\\\frac{M_{n,k}}{n} , ~\\\\mbox{ valid for any } k\\\\ge 1, $ where $\\\\kappa = \\\\sqrt{e} / (2\\\\sqrt{e}-2)$ and $M_{n,k} = \\\\lbrace (\\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2^{2k})\\\\rbrace ^{1/(2k)}$ .\",\n \"Next we bound $(Z)$ via a maximal inequality specialized to VC type classes.\",\n \"Note that the function class $_1$ is a $p$ -dimensional vector space, and $_2$ is included in a $(p + 1)$ -dimensional vector space.\",\n \"By Lemma 2.6.15 of [50], both $_1$ and $_2$ have VC-subgraph of index smaller than or equal to $p+2$ .\",\n \"Moreover, since $\\\\phi (\\\\cdot ) = K(\\\\cdot /h)$ is an non-decreasing function, it follows from Lemma 9.9 of [63] that $\\\\phi \\\\circ _2$ is VC with index less than or equal $ p+2$ , and so is $\\\\phi \\\\circ _2 - f_0$ .\",\n \"Let $F_1(, \\\\varepsilon ) = r \\\\Vert \\\\Vert _2$ and $F_2 (, \\\\varepsilon ) \\\\equiv 1$ be the envelops of $_1$ and $\\\\phi \\\\circ _2 - f_0$ , respectively.\",\n \"Hence, by Theorem 9.3 in [63] (or Theorem 2.6.7 in [50]), we obtain the following bounds on the covering numbers of function classes $_1$ and $\\\\phi \\\\circ _2 - f_0$ : there exists a universal constant $A_0>0$ such that, for any probability measure $Q$ with $\\\\Vert F_1 \\\\Vert _{Q,2} >0$ and any $\\\\epsilon \\\\in (0,1)$ , $N( _1, L_2(Q) , \\\\epsilon \\\\Vert F_1 \\\\Vert _{Q,2} ) \\\\le A_0 (p+2)(4e)^{p+2} \\\\biggl ( \\\\frac{2}{\\\\epsilon } \\\\biggr )^{2(p+1)}; \\\\nonumber \\\\\\\\N( \\\\phi \\\\circ _2 - f_0 , L_2(Q) , \\\\epsilon \\\\Vert F_2 \\\\Vert _{Q,2} ) \\\\le A_0 (p+2)(4e)^{p+2} \\\\biggl ( \\\\frac{2}{\\\\epsilon } \\\\biggr )^{2(p+1)} .\\\\nonumber $ For the product class $$ with envelop $G(, \\\\varepsilon )= F_1(, \\\\varepsilon ) F_2(, \\\\varepsilon ) = r \\\\Vert \\\\Vert _2$ , applying Corollary A.1 in the supplement of [61] yields that, for any $\\\\epsilon \\\\in (0 ,1)$ , $& N( , L_2(Q) , \\\\epsilon \\\\Vert G \\\\Vert _{Q,2} ) \\\\nonumber \\\\\\\\& \\\\le N( _1, L_2(Q) , 2^{-1/2} \\\\epsilon \\\\Vert F_1 \\\\Vert _{Q,2} ) N( \\\\phi \\\\circ _2 - f_0 , L_2(Q) , 2^{-1/2} \\\\epsilon \\\\Vert F_2 \\\\Vert _{Q,2} ) \\\\nonumber \\\\\\\\& \\\\le A_0^2 (p+2)^2 (4 e)^{2(p+2)} \\\\biggl ( \\\\frac{2^{3/2}}{\\\\epsilon } \\\\biggr )^{4(p+1)} \\\\le \\\\biggl ( \\\\frac{A_1}{\\\\epsilon } \\\\biggr )^{4(p+2)} \\\\nonumber $ for some $A_1 > e$ .\",\n \"Another key quantity is the variance proxy $\\\\sup _{g\\\\in } \\\\lbrace g^2(, \\\\varepsilon )\\\\rbrace $ .\",\n \"Given some $g = g_{, } \\\\in $ with $, \\\\in \\\\mathbb {B}^p(r)$ , we have $\\\\lbrace g^2(, \\\\varepsilon )\\\\rbrace & = ( \\\\langle , \\\\rangle ^2 [K\\\\lbrace ( \\\\langle , \\\\rangle - \\\\varepsilon ) /h\\\\rbrace - K(-\\\\varepsilon /h) ]^2) \\\\nonumber \\\\\\\\& = \\\\biggl (\\\\langle , \\\\rangle ^2 \\\\int _{-\\\\infty }^\\\\infty \\\\bigl \\\\lbrace K\\\\bigl ( ( \\\\langle , \\\\rangle -u ) /h \\\\bigr ) - K(-u/h) \\\\bigr \\\\rbrace ^2 f_{\\\\varepsilon | } (u) {\\\\rm d} u\\\\biggr ) \\\\nonumber \\\\\\\\& = h \\\\, \\\\biggl (\\\\langle , \\\\rangle ^2 \\\\int _{-\\\\infty }^\\\\infty \\\\bigl \\\\lbrace K \\\\bigl ( \\\\langle , \\\\rangle /h + v \\\\bigr ) - K(v) \\\\bigr \\\\rbrace ^2 f_{\\\\varepsilon | } (-vh) {\\\\rm d} v\\\\nonumber \\\\biggr ) \\\\\\\\& \\\\le \\\\bar{f} h^{-1} \\\\biggl (\\\\langle , \\\\rangle ^2 \\\\langle , \\\\rangle ^2 \\\\int _{-\\\\infty }^\\\\infty \\\\biggl \\\\lbrace \\\\int _0^1 K(v+ s \\\\langle ,\\\\rangle /h ) {\\\\rm d} s \\\\biggr \\\\rbrace ^2 {\\\\rm d} v\\\\biggr ) \\\\nonumber \\\\\\\\& \\\\le \\\\bar{f} h^{-1} \\\\biggl (\\\\langle , \\\\rangle ^2 \\\\langle , \\\\rangle ^2\\\\Biggl [ \\\\int _0^1 \\\\biggl \\\\lbrace \\\\int _{-\\\\infty }^\\\\infty K^2(v+ s \\\\langle ,\\\\rangle /h ) {\\\\rm d} v \\\\biggr \\\\rbrace ^{1/2} {\\\\rm d}s \\\\Biggr ]^2 \\\\nonumber \\\\biggr ) \\\\\\\\& \\\\le \\\\kappa _u \\\\bar{f} h^{-1} (\\\\langle , \\\\rangle ^2 \\\\langle , \\\\rangle ^2 ), \\\\nonumber $ where the second inequality follows from an application of generalized Minkowski's integral inequality.\",\n \"Hence, $\\\\sup _{g\\\\in } \\\\lbrace g^2(, \\\\varepsilon )\\\\rbrace \\\\le \\\\kappa _u \\\\bar{f} \\\\mu _4 r^4h^{-1}$ .\",\n \"For the envelop $G$ for $$ , it can be shown that $\\\\lbrace G^2(, \\\\varepsilon )\\\\rbrace = r^2 p$ , $\\\\max _{1\\\\le i\\\\le n}\\\\sup _{g\\\\in } | g(_i) | \\\\le r\\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2$ , and $\\\\lbrace \\\\max _{1\\\\le i\\\\le n}G^2(_i, \\\\varepsilon _i)\\\\rbrace = r^2 M_{n,1}^2$ with $M_{n,1}$ given in (REF ).\",\n \"Equipped with the above bounds, we apply Corollary 5.1 in [61] to obtain $(Z) \\\\lesssim r^2 \\\\sqrt{\\\\frac{p}{nh} \\\\log ( A_2 p h /r^2 ) } + r M_{n,1} \\\\log ( A_2 p h /r^2 ) \\\\frac{ p }{n} , $ where $A_2>0$ depends on $(\\\\bar{f} , \\\\kappa _u, \\\\mu _4)$ and $A_1$ .\",\n \"It remains to bound $M_{n,1}$ , which appears in both (REF ) and (REF ).\",\n \"In fact, the exponential tail probability (REF ) can be directly used to bound the expectation.\",\n \"Given a non-negative random variable $X$ and constants $A, a>0$ , we have $(X) & = \\\\int _0^\\\\infty (X\\\\ge t) {\\\\rm d} t \\\\le A + \\\\int _A^\\\\infty (X\\\\ge t) {\\\\rm d} t \\\\nonumber \\\\\\\\& = A + \\\\int _0^\\\\infty (X\\\\ge A+ t) {\\\\rm d} t = A + a \\\\int _0^\\\\infty (X\\\\ge A + a s ) {\\\\rm d} s .\",\n \"\\\\nonumber $ Taking $X= \\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2^2$ , $A= 2 \\\\upsilon _1^2 \\\\lbrace 3.7 p + \\\\log (2n) \\\\rbrace $ and $a=2 \\\\upsilon _1^2 $ , we obtain $M_{n,1}^2 = \\\\Biggl ( \\\\max _{1\\\\le i\\\\le n} \\\\Vert _i \\\\Vert _2^2 \\\\Biggr ) \\\\le 2 \\\\upsilon _1^2 \\\\bigl \\\\lbrace 3.7 p + \\\\log (2e n) \\\\bigr \\\\rbrace .\",\n \"$ Back to the supremum $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert G^\\\\flat (\\\\beta ) - G^\\\\flat (\\\\beta ^*) \\\\Vert _2$ , first by (REF ), (REF ), and the definition of $Z$ below (REF ), we obtain $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert G^\\\\flat (\\\\beta ) - G^\\\\flat (\\\\beta ^*) \\\\Vert _2 = O_{^*} (r^{-1} Z ) .\",\n \"$ For the conditional Rademacher average $Z$ , it follows from Markov's inequality that $\\\\lbrace Z \\\\ge (Z) + u \\\\rbrace \\\\le u^{-2} \\\\lbrace Z- (Z)\\\\rbrace _+^2$ , valid for any $u >0$ .\",\n \"Applying the bound (REF ) with $k=1$ , (REF ) and (REF ) yields $& \\\\bigl [ \\\\lbrace Z- (Z)\\\\rbrace _+^2 \\\\bigr ]^{1/2} \\\\lesssim \\\\sqrt{ (Z) \\\\cdot (p +\\\\log n)^{1/2} \\\\frac{r}{n} } + (p+\\\\log n)^{1/2} \\\\frac{r}{n}; \\\\nonumber \\\\\\\\&(Z) \\\\lesssim r^2 \\\\sqrt{ \\\\frac{p}{n h} \\\\log (A_2ph / r^2 ) } + r \\\\log (A_2ph / r^2 ) (p+\\\\log n)^{1/2} \\\\frac{p}{n} , \\\\nonumber $ from which we conclude that $r^{-1 }Z = O_{} \\\\Biggl ( r \\\\sqrt{ \\\\frac{p}{n h} \\\\log (A_2ph / r^2 ) } + \\\\log (A_2ph / r^2 ) (p+\\\\log n)^{1/2} \\\\frac{p}{n} \\\\Biggr ).$ Turning to the second term on the right-hand side of (REF ), the earlier bound (REF ) implies that there exists some event $_1(t)$ with $\\\\lbrace _1(t) \\\\rbrace \\\\ge 1-e^{-t}$ such that $\\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert \\\\Delta (\\\\beta ) \\\\Vert _2 \\\\le C_1 \\\\Biggl ( r \\\\sqrt{\\\\frac{p+t}{n h}} + h r + r^2 \\\\Biggr ) ~\\\\mbox{ on } _1 (t), $ where $C_1>0$ depends on $( \\\\upsilon _1, \\\\kappa _1, \\\\kappa _u, l_0, \\\\bar{f} , {f})$ .\",\n \"With the above preparations, we are ready to prove the claim.\",\n \"Assume that the triplet $(n,p,h)$ satisfies the scaling $ \\\\sqrt{(p+\\\\log n)/n} \\\\lesssim h \\\\lesssim 1$ .\",\n \"By Theorems REF and REF , there exist some event $_n$ with $(_n ) \\\\ge 1-5n^{-1}$ and some $r=r(n,p,h)\\\\asymp \\\\sqrt{(p+\\\\log n)/n} + h^2>0$ such that, $\\\\Vert \\\\hat{\\\\beta }_h - \\\\beta ^* \\\\Vert _{} \\\\le r$ on the event $_n$ , $ \\\\Vert \\\\hat{\\\\beta }_h^\\\\flat - \\\\beta ^* \\\\Vert _{} \\\\le r$ with $^*$ -probability at least $1-2 n^{-1}$ conditioned on $_n$ .\",\n \"Taking $t=\\\\log n$ and $_{1n} = _1(\\\\log n)$ in (REF ), we have on the event $_{1n} \\\\cap _n$ that $\\\\chi _{1n} := \\\\Vert \\\\Delta (\\\\hat{\\\\beta }_h ) \\\\Vert _2 \\\\lesssim \\\\frac{p + \\\\log n}{ h^{1/2} n } + h \\\\sqrt{\\\\frac{p + \\\\log n}{n}} + h^3.\",\n \"\\\\nonumber $ Moreover, with $^*$ -probability at least $1-2n^{-1}$ conditioned on $_n$ , $\\\\Biggl \\\\Vert ^{-1/2} \\\\Biggl [ ( \\\\hat{\\\\beta }_h^\\\\flat - \\\\beta ^* ) - \\\\frac{1}{n} \\\\sum _{i=1}^nw_i \\\\bigl \\\\lbrace \\\\tau - K(- \\\\varepsilon _i /h) \\\\bigr \\\\rbrace _i \\\\Biggr ] \\\\Biggr \\\\Vert _2 \\\\le \\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert \\\\Delta ^\\\\flat (\\\\beta ) \\\\Vert _2 .\",\n \"\\\\nonumber $ Let $\\\\chi _{2n} = ^* \\\\lbrace \\\\sup _{\\\\beta \\\\in \\\\beta ^* + \\\\Theta (r)} \\\\Vert \\\\Delta ^\\\\flat (\\\\beta ) \\\\Vert _2 \\\\rbrace $ , for which it follows from (REF ), (REF ), and (REF ) with $t=\\\\log n$ that $\\\\chi _{2n} = O_{} \\\\Biggl \\\\lbrace \\\\frac{\\\\sqrt{(p+\\\\log n)p \\\\log n}}{h^{1/2} n} + h^{3/2} \\\\sqrt{\\\\frac{p \\\\log n }{n}} + h \\\\sqrt{\\\\frac{p + \\\\log n}{n}} + h^3 + (p+\\\\log n)^{1/2} \\\\frac{p \\\\log n}{n} \\\\Biggr \\\\rbrace .$ With $h=h_n\\\\asymp \\\\lbrace (p+\\\\log n)/n \\\\rbrace ^{\\\\delta }$ for some $\\\\delta \\\\in (1/3 , 1/2]$ , the first term on the right-hand side dominates the rest except the last one.\",\n \"Then, putting together the pieces establishes the claim.\",\n \"$\\\\Box $\"\n ],\n [\n \"Theoretical Properties of One-step Conquer\",\n \"In this section, we provide theoretical properties of the one-step conquer estimator $\\\\hat{\\\\beta }$ , defined in Section REF .\",\n \"The key message is that, when higher-order kernels are used (and if the conditional density $f_{\\\\varepsilon |}(\\\\cdot )$ has enough derivatives), the asymptotic normality of the one-step estimator holds under weaker growth conditions on $p$ .\",\n \"For example, the scaling condition $p=o(n^{3/8})$ that is required for the conquer estimator can be reduced to roughly $p=o(n^{7/16})$ for the one-step conquer estimator using a kernel of order 4.\",\n \"Let $G(\\\\cdot )$ be a symmetric kernel of order $\\\\nu >2$ , that is, $\\\\int _{-\\\\infty }^\\\\infty u^k G(u) \\\\, {\\\\rm d}u=0$ for $k=1,\\\\ldots , \\\\nu -1$ and $\\\\int _{-\\\\infty }^\\\\infty u^\\\\nu G(u) \\\\, {\\\\rm d} u \\\\ne 0$ .\",\n \"Moreover, $\\\\kappa ^G_k = \\\\int _{-\\\\infty }^\\\\infty |u^k G(u) | \\\\, {\\\\rm d}u <\\\\infty $ for $1\\\\le k\\\\le \\\\nu $ , $G$ is uniformly bounded with $\\\\kappa ^G_u = \\\\sup _{u\\\\in } |G(u) | <\\\\infty $ and is $L^G$ -Lipschitz continuous for some $L^G>0$ .\",\n \"The use of a higher-order kernel does not necessarily reduce bias unless the conditional density $ f_{\\\\varepsilon | }(\\\\cdot )$ of $\\\\varepsilon $ given $$ is sufficiently smooth.\",\n \"Therefore, we further impose the following smoothness conditions on $f_{\\\\varepsilon | }(\\\\cdot )$ .\",\n \"Let $\\\\nu \\\\ge 4$ be the integer in Condition .\",\n \"The conditional density $ f_{\\\\varepsilon | }(\\\\cdot )$ is $(\\\\nu -1)$ -times differentiable, and satisfies $| f_{\\\\varepsilon | }^{(\\\\nu -2)} (u) - f_{\\\\varepsilon | }^{(\\\\nu -2)} (0) | \\\\le L_{\\\\nu -2} |u|$ for all $u\\\\in $ almost surely (over the random vector $$ ), where $L_{\\\\nu -2}>0$ is a constant.\",\n \"Also, there exists some constant $C_G>0$ such that $ \\\\int _{-\\\\infty }^\\\\infty | u^{\\\\nu -1} G(u)| \\\\cdot \\\\sup _{|t| \\\\le |u|} | f_{\\\\varepsilon | }^{(\\\\nu -1)} (t)- f_{\\\\varepsilon | }^{(\\\\nu -1)} (0) | \\\\, {\\\\rm d}u \\\\le C_G$ almost surely.\",\n \"Notably, we have $\\\\nabla Q^G_b( \\\\beta ) = \\\\bigl \\\\lbrace G\\\\big ( ( \\\\langle , \\\\beta \\\\rangle - y) /b \\\\big ) - \\\\tau \\\\bigr \\\\rbrace ~~\\\\mbox{ and }~~\\\\nabla ^2 Q^G_b (\\\\beta ) = \\\\big \\\\lbrace G_b( y- \\\\langle , \\\\beta \\\\rangle ) ^\\\\rbrace ,$ representing the population score and Hessian of $Q^G_b (\\\\cdot ) = \\\\hat{Q}^G_b(\\\\cdot )$ .\",\n \"As $b \\\\rightarrow 0$ , we expect $\\\\nabla Q^G_b( \\\\beta ^*)$ and $\\\\nabla ^2 Q^G_b (\\\\beta ^*)$ to converge to $\\\\textbf {0}$ (zero vector in $^p$ ) and $= \\\\lbrace f_{\\\\varepsilon | } (0) ^$ , respectively.\",\n \"The following proposition validates this claim by providing explicit error bounds.\",\n \"Let $b\\\\in (0, 1)$ be a bandwidth.\",\n \"Under Conditions and , we have $\\\\big \\\\Vert ^{-1/2} \\\\nabla Q^G_b(\\\\beta ^*) \\\\big \\\\Vert _2 & \\\\le L_{\\\\nu -2} \\\\kappa ^G_\\\\nu \\\\, b^\\\\nu / \\\\nu !\",\n \"\\\\nonumber \\\\\\\\~~\\\\mbox{ and }~~\\\\big \\\\Vert ^{-1/2} \\\\nabla ^2 Q^G_b(\\\\beta ^*) ^{-1/2} - _0 \\\\big \\\\Vert _2 & \\\\le C_G \\\\, b^{\\\\nu -1} / (\\\\nu -1)!\",\n \", \\\\nonumber $ where $_0 = ^{-1/2} ^{-1/2} = \\\\lbrace f_{\\\\varepsilon |} (0) ^$ with $= ^{-1/2} $ .\",\n \"Proposition shows that when a higher-order kernel is used, the bias is significantly reduced in the sense that $\\\\Vert \\\\nabla Q^G_b(\\\\beta ^*) \\\\Vert _2 = ( b^\\\\nu )$ and $\\\\Vert \\\\nabla ^2 Q^G_b(\\\\beta ^*) - \\\\Vert _2 = (b^{\\\\nu -1})$ , where $\\\\nu \\\\ge 4$ is an even integer.\",\n \"Notably, even if the kernel $H$ has negative parts, the population Hessian $\\\\nabla ^2 Q^G_b(\\\\beta ^*) $ preserves the positive definiteness of $$ as long as the bandwidth $g$ is sufficient small, To construct the one-step conquer estimator, two key quantities are the sample Hessian $\\\\nabla ^2 \\\\hat{Q}^G_b( \\\\cdot ) $ and sample gradient $\\\\nabla \\\\hat{Q}^G_b(\\\\cdot )$ , both evaluated at $\\\\beta $ , a consistent initial estimate.\",\n \"In the next two propositions, we establish uniform convergence results of the Hessian and gradient of the empirical smoothed loss to their population counterparts.\",\n \"As a direct consequence, $\\\\nabla ^2 \\\\hat{Q}^G_b( \\\\beta )$ is positive definite with high probability, provided that $\\\\beta $ is consistent (i.e., in a local vicinity of $\\\\beta ^*$ ).\",\n \"To be more specific, for $r>0$ , we define the local neighborhood $\\\\Theta ^*(r) = \\\\bigl \\\\lbrace \\\\beta \\\\in ^p : \\\\Vert \\\\beta - \\\\beta ^* \\\\Vert _{} \\\\le r \\\\bigr \\\\rbrace .\",\n \"$ Conditions , and REF ensure that, with probability at least $1-e^{-t}$ , $& \\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)}\\\\big \\\\Vert ^{-1/2} \\\\big \\\\lbrace \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla ^2 Q^G_b(\\\\beta ) \\\\big \\\\rbrace \\\\nonumber ^{-1/2} \\\\big \\\\Vert _2 \\\\lesssim \\\\sqrt{\\\\frac{p \\\\log n + t}{n b }} + \\\\frac{p\\\\log n+t}{n b } + \\\\frac{(p + t)^{1/2} r }{n b^2} \\\\nonumber $ as long as $n\\\\gtrsim p+t$ .\",\n \"Conditions , and REF ensure that, with probability at least $1- e^{-t}$ , $& \\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)} \\\\big \\\\Vert ^{-1/2} \\\\big \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) - ( \\\\beta - \\\\beta ^*) \\\\big \\\\rbrace \\\\big \\\\Vert _2 \\\\lesssim r \\\\,\\\\Biggl ( \\\\sqrt{\\\\frac{p+ t}{n b }} + r +b^{\\\\nu -1} \\\\Bigg ) $ as long as $\\\\sqrt{(p+t) /n } \\\\lesssim b \\\\lesssim 1$ .\",\n \"With the above preparations, we are ready to present the Bahadur representation for the one-step conquer estimator $\\\\hat{\\\\beta }$ .\",\n \"Assume Conditions REF , REF and REF in the main text and Conditions and hold.\",\n \"For any $t>0$ , let the sample size $n$ , dimension $p$ and the bandwidths $h , b>0$ satisfy $n\\\\gtrsim p(\\\\log n)^2 +t$ , $\\\\sqrt{(p+t)/n} \\\\lesssim h \\\\lesssim \\\\lbrace (p+t)/n \\\\rbrace ^{1/4}$ and $\\\\sqrt{(p+t)/n } \\\\lesssim b \\\\lesssim \\\\lbrace (p+t)/n \\\\rbrace ^{1/(2\\\\nu )}$ .\",\n \"Then, the one-step conquer estimator $\\\\hat{\\\\beta }$ satisfies the bound $ \\\\Bigg \\\\Vert \\\\, \\\\hat{\\\\beta }- \\\\beta ^* - ^{-1} \\\\frac{ 1}{n} \\\\sum _{i=1}^n\\\\bigl \\\\lbrace \\\\tau - G(-\\\\varepsilon _i/b ) \\\\bigr \\\\rbrace _i \\\\Bigg \\\\Vert _2 \\\\lesssim \\\\Biggl \\\\lbrace \\\\underbrace{ \\\\sqrt{ (p\\\\log n + t)/(n b ) } }_{{\\\\rm variance~ term}}+ \\\\underbrace{ b^{\\\\nu -1} }_{{\\\\rm bias~term}} \\\\Biggr \\\\rbrace \\\\sqrt{\\\\frac{p+t}{n}}$ with probability at least $1-5e^{-t}$ , where $G(u) = \\\\int _{-\\\\infty }^u G(v)\\\\,{\\\\rm d}v$ .\",\n \"Theorem shows that using a higher-order kernel ($\\\\nu \\\\ge 4$ ) allows one to choose larger bandwidth, thereby reducing the “variance\\\" and the total Bahadur linearization error.\",\n \"Similarly to Theorem REF in the main text, the following asymptotic normal approximation result for linear projections of one-step conquer is a direct consequence of Theorem .\",\n \"Assume Conditions REF , REF and REF in the main text and Conditions and hold.\",\n \"Let the bandwidths satisfy $( \\\\frac{p+\\\\log n}{n} )^{1/2} \\\\lesssim h \\\\lesssim ( \\\\frac{p+\\\\log n}{n} )^{1/4}$ and $( \\\\frac{p+\\\\log n}{n} )^{1/2} \\\\lesssim b \\\\lesssim (\\\\frac{p+\\\\log n}{n})^{1/(2\\\\nu )}$ .\",\n \"Then, $\\\\sup _{x\\\\in , \\\\, \\\\in ^p } \\\\bigl | \\\\bigl ( n^{1/2} \\\\langle , \\\\hat{\\\\beta }- \\\\beta ^* \\\\rangle \\\\le \\\\sigma _\\\\tau x \\\\bigr ) - \\\\Phi (x ) \\\\bigr | \\\\lesssim \\\\sqrt{\\\\frac{ (p+\\\\log n)p \\\\log n}{n b}} + n^{1/2} b^{\\\\nu } , $ where $\\\\sigma _\\\\tau ^2 = \\\\tau (1-\\\\tau ) \\\\Vert ^{-1} \\\\Vert _{ }^2$ .\",\n \"In particular, with a choice of bandwidth $b \\\\asymp (\\\\frac{p+\\\\log n}{n})^{2/(2\\\\nu +1)}$ , $\\\\sup _{x\\\\in , \\\\, \\\\in ^p } \\\\bigl | \\\\bigl ( n^{1/2} \\\\langle , \\\\hat{\\\\beta }- \\\\beta ^* \\\\rangle \\\\le \\\\sigma _\\\\tau x \\\\bigr ) - \\\\Phi (x) \\\\bigr | \\\\rightarrow 0 \\\\nonumber $ as $n, p \\\\rightarrow \\\\infty $ under the scaling $p^{4\\\\nu /(2\\\\nu -1)} (\\\\log n)^{(2\\\\nu +1 )/(2\\\\nu -1)} =o(n)$ .\",\n \"Let $G(\\\\cdot )$ be a kernel of order $\\\\nu =4$ .\",\n \"In view of Theorem , we take $h \\\\asymp \\\\lbrace (p+\\\\log n)/n\\\\rbrace ^{2/5}$ as in the main text and $b = \\\\lbrace (p+\\\\log n)/n\\\\rbrace ^{2/9}$ , thereby obtaining that $n^{1/2}\\\\langle , \\\\hat{\\\\beta }-\\\\beta \\\\rangle $ , for an arbitrary $\\\\in ^p$ , is asymptotically normally distributed as long as $p^{16/7} (\\\\log n)^{9/7} = o(n)$ as $n\\\\rightarrow \\\\infty $ .\"\n ],\n [\n \"Proof of Proposition \",\n \"We start from the gradient $^{-1/2}\\\\nabla Q^G_b(\\\\beta ^*)=\\\\lbrace G ( - \\\\varepsilon /b ) - \\\\tau \\\\rbrace $ with $=^{-1/2} $ .\",\n \"Let $_{}$ be the conditional expectation given $$ .\",\n \"By integration by parts, $_{}G ( -\\\\varepsilon /b ) & = \\\\int _{-\\\\infty }^\\\\infty G(-t/b ) \\\\, {\\\\rm d} F_{\\\\varepsilon | } (t) = \\\\int _{-\\\\infty }^\\\\infty G(u) F_{\\\\varepsilon | } (- b u ) \\\\, {\\\\rm d} u.\",\n \"$ Applying a Taylor expansion with integral remainder on $F_{\\\\varepsilon | } (- b u )$ yields $F_{\\\\varepsilon | } (- b u ) &= F_{\\\\varepsilon | }(0) + \\\\sum _{\\\\ell =1}^{\\\\nu -1} F_{\\\\varepsilon | }^{(\\\\ell )} (0) \\\\frac{(- b u)^\\\\ell }{\\\\ell !}\",\n \"+ \\\\frac{(-b u)^{\\\\nu -1}}{(\\\\nu -2)!}\",\n \"\\\\int _0^1 (1-w)^{\\\\nu -2} \\\\big \\\\lbrace F_{\\\\varepsilon | }^{(\\\\nu -1)} (- b u w) - F_{\\\\varepsilon | }^{(\\\\nu -1)} (0) \\\\bigr \\\\rbrace {\\\\rm d} w \\\\\\\\&= \\\\tau + \\\\sum _{\\\\ell =0}^{\\\\nu -2} f_{\\\\varepsilon | }^{(\\\\ell )} (0) \\\\frac{(- b u)^{\\\\ell +1}}{(\\\\ell +1)!}\",\n \"+ \\\\frac{(- b u)^{\\\\nu -1}}{(\\\\nu -2)!}\",\n \"\\\\int _0^1 (1-w)^{\\\\nu -2} \\\\big \\\\lbrace f_{\\\\varepsilon | }^{(\\\\nu -2)} (- b u w) - f_{\\\\varepsilon | }^{(\\\\nu -2)} (0) \\\\bigr \\\\rbrace {\\\\rm d} w .$ Recall that $G$ is a kernel of order $\\\\nu \\\\ge 4$ (an even integer) and $\\\\kappa ^G_\\\\nu = \\\\int _{-\\\\infty }^\\\\infty |u^\\\\nu G (u) |\\\\, {\\\\rm d}u <\\\\infty $ .\",\n \"Substituting the above expansion into (REF ), we obtain $_{}G ( -\\\\varepsilon /b ) = \\\\tau - \\\\frac{b^{\\\\nu -1}}{(\\\\nu -2)!}\",\n \"\\\\int _{-\\\\infty }^\\\\infty \\\\int _0^1 u^{\\\\nu -1} G(u) (1-w)^{\\\\nu -2} \\\\big \\\\lbrace f_{\\\\varepsilon | }^{(\\\\nu -2)} (- b u w) - f_{\\\\varepsilon | }^{(\\\\nu -2)} (0) \\\\bigr \\\\rbrace {\\\\rm d} w {\\\\rm d} u .\\\\nonumber $ Furthermore, by the Lipschitz continuity of $f_{\\\\varepsilon | }^{(\\\\nu -2)}(\\\\cdot )$ around 0, $| _{}G ( -\\\\varepsilon /b ) - \\\\tau | & \\\\le \\\\frac{L_{\\\\nu -2} b^{\\\\nu } }{(\\\\nu -2)!}\",\n \"\\\\int _{-\\\\infty }^\\\\infty \\\\int _0^1 | u^{\\\\nu } G(u)| (1-w)^{\\\\nu -2} w \\\\,{\\\\rm d} w {\\\\rm d} u \\\\nonumber \\\\\\\\& = B(2,\\\\nu -1) L_{\\\\nu -2} \\\\kappa ^G_\\\\nu \\\\, b^{\\\\nu } / (\\\\nu -2)!\",\n \", \\\\nonumber $ where $B(x,y) := \\\\int _0^1 t^{x-1} (1-t)^{y-1} {\\\\rm d}t$ denotes the beta function.\",\n \"In particular, $B(2,\\\\nu -1)= \\\\Gamma (2)\\\\Gamma (\\\\nu -1)/\\\\Gamma (\\\\nu +1)= (\\\\nu -2)!/\\\\nu !$ .\",\n \"Putting together the pieces yields $\\\\Vert ^{-1/2} \\\\nabla Q^G_b(\\\\beta ^*) \\\\Vert _2 & = \\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\, _{} \\\\big \\\\lbrace G(-\\\\varepsilon /b ) - \\\\tau \\\\big \\\\rbrace ^\\\\le L_{\\\\nu -2} \\\\kappa ^G_\\\\nu \\\\, b^\\\\nu / \\\\nu !\",\n \".\",\n \"\\\\nonumber $ Turning to the Hessian, note that $& \\\\big \\\\Vert ^{-1/2} \\\\nabla ^2 Q^G_b(\\\\beta ^*) ^{-1/2} - _0 \\\\big \\\\Vert _2 = \\\\Bigg \\\\Vert \\\\int _{-\\\\infty }^\\\\infty G(u) \\\\big \\\\lbrace f_{\\\\varepsilon | } (- b u) - f_{\\\\varepsilon | }(0) \\\\big \\\\rbrace {\\\\rm d}u \\\\, ^\\\\Vert _2 .\\\\nonumber $ Applying a similar Taylor expansion as above, we have $f_{\\\\varepsilon | } (t) &= f_{\\\\varepsilon | }(0) + \\\\sum _{\\\\ell =1}^{\\\\nu -1} f_{\\\\varepsilon | }^{(\\\\ell )} (0) \\\\frac{t^\\\\ell }{\\\\ell !}\",\n \"+ \\\\frac{t^{\\\\nu -1}}{(\\\\nu -2)!}\",\n \"\\\\int _0^1 (1-w)^{\\\\nu -2} \\\\big \\\\lbrace f_{\\\\varepsilon | }^{(\\\\nu -1)} (t w ) - f_{\\\\varepsilon | }^{(\\\\nu -1)} (0) \\\\bigr \\\\rbrace {\\\\rm d} w .\",\n \"$ Under Conditions and , it follows that $& \\\\big \\\\Vert ^{-1/2} \\\\nabla ^2 Q^G_b(\\\\beta ^*) ^{-1/2} - _0 \\\\big \\\\Vert _2 \\\\\\\\& \\\\le \\\\frac{b^{\\\\nu -1}}{(\\\\nu -2)!}\",\n \"\\\\sup _{, \\\\delta \\\\in \\\\mathbb {S}^{p-1}} \\\\int _{-\\\\infty }^\\\\infty \\\\int _0^1 u^{\\\\nu -1} G(u) (1-w)^{\\\\nu -2}\\\\big \\\\lbrace f_{\\\\varepsilon | }^{(\\\\nu -1)} (- b u w) - f_{\\\\varepsilon | }^{(\\\\nu -1)} (0) \\\\bigr \\\\rbrace {\\\\rm d} w\\\\, {\\\\rm d}u \\\\, \\\\big \\\\langle , \\\\big \\\\rangle \\\\big \\\\langle , \\\\delta \\\\big \\\\rangle \\\\\\\\& \\\\le \\\\frac{b^{\\\\nu -1}}{(\\\\nu -1)!}\",\n \"\\\\sup _{, \\\\delta \\\\in \\\\mathbb {S}^{p-1}} \\\\int _{-\\\\infty }^\\\\infty | u^{\\\\nu -1} G(u) | \\\\sup _{|t| \\\\le b |u|} \\\\big | f_{\\\\varepsilon | }^{(\\\\nu -1)} (t) - f_{\\\\varepsilon | }^{(\\\\nu -1)} (0) \\\\bigr | \\\\, {\\\\rm d}u \\\\cdot | \\\\langle , \\\\rangle \\\\langle , \\\\delta \\\\big \\\\rangle | \\\\\\\\& \\\\le \\\\frac{C_G b^{\\\\nu -1 }}{(\\\\nu -1)!}\",\n \"\\\\sup _{ \\\\in \\\\mathbb {S}^{p-1}} \\\\big \\\\langle , \\\\big \\\\rangle ^2 = \\\\frac{C_G}{(\\\\nu -1)!}\",\n \"b^{\\\\nu -1 } .$ This completes the proof.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Proposition \",\n \"Consider the change of variable $\\\\delta = ^{1/2} (\\\\beta - \\\\beta ^*)$ , so that $\\\\beta \\\\in \\\\Theta ^*(r)$ is equivalent to $\\\\delta \\\\in \\\\mathbb {B}^p(r)$ .\",\n \"Write $_i = ^{-1/2} _i \\\\in ^p$ , which are isotropic random vectors, and define $_n(\\\\delta ) = ^{-1/2} \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) ^{-1/2} = \\\\frac{1}{n} \\\\sum _{i=1}^nG_b(\\\\varepsilon _i - _i^) _i _i^ \\\\quad (\\\\delta ) = \\\\bigl \\\\lbrace _n(\\\\delta ) \\\\bigr \\\\rbrace .$ For any $\\\\epsilon \\\\in (0,r)$ , there exists an $\\\\epsilon $ -net $\\\\lbrace \\\\delta _1, \\\\ldots , \\\\delta _{d_\\\\epsilon }\\\\rbrace $ with $d_\\\\epsilon \\\\le (1+ 2r/\\\\epsilon )^p$ satisfying that, for each $\\\\delta \\\\in \\\\mathbb {B}^p(r)$ , there exists some $1\\\\le j\\\\le d_\\\\epsilon $ such that $\\\\Vert \\\\delta - \\\\delta _j \\\\Vert _2 \\\\le \\\\epsilon $ .\",\n \"Hence, $& \\\\Vert _n(\\\\delta ) - (\\\\delta ) \\\\Vert _2 \\\\nonumber \\\\\\\\& \\\\le \\\\Vert _n(\\\\delta ) - _n(\\\\delta _j) \\\\Vert _2 + \\\\Vert _n(\\\\delta _j) - (\\\\delta _j) \\\\Vert _2 + \\\\Vert (\\\\delta _j) - (\\\\delta ) \\\\Vert _2 \\\\nonumber \\\\\\\\& =: I_1(\\\\delta ) + I_2 (\\\\delta _j)+ I_3(\\\\delta ).\",\n \"\\\\nonumber $ For $I_1(\\\\delta )$ , note that $G_b(u) = (1/b)G(u/b)$ is Lipschitz continuous, i.e.\",\n \"$|G_b(u) -G_b(v)| \\\\le L^G b^{-2} |u-v|$ for all $u,v \\\\in $ .\",\n \"It follows that $I_1(\\\\delta ) & \\\\le \\\\sup _{, \\\\in \\\\mathbb {S}^{p-1}} \\\\frac{1}{n} \\\\sum _{i=1}^n| G_b(\\\\varepsilon _i - _i^) - G_b(\\\\varepsilon _i -_i^_j ) | \\\\cdot | _i^\\\\cdot _i^| \\\\nonumber \\\\\\\\& \\\\le L^G b^{-2}\\\\sup _{, \\\\in \\\\mathbb {S}^{p-1}} \\\\frac{1}{n} \\\\sum _{i=1}^n| _i^ \\\\delta -\\\\delta _j ) \\\\cdot _i^\\\\cdot _i^| \\\\nonumber \\\\\\\\& \\\\le L^G b^{-2} \\\\epsilon \\\\underbrace{ \\\\sup _{\\\\in \\\\mathbb {S}^{p-1} } \\\\frac{1}{n} \\\\sum _{i=1}^n| _i^|^3 }_{=: M_{n,3}} .\",\n \"$ Next, we use the standard covering argument to bound $M_{n,3}$ .\",\n \"Given $\\\\epsilon _1 \\\\in (0,1)$ , let $_1$ be an $\\\\epsilon _1$ -net of the unit sphere $\\\\mathbb {S}^{p-1}$ with $d_{\\\\epsilon _1} := | _1 | \\\\le (1+2/\\\\epsilon _1)^p$ such that for every $\\\\in \\\\mathbb {S}^{p-1}$ , there exists some $\\\\in _1$ satisfying $\\\\Vert - \\\\Vert _2 \\\\le \\\\epsilon _1$ .\",\n \"Define the (standardized) design matrix $_n = n^{-1/3}(_1,\\\\ldots , _n)^^{n\\\\times p}$ , so that $M_{n,3} = \\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\Vert _n \\\\Vert _3^3$ .\",\n \"By the triangle inequality, $\\\\Vert _n \\\\Vert _3 & \\\\le \\\\Vert _n \\\\Vert _3 + \\\\Vert _n ( - ) \\\\Vert _3 \\\\nonumber \\\\\\\\& = \\\\Vert _n \\\\delta \\\\Vert _3 +\\\\bigg ( \\\\frac{1}{n} \\\\sum _{i=1}^n| _i^ - ) |^3 \\\\bigg )^{1/3} \\\\le \\\\Vert _n \\\\Vert _3 + \\\\epsilon _1 M_{n,3}^{1/3} .\",\n \"\\\\nonumber $ Taking the maximum over $\\\\in _1$ , and then taking the supremum over $\\\\in \\\\mathbb {S}^{p-1}$ , we arrive at $M_{n,3} \\\\le (1-\\\\epsilon _1)^{-3} \\\\widetilde{M}_{n, 3} , $ where $ \\\\widetilde{M}_{n, 3} = \\\\max _{\\\\in _1 } (1/n) \\\\sum _{i=1}^n| _i^|^3$ .\",\n \"For every $\\\\in _1$ , note that $( |_i^|^3 \\\\ge y\\\\rbrace \\\\le 2e^{-y^{2/3}/(2\\\\upsilon _1^2)}$ for any $y>0$ .\",\n \"Hence, by inequality (3.6) in [58] with $s=2/3$ , we obtain that for any $z\\\\ge 3$ , $\\\\frac{1}{n} \\\\sum _{i=1}^n|_i^|^3 \\\\le |^|^3 + C_1 \\\\Biggl ( \\\\sqrt{\\\\frac{z }{n}} + \\\\frac{z^{3/2}}{n} \\\\Bigg )$ with probability at least $1-e^{3-z}$ , where $C_1>0$ is a constant depending only on $\\\\upsilon _1$ .\",\n \"Taking the union bound over all vectors $$ in $_1$ yields that, with probability at least $1-d_{\\\\epsilon _1} e^{3-z} \\\\ge 1- e^{3+p\\\\log (1+2/\\\\epsilon _1) - z}$ , $\\\\widetilde{M}_{n,3} \\\\le \\\\mu _3 + C_1\\\\Biggl ( \\\\sqrt{\\\\frac{z }{n}} + \\\\frac{z^{3/2}}{n} \\\\Bigg )$ where $\\\\mu _3 = \\\\sup _{\\\\in \\\\mathbb {S}^{p-1} } |^|^3$ .\",\n \"Reorganizing the terms, we have $\\\\widetilde{M}_{n,3} \\\\le \\\\mu _3 + C_1 \\\\Bigg [ \\\\sqrt{\\\\frac{p\\\\log (1+2/\\\\epsilon _1) +4+t }{n}} + \\\\frac{\\\\lbrace p\\\\log (1+2/\\\\epsilon _1) + 4 + t \\\\rbrace ^{3/2}}{n} \\\\Bigg ] $ with probability at least $1- e^{-t}/2$ .\",\n \"Hence, taking $\\\\epsilon =1/8$ in (REF ) and (REF ) implies $M_{n,3} \\\\le 1.5\\\\mu _3 + 1.5 C_1 \\\\Bigg \\\\lbrace \\\\sqrt{\\\\frac{3p+ 4 +t }{n}} + \\\\frac{ ( 3p + 4+ t )^{3/2}}{n} \\\\Bigg \\\\rbrace .\",\n \"\\\\nonumber $ Plugging the above bound into (REF ) yields that as long as $n \\\\gtrsim p+t$ , $\\\\sup _{\\\\delta \\\\in \\\\mathbb {B}^p(r)} I_1(\\\\delta ) \\\\lesssim (p + t)^{1/2} b^{-2} \\\\epsilon $ with probability at least $1- e^{-t}/2$ .\",\n \"For $I_3(\\\\delta )$ , it can be similarly obtained that $I_3(\\\\delta ) \\\\le L^G b^{-2} \\\\sup _{, \\\\in \\\\mathbb {S}^{p-1}} | ^ \\\\delta - \\\\delta _j ) \\\\cdot ^\\\\cdot ^| \\\\le L^G \\\\mu _3 \\\\, b^{-2}\\\\epsilon $ uniformly over all $\\\\delta \\\\in \\\\mathbb {B}^p(r)$ .\",\n \"Turning to $I_2(\\\\delta _j)$ , note that $_n(\\\\delta _j) - (\\\\delta _j ) = (1/n) \\\\sum _{i=1}^n(1-) \\\\phi _{ij} _i _i^, where $ ij = Gb(i - ij )$ satisfy $ |ij| Gu b-1$ and{\\\\begin{@align}{1}{-1}\\\\bigl ( \\\\phi _{ij}^2 | _i \\\\bigr ) = \\\\frac{1}{b^2} \\\\int _{-\\\\infty }^\\\\infty G^2 \\\\biggl ( \\\\frac{ \\\\langle _i , \\\\delta \\\\rangle - t}{b} \\\\biggr ) f_{\\\\varepsilon _i | _i } (t) \\\\, {\\\\rm d} t = \\\\frac{1}{b} \\\\int _{-\\\\infty }^\\\\infty G^2(u) f_{\\\\varepsilon _i | _i } (_i^- b u ) \\\\, {\\\\rm d} u \\\\le \\\\frac{ \\\\bar{f} m_{G^2 } }{b} \\\\nonumber \\\\end{@align}}almost surely, where $ mG2 := -G2(u) du <$.Given $ 2(0,1/2)$, there exits an $ 2$-net $$ of the sphere $ Sp-1$ with $ ||(1+2/2)p$ such that $ n($\\\\delta $ j) - ($\\\\delta $ j )2 (1-22)-1 | n($\\\\delta $ j) - ($\\\\delta $ j ) }|$.\",\n \"Given $$ and $ k=2,3,...$, we bound the higher order moments of $ ij (i)2$ by{\\\\begin{@align}{1}{-1}|\\\\phi _{ij} (_i^)^2 |^k & \\\\le \\\\bar{f} m_{G^2 } b^{-1} \\\\cdot (\\\\kappa ^G_u b^{-1})^{k-2} \\\\upsilon _1^{2k} \\\\cdot 2k \\\\int _0^\\\\infty \\\\bigl ( |_i^| \\\\ge \\\\upsilon _1 u \\\\bigr ) u^{2k-1} {\\\\rm d} u \\\\nonumber \\\\\\\\& \\\\le \\\\bar{f} m_{G^2 } b^{-1} \\\\cdot ( \\\\kappa ^G_u b^{-1})^{k-2} \\\\upsilon _1^{2k} \\\\cdot 4k \\\\int _0^\\\\infty u^{2k-1} e^{-u^2/2} {\\\\rm d} u \\\\nonumber \\\\\\\\& \\\\le \\\\bar{f} m_{G^2} b^{-1} \\\\cdot ( \\\\kappa ^G_u b^{-1})^{k-2} \\\\upsilon _1^{2k} \\\\cdot 2^{k+1} k!\",\n \".\\\\nonumber \\\\end{@align}}In particular, $ ij2 (i)4 16 14 f mG2 b-1 $, and $ |ij (i)2 |kk!2 16 14 f mG2 b-1 (212Gu b-1 )k-2$ for $ k3$.\",\n \"Applying Bernstein^{\\\\prime }s inequality and the union bound, we find that for any $ u0$,{\\\\begin{@align}{1}{-1}& \\\\Vert _n(\\\\delta _j) - (\\\\delta _j ) \\\\Vert _2 \\\\nonumber \\\\\\\\& \\\\le \\\\frac{1}{1-2\\\\epsilon _2} \\\\max _{\\\\in } \\\\Biggl | \\\\frac{1}{n} \\\\sum _{i=1}^n(1-) \\\\phi _{ij} (_i^)^2 \\\\Biggr | \\\\le \\\\frac{ 2\\\\upsilon _1^2}{1- 2\\\\epsilon _2} \\\\Biggl ( 2 \\\\sqrt{2 \\\\bar{f} m_{G^2 } \\\\frac{u}{n b }} +\\\\kappa ^G_u \\\\frac{u}{n b} \\\\Biggr ) \\\\nonumber \\\\end{@align}}with probability at least $ 1-2 (1+2/2)p e-u = 1 - (1/2) e(4) + p(1+2/2) - u$.\",\n \"Setting $ 2=2/(e3-1)$ and $ u= (4) + 3p + v$, it follows that with probability at least $ 1 -e-v/2$,{\\\\begin{@align}{1}{-1}I_2(\\\\delta _j) \\\\lesssim \\\\sqrt{ \\\\frac{ p + v }{n b } } + \\\\frac{ p + v }{n b} .\",\n \"\\\\nonumber \\\\end{@align}}Once again, taking the union bound over $ j=1,..., d$ and setting $ v= p (1 + 2r/)+ t$, we obtain that with probability at least $ 1-de-v 1-e-t/2$,{\\\\begin{@align}{1}{-1}\\\\max _{1\\\\le j\\\\le N_{\\\\epsilon }} I_2(\\\\delta _j) \\\\lesssim \\\\sqrt{\\\\frac{p\\\\log (3 e r/\\\\epsilon )+t}{n b }} + \\\\frac{p\\\\log (3e r/\\\\epsilon )+t}{n b} .\",\n \"\\\\end{@align}}$ Finally, combining (REF ), (REF ) and (), and taking $\\\\epsilon = r/n \\\\in (0, r)$ in the beginning of the proof, we conclude that with probability at least $1- e^{-t}$ , $& \\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)}\\\\big \\\\Vert ^{-1/2} \\\\big \\\\lbrace \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla ^2 Q^G_b(\\\\beta ) \\\\big \\\\rbrace \\\\nonumber ^{-1/2} \\\\big \\\\Vert _2 \\\\lesssim \\\\sqrt{\\\\frac{p \\\\log n + t}{n b }} + \\\\frac{p\\\\log n+t}{n b } + \\\\frac{(p + t)^{1/2} r }{n b^2} \\\\nonumber $ as long as $n\\\\gtrsim p+t$ .\",\n \"This completes the proof.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Proposition \",\n \"Define the stochastic process $\\\\Delta _b(\\\\beta ) = ^{-1/2} [ \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\rbrace - ( \\\\beta - \\\\beta ^*) ]$ .\",\n \"By the triangle inequality, $\\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)} \\\\Vert \\\\Delta _b(\\\\beta ) \\\\Vert _2 \\\\le \\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)} \\\\Vert \\\\Delta _b(\\\\beta ) \\\\Vert _2 + \\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)} \\\\Vert \\\\Delta _b(\\\\beta ) - \\\\Delta _b(\\\\beta ) \\\\Vert _2$ Recall that $_0 = ^{-1/2} ^{-1/2} = \\\\lbrace f_{\\\\varepsilon |}(0) ^$ .\",\n \"For the first term on the right-hand side, using the mean value theorem for vector-valued functions yields $\\\\Delta _b(\\\\beta ) = \\\\Biggl \\\\lbrace ^{-1/2} \\\\int _0^1 \\\\nabla ^2 Q^G_b\\\\bigl ( (1-s) \\\\beta ^* + s \\\\beta \\\\bigr ) \\\\, {\\\\rm d}s \\\\, ^{-1/2} - _0 \\\\Biggr \\\\rbrace ^{1/2} ( \\\\beta -\\\\beta ^* ) .$ By a change of variable $\\\\delta = ^{1/2}(\\\\beta - \\\\beta ^*)$ , $\\\\nabla ^2 Q^G_b\\\\bigl ( (1-s) \\\\beta ^* + s \\\\beta \\\\bigr ) = \\\\int _{-\\\\infty }^\\\\infty G(u) f_{\\\\varepsilon | } (s ^- b u ) \\\\, {\\\\rm d}u \\\\cdot ^$ For every $s \\\\in [0,1]$ and $u\\\\in $ , it ensures from $f_{\\\\varepsilon | }(\\\\cdot )$ being Lipschitz that $| f_{\\\\varepsilon | } (s ^- b u ) - f_{\\\\varepsilon | } ( - b u ) | \\\\le l_0 s \\\\cdot | ^| .$ Moreover, by the Taylor expansion (REF ), $f_{\\\\varepsilon | } ( - b u ) & = f_{\\\\varepsilon | }(0) + \\\\sum _{\\\\ell =1}^{\\\\nu -1} f_{\\\\varepsilon | }^{(\\\\ell )}(0) \\\\frac{( - b u )^\\\\ell }{\\\\ell !}\",\n \"+ \\\\frac{( - b u)^{\\\\nu -1}}{(\\\\nu -2)!}\",\n \"\\\\int _0^1 (1-w)^{\\\\nu -2} \\\\bigl \\\\lbrace f_{\\\\varepsilon | }^{(\\\\nu -1)}( - b u w ) -f_{\\\\varepsilon | }^{(\\\\nu -1)}(0)\\\\bigr \\\\rbrace \\\\, {\\\\rm d}w.$ Consequently, $& \\\\Bigg \\\\Vert ^{-1/2} \\\\int _0^1 \\\\nabla ^2 Q^G_b\\\\bigl ( (1-s) \\\\beta ^* + s\\\\beta \\\\bigr )\\\\, {\\\\rm d}s \\\\, ^{-1/2} - _0 \\\\Bigg \\\\Vert _2 \\\\\\\\&\\\\le \\\\Bigg \\\\Vert \\\\frac{b^{\\\\nu -1}}{(\\\\nu -2)!}\",\n \"\\\\int _{-\\\\infty }^\\\\infty \\\\int _0^1 u^{\\\\nu -1} G(u) (1-w)^{\\\\nu -2} \\\\bigl \\\\lbrace f_{\\\\varepsilon | }^{(\\\\nu -1)}( - b u w ) -f_{\\\\varepsilon | }^{(\\\\nu -1)}(0)\\\\bigr \\\\rbrace \\\\, {\\\\rm d}w {\\\\rm d}u \\\\cdot ^\\\\Vert _2 \\\\\\\\&~~~~ + \\\\frac{l_0}{2} \\\\cdot \\\\big \\\\Vert \\\\, |^| \\\\cdot ^\\\\Vert _2 \\\\\\\\& \\\\le \\\\frac{C_G b^{\\\\nu -1}}{(\\\\nu -1)!}\",\n \"\\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\big \\\\langle , \\\\big \\\\rangle ^2 + \\\\frac{l_0}{2} \\\\sup _{\\\\in \\\\mathbb {S}^{p-1}} \\\\big ( | ^| \\\\langle , \\\\rangle ^2 \\\\big ) \\\\\\\\& \\\\le \\\\Bigg ( \\\\frac{ C_G }{(\\\\nu -1)! }\",\n \"b^{\\\\nu -1} + 0.5 l_0 \\\\mu _3\\\\Vert \\\\delta \\\\Vert _2 \\\\Bigg ) .$ Taking the supremum over $\\\\beta \\\\in \\\\Theta ^*(r)$ , or equivalently $\\\\delta \\\\in \\\\mathbb {B}^p(r)$ , yields $\\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)} \\\\Vert \\\\Delta _b(\\\\beta ) \\\\Vert \\\\le \\\\Bigg ( \\\\frac{ C_G }{(\\\\nu -1)! }\",\n \"b^{\\\\nu -1} + 0.5 l_0 \\\\mu _3 r \\\\Bigg ) r \\\\lesssim (b^{\\\\nu -1} + r ) r .\",\n \"\\\\nonumber $ For the stochastic term $ \\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)} \\\\Vert \\\\Delta _b(\\\\beta ) - \\\\Delta _b(\\\\beta ) \\\\Vert _2$ , following the proof of Theorem REF , it can be similarly shown that with probability at least $1-e^{-t}$ , $\\\\sup _{\\\\beta \\\\in \\\\Theta ^*(r)} \\\\Vert \\\\Delta _b(\\\\beta ) - \\\\Delta _b(\\\\beta ) \\\\Vert _2 \\\\lesssim r \\\\sqrt{\\\\frac{p+t}{n b}} \\\\nonumber $ as long as $\\\\sqrt{(p+t)/n} \\\\lesssim b\\\\lesssim 1$ .\",\n \"Combining the last two displays completes the proof of the claim (REF ).\",\n \"$\\\\Box $\"\n ],\n [\n \"Proof of Theorem \",\n \"Step 1 (Consistency of the initial estimate).\",\n \"First, note that the consistency of the initial estimator $\\\\beta $ —namely, $\\\\beta $ lies in a local neighborhood of $\\\\beta ^*$ with high probability, is a direct consequence of Theorem REF .\",\n \"Given a non-negative kernel $K(\\\\cdot )$ and for any $t>0$ , the ensuing estimator $\\\\beta $ satisfies $\\\\Vert \\\\beta - \\\\beta ^* \\\\Vert _{} \\\\le r_n \\\\asymp \\\\sqrt{\\\\frac{p+t}{n}} $ with probability at least $1-2e^{-t}$ as long as $ (\\\\frac{p+t}{n})^{1/2} \\\\lesssim h \\\\lesssim (\\\\frac{p+t}{n})^{1/4}$ .\",\n \"Provided that the sample Hessian $\\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta )$ is invertible, by the definition of $\\\\hat{\\\\beta }$ we obtain $^{1/2} ( \\\\hat{\\\\beta }- \\\\beta ) & = - \\\\bigl \\\\lbrace ^{-1/2} \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) ^{-1/2} \\\\bigr \\\\rbrace ^{-1} ^{-1/2}\\\\nabla \\\\hat{Q}^G_b(\\\\beta ) \\\\nonumber \\\\\\\\& = - \\\\bigl \\\\lbrace ^{-1/2} \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) ^{-1/2} \\\\bigr \\\\rbrace ^{-1} \\\\bigl [ ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\bigr \\\\rbrace - ^{-1/2} (\\\\beta - \\\\beta ^* ) \\\\bigr ] \\\\nonumber \\\\\\\\& ~~~~~ - \\\\bigl \\\\lbrace ^{-1/2} \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) ^{-1/2} \\\\bigr \\\\rbrace ^{-1} \\\\bigl \\\\lbrace ^{-1/2} (\\\\beta - \\\\beta ^* ) + ^{-1/2} \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* )\\\\bigr \\\\rbrace ,\\\\nonumber $ or equivalently, $^{1/2} (\\\\hat{\\\\beta }- \\\\beta ^*) & = ^{1/2} (\\\\beta -\\\\beta ^* ) - _0^{-1} _0 ^{1/2} (\\\\beta - \\\\beta ^* ) - _0^{-1} ^{-1/2} \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\nonumber \\\\\\\\& ~~~~ - _0^{-1} ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) - (\\\\beta - \\\\beta ^* ) \\\\bigr \\\\rbrace , \\\\nonumber $ where $_0 = \\\\lbrace f_{\\\\varepsilon |}(0) ^ = ^{-1/2} ^{-1/2}$ and $_0 := ^{-1/2} \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) ^{-1/2} $ .\",\n \"It follows that $& \\\\bigl \\\\Vert ^{1/2} ( \\\\hat{\\\\beta }- \\\\beta ^* ) + ^{1/2} ^{-1} \\\\nabla \\\\hat{}_b^G(\\\\beta ^* ) \\\\bigr \\\\Vert _2= \\\\bigl \\\\Vert ^{1/2} (\\\\hat{\\\\beta }- \\\\beta ^*) + _0^{-1} ^{-1/2}\\\\nabla \\\\hat{}_b^G(\\\\beta ^* ) \\\\bigr \\\\Vert _2 \\\\nonumber \\\\\\\\& \\\\le \\\\underbrace{ \\\\Vert _p - _0^{-1} _0 \\\\Vert _2 }_{ {\\\\rm see~Proposition~\\\\ref {prop:Hessian.uniform}} }\\\\cdot \\\\bigl \\\\lbrace \\\\Vert \\\\beta -\\\\beta ^* \\\\Vert _{} + \\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\Vert _2 \\\\bigr \\\\rbrace \\\\\\\\&~~~~~ + \\\\Vert _0^{-1} \\\\Vert _2 \\\\cdot \\\\underbrace{ \\\\big \\\\Vert ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) - (\\\\beta - \\\\beta ^* ) \\\\bigr \\\\rbrace \\\\big \\\\Vert _2 }_{ {\\\\rm see~Proposition~\\\\ref {prop:gradient.uniform}}}.\",\n \"\\\\nonumber $ According to the above bound, we need to further control the following three terms: $\\\\Vert _p - _0^{-1} _0 \\\\Vert _2 , ~~\\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\Vert _2~\\\\mbox{ and }~\\\\big \\\\Vert ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) -(\\\\beta - \\\\beta ^* ) \\\\bigr \\\\rbrace \\\\big \\\\Vert _2.$ Step 2 (Invertibility of the sample Hessian $\\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta )$ ).\",\n \"Recall that $_0 = ^{-1/2} \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) ^{-1/2} $ and $_0 = ^{-1/2} ^{-1/2}$ .\",\n \"By the triangle inequality, $& \\\\Vert _0 - _0 \\\\Vert _2 \\\\le \\\\big \\\\Vert ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla ^2 \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla ^2 Q^G_b( \\\\beta ^*) \\\\bigr \\\\rbrace ^{-1/2} \\\\big \\\\Vert _2 + \\\\big \\\\Vert ^{-1/2} \\\\nabla ^2 Q^G_b( \\\\beta ^*) ^{-1/2} - _0 \\\\big \\\\Vert _2 .\",\n \"\\\\nonumber $ Let the bandwidth $b$ satisfy $ \\\\max \\\\lbrace \\\\frac{p\\\\log n + t}{n} , n^{-1/2}\\\\rbrace \\\\lesssim b \\\\lesssim 1$ .\",\n \"Then, applying Propositions and with $r=r_n$ yields that with probability $1-3e^{-t}$ , $& \\\\Vert _0 - _0 \\\\Vert _2 \\\\le \\\\delta _n \\\\asymp \\\\sqrt{\\\\frac{p\\\\log n + t}{ nb }} + b^{\\\\nu -1} .\",\n \"\\\\nonumber $ Note that, under Condition REF , $0< {f} \\\\le \\\\lambda _{\\\\min }(_0) \\\\le \\\\lambda _{\\\\max }(_0) \\\\le \\\\bar{f}$ .\",\n \"For sufficiently large $n$ , this implies $\\\\Vert _0^{-1} _0 - _p \\\\Vert _2 \\\\le {f}^{-1} \\\\delta _n < 1$ , and hence $\\\\Vert _0^{-1} _0 - _p \\\\Vert _2 \\\\le \\\\frac{\\\\delta _n}{ {f} - \\\\delta _n } ~~\\\\mbox{ and }~~ \\\\Vert _0^{-1} \\\\Vert _2 \\\\le \\\\frac{1}{{f} - \\\\delta _n} .$ Step 3 (Controlling the score).\",\n \"In view of (REF ), it remains to control $\\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* )\\\\Vert _2$ and $\\\\Vert ^{-1/2} \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\rbrace - ^{-1/2} (\\\\beta - \\\\beta ^* ) \\\\Vert _2$ .\",\n \"For the latter, applying the concentration bound (REF ) and Proposition yields that, with probability at least $1-3e^{-t}$ , $\\\\big \\\\Vert ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\bigr \\\\rbrace - ^{-1/2} (\\\\beta - \\\\beta ^* ) \\\\big \\\\Vert _2 \\\\lesssim \\\\frac{p+ t}{n b^{1/2}} + b^{\\\\nu -1} \\\\sqrt{\\\\frac{p+t}{n}} $ as long as $ (\\\\frac{p+t}{n})^{1/2} \\\\lesssim b \\\\lesssim 1$ .\",\n \"Turning to $\\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* )\\\\Vert _2$ , it follows from Lemma REF and Proposition that with probability at least $1-e^{-t}$ , $\\\\Vert ^{-1/2} \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\Vert _2 \\\\lesssim \\\\sqrt{\\\\frac{p+ t}{n}} + b^\\\\nu .\",\n \"$ Finally, combining the bounds (REF )–(REF ), we conclude that with probability at least $1-5e^{-t}$ , $\\\\big \\\\Vert ^{-1/2} \\\\bigl \\\\lbrace \\\\nabla \\\\hat{Q}^G_b(\\\\beta ) - \\\\nabla \\\\hat{Q}^G_b(\\\\beta ^* ) \\\\bigr \\\\rbrace - ^{-1/2} (\\\\beta - \\\\beta ^* ) \\\\big \\\\Vert _2\\\\lesssim \\\\Biggl ( \\\\sqrt{\\\\frac{p\\\\log n+ t}{n b}}+ b^{\\\\nu -1} \\\\Biggr ) \\\\Bigg ( \\\\sqrt{\\\\frac{p+t}{n}} + b^\\\\nu \\\\Bigg ) , \\\\nonumber $ provided that $ \\\\max \\\\bigl \\\\lbrace \\\\frac{p\\\\log n + t}{n} , (\\\\frac{p + t}{n})^{1/2} \\\\bigr \\\\rbrace \\\\lesssim b \\\\lesssim 1$ .\",\n \"Under the sample size requirement $n\\\\gtrsim p \\\\,(\\\\log n)^2+t$ and additional constraint $b \\\\lesssim (\\\\frac{p+t}{n})^{1/(2\\\\nu )}$ , this leads to the claimed bound (REF ).\",\n \"$\\\\Box $\"\n ],\n [\n \"Additional Simulation Results\",\n \"In this section, we present additional results for the numerical studies as described in Section in Figures REF –REF .\"\n ]\n]"},"id":{"kind":"string","value":"2012.05187"}}},{"rowIdx":3687,"cells":{"text":{"kind":"list like","value":[["First observation of the decay $B_s^0 \\to K^-\\mu^+\\nu_\\mu$ and a\n measurement of $|V_{ub}|/|V_{cb}|$"],["Abstract The first observation of the suppressed semileptonic $B_s^0 \\to K^-\\mu^+\\nu_\\mu$ decay is reported.","Using a data sample recorded in {\\it pp} collisions in 2012 with the LHCb detector, corresponding to an integrated luminosity of 2 $\\mathrm{fb}^{-1}$, the branching fraction \\mbox{$\\mathcal{B}(B_s^0 \\to K^-\\mu^+\\nu_\\mu)$} is measured to be $(1.06\\pm0.05~(\\mathrm{stat})\\pm0.08~(\\mathrm{syst}))\\times 10^{-4}$, where the first uncertainty is statistical and the second one represents the combined systematic uncertainties.","The decay $B_s^0 \\to D_s^-\\mu^+\\nu_\\mu$, where $D_s^-$ is reconstructed in the final state $K^+K^-\\pi^-$, is used as a normalization channel to minimize the experimental systematic uncertainty.","Theoretical calculations on the form factors of the $B_s^0 \\to K^-$ and $B_s^0 \\to D_s^-$ transitions are employed to determine the ratio of the CKM matrix elements ${|V_{ub}|}/{|V_{cb}|}$ at low and high $B_s^0 \\to K^-$ momentum transfer."],["0.5em"]],"string":"[\n [\n \"First observation of the decay $B_s^0 \\\\to K^-\\\\mu^+\\\\nu_\\\\mu$ and a\\n measurement of $|V_{ub}|/|V_{cb}|$\"\n ],\n [\n \"Abstract The first observation of the suppressed semileptonic $B_s^0 \\\\to K^-\\\\mu^+\\\\nu_\\\\mu$ decay is reported.\",\n \"Using a data sample recorded in {\\\\it pp} collisions in 2012 with the LHCb detector, corresponding to an integrated luminosity of 2 $\\\\mathrm{fb}^{-1}$, the branching fraction \\\\mbox{$\\\\mathcal{B}(B_s^0 \\\\to K^-\\\\mu^+\\\\nu_\\\\mu)$} is measured to be $(1.06\\\\pm0.05~(\\\\mathrm{stat})\\\\pm0.08~(\\\\mathrm{syst}))\\\\times 10^{-4}$, where the first uncertainty is statistical and the second one represents the combined systematic uncertainties.\",\n \"The decay $B_s^0 \\\\to D_s^-\\\\mu^+\\\\nu_\\\\mu$, where $D_s^-$ is reconstructed in the final state $K^+K^-\\\\pi^-$, is used as a normalization channel to minimize the experimental systematic uncertainty.\",\n \"Theoretical calculations on the form factors of the $B_s^0 \\\\to K^-$ and $B_s^0 \\\\to D_s^-$ transitions are employed to determine the ratio of the CKM matrix elements ${|V_{ub}|}/{|V_{cb}|}$ at low and high $B_s^0 \\\\to K^-$ momentum transfer.\"\n ],\n [\n \"0.5em\"\n ]\n]"},"id":{"kind":"string","value":"2012.05143"}}},{"rowIdx":3688,"cells":{"text":{"kind":"list like","value":[["Long-time behaviour of a model for p62-ubiquitin aggregation in cellular\n autophagy"],["Abstract The qualitative behavior of a recently formulated ODE model for the dynamics of heterogenous aggregates is analyzed.","Aggregates contain two types of particles, oligomers and cross-linkers.","The motivation is a preparatory step of cellular autophagy, the aggregation of oligomers of the protein p62 in the presence of ubiquitin cross-linkers.","A combination of explicit computations, formal asymptotics, and numerical simulations has led to conjectures on the bifurcation behavior, certain aspects of which are proven rigorously in this work.","In particular, the stability of the zero state, where the model has a smoothness deficit is analyzed by a combination of regularizing transformations and blow-up techniques.","On the other hand, in a different parameter regime, the existence of polynomially growing solutions is shown by Poincar\\'e compactification, combined with a singular perturbation analysis ."],["Introduction","A preparatory step of cellular autophagy is the aggregation of cellular waste material before inclusion in an autophagosome and, later, a lysosome, which are vesicular structures, where the waste is eventually decomposed.","In vitro studies of the evolution of heterogeneous aggregates of the proteins p62 and ubiquitin [8] have motivated the formulation of a mathematical model of this process [1].","The model has the form of an ODE system, which shows different qualitative behaviour in three different regions of parameter space.","This statement is based on formal asymptotics and numerical simulations carried out in [1].","Since some of these observations are not accessible to standard dynamical systems methods, it is the purpose of this work to provide a rigorous analysis.","The model is based on the assumption that ubiquitinated waste material serves as a cross-linker between p62 oligomers of a fixed size $n\\ge 3$ , where each monomer can serve as a binding site for cross-linking.","It describes the evolution of the size of aggregates through the evolution of three parameters $p$ , $q$ and $r$ , where $p$ represents the number of one-hand bound ubiquitin links in the aggregate (in green in Figs.","REF and REF ), $q$ represents the number of both-hand bound cross-links (in red in Figs.","REF and REF ), and $r$ represents the number of p62$_n$ oligomers in the aggregate (in black in Figs.","REF and REF with $n=5$ ).","Since we are interested in large aggregates, the variables $(p,q,r)$ are considered as continuous after an appropriate scaling.","The state space is a subset of the positive octant of $\\mathbb {R}^3$ determined by two constraints: For a connected aggregate the number of two-hand bound cross-links has to be at least the number of p62 oligomers minus one.","In the continuous description this becomes the constraint $q\\ge r$ .","Since the total number of binding sites on the p62 oligomers in an aggregate is $nr$ , the number of free binding sites is equal to $nr-p-2q$ , which has to be nonnegative.","The dynamics of an aggregate is governed by the basic binding and unbinding reactions between cross-linkers and p62 oligomers.","Since the reaction rates depend on the state of the reaction partners and of the aggregate, six different reactions have been considered in [1] (see Figs.","REF and REF ).","The models for the reaction rates are based on the law of mass action.","However, since the shape of an aggregate is not described unambiguously by the parameters $(p,q,r)$ , some additional empirical modeling assumptions are required.","Figure: Left: Reaction 1: addition of a free cross-linker to the aggregate.","Right: Reaction 2: addition of a p62 5 _5 oligomer to the aggregate.Figure: Left: Reaction 3 is a rearrangement making the aggregate more cohesive.","Right: the breaking of a cross-link bound can lead either to the reverse of Reaction 2 of of Reaction 3.Reaction 1 is the addition of a free cross-linker to the aggregate.","This is a second order reaction with a rate proportional to the concentration of free cross-linkers and to the number of free binding sites on oligomers.","Since the supply of free cross-linkers and oligomers has been modeled as not limiting in [1], the reaction rate is written as $\\kappa _1(nr-p-2q)$ with rate constant $\\kappa _1$ , which can be seen as proportional to the cross-linker concentration, modeled as constant.","Similarly Reaction 2, the addition of a free oligomer to the aggregate, is modeled as a first order reaction with rate $\\kappa _2 p$ proportional to the number of one-hand bound cross-linkers.","Reaction 3 is consolidating the aggregate by building an additional cross-link using a so far only one-hand bound cross-linker.","Its rate is $\\kappa _3 p(nr-p-2q)$ .","For the reverse of Reaction 1, the rate $\\kappa _{-1}p$ should not be a surprise.","The reverses of Reactions 2 and 3 are actually the same reaction with rate $\\kappa _- q$ , but with possibly different outcomes.","Therefore we write their rate constants as $\\kappa _{-2}:= \\kappa _- \\alpha $ and $\\kappa _{-3}:= \\kappa _- (1-\\alpha )$ , with $\\alpha =\\alpha (q,r)\\in [0,1]$ .","The reverse of Reaction 2, i.e.","loss of an oligomer, never happens in a fully connected aggregate with $nr=2q$ .","It always happens in a minimally connected aggregate with $q=r$ .","This motivates the choice $\\alpha = \\frac{nr-2q}{(n-2)r}$ .","Concerning the outcome of this reverse reaction, it has to be taken into account that the loss of an oligomer might also mean a loss of one-hand bound cross-links attached to it.","This produces the loss term $\\kappa _- q \\frac{(n-1)p}{(n-2)r}$ (see [1] for details).","It is now straightforward to write down the ODE problem governing the evolution of the state variables: $ \\begin{aligned}\\dot{p} &= (\\kappa _1 - \\kappa _3 p) (nr-p-2q) + \\kappa _{-}q \\left( 1 - \\frac{(n-1)p}{(n-2)r}\\right) - (\\kappa _2 + \\kappa _{-1})p \\,, &\\qquad p(0)=p_0\\,,\\\\\\dot{q} &= \\kappa _2 p + \\kappa _3 p(nr-p-2q) - \\kappa _{-}q\\,, &\\qquad q(0)=q_0\\,, \\\\\\dot{r} &= \\kappa _2 p - \\kappa _{-} q \\alpha (q,r), \\qquad \\alpha (q,r) = \\frac{nr -2q}{(n-2) r}\\,, &\\qquad r(0)=r_0\\,,\\end{aligned}$ with the inequalities $ nr - p - 2q \\ge 0 \\,, \\qquad q\\ge r \\,,$ implying $0 \\le \\alpha (q,r) \\le 1 \\,.$ We recall from [1] that for initial data $p_0,q_0,r_0>0$ satisfying (REF ), which we assume in the following, the initial value problem (REF ) has a unique, global solution propagating (REF ), the nonnegativity of the components, and in particular $r(t),q(t) > 0 \\,,\\qquad t\\ge 0 \\,.$ Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5,κ 1 =κ 2 =κ 3 =κ -1 =1\\kappa _1= \\kappa _2=\\kappa _3=\\kappa _{-1}=1, and κ - =0.6\\kappa _{-}=0.6, giving 0<α ¯<10<\\bar{\\alpha }< 1 and convergence to the nontrivial equilibrium.Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5, κ 1 =κ 2 =κ 3 =κ -1 =1\\kappa _1= \\kappa _2=\\kappa _3=\\kappa _{-1} = 1 ,and κ - =0.93\\kappa _{-}=0.93, giving α ¯>1\\bar{\\alpha }>1 and convergence to the zero steady state.Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5,κ 1 =κ 2 =κ 3 =κ -1 =1\\kappa _1= \\kappa _2=\\kappa _3=\\kappa _{-1} = 1, and κ - =0.2\\kappa _{-}=0.2, giving α ¯<0\\bar{\\alpha }<0 and a polynomially growing aggregate.The search for steady states [1] has suggested a splitting of the parameter space into three regions.","Besides the trivial steady state $(p,q,r)=(0,0,0)$ , only one other equilibrium may exist, which can be computed explicitly: $& \\bar{p} =\\frac{\\kappa _- A}{\\kappa _2} \\frac{1-\\bar{\\alpha }}{\\bar{\\alpha }} \\,,\\qquad \\bar{q} = A \\frac{1-\\bar{\\alpha }}{\\bar{\\alpha }^2} \\,, \\qquad \\bar{r} = \\frac{2A}{n-(n-2)\\bar{\\alpha }} \\frac{1-\\bar{\\alpha }}{\\bar{\\alpha }^2}\\,, \\\\&\\mbox{with}\\quad \\bar{\\alpha }= \\frac{n}{n-2} + \\frac{\\kappa _{-1} + \\kappa _1 - \\sqrt{(\\kappa _1 + \\kappa _{-1})^2 + 4 \\kappa _1 \\kappa _2 (n-1)}}{\\kappa _{-} (n-1)} \\,, \\quad A = \\frac{2\\kappa _1 \\kappa _2^2 (n-2)}{\\kappa _3 \\kappa _- (\\kappa _-(n-1)(n - (n-2)\\bar{\\alpha }) + 2\\kappa _{-1}(n-2))} \\,.\\nonumber $ Since $\\bar{\\alpha }= \\alpha (\\bar{q},\\bar{r})$ is the equilibrium value of $\\alpha $ , the nontrivial steady state is relevant only in the parameter region defined by $0<\\bar{\\alpha }<1$ .","It has been conjectured in [1] that in this parameter region $(\\bar{p},\\bar{q}, \\bar{r})$ is globally attracting, which has been supported by numerical simulations (see also Fig.","REF ).","Local stability could in principle be examined by linearization.","However, the complexity of the resulting formulas has been prohibitive.","Since $(\\bar{p},\\bar{q},\\bar{r})\\rightarrow (0,0,0)$ as $\\bar{\\alpha }\\rightarrow 1-$ , it seems natural to expect a transcritical bifurcation at $\\bar{\\alpha }=1$ with stability of the trivial steady state for $\\bar{\\alpha }>1$ .","Again the conjecture of global asymptotic stability of $(0,0,0)$ for $\\bar{\\alpha }>1$ has been supported by simulations (see for example Fig.","REF ).","The right hand sides of (REF ) are continuous up to the origin (when considered as an element of the set of admissible states), since $0\\le \\alpha (q,r)\\le 1$ and $p/r\\le n$ .","However, their nonsmoothness prohibits a standard local stability or bifurcation analysis.","The expected local stability behaviour (asymptotic stability for $\\bar{\\alpha }>1$ , instability for $\\bar{\\alpha }<1$ ) is proven in Section .","The analysis is based on a regularizing transformation, which makes the steady state very degenerate, combined with a blow-up analysis [2].","The fact that the components of the nontrivial equilibrium tend to infinity when $\\bar{\\alpha }\\rightarrow 0+$ suggests that solutions might be unbounded for $\\bar{\\alpha }<0$ .","In this parameter region approximate solutions with polynomial growth of the form $p(t) = p_1 t+ \\textit {o}(t) \\,,\\qquad q(t) = q_2 t^2 + \\textit {o}(t^2) \\,,\\qquad r(t) = \\frac{2q_2}{n} t^2+ \\textit {o}(t^2) \\,,\\qquad \\mbox{as } t\\rightarrow \\infty \\,,$ have been constructed in [1] by formal asymptotic methods.","It has also been shown that no other growth behaviour (polynomial with other powers or exponential) should be expected, and the conjecture that all solutions have the constructed asymptotic behaviour is again verified by simulations (see Fig.","for example REF ).","We justify the formal asymptotics in Section .","A variant of Poincaré compactification [4] produces a problem with bounded solutions and with three different time scales, which is analyzed by singular perturbation methods [3].","The final result is existence and semi-local stability of the polynomially growing solutions, where 'semi-local' means that initial data have to be large with relative sizes as in (REF ).","The article is concluded by a discussion section about biological interpretation of our results as well as perspectives."],["Local stability of the zero steady state","In this section, we study under which conditions small aggregates tend to disaggregate.","This is equivalent to studying the stability of the zero-steady-state $(p,q,r)=(0,0,0)$ of the system (REF ).","Because of the appearance of the ratios $\\frac{p}{r}$ and $\\frac{q}{r}$ , the Jacobian of the right hand side of (REF ) is not defined there.","As a consequence of (REF ) the regularizing transformation $\\tau := \\int _0^t r(s)^{-1} \\mathrm {d}s$ is well defined and leads to $ \\begin{aligned}\\frac{dp}{d\\tau } &= r(\\kappa _1 - \\kappa _3 p) (nr-p-2q) + \\kappa _{-}q \\left( r - \\frac{(n-1)p}{n-2}\\right) - (\\kappa _2 + \\kappa _{-1})pr \\,, \\\\\\frac{dq}{d\\tau } &= \\kappa _2 pr + \\kappa _3 pr(nr-p-2q) - \\kappa _{-}qr\\,, \\\\\\frac{dr}{d\\tau } &= \\kappa _2 pr - \\kappa _{-} q \\frac{nr -2q}{n-2}\\,.\\end{aligned}$ The regularization came at the expense that the zero steady state is degenerate in (REF ), since the right hand side is of second order in terms of the densities.","A classical approach to study such non-hyperbolic points is blow-up [2].","The standard blow-up transformation would be the introduction of spherical coordinates, blowing up the origin to the part of $\\mathbb {S}^2$ in the positive octant.","It is also common to work with charts instead.","In our case this preserves the polynomial form of the right hand side.","Although the charts in the different coordinate directions are equivalent, since the state space is a subset of the positive octant, it has turned out to be convenient to use the $q$ -chart, whence the blow-up transformation $(p,q,r) \\rightarrow (p_1,q_1,r_1)$ is given by $p= p_1q_1\\,, \\qquad q=q_1\\,,\\qquad r=r_1 q_1 \\,,$ and we also introduce another change of time scale: $T := \\int _0^\\tau q_1(\\sigma )d\\sigma $ , again justified by (REF ), leading to $\\frac{dq_1}{dT} &=& q_1 r_1\\left( \\kappa _2 p_1 - \\kappa _- \\right) + \\kappa _3 p_1 r_1 q_1^2(nr_1-p_1-2)\\,,\\nonumber \\\\\\frac{dp_1}{dT} &=& r_1(\\kappa _1 - \\kappa _3 p_1 q_1)(nr_1-p_1-2) + \\kappa _- \\left(r_1 - \\frac{n-1}{n-2}p_1\\right) - (\\kappa _2 + \\kappa _{-1})p_1r_1- p_1 r_1(\\kappa _2 p_1 - \\kappa _-) \\nonumber \\\\&& -\\kappa _3 p_1^2 r_1 q_1 (nr_1 - p_1 - 2) \\,,\\\\\\frac{dr_1}{dT} &=& (1-r_1)\\left( \\kappa _2 p_1 r_1 + \\kappa _- \\left( \\frac{2}{n-2} - r_1\\right) \\right) - \\kappa _3 p_1 r_1^2 q_1 (nr_1 - p_1 - 2)\\,.\\nonumber $ The invariant manifold $q_1=0$ of this system corresponds to the zero steady state of (REF ).","The inequalities (REF ) become $r_1 \\le 1 \\,,\\qquad 0\\le p_1 \\le nr_1-2 \\,,$ in terms of the new variables, i.e.","the dynamics of $(p_1,r_1)$ remains in the triangle depicted in Fig.","REF .","Since $r_1\\ge 2/n$ , we conclude from the equation for $q_1$ that the invariant manifold is locally exponentially attracting in the region to the left of the line $p_1 = \\kappa _-/\\kappa _2$ .","Since $p_1\\le n-2$ , the inequality $\\kappa _- > (n-2)\\kappa _2$ already implies local asymptotic stability of the invariant manifold $q_1=0$ of (REF ) and therefore of the zero steady state of (REF ).","Note that $\\kappa _- > (n-2)\\kappa _2$ also implies $\\bar{\\alpha }> 1$ for $\\bar{\\alpha }$ defined by (REF ).","Figure: The dynamics in the (p 1 ,r 1 )(p_1,r_1)-plane is limited to the shaded triangle because of the inequalities ().In the following we therefore consider the case $\\kappa _- \\le (n-2)\\kappa _2$ (see Fig.","REF ) and $\\bar{\\alpha }> 1$ , where the latter is equivalent to $\\kappa _1\\kappa _2(n-2)^2 < \\kappa _-(\\kappa _1 + \\kappa _{-1})(n-2) + \\kappa _-^2(n-1) \\,,$ see also [1].","The flow on the invariant manifold $q_1=0$ of (REF ) is governed by the system $\\frac{dp_1}{dT} &=& r_1\\kappa _1 (nr_1-p_1-2) + \\kappa _- \\left(r_1 - \\frac{n-1}{n-2}p_1\\right) - (\\kappa _2 + \\kappa _{-1})p_1r_1- p_1 r_1(\\kappa _2 p_1 - \\kappa _-) \\nonumber \\\\\\frac{dr_1}{dT} &=& (1-r_1)\\left( \\kappa _2 p_1 r_1 + \\kappa _- \\left( \\frac{2}{n-2} - r_1\\right) \\right) \\,.$ In the right part of the triangle, i.e.","for $r_1 \\le 1 \\,,\\qquad \\frac{\\kappa _-}{\\kappa _2}\\le p_1 \\le nr_1-2 \\,,$ we have $\\frac{dp_1}{dT} &\\le & r_1\\kappa _1 \\left( n - \\frac{\\kappa _-}{\\kappa _2} - 2\\right) + \\kappa _- \\left( r_1 - \\frac{n-1}{n-2}\\,\\frac{\\kappa _-}{\\kappa _2}\\right)- (\\kappa _2 + \\kappa _{-1})\\frac{\\kappa _-}{\\kappa _2}r_1 \\\\&=& \\frac{r_1\\bigl (\\kappa _1\\kappa _2(n-2)^2 - \\kappa _-(\\kappa _1 + \\kappa _{-1})(n-2)\\bigr ) - \\kappa _-^2(n-1)}{\\kappa _2(n-2)}< \\frac{(r_1-1)\\kappa _-^2(n-1)}{\\kappa _2(n-2)} \\le 0 \\,,$ where the strict inequality is due to (REF ).","This implies that all trajectories reach the left part of the triangle, i.e.","$p_1 < \\kappa _-/\\kappa _2$ in finite time.","By standard regular perturbation theory the dynamics for the full system (REF ), when started close to the invariant manifold $q_1=0$ , remains close to the dynamics on the invariant manifold for finite time, until the region $p_1 < \\kappa _-/\\kappa _2$ is reached, where the invariant manifold is attracting.","Thus $q=q_1$ tends to zero and, by the inequalities (REF ), the same is true for $p$ and $r$ .","Now we consider the case $\\bar{\\alpha }<1$ , i.e.","the opposite of inequality (REF ), and look for a steady state on the invariant manifold $r_1=1$ of the system (REF ).","Since $\\frac{dp_1}{dT}\\Bigm |_{r_1=1,p_1=\\kappa _-/\\kappa _2} &=& \\frac{\\kappa _1\\kappa _2(n-2)^2 - \\kappa _-(\\kappa _1 + \\kappa _{-1})(n-2) - \\kappa _-^2(n-1)}{\\kappa _2(n-2)} >0 \\,,\\\\\\frac{dp_1}{dT}\\Bigm |_{r_1=1,p_1=n-2} &=& -(n-2)(n\\kappa _2 + \\kappa _{-1}) < 0 \\,,$ there exists a steady state $(p_1,r_1) = (p_1^*,1)$ with $\\kappa _-/\\kappa _2 < p_1^* < n-2$ , which is stable under the flow along $r_1=1$ .","On the other hand $\\frac{1}{1-r_1}\\, \\frac{dr_1}{dT}\\Bigm |_{r_1=1,p_1=p_1^*} = \\left( \\kappa _2 p_1^* + \\kappa _- \\left( \\frac{2}{n-2} - 1\\right) \\right) > \\frac{2\\kappa _-}{n-2} >0 \\,,$ which implies stability of the manifold $r_1=1$ close to the steady state, and therefore stability of the steady state.","The existence of a stable steady state on the invariant manifold $q_1=0$ of (REF ) in the region, where the manifold is repulsive, implies instability of the manifold and therefore also of the zero steady state of (REF ).","This completes the proof of the main result of this section.","Theorem 1 Let $\\bar{\\alpha }$ be defined by (REF ).","Then the steady state $(0,0,0)$ of the system (REF ) is locally asymptotically stable for $\\bar{\\alpha }>1$ and unstable for $\\bar{\\alpha }<1$ ."],["Polynomially growing regime","The goal of this section is a rigorous justification of the formal asymptotics (REF ) (see [1]) under the assumption $\\bar{\\alpha }<0$ with $\\bar{\\alpha }$ defined in (REF ), i.e.","$4\\kappa _1\\kappa _2(n-2)^2 > n\\kappa _-\\bigl (2(\\kappa _1 + \\kappa _{-1})(n-2) + \\kappa _-n(n-1) \\bigr )\\,,$ see also [1].","Considering (REF ), it would be natural to write an equation for $p(t)/t$ .","It is easily seen from (REF ) that its derivative contains terms of the order of $t^2$ .","Similarly the derivative of $q(t)/t^2$ has contributions up to the order of $t$ , whereas the derivative of $r(t)/t^2$ is a combination of terms bounded as $t\\rightarrow \\infty $ .","This shows that we are confronted with a problem with different time scales, which will put us into the realm of singular perturbation theory (see, e.g.","[3], [6]).","The leading order term in the fastest equation, i.e.","the $p$ -equation, is $-\\kappa _3 p (nr-2q)$ , from which it has been concluded in [1] that $nr(t)\\approx 2q(t)$ as $t\\rightarrow \\infty $ .","In a standard singular perturbation setting, it should be possible to express $p(t)$ from this relation.","Since this is not the case, our problem belongs to the family of singular singularly perturbed problems (see e.g.","[5]) which, however, can be transformed to the standard regular form in many cases.","The introduction of $p(t)/t$ , $q(t)/t^2$ , $r(t)/t^2$ , as new variables would lead to a study of bounded solutions, but to a non-autonomous system.","We shall use a variant of the Poincaré compactification method [4] instead.","The previous observations led us to the introduction of the new variables $u= \\frac{p}{\\sqrt{p+q}} \\,,\\qquad v=\\frac{2p+2q -nr}{\\sqrt{p+q}} \\,,\\qquad w=\\frac{1}{\\sqrt{p+q}} \\,,$ where we expect that $w(t)$ tends to zero as $t^{-1}$ , and that $u(t)$ and $v(t)$ converge to nontrivial limits.","Since this coordinate change produces a singularity at $w=0$ , we also change the time variable by $\\tau = \\int _0^t ds/w(s)$ .","In terms of the new variables system (REF ) becomes $\\frac{du}{d\\tau } &=& (\\kappa _1 w - \\kappa _3 u)(u-v) + \\kappa _-(1-uw)\\left( 1 - \\frac{n(n-1)uw}{(n-2)(2-vw)}\\right) - (\\kappa _2+\\kappa _{-1})uw \\nonumber \\\\&& - uw^2A(u,v,w) \\,,\\nonumber \\\\\\frac{dv}{d\\tau } &=& w\\left(2\\kappa _1(u-v) - (2\\kappa _{-1} + n\\kappa _2)u + \\kappa _- (1-uw)n \\frac{2u-nv}{(n-2)(2-vw)}\\right) - vw^2A(u,v,w) \\,,\\\\\\frac{dw}{d\\tau } &=& - w^3 A(u,v,w) \\,,\\quad A(u,v,w) := \\frac{1}{2}\\left(\\kappa _1(u-v) - \\kappa _{-1}u - \\kappa _-(1-uw)\\frac{n(n-1)u}{(n-2)(2-vw)}\\right) \\,.\\nonumber $ Our goal is to prove that solutions converge to a steady state $(u^*,v^*,w^*)$ with $w^*=0$ , which obviously has to satisfy $- \\kappa _3 u^*(u^*-v^*) + \\kappa _-=0$ , implying $u^* = U(v^*) := \\frac{1}{2} \\left( v^* +\\sqrt{(v^*)^2 + 4\\kappa _-/\\kappa _3}\\right) \\,,$ since we need $u^*>0$ .","We intend to show that $v^*$ is determined from the requirement that the large parenthesis in the $v$ -equation vanishes.","The argument is essentially that for small values of $w$ , the variable $v$ evolves much faster than $w$ .","In order to make the slow-fast structure of this system more apparent and to allow the application of basic results from singular perturbation theory, we assume that the initial value for $w$ is small and define $\\varepsilon := (p_0+q_0)^{-1/2}\\ll 1$ and the rescaled variable $W= w/\\varepsilon $ , leading to $\\frac{du}{d\\tau } &=& - \\kappa _3 u(u-v) + \\kappa _- + O(\\varepsilon ) \\,,\\nonumber \\\\\\frac{dv}{d\\tau } &=& \\varepsilon W\\left(2\\kappa _1(u-v) - (2\\kappa _{-1} + n\\kappa _2)u + \\kappa _- n \\frac{2u-nv}{2(n-2)}\\right) +O(\\varepsilon ^2) \\,,\\\\\\frac{dW}{d\\tau } &=& - \\varepsilon ^2 W^3 A(u,v,0) + O(\\varepsilon ^3) \\,.\\nonumber $ The initial data are denoted by $u(0)=u_0 := \\frac{p_0}{\\sqrt{p_0+q_0}} >0\\,,\\qquad v(0)=v_0 := \\frac{2p_0+2q_0 -nr_0}{\\sqrt{p_0+q_0}}\\,,\\qquad W(0)=1\\,,$ where in the following we consider $u_0$ and $v_0$ as fixed when $\\varepsilon \\rightarrow 0$ .","This is a singular perturbation problem in standard form, where $\\tau $ plays the role of an initial layer variable.","We pass to the limit $\\varepsilon \\rightarrow 0$ to obtain the initial layer problem $\\frac{d\\hat{u}}{d\\tau } &=& - \\kappa _3 \\hat{u}(\\hat{u}-\\hat{v}) + \\kappa _- \\,,\\\\\\frac{d\\hat{v}}{d\\tau } &=& \\frac{d\\hat{W}}{d\\tau } = 0 \\,,\\nonumber $ subject to the initial conditions.","By the qualitative behaviour of the right hand side of the first equation, the solution satisfies $\\hat{v}(\\tau )=v_0$ , $\\hat{W}(\\tau )=1$ , and $\\lim _{\\tau \\rightarrow \\infty } \\hat{u}(\\tau ) = U(v_0) \\,,$ with exponential convergence, where $U$ has been defined in (REF ).","The equation $u=U(v)$ defines the so called reduced manifold.","Since it is exponentially attracting, the Tikhonov theorem [7] (or rather its extension [3]) implies that, after the initial layer, i.e.","when written in terms of the slow variable $\\sigma = \\varepsilon \\tau $ , the solution trajectory remains exponentially close to the slow manifold, which is approximated by the reduced manifold, and the flow on the slow manifold satisfies $\\frac{dv}{d\\sigma } &=& W\\left(2\\kappa _1(U(v)-v) - (2\\kappa _{-1} + n\\kappa _2)U(v) + \\kappa _- n \\frac{2U(v)-nv}{2(n-2)}\\right) +O(\\varepsilon ) \\,,\\nonumber \\\\\\frac{dW}{d\\sigma } &=& - \\varepsilon W^3 A(U(v),v,0) + O(\\varepsilon ^2) \\,,$ with $v(0)=v_0$ , $W(0)=1$ .","This is again a singular perturbation problem in standard form, where now $\\sigma $ is the initial layer variable.","We repeat the above procedure and consider the limiting layer problem $\\frac{d\\tilde{v}}{d\\sigma } &=& \\tilde{W}\\left(2\\kappa _1(U(y\\tilde{v})-\\tilde{v}) - (2\\kappa _{-1} + n\\kappa _2)U(\\tilde{v})+ \\kappa _- n \\frac{2U(\\tilde{v})-n\\tilde{v}}{2(n-2)}\\right) \\,,\\\\\\frac{d\\tilde{W}}{d\\sigma } &=& 0 \\,.\\nonumber $ The observations $U(-\\infty ) = 0 \\,,\\qquad U(\\infty )=\\infty \\,,\\qquad 0< U^{\\prime }(v) < 1 \\,,$ suffice to show that the right hand side of the first equation is a strictly decreasing function of $v$ with a unique zero $v^*$ , which can actually be computed explicitly: $v^* = B\\left(\\kappa _1 - \\kappa _{-1} - \\frac{n}{2}\\kappa _2 + \\frac{n}{2(n-2)}\\kappa _-\\right)$ $\\mbox{with}\\quad B = 2\\sqrt{\\frac{\\kappa _-}{\\kappa _3}} \\left( \\frac{n^3}{4(n-2)}\\kappa _-^2 + 4\\kappa _1 \\kappa _{-1} + 2n \\kappa _1\\kappa _2+ n\\kappa _1 \\kappa _- + \\frac{n^2}{n-2} \\kappa _{-1}\\kappa _- + \\frac{n^3}{2(n-2)\\kappa _2 \\kappa _-}\\right)^{-1/2}$ The solution of (REF ) with $\\tilde{v}(0)=v_0$ satisfies $\\lim _{\\sigma \\rightarrow \\infty } \\tilde{v}(\\sigma ) = v^*$ with exponential convergence.","Another application of the Tikhonov theorem shows that the slowest part of the dynamics with $t=O(\\varepsilon ^{-1})$ can be approximated by $\\frac{dW}{d\\sigma } = - \\varepsilon W^3 A^* \\,,\\qquad W(0)=1 \\,,$ with $A^* := A(U(v^*),v^*,0) = \\frac{nB}{16(n-2)^2}(4(n-2)^2\\kappa _1\\kappa _2 - 2n(n-2)\\kappa _-(\\kappa _1+\\kappa _{-1}) - n^2(n-1)\\kappa _-^2) >0 \\,,$ by (REF ).","This gives the approximation $W(\\sigma ) = (1+2 A^*\\varepsilon \\sigma )^{-1/2} \\,.$ The results of [3] imply that the approximations are accurate with errors of order $\\varepsilon $ uniformly with respect to time.","Theorem 2 Let (REF ) hold.","Then, for $\\varepsilon >0$ small enough, the solution of (REF ) with initial conditions $u(0) = u_0>0 \\,,\\qquad v(0) = v_0 \\in \\mathbb {R} \\,, \\qquad w(0)=\\varepsilon \\,,$ satisfies $u(\\tau ) &=& \\hat{u}(\\tau ) - U(v_0) + U(\\tilde{v}(\\varepsilon \\tau )) + O(\\varepsilon ) \\,,\\\\v(\\tau ) &=& \\tilde{v}(\\varepsilon \\tau ) + O(\\varepsilon ) \\,,\\\\w(\\tau ) &=& \\varepsilon (1 + 2A^* \\varepsilon ^2\\tau )^{-1/2} + O(\\varepsilon ^2) \\,,$ uniformly in $\\tau \\ge 0$ , where $U$ is given in (REF ), $\\hat{u}$ solves (REF ), $\\tilde{v}$ solves (REF ), and $A^*$ is given in (REF ).","Actually more can be deduced.","In terms of the original time variable $t$ , the equation for $w$ in (REF ) becomes $\\dot{w} = -w^2 A(u,v,w) \\,.$ Under the assumptions of Theorem REF , $A(u,v,w)$ is uniformly close to the positive constant $A^*$ and therefore uniformly positive for large enough $t$ .","This implies that $w$ tends to zero as $t\\rightarrow \\infty $ .","The slow manifold of the system (REF ) reduces to the steady state $(v,W)=(v^*,0)$ for $W=0$ .","Therefore $v$ tends to $v^*$ as $t\\rightarrow \\infty $ .","Analogously, the slow manifold of (REF ) reduces to the steady state $(u,v,W)=(u^*=U(v^*),v^*,0)$ at $W=0$ , implying convergence of $u$ to $u^*$ .","This in turn implies convergence of $A(u,v,w)$ to $A^*$ , which can be used in (REF ).","Corollary 1 Let the assumptions of Theorem REF hold.","Then $\\lim _{t\\rightarrow \\infty } u(t) = u^* \\,,\\qquad \\lim _{t\\rightarrow \\infty } v(t) = v^* \\,,\\qquad w(t) = \\frac{1}{A^* t} + O\\left( \\frac{1}{t^2}\\right) \\quad \\mbox{as } t\\rightarrow \\infty \\,.$ Finally, we reformulate these results in terms of the original variables, verifying the formal asymptotics of [1] for initial data, which are in a sense already 'close enough' to the polynomially growing solutions.","Theorem 3 Let (REF ) hold, let $c_2\\ge c_1>0$ , and let $\\delta >0$ be small enough.","Let the initial data satisfy $p_0 = \\frac{c_1}{\\delta } \\,,\\qquad q_0 = \\frac{1}{\\delta ^2} \\,,\\qquad r_0 = \\frac{2}{n\\delta ^2} + \\frac{c_2}{n\\delta }$ Then the solution of (REF ) with $(p(0),q(0),r(0)) = (p_0,q_0,r_0)$ satisfies $p(t) = u^*A^* t + o(t)\\,,\\qquad q(t) = (A^*)^2 t^2 + o(t^2) \\,,\\qquad r(t) = \\frac{2}{n}(A^*)^2 t^2 + o(t^2) \\,,\\qquad \\mbox{as } t\\rightarrow \\infty \\,.$ We just need to verify that the assumptions of this theorem imply the assumptions of Theorem REF .","The result is then a direct consequence of Corollary REF .","Actually the assumptions of Theorem REF hold with $\\varepsilon \\approx \\delta $ , since $u_0 = \\frac{c_1}{\\sqrt{1+c_1\\delta }} \\,,\\qquad v_0 = \\frac{2c_1-c_2}{\\sqrt{1+c_1\\delta }} \\,,\\qquad w_0 = \\frac{\\delta }{\\sqrt{1+c_1\\delta }} \\,.$"],["Discussion","In this work a mathematical model for aggregation via cross-linking has been analyzed.","Besides the basic assumption that aggregating particles (here p62 oligomers) need to have at least $n=3$ binding sites for cross-linkers (here ubiqutinated cargo), the rate constants for binding reactions need to be large enough compared to those for the unbinding reactions (the opposite of inequality (REF )) for stable aggregates to exist.","Under a stronger condition (inequality (REF )) aggregates grow indefinitely in the presence of an unlimited supply of free particles and cross-linkers.","These conjectures from [1], where the model has been formulated, have been partially proven in this work.","It has been shown in Section that small aggregates get completely degraded under the condition (REF ) and that they grow under the opposite condition.","In the latter case, but when (REF ) does not hold, there exists an equilibrium configuration with positive aggregate size.","Finally, it has been shown in Section that under the condition (REF ) aggregate size grows polynomially with time (actually like $t^2$ ) for appropriate initial states.","Figure: Bifurcation diagram obtained for κ -1 =κ - =1\\kappa _{-1}=\\kappa _{-}=1The constants $\\kappa _1$ , $\\kappa _2$ in the model have to be interpreted as the products of rate constants with the concentrations of free cross-linkers and, respectively, of free particles.","This means that the conditions (REF ) and (REF ) are actually conditions for these concentrations.","Fig.","REF shows a bifurcation diagram in terms of $\\kappa _1$ and $\\kappa _2$ with the curves $\\bar{\\alpha }=1$ , corresponding to equality in (REF ), and $\\bar{\\alpha }=0$ , corresponding to equality in (REF ).","The qualitative behaviour is no surprise: Close to the origin, i.e.","for small concentrations of free particles and cross-linkers, aggregates are unstable.","Moving to the right and/or up we pass through two bifurcations to stable finite aggregate size and, subsequently, to polynomial growth of aggregates.","Less obvious is the fact that the picture is rather unsymmetric with respect to the two parameters.","The condition $(n-2)\\kappa _2> \\kappa _-$ is necessary for the existence of stable aggregates, regardless of the value of $\\kappa _1$ , whereas arbitrarily small values of $\\kappa _1$ can be compensated by large enough $\\kappa _2$ .","This means that, if the concentration of free particles is below a threshold, even a large concentration of cross-linkers does not lead to aggregation, whereas arbitrarily small numbers of cross-linkers are used for aggregation if the particle concentration is high.","For the application in cellular autophagy this means that aggregation will only happen for large enough concentrations of p62 oligomers.","However, arbitrarily small amounts of ubiquitinated cargo can be aggregated in the presence of a large enough supply of oligomers.","This work has been motivated by the experimental results of [8], where aggregates have been detected by light microscopy.","If the evolution of single aggregates can be followed, the growth like $t^2$ might be observed as a fluorescence signal of tagged oligomers, which goes like $t^2$ , or cross section areas going like $t^{4/3}$ , if a roughly spherical shape of aggregates is assumed.","For quantitative predictions of such experiments, the model should be extended in various ways.","First, the limited supply of free p62 oligomers and of free cross-linkers should be taken into account.","This is straightforward for the modeling of a single aggregate, but if many aggregates develop simultaneously, they will compete for the free particles.","Apart from that the number of aggregates has to be predicted, which requires modeling of the nucleation process.","Finally, it is very likely that the coagulation of aggregates plays an important role.","A growth-coagulation model for distributions of aggregates, based on the growth model (REF ) would be prohibitively complex.","It is therefore the subject of ongoing work to formulate, analyze, and simulate a growth-coagulation model based on the multiscale analysis of Section , where aggregates are only described by the size parameter $r$ (number of p62 oligomers in the aggregate), whose evolution is determined by the slow dynamics (REF ), which translates to an equation of the form $\\dot{r} = C\\sqrt{r}$ for $r$ .","This approach raises several challenging issues such as the development of an efficient simulation algorithm or the existence and stability of equilibrium aggregate distributions."]],"string":"[\n [\n \"Long-time behaviour of a model for p62-ubiquitin aggregation in cellular\\n autophagy\"\n ],\n [\n \"Abstract The qualitative behavior of a recently formulated ODE model for the dynamics of heterogenous aggregates is analyzed.\",\n \"Aggregates contain two types of particles, oligomers and cross-linkers.\",\n \"The motivation is a preparatory step of cellular autophagy, the aggregation of oligomers of the protein p62 in the presence of ubiquitin cross-linkers.\",\n \"A combination of explicit computations, formal asymptotics, and numerical simulations has led to conjectures on the bifurcation behavior, certain aspects of which are proven rigorously in this work.\",\n \"In particular, the stability of the zero state, where the model has a smoothness deficit is analyzed by a combination of regularizing transformations and blow-up techniques.\",\n \"On the other hand, in a different parameter regime, the existence of polynomially growing solutions is shown by Poincar\\\\'e compactification, combined with a singular perturbation analysis .\"\n ],\n [\n \"Introduction\",\n \"A preparatory step of cellular autophagy is the aggregation of cellular waste material before inclusion in an autophagosome and, later, a lysosome, which are vesicular structures, where the waste is eventually decomposed.\",\n \"In vitro studies of the evolution of heterogeneous aggregates of the proteins p62 and ubiquitin [8] have motivated the formulation of a mathematical model of this process [1].\",\n \"The model has the form of an ODE system, which shows different qualitative behaviour in three different regions of parameter space.\",\n \"This statement is based on formal asymptotics and numerical simulations carried out in [1].\",\n \"Since some of these observations are not accessible to standard dynamical systems methods, it is the purpose of this work to provide a rigorous analysis.\",\n \"The model is based on the assumption that ubiquitinated waste material serves as a cross-linker between p62 oligomers of a fixed size $n\\\\ge 3$ , where each monomer can serve as a binding site for cross-linking.\",\n \"It describes the evolution of the size of aggregates through the evolution of three parameters $p$ , $q$ and $r$ , where $p$ represents the number of one-hand bound ubiquitin links in the aggregate (in green in Figs.\",\n \"REF and REF ), $q$ represents the number of both-hand bound cross-links (in red in Figs.\",\n \"REF and REF ), and $r$ represents the number of p62$_n$ oligomers in the aggregate (in black in Figs.\",\n \"REF and REF with $n=5$ ).\",\n \"Since we are interested in large aggregates, the variables $(p,q,r)$ are considered as continuous after an appropriate scaling.\",\n \"The state space is a subset of the positive octant of $\\\\mathbb {R}^3$ determined by two constraints: For a connected aggregate the number of two-hand bound cross-links has to be at least the number of p62 oligomers minus one.\",\n \"In the continuous description this becomes the constraint $q\\\\ge r$ .\",\n \"Since the total number of binding sites on the p62 oligomers in an aggregate is $nr$ , the number of free binding sites is equal to $nr-p-2q$ , which has to be nonnegative.\",\n \"The dynamics of an aggregate is governed by the basic binding and unbinding reactions between cross-linkers and p62 oligomers.\",\n \"Since the reaction rates depend on the state of the reaction partners and of the aggregate, six different reactions have been considered in [1] (see Figs.\",\n \"REF and REF ).\",\n \"The models for the reaction rates are based on the law of mass action.\",\n \"However, since the shape of an aggregate is not described unambiguously by the parameters $(p,q,r)$ , some additional empirical modeling assumptions are required.\",\n \"Figure: Left: Reaction 1: addition of a free cross-linker to the aggregate.\",\n \"Right: Reaction 2: addition of a p62 5 _5 oligomer to the aggregate.Figure: Left: Reaction 3 is a rearrangement making the aggregate more cohesive.\",\n \"Right: the breaking of a cross-link bound can lead either to the reverse of Reaction 2 of of Reaction 3.Reaction 1 is the addition of a free cross-linker to the aggregate.\",\n \"This is a second order reaction with a rate proportional to the concentration of free cross-linkers and to the number of free binding sites on oligomers.\",\n \"Since the supply of free cross-linkers and oligomers has been modeled as not limiting in [1], the reaction rate is written as $\\\\kappa _1(nr-p-2q)$ with rate constant $\\\\kappa _1$ , which can be seen as proportional to the cross-linker concentration, modeled as constant.\",\n \"Similarly Reaction 2, the addition of a free oligomer to the aggregate, is modeled as a first order reaction with rate $\\\\kappa _2 p$ proportional to the number of one-hand bound cross-linkers.\",\n \"Reaction 3 is consolidating the aggregate by building an additional cross-link using a so far only one-hand bound cross-linker.\",\n \"Its rate is $\\\\kappa _3 p(nr-p-2q)$ .\",\n \"For the reverse of Reaction 1, the rate $\\\\kappa _{-1}p$ should not be a surprise.\",\n \"The reverses of Reactions 2 and 3 are actually the same reaction with rate $\\\\kappa _- q$ , but with possibly different outcomes.\",\n \"Therefore we write their rate constants as $\\\\kappa _{-2}:= \\\\kappa _- \\\\alpha $ and $\\\\kappa _{-3}:= \\\\kappa _- (1-\\\\alpha )$ , with $\\\\alpha =\\\\alpha (q,r)\\\\in [0,1]$ .\",\n \"The reverse of Reaction 2, i.e.\",\n \"loss of an oligomer, never happens in a fully connected aggregate with $nr=2q$ .\",\n \"It always happens in a minimally connected aggregate with $q=r$ .\",\n \"This motivates the choice $\\\\alpha = \\\\frac{nr-2q}{(n-2)r}$ .\",\n \"Concerning the outcome of this reverse reaction, it has to be taken into account that the loss of an oligomer might also mean a loss of one-hand bound cross-links attached to it.\",\n \"This produces the loss term $\\\\kappa _- q \\\\frac{(n-1)p}{(n-2)r}$ (see [1] for details).\",\n \"It is now straightforward to write down the ODE problem governing the evolution of the state variables: $ \\\\begin{aligned}\\\\dot{p} &= (\\\\kappa _1 - \\\\kappa _3 p) (nr-p-2q) + \\\\kappa _{-}q \\\\left( 1 - \\\\frac{(n-1)p}{(n-2)r}\\\\right) - (\\\\kappa _2 + \\\\kappa _{-1})p \\\\,, &\\\\qquad p(0)=p_0\\\\,,\\\\\\\\\\\\dot{q} &= \\\\kappa _2 p + \\\\kappa _3 p(nr-p-2q) - \\\\kappa _{-}q\\\\,, &\\\\qquad q(0)=q_0\\\\,, \\\\\\\\\\\\dot{r} &= \\\\kappa _2 p - \\\\kappa _{-} q \\\\alpha (q,r), \\\\qquad \\\\alpha (q,r) = \\\\frac{nr -2q}{(n-2) r}\\\\,, &\\\\qquad r(0)=r_0\\\\,,\\\\end{aligned}$ with the inequalities $ nr - p - 2q \\\\ge 0 \\\\,, \\\\qquad q\\\\ge r \\\\,,$ implying $0 \\\\le \\\\alpha (q,r) \\\\le 1 \\\\,.$ We recall from [1] that for initial data $p_0,q_0,r_0>0$ satisfying (REF ), which we assume in the following, the initial value problem (REF ) has a unique, global solution propagating (REF ), the nonnegativity of the components, and in particular $r(t),q(t) > 0 \\\\,,\\\\qquad t\\\\ge 0 \\\\,.$ Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5,κ 1 =κ 2 =κ 3 =κ -1 =1\\\\kappa _1= \\\\kappa _2=\\\\kappa _3=\\\\kappa _{-1}=1, and κ - =0.6\\\\kappa _{-}=0.6, giving 0<α ¯<10<\\\\bar{\\\\alpha }< 1 and convergence to the nontrivial equilibrium.Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5, κ 1 =κ 2 =κ 3 =κ -1 =1\\\\kappa _1= \\\\kappa _2=\\\\kappa _3=\\\\kappa _{-1} = 1 ,and κ - =0.93\\\\kappa _{-}=0.93, giving α ¯>1\\\\bar{\\\\alpha }>1 and convergence to the zero steady state.Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5,κ 1 =κ 2 =κ 3 =κ -1 =1\\\\kappa _1= \\\\kappa _2=\\\\kappa _3=\\\\kappa _{-1} = 1, and κ - =0.2\\\\kappa _{-}=0.2, giving α ¯<0\\\\bar{\\\\alpha }<0 and a polynomially growing aggregate.The search for steady states [1] has suggested a splitting of the parameter space into three regions.\",\n \"Besides the trivial steady state $(p,q,r)=(0,0,0)$ , only one other equilibrium may exist, which can be computed explicitly: $& \\\\bar{p} =\\\\frac{\\\\kappa _- A}{\\\\kappa _2} \\\\frac{1-\\\\bar{\\\\alpha }}{\\\\bar{\\\\alpha }} \\\\,,\\\\qquad \\\\bar{q} = A \\\\frac{1-\\\\bar{\\\\alpha }}{\\\\bar{\\\\alpha }^2} \\\\,, \\\\qquad \\\\bar{r} = \\\\frac{2A}{n-(n-2)\\\\bar{\\\\alpha }} \\\\frac{1-\\\\bar{\\\\alpha }}{\\\\bar{\\\\alpha }^2}\\\\,, \\\\\\\\&\\\\mbox{with}\\\\quad \\\\bar{\\\\alpha }= \\\\frac{n}{n-2} + \\\\frac{\\\\kappa _{-1} + \\\\kappa _1 - \\\\sqrt{(\\\\kappa _1 + \\\\kappa _{-1})^2 + 4 \\\\kappa _1 \\\\kappa _2 (n-1)}}{\\\\kappa _{-} (n-1)} \\\\,, \\\\quad A = \\\\frac{2\\\\kappa _1 \\\\kappa _2^2 (n-2)}{\\\\kappa _3 \\\\kappa _- (\\\\kappa _-(n-1)(n - (n-2)\\\\bar{\\\\alpha }) + 2\\\\kappa _{-1}(n-2))} \\\\,.\\\\nonumber $ Since $\\\\bar{\\\\alpha }= \\\\alpha (\\\\bar{q},\\\\bar{r})$ is the equilibrium value of $\\\\alpha $ , the nontrivial steady state is relevant only in the parameter region defined by $0<\\\\bar{\\\\alpha }<1$ .\",\n \"It has been conjectured in [1] that in this parameter region $(\\\\bar{p},\\\\bar{q}, \\\\bar{r})$ is globally attracting, which has been supported by numerical simulations (see also Fig.\",\n \"REF ).\",\n \"Local stability could in principle be examined by linearization.\",\n \"However, the complexity of the resulting formulas has been prohibitive.\",\n \"Since $(\\\\bar{p},\\\\bar{q},\\\\bar{r})\\\\rightarrow (0,0,0)$ as $\\\\bar{\\\\alpha }\\\\rightarrow 1-$ , it seems natural to expect a transcritical bifurcation at $\\\\bar{\\\\alpha }=1$ with stability of the trivial steady state for $\\\\bar{\\\\alpha }>1$ .\",\n \"Again the conjecture of global asymptotic stability of $(0,0,0)$ for $\\\\bar{\\\\alpha }>1$ has been supported by simulations (see for example Fig.\",\n \"REF ).\",\n \"The right hand sides of (REF ) are continuous up to the origin (when considered as an element of the set of admissible states), since $0\\\\le \\\\alpha (q,r)\\\\le 1$ and $p/r\\\\le n$ .\",\n \"However, their nonsmoothness prohibits a standard local stability or bifurcation analysis.\",\n \"The expected local stability behaviour (asymptotic stability for $\\\\bar{\\\\alpha }>1$ , instability for $\\\\bar{\\\\alpha }<1$ ) is proven in Section .\",\n \"The analysis is based on a regularizing transformation, which makes the steady state very degenerate, combined with a blow-up analysis [2].\",\n \"The fact that the components of the nontrivial equilibrium tend to infinity when $\\\\bar{\\\\alpha }\\\\rightarrow 0+$ suggests that solutions might be unbounded for $\\\\bar{\\\\alpha }<0$ .\",\n \"In this parameter region approximate solutions with polynomial growth of the form $p(t) = p_1 t+ \\\\textit {o}(t) \\\\,,\\\\qquad q(t) = q_2 t^2 + \\\\textit {o}(t^2) \\\\,,\\\\qquad r(t) = \\\\frac{2q_2}{n} t^2+ \\\\textit {o}(t^2) \\\\,,\\\\qquad \\\\mbox{as } t\\\\rightarrow \\\\infty \\\\,,$ have been constructed in [1] by formal asymptotic methods.\",\n \"It has also been shown that no other growth behaviour (polynomial with other powers or exponential) should be expected, and the conjecture that all solutions have the constructed asymptotic behaviour is again verified by simulations (see Fig.\",\n \"for example REF ).\",\n \"We justify the formal asymptotics in Section .\",\n \"A variant of Poincaré compactification [4] produces a problem with bounded solutions and with three different time scales, which is analyzed by singular perturbation methods [3].\",\n \"The final result is existence and semi-local stability of the polynomially growing solutions, where 'semi-local' means that initial data have to be large with relative sizes as in (REF ).\",\n \"The article is concluded by a discussion section about biological interpretation of our results as well as perspectives.\"\n ],\n [\n \"Local stability of the zero steady state\",\n \"In this section, we study under which conditions small aggregates tend to disaggregate.\",\n \"This is equivalent to studying the stability of the zero-steady-state $(p,q,r)=(0,0,0)$ of the system (REF ).\",\n \"Because of the appearance of the ratios $\\\\frac{p}{r}$ and $\\\\frac{q}{r}$ , the Jacobian of the right hand side of (REF ) is not defined there.\",\n \"As a consequence of (REF ) the regularizing transformation $\\\\tau := \\\\int _0^t r(s)^{-1} \\\\mathrm {d}s$ is well defined and leads to $ \\\\begin{aligned}\\\\frac{dp}{d\\\\tau } &= r(\\\\kappa _1 - \\\\kappa _3 p) (nr-p-2q) + \\\\kappa _{-}q \\\\left( r - \\\\frac{(n-1)p}{n-2}\\\\right) - (\\\\kappa _2 + \\\\kappa _{-1})pr \\\\,, \\\\\\\\\\\\frac{dq}{d\\\\tau } &= \\\\kappa _2 pr + \\\\kappa _3 pr(nr-p-2q) - \\\\kappa _{-}qr\\\\,, \\\\\\\\\\\\frac{dr}{d\\\\tau } &= \\\\kappa _2 pr - \\\\kappa _{-} q \\\\frac{nr -2q}{n-2}\\\\,.\\\\end{aligned}$ The regularization came at the expense that the zero steady state is degenerate in (REF ), since the right hand side is of second order in terms of the densities.\",\n \"A classical approach to study such non-hyperbolic points is blow-up [2].\",\n \"The standard blow-up transformation would be the introduction of spherical coordinates, blowing up the origin to the part of $\\\\mathbb {S}^2$ in the positive octant.\",\n \"It is also common to work with charts instead.\",\n \"In our case this preserves the polynomial form of the right hand side.\",\n \"Although the charts in the different coordinate directions are equivalent, since the state space is a subset of the positive octant, it has turned out to be convenient to use the $q$ -chart, whence the blow-up transformation $(p,q,r) \\\\rightarrow (p_1,q_1,r_1)$ is given by $p= p_1q_1\\\\,, \\\\qquad q=q_1\\\\,,\\\\qquad r=r_1 q_1 \\\\,,$ and we also introduce another change of time scale: $T := \\\\int _0^\\\\tau q_1(\\\\sigma )d\\\\sigma $ , again justified by (REF ), leading to $\\\\frac{dq_1}{dT} &=& q_1 r_1\\\\left( \\\\kappa _2 p_1 - \\\\kappa _- \\\\right) + \\\\kappa _3 p_1 r_1 q_1^2(nr_1-p_1-2)\\\\,,\\\\nonumber \\\\\\\\\\\\frac{dp_1}{dT} &=& r_1(\\\\kappa _1 - \\\\kappa _3 p_1 q_1)(nr_1-p_1-2) + \\\\kappa _- \\\\left(r_1 - \\\\frac{n-1}{n-2}p_1\\\\right) - (\\\\kappa _2 + \\\\kappa _{-1})p_1r_1- p_1 r_1(\\\\kappa _2 p_1 - \\\\kappa _-) \\\\nonumber \\\\\\\\&& -\\\\kappa _3 p_1^2 r_1 q_1 (nr_1 - p_1 - 2) \\\\,,\\\\\\\\\\\\frac{dr_1}{dT} &=& (1-r_1)\\\\left( \\\\kappa _2 p_1 r_1 + \\\\kappa _- \\\\left( \\\\frac{2}{n-2} - r_1\\\\right) \\\\right) - \\\\kappa _3 p_1 r_1^2 q_1 (nr_1 - p_1 - 2)\\\\,.\\\\nonumber $ The invariant manifold $q_1=0$ of this system corresponds to the zero steady state of (REF ).\",\n \"The inequalities (REF ) become $r_1 \\\\le 1 \\\\,,\\\\qquad 0\\\\le p_1 \\\\le nr_1-2 \\\\,,$ in terms of the new variables, i.e.\",\n \"the dynamics of $(p_1,r_1)$ remains in the triangle depicted in Fig.\",\n \"REF .\",\n \"Since $r_1\\\\ge 2/n$ , we conclude from the equation for $q_1$ that the invariant manifold is locally exponentially attracting in the region to the left of the line $p_1 = \\\\kappa _-/\\\\kappa _2$ .\",\n \"Since $p_1\\\\le n-2$ , the inequality $\\\\kappa _- > (n-2)\\\\kappa _2$ already implies local asymptotic stability of the invariant manifold $q_1=0$ of (REF ) and therefore of the zero steady state of (REF ).\",\n \"Note that $\\\\kappa _- > (n-2)\\\\kappa _2$ also implies $\\\\bar{\\\\alpha }> 1$ for $\\\\bar{\\\\alpha }$ defined by (REF ).\",\n \"Figure: The dynamics in the (p 1 ,r 1 )(p_1,r_1)-plane is limited to the shaded triangle because of the inequalities ().In the following we therefore consider the case $\\\\kappa _- \\\\le (n-2)\\\\kappa _2$ (see Fig.\",\n \"REF ) and $\\\\bar{\\\\alpha }> 1$ , where the latter is equivalent to $\\\\kappa _1\\\\kappa _2(n-2)^2 < \\\\kappa _-(\\\\kappa _1 + \\\\kappa _{-1})(n-2) + \\\\kappa _-^2(n-1) \\\\,,$ see also [1].\",\n \"The flow on the invariant manifold $q_1=0$ of (REF ) is governed by the system $\\\\frac{dp_1}{dT} &=& r_1\\\\kappa _1 (nr_1-p_1-2) + \\\\kappa _- \\\\left(r_1 - \\\\frac{n-1}{n-2}p_1\\\\right) - (\\\\kappa _2 + \\\\kappa _{-1})p_1r_1- p_1 r_1(\\\\kappa _2 p_1 - \\\\kappa _-) \\\\nonumber \\\\\\\\\\\\frac{dr_1}{dT} &=& (1-r_1)\\\\left( \\\\kappa _2 p_1 r_1 + \\\\kappa _- \\\\left( \\\\frac{2}{n-2} - r_1\\\\right) \\\\right) \\\\,.$ In the right part of the triangle, i.e.\",\n \"for $r_1 \\\\le 1 \\\\,,\\\\qquad \\\\frac{\\\\kappa _-}{\\\\kappa _2}\\\\le p_1 \\\\le nr_1-2 \\\\,,$ we have $\\\\frac{dp_1}{dT} &\\\\le & r_1\\\\kappa _1 \\\\left( n - \\\\frac{\\\\kappa _-}{\\\\kappa _2} - 2\\\\right) + \\\\kappa _- \\\\left( r_1 - \\\\frac{n-1}{n-2}\\\\,\\\\frac{\\\\kappa _-}{\\\\kappa _2}\\\\right)- (\\\\kappa _2 + \\\\kappa _{-1})\\\\frac{\\\\kappa _-}{\\\\kappa _2}r_1 \\\\\\\\&=& \\\\frac{r_1\\\\bigl (\\\\kappa _1\\\\kappa _2(n-2)^2 - \\\\kappa _-(\\\\kappa _1 + \\\\kappa _{-1})(n-2)\\\\bigr ) - \\\\kappa _-^2(n-1)}{\\\\kappa _2(n-2)}< \\\\frac{(r_1-1)\\\\kappa _-^2(n-1)}{\\\\kappa _2(n-2)} \\\\le 0 \\\\,,$ where the strict inequality is due to (REF ).\",\n \"This implies that all trajectories reach the left part of the triangle, i.e.\",\n \"$p_1 < \\\\kappa _-/\\\\kappa _2$ in finite time.\",\n \"By standard regular perturbation theory the dynamics for the full system (REF ), when started close to the invariant manifold $q_1=0$ , remains close to the dynamics on the invariant manifold for finite time, until the region $p_1 < \\\\kappa _-/\\\\kappa _2$ is reached, where the invariant manifold is attracting.\",\n \"Thus $q=q_1$ tends to zero and, by the inequalities (REF ), the same is true for $p$ and $r$ .\",\n \"Now we consider the case $\\\\bar{\\\\alpha }<1$ , i.e.\",\n \"the opposite of inequality (REF ), and look for a steady state on the invariant manifold $r_1=1$ of the system (REF ).\",\n \"Since $\\\\frac{dp_1}{dT}\\\\Bigm |_{r_1=1,p_1=\\\\kappa _-/\\\\kappa _2} &=& \\\\frac{\\\\kappa _1\\\\kappa _2(n-2)^2 - \\\\kappa _-(\\\\kappa _1 + \\\\kappa _{-1})(n-2) - \\\\kappa _-^2(n-1)}{\\\\kappa _2(n-2)} >0 \\\\,,\\\\\\\\\\\\frac{dp_1}{dT}\\\\Bigm |_{r_1=1,p_1=n-2} &=& -(n-2)(n\\\\kappa _2 + \\\\kappa _{-1}) < 0 \\\\,,$ there exists a steady state $(p_1,r_1) = (p_1^*,1)$ with $\\\\kappa _-/\\\\kappa _2 < p_1^* < n-2$ , which is stable under the flow along $r_1=1$ .\",\n \"On the other hand $\\\\frac{1}{1-r_1}\\\\, \\\\frac{dr_1}{dT}\\\\Bigm |_{r_1=1,p_1=p_1^*} = \\\\left( \\\\kappa _2 p_1^* + \\\\kappa _- \\\\left( \\\\frac{2}{n-2} - 1\\\\right) \\\\right) > \\\\frac{2\\\\kappa _-}{n-2} >0 \\\\,,$ which implies stability of the manifold $r_1=1$ close to the steady state, and therefore stability of the steady state.\",\n \"The existence of a stable steady state on the invariant manifold $q_1=0$ of (REF ) in the region, where the manifold is repulsive, implies instability of the manifold and therefore also of the zero steady state of (REF ).\",\n \"This completes the proof of the main result of this section.\",\n \"Theorem 1 Let $\\\\bar{\\\\alpha }$ be defined by (REF ).\",\n \"Then the steady state $(0,0,0)$ of the system (REF ) is locally asymptotically stable for $\\\\bar{\\\\alpha }>1$ and unstable for $\\\\bar{\\\\alpha }<1$ .\"\n ],\n [\n \"Polynomially growing regime\",\n \"The goal of this section is a rigorous justification of the formal asymptotics (REF ) (see [1]) under the assumption $\\\\bar{\\\\alpha }<0$ with $\\\\bar{\\\\alpha }$ defined in (REF ), i.e.\",\n \"$4\\\\kappa _1\\\\kappa _2(n-2)^2 > n\\\\kappa _-\\\\bigl (2(\\\\kappa _1 + \\\\kappa _{-1})(n-2) + \\\\kappa _-n(n-1) \\\\bigr )\\\\,,$ see also [1].\",\n \"Considering (REF ), it would be natural to write an equation for $p(t)/t$ .\",\n \"It is easily seen from (REF ) that its derivative contains terms of the order of $t^2$ .\",\n \"Similarly the derivative of $q(t)/t^2$ has contributions up to the order of $t$ , whereas the derivative of $r(t)/t^2$ is a combination of terms bounded as $t\\\\rightarrow \\\\infty $ .\",\n \"This shows that we are confronted with a problem with different time scales, which will put us into the realm of singular perturbation theory (see, e.g.\",\n \"[3], [6]).\",\n \"The leading order term in the fastest equation, i.e.\",\n \"the $p$ -equation, is $-\\\\kappa _3 p (nr-2q)$ , from which it has been concluded in [1] that $nr(t)\\\\approx 2q(t)$ as $t\\\\rightarrow \\\\infty $ .\",\n \"In a standard singular perturbation setting, it should be possible to express $p(t)$ from this relation.\",\n \"Since this is not the case, our problem belongs to the family of singular singularly perturbed problems (see e.g.\",\n \"[5]) which, however, can be transformed to the standard regular form in many cases.\",\n \"The introduction of $p(t)/t$ , $q(t)/t^2$ , $r(t)/t^2$ , as new variables would lead to a study of bounded solutions, but to a non-autonomous system.\",\n \"We shall use a variant of the Poincaré compactification method [4] instead.\",\n \"The previous observations led us to the introduction of the new variables $u= \\\\frac{p}{\\\\sqrt{p+q}} \\\\,,\\\\qquad v=\\\\frac{2p+2q -nr}{\\\\sqrt{p+q}} \\\\,,\\\\qquad w=\\\\frac{1}{\\\\sqrt{p+q}} \\\\,,$ where we expect that $w(t)$ tends to zero as $t^{-1}$ , and that $u(t)$ and $v(t)$ converge to nontrivial limits.\",\n \"Since this coordinate change produces a singularity at $w=0$ , we also change the time variable by $\\\\tau = \\\\int _0^t ds/w(s)$ .\",\n \"In terms of the new variables system (REF ) becomes $\\\\frac{du}{d\\\\tau } &=& (\\\\kappa _1 w - \\\\kappa _3 u)(u-v) + \\\\kappa _-(1-uw)\\\\left( 1 - \\\\frac{n(n-1)uw}{(n-2)(2-vw)}\\\\right) - (\\\\kappa _2+\\\\kappa _{-1})uw \\\\nonumber \\\\\\\\&& - uw^2A(u,v,w) \\\\,,\\\\nonumber \\\\\\\\\\\\frac{dv}{d\\\\tau } &=& w\\\\left(2\\\\kappa _1(u-v) - (2\\\\kappa _{-1} + n\\\\kappa _2)u + \\\\kappa _- (1-uw)n \\\\frac{2u-nv}{(n-2)(2-vw)}\\\\right) - vw^2A(u,v,w) \\\\,,\\\\\\\\\\\\frac{dw}{d\\\\tau } &=& - w^3 A(u,v,w) \\\\,,\\\\quad A(u,v,w) := \\\\frac{1}{2}\\\\left(\\\\kappa _1(u-v) - \\\\kappa _{-1}u - \\\\kappa _-(1-uw)\\\\frac{n(n-1)u}{(n-2)(2-vw)}\\\\right) \\\\,.\\\\nonumber $ Our goal is to prove that solutions converge to a steady state $(u^*,v^*,w^*)$ with $w^*=0$ , which obviously has to satisfy $- \\\\kappa _3 u^*(u^*-v^*) + \\\\kappa _-=0$ , implying $u^* = U(v^*) := \\\\frac{1}{2} \\\\left( v^* +\\\\sqrt{(v^*)^2 + 4\\\\kappa _-/\\\\kappa _3}\\\\right) \\\\,,$ since we need $u^*>0$ .\",\n \"We intend to show that $v^*$ is determined from the requirement that the large parenthesis in the $v$ -equation vanishes.\",\n \"The argument is essentially that for small values of $w$ , the variable $v$ evolves much faster than $w$ .\",\n \"In order to make the slow-fast structure of this system more apparent and to allow the application of basic results from singular perturbation theory, we assume that the initial value for $w$ is small and define $\\\\varepsilon := (p_0+q_0)^{-1/2}\\\\ll 1$ and the rescaled variable $W= w/\\\\varepsilon $ , leading to $\\\\frac{du}{d\\\\tau } &=& - \\\\kappa _3 u(u-v) + \\\\kappa _- + O(\\\\varepsilon ) \\\\,,\\\\nonumber \\\\\\\\\\\\frac{dv}{d\\\\tau } &=& \\\\varepsilon W\\\\left(2\\\\kappa _1(u-v) - (2\\\\kappa _{-1} + n\\\\kappa _2)u + \\\\kappa _- n \\\\frac{2u-nv}{2(n-2)}\\\\right) +O(\\\\varepsilon ^2) \\\\,,\\\\\\\\\\\\frac{dW}{d\\\\tau } &=& - \\\\varepsilon ^2 W^3 A(u,v,0) + O(\\\\varepsilon ^3) \\\\,.\\\\nonumber $ The initial data are denoted by $u(0)=u_0 := \\\\frac{p_0}{\\\\sqrt{p_0+q_0}} >0\\\\,,\\\\qquad v(0)=v_0 := \\\\frac{2p_0+2q_0 -nr_0}{\\\\sqrt{p_0+q_0}}\\\\,,\\\\qquad W(0)=1\\\\,,$ where in the following we consider $u_0$ and $v_0$ as fixed when $\\\\varepsilon \\\\rightarrow 0$ .\",\n \"This is a singular perturbation problem in standard form, where $\\\\tau $ plays the role of an initial layer variable.\",\n \"We pass to the limit $\\\\varepsilon \\\\rightarrow 0$ to obtain the initial layer problem $\\\\frac{d\\\\hat{u}}{d\\\\tau } &=& - \\\\kappa _3 \\\\hat{u}(\\\\hat{u}-\\\\hat{v}) + \\\\kappa _- \\\\,,\\\\\\\\\\\\frac{d\\\\hat{v}}{d\\\\tau } &=& \\\\frac{d\\\\hat{W}}{d\\\\tau } = 0 \\\\,,\\\\nonumber $ subject to the initial conditions.\",\n \"By the qualitative behaviour of the right hand side of the first equation, the solution satisfies $\\\\hat{v}(\\\\tau )=v_0$ , $\\\\hat{W}(\\\\tau )=1$ , and $\\\\lim _{\\\\tau \\\\rightarrow \\\\infty } \\\\hat{u}(\\\\tau ) = U(v_0) \\\\,,$ with exponential convergence, where $U$ has been defined in (REF ).\",\n \"The equation $u=U(v)$ defines the so called reduced manifold.\",\n \"Since it is exponentially attracting, the Tikhonov theorem [7] (or rather its extension [3]) implies that, after the initial layer, i.e.\",\n \"when written in terms of the slow variable $\\\\sigma = \\\\varepsilon \\\\tau $ , the solution trajectory remains exponentially close to the slow manifold, which is approximated by the reduced manifold, and the flow on the slow manifold satisfies $\\\\frac{dv}{d\\\\sigma } &=& W\\\\left(2\\\\kappa _1(U(v)-v) - (2\\\\kappa _{-1} + n\\\\kappa _2)U(v) + \\\\kappa _- n \\\\frac{2U(v)-nv}{2(n-2)}\\\\right) +O(\\\\varepsilon ) \\\\,,\\\\nonumber \\\\\\\\\\\\frac{dW}{d\\\\sigma } &=& - \\\\varepsilon W^3 A(U(v),v,0) + O(\\\\varepsilon ^2) \\\\,,$ with $v(0)=v_0$ , $W(0)=1$ .\",\n \"This is again a singular perturbation problem in standard form, where now $\\\\sigma $ is the initial layer variable.\",\n \"We repeat the above procedure and consider the limiting layer problem $\\\\frac{d\\\\tilde{v}}{d\\\\sigma } &=& \\\\tilde{W}\\\\left(2\\\\kappa _1(U(y\\\\tilde{v})-\\\\tilde{v}) - (2\\\\kappa _{-1} + n\\\\kappa _2)U(\\\\tilde{v})+ \\\\kappa _- n \\\\frac{2U(\\\\tilde{v})-n\\\\tilde{v}}{2(n-2)}\\\\right) \\\\,,\\\\\\\\\\\\frac{d\\\\tilde{W}}{d\\\\sigma } &=& 0 \\\\,.\\\\nonumber $ The observations $U(-\\\\infty ) = 0 \\\\,,\\\\qquad U(\\\\infty )=\\\\infty \\\\,,\\\\qquad 0< U^{\\\\prime }(v) < 1 \\\\,,$ suffice to show that the right hand side of the first equation is a strictly decreasing function of $v$ with a unique zero $v^*$ , which can actually be computed explicitly: $v^* = B\\\\left(\\\\kappa _1 - \\\\kappa _{-1} - \\\\frac{n}{2}\\\\kappa _2 + \\\\frac{n}{2(n-2)}\\\\kappa _-\\\\right)$ $\\\\mbox{with}\\\\quad B = 2\\\\sqrt{\\\\frac{\\\\kappa _-}{\\\\kappa _3}} \\\\left( \\\\frac{n^3}{4(n-2)}\\\\kappa _-^2 + 4\\\\kappa _1 \\\\kappa _{-1} + 2n \\\\kappa _1\\\\kappa _2+ n\\\\kappa _1 \\\\kappa _- + \\\\frac{n^2}{n-2} \\\\kappa _{-1}\\\\kappa _- + \\\\frac{n^3}{2(n-2)\\\\kappa _2 \\\\kappa _-}\\\\right)^{-1/2}$ The solution of (REF ) with $\\\\tilde{v}(0)=v_0$ satisfies $\\\\lim _{\\\\sigma \\\\rightarrow \\\\infty } \\\\tilde{v}(\\\\sigma ) = v^*$ with exponential convergence.\",\n \"Another application of the Tikhonov theorem shows that the slowest part of the dynamics with $t=O(\\\\varepsilon ^{-1})$ can be approximated by $\\\\frac{dW}{d\\\\sigma } = - \\\\varepsilon W^3 A^* \\\\,,\\\\qquad W(0)=1 \\\\,,$ with $A^* := A(U(v^*),v^*,0) = \\\\frac{nB}{16(n-2)^2}(4(n-2)^2\\\\kappa _1\\\\kappa _2 - 2n(n-2)\\\\kappa _-(\\\\kappa _1+\\\\kappa _{-1}) - n^2(n-1)\\\\kappa _-^2) >0 \\\\,,$ by (REF ).\",\n \"This gives the approximation $W(\\\\sigma ) = (1+2 A^*\\\\varepsilon \\\\sigma )^{-1/2} \\\\,.$ The results of [3] imply that the approximations are accurate with errors of order $\\\\varepsilon $ uniformly with respect to time.\",\n \"Theorem 2 Let (REF ) hold.\",\n \"Then, for $\\\\varepsilon >0$ small enough, the solution of (REF ) with initial conditions $u(0) = u_0>0 \\\\,,\\\\qquad v(0) = v_0 \\\\in \\\\mathbb {R} \\\\,, \\\\qquad w(0)=\\\\varepsilon \\\\,,$ satisfies $u(\\\\tau ) &=& \\\\hat{u}(\\\\tau ) - U(v_0) + U(\\\\tilde{v}(\\\\varepsilon \\\\tau )) + O(\\\\varepsilon ) \\\\,,\\\\\\\\v(\\\\tau ) &=& \\\\tilde{v}(\\\\varepsilon \\\\tau ) + O(\\\\varepsilon ) \\\\,,\\\\\\\\w(\\\\tau ) &=& \\\\varepsilon (1 + 2A^* \\\\varepsilon ^2\\\\tau )^{-1/2} + O(\\\\varepsilon ^2) \\\\,,$ uniformly in $\\\\tau \\\\ge 0$ , where $U$ is given in (REF ), $\\\\hat{u}$ solves (REF ), $\\\\tilde{v}$ solves (REF ), and $A^*$ is given in (REF ).\",\n \"Actually more can be deduced.\",\n \"In terms of the original time variable $t$ , the equation for $w$ in (REF ) becomes $\\\\dot{w} = -w^2 A(u,v,w) \\\\,.$ Under the assumptions of Theorem REF , $A(u,v,w)$ is uniformly close to the positive constant $A^*$ and therefore uniformly positive for large enough $t$ .\",\n \"This implies that $w$ tends to zero as $t\\\\rightarrow \\\\infty $ .\",\n \"The slow manifold of the system (REF ) reduces to the steady state $(v,W)=(v^*,0)$ for $W=0$ .\",\n \"Therefore $v$ tends to $v^*$ as $t\\\\rightarrow \\\\infty $ .\",\n \"Analogously, the slow manifold of (REF ) reduces to the steady state $(u,v,W)=(u^*=U(v^*),v^*,0)$ at $W=0$ , implying convergence of $u$ to $u^*$ .\",\n \"This in turn implies convergence of $A(u,v,w)$ to $A^*$ , which can be used in (REF ).\",\n \"Corollary 1 Let the assumptions of Theorem REF hold.\",\n \"Then $\\\\lim _{t\\\\rightarrow \\\\infty } u(t) = u^* \\\\,,\\\\qquad \\\\lim _{t\\\\rightarrow \\\\infty } v(t) = v^* \\\\,,\\\\qquad w(t) = \\\\frac{1}{A^* t} + O\\\\left( \\\\frac{1}{t^2}\\\\right) \\\\quad \\\\mbox{as } t\\\\rightarrow \\\\infty \\\\,.$ Finally, we reformulate these results in terms of the original variables, verifying the formal asymptotics of [1] for initial data, which are in a sense already 'close enough' to the polynomially growing solutions.\",\n \"Theorem 3 Let (REF ) hold, let $c_2\\\\ge c_1>0$ , and let $\\\\delta >0$ be small enough.\",\n \"Let the initial data satisfy $p_0 = \\\\frac{c_1}{\\\\delta } \\\\,,\\\\qquad q_0 = \\\\frac{1}{\\\\delta ^2} \\\\,,\\\\qquad r_0 = \\\\frac{2}{n\\\\delta ^2} + \\\\frac{c_2}{n\\\\delta }$ Then the solution of (REF ) with $(p(0),q(0),r(0)) = (p_0,q_0,r_0)$ satisfies $p(t) = u^*A^* t + o(t)\\\\,,\\\\qquad q(t) = (A^*)^2 t^2 + o(t^2) \\\\,,\\\\qquad r(t) = \\\\frac{2}{n}(A^*)^2 t^2 + o(t^2) \\\\,,\\\\qquad \\\\mbox{as } t\\\\rightarrow \\\\infty \\\\,.$ We just need to verify that the assumptions of this theorem imply the assumptions of Theorem REF .\",\n \"The result is then a direct consequence of Corollary REF .\",\n \"Actually the assumptions of Theorem REF hold with $\\\\varepsilon \\\\approx \\\\delta $ , since $u_0 = \\\\frac{c_1}{\\\\sqrt{1+c_1\\\\delta }} \\\\,,\\\\qquad v_0 = \\\\frac{2c_1-c_2}{\\\\sqrt{1+c_1\\\\delta }} \\\\,,\\\\qquad w_0 = \\\\frac{\\\\delta }{\\\\sqrt{1+c_1\\\\delta }} \\\\,.$\"\n ],\n [\n \"Discussion\",\n \"In this work a mathematical model for aggregation via cross-linking has been analyzed.\",\n \"Besides the basic assumption that aggregating particles (here p62 oligomers) need to have at least $n=3$ binding sites for cross-linkers (here ubiqutinated cargo), the rate constants for binding reactions need to be large enough compared to those for the unbinding reactions (the opposite of inequality (REF )) for stable aggregates to exist.\",\n \"Under a stronger condition (inequality (REF )) aggregates grow indefinitely in the presence of an unlimited supply of free particles and cross-linkers.\",\n \"These conjectures from [1], where the model has been formulated, have been partially proven in this work.\",\n \"It has been shown in Section that small aggregates get completely degraded under the condition (REF ) and that they grow under the opposite condition.\",\n \"In the latter case, but when (REF ) does not hold, there exists an equilibrium configuration with positive aggregate size.\",\n \"Finally, it has been shown in Section that under the condition (REF ) aggregate size grows polynomially with time (actually like $t^2$ ) for appropriate initial states.\",\n \"Figure: Bifurcation diagram obtained for κ -1 =κ - =1\\\\kappa _{-1}=\\\\kappa _{-}=1The constants $\\\\kappa _1$ , $\\\\kappa _2$ in the model have to be interpreted as the products of rate constants with the concentrations of free cross-linkers and, respectively, of free particles.\",\n \"This means that the conditions (REF ) and (REF ) are actually conditions for these concentrations.\",\n \"Fig.\",\n \"REF shows a bifurcation diagram in terms of $\\\\kappa _1$ and $\\\\kappa _2$ with the curves $\\\\bar{\\\\alpha }=1$ , corresponding to equality in (REF ), and $\\\\bar{\\\\alpha }=0$ , corresponding to equality in (REF ).\",\n \"The qualitative behaviour is no surprise: Close to the origin, i.e.\",\n \"for small concentrations of free particles and cross-linkers, aggregates are unstable.\",\n \"Moving to the right and/or up we pass through two bifurcations to stable finite aggregate size and, subsequently, to polynomial growth of aggregates.\",\n \"Less obvious is the fact that the picture is rather unsymmetric with respect to the two parameters.\",\n \"The condition $(n-2)\\\\kappa _2> \\\\kappa _-$ is necessary for the existence of stable aggregates, regardless of the value of $\\\\kappa _1$ , whereas arbitrarily small values of $\\\\kappa _1$ can be compensated by large enough $\\\\kappa _2$ .\",\n \"This means that, if the concentration of free particles is below a threshold, even a large concentration of cross-linkers does not lead to aggregation, whereas arbitrarily small numbers of cross-linkers are used for aggregation if the particle concentration is high.\",\n \"For the application in cellular autophagy this means that aggregation will only happen for large enough concentrations of p62 oligomers.\",\n \"However, arbitrarily small amounts of ubiquitinated cargo can be aggregated in the presence of a large enough supply of oligomers.\",\n \"This work has been motivated by the experimental results of [8], where aggregates have been detected by light microscopy.\",\n \"If the evolution of single aggregates can be followed, the growth like $t^2$ might be observed as a fluorescence signal of tagged oligomers, which goes like $t^2$ , or cross section areas going like $t^{4/3}$ , if a roughly spherical shape of aggregates is assumed.\",\n \"For quantitative predictions of such experiments, the model should be extended in various ways.\",\n \"First, the limited supply of free p62 oligomers and of free cross-linkers should be taken into account.\",\n \"This is straightforward for the modeling of a single aggregate, but if many aggregates develop simultaneously, they will compete for the free particles.\",\n \"Apart from that the number of aggregates has to be predicted, which requires modeling of the nucleation process.\",\n \"Finally, it is very likely that the coagulation of aggregates plays an important role.\",\n \"A growth-coagulation model for distributions of aggregates, based on the growth model (REF ) would be prohibitively complex.\",\n \"It is therefore the subject of ongoing work to formulate, analyze, and simulate a growth-coagulation model based on the multiscale analysis of Section , where aggregates are only described by the size parameter $r$ (number of p62 oligomers in the aggregate), whose evolution is determined by the slow dynamics (REF ), which translates to an equation of the form $\\\\dot{r} = C\\\\sqrt{r}$ for $r$ .\",\n \"This approach raises several challenging issues such as the development of an efficient simulation algorithm or the existence and stability of equilibrium aggregate distributions.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05201"}}},{"rowIdx":3689,"cells":{"text":{"kind":"list like","value":[["1000 days of lowest frequency emission from the low-luminosity GRB\n 171205A"],["Abstract We report the lowest frequency measurements of gamma-ray burst (GRB) 171205A with the upgraded Giant Metrewave Radio Telescope (uGMRT) covering a frequency range from 250--1450 MHz and a period of $4-937$ days.","It is the first GRB afterglow detected at 250--500 MHz frequency range and the second brightest GRB detected with the uGMRT.","Even though the GRB is observed for nearly 1000 days, there is no evidence of transition to non-relativistic regime.","We also analyse the archival ${\\it Chandra}$ X-ray data on day $\\sim 70$ and day $\\sim 200$.","We also find no evidence of a jet break from the analysis of combined data.","We fit synchrotron afterglow emission arising from a relativistic, isotropic, self-similar deceleration as well as from a shock-breakout of wide-angle cocoon.","Our data also allow us to discern the nature and the density of the circumburst medium.","We find that the density profile deviates from a standard constant density medium and suggests that the GRB exploded in a stratified wind like medium.","Our analysis shows that the lowest frequency measurements covering the absorbed part of the light curves are critical to unravel the GRB environment.","Our data combined with other published measurements indicate that the radio afterglow has contribution from two components: a weak, possibly slightly off-axis jet and a surrounding wider cocoon, consistent with the results of Izzo et al.","(2019).","The cocoon emission likely dominates at early epochs, whereas the jet starts to dominate at later epochs, resulting in flatter radio lightcurves."],["Gamma Ray Bursts (GRBs) are the most energetic flashes of gamma rays, with $T_{90}$ duration (time interval between which 5% to 95% of fluence is collected by the detector) ranging between a few milli-seconds to thousands of seconds [87].","The GRBs can be classified in two broad classes i.e short/hard GRBs with duration less than 2 secs, and long/soft GRBs of duration greater than 2 secs [42].","According to well accepted theories, most of the long soft GRBs originate from gravitational collapse of massive stars (collapsar model); and short hard GRBs result from explosive binary compact object mergers [86].","The GRBs from both channels of formation power relativistic collimated jets which give Doppler-boosted high luminosities in gamma-rays.","GRBs are cosmological events with an average redshift $z\\approx 2.2$ [29], and have isotropic-equivalent gamma ray luminosities ($L_{\\rm iso}$ ) of the order of $10^{51}$ to $10^{53}$ erg s$^{-1}$ .","However, a handful of long/soft GRBs with spectroscopically identified supernovae have been discovered with luminosities that are 3–5 orders of magnitude lower than the average, i.e.","[69], [10].","Their low luminosities allow them to be detected only at low redshifts, though they may be 10–100 times more abundant than regular GRBs [68].","The prompt light-curves of typical low-luminosity GRBs are smooth and spectra have a single peak with the peak energy generally below $\\sim $ 50 keV, which softens further with time [56], [10].","The radio afterglow of these GRBs tend to indicate similar energy content in mildly relativistic ejecta [43].","Many of these GRBs are associated with broad-line Type Ic supernovae.","The list of such GRBs-supernovae include some of the well studied cases such as, GRB 980425/SN 1998bw [31], GRB 030329/SN 2003dh [36], GRB 031203/SN 2003lw [48], GRB 060218/SN 2006aj [9], GRB 100316D/SN 2010bh [74], GRB 111209A/SN 2011kl [32], GRB 120422A/SN 2012bz [50], GRB 130427A/SN 2013cq [51], GRB 130702A/SN 2013dx [12], GRB 161219B/SN 2016jca [11], GRB 171010A/SN 2017htp [52], GRB 190829A/ SN 2019oyu [80], although only the five are nearby, $z\\lesssim 0.1$ [10].","A significant amount of work has gone towards understanding whether the low-luminosity GRBs are simply the low-energy counterparts of the cosmological GRBs, or have a different emission mechanism.","Many low-luminosity GRBs do not follow the $E_{\\rm iso}-E_p$ Amati relation [1], indicating that their emission mechanism should be different from that of canonical, more distant GRBs.","Technically an off-axis jet can also explain low-luminosity emission from GRB, but predicts an achromatic steepening of the light curve, absent in many low-luminosity GRBs.","There have been suggestions that in contrast to the emission from an ultra-relativistic jet driven by a central engine, these low-luminosity GRBs are powered by shock breakouts [43], [56], [54], [5], [76].","In some cases observations have suggested a mildly relativistic blast wave being responsible for producing the radio afterglow.","This has supported the relativistic shock breakout model, e.g.","GRB 980425 [43].","The shock-break out model got further support in case of GRB 060218, in which a thermal component was also seen which cooled and shifted to optical/UV band with time.","This was interpreted to be arising from the break out of a shock driven by a mildly relativistic supernova shell in the progenitor wind [9], although late time photospheric emission from a jet [27], or thermal emission from a cocoon [78] can also explain it.","[8] have investigated whether the low-luminosity GRBs launch relativistic jets like their high energy counterparts, but incur resistance by the stellar envelopes surrounding their progenitor stars.","They found that some low-luminosity GRBs have much shorter durations compared to the jet breakout time, This is inconsistent with the collapsar model which is largely successful in explaining the cosmological GRBs [87].","While [5] and [56] have developed spherical relativistic shock break out models in context of low-luminosity GRBs, [54] addressed some of the problems of spherical shock-break out model by introducing a low mass optically thick stellar envelope surrounding the progenitor star.","In this model, the explosion powering the low-luminosity GRBs was not the spherical breakout of the supernova shock, but by a jet that gets choked in the envelope and powers quasi-spherical explosion.","To extend this idea further, [55] considered a cocoon breakout model.","In this model, as the GRB jet pushes through the stellar material, it heats the surrounding gas and produces a high-pressure sub-relativistic cocoon, which at the time of breakout, produces a relatively faint flare of $\\gamma $ -rays.","This break out will not be as spherical as a supernova break-out, but will be wider than a jet.","In this model, interaction of the cocoon with the surrounding medium can give rise to a late time radio and X-ray afterglow.","However, [37] have provided an alternative mechanism, where the composite emission of GRB 060218/SN 2006aj could be explained by a weak jet, along with a quasi-spherical supernova ejecta.","llGBRs also have a radio afterglow, which indicates a comparable energy in mildly relativistic ejecta (Kulkarni et al.","1998; Soderberg et al.","2004, 2006; Margutti et al.","2013).","The lack of bright, late-time, radio emission from ll-GRBs strongly constrain the total energy of any relativistic outflow involved in these events (Waxman 2004; Soderberg et al.","2004, 2006b).","Additionally, statistical arguments rule out the possibility that ll-GRBs are regular LGRBs viewed at a large angle (e.g., Daigne & Mochkovitch 2007).","Thus, if ll-GRBs are generated by relativistic jets these jets must be weak and have a large opening angle.","The bursts with $L_\\gamma > 2 \\times 10^{48}$ erg s$^{-1}$ are considered as regular GRBs and are separated into LGRBs and SGRBs according to the standard criterion of whether T90 in the observer frame is above or below 2 s (Bromberg, Nakar, Piran 2011).","GRB 171205A is a nearby low-luminosity GRB with a $T_{90}$ duration of almost 189.4 secs [24].","It was first discovered by the Burst Alert Telescope (BAT) onboard Swift on 5th December, 2017 [23].","It has a redshift of 0.0368 [23], [38].","The isotropic energy release in gamma ray band at GRB rest frame was $2.18_{-0.50}^{+0.63}\\times 10^{49}$ ergs [24].","The host of this event was a bright spiral galaxy named 2MASX J11093966-1235116 [38], with a mass of the order of $10^{10}M_{\\odot }$ and a star formation rate of $3\\pm 1M_{\\odot }/\\rm {yr}$ [63].","An emergent supernova event (SN 2017iuk) was seen three days after the burst [24].","There are a handful of observations for GRB 171205A from mm to radio bands.","It was detected by [22] using Northern Extended Millimeter Array (NOEMA) with a flux density of $\\sim 35$ mJy at 150 GHz after 20.2 hours of the burst.","Atacama Large Millimeter/Submillimeter Array (ALMA) detected a bright afterglow of significance more than 100$\\sigma $ at 92 GHz and 340 GHz on 10-11 December, 2017 [62].","RATAN-600 radio telescope detected it at 4.7 and 8.2 GHz bands during 9th December to 16th December, 2017 [81].","Karl G. Jansky Very Large Array (VLA) also observed the afterglow in the frequency range 4.5 to 16.5 GHz [44].","This GRB also had a Very Long Baseline Array (VLBA) detection at different frequencies [61].","These observations showed a steeply rising spectrum($\\propto \\nu ^{2}$ ) at low frequency which indicates a synchrotron self-absorbed spectrum.","This is the first GRB for which [82] claimed to have detected polarization at the ALMA 90GHz frequency, though this claim has been disputed by [45].","The upgraded Giant Metrewave Radio Telescope (uGMRT) first detected it on 20th December, 2017 at 1400 MHz [15] after a non detection on 10th and 11th December, 2017 [16].","The observed flux density at that epoch was $782\\pm 57 \\rm {\\mu Jy}$ .","Radio afterglow emission from GRBs evolve slowly which gives us the opportunity to observe it for a long time and obtain the distribution of the kinetic energy in the velocity space.","Since this distribution is different for various models, especially central engine driven versus shock break out, radio observations provide unique opportunity to distinguish between various emission models [43].","Additionally, the early radio emission is likely to be absorbed via synchrotron self-absorption (SSA), and thus constrain the circumburst medium (CBM) density [17].","The late time radio observations in the Newtonian limit, when the jet becomes sub-relativistic, are nearly independent of jet geometry and measure the kinetic energy of the afterglow accurately [26].","In this paper we present low frequency observations of GRB 171205A taken with the uGMRT for around 1000 days.","We summarize our observations and data analysis in §.","We discuss our model in §REF and results in §REF .","In §, we discuss the properties of GRB 171205A in conjunction with published measurements at higher frequencies and present our main conclusions.","Unless otherwise stated, we assume a cosmology with $H_0=67.3$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _m=0.315$ , $\\Omega _\\Lambda =0.685$ [64]."],["The GMRT observed GRB 171205A starting 2017 December 10 and continued observing until 2020 June 26.","The observations were taken in band 5 (1000–1450 MHz), band 4 (550–900 MHz) and band 3 (250–500 MHz).","The bandwidth for band 4 and band 5 was 400 MHz while for band 3 it was 200 MHz.","The duration of each observation was around 2-3 hours including overheads (on source time 1.5 hours).","We observed flux density calibrators 3C286 and 3C48; and a phase calibrator J1130-148.","Flux calibrators were also used as bandpass calibrators.","We use a package Common Astronomy Software Applications( CASA) for data analysis.","The data were analyzed in three major steps, i.e flagging, calibration and imaging.","The CASA task `flagdata' was used to remove dead antennas and bad data.","In addition, the tasks `tfcrop'(http://www.aoc.nrao.edu/~rurvashi/TFCrop/TFCropV1/node2.html) and `rflag'(https://casa.nrao.edu/Release4.2.2/docs/userman/UserMansu167.html) were used to flag the radio frequency interference (RFI).","The calibration [30] was performed to remove the instrumental and atmospheric effects from the measurement.","The final part of processing was imaging.","The continuum imaging of the target source was done using CASA task `tclean'.","Finally, a few rounds of `phase only' mode and two rounds of `amplitude-phase' self-calibrations were run.","We fit a Gaussian to determine the GRB flux density at the GRB position.","The flux densities are shown in Table .","Sample radio images of GRB171205A at bands 5, 4 and 3 are shown in Figure REF .","The errors in flux densities in Table show only the statistical errors.","We also add 15% of flux densities in quadrature to account the uncertainties due to calibration and other systematics for GMRT bands during our model fit.","We closely follow the procedure shown in [14].","The table list the details of observations and flux densities at various epochs.","Figure: GMRT images of GRB171205A.","Panel a) Band 5 detection on 2019 May 13.","Panel b) Band 4 detection on 2019September 10.Panel c) Band 3 detection on 2019 May 13.","The contours in black lines show detection significance and are at 20σ\\sigma , 40σ\\sigma and 60σ\\sigma for bands 4 and 5; and at 6σ\\sigma , 8σ\\sigma , and 10σ\\sigma for band 3, where σ\\sigma is the map rms of corresponding images.","For the images displayed, they are following: Band 5, 17 μ\\mu Jy, Band 4, 30 μ\\mu Jy and Band 5, 80 μ\\mu Jy.We also extract Swift-XRT 0.3–10 keV flux light curve from the the Swift online repositoryhttps://www.swift.ac.uk/xrt_products/00794972.","The light curve post day 1 indicate a photon index $\\Gamma =1.94^{+0.23}_{-0.22}$ and column density $N_H=(1.2^{+0.8}_{-0.7} \\times 10^{21}$ cm$^{-2}$https://www.swift.ac.uk/xrt_live_cat/00794972/.","This is in addition to Galactic column density of $5.89\\times 10^{20}$ cm$^{-2}$ .","We converted X-ray flux into into 1 keV spectral flux density using this photon index.","Swift data covered observations until 2020 May 27.","In addition, we analysed two archival data from Chandra ACIS-S on 2018 Feb 14 and 2018 June 29 (PI: Margutti).","We used Chandra Interactive Analysis of Observations software [28] task specextractor to extract the spectra, response and ancillary matrices.","We used CIAO version 4.6 along with CALDB version 4.5.9.","The HEAsofthttp://heasarc.gsfc.nasa.gov/docs/software/lheasoft/ package Xspec version 12.1 [3] was used to carry out the analysis of the Chandra spectra.","The GRB was detected in the first obserations at 71 days with 0.3–10 keV unabsorbed flux of $(11.25\\pm 3.38)\\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ .","The second Chandra observation on second epoch, i.e 206 days, resulted in a 3-$\\sigma $ upper limit $< 3.98\\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ .","lcccrc 1 Flux Densities of GRB171205A 0pt Date of Observation Band Frequency Days since explosion Flux Densitya Map RMS MHz (mJy) $\\mu $ Jy/beam 2017 Dec 10.10 5 1255 4.79 $<0.07$ 24 2017 Dec 11.02 4 648 5.71 $<$ 0.06 20 2017 Dec 19.91 5 1265 14.60 $0.64\\pm 0.05$ 17 2017 Dec 26.90 5 1265 21.59 1.00$\\pm $ 0.07 17 2017 Dec 28.91 4 607 23.60 $<$ 0.39 130 2018 Jan 16.94 5 1265 42.63 1.75$\\pm $ 0.05 17 2018 Feb 12.85 5 1370 68.54 3.04$\\pm $ 0.09 41 2018 Feb 17.85 4 607 73.54 1.37$\\pm $ 0.15 115 2018 Mar 20.68 5 1352 105.37 5.79$\\pm $ 0.08 34 2018 Jun 08.44 3 402 185.13 $2.94\\pm 0.41$ 106 2018 Jun 10.68 5 1255 187.37 3.07$\\pm $ 0.11 26 2018 Jun 11.42 4 745 188.11 $2.46\\pm 0.49$ 141 2018 Jul 13.47 5 1250 220.16 3.55$\\pm $ 0.12 17 2018 Jul 15.35 4 610 222.043.13$\\pm $ 0.18 66 2018 Jul 23.35 3 402 230.04 $2.30\\pm 0.22$ 64 2018 Jul 26.57 5 1265 233.26 $2.60\\pm 0.08$ 26 2018 Jul 28.35 4 750 235 2.92$\\pm $ 0.31 62 2018 Aug 24.32 5 1265 262.01 $3.32\\pm 0.06$ 33 2018 Aug 25.22 4 607 262.91 $1.06\\pm 0.16$ 81 2018 Sep 21.35 5 1265 290.04 $3.03\\pm 0.10$ 59 2018 Sep 23.16 3 402 291.85 $2.01\\pm 0.15$ 63 2018 Oct 20.28 5 1250 319 3.18 $\\pm $ 0.13 52 2018 Oct 26.09 3 400 324.78 1.77 $ \\pm $ 0.27 131 2018 Oct 26.33 4 607 325.02 $2.61\\pm 0.26$ 143 2018 Dec 21.88 3 402 381.57 $1.61\\pm 0.31$ 49 2018 Dec 22.02 5 1255381.71 $2.83\\pm 0.06$ 21 2018 Dec 22.14 4 610 381.83 $1.52\\pm 0.17$ 52 2019 Feb 26.98 4 610 448.67 1.52$\\pm $ 0.21 79 2019 Feb 26.70 3 402 448.39 1.05$\\pm $ 0.16 40 2019 Feb 26.87 5 1250 448.56 1.87$\\pm $ 0.06 17 2019 May 13.49 5 1250 524.18 1.74$\\pm $ 0.04 17 2019 May 13.62 3 402 524.31 1.43 $\\pm $ 0.26 79 2019 May 13.77 4 607 524.46 2.09 $\\pm $ 0.18 81 2019 Sep 10.16 5 1250 643.85 1.74 $\\pm $ 0.03 19 2019 Sep 10.31 3 402 644.00 1.67 $\\pm $ 0.21 50 2019 Sep 10.43 4 750 644.12 1.93 $\\pm $ 0.10 30 2019 Dec 09.91 4 647 734.60 1.38 $\\pm $ 0.12 37 2019 Dec 10.02 5 1265 734.71 1.71 $\\pm $ 0.04 20 2019 Dec 10.19 3 402 734.88 1.30 $\\pm $ 0.17 59 2020 June 26.37 4 648 934.06 $1.29\\pm 0.11$ 20 2020 June 26.49 5 1255 934.18 $1.23\\pm 0.04$ 21 2020 June 29.44 3 402 937.13 $1.12\\pm 0.16$ 55 aThe uncertainties reflect the statistical errors."],["GRB afterglow Model","We use external synchrotron model for the GRB afterglow emission, which arises due to the interaction between the GRB outflow and the surrounding CBM [34].","As the outflow moves into the CBM, a `forward shock' or a `blast wave' shock moving into the CBM, and a `reverse shock' moving into the ejected outflow are created.","These shocks have the ability to accelerate charged particles to relativistic speeds via Fermi acceleration [47].","Radio afterglow emission is expected to be synchrotron emission arising due to these relativistic charged particles in the shocks in the presence of magnetic fields.","The evolution of the blast wave is `self-similar' [7], and the dynamics depends only on the density of the CBM and the blast wave energy.","The CBM is usually modelled to be one of the two forms, a constant density medium and a wind like density medium [18].","The number density profile of the ambient medium is usually modelled as a powerlaw $n \\propto r^{-k}$ .","For constant density case, the parameter $k=0$ and $n=n_0$ .","For wind like case, the mass flows radially outwards at uniform speed and rate from the of GRB progenitor giving $k=2$ ; hence for a mass-loss rate from the progenitor $\\dot{M}_W$ and the progenitor wind velocity $V_W$ , the density can be defined as [33], [19]: $n=\\frac{\\dot{M}_W}{4\\pi r^{2} m_p V_W}=3\\times 10^{35} A_* r^{-2}$ Where $m_p$ is the mass of the proton, $A_*$ is in units of $3\\times 10^{35}$ cm$^{-1}$ (or $5\\times 10^{11}$ g cm$^{-1}$ for mass-density), corresponding to $\\dot{M}_{W,-5}/V_{W,3}$ .","Here $\\dot{M}_{W,-5}$ is mass loss rate in $10^{-5}M_{\\odot }/\\rm {yr}$ and $V_{W,3}=10^3\\,\\rm {km/s}$ .","The GRB wideband afterglow spectrum has several breaks characterized by various characteristic frequencies, namely $\\nu _a$ (the transition from optically thick to thin region, i.e.","synchrotron self-absorption (SSA) peak), $\\nu _c$ (synchrotron cooling frequency) and $\\nu _m$ (frequency corresponding to minimum injected Lorentz factor).","In the fast cooling regime the frequency ordering is $\\nu _m > \\nu _c$ , while in the slow cooling regime the ordering is opposite.","Generally afterglow modelling is done in the slow cooling regime, where the most relevant ordering in radio frequencies in first few days are $\\nu _a \\le \\nu _m \\le \\nu _c$ ; and then $\\nu _m \\le \\nu _a \\le \\nu _c$ at later times [35].","However, there has been evidence for fast cooling in some GRBs with high density environments [17].","The afterglow spectra evolves as $F_{\\nu } \\propto \\nu ^2$ ($\\nu <\\nu _a$ ) and $F_{\\nu } \\propto \\nu ^{1/3}$ ($\\nu _a < \\nu < \\min (\\nu _m,\\nu _c)$ ) for both fast as well as the slow cooling.","In the regime $\\min (\\nu _m,\\nu _c) < \\nu < \\max (\\nu _m,\\nu _c) $ , the evolution changes to $F_{\\nu } \\propto \\nu ^{-1/2}$ and $F_{\\nu } \\propto \\nu ^{-(p-1)/2}$ for fast and slow cooling regimes, respectively, and then evolves as $F_{\\nu } \\propto \\nu ^{-p/2}$ for $ \\nu > \\max (\\nu _m,\\nu _c) $ , where $p$ is the usual power law index showing particle number distribution with energy in non-thermal emission case.","In addition to standard afterglow models, shock breakout model too has been favoured for low-luminosity GRBs, where the breakout of a shock travelling through the stellar envelope may be responsible for gamma-ray emission.","Due to decreasing density of the stellar matter outwards, the shock breakout velocity increases and may become relativistic.","[56] have defined a ‘relativistic breakout closure relation’ between the breakout energy $E_{\\rm bo}$ , temperature $T_{\\rm bo}$ and duration $t_{\\rm bo}^{\\rm obs}$ , i.e.","$( t_{\\rm bo}^{\\rm obs}/20 \\,\\rm s) \\sim (E_{\\rm bo}/10^{46}\\, \\rm erg)^{1/2} (T_{bo}/50 \\, \\rm keV)^{-2.68}$ .","This relation has been found to be followed by several low luminosity GRBs.","For GRB 171205A, the relation gives $\\sim 80$ s, which is roughly 1/3rd of the observed duration.","However, the large uncertainties in the $E_p$ and $E_{\\rm iso}$ [24] do not rule out this model."],["Inputs from high frequency data","The Swift-XRT light curve covers epoch until day $\\sim 902$ .","In addition, Chandra observations are on day 71 and day 205 (Fig.","REF ).","We fit a power law to the X-ray data post day 1.","This is to avoid possible energy injection due to central engine activities at $\\le 1$ d. The light curve is fit with power law index $\\alpha =1.06\\pm 0.06$ .","[24] also find a power law fit with index $1.08$ to the light curve post $\\sim $ 1 day, consistent with our value.","This is typical value for the standard X-ray light curve decay before jet break [59], suggesting that no jet-break was seen until the last Swift-XRT epoch.","The jet break time is thus constrained to $t_{\\rm jet}\\ge 71\\,\\rm d$ [66], i.e.","the last detected Chandra epoch.","The photon index of the Swift-XRT data is a photon index $\\Gamma =1.94^{+0.23}_{-0.22}$ https://www.swift.ac.uk/xrt_live_cat/00794972/.","This suggests $\\beta =0.94^{+0.23}_{-0.22}$ .","The values of $\\alpha $ and $\\beta $ are consistent with X-ray frequency ($\\nu _X$ ) being above the cooling frequency ($\\nu _X>\\nu _c$ ), for both wind as well as ISM density profiles.","The X-ray temporal index is consistent with $\\alpha =(3p-2)/4$ and spectral index $\\beta =p/2$ .","These values gives $p$ to be $2.08\\pm 0.08$ and $1.88\\pm 0.46$ , respectively, which, within the errorbars, are consistent with each other.","Within the errorbars, this is also consistent with $\\nu _X<\\nu _c$ for a ISM like medium where $\\alpha =3(p-1)/4$ and $\\beta =(p-1)/2$ .","We also plot early time 230 and 345 GHz light curves GRB 171205A, taken from [82].","We estimate the realistic errorbars in the flux density values by adding 5% of the flux density in quadrature to the map rms to account for the systematic errors.","The mm highest flux density is significantly higher than the peak flux densities at uGMRT bands, indicating most likely presence of wind like medium.","The mm light curves are jointly fit with a broken powerlaw with common post peak index.","The data are best fit with post break index $-1.37\\pm 0.07$ (bottom panel of Fig.","REF ).","For above values of $p$ , this is consistent with an evolution of $(3p-1)/4$ , for $\\nu _m<\\nu _{\\rm mm}<\\nu _c$ for wind density profile.","The pre-break indices $-0.17\\pm 0.14$ and $-0.54 \\pm 0.31$ for 230 and 345 GHz bands, respectively.","The breaks in 230 and 345 GHz are at $3.14\\pm 0.28$ day and $6.12\\pm 1.17$ day, respectively.","At the epochs of the breaks, the flux density of the 230 and 345 GHz light curves are $43.83\\pm 3.29$ and $14.85\\pm 2.86$ mJy, respectively.","We also fit the 345 GHz data with a single powerlaw model to determine the significance of the break.","The single powerlaw model fits with an index of $-1.05\\pm 0.14$ , however results in a much larger reduced-$\\chi ^2$ value of 2.47 as compared to 0.81 in the broken power law case.","For the broken powerlaw model, the characteristic frequency evolves with an index of $+0.71\\pm 0.12$ ., hence the breaks can not be due to passage of $\\nu _m$ , which decreases with time.","The only possibility is the breaks are being due to $\\nu _c$ in the wind medium, where $\\nu _c \\propto t^{1/2}$ .","Thus the data indicate.","$\\nu _c\\approx 230$ GHz on day 3.14 and $\\nu _c\\approx 345$ GHz on day 6.12.","However, there is one concern.","The pre-break evolution is rather flat.","This could be reconciled if mm bands are close to passage of $\\nu _m$ .","Another possibility is that if there is a reverse shock component which is contributing to the mm band at the early epoch.","The thick shell reverse shock model during the reverse shock crossing phase will evolve as $-(p-2)/2$ , for $\\nu _{\\rm mm}>\\nu _c$ in the slow cooling phase and $\\nu _{\\rm mm}>\\nu _m$ in fast cooling phase.","The characteristic frequency evolves with an index of $+0.71\\pm 0.12$ , this is in between evolution of the $\\nu _c\\, (\\propto t^{1/2}) $ in the forward shock and the evolution of $\\nu _c\\, (\\propto t^{1}) $ in the reverse shock.","This also suggests the possibility of reverse shock contributing to the afterglow model.","If so, then the breaks at day 3.14 and 6.12 in 230 and 345 GHz are artificial breaks and may not reflect the cooling break.","Figure: Top panel: Near-simultaneous spectra on day ∼4\\sim 4, day ∼11\\sim 11 and day ∼909\\sim 909.","Here we use only the cm and mmdata.","Bottom panel: The same as top panel but including the X-ray data as well.","The spectra are fit with smoothed single broken powerlaw fits.","For day 4.2, we also fit the data witha powerlaw with two breaks.","The double broken powerlaw fit is indicated with a black continuous line and the single one with dashed blue line.We also combine the early epoch published cm and mm data from [82], late epoch Australian Telescope Compact Array (ATCA) data from [46] with the uGMRT and the X-ray data, to obtain near simultaneous spectra on around day 4, day 11 and day 909 (Fig.","REF ).","For the early epoch spectrum, the VLA data are on day 4.3 and the ALMA data are on day 5.2.","We use the nearest epoch temporal evolution to derive the ALMA values on day 4.3.","In the figure, we plot the values on day 5.2 as well as their derived values on day 4.3 along with the VLA data.","We do not use optical data as supernova signatures appeared by day 3 and hence the optical data are likely to be heavily contaminated by the underlying supernova.","We fit a smoothed single broken powerlaw (SBPL) and smoothed double broken powerlaw (DBPL) fits to day 4.3 spectrum.","We used formalism of [34] for the treatment of smoothening of powerlaws at the break frequencies.","We first fit only the cm and mm data.","For day 4.3, the best fit SBPL model peaks at $30.41\\pm 4.08$ GHz and afterwards evolves as $-0.41\\pm 0.25$ .","Here we have fixed the pre-break spectral index to 2.","However, even when we use this as a free parameter, the best-fit index is consistent with 2 within errorbars.","The DBPL model fits the data well and give the breaks at $7.25\\pm 2.11$ GHz and $44.41\\pm 3.62$ GHz with post break indices $1.26\\pm 0.12$ , and $-0.41\\pm 0.02$ , respectively.","The peak flux density is $43.80\\pm 1.12$ mJy.","We also fit SBPL for the spectrum on day 11 and 909.","The data indicates a fit $-0.68\\pm 0.02$ with peak $<90$ GHz, with the peak flux density $>16.2$ mJy.","The pre-break index is $0.43\\pm 0.46$ .","The spectrum on day 909 are fit with pre-break and post-break indices of $0.15\\pm 0.13$ and $-0.90\\pm 0.04$ , respectively, and a peak at $1.55\\pm 0.48$ GHz.","The peak flux at day 909 is $1.15\\pm 0.16$ mJy.","Now we carry out the above fits including the X-ray data as well.","The data to mm to X-ray data are fit by indices close to $-1$ .","These values are $-1.01\\pm 0.18$ and $-1.01\\pm 0.02$ for the first two spectra, respectively.","For the day 909, the X-ray upper limit do not constrain the model.","The mm to X-ray indices consistent with the X-ray spectral indices between 0.3–10 keV.","This indicates that the cooling frequency ($\\nu _c$ ) is probably close to mm values if $ \\nu _X > \\nu _c$ .","Due to lack of optical data, we cannot constrain the cooling frequencies more precisely.","The peak flux density and the frequency of the peak in the three cases are $37.79\\pm 3.95$ mJy at $160.93\\pm 35.31$ GHz, $>16.2$ mJy at $<90$ GHz and $1.15\\pm 0.16$ mJy at $1.55\\pm 0.48$ GHz., respectively.","Our analysis indicate that the peak of the spectra on day 4 and day 909 are due to $\\nu _m$ or $\\nu _a$ .","Between day 4 and 909, the peak flux density evolves as $-0.63\\pm 0.02$ .","This clearly rules out ISM model and supports the wind model.","[88] has estimated kinetic energy in the synchrotron afterglow, $E_K$ , from the X-ray data at the time of shallow to normal decay, which for GRB 171205A is 1.05 day.","For $\\nu _X>\\nu _c$ , $E_K$ is independent of density and is only weakly depends on $B$ and $p$ , and therefore an ideal regime to measure $E_K$ .","One can then derive $E_K$ from the X-ray band using Eq 9 of [88], which in this case is $E_K\\approx 1.4\\times 10^{50} (\\epsilon _B/0.01)^{2-p/2+p} (\\epsilon _e/0.1)^{4(1-p)/2+p}$ erg."],["Initial inferences from uGMRT data","We plot uGMRT radio light curves and fit them jointly with a smoothed broken powerlaw (SBPL) model.","We allow the normalization to vary but fix the indices before and after the peak.","We show the light curves in Fig.","REF .","In the figure, the band 4 and 5 values are scaled by factors of 10 and 100, respectively, for clarity.","The indicies before and after the peak are $1.37\\pm 0.20$ and $-0.72\\pm 0.06$ .","The evolution before peak at uGMRT frequencies ($\\nu _{\\rm radio}$ ) is in rough agreement with wind slow cooling case for $\\nu _{\\rm radio}<\\nu _a$ for $\\nu _a< \\rm min(\\nu _m,\\nu _c)$ ; the wind fast cooling phase for $\\nu _a<\\nu _c<\\nu _m$ if the observing frequency is in the transition zone between $\\nu _{\\rm radio}<\\nu _a$ to $\\nu _a<\\nu _{\\rm radio}<\\nu _c$ , as well as wind slow cooling phase for $\\nu _m<\\nu _a<\\nu _c$ for transition between $\\nu _{\\rm radio}<\\nu _m$ to $\\nu _m<\\nu _{\\rm radio}<\\nu _a$ .","The post peak index is rather shallow and is consistent only with the wind fast cooling in the regime $\\nu _a<\\nu _{\\rm radio}<\\nu _c$ .","It is rather shallow for the post jet-break or non-relativistic evolution.","However, as we discuss in the next section, the shallow decline of radio light curve in seen in other GRBs as well, and other emission components may contribute to it.","From our fits, the epochs of the peak flux densities are $5.19\\pm 0.05$ and $3.23\\pm 0.15$ mJy, respectively, in bands 5 and 4 on day $101.08\\pm 9.58$ and day $153.71\\pm 40.31$ , respectively.","In band 3, the data are optically thin, which constraints the peak to be $<185.13$ day and flux $>2.81$ mJy.","The peak flux density evolves as $-1.13 \\pm 0.76$ .","While this value has a large error, it is consistent with the evolution in the stratified wind within 2-$\\sigma $ and most likely rule out all the models involving ISM, where $F_{\\rm max} \\propto t^0$ .","This also rules out $1\\nu _c\\end{array}\\right.","}$ Case 2 ($\\nu _m<\\nu _a<\\nu _c$ ): $F_{\\nu }(t) \\propto {\\left\\lbrace \\begin{array}{ll} \\nu ^2 t^{s+1}, & \\text{for} \\ \\nu <\\nu _m \\\\\\nu ^{5/2} t^{\\frac{7+3s}{4}}, & \\text{for} \\ \\nu _m<\\nu <\\nu _a \\ \\\\\\nu ^{-(p-1)/2} t^{-\\frac{3p-1 -s(p+1)}{4}} , & \\text{for} \\ \\nu _a<\\nu <\\nu _c \\\\\\nu ^{-p/2} t^{-\\frac{3p-2 -s(p+2)}{4}}, & \\text{for}\\ \\nu >\\nu _c\\end{array}\\right.","}$ Case 3 ($\\nu _a<\\nu _c<\\nu _m$ ): $F_{\\nu }(t) \\propto {\\left\\lbrace \\begin{array}{ll} \\nu ^2 t^{2+s}, & \\text{for} \\ \\nu <\\nu _a \\\\\\nu ^{1/3} t^{-\\frac{2(s+1)}{3}}, & \\text{for} \\ \\nu _a<\\nu <\\nu _c \\ \\\\\\nu ^{-1/2} t^{-\\frac{s+1}{4}} , & \\text{for} \\ \\nu _c<\\nu <\\nu _m \\\\\\nu ^{-p/2} t^{-\\frac{3p-2 +(p+2)s}{4}}, & \\text{for}\\ \\nu >\\nu _m\\end{array}\\right.","}$ We now fit the uGMRT data with both standard isotropic afterglow and shock break-out afterglow models.","We use smoothed broken powerlaw models for various regimes following the recipe of [34].","We use $z=0.036$ , $D=163$ Mpc.","We define $E_{\\rm KE}=E_{\\rm \\gamma .iso}/\\eta $ , and keep $\\eta $ ($\\eta _{\\rm eff}$ for SBO) as the free parameter.","The parameters $p$ , $A^*$ , $\\epsilon _B$ and $\\epsilon _e$ are also free parameters.","With the inputs above, we carry out the detailed modelling using Markov chain Monte Carlo (MCMC) fitting using the Python package emcee [25].","We choose 150 walkers, 2000 steps.","Even though the analytical modelling suggests wind like medium, we still start with fits to a constant density medium.","The fit results in a high values of reduced-$\\chi ^2$ further ruling out the constant density model.","We fit the standard wind model with $k=2$ for all the cases.","In addition, we also account for non-standard wind density medium keeping $k$ as a free parameter.","Table REF shows the fit statistics for different parameters using the above mentioned models.","For $\\nu _a<\\nu _m<\\nu _c$ case, we do not list general $k$ model as this model performed quite poorly for both afterglow as well as shock breakout.","$\\nu _a<\\nu _m<\\nu _c$ generally performs very poorly, with shock breakout model performing slightly better than the isotropic afterglow model.","In addition, the parameters obtained in this model are rather unphysical.","While the fast cooling model gives best reduced $\\chi _{\\nu }^{2}$ , this case is unlikely to be true.","The analysis of mm and X-ray light published data have already revealed that $\\nu _c$ lies between the mm and X-ray frequencies.","Since $\\nu _c \\propto t^{1/2}$ in the wind model, uGMRT radio frequencies cannot be in the fast cooling regime.","The most viable model fits are obtained for $\\nu _m<\\nu _a<\\nu _c$ case.","This is quite viable since in wind density provide, $\\nu _m$ evolves faster than $\\nu _a$ and may reach $\\nu _m<\\nu _a$ regime at late epochs [35].","Here keeping $k$ as free parameter also results in $k \\sim 2$ .","Our model fits are equally good for the standard afterglow model and the shock breakout model, and uGMRT data alone cannot differentiate between the two.","In Fig.","REF , we show the light curve for a standard wind $k=2$ model for $\\nu _m<\\nu _a<\\nu _c$ case for both isotropic forward shock afterglow as well as the shock breakout afterglow model.","|l|ll|llll|llll| Best fit parameters for GRB 171205A uGMRT data 9 0pt Param.","2|c|$\\nu _a<\\nu _m<\\nu _c$ 4|c|$\\nu _m<\\nu _a<\\nu _c$ 4|c|$\\nu _a<\\nu _c<\\nu _m$ AG SBO 2|c|AG 2|c|SBO 2|c|AG 2|c|SBO $k=2$ $k=2$ $k=2$ general $k$ $k=2$ general $k$ $k=2$ general $k$ $k=2$ general $k$ $A_*$ $7.37^{+0.95}_{-0.80}$ $2.82^{+0.46}_{-0.39}$ $1.58^{+0.11}_{-0.75}$ $1.11^{+1.03}_{-0.54}$ $2.89^{+1.95}_{-1.26}$ $3.54^{+2.46}_{-1.55}$ $1.69^{+1.15}_{-0.51}$ $0.17^{+0.19}_{-0.09}$ $2.15^{+1.53}_{-0.78}$ $0.22^{+0.16}_{-0.11}$ $\\eta (\\eta _{\\rm eff}$ for SBO) $0.005^{+0.0004}_{-0.0004}$ $0.30^{+0.13}_{-0.12}$ $0.02^{+0.02}_{-0.01}$ $0.02^{+0.03}_{-0.02}$ $0.07^{+0.09}_{-0.05}$ $0.13^{+0.12}_{-0.07}$ $0.014^{+0.002}_{-0.002}$ $0.03^{+0.003}_{-0.003}$ $0.06^{+0.05}_{-0.03}$ $0.03^{+0.02}_{-0.01}$ $\\epsilon _B$ $ 0.94^{+0.05}_{-0.09}$ $0.70^{+0.20}_{-0.20}$ $0.21^{+0.24}_{-0.13}$ $0.11^{+0.17}_{-0.06}$ $0.03^{+0.03}_{-0.02}$ $0.01^{+0.01}_{-0.01}$ $0.17^{+0.14}_{-0.10}$ $0.02^{+0.01}_{-0.01}$ $0.08^{+0.09}_{-0.05}$ $0.01^{+0.01}_{-0.01}$ $\\epsilon _e$ $0.99^{+0.01}_{-0.02}$ $0.78^{+0.16}_{-0.23}$ $0.12^{+0.09}_{-0.06}$ $0.13^{+0.11}_{-0.07}$ $0.24^{+0.17}_{-0.11}$ $0.12^{+0.07}_{-0.06}$ $\\cdots $ $\\cdots $ $\\cdots $ $\\cdots $ $p$ $2.55^{+0.08}_{-0.06}$ $3.87^{+0.10}_{-0.19}$ $2.22^{+0.04}_{-0.04}$ $2.18^{+0.14}_{-0.09}$ $2.23^{+0.04}_{-0.05}$ $2.22^{+0.16}_{-0.13}$ $\\cdots $ $\\cdots $ $\\cdots $ $\\cdots $ $k$ $\\cdots $ $\\cdots $ $\\cdots $ $1.99^{+0.01}_{-0.02} $ $\\cdots $ $1.99^{+0.01}_{-0.01} $ $\\cdots $ $1.91^{+0.02}_{-0.01} $ $\\cdots $ $1.92^{+0.02}_{-0.02} $ $s$ $\\cdots $ $0.55^{+0.04}_{-0.04}$ $\\cdots $ $\\cdots $ $0.13^{+0.08}_{-0.07}$ $0.15^{+0.08}_{-0.08}$ $\\cdots $ $\\cdots $ $0.35^{+0.13}_{-0.16}$ $0.08^{+0.16}_{-0.18}$ $\\chi _{\\nu }^{2}= 7.55$ $\\chi _{\\nu }^{2}=2.58$ $\\chi _{\\nu }^{2}=1.74$ $\\chi _{\\nu }^{2}=1.80$ $\\chi _{\\nu }^{2}=1.72$ $\\chi _{\\nu }^{2}=1.78$ $\\chi _{\\nu }^{2}=1.60$ $\\chi _{\\nu }^{2}=1.52$ $\\chi _{\\nu }^{2}=1.58$ $\\chi _{\\nu }^{2}=1.57$ Here AG is the standard isotropic afterglow model and SBO is the shock breakout model.","In case of fast cooling, the data are only in the regime $\\nu _a$ to $\\nu _c$ ( 2 to 1/3 transition of spectra), which we don't have $p$ dependencies in temporal or spectral slopes.","The only $p$ dependency is in the expression of $\\nu _a$ via that ratio of $G(p)$ parameter which we have taken to be of order unity for $p\\sim 2.1$ .","The $\\epsilon _e$ dependency is also not there as $\\nu _m$ is unconstrained.","Figure: Upper left panel: Light curves of GRB171205A in radio regime using the slow cooling model (ν m <ν a <ν c \\nu _m<\\nu _a<\\nu _c) and standard wind (k=2k=2).","Upper right panel: Posterior Distributions of parameters for this model.","Lower left panel: Light curves of GRB171205A in radio regime using the slow cooling model (ν m <ν a <ν c \\nu _m<\\nu _a<\\nu _c) with standard wind (k=2k=2) and shock breakout scenario.","Lower right panel: Posterior Distributions of parameters for this model.In the left panels, the blue, red and green points are observed data in band 5 , band 4 and band 3 respectively which are included in the fit.","The points with arrow have only the upper limit of flux.","The lines are the best fits.","In the right panels, the 2D plots show the joint probability distribution of any two parameters.","The contours are at 0.5σ\\sigma , 1σ\\sigma , 2σ\\sigma , 3σ\\sigma .","The middle dotted lines in the 1D parameter distribution is the median value of posterior followed by 1σ\\sigma and 2σ\\sigma lines on both sides.","σ\\sigma is standard deviation of the corresponding distribution."],["Properties of GRB 171205A from radio modelling","The peak radio flux density of GRB 171205A at 1.3 GHz is $\\sim 10^{29}$ erg s$^{-1}$ Hz$^{-1}$ .","This is two orders of magnitude fainter than cosmological GRBs at this frequency (Fig.","REF ).","However, these values are comparable to other low-luminosity GRBs, e.g.","GRB 031203 [70], 980425 [43].","Figure: Plot of 1.4 GHz luminosities of canonical GRBs taken from .","Here we overlay the uGMRT 1.3 GHz measurementsfor GRB 171205A.","Our values are at least two orders of magnitude smaller than canonical GRBs.The value of $A_*$ in the standard afterglow and the SBO models are $1.58^{+0.11}_{-0.75}$ and $2.89^{+1.95}_{-1.26}$ , respectively, which, assuming a wind velocity of 1000 km s$^{-1}$ , translate to mass-loss rates of $1.58^{+0.11}_{-0.75}\\times 10^{-6} $ M$_\\odot \\,\\rm yr^{-1}$ and $2.89^{+1.95}_{-1.26}\\times 10^{-6} $ M$_\\odot \\,\\rm yr^{-1}$ , respectively, for the two models.","The nature of the surrounding ambient medium reflects on the progenitor nature of GRBs.","It is expected that the progenitors of long GRBs are massive stars (Wolf-Rayet) and in most of the cases a long GRB is associated with a supernova [43], [86].","Another evidence for massive star progenitors is that the long GRBs generally have star-forming host galaxies [87].","In such a case, one expects the association with a wind like circumburst medium.","However, several GRBs from massive stars collapse have shown homogeneous density [58], [59].","A constant density medium can be produced around a massive star if the wind faces a shock termination [19].","The low frequency observations present here provide a unique opportunity to determine the nature of the circumburst medium of GRB 171205A and establish that GRB 171205A exploded in a wind like environment.","At uGMRT frequencies, the optically thick to thin transition peak arises at a long time ($t > 100$ d) after the burst, indicating a relatively high density medium.","This may be created due to a large stellar mass-loss rate or a low wind velocity.","Some previous works [20] have shown that the large mass-loss rate for Wolf-Rayet stars are associated with large metallicity of the medium.","Thus GRBs in wind medium can be potential tools for studying metallicity variation at different redshifts.","The uGMRT light curve declines as $\\sim t^{-0.7}$ .","This indicates that there is no jet break until 3 years.","There are several explanations for the lack of jet breaks in some GRBs.","In the cases of GRB 980326 and GRB 980519, [35] have argued that a wind medium can dilute the jet break even for highly collimated bursts.","The jet break will be absent if the radio emission indeed arises from a quasi-spherical afterglow, such as that due to shock breakout [56] or cocoon [54].","In case of GRB 030329, [6] have argued that radio emission may be arising from two components, a narrow jet, surrounded by a wider component (e.g.","cocoon) and the radio emission is being dominated by the wider component.","However, the requirement of this model is that the contribution to the radio afterglow from the narrow jet may be negligible.","X-ray observations cover the period of $\\sim 200$ days and show no indication of a jet break at least until the last detection on day $\\sim 70$ .","Using $t_j >71$ d, gives a limit $\\theta _j > 1.2$ radians for the AG model and $\\theta _j > 1.9$ for SBO model [57], [84].","[49] have shown that ejecta kinetic energy profiles in stripped enveloped supernovae vary based on different explosion mechanisms.","While stripped-envelope supernovae have a steep dependence $E_K \\propto (\\Gamma \\beta )^{-5.2}$ indicating no central engine activity, relativistic supernovae, sub-energetic GRBs with SBO mechanisms are flatter with $E_K \\propto (\\Gamma \\beta )^{-2.4}$ showing weak activity from central engine.","Canonical GRBs, on the other hand, follow $E_K \\propto (\\Gamma \\beta )^{-0.4}$ typical of jet-driven explosions with long-lasting central engines.","Our radio modeling and the relativistic treatment of [4], results in $E_K \\approx 1.1 \\times 10^{51}$ ergs and $\\Gamma \\beta \\sim 1 $ .","For the non-relativistic supernova component, we use values from [39], i.e.","supernova kinetic energy $E_K=2.4 \\times 10^{52}$ ergs and ejecta velocity 55000 km s$^{-1}$ .","Using these values, $E_K \\sim (\\Gamma \\beta )^{-1.9}$ .","While GRB 171205A is a sub-energetic GRB, it follows the energy-velocity profile somewhere between canonical GRBs and the SBOs.","These arguments suggest that the jet and shock-breakout both may play an important role in the late time afterglow emission in GRB 171205A.","Since we have radio light curve peaks at two uGMRT frequencies, we could also estimate some more parameter evolutions.","The relativistic energy under equipartition assumption at the two epochs are $E_{\\rm Eq}=3.6\\times 10^{48}$ ergs and $4.9\\times 10^{48}$ ergs [4].","Thus there is an indication of enhancement of energy $E \\propto t^{0.48}$ .","This along with flatter light curve decays may also be explained if there is an energy injection from the central engine to the shock.","For an injection luminosity $L(t)=L_0(t/t_0)^{-q}$ , $E \\propto t^{1-q}$ .","This implies $q=0.52$ .","However, our best fits result in much larger value of $q$ ($s=0.13$ , $q=1-s\\equiv 0.87$ .","This may imply that if energy injection, it is not continuous and probably lasted for a small amount of time.","The equipartition size obtained from the above formulation [4] at the epochs of two peaks in band 5 and band 4 follow R(t) $\\propto $ $t^{0.49}$ .","We note that for ISM and Wind, density profiles, $R$ follows as $R\\propto t^{1/4}$ and $R\\propto t^{1/2}$ , so it also points towards a wind like medium surrounding GRB171205A.","Our uGMRT observations cover the period of around 1000 days.","However, our data does not suggest the GRB to be in the Newtonian regime yet (Fig.","REF ).","This is not uncommon for low-luminosity GRBs [37].","We note that the value of $\\Gamma \\beta $ indicates mildly relativistic outflow.","Hence it is likely that the GRB is making a transition into Newtonian regime soon."],["Shallow decay of radio afterglow","We note that the decay of the radio afterglow is much shallower than that of the X-ray afterglow.","The shallowness of radio lightcurves was first pointed out by [60].","For a reasonable afterglow parameters, they estimated that the afterglow is supposed to cross $\\nu _m$ at around 10 days for 10 GHz and follow a decay slope of $(3p-1)/4$ for a wind medium, which was not the case for some GRBs, e.g.","GRB 991216 and GRB 000926.","They explored the difference between the radio and the optical decay indices could be caused by the fact that the injection frequency remains above the radio domain ($\\sim $ 10 GHz), or a different population of electrons, or a variability of microparameters.","Finally, they concluded that a long-lived reverse shock in the radio regime could cause this flattening.","However, this is unlikely in GRB 171205A as a strong persistent reverse shock requires a low wind density [65].","Here the late rising of the radio afterglow suggests a comparatively high density medium which is against the previous statement.","So, this scenario can be excluded.","[40] noted that radio afterglows of some GRBs deviate at late times and low frequencies from the standard model, and attempted to explain it with the two-component jet model, a narrow jet core and a wider cocoon surrounding the jet.","[34] have explained this flattening due to counter-jet which becomes visible when turning sub-relativistic.","While such two components should result in a bump, for stratified wind medium, the revelation of counter-jet is more gradual, causing mild flattening.","An energy injection event or a different component dominating the radio emission can also produce this flattening.","In case of slightly off-axis jet, the early radio emission could possibly be from the cocoon, the accelerated polar ejecta and at a late phase the contribution from the off-axis jet coming to the line of sight can increase the total radio flux [21].","The lack of X-ray data at such late times prevents us from directly distinguishing between these scenarios.","A population of quasi-thermal electrons has also been argued as one of the reasons [85], which would mainly dominate at radio frequencies, as it would result in increased $\\nu _a$ and suppressing radio emission below this.","In GRB 171205A, this has been tentatively supported from the polarization measurements.","As per [82], the mm data revealed 0.27% level linear-polarisation which is a factor of 4 smaller than optical polarization measurements.","This has been explained as Faraday depolarization by non-accelerated, cool electrons in the shocked region.","However, one cannot rely on these results due to dispute of the detection claimed by [45]."],["Origin of radio emission","There have been suggestions that sub-energetic bursts are simply canonical GRBs viewed off-axis [53].","However, such bursts will have two distinguishing characters, a) low $E_p$ b) a rise in the afterglow energy while the shocked ejecta gradually comes into our line of sight.","In GRB 171205A, the afterglow energy increases slightly from $3.6\\times 10^{48}$ erg to $4.9\\times 10^{48}$ erg between $\\sim 100$ and $\\sim 200$ days.","However, the $E_p$ is comparable to that of canonical GRBs.","Additionally, off-axis jet is a geometric effect, which would result in a frequency independent break in the light curve, which has not been seen for GRB 171205A, ruling out the off-axis model [24].","Though a jet somewhat off-axis is not ruled out.","[24] found that GRB 171205A is an outlier of the Amati relation, as are some other low redshift GRBs, and its emission mechanism should be different from that of canonical, more distant GRBs.","There are two models to explain the electromagnetic emission in low-luminosity GRBs, central engine driven [49], [37] and shock-breakout driven [43], [56], [4], [75].","An issue with a purely shock breakout model is the requirement of high $\\gamma $ -ray efficiency, for a quasi-spherical outflow.","Another issue is generation of such relativistic quasi-spherical outflow.","One can envisage a situation where some fraction of the supernova ejecta is accelerated to relativistic speeds to provide this quasi-spherical relativistic outflow.","However, [79] have shown that only a fraction ($\\sim 10^{-4}$ ) of supernova energy goes in relativistic ejecta.","[54] has suggested an alternative scenario where a choked jet in a low-mass envelope can put a significant energy into a quasi-spherical, relativistic flow.","Our uGMRT model fits are incapable of distinguishing between canonical afterglow versus SBO afterglow models.","We check the applicability of this model for GRB 171205A.","In this model, the expanding outflow is considered to harbour a series of successive shocks, which accelerate and catch up to the boundary and hence explain the increasing afterglow energy via continuous injection.","In this model, eventually the total energy should reach around $2.4 \\times 10^{52}$ erg [39], the kinetic energy of the associated supernova.","Generally one does not see such a large amount of energy as radio observations do not cover epochs late enough.","However, our radio observations cover a period of nearly 1000 days.","The SBO model fit gives $E_K\\approx 3.4\\times 10^{50}$ erg, two orders of magnitude smaller than the one predicted in pure SBO model.","[77] carried out multiwavelength hydrodynamical modelling of GRB 171205A in the framework of post shock break-out relativistic SN ejecta-CBM interaction scenario.","While they claimed that this model worked well for GRB 171205A and favour the wind model, we note some problems with their model.","They had to introduce a centrally concentrated CBM with a sudden density drop to explain the available radio and X-ray data.","however, our work includes radio measurements upto 3 years and do not show a sudden density drop.","The pure shock breakout model also has some other problems.","A shock breakout model predicts a $\\gamma $ -ray emission lasting $\\ge 1000 s$ , lower $E_p$ (not exceeding 50 keV), a large absorption column density, a late time soft X-ray emission, and comparable energy in the X-ray emission and the prompt $\\gamma $ -ray flare.","We note that the X-ray spectrum shows an intrinsic hydrogen column density $N_H=7.4^{+4.1}_{-3.6} \\times 10^{20}$ cm$^{-2}$ [24].","This intrinsic column density is at the low-end of low-luminosity GRB distribution, even among low redshift Swift-XRT GRBs where the mean is $N_H=2.4 \\times 10^{21}$ cm$^{-2}$ at $z < 0.2$ [2].","For the observations post day 1 onwards, there may be indication of a slightly higher column density $N_H=1.2^{+0.8}_{-1.7} \\times 10^{21}$ cm$^{-2}$ https://www.swift.ac.uk/xrt_live_cat/00794972/.","But there is no particular indication of significant spectral softening, except of a slight indication of $\\Gamma =1.63\\pm 0.30$ to $\\Gamma =1.94\\pm 0.23$ .","The total X-ray energy from the first observation onwards until the last detection is at least an order of magnitude smaller than the prompt energy.","[24] have claimed the presence of a thermal component.","While shock breakout from supernovae is the most favourable model for a thermal component [56], [75], late time photospheric emission from jet [27], or thermal emission from cocoon [78] can also explain this component.","If we assume the blackbody component to be significant, it comprises of 20% flux and has a temperature of 89 eV [24].","This corresponds to a radius $R =(E_{\\rm iso} /aT_{BB}^4)^{1/3} \\approx 1.4 \\times 10^{13}$ cm ($a$ is the radiation density constant).","This is much larger than the typical Wolf-Rayet star radius, but can be explained if the shock expands in a non-spherical manner.","Alternatively, [54] suggested the presence of an optically thick stellar envelope further away from the star, from where the breakout happens.","We have seen that our observations are inconsistent with a pure shock breakout due to the short duration, higher $E_p$ , shallow $E$ vs $\\Gamma \\beta $ relation, low column density and a much larger break-out radius predicted by the thermal component.","It can also not be explained as merely a canonical GRB seen off-axis.","We show below that both these components contribute towards the radio afterglow.","GRB 171205A is the first GRB in which direct signatures of a cocoon has been seen [39].","This is rare because generally a line-of-sight jet is much brighter than the associated cocoon, hiding the cocoon signatures.","An off-axis jet can reveal itself by the associated supernova, but the cocoon signatures are long gone by the time supernova may be discovered.","Ideally cocoon signatures are visible only in slightly off-axis GRBs.","In GRB 171205A, the cocoon was identified by the broad absorption features overlapped on the supernova spectrum [39] They also estimated, from the energy deposited in cocoon, that the jet was quite energetic [39] .","This may imply that we may be seeing a slightly off-axis jet, which enabled us to reveal the cocoon, not overshadowed by the bright jet.","With this knowledge, it is likely that radio afterglow has from both components, the sub-relativistic wider cocoon and a slightly off-axis jet.","The cocoon radio emission dominates the GRB emission at early times when the GRB jet is off-axis.","Later the additional flux contribution comes when the jet spreads sideways and comes in our line of sight.","In such a case, the total radio flux can also be large compared to on-axis GRBs since the cocoon and the jet carry comparable energy [21].","[39] could not distinguish between the cocoon along with slightly off-axis jet versus only cocoon emission, where the faint $\\gamma $ -rays are the predicted signal of the cocoon breaking out of the stellar envelope.","However, our radio modelling rules out the pure shock break-out model in favour of cocoon along with a slightly off-axis jet.","As per theoretical models, the indicated speed of the ejecta is also consistent with the sub-relativistic speeds expected in this model."],["Comparison with other low-$z$ low-luminosity GRBs","With $E_{\\rm iso}=2.3\\times 10^{49}$ erg and $z=0.0368$ , GRB 171205A is one of the few low-$z$ ($z\\lesssim 0.1$ ), low-luminosity GRBs.","Other GRBs in this category are GRB 980425, [31], GRB 031203, [67], GRB 060218 [9] and GRB 100316D [74].","Like other low-luminosity GRBs, the spectrum of GRB 171205A can be fit by a simple power-law model, however, the photon index $\\Gamma \\sim 1.94$ is harder than these GRBs, except that of GRB 031203 [67].","Even amongst the intrinsically sub-energetic bursts, GRB 171205A resembles more closely with GRBs 980425 and 031203 and not GRBs 060218 and 100316D.","GRBs 980425 and 031203 were difficult to realise in the shock break out models due to their relatively hard spectra, shorter durations and larger $E_p$ , though lack of thermal equilibrium may accommodate it [41].","GRBs 060218 and 100316D stand out due to their large durations of a few thousand seconds and lower peak energies $E_p < 50$ keV.","The shorter $T_{90}$ and higher $E_{\\rm p}$ for GRB 171205A are are comparable to the respective values for GRBs 980425 and 031203.","A major difference between GRB 171205A and other low-luminosity GRBs is the absence of intrinsic absorption column density.","All other GRBs in this class show significant neutral hydrogen column density $N_H \\sim (6-7)\\times 10^{21}$ cm$^{-2}$ , as opposed to an order of magnitude lower $N_H$ for GRB 171205A [24].","The higher column density is considered an essential feature for supernova shock break out models in low-luminosity GRBs.","The peak radio flux density of GRB 171205A at 1.3 GHz is $\\sim 10^{29}$ erg s$^{-1}$ Hz$^{-1}$ .","This value is comparable to other low-luminosity GRBs, e.g.","GRB 031203 [70].","However, the X-ray luminosity at 10 hr is $L_x\\approx 2\\times 10^{42}$ erg s$^{-1}$ , which is 4 times smaller than that of GRB 031203 [70].","This discrepancy is even more significant for a wind-like medium, since the measured peak radio luminosities are at 1.3 GHz and 8.5 GHz bands for GRB 171205A and GRB 031203, respectively.","It has been found that the X-ray afterglow decays slowly in low-luminosity GRBs.","For example, the X-ray emission in GRB 031203 followed $F_{\\rm xray}(t) \\propto \\nu ^{-0.8} t^{-0.4}$ [70], and for GRB 980425 $F_{\\rm xray}(t) \\propto \\nu ^{-1} t^{-0.2}$ [43].","This is opposed to the canonical GRBs with $F_{\\rm xray}(t) \\propto \\nu ^{-1.3} t^{-1}$ .","[5] have shown that a flat temporal decay of the X-ray light curve can be explained in the shock-break out model, although dominance of an underlying supernova has also been suggested as a possible reason for this flatness [70].","With $F_{\\rm xray}(t) \\propto \\nu ^{-0.94} t^{-1.1}$ , GRB 171205A is more like canonical GRBs.","On the contrary, the X-ray spectral index was very soft for GRB 060218A with $F_{\\rm xray}(t) \\propto \\nu ^{-2.2} t^{-1.1}$ , which softened even further at later epochs [71].","GRB 171205A did not so that much spectral softening.","In addition, contrary to smooth light curves of low-luminosity GRBs, the X-ray light curve of GRB 171205A revealed three temporal breaks [24].","The lack of jet break seems to be a common feature of low-luminosity GRBs.","This may either indicate much wider angle ejecta responsible for afterglow emission, or stratified medium which can dilute the jet break in GRBs and hide its signatures.","Another defining feature of these GRBs is the presence of a thermal component, which has also been seen in GRB 171205A [24].","Though this feature is quite common in GRBs associated with SNe [9], [74], their origin is still a matter of debate.","Shock breakout from the relativistic supernova shell is the most favourable model [75], however, late time photospheric emission from a jet [27], or a thermal emission from a cocoon [78] can also explain this component.","However, it appears that the presence of a thermal component is not a unique feature of low-luminosity GRBs.","[73], [72] analysed a sample of canonical GRBs and found the evidence of thermal component in a significant number of GRBs, though large radii associated with the blackbody emission argue against the shock breakout model.","Finally while low-luminosity GRBs are considered to be a different class, [54] has provided a unified picture of low-luminosity GRBs and cosmological GRBs, in terms of a key difference, namely, the existence of an extended low-mass envelope.","In the unified model, the envelope is present in the low-luminosity GRBs, but absent in cosmological GRBs.","The lack of envelope allows a jet to launch without any resistance in cosmological GRBs, but the extended envelope in low-luminosity GRBs smothers the jet and deposits a large amount of energy in the stellar envelope, driving a mildly relativistic shock producing low-luminosity GRBs via shock breakout.","Our model does not support this picture."],["Summary and Conclusions","In this paper, we have presented the sub-GHz observations of a low-luminosity GRB 171205A upto around 1000 days.","These are the best sampled low frequency light curves of any GRB.","For the first time we report a lowest frequency (250–500 MHz) detection of a GRB.","Our light curves cover a period of two years.","While we are able to see the light curve peak transitions in bands 5 and 4, we missed the peak in band 3 due to the lack of early data.","The radio data suggest that the GRB exploded in a wind medium.","At sub-GHz frequencies at late time, the afterglow is in $\\nu _m<\\nu _a<\\nu _c$ regime, a common phenomenon seen in late time radio afterglows.","The late time Chandra X-ray measurements constrain the jet break to be $t_{\\rm jet}>71$ days.","Even though GRB 171205A has significant similarities with other low-luminosity GRBs, it deviates from this class in many respects.","We suggest that the radio emission arises from both a cocoon and a jet, where the jet is slightly off-axis [39].","The early epoch radio emission is dominated by the cocoon surrounding the jet, while the late time radio emission has contribution from the jet.","The flatter radio light curves, harder GRB X-ray spectrum, large $E_p$ , shorter $T_{90}$ , kinetic energy - ejecta velocity relation and time dependence of various parameters are consistent with this picture.","Our work emphasises the importance of the nature of CBM, which is critical in understanding the evolution of GRB afterglows, and are best revealed by low-frequency radio measurements.","We thank the referee for very insightful comments which helped improve the manuscript significantly.","We thank Dipankar Bhattacharya for carefully reading the manuscript and providing critical suggestions.","B.M.","and P.C.","acknowledge support of the Department of Atomic Energy, Government of India, under the project no.","12-R&D-TFR-5.02-0700.","P.C.","acknowledges support from the Department of Science and Technology via SwaranaJayanti Fellowship award (file no.DST/SJF/PSA-01/2014-15).","We thank the staff of the GMRT that made these observations possible.","GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research."]],"string":"[\n [\n \"1000 days of lowest frequency emission from the low-luminosity GRB\\n 171205A\"\n ],\n [\n \"Abstract We report the lowest frequency measurements of gamma-ray burst (GRB) 171205A with the upgraded Giant Metrewave Radio Telescope (uGMRT) covering a frequency range from 250--1450 MHz and a period of $4-937$ days.\",\n \"It is the first GRB afterglow detected at 250--500 MHz frequency range and the second brightest GRB detected with the uGMRT.\",\n \"Even though the GRB is observed for nearly 1000 days, there is no evidence of transition to non-relativistic regime.\",\n \"We also analyse the archival ${\\\\it Chandra}$ X-ray data on day $\\\\sim 70$ and day $\\\\sim 200$.\",\n \"We also find no evidence of a jet break from the analysis of combined data.\",\n \"We fit synchrotron afterglow emission arising from a relativistic, isotropic, self-similar deceleration as well as from a shock-breakout of wide-angle cocoon.\",\n \"Our data also allow us to discern the nature and the density of the circumburst medium.\",\n \"We find that the density profile deviates from a standard constant density medium and suggests that the GRB exploded in a stratified wind like medium.\",\n \"Our analysis shows that the lowest frequency measurements covering the absorbed part of the light curves are critical to unravel the GRB environment.\",\n \"Our data combined with other published measurements indicate that the radio afterglow has contribution from two components: a weak, possibly slightly off-axis jet and a surrounding wider cocoon, consistent with the results of Izzo et al.\",\n \"(2019).\",\n \"The cocoon emission likely dominates at early epochs, whereas the jet starts to dominate at later epochs, resulting in flatter radio lightcurves.\"\n ],\n [\n \"Gamma Ray Bursts (GRBs) are the most energetic flashes of gamma rays, with $T_{90}$ duration (time interval between which 5% to 95% of fluence is collected by the detector) ranging between a few milli-seconds to thousands of seconds [87].\",\n \"The GRBs can be classified in two broad classes i.e short/hard GRBs with duration less than 2 secs, and long/soft GRBs of duration greater than 2 secs [42].\",\n \"According to well accepted theories, most of the long soft GRBs originate from gravitational collapse of massive stars (collapsar model); and short hard GRBs result from explosive binary compact object mergers [86].\",\n \"The GRBs from both channels of formation power relativistic collimated jets which give Doppler-boosted high luminosities in gamma-rays.\",\n \"GRBs are cosmological events with an average redshift $z\\\\approx 2.2$ [29], and have isotropic-equivalent gamma ray luminosities ($L_{\\\\rm iso}$ ) of the order of $10^{51}$ to $10^{53}$ erg s$^{-1}$ .\",\n \"However, a handful of long/soft GRBs with spectroscopically identified supernovae have been discovered with luminosities that are 3–5 orders of magnitude lower than the average, i.e.\",\n \"[69], [10].\",\n \"Their low luminosities allow them to be detected only at low redshifts, though they may be 10–100 times more abundant than regular GRBs [68].\",\n \"The prompt light-curves of typical low-luminosity GRBs are smooth and spectra have a single peak with the peak energy generally below $\\\\sim $ 50 keV, which softens further with time [56], [10].\",\n \"The radio afterglow of these GRBs tend to indicate similar energy content in mildly relativistic ejecta [43].\",\n \"Many of these GRBs are associated with broad-line Type Ic supernovae.\",\n \"The list of such GRBs-supernovae include some of the well studied cases such as, GRB 980425/SN 1998bw [31], GRB 030329/SN 2003dh [36], GRB 031203/SN 2003lw [48], GRB 060218/SN 2006aj [9], GRB 100316D/SN 2010bh [74], GRB 111209A/SN 2011kl [32], GRB 120422A/SN 2012bz [50], GRB 130427A/SN 2013cq [51], GRB 130702A/SN 2013dx [12], GRB 161219B/SN 2016jca [11], GRB 171010A/SN 2017htp [52], GRB 190829A/ SN 2019oyu [80], although only the five are nearby, $z\\\\lesssim 0.1$ [10].\",\n \"A significant amount of work has gone towards understanding whether the low-luminosity GRBs are simply the low-energy counterparts of the cosmological GRBs, or have a different emission mechanism.\",\n \"Many low-luminosity GRBs do not follow the $E_{\\\\rm iso}-E_p$ Amati relation [1], indicating that their emission mechanism should be different from that of canonical, more distant GRBs.\",\n \"Technically an off-axis jet can also explain low-luminosity emission from GRB, but predicts an achromatic steepening of the light curve, absent in many low-luminosity GRBs.\",\n \"There have been suggestions that in contrast to the emission from an ultra-relativistic jet driven by a central engine, these low-luminosity GRBs are powered by shock breakouts [43], [56], [54], [5], [76].\",\n \"In some cases observations have suggested a mildly relativistic blast wave being responsible for producing the radio afterglow.\",\n \"This has supported the relativistic shock breakout model, e.g.\",\n \"GRB 980425 [43].\",\n \"The shock-break out model got further support in case of GRB 060218, in which a thermal component was also seen which cooled and shifted to optical/UV band with time.\",\n \"This was interpreted to be arising from the break out of a shock driven by a mildly relativistic supernova shell in the progenitor wind [9], although late time photospheric emission from a jet [27], or thermal emission from a cocoon [78] can also explain it.\",\n \"[8] have investigated whether the low-luminosity GRBs launch relativistic jets like their high energy counterparts, but incur resistance by the stellar envelopes surrounding their progenitor stars.\",\n \"They found that some low-luminosity GRBs have much shorter durations compared to the jet breakout time, This is inconsistent with the collapsar model which is largely successful in explaining the cosmological GRBs [87].\",\n \"While [5] and [56] have developed spherical relativistic shock break out models in context of low-luminosity GRBs, [54] addressed some of the problems of spherical shock-break out model by introducing a low mass optically thick stellar envelope surrounding the progenitor star.\",\n \"In this model, the explosion powering the low-luminosity GRBs was not the spherical breakout of the supernova shock, but by a jet that gets choked in the envelope and powers quasi-spherical explosion.\",\n \"To extend this idea further, [55] considered a cocoon breakout model.\",\n \"In this model, as the GRB jet pushes through the stellar material, it heats the surrounding gas and produces a high-pressure sub-relativistic cocoon, which at the time of breakout, produces a relatively faint flare of $\\\\gamma $ -rays.\",\n \"This break out will not be as spherical as a supernova break-out, but will be wider than a jet.\",\n \"In this model, interaction of the cocoon with the surrounding medium can give rise to a late time radio and X-ray afterglow.\",\n \"However, [37] have provided an alternative mechanism, where the composite emission of GRB 060218/SN 2006aj could be explained by a weak jet, along with a quasi-spherical supernova ejecta.\",\n \"llGBRs also have a radio afterglow, which indicates a comparable energy in mildly relativistic ejecta (Kulkarni et al.\",\n \"1998; Soderberg et al.\",\n \"2004, 2006; Margutti et al.\",\n \"2013).\",\n \"The lack of bright, late-time, radio emission from ll-GRBs strongly constrain the total energy of any relativistic outflow involved in these events (Waxman 2004; Soderberg et al.\",\n \"2004, 2006b).\",\n \"Additionally, statistical arguments rule out the possibility that ll-GRBs are regular LGRBs viewed at a large angle (e.g., Daigne & Mochkovitch 2007).\",\n \"Thus, if ll-GRBs are generated by relativistic jets these jets must be weak and have a large opening angle.\",\n \"The bursts with $L_\\\\gamma > 2 \\\\times 10^{48}$ erg s$^{-1}$ are considered as regular GRBs and are separated into LGRBs and SGRBs according to the standard criterion of whether T90 in the observer frame is above or below 2 s (Bromberg, Nakar, Piran 2011).\",\n \"GRB 171205A is a nearby low-luminosity GRB with a $T_{90}$ duration of almost 189.4 secs [24].\",\n \"It was first discovered by the Burst Alert Telescope (BAT) onboard Swift on 5th December, 2017 [23].\",\n \"It has a redshift of 0.0368 [23], [38].\",\n \"The isotropic energy release in gamma ray band at GRB rest frame was $2.18_{-0.50}^{+0.63}\\\\times 10^{49}$ ergs [24].\",\n \"The host of this event was a bright spiral galaxy named 2MASX J11093966-1235116 [38], with a mass of the order of $10^{10}M_{\\\\odot }$ and a star formation rate of $3\\\\pm 1M_{\\\\odot }/\\\\rm {yr}$ [63].\",\n \"An emergent supernova event (SN 2017iuk) was seen three days after the burst [24].\",\n \"There are a handful of observations for GRB 171205A from mm to radio bands.\",\n \"It was detected by [22] using Northern Extended Millimeter Array (NOEMA) with a flux density of $\\\\sim 35$ mJy at 150 GHz after 20.2 hours of the burst.\",\n \"Atacama Large Millimeter/Submillimeter Array (ALMA) detected a bright afterglow of significance more than 100$\\\\sigma $ at 92 GHz and 340 GHz on 10-11 December, 2017 [62].\",\n \"RATAN-600 radio telescope detected it at 4.7 and 8.2 GHz bands during 9th December to 16th December, 2017 [81].\",\n \"Karl G. Jansky Very Large Array (VLA) also observed the afterglow in the frequency range 4.5 to 16.5 GHz [44].\",\n \"This GRB also had a Very Long Baseline Array (VLBA) detection at different frequencies [61].\",\n \"These observations showed a steeply rising spectrum($\\\\propto \\\\nu ^{2}$ ) at low frequency which indicates a synchrotron self-absorbed spectrum.\",\n \"This is the first GRB for which [82] claimed to have detected polarization at the ALMA 90GHz frequency, though this claim has been disputed by [45].\",\n \"The upgraded Giant Metrewave Radio Telescope (uGMRT) first detected it on 20th December, 2017 at 1400 MHz [15] after a non detection on 10th and 11th December, 2017 [16].\",\n \"The observed flux density at that epoch was $782\\\\pm 57 \\\\rm {\\\\mu Jy}$ .\",\n \"Radio afterglow emission from GRBs evolve slowly which gives us the opportunity to observe it for a long time and obtain the distribution of the kinetic energy in the velocity space.\",\n \"Since this distribution is different for various models, especially central engine driven versus shock break out, radio observations provide unique opportunity to distinguish between various emission models [43].\",\n \"Additionally, the early radio emission is likely to be absorbed via synchrotron self-absorption (SSA), and thus constrain the circumburst medium (CBM) density [17].\",\n \"The late time radio observations in the Newtonian limit, when the jet becomes sub-relativistic, are nearly independent of jet geometry and measure the kinetic energy of the afterglow accurately [26].\",\n \"In this paper we present low frequency observations of GRB 171205A taken with the uGMRT for around 1000 days.\",\n \"We summarize our observations and data analysis in §.\",\n \"We discuss our model in §REF and results in §REF .\",\n \"In §, we discuss the properties of GRB 171205A in conjunction with published measurements at higher frequencies and present our main conclusions.\",\n \"Unless otherwise stated, we assume a cosmology with $H_0=67.3$ km s$^{-1}$ Mpc$^{-1}$ , $\\\\Omega _m=0.315$ , $\\\\Omega _\\\\Lambda =0.685$ [64].\"\n ],\n [\n \"The GMRT observed GRB 171205A starting 2017 December 10 and continued observing until 2020 June 26.\",\n \"The observations were taken in band 5 (1000–1450 MHz), band 4 (550–900 MHz) and band 3 (250–500 MHz).\",\n \"The bandwidth for band 4 and band 5 was 400 MHz while for band 3 it was 200 MHz.\",\n \"The duration of each observation was around 2-3 hours including overheads (on source time 1.5 hours).\",\n \"We observed flux density calibrators 3C286 and 3C48; and a phase calibrator J1130-148.\",\n \"Flux calibrators were also used as bandpass calibrators.\",\n \"We use a package Common Astronomy Software Applications( CASA) for data analysis.\",\n \"The data were analyzed in three major steps, i.e flagging, calibration and imaging.\",\n \"The CASA task `flagdata' was used to remove dead antennas and bad data.\",\n \"In addition, the tasks `tfcrop'(http://www.aoc.nrao.edu/~rurvashi/TFCrop/TFCropV1/node2.html) and `rflag'(https://casa.nrao.edu/Release4.2.2/docs/userman/UserMansu167.html) were used to flag the radio frequency interference (RFI).\",\n \"The calibration [30] was performed to remove the instrumental and atmospheric effects from the measurement.\",\n \"The final part of processing was imaging.\",\n \"The continuum imaging of the target source was done using CASA task `tclean'.\",\n \"Finally, a few rounds of `phase only' mode and two rounds of `amplitude-phase' self-calibrations were run.\",\n \"We fit a Gaussian to determine the GRB flux density at the GRB position.\",\n \"The flux densities are shown in Table .\",\n \"Sample radio images of GRB171205A at bands 5, 4 and 3 are shown in Figure REF .\",\n \"The errors in flux densities in Table show only the statistical errors.\",\n \"We also add 15% of flux densities in quadrature to account the uncertainties due to calibration and other systematics for GMRT bands during our model fit.\",\n \"We closely follow the procedure shown in [14].\",\n \"The table list the details of observations and flux densities at various epochs.\",\n \"Figure: GMRT images of GRB171205A.\",\n \"Panel a) Band 5 detection on 2019 May 13.\",\n \"Panel b) Band 4 detection on 2019September 10.Panel c) Band 3 detection on 2019 May 13.\",\n \"The contours in black lines show detection significance and are at 20σ\\\\sigma , 40σ\\\\sigma and 60σ\\\\sigma for bands 4 and 5; and at 6σ\\\\sigma , 8σ\\\\sigma , and 10σ\\\\sigma for band 3, where σ\\\\sigma is the map rms of corresponding images.\",\n \"For the images displayed, they are following: Band 5, 17 μ\\\\mu Jy, Band 4, 30 μ\\\\mu Jy and Band 5, 80 μ\\\\mu Jy.We also extract Swift-XRT 0.3–10 keV flux light curve from the the Swift online repositoryhttps://www.swift.ac.uk/xrt_products/00794972.\",\n \"The light curve post day 1 indicate a photon index $\\\\Gamma =1.94^{+0.23}_{-0.22}$ and column density $N_H=(1.2^{+0.8}_{-0.7} \\\\times 10^{21}$ cm$^{-2}$https://www.swift.ac.uk/xrt_live_cat/00794972/.\",\n \"This is in addition to Galactic column density of $5.89\\\\times 10^{20}$ cm$^{-2}$ .\",\n \"We converted X-ray flux into into 1 keV spectral flux density using this photon index.\",\n \"Swift data covered observations until 2020 May 27.\",\n \"In addition, we analysed two archival data from Chandra ACIS-S on 2018 Feb 14 and 2018 June 29 (PI: Margutti).\",\n \"We used Chandra Interactive Analysis of Observations software [28] task specextractor to extract the spectra, response and ancillary matrices.\",\n \"We used CIAO version 4.6 along with CALDB version 4.5.9.\",\n \"The HEAsofthttp://heasarc.gsfc.nasa.gov/docs/software/lheasoft/ package Xspec version 12.1 [3] was used to carry out the analysis of the Chandra spectra.\",\n \"The GRB was detected in the first obserations at 71 days with 0.3–10 keV unabsorbed flux of $(11.25\\\\pm 3.38)\\\\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ .\",\n \"The second Chandra observation on second epoch, i.e 206 days, resulted in a 3-$\\\\sigma $ upper limit $< 3.98\\\\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ .\",\n \"lcccrc 1 Flux Densities of GRB171205A 0pt Date of Observation Band Frequency Days since explosion Flux Densitya Map RMS MHz (mJy) $\\\\mu $ Jy/beam 2017 Dec 10.10 5 1255 4.79 $<0.07$ 24 2017 Dec 11.02 4 648 5.71 $<$ 0.06 20 2017 Dec 19.91 5 1265 14.60 $0.64\\\\pm 0.05$ 17 2017 Dec 26.90 5 1265 21.59 1.00$\\\\pm $ 0.07 17 2017 Dec 28.91 4 607 23.60 $<$ 0.39 130 2018 Jan 16.94 5 1265 42.63 1.75$\\\\pm $ 0.05 17 2018 Feb 12.85 5 1370 68.54 3.04$\\\\pm $ 0.09 41 2018 Feb 17.85 4 607 73.54 1.37$\\\\pm $ 0.15 115 2018 Mar 20.68 5 1352 105.37 5.79$\\\\pm $ 0.08 34 2018 Jun 08.44 3 402 185.13 $2.94\\\\pm 0.41$ 106 2018 Jun 10.68 5 1255 187.37 3.07$\\\\pm $ 0.11 26 2018 Jun 11.42 4 745 188.11 $2.46\\\\pm 0.49$ 141 2018 Jul 13.47 5 1250 220.16 3.55$\\\\pm $ 0.12 17 2018 Jul 15.35 4 610 222.043.13$\\\\pm $ 0.18 66 2018 Jul 23.35 3 402 230.04 $2.30\\\\pm 0.22$ 64 2018 Jul 26.57 5 1265 233.26 $2.60\\\\pm 0.08$ 26 2018 Jul 28.35 4 750 235 2.92$\\\\pm $ 0.31 62 2018 Aug 24.32 5 1265 262.01 $3.32\\\\pm 0.06$ 33 2018 Aug 25.22 4 607 262.91 $1.06\\\\pm 0.16$ 81 2018 Sep 21.35 5 1265 290.04 $3.03\\\\pm 0.10$ 59 2018 Sep 23.16 3 402 291.85 $2.01\\\\pm 0.15$ 63 2018 Oct 20.28 5 1250 319 3.18 $\\\\pm $ 0.13 52 2018 Oct 26.09 3 400 324.78 1.77 $ \\\\pm $ 0.27 131 2018 Oct 26.33 4 607 325.02 $2.61\\\\pm 0.26$ 143 2018 Dec 21.88 3 402 381.57 $1.61\\\\pm 0.31$ 49 2018 Dec 22.02 5 1255381.71 $2.83\\\\pm 0.06$ 21 2018 Dec 22.14 4 610 381.83 $1.52\\\\pm 0.17$ 52 2019 Feb 26.98 4 610 448.67 1.52$\\\\pm $ 0.21 79 2019 Feb 26.70 3 402 448.39 1.05$\\\\pm $ 0.16 40 2019 Feb 26.87 5 1250 448.56 1.87$\\\\pm $ 0.06 17 2019 May 13.49 5 1250 524.18 1.74$\\\\pm $ 0.04 17 2019 May 13.62 3 402 524.31 1.43 $\\\\pm $ 0.26 79 2019 May 13.77 4 607 524.46 2.09 $\\\\pm $ 0.18 81 2019 Sep 10.16 5 1250 643.85 1.74 $\\\\pm $ 0.03 19 2019 Sep 10.31 3 402 644.00 1.67 $\\\\pm $ 0.21 50 2019 Sep 10.43 4 750 644.12 1.93 $\\\\pm $ 0.10 30 2019 Dec 09.91 4 647 734.60 1.38 $\\\\pm $ 0.12 37 2019 Dec 10.02 5 1265 734.71 1.71 $\\\\pm $ 0.04 20 2019 Dec 10.19 3 402 734.88 1.30 $\\\\pm $ 0.17 59 2020 June 26.37 4 648 934.06 $1.29\\\\pm 0.11$ 20 2020 June 26.49 5 1255 934.18 $1.23\\\\pm 0.04$ 21 2020 June 29.44 3 402 937.13 $1.12\\\\pm 0.16$ 55 aThe uncertainties reflect the statistical errors.\"\n ],\n [\n \"GRB afterglow Model\",\n \"We use external synchrotron model for the GRB afterglow emission, which arises due to the interaction between the GRB outflow and the surrounding CBM [34].\",\n \"As the outflow moves into the CBM, a `forward shock' or a `blast wave' shock moving into the CBM, and a `reverse shock' moving into the ejected outflow are created.\",\n \"These shocks have the ability to accelerate charged particles to relativistic speeds via Fermi acceleration [47].\",\n \"Radio afterglow emission is expected to be synchrotron emission arising due to these relativistic charged particles in the shocks in the presence of magnetic fields.\",\n \"The evolution of the blast wave is `self-similar' [7], and the dynamics depends only on the density of the CBM and the blast wave energy.\",\n \"The CBM is usually modelled to be one of the two forms, a constant density medium and a wind like density medium [18].\",\n \"The number density profile of the ambient medium is usually modelled as a powerlaw $n \\\\propto r^{-k}$ .\",\n \"For constant density case, the parameter $k=0$ and $n=n_0$ .\",\n \"For wind like case, the mass flows radially outwards at uniform speed and rate from the of GRB progenitor giving $k=2$ ; hence for a mass-loss rate from the progenitor $\\\\dot{M}_W$ and the progenitor wind velocity $V_W$ , the density can be defined as [33], [19]: $n=\\\\frac{\\\\dot{M}_W}{4\\\\pi r^{2} m_p V_W}=3\\\\times 10^{35} A_* r^{-2}$ Where $m_p$ is the mass of the proton, $A_*$ is in units of $3\\\\times 10^{35}$ cm$^{-1}$ (or $5\\\\times 10^{11}$ g cm$^{-1}$ for mass-density), corresponding to $\\\\dot{M}_{W,-5}/V_{W,3}$ .\",\n \"Here $\\\\dot{M}_{W,-5}$ is mass loss rate in $10^{-5}M_{\\\\odot }/\\\\rm {yr}$ and $V_{W,3}=10^3\\\\,\\\\rm {km/s}$ .\",\n \"The GRB wideband afterglow spectrum has several breaks characterized by various characteristic frequencies, namely $\\\\nu _a$ (the transition from optically thick to thin region, i.e.\",\n \"synchrotron self-absorption (SSA) peak), $\\\\nu _c$ (synchrotron cooling frequency) and $\\\\nu _m$ (frequency corresponding to minimum injected Lorentz factor).\",\n \"In the fast cooling regime the frequency ordering is $\\\\nu _m > \\\\nu _c$ , while in the slow cooling regime the ordering is opposite.\",\n \"Generally afterglow modelling is done in the slow cooling regime, where the most relevant ordering in radio frequencies in first few days are $\\\\nu _a \\\\le \\\\nu _m \\\\le \\\\nu _c$ ; and then $\\\\nu _m \\\\le \\\\nu _a \\\\le \\\\nu _c$ at later times [35].\",\n \"However, there has been evidence for fast cooling in some GRBs with high density environments [17].\",\n \"The afterglow spectra evolves as $F_{\\\\nu } \\\\propto \\\\nu ^2$ ($\\\\nu <\\\\nu _a$ ) and $F_{\\\\nu } \\\\propto \\\\nu ^{1/3}$ ($\\\\nu _a < \\\\nu < \\\\min (\\\\nu _m,\\\\nu _c)$ ) for both fast as well as the slow cooling.\",\n \"In the regime $\\\\min (\\\\nu _m,\\\\nu _c) < \\\\nu < \\\\max (\\\\nu _m,\\\\nu _c) $ , the evolution changes to $F_{\\\\nu } \\\\propto \\\\nu ^{-1/2}$ and $F_{\\\\nu } \\\\propto \\\\nu ^{-(p-1)/2}$ for fast and slow cooling regimes, respectively, and then evolves as $F_{\\\\nu } \\\\propto \\\\nu ^{-p/2}$ for $ \\\\nu > \\\\max (\\\\nu _m,\\\\nu _c) $ , where $p$ is the usual power law index showing particle number distribution with energy in non-thermal emission case.\",\n \"In addition to standard afterglow models, shock breakout model too has been favoured for low-luminosity GRBs, where the breakout of a shock travelling through the stellar envelope may be responsible for gamma-ray emission.\",\n \"Due to decreasing density of the stellar matter outwards, the shock breakout velocity increases and may become relativistic.\",\n \"[56] have defined a ‘relativistic breakout closure relation’ between the breakout energy $E_{\\\\rm bo}$ , temperature $T_{\\\\rm bo}$ and duration $t_{\\\\rm bo}^{\\\\rm obs}$ , i.e.\",\n \"$( t_{\\\\rm bo}^{\\\\rm obs}/20 \\\\,\\\\rm s) \\\\sim (E_{\\\\rm bo}/10^{46}\\\\, \\\\rm erg)^{1/2} (T_{bo}/50 \\\\, \\\\rm keV)^{-2.68}$ .\",\n \"This relation has been found to be followed by several low luminosity GRBs.\",\n \"For GRB 171205A, the relation gives $\\\\sim 80$ s, which is roughly 1/3rd of the observed duration.\",\n \"However, the large uncertainties in the $E_p$ and $E_{\\\\rm iso}$ [24] do not rule out this model.\"\n ],\n [\n \"Inputs from high frequency data\",\n \"The Swift-XRT light curve covers epoch until day $\\\\sim 902$ .\",\n \"In addition, Chandra observations are on day 71 and day 205 (Fig.\",\n \"REF ).\",\n \"We fit a power law to the X-ray data post day 1.\",\n \"This is to avoid possible energy injection due to central engine activities at $\\\\le 1$ d. The light curve is fit with power law index $\\\\alpha =1.06\\\\pm 0.06$ .\",\n \"[24] also find a power law fit with index $1.08$ to the light curve post $\\\\sim $ 1 day, consistent with our value.\",\n \"This is typical value for the standard X-ray light curve decay before jet break [59], suggesting that no jet-break was seen until the last Swift-XRT epoch.\",\n \"The jet break time is thus constrained to $t_{\\\\rm jet}\\\\ge 71\\\\,\\\\rm d$ [66], i.e.\",\n \"the last detected Chandra epoch.\",\n \"The photon index of the Swift-XRT data is a photon index $\\\\Gamma =1.94^{+0.23}_{-0.22}$ https://www.swift.ac.uk/xrt_live_cat/00794972/.\",\n \"This suggests $\\\\beta =0.94^{+0.23}_{-0.22}$ .\",\n \"The values of $\\\\alpha $ and $\\\\beta $ are consistent with X-ray frequency ($\\\\nu _X$ ) being above the cooling frequency ($\\\\nu _X>\\\\nu _c$ ), for both wind as well as ISM density profiles.\",\n \"The X-ray temporal index is consistent with $\\\\alpha =(3p-2)/4$ and spectral index $\\\\beta =p/2$ .\",\n \"These values gives $p$ to be $2.08\\\\pm 0.08$ and $1.88\\\\pm 0.46$ , respectively, which, within the errorbars, are consistent with each other.\",\n \"Within the errorbars, this is also consistent with $\\\\nu _X<\\\\nu _c$ for a ISM like medium where $\\\\alpha =3(p-1)/4$ and $\\\\beta =(p-1)/2$ .\",\n \"We also plot early time 230 and 345 GHz light curves GRB 171205A, taken from [82].\",\n \"We estimate the realistic errorbars in the flux density values by adding 5% of the flux density in quadrature to the map rms to account for the systematic errors.\",\n \"The mm highest flux density is significantly higher than the peak flux densities at uGMRT bands, indicating most likely presence of wind like medium.\",\n \"The mm light curves are jointly fit with a broken powerlaw with common post peak index.\",\n \"The data are best fit with post break index $-1.37\\\\pm 0.07$ (bottom panel of Fig.\",\n \"REF ).\",\n \"For above values of $p$ , this is consistent with an evolution of $(3p-1)/4$ , for $\\\\nu _m<\\\\nu _{\\\\rm mm}<\\\\nu _c$ for wind density profile.\",\n \"The pre-break indices $-0.17\\\\pm 0.14$ and $-0.54 \\\\pm 0.31$ for 230 and 345 GHz bands, respectively.\",\n \"The breaks in 230 and 345 GHz are at $3.14\\\\pm 0.28$ day and $6.12\\\\pm 1.17$ day, respectively.\",\n \"At the epochs of the breaks, the flux density of the 230 and 345 GHz light curves are $43.83\\\\pm 3.29$ and $14.85\\\\pm 2.86$ mJy, respectively.\",\n \"We also fit the 345 GHz data with a single powerlaw model to determine the significance of the break.\",\n \"The single powerlaw model fits with an index of $-1.05\\\\pm 0.14$ , however results in a much larger reduced-$\\\\chi ^2$ value of 2.47 as compared to 0.81 in the broken power law case.\",\n \"For the broken powerlaw model, the characteristic frequency evolves with an index of $+0.71\\\\pm 0.12$ ., hence the breaks can not be due to passage of $\\\\nu _m$ , which decreases with time.\",\n \"The only possibility is the breaks are being due to $\\\\nu _c$ in the wind medium, where $\\\\nu _c \\\\propto t^{1/2}$ .\",\n \"Thus the data indicate.\",\n \"$\\\\nu _c\\\\approx 230$ GHz on day 3.14 and $\\\\nu _c\\\\approx 345$ GHz on day 6.12.\",\n \"However, there is one concern.\",\n \"The pre-break evolution is rather flat.\",\n \"This could be reconciled if mm bands are close to passage of $\\\\nu _m$ .\",\n \"Another possibility is that if there is a reverse shock component which is contributing to the mm band at the early epoch.\",\n \"The thick shell reverse shock model during the reverse shock crossing phase will evolve as $-(p-2)/2$ , for $\\\\nu _{\\\\rm mm}>\\\\nu _c$ in the slow cooling phase and $\\\\nu _{\\\\rm mm}>\\\\nu _m$ in fast cooling phase.\",\n \"The characteristic frequency evolves with an index of $+0.71\\\\pm 0.12$ , this is in between evolution of the $\\\\nu _c\\\\, (\\\\propto t^{1/2}) $ in the forward shock and the evolution of $\\\\nu _c\\\\, (\\\\propto t^{1}) $ in the reverse shock.\",\n \"This also suggests the possibility of reverse shock contributing to the afterglow model.\",\n \"If so, then the breaks at day 3.14 and 6.12 in 230 and 345 GHz are artificial breaks and may not reflect the cooling break.\",\n \"Figure: Top panel: Near-simultaneous spectra on day ∼4\\\\sim 4, day ∼11\\\\sim 11 and day ∼909\\\\sim 909.\",\n \"Here we use only the cm and mmdata.\",\n \"Bottom panel: The same as top panel but including the X-ray data as well.\",\n \"The spectra are fit with smoothed single broken powerlaw fits.\",\n \"For day 4.2, we also fit the data witha powerlaw with two breaks.\",\n \"The double broken powerlaw fit is indicated with a black continuous line and the single one with dashed blue line.We also combine the early epoch published cm and mm data from [82], late epoch Australian Telescope Compact Array (ATCA) data from [46] with the uGMRT and the X-ray data, to obtain near simultaneous spectra on around day 4, day 11 and day 909 (Fig.\",\n \"REF ).\",\n \"For the early epoch spectrum, the VLA data are on day 4.3 and the ALMA data are on day 5.2.\",\n \"We use the nearest epoch temporal evolution to derive the ALMA values on day 4.3.\",\n \"In the figure, we plot the values on day 5.2 as well as their derived values on day 4.3 along with the VLA data.\",\n \"We do not use optical data as supernova signatures appeared by day 3 and hence the optical data are likely to be heavily contaminated by the underlying supernova.\",\n \"We fit a smoothed single broken powerlaw (SBPL) and smoothed double broken powerlaw (DBPL) fits to day 4.3 spectrum.\",\n \"We used formalism of [34] for the treatment of smoothening of powerlaws at the break frequencies.\",\n \"We first fit only the cm and mm data.\",\n \"For day 4.3, the best fit SBPL model peaks at $30.41\\\\pm 4.08$ GHz and afterwards evolves as $-0.41\\\\pm 0.25$ .\",\n \"Here we have fixed the pre-break spectral index to 2.\",\n \"However, even when we use this as a free parameter, the best-fit index is consistent with 2 within errorbars.\",\n \"The DBPL model fits the data well and give the breaks at $7.25\\\\pm 2.11$ GHz and $44.41\\\\pm 3.62$ GHz with post break indices $1.26\\\\pm 0.12$ , and $-0.41\\\\pm 0.02$ , respectively.\",\n \"The peak flux density is $43.80\\\\pm 1.12$ mJy.\",\n \"We also fit SBPL for the spectrum on day 11 and 909.\",\n \"The data indicates a fit $-0.68\\\\pm 0.02$ with peak $<90$ GHz, with the peak flux density $>16.2$ mJy.\",\n \"The pre-break index is $0.43\\\\pm 0.46$ .\",\n \"The spectrum on day 909 are fit with pre-break and post-break indices of $0.15\\\\pm 0.13$ and $-0.90\\\\pm 0.04$ , respectively, and a peak at $1.55\\\\pm 0.48$ GHz.\",\n \"The peak flux at day 909 is $1.15\\\\pm 0.16$ mJy.\",\n \"Now we carry out the above fits including the X-ray data as well.\",\n \"The data to mm to X-ray data are fit by indices close to $-1$ .\",\n \"These values are $-1.01\\\\pm 0.18$ and $-1.01\\\\pm 0.02$ for the first two spectra, respectively.\",\n \"For the day 909, the X-ray upper limit do not constrain the model.\",\n \"The mm to X-ray indices consistent with the X-ray spectral indices between 0.3–10 keV.\",\n \"This indicates that the cooling frequency ($\\\\nu _c$ ) is probably close to mm values if $ \\\\nu _X > \\\\nu _c$ .\",\n \"Due to lack of optical data, we cannot constrain the cooling frequencies more precisely.\",\n \"The peak flux density and the frequency of the peak in the three cases are $37.79\\\\pm 3.95$ mJy at $160.93\\\\pm 35.31$ GHz, $>16.2$ mJy at $<90$ GHz and $1.15\\\\pm 0.16$ mJy at $1.55\\\\pm 0.48$ GHz., respectively.\",\n \"Our analysis indicate that the peak of the spectra on day 4 and day 909 are due to $\\\\nu _m$ or $\\\\nu _a$ .\",\n \"Between day 4 and 909, the peak flux density evolves as $-0.63\\\\pm 0.02$ .\",\n \"This clearly rules out ISM model and supports the wind model.\",\n \"[88] has estimated kinetic energy in the synchrotron afterglow, $E_K$ , from the X-ray data at the time of shallow to normal decay, which for GRB 171205A is 1.05 day.\",\n \"For $\\\\nu _X>\\\\nu _c$ , $E_K$ is independent of density and is only weakly depends on $B$ and $p$ , and therefore an ideal regime to measure $E_K$ .\",\n \"One can then derive $E_K$ from the X-ray band using Eq 9 of [88], which in this case is $E_K\\\\approx 1.4\\\\times 10^{50} (\\\\epsilon _B/0.01)^{2-p/2+p} (\\\\epsilon _e/0.1)^{4(1-p)/2+p}$ erg.\"\n ],\n [\n \"Initial inferences from uGMRT data\",\n \"We plot uGMRT radio light curves and fit them jointly with a smoothed broken powerlaw (SBPL) model.\",\n \"We allow the normalization to vary but fix the indices before and after the peak.\",\n \"We show the light curves in Fig.\",\n \"REF .\",\n \"In the figure, the band 4 and 5 values are scaled by factors of 10 and 100, respectively, for clarity.\",\n \"The indicies before and after the peak are $1.37\\\\pm 0.20$ and $-0.72\\\\pm 0.06$ .\",\n \"The evolution before peak at uGMRT frequencies ($\\\\nu _{\\\\rm radio}$ ) is in rough agreement with wind slow cooling case for $\\\\nu _{\\\\rm radio}<\\\\nu _a$ for $\\\\nu _a< \\\\rm min(\\\\nu _m,\\\\nu _c)$ ; the wind fast cooling phase for $\\\\nu _a<\\\\nu _c<\\\\nu _m$ if the observing frequency is in the transition zone between $\\\\nu _{\\\\rm radio}<\\\\nu _a$ to $\\\\nu _a<\\\\nu _{\\\\rm radio}<\\\\nu _c$ , as well as wind slow cooling phase for $\\\\nu _m<\\\\nu _a<\\\\nu _c$ for transition between $\\\\nu _{\\\\rm radio}<\\\\nu _m$ to $\\\\nu _m<\\\\nu _{\\\\rm radio}<\\\\nu _a$ .\",\n \"The post peak index is rather shallow and is consistent only with the wind fast cooling in the regime $\\\\nu _a<\\\\nu _{\\\\rm radio}<\\\\nu _c$ .\",\n \"It is rather shallow for the post jet-break or non-relativistic evolution.\",\n \"However, as we discuss in the next section, the shallow decline of radio light curve in seen in other GRBs as well, and other emission components may contribute to it.\",\n \"From our fits, the epochs of the peak flux densities are $5.19\\\\pm 0.05$ and $3.23\\\\pm 0.15$ mJy, respectively, in bands 5 and 4 on day $101.08\\\\pm 9.58$ and day $153.71\\\\pm 40.31$ , respectively.\",\n \"In band 3, the data are optically thin, which constraints the peak to be $<185.13$ day and flux $>2.81$ mJy.\",\n \"The peak flux density evolves as $-1.13 \\\\pm 0.76$ .\",\n \"While this value has a large error, it is consistent with the evolution in the stratified wind within 2-$\\\\sigma $ and most likely rule out all the models involving ISM, where $F_{\\\\rm max} \\\\propto t^0$ .\",\n \"This also rules out $1\\\\nu _c\\\\end{array}\\\\right.\",\n \"}$ Case 2 ($\\\\nu _m<\\\\nu _a<\\\\nu _c$ ): $F_{\\\\nu }(t) \\\\propto {\\\\left\\\\lbrace \\\\begin{array}{ll} \\\\nu ^2 t^{s+1}, & \\\\text{for} \\\\ \\\\nu <\\\\nu _m \\\\\\\\\\\\nu ^{5/2} t^{\\\\frac{7+3s}{4}}, & \\\\text{for} \\\\ \\\\nu _m<\\\\nu <\\\\nu _a \\\\ \\\\\\\\\\\\nu ^{-(p-1)/2} t^{-\\\\frac{3p-1 -s(p+1)}{4}} , & \\\\text{for} \\\\ \\\\nu _a<\\\\nu <\\\\nu _c \\\\\\\\\\\\nu ^{-p/2} t^{-\\\\frac{3p-2 -s(p+2)}{4}}, & \\\\text{for}\\\\ \\\\nu >\\\\nu _c\\\\end{array}\\\\right.\",\n \"}$ Case 3 ($\\\\nu _a<\\\\nu _c<\\\\nu _m$ ): $F_{\\\\nu }(t) \\\\propto {\\\\left\\\\lbrace \\\\begin{array}{ll} \\\\nu ^2 t^{2+s}, & \\\\text{for} \\\\ \\\\nu <\\\\nu _a \\\\\\\\\\\\nu ^{1/3} t^{-\\\\frac{2(s+1)}{3}}, & \\\\text{for} \\\\ \\\\nu _a<\\\\nu <\\\\nu _c \\\\ \\\\\\\\\\\\nu ^{-1/2} t^{-\\\\frac{s+1}{4}} , & \\\\text{for} \\\\ \\\\nu _c<\\\\nu <\\\\nu _m \\\\\\\\\\\\nu ^{-p/2} t^{-\\\\frac{3p-2 +(p+2)s}{4}}, & \\\\text{for}\\\\ \\\\nu >\\\\nu _m\\\\end{array}\\\\right.\",\n \"}$ We now fit the uGMRT data with both standard isotropic afterglow and shock break-out afterglow models.\",\n \"We use smoothed broken powerlaw models for various regimes following the recipe of [34].\",\n \"We use $z=0.036$ , $D=163$ Mpc.\",\n \"We define $E_{\\\\rm KE}=E_{\\\\rm \\\\gamma .iso}/\\\\eta $ , and keep $\\\\eta $ ($\\\\eta _{\\\\rm eff}$ for SBO) as the free parameter.\",\n \"The parameters $p$ , $A^*$ , $\\\\epsilon _B$ and $\\\\epsilon _e$ are also free parameters.\",\n \"With the inputs above, we carry out the detailed modelling using Markov chain Monte Carlo (MCMC) fitting using the Python package emcee [25].\",\n \"We choose 150 walkers, 2000 steps.\",\n \"Even though the analytical modelling suggests wind like medium, we still start with fits to a constant density medium.\",\n \"The fit results in a high values of reduced-$\\\\chi ^2$ further ruling out the constant density model.\",\n \"We fit the standard wind model with $k=2$ for all the cases.\",\n \"In addition, we also account for non-standard wind density medium keeping $k$ as a free parameter.\",\n \"Table REF shows the fit statistics for different parameters using the above mentioned models.\",\n \"For $\\\\nu _a<\\\\nu _m<\\\\nu _c$ case, we do not list general $k$ model as this model performed quite poorly for both afterglow as well as shock breakout.\",\n \"$\\\\nu _a<\\\\nu _m<\\\\nu _c$ generally performs very poorly, with shock breakout model performing slightly better than the isotropic afterglow model.\",\n \"In addition, the parameters obtained in this model are rather unphysical.\",\n \"While the fast cooling model gives best reduced $\\\\chi _{\\\\nu }^{2}$ , this case is unlikely to be true.\",\n \"The analysis of mm and X-ray light published data have already revealed that $\\\\nu _c$ lies between the mm and X-ray frequencies.\",\n \"Since $\\\\nu _c \\\\propto t^{1/2}$ in the wind model, uGMRT radio frequencies cannot be in the fast cooling regime.\",\n \"The most viable model fits are obtained for $\\\\nu _m<\\\\nu _a<\\\\nu _c$ case.\",\n \"This is quite viable since in wind density provide, $\\\\nu _m$ evolves faster than $\\\\nu _a$ and may reach $\\\\nu _m<\\\\nu _a$ regime at late epochs [35].\",\n \"Here keeping $k$ as free parameter also results in $k \\\\sim 2$ .\",\n \"Our model fits are equally good for the standard afterglow model and the shock breakout model, and uGMRT data alone cannot differentiate between the two.\",\n \"In Fig.\",\n \"REF , we show the light curve for a standard wind $k=2$ model for $\\\\nu _m<\\\\nu _a<\\\\nu _c$ case for both isotropic forward shock afterglow as well as the shock breakout afterglow model.\",\n \"|l|ll|llll|llll| Best fit parameters for GRB 171205A uGMRT data 9 0pt Param.\",\n \"2|c|$\\\\nu _a<\\\\nu _m<\\\\nu _c$ 4|c|$\\\\nu _m<\\\\nu _a<\\\\nu _c$ 4|c|$\\\\nu _a<\\\\nu _c<\\\\nu _m$ AG SBO 2|c|AG 2|c|SBO 2|c|AG 2|c|SBO $k=2$ $k=2$ $k=2$ general $k$ $k=2$ general $k$ $k=2$ general $k$ $k=2$ general $k$ $A_*$ $7.37^{+0.95}_{-0.80}$ $2.82^{+0.46}_{-0.39}$ $1.58^{+0.11}_{-0.75}$ $1.11^{+1.03}_{-0.54}$ $2.89^{+1.95}_{-1.26}$ $3.54^{+2.46}_{-1.55}$ $1.69^{+1.15}_{-0.51}$ $0.17^{+0.19}_{-0.09}$ $2.15^{+1.53}_{-0.78}$ $0.22^{+0.16}_{-0.11}$ $\\\\eta (\\\\eta _{\\\\rm eff}$ for SBO) $0.005^{+0.0004}_{-0.0004}$ $0.30^{+0.13}_{-0.12}$ $0.02^{+0.02}_{-0.01}$ $0.02^{+0.03}_{-0.02}$ $0.07^{+0.09}_{-0.05}$ $0.13^{+0.12}_{-0.07}$ $0.014^{+0.002}_{-0.002}$ $0.03^{+0.003}_{-0.003}$ $0.06^{+0.05}_{-0.03}$ $0.03^{+0.02}_{-0.01}$ $\\\\epsilon _B$ $ 0.94^{+0.05}_{-0.09}$ $0.70^{+0.20}_{-0.20}$ $0.21^{+0.24}_{-0.13}$ $0.11^{+0.17}_{-0.06}$ $0.03^{+0.03}_{-0.02}$ $0.01^{+0.01}_{-0.01}$ $0.17^{+0.14}_{-0.10}$ $0.02^{+0.01}_{-0.01}$ $0.08^{+0.09}_{-0.05}$ $0.01^{+0.01}_{-0.01}$ $\\\\epsilon _e$ $0.99^{+0.01}_{-0.02}$ $0.78^{+0.16}_{-0.23}$ $0.12^{+0.09}_{-0.06}$ $0.13^{+0.11}_{-0.07}$ $0.24^{+0.17}_{-0.11}$ $0.12^{+0.07}_{-0.06}$ $\\\\cdots $ $\\\\cdots $ $\\\\cdots $ $\\\\cdots $ $p$ $2.55^{+0.08}_{-0.06}$ $3.87^{+0.10}_{-0.19}$ $2.22^{+0.04}_{-0.04}$ $2.18^{+0.14}_{-0.09}$ $2.23^{+0.04}_{-0.05}$ $2.22^{+0.16}_{-0.13}$ $\\\\cdots $ $\\\\cdots $ $\\\\cdots $ $\\\\cdots $ $k$ $\\\\cdots $ $\\\\cdots $ $\\\\cdots $ $1.99^{+0.01}_{-0.02} $ $\\\\cdots $ $1.99^{+0.01}_{-0.01} $ $\\\\cdots $ $1.91^{+0.02}_{-0.01} $ $\\\\cdots $ $1.92^{+0.02}_{-0.02} $ $s$ $\\\\cdots $ $0.55^{+0.04}_{-0.04}$ $\\\\cdots $ $\\\\cdots $ $0.13^{+0.08}_{-0.07}$ $0.15^{+0.08}_{-0.08}$ $\\\\cdots $ $\\\\cdots $ $0.35^{+0.13}_{-0.16}$ $0.08^{+0.16}_{-0.18}$ $\\\\chi _{\\\\nu }^{2}= 7.55$ $\\\\chi _{\\\\nu }^{2}=2.58$ $\\\\chi _{\\\\nu }^{2}=1.74$ $\\\\chi _{\\\\nu }^{2}=1.80$ $\\\\chi _{\\\\nu }^{2}=1.72$ $\\\\chi _{\\\\nu }^{2}=1.78$ $\\\\chi _{\\\\nu }^{2}=1.60$ $\\\\chi _{\\\\nu }^{2}=1.52$ $\\\\chi _{\\\\nu }^{2}=1.58$ $\\\\chi _{\\\\nu }^{2}=1.57$ Here AG is the standard isotropic afterglow model and SBO is the shock breakout model.\",\n \"In case of fast cooling, the data are only in the regime $\\\\nu _a$ to $\\\\nu _c$ ( 2 to 1/3 transition of spectra), which we don't have $p$ dependencies in temporal or spectral slopes.\",\n \"The only $p$ dependency is in the expression of $\\\\nu _a$ via that ratio of $G(p)$ parameter which we have taken to be of order unity for $p\\\\sim 2.1$ .\",\n \"The $\\\\epsilon _e$ dependency is also not there as $\\\\nu _m$ is unconstrained.\",\n \"Figure: Upper left panel: Light curves of GRB171205A in radio regime using the slow cooling model (ν m <ν a <ν c \\\\nu _m<\\\\nu _a<\\\\nu _c) and standard wind (k=2k=2).\",\n \"Upper right panel: Posterior Distributions of parameters for this model.\",\n \"Lower left panel: Light curves of GRB171205A in radio regime using the slow cooling model (ν m <ν a <ν c \\\\nu _m<\\\\nu _a<\\\\nu _c) with standard wind (k=2k=2) and shock breakout scenario.\",\n \"Lower right panel: Posterior Distributions of parameters for this model.In the left panels, the blue, red and green points are observed data in band 5 , band 4 and band 3 respectively which are included in the fit.\",\n \"The points with arrow have only the upper limit of flux.\",\n \"The lines are the best fits.\",\n \"In the right panels, the 2D plots show the joint probability distribution of any two parameters.\",\n \"The contours are at 0.5σ\\\\sigma , 1σ\\\\sigma , 2σ\\\\sigma , 3σ\\\\sigma .\",\n \"The middle dotted lines in the 1D parameter distribution is the median value of posterior followed by 1σ\\\\sigma and 2σ\\\\sigma lines on both sides.\",\n \"σ\\\\sigma is standard deviation of the corresponding distribution.\"\n ],\n [\n \"Properties of GRB 171205A from radio modelling\",\n \"The peak radio flux density of GRB 171205A at 1.3 GHz is $\\\\sim 10^{29}$ erg s$^{-1}$ Hz$^{-1}$ .\",\n \"This is two orders of magnitude fainter than cosmological GRBs at this frequency (Fig.\",\n \"REF ).\",\n \"However, these values are comparable to other low-luminosity GRBs, e.g.\",\n \"GRB 031203 [70], 980425 [43].\",\n \"Figure: Plot of 1.4 GHz luminosities of canonical GRBs taken from .\",\n \"Here we overlay the uGMRT 1.3 GHz measurementsfor GRB 171205A.\",\n \"Our values are at least two orders of magnitude smaller than canonical GRBs.The value of $A_*$ in the standard afterglow and the SBO models are $1.58^{+0.11}_{-0.75}$ and $2.89^{+1.95}_{-1.26}$ , respectively, which, assuming a wind velocity of 1000 km s$^{-1}$ , translate to mass-loss rates of $1.58^{+0.11}_{-0.75}\\\\times 10^{-6} $ M$_\\\\odot \\\\,\\\\rm yr^{-1}$ and $2.89^{+1.95}_{-1.26}\\\\times 10^{-6} $ M$_\\\\odot \\\\,\\\\rm yr^{-1}$ , respectively, for the two models.\",\n \"The nature of the surrounding ambient medium reflects on the progenitor nature of GRBs.\",\n \"It is expected that the progenitors of long GRBs are massive stars (Wolf-Rayet) and in most of the cases a long GRB is associated with a supernova [43], [86].\",\n \"Another evidence for massive star progenitors is that the long GRBs generally have star-forming host galaxies [87].\",\n \"In such a case, one expects the association with a wind like circumburst medium.\",\n \"However, several GRBs from massive stars collapse have shown homogeneous density [58], [59].\",\n \"A constant density medium can be produced around a massive star if the wind faces a shock termination [19].\",\n \"The low frequency observations present here provide a unique opportunity to determine the nature of the circumburst medium of GRB 171205A and establish that GRB 171205A exploded in a wind like environment.\",\n \"At uGMRT frequencies, the optically thick to thin transition peak arises at a long time ($t > 100$ d) after the burst, indicating a relatively high density medium.\",\n \"This may be created due to a large stellar mass-loss rate or a low wind velocity.\",\n \"Some previous works [20] have shown that the large mass-loss rate for Wolf-Rayet stars are associated with large metallicity of the medium.\",\n \"Thus GRBs in wind medium can be potential tools for studying metallicity variation at different redshifts.\",\n \"The uGMRT light curve declines as $\\\\sim t^{-0.7}$ .\",\n \"This indicates that there is no jet break until 3 years.\",\n \"There are several explanations for the lack of jet breaks in some GRBs.\",\n \"In the cases of GRB 980326 and GRB 980519, [35] have argued that a wind medium can dilute the jet break even for highly collimated bursts.\",\n \"The jet break will be absent if the radio emission indeed arises from a quasi-spherical afterglow, such as that due to shock breakout [56] or cocoon [54].\",\n \"In case of GRB 030329, [6] have argued that radio emission may be arising from two components, a narrow jet, surrounded by a wider component (e.g.\",\n \"cocoon) and the radio emission is being dominated by the wider component.\",\n \"However, the requirement of this model is that the contribution to the radio afterglow from the narrow jet may be negligible.\",\n \"X-ray observations cover the period of $\\\\sim 200$ days and show no indication of a jet break at least until the last detection on day $\\\\sim 70$ .\",\n \"Using $t_j >71$ d, gives a limit $\\\\theta _j > 1.2$ radians for the AG model and $\\\\theta _j > 1.9$ for SBO model [57], [84].\",\n \"[49] have shown that ejecta kinetic energy profiles in stripped enveloped supernovae vary based on different explosion mechanisms.\",\n \"While stripped-envelope supernovae have a steep dependence $E_K \\\\propto (\\\\Gamma \\\\beta )^{-5.2}$ indicating no central engine activity, relativistic supernovae, sub-energetic GRBs with SBO mechanisms are flatter with $E_K \\\\propto (\\\\Gamma \\\\beta )^{-2.4}$ showing weak activity from central engine.\",\n \"Canonical GRBs, on the other hand, follow $E_K \\\\propto (\\\\Gamma \\\\beta )^{-0.4}$ typical of jet-driven explosions with long-lasting central engines.\",\n \"Our radio modeling and the relativistic treatment of [4], results in $E_K \\\\approx 1.1 \\\\times 10^{51}$ ergs and $\\\\Gamma \\\\beta \\\\sim 1 $ .\",\n \"For the non-relativistic supernova component, we use values from [39], i.e.\",\n \"supernova kinetic energy $E_K=2.4 \\\\times 10^{52}$ ergs and ejecta velocity 55000 km s$^{-1}$ .\",\n \"Using these values, $E_K \\\\sim (\\\\Gamma \\\\beta )^{-1.9}$ .\",\n \"While GRB 171205A is a sub-energetic GRB, it follows the energy-velocity profile somewhere between canonical GRBs and the SBOs.\",\n \"These arguments suggest that the jet and shock-breakout both may play an important role in the late time afterglow emission in GRB 171205A.\",\n \"Since we have radio light curve peaks at two uGMRT frequencies, we could also estimate some more parameter evolutions.\",\n \"The relativistic energy under equipartition assumption at the two epochs are $E_{\\\\rm Eq}=3.6\\\\times 10^{48}$ ergs and $4.9\\\\times 10^{48}$ ergs [4].\",\n \"Thus there is an indication of enhancement of energy $E \\\\propto t^{0.48}$ .\",\n \"This along with flatter light curve decays may also be explained if there is an energy injection from the central engine to the shock.\",\n \"For an injection luminosity $L(t)=L_0(t/t_0)^{-q}$ , $E \\\\propto t^{1-q}$ .\",\n \"This implies $q=0.52$ .\",\n \"However, our best fits result in much larger value of $q$ ($s=0.13$ , $q=1-s\\\\equiv 0.87$ .\",\n \"This may imply that if energy injection, it is not continuous and probably lasted for a small amount of time.\",\n \"The equipartition size obtained from the above formulation [4] at the epochs of two peaks in band 5 and band 4 follow R(t) $\\\\propto $ $t^{0.49}$ .\",\n \"We note that for ISM and Wind, density profiles, $R$ follows as $R\\\\propto t^{1/4}$ and $R\\\\propto t^{1/2}$ , so it also points towards a wind like medium surrounding GRB171205A.\",\n \"Our uGMRT observations cover the period of around 1000 days.\",\n \"However, our data does not suggest the GRB to be in the Newtonian regime yet (Fig.\",\n \"REF ).\",\n \"This is not uncommon for low-luminosity GRBs [37].\",\n \"We note that the value of $\\\\Gamma \\\\beta $ indicates mildly relativistic outflow.\",\n \"Hence it is likely that the GRB is making a transition into Newtonian regime soon.\"\n ],\n [\n \"Shallow decay of radio afterglow\",\n \"We note that the decay of the radio afterglow is much shallower than that of the X-ray afterglow.\",\n \"The shallowness of radio lightcurves was first pointed out by [60].\",\n \"For a reasonable afterglow parameters, they estimated that the afterglow is supposed to cross $\\\\nu _m$ at around 10 days for 10 GHz and follow a decay slope of $(3p-1)/4$ for a wind medium, which was not the case for some GRBs, e.g.\",\n \"GRB 991216 and GRB 000926.\",\n \"They explored the difference between the radio and the optical decay indices could be caused by the fact that the injection frequency remains above the radio domain ($\\\\sim $ 10 GHz), or a different population of electrons, or a variability of microparameters.\",\n \"Finally, they concluded that a long-lived reverse shock in the radio regime could cause this flattening.\",\n \"However, this is unlikely in GRB 171205A as a strong persistent reverse shock requires a low wind density [65].\",\n \"Here the late rising of the radio afterglow suggests a comparatively high density medium which is against the previous statement.\",\n \"So, this scenario can be excluded.\",\n \"[40] noted that radio afterglows of some GRBs deviate at late times and low frequencies from the standard model, and attempted to explain it with the two-component jet model, a narrow jet core and a wider cocoon surrounding the jet.\",\n \"[34] have explained this flattening due to counter-jet which becomes visible when turning sub-relativistic.\",\n \"While such two components should result in a bump, for stratified wind medium, the revelation of counter-jet is more gradual, causing mild flattening.\",\n \"An energy injection event or a different component dominating the radio emission can also produce this flattening.\",\n \"In case of slightly off-axis jet, the early radio emission could possibly be from the cocoon, the accelerated polar ejecta and at a late phase the contribution from the off-axis jet coming to the line of sight can increase the total radio flux [21].\",\n \"The lack of X-ray data at such late times prevents us from directly distinguishing between these scenarios.\",\n \"A population of quasi-thermal electrons has also been argued as one of the reasons [85], which would mainly dominate at radio frequencies, as it would result in increased $\\\\nu _a$ and suppressing radio emission below this.\",\n \"In GRB 171205A, this has been tentatively supported from the polarization measurements.\",\n \"As per [82], the mm data revealed 0.27% level linear-polarisation which is a factor of 4 smaller than optical polarization measurements.\",\n \"This has been explained as Faraday depolarization by non-accelerated, cool electrons in the shocked region.\",\n \"However, one cannot rely on these results due to dispute of the detection claimed by [45].\"\n ],\n [\n \"Origin of radio emission\",\n \"There have been suggestions that sub-energetic bursts are simply canonical GRBs viewed off-axis [53].\",\n \"However, such bursts will have two distinguishing characters, a) low $E_p$ b) a rise in the afterglow energy while the shocked ejecta gradually comes into our line of sight.\",\n \"In GRB 171205A, the afterglow energy increases slightly from $3.6\\\\times 10^{48}$ erg to $4.9\\\\times 10^{48}$ erg between $\\\\sim 100$ and $\\\\sim 200$ days.\",\n \"However, the $E_p$ is comparable to that of canonical GRBs.\",\n \"Additionally, off-axis jet is a geometric effect, which would result in a frequency independent break in the light curve, which has not been seen for GRB 171205A, ruling out the off-axis model [24].\",\n \"Though a jet somewhat off-axis is not ruled out.\",\n \"[24] found that GRB 171205A is an outlier of the Amati relation, as are some other low redshift GRBs, and its emission mechanism should be different from that of canonical, more distant GRBs.\",\n \"There are two models to explain the electromagnetic emission in low-luminosity GRBs, central engine driven [49], [37] and shock-breakout driven [43], [56], [4], [75].\",\n \"An issue with a purely shock breakout model is the requirement of high $\\\\gamma $ -ray efficiency, for a quasi-spherical outflow.\",\n \"Another issue is generation of such relativistic quasi-spherical outflow.\",\n \"One can envisage a situation where some fraction of the supernova ejecta is accelerated to relativistic speeds to provide this quasi-spherical relativistic outflow.\",\n \"However, [79] have shown that only a fraction ($\\\\sim 10^{-4}$ ) of supernova energy goes in relativistic ejecta.\",\n \"[54] has suggested an alternative scenario where a choked jet in a low-mass envelope can put a significant energy into a quasi-spherical, relativistic flow.\",\n \"Our uGMRT model fits are incapable of distinguishing between canonical afterglow versus SBO afterglow models.\",\n \"We check the applicability of this model for GRB 171205A.\",\n \"In this model, the expanding outflow is considered to harbour a series of successive shocks, which accelerate and catch up to the boundary and hence explain the increasing afterglow energy via continuous injection.\",\n \"In this model, eventually the total energy should reach around $2.4 \\\\times 10^{52}$ erg [39], the kinetic energy of the associated supernova.\",\n \"Generally one does not see such a large amount of energy as radio observations do not cover epochs late enough.\",\n \"However, our radio observations cover a period of nearly 1000 days.\",\n \"The SBO model fit gives $E_K\\\\approx 3.4\\\\times 10^{50}$ erg, two orders of magnitude smaller than the one predicted in pure SBO model.\",\n \"[77] carried out multiwavelength hydrodynamical modelling of GRB 171205A in the framework of post shock break-out relativistic SN ejecta-CBM interaction scenario.\",\n \"While they claimed that this model worked well for GRB 171205A and favour the wind model, we note some problems with their model.\",\n \"They had to introduce a centrally concentrated CBM with a sudden density drop to explain the available radio and X-ray data.\",\n \"however, our work includes radio measurements upto 3 years and do not show a sudden density drop.\",\n \"The pure shock breakout model also has some other problems.\",\n \"A shock breakout model predicts a $\\\\gamma $ -ray emission lasting $\\\\ge 1000 s$ , lower $E_p$ (not exceeding 50 keV), a large absorption column density, a late time soft X-ray emission, and comparable energy in the X-ray emission and the prompt $\\\\gamma $ -ray flare.\",\n \"We note that the X-ray spectrum shows an intrinsic hydrogen column density $N_H=7.4^{+4.1}_{-3.6} \\\\times 10^{20}$ cm$^{-2}$ [24].\",\n \"This intrinsic column density is at the low-end of low-luminosity GRB distribution, even among low redshift Swift-XRT GRBs where the mean is $N_H=2.4 \\\\times 10^{21}$ cm$^{-2}$ at $z < 0.2$ [2].\",\n \"For the observations post day 1 onwards, there may be indication of a slightly higher column density $N_H=1.2^{+0.8}_{-1.7} \\\\times 10^{21}$ cm$^{-2}$ https://www.swift.ac.uk/xrt_live_cat/00794972/.\",\n \"But there is no particular indication of significant spectral softening, except of a slight indication of $\\\\Gamma =1.63\\\\pm 0.30$ to $\\\\Gamma =1.94\\\\pm 0.23$ .\",\n \"The total X-ray energy from the first observation onwards until the last detection is at least an order of magnitude smaller than the prompt energy.\",\n \"[24] have claimed the presence of a thermal component.\",\n \"While shock breakout from supernovae is the most favourable model for a thermal component [56], [75], late time photospheric emission from jet [27], or thermal emission from cocoon [78] can also explain this component.\",\n \"If we assume the blackbody component to be significant, it comprises of 20% flux and has a temperature of 89 eV [24].\",\n \"This corresponds to a radius $R =(E_{\\\\rm iso} /aT_{BB}^4)^{1/3} \\\\approx 1.4 \\\\times 10^{13}$ cm ($a$ is the radiation density constant).\",\n \"This is much larger than the typical Wolf-Rayet star radius, but can be explained if the shock expands in a non-spherical manner.\",\n \"Alternatively, [54] suggested the presence of an optically thick stellar envelope further away from the star, from where the breakout happens.\",\n \"We have seen that our observations are inconsistent with a pure shock breakout due to the short duration, higher $E_p$ , shallow $E$ vs $\\\\Gamma \\\\beta $ relation, low column density and a much larger break-out radius predicted by the thermal component.\",\n \"It can also not be explained as merely a canonical GRB seen off-axis.\",\n \"We show below that both these components contribute towards the radio afterglow.\",\n \"GRB 171205A is the first GRB in which direct signatures of a cocoon has been seen [39].\",\n \"This is rare because generally a line-of-sight jet is much brighter than the associated cocoon, hiding the cocoon signatures.\",\n \"An off-axis jet can reveal itself by the associated supernova, but the cocoon signatures are long gone by the time supernova may be discovered.\",\n \"Ideally cocoon signatures are visible only in slightly off-axis GRBs.\",\n \"In GRB 171205A, the cocoon was identified by the broad absorption features overlapped on the supernova spectrum [39] They also estimated, from the energy deposited in cocoon, that the jet was quite energetic [39] .\",\n \"This may imply that we may be seeing a slightly off-axis jet, which enabled us to reveal the cocoon, not overshadowed by the bright jet.\",\n \"With this knowledge, it is likely that radio afterglow has from both components, the sub-relativistic wider cocoon and a slightly off-axis jet.\",\n \"The cocoon radio emission dominates the GRB emission at early times when the GRB jet is off-axis.\",\n \"Later the additional flux contribution comes when the jet spreads sideways and comes in our line of sight.\",\n \"In such a case, the total radio flux can also be large compared to on-axis GRBs since the cocoon and the jet carry comparable energy [21].\",\n \"[39] could not distinguish between the cocoon along with slightly off-axis jet versus only cocoon emission, where the faint $\\\\gamma $ -rays are the predicted signal of the cocoon breaking out of the stellar envelope.\",\n \"However, our radio modelling rules out the pure shock break-out model in favour of cocoon along with a slightly off-axis jet.\",\n \"As per theoretical models, the indicated speed of the ejecta is also consistent with the sub-relativistic speeds expected in this model.\"\n ],\n [\n \"Comparison with other low-$z$ low-luminosity GRBs\",\n \"With $E_{\\\\rm iso}=2.3\\\\times 10^{49}$ erg and $z=0.0368$ , GRB 171205A is one of the few low-$z$ ($z\\\\lesssim 0.1$ ), low-luminosity GRBs.\",\n \"Other GRBs in this category are GRB 980425, [31], GRB 031203, [67], GRB 060218 [9] and GRB 100316D [74].\",\n \"Like other low-luminosity GRBs, the spectrum of GRB 171205A can be fit by a simple power-law model, however, the photon index $\\\\Gamma \\\\sim 1.94$ is harder than these GRBs, except that of GRB 031203 [67].\",\n \"Even amongst the intrinsically sub-energetic bursts, GRB 171205A resembles more closely with GRBs 980425 and 031203 and not GRBs 060218 and 100316D.\",\n \"GRBs 980425 and 031203 were difficult to realise in the shock break out models due to their relatively hard spectra, shorter durations and larger $E_p$ , though lack of thermal equilibrium may accommodate it [41].\",\n \"GRBs 060218 and 100316D stand out due to their large durations of a few thousand seconds and lower peak energies $E_p < 50$ keV.\",\n \"The shorter $T_{90}$ and higher $E_{\\\\rm p}$ for GRB 171205A are are comparable to the respective values for GRBs 980425 and 031203.\",\n \"A major difference between GRB 171205A and other low-luminosity GRBs is the absence of intrinsic absorption column density.\",\n \"All other GRBs in this class show significant neutral hydrogen column density $N_H \\\\sim (6-7)\\\\times 10^{21}$ cm$^{-2}$ , as opposed to an order of magnitude lower $N_H$ for GRB 171205A [24].\",\n \"The higher column density is considered an essential feature for supernova shock break out models in low-luminosity GRBs.\",\n \"The peak radio flux density of GRB 171205A at 1.3 GHz is $\\\\sim 10^{29}$ erg s$^{-1}$ Hz$^{-1}$ .\",\n \"This value is comparable to other low-luminosity GRBs, e.g.\",\n \"GRB 031203 [70].\",\n \"However, the X-ray luminosity at 10 hr is $L_x\\\\approx 2\\\\times 10^{42}$ erg s$^{-1}$ , which is 4 times smaller than that of GRB 031203 [70].\",\n \"This discrepancy is even more significant for a wind-like medium, since the measured peak radio luminosities are at 1.3 GHz and 8.5 GHz bands for GRB 171205A and GRB 031203, respectively.\",\n \"It has been found that the X-ray afterglow decays slowly in low-luminosity GRBs.\",\n \"For example, the X-ray emission in GRB 031203 followed $F_{\\\\rm xray}(t) \\\\propto \\\\nu ^{-0.8} t^{-0.4}$ [70], and for GRB 980425 $F_{\\\\rm xray}(t) \\\\propto \\\\nu ^{-1} t^{-0.2}$ [43].\",\n \"This is opposed to the canonical GRBs with $F_{\\\\rm xray}(t) \\\\propto \\\\nu ^{-1.3} t^{-1}$ .\",\n \"[5] have shown that a flat temporal decay of the X-ray light curve can be explained in the shock-break out model, although dominance of an underlying supernova has also been suggested as a possible reason for this flatness [70].\",\n \"With $F_{\\\\rm xray}(t) \\\\propto \\\\nu ^{-0.94} t^{-1.1}$ , GRB 171205A is more like canonical GRBs.\",\n \"On the contrary, the X-ray spectral index was very soft for GRB 060218A with $F_{\\\\rm xray}(t) \\\\propto \\\\nu ^{-2.2} t^{-1.1}$ , which softened even further at later epochs [71].\",\n \"GRB 171205A did not so that much spectral softening.\",\n \"In addition, contrary to smooth light curves of low-luminosity GRBs, the X-ray light curve of GRB 171205A revealed three temporal breaks [24].\",\n \"The lack of jet break seems to be a common feature of low-luminosity GRBs.\",\n \"This may either indicate much wider angle ejecta responsible for afterglow emission, or stratified medium which can dilute the jet break in GRBs and hide its signatures.\",\n \"Another defining feature of these GRBs is the presence of a thermal component, which has also been seen in GRB 171205A [24].\",\n \"Though this feature is quite common in GRBs associated with SNe [9], [74], their origin is still a matter of debate.\",\n \"Shock breakout from the relativistic supernova shell is the most favourable model [75], however, late time photospheric emission from a jet [27], or a thermal emission from a cocoon [78] can also explain this component.\",\n \"However, it appears that the presence of a thermal component is not a unique feature of low-luminosity GRBs.\",\n \"[73], [72] analysed a sample of canonical GRBs and found the evidence of thermal component in a significant number of GRBs, though large radii associated with the blackbody emission argue against the shock breakout model.\",\n \"Finally while low-luminosity GRBs are considered to be a different class, [54] has provided a unified picture of low-luminosity GRBs and cosmological GRBs, in terms of a key difference, namely, the existence of an extended low-mass envelope.\",\n \"In the unified model, the envelope is present in the low-luminosity GRBs, but absent in cosmological GRBs.\",\n \"The lack of envelope allows a jet to launch without any resistance in cosmological GRBs, but the extended envelope in low-luminosity GRBs smothers the jet and deposits a large amount of energy in the stellar envelope, driving a mildly relativistic shock producing low-luminosity GRBs via shock breakout.\",\n \"Our model does not support this picture.\"\n ],\n [\n \"Summary and Conclusions\",\n \"In this paper, we have presented the sub-GHz observations of a low-luminosity GRB 171205A upto around 1000 days.\",\n \"These are the best sampled low frequency light curves of any GRB.\",\n \"For the first time we report a lowest frequency (250–500 MHz) detection of a GRB.\",\n \"Our light curves cover a period of two years.\",\n \"While we are able to see the light curve peak transitions in bands 5 and 4, we missed the peak in band 3 due to the lack of early data.\",\n \"The radio data suggest that the GRB exploded in a wind medium.\",\n \"At sub-GHz frequencies at late time, the afterglow is in $\\\\nu _m<\\\\nu _a<\\\\nu _c$ regime, a common phenomenon seen in late time radio afterglows.\",\n \"The late time Chandra X-ray measurements constrain the jet break to be $t_{\\\\rm jet}>71$ days.\",\n \"Even though GRB 171205A has significant similarities with other low-luminosity GRBs, it deviates from this class in many respects.\",\n \"We suggest that the radio emission arises from both a cocoon and a jet, where the jet is slightly off-axis [39].\",\n \"The early epoch radio emission is dominated by the cocoon surrounding the jet, while the late time radio emission has contribution from the jet.\",\n \"The flatter radio light curves, harder GRB X-ray spectrum, large $E_p$ , shorter $T_{90}$ , kinetic energy - ejecta velocity relation and time dependence of various parameters are consistent with this picture.\",\n \"Our work emphasises the importance of the nature of CBM, which is critical in understanding the evolution of GRB afterglows, and are best revealed by low-frequency radio measurements.\",\n \"We thank the referee for very insightful comments which helped improve the manuscript significantly.\",\n \"We thank Dipankar Bhattacharya for carefully reading the manuscript and providing critical suggestions.\",\n \"B.M.\",\n \"and P.C.\",\n \"acknowledge support of the Department of Atomic Energy, Government of India, under the project no.\",\n \"12-R&D-TFR-5.02-0700.\",\n \"P.C.\",\n \"acknowledges support from the Department of Science and Technology via SwaranaJayanti Fellowship award (file no.DST/SJF/PSA-01/2014-15).\",\n \"We thank the staff of the GMRT that made these observations possible.\",\n \"GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05166"}}},{"rowIdx":3690,"cells":{"text":{"kind":"list like","value":[["MetaInfoNet: Learning Task-Guided Information for Sample Reweighting"],["Abstract Deep neural networks have been shown to easily overfit to biased training data with label noise or class imbalance.","Meta-learning algorithms are commonly designed to alleviate this issue in the form of sample reweighting, by learning a meta weighting network that takes training losses as inputs to generate sample weights.","In this paper, we advocate that choosing proper inputs for the meta weighting network is crucial for desired sample weights in a specific task, while training loss is not always the correct answer.","In view of this, we propose a novel meta-learning algorithm, MetaInfoNet, which automatically learns effective representations as inputs for the meta weighting network by emphasizing task-related information with an information bottleneck strategy.","Extensive experimental results on benchmark datasets with label noise or class imbalance validate that MetaInfoNet is superior to many state-of-the-art methods."],["Introduction","Deep Neural Networks (DNNs) have achieved remarkable success on various computer vision tasks due to their powerful capacity for modeling complex input patterns.","Despite their success, the vulnerability of DNNs has been extensively illustrated in many previous studies [2], [5], [10], [17], [28], [30].","One important drawback of DNNs is that DNNs could easily overfit to biased training data, where the distribution of training data is inconsistent with that of the evaluation data.","There are many different forms of distribution mismatch, leading to poor performance in generalization.","A typical example is class imbalance [7], [15], where the distribution of data across the classes is not equal in the training set.","This issue will sometimes lead to biased training models that does not perform well in practice [3], [31], [42].","Another popular type of distribution mismatch is label noise, which usually happens when the training set is collected from a crowdsourcing system [49] or search engines [4].","It has been shown that a standard CNN can fit any ratio of label noise in the training set and eventually leads to poor generalization performance [2], [53].","Therefore, robust learning from these biased data has become an important and challenging problem in machine learning and computer vision.","Figure: Sample weight distributions on trainingdata of different instatiations (different inputs) of meta weighting network under the 40% Flip-1 label noise case.","It is ideally expected that higher weights are assigned to clean data (green) while lower weights are assigned to noisy data (red).Sample reweighting algorithms [20], [23], [26] are widely studied for robust learning from biased data.","The main idea is to generate different weights for training losses to different samples.","Some existing algorithms design specific weighting function with training loss by simple rules, such as monotonically increasing [23], [26] or monotonically decreasing [8], [20], [53], which means taking samples with larger or smaller loss values as more important ones.","However, these methods need to manually design a specific form of weighting function based on certain assumptions on training data.","To make the learning more automatic and reliable across various biased settings, a popular research line is meta sample weighting, which learns weights for each sample implicitly [35] or automatically learns an explicit weighting function [37].","Specifically, the weighting function could be parameterized as a multilayer perceptron (MLP) network that maps the training loss to sample weight [37], and the training of the parameters could by guided by a small unbiased validation set.","For easy reference, we call this instantiation of meta weighting network (MW-Net) as LossNet [37].","In this paper, we advocate that choosing proper inputs for MW-Net is crucial for desired sample weights in a specific task, while training loss is not always the correct answer.","As shown in Figure REF , we illustrate the sample weight distributions on training data of different instantiations (different inputs) of MW-Net.","We can observe that LossNet tends to produce polarized results, and high weights could be assigned to some noisy data.","This is because the commonly used cross-entropy loss is known to be highly overconfident [27], [40].","Besides, the training loss may not contain rich and flexible information of the original sample for capturing meaningful sample weights, due to the restriction on the fixed loss function.","To alleviate this issue, an intuitive method is to use the logits and labels as inputs (instead of training loss with a specified loss function) to build MW-Net (we call such a instantiation of MW-Net as LogitNet).","As shown in Figure REF , LogitNet would not produce polarized results, while it could still assign high sample weights to some noisy data.","This is because LogitNet may contain too much redundant information with the unprocessed logits.","To address this problem, we propose a novel meta-learning algorithm, MetaInfoNet, which automatically learns effective representations as inputs for MW-Net by emphasizing task-related information with an information bottleneck strategy.","As shown in Figure REF , our proposed MetaInfoNet can learn smooth and meaningful sample weights, and the trend that higher weights are assigned to clean data while lower weights are assigned to noisy data is clearly demonstrated.","Finally, we conduct extensive experiments on simulated and real-world datasets with class imbalance or label noise, to show that MetaInfoNet significantly improves the robustness of deep learning on training data under various biased settings.","Empirical results demonstrate that the robustness of deep models trained by our proposed MetaInfoNet is superior to many state-of-the-art methods."],["Related Work","Sample Reweighting Methods.","The idea of sample reweighting has been commonly used in the machine learning literature.","For example, hard example mining downsamples the majority class and exploits the most challenging examples [26].","Similarly, Focal loss emphasizes harder examples by soft weighting [23].","On the contrary, self-paced learning (SPL) takes samples with smaller loss values as more important ones firstly [20].","Despite their success under some specific settings, these methods need to manually design a particular form of weighting function based on certain assumptions on training data, which might be impractical.","Rather than predefined by human experts, MentorNet uses a bidirectional LSTM network to learn a curriculum from data with label noises [16].","However, the weighting function of MentorNet is too complicated and would overfit to the biased data.","Meta Learning Methods for Robustness.","Meta-learning algorithms are introduced to improve robustness of deep learning in the form of sample reweighting.","The first work is L2RW [35], which implicitly learns the weights without a pre-defined weighting function, then uses a small unbiased validation set to guide the training of its parameters.","In contrast, LossNet [37] parameterizes the weighting function as an MLP network explicitly, mapping from training loss to sample weight.","However, the capacity of LossNet is limited by its input with the fixed loss function.","Learning with Class Imbalance.","In addition to sample reweighting, there are other methods to handle the class imbalance issue in deep learning.","For example, some methods try to transfer the knowledge learned from major classes to minor classes [7], [46].","The metric learning based methods have also been developed to effectively exploit the tailed data to improve the generalization ability, e.g., triple-header loss [15] and range loss [52].","Learning with Label Noise.","For handling noisy label issues [9], [13], [21], [24], [29], [39], [44], [45], some other algorithms focus on estimating the label transition matrix.","For example, F-correction [32] proposed a loss correction approach by heuristically estimating the noise transition matrix.","In these approaches, the quality of noise rate estimation is a critical factor for improving robustness.","However, noise rate estimation is challenging, especially on datasets with a large number of classes.","Another popular research line of handling noisy labels is to train models on small-loss instances, which can be viewed as a hard version of sample weighting [12], [16], [47].","Information Bottleneck.","Information Bottleneck (IB) method was initially proposed in [41], where the idea can be formulated as a variational principle of minimizing the mutual information between the input and the learned representation, while preserving the information about the learning task.","The IB method has been successfully applied to supervised learning [1], generative modeling [33] and reinforcement learning [33], [43].","In this work, we apply the IB algorithm to enforce the input of MW-Net to focus on the relevant information implicitly defined by the learning task."],["Preliminaries","In this section, we introduce the problem setting of learning from biased training data with sample reweighting and the formulation of LossNet [37], which is a representative related work."],["Problem Setting","For multi-class classification with $K$ classes, we suppose the biased training dataset with $N$ samples is given as $\\mathcal {D}^{\\mathrm {train}} = \\lbrace x_i, y_i\\rbrace ^N_{i=1}$ , where $x_i$ is the $i$ -th instance with its observed label as $y_i \\in \\lbrace 1, \\dots , K\\rbrace $ .","Similar to L2RW and LossNet, we assume that there is a small unbiased and clean meta dataset with $M$ samples $\\mathcal {D}^{\\mathrm {val}} = \\lbrace x^{\\mathrm {val}}_i, y^{\\mathrm {val}}_i\\rbrace ^M_{i=1}$ and $M \\ll N$ , representing the meta-knowledge of ground-truth sample-label association.","For the classifier to be trained, we denote it as $f_{\\Theta }(x)$ , where $\\Theta $ is the parameters of the classifier.","Generally, we optimize the classifier network by minimizing the training loss: $\\frac{1}{N} \\sum ^N_{i=1} L^{\\mathrm {train}}_i(\\Theta ) = \\frac{1}{N} \\sum ^N_{i=1} \\ell (f_{\\Theta }(x_i), y_i)$ , where $\\ell $ denotes the employed loss function (e.g., cross entropy loss), and each input example is weighted equally.","For enhancing the robustness of training on the biased training data, we aim to assign weight $w_i$ on the loss of the $i$ -th sample.","Without loss of generality, the weighting function can be formulated as $\\mathcal {V}_{\\Phi }((x_i, y_i), f_{\\Theta })$ where $\\Phi $ denotes the parameters of the weighting function.","Then the optimal parameter $\\Theta $ of classifier is calculated by minimizing the following weighted loss: $\\begin{split}\\Theta ^{*}(\\Phi ) &= \\underset{\\Theta }{\\arg \\min } \\ \\mathcal {L}^{\\mathrm {train}}(\\Theta ;\\Phi )\\\\&\\triangleq \\underset{\\Theta }{\\arg \\min } \\frac{1}{N} \\sum _{i=1}^{N} w_i L^{\\mathrm {train}}_i(\\Theta ),\\end{split}$ where $w_i = \\mathcal {V}((x_i, y_i), f_{\\Theta }; \\Phi )$ denotes the example weight and $L^{\\mathrm {train}}_i(\\Theta ) = \\ell (f_{\\Theta }(x_i), y_i)$ denotes the loss on the training example $(x_i,y_i)$ ."],["Formulation of LossNet","LossNet [37] formulates the weighting function $\\mathcal {V}$ as an MLP network, and automatically trains it in a meta-learning manner.","Specifically, the MLP network learns a mapping from training loss to sample weight, $w_i = \\mathcal {V}_{\\Phi }(L_i^{\\mathrm {train}}(f_{\\Theta }(x_i), y_i))$ , where $\\Phi $ denotes the parameters of the MLP net.","In the training process, the optimal parameter $\\Phi ^{*}$ of LossNet is calculated by minimizing the following meta loss: $\\begin{split}\\Phi ^{*} &= \\underset{\\Phi }{\\arg \\min } \\ \\mathcal {L}^{\\mathrm {meta}}(\\Theta ^*(\\Phi ))\\\\&\\triangleq \\underset{\\Phi }{\\arg \\min } \\frac{1}{M} \\sum _{i=1}^{M} L_{i}^{\\mathrm {meta}}(\\Theta ^*(\\Phi )),\\end{split}$ where $L_{i}^{\\mathrm {meta}}(\\Theta ^*) = \\ell (f_{\\Theta ^*}(x^{\\mathrm {val}}_i), y_i^{\\mathrm {val}}))$ denotes the meta loss on the validation example $(x^{\\mathrm {val}}_i,y^{\\mathrm {val}}_i)$ .","As we analyzed before, the training loss may not contain rich and flexible information of the original sample for capturing meaningful sample weights, due to the restriction on the fixed loss function.","As the commonly used cross-entropy loss is known to be highly overconfident [27], [40], LossNet tends to produce polarized results, and high weights could be assigned to some noisy data."],["The Proposed Approach","In this section, we propose to automatically learn proper inputs for MW-Net, thereby adapting to various biased settings.","[t] The Meta Weighting Algorithm 1.1 Training dataset $\\mathcal {D}$ , meta dataset $\\mathcal {D}_m$ , batch size $n,m$ , max iterations $T$ .","Parameter $\\Theta $ of classifier $f$ .","Initialize parameters of Classifier and MW-Net $t=0$ $\\mathrm {to}$ $T$ $\\lbrace x_i,y_i\\rbrace _{i=1}^n \\leftarrow \\text{SampleMiniBatch}(\\mathcal {D},n)$ $\\lbrace x^{\\mathrm {val}}_i,y^{\\mathrm {val}}_i\\rbrace _{i=1}^m \\leftarrow \\text{SampleMiniBatch}(\\mathcal {D}^{\\mathrm {val}},m)$ $ z_i = f_{\\Theta }(x_i), \\tilde{w} = \\mathcal {V}_{\\Phi ^{(t)}}(z,y)$ $w_i \\leftarrow \\frac{\\tilde{w}_i}{\\sum _{j} \\tilde{w}_j +\\delta \\left(\\sum _{j} \\tilde{w}_j\\right)}$ $\\mathcal {L}^{\\mathrm {train}} = \\frac{1}{n} \\sum _{i=1}^{n} w_i \\cdot \\ell (z_i, y_i)$ $\\nabla \\Theta \\leftarrow \\text{ BackwardAD }\\left(\\mathcal {L}^{\\mathrm {train}}, \\Theta \\right)$ $\\widehat{\\Theta } \\leftarrow \\Theta -\\alpha \\nabla \\Theta $ $\\mathcal {L}^{\\mathrm {meta}} = \\frac{1}{m} \\sum _{i=1}^{m} \\ell (f_{\\widehat{\\Theta }}(x^{\\mathrm {val}}_i), y^{\\mathrm {val}}_i)$ Update MW-Net $\\mathcal {V}$ to $\\Phi ^{(t+1)}$ by $\\mathcal {L}^{\\mathrm {meta}}$ $\\tilde{w} = \\mathcal {V}_{\\Phi ^{(t+1)}}(z,y); w_i \\leftarrow \\frac{\\tilde{w}_i}{\\sum _{j} \\tilde{w}_j+\\delta \\left(\\sum _{j} \\tilde{w}_j\\right)}$ $\\mathcal {L}^{\\mathrm {train}} = \\frac{1}{n} \\sum _{i=1}^{n} w_i L^{\\mathrm {train}}_i(z_i, y_i)$ Update Classifier $f$ to $\\Theta ^{(t+1)}$ by $\\mathcal {L}^{\\mathrm {train}}$"],["A General Framework for Meta Weighting","As shown in Figure REF , we define a general framework for meta weighting, which contains a MW-Net that takes information from the samples $x_i$ and its given labels $y_i$ (instead of training loss) as inputs.","To abstract information from the sample $x_i$ , we adopt the outputs of the classifier $f$ , (i.e., $f_{\\Theta }(x_i)$ ) for MW-Net.","In this way, LossNet can be interpreted as an instantiation of MW-Net.","Intuitively, if we calculate the cross-entropy loss with the logit and its corresponding label, we can exactly recover the LossNet model.","The training loss can be seen as a representation of information abstracted from samples $x_i$ and its given labels $y_i$ .","The left of Figure REF shows that LossNet is an instantiation of MW-Net.","Under the general framework in Figure REF , LossNet could be replaced by a more powerful instantiation of MW-Net, thereby achieving better performance.","Similar to L2RW [35] and LossNet [37], calculating the optimal parameters $\\Theta ^*$ and $\\Phi ^*$ requires two nested loops of optimization.","To improve the optimization efficiency, we adopt an online strategy to alternatively update the classifier $f$ with $\\Theta $ and the MW-Net $\\mathcal {V}$ with $\\Phi $ through a single optimization loop.","We describe the details of the training process in Algorithm .","In general, the training process can be separated into three parts: Virtually updating parameters of classifier.","In the first part, we randomly sample a mini-batch of training samples ${(x_i, y_i), 1 \\le i \\le n}$ from the training dataset, where $n$ is the mini-batch size.","Then the virtual updating of the classifier network parameter can be formulated by moving the current $\\Theta ^{(t)}$ along the descent direction of the objective loss in Eq.","(REF ) on a mini-batch training data (lines 5-9) : $\\nonumber \\widehat{\\Theta }^{(t)}(\\Phi ) = \\Theta ^{(t)} - \\alpha \\frac{1}{n} \\left.","\\sum _{i=1}^{n} \\mathcal {V}_{\\Phi }((f_{\\Theta }(x_i), y_i)) \\nabla _{\\Theta } L^{\\mathrm {train}}_{i}\\right|_{\\Theta ^{(t)}},$ where $\\alpha $ is the step size.","Updating parameters of MW-Net.","After receiving the feedback of the virtually updated classifier, i.e., $\\widehat{\\Theta }^{(t)}(\\Phi )$ , the parameter $\\Phi $ of the MW-Net can then be readily updated with Eq.","(REF ) calculated on the meta dataset (lines 10-11): $\\Phi ^{(t+1)}=\\Phi ^{(t)}-\\beta \\frac{1}{m} \\left.\\sum _{i=1}^{m} \\nabla _{\\Phi } L_{i}^{\\mathrm {meta}}(\\widehat{\\Theta }^{(t)}(\\Phi ))\\right|_{\\Phi ^{(t)}},$ where $\\beta $ is the step size.","Updating parameters of classifier.","Then, the updated $\\Phi ^{(t+1)}$ is employed to ameliorate the parameter $\\Theta $ of the classifier (lines 12-14), i.e., $\\nonumber &\\Theta ^{(t+1)}=\\Theta ^{(t)}\\\\\\nonumber &\\quad \\quad \\quad \\quad -\\alpha \\frac{1}{n}\\left.\\sum _{i=1}^{n} \\mathcal {V}_{\\Phi ^{(t+1)}}\\left((f_{\\Theta }(x_i), y_i)\\right) \\nabla _{\\Theta } L_{i}^{\\mathrm {train}}\\right|_{\\Theta ^{(t)}},$ Figure: Three different instantiations of MW-Net, including LossNet, LogitNet, and MetaInfoNet."],["LogitNet: An Instatiation of MW-Net","Now the question becomes how to design MW-Net, for mapping information from the sample and its given label to weight.","An intuitive idea is to simply take the model output of $x_i$ , i.e., $f_{\\Theta }(x)$ .","For the given label $y_i$ , we employ a label embedding layer to produce an embedding vector $e(y_i)\\in \\mathbb {R}^K$ , which has the same dimension as $f_{\\Theta }(x)$ .","It is a fully connected layer that projects the given label to a dense vector.","The obtained embedding vector can be seen as a latent vector for describing the properties of a specific class, in the context of MW-Net.","Since MW-Net adopts two pathways to extract information from $x_i$ and $y_i$ , it is intuitive to combine the vectors from the two pathways (i.e., $f_{\\Theta }(x)$ and $e(y_i)$ ) by concatenating them.","However, a vector concatenation cannot account for any interactions between samples and its given labels.","Therefore, we apply an element-wise product to combine them as the input for an MLP in MW-Net, that is, $w_i=\\text{MLP}(f_{\\Theta }(x_i) \\odot e(y_i))$ , where $\\odot $ denotes the element-wise product.","In this way, the combined vector can capture the interaction information between the sample and its given label before we feed it into an MLP.","For easy reference, we call this instantiation of MW-Net as LogitNet.","The structure of LogitNet is shown on the center of Figure REF ."],["MetaInfoNet: An Instatiation of MW-Net with Information Bottleneck","Although LogitNet introduces more information to the weighting function, it also leads to a new problem: the unprocessed logits may contain too much redundant information for the weighting task.","To “squeeze out\" the redundant information of the inputs for LogitNet, we propose a novel algorithm by information bottleneck (IB) principle [1], [36], [41], called MetaInfoNet.","Instead of directly using the output of classifier $z = f_{\\Theta }(x)$ in LogitNet, we aim to learn a representation $\\psi $ that captures the most relevant information from the logit $z$ and removes redundant formation, with respect to the weighting task.","In this way, the weight of sample $x_i$ can be generated by $w_i=\\text{MLP}(\\psi _i \\odot e(y_i))$ .","Figure: The detailed structure of IB layer.Concretely, we propose an information-theoretic regularization by limiting the mutual information between the logit and the learned representation: $I(Z;\\Psi ) \\le I_c,$ , where $Z$ is a random variable (for the logit vector) with a probability density function $p_{Z}(z)$ , $\\Psi $ is a random variable (for the learned representation vector) with a probability density function $p_{\\Psi }(\\psi )$ and $I_c$ is a constant that represents threshold of the mutual information.","The constraint on mutual information can be viewed as a penalty term restricting the information from the logit vector to the learned representation vector.","Consequently, for learning an effective representation $\\psi $ , we add the mutual information constraint for optimizing MW-Net: $\\mathcal {L}^{\\mathrm {meta}}(\\Theta (\\Phi )) = \\sum \\nolimits _{i=1}^M L^{\\mathrm {meta}}_i(\\Theta (\\Phi )), \\text{s.t.}","\\ I(Z;\\Psi ) \\le I_c,$ In practice, we propose to minimize the following objective using the information bottleneck Lagrangian: $\\sum \\nolimits _{i=1}^M L^{\\mathrm {meta}}_i(\\Theta (\\Phi )) + \\lambda I(Z;\\Psi ),$ where $\\lambda $ is the Lagrange multiplier.","As $\\lambda $ increases, the information from $Z$ to $\\Psi $ becomes denser while keeping relevant to the meta learning task.","In the above equation, $I(Z;\\Psi )$ is defined as: $I(Z;\\Psi ) &=\\iint p\\left(z, \\psi \\right) \\log \\frac{p\\left(z, \\psi \\right)}{p\\left(z\\right) p\\left(\\psi \\right)} \\mathrm {d} z \\mathrm {d} \\psi \\\\&=\\iint p\\left(z\\right) p_{ib}\\left(\\psi \\mid z\\right) \\log \\frac{p_{ib}\\left(\\psi \\mid z\\right)}{p\\left(\\psi \\right)} \\mathrm {d} z \\mathrm {d} \\psi \\nonumber $ where $p(z, \\psi )$ is the joint probability of $z$ and $\\psi $ , and $p_{ib}$ denotes the information bottleneck (IB) layer.","Unfortunately, computing the marginal distribution $p(\\psi ) = \\int p\\left(\\psi \\mid z\\right) p\\left(z\\right) d z$ is quite challenging as we do not know the prior distribution of $p(z)$ .","With the help of variational information bottleneck [1], we use a Gaussian approximation $h(\\psi )$ of the marginal distribution $p(\\psi )$ and view $p_{ib}$ as multivariate variational encoders (see Figure REF ).","Since $D_{\\mathrm {KL}}\\left[p\\left(\\psi \\right) \\Vert h\\left(\\psi \\right)\\right] \\ge 0$ , where the $D_{\\mathrm {KL}}$ is the Kullback-Leibler divergence, we expand the KL term and get $\\int p\\left(\\psi \\right) \\log p\\left(\\psi \\right) d \\psi \\ge \\int p\\left(\\psi \\right) \\log h\\left(\\psi \\right) d \\psi $ , an upper bound on the mutual information $I(Z;\\Psi )$ can be obtained via the KL divergence: $\\nonumber I\\left(Z ; \\Psi \\right) & \\le \\int p\\left(z\\right) p_{ib}\\left(\\psi \\mid z\\right) \\log \\frac{p_{ib}\\left(\\psi \\mid z\\right)}{h\\left(\\psi \\right)} d z d \\psi \\\\\\nonumber &=\\mathbb {E}_{z \\sim p\\left(z\\right)}\\left[D_{\\mathrm {KL}}\\left[p_{ib}\\left(\\psi \\mid z\\right) \\Vert h\\left(\\psi \\right)\\right]\\right],$ which further provides an upper bound $\\widetilde{\\mathcal {L}}^{\\mathrm {meta}}(\\Theta (\\Phi ))$ on the regularized empirical objective that we minimize: $\\widetilde{\\mathcal {L}}^{\\mathrm {meta}}(\\Theta (\\Phi )) & = \\sum \\nolimits _{i=1}^M L^{\\mathrm {meta}}_i(\\Theta (\\Phi )) \\\\\\nonumber & + \\lambda \\sum \\nolimits _{j=1}^N\\left[D_{\\mathrm {KL}}\\left[p_{ib}\\left(\\psi _j \\mid z_j\\right) \\Vert h\\left(\\psi _j\\right)\\right]\\right].$ Recall that the learned representation $\\psi $ via IB layer is used to generate the weight $w_i=\\text{MLP}(\\psi _i \\odot e(y_i))$ , then we optimize the MW-Net with the new meta loss in Eq.","(REF ) to control the information capacity of the learned representation $\\psi $ , while keeping the task-relevant information.","The network structure of the IB layer is shown in Figure REF .","Specifically, a linear layer is applied to map logit $z$ to a multivariate normal distribution.","Then we use the “re-parameterization\" trick [18] to sample a representation $\\psi $ from the learned distribution, parameterized by a mean vector $\\mu $ and a variance vector $\\sigma $ .","Another alternative trick is to implement skip connection by concatenating the representation $\\phi $ and the origin logit $z$ .","In such a manner, we emphasize the most relevant information in $\\psi $ , with respect to the learning task, while keeping intact information from the logit $z$ .","To keep aligned with the dimension of label embedding for the element-wise product, we add a linear layer to scale the dimension of the learned representation $\\psi $ ."],["Convergence Analysis","Here, we further provide the convergence analysis of MetaInfoNet.","We show that MetaInfoNet converges to the critical points of both the meta and training loss function under some mild conditions in the following two theorems.","Theorem 1 Suppose the loss function $L$ is Lipshchitz smooth with constant $\\rho $ , and $\\mathcal {V}(\\cdot )$ is differential with a $\\delta $ -bounded gradients and twice differential with its Hessian bounded by $\\mathfrak {B}$ , and the loss function $L$ has $\\gamma $ -bounded gradients with respect to training/meta data.","Let the learning rate $\\alpha _t$ (for updating classifier) satisfies $\\alpha _t=\\min \\lbrace 1,\\frac{k}{T}\\rbrace $ , for some constant $k>0$ , such that $\\frac{k}{T}<1$ , and the learning rate $\\beta _t$ (for updating meta weight net) satisfies $\\beta _t=\\min \\lbrace \\frac{1}{L},\\frac{c}{\\sigma \\sqrt{T}}\\rbrace $ for some constant $c>0$ where $\\sigma $ is the variance of drawing uniformly mini-batch sample at random, such that $\\frac{\\sigma \\sqrt{T}}{c}>\\rho $ and $\\sum _{t=1}^{\\infty }\\beta _t\\le \\infty ,\\sum _{t=1}^{\\infty }\\beta _t^2\\le \\infty $ .","Then we have $\\nonumber \\lim \\limits _{t\\rightarrow \\infty }\\mathbb {E}\\Big [\\left\\Vert \\nabla _{\\Theta }\\mathcal {L}^{\\mathrm {train}}(\\Theta ^{(t)};\\Phi ^{(t)})\\right\\Vert ^2_2\\Big ] = 0.$ Theorem 2 When the conditions in Theorem REF holds, our proposed algorithm can achieve $\\mathbb {E}[\\left\\Vert \\nabla \\mathcal {L}^{\\mathrm {meta}}(\\Theta ^{(t)})\\right\\Vert _2^2]\\le \\epsilon $ in $\\mathcal {O}(1/\\epsilon ^2)$ steps, more specifically, $\\nonumber \\min _{0\\le t\\le T}\\mathbb {E}[\\left\\Vert \\nabla _{\\Theta } \\mathcal {L}^{\\mathrm {meta}}(\\Theta ^{(t)})\\right\\Vert _2^2]\\le \\mathcal {O}(\\frac{O}{\\sqrt{T}}),$ where $C$ is some constant independent of the convergence process.","It is worth noting that we have shown that our proposed method MetaInfoNet and LossNet [37] are different instantiations of MW-Net.","Therefore, MetaInfoNet shares the same convergence properties as LossNet if we adopt the same assumptions for analyzing the two methods.","We omit the proofs of the two theorems, as they are very similar to those proofs in [37]."],["Experiments","In this section, we first implement experiments on simulated datasets with class imbalance and noisy labels.","Then two real-world datasets with biased training data are also used to verify the effectiveness of our proposed algorithms."],["Experiment setup","Datasets.","For the class imbalance setting, we verify the effectiveness of our proposed algorithm on the Long-Tailed version of CIFAR10 and CIFAR100 datasets [6].","Specifically, we reduce the number of training examples per class according to an exponential function $n = n_i\\mu ^i$ , where $i$ is the class index, $n_i$ is the original number of training samples in the largest class divided by the smallest.","Following LossNet [37], we randomly select ten images per class in the validation set as the meta dataset.","For noisy label settings, we conduct experiments on simulated noisy datasets, including CIFAR10 and CIFAR100 [19].","We study two types of label noise on the training set: 1) Flip noise.","Following LossNet [37], we mainly consider the case that the label of each example is flipped to two similar classes with equal probability $\\frac{p}{2}$ (Flip-2).","2) Uniform noise.","The label of each example is changed to a random class with probability $p$ independently.","To verify the effectiveness of our algorithms in various settings, we also compare our algorithms with existing meta reweighting algorithms in a harder case: One class is randomly selected as the similar class with probability $p$ (Flip-1).","In these cases, 1000 images with clean labels in the validation set are randomly selected as the meta dataset.","To verify the effectiveness of the proposed method on real-world data, we conduct experiments on the Clothing1M dataset [48], containing 1 million images of clothing obtained from online shopping websites with 14 categories, e.g., T-shirt, Shirt, Knitwear.","The labels are generated by using surrounding texts of the images provided by the sellers, and therefore contain many errors.","We use the 7k clean data as the meta dataset.","For preprocessing, we resize the image to $256 \\times 256$ , crop the middle $224 \\times 224$ as input, and perform normalization.","Moreover, we also compare our proposed algorithms with existing meta reweighting algorithms on ANIMAL10N dataset [38], which contains 5 pairs of confusing animals with a total of 55,000 images.","In our experiments, the training dataset contains 50,000 pictures with noisy labels, and 1000 pictures with clean labels are randomly selected as the meta dataset.","The rest 4000 images are set as the test dataset.","Network Structure and Optimizer.","For experiments on CIFAR10 and CIFAR100, we train ResNet-32 [14] for the class imbalance settings.","For the settings of label noise, we adopt WRN-28-10 [51] for uniform noise and ResNet-32 for flip noise.","Specifically, SGD optimizer is applied with a momentum 0.9, a weight decay $5 \\times 10^{-4}$ , an initial learning rate 0.1, and the batch size is set to 100.","The learning rate of the classifier network is divided by 10 after 80 epochs and 100 epochs (for a total of 120 epochs), and after 30 epochs and 40 epochs (for a total of 50 epochs).","For experiments on Clothing1M, we use ResNet-50 pre-trained on ImageNet, following the previous works [32], [37], [39].","About the optimizer, we use SGD with a momentum 0.9, a weight decay $10^{-3}$ , and an initial learning rate 0.01, and batch size 32.","The learning rate of ResNet-50 is divided by 10 after 5 epochs (for a total 10 epochs).","For experiments on ANIMAL10N, we also use ResNet-32.","The setting of the SGD optimizer is the same as that on CIFAR10.","We run 100 epochs in total, and the learning rate of the classifier network is divided by 10 after 50 epochs and 75 epochs.","For the training of meta weighting network, we set the learning rate to $10^{-3}$ and set the weight decay to $5 \\times 10^{-4}$ .","About the $\\lambda $ in Eq.","(REF ), we search it in $[0, 0.01, 0.03, 0.1, 0.3, 1]$ .","Additionally, we can set an interval parameter to control the updating of MW-Net, e.g., when the interval is 10, we update the MW-net every 10 iterations.","We implement all methods with default parameters by PyTorch, and conduct all the experiments on NVIDIA Tesla V100 GPUs.","We repeated the experiments 5 times with different random seeds for network initialization and label noise generation.","Compared Methods.","For class imbalance setting, we compare our proposed algorithm with: 1) Standard, which simply uses CE loss to train the DNNs with equal weights; 2) Focal loss [23], which emphasizes harder examples by sample reweighting; 3) Class-balanced [6], which represents one of the state-of-the-arts of the predefined sample reweighting methods; 4) Fine-tuning, which finetunes the standard model on the meta dataset to further enhance its performance; 5) L2RW [35], which leverages an additional meta dataset to adaptively assign weights on training samples; 6) LossNet [37], which automatically learns an explicit loss-weight function in a meta-learning manner.","For noisy label setting, the compared methods include: 1) Standard; 2) Bootstrap [34]; 3) S-Model [11]; 4) SPL [20]; 5) Focal Loss [23]; 6) Co-teaching [12]; 7) D2L [25]; 6) Fine-tuning; 7) MentorNet [16]; 8) L2RW [35]; 9) LossNet [37].","Table: Test accuracy (%) on long-tailed CIFAR-10 and CIFAR-100.","The best results are highlighted in bold.Table: Average test accuracy (%) with standard deviation on synthesized datasets under Flip-2 noise (over 5 trials).","The best results are highlighted in bold."],["Experiments with Class Imbalance","Table REF shows the test accuracy of different algorithms on long-tailed CIFAR10 and long-tailed CIFAR100.","As we can see, our proposed algorithms work better than all the baselines in most of the long-tailed cases, which demonstrates their robustness against class imbalance.","When imbalance factor is 1, e.g., there are the same number of samples in all classes, the fine-tuning method obtains the best performance on both CIFAR10 and CIFAR100, and our algorithms also get comparable performance.","As the imbalance factor rises from 10 to 200, the datasets become more and more unbalanced and the accuracy of all algorithms decreases gradually.","We observe that LogitNet achieves better or comparable results compared to LossNet in most cases, which indicates introducing more information can improve the effectiveness of MW-Net.","Moreover, MetaInfoNet consistently performs better than LogitNet in most cases, verifying the effectiveness of the information bottleneck method while learning the representation.","Table: Average test accuracy (%) with standard deviation on CIFAR10 with WRN-28-10 under Unif noise (over 5 trials).","The best results are highlighted in bold."],["Experiments with Noisy Labels","Table REF shows the test accuracy of ResNet-32 averaged over 5 repetitions on both CIFAR10 and CIFAR100 under Flip-2 noise.","We can observe that MetaInfoNet achieves the best performance across both datasets and all noise rates, indicating the effectiveness of our proposed architecture.","When the noise rate is 0%, MetaInfoNet also gets comparable results compared to the Fine-tune baseline while performs better than other meta-reweighting algorithms.","Additionally, the results of LogitNet are similar to those of LossNet, which shows that simply introducing more information cannot improve the capacity of meta sample weighting in this case.","Table REF presents the test accuracy of WRN-28-10 averaged over 5 repetitions on CIFAR10 under Unif noise.","It can be observed that all the meta weighting algorithms, including MetaInforNet, LogitNet, LossNet and L2RW, perform the best among all algorithms, which demonstrates their robustness.","The results also indicate that uniform noise is a relatively simple biased setting for meta weighting algorithms.","Now we turn to a more difficult noisy setting, Flip-1 noise, that each example's label is flipped to one similar class with total probability $p$ .","As shown in Table REF , Best denotes the scores of the epoch where the validation accuracy is optimal, and Last denotes the average accuracy over the last 10 epochs at the end of the training stage.","As we can see, LossNet performs poorly even gets lower test accuracy than L2RW, while MetaInfoNet attains more than 8% improvement over LossNet.","Moreover, MetaInfoNet keeps the best performance in both CIFAR10 and CIFAR100 and all noise rates.","Specifically, in the 20% label noise cases, all four meta weighting algorithms could maintain almost the $best$ accuracy in the last 10 epochs.","When it improves to the 40% label noise cases, MetaInfoNet keeps the advantage while the other three algorithms' performance decrease dramatically after reaching the top.","The result shows that MetaInfoNet is more robust to label noise compared to the other meta-weighting algorithms and has a better capacity to adapt to different biased settings in training data."],["Experiments on Real-world Datasets","Table REF shows the classification accuracy on the Clothing1M test set.","Specifically, MetaInfoNet achieves the best accuracy, while LogitNet performs worse than LossNet.","The results show that the IB layer is an efficient way to improving the meta-weighting framework with logits and labels as inputs.","Table: Classification accuracy (%) on Clothing1M.We also compare our proposed algorithms to the Finetune baseline and the other meta-weighting algorithms on the ANIMAL-10N dataset, as shown in Table REF .","Note that the noise rate is about 8% but it is difficult for even people to distinguish between categories, the performance gains of meta weighting baselines are not huge compared to the Finetune method.","In this case, MetaInfoNet still obtains nearly 2% improvement over the other methods on best.","After all epochs, MetaInfoNet maintains this advantage over the best baseline method.","Table: Classification accuracy (%) on ANIMAL-10N."],["Conclusion","In this paper, we propose an effective approach called MetaInfoNet to improve the robustness of deep neural networks under various biased settings in training data.","Compared with current meta reweighting algorithms that directly map loss to weight for each sample, MetaInfoNet could automatically learn effective representations as inputs for the meta weighting network by emphasizing task-related information with an information bottleneck strategy.","The Empirical results on simulated and real-world datasets demonstrate that the robustness of deep models trained by our proposed approach is superior to many state-of-the-art approaches in general biased settings, such as class imbalance, label noise, and more complicated real cases."]],"string":"[\n [\n \"MetaInfoNet: Learning Task-Guided Information for Sample Reweighting\"\n ],\n [\n \"Abstract Deep neural networks have been shown to easily overfit to biased training data with label noise or class imbalance.\",\n \"Meta-learning algorithms are commonly designed to alleviate this issue in the form of sample reweighting, by learning a meta weighting network that takes training losses as inputs to generate sample weights.\",\n \"In this paper, we advocate that choosing proper inputs for the meta weighting network is crucial for desired sample weights in a specific task, while training loss is not always the correct answer.\",\n \"In view of this, we propose a novel meta-learning algorithm, MetaInfoNet, which automatically learns effective representations as inputs for the meta weighting network by emphasizing task-related information with an information bottleneck strategy.\",\n \"Extensive experimental results on benchmark datasets with label noise or class imbalance validate that MetaInfoNet is superior to many state-of-the-art methods.\"\n ],\n [\n \"Introduction\",\n \"Deep Neural Networks (DNNs) have achieved remarkable success on various computer vision tasks due to their powerful capacity for modeling complex input patterns.\",\n \"Despite their success, the vulnerability of DNNs has been extensively illustrated in many previous studies [2], [5], [10], [17], [28], [30].\",\n \"One important drawback of DNNs is that DNNs could easily overfit to biased training data, where the distribution of training data is inconsistent with that of the evaluation data.\",\n \"There are many different forms of distribution mismatch, leading to poor performance in generalization.\",\n \"A typical example is class imbalance [7], [15], where the distribution of data across the classes is not equal in the training set.\",\n \"This issue will sometimes lead to biased training models that does not perform well in practice [3], [31], [42].\",\n \"Another popular type of distribution mismatch is label noise, which usually happens when the training set is collected from a crowdsourcing system [49] or search engines [4].\",\n \"It has been shown that a standard CNN can fit any ratio of label noise in the training set and eventually leads to poor generalization performance [2], [53].\",\n \"Therefore, robust learning from these biased data has become an important and challenging problem in machine learning and computer vision.\",\n \"Figure: Sample weight distributions on trainingdata of different instatiations (different inputs) of meta weighting network under the 40% Flip-1 label noise case.\",\n \"It is ideally expected that higher weights are assigned to clean data (green) while lower weights are assigned to noisy data (red).Sample reweighting algorithms [20], [23], [26] are widely studied for robust learning from biased data.\",\n \"The main idea is to generate different weights for training losses to different samples.\",\n \"Some existing algorithms design specific weighting function with training loss by simple rules, such as monotonically increasing [23], [26] or monotonically decreasing [8], [20], [53], which means taking samples with larger or smaller loss values as more important ones.\",\n \"However, these methods need to manually design a specific form of weighting function based on certain assumptions on training data.\",\n \"To make the learning more automatic and reliable across various biased settings, a popular research line is meta sample weighting, which learns weights for each sample implicitly [35] or automatically learns an explicit weighting function [37].\",\n \"Specifically, the weighting function could be parameterized as a multilayer perceptron (MLP) network that maps the training loss to sample weight [37], and the training of the parameters could by guided by a small unbiased validation set.\",\n \"For easy reference, we call this instantiation of meta weighting network (MW-Net) as LossNet [37].\",\n \"In this paper, we advocate that choosing proper inputs for MW-Net is crucial for desired sample weights in a specific task, while training loss is not always the correct answer.\",\n \"As shown in Figure REF , we illustrate the sample weight distributions on training data of different instantiations (different inputs) of MW-Net.\",\n \"We can observe that LossNet tends to produce polarized results, and high weights could be assigned to some noisy data.\",\n \"This is because the commonly used cross-entropy loss is known to be highly overconfident [27], [40].\",\n \"Besides, the training loss may not contain rich and flexible information of the original sample for capturing meaningful sample weights, due to the restriction on the fixed loss function.\",\n \"To alleviate this issue, an intuitive method is to use the logits and labels as inputs (instead of training loss with a specified loss function) to build MW-Net (we call such a instantiation of MW-Net as LogitNet).\",\n \"As shown in Figure REF , LogitNet would not produce polarized results, while it could still assign high sample weights to some noisy data.\",\n \"This is because LogitNet may contain too much redundant information with the unprocessed logits.\",\n \"To address this problem, we propose a novel meta-learning algorithm, MetaInfoNet, which automatically learns effective representations as inputs for MW-Net by emphasizing task-related information with an information bottleneck strategy.\",\n \"As shown in Figure REF , our proposed MetaInfoNet can learn smooth and meaningful sample weights, and the trend that higher weights are assigned to clean data while lower weights are assigned to noisy data is clearly demonstrated.\",\n \"Finally, we conduct extensive experiments on simulated and real-world datasets with class imbalance or label noise, to show that MetaInfoNet significantly improves the robustness of deep learning on training data under various biased settings.\",\n \"Empirical results demonstrate that the robustness of deep models trained by our proposed MetaInfoNet is superior to many state-of-the-art methods.\"\n ],\n [\n \"Related Work\",\n \"Sample Reweighting Methods.\",\n \"The idea of sample reweighting has been commonly used in the machine learning literature.\",\n \"For example, hard example mining downsamples the majority class and exploits the most challenging examples [26].\",\n \"Similarly, Focal loss emphasizes harder examples by soft weighting [23].\",\n \"On the contrary, self-paced learning (SPL) takes samples with smaller loss values as more important ones firstly [20].\",\n \"Despite their success under some specific settings, these methods need to manually design a particular form of weighting function based on certain assumptions on training data, which might be impractical.\",\n \"Rather than predefined by human experts, MentorNet uses a bidirectional LSTM network to learn a curriculum from data with label noises [16].\",\n \"However, the weighting function of MentorNet is too complicated and would overfit to the biased data.\",\n \"Meta Learning Methods for Robustness.\",\n \"Meta-learning algorithms are introduced to improve robustness of deep learning in the form of sample reweighting.\",\n \"The first work is L2RW [35], which implicitly learns the weights without a pre-defined weighting function, then uses a small unbiased validation set to guide the training of its parameters.\",\n \"In contrast, LossNet [37] parameterizes the weighting function as an MLP network explicitly, mapping from training loss to sample weight.\",\n \"However, the capacity of LossNet is limited by its input with the fixed loss function.\",\n \"Learning with Class Imbalance.\",\n \"In addition to sample reweighting, there are other methods to handle the class imbalance issue in deep learning.\",\n \"For example, some methods try to transfer the knowledge learned from major classes to minor classes [7], [46].\",\n \"The metric learning based methods have also been developed to effectively exploit the tailed data to improve the generalization ability, e.g., triple-header loss [15] and range loss [52].\",\n \"Learning with Label Noise.\",\n \"For handling noisy label issues [9], [13], [21], [24], [29], [39], [44], [45], some other algorithms focus on estimating the label transition matrix.\",\n \"For example, F-correction [32] proposed a loss correction approach by heuristically estimating the noise transition matrix.\",\n \"In these approaches, the quality of noise rate estimation is a critical factor for improving robustness.\",\n \"However, noise rate estimation is challenging, especially on datasets with a large number of classes.\",\n \"Another popular research line of handling noisy labels is to train models on small-loss instances, which can be viewed as a hard version of sample weighting [12], [16], [47].\",\n \"Information Bottleneck.\",\n \"Information Bottleneck (IB) method was initially proposed in [41], where the idea can be formulated as a variational principle of minimizing the mutual information between the input and the learned representation, while preserving the information about the learning task.\",\n \"The IB method has been successfully applied to supervised learning [1], generative modeling [33] and reinforcement learning [33], [43].\",\n \"In this work, we apply the IB algorithm to enforce the input of MW-Net to focus on the relevant information implicitly defined by the learning task.\"\n ],\n [\n \"Preliminaries\",\n \"In this section, we introduce the problem setting of learning from biased training data with sample reweighting and the formulation of LossNet [37], which is a representative related work.\"\n ],\n [\n \"Problem Setting\",\n \"For multi-class classification with $K$ classes, we suppose the biased training dataset with $N$ samples is given as $\\\\mathcal {D}^{\\\\mathrm {train}} = \\\\lbrace x_i, y_i\\\\rbrace ^N_{i=1}$ , where $x_i$ is the $i$ -th instance with its observed label as $y_i \\\\in \\\\lbrace 1, \\\\dots , K\\\\rbrace $ .\",\n \"Similar to L2RW and LossNet, we assume that there is a small unbiased and clean meta dataset with $M$ samples $\\\\mathcal {D}^{\\\\mathrm {val}} = \\\\lbrace x^{\\\\mathrm {val}}_i, y^{\\\\mathrm {val}}_i\\\\rbrace ^M_{i=1}$ and $M \\\\ll N$ , representing the meta-knowledge of ground-truth sample-label association.\",\n \"For the classifier to be trained, we denote it as $f_{\\\\Theta }(x)$ , where $\\\\Theta $ is the parameters of the classifier.\",\n \"Generally, we optimize the classifier network by minimizing the training loss: $\\\\frac{1}{N} \\\\sum ^N_{i=1} L^{\\\\mathrm {train}}_i(\\\\Theta ) = \\\\frac{1}{N} \\\\sum ^N_{i=1} \\\\ell (f_{\\\\Theta }(x_i), y_i)$ , where $\\\\ell $ denotes the employed loss function (e.g., cross entropy loss), and each input example is weighted equally.\",\n \"For enhancing the robustness of training on the biased training data, we aim to assign weight $w_i$ on the loss of the $i$ -th sample.\",\n \"Without loss of generality, the weighting function can be formulated as $\\\\mathcal {V}_{\\\\Phi }((x_i, y_i), f_{\\\\Theta })$ where $\\\\Phi $ denotes the parameters of the weighting function.\",\n \"Then the optimal parameter $\\\\Theta $ of classifier is calculated by minimizing the following weighted loss: $\\\\begin{split}\\\\Theta ^{*}(\\\\Phi ) &= \\\\underset{\\\\Theta }{\\\\arg \\\\min } \\\\ \\\\mathcal {L}^{\\\\mathrm {train}}(\\\\Theta ;\\\\Phi )\\\\\\\\&\\\\triangleq \\\\underset{\\\\Theta }{\\\\arg \\\\min } \\\\frac{1}{N} \\\\sum _{i=1}^{N} w_i L^{\\\\mathrm {train}}_i(\\\\Theta ),\\\\end{split}$ where $w_i = \\\\mathcal {V}((x_i, y_i), f_{\\\\Theta }; \\\\Phi )$ denotes the example weight and $L^{\\\\mathrm {train}}_i(\\\\Theta ) = \\\\ell (f_{\\\\Theta }(x_i), y_i)$ denotes the loss on the training example $(x_i,y_i)$ .\"\n ],\n [\n \"Formulation of LossNet\",\n \"LossNet [37] formulates the weighting function $\\\\mathcal {V}$ as an MLP network, and automatically trains it in a meta-learning manner.\",\n \"Specifically, the MLP network learns a mapping from training loss to sample weight, $w_i = \\\\mathcal {V}_{\\\\Phi }(L_i^{\\\\mathrm {train}}(f_{\\\\Theta }(x_i), y_i))$ , where $\\\\Phi $ denotes the parameters of the MLP net.\",\n \"In the training process, the optimal parameter $\\\\Phi ^{*}$ of LossNet is calculated by minimizing the following meta loss: $\\\\begin{split}\\\\Phi ^{*} &= \\\\underset{\\\\Phi }{\\\\arg \\\\min } \\\\ \\\\mathcal {L}^{\\\\mathrm {meta}}(\\\\Theta ^*(\\\\Phi ))\\\\\\\\&\\\\triangleq \\\\underset{\\\\Phi }{\\\\arg \\\\min } \\\\frac{1}{M} \\\\sum _{i=1}^{M} L_{i}^{\\\\mathrm {meta}}(\\\\Theta ^*(\\\\Phi )),\\\\end{split}$ where $L_{i}^{\\\\mathrm {meta}}(\\\\Theta ^*) = \\\\ell (f_{\\\\Theta ^*}(x^{\\\\mathrm {val}}_i), y_i^{\\\\mathrm {val}}))$ denotes the meta loss on the validation example $(x^{\\\\mathrm {val}}_i,y^{\\\\mathrm {val}}_i)$ .\",\n \"As we analyzed before, the training loss may not contain rich and flexible information of the original sample for capturing meaningful sample weights, due to the restriction on the fixed loss function.\",\n \"As the commonly used cross-entropy loss is known to be highly overconfident [27], [40], LossNet tends to produce polarized results, and high weights could be assigned to some noisy data.\"\n ],\n [\n \"The Proposed Approach\",\n \"In this section, we propose to automatically learn proper inputs for MW-Net, thereby adapting to various biased settings.\",\n \"[t] The Meta Weighting Algorithm 1.1 Training dataset $\\\\mathcal {D}$ , meta dataset $\\\\mathcal {D}_m$ , batch size $n,m$ , max iterations $T$ .\",\n \"Parameter $\\\\Theta $ of classifier $f$ .\",\n \"Initialize parameters of Classifier and MW-Net $t=0$ $\\\\mathrm {to}$ $T$ $\\\\lbrace x_i,y_i\\\\rbrace _{i=1}^n \\\\leftarrow \\\\text{SampleMiniBatch}(\\\\mathcal {D},n)$ $\\\\lbrace x^{\\\\mathrm {val}}_i,y^{\\\\mathrm {val}}_i\\\\rbrace _{i=1}^m \\\\leftarrow \\\\text{SampleMiniBatch}(\\\\mathcal {D}^{\\\\mathrm {val}},m)$ $ z_i = f_{\\\\Theta }(x_i), \\\\tilde{w} = \\\\mathcal {V}_{\\\\Phi ^{(t)}}(z,y)$ $w_i \\\\leftarrow \\\\frac{\\\\tilde{w}_i}{\\\\sum _{j} \\\\tilde{w}_j +\\\\delta \\\\left(\\\\sum _{j} \\\\tilde{w}_j\\\\right)}$ $\\\\mathcal {L}^{\\\\mathrm {train}} = \\\\frac{1}{n} \\\\sum _{i=1}^{n} w_i \\\\cdot \\\\ell (z_i, y_i)$ $\\\\nabla \\\\Theta \\\\leftarrow \\\\text{ BackwardAD }\\\\left(\\\\mathcal {L}^{\\\\mathrm {train}}, \\\\Theta \\\\right)$ $\\\\widehat{\\\\Theta } \\\\leftarrow \\\\Theta -\\\\alpha \\\\nabla \\\\Theta $ $\\\\mathcal {L}^{\\\\mathrm {meta}} = \\\\frac{1}{m} \\\\sum _{i=1}^{m} \\\\ell (f_{\\\\widehat{\\\\Theta }}(x^{\\\\mathrm {val}}_i), y^{\\\\mathrm {val}}_i)$ Update MW-Net $\\\\mathcal {V}$ to $\\\\Phi ^{(t+1)}$ by $\\\\mathcal {L}^{\\\\mathrm {meta}}$ $\\\\tilde{w} = \\\\mathcal {V}_{\\\\Phi ^{(t+1)}}(z,y); w_i \\\\leftarrow \\\\frac{\\\\tilde{w}_i}{\\\\sum _{j} \\\\tilde{w}_j+\\\\delta \\\\left(\\\\sum _{j} \\\\tilde{w}_j\\\\right)}$ $\\\\mathcal {L}^{\\\\mathrm {train}} = \\\\frac{1}{n} \\\\sum _{i=1}^{n} w_i L^{\\\\mathrm {train}}_i(z_i, y_i)$ Update Classifier $f$ to $\\\\Theta ^{(t+1)}$ by $\\\\mathcal {L}^{\\\\mathrm {train}}$\"\n ],\n [\n \"A General Framework for Meta Weighting\",\n \"As shown in Figure REF , we define a general framework for meta weighting, which contains a MW-Net that takes information from the samples $x_i$ and its given labels $y_i$ (instead of training loss) as inputs.\",\n \"To abstract information from the sample $x_i$ , we adopt the outputs of the classifier $f$ , (i.e., $f_{\\\\Theta }(x_i)$ ) for MW-Net.\",\n \"In this way, LossNet can be interpreted as an instantiation of MW-Net.\",\n \"Intuitively, if we calculate the cross-entropy loss with the logit and its corresponding label, we can exactly recover the LossNet model.\",\n \"The training loss can be seen as a representation of information abstracted from samples $x_i$ and its given labels $y_i$ .\",\n \"The left of Figure REF shows that LossNet is an instantiation of MW-Net.\",\n \"Under the general framework in Figure REF , LossNet could be replaced by a more powerful instantiation of MW-Net, thereby achieving better performance.\",\n \"Similar to L2RW [35] and LossNet [37], calculating the optimal parameters $\\\\Theta ^*$ and $\\\\Phi ^*$ requires two nested loops of optimization.\",\n \"To improve the optimization efficiency, we adopt an online strategy to alternatively update the classifier $f$ with $\\\\Theta $ and the MW-Net $\\\\mathcal {V}$ with $\\\\Phi $ through a single optimization loop.\",\n \"We describe the details of the training process in Algorithm .\",\n \"In general, the training process can be separated into three parts: Virtually updating parameters of classifier.\",\n \"In the first part, we randomly sample a mini-batch of training samples ${(x_i, y_i), 1 \\\\le i \\\\le n}$ from the training dataset, where $n$ is the mini-batch size.\",\n \"Then the virtual updating of the classifier network parameter can be formulated by moving the current $\\\\Theta ^{(t)}$ along the descent direction of the objective loss in Eq.\",\n \"(REF ) on a mini-batch training data (lines 5-9) : $\\\\nonumber \\\\widehat{\\\\Theta }^{(t)}(\\\\Phi ) = \\\\Theta ^{(t)} - \\\\alpha \\\\frac{1}{n} \\\\left.\",\n \"\\\\sum _{i=1}^{n} \\\\mathcal {V}_{\\\\Phi }((f_{\\\\Theta }(x_i), y_i)) \\\\nabla _{\\\\Theta } L^{\\\\mathrm {train}}_{i}\\\\right|_{\\\\Theta ^{(t)}},$ where $\\\\alpha $ is the step size.\",\n \"Updating parameters of MW-Net.\",\n \"After receiving the feedback of the virtually updated classifier, i.e., $\\\\widehat{\\\\Theta }^{(t)}(\\\\Phi )$ , the parameter $\\\\Phi $ of the MW-Net can then be readily updated with Eq.\",\n \"(REF ) calculated on the meta dataset (lines 10-11): $\\\\Phi ^{(t+1)}=\\\\Phi ^{(t)}-\\\\beta \\\\frac{1}{m} \\\\left.\\\\sum _{i=1}^{m} \\\\nabla _{\\\\Phi } L_{i}^{\\\\mathrm {meta}}(\\\\widehat{\\\\Theta }^{(t)}(\\\\Phi ))\\\\right|_{\\\\Phi ^{(t)}},$ where $\\\\beta $ is the step size.\",\n \"Updating parameters of classifier.\",\n \"Then, the updated $\\\\Phi ^{(t+1)}$ is employed to ameliorate the parameter $\\\\Theta $ of the classifier (lines 12-14), i.e., $\\\\nonumber &\\\\Theta ^{(t+1)}=\\\\Theta ^{(t)}\\\\\\\\\\\\nonumber &\\\\quad \\\\quad \\\\quad \\\\quad -\\\\alpha \\\\frac{1}{n}\\\\left.\\\\sum _{i=1}^{n} \\\\mathcal {V}_{\\\\Phi ^{(t+1)}}\\\\left((f_{\\\\Theta }(x_i), y_i)\\\\right) \\\\nabla _{\\\\Theta } L_{i}^{\\\\mathrm {train}}\\\\right|_{\\\\Theta ^{(t)}},$ Figure: Three different instantiations of MW-Net, including LossNet, LogitNet, and MetaInfoNet.\"\n ],\n [\n \"LogitNet: An Instatiation of MW-Net\",\n \"Now the question becomes how to design MW-Net, for mapping information from the sample and its given label to weight.\",\n \"An intuitive idea is to simply take the model output of $x_i$ , i.e., $f_{\\\\Theta }(x)$ .\",\n \"For the given label $y_i$ , we employ a label embedding layer to produce an embedding vector $e(y_i)\\\\in \\\\mathbb {R}^K$ , which has the same dimension as $f_{\\\\Theta }(x)$ .\",\n \"It is a fully connected layer that projects the given label to a dense vector.\",\n \"The obtained embedding vector can be seen as a latent vector for describing the properties of a specific class, in the context of MW-Net.\",\n \"Since MW-Net adopts two pathways to extract information from $x_i$ and $y_i$ , it is intuitive to combine the vectors from the two pathways (i.e., $f_{\\\\Theta }(x)$ and $e(y_i)$ ) by concatenating them.\",\n \"However, a vector concatenation cannot account for any interactions between samples and its given labels.\",\n \"Therefore, we apply an element-wise product to combine them as the input for an MLP in MW-Net, that is, $w_i=\\\\text{MLP}(f_{\\\\Theta }(x_i) \\\\odot e(y_i))$ , where $\\\\odot $ denotes the element-wise product.\",\n \"In this way, the combined vector can capture the interaction information between the sample and its given label before we feed it into an MLP.\",\n \"For easy reference, we call this instantiation of MW-Net as LogitNet.\",\n \"The structure of LogitNet is shown on the center of Figure REF .\"\n ],\n [\n \"MetaInfoNet: An Instatiation of MW-Net with Information Bottleneck\",\n \"Although LogitNet introduces more information to the weighting function, it also leads to a new problem: the unprocessed logits may contain too much redundant information for the weighting task.\",\n \"To “squeeze out\\\" the redundant information of the inputs for LogitNet, we propose a novel algorithm by information bottleneck (IB) principle [1], [36], [41], called MetaInfoNet.\",\n \"Instead of directly using the output of classifier $z = f_{\\\\Theta }(x)$ in LogitNet, we aim to learn a representation $\\\\psi $ that captures the most relevant information from the logit $z$ and removes redundant formation, with respect to the weighting task.\",\n \"In this way, the weight of sample $x_i$ can be generated by $w_i=\\\\text{MLP}(\\\\psi _i \\\\odot e(y_i))$ .\",\n \"Figure: The detailed structure of IB layer.Concretely, we propose an information-theoretic regularization by limiting the mutual information between the logit and the learned representation: $I(Z;\\\\Psi ) \\\\le I_c,$ , where $Z$ is a random variable (for the logit vector) with a probability density function $p_{Z}(z)$ , $\\\\Psi $ is a random variable (for the learned representation vector) with a probability density function $p_{\\\\Psi }(\\\\psi )$ and $I_c$ is a constant that represents threshold of the mutual information.\",\n \"The constraint on mutual information can be viewed as a penalty term restricting the information from the logit vector to the learned representation vector.\",\n \"Consequently, for learning an effective representation $\\\\psi $ , we add the mutual information constraint for optimizing MW-Net: $\\\\mathcal {L}^{\\\\mathrm {meta}}(\\\\Theta (\\\\Phi )) = \\\\sum \\\\nolimits _{i=1}^M L^{\\\\mathrm {meta}}_i(\\\\Theta (\\\\Phi )), \\\\text{s.t.}\",\n \"\\\\ I(Z;\\\\Psi ) \\\\le I_c,$ In practice, we propose to minimize the following objective using the information bottleneck Lagrangian: $\\\\sum \\\\nolimits _{i=1}^M L^{\\\\mathrm {meta}}_i(\\\\Theta (\\\\Phi )) + \\\\lambda I(Z;\\\\Psi ),$ where $\\\\lambda $ is the Lagrange multiplier.\",\n \"As $\\\\lambda $ increases, the information from $Z$ to $\\\\Psi $ becomes denser while keeping relevant to the meta learning task.\",\n \"In the above equation, $I(Z;\\\\Psi )$ is defined as: $I(Z;\\\\Psi ) &=\\\\iint p\\\\left(z, \\\\psi \\\\right) \\\\log \\\\frac{p\\\\left(z, \\\\psi \\\\right)}{p\\\\left(z\\\\right) p\\\\left(\\\\psi \\\\right)} \\\\mathrm {d} z \\\\mathrm {d} \\\\psi \\\\\\\\&=\\\\iint p\\\\left(z\\\\right) p_{ib}\\\\left(\\\\psi \\\\mid z\\\\right) \\\\log \\\\frac{p_{ib}\\\\left(\\\\psi \\\\mid z\\\\right)}{p\\\\left(\\\\psi \\\\right)} \\\\mathrm {d} z \\\\mathrm {d} \\\\psi \\\\nonumber $ where $p(z, \\\\psi )$ is the joint probability of $z$ and $\\\\psi $ , and $p_{ib}$ denotes the information bottleneck (IB) layer.\",\n \"Unfortunately, computing the marginal distribution $p(\\\\psi ) = \\\\int p\\\\left(\\\\psi \\\\mid z\\\\right) p\\\\left(z\\\\right) d z$ is quite challenging as we do not know the prior distribution of $p(z)$ .\",\n \"With the help of variational information bottleneck [1], we use a Gaussian approximation $h(\\\\psi )$ of the marginal distribution $p(\\\\psi )$ and view $p_{ib}$ as multivariate variational encoders (see Figure REF ).\",\n \"Since $D_{\\\\mathrm {KL}}\\\\left[p\\\\left(\\\\psi \\\\right) \\\\Vert h\\\\left(\\\\psi \\\\right)\\\\right] \\\\ge 0$ , where the $D_{\\\\mathrm {KL}}$ is the Kullback-Leibler divergence, we expand the KL term and get $\\\\int p\\\\left(\\\\psi \\\\right) \\\\log p\\\\left(\\\\psi \\\\right) d \\\\psi \\\\ge \\\\int p\\\\left(\\\\psi \\\\right) \\\\log h\\\\left(\\\\psi \\\\right) d \\\\psi $ , an upper bound on the mutual information $I(Z;\\\\Psi )$ can be obtained via the KL divergence: $\\\\nonumber I\\\\left(Z ; \\\\Psi \\\\right) & \\\\le \\\\int p\\\\left(z\\\\right) p_{ib}\\\\left(\\\\psi \\\\mid z\\\\right) \\\\log \\\\frac{p_{ib}\\\\left(\\\\psi \\\\mid z\\\\right)}{h\\\\left(\\\\psi \\\\right)} d z d \\\\psi \\\\\\\\\\\\nonumber &=\\\\mathbb {E}_{z \\\\sim p\\\\left(z\\\\right)}\\\\left[D_{\\\\mathrm {KL}}\\\\left[p_{ib}\\\\left(\\\\psi \\\\mid z\\\\right) \\\\Vert h\\\\left(\\\\psi \\\\right)\\\\right]\\\\right],$ which further provides an upper bound $\\\\widetilde{\\\\mathcal {L}}^{\\\\mathrm {meta}}(\\\\Theta (\\\\Phi ))$ on the regularized empirical objective that we minimize: $\\\\widetilde{\\\\mathcal {L}}^{\\\\mathrm {meta}}(\\\\Theta (\\\\Phi )) & = \\\\sum \\\\nolimits _{i=1}^M L^{\\\\mathrm {meta}}_i(\\\\Theta (\\\\Phi )) \\\\\\\\\\\\nonumber & + \\\\lambda \\\\sum \\\\nolimits _{j=1}^N\\\\left[D_{\\\\mathrm {KL}}\\\\left[p_{ib}\\\\left(\\\\psi _j \\\\mid z_j\\\\right) \\\\Vert h\\\\left(\\\\psi _j\\\\right)\\\\right]\\\\right].$ Recall that the learned representation $\\\\psi $ via IB layer is used to generate the weight $w_i=\\\\text{MLP}(\\\\psi _i \\\\odot e(y_i))$ , then we optimize the MW-Net with the new meta loss in Eq.\",\n \"(REF ) to control the information capacity of the learned representation $\\\\psi $ , while keeping the task-relevant information.\",\n \"The network structure of the IB layer is shown in Figure REF .\",\n \"Specifically, a linear layer is applied to map logit $z$ to a multivariate normal distribution.\",\n \"Then we use the “re-parameterization\\\" trick [18] to sample a representation $\\\\psi $ from the learned distribution, parameterized by a mean vector $\\\\mu $ and a variance vector $\\\\sigma $ .\",\n \"Another alternative trick is to implement skip connection by concatenating the representation $\\\\phi $ and the origin logit $z$ .\",\n \"In such a manner, we emphasize the most relevant information in $\\\\psi $ , with respect to the learning task, while keeping intact information from the logit $z$ .\",\n \"To keep aligned with the dimension of label embedding for the element-wise product, we add a linear layer to scale the dimension of the learned representation $\\\\psi $ .\"\n ],\n [\n \"Convergence Analysis\",\n \"Here, we further provide the convergence analysis of MetaInfoNet.\",\n \"We show that MetaInfoNet converges to the critical points of both the meta and training loss function under some mild conditions in the following two theorems.\",\n \"Theorem 1 Suppose the loss function $L$ is Lipshchitz smooth with constant $\\\\rho $ , and $\\\\mathcal {V}(\\\\cdot )$ is differential with a $\\\\delta $ -bounded gradients and twice differential with its Hessian bounded by $\\\\mathfrak {B}$ , and the loss function $L$ has $\\\\gamma $ -bounded gradients with respect to training/meta data.\",\n \"Let the learning rate $\\\\alpha _t$ (for updating classifier) satisfies $\\\\alpha _t=\\\\min \\\\lbrace 1,\\\\frac{k}{T}\\\\rbrace $ , for some constant $k>0$ , such that $\\\\frac{k}{T}<1$ , and the learning rate $\\\\beta _t$ (for updating meta weight net) satisfies $\\\\beta _t=\\\\min \\\\lbrace \\\\frac{1}{L},\\\\frac{c}{\\\\sigma \\\\sqrt{T}}\\\\rbrace $ for some constant $c>0$ where $\\\\sigma $ is the variance of drawing uniformly mini-batch sample at random, such that $\\\\frac{\\\\sigma \\\\sqrt{T}}{c}>\\\\rho $ and $\\\\sum _{t=1}^{\\\\infty }\\\\beta _t\\\\le \\\\infty ,\\\\sum _{t=1}^{\\\\infty }\\\\beta _t^2\\\\le \\\\infty $ .\",\n \"Then we have $\\\\nonumber \\\\lim \\\\limits _{t\\\\rightarrow \\\\infty }\\\\mathbb {E}\\\\Big [\\\\left\\\\Vert \\\\nabla _{\\\\Theta }\\\\mathcal {L}^{\\\\mathrm {train}}(\\\\Theta ^{(t)};\\\\Phi ^{(t)})\\\\right\\\\Vert ^2_2\\\\Big ] = 0.$ Theorem 2 When the conditions in Theorem REF holds, our proposed algorithm can achieve $\\\\mathbb {E}[\\\\left\\\\Vert \\\\nabla \\\\mathcal {L}^{\\\\mathrm {meta}}(\\\\Theta ^{(t)})\\\\right\\\\Vert _2^2]\\\\le \\\\epsilon $ in $\\\\mathcal {O}(1/\\\\epsilon ^2)$ steps, more specifically, $\\\\nonumber \\\\min _{0\\\\le t\\\\le T}\\\\mathbb {E}[\\\\left\\\\Vert \\\\nabla _{\\\\Theta } \\\\mathcal {L}^{\\\\mathrm {meta}}(\\\\Theta ^{(t)})\\\\right\\\\Vert _2^2]\\\\le \\\\mathcal {O}(\\\\frac{O}{\\\\sqrt{T}}),$ where $C$ is some constant independent of the convergence process.\",\n \"It is worth noting that we have shown that our proposed method MetaInfoNet and LossNet [37] are different instantiations of MW-Net.\",\n \"Therefore, MetaInfoNet shares the same convergence properties as LossNet if we adopt the same assumptions for analyzing the two methods.\",\n \"We omit the proofs of the two theorems, as they are very similar to those proofs in [37].\"\n ],\n [\n \"Experiments\",\n \"In this section, we first implement experiments on simulated datasets with class imbalance and noisy labels.\",\n \"Then two real-world datasets with biased training data are also used to verify the effectiveness of our proposed algorithms.\"\n ],\n [\n \"Experiment setup\",\n \"Datasets.\",\n \"For the class imbalance setting, we verify the effectiveness of our proposed algorithm on the Long-Tailed version of CIFAR10 and CIFAR100 datasets [6].\",\n \"Specifically, we reduce the number of training examples per class according to an exponential function $n = n_i\\\\mu ^i$ , where $i$ is the class index, $n_i$ is the original number of training samples in the largest class divided by the smallest.\",\n \"Following LossNet [37], we randomly select ten images per class in the validation set as the meta dataset.\",\n \"For noisy label settings, we conduct experiments on simulated noisy datasets, including CIFAR10 and CIFAR100 [19].\",\n \"We study two types of label noise on the training set: 1) Flip noise.\",\n \"Following LossNet [37], we mainly consider the case that the label of each example is flipped to two similar classes with equal probability $\\\\frac{p}{2}$ (Flip-2).\",\n \"2) Uniform noise.\",\n \"The label of each example is changed to a random class with probability $p$ independently.\",\n \"To verify the effectiveness of our algorithms in various settings, we also compare our algorithms with existing meta reweighting algorithms in a harder case: One class is randomly selected as the similar class with probability $p$ (Flip-1).\",\n \"In these cases, 1000 images with clean labels in the validation set are randomly selected as the meta dataset.\",\n \"To verify the effectiveness of the proposed method on real-world data, we conduct experiments on the Clothing1M dataset [48], containing 1 million images of clothing obtained from online shopping websites with 14 categories, e.g., T-shirt, Shirt, Knitwear.\",\n \"The labels are generated by using surrounding texts of the images provided by the sellers, and therefore contain many errors.\",\n \"We use the 7k clean data as the meta dataset.\",\n \"For preprocessing, we resize the image to $256 \\\\times 256$ , crop the middle $224 \\\\times 224$ as input, and perform normalization.\",\n \"Moreover, we also compare our proposed algorithms with existing meta reweighting algorithms on ANIMAL10N dataset [38], which contains 5 pairs of confusing animals with a total of 55,000 images.\",\n \"In our experiments, the training dataset contains 50,000 pictures with noisy labels, and 1000 pictures with clean labels are randomly selected as the meta dataset.\",\n \"The rest 4000 images are set as the test dataset.\",\n \"Network Structure and Optimizer.\",\n \"For experiments on CIFAR10 and CIFAR100, we train ResNet-32 [14] for the class imbalance settings.\",\n \"For the settings of label noise, we adopt WRN-28-10 [51] for uniform noise and ResNet-32 for flip noise.\",\n \"Specifically, SGD optimizer is applied with a momentum 0.9, a weight decay $5 \\\\times 10^{-4}$ , an initial learning rate 0.1, and the batch size is set to 100.\",\n \"The learning rate of the classifier network is divided by 10 after 80 epochs and 100 epochs (for a total of 120 epochs), and after 30 epochs and 40 epochs (for a total of 50 epochs).\",\n \"For experiments on Clothing1M, we use ResNet-50 pre-trained on ImageNet, following the previous works [32], [37], [39].\",\n \"About the optimizer, we use SGD with a momentum 0.9, a weight decay $10^{-3}$ , and an initial learning rate 0.01, and batch size 32.\",\n \"The learning rate of ResNet-50 is divided by 10 after 5 epochs (for a total 10 epochs).\",\n \"For experiments on ANIMAL10N, we also use ResNet-32.\",\n \"The setting of the SGD optimizer is the same as that on CIFAR10.\",\n \"We run 100 epochs in total, and the learning rate of the classifier network is divided by 10 after 50 epochs and 75 epochs.\",\n \"For the training of meta weighting network, we set the learning rate to $10^{-3}$ and set the weight decay to $5 \\\\times 10^{-4}$ .\",\n \"About the $\\\\lambda $ in Eq.\",\n \"(REF ), we search it in $[0, 0.01, 0.03, 0.1, 0.3, 1]$ .\",\n \"Additionally, we can set an interval parameter to control the updating of MW-Net, e.g., when the interval is 10, we update the MW-net every 10 iterations.\",\n \"We implement all methods with default parameters by PyTorch, and conduct all the experiments on NVIDIA Tesla V100 GPUs.\",\n \"We repeated the experiments 5 times with different random seeds for network initialization and label noise generation.\",\n \"Compared Methods.\",\n \"For class imbalance setting, we compare our proposed algorithm with: 1) Standard, which simply uses CE loss to train the DNNs with equal weights; 2) Focal loss [23], which emphasizes harder examples by sample reweighting; 3) Class-balanced [6], which represents one of the state-of-the-arts of the predefined sample reweighting methods; 4) Fine-tuning, which finetunes the standard model on the meta dataset to further enhance its performance; 5) L2RW [35], which leverages an additional meta dataset to adaptively assign weights on training samples; 6) LossNet [37], which automatically learns an explicit loss-weight function in a meta-learning manner.\",\n \"For noisy label setting, the compared methods include: 1) Standard; 2) Bootstrap [34]; 3) S-Model [11]; 4) SPL [20]; 5) Focal Loss [23]; 6) Co-teaching [12]; 7) D2L [25]; 6) Fine-tuning; 7) MentorNet [16]; 8) L2RW [35]; 9) LossNet [37].\",\n \"Table: Test accuracy (%) on long-tailed CIFAR-10 and CIFAR-100.\",\n \"The best results are highlighted in bold.Table: Average test accuracy (%) with standard deviation on synthesized datasets under Flip-2 noise (over 5 trials).\",\n \"The best results are highlighted in bold.\"\n ],\n [\n \"Experiments with Class Imbalance\",\n \"Table REF shows the test accuracy of different algorithms on long-tailed CIFAR10 and long-tailed CIFAR100.\",\n \"As we can see, our proposed algorithms work better than all the baselines in most of the long-tailed cases, which demonstrates their robustness against class imbalance.\",\n \"When imbalance factor is 1, e.g., there are the same number of samples in all classes, the fine-tuning method obtains the best performance on both CIFAR10 and CIFAR100, and our algorithms also get comparable performance.\",\n \"As the imbalance factor rises from 10 to 200, the datasets become more and more unbalanced and the accuracy of all algorithms decreases gradually.\",\n \"We observe that LogitNet achieves better or comparable results compared to LossNet in most cases, which indicates introducing more information can improve the effectiveness of MW-Net.\",\n \"Moreover, MetaInfoNet consistently performs better than LogitNet in most cases, verifying the effectiveness of the information bottleneck method while learning the representation.\",\n \"Table: Average test accuracy (%) with standard deviation on CIFAR10 with WRN-28-10 under Unif noise (over 5 trials).\",\n \"The best results are highlighted in bold.\"\n ],\n [\n \"Experiments with Noisy Labels\",\n \"Table REF shows the test accuracy of ResNet-32 averaged over 5 repetitions on both CIFAR10 and CIFAR100 under Flip-2 noise.\",\n \"We can observe that MetaInfoNet achieves the best performance across both datasets and all noise rates, indicating the effectiveness of our proposed architecture.\",\n \"When the noise rate is 0%, MetaInfoNet also gets comparable results compared to the Fine-tune baseline while performs better than other meta-reweighting algorithms.\",\n \"Additionally, the results of LogitNet are similar to those of LossNet, which shows that simply introducing more information cannot improve the capacity of meta sample weighting in this case.\",\n \"Table REF presents the test accuracy of WRN-28-10 averaged over 5 repetitions on CIFAR10 under Unif noise.\",\n \"It can be observed that all the meta weighting algorithms, including MetaInforNet, LogitNet, LossNet and L2RW, perform the best among all algorithms, which demonstrates their robustness.\",\n \"The results also indicate that uniform noise is a relatively simple biased setting for meta weighting algorithms.\",\n \"Now we turn to a more difficult noisy setting, Flip-1 noise, that each example's label is flipped to one similar class with total probability $p$ .\",\n \"As shown in Table REF , Best denotes the scores of the epoch where the validation accuracy is optimal, and Last denotes the average accuracy over the last 10 epochs at the end of the training stage.\",\n \"As we can see, LossNet performs poorly even gets lower test accuracy than L2RW, while MetaInfoNet attains more than 8% improvement over LossNet.\",\n \"Moreover, MetaInfoNet keeps the best performance in both CIFAR10 and CIFAR100 and all noise rates.\",\n \"Specifically, in the 20% label noise cases, all four meta weighting algorithms could maintain almost the $best$ accuracy in the last 10 epochs.\",\n \"When it improves to the 40% label noise cases, MetaInfoNet keeps the advantage while the other three algorithms' performance decrease dramatically after reaching the top.\",\n \"The result shows that MetaInfoNet is more robust to label noise compared to the other meta-weighting algorithms and has a better capacity to adapt to different biased settings in training data.\"\n ],\n [\n \"Experiments on Real-world Datasets\",\n \"Table REF shows the classification accuracy on the Clothing1M test set.\",\n \"Specifically, MetaInfoNet achieves the best accuracy, while LogitNet performs worse than LossNet.\",\n \"The results show that the IB layer is an efficient way to improving the meta-weighting framework with logits and labels as inputs.\",\n \"Table: Classification accuracy (%) on Clothing1M.We also compare our proposed algorithms to the Finetune baseline and the other meta-weighting algorithms on the ANIMAL-10N dataset, as shown in Table REF .\",\n \"Note that the noise rate is about 8% but it is difficult for even people to distinguish between categories, the performance gains of meta weighting baselines are not huge compared to the Finetune method.\",\n \"In this case, MetaInfoNet still obtains nearly 2% improvement over the other methods on best.\",\n \"After all epochs, MetaInfoNet maintains this advantage over the best baseline method.\",\n \"Table: Classification accuracy (%) on ANIMAL-10N.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we propose an effective approach called MetaInfoNet to improve the robustness of deep neural networks under various biased settings in training data.\",\n \"Compared with current meta reweighting algorithms that directly map loss to weight for each sample, MetaInfoNet could automatically learn effective representations as inputs for the meta weighting network by emphasizing task-related information with an information bottleneck strategy.\",\n \"The Empirical results on simulated and real-world datasets demonstrate that the robustness of deep models trained by our proposed approach is superior to many state-of-the-art approaches in general biased settings, such as class imbalance, label noise, and more complicated real cases.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05273"}}},{"rowIdx":3691,"cells":{"text":{"kind":"list like","value":[["Quantum Discrimination of Two Noisy Displaced Number States"],["Abstract The quantum discrimination of two non-coherent states draws much attention recently.","In this letter, we first consider the quantum discrimination of two noiseless displaced number states.","Then we derive the Fock representation of noisy displaced number states and address the problem of discriminating between two noisy displaced number states.","We further prove that the optimal quantum discrimination of two noisy displaced number states can be achieved by the Kennedy receiver with threshold detection.","Simulation results verify the theoretical derivations and show that the error probability of on-off keying modulation using a displaced number state is significantly less than that of on-off keying modulation using a coherent state with the same average energy."],["Introduction","Quantum discrimination of two optical quantum states plays a crucial rule in quantum information processing tasks, e.g., continuous-variable quantum key distributions and optical quantum communications [1], [2], [3], [4].","Due to the good compatibility with classical optical infrastructures, coherent states [5] generated by lasers are usually employed as the information carriers in quantum communication systems.","However, the minimum discrimination error probability (MDEP) of discriminating two coherent states cannot be zero because of the non-orthogonal property of two coherent states [5].","To improve the performance of the quantum discrimination, new information carriers using non-coherent states draw much attention recently [6], [7].","For example, the problem of discriminating between two noisy photon-added coherent states (PACSs) was addressed in [7].","The PACS is generated by sequentially applying the displacement operator and the creation operator on a vacuum state.","It was demonstrated that the error probability can be significantly reduced when PACSs instead of coherent states are employed in pulse position modulations [6].","Inspired by [7], we focus on the quantum discrimination between two noisy displaced number states (DNSs).","The DNS is generated by sequentially applying the creation operator and the displacement operator on a vacuum state.","The properties of noiseless DNS were discussed in [8], [9], [10].","However, the thermal noise is inevitable in preparing a DNS and the property of a noisy DNS has not been studied yet.","Besides, to the best of the authors' knowledge, the problem of discriminating between two noisy DNSs has not been addressed.","In this letter, we first address the problem of discriminating between two noiseless DNSs.","Then we derive the Fock representation of noisy DNSs and address the problem of discriminating between two noisy DNSs.","Using the Fock representation of noisy DNSs, we further prove that the optimal quantum discrimination of two noisy DNSs can be achieved using a Kennedy receiver with a threshold detection.","The simulation results verify our theoretical derivations.","We also explore the possibility of employing DNSs instead of coherent states in on-off keying (OOK) modulations; and find that the error probability of OOK modulation using a DNS can be significantly reduced compared with the error probability of OOK modulation using a coherent state with the same average energy.","The DNS is generated by sequentially applying the creation operator and the displacement operator on a vacuum state; and it can be written as [8], [9] $\\mathinner {|{\\mu ,k}\\rangle }= \\hat{D}(\\mu )\\mathinner {|{k}\\rangle }$ where $\\hat{D}(\\mu )$ is the displacement operator; and $\\mathinner {|{k}\\rangle }$ is the number state containing $k$ photons.","The number state decomposition of a DNS can be obtained as [9] $\\mathinner {|{\\mu ,k}\\rangle }=\\sum _{n=0}^{\\infty }b_n\\mathinner {|{n}\\rangle }$ where the coefficient $b_n$ is given by [9] $b_n=\\left\\lbrace \\begin{array}{ll}\\sqrt{\\frac{n!}{k!","}}(-\\mu ^*)^{k-n}e^{-\\frac{|\\mu |^2}{2}}L_n^{(k-n)}(|\\mu |^2), \\text{ for } n0}\\lambda _n\\end{aligned}$ where $\\Vert \\hat{A}\\Vert _1=\\text{tr}\\lbrace \\sqrt{\\hat{A}^{\\dagger }\\hat{A}}\\rbrace $ denotes the trace norm of the operator $\\hat{A}$ .","For discriminating between two pure states $\\hat{\\rho }_0=\\mathinner {|{\\psi _0}\\rangle }\\mathinner {\\langle {\\psi _0}|}$ and $\\hat{\\rho }_1=\\mathinner {|{\\psi _1}\\rangle }\\mathinner {\\langle {\\psi _1}|}$ , the Helstrom bound (REF ) can be rewritten as $P_e=\\frac{1}{2}-\\frac{1}{2}\\sqrt{1-4p_0p_1|\\left\\langle \\psi _0|\\psi _1 \\right\\rangle |^2}.$ Therefore, the MDEP for discriminating between two noiseless DNSs $\\mathinner {|{\\xi ,h}\\rangle }$ and $\\mathinner {|{\\mu ,k}\\rangle }$ is determined by the inner product $\\langle \\xi ,h|\\mu ,k \\rangle $ .","Substituting (REF ) into (REF ), we can obtain the MDEP as $\\begin{aligned}P_e=\\frac{1}{2}-\\frac{1}{2}&\\left\\lbrace 1-4p_0p_1\\frac{k!}{h!","}|\\mu -\\xi |^{2(h-k)}e^{-|\\mu -\\xi |^2}\\right.\\\\&\\quad \\times \\left.\\left[L_k^{(h-k)}(|\\mu -\\xi |^2)\\right]^2\\right\\rbrace ^{\\frac{1}{2}}.\\end{aligned}$ From (REF ), we can observe that the perfect discrimination with zero error probability happens in the following two situations: (i) $\\mu =\\xi $ and $h\\ne k$ ; (ii) $L_k(|\\mu -\\xi |^2)=0$ and $h=k\\ne 0$ .","Notice that when $h=k=0$ , the two DNSs becomes two non-orthogonal coherent states $\\mathinner {|{\\xi }\\rangle }$ and $\\mathinner {|{\\mu }\\rangle }$ .","Then the MDEP approaches zero when $|\\mu -\\xi |^2$ approaches $\\infty $ ."],["Noisy DNSs","A noisy number state $\\hat{\\rho }(k)$ is obtained by applying creation operators on a thermal state $\\hat{\\rho }_{th}$ , which results in $\\hat{\\rho }(k)=\\frac{(\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k}{\\text{tr}\\lbrace (\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k\\rbrace }$ where $\\text{tr}\\lbrace \\cdot \\rbrace $ denotes the trace operation.","Then the noisy DNS $\\hat{\\rho }(\\mu ,k)$ is defined as $\\hat{\\rho }(\\mu ,k)\\triangleq \\frac{\\hat{D}(\\mu )(\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k\\hat{D}^{\\dagger }(\\mu )}{N_k}$ where $N_k$ can be obtained as $\\begin{aligned}N_k=\\text{tr}\\lbrace \\hat{D}(\\mu )(\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k\\hat{D}^{\\dagger }(\\mu )\\rbrace =k!","(n_t+1)^k\\end{aligned}$ and where $n_t$ is the average number of thermal photons due to the presence of thermal noise.","The following theorem presents the Fock representation for a noisy DNS.","Theorem 1 (Fock representation) The Fock representation of a noisy DNS $\\hat{\\rho }(\\mu ,k)$ is found to be $\\begin{aligned}\\mathinner {\\langle {n}|}&\\hat{\\rho }(\\mu ,k)\\mathinner {|{m}\\rangle }\\\\&=\\sum _{i=0}^{k}\\sum _{j=0}^{k}I(n\\ge i;m\\ge j)\\frac{(-1)^{i+j}\\binom{m}{j}\\binom{k}{i}}{(k-j)!}\\sqrt{\\frac{n!}{m!","}}e^{-\\frac{|\\mu |^2}{n_t+1}}\\\\&\\quad \\times \\frac{|\\mu |^{2(k-j)}(\\mu ^*)^{m-n} n_t^{n-i}}{(n_t+1)^{m+k-j+1}}L_{n-i}^{(m-n+i-j)}\\left(-\\frac{|\\mu |^2}{n_t(n_t+1)}\\right)\\end{aligned}$ where $I(n\\ge i;m\\ge j)$ is an indicator function defined as $\\begin{aligned}I(n\\ge i;m\\ge j)\\triangleq \\left\\lbrace \\begin{array}{ll}1,\\text{ for } n\\ge i \\text{ and } m\\ge j \\\\0,\\text{ otherwise.","}\\end{array}\\right.\\end{aligned}$ See Appendix .","Using the Fock representation in (REF ), we can obtain the photon statistics of a noisy DNS as $\\begin{aligned}p(n)&= \\mathinner {\\langle {n}|}\\hat{\\rho }(\\mu ,k)\\mathinner {|{n}\\rangle }\\\\&=\\sum _{i=0}^{k}\\sum _{j=0}^{k}I(n\\ge i;n\\ge j)\\frac{(-1)^{i+j}\\binom{n}{j}\\binom{k}{i}}{(k-j)!","}e^{-\\frac{|\\mu |^2}{n_t+1}}\\\\&\\quad \\times \\frac{|\\mu |^{2(k-j)} n_t^{n-i}}{(n_t+1)^{n+k-j+1}}L_{n-i}^{(i-j)}\\left(-\\frac{|\\mu |^2}{n_t(n_t+1)}\\right).\\end{aligned}$ Then the average number of photons $n_p$ contained in a noisy DNS can be obtained in the following lemma.","Lemma 1 The average number of photons $n_p(\\mu ,k)$ contained in a noisy DNS $\\hat{\\rho }(\\mu ,k)$ is found to be $n_p(\\mu ,k)=|\\mu |^2+k(n_t+1)+n_t.$ See Appendix ."],["Discriminate Two Noisy DNSs","Now we consider the discrimination of two noisy DNSs $\\hat{\\rho }_0=\\hat{\\rho }(\\xi ,h)$ and $\\hat{\\rho }_1=\\hat{\\rho }(\\mu ,k)$ with prior probabilities $p_0$ and $p_1$ , respectively.","Here we let $h\\ge k$ without loss of generality.","According to the Helstrom bound (REF ), the MDEP is determined by all the positive eigenvalues $\\lambda _n$ of the decision operator $\\hat{\\Delta }=p_1\\hat{\\rho }(\\mu ,k)-p_0\\hat{\\rho }(\\xi ,h)$ .","Because the displacement operator is an unitary operator, $\\hat{\\Delta }$ and $\\hat{D}(-\\xi )\\hat{\\Delta }\\hat{D}^{\\dagger }(-\\xi )$ share the same eigenvalues.","Then it is readable to show that $\\Vert p_1\\hat{\\rho }(\\mu ,k)-p_0\\hat{\\rho }(\\xi ,h)\\Vert _1=\\Vert p_1\\hat{\\rho }(\\mu -\\xi ,k)-p_0\\hat{\\rho }(0,h)\\Vert _1.$ It is challenging to obtain an analytical form of the eigenvalues $\\lambda _n$ for an arbitrary $\\hat{\\Delta }$ .","However, eq.","(REF ) indicates that when $\\mu =\\xi $ , we only need to obtain the eigenvalues of $p_1\\hat{\\rho }(0,k)-p_0\\hat{\\rho }(0,h)$ .","Using the Fock representation (REF ), we can obtain the eigenvalues of $p_1\\hat{\\rho }(0,k)-p_0\\hat{\\rho }(0,h)$ as $\\begin{aligned}\\lambda _n&=p_1\\mathinner {\\langle {n}|}\\hat{\\rho }(0,k)\\mathinner {|{n}\\rangle }-p_0\\mathinner {\\langle {n}|}\\hat{\\rho }(0,h)\\mathinner {|{n}\\rangle }\\\\&=\\left\\lbrace \\begin{array}{ll}0, \\text{ for } nn_{th}.\\end{array}\\right.$ Therefore, by optimizing the threshold in (REF ), we can always achieve the MDEP in (REF ), i.e., we have $\\sum _{\\lambda _n>0}\\lambda _n=\\sum _{n=0}^{n_{th}}\\lambda _n.$ Accordingly, the optimal threshold $n_{th}$ is the maximum $n$ satisfying (REF )."]],"string":"[\n [\n \"Quantum Discrimination of Two Noisy Displaced Number States\"\n ],\n [\n \"Abstract The quantum discrimination of two non-coherent states draws much attention recently.\",\n \"In this letter, we first consider the quantum discrimination of two noiseless displaced number states.\",\n \"Then we derive the Fock representation of noisy displaced number states and address the problem of discriminating between two noisy displaced number states.\",\n \"We further prove that the optimal quantum discrimination of two noisy displaced number states can be achieved by the Kennedy receiver with threshold detection.\",\n \"Simulation results verify the theoretical derivations and show that the error probability of on-off keying modulation using a displaced number state is significantly less than that of on-off keying modulation using a coherent state with the same average energy.\"\n ],\n [\n \"Introduction\",\n \"Quantum discrimination of two optical quantum states plays a crucial rule in quantum information processing tasks, e.g., continuous-variable quantum key distributions and optical quantum communications [1], [2], [3], [4].\",\n \"Due to the good compatibility with classical optical infrastructures, coherent states [5] generated by lasers are usually employed as the information carriers in quantum communication systems.\",\n \"However, the minimum discrimination error probability (MDEP) of discriminating two coherent states cannot be zero because of the non-orthogonal property of two coherent states [5].\",\n \"To improve the performance of the quantum discrimination, new information carriers using non-coherent states draw much attention recently [6], [7].\",\n \"For example, the problem of discriminating between two noisy photon-added coherent states (PACSs) was addressed in [7].\",\n \"The PACS is generated by sequentially applying the displacement operator and the creation operator on a vacuum state.\",\n \"It was demonstrated that the error probability can be significantly reduced when PACSs instead of coherent states are employed in pulse position modulations [6].\",\n \"Inspired by [7], we focus on the quantum discrimination between two noisy displaced number states (DNSs).\",\n \"The DNS is generated by sequentially applying the creation operator and the displacement operator on a vacuum state.\",\n \"The properties of noiseless DNS were discussed in [8], [9], [10].\",\n \"However, the thermal noise is inevitable in preparing a DNS and the property of a noisy DNS has not been studied yet.\",\n \"Besides, to the best of the authors' knowledge, the problem of discriminating between two noisy DNSs has not been addressed.\",\n \"In this letter, we first address the problem of discriminating between two noiseless DNSs.\",\n \"Then we derive the Fock representation of noisy DNSs and address the problem of discriminating between two noisy DNSs.\",\n \"Using the Fock representation of noisy DNSs, we further prove that the optimal quantum discrimination of two noisy DNSs can be achieved using a Kennedy receiver with a threshold detection.\",\n \"The simulation results verify our theoretical derivations.\",\n \"We also explore the possibility of employing DNSs instead of coherent states in on-off keying (OOK) modulations; and find that the error probability of OOK modulation using a DNS can be significantly reduced compared with the error probability of OOK modulation using a coherent state with the same average energy.\",\n \"The DNS is generated by sequentially applying the creation operator and the displacement operator on a vacuum state; and it can be written as [8], [9] $\\\\mathinner {|{\\\\mu ,k}\\\\rangle }= \\\\hat{D}(\\\\mu )\\\\mathinner {|{k}\\\\rangle }$ where $\\\\hat{D}(\\\\mu )$ is the displacement operator; and $\\\\mathinner {|{k}\\\\rangle }$ is the number state containing $k$ photons.\",\n \"The number state decomposition of a DNS can be obtained as [9] $\\\\mathinner {|{\\\\mu ,k}\\\\rangle }=\\\\sum _{n=0}^{\\\\infty }b_n\\\\mathinner {|{n}\\\\rangle }$ where the coefficient $b_n$ is given by [9] $b_n=\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\sqrt{\\\\frac{n!}{k!\",\n \"}}(-\\\\mu ^*)^{k-n}e^{-\\\\frac{|\\\\mu |^2}{2}}L_n^{(k-n)}(|\\\\mu |^2), \\\\text{ for } n0}\\\\lambda _n\\\\end{aligned}$ where $\\\\Vert \\\\hat{A}\\\\Vert _1=\\\\text{tr}\\\\lbrace \\\\sqrt{\\\\hat{A}^{\\\\dagger }\\\\hat{A}}\\\\rbrace $ denotes the trace norm of the operator $\\\\hat{A}$ .\",\n \"For discriminating between two pure states $\\\\hat{\\\\rho }_0=\\\\mathinner {|{\\\\psi _0}\\\\rangle }\\\\mathinner {\\\\langle {\\\\psi _0}|}$ and $\\\\hat{\\\\rho }_1=\\\\mathinner {|{\\\\psi _1}\\\\rangle }\\\\mathinner {\\\\langle {\\\\psi _1}|}$ , the Helstrom bound (REF ) can be rewritten as $P_e=\\\\frac{1}{2}-\\\\frac{1}{2}\\\\sqrt{1-4p_0p_1|\\\\left\\\\langle \\\\psi _0|\\\\psi _1 \\\\right\\\\rangle |^2}.$ Therefore, the MDEP for discriminating between two noiseless DNSs $\\\\mathinner {|{\\\\xi ,h}\\\\rangle }$ and $\\\\mathinner {|{\\\\mu ,k}\\\\rangle }$ is determined by the inner product $\\\\langle \\\\xi ,h|\\\\mu ,k \\\\rangle $ .\",\n \"Substituting (REF ) into (REF ), we can obtain the MDEP as $\\\\begin{aligned}P_e=\\\\frac{1}{2}-\\\\frac{1}{2}&\\\\left\\\\lbrace 1-4p_0p_1\\\\frac{k!}{h!\",\n \"}|\\\\mu -\\\\xi |^{2(h-k)}e^{-|\\\\mu -\\\\xi |^2}\\\\right.\\\\\\\\&\\\\quad \\\\times \\\\left.\\\\left[L_k^{(h-k)}(|\\\\mu -\\\\xi |^2)\\\\right]^2\\\\right\\\\rbrace ^{\\\\frac{1}{2}}.\\\\end{aligned}$ From (REF ), we can observe that the perfect discrimination with zero error probability happens in the following two situations: (i) $\\\\mu =\\\\xi $ and $h\\\\ne k$ ; (ii) $L_k(|\\\\mu -\\\\xi |^2)=0$ and $h=k\\\\ne 0$ .\",\n \"Notice that when $h=k=0$ , the two DNSs becomes two non-orthogonal coherent states $\\\\mathinner {|{\\\\xi }\\\\rangle }$ and $\\\\mathinner {|{\\\\mu }\\\\rangle }$ .\",\n \"Then the MDEP approaches zero when $|\\\\mu -\\\\xi |^2$ approaches $\\\\infty $ .\"\n ],\n [\n \"Noisy DNSs\",\n \"A noisy number state $\\\\hat{\\\\rho }(k)$ is obtained by applying creation operators on a thermal state $\\\\hat{\\\\rho }_{th}$ , which results in $\\\\hat{\\\\rho }(k)=\\\\frac{(\\\\hat{A}^{\\\\dagger })^k\\\\hat{\\\\rho }_{th}\\\\hat{A}^k}{\\\\text{tr}\\\\lbrace (\\\\hat{A}^{\\\\dagger })^k\\\\hat{\\\\rho }_{th}\\\\hat{A}^k\\\\rbrace }$ where $\\\\text{tr}\\\\lbrace \\\\cdot \\\\rbrace $ denotes the trace operation.\",\n \"Then the noisy DNS $\\\\hat{\\\\rho }(\\\\mu ,k)$ is defined as $\\\\hat{\\\\rho }(\\\\mu ,k)\\\\triangleq \\\\frac{\\\\hat{D}(\\\\mu )(\\\\hat{A}^{\\\\dagger })^k\\\\hat{\\\\rho }_{th}\\\\hat{A}^k\\\\hat{D}^{\\\\dagger }(\\\\mu )}{N_k}$ where $N_k$ can be obtained as $\\\\begin{aligned}N_k=\\\\text{tr}\\\\lbrace \\\\hat{D}(\\\\mu )(\\\\hat{A}^{\\\\dagger })^k\\\\hat{\\\\rho }_{th}\\\\hat{A}^k\\\\hat{D}^{\\\\dagger }(\\\\mu )\\\\rbrace =k!\",\n \"(n_t+1)^k\\\\end{aligned}$ and where $n_t$ is the average number of thermal photons due to the presence of thermal noise.\",\n \"The following theorem presents the Fock representation for a noisy DNS.\",\n \"Theorem 1 (Fock representation) The Fock representation of a noisy DNS $\\\\hat{\\\\rho }(\\\\mu ,k)$ is found to be $\\\\begin{aligned}\\\\mathinner {\\\\langle {n}|}&\\\\hat{\\\\rho }(\\\\mu ,k)\\\\mathinner {|{m}\\\\rangle }\\\\\\\\&=\\\\sum _{i=0}^{k}\\\\sum _{j=0}^{k}I(n\\\\ge i;m\\\\ge j)\\\\frac{(-1)^{i+j}\\\\binom{m}{j}\\\\binom{k}{i}}{(k-j)!}\\\\sqrt{\\\\frac{n!}{m!\",\n \"}}e^{-\\\\frac{|\\\\mu |^2}{n_t+1}}\\\\\\\\&\\\\quad \\\\times \\\\frac{|\\\\mu |^{2(k-j)}(\\\\mu ^*)^{m-n} n_t^{n-i}}{(n_t+1)^{m+k-j+1}}L_{n-i}^{(m-n+i-j)}\\\\left(-\\\\frac{|\\\\mu |^2}{n_t(n_t+1)}\\\\right)\\\\end{aligned}$ where $I(n\\\\ge i;m\\\\ge j)$ is an indicator function defined as $\\\\begin{aligned}I(n\\\\ge i;m\\\\ge j)\\\\triangleq \\\\left\\\\lbrace \\\\begin{array}{ll}1,\\\\text{ for } n\\\\ge i \\\\text{ and } m\\\\ge j \\\\\\\\0,\\\\text{ otherwise.\",\n \"}\\\\end{array}\\\\right.\\\\end{aligned}$ See Appendix .\",\n \"Using the Fock representation in (REF ), we can obtain the photon statistics of a noisy DNS as $\\\\begin{aligned}p(n)&= \\\\mathinner {\\\\langle {n}|}\\\\hat{\\\\rho }(\\\\mu ,k)\\\\mathinner {|{n}\\\\rangle }\\\\\\\\&=\\\\sum _{i=0}^{k}\\\\sum _{j=0}^{k}I(n\\\\ge i;n\\\\ge j)\\\\frac{(-1)^{i+j}\\\\binom{n}{j}\\\\binom{k}{i}}{(k-j)!\",\n \"}e^{-\\\\frac{|\\\\mu |^2}{n_t+1}}\\\\\\\\&\\\\quad \\\\times \\\\frac{|\\\\mu |^{2(k-j)} n_t^{n-i}}{(n_t+1)^{n+k-j+1}}L_{n-i}^{(i-j)}\\\\left(-\\\\frac{|\\\\mu |^2}{n_t(n_t+1)}\\\\right).\\\\end{aligned}$ Then the average number of photons $n_p$ contained in a noisy DNS can be obtained in the following lemma.\",\n \"Lemma 1 The average number of photons $n_p(\\\\mu ,k)$ contained in a noisy DNS $\\\\hat{\\\\rho }(\\\\mu ,k)$ is found to be $n_p(\\\\mu ,k)=|\\\\mu |^2+k(n_t+1)+n_t.$ See Appendix .\"\n ],\n [\n \"Discriminate Two Noisy DNSs\",\n \"Now we consider the discrimination of two noisy DNSs $\\\\hat{\\\\rho }_0=\\\\hat{\\\\rho }(\\\\xi ,h)$ and $\\\\hat{\\\\rho }_1=\\\\hat{\\\\rho }(\\\\mu ,k)$ with prior probabilities $p_0$ and $p_1$ , respectively.\",\n \"Here we let $h\\\\ge k$ without loss of generality.\",\n \"According to the Helstrom bound (REF ), the MDEP is determined by all the positive eigenvalues $\\\\lambda _n$ of the decision operator $\\\\hat{\\\\Delta }=p_1\\\\hat{\\\\rho }(\\\\mu ,k)-p_0\\\\hat{\\\\rho }(\\\\xi ,h)$ .\",\n \"Because the displacement operator is an unitary operator, $\\\\hat{\\\\Delta }$ and $\\\\hat{D}(-\\\\xi )\\\\hat{\\\\Delta }\\\\hat{D}^{\\\\dagger }(-\\\\xi )$ share the same eigenvalues.\",\n \"Then it is readable to show that $\\\\Vert p_1\\\\hat{\\\\rho }(\\\\mu ,k)-p_0\\\\hat{\\\\rho }(\\\\xi ,h)\\\\Vert _1=\\\\Vert p_1\\\\hat{\\\\rho }(\\\\mu -\\\\xi ,k)-p_0\\\\hat{\\\\rho }(0,h)\\\\Vert _1.$ It is challenging to obtain an analytical form of the eigenvalues $\\\\lambda _n$ for an arbitrary $\\\\hat{\\\\Delta }$ .\",\n \"However, eq.\",\n \"(REF ) indicates that when $\\\\mu =\\\\xi $ , we only need to obtain the eigenvalues of $p_1\\\\hat{\\\\rho }(0,k)-p_0\\\\hat{\\\\rho }(0,h)$ .\",\n \"Using the Fock representation (REF ), we can obtain the eigenvalues of $p_1\\\\hat{\\\\rho }(0,k)-p_0\\\\hat{\\\\rho }(0,h)$ as $\\\\begin{aligned}\\\\lambda _n&=p_1\\\\mathinner {\\\\langle {n}|}\\\\hat{\\\\rho }(0,k)\\\\mathinner {|{n}\\\\rangle }-p_0\\\\mathinner {\\\\langle {n}|}\\\\hat{\\\\rho }(0,h)\\\\mathinner {|{n}\\\\rangle }\\\\\\\\&=\\\\left\\\\lbrace \\\\begin{array}{ll}0, \\\\text{ for } nn_{th}.\\\\end{array}\\\\right.$ Therefore, by optimizing the threshold in (REF ), we can always achieve the MDEP in (REF ), i.e., we have $\\\\sum _{\\\\lambda _n>0}\\\\lambda _n=\\\\sum _{n=0}^{n_{th}}\\\\lambda _n.$ Accordingly, the optimal threshold $n_{th}$ is the maximum $n$ satisfying (REF ).\"\n ]\n]"},"id":{"kind":"string","value":"2012.05165"}}},{"rowIdx":3692,"cells":{"text":{"kind":"list like","value":[["Sneak Preview: PowerDynamics.jl -= An Open-Source library for analyzing\n dynamic stability in power grids with high shares of renewable energy"],["Abstract PowerDynamics.jl is an Open-Source library for dynamic power grid modeling built in the latest scientific programming language, Julia.","It provides all the tools necessary to analyze the dynamical stability of power grids with high share of renewable energy.","In contrast to conventional tools, it makes full use of the simplicity and generality that Julia combines with highly-optimized just-in-time compiled Code.","Additionally, its ecosystem provides DifferentialEquations.jl, a high-performance library for solving differential equations with built-in solvers and interfaces to industrial grade solvers like Sundials.","PowerDynamics.jl provides a multitude of dynamics for different node/bus-types, e.g.","rotating masses, droop-control in inverters, and is able to explicitly model time delays of inverters.","Furthermore, it includes realistic models of fluctuations from renewable energy sources.","In this paper, we demonstrate how to use PowerDynamics.jl for the IEEE 14-bus distribution grid feeder."],["Introduction","Dynamic stability analysis of power grids concerns the investigation of transient stability and power quality.","Especially with more intermittent renewable generation sudden changes in power followed by the transient reaction of frequency and voltage dynamics may present a challenge to power system stability.","At the same time, fluctuations also challenge power quality, which refers to whether voltage and frequency stay within safety bounds and whether the waveform has undesired harmonic distortions.","Furthermore, the wide deployment of power electronics, witch RES are connected to the grid, introduces new features such as delayed reaction into future power grids.","These challenges have been the topic of recent research on the stability of renewable power grids [1], [2], [3].","However, the usage of state-of-the-art Open-Source software repeatedly posed limitations to the possible modeling scope.","One reason is the lack of solvers for the different types of dynamics which need to be described a stochastic differential equations, delay differential equations and differential algebraic equations.","At the same time to network size was constrained by computational performance.","This means a realistic model implementation and hence reliable stability analysis has been constrained by the available OS software framework.","In this conference paper, we give a sneak preview to PowerDynamics.jl, an new Open-Source approach to dynamic power grid modelingThis paper is meant as a sneak preview, as the planned publication date is Oct 15, 2018, i.e.","before the Wind Integration Workshop 2018 will take place..","While there are multiple modeling suites available they are either proprietary (e.g.","PSS® [4]) or need proprietary tools to be used (e.g.","PSAT [5], [6]).","With PowerDynamics.jl, we aim to provide a modeling framework that tackles all the needs a modeler for power grids with high shares of renewables to execute dynamic analyses and the first publication will provide tools for time-domain modeling.","At the moment, PowerDynamics.jl is under high development, with the basic software architecture already being developed and implemented.","We are currently adding new types of dynamics and controls to represent different buses, e.g.","droop control, multiple inverter types, higher-order representations of synchronous machines, and in particular intermittency representations of renewable energy sources.","A second focus of effort goes into usability, in order to improve the experience of modelers and decrease their time they have to concern with software-related problems so they can focus on their modeling instead.","Our vision is to provide a modular, fully flexible library where modelers can decide whether they (a) want to simply use repdefined bus models to analyze a power grid quickly, (b) want to into all the detail of modeling each bus type, or (c) want to find their right position in-between (a) and (b).","All of that is done in the context of an Open-Source context in order to make it available to and invitre contributions from research and industry.","PowerDynamics.jl is written in Julia [7], a new, fast developing programming language targeting the scientific computing community.","With its recent release to Julia 1.0, it has matured enough to become a powerful player competing with more traditional programming languages like Python and C in the modeling world.","In the wider context, there is a current movement within the Open Energy Modeling community [8] to switch to Julia and unite modeling efforts.","With PowerDynamics.jl we are aiming to support these efforts by contributing to dynamic power system analysis.","The rest of the paper is structured as follows: in sec:julia “sec:julia” we give a detailed reasoning why we find Julia to be the correct choice as the programming language for our modeling effort; in sec:powerdynamics “sec:powerdynamics” we introduce the basic mathematical notions we use in PowerDynamics.jl; in sec:ieee14 “sec:ieee14” we provide the details of the example system used in this paper; in sec:implementation “sec:implementation” we show how to implement the IEEE 14-bus feeder in PowerDynamics.jl; in sec:modeling “sec:modeling” we present the results for our example model; and in sec:conclusion “sec:conclusion” we conclude the paper."],["Convention & Notation","Within this paper, we take up the convention of PowerDynamics.jl where minor letters are used as variables for dynamic variables and capital letters denote (constant) parameters and all variables are in defined in the co-rotating frame.","Further, we take $u_a$ to to be the complex voltage, $v_a = |u_a|$ to be the voltage magnitude, $\\varphi _a = \\arg (u_a)$ to be the voltage angle, $i_a = \\sum _b Y^L_{ab}u_b$ to be the complex current, $s_a = u_a \\cdot i_a^*$ to be the complex power, $p_a = \\text{Re}(s_a)$ the active power and $q_a = \\text{Im}(s_a)$ to be the reactive power of the $$ ath bus.","$Y^L_{}$ is the admittance Laplacian.","The imaginary element is denoted as $j = \\sqrt{-1}$ ."],["Julia for Scientific Modeling","The advantages of Julia for scientific modeling can be summarized as (adapted from [9]): Julia is fast!","It was designed from the beginning as a high-performance language.","Using the just-in-time (JIT) compilation method, it produces efficient native code for multiple platforms.","Julia allows for rapid development!","It uses dynamic typing, so it is easy to use, feels like a scripting language and has good interactive use without losing any performance.","That way, it saves the developer a lot of time.","Julia is technical!","It is developed for numerical computing, the syntax is focusing on precise mathematics and many datatypes and even parallelism is available out of the box.","Julia is general!","Using multiple dispatch as the fundamental paradigm, it is easy to express object-oriented and functional programming patterns at the same time.","Julia is composable!","Julia has been designed such that independent packages work well together.","Hence, we can use matrices of unit quantities or differentiation with sparse matrices without having these types ever defined explicitly before.","Julia has a growing ecosystem!","The aforementioned advantages attract a growing communityThe community growth recently reached the point where the organizers had to cut off ticket sales for this year's JuliaCon at some point.","and with that a growing ecosystem of Julia packages.","Particularly important for modeling is the DifferentialEquations.jl library which provides high-performance solvers and interfaces to industrial grade solvers like SUNDIALS [10] for solving multiple kinds of differential equations (ordinary, algebraic, delayed, stochastic).","Julia allows for metaprogramming!","This means, at execution time, the source code is available as data and one can easily modify it.","Furthermore, one can even add own syntax to Julia.","Within PowerDynamics.jl, we make heavy use of metaprogramming to make it as simple as possible for a user to implement her/his own kind of bus model (see sec:implementation-nodes).","Also, it allows to write very simple code that can automatically be modified to highly optimized code.","Of course, this is just a simply overview and we would strongly recommend the reader to try out Julia and convince herself/himselfhttps://julialang.org/learning/."],["Mathematics of PowerDynamics.jl","Within this paper, we take the approach to describe the dynamics of a power grid by a (set of) semi-explicit differential algebraic equation(s).","We explicitly distinguish between the complex voltages $u_a$ (of the $$ ath bus), whose information is transmitted between buses directly through the complex current $i$ , and internal variables $x_{ab}$ (the $$ bth internal variable of the $$ ath bus), that exist locally at a node.","An example for an internal variable would be the frequency of a synchronous machine.","While the phase angle $\\varphi $ is transmitted directly via the current, its derivative is the angle velocity whose dynamics is defined locally by the the rotating mass.","We are aware that a multitude of approaches are possible to write down the dynamics and we have chosen this particular one as it can be easily translated into source code.","Writing this idea in formulas yields $i_a = \\sum _c Y^L_{ac} \\cdot u_c , \\\\m^u_a \\dot{u}_a = f_a(u_a, x_a, i_a) , \\\\m^x_{ab} \\dot{x}_{ab} = g_{ab}(u_a, x_a, i_a) , $ where $m^u$ / $m^x$ are the masses for the voltages / internal variables respectivelyWithin PowerDynamics.jl we allow masses to be 1 or 0.. Also, $f_a$ encodes the voltage dynamics of the $$ ath bus and $g_{ab}$ the dynamics of the $$ bth internal variable of the $$ ath bus.","Note the convention of using minor letters for dynamical variables.","$Y^L_{}$ is the admittance Laplacian.","So having the admittances encoded as $Y_{ab}$ where $Y_{ab} = 0$ if there is no line, or $Y_{ab}$ with a complex value (with $\\text{Re}(Y_{ab})\\ge 0$ ) for existing lines, then the admittance Laplacian is defined as $Y^L_{ab} = \\delta _{ab}\\sum _c Y_{ac} - Y_{ab},$ where $\\delta _{ab}$ is Kronecker-$\\delta $ .","Particular examples would be an algebraic slack bus as the $$ ath bus with a fixed complex voltage $U_a$ .","There are no internal variables and the voltage mass is $m_a^u = 0$ (as its an algebraic equation), so it can simply be written as $0\\cdot \\dot{u}_a = f_a(u_a, x_a, i_a) = u_a - U_a , $ where we have explicitly not removed $\\dot{u}$ in order to keep a strong analogy the the implementation shown later.","Similarly, we can treat an algebraic load at the $$ ath bus with a constant complex power $-S_a$ flowing outThe $-$ is due to the definition of all variables such that the power flows always from the respective node into the power grid.. Again, there are no internal variables and the voltage mass is 0 so it reduces to $0\\cdot \\dot{u}_a = f_a(u_a, x_a, i_a) = s_a - S_a = u_a\\cdot i_a^* - S_a .$ A synchronous machine as the $$ ath bus can be represented by the Swing equation (or 2nd-order synchronous machine model) [11], [12] with a produced active power $P_a$ , the damping constant $D_a$ , the rated frequency $\\Omega $ and the inertia $H_a$ .","In this case, the voltage mass is 1, there is one internal variable, the frequency, $x_{a1} = \\omega _a$ with a mass of 1 because it is dynamic and not an algebraic constraintIn vector notation this gives $x_{a} = (\\omega _a)$ for the Swing equation..","Hence, the equations reduce to $&\\dot{u}_a = f_a(u_a, (\\omega _a), i_a) = u_aj\\omega _a , \\\\&\\begin{aligned}\\dot{x}_{a1} = \\dot{\\omega }_a &= g_{a1}(u_a, (\\omega _a), i_a)\\\\&= \\frac{2\\pi \\Omega }{H_a}\\left( P_a - D_a\\omega _a - \\text{Re}(u_a\\cdot i_a^*) \\right)\\end{aligned} $ where we would like to remind the reader of $j$ being used as the imaginary element in this paper.","Note that with $u_a = e^{j\\varphi _a}\\ \\Rightarrow \\ du_a = ju_ad\\varphi _a$ , eq:swing-u reduces to $\\dot{\\varphi }_a = \\omega _a$ and one recovers the usual version in terms of the voltage angle.","In the following section, we will introduce the example system in order to demonstrate afterwards, how these kind of dynamics are then described in PowerDynamics.jl."],["The IEEE 14-bus distribution grid feeder","The IEEE 14-bus system is a representation of a medium-voltage distribution grid.","A graphical representation is in fig:ieee14grid, which is extracted from [13].","There are two generators, where the one at bus 2 is the slack bus, three synchronous compensators, and eight (complex) loads.","Bus 7 is for the representation of the transformer only.","The generator at bus 1 and the synchronous compensators will be modeled with the swing equation, the slack bus 2 with the algebraic constraint introduced in sec:powerdynamics and the loads with the algebraic constraints for complex loads (see sec:powerdynamics also).","The bus parameters haven been extracted from [13] and are presented in tab:ieee14-bus-parameters.","The resistance $R$ and reactance $X$ for the lines (taken from [13] as well) are summarized in tab:ieee14-line-parameters.","Having a description of the power grid in question enables us to have a look at the implementation in PowerDynamics.jl now.","Table: Bus parameters of the IEEE 14-bus test system.","INERTIA UNIT IS?Table: Line parameters of the IEEE 14-bus test system."],["Implementation in PowerDynamics.jl","In this section, we will show first how to implement the dynamics for the different buses presented in sec:powerdynamics in PowerDynamics.jl and then how create a power grid model from these bus dynamics definitions, in this case the IEEE 14-bus system."],["Defining the dynamics","The implementation of such a power grid in PowerDynamics.jl is strongly based on the $\\mathtt {@DynamicNode}$ macro.","A macro is a metaprogramming function that takes part of the source codes and modifies it at parsing time before giving it to the compiler, that creates the actual bitcode.","In case of $\\mathtt {@DynamicNode}$ , this is used to hide all the complicated internals of PowerDynamics.jl and make the definition of a type of dynamics as easy as possible.","The following examples are already provided by default in PowerDynamics.jl but using them as an example, any kind of dynamics can be implemented.","Using $\\mathtt {@DynamicNode}$ , the slack bus constraint from eq:slack-bus is implemented as [linewidth=] @DynamicNode SlackAlgebraic(U) <: OrdinaryNodeDynamicsWithMass(mu=false) begin end [] begin du = u - U end where line breaks and line numbers have been added for easier orientation.","The left part on the first line states the name of the new type $\\mathtt {SlackAlgebraic}$ and the parameter name $\\mathtt {U}$ .","Separated by the subtyping operator ($\\mathtt {<:}$ ) comes the statement that it can be represented by an ordinary differential equation with a mass term, where the mass for the voltage m_u is set to $\\mathtt {false}$ , meaning $m^u = 0$ from eq:u-general as it is a constraint.","The [] in line 2 states that there are no internal variables.","And line 3 simply provides the formula for the dynamics and assigns it to $\\mathtt {du}$ .","As m_u = false is given, this simply translates it to a constraint.","In a similar manner, the load is defined as an algebraic PQ-constraint [linewidth=] @DynamicNode PQAlgebraic(S) <: OrdinaryNodeDynamicsWithMass(mu=false) begin end [] begin s = u*conj(i) du = S - s end In this case, an additional line was added to calculate the outflowing complex power $\\mathtt {s}$ from the voltage $\\mathtt {u}$ and the current $\\mathtt {i}$ and then the definition of the constraint is written in line 4.","Concerning the generator and the synchronous compensators, we implemented the swing equation as [linewidth=] @DynamicNode SwingEq(H, P, D, SOmega) <: OrdinaryNodeDynamics() begin @assert D > 0 \"damping (D) should be >0\" @assert H > 0 \"inertia (H) should be >0\" SOmegaH = SOmega * 2pi / H end [[Somega, dSomega]] begin p = real(u * conj(i)) du = u * im * Somega dSomega = (P - D*Somega - p)*SOmegaH end Again, the left part of the first line provides the name of the new type and the name of the parameters.","Lines 2 to 4 are code that should be run only once.","In this case these are consistency checks, that the damping and inertia are positive, and the reduction of the rated frequency $\\mathtt {\\Omega }$ and the inertia $\\mathtt {H}$ to a single variable $\\mathtt {\\Omega \\_H}$ .","In line 5, the variable name of the internal variable $\\mathtt {\\omega }$ and the name for its derivative $\\mathtt {d\\omega }$ are given.","Finally, lines 6 to 8 implement eq:swing-u,eq:swing-omega by simply writing down the mathematical terms."],["Instantiating the power grid model","In order to create the grid model, we first have to instantiate the bus models simply by calling them with the corresponding parameter values from tab:ieee14-bus-parameters, e.g.",": [linewidth=] SwingEq(H=5.148, P=2.32, D=2, SOmega=50) for bus 1 PQAlgebraic(S=-0.295-0.166im) for bus 9 Within the actual code we simply loaded the data from a $\\mathtt {.csv}$ file into and automatized this instantiationThe full source code is not part of this paper as it would be too long, but it will be published along with PowerDynamics.jl..","The instantiated bus models should then be saved in an array called e.g.","$\\mathtt {nodes}$ .","Similarly, the admittance Laplacian should be generated from the line data in tab:ieee14-line-parameters and saved in a matrix called e.g.","$\\mathtt {LY}$ .","The actual grid model instatiation is then simply one line where the model is saved in the variable $\\mathtt {g}$ : [linewidth=] g = GridDynamics(nodelist, LY) Now, $\\mathtt {g}$ contains all the information of the power grid and in the following section we will show how to solve it."],["Modeling Results","Within this section, we will analyze two simple cases: (a) a frequency perturbation at the largest generator at bus 1 and (b) a line tripping event at of line 2 (between bus 1 and 5) (cmp.","fig:ieee14grid).","Before we find the normal point of operation for the power grid, i.e.","the fixed point of eq:i-definition,eq:u-general,eq:x-general or synchronous state for the IEEE 14-bus system.","For that, we use the grid model $\\mathtt {g}$ generated in the previous sec:implementation and use the function provided by PowerDynamics.jl: [linewidth=] fp = operationpoint(g, ones(SystemSize(g))) where $\\mathtt {ones(SystemSize(g))}$ is a vector of the correct length for the initial condition of the fixed point search.","$\\mathtt {fp}$ is now a $\\mathtt {State}$ object that we can use as initial condition for the solving the differential equations corresponding to the power grid."],["Frequency perturbation","In order to model a frequency perturbation, one can simply take a copy of the fixed point as found before and adjust the initial frequency value: [linewidth=] x0 = copy(fp) x0[1, :int, 1] += 0.2 The second line can be read as “add 0.2 to the 1$^\\text{st}$ internal variable of the 1$^\\text{st}$ node” which is the frequency $\\omega $ as there is only one internal variable.","Note that the first $\\mathtt {1}$ refers to the node and then second $\\mathtt {1}$ to the internal variable counter.","The power grid with the initial condition $\\mathtt {x0}$ can then be solved for a time span of 0.5 seconds by calling: [linewidth=] sol = solve(g, x0, (0.0,.5)) The solution is shown in fig:ieee14-frequency-perturbation.","It shows that the system is very stable against frequency perturbations.","The actual dynamics is not so exciting as the system is very stable.","Please note that this system was taken as an example to present on how easily one can model a power grid using PowerDynamics.jl, not to find new exciting dynamics."],["Line tripping","To show some more dynamic behavior we simulated a line tripping as well.","We model this effect by taking the operation point of the full power grid ($\\mathtt {fp}$ ) as initial condition but defining a new admittance Laplacian where line 2 (between bus 1 and 5, see tab:ieee14-line-parameters,fig:ieee14grid) has been taken out (i.e.","the admittance is set to 0).","Running the model with this new Laplacian yields fig:ieee14-line-tripping.","In the frequency plot we identify how the frequency of bus 1, where the line tripping happened, compensates for the momentarily excess power at the bus.","The lacking power in the rest of the grid is matched by the synchronous compensators whose frequency decreases in turn.","Note that the angular frequency $\\omega $ is shown, so with a division by $2\\pi $ the maximal frequency deviation $f$ is $\\approx 0.05\\,$ Hz.","After about one second, the system recovers to the normal state of operation."],["Conclusion & Outlook","Within this paper, we have seen how one one can use the Open-Source library PowerDynamics.jl in order to model the dynamics of a power grid with just a few lines of code.","We have seen how the fundamental mathematical equations (given in sec:powerdynamics) translate to source code that reads exactly the same.","We employed PowerDynamics.jl for the IEEE 14-bus distribution grid feeder in order to demonstrate how one can easily simulate faults and analyze the transient reaction of the power grid dynamics.","As example scenarios we used a frequency perturbation and a line tripping.","Finally, this paper is really just a sneak preview with a simple example the publication of PowerDynamics.jl is planned for October 15, 2018.","By then, we will have added more inverter control schemes and stochastic descriptions of intermittency due to renewable energy sources."],["Acknowledgment","This paper was presented at the 19th Wind Integration Workshop and published in the workshop’s proceedings.","We would like to thank the German Academic Exchange Service for the opportunity to participate at the Wind Integration Workshop 2018 in Stockholm via the funding program “Kongressreisen 2018”.","This paper is based on work developed within the Climate-KIC Pathfinder project “elena – electricity network analysis” funded by the European Institute of Innovation & Technology.","This work was conducted in the framework of the Complex Energy Networks research group at the Potsdam Institute for Climate Impact Research.","We would like to thank Frank Hellmann and Paul Schultz for the discussions on structuring an Open-Source library for dynamic power grid modeling."]],"string":"[\n [\n \"Sneak Preview: PowerDynamics.jl -= An Open-Source library for analyzing\\n dynamic stability in power grids with high shares of renewable energy\"\n ],\n [\n \"Abstract PowerDynamics.jl is an Open-Source library for dynamic power grid modeling built in the latest scientific programming language, Julia.\",\n \"It provides all the tools necessary to analyze the dynamical stability of power grids with high share of renewable energy.\",\n \"In contrast to conventional tools, it makes full use of the simplicity and generality that Julia combines with highly-optimized just-in-time compiled Code.\",\n \"Additionally, its ecosystem provides DifferentialEquations.jl, a high-performance library for solving differential equations with built-in solvers and interfaces to industrial grade solvers like Sundials.\",\n \"PowerDynamics.jl provides a multitude of dynamics for different node/bus-types, e.g.\",\n \"rotating masses, droop-control in inverters, and is able to explicitly model time delays of inverters.\",\n \"Furthermore, it includes realistic models of fluctuations from renewable energy sources.\",\n \"In this paper, we demonstrate how to use PowerDynamics.jl for the IEEE 14-bus distribution grid feeder.\"\n ],\n [\n \"Introduction\",\n \"Dynamic stability analysis of power grids concerns the investigation of transient stability and power quality.\",\n \"Especially with more intermittent renewable generation sudden changes in power followed by the transient reaction of frequency and voltage dynamics may present a challenge to power system stability.\",\n \"At the same time, fluctuations also challenge power quality, which refers to whether voltage and frequency stay within safety bounds and whether the waveform has undesired harmonic distortions.\",\n \"Furthermore, the wide deployment of power electronics, witch RES are connected to the grid, introduces new features such as delayed reaction into future power grids.\",\n \"These challenges have been the topic of recent research on the stability of renewable power grids [1], [2], [3].\",\n \"However, the usage of state-of-the-art Open-Source software repeatedly posed limitations to the possible modeling scope.\",\n \"One reason is the lack of solvers for the different types of dynamics which need to be described a stochastic differential equations, delay differential equations and differential algebraic equations.\",\n \"At the same time to network size was constrained by computational performance.\",\n \"This means a realistic model implementation and hence reliable stability analysis has been constrained by the available OS software framework.\",\n \"In this conference paper, we give a sneak preview to PowerDynamics.jl, an new Open-Source approach to dynamic power grid modelingThis paper is meant as a sneak preview, as the planned publication date is Oct 15, 2018, i.e.\",\n \"before the Wind Integration Workshop 2018 will take place..\",\n \"While there are multiple modeling suites available they are either proprietary (e.g.\",\n \"PSS® [4]) or need proprietary tools to be used (e.g.\",\n \"PSAT [5], [6]).\",\n \"With PowerDynamics.jl, we aim to provide a modeling framework that tackles all the needs a modeler for power grids with high shares of renewables to execute dynamic analyses and the first publication will provide tools for time-domain modeling.\",\n \"At the moment, PowerDynamics.jl is under high development, with the basic software architecture already being developed and implemented.\",\n \"We are currently adding new types of dynamics and controls to represent different buses, e.g.\",\n \"droop control, multiple inverter types, higher-order representations of synchronous machines, and in particular intermittency representations of renewable energy sources.\",\n \"A second focus of effort goes into usability, in order to improve the experience of modelers and decrease their time they have to concern with software-related problems so they can focus on their modeling instead.\",\n \"Our vision is to provide a modular, fully flexible library where modelers can decide whether they (a) want to simply use repdefined bus models to analyze a power grid quickly, (b) want to into all the detail of modeling each bus type, or (c) want to find their right position in-between (a) and (b).\",\n \"All of that is done in the context of an Open-Source context in order to make it available to and invitre contributions from research and industry.\",\n \"PowerDynamics.jl is written in Julia [7], a new, fast developing programming language targeting the scientific computing community.\",\n \"With its recent release to Julia 1.0, it has matured enough to become a powerful player competing with more traditional programming languages like Python and C in the modeling world.\",\n \"In the wider context, there is a current movement within the Open Energy Modeling community [8] to switch to Julia and unite modeling efforts.\",\n \"With PowerDynamics.jl we are aiming to support these efforts by contributing to dynamic power system analysis.\",\n \"The rest of the paper is structured as follows: in sec:julia “sec:julia” we give a detailed reasoning why we find Julia to be the correct choice as the programming language for our modeling effort; in sec:powerdynamics “sec:powerdynamics” we introduce the basic mathematical notions we use in PowerDynamics.jl; in sec:ieee14 “sec:ieee14” we provide the details of the example system used in this paper; in sec:implementation “sec:implementation” we show how to implement the IEEE 14-bus feeder in PowerDynamics.jl; in sec:modeling “sec:modeling” we present the results for our example model; and in sec:conclusion “sec:conclusion” we conclude the paper.\"\n ],\n [\n \"Convention & Notation\",\n \"Within this paper, we take up the convention of PowerDynamics.jl where minor letters are used as variables for dynamic variables and capital letters denote (constant) parameters and all variables are in defined in the co-rotating frame.\",\n \"Further, we take $u_a$ to to be the complex voltage, $v_a = |u_a|$ to be the voltage magnitude, $\\\\varphi _a = \\\\arg (u_a)$ to be the voltage angle, $i_a = \\\\sum _b Y^L_{ab}u_b$ to be the complex current, $s_a = u_a \\\\cdot i_a^*$ to be the complex power, $p_a = \\\\text{Re}(s_a)$ the active power and $q_a = \\\\text{Im}(s_a)$ to be the reactive power of the $$ ath bus.\",\n \"$Y^L_{}$ is the admittance Laplacian.\",\n \"The imaginary element is denoted as $j = \\\\sqrt{-1}$ .\"\n ],\n [\n \"Julia for Scientific Modeling\",\n \"The advantages of Julia for scientific modeling can be summarized as (adapted from [9]): Julia is fast!\",\n \"It was designed from the beginning as a high-performance language.\",\n \"Using the just-in-time (JIT) compilation method, it produces efficient native code for multiple platforms.\",\n \"Julia allows for rapid development!\",\n \"It uses dynamic typing, so it is easy to use, feels like a scripting language and has good interactive use without losing any performance.\",\n \"That way, it saves the developer a lot of time.\",\n \"Julia is technical!\",\n \"It is developed for numerical computing, the syntax is focusing on precise mathematics and many datatypes and even parallelism is available out of the box.\",\n \"Julia is general!\",\n \"Using multiple dispatch as the fundamental paradigm, it is easy to express object-oriented and functional programming patterns at the same time.\",\n \"Julia is composable!\",\n \"Julia has been designed such that independent packages work well together.\",\n \"Hence, we can use matrices of unit quantities or differentiation with sparse matrices without having these types ever defined explicitly before.\",\n \"Julia has a growing ecosystem!\",\n \"The aforementioned advantages attract a growing communityThe community growth recently reached the point where the organizers had to cut off ticket sales for this year's JuliaCon at some point.\",\n \"and with that a growing ecosystem of Julia packages.\",\n \"Particularly important for modeling is the DifferentialEquations.jl library which provides high-performance solvers and interfaces to industrial grade solvers like SUNDIALS [10] for solving multiple kinds of differential equations (ordinary, algebraic, delayed, stochastic).\",\n \"Julia allows for metaprogramming!\",\n \"This means, at execution time, the source code is available as data and one can easily modify it.\",\n \"Furthermore, one can even add own syntax to Julia.\",\n \"Within PowerDynamics.jl, we make heavy use of metaprogramming to make it as simple as possible for a user to implement her/his own kind of bus model (see sec:implementation-nodes).\",\n \"Also, it allows to write very simple code that can automatically be modified to highly optimized code.\",\n \"Of course, this is just a simply overview and we would strongly recommend the reader to try out Julia and convince herself/himselfhttps://julialang.org/learning/.\"\n ],\n [\n \"Mathematics of PowerDynamics.jl\",\n \"Within this paper, we take the approach to describe the dynamics of a power grid by a (set of) semi-explicit differential algebraic equation(s).\",\n \"We explicitly distinguish between the complex voltages $u_a$ (of the $$ ath bus), whose information is transmitted between buses directly through the complex current $i$ , and internal variables $x_{ab}$ (the $$ bth internal variable of the $$ ath bus), that exist locally at a node.\",\n \"An example for an internal variable would be the frequency of a synchronous machine.\",\n \"While the phase angle $\\\\varphi $ is transmitted directly via the current, its derivative is the angle velocity whose dynamics is defined locally by the the rotating mass.\",\n \"We are aware that a multitude of approaches are possible to write down the dynamics and we have chosen this particular one as it can be easily translated into source code.\",\n \"Writing this idea in formulas yields $i_a = \\\\sum _c Y^L_{ac} \\\\cdot u_c , \\\\\\\\m^u_a \\\\dot{u}_a = f_a(u_a, x_a, i_a) , \\\\\\\\m^x_{ab} \\\\dot{x}_{ab} = g_{ab}(u_a, x_a, i_a) , $ where $m^u$ / $m^x$ are the masses for the voltages / internal variables respectivelyWithin PowerDynamics.jl we allow masses to be 1 or 0.. Also, $f_a$ encodes the voltage dynamics of the $$ ath bus and $g_{ab}$ the dynamics of the $$ bth internal variable of the $$ ath bus.\",\n \"Note the convention of using minor letters for dynamical variables.\",\n \"$Y^L_{}$ is the admittance Laplacian.\",\n \"So having the admittances encoded as $Y_{ab}$ where $Y_{ab} = 0$ if there is no line, or $Y_{ab}$ with a complex value (with $\\\\text{Re}(Y_{ab})\\\\ge 0$ ) for existing lines, then the admittance Laplacian is defined as $Y^L_{ab} = \\\\delta _{ab}\\\\sum _c Y_{ac} - Y_{ab},$ where $\\\\delta _{ab}$ is Kronecker-$\\\\delta $ .\",\n \"Particular examples would be an algebraic slack bus as the $$ ath bus with a fixed complex voltage $U_a$ .\",\n \"There are no internal variables and the voltage mass is $m_a^u = 0$ (as its an algebraic equation), so it can simply be written as $0\\\\cdot \\\\dot{u}_a = f_a(u_a, x_a, i_a) = u_a - U_a , $ where we have explicitly not removed $\\\\dot{u}$ in order to keep a strong analogy the the implementation shown later.\",\n \"Similarly, we can treat an algebraic load at the $$ ath bus with a constant complex power $-S_a$ flowing outThe $-$ is due to the definition of all variables such that the power flows always from the respective node into the power grid.. Again, there are no internal variables and the voltage mass is 0 so it reduces to $0\\\\cdot \\\\dot{u}_a = f_a(u_a, x_a, i_a) = s_a - S_a = u_a\\\\cdot i_a^* - S_a .$ A synchronous machine as the $$ ath bus can be represented by the Swing equation (or 2nd-order synchronous machine model) [11], [12] with a produced active power $P_a$ , the damping constant $D_a$ , the rated frequency $\\\\Omega $ and the inertia $H_a$ .\",\n \"In this case, the voltage mass is 1, there is one internal variable, the frequency, $x_{a1} = \\\\omega _a$ with a mass of 1 because it is dynamic and not an algebraic constraintIn vector notation this gives $x_{a} = (\\\\omega _a)$ for the Swing equation..\",\n \"Hence, the equations reduce to $&\\\\dot{u}_a = f_a(u_a, (\\\\omega _a), i_a) = u_aj\\\\omega _a , \\\\\\\\&\\\\begin{aligned}\\\\dot{x}_{a1} = \\\\dot{\\\\omega }_a &= g_{a1}(u_a, (\\\\omega _a), i_a)\\\\\\\\&= \\\\frac{2\\\\pi \\\\Omega }{H_a}\\\\left( P_a - D_a\\\\omega _a - \\\\text{Re}(u_a\\\\cdot i_a^*) \\\\right)\\\\end{aligned} $ where we would like to remind the reader of $j$ being used as the imaginary element in this paper.\",\n \"Note that with $u_a = e^{j\\\\varphi _a}\\\\ \\\\Rightarrow \\\\ du_a = ju_ad\\\\varphi _a$ , eq:swing-u reduces to $\\\\dot{\\\\varphi }_a = \\\\omega _a$ and one recovers the usual version in terms of the voltage angle.\",\n \"In the following section, we will introduce the example system in order to demonstrate afterwards, how these kind of dynamics are then described in PowerDynamics.jl.\"\n ],\n [\n \"The IEEE 14-bus distribution grid feeder\",\n \"The IEEE 14-bus system is a representation of a medium-voltage distribution grid.\",\n \"A graphical representation is in fig:ieee14grid, which is extracted from [13].\",\n \"There are two generators, where the one at bus 2 is the slack bus, three synchronous compensators, and eight (complex) loads.\",\n \"Bus 7 is for the representation of the transformer only.\",\n \"The generator at bus 1 and the synchronous compensators will be modeled with the swing equation, the slack bus 2 with the algebraic constraint introduced in sec:powerdynamics and the loads with the algebraic constraints for complex loads (see sec:powerdynamics also).\",\n \"The bus parameters haven been extracted from [13] and are presented in tab:ieee14-bus-parameters.\",\n \"The resistance $R$ and reactance $X$ for the lines (taken from [13] as well) are summarized in tab:ieee14-line-parameters.\",\n \"Having a description of the power grid in question enables us to have a look at the implementation in PowerDynamics.jl now.\",\n \"Table: Bus parameters of the IEEE 14-bus test system.\",\n \"INERTIA UNIT IS?Table: Line parameters of the IEEE 14-bus test system.\"\n ],\n [\n \"Implementation in PowerDynamics.jl\",\n \"In this section, we will show first how to implement the dynamics for the different buses presented in sec:powerdynamics in PowerDynamics.jl and then how create a power grid model from these bus dynamics definitions, in this case the IEEE 14-bus system.\"\n ],\n [\n \"Defining the dynamics\",\n \"The implementation of such a power grid in PowerDynamics.jl is strongly based on the $\\\\mathtt {@DynamicNode}$ macro.\",\n \"A macro is a metaprogramming function that takes part of the source codes and modifies it at parsing time before giving it to the compiler, that creates the actual bitcode.\",\n \"In case of $\\\\mathtt {@DynamicNode}$ , this is used to hide all the complicated internals of PowerDynamics.jl and make the definition of a type of dynamics as easy as possible.\",\n \"The following examples are already provided by default in PowerDynamics.jl but using them as an example, any kind of dynamics can be implemented.\",\n \"Using $\\\\mathtt {@DynamicNode}$ , the slack bus constraint from eq:slack-bus is implemented as [linewidth=] @DynamicNode SlackAlgebraic(U) <: OrdinaryNodeDynamicsWithMass(mu=false) begin end [] begin du = u - U end where line breaks and line numbers have been added for easier orientation.\",\n \"The left part on the first line states the name of the new type $\\\\mathtt {SlackAlgebraic}$ and the parameter name $\\\\mathtt {U}$ .\",\n \"Separated by the subtyping operator ($\\\\mathtt {<:}$ ) comes the statement that it can be represented by an ordinary differential equation with a mass term, where the mass for the voltage m_u is set to $\\\\mathtt {false}$ , meaning $m^u = 0$ from eq:u-general as it is a constraint.\",\n \"The [] in line 2 states that there are no internal variables.\",\n \"And line 3 simply provides the formula for the dynamics and assigns it to $\\\\mathtt {du}$ .\",\n \"As m_u = false is given, this simply translates it to a constraint.\",\n \"In a similar manner, the load is defined as an algebraic PQ-constraint [linewidth=] @DynamicNode PQAlgebraic(S) <: OrdinaryNodeDynamicsWithMass(mu=false) begin end [] begin s = u*conj(i) du = S - s end In this case, an additional line was added to calculate the outflowing complex power $\\\\mathtt {s}$ from the voltage $\\\\mathtt {u}$ and the current $\\\\mathtt {i}$ and then the definition of the constraint is written in line 4.\",\n \"Concerning the generator and the synchronous compensators, we implemented the swing equation as [linewidth=] @DynamicNode SwingEq(H, P, D, SOmega) <: OrdinaryNodeDynamics() begin @assert D > 0 \\\"damping (D) should be >0\\\" @assert H > 0 \\\"inertia (H) should be >0\\\" SOmegaH = SOmega * 2pi / H end [[Somega, dSomega]] begin p = real(u * conj(i)) du = u * im * Somega dSomega = (P - D*Somega - p)*SOmegaH end Again, the left part of the first line provides the name of the new type and the name of the parameters.\",\n \"Lines 2 to 4 are code that should be run only once.\",\n \"In this case these are consistency checks, that the damping and inertia are positive, and the reduction of the rated frequency $\\\\mathtt {\\\\Omega }$ and the inertia $\\\\mathtt {H}$ to a single variable $\\\\mathtt {\\\\Omega \\\\_H}$ .\",\n \"In line 5, the variable name of the internal variable $\\\\mathtt {\\\\omega }$ and the name for its derivative $\\\\mathtt {d\\\\omega }$ are given.\",\n \"Finally, lines 6 to 8 implement eq:swing-u,eq:swing-omega by simply writing down the mathematical terms.\"\n ],\n [\n \"Instantiating the power grid model\",\n \"In order to create the grid model, we first have to instantiate the bus models simply by calling them with the corresponding parameter values from tab:ieee14-bus-parameters, e.g.\",\n \": [linewidth=] SwingEq(H=5.148, P=2.32, D=2, SOmega=50) for bus 1 PQAlgebraic(S=-0.295-0.166im) for bus 9 Within the actual code we simply loaded the data from a $\\\\mathtt {.csv}$ file into and automatized this instantiationThe full source code is not part of this paper as it would be too long, but it will be published along with PowerDynamics.jl..\",\n \"The instantiated bus models should then be saved in an array called e.g.\",\n \"$\\\\mathtt {nodes}$ .\",\n \"Similarly, the admittance Laplacian should be generated from the line data in tab:ieee14-line-parameters and saved in a matrix called e.g.\",\n \"$\\\\mathtt {LY}$ .\",\n \"The actual grid model instatiation is then simply one line where the model is saved in the variable $\\\\mathtt {g}$ : [linewidth=] g = GridDynamics(nodelist, LY) Now, $\\\\mathtt {g}$ contains all the information of the power grid and in the following section we will show how to solve it.\"\n ],\n [\n \"Modeling Results\",\n \"Within this section, we will analyze two simple cases: (a) a frequency perturbation at the largest generator at bus 1 and (b) a line tripping event at of line 2 (between bus 1 and 5) (cmp.\",\n \"fig:ieee14grid).\",\n \"Before we find the normal point of operation for the power grid, i.e.\",\n \"the fixed point of eq:i-definition,eq:u-general,eq:x-general or synchronous state for the IEEE 14-bus system.\",\n \"For that, we use the grid model $\\\\mathtt {g}$ generated in the previous sec:implementation and use the function provided by PowerDynamics.jl: [linewidth=] fp = operationpoint(g, ones(SystemSize(g))) where $\\\\mathtt {ones(SystemSize(g))}$ is a vector of the correct length for the initial condition of the fixed point search.\",\n \"$\\\\mathtt {fp}$ is now a $\\\\mathtt {State}$ object that we can use as initial condition for the solving the differential equations corresponding to the power grid.\"\n ],\n [\n \"Frequency perturbation\",\n \"In order to model a frequency perturbation, one can simply take a copy of the fixed point as found before and adjust the initial frequency value: [linewidth=] x0 = copy(fp) x0[1, :int, 1] += 0.2 The second line can be read as “add 0.2 to the 1$^\\\\text{st}$ internal variable of the 1$^\\\\text{st}$ node” which is the frequency $\\\\omega $ as there is only one internal variable.\",\n \"Note that the first $\\\\mathtt {1}$ refers to the node and then second $\\\\mathtt {1}$ to the internal variable counter.\",\n \"The power grid with the initial condition $\\\\mathtt {x0}$ can then be solved for a time span of 0.5 seconds by calling: [linewidth=] sol = solve(g, x0, (0.0,.5)) The solution is shown in fig:ieee14-frequency-perturbation.\",\n \"It shows that the system is very stable against frequency perturbations.\",\n \"The actual dynamics is not so exciting as the system is very stable.\",\n \"Please note that this system was taken as an example to present on how easily one can model a power grid using PowerDynamics.jl, not to find new exciting dynamics.\"\n ],\n [\n \"Line tripping\",\n \"To show some more dynamic behavior we simulated a line tripping as well.\",\n \"We model this effect by taking the operation point of the full power grid ($\\\\mathtt {fp}$ ) as initial condition but defining a new admittance Laplacian where line 2 (between bus 1 and 5, see tab:ieee14-line-parameters,fig:ieee14grid) has been taken out (i.e.\",\n \"the admittance is set to 0).\",\n \"Running the model with this new Laplacian yields fig:ieee14-line-tripping.\",\n \"In the frequency plot we identify how the frequency of bus 1, where the line tripping happened, compensates for the momentarily excess power at the bus.\",\n \"The lacking power in the rest of the grid is matched by the synchronous compensators whose frequency decreases in turn.\",\n \"Note that the angular frequency $\\\\omega $ is shown, so with a division by $2\\\\pi $ the maximal frequency deviation $f$ is $\\\\approx 0.05\\\\,$ Hz.\",\n \"After about one second, the system recovers to the normal state of operation.\"\n ],\n [\n \"Conclusion & Outlook\",\n \"Within this paper, we have seen how one one can use the Open-Source library PowerDynamics.jl in order to model the dynamics of a power grid with just a few lines of code.\",\n \"We have seen how the fundamental mathematical equations (given in sec:powerdynamics) translate to source code that reads exactly the same.\",\n \"We employed PowerDynamics.jl for the IEEE 14-bus distribution grid feeder in order to demonstrate how one can easily simulate faults and analyze the transient reaction of the power grid dynamics.\",\n \"As example scenarios we used a frequency perturbation and a line tripping.\",\n \"Finally, this paper is really just a sneak preview with a simple example the publication of PowerDynamics.jl is planned for October 15, 2018.\",\n \"By then, we will have added more inverter control schemes and stochastic descriptions of intermittency due to renewable energy sources.\"\n ],\n [\n \"Acknowledgment\",\n \"This paper was presented at the 19th Wind Integration Workshop and published in the workshop’s proceedings.\",\n \"We would like to thank the German Academic Exchange Service for the opportunity to participate at the Wind Integration Workshop 2018 in Stockholm via the funding program “Kongressreisen 2018”.\",\n \"This paper is based on work developed within the Climate-KIC Pathfinder project “elena – electricity network analysis” funded by the European Institute of Innovation & Technology.\",\n \"This work was conducted in the framework of the Complex Energy Networks research group at the Potsdam Institute for Climate Impact Research.\",\n \"We would like to thank Frank Hellmann and Paul Schultz for the discussions on structuring an Open-Source library for dynamic power grid modeling.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05175"}}},{"rowIdx":3693,"cells":{"text":{"kind":"list like","value":[["Symmetry and Quantum Query-to-Communication Simulation"],["Abstract Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) < 2 R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively.","Chakraborty et al.","(CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required.","We improve upon their result in several ways.","We show that the log n overhead is not required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05).","This upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication.","To prove this, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function.","In view of our first result, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive.","We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2.","We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free."],["Motivation and main results","The classical model of communication complexity was introduced by Yao [33], who also subsequently introduced its quantum analogue [34].","Communication complexity has important applications in several disciplines, in particular for lower bounds on circuits, data structures, streaming algorithms, and many other complexity measures (see, for example, [22] and the references therein).","A natural way to derive a communication problem from a Boolean function $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is via composition.","Let $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function and let $G: \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a “two-party function”.","Then $F = f \\circ G : \\left\\lbrace -1,1 \\right\\rbrace ^{nj} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{nk} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ denotes the function corresponding to the communication problem in which Alice is given input $X = (X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nj}$ , Bob is given $Y = (Y_1, \\dots , Y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nk}$ , and their task is to compute $F(X, Y) = f(G(X_1, Y_1), \\dots , G(X_n, Y_n))$ .","Many well-known functions in communication complexity are derived in this way, such as Set-Disjointness ($\\mathsf {DISJ}_n := \\mathsf {NOR}_n \\circ \\mathsf {AND}_2$ ), Inner Product ($\\mathsf {IP}_{n} : = \\mathsf {PARITY}_n \\circ \\mathsf {AND}_2$ ) and Equality ($\\mathsf {EQ}_{n} : = \\mathsf {NOR}_n \\circ \\mathsf {XOR}_2$ ).","A natural approach to obtain efficient quantum communication protocols for $f \\circ G$ is to “simulate” a quantum query algorithm for $f$ , where a query to the $i$ th input bit of $f$ is simulated by a communication protocol that computes $G(X_i,Y_i)$ .","Buhrman, Cleve and Wigderson [6] observed that such a simulation is indeed possible if $G$ is bitwise AND or XOR.","Theorem 1.1 ([6]) For every Boolean function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ , we have $\\mathsf {Q}^{cc}\\left(f \\circ \\square \\right) = O\\left(\\mathsf {Q}(f) \\log n\\right).$ Here $\\mathsf {Q}(f)$ denotes the bounded-error quantum query complexity of $f$ , and $\\mathsf {Q}^{cc}(f\\circ \\square )$ denotes the bounded-error quantum communication complexity for computing $f\\circ \\square $ .","Throughout this paper, we refer to Theorem REF as the BCW simulation.","[6] used this, for instance, to show that the bounded-error quantum communication complexity of the Set-Disjointness function is $O(\\sqrt{n}\\log n)$ , using Grover's $O(\\sqrt{n})$ -query search algorithm [16] for the $\\mathsf {NOR}_n$ function.","It is folklore in the classical world that the analogous simulation does not incur a $\\log n$ factor overhead.","That is, $\\mathsf {R}^{cc}\\left(f \\circ \\square \\right) \\le 2\\mathsf {R}(f),$ where $\\mathsf {R}(f)$ denotes the bounded-error randomized query complexity of $f$ and $\\mathsf {R}^{cc}(f\\circ \\square )$ denotes the bounded-error randomized communication complexity for computing $f\\circ \\square $ .","Thus, a natural question is whether the multiplicative $\\log n$ blow-up in the communication cost in the BCW simulation is necessary.","Chakraborty et al.","[12] answered this question and exhibited a total function for which the $\\log n$ blow-up is indeed necessary when $\\mathsf {XOR}_2$ is the inner function.","Theorem 1.2 ([12]) There exists a function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ such that $\\mathsf {Q}^{cc,*}(f \\circ \\mathsf {XOR}_2) = \\Omega (\\mathsf {Q}(f) \\log n).$ Here $\\mathsf {Q}^{cc,*}(F)$ denotes the bounded-error quantum communication complexity of two-party function $F$ when Alice and Bob shared an entangled state at the start of the protocol for free.","This gives rise to the following basic question: is there a natural class of functions for which the $\\log n$ overhead in the BCW simulation is not required?","Improving upon Høyer and de Wolf [17], Aaronson and Ambainis [1] showed that for the canonical problem of Set-Disjointness, the $\\log n$ overhead in the BCW simulation can be avoided.","Since the outer function $\\mathsf {NOR}_n$ is symmetric (i.e., it only depends on the Hamming weight of its input, its number of $-1$ s), a natural question is whether the $\\log n$ overhead can be avoided whenever the outer function is symmetric.","Our first result gives a positive answer to this question.","Theorem 1.3 For every symmetric Boolean function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and two-party function $G:\\left\\lbrace -1,1 \\right\\rbrace ^j\\times \\left\\lbrace -1,1 \\right\\rbrace ^k\\rightarrow \\lbrace 0,1\\rbrace $ , we have $\\mathsf {Q}^{cc,*}(f \\circ G)=O(\\mathsf {Q}(f)\\mathsf {Q}_E^{cc}(G)).$ Here $\\mathsf {Q}_E^{cc}(G)$ denotes the exact quantum communication complexity of $G$ , where the error probability is 0.","In particular, if $G\\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ then $\\mathsf {Q}_E^{cc}(G)=1$ and hence $\\mathsf {Q}^{cc,*}(f \\circ G)=O(\\mathsf {Q}(f))$ .","Remark 1.4 If $\\mathsf {Q}(f)=\\Theta (\\sqrt{tn})$ , then our protocol in the proof of Theorem REF starts from a shared entangled state of $O(t\\log n)$ EPR-pairs.","Note that if $t\\le n\\mathsf {Q}_E^{cc}(G)^2/(\\log n)^2$ (this condition holds for instance if $\\mathsf {Q}_E^{cc}(G)\\ge \\log n$ ) then this number of EPR-pairs is no more than the amount of communication and hence might as well be established in the first message, giving asymptotically the same upper bound $\\mathsf {Q}^{cc}(f \\circ G)=O(\\mathsf {Q}(f)\\mathsf {Q}_E^{cc}(G))$ for the model without prior entanglement.","The next question one might ask is whether one can weaken the notion of symmetry required in Theorem REF .","A natural generalization of the class of symmetric functions is the class of transitive-symmetric functions.","A function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is said to be transitive-symmetric if for all $i, j \\in [n]$ , there exists $\\sigma \\in S_n$ such that $\\sigma (i) = j$ , and $f(x) = f(\\sigma (x))$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","Henceforth we refer to transitive-symmetric functions as simply transitive functions.","Can the $\\log n$ overhead in the BCW simulation be avoided whenever the outer function is transitive?","We give a negative answer to this question in a strong sense: the $\\log n$ overhead is still necessary even when we allow the quantum communication protocol an error probability that can be arbitrarily close to $1/2$ .","Theorem 1.5 There exists a transitive and total function $f : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , such that $\\mathsf {UPP}^{cc}(f \\circ \\square ) & = \\Omega (\\mathsf {Q}(f) \\log n)$ for every $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .","Here $\\mathsf {UPP}^{cc}(f \\circ \\square )$ denotes the unbounded-error quantum communication complexity of $f \\circ \\square $ (adding “quantum” here only changes the communication complexity by a constant factor).","The unbounded-error model of communication was introduced by Paturi and Simon [30] and is the strongest communication complexity model against which we know how to prove explicit lower bounds.","This model is known to be strictly stronger than the bounded-error quantum model.","For instance, the Set-Disjointness function on $n$ inputs requires $\\Omega (n)$ bits or $\\Omega (\\sqrt{n})$ qubits of communication in the bounded-error model, but only requires $O(\\log n)$ bits of communication in the unbounded-error model.","In fact, it follows from a recent result of Hatami, Hosseini and Lovett [18] that there exists a function $F : \\left\\lbrace -1,1 \\right\\rbrace ^n\\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ with $\\mathsf {Q}^{cc,*}(F) = \\Omega (n)$ while $\\mathsf {UPP}^{cc}(F) = O(1)$ .","Theorem REF and Theorem REF clearly demonstrate the role of symmetry in determining the presence of the $\\log n$ overhead in the BCW query-to-communication simulation: this overhead is absent for symmetric functions (Theorem REF ), but present for a transitive function even when the model of communication under consideration is as strong as the unbounded-error model (Theorem REF ).","We also give a general recipe to construct functions for which the $\\log n$ overhead is required in the BCW simulation in the bounded-error communication model (see Theorem REF )."],["Overview of our approach and techniques","In this section we discuss the ideas that go into the proofs of Theorem REF and Theorem REF ."],["Communication complexity upper bound for symmetric functions","To prove Theorem REF we use the well-known fact that every symmetric function $f$ has an interval around Hamming weight $n/2$ where the function is constant; for $\\mathsf {NOR}_n$ the length of this interval would be essentially $n$ , while for $\\mathsf {PARITY}_n$ it would be 1.","To compute $f$ , it suffices to either determine that the Hamming weight of the input lies in that interval (because the function value is the same throughout that interval) or to count the Hamming weight exactly.","For two-party functions of the form $f\\circ G$ , we want to do this type of counting on the $n$ -bit string $z=(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","We show how this can be done with $O(\\mathsf {Q}(f)\\,\\mathsf {Q}^{cc}_E(G))$ qubits of communication if we had a quantum protocol that can find $-1$ s in the string $z$ at a cost of $O(\\sqrt{n}\\,\\mathsf {Q}^{cc}_E(G))$ qubits.","Such a protocol was already given by Aaronson and Ambainis for the special case where $G=\\mathsf {AND}_2$ for their optimal quantum protocol for Set-Disjointness, as a corollary of their quantum walk algorithm for search on a grid [1].","In this paper we give an alternative $O(\\sqrt{n}\\,\\mathsf {Q}^{cc}_E(G))$ -qubit protocol.","This implies the result of Aaronson and Ambainis as a special case, but it is arguably simpler and may be of independent interest.","Our protocol can be viewed as an efficient distributed implementation of amplitude amplification with faulty components.","In particular, we replace the usual reflection about the uniform superposition by an imperfect reflection about the $n$ -dimensional maximally entangled state ($=\\log n$ EPR-pairs if $n$ is a power of 2).","Such a reflection would require $O(\\log n)$ qubits of communication to implement perfectly, but can be implemented with small error using only $O(1)$ qubits of communication, by invoking the efficient protocol of Aharonov et al.","[4] that tests whether a given bipartite state equals the $n$ -dimensional maximally entangled state.","Still, at the start of this protocol we need to assume (or establish by means of quantum communication) a shared state of $\\log n$ EPR-pairs.","If $\\mathsf {Q}(f)=\\Theta (\\sqrt{tn})$ then our protocol for $f\\circ G$ will run the $-1$ -finding protocol $O(t)$ times, which accounts for our assumption that we share $O(t\\log n)$ EPR-pairs at the start of the protocol."],["Communication complexity lower bound for transitive functions","For proving Theorem REF , we exhibit a transitive function $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ whose bounded-error quantum query complexity is $O(n)$ and the unbounded-error communication complexity of $f \\circ \\square $ is $\\Omega (n \\log n)$ for $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .","Function construction and transitivity: For the construction of $f$ we first require the definition of Hadamard codewords.","The Hadamard codeword of $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , denoted by $H(s) \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , is a list of all parities of $s$ .","See Figure REF for a graphical visualization of $f$ .","Figure: If the inputs to the jj-th h 𝖨𝖯 logn h_{\\mathsf {IP}_{\\log n}} are the Hadamard codewords H(s j )H(s_j) and H(t j )H(t_j) for all j∈[n]j \\in [n] and some s j ,t j ∈-1,1 logn s_j, t_j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}, then f=𝖯𝖠𝖱𝖨𝖳𝖸(𝖨𝖯 logn (s 1 ,t 1 ),⋯,𝖨𝖯 logn (s n ,t n ))f = \\mathsf {PARITY}(\\mathsf {IP}_{\\log n}(s_1,t_1), \\dots , \\mathsf {IP}_{\\log n}(s_n, t_n)).","If there exists at least one j∈[n]j \\in [n] for which either x j1 ,⋯,x jn x_{j1}, \\dots , x_{jn} or y j1 ,⋯,y jn y_{j1}, \\dots , y_{jn} is not a Hadamard codeword, then ff outputs -1-1.Using properties of $\\mathsf {IP}$ and Hadamard codewords, and the symmetry of $\\mathsf {PARITY}_n$ , we are able to show that $f$ is transitive (see Claim REF ).","Query upper bound: The query upper bound of $O(n)$ follows along the lines of [12], using the Bernstein-Vazirani algorithm to decode the Hadamard codewords, and Grover's algorithm to check that they actually are Hadamard codewords.","This approach was in turn inspired by a query upper bound due to Ambainis and de Wolf [3].","See the proof of Theorem REF for the query algorithm and its analysis.","Communication lower bound: Towards the unbounded-error communication lower bound, we first recall that each input block of $f$ equals $\\mathsf {IP}_{\\log n}$ if the inputs to each block are promised to be Hadamard codewords.","Hence $f$ equals $\\mathsf {IP}_{n \\log n}$ under this promise, since $\\mathsf {PARITY}_n \\circ \\mathsf {IP}_{\\log n} = \\mathsf {IP}_{n \\log n}$ .","Thus by setting certain inputs to Alice and Bob suitably, $f \\circ \\square $ is at least as hard as $\\mathsf {IP}_{n \\log n}$ for $\\square \\in \\lbrace \\mathsf {AND}_2, \\mathsf {XOR}_2\\rbrace $ (for a formal statement, see Lemma REF with $r = \\mathsf {PARITY}_n$ and $g = \\mathsf {IP}_{\\log n}$ ).","It is known from a seminal result of Forster [15] that the unbounded-error communication complexity of $\\mathsf {IP}_{n \\log n}$ equals $\\Omega (n \\log n)$ , completing the proof of the lower bound.","This proof is more general than and arguably simpler than the proof of the lower bound for bounded-error quantum communication complexity in [12]."],["Other results"," We give a general recipe for constructing a class of functions that witness tightness of the BCW simulation where the inner gadget is either $\\mathsf {AND}_2$ or $\\mathsf {XOR}_2$ .","However, the communication lower bound we obtain here is in the bounded-error model in contrast to Theorem REF , where the communication lower bound is proven in the unbounded-error model.","The functions $f$ constructed for this purpose are composed functions similar to the construction in Figure REF , except that we are able to use a more general class of functions in place of the outer $\\mathsf {PARITY}$ function, and also a more general class of functions in place of the inner $\\mathsf {IP}_{\\log n}$ functions.","See Figure REF and its caption for an illustration and a more precise definition.","Figure: In this figure, G:-1,1 log ×-1,1 logn →-1,1G:\\left\\lbrace -1,1 \\right\\rbrace ^{\\log } \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace .","If the inputs to the jj-th h G h_{G} are Hadamard codewords in ±H(s j )\\pm H(s_j) and ±H(t j )\\pm H(t_j) for all j∈[n]j \\in [n] and some s j ,t j ∈-1,1 logn s_j, t_j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}, then f=r(G(s 1 ,t 1 ),⋯,G(s n ,t n ))f = r(G(s_1,t_1), \\dots , G(s_n, t_n)).","If there exists at least one j∈[n]j \\in [n] for which either x j1 ,⋯,x jn x_{j1}, \\dots , x_{jn} or y j1 ,⋯,y jn y_{j1}, \\dots , y_{jn} is not a Hadamard codeword, then ff outputs -1-1.We require some additional constraints on the outer and inner functions.","First, the approximate degree of $r$ should be $\\Omega (n)$ .","Second, the discrepancy of $G$ should be small with respect to some “balanced” probability distribution (see Definition REF and Definition REF for formal definitions of these notions).","Theorem 1.6 (Informal version of Theorem REF ) Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be such that $\\widetilde{\\textnormal {deg}}(r) = \\Omega (n)$ and let $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function.","Define $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^{2}} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ as in Figure REF .","If there exists $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\mathbb {R}$ that is a balanced probability distribution with respect to $G$ and $\\textnormal {disc}_{\\mu }(G) = n^{-\\Omega (1)}$ , then $\\mathsf {Q}(f) & = O(n),\\\\\\mathsf {Q}^{cc, *}(f \\circ \\square ) & = \\Omega (n \\log n),$ for every $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .","The query upper bound follows along similar lines as that of Theorem REF .","For the lower bound, we first show via a reduction that for $f$ as described in Figure REF and $\\square \\in \\lbrace \\mathsf {AND}_2, \\mathsf {XOR}_2\\rbrace $ , the communication problem $f \\circ \\square $ is at least as hard as $r \\circ G$ (see Lemma REF ).","This part of the lower bound proof is the same as in the proof of Theorem REF .","For the hardness of $r \\circ G$ (which in the case of Theorem REF turned out to be $\\mathsf {IP}_{n \\log n}$ , for which Forster's theorem yields an unbounded-error communication lower bound), we are able to use a theorem implicit in a work of Lee and Zhang [25].","This theorem gives a lower bound on the bounded-error communication complexity of $r \\circ G$ in terms of the approximate degree of $r$ and the discrepancy of $G$ under a balanced distribution.","For completeness, we provide an explicit proof in Appendix .","We recover the result of Chakraborty et al.","(Theorem REF ) using a more general technique, and additionally show that $\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {AND}_2) = \\Omega (\\mathsf {Q}(f) \\log n)$ , where $f$ is as in Theorem REF .","This is discussed in Appendix .","For a Boolean function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , let $\\log \\Vert \\widehat{f}\\Vert _{1, 1/3}$ denote the log-approximate-spectral norm of $f$ , and $\\widetilde{\\textnormal {deg}}_{1/3}(f)$ denote its approximate degree.","As an approach towards proving the Fourier entropy-influence conjecture, Arunachalam et al.","[2] asked whether it is true for all total functions $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ that $\\log \\Vert \\widehat{f}\\Vert _{1, 1/3} = O(\\widetilde{\\textnormal {deg}}_{1/3}(f))$ .","Chakraborty et al.","gave a negative answer to this question by showing that for the function $f$ in Theorem REF , $\\log \\Vert \\widehat{f}\\Vert _{1, 1/3} = \\Omega (\\widetilde{\\textnormal {deg}}_{1/3}(f) \\log n)$ .","They also showed that the $\\log n$ overhead is not required for the class of symmetric functions [12].","In this paper we complement this result by showing that the $\\log n$ overhead is required for the class of transitive functions.","This is discussed in Appendix ."],["Organization","Section introduces some notation and preliminaries.","In Section we construct our new one-sided error protocol for finding solutions in the string $z=(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , as a corollary of our distributed version of amplitude amplification.","In Section we prove Theorem REF , which shows that the $\\log n$ overhead in the BCW simulation can be avoided when the outer function is symmetric; this relies on the protocol from Section .","In Section we prove Theorem REF and Theorem REF , which are our results regarding necessity of the $\\log n$ overhead in the BCW simulation in the unbounded-error and the bounded-error models of communication, respectively.","In Appendix we show that the function constructed in [12] also requires a $\\log n$ overhead in the BCW simulation when the inner function is $\\mathsf {AND}_2$ .","In Appendix we exhibit a transitive function $f$ on $n$ bits for which $\\log (\\Vert \\widehat{f}\\Vert _{1, 1/3}) = \\Omega (\\widetilde{\\textnormal {deg}}(f) \\log n)$ .","Finally in Appendix we prove Theorem REF , which shows a lower bound on the bounded-error quantum communication complexity of $f \\circ G$ in terms of the approximate degree of $f$ and the discrepancy of $G$ under balanced distributions."],["Notation and preliminaries","Without loss of generality, we assume $n$ to be a power of 2 in this paper, unless explicitly stated otherwise.","All logarithms in this paper are base 2.","Let $S_n$ denote the symmetric group over the set $[n]=\\lbrace 1,\\ldots ,n\\rbrace $ .","For a string $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ and $\\sigma \\in S_n$ , let $\\sigma (x)$ denote the string $x_{\\sigma (1)}, \\dots , x_{\\sigma (n)} \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","Consider an arbitrary but fixed bijection between subsets of $[\\log n]$ and elements of $[n]$ .","For a string $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , we abuse notation and also use $s$ to denote the equivalent element of $[n]$ .","The view we take will be clear from context.","For a string $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ and set $S \\subseteq [n]$ , define the string $x_S \\in \\left\\lbrace -1,1 \\right\\rbrace ^{S}$ to be the restriction of $x$ to the coordinates in $S$ .","Let $1^n$ and $(-1)^n$ denote the $n$ -bit string $(1, 1, \\dots , 1)$ and $(-1, -1, \\dots , -1)$ , respectively."],["Boolean functions","For strings $x, y \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , let $\\langle x, y \\rangle $ denote the inner product (mod 2) of $x$ and $y$ .","That is, $\\langle x, y \\rangle = \\prod _{i = 1}^n (x_i \\wedge y_i).$ For every positive integer $n$ , let $\\mathsf {PARITY}_n : \\lbrace -1,1\\rbrace ^n \\rightarrow \\lbrace -1,1\\rbrace $ be defined as: $\\mathsf {PARITY}_n(x_1, \\dots , x_n) = \\prod _{i \\in [n]} x_i.$ Definition 2.1 (Symmetric functions) A function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is symmetric if for all $\\sigma \\in S_n$ and for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ we have $f(x) = f(\\sigma (x))$ .","Definition 2.2 (Transitive functions) A function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive if for all $i,j \\in [n]$ there exists a permutation $\\sigma \\in S_n$ such that $\\sigma (i) = j$ , and $f(x) = f(\\sigma (x))$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","We next discuss function composition.","For total functions $f, g$ , let $f \\circ g$ denote the standard composition of the functions $f$ and $g$ .","We also require the following notion of composition of a total function $f$ with a partial function $g$ .","Definition 2.3 (Composition with partial functions) Let $f : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function and let $g : \\left\\lbrace -1,1 \\right\\rbrace ^{m} \\rightarrow \\left\\lbrace -1, 1, \\star \\right\\rbrace $ be a partial function.","Let $f ~\\widetilde{\\circ }~g : \\left\\lbrace -1,1 \\right\\rbrace ^{nm} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ denote the total function that is defined as follows on input $(X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nm}$ , where $X_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^m$ for all $i \\in [n]$ .","$f ~\\widetilde{\\circ }~g (X_1, \\dots , X_n) = {\\left\\lbrace \\begin{array}{ll}f(g(X_1), \\dots , g(X_n)) & \\text{if}~g(X_i) \\in \\left\\lbrace -1,1 \\right\\rbrace ~\\text{for all}~i \\in [n],\\\\-1 & \\text{otherwise}.\\end{array}\\right.","}$ That is, we use $f ~\\widetilde{\\circ }~g$ to denote the total function that equals $f \\circ g$ on inputs when each copy of $g$ outputs a value in $\\left\\lbrace -1,1 \\right\\rbrace $ , and equals $-1$ otherwise.","Definition 2.4 (Approximate degree) For every $\\varepsilon \\ge 0$ , the $\\varepsilon $ -approximate degree of a function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is defined to be the minimum degree of a real polynomial $p : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ that uniformly approximates $f$ to error $\\varepsilon $ .","That is, $\\widetilde{\\textnormal {deg}}_{\\varepsilon }(f) = \\min \\left\\lbrace \\textnormal {deg}(p) : |p(x) - f(x)| \\le \\varepsilon ~\\text{for all}~x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n \\right\\rbrace .$ Unless specified otherwise, we drop $\\varepsilon $ from the subscript and assume $\\varepsilon = 1/3$ .","We assume familiarity with quantum computing [26], and use $\\mathsf {Q}_{\\varepsilon }(f)$ to denote the $\\varepsilon $ -error query complexity of $f$ .","Unless specified otherwise, we drop $\\varepsilon $ from the subscript and assume $\\varepsilon = 1/3$ .","Theorem 2.5 ([5]) Let $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function.","Then $\\mathsf {Q}(f) \\ge \\widetilde{\\textnormal {deg}}(f)/2$ ."],["Communication complexity","We assume familiarity with communication complexity [22].","Definition 2.6 (Two-party function) We call a function $G: \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ a two-party function to indicate that it corresponds to a communication problem in which Alice is given input $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^j$ , Bob is given input $y \\in \\left\\lbrace -1,1 \\right\\rbrace ^k$ , and their task is to compute $G(x, y)$ .","Remark 2.7 Throughout this paper, we use uppercase letters to denote two-party functions, and lowercase letters to denote functions which are not two-party functions.","Definition 2.8 (Composition with two-party functions) Let $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function and let $G: \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a two-party function.","Then $F = f \\circ G : \\left\\lbrace -1,1 \\right\\rbrace ^{nj} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{nk} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ denotes the two-party function corresponding to the communication problem in which Alice is given input $X = (X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nj}$ , Bob is given $Y = (Y_1, \\dots , Y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nk}$ , and their task is compute $F(X, Y) = f(G(X_1, Y_1), \\dots , G(X_n, Y_n))$ .","Definition 2.9 (Inner Product function) For every positive integer $n$ , define the function $\\mathsf {IP}_n : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ by $\\mathsf {IP}_n(x,y) = \\langle x,y \\rangle .$ In other words, $\\mathsf {IP}_n = \\mathsf {PARITY}_n \\circ \\mathsf {AND}_2$ .","Observation 2.10 For all positive integers $k, t$ , $\\mathsf {PARITY}_k \\circ \\mathsf {IP}_t = \\mathsf {IP}_{kt}$ .","We also assume familiarity with quantum communication complexity [32].","We use $\\mathsf {Q}_{\\varepsilon }^{cc}(G)$ and $\\mathsf {Q}_{\\varepsilon }^{cc,*}(G)$ to represent the $\\varepsilon $ -error quantum communication complexity of a two-party function $G$ in the models without and with unlimited shared entanglement, respectively.","Unless specified otherwise, we drop $\\varepsilon $ from the subscript and assume $\\varepsilon = 1/3$ .","Definition 2.11 (Balanced probability distribution) We call a probability distribution $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ balanced with respect to a function $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ if $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} f(x) \\mu (x) = 0$ .","Definition 2.12 (Discrepancy) Let $G : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function and $\\lambda $ be a distribution on $\\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k$ .","For every $S \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^j$ and $T \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^k$ , define $\\textnormal {disc}_\\lambda (S \\times T, G) = \\left|\\sum _{x, y \\in S \\times T}G(x, y)\\lambda (x, y) \\right|.$ The discrepancy of $G$ under the distribution $\\lambda $ is defined to be $\\textnormal {disc}_{\\lambda }(G) = \\max _{S \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^j, T \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^k} \\textnormal {disc}_{\\lambda }(S \\times T, G),$ and the discrepancy of $f$ is defined to be $\\textnormal {disc}(G) = \\min _{\\lambda }\\textnormal {disc}_{\\lambda }(G).$ Definition 2.13 (Balanced-discrepancy) Let $G : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function and $\\Lambda $ be the set of all balanced distributions on $\\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k$ with respect to $G$ .","The balanced-discrepancy of $G$ is defined to be $\\textnormal {bdisc}(G) = \\min _{\\lambda \\in \\Lambda }\\textnormal {disc}_{\\lambda }(G).$ The following theorem is implicit in [25].","We prove it in Appendix REF for completeness.","Theorem 2.14 Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be functions.","Let $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\mathbb {R}$ be a balanced distribution with respect to $G$ and $\\textnormal {disc}_{\\mu }(G) = o(1)$ .","If $\\frac{8en}{\\widetilde{\\textnormal {deg}}(r)}\\le \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)}\\right)^{1-\\beta }$ for some constant $\\beta \\in (0,1)$ , then $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)}\\right)\\right).$ In particular, $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)\\right).$"],["Hadamard encoding","Recall that we index coordinates of $n$ -bit strings by integers in $[n]$ , and also interchangeably by strings in $\\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ via the natural correspondence.","For $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , let $-x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ be defined as $(-x)_i = -x_i$ for all $i \\in [n]$ .","We use the notation $\\pm x$ to denote the set $\\lbrace x, -x\\rbrace $ .","Definition 2.15 (Hadamard Codewords) For every positive integer $n$ and $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , let $H(s) \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ be defined as $(H(s))_t = \\prod _{i: s_i = -1} t_i~\\text{for all}~t \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}.$ If $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ is such that $x = H(s)$ for some $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , we say $x$ is a Hadamard codeword corresponding to $s$ .","That is, for every $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , there is an $n$ -bit Hadamard codeword corresponding to $s$ .","This represents the enumeration of all parities of $s$ .","We now define how to encode a two-party total function $G$ on $(\\log j + \\log k)$ input bits to a partial function $h_G$ on $(j+k)$ input bits, using Hadamard encoding.","Definition 2.16 (Hadamardization of functions) Let $j, k \\ge 1$ be powers of 2, and let $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function.","Define a partial function $h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{j + k} \\rightarrow \\left\\lbrace -1, 1, \\star \\right\\rbrace $ by $h_G(x, y) = {\\left\\lbrace \\begin{array}{ll}G(s, t) & \\text{if}~x \\in \\pm H(s), y \\in \\pm H(t)~\\text{for some}~s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}, t \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}\\\\\\star & \\text{otherwise}.\\end{array}\\right.","}$"],["Additional concepts from quantum computing","The Bernstein-Vazirani algorithm [11] is a quantum query algorithm that takes an $n$ -bit string as input and outputs a $(\\log n)$ -bit string.","The algorithm has the following properties: the algorithm makes one quantum query to the input and if the input $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ satisfies $x\\in \\pm H(s)$ for some $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , then the algorithm returns $s$ with probability 1.","Consider a symmetric Boolean function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","Define the quantity $\\Gamma (f)=\\min \\lbrace |2k-n+1| : f(x)\\ne f(y)\\text{ if }|x|=k\\text{ and } |y|=k+1\\rbrace $ from [29].","One can think of $\\Gamma (f)$ as essentially the length of the interval of Hamming weights around $n/2$ where $f$ is constant (for example, for the majority and parity functions this would be 1, and for $\\mathsf {OR}_n$ this would be $n-1$ ).","Theorem 2.17 ([5]) For every symmetric function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , we have $\\mathsf {Q}(f) = \\Theta (\\sqrt{(n-\\Gamma (f))n}).$ The upper bound follows from a quantum algorithm that exactly counts the Hamming weight $|x|$ of the input if $|x|\\le t$ or $|x|\\ge n-t$ for $t=\\left\\lceil (n-\\Gamma (f))/2 \\right\\rceil $ , and that otherwise learns $|x|$ is in the interval $[t+1,n-t-1]$ (which is an interval around $n/2$ where $f(x)$ is constant).","By the definition of $\\Gamma (f)$ , this information about $|x|$ suffices to compute $f(x)$ .","In Section we use this observation to give an efficient quantum communication protocol for a two-party function $f\\circ G$ .","We will need a unitary protocol that allows Alice and Bob to implement an approximate reflection about the $n$ -dimensional maximally entangled state $|\\psi \\rangle =\\frac{1}{\\sqrt{n}}\\sum _{i\\in \\lbrace 0,1\\rbrace ^{\\log n}}|i\\rangle |i\\rangle .$ Ideally, such a reflection would map $|\\psi \\rangle $ to itself, and put a minus sign in front of all states orthogonal to $|\\psi \\rangle $ .","Doing this perfectly would requires $O(\\log n)$ qubits of communication.","Fortunately we can derive a cheaper protocol from a test that Aharonov et al.","[4] designed, which uses $O(\\log (1/\\varepsilon ))$ qubits of communication and checks whether a given bipartite state equals $|\\psi \\rangle $ , with one-sided error probability $\\varepsilon $ .","By the usual trick of running this protocol, applying a $Z$ -gate to the answer qubit, and then reversing the protocol, we can implement the desired reflection approximately.Possibly with some auxiliary qubits on Alice and Bob's side which start in $|0\\rangle $ and end in $|0\\rangle $ , except in a part of the final state that has norm at most $\\varepsilon $ .","A bit more precisely: Theorem 2.18 Let $R_{\\psi }=2|\\psi \\rangle \\!","\\langle \\psi |-I$ be the reflection about the maximally entangled state shared between Alice and Bob.","There exists a protocol that uses $O(\\log (1/\\varepsilon ))$ qubits of communication and that implements a unitary $R^\\varepsilon _{\\psi }$ such that $\\left\\Vert R^\\varepsilon _{\\psi } - R_{\\psi } \\right\\Vert \\le \\varepsilon $ and $R^\\varepsilon _{\\psi }|\\psi \\rangle =|\\psi \\rangle $ .","We use $\\mathsf {UPP}^{cc}(F)$ to denote unbounded-error quantum communication complexity of two-party function $F$ .","It is folklore (see for example [20]) that the unbounded-error quantum communication complexityThe unbounded-error model does not allow shared randomness or prior shared entanglement (which yields shared randomness by measuring) between Alice and Bob, since any two-party function $F$ would have constant communication complexity in that setting.","of $F$ equals its classical counterpart up to a factor of at most 2 so it does not really matter much whether we use $\\mathsf {UPP}^{cc}$ for classical unbounded-error communication complexity (as it is commonly used) or for quantum unbounded-error complexity.","Crucially, for both the complexity of $\\mathsf {IP}_n$ is linear in $n$ : Theorem 2.19 ([15]) Let $n$ be a positive integer.","Then, $\\mathsf {UPP}^{cc}(\\mathsf {IP}_n) = \\Omega (n).$"],["Noisy amplitude amplification and a new distributed-search protocol","In this section we present a version of quantum amplitude amplification that still works if the reflections involved are not perfectly implemented.","In particular, the usual reflection about the uniform superposition will be replaced in the communication setting by an imperfect reflection about the $n$ -dimensional maximally entangled state, based on the communication-efficient protocol of Aharonov et al.","[4] for testing whether Alice and Bob share that state.","This allows us to avoid the $\\log n$ factor that would be incurred if we instead used a BCW-style distributed implementation of standard amplitude amplification, with $O(\\log n)$ qubits of communication to implement each query.","Our main result in this section is the following general theorem, which allows us to search among a sequence of two-party instances $(X_1,Y_1),\\ldots ,(X_n,Y_n)$ for an index $i \\in [n]$ where $G(X_i,Y_i)=-1$ , for any two-party function $G$ .","Theorem 3.1 Let $G: \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a two-party function, $X = (X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nj}$ and $Y = (Y_1, \\dots , Y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nk}$ .","Define $z=(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","Assume Alice and Bob start with $\\left\\lceil \\log n \\right\\rceil $ shared EPR-pairs.","There exists a quantum protocol using $O(\\sqrt{n}\\,\\mathsf {Q}_E^{cc}(G))$ qubits of communication that finds (with success probability $\\ge 0.99$ ) an $i\\in [n]$ such that $z_i=-1$ if such an $i$ exists, and says “no” with probability 1 if no such $i$ exists.","If the number of $-1$ s in $z$ is within a factor of 2 from a known integer $t$ , then the communication can be reduced to $O(\\sqrt{n/t}\\,\\mathsf {Q}_E^{cc}(G))$ qubits.","Remark 3.2 The $\\log n$ shared EPR-pairs that we assume Alice and Bob share at the start could also be established by means of $\\log n$ qubits of communication at the start of the protocol.","For the result in the first bullet, this additional communication does not change the asymptotic bound.","For the result of the second bullet, if $t\\le n\\mathsf {Q}_E^{cc}(G)^2/(\\log n)^2$ , then this additional communication does not change the asymptotic bound either.","However, if $t=\\omega (n/(\\log n)^2)$ and $\\mathsf {Q}_E^{cc}(G)=O(1)$ then the quantum communication $O(\\sqrt{n/t}\\,\\mathsf {Q}_E^{cc}(G))$ is $o(\\log n)$ and establishing the $\\log n$ EPR-pairs by means of a first message makes a difference.","As a corollary, we obtain a new $O(\\sqrt{n})$ -qubit protocol for the distributed search problem composed with $G=\\mathsf {AND}_2$ (whose decision version is the Set-Disjointness problem)."],["Amplitude amplification with perfect reflections","We first describe basic amplitude amplification in a slightly unusual recursive manner, similar to [19].","We are dealing with a search problem where some set $\\cal G$ of basis states are deemed “good” and the other basis states are deemed “bad.” Let $P_{\\cal G}=\\sum _{g\\in {\\cal G}}|g\\rangle \\!","\\langle g|$ be the projector onto the span of the good basis states, and $O_{\\cal G}=I-2P_{\\cal G}$ be the reflection that puts a `$-$ ' in front of the good basis states: $O_{\\cal G}|g\\rangle =-|g\\rangle $ for all basis states $g\\in {\\cal G}$ , and $O_{\\cal G}|b\\rangle =|b\\rangle $ for all basis states $b\\notin {\\cal G}$ .","Suppose we have an initial state $|\\psi \\rangle $ which is a superposition of a good state and a bad state: $|\\psi \\rangle =\\sin (\\theta )|G\\rangle +\\cos (\\theta )|B\\rangle ,$ where $|G\\rangle =P_{\\cal G}|\\psi \\rangle /\\left\\Vert P_{\\cal G}|\\psi \\rangle \\right\\Vert $ and $|B\\rangle =(I-P_{\\cal G})|\\psi \\rangle /\\left\\Vert (I-P_{\\cal G})|\\psi \\rangle \\right\\Vert $ .","For example in Grover's algorithm, with a search space of size $n$ containing $t$ solutions, the initial state $|\\psi \\rangle $ would be the uniform superposition, and its overlap (inner product) with the good subspace spanned by the $t$ “good” (sometimes called “marked”) basis states would be $\\sin (\\theta )=\\sqrt{t/n}$ .","We'd like to increase the weight of the good state, i.e., move the angle $\\theta $ closer to $\\pi /2$ .","Let $R_\\psi $ denote the reflection about the state $|\\psi \\rangle $ , i.e., $R_\\psi |\\psi \\rangle =|\\psi \\rangle $ and $R_\\psi {|\\phi \\rangle }=-|\\phi \\rangle $ for every $|\\phi \\rangle $ that is orthogonal to $|\\psi \\rangle $ .","Then the algorithm $A_1=R_\\psi \\cdot O_{\\cal G}$ is the product of two reflections, which (in the 2-dimensional space spanned by $|G\\rangle $ and $|B\\rangle $ ) corresponds to a rotation by an angle $2\\theta $ , thus increasing our angle from $\\theta $ to $3\\theta $ .","This is the basic amplitude amplification step.","It maps $|\\psi \\rangle \\mapsto A_1|\\psi \\rangle =\\sin (3\\theta )|G\\rangle +\\cos (3\\theta )|B\\rangle .$ We can now repeat this step recursively, defining $A_2=A_1R_\\psi A^*_1\\cdot O_{\\cal G}\\cdot A_1.$ Note that $A_1R_\\psi A^*_1$ is a reflection about the state $A_1|\\psi \\rangle $ .","Thus $A_2$ triples the angle between $A_1|\\psi \\rangle $ and $|B\\rangle $ , mapping $|\\psi \\rangle \\mapsto A_2|\\psi \\rangle =\\sin (9\\theta )|G\\rangle +\\cos (9\\theta )|B\\rangle .$ Continuing recursively in this fashion, define the algorithm $A_{j+1}=A_j R_\\psi A_j^*\\cdot O_{\\cal G}\\cdot A_j.$ The last algorithm $A_k$ will map $|\\psi \\rangle \\mapsto A_k|\\psi \\rangle =\\sin (3^k\\theta )|G\\rangle +\\cos (3^k\\theta )|B\\rangle .$ Hence after $k$ recursive amplitude amplification steps, we have angle $3^k\\theta $ .","Since we want to end up with angle $\\approx \\pi /2$ , if we know $\\theta $ then we can choose $k=\\left\\lfloor \\log _3(\\pi /(2\\theta )) \\right\\rfloor .$ This gives us an angle $3^k\\theta \\in (\\pi /6,\\pi /2]$ , so the final state $A_k|\\psi \\rangle $ has overlap $\\sin (\\theta _k)>1/2$ with the good state $|G\\rangle $ .","Let $C_k$ denote the “cost” (in whatever measure, for example query complexity, or communication complexity, or circuit size) of algorithm $A_k$ .","Looking at its recursive definition (Equation (REF )), $C_k$ is 3 times $C_{k-1}$ , plus the cost of $R_\\psi $ plus the cost of $O_{\\cal G}$ .","If we just count applications of $O_{\\cal G}$ (“queries”), considering $R_\\psi $ to be free, then $C_{k+1}=3C_k+1$ .","This recursion has the closed form $C_k=\\sum _{i=0}^{k-1} 3^i< 3^k$ .","With the above choice of $k$ we get $C_k=O(1/\\theta )$ .","In the case of Grover's algorithm, where $\\theta =\\arcsin (\\sqrt{t/n})\\approx \\sqrt{t/n}$ , the cost is $C_k=O(\\sqrt{n/t})$ ."],["Amplitude amplification with imperfect reflections","Now we consider the situation where we do not implement the reflections $R_\\psi $ perfectly, but instead implement another unitary $R^\\varepsilon _\\psi $ at operator-norm distance $\\left\\Vert R^\\varepsilon _\\psi -R_\\psi \\right\\Vert \\le \\varepsilon $ from $R_\\psi $ , with the additional property that $R^\\varepsilon _\\psi |\\psi \\rangle =|\\psi \\rangle $ (this one-sided error property will be important for the proof).","We can control this error $\\varepsilon $ , but smaller $\\varepsilon $ will typically correspond to higher cost of $R^\\varepsilon _\\psi $ .","The reflection $O_{\\cal G}$ will still be implemented perfectly below.","We again start with the initial state $|\\psi \\rangle =\\sin (\\theta )|G\\rangle +\\cos (\\theta )|B\\rangle .$ For errors $\\varepsilon _1,\\ldots ,\\varepsilon _k$ that we will specify later, recursively define the following algorithms.","$A_1=R^{\\varepsilon _1}_\\psi \\cdot O_{\\cal G}\\mbox{~~and~~}A_{j+1}=A_jR^{\\varepsilon _{j+1}}_\\psi A_j^*\\cdot O_{\\cal G}\\cdot A_j.$ These algorithms will map the initial state as follows: $|\\psi \\rangle \\mapsto |\\psi _j\\rangle =A_j|\\psi \\rangle =\\sin (3^j\\theta )|G\\rangle +\\cos (3^j\\theta )|B\\rangle +|E_j\\rangle ,$ where $|E_j\\rangle $ is some unnormalized error state defined by the above equation; its norm $\\eta _j$ quantifies the extent to which we deviate from perfect amplitude amplification.","Our goal here is to upper bound this $\\eta _j$ .","In order to see how $\\eta _j$ can grow, let us see how $A_jR^{\\varepsilon _{j+1}}_\\psi A_j^*\\cdot O_{\\cal G}$ acts on $\\sin (3^j\\theta )|G\\rangle +\\cos (3^j\\theta )|B\\rangle $ (we'll take into account the effects of the error term $|E_j\\rangle $ later).","If $R^{\\varepsilon _{j+1}}_\\psi $ were equal to $R_\\psi $ , then we would have one perfect round of amplitude amplification and obtain $\\sin (3^{j+1}\\theta )|G\\rangle +\\cos (3^{j+1}\\theta )|B\\rangle $ ; but since $R^{\\varepsilon _{j+1}}_\\psi $ is only $\\varepsilon _{j+1}$ -close to $R_\\psi $ , additional errors can appear.","First we apply $O_{\\cal G}$ , which negates $|G\\rangle $ and hence changes the state to $-\\sin (3^j\\theta )|G\\rangle +\\cos (3^j\\theta )|B\\rangle =|\\psi _j\\rangle -|E_j\\rangle -2\\sin (3^j\\theta )|G\\rangle .$ Second we apply $V=A_jR^{\\varepsilon _{j+1}}_\\psi A_j^*$ .","Let $V^{\\prime }=A_jR_\\psi A_j^*$ , and note that $V|\\psi _j\\rangle =V^{\\prime }|\\psi _j\\rangle =|\\psi _j\\rangle $ and $\\left\\Vert V^{\\prime }-V \\right\\Vert =\\left\\Vert R_\\psi -R^{\\varepsilon _{j+1}}_\\psi \\right\\Vert \\le \\varepsilon _{j+1}$ .","The new state is $V(|\\psi _j\\rangle -|E_j\\rangle -2\\sin (3^j\\theta )|G\\rangle )& = V^{\\prime }(|\\psi _j\\rangle -|E_j\\rangle -2\\sin (3^j\\theta )|G\\rangle ) +(V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle )\\\\& = V^{\\prime }(-\\sin (3^j\\theta )|G\\rangle +\\cos (3^j\\theta )|B\\rangle )+(V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle )\\\\& = \\sin (3^{j+1}\\theta )|G\\rangle +\\cos (3^{j+1}\\theta )|B\\rangle +(V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle ).$ Putting back also the earlier error term $|E_j\\rangle $ from Equation (REF ) (to which the unitary $VO_{\\cal G}$ is applied as well), it follows that the new error state is $|E_{j+1}\\rangle =|\\psi _{j+1}\\rangle -(\\sin (3^{j+1}\\theta )|G\\rangle +\\cos (3^{j+1}\\theta )|B\\rangle )=VO_{\\cal G}|E_j\\rangle +(V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle ).$ Its norm is $\\eta _{j+1} & \\le \\left\\Vert VO_{\\cal G}|E_j\\rangle \\right\\Vert +\\left\\Vert (V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle ) \\right\\Vert \\\\& \\le \\eta _j + \\varepsilon _{j+1}(\\eta _j+2\\sin (3^j\\theta ))= (1+\\varepsilon _{j+1})\\eta _j + 2\\varepsilon _{j+1}\\sin (3^j\\theta ).$ Since $\\eta _0=0$ , we can “unfold” the above recursive upper bound to the following, which is easy to verify by induction on $k$ : $\\eta _k \\le \\sum _{j=1}^k\\prod _{\\ell =j+1}^k(1+\\varepsilon _\\ell )2\\varepsilon _j\\sin (3^{j-1}\\theta ).$ For each $1 \\le j \\le k$ , choose $\\varepsilon _j=\\frac{1}{100\\cdot 4^j}.$ Note that $\\sigma =\\sum _{j=1}^k\\varepsilon _k\\le 1/300$ .","With this choice of $\\varepsilon _j$ 's, and the inequalities $1+x\\le e^x$ , $e^\\sigma \\le 1.5$ and $\\sin (x)\\le x$ for $x \\le \\pi /2$ (which is the case here), we can upper bound the norm of the error term $|E_k\\rangle $ after $k$ iterations (see Equation (REF )) as $\\eta _k \\le \\sum _{j=1}^k e^\\sigma 2\\varepsilon _j 3^{j-1}\\theta \\le \\frac{3\\theta }{400}\\sum _{j=1}^k (3/4)^{j-1}\\le \\frac{3\\theta }{100}.$ Accordingly, up to very small error we have done perfect amplitude amplification."],["Distributed amplitude amplification with imperfect reflection","We will now instantiate the above scheme to the case of distributed search, where our measure of cost is communication, that is, the number of qubits sent between Alice and Bob.","Specifically, consider the intersection problem where Alice and Bob have inputs $x\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ and $y\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , respectively.","Assume for simplicity that $n$ is a power of 2, so $\\log n$ is an integer.","Alice and Bob want to find an $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace =\\lbrace 0,1\\rbrace ^{\\log n}$ such that $x_i=y_i=-1$ , if such an $i$ exists.","The basis states in this distributed problem are $|i\\rangle |j\\rangle $ , and we define the set of “good” basis states as ${\\cal G}=\\lbrace |i\\rangle |j\\rangle \\mid x_i=y_j=-1\\rbrace ,$ even though we are only looking for $i,j$ where $i=j$ (it's easier to implement $O_{\\cal G}$ with this more liberal definition of $\\cal G$ ).","Our protocol will start with the maximally entangled initial state $|\\psi \\rangle $ in $n$ dimensions, which corresponds to $\\log n$ EPR-pairs: $|\\psi \\rangle =\\frac{1}{\\sqrt{n}}\\sum _{i\\in \\lbrace 0,1\\rbrace ^{\\log n}}|i\\rangle |i\\rangle =\\sin (\\theta )|G\\rangle +\\cos (\\theta )|B\\rangle ,$ where we assume there are $t$ $i$ 's where $x_i=y_i=-1$ , i.e., $t$ solutions to the intersection problem, so $\\theta =\\arcsin (\\sqrt{t/n}).$ and $|G\\rangle =\\frac{1}{\\sqrt{t}}\\sum _{(i,i)\\in {\\cal G}}|i\\rangle |i\\rangle .$ It costs $\\left\\lceil \\log n \\right\\rceil $ qubits of communication between Alice and Bob to establish this initial shared state, or it costs nothing if we assume pre-shared entanglement.","Our goal is to end up with a state that has large inner product with $|G\\rangle $ .","In order to be able to use amplitude amplification, we would like to be able to reflect about the above state $|\\psi \\rangle $ .","However, in general this perfect reflection $R_{\\psi }$ costs a lot of communication: Alice would send her $\\log n$ qubits to Bob, who would unitarily put a $-1$ in front of all states orthogonal to $|\\psi \\rangle $ , and then sends back Alice's qubits.","This has a communication cost of $O(\\log n)$ qubits, which is too much for our purposes.","Fortunately, Theorem REF gives us a way to implement a one-sided $\\varepsilon $ -error reflection protocol $R^\\varepsilon _{\\psi }$ that only costs $O(\\log (1/\\varepsilon ))$ qubits of communication.","The reflection $O_{\\cal G}$ puts a `$-$ ' in front of the basis states $|i\\rangle |j\\rangle $ in $\\cal G$ .","This can be implemented perfectly using only 2 qubits of communication, as follows.","To implement this reflection on her basis state $|i\\rangle $ , Alice XORs $|x_i\\rangle $ into a fresh auxiliary $|0\\rangle $ -qubit and sends this qubit to Bob.","Bob receives this qubit and applies the following unitary map: $|b\\rangle |j\\rangle \\mapsto y_j^{b}|b\\rangle |j\\rangle ,~~~b\\in \\lbrace 0,1\\rbrace ,j\\in [n].$ He sends back the auxiliary qubit.","Alice sets the auxiliary qubit back to $|0\\rangle $ by XOR-ing $x_i$ into it.","Ignoring the auxiliary qubit (which starts and ends in state $|0\\rangle $ ), this maps $|i\\rangle |j\\rangle \\mapsto (-1)^{[x_i=y_j=-1]}|i\\rangle |j\\rangle $ .","Hence we have implemented $O_{\\cal G}$ correctly: a minus sign is applied exactly for the good basis states, the ones where $x_i=y_j=-1$ .","Now consider the algorithms (more precisely, communication protocols): $A_1=R^{\\varepsilon _1}_\\psi \\cdot O_{\\cal G}\\mbox{~~and~~}A_{j+1}=A_jR^{\\varepsilon _{j+1}}_\\psi A_j^*\\cdot O_{\\cal G}\\cdot A_j$ with the choice of $\\varepsilon _j$ 's from Equation (REF ).","If we pick $k=\\left\\lfloor \\log _3(\\pi /(2\\theta )) \\right\\rfloor $ , like in Equation (REF ), then $3^k\\theta \\in (\\pi /6,\\pi /2]$ .","Hence by Equation (REF ) and Equation (REF ), the inner product of our final state with $|G\\rangle $ will be between $\\sin (3^k\\theta )-3\\theta /100\\ge 0.4$ and 1.","At this point Alice and Bob can measure, and with probability $\\ge 0.4^2$ they will each see the same $i$ , with the property that $x_i=y_i=-1$ .","From Equation (REF ) and Theorem REF , the recursion for the communication costs of these algorithms is $C_{j+1}=3C_j+O(\\log (1/\\varepsilon _{j+1}))+2.$ Solving this recurrence with our $\\varepsilon _j$ 's from Equation (REF ) and the value of $\\theta $ from Equation (REF ) we obtain $C_k=\\sum _{j=1}^k 3^{k-j} (O(\\log (1/\\varepsilon _j))+2)=\\sum _{j=1}^k 3^{k-j}O(j)= O(3^k)=O(\\sqrt{n/t}).$ Thus, using $O(\\sqrt{n/t})$ qubits of communication we can find (with constant success probability) an intersection point $i$ .","This also allows us to solve the Set-Disjointness problem (the decision problem whose output is 1 if there is no intersection between $x$ and $y$ ).","Note that if the $t$ we used equals the actual number of solutions only up to a factor of 2, the above protocol still has $\\Omega (1)$ probability to find a solution, and $O(1)$ repetitions will boost this success probability to $0.99$ .","In case we do not even know $t$ approximately, we can use the standard technique of trying exponentially decreasing guesses for $t$ to find an intersection point with communication $O(\\sqrt{n})$ .","Note that there is no log-factor in the communication complexity, in contrast to the original $O(\\sqrt{n}\\log n)$ -qubit Grover-based quantum protocol for the intersection problem of Buhrman et al. [6].","Aaronson and Ambainis [1] earlier already managed to remove the log-factor, giving an $O(\\sqrt{n})$ -qubit protocol for Set-Disjointness as a consequence of their local version of quantum search on a grid graph (which is optimal [31]).","We have just reproved this result of [1] in a different and arguably simpler way.","The above description is geared towards the intersection problem, where the “inner” function is $G=\\mathsf {AND}_2$ : we called a basis state $|i\\rangle |j\\rangle $ “good” if $x_i=y_j=-1$ .","However, this can easily be generalized to the situation where Alice and Bob's respective inputs are $X=(X_1,\\ldots ,X_n)$ and $Y=(Y_1,\\ldots ,Y_n)$ and we want to find an $i\\in [n]$ where $G(X_i,Y_i)=-1$ for some two-party function $G$ , and define the set of “good” basis states as ${\\cal G}=\\lbrace |i\\rangle |j\\rangle \\mid G(X_i,Y_j)=-1\\rbrace $ .We intentionally use the letter `$G$ ' to mean “good” in $\\cal G$ and and to refer to the two-party function $G$ , since $G$ determines which basis states $|i\\rangle |j\\rangle $ are “good.” The only thing that changes in the above is the implementation of the reflection $O_{\\cal G}$ , which would now be computed by means of an exact quantum communication protocol for $G(X_i,Y_j)$ , at a cost of $2\\mathsf {Q}_E^{cc}(G)$ qubits of communication.The factor of 2 is to reverse the protocol after the phase $G(X_i,Y_j)$ has been added to basis state $|i\\rangle |j\\rangle $ , in order to set any workspace qubits back to $|0\\rangle $ .","Note that because we can check (at the expense of another $\\mathsf {Q}_E^{cc}(G)$ qubits of communication) whether the output index $i$ actually satisfies $G(X_i,Y_i)=-1$ , we may assume the protocol has one-sided error: it always outputs “no” if there is no such $i$ .","This concludes the proof of Theorem REF ."],["No log-factor needed for symmetric functions","In this section we prove Theorem REF from the introduction.","Consider a symmetric Boolean function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","As explained in Section REF , there is an integer $t=\\left\\lceil (n-\\Gamma (f))/2 \\right\\rceil $ such that we can compute $f$ if we learn the Hamming weight $|z|$ of the input $z\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ or learn that $|z|\\in [t+1,n-t-1]$ .","The bounded-error quantum query complexity is $\\mathsf {Q}(f)=\\Theta (\\sqrt{tn})$ (Theorem REF ).","For a given two-party function $G:\\left\\lbrace -1,1 \\right\\rbrace ^j\\times \\left\\lbrace -1,1 \\right\\rbrace ^k\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , we have an induced two-party function $F:\\left\\lbrace -1,1 \\right\\rbrace ^{nj}\\times \\left\\lbrace -1,1 \\right\\rbrace ^{nk}\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ defined as $F(X_1,\\ldots ,X_n,Y_1,\\ldots ,Y_n)=f(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))$ .","Define $z=(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))\\in \\left\\lbrace -1,1 \\right\\rbrace ^n.$ Then $F(X,Y)=f(z)$ only depends on the number of $-1$ s in $z$ .","The following theorem allows us to count this number using $O(\\mathsf {Q}(f)\\,\\mathsf {Q}_E^{cc}(G))$ qubits of communication.","Theorem 4.1 For every $t$ between 1 and $n/2$ , there exists a quantum protocol that starts from $O(t\\log n)$ EPR-pairs, communicates $O(\\sqrt{tn}\\,\\mathsf {Q}_E^{cc}(G))$ qubits, and tells us $|z|$ or tells us that $|z|>t$ , with error probability $\\le 1/8$ .","Abbreviate $q=\\mathsf {Q}_E^{cc}(G)$ .","Our protocol has two parts: the first filters out the case $|z|\\ge 2t$ , while the second finds all solutions if $|z|<2t$ ."],["Part 1.","First Alice and Bob decide between the case (1) $|z|\\ge 2t$ and (2) $|z|\\le t$ (even though $|z|$ might also lie in $\\lbrace t+1,\\ldots ,2t-1\\rbrace $ ) using $O(\\sqrt{n}q)$ qubits of communication, as follows.","They use shared randomness to choose a uniformly random subset $S\\subseteq [n]$ of $\\left\\lceil n/(2t) \\right\\rceil $ elements.","Let $E$ be the event that $z_i=-1$ for at least one $i\\in S$ .","By standard calculations there exist $p_1,p_2\\in [0,1]$ with $p_1=p_2+\\Omega (1)$ such that $\\Pr [E]\\ge p_1$ in case (1) and $\\Pr [E]\\le p_2$ in case (2).","Alice and Bob use the distributed-search protocol from the first bullet of Theorem REF to decide $E$ , with $O(\\sqrt{|S|}\\,q)=O(\\sqrt{n}\\,q)$ qubits of communication (plus a negligible $O(\\log n)$ EPR-pairs) and error probability much smaller than $p_1-p_2$ .","By repeating this a sufficiently large constant number of times and seeing whether the fraction of successes was larger or smaller than $(p_1+p_2)/2$ , they can distinguish between cases (1) and (2) with success probability $\\ge 15/16$ .","If they conclude they're in case (1) then they output “$|z|>t$ ” and otherwise they proceed to the second part of the protocol.","Note that if $|z|\\in \\lbrace t+1,\\ldots ,2t-1\\rbrace $ (the “grey zone” in between cases (1) and (2)), then we can't give high-probability guarantees for one output or the other, but concluding (1) leads to the correct output “$|z|>t$ ” in this case, while concluding (2) means the protocol proceeds to Part 2.","So either course of action is fine if $|z|\\in \\lbrace t+1,\\ldots ,2t-1\\rbrace $ .","By Newman's theorem [27] the shared randomness used for choosing $S$ can be replaced by $O(\\log n)$ bits of private randomness on Alice's part, which she can send to Bob in her first message, so Part 1 communicates $O(\\sqrt{n}\\,q)$ qubits in total."],["Part 2.","We condition on Part 1 successfully filtering out case (1), so from now on assume $|z|<2t$ .","Our goal in this second part of the protocol is to find all indices $i$ such that $z_i=-1$ (we call such $i$ “solutions”), with probability $\\ge 15/16$ , using $O(\\sqrt{tn}\\,q)$ qubits of communication.","This will imply that the overall protocol is correct with probability $1-1/16-1/16=7/8$ , and uses $O(\\sqrt{tn}\\,q)$ qubits of communication in total.","For an integer $k\\ge 1$ , consider the following protocol $P_k$ .","[H] Input: An integer $k \\ge 1$ $200\\sqrt{2^k n}\\,q$ qubits have been sent Run the protocol from the last bullet of Theorem REF with $t=2^{k-1}$ .","(suppressing some constant factors, assume for simplicity that this uses $\\sqrt{n/2^k}\\,q$ qubits of communication, $\\log n$ shared EPR-pairs at the start, and has probability $\\ge 1/100$ to find a solution if the actual number of solutions is in $[t/2,2t]$ ).","Alice measures and gets outcome $i\\in [n]$ and Bob measures and gets outcome $j\\in [n]$ , respectively.","Alice sends $i$ to Bob, Bob sends $j$ to Alice.","If $i=j$ then they verify that $G(X_i,Y_i)=-1$ by one run of the protocol for $G$ , and if so then they replace $X_i,Y_i$ by some pre-agreed inputs $X^{\\prime }_i,Y^{\\prime }_i$ , respectively, such that $G(X^{\\prime }_i,Y^{\\prime }_i)=1$ (this reduces the number of $-1$ s in $z$ by 1) Protocol $P_k$ Claim 4.2 Suppose $|z|\\in [2^{k-1},2^k)$ .","Then protocol $P_k$ uses $O(\\sqrt{2^k n}\\,q)$ qubits of communication, assumes $O(2^k\\log n)$ EPR-pairs at the start of the protocol, and finds at least $|z|-2^{k-1}+1$ solutions, except with probability $\\le 1/2$ .","The upper bound on the communication is obvious from the stopping criterion of $P_k$ .","As long as the remaining number of solutions is $\\ge 2^{k-1}$ , each run of the protocol has probability $\\ge 1/100$ to find another solution.","Hence the expected number of runs of the protocol of Theorem REF to find at least $|z|-2^{k-1}+1$ solutions, is $\\le 100(|z|-2^{k-1}+1)$ .","By Markov's inequality, the probability that we haven't yet found $|z|-2^{k-1}+1$ solutions after $\\le 200(|z|-2^{k-1}+1)\\le 100\\cdot 2^k$ runs, is $\\le 1/2$ .","The communication cost of so many runs is $100\\cdot 2^k (\\sqrt{n/2^k}\\,q + \\log n)\\le 200 \\sqrt{2^k n}\\,q$ qubits.","Hence by the time that the number of qubits of the stopping criterion have been communicated, we have probability $\\ge 1/2$ of having found at least $|z|-2^{k-1}+1$ solutions.","The assumed number of EPR-pairs at the start is $\\log n$ per run, so $O(2^k\\log n)$ in total.","Note that if we start with a number of solutions $|z|\\in [2^{k-1},2^k)$ , and $P_k$ succeeds in finding at least $|z|-2^{k-1}+1$ new solutions, then afterwards we have $< 2^{k-1}$ solutions left.","The following protocol runs these $P_k$ in sequence, pushing down the remaining number of solutions to 0.","[H] $k=\\left\\lceil \\log _2 (2t) \\right\\rceil $ downto 1 Run $P_k$ a total of $r_k=\\left\\lceil \\log _2 (2t) \\right\\rceil -k+5$ times (replacing all $-1$ s found by $+1s$ in $z$ ).","Output the total number of solutions found.","Protocol $P$ Claim 4.3 If $|z|<2t$ then protocol $\\cal P$ uses $O(\\sqrt{tn}\\,q)$ qubits of communication, assumes $O(t\\log n)$ EPR-pairs at the start of the protocol, and outputs $|z|$ , except with probability $\\le 1/16$ .","First, by Claim REF , the total number of qubits communicated is $\\sum _{k=1}^{\\left\\lceil \\log _2 (2t) \\right\\rceil } r_k\\cdot O(\\sqrt{2^k n}\\,q)=O(\\sqrt{tn}\\,q)\\cdot \\sum _{\\ell =0}^{\\left\\lceil \\log _2 (2t) \\right\\rceil -1} (\\ell +5) /\\sqrt{2^\\ell }=O(\\sqrt{tn}\\,q),$ where we used a variable substitution $k=\\left\\lceil \\log _2 (2t) \\right\\rceil -\\ell $ .","Second, the number of EPR-pairs we're starting from is $\\sum _{k=1}^{\\left\\lceil \\log _2 (2t) \\right\\rceil }r_k\\cdot O(2^k\\log n)=O(t\\log n)\\cdot \\sum _{\\ell =0}^{\\left\\lceil \\log _2 (2t) \\right\\rceil -1}(\\ell +5) /2^\\ell =O(t\\log n).$ Third, by Claim REF and the fact that we are performing $r_k$ repetitions of $P_k$ , if the $k$ th round of $\\cal P$ starts with a remaining number of solutions that is in the interval $[2^{k-1},2^k)$ then that round ends with $<2^{k-1}$ remaining solutions, except with probability at most $1/2^{r_k}$ .","By the union bound, the probability that any one of the $\\left\\lceil \\log _2 (2t) \\right\\rceil $ rounds does not succeed at this, is at most $\\sum _{k=1}^{\\left\\lceil \\log _2 (2t) \\right\\rceil } \\frac{1}{2^{r_k}}=\\sum _{\\ell =0}^{\\left\\lceil \\log _2 (2t) \\right\\rceil -1}\\frac{1}{2^{\\ell +5}}\\le \\frac{1}{16}.$ Since $2^{\\left\\lceil \\log _2 (2t) \\right\\rceil }\\ge 2t$ and we start with $|z|<2t$ , if each round succeeds, then by the end of $\\cal P$ there are no remaining solutions left.","Thus, the protocol $\\cal P$ finds all solutions and learns $|z|$ with probability at least $15/16$ .","Part 1 and Part 2 each have error probability $\\le 1/16$ , so by the union bound the protocol succeeds except with probability $1/8$ .","If $|z|\\ge 2t$ then Part 1 outputs the correct answer “$|z|>t$ ”; if $|z|\\le t$ then all solutions (and hence $|z|$ ) are found by Part 2; and if $|z|\\in \\lbrace t+1,\\ldots ,2t-1\\rbrace $ then either Part 1 already outputs the correct answer “$|z|>t$ ” or the protocol proceeds to Part 2 which then finds all solutions.","We can use the above theorem twice: once to count the number of $-1$ s in $z$ (up to $t$ ) and once to count the number of 1s in $z$ (up to $t$ ).","This uses $O(\\sqrt{tn}\\,\\mathsf {Q}^{cc}_E(G))=O(\\mathsf {Q}(f)\\,\\mathsf {Q}_E^{cc}(G))$ qubits of communication, assumes $O(t\\log n)$ shared EPR-pairs at the start of the protocol, and gives us enough information about $|z|$ to compute $f(z)=F(X,Y)$ .","This concludes the proof of Theorem REF from the introduction, restated below.","Theorem 4.4 (Restatement of Theorem REF ) For every symmetric Boolean function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and two-party function $G:\\left\\lbrace -1,1 \\right\\rbrace ^j\\times \\left\\lbrace -1,1 \\right\\rbrace ^k\\rightarrow \\lbrace 0,1\\rbrace $ , we have $\\mathsf {Q}^{cc,*}(f \\circ G)=O(\\mathsf {Q}(f)\\mathsf {Q}_E^{cc}(G)).$ If $\\mathsf {Q}(f)=\\Theta (\\sqrt{tn})$ , then our protocol assumes a shared state of $O(t\\log n)$ EPR-pairs at the start.","We remark that for the special case where $G=\\mathsf {AND}_2$ , our upper bound matches the lower bound proved by Razborov [31], except for symmetric functions $f$ where the first switch of function value happens at Hamming weights very close to $n$ .","In particular, if $f=\\mathsf {AND}_n$ and $G=\\mathsf {AND}_2$ , then $\\mathsf {Q}^{cc}(f \\circ G)=1$ but $\\mathsf {Q}(f)=\\Theta (\\sqrt{n})$ ."],["Necessity of the log-factor overhead in the BCW simulation","In this section we prove Theorem REF and Theorem REF .","For Theorem REF we exhibit a function $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ for which $\\mathsf {Q}(f)= O(n)$ and $\\mathsf {UPP}(f \\circ \\square ) = \\Omega (n \\log n)$ for $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .","In Theorem REF we show a general recipe for constructing total functions $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ such that $\\mathsf {Q}(f) = O(n)$ and $\\mathsf {Q}^{cc, *}(f \\circ \\square ) = \\Omega (n \\log n)$ for $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .","We first give a formal statement of Theorem REF .","Theorem 5.1 (Restatement of Theorem REF ) Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function with $\\widetilde{\\textnormal {deg}}(r) = \\Omega (n)$ , $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function and $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .","Define $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","If there exists $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\mathbb {R}$ that is a balanced probability distribution with respect to $G$ and $\\textnormal {disc}_{\\mu }(G) = n^{-\\Omega (1)}$ , then $\\mathsf {Q}(f) & = O(n),\\\\\\mathsf {Q}^{cc, *}(f \\circ \\square ) & = \\Omega (n \\log n).$ The proofs of Theorem REF and Theorem REF each involve proving a query complexity upper bound and a communication complexity lower bound.","The proofs of the query complexity upper bounds are along similar lines and follow from Theorem REF and Corollary REF (see Section REF ).","The proofs of the communication complexity lower bounds each involve a reduction from a problem whose communication complexity is easier to analyze (see Lemma REF in Section REF ).","Finally, we complete the proofs of Theorem REF and Theorem REF in Section REF ."],["Quantum query complexity upper bound","We start by stating the main theorem in this section.","Theorem 5.2 Let $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $r : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","Then the quantum query complexity of the function $r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{n(j+k)} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is given by $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) = O(n + \\sqrt{n(j+k)}).$ As a corollary we obtain the following on instantiating $j = k = n$ and $r$ as a Boolean function with quantum query complexity $\\Theta (n)$ in Theorem REF .","Corollary 5.3 Let $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a non-constant function and let $r : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function with $\\mathsf {Q}(r) = \\Theta (n)$ .","Then the quantum query complexity of the total function $r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) = \\Theta (n).$ The upper bound $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) = O(n)$ follows by plugging in parameters in Theorem REF .","For the lower bound, we show that $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) \\ge \\mathsf {Q}(r)$ .","Since $G$ is non-constant, there exist $x_1, y_1, x_2, y_2 \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ such that $G(x_1, y_1) = -1$ and $G(x_2, y_2) = 1$ .","Let $X_1 = H(x_1), Y_1 = H(y_1)$ , $X_2 = H(x_2)$ and $X_2 = H(y_2)$ .","Consider $r ~\\widetilde{\\circ }~h_G$ only restricted to inputs where the inputs to each copy of $h_G$ are either $(X_1, Y_1)$ or $(X_2, Y_2)$ .","Under this restriction, $r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is the same as $r : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","Thus $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) \\ge \\mathsf {Q}(r) = \\Omega (n)$ .","We now prove Theorem REF .","[Proof of Theorem REF ] Recall from Definition REF that the function $h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{j + k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is defined as $h_G(x, y) = G(s, t)$ if $x \\in \\pm H(s)$ and $y \\in \\pm H(t)$ for some $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}$ and $t \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}$ , and $h_G(x, y) = \\star $ otherwise.","Also recall from Definition REF that the function $r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{n(j+k)} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is defined as $r ~\\widetilde{\\circ }~h_G((X_1, Y_1), \\dots , (X_n, Y_n)) = r \\circ h_G((X_1, Y_1), \\dots , (X_n, Y_n))$ if $h_G((X_i, Y_i)) \\in \\left\\lbrace -1,1 \\right\\rbrace $ for all $i \\in [n]$ , and $-1$ otherwise."],["Quantum query algorithm:","View inputs to $r ~\\widetilde{\\circ }~h_G$ as $(X_1, Y_1, \\dots , X_n, Y_n)$ , where $X_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^j$ for all $i \\in [n]$ and $Y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^k$ for all $i \\in [n]$ .","We give a quantum algorithm and its analysis below.","Run $2n$ instances of the Bernstein-Vazirani algorithm: 1 instance on each $X_i$ and 1 instance on each $Y_i$ , to obtain $2n$ strings $x_1, \\dots , x_n, y_1, \\dots , y_n$ , where each $x_i$ is a $(\\log j)$ -bit string and each $y_i$ is a $(\\log k)$ -bit string.","For each $X_i$ and $Y_i$ , query $(X_i)_{1^{\\log j}}$ and $(Y_i)_{1^{\\log k}}$ to obtain bits $b_i, c_i \\in \\left\\lbrace -1,1 \\right\\rbrace $ for all $i \\in [n]$ .","Run Grover's search [16], [7] to check equality of the following two $(nj + nk)$ -bit strings: $(b_1 H(x_1), \\dots , b_n H(x_n), c_1 H(y_1), \\dots , c_n H(y_n))$ and $(X_1, \\dots , X_n, Y_1, \\dots , Y_n)$ .","If the step above outputs that the strings are equal, then output $r(G(x_1, y_1), \\dots , G(x_n, y_n))$ .","Else, output $-1$ ."],["Analysis of the algorithm:"," If the input is indeed of the form $(X_1, Y_1), \\dots , (X_n, Y_n)$ where each $X_i \\in \\pm H(x_i)$ and $Y_i \\in \\pm H(y_i)$ for some $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}$ and $y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}$ , then Step REF outputs the correct strings $x_{1}, \\dots , x_{n}, y_1, \\dots , y_n$ with probability 1 by the properties of the Bernstein-Vazirani algorithm.","Step REF then implies that $X_i = b_i H(x_i)$ and $Y_i = c_i H(y_i)$ for all $i \\in [n]$ .","Next, Step REF outputs that the strings are equal with probability 1 (since the strings whose equality are to be checked are equal).","Hence the algorithm is correct with probability 1 in this case, since $(r ~\\widetilde{\\circ }~h_G)(X_1, Y_1, \\dots , X_n, Y_n) = r(G(x_1, y_1), \\dots , G(x_n, y_n))$ .","If the input is such that there exists an index $i \\in [n]$ for which $X_i \\notin \\pm H(x_i)$ for every $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}$ or $Y_i \\notin \\pm H(y_i)$ for every $y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}$ , then the two strings for which equality is to be checked in the Step REF are not equal.","Grover's search catches a discrepancy with probability at least $2/3$ .","Hence, the algorithm outputs $-1$ (as does $r ~\\widetilde{\\circ }~h_G$ ), and is correct with probability at least $2/3$ in this case."],["Cost of the algorithm:","Step REF accounts for $2n$ quantum queries.","Step REF accounts for $2n$ quantum queries.","Step REF accounts for $O(\\sqrt{n(j+k)})$ quantum queries.","Thus, $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) = O(n + \\sqrt{n(j+k)}).$"],["Quantum communication complexity lower bound","In this section we first show a communication lower bound (under some model) on $(r ~\\widetilde{\\circ }~h_G) \\circ \\square $ in terms of the communication complexity of $r \\circ G$ (in the same model of communication) using a simple reduction.","We state the lemma below (Lemma REF ) for the case where the models under consideration are the bounded-error and unbounded-error quantum models, since these are the models of interest to us.","Lemma 5.4 Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ and $CC \\in \\lbrace \\mathsf {Q}^{cc, *}, \\mathsf {UPP}^{cc}\\rbrace $ .","Then, $CC((r ~\\widetilde{\\circ }~h_G) \\circ \\square ) \\ge CC(r \\circ G).$ We first consider the case $\\square = \\mathsf {AND}_2$ .","Consider a protocol $\\Pi $ of cost $\\ell $ that solves $(r ~\\widetilde{\\circ }~h_G) \\circ \\square $ in the $CC$ -model.","We exhibit below a protocol of cost $\\ell $ that solves $r \\circ G$ in the same model.","Suppose Alice is given input $x = (x_1, \\dots , x_{n}) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n \\log j}$ and Bob is given input $y = (y_1, \\dots , y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n \\log k}$ , where $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}, y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}$ for each $i \\in [n]$ ."],["Preprocessing step:","Alice constructs the $(n(j+k))$ -bit string $X = ((H(x_1), (-1)^{k}), \\dots , (H(x_n), (-1)^k)) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n(j+k)},$ and Bob constructs the $(n(j+k))$ -bit string $Y = (((-1)^j, H(y_1)), \\dots , ((-1)^j, H(y_n))) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n(j+k)}.$"],["Protocol:","Alice and Bob run the protocol $\\Pi $ with input $(X, Y)$ and output $\\Pi (X, Y)$ ."],["Cost:","The preprocessing of the inputs to obtain $X$ from $x$ and $Y$ from $y$ takes no communication.","Hence the total amount of communication is at most the cost of $\\Pi $ ."],["Correctness:","For $X$ and $Y$ constructed in Equation (REF ) and Equation (REF ), respectively, we now argue that $(r \\circ G)(x, y) = ((r ~\\widetilde{\\circ }~h_G) \\circ \\mathsf {AND}_2) (X, Y)$ , which would conclude the proof for $\\square = \\mathsf {AND}_2$ .","$((r ~\\widetilde{\\circ }~h_G) \\circ \\mathsf {AND}_2)(X, Y) & = (r ~\\widetilde{\\circ }~h_G)((H(x_1), H(y_1)), \\dots , (H(x_n), H(y_n)))\\\\& = r(G(x_1, y_1), \\dots , G(x_n, y_n)) \\\\& = (r \\circ G)(x, y).$ Thus, $CC((r ~\\widetilde{\\circ }~h_G) \\circ \\mathsf {AND}_2) \\ge CC(r \\circ G).$ The argument for $\\square = \\mathsf {XOR}_2$ follows along the same lines, with the strings $(-1)^j$ and $(-1)^k$ replaced by $1^j$ and $1^k$ , respectively, in the preprocessing step."],["On the tightness of the BCW simulation","We prove Theorem REF in Section REF and Theorem REF in Section REF ."],["Proof of Theorem ","Towards proving Theorem REF , we first observe how to obtain bounded-error quantum communication complexity lower bounds using Theorem REF and Lemma REF .","Lemma 5.5 Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be functions such that $\\textnormal {bdisc}(G) = o(1)$ and $\\frac{8en}{\\widetilde{\\textnormal {deg}}(r)} \\le \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)^{1-\\beta }$ for some constant $\\beta \\in (0,1)$ .","Let $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .","Then, $\\mathsf {Q}^{cc, *} ((r ~\\widetilde{\\circ }~h_G) \\circ \\square ) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)\\right).$ By Lemma REF we have $\\mathsf {Q}^{cc, *} ((r ~\\widetilde{\\circ }~h_G) \\circ \\square ) \\ge \\mathsf {Q}^{cc, *}(r \\circ G)$ .","By Theorem REF , $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)\\right)$ .","We now prove Theorem REF .","[Proof of Theorem REF ] Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be as in the statement of the theorem.","We have $\\mathsf {Q}(r) \\ge \\widetilde{\\textnormal {deg}}(r)/2 = \\Omega (n)$ , where the first inequality follows by Theorem REF and the second equality follows from the assumption that $\\widetilde{\\textnormal {deg}}(r) = \\Omega (n)$ .","Moreover, $\\mathsf {Q}(r) \\le n$ since $r$ is a function on $n$ input variables.","Hence $\\mathsf {Q}(r) = \\Theta (n)$ .","Thus, Corollary REF is applicable, and we have $\\mathsf {Q}(f) = \\Theta (n).$ For the lower bound, $\\widetilde{\\textnormal {deg}}(r) = \\Omega (n)$ by assumption.","Thus $\\frac{2en}{\\widetilde{\\textnormal {deg}}(r)} = O(1).$ Also, since by assumption $\\frac{1}{\\textnormal {bdisc}(G)} = n^{\\Omega (1)} = \\omega (1)$ , we have $\\frac{2en}{\\widetilde{\\textnormal {deg}}(r)} \\le \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)^{1-\\beta }$ for every constant $\\beta \\in (0,1)$ .","Lemma REF implies $\\mathsf {Q}^{cc, *}(f \\circ \\square ) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right) \\right) = \\Omega (n \\log n).$"],["Proof of Theorem ","The total function $f: \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ that we use to prove Theorem REF is $f = r ~\\widetilde{\\circ }~h_G$ , where $r = \\mathsf {PARITY}_n$ and $G = \\mathsf {IP}_{\\log n}$ .","Note that Theorem REF implies $\\mathsf {Q}(f) = O(n)$ .","For the quantum communication complexity lower bound in Theorem REF , we are able to show not only $\\mathsf {Q}^{cc, *}(f \\circ \\square ) = \\Omega (n \\log n)$ , but $\\mathsf {UPP}^{cc}(f \\circ \\square ) = \\Omega (n \\log n)$ for $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .","The following claim shows that $f$ is transitive.","Claim 5.6 Let $n > 0$ be a power of 2.","Let $r = \\mathsf {PARITY}_n : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G = \\mathsf {IP}_{\\log n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","The function $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive.","We first show that $h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive.","We next observe that $s ~\\widetilde{\\circ }~t$ is transitive whenever $s$ is symmetric and $t$ is transitive ($s$ can be assumed to just be transitive rather than symmetric, as noted in Remark REF ).","The theorem then follows since $\\mathsf {PARITY}_n$ is symmetric.","Towards showing transitivity of $h_G$ , let $\\pi \\in S_{2n}$ , and $(\\sigma _{\\ell }, \\sigma _\\ell ) \\in S_{2n}$ for $\\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ be defined as follows.","(Here $\\sigma _\\ell \\in S_n$ ; the first copy acts on the first $n$ coordinates, and the second copy acts on the next $n$ coordinates.)","$\\pi (k) = {\\left\\lbrace \\begin{array}{ll}k + n & k \\le n\\\\k - n & k > n.\\end{array}\\right.","}$ That is, on every string $(x, y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ , the permutation $\\pi $ maps $(x, y)$ to $(y, x)$ .","For every $\\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , the permutation $\\sigma _\\ell \\in S_n$ is defined as $\\sigma _{\\ell }(i) = i \\oplus \\ell ,$ where $i \\oplus \\ell $ denotes the bitwise XOR of the strings $i$ and $\\ell $ .","That is, for every input $(x, y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ and every $k \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , the input bit $x_{k}$ is mapped to $x_{k \\oplus \\ell }$ and $y_k$ is mapped to $y_{k\\oplus \\ell }$ .","For every $(x,y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ and $i,j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , the permutation $\\sigma _{i \\oplus j}(x,y)$ swaps $x_i$ and $x_j$ , and also swaps $y_i$ and $y_j$ .","If for $i, j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , our task was to swap the $i$ 'th index of the first $n$ variables with the $j$ 'th index of the second $n$ variables, then the permutation $\\sigma _{i\\oplus j} \\circ \\pi $ does the job.","That is, for every $(x,y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ and $i,j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , the permutation $\\sigma _{i\\oplus j} \\circ \\pi $ maps $x_i$ to $y_j$ .","Thus the set of permutations $\\lbrace \\pi , \\lbrace \\sigma _\\ell : \\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}\\rbrace \\rbrace $ acts transitively on $S_{2n}$ .","Now we show that for all $x,y \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ and all $\\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , we have $h_G(\\sigma _{\\ell }(x), \\sigma _\\ell (y)) = h_G(x,y)$ .","Fix $\\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ .","If $x \\in \\pm H(s)$ and $y \\in \\pm H(t)$ are Hadamard codewords, then $x_k = \\langle k, s \\rangle $ and $y_k = \\langle k, t \\rangle $ for all $k \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , and $G(x,y) = \\langle s, t \\rangle $ .","Thus, for every $k \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ we have $\\sigma _{\\ell }(x_k) = x_{k \\oplus \\ell } = \\langle k \\oplus \\ell , s \\rangle = \\langle \\ell , s \\rangle \\cdot \\langle k , s \\rangle $ .","Hence $\\sigma _{\\ell }(x) \\in \\pm H(s)$ (since $\\langle \\ell , s \\rangle $ does not depend on $k$ , and takes value either 1 or $-1$ ).","Similarly, $\\sigma _{\\ell }(y) \\in \\pm H(t)$ .","Thus $h_G(\\sigma _{\\ell }(x,y)) = h_G(x,y)$ .","If $x$ ($y$ , respectively) is not a Hadamard codeword, then a similar argument shows that for all $\\ell \\in [n]$ , $\\sigma _{\\ell }(x)$ ($\\sigma _{\\ell }(y)$ , respectively) is also not a Hadamard codeword.","Using the fact that $\\langle s, t \\rangle = \\langle t, s \\rangle $ for every $s, t \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , one may verify that $h_G(\\pi (x,y)) = h_G(x,y)$ for all $x,y \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ .","Along with the observation that $\\mathsf {PARITY}_n$ is a symmetric function, we have that $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive under the following permutations: $S_n$ acting on the inputs of $\\mathsf {PARITY}_n$ , and The group generated by $\\left\\lbrace \\pi \\right\\rbrace \\cup \\left\\lbrace (\\sigma _\\ell , \\sigma _\\ell ) : \\ell \\in [n] \\right\\rbrace $ acting independently on the inputs of each copy of $h_G$ , where $\\sigma _{\\ell }$ is as in Equation (REF ).","We now prove Theorem REF .","[Proof of Theorem REF ] Let $n > 0$ be a power of 2.","Let $r = \\mathsf {PARITY}_n : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G = \\mathsf {IP}_{\\log n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","Let $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","By Claim REF , $f$ is transitive.","By Corollary REF we have $\\mathsf {Q}(f) = \\Theta (n).$ For the communication lower bound we have $\\mathsf {UPP}^{cc}(f \\circ \\square ) & = \\mathsf {UPP}^{cc}((r ~\\widetilde{\\circ }~h_G) \\circ \\square ) \\\\& \\ge \\mathsf {UPP}^{cc}(\\mathsf {PARITY}_n \\circ \\mathsf {IP}_{\\log n}) \\\\& = \\mathsf {UPP}^{cc}(\\mathsf {IP}_{n \\log n}) \\\\& = \\Omega (n \\log n) .$ Remark 5.7 The proof of transitivity of $f$ in Theorem REF can also be used to prove that if $r : \\left\\lbrace -1,1 \\right\\rbrace ^n$ is transitive and $G = \\mathsf {IP}_{\\log n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , then $r ~\\widetilde{\\circ }~h_G$ is transitive as well.","By instantiating $r$ to a transitive function with approximate degree $\\Omega (n)$ (e.g.","Majority), Theorem REF implies that the BCW simulation is tight w.r.t.","the bounded-error communication model for a wide class of transitive functions."],["Hardness of composing the Chakraborty et al. function with $\\mathsf {AND}_2$","Chakraborty et al.","[12] exhibited a total function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ for which $\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {XOR}_2) = \\Omega (\\mathsf {Q}(f) \\log n)$ , that is, the log-factor overhead in the BCW simulation is necessary for $f$ when the inner function is $\\mathsf {XOR}_2$ .","In this section we show that the log-factor overhead is necessary for $f$ even when the inner function is $\\mathsf {AND}_2$ .","Definition A.1 (Addressing function) For an integer $n>0$ that is a power of 2, let the Addressing function, denoted $\\mathsf {ADDR}_{n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , be a two-party function defined as follows.","$\\mathsf {ADDR}_n(x_1, \\dots , x_{\\log n}, y_1, \\dots , y_n) = y_{\\textnormal {bin}(i)},$ where $\\textnormal {bin}(i)$ represents the integer in $[n]$ whose binary representation is $i$ .","In the corresponding communication problem, Alice holds the inputs $x_1, \\dots , x_{\\log n}$ , and Bob holds the inputs $y_1, \\dots , y_n$ .","In [12] a subset of the authors defined the following function $f: \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","Let $X_i, Y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ for all $i \\in [n]$ .","$f(X_1, Y_1, \\dots , X_n, Y_n) ={\\left\\lbrace \\begin{array}{ll}\\mathsf {PARITY}(\\mathsf {ADDR}_n(x_1, Y_1), \\dots , \\mathsf {ADDR}_n(x_n, Y_n)) \\textnormal { if } \\forall i \\in [n], X_i \\in \\pm H(x_i)\\\\-1 \\textnormal { otherwise},\\end{array}\\right.","}$ That is, if there exist $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ for all $i \\in [n]$ such that $X_i \\in \\pm H(x_i)$ , then $f(X_1, Y_1, \\dots , X_n, Y_n)$ equals the parity of $\\mathsf {ADDR}_n(x_1, Y_1), \\dots , \\mathsf {ADDR}_n(x_n, Y_n)$ , and $f$ equals $-1$ otherwise.","Chakraborty et al.","proved the following quantum query upper bound on $f$ (set $\\delta = 1/2$ in [12]).","Theorem A.2 ([12]) For $f: \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ as defined in Equation (REF ), $\\mathsf {Q}(f) = O(n)$ .","Remark A.3 The function considered in [12] is not exactly as in Equation (REF ) because the former one did not consider negations of Hadamard codewords as done in Equation (REF ).","However, Theorem REF can still be seen to hold by incorporating a step similar to Step REF in the algorithm given in the proof of [12].","Chakraborty et al.","showed the BCW simulation theorem is tight for $f$ when the inner function is $\\mathsf {XOR}_2$ , that is $\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {XOR}_2) = \\Omega (n \\log n)$ .","They left open the question whether the BCW simulation theorem is tight for $f$ even when the inner function is $\\mathsf {AND}_2$ .","We show that this is indeed the case.","The quantum query upper bound holds by Theorem REF .","For the communication complexity lower bound we use techniques similar to those in the proof of Lemma REF .","Lemma A.4 Let $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be as in Equation (REF ), and $CC \\in \\lbrace \\mathsf {Q}^{cc, *}, \\mathsf {UPP}^{cc} \\rbrace $ .","Then, $CC(r \\circ \\mathsf {AND}_2) \\ge CC(\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}).$ We require the following lemma.","Lemma A.5 Let $n > 0$ be a power of 2, and let $U$ be the uniform distribution on $\\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{n}$ .","Then, $\\textnormal {disc}_U(\\mathsf {ADDR}_{n}) \\le \\frac{1}{\\sqrt{n}}.$ We provide a proof for completeness.","Consider the uniform distribution $U: \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ defined as $U(x,y) = \\frac{1}{n2^n}$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ and $y \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","Let $A_{n \\times 2^n}$ be the communication matrix of $\\mathsf {ADDR}_n: \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","Observe that for every $i,j \\in [n]$ , $(AA^T)_{ij}= \\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} x_i x_j={\\left\\lbrace \\begin{array}{ll}2^n \\text{ if } i = j \\\\0 \\text{ otherwise}.\\end{array}\\right.","}$ Hence $AA^T = 2^n I_n$ , where $I_n$ denotes the $n \\times n$ identity matrix.","Thus, $\\Vert A\\Vert = \\sqrt{2^n}$ .","For $S \\subseteq [n]$ and $T \\subseteq [2^n]$ , let $1_S: [n] \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ and $1_T: [2^n] \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ be the indicator function of corresponding rows and columns of $A$ , respectively.","We now upper bound the discrepancy with respect to the uniform distribution: $\\textnormal {disc}_U(\\mathsf {ADDR}_n)&= \\max _{S \\subseteq [n], T\\subseteq [2^n]} \\left|(1_S)^T \\cdot (U \\circ A) \\cdot 1_T \\right| \\\\&\\le \\sqrt{n} \\cdot \\Vert U \\circ A \\Vert \\cdot \\sqrt{2^n} \\\\&= \\frac{1}{\\sqrt{n2^n}} \\Vert A \\Vert \\\\&= \\frac{1}{\\sqrt{n}}.$ We now prove Lemma REF using Lemma REF .","[Proof of Lemma REF ] Consider a protocol $\\Pi $ of cost $\\ell $ that solves $f \\circ \\mathsf {AND}_2$ in the $CC$ -model.","We exhibit below a protocol of cost $\\ell $ that solves $\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}$ in the same model.","Recall from Definitions REF and REF that in the communication problem $\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}$ , Alice is given the address bits and Bob is given the target bits for each copy of the inner gadget $\\mathsf {ADDR}_{n}$ .","Suppose Alice is given input $x = (x_1, \\dots , x_{n}) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n \\log n}$ and Bob is given input $y = (y_1, \\dots , y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n^2}$ , where $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}, y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n}$ for each $i \\in [n]$ ."],["Preprocessing step:","Alice constructs the following $(2n^2)$ -bit string $X = (H(x_1), (-1)^{n}), \\dots , (H(x_n), (-1)^n)) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2},$ and Bob constructs the $(2n^2)$ -bit string $Y = (((-1)^n, y_1), \\dots , ((-1)^n, y_n)) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2}.$"],["Protocol:","Alice and Bob run the protocol $\\Pi $ with input $(X, Y)$ , and output $\\Pi (X, Y)$ ."],["Cost:","The preprocessing of the inputs to obtain $X$ from $x$ and $Y$ from $y$ takes no communication.","Hence the total amount of communication is at most the cost of $\\Pi $ ."],["Correctness:","We now argue that $\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}(x, y) = (f \\circ \\mathsf {AND}_2) (X, Y)$ , which would conclude the proof.","We have from Equation (REF ) and Equation (REF ) that $(f \\circ \\mathsf {AND}_2)(X, Y) & = f((H(x_1), y_1), \\dots , (H(x_n), y_n))\\\\& = \\mathsf {PARITY}_n(\\mathsf {ADDR}_n(x_1, y_1), \\dots , \\mathsf {ADDR}_n(x_n, y_n)) \\\\& = \\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}(x, y).$ Thus, $CC(f \\circ \\mathsf {AND}_2) \\ge CC(\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}),$ which proves the lemma.","Theorem A.6 Let $n > 0$ be a sufficiently large power of 2.","Let $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be as in Equation (REF ).","Then, $\\mathsf {Q}(f) & = O(n),\\\\\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {AND}_2) & = \\Omega (n \\log n).$ The upper bound follows from Theorem REF .","For the lower bound, we first note that for sufficiently large $n$ and every constant $\\beta \\in (0, 1)$ , we have $\\frac{2en}{\\widetilde{\\textnormal {deg}}(\\mathsf {PARITY}_n)} = 2e < (\\sqrt{n})^{1 - \\beta } = \\left(\\frac{1}{\\textnormal {disc}_{U}(\\mathsf {ADDR}_n)}\\right)^{1-\\beta },$ where $U$ denotes the uniform distribution over $\\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","The first equality above holds since $\\widetilde{\\textnormal {deg}}(\\mathsf {PARITY}_n) = n$ , and the last equality holds by Lemma REF .","It is easy to verify that $U$ is a balanced distribution w.r.t.","$\\mathsf {ADDR}_n$ .","Hence, $\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {AND}_2) & \\ge \\mathsf {Q}^{cc, *}(\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_n) \\\\& = \\Omega \\left(\\widetilde{\\textnormal {deg}}(\\mathsf {PARITY}_n) \\log \\left(\\frac{1}{\\textnormal {disc}_U(\\mathsf {ADDR}_{n})}\\right)\\right) \\\\& = \\Omega \\left(n \\log \\left(\\frac{1}{\\textnormal {disc}_U(\\mathsf {ADDR}_{n})}\\right)\\right) \\\\& = \\Omega (n \\log n).","$"],["A separation between log-approximate-spectral norm and approximate degree for a transitive function","In this section, we exhibit a transitive function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ for which the logarithm of its approximate spectral norm ($\\log (\\Vert \\widehat{f}\\Vert _{1, 1/3})$ ) is at least $\\Omega (\\widetilde{\\textnormal {deg}}(f) \\log n)$ .","We refer the reader to [28] for preliminaries on Fourier analysis of Boolean functions.","Definition B.1 (Spectral norm) Let $p : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be a function, and let $p = \\sum _{S \\subseteq [n]}\\widehat{p}(S)\\chi _S$ denote its Fourier expansion.","The spectral norm of $p$ is define by $\\Vert \\widehat{p}\\Vert _1 := \\sum _{S \\subseteq [n]} \\left|\\widehat{p}(S) \\right|.$ Definition B.2 (Approximate Spectral Norm) The approximate spectral norm of a function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\lbrace -1,1\\rbrace $ , denoted by $\\Vert \\widehat{f}\\Vert _{1,\\varepsilon }$ is defined to be the minimum spectral norm of a real polynomial $p : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ that satisfies $\\left|p(x) - f(x) \\right| \\le \\varepsilon $ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","That is, $\\Vert \\widehat{f}\\Vert _{1,\\varepsilon } := \\min \\left\\lbrace \\Vert \\widehat{p}\\Vert _1 : \\left|p(x) - f(x) \\right| \\le \\varepsilon ~\\text{for all~} x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n \\right\\rbrace .$ In [12] the following question raised in [2] was shown to have a negative answer, using the function $f$ of Equation (REF ) to witness this.","Question B.3 ([2]) Is it true that for all Boolean functions $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $\\log (\\Vert \\widehat{f}\\Vert _{1,\\varepsilon }) = O(\\widetilde{\\textnormal {deg}}(f))?$ In this section we show that Question REF is false even for the special class of transitive Boolean functions.","This is in stark contrast with the class of symmetric functions, for which Question REF is true (see [12]).","Claim B.4 There exists a transitive function $f: \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ such that $\\log \\left(\\Vert \\widehat{f}\\Vert _{1, 1/3} \\right) = \\Omega (\\widetilde{\\textnormal {deg}}(f) \\log n)$ .","We first state some required preliminaries.","Definition B.5 (Monomial projection) We call a function $g:\\left\\lbrace -1,1 \\right\\rbrace ^m \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ a monomial projection of a function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ if $g$ can be expressed as $g(x_1,\\dots , x_m) = f(M_1,\\dots , M_n)$ , where each $M_i$ is a monomial in the variables $x_1,\\dots , x_m$ .","It is known that the approximate spectral norm of a function can only decrease upon monomial projections (see, for example, [14]).","Observation B.6 ([14]) For $f : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $g : \\left\\lbrace -1,1 \\right\\rbrace ^{m} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ such that $g$ is a monomial projection of $f$ , $\\Vert \\widehat{g}\\Vert _{1, 1/3} \\le \\Vert \\widehat{f}\\Vert _{1, 1/3}.$ Fact B.7 (Fourier coefficients of $\\mathsf {IP}_{n}$ ) Let $\\mathsf {IP}_n: \\left\\lbrace -1,1 \\right\\rbrace ^{2n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be as in Definition REF .","Then for all $S \\subseteq [2n]$ we have, $\\left|\\widehat{\\mathsf {IP}_n}(S) \\right| = \\frac{1}{2^n}.$ Fact B.8 (Plancherel's Theorem) Let $f,g: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be functions.","Then, $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n}f(x)g(x) = \\sum _{S \\subseteq [n]} \\widehat{f}(S) \\widehat{g}(S).$ [Proof of Claim REF ] Let $n > 0$ be a power of 2.","Let $r = \\mathsf {PARITY}_n : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G = \\mathsf {IP}_{\\log n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","From Claim REF , the function $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive.","By Corollary REF we have $\\mathsf {Q}(f) = \\Theta (n).","$ By Theorem REF we have $\\mathsf {Q}(f) \\ge \\widetilde{\\textnormal {deg}}(f)/2$ and together with Equation (REF ), this implies $\\widetilde{\\textnormal {deg}}(f) = O(n)$ .","Thus to complete the proof of the claim, it suffices to show $\\log \\left(\\Vert \\widehat{f}\\Vert _{1, 1/3} \\right) = \\Omega (n \\log n)$ .","We first note that $\\mathsf {IP}_{n \\log n}$ is a monomial projection of $f$ .","Consider the function $f$ acting on the input variables $x^{(1)}, \\dots , x^{(n)}, y^{(1)}, \\dots , y^{(n)}$ , where $x^{(i)}, y^{(i)} \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n}$ for all $i \\in [n]$ .","For $i \\in [n]$ , set $x^{(i)}_{1^{\\log n}} = y^{(i)}_{1^{\\log n}} = 1$ .","For $i \\in [n]$ and string $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\setminus \\left\\lbrace 1^{\\log n} \\right\\rbrace $ , set $x^{(i)}_s = \\prod _{j: s_j = -1}x^{(i)}_j $ and $y^{(i)}_s = \\prod _{j: s_j = -1}y^{(i)}_j$ .","That is, in each block of inputs $x^{(i)}$ and $y^{(i)}$ , the coordinate corresponding to $1^{\\log n}$ equals 1, the coordinates corresponding to $j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ with $|j| = 1$ are free variables, and all other variables are replaced by monomials in these variables.","Under the above monomial projection there are $2n \\log n$ free variables, namely $\\left\\lbrace x^{(i)}_{s}, y^{(i)}_{s} : i \\in [n], s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}, |s| = 1\\right\\rbrace $ .","Also note that under this projection and every setting of the free variables, the blocks $x^{(i)}$ and $y^{(i)}$ , for $i \\in [n]$ , are always Hadamard codewords.","Let $f^{\\prime }$ be the monomial projection (see Definition REF ) of $f$ under the projection defined in this paragraph.","For the purpose of the next equality we abbreviate strings $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ of Hamming weight 1 by the set $\\left\\lbrace i \\right\\rbrace $ , where $i \\in [\\log n]$ is such that $s_i = -1$ .","On the free variables, the projected function $f^{\\prime }$ equals $\\mathsf {PARITY}_n \\left(\\langle (x^{(1)}_{\\lbrace 1\\rbrace }, \\dots , x^{(1)}_{\\lbrace \\log n\\rbrace }), (y^{(1)}_{\\lbrace 1\\rbrace }, \\dots , y^{(1)}_{\\lbrace \\log n\\rbrace })\\rangle , \\dots , \\langle (x^{(n)}_{\\lbrace 1\\rbrace }, \\dots , x^{(n)}_{\\lbrace \\log n\\rbrace }), (y^{(n)}_{\\lbrace 1\\rbrace }, \\dots , y^{(n)}_{\\lbrace \\log n\\rbrace })\\rangle \\right).$ Thus, $f^{\\prime } = \\mathsf {IP}_{n\\log n}.$ It follows from earlier works [8], [9] that every polynomial that approximates $\\mathsf {IP}_{n\\log n}$ to error $1/3$ must have spectral norm $2^{\\Omega (n \\log n)}$ .","We include a short proof below for completeness.","Let $P$ be a polynomial that approximates $\\mathsf {IP}_{n\\log n}$ to error $1/3$ .","Since we have $P(x, y)\\mathsf {IP}_{n\\log n}(x, y) \\ge 2/3$ for all $(x, y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n\\log n}$ , $2/3 & \\le \\mathbb {E}_{x, y \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n\\log n}} [P(x, y)\\mathsf {IP}_{n\\log n}(x, y)]\\\\& = \\sum _{S \\subseteq [2n\\log n]}\\widehat{P}(S)\\widehat{\\mathsf {IP}_{n\\log n}}(S) \\\\& \\le \\sum _{S \\subseteq [2n\\log n]}\\frac{|\\widehat{P}(S)|}{2^{n \\log n}} \\\\& = \\frac{\\Vert \\widehat{P}\\Vert _1}{2^{n \\log n}}\\\\& \\Rightarrow \\log (\\Vert \\widehat{P}\\Vert _1) = \\Omega (n \\log n)\\\\& \\Rightarrow \\log (\\Vert \\widehat{\\mathsf {IP}_{n \\log n}}\\Vert _{1, 1/3}) = \\Omega (n \\log n).$ This yields the desired contradiction by Observation REF ."],["Quantum communication lower bound via the generalized discrepancy method","In this section we prove Theorem REF , which gives a lower bound on the quantum communication complexity of a composed function in terms of the approximate degree of the outer function and the discrepancy of the inner function.","This result is implicit in [25].","Their result is stated in the more general setting of matrix norms, and the $\\log $ factor we require on the right-hand side of Theorem REF is not included in their statement.","Theorem REF follows implicitly from the proof of [25] along with the fact that $\\gamma _2^*$ -norm characterizes discrepancy [23].","For completeness and clarity, we prove Theorem REF below from first principles.","Definition C.1 For functions $f,g : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ and a probability distribution $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ the correlation between $f$ and $g$ with respect to $\\mu $ is defined to be $\\textnormal {corr}_{\\mu }(f, g) = \\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} f(x)g(x) \\mu (x).$ For a Boolean function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , its approximate degree can be captured by a certain linear program.","Writing out the dual of this program and analyzing its optimum yields the following theorem (see, for example, [10]).","Theorem C.2 (Dual witness for $\\varepsilon $ -approximate degree) For every $\\varepsilon \\ge 0$ and $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $\\widetilde{\\textnormal {deg}}_{\\varepsilon }(f) \\ge d$ if and only if there exists a polynomial $\\psi :\\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ such that $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} f(x) \\psi (x) > \\varepsilon $ , $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} |\\psi (x)| = 1$ and $\\widehat{\\psi }(S) = 0$ for all $|S| < d$ .","We require the following fact, which follows immediately from the fact that $\\widehat{f}(S)$ is a uniform average of different signed values of $f(x)$ .","Fact C.3 (Folklore) For every function $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ , $2^n \\max _{S \\subseteq [n]} \\left|\\widehat{f}(S) \\right| \\le \\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\left|f(x) \\right|.$ The following theorem shows that if a two-party function $F$ correlates well with a two-party function $G$ under some distribution $\\lambda $ , and the discrepancy of $G$ under $\\lambda $ is small, then the bounded-error quantum communication complexity of $F$ must be large.","This is referred to as the generalized discrepancy method, and was first proposed by Klauck [21].","The following version can be found in [13], for example, stated as a lower bound on randomized communication complexity.","However, the generalized discrepancy method is also known to give lower bounds on bounded-error quantum communication complexity, even in the model where the parties can share an arbitrary prior entangled state for free.","Theorem C.4 (Generalized Discrepancy Bound) Consider functions $E, F : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .","If there exists a distribution $\\lambda : \\left\\lbrace -1,1 \\right\\rbrace ^{m} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ such that $\\textnormal {corr}_{\\lambda }(E, F) \\ge \\delta $ , then $\\mathsf {Q}^{cc, *}_{\\frac{1}{2} - \\varepsilon }(E)=\\Omega \\left( \\log \\left(\\frac{\\delta + 2\\varepsilon - 1}{\\textnormal {disc}_{\\lambda }(F)} \\right)\\right).$ Definition C.5 For a distribution $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ and integer $k>0$ , define the distribution $\\mu ^{\\otimes k} : \\left\\lbrace -1,1 \\right\\rbrace ^{mk} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{nk} \\rightarrow \\mathbb {R}$ by $\\mu ^{\\otimes k}((X_1, Y_1) \\dots , (X_k, Y_k)) = \\prod _{i \\in [k]}\\mu (X_i, Y_i),$ where $X_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{m}$ and $Y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ for all $i \\in [k]$ .","We require the following XOR lemma for discrepancy due to Lee, Shraibman and Špalek [24].","Theorem C.6 ([24]) Let $P : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a two-party function and $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be a distribution.","For every integer $k > 0$ , $\\textnormal {disc}_{\\mu ^{\\otimes k}}(\\mathsf {PARITY}_k \\circ P) \\le (8 \\textnormal {disc}_{\\mu }(P))^{k}.$ We recall Theorem REF below.","Theorem C.7 (Restatement of Theorem REF ) Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be functions.","Let $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\mathbb {R}$ be a balanced distribution with respect to $G$ and $\\textnormal {disc}_{\\mu }(G) = o(1)$ .","If $\\frac{8en}{\\widetilde{\\textnormal {deg}}(r)} \\le \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)}\\right)^{1-\\beta }$ for some constant $\\beta \\in (0,1)$ , then $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)}\\right)\\right).$ In particular, $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)\\right).$ For simplicity we assume $j = k = m$ .","The general proof follows along similar lines.","Let $\\widetilde{\\textnormal {deg}}(r) = d$ , and let $\\psi : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be a dual witness for this as given by Theorem REF .","Let $\\nu : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be defined as $\\nu (x) = |\\psi (x)|$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ and also define $h: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ as $h(x) = \\text{sign}(\\psi (x))$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","First note that $\\nu $ is a distribution since $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\nu (x) = \\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} |\\psi (x)| = 1$ by Theorem REF .","From Theorem REF we also have $\\textnormal {corr}_{\\nu }(r, h) & > 1/3 \\\\\\widehat{h\\nu }(S) & = 0 \\quad \\text{for all~} |S| < d,$ where $h\\nu (x) := h(x)\\nu (x) = \\psi (x)$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","We will construct a probability distribution $\\lambda : \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\rightarrow \\mathbb {R}$ using $\\nu : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ and $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^m \\rightarrow \\mathbb {R}$ such that $r \\circ G$ and $h \\circ G$ have large correlation under $\\lambda $ .","We will also show that $\\textnormal {disc}_{\\lambda }(h \\circ G)$ is small.","The proof of the theorem would then follow from Theorem REF .","Let $X,Y \\in \\left\\lbrace -1,1 \\right\\rbrace ^{mn}$ be such that $X = (X_1, \\dots , X_n)$ , $Y = (Y_1, \\dots , Y_n)$ and $X_i, Y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^m$ for all $i \\in [n]$ .","Also, let $G(X,Y) = (G(X_1, Y_1), \\dots , G(X_n, Y_n)) \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .","Define $\\lambda : \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\rightarrow \\mathbb {R}$ as follows.","$\\lambda (X,Y)&= 2^n \\cdot \\nu (G(X, Y)) \\cdot \\prod _{i \\in [n]} \\mu (X_i,Y_i).$ Observe that for any $z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , $\\sum _{X,Y : G(X, Y) = z} \\lambda (X,Y)&= \\sum _{X,Y: G(X,Y) = z} 2^n \\cdot \\nu (G(X, Y)) \\cdot \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\nonumber \\\\&= 2^n \\nu (z) \\prod _{i \\in [n]}\\left(\\sum _{X_i, Y_i : G(X_i, Y_i) = z_i} \\mu (X_i, Y_i)\\right)\\nonumber \\\\&= \\nu (z), $ where the last equality follows since $\\mu $ is balanced w.r.t.","$G$ by assumption.","Thus $\\sum _{X,Y} \\lambda (X,Y) = \\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\nu (z) = 1.$ We next observe that $\\textnormal {corr}_{\\lambda }(r \\circ G, h \\circ G)$ is large.","$\\textnormal {corr}_{\\lambda }(r \\circ G, h \\circ G)&= \\sum _{X,Y} {r \\circ G}(X,Y) \\cdot {h \\circ G}(X,Y) \\cdot \\lambda (X,Y) \\nonumber \\\\&= \\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\sum _{X,Y: G(X,Y) = z} r(z)h(z) \\cdot \\lambda (X,Y) \\nonumber \\\\&= \\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} r(z)h(z) \\nu (z) \\nonumber \\\\&= \\textnormal {corr}_{\\nu }(r, h) > 1/3.","$ where the last equality follows from Equation (REF ).","We now upper bound the discrepancy of $h \\circ G$ with respect to a rectangle $R$ , under the distribution $\\lambda $ .","Let $R \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{mn}$ be any rectangle of the form $R(X, Y) = A(X)B(Y)$ for $A : \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ and $B : \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ .","$\\textnormal {\\textnormal {disc}}_{\\lambda }(h \\circ G, R)&= \\left|\\sum _{X,Y} {h \\circ G}(X,Y) \\cdot R(X,Y) \\cdot \\lambda (X,Y) \\right| \\\\&= \\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\sum _{X,Y: G(X,Y) = z} {h \\circ G}(X,Y) \\cdot R(X,Y) \\cdot \\lambda (X,Y) \\right|\\\\&= 2^n \\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} h(z) \\nu (z) \\sum _{X,Y: G(X,Y) = z} R(X,Y) \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\right| \\\\&= 2^n \\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\left(\\sum _{S \\subseteq [n]} \\widehat{h\\nu }(S)\\chi _S(z)\\right) \\sum _{X,Y: G(X,Y) = z} R(X,Y) \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\right|\\\\&= 2^n\\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\sum _{S \\subseteq [n]} \\widehat{h\\nu }(S) \\sum _{X,Y: G(X,Y) = z} R(X,Y) \\cdot \\chi _S(z) \\cdot \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\right| \\\\&= 2^n\\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\sum _{S \\subseteq [n]} \\widehat{h\\nu }(S) \\sum _{X,Y: G(X,Y) = z} R(X,Y) \\cdot \\prod _{i \\in S}G(X_i,Y_i) \\cdot \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\right| \\\\& \\le 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\cdot \\left|\\sum _{X,Y} R(X,Y) \\cdot \\prod _{i \\in S}G(X_i,Y_i)\\mu (X_i, Y_i) \\cdot \\prod _{j \\notin S}\\mu (X_j,Y_j) \\right| \\\\& \\le 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\cdot \\left|\\sum _{X_j,Y_j : j \\notin S} \\prod _{j \\notin S}\\mu (X_j,Y_j) \\left(\\sum _{X_i,Y_i : i \\in S} A(X) B(Y) \\cdot \\prod _{i \\in S}G\\mu (X_i, Y_i)\\right) \\right|.$ For any $X = (X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{mn}$ (respectively, $Y = (Y_1, \\dots , Y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{mn}$ ) and set $S \\subseteq [n]$ define the string $X_S \\in \\left\\lbrace -1,1 \\right\\rbrace ^{m|S|}$ (respectively, $Y_S \\in \\left\\lbrace -1,1 \\right\\rbrace ^{m|S|}$ ) by $X_S = (\\dots , X_i, \\dots )$ (respectively, $Y_S = (\\dots , Y_i, \\dots )$ ), where $i$ ranges over all elements of $S$ .","For any $S$ and fixed $\\lbrace X_j : j \\notin S\\rbrace $ (respectively, $\\lbrace Y_j : j \\notin S\\rbrace $ ) note that $A(X) = A^{\\prime }(X_S)$ (respectively, $B(X) = B^{\\prime }(X_S)$ ) for some $A^{\\prime }: \\left\\lbrace -1,1 \\right\\rbrace ^{m|S|} \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ (respectively, $B^{\\prime }: \\left\\lbrace -1,1 \\right\\rbrace ^{m|S|} \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ ) which is a function of the fixed values $\\lbrace X_j : j \\notin S\\rbrace $ (respectively, $\\lbrace Y_j : j \\notin S\\rbrace $ ).","Let $R^{\\prime }(X_S, Y_S) = A^{\\prime }(X_S) B^{\\prime }(Y_S)$ .","Continuing from the above we obtain, $\\textnormal {\\textnormal {disc}}_{\\lambda }(h \\circ G, R)&\\le 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\left|\\sum _{X_j,Y_j : j \\notin S} \\prod _{j \\notin S}\\mu (X_j,Y_j) \\left(\\sum _{X_i,Y_i : i \\in S} R^{\\prime }(X_S, Y_S) \\prod _{i \\in S}G\\mu (X_i, Y_i)\\right) \\right|\\\\& \\le 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\left| \\sum _{X_j,Y_j : j \\notin S} \\prod _{j \\notin S}\\mu _j(X_j,Y_j) \\left(\\textnormal {disc}_{\\mu ^{\\otimes |S|}}(\\mathsf {XOR}_{|S|} \\circ G)\\right) \\right| \\\\&= 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\cdot \\textnormal {disc}_{\\mu ^{\\otimes |S|}}(\\mathsf {XOR}_{|S|} \\circ G)\\\\&\\le \\sum _{S \\subseteq [n], |S| \\ge d} \\left(8 \\textnormal {disc}_{\\mu }(G)\\right)^{|S|}\\\\&= \\sum _{k = d}^n \\binom{n}{k} \\left(8 \\textnormal {disc}_{\\mu }(G)\\right)^{k} \\le \\sum _{k = d}^n \\left( \\frac{8en}{k} \\textnormal {disc}_{\\mu }(G)\\right)^{k}\\\\&\\le \\sum _{k = d}^n \\left(\\textnormal {disc}_{\\mu }(G)^{\\beta }\\right)^{k}\\\\&\\le \\frac{\\textnormal {disc}_{\\mu }(G)^{d \\cdot \\beta }}{1 - \\textnormal {disc}_{\\mu }(G)^{\\beta }}$ Since $\\textnormal {disc}_{\\mu }(G) = o(1)$ by assumption ($\\textnormal {disc}_{\\mu }(G) = 1 - \\Omega (1)$ suffices, but we assume $\\textnormal {disc}_{\\mu }(G) = o(1)$ in the statement for readability) and $\\beta \\in (0, 1)$ is a constant, we have $1 - \\textnormal {disc}_\\mu (G)^\\beta = \\Omega (1)$ , and hence $\\textnormal {disc}_{\\lambda }(h \\circ G) = O(\\textnormal {disc}_{\\mu }(G)^{d \\beta }).$ Hence, $\\mathsf {Q}^{cc}_{1/2 - 2/5}(r \\circ G)&\\ge \\log \\left(\\frac{1/3 + 4/5 - 1}{\\textnormal {disc}_{\\lambda }(h \\circ G)}\\right) \\\\&= \\Omega \\left( \\log \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)^{d\\beta }}\\right) \\right) \\\\&= \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left( \\frac{1}{\\textnormal {disc}_{\\mu }(g)}\\right)\\right) .$ The theorem follows since $\\mathsf {Q}^{cc}(F) = \\Theta (\\mathsf {Q}^{cc}_{\\varepsilon }(F))$ for all constants $\\varepsilon \\in (0, 1/2)$ ."]],"string":"[\n [\n \"Symmetry and Quantum Query-to-Communication Simulation\"\n ],\n [\n \"Abstract Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) < 2 R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively.\",\n \"Chakraborty et al.\",\n \"(CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required.\",\n \"We improve upon their result in several ways.\",\n \"We show that the log n overhead is not required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05).\",\n \"This upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication.\",\n \"To prove this, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function.\",\n \"In view of our first result, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive.\",\n \"We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2.\",\n \"We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.\"\n ],\n [\n \"Motivation and main results\",\n \"The classical model of communication complexity was introduced by Yao [33], who also subsequently introduced its quantum analogue [34].\",\n \"Communication complexity has important applications in several disciplines, in particular for lower bounds on circuits, data structures, streaming algorithms, and many other complexity measures (see, for example, [22] and the references therein).\",\n \"A natural way to derive a communication problem from a Boolean function $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is via composition.\",\n \"Let $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a function and let $G: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a “two-party function”.\",\n \"Then $F = f \\\\circ G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nj} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nk} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ denotes the function corresponding to the communication problem in which Alice is given input $X = (X_1, \\\\dots , X_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nj}$ , Bob is given $Y = (Y_1, \\\\dots , Y_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nk}$ , and their task is to compute $F(X, Y) = f(G(X_1, Y_1), \\\\dots , G(X_n, Y_n))$ .\",\n \"Many well-known functions in communication complexity are derived in this way, such as Set-Disjointness ($\\\\mathsf {DISJ}_n := \\\\mathsf {NOR}_n \\\\circ \\\\mathsf {AND}_2$ ), Inner Product ($\\\\mathsf {IP}_{n} : = \\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {AND}_2$ ) and Equality ($\\\\mathsf {EQ}_{n} : = \\\\mathsf {NOR}_n \\\\circ \\\\mathsf {XOR}_2$ ).\",\n \"A natural approach to obtain efficient quantum communication protocols for $f \\\\circ G$ is to “simulate” a quantum query algorithm for $f$ , where a query to the $i$ th input bit of $f$ is simulated by a communication protocol that computes $G(X_i,Y_i)$ .\",\n \"Buhrman, Cleve and Wigderson [6] observed that such a simulation is indeed possible if $G$ is bitwise AND or XOR.\",\n \"Theorem 1.1 ([6]) For every Boolean function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ , we have $\\\\mathsf {Q}^{cc}\\\\left(f \\\\circ \\\\square \\\\right) = O\\\\left(\\\\mathsf {Q}(f) \\\\log n\\\\right).$ Here $\\\\mathsf {Q}(f)$ denotes the bounded-error quantum query complexity of $f$ , and $\\\\mathsf {Q}^{cc}(f\\\\circ \\\\square )$ denotes the bounded-error quantum communication complexity for computing $f\\\\circ \\\\square $ .\",\n \"Throughout this paper, we refer to Theorem REF as the BCW simulation.\",\n \"[6] used this, for instance, to show that the bounded-error quantum communication complexity of the Set-Disjointness function is $O(\\\\sqrt{n}\\\\log n)$ , using Grover's $O(\\\\sqrt{n})$ -query search algorithm [16] for the $\\\\mathsf {NOR}_n$ function.\",\n \"It is folklore in the classical world that the analogous simulation does not incur a $\\\\log n$ factor overhead.\",\n \"That is, $\\\\mathsf {R}^{cc}\\\\left(f \\\\circ \\\\square \\\\right) \\\\le 2\\\\mathsf {R}(f),$ where $\\\\mathsf {R}(f)$ denotes the bounded-error randomized query complexity of $f$ and $\\\\mathsf {R}^{cc}(f\\\\circ \\\\square )$ denotes the bounded-error randomized communication complexity for computing $f\\\\circ \\\\square $ .\",\n \"Thus, a natural question is whether the multiplicative $\\\\log n$ blow-up in the communication cost in the BCW simulation is necessary.\",\n \"Chakraborty et al.\",\n \"[12] answered this question and exhibited a total function for which the $\\\\log n$ blow-up is indeed necessary when $\\\\mathsf {XOR}_2$ is the inner function.\",\n \"Theorem 1.2 ([12]) There exists a function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ such that $\\\\mathsf {Q}^{cc,*}(f \\\\circ \\\\mathsf {XOR}_2) = \\\\Omega (\\\\mathsf {Q}(f) \\\\log n).$ Here $\\\\mathsf {Q}^{cc,*}(F)$ denotes the bounded-error quantum communication complexity of two-party function $F$ when Alice and Bob shared an entangled state at the start of the protocol for free.\",\n \"This gives rise to the following basic question: is there a natural class of functions for which the $\\\\log n$ overhead in the BCW simulation is not required?\",\n \"Improving upon Høyer and de Wolf [17], Aaronson and Ambainis [1] showed that for the canonical problem of Set-Disjointness, the $\\\\log n$ overhead in the BCW simulation can be avoided.\",\n \"Since the outer function $\\\\mathsf {NOR}_n$ is symmetric (i.e., it only depends on the Hamming weight of its input, its number of $-1$ s), a natural question is whether the $\\\\log n$ overhead can be avoided whenever the outer function is symmetric.\",\n \"Our first result gives a positive answer to this question.\",\n \"Theorem 1.3 For every symmetric Boolean function $f:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n\\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and two-party function $G:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j\\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ , we have $\\\\mathsf {Q}^{cc,*}(f \\\\circ G)=O(\\\\mathsf {Q}(f)\\\\mathsf {Q}_E^{cc}(G)).$ Here $\\\\mathsf {Q}_E^{cc}(G)$ denotes the exact quantum communication complexity of $G$ , where the error probability is 0.\",\n \"In particular, if $G\\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ then $\\\\mathsf {Q}_E^{cc}(G)=1$ and hence $\\\\mathsf {Q}^{cc,*}(f \\\\circ G)=O(\\\\mathsf {Q}(f))$ .\",\n \"Remark 1.4 If $\\\\mathsf {Q}(f)=\\\\Theta (\\\\sqrt{tn})$ , then our protocol in the proof of Theorem REF starts from a shared entangled state of $O(t\\\\log n)$ EPR-pairs.\",\n \"Note that if $t\\\\le n\\\\mathsf {Q}_E^{cc}(G)^2/(\\\\log n)^2$ (this condition holds for instance if $\\\\mathsf {Q}_E^{cc}(G)\\\\ge \\\\log n$ ) then this number of EPR-pairs is no more than the amount of communication and hence might as well be established in the first message, giving asymptotically the same upper bound $\\\\mathsf {Q}^{cc}(f \\\\circ G)=O(\\\\mathsf {Q}(f)\\\\mathsf {Q}_E^{cc}(G))$ for the model without prior entanglement.\",\n \"The next question one might ask is whether one can weaken the notion of symmetry required in Theorem REF .\",\n \"A natural generalization of the class of symmetric functions is the class of transitive-symmetric functions.\",\n \"A function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is said to be transitive-symmetric if for all $i, j \\\\in [n]$ , there exists $\\\\sigma \\\\in S_n$ such that $\\\\sigma (i) = j$ , and $f(x) = f(\\\\sigma (x))$ for all $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"Henceforth we refer to transitive-symmetric functions as simply transitive functions.\",\n \"Can the $\\\\log n$ overhead in the BCW simulation be avoided whenever the outer function is transitive?\",\n \"We give a negative answer to this question in a strong sense: the $\\\\log n$ overhead is still necessary even when we allow the quantum communication protocol an error probability that can be arbitrarily close to $1/2$ .\",\n \"Theorem 1.5 There exists a transitive and total function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , such that $\\\\mathsf {UPP}^{cc}(f \\\\circ \\\\square ) & = \\\\Omega (\\\\mathsf {Q}(f) \\\\log n)$ for every $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ .\",\n \"Here $\\\\mathsf {UPP}^{cc}(f \\\\circ \\\\square )$ denotes the unbounded-error quantum communication complexity of $f \\\\circ \\\\square $ (adding “quantum” here only changes the communication complexity by a constant factor).\",\n \"The unbounded-error model of communication was introduced by Paturi and Simon [30] and is the strongest communication complexity model against which we know how to prove explicit lower bounds.\",\n \"This model is known to be strictly stronger than the bounded-error quantum model.\",\n \"For instance, the Set-Disjointness function on $n$ inputs requires $\\\\Omega (n)$ bits or $\\\\Omega (\\\\sqrt{n})$ qubits of communication in the bounded-error model, but only requires $O(\\\\log n)$ bits of communication in the unbounded-error model.\",\n \"In fact, it follows from a recent result of Hatami, Hosseini and Lovett [18] that there exists a function $F : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n\\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ with $\\\\mathsf {Q}^{cc,*}(F) = \\\\Omega (n)$ while $\\\\mathsf {UPP}^{cc}(F) = O(1)$ .\",\n \"Theorem REF and Theorem REF clearly demonstrate the role of symmetry in determining the presence of the $\\\\log n$ overhead in the BCW query-to-communication simulation: this overhead is absent for symmetric functions (Theorem REF ), but present for a transitive function even when the model of communication under consideration is as strong as the unbounded-error model (Theorem REF ).\",\n \"We also give a general recipe to construct functions for which the $\\\\log n$ overhead is required in the BCW simulation in the bounded-error communication model (see Theorem REF ).\"\n ],\n [\n \"Overview of our approach and techniques\",\n \"In this section we discuss the ideas that go into the proofs of Theorem REF and Theorem REF .\"\n ],\n [\n \"Communication complexity upper bound for symmetric functions\",\n \"To prove Theorem REF we use the well-known fact that every symmetric function $f$ has an interval around Hamming weight $n/2$ where the function is constant; for $\\\\mathsf {NOR}_n$ the length of this interval would be essentially $n$ , while for $\\\\mathsf {PARITY}_n$ it would be 1.\",\n \"To compute $f$ , it suffices to either determine that the Hamming weight of the input lies in that interval (because the function value is the same throughout that interval) or to count the Hamming weight exactly.\",\n \"For two-party functions of the form $f\\\\circ G$ , we want to do this type of counting on the $n$ -bit string $z=(G(X_1,Y_1),\\\\ldots ,G(X_n,Y_n))\\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"We show how this can be done with $O(\\\\mathsf {Q}(f)\\\\,\\\\mathsf {Q}^{cc}_E(G))$ qubits of communication if we had a quantum protocol that can find $-1$ s in the string $z$ at a cost of $O(\\\\sqrt{n}\\\\,\\\\mathsf {Q}^{cc}_E(G))$ qubits.\",\n \"Such a protocol was already given by Aaronson and Ambainis for the special case where $G=\\\\mathsf {AND}_2$ for their optimal quantum protocol for Set-Disjointness, as a corollary of their quantum walk algorithm for search on a grid [1].\",\n \"In this paper we give an alternative $O(\\\\sqrt{n}\\\\,\\\\mathsf {Q}^{cc}_E(G))$ -qubit protocol.\",\n \"This implies the result of Aaronson and Ambainis as a special case, but it is arguably simpler and may be of independent interest.\",\n \"Our protocol can be viewed as an efficient distributed implementation of amplitude amplification with faulty components.\",\n \"In particular, we replace the usual reflection about the uniform superposition by an imperfect reflection about the $n$ -dimensional maximally entangled state ($=\\\\log n$ EPR-pairs if $n$ is a power of 2).\",\n \"Such a reflection would require $O(\\\\log n)$ qubits of communication to implement perfectly, but can be implemented with small error using only $O(1)$ qubits of communication, by invoking the efficient protocol of Aharonov et al.\",\n \"[4] that tests whether a given bipartite state equals the $n$ -dimensional maximally entangled state.\",\n \"Still, at the start of this protocol we need to assume (or establish by means of quantum communication) a shared state of $\\\\log n$ EPR-pairs.\",\n \"If $\\\\mathsf {Q}(f)=\\\\Theta (\\\\sqrt{tn})$ then our protocol for $f\\\\circ G$ will run the $-1$ -finding protocol $O(t)$ times, which accounts for our assumption that we share $O(t\\\\log n)$ EPR-pairs at the start of the protocol.\"\n ],\n [\n \"Communication complexity lower bound for transitive functions\",\n \"For proving Theorem REF , we exhibit a transitive function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ whose bounded-error quantum query complexity is $O(n)$ and the unbounded-error communication complexity of $f \\\\circ \\\\square $ is $\\\\Omega (n \\\\log n)$ for $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ .\",\n \"Function construction and transitivity: For the construction of $f$ we first require the definition of Hadamard codewords.\",\n \"The Hadamard codeword of $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , denoted by $H(s) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ , is a list of all parities of $s$ .\",\n \"See Figure REF for a graphical visualization of $f$ .\",\n \"Figure: If the inputs to the jj-th h 𝖨𝖯 logn h_{\\\\mathsf {IP}_{\\\\log n}} are the Hadamard codewords H(s j )H(s_j) and H(t j )H(t_j) for all j∈[n]j \\\\in [n] and some s j ,t j ∈-1,1 logn s_j, t_j \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}, then f=𝖯𝖠𝖱𝖨𝖳𝖸(𝖨𝖯 logn (s 1 ,t 1 ),⋯,𝖨𝖯 logn (s n ,t n ))f = \\\\mathsf {PARITY}(\\\\mathsf {IP}_{\\\\log n}(s_1,t_1), \\\\dots , \\\\mathsf {IP}_{\\\\log n}(s_n, t_n)).\",\n \"If there exists at least one j∈[n]j \\\\in [n] for which either x j1 ,⋯,x jn x_{j1}, \\\\dots , x_{jn} or y j1 ,⋯,y jn y_{j1}, \\\\dots , y_{jn} is not a Hadamard codeword, then ff outputs -1-1.Using properties of $\\\\mathsf {IP}$ and Hadamard codewords, and the symmetry of $\\\\mathsf {PARITY}_n$ , we are able to show that $f$ is transitive (see Claim REF ).\",\n \"Query upper bound: The query upper bound of $O(n)$ follows along the lines of [12], using the Bernstein-Vazirani algorithm to decode the Hadamard codewords, and Grover's algorithm to check that they actually are Hadamard codewords.\",\n \"This approach was in turn inspired by a query upper bound due to Ambainis and de Wolf [3].\",\n \"See the proof of Theorem REF for the query algorithm and its analysis.\",\n \"Communication lower bound: Towards the unbounded-error communication lower bound, we first recall that each input block of $f$ equals $\\\\mathsf {IP}_{\\\\log n}$ if the inputs to each block are promised to be Hadamard codewords.\",\n \"Hence $f$ equals $\\\\mathsf {IP}_{n \\\\log n}$ under this promise, since $\\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {IP}_{\\\\log n} = \\\\mathsf {IP}_{n \\\\log n}$ .\",\n \"Thus by setting certain inputs to Alice and Bob suitably, $f \\\\circ \\\\square $ is at least as hard as $\\\\mathsf {IP}_{n \\\\log n}$ for $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2, \\\\mathsf {XOR}_2\\\\rbrace $ (for a formal statement, see Lemma REF with $r = \\\\mathsf {PARITY}_n$ and $g = \\\\mathsf {IP}_{\\\\log n}$ ).\",\n \"It is known from a seminal result of Forster [15] that the unbounded-error communication complexity of $\\\\mathsf {IP}_{n \\\\log n}$ equals $\\\\Omega (n \\\\log n)$ , completing the proof of the lower bound.\",\n \"This proof is more general than and arguably simpler than the proof of the lower bound for bounded-error quantum communication complexity in [12].\"\n ],\n [\n \"Other results\",\n \" We give a general recipe for constructing a class of functions that witness tightness of the BCW simulation where the inner gadget is either $\\\\mathsf {AND}_2$ or $\\\\mathsf {XOR}_2$ .\",\n \"However, the communication lower bound we obtain here is in the bounded-error model in contrast to Theorem REF , where the communication lower bound is proven in the unbounded-error model.\",\n \"The functions $f$ constructed for this purpose are composed functions similar to the construction in Figure REF , except that we are able to use a more general class of functions in place of the outer $\\\\mathsf {PARITY}$ function, and also a more general class of functions in place of the inner $\\\\mathsf {IP}_{\\\\log n}$ functions.\",\n \"See Figure REF and its caption for an illustration and a more precise definition.\",\n \"Figure: In this figure, G:-1,1 log ×-1,1 logn →-1,1G:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log } \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace .\",\n \"If the inputs to the jj-th h G h_{G} are Hadamard codewords in ±H(s j )\\\\pm H(s_j) and ±H(t j )\\\\pm H(t_j) for all j∈[n]j \\\\in [n] and some s j ,t j ∈-1,1 logn s_j, t_j \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}, then f=r(G(s 1 ,t 1 ),⋯,G(s n ,t n ))f = r(G(s_1,t_1), \\\\dots , G(s_n, t_n)).\",\n \"If there exists at least one j∈[n]j \\\\in [n] for which either x j1 ,⋯,x jn x_{j1}, \\\\dots , x_{jn} or y j1 ,⋯,y jn y_{j1}, \\\\dots , y_{jn} is not a Hadamard codeword, then ff outputs -1-1.We require some additional constraints on the outer and inner functions.\",\n \"First, the approximate degree of $r$ should be $\\\\Omega (n)$ .\",\n \"Second, the discrepancy of $G$ should be small with respect to some “balanced” probability distribution (see Definition REF and Definition REF for formal definitions of these notions).\",\n \"Theorem 1.6 (Informal version of Theorem REF ) Let $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be such that $\\\\widetilde{\\\\textnormal {deg}}(r) = \\\\Omega (n)$ and let $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a total function.\",\n \"Define $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^{2}} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ as in Figure REF .\",\n \"If there exists $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\mathbb {R}$ that is a balanced probability distribution with respect to $G$ and $\\\\textnormal {disc}_{\\\\mu }(G) = n^{-\\\\Omega (1)}$ , then $\\\\mathsf {Q}(f) & = O(n),\\\\\\\\\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\square ) & = \\\\Omega (n \\\\log n),$ for every $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ .\",\n \"The query upper bound follows along similar lines as that of Theorem REF .\",\n \"For the lower bound, we first show via a reduction that for $f$ as described in Figure REF and $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2, \\\\mathsf {XOR}_2\\\\rbrace $ , the communication problem $f \\\\circ \\\\square $ is at least as hard as $r \\\\circ G$ (see Lemma REF ).\",\n \"This part of the lower bound proof is the same as in the proof of Theorem REF .\",\n \"For the hardness of $r \\\\circ G$ (which in the case of Theorem REF turned out to be $\\\\mathsf {IP}_{n \\\\log n}$ , for which Forster's theorem yields an unbounded-error communication lower bound), we are able to use a theorem implicit in a work of Lee and Zhang [25].\",\n \"This theorem gives a lower bound on the bounded-error communication complexity of $r \\\\circ G$ in terms of the approximate degree of $r$ and the discrepancy of $G$ under a balanced distribution.\",\n \"For completeness, we provide an explicit proof in Appendix .\",\n \"We recover the result of Chakraborty et al.\",\n \"(Theorem REF ) using a more general technique, and additionally show that $\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\mathsf {AND}_2) = \\\\Omega (\\\\mathsf {Q}(f) \\\\log n)$ , where $f$ is as in Theorem REF .\",\n \"This is discussed in Appendix .\",\n \"For a Boolean function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , let $\\\\log \\\\Vert \\\\widehat{f}\\\\Vert _{1, 1/3}$ denote the log-approximate-spectral norm of $f$ , and $\\\\widetilde{\\\\textnormal {deg}}_{1/3}(f)$ denote its approximate degree.\",\n \"As an approach towards proving the Fourier entropy-influence conjecture, Arunachalam et al.\",\n \"[2] asked whether it is true for all total functions $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ that $\\\\log \\\\Vert \\\\widehat{f}\\\\Vert _{1, 1/3} = O(\\\\widetilde{\\\\textnormal {deg}}_{1/3}(f))$ .\",\n \"Chakraborty et al.\",\n \"gave a negative answer to this question by showing that for the function $f$ in Theorem REF , $\\\\log \\\\Vert \\\\widehat{f}\\\\Vert _{1, 1/3} = \\\\Omega (\\\\widetilde{\\\\textnormal {deg}}_{1/3}(f) \\\\log n)$ .\",\n \"They also showed that the $\\\\log n$ overhead is not required for the class of symmetric functions [12].\",\n \"In this paper we complement this result by showing that the $\\\\log n$ overhead is required for the class of transitive functions.\",\n \"This is discussed in Appendix .\"\n ],\n [\n \"Organization\",\n \"Section introduces some notation and preliminaries.\",\n \"In Section we construct our new one-sided error protocol for finding solutions in the string $z=(G(X_1,Y_1),\\\\ldots ,G(X_n,Y_n))\\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ , as a corollary of our distributed version of amplitude amplification.\",\n \"In Section we prove Theorem REF , which shows that the $\\\\log n$ overhead in the BCW simulation can be avoided when the outer function is symmetric; this relies on the protocol from Section .\",\n \"In Section we prove Theorem REF and Theorem REF , which are our results regarding necessity of the $\\\\log n$ overhead in the BCW simulation in the unbounded-error and the bounded-error models of communication, respectively.\",\n \"In Appendix we show that the function constructed in [12] also requires a $\\\\log n$ overhead in the BCW simulation when the inner function is $\\\\mathsf {AND}_2$ .\",\n \"In Appendix we exhibit a transitive function $f$ on $n$ bits for which $\\\\log (\\\\Vert \\\\widehat{f}\\\\Vert _{1, 1/3}) = \\\\Omega (\\\\widetilde{\\\\textnormal {deg}}(f) \\\\log n)$ .\",\n \"Finally in Appendix we prove Theorem REF , which shows a lower bound on the bounded-error quantum communication complexity of $f \\\\circ G$ in terms of the approximate degree of $f$ and the discrepancy of $G$ under balanced distributions.\"\n ],\n [\n \"Notation and preliminaries\",\n \"Without loss of generality, we assume $n$ to be a power of 2 in this paper, unless explicitly stated otherwise.\",\n \"All logarithms in this paper are base 2.\",\n \"Let $S_n$ denote the symmetric group over the set $[n]=\\\\lbrace 1,\\\\ldots ,n\\\\rbrace $ .\",\n \"For a string $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ and $\\\\sigma \\\\in S_n$ , let $\\\\sigma (x)$ denote the string $x_{\\\\sigma (1)}, \\\\dots , x_{\\\\sigma (n)} \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"Consider an arbitrary but fixed bijection between subsets of $[\\\\log n]$ and elements of $[n]$ .\",\n \"For a string $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , we abuse notation and also use $s$ to denote the equivalent element of $[n]$ .\",\n \"The view we take will be clear from context.\",\n \"For a string $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ and set $S \\\\subseteq [n]$ , define the string $x_S \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{S}$ to be the restriction of $x$ to the coordinates in $S$ .\",\n \"Let $1^n$ and $(-1)^n$ denote the $n$ -bit string $(1, 1, \\\\dots , 1)$ and $(-1, -1, \\\\dots , -1)$ , respectively.\"\n ],\n [\n \"Boolean functions\",\n \"For strings $x, y \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ , let $\\\\langle x, y \\\\rangle $ denote the inner product (mod 2) of $x$ and $y$ .\",\n \"That is, $\\\\langle x, y \\\\rangle = \\\\prod _{i = 1}^n (x_i \\\\wedge y_i).$ For every positive integer $n$ , let $\\\\mathsf {PARITY}_n : \\\\lbrace -1,1\\\\rbrace ^n \\\\rightarrow \\\\lbrace -1,1\\\\rbrace $ be defined as: $\\\\mathsf {PARITY}_n(x_1, \\\\dots , x_n) = \\\\prod _{i \\\\in [n]} x_i.$ Definition 2.1 (Symmetric functions) A function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is symmetric if for all $\\\\sigma \\\\in S_n$ and for all $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ we have $f(x) = f(\\\\sigma (x))$ .\",\n \"Definition 2.2 (Transitive functions) A function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is transitive if for all $i,j \\\\in [n]$ there exists a permutation $\\\\sigma \\\\in S_n$ such that $\\\\sigma (i) = j$ , and $f(x) = f(\\\\sigma (x))$ for all $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"We next discuss function composition.\",\n \"For total functions $f, g$ , let $f \\\\circ g$ denote the standard composition of the functions $f$ and $g$ .\",\n \"We also require the following notion of composition of a total function $f$ with a partial function $g$ .\",\n \"Definition 2.3 (Composition with partial functions) Let $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a total function and let $g : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{m} \\\\rightarrow \\\\left\\\\lbrace -1, 1, \\\\star \\\\right\\\\rbrace $ be a partial function.\",\n \"Let $f ~\\\\widetilde{\\\\circ }~g : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nm} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ denote the total function that is defined as follows on input $(X_1, \\\\dots , X_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nm}$ , where $X_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m$ for all $i \\\\in [n]$ .\",\n \"$f ~\\\\widetilde{\\\\circ }~g (X_1, \\\\dots , X_n) = {\\\\left\\\\lbrace \\\\begin{array}{ll}f(g(X_1), \\\\dots , g(X_n)) & \\\\text{if}~g(X_i) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ~\\\\text{for all}~i \\\\in [n],\\\\\\\\-1 & \\\\text{otherwise}.\\\\end{array}\\\\right.\",\n \"}$ That is, we use $f ~\\\\widetilde{\\\\circ }~g$ to denote the total function that equals $f \\\\circ g$ on inputs when each copy of $g$ outputs a value in $\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , and equals $-1$ otherwise.\",\n \"Definition 2.4 (Approximate degree) For every $\\\\varepsilon \\\\ge 0$ , the $\\\\varepsilon $ -approximate degree of a function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is defined to be the minimum degree of a real polynomial $p : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ that uniformly approximates $f$ to error $\\\\varepsilon $ .\",\n \"That is, $\\\\widetilde{\\\\textnormal {deg}}_{\\\\varepsilon }(f) = \\\\min \\\\left\\\\lbrace \\\\textnormal {deg}(p) : |p(x) - f(x)| \\\\le \\\\varepsilon ~\\\\text{for all}~x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\right\\\\rbrace .$ Unless specified otherwise, we drop $\\\\varepsilon $ from the subscript and assume $\\\\varepsilon = 1/3$ .\",\n \"We assume familiarity with quantum computing [26], and use $\\\\mathsf {Q}_{\\\\varepsilon }(f)$ to denote the $\\\\varepsilon $ -error query complexity of $f$ .\",\n \"Unless specified otherwise, we drop $\\\\varepsilon $ from the subscript and assume $\\\\varepsilon = 1/3$ .\",\n \"Theorem 2.5 ([5]) Let $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a function.\",\n \"Then $\\\\mathsf {Q}(f) \\\\ge \\\\widetilde{\\\\textnormal {deg}}(f)/2$ .\"\n ],\n [\n \"Communication complexity\",\n \"We assume familiarity with communication complexity [22].\",\n \"Definition 2.6 (Two-party function) We call a function $G: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ a two-party function to indicate that it corresponds to a communication problem in which Alice is given input $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j$ , Bob is given input $y \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k$ , and their task is to compute $G(x, y)$ .\",\n \"Remark 2.7 Throughout this paper, we use uppercase letters to denote two-party functions, and lowercase letters to denote functions which are not two-party functions.\",\n \"Definition 2.8 (Composition with two-party functions) Let $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a function and let $G: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a two-party function.\",\n \"Then $F = f \\\\circ G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nj} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nk} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ denotes the two-party function corresponding to the communication problem in which Alice is given input $X = (X_1, \\\\dots , X_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nj}$ , Bob is given $Y = (Y_1, \\\\dots , Y_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nk}$ , and their task is compute $F(X, Y) = f(G(X_1, Y_1), \\\\dots , G(X_n, Y_n))$ .\",\n \"Definition 2.9 (Inner Product function) For every positive integer $n$ , define the function $\\\\mathsf {IP}_n : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ by $\\\\mathsf {IP}_n(x,y) = \\\\langle x,y \\\\rangle .$ In other words, $\\\\mathsf {IP}_n = \\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {AND}_2$ .\",\n \"Observation 2.10 For all positive integers $k, t$ , $\\\\mathsf {PARITY}_k \\\\circ \\\\mathsf {IP}_t = \\\\mathsf {IP}_{kt}$ .\",\n \"We also assume familiarity with quantum communication complexity [32].\",\n \"We use $\\\\mathsf {Q}_{\\\\varepsilon }^{cc}(G)$ and $\\\\mathsf {Q}_{\\\\varepsilon }^{cc,*}(G)$ to represent the $\\\\varepsilon $ -error quantum communication complexity of a two-party function $G$ in the models without and with unlimited shared entanglement, respectively.\",\n \"Unless specified otherwise, we drop $\\\\varepsilon $ from the subscript and assume $\\\\varepsilon = 1/3$ .\",\n \"Definition 2.11 (Balanced probability distribution) We call a probability distribution $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ balanced with respect to a function $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ if $\\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} f(x) \\\\mu (x) = 0$ .\",\n \"Definition 2.12 (Discrepancy) Let $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a function and $\\\\lambda $ be a distribution on $\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k$ .\",\n \"For every $S \\\\subseteq \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j$ and $T \\\\subseteq \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k$ , define $\\\\textnormal {disc}_\\\\lambda (S \\\\times T, G) = \\\\left|\\\\sum _{x, y \\\\in S \\\\times T}G(x, y)\\\\lambda (x, y) \\\\right|.$ The discrepancy of $G$ under the distribution $\\\\lambda $ is defined to be $\\\\textnormal {disc}_{\\\\lambda }(G) = \\\\max _{S \\\\subseteq \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j, T \\\\subseteq \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k} \\\\textnormal {disc}_{\\\\lambda }(S \\\\times T, G),$ and the discrepancy of $f$ is defined to be $\\\\textnormal {disc}(G) = \\\\min _{\\\\lambda }\\\\textnormal {disc}_{\\\\lambda }(G).$ Definition 2.13 (Balanced-discrepancy) Let $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a function and $\\\\Lambda $ be the set of all balanced distributions on $\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k$ with respect to $G$ .\",\n \"The balanced-discrepancy of $G$ is defined to be $\\\\textnormal {bdisc}(G) = \\\\min _{\\\\lambda \\\\in \\\\Lambda }\\\\textnormal {disc}_{\\\\lambda }(G).$ The following theorem is implicit in [25].\",\n \"We prove it in Appendix REF for completeness.\",\n \"Theorem 2.14 Let $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be functions.\",\n \"Let $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\mathbb {R}$ be a balanced distribution with respect to $G$ and $\\\\textnormal {disc}_{\\\\mu }(G) = o(1)$ .\",\n \"If $\\\\frac{8en}{\\\\widetilde{\\\\textnormal {deg}}(r)}\\\\le \\\\left(\\\\frac{1}{\\\\textnormal {disc}_{\\\\mu }(G)}\\\\right)^{1-\\\\beta }$ for some constant $\\\\beta \\\\in (0,1)$ , then $\\\\mathsf {Q}^{cc, *}(r \\\\circ G) = \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(r) \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {disc}_{\\\\mu }(G)}\\\\right)\\\\right).$ In particular, $\\\\mathsf {Q}^{cc, *}(r \\\\circ G) = \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(r) \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {bdisc}(G)}\\\\right)\\\\right).$\"\n ],\n [\n \"Hadamard encoding\",\n \"Recall that we index coordinates of $n$ -bit strings by integers in $[n]$ , and also interchangeably by strings in $\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ via the natural correspondence.\",\n \"For $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ , let $-x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ be defined as $(-x)_i = -x_i$ for all $i \\\\in [n]$ .\",\n \"We use the notation $\\\\pm x$ to denote the set $\\\\lbrace x, -x\\\\rbrace $ .\",\n \"Definition 2.15 (Hadamard Codewords) For every positive integer $n$ and $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , let $H(s) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ be defined as $(H(s))_t = \\\\prod _{i: s_i = -1} t_i~\\\\text{for all}~t \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}.$ If $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ is such that $x = H(s)$ for some $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , we say $x$ is a Hadamard codeword corresponding to $s$ .\",\n \"That is, for every $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , there is an $n$ -bit Hadamard codeword corresponding to $s$ .\",\n \"This represents the enumeration of all parities of $s$ .\",\n \"We now define how to encode a two-party total function $G$ on $(\\\\log j + \\\\log k)$ input bits to a partial function $h_G$ on $(j+k)$ input bits, using Hadamard encoding.\",\n \"Definition 2.16 (Hadamardization of functions) Let $j, k \\\\ge 1$ be powers of 2, and let $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a function.\",\n \"Define a partial function $h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{j + k} \\\\rightarrow \\\\left\\\\lbrace -1, 1, \\\\star \\\\right\\\\rbrace $ by $h_G(x, y) = {\\\\left\\\\lbrace \\\\begin{array}{ll}G(s, t) & \\\\text{if}~x \\\\in \\\\pm H(s), y \\\\in \\\\pm H(t)~\\\\text{for some}~s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j}, t \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k}\\\\\\\\\\\\star & \\\\text{otherwise}.\\\\end{array}\\\\right.\",\n \"}$\"\n ],\n [\n \"Additional concepts from quantum computing\",\n \"The Bernstein-Vazirani algorithm [11] is a quantum query algorithm that takes an $n$ -bit string as input and outputs a $(\\\\log n)$ -bit string.\",\n \"The algorithm has the following properties: the algorithm makes one quantum query to the input and if the input $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ satisfies $x\\\\in \\\\pm H(s)$ for some $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , then the algorithm returns $s$ with probability 1.\",\n \"Consider a symmetric Boolean function $f:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n\\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"Define the quantity $\\\\Gamma (f)=\\\\min \\\\lbrace |2k-n+1| : f(x)\\\\ne f(y)\\\\text{ if }|x|=k\\\\text{ and } |y|=k+1\\\\rbrace $ from [29].\",\n \"One can think of $\\\\Gamma (f)$ as essentially the length of the interval of Hamming weights around $n/2$ where $f$ is constant (for example, for the majority and parity functions this would be 1, and for $\\\\mathsf {OR}_n$ this would be $n-1$ ).\",\n \"Theorem 2.17 ([5]) For every symmetric function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , we have $\\\\mathsf {Q}(f) = \\\\Theta (\\\\sqrt{(n-\\\\Gamma (f))n}).$ The upper bound follows from a quantum algorithm that exactly counts the Hamming weight $|x|$ of the input if $|x|\\\\le t$ or $|x|\\\\ge n-t$ for $t=\\\\left\\\\lceil (n-\\\\Gamma (f))/2 \\\\right\\\\rceil $ , and that otherwise learns $|x|$ is in the interval $[t+1,n-t-1]$ (which is an interval around $n/2$ where $f(x)$ is constant).\",\n \"By the definition of $\\\\Gamma (f)$ , this information about $|x|$ suffices to compute $f(x)$ .\",\n \"In Section we use this observation to give an efficient quantum communication protocol for a two-party function $f\\\\circ G$ .\",\n \"We will need a unitary protocol that allows Alice and Bob to implement an approximate reflection about the $n$ -dimensional maximally entangled state $|\\\\psi \\\\rangle =\\\\frac{1}{\\\\sqrt{n}}\\\\sum _{i\\\\in \\\\lbrace 0,1\\\\rbrace ^{\\\\log n}}|i\\\\rangle |i\\\\rangle .$ Ideally, such a reflection would map $|\\\\psi \\\\rangle $ to itself, and put a minus sign in front of all states orthogonal to $|\\\\psi \\\\rangle $ .\",\n \"Doing this perfectly would requires $O(\\\\log n)$ qubits of communication.\",\n \"Fortunately we can derive a cheaper protocol from a test that Aharonov et al.\",\n \"[4] designed, which uses $O(\\\\log (1/\\\\varepsilon ))$ qubits of communication and checks whether a given bipartite state equals $|\\\\psi \\\\rangle $ , with one-sided error probability $\\\\varepsilon $ .\",\n \"By the usual trick of running this protocol, applying a $Z$ -gate to the answer qubit, and then reversing the protocol, we can implement the desired reflection approximately.Possibly with some auxiliary qubits on Alice and Bob's side which start in $|0\\\\rangle $ and end in $|0\\\\rangle $ , except in a part of the final state that has norm at most $\\\\varepsilon $ .\",\n \"A bit more precisely: Theorem 2.18 Let $R_{\\\\psi }=2|\\\\psi \\\\rangle \\\\!\",\n \"\\\\langle \\\\psi |-I$ be the reflection about the maximally entangled state shared between Alice and Bob.\",\n \"There exists a protocol that uses $O(\\\\log (1/\\\\varepsilon ))$ qubits of communication and that implements a unitary $R^\\\\varepsilon _{\\\\psi }$ such that $\\\\left\\\\Vert R^\\\\varepsilon _{\\\\psi } - R_{\\\\psi } \\\\right\\\\Vert \\\\le \\\\varepsilon $ and $R^\\\\varepsilon _{\\\\psi }|\\\\psi \\\\rangle =|\\\\psi \\\\rangle $ .\",\n \"We use $\\\\mathsf {UPP}^{cc}(F)$ to denote unbounded-error quantum communication complexity of two-party function $F$ .\",\n \"It is folklore (see for example [20]) that the unbounded-error quantum communication complexityThe unbounded-error model does not allow shared randomness or prior shared entanglement (which yields shared randomness by measuring) between Alice and Bob, since any two-party function $F$ would have constant communication complexity in that setting.\",\n \"of $F$ equals its classical counterpart up to a factor of at most 2 so it does not really matter much whether we use $\\\\mathsf {UPP}^{cc}$ for classical unbounded-error communication complexity (as it is commonly used) or for quantum unbounded-error complexity.\",\n \"Crucially, for both the complexity of $\\\\mathsf {IP}_n$ is linear in $n$ : Theorem 2.19 ([15]) Let $n$ be a positive integer.\",\n \"Then, $\\\\mathsf {UPP}^{cc}(\\\\mathsf {IP}_n) = \\\\Omega (n).$\"\n ],\n [\n \"Noisy amplitude amplification and a new distributed-search protocol\",\n \"In this section we present a version of quantum amplitude amplification that still works if the reflections involved are not perfectly implemented.\",\n \"In particular, the usual reflection about the uniform superposition will be replaced in the communication setting by an imperfect reflection about the $n$ -dimensional maximally entangled state, based on the communication-efficient protocol of Aharonov et al.\",\n \"[4] for testing whether Alice and Bob share that state.\",\n \"This allows us to avoid the $\\\\log n$ factor that would be incurred if we instead used a BCW-style distributed implementation of standard amplitude amplification, with $O(\\\\log n)$ qubits of communication to implement each query.\",\n \"Our main result in this section is the following general theorem, which allows us to search among a sequence of two-party instances $(X_1,Y_1),\\\\ldots ,(X_n,Y_n)$ for an index $i \\\\in [n]$ where $G(X_i,Y_i)=-1$ , for any two-party function $G$ .\",\n \"Theorem 3.1 Let $G: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a two-party function, $X = (X_1, \\\\dots , X_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nj}$ and $Y = (Y_1, \\\\dots , Y_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nk}$ .\",\n \"Define $z=(G(X_1,Y_1),\\\\ldots ,G(X_n,Y_n))\\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"Assume Alice and Bob start with $\\\\left\\\\lceil \\\\log n \\\\right\\\\rceil $ shared EPR-pairs.\",\n \"There exists a quantum protocol using $O(\\\\sqrt{n}\\\\,\\\\mathsf {Q}_E^{cc}(G))$ qubits of communication that finds (with success probability $\\\\ge 0.99$ ) an $i\\\\in [n]$ such that $z_i=-1$ if such an $i$ exists, and says “no” with probability 1 if no such $i$ exists.\",\n \"If the number of $-1$ s in $z$ is within a factor of 2 from a known integer $t$ , then the communication can be reduced to $O(\\\\sqrt{n/t}\\\\,\\\\mathsf {Q}_E^{cc}(G))$ qubits.\",\n \"Remark 3.2 The $\\\\log n$ shared EPR-pairs that we assume Alice and Bob share at the start could also be established by means of $\\\\log n$ qubits of communication at the start of the protocol.\",\n \"For the result in the first bullet, this additional communication does not change the asymptotic bound.\",\n \"For the result of the second bullet, if $t\\\\le n\\\\mathsf {Q}_E^{cc}(G)^2/(\\\\log n)^2$ , then this additional communication does not change the asymptotic bound either.\",\n \"However, if $t=\\\\omega (n/(\\\\log n)^2)$ and $\\\\mathsf {Q}_E^{cc}(G)=O(1)$ then the quantum communication $O(\\\\sqrt{n/t}\\\\,\\\\mathsf {Q}_E^{cc}(G))$ is $o(\\\\log n)$ and establishing the $\\\\log n$ EPR-pairs by means of a first message makes a difference.\",\n \"As a corollary, we obtain a new $O(\\\\sqrt{n})$ -qubit protocol for the distributed search problem composed with $G=\\\\mathsf {AND}_2$ (whose decision version is the Set-Disjointness problem).\"\n ],\n [\n \"Amplitude amplification with perfect reflections\",\n \"We first describe basic amplitude amplification in a slightly unusual recursive manner, similar to [19].\",\n \"We are dealing with a search problem where some set $\\\\cal G$ of basis states are deemed “good” and the other basis states are deemed “bad.” Let $P_{\\\\cal G}=\\\\sum _{g\\\\in {\\\\cal G}}|g\\\\rangle \\\\!\",\n \"\\\\langle g|$ be the projector onto the span of the good basis states, and $O_{\\\\cal G}=I-2P_{\\\\cal G}$ be the reflection that puts a `$-$ ' in front of the good basis states: $O_{\\\\cal G}|g\\\\rangle =-|g\\\\rangle $ for all basis states $g\\\\in {\\\\cal G}$ , and $O_{\\\\cal G}|b\\\\rangle =|b\\\\rangle $ for all basis states $b\\\\notin {\\\\cal G}$ .\",\n \"Suppose we have an initial state $|\\\\psi \\\\rangle $ which is a superposition of a good state and a bad state: $|\\\\psi \\\\rangle =\\\\sin (\\\\theta )|G\\\\rangle +\\\\cos (\\\\theta )|B\\\\rangle ,$ where $|G\\\\rangle =P_{\\\\cal G}|\\\\psi \\\\rangle /\\\\left\\\\Vert P_{\\\\cal G}|\\\\psi \\\\rangle \\\\right\\\\Vert $ and $|B\\\\rangle =(I-P_{\\\\cal G})|\\\\psi \\\\rangle /\\\\left\\\\Vert (I-P_{\\\\cal G})|\\\\psi \\\\rangle \\\\right\\\\Vert $ .\",\n \"For example in Grover's algorithm, with a search space of size $n$ containing $t$ solutions, the initial state $|\\\\psi \\\\rangle $ would be the uniform superposition, and its overlap (inner product) with the good subspace spanned by the $t$ “good” (sometimes called “marked”) basis states would be $\\\\sin (\\\\theta )=\\\\sqrt{t/n}$ .\",\n \"We'd like to increase the weight of the good state, i.e., move the angle $\\\\theta $ closer to $\\\\pi /2$ .\",\n \"Let $R_\\\\psi $ denote the reflection about the state $|\\\\psi \\\\rangle $ , i.e., $R_\\\\psi |\\\\psi \\\\rangle =|\\\\psi \\\\rangle $ and $R_\\\\psi {|\\\\phi \\\\rangle }=-|\\\\phi \\\\rangle $ for every $|\\\\phi \\\\rangle $ that is orthogonal to $|\\\\psi \\\\rangle $ .\",\n \"Then the algorithm $A_1=R_\\\\psi \\\\cdot O_{\\\\cal G}$ is the product of two reflections, which (in the 2-dimensional space spanned by $|G\\\\rangle $ and $|B\\\\rangle $ ) corresponds to a rotation by an angle $2\\\\theta $ , thus increasing our angle from $\\\\theta $ to $3\\\\theta $ .\",\n \"This is the basic amplitude amplification step.\",\n \"It maps $|\\\\psi \\\\rangle \\\\mapsto A_1|\\\\psi \\\\rangle =\\\\sin (3\\\\theta )|G\\\\rangle +\\\\cos (3\\\\theta )|B\\\\rangle .$ We can now repeat this step recursively, defining $A_2=A_1R_\\\\psi A^*_1\\\\cdot O_{\\\\cal G}\\\\cdot A_1.$ Note that $A_1R_\\\\psi A^*_1$ is a reflection about the state $A_1|\\\\psi \\\\rangle $ .\",\n \"Thus $A_2$ triples the angle between $A_1|\\\\psi \\\\rangle $ and $|B\\\\rangle $ , mapping $|\\\\psi \\\\rangle \\\\mapsto A_2|\\\\psi \\\\rangle =\\\\sin (9\\\\theta )|G\\\\rangle +\\\\cos (9\\\\theta )|B\\\\rangle .$ Continuing recursively in this fashion, define the algorithm $A_{j+1}=A_j R_\\\\psi A_j^*\\\\cdot O_{\\\\cal G}\\\\cdot A_j.$ The last algorithm $A_k$ will map $|\\\\psi \\\\rangle \\\\mapsto A_k|\\\\psi \\\\rangle =\\\\sin (3^k\\\\theta )|G\\\\rangle +\\\\cos (3^k\\\\theta )|B\\\\rangle .$ Hence after $k$ recursive amplitude amplification steps, we have angle $3^k\\\\theta $ .\",\n \"Since we want to end up with angle $\\\\approx \\\\pi /2$ , if we know $\\\\theta $ then we can choose $k=\\\\left\\\\lfloor \\\\log _3(\\\\pi /(2\\\\theta )) \\\\right\\\\rfloor .$ This gives us an angle $3^k\\\\theta \\\\in (\\\\pi /6,\\\\pi /2]$ , so the final state $A_k|\\\\psi \\\\rangle $ has overlap $\\\\sin (\\\\theta _k)>1/2$ with the good state $|G\\\\rangle $ .\",\n \"Let $C_k$ denote the “cost” (in whatever measure, for example query complexity, or communication complexity, or circuit size) of algorithm $A_k$ .\",\n \"Looking at its recursive definition (Equation (REF )), $C_k$ is 3 times $C_{k-1}$ , plus the cost of $R_\\\\psi $ plus the cost of $O_{\\\\cal G}$ .\",\n \"If we just count applications of $O_{\\\\cal G}$ (“queries”), considering $R_\\\\psi $ to be free, then $C_{k+1}=3C_k+1$ .\",\n \"This recursion has the closed form $C_k=\\\\sum _{i=0}^{k-1} 3^i< 3^k$ .\",\n \"With the above choice of $k$ we get $C_k=O(1/\\\\theta )$ .\",\n \"In the case of Grover's algorithm, where $\\\\theta =\\\\arcsin (\\\\sqrt{t/n})\\\\approx \\\\sqrt{t/n}$ , the cost is $C_k=O(\\\\sqrt{n/t})$ .\"\n ],\n [\n \"Amplitude amplification with imperfect reflections\",\n \"Now we consider the situation where we do not implement the reflections $R_\\\\psi $ perfectly, but instead implement another unitary $R^\\\\varepsilon _\\\\psi $ at operator-norm distance $\\\\left\\\\Vert R^\\\\varepsilon _\\\\psi -R_\\\\psi \\\\right\\\\Vert \\\\le \\\\varepsilon $ from $R_\\\\psi $ , with the additional property that $R^\\\\varepsilon _\\\\psi |\\\\psi \\\\rangle =|\\\\psi \\\\rangle $ (this one-sided error property will be important for the proof).\",\n \"We can control this error $\\\\varepsilon $ , but smaller $\\\\varepsilon $ will typically correspond to higher cost of $R^\\\\varepsilon _\\\\psi $ .\",\n \"The reflection $O_{\\\\cal G}$ will still be implemented perfectly below.\",\n \"We again start with the initial state $|\\\\psi \\\\rangle =\\\\sin (\\\\theta )|G\\\\rangle +\\\\cos (\\\\theta )|B\\\\rangle .$ For errors $\\\\varepsilon _1,\\\\ldots ,\\\\varepsilon _k$ that we will specify later, recursively define the following algorithms.\",\n \"$A_1=R^{\\\\varepsilon _1}_\\\\psi \\\\cdot O_{\\\\cal G}\\\\mbox{~~and~~}A_{j+1}=A_jR^{\\\\varepsilon _{j+1}}_\\\\psi A_j^*\\\\cdot O_{\\\\cal G}\\\\cdot A_j.$ These algorithms will map the initial state as follows: $|\\\\psi \\\\rangle \\\\mapsto |\\\\psi _j\\\\rangle =A_j|\\\\psi \\\\rangle =\\\\sin (3^j\\\\theta )|G\\\\rangle +\\\\cos (3^j\\\\theta )|B\\\\rangle +|E_j\\\\rangle ,$ where $|E_j\\\\rangle $ is some unnormalized error state defined by the above equation; its norm $\\\\eta _j$ quantifies the extent to which we deviate from perfect amplitude amplification.\",\n \"Our goal here is to upper bound this $\\\\eta _j$ .\",\n \"In order to see how $\\\\eta _j$ can grow, let us see how $A_jR^{\\\\varepsilon _{j+1}}_\\\\psi A_j^*\\\\cdot O_{\\\\cal G}$ acts on $\\\\sin (3^j\\\\theta )|G\\\\rangle +\\\\cos (3^j\\\\theta )|B\\\\rangle $ (we'll take into account the effects of the error term $|E_j\\\\rangle $ later).\",\n \"If $R^{\\\\varepsilon _{j+1}}_\\\\psi $ were equal to $R_\\\\psi $ , then we would have one perfect round of amplitude amplification and obtain $\\\\sin (3^{j+1}\\\\theta )|G\\\\rangle +\\\\cos (3^{j+1}\\\\theta )|B\\\\rangle $ ; but since $R^{\\\\varepsilon _{j+1}}_\\\\psi $ is only $\\\\varepsilon _{j+1}$ -close to $R_\\\\psi $ , additional errors can appear.\",\n \"First we apply $O_{\\\\cal G}$ , which negates $|G\\\\rangle $ and hence changes the state to $-\\\\sin (3^j\\\\theta )|G\\\\rangle +\\\\cos (3^j\\\\theta )|B\\\\rangle =|\\\\psi _j\\\\rangle -|E_j\\\\rangle -2\\\\sin (3^j\\\\theta )|G\\\\rangle .$ Second we apply $V=A_jR^{\\\\varepsilon _{j+1}}_\\\\psi A_j^*$ .\",\n \"Let $V^{\\\\prime }=A_jR_\\\\psi A_j^*$ , and note that $V|\\\\psi _j\\\\rangle =V^{\\\\prime }|\\\\psi _j\\\\rangle =|\\\\psi _j\\\\rangle $ and $\\\\left\\\\Vert V^{\\\\prime }-V \\\\right\\\\Vert =\\\\left\\\\Vert R_\\\\psi -R^{\\\\varepsilon _{j+1}}_\\\\psi \\\\right\\\\Vert \\\\le \\\\varepsilon _{j+1}$ .\",\n \"The new state is $V(|\\\\psi _j\\\\rangle -|E_j\\\\rangle -2\\\\sin (3^j\\\\theta )|G\\\\rangle )& = V^{\\\\prime }(|\\\\psi _j\\\\rangle -|E_j\\\\rangle -2\\\\sin (3^j\\\\theta )|G\\\\rangle ) +(V^{\\\\prime }-V)(|E_j\\\\rangle +2\\\\sin (3^j\\\\theta )|G\\\\rangle )\\\\\\\\& = V^{\\\\prime }(-\\\\sin (3^j\\\\theta )|G\\\\rangle +\\\\cos (3^j\\\\theta )|B\\\\rangle )+(V^{\\\\prime }-V)(|E_j\\\\rangle +2\\\\sin (3^j\\\\theta )|G\\\\rangle )\\\\\\\\& = \\\\sin (3^{j+1}\\\\theta )|G\\\\rangle +\\\\cos (3^{j+1}\\\\theta )|B\\\\rangle +(V^{\\\\prime }-V)(|E_j\\\\rangle +2\\\\sin (3^j\\\\theta )|G\\\\rangle ).$ Putting back also the earlier error term $|E_j\\\\rangle $ from Equation (REF ) (to which the unitary $VO_{\\\\cal G}$ is applied as well), it follows that the new error state is $|E_{j+1}\\\\rangle =|\\\\psi _{j+1}\\\\rangle -(\\\\sin (3^{j+1}\\\\theta )|G\\\\rangle +\\\\cos (3^{j+1}\\\\theta )|B\\\\rangle )=VO_{\\\\cal G}|E_j\\\\rangle +(V^{\\\\prime }-V)(|E_j\\\\rangle +2\\\\sin (3^j\\\\theta )|G\\\\rangle ).$ Its norm is $\\\\eta _{j+1} & \\\\le \\\\left\\\\Vert VO_{\\\\cal G}|E_j\\\\rangle \\\\right\\\\Vert +\\\\left\\\\Vert (V^{\\\\prime }-V)(|E_j\\\\rangle +2\\\\sin (3^j\\\\theta )|G\\\\rangle ) \\\\right\\\\Vert \\\\\\\\& \\\\le \\\\eta _j + \\\\varepsilon _{j+1}(\\\\eta _j+2\\\\sin (3^j\\\\theta ))= (1+\\\\varepsilon _{j+1})\\\\eta _j + 2\\\\varepsilon _{j+1}\\\\sin (3^j\\\\theta ).$ Since $\\\\eta _0=0$ , we can “unfold” the above recursive upper bound to the following, which is easy to verify by induction on $k$ : $\\\\eta _k \\\\le \\\\sum _{j=1}^k\\\\prod _{\\\\ell =j+1}^k(1+\\\\varepsilon _\\\\ell )2\\\\varepsilon _j\\\\sin (3^{j-1}\\\\theta ).$ For each $1 \\\\le j \\\\le k$ , choose $\\\\varepsilon _j=\\\\frac{1}{100\\\\cdot 4^j}.$ Note that $\\\\sigma =\\\\sum _{j=1}^k\\\\varepsilon _k\\\\le 1/300$ .\",\n \"With this choice of $\\\\varepsilon _j$ 's, and the inequalities $1+x\\\\le e^x$ , $e^\\\\sigma \\\\le 1.5$ and $\\\\sin (x)\\\\le x$ for $x \\\\le \\\\pi /2$ (which is the case here), we can upper bound the norm of the error term $|E_k\\\\rangle $ after $k$ iterations (see Equation (REF )) as $\\\\eta _k \\\\le \\\\sum _{j=1}^k e^\\\\sigma 2\\\\varepsilon _j 3^{j-1}\\\\theta \\\\le \\\\frac{3\\\\theta }{400}\\\\sum _{j=1}^k (3/4)^{j-1}\\\\le \\\\frac{3\\\\theta }{100}.$ Accordingly, up to very small error we have done perfect amplitude amplification.\"\n ],\n [\n \"Distributed amplitude amplification with imperfect reflection\",\n \"We will now instantiate the above scheme to the case of distributed search, where our measure of cost is communication, that is, the number of qubits sent between Alice and Bob.\",\n \"Specifically, consider the intersection problem where Alice and Bob have inputs $x\\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ and $y\\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ , respectively.\",\n \"Assume for simplicity that $n$ is a power of 2, so $\\\\log n$ is an integer.\",\n \"Alice and Bob want to find an $i\\\\in \\\\lbrace 0,\\\\ldots ,n-1\\\\rbrace =\\\\lbrace 0,1\\\\rbrace ^{\\\\log n}$ such that $x_i=y_i=-1$ , if such an $i$ exists.\",\n \"The basis states in this distributed problem are $|i\\\\rangle |j\\\\rangle $ , and we define the set of “good” basis states as ${\\\\cal G}=\\\\lbrace |i\\\\rangle |j\\\\rangle \\\\mid x_i=y_j=-1\\\\rbrace ,$ even though we are only looking for $i,j$ where $i=j$ (it's easier to implement $O_{\\\\cal G}$ with this more liberal definition of $\\\\cal G$ ).\",\n \"Our protocol will start with the maximally entangled initial state $|\\\\psi \\\\rangle $ in $n$ dimensions, which corresponds to $\\\\log n$ EPR-pairs: $|\\\\psi \\\\rangle =\\\\frac{1}{\\\\sqrt{n}}\\\\sum _{i\\\\in \\\\lbrace 0,1\\\\rbrace ^{\\\\log n}}|i\\\\rangle |i\\\\rangle =\\\\sin (\\\\theta )|G\\\\rangle +\\\\cos (\\\\theta )|B\\\\rangle ,$ where we assume there are $t$ $i$ 's where $x_i=y_i=-1$ , i.e., $t$ solutions to the intersection problem, so $\\\\theta =\\\\arcsin (\\\\sqrt{t/n}).$ and $|G\\\\rangle =\\\\frac{1}{\\\\sqrt{t}}\\\\sum _{(i,i)\\\\in {\\\\cal G}}|i\\\\rangle |i\\\\rangle .$ It costs $\\\\left\\\\lceil \\\\log n \\\\right\\\\rceil $ qubits of communication between Alice and Bob to establish this initial shared state, or it costs nothing if we assume pre-shared entanglement.\",\n \"Our goal is to end up with a state that has large inner product with $|G\\\\rangle $ .\",\n \"In order to be able to use amplitude amplification, we would like to be able to reflect about the above state $|\\\\psi \\\\rangle $ .\",\n \"However, in general this perfect reflection $R_{\\\\psi }$ costs a lot of communication: Alice would send her $\\\\log n$ qubits to Bob, who would unitarily put a $-1$ in front of all states orthogonal to $|\\\\psi \\\\rangle $ , and then sends back Alice's qubits.\",\n \"This has a communication cost of $O(\\\\log n)$ qubits, which is too much for our purposes.\",\n \"Fortunately, Theorem REF gives us a way to implement a one-sided $\\\\varepsilon $ -error reflection protocol $R^\\\\varepsilon _{\\\\psi }$ that only costs $O(\\\\log (1/\\\\varepsilon ))$ qubits of communication.\",\n \"The reflection $O_{\\\\cal G}$ puts a `$-$ ' in front of the basis states $|i\\\\rangle |j\\\\rangle $ in $\\\\cal G$ .\",\n \"This can be implemented perfectly using only 2 qubits of communication, as follows.\",\n \"To implement this reflection on her basis state $|i\\\\rangle $ , Alice XORs $|x_i\\\\rangle $ into a fresh auxiliary $|0\\\\rangle $ -qubit and sends this qubit to Bob.\",\n \"Bob receives this qubit and applies the following unitary map: $|b\\\\rangle |j\\\\rangle \\\\mapsto y_j^{b}|b\\\\rangle |j\\\\rangle ,~~~b\\\\in \\\\lbrace 0,1\\\\rbrace ,j\\\\in [n].$ He sends back the auxiliary qubit.\",\n \"Alice sets the auxiliary qubit back to $|0\\\\rangle $ by XOR-ing $x_i$ into it.\",\n \"Ignoring the auxiliary qubit (which starts and ends in state $|0\\\\rangle $ ), this maps $|i\\\\rangle |j\\\\rangle \\\\mapsto (-1)^{[x_i=y_j=-1]}|i\\\\rangle |j\\\\rangle $ .\",\n \"Hence we have implemented $O_{\\\\cal G}$ correctly: a minus sign is applied exactly for the good basis states, the ones where $x_i=y_j=-1$ .\",\n \"Now consider the algorithms (more precisely, communication protocols): $A_1=R^{\\\\varepsilon _1}_\\\\psi \\\\cdot O_{\\\\cal G}\\\\mbox{~~and~~}A_{j+1}=A_jR^{\\\\varepsilon _{j+1}}_\\\\psi A_j^*\\\\cdot O_{\\\\cal G}\\\\cdot A_j$ with the choice of $\\\\varepsilon _j$ 's from Equation (REF ).\",\n \"If we pick $k=\\\\left\\\\lfloor \\\\log _3(\\\\pi /(2\\\\theta )) \\\\right\\\\rfloor $ , like in Equation (REF ), then $3^k\\\\theta \\\\in (\\\\pi /6,\\\\pi /2]$ .\",\n \"Hence by Equation (REF ) and Equation (REF ), the inner product of our final state with $|G\\\\rangle $ will be between $\\\\sin (3^k\\\\theta )-3\\\\theta /100\\\\ge 0.4$ and 1.\",\n \"At this point Alice and Bob can measure, and with probability $\\\\ge 0.4^2$ they will each see the same $i$ , with the property that $x_i=y_i=-1$ .\",\n \"From Equation (REF ) and Theorem REF , the recursion for the communication costs of these algorithms is $C_{j+1}=3C_j+O(\\\\log (1/\\\\varepsilon _{j+1}))+2.$ Solving this recurrence with our $\\\\varepsilon _j$ 's from Equation (REF ) and the value of $\\\\theta $ from Equation (REF ) we obtain $C_k=\\\\sum _{j=1}^k 3^{k-j} (O(\\\\log (1/\\\\varepsilon _j))+2)=\\\\sum _{j=1}^k 3^{k-j}O(j)= O(3^k)=O(\\\\sqrt{n/t}).$ Thus, using $O(\\\\sqrt{n/t})$ qubits of communication we can find (with constant success probability) an intersection point $i$ .\",\n \"This also allows us to solve the Set-Disjointness problem (the decision problem whose output is 1 if there is no intersection between $x$ and $y$ ).\",\n \"Note that if the $t$ we used equals the actual number of solutions only up to a factor of 2, the above protocol still has $\\\\Omega (1)$ probability to find a solution, and $O(1)$ repetitions will boost this success probability to $0.99$ .\",\n \"In case we do not even know $t$ approximately, we can use the standard technique of trying exponentially decreasing guesses for $t$ to find an intersection point with communication $O(\\\\sqrt{n})$ .\",\n \"Note that there is no log-factor in the communication complexity, in contrast to the original $O(\\\\sqrt{n}\\\\log n)$ -qubit Grover-based quantum protocol for the intersection problem of Buhrman et al. [6].\",\n \"Aaronson and Ambainis [1] earlier already managed to remove the log-factor, giving an $O(\\\\sqrt{n})$ -qubit protocol for Set-Disjointness as a consequence of their local version of quantum search on a grid graph (which is optimal [31]).\",\n \"We have just reproved this result of [1] in a different and arguably simpler way.\",\n \"The above description is geared towards the intersection problem, where the “inner” function is $G=\\\\mathsf {AND}_2$ : we called a basis state $|i\\\\rangle |j\\\\rangle $ “good” if $x_i=y_j=-1$ .\",\n \"However, this can easily be generalized to the situation where Alice and Bob's respective inputs are $X=(X_1,\\\\ldots ,X_n)$ and $Y=(Y_1,\\\\ldots ,Y_n)$ and we want to find an $i\\\\in [n]$ where $G(X_i,Y_i)=-1$ for some two-party function $G$ , and define the set of “good” basis states as ${\\\\cal G}=\\\\lbrace |i\\\\rangle |j\\\\rangle \\\\mid G(X_i,Y_j)=-1\\\\rbrace $ .We intentionally use the letter `$G$ ' to mean “good” in $\\\\cal G$ and and to refer to the two-party function $G$ , since $G$ determines which basis states $|i\\\\rangle |j\\\\rangle $ are “good.” The only thing that changes in the above is the implementation of the reflection $O_{\\\\cal G}$ , which would now be computed by means of an exact quantum communication protocol for $G(X_i,Y_j)$ , at a cost of $2\\\\mathsf {Q}_E^{cc}(G)$ qubits of communication.The factor of 2 is to reverse the protocol after the phase $G(X_i,Y_j)$ has been added to basis state $|i\\\\rangle |j\\\\rangle $ , in order to set any workspace qubits back to $|0\\\\rangle $ .\",\n \"Note that because we can check (at the expense of another $\\\\mathsf {Q}_E^{cc}(G)$ qubits of communication) whether the output index $i$ actually satisfies $G(X_i,Y_i)=-1$ , we may assume the protocol has one-sided error: it always outputs “no” if there is no such $i$ .\",\n \"This concludes the proof of Theorem REF .\"\n ],\n [\n \"No log-factor needed for symmetric functions\",\n \"In this section we prove Theorem REF from the introduction.\",\n \"Consider a symmetric Boolean function $f:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n\\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"As explained in Section REF , there is an integer $t=\\\\left\\\\lceil (n-\\\\Gamma (f))/2 \\\\right\\\\rceil $ such that we can compute $f$ if we learn the Hamming weight $|z|$ of the input $z\\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ or learn that $|z|\\\\in [t+1,n-t-1]$ .\",\n \"The bounded-error quantum query complexity is $\\\\mathsf {Q}(f)=\\\\Theta (\\\\sqrt{tn})$ (Theorem REF ).\",\n \"For a given two-party function $G:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j\\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k\\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , we have an induced two-party function $F:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nj}\\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nk}\\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ defined as $F(X_1,\\\\ldots ,X_n,Y_1,\\\\ldots ,Y_n)=f(G(X_1,Y_1),\\\\ldots ,G(X_n,Y_n))$ .\",\n \"Define $z=(G(X_1,Y_1),\\\\ldots ,G(X_n,Y_n))\\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n.$ Then $F(X,Y)=f(z)$ only depends on the number of $-1$ s in $z$ .\",\n \"The following theorem allows us to count this number using $O(\\\\mathsf {Q}(f)\\\\,\\\\mathsf {Q}_E^{cc}(G))$ qubits of communication.\",\n \"Theorem 4.1 For every $t$ between 1 and $n/2$ , there exists a quantum protocol that starts from $O(t\\\\log n)$ EPR-pairs, communicates $O(\\\\sqrt{tn}\\\\,\\\\mathsf {Q}_E^{cc}(G))$ qubits, and tells us $|z|$ or tells us that $|z|>t$ , with error probability $\\\\le 1/8$ .\",\n \"Abbreviate $q=\\\\mathsf {Q}_E^{cc}(G)$ .\",\n \"Our protocol has two parts: the first filters out the case $|z|\\\\ge 2t$ , while the second finds all solutions if $|z|<2t$ .\"\n ],\n [\n \"Part 1.\",\n \"First Alice and Bob decide between the case (1) $|z|\\\\ge 2t$ and (2) $|z|\\\\le t$ (even though $|z|$ might also lie in $\\\\lbrace t+1,\\\\ldots ,2t-1\\\\rbrace $ ) using $O(\\\\sqrt{n}q)$ qubits of communication, as follows.\",\n \"They use shared randomness to choose a uniformly random subset $S\\\\subseteq [n]$ of $\\\\left\\\\lceil n/(2t) \\\\right\\\\rceil $ elements.\",\n \"Let $E$ be the event that $z_i=-1$ for at least one $i\\\\in S$ .\",\n \"By standard calculations there exist $p_1,p_2\\\\in [0,1]$ with $p_1=p_2+\\\\Omega (1)$ such that $\\\\Pr [E]\\\\ge p_1$ in case (1) and $\\\\Pr [E]\\\\le p_2$ in case (2).\",\n \"Alice and Bob use the distributed-search protocol from the first bullet of Theorem REF to decide $E$ , with $O(\\\\sqrt{|S|}\\\\,q)=O(\\\\sqrt{n}\\\\,q)$ qubits of communication (plus a negligible $O(\\\\log n)$ EPR-pairs) and error probability much smaller than $p_1-p_2$ .\",\n \"By repeating this a sufficiently large constant number of times and seeing whether the fraction of successes was larger or smaller than $(p_1+p_2)/2$ , they can distinguish between cases (1) and (2) with success probability $\\\\ge 15/16$ .\",\n \"If they conclude they're in case (1) then they output “$|z|>t$ ” and otherwise they proceed to the second part of the protocol.\",\n \"Note that if $|z|\\\\in \\\\lbrace t+1,\\\\ldots ,2t-1\\\\rbrace $ (the “grey zone” in between cases (1) and (2)), then we can't give high-probability guarantees for one output or the other, but concluding (1) leads to the correct output “$|z|>t$ ” in this case, while concluding (2) means the protocol proceeds to Part 2.\",\n \"So either course of action is fine if $|z|\\\\in \\\\lbrace t+1,\\\\ldots ,2t-1\\\\rbrace $ .\",\n \"By Newman's theorem [27] the shared randomness used for choosing $S$ can be replaced by $O(\\\\log n)$ bits of private randomness on Alice's part, which she can send to Bob in her first message, so Part 1 communicates $O(\\\\sqrt{n}\\\\,q)$ qubits in total.\"\n ],\n [\n \"Part 2.\",\n \"We condition on Part 1 successfully filtering out case (1), so from now on assume $|z|<2t$ .\",\n \"Our goal in this second part of the protocol is to find all indices $i$ such that $z_i=-1$ (we call such $i$ “solutions”), with probability $\\\\ge 15/16$ , using $O(\\\\sqrt{tn}\\\\,q)$ qubits of communication.\",\n \"This will imply that the overall protocol is correct with probability $1-1/16-1/16=7/8$ , and uses $O(\\\\sqrt{tn}\\\\,q)$ qubits of communication in total.\",\n \"For an integer $k\\\\ge 1$ , consider the following protocol $P_k$ .\",\n \"[H] Input: An integer $k \\\\ge 1$ $200\\\\sqrt{2^k n}\\\\,q$ qubits have been sent Run the protocol from the last bullet of Theorem REF with $t=2^{k-1}$ .\",\n \"(suppressing some constant factors, assume for simplicity that this uses $\\\\sqrt{n/2^k}\\\\,q$ qubits of communication, $\\\\log n$ shared EPR-pairs at the start, and has probability $\\\\ge 1/100$ to find a solution if the actual number of solutions is in $[t/2,2t]$ ).\",\n \"Alice measures and gets outcome $i\\\\in [n]$ and Bob measures and gets outcome $j\\\\in [n]$ , respectively.\",\n \"Alice sends $i$ to Bob, Bob sends $j$ to Alice.\",\n \"If $i=j$ then they verify that $G(X_i,Y_i)=-1$ by one run of the protocol for $G$ , and if so then they replace $X_i,Y_i$ by some pre-agreed inputs $X^{\\\\prime }_i,Y^{\\\\prime }_i$ , respectively, such that $G(X^{\\\\prime }_i,Y^{\\\\prime }_i)=1$ (this reduces the number of $-1$ s in $z$ by 1) Protocol $P_k$ Claim 4.2 Suppose $|z|\\\\in [2^{k-1},2^k)$ .\",\n \"Then protocol $P_k$ uses $O(\\\\sqrt{2^k n}\\\\,q)$ qubits of communication, assumes $O(2^k\\\\log n)$ EPR-pairs at the start of the protocol, and finds at least $|z|-2^{k-1}+1$ solutions, except with probability $\\\\le 1/2$ .\",\n \"The upper bound on the communication is obvious from the stopping criterion of $P_k$ .\",\n \"As long as the remaining number of solutions is $\\\\ge 2^{k-1}$ , each run of the protocol has probability $\\\\ge 1/100$ to find another solution.\",\n \"Hence the expected number of runs of the protocol of Theorem REF to find at least $|z|-2^{k-1}+1$ solutions, is $\\\\le 100(|z|-2^{k-1}+1)$ .\",\n \"By Markov's inequality, the probability that we haven't yet found $|z|-2^{k-1}+1$ solutions after $\\\\le 200(|z|-2^{k-1}+1)\\\\le 100\\\\cdot 2^k$ runs, is $\\\\le 1/2$ .\",\n \"The communication cost of so many runs is $100\\\\cdot 2^k (\\\\sqrt{n/2^k}\\\\,q + \\\\log n)\\\\le 200 \\\\sqrt{2^k n}\\\\,q$ qubits.\",\n \"Hence by the time that the number of qubits of the stopping criterion have been communicated, we have probability $\\\\ge 1/2$ of having found at least $|z|-2^{k-1}+1$ solutions.\",\n \"The assumed number of EPR-pairs at the start is $\\\\log n$ per run, so $O(2^k\\\\log n)$ in total.\",\n \"Note that if we start with a number of solutions $|z|\\\\in [2^{k-1},2^k)$ , and $P_k$ succeeds in finding at least $|z|-2^{k-1}+1$ new solutions, then afterwards we have $< 2^{k-1}$ solutions left.\",\n \"The following protocol runs these $P_k$ in sequence, pushing down the remaining number of solutions to 0.\",\n \"[H] $k=\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil $ downto 1 Run $P_k$ a total of $r_k=\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil -k+5$ times (replacing all $-1$ s found by $+1s$ in $z$ ).\",\n \"Output the total number of solutions found.\",\n \"Protocol $P$ Claim 4.3 If $|z|<2t$ then protocol $\\\\cal P$ uses $O(\\\\sqrt{tn}\\\\,q)$ qubits of communication, assumes $O(t\\\\log n)$ EPR-pairs at the start of the protocol, and outputs $|z|$ , except with probability $\\\\le 1/16$ .\",\n \"First, by Claim REF , the total number of qubits communicated is $\\\\sum _{k=1}^{\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil } r_k\\\\cdot O(\\\\sqrt{2^k n}\\\\,q)=O(\\\\sqrt{tn}\\\\,q)\\\\cdot \\\\sum _{\\\\ell =0}^{\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil -1} (\\\\ell +5) /\\\\sqrt{2^\\\\ell }=O(\\\\sqrt{tn}\\\\,q),$ where we used a variable substitution $k=\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil -\\\\ell $ .\",\n \"Second, the number of EPR-pairs we're starting from is $\\\\sum _{k=1}^{\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil }r_k\\\\cdot O(2^k\\\\log n)=O(t\\\\log n)\\\\cdot \\\\sum _{\\\\ell =0}^{\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil -1}(\\\\ell +5) /2^\\\\ell =O(t\\\\log n).$ Third, by Claim REF and the fact that we are performing $r_k$ repetitions of $P_k$ , if the $k$ th round of $\\\\cal P$ starts with a remaining number of solutions that is in the interval $[2^{k-1},2^k)$ then that round ends with $<2^{k-1}$ remaining solutions, except with probability at most $1/2^{r_k}$ .\",\n \"By the union bound, the probability that any one of the $\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil $ rounds does not succeed at this, is at most $\\\\sum _{k=1}^{\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil } \\\\frac{1}{2^{r_k}}=\\\\sum _{\\\\ell =0}^{\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil -1}\\\\frac{1}{2^{\\\\ell +5}}\\\\le \\\\frac{1}{16}.$ Since $2^{\\\\left\\\\lceil \\\\log _2 (2t) \\\\right\\\\rceil }\\\\ge 2t$ and we start with $|z|<2t$ , if each round succeeds, then by the end of $\\\\cal P$ there are no remaining solutions left.\",\n \"Thus, the protocol $\\\\cal P$ finds all solutions and learns $|z|$ with probability at least $15/16$ .\",\n \"Part 1 and Part 2 each have error probability $\\\\le 1/16$ , so by the union bound the protocol succeeds except with probability $1/8$ .\",\n \"If $|z|\\\\ge 2t$ then Part 1 outputs the correct answer “$|z|>t$ ”; if $|z|\\\\le t$ then all solutions (and hence $|z|$ ) are found by Part 2; and if $|z|\\\\in \\\\lbrace t+1,\\\\ldots ,2t-1\\\\rbrace $ then either Part 1 already outputs the correct answer “$|z|>t$ ” or the protocol proceeds to Part 2 which then finds all solutions.\",\n \"We can use the above theorem twice: once to count the number of $-1$ s in $z$ (up to $t$ ) and once to count the number of 1s in $z$ (up to $t$ ).\",\n \"This uses $O(\\\\sqrt{tn}\\\\,\\\\mathsf {Q}^{cc}_E(G))=O(\\\\mathsf {Q}(f)\\\\,\\\\mathsf {Q}_E^{cc}(G))$ qubits of communication, assumes $O(t\\\\log n)$ shared EPR-pairs at the start of the protocol, and gives us enough information about $|z|$ to compute $f(z)=F(X,Y)$ .\",\n \"This concludes the proof of Theorem REF from the introduction, restated below.\",\n \"Theorem 4.4 (Restatement of Theorem REF ) For every symmetric Boolean function $f:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n\\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and two-party function $G:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j\\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k\\\\rightarrow \\\\lbrace 0,1\\\\rbrace $ , we have $\\\\mathsf {Q}^{cc,*}(f \\\\circ G)=O(\\\\mathsf {Q}(f)\\\\mathsf {Q}_E^{cc}(G)).$ If $\\\\mathsf {Q}(f)=\\\\Theta (\\\\sqrt{tn})$ , then our protocol assumes a shared state of $O(t\\\\log n)$ EPR-pairs at the start.\",\n \"We remark that for the special case where $G=\\\\mathsf {AND}_2$ , our upper bound matches the lower bound proved by Razborov [31], except for symmetric functions $f$ where the first switch of function value happens at Hamming weights very close to $n$ .\",\n \"In particular, if $f=\\\\mathsf {AND}_n$ and $G=\\\\mathsf {AND}_2$ , then $\\\\mathsf {Q}^{cc}(f \\\\circ G)=1$ but $\\\\mathsf {Q}(f)=\\\\Theta (\\\\sqrt{n})$ .\"\n ],\n [\n \"Necessity of the log-factor overhead in the BCW simulation\",\n \"In this section we prove Theorem REF and Theorem REF .\",\n \"For Theorem REF we exhibit a function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ for which $\\\\mathsf {Q}(f)= O(n)$ and $\\\\mathsf {UPP}(f \\\\circ \\\\square ) = \\\\Omega (n \\\\log n)$ for $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ .\",\n \"In Theorem REF we show a general recipe for constructing total functions $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ such that $\\\\mathsf {Q}(f) = O(n)$ and $\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\square ) = \\\\Omega (n \\\\log n)$ for $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ .\",\n \"We first give a formal statement of Theorem REF .\",\n \"Theorem 5.1 (Restatement of Theorem REF ) Let $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a total function with $\\\\widetilde{\\\\textnormal {deg}}(r) = \\\\Omega (n)$ , $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a total function and $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ .\",\n \"Define $f = r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"If there exists $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\mathbb {R}$ that is a balanced probability distribution with respect to $G$ and $\\\\textnormal {disc}_{\\\\mu }(G) = n^{-\\\\Omega (1)}$ , then $\\\\mathsf {Q}(f) & = O(n),\\\\\\\\\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\square ) & = \\\\Omega (n \\\\log n).$ The proofs of Theorem REF and Theorem REF each involve proving a query complexity upper bound and a communication complexity lower bound.\",\n \"The proofs of the query complexity upper bounds are along similar lines and follow from Theorem REF and Corollary REF (see Section REF ).\",\n \"The proofs of the communication complexity lower bounds each involve a reduction from a problem whose communication complexity is easier to analyze (see Lemma REF in Section REF ).\",\n \"Finally, we complete the proofs of Theorem REF and Theorem REF in Section REF .\"\n ],\n [\n \"Quantum query complexity upper bound\",\n \"We start by stating the main theorem in this section.\",\n \"Theorem 5.2 Let $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"Then the quantum query complexity of the function $r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n(j+k)} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is given by $\\\\mathsf {Q}(r ~\\\\widetilde{\\\\circ }~h_G) = O(n + \\\\sqrt{n(j+k)}).$ As a corollary we obtain the following on instantiating $j = k = n$ and $r$ as a Boolean function with quantum query complexity $\\\\Theta (n)$ in Theorem REF .\",\n \"Corollary 5.3 Let $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a non-constant function and let $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a total function with $\\\\mathsf {Q}(r) = \\\\Theta (n)$ .\",\n \"Then the quantum query complexity of the total function $r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is $\\\\mathsf {Q}(r ~\\\\widetilde{\\\\circ }~h_G) = \\\\Theta (n).$ The upper bound $\\\\mathsf {Q}(r ~\\\\widetilde{\\\\circ }~h_G) = O(n)$ follows by plugging in parameters in Theorem REF .\",\n \"For the lower bound, we show that $\\\\mathsf {Q}(r ~\\\\widetilde{\\\\circ }~h_G) \\\\ge \\\\mathsf {Q}(r)$ .\",\n \"Since $G$ is non-constant, there exist $x_1, y_1, x_2, y_2 \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ such that $G(x_1, y_1) = -1$ and $G(x_2, y_2) = 1$ .\",\n \"Let $X_1 = H(x_1), Y_1 = H(y_1)$ , $X_2 = H(x_2)$ and $X_2 = H(y_2)$ .\",\n \"Consider $r ~\\\\widetilde{\\\\circ }~h_G$ only restricted to inputs where the inputs to each copy of $h_G$ are either $(X_1, Y_1)$ or $(X_2, Y_2)$ .\",\n \"Under this restriction, $r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is the same as $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"Thus $\\\\mathsf {Q}(r ~\\\\widetilde{\\\\circ }~h_G) \\\\ge \\\\mathsf {Q}(r) = \\\\Omega (n)$ .\",\n \"We now prove Theorem REF .\",\n \"[Proof of Theorem REF ] Recall from Definition REF that the function $h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{j + k} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is defined as $h_G(x, y) = G(s, t)$ if $x \\\\in \\\\pm H(s)$ and $y \\\\in \\\\pm H(t)$ for some $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j}$ and $t \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k}$ , and $h_G(x, y) = \\\\star $ otherwise.\",\n \"Also recall from Definition REF that the function $r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n(j+k)} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is defined as $r ~\\\\widetilde{\\\\circ }~h_G((X_1, Y_1), \\\\dots , (X_n, Y_n)) = r \\\\circ h_G((X_1, Y_1), \\\\dots , (X_n, Y_n))$ if $h_G((X_i, Y_i)) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ for all $i \\\\in [n]$ , and $-1$ otherwise.\"\n ],\n [\n \"Quantum query algorithm:\",\n \"View inputs to $r ~\\\\widetilde{\\\\circ }~h_G$ as $(X_1, Y_1, \\\\dots , X_n, Y_n)$ , where $X_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j$ for all $i \\\\in [n]$ and $Y_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k$ for all $i \\\\in [n]$ .\",\n \"We give a quantum algorithm and its analysis below.\",\n \"Run $2n$ instances of the Bernstein-Vazirani algorithm: 1 instance on each $X_i$ and 1 instance on each $Y_i$ , to obtain $2n$ strings $x_1, \\\\dots , x_n, y_1, \\\\dots , y_n$ , where each $x_i$ is a $(\\\\log j)$ -bit string and each $y_i$ is a $(\\\\log k)$ -bit string.\",\n \"For each $X_i$ and $Y_i$ , query $(X_i)_{1^{\\\\log j}}$ and $(Y_i)_{1^{\\\\log k}}$ to obtain bits $b_i, c_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ for all $i \\\\in [n]$ .\",\n \"Run Grover's search [16], [7] to check equality of the following two $(nj + nk)$ -bit strings: $(b_1 H(x_1), \\\\dots , b_n H(x_n), c_1 H(y_1), \\\\dots , c_n H(y_n))$ and $(X_1, \\\\dots , X_n, Y_1, \\\\dots , Y_n)$ .\",\n \"If the step above outputs that the strings are equal, then output $r(G(x_1, y_1), \\\\dots , G(x_n, y_n))$ .\",\n \"Else, output $-1$ .\"\n ],\n [\n \"Analysis of the algorithm:\",\n \" If the input is indeed of the form $(X_1, Y_1), \\\\dots , (X_n, Y_n)$ where each $X_i \\\\in \\\\pm H(x_i)$ and $Y_i \\\\in \\\\pm H(y_i)$ for some $x_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j}$ and $y_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k}$ , then Step REF outputs the correct strings $x_{1}, \\\\dots , x_{n}, y_1, \\\\dots , y_n$ with probability 1 by the properties of the Bernstein-Vazirani algorithm.\",\n \"Step REF then implies that $X_i = b_i H(x_i)$ and $Y_i = c_i H(y_i)$ for all $i \\\\in [n]$ .\",\n \"Next, Step REF outputs that the strings are equal with probability 1 (since the strings whose equality are to be checked are equal).\",\n \"Hence the algorithm is correct with probability 1 in this case, since $(r ~\\\\widetilde{\\\\circ }~h_G)(X_1, Y_1, \\\\dots , X_n, Y_n) = r(G(x_1, y_1), \\\\dots , G(x_n, y_n))$ .\",\n \"If the input is such that there exists an index $i \\\\in [n]$ for which $X_i \\\\notin \\\\pm H(x_i)$ for every $x_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j}$ or $Y_i \\\\notin \\\\pm H(y_i)$ for every $y_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k}$ , then the two strings for which equality is to be checked in the Step REF are not equal.\",\n \"Grover's search catches a discrepancy with probability at least $2/3$ .\",\n \"Hence, the algorithm outputs $-1$ (as does $r ~\\\\widetilde{\\\\circ }~h_G$ ), and is correct with probability at least $2/3$ in this case.\"\n ],\n [\n \"Cost of the algorithm:\",\n \"Step REF accounts for $2n$ quantum queries.\",\n \"Step REF accounts for $2n$ quantum queries.\",\n \"Step REF accounts for $O(\\\\sqrt{n(j+k)})$ quantum queries.\",\n \"Thus, $\\\\mathsf {Q}(r ~\\\\widetilde{\\\\circ }~h_G) = O(n + \\\\sqrt{n(j+k)}).$\"\n ],\n [\n \"Quantum communication complexity lower bound\",\n \"In this section we first show a communication lower bound (under some model) on $(r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\square $ in terms of the communication complexity of $r \\\\circ G$ (in the same model of communication) using a simple reduction.\",\n \"We state the lemma below (Lemma REF ) for the case where the models under consideration are the bounded-error and unbounded-error quantum models, since these are the models of interest to us.\",\n \"Lemma 5.4 Let $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ and $CC \\\\in \\\\lbrace \\\\mathsf {Q}^{cc, *}, \\\\mathsf {UPP}^{cc}\\\\rbrace $ .\",\n \"Then, $CC((r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\square ) \\\\ge CC(r \\\\circ G).$ We first consider the case $\\\\square = \\\\mathsf {AND}_2$ .\",\n \"Consider a protocol $\\\\Pi $ of cost $\\\\ell $ that solves $(r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\square $ in the $CC$ -model.\",\n \"We exhibit below a protocol of cost $\\\\ell $ that solves $r \\\\circ G$ in the same model.\",\n \"Suppose Alice is given input $x = (x_1, \\\\dots , x_{n}) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n \\\\log j}$ and Bob is given input $y = (y_1, \\\\dots , y_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n \\\\log k}$ , where $x_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j}, y_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k}$ for each $i \\\\in [n]$ .\"\n ],\n [\n \"Preprocessing step:\",\n \"Alice constructs the $(n(j+k))$ -bit string $X = ((H(x_1), (-1)^{k}), \\\\dots , (H(x_n), (-1)^k)) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n(j+k)},$ and Bob constructs the $(n(j+k))$ -bit string $Y = (((-1)^j, H(y_1)), \\\\dots , ((-1)^j, H(y_n))) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n(j+k)}.$\"\n ],\n [\n \"Protocol:\",\n \"Alice and Bob run the protocol $\\\\Pi $ with input $(X, Y)$ and output $\\\\Pi (X, Y)$ .\"\n ],\n [\n \"Cost:\",\n \"The preprocessing of the inputs to obtain $X$ from $x$ and $Y$ from $y$ takes no communication.\",\n \"Hence the total amount of communication is at most the cost of $\\\\Pi $ .\"\n ],\n [\n \"Correctness:\",\n \"For $X$ and $Y$ constructed in Equation (REF ) and Equation (REF ), respectively, we now argue that $(r \\\\circ G)(x, y) = ((r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\mathsf {AND}_2) (X, Y)$ , which would conclude the proof for $\\\\square = \\\\mathsf {AND}_2$ .\",\n \"$((r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\mathsf {AND}_2)(X, Y) & = (r ~\\\\widetilde{\\\\circ }~h_G)((H(x_1), H(y_1)), \\\\dots , (H(x_n), H(y_n)))\\\\\\\\& = r(G(x_1, y_1), \\\\dots , G(x_n, y_n)) \\\\\\\\& = (r \\\\circ G)(x, y).$ Thus, $CC((r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\mathsf {AND}_2) \\\\ge CC(r \\\\circ G).$ The argument for $\\\\square = \\\\mathsf {XOR}_2$ follows along the same lines, with the strings $(-1)^j$ and $(-1)^k$ replaced by $1^j$ and $1^k$ , respectively, in the preprocessing step.\"\n ],\n [\n \"On the tightness of the BCW simulation\",\n \"We prove Theorem REF in Section REF and Theorem REF in Section REF .\"\n ],\n [\n \"Proof of Theorem \",\n \"Towards proving Theorem REF , we first observe how to obtain bounded-error quantum communication complexity lower bounds using Theorem REF and Lemma REF .\",\n \"Lemma 5.5 Let $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log j} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log k} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be functions such that $\\\\textnormal {bdisc}(G) = o(1)$ and $\\\\frac{8en}{\\\\widetilde{\\\\textnormal {deg}}(r)} \\\\le \\\\left(\\\\frac{1}{\\\\textnormal {bdisc}(G)}\\\\right)^{1-\\\\beta }$ for some constant $\\\\beta \\\\in (0,1)$ .\",\n \"Let $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ .\",\n \"Then, $\\\\mathsf {Q}^{cc, *} ((r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\square ) = \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(r) \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {bdisc}(G)}\\\\right)\\\\right).$ By Lemma REF we have $\\\\mathsf {Q}^{cc, *} ((r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\square ) \\\\ge \\\\mathsf {Q}^{cc, *}(r \\\\circ G)$ .\",\n \"By Theorem REF , $\\\\mathsf {Q}^{cc, *}(r \\\\circ G) = \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(r) \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {bdisc}(G)}\\\\right)\\\\right)$ .\",\n \"We now prove Theorem REF .\",\n \"[Proof of Theorem REF ] Let $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $f = r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be as in the statement of the theorem.\",\n \"We have $\\\\mathsf {Q}(r) \\\\ge \\\\widetilde{\\\\textnormal {deg}}(r)/2 = \\\\Omega (n)$ , where the first inequality follows by Theorem REF and the second equality follows from the assumption that $\\\\widetilde{\\\\textnormal {deg}}(r) = \\\\Omega (n)$ .\",\n \"Moreover, $\\\\mathsf {Q}(r) \\\\le n$ since $r$ is a function on $n$ input variables.\",\n \"Hence $\\\\mathsf {Q}(r) = \\\\Theta (n)$ .\",\n \"Thus, Corollary REF is applicable, and we have $\\\\mathsf {Q}(f) = \\\\Theta (n).$ For the lower bound, $\\\\widetilde{\\\\textnormal {deg}}(r) = \\\\Omega (n)$ by assumption.\",\n \"Thus $\\\\frac{2en}{\\\\widetilde{\\\\textnormal {deg}}(r)} = O(1).$ Also, since by assumption $\\\\frac{1}{\\\\textnormal {bdisc}(G)} = n^{\\\\Omega (1)} = \\\\omega (1)$ , we have $\\\\frac{2en}{\\\\widetilde{\\\\textnormal {deg}}(r)} \\\\le \\\\left(\\\\frac{1}{\\\\textnormal {bdisc}(G)}\\\\right)^{1-\\\\beta }$ for every constant $\\\\beta \\\\in (0,1)$ .\",\n \"Lemma REF implies $\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\square ) = \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(r) \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {bdisc}(G)}\\\\right) \\\\right) = \\\\Omega (n \\\\log n).$\"\n ],\n [\n \"Proof of Theorem \",\n \"The total function $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ that we use to prove Theorem REF is $f = r ~\\\\widetilde{\\\\circ }~h_G$ , where $r = \\\\mathsf {PARITY}_n$ and $G = \\\\mathsf {IP}_{\\\\log n}$ .\",\n \"Note that Theorem REF implies $\\\\mathsf {Q}(f) = O(n)$ .\",\n \"For the quantum communication complexity lower bound in Theorem REF , we are able to show not only $\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\square ) = \\\\Omega (n \\\\log n)$ , but $\\\\mathsf {UPP}^{cc}(f \\\\circ \\\\square ) = \\\\Omega (n \\\\log n)$ for $\\\\square \\\\in \\\\lbrace \\\\mathsf {AND}_2,\\\\mathsf {XOR}_2\\\\rbrace $ .\",\n \"The following claim shows that $f$ is transitive.\",\n \"Claim 5.6 Let $n > 0$ be a power of 2.\",\n \"Let $r = \\\\mathsf {PARITY}_n : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $G = \\\\mathsf {IP}_{\\\\log n} : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"The function $f = r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is transitive.\",\n \"We first show that $h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is transitive.\",\n \"We next observe that $s ~\\\\widetilde{\\\\circ }~t$ is transitive whenever $s$ is symmetric and $t$ is transitive ($s$ can be assumed to just be transitive rather than symmetric, as noted in Remark REF ).\",\n \"The theorem then follows since $\\\\mathsf {PARITY}_n$ is symmetric.\",\n \"Towards showing transitivity of $h_G$ , let $\\\\pi \\\\in S_{2n}$ , and $(\\\\sigma _{\\\\ell }, \\\\sigma _\\\\ell ) \\\\in S_{2n}$ for $\\\\ell \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ be defined as follows.\",\n \"(Here $\\\\sigma _\\\\ell \\\\in S_n$ ; the first copy acts on the first $n$ coordinates, and the second copy acts on the next $n$ coordinates.)\",\n \"$\\\\pi (k) = {\\\\left\\\\lbrace \\\\begin{array}{ll}k + n & k \\\\le n\\\\\\\\k - n & k > n.\\\\end{array}\\\\right.\",\n \"}$ That is, on every string $(x, y) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n}$ , the permutation $\\\\pi $ maps $(x, y)$ to $(y, x)$ .\",\n \"For every $\\\\ell \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , the permutation $\\\\sigma _\\\\ell \\\\in S_n$ is defined as $\\\\sigma _{\\\\ell }(i) = i \\\\oplus \\\\ell ,$ where $i \\\\oplus \\\\ell $ denotes the bitwise XOR of the strings $i$ and $\\\\ell $ .\",\n \"That is, for every input $(x, y) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n}$ and every $k \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , the input bit $x_{k}$ is mapped to $x_{k \\\\oplus \\\\ell }$ and $y_k$ is mapped to $y_{k\\\\oplus \\\\ell }$ .\",\n \"For every $(x,y) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n}$ and $i,j \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , the permutation $\\\\sigma _{i \\\\oplus j}(x,y)$ swaps $x_i$ and $x_j$ , and also swaps $y_i$ and $y_j$ .\",\n \"If for $i, j \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , our task was to swap the $i$ 'th index of the first $n$ variables with the $j$ 'th index of the second $n$ variables, then the permutation $\\\\sigma _{i\\\\oplus j} \\\\circ \\\\pi $ does the job.\",\n \"That is, for every $(x,y) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n}$ and $i,j \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , the permutation $\\\\sigma _{i\\\\oplus j} \\\\circ \\\\pi $ maps $x_i$ to $y_j$ .\",\n \"Thus the set of permutations $\\\\lbrace \\\\pi , \\\\lbrace \\\\sigma _\\\\ell : \\\\ell \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}\\\\rbrace \\\\rbrace $ acts transitively on $S_{2n}$ .\",\n \"Now we show that for all $x,y \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n}$ and all $\\\\ell \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , we have $h_G(\\\\sigma _{\\\\ell }(x), \\\\sigma _\\\\ell (y)) = h_G(x,y)$ .\",\n \"Fix $\\\\ell \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ .\",\n \"If $x \\\\in \\\\pm H(s)$ and $y \\\\in \\\\pm H(t)$ are Hadamard codewords, then $x_k = \\\\langle k, s \\\\rangle $ and $y_k = \\\\langle k, t \\\\rangle $ for all $k \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , and $G(x,y) = \\\\langle s, t \\\\rangle $ .\",\n \"Thus, for every $k \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ we have $\\\\sigma _{\\\\ell }(x_k) = x_{k \\\\oplus \\\\ell } = \\\\langle k \\\\oplus \\\\ell , s \\\\rangle = \\\\langle \\\\ell , s \\\\rangle \\\\cdot \\\\langle k , s \\\\rangle $ .\",\n \"Hence $\\\\sigma _{\\\\ell }(x) \\\\in \\\\pm H(s)$ (since $\\\\langle \\\\ell , s \\\\rangle $ does not depend on $k$ , and takes value either 1 or $-1$ ).\",\n \"Similarly, $\\\\sigma _{\\\\ell }(y) \\\\in \\\\pm H(t)$ .\",\n \"Thus $h_G(\\\\sigma _{\\\\ell }(x,y)) = h_G(x,y)$ .\",\n \"If $x$ ($y$ , respectively) is not a Hadamard codeword, then a similar argument shows that for all $\\\\ell \\\\in [n]$ , $\\\\sigma _{\\\\ell }(x)$ ($\\\\sigma _{\\\\ell }(y)$ , respectively) is also not a Hadamard codeword.\",\n \"Using the fact that $\\\\langle s, t \\\\rangle = \\\\langle t, s \\\\rangle $ for every $s, t \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ , one may verify that $h_G(\\\\pi (x,y)) = h_G(x,y)$ for all $x,y \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n}$ .\",\n \"Along with the observation that $\\\\mathsf {PARITY}_n$ is a symmetric function, we have that $f = r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is transitive under the following permutations: $S_n$ acting on the inputs of $\\\\mathsf {PARITY}_n$ , and The group generated by $\\\\left\\\\lbrace \\\\pi \\\\right\\\\rbrace \\\\cup \\\\left\\\\lbrace (\\\\sigma _\\\\ell , \\\\sigma _\\\\ell ) : \\\\ell \\\\in [n] \\\\right\\\\rbrace $ acting independently on the inputs of each copy of $h_G$ , where $\\\\sigma _{\\\\ell }$ is as in Equation (REF ).\",\n \"We now prove Theorem REF .\",\n \"[Proof of Theorem REF ] Let $n > 0$ be a power of 2.\",\n \"Let $r = \\\\mathsf {PARITY}_n : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $G = \\\\mathsf {IP}_{\\\\log n} : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"Let $f = r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"By Claim REF , $f$ is transitive.\",\n \"By Corollary REF we have $\\\\mathsf {Q}(f) = \\\\Theta (n).$ For the communication lower bound we have $\\\\mathsf {UPP}^{cc}(f \\\\circ \\\\square ) & = \\\\mathsf {UPP}^{cc}((r ~\\\\widetilde{\\\\circ }~h_G) \\\\circ \\\\square ) \\\\\\\\& \\\\ge \\\\mathsf {UPP}^{cc}(\\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {IP}_{\\\\log n}) \\\\\\\\& = \\\\mathsf {UPP}^{cc}(\\\\mathsf {IP}_{n \\\\log n}) \\\\\\\\& = \\\\Omega (n \\\\log n) .$ Remark 5.7 The proof of transitivity of $f$ in Theorem REF can also be used to prove that if $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ is transitive and $G = \\\\mathsf {IP}_{\\\\log n} : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , then $r ~\\\\widetilde{\\\\circ }~h_G$ is transitive as well.\",\n \"By instantiating $r$ to a transitive function with approximate degree $\\\\Omega (n)$ (e.g.\",\n \"Majority), Theorem REF implies that the BCW simulation is tight w.r.t.\",\n \"the bounded-error communication model for a wide class of transitive functions.\"\n ],\n [\n \"Hardness of composing the Chakraborty et al. function with $\\\\mathsf {AND}_2$\",\n \"Chakraborty et al.\",\n \"[12] exhibited a total function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ for which $\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\mathsf {XOR}_2) = \\\\Omega (\\\\mathsf {Q}(f) \\\\log n)$ , that is, the log-factor overhead in the BCW simulation is necessary for $f$ when the inner function is $\\\\mathsf {XOR}_2$ .\",\n \"In this section we show that the log-factor overhead is necessary for $f$ even when the inner function is $\\\\mathsf {AND}_2$ .\",\n \"Definition A.1 (Addressing function) For an integer $n>0$ that is a power of 2, let the Addressing function, denoted $\\\\mathsf {ADDR}_{n} : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , be a two-party function defined as follows.\",\n \"$\\\\mathsf {ADDR}_n(x_1, \\\\dots , x_{\\\\log n}, y_1, \\\\dots , y_n) = y_{\\\\textnormal {bin}(i)},$ where $\\\\textnormal {bin}(i)$ represents the integer in $[n]$ whose binary representation is $i$ .\",\n \"In the corresponding communication problem, Alice holds the inputs $x_1, \\\\dots , x_{\\\\log n}$ , and Bob holds the inputs $y_1, \\\\dots , y_n$ .\",\n \"In [12] a subset of the authors defined the following function $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"Let $X_i, Y_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ for all $i \\\\in [n]$ .\",\n \"$f(X_1, Y_1, \\\\dots , X_n, Y_n) ={\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\mathsf {PARITY}(\\\\mathsf {ADDR}_n(x_1, Y_1), \\\\dots , \\\\mathsf {ADDR}_n(x_n, Y_n)) \\\\textnormal { if } \\\\forall i \\\\in [n], X_i \\\\in \\\\pm H(x_i)\\\\\\\\-1 \\\\textnormal { otherwise},\\\\end{array}\\\\right.\",\n \"}$ That is, if there exist $x_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ for all $i \\\\in [n]$ such that $X_i \\\\in \\\\pm H(x_i)$ , then $f(X_1, Y_1, \\\\dots , X_n, Y_n)$ equals the parity of $\\\\mathsf {ADDR}_n(x_1, Y_1), \\\\dots , \\\\mathsf {ADDR}_n(x_n, Y_n)$ , and $f$ equals $-1$ otherwise.\",\n \"Chakraborty et al.\",\n \"proved the following quantum query upper bound on $f$ (set $\\\\delta = 1/2$ in [12]).\",\n \"Theorem A.2 ([12]) For $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ as defined in Equation (REF ), $\\\\mathsf {Q}(f) = O(n)$ .\",\n \"Remark A.3 The function considered in [12] is not exactly as in Equation (REF ) because the former one did not consider negations of Hadamard codewords as done in Equation (REF ).\",\n \"However, Theorem REF can still be seen to hold by incorporating a step similar to Step REF in the algorithm given in the proof of [12].\",\n \"Chakraborty et al.\",\n \"showed the BCW simulation theorem is tight for $f$ when the inner function is $\\\\mathsf {XOR}_2$ , that is $\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\mathsf {XOR}_2) = \\\\Omega (n \\\\log n)$ .\",\n \"They left open the question whether the BCW simulation theorem is tight for $f$ even when the inner function is $\\\\mathsf {AND}_2$ .\",\n \"We show that this is indeed the case.\",\n \"The quantum query upper bound holds by Theorem REF .\",\n \"For the communication complexity lower bound we use techniques similar to those in the proof of Lemma REF .\",\n \"Lemma A.4 Let $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be as in Equation (REF ), and $CC \\\\in \\\\lbrace \\\\mathsf {Q}^{cc, *}, \\\\mathsf {UPP}^{cc} \\\\rbrace $ .\",\n \"Then, $CC(r \\\\circ \\\\mathsf {AND}_2) \\\\ge CC(\\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {ADDR}_{n}).$ We require the following lemma.\",\n \"Lemma A.5 Let $n > 0$ be a power of 2, and let $U$ be the uniform distribution on $\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n}$ .\",\n \"Then, $\\\\textnormal {disc}_U(\\\\mathsf {ADDR}_{n}) \\\\le \\\\frac{1}{\\\\sqrt{n}}.$ We provide a proof for completeness.\",\n \"Consider the uniform distribution $U: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ defined as $U(x,y) = \\\\frac{1}{n2^n}$ for all $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ and $y \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"Let $A_{n \\\\times 2^n}$ be the communication matrix of $\\\\mathsf {ADDR}_n: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"Observe that for every $i,j \\\\in [n]$ , $(AA^T)_{ij}= \\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} x_i x_j={\\\\left\\\\lbrace \\\\begin{array}{ll}2^n \\\\text{ if } i = j \\\\\\\\0 \\\\text{ otherwise}.\\\\end{array}\\\\right.\",\n \"}$ Hence $AA^T = 2^n I_n$ , where $I_n$ denotes the $n \\\\times n$ identity matrix.\",\n \"Thus, $\\\\Vert A\\\\Vert = \\\\sqrt{2^n}$ .\",\n \"For $S \\\\subseteq [n]$ and $T \\\\subseteq [2^n]$ , let $1_S: [n] \\\\rightarrow \\\\left\\\\lbrace 0,1 \\\\right\\\\rbrace $ and $1_T: [2^n] \\\\rightarrow \\\\left\\\\lbrace 0,1 \\\\right\\\\rbrace $ be the indicator function of corresponding rows and columns of $A$ , respectively.\",\n \"We now upper bound the discrepancy with respect to the uniform distribution: $\\\\textnormal {disc}_U(\\\\mathsf {ADDR}_n)&= \\\\max _{S \\\\subseteq [n], T\\\\subseteq [2^n]} \\\\left|(1_S)^T \\\\cdot (U \\\\circ A) \\\\cdot 1_T \\\\right| \\\\\\\\&\\\\le \\\\sqrt{n} \\\\cdot \\\\Vert U \\\\circ A \\\\Vert \\\\cdot \\\\sqrt{2^n} \\\\\\\\&= \\\\frac{1}{\\\\sqrt{n2^n}} \\\\Vert A \\\\Vert \\\\\\\\&= \\\\frac{1}{\\\\sqrt{n}}.$ We now prove Lemma REF using Lemma REF .\",\n \"[Proof of Lemma REF ] Consider a protocol $\\\\Pi $ of cost $\\\\ell $ that solves $f \\\\circ \\\\mathsf {AND}_2$ in the $CC$ -model.\",\n \"We exhibit below a protocol of cost $\\\\ell $ that solves $\\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {ADDR}_{n}$ in the same model.\",\n \"Recall from Definitions REF and REF that in the communication problem $\\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {ADDR}_{n}$ , Alice is given the address bits and Bob is given the target bits for each copy of the inner gadget $\\\\mathsf {ADDR}_{n}$ .\",\n \"Suppose Alice is given input $x = (x_1, \\\\dots , x_{n}) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n \\\\log n}$ and Bob is given input $y = (y_1, \\\\dots , y_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n^2}$ , where $x_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}, y_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n}$ for each $i \\\\in [n]$ .\"\n ],\n [\n \"Preprocessing step:\",\n \"Alice constructs the following $(2n^2)$ -bit string $X = (H(x_1), (-1)^{n}), \\\\dots , (H(x_n), (-1)^n)) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2},$ and Bob constructs the $(2n^2)$ -bit string $Y = (((-1)^n, y_1), \\\\dots , ((-1)^n, y_n)) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2}.$\"\n ],\n [\n \"Protocol:\",\n \"Alice and Bob run the protocol $\\\\Pi $ with input $(X, Y)$ , and output $\\\\Pi (X, Y)$ .\"\n ],\n [\n \"Cost:\",\n \"The preprocessing of the inputs to obtain $X$ from $x$ and $Y$ from $y$ takes no communication.\",\n \"Hence the total amount of communication is at most the cost of $\\\\Pi $ .\"\n ],\n [\n \"Correctness:\",\n \"We now argue that $\\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {ADDR}_{n}(x, y) = (f \\\\circ \\\\mathsf {AND}_2) (X, Y)$ , which would conclude the proof.\",\n \"We have from Equation (REF ) and Equation (REF ) that $(f \\\\circ \\\\mathsf {AND}_2)(X, Y) & = f((H(x_1), y_1), \\\\dots , (H(x_n), y_n))\\\\\\\\& = \\\\mathsf {PARITY}_n(\\\\mathsf {ADDR}_n(x_1, y_1), \\\\dots , \\\\mathsf {ADDR}_n(x_n, y_n)) \\\\\\\\& = \\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {ADDR}_{n}(x, y).$ Thus, $CC(f \\\\circ \\\\mathsf {AND}_2) \\\\ge CC(\\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {ADDR}_{n}),$ which proves the lemma.\",\n \"Theorem A.6 Let $n > 0$ be a sufficiently large power of 2.\",\n \"Let $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be as in Equation (REF ).\",\n \"Then, $\\\\mathsf {Q}(f) & = O(n),\\\\\\\\\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\mathsf {AND}_2) & = \\\\Omega (n \\\\log n).$ The upper bound follows from Theorem REF .\",\n \"For the lower bound, we first note that for sufficiently large $n$ and every constant $\\\\beta \\\\in (0, 1)$ , we have $\\\\frac{2en}{\\\\widetilde{\\\\textnormal {deg}}(\\\\mathsf {PARITY}_n)} = 2e < (\\\\sqrt{n})^{1 - \\\\beta } = \\\\left(\\\\frac{1}{\\\\textnormal {disc}_{U}(\\\\mathsf {ADDR}_n)}\\\\right)^{1-\\\\beta },$ where $U$ denotes the uniform distribution over $\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"The first equality above holds since $\\\\widetilde{\\\\textnormal {deg}}(\\\\mathsf {PARITY}_n) = n$ , and the last equality holds by Lemma REF .\",\n \"It is easy to verify that $U$ is a balanced distribution w.r.t.\",\n \"$\\\\mathsf {ADDR}_n$ .\",\n \"Hence, $\\\\mathsf {Q}^{cc, *}(f \\\\circ \\\\mathsf {AND}_2) & \\\\ge \\\\mathsf {Q}^{cc, *}(\\\\mathsf {PARITY}_n \\\\circ \\\\mathsf {ADDR}_n) \\\\\\\\& = \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(\\\\mathsf {PARITY}_n) \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {disc}_U(\\\\mathsf {ADDR}_{n})}\\\\right)\\\\right) \\\\\\\\& = \\\\Omega \\\\left(n \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {disc}_U(\\\\mathsf {ADDR}_{n})}\\\\right)\\\\right) \\\\\\\\& = \\\\Omega (n \\\\log n).\",\n \"$\"\n ],\n [\n \"A separation between log-approximate-spectral norm and approximate degree for a transitive function\",\n \"In this section, we exhibit a transitive function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ for which the logarithm of its approximate spectral norm ($\\\\log (\\\\Vert \\\\widehat{f}\\\\Vert _{1, 1/3})$ ) is at least $\\\\Omega (\\\\widetilde{\\\\textnormal {deg}}(f) \\\\log n)$ .\",\n \"We refer the reader to [28] for preliminaries on Fourier analysis of Boolean functions.\",\n \"Definition B.1 (Spectral norm) Let $p : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ be a function, and let $p = \\\\sum _{S \\\\subseteq [n]}\\\\widehat{p}(S)\\\\chi _S$ denote its Fourier expansion.\",\n \"The spectral norm of $p$ is define by $\\\\Vert \\\\widehat{p}\\\\Vert _1 := \\\\sum _{S \\\\subseteq [n]} \\\\left|\\\\widehat{p}(S) \\\\right|.$ Definition B.2 (Approximate Spectral Norm) The approximate spectral norm of a function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\lbrace -1,1\\\\rbrace $ , denoted by $\\\\Vert \\\\widehat{f}\\\\Vert _{1,\\\\varepsilon }$ is defined to be the minimum spectral norm of a real polynomial $p : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ that satisfies $\\\\left|p(x) - f(x) \\\\right| \\\\le \\\\varepsilon $ for all $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"That is, $\\\\Vert \\\\widehat{f}\\\\Vert _{1,\\\\varepsilon } := \\\\min \\\\left\\\\lbrace \\\\Vert \\\\widehat{p}\\\\Vert _1 : \\\\left|p(x) - f(x) \\\\right| \\\\le \\\\varepsilon ~\\\\text{for all~} x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\right\\\\rbrace .$ In [12] the following question raised in [2] was shown to have a negative answer, using the function $f$ of Equation (REF ) to witness this.\",\n \"Question B.3 ([2]) Is it true that for all Boolean functions $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , $\\\\log (\\\\Vert \\\\widehat{f}\\\\Vert _{1,\\\\varepsilon }) = O(\\\\widetilde{\\\\textnormal {deg}}(f))?$ In this section we show that Question REF is false even for the special class of transitive Boolean functions.\",\n \"This is in stark contrast with the class of symmetric functions, for which Question REF is true (see [12]).\",\n \"Claim B.4 There exists a transitive function $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ such that $\\\\log \\\\left(\\\\Vert \\\\widehat{f}\\\\Vert _{1, 1/3} \\\\right) = \\\\Omega (\\\\widetilde{\\\\textnormal {deg}}(f) \\\\log n)$ .\",\n \"We first state some required preliminaries.\",\n \"Definition B.5 (Monomial projection) We call a function $g:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ a monomial projection of a function $f:\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ if $g$ can be expressed as $g(x_1,\\\\dots , x_m) = f(M_1,\\\\dots , M_n)$ , where each $M_i$ is a monomial in the variables $x_1,\\\\dots , x_m$ .\",\n \"It is known that the approximate spectral norm of a function can only decrease upon monomial projections (see, for example, [14]).\",\n \"Observation B.6 ([14]) For $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $g : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{m} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ such that $g$ is a monomial projection of $f$ , $\\\\Vert \\\\widehat{g}\\\\Vert _{1, 1/3} \\\\le \\\\Vert \\\\widehat{f}\\\\Vert _{1, 1/3}.$ Fact B.7 (Fourier coefficients of $\\\\mathsf {IP}_{n}$ ) Let $\\\\mathsf {IP}_n: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be as in Definition REF .\",\n \"Then for all $S \\\\subseteq [2n]$ we have, $\\\\left|\\\\widehat{\\\\mathsf {IP}_n}(S) \\\\right| = \\\\frac{1}{2^n}.$ Fact B.8 (Plancherel's Theorem) Let $f,g: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ be functions.\",\n \"Then, $\\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n}f(x)g(x) = \\\\sum _{S \\\\subseteq [n]} \\\\widehat{f}(S) \\\\widehat{g}(S).$ [Proof of Claim REF ] Let $n > 0$ be a power of 2.\",\n \"Let $r = \\\\mathsf {PARITY}_n : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $G = \\\\mathsf {IP}_{\\\\log n} : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"From Claim REF , the function $f = r ~\\\\widetilde{\\\\circ }~h_G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n^2} \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ is transitive.\",\n \"By Corollary REF we have $\\\\mathsf {Q}(f) = \\\\Theta (n).\",\n \"$ By Theorem REF we have $\\\\mathsf {Q}(f) \\\\ge \\\\widetilde{\\\\textnormal {deg}}(f)/2$ and together with Equation (REF ), this implies $\\\\widetilde{\\\\textnormal {deg}}(f) = O(n)$ .\",\n \"Thus to complete the proof of the claim, it suffices to show $\\\\log \\\\left(\\\\Vert \\\\widehat{f}\\\\Vert _{1, 1/3} \\\\right) = \\\\Omega (n \\\\log n)$ .\",\n \"We first note that $\\\\mathsf {IP}_{n \\\\log n}$ is a monomial projection of $f$ .\",\n \"Consider the function $f$ acting on the input variables $x^{(1)}, \\\\dots , x^{(n)}, y^{(1)}, \\\\dots , y^{(n)}$ , where $x^{(i)}, y^{(i)} \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{n}$ for all $i \\\\in [n]$ .\",\n \"For $i \\\\in [n]$ , set $x^{(i)}_{1^{\\\\log n}} = y^{(i)}_{1^{\\\\log n}} = 1$ .\",\n \"For $i \\\\in [n]$ and string $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n} \\\\setminus \\\\left\\\\lbrace 1^{\\\\log n} \\\\right\\\\rbrace $ , set $x^{(i)}_s = \\\\prod _{j: s_j = -1}x^{(i)}_j $ and $y^{(i)}_s = \\\\prod _{j: s_j = -1}y^{(i)}_j$ .\",\n \"That is, in each block of inputs $x^{(i)}$ and $y^{(i)}$ , the coordinate corresponding to $1^{\\\\log n}$ equals 1, the coordinates corresponding to $j \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ with $|j| = 1$ are free variables, and all other variables are replaced by monomials in these variables.\",\n \"Under the above monomial projection there are $2n \\\\log n$ free variables, namely $\\\\left\\\\lbrace x^{(i)}_{s}, y^{(i)}_{s} : i \\\\in [n], s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}, |s| = 1\\\\right\\\\rbrace $ .\",\n \"Also note that under this projection and every setting of the free variables, the blocks $x^{(i)}$ and $y^{(i)}$ , for $i \\\\in [n]$ , are always Hadamard codewords.\",\n \"Let $f^{\\\\prime }$ be the monomial projection (see Definition REF ) of $f$ under the projection defined in this paragraph.\",\n \"For the purpose of the next equality we abbreviate strings $s \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{\\\\log n}$ of Hamming weight 1 by the set $\\\\left\\\\lbrace i \\\\right\\\\rbrace $ , where $i \\\\in [\\\\log n]$ is such that $s_i = -1$ .\",\n \"On the free variables, the projected function $f^{\\\\prime }$ equals $\\\\mathsf {PARITY}_n \\\\left(\\\\langle (x^{(1)}_{\\\\lbrace 1\\\\rbrace }, \\\\dots , x^{(1)}_{\\\\lbrace \\\\log n\\\\rbrace }), (y^{(1)}_{\\\\lbrace 1\\\\rbrace }, \\\\dots , y^{(1)}_{\\\\lbrace \\\\log n\\\\rbrace })\\\\rangle , \\\\dots , \\\\langle (x^{(n)}_{\\\\lbrace 1\\\\rbrace }, \\\\dots , x^{(n)}_{\\\\lbrace \\\\log n\\\\rbrace }), (y^{(n)}_{\\\\lbrace 1\\\\rbrace }, \\\\dots , y^{(n)}_{\\\\lbrace \\\\log n\\\\rbrace })\\\\rangle \\\\right).$ Thus, $f^{\\\\prime } = \\\\mathsf {IP}_{n\\\\log n}.$ It follows from earlier works [8], [9] that every polynomial that approximates $\\\\mathsf {IP}_{n\\\\log n}$ to error $1/3$ must have spectral norm $2^{\\\\Omega (n \\\\log n)}$ .\",\n \"We include a short proof below for completeness.\",\n \"Let $P$ be a polynomial that approximates $\\\\mathsf {IP}_{n\\\\log n}$ to error $1/3$ .\",\n \"Since we have $P(x, y)\\\\mathsf {IP}_{n\\\\log n}(x, y) \\\\ge 2/3$ for all $(x, y) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n\\\\log n}$ , $2/3 & \\\\le \\\\mathbb {E}_{x, y \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{2n\\\\log n}} [P(x, y)\\\\mathsf {IP}_{n\\\\log n}(x, y)]\\\\\\\\& = \\\\sum _{S \\\\subseteq [2n\\\\log n]}\\\\widehat{P}(S)\\\\widehat{\\\\mathsf {IP}_{n\\\\log n}}(S) \\\\\\\\& \\\\le \\\\sum _{S \\\\subseteq [2n\\\\log n]}\\\\frac{|\\\\widehat{P}(S)|}{2^{n \\\\log n}} \\\\\\\\& = \\\\frac{\\\\Vert \\\\widehat{P}\\\\Vert _1}{2^{n \\\\log n}}\\\\\\\\& \\\\Rightarrow \\\\log (\\\\Vert \\\\widehat{P}\\\\Vert _1) = \\\\Omega (n \\\\log n)\\\\\\\\& \\\\Rightarrow \\\\log (\\\\Vert \\\\widehat{\\\\mathsf {IP}_{n \\\\log n}}\\\\Vert _{1, 1/3}) = \\\\Omega (n \\\\log n).$ This yields the desired contradiction by Observation REF .\"\n ],\n [\n \"Quantum communication lower bound via the generalized discrepancy method\",\n \"In this section we prove Theorem REF , which gives a lower bound on the quantum communication complexity of a composed function in terms of the approximate degree of the outer function and the discrepancy of the inner function.\",\n \"This result is implicit in [25].\",\n \"Their result is stated in the more general setting of matrix norms, and the $\\\\log $ factor we require on the right-hand side of Theorem REF is not included in their statement.\",\n \"Theorem REF follows implicitly from the proof of [25] along with the fact that $\\\\gamma _2^*$ -norm characterizes discrepancy [23].\",\n \"For completeness and clarity, we prove Theorem REF below from first principles.\",\n \"Definition C.1 For functions $f,g : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ and a probability distribution $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ the correlation between $f$ and $g$ with respect to $\\\\mu $ is defined to be $\\\\textnormal {corr}_{\\\\mu }(f, g) = \\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} f(x)g(x) \\\\mu (x).$ For a Boolean function $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , its approximate degree can be captured by a certain linear program.\",\n \"Writing out the dual of this program and analyzing its optimum yields the following theorem (see, for example, [10]).\",\n \"Theorem C.2 (Dual witness for $\\\\varepsilon $ -approximate degree) For every $\\\\varepsilon \\\\ge 0$ and $f : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ , $\\\\widetilde{\\\\textnormal {deg}}_{\\\\varepsilon }(f) \\\\ge d$ if and only if there exists a polynomial $\\\\psi :\\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ such that $\\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} f(x) \\\\psi (x) > \\\\varepsilon $ , $\\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} |\\\\psi (x)| = 1$ and $\\\\widehat{\\\\psi }(S) = 0$ for all $|S| < d$ .\",\n \"We require the following fact, which follows immediately from the fact that $\\\\widehat{f}(S)$ is a uniform average of different signed values of $f(x)$ .\",\n \"Fact C.3 (Folklore) For every function $f: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ , $2^n \\\\max _{S \\\\subseteq [n]} \\\\left|\\\\widehat{f}(S) \\\\right| \\\\le \\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} \\\\left|f(x) \\\\right|.$ The following theorem shows that if a two-party function $F$ correlates well with a two-party function $G$ under some distribution $\\\\lambda $ , and the discrepancy of $G$ under $\\\\lambda $ is small, then the bounded-error quantum communication complexity of $F$ must be large.\",\n \"This is referred to as the generalized discrepancy method, and was first proposed by Klauck [21].\",\n \"The following version can be found in [13], for example, stated as a lower bound on randomized communication complexity.\",\n \"However, the generalized discrepancy method is also known to give lower bounds on bounded-error quantum communication complexity, even in the model where the parties can share an arbitrary prior entangled state for free.\",\n \"Theorem C.4 (Generalized Discrepancy Bound) Consider functions $E, F : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ .\",\n \"If there exists a distribution $\\\\lambda : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{m} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ such that $\\\\textnormal {corr}_{\\\\lambda }(E, F) \\\\ge \\\\delta $ , then $\\\\mathsf {Q}^{cc, *}_{\\\\frac{1}{2} - \\\\varepsilon }(E)=\\\\Omega \\\\left( \\\\log \\\\left(\\\\frac{\\\\delta + 2\\\\varepsilon - 1}{\\\\textnormal {disc}_{\\\\lambda }(F)} \\\\right)\\\\right).$ Definition C.5 For a distribution $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ and integer $k>0$ , define the distribution $\\\\mu ^{\\\\otimes k} : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mk} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{nk} \\\\rightarrow \\\\mathbb {R}$ by $\\\\mu ^{\\\\otimes k}((X_1, Y_1) \\\\dots , (X_k, Y_k)) = \\\\prod _{i \\\\in [k]}\\\\mu (X_i, Y_i),$ where $X_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{m}$ and $Y_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ for all $i \\\\in [k]$ .\",\n \"We require the following XOR lemma for discrepancy due to Lee, Shraibman and Špalek [24].\",\n \"Theorem C.6 ([24]) Let $P : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be a two-party function and $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ be a distribution.\",\n \"For every integer $k > 0$ , $\\\\textnormal {disc}_{\\\\mu ^{\\\\otimes k}}(\\\\mathsf {PARITY}_k \\\\circ P) \\\\le (8 \\\\textnormal {disc}_{\\\\mu }(P))^{k}.$ We recall Theorem REF below.\",\n \"Theorem C.7 (Restatement of Theorem REF ) Let $r : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ and $G : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ be functions.\",\n \"Let $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^j \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^k \\\\rightarrow \\\\mathbb {R}$ be a balanced distribution with respect to $G$ and $\\\\textnormal {disc}_{\\\\mu }(G) = o(1)$ .\",\n \"If $\\\\frac{8en}{\\\\widetilde{\\\\textnormal {deg}}(r)} \\\\le \\\\left(\\\\frac{1}{\\\\textnormal {disc}_{\\\\mu }(G)}\\\\right)^{1-\\\\beta }$ for some constant $\\\\beta \\\\in (0,1)$ , then $\\\\mathsf {Q}^{cc, *}(r \\\\circ G) = \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(r) \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {disc}_{\\\\mu }(G)}\\\\right)\\\\right).$ In particular, $\\\\mathsf {Q}^{cc, *}(r \\\\circ G) = \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(r) \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {bdisc}(G)}\\\\right)\\\\right).$ For simplicity we assume $j = k = m$ .\",\n \"The general proof follows along similar lines.\",\n \"Let $\\\\widetilde{\\\\textnormal {deg}}(r) = d$ , and let $\\\\psi : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ be a dual witness for this as given by Theorem REF .\",\n \"Let $\\\\nu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ be defined as $\\\\nu (x) = |\\\\psi (x)|$ for all $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ and also define $h: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace $ as $h(x) = \\\\text{sign}(\\\\psi (x))$ for all $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"First note that $\\\\nu $ is a distribution since $\\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} \\\\nu (x) = \\\\sum _{x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} |\\\\psi (x)| = 1$ by Theorem REF .\",\n \"From Theorem REF we also have $\\\\textnormal {corr}_{\\\\nu }(r, h) & > 1/3 \\\\\\\\\\\\widehat{h\\\\nu }(S) & = 0 \\\\quad \\\\text{for all~} |S| < d,$ where $h\\\\nu (x) := h(x)\\\\nu (x) = \\\\psi (x)$ for all $x \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"We will construct a probability distribution $\\\\lambda : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn} \\\\rightarrow \\\\mathbb {R}$ using $\\\\nu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n \\\\rightarrow \\\\mathbb {R}$ and $\\\\mu : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m \\\\rightarrow \\\\mathbb {R}$ such that $r \\\\circ G$ and $h \\\\circ G$ have large correlation under $\\\\lambda $ .\",\n \"We will also show that $\\\\textnormal {disc}_{\\\\lambda }(h \\\\circ G)$ is small.\",\n \"The proof of the theorem would then follow from Theorem REF .\",\n \"Let $X,Y \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn}$ be such that $X = (X_1, \\\\dots , X_n)$ , $Y = (Y_1, \\\\dots , Y_n)$ and $X_i, Y_i \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^m$ for all $i \\\\in [n]$ .\",\n \"Also, let $G(X,Y) = (G(X_1, Y_1), \\\\dots , G(X_n, Y_n)) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ .\",\n \"Define $\\\\lambda : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn} \\\\rightarrow \\\\mathbb {R}$ as follows.\",\n \"$\\\\lambda (X,Y)&= 2^n \\\\cdot \\\\nu (G(X, Y)) \\\\cdot \\\\prod _{i \\\\in [n]} \\\\mu (X_i,Y_i).$ Observe that for any $z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n$ , $\\\\sum _{X,Y : G(X, Y) = z} \\\\lambda (X,Y)&= \\\\sum _{X,Y: G(X,Y) = z} 2^n \\\\cdot \\\\nu (G(X, Y)) \\\\cdot \\\\prod _{i \\\\in [n]} \\\\mu (X_i,Y_i) \\\\nonumber \\\\\\\\&= 2^n \\\\nu (z) \\\\prod _{i \\\\in [n]}\\\\left(\\\\sum _{X_i, Y_i : G(X_i, Y_i) = z_i} \\\\mu (X_i, Y_i)\\\\right)\\\\nonumber \\\\\\\\&= \\\\nu (z), $ where the last equality follows since $\\\\mu $ is balanced w.r.t.\",\n \"$G$ by assumption.\",\n \"Thus $\\\\sum _{X,Y} \\\\lambda (X,Y) = \\\\sum _{z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} \\\\nu (z) = 1.$ We next observe that $\\\\textnormal {corr}_{\\\\lambda }(r \\\\circ G, h \\\\circ G)$ is large.\",\n \"$\\\\textnormal {corr}_{\\\\lambda }(r \\\\circ G, h \\\\circ G)&= \\\\sum _{X,Y} {r \\\\circ G}(X,Y) \\\\cdot {h \\\\circ G}(X,Y) \\\\cdot \\\\lambda (X,Y) \\\\nonumber \\\\\\\\&= \\\\sum _{z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} \\\\sum _{X,Y: G(X,Y) = z} r(z)h(z) \\\\cdot \\\\lambda (X,Y) \\\\nonumber \\\\\\\\&= \\\\sum _{z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} r(z)h(z) \\\\nu (z) \\\\nonumber \\\\\\\\&= \\\\textnormal {corr}_{\\\\nu }(r, h) > 1/3.\",\n \"$ where the last equality follows from Equation (REF ).\",\n \"We now upper bound the discrepancy of $h \\\\circ G$ with respect to a rectangle $R$ , under the distribution $\\\\lambda $ .\",\n \"Let $R \\\\subseteq \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn} \\\\times \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn}$ be any rectangle of the form $R(X, Y) = A(X)B(Y)$ for $A : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn} \\\\rightarrow \\\\left\\\\lbrace 0,1 \\\\right\\\\rbrace $ and $B : \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn} \\\\rightarrow \\\\left\\\\lbrace 0,1 \\\\right\\\\rbrace $ .\",\n \"$\\\\textnormal {\\\\textnormal {disc}}_{\\\\lambda }(h \\\\circ G, R)&= \\\\left|\\\\sum _{X,Y} {h \\\\circ G}(X,Y) \\\\cdot R(X,Y) \\\\cdot \\\\lambda (X,Y) \\\\right| \\\\\\\\&= \\\\left|\\\\sum _{z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} \\\\sum _{X,Y: G(X,Y) = z} {h \\\\circ G}(X,Y) \\\\cdot R(X,Y) \\\\cdot \\\\lambda (X,Y) \\\\right|\\\\\\\\&= 2^n \\\\left|\\\\sum _{z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} h(z) \\\\nu (z) \\\\sum _{X,Y: G(X,Y) = z} R(X,Y) \\\\prod _{i \\\\in [n]} \\\\mu (X_i,Y_i) \\\\right| \\\\\\\\&= 2^n \\\\left|\\\\sum _{z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} \\\\left(\\\\sum _{S \\\\subseteq [n]} \\\\widehat{h\\\\nu }(S)\\\\chi _S(z)\\\\right) \\\\sum _{X,Y: G(X,Y) = z} R(X,Y) \\\\prod _{i \\\\in [n]} \\\\mu (X_i,Y_i) \\\\right|\\\\\\\\&= 2^n\\\\left|\\\\sum _{z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} \\\\sum _{S \\\\subseteq [n]} \\\\widehat{h\\\\nu }(S) \\\\sum _{X,Y: G(X,Y) = z} R(X,Y) \\\\cdot \\\\chi _S(z) \\\\cdot \\\\prod _{i \\\\in [n]} \\\\mu (X_i,Y_i) \\\\right| \\\\\\\\&= 2^n\\\\left|\\\\sum _{z \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^n} \\\\sum _{S \\\\subseteq [n]} \\\\widehat{h\\\\nu }(S) \\\\sum _{X,Y: G(X,Y) = z} R(X,Y) \\\\cdot \\\\prod _{i \\\\in S}G(X_i,Y_i) \\\\cdot \\\\prod _{i \\\\in [n]} \\\\mu (X_i,Y_i) \\\\right| \\\\\\\\& \\\\le 2^n \\\\sum _{S \\\\subseteq [n], |S| \\\\ge d}\\\\left|\\\\widehat{h\\\\nu }(S) \\\\right| \\\\cdot \\\\left|\\\\sum _{X,Y} R(X,Y) \\\\cdot \\\\prod _{i \\\\in S}G(X_i,Y_i)\\\\mu (X_i, Y_i) \\\\cdot \\\\prod _{j \\\\notin S}\\\\mu (X_j,Y_j) \\\\right| \\\\\\\\& \\\\le 2^n \\\\sum _{S \\\\subseteq [n], |S| \\\\ge d}\\\\left|\\\\widehat{h\\\\nu }(S) \\\\right| \\\\cdot \\\\left|\\\\sum _{X_j,Y_j : j \\\\notin S} \\\\prod _{j \\\\notin S}\\\\mu (X_j,Y_j) \\\\left(\\\\sum _{X_i,Y_i : i \\\\in S} A(X) B(Y) \\\\cdot \\\\prod _{i \\\\in S}G\\\\mu (X_i, Y_i)\\\\right) \\\\right|.$ For any $X = (X_1, \\\\dots , X_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn}$ (respectively, $Y = (Y_1, \\\\dots , Y_n) \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{mn}$ ) and set $S \\\\subseteq [n]$ define the string $X_S \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{m|S|}$ (respectively, $Y_S \\\\in \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{m|S|}$ ) by $X_S = (\\\\dots , X_i, \\\\dots )$ (respectively, $Y_S = (\\\\dots , Y_i, \\\\dots )$ ), where $i$ ranges over all elements of $S$ .\",\n \"For any $S$ and fixed $\\\\lbrace X_j : j \\\\notin S\\\\rbrace $ (respectively, $\\\\lbrace Y_j : j \\\\notin S\\\\rbrace $ ) note that $A(X) = A^{\\\\prime }(X_S)$ (respectively, $B(X) = B^{\\\\prime }(X_S)$ ) for some $A^{\\\\prime }: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{m|S|} \\\\rightarrow \\\\left\\\\lbrace 0,1 \\\\right\\\\rbrace $ (respectively, $B^{\\\\prime }: \\\\left\\\\lbrace -1,1 \\\\right\\\\rbrace ^{m|S|} \\\\rightarrow \\\\left\\\\lbrace 0,1 \\\\right\\\\rbrace $ ) which is a function of the fixed values $\\\\lbrace X_j : j \\\\notin S\\\\rbrace $ (respectively, $\\\\lbrace Y_j : j \\\\notin S\\\\rbrace $ ).\",\n \"Let $R^{\\\\prime }(X_S, Y_S) = A^{\\\\prime }(X_S) B^{\\\\prime }(Y_S)$ .\",\n \"Continuing from the above we obtain, $\\\\textnormal {\\\\textnormal {disc}}_{\\\\lambda }(h \\\\circ G, R)&\\\\le 2^n \\\\sum _{S \\\\subseteq [n], |S| \\\\ge d}\\\\left|\\\\widehat{h\\\\nu }(S) \\\\right| \\\\left|\\\\sum _{X_j,Y_j : j \\\\notin S} \\\\prod _{j \\\\notin S}\\\\mu (X_j,Y_j) \\\\left(\\\\sum _{X_i,Y_i : i \\\\in S} R^{\\\\prime }(X_S, Y_S) \\\\prod _{i \\\\in S}G\\\\mu (X_i, Y_i)\\\\right) \\\\right|\\\\\\\\& \\\\le 2^n \\\\sum _{S \\\\subseteq [n], |S| \\\\ge d}\\\\left|\\\\widehat{h\\\\nu }(S) \\\\right| \\\\left| \\\\sum _{X_j,Y_j : j \\\\notin S} \\\\prod _{j \\\\notin S}\\\\mu _j(X_j,Y_j) \\\\left(\\\\textnormal {disc}_{\\\\mu ^{\\\\otimes |S|}}(\\\\mathsf {XOR}_{|S|} \\\\circ G)\\\\right) \\\\right| \\\\\\\\&= 2^n \\\\sum _{S \\\\subseteq [n], |S| \\\\ge d}\\\\left|\\\\widehat{h\\\\nu }(S) \\\\right| \\\\cdot \\\\textnormal {disc}_{\\\\mu ^{\\\\otimes |S|}}(\\\\mathsf {XOR}_{|S|} \\\\circ G)\\\\\\\\&\\\\le \\\\sum _{S \\\\subseteq [n], |S| \\\\ge d} \\\\left(8 \\\\textnormal {disc}_{\\\\mu }(G)\\\\right)^{|S|}\\\\\\\\&= \\\\sum _{k = d}^n \\\\binom{n}{k} \\\\left(8 \\\\textnormal {disc}_{\\\\mu }(G)\\\\right)^{k} \\\\le \\\\sum _{k = d}^n \\\\left( \\\\frac{8en}{k} \\\\textnormal {disc}_{\\\\mu }(G)\\\\right)^{k}\\\\\\\\&\\\\le \\\\sum _{k = d}^n \\\\left(\\\\textnormal {disc}_{\\\\mu }(G)^{\\\\beta }\\\\right)^{k}\\\\\\\\&\\\\le \\\\frac{\\\\textnormal {disc}_{\\\\mu }(G)^{d \\\\cdot \\\\beta }}{1 - \\\\textnormal {disc}_{\\\\mu }(G)^{\\\\beta }}$ Since $\\\\textnormal {disc}_{\\\\mu }(G) = o(1)$ by assumption ($\\\\textnormal {disc}_{\\\\mu }(G) = 1 - \\\\Omega (1)$ suffices, but we assume $\\\\textnormal {disc}_{\\\\mu }(G) = o(1)$ in the statement for readability) and $\\\\beta \\\\in (0, 1)$ is a constant, we have $1 - \\\\textnormal {disc}_\\\\mu (G)^\\\\beta = \\\\Omega (1)$ , and hence $\\\\textnormal {disc}_{\\\\lambda }(h \\\\circ G) = O(\\\\textnormal {disc}_{\\\\mu }(G)^{d \\\\beta }).$ Hence, $\\\\mathsf {Q}^{cc}_{1/2 - 2/5}(r \\\\circ G)&\\\\ge \\\\log \\\\left(\\\\frac{1/3 + 4/5 - 1}{\\\\textnormal {disc}_{\\\\lambda }(h \\\\circ G)}\\\\right) \\\\\\\\&= \\\\Omega \\\\left( \\\\log \\\\left(\\\\frac{1}{\\\\textnormal {disc}_{\\\\mu }(G)^{d\\\\beta }}\\\\right) \\\\right) \\\\\\\\&= \\\\Omega \\\\left(\\\\widetilde{\\\\textnormal {deg}}(r) \\\\log \\\\left( \\\\frac{1}{\\\\textnormal {disc}_{\\\\mu }(g)}\\\\right)\\\\right) .$ The theorem follows since $\\\\mathsf {Q}^{cc}(F) = \\\\Theta (\\\\mathsf {Q}^{cc}_{\\\\varepsilon }(F))$ for all constants $\\\\varepsilon \\\\in (0, 1/2)$ .\"\n ]\n]"},"id":{"kind":"string","value":"2012.05233"}}},{"rowIdx":3694,"cells":{"text":{"kind":"list like","value":[["Jets and elliptic flow correlations at low and high transverse momenta\n in ultrarelativistic A+A collisions"],["Abstract Data from the Large Hadron Collider on elliptic flow correlations at low and high $p_T$ from Pb+Pb collisions at $\\sqrt{s_{NN}} = 5.02$~TeV are analyzed and interpreted in the framework of the HYDJET++ model.","This model allows us to describe simultaneously the region of both low and high transverse momenta and, therefore, to reproduce the experimentally observed nontrivial centrality dependence of elliptic flow correlations.","The origin of the correlations between low and high-$p_T$ flow components in peripheral lead-lead collisions is traced to correlations of particles in jets."],["Introduction","A number of exquisite and intriguing phenomena, which have never been systematically studied at the accelerators of previous generation, has been observed after the start of Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) heavy-ion program.","Azimuthal anisotropy of multi-particle production is among them as a powerful probe of collective properties of a new state of matter, quark-gluon plasma (QGP), see, e.g., [1].","It is commonly described by the Fourier decomposition of the invariant cross section in a form [2], [3]: $\\displaystyle E\\frac{d^3N}{dp^3} &=&\\frac{d^2N}{2\\pi p_{T}dp_{T}d\\eta }\\nonumber \\\\&\\times & \\left\\lbrace 1+2\\sum \\limits _{n = 1}^\\infty v_{n}(p_{T},\\eta )\\cos { \\left[ n(\\varphi -\\Psi _{n}^{PP}) \\right] }\\right\\rbrace ,$ where $p_{T}$ is the transverse momentum, $\\eta $ is the pseudorapidity, $\\varphi $ is the azimuthal angle with respect to the participant plane $\\Psi _{n}^{PP}$ of $n$ -th order, and $v_{n}$ are the Fourier coefficients: $v_n = \\langle \\langle \\cos { \\left[ n(\\varphi -\\Psi _{n}^{PP}) \\right]}\\rangle \\rangle .$ The averaging in the last equation is performed over all particles in a single event and over all events.","The second harmonic, $v_2$ , typically referred to as elliptic flow, is the most thoroughly investigated one, for review see [4] and references therein.","The reason is obvious: it directly relates the anisotropic shape of the overlap region of the colliding nuclei to the corresponding anisotropy of the outgoing momentum distribution.","At relatively low transverse momenta, $p_{T} < 3\\div 4$ ${\\rm GeV}/c$ , the azimuthal anisotropy results from a pressure-driven anisotropic expansion of the created matter, with more particles emitted in the direction of the largest pressure gradients [5].","At higher transverse momenta, this anisotropy is understood to arise from the path-length dependent energy loss of partonic jets as they traverse the matter, with more jet particles emitted in the direction of shortest path-length [6].","The correlations between soft and hard contributions to anisotropic flow has attracted a lot of attention, see, e.g., [7], [8], [9] and references therein.","Recently, the interesting correlations between $v_2$ values at high and low transverse momenta for different centralities has been reported by the CMS Collaboration [10].","These correlations can also be seen in the ATLAS data [11].","In the present paper we analyze and interpret this intriguing experimental observation within the hydjet++ model, which allows us to perform such analysis due to its remarkable feature, namely the presence of soft and hard physics simultaneously.","The paper is organized as follows.","Basic features of the model are sketched in Sec. .","Here the origin of elliptic flow in the model, and interplay between the soft and hard physics are discussed.","Section presents the results of model calculations of hadronic elliptic flow in Pb+Pb collisions at $\\sqrt{s_{\\mbox{\\tiny {\\it {NN}}}}}= 5.02$ TeV.","The calculations are in fair agreement with the experimental data, and the role of jets is clarified.","Finally, conclusions are drawn in Sec.","."],["The model hydjet++ [12] is the popular and known event generator, which describes successfully the large number of physical observables measured in heavy-ion collisions during the RHIC and LHC operation.","Among them are centrality and pseudorapidity dependence of inclusive charged particle multiplicity, transverse momentum spectra and $\\pi ^\\pm \\pi ^\\pm $ correlation radii in central Pb+Pb collisions [13], momentum and centrality dependence of elliptic and higher-order harmonic coefficients [14], [15], [16], [17], [18], flow fluctuations [19], angular dihadron correlations [20], forward-backward multiplicity correlations [21], jet quenching effects [22], [23] and heavy meson production [24], [25], [26].","Details of the model can be found in the hydjet++ manual [12].","The event generator includes two independent components: the soft, hydro-type state and the hard state resulting from in-medium multi-parton fragmentation.","The soft component is the thermal hadronic state generated on the chemical and thermal freeze-out hypersurfaces prescribed by the parametrization of relativistic hydrodynamics with preset freeze-out conditions.","It represents the adapted version of the event generator fastmc [27], [28].","Particle multiplicities are calculated using the effective thermal volume approach and Poisson multiplicity distribution around its mean value, which is supposed to be proportional to the number of participating nucleons for a given impact parameter in a $A\\mbox{+}A$ collision.","To simulate the elliptic flow effect, the hydro-inspired parametrization for the momentum and spatial anisotropy of soft hadron emission source is implemented [12], [29].","Note that there are two parameters which govern the strength and the direction of the elliptic flow in original hydjet++ version [12].","The first one is the spatial anisotropy $\\epsilon (b)$ .","It is responsible for the elliptic modulation of the final freeze-out hyper-surface at a given impact parameter $b$ .","The second one is the momentum anisotropy $\\delta (b)$ , dealing with the modulation of the flow velocity profile.","One can treat these two parameters as independent ones and, therefore, adjust it separately for each centrality by comparing to data.","Although providing better agreement with the data, this procedure leads to significant increase of the parameters to be tuned.","Therefore, for the sake of simplicity we opted for another scenario, which is implemented in basic version of the model.","According to it, both parameters are correlated through the dependence of the elliptic flow coefficient $v_2$ on both $\\epsilon (b)$ and $\\delta (b)$ , obtained in the hydrodynamic approach [29]: $v_2(\\epsilon , \\delta ) \\propto \\frac{2(\\delta -\\epsilon )}{(1-\\delta ^2)(1-\\epsilon ^{2})}~.$ Because $v_2(b)$ is proportional to the initial ellipticity $\\epsilon _0 (b)=b/2R_A$ , where $R_A$ is the radius of colliding nucleus, the relation between $\\epsilon (b)$ and $\\delta (b)$ reads [12]: $\\delta = \\frac{\\sqrt{1+4B(\\epsilon +B)}-1}{2B}~, \\quad B=C(1-\\epsilon ^2)\\epsilon ~, \\quad \\epsilon =k \\epsilon _0~.$ Two new parameters $C$ and $k$ , entering the last expression, are independent on centrality and, therefore, should be obtained from the fit to the experimental data.","In hard sector the model propagates the hard partons through the expanding quark-gluon plasma and takes into account both collisional loss and gluon radiation due to parton rescattering.","It is based on the pyquen partonic energy loss model [30].","The number of jets is generated according to the binomial distribution.","Their mean number in an $A\\mbox{+}A$ event is calculated as a product of the number of binary nucleon-nucleon (NN) sub-collisions at a given impact parameter and the integral cross section of the hard process in NN collisions with the minimum transverse momentum transfer $p_T^{\\rm min}$ .","The latter is the input parameter of the model.","In the hydjet++ framework, partons produced in (semi)hard processes with the momentum transfer lower than $p_T^{\\rm min}$ are considered as being “thermalized”, and their hadronization products are automatically included in the soft component of the event.","Recall that there are many competing event generators successfully describing the soft [31], [32], [33], [34], [35], [36], [37], [38] and hard [39], [40], [41] momentum components of particle production in heavy-ion collisions separately.","The hydjet++ is among the few ones [42], [43] which allows us to study the soft and hard physics simultaneously."],["Results","To measure azimuthal correlations and to extract the Fourier coefficients the CMS Collaboration employs the cumulant and the scalar product (SP) methods.","The 2- and 4-particle correlations are defined as: $\\langle \\langle 2\\rangle \\rangle = \\langle \\langle e^{in(\\varphi _1-\\varphi _2)}\\rangle \\rangle , \\quad \\quad \\langle \\langle 4\\rangle \\rangle = \\langle \\langle e^{in(\\varphi _1+\\varphi _2-\\varphi _3-\\varphi _4)}\\rangle \\rangle .$ Here the double averaging is performed over all particle combinations and over all events.","The multi-particle cumulant method is applied to measure $v_2$ from 4-particle correlations.","The 2-nd order and 4-th order cumulants, $c_n\\lbrace 2\\rbrace $ and $c_n\\lbrace 4\\rbrace $ , respectively, are given as [44] $c_n\\lbrace 2\\rbrace = \\langle \\langle 2\\rangle \\rangle , \\quad \\quad c_n\\lbrace 4\\rbrace = \\langle \\langle 4\\rangle \\rangle -2 \\cdot \\langle \\langle 2\\rangle \\rangle ^{2}.$ The cumulants are expressed in terms of the corresponding $Q_n$ vectors [45].","For differential flow calculations the restricted 2- and 4-particle correlations, $\\langle \\langle 2^\\prime \\rangle \\rangle $ and $\\langle \\langle 4^\\prime \\rangle \\rangle $ , are defined, where transverse momentum of one of the particles is limited to being within a certain $p_T$ bin.","The differential cumulant $d_n\\lbrace 4\\rbrace $ reads $d_n\\lbrace 4\\rbrace = \\langle \\langle 4^\\prime \\rangle \\rangle -2 \\cdot \\langle \\langle 2^\\prime \\rangle \\rangle \\cdot \\langle \\langle 2\\rangle \\rangle .$ Finally, the differential $v_n\\lbrace 4\\rbrace (p_T)$ coefficients are derived as $v_n\\lbrace 4\\rbrace (p_T) = -d_n\\lbrace 4\\rbrace \\cdot (-c_n\\lbrace 4\\rbrace )^{-3/4}.$ This technique is used by the CMS Collaboration [10].","The same technique is applied in the present paper to calculate the corresponding differential $v_2\\lbrace 4\\rbrace (p_T)$ coefficients in the hydjet++ model.","Figure: (Color online).The comparison of CMS data for v 2 {4}(p T )v_2\\lbrace 4\\rbrace (p_T) (triangles) and hydjet++ calculations for v 2 {4}(p T )v_2\\lbrace 4\\rbrace (p_T) (squares) andv 2 RP (p T )v_2^{RP}(p_T) (circles) in Pb+Pb collisions at s NN =5.02\\sqrt{s_{\\mbox{\\tiny {\\it {NN}}}}}= 5.02 TeVfor the centrality 30–40%.","Lines are drawn to guide the eye.To compare model calculations with the data we generated Pb+Pb collisions at $\\sqrt{s_{\\mbox{\\tiny {\\it {NN}}}}}= 5.02$ TeV in seven centrality bins: $\\sigma /\\sigma _{geo}=$ 5–10,10–15,15–20,20–30,30–40,40–50 and 50–60%.","Statistics of generated events varies from 2 million for semi-central to 7 million for very periperal collisions, respectively.","Figure REF shows the elliptic flow restored by 4-cumulant method from the hydjet++ calculated events in comparison with the CMS data [10].","The model generated “true” value of the elliptic flow coefficient $v_2^{RP}(p_T)$ is also presented.","It is calculated with respect to the reaction plane ($RP$ ), which is exactly known and is directed along the impact parameter ${\\bf b}$ in the model.","At relatively low transverse momenta, $p_{T}<3\\div 4$ GeV/$c$ , the measured elliptic flow, $v_2\\lbrace 4\\rbrace (p_T)$ (CMS), and the restored one, $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), are practically equal to the generated original elliptic coefficient $v_2^{RP}(p_T)$ .","It is not surprising, because in this momentum region the bulk of produced particles are mainly correlated with the reaction plane only, whereas nonflow correlations are relatively small.","At higher transverse momenta, $p_{T}>10$ GeV/$c$ , which is the region of the special interest here, the values of the measured elliptic flow and the flow restored from the model calculations are close to each other, but noticeably larger than the generated elliptic flow $v_2^{RP}(p_T)$ .","In the intermediate region, $4< p_{T}<10$ GeV/$c$ , the both model calculated coefficients, $v_2^{RP}(p_T)$ and $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), lie below the CMS data.","Figure: (Color online).The elliptic flow soft (triangles) and hard (squares) components andresulting total flow v 2 RP (p T )v_2^{RP}(p_T) (circles) in hydjet++ for Pb+Pb collisions at s NN =5.02\\sqrt{s_{NN}}=5.02 TeV with the centrality 30–40%.Figure REF illustrates the structure of elliptic flow in hydjet++ .","Here the coefficients $v_2^{RP}(p_T)$ are calculated with respect to the reaction plane separately for each component, soft and hard, together with the resulting total value.","This illustration shows the slight model drawback in the intermediate momentum region.","Originally, the model includes the adequate description of soft and hard physics.","In the intermediate $p_T$ region the result is obtained by a simple superposition of two independent contributions.","The crosslinking is regulated by one parameter, $p_T^{\\rm min}$ , which is the minimum transverse momentum transfer in a hard parton subprocess.","For the transverse momentum spectra this procedure is painless since the crosslinking takes place for a continuous function, and its fracture is smoothed out by the overlapping of particles from both contributions.","As a result, we describe effectively the transverse momentum spectra even in the intermediate region also [13] without any additional mechanism.","For elliptic flow coefficients the crosslinking takes place for a “discontinuous” function, see the crosslinking region of $p_T \\simeq 10$ GeV/$c$ , where $v_2^{RP}(p_T)$ (soft) and $v_2^{RP}(p_T)$ (hard) have the quite different values.","Thus, the simple smoothing is not enough to describe this region by a simple superposition.","It means that some improvements of the model are required to describe successfully the whole $p_T$ -region.","For instance, it can be a minijet production or some other mechanism.","Fortunately, this “problematic” region of the model is not used in the present flow correlation analysis.","Now we focus on the high transverse momentum region, to study which the large number of events should be generated.","In this region the hydjet++ flow, restored by cumulant method, is close to the measured one, but it is visibly larger than the original elliptic flow generated in the model.","This observation requires an explanation.","The azimuthal anisotropy arises in the model as a result of jet quenching, see [22], [23].","Due to the path-length dependent energy loss of partonic jets as they traverse the matter, the jet particles become to be correlated with the reaction plane.","However, the particle correlations relative to the jet axis remain also.","Unlike the region of small transverse momentum, there are at least two singled out directions, the reaction plane and the jet axis, respectively, relative to which particles are correlated.","In this case we can decompose the azimuthal distribution in a form $\\displaystyle \\frac{dN}{d\\varphi } &=& N_0 \\big (1+ 2 v_2^{RP} \\cos { \\left[2(\\varphi -\\Psi _{2}^{RP}) \\right] }\\nonumber \\\\&+& 2 v_2^{jet} \\cos { \\left[ 2(\\varphi -\\Psi _{2}^{jet}) \\right] } \\big ),$ where the direction of jet axis $\\Psi _{2}^{jet}$ is randomly oriented with respect to the reaction plane $\\Psi _{2}^{RP}$ , which is set to zero in the model calculations.","$N_0$ is the normalization factor.","The 2- and 4-particle correlations are estimated as $\\langle \\langle 2\\rangle \\rangle \\simeq (v_2^{RP})^2 + (v_2^{jet})^2 ,\\quad \\quad \\langle \\langle 4\\rangle \\rangle \\simeq [(v_2^{RP})^2 + (v_2^{jet})^2]^2$ and they are sensitive to both types of correlations discussed above.","Herewith $\\langle \\langle 2\\rangle \\rangle ^{1/2}$ and $\\langle \\langle 4\\rangle \\rangle ^{1/4}$ are always larger than the “true” elliptic coefficient $v_2^{RP}(p_T)$ which we are interested in.","Figure: (Color online).The correlation between v 2 v_2 values at low and highp T p_T in Pb+Pb collisions at s NN =5.02\\sqrt{s_{\\mbox{\\tiny {\\it {NN}}}}}= 5.02 TeVas function of centrality.The points represent the centrality bins 5–10,10–15,15–20,20–30,30–40,40–50, and 50–60%.The model results are presented for v 2 RP v_2^{RP} (circles) andv 2 {4}v_2\\lbrace 4\\rbrace (squares).","Data (triangles) are taken from .Model calculations are performed (a) with and(b) without the jet quenching.The correlations between the low-$p_T$ and high-$p_T$ elliptic flow which are the main issue of our study are displayed in Fig.","REF (a).","The picked up intervals are $14< p_{T}<20$ GeV/$c$ for high and $1.0< p_{T}<1.25$ GeV/$c$ for low transverse momenta, respectively.","One can see that the model calculations, $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), reproduce the experimentally observed centrality dependence of the flow correlations fairly well except the last centrality bin, while the generated original elliptic flow $v_2^{RP}(p_T)$ at high $p_T$ goes always lower than the restored flow in accordance with the explanation above.","The deviations become significant for the centralities larger than 30–40%.","In this centrality interval the anisotropy caused by the jet quenching becomes to die out, but particles remain correlated with the axes of jets.","Note that in the low-$p_T$ region the integrated values of $v_2$ in the model are lower than the data.","This is due to simplification, discussed in Sec.",", aiming to reduce the number of free parameters in the model.","To reach better quantitative agreement with the data one has to treat both anisotropy parameters, $\\varepsilon (b)$ and $\\delta (b)$ , as independent ones, see [19].","This circumstance, however, does not affect the main results associated with identifying the role of jets on the behavior of cumulants.","Note also, that in the high-$p_T$ sector the agreement between the model results and the data is good.","Figure REF (b) demonstrates the comparison of model calculations without jet quenching effects with the data.","In this case $v_2^{RP} =0$ and the correlations relative to the jet axis contributes to the 4-th order cumulant only, but the magnitude of jet correlations is not strong enough to reproduce data.","Thus, in the collisions with centrality up to 30–40% the azimuthal anisotropy due to jet quenching reveals itself, whereas in more peripheral collisions the jet correlations contribute significantly to the 4-th order cumulant.","The reason is as follows.","Since in peripheral heavy-ion collisions there are simply quantitatively fewer nucleon-nucleon collisions, then, fewer jets are produced.","In the limiting case there is one back-to-back pair, and the method sees this axis and anisotropy.","In more central collisions there are many jet pairs.","They are all distributed randomly in azimuth, therefore, the anisotropy caused by jets tends to zero.","It is worth noting that jets were the main source of violation of the number-of-constituent quark (NCQ) scaling in hydjet++ calculations of differential elliptic [14], [15] and triangular [18] flow.","The linear fit [10], performed by CMS Collaboration to data on elliptic flow correlations at low and high transverse momenta, also indicates some sort of scaling behavior.","In contrast to situation with the NCQ scaling, here jets work toward the scaling fulfillment."],["Conclusions","The phenomenological analysis of elliptic flow correlations at low and high $p_T$ in Pb+Pb collisions at center-of-mass energy 5.02 TeV per nucleon pair has been performed within the two-component hydjet++ model.","These correlations are stipulated by the fact that the magnitudes of anisotropy at low and high $p_T$ are mainly determined by the value of initial ellipticity of the nuclei overlapping.","At relatively low transverse momenta, $p_{T}<3\\div 4$ GeV/$c$ , the model generated elliptic flow $v_2^{RP}(p_T)$ and its value restored by 4-cumulant method, $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), are very close to the differential elliptic flow $v_2\\lbrace 4\\rbrace (p_T)$ measured by the CMS Collaboration.","At high transverse momenta $p_{T}> 10$ GeV/$c$ the cumulants are sensitive to both the anisotropy due to jet quenching, $v_2^{RP}$ , and the particle correlations with the jet axis, $v_2^{jet}$ .","In the collisions with centrality up to 30–40% the 4-cumulant method “measures” mainly the azimuthal anisotropy due to jet quenching, whereas in more peripheral collisions it is affected by the jet correlations primarily.","The model calculations restored by this method, $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), reproduce the experimentally observed centrality dependence of elliptic flow correlations rather well without any additional tunes of model parameters.","Fruitful discussions with A.I.","Demyanov and L.V.","Malinina are gratefully acknowledged.","This work was supported in parts by Russian Foundation for Basic Research (RFBR) under Grant No.","18-02-00155, Grant No.","18-02-40084 and Grant No.","18-02-40085.","L.B.","and E.Z.","acknowledge support of the Norwegian Research Council (NFR) under grant No.","255253/F50 - “CERN Heavy Ion Theory\"."]],"string":"[\n [\n \"Jets and elliptic flow correlations at low and high transverse momenta\\n in ultrarelativistic A+A collisions\"\n ],\n [\n \"Abstract Data from the Large Hadron Collider on elliptic flow correlations at low and high $p_T$ from Pb+Pb collisions at $\\\\sqrt{s_{NN}} = 5.02$~TeV are analyzed and interpreted in the framework of the HYDJET++ model.\",\n \"This model allows us to describe simultaneously the region of both low and high transverse momenta and, therefore, to reproduce the experimentally observed nontrivial centrality dependence of elliptic flow correlations.\",\n \"The origin of the correlations between low and high-$p_T$ flow components in peripheral lead-lead collisions is traced to correlations of particles in jets.\"\n ],\n [\n \"Introduction\",\n \"A number of exquisite and intriguing phenomena, which have never been systematically studied at the accelerators of previous generation, has been observed after the start of Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) heavy-ion program.\",\n \"Azimuthal anisotropy of multi-particle production is among them as a powerful probe of collective properties of a new state of matter, quark-gluon plasma (QGP), see, e.g., [1].\",\n \"It is commonly described by the Fourier decomposition of the invariant cross section in a form [2], [3]: $\\\\displaystyle E\\\\frac{d^3N}{dp^3} &=&\\\\frac{d^2N}{2\\\\pi p_{T}dp_{T}d\\\\eta }\\\\nonumber \\\\\\\\&\\\\times & \\\\left\\\\lbrace 1+2\\\\sum \\\\limits _{n = 1}^\\\\infty v_{n}(p_{T},\\\\eta )\\\\cos { \\\\left[ n(\\\\varphi -\\\\Psi _{n}^{PP}) \\\\right] }\\\\right\\\\rbrace ,$ where $p_{T}$ is the transverse momentum, $\\\\eta $ is the pseudorapidity, $\\\\varphi $ is the azimuthal angle with respect to the participant plane $\\\\Psi _{n}^{PP}$ of $n$ -th order, and $v_{n}$ are the Fourier coefficients: $v_n = \\\\langle \\\\langle \\\\cos { \\\\left[ n(\\\\varphi -\\\\Psi _{n}^{PP}) \\\\right]}\\\\rangle \\\\rangle .$ The averaging in the last equation is performed over all particles in a single event and over all events.\",\n \"The second harmonic, $v_2$ , typically referred to as elliptic flow, is the most thoroughly investigated one, for review see [4] and references therein.\",\n \"The reason is obvious: it directly relates the anisotropic shape of the overlap region of the colliding nuclei to the corresponding anisotropy of the outgoing momentum distribution.\",\n \"At relatively low transverse momenta, $p_{T} < 3\\\\div 4$ ${\\\\rm GeV}/c$ , the azimuthal anisotropy results from a pressure-driven anisotropic expansion of the created matter, with more particles emitted in the direction of the largest pressure gradients [5].\",\n \"At higher transverse momenta, this anisotropy is understood to arise from the path-length dependent energy loss of partonic jets as they traverse the matter, with more jet particles emitted in the direction of shortest path-length [6].\",\n \"The correlations between soft and hard contributions to anisotropic flow has attracted a lot of attention, see, e.g., [7], [8], [9] and references therein.\",\n \"Recently, the interesting correlations between $v_2$ values at high and low transverse momenta for different centralities has been reported by the CMS Collaboration [10].\",\n \"These correlations can also be seen in the ATLAS data [11].\",\n \"In the present paper we analyze and interpret this intriguing experimental observation within the hydjet++ model, which allows us to perform such analysis due to its remarkable feature, namely the presence of soft and hard physics simultaneously.\",\n \"The paper is organized as follows.\",\n \"Basic features of the model are sketched in Sec. .\",\n \"Here the origin of elliptic flow in the model, and interplay between the soft and hard physics are discussed.\",\n \"Section presents the results of model calculations of hadronic elliptic flow in Pb+Pb collisions at $\\\\sqrt{s_{\\\\mbox{\\\\tiny {\\\\it {NN}}}}}= 5.02$ TeV.\",\n \"The calculations are in fair agreement with the experimental data, and the role of jets is clarified.\",\n \"Finally, conclusions are drawn in Sec.\",\n \".\"\n ],\n [\n \"The model hydjet++ [12] is the popular and known event generator, which describes successfully the large number of physical observables measured in heavy-ion collisions during the RHIC and LHC operation.\",\n \"Among them are centrality and pseudorapidity dependence of inclusive charged particle multiplicity, transverse momentum spectra and $\\\\pi ^\\\\pm \\\\pi ^\\\\pm $ correlation radii in central Pb+Pb collisions [13], momentum and centrality dependence of elliptic and higher-order harmonic coefficients [14], [15], [16], [17], [18], flow fluctuations [19], angular dihadron correlations [20], forward-backward multiplicity correlations [21], jet quenching effects [22], [23] and heavy meson production [24], [25], [26].\",\n \"Details of the model can be found in the hydjet++ manual [12].\",\n \"The event generator includes two independent components: the soft, hydro-type state and the hard state resulting from in-medium multi-parton fragmentation.\",\n \"The soft component is the thermal hadronic state generated on the chemical and thermal freeze-out hypersurfaces prescribed by the parametrization of relativistic hydrodynamics with preset freeze-out conditions.\",\n \"It represents the adapted version of the event generator fastmc [27], [28].\",\n \"Particle multiplicities are calculated using the effective thermal volume approach and Poisson multiplicity distribution around its mean value, which is supposed to be proportional to the number of participating nucleons for a given impact parameter in a $A\\\\mbox{+}A$ collision.\",\n \"To simulate the elliptic flow effect, the hydro-inspired parametrization for the momentum and spatial anisotropy of soft hadron emission source is implemented [12], [29].\",\n \"Note that there are two parameters which govern the strength and the direction of the elliptic flow in original hydjet++ version [12].\",\n \"The first one is the spatial anisotropy $\\\\epsilon (b)$ .\",\n \"It is responsible for the elliptic modulation of the final freeze-out hyper-surface at a given impact parameter $b$ .\",\n \"The second one is the momentum anisotropy $\\\\delta (b)$ , dealing with the modulation of the flow velocity profile.\",\n \"One can treat these two parameters as independent ones and, therefore, adjust it separately for each centrality by comparing to data.\",\n \"Although providing better agreement with the data, this procedure leads to significant increase of the parameters to be tuned.\",\n \"Therefore, for the sake of simplicity we opted for another scenario, which is implemented in basic version of the model.\",\n \"According to it, both parameters are correlated through the dependence of the elliptic flow coefficient $v_2$ on both $\\\\epsilon (b)$ and $\\\\delta (b)$ , obtained in the hydrodynamic approach [29]: $v_2(\\\\epsilon , \\\\delta ) \\\\propto \\\\frac{2(\\\\delta -\\\\epsilon )}{(1-\\\\delta ^2)(1-\\\\epsilon ^{2})}~.$ Because $v_2(b)$ is proportional to the initial ellipticity $\\\\epsilon _0 (b)=b/2R_A$ , where $R_A$ is the radius of colliding nucleus, the relation between $\\\\epsilon (b)$ and $\\\\delta (b)$ reads [12]: $\\\\delta = \\\\frac{\\\\sqrt{1+4B(\\\\epsilon +B)}-1}{2B}~, \\\\quad B=C(1-\\\\epsilon ^2)\\\\epsilon ~, \\\\quad \\\\epsilon =k \\\\epsilon _0~.$ Two new parameters $C$ and $k$ , entering the last expression, are independent on centrality and, therefore, should be obtained from the fit to the experimental data.\",\n \"In hard sector the model propagates the hard partons through the expanding quark-gluon plasma and takes into account both collisional loss and gluon radiation due to parton rescattering.\",\n \"It is based on the pyquen partonic energy loss model [30].\",\n \"The number of jets is generated according to the binomial distribution.\",\n \"Their mean number in an $A\\\\mbox{+}A$ event is calculated as a product of the number of binary nucleon-nucleon (NN) sub-collisions at a given impact parameter and the integral cross section of the hard process in NN collisions with the minimum transverse momentum transfer $p_T^{\\\\rm min}$ .\",\n \"The latter is the input parameter of the model.\",\n \"In the hydjet++ framework, partons produced in (semi)hard processes with the momentum transfer lower than $p_T^{\\\\rm min}$ are considered as being “thermalized”, and their hadronization products are automatically included in the soft component of the event.\",\n \"Recall that there are many competing event generators successfully describing the soft [31], [32], [33], [34], [35], [36], [37], [38] and hard [39], [40], [41] momentum components of particle production in heavy-ion collisions separately.\",\n \"The hydjet++ is among the few ones [42], [43] which allows us to study the soft and hard physics simultaneously.\"\n ],\n [\n \"Results\",\n \"To measure azimuthal correlations and to extract the Fourier coefficients the CMS Collaboration employs the cumulant and the scalar product (SP) methods.\",\n \"The 2- and 4-particle correlations are defined as: $\\\\langle \\\\langle 2\\\\rangle \\\\rangle = \\\\langle \\\\langle e^{in(\\\\varphi _1-\\\\varphi _2)}\\\\rangle \\\\rangle , \\\\quad \\\\quad \\\\langle \\\\langle 4\\\\rangle \\\\rangle = \\\\langle \\\\langle e^{in(\\\\varphi _1+\\\\varphi _2-\\\\varphi _3-\\\\varphi _4)}\\\\rangle \\\\rangle .$ Here the double averaging is performed over all particle combinations and over all events.\",\n \"The multi-particle cumulant method is applied to measure $v_2$ from 4-particle correlations.\",\n \"The 2-nd order and 4-th order cumulants, $c_n\\\\lbrace 2\\\\rbrace $ and $c_n\\\\lbrace 4\\\\rbrace $ , respectively, are given as [44] $c_n\\\\lbrace 2\\\\rbrace = \\\\langle \\\\langle 2\\\\rangle \\\\rangle , \\\\quad \\\\quad c_n\\\\lbrace 4\\\\rbrace = \\\\langle \\\\langle 4\\\\rangle \\\\rangle -2 \\\\cdot \\\\langle \\\\langle 2\\\\rangle \\\\rangle ^{2}.$ The cumulants are expressed in terms of the corresponding $Q_n$ vectors [45].\",\n \"For differential flow calculations the restricted 2- and 4-particle correlations, $\\\\langle \\\\langle 2^\\\\prime \\\\rangle \\\\rangle $ and $\\\\langle \\\\langle 4^\\\\prime \\\\rangle \\\\rangle $ , are defined, where transverse momentum of one of the particles is limited to being within a certain $p_T$ bin.\",\n \"The differential cumulant $d_n\\\\lbrace 4\\\\rbrace $ reads $d_n\\\\lbrace 4\\\\rbrace = \\\\langle \\\\langle 4^\\\\prime \\\\rangle \\\\rangle -2 \\\\cdot \\\\langle \\\\langle 2^\\\\prime \\\\rangle \\\\rangle \\\\cdot \\\\langle \\\\langle 2\\\\rangle \\\\rangle .$ Finally, the differential $v_n\\\\lbrace 4\\\\rbrace (p_T)$ coefficients are derived as $v_n\\\\lbrace 4\\\\rbrace (p_T) = -d_n\\\\lbrace 4\\\\rbrace \\\\cdot (-c_n\\\\lbrace 4\\\\rbrace )^{-3/4}.$ This technique is used by the CMS Collaboration [10].\",\n \"The same technique is applied in the present paper to calculate the corresponding differential $v_2\\\\lbrace 4\\\\rbrace (p_T)$ coefficients in the hydjet++ model.\",\n \"Figure: (Color online).The comparison of CMS data for v 2 {4}(p T )v_2\\\\lbrace 4\\\\rbrace (p_T) (triangles) and hydjet++ calculations for v 2 {4}(p T )v_2\\\\lbrace 4\\\\rbrace (p_T) (squares) andv 2 RP (p T )v_2^{RP}(p_T) (circles) in Pb+Pb collisions at s NN =5.02\\\\sqrt{s_{\\\\mbox{\\\\tiny {\\\\it {NN}}}}}= 5.02 TeVfor the centrality 30–40%.\",\n \"Lines are drawn to guide the eye.To compare model calculations with the data we generated Pb+Pb collisions at $\\\\sqrt{s_{\\\\mbox{\\\\tiny {\\\\it {NN}}}}}= 5.02$ TeV in seven centrality bins: $\\\\sigma /\\\\sigma _{geo}=$ 5–10,10–15,15–20,20–30,30–40,40–50 and 50–60%.\",\n \"Statistics of generated events varies from 2 million for semi-central to 7 million for very periperal collisions, respectively.\",\n \"Figure REF shows the elliptic flow restored by 4-cumulant method from the hydjet++ calculated events in comparison with the CMS data [10].\",\n \"The model generated “true” value of the elliptic flow coefficient $v_2^{RP}(p_T)$ is also presented.\",\n \"It is calculated with respect to the reaction plane ($RP$ ), which is exactly known and is directed along the impact parameter ${\\\\bf b}$ in the model.\",\n \"At relatively low transverse momenta, $p_{T}<3\\\\div 4$ GeV/$c$ , the measured elliptic flow, $v_2\\\\lbrace 4\\\\rbrace (p_T)$ (CMS), and the restored one, $v_2\\\\lbrace 4\\\\rbrace (p_T)$ (hydjet++), are practically equal to the generated original elliptic coefficient $v_2^{RP}(p_T)$ .\",\n \"It is not surprising, because in this momentum region the bulk of produced particles are mainly correlated with the reaction plane only, whereas nonflow correlations are relatively small.\",\n \"At higher transverse momenta, $p_{T}>10$ GeV/$c$ , which is the region of the special interest here, the values of the measured elliptic flow and the flow restored from the model calculations are close to each other, but noticeably larger than the generated elliptic flow $v_2^{RP}(p_T)$ .\",\n \"In the intermediate region, $4< p_{T}<10$ GeV/$c$ , the both model calculated coefficients, $v_2^{RP}(p_T)$ and $v_2\\\\lbrace 4\\\\rbrace (p_T)$ (hydjet++), lie below the CMS data.\",\n \"Figure: (Color online).The elliptic flow soft (triangles) and hard (squares) components andresulting total flow v 2 RP (p T )v_2^{RP}(p_T) (circles) in hydjet++ for Pb+Pb collisions at s NN =5.02\\\\sqrt{s_{NN}}=5.02 TeV with the centrality 30–40%.Figure REF illustrates the structure of elliptic flow in hydjet++ .\",\n \"Here the coefficients $v_2^{RP}(p_T)$ are calculated with respect to the reaction plane separately for each component, soft and hard, together with the resulting total value.\",\n \"This illustration shows the slight model drawback in the intermediate momentum region.\",\n \"Originally, the model includes the adequate description of soft and hard physics.\",\n \"In the intermediate $p_T$ region the result is obtained by a simple superposition of two independent contributions.\",\n \"The crosslinking is regulated by one parameter, $p_T^{\\\\rm min}$ , which is the minimum transverse momentum transfer in a hard parton subprocess.\",\n \"For the transverse momentum spectra this procedure is painless since the crosslinking takes place for a continuous function, and its fracture is smoothed out by the overlapping of particles from both contributions.\",\n \"As a result, we describe effectively the transverse momentum spectra even in the intermediate region also [13] without any additional mechanism.\",\n \"For elliptic flow coefficients the crosslinking takes place for a “discontinuous” function, see the crosslinking region of $p_T \\\\simeq 10$ GeV/$c$ , where $v_2^{RP}(p_T)$ (soft) and $v_2^{RP}(p_T)$ (hard) have the quite different values.\",\n \"Thus, the simple smoothing is not enough to describe this region by a simple superposition.\",\n \"It means that some improvements of the model are required to describe successfully the whole $p_T$ -region.\",\n \"For instance, it can be a minijet production or some other mechanism.\",\n \"Fortunately, this “problematic” region of the model is not used in the present flow correlation analysis.\",\n \"Now we focus on the high transverse momentum region, to study which the large number of events should be generated.\",\n \"In this region the hydjet++ flow, restored by cumulant method, is close to the measured one, but it is visibly larger than the original elliptic flow generated in the model.\",\n \"This observation requires an explanation.\",\n \"The azimuthal anisotropy arises in the model as a result of jet quenching, see [22], [23].\",\n \"Due to the path-length dependent energy loss of partonic jets as they traverse the matter, the jet particles become to be correlated with the reaction plane.\",\n \"However, the particle correlations relative to the jet axis remain also.\",\n \"Unlike the region of small transverse momentum, there are at least two singled out directions, the reaction plane and the jet axis, respectively, relative to which particles are correlated.\",\n \"In this case we can decompose the azimuthal distribution in a form $\\\\displaystyle \\\\frac{dN}{d\\\\varphi } &=& N_0 \\\\big (1+ 2 v_2^{RP} \\\\cos { \\\\left[2(\\\\varphi -\\\\Psi _{2}^{RP}) \\\\right] }\\\\nonumber \\\\\\\\&+& 2 v_2^{jet} \\\\cos { \\\\left[ 2(\\\\varphi -\\\\Psi _{2}^{jet}) \\\\right] } \\\\big ),$ where the direction of jet axis $\\\\Psi _{2}^{jet}$ is randomly oriented with respect to the reaction plane $\\\\Psi _{2}^{RP}$ , which is set to zero in the model calculations.\",\n \"$N_0$ is the normalization factor.\",\n \"The 2- and 4-particle correlations are estimated as $\\\\langle \\\\langle 2\\\\rangle \\\\rangle \\\\simeq (v_2^{RP})^2 + (v_2^{jet})^2 ,\\\\quad \\\\quad \\\\langle \\\\langle 4\\\\rangle \\\\rangle \\\\simeq [(v_2^{RP})^2 + (v_2^{jet})^2]^2$ and they are sensitive to both types of correlations discussed above.\",\n \"Herewith $\\\\langle \\\\langle 2\\\\rangle \\\\rangle ^{1/2}$ and $\\\\langle \\\\langle 4\\\\rangle \\\\rangle ^{1/4}$ are always larger than the “true” elliptic coefficient $v_2^{RP}(p_T)$ which we are interested in.\",\n \"Figure: (Color online).The correlation between v 2 v_2 values at low and highp T p_T in Pb+Pb collisions at s NN =5.02\\\\sqrt{s_{\\\\mbox{\\\\tiny {\\\\it {NN}}}}}= 5.02 TeVas function of centrality.The points represent the centrality bins 5–10,10–15,15–20,20–30,30–40,40–50, and 50–60%.The model results are presented for v 2 RP v_2^{RP} (circles) andv 2 {4}v_2\\\\lbrace 4\\\\rbrace (squares).\",\n \"Data (triangles) are taken from .Model calculations are performed (a) with and(b) without the jet quenching.The correlations between the low-$p_T$ and high-$p_T$ elliptic flow which are the main issue of our study are displayed in Fig.\",\n \"REF (a).\",\n \"The picked up intervals are $14< p_{T}<20$ GeV/$c$ for high and $1.0< p_{T}<1.25$ GeV/$c$ for low transverse momenta, respectively.\",\n \"One can see that the model calculations, $v_2\\\\lbrace 4\\\\rbrace (p_T)$ (hydjet++), reproduce the experimentally observed centrality dependence of the flow correlations fairly well except the last centrality bin, while the generated original elliptic flow $v_2^{RP}(p_T)$ at high $p_T$ goes always lower than the restored flow in accordance with the explanation above.\",\n \"The deviations become significant for the centralities larger than 30–40%.\",\n \"In this centrality interval the anisotropy caused by the jet quenching becomes to die out, but particles remain correlated with the axes of jets.\",\n \"Note that in the low-$p_T$ region the integrated values of $v_2$ in the model are lower than the data.\",\n \"This is due to simplification, discussed in Sec.\",\n \", aiming to reduce the number of free parameters in the model.\",\n \"To reach better quantitative agreement with the data one has to treat both anisotropy parameters, $\\\\varepsilon (b)$ and $\\\\delta (b)$ , as independent ones, see [19].\",\n \"This circumstance, however, does not affect the main results associated with identifying the role of jets on the behavior of cumulants.\",\n \"Note also, that in the high-$p_T$ sector the agreement between the model results and the data is good.\",\n \"Figure REF (b) demonstrates the comparison of model calculations without jet quenching effects with the data.\",\n \"In this case $v_2^{RP} =0$ and the correlations relative to the jet axis contributes to the 4-th order cumulant only, but the magnitude of jet correlations is not strong enough to reproduce data.\",\n \"Thus, in the collisions with centrality up to 30–40% the azimuthal anisotropy due to jet quenching reveals itself, whereas in more peripheral collisions the jet correlations contribute significantly to the 4-th order cumulant.\",\n \"The reason is as follows.\",\n \"Since in peripheral heavy-ion collisions there are simply quantitatively fewer nucleon-nucleon collisions, then, fewer jets are produced.\",\n \"In the limiting case there is one back-to-back pair, and the method sees this axis and anisotropy.\",\n \"In more central collisions there are many jet pairs.\",\n \"They are all distributed randomly in azimuth, therefore, the anisotropy caused by jets tends to zero.\",\n \"It is worth noting that jets were the main source of violation of the number-of-constituent quark (NCQ) scaling in hydjet++ calculations of differential elliptic [14], [15] and triangular [18] flow.\",\n \"The linear fit [10], performed by CMS Collaboration to data on elliptic flow correlations at low and high transverse momenta, also indicates some sort of scaling behavior.\",\n \"In contrast to situation with the NCQ scaling, here jets work toward the scaling fulfillment.\"\n ],\n [\n \"Conclusions\",\n \"The phenomenological analysis of elliptic flow correlations at low and high $p_T$ in Pb+Pb collisions at center-of-mass energy 5.02 TeV per nucleon pair has been performed within the two-component hydjet++ model.\",\n \"These correlations are stipulated by the fact that the magnitudes of anisotropy at low and high $p_T$ are mainly determined by the value of initial ellipticity of the nuclei overlapping.\",\n \"At relatively low transverse momenta, $p_{T}<3\\\\div 4$ GeV/$c$ , the model generated elliptic flow $v_2^{RP}(p_T)$ and its value restored by 4-cumulant method, $v_2\\\\lbrace 4\\\\rbrace (p_T)$ (hydjet++), are very close to the differential elliptic flow $v_2\\\\lbrace 4\\\\rbrace (p_T)$ measured by the CMS Collaboration.\",\n \"At high transverse momenta $p_{T}> 10$ GeV/$c$ the cumulants are sensitive to both the anisotropy due to jet quenching, $v_2^{RP}$ , and the particle correlations with the jet axis, $v_2^{jet}$ .\",\n \"In the collisions with centrality up to 30–40% the 4-cumulant method “measures” mainly the azimuthal anisotropy due to jet quenching, whereas in more peripheral collisions it is affected by the jet correlations primarily.\",\n \"The model calculations restored by this method, $v_2\\\\lbrace 4\\\\rbrace (p_T)$ (hydjet++), reproduce the experimentally observed centrality dependence of elliptic flow correlations rather well without any additional tunes of model parameters.\",\n \"Fruitful discussions with A.I.\",\n \"Demyanov and L.V.\",\n \"Malinina are gratefully acknowledged.\",\n \"This work was supported in parts by Russian Foundation for Basic Research (RFBR) under Grant No.\",\n \"18-02-00155, Grant No.\",\n \"18-02-40084 and Grant No.\",\n \"18-02-40085.\",\n \"L.B.\",\n \"and E.Z.\",\n \"acknowledge support of the Norwegian Research Council (NFR) under grant No.\",\n \"255253/F50 - “CERN Heavy Ion Theory\\\".\"\n ]\n]"},"id":{"kind":"string","value":"2012.05139"}}},{"rowIdx":3695,"cells":{"text":{"kind":"list like","value":[["Extracting the signed backbone of intrinsically dense weighted networks"],["Abstract Networks provide useful tools for analyzing diverse complex systems from natural, social, and technological domains.","Growing size and variety of data such as more nodes and links and associated weights, directions, and signs can provide accessory information.","Link and weight abundance, on the other hand, results in denser networks with noisy, insignificant, or otherwise redundant data.","Moreover, typical network analysis and visualization techniques presuppose sparsity and are not appropriate or scalable for dense and weighted networks.","As a remedy, network backbone extraction methods aim to retain only the important links while preserving the useful and elucidative structure of the original networks for further analyses.","Here, we provide the first methods for extracting signed network backbones from intrinsically dense unsigned unipartite weighted networks.","Utilizing a null model based on statistical techniques, the proposed significance filter and vigor filter allow inferring edge signs.","Empirical analysis on migration, voting, temporal interaction, and species similarity networks reveals that the proposed filters extract meaningful and sparse signed backbones while preserving the multiscale nature of the network.","The resulting backbones exhibit characteristics typically associated with signed networks such as reciprocity, structural balance, and community structure.","The developed tool is provided as a free, open-source software package."],["Introduction","Networks are increasingly useful in modeling and studying many problems in seemingly unrelated domains from social sciences [9], natural sciences and engineering [74], and arts and humanities [68].","A very simple network consists of nodes (vertices, actors) and links (edges, ties) connecting them.","The network science tools utilize the information from the interdependence between entities given by the network structure.","The data that can be modeled in the network form is not limited only to the node and edge structure.","More complex networks have attributes associated with edges and nodes, especially with the advent of advanced data collection methods.","In weighted networks, edges have numeric weights indicating its intensity (e.g., coupling strength, amount, similarity, etc.).","In directed networks, edges have distinguishable source and target nodes, indicating the direction of the relation (e.g., flow, following, etc.).","For instance, the air traffic can be modeled as a very simple network such that a node represents an airfield and an edge between two nodes indicates a flight between them.","In this case, the number of connections an airfield has is given by the degree of the corresponding node.","The same system can also be modeled as a directed weighted network where an edge carries the take-off and landing airfield data as its direction and passenger capacity data as its weight.","In this way, the outgoing and incoming capacity of an airfield is given by the out- and in-strength of the node, that is the total weight of edges leaving and entering the node, respectively.","This very simplified example demonstrates the view that as networks get more complex, richer analyses can be made.","Another type of networks, which is explored relatively less in the literature, is signed networks where edges have negative or positive signs respectively indicating antagonism (e.g., dislike, distrust, foes, dissimilarity, voting against, inhibition, etc.)","or rapport (e.g., like, trust, friendship, similarity, voting for, activation, etc.).","Negative edges are not simply negation of positive edges [81], they show distinct behavior in networks [76], and their inclusion enrich the typical tasks on networks such as link prediction [50], [36], recommender systems [79], [89], node classification [60], [78], node centrality and ranking [31], [85], representation learning [46], [21], information diffusion and influence maximization [39], [44], finding cliques [51], community detection, graph partitioning and blockmodels [27], [55], [54], and polarization [8], [87].","In addition to the typical and prevalent applications in social media [80], signed networks also find application areas in politics [3], international relations [24], finance [37], biology [42], [63], [40], and ecology [40].","The density of a network is usually defined by the ratio of the number of observed links to the number of possible links.","Formally, sparse networks are those where density asymptotically goes to zero in the limit of large number of nodes.","However, in most empirical networks, it is impossible to evaluate this [61].","As a result, in general, an empirical network is called sparse if it has a low density.","The majority of real-world networks are sparse [62].","For instance, the number of stable social relations of a human is limited by cognitive constraints [34], stations are usually only connected to nearest stations in power grid networks, a webpage points out only so many others in web networks, and so on.","On the other hand, some networks might be very dense or even almost completely connected by its own nature.","Most countries trade with most other countries, humans move/migrate from many locations to many other locations, many species have many predators and preys [25], most people interact with most others in certain social settings, and so on.","The earlier view that additional data allowing richer analyses is restricted by the amount and structure of the data itself, particularly in the case where the additional data is provided by more links between the nodes.","In addition to the fact that sparsity is desirable due to the computational complexity of many network algorithms, most of the typical network analysis and visualization methods assume the networks to be sufficiently sparse [61].","Apart from computational and methodological concerns, dense networks might have noisy, uninformative, insignificant, or otherwise redundant links.","This further worsens the application and interpretation capacity in many network tasks e.g., those relating to node centrality, cliques and communities, and diffusion.","Therefore, the relevant and significant information should be extracted from such dense networks such that the original rich data is reduced into a network that is sparse and simpler but maintains adequate structural information for efficient and effective analyses.","There is a body of literature for extracting backbones of usually weighted and relatively dense networks, which we review in the next section.","Such information filtering task is usually referred to as backbone extraction or network sparsification and aims to remove statistically insignificant or otherwise redundant links while maintaining the informative structural properties for further analysis.","To the best of our knowledge, this is the first study in the literature that provides methods for extracting signed network backbones from weighted dense networks.","Unlike the earlier studies, our method relies on and requires the input network to be intrinsically dense; which in turn enables inferring the link signs.","We use the term intrinsically dense for characterizing networks where all nodes, in a sense, are aware of all other nodes and can interact with them without obvious natural limits such as those mentioned earlier for sparse networks.","This definition follows that an edge does not necessarily represent the existence of a positive relationship but might be an artifact of the randomness or even a negative relationship depending on its weight.","Such definition of intrinsically dense networks provides a distinction not only from sparse networks but also from certain dense networks.","For instance, human interaction network in a workshop with two parallel sessions with a single mutual short break is not intrinsically dense because the participants in different sessions are not given sufficient time to meet with others, thus, absence of a link cannot reasonably indicate if two participants from the different sessions are avoiding each other.","As a demonstrating example of intrinsically dense networks, consider a network where link weights denote similarity of respective nodes.","Lower similarity does not only mean a lack of similarity but often indicates dissimilarity, a negative underlying link.","In the same vein, voting for a set of candidates indicates support for those candidates while also suggesting opposition for the other candidates especially those who are otherwise popular.","Finally, we should highlight that whether a network is intrinsically dense or not is a matter of extent and not a strictly binary decision.","We suggest that positive and negative links are those with weights significantly and substantially deviating from the random expectation under a suitable null model.","We develop an appropriate null model based on hypergeometric distribution and iterative proportional fitting procedure and offer significance filter and vigor filter for extracting signed backbones of intrinsically dense networks.","The proposed methodology is capable of handling directed or undirected networks as input and producing weighted or unweighted signed backbones.","Our contribution can be summarized as follows.","[itemsep=0pt] The first extensive literature review on network backbone extraction methods is provided (Section ).","The first methods to extract signed backbones of intrinsically dense weighted networks are proposed (Section ).","The methodology is empirically evaluated on real-world networks of different characteristics and its feasibility and usefulness is shown (Section )."],["Backbone extraction methods for unsigned networks","The simplest yet a popular approach is to apply a global threshold where only those edges with weights satisfying the predetermined threshold are retained.","There are two major problems with this approach.","First, the choice of the threshold is rather arbitrary and non-impartial unless the physical meaning of the weights in the domain of the particular network allows an explanation.","The second and more important problem is the case of multiscale networks where the weights are distributed over a broad range of scales.","Such characteristic is definitely not an exception but an observed phenomenon in many real-world networks [5].","For instance, in many networks where rich club effects [72] are prevalent, nodes with higher degrees/strengths tend to form links with nodes of the same or higher degree/strength which results in a hierarchical and multiscale network structure.","Rich club effects and such structures are present in many weighted networks including the global airline traffic network [1], worldwide maritime transportation network [41], world trade network [7], [67], international credit-debt networks [18], population flow network during a national holiday [86], patient referral networks [77], and communication networks in the brains of humans [1] and rats [52].","Therefore, weights with small magnitudes are not necessarily noise and weights with large magnitudes are not necessarily important but might be the result of such multiscale weight characteristics.","In multiscale networks, the application of a global threshold tends to eliminate the edges of low-strength nodes regardless of the local significance of those edges.","As a result, the structure of high-strength nodes can be preserved but local regions characterized by relatively low-strength nodes are underestimated, might become disconnected, and might even be completely eliminated.","In summary, increasing the value of a global threshold loses local structures at the bottom of the hierarchy whereas decreasing it results in retained noise in the upper levels.","Consequently, global thresholding would only work under the often-unrealistic assumption that edge weights are independent and identically distributed random variables.","As a remedy, the global threshold method can be improved to work on weights represented not in universal units but as fractions of the node degrees/strengths.","Yet, an edge is associated with two nodes and it again becomes relatively arbitrary to choose the normalization factor.","Such normalization might not be robust to nodes with very low and high degrees as well since it potentially overestimates the edges of low-degree nodes and underestimates the edges of high-degree nodes.","Overall, it could serve as a practical remedy for certain problems but the need for more statistically-sound methods is clear.","Another approach is to extract the spanning tree with maximal weight by appropriately transforming the weights and finding the minimum spanning tree.","This method does not require any extrinsic input and ensures that the extracted backbone is connected.","However, the backbone will be acyclic and local clustering and community structure cannot be preserved.","Using a similar spanning tree approach, Tumminello et al.","[82] propose a technique for controlling the genus of the backbone which, in turn, enables preservation of some local structure.","Grady et al.","[35] extracts the high-salience skeleton of the network which consists of only the salient edges.","For each node, they construct the shortest path tree by merging all shortest paths from the node to all other nodes.","Then, the saliency of an edge is defined as the fraction of the all shortest path trees where the edge is a member.","They show that edge saliency shows a bimodal distribution near the boundaries.","Only the edges with saliency near 1 are retained; effectively eliminating the need for choosing an arbitrary threshold.","By design, it ensures the connectivity of the extracted backbone.","Edge saliency is different from edge betweenness where the former tends to award the edges in the periphery (e.g., low-degree nodes) whereas the latter tends to award the edges in the core (e.g., high-degree nodes).","The methods described so far do not assume any underlying null model and do not compare observed weights to expected weights for statistical evaluation.","Disparity filter [70] assumes that the normalized weights of the links of a node follow a uniform distribution.","Comparing the observed weights to this null model at a desired significance level, the network backbone including only the statistically significant links can be obtained.","As mentioned earlier, a weight can be normalized and evaluated separately for the two nodes it connects.","Thus, a link can be significant from the viewpoint of one node and not the other.","This problem is tackled by retaining a link if it satisfies the significance condition for at least one of the two nodes.","Adopting a loosely similar weight normalization scheme, bistochastic filter [73] first scales the weight matrix such that its marginals (i.e., the row and column totals, in- and out-strength sequence) are equal to 1 resulting in a doubly stochastic matrix.","Then, the links with highest weights are retained until the network is strongly connected; or until any other stopping criteria which can result in sparser backbones with disconnected components or even denser backbones.","In the same line of research, LANS [30] does not assume an underlying distribution for the weights of a node and employs the empirical cumulative density function instead.","It is stated that it is more robust to the highly heterogeneous local weight distributions than disparity filter and bistochastic filter are.","Rather than adopting a local approach in developing null models, GloSS filter [64] utilizes a global null model.","The model considers both the strengths and degrees and aim to preserve the global weight distribution and the topology.","GLANB [90] evaluates the statistical significance of link saliency to retain satisfactory links; loosely combining disparity filter and high-salience skeleton.","Dianati et al.","[22] introduce two interrelated filters.","Treating an integer-weighted network as a multiedge network, they assume that each unit edge chooses two nodes in random respecting the degree sequences which results in a binomial distribution for weights.","The null model is indicated to blend the approaches of disparity filter and GloSS.","Marginal likelihood filter operates by evaluating each edge separately against a chosen significance level whereas global likelihood filter incorporates exponential random graph model with a Monte Carlo simulation scheme to consider all links at once.","ECM filter [33] enhances the null model of Dianati et al.","[22] with the purpose of retaining the relation between strengths and degrees.","Polya filter [58] assumes an underlying null with a self-reinforcing mechanism in which the generation of link weight increases the probability of further weight generations for that link.","Tumminello et al.","[83] introduce hypergeometric filtering originally for bipartite weighted networks.","It is employed for unipartite weighted networks [65], [71] as well.","Assuming integer weights and treating weighted links as multiple links, every time a node increases its degree by 1 by adding a new link, the node at the other end of the link is selected randomly among all possible nodes.","Then, the edge weight generation process can be described by a hypergeometric distribution which becomes the null model for the filter.","Comparing observed values with the null model, statistically significant links can be extracted at desired levels of confidence.","Noise-Corrected Bayesian filter [19] assumes edge weight generation follows a binomial distribution where a null model based on the hypergeometric distribution is used in determining Bayesian priors.","Lift value for each link (i.e., the ratio of the observed value to the expected value) is calculated, transformed to the $[-1, 1]$ range, and the associated variance is estimated with a Bayesian inference schema.","Using the appropriate posterior variances, the links which satisfy a desired significance level are retained.","Although more specific approaches exist with different purposes (e.g., to retain the most consistent links in brain networks based on diffusion images [66], to preserve functional backbones based on network motifs [13]), the state-of-the-art concentrates on developing appropriate null models to evaluate link weights against and retain only those satisfying a desired level of statistical significance."],["Formal Problem Definition","We denote an intrinsically dense, undirected or directed, and non-negative weighted network without self-loops by $G := (V, W)$ .","$V$ is the set of nodes with $i, j \\in V$ as its general elements and its cardinality is $n = |V|$ .","$W$ is the corresponding weight matrix.","When $G$ is undirected $W_{ij}$ denotes the weight of the link between $i$ and $j$ and $W_{ij} = W_{ji}$ .","When $G$ is directed, $W_{ij}$ denotes the weight of the link from $i$ to $j$ .","Self-loops are not allowed hence $W_{ii} = 0 \\; \\forall i$ .","Total outflow of $i$ is denoted with $W_{i.}","= \\sum _{j} W_{ij}$ .","Total inflow to $j$ is denoted with $W_{.j} = \\sum _{i} W_{ij}$ .","The sum of all weights is denoted with $W_{..} = \\sum _{i} \\sum _{j} W_{ij}$ .","We employ the same definitions of $W_{i.","}$ , $W_{.j}$ , and $W_{..}$ for directed and undirected networks, i.e., treating an undirected link as two reciprocal directed links with equal weights.","Given $G$ , the aim is to extract a meaningful signed network $\\hat{G} := (\\hat{V}, \\hat{A}, \\hat{W})$ where $\\hat{A}$ is a sparse adjacency matrix in general such that $\\hat{A}_{ij} \\in \\lbrace -1, 0, 1\\rbrace $ where $-1$ denotes a negative link, 1 denotes a positive link, and 0 indicates the absence of a link.","$\\hat{W}$ is the corresponding optional weight matrix.","If $\\hat{A}_{ij} = 0$ , then $\\hat{W}_{ij} = 0$ ; otherwise $\\hat{W}_{ij}$ has the same sign as $\\hat{A}_{ij}$ ."],["Proposed Solution","The null model.","To extract the signed edges, it is necessary to have a null model which empirical edge weights are compared against to distinguish edge weights that are due to chance and edge weights that significantly deviate from the expected values in either positive or negative direction.","We make the assumption that every time a node increases its out-strength by one, it randomly chooses the other node with probabilities proportional to their in-strengths.","In this way, the process is reduced to sampling without replacement problem, of which the urn problem is a famous example.","We describe the process with a simple analogy.","For each node $i$ , there are $W_{..} - W_{.i}$ marbles in the urn (not allowing self-loops), $W_{i.","}$ marbles are chosen from the urn without replacement, and there are $W_{.j}$ marbles in the urn for each $j$ .","Such process is well-characterized by hypergeometric distribution and allows us to calculate statistical quantities for links between all $i$ and $j$ .","The mean of the hypergeometric distribution associated with the link from $i$ to $j$ is given by Eq.","REF .","However, this formulation does not preserve the in-strength and out-strength sequences in the system, thus, is inadequate for our purposes.","Given that $N$ is the weight matrix under null model with $N_{ij}$ denoting the expected weight from $i$ to $j$ , the following should be satisfied: $N_{i.}","= W_{i.","}$ , $N_{.j} = W_{.j}$ , and $N_{..} = W_{..}$ .","For this purpose, we utilize the iterative proportional fitting procedure (IPFP) described in [4], [59] which is shown to converge to an optimal solution [10].","Given the column and row marginals (e.g., in- and out-strength sequences) and an appropriate prior matrix with certain elements as zero (e.g., diagonal elements to disallow self-loops) and other elements as akin to Eq.","REF , IPFP estimates the values for non-zero elements of the matrix through a series of row-scaling and column-scaling operations until achieving a desired precision.","Given out- and in-strength ($W_{i.","}$ and $W_{.j}$ ) sequences as the marginals and the values obtained from Eq.","REF as the elements prior matrix with its diagonal elements set to 0, IPFP finds the suitable matrix $N$ representing the expected weights under our proposed null model.","$\\frac{W_{i.}","W_{.j}}{W_{..} - W_{.i}}$ In evaluating the statistical significance of an edge weight, only considering the difference of observed and expected values without referencing to a dispersion measure is not viable.","For each edge, dispersion of its expected weight must be known to produce confidence intervals.","Accordingly, we estimate the variance of $N_{ij}$ based on the associated hypergeometric distribution and obtain its standard deviation as given by Eq.","REF .","$\\sigma _{ij} = \\left( W_{i.}","\\; \\frac{W_{.j}}{W_{..} - W_{.i}} \\; \\frac{(W_{..} - W_{.i}-W_{.j})}{W_{..} - W_{.i}} \\; \\frac{W_{..} - W_{.i}-W_{i.}}{W_{..}","- W_{.i}-1} \\right) ^ {\\frac{1}{2}}$ Significance filter.","The first filter we propose functions to eliminate links whose weights do not significantly deviate from their expected values.","Significance filter retains only those links satisfying the condition $\\sigma _{ij} \\: \\alpha ^- \\: \\le W_{ij} - N_{ij} \\le \\: \\sigma _{ij} \\: \\alpha ^+$ where $\\alpha ^-$ and $\\alpha ^+$ respectively take non-positive and non-negative values and are user-defined hyperparameters specifying the desired significance thresholds for negative and positive signed edges.","It should be highlighted that the hypergeometric distribution is a distribution for discrete events.","In general, we can slightly abuse it by allowing continuous weights or rounding weights to the nearest integers without much impact on the resulting backbone.","However, a larger issue that is not adequately discoursed in the hypergeometric filtering literature is the impact of the magnitude of weights on the statistical significance evaluation.","As the magnitude of weights increases, the confidence intervals relative to the weights become narrower, as a consequence of the assumption that the weights are discrete events.","For instance, the same monetary network represented in dollars or cents would result in different network backbones for the same values of $\\alpha $ .","This issue is less pressing when the link weights essentially correspond to discrete events such as human migration networks or voting networks.","Taking these into account, it can be concluded that the values chosen for $\\alpha $ might not accurately translate into traditional p-values in many networks due to their non-discrete nature.","Hence, the choice of $\\alpha $ should not necessarily be limited to traditionally employed values especially when the weights do not correspond to discrete events.","Vigor filter.","The significance filter might retain statistically significant but otherwise very weak links.","Yet, an edge can be evaluated also based on whether it is sufficiently strong in terms of intensity.","This is useful for multiple reasons: (i) the employed significance filter might be too permissive in certain circumstances, (ii) the signed network backbone is needed to reflect only binary links where opinions are rather strong, or (iii) higher sparsity is desired.","We define vigor of an edge from $i$ to $j$ as $\\beta _{ij}$ (pronounced as víta in Modern Greek) in Eq.","REF .","It takes values in the range $[-1, 1]$ , its magnitude indicates the intensity of the link and $\\beta _{ij} = 0$ when $W_{ij} = N_{ij}$ .","Vigor filter retains only those links satisfying the condition $\\beta ^- \\le \\beta _{ij} \\le \\beta ^+$ where $\\beta ^+$ and $\\beta ^-$ respectively take non-positive and non-negative values and are user-defined hyperparameters specifying the desired vigor thresholds for positive and negative signed edges.","$\\beta _{ij} = \\frac{W_{ij}/N_{ij} \\; -1}{W_{ij}/N_{ij} \\; +1}$ Signed backbone extraction.","The signed backbone of a network can be extracted using a combination of significance filter and vigor filter.","We suggest always employing the former since $\\beta $ values might be unreliable in the small-magnitude weight regime characterized by small $N_{ij}$ , relatively large $\\sigma _{ij}$ , and small $W_{ij}$ .","Significance filter would function to eliminate such links, therefore, the retained links would have reliable vigor.","Vigor values of the remaining edges can be utilized as the signed edge weights in the extracted backbone.","In the case of weighted backbone, the use of vigor filter is rather optional since the vigor information is directly carried onto the network.","Alternatively, the weights can be ignored to produce a network with binary opinions.","In this case, we recommend utilizing the vigor filter to retain only the edges that exhibit appropriate levels of intensity.","Overall, for setting appropriate values to the hyperparameters, one should consider (i) the physical meaning of hyperparameters in the context of the specific problem, (ii) the desired level of sparsity for the extracted backbone.","Undirected networks.","The described method is defined well on the directed networks.","The generalization to the undirected networks is ensured in the following way.","An undirected link is replaced with two reciprocal directed links of the same weight; effectively transforming the network into a directed network.","The null model and the filters produce the same null expectation and nearly the same varianceFor sufficiently large networks, $W_{..} - W_{.j} \\approx W_{..} - W_{.i}$ in In Eq.","REF for such reciprocal links.","Finally, the directed backbone can be transformed back into an undirected backbone by treating reciprocal directed links as undirected links and removing the redundancy by keeping the link with the highest absolute vigor."],["Empirical Analysis","In this section, relevance and diverse characteristics of the employed network datasets are described, and proposed filters are empirically analyzed in terms of resulting backbone sizes, robustness to multiscale networks, and structures of the extracted signed network backbones.The datasets and code for producing the analyses in this section can be made available upon request."],["Datasets.","The proposed method for signed network backbone extraction is experimented on four real-world networks from different domains and of varying statistical properties.","For each network; the number of nodes $n$ , number of nonzero links $m$ , density, five-point summary of weights (minimum, first quantile, median, third quantile, and maximum values), and three-point summary (minimum, median, and maximum values) of node strengths (out- and in-strengths if networks are directed) are presented in Table REF .","Table: Employed NetworksMigration network [11] reports interstate migration flows (in terms of people) among the US statesThe dataset contains District of Columbia in addition to the 50 states.","in 2018.","Such internal migration data is explored for its correlation with economic productivity differences, geographical proximity, political preferences, and other cultural and historical factors [15], [53], [17].","Eurovision network [84] represents the votes between participant countries of the song contest in 20032003 was selected because it was the last year that all countries competed in a single round.. Each country, via public votesIreland, Russia, and Bosnia and Herzegovina exercised jury voting instead of public voting., awards the set of points {1,2,3,4,5,6,7,8,10,12} to 10 other countries.","Studies employing a network analysis perspective [28], [16], [32], [56], [57], [26], [75] show that voting behavior is not determined only by the music/performance quality but affected by political factors, geographical and cultural similarity, diasporas, and others.","Contact network data [43] is gathered via wearable sensors at an academic conferenceThe temporal face-to-face interaction dataset was collected at Hypertext conference in 2009 as part of SocioPatterns research collaboration.","with 113 attendees over 2.5 days.","The edge weights are generated by aggregating the number of 20-second intervals the respective two participants spent face-to-face over the course of the conference.","The networks of such temporal ties are utilized for temporal backbone extraction, investigation of spreading processes, and other behavioral and structural analysis [47], [88], [49], [6].","Species network [48] is generated based on the cohabitation patterns of 62 marine species in South Florida.","Edge weights represent the similarity of species based on the habitats they co-occupy during the same life stagesGiven the species v. habitat-life stage bipartite matrix $B$ , $W_{ij} := \\sum _{k} B_{ik} B_{jk}$ where $i, j$ are the species and $k$ is the habitat-life stage pair, and $B_{ik}$ is the respective species-habitat-life stage score.","More information on the original dataset can be found at https://atlanticfishhabitat.org/species-habitat-matrix/.","Information derived from such cohabitation networks is useful in various ecological studies [45], [12], [29].","Moreover, none of the employed networks have natural self-loops; which make them particularly suitable for our study.","Our proposed method considers the propensity of nodes for incoming and outgoing edge weights by explicitly including the in-strengths and out-strengths in the null model.","For instance, high economic productivity for a state would result in more immigration into it.","Similarly, low productivity would cause emigration from that state.","The effect of such productivity is homogeneous in the sense that all other nodes perceive it in the same way.","Such homogeneous effects are eliminated for the large part by our null model.","However, for instance, the effect of political affiliation or geographic location is perceived differently by other nodes.","Hence, the extracted backbone is expected to reflect non-economic factors in Migration network.","With a simple analogy, the quality of the songs and performances in Eurovision song contest is equivalent to productivity.","Hence, the elimination of such effect by the null model would result in an extracted signed network representing the non-quality factors in voting behavior.","Similarly, the null model largely eliminates the effects of participants' tendency to engage in conversations (e.g., popularity, extroversion) and the extent of species' ability to live in many different habitats."],["Backbone Size","In general, a signed network backbone should be sufficiently sparse such that only the important edges are preserved.","At the same time, unlike edges, most nodes should be retained since they are often the subject of the analysis and are largely indispensable in understanding the global network structure.","Here, we extract the signed backbones of the four networks under various hyperparameter regimes and observe the size of the resulting backbone in terms of signed edges and nodes.","Effects of significance threshold.","Figure REF visualizes the fraction of links and nodes retained for meaningful continuous ranges of the significance threshold under different vigor filters.","The suitable range of $\\alpha $ differs between networks mostly due to the varying overall magnitude of edge weights.","The significance threshold is represented with $\\alpha = \\alpha ^+ = - \\alpha ^-$ on x-axes.","Three different vigor threshold settings are represented in columns.","For edges, percentage values on y-axes are with respect to all possible edges, that is $n^2 - n$ edges for directed networks and half of it for undirected networks.","Figure: Effects of significance filter on the backbone size.Figure: Effects of vigor filter on the backbone size.Except for Species network, all nodes are retained in the backbones even under very strict settings where 80% to 85% of all possible edges are eliminated.","When edges are inspected, apart from the Species network where edges represent similarity, the negative edges are more frequent than the positive edges when $\\alpha = 0, (\\beta ^-, \\beta ^+) = (0, 0)$ .","This is due to the case that edges with zero weights (i.e., nonexistent edges) or trivially small values are treated as negative links regardless of their statistical significance or vigor.","As the values of $\\alpha $ increase, the number of negative links in the extracted backbone decreases faster than the positive links since many of the zero or trivially small weights are rather statistically insignificant.","On the other hand, utilizing vigor filter without the significance filter does not eliminate such zero-weighted links since those edges have vigor values of $-1$ (since 0 divided by any null expectation is 0, which maps to vigor of $-1$ ).","Hence, the decrease in negative edges in second and third columns in comparison to the first column of Figure REF when $\\alpha = 0$ is not provided by eliminating zero-weighted edges but other weak edges.","Therefore, significance filter is always necessary when insignificant zero or trivially small edge weights exist.","Employing very large $\\alpha $ values eliminates negative links almost completely.","This is largely because the edges with relatively small expected values under the null model have relatively large variances.","Increasing $\\alpha $ too much, thus, tends to eliminate negative edges first as well as other edges in local regions characterized by edge weights of small magnitude.","In the case of high-magnitude regions, an edge between two high-strength nodes is expected to have a large expectation under the null model.","It is much harder for such edges to empirically observe much larger weights than the null expectation since the total weight in the network is fixed.","This implies an upper boundary on vigor values for edges connecting to high-strength nodes.","Therefore, increasing $\\beta $ too much, especially on its positive range, tends to eliminate edges between central, high-strength nodes.","Effects of vigor threshold.","In a similar fashion, we have also explored the whole continuous range of vigor threshold under different significance threshold filter in Figure REF .","The significance threshold is represented with $\\beta = \\beta ^+ = -\\beta ^-$ on x-axes.","Three different significance threshold settings are represented in columns.","Except for Species network and except under strict vigor thresholds, all or almost all nodes remain in the backbone.","When edges are inspected, in line with the relevant conclusions derived from Figure REF , increasing $\\beta $ does not sufficiently eliminate the negative edges which would be otherwise statistically insignificant.","In very large values of $\\beta $ , positive links are largely eliminated.","When $\\beta = 1$ , the edges remaining in the backbone are the zero-weighted (i.e., nonexistent) edges.","Accordingly, employing very large vigor thresholds should be avoided.","Conclusion.","Overall, we have shown that the proposed method is able to reduce an intrinsically dense network to a signed backbone of the relative size of $10\\%$ to $20\\%$ such that the resulting backbone contains comparable portions of negative and positive edges.","Yet, the hyperparameter selection largely lies with the user and can be changed according to the purpose of analysis and the nature of the network.","For instance, Species network is constructed with similarity values as edge weights and the edge weights do not follow a skewed distribution unlike other networks.","Therefore, its behavior under the proposed filters partially deviates from the behavior of the other employed networks.","Principally, we can conclude that (i) significance filter should almost always be used with a fair threshold, (ii) very large threshold values for both filters should be avoided, and (iii) a sufficiently balanced, statistically meaningful, and sparse backbone can be extracted with a balanced utilization of the two proposed filters."],["Heterogeneity","As discussed in Section , an enviable backbone extraction method should respect the weight and strength heterogeneity in the original network (i.e., its multiscale, hierarchical nature).","That is to say, the retained edges should not be only those originally with large weights or those connecting high-strength nodes.","Heterogeneity of edge weights.","Figure REF presents the original weight distributionSmall random noise is added for Eurovision network for visualization purposes.","of the retained edges in backbones of different sizes The backbones are extracted with the following ($\\alpha ^-, \\alpha ^+$ ), ($\\beta ^-, \\beta ^+$ ) settings for the respective backbone sizes: Migration $\\rightarrow $ 25%: (-33, 33), (-0.33, 0.33); 10%: (-40, 40), (-0.57, 0.57); 5%: (-40, 40), (-0.72, 0.72).","Eurovision $\\rightarrow $ 25%: (-1.5, 2.25), (0, 0); 10%: (-2, 4.5), (0, 0); 5%: (-2.5, 5), (0, 0).","Contact $\\rightarrow $ 25%: (-2, 2), (0, 0); 10%: (-3, 3), (-0.66, 0.66); 5%: (-5, 5), (-0.55, 0.55).","Species $\\rightarrow $ 25%: (-3.5, 3.5), (0.15, 0.15); 10%: (-5, 5), (-0.2, 0.2); 5%: (-5.5, 5.5), (-0.33, 0.33).",".","The figure shows that even when the extracted backbone retains only $5\\%$ of all possible edges, the heterogeneity of edge weights are respected.","It also visually depicts that there is no global cutoff value for determining the sign of edges as those cutoff values are established individually for each edge by the null model.","Figure: Original weights of the retained edgesHeterogeneity of node strengths.","Figure REF show the strength distributionSmall random noise is added for Eurovision network for visualization purposes.","of the dyads for the original networks in transparent color and the extracted backbones with relative size of $\\approx 20\\%$The backbones in Figure REF are extracted with the following ($\\alpha ^-, \\alpha ^+$ ), ($\\beta ^-, \\beta ^+$ ) settings.","Migration $\\rightarrow $ (-40, 40), (-0.25, 0.25).","Eurovision $\\rightarrow $ (-1.8, 2.4), (-0.5, 0.3).","Contact $\\rightarrow $ (-2.25, 2.25), (-0.25, 0.25).","Species $\\rightarrow $ (-3.6, 3.6), (-0.25, 0.25).","in opaque color.","Specifically, the edge between nodes $i$ and $j$ is represented with a point colored based on its sign.","X-axis denotes the (out-)strength of $i$ and y-axis denotes the (in-)strength of $j$For an undirected network, $i$ and $j$ might switch arbitrarily.","As demonstrated by the figure, the retained edges are rather evenly distributed in the plotting space and the original heterogeneity is respected.","The retained edges between very low-strength nodes are usually positive since null expectations are usually low in that regime and it is difficult to conclude whether small-weighted edges in low-strength regimes are statistically significant.","Likewise, the edges between very high-strength nodes are usually negative due to the upper bound on vigor implied by the large null expectation and the fixed amount of total weights.","On the other hand, in the majority of strength-strength regions, the edge signs exhibit a mixed distribution.","Therefore, this is generally a desired property since the proposed method, by itself, does not allow deducing strong conclusions when the data is limited.","Figure: Original strength distribution of the retained dyadsConclusion.","We can conclude that the heterogeneity given by (i) the multiscale nature of edge weights and (ii) heterogeneous nature of node strengths are respected, and (iii) the proposed filters do not draw strong conclusions when the data is limited."],["Structure of the backbones","Generally speaking, we expect a real-world signed network to show some extent of reciprocity and have its reciprocal edges be of the same sign, and demonstrate sufficient levels of structural balance.","Moreover, most real-world networks exhibit a community structure regardless of whether they are signed or not.","Therefore, an informative signed backbone should have these characteristics in general and up to a certain extent.","We analyze the extracted backbonesThe backbones in Figure REF are extracted with the following ($\\alpha ^-, \\alpha ^+$ ), ($\\beta ^-, \\beta ^+$ ) settings.","Migration $\\rightarrow $ (-40, 40), (-0.33, 0.33).","Eurovision $\\rightarrow $ (-1, 0.5), (-0.33, 0.1).","Contact $\\rightarrow $ (-3, 3), (-0.33, 0.33).","Species $\\rightarrow $ (-5,3), (-0.5, 0.1).","of the employed networks in terms of reciprocity, structural balance, and community structure.",".","Reciprocity.","Reciprocity in unsigned networks is generally defined as the ratio of the number of reciprocal edges to the number of all edges.","In signed networks, however, the edge signs should also be considered in the analyses.","Table REF presents the number of nodes, edges, nonreciprocal edges ({$\\times $ , $\\cdot $ }), reciprocated positive edges ({+, +}), reciprocated negative edges ({-, -}), and reciprocated edges with sign conflict ({+, -}) for the directed backbones extracted from directed networks.","In Migration backbone, there is considerable reciprocity with no conflicting edge signs between any node pair.","In contrast, Eurovision backbone discloses a substantial amount of conflicting edge pairs which might be of interest for further investigation into its voting dynamics (e.g., the influence of diasporas).","Table: Reciprocity in extracted directed backbonesFor the rest of the analysis, the directed backbones are transformed into undirected backbones in the following way.","For Migration backbone, directed edges are transformed into undirected edges of the same sign with positive signs having a priority over negative signs when in conflict.","For Eurovision backbone, a positive (or negative) edge is created between two nodes when there are exactly two positive (or negative) directed edges between them (i.e., reciprocal edges of the same sign).","Structural Balance.","As put forward by Heider [38] and formalized for signed networks by Cartwright and Harary [14], an undirected triple is said to be balanced if its edges have the signs {+, +, +} or {+, -, -} and unbalanced if its edges have the signs {+, +, -} or {-, -, -}.","The balanced triples can be simply described with the following expressions: friend of my friend is my friend and enemy of my friend is my enemy.","Davis [20] defines a weaker notion and proposes that enemy of my friend is my enemy is not necessarily required for balance and the only unbalanced triple among the four possible settings is the one with edge signs {+, +, -}.","Accordingly, structural balance (SB) and weak structural balance (WSB) of a network can be defined as the ratio of the number of balanced triples to the number of all triples.","Frustration index (FI) [2] is a measure of the minimum number of required edges such that upon their deletion the network is partitioned into two communities with no positive edges between the communities and with no negative edges within communities.","The number of such edges is scaled so that a fully balanced network of two communities results in a FI of 1.","Table REF shows the count of nodes, edges, and four possible triple settings; and three different structural balance measures for the undirected backbones.The number of triples and frustration index is calculated using signnet [69] package in R. The backbones for Migration and Species shows a strong balance for all three measures.","The other two backbones are also highly balanced in terms of weak structural balance which is shown to be more appropriate in real-world networks.","Moreover, it should be highlighted that in the calculation of the frustration index the network is partitioned into only two communities whereas there are multiple underlying communities for three of the networks as discussed next.","Table: Structural characteristics of extracted backbonesCommunity Structure.","Based on the partitioning methods for signed networks [23], the extracted backbones are partitioned into communities where the number of negative edges within communities and number of positive edges between communities are minimized via heuristicsCommunities are found and visualized using signnet [69] package in R. Figure REF depicts the community structure via a visual block matrix where the order of nodes is the same for its rows and columns.","Diagonal blocks are expected to cover the positive edges and non-diagonal blocks are expected to cover the negative edges.","Overall, the extracted backbones manifest community structures.","The backbone of Migration has 5 clear densely connected communities and two outlier individual nodes whereas the communities in Eurovision are not as dense.","The backbone of Contact consists of one large sparse group and several relatively-small denser groups.","The backbone of Species has a very clear structure with two similar-sized communities, visually confirming the very high frustration index ($0.96$ ) obtained for it.","Figure: Block models of extracted backbonesTable REF shows members of each group in the order of diagonal blocks where nodes are also listed in the same order as they appear in the rows and columns.","We can observe that the states are grouped mostly based on their geographical proximity except for two outlier states which are not part of any groups.","Similarly, countries in the Eurovision backbone are grouped mainly based on geographic, cultural, and historical ties.","Although the results are seemingly plausible, a sophisticated and conclusive discussion on the found communities and its usefulness is spared for future studies in those domains.","Table: Block membershipsConclusion.","We showed that the extracted backbones usually (i) reveal interesting reciprocal structures, (ii) are structurally balanced, and (iii) exhibit clear community structure."],["Conclusion and Future Work","In this study, we put forward an initial discussion on intrinsically dense networks and provide significance filter and vigor filter for extracting signed backbones of such networks.","Empirical evaluations that utilize the proposed filters on a variety of real-world networks show that sparse backbones can be obtained while maintaining comparable numbers of positive and negative links and respecting the original weight and strength heterogeneity.","In general, the obtained backbones exhibits characteristics associated with signed networks such as reciprocity, structural balance, and community structure.","The extraction method, by design, does not prematurely arrive at conclusions regarding the existence of signed links between nodes when the data is limited.","On the other hand, choosing appropriate hyperparameter values lies with the user and should be guided by empirical analysis and recommendations provided in this work as well as the purpose and the nature of the specific problem.","Further studies can improve this work in two major ways.","As we presented many examples throughout the manuscript, intrinsically dense networks exist in many different domains.","First, a stream of its applications in different fields and utilization of the resulting backbones in different tasks would provide important feedback regarding its usefulness, weaknesses, and strengths.","Second, hypergeometric distribution characterizes a discrete process and such discrete distributions are being utilized in the state-of-the-art backbone extraction methods including ours.","As we discussed earlier, null models developed based on it can have certain undesirable properties for certain networks.","Although such issues are easy to handle for practical purposes, a natural avenue is to develop new null models that are more appropriate for the case of link weights which are continuous or that can be equivalently represented in different units."]],"string":"[\n [\n \"Extracting the signed backbone of intrinsically dense weighted networks\"\n ],\n [\n \"Abstract Networks provide useful tools for analyzing diverse complex systems from natural, social, and technological domains.\",\n \"Growing size and variety of data such as more nodes and links and associated weights, directions, and signs can provide accessory information.\",\n \"Link and weight abundance, on the other hand, results in denser networks with noisy, insignificant, or otherwise redundant data.\",\n \"Moreover, typical network analysis and visualization techniques presuppose sparsity and are not appropriate or scalable for dense and weighted networks.\",\n \"As a remedy, network backbone extraction methods aim to retain only the important links while preserving the useful and elucidative structure of the original networks for further analyses.\",\n \"Here, we provide the first methods for extracting signed network backbones from intrinsically dense unsigned unipartite weighted networks.\",\n \"Utilizing a null model based on statistical techniques, the proposed significance filter and vigor filter allow inferring edge signs.\",\n \"Empirical analysis on migration, voting, temporal interaction, and species similarity networks reveals that the proposed filters extract meaningful and sparse signed backbones while preserving the multiscale nature of the network.\",\n \"The resulting backbones exhibit characteristics typically associated with signed networks such as reciprocity, structural balance, and community structure.\",\n \"The developed tool is provided as a free, open-source software package.\"\n ],\n [\n \"Introduction\",\n \"Networks are increasingly useful in modeling and studying many problems in seemingly unrelated domains from social sciences [9], natural sciences and engineering [74], and arts and humanities [68].\",\n \"A very simple network consists of nodes (vertices, actors) and links (edges, ties) connecting them.\",\n \"The network science tools utilize the information from the interdependence between entities given by the network structure.\",\n \"The data that can be modeled in the network form is not limited only to the node and edge structure.\",\n \"More complex networks have attributes associated with edges and nodes, especially with the advent of advanced data collection methods.\",\n \"In weighted networks, edges have numeric weights indicating its intensity (e.g., coupling strength, amount, similarity, etc.).\",\n \"In directed networks, edges have distinguishable source and target nodes, indicating the direction of the relation (e.g., flow, following, etc.).\",\n \"For instance, the air traffic can be modeled as a very simple network such that a node represents an airfield and an edge between two nodes indicates a flight between them.\",\n \"In this case, the number of connections an airfield has is given by the degree of the corresponding node.\",\n \"The same system can also be modeled as a directed weighted network where an edge carries the take-off and landing airfield data as its direction and passenger capacity data as its weight.\",\n \"In this way, the outgoing and incoming capacity of an airfield is given by the out- and in-strength of the node, that is the total weight of edges leaving and entering the node, respectively.\",\n \"This very simplified example demonstrates the view that as networks get more complex, richer analyses can be made.\",\n \"Another type of networks, which is explored relatively less in the literature, is signed networks where edges have negative or positive signs respectively indicating antagonism (e.g., dislike, distrust, foes, dissimilarity, voting against, inhibition, etc.)\",\n \"or rapport (e.g., like, trust, friendship, similarity, voting for, activation, etc.).\",\n \"Negative edges are not simply negation of positive edges [81], they show distinct behavior in networks [76], and their inclusion enrich the typical tasks on networks such as link prediction [50], [36], recommender systems [79], [89], node classification [60], [78], node centrality and ranking [31], [85], representation learning [46], [21], information diffusion and influence maximization [39], [44], finding cliques [51], community detection, graph partitioning and blockmodels [27], [55], [54], and polarization [8], [87].\",\n \"In addition to the typical and prevalent applications in social media [80], signed networks also find application areas in politics [3], international relations [24], finance [37], biology [42], [63], [40], and ecology [40].\",\n \"The density of a network is usually defined by the ratio of the number of observed links to the number of possible links.\",\n \"Formally, sparse networks are those where density asymptotically goes to zero in the limit of large number of nodes.\",\n \"However, in most empirical networks, it is impossible to evaluate this [61].\",\n \"As a result, in general, an empirical network is called sparse if it has a low density.\",\n \"The majority of real-world networks are sparse [62].\",\n \"For instance, the number of stable social relations of a human is limited by cognitive constraints [34], stations are usually only connected to nearest stations in power grid networks, a webpage points out only so many others in web networks, and so on.\",\n \"On the other hand, some networks might be very dense or even almost completely connected by its own nature.\",\n \"Most countries trade with most other countries, humans move/migrate from many locations to many other locations, many species have many predators and preys [25], most people interact with most others in certain social settings, and so on.\",\n \"The earlier view that additional data allowing richer analyses is restricted by the amount and structure of the data itself, particularly in the case where the additional data is provided by more links between the nodes.\",\n \"In addition to the fact that sparsity is desirable due to the computational complexity of many network algorithms, most of the typical network analysis and visualization methods assume the networks to be sufficiently sparse [61].\",\n \"Apart from computational and methodological concerns, dense networks might have noisy, uninformative, insignificant, or otherwise redundant links.\",\n \"This further worsens the application and interpretation capacity in many network tasks e.g., those relating to node centrality, cliques and communities, and diffusion.\",\n \"Therefore, the relevant and significant information should be extracted from such dense networks such that the original rich data is reduced into a network that is sparse and simpler but maintains adequate structural information for efficient and effective analyses.\",\n \"There is a body of literature for extracting backbones of usually weighted and relatively dense networks, which we review in the next section.\",\n \"Such information filtering task is usually referred to as backbone extraction or network sparsification and aims to remove statistically insignificant or otherwise redundant links while maintaining the informative structural properties for further analysis.\",\n \"To the best of our knowledge, this is the first study in the literature that provides methods for extracting signed network backbones from weighted dense networks.\",\n \"Unlike the earlier studies, our method relies on and requires the input network to be intrinsically dense; which in turn enables inferring the link signs.\",\n \"We use the term intrinsically dense for characterizing networks where all nodes, in a sense, are aware of all other nodes and can interact with them without obvious natural limits such as those mentioned earlier for sparse networks.\",\n \"This definition follows that an edge does not necessarily represent the existence of a positive relationship but might be an artifact of the randomness or even a negative relationship depending on its weight.\",\n \"Such definition of intrinsically dense networks provides a distinction not only from sparse networks but also from certain dense networks.\",\n \"For instance, human interaction network in a workshop with two parallel sessions with a single mutual short break is not intrinsically dense because the participants in different sessions are not given sufficient time to meet with others, thus, absence of a link cannot reasonably indicate if two participants from the different sessions are avoiding each other.\",\n \"As a demonstrating example of intrinsically dense networks, consider a network where link weights denote similarity of respective nodes.\",\n \"Lower similarity does not only mean a lack of similarity but often indicates dissimilarity, a negative underlying link.\",\n \"In the same vein, voting for a set of candidates indicates support for those candidates while also suggesting opposition for the other candidates especially those who are otherwise popular.\",\n \"Finally, we should highlight that whether a network is intrinsically dense or not is a matter of extent and not a strictly binary decision.\",\n \"We suggest that positive and negative links are those with weights significantly and substantially deviating from the random expectation under a suitable null model.\",\n \"We develop an appropriate null model based on hypergeometric distribution and iterative proportional fitting procedure and offer significance filter and vigor filter for extracting signed backbones of intrinsically dense networks.\",\n \"The proposed methodology is capable of handling directed or undirected networks as input and producing weighted or unweighted signed backbones.\",\n \"Our contribution can be summarized as follows.\",\n \"[itemsep=0pt] The first extensive literature review on network backbone extraction methods is provided (Section ).\",\n \"The first methods to extract signed backbones of intrinsically dense weighted networks are proposed (Section ).\",\n \"The methodology is empirically evaluated on real-world networks of different characteristics and its feasibility and usefulness is shown (Section ).\"\n ],\n [\n \"Backbone extraction methods for unsigned networks\",\n \"The simplest yet a popular approach is to apply a global threshold where only those edges with weights satisfying the predetermined threshold are retained.\",\n \"There are two major problems with this approach.\",\n \"First, the choice of the threshold is rather arbitrary and non-impartial unless the physical meaning of the weights in the domain of the particular network allows an explanation.\",\n \"The second and more important problem is the case of multiscale networks where the weights are distributed over a broad range of scales.\",\n \"Such characteristic is definitely not an exception but an observed phenomenon in many real-world networks [5].\",\n \"For instance, in many networks where rich club effects [72] are prevalent, nodes with higher degrees/strengths tend to form links with nodes of the same or higher degree/strength which results in a hierarchical and multiscale network structure.\",\n \"Rich club effects and such structures are present in many weighted networks including the global airline traffic network [1], worldwide maritime transportation network [41], world trade network [7], [67], international credit-debt networks [18], population flow network during a national holiday [86], patient referral networks [77], and communication networks in the brains of humans [1] and rats [52].\",\n \"Therefore, weights with small magnitudes are not necessarily noise and weights with large magnitudes are not necessarily important but might be the result of such multiscale weight characteristics.\",\n \"In multiscale networks, the application of a global threshold tends to eliminate the edges of low-strength nodes regardless of the local significance of those edges.\",\n \"As a result, the structure of high-strength nodes can be preserved but local regions characterized by relatively low-strength nodes are underestimated, might become disconnected, and might even be completely eliminated.\",\n \"In summary, increasing the value of a global threshold loses local structures at the bottom of the hierarchy whereas decreasing it results in retained noise in the upper levels.\",\n \"Consequently, global thresholding would only work under the often-unrealistic assumption that edge weights are independent and identically distributed random variables.\",\n \"As a remedy, the global threshold method can be improved to work on weights represented not in universal units but as fractions of the node degrees/strengths.\",\n \"Yet, an edge is associated with two nodes and it again becomes relatively arbitrary to choose the normalization factor.\",\n \"Such normalization might not be robust to nodes with very low and high degrees as well since it potentially overestimates the edges of low-degree nodes and underestimates the edges of high-degree nodes.\",\n \"Overall, it could serve as a practical remedy for certain problems but the need for more statistically-sound methods is clear.\",\n \"Another approach is to extract the spanning tree with maximal weight by appropriately transforming the weights and finding the minimum spanning tree.\",\n \"This method does not require any extrinsic input and ensures that the extracted backbone is connected.\",\n \"However, the backbone will be acyclic and local clustering and community structure cannot be preserved.\",\n \"Using a similar spanning tree approach, Tumminello et al.\",\n \"[82] propose a technique for controlling the genus of the backbone which, in turn, enables preservation of some local structure.\",\n \"Grady et al.\",\n \"[35] extracts the high-salience skeleton of the network which consists of only the salient edges.\",\n \"For each node, they construct the shortest path tree by merging all shortest paths from the node to all other nodes.\",\n \"Then, the saliency of an edge is defined as the fraction of the all shortest path trees where the edge is a member.\",\n \"They show that edge saliency shows a bimodal distribution near the boundaries.\",\n \"Only the edges with saliency near 1 are retained; effectively eliminating the need for choosing an arbitrary threshold.\",\n \"By design, it ensures the connectivity of the extracted backbone.\",\n \"Edge saliency is different from edge betweenness where the former tends to award the edges in the periphery (e.g., low-degree nodes) whereas the latter tends to award the edges in the core (e.g., high-degree nodes).\",\n \"The methods described so far do not assume any underlying null model and do not compare observed weights to expected weights for statistical evaluation.\",\n \"Disparity filter [70] assumes that the normalized weights of the links of a node follow a uniform distribution.\",\n \"Comparing the observed weights to this null model at a desired significance level, the network backbone including only the statistically significant links can be obtained.\",\n \"As mentioned earlier, a weight can be normalized and evaluated separately for the two nodes it connects.\",\n \"Thus, a link can be significant from the viewpoint of one node and not the other.\",\n \"This problem is tackled by retaining a link if it satisfies the significance condition for at least one of the two nodes.\",\n \"Adopting a loosely similar weight normalization scheme, bistochastic filter [73] first scales the weight matrix such that its marginals (i.e., the row and column totals, in- and out-strength sequence) are equal to 1 resulting in a doubly stochastic matrix.\",\n \"Then, the links with highest weights are retained until the network is strongly connected; or until any other stopping criteria which can result in sparser backbones with disconnected components or even denser backbones.\",\n \"In the same line of research, LANS [30] does not assume an underlying distribution for the weights of a node and employs the empirical cumulative density function instead.\",\n \"It is stated that it is more robust to the highly heterogeneous local weight distributions than disparity filter and bistochastic filter are.\",\n \"Rather than adopting a local approach in developing null models, GloSS filter [64] utilizes a global null model.\",\n \"The model considers both the strengths and degrees and aim to preserve the global weight distribution and the topology.\",\n \"GLANB [90] evaluates the statistical significance of link saliency to retain satisfactory links; loosely combining disparity filter and high-salience skeleton.\",\n \"Dianati et al.\",\n \"[22] introduce two interrelated filters.\",\n \"Treating an integer-weighted network as a multiedge network, they assume that each unit edge chooses two nodes in random respecting the degree sequences which results in a binomial distribution for weights.\",\n \"The null model is indicated to blend the approaches of disparity filter and GloSS.\",\n \"Marginal likelihood filter operates by evaluating each edge separately against a chosen significance level whereas global likelihood filter incorporates exponential random graph model with a Monte Carlo simulation scheme to consider all links at once.\",\n \"ECM filter [33] enhances the null model of Dianati et al.\",\n \"[22] with the purpose of retaining the relation between strengths and degrees.\",\n \"Polya filter [58] assumes an underlying null with a self-reinforcing mechanism in which the generation of link weight increases the probability of further weight generations for that link.\",\n \"Tumminello et al.\",\n \"[83] introduce hypergeometric filtering originally for bipartite weighted networks.\",\n \"It is employed for unipartite weighted networks [65], [71] as well.\",\n \"Assuming integer weights and treating weighted links as multiple links, every time a node increases its degree by 1 by adding a new link, the node at the other end of the link is selected randomly among all possible nodes.\",\n \"Then, the edge weight generation process can be described by a hypergeometric distribution which becomes the null model for the filter.\",\n \"Comparing observed values with the null model, statistically significant links can be extracted at desired levels of confidence.\",\n \"Noise-Corrected Bayesian filter [19] assumes edge weight generation follows a binomial distribution where a null model based on the hypergeometric distribution is used in determining Bayesian priors.\",\n \"Lift value for each link (i.e., the ratio of the observed value to the expected value) is calculated, transformed to the $[-1, 1]$ range, and the associated variance is estimated with a Bayesian inference schema.\",\n \"Using the appropriate posterior variances, the links which satisfy a desired significance level are retained.\",\n \"Although more specific approaches exist with different purposes (e.g., to retain the most consistent links in brain networks based on diffusion images [66], to preserve functional backbones based on network motifs [13]), the state-of-the-art concentrates on developing appropriate null models to evaluate link weights against and retain only those satisfying a desired level of statistical significance.\"\n ],\n [\n \"Formal Problem Definition\",\n \"We denote an intrinsically dense, undirected or directed, and non-negative weighted network without self-loops by $G := (V, W)$ .\",\n \"$V$ is the set of nodes with $i, j \\\\in V$ as its general elements and its cardinality is $n = |V|$ .\",\n \"$W$ is the corresponding weight matrix.\",\n \"When $G$ is undirected $W_{ij}$ denotes the weight of the link between $i$ and $j$ and $W_{ij} = W_{ji}$ .\",\n \"When $G$ is directed, $W_{ij}$ denotes the weight of the link from $i$ to $j$ .\",\n \"Self-loops are not allowed hence $W_{ii} = 0 \\\\; \\\\forall i$ .\",\n \"Total outflow of $i$ is denoted with $W_{i.}\",\n \"= \\\\sum _{j} W_{ij}$ .\",\n \"Total inflow to $j$ is denoted with $W_{.j} = \\\\sum _{i} W_{ij}$ .\",\n \"The sum of all weights is denoted with $W_{..} = \\\\sum _{i} \\\\sum _{j} W_{ij}$ .\",\n \"We employ the same definitions of $W_{i.\",\n \"}$ , $W_{.j}$ , and $W_{..}$ for directed and undirected networks, i.e., treating an undirected link as two reciprocal directed links with equal weights.\",\n \"Given $G$ , the aim is to extract a meaningful signed network $\\\\hat{G} := (\\\\hat{V}, \\\\hat{A}, \\\\hat{W})$ where $\\\\hat{A}$ is a sparse adjacency matrix in general such that $\\\\hat{A}_{ij} \\\\in \\\\lbrace -1, 0, 1\\\\rbrace $ where $-1$ denotes a negative link, 1 denotes a positive link, and 0 indicates the absence of a link.\",\n \"$\\\\hat{W}$ is the corresponding optional weight matrix.\",\n \"If $\\\\hat{A}_{ij} = 0$ , then $\\\\hat{W}_{ij} = 0$ ; otherwise $\\\\hat{W}_{ij}$ has the same sign as $\\\\hat{A}_{ij}$ .\"\n ],\n [\n \"Proposed Solution\",\n \"The null model.\",\n \"To extract the signed edges, it is necessary to have a null model which empirical edge weights are compared against to distinguish edge weights that are due to chance and edge weights that significantly deviate from the expected values in either positive or negative direction.\",\n \"We make the assumption that every time a node increases its out-strength by one, it randomly chooses the other node with probabilities proportional to their in-strengths.\",\n \"In this way, the process is reduced to sampling without replacement problem, of which the urn problem is a famous example.\",\n \"We describe the process with a simple analogy.\",\n \"For each node $i$ , there are $W_{..} - W_{.i}$ marbles in the urn (not allowing self-loops), $W_{i.\",\n \"}$ marbles are chosen from the urn without replacement, and there are $W_{.j}$ marbles in the urn for each $j$ .\",\n \"Such process is well-characterized by hypergeometric distribution and allows us to calculate statistical quantities for links between all $i$ and $j$ .\",\n \"The mean of the hypergeometric distribution associated with the link from $i$ to $j$ is given by Eq.\",\n \"REF .\",\n \"However, this formulation does not preserve the in-strength and out-strength sequences in the system, thus, is inadequate for our purposes.\",\n \"Given that $N$ is the weight matrix under null model with $N_{ij}$ denoting the expected weight from $i$ to $j$ , the following should be satisfied: $N_{i.}\",\n \"= W_{i.\",\n \"}$ , $N_{.j} = W_{.j}$ , and $N_{..} = W_{..}$ .\",\n \"For this purpose, we utilize the iterative proportional fitting procedure (IPFP) described in [4], [59] which is shown to converge to an optimal solution [10].\",\n \"Given the column and row marginals (e.g., in- and out-strength sequences) and an appropriate prior matrix with certain elements as zero (e.g., diagonal elements to disallow self-loops) and other elements as akin to Eq.\",\n \"REF , IPFP estimates the values for non-zero elements of the matrix through a series of row-scaling and column-scaling operations until achieving a desired precision.\",\n \"Given out- and in-strength ($W_{i.\",\n \"}$ and $W_{.j}$ ) sequences as the marginals and the values obtained from Eq.\",\n \"REF as the elements prior matrix with its diagonal elements set to 0, IPFP finds the suitable matrix $N$ representing the expected weights under our proposed null model.\",\n \"$\\\\frac{W_{i.}\",\n \"W_{.j}}{W_{..} - W_{.i}}$ In evaluating the statistical significance of an edge weight, only considering the difference of observed and expected values without referencing to a dispersion measure is not viable.\",\n \"For each edge, dispersion of its expected weight must be known to produce confidence intervals.\",\n \"Accordingly, we estimate the variance of $N_{ij}$ based on the associated hypergeometric distribution and obtain its standard deviation as given by Eq.\",\n \"REF .\",\n \"$\\\\sigma _{ij} = \\\\left( W_{i.}\",\n \"\\\\; \\\\frac{W_{.j}}{W_{..} - W_{.i}} \\\\; \\\\frac{(W_{..} - W_{.i}-W_{.j})}{W_{..} - W_{.i}} \\\\; \\\\frac{W_{..} - W_{.i}-W_{i.}}{W_{..}\",\n \"- W_{.i}-1} \\\\right) ^ {\\\\frac{1}{2}}$ Significance filter.\",\n \"The first filter we propose functions to eliminate links whose weights do not significantly deviate from their expected values.\",\n \"Significance filter retains only those links satisfying the condition $\\\\sigma _{ij} \\\\: \\\\alpha ^- \\\\: \\\\le W_{ij} - N_{ij} \\\\le \\\\: \\\\sigma _{ij} \\\\: \\\\alpha ^+$ where $\\\\alpha ^-$ and $\\\\alpha ^+$ respectively take non-positive and non-negative values and are user-defined hyperparameters specifying the desired significance thresholds for negative and positive signed edges.\",\n \"It should be highlighted that the hypergeometric distribution is a distribution for discrete events.\",\n \"In general, we can slightly abuse it by allowing continuous weights or rounding weights to the nearest integers without much impact on the resulting backbone.\",\n \"However, a larger issue that is not adequately discoursed in the hypergeometric filtering literature is the impact of the magnitude of weights on the statistical significance evaluation.\",\n \"As the magnitude of weights increases, the confidence intervals relative to the weights become narrower, as a consequence of the assumption that the weights are discrete events.\",\n \"For instance, the same monetary network represented in dollars or cents would result in different network backbones for the same values of $\\\\alpha $ .\",\n \"This issue is less pressing when the link weights essentially correspond to discrete events such as human migration networks or voting networks.\",\n \"Taking these into account, it can be concluded that the values chosen for $\\\\alpha $ might not accurately translate into traditional p-values in many networks due to their non-discrete nature.\",\n \"Hence, the choice of $\\\\alpha $ should not necessarily be limited to traditionally employed values especially when the weights do not correspond to discrete events.\",\n \"Vigor filter.\",\n \"The significance filter might retain statistically significant but otherwise very weak links.\",\n \"Yet, an edge can be evaluated also based on whether it is sufficiently strong in terms of intensity.\",\n \"This is useful for multiple reasons: (i) the employed significance filter might be too permissive in certain circumstances, (ii) the signed network backbone is needed to reflect only binary links where opinions are rather strong, or (iii) higher sparsity is desired.\",\n \"We define vigor of an edge from $i$ to $j$ as $\\\\beta _{ij}$ (pronounced as víta in Modern Greek) in Eq.\",\n \"REF .\",\n \"It takes values in the range $[-1, 1]$ , its magnitude indicates the intensity of the link and $\\\\beta _{ij} = 0$ when $W_{ij} = N_{ij}$ .\",\n \"Vigor filter retains only those links satisfying the condition $\\\\beta ^- \\\\le \\\\beta _{ij} \\\\le \\\\beta ^+$ where $\\\\beta ^+$ and $\\\\beta ^-$ respectively take non-positive and non-negative values and are user-defined hyperparameters specifying the desired vigor thresholds for positive and negative signed edges.\",\n \"$\\\\beta _{ij} = \\\\frac{W_{ij}/N_{ij} \\\\; -1}{W_{ij}/N_{ij} \\\\; +1}$ Signed backbone extraction.\",\n \"The signed backbone of a network can be extracted using a combination of significance filter and vigor filter.\",\n \"We suggest always employing the former since $\\\\beta $ values might be unreliable in the small-magnitude weight regime characterized by small $N_{ij}$ , relatively large $\\\\sigma _{ij}$ , and small $W_{ij}$ .\",\n \"Significance filter would function to eliminate such links, therefore, the retained links would have reliable vigor.\",\n \"Vigor values of the remaining edges can be utilized as the signed edge weights in the extracted backbone.\",\n \"In the case of weighted backbone, the use of vigor filter is rather optional since the vigor information is directly carried onto the network.\",\n \"Alternatively, the weights can be ignored to produce a network with binary opinions.\",\n \"In this case, we recommend utilizing the vigor filter to retain only the edges that exhibit appropriate levels of intensity.\",\n \"Overall, for setting appropriate values to the hyperparameters, one should consider (i) the physical meaning of hyperparameters in the context of the specific problem, (ii) the desired level of sparsity for the extracted backbone.\",\n \"Undirected networks.\",\n \"The described method is defined well on the directed networks.\",\n \"The generalization to the undirected networks is ensured in the following way.\",\n \"An undirected link is replaced with two reciprocal directed links of the same weight; effectively transforming the network into a directed network.\",\n \"The null model and the filters produce the same null expectation and nearly the same varianceFor sufficiently large networks, $W_{..} - W_{.j} \\\\approx W_{..} - W_{.i}$ in In Eq.\",\n \"REF for such reciprocal links.\",\n \"Finally, the directed backbone can be transformed back into an undirected backbone by treating reciprocal directed links as undirected links and removing the redundancy by keeping the link with the highest absolute vigor.\"\n ],\n [\n \"Empirical Analysis\",\n \"In this section, relevance and diverse characteristics of the employed network datasets are described, and proposed filters are empirically analyzed in terms of resulting backbone sizes, robustness to multiscale networks, and structures of the extracted signed network backbones.The datasets and code for producing the analyses in this section can be made available upon request.\"\n ],\n [\n \"Datasets.\",\n \"The proposed method for signed network backbone extraction is experimented on four real-world networks from different domains and of varying statistical properties.\",\n \"For each network; the number of nodes $n$ , number of nonzero links $m$ , density, five-point summary of weights (minimum, first quantile, median, third quantile, and maximum values), and three-point summary (minimum, median, and maximum values) of node strengths (out- and in-strengths if networks are directed) are presented in Table REF .\",\n \"Table: Employed NetworksMigration network [11] reports interstate migration flows (in terms of people) among the US statesThe dataset contains District of Columbia in addition to the 50 states.\",\n \"in 2018.\",\n \"Such internal migration data is explored for its correlation with economic productivity differences, geographical proximity, political preferences, and other cultural and historical factors [15], [53], [17].\",\n \"Eurovision network [84] represents the votes between participant countries of the song contest in 20032003 was selected because it was the last year that all countries competed in a single round.. Each country, via public votesIreland, Russia, and Bosnia and Herzegovina exercised jury voting instead of public voting., awards the set of points {1,2,3,4,5,6,7,8,10,12} to 10 other countries.\",\n \"Studies employing a network analysis perspective [28], [16], [32], [56], [57], [26], [75] show that voting behavior is not determined only by the music/performance quality but affected by political factors, geographical and cultural similarity, diasporas, and others.\",\n \"Contact network data [43] is gathered via wearable sensors at an academic conferenceThe temporal face-to-face interaction dataset was collected at Hypertext conference in 2009 as part of SocioPatterns research collaboration.\",\n \"with 113 attendees over 2.5 days.\",\n \"The edge weights are generated by aggregating the number of 20-second intervals the respective two participants spent face-to-face over the course of the conference.\",\n \"The networks of such temporal ties are utilized for temporal backbone extraction, investigation of spreading processes, and other behavioral and structural analysis [47], [88], [49], [6].\",\n \"Species network [48] is generated based on the cohabitation patterns of 62 marine species in South Florida.\",\n \"Edge weights represent the similarity of species based on the habitats they co-occupy during the same life stagesGiven the species v. habitat-life stage bipartite matrix $B$ , $W_{ij} := \\\\sum _{k} B_{ik} B_{jk}$ where $i, j$ are the species and $k$ is the habitat-life stage pair, and $B_{ik}$ is the respective species-habitat-life stage score.\",\n \"More information on the original dataset can be found at https://atlanticfishhabitat.org/species-habitat-matrix/.\",\n \"Information derived from such cohabitation networks is useful in various ecological studies [45], [12], [29].\",\n \"Moreover, none of the employed networks have natural self-loops; which make them particularly suitable for our study.\",\n \"Our proposed method considers the propensity of nodes for incoming and outgoing edge weights by explicitly including the in-strengths and out-strengths in the null model.\",\n \"For instance, high economic productivity for a state would result in more immigration into it.\",\n \"Similarly, low productivity would cause emigration from that state.\",\n \"The effect of such productivity is homogeneous in the sense that all other nodes perceive it in the same way.\",\n \"Such homogeneous effects are eliminated for the large part by our null model.\",\n \"However, for instance, the effect of political affiliation or geographic location is perceived differently by other nodes.\",\n \"Hence, the extracted backbone is expected to reflect non-economic factors in Migration network.\",\n \"With a simple analogy, the quality of the songs and performances in Eurovision song contest is equivalent to productivity.\",\n \"Hence, the elimination of such effect by the null model would result in an extracted signed network representing the non-quality factors in voting behavior.\",\n \"Similarly, the null model largely eliminates the effects of participants' tendency to engage in conversations (e.g., popularity, extroversion) and the extent of species' ability to live in many different habitats.\"\n ],\n [\n \"Backbone Size\",\n \"In general, a signed network backbone should be sufficiently sparse such that only the important edges are preserved.\",\n \"At the same time, unlike edges, most nodes should be retained since they are often the subject of the analysis and are largely indispensable in understanding the global network structure.\",\n \"Here, we extract the signed backbones of the four networks under various hyperparameter regimes and observe the size of the resulting backbone in terms of signed edges and nodes.\",\n \"Effects of significance threshold.\",\n \"Figure REF visualizes the fraction of links and nodes retained for meaningful continuous ranges of the significance threshold under different vigor filters.\",\n \"The suitable range of $\\\\alpha $ differs between networks mostly due to the varying overall magnitude of edge weights.\",\n \"The significance threshold is represented with $\\\\alpha = \\\\alpha ^+ = - \\\\alpha ^-$ on x-axes.\",\n \"Three different vigor threshold settings are represented in columns.\",\n \"For edges, percentage values on y-axes are with respect to all possible edges, that is $n^2 - n$ edges for directed networks and half of it for undirected networks.\",\n \"Figure: Effects of significance filter on the backbone size.Figure: Effects of vigor filter on the backbone size.Except for Species network, all nodes are retained in the backbones even under very strict settings where 80% to 85% of all possible edges are eliminated.\",\n \"When edges are inspected, apart from the Species network where edges represent similarity, the negative edges are more frequent than the positive edges when $\\\\alpha = 0, (\\\\beta ^-, \\\\beta ^+) = (0, 0)$ .\",\n \"This is due to the case that edges with zero weights (i.e., nonexistent edges) or trivially small values are treated as negative links regardless of their statistical significance or vigor.\",\n \"As the values of $\\\\alpha $ increase, the number of negative links in the extracted backbone decreases faster than the positive links since many of the zero or trivially small weights are rather statistically insignificant.\",\n \"On the other hand, utilizing vigor filter without the significance filter does not eliminate such zero-weighted links since those edges have vigor values of $-1$ (since 0 divided by any null expectation is 0, which maps to vigor of $-1$ ).\",\n \"Hence, the decrease in negative edges in second and third columns in comparison to the first column of Figure REF when $\\\\alpha = 0$ is not provided by eliminating zero-weighted edges but other weak edges.\",\n \"Therefore, significance filter is always necessary when insignificant zero or trivially small edge weights exist.\",\n \"Employing very large $\\\\alpha $ values eliminates negative links almost completely.\",\n \"This is largely because the edges with relatively small expected values under the null model have relatively large variances.\",\n \"Increasing $\\\\alpha $ too much, thus, tends to eliminate negative edges first as well as other edges in local regions characterized by edge weights of small magnitude.\",\n \"In the case of high-magnitude regions, an edge between two high-strength nodes is expected to have a large expectation under the null model.\",\n \"It is much harder for such edges to empirically observe much larger weights than the null expectation since the total weight in the network is fixed.\",\n \"This implies an upper boundary on vigor values for edges connecting to high-strength nodes.\",\n \"Therefore, increasing $\\\\beta $ too much, especially on its positive range, tends to eliminate edges between central, high-strength nodes.\",\n \"Effects of vigor threshold.\",\n \"In a similar fashion, we have also explored the whole continuous range of vigor threshold under different significance threshold filter in Figure REF .\",\n \"The significance threshold is represented with $\\\\beta = \\\\beta ^+ = -\\\\beta ^-$ on x-axes.\",\n \"Three different significance threshold settings are represented in columns.\",\n \"Except for Species network and except under strict vigor thresholds, all or almost all nodes remain in the backbone.\",\n \"When edges are inspected, in line with the relevant conclusions derived from Figure REF , increasing $\\\\beta $ does not sufficiently eliminate the negative edges which would be otherwise statistically insignificant.\",\n \"In very large values of $\\\\beta $ , positive links are largely eliminated.\",\n \"When $\\\\beta = 1$ , the edges remaining in the backbone are the zero-weighted (i.e., nonexistent) edges.\",\n \"Accordingly, employing very large vigor thresholds should be avoided.\",\n \"Conclusion.\",\n \"Overall, we have shown that the proposed method is able to reduce an intrinsically dense network to a signed backbone of the relative size of $10\\\\%$ to $20\\\\%$ such that the resulting backbone contains comparable portions of negative and positive edges.\",\n \"Yet, the hyperparameter selection largely lies with the user and can be changed according to the purpose of analysis and the nature of the network.\",\n \"For instance, Species network is constructed with similarity values as edge weights and the edge weights do not follow a skewed distribution unlike other networks.\",\n \"Therefore, its behavior under the proposed filters partially deviates from the behavior of the other employed networks.\",\n \"Principally, we can conclude that (i) significance filter should almost always be used with a fair threshold, (ii) very large threshold values for both filters should be avoided, and (iii) a sufficiently balanced, statistically meaningful, and sparse backbone can be extracted with a balanced utilization of the two proposed filters.\"\n ],\n [\n \"Heterogeneity\",\n \"As discussed in Section , an enviable backbone extraction method should respect the weight and strength heterogeneity in the original network (i.e., its multiscale, hierarchical nature).\",\n \"That is to say, the retained edges should not be only those originally with large weights or those connecting high-strength nodes.\",\n \"Heterogeneity of edge weights.\",\n \"Figure REF presents the original weight distributionSmall random noise is added for Eurovision network for visualization purposes.\",\n \"of the retained edges in backbones of different sizes The backbones are extracted with the following ($\\\\alpha ^-, \\\\alpha ^+$ ), ($\\\\beta ^-, \\\\beta ^+$ ) settings for the respective backbone sizes: Migration $\\\\rightarrow $ 25%: (-33, 33), (-0.33, 0.33); 10%: (-40, 40), (-0.57, 0.57); 5%: (-40, 40), (-0.72, 0.72).\",\n \"Eurovision $\\\\rightarrow $ 25%: (-1.5, 2.25), (0, 0); 10%: (-2, 4.5), (0, 0); 5%: (-2.5, 5), (0, 0).\",\n \"Contact $\\\\rightarrow $ 25%: (-2, 2), (0, 0); 10%: (-3, 3), (-0.66, 0.66); 5%: (-5, 5), (-0.55, 0.55).\",\n \"Species $\\\\rightarrow $ 25%: (-3.5, 3.5), (0.15, 0.15); 10%: (-5, 5), (-0.2, 0.2); 5%: (-5.5, 5.5), (-0.33, 0.33).\",\n \".\",\n \"The figure shows that even when the extracted backbone retains only $5\\\\%$ of all possible edges, the heterogeneity of edge weights are respected.\",\n \"It also visually depicts that there is no global cutoff value for determining the sign of edges as those cutoff values are established individually for each edge by the null model.\",\n \"Figure: Original weights of the retained edgesHeterogeneity of node strengths.\",\n \"Figure REF show the strength distributionSmall random noise is added for Eurovision network for visualization purposes.\",\n \"of the dyads for the original networks in transparent color and the extracted backbones with relative size of $\\\\approx 20\\\\%$The backbones in Figure REF are extracted with the following ($\\\\alpha ^-, \\\\alpha ^+$ ), ($\\\\beta ^-, \\\\beta ^+$ ) settings.\",\n \"Migration $\\\\rightarrow $ (-40, 40), (-0.25, 0.25).\",\n \"Eurovision $\\\\rightarrow $ (-1.8, 2.4), (-0.5, 0.3).\",\n \"Contact $\\\\rightarrow $ (-2.25, 2.25), (-0.25, 0.25).\",\n \"Species $\\\\rightarrow $ (-3.6, 3.6), (-0.25, 0.25).\",\n \"in opaque color.\",\n \"Specifically, the edge between nodes $i$ and $j$ is represented with a point colored based on its sign.\",\n \"X-axis denotes the (out-)strength of $i$ and y-axis denotes the (in-)strength of $j$For an undirected network, $i$ and $j$ might switch arbitrarily.\",\n \"As demonstrated by the figure, the retained edges are rather evenly distributed in the plotting space and the original heterogeneity is respected.\",\n \"The retained edges between very low-strength nodes are usually positive since null expectations are usually low in that regime and it is difficult to conclude whether small-weighted edges in low-strength regimes are statistically significant.\",\n \"Likewise, the edges between very high-strength nodes are usually negative due to the upper bound on vigor implied by the large null expectation and the fixed amount of total weights.\",\n \"On the other hand, in the majority of strength-strength regions, the edge signs exhibit a mixed distribution.\",\n \"Therefore, this is generally a desired property since the proposed method, by itself, does not allow deducing strong conclusions when the data is limited.\",\n \"Figure: Original strength distribution of the retained dyadsConclusion.\",\n \"We can conclude that the heterogeneity given by (i) the multiscale nature of edge weights and (ii) heterogeneous nature of node strengths are respected, and (iii) the proposed filters do not draw strong conclusions when the data is limited.\"\n ],\n [\n \"Structure of the backbones\",\n \"Generally speaking, we expect a real-world signed network to show some extent of reciprocity and have its reciprocal edges be of the same sign, and demonstrate sufficient levels of structural balance.\",\n \"Moreover, most real-world networks exhibit a community structure regardless of whether they are signed or not.\",\n \"Therefore, an informative signed backbone should have these characteristics in general and up to a certain extent.\",\n \"We analyze the extracted backbonesThe backbones in Figure REF are extracted with the following ($\\\\alpha ^-, \\\\alpha ^+$ ), ($\\\\beta ^-, \\\\beta ^+$ ) settings.\",\n \"Migration $\\\\rightarrow $ (-40, 40), (-0.33, 0.33).\",\n \"Eurovision $\\\\rightarrow $ (-1, 0.5), (-0.33, 0.1).\",\n \"Contact $\\\\rightarrow $ (-3, 3), (-0.33, 0.33).\",\n \"Species $\\\\rightarrow $ (-5,3), (-0.5, 0.1).\",\n \"of the employed networks in terms of reciprocity, structural balance, and community structure.\",\n \".\",\n \"Reciprocity.\",\n \"Reciprocity in unsigned networks is generally defined as the ratio of the number of reciprocal edges to the number of all edges.\",\n \"In signed networks, however, the edge signs should also be considered in the analyses.\",\n \"Table REF presents the number of nodes, edges, nonreciprocal edges ({$\\\\times $ , $\\\\cdot $ }), reciprocated positive edges ({+, +}), reciprocated negative edges ({-, -}), and reciprocated edges with sign conflict ({+, -}) for the directed backbones extracted from directed networks.\",\n \"In Migration backbone, there is considerable reciprocity with no conflicting edge signs between any node pair.\",\n \"In contrast, Eurovision backbone discloses a substantial amount of conflicting edge pairs which might be of interest for further investigation into its voting dynamics (e.g., the influence of diasporas).\",\n \"Table: Reciprocity in extracted directed backbonesFor the rest of the analysis, the directed backbones are transformed into undirected backbones in the following way.\",\n \"For Migration backbone, directed edges are transformed into undirected edges of the same sign with positive signs having a priority over negative signs when in conflict.\",\n \"For Eurovision backbone, a positive (or negative) edge is created between two nodes when there are exactly two positive (or negative) directed edges between them (i.e., reciprocal edges of the same sign).\",\n \"Structural Balance.\",\n \"As put forward by Heider [38] and formalized for signed networks by Cartwright and Harary [14], an undirected triple is said to be balanced if its edges have the signs {+, +, +} or {+, -, -} and unbalanced if its edges have the signs {+, +, -} or {-, -, -}.\",\n \"The balanced triples can be simply described with the following expressions: friend of my friend is my friend and enemy of my friend is my enemy.\",\n \"Davis [20] defines a weaker notion and proposes that enemy of my friend is my enemy is not necessarily required for balance and the only unbalanced triple among the four possible settings is the one with edge signs {+, +, -}.\",\n \"Accordingly, structural balance (SB) and weak structural balance (WSB) of a network can be defined as the ratio of the number of balanced triples to the number of all triples.\",\n \"Frustration index (FI) [2] is a measure of the minimum number of required edges such that upon their deletion the network is partitioned into two communities with no positive edges between the communities and with no negative edges within communities.\",\n \"The number of such edges is scaled so that a fully balanced network of two communities results in a FI of 1.\",\n \"Table REF shows the count of nodes, edges, and four possible triple settings; and three different structural balance measures for the undirected backbones.The number of triples and frustration index is calculated using signnet [69] package in R. The backbones for Migration and Species shows a strong balance for all three measures.\",\n \"The other two backbones are also highly balanced in terms of weak structural balance which is shown to be more appropriate in real-world networks.\",\n \"Moreover, it should be highlighted that in the calculation of the frustration index the network is partitioned into only two communities whereas there are multiple underlying communities for three of the networks as discussed next.\",\n \"Table: Structural characteristics of extracted backbonesCommunity Structure.\",\n \"Based on the partitioning methods for signed networks [23], the extracted backbones are partitioned into communities where the number of negative edges within communities and number of positive edges between communities are minimized via heuristicsCommunities are found and visualized using signnet [69] package in R. Figure REF depicts the community structure via a visual block matrix where the order of nodes is the same for its rows and columns.\",\n \"Diagonal blocks are expected to cover the positive edges and non-diagonal blocks are expected to cover the negative edges.\",\n \"Overall, the extracted backbones manifest community structures.\",\n \"The backbone of Migration has 5 clear densely connected communities and two outlier individual nodes whereas the communities in Eurovision are not as dense.\",\n \"The backbone of Contact consists of one large sparse group and several relatively-small denser groups.\",\n \"The backbone of Species has a very clear structure with two similar-sized communities, visually confirming the very high frustration index ($0.96$ ) obtained for it.\",\n \"Figure: Block models of extracted backbonesTable REF shows members of each group in the order of diagonal blocks where nodes are also listed in the same order as they appear in the rows and columns.\",\n \"We can observe that the states are grouped mostly based on their geographical proximity except for two outlier states which are not part of any groups.\",\n \"Similarly, countries in the Eurovision backbone are grouped mainly based on geographic, cultural, and historical ties.\",\n \"Although the results are seemingly plausible, a sophisticated and conclusive discussion on the found communities and its usefulness is spared for future studies in those domains.\",\n \"Table: Block membershipsConclusion.\",\n \"We showed that the extracted backbones usually (i) reveal interesting reciprocal structures, (ii) are structurally balanced, and (iii) exhibit clear community structure.\"\n ],\n [\n \"Conclusion and Future Work\",\n \"In this study, we put forward an initial discussion on intrinsically dense networks and provide significance filter and vigor filter for extracting signed backbones of such networks.\",\n \"Empirical evaluations that utilize the proposed filters on a variety of real-world networks show that sparse backbones can be obtained while maintaining comparable numbers of positive and negative links and respecting the original weight and strength heterogeneity.\",\n \"In general, the obtained backbones exhibits characteristics associated with signed networks such as reciprocity, structural balance, and community structure.\",\n \"The extraction method, by design, does not prematurely arrive at conclusions regarding the existence of signed links between nodes when the data is limited.\",\n \"On the other hand, choosing appropriate hyperparameter values lies with the user and should be guided by empirical analysis and recommendations provided in this work as well as the purpose and the nature of the specific problem.\",\n \"Further studies can improve this work in two major ways.\",\n \"As we presented many examples throughout the manuscript, intrinsically dense networks exist in many different domains.\",\n \"First, a stream of its applications in different fields and utilization of the resulting backbones in different tasks would provide important feedback regarding its usefulness, weaknesses, and strengths.\",\n \"Second, hypergeometric distribution characterizes a discrete process and such discrete distributions are being utilized in the state-of-the-art backbone extraction methods including ours.\",\n \"As we discussed earlier, null models developed based on it can have certain undesirable properties for certain networks.\",\n \"Although such issues are easy to handle for practical purposes, a natural avenue is to develop new null models that are more appropriate for the case of link weights which are continuous or that can be equivalently represented in different units.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05216"}}},{"rowIdx":3696,"cells":{"text":{"kind":"list like","value":[["Simple is not Easy: A Simple Strong Baseline for TextVQA and TextCaps"],["Abstract Texts appearing in daily scenes that can be recognized by OCR (Optical Character Recognition) tools contain significant information, such as street name, product brand and prices.","Two tasks -- text-based visual question answering and text-based image captioning, with a text extension from existing vision-language applications, are catching on rapidly.","To address these problems, many sophisticated multi-modality encoding frameworks (such as heterogeneous graph structure) are being used.","In this paper, we argue that a simple attention mechanism can do the same or even better job without any bells and whistles.","Under this mechanism, we simply split OCR token features into separate visual- and linguistic-attention branches, and send them to a popular Transformer decoder to generate answers or captions.","Surprisingly, we find this simple baseline model is rather strong -- it consistently outperforms state-of-the-art (SOTA) models on two popular benchmarks, TextVQA and all three tasks of ST-VQA, although these SOTA models use far more complex encoding mechanisms.","Transferring it to text-based image captioning, we also surpass the TextCaps Challenge 2020 winner.","We wish this work to set the new baseline for this two OCR text related applications and to inspire new thinking of multi-modality encoder design.","Code is available at https://github.com/ZephyrZhuQi/ssbaseline"],["Introduction","To automatically answer a question or generate a description for images that require scene text understanding and reasoning has broad prospects for commercial applications, such as assisted driving and online shopping.","Equipped with these abilities, a model can help drivers decide distance to the next street or help customers get more details about a product.","Two kinds of tasks that focus on text in images have recently been introduced, which are text-based visual question answering (TextVQA) [34], [7] and text-based image captioning (TextCaps) [33].","For example, in Figure REF , a model is required to answer a question or generate a description by reading and reasoning the texts “tellus mater inc.” in the image.","These two tasks pose a challenge to current VQA or image captioning models as they explicitly require understanding of a new modality – Optical Character Recognition (OCR).","A model must efficiently utilize text-related features to solve these problems.","Figure: An example of TextVQA and TextCaps tasks.","The answer and description are generated by our model.","Our simple baseline is able to read texts and answer related questions.","Besides, it can also observe the image and generate a description with texts embedded in it.Figure: Encoders of different models.","(a) Current state-of-the-art model M4C on TextVQA task forwards each feature vector of all modalities indiscriminately into transformer layers, which exhaust tremendous computation.","(b) MM-GNN handcrafts three graphs to represent the image and applies three aggregators step by step to pass messages between graphs.","(c) SMA introduces a heterogeneous graph and considers object-object, object-text and text-text relationships, upon which a graph attention network is then used to reason over them.","(d) Our baseline uses three vanilla attention blocks to highlight most relevant features and combines them into six individually-functioned vectors, which is then sent into transformer-based fusion encoders.","The considerably fewer parameters of six vectors save computation.For TextVQA task, the current state-of-the-art model M4C [17] handles all modalities (questions, visual objects and OCR tokens) over a joint embedding space.","Although this homogeneous method is easy to implement, fast to train and has made great headway, it considers that texts and visual objects contribute indiscriminately to this problem and uses text features as a whole.","For TextCaps problem, the only difference is that it only has two modalities: visual objects and OCR tokens.","However, these limitations remain.","Some other works proposed even more complex structures to encode and fuse multi-modality features of this task, i.e., questions, OCR tokens and images.","For example, SMA [13] uses a heterogeneous graph to encode object-object, object-text and text-text relationships in the image, and then designs a graph attention network to reason over it.","MM-GNN [14] represents an image as three graphs and introduces three aggregators to guide message passing from one graph to another.","In this paper, we use the vanilla attention mechanism to fuse pairwise modalities.","Under this mechanism, we further use a more reasonable method to utilize text features which leads to a higher performance, that is by splitting text features into two functionally different parts, i.e., linguistic- and visual-part which flows into corresponding attention branch.","The encoded features are then sent to a popularly-used Transformer-based decoder to generate answers or captions.","As compared to the aforementioned M4C models (shown in Figure REF a) that throw each instance of every modality into transformer layers, our model (in Figure REF d) first uses three attention blocks to filter out irrelevant or redundant features and aggregate them into six individually-functioned vectors.","In contrast to hundreds of feature vectors in M4C, the six vectors consume much less computation.","Moreover, to group text features into visual- and linguistic- parts is more reasonable.","When comparing with the graph-based multi-modal encoders such as MM-GNN (Figure REF b) and SMA (Figure REF c), our baseline is extremely simple in design and reduces much space and time complexity.","Figure: A simple baseline model for TextVQA.","Given an image and a question, we prepare three features (OCR visual-part, OCR linguistic-part and object features) and three question self-attention outputs.","The six sequences are put into attention block and fused into six vectors, upon which we calculate element-wise product two by two to get concatenated embeddings.","The encoder outputs predict the first word and the rest of answer is produced by an iterative decoder.In addition, for the first time we ask the question: to what extent OCRs contribute to the final performance of TextVQA, in contrast to the other modality - visual contents such as objects and scenes?","An interesting phenomenon is observed that OCRs play an almost key role in this special problem while visual contents only serve as assisting factors.","A strong model without the use of visual contents surpasses current state-of-the-art model, demonstrating the power of proposed pairwise fusion mechanism and primary role of texts.","To demonstrate the effectiveness of our proposed simple baseline model, we test it on both TextVQA [34] and TextCaps [33] tasks.","For the TextVQA, we outperforms the state-of-the-art (SOTA) on TextVQA dataset and all three tasks of ST-VQA, and rank the first on both leaderboardshttps://eval.ai/web/challenges/challenge-page/551/leaderboard/1575https://rrc.cvc.uab.es/?ch=11&com=evaluation&task=1.","More importantly, all compared SOTA models use the similar Transformer decoder with ours, but with far more complex encoding mechanisms.","For TextCaps, we surpass the TextCaps Challenge 2020 winner and now rank the first place on the leaderboardhttps://eval.ai/web/challenges/challenge-page/573/leaderboard/1617.","Overall, the major contribution of this work is to provide a simple but rather strong baseline for the text-based vision-and-language research.","This could be the new baseline (backbone) model for both TextVQA and TextCaps.","More importantly, we wish this work to inspire a new thinking of multi-modality encoder design – simple is not easy."],["Related Work","Text based visual question answering.","VQA [5], [18], [19], [39] has seen rapid development in recent years.","A new task – TextVQA goes one step further and aims at the understanding and reasoning of scene texts in images.","A model needs to read texts first and then answer related questions in natural everyday situations.","Two datasets, TextVQA [34] and ST-VQA [7] are introduced concurrently to benchmark progress in this field.","To solve this problem, various methods have also been proposed.","LoRRA, the baseline model in TextVQA, uses bottom-up and top-down [3] attention on visual objects and texts to select an answer from either vocabulary or fixed-index OCR.","M4C [17] is equipped with a vanilla transformer decoder to iteratively generate answers and a flexible pointer network to point back at most probable OCR token at one decoding step.","MM-GNN [14] designs a representation of three graphs and introduces three aggregators to update message passing for question answering.","Text based image captioning.","Image captioning challenges a model to automatically generate a natural language description based on the contents in an image.","Existing datasets, e.g., COCO Captions [8] and Flickr30k [43], focus more on visual objects.","To enhance text comprehension in the context of an image, a new dataset called TextCaps [33] is proposed.","It requires a model to read and reason about texts and generate coherent descriptions.","The baseline model in TextCaps is modified from aforementioned M4C slightly, by removing question input directly.","Generative transformer decoder.","To address the problem that answers in these two text-based tasks are usually concatenated by more than one word, we use the structure of transformer [11] decoder in answer module.","Following previous work, we also use the generative transformer decoder for fair comparison."],["Proposed Method","Given the three modalities (questions, OCR tokens, visual objects), the first step of our model is to prepare the features by projecting them into the same dimension.","Then we describe the formulation of Attention Block for feature summarizing.","Stacking the blocks together yields encoder for downstream tasks.","Using encoder output to produce the first word in answer, we then use an iterative decoder to predict the rest of words.","This whole process is shown in Figure REF .","When transferring to TextCaps, we only make minimal modifications which will be detailed in Section REF .","Notation In the remainder of this paper, all $\\mathbf {W}$ are learned linear transformations, with different symbols to denote independent parameters, e.g., $\\mathbf {W}_{fr}$ .","$\\mathrm {LN}$ is Layer Normalization [6].","$\\circ $ represents element-wise product."],["Feature Preparation","Question features.","For a question with $L$ words, we first use a three-layer BERT [11] model to embed it into $Q = \\lbrace q_{i}\\rbrace _{i=1}^{L}$ .","This BERT model is finetuned during training.","OCR features.","In text-based VQA and image captioning problem, texts are of key importance.","Simply gather every feature of text together is not efficient enough.","When faced with a bunch of OCRs, human recognition system tends to use two complementary methods to select subsequent words, either by finding similarly-looking and spatially-close words or choosing words coherent in linguistic meaning.","For this intuitive purpose, we split features of $N$ OCR tokens into two parts: Visual and Linguistic.","1) OCR visual-part.","Visual features are combined by appearance feature and spatial feature as they show what eyes catch of, without further processing of natural language system.","From this part, a model can get visual information such as word font, color and background.","These two are extracted by an off-the-shelf Faster R-CNN [30] detector.","${\\begin{split}& \\mathbf {x}_{i}^{ocr,v} = \\mathrm {LN}( \\mathbf {W}_{fr}\\mathbf {x}_{i}^{ocr,fr})+ \\mathrm {LN}(\\mathbf {W}_{bx}\\mathbf {x}_{i}^{ocr,bx}),\\end{split}}$ where $\\mathbf {x}_{i}^{ocr,fr}$ is the appearance feature extracted from the fc6 layer of Faster R-CNN detector.","The fc7 weights are finetuned on our task.","$\\mathbf {x}_{i}^{ocr,bx}$ is the bounding box feature in the format of $[x_{tl},y_{tl},x_{br},y_{br}]$ , where $tl$ and $br$ denotes top left and bottom right coordinates respectively.","2) OCR linguistic-part.","Linguistic features are made up of 1) FastText feature $\\mathbf {x}_{i}^{ocr,ft}$ , which is extracted from a pretrained word embedding and 2) character-level Pyramidal Histogram of Characters (PHOC) [1] feature $x_{i}^{ocr,ph}$ as they contain natural language related information.","${\\begin{split}& \\mathbf {x}_{i}^{ocr,l} = \\mathrm {LN}( \\mathbf {W}_{ft}\\mathbf {x}_{i}^{ocr,ft} + \\mathbf {W}_{ph}\\mathbf {x}_{i}^{ocr,ph})\\end{split}}$ 3) OCR additional features.","In the SBD-Trans [25], [40] that we use to recognize OCR tokens, the holistic representations in a specific text region are themselves visual features, however, are employed for linguistic word classification purpose.","Therefore, they cover both visual and linguistic context of the OCR token and we thus introduce the Recog-CNN feature $\\mathbf {x}_{i}^{ocr,rg}$ from this network to enrich text features.","Finally, Recog-CNN features are added to OCR visual- and linguistic-part simultaneously.","${\\begin{split}& \\mathbf {x}_{i}^{ocr,v} = \\mathrm {LN}( \\mathbf {W}_{fr}\\mathbf {x}_{i}^{ocr,fr} + \\mathbf {W}_{rg}\\mathbf {x}_{i}^{ocr,rg})+ \\mathrm {LN}(\\mathbf {W}_{bx}\\mathbf {x}_{i}^{ocr,bx})\\\\& \\mathbf {x}_{i}^{ocr,s} = \\mathrm {LN}( \\mathbf {W}_{ft}\\mathbf {x}_{i}^{ocr,ft} + \\mathbf {W}_{ph}\\mathbf {x}_{i}^{ocr,ph} + \\mathbf {W}_{rg}\\mathbf {x}_{i}^{ocr,rg})\\end{split}}$ Visual features.","In text-based tasks, visual contents in an image can be utilized to assist textual information in the reasoning process.","To prove that our simple attention block has the power of using visual features in various forms, we adopt either grid-based global features or region-based object features.","1) Global features.","We obtain image global features $x_{i}^{glob}$ from a ResNet-152 [15] model pretrained on ImageNet, by average pooling $2048D$ features from the res-5c block, yielding a $14 \\times 14 \\times 2048$ feature for one image.","To be consistent with other features, we resize the feature into $196 \\times 2048$ , a total of 196 uniformly-cut grids.","${\\begin{split}& \\mathbf {x}_{i}^{glob} = \\mathrm {LN}( \\mathbf {W}_{g}\\mathbf {x}_{i}^{glob})\\end{split}}$ 2) Object features.","The region-based object features are extracted from the same Faster F-CNN model as mentioned in OCR features part.","${\\begin{split}& \\mathbf {x}_{i}^{obj} = \\mathrm {LN}( \\mathbf {W}_{fr}^{^{\\prime }}\\mathbf {x}_{i}^{obj,fr})+ \\mathrm {LN}(\\mathbf {W}_{bx}^{^{\\prime }}\\mathbf {x}_{i}^{obj,bx}),\\end{split}}$ where $\\mathbf {x}_{i}^{obj,fr}$ is the appearance feature and $\\mathbf {x}_{i}^{obj,bx}$ is the bounding box feature."],["Attention Block as Feature Summarizing","In tasks that cross the fields of computer vision and natural language processing, modality fusion is of superior importance.","Treating them as homogeneous entities in a joint embedding space might be easy to implement, however, is not carefully tailored to a specific problem.","Moreover, the many parameters of all entities in the large model (for example, a Transformer) consume much computation.","To grasp interaction between modalities for maximum benefit and filter out irrelevant or redundant features before the entering into a large fusion layer, we use a simple attention block to input two sequences of entities and output two processed vectors, which is shown as the Attention Block in Figure REF .","The two sequences of entities might be any sequence we want.","For TextVQA problem, question changes in real-time and plays a dominant role in final answering.","The design of question needs careful consideration and its existence should contribute throughout the process.","For example, here we use question as one input of query in attention block.","The sequence of question words goes through a self-attention process before forwarding into attention block.","First we put the question word sequence $Q = \\lbrace q_{i}\\rbrace _{i=1}^{L}$ through a fully connected feed-forward network, which consists of two linear transformations (or two convolutions with 1 as kernel size) and one ReLU activation function between them.","$\\begin{split}&q_{i}^{fc} = \\mathrm {conv}\\lbrace \\mathrm {ReLU}[\\mathrm {conv}(q_{i})]\\rbrace , \\,\\, i = 1, \\dots , L ;\\end{split}$ A softmax layer is the used to compute the attention on each word in the question.","$\\begin{split}&a_i = \\mathrm {Softmax}({q_{i}^{fc}}), \\,\\, i = 1, \\dots , L ;\\\\\\end{split}$ This is known as self-attention and these weights are multiplied with original question embedding to get weighted sum of word embeddings.","$\\begin{split}&Q^s = \\textstyle {\\sum _{i=1}^{L}} a_i {q}_{i}.\\end{split}$ If we have several individual entities to combine with question, the corresponding number of parallel self-attention processes are performed on the same question with independent parameters.","For example, we can get $Q^s_{v}$ , $Q^s_{l}$ and $Q^s_{o}$ for OCR visual-part, OCR linguistic-part and object regions respectively.","Then the $Q^s$ are used as query for corresponding features.","We calculate the attention weights under the guidance of $Q^s$ , which are then put into a softmax layer.","Finally the weights are multiplied with original queried features to get a filtered vector.","Here we take the pair of $Q^s_{v}$ and ${\\mathbf {x}}_{i}^{ocr,v}$ as an example: ${\\begin{split}&p_i = \\mathbf {W}[\\mathrm {ReLU}(\\mathbf {W}_{s} Q^s_{v})\\circ \\mathrm {ReLU}(\\mathbf {W}_{x}{x}_i^{ocr,v})],\\\\&s_i = \\mathrm {Softmax}(p_i), \\quad i=1,\\dots ,N,\\\\&g^{ocr,v} = \\textstyle {\\sum _{i=1}^{N}} s_i {\\mathbf {x}}_{i}^{ocr,v}\\end{split}}$ where $g^{ocr,v}$ is the output of attention block.","Similarly, we can get $g^{ocr,l}$ for the OCR-linguistic summarizing feature, $g^{obj}$ for the object summarizing feature.","Different from M4C sending every single question tokens, OCR tokens and objects into the transformer feature fusion layer, here we only have 6 feature vectors ($Q^s_{v}$ , $Q^s_{l}$ , $Q^s_{o}$ , $g^{ocr,v}$ , $g^{ocr,l}$ and $g^{obj}$ ) which are sent to the following process.","This largely decreases the computation complexity and burden, considering that the transformer is a parameter-heavy network."],["Stacked-Block Encoder","The attention block in Section REF can be stacked together as an encoder which produces combined embedding for downstream tasks.","TextVQA baseline model.","As presented in the above module, questions are sent through self-attention to output $Q^s_{v}$ , $Q^s_{l}$ and $Q^s_{o}$ .","OCR features in images are splitted into visual- and linguistic part, which are $x^{ocr,v}$ and $x^{ocr,l}$ .","We also have object features $x^{obj}$ .","The six sequences are put into three attention blocks and we get six $768D$ vectors which are then forwarded into a fusion encoder.","The fusion encoder, OCR encoder and generative decoder in Figure REF are in the same transformer model but undertaking different roles.","After fusion encoder processing, the six vectors conduct element-wise multiplication in a pairwise way to get corresponding embeddings which are concatenated together.","Then we use a fully-connected layer to transform the concatenated embeddings to a context embedding with appropriate dimension, upon which we generate the first answer output.","Given the first answer word, a generative decoder is then used to select the rest of answer, which will be detailed in Section REF and Supplementary Material.","Figure: TextCaps baseline model.","It has a same structure as TextVQA baseline model.Table: We ablate our model on TextVQA dataset by testing number of attention blocks, forms of visual object features and addition of OCR representations.Table: Comparison to previous work on TextVQA dataset.","Our model sets new state-of-the-art with an extremely simple design.TextCaps baseline model.","As there are no questions in TextCaps, we use objects to guide OCR visual- and linguistic-part and use OCRs to guide object features.","Technically we simply replace question word sequence with OCR token sequence or object proposal feature sequence.","The other settings are the same with TextVQA.","Figure REF illustrates our Textcaps baseline model.","To easily transfer to another task demonstrates the generalization ability and simplicity of our method."],["Answer Generation Module","To answer a question or generate a caption, we use a generative decoder based on transformer.","It takes as input the `context embedding' from the above encoder and select the first word of the answer.","Based on the first output word, we then use the decoder to find the next word token either from a pre-built vocabulary or the candidate OCR tokens extracted from the given image, based on a scoring function.","Training Loss.","Considering that the answer may come from two sources, we use multi-label binary cross-entropy (bce) loss: ${\\begin{split}pred &= \\frac{1}{1+\\exp {(-y_{pred})}}, \\\\\\mathcal {L}_{bce} &= -y_{gt} \\mathrm {log}(pred)-(1-y_{gt}) \\mathrm {log}(1-pred),\\end{split}}$ where $y_{pred}$ is prediction and $y_{gt}$ is ground-truth target.","Additional Training Loss.","In some cases, the model reasons correctly, however, picks slightly different words than what we expected due to defective reading (OCR) ability.","To take advantage of these predictions, we introduce a new policy gradient loss as an auxilliary task inspired by reinforcement learning.","In this task, the greater reward, the better.","We take Average Normalized Levenshtein Similarity (ANLS) metric$\\mathrm {ANLS}(s_{1},s_{2})=1-d(s_{1},s_{2})/\\mathrm {max}(\\mathrm {len}(s_{1}),\\mathrm {len}(s_{2}))$ , d(,) is edit distance.","as the reward which measures the character similarity between predicted answer and ground-truth answer.","$\\begin{split}&r = \\mathrm {ANLS}(\\phi (y_{gt}),\\phi (y_{pred})), \\\\&y = \\mathbb {I}(\\mathrm {softmax}(y_{pred})), \\\\&\\mathcal {L}_{pg} = (0.5 - r) (y_{gt} \\mathrm {log}(y) + (1-y_{gt}) \\mathrm {log}(1-y)),\\\\&\\mathcal {L} = \\mathcal {L}_{bce}+\\alpha \\cdot \\mathcal {L}_{pg},\\end{split}$ where $\\phi $ is a mapping function that returns a sentence given predicting score (e.g., , $y_{pred}$ ), ANLS($\\cdot $ ) is used to calculate similarity between two phrases, $\\mathbb {I}$ is an indicator function to choose the maximum probability element.","The additional training loss is a weighted sum of $\\mathcal {L}_{bce}$ and $\\mathcal {L}_{pg}$ , where $\\alpha $ is a hyper-parameter to control the trade-off of $\\mathcal {L}_{pg}$ .","After introducing policy gradient loss, our model is able to learn fine-grained character composition alongside linguistic information.","We only apply this additional loss on ST-VQA dataset, which brings roughly $1\\%$ improvement."],["Experiments","Extensive experiments are conducted across two categories of tasks: TextVQA and TextCap.","For TextVQA we set new state-of-the-art on TextVQA dataset and all three tasks of ST-VQA.","For TextCaps we surpass 2020 TextCaps Challenge winner.","See more experiments details below."],["Implementation Details","The set of methods are built on top of PyTorch.","We use Adam as the optimizer.","The learning rate for TextVQA and TexCaps is set to $1e-4$ .","For TextVQA we multiply the learning rate with a factor of $0.1$ at the $14,000$ and $15,000$ iterations in a total of $24,000$ iterations.","For TextCaps the multiplication is done at the $3,000$ and $4,000$ iterations, with $12,000$ total iterations.","We set the maximum length of questions to $L = 20$ .","We recognize at most $N = 50$ OCR tokens and detect at most $M = 100$ objects.","The maximum number of decoding steps is set to 12.","Transformer layer in our model uses 12 attention heads.","The other hyper-parameters are the same with BERT-BASE [11].","We use the same model on TextVQA and three tasks of ST-VQA, only with different answer vocabulary, both with a fixed size of 5000."],["Ablation Study on TextVQA Dataset","TextVQA [34] is a popular benchmark dataset to test scene text understanding and reasoning ability, which contains $45,336$ questions on $28,408$ images.","Diverse questions involving inquires about time, names, brands, authors, etc., and dynamic OCR tokens that might be rotated, casual or partially occluded make it a challenging task.","We first conduct an experiment of building only one block, with one question self-attention output guiding the whole set of text features as a comparison.","This is a one-block model in TableREF which does not perform as good as the state-of-the-art.","To investigate how well text features can perform without the usage of visual grid-based or region-based features, we build a two-block model.","Given the two categories of OCR features – visual and linguistic, we find that our simple model is already able to perform promisingly on TextVQA problem.","Line 1 and Line 2 tell clearly the validity of sorting text features into two groups ($1.77\\%$ difference).","When building our third block on the basis of visual contents, either global features or object-level features are at our disposal.","The incorporation of a third block has modest improvements ($0.24\\%$ for global and $0.63\\%$ for object features).","From Line 4 to Line 5, a new Recog-CNN feature is added to enrich text representation and brings $0.37\\%$ improvement.","We also use more transformer layers (from 4 to 8) and get $0.1\\%$ higher result.","Then we use a much better OCR system (especially on recognition part) and obtain large performance boost (from $40.38\\%$ to $43.95\\%$ ).","Table: Comparison to previous work on ST-VQA dataset.","With TextVQA pretraining, our model outperforms current approaches by a large margin.Figure: Qualitative examples of our baseline model in contrast to M4C.","While our model can read texts in accordance with written language system, M4C can only pick tokens in a random and errorneous way.Qualitative examples.","We present four examples in Figure REF where our model shows the ability to really read scene texts in a way similar to humans, i.e., left-to-right then top-to-bottom.","In contrast, current state-of-the-art model M4C fails to read tokens in a correct order."],["Comparison with state-of-the-art","TextVQA dataset.","Even stripped of the usage of visual contents in the image, our two-block model already surpasses current state-of-the-art M4C by $0.98\\%$ on test set (Line 2 in Table REF VS. Line 4 in Table REF ).","Using the same OCR system, our baseline model further improves upon M4C by $0.98\\%$ on val and $1.91\\%$ on test (Line 7 VS. Line 4 in Table REF ).","Compared to the top entries in TextVQA Challenge 2020, our baseline has a significantly simpler model design, especially on the encoder side.","M4C and SA-M4C take all parameters of entities into transformer layers and join large amout of computation.","SMA uses a heterogeneous graph to explicitly consider different nodes and compute attention weights on 5-neighbored graph.","Our model that surpasses all of them only sends six holistic vectors two-by-two into transformer layers, which tremendously saves computation.","ST-VQA dataset.","The ST-VQA dataset [7] is another popular dataset with three tasks, which gradually increase in difficulty.","Task 1 provides a dynamic candidate dictionary of 100 words per image, while Task 2 provides a fixed answer dictionary of $30,000$ words for the whole dataset.","As for Task 3, however, the model are supposed to generate answer without extra information.","We also evaluate our model on ST-VQA dataset, using the same model from TextVQA for all three tasks.","Without any additional training data, our model achieved the highest on Task 2 and Task 3 (Line 4 in Table REF ).","Using TextVQA dataset as additional training data, our model sets new state-of-the-art on all three tasks and outperforms current approaches by a large margin.","Table: Results on TextCaps dataset.","(B: BLEU-4; M: METEOR; R: ROUGE_L; S: SPICE; C: CIDEr)"],["TextCaps Dataset","TextCaps is a new dataset that requires a model to read texts in images and generate descriptions based on scene text understanding and reasoning.","In TextCaps, automatic captioning metrics (BLEU [28], METEOR [10], ROUGE_L [23], SPICE [2] and CIDEr [36]) are compared with human evaluation scores.","All automatic metrics show high correlation with human scores, among which CIDEr and METEOR have the highest.","M4C-Captioner is the method provided in TextCaps, which is modified from M4C model by simply removing question input.","Similarly, simply replacing question in our TextVQA baseline model with object or OCR sequence yields our TextCaps baseline model.","Using exactly the same OCR system, OCR representations, our baseline with Rosetta-en OCR (Line 2 on Table REF ) already surpass M4C-Captioner (Line 1 on Table REF ), especially on BLUE-4 and CIDEr metric.","By upgrading our OCR system to SBD-Trans and using 6-layer transformer in our encoder-decoder structure, our baseline further exceeds TextCaps Challenge Winner on BLUE-4 and CIDEr metric as shown in Line 5 and Line 6 of Table REF .","[h] Answer generation in TextVQA Question features $Q$ , OCR features $x^{ocr}$ , Object features $x^{obj}$ An answer with a length of $T$ words $A=\\lbrace a_{i}\\rbrace _{i=0}^{T-1}$ Obtain three independent question signals: $(Q^s_{v}, Q^s_{l}, Q^s_{o}) = \\mathrm {Self\\_Attention}^{\\lbrace v,l,o\\rbrace }(Q)$ Split OCR features into OCR visual- and linguistic-part ($x^{ocr,v}$ , $x^{ocr,l}$ ) Forward question signals and three features into three Attention Blocks in a pairwise way: $g^{ocr,v} = \\mathrm {Att\\_Blk}(Q^s_{v}, x^{ocr,v});\\ g^{ocr,l} = \\mathrm {Att\\_Blk}(Q^s_{l}, x^{ocr,l});\\ g^{obj} = \\mathrm {Att\\_Blk}(Q^s_{o}, x^{obj})$ Forward six vectors into fusion encoder: $\\bar{Q}^s_{v}, \\bar{Q}^s_{l}, \\bar{Q}^s_{o}, \\bar{g}^{ocr,v}, \\bar{g}^{ocr,l}, \\bar{g}^{obj} = \\mathrm {Fusion\\_Encoder}(Q^s_{v}, Q^s_{l}, Q^s_{o}, g^{ocr,v}, g^{ocr,l}, g^{obj})$ Element-wise multiply the six vectors two by two: $\\mathbf {e}^{ocr,v}=\\bar{Q}^s_{v}\\circ \\bar{g}^{ocr,v};\\ \\mathbf {e}^{ocr,l}=\\bar{Q}^s_{l}\\circ \\bar{g}^{ocr,l};\\ \\mathbf {e}^{obj}=\\bar{Q}^s_{o}\\circ \\bar{g}^{obj}$ Concatenate the three results together and transform it into context embedding $\\mathrm {context\\, embedding}=\\mathrm {Linear}([\\mathbf {e}^{ocr,v}; \\mathbf {e}^{ocr,l}; \\mathbf {e}^{obj}])$ $t = 0$ Based on context embedding, calculate the first word $a_{t}$ in the answer: 1) Map context embedding to a $5000D$ vector through a linear layer as scores for vocabulary 2) Get $50D$ dot product between context embedding and all 50 OCR encoder outputs as scores for OCR tokens 3) Concatenate $5000D$ vector and $50D$ vector together, from which select the highest-score index as $a_{t}$ $t=1;t \\le T-1;t++$ the previous output $a_{t-1}$ is from vocabulary Forward linear layer weights corresponding to the previous output $a_{t-1}$ into generative decoder and obtain decoder embedding the previous output $a_{t-1}$ is from OCR tokens Forward OCR features of the previous output $a_{t-1}$ into generative decoder and obtain decoder embedding Based on decoder embedding, calculated the current word $a_{t}$ in the answer: 1) Map decoder embedding to a $5000D$ vector through a linear layer as scores for vocabulary 2) Get $50D$ dot product between decoder embedding and all 50 OCR encoder outputs as scores for OCR tokens 3) Concatenate $5000D$ vector and $50D$ vector together, from which select the highest-score index as $a_{t}$ return $A=\\lbrace a_{i}\\rbrace _{i=0}^{T-1}$ Table: Computational complexity of two model encoders.","L=20L=20 is the length of question; N=50N=50 is the number of OCR tokens; M=100M=100 is the number of detected objects.","Here we omit all vector dimensions DD for simplicity.","In transformer encoder of our model, apart from the six vectors, we also input 50 OCR tokens to compute dot product value in answer choosing."],["Conclusion","In this paper, we provide a simple but rather strong baseline for the text-based vision-and-language research.","Instead of handling all modalities over a joint embedding space or via complicated graph structural encoding, we use the vanilla attention mechanism to fuse pairwise modalities.","We further split text features into two functionally different parts, i.e., linguistic- and visual-part which flow into corresponding attention branch.","We evaluate our simple baseline model on TextVQA, ST-VQA and TextCaps, all leading to the best performance on the public leaderboards.","This sets the new state-of-the-art and our model could be the new backbone model for both TextVQA and TextCaps.","What's more, we believe this work inspires a new thinking of the multi-modality encoder design."],["Algorithm","Algorithm REF provides the pseudo-code for the process of answer generation in the TextVQA problem.","The description generation process in the TextCaps problem is roughly the same, except that the decoding step $T$ is set to 30 rather than 12 for a longer and more expressive image description.","Besides, the question features are also replaced by OCR features and Object features to guide each other."],["Comparison of Computation Complexity between Encoders","As our model and M4C have the same generative decoder, we only compare encoders in Table REF .","In one Attention Block, main operations are element-wise multiplication and final matrix multiplication, which yield $O(n \\cdot d)=2n \\cdot d$ operations, where n is number of queried features.","In Transformer encoder, the complexity is $O(n^2 \\cdot d)$ , where $n$ is the sequence length and $d$ is the representation dimension."],["Additional Experiment Details","OCRs and Objects self-attention in TextCaps.","Different from question self-attention in TextVQA problem that uses a BERT model to capture the sequence information, for OCRs and Objects self-attention in TextCaps we use an LSTM [16] layer to capture the recurrent features of isolate regions before the self-attention calculation.","SBD-Trans training data.","The SBD model is pretrained on a $60k$ dataset, which consists of $30,000$ images from LSVT [35] training set, $10,000$ images from MLT 2019 [27] training set, $5,603$ images from ArT [9] , and $14,859$ images selected from a bunch of datasets – RCTW-17 [32], ICDAR 2013 [22], ICDAR 2015 [21], MSRA-TD500 [41], COCOText [37], and USTB-SV1K [42].","The model was finally finetuned on MLT 2019 training set.","The robust transformer based network is trained on the following datasets: IIIT 5K-Words [26], Street View Text [38], ICDAR 2013, ICDAR 2015, Street View Text Perspective [29], CUTE80 [31] and ArT."],["Additional Qualitative Examples","To demonstrate that our model not only learns fixed superficial correlation between questions and prominent OCR tokens, we present several images in the dataset with two different questions to which our model both answers correctly, while other models such as M4C fails, in Figure REF .","TextCaps examples are in Figure REF .","Figure: Qualitative examples of our model on TextCaps.Figure: Examples that our model fails at on TextVQA."],["Failure Cases on TextVQA","The questions that our model struggles at can be categorized into several groups.","Firstly, due to the inappropriate tricky design of the dataset, models are required to read the time, find a specific letter in a word, or read a ruler, e.g., Figure REF a), b) and c).","Secondly, even though our model answers reasonably, it does not lie in the region of ground-truth answers as shown in Figure REF d).","Thirdly, the mistake might occur because of defective reading ability that OCR tokens do not include answer candidates at all, as shown in Figure REF e).","Fourthly, it is difficult to discern the color of tokens as shown in Figure REF f).","Finally, when it comes to nuanced relationship understanding, our model is not yet complex enough to reason as shown in Figure REF g) and h)."]],"string":"[\n [\n \"Simple is not Easy: A Simple Strong Baseline for TextVQA and TextCaps\"\n ],\n [\n \"Abstract Texts appearing in daily scenes that can be recognized by OCR (Optical Character Recognition) tools contain significant information, such as street name, product brand and prices.\",\n \"Two tasks -- text-based visual question answering and text-based image captioning, with a text extension from existing vision-language applications, are catching on rapidly.\",\n \"To address these problems, many sophisticated multi-modality encoding frameworks (such as heterogeneous graph structure) are being used.\",\n \"In this paper, we argue that a simple attention mechanism can do the same or even better job without any bells and whistles.\",\n \"Under this mechanism, we simply split OCR token features into separate visual- and linguistic-attention branches, and send them to a popular Transformer decoder to generate answers or captions.\",\n \"Surprisingly, we find this simple baseline model is rather strong -- it consistently outperforms state-of-the-art (SOTA) models on two popular benchmarks, TextVQA and all three tasks of ST-VQA, although these SOTA models use far more complex encoding mechanisms.\",\n \"Transferring it to text-based image captioning, we also surpass the TextCaps Challenge 2020 winner.\",\n \"We wish this work to set the new baseline for this two OCR text related applications and to inspire new thinking of multi-modality encoder design.\",\n \"Code is available at https://github.com/ZephyrZhuQi/ssbaseline\"\n ],\n [\n \"Introduction\",\n \"To automatically answer a question or generate a description for images that require scene text understanding and reasoning has broad prospects for commercial applications, such as assisted driving and online shopping.\",\n \"Equipped with these abilities, a model can help drivers decide distance to the next street or help customers get more details about a product.\",\n \"Two kinds of tasks that focus on text in images have recently been introduced, which are text-based visual question answering (TextVQA) [34], [7] and text-based image captioning (TextCaps) [33].\",\n \"For example, in Figure REF , a model is required to answer a question or generate a description by reading and reasoning the texts “tellus mater inc.” in the image.\",\n \"These two tasks pose a challenge to current VQA or image captioning models as they explicitly require understanding of a new modality – Optical Character Recognition (OCR).\",\n \"A model must efficiently utilize text-related features to solve these problems.\",\n \"Figure: An example of TextVQA and TextCaps tasks.\",\n \"The answer and description are generated by our model.\",\n \"Our simple baseline is able to read texts and answer related questions.\",\n \"Besides, it can also observe the image and generate a description with texts embedded in it.Figure: Encoders of different models.\",\n \"(a) Current state-of-the-art model M4C on TextVQA task forwards each feature vector of all modalities indiscriminately into transformer layers, which exhaust tremendous computation.\",\n \"(b) MM-GNN handcrafts three graphs to represent the image and applies three aggregators step by step to pass messages between graphs.\",\n \"(c) SMA introduces a heterogeneous graph and considers object-object, object-text and text-text relationships, upon which a graph attention network is then used to reason over them.\",\n \"(d) Our baseline uses three vanilla attention blocks to highlight most relevant features and combines them into six individually-functioned vectors, which is then sent into transformer-based fusion encoders.\",\n \"The considerably fewer parameters of six vectors save computation.For TextVQA task, the current state-of-the-art model M4C [17] handles all modalities (questions, visual objects and OCR tokens) over a joint embedding space.\",\n \"Although this homogeneous method is easy to implement, fast to train and has made great headway, it considers that texts and visual objects contribute indiscriminately to this problem and uses text features as a whole.\",\n \"For TextCaps problem, the only difference is that it only has two modalities: visual objects and OCR tokens.\",\n \"However, these limitations remain.\",\n \"Some other works proposed even more complex structures to encode and fuse multi-modality features of this task, i.e., questions, OCR tokens and images.\",\n \"For example, SMA [13] uses a heterogeneous graph to encode object-object, object-text and text-text relationships in the image, and then designs a graph attention network to reason over it.\",\n \"MM-GNN [14] represents an image as three graphs and introduces three aggregators to guide message passing from one graph to another.\",\n \"In this paper, we use the vanilla attention mechanism to fuse pairwise modalities.\",\n \"Under this mechanism, we further use a more reasonable method to utilize text features which leads to a higher performance, that is by splitting text features into two functionally different parts, i.e., linguistic- and visual-part which flows into corresponding attention branch.\",\n \"The encoded features are then sent to a popularly-used Transformer-based decoder to generate answers or captions.\",\n \"As compared to the aforementioned M4C models (shown in Figure REF a) that throw each instance of every modality into transformer layers, our model (in Figure REF d) first uses three attention blocks to filter out irrelevant or redundant features and aggregate them into six individually-functioned vectors.\",\n \"In contrast to hundreds of feature vectors in M4C, the six vectors consume much less computation.\",\n \"Moreover, to group text features into visual- and linguistic- parts is more reasonable.\",\n \"When comparing with the graph-based multi-modal encoders such as MM-GNN (Figure REF b) and SMA (Figure REF c), our baseline is extremely simple in design and reduces much space and time complexity.\",\n \"Figure: A simple baseline model for TextVQA.\",\n \"Given an image and a question, we prepare three features (OCR visual-part, OCR linguistic-part and object features) and three question self-attention outputs.\",\n \"The six sequences are put into attention block and fused into six vectors, upon which we calculate element-wise product two by two to get concatenated embeddings.\",\n \"The encoder outputs predict the first word and the rest of answer is produced by an iterative decoder.In addition, for the first time we ask the question: to what extent OCRs contribute to the final performance of TextVQA, in contrast to the other modality - visual contents such as objects and scenes?\",\n \"An interesting phenomenon is observed that OCRs play an almost key role in this special problem while visual contents only serve as assisting factors.\",\n \"A strong model without the use of visual contents surpasses current state-of-the-art model, demonstrating the power of proposed pairwise fusion mechanism and primary role of texts.\",\n \"To demonstrate the effectiveness of our proposed simple baseline model, we test it on both TextVQA [34] and TextCaps [33] tasks.\",\n \"For the TextVQA, we outperforms the state-of-the-art (SOTA) on TextVQA dataset and all three tasks of ST-VQA, and rank the first on both leaderboardshttps://eval.ai/web/challenges/challenge-page/551/leaderboard/1575https://rrc.cvc.uab.es/?ch=11&com=evaluation&task=1.\",\n \"More importantly, all compared SOTA models use the similar Transformer decoder with ours, but with far more complex encoding mechanisms.\",\n \"For TextCaps, we surpass the TextCaps Challenge 2020 winner and now rank the first place on the leaderboardhttps://eval.ai/web/challenges/challenge-page/573/leaderboard/1617.\",\n \"Overall, the major contribution of this work is to provide a simple but rather strong baseline for the text-based vision-and-language research.\",\n \"This could be the new baseline (backbone) model for both TextVQA and TextCaps.\",\n \"More importantly, we wish this work to inspire a new thinking of multi-modality encoder design – simple is not easy.\"\n ],\n [\n \"Related Work\",\n \"Text based visual question answering.\",\n \"VQA [5], [18], [19], [39] has seen rapid development in recent years.\",\n \"A new task – TextVQA goes one step further and aims at the understanding and reasoning of scene texts in images.\",\n \"A model needs to read texts first and then answer related questions in natural everyday situations.\",\n \"Two datasets, TextVQA [34] and ST-VQA [7] are introduced concurrently to benchmark progress in this field.\",\n \"To solve this problem, various methods have also been proposed.\",\n \"LoRRA, the baseline model in TextVQA, uses bottom-up and top-down [3] attention on visual objects and texts to select an answer from either vocabulary or fixed-index OCR.\",\n \"M4C [17] is equipped with a vanilla transformer decoder to iteratively generate answers and a flexible pointer network to point back at most probable OCR token at one decoding step.\",\n \"MM-GNN [14] designs a representation of three graphs and introduces three aggregators to update message passing for question answering.\",\n \"Text based image captioning.\",\n \"Image captioning challenges a model to automatically generate a natural language description based on the contents in an image.\",\n \"Existing datasets, e.g., COCO Captions [8] and Flickr30k [43], focus more on visual objects.\",\n \"To enhance text comprehension in the context of an image, a new dataset called TextCaps [33] is proposed.\",\n \"It requires a model to read and reason about texts and generate coherent descriptions.\",\n \"The baseline model in TextCaps is modified from aforementioned M4C slightly, by removing question input directly.\",\n \"Generative transformer decoder.\",\n \"To address the problem that answers in these two text-based tasks are usually concatenated by more than one word, we use the structure of transformer [11] decoder in answer module.\",\n \"Following previous work, we also use the generative transformer decoder for fair comparison.\"\n ],\n [\n \"Proposed Method\",\n \"Given the three modalities (questions, OCR tokens, visual objects), the first step of our model is to prepare the features by projecting them into the same dimension.\",\n \"Then we describe the formulation of Attention Block for feature summarizing.\",\n \"Stacking the blocks together yields encoder for downstream tasks.\",\n \"Using encoder output to produce the first word in answer, we then use an iterative decoder to predict the rest of words.\",\n \"This whole process is shown in Figure REF .\",\n \"When transferring to TextCaps, we only make minimal modifications which will be detailed in Section REF .\",\n \"Notation In the remainder of this paper, all $\\\\mathbf {W}$ are learned linear transformations, with different symbols to denote independent parameters, e.g., $\\\\mathbf {W}_{fr}$ .\",\n \"$\\\\mathrm {LN}$ is Layer Normalization [6].\",\n \"$\\\\circ $ represents element-wise product.\"\n ],\n [\n \"Feature Preparation\",\n \"Question features.\",\n \"For a question with $L$ words, we first use a three-layer BERT [11] model to embed it into $Q = \\\\lbrace q_{i}\\\\rbrace _{i=1}^{L}$ .\",\n \"This BERT model is finetuned during training.\",\n \"OCR features.\",\n \"In text-based VQA and image captioning problem, texts are of key importance.\",\n \"Simply gather every feature of text together is not efficient enough.\",\n \"When faced with a bunch of OCRs, human recognition system tends to use two complementary methods to select subsequent words, either by finding similarly-looking and spatially-close words or choosing words coherent in linguistic meaning.\",\n \"For this intuitive purpose, we split features of $N$ OCR tokens into two parts: Visual and Linguistic.\",\n \"1) OCR visual-part.\",\n \"Visual features are combined by appearance feature and spatial feature as they show what eyes catch of, without further processing of natural language system.\",\n \"From this part, a model can get visual information such as word font, color and background.\",\n \"These two are extracted by an off-the-shelf Faster R-CNN [30] detector.\",\n \"${\\\\begin{split}& \\\\mathbf {x}_{i}^{ocr,v} = \\\\mathrm {LN}( \\\\mathbf {W}_{fr}\\\\mathbf {x}_{i}^{ocr,fr})+ \\\\mathrm {LN}(\\\\mathbf {W}_{bx}\\\\mathbf {x}_{i}^{ocr,bx}),\\\\end{split}}$ where $\\\\mathbf {x}_{i}^{ocr,fr}$ is the appearance feature extracted from the fc6 layer of Faster R-CNN detector.\",\n \"The fc7 weights are finetuned on our task.\",\n \"$\\\\mathbf {x}_{i}^{ocr,bx}$ is the bounding box feature in the format of $[x_{tl},y_{tl},x_{br},y_{br}]$ , where $tl$ and $br$ denotes top left and bottom right coordinates respectively.\",\n \"2) OCR linguistic-part.\",\n \"Linguistic features are made up of 1) FastText feature $\\\\mathbf {x}_{i}^{ocr,ft}$ , which is extracted from a pretrained word embedding and 2) character-level Pyramidal Histogram of Characters (PHOC) [1] feature $x_{i}^{ocr,ph}$ as they contain natural language related information.\",\n \"${\\\\begin{split}& \\\\mathbf {x}_{i}^{ocr,l} = \\\\mathrm {LN}( \\\\mathbf {W}_{ft}\\\\mathbf {x}_{i}^{ocr,ft} + \\\\mathbf {W}_{ph}\\\\mathbf {x}_{i}^{ocr,ph})\\\\end{split}}$ 3) OCR additional features.\",\n \"In the SBD-Trans [25], [40] that we use to recognize OCR tokens, the holistic representations in a specific text region are themselves visual features, however, are employed for linguistic word classification purpose.\",\n \"Therefore, they cover both visual and linguistic context of the OCR token and we thus introduce the Recog-CNN feature $\\\\mathbf {x}_{i}^{ocr,rg}$ from this network to enrich text features.\",\n \"Finally, Recog-CNN features are added to OCR visual- and linguistic-part simultaneously.\",\n \"${\\\\begin{split}& \\\\mathbf {x}_{i}^{ocr,v} = \\\\mathrm {LN}( \\\\mathbf {W}_{fr}\\\\mathbf {x}_{i}^{ocr,fr} + \\\\mathbf {W}_{rg}\\\\mathbf {x}_{i}^{ocr,rg})+ \\\\mathrm {LN}(\\\\mathbf {W}_{bx}\\\\mathbf {x}_{i}^{ocr,bx})\\\\\\\\& \\\\mathbf {x}_{i}^{ocr,s} = \\\\mathrm {LN}( \\\\mathbf {W}_{ft}\\\\mathbf {x}_{i}^{ocr,ft} + \\\\mathbf {W}_{ph}\\\\mathbf {x}_{i}^{ocr,ph} + \\\\mathbf {W}_{rg}\\\\mathbf {x}_{i}^{ocr,rg})\\\\end{split}}$ Visual features.\",\n \"In text-based tasks, visual contents in an image can be utilized to assist textual information in the reasoning process.\",\n \"To prove that our simple attention block has the power of using visual features in various forms, we adopt either grid-based global features or region-based object features.\",\n \"1) Global features.\",\n \"We obtain image global features $x_{i}^{glob}$ from a ResNet-152 [15] model pretrained on ImageNet, by average pooling $2048D$ features from the res-5c block, yielding a $14 \\\\times 14 \\\\times 2048$ feature for one image.\",\n \"To be consistent with other features, we resize the feature into $196 \\\\times 2048$ , a total of 196 uniformly-cut grids.\",\n \"${\\\\begin{split}& \\\\mathbf {x}_{i}^{glob} = \\\\mathrm {LN}( \\\\mathbf {W}_{g}\\\\mathbf {x}_{i}^{glob})\\\\end{split}}$ 2) Object features.\",\n \"The region-based object features are extracted from the same Faster F-CNN model as mentioned in OCR features part.\",\n \"${\\\\begin{split}& \\\\mathbf {x}_{i}^{obj} = \\\\mathrm {LN}( \\\\mathbf {W}_{fr}^{^{\\\\prime }}\\\\mathbf {x}_{i}^{obj,fr})+ \\\\mathrm {LN}(\\\\mathbf {W}_{bx}^{^{\\\\prime }}\\\\mathbf {x}_{i}^{obj,bx}),\\\\end{split}}$ where $\\\\mathbf {x}_{i}^{obj,fr}$ is the appearance feature and $\\\\mathbf {x}_{i}^{obj,bx}$ is the bounding box feature.\"\n ],\n [\n \"Attention Block as Feature Summarizing\",\n \"In tasks that cross the fields of computer vision and natural language processing, modality fusion is of superior importance.\",\n \"Treating them as homogeneous entities in a joint embedding space might be easy to implement, however, is not carefully tailored to a specific problem.\",\n \"Moreover, the many parameters of all entities in the large model (for example, a Transformer) consume much computation.\",\n \"To grasp interaction between modalities for maximum benefit and filter out irrelevant or redundant features before the entering into a large fusion layer, we use a simple attention block to input two sequences of entities and output two processed vectors, which is shown as the Attention Block in Figure REF .\",\n \"The two sequences of entities might be any sequence we want.\",\n \"For TextVQA problem, question changes in real-time and plays a dominant role in final answering.\",\n \"The design of question needs careful consideration and its existence should contribute throughout the process.\",\n \"For example, here we use question as one input of query in attention block.\",\n \"The sequence of question words goes through a self-attention process before forwarding into attention block.\",\n \"First we put the question word sequence $Q = \\\\lbrace q_{i}\\\\rbrace _{i=1}^{L}$ through a fully connected feed-forward network, which consists of two linear transformations (or two convolutions with 1 as kernel size) and one ReLU activation function between them.\",\n \"$\\\\begin{split}&q_{i}^{fc} = \\\\mathrm {conv}\\\\lbrace \\\\mathrm {ReLU}[\\\\mathrm {conv}(q_{i})]\\\\rbrace , \\\\,\\\\, i = 1, \\\\dots , L ;\\\\end{split}$ A softmax layer is the used to compute the attention on each word in the question.\",\n \"$\\\\begin{split}&a_i = \\\\mathrm {Softmax}({q_{i}^{fc}}), \\\\,\\\\, i = 1, \\\\dots , L ;\\\\\\\\\\\\end{split}$ This is known as self-attention and these weights are multiplied with original question embedding to get weighted sum of word embeddings.\",\n \"$\\\\begin{split}&Q^s = \\\\textstyle {\\\\sum _{i=1}^{L}} a_i {q}_{i}.\\\\end{split}$ If we have several individual entities to combine with question, the corresponding number of parallel self-attention processes are performed on the same question with independent parameters.\",\n \"For example, we can get $Q^s_{v}$ , $Q^s_{l}$ and $Q^s_{o}$ for OCR visual-part, OCR linguistic-part and object regions respectively.\",\n \"Then the $Q^s$ are used as query for corresponding features.\",\n \"We calculate the attention weights under the guidance of $Q^s$ , which are then put into a softmax layer.\",\n \"Finally the weights are multiplied with original queried features to get a filtered vector.\",\n \"Here we take the pair of $Q^s_{v}$ and ${\\\\mathbf {x}}_{i}^{ocr,v}$ as an example: ${\\\\begin{split}&p_i = \\\\mathbf {W}[\\\\mathrm {ReLU}(\\\\mathbf {W}_{s} Q^s_{v})\\\\circ \\\\mathrm {ReLU}(\\\\mathbf {W}_{x}{x}_i^{ocr,v})],\\\\\\\\&s_i = \\\\mathrm {Softmax}(p_i), \\\\quad i=1,\\\\dots ,N,\\\\\\\\&g^{ocr,v} = \\\\textstyle {\\\\sum _{i=1}^{N}} s_i {\\\\mathbf {x}}_{i}^{ocr,v}\\\\end{split}}$ where $g^{ocr,v}$ is the output of attention block.\",\n \"Similarly, we can get $g^{ocr,l}$ for the OCR-linguistic summarizing feature, $g^{obj}$ for the object summarizing feature.\",\n \"Different from M4C sending every single question tokens, OCR tokens and objects into the transformer feature fusion layer, here we only have 6 feature vectors ($Q^s_{v}$ , $Q^s_{l}$ , $Q^s_{o}$ , $g^{ocr,v}$ , $g^{ocr,l}$ and $g^{obj}$ ) which are sent to the following process.\",\n \"This largely decreases the computation complexity and burden, considering that the transformer is a parameter-heavy network.\"\n ],\n [\n \"Stacked-Block Encoder\",\n \"The attention block in Section REF can be stacked together as an encoder which produces combined embedding for downstream tasks.\",\n \"TextVQA baseline model.\",\n \"As presented in the above module, questions are sent through self-attention to output $Q^s_{v}$ , $Q^s_{l}$ and $Q^s_{o}$ .\",\n \"OCR features in images are splitted into visual- and linguistic part, which are $x^{ocr,v}$ and $x^{ocr,l}$ .\",\n \"We also have object features $x^{obj}$ .\",\n \"The six sequences are put into three attention blocks and we get six $768D$ vectors which are then forwarded into a fusion encoder.\",\n \"The fusion encoder, OCR encoder and generative decoder in Figure REF are in the same transformer model but undertaking different roles.\",\n \"After fusion encoder processing, the six vectors conduct element-wise multiplication in a pairwise way to get corresponding embeddings which are concatenated together.\",\n \"Then we use a fully-connected layer to transform the concatenated embeddings to a context embedding with appropriate dimension, upon which we generate the first answer output.\",\n \"Given the first answer word, a generative decoder is then used to select the rest of answer, which will be detailed in Section REF and Supplementary Material.\",\n \"Figure: TextCaps baseline model.\",\n \"It has a same structure as TextVQA baseline model.Table: We ablate our model on TextVQA dataset by testing number of attention blocks, forms of visual object features and addition of OCR representations.Table: Comparison to previous work on TextVQA dataset.\",\n \"Our model sets new state-of-the-art with an extremely simple design.TextCaps baseline model.\",\n \"As there are no questions in TextCaps, we use objects to guide OCR visual- and linguistic-part and use OCRs to guide object features.\",\n \"Technically we simply replace question word sequence with OCR token sequence or object proposal feature sequence.\",\n \"The other settings are the same with TextVQA.\",\n \"Figure REF illustrates our Textcaps baseline model.\",\n \"To easily transfer to another task demonstrates the generalization ability and simplicity of our method.\"\n ],\n [\n \"Answer Generation Module\",\n \"To answer a question or generate a caption, we use a generative decoder based on transformer.\",\n \"It takes as input the `context embedding' from the above encoder and select the first word of the answer.\",\n \"Based on the first output word, we then use the decoder to find the next word token either from a pre-built vocabulary or the candidate OCR tokens extracted from the given image, based on a scoring function.\",\n \"Training Loss.\",\n \"Considering that the answer may come from two sources, we use multi-label binary cross-entropy (bce) loss: ${\\\\begin{split}pred &= \\\\frac{1}{1+\\\\exp {(-y_{pred})}}, \\\\\\\\\\\\mathcal {L}_{bce} &= -y_{gt} \\\\mathrm {log}(pred)-(1-y_{gt}) \\\\mathrm {log}(1-pred),\\\\end{split}}$ where $y_{pred}$ is prediction and $y_{gt}$ is ground-truth target.\",\n \"Additional Training Loss.\",\n \"In some cases, the model reasons correctly, however, picks slightly different words than what we expected due to defective reading (OCR) ability.\",\n \"To take advantage of these predictions, we introduce a new policy gradient loss as an auxilliary task inspired by reinforcement learning.\",\n \"In this task, the greater reward, the better.\",\n \"We take Average Normalized Levenshtein Similarity (ANLS) metric$\\\\mathrm {ANLS}(s_{1},s_{2})=1-d(s_{1},s_{2})/\\\\mathrm {max}(\\\\mathrm {len}(s_{1}),\\\\mathrm {len}(s_{2}))$ , d(,) is edit distance.\",\n \"as the reward which measures the character similarity between predicted answer and ground-truth answer.\",\n \"$\\\\begin{split}&r = \\\\mathrm {ANLS}(\\\\phi (y_{gt}),\\\\phi (y_{pred})), \\\\\\\\&y = \\\\mathbb {I}(\\\\mathrm {softmax}(y_{pred})), \\\\\\\\&\\\\mathcal {L}_{pg} = (0.5 - r) (y_{gt} \\\\mathrm {log}(y) + (1-y_{gt}) \\\\mathrm {log}(1-y)),\\\\\\\\&\\\\mathcal {L} = \\\\mathcal {L}_{bce}+\\\\alpha \\\\cdot \\\\mathcal {L}_{pg},\\\\end{split}$ where $\\\\phi $ is a mapping function that returns a sentence given predicting score (e.g., , $y_{pred}$ ), ANLS($\\\\cdot $ ) is used to calculate similarity between two phrases, $\\\\mathbb {I}$ is an indicator function to choose the maximum probability element.\",\n \"The additional training loss is a weighted sum of $\\\\mathcal {L}_{bce}$ and $\\\\mathcal {L}_{pg}$ , where $\\\\alpha $ is a hyper-parameter to control the trade-off of $\\\\mathcal {L}_{pg}$ .\",\n \"After introducing policy gradient loss, our model is able to learn fine-grained character composition alongside linguistic information.\",\n \"We only apply this additional loss on ST-VQA dataset, which brings roughly $1\\\\%$ improvement.\"\n ],\n [\n \"Experiments\",\n \"Extensive experiments are conducted across two categories of tasks: TextVQA and TextCap.\",\n \"For TextVQA we set new state-of-the-art on TextVQA dataset and all three tasks of ST-VQA.\",\n \"For TextCaps we surpass 2020 TextCaps Challenge winner.\",\n \"See more experiments details below.\"\n ],\n [\n \"Implementation Details\",\n \"The set of methods are built on top of PyTorch.\",\n \"We use Adam as the optimizer.\",\n \"The learning rate for TextVQA and TexCaps is set to $1e-4$ .\",\n \"For TextVQA we multiply the learning rate with a factor of $0.1$ at the $14,000$ and $15,000$ iterations in a total of $24,000$ iterations.\",\n \"For TextCaps the multiplication is done at the $3,000$ and $4,000$ iterations, with $12,000$ total iterations.\",\n \"We set the maximum length of questions to $L = 20$ .\",\n \"We recognize at most $N = 50$ OCR tokens and detect at most $M = 100$ objects.\",\n \"The maximum number of decoding steps is set to 12.\",\n \"Transformer layer in our model uses 12 attention heads.\",\n \"The other hyper-parameters are the same with BERT-BASE [11].\",\n \"We use the same model on TextVQA and three tasks of ST-VQA, only with different answer vocabulary, both with a fixed size of 5000.\"\n ],\n [\n \"Ablation Study on TextVQA Dataset\",\n \"TextVQA [34] is a popular benchmark dataset to test scene text understanding and reasoning ability, which contains $45,336$ questions on $28,408$ images.\",\n \"Diverse questions involving inquires about time, names, brands, authors, etc., and dynamic OCR tokens that might be rotated, casual or partially occluded make it a challenging task.\",\n \"We first conduct an experiment of building only one block, with one question self-attention output guiding the whole set of text features as a comparison.\",\n \"This is a one-block model in TableREF which does not perform as good as the state-of-the-art.\",\n \"To investigate how well text features can perform without the usage of visual grid-based or region-based features, we build a two-block model.\",\n \"Given the two categories of OCR features – visual and linguistic, we find that our simple model is already able to perform promisingly on TextVQA problem.\",\n \"Line 1 and Line 2 tell clearly the validity of sorting text features into two groups ($1.77\\\\%$ difference).\",\n \"When building our third block on the basis of visual contents, either global features or object-level features are at our disposal.\",\n \"The incorporation of a third block has modest improvements ($0.24\\\\%$ for global and $0.63\\\\%$ for object features).\",\n \"From Line 4 to Line 5, a new Recog-CNN feature is added to enrich text representation and brings $0.37\\\\%$ improvement.\",\n \"We also use more transformer layers (from 4 to 8) and get $0.1\\\\%$ higher result.\",\n \"Then we use a much better OCR system (especially on recognition part) and obtain large performance boost (from $40.38\\\\%$ to $43.95\\\\%$ ).\",\n \"Table: Comparison to previous work on ST-VQA dataset.\",\n \"With TextVQA pretraining, our model outperforms current approaches by a large margin.Figure: Qualitative examples of our baseline model in contrast to M4C.\",\n \"While our model can read texts in accordance with written language system, M4C can only pick tokens in a random and errorneous way.Qualitative examples.\",\n \"We present four examples in Figure REF where our model shows the ability to really read scene texts in a way similar to humans, i.e., left-to-right then top-to-bottom.\",\n \"In contrast, current state-of-the-art model M4C fails to read tokens in a correct order.\"\n ],\n [\n \"Comparison with state-of-the-art\",\n \"TextVQA dataset.\",\n \"Even stripped of the usage of visual contents in the image, our two-block model already surpasses current state-of-the-art M4C by $0.98\\\\%$ on test set (Line 2 in Table REF VS. Line 4 in Table REF ).\",\n \"Using the same OCR system, our baseline model further improves upon M4C by $0.98\\\\%$ on val and $1.91\\\\%$ on test (Line 7 VS. Line 4 in Table REF ).\",\n \"Compared to the top entries in TextVQA Challenge 2020, our baseline has a significantly simpler model design, especially on the encoder side.\",\n \"M4C and SA-M4C take all parameters of entities into transformer layers and join large amout of computation.\",\n \"SMA uses a heterogeneous graph to explicitly consider different nodes and compute attention weights on 5-neighbored graph.\",\n \"Our model that surpasses all of them only sends six holistic vectors two-by-two into transformer layers, which tremendously saves computation.\",\n \"ST-VQA dataset.\",\n \"The ST-VQA dataset [7] is another popular dataset with three tasks, which gradually increase in difficulty.\",\n \"Task 1 provides a dynamic candidate dictionary of 100 words per image, while Task 2 provides a fixed answer dictionary of $30,000$ words for the whole dataset.\",\n \"As for Task 3, however, the model are supposed to generate answer without extra information.\",\n \"We also evaluate our model on ST-VQA dataset, using the same model from TextVQA for all three tasks.\",\n \"Without any additional training data, our model achieved the highest on Task 2 and Task 3 (Line 4 in Table REF ).\",\n \"Using TextVQA dataset as additional training data, our model sets new state-of-the-art on all three tasks and outperforms current approaches by a large margin.\",\n \"Table: Results on TextCaps dataset.\",\n \"(B: BLEU-4; M: METEOR; R: ROUGE_L; S: SPICE; C: CIDEr)\"\n ],\n [\n \"TextCaps Dataset\",\n \"TextCaps is a new dataset that requires a model to read texts in images and generate descriptions based on scene text understanding and reasoning.\",\n \"In TextCaps, automatic captioning metrics (BLEU [28], METEOR [10], ROUGE_L [23], SPICE [2] and CIDEr [36]) are compared with human evaluation scores.\",\n \"All automatic metrics show high correlation with human scores, among which CIDEr and METEOR have the highest.\",\n \"M4C-Captioner is the method provided in TextCaps, which is modified from M4C model by simply removing question input.\",\n \"Similarly, simply replacing question in our TextVQA baseline model with object or OCR sequence yields our TextCaps baseline model.\",\n \"Using exactly the same OCR system, OCR representations, our baseline with Rosetta-en OCR (Line 2 on Table REF ) already surpass M4C-Captioner (Line 1 on Table REF ), especially on BLUE-4 and CIDEr metric.\",\n \"By upgrading our OCR system to SBD-Trans and using 6-layer transformer in our encoder-decoder structure, our baseline further exceeds TextCaps Challenge Winner on BLUE-4 and CIDEr metric as shown in Line 5 and Line 6 of Table REF .\",\n \"[h] Answer generation in TextVQA Question features $Q$ , OCR features $x^{ocr}$ , Object features $x^{obj}$ An answer with a length of $T$ words $A=\\\\lbrace a_{i}\\\\rbrace _{i=0}^{T-1}$ Obtain three independent question signals: $(Q^s_{v}, Q^s_{l}, Q^s_{o}) = \\\\mathrm {Self\\\\_Attention}^{\\\\lbrace v,l,o\\\\rbrace }(Q)$ Split OCR features into OCR visual- and linguistic-part ($x^{ocr,v}$ , $x^{ocr,l}$ ) Forward question signals and three features into three Attention Blocks in a pairwise way: $g^{ocr,v} = \\\\mathrm {Att\\\\_Blk}(Q^s_{v}, x^{ocr,v});\\\\ g^{ocr,l} = \\\\mathrm {Att\\\\_Blk}(Q^s_{l}, x^{ocr,l});\\\\ g^{obj} = \\\\mathrm {Att\\\\_Blk}(Q^s_{o}, x^{obj})$ Forward six vectors into fusion encoder: $\\\\bar{Q}^s_{v}, \\\\bar{Q}^s_{l}, \\\\bar{Q}^s_{o}, \\\\bar{g}^{ocr,v}, \\\\bar{g}^{ocr,l}, \\\\bar{g}^{obj} = \\\\mathrm {Fusion\\\\_Encoder}(Q^s_{v}, Q^s_{l}, Q^s_{o}, g^{ocr,v}, g^{ocr,l}, g^{obj})$ Element-wise multiply the six vectors two by two: $\\\\mathbf {e}^{ocr,v}=\\\\bar{Q}^s_{v}\\\\circ \\\\bar{g}^{ocr,v};\\\\ \\\\mathbf {e}^{ocr,l}=\\\\bar{Q}^s_{l}\\\\circ \\\\bar{g}^{ocr,l};\\\\ \\\\mathbf {e}^{obj}=\\\\bar{Q}^s_{o}\\\\circ \\\\bar{g}^{obj}$ Concatenate the three results together and transform it into context embedding $\\\\mathrm {context\\\\, embedding}=\\\\mathrm {Linear}([\\\\mathbf {e}^{ocr,v}; \\\\mathbf {e}^{ocr,l}; \\\\mathbf {e}^{obj}])$ $t = 0$ Based on context embedding, calculate the first word $a_{t}$ in the answer: 1) Map context embedding to a $5000D$ vector through a linear layer as scores for vocabulary 2) Get $50D$ dot product between context embedding and all 50 OCR encoder outputs as scores for OCR tokens 3) Concatenate $5000D$ vector and $50D$ vector together, from which select the highest-score index as $a_{t}$ $t=1;t \\\\le T-1;t++$ the previous output $a_{t-1}$ is from vocabulary Forward linear layer weights corresponding to the previous output $a_{t-1}$ into generative decoder and obtain decoder embedding the previous output $a_{t-1}$ is from OCR tokens Forward OCR features of the previous output $a_{t-1}$ into generative decoder and obtain decoder embedding Based on decoder embedding, calculated the current word $a_{t}$ in the answer: 1) Map decoder embedding to a $5000D$ vector through a linear layer as scores for vocabulary 2) Get $50D$ dot product between decoder embedding and all 50 OCR encoder outputs as scores for OCR tokens 3) Concatenate $5000D$ vector and $50D$ vector together, from which select the highest-score index as $a_{t}$ return $A=\\\\lbrace a_{i}\\\\rbrace _{i=0}^{T-1}$ Table: Computational complexity of two model encoders.\",\n \"L=20L=20 is the length of question; N=50N=50 is the number of OCR tokens; M=100M=100 is the number of detected objects.\",\n \"Here we omit all vector dimensions DD for simplicity.\",\n \"In transformer encoder of our model, apart from the six vectors, we also input 50 OCR tokens to compute dot product value in answer choosing.\"\n ],\n [\n \"Conclusion\",\n \"In this paper, we provide a simple but rather strong baseline for the text-based vision-and-language research.\",\n \"Instead of handling all modalities over a joint embedding space or via complicated graph structural encoding, we use the vanilla attention mechanism to fuse pairwise modalities.\",\n \"We further split text features into two functionally different parts, i.e., linguistic- and visual-part which flow into corresponding attention branch.\",\n \"We evaluate our simple baseline model on TextVQA, ST-VQA and TextCaps, all leading to the best performance on the public leaderboards.\",\n \"This sets the new state-of-the-art and our model could be the new backbone model for both TextVQA and TextCaps.\",\n \"What's more, we believe this work inspires a new thinking of the multi-modality encoder design.\"\n ],\n [\n \"Algorithm\",\n \"Algorithm REF provides the pseudo-code for the process of answer generation in the TextVQA problem.\",\n \"The description generation process in the TextCaps problem is roughly the same, except that the decoding step $T$ is set to 30 rather than 12 for a longer and more expressive image description.\",\n \"Besides, the question features are also replaced by OCR features and Object features to guide each other.\"\n ],\n [\n \"Comparison of Computation Complexity between Encoders\",\n \"As our model and M4C have the same generative decoder, we only compare encoders in Table REF .\",\n \"In one Attention Block, main operations are element-wise multiplication and final matrix multiplication, which yield $O(n \\\\cdot d)=2n \\\\cdot d$ operations, where n is number of queried features.\",\n \"In Transformer encoder, the complexity is $O(n^2 \\\\cdot d)$ , where $n$ is the sequence length and $d$ is the representation dimension.\"\n ],\n [\n \"Additional Experiment Details\",\n \"OCRs and Objects self-attention in TextCaps.\",\n \"Different from question self-attention in TextVQA problem that uses a BERT model to capture the sequence information, for OCRs and Objects self-attention in TextCaps we use an LSTM [16] layer to capture the recurrent features of isolate regions before the self-attention calculation.\",\n \"SBD-Trans training data.\",\n \"The SBD model is pretrained on a $60k$ dataset, which consists of $30,000$ images from LSVT [35] training set, $10,000$ images from MLT 2019 [27] training set, $5,603$ images from ArT [9] , and $14,859$ images selected from a bunch of datasets – RCTW-17 [32], ICDAR 2013 [22], ICDAR 2015 [21], MSRA-TD500 [41], COCOText [37], and USTB-SV1K [42].\",\n \"The model was finally finetuned on MLT 2019 training set.\",\n \"The robust transformer based network is trained on the following datasets: IIIT 5K-Words [26], Street View Text [38], ICDAR 2013, ICDAR 2015, Street View Text Perspective [29], CUTE80 [31] and ArT.\"\n ],\n [\n \"Additional Qualitative Examples\",\n \"To demonstrate that our model not only learns fixed superficial correlation between questions and prominent OCR tokens, we present several images in the dataset with two different questions to which our model both answers correctly, while other models such as M4C fails, in Figure REF .\",\n \"TextCaps examples are in Figure REF .\",\n \"Figure: Qualitative examples of our model on TextCaps.Figure: Examples that our model fails at on TextVQA.\"\n ],\n [\n \"Failure Cases on TextVQA\",\n \"The questions that our model struggles at can be categorized into several groups.\",\n \"Firstly, due to the inappropriate tricky design of the dataset, models are required to read the time, find a specific letter in a word, or read a ruler, e.g., Figure REF a), b) and c).\",\n \"Secondly, even though our model answers reasonably, it does not lie in the region of ground-truth answers as shown in Figure REF d).\",\n \"Thirdly, the mistake might occur because of defective reading ability that OCR tokens do not include answer candidates at all, as shown in Figure REF e).\",\n \"Fourthly, it is difficult to discern the color of tokens as shown in Figure REF f).\",\n \"Finally, when it comes to nuanced relationship understanding, our model is not yet complex enough to reason as shown in Figure REF g) and h).\"\n ]\n]"},"id":{"kind":"string","value":"2012.05153"}}},{"rowIdx":3697,"cells":{"text":{"kind":"list like","value":[["Cosmological constraints with the Effective Fluid approach for Modified\n Gravity"],["Abstract Cosmological constraints of Modified Gravity (MG) models are seldom carried out rigorously.","First, even though general MG models evolve differently (i.e., background and perturbations) to the standard cosmological model, it is usual to assume a $\\Lambda$CDM background.","This treatment is not correct and in the era of precision cosmology could induce undesired biases in cosmological parameters.","Second, neutrino mass is usually held fixed in the analyses which could obscure its relation to MG parameters.","In a couple of previous papers we showed that by using the Effective Fluid Approach we can accurately compute observables in fairly general MG models.","An appealing advantage of our approach is that it allows a pretty easy implementation of this kinds of models in Boltzmann solvers (i.e., less error--prone) while having a useful analytical description of the effective fluid to understand the underlying physics.","This paper illustrates how an effective fluid approach can be used to carry out proper analyses of cosmological constraints in MG models.","We investigated three MG models including the sum of neutrino masses as a varying parameter in our Markov Chain Monte Carlo analyses.","Two models (i.e., Designer $f(R)$ [DES-fR] and Designer Horndeski [HDES]) have a background matching $\\Lambda$CDM, while in a third model (i.e., Hu $\\&$ Sawicki $f(R)$ model [HS]) the background differs from the standard model.","In this way we estimate how relevant the background is when constraining MG parameters along with neutrinos' masses.","We implement the models in the popular Boltzmann solver CLASS and use recent, available data (i.e., Planck 2018, CMB lensing, BAO, SNIa Pantheon compilation, $H_0$ from SHOES, and RSD Gold-18 compilation) to compute tight cosmological constraints in the MG parameters that account for deviation from the $\\Lambda$CDM model.","[abridged]"],["Introduction ","The growing evidence for the late-time accelerating expansion of the Universe represents a major milestone in cosmology [1], [2], [3], [4] and investigations using Machine Learning algorithms have confirmed this fact through model independent analyses [5], [6], [7].","Bayesian analyses indicate that the standard model of cosmology $\\Lambda $ CDM is preferred over its alternatives due to its simplicity and lower number of free parameters [8].","The concordance model is relatively simple and having just six free parameters is able to make predictions which agree remarkably well with most observations [3], [4], [9], [10].","Nevertheless, the cosmological constant problem, our current ignorance on the nature of Dark Matter (DM) as well as a worrying discordance in a few cosmological parameters represent big disadvantages for the $\\Lambda $ CDM model.","Over time the disagreement between the Hubble constant determined via distance ladder and the $H_0$ value obtained through analyses of the CMB has become more interesting [11].","Although the discrepancy could be due to unaccounted-for systematic errors, there also exists the more appealing possibility of new physics (e.g.","Modified Gravity, early Dark Energy).","With the coming of latest data analyses disagreements on the values of $\\Omega _m$ and $\\sigma _8$ also became apparent [12] or for example, a $\\sim 4\\sigma $ deviation of the dark energy equation of state $w(z)$ from the $\\Lambda $ CDM model using quasars at high redshift up to $z\\sim 7.5$ [13].","The curvature of the Universe has also given rise to a discussion on recent analyses [14], [3], [15], [16].","There are some mild hints ($\\approx 2\\sigma $ ) of Modified Gravity (MG) [3], [17], which challenge assumptions made in the standard model.","Here we will investigate some viable MG models in light of latest data releases.","Dark Energy (DE) and MG have emerged as alternatives to the cosmological constant model.","In the case of DE, some of the simplest models include minimally coupled scalar fields in the form of quintessence, that has a standard kinetic term, or k-essence which has a generalized kinetic-term [18].","On the other hand, MG models are covariant modifications to General Relativity (GR) that extend the Einstein-Hilbert action in various ways: either by promoting it to a function as in the $f(R)$ models and by introducing higher order curvature invariants (see for example [19]) or by adding non-minimally coupled scalar fields, as in the case of Horndeski theory [20].","MG models have the advantage that they are also inspired from high-energy physics, as covariant modifications to GR of a similar form appear naturally when one tries to renormalize GR at one loop order [21].","The advantages of these alternatives, especially of the MG models, are clear: there is no need for a cosmological constant as the accelerating expansion of the Universe can be explained by the presence of the extra terms in the modified Friedmann equations due to the new degrees of freedom.","Both DE and MG models are also able to describe well the cosmological observations and give equally good fits to the data as the $\\Lambda $ CDM model.","On the other hand, there are also some disadvantages for these models, for example the presence of the additional parameters may penalize some of the models when one calculates the Bayesian evidence and uses the Jeffreys' scale, albeit it has been shown that the latter has to be interpreted with care [22].","Furthermore, by using N-body simulations it has been shown that a compelling $f(R)$ model fails to reproduce the observed redshift-space clustering on scales $\\sim 1-10 ~ \\mathrm {Mpc}~ h^{-1}$ [23].","Finally, as both the extra scalar field degrees of freedom and the higher order corrections to GR are as yet unobserved in a laboratory or in an astrophysical setting, their motivation is obviously somewhat weakened.","Not only that, but recently several MG models of the Horndeski type have been ruled out, via the measurement of the speed of propagation of the gravitational waves by the event GW170817 and its optical counterpart GRB170817A [24], [25], [26].","Thus, as the available parameter space has shrank remarkably, there are a few remaining models which deserve attention as well as proper analyses.","However, many analyses of the remaining models, especially the ones where the background expansion differs significantly from the $\\Lambda $ CDM model, do not consider the background expansion properly and just fix it to either the $\\Lambda $ CDM or a constant $w$ model, as was observed in Refs.","[27], [28].","This obviously biases the results as it introduces biases in the cosmological parameters and spurious tensions with the data.","However, some recent analyses have also acknowledged this discrepancy and newer versions of the Boltzmann solvers now have support for the correct backgrounds in some cases [29].","On the other hand, the so-called Effective Fluid Approach has the advantage that it presents a unified approach to analyse all models under the same umbrella, allows for the correct background expansion in the models, all without sacrificing the accuracy of the results [27], [28], [30].","In a nutshell, the Effective Fluid Approach works by rewriting the field equations of the MG model as GR and a DE fluid with an equation of state $w$ , a pressure perturbation $\\delta P$ and an anisotropic stress $\\sigma $ .","Especially the latter is crucial as sometimes it is ignored in analyses of MG models [31], something which might bias the results [32].","Moreover, through a joint Machine Learning analysis applied to the latest cosmological data hints of dark energy anisotropic stress were found [7].","In the Effective Fluid Approach we also assume that in the relevant scales, where linear theory applies, the sub-horizon and quasi-static approximations hold.","With these, general analytical expressions for the equation of state, the pressure perturbation, and the anisotropic stress were found in Refs.","[27], [28].","With the latter, one may then just solve numerically the evolution equations for the perturbations, found for example in Ref. [33].","Thus, the main advantage of the Effective fluid approach is that once one has the expressions for the variables $w$ , $\\delta P$ and $\\sigma $ , it is very straightforward to also implement them in standard Boltzmann codes, such as CLASS, with very minimal modifications.","In fact, in [27], [28], this was done with the EFCLASS code, which implements the aforementioned approach, where it was found that EFCLASS and hi-CLASS [34], a modification of CLASS that solves numerically the whole set of perturbation equations for Horndeski, agree to better than $0.1\\%$ [28].","Recently, a comparison of different approaches to the quasi-static approximation in Horndeski models was made by Ref.","[35], by applying this approximation to either the field equations, as done in the Effective fluid approach, or the equations for the two metric potentials $\\Phi $ and $\\Psi $ and finally, the use of the attractor solution derived within the Equation of State approach [36].","It was found that all three approaches agree exactly on small scales and that in general, this approach is valuable in future model selections analyses for models beyond the $\\Lambda $ CDM model.","In this analysis we use the Effective Fluid approach and our EFCLASS code, for the background and first order perturbations, to obtain cosmological constraints with the latest cosmological data sets: we include the Pantheon SNe compilation [37], the Planck 2018 CMB data [3], the $H_0$ Riess measurement [38], various BAO points [39], [40], [41], and a new redshift space distortions (RSD) likelihood (see Ref.","[42] for the “Gold 2018\" compilation of Ref. [43]).","The paper is organised as follows.","In Sec.","we briefly summarize our theoretical framework and present the models we consider, in Sec.","we present the results of our MCMC analysis with EFCLASS, in Sec.","we conclude and lastly, in Appendix we present some details on our RSD likelihood."],["Theoretical framework ","The standard cosmological model $\\Lambda $ CDM assumes the Einstein-Hilbert action $\\mathcal {L} = \\frac{1}{2\\kappa }f(R) + \\mathcal {L}_m$ with $f(R)=R$ , $R$ the Ricci scalar, $\\mathcal {L}_m$ the Lagrangian for matter fields, and the constant $\\kappa \\equiv 8\\pi G_N$ with $G_N$ being the bare Newton's constant.","We can derive the field equations by applying the Principle of Least Action; they read $G_{\\mu \\nu } = \\kappa T^{(m)}_{\\mu \\nu }$ where $G_{\\mu \\nu }\\equiv R_{\\mu \\nu }-\\frac{1}{2}g_{\\mu \\nu }R$ is the Einstein tensor, $R_{\\mu \\nu }$ is the Ricci tensor, $g_{\\mu \\nu }$ is the metric, and $T^{(m)}_{\\mu \\nu }$ is the energy-momentum tensor of matter fields.Throughout this paper our conventions are: $(-+++)$ for the metric signature, the Riemann and Ricci tensors are given respectively by $V_{b;cd}-V_{b;dc}=V_{a}R^{a}_{\\;bcd}$ and $R_{ab}=R^{s}_{\\;asb}$ .","It is also possible to consider more general theories (e.g., $f(R)\\ne R$ , Horndeski theories) and derive similar field equations.","As we clearly explained in Refs.","[27], [28], modifications to GR in these kinds of MG models can be interpreted as an effective fluid yielding field equations which schematically look like $G_{\\mu \\nu }&=&\\kappa \\left(T_{\\mu \\nu }^{(m)}+T_{\\mu \\nu }^{(\\textrm {DE})}\\right).$ Here the tensor $T_{\\mu \\nu }^{(\\textrm {DE})}$ depends on the metric and its derivatives, and in the case of general scalar-tensor theories, also on the scalar field and its derivatives (see [28]).","Observational evidence indicates that the Universe is statistically homogeneous and isotropic on large scales [44], [45], [46], [47].","Therefore we will as usual assume a flat linearly perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) metric $ds^2=a(\\tau )^2\\left[-(1+2\\Psi (\\vec{x},\\tau ))d\\tau ^2+(1-2\\Phi (\\vec{x},\\tau ))d\\vec{x}^2\\right],$ where $a$ is the scale factor, $\\vec{x}$ represents spatial coordinates, $\\tau $ is the conformal time, and $\\Psi $ and $\\Phi $ are the gravitational potentials in the Newtonian gauge.","In order to describe matter fields, we will consider them as ideal fluids having small perturbations.","We will take into consideration an Effective Fluid Approach (EFA) when dealing with MG models: modifications to GR will be described by an effective fluid having equation of state, pressure perturbation, and anisotropic stress.","In our approach, and assuming a metric (REF ), the background evolution is governed by the Friedmann equations $\\mathcal {H}^2&=&\\frac{\\kappa }{3}a^2 \\left(\\bar{\\rho }_{m}+\\bar{\\rho }_{\\textrm {DE}}\\right), \\\\\\dot{\\mathcal {H}}&=&-\\frac{\\kappa }{6}a^2 \\left(\\left(\\bar{\\rho }_{m}+3\\bar{P}_{m}\\right)+\\left(\\bar{\\rho }_{\\textrm {DE}}+3\\bar{P}_{\\textrm {DE}}\\right)\\right),$ where $\\mathcal {H}\\equiv \\dot{a}/a$ is the conformal Hubble parameter.In our notation a dot over a function denote derivative respect to the conformal time $\\dot{f}=df/d\\tau $ .","The Hubble parameter $H$ and the conformal Hubble parameter $H$ are related via $\\mathcal {H}=aH$ .","Here $\\bar{\\rho }_{\\textrm {DE}}$ and $\\bar{P}_{\\textrm {DE}}$ respectively denote density and pressure of the DE effective fluid.","The effective DE equation of state $w_{\\textrm {DE}} \\equiv \\bar{P}_{\\textrm {DE}}/\\bar{\\rho }_{\\textrm {DE}}$ allows us to describe the background evolution in these kinds of models.","Another ingredient that we need is the evolution equations for the perturbations obtained through the energy-momentum conservation $T^{\\mu \\nu }_{;\\nu }=0$ [27]: $\\delta ^{\\prime } &=& 3(1+w) \\Phi ^{\\prime }-\\frac{V}{a^2 H}-\\frac{3}{a}\\left(\\frac{\\delta P}{\\bar{\\rho }}-w\\delta \\right),$ $V^{\\prime } &=& -(1-3w)\\frac{V}{a}+\\frac{k^2}{a^2 H}\\frac{\\delta P}{\\bar{\\rho }} +(1+w)\\frac{k^2}{a^2 H} \\Psi \\nonumber \\\\&-&\\frac{2}{3} \\frac{k^2}{a^2 H} \\pi ,$ where $V$ is the scalar velocity perturbation, $\\delta P$ is the pressure perturbation, $\\pi $ is related to the anisotropic stress as $\\pi \\equiv \\frac{3}{2}(1+w)\\sigma $ and the prime $^{\\prime }$ is the derivative with respect to the scale factor $a$ .","In [28] we extensively used the EFA on the remaining Horndeski Lagrangian.","By applying both sub-horizon and quasi-static approximations, we managed to find analytical expressions for several quantities describing Horndeski models as an effective DE fluid, namely, DE pressure perturbation $\\frac{\\delta P_{\\textrm {DE}}}{\\overline{\\rho }_{DE}}(a,k) \\nonumber ,$ DE scalar velocity perturbation $V_{\\textrm {DE}}(a,k) \\nonumber ,$ and DE anisotropic stress $\\pi _{\\textrm {DE}}(a,k) \\nonumber .$ Having mapped remaining Horndeski models into an effective DE fluid, one can easily implement them in Boltzmann codes which compute observables such as the CMB angular power spectrum and the matter power spectrum.","In the following subsections we will provide a few details on the specific models that we study in this work."],["DES-fR model","The remaining Horndeski Lagrangian includes a very important sort of MG models, namely, $f(R)$ .","In [27] we showed there is a correspondence between a given $f(R)$ model and its effective DE equation of state $w_{DE}$ .","Then it becomes clear that by specifying a background (i.e., an equation of state), it is in principle possible to obtain a corresponding $f(R)$ .","The models obtained in this way are the so-called designer f(R) models [48], [49], [50].","An interesting case is the $w_{DE}=-1$ designer $f(R)$ model (DES-fR, henceforth) that mimics the standard $\\Lambda $ CDM model at the background, while exhibiting differences in the evolution of the linear perturbations.","The DES-fR model satisfying all viability conditions (see, for instance, [51]) is given by [50] $f(R)&=&R-2\\Lambda +\\alpha ~H_0^2\\left(\\frac{\\Lambda }{R-3 \\Lambda }\\right)^{c_{0}} \\times \\nonumber \\\\& & {}_2F_1\\left(c_{0},\\frac{3}{2}+c_{0},\\frac{13}{6}+2c_{0},\\frac{\\Lambda }{R-3 \\Lambda }\\right)\\;,$ where $c_{0}=\\frac{1}{12} \\left(-7+\\sqrt{73}\\right)$ , $\\alpha $ is a free dimensionless parameter, $H_0$ is the Hubble constant, $\\Lambda $ is a constant, and ${}_2F_1$ is a hypergeometric function.","In the literature it is common to define $F\\equiv f^{\\prime }(R)$ and we will follow this convention.","Instead of using $\\alpha $ in Eq.","(REF ) we can parametrise our expressions in terms of $b_\\pi = f_{R,0} \\equiv F(a=1)-1$ .","Furthermore, in Ref.","[27] we found that in the range $a\\in [0,1]$ the following approximation around $a\\simeq 0$ is equally accurate $&F(a)&\\simeq 1 + \\nonumber \\\\& & f_{R,0}\\frac{\\Omega _{m0}^{-c_0-1}}{\\, _2F_1\\left(c_0+1,c_0+\\frac{3}{2};2 c_0+\\frac{13}{6};1-\\Omega _{m0}\\right)}a^{3 (1 + c_0)} \\nonumber \\\\&+&\\mathcal {O}(a^{3(2+c_0)}) , $ Then, when we rewrite our expressions for the DE effective fluid in terms of $f_{R,0}$ we find that they depend on $g(x) = {}_2F_1(\\frac{3}{2}+c_0,2+c_0;2c_0+\\frac{13}{6},x),$ where $x \\equiv \\frac{a^3(\\Omega _{m,0}-1)}{a^3(\\Omega _{m,0}-1)-\\Omega _{m,0}}.$ Implementing special functions in EFCLASS is not an easy task.","Therefore, we used a Taylor approximation for $g(x)$ in Eq.","(REF ) which works really well around $x=0$ while keeping 30 terms in the expansion."],["HS model","The popular Hu $\\&$ Sawicki model (HS, henceforward) [52] $f(R)=R-m^2 \\frac{c_1 (R/m^2)^n}{1+c_2 (R/m^2)^n},$ can actually be rewritten, after some algebraic manipulations, as [53] $f(R)= R- \\frac{2\\Lambda }{1+\\left(\\frac{b_{hs} \\Lambda }{R}\\right)^n},$ where $\\Lambda = \\frac{m^2 c_1}{2c_2}$ and $b_{hs}=\\frac{2 c_2^{1-1/n}}{c_1}$ .","In [53] the authors found that when written in the form (REF ), it is clear the reason why the HS model satisfies solar system tests: if $b_{hs} \\rightarrow 0$ $\\Lambda $ CDM is recovered, and if $b_{hs} \\rightarrow \\infty $ a matter dominated universe is obtained i.e., $\\lim _{b_{hs}\\rightarrow 0}f(R)&=&R-2\\Lambda , \\nonumber \\\\\\lim _{b_{hs}\\rightarrow \\infty }f(R)&=&R.$ If the parameter $b_{hs}$ is small enough, the HS model can be regarded as a “perturbation” around $\\Lambda $ CDM .","This is key in our work since we do not assume the usual approximation of fixing the background to $\\Lambda $ CDM when investigating the HS model.","However, solving numerically the equation for the Hubble parameter is not trivial in the HS model and we worked out a different approach.","Instead of approximating the background, we solve the field equations and find an approximate, accurate analytical expression for the Hubble parameter.","We followed the treatment in [53], where it was shown that HS Hubble parameter can be written as $H_{HS}(a)^2 = H_{\\Lambda }(a)^2 + b_{hs} \\, \\delta H_1(a)^2 + b_{hs}^2\\, \\delta H_2(a)^2 + \\hdots ,~~$ which is an analytical approximation that works extremely well, for example for $b_{hs}\\le 0.1$ the average error with respect to the numerical solution is $10^{-5}\\%$ for redshifts $z\\le 30$ ."],["HDES model","In what follows, we will present a brief overview of a family of designer Horndeski (HDES) models that have been already studied in Ref. [28].","These are models whose background is exactly that of the $\\Lambda $ CDM model but at the perturbation level it is dictated by the Horndeski theory.","In its full form, Horndeski theory constitutes as the most general Lorentz-invariant extension of GR in four dimensions and contains a few DE and MG models.","Due to the recent discovery of gravitational waves by the LIGO Collaboration the Horndeski Lagrangian has been severely reduced.","In particular, it has been found the following constrain on the speed of GWs [25] $-3 \\cdot 10^{-15} \\le c_g/c-1 \\le 7 \\cdot 10^{-16},$ which implies that for Horndeski theories $G_{4X}\\approx 0, \\hspace{5.69054pt} G_5 \\approx \\text{constant}.$ Then, the surviving part of the Horndeski Lagrangian reads, $S[g_{\\mu \\nu }, \\phi ] = \\int d^{4}x\\sqrt{-g}\\left[\\sum ^{4}_{i=2} \\mathcal {L}_i\\left[g_{\\mu \\nu },\\phi \\right] + \\mathcal {L}_m \\right],$ where $\\mathcal {L}_2&=& G_2\\left(\\phi ,X\\right) \\equiv K\\left(\\phi ,X\\right),\\\\\\mathcal {L}_3&=&-G_3\\left(\\phi ,X\\right)\\Box \\phi ,\\\\\\mathcal {L}_4&=&G_4\\left(\\phi \\right) R,$ and $\\phi $ is a scalar field, $X \\equiv -\\frac{1}{2}\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi $ is a kinetic term, and $\\Box \\phi \\equiv g^{\\mu \\nu }\\nabla _{\\mu }\\nabla _{\\nu }\\phi $ ; $K$ , $G_3$ and $G_4$ are free functions of $\\phi $ and $X$ .","From the action (REF ) one can find several theories, for example $f(R)$ theories [54], Brans-Dicke theories, [55] and Cubic Galileon [56].","The HDES family of models that where constructed in Ref.","[28] limit to the Kinetic Gravity Braiding (KGB) which is distinguished by the following functions $K=K(X), \\hspace{14.22636pt} G_3=G_3(X), \\hspace{14.22636pt} G_4=\\frac{1}{2 \\kappa }.$ With Eq.","(REF ), we will present how to find a specific family of designer models such that $w_{\\textrm {DE}}=-1$ , i.e., the background is always that of the $\\Lambda $ CDM model but at the perturbation level it follows Horndeski's theory.","The usefulness of this designer model comes from allowing one to detect deviations from $\\Lambda $ CDM at the perturbations level and is a natural expansion of our previous work [50], [27].","For our HDES model we need two functions, the modified Friedmann equation and the scalar field conservation equation, both of which have been presented and analyzed in detail in [28].","The modified Friedmann equation reads, $&-H(a)^2-\\frac{K(X)}{3}+H^2_0\\Omega _m(a)+\\nonumber \\\\&+2\\sqrt{2}X^{3/2}H(a)G_{3X}+\\frac{2}{3}X K_X=0,$ where $\\Omega _m(a)$ is the matter density and $H_0$ is the Hubble parameter.","The scalar field conservation equation is $\\frac{J_c}{a^3}-6XH(a)G_{3X}-\\sqrt{2}\\sqrt{X}K_X=0.$ The constant $J_c$ quantifies our deviation from the attractor given the KGB model [57].","By looking at Eqs.","(REF ) and (REF ) we see that we have three unknown functions $(G_{3X}(X),K(X), H(a))$ hence the system is undetermined.","Then, we need to describe one of the three unknown functions $(G_{3X}(X),K(X), H(a))$ and find out the other two using Eqs.","(REF ) and (REF ).","For convenience, we write the Hubble parameter as a function of the kinetic term $X$ , i.e., $H=H(X)$ and then solve the previous equations to find $(G_{3X}(X),K(X))$ .","Then we find $K(X) &=& -3 H_0^2 \\Omega _{\\Lambda ,0}+\\frac{J_c \\sqrt{2X} H(X)^2}{H_0^2 \\Omega _{m,0}}-\\frac{J_c \\sqrt{2X} \\Omega _{\\Lambda ,0}}{\\Omega _{m,0}}, \\nonumber \\\\G_{3X}(X) &=& -\\frac{2 J_c H^{\\prime }(X)}{3 H_0^2 \\Omega _{m,0}},$ where $\\Omega _{m,0}$ is the matter density at redshift $z=0$ and $\\Omega _{\\Lambda ,0}=1-\\Omega _{m,0}$ .","With Eqs.","(REF ) we can create a whole family of designer models.","A specific model found, dubbed HDES [28], that has a smooth limit to $\\Lambda $ CDM and also recovers GR when $J_c\\sim 0$ is the following.","First, we demand that the kinetic term behaves as $X= \\frac{c_0}{H(a)^n}$ , where $c_0>0$ and $n>0$ .","Then, from Eqs.","(REF ) and (REF ) we have $G_{3}(X)&=&-\\frac{2 J_c c_0^{1/n} X^{-1/n}}{3 H_0^2 \\Omega _{m,0}},\\\\K(X)&=&\\frac{\\sqrt{2} J_c c_0^{2/n} X^{\\frac{1}{2}-\\frac{2}{n}}}{H_0^2 \\Omega _{m,0}}-3 H_0^2 \\Omega _{\\Lambda ,0}-\\frac{\\sqrt{2} J_c \\sqrt{X} \\Omega _{\\Lambda ,0}}{\\Omega _{m,0}}.\\nonumber $ A particular HDES model that we have selected for our MCMC runs and we also used it in our comparison with hi-CLASS previously [28] comes by setting $n=1$ in Eq.","(REF ).","In the effective fluid approach one needs to specify three functions: the equation of state, the pressure perturbation and the anisotropic stress $(w,\\delta P,\\sigma )$ to modify properly the background and the linear perturbations in the CLASS code.","With the HDES model we only need to specify $\\delta P$ since $w=-1$ and $\\sigma =0$ for this particular model.","As it was shown in [28], computing $\\delta P$ through the effective fluid approach and then introducing this function into the evolution equation for the perturbations Eqs.","(REF ) and (REF ) we can specify the DE scalar velocity perturbation $V_{\\textrm {DE}}$ as $V_{\\textrm {DE}}\\simeq \\left(-\\frac{14 \\sqrt{2}}{3} \\Omega _{m,0}^{-3/4} J_c~ a^{1/4}\\right)\\frac{\\bar{\\rho }_m}{\\bar{\\rho }_{\\textrm {DE}}} \\delta _m.$ which is what we implement later in our modified version of CLASS."],["Methodology","In order to compute cosmological constraints we use the following data sets.","Firstly, we utilise the 2018 release by the Planck Collaboration including temperature and polarisation anisotropies of the CMB (TTTEEE) as well as CMB lensing (lensing) [3].","Secondly, we include measurements of Baryon Acoustic Oscillations (BAO) from Refs.","[39], [40], [41].","Thirdly, Pantheon supernovae (SNe) from [37] were also incorporated in the analysis.","Fourthly, we employed local Hubble measurement (H0) from Ref.","[38] as a Gaussian prior.","Finally, we coded a new likelihood for a compilation of Redshift-Space-Distortions (RSD) measurements (see Appendix and Ref.","[42]).","Table: Flat prior bounds used in the MCMC analyses.","Prior range for other parameters is set as in Table 1 of Ref.","The cosmological models previously discussed in Section were implemented in our Boltzmann solver EFCLASS.","For a given cosmological model and a set of cosmological parameters we can compute the solution for both background and linear perturbations, that is, we can predict observables such as the matter power spectrum and the CMB angular power spectrum.","Since the parameter space not only includes cosmological parameters, but also several nuissance parameters, it becomes hard to find the best fit model as well as the relevant statistical information.","The usual approach is to use Markov Chain Monte Carlo (MCMC) techniques [59], [60] and we will do so.","We explore the parameter space of the cosmological models with the code Montepython [61], [62] which works along with EFCLASS: theoretical predictions are computed and compared to observations through likelihood functions $\\sim 10^5$ times.","The MCMC procedure allows us not only to find the best fit model parameters, but also to obtain the relevant countours confidence.","In our analysis we use the set of flat priors in Table REF .","In Fig.","REF we show the $68\\%$ and $95\\%$ confidence contours for the DES-fR model.","Vertical dashed and horizontal dotted lines indicate the values obtained by the Planck Collaboration in their analyses for the standard cosmological model $\\Lambda $ CDM (last column in Table 2 of Ref.","[3]).","The relevant statistical information (i.e., mean values and $68\\%$ confidence limits) is shown in Table REF .","We see there is good agreement for common parameters in both DES-fR and $\\Lambda $ CDM models.","Although error on neutrino masses get significantly reduced as we add more data, we can only set an upper limit when combining all data sets.","In the case of the MG parameter $b_\\pi $ we do not observe any degeneracy with other parameters in the model.","It is interesting that the constraints on MG, although still prior dominated, present different tendencies according to the combination of data sets: i) RSD push the MG constraints towards GR, while the $H_0$ tension remains unresolved; ii) if we exclude RSD from the data sets, we notice a preference for a MG scenario, but still hitting the prior bound on the right and not solving the problem with $H_0$ ; iii) a similar situation occurs when we exclude supernovae, $H_0$ , and RSD from the data sets, because there is a preference for MG (prior dominated though) while obtaining a $H_0$ value that agrees very well with Planck Collaboration results for $\\Lambda $ CDM .","Finally, we note that our derived value for the parameter $\\sigma _8 = 0.815^{+0.009}_{-0.007} \\qquad (68\\%),$ when including the whole data set, agrees very well with the value found by DES Collaboration $\\sigma _8 = 0.807^{+0.062}_{-0.041}$ for the $\\Lambda $ CDM model [63].","Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\% and 95%95\\%) for the DES-fR model.","The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\Lambda CDM parameters (last column in Table 2 of Ref.",").","TTTEEE \\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\rm {lensing} stands for CMB lensing, BAO \\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\rm {SNe} stands for supernovae, H0\\rm {H0} stands for the Hubble constant, and RSD \\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\% confidence limits on cosmological parameters for the DES-fR model.","Here ⋯\\left\\lbrace \\dots \\right\\rbrace stands for the inclusion of data from column on the left."],["HS model","In Fig.","REF we depict $68\\%$ and $95\\%$ confidence contours for the HS model using a number of data sets.","Dashed-vertical and dotted-horizontal lines are the parameter values that the Planck Collaboration reported for its analysis using $\\Lambda $ CDM model (last column in Table 2 of Ref.","[3]).","Table REF contains relevant statistical information for our analysis: we show mean values and $68\\%$ limits for the HS model.","Again, cosmological parameters which are common to both $\\Lambda $ CDM and HS models are in good agreement with Planck Collaboration's results.","As in the case for the DES-fR model we can only find an upper limit for the neutrino masses which is slightly smaller for the HS model.","Also in this case MG constraints are prior dominated and we observe a preference for departure from GR in most probe combination, the exception being the case including RSD.","The latter again goes towards GR while not solving the $H_0$ discrepancy with the local value.","Interestingly, in Ref.","[64] the authors analyzed galaxy morphology and placed the following constraint for the HS model $f_{R0}<1.4\\times 10^{-8}$ .","By using the whole data set we find $\\sigma _8 = 0.816^{+0.008}_{-0.007} \\qquad (68\\%),$ which perfectly agrees with the value found for the DES-fR model.","Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\% and 95%95\\%) for the HS model.","The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\Lambda CDM parameters (last column in Table 2 of Ref.",").","TTTEEE \\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\rm {lensing} stands for CMB lensing, BAO \\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\rm {SNe} stands for supernovae, H0\\rm {H0} stands for the Hubble constant, and RSD \\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\% confidence limits on cosmological parameters for the HS model.","Here ⋯\\left\\lbrace \\dots \\right\\rbrace stands for the inclusion of data from column on the left."],["HDES model","Fig.","REF shows confidence contours for the cosmological parameters in the HDES model.","We see good agreement in parameters that also play a part in $\\Lambda $ CDM model; the values found by the Planck Collaboration (last column in Table 2 of Ref.","[3]) are depicted as vertical-dashed and horizontal-dotted lines in Fig.","REF .","As for the DES-fR and HS models, in this case the neutrino masses remain unconstrained in our analysis and we can only set an upper limit.","Concerning the MG parameter we observe that results are not decisive since posteriors are mostly affected by the prior distribution.","Although there exist preference for departure from GR when including $H_0$ and RSD in the data set, the constraints hit the prior bound on the right.","Interesting in this case RSD push the constraints far from GR, whereas in the case of DES-fR and HS models the whole data set prefer the GR limit.","Finally we note that our derived $\\sigma _8 = 0.814^{+0.009}_{-0.007} \\qquad (68\\%),$ taking into consideration the full data set agrees well with values found for DES-fR and HS models.","In Table REF we show mean values and $68\\%$ confidence bounds for the cosmological parameters in the HDES model.","Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\% and 95%95\\%) for the HDES model.","The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\Lambda CDM parameters (last column in Table 2 of Ref.",").","TTTEEE \\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\rm {lensing} stands for CMB lensing, BAO \\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\rm {SNe} stands for supernovae, H0\\rm {H0} stands for the Hubble constant, and RSD \\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\% confidence limits on cosmological parameters for the HDES model.","Here ⋯\\left\\lbrace \\dots \\right\\rbrace stands for the inclusion of data from column on the left."],["Conclusions","Over the past decades several cosmological models have emerged as a plausible explanation for the late-time accelerating expansion of the Universe.","In this paper we investigated three MG models which satisfy solar system tests and also fulfil constraints on the speed of propagation of GWs, namely: DES-fR, HS, and the HDES models.","It is possible to interpret MG models as an effective fluid and we followed this approach in this work.","We implemented DES-fR, HS, and HDES models in the Boltzmann solver EFCLASS which uses sub-horizon and quasi-static approximations when solving the perturbation equations.","We showed in previous works that the observables are accurately computed (i.e., better than $0.1\\%$ as compared to outputs from codes which do not use any approximation) while having the advantage of analytical expressions describing MG as an effective fluid.","When constraining the parameter space for the HS model is usual to assume a $\\Lambda $ CDM background.","This is however incorrect as the background for the HS model is in general different from the $\\Lambda $ CDM model.","In this paper we dropped this assumption and solved the perturbations equations taking into consideration the background evolution too.","We found constraints which are in good agreement with results by the Planck Collaboration when the parameter spaces overlap.","We also note that the constraints on the MG parameter are dominated by the prior hence unconstrained by current data sets.","As the HS model has an additional parameter than the $\\Lambda $ CDM model, the former will be severely penalized in any Bayesian model comparison.","Since data indicate a preference for the standard model it is interesting to study models which exactly match the $\\Lambda $ CDM background.","These models might rely on new physics while also behaving differently at the perturbations level with respect to the $\\Lambda $ CDM model.","By investigating these kinds of models we can also reveal whether or not current data sets can discriminate alternative models from the concordance model.","In this paper we investigated two models, namely, a designer $f(R)$ (DES-fR) and a designer Horndeski model (HDES).","When considering common cosmological parameters, constraints for the DES-fR model do not exhibit significant discrepancies with results by the Planck Collaboration for the standard model.","Concerning the MG parameter we note the results depend on the probe combination.","Most cases are dominated by the prior and hence unconstrained.","The full data set however, prefers the GR limit.","One reason for this might be the strong constraints from the RSD likelihood.","As the surveys that make the RSD measurements assume a $\\Lambda $ CDM model in their analysis, the data themselves maybe a bit biased.","While this in general can be corrected, up to a point, with the AP correction as mentioned in the Appendix, some residual bias may remain.","While this is an important point, it is however outside the scope of our present analysis, thus we leave it for future work.","Regarding the constraints for the HDES model we also find good agreement with parameters also appearing in the $\\Lambda $ CDM model.","Here constraints on MG are also inconclusive as the posteriors are prior dominated.","Interestingly, in this case the full data set shows a slight preference for a departure from GR.","In summary, our results do not conclusively indicate the presence of modifications to GR.","Since our MG constraints are prior dominated we conclude $\\Lambda $ CDM is still the preferred model.","Where data sets overlap, our results fully agree with the investigation carried out by the Planck Collaboration [17]."],["Acknowledgements","The authors would like to thank M. Martinelli for useful discussions.","W.C. acknowledges financial support from the Departamento Administrativo de Ciencia, Tecnología e Innovación (COLCIENCIAS) under the project “Discriminación de modelos de energía oscura y gravedad modificada con futuros datos de cartografiado galáctico\" and from Universidad del Valle under the contract 449-2019.","S.N.","and R.A. acknowledge support from the Research Projects PGC2018-094773-B-C32 and the Centro de Excelencia Severo Ochoa Program SEV-2016-0597.","S.N.","also acknowledges support from the Ramón y Cajal program through Grant No.","RYC-2014-15843.","The calculations for this article were carried out on the Datacenter CIBioFi.","The statistical analyses as well as the plots were made with the Python package GetDist https://github.com/cmbant/getdist.","The numerical codes used by the authors in the analysis of the paper and our modifications to the CLASS code, which we call EFCLASS, can be found on the websites of the EFCLASS here, here and here.","The publicly available RSD Montepython likelihood for the growth-rate $f\\sigma _8$ data set can be found at https://github.com/snesseris/RSD-growth."],["The RSD likelihood ","In this appendix we will present some details on the growth RSD likelihood implemented in MontePython and used to analyse the growth $f\\sigma _8$ data of the “Gold 2018\" compilation, namely, the $N=22$ data points of Ref. [43].","This likelihood was first presented in Ref.","[42], but we summarize here again some of the key details for completeness.","The growth data we use in our analysis are based on measurements of the RSD, which is a direct probe of the LSS.","In essence, these data directly measure $f\\sigma _8(a)\\equiv f(a)\\cdot \\sigma (a)$ , where $f(a) \\equiv \\frac{d ln\\delta }{d lna}$ is the growth rate and $\\sigma (a) \\equiv \\sigma _{8,0}\\frac{\\delta (a)}{\\delta (1)}$ are the redshift-dependent rms fluctuations of the linear density field within spheres of radius $R=8$ Mpc $h^{-1}$ .","Note that by $\\sigma _{8,0}$ we denote the present value of this parameter.","This dataset has been shown to be internally robust and unbiased by the authors of Ref.","[43] that used the “robustness” criterion of Ref. [65].","The latter uses combinations of subsets in the data to perform a Bayesian analysis and establish the dataset's overall consistency.","The value of the parameter $f\\sigma _8(a)$ can be measured directly by using the ratio of the monopole to the quadrupole of the redshift-space power spectrum.","Thus, one can show with linear theory that $f\\sigma _8(a)$ does not depend on the bias $b(k,z)$ , which maybe both scale and redshift dependent, as it drops from the equations due to the particular combination of variables [66], [67], [68].","Furthermore, $f\\sigma _8(a)$ has been show to be a good discriminator of DE models [67].","The RSD data points are given explicitly in Ref.","[43] as $ f\\sigma _8^\\textrm {obs,i}=\\Big (f\\sigma _8^\\textrm {obs}(z_1),\\dots , f\\sigma _8^\\textrm {obs}(z_n)\\Big )$ .","It should be noted that a few of the growth points are correlated, while most also require a fiducial cosmology that has to be adjusted for the Alcock-Paczynski effect (see Refs.","[43], [69], [70] as well as Refs.","[71], [72], [73] for earlier analyses).","The data points that are correlated are the four points from SDSS [74] and the three WiggleZ points from Ref. [75].","Their covariance matrices are given by $\\mathbf {C}_{\\text{WiggleZ}}= 10^{-3}\\left(\\begin{array}{ccc}6.400 & 2.570 & 0.000 \\\\2.570 & 3.969 & 2.540 \\\\0.000 & 2.540 & 5.184 \\\\\\end{array}\\right),$ for the WiggleZ data and for the SDSS points by $\\mathbf {C}_{\\text{SDSS-IV}}= 10^{-2}\\left(\\begin{array}{cccc}3.098 & 0.892 & 0.329 & -0.021\\\\0.892 & 0.980 & 0.436 & 0.076\\\\0.329 & 0.436 & 0.490 & 0.350 \\\\-0.021 & 0.076 & 0.350 & 1.124\\end{array}\\right).$ In order to perform the correction for the Alcock-Paczynski effect, we follow the prescription of Ref.","[69], which requires the use of a correction factor given by $\\text{fac}(z^i)= \\frac{H(z^i)\\,d_A(z^i)}{H^{\\text{ref},i}(z^i) \\, d_A^{\\text{ref},i}(z^i)}\\;,$ where the label “$\\text{ref},i$ ” stands for the fiducial cosmology at the redshift $z^i$ .","Then, the corrected data will be given by [76] $f\\sigma _8^{\\textrm {th,i}}\\rightarrow \\frac{f\\sigma _8^{\\textrm {th,i}}}{\\text{fac}(z^i)}\\;.$ By defining the vector $\\mathbf {V}$ for the data via: $\\mathbf {V} = \\mathbf {f\\sigma _8^\\textrm {obs,i}} - \\frac{f\\sigma _8^{\\textrm {th,i}}}{\\text{fac}(z^i)},$ the chi-squared is then given by $\\chi ^2= \\mathbf {V}^T \\mathbf {C}^{-1} \\mathbf {V}\\;.$ To conclude, we also need the theoretical prediction for the growth $\\delta (k,z)$ at each redshift, which in CLASS can be estimated from the matter power spectrum as $\\delta (k,z)=\\sqrt{\\frac{P(k,z)}{P(k,0)}}$ , where we can obtain $P(k,z)$ via the function cosmo.pk(k,z).","Finally, the exact value of the growth-rate $f\\sigma _8(k,z)$ can be calculated numerically with direct differentiation and cubic interpolations."]],"string":"[\n [\n \"Cosmological constraints with the Effective Fluid approach for Modified\\n Gravity\"\n ],\n [\n \"Abstract Cosmological constraints of Modified Gravity (MG) models are seldom carried out rigorously.\",\n \"First, even though general MG models evolve differently (i.e., background and perturbations) to the standard cosmological model, it is usual to assume a $\\\\Lambda$CDM background.\",\n \"This treatment is not correct and in the era of precision cosmology could induce undesired biases in cosmological parameters.\",\n \"Second, neutrino mass is usually held fixed in the analyses which could obscure its relation to MG parameters.\",\n \"In a couple of previous papers we showed that by using the Effective Fluid Approach we can accurately compute observables in fairly general MG models.\",\n \"An appealing advantage of our approach is that it allows a pretty easy implementation of this kinds of models in Boltzmann solvers (i.e., less error--prone) while having a useful analytical description of the effective fluid to understand the underlying physics.\",\n \"This paper illustrates how an effective fluid approach can be used to carry out proper analyses of cosmological constraints in MG models.\",\n \"We investigated three MG models including the sum of neutrino masses as a varying parameter in our Markov Chain Monte Carlo analyses.\",\n \"Two models (i.e., Designer $f(R)$ [DES-fR] and Designer Horndeski [HDES]) have a background matching $\\\\Lambda$CDM, while in a third model (i.e., Hu $\\\\&$ Sawicki $f(R)$ model [HS]) the background differs from the standard model.\",\n \"In this way we estimate how relevant the background is when constraining MG parameters along with neutrinos' masses.\",\n \"We implement the models in the popular Boltzmann solver CLASS and use recent, available data (i.e., Planck 2018, CMB lensing, BAO, SNIa Pantheon compilation, $H_0$ from SHOES, and RSD Gold-18 compilation) to compute tight cosmological constraints in the MG parameters that account for deviation from the $\\\\Lambda$CDM model.\",\n \"[abridged]\"\n ],\n [\n \"Introduction \",\n \"The growing evidence for the late-time accelerating expansion of the Universe represents a major milestone in cosmology [1], [2], [3], [4] and investigations using Machine Learning algorithms have confirmed this fact through model independent analyses [5], [6], [7].\",\n \"Bayesian analyses indicate that the standard model of cosmology $\\\\Lambda $ CDM is preferred over its alternatives due to its simplicity and lower number of free parameters [8].\",\n \"The concordance model is relatively simple and having just six free parameters is able to make predictions which agree remarkably well with most observations [3], [4], [9], [10].\",\n \"Nevertheless, the cosmological constant problem, our current ignorance on the nature of Dark Matter (DM) as well as a worrying discordance in a few cosmological parameters represent big disadvantages for the $\\\\Lambda $ CDM model.\",\n \"Over time the disagreement between the Hubble constant determined via distance ladder and the $H_0$ value obtained through analyses of the CMB has become more interesting [11].\",\n \"Although the discrepancy could be due to unaccounted-for systematic errors, there also exists the more appealing possibility of new physics (e.g.\",\n \"Modified Gravity, early Dark Energy).\",\n \"With the coming of latest data analyses disagreements on the values of $\\\\Omega _m$ and $\\\\sigma _8$ also became apparent [12] or for example, a $\\\\sim 4\\\\sigma $ deviation of the dark energy equation of state $w(z)$ from the $\\\\Lambda $ CDM model using quasars at high redshift up to $z\\\\sim 7.5$ [13].\",\n \"The curvature of the Universe has also given rise to a discussion on recent analyses [14], [3], [15], [16].\",\n \"There are some mild hints ($\\\\approx 2\\\\sigma $ ) of Modified Gravity (MG) [3], [17], which challenge assumptions made in the standard model.\",\n \"Here we will investigate some viable MG models in light of latest data releases.\",\n \"Dark Energy (DE) and MG have emerged as alternatives to the cosmological constant model.\",\n \"In the case of DE, some of the simplest models include minimally coupled scalar fields in the form of quintessence, that has a standard kinetic term, or k-essence which has a generalized kinetic-term [18].\",\n \"On the other hand, MG models are covariant modifications to General Relativity (GR) that extend the Einstein-Hilbert action in various ways: either by promoting it to a function as in the $f(R)$ models and by introducing higher order curvature invariants (see for example [19]) or by adding non-minimally coupled scalar fields, as in the case of Horndeski theory [20].\",\n \"MG models have the advantage that they are also inspired from high-energy physics, as covariant modifications to GR of a similar form appear naturally when one tries to renormalize GR at one loop order [21].\",\n \"The advantages of these alternatives, especially of the MG models, are clear: there is no need for a cosmological constant as the accelerating expansion of the Universe can be explained by the presence of the extra terms in the modified Friedmann equations due to the new degrees of freedom.\",\n \"Both DE and MG models are also able to describe well the cosmological observations and give equally good fits to the data as the $\\\\Lambda $ CDM model.\",\n \"On the other hand, there are also some disadvantages for these models, for example the presence of the additional parameters may penalize some of the models when one calculates the Bayesian evidence and uses the Jeffreys' scale, albeit it has been shown that the latter has to be interpreted with care [22].\",\n \"Furthermore, by using N-body simulations it has been shown that a compelling $f(R)$ model fails to reproduce the observed redshift-space clustering on scales $\\\\sim 1-10 ~ \\\\mathrm {Mpc}~ h^{-1}$ [23].\",\n \"Finally, as both the extra scalar field degrees of freedom and the higher order corrections to GR are as yet unobserved in a laboratory or in an astrophysical setting, their motivation is obviously somewhat weakened.\",\n \"Not only that, but recently several MG models of the Horndeski type have been ruled out, via the measurement of the speed of propagation of the gravitational waves by the event GW170817 and its optical counterpart GRB170817A [24], [25], [26].\",\n \"Thus, as the available parameter space has shrank remarkably, there are a few remaining models which deserve attention as well as proper analyses.\",\n \"However, many analyses of the remaining models, especially the ones where the background expansion differs significantly from the $\\\\Lambda $ CDM model, do not consider the background expansion properly and just fix it to either the $\\\\Lambda $ CDM or a constant $w$ model, as was observed in Refs.\",\n \"[27], [28].\",\n \"This obviously biases the results as it introduces biases in the cosmological parameters and spurious tensions with the data.\",\n \"However, some recent analyses have also acknowledged this discrepancy and newer versions of the Boltzmann solvers now have support for the correct backgrounds in some cases [29].\",\n \"On the other hand, the so-called Effective Fluid Approach has the advantage that it presents a unified approach to analyse all models under the same umbrella, allows for the correct background expansion in the models, all without sacrificing the accuracy of the results [27], [28], [30].\",\n \"In a nutshell, the Effective Fluid Approach works by rewriting the field equations of the MG model as GR and a DE fluid with an equation of state $w$ , a pressure perturbation $\\\\delta P$ and an anisotropic stress $\\\\sigma $ .\",\n \"Especially the latter is crucial as sometimes it is ignored in analyses of MG models [31], something which might bias the results [32].\",\n \"Moreover, through a joint Machine Learning analysis applied to the latest cosmological data hints of dark energy anisotropic stress were found [7].\",\n \"In the Effective Fluid Approach we also assume that in the relevant scales, where linear theory applies, the sub-horizon and quasi-static approximations hold.\",\n \"With these, general analytical expressions for the equation of state, the pressure perturbation, and the anisotropic stress were found in Refs.\",\n \"[27], [28].\",\n \"With the latter, one may then just solve numerically the evolution equations for the perturbations, found for example in Ref. [33].\",\n \"Thus, the main advantage of the Effective fluid approach is that once one has the expressions for the variables $w$ , $\\\\delta P$ and $\\\\sigma $ , it is very straightforward to also implement them in standard Boltzmann codes, such as CLASS, with very minimal modifications.\",\n \"In fact, in [27], [28], this was done with the EFCLASS code, which implements the aforementioned approach, where it was found that EFCLASS and hi-CLASS [34], a modification of CLASS that solves numerically the whole set of perturbation equations for Horndeski, agree to better than $0.1\\\\%$ [28].\",\n \"Recently, a comparison of different approaches to the quasi-static approximation in Horndeski models was made by Ref.\",\n \"[35], by applying this approximation to either the field equations, as done in the Effective fluid approach, or the equations for the two metric potentials $\\\\Phi $ and $\\\\Psi $ and finally, the use of the attractor solution derived within the Equation of State approach [36].\",\n \"It was found that all three approaches agree exactly on small scales and that in general, this approach is valuable in future model selections analyses for models beyond the $\\\\Lambda $ CDM model.\",\n \"In this analysis we use the Effective Fluid approach and our EFCLASS code, for the background and first order perturbations, to obtain cosmological constraints with the latest cosmological data sets: we include the Pantheon SNe compilation [37], the Planck 2018 CMB data [3], the $H_0$ Riess measurement [38], various BAO points [39], [40], [41], and a new redshift space distortions (RSD) likelihood (see Ref.\",\n \"[42] for the “Gold 2018\\\" compilation of Ref. [43]).\",\n \"The paper is organised as follows.\",\n \"In Sec.\",\n \"we briefly summarize our theoretical framework and present the models we consider, in Sec.\",\n \"we present the results of our MCMC analysis with EFCLASS, in Sec.\",\n \"we conclude and lastly, in Appendix we present some details on our RSD likelihood.\"\n ],\n [\n \"Theoretical framework \",\n \"The standard cosmological model $\\\\Lambda $ CDM assumes the Einstein-Hilbert action $\\\\mathcal {L} = \\\\frac{1}{2\\\\kappa }f(R) + \\\\mathcal {L}_m$ with $f(R)=R$ , $R$ the Ricci scalar, $\\\\mathcal {L}_m$ the Lagrangian for matter fields, and the constant $\\\\kappa \\\\equiv 8\\\\pi G_N$ with $G_N$ being the bare Newton's constant.\",\n \"We can derive the field equations by applying the Principle of Least Action; they read $G_{\\\\mu \\\\nu } = \\\\kappa T^{(m)}_{\\\\mu \\\\nu }$ where $G_{\\\\mu \\\\nu }\\\\equiv R_{\\\\mu \\\\nu }-\\\\frac{1}{2}g_{\\\\mu \\\\nu }R$ is the Einstein tensor, $R_{\\\\mu \\\\nu }$ is the Ricci tensor, $g_{\\\\mu \\\\nu }$ is the metric, and $T^{(m)}_{\\\\mu \\\\nu }$ is the energy-momentum tensor of matter fields.Throughout this paper our conventions are: $(-+++)$ for the metric signature, the Riemann and Ricci tensors are given respectively by $V_{b;cd}-V_{b;dc}=V_{a}R^{a}_{\\\\;bcd}$ and $R_{ab}=R^{s}_{\\\\;asb}$ .\",\n \"It is also possible to consider more general theories (e.g., $f(R)\\\\ne R$ , Horndeski theories) and derive similar field equations.\",\n \"As we clearly explained in Refs.\",\n \"[27], [28], modifications to GR in these kinds of MG models can be interpreted as an effective fluid yielding field equations which schematically look like $G_{\\\\mu \\\\nu }&=&\\\\kappa \\\\left(T_{\\\\mu \\\\nu }^{(m)}+T_{\\\\mu \\\\nu }^{(\\\\textrm {DE})}\\\\right).$ Here the tensor $T_{\\\\mu \\\\nu }^{(\\\\textrm {DE})}$ depends on the metric and its derivatives, and in the case of general scalar-tensor theories, also on the scalar field and its derivatives (see [28]).\",\n \"Observational evidence indicates that the Universe is statistically homogeneous and isotropic on large scales [44], [45], [46], [47].\",\n \"Therefore we will as usual assume a flat linearly perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) metric $ds^2=a(\\\\tau )^2\\\\left[-(1+2\\\\Psi (\\\\vec{x},\\\\tau ))d\\\\tau ^2+(1-2\\\\Phi (\\\\vec{x},\\\\tau ))d\\\\vec{x}^2\\\\right],$ where $a$ is the scale factor, $\\\\vec{x}$ represents spatial coordinates, $\\\\tau $ is the conformal time, and $\\\\Psi $ and $\\\\Phi $ are the gravitational potentials in the Newtonian gauge.\",\n \"In order to describe matter fields, we will consider them as ideal fluids having small perturbations.\",\n \"We will take into consideration an Effective Fluid Approach (EFA) when dealing with MG models: modifications to GR will be described by an effective fluid having equation of state, pressure perturbation, and anisotropic stress.\",\n \"In our approach, and assuming a metric (REF ), the background evolution is governed by the Friedmann equations $\\\\mathcal {H}^2&=&\\\\frac{\\\\kappa }{3}a^2 \\\\left(\\\\bar{\\\\rho }_{m}+\\\\bar{\\\\rho }_{\\\\textrm {DE}}\\\\right), \\\\\\\\\\\\dot{\\\\mathcal {H}}&=&-\\\\frac{\\\\kappa }{6}a^2 \\\\left(\\\\left(\\\\bar{\\\\rho }_{m}+3\\\\bar{P}_{m}\\\\right)+\\\\left(\\\\bar{\\\\rho }_{\\\\textrm {DE}}+3\\\\bar{P}_{\\\\textrm {DE}}\\\\right)\\\\right),$ where $\\\\mathcal {H}\\\\equiv \\\\dot{a}/a$ is the conformal Hubble parameter.In our notation a dot over a function denote derivative respect to the conformal time $\\\\dot{f}=df/d\\\\tau $ .\",\n \"The Hubble parameter $H$ and the conformal Hubble parameter $H$ are related via $\\\\mathcal {H}=aH$ .\",\n \"Here $\\\\bar{\\\\rho }_{\\\\textrm {DE}}$ and $\\\\bar{P}_{\\\\textrm {DE}}$ respectively denote density and pressure of the DE effective fluid.\",\n \"The effective DE equation of state $w_{\\\\textrm {DE}} \\\\equiv \\\\bar{P}_{\\\\textrm {DE}}/\\\\bar{\\\\rho }_{\\\\textrm {DE}}$ allows us to describe the background evolution in these kinds of models.\",\n \"Another ingredient that we need is the evolution equations for the perturbations obtained through the energy-momentum conservation $T^{\\\\mu \\\\nu }_{;\\\\nu }=0$ [27]: $\\\\delta ^{\\\\prime } &=& 3(1+w) \\\\Phi ^{\\\\prime }-\\\\frac{V}{a^2 H}-\\\\frac{3}{a}\\\\left(\\\\frac{\\\\delta P}{\\\\bar{\\\\rho }}-w\\\\delta \\\\right),$ $V^{\\\\prime } &=& -(1-3w)\\\\frac{V}{a}+\\\\frac{k^2}{a^2 H}\\\\frac{\\\\delta P}{\\\\bar{\\\\rho }} +(1+w)\\\\frac{k^2}{a^2 H} \\\\Psi \\\\nonumber \\\\\\\\&-&\\\\frac{2}{3} \\\\frac{k^2}{a^2 H} \\\\pi ,$ where $V$ is the scalar velocity perturbation, $\\\\delta P$ is the pressure perturbation, $\\\\pi $ is related to the anisotropic stress as $\\\\pi \\\\equiv \\\\frac{3}{2}(1+w)\\\\sigma $ and the prime $^{\\\\prime }$ is the derivative with respect to the scale factor $a$ .\",\n \"In [28] we extensively used the EFA on the remaining Horndeski Lagrangian.\",\n \"By applying both sub-horizon and quasi-static approximations, we managed to find analytical expressions for several quantities describing Horndeski models as an effective DE fluid, namely, DE pressure perturbation $\\\\frac{\\\\delta P_{\\\\textrm {DE}}}{\\\\overline{\\\\rho }_{DE}}(a,k) \\\\nonumber ,$ DE scalar velocity perturbation $V_{\\\\textrm {DE}}(a,k) \\\\nonumber ,$ and DE anisotropic stress $\\\\pi _{\\\\textrm {DE}}(a,k) \\\\nonumber .$ Having mapped remaining Horndeski models into an effective DE fluid, one can easily implement them in Boltzmann codes which compute observables such as the CMB angular power spectrum and the matter power spectrum.\",\n \"In the following subsections we will provide a few details on the specific models that we study in this work.\"\n ],\n [\n \"DES-fR model\",\n \"The remaining Horndeski Lagrangian includes a very important sort of MG models, namely, $f(R)$ .\",\n \"In [27] we showed there is a correspondence between a given $f(R)$ model and its effective DE equation of state $w_{DE}$ .\",\n \"Then it becomes clear that by specifying a background (i.e., an equation of state), it is in principle possible to obtain a corresponding $f(R)$ .\",\n \"The models obtained in this way are the so-called designer f(R) models [48], [49], [50].\",\n \"An interesting case is the $w_{DE}=-1$ designer $f(R)$ model (DES-fR, henceforth) that mimics the standard $\\\\Lambda $ CDM model at the background, while exhibiting differences in the evolution of the linear perturbations.\",\n \"The DES-fR model satisfying all viability conditions (see, for instance, [51]) is given by [50] $f(R)&=&R-2\\\\Lambda +\\\\alpha ~H_0^2\\\\left(\\\\frac{\\\\Lambda }{R-3 \\\\Lambda }\\\\right)^{c_{0}} \\\\times \\\\nonumber \\\\\\\\& & {}_2F_1\\\\left(c_{0},\\\\frac{3}{2}+c_{0},\\\\frac{13}{6}+2c_{0},\\\\frac{\\\\Lambda }{R-3 \\\\Lambda }\\\\right)\\\\;,$ where $c_{0}=\\\\frac{1}{12} \\\\left(-7+\\\\sqrt{73}\\\\right)$ , $\\\\alpha $ is a free dimensionless parameter, $H_0$ is the Hubble constant, $\\\\Lambda $ is a constant, and ${}_2F_1$ is a hypergeometric function.\",\n \"In the literature it is common to define $F\\\\equiv f^{\\\\prime }(R)$ and we will follow this convention.\",\n \"Instead of using $\\\\alpha $ in Eq.\",\n \"(REF ) we can parametrise our expressions in terms of $b_\\\\pi = f_{R,0} \\\\equiv F(a=1)-1$ .\",\n \"Furthermore, in Ref.\",\n \"[27] we found that in the range $a\\\\in [0,1]$ the following approximation around $a\\\\simeq 0$ is equally accurate $&F(a)&\\\\simeq 1 + \\\\nonumber \\\\\\\\& & f_{R,0}\\\\frac{\\\\Omega _{m0}^{-c_0-1}}{\\\\, _2F_1\\\\left(c_0+1,c_0+\\\\frac{3}{2};2 c_0+\\\\frac{13}{6};1-\\\\Omega _{m0}\\\\right)}a^{3 (1 + c_0)} \\\\nonumber \\\\\\\\&+&\\\\mathcal {O}(a^{3(2+c_0)}) , $ Then, when we rewrite our expressions for the DE effective fluid in terms of $f_{R,0}$ we find that they depend on $g(x) = {}_2F_1(\\\\frac{3}{2}+c_0,2+c_0;2c_0+\\\\frac{13}{6},x),$ where $x \\\\equiv \\\\frac{a^3(\\\\Omega _{m,0}-1)}{a^3(\\\\Omega _{m,0}-1)-\\\\Omega _{m,0}}.$ Implementing special functions in EFCLASS is not an easy task.\",\n \"Therefore, we used a Taylor approximation for $g(x)$ in Eq.\",\n \"(REF ) which works really well around $x=0$ while keeping 30 terms in the expansion.\"\n ],\n [\n \"HS model\",\n \"The popular Hu $\\\\&$ Sawicki model (HS, henceforward) [52] $f(R)=R-m^2 \\\\frac{c_1 (R/m^2)^n}{1+c_2 (R/m^2)^n},$ can actually be rewritten, after some algebraic manipulations, as [53] $f(R)= R- \\\\frac{2\\\\Lambda }{1+\\\\left(\\\\frac{b_{hs} \\\\Lambda }{R}\\\\right)^n},$ where $\\\\Lambda = \\\\frac{m^2 c_1}{2c_2}$ and $b_{hs}=\\\\frac{2 c_2^{1-1/n}}{c_1}$ .\",\n \"In [53] the authors found that when written in the form (REF ), it is clear the reason why the HS model satisfies solar system tests: if $b_{hs} \\\\rightarrow 0$ $\\\\Lambda $ CDM is recovered, and if $b_{hs} \\\\rightarrow \\\\infty $ a matter dominated universe is obtained i.e., $\\\\lim _{b_{hs}\\\\rightarrow 0}f(R)&=&R-2\\\\Lambda , \\\\nonumber \\\\\\\\\\\\lim _{b_{hs}\\\\rightarrow \\\\infty }f(R)&=&R.$ If the parameter $b_{hs}$ is small enough, the HS model can be regarded as a “perturbation” around $\\\\Lambda $ CDM .\",\n \"This is key in our work since we do not assume the usual approximation of fixing the background to $\\\\Lambda $ CDM when investigating the HS model.\",\n \"However, solving numerically the equation for the Hubble parameter is not trivial in the HS model and we worked out a different approach.\",\n \"Instead of approximating the background, we solve the field equations and find an approximate, accurate analytical expression for the Hubble parameter.\",\n \"We followed the treatment in [53], where it was shown that HS Hubble parameter can be written as $H_{HS}(a)^2 = H_{\\\\Lambda }(a)^2 + b_{hs} \\\\, \\\\delta H_1(a)^2 + b_{hs}^2\\\\, \\\\delta H_2(a)^2 + \\\\hdots ,~~$ which is an analytical approximation that works extremely well, for example for $b_{hs}\\\\le 0.1$ the average error with respect to the numerical solution is $10^{-5}\\\\%$ for redshifts $z\\\\le 30$ .\"\n ],\n [\n \"HDES model\",\n \"In what follows, we will present a brief overview of a family of designer Horndeski (HDES) models that have been already studied in Ref. [28].\",\n \"These are models whose background is exactly that of the $\\\\Lambda $ CDM model but at the perturbation level it is dictated by the Horndeski theory.\",\n \"In its full form, Horndeski theory constitutes as the most general Lorentz-invariant extension of GR in four dimensions and contains a few DE and MG models.\",\n \"Due to the recent discovery of gravitational waves by the LIGO Collaboration the Horndeski Lagrangian has been severely reduced.\",\n \"In particular, it has been found the following constrain on the speed of GWs [25] $-3 \\\\cdot 10^{-15} \\\\le c_g/c-1 \\\\le 7 \\\\cdot 10^{-16},$ which implies that for Horndeski theories $G_{4X}\\\\approx 0, \\\\hspace{5.69054pt} G_5 \\\\approx \\\\text{constant}.$ Then, the surviving part of the Horndeski Lagrangian reads, $S[g_{\\\\mu \\\\nu }, \\\\phi ] = \\\\int d^{4}x\\\\sqrt{-g}\\\\left[\\\\sum ^{4}_{i=2} \\\\mathcal {L}_i\\\\left[g_{\\\\mu \\\\nu },\\\\phi \\\\right] + \\\\mathcal {L}_m \\\\right],$ where $\\\\mathcal {L}_2&=& G_2\\\\left(\\\\phi ,X\\\\right) \\\\equiv K\\\\left(\\\\phi ,X\\\\right),\\\\\\\\\\\\mathcal {L}_3&=&-G_3\\\\left(\\\\phi ,X\\\\right)\\\\Box \\\\phi ,\\\\\\\\\\\\mathcal {L}_4&=&G_4\\\\left(\\\\phi \\\\right) R,$ and $\\\\phi $ is a scalar field, $X \\\\equiv -\\\\frac{1}{2}\\\\partial _{\\\\mu }\\\\phi \\\\partial ^{\\\\mu }\\\\phi $ is a kinetic term, and $\\\\Box \\\\phi \\\\equiv g^{\\\\mu \\\\nu }\\\\nabla _{\\\\mu }\\\\nabla _{\\\\nu }\\\\phi $ ; $K$ , $G_3$ and $G_4$ are free functions of $\\\\phi $ and $X$ .\",\n \"From the action (REF ) one can find several theories, for example $f(R)$ theories [54], Brans-Dicke theories, [55] and Cubic Galileon [56].\",\n \"The HDES family of models that where constructed in Ref.\",\n \"[28] limit to the Kinetic Gravity Braiding (KGB) which is distinguished by the following functions $K=K(X), \\\\hspace{14.22636pt} G_3=G_3(X), \\\\hspace{14.22636pt} G_4=\\\\frac{1}{2 \\\\kappa }.$ With Eq.\",\n \"(REF ), we will present how to find a specific family of designer models such that $w_{\\\\textrm {DE}}=-1$ , i.e., the background is always that of the $\\\\Lambda $ CDM model but at the perturbation level it follows Horndeski's theory.\",\n \"The usefulness of this designer model comes from allowing one to detect deviations from $\\\\Lambda $ CDM at the perturbations level and is a natural expansion of our previous work [50], [27].\",\n \"For our HDES model we need two functions, the modified Friedmann equation and the scalar field conservation equation, both of which have been presented and analyzed in detail in [28].\",\n \"The modified Friedmann equation reads, $&-H(a)^2-\\\\frac{K(X)}{3}+H^2_0\\\\Omega _m(a)+\\\\nonumber \\\\\\\\&+2\\\\sqrt{2}X^{3/2}H(a)G_{3X}+\\\\frac{2}{3}X K_X=0,$ where $\\\\Omega _m(a)$ is the matter density and $H_0$ is the Hubble parameter.\",\n \"The scalar field conservation equation is $\\\\frac{J_c}{a^3}-6XH(a)G_{3X}-\\\\sqrt{2}\\\\sqrt{X}K_X=0.$ The constant $J_c$ quantifies our deviation from the attractor given the KGB model [57].\",\n \"By looking at Eqs.\",\n \"(REF ) and (REF ) we see that we have three unknown functions $(G_{3X}(X),K(X), H(a))$ hence the system is undetermined.\",\n \"Then, we need to describe one of the three unknown functions $(G_{3X}(X),K(X), H(a))$ and find out the other two using Eqs.\",\n \"(REF ) and (REF ).\",\n \"For convenience, we write the Hubble parameter as a function of the kinetic term $X$ , i.e., $H=H(X)$ and then solve the previous equations to find $(G_{3X}(X),K(X))$ .\",\n \"Then we find $K(X) &=& -3 H_0^2 \\\\Omega _{\\\\Lambda ,0}+\\\\frac{J_c \\\\sqrt{2X} H(X)^2}{H_0^2 \\\\Omega _{m,0}}-\\\\frac{J_c \\\\sqrt{2X} \\\\Omega _{\\\\Lambda ,0}}{\\\\Omega _{m,0}}, \\\\nonumber \\\\\\\\G_{3X}(X) &=& -\\\\frac{2 J_c H^{\\\\prime }(X)}{3 H_0^2 \\\\Omega _{m,0}},$ where $\\\\Omega _{m,0}$ is the matter density at redshift $z=0$ and $\\\\Omega _{\\\\Lambda ,0}=1-\\\\Omega _{m,0}$ .\",\n \"With Eqs.\",\n \"(REF ) we can create a whole family of designer models.\",\n \"A specific model found, dubbed HDES [28], that has a smooth limit to $\\\\Lambda $ CDM and also recovers GR when $J_c\\\\sim 0$ is the following.\",\n \"First, we demand that the kinetic term behaves as $X= \\\\frac{c_0}{H(a)^n}$ , where $c_0>0$ and $n>0$ .\",\n \"Then, from Eqs.\",\n \"(REF ) and (REF ) we have $G_{3}(X)&=&-\\\\frac{2 J_c c_0^{1/n} X^{-1/n}}{3 H_0^2 \\\\Omega _{m,0}},\\\\\\\\K(X)&=&\\\\frac{\\\\sqrt{2} J_c c_0^{2/n} X^{\\\\frac{1}{2}-\\\\frac{2}{n}}}{H_0^2 \\\\Omega _{m,0}}-3 H_0^2 \\\\Omega _{\\\\Lambda ,0}-\\\\frac{\\\\sqrt{2} J_c \\\\sqrt{X} \\\\Omega _{\\\\Lambda ,0}}{\\\\Omega _{m,0}}.\\\\nonumber $ A particular HDES model that we have selected for our MCMC runs and we also used it in our comparison with hi-CLASS previously [28] comes by setting $n=1$ in Eq.\",\n \"(REF ).\",\n \"In the effective fluid approach one needs to specify three functions: the equation of state, the pressure perturbation and the anisotropic stress $(w,\\\\delta P,\\\\sigma )$ to modify properly the background and the linear perturbations in the CLASS code.\",\n \"With the HDES model we only need to specify $\\\\delta P$ since $w=-1$ and $\\\\sigma =0$ for this particular model.\",\n \"As it was shown in [28], computing $\\\\delta P$ through the effective fluid approach and then introducing this function into the evolution equation for the perturbations Eqs.\",\n \"(REF ) and (REF ) we can specify the DE scalar velocity perturbation $V_{\\\\textrm {DE}}$ as $V_{\\\\textrm {DE}}\\\\simeq \\\\left(-\\\\frac{14 \\\\sqrt{2}}{3} \\\\Omega _{m,0}^{-3/4} J_c~ a^{1/4}\\\\right)\\\\frac{\\\\bar{\\\\rho }_m}{\\\\bar{\\\\rho }_{\\\\textrm {DE}}} \\\\delta _m.$ which is what we implement later in our modified version of CLASS.\"\n ],\n [\n \"Methodology\",\n \"In order to compute cosmological constraints we use the following data sets.\",\n \"Firstly, we utilise the 2018 release by the Planck Collaboration including temperature and polarisation anisotropies of the CMB (TTTEEE) as well as CMB lensing (lensing) [3].\",\n \"Secondly, we include measurements of Baryon Acoustic Oscillations (BAO) from Refs.\",\n \"[39], [40], [41].\",\n \"Thirdly, Pantheon supernovae (SNe) from [37] were also incorporated in the analysis.\",\n \"Fourthly, we employed local Hubble measurement (H0) from Ref.\",\n \"[38] as a Gaussian prior.\",\n \"Finally, we coded a new likelihood for a compilation of Redshift-Space-Distortions (RSD) measurements (see Appendix and Ref.\",\n \"[42]).\",\n \"Table: Flat prior bounds used in the MCMC analyses.\",\n \"Prior range for other parameters is set as in Table 1 of Ref.\",\n \"The cosmological models previously discussed in Section were implemented in our Boltzmann solver EFCLASS.\",\n \"For a given cosmological model and a set of cosmological parameters we can compute the solution for both background and linear perturbations, that is, we can predict observables such as the matter power spectrum and the CMB angular power spectrum.\",\n \"Since the parameter space not only includes cosmological parameters, but also several nuissance parameters, it becomes hard to find the best fit model as well as the relevant statistical information.\",\n \"The usual approach is to use Markov Chain Monte Carlo (MCMC) techniques [59], [60] and we will do so.\",\n \"We explore the parameter space of the cosmological models with the code Montepython [61], [62] which works along with EFCLASS: theoretical predictions are computed and compared to observations through likelihood functions $\\\\sim 10^5$ times.\",\n \"The MCMC procedure allows us not only to find the best fit model parameters, but also to obtain the relevant countours confidence.\",\n \"In our analysis we use the set of flat priors in Table REF .\",\n \"In Fig.\",\n \"REF we show the $68\\\\%$ and $95\\\\%$ confidence contours for the DES-fR model.\",\n \"Vertical dashed and horizontal dotted lines indicate the values obtained by the Planck Collaboration in their analyses for the standard cosmological model $\\\\Lambda $ CDM (last column in Table 2 of Ref.\",\n \"[3]).\",\n \"The relevant statistical information (i.e., mean values and $68\\\\%$ confidence limits) is shown in Table REF .\",\n \"We see there is good agreement for common parameters in both DES-fR and $\\\\Lambda $ CDM models.\",\n \"Although error on neutrino masses get significantly reduced as we add more data, we can only set an upper limit when combining all data sets.\",\n \"In the case of the MG parameter $b_\\\\pi $ we do not observe any degeneracy with other parameters in the model.\",\n \"It is interesting that the constraints on MG, although still prior dominated, present different tendencies according to the combination of data sets: i) RSD push the MG constraints towards GR, while the $H_0$ tension remains unresolved; ii) if we exclude RSD from the data sets, we notice a preference for a MG scenario, but still hitting the prior bound on the right and not solving the problem with $H_0$ ; iii) a similar situation occurs when we exclude supernovae, $H_0$ , and RSD from the data sets, because there is a preference for MG (prior dominated though) while obtaining a $H_0$ value that agrees very well with Planck Collaboration results for $\\\\Lambda $ CDM .\",\n \"Finally, we note that our derived value for the parameter $\\\\sigma _8 = 0.815^{+0.009}_{-0.007} \\\\qquad (68\\\\%),$ when including the whole data set, agrees very well with the value found by DES Collaboration $\\\\sigma _8 = 0.807^{+0.062}_{-0.041}$ for the $\\\\Lambda $ CDM model [63].\",\n \"Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\\\% and 95%95\\\\%) for the DES-fR model.\",\n \"The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\\\Lambda CDM parameters (last column in Table 2 of Ref.\",\n \").\",\n \"TTTEEE \\\\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\\\rm {lensing} stands for CMB lensing, BAO \\\\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\\\rm {SNe} stands for supernovae, H0\\\\rm {H0} stands for the Hubble constant, and RSD \\\\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\\\% confidence limits on cosmological parameters for the DES-fR model.\",\n \"Here ⋯\\\\left\\\\lbrace \\\\dots \\\\right\\\\rbrace stands for the inclusion of data from column on the left.\"\n ],\n [\n \"HS model\",\n \"In Fig.\",\n \"REF we depict $68\\\\%$ and $95\\\\%$ confidence contours for the HS model using a number of data sets.\",\n \"Dashed-vertical and dotted-horizontal lines are the parameter values that the Planck Collaboration reported for its analysis using $\\\\Lambda $ CDM model (last column in Table 2 of Ref.\",\n \"[3]).\",\n \"Table REF contains relevant statistical information for our analysis: we show mean values and $68\\\\%$ limits for the HS model.\",\n \"Again, cosmological parameters which are common to both $\\\\Lambda $ CDM and HS models are in good agreement with Planck Collaboration's results.\",\n \"As in the case for the DES-fR model we can only find an upper limit for the neutrino masses which is slightly smaller for the HS model.\",\n \"Also in this case MG constraints are prior dominated and we observe a preference for departure from GR in most probe combination, the exception being the case including RSD.\",\n \"The latter again goes towards GR while not solving the $H_0$ discrepancy with the local value.\",\n \"Interestingly, in Ref.\",\n \"[64] the authors analyzed galaxy morphology and placed the following constraint for the HS model $f_{R0}<1.4\\\\times 10^{-8}$ .\",\n \"By using the whole data set we find $\\\\sigma _8 = 0.816^{+0.008}_{-0.007} \\\\qquad (68\\\\%),$ which perfectly agrees with the value found for the DES-fR model.\",\n \"Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\\\% and 95%95\\\\%) for the HS model.\",\n \"The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\\\Lambda CDM parameters (last column in Table 2 of Ref.\",\n \").\",\n \"TTTEEE \\\\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\\\rm {lensing} stands for CMB lensing, BAO \\\\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\\\rm {SNe} stands for supernovae, H0\\\\rm {H0} stands for the Hubble constant, and RSD \\\\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\\\% confidence limits on cosmological parameters for the HS model.\",\n \"Here ⋯\\\\left\\\\lbrace \\\\dots \\\\right\\\\rbrace stands for the inclusion of data from column on the left.\"\n ],\n [\n \"HDES model\",\n \"Fig.\",\n \"REF shows confidence contours for the cosmological parameters in the HDES model.\",\n \"We see good agreement in parameters that also play a part in $\\\\Lambda $ CDM model; the values found by the Planck Collaboration (last column in Table 2 of Ref.\",\n \"[3]) are depicted as vertical-dashed and horizontal-dotted lines in Fig.\",\n \"REF .\",\n \"As for the DES-fR and HS models, in this case the neutrino masses remain unconstrained in our analysis and we can only set an upper limit.\",\n \"Concerning the MG parameter we observe that results are not decisive since posteriors are mostly affected by the prior distribution.\",\n \"Although there exist preference for departure from GR when including $H_0$ and RSD in the data set, the constraints hit the prior bound on the right.\",\n \"Interesting in this case RSD push the constraints far from GR, whereas in the case of DES-fR and HS models the whole data set prefer the GR limit.\",\n \"Finally we note that our derived $\\\\sigma _8 = 0.814^{+0.009}_{-0.007} \\\\qquad (68\\\\%),$ taking into consideration the full data set agrees well with values found for DES-fR and HS models.\",\n \"In Table REF we show mean values and $68\\\\%$ confidence bounds for the cosmological parameters in the HDES model.\",\n \"Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\\\% and 95%95\\\\%) for the HDES model.\",\n \"The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\\\Lambda CDM parameters (last column in Table 2 of Ref.\",\n \").\",\n \"TTTEEE \\\\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\\\rm {lensing} stands for CMB lensing, BAO \\\\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\\\rm {SNe} stands for supernovae, H0\\\\rm {H0} stands for the Hubble constant, and RSD \\\\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\\\% confidence limits on cosmological parameters for the HDES model.\",\n \"Here ⋯\\\\left\\\\lbrace \\\\dots \\\\right\\\\rbrace stands for the inclusion of data from column on the left.\"\n ],\n [\n \"Conclusions\",\n \"Over the past decades several cosmological models have emerged as a plausible explanation for the late-time accelerating expansion of the Universe.\",\n \"In this paper we investigated three MG models which satisfy solar system tests and also fulfil constraints on the speed of propagation of GWs, namely: DES-fR, HS, and the HDES models.\",\n \"It is possible to interpret MG models as an effective fluid and we followed this approach in this work.\",\n \"We implemented DES-fR, HS, and HDES models in the Boltzmann solver EFCLASS which uses sub-horizon and quasi-static approximations when solving the perturbation equations.\",\n \"We showed in previous works that the observables are accurately computed (i.e., better than $0.1\\\\%$ as compared to outputs from codes which do not use any approximation) while having the advantage of analytical expressions describing MG as an effective fluid.\",\n \"When constraining the parameter space for the HS model is usual to assume a $\\\\Lambda $ CDM background.\",\n \"This is however incorrect as the background for the HS model is in general different from the $\\\\Lambda $ CDM model.\",\n \"In this paper we dropped this assumption and solved the perturbations equations taking into consideration the background evolution too.\",\n \"We found constraints which are in good agreement with results by the Planck Collaboration when the parameter spaces overlap.\",\n \"We also note that the constraints on the MG parameter are dominated by the prior hence unconstrained by current data sets.\",\n \"As the HS model has an additional parameter than the $\\\\Lambda $ CDM model, the former will be severely penalized in any Bayesian model comparison.\",\n \"Since data indicate a preference for the standard model it is interesting to study models which exactly match the $\\\\Lambda $ CDM background.\",\n \"These models might rely on new physics while also behaving differently at the perturbations level with respect to the $\\\\Lambda $ CDM model.\",\n \"By investigating these kinds of models we can also reveal whether or not current data sets can discriminate alternative models from the concordance model.\",\n \"In this paper we investigated two models, namely, a designer $f(R)$ (DES-fR) and a designer Horndeski model (HDES).\",\n \"When considering common cosmological parameters, constraints for the DES-fR model do not exhibit significant discrepancies with results by the Planck Collaboration for the standard model.\",\n \"Concerning the MG parameter we note the results depend on the probe combination.\",\n \"Most cases are dominated by the prior and hence unconstrained.\",\n \"The full data set however, prefers the GR limit.\",\n \"One reason for this might be the strong constraints from the RSD likelihood.\",\n \"As the surveys that make the RSD measurements assume a $\\\\Lambda $ CDM model in their analysis, the data themselves maybe a bit biased.\",\n \"While this in general can be corrected, up to a point, with the AP correction as mentioned in the Appendix, some residual bias may remain.\",\n \"While this is an important point, it is however outside the scope of our present analysis, thus we leave it for future work.\",\n \"Regarding the constraints for the HDES model we also find good agreement with parameters also appearing in the $\\\\Lambda $ CDM model.\",\n \"Here constraints on MG are also inconclusive as the posteriors are prior dominated.\",\n \"Interestingly, in this case the full data set shows a slight preference for a departure from GR.\",\n \"In summary, our results do not conclusively indicate the presence of modifications to GR.\",\n \"Since our MG constraints are prior dominated we conclude $\\\\Lambda $ CDM is still the preferred model.\",\n \"Where data sets overlap, our results fully agree with the investigation carried out by the Planck Collaboration [17].\"\n ],\n [\n \"Acknowledgements\",\n \"The authors would like to thank M. Martinelli for useful discussions.\",\n \"W.C. acknowledges financial support from the Departamento Administrativo de Ciencia, Tecnología e Innovación (COLCIENCIAS) under the project “Discriminación de modelos de energía oscura y gravedad modificada con futuros datos de cartografiado galáctico\\\" and from Universidad del Valle under the contract 449-2019.\",\n \"S.N.\",\n \"and R.A. acknowledge support from the Research Projects PGC2018-094773-B-C32 and the Centro de Excelencia Severo Ochoa Program SEV-2016-0597.\",\n \"S.N.\",\n \"also acknowledges support from the Ramón y Cajal program through Grant No.\",\n \"RYC-2014-15843.\",\n \"The calculations for this article were carried out on the Datacenter CIBioFi.\",\n \"The statistical analyses as well as the plots were made with the Python package GetDist https://github.com/cmbant/getdist.\",\n \"The numerical codes used by the authors in the analysis of the paper and our modifications to the CLASS code, which we call EFCLASS, can be found on the websites of the EFCLASS here, here and here.\",\n \"The publicly available RSD Montepython likelihood for the growth-rate $f\\\\sigma _8$ data set can be found at https://github.com/snesseris/RSD-growth.\"\n ],\n [\n \"The RSD likelihood \",\n \"In this appendix we will present some details on the growth RSD likelihood implemented in MontePython and used to analyse the growth $f\\\\sigma _8$ data of the “Gold 2018\\\" compilation, namely, the $N=22$ data points of Ref. [43].\",\n \"This likelihood was first presented in Ref.\",\n \"[42], but we summarize here again some of the key details for completeness.\",\n \"The growth data we use in our analysis are based on measurements of the RSD, which is a direct probe of the LSS.\",\n \"In essence, these data directly measure $f\\\\sigma _8(a)\\\\equiv f(a)\\\\cdot \\\\sigma (a)$ , where $f(a) \\\\equiv \\\\frac{d ln\\\\delta }{d lna}$ is the growth rate and $\\\\sigma (a) \\\\equiv \\\\sigma _{8,0}\\\\frac{\\\\delta (a)}{\\\\delta (1)}$ are the redshift-dependent rms fluctuations of the linear density field within spheres of radius $R=8$ Mpc $h^{-1}$ .\",\n \"Note that by $\\\\sigma _{8,0}$ we denote the present value of this parameter.\",\n \"This dataset has been shown to be internally robust and unbiased by the authors of Ref.\",\n \"[43] that used the “robustness” criterion of Ref. [65].\",\n \"The latter uses combinations of subsets in the data to perform a Bayesian analysis and establish the dataset's overall consistency.\",\n \"The value of the parameter $f\\\\sigma _8(a)$ can be measured directly by using the ratio of the monopole to the quadrupole of the redshift-space power spectrum.\",\n \"Thus, one can show with linear theory that $f\\\\sigma _8(a)$ does not depend on the bias $b(k,z)$ , which maybe both scale and redshift dependent, as it drops from the equations due to the particular combination of variables [66], [67], [68].\",\n \"Furthermore, $f\\\\sigma _8(a)$ has been show to be a good discriminator of DE models [67].\",\n \"The RSD data points are given explicitly in Ref.\",\n \"[43] as $ f\\\\sigma _8^\\\\textrm {obs,i}=\\\\Big (f\\\\sigma _8^\\\\textrm {obs}(z_1),\\\\dots , f\\\\sigma _8^\\\\textrm {obs}(z_n)\\\\Big )$ .\",\n \"It should be noted that a few of the growth points are correlated, while most also require a fiducial cosmology that has to be adjusted for the Alcock-Paczynski effect (see Refs.\",\n \"[43], [69], [70] as well as Refs.\",\n \"[71], [72], [73] for earlier analyses).\",\n \"The data points that are correlated are the four points from SDSS [74] and the three WiggleZ points from Ref. [75].\",\n \"Their covariance matrices are given by $\\\\mathbf {C}_{\\\\text{WiggleZ}}= 10^{-3}\\\\left(\\\\begin{array}{ccc}6.400 & 2.570 & 0.000 \\\\\\\\2.570 & 3.969 & 2.540 \\\\\\\\0.000 & 2.540 & 5.184 \\\\\\\\\\\\end{array}\\\\right),$ for the WiggleZ data and for the SDSS points by $\\\\mathbf {C}_{\\\\text{SDSS-IV}}= 10^{-2}\\\\left(\\\\begin{array}{cccc}3.098 & 0.892 & 0.329 & -0.021\\\\\\\\0.892 & 0.980 & 0.436 & 0.076\\\\\\\\0.329 & 0.436 & 0.490 & 0.350 \\\\\\\\-0.021 & 0.076 & 0.350 & 1.124\\\\end{array}\\\\right).$ In order to perform the correction for the Alcock-Paczynski effect, we follow the prescription of Ref.\",\n \"[69], which requires the use of a correction factor given by $\\\\text{fac}(z^i)= \\\\frac{H(z^i)\\\\,d_A(z^i)}{H^{\\\\text{ref},i}(z^i) \\\\, d_A^{\\\\text{ref},i}(z^i)}\\\\;,$ where the label “$\\\\text{ref},i$ ” stands for the fiducial cosmology at the redshift $z^i$ .\",\n \"Then, the corrected data will be given by [76] $f\\\\sigma _8^{\\\\textrm {th,i}}\\\\rightarrow \\\\frac{f\\\\sigma _8^{\\\\textrm {th,i}}}{\\\\text{fac}(z^i)}\\\\;.$ By defining the vector $\\\\mathbf {V}$ for the data via: $\\\\mathbf {V} = \\\\mathbf {f\\\\sigma _8^\\\\textrm {obs,i}} - \\\\frac{f\\\\sigma _8^{\\\\textrm {th,i}}}{\\\\text{fac}(z^i)},$ the chi-squared is then given by $\\\\chi ^2= \\\\mathbf {V}^T \\\\mathbf {C}^{-1} \\\\mathbf {V}\\\\;.$ To conclude, we also need the theoretical prediction for the growth $\\\\delta (k,z)$ at each redshift, which in CLASS can be estimated from the matter power spectrum as $\\\\delta (k,z)=\\\\sqrt{\\\\frac{P(k,z)}{P(k,0)}}$ , where we can obtain $P(k,z)$ via the function cosmo.pk(k,z).\",\n \"Finally, the exact value of the growth-rate $f\\\\sigma _8(k,z)$ can be calculated numerically with direct differentiation and cubic interpolations.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05282"}}},{"rowIdx":3698,"cells":{"text":{"kind":"list like","value":[["Product of Matrix Valued Truncated Toeplitz Operators"],["Abstract Let $A_\\Phi$ be a matrix valued truncated Toeplitz operator-the compression of multiplication operator to vector-valued model space $H^2(E)\\ominus \\Theta H^2(E)$, where $\\Theta$ is a matrix valued non constant inner function.","Under supplementary assumptions, we find necessary and sufficient condition that the product $A_\\Phi A_\\Psi$ is itself a matrix valued truncated Toeplitz operator."],["Introduction","Toeplitz operators are the compressions of multiplication operator to the usual Hardy Hilbert space $H^2$ .","In [3], Brown and Halmos describe the algebraic properties of Toeplitz operators.","Among other things, they found necessary and sufficient conditions for the product of two Toeplitz operators to itself be a Toeplitz operator, namely that either the first operator’s symbol is antiholomorphic or the second operator’s symbol is holomorphic.","In either case, the symbol of the product is the product of the symbols.","In the last decade, a large amount of research has concentrated on a generalization of Toeplitz matrices, namely truncated Toeplitz operators.","These are the compressions of multiplication operator to subspaces of the Hardy space which are invariant under the backward shift operator.","They have been formally introduced in [10]; see [5] for a more recent survey.","Sarason [10] found equivalents to several of Brown and Halmos’s results for truncated Toeplitz operators on the model spaces $H^2\\ominus \\theta H^2$ , where $\\theta $ is some non-constant inner function.","The model spaces are the backward-shift invariant subspaces of $H^2$ (that they are backward shift invariant follows easily from the fact that $\\theta H^2$ is clearly shift invariant).","We refer the reader to [1], [2], [5] (see also [6]) for the general theory of these operators.","It is well know that the product of two truncated Toeplitz operators is not a truncated Toeplitz operator.","In particular, in [11] Sedlock has investigated when a product of truncated Toeplitz operators is itself a truncated Toeplitz operator.","Most recently, the basics of corresponding matrix valued truncated Toeplitz operators (MTTOs), which are compressions to ${\\mathcal {K}}_\\Theta $ of multiplications with matrix valued functions on $H^2(E)$ has been developed in [9].","In view of the result of [11], there arises a basic question related to the product of matrix valued truncated Toeplitz operators that is when the product of two MTTOs is still an MTTO?","But there is no such simple result in the general case, and we need some supplementary assumptions to obtain the main result Theorem 4.6.","The purpose of the present paper is to adapt the approach in [4] to the case of MTTOs on an arbitrary model space.","The plan of the paper is following: By means of Section 2, we want to make sure that the reader has become acquainted to model spaces and their operators and other useful facts from this area, needed when we are going to start the main work in section 4.","In Section 3 we will introduce truncated Toeplitz operators (TTOs) and matrix valued truncated Toeplitz operators (MTTOs).","The last section contains a particular case of MTTOs namely the Block Toeplitz matrices."],["Model Spaces and Operators","Let $\\mathbb {C}$ denote the complex plane, $\\mathbb {D}$ the unit disc in $\\mathbb {C}$ and $\\mathbb {T}$ one dimensional torus in $\\mathbb {C}$ .","In the sequel $E$ will denote a fixed Hilbert space of dimension $d$ .","We designate the algebra of bounded linear operators on $E$ by $\\mathcal {L}(E)$ and by ${\\mathcal {L}}(E,K)$ the space of all bounded linear operators from Hilbert space $E$ to a Hilbert space $K$ .","The space $L^2(E)$ is defined as usual, by $L^2(E):=\\left\\lbrace f:\\mathbb {T}\\longrightarrow E: f(e^{it})=\\sum _{n=-\\infty }^{\\infty } a_ne^{int}: a_n\\in E,\\sum _{n=-\\infty }^{\\infty }\\Vert a_n\\Vert ^2<\\infty \\right\\rbrace $ endowed with the inner product $\\langle f , g \\rangle _{L^2 (E)} =\\int _{\\mathbb {T}}\\langle f(e^{it}), g(e^{it})\\rangle _E dm$ where $dm=\\frac{dt}{2\\pi }$ is the normalized Lebesgue measure on $\\mathbb {T}$ .","The norm induced by the inner product is given by $\\Vert f\\Vert _{L^2(E)}=\\int _{\\mathbb {T}}\\Vert f\\Vert _{E}^2dm.$ If $\\dim E=1$ (i.e., $E = \\mathbb {C}$ ) then $L^2 (E)$ consists of scalar-valued functions and is denoted by $L^2$ .","The Hardy space $H^2(E)$ is the subspace of $L^2(E)$ formed by the functions with vanishing negative Fourier coefficients; it can be identified with a space of $E$ - valued functions analytic in $\\mathbb {D}$ , from which the boundary values can be recovered almost everywhere through radial limits.","One can also view $H^2(E)$ as the direct sum of $d$ standard $H^2$ spaces.","We have the orthogonal decomposition $L^2(E)=H^2(E)\\oplus (zH^2(E))^*.$ The spaces $ L^\\infty (E)\\subset L^2(E) $ is formed by the essentially bounded functions with values in $ E $ ; then $ H^\\infty (E)\\subset H^2(E) $ are the functions in $ L^\\infty (E) $ with vanishing negative Fourier coefficients.","Taking into account that $\\mathcal {L}(E)$ is a Hilbert space endowed with the Hilbert-Schmidt norm, we may similarly define $ H^2({\\mathcal {L}}(E)) \\subset L^2({\\mathcal {L}}(E))$ and $ H^\\infty ({\\mathcal {L}}(E)) \\subset L^\\infty ({\\mathcal {L}}(E)) $ .","Note, however, that we prefer to consider on $ L^\\infty ({\\mathcal {L}}(E)) $ and $ H^\\infty ({\\mathcal {L}}(E)) $ the equivalent norm obtained by considering on $ {\\mathcal {L}}(E) $ the operator norm instead of the Hilbert-Schmidt norm.","The space $L^2(\\mathcal {L}(E))$ may be identified with the matrices with all the entries in $L^2$ .","We have an orthogonal decomposition $L^2 (\\mathcal {L}(E)) =[zH^2 \\mathcal {L}(E)]^*\\oplus H^2(\\mathcal {L}(E)).$ The space $ L^\\infty ({\\mathcal {L}}(E)) $ acts on $L^2 (E)$ by means of multiplication: to $\\Phi \\in \\ L^\\infty (\\mathcal {L}(E))$ we associate the operator $M_\\Phi $ defined by $M_\\Phi (f)=\\Phi f$ for all $f\\in L^2(E)$ .","Let $S$ denote the forward shift operator $(Sf)( z ) = zf (z)$ on $H^2 (E)$ ; it is the restriction of $M_z$ to $H^2(E)$ .","Its adjoint (the backward shift) is the operator $(S^*f)(z) =\\frac{f(z )-f(0)}{z}$ One sees easily that $I-SS^*$ is precisely the orthogonal projection onto the space of constant functions.","The main object of study is formed by the model spaces and the operators acting on them.","These are defined as follows.","First, an inner function is an element $ \\Theta \\in H^2(\\mathcal {L}(E))$ whose boundary values are almost everywhere unitary operators in $\\mathcal {L}(E)$ .","The inner function is called pure if $ \\Vert \\Theta (0)\\Vert <1 $ ; this is equivalent to requiring that $ \\Theta $ has no constant unitary part.","Consider then a pure inner function $\\Theta $ , with values in $\\mathcal {L}(E)$ .","The model space associated to a pure inner function $\\Theta $ , denoted by $\\mathcal {K}_{\\Theta }$ and is defined by $\\mathcal {K}_{\\Theta }=H^{2}(E)\\ominus \\Theta H^{2}(E).$ Just like the Beurling-type subspaces $\\Theta H^{2}(E)$ constitute nontrivial invariant subspaces for the unilateral shift $S$ on $H^{2}(E)$ , the subspaces $\\mathcal {K}_{\\Theta }$ play an analogous role for the backward shift $S^{*}$ .","The orthogonal projection onto $\\mathcal {K}_{\\Theta }$ will be denoted by $P_{\\Theta }$ .","It is also known that $ {\\mathcal {K}}_\\Theta \\cap H^\\infty (E) $ is dense in $ {\\mathcal {K}}_\\Theta $ .","The analogous space of matrix-valued functions is denoted by $\\mathcal {M}_\\Theta $ ; it is the orthogonal complement of $\\Theta H^2(\\mathcal {L}(E))$ in $H^2(\\mathcal {L}(E))$ .","The model operator $S_\\Theta \\in \\mathcal {L}(\\mathcal {K}_\\Theta )$ is defined by the formula $(S_\\Theta f)(z)=P_\\Theta (zf).$ The adjoint of $S_\\Theta $ is given by $(S_\\Theta ^*f)(z)=\\frac{f(z)-f(0)}{z};$ it is the restriction of the left shift in $H^2(E)$ to the $S^*$ -invariant subspace $\\mathcal {K}_\\Theta $ .","Let us assume that $ \\Theta (0)=0 $ , so $ \\Theta =z\\Theta _1 $ , which is the case we will consider in the sequel.","Then $ I-S_\\Theta S_\\Theta ^* $ is the projection $ P_0 $ onto the constant functions, which are contained in $ {\\mathcal {K}}_\\Theta $ , while $ I-S_\\Theta ^* S_\\Theta $ is the projection $P_{D_*}$ onto the space $ {\\mathcal {D}}_*=\\lbrace \\Theta _1 x: x\\in E \\rbrace $ (which is also contained in $ {\\mathcal {K}}_\\Theta $ ).","The scalar valued model spaces and operators are obtained when $ \\dim E=1 $ ; that is, when $ E=\\mathbb {C}$ .","We have then the classical spaces $ H^2\\subset L^2 $ and $L^\\infty $ .","The inner function is a scalar inner function $ \\theta $ , and the model space is $ \\mathcal {K}_{\\theta }=H^{2}\\ominus \\theta H^{2} $ .","In particular, in case $ \\theta (z)=z^n $ , $ {\\mathcal {K}}_\\theta $ becomes the $ n $ -dimensional space of polynomials of degree at most $ n-1 $ .","Definition 2.1 A conjugation on a complex Hilbert space $\\mathcal {H}$ is a function $C:\\mathcal {H}\\longrightarrow \\mathcal {H}$ that is (i) conjugate linear: that is $C(\\alpha x+\\beta y)=\\overline{\\alpha }Cx +\\overline{\\beta }Cy$ for all $x,y\\in \\mathcal {H}$ and $\\alpha ,\\beta \\in \\mathbb {C}$ , (ii) involutive: $C^{2}=I$ , (iii) isometric: $\\Vert Cx\\Vert =\\Vert x\\Vert $ for all $x\\in \\mathcal {H}$ .","The following result is an immediate consequence of a theorem in [9].","Lemma 2.1 Suppose $ \\Theta (0)=0 $ , so $ \\Theta =z\\Theta _1 $ .","Let $\\Gamma $ be a conjugation on $E$ , and suppose that $\\Theta (e^{it})^{*}=\\Gamma \\Theta (e^{it})\\Gamma $ a.e.","on $\\mathbb {T}$ .","Then the map $C_{\\Gamma }$ defined by $C_\\Gamma (f)=z\\Theta _{1}\\Gamma f$ is a conjugation on $z\\mathcal {K}_{\\Theta _1}$ ."],["Truncated Toeplitz Operators and Matrix valued Truncated Toeplitz Operators","If $ \\phi \\in L^\\infty $ , then the compression of the multiplication operator $ M_\\phi $ to $ H^2 $ is called a Toeplitz operator and is denoted by $ T_\\phi $ .","That means that $ T_\\phi =P_{H^2}M_\\phi |H^2 $ , where $P_H^2$ is the orthogonal projection of $L^2$ onto $H^2$ .","More than a decade ago, Sarason has introduced in [10] the so-called truncated Toeplitz operators.","Remember that $ P_\\theta $ is the orthogonal projection onto the model space $ {\\mathcal {K}}_\\theta $ .","If $ \\phi \\in L^\\infty $ , then the truncated Toeplitz operator $ A^\\theta _\\phi $ is defined to be the compression of $ M_\\phi $ to $ {\\mathcal {K}}_\\theta $ .","That means that $ A^\\theta _\\phi =P_\\theta M_\\phi |{\\mathcal {K}}_\\theta $ .","In particular, we see that with this notation $ S_\\theta = A^\\theta _z $ .","Let us now remember that for $ \\theta (z)=z^n $ the space $ {\\mathcal {K}}_\\theta $ is formed by the polynomials of degree not greater than $ n-1 $ .","The monomials $ z^k $ , $ k=0,\\dots , n-1 $ form an orthonormal basis of $ {\\mathcal {K}}_{z^n} $ .","If we write the matrix of an operator with respect to this basis, then one can see that truncated Toeplitz operators correspond precisely to Toeplitz matrices.","Passing now beyond the scalar case, let us suppose that $\\Theta $ is a pure inner function.","The analogue of truncated Toeplitz operators have been defined in [9], where they are called matrix valued truncated Toeplitz operators.","Suppose then that $\\Phi \\in L^2(\\mathcal {L}(E))$ .","Consider the linear map $f\\longrightarrow P_\\Theta (\\Phi f)$ , defined on $\\mathcal {K}_\\Theta \\cap H^\\infty (E)$ .","If it is bounded, then it uniquely determines an operator in $\\mathcal {L}(\\mathcal {K}_\\Theta )$ , denoted by $A_\\Phi ^\\Theta $ , and called a matrix valued truncated Toeplitz operator (MTTO).","The function $\\Phi $ is then called a symbol of the operator.","We will usually drop the subscript $\\Theta $ , as we consider a fixed inner function.","We denote by $\\mathcal {T}(\\mathcal {K}_{\\Theta })$ the space of all MTTOs on the model space $\\mathcal {K}_\\Theta $ .","Note that if $ \\Phi \\in L^\\infty ({\\mathcal {L}}(E)) $ (that is, it is bounded), then it follows that $f\\longrightarrow P_\\Theta (\\Phi f)$ defines a bounded linear operator on the whole of $ {\\mathcal {K}}_\\Theta $ , and thus $A_\\Phi ^\\Theta \\in {\\mathcal {T}}({\\mathcal {K}}_\\Theta )$ .","But we may have bounded MTTOs which have no bounded symbols, which is one of the complications of the theory.","However, a result of [9] tells that any operator in $ {\\mathcal {T}}({\\mathcal {K}}_\\Theta ) $ has a symbol in $ {\\mathcal {M}}_\\Theta +({\\mathcal {M}}_\\Theta )^* $ ; this is why we will restrict in the sequel to considering operators $A_\\Phi ^\\Theta $ with $ \\Phi \\in {\\mathcal {M}}_\\Theta +({\\mathcal {M}}_\\Theta )^* $ .","The operator $S_\\Theta $ is a simple example of MTTO; it is obtained by taking $\\Phi (z)=zI_E$ .","This example is rather special because the symbol is scalar valued.","It is immediate that $A_\\Phi ^*=A_{\\Phi ^*};$ so $\\mathcal {T}(\\mathcal {K}_{\\Theta })$ is a self adjoint linear space.","In section 5 we will see that if $\\Theta (z)=z^N I_E$ , then the MTTOs obtained are actually familiar objects, namely block Toeplitz matrices of dimension $N$ , in which the entries are the matrices of dimension $d$ .","The theory of Block Toeplitz matrices has been an inspiration for research in the domain of matrix valued truncated Toeplitz operators.","In particular, it should be mentioned that some of the classes of block Toeplitz matrices which are closed to multiplication are found in [7] and [8].","As supposed above, we will consider a fixed inner function $ \\Theta $ and different MTTOs acting on $ {\\mathcal {K}}_\\Theta $ .","The symbols of these operators will be $ \\Phi , \\Psi ,\\dots $ .","Before the ending of this section we need to quote result from [12].","Proposition 3.1 Suppose $\\Theta $ be an inner function, and $\\Phi \\in H^\\infty (\\mathcal {L}(E))$ such that $\\Phi \\Theta =\\Theta \\Phi $ .","Then (1) $\\Theta H^2(\\mathcal {L}(E))$ is invariant with respect to $M_\\Phi $ (and consequently $\\mathcal {K}_\\Theta )$ is invariant under $M_{\\Phi }^*$ .","(2) $A_\\Phi S_\\Theta =S_\\Theta A_\\Phi $ and consequently $A_\\Phi ^*S_\\Theta ^*=S_\\Theta ^*$ Set $\\Delta =I-S_\\Theta S_\\Theta ^*$ .","Note that $P_0$ is the orthogonal projection onto the constant functions contained in $\\mathcal {K}_\\Theta $ .","The next result from [9] characterizes MTTOs among all operators on $ {\\mathcal {K}}_\\Theta $ .","Proposition 3.2 A bounded operator $A$ on $\\mathcal {K}_\\Theta $ belongs to $\\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only if there exist operators $B,B^\\prime $ on $\\mathcal {K}_\\Theta $ such that $\\Delta (A)=BP_0+P_0{B^\\prime }$ In this case $A=A_{\\Phi +{\\Phi ^\\prime }^*}$ , where $\\Phi ,\\Phi ^\\prime \\in H^2(\\mathcal {L}(E))$ ."],["Main results","In view of the result of Sedlock [11], a natural question is to determine when is the product of two MTTOs still an MTTO.","However, there is no such simple result in the general case, and we need some supplementary assumptions to obtain the main result, Theorem REF .","The path we take is suggested by [4], but the matrix valued situation is much more complicated.","We will consider in the rest of this section a fixed inner function $\\Theta \\in H^\\infty (E)$ subjected to the condition $\\Theta (0)=0$ .","Then $\\Theta (z)=z\\Theta _{1}(z)$ , where $\\Theta _{1}\\in H^\\infty (E)$ is also inner.","We have the orthogonal decomposition ${\\mathcal {K}}_\\Theta =E\\oplus z{\\mathcal {K}}_{\\Theta _1}.$ Take now $\\Phi \\in {\\mathcal {M}}_\\Theta +({\\mathcal {M}}_\\Theta )^* $ .","We can write then $\\Phi =z\\Phi _{+}+\\bar{z}\\Phi _{-}^*+\\Phi _0$ with $\\Phi _\\pm \\in \\mathcal {M}_{\\Theta _1}$ and $\\Phi _0\\in \\mathcal {L}(E)$ .","If $ \\Phi (e^{it})=\\displaystyle \\sum _{n=-\\infty }^{\\infty }\\Phi _n e^{int}$ with $\\Phi _n\\in \\mathcal {L}(E)$ , then $\\begin{split}\\Phi _+(z)&=\\sum _{n=1}^{\\infty }\\Phi _n z^n=\\sum _{n=1}^{\\infty } \\left( \\int \\Phi (e^{it})e^{-int}\\, dt \\right) z^n\\\\&=\\int \\Phi (e^{it}) \\left( \\sum _{n=1}^{\\infty } e^{-int}z^n \\right)dt =\\int \\Phi (e^{it}) \\frac{e^{it}z}{1-e^{it}z}dt.\\end{split}$ Remember that two operators $ A,B $ are said to doubly commute if $ AB=BA $ and $ AB^*=B^*A $ (whence it follows that also $ A^*B^*=B^*A^* $ and $ A^*B=BA^* $ ).","Lemma 4.1 Suppose that $\\Theta (0)=0$ and $\\Phi ,\\Psi \\in \\mathcal {M}_\\Theta +({\\mathcal {M}}_\\Theta )^*$ such that $\\Phi (e^{it})\\Psi (e^{is})=\\Psi (e^{is})\\Phi (e^{it})$ for any $ t,s $ .","(i) For any $ s,t $ we have $\\Phi _+(e^{it})\\Psi (e^{is})=\\Psi (e^{is})\\Phi _+(e^{it})$ and $\\Phi _-(e^{it})\\Psi (e^{is})=\\Psi (e^{is})\\Phi _-(e^{it})$ .","(ii) If the values of $\\Phi ,\\Psi $ doubly commute with those of $\\Theta $ , then the same is true for $ \\Phi _{\\pm }, \\Psi _{\\pm } $ .","(iii) If $\\Gamma $ is a conjugation on $E$ such that $\\Phi (e^{it})^*=\\Gamma \\Phi (e^{it})\\Gamma $ , then $\\Phi _\\pm (e^{it})^*=\\Gamma \\Phi _\\pm (e^{it})\\Gamma $ .","We will give the proof only for one of the equalities in (ii); the rest are similar.","Using (REF ), we have $\\begin{split}\\Psi (e^{is}) \\Phi _+(z)&= \\Psi (e^{is})\\int \\Phi (e^{it}) \\frac{e^{it}z}{1-e^{it}z}dt=\\int \\Psi (e^{is}) \\Phi (e^{it}) \\frac{e^{it}z}{1-e^{it}z}dt\\\\&=\\int \\Phi (e^{it})\\Psi (e^{is}) \\frac{e^{it}z}{1-e^{it}z}dt=\\left(\\int \\Phi (e^{it}) \\frac{e^{it}z}{1-e^{it}z}dt\\right)\\Psi (e^{is})=\\Phi _+(z)\\Psi (e^{is}).\\end{split}$ By taking radial limits a.e., one obtains the required commutativity.","The next lemma gives an identification of elements in $ {\\mathcal {M}}_\\Theta $ .","Lemma 4.2 The map $ \\Phi \\mapsto J_\\Phi $ , defined by $J_\\Phi (x)(z)=\\Phi (z)x.$ is a bijection between $ {\\mathcal {M}}_\\Theta $ and $ {\\mathcal {L}}(E, {\\mathcal {K}}_\\Theta ) $ .","Fixing a basis $e_1,\\cdots ,e_d$ in $E$ and defining the transformation $J:E\\longrightarrow \\mathcal {K}_\\Theta $ as follow $J(e_k)=\\phi _k,\\qquad 1\\le k\\le d,$ where $\\phi _k\\in \\mathcal {K}_\\Theta $ for every $1\\le k\\le d$ .","If we arrange $\\phi _k$ as a column vectors then we obtain $\\Phi \\in \\mathcal {M}_\\Theta $ .","Conversely if we have $\\Phi \\in \\mathcal {M}_\\Theta $ then we obtain the map $J_\\Phi : E\\longrightarrow \\mathcal {K}_\\Theta $ which sends $e_k$ to $k$ th column of $\\Phi $ for every $1\\le k\\le d$ .","Note that, if $\\Phi \\in \\mathcal {M}_\\Theta $ , then the relation between $J_\\Phi $ and $\\Phi $ is simply $J_\\Phi (x)(z)=\\Phi (z)x.$ We denote by $J_0$ the embedding of $E$ in $K_\\Theta $ ; that is for every $x\\in E$ , $J_0(x)=x$ .","It is easy to see that in a given basis the matrices of functions in $z\\mathcal {M}_{\\Theta _1}$ are characterized by the fact that columns are functions in $z\\mathcal {K}_{\\Theta _1}$ .","Finally, we define $\\mathbf {C}_\\Gamma (\\Phi ) $ by giving the action of $J_\\Phi $ on $x\\in E$ as $J_{\\mathbf {C}_\\Gamma (\\Phi )} x=C_\\Gamma (\\Phi \\Gamma x),$ where $ C_\\Gamma $ is defined by (REF ).","In the rest of this section we assume that $\\mathfrak {F}$ is a commutative algebra of functions contained in ${\\mathcal {M}}_\\Theta +({\\mathcal {M}}_\\Theta )^*$ , such that all the elements of $\\mathfrak {F}$ doubly commute with those of $\\Theta $ .","The next lemma is the main technical result of this section.","Lemma 4.3 Suppose that $\\Theta (0)=0$ and $\\Phi ,\\Psi \\in \\mathfrak {F}$ .","Then there exist operators $ X, Y\\in \\mathcal {M}_\\Theta $ such that $\\Delta (A_\\Phi A_\\Psi )=J_{z\\Phi _+}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*+XP_0+P_0Y$ For any $\\Phi \\in \\mathcal {M}_{\\Theta }+({\\mathcal {M}}_\\Theta )^*$ we will denote $\\hat{\\Phi }=\\Phi -\\Phi _0$ .","It follows easily from Lemma REF that $\\Phi _0\\Psi (e^{is})=\\Psi (e^{is})\\Phi _0$ and $\\Phi _0\\hat{\\Psi }(e^{is})=\\hat{\\Psi }(e^{is})\\Phi _0$ .","In the same way one can obtain $\\Psi _0\\Phi (e^{is})=\\Phi (e^{is})\\Psi _0$ and $\\Psi _0\\hat{\\Phi }(e^{is})=\\hat{\\Phi }(e^{is})\\Psi _0$ .","A similar argument works for double commutation with $\\Theta $ .","Since $\\Phi _0,\\Psi _0\\in H^2(\\mathcal {L}(E))$ commutes with $\\Theta $ then by using Proposition REF $S_\\Theta $ commutes with $A_{\\Phi _0}$ and $A_{\\Psi _0}$ , and therefore $\\Delta (A_{\\Phi } A_{\\Psi })&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+\\Delta (A_{\\hat{\\Phi }}A_{\\Psi _0})+\\Delta (A_{\\Phi _0}A_{\\hat{\\Psi }})+\\Delta (A_{\\Phi _0}A_{\\Psi _0}).\\\\&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+\\Psi _0\\Delta (A_{\\hat{\\Phi }})+\\Phi _0\\Delta (A_{\\hat{\\Psi }})+\\Phi _0\\Psi _0\\Delta (I)\\\\&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}) +\\Psi _0\\Delta (A_{\\hat{\\Phi }})+\\Phi _0\\Delta (A_{\\hat{\\Psi }})+\\Phi _0\\Psi _0P_0.$ By Proposition REF there exist operators $B=z\\Phi _{+}$ and $B^\\prime =z\\Phi _{-}$ such that $\\Delta (A_{\\hat{\\Phi }})=z\\Phi _{+}P_0+P_0\\bar{z}\\Phi _{-}^*$ with $\\Phi _{\\pm }\\in \\mathcal {M}_{\\Theta _{1}}$ .","Similarly $\\Delta (A_{\\hat{\\Psi }})=z\\Psi _{+}P_0+P_0\\bar{z}\\Psi _{-}^*$ and $\\Psi _{\\pm }\\in \\mathcal {M}_{\\Theta _{1}}$ .","Using Lemma REF , we have $\\Delta (A_\\Phi A_\\Psi )&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+\\Psi _0(z\\Phi _{+}P_0+P_0\\bar{z}\\Phi _{-}^*)+\\Phi _0(z\\Psi _{+}P_0+P_0\\bar{z}\\Psi _{-}^*)+\\Phi _0\\Psi _0P_0\\\\&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+(\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0 \\Psi _0)P_0+(\\Psi _0P_0 \\bar{z}\\Phi _{-}^*+\\Phi _0P_0\\bar{z}\\Psi _{-}^*)$ Since $P_0$ is the projection onto the constants then it must commute with $\\Phi _0$ and $\\Psi _0$ .","Therefore $\\Delta (A_\\Phi A_\\Psi )=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+(\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0)P_0+P_0(\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*)$ Now, by using the definition of $\\Delta $ , $\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})&=A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}-S_\\Theta A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}S_\\Theta ^*\\\\&=A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}-A_{\\hat{\\Phi }}S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*+A_{\\hat{\\Phi }}S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-S_\\Theta A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}S_\\Theta ^*\\\\&=A_{\\hat{\\Phi }}\\Delta (A_{\\hat{\\Psi }})+\\Delta (A_{\\hat{\\Phi }})S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*\\\\&=A_{\\hat{\\Psi }}(z\\Phi _{+}P_0+P_0\\bar{z}\\Psi _{-}^*)+(z\\Phi _{+}P_0+P_0\\bar{z}\\Phi _{-}^*)S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*}-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*$ or $\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})= A_{\\hat{\\Phi }}z\\Psi _{+}P_0+A_{\\hat{\\Phi }}P_0\\bar{z}\\Psi _{-}^*+z\\Phi _{+}P_0 S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*} +P_0\\bar{z}\\Phi _{-}^*S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*}-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*.$ Since the constant functions are in $\\mathcal {K}_{\\Theta }$ , we have $ P_0P_\\Theta =P_\\Theta P_0 $ .","Also since $ P_0zf=0 $ , then $P_0S_\\Theta f=P_0P_\\Theta zf= P_\\Theta P_0 zf=0,$ So the third term in the left hand side of (REF ) is 0.","The second term is $A_{\\hat{\\Phi }}P_0\\bar{z}\\Psi _{-}^*=P_\\Theta (z\\Phi _+ +\\bar{z}\\Phi _-^*)P_0\\bar{z}\\Psi _{-}^*.$ But, since $ \\Phi _+\\in \\mathcal {M}_{\\Theta _{1}} $ , we have, for any constant function $ x $ , $ \\Phi _+ x\\in \\mathcal {K}_{\\Theta _1} $ , $ z\\Phi _+ x\\in \\mathcal {K}_{\\Theta } $ , and therefore $ P_\\Theta z\\Phi _+P_0=z\\Phi _+P_0 $ .","Also, for any constant function $ x $ , $ \\bar{z}\\Phi _-^*x\\perp H^2(E) $ , so $ P_\\Theta \\bar{z}\\Phi _-^*P_0=0 $ .","So $\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})= A_{\\hat{\\Phi }}z\\Psi _{+}P_0+z\\Phi _+ P_0\\bar{z}\\Psi _{-}^*+P_0\\bar{z}\\Phi _{-}^*S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*}-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*.$ Since $ J_0 $ is the embedding of the constants into $ \\mathcal {K}_\\Theta $ , we have, for $ f\\in \\mathcal {K}_\\Theta $ and $ x\\in E $ , $\\langle J_0^*f, x\\rangle =\\langle f,J_0 x\\rangle =\\langle f(0), x\\rangle $ whence $ J^*_0f=f(0) $ .","So $J_\\Phi J_0^*f=J_\\Phi f(0)= \\Phi (z)f(0)=\\Phi P_0 f,\\quad \\hbox{for any}\\quad f\\in \\mathcal {K}_\\Theta .$ By taking adjoints we have $J_0 J_\\Phi ^*=P_0\\Phi ^*$ .","Therefore we can write (REF ) as $\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})= A_{\\hat{\\Phi }} J_{z\\Psi _{+}}J_0^*+J_{z\\Phi _{+}}P_0 J_{z\\Psi _{-}}^*+P_0\\bar{z}\\Phi _{-}^*S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*}-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*.$ Since $ \\mathcal {D}_* $ is the space spanned by $ \\Theta _1 E $ , we can define an isometry $ V:E\\rightarrow \\mathcal {K}_\\Theta $ by the formula $ Vx=\\Theta _1 x $ , and, moreover, $P_{\\mathcal {D}_*}=VV^* $ .","Also, we have $A_{\\hat{\\Phi }}Vx=P_\\Theta \\hat{\\Phi }Vx=P_\\Theta z\\Phi _+ Vx +P_\\Theta \\bar{z}\\Phi _-^* Vx.$ Then, using the commutativity between $ \\Theta $ and $ \\Phi _+ $ , $z\\Phi _+ V x=z\\Phi _+ \\Theta _1 x= z\\Theta _1\\Phi _+ x=\\Theta \\Phi _+ x\\perp \\mathcal {K}_\\Theta ,$ and so the first term in (REF ) is 0.","We have also $\\bar{z}\\Phi _-^*V x&=\\bar{z}\\Phi _-^*\\Theta _1 x=\\Theta _1 \\bar{z} \\Phi _-^* x= \\Theta _1 \\bar{z} \\Phi _-^* \\Gamma \\Gamma x\\\\&=\\bar{z}( z\\Theta _1 \\Gamma (z \\Phi _-\\Gamma x))=\\bar{z} C_\\Gamma ( z \\Phi _-\\Gamma x).=\\bar{z}J_{\\mathbf {C}_\\Gamma }(z\\Phi _{-})x.$ Since $ C_\\Gamma $ is a conjugation on $ z\\mathcal {K}_{\\Theta _1} $ , $ \\bar{z} C_\\Gamma (z \\Phi _-\\Gamma x)\\in \\mathcal {K}_{\\Theta }$ , and therefore $P_\\Theta \\bar{z}\\Phi _-^*V x=\\bar{z}\\Phi _-^*V x= \\bar{z} C_\\Gamma z \\Phi _-\\Gamma x=\\bar{z}J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}x.$ So $S_\\Theta A_{\\hat{\\Phi }}Vx= z \\bar{z} C_\\Gamma z \\Phi _-\\Gamma x= C_\\Gamma z \\Phi _-\\Gamma x=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}x.$ Similarly, we obtain $S_\\Theta A_{\\hat{\\Psi }^*}Vx=C_\\Gamma z \\Psi _+\\Gamma x=J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})} x.$ Consequently, $S_\\Theta A_{\\hat{\\Phi }}V=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}, \\qquad S_\\Theta A_{\\hat{\\Psi }^*}V=J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})},$ Finally, the last term in (REF ) is $S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*=S_\\Theta A_{\\hat{\\Phi }}VV^*A_{\\hat{\\Psi }}S_\\Theta ^*=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*.$ Combining (REF ) and (REF ) we get $\\Delta (A_{\\hat{\\Phi }} A_{\\hat{\\Psi }})=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*+J_{z\\Phi _+}P_0J_{z\\Psi _{-}}^*+J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*$ so we have $\\begin{split}\\Delta (A_\\Phi A_\\Psi )&=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*+J_{z\\Phi _+}P_0J_{z\\Psi _{-}}^*+J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*\\\\&+(\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0)P_0+P_0(\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*)\\end{split}$ Now we have the relations $J_\\phi J_0^*=\\Phi P_0, J_0J_\\Phi =P_0\\Phi ^*$ So the second term in (REF ) becomes $z\\Phi _{+}P_0(z\\Phi )^*=J_{z\\Phi _{+}}J_0^*J_0J_{z\\Phi _{-}}^*=J_{z\\Phi _{+}}J_{z\\Phi _{-}}^*$ Also, since $J_0: E\\longrightarrow \\mathcal {K}_{\\Theta }$ is the embedding of the constants, while $P_0:\\mathcal {K}_\\Theta \\longrightarrow \\mathcal {K}_\\Theta $ is the projection onto the constant functions, it follows immediately that $P_0J_0=J_0, J_0^*P_0=J_0^*$ Therefore we can write the first and third term in (REF ) as $A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*P_0,\\quad J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*=P_0J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*.$ So we have $\\begin{split}\\Delta (A_\\Phi A_\\Psi )&=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*P_0+J_{z\\Phi _+}J_{z\\Psi _{-}}^*+P_0J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*\\\\&+(\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0)P_0+P_0(\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*)\\\\&=J_{z\\Phi _+}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*+[A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*+\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0]P_0\\\\&+P_0[J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*+\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*]\\\\&=J_{z\\Phi _+}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*+XP_0+P_0Y\\end{split}$ where, $X=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*+\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0$ and $Y=J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*+\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*$ .","Theorem 4.4 Suppose $\\Theta (0)=0$ and $\\Phi ,\\Psi ,\\chi , \\zeta \\in \\mathfrak {F}$ .","Then $A_\\Phi A_\\Psi -A_\\chi A_\\zeta \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only if $ J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _-)}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*=J_{z\\chi _+}J_{z\\zeta _-}^*-J_{\\mathbf {C}_\\Gamma (z\\chi _-)}J_{\\mathbf {C}_\\Gamma (z\\zeta _+)}^*.$ By Lemma REF there exists operators $X, Y\\in \\mathcal {M}_\\Theta $ such that $\\Delta (A_\\Phi A_\\Psi -A_\\chi A_\\zeta )=J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}-J_{z\\chi _+}J_{z\\zeta _-}^*+J_{\\mathbf {C}_\\Gamma (z\\chi _-)}J_{\\mathbf {C}_\\Gamma (z\\zeta _+)}+XP_0+P_0Y.$ By Proposition REF , we have $A_\\Phi A_\\Psi -A_\\chi A_\\zeta \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only there exist operators $B, B^\\prime $ such that $\\Delta (A_\\Phi A_\\Psi -A_\\chi A_\\zeta )=BP_0+P_0B.$ The last two equations say that $A_\\Phi A_\\Psi -A_\\chi A_\\zeta \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only if there exist $ X^{\\prime },Y^{\\prime } $ such that $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}-J_{z\\chi _+}J_{z\\zeta _-}^*+J_{\\mathbf {C}_\\Gamma (z\\chi _-)}J_{\\mathbf {C}_\\Gamma (z\\zeta _+)}=X^{\\prime }P_0+P_0Y^\\prime .$ Since $\\Theta (0)=0$ then we can write $\\mathcal {K}_\\Theta =E\\oplus \\mathcal {K}_{\\Theta _{1}}$ .","With respect to this decomposition, the left hand side of (REF ) has zeros on the first row and column, while the right hand side is the general form of an operator that has zeros in the lower right corner.","Now it is clear that for (REF ) to be true both sides have to be zero, which proves the theorem.","The following result is the main result of this paper: it gives the answer to the question stated at the beginning of this section, namelu when is the product of two MTTOs also an MTTO.","Theorem 4.5 Suppose $\\Theta (0)=0$ , $\\Phi ,\\Psi \\in \\mathfrak {F}$ and $A_\\Phi , A_\\Psi , \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ .","Then $A_\\Phi A_\\Psi \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only if $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*=J_{\\mathbf {C}_\\Gamma (z\\Phi _-)}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*.$ Applying Theorem REF to the case $\\chi =\\zeta =0$ ."],["A particular case: Block Toeplitz Matrices","In this section let $\\Theta (z)=z^NI_E$ for some fixed positive integer $N$ .","Then ${\\mathcal {K}}_\\Theta $ is the Hilbert space of all polynomials in $z$ of degree at most $N-1$ with coefficients from $E$ ,i.e., ${\\mathcal {K}}_\\Theta =\\lbrace a_0+a_1z+\\cdots a_{N-1}z^{N-1};a_0,a_1,\\cdots ,a_{N-1}\\in E\\rbrace .$ One can also identify this space with N-copies of $E$ by mapping $\\displaystyle \\sum _{k=0}^{N-1}a_kz^k$ into $\\displaystyle \\otimes _{k=0}^{N-1}a_k$ .","Now if we take $A_\\Phi $ corresponding to function $\\Phi \\in L^{2}({\\mathcal {L}}(E))$ having Fourier expansion $\\Phi (e^{it})=\\displaystyle \\sum _{n=-\\infty }^{\\infty }\\Phi _{n}e^{int}$ , with $\\Phi _n\\in {\\mathcal {L}}(E)$ .","Then it can be easily seen that with respect to the direct decomposition given above $A_\\Phi $ has a natural representation as a matrices: Let $\\Phi ({k})$ be the $k$ -th Fourier coefficient of $\\Phi $ .","Then ${\\mathcal {T}}({\\mathcal {K}}_\\Theta )=\\Biggl \\lbrace A_\\Phi =\\begin{pmatrix}\\Phi (0) & \\Phi (1)&\\cdots &\\Phi (N-1)\\\\\\Phi (-1)& \\Phi (0)&\\cdots & \\Phi (N-2)\\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\\\Phi (1-N)& \\Phi (2-N)& \\cdots &\\Phi (0)\\end{pmatrix}; \\Phi ({k})\\in {\\mathcal {L}}(E)\\Biggr \\rbrace $ Suppose now that $\\Phi ,\\Psi $ belong to a commutative algebra $\\mathfrak {F}$ .","Since $\\Theta $ is a scalar valued inner function the double commutation assumption on $\\Phi ,\\Psi $ is satisfied.","As stated in section 4, we can write the symbol $\\Phi \\in \\mathfrak {F}$ of any MTTO as $\\Phi =z\\Phi _{+}+\\bar{z}\\Phi _{-}+\\Phi _{0},$ where $\\Phi _{\\pm }\\in {\\mathcal {M}}_{\\Theta _1}$ and $\\Phi _{0}\\in {\\mathcal {L}}(E)$ .","Let $\\Phi _+(k)$ and $\\Phi _{-}(k)$ denote the Fourier coefficients of $\\Phi _{+}$ and $\\Phi _{-}$ respectively then we have $A_{z\\Phi _{+}}=\\begin{pmatrix}0 & 0 &\\cdots & 0\\\\\\Phi _+(0) & 0&\\cdots & 0 \\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\\\Phi _+(N-2) & \\Phi _+(N-3)& \\cdots &0\\end{pmatrix}$ and $A_{\\bar{z}\\Phi _{-}^{*}}=\\begin{pmatrix}0 & \\Phi _{-}^{*}(0) &\\cdots & \\Phi _{-}^{*}(N-2) \\\\0 & 0&\\cdots & \\Phi _{-}^{*}(N-3) \\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\0 & 0 & \\cdots &0\\end{pmatrix}$ We have therefore $A_\\Phi =A_{z\\Phi _{+}+\\bar{z}\\Phi _{-}+\\Phi _{0}}=\\begin{pmatrix}\\Phi _{0} & \\Phi _{-}^{*}(0) &\\cdots & \\Phi _{-}^{*}(N-2) \\\\\\Phi _{+}(0) & \\Phi _{0}&\\cdots & \\Phi _{-}^{*}(N-3) \\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\\\Phi _{+}(N-2) & \\Phi _{+}(N-3) & \\cdots &\\Phi _{0}\\end{pmatrix}$ Similarly, $A_\\Psi =A_{z\\Psi _{+}+\\bar{z}\\Psi _{-}+\\Psi _{0}}=\\begin{pmatrix}\\Psi _{0} & \\Psi _{-}^{*}(0) &\\cdots & \\Psi _{-}^{*}(N-2) \\\\\\Psi _{+}(0) & \\Psi _{0}&\\cdots & \\Psi _{-}^{*}(N-3) \\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\\\Psi _{+}(N-2) & \\Psi _{+}(N-3) & \\cdots &\\Psi _{0}\\end{pmatrix}$ Since $\\Theta (z)=z^NI_E$ , ${\\mathcal {M}}_\\Theta $ is the space of all polynomials in $z$ of degree at most $N-1$ with coefficients from ${\\mathcal {L}}(E)$ ; also, $\\Theta (0)=0$ imply that $\\Theta (z)=z\\Theta _1(z)$ , where $\\Theta _1$ is also inner.","Then ${\\mathcal {M}}_{\\Theta _1}$ is also the space of polynomials in $z$ with coefficients from ${\\mathcal {L}}(E)$ but of degree not greater than $N-2$ .","For any $x\\in E$ , $J_{z\\Phi _{+}}x=z\\Phi _{+}x$ , where $\\Phi _{+}\\in {\\mathcal {L}}(E)$ , and for any $f\\in {\\mathcal {K}}_\\Theta $ , $J_{z\\Psi _{-}}^*f=\\bar{z}\\Psi _{-}^*f$ , with $\\Psi _{-}\\in {\\mathcal {L}}(E)$ .","Now we have $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*f&=z\\Phi _{+}\\bar{z}\\Psi _{-}^*f=\\Phi _{+}\\Psi _{-}^*f=\\left(\\sum _{k=0}^{N-2}\\Phi _{+}(k)z^k\\right)\\left(\\sum _{k=0}^{N-2}\\Psi _{-}^*(k)\\bar{z}^k\\right)f\\\\&=\\left(\\sum _{k=0}^{N-2}\\Phi _{+}(k)\\Psi _{-}^*(k)+\\sum _{k=0}^{N-3}\\Phi _{+}(k)\\Psi _{-}^*(k+1)\\bar{z}+\\cdots \\Phi _{+}(0)\\Psi _{-}^*(N-2)\\bar{z}^{N-2}\\right)f+\\\\&\\left(\\sum _{k=0}^{N-3}\\Phi _{+}(k+1)\\Psi _{-}^*(k)z+\\sum _{k=0}^{N-4}\\Phi _{+}(k+2)\\Psi _{-}^*(k)z^2+\\cdots \\Phi _{+}(N-2)\\Psi _{-}^*(0)z^{N-2}\\right)f\\\\&=\\sum _{m=N-2}^{0}\\sum _{k=0}^{m}\\Phi _{+}(k)\\Psi _{-}^*(k+(N-2)-m)\\bar{z}^{N-2-m}f+\\\\&\\sum _{m=N-3}^{0}\\sum _{k=0}^{m}\\Phi _{+}(k-m+N-2)\\Psi _{-}^*(k)z^{N-2-m}f$ $\\begin{split}J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*f=\\sum _{m=N-2}^{0}\\sum _{k=0}^{m}\\Phi _{+}(k)\\Psi _{-}^*(k+(N-2)-m)\\bar{z}^{N-2-m}f+\\\\+\\sum _{m=N-3}^{0}\\sum _{k=0}^{m}\\Phi _{+}(k+(N-2)-m)\\Psi _{-}^*(k)z^{N-2-m}f\\end{split}$ and for any $f$ in ${\\mathcal {K}}_\\Theta $ $\\begin{split}J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})} J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*f=\\Phi _{-}^*\\Psi _{+}f=\\sum _{m=N-2}^{0}\\sum _{k=0}^{m}\\Phi _{-}^*(k)\\Psi _{+}(k+(N-2)-m)\\bar{z}^{N-2-m}f+\\\\+\\sum _{m=N-3}^{0}\\sum _{k=0}^{m}\\Phi _{-}^*(k-m+N-2)\\Psi _{+}(k)z^{N-2-m}f\\end{split}$ By Theorem REF $A_\\Phi A_\\Psi \\in {\\mathcal {T}}({\\mathcal {K}}_\\Theta )$ if and only if $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})} J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*$ .","Comparing corresponding coefficients we get $\\bar{z}^{N-2}:\\Phi _{+}(0)\\Psi _{-}^*(N-2)=\\Phi _{-}(0)^*\\Psi _{+}(N-2)\\Rightarrow \\Phi _{+}^*(0)\\Psi _{-}(N-2)=\\Phi _{-}^*(0)\\Psi _{+}(N-2)\\\\$ $\\bar{z}^{N-3}:\\Phi _{+}(0)\\Psi _{-}^*(N-2)+\\Phi _{+}(1)\\Psi _{-}^*(N-3)=\\Phi _{-}^*(0)\\Psi _{+}(N-2)+\\Phi _{-}^*(1)\\Psi _{+}(N-3)$ Using (REF ) in (REF ) we have $\\Phi _{+}(1)\\Psi _{-}^*(N-3)=\\Phi _{-}^*(1)\\Psi _{+}(N-3)$ .","In general $\\Phi _{+}(i)\\Psi _{-}^*(N-2-i)=\\Psi _{+}(N-2-i)\\Phi _{-}^*(0)\\quad \\hbox{for every}\\quad i=0,\\cdots ,N-2.$ In the same way compairing coefficients of $z,z^2,\\cdots z^{N-2}$ we obtain $\\Psi _{+}(i)\\Phi _{-}^*(N-2-i)=\\Phi _{+}(N-2-i)\\Psi _{-}^*(i)\\quad \\hbox{for every}\\quad i=0,1,\\cdots ,N-2$ If we take $A_i=\\Phi _{-}^*(i-1)$ , $A_{i-N}=\\Phi _{+}(i-(N-1))$ and $B_i=\\Psi _{-}^*(i-1)$ , $B_{i-N}=\\Psi _{+}(i-(N-1))$ for every $i=1,2,\\cdots , N-1$ then Lemma 3.1(i) of [8] imply that $A_\\Phi A_{\\Psi }$ is a block Toeplitz matrix.","Thus the condition $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})} J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*$ is equivalent to the condition Lemma 3.1(i) of [8]."],["Acknowledgements","The author is highly grateful to Dr. Dan Timotin for his valuable suggestions and comments."]],"string":"[\n [\n \"Product of Matrix Valued Truncated Toeplitz Operators\"\n ],\n [\n \"Abstract Let $A_\\\\Phi$ be a matrix valued truncated Toeplitz operator-the compression of multiplication operator to vector-valued model space $H^2(E)\\\\ominus \\\\Theta H^2(E)$, where $\\\\Theta$ is a matrix valued non constant inner function.\",\n \"Under supplementary assumptions, we find necessary and sufficient condition that the product $A_\\\\Phi A_\\\\Psi$ is itself a matrix valued truncated Toeplitz operator.\"\n ],\n [\n \"Introduction\",\n \"Toeplitz operators are the compressions of multiplication operator to the usual Hardy Hilbert space $H^2$ .\",\n \"In [3], Brown and Halmos describe the algebraic properties of Toeplitz operators.\",\n \"Among other things, they found necessary and sufficient conditions for the product of two Toeplitz operators to itself be a Toeplitz operator, namely that either the first operator’s symbol is antiholomorphic or the second operator’s symbol is holomorphic.\",\n \"In either case, the symbol of the product is the product of the symbols.\",\n \"In the last decade, a large amount of research has concentrated on a generalization of Toeplitz matrices, namely truncated Toeplitz operators.\",\n \"These are the compressions of multiplication operator to subspaces of the Hardy space which are invariant under the backward shift operator.\",\n \"They have been formally introduced in [10]; see [5] for a more recent survey.\",\n \"Sarason [10] found equivalents to several of Brown and Halmos’s results for truncated Toeplitz operators on the model spaces $H^2\\\\ominus \\\\theta H^2$ , where $\\\\theta $ is some non-constant inner function.\",\n \"The model spaces are the backward-shift invariant subspaces of $H^2$ (that they are backward shift invariant follows easily from the fact that $\\\\theta H^2$ is clearly shift invariant).\",\n \"We refer the reader to [1], [2], [5] (see also [6]) for the general theory of these operators.\",\n \"It is well know that the product of two truncated Toeplitz operators is not a truncated Toeplitz operator.\",\n \"In particular, in [11] Sedlock has investigated when a product of truncated Toeplitz operators is itself a truncated Toeplitz operator.\",\n \"Most recently, the basics of corresponding matrix valued truncated Toeplitz operators (MTTOs), which are compressions to ${\\\\mathcal {K}}_\\\\Theta $ of multiplications with matrix valued functions on $H^2(E)$ has been developed in [9].\",\n \"In view of the result of [11], there arises a basic question related to the product of matrix valued truncated Toeplitz operators that is when the product of two MTTOs is still an MTTO?\",\n \"But there is no such simple result in the general case, and we need some supplementary assumptions to obtain the main result Theorem 4.6.\",\n \"The purpose of the present paper is to adapt the approach in [4] to the case of MTTOs on an arbitrary model space.\",\n \"The plan of the paper is following: By means of Section 2, we want to make sure that the reader has become acquainted to model spaces and their operators and other useful facts from this area, needed when we are going to start the main work in section 4.\",\n \"In Section 3 we will introduce truncated Toeplitz operators (TTOs) and matrix valued truncated Toeplitz operators (MTTOs).\",\n \"The last section contains a particular case of MTTOs namely the Block Toeplitz matrices.\"\n ],\n [\n \"Model Spaces and Operators\",\n \"Let $\\\\mathbb {C}$ denote the complex plane, $\\\\mathbb {D}$ the unit disc in $\\\\mathbb {C}$ and $\\\\mathbb {T}$ one dimensional torus in $\\\\mathbb {C}$ .\",\n \"In the sequel $E$ will denote a fixed Hilbert space of dimension $d$ .\",\n \"We designate the algebra of bounded linear operators on $E$ by $\\\\mathcal {L}(E)$ and by ${\\\\mathcal {L}}(E,K)$ the space of all bounded linear operators from Hilbert space $E$ to a Hilbert space $K$ .\",\n \"The space $L^2(E)$ is defined as usual, by $L^2(E):=\\\\left\\\\lbrace f:\\\\mathbb {T}\\\\longrightarrow E: f(e^{it})=\\\\sum _{n=-\\\\infty }^{\\\\infty } a_ne^{int}: a_n\\\\in E,\\\\sum _{n=-\\\\infty }^{\\\\infty }\\\\Vert a_n\\\\Vert ^2<\\\\infty \\\\right\\\\rbrace $ endowed with the inner product $\\\\langle f , g \\\\rangle _{L^2 (E)} =\\\\int _{\\\\mathbb {T}}\\\\langle f(e^{it}), g(e^{it})\\\\rangle _E dm$ where $dm=\\\\frac{dt}{2\\\\pi }$ is the normalized Lebesgue measure on $\\\\mathbb {T}$ .\",\n \"The norm induced by the inner product is given by $\\\\Vert f\\\\Vert _{L^2(E)}=\\\\int _{\\\\mathbb {T}}\\\\Vert f\\\\Vert _{E}^2dm.$ If $\\\\dim E=1$ (i.e., $E = \\\\mathbb {C}$ ) then $L^2 (E)$ consists of scalar-valued functions and is denoted by $L^2$ .\",\n \"The Hardy space $H^2(E)$ is the subspace of $L^2(E)$ formed by the functions with vanishing negative Fourier coefficients; it can be identified with a space of $E$ - valued functions analytic in $\\\\mathbb {D}$ , from which the boundary values can be recovered almost everywhere through radial limits.\",\n \"One can also view $H^2(E)$ as the direct sum of $d$ standard $H^2$ spaces.\",\n \"We have the orthogonal decomposition $L^2(E)=H^2(E)\\\\oplus (zH^2(E))^*.$ The spaces $ L^\\\\infty (E)\\\\subset L^2(E) $ is formed by the essentially bounded functions with values in $ E $ ; then $ H^\\\\infty (E)\\\\subset H^2(E) $ are the functions in $ L^\\\\infty (E) $ with vanishing negative Fourier coefficients.\",\n \"Taking into account that $\\\\mathcal {L}(E)$ is a Hilbert space endowed with the Hilbert-Schmidt norm, we may similarly define $ H^2({\\\\mathcal {L}}(E)) \\\\subset L^2({\\\\mathcal {L}}(E))$ and $ H^\\\\infty ({\\\\mathcal {L}}(E)) \\\\subset L^\\\\infty ({\\\\mathcal {L}}(E)) $ .\",\n \"Note, however, that we prefer to consider on $ L^\\\\infty ({\\\\mathcal {L}}(E)) $ and $ H^\\\\infty ({\\\\mathcal {L}}(E)) $ the equivalent norm obtained by considering on $ {\\\\mathcal {L}}(E) $ the operator norm instead of the Hilbert-Schmidt norm.\",\n \"The space $L^2(\\\\mathcal {L}(E))$ may be identified with the matrices with all the entries in $L^2$ .\",\n \"We have an orthogonal decomposition $L^2 (\\\\mathcal {L}(E)) =[zH^2 \\\\mathcal {L}(E)]^*\\\\oplus H^2(\\\\mathcal {L}(E)).$ The space $ L^\\\\infty ({\\\\mathcal {L}}(E)) $ acts on $L^2 (E)$ by means of multiplication: to $\\\\Phi \\\\in \\\\ L^\\\\infty (\\\\mathcal {L}(E))$ we associate the operator $M_\\\\Phi $ defined by $M_\\\\Phi (f)=\\\\Phi f$ for all $f\\\\in L^2(E)$ .\",\n \"Let $S$ denote the forward shift operator $(Sf)( z ) = zf (z)$ on $H^2 (E)$ ; it is the restriction of $M_z$ to $H^2(E)$ .\",\n \"Its adjoint (the backward shift) is the operator $(S^*f)(z) =\\\\frac{f(z )-f(0)}{z}$ One sees easily that $I-SS^*$ is precisely the orthogonal projection onto the space of constant functions.\",\n \"The main object of study is formed by the model spaces and the operators acting on them.\",\n \"These are defined as follows.\",\n \"First, an inner function is an element $ \\\\Theta \\\\in H^2(\\\\mathcal {L}(E))$ whose boundary values are almost everywhere unitary operators in $\\\\mathcal {L}(E)$ .\",\n \"The inner function is called pure if $ \\\\Vert \\\\Theta (0)\\\\Vert <1 $ ; this is equivalent to requiring that $ \\\\Theta $ has no constant unitary part.\",\n \"Consider then a pure inner function $\\\\Theta $ , with values in $\\\\mathcal {L}(E)$ .\",\n \"The model space associated to a pure inner function $\\\\Theta $ , denoted by $\\\\mathcal {K}_{\\\\Theta }$ and is defined by $\\\\mathcal {K}_{\\\\Theta }=H^{2}(E)\\\\ominus \\\\Theta H^{2}(E).$ Just like the Beurling-type subspaces $\\\\Theta H^{2}(E)$ constitute nontrivial invariant subspaces for the unilateral shift $S$ on $H^{2}(E)$ , the subspaces $\\\\mathcal {K}_{\\\\Theta }$ play an analogous role for the backward shift $S^{*}$ .\",\n \"The orthogonal projection onto $\\\\mathcal {K}_{\\\\Theta }$ will be denoted by $P_{\\\\Theta }$ .\",\n \"It is also known that $ {\\\\mathcal {K}}_\\\\Theta \\\\cap H^\\\\infty (E) $ is dense in $ {\\\\mathcal {K}}_\\\\Theta $ .\",\n \"The analogous space of matrix-valued functions is denoted by $\\\\mathcal {M}_\\\\Theta $ ; it is the orthogonal complement of $\\\\Theta H^2(\\\\mathcal {L}(E))$ in $H^2(\\\\mathcal {L}(E))$ .\",\n \"The model operator $S_\\\\Theta \\\\in \\\\mathcal {L}(\\\\mathcal {K}_\\\\Theta )$ is defined by the formula $(S_\\\\Theta f)(z)=P_\\\\Theta (zf).$ The adjoint of $S_\\\\Theta $ is given by $(S_\\\\Theta ^*f)(z)=\\\\frac{f(z)-f(0)}{z};$ it is the restriction of the left shift in $H^2(E)$ to the $S^*$ -invariant subspace $\\\\mathcal {K}_\\\\Theta $ .\",\n \"Let us assume that $ \\\\Theta (0)=0 $ , so $ \\\\Theta =z\\\\Theta _1 $ , which is the case we will consider in the sequel.\",\n \"Then $ I-S_\\\\Theta S_\\\\Theta ^* $ is the projection $ P_0 $ onto the constant functions, which are contained in $ {\\\\mathcal {K}}_\\\\Theta $ , while $ I-S_\\\\Theta ^* S_\\\\Theta $ is the projection $P_{D_*}$ onto the space $ {\\\\mathcal {D}}_*=\\\\lbrace \\\\Theta _1 x: x\\\\in E \\\\rbrace $ (which is also contained in $ {\\\\mathcal {K}}_\\\\Theta $ ).\",\n \"The scalar valued model spaces and operators are obtained when $ \\\\dim E=1 $ ; that is, when $ E=\\\\mathbb {C}$ .\",\n \"We have then the classical spaces $ H^2\\\\subset L^2 $ and $L^\\\\infty $ .\",\n \"The inner function is a scalar inner function $ \\\\theta $ , and the model space is $ \\\\mathcal {K}_{\\\\theta }=H^{2}\\\\ominus \\\\theta H^{2} $ .\",\n \"In particular, in case $ \\\\theta (z)=z^n $ , $ {\\\\mathcal {K}}_\\\\theta $ becomes the $ n $ -dimensional space of polynomials of degree at most $ n-1 $ .\",\n \"Definition 2.1 A conjugation on a complex Hilbert space $\\\\mathcal {H}$ is a function $C:\\\\mathcal {H}\\\\longrightarrow \\\\mathcal {H}$ that is (i) conjugate linear: that is $C(\\\\alpha x+\\\\beta y)=\\\\overline{\\\\alpha }Cx +\\\\overline{\\\\beta }Cy$ for all $x,y\\\\in \\\\mathcal {H}$ and $\\\\alpha ,\\\\beta \\\\in \\\\mathbb {C}$ , (ii) involutive: $C^{2}=I$ , (iii) isometric: $\\\\Vert Cx\\\\Vert =\\\\Vert x\\\\Vert $ for all $x\\\\in \\\\mathcal {H}$ .\",\n \"The following result is an immediate consequence of a theorem in [9].\",\n \"Lemma 2.1 Suppose $ \\\\Theta (0)=0 $ , so $ \\\\Theta =z\\\\Theta _1 $ .\",\n \"Let $\\\\Gamma $ be a conjugation on $E$ , and suppose that $\\\\Theta (e^{it})^{*}=\\\\Gamma \\\\Theta (e^{it})\\\\Gamma $ a.e.\",\n \"on $\\\\mathbb {T}$ .\",\n \"Then the map $C_{\\\\Gamma }$ defined by $C_\\\\Gamma (f)=z\\\\Theta _{1}\\\\Gamma f$ is a conjugation on $z\\\\mathcal {K}_{\\\\Theta _1}$ .\"\n ],\n [\n \"Truncated Toeplitz Operators and Matrix valued Truncated Toeplitz Operators\",\n \"If $ \\\\phi \\\\in L^\\\\infty $ , then the compression of the multiplication operator $ M_\\\\phi $ to $ H^2 $ is called a Toeplitz operator and is denoted by $ T_\\\\phi $ .\",\n \"That means that $ T_\\\\phi =P_{H^2}M_\\\\phi |H^2 $ , where $P_H^2$ is the orthogonal projection of $L^2$ onto $H^2$ .\",\n \"More than a decade ago, Sarason has introduced in [10] the so-called truncated Toeplitz operators.\",\n \"Remember that $ P_\\\\theta $ is the orthogonal projection onto the model space $ {\\\\mathcal {K}}_\\\\theta $ .\",\n \"If $ \\\\phi \\\\in L^\\\\infty $ , then the truncated Toeplitz operator $ A^\\\\theta _\\\\phi $ is defined to be the compression of $ M_\\\\phi $ to $ {\\\\mathcal {K}}_\\\\theta $ .\",\n \"That means that $ A^\\\\theta _\\\\phi =P_\\\\theta M_\\\\phi |{\\\\mathcal {K}}_\\\\theta $ .\",\n \"In particular, we see that with this notation $ S_\\\\theta = A^\\\\theta _z $ .\",\n \"Let us now remember that for $ \\\\theta (z)=z^n $ the space $ {\\\\mathcal {K}}_\\\\theta $ is formed by the polynomials of degree not greater than $ n-1 $ .\",\n \"The monomials $ z^k $ , $ k=0,\\\\dots , n-1 $ form an orthonormal basis of $ {\\\\mathcal {K}}_{z^n} $ .\",\n \"If we write the matrix of an operator with respect to this basis, then one can see that truncated Toeplitz operators correspond precisely to Toeplitz matrices.\",\n \"Passing now beyond the scalar case, let us suppose that $\\\\Theta $ is a pure inner function.\",\n \"The analogue of truncated Toeplitz operators have been defined in [9], where they are called matrix valued truncated Toeplitz operators.\",\n \"Suppose then that $\\\\Phi \\\\in L^2(\\\\mathcal {L}(E))$ .\",\n \"Consider the linear map $f\\\\longrightarrow P_\\\\Theta (\\\\Phi f)$ , defined on $\\\\mathcal {K}_\\\\Theta \\\\cap H^\\\\infty (E)$ .\",\n \"If it is bounded, then it uniquely determines an operator in $\\\\mathcal {L}(\\\\mathcal {K}_\\\\Theta )$ , denoted by $A_\\\\Phi ^\\\\Theta $ , and called a matrix valued truncated Toeplitz operator (MTTO).\",\n \"The function $\\\\Phi $ is then called a symbol of the operator.\",\n \"We will usually drop the subscript $\\\\Theta $ , as we consider a fixed inner function.\",\n \"We denote by $\\\\mathcal {T}(\\\\mathcal {K}_{\\\\Theta })$ the space of all MTTOs on the model space $\\\\mathcal {K}_\\\\Theta $ .\",\n \"Note that if $ \\\\Phi \\\\in L^\\\\infty ({\\\\mathcal {L}}(E)) $ (that is, it is bounded), then it follows that $f\\\\longrightarrow P_\\\\Theta (\\\\Phi f)$ defines a bounded linear operator on the whole of $ {\\\\mathcal {K}}_\\\\Theta $ , and thus $A_\\\\Phi ^\\\\Theta \\\\in {\\\\mathcal {T}}({\\\\mathcal {K}}_\\\\Theta )$ .\",\n \"But we may have bounded MTTOs which have no bounded symbols, which is one of the complications of the theory.\",\n \"However, a result of [9] tells that any operator in $ {\\\\mathcal {T}}({\\\\mathcal {K}}_\\\\Theta ) $ has a symbol in $ {\\\\mathcal {M}}_\\\\Theta +({\\\\mathcal {M}}_\\\\Theta )^* $ ; this is why we will restrict in the sequel to considering operators $A_\\\\Phi ^\\\\Theta $ with $ \\\\Phi \\\\in {\\\\mathcal {M}}_\\\\Theta +({\\\\mathcal {M}}_\\\\Theta )^* $ .\",\n \"The operator $S_\\\\Theta $ is a simple example of MTTO; it is obtained by taking $\\\\Phi (z)=zI_E$ .\",\n \"This example is rather special because the symbol is scalar valued.\",\n \"It is immediate that $A_\\\\Phi ^*=A_{\\\\Phi ^*};$ so $\\\\mathcal {T}(\\\\mathcal {K}_{\\\\Theta })$ is a self adjoint linear space.\",\n \"In section 5 we will see that if $\\\\Theta (z)=z^N I_E$ , then the MTTOs obtained are actually familiar objects, namely block Toeplitz matrices of dimension $N$ , in which the entries are the matrices of dimension $d$ .\",\n \"The theory of Block Toeplitz matrices has been an inspiration for research in the domain of matrix valued truncated Toeplitz operators.\",\n \"In particular, it should be mentioned that some of the classes of block Toeplitz matrices which are closed to multiplication are found in [7] and [8].\",\n \"As supposed above, we will consider a fixed inner function $ \\\\Theta $ and different MTTOs acting on $ {\\\\mathcal {K}}_\\\\Theta $ .\",\n \"The symbols of these operators will be $ \\\\Phi , \\\\Psi ,\\\\dots $ .\",\n \"Before the ending of this section we need to quote result from [12].\",\n \"Proposition 3.1 Suppose $\\\\Theta $ be an inner function, and $\\\\Phi \\\\in H^\\\\infty (\\\\mathcal {L}(E))$ such that $\\\\Phi \\\\Theta =\\\\Theta \\\\Phi $ .\",\n \"Then (1) $\\\\Theta H^2(\\\\mathcal {L}(E))$ is invariant with respect to $M_\\\\Phi $ (and consequently $\\\\mathcal {K}_\\\\Theta )$ is invariant under $M_{\\\\Phi }^*$ .\",\n \"(2) $A_\\\\Phi S_\\\\Theta =S_\\\\Theta A_\\\\Phi $ and consequently $A_\\\\Phi ^*S_\\\\Theta ^*=S_\\\\Theta ^*$ Set $\\\\Delta =I-S_\\\\Theta S_\\\\Theta ^*$ .\",\n \"Note that $P_0$ is the orthogonal projection onto the constant functions contained in $\\\\mathcal {K}_\\\\Theta $ .\",\n \"The next result from [9] characterizes MTTOs among all operators on $ {\\\\mathcal {K}}_\\\\Theta $ .\",\n \"Proposition 3.2 A bounded operator $A$ on $\\\\mathcal {K}_\\\\Theta $ belongs to $\\\\mathcal {T}(\\\\mathcal {K}_\\\\Theta )$ if and only if there exist operators $B,B^\\\\prime $ on $\\\\mathcal {K}_\\\\Theta $ such that $\\\\Delta (A)=BP_0+P_0{B^\\\\prime }$ In this case $A=A_{\\\\Phi +{\\\\Phi ^\\\\prime }^*}$ , where $\\\\Phi ,\\\\Phi ^\\\\prime \\\\in H^2(\\\\mathcal {L}(E))$ .\"\n ],\n [\n \"Main results\",\n \"In view of the result of Sedlock [11], a natural question is to determine when is the product of two MTTOs still an MTTO.\",\n \"However, there is no such simple result in the general case, and we need some supplementary assumptions to obtain the main result, Theorem REF .\",\n \"The path we take is suggested by [4], but the matrix valued situation is much more complicated.\",\n \"We will consider in the rest of this section a fixed inner function $\\\\Theta \\\\in H^\\\\infty (E)$ subjected to the condition $\\\\Theta (0)=0$ .\",\n \"Then $\\\\Theta (z)=z\\\\Theta _{1}(z)$ , where $\\\\Theta _{1}\\\\in H^\\\\infty (E)$ is also inner.\",\n \"We have the orthogonal decomposition ${\\\\mathcal {K}}_\\\\Theta =E\\\\oplus z{\\\\mathcal {K}}_{\\\\Theta _1}.$ Take now $\\\\Phi \\\\in {\\\\mathcal {M}}_\\\\Theta +({\\\\mathcal {M}}_\\\\Theta )^* $ .\",\n \"We can write then $\\\\Phi =z\\\\Phi _{+}+\\\\bar{z}\\\\Phi _{-}^*+\\\\Phi _0$ with $\\\\Phi _\\\\pm \\\\in \\\\mathcal {M}_{\\\\Theta _1}$ and $\\\\Phi _0\\\\in \\\\mathcal {L}(E)$ .\",\n \"If $ \\\\Phi (e^{it})=\\\\displaystyle \\\\sum _{n=-\\\\infty }^{\\\\infty }\\\\Phi _n e^{int}$ with $\\\\Phi _n\\\\in \\\\mathcal {L}(E)$ , then $\\\\begin{split}\\\\Phi _+(z)&=\\\\sum _{n=1}^{\\\\infty }\\\\Phi _n z^n=\\\\sum _{n=1}^{\\\\infty } \\\\left( \\\\int \\\\Phi (e^{it})e^{-int}\\\\, dt \\\\right) z^n\\\\\\\\&=\\\\int \\\\Phi (e^{it}) \\\\left( \\\\sum _{n=1}^{\\\\infty } e^{-int}z^n \\\\right)dt =\\\\int \\\\Phi (e^{it}) \\\\frac{e^{it}z}{1-e^{it}z}dt.\\\\end{split}$ Remember that two operators $ A,B $ are said to doubly commute if $ AB=BA $ and $ AB^*=B^*A $ (whence it follows that also $ A^*B^*=B^*A^* $ and $ A^*B=BA^* $ ).\",\n \"Lemma 4.1 Suppose that $\\\\Theta (0)=0$ and $\\\\Phi ,\\\\Psi \\\\in \\\\mathcal {M}_\\\\Theta +({\\\\mathcal {M}}_\\\\Theta )^*$ such that $\\\\Phi (e^{it})\\\\Psi (e^{is})=\\\\Psi (e^{is})\\\\Phi (e^{it})$ for any $ t,s $ .\",\n \"(i) For any $ s,t $ we have $\\\\Phi _+(e^{it})\\\\Psi (e^{is})=\\\\Psi (e^{is})\\\\Phi _+(e^{it})$ and $\\\\Phi _-(e^{it})\\\\Psi (e^{is})=\\\\Psi (e^{is})\\\\Phi _-(e^{it})$ .\",\n \"(ii) If the values of $\\\\Phi ,\\\\Psi $ doubly commute with those of $\\\\Theta $ , then the same is true for $ \\\\Phi _{\\\\pm }, \\\\Psi _{\\\\pm } $ .\",\n \"(iii) If $\\\\Gamma $ is a conjugation on $E$ such that $\\\\Phi (e^{it})^*=\\\\Gamma \\\\Phi (e^{it})\\\\Gamma $ , then $\\\\Phi _\\\\pm (e^{it})^*=\\\\Gamma \\\\Phi _\\\\pm (e^{it})\\\\Gamma $ .\",\n \"We will give the proof only for one of the equalities in (ii); the rest are similar.\",\n \"Using (REF ), we have $\\\\begin{split}\\\\Psi (e^{is}) \\\\Phi _+(z)&= \\\\Psi (e^{is})\\\\int \\\\Phi (e^{it}) \\\\frac{e^{it}z}{1-e^{it}z}dt=\\\\int \\\\Psi (e^{is}) \\\\Phi (e^{it}) \\\\frac{e^{it}z}{1-e^{it}z}dt\\\\\\\\&=\\\\int \\\\Phi (e^{it})\\\\Psi (e^{is}) \\\\frac{e^{it}z}{1-e^{it}z}dt=\\\\left(\\\\int \\\\Phi (e^{it}) \\\\frac{e^{it}z}{1-e^{it}z}dt\\\\right)\\\\Psi (e^{is})=\\\\Phi _+(z)\\\\Psi (e^{is}).\\\\end{split}$ By taking radial limits a.e., one obtains the required commutativity.\",\n \"The next lemma gives an identification of elements in $ {\\\\mathcal {M}}_\\\\Theta $ .\",\n \"Lemma 4.2 The map $ \\\\Phi \\\\mapsto J_\\\\Phi $ , defined by $J_\\\\Phi (x)(z)=\\\\Phi (z)x.$ is a bijection between $ {\\\\mathcal {M}}_\\\\Theta $ and $ {\\\\mathcal {L}}(E, {\\\\mathcal {K}}_\\\\Theta ) $ .\",\n \"Fixing a basis $e_1,\\\\cdots ,e_d$ in $E$ and defining the transformation $J:E\\\\longrightarrow \\\\mathcal {K}_\\\\Theta $ as follow $J(e_k)=\\\\phi _k,\\\\qquad 1\\\\le k\\\\le d,$ where $\\\\phi _k\\\\in \\\\mathcal {K}_\\\\Theta $ for every $1\\\\le k\\\\le d$ .\",\n \"If we arrange $\\\\phi _k$ as a column vectors then we obtain $\\\\Phi \\\\in \\\\mathcal {M}_\\\\Theta $ .\",\n \"Conversely if we have $\\\\Phi \\\\in \\\\mathcal {M}_\\\\Theta $ then we obtain the map $J_\\\\Phi : E\\\\longrightarrow \\\\mathcal {K}_\\\\Theta $ which sends $e_k$ to $k$ th column of $\\\\Phi $ for every $1\\\\le k\\\\le d$ .\",\n \"Note that, if $\\\\Phi \\\\in \\\\mathcal {M}_\\\\Theta $ , then the relation between $J_\\\\Phi $ and $\\\\Phi $ is simply $J_\\\\Phi (x)(z)=\\\\Phi (z)x.$ We denote by $J_0$ the embedding of $E$ in $K_\\\\Theta $ ; that is for every $x\\\\in E$ , $J_0(x)=x$ .\",\n \"It is easy to see that in a given basis the matrices of functions in $z\\\\mathcal {M}_{\\\\Theta _1}$ are characterized by the fact that columns are functions in $z\\\\mathcal {K}_{\\\\Theta _1}$ .\",\n \"Finally, we define $\\\\mathbf {C}_\\\\Gamma (\\\\Phi ) $ by giving the action of $J_\\\\Phi $ on $x\\\\in E$ as $J_{\\\\mathbf {C}_\\\\Gamma (\\\\Phi )} x=C_\\\\Gamma (\\\\Phi \\\\Gamma x),$ where $ C_\\\\Gamma $ is defined by (REF ).\",\n \"In the rest of this section we assume that $\\\\mathfrak {F}$ is a commutative algebra of functions contained in ${\\\\mathcal {M}}_\\\\Theta +({\\\\mathcal {M}}_\\\\Theta )^*$ , such that all the elements of $\\\\mathfrak {F}$ doubly commute with those of $\\\\Theta $ .\",\n \"The next lemma is the main technical result of this section.\",\n \"Lemma 4.3 Suppose that $\\\\Theta (0)=0$ and $\\\\Phi ,\\\\Psi \\\\in \\\\mathfrak {F}$ .\",\n \"Then there exist operators $ X, Y\\\\in \\\\mathcal {M}_\\\\Theta $ such that $\\\\Delta (A_\\\\Phi A_\\\\Psi )=J_{z\\\\Phi _+}J_{z\\\\Psi _{-}}^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*+XP_0+P_0Y$ For any $\\\\Phi \\\\in \\\\mathcal {M}_{\\\\Theta }+({\\\\mathcal {M}}_\\\\Theta )^*$ we will denote $\\\\hat{\\\\Phi }=\\\\Phi -\\\\Phi _0$ .\",\n \"It follows easily from Lemma REF that $\\\\Phi _0\\\\Psi (e^{is})=\\\\Psi (e^{is})\\\\Phi _0$ and $\\\\Phi _0\\\\hat{\\\\Psi }(e^{is})=\\\\hat{\\\\Psi }(e^{is})\\\\Phi _0$ .\",\n \"In the same way one can obtain $\\\\Psi _0\\\\Phi (e^{is})=\\\\Phi (e^{is})\\\\Psi _0$ and $\\\\Psi _0\\\\hat{\\\\Phi }(e^{is})=\\\\hat{\\\\Phi }(e^{is})\\\\Psi _0$ .\",\n \"A similar argument works for double commutation with $\\\\Theta $ .\",\n \"Since $\\\\Phi _0,\\\\Psi _0\\\\in H^2(\\\\mathcal {L}(E))$ commutes with $\\\\Theta $ then by using Proposition REF $S_\\\\Theta $ commutes with $A_{\\\\Phi _0}$ and $A_{\\\\Psi _0}$ , and therefore $\\\\Delta (A_{\\\\Phi } A_{\\\\Psi })&=\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})+\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\Psi _0})+\\\\Delta (A_{\\\\Phi _0}A_{\\\\hat{\\\\Psi }})+\\\\Delta (A_{\\\\Phi _0}A_{\\\\Psi _0}).\\\\\\\\&=\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})+\\\\Psi _0\\\\Delta (A_{\\\\hat{\\\\Phi }})+\\\\Phi _0\\\\Delta (A_{\\\\hat{\\\\Psi }})+\\\\Phi _0\\\\Psi _0\\\\Delta (I)\\\\\\\\&=\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }}) +\\\\Psi _0\\\\Delta (A_{\\\\hat{\\\\Phi }})+\\\\Phi _0\\\\Delta (A_{\\\\hat{\\\\Psi }})+\\\\Phi _0\\\\Psi _0P_0.$ By Proposition REF there exist operators $B=z\\\\Phi _{+}$ and $B^\\\\prime =z\\\\Phi _{-}$ such that $\\\\Delta (A_{\\\\hat{\\\\Phi }})=z\\\\Phi _{+}P_0+P_0\\\\bar{z}\\\\Phi _{-}^*$ with $\\\\Phi _{\\\\pm }\\\\in \\\\mathcal {M}_{\\\\Theta _{1}}$ .\",\n \"Similarly $\\\\Delta (A_{\\\\hat{\\\\Psi }})=z\\\\Psi _{+}P_0+P_0\\\\bar{z}\\\\Psi _{-}^*$ and $\\\\Psi _{\\\\pm }\\\\in \\\\mathcal {M}_{\\\\Theta _{1}}$ .\",\n \"Using Lemma REF , we have $\\\\Delta (A_\\\\Phi A_\\\\Psi )&=\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})+\\\\Psi _0(z\\\\Phi _{+}P_0+P_0\\\\bar{z}\\\\Phi _{-}^*)+\\\\Phi _0(z\\\\Psi _{+}P_0+P_0\\\\bar{z}\\\\Psi _{-}^*)+\\\\Phi _0\\\\Psi _0P_0\\\\\\\\&=\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})+(\\\\Psi _0 z\\\\Phi _{+}+\\\\Phi _0z\\\\Psi _{+}+\\\\Phi _0 \\\\Psi _0)P_0+(\\\\Psi _0P_0 \\\\bar{z}\\\\Phi _{-}^*+\\\\Phi _0P_0\\\\bar{z}\\\\Psi _{-}^*)$ Since $P_0$ is the projection onto the constants then it must commute with $\\\\Phi _0$ and $\\\\Psi _0$ .\",\n \"Therefore $\\\\Delta (A_\\\\Phi A_\\\\Psi )=\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})+(\\\\Psi _0 z\\\\Phi _{+}+\\\\Phi _0z\\\\Psi _{+}+\\\\Phi _0\\\\Psi _0)P_0+P_0(\\\\Psi _0\\\\bar{z}\\\\Phi _{-}^*+\\\\Phi _0\\\\bar{z}\\\\Psi _{-}^*)$ Now, by using the definition of $\\\\Delta $ , $\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})&=A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }}-S_\\\\Theta A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*\\\\\\\\&=A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }}-A_{\\\\hat{\\\\Phi }}S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*+A_{\\\\hat{\\\\Phi }}S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*-S_\\\\Theta A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*\\\\\\\\&=A_{\\\\hat{\\\\Phi }}\\\\Delta (A_{\\\\hat{\\\\Psi }})+\\\\Delta (A_{\\\\hat{\\\\Phi }})S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*-S_\\\\Theta A_{\\\\hat{\\\\Phi }}P_{{\\\\mathcal {D}}_*}A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*\\\\\\\\&=A_{\\\\hat{\\\\Psi }}(z\\\\Phi _{+}P_0+P_0\\\\bar{z}\\\\Psi _{-}^*)+(z\\\\Phi _{+}P_0+P_0\\\\bar{z}\\\\Phi _{-}^*)S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_{\\\\Theta }^{*}-S_\\\\Theta A_{\\\\hat{\\\\Phi }}P_{{\\\\mathcal {D}}_*}A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*$ or $\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})= A_{\\\\hat{\\\\Phi }}z\\\\Psi _{+}P_0+A_{\\\\hat{\\\\Phi }}P_0\\\\bar{z}\\\\Psi _{-}^*+z\\\\Phi _{+}P_0 S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_{\\\\Theta }^{*} +P_0\\\\bar{z}\\\\Phi _{-}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_{\\\\Theta }^{*}-S_\\\\Theta A_{\\\\hat{\\\\Phi }}P_{{\\\\mathcal {D}}_*}A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*.$ Since the constant functions are in $\\\\mathcal {K}_{\\\\Theta }$ , we have $ P_0P_\\\\Theta =P_\\\\Theta P_0 $ .\",\n \"Also since $ P_0zf=0 $ , then $P_0S_\\\\Theta f=P_0P_\\\\Theta zf= P_\\\\Theta P_0 zf=0,$ So the third term in the left hand side of (REF ) is 0.\",\n \"The second term is $A_{\\\\hat{\\\\Phi }}P_0\\\\bar{z}\\\\Psi _{-}^*=P_\\\\Theta (z\\\\Phi _+ +\\\\bar{z}\\\\Phi _-^*)P_0\\\\bar{z}\\\\Psi _{-}^*.$ But, since $ \\\\Phi _+\\\\in \\\\mathcal {M}_{\\\\Theta _{1}} $ , we have, for any constant function $ x $ , $ \\\\Phi _+ x\\\\in \\\\mathcal {K}_{\\\\Theta _1} $ , $ z\\\\Phi _+ x\\\\in \\\\mathcal {K}_{\\\\Theta } $ , and therefore $ P_\\\\Theta z\\\\Phi _+P_0=z\\\\Phi _+P_0 $ .\",\n \"Also, for any constant function $ x $ , $ \\\\bar{z}\\\\Phi _-^*x\\\\perp H^2(E) $ , so $ P_\\\\Theta \\\\bar{z}\\\\Phi _-^*P_0=0 $ .\",\n \"So $\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})= A_{\\\\hat{\\\\Phi }}z\\\\Psi _{+}P_0+z\\\\Phi _+ P_0\\\\bar{z}\\\\Psi _{-}^*+P_0\\\\bar{z}\\\\Phi _{-}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_{\\\\Theta }^{*}-S_\\\\Theta A_{\\\\hat{\\\\Phi }}P_{{\\\\mathcal {D}}_*}A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*.$ Since $ J_0 $ is the embedding of the constants into $ \\\\mathcal {K}_\\\\Theta $ , we have, for $ f\\\\in \\\\mathcal {K}_\\\\Theta $ and $ x\\\\in E $ , $\\\\langle J_0^*f, x\\\\rangle =\\\\langle f,J_0 x\\\\rangle =\\\\langle f(0), x\\\\rangle $ whence $ J^*_0f=f(0) $ .\",\n \"So $J_\\\\Phi J_0^*f=J_\\\\Phi f(0)= \\\\Phi (z)f(0)=\\\\Phi P_0 f,\\\\quad \\\\hbox{for any}\\\\quad f\\\\in \\\\mathcal {K}_\\\\Theta .$ By taking adjoints we have $J_0 J_\\\\Phi ^*=P_0\\\\Phi ^*$ .\",\n \"Therefore we can write (REF ) as $\\\\Delta (A_{\\\\hat{\\\\Phi }}A_{\\\\hat{\\\\Psi }})= A_{\\\\hat{\\\\Phi }} J_{z\\\\Psi _{+}}J_0^*+J_{z\\\\Phi _{+}}P_0 J_{z\\\\Psi _{-}}^*+P_0\\\\bar{z}\\\\Phi _{-}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_{\\\\Theta }^{*}-S_\\\\Theta A_{\\\\hat{\\\\Phi }}P_{{\\\\mathcal {D}}_*}A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*.$ Since $ \\\\mathcal {D}_* $ is the space spanned by $ \\\\Theta _1 E $ , we can define an isometry $ V:E\\\\rightarrow \\\\mathcal {K}_\\\\Theta $ by the formula $ Vx=\\\\Theta _1 x $ , and, moreover, $P_{\\\\mathcal {D}_*}=VV^* $ .\",\n \"Also, we have $A_{\\\\hat{\\\\Phi }}Vx=P_\\\\Theta \\\\hat{\\\\Phi }Vx=P_\\\\Theta z\\\\Phi _+ Vx +P_\\\\Theta \\\\bar{z}\\\\Phi _-^* Vx.$ Then, using the commutativity between $ \\\\Theta $ and $ \\\\Phi _+ $ , $z\\\\Phi _+ V x=z\\\\Phi _+ \\\\Theta _1 x= z\\\\Theta _1\\\\Phi _+ x=\\\\Theta \\\\Phi _+ x\\\\perp \\\\mathcal {K}_\\\\Theta ,$ and so the first term in (REF ) is 0.\",\n \"We have also $\\\\bar{z}\\\\Phi _-^*V x&=\\\\bar{z}\\\\Phi _-^*\\\\Theta _1 x=\\\\Theta _1 \\\\bar{z} \\\\Phi _-^* x= \\\\Theta _1 \\\\bar{z} \\\\Phi _-^* \\\\Gamma \\\\Gamma x\\\\\\\\&=\\\\bar{z}( z\\\\Theta _1 \\\\Gamma (z \\\\Phi _-\\\\Gamma x))=\\\\bar{z} C_\\\\Gamma ( z \\\\Phi _-\\\\Gamma x).=\\\\bar{z}J_{\\\\mathbf {C}_\\\\Gamma }(z\\\\Phi _{-})x.$ Since $ C_\\\\Gamma $ is a conjugation on $ z\\\\mathcal {K}_{\\\\Theta _1} $ , $ \\\\bar{z} C_\\\\Gamma (z \\\\Phi _-\\\\Gamma x)\\\\in \\\\mathcal {K}_{\\\\Theta }$ , and therefore $P_\\\\Theta \\\\bar{z}\\\\Phi _-^*V x=\\\\bar{z}\\\\Phi _-^*V x= \\\\bar{z} C_\\\\Gamma z \\\\Phi _-\\\\Gamma x=\\\\bar{z}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}x.$ So $S_\\\\Theta A_{\\\\hat{\\\\Phi }}Vx= z \\\\bar{z} C_\\\\Gamma z \\\\Phi _-\\\\Gamma x= C_\\\\Gamma z \\\\Phi _-\\\\Gamma x=J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}x.$ Similarly, we obtain $S_\\\\Theta A_{\\\\hat{\\\\Psi }^*}Vx=C_\\\\Gamma z \\\\Psi _+\\\\Gamma x=J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})} x.$ Consequently, $S_\\\\Theta A_{\\\\hat{\\\\Phi }}V=J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}, \\\\qquad S_\\\\Theta A_{\\\\hat{\\\\Psi }^*}V=J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})},$ Finally, the last term in (REF ) is $S_\\\\Theta A_{\\\\hat{\\\\Phi }}P_{{\\\\mathcal {D}}_*}A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*=S_\\\\Theta A_{\\\\hat{\\\\Phi }}VV^*A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*=J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*.$ Combining (REF ) and (REF ) we get $\\\\Delta (A_{\\\\hat{\\\\Phi }} A_{\\\\hat{\\\\Psi }})=A_{\\\\hat{\\\\Phi }}J_{z\\\\Psi _+}J_0^*+J_{z\\\\Phi _+}P_0J_{z\\\\Psi _{-}}^*+J_0J_{z\\\\Phi _{-}}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*$ so we have $\\\\begin{split}\\\\Delta (A_\\\\Phi A_\\\\Psi )&=A_{\\\\hat{\\\\Phi }}J_{z\\\\Psi _+}J_0^*+J_{z\\\\Phi _+}P_0J_{z\\\\Psi _{-}}^*+J_0J_{z\\\\Phi _{-}}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*\\\\\\\\&+(\\\\Psi _0 z\\\\Phi _{+}+\\\\Phi _0z\\\\Psi _{+}+\\\\Phi _0\\\\Psi _0)P_0+P_0(\\\\Psi _0\\\\bar{z}\\\\Phi _{-}^*+\\\\Phi _0\\\\bar{z}\\\\Psi _{-}^*)\\\\end{split}$ Now we have the relations $J_\\\\phi J_0^*=\\\\Phi P_0, J_0J_\\\\Phi =P_0\\\\Phi ^*$ So the second term in (REF ) becomes $z\\\\Phi _{+}P_0(z\\\\Phi )^*=J_{z\\\\Phi _{+}}J_0^*J_0J_{z\\\\Phi _{-}}^*=J_{z\\\\Phi _{+}}J_{z\\\\Phi _{-}}^*$ Also, since $J_0: E\\\\longrightarrow \\\\mathcal {K}_{\\\\Theta }$ is the embedding of the constants, while $P_0:\\\\mathcal {K}_\\\\Theta \\\\longrightarrow \\\\mathcal {K}_\\\\Theta $ is the projection onto the constant functions, it follows immediately that $P_0J_0=J_0, J_0^*P_0=J_0^*$ Therefore we can write the first and third term in (REF ) as $A_{\\\\hat{\\\\Phi }}J_{z\\\\Psi _+}J_0^*=A_{\\\\hat{\\\\Phi }}J_{z\\\\Psi _+}J_0^*P_0,\\\\quad J_0J_{z\\\\Phi _{-}}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*=P_0J_0J_{z\\\\Phi _{-}}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*.$ So we have $\\\\begin{split}\\\\Delta (A_\\\\Phi A_\\\\Psi )&=A_{\\\\hat{\\\\Phi }}J_{z\\\\Psi _+}J_0^*P_0+J_{z\\\\Phi _+}J_{z\\\\Psi _{-}}^*+P_0J_0J_{z\\\\Phi _{-}}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*\\\\\\\\&+(\\\\Psi _0 z\\\\Phi _{+}+\\\\Phi _0z\\\\Psi _{+}+\\\\Phi _0\\\\Psi _0)P_0+P_0(\\\\Psi _0\\\\bar{z}\\\\Phi _{-}^*+\\\\Phi _0\\\\bar{z}\\\\Psi _{-}^*)\\\\\\\\&=J_{z\\\\Phi _+}J_{z\\\\Psi _{-}}^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*+[A_{\\\\hat{\\\\Phi }}J_{z\\\\Psi _+}J_0^*+\\\\Psi _0 z\\\\Phi _{+}+\\\\Phi _0z\\\\Psi _{+}+\\\\Phi _0\\\\Psi _0]P_0\\\\\\\\&+P_0[J_0J_{z\\\\Phi _{-}}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*+\\\\Psi _0\\\\bar{z}\\\\Phi _{-}^*+\\\\Phi _0\\\\bar{z}\\\\Psi _{-}^*]\\\\\\\\&=J_{z\\\\Phi _+}J_{z\\\\Psi _{-}}^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*+XP_0+P_0Y\\\\end{split}$ where, $X=A_{\\\\hat{\\\\Phi }}J_{z\\\\Psi _+}J_0^*+\\\\Psi _0 z\\\\Phi _{+}+\\\\Phi _0z\\\\Psi _{+}+\\\\Phi _0\\\\Psi _0$ and $Y=J_0J_{z\\\\Phi _{-}}^*S_\\\\Theta A_{\\\\hat{\\\\Psi }}S_\\\\Theta ^*+\\\\Psi _0\\\\bar{z}\\\\Phi _{-}^*+\\\\Phi _0\\\\bar{z}\\\\Psi _{-}^*$ .\",\n \"Theorem 4.4 Suppose $\\\\Theta (0)=0$ and $\\\\Phi ,\\\\Psi ,\\\\chi , \\\\zeta \\\\in \\\\mathfrak {F}$ .\",\n \"Then $A_\\\\Phi A_\\\\Psi -A_\\\\chi A_\\\\zeta \\\\in \\\\mathcal {T}(\\\\mathcal {K}_\\\\Theta )$ if and only if $ J_{z\\\\Phi _{+}}J_{z\\\\Psi _{-}}^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _-)}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*=J_{z\\\\chi _+}J_{z\\\\zeta _-}^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\chi _-)}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\zeta _+)}^*.$ By Lemma REF there exists operators $X, Y\\\\in \\\\mathcal {M}_\\\\Theta $ such that $\\\\Delta (A_\\\\Phi A_\\\\Psi -A_\\\\chi A_\\\\zeta )=J_{z\\\\Phi _{+}}J_{z\\\\Psi _{-}}^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}-J_{z\\\\chi _+}J_{z\\\\zeta _-}^*+J_{\\\\mathbf {C}_\\\\Gamma (z\\\\chi _-)}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\zeta _+)}+XP_0+P_0Y.$ By Proposition REF , we have $A_\\\\Phi A_\\\\Psi -A_\\\\chi A_\\\\zeta \\\\in \\\\mathcal {T}(\\\\mathcal {K}_\\\\Theta )$ if and only there exist operators $B, B^\\\\prime $ such that $\\\\Delta (A_\\\\Phi A_\\\\Psi -A_\\\\chi A_\\\\zeta )=BP_0+P_0B.$ The last two equations say that $A_\\\\Phi A_\\\\Psi -A_\\\\chi A_\\\\zeta \\\\in \\\\mathcal {T}(\\\\mathcal {K}_\\\\Theta )$ if and only if there exist $ X^{\\\\prime },Y^{\\\\prime } $ such that $J_{z\\\\Phi _{+}}J_{z\\\\Psi _{-}}^*-J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}-J_{z\\\\chi _+}J_{z\\\\zeta _-}^*+J_{\\\\mathbf {C}_\\\\Gamma (z\\\\chi _-)}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\zeta _+)}=X^{\\\\prime }P_0+P_0Y^\\\\prime .$ Since $\\\\Theta (0)=0$ then we can write $\\\\mathcal {K}_\\\\Theta =E\\\\oplus \\\\mathcal {K}_{\\\\Theta _{1}}$ .\",\n \"With respect to this decomposition, the left hand side of (REF ) has zeros on the first row and column, while the right hand side is the general form of an operator that has zeros in the lower right corner.\",\n \"Now it is clear that for (REF ) to be true both sides have to be zero, which proves the theorem.\",\n \"The following result is the main result of this paper: it gives the answer to the question stated at the beginning of this section, namelu when is the product of two MTTOs also an MTTO.\",\n \"Theorem 4.5 Suppose $\\\\Theta (0)=0$ , $\\\\Phi ,\\\\Psi \\\\in \\\\mathfrak {F}$ and $A_\\\\Phi , A_\\\\Psi , \\\\in \\\\mathcal {T}(\\\\mathcal {K}_\\\\Theta )$ .\",\n \"Then $A_\\\\Phi A_\\\\Psi \\\\in \\\\mathcal {T}(\\\\mathcal {K}_\\\\Theta )$ if and only if $J_{z\\\\Phi _{+}}J_{z\\\\Psi _{-}}^*=J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _-)}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*.$ Applying Theorem REF to the case $\\\\chi =\\\\zeta =0$ .\"\n ],\n [\n \"A particular case: Block Toeplitz Matrices\",\n \"In this section let $\\\\Theta (z)=z^NI_E$ for some fixed positive integer $N$ .\",\n \"Then ${\\\\mathcal {K}}_\\\\Theta $ is the Hilbert space of all polynomials in $z$ of degree at most $N-1$ with coefficients from $E$ ,i.e., ${\\\\mathcal {K}}_\\\\Theta =\\\\lbrace a_0+a_1z+\\\\cdots a_{N-1}z^{N-1};a_0,a_1,\\\\cdots ,a_{N-1}\\\\in E\\\\rbrace .$ One can also identify this space with N-copies of $E$ by mapping $\\\\displaystyle \\\\sum _{k=0}^{N-1}a_kz^k$ into $\\\\displaystyle \\\\otimes _{k=0}^{N-1}a_k$ .\",\n \"Now if we take $A_\\\\Phi $ corresponding to function $\\\\Phi \\\\in L^{2}({\\\\mathcal {L}}(E))$ having Fourier expansion $\\\\Phi (e^{it})=\\\\displaystyle \\\\sum _{n=-\\\\infty }^{\\\\infty }\\\\Phi _{n}e^{int}$ , with $\\\\Phi _n\\\\in {\\\\mathcal {L}}(E)$ .\",\n \"Then it can be easily seen that with respect to the direct decomposition given above $A_\\\\Phi $ has a natural representation as a matrices: Let $\\\\Phi ({k})$ be the $k$ -th Fourier coefficient of $\\\\Phi $ .\",\n \"Then ${\\\\mathcal {T}}({\\\\mathcal {K}}_\\\\Theta )=\\\\Biggl \\\\lbrace A_\\\\Phi =\\\\begin{pmatrix}\\\\Phi (0) & \\\\Phi (1)&\\\\cdots &\\\\Phi (N-1)\\\\\\\\\\\\Phi (-1)& \\\\Phi (0)&\\\\cdots & \\\\Phi (N-2)\\\\\\\\\\\\vdots & \\\\vdots & \\\\ddots &\\\\vdots \\\\\\\\\\\\Phi (1-N)& \\\\Phi (2-N)& \\\\cdots &\\\\Phi (0)\\\\end{pmatrix}; \\\\Phi ({k})\\\\in {\\\\mathcal {L}}(E)\\\\Biggr \\\\rbrace $ Suppose now that $\\\\Phi ,\\\\Psi $ belong to a commutative algebra $\\\\mathfrak {F}$ .\",\n \"Since $\\\\Theta $ is a scalar valued inner function the double commutation assumption on $\\\\Phi ,\\\\Psi $ is satisfied.\",\n \"As stated in section 4, we can write the symbol $\\\\Phi \\\\in \\\\mathfrak {F}$ of any MTTO as $\\\\Phi =z\\\\Phi _{+}+\\\\bar{z}\\\\Phi _{-}+\\\\Phi _{0},$ where $\\\\Phi _{\\\\pm }\\\\in {\\\\mathcal {M}}_{\\\\Theta _1}$ and $\\\\Phi _{0}\\\\in {\\\\mathcal {L}}(E)$ .\",\n \"Let $\\\\Phi _+(k)$ and $\\\\Phi _{-}(k)$ denote the Fourier coefficients of $\\\\Phi _{+}$ and $\\\\Phi _{-}$ respectively then we have $A_{z\\\\Phi _{+}}=\\\\begin{pmatrix}0 & 0 &\\\\cdots & 0\\\\\\\\\\\\Phi _+(0) & 0&\\\\cdots & 0 \\\\\\\\\\\\vdots & \\\\vdots & \\\\ddots &\\\\vdots \\\\\\\\\\\\Phi _+(N-2) & \\\\Phi _+(N-3)& \\\\cdots &0\\\\end{pmatrix}$ and $A_{\\\\bar{z}\\\\Phi _{-}^{*}}=\\\\begin{pmatrix}0 & \\\\Phi _{-}^{*}(0) &\\\\cdots & \\\\Phi _{-}^{*}(N-2) \\\\\\\\0 & 0&\\\\cdots & \\\\Phi _{-}^{*}(N-3) \\\\\\\\\\\\vdots & \\\\vdots & \\\\ddots &\\\\vdots \\\\\\\\0 & 0 & \\\\cdots &0\\\\end{pmatrix}$ We have therefore $A_\\\\Phi =A_{z\\\\Phi _{+}+\\\\bar{z}\\\\Phi _{-}+\\\\Phi _{0}}=\\\\begin{pmatrix}\\\\Phi _{0} & \\\\Phi _{-}^{*}(0) &\\\\cdots & \\\\Phi _{-}^{*}(N-2) \\\\\\\\\\\\Phi _{+}(0) & \\\\Phi _{0}&\\\\cdots & \\\\Phi _{-}^{*}(N-3) \\\\\\\\\\\\vdots & \\\\vdots & \\\\ddots &\\\\vdots \\\\\\\\\\\\Phi _{+}(N-2) & \\\\Phi _{+}(N-3) & \\\\cdots &\\\\Phi _{0}\\\\end{pmatrix}$ Similarly, $A_\\\\Psi =A_{z\\\\Psi _{+}+\\\\bar{z}\\\\Psi _{-}+\\\\Psi _{0}}=\\\\begin{pmatrix}\\\\Psi _{0} & \\\\Psi _{-}^{*}(0) &\\\\cdots & \\\\Psi _{-}^{*}(N-2) \\\\\\\\\\\\Psi _{+}(0) & \\\\Psi _{0}&\\\\cdots & \\\\Psi _{-}^{*}(N-3) \\\\\\\\\\\\vdots & \\\\vdots & \\\\ddots &\\\\vdots \\\\\\\\\\\\Psi _{+}(N-2) & \\\\Psi _{+}(N-3) & \\\\cdots &\\\\Psi _{0}\\\\end{pmatrix}$ Since $\\\\Theta (z)=z^NI_E$ , ${\\\\mathcal {M}}_\\\\Theta $ is the space of all polynomials in $z$ of degree at most $N-1$ with coefficients from ${\\\\mathcal {L}}(E)$ ; also, $\\\\Theta (0)=0$ imply that $\\\\Theta (z)=z\\\\Theta _1(z)$ , where $\\\\Theta _1$ is also inner.\",\n \"Then ${\\\\mathcal {M}}_{\\\\Theta _1}$ is also the space of polynomials in $z$ with coefficients from ${\\\\mathcal {L}}(E)$ but of degree not greater than $N-2$ .\",\n \"For any $x\\\\in E$ , $J_{z\\\\Phi _{+}}x=z\\\\Phi _{+}x$ , where $\\\\Phi _{+}\\\\in {\\\\mathcal {L}}(E)$ , and for any $f\\\\in {\\\\mathcal {K}}_\\\\Theta $ , $J_{z\\\\Psi _{-}}^*f=\\\\bar{z}\\\\Psi _{-}^*f$ , with $\\\\Psi _{-}\\\\in {\\\\mathcal {L}}(E)$ .\",\n \"Now we have $J_{z\\\\Phi _{+}}J_{z\\\\Psi _{-}}^*f&=z\\\\Phi _{+}\\\\bar{z}\\\\Psi _{-}^*f=\\\\Phi _{+}\\\\Psi _{-}^*f=\\\\left(\\\\sum _{k=0}^{N-2}\\\\Phi _{+}(k)z^k\\\\right)\\\\left(\\\\sum _{k=0}^{N-2}\\\\Psi _{-}^*(k)\\\\bar{z}^k\\\\right)f\\\\\\\\&=\\\\left(\\\\sum _{k=0}^{N-2}\\\\Phi _{+}(k)\\\\Psi _{-}^*(k)+\\\\sum _{k=0}^{N-3}\\\\Phi _{+}(k)\\\\Psi _{-}^*(k+1)\\\\bar{z}+\\\\cdots \\\\Phi _{+}(0)\\\\Psi _{-}^*(N-2)\\\\bar{z}^{N-2}\\\\right)f+\\\\\\\\&\\\\left(\\\\sum _{k=0}^{N-3}\\\\Phi _{+}(k+1)\\\\Psi _{-}^*(k)z+\\\\sum _{k=0}^{N-4}\\\\Phi _{+}(k+2)\\\\Psi _{-}^*(k)z^2+\\\\cdots \\\\Phi _{+}(N-2)\\\\Psi _{-}^*(0)z^{N-2}\\\\right)f\\\\\\\\&=\\\\sum _{m=N-2}^{0}\\\\sum _{k=0}^{m}\\\\Phi _{+}(k)\\\\Psi _{-}^*(k+(N-2)-m)\\\\bar{z}^{N-2-m}f+\\\\\\\\&\\\\sum _{m=N-3}^{0}\\\\sum _{k=0}^{m}\\\\Phi _{+}(k-m+N-2)\\\\Psi _{-}^*(k)z^{N-2-m}f$ $\\\\begin{split}J_{z\\\\Phi _{+}}J_{z\\\\Psi _{-}}^*f=\\\\sum _{m=N-2}^{0}\\\\sum _{k=0}^{m}\\\\Phi _{+}(k)\\\\Psi _{-}^*(k+(N-2)-m)\\\\bar{z}^{N-2-m}f+\\\\\\\\+\\\\sum _{m=N-3}^{0}\\\\sum _{k=0}^{m}\\\\Phi _{+}(k+(N-2)-m)\\\\Psi _{-}^*(k)z^{N-2-m}f\\\\end{split}$ and for any $f$ in ${\\\\mathcal {K}}_\\\\Theta $ $\\\\begin{split}J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})} J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*f=\\\\Phi _{-}^*\\\\Psi _{+}f=\\\\sum _{m=N-2}^{0}\\\\sum _{k=0}^{m}\\\\Phi _{-}^*(k)\\\\Psi _{+}(k+(N-2)-m)\\\\bar{z}^{N-2-m}f+\\\\\\\\+\\\\sum _{m=N-3}^{0}\\\\sum _{k=0}^{m}\\\\Phi _{-}^*(k-m+N-2)\\\\Psi _{+}(k)z^{N-2-m}f\\\\end{split}$ By Theorem REF $A_\\\\Phi A_\\\\Psi \\\\in {\\\\mathcal {T}}({\\\\mathcal {K}}_\\\\Theta )$ if and only if $J_{z\\\\Phi _{+}}J_{z\\\\Psi _{-}}^*=J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})} J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*$ .\",\n \"Comparing corresponding coefficients we get $\\\\bar{z}^{N-2}:\\\\Phi _{+}(0)\\\\Psi _{-}^*(N-2)=\\\\Phi _{-}(0)^*\\\\Psi _{+}(N-2)\\\\Rightarrow \\\\Phi _{+}^*(0)\\\\Psi _{-}(N-2)=\\\\Phi _{-}^*(0)\\\\Psi _{+}(N-2)\\\\\\\\$ $\\\\bar{z}^{N-3}:\\\\Phi _{+}(0)\\\\Psi _{-}^*(N-2)+\\\\Phi _{+}(1)\\\\Psi _{-}^*(N-3)=\\\\Phi _{-}^*(0)\\\\Psi _{+}(N-2)+\\\\Phi _{-}^*(1)\\\\Psi _{+}(N-3)$ Using (REF ) in (REF ) we have $\\\\Phi _{+}(1)\\\\Psi _{-}^*(N-3)=\\\\Phi _{-}^*(1)\\\\Psi _{+}(N-3)$ .\",\n \"In general $\\\\Phi _{+}(i)\\\\Psi _{-}^*(N-2-i)=\\\\Psi _{+}(N-2-i)\\\\Phi _{-}^*(0)\\\\quad \\\\hbox{for every}\\\\quad i=0,\\\\cdots ,N-2.$ In the same way compairing coefficients of $z,z^2,\\\\cdots z^{N-2}$ we obtain $\\\\Psi _{+}(i)\\\\Phi _{-}^*(N-2-i)=\\\\Phi _{+}(N-2-i)\\\\Psi _{-}^*(i)\\\\quad \\\\hbox{for every}\\\\quad i=0,1,\\\\cdots ,N-2$ If we take $A_i=\\\\Phi _{-}^*(i-1)$ , $A_{i-N}=\\\\Phi _{+}(i-(N-1))$ and $B_i=\\\\Psi _{-}^*(i-1)$ , $B_{i-N}=\\\\Psi _{+}(i-(N-1))$ for every $i=1,2,\\\\cdots , N-1$ then Lemma 3.1(i) of [8] imply that $A_\\\\Phi A_{\\\\Psi }$ is a block Toeplitz matrix.\",\n \"Thus the condition $J_{z\\\\Phi _{+}}J_{z\\\\Psi _{-}}^*=J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Phi _{-})} J_{\\\\mathbf {C}_\\\\Gamma (z\\\\Psi _{+})}^*$ is equivalent to the condition Lemma 3.1(i) of [8].\"\n ],\n [\n \"Acknowledgements\",\n \"The author is highly grateful to Dr. Dan Timotin for his valuable suggestions and comments.\"\n ]\n]"},"id":{"kind":"string","value":"2012.05279"}}},{"rowIdx":3699,"cells":{"text":{"kind":"list like","value":[["Off-shell Higgs Couplings in $H^*\\to ZZ\\to \\ell\\ell\\nu\\nu$"],["Abstract We explore the new physics reach for the off-shell Higgs boson measurement in the ${pp \\to H^* \\rightarrow Z(\\ell^{+}\\ell^{-})Z(\\nu\\bar{\\nu})}$ channel at the high-luminosity LHC.","The new physics sensitivity is parametrized in terms of the Higgs boson width, effective field theory framework, and a non-local Higgs-top coupling form factor.","Adopting Machine-learning techniques, we demonstrate that the combination of a large signal rate and a precise phenomenological probe for the process energy scale, due to the transverse $ZZ$ mass, leads to significant sensitivities beyond the existing results in the literature for the new physics scenarios considered."],["Introduction","After the Higgs boson discovery at the Large Hadron Collider (LHC) [1], [2], [3], [4], [5], the study of the Higgs properties has been one of the top priorities in searching for new physics beyond the Standard Model (BSM).","Indeed, the Higgs boson is a unique class in the SM particle spectrum and is most mysterious in many aspects.","The puzzles associated with the Higgs boson include the mass hierarchy between the unprotected electroweak (EW) scale ($v$ ) and the Planck scale ($M_{PL}$ ), the neutrino mass generation, the possible connection to dark matter, the nature of the electroweak phase transition in the early universe, to name a few.","Precision studies of the Higgs boson properties can be sensitive to new physics at a higher scale.","Parametrically, new physics at a scale $\\Lambda $ may result in the effects of the order $v^2/\\Lambda ^2$ .","So far, the measurements at the LHC based on the Higgs signal strength are in full agreement with the SM predictions.","However, these measurements mostly focus on the on-shell Higgs boson production, exploring the Higgs properties at low energy scales of the order $v$ .","It has been argued that if we explore the Higgs physics at a higher scale $Q$ , the sensitivity can be enhanced as $Q^2/\\Lambda ^2$ .","A particularly interesting option is to examine the Higgs sector across different energy scales, using the sizable off-shell Higgs boson rates at the LHC [6], [7], [8], [9], [10].","While the off-shell Higgs new physics sensitivity is typically derived at the LHC with the $H^* \\rightarrow ZZ \\rightarrow 4\\ell $ channel [11], [12], [13], [14], [15], [16], [17], [18], we demonstrate in this work that the extension to the channel $ZZ \\rightarrow \\ell \\ell \\nu \\nu $ can significantly contribute to the potential discoveries.","This channel provides two key ingredients to probe the high energy regime with enough statistics despite of the presence of two missing neutrinos in the final state.","First, it displays a larger event rate by a factor of six than the four charged lepton channel.","Second, the transverse mass for the $ZZ$ system sets the physical scale $Q^2$ and results in a precise phenomenological probe to the underlying physics.","In this paper, we extend the existing studies and carry out comprehensive analyses for an off-shell channel in the Higgs decay $pp \\rightarrow H^* \\rightarrow ZZ \\rightarrow \\ell ^+ \\ell ^-\\ \\nu \\bar{\\nu },$ where $\\ell =e,\\mu $ and $\\nu =\\nu _e,\\nu _\\mu ,\\nu _\\tau $ .","Because of the rather clean decay modes, we focus on the leading production channel of the Higgs boson via the gluon fusion.","First, we phenomenologically explore a theoretical scenario with additional unobserved Higgs decay channels leading to an increase in the Higgs boson width, $\\Gamma _H/\\Gamma _H^{SM}>1$ .","The distinctive dependence for the on-shell and off-shell cross-sections with the Higgs boson width foster the conditions for a precise measurement for this key ingredient of the Higgs sector.","We adopt the Machine-learning techniques in the form of Boosted Decision Tree (BDT) to enhance the signal sensitivity.","This analysis sets the stage for our followup explorations.","Second, we study the effective field theory framework, taking advantage of the characteristic energy-dependence from some of the operators.","Finally, we address a more general hypothesis that features a non-local momentum-dependent Higgs-top interaction [18], namely, a form factor, that generically represents the composite substructure.","Overall, the purpose of this paper is to highlight the complementarity across a multitude of frameworks [14], [13], [19], [15], [16], [17], [18] via the promising process at the LHC $H^*\\rightarrow Z(\\ell \\ell )Z(\\nu \\nu )$ , from models that predict invisible Higgs decays, passing by the effective field theory, and a non-local form-factor scenario.","Our results demonstrate significant sensitivities at the High-Luminosity LHC (HL-LHC) to the new physics scenarios considered here beyond the existing literature.","The rest of the paper is organized as follows.","In Sec.",", we derive the Higgs width limit at HL-LHC.","Next, in Sec.",", we study the new physics sensitivity within effective field theory framework.","In Sec.",", we scrutinize the effects of a non-local Higgs-top form-factor.","Finally, we present a summary in Sec.","."],["Higgs Boson Width","The combination of on-shell and off-shell Higgs boson rates addresses one of the major shortcomings of the LHC, namely the Higgs boson width measurement [6], [7].","This method breaks the degeneracy present on the on-shell Higgs coupling studies $\\sigma _{i\\rightarrow H \\rightarrow f}^{\\text{on-shell}}\\propto \\frac{g_i^2(m_H)g_f^2(m_H)}{\\Gamma _H}\\,,$ where the total on-shell rate can be kept constant under the transformation $g_{i,f}(m_H)\\rightarrow \\xi g_{i,f}(m_H)$ with $\\Gamma _H\\rightarrow \\xi ^4\\Gamma _H$ .","The off-shell Higgs rate, due to a sub-leading dependence on the Higgs boson width $\\Gamma _H$ $\\sigma _{i\\rightarrow H^* \\rightarrow f}^{\\text{off-shell}}\\propto g_i^2(\\sqrt{\\hat{s}})g_f^2(\\sqrt{\\hat{s}})\\,,$ breaks this degeneracy, where $\\sqrt{\\hat{s}}$ is the partonic c.m.","energy that characterizes the scale of the off-shell Higgs.","In particular, if the new physics effects result in the same coupling modifiers at both kinematical regimes [14], [13], [15], [16], the relative measurement of the on-shell and off-shell signal strengths can uncover the Higgs boson width, $\\mu _{\\text{off-shell}}/\\mu _{\\text{on-shell}}=\\Gamma _H/\\Gamma _H^{SM}$ .","In this section, we derive a projection for the Higgs boson width measurement at the ${\\sqrt{s}=14}$ TeV high-luminosity LHC, exploring the $ZZ\\rightarrow 2\\ell 2\\nu $ final state.","We consider the signal channel as in Eq.","(REF ).","The signal is characterized by two same-flavor opposite sign leptons, $\\ell =e$ or $\\mu $ , which reconstruct a $Z$ boson and recoil against a large missing transverse momentum from $Z\\rightarrow \\nu \\bar{\\nu }$ .","The major backgrounds for this search are the Drell-Yan (DY) processes $q\\bar{q}\\rightarrow ZZ, ZW$ and gluon fusion (GF) $gg\\rightarrow ZZ$ process, see Fig.","REF for a sample of the Feynman diagrams.","While the Drell-Yan component displays the largest rate, the gluon fusion box diagrams interfere with the Higgs signal, resulting in important contributions mostly at the off-shell Higgs regime [6].","In our calculations, the signal and background samples are generated with MadGraph5_aMC@NLO [20], [21].","The Drell-Yan background is generated at the NLO with the MC@NLO algorithm [22].","Higher order QCD effects to the loop-induced gluon fusion component are included via a universal $K$ -factor [8], [23].","Spin correlation effects for the $Z$ and $W$ bosons decays are obtained in our simulations with the MadSpin package [24].","The renormalization and factorization scales are set by the invariant mass of the gauge boson pair $Q= m_{VV}/2$ , using the PDF set nn23nlo [25].","Hadronization and underlying event effects are simulated with Pythia8 [26], and detector effects are accounted for with the Delphes3 package [27].","We start our analysis with some basic lepton selections.","We require two same-flavor and opposite sign leptons with $|\\eta _\\ell |<2.5$ and $p_{T\\ell }>10$ GeV in the invariant mass window $76~\\text{GeV} 175$ GeV and a minimum transverse mass for the $ZZ$ system ${m_T^{ZZ}>250}$ GeV, defined as $m_T ^{ZZ}= \\sqrt{\\left(\\sqrt{m_Z^2 + p_{T(\\ell \\ell )}^2} + \\sqrt{m_Z^2 + (E_T^{\\mathrm {miss}})^2}\\right)^2 - \\left| \\overrightarrow{p}_{TZ} + \\overrightarrow{E}_T^{\\mathrm {miss}}\\right|^2} \\, .$ The consistency of our event simulation and analysis setup is confirmed through a cross-check with the ATLAS study in Ref. [9].","To further control the large Drell-Yan background, a Boosted Decision Tree (BDT) is implemented via the Toolkit for Multivariate Data Analysis with ROOT (TMVA) [28].","The BDT is trained to distinguish the full background events from the $s$ -channel Higgs production.","The variables used in the BDT are missing transverse energy, the momenta and rapidity for the leading and sub-leading leptons $(p_T^{\\ell 1}, \\eta ^{\\ell 1}, p_T^{\\ell 2},\\eta ^{\\ell 2})$ , the leading jet $(p_T^{j1},\\eta ^{j1})$ , the separation between the two charged leptons $\\Delta R_{\\ell \\ell }$ , the azimuthal angle difference between the di-lepton system and the missing transverse energy $\\Delta \\phi (\\vec{p}_{T}^{~\\ell \\ell },\\vec{E}^{\\mathrm {miss}}_T)$ , and the scalar sum of jets and lepton transverse momenta $H_T$ .","Finally, we also include the polar $\\theta $ and azimuthal $\\phi $ angles of the charged lepton $\\ell ^- $ in the $Z$ rest frame [29], [30].","We choose the coordinate system for the $Z$ rest frame following Collins and Soper (Collins-Soper frame) [31].","The signal and background distributions for these observables are illustrated in Fig.","REF .","We observe significant differences between the $s$ -channel signal and background in the $(\\theta ,\\phi )$ angle distributions.","These kinematic features arise from the different $Z$ boson polarizations for the signal and background components at the large di-boson invariant mass $m_{T}^{ZZ}$ [15], [32].","Whereas the $s$ -channel Higgs tends to have $Z_L$ dominance, the DY background is mostly $Z_T$ dominated.","Figure: BDT distribution for the ss-channel Higgs signal (red) and background (blue).We would like to illustrate the power of the implemented BDT analysis to separate the $s$ -channel Higgs from the background contributions in Fig.","REF .","The BDT discriminator is defined in the range $[-1,1]$ .","The events with discriminant close to $-1$ are classified as background-like and those close to 1 are signal-like.","The optimal BDT score selection has been performed with TMVA.","To estimate the effectiveness of the BDT treatment, we note that one can reach $S/\\sqrt{S+B}=5$ at an integrated luminosity of $273~\\text{fb}^{-1}$ with signal efficiency 88% and background rejection of 34%, by requiring ${\\text{BDT}_{\\text{response}}>-0.26}$ .","Now that we have tamed the dominant backgrounds ${q\\bar{q}\\rightarrow ZZ,ZW}$ , we move on to the new physics sensitivity study.","To maximize the sensitivity of the Higgs width measurement, we explore the most sensitive variable, $m_{T}^{ZZ}$ distribution, and perform a binned log-likelihood ratio analysis.","In Fig.","REF , we display the 95% CL on the Higgs width $\\Gamma _H/ \\Gamma _H^{SM}$ as a function of the $\\sqrt{s}=14$ TeV LHC luminosity.","To infer the relevance of the multivariate analysis, that particularly explore the observables $(E_T^{\\mathrm {miss}},\\theta ,\\phi )$ depicted in Fig.","REF , we display the results in two analysis scenarios: in blue we show the cut-based analysis and in red the results accounting for the BDT-based framework.","The significant sensitivity enhancement due to the BDT highlights the importance of accounting for the full kinematic dependence, including the $Z$ -boson spin correlation effects.","Whereas the Higgs width can be constrained to $\\Gamma _H/ \\Gamma _H^{SM}<1.35$ at 95% CL level following the cut-based analysis, $\\Gamma _H/ \\Gamma _H^{SM}<1.31$ in the BDT-based study assuming $\\mathcal {L}=3~\\text{ab}^{-1}$ of data.","Hence, the BDT limits result in an improvement of $\\mathcal {O}(5\\%)$ on the final Higgs width sensitivity.","These results are competitive to the HL-LHC estimates for the four charged lepton final state derived by ATLAS and CMS, where the respective limits are $\\Gamma _H/ \\Gamma _H^{SM}<\\mathcal {O}(1.3)$ and $\\mathcal {O}(1.5)$ at 68% CL [33], [34].","Figure: 95% CL bound on the Higgs width Γ H /Γ H SM \\Gamma _H/ \\Gamma _H^{SM} as a function of the s=14\\sqrt{s}=14 TeV LHC luminosity.","We display the results forthe cut-based study (blue) and BDT-based analysis (red)."],["Effective Field Theory","The Effective Field Theory (EFT) provides a consistent framework to parametrize beyond the SM effects in the presence of a mass gap between the SM and new physics states.","In this context, the new physics states can be integrated out and parametrized in terms of higher dimension operators [35].","In this section we parametrize the new physics effects in terms of the EFT framework [36], [37].","Instead of performing a global coupling fit, we will focus on a relevant subset of higher dimension operators that affect the Higgs production via gluon fusion.","This will shed light on the new physics sensitivity for the off-shell $pp\\rightarrow H^*\\rightarrow Z(\\ell \\ell )Z(\\nu \\nu )$ channel.","Our effective Lagrangian can be written as $\\mathcal {L}\\supset & c_g \\frac{\\alpha _s}{12\\pi v^2}|\\mathcal {H}|^2G_{\\mu \\nu }G^{\\mu \\nu }+c_t\\frac{y_t}{v^2}|\\mathcal {H}|^2 \\bar{Q}_L\\tilde{\\mathcal {H}}t_R+\\text{h.c.} \\,\\,,$ where $\\mathcal {H}$ is the SM Higgs doublet and $v=246$ GeV is the vacuum expectation value of the SM Higgs field.","The couplings are normalized in such a way for future convenience.","If we wish to make connection with the new physics scale $\\Lambda $ , we would have the scaling as $c_g, c_t \\sim v^2/\\Lambda ^2$ .","After electroweak symmetry breaking, Eq.","(REF ) renders into the following interaction terms with a single Higgs boson $\\mathcal {L}\\supset & \\kappa _g \\frac{\\alpha _s}{12\\pi v}H G_{\\mu \\nu }G^{\\mu \\nu }-\\kappa _t \\frac{m_t}{v}H\\left(\\bar{t}_Rt_L+\\text{h.c.}\\right) \\,\\,,$ where the coupling modifiers $\\kappa _{g,t}$ and the Wilson coefficients $c_{g,t}$ are related by $\\kappa _g=c_g$ and $\\kappa _t=1-\\text{Re}(c_t)$ .","We depict in Fig.","REF the $gg\\rightarrow ZZ$ Feynman diagrams that account for these new physics effects.","Whereas Eq.","(REF ) represents only a sub-set of high dimensional operators affecting the Higgs interactions [36], [37], we focus on it to highlight the effectiveness for the off-shell Higgs measurements to resolve a notorious degeneracy involving these terms.","The gluon fusion Higgs production at low energy regime can be well approximated by the Higgs Low Energy Theorem [38], [39], where the total Higgs production cross-section scales as ${\\sigma _{\\text{GF}}\\propto |\\kappa _t+\\kappa _g|^2}$ .","Therefore, low energy measurements, such as on-shell and non-boosted Higgs production [40], [41], [42], [43], [44], [13], [45], [15], [46], are unable to resolve the ${|\\kappa _t+\\kappa _g|=\\text{constant}}$ degeneracy.","While the combination between the $t\\bar{t}H$ and gluon fusion Higgs production have the potential to break this blind direction [47], we will illustrate that the Higgs production at the off-shell regime can also result into relevant contributions to resolve this degeneracy.","Figure: Feynman diagrams for the GF gg→ZZgg\\rightarrow ZZprocess.","The new physics effects from Eq.","() display deviations on thecoefficients κ t \\kappa _t and κ g \\kappa _g from the SM point (κ t ,κ g )=(1,0)(\\kappa _t,\\kappa _g)=(1,0).Figure: Transverse mass distributions m T ZZ m_T^{ZZ} for the DY and GF Z(ℓℓ)Z(νν)Z(\\ell \\ell )Z(\\nu \\nu ) processes.","The new physics effectsare parametrized by deviations from SM point (κ t ,κ g )=(1,0)(\\kappa _t,\\kappa _g)=(1,0).","We follow the benchmark analysis defined in Sec .Since the Higgs boson decays mostly to longitudinal gauge bosons at the high energy regime, it is enlightening to inspect the signal amplitude for the longitudinal components.","The amplitudes associated to each contribution presented in Fig.","REF can be approximated at $m_{ZZ} \\gg m_t,m_H,m_Z$ by [48], [13], [15] $\\mathcal {M}_t^{++00} &\\approx + \\frac{m_t^2}{2m_Z^2} \\log ^2\\frac{m_{ZZ}^2}{m_t^2}\\,, \\\\\\mathcal {M}_g^{++00} &\\approx - \\frac{m_{ZZ}^2}{2m_Z^2}\\,, \\\\\\mathcal {M}_c^{++00} &\\approx - \\frac{m_t^2}{2m_Z^2} \\log ^2\\frac{m_{ZZ}^2}{m_t^2} \\,.$ Two comments are in order.","First, both the $s$ -channel top loop $\\mathcal {M}_t$ and the continuum $\\mathcal {M}_c$ amplitudes display logarithmic dependences on $m_{ZZ}/m_t$ at the far off-shell regime.","In the SM scenario the ultraviolet logarithm between these two amplitudes cancel, ensuring a proper high energy behavior when calculating the full amplitude.","Second, it is worth noting the difference in sign between the $s$ -channel contributions $\\mathcal {M}_t$ and $\\mathcal {M}_g$ .","This results into a destructive interference between $\\mathcal {M}_t$ and $\\mathcal {M}_c$ , contrasting to a constructive interference between $\\mathcal {M}_g$ and $\\mathcal {M}_c$ .","In the following, we will explore these phenomenological effects pinning down the new physics sensitivity with a higher precision.","Figure: 95% CL bound on the coupling modifiers κ t \\kappa _t and κ g \\kappa _g when accounting for the off-shell Higgs measurementin the Z(ℓℓ)Z(νν)Z(\\ell \\ell )Z(\\nu \\nu ) channel.","We assume the 14 TeV LHC with 3ab -1 3~\\text{ab}^{-1} of data.Exploiting the larger rate for $ZZ\\rightarrow \\ell \\ell \\nu \\nu $ than that for $ZZ\\rightarrow 4\\ell $ [13], [14], [15], we explore the off-shell Higgs physics at the HL-LHC.","To simulate the full loop-induced effects, we implemented Eq.","(REF ) into FeynRules/NLOCT [49], [50] through a new fermion state, and adjusting its parameters to match the low-energy Higgs interaction $HG_{\\mu \\nu }G^{\\mu \\nu }$ [38], [39].","Feynman rules are exported to a Universal FeynRules Output (UFO) [51] and the Monte Carlo event generation is performed with MadGraph5aMC@NLO [20].","In Fig.","REF , we present the Drell-Yan (DY) and the gluon-fusion (GF) $m_{T}^{ZZ}$ distributions for different signal hypotheses.","In the bottom panel, we display the ratio between the GF beyond the SM (BSM) scenarios with respect to the GF SM.","In agreement with Eq.","(REF ), we observe a suppression for the full process when accounting for the $s$ -channel top loop contributions and an enhancement when including the new physics terms associated to $\\mathcal {M}_g$ at high energies.","We follow the benchmark analysis defined in Sec. .","After the BDT study, the resulting events are used in a binned log-likelihood analysis with the $m_T^{ZZ}$ distribution.","This approach explores the characteristic high energy behavior for the new physics terms highlighted in Eq.","(REF ) and illustrated in Fig.","REF .","We present in Fig.","REF the resulting 95% CL sensitivity to the $(\\kappa _t,\\kappa _g)$ new physics parameters at the high-luminosity LHC.","In particular, we observe that the LHC can bound the top Yukawa within $\\kappa _t\\approx [0.4,1.1]$ at 95% CL, using this single off-shell channel.","The observed asymmetry in the limit, in respect to the SM point, arises from the large and negative interference term between the $s$ -channel and the continuum amplitudes.","The upper bound on $\\kappa _t$ is complementary to the direct Yukawa measurement via $ttH$ [52] and can be further improved through a combination with the additional relevant off-shell Higgs final states.","The results derived in this section are competitive to the CMS HL-LHC prediction that considers the boosted Higgs production combining the $H\\rightarrow 4\\ell $ and $H\\rightarrow \\gamma \\gamma $ channels [34].","The CMS projection results into an upper bound on the top Yukawa of $\\kappa _t\\lesssim 1.2$ at 95% CL."],["Higgs-Top Form Factor","The fact that the observed Higgs boson mass is much lighter than the Planck scale implies that there is an unnatural cancellation between the bare mass and the quantum corrections.","Since the mass of the Higgs particle is not protected from quantum corrections, it is well-motivated to consider that it may not be fundamental, but composite in nature [53], [54], [55], [56].","In such a scenario, the Higgs boson is proposed as a bound state of a strongly interacting sector with a composite scale $\\Lambda $ .","In addition, the top quark, which is the heaviest particle in the SM, can also be composite.","In this case, the top Yukawa coupling will be modified by a momentum-dependent form factor at a scale $q^2$ close to or above the new physics scale $\\Lambda ^2$ .","It is challenging to find a general construction for such form factor without knowing the underlying dynamics.","Here, we will adopt a phenomenological ansatz motivated by the nucleon form factor [57].","It is defined as $\\Gamma (q^2/\\Lambda ^2) = \\frac{1}{(1+q^2/\\Lambda ^2)^n}\\,,$ where $q^2$ is the virtuality of the Higgs boson.","For $n=2$ , it is a dipole-form factor and corresponds to an exponential spacial distribution.","Building upon Ref.","[18], we study the impact of this form factor on $ g g\\rightarrow H^{*} \\rightarrow Z Z $ process now with the complementary final state $ \\ell ^+ \\ell ^- \\nu \\overline{\\nu }$ .","Figure: 95% CL sensitivity on the new physics scale Λ\\Lambda as a function of the LHC luminosity.","We assume the form factor in Eq.","()with n=2n=2 (dashed line) and n=3n=3 (solid line) at the 14 TeV LHC.In Fig.","REF , we illustrate the $m_{T}^{ZZ}$ distribution for the full gluon fusion $ g g ( \\rightarrow H^* ) \\rightarrow Z Z $ process.","We show the Standard Model (black) and the form factor scenario (red).","We assume $n=2$ or 3 and $\\Lambda =1.5$ TeV for the depicted form factor scenarios.","The differences between Standard Model and form factor cases become larger when the energy scales are comparable or above $\\Lambda $ due to the suppression of destructive interference between Higgs signal and continuum background.","Thus, we perform the same BDT procedure introduced in Sec.","followed by a binned log-likelihood ratio test in the $m_T^{ZZ}$ distribution to fully explore this effect.","In Fig.","REF , we display the sensitivity reach for the LHC in the Higgs-top form factor.","We observe that the LHC can bound these new physics effects up to $\\Lambda = 1.5$ TeV for $n = 2 $ and $\\Lambda = 2.1$ TeV for $n=3$ at 95% CL.","The large event rate for the $H^*\\rightarrow ZZ\\rightarrow \\ell \\ell \\nu \\nu $ signal results in a more precise probe to the ultraviolet regime than for the $H^*\\rightarrow ZZ\\rightarrow 4\\ell $ channel, where the limits on the new physics scale are $\\Lambda = 0.8$ TeV for $n = 2 $ and $\\Lambda = 1.1$ TeV for $n=3$ at 95% CL [18]."],["Summary","We have systematically studied the off-shell Higgs production in the $pp\\rightarrow H^{*}\\rightarrow Z(\\ell \\ell )Z(\\nu \\nu )$ channel at the high-luminosity LHC.","We showed that this signature is crucial to probe the Higgs couplings across different energy scales potentially shedding light on new physics at the ultraviolet regime.","To illustrate its physics potential, we derived the LHC sensitivity to three BSM benchmark scenarios where the new physics effects are parametrized in terms of the Higgs boson width, the effective field theory framework, and a non-local Higgs-top coupling form factor.","The combination of a large signal rate and a precise phenomenological probe for the process energy scale, due to the transverse $ZZ$ mass, renders strong limits for all considered BSM scenarios.","A summary table and comparison with the existing results in the literature are provided in Table REF .","Adopting Machine-learning techniques, we demonstrated in the form of BDT that the HL-LHC, with $\\mathcal {L}=3~\\text{ab}^{-1}$ of data, will display large sensitivity to the Higgs boson width, $\\Gamma _H/ \\Gamma _H^{SM}<1.31$ .","In addition, the characteristic high energy behavior for the new physics terms within the EFT framework results in relevant bounds on the $(\\kappa _t,\\kappa _g)$ new physics parameters, resolving the low energy degeneracy in the gluon fusion Higgs production.","In particular, we observe that the LHC can bound the top Yukawa within $\\kappa _t\\approx [0.4,1.1]$ at 95% CL.","The upper bound on $\\kappa _t$ is complementary to the direct Yukawa measurement via $ttH$ and can be further improved in conjunction with additional relevant off-shell Higgs channels.","Finally, when considering a more general hypothesis that features a non-local momentum-dependent Higgs-top interaction, we obtain that the HL-LHC is sensitive to new physics effects at large energies with $\\Lambda = 1.5$ TeV for $n = 2 $ and $\\Lambda = 2.1$ TeV for $n=3$ at 95% CL.","We conclude that, utilizing the promising $H^*\\rightarrow Z(\\ell ^+\\ell ^-)Z(\\nu \\bar{\\nu })$ channel at the HL-LHC and adopting the Machine-Learning techniques, the combination of a large signal rate and a precise phenomenological probe for the process energy scale renders improved sensitivities beyond the existing literature, to all the three BSM scenarios considered in this work.","This work was supported by the U.S. Department of Energy under grant No.","DE-FG02- 95ER40896 and by the PITT PACC.","DG was supported by the U.S. Department of Energy under grant number DE-SC 0016013."]],"string":"[\n [\n \"Off-shell Higgs Couplings in $H^*\\\\to ZZ\\\\to \\\\ell\\\\ell\\\\nu\\\\nu$\"\n ],\n [\n \"Abstract We explore the new physics reach for the off-shell Higgs boson measurement in the ${pp \\\\to H^* \\\\rightarrow Z(\\\\ell^{+}\\\\ell^{-})Z(\\\\nu\\\\bar{\\\\nu})}$ channel at the high-luminosity LHC.\",\n \"The new physics sensitivity is parametrized in terms of the Higgs boson width, effective field theory framework, and a non-local Higgs-top coupling form factor.\",\n \"Adopting Machine-learning techniques, we demonstrate that the combination of a large signal rate and a precise phenomenological probe for the process energy scale, due to the transverse $ZZ$ mass, leads to significant sensitivities beyond the existing results in the literature for the new physics scenarios considered.\"\n ],\n [\n \"Introduction\",\n \"After the Higgs boson discovery at the Large Hadron Collider (LHC) [1], [2], [3], [4], [5], the study of the Higgs properties has been one of the top priorities in searching for new physics beyond the Standard Model (BSM).\",\n \"Indeed, the Higgs boson is a unique class in the SM particle spectrum and is most mysterious in many aspects.\",\n \"The puzzles associated with the Higgs boson include the mass hierarchy between the unprotected electroweak (EW) scale ($v$ ) and the Planck scale ($M_{PL}$ ), the neutrino mass generation, the possible connection to dark matter, the nature of the electroweak phase transition in the early universe, to name a few.\",\n \"Precision studies of the Higgs boson properties can be sensitive to new physics at a higher scale.\",\n \"Parametrically, new physics at a scale $\\\\Lambda $ may result in the effects of the order $v^2/\\\\Lambda ^2$ .\",\n \"So far, the measurements at the LHC based on the Higgs signal strength are in full agreement with the SM predictions.\",\n \"However, these measurements mostly focus on the on-shell Higgs boson production, exploring the Higgs properties at low energy scales of the order $v$ .\",\n \"It has been argued that if we explore the Higgs physics at a higher scale $Q$ , the sensitivity can be enhanced as $Q^2/\\\\Lambda ^2$ .\",\n \"A particularly interesting option is to examine the Higgs sector across different energy scales, using the sizable off-shell Higgs boson rates at the LHC [6], [7], [8], [9], [10].\",\n \"While the off-shell Higgs new physics sensitivity is typically derived at the LHC with the $H^* \\\\rightarrow ZZ \\\\rightarrow 4\\\\ell $ channel [11], [12], [13], [14], [15], [16], [17], [18], we demonstrate in this work that the extension to the channel $ZZ \\\\rightarrow \\\\ell \\\\ell \\\\nu \\\\nu $ can significantly contribute to the potential discoveries.\",\n \"This channel provides two key ingredients to probe the high energy regime with enough statistics despite of the presence of two missing neutrinos in the final state.\",\n \"First, it displays a larger event rate by a factor of six than the four charged lepton channel.\",\n \"Second, the transverse mass for the $ZZ$ system sets the physical scale $Q^2$ and results in a precise phenomenological probe to the underlying physics.\",\n \"In this paper, we extend the existing studies and carry out comprehensive analyses for an off-shell channel in the Higgs decay $pp \\\\rightarrow H^* \\\\rightarrow ZZ \\\\rightarrow \\\\ell ^+ \\\\ell ^-\\\\ \\\\nu \\\\bar{\\\\nu },$ where $\\\\ell =e,\\\\mu $ and $\\\\nu =\\\\nu _e,\\\\nu _\\\\mu ,\\\\nu _\\\\tau $ .\",\n \"Because of the rather clean decay modes, we focus on the leading production channel of the Higgs boson via the gluon fusion.\",\n \"First, we phenomenologically explore a theoretical scenario with additional unobserved Higgs decay channels leading to an increase in the Higgs boson width, $\\\\Gamma _H/\\\\Gamma _H^{SM}>1$ .\",\n \"The distinctive dependence for the on-shell and off-shell cross-sections with the Higgs boson width foster the conditions for a precise measurement for this key ingredient of the Higgs sector.\",\n \"We adopt the Machine-learning techniques in the form of Boosted Decision Tree (BDT) to enhance the signal sensitivity.\",\n \"This analysis sets the stage for our followup explorations.\",\n \"Second, we study the effective field theory framework, taking advantage of the characteristic energy-dependence from some of the operators.\",\n \"Finally, we address a more general hypothesis that features a non-local momentum-dependent Higgs-top interaction [18], namely, a form factor, that generically represents the composite substructure.\",\n \"Overall, the purpose of this paper is to highlight the complementarity across a multitude of frameworks [14], [13], [19], [15], [16], [17], [18] via the promising process at the LHC $H^*\\\\rightarrow Z(\\\\ell \\\\ell )Z(\\\\nu \\\\nu )$ , from models that predict invisible Higgs decays, passing by the effective field theory, and a non-local form-factor scenario.\",\n \"Our results demonstrate significant sensitivities at the High-Luminosity LHC (HL-LHC) to the new physics scenarios considered here beyond the existing literature.\",\n \"The rest of the paper is organized as follows.\",\n \"In Sec.\",\n \", we derive the Higgs width limit at HL-LHC.\",\n \"Next, in Sec.\",\n \", we study the new physics sensitivity within effective field theory framework.\",\n \"In Sec.\",\n \", we scrutinize the effects of a non-local Higgs-top form-factor.\",\n \"Finally, we present a summary in Sec.\",\n \".\"\n ],\n [\n \"Higgs Boson Width\",\n \"The combination of on-shell and off-shell Higgs boson rates addresses one of the major shortcomings of the LHC, namely the Higgs boson width measurement [6], [7].\",\n \"This method breaks the degeneracy present on the on-shell Higgs coupling studies $\\\\sigma _{i\\\\rightarrow H \\\\rightarrow f}^{\\\\text{on-shell}}\\\\propto \\\\frac{g_i^2(m_H)g_f^2(m_H)}{\\\\Gamma _H}\\\\,,$ where the total on-shell rate can be kept constant under the transformation $g_{i,f}(m_H)\\\\rightarrow \\\\xi g_{i,f}(m_H)$ with $\\\\Gamma _H\\\\rightarrow \\\\xi ^4\\\\Gamma _H$ .\",\n \"The off-shell Higgs rate, due to a sub-leading dependence on the Higgs boson width $\\\\Gamma _H$ $\\\\sigma _{i\\\\rightarrow H^* \\\\rightarrow f}^{\\\\text{off-shell}}\\\\propto g_i^2(\\\\sqrt{\\\\hat{s}})g_f^2(\\\\sqrt{\\\\hat{s}})\\\\,,$ breaks this degeneracy, where $\\\\sqrt{\\\\hat{s}}$ is the partonic c.m.\",\n \"energy that characterizes the scale of the off-shell Higgs.\",\n \"In particular, if the new physics effects result in the same coupling modifiers at both kinematical regimes [14], [13], [15], [16], the relative measurement of the on-shell and off-shell signal strengths can uncover the Higgs boson width, $\\\\mu _{\\\\text{off-shell}}/\\\\mu _{\\\\text{on-shell}}=\\\\Gamma _H/\\\\Gamma _H^{SM}$ .\",\n \"In this section, we derive a projection for the Higgs boson width measurement at the ${\\\\sqrt{s}=14}$ TeV high-luminosity LHC, exploring the $ZZ\\\\rightarrow 2\\\\ell 2\\\\nu $ final state.\",\n \"We consider the signal channel as in Eq.\",\n \"(REF ).\",\n \"The signal is characterized by two same-flavor opposite sign leptons, $\\\\ell =e$ or $\\\\mu $ , which reconstruct a $Z$ boson and recoil against a large missing transverse momentum from $Z\\\\rightarrow \\\\nu \\\\bar{\\\\nu }$ .\",\n \"The major backgrounds for this search are the Drell-Yan (DY) processes $q\\\\bar{q}\\\\rightarrow ZZ, ZW$ and gluon fusion (GF) $gg\\\\rightarrow ZZ$ process, see Fig.\",\n \"REF for a sample of the Feynman diagrams.\",\n \"While the Drell-Yan component displays the largest rate, the gluon fusion box diagrams interfere with the Higgs signal, resulting in important contributions mostly at the off-shell Higgs regime [6].\",\n \"In our calculations, the signal and background samples are generated with MadGraph5_aMC@NLO [20], [21].\",\n \"The Drell-Yan background is generated at the NLO with the MC@NLO algorithm [22].\",\n \"Higher order QCD effects to the loop-induced gluon fusion component are included via a universal $K$ -factor [8], [23].\",\n \"Spin correlation effects for the $Z$ and $W$ bosons decays are obtained in our simulations with the MadSpin package [24].\",\n \"The renormalization and factorization scales are set by the invariant mass of the gauge boson pair $Q= m_{VV}/2$ , using the PDF set nn23nlo [25].\",\n \"Hadronization and underlying event effects are simulated with Pythia8 [26], and detector effects are accounted for with the Delphes3 package [27].\",\n \"We start our analysis with some basic lepton selections.\",\n \"We require two same-flavor and opposite sign leptons with $|\\\\eta _\\\\ell |<2.5$ and $p_{T\\\\ell }>10$ GeV in the invariant mass window $76~\\\\text{GeV} 175$ GeV and a minimum transverse mass for the $ZZ$ system ${m_T^{ZZ}>250}$ GeV, defined as $m_T ^{ZZ}= \\\\sqrt{\\\\left(\\\\sqrt{m_Z^2 + p_{T(\\\\ell \\\\ell )}^2} + \\\\sqrt{m_Z^2 + (E_T^{\\\\mathrm {miss}})^2}\\\\right)^2 - \\\\left| \\\\overrightarrow{p}_{TZ} + \\\\overrightarrow{E}_T^{\\\\mathrm {miss}}\\\\right|^2} \\\\, .$ The consistency of our event simulation and analysis setup is confirmed through a cross-check with the ATLAS study in Ref. [9].\",\n \"To further control the large Drell-Yan background, a Boosted Decision Tree (BDT) is implemented via the Toolkit for Multivariate Data Analysis with ROOT (TMVA) [28].\",\n \"The BDT is trained to distinguish the full background events from the $s$ -channel Higgs production.\",\n \"The variables used in the BDT are missing transverse energy, the momenta and rapidity for the leading and sub-leading leptons $(p_T^{\\\\ell 1}, \\\\eta ^{\\\\ell 1}, p_T^{\\\\ell 2},\\\\eta ^{\\\\ell 2})$ , the leading jet $(p_T^{j1},\\\\eta ^{j1})$ , the separation between the two charged leptons $\\\\Delta R_{\\\\ell \\\\ell }$ , the azimuthal angle difference between the di-lepton system and the missing transverse energy $\\\\Delta \\\\phi (\\\\vec{p}_{T}^{~\\\\ell \\\\ell },\\\\vec{E}^{\\\\mathrm {miss}}_T)$ , and the scalar sum of jets and lepton transverse momenta $H_T$ .\",\n \"Finally, we also include the polar $\\\\theta $ and azimuthal $\\\\phi $ angles of the charged lepton $\\\\ell ^- $ in the $Z$ rest frame [29], [30].\",\n \"We choose the coordinate system for the $Z$ rest frame following Collins and Soper (Collins-Soper frame) [31].\",\n \"The signal and background distributions for these observables are illustrated in Fig.\",\n \"REF .\",\n \"We observe significant differences between the $s$ -channel signal and background in the $(\\\\theta ,\\\\phi )$ angle distributions.\",\n \"These kinematic features arise from the different $Z$ boson polarizations for the signal and background components at the large di-boson invariant mass $m_{T}^{ZZ}$ [15], [32].\",\n \"Whereas the $s$ -channel Higgs tends to have $Z_L$ dominance, the DY background is mostly $Z_T$ dominated.\",\n \"Figure: BDT distribution for the ss-channel Higgs signal (red) and background (blue).We would like to illustrate the power of the implemented BDT analysis to separate the $s$ -channel Higgs from the background contributions in Fig.\",\n \"REF .\",\n \"The BDT discriminator is defined in the range $[-1,1]$ .\",\n \"The events with discriminant close to $-1$ are classified as background-like and those close to 1 are signal-like.\",\n \"The optimal BDT score selection has been performed with TMVA.\",\n \"To estimate the effectiveness of the BDT treatment, we note that one can reach $S/\\\\sqrt{S+B}=5$ at an integrated luminosity of $273~\\\\text{fb}^{-1}$ with signal efficiency 88% and background rejection of 34%, by requiring ${\\\\text{BDT}_{\\\\text{response}}>-0.26}$ .\",\n \"Now that we have tamed the dominant backgrounds ${q\\\\bar{q}\\\\rightarrow ZZ,ZW}$ , we move on to the new physics sensitivity study.\",\n \"To maximize the sensitivity of the Higgs width measurement, we explore the most sensitive variable, $m_{T}^{ZZ}$ distribution, and perform a binned log-likelihood ratio analysis.\",\n \"In Fig.\",\n \"REF , we display the 95% CL on the Higgs width $\\\\Gamma _H/ \\\\Gamma _H^{SM}$ as a function of the $\\\\sqrt{s}=14$ TeV LHC luminosity.\",\n \"To infer the relevance of the multivariate analysis, that particularly explore the observables $(E_T^{\\\\mathrm {miss}},\\\\theta ,\\\\phi )$ depicted in Fig.\",\n \"REF , we display the results in two analysis scenarios: in blue we show the cut-based analysis and in red the results accounting for the BDT-based framework.\",\n \"The significant sensitivity enhancement due to the BDT highlights the importance of accounting for the full kinematic dependence, including the $Z$ -boson spin correlation effects.\",\n \"Whereas the Higgs width can be constrained to $\\\\Gamma _H/ \\\\Gamma _H^{SM}<1.35$ at 95% CL level following the cut-based analysis, $\\\\Gamma _H/ \\\\Gamma _H^{SM}<1.31$ in the BDT-based study assuming $\\\\mathcal {L}=3~\\\\text{ab}^{-1}$ of data.\",\n \"Hence, the BDT limits result in an improvement of $\\\\mathcal {O}(5\\\\%)$ on the final Higgs width sensitivity.\",\n \"These results are competitive to the HL-LHC estimates for the four charged lepton final state derived by ATLAS and CMS, where the respective limits are $\\\\Gamma _H/ \\\\Gamma _H^{SM}<\\\\mathcal {O}(1.3)$ and $\\\\mathcal {O}(1.5)$ at 68% CL [33], [34].\",\n \"Figure: 95% CL bound on the Higgs width Γ H /Γ H SM \\\\Gamma _H/ \\\\Gamma _H^{SM} as a function of the s=14\\\\sqrt{s}=14 TeV LHC luminosity.\",\n \"We display the results forthe cut-based study (blue) and BDT-based analysis (red).\"\n ],\n [\n \"Effective Field Theory\",\n \"The Effective Field Theory (EFT) provides a consistent framework to parametrize beyond the SM effects in the presence of a mass gap between the SM and new physics states.\",\n \"In this context, the new physics states can be integrated out and parametrized in terms of higher dimension operators [35].\",\n \"In this section we parametrize the new physics effects in terms of the EFT framework [36], [37].\",\n \"Instead of performing a global coupling fit, we will focus on a relevant subset of higher dimension operators that affect the Higgs production via gluon fusion.\",\n \"This will shed light on the new physics sensitivity for the off-shell $pp\\\\rightarrow H^*\\\\rightarrow Z(\\\\ell \\\\ell )Z(\\\\nu \\\\nu )$ channel.\",\n \"Our effective Lagrangian can be written as $\\\\mathcal {L}\\\\supset & c_g \\\\frac{\\\\alpha _s}{12\\\\pi v^2}|\\\\mathcal {H}|^2G_{\\\\mu \\\\nu }G^{\\\\mu \\\\nu }+c_t\\\\frac{y_t}{v^2}|\\\\mathcal {H}|^2 \\\\bar{Q}_L\\\\tilde{\\\\mathcal {H}}t_R+\\\\text{h.c.} \\\\,\\\\,,$ where $\\\\mathcal {H}$ is the SM Higgs doublet and $v=246$ GeV is the vacuum expectation value of the SM Higgs field.\",\n \"The couplings are normalized in such a way for future convenience.\",\n \"If we wish to make connection with the new physics scale $\\\\Lambda $ , we would have the scaling as $c_g, c_t \\\\sim v^2/\\\\Lambda ^2$ .\",\n \"After electroweak symmetry breaking, Eq.\",\n \"(REF ) renders into the following interaction terms with a single Higgs boson $\\\\mathcal {L}\\\\supset & \\\\kappa _g \\\\frac{\\\\alpha _s}{12\\\\pi v}H G_{\\\\mu \\\\nu }G^{\\\\mu \\\\nu }-\\\\kappa _t \\\\frac{m_t}{v}H\\\\left(\\\\bar{t}_Rt_L+\\\\text{h.c.}\\\\right) \\\\,\\\\,,$ where the coupling modifiers $\\\\kappa _{g,t}$ and the Wilson coefficients $c_{g,t}$ are related by $\\\\kappa _g=c_g$ and $\\\\kappa _t=1-\\\\text{Re}(c_t)$ .\",\n \"We depict in Fig.\",\n \"REF the $gg\\\\rightarrow ZZ$ Feynman diagrams that account for these new physics effects.\",\n \"Whereas Eq.\",\n \"(REF ) represents only a sub-set of high dimensional operators affecting the Higgs interactions [36], [37], we focus on it to highlight the effectiveness for the off-shell Higgs measurements to resolve a notorious degeneracy involving these terms.\",\n \"The gluon fusion Higgs production at low energy regime can be well approximated by the Higgs Low Energy Theorem [38], [39], where the total Higgs production cross-section scales as ${\\\\sigma _{\\\\text{GF}}\\\\propto |\\\\kappa _t+\\\\kappa _g|^2}$ .\",\n \"Therefore, low energy measurements, such as on-shell and non-boosted Higgs production [40], [41], [42], [43], [44], [13], [45], [15], [46], are unable to resolve the ${|\\\\kappa _t+\\\\kappa _g|=\\\\text{constant}}$ degeneracy.\",\n \"While the combination between the $t\\\\bar{t}H$ and gluon fusion Higgs production have the potential to break this blind direction [47], we will illustrate that the Higgs production at the off-shell regime can also result into relevant contributions to resolve this degeneracy.\",\n \"Figure: Feynman diagrams for the GF gg→ZZgg\\\\rightarrow ZZprocess.\",\n \"The new physics effects from Eq.\",\n \"() display deviations on thecoefficients κ t \\\\kappa _t and κ g \\\\kappa _g from the SM point (κ t ,κ g )=(1,0)(\\\\kappa _t,\\\\kappa _g)=(1,0).Figure: Transverse mass distributions m T ZZ m_T^{ZZ} for the DY and GF Z(ℓℓ)Z(νν)Z(\\\\ell \\\\ell )Z(\\\\nu \\\\nu ) processes.\",\n \"The new physics effectsare parametrized by deviations from SM point (κ t ,κ g )=(1,0)(\\\\kappa _t,\\\\kappa _g)=(1,0).\",\n \"We follow the benchmark analysis defined in Sec .Since the Higgs boson decays mostly to longitudinal gauge bosons at the high energy regime, it is enlightening to inspect the signal amplitude for the longitudinal components.\",\n \"The amplitudes associated to each contribution presented in Fig.\",\n \"REF can be approximated at $m_{ZZ} \\\\gg m_t,m_H,m_Z$ by [48], [13], [15] $\\\\mathcal {M}_t^{++00} &\\\\approx + \\\\frac{m_t^2}{2m_Z^2} \\\\log ^2\\\\frac{m_{ZZ}^2}{m_t^2}\\\\,, \\\\\\\\\\\\mathcal {M}_g^{++00} &\\\\approx - \\\\frac{m_{ZZ}^2}{2m_Z^2}\\\\,, \\\\\\\\\\\\mathcal {M}_c^{++00} &\\\\approx - \\\\frac{m_t^2}{2m_Z^2} \\\\log ^2\\\\frac{m_{ZZ}^2}{m_t^2} \\\\,.$ Two comments are in order.\",\n \"First, both the $s$ -channel top loop $\\\\mathcal {M}_t$ and the continuum $\\\\mathcal {M}_c$ amplitudes display logarithmic dependences on $m_{ZZ}/m_t$ at the far off-shell regime.\",\n \"In the SM scenario the ultraviolet logarithm between these two amplitudes cancel, ensuring a proper high energy behavior when calculating the full amplitude.\",\n \"Second, it is worth noting the difference in sign between the $s$ -channel contributions $\\\\mathcal {M}_t$ and $\\\\mathcal {M}_g$ .\",\n \"This results into a destructive interference between $\\\\mathcal {M}_t$ and $\\\\mathcal {M}_c$ , contrasting to a constructive interference between $\\\\mathcal {M}_g$ and $\\\\mathcal {M}_c$ .\",\n \"In the following, we will explore these phenomenological effects pinning down the new physics sensitivity with a higher precision.\",\n \"Figure: 95% CL bound on the coupling modifiers κ t \\\\kappa _t and κ g \\\\kappa _g when accounting for the off-shell Higgs measurementin the Z(ℓℓ)Z(νν)Z(\\\\ell \\\\ell )Z(\\\\nu \\\\nu ) channel.\",\n \"We assume the 14 TeV LHC with 3ab -1 3~\\\\text{ab}^{-1} of data.Exploiting the larger rate for $ZZ\\\\rightarrow \\\\ell \\\\ell \\\\nu \\\\nu $ than that for $ZZ\\\\rightarrow 4\\\\ell $ [13], [14], [15], we explore the off-shell Higgs physics at the HL-LHC.\",\n \"To simulate the full loop-induced effects, we implemented Eq.\",\n \"(REF ) into FeynRules/NLOCT [49], [50] through a new fermion state, and adjusting its parameters to match the low-energy Higgs interaction $HG_{\\\\mu \\\\nu }G^{\\\\mu \\\\nu }$ [38], [39].\",\n \"Feynman rules are exported to a Universal FeynRules Output (UFO) [51] and the Monte Carlo event generation is performed with MadGraph5aMC@NLO [20].\",\n \"In Fig.\",\n \"REF , we present the Drell-Yan (DY) and the gluon-fusion (GF) $m_{T}^{ZZ}$ distributions for different signal hypotheses.\",\n \"In the bottom panel, we display the ratio between the GF beyond the SM (BSM) scenarios with respect to the GF SM.\",\n \"In agreement with Eq.\",\n \"(REF ), we observe a suppression for the full process when accounting for the $s$ -channel top loop contributions and an enhancement when including the new physics terms associated to $\\\\mathcal {M}_g$ at high energies.\",\n \"We follow the benchmark analysis defined in Sec. .\",\n \"After the BDT study, the resulting events are used in a binned log-likelihood analysis with the $m_T^{ZZ}$ distribution.\",\n \"This approach explores the characteristic high energy behavior for the new physics terms highlighted in Eq.\",\n \"(REF ) and illustrated in Fig.\",\n \"REF .\",\n \"We present in Fig.\",\n \"REF the resulting 95% CL sensitivity to the $(\\\\kappa _t,\\\\kappa _g)$ new physics parameters at the high-luminosity LHC.\",\n \"In particular, we observe that the LHC can bound the top Yukawa within $\\\\kappa _t\\\\approx [0.4,1.1]$ at 95% CL, using this single off-shell channel.\",\n \"The observed asymmetry in the limit, in respect to the SM point, arises from the large and negative interference term between the $s$ -channel and the continuum amplitudes.\",\n \"The upper bound on $\\\\kappa _t$ is complementary to the direct Yukawa measurement via $ttH$ [52] and can be further improved through a combination with the additional relevant off-shell Higgs final states.\",\n \"The results derived in this section are competitive to the CMS HL-LHC prediction that considers the boosted Higgs production combining the $H\\\\rightarrow 4\\\\ell $ and $H\\\\rightarrow \\\\gamma \\\\gamma $ channels [34].\",\n \"The CMS projection results into an upper bound on the top Yukawa of $\\\\kappa _t\\\\lesssim 1.2$ at 95% CL.\"\n ],\n [\n \"Higgs-Top Form Factor\",\n \"The fact that the observed Higgs boson mass is much lighter than the Planck scale implies that there is an unnatural cancellation between the bare mass and the quantum corrections.\",\n \"Since the mass of the Higgs particle is not protected from quantum corrections, it is well-motivated to consider that it may not be fundamental, but composite in nature [53], [54], [55], [56].\",\n \"In such a scenario, the Higgs boson is proposed as a bound state of a strongly interacting sector with a composite scale $\\\\Lambda $ .\",\n \"In addition, the top quark, which is the heaviest particle in the SM, can also be composite.\",\n \"In this case, the top Yukawa coupling will be modified by a momentum-dependent form factor at a scale $q^2$ close to or above the new physics scale $\\\\Lambda ^2$ .\",\n \"It is challenging to find a general construction for such form factor without knowing the underlying dynamics.\",\n \"Here, we will adopt a phenomenological ansatz motivated by the nucleon form factor [57].\",\n \"It is defined as $\\\\Gamma (q^2/\\\\Lambda ^2) = \\\\frac{1}{(1+q^2/\\\\Lambda ^2)^n}\\\\,,$ where $q^2$ is the virtuality of the Higgs boson.\",\n \"For $n=2$ , it is a dipole-form factor and corresponds to an exponential spacial distribution.\",\n \"Building upon Ref.\",\n \"[18], we study the impact of this form factor on $ g g\\\\rightarrow H^{*} \\\\rightarrow Z Z $ process now with the complementary final state $ \\\\ell ^+ \\\\ell ^- \\\\nu \\\\overline{\\\\nu }$ .\",\n \"Figure: 95% CL sensitivity on the new physics scale Λ\\\\Lambda as a function of the LHC luminosity.\",\n \"We assume the form factor in Eq.\",\n \"()with n=2n=2 (dashed line) and n=3n=3 (solid line) at the 14 TeV LHC.In Fig.\",\n \"REF , we illustrate the $m_{T}^{ZZ}$ distribution for the full gluon fusion $ g g ( \\\\rightarrow H^* ) \\\\rightarrow Z Z $ process.\",\n \"We show the Standard Model (black) and the form factor scenario (red).\",\n \"We assume $n=2$ or 3 and $\\\\Lambda =1.5$ TeV for the depicted form factor scenarios.\",\n \"The differences between Standard Model and form factor cases become larger when the energy scales are comparable or above $\\\\Lambda $ due to the suppression of destructive interference between Higgs signal and continuum background.\",\n \"Thus, we perform the same BDT procedure introduced in Sec.\",\n \"followed by a binned log-likelihood ratio test in the $m_T^{ZZ}$ distribution to fully explore this effect.\",\n \"In Fig.\",\n \"REF , we display the sensitivity reach for the LHC in the Higgs-top form factor.\",\n \"We observe that the LHC can bound these new physics effects up to $\\\\Lambda = 1.5$ TeV for $n = 2 $ and $\\\\Lambda = 2.1$ TeV for $n=3$ at 95% CL.\",\n \"The large event rate for the $H^*\\\\rightarrow ZZ\\\\rightarrow \\\\ell \\\\ell \\\\nu \\\\nu $ signal results in a more precise probe to the ultraviolet regime than for the $H^*\\\\rightarrow ZZ\\\\rightarrow 4\\\\ell $ channel, where the limits on the new physics scale are $\\\\Lambda = 0.8$ TeV for $n = 2 $ and $\\\\Lambda = 1.1$ TeV for $n=3$ at 95% CL [18].\"\n ],\n [\n \"Summary\",\n \"We have systematically studied the off-shell Higgs production in the $pp\\\\rightarrow H^{*}\\\\rightarrow Z(\\\\ell \\\\ell )Z(\\\\nu \\\\nu )$ channel at the high-luminosity LHC.\",\n \"We showed that this signature is crucial to probe the Higgs couplings across different energy scales potentially shedding light on new physics at the ultraviolet regime.\",\n \"To illustrate its physics potential, we derived the LHC sensitivity to three BSM benchmark scenarios where the new physics effects are parametrized in terms of the Higgs boson width, the effective field theory framework, and a non-local Higgs-top coupling form factor.\",\n \"The combination of a large signal rate and a precise phenomenological probe for the process energy scale, due to the transverse $ZZ$ mass, renders strong limits for all considered BSM scenarios.\",\n \"A summary table and comparison with the existing results in the literature are provided in Table REF .\",\n \"Adopting Machine-learning techniques, we demonstrated in the form of BDT that the HL-LHC, with $\\\\mathcal {L}=3~\\\\text{ab}^{-1}$ of data, will display large sensitivity to the Higgs boson width, $\\\\Gamma _H/ \\\\Gamma _H^{SM}<1.31$ .\",\n \"In addition, the characteristic high energy behavior for the new physics terms within the EFT framework results in relevant bounds on the $(\\\\kappa _t,\\\\kappa _g)$ new physics parameters, resolving the low energy degeneracy in the gluon fusion Higgs production.\",\n \"In particular, we observe that the LHC can bound the top Yukawa within $\\\\kappa _t\\\\approx [0.4,1.1]$ at 95% CL.\",\n \"The upper bound on $\\\\kappa _t$ is complementary to the direct Yukawa measurement via $ttH$ and can be further improved in conjunction with additional relevant off-shell Higgs channels.\",\n \"Finally, when considering a more general hypothesis that features a non-local momentum-dependent Higgs-top interaction, we obtain that the HL-LHC is sensitive to new physics effects at large energies with $\\\\Lambda = 1.5$ TeV for $n = 2 $ and $\\\\Lambda = 2.1$ TeV for $n=3$ at 95% CL.\",\n \"We conclude that, utilizing the promising $H^*\\\\rightarrow Z(\\\\ell ^+\\\\ell ^-)Z(\\\\nu \\\\bar{\\\\nu })$ channel at the HL-LHC and adopting the Machine-Learning techniques, the combination of a large signal rate and a precise phenomenological probe for the process energy scale renders improved sensitivities beyond the existing literature, to all the three BSM scenarios considered in this work.\",\n \"This work was supported by the U.S. Department of Energy under grant No.\",\n \"DE-FG02- 95ER40896 and by the PITT PACC.\",\n \"DG was supported by the U.S. Department of Energy under grant number DE-SC 0016013.\"\n 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"Alternating Projections Methods for Discrete-time Stabilization of\n Quantum States"
],
[
"Abstract We study sequences (both cyclic and randomized) of idempotent completely-positive trace-preserving quantum maps, and show how they asymptotically converge to the intersection of their fixed point sets via alternating projection methods.",
"We characterize the robustness features of the protocol against randomization and provide basic bounds on its convergence speed.",
"The general results are then specialized to stabilizing en- tangled states in finite-dimensional multipartite quantum systems subject to a resource constraint, a problem of key interest for quantum information applications.",
"We conclude by suggesting further developments, including techniques to enlarge the set of stabilizable states and ensure efficient, finite-time preparation."
],
[
"Introduction",
"Driving a quantum system to a desired state is a prerequisite for quantum control applications ranging from quantum chemistry to quantum computation [1].",
"An important distinction arises depending on whether the target state is to be obtained starting from a known initial state – in which case such knowledge may be leveraged to design a control law that effects a “one-to-one” state transfer; or, the initial state is arbitrary (possibly unknown) – in which case the dynamics must allow for “all-to-one” transitions.",
"We shall refer to the latter as a state preparation task.",
"The feasibility of these tasks, as well as the robustness and efficiency of the control protocol itself, are heavily influenced by the control resources that are permitted.",
"Unitary control can only allow for transfer between states (represented by density operators), that are iso-spectral [2].",
"For both more general quantum state transfers, and for all state preparation tasks, access to non-unitary control (in the form of either coupling to an external reservoir or employing measurement and feedback) becomes imperative.",
"If a dissipative mechanism that “cools” the system to a known pure state is available, the combination of this all-to-one initialization step with state-controllable one-to-one unitary dynamics [3] is the simplest approach for achieving pure-state preparation, with methods from optimal control theory being typically employed for synthesizing the desired unitary dynamics [1], [4].",
"In the same spirit, in the circuit model of quantum computation [5], preparation of arbitrary pure states is attained by initializing the quantum register in a known factorized pure state, and then implementing a sequence of unitary transformations (“quantum gates”) drawn from a universal set.",
"Sampling from a mixed target state can be obtained by allowing for randomization of the applied quantum gates in conjunction with Metropolis-type algorithms [6].",
"Additional possibilities for state preparation arise if the target system is allowed to couple to a quantum auxiliary system, so that the pair can be jointly initialized and controlled, and the ancilla reset or traced over [7], [8].",
"For example, in [9] it is shown how sequential unitary coupling to an ancilla may be used to design a sequence of non-unitary transformations (“quantum channels”) on a target multi-qubit system that dissipatively prepare it in a matrix product state.",
"In the above-mentioned state-preparation methods, the dissipative mechanism is fixed (other than being turned on and off as needed), and the control design happens at the unitary level, directly on the target space or an enlarged one.",
"A more powerful setting is to allow dissipative control design from the outset [10], [7], [11].",
"This opens up the possibility to synthesize all-to-one open-system dynamics that not only prepare the target state of interest but, additionally, leave it invariant throughout – that is, achieves stabilization, which is the task we focus on in this work.",
"Quantum state stabilization has been investigated from different perspectives, including feedback design with classical [12], [13], [14], [15], [16], [17], [18], [19] and quantum [20], [21], [22], [23] controllers, as well as open-loop reservoir engineering techniques with both time-independent dynamics and switching control [24], [25], [26], [27], [28], [29], [30], [31], [32].",
"Most of this research effort, however, has focused on continuous-time models, with fewer studies addressing discrete-time quantum dynamics.",
"With “digital” open-system quantum simulators being now experimentally accessible [33], [34], investigating quantum stabilization problems in discrete time becomes both natural and important.",
"Thanks to the invariance requirement, “dissipative quantum circuits” bring distinctive advantages toward preparing pure or mixed target states on-demand, notably: While in a unitary quantum circuit or a sequential generation scheme, the desired state is only available at the completion of the full protocol, invariance of the target ensures that repeating a stabilizing protocol or even portions of it, will further maintain the system in the target state (if so desired), without disruption.",
"The order of the applied control operations need no longer be crucial, allowing for the target state to still be reached probabilistically (in a suitable sense), while incorporating robustness against the implementation order.",
"If at a certain instant a wrong map is implemented, or some transient noise perturbs the dynamics, these unwanted effects can be re-absorbed without requiring active intervention or the whole preparation protocol having to be re-implemented correctly.",
"Discrete-time quantum Markov dynamics are described by sequences of quantum channels, namely, completely-positive, trace-preserving (CPTP) maps [35].",
"This give rise to a rich stability theory that can be seen as the non-commutative generalization of the asymptotic analysis of classical Markov chains, and that thus far has being studied in depth only in the time-homogeneous case [36], [37], including elementary feedback stabilizability and reachability problems [38], [39].",
"In this work, we show that time-dependent sequences of CPTP maps can be used to make their common fixed states the minimal asymptotically stable sets, which are reached by iterating cyclically a finite subsequence.",
"The methods we introduce employ a finite number of idempotent CPTP maps, which we call CPTP projections, and can be considered a quantum version of alternated projections methods.",
"The latter, stemming from seminal results by von Neumann [40] and extended by Halperin [41] and others [42], [43], are a family of (classical) algorithms that, loosely speaking, aim to select an element in the intersection of a number of sets that minimizes a natural (quadratic) distance with respect to the input.",
"The numerous applications of such classical algorithms include estimation [44] and control [45] and, recently, specific tasks in quantum information, such as quantum channel construction [46].",
"In the context of quantum stabilization, we show that instead of working with the standard (Hilbert-Schmidt) inner product, it is natural to resort to a different inner product, a weighted inner product for which the CPTP projections become orthogonal, and the original results apply.",
"When, depending on the structure of the fixed-point set, this strategy is not viable, we establish convergence by a different proof that does not directly build on existing alternating projection theorems.",
"For all the proposed sequences, the order of implementation is not crucial, and convergence in probability is guaranteed even when the sequence is randomized, under very mild hypotheses on the distribution.",
"Section introduces the models of interest, and recalls some key results regarding stability and fixed points of CPTP maps.",
"Our general results on quantum alternating projections are presented in Section , after proving that CPTP projections can be seen as orthogonal projections with respect to a weighted inner product.",
"Basic bounds on the speed of convergence and robustness of the algorithms are discussed in Sections REF and REF , respectively.",
"In Section we specialize these results to distributed stabilization of entangled states on multipartite quantum systems, where the robustness properties imply that the target can be reached by unsupervised randomized applications of dissipative quantum maps.",
"We consider a finite-dimensional quantum system, associated to a Hilbert space $\\mathcal {H}\\approx d.$ Let ${\\cal B}(\\mathcal {H})$ denote the space of linear bounded operators on $\\mathcal {H},$ with $\\dag $ being the adjoint operation.",
"The state of the system at each time $t \\ge 0$ is a density matrix in ${\\mathfrak {D}}(\\mathcal {H}),$ namely a positive-semidefinite, trace one matrix.",
"Let $\\rho _0$ be the initial state.",
"We consider time inhomogenous Markov dynamics, namely, sequences of CPTP maps $\\lbrace \\mathcal {E}_t\\rbrace ,$ defining the state evolution for through the following dynamical equation: $\\rho _{t+1}=\\mathcal {E}_t(\\rho _t), \\quad t\\ge 0.$ Recall that a linear map $\\mathcal {E}$ is CPTP if and only if it admits an operator-sum representation (OSR) [35]: $\\mathcal {E}(\\rho )=\\sum _k M_k\\rho M_k^\\dag ,$ where the (Hellwig-Kraus) operators $\\lbrace M_k\\rbrace \\subset {\\cal B}(\\mathcal {H})$ satisfy $\\sum _k M_k^\\dag M_k=I.$ We shall assume that for all $t>0$ the map $\\mathcal {E}_t=\\mathcal {E}_{j(t)}$ is chosen from a set of “available” maps, to be designed within the available control capabilities.",
"In particular, in Section we will focus on locality-constrained dynamics.",
"For any $t \\ge s\\ge 0,$ we shall denote by $\\mathcal {E}_{t,s} \\equiv \\mathcal {E}_{t-1}\\circ \\mathcal {E}_{t-2}\\circ \\ldots \\circ \\mathcal {E}_s, \\quad (\\mathcal {E}_{t,t}= {\\cal I}), $ the evolution map, or “propagator”, from $s$ to $t.$ A set $\\mathcal {S}$ is invariant for the dynamics if $\\mathcal {E}_{t,s}(\\tau )\\in \\mathcal {S}$ for all $\\tau \\in \\mathcal {S}$ .",
"Define the distance of an operator $\\rho $ from a set $\\mathcal {S}$ as $d(\\rho ,\\mathcal {S}) \\equiv \\inf _{\\tau \\in \\mathcal {S}}\\Vert \\rho -\\tau \\Vert _1,$ with $\\Vert \\cdot \\Vert _1$ being the trace norm.",
"The following definitions are straightforward adaptations of the standard ones [47]: Definition 1 (i) An invariant set $\\mathcal {S}$ is (uniformly) simply stable if for any $\\varepsilon >0$ there exists $\\delta > 0$ such that $d(\\tau ,\\mathcal {S})<\\delta $ ensures $d(\\mathcal {E}_{t,s}(\\tau ),\\mathcal {S})<\\varepsilon $ for all $t\\ge s\\ge 0.$ (ii) An invariant set $\\mathcal {S}$ is globally asymptotically stable (GAS) if it is simply stable and $&&\\lim _{t\\rightarrow \\infty }d(\\mathcal {E}_{t,s}(\\rho ),\\mathcal {S})=0,\\quad \\forall \\rho ,\\; s\\ge 0.$ Notice that, since we are dealing with finite-dimensional systems, convergence in any matrix norm is equivalent.",
"Furthermore, since CPTP maps are trace-norm contractions [5], we have that simple stability is always guaranteed (and actually the distance is monotonically non-increasing): Proposition 1 If a set $\\mathcal {S}$ is invariant for the dynamics $\\lbrace \\mathcal {E}_{t,s}\\rbrace _{t,s\\ge 0}$ , then it is simply stable.",
"Proof.",
"We have, for all ${t,s\\ge 0}$ : $d(\\mathcal {E}_{t,s}(\\rho ),\\mathcal {S})&\\le & d(\\mathcal {E}_{t,s}(\\rho ),\\mathcal {E}_{t,s}(\\tau _{t,s}^*))\\\\&\\le & d(\\rho ,\\tau _{t,s}^*) \\\\&=&d(\\rho ,\\mathcal {S}).$ The first inequality is true, by definition, for all $ \\tau _{t,s} \\in \\mathcal {S},$ and also on the closure $\\bar{\\mathcal {S}}$ , thanks to continuity of $\\mathcal {E}_{t,s}$ ; the second holds due to contractivity of $\\mathcal {E},$ and the last equality follows by letting $\\tau _{t,s}^* \\equiv {\\rm arg}\\min _{\\tau \\in \\bar{\\mathcal {S}}}\\Vert \\rho -\\tau \\Vert _1,$ where we can take the min since $\\bar{\\mathcal {S}}$ is closed and compact.",
"$\\Box $"
],
[
"Fixed points of CPTP maps",
"We collect in this section some relevant results on the structure of fixed-point sets for a CPTP map $\\mathcal {E}$ , denoted by ${\\rm fix}(\\mathcal {E}).$ More details can be found in [48], [49], [36], [50].",
"Let ${\\rm alg}(\\mathcal {E})$ denote the $\\dag $ -closed algebra generated by the operators in the OSR of $\\mathcal {E},$ and $\\mathcal {A}^{\\prime }$ denote the commutant of $\\mathcal {A}$ , respectively.",
"For unital CP maps, ${\\rm fix}(\\mathcal {E})$ is a $\\dag $ -closed algebra, ${\\rm fix}(\\mathcal {E})={\\rm alg}(\\mathcal {E})^{\\prime }={\\rm fix}(\\mathcal {E}^\\dag )$ [36], [50].",
"This implies that it admits a (Wedderburn) block decomposition [51]: ${\\rm fix}(\\mathcal {E}) = \\bigoplus _\\ell {\\cal B}(\\mathcal {H}_{S,\\ell }) \\otimes I_{F,\\ell },$ with respect to a Hilbert space decomposition: $\\mathcal {H}=\\bigoplus _\\ell \\mathcal {H}_{S,\\ell } \\otimes \\mathcal {H}_{F,\\ell }.$ For general (not necessarily unital) CPTP maps the following holds [50], [36], [48]: Theorem 1 (Fixed-point sets, generic case) Given a CPTP map $\\mathcal {E}$ which admits a full-rank fixed point $\\rho $ , we have ${\\rm fix}(\\mathcal {E})=\\rho ^\\frac{1}{2}\\,{\\rm fix}(\\mathcal {E}^\\dag )\\,\\rho ^\\frac{1}{2}.$ Moreover, with respect to the decomposition of ${\\rm fix}(\\mathcal {E}^\\dag )=\\bigoplus _\\ell {\\cal B}(\\mathcal {H}_{S,\\ell }) \\otimes {I}_{F,\\ell }$ , the fixed state has the structure: $\\rho = \\bigoplus _\\ell p_\\ell \\rho _{S,\\ell }\\otimes \\tau _{F,\\ell },$ where $\\rho _{S,\\ell }$ and $\\tau _{F,\\ell }$ are full-rank density operators of appropriate dimension, and $p_\\ell $ a set of convex weights.",
"This means that, given a CPTP map admitting a full-rank invariant state $\\rho $ , the fixed-point sets ${\\rm fix}(\\mathcal {E})$ is a $\\rho $ -distorted algebra, namely, an associative algebra with respect to a modified product (i.e.",
"$X \\times _\\rho Y= X\\rho ^{-1} Y$ ), with structure ${\\cal A}_\\rho =\\bigoplus _\\ell {\\cal B}(\\mathcal {H}_{S,\\ell }) \\otimes \\tau _{F,\\ell }, $ where $\\tau _{F,\\ell }$ are a set of density operators of appropriate dimension (the same for every element in ${\\rm fix}(\\mathcal {E})$ ).",
"In addition, since $\\rho $ has the same block structure (REF ), ${\\rm fix}(\\mathcal {E})$ is clearly invariant with respect to the action of the linear map ${\\cal M}_{\\rho ,\\lambda }(X) \\equiv \\rho ^{\\lambda } X \\rho ^{-\\lambda }$ for any $\\lambda \\in \\mathbb {C}$ , and in particular for the modular group $\\lbrace {\\cal M}_{\\rho ,i\\phi }\\rbrace $ [49].",
"The same holds for the fixed points of the dual dynamics.",
"In [48], the following result has been proved using Takesaki's theorem, showing that commutativity with a modular-type operator is actually sufficient to ensure that a distorted algebra is a valid fixed-point set.",
"Theorem 2 (Existence of $\\rho $ -preserving dynamics) Let $\\rho $ be a full-rank density operator.",
"A distorted algebra ${\\cal A}_\\rho ,$ such that $\\rho \\in {\\cal A}_\\rho ,$ admits a CPTP map $\\mathcal {E}$ such that ${\\rm fix}(\\mathcal {E})={\\cal A}_\\rho $ if and only if it is invariant for ${\\cal M}_{\\rho ,\\frac{1}{2}}.$ To our present aim, it is worth remarking that in the proof of the above result, a CPTP idempotent map is derived as the dual of a conditional expectation map, namely, the orthogonal projection onto the (standard) algebra ${\\rm fix}(\\mathcal {E}^\\dag ).$ If the CPTP map $\\mathcal {E}$ does not admit a full rank invariant state, then it is possible to characterize the fixed-point set by first reducing to the support of the invariant states.",
"This leads to the following structure theorem [50], [36], [48]: Theorem 3 (Fixed-point sets, general case) Given a CPTP map $\\mathcal {E},$ and a maximal-rank fixed point $\\rho $ with $\\tilde{\\mathcal {H}}\\equiv {\\rm supp}(\\rho )$ , let $\\tilde{\\mathcal {E}}$ denote the reduction of $\\mathcal {E}$ to $\\mathcal {B}(\\tilde{\\mathcal {H}})$ .",
"We then have ${\\rm fix}(\\mathcal {E})=\\rho ^\\frac{1}{2}\\,({\\rm ker}(\\tilde{\\mathcal {E}}^\\dag )\\oplus {\\mathbb {O}}) \\,\\rho ^\\frac{1}{2},$ where ${\\mathbb {O}}$ is the zero operator on the complement of $\\tilde{\\mathcal {H}}.$"
],
[
"von Neumann-Halperin Theorem",
"Many of the ideas we use in this paper are inspired by a classical result originally due to von Neumann [40], and later extended by Halperin to multiple projectors: Theorem 4 (von Neumann-Halperin alternating projections) If $\\mathcal {M}_1$ ,...,$\\mathcal {M}_r$ are closed subspaces in a Hilbert space $\\mathcal {H}$ , and $P_{\\mathcal {M}_j}$ are the corresponding orthogonal projections, then $\\lim _{n\\rightarrow \\infty }(P_{\\mathcal {M}_1}...P_{\\mathcal {M}_r})^nx=Px, \\quad \\forall x\\in \\mathcal {H}, $ where $P$ is the orthogonal projection onto ${\\bigcap _{i=1}^r\\mathcal {M}_i}$ .",
"A proof for this theorem can be found in Halperin's original work [41].",
"Since then, the result has been refined in many ways, has inspired similar convergence results that use information projections [52] and, in full generality, projections in the sense of Bregman divergences [53], [42].",
"The applications of the results are manifold, especially in algorithms: while it is beyond the scope of this work to attempt a review, a good collection is presented in [43].",
"Some bounds on the convergence rate for the alternating projection methods can be derived by looking at the angles between the subspaces we are projecting on.",
"We recall their definition and basic properties in Appendix REF , see again [43] for more details."
],
[
"CPTP projections and orthogonality",
"We call an idempotent CPTP map, namely, one that satisfies $\\mathcal {E}^2=\\mathcal {E}$ , a CPTP projection.",
"As any linear idempotent map, $\\mathcal {E}$ has only $0,1$ eigenvalues and maps any operator $X$ onto the set of its fixed points, ${\\rm fix}(\\mathcal {E})$ .",
"Recall that ${\\rm fix}(\\mathcal {E})=\\bigoplus _\\ell [ \\mathcal {B}(\\mathcal {H}_{S,\\ell })\\otimes \\tau _{F,\\ell } ] \\oplus \\mathbb {O}_R,$ for some Hilbert-space decomposition: $\\mathcal {H}= \\bigoplus _\\ell (\\mathcal {H}_{S,\\ell }\\otimes \\mathcal {H}_{F,\\ell })\\oplus \\mathcal {H}_R,$ where the last zero-block is not present if there exists a $\\rho >0$ in ${\\rm fix}(\\mathcal {E}).$ We next give the structure of the CPTP projection associated to ${\\rm fix}(\\mathcal {E})$ : The result is essentially known (see e.g.",
"[36], [49]) but we provide a short proof for completeness): Proposition 2 Given a CPTP map $\\mathcal {E}$ with $\\rho $ a fixed point of maximal rank, a CPTP projection onto $\\mathcal {A}_\\rho ={\\rm fix}(\\mathcal {E})$ exists and is given by $\\mathcal {E}_{\\mathcal {A}_\\rho }(X)=\\lim _{T\\rightarrow +\\infty }\\frac{1}{T}\\sum _{i=0}^{T-1}\\mathcal {E}^i(X).$ If the fixed point $\\rho $ is full rank, then the CPTP projection onto $\\mathcal {A}_\\rho = \\oplus _\\ell {\\cal B}(\\mathcal {H}_{S,\\ell }) \\otimes \\tau _{F,\\ell } $ is equivalently given by ${\\mathcal {E}_{\\mathcal {A}_\\rho }}(X)=\\bigoplus _\\ell {\\rm Tr}_{F,\\ell }(\\Pi _{SF,\\ell }X\\, \\Pi _{SF,\\ell })\\otimes {{\\tau _{F,\\ell }}},$ where $\\Pi _{SF,\\ell }$ is the orthogonal projection from $\\mathcal {H}$ onto the subspace $\\mathcal {H}_{S,\\ell } \\otimes \\mathcal {H}_{F,\\ell }$ .",
"Proof.",
"Recalling that $\\Vert \\mathcal {E}\\Vert _\\infty =1$ , it is straightforward to prove that the limit in Eq.",
"(REF ) exists and is a CPTP map.",
"Furthermore, clearly ${\\rm fix}(\\mathcal {E})\\subseteq {\\rm fix}(\\mathcal {E}_{{\\cal A}_\\rho }),$ and $\\mathcal {E}_{{\\cal A}_\\rho }\\mathcal {E}=\\mathcal {E}\\mathcal {E}_{{\\cal A}_\\rho }=\\mathcal {E}_{{\\cal A}_\\rho }.$ It follows that $\\mathcal {E}_{{\\cal A}_\\rho }^2=\\mathcal {E}_{{\\cal A}_\\rho }.$ On the other hand, it is immediate to verify that the right hand side of Eq.",
"(REF ) is CPTP, has image equal to its fixed points ${\\mathcal {A}}_\\rho = {\\rm fix}({\\mathcal {E}}),$ and is idempotent.",
"Hence, it coincides with $\\mathcal {E}_{\\mathcal {A}_\\rho }.$ $\\Box $ For a full-rank fixed-point set, CPTP projections are not orthogonal projections onto ${\\rm fix}(\\mathcal {E})$ , at least with respect to the Hilbert-Schmidt inner product.",
"The proof is given in Appendix REF .",
"We are nonetheless going to show that $\\mathcal {E}_\\mathcal {A}$ is an orthogonal projection with respect to a different inner product.",
"This proves that the map in Eq.",
"(REF ) is the unique CPTP projection onto $\\mathcal {A}_\\rho $ .",
"If the fixed-point set does not contains a full-rank state, Eq.",
"(REF ) still defines a valid CPTP projection onto ${\\rm fix}(\\mathcal {E})$ ; however, this need not be unique.",
"We will exploit this fact in the proof of Theorem REF , where we choose a particular one.",
"Definition 2 Let $\\xi $ be a positive-definite operator.",
"(i) We define the $\\xi $ -inner product as $\\langle X,Y\\rangle _\\xi \\equiv {\\rm Tr}(X\\xi Y);$ (ii) We define the symmetric $\\xi $ -inner product as $\\langle X,Y\\rangle _{\\xi ,s} \\equiv {\\rm Tr}(X\\xi ^\\frac{1}{2}Y\\xi ^\\frac{1}{2}).$ It is straightforward to verify that both (REF ) and (REF ) are valid inner products.",
"We next show that $\\mathcal {E}_\\mathcal {A}$ is an orthogonal projection with respect to (REF ) and (REF ), when $\\xi =\\rho ^{-1}$ for a full rank fixed point $\\rho .$ We will need a preliminary lemma.",
"With $W \\equiv \\bigoplus W_i$ we will denote an operator that acts as $W_i$ on $\\mathcal {H}_i$ , for a direct-sum decomposition of $\\mathcal {H}=\\bigoplus _i \\mathcal {H}_i$ .",
"Lemma 1 Consider $Y,W\\in \\mathcal {B}(\\mathcal {H}),$ where $W$ admits an orthogonal block-diagonal representation $W=\\bigoplus _\\ell W_\\ell $ .",
"Then ${\\rm Tr}(WY)=\\sum _\\ell {\\rm Tr}( W_\\ell Y_\\ell )$ , where $Y_\\ell =\\Pi _\\ell Y\\Pi _\\ell $ .",
"Proof.",
"Let $\\Pi _\\ell $ be the projector onto $\\mathcal {H}_\\ell $ .",
"Remembering that $\\sum _\\ell \\Pi _\\ell =I$ and $\\Pi _\\ell =\\Pi _\\ell ^2$ , it follows that ${\\rm Tr}(X) = \\sum _\\ell {\\rm Tr}(\\Pi _\\ell X)=\\sum _\\ell {\\rm Tr}(\\Pi _\\ell X\\Pi _\\ell ).$ Therefore, we obtain: ${\\rm Tr}(WY)&=&{\\rm Tr}(\\sum _\\ell \\Pi _\\ell \\bigoplus _j W_j Y)=\\sum _\\ell {\\rm Tr}( \\Pi _\\ell W_\\ell Y)\\\\&=&\\sum _\\ell {\\rm Tr}( \\Pi _\\ell W_\\ell \\Pi _\\ell Y)=\\sum _\\ell {\\rm Tr}( W_\\ell \\Pi _\\ell Y\\Pi _\\ell )\\\\&=&\\sum _\\ell {\\rm Tr}( W_\\ell Y_\\ell ).",
"\\hspace{119.50148pt} \\Box $ Proposition 3 Let $\\xi =\\rho ^{-1}$ , where $\\rho $ is a full-rank fixed state in $\\mathcal {A}_\\rho $ , which is invariant for ${\\cal M}_{\\rho ,\\frac{1}{2}}.$ Then $\\mathcal {E}_{\\mathcal {A}_\\rho }$ is an orthogonal projection with respect to the inner products in (REF ) and (REF ).",
"Proof.",
"We already know that $\\mathcal {E}$ is linear and idempotent.",
"In order to show that $\\mathcal {E}$ is an orthogonal projection, we need to show that it is self-adjoint relative to the relevant inner product.",
"Let us consider $\\rho =\\bigoplus \\rho _\\ell \\otimes \\tau _\\ell $ and, as above: $X_\\ell =&\\Pi _{SF,\\ell } X\\,\\Pi _{SF,\\ell }=\\sum _k A_{k,\\ell }\\otimes B_{k,\\ell },\\\\Y_\\ell =&\\Pi _{SF,\\ell } Y\\,\\Pi _{SF,\\ell }=\\sum _j C_{j,\\ell }\\otimes D_{j,\\ell }.$ If we apply Lemma REF to the operator $W = \\mathcal {E}_{\\mathcal {A}_\\rho }(X)\\rho ^{-1} = \\bigoplus _\\ell ( [{\\rm Tr}_{F,\\ell }(X_\\ell )\\otimes \\tau _\\ell ](\\rho _\\ell ^{-1}\\otimes \\tau _\\ell ^{-1})),$ we obtain: $\\langle \\mathcal {E}(X),Y \\rangle _{\\xi }&\\hspace{-8.53581pt}=&\\hspace{-8.53581pt}{\\rm Tr}(\\mathcal {E}_{\\mathcal {A}_\\rho }(X)\\rho ^{-1}Y) \\\\&\\hspace{-8.53581pt}=& \\hspace{-8.53581pt}{\\rm Tr}(\\bigoplus _\\ell {\\rm Tr}_{F,\\ell }(X_\\ell )\\otimes \\tau _\\ell (\\rho _\\ell ^{-1}\\otimes \\tau _\\ell ^{-1})Y_\\ell )\\\\&\\hspace{-8.53581pt}=& \\hspace{-8.53581pt}\\sum _{\\ell ,k,j}{\\rm Tr}([A_{k,\\ell }{\\rm Tr}(B_{k,\\ell })\\rho _\\ell ^{-1}\\otimes I][C_{j,\\ell }\\otimes D_{j,\\ell }])\\\\&\\hspace{-8.53581pt}=&\\hspace{-8.53581pt}\\sum _{\\ell ,k,j}{\\rm Tr}(B_{k,\\ell }){\\rm Tr}(A_{k,\\ell }\\rho _\\ell ^{-1}C_{j,\\ell }){\\rm Tr}( D_{j,\\ell }).$ By similar calculation, $\\langle X,\\mathcal {E}(Y)\\rangle _{\\xi } &=& {\\rm Tr}(X\\rho ^{-1}\\mathcal {E}(Y)) \\\\&=& \\sum _{\\ell ,k,j}{\\rm Tr}(B_{k,\\ell }){\\rm Tr}(A_{k,\\ell }\\rho _\\ell ^{-1}C_{j,\\ell }){\\rm Tr}(D_{j,\\ell }).$ By comparison, we infer that $\\langle \\mathcal {E}(X),Y\\rangle _\\xi =\\langle X,\\mathcal {E}(Y)\\rangle _\\xi .$ A similar proof can be carried over using the symmetric $\\xi $ -inner product of Eq.",
"(REF ).$\\Box $ We are now ready to prove the main results of this section.",
"The first shows that the set of states with support on a target subspace can be made GAS by sequences of CPTP projections on larger subspaces that have the target as intersection.",
"Theorem 5 (Subspace stabilization) Let $\\mathcal {H}_j$ , $j=1,\\ldots ,r,$ be subspaces such that $\\bigcap _j\\mathcal {H}_j \\equiv \\hat{\\mathcal {H}}.$ Then there exists CPTP projections ${\\mathcal {E}}_1,\\ldots ,{\\mathcal {E}}_r$ onto ${\\cal B}(\\mathcal {H}_j)$ , $j=1,\\ldots ,r,$ such that $\\forall \\tau \\in \\cal {D}(\\mathcal {H})$ : $\\lim _{n\\rightarrow \\infty }({\\mathcal {E}}_r\\ldots {\\mathcal {E}}_1)^n(\\tau )={\\mathcal {E}}_{{\\cal B}(\\hat{\\mathcal {H}})}(\\tau ),$ where ${\\mathcal {E}}_{{\\cal B}(\\hat{\\mathcal {H}})}$ is a CPTP projection onto ${{\\cal B}(\\hat{\\mathcal {H}})}.$ Proof.",
"We shall explicitly construct CPTP maps whose cyclic application ensures stabilization.",
"Define $P_{j} $ to be the projector onto $\\mathcal {H}_j$ , and the CPTP maps: $\\mathcal {E}_{j}(\\cdot ) \\equiv P_{j} (\\cdot ) P_{j} + \\frac{ P_{j} }{ {\\rm Tr}({P_{j}}) } \\,{\\rm Tr}\\,(P^\\perp _{j}\\cdot ).$ Consider $\\hat{P}$ the orthogonal projection onto $\\hat{\\mathcal {H}}$ and the positive-semidefinite function $V(\\tau )=1-{\\rm Tr}{}({\\hat{P} \\, \\tau }),$ $\\tau \\in {\\mathcal {B}}(\\mathcal {H}).$ The variation of $V,$ when a $\\mathcal {E}_j$ is applied, is $\\Delta V (\\tau ) {\\equiv V(\\mathcal {E}_j(\\tau )) - V(\\tau )}=-{\\rm Tr}{} [{\\hat{P} \\, (\\mathcal {E}_j(\\tau )-\\tau )}]\\equiv \\Delta V_j(\\tau ) .$ If we show that this function is non-increasing along the trajectories generated by repetitions of the cycle of all maps, namely, $\\mathcal {E}_{\\rm cycle}\\equiv \\mathcal {E}_r\\circ \\ldots \\circ \\mathcal {E}_1$ , the system is periodic thus its stability can be studied as a time-invariant one.",
"Hence, by LaSalle-Krasowskii theorem [54], the trajectories (being all bounded) will converge to the largest invariant set contained in the set of $\\tau $ such that on a cycle $\\Delta V_{\\rm cycle}(\\tau )=0$ .",
"We next show that this set must have support only on $\\hat{\\mathcal {H}}$ .",
"If an operator $\\rho $ has support on $\\hat{\\mathcal {H}},$ it is clearly invariant and $\\Delta V(\\rho )=0$ .",
"Assume now that $\\textup {supp}(\\tau )\\nsubseteq \\mathcal {H}_j$ for some $j$ , that is, ${\\rm Tr}{}{ (\\tau P_j^\\perp )} >0$ .",
"By using the form of the map $\\mathcal {E}_j$ given in Eq.",
"(REF ), we have $\\Delta V_j(\\tau )&=&-{\\rm Tr}{}{(\\hat{P} (P_j \\tau P_j ) )} - {\\rm Tr}{}{(\\tau P_j^\\perp )}\\frac{ {\\rm Tr}{}{(\\hat{P} (P_j))} }{ \\text{Tr}_{}\\left(P_j \\right) }\\nonumber \\\\&+ & \\hspace*{8.53581pt} {\\rm Tr}{}{(\\hat{P}\\tau )}$ The sum of the first and the third term in the above equation is zero since $\\hat{P}\\le P_j,$ .",
"The second term, on the other hand, is strictly negative.",
"This is because: (i) we assumed that ${\\rm Tr}( {\\tau P_j^\\perp } ) >0$ ; (ii) with $\\hat{P}\\le P_j,$ and ${\\cal E}_j (P_j)$ having the same support of $P_j $ by construction, it also follows that $\\text{Tr}_{}\\left(\\Pi {\\cal E}_j (P_j )\\right)>0.$ This implies that $\\mathcal {E}_j$ either leaves $\\tau $ (and hence $V(\\tau )$ ) invariant, or $\\Delta V_j(\\rho )<0.$ Hence, each cycle $\\mathcal {E}_{\\rm cycle}$ is such that $\\Delta V_{\\rm cycle}(\\tau )=\\sum _{j=1}^r\\Delta V_j(\\tau )<0$ for all $\\tau \\notin {D}(\\hat{\\mathcal {H}}).$ We thus showed that no state $\\tau $ with support outside of $\\hat{\\mathcal {H}}$ can be in the attractive set for the dynamics.",
"Hence, the dynamics asymptotically converges onto ${D}(\\hat{\\mathcal {H}})$ which is the only invariant set for all the $\\mathcal {E}_j$ .",
"$\\Box $ The second result shows that the a similar property holds for more general fixed-point sets, as long as they contain a full-rank state: Theorem 6 (Full-rank fixed-set stabilization) Let the maps ${\\mathcal {E}}_1,\\ldots , {\\mathcal {E}}_r$ be CPTP projections onto $\\mathcal {A}_i$ , $i=1,\\ldots ,r,$ and assume that $\\hat{\\mathcal {A}}\\equiv \\bigcap _{i=1}^r \\mathcal {A}_i$ contains a full-rank state $\\rho .$ Then $\\forall \\tau \\in \\mathcal {D}(\\mathcal {H})$ : $\\lim _{n\\rightarrow \\infty }({\\mathcal {E}}_r\\ldots {\\mathcal {E}}_1)^n(\\tau )={\\mathcal {E}}_{\\hat{\\mathcal {A}}}(\\tau ),$ where ${\\mathcal {E}_{\\hat{\\mathcal {A}}}}$ is the CPTP projection onto $\\hat{\\mathcal {A}}.$ Proof.",
"Let us consider $\\xi =\\rho ^{-1}$ ; then $\\rho \\in \\hat{\\cal A}$ implies that the maps $\\hat{\\mathcal {E}}_i$ are all orthogonal projections with respect to the same $\\rho ^{-1}$ -modified inner product (Propositions REF , REF ).",
"Hence, it suffice to apply von Neumann-Halperin, Theorem REF : asymptotically, the cyclic application of orthogonal projections onto subsets converges to the projection onto the intersection of the subsets; in our case, the latter is $\\hat{\\mathcal {A}}$ .",
"$\\Box $ Together with Theorem REF , the above result implies that the intersection of fixed-point sets is still a fixed-point set of some map, as long as it contains a full-rank state: Corollary 1 If $\\mathcal {A}_{i}$ , $i=1,\\ldots ,r,$ are $\\rho $ -distorted algebras, with $\\rho $ full rank, and are invariant for ${\\cal M}_{\\rho ,\\frac{1}{2}},$ then $\\hat{\\mathcal {A}}=\\bigcap _{i=1}^r \\mathcal {A}_i$ is also a $\\rho $ -distorted algebra, invariant for ${\\cal M}_{\\rho ,\\frac{1}{2}}.$ Proof.",
"$\\hat{\\mathcal {A}}$ contains $\\rho $ and the previous Theorem ensures that a CPTP projection onto it exists.",
"Then by Theorem REF it is invariant for ${\\cal M}_{\\rho ,\\frac{1}{2}}.$ $\\Box $ Lastly, combining the ideas of the proof of Theorem REF and REF , we obtain sufficient conditions for general fixed-point sets (that is, neither full algebras on a subspace nor containing a full-rank fixed-state).",
"Theorem 7 (General fixed-point set stabilization) Assume that the CPTP fixed-point sets $\\mathcal {A}_i$ , $i=1,\\ldots ,r,$ are such that $\\hat{\\mathcal {A}}\\equiv \\bigcap _{i=1}^r \\mathcal {A}_i$ satisfies ${\\rm supp}(\\hat{\\mathcal {A}})=\\bigcap _{i=1}^r {\\rm supp}(\\mathcal {A}_i).$ Then there exist CPTP projections ${\\mathcal {E}}_1,\\ldots ,{\\mathcal {E}}_r$ onto $\\mathcal {A}_i$ , $i=1,\\ldots ,r,$ such that $\\forall \\tau \\in \\mathcal {D}(\\mathcal {H})$ : $\\lim _{n\\rightarrow \\infty }({\\mathcal {E}}_r\\ldots {\\mathcal {E}}_1)^n(\\tau )={\\mathcal {E}}_{\\hat{\\mathcal {A}}}(\\tau ),$ where ${\\mathcal {E}}$ is a CPTP projection onto $\\hat{\\mathcal {A}}.$ Proof.",
"To prove the claim, we explicitly construct the maps combining the ideas from the two previous theorems.",
"Define $P_{j} $ to be the projector onto ${\\rm supp}(\\mathcal {A}_j)$ , and the maps $\\mathcal {E}^0_{j}(\\cdot ) &\\equiv & P_{j} (\\cdot ) P_{j} + \\frac{ P_{j} }{ {\\rm Tr}({P_{j}}) } \\,{\\rm Tr}\\,(P^\\perp _{j}\\cdot ), \\\\\\mathcal {E}^1_{j} & \\equiv &\\mathcal {E}_{\\mathcal {A}_j}\\oplus {{\\mathcal {I}}_{\\mathcal {A}_j^\\perp }, }$ where $\\mathcal {E}_{\\mathcal {A}_j}:\\mathcal {B}({\\rm supp}(\\mathcal {A}_j))\\rightarrow \\mathcal {B}({\\rm supp}(\\mathcal {A}_j))$ is the unique CPTP projection onto $\\mathcal {A}_j$ (notice that on its own support $\\mathcal {A}_j$ includes a full-rank state), and ${\\mathcal {I}}_{\\mathcal {A}_j^\\perp }$ denotes the identity map on operators on ${\\rm supp}(\\mathcal {A}_j)^\\perp $ .",
"Now construct $\\mathcal {E}_j (\\cdot )\\equiv \\mathcal {E}^1_{j}\\circ \\mathcal {E}^0_{j}(\\cdot ).$ Since each map $\\mathcal {E}^1_{j}$ leaves the support of $P_j$ invariant, the same Lyapunov argument of Theorem REF shows that: ${\\rm supp}(\\lim _{n\\rightarrow \\infty }({\\mathcal {E}}_r\\ldots {\\mathcal {E}}_1)^n(\\rho ))\\subseteq {\\rm supp}({\\mathcal {E}}_{{\\cal B}(\\hat{\\mathcal {H}})}(\\tau )).$ We thus have that the largest invariant set for a cycle of maps ${\\mathcal {E}}_r\\ldots {\\mathcal {E}}_1$ has support equal to $\\hat{\\mathcal {A}},$ and by the discrete-time invariance principle [54], the dynamics converge to that.",
"Now notice that, since $\\hat{\\mathcal {A}}$ is contained in each of the $\\mathcal {A}_j={\\rm fix}(\\mathcal {E}_j)$ , such is any maximum-rank operator in $\\hat{\\mathcal {A}},$ which implies (see e.g.",
"Lemma 1 in [37]) that ${\\rm supp}(\\hat{\\mathcal {A}})$ is an invariant subspace for each $\\mathcal {E}_j.$ Hence, $\\mathcal {E}_j$ restricted to ${\\cal B}({\\rm supp}(\\hat{\\mathcal {A}}))$ is still CPTP, and by construction projects onto the elements of $\\mathcal {A}_j$ that have support contained in ${\\rm supp}(\\hat{\\mathcal {A}}).$ Such a set, call it $\\hat{\\mathcal {A}}_j$ , is thus a valid fixed-point set.",
"By Theorem REF , we have that on the support of $\\hat{\\mathcal {A}}$ the limit in Eq.",
"(REF ) converges to $\\hat{\\mathcal {A}}.$ This shows that the largest invariant set for the cycle is exactly $\\hat{\\mathcal {A}},$ hence the claim is proved.",
"$\\Box $ Remark: In order for the proposed quantum alternating projection methods to be effective, it is important that the relevant CPTP maps be sufficiently simple to evaluate and implement.",
"Assuming that the map $\\mathcal {E}$ is easily achievable, it is useful to note that the projection map $\\mathcal {E}_{{\\cal A}_\\rho }$ defined in Eq.",
"(REF ) may be approximated through iteration of a map $\\tilde{\\mathcal {E}}_\\lambda \\equiv (1-\\lambda ) \\mathcal {E}+ \\lambda {\\cal I}$ , where $\\lambda \\in (0,1)$ .",
"Since $\\tilde{\\mathcal {E}}_\\lambda $ has 1 as the only eigenvalue on the unit circle [55], it is easy to show that $ \\lim _{n \\rightarrow \\infty } \\tilde{\\mathcal {E}}_\\lambda ^n = \\mathcal {E}_{{\\cal A}_\\rho }$ , $\\mathcal {E}_{{\\cal A}_\\rho } \\approx \\tilde{\\mathcal {E}}_\\lambda ^n$ for a sufficiently large number of iterations."
],
[
"Convergence rate",
"For practical applications, a relevant aspect of stabilizing a target set is provided by the rate of asymptotic convergence.",
"In our case, focusing for concreteness on state stabilization, the key to Theorems REF and REF (and to Theorem REF that will be given in Sec.",
"REF ) is Halperin's alternating projection result.",
"Thus, if we are interested in the speed of convergence of stabilizing dynamics for a full-rank state $\\rho $ , this is just the speed of convergence of the classical alternating projection method.",
"As mentioned, and as we recall in Appendix REF , the rate is related to the angles between the subspaces.",
"In fact, an upper bound for the distance decrease from the target attained in $n$ repetitions of a cycle of maps $\\mathcal {E}_{\\rm cycle}= \\mathcal {E}_r\\circ \\ldots \\circ \\mathcal {E}_1$ is given in terms of the contraction coefficient $c^{\\frac{n}{2}},$ where $c \\equiv 1-\\prod _{i=1}^{r-1} \\sin ^2{\\theta _i},$ and $\\theta _i$ is the angle between $\\mathcal {A}_i$ and the intersection of the fixed points of the following maps.",
"In particular, by Theorem REF in the Appendix, it follows that a sufficient condition for finite-time convergence of iterated projections is given by $c=0,$ which is satisfied for example if $c(\\mathcal {A}_i,\\mathcal {A}_j)=0$ for all $1\\le i,j\\le r$ .",
"That is, equivalently, $\\big [\\mathcal {A}_i\\cap \\big (\\bigcap _{t=1}^r\\mathcal {A}_t\\big )^\\perp \\big ]\\perp _\\rho \\big [\\mathcal {A}_j\\cap \\big (\\bigcap _{t=1}^r\\mathcal {A}_t\\big )^\\perp \\big ],$ for every $i,j=i+1,\\ldots ,r$ , where orthogonality is with respect to the $\\rho ^{-1}$ -inner product, either symmetric or not.",
"However, this condition is clearly not necessary.",
"For a pure target state $\\rho $ , a natural way to quantify the convergence rate is to consider the decrease of a suitable Lyapunov function.",
"Given the form of the projection maps we propose, a natural choice is the same $V$ we use in the proof of Theorem REF , namely, $V(\\tau ) = 1-{\\rm Tr}{}({\\rho \\, \\tau }).$ The variation of $V,$ when $\\mathcal {E}_{\\rm cycle}$ is applied, is $\\Delta V(\\tau )=-{\\rm Tr}{} [{\\rho (\\mathcal {E}_{\\rm cycle}(\\tau )-\\tau )}] <0, \\quad \\forall \\tau .",
"$ The contraction coefficient in the pseudo-distance $V$ is then: $c=\\max _{\\tau \\ge 0,{\\rm Tr}(\\tau )=1, {\\rm Tr}({\\tau \\rho )=0} }\\Delta V(\\tau ).$ In this way, we select the rate corresponding to the worst-case state with support orthogonal to the target (notice that maximization over all states would have just given zero)."
],
[
"Robustness with respect to randomization",
"While Theorem REF and Theorem REF require deterministic cyclic repetition of the CPTP projections, the order is not critical for convergence.",
"Randomizing the order of the maps still leads to asymptotic convergence, albeit in probability.",
"We say that an operator-valued process $X(t)$ converges in probability to $X^*$ if, for any $\\delta ,\\varepsilon > 0$ , there exists a time $T>0$ such that $\\mathbb {P}[\\, {\\rm Tr}((X(T)-X_*)^2) \\, > \\varepsilon \\,] \\; < \\delta \\, .$ Likewise, $X(t)$ converges in expectation if $\\mathbb {E}(\\rho (t))\\rightarrow \\rho ^*$ when $t\\rightarrow +\\infty .$ Establishing convergence in probability uses the following lemma, adapted from [56], a consequence of the second Borel-Cantelli lemma: Lemma 2 (Convergence in probability) Consider a finite number of CPTP maps $\\lbrace \\mathcal {E}_j\\rbrace _{j=1}^M,$ and a (Lyapunov) function $V(\\rho ),$ such that $V(\\rho )\\ge 0$ and $V(\\rho )=0$ if and only if $\\rho \\in {\\cal S},$ with $\\cal S\\subset {\\mathcal {D}}(\\mathcal {H})$ some set of density operators.",
"Assume, furthermore that: (i) For each $j$ and state $\\rho $ , $V(\\mathcal {E}_j(\\rho ))\\le V(\\rho )$ .",
"(ii) For each $\\varepsilon > 0$ there exists a finite sequence of maps $\\mathcal {E}_{\\varepsilon }=\\mathcal {E}_{j_K}\\circ \\ldots \\circ \\mathcal {E}_{{j_1}},$ with $j_\\ell \\in \\lbrace 1,\\ldots , M \\rbrace $ for all $\\ell $ , such that $V(\\mathcal {E}_{\\varepsilon }(\\rho ))< \\varepsilon $ for all $\\rho \\ne {\\cal S}.$ Assume that the maps are selected at random, with independent probability distribution $\\mathbb {P}_t[\\mathcal {E}_{j}]$ at each time $t$ , and that there exists $\\varepsilon >0$ for which $\\mathbb {P}_t[\\mathcal {E}_{j}] > \\varepsilon $ for all $t$ .",
"Then, for any $\\gamma >0$ , the probability of having $V(\\rho (t))<\\gamma $ converges to 1 as $t\\rightarrow +\\infty $ .",
"Using the above result, we can prove the following: Corollary 2 Let ${\\mathcal {E}}_1,\\ldots ,{\\mathcal {E}}_r$ CPTP projections onto $\\mathcal {A}_i={\\mathfrak {B}}(\\mathcal {H}_i)$ , $i=1,\\ldots , r$ .",
"Assume that at each step $t\\ge 0$ the map $\\mathcal {E}_{j(t)}$ is selected randomly from a probability distribution $ \\Big \\lbrace q_j(t)=\\mathbb {P}[{{\\cal E}_{j(t)}}] > 0 \\vert \\sum _{j} q_{j}(t) = 1 \\Big \\rbrace ,$ and that $q_j(t)>\\epsilon >0$ for all $j$ and $t\\ge 0.$ For all $ \\tau \\in \\mathfrak {D}(\\mathcal {H})$ , let $\\tau (t) \\equiv \\mathcal {E}_{j(t)}\\circ \\ldots \\circ \\mathcal {E}_{j(1)}(\\tau ).$ Then $\\tau (t)$ converges in probability and in expectation to $\\tau ^*={\\mathcal {E}}_{\\hat{\\mathcal {A}}}(\\tau ),$ where ${\\mathcal {E}}$ is the CPTP projection onto $\\hat{\\mathcal {A}}.$ Proof.",
"Given Lemma REF , it suffices to consider $V(\\tau )\\equiv 1-\\rm {Tr}(\\hat{P} \\, \\tau ).$ It is non-increasing, and Theorem REF also ensures that for every $\\varepsilon >0$ , there exists a finite number of cycles of the maps that makes $V(\\tau )< \\varepsilon .$ $\\Box $ A similar result holds for the full-rank case: Corollary 3 Let ${\\mathcal {E}}_1,\\ldots ,{\\mathcal {E}}_r$ CPTP projections onto $\\mathcal {A}_i$ , $i=1,\\ldots ,r,$ and assume that $\\hat{\\mathcal {A}}=\\bigcap _{i=1}^r \\mathcal {A}_i$ contains a full-rank state $\\rho .$ Assume that at each step $t\\ge 0$ the map $\\mathcal {E}_{j(t)}$ is selected randomly from a probability distribution $\\Big \\lbrace q_j(t)=\\mathbb {P}[{{\\cal E}_{j(t)}}] > 0 \\vert \\sum _{j} q_{j}(t) = 1 \\Big \\rbrace ,$ and that $q_j(t)>\\epsilon >0$ for all $j$ and $t\\ge 0.$ For all $ \\tau \\in \\mathfrak {D}(\\mathcal {H})$ , let $\\tau (t) \\equiv \\mathcal {E}_j(t)\\circ \\ldots \\circ \\mathcal {E}_j(1)(\\tau ).$ Then $\\tau (t)$ converges in probability and in expectation to $\\tau ^*={\\mathcal {E}}_{\\hat{\\mathcal {A}}}(\\tau ),$ where ${\\mathcal {E}}$ is the CPTP projection onto $\\hat{\\mathcal {A}}.$ Proof.",
"Given the Lemma REF , it suffices to consider $V(\\tau )\\equiv \\langle (\\tau -\\tau ^*),(\\tau -\\tau ^*)\\rangle _{\\rho ^{-1}}.$ It is non-increasing, and Theorem REF ensures that for every $\\varepsilon >0$ there exists a finite number of cycles of the maps that makes $V(\\tau )< \\varepsilon .$ $\\Box $"
],
[
"Locality notion",
"In this section we specialize to a multipartite quantum system consisting of $n$ (distinguishable) subsystems, or “qudits”, defined on a tensor-product Hilbert space $\\mathcal {H}\\equiv \\bigotimes _{a=1}^n\\mathcal {H}_a, \\quad a=1,\\ldots ,n,\\;\\text{dim}(\\mathcal {H}_a)=d_a, \\, \\text{dim}(\\mathcal {H})=d.",
"$ In order to impose quasi-locality constraints on operators and dynamics on $\\mathcal {H},$ we introduce neighborhoods.",
"Following [27], [28], [48], neighborhoods $\\lbrace {\\cal N}_j \\rbrace $ are subsets of indexes labeling the subsystems, that is, ${\\cal N}_j\\subsetneq \\lbrace 1,\\ldots ,n\\rbrace , \\quad {j=1,\\ldots , K.}$ A neighborhood operator $M$ is an operator on $\\mathcal {H}$ such that there exists a neighborhood ${\\cal N}_j$ for which we may write $ M \\equiv M_{{\\cal N}_j}\\otimes I_{\\overline{\\cal N}_j},$ where $M_{{\\cal N}_j}$ accounts for the action of $M$ on subsystems in ${\\cal N}_j$ , and $I_{\\overline{\\cal N}_j}\\equiv \\bigotimes _{a\\notin {\\cal N}_j}I_a$ is the identity on the remaining ones.",
"Once a state $\\rho \\in {\\cal D}(\\mathcal {H})$ and a neighborhood structure are assigned on $\\mathcal {H},$ reduced neighborhood states may be computed via partial trace as usual: $\\rho _{{\\cal N}_j} \\equiv \\mbox{Tr}_{\\overline{\\cal N}_j}(\\rho ),\\quad \\rho \\in {\\mathfrak {D}}(\\mathcal {H}), \\;\\; j=1,\\ldots , K,$ where ${\\rm Tr}_{\\overline{\\cal N}_j}$ indicates the partial trace over the tensor complement of the neighborhood ${\\cal N}_j,$ namely, $\\mathcal {H}_{\\overline{\\cal N}_j} \\equiv \\bigotimes _{a\\notin {\\cal N}_j}\\mathcal {H}_a$ .",
"A strictly “local” setting corresponds to the case where ${\\cal N}_j \\equiv \\lbrace j\\rbrace $ , that is, each subsystem forms a distinct neighborhood.",
"Assume that some quasi-locality notion is fixed by specifying a set of neighborhoods, ${\\cal N}\\equiv \\lbrace {\\cal N}_j\\rbrace $ .",
"A CP map $\\mathcal {E}$ is a neighborhood map relative to ${\\cal N}$ if, for some $j$ , $\\mathcal {E}=\\mathcal {E}_{{\\cal N}_j}\\otimes {\\cal I}_{\\overline{\\cal N}_j},\\ $ where $\\mathcal {E}_{{\\cal N}_j}$ is the restriction of $\\mathcal {E}$ to operators on the subsystems in ${\\cal N}_j$ and ${\\cal I}_{\\overline{\\cal N}_j}$ is the identity map for operators on $\\mathcal {H}_{\\overline{\\cal N}_j}$ .",
"An equivalent formulation can be given in terms of the OSR: that is, $\\mathcal {E}(\\rho )=\\sum _k M_k \\rho M_k^\\dag $ is a neighborhood map relative to ${\\cal N}$ if there exists a neighborhood ${\\cal N}_j$ such that, for all $k,$ $M_k = M_{{\\cal N}_j,k}\\otimes I_{\\overline{\\cal N}_j}.$ The reduced map on the neighborhood is then $\\mathcal {E}_{{\\cal N}_j}(\\cdot )=\\sum _k M_{{\\cal N}_j,k}\\,\\cdot \\,M_{{\\cal N}_j,k}^\\dag .$ Since the identity factor is preserved by sums (and products) of the $M_k$ , it is immediate to verify that the property of $\\mathcal {E}$ being a neighborhood map is well-defined with respect to the freedom in the OSR [5].",
"Definition 3 (i) A state $\\rho $ is discrete-time Quasi-Locally Stabilizable (QLS) if there exists a sequence $\\lbrace \\mathcal {E}_t\\rbrace _{t\\ge 0}$ of neighborhood maps such that $\\rho $ is GAS for the associated propagator $\\mathcal {E}_{t,s}=\\mathcal {E}_{t-1}\\circ \\ldots \\circ \\mathcal {E}_s$ , namely: $&&\\hspace*{-14.22636pt} \\mathcal {E}_{t,s}(\\rho )=\\rho ,\\quad \\forall t\\ge s\\ge 0;\\\\&&\\hspace*{-14.22636pt}\\lim _{t\\rightarrow \\infty } \\Vert \\mathcal {E}_{t,s}(\\sigma ),\\rho \\Vert =0,\\quad \\forall \\sigma \\in {\\mathfrak {D}}(\\mathcal {H}),\\, \\forall s\\ge 0.$ (ii) The state is QLS in finite time (or finite-time QLS) if there exists a finite sequence of $T$ maps whose propagator satisfies the invariance requirement of Eq.",
"(REF ) and ${\\mathcal {E}_{T,0}(\\sigma )=\\rho , \\quad \\forall \\sigma \\in {\\mathfrak {D}}(\\mathcal {H}).",
"}$ Remark: With respect to the definition of quasi-locality that naturally emerges for continuous-time Markov dynamics [27], [28], [48], it is important to appreciate that constraining discrete-time dynamics to be QL in the above sense is more restrictive.",
"In fact, even if a generator ${\\cal L}$ of a continuous-time (homogeneous) semigroup can be written as a sum of neighborhood generators, namely, ${\\cal L}=\\sum _k {\\cal L}_k$ , the generated semigroup $\\mathcal {E}_t \\equiv e^{\\mathcal {L}t}, t\\ge 0,$ is not, in general, QL in the sense of Eq.",
"(REF ) at any time.",
"In some sense, one may think of the different noise components $\\mathcal {L}_1,\\ldots ,\\mathcal {L}_k$ of the continuous-time generator as acting “in parallel”.",
"On the other hand, were the maps $\\mathcal {E}_j$ we consider in this paper each generated by some corresponding neighborhood generator ${\\mathcal {L}_j},$ then by QL discrete-time dynamics we would be requesting that, on each time interval, a single noise operator is active, thus obtaining global switching dynamics [32] of the form $ e^{\\mathcal {L}_k T_k}\\circ e^{\\mathcal {L}_{k-1} T_{k-1}}\\circ \\ldots \\circ e^{\\mathcal {L}_1 T_1}.$ We could have requested each $\\mathcal {E}_t$ to be a convex combination of neighborhood maps acting on different neighborhoods, however it is not difficult to see that this case can be studied as the convergence in expectation for a randomized sequence.",
"Hence, we are focusing on the most restrictive definition of QL constraint for discrete-time Markov dynamics.",
"With respect to the continuous dynamics, however, we allow for the evolution to be time-inhomogeneous.",
"Remarkably, we shall find a characterization of QLS pure states that is equivalent to the continuous-time case, when the latter dynamics are required to be frustration free (see Section REF )."
],
[
"Invariance conditions and minimal fixed point sets",
"In this section, we build on the invariance requirement of Eq.",
"(REF ) to find necessary conditions that the discrete-time dynamics must satisfy in order to have a given state $\\rho $ as its unique and attracting equilibrium.",
"These impose a certain minimal fixed-point set, and hence suggest a structure for the stabilizing dynamics.",
"Following [48], given an operator $X\\in \\mathcal {B}(\\mathcal {H}_A\\otimes \\mathcal {H}_B)$ , with corresponding (operator) Schmidt decomposition $X=\\sum _j A_j\\otimes B_j,$ we define its Schmidt span as: $\\Sigma _A(X) \\equiv {\\rm span}(\\lbrace A_j\\rbrace ).$ The Schmidt span is important because, if we want to leave an operator invariant with a neighborhood map, this also imposes the invariance of its Schmidt span.",
"The following lemma, proved in [48], makes this precise: Lemma 3 Given a vector $v\\in V_{A}\\otimes V_{B}$ and $M_A\\in \\mathcal {B}(V_A)$ , if $(M_A\\otimes \\mathbb {I}_B) v=\\lambda v$ , then $(M_A\\otimes \\mathbb {I}_B) v^{\\prime }=\\lambda v^{\\prime }$ for all $v^{\\prime }\\in \\Sigma _{A}(v)\\otimes V_B$ .",
"What we need here can be obtained by adapting this result to our case, specifically: Corollary 4 Given a $\\rho \\in \\mathcal {D}(\\mathcal {H}_{\\mathcal {N}_j}\\otimes \\mathcal {H}_{\\overline{{\\mathcal {N}}}_j})$ and a neighborhood $\\mathcal {E}=\\mathcal {E}_{\\mathcal {N}_j}\\otimes I_{\\overline{\\mathcal {N}}_j}$ , then ${\\rm span}(\\rho )\\subseteq {\\rm fix}(\\mathcal {E})$ implies $\\Sigma _{\\mathcal {N}_j}(\\rho )\\otimes \\mathcal {B}(\\mathcal {H}_{\\overline{\\mathcal {N}}_j})\\subseteq {\\rm fix}(\\mathcal {E}_{{\\cal N}_j}).$ A Schmidt span need not be a valid fixed-point set, namely, a $\\rho $ -distorted algebra that is invariant for ${\\cal M}_{\\rho ,\\frac{1}{2}}.$ In general, we need to further enlarge the QL fixed-point sets from the Schmidt span to suitable algebras.",
"We discuss separately two relevant cases.",
"$\\bullet $ Pure states.— Let $\\rho =| \\psi \\rangle \\langle \\psi |$ be a pure state and assume that, with respect to the factorization $\\mathcal {H}_{\\mathcal {N}_j}\\otimes \\mathcal {H}_{\\overline{\\mathcal {N}}_j},$ its Schmidt decomposition $| \\psi \\rangle =\\sum _k c_k| \\psi _k \\rangle \\otimes | \\phi _k \\rangle .$ Let $\\mathcal {H}^0_{\\mathcal {N}_j} \\equiv {\\rm span}\\lbrace | \\psi _k \\rangle \\rbrace = \\text{supp} (\\rho _{\\mathcal {N}_j} )$ .",
"Then we have [48]: $\\Sigma _{\\mathcal {N}_j}(\\rho )={\\cal B}(\\mathcal {H}^0_{\\mathcal {N}_j}).$ In this case the Schmidt span is indeed a valid fixed-point set, and no further enlargement is needed.",
"The minimal fixed-point set for neighborhood maps required to preserve $\\rho $ is thus $\\mathcal {F}_j \\equiv {\\cal B}(\\mathcal {H}^0_{\\mathcal {N}_j}) \\otimes \\mathcal {B}(\\mathcal {H}_{\\overline{\\mathcal {N}}_j}) .$ By construction, each $\\mathcal {F}_j$ contains $\\rho .$ Notice that their intersection is just $\\rho $ if and only if ${\\rm span}\\lbrace | \\psi \\rangle \\rbrace =\\bigcap _j \\mathcal {H}^0_{\\mathcal {N}_j} \\otimes \\mathcal {H}_{\\overline{\\mathcal {N}}_j}=\\bigcap _j {\\mathcal {H}^0_j, }$ where we have defined $\\mathcal {H}^0_j\\equiv \\mathcal {H}^0_{\\mathcal {N}_j} \\otimes \\mathcal {H}_{\\overline{\\mathcal {N}}_j}.$ $\\bullet $ Full rank states.— If $\\rho $ is a full-rank state, and $W$ a set of operators, the minimal fixed-point set generated by $\\rho $ and $W$, by Theorem REF , is the smallest $\\rho $ -distorted algebra generated by $W$ which is invariant with respect to ${\\cal M}_{\\rho ,\\frac{1}{2}}$ .",
"Notice that, since $\\rho $ is full rank, its reduced states $\\rho _{\\mathcal {N}_j}$ are also full rank.",
"Denote by ${\\rm alg}_\\rho (W)$ the $\\dag $ -closed $\\rho $ -distorted algebra generated by $W.$ Call $W_j \\equiv \\Sigma _{\\mathcal {N}_j}(\\rho )$ .",
"The minimal fixed-point sets $\\mathcal {F}_{\\rho _{\\mathcal {N}_j}}(W_j )$ can then be constructed iteratively from $\\mathcal {F}_j^{(0)} \\equiv {\\rm alg}_{\\rho _{\\mathcal {N}_j}}(W_j)$ , with the $k$ -th step given by [48]: $\\mathcal {F}_j^{(k+1)} \\equiv {\\rm alg}_{\\rho _{\\mathcal {N}_j}}{({\\cal M}_{ \\rho _{\\mathcal {N}_j}, \\frac{1}{2} } (\\mathcal {F}_j^{(k)}),{\\mathcal {F}_j^{(k)}}).",
"}$ We keep iterating until $\\mathcal {F}_j^{(k+1)}=\\mathcal {F}_j^{(k)} = \\mathcal {F}_{\\rho _{{\\cal N}_j}} (W_j).$ When that happens, define $\\mathcal {F}_j \\equiv \\mathcal {F}_{\\rho _{{\\cal N}_j}} ( \\Sigma _{\\mathcal {N}_j}(\\rho ) ) \\otimes {\\cal B}(\\mathcal {H}_{\\overline{\\mathcal {N}}_j}).$ Since the ${\\cal F}_j$ are constructed to be the minimal sets for neighborhood maps that contain the given state and its corresponding Schmidt span, then clearly: ${\\rm span}(\\rho )\\subset \\bigcap _j \\mathcal {F}_j.$"
],
[
"Stabilizability under quasi-locality constraints",
"In the case of a pure target state, we can prove the following: Theorem 8 (QLS pure states) A pure state $\\rho =|\\psi \\rangle \\langle \\psi |$ is QLS by discrete-time dynamics if and only if ${ {\\rm supp}(\\rho )= \\bigcap _j {{\\mathcal {H}}^0_{j}.}",
"}$ Proof.",
"Given Corollary REF , any dynamics that make $\\rho $ QLS (and hence leaves it invariant) must consist of neighborhood maps $\\lbrace \\mathcal {E}_j\\rbrace $ with corresponding fixed points such that: $\\mathcal {F}_{k}\\subseteq {\\rm fix}(\\mathcal {E}_j),$ whenever $\\mathcal {E}_j$ is a $\\mathcal {N}_{k}$ -neighborhood map.",
"If the intersection of the fixed-point sets is not unique, then $\\rho $ cannot be GAS, since there would be another state that is not attracted to it.",
"Given Eq.",
"(REF ), we have $ {{\\rm span}(\\rho )=\\bigcap _k \\mathcal {F}_k \\quad \\Longleftrightarrow \\quad {\\rm supp}(\\rho )= \\bigcap _j {{\\mathcal {H}}^0_{j}.",
"}, }$ which proves necessity.",
"For sufficiency, we explicitly construct neighborhood maps whose cyclic application ensures stabilization.",
"Define$P_{\\mathcal {N}_j} $ to be the projector onto $\\textup {supp}(\\rho _{\\mathcal {N}_j} )$ , and the CPTP maps: $\\mathcal {E}_{\\mathcal {N}_j}(\\cdot ) \\equiv P_{\\mathcal {N}_j} (\\cdot ) P_{\\mathcal {N}_j} + \\frac{ P_{\\mathcal {N}_j} }{ {\\rm Tr}({P_{\\mathcal {N}_j}}) } \\,{\\rm Tr}\\,(P^\\perp _{\\mathcal {N}_j}\\cdot ),$ with $\\mathcal {E}_j \\equiv \\mathcal {E}_{\\mathcal {N}_j}\\otimes {\\mathcal {I}}_{\\overline{\\mathcal {N}}_j}.$ Consider the positive-semidefinite function $V(\\tau )=1-\\text{Tr}_{}\\left(\\rho \\, \\tau \\right),$ $\\tau \\in {\\mathcal {B}}(\\mathcal {H}).$ The result then follows from Theorem REF .",
"$\\Box $ An equivalent characterization can be given for full-rank target states: Theorem 9 (QLS full-rank states) A full-rank state $\\rho \\in {\\cal D}(\\mathcal {H})$ is QLS by discrete-time dynamics if and only if ${\\rm span}(\\rho )=\\bigcap _k \\mathcal {F}_k$ Proof.",
"As before, by contradiction, suppose that $\\rho _2\\in \\bigcap _k \\mathcal {F}_k$ exists, such that $\\rho _2\\ne \\rho $ .",
"This clearly implies that $\\rho $ cannot be GAS because there would exist another invariant state, which is not attracted to $\\rho $ .",
"This proves necessity.",
"Sufficiency derives from the alternating CPTP projection theorem.",
"Specifically, let $\\mathcal {E}_{\\mathcal {N}_k}$ be the CPTP projection onto ${\\cal F}_k,$ and $\\mathcal {E}_k\\equiv \\mathcal {E}_{\\mathcal {N}_k}\\otimes {\\rm Id}_{\\overline{\\mathcal {N}}_k}.$ By Theorem REF , we already know that for every $\\rho $ , $(\\mathcal {E}_M\\ldots \\mathcal {E}_1)^k(\\rho )\\rightarrow \\bigcap _k \\mathcal {F}_k$ for $k\\rightarrow \\infty $ .",
"Now, by hypothesis, $\\bigcap _k \\mathcal {F}_k={\\rm span}(\\rho )$ and, being $\\rho $ the only (trace one) state in his own span, $\\rho $ is GAS.",
"$\\Box $ A set of sufficient conditions, stemming from Theorem REF , can be also derived in an analogous way for a general target state."
],
[
"Physical interpretation: discrete-time quasi-local stabilizability is equivalent to\ncooling without frustration",
"Consider a quasi-local Hamiltonian, that is, $H=\\sum _k H_k,$ $H_k=H_{{\\cal N}_k}\\otimes I_{\\overline{\\cal N}_k}.$ $H$ is called a parent Hamiltonian for a pure state $| \\psi \\rangle $ if it admits $| \\psi \\rangle $ as a ground state, and it is called a is called a frustration-free (FF) Hamiltonian if any global ground state is also a local ground state [57], that is, $ \\text{argmin}_{| \\psi \\rangle \\in \\mathcal {H}} \\langle \\psi | H | \\psi \\rangle \\subseteq \\text{argmin}_{| \\psi \\rangle \\in \\mathcal {H}} \\langle \\psi | H_k | \\psi \\rangle , \\forall k. $ Suppose that a target state $| \\psi \\rangle $ admits a FF QL parent Hamiltonian $H$ for which it is the unique ground state.",
"Then, similarly to what has been done for continuous-time dissipative preparation [31], [27], the structure of $H$ may be naturally used to derive a stabilizing discrete-time dynamics: it suffices to implement neighborhood maps $M_k$ that stabilize the eigenspace associated to the minimum eigenvalue of each $H_k$ .",
"These can thought as maps that locally “cool” the system.",
"In view of Theorem REF , it is easy to show that this condition is also necessary: Corollary 5 A state $\\rho =|\\psi \\rangle \\langle \\psi |$ is QLS by discrete-time dynamics if and only if it is the unique ground state of a FF QL parent Hamiltonian.",
"Proof.",
"Without loss of generality we can consider FF QL Hamiltonians $H=\\sum _k H_k,$ where each $H_k$ is a projection.",
"Let $\\rho $ satisfy Eq.",
"(REF ), which is equivalent to be QLS, and define $H_k \\equiv \\Pi ^\\perp _{{\\cal N}_k}\\otimes I_{\\overline{\\cal N}_k}$ , with $\\Pi ^\\perp _{{\\cal N}_k}$ being the orthogonal projector onto the orthogonal complement of the support of $\\rho _{{\\cal N}_k}$ , that is, $\\mathcal {H}_{{\\cal N}_k}\\ominus {\\rm supp}({\\rho _{{\\cal N}_k}})$ .",
"Given Theorem REF , $|\\psi \\rangle $ is the unique pure state in $\\bigcap _k{\\rm supp}(\\rho _{{\\cal N}_k}\\otimes I_{\\overline{\\cal N}_k}),$ and thus the unique state in the kernel of all the $H_k$ .",
"Conversely, if a FF QL parent Hamiltonian exists, the kernels of the $H_k$ satisfy the QLS condition and to each $H_k$ we can associate a CPTP map as in (REF ) that projects onto its kernel.",
"$\\Box $ An equivalent characterization works for generic, full-rank target states, but we need to move from Hamiltonians to semigroup generators, while maintaining frustration-freeness in a suitable sense.",
"If, as before, ${\\cal E}_{t}=e^{{\\cal L}}$ is the propagator arising from a time-invariant QL generator ${\\cal L}$ , we are interested in QL generators whose neighborhood components drive the system to a global equilibrium which is also a local equilibrium for each of them separately.",
"That is, following [58], [48]: Definition 4 A QL generator $\\mathcal {L}=\\sum _{j}\\mathcal {L}_{j},$ where ${\\cal L}_j$ are neighborhood generators, is Frustration Free (FF) relative to a neighborhood structure $\\mathcal {N}=\\lbrace \\mathcal {N}_j\\rbrace $ if $\\rho \\in {\\rm ker}(\\mathcal {L})\\quad \\iff \\quad \\rho \\in {\\rm ker}(\\mathcal {L}_j), \\quad \\forall j.$ It is worth noting that a state which is invariant for all the local generators is always an equilibrium: the real requirement is that these states are all the equilibria.",
"We then have: Proposition 4 A pure or full-rank state $\\rho $ is discrete-time QLS if and only if it is QLS via FF continuous-time dynamics, that is, there exists a FF generator ${\\cal L}$ with respect to the same neighborhood structure $\\mathcal {N}$ such that $\\lim _{t\\rightarrow +\\infty }e^{{\\cal L}t}\\rho _0=\\rho ,\\quad \\forall \\rho _0.$ Proof.",
"The claim follows from Theorems 7 and 8 in [48], which characterize the states that are unique fixed points of a FF generator as precisely the states that satisfy Eq.",
"(REF ).",
"$\\Box $ Remark: Based on the above results, the conditions that guarantee either a pure or a full-rank state to be QLS in discrete time are the same that guarantee existence of a FF stabilizing generator in continuous time.",
"We stress that if more general continuous-time generators are allowed, namely, frustration is permitted as in Hamiltonian-assisted stabilization [28], then the continuous-time setting can be more powerful.",
"On the one hand, considering the stricter nature of the QL constraint for the discrete-time setting, this is not surprising.",
"On the other hand, if Liouvillian is no longer FF, then the target is globally invariant for ${\\cal L}$ but no longer invariant for individual QL components ${\\cal L}_j$ , suggesting that a weaker (“stroboscopic”) invariance requirement could be more appropriate to “mimic” the effect of frustration in the discrete-time QL setting."
],
[
"Classes of discrete-time QLS states",
"Being the conditions for discrete-time QLS states the same as in continuous time with FF dynamics, we may conclude whether certain classes of states are QLS or not by exploiting the results already established in [27], [28], [48].",
"While we refer to the original references for additional detail and context, some notable examples are summarized in what follows.",
"Among pure states: Graph states [59] (more generally, stabilizer states [5] that admit stabilizer group generators that are neighborhood operators) are discrete-time QLS.",
"These states are entangled, and are a key resource for one-way quantum computation.",
"Certain, but not all, Dicke states are discrete-time QLS.",
"Dicke states are symmetric with respect to subsystem permutations, and have a specified “excitation number” [60].",
"Dicke states exhibit entanglement properties that are, in some sense, robust: some entanglement is preserved even if some subsystems are measured or traced out.",
"The $n$ -qubit single-excitation Dicke state, also known as W-state, $| \\psi ^n_{\\text{W}} \\rangle \\equiv \\frac{1}{\\sqrt{n}} (| 100\\ldots 0 \\rangle +| 010\\ldots 0 \\rangle +\\ldots +| 000\\ldots 1 \\rangle ), $ fails to satisfy the conditions of Theorem REF , so $\\rho _{\\text{W}}$ is not discrete-time QLS for non-trivial neighborhood structures (that is, unless there is a neighborhood that covers the whole network).",
"On the other hand, for example, the two-excitation Dicke state on $n=4$ qubits, $&&\\hspace{-22.76219pt}| \\psi _D^4 \\rangle \\equiv \\\\&&\\hspace{-22.76219pt}\\frac{| 1100 \\rangle +| 1010 \\rangle +| 1001 \\rangle +| 0110 \\rangle +| 0101 \\rangle +| 0011 \\rangle }{\\sqrt{6}},$ is QLS.",
"A more general class of QLS Dicke states on qudits is presented in [48].",
"Among full-rank states: Commuting Gibbs states are discrete-time QLS with respect to a suitable locality notion [48].",
"A Gibbs state represent the canonical thermal equilibrium state for a statistical system at temperature $\\beta ^{-1}$ : if a chain of qudits is associated to a nearest-neighbor (NN) Hamiltonian $H=\\sum _k H_k$ , its Gibbs state is $\\rho _\\beta \\equiv \\frac{e^{-\\beta H}}{{\\rm Tr}(e^{-\\beta H})}.$ If the Hamiltonian is commuting, namely, $[H_j,H_k]=0$ for all $j,k,$ then $\\rho $ is QLS with respect to an enlarged QL notion, where ${\\cal N}^0_j$ contains all subsystems that belong to a NN neighborhood ${\\cal N}_k$ such that ${\\cal N}_k\\cap {\\cal N}_j\\ne \\emptyset .$ In analogy to the continuous-time case [58], this shows that Gibbs samplers based on QL discrete-time dissipative dynamics are also viable, at least in the commuting case.",
"Certain mixtures of factorized and entangled states are discrete-time QLS.",
"For example, consider a 4-qubits system and the family of states parametrized by $\\epsilon \\in (0,1)$ : $ \\rho _\\epsilon \\equiv (1-\\epsilon ) \\, | \\psi _D^4 \\rangle \\langle \\psi _D^4 |+ \\epsilon \\, | \\textup {GHZ}^4 \\rangle \\langle \\textup {GHZ}^4 |, $ where $| \\textup {GHZ}^n \\rangle \\equiv \\left(| 0 \\rangle ^{\\otimes n}+| 1 \\rangle ^{\\otimes n}\\right)/\\sqrt{2}$ denotes the maximally-entangled Greenberger-Horne-Zeilinger (GHZ) states on $n$ qubits, and $\\mathcal {N}_1 = \\lbrace 1,2,3\\rbrace ,$ $\\mathcal {N}_2=\\lbrace 2,3,4\\rbrace $ .",
"This shows that we can stabilize states that are arbitrarily close to states that are provably not QLS, as the GHZ states [28], thereby achieving practical stabilization of the latter.",
"We have introduced alternating projection methods based on sequences of CPTP projections, and used them in designing discrete-time stabilizing dynamics for entangled states in multipartite quantum systems subject to realistic quasi-locality constraints.",
"When feasible, pursuing stabilization instead of preparation offers important advantages, including the possibility to retrieve the target state on-demand, at any (discrete) time after sufficient convergence is attained, since the invariance of the latter ensures that it is not ruined by subsequent maps.",
"We show that the proposed methods are also suitable for distributed, randomized and unsupervised implementations on large networks.",
"While the locality constraints we impose on the discrete-time dynamics are stricter, the stabilizable states are, remarkably, the same that are stabilizable for continuous-time frustration-free generators.",
"From a methodological standpoint, our results shed further light on the structure and intersection of fixed-point sets of CPTP maps, a structure of interest not only in control, but also in operator-algebraic approaches to quantum systems [61], quantum statistics [49] and quantum error correction theory [62], [63], [64].",
"In particular, we show that the intersection of fixed-point sets is still a fixed-point set, as long as it contains a full-rank state.",
"In developing our results, we use both standard results from classical alternating projections and Lyapunov methods tailored to the positive linear maps at hand.",
"Towards applications, the proposed alternating projection methods are in principle suitable for implementation in digital open-quantum system simulators, such as demonstrated in proof-of-principle trapped-ion experiments [33].",
"Beside providing protocols for stabilizing relevant classes of pure entangles states, our methods point to an alternative approach for constructing quantum samplers using quasi-local resources.",
"Some developments of this line of research are worth highlighting.",
"First, in order to extend the applicability of the proposed methods to more general classes of states, as well as to establish a tighter link to quantum error correction and dissipative code preparation, it is natural to look at discrete-time conditional stabilization, in the spirit of [28].",
"Notably, in [56], it has been shown that GHZ states and all Dicke states can be made conditionally asymptotically stable for QL discrete-time dynamics, with a suitable basin of attraction.",
"Second, while we recalled some basic classical bounds on the convergence speed, that apply to the stabilization of full-rank states, their geometric nature makes it hard to obtain useful insight from them.",
"A more intuitive approach to convergence speed and its optimization, following e.g.",
"[26], [37], may offer a more promising venue in that respect.",
"It has also been recently shown that linear Lyapunov functions can not only be used to prove convergence, but also provide sharp bounds on the convergence speed in continuous-time dynamics [65].",
"It would be interesting to extend the analysis to the non-homogeneous, discrete-time cases considered in this work.",
"Lastly, the characterization of physically relevant scenarios in which finite-time stabilization is possible under locality constraints is a challenging open problem, which we plan to address elsewhere [66]."
],
[
"Acknowledgements",
"It is a pleasure to acknowledge stimulating discussions on the topics of this work with A. Ferrante e L. Finesso.",
"F.T.",
"is especially grateful to V. Umanità and E. Sasso for pointing him towards Takesaki's theorem.",
"Work at Dartmouth was supported by the National Science Foundation through grant No.",
"PHY-1620541."
],
[
"Angles between subspaces",
"Define the function $\\arccos :[-1,1]\\rightarrow [-\\frac{\\pi }{2},\\frac{\\pi }{2} ]$ .",
"We will use only the elements in interval $[0,1]$ .",
"Then the angle $\\theta (\\mathcal {M},\\mathcal {N})$ between two closed subspaces $\\mathcal {M}$ and $\\mathcal {N}$ of $\\mathcal {H}$ is an element of $[0,\\frac{\\pi }{2}]$ .",
"We have the following: Definition 5 The cosine $c(\\mathcal {M},\\mathcal {N})$ between two closed subspaces $\\mathcal {M}$ and $\\mathcal {N}$ of $\\mathcal {H}$ is given by $c(\\mathcal {M},\\mathcal {N})&\\equiv &\\sup \\Big \\lbrace |\\langle x,y\\rangle |:x\\in \\mathcal {M}\\cap (\\mathcal {M}\\cap \\mathcal {N})^\\perp ,\\\\ &&\\Vert x\\Vert \\le 1, y\\in \\mathcal {N}\\cap (\\mathcal {M}\\cap \\mathcal {N})^\\perp , \\Vert y\\Vert \\le 1 \\Big \\rbrace .$ Then the angle is given by: $\\theta (\\mathcal {M},\\mathcal {N})=\\arccos (c(\\mathcal {M},\\mathcal {N})).$ Some consequences of the above definitions are the following: $0\\le c(\\mathcal {M},\\mathcal {N})\\le 1$ ; $c(\\mathcal {M},\\mathcal {N})=c(\\mathcal {N},\\mathcal {M})$ ; $c(\\mathcal {M},\\mathcal {N})=\\Vert P_\\mathcal {M}P_\\mathcal {N}-P_{\\mathcal {M}\\cap \\mathcal {N}}\\Vert =\\Vert P_\\mathcal {M}P_\\mathcal {N}P_{(\\mathcal {M}\\cap \\mathcal {N})^\\perp }\\Vert $ .",
"We next state the result that gives the exact rate in case of projection onto two subspaces [43]: Theorem 10 In the norm induced by the inner product, and for each $n$ , the following equality holds: $\\Vert (P_{\\mathcal {M}_2}P_{\\mathcal {M}_1})^n-P_{\\mathcal {M}_1\\cap \\mathcal {M}_2}\\Vert = c(\\mathcal {M}_1,\\mathcal {M}_2)^{2n-1}.$ In case of alternating projections on the intersection of more than two subspaces, an exact expression is no longer available, however an upper bound may be given [43]: Theorem 11 For each $i=1,2,\\ldots ,r$ , let $\\mathcal {M}_i$ be a closed subspace of $\\mathcal {H}$ .",
"Then, for each $x\\in \\mathcal {H}$ , and for any integer $n\\ge 1$ it holds: $\\Vert (P_{\\mathcal {M}_r}...P_{\\mathcal {M}_1})^n x-P_{\\bigcap _{i=1}^r\\mathcal {M}_i}x\\Vert \\le c^{\\frac{n}{2}}\\Vert x-P_{\\bigcap _{i=1}^r\\mathcal {M}_i}x\\Vert ,$ where the contraction coefficient $c=1-\\prod _{i=1}^{r-1}\\sin ^2\\theta _i,$ and $\\theta _i$ is the angle between $\\mathcal {M}_i$ and $\\bigcap _{j=i+1}^r\\mathcal {M}_j$ ."
],
[
"Non-orthogonality of ${\\cal E}_{\\cal A}$ with respect to the Hilbert-Schmidt inner product",
"Let us decompose a full-rank fixed point set $\\mathcal {A}_\\rho =\\bigoplus _\\ell \\mathcal {A}_\\ell =\\bigoplus _\\ell \\mathcal {B}(\\mathcal {H}_{S,\\ell })\\otimes \\tau _\\ell $ , (where $\\tau _\\ell \\equiv \\tau _{F,\\ell } $ ).",
"By definition, the orthogonal projection of $X$ onto $\\mathcal {A}_i$ is given by $P_\\mathcal {A}(X) \\equiv \\sum _{\\ell ,i}\\langle \\sigma _{\\ell ,i }\\otimes { \\tau _\\ell } ,X \\rangle _{HS} \\,\\sigma _{\\ell ,i}\\otimes \\tau _\\ell , $ where $\\sigma _{\\ell ,i}\\otimes \\tau _\\ell $ is an orthonormal basis for $\\mathcal {A}_\\ell $ .",
"Note that the outcome only depends on the restrictions of $X$ to the supports of the $\\mathcal {A}_\\ell .$ Hence, decompose $X \\equiv \\sum _\\ell X_\\ell +\\Delta X$ , where $X_\\ell =\\Pi _{SF,\\ell }X\\Pi _{SF,\\ell },$ and further decompose $X_\\ell \\equiv \\sum _k A_{\\ell ,k}\\otimes B_{\\ell ,k},$ so we can write: $P_\\mathcal {A}(X)&\\hspace*{-5.69054pt}=\\hspace*{-5.69054pt}&\\bigoplus _i\\sum _{j,\\ell }\\Big (\\sum _k {\\rm Tr}[(\\sigma _j\\otimes \\tau _\\ell )(A_{\\ell ,k}\\otimes B_{\\ell ,k})]\\sigma _j\\otimes \\tau _\\ell \\Big )\\\\&\\hspace*{-5.69054pt}=\\hspace*{-5.69054pt}&\\bigoplus _\\ell \\sum _{j,\\ell }\\Big ({\\rm Tr}[\\sigma _j \\sum _k(A_{\\ell ,k} {\\rm Tr}(\\tau _\\ell B_{\\ell ,k}))]\\sigma _j\\otimes \\tau _\\ell )\\Big ).$ By comparing the latter equation with Eq.",
"(REF ), we have that $P_\\mathcal {A}=\\mathcal {E}_\\mathcal {A}$ if and only if $\\sum _k(A_k {\\rm Tr}(\\tau _jB_k))={\\rm Tr}_{F,\\ell }(X_\\ell ),$ which is equivalent to request that $\\tau _j=\\lambda _\\ell I.$ Hence, unless $\\cal {A}_\\rho $ contains the completely mixed state, ${\\mathcal {E}}_\\mathcal {A}$ in Eq.",
"(REF ) is not an orthogonal projection with respect to the Hilbert-Schmidt inner product.",
"$\\Box $"
],
[
"Locality notion",
"In this section we specialize to a multipartite quantum system consisting of $n$ (distinguishable) subsystems, or “qudits”, defined on a tensor-product Hilbert space $\\mathcal {H}\\equiv \\bigotimes _{a=1}^n\\mathcal {H}_a, \\quad a=1,\\ldots ,n,\\;\\text{dim}(\\mathcal {H}_a)=d_a, \\, \\text{dim}(\\mathcal {H})=d.",
"$ In order to impose quasi-locality constraints on operators and dynamics on $\\mathcal {H},$ we introduce neighborhoods.",
"Following [27], [28], [48], neighborhoods $\\lbrace {\\cal N}_j \\rbrace $ are subsets of indexes labeling the subsystems, that is, ${\\cal N}_j\\subsetneq \\lbrace 1,\\ldots ,n\\rbrace , \\quad {j=1,\\ldots , K.}$ A neighborhood operator $M$ is an operator on $\\mathcal {H}$ such that there exists a neighborhood ${\\cal N}_j$ for which we may write $ M \\equiv M_{{\\cal N}_j}\\otimes I_{\\overline{\\cal N}_j},$ where $M_{{\\cal N}_j}$ accounts for the action of $M$ on subsystems in ${\\cal N}_j$ , and $I_{\\overline{\\cal N}_j}\\equiv \\bigotimes _{a\\notin {\\cal N}_j}I_a$ is the identity on the remaining ones.",
"Once a state $\\rho \\in {\\cal D}(\\mathcal {H})$ and a neighborhood structure are assigned on $\\mathcal {H},$ reduced neighborhood states may be computed via partial trace as usual: $\\rho _{{\\cal N}_j} \\equiv \\mbox{Tr}_{\\overline{\\cal N}_j}(\\rho ),\\quad \\rho \\in {\\mathfrak {D}}(\\mathcal {H}), \\;\\; j=1,\\ldots , K,$ where ${\\rm Tr}_{\\overline{\\cal N}_j}$ indicates the partial trace over the tensor complement of the neighborhood ${\\cal N}_j,$ namely, $\\mathcal {H}_{\\overline{\\cal N}_j} \\equiv \\bigotimes _{a\\notin {\\cal N}_j}\\mathcal {H}_a$ .",
"A strictly “local” setting corresponds to the case where ${\\cal N}_j \\equiv \\lbrace j\\rbrace $ , that is, each subsystem forms a distinct neighborhood.",
"Assume that some quasi-locality notion is fixed by specifying a set of neighborhoods, ${\\cal N}\\equiv \\lbrace {\\cal N}_j\\rbrace $ .",
"A CP map $\\mathcal {E}$ is a neighborhood map relative to ${\\cal N}$ if, for some $j$ , $\\mathcal {E}=\\mathcal {E}_{{\\cal N}_j}\\otimes {\\cal I}_{\\overline{\\cal N}_j},\\ $ where $\\mathcal {E}_{{\\cal N}_j}$ is the restriction of $\\mathcal {E}$ to operators on the subsystems in ${\\cal N}_j$ and ${\\cal I}_{\\overline{\\cal N}_j}$ is the identity map for operators on $\\mathcal {H}_{\\overline{\\cal N}_j}$ .",
"An equivalent formulation can be given in terms of the OSR: that is, $\\mathcal {E}(\\rho )=\\sum _k M_k \\rho M_k^\\dag $ is a neighborhood map relative to ${\\cal N}$ if there exists a neighborhood ${\\cal N}_j$ such that, for all $k,$ $M_k = M_{{\\cal N}_j,k}\\otimes I_{\\overline{\\cal N}_j}.$ The reduced map on the neighborhood is then $\\mathcal {E}_{{\\cal N}_j}(\\cdot )=\\sum _k M_{{\\cal N}_j,k}\\,\\cdot \\,M_{{\\cal N}_j,k}^\\dag .$ Since the identity factor is preserved by sums (and products) of the $M_k$ , it is immediate to verify that the property of $\\mathcal {E}$ being a neighborhood map is well-defined with respect to the freedom in the OSR [5].",
"Definition 3 (i) A state $\\rho $ is discrete-time Quasi-Locally Stabilizable (QLS) if there exists a sequence $\\lbrace \\mathcal {E}_t\\rbrace _{t\\ge 0}$ of neighborhood maps such that $\\rho $ is GAS for the associated propagator $\\mathcal {E}_{t,s}=\\mathcal {E}_{t-1}\\circ \\ldots \\circ \\mathcal {E}_s$ , namely: $&&\\hspace*{-14.22636pt} \\mathcal {E}_{t,s}(\\rho )=\\rho ,\\quad \\forall t\\ge s\\ge 0;\\\\&&\\hspace*{-14.22636pt}\\lim _{t\\rightarrow \\infty } \\Vert \\mathcal {E}_{t,s}(\\sigma ),\\rho \\Vert =0,\\quad \\forall \\sigma \\in {\\mathfrak {D}}(\\mathcal {H}),\\, \\forall s\\ge 0.$ (ii) The state is QLS in finite time (or finite-time QLS) if there exists a finite sequence of $T$ maps whose propagator satisfies the invariance requirement of Eq.",
"(REF ) and ${\\mathcal {E}_{T,0}(\\sigma )=\\rho , \\quad \\forall \\sigma \\in {\\mathfrak {D}}(\\mathcal {H}).",
"}$ Remark: With respect to the definition of quasi-locality that naturally emerges for continuous-time Markov dynamics [27], [28], [48], it is important to appreciate that constraining discrete-time dynamics to be QL in the above sense is more restrictive.",
"In fact, even if a generator ${\\cal L}$ of a continuous-time (homogeneous) semigroup can be written as a sum of neighborhood generators, namely, ${\\cal L}=\\sum _k {\\cal L}_k$ , the generated semigroup $\\mathcal {E}_t \\equiv e^{\\mathcal {L}t}, t\\ge 0,$ is not, in general, QL in the sense of Eq.",
"(REF ) at any time.",
"In some sense, one may think of the different noise components $\\mathcal {L}_1,\\ldots ,\\mathcal {L}_k$ of the continuous-time generator as acting “in parallel”.",
"On the other hand, were the maps $\\mathcal {E}_j$ we consider in this paper each generated by some corresponding neighborhood generator ${\\mathcal {L}_j},$ then by QL discrete-time dynamics we would be requesting that, on each time interval, a single noise operator is active, thus obtaining global switching dynamics [32] of the form $ e^{\\mathcal {L}_k T_k}\\circ e^{\\mathcal {L}_{k-1} T_{k-1}}\\circ \\ldots \\circ e^{\\mathcal {L}_1 T_1}.$ We could have requested each $\\mathcal {E}_t$ to be a convex combination of neighborhood maps acting on different neighborhoods, however it is not difficult to see that this case can be studied as the convergence in expectation for a randomized sequence.",
"Hence, we are focusing on the most restrictive definition of QL constraint for discrete-time Markov dynamics.",
"With respect to the continuous dynamics, however, we allow for the evolution to be time-inhomogeneous.",
"Remarkably, we shall find a characterization of QLS pure states that is equivalent to the continuous-time case, when the latter dynamics are required to be frustration free (see Section REF )."
],
[
"Invariance conditions and minimal fixed point sets",
"In this section, we build on the invariance requirement of Eq.",
"(REF ) to find necessary conditions that the discrete-time dynamics must satisfy in order to have a given state $\\rho $ as its unique and attracting equilibrium.",
"These impose a certain minimal fixed-point set, and hence suggest a structure for the stabilizing dynamics.",
"Following [48], given an operator $X\\in \\mathcal {B}(\\mathcal {H}_A\\otimes \\mathcal {H}_B)$ , with corresponding (operator) Schmidt decomposition $X=\\sum _j A_j\\otimes B_j,$ we define its Schmidt span as: $\\Sigma _A(X) \\equiv {\\rm span}(\\lbrace A_j\\rbrace ).$ The Schmidt span is important because, if we want to leave an operator invariant with a neighborhood map, this also imposes the invariance of its Schmidt span.",
"The following lemma, proved in [48], makes this precise: Lemma 3 Given a vector $v\\in V_{A}\\otimes V_{B}$ and $M_A\\in \\mathcal {B}(V_A)$ , if $(M_A\\otimes \\mathbb {I}_B) v=\\lambda v$ , then $(M_A\\otimes \\mathbb {I}_B) v^{\\prime }=\\lambda v^{\\prime }$ for all $v^{\\prime }\\in \\Sigma _{A}(v)\\otimes V_B$ .",
"What we need here can be obtained by adapting this result to our case, specifically: Corollary 4 Given a $\\rho \\in \\mathcal {D}(\\mathcal {H}_{\\mathcal {N}_j}\\otimes \\mathcal {H}_{\\overline{{\\mathcal {N}}}_j})$ and a neighborhood $\\mathcal {E}=\\mathcal {E}_{\\mathcal {N}_j}\\otimes I_{\\overline{\\mathcal {N}}_j}$ , then ${\\rm span}(\\rho )\\subseteq {\\rm fix}(\\mathcal {E})$ implies $\\Sigma _{\\mathcal {N}_j}(\\rho )\\otimes \\mathcal {B}(\\mathcal {H}_{\\overline{\\mathcal {N}}_j})\\subseteq {\\rm fix}(\\mathcal {E}_{{\\cal N}_j}).$ A Schmidt span need not be a valid fixed-point set, namely, a $\\rho $ -distorted algebra that is invariant for ${\\cal M}_{\\rho ,\\frac{1}{2}}.$ In general, we need to further enlarge the QL fixed-point sets from the Schmidt span to suitable algebras.",
"We discuss separately two relevant cases.",
"$\\bullet $ Pure states.— Let $\\rho =| \\psi \\rangle \\langle \\psi |$ be a pure state and assume that, with respect to the factorization $\\mathcal {H}_{\\mathcal {N}_j}\\otimes \\mathcal {H}_{\\overline{\\mathcal {N}}_j},$ its Schmidt decomposition $| \\psi \\rangle =\\sum _k c_k| \\psi _k \\rangle \\otimes | \\phi _k \\rangle .$ Let $\\mathcal {H}^0_{\\mathcal {N}_j} \\equiv {\\rm span}\\lbrace | \\psi _k \\rangle \\rbrace = \\text{supp} (\\rho _{\\mathcal {N}_j} )$ .",
"Then we have [48]: $\\Sigma _{\\mathcal {N}_j}(\\rho )={\\cal B}(\\mathcal {H}^0_{\\mathcal {N}_j}).$ In this case the Schmidt span is indeed a valid fixed-point set, and no further enlargement is needed.",
"The minimal fixed-point set for neighborhood maps required to preserve $\\rho $ is thus $\\mathcal {F}_j \\equiv {\\cal B}(\\mathcal {H}^0_{\\mathcal {N}_j}) \\otimes \\mathcal {B}(\\mathcal {H}_{\\overline{\\mathcal {N}}_j}) .$ By construction, each $\\mathcal {F}_j$ contains $\\rho .$ Notice that their intersection is just $\\rho $ if and only if ${\\rm span}\\lbrace | \\psi \\rangle \\rbrace =\\bigcap _j \\mathcal {H}^0_{\\mathcal {N}_j} \\otimes \\mathcal {H}_{\\overline{\\mathcal {N}}_j}=\\bigcap _j {\\mathcal {H}^0_j, }$ where we have defined $\\mathcal {H}^0_j\\equiv \\mathcal {H}^0_{\\mathcal {N}_j} \\otimes \\mathcal {H}_{\\overline{\\mathcal {N}}_j}.$ $\\bullet $ Full rank states.— If $\\rho $ is a full-rank state, and $W$ a set of operators, the minimal fixed-point set generated by $\\rho $ and $W$, by Theorem REF , is the smallest $\\rho $ -distorted algebra generated by $W$ which is invariant with respect to ${\\cal M}_{\\rho ,\\frac{1}{2}}$ .",
"Notice that, since $\\rho $ is full rank, its reduced states $\\rho _{\\mathcal {N}_j}$ are also full rank.",
"Denote by ${\\rm alg}_\\rho (W)$ the $\\dag $ -closed $\\rho $ -distorted algebra generated by $W.$ Call $W_j \\equiv \\Sigma _{\\mathcal {N}_j}(\\rho )$ .",
"The minimal fixed-point sets $\\mathcal {F}_{\\rho _{\\mathcal {N}_j}}(W_j )$ can then be constructed iteratively from $\\mathcal {F}_j^{(0)} \\equiv {\\rm alg}_{\\rho _{\\mathcal {N}_j}}(W_j)$ , with the $k$ -th step given by [48]: $\\mathcal {F}_j^{(k+1)} \\equiv {\\rm alg}_{\\rho _{\\mathcal {N}_j}}{({\\cal M}_{ \\rho _{\\mathcal {N}_j}, \\frac{1}{2} } (\\mathcal {F}_j^{(k)}),{\\mathcal {F}_j^{(k)}}).",
"}$ We keep iterating until $\\mathcal {F}_j^{(k+1)}=\\mathcal {F}_j^{(k)} = \\mathcal {F}_{\\rho _{{\\cal N}_j}} (W_j).$ When that happens, define $\\mathcal {F}_j \\equiv \\mathcal {F}_{\\rho _{{\\cal N}_j}} ( \\Sigma _{\\mathcal {N}_j}(\\rho ) ) \\otimes {\\cal B}(\\mathcal {H}_{\\overline{\\mathcal {N}}_j}).$ Since the ${\\cal F}_j$ are constructed to be the minimal sets for neighborhood maps that contain the given state and its corresponding Schmidt span, then clearly: ${\\rm span}(\\rho )\\subset \\bigcap _j \\mathcal {F}_j.$"
],
[
"Stabilizability under quasi-locality constraints",
"In the case of a pure target state, we can prove the following: Theorem 8 (QLS pure states) A pure state $\\rho =|\\psi \\rangle \\langle \\psi |$ is QLS by discrete-time dynamics if and only if ${ {\\rm supp}(\\rho )= \\bigcap _j {{\\mathcal {H}}^0_{j}.}",
"}$ Proof.",
"Given Corollary REF , any dynamics that make $\\rho $ QLS (and hence leaves it invariant) must consist of neighborhood maps $\\lbrace \\mathcal {E}_j\\rbrace $ with corresponding fixed points such that: $\\mathcal {F}_{k}\\subseteq {\\rm fix}(\\mathcal {E}_j),$ whenever $\\mathcal {E}_j$ is a $\\mathcal {N}_{k}$ -neighborhood map.",
"If the intersection of the fixed-point sets is not unique, then $\\rho $ cannot be GAS, since there would be another state that is not attracted to it.",
"Given Eq.",
"(REF ), we have $ {{\\rm span}(\\rho )=\\bigcap _k \\mathcal {F}_k \\quad \\Longleftrightarrow \\quad {\\rm supp}(\\rho )= \\bigcap _j {{\\mathcal {H}}^0_{j}.",
"}, }$ which proves necessity.",
"For sufficiency, we explicitly construct neighborhood maps whose cyclic application ensures stabilization.",
"Define$P_{\\mathcal {N}_j} $ to be the projector onto $\\textup {supp}(\\rho _{\\mathcal {N}_j} )$ , and the CPTP maps: $\\mathcal {E}_{\\mathcal {N}_j}(\\cdot ) \\equiv P_{\\mathcal {N}_j} (\\cdot ) P_{\\mathcal {N}_j} + \\frac{ P_{\\mathcal {N}_j} }{ {\\rm Tr}({P_{\\mathcal {N}_j}}) } \\,{\\rm Tr}\\,(P^\\perp _{\\mathcal {N}_j}\\cdot ),$ with $\\mathcal {E}_j \\equiv \\mathcal {E}_{\\mathcal {N}_j}\\otimes {\\mathcal {I}}_{\\overline{\\mathcal {N}}_j}.$ Consider the positive-semidefinite function $V(\\tau )=1-\\text{Tr}_{}\\left(\\rho \\, \\tau \\right),$ $\\tau \\in {\\mathcal {B}}(\\mathcal {H}).$ The result then follows from Theorem REF .",
"$\\Box $ An equivalent characterization can be given for full-rank target states: Theorem 9 (QLS full-rank states) A full-rank state $\\rho \\in {\\cal D}(\\mathcal {H})$ is QLS by discrete-time dynamics if and only if ${\\rm span}(\\rho )=\\bigcap _k \\mathcal {F}_k$ Proof.",
"As before, by contradiction, suppose that $\\rho _2\\in \\bigcap _k \\mathcal {F}_k$ exists, such that $\\rho _2\\ne \\rho $ .",
"This clearly implies that $\\rho $ cannot be GAS because there would exist another invariant state, which is not attracted to $\\rho $ .",
"This proves necessity.",
"Sufficiency derives from the alternating CPTP projection theorem.",
"Specifically, let $\\mathcal {E}_{\\mathcal {N}_k}$ be the CPTP projection onto ${\\cal F}_k,$ and $\\mathcal {E}_k\\equiv \\mathcal {E}_{\\mathcal {N}_k}\\otimes {\\rm Id}_{\\overline{\\mathcal {N}}_k}.$ By Theorem REF , we already know that for every $\\rho $ , $(\\mathcal {E}_M\\ldots \\mathcal {E}_1)^k(\\rho )\\rightarrow \\bigcap _k \\mathcal {F}_k$ for $k\\rightarrow \\infty $ .",
"Now, by hypothesis, $\\bigcap _k \\mathcal {F}_k={\\rm span}(\\rho )$ and, being $\\rho $ the only (trace one) state in his own span, $\\rho $ is GAS.",
"$\\Box $ A set of sufficient conditions, stemming from Theorem REF , can be also derived in an analogous way for a general target state."
],
[
"Physical interpretation: discrete-time quasi-local stabilizability is equivalent to\ncooling without frustration",
"Consider a quasi-local Hamiltonian, that is, $H=\\sum _k H_k,$ $H_k=H_{{\\cal N}_k}\\otimes I_{\\overline{\\cal N}_k}.$ $H$ is called a parent Hamiltonian for a pure state $| \\psi \\rangle $ if it admits $| \\psi \\rangle $ as a ground state, and it is called a is called a frustration-free (FF) Hamiltonian if any global ground state is also a local ground state [57], that is, $ \\text{argmin}_{| \\psi \\rangle \\in \\mathcal {H}} \\langle \\psi | H | \\psi \\rangle \\subseteq \\text{argmin}_{| \\psi \\rangle \\in \\mathcal {H}} \\langle \\psi | H_k | \\psi \\rangle , \\forall k. $ Suppose that a target state $| \\psi \\rangle $ admits a FF QL parent Hamiltonian $H$ for which it is the unique ground state.",
"Then, similarly to what has been done for continuous-time dissipative preparation [31], [27], the structure of $H$ may be naturally used to derive a stabilizing discrete-time dynamics: it suffices to implement neighborhood maps $M_k$ that stabilize the eigenspace associated to the minimum eigenvalue of each $H_k$ .",
"These can thought as maps that locally “cool” the system.",
"In view of Theorem REF , it is easy to show that this condition is also necessary: Corollary 5 A state $\\rho =|\\psi \\rangle \\langle \\psi |$ is QLS by discrete-time dynamics if and only if it is the unique ground state of a FF QL parent Hamiltonian.",
"Proof.",
"Without loss of generality we can consider FF QL Hamiltonians $H=\\sum _k H_k,$ where each $H_k$ is a projection.",
"Let $\\rho $ satisfy Eq.",
"(REF ), which is equivalent to be QLS, and define $H_k \\equiv \\Pi ^\\perp _{{\\cal N}_k}\\otimes I_{\\overline{\\cal N}_k}$ , with $\\Pi ^\\perp _{{\\cal N}_k}$ being the orthogonal projector onto the orthogonal complement of the support of $\\rho _{{\\cal N}_k}$ , that is, $\\mathcal {H}_{{\\cal N}_k}\\ominus {\\rm supp}({\\rho _{{\\cal N}_k}})$ .",
"Given Theorem REF , $|\\psi \\rangle $ is the unique pure state in $\\bigcap _k{\\rm supp}(\\rho _{{\\cal N}_k}\\otimes I_{\\overline{\\cal N}_k}),$ and thus the unique state in the kernel of all the $H_k$ .",
"Conversely, if a FF QL parent Hamiltonian exists, the kernels of the $H_k$ satisfy the QLS condition and to each $H_k$ we can associate a CPTP map as in (REF ) that projects onto its kernel.",
"$\\Box $ An equivalent characterization works for generic, full-rank target states, but we need to move from Hamiltonians to semigroup generators, while maintaining frustration-freeness in a suitable sense.",
"If, as before, ${\\cal E}_{t}=e^{{\\cal L}}$ is the propagator arising from a time-invariant QL generator ${\\cal L}$ , we are interested in QL generators whose neighborhood components drive the system to a global equilibrium which is also a local equilibrium for each of them separately.",
"That is, following [58], [48]: Definition 4 A QL generator $\\mathcal {L}=\\sum _{j}\\mathcal {L}_{j},$ where ${\\cal L}_j$ are neighborhood generators, is Frustration Free (FF) relative to a neighborhood structure $\\mathcal {N}=\\lbrace \\mathcal {N}_j\\rbrace $ if $\\rho \\in {\\rm ker}(\\mathcal {L})\\quad \\iff \\quad \\rho \\in {\\rm ker}(\\mathcal {L}_j), \\quad \\forall j.$ It is worth noting that a state which is invariant for all the local generators is always an equilibrium: the real requirement is that these states are all the equilibria.",
"We then have: Proposition 4 A pure or full-rank state $\\rho $ is discrete-time QLS if and only if it is QLS via FF continuous-time dynamics, that is, there exists a FF generator ${\\cal L}$ with respect to the same neighborhood structure $\\mathcal {N}$ such that $\\lim _{t\\rightarrow +\\infty }e^{{\\cal L}t}\\rho _0=\\rho ,\\quad \\forall \\rho _0.$ Proof.",
"The claim follows from Theorems 7 and 8 in [48], which characterize the states that are unique fixed points of a FF generator as precisely the states that satisfy Eq.",
"(REF ).",
"$\\Box $ Remark: Based on the above results, the conditions that guarantee either a pure or a full-rank state to be QLS in discrete time are the same that guarantee existence of a FF stabilizing generator in continuous time.",
"We stress that if more general continuous-time generators are allowed, namely, frustration is permitted as in Hamiltonian-assisted stabilization [28], then the continuous-time setting can be more powerful.",
"On the one hand, considering the stricter nature of the QL constraint for the discrete-time setting, this is not surprising.",
"On the other hand, if Liouvillian is no longer FF, then the target is globally invariant for ${\\cal L}$ but no longer invariant for individual QL components ${\\cal L}_j$ , suggesting that a weaker (“stroboscopic”) invariance requirement could be more appropriate to “mimic” the effect of frustration in the discrete-time QL setting."
],
[
"Classes of discrete-time QLS states",
"Being the conditions for discrete-time QLS states the same as in continuous time with FF dynamics, we may conclude whether certain classes of states are QLS or not by exploiting the results already established in [27], [28], [48].",
"While we refer to the original references for additional detail and context, some notable examples are summarized in what follows.",
"Among pure states: Graph states [59] (more generally, stabilizer states [5] that admit stabilizer group generators that are neighborhood operators) are discrete-time QLS.",
"These states are entangled, and are a key resource for one-way quantum computation.",
"Certain, but not all, Dicke states are discrete-time QLS.",
"Dicke states are symmetric with respect to subsystem permutations, and have a specified “excitation number” [60].",
"Dicke states exhibit entanglement properties that are, in some sense, robust: some entanglement is preserved even if some subsystems are measured or traced out.",
"The $n$ -qubit single-excitation Dicke state, also known as W-state, $| \\psi ^n_{\\text{W}} \\rangle \\equiv \\frac{1}{\\sqrt{n}} (| 100\\ldots 0 \\rangle +| 010\\ldots 0 \\rangle +\\ldots +| 000\\ldots 1 \\rangle ), $ fails to satisfy the conditions of Theorem REF , so $\\rho _{\\text{W}}$ is not discrete-time QLS for non-trivial neighborhood structures (that is, unless there is a neighborhood that covers the whole network).",
"On the other hand, for example, the two-excitation Dicke state on $n=4$ qubits, $&&\\hspace{-22.76219pt}| \\psi _D^4 \\rangle \\equiv \\\\&&\\hspace{-22.76219pt}\\frac{| 1100 \\rangle +| 1010 \\rangle +| 1001 \\rangle +| 0110 \\rangle +| 0101 \\rangle +| 0011 \\rangle }{\\sqrt{6}},$ is QLS.",
"A more general class of QLS Dicke states on qudits is presented in [48].",
"Among full-rank states: Commuting Gibbs states are discrete-time QLS with respect to a suitable locality notion [48].",
"A Gibbs state represent the canonical thermal equilibrium state for a statistical system at temperature $\\beta ^{-1}$ : if a chain of qudits is associated to a nearest-neighbor (NN) Hamiltonian $H=\\sum _k H_k$ , its Gibbs state is $\\rho _\\beta \\equiv \\frac{e^{-\\beta H}}{{\\rm Tr}(e^{-\\beta H})}.$ If the Hamiltonian is commuting, namely, $[H_j,H_k]=0$ for all $j,k,$ then $\\rho $ is QLS with respect to an enlarged QL notion, where ${\\cal N}^0_j$ contains all subsystems that belong to a NN neighborhood ${\\cal N}_k$ such that ${\\cal N}_k\\cap {\\cal N}_j\\ne \\emptyset .$ In analogy to the continuous-time case [58], this shows that Gibbs samplers based on QL discrete-time dissipative dynamics are also viable, at least in the commuting case.",
"Certain mixtures of factorized and entangled states are discrete-time QLS.",
"For example, consider a 4-qubits system and the family of states parametrized by $\\epsilon \\in (0,1)$ : $ \\rho _\\epsilon \\equiv (1-\\epsilon ) \\, | \\psi _D^4 \\rangle \\langle \\psi _D^4 |+ \\epsilon \\, | \\textup {GHZ}^4 \\rangle \\langle \\textup {GHZ}^4 |, $ where $| \\textup {GHZ}^n \\rangle \\equiv \\left(| 0 \\rangle ^{\\otimes n}+| 1 \\rangle ^{\\otimes n}\\right)/\\sqrt{2}$ denotes the maximally-entangled Greenberger-Horne-Zeilinger (GHZ) states on $n$ qubits, and $\\mathcal {N}_1 = \\lbrace 1,2,3\\rbrace ,$ $\\mathcal {N}_2=\\lbrace 2,3,4\\rbrace $ .",
"This shows that we can stabilize states that are arbitrarily close to states that are provably not QLS, as the GHZ states [28], thereby achieving practical stabilization of the latter.",
"We have introduced alternating projection methods based on sequences of CPTP projections, and used them in designing discrete-time stabilizing dynamics for entangled states in multipartite quantum systems subject to realistic quasi-locality constraints.",
"When feasible, pursuing stabilization instead of preparation offers important advantages, including the possibility to retrieve the target state on-demand, at any (discrete) time after sufficient convergence is attained, since the invariance of the latter ensures that it is not ruined by subsequent maps.",
"We show that the proposed methods are also suitable for distributed, randomized and unsupervised implementations on large networks.",
"While the locality constraints we impose on the discrete-time dynamics are stricter, the stabilizable states are, remarkably, the same that are stabilizable for continuous-time frustration-free generators.",
"From a methodological standpoint, our results shed further light on the structure and intersection of fixed-point sets of CPTP maps, a structure of interest not only in control, but also in operator-algebraic approaches to quantum systems [61], quantum statistics [49] and quantum error correction theory [62], [63], [64].",
"In particular, we show that the intersection of fixed-point sets is still a fixed-point set, as long as it contains a full-rank state.",
"In developing our results, we use both standard results from classical alternating projections and Lyapunov methods tailored to the positive linear maps at hand.",
"Towards applications, the proposed alternating projection methods are in principle suitable for implementation in digital open-quantum system simulators, such as demonstrated in proof-of-principle trapped-ion experiments [33].",
"Beside providing protocols for stabilizing relevant classes of pure entangles states, our methods point to an alternative approach for constructing quantum samplers using quasi-local resources.",
"Some developments of this line of research are worth highlighting.",
"First, in order to extend the applicability of the proposed methods to more general classes of states, as well as to establish a tighter link to quantum error correction and dissipative code preparation, it is natural to look at discrete-time conditional stabilization, in the spirit of [28].",
"Notably, in [56], it has been shown that GHZ states and all Dicke states can be made conditionally asymptotically stable for QL discrete-time dynamics, with a suitable basin of attraction.",
"Second, while we recalled some basic classical bounds on the convergence speed, that apply to the stabilization of full-rank states, their geometric nature makes it hard to obtain useful insight from them.",
"A more intuitive approach to convergence speed and its optimization, following e.g.",
"[26], [37], may offer a more promising venue in that respect.",
"It has also been recently shown that linear Lyapunov functions can not only be used to prove convergence, but also provide sharp bounds on the convergence speed in continuous-time dynamics [65].",
"It would be interesting to extend the analysis to the non-homogeneous, discrete-time cases considered in this work.",
"Lastly, the characterization of physically relevant scenarios in which finite-time stabilization is possible under locality constraints is a challenging open problem, which we plan to address elsewhere [66]."
],
[
"Acknowledgements",
"It is a pleasure to acknowledge stimulating discussions on the topics of this work with A. Ferrante e L. Finesso.",
"F.T.",
"is especially grateful to V. Umanità and E. Sasso for pointing him towards Takesaki's theorem.",
"Work at Dartmouth was supported by the National Science Foundation through grant No.",
"PHY-1620541."
],
[
"Angles between subspaces",
"Define the function $\\arccos :[-1,1]\\rightarrow [-\\frac{\\pi }{2},\\frac{\\pi }{2} ]$ .",
"We will use only the elements in interval $[0,1]$ .",
"Then the angle $\\theta (\\mathcal {M},\\mathcal {N})$ between two closed subspaces $\\mathcal {M}$ and $\\mathcal {N}$ of $\\mathcal {H}$ is an element of $[0,\\frac{\\pi }{2}]$ .",
"We have the following: Definition 5 The cosine $c(\\mathcal {M},\\mathcal {N})$ between two closed subspaces $\\mathcal {M}$ and $\\mathcal {N}$ of $\\mathcal {H}$ is given by $c(\\mathcal {M},\\mathcal {N})&\\equiv &\\sup \\Big \\lbrace |\\langle x,y\\rangle |:x\\in \\mathcal {M}\\cap (\\mathcal {M}\\cap \\mathcal {N})^\\perp ,\\\\ &&\\Vert x\\Vert \\le 1, y\\in \\mathcal {N}\\cap (\\mathcal {M}\\cap \\mathcal {N})^\\perp , \\Vert y\\Vert \\le 1 \\Big \\rbrace .$ Then the angle is given by: $\\theta (\\mathcal {M},\\mathcal {N})=\\arccos (c(\\mathcal {M},\\mathcal {N})).$ Some consequences of the above definitions are the following: $0\\le c(\\mathcal {M},\\mathcal {N})\\le 1$ ; $c(\\mathcal {M},\\mathcal {N})=c(\\mathcal {N},\\mathcal {M})$ ; $c(\\mathcal {M},\\mathcal {N})=\\Vert P_\\mathcal {M}P_\\mathcal {N}-P_{\\mathcal {M}\\cap \\mathcal {N}}\\Vert =\\Vert P_\\mathcal {M}P_\\mathcal {N}P_{(\\mathcal {M}\\cap \\mathcal {N})^\\perp }\\Vert $ .",
"We next state the result that gives the exact rate in case of projection onto two subspaces [43]: Theorem 10 In the norm induced by the inner product, and for each $n$ , the following equality holds: $\\Vert (P_{\\mathcal {M}_2}P_{\\mathcal {M}_1})^n-P_{\\mathcal {M}_1\\cap \\mathcal {M}_2}\\Vert = c(\\mathcal {M}_1,\\mathcal {M}_2)^{2n-1}.$ In case of alternating projections on the intersection of more than two subspaces, an exact expression is no longer available, however an upper bound may be given [43]: Theorem 11 For each $i=1,2,\\ldots ,r$ , let $\\mathcal {M}_i$ be a closed subspace of $\\mathcal {H}$ .",
"Then, for each $x\\in \\mathcal {H}$ , and for any integer $n\\ge 1$ it holds: $\\Vert (P_{\\mathcal {M}_r}...P_{\\mathcal {M}_1})^n x-P_{\\bigcap _{i=1}^r\\mathcal {M}_i}x\\Vert \\le c^{\\frac{n}{2}}\\Vert x-P_{\\bigcap _{i=1}^r\\mathcal {M}_i}x\\Vert ,$ where the contraction coefficient $c=1-\\prod _{i=1}^{r-1}\\sin ^2\\theta _i,$ and $\\theta _i$ is the angle between $\\mathcal {M}_i$ and $\\bigcap _{j=i+1}^r\\mathcal {M}_j$ ."
],
[
"Non-orthogonality of ${\\cal E}_{\\cal A}$ with respect to the Hilbert-Schmidt inner product",
"Let us decompose a full-rank fixed point set $\\mathcal {A}_\\rho =\\bigoplus _\\ell \\mathcal {A}_\\ell =\\bigoplus _\\ell \\mathcal {B}(\\mathcal {H}_{S,\\ell })\\otimes \\tau _\\ell $ , (where $\\tau _\\ell \\equiv \\tau _{F,\\ell } $ ).",
"By definition, the orthogonal projection of $X$ onto $\\mathcal {A}_i$ is given by $P_\\mathcal {A}(X) \\equiv \\sum _{\\ell ,i}\\langle \\sigma _{\\ell ,i }\\otimes { \\tau _\\ell } ,X \\rangle _{HS} \\,\\sigma _{\\ell ,i}\\otimes \\tau _\\ell , $ where $\\sigma _{\\ell ,i}\\otimes \\tau _\\ell $ is an orthonormal basis for $\\mathcal {A}_\\ell $ .",
"Note that the outcome only depends on the restrictions of $X$ to the supports of the $\\mathcal {A}_\\ell .$ Hence, decompose $X \\equiv \\sum _\\ell X_\\ell +\\Delta X$ , where $X_\\ell =\\Pi _{SF,\\ell }X\\Pi _{SF,\\ell },$ and further decompose $X_\\ell \\equiv \\sum _k A_{\\ell ,k}\\otimes B_{\\ell ,k},$ so we can write: $P_\\mathcal {A}(X)&\\hspace*{-5.69054pt}=\\hspace*{-5.69054pt}&\\bigoplus _i\\sum _{j,\\ell }\\Big (\\sum _k {\\rm Tr}[(\\sigma _j\\otimes \\tau _\\ell )(A_{\\ell ,k}\\otimes B_{\\ell ,k})]\\sigma _j\\otimes \\tau _\\ell \\Big )\\\\&\\hspace*{-5.69054pt}=\\hspace*{-5.69054pt}&\\bigoplus _\\ell \\sum _{j,\\ell }\\Big ({\\rm Tr}[\\sigma _j \\sum _k(A_{\\ell ,k} {\\rm Tr}(\\tau _\\ell B_{\\ell ,k}))]\\sigma _j\\otimes \\tau _\\ell )\\Big ).$ By comparing the latter equation with Eq.",
"(REF ), we have that $P_\\mathcal {A}=\\mathcal {E}_\\mathcal {A}$ if and only if $\\sum _k(A_k {\\rm Tr}(\\tau _jB_k))={\\rm Tr}_{F,\\ell }(X_\\ell ),$ which is equivalent to request that $\\tau _j=\\lambda _\\ell I.$ Hence, unless $\\cal {A}_\\rho $ contains the completely mixed state, ${\\mathcal {E}}_\\mathcal {A}$ in Eq.",
"(REF ) is not an orthogonal projection with respect to the Hilbert-Schmidt inner product.",
"$\\Box $"
]
] | 1612.05554 |
[
[
"Algebraic isomonodromic deformations and the mapping class group"
],
[
"Abstract The germ of the universal isomonodromic deformation of a logarithmic connection on a stable n-pointed genus g curve always exists in the analytic category.",
"The first part of this paper investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation.",
"Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group.",
"The second part of this paper studies the dynamics of this action in the particular case of reducible rank 2 representations and genus g > 0, allowing to classify all finite orbits.",
"Both of these results extend recent ones concerning the genus 0 case."
],
[
"Introduction",
"The mapping class group.",
"Let $g$ and $n$ be nonnegative integers.",
"Let $\\Sigma _g$ be a compact oriented real surface of genus $g$ , let $y^n=(y_1, \\ldots , y_n)$ be a sequence of $n$ distinct points in $\\Sigma _g$ .",
"We shall denote by $Y^n:=\\lbrace y_1, \\ldots , y_n\\rbrace $ the corresponding (unordered) set of points.",
"The (pure) mapping class group of $(\\Sigma _g,y^n)$ is defined to be the set of orientation preserving homeomorphisms $h$ of $\\Sigma _g$ such that $h(y_i)=y_i$ for all $i\\in \\llbracket 1, n\\rrbracket :=\\left\\lbrace k \\in \\mathbb {Z}~|~1\\le k\\le n\\right\\rbrace $ , quotiented by isotopies: $\\Gamma _{g,n}:={\\raisebox {.0em}{\\mathrm {Homeo}_+(\\Sigma _g, y^n)}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ \\lbrace \\mbox{isotopies relative to $Y^n$}\\rbrace \\, .",
"}}$ We can also consider homeomorphisms of $\\Sigma _g$ that preserve the set $Y^n$ , but do not necessarily preserve the labelling of the punctures.",
"This leads to the full mapping class group $\\hat{\\Gamma }_{g,n}:={\\raisebox {.0em}{\\mathrm {Homeo}_+(\\Sigma _g,Y^n)}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ \\lbrace \\mbox{isotopies relative to $Y^n$}\\rbrace \\, .",
"}}$ Note that we have an exact sequence of groups $1\\longrightarrow \\Gamma _{g,n}\\longrightarrow \\hat{\\Gamma }_{g,n}\\longrightarrow \\mathfrak {S}_n \\longrightarrow 1,$ where $\\mathfrak {S}_n$ denotes the symmetric group of degree $n$ .",
"In particular, $\\Gamma _{g,n}$ is a subgroup of $\\hat{\\Gamma }_{g,n}$ of finite index $n!$ .",
"Let now $y_0 \\in \\Sigma _g\\setminus Y^n$ be a point.",
"We denote the fundamental group of $\\Sigma _g\\setminus Y^n$ with respect to the base point $y_0$ by $\\Lambda _{g,n}:=\\pi _1(\\Sigma _g \\setminus Y^n,y_{0})\\, .$ The composition $\\alpha \\, .\\, \\alpha ^{\\prime }$ of two paths $\\alpha , \\alpha ^{\\prime }\\in \\Lambda _{g,n}$ shall denote the usual concatenation “first $\\alpha $ , then $\\alpha ^{\\prime }$ ”.",
"For any group $G$ , we may consider the space $\\operatorname{Hom}(\\Lambda _{g,n}, G)$ of representations as well as the space of representations modulo conjugation, which we shall denote $\\chi _{g,n}(G):= {\\raisebox {.0em}{\\operatorname{Hom}(\\Lambda _{g,n}, G)}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ G }}\\, .$ The mapping class group acts on $\\chi _{g,n}(G)$ .",
"Define the groups of orientation preserving homeomorphisms $h$ of $\\Sigma _g$ such that $h(y_0)=y_0$ and $h(y^n)=y^n$ , respectively $h(Y^n)=Y^n$ , modulo isotopy: $\\begin{array}{rll}\\Gamma _{g,n+1}&:=&{\\raisebox {.0em}{\\mathrm {Homeo}_+(\\Sigma _g,y^n, y_0)}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ \\lbrace \\mbox{isotopies relative to $Y^n \\cup \\lbrace y_0\\rbrace $}\\rbrace \\, , }}\\vspace{8.5359pt}\\\\\\hat{\\Gamma }_{g,n}^{\\bullet }&:=&{\\raisebox {.0em}{\\mathrm {Homeo}_+(\\Sigma _g, Y^n, y_0)}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ \\lbrace \\mbox{isotopies relative to $Y^n \\cup \\lbrace y_0\\rbrace $}\\rbrace \\, . }}",
"\\end{array}$ Now $\\hat{\\Gamma }_{g,n}^{\\bullet }$ naturally acts on the fundamental group $\\Lambda _{g,n}$ : for $h \\in \\hat{\\Gamma }_{g,n}^{\\bullet }$ and $\\alpha \\in \\Lambda _{g,n}$ , we set $\\mathfrak {a}(h)(\\alpha ) :=h_*\\alpha \\, .$ Via the forgetful maps $\\Gamma _{g,n+1}\\rightarrow \\Gamma _{g,n}$ and $\\hat{\\Gamma }_{g,n}^{\\bullet }\\rightarrow \\hat{\\Gamma }_{g,n}$ we obtain a commutative diagram $ \\hspace{85.35826pt} \\begin{xy}{\\Gamma _{g,n+1} @{->>}[d]@{^{(}->}[r]&\\hat{\\Gamma }_{g,n}^{\\bullet } [r]^{\\mathfrak {a} \\textrm {~~~~~}}@{->>}[d]& \\mathrm {Aut}(\\Lambda _{g,n})@{->>}[d]&\\\\ \\Gamma _{g,n}@{^{(}->}[r]&\\hat{\\Gamma }_{g,n}[r]& \\mathrm {Out}(\\Lambda _{g,n}):@{=}[r]&{\\raisebox {.0em}{\\mathrm {Aut}(\\Lambda _{g,n})}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ \\mathrm {Inn}(\\Lambda _{g,n})\\, .",
"}}}\\end{xy}$ Indeed, any element $h\\in \\mathrm {Homeo}_+(\\Sigma _g, y^n)$ may be lifted to an element $h_0\\in \\mathrm {Homeo}_+(\\Sigma _g,y^n, y_0)$ .",
"Let $h_1\\in \\mathrm {Homeo}_+(\\Sigma _g,y^n, y_0)$ be another representative.",
"Then they are the extremities of an isotopy $(h_t)_{t\\in [0,1]}$ relative to $Y^n$ .",
"We have a loop $\\gamma \\in \\Lambda _{g,n}$ defined by $\\gamma (t)=h_t(y_0)$ .",
"Then for any $\\alpha \\in \\Lambda _{g,n}$ , we have $\\mathfrak {a}(h_1)(\\alpha )=\\gamma ^{-1}\\, .\\, \\mathfrak {a}(h_0)(\\alpha )\\, .\\, \\gamma $ .",
"In particular, for any group $G$ , the mapping class group $\\hat{\\Gamma }_{g,n}$ acts on the space $\\chi _{g,n}(G)$ , and this action lifts to an action of $\\hat{\\Gamma }_{g,n}^{\\bullet }$ on the space $\\operatorname{Hom}(\\Lambda _{g,n}, G)$ .",
"More precisely, for all $\\rho \\in \\operatorname{Hom}(\\Lambda _{g,n}, G), h\\in \\hat{\\Gamma }_{g,n}^{\\bullet }$ and $\\alpha \\in \\Lambda _{g,n}$ , we define $ ([h]\\cdot \\rho )(\\alpha ): = \\rho (\\mathfrak {a}(h^{-1})(\\alpha ))\\, .$ Application to isomonodromic deformations and a dynamical study.",
"In this paper, we establish two results about finite orbits of the mapping class group action on $\\chi _{g,n}(G)$ for $G=\\mathrm {GL}_r.",
"These results and their respective proofs can be read independently.",
"In Theorem \\ref {algebrization thm}, which will be stated in Section \\ref {SecIntroAlg} and proven in Part \\ref {partAlg}, we relate such finite orbits to the existence of an algebraic universal isomonodromic deformation of a logarithmic connection over a curve, whose monodromy belongs to that orbit.",
"This motivates Theorem \\ref {mainthm dynamics}, which will be stated in Section \\ref {SecIntroDyn}and proven in Part \\ref {partDyn}, classifying conjugacy classes of reducible rank 2 representations with finite orbit.",
"To that end, we introduce a specific presentation of $ g,n$ and explicit \\textit {formulae} for the mapping class group action.$ Remark 1.0.1 Recall that a representation $\\rho \\in \\operatorname{Hom}(\\Lambda _{g,n}, \\mathrm {GL}_r$ is called irreducible if the only subvector spaces $V\\subset r$ stable under $\\mathrm {Im}(\\rho )$ are $\\lbrace 0\\rbrace $ and $r$ .",
"A semisimple representation is a direct sum of irreducible representations."
],
[
"Algebraization of universal isomonodromic deformations",
"We need to introduce some additional vocabulary before stating our main result, which can be seen as a criterion under which a GAGA-type theorem holds for isomonodromic deformations.",
"In order to avoid having to introduce each definition twice, we adopt the $$ -notation described in Table REF .",
"Table: NO_CAPTIONLogarithmic connections.",
"Let $X$ be a $$ -manifold and let $D$ be a (possibly empty) reduced divisor on $X$ .",
"Denote by $D_1, \\ldots , D_n$ the irreducible components of $D$ .",
"A logarithmic $$ -connection of rank $r$ over $X$ with polar divisor $D$ is a pair $(E,\\nabla )$ , where $E\\rightarrow X$ is a $$ -vector bundle of rank $r$ over $X$ , whose sheaf of sections we shall denote by $\\mathcal {E}$ , and $\\nabla $ is a $-linear morphism$$\\nabla : \\mathcal {E} \\rightarrow \\mathcal {E}\\otimes \\Omega ^1_X(\\mathrm {log}\\, D)\\, , $$which satisfies the Leibniz rule$$\\nabla (f\\cdot e) =f\\cdot \\nabla (e)+e\\otimes \\mathrm {d}f\\, $$for any $ fOX() , eE()$, where $ X$ is any $$-open subset.We require $ D$ to be minimal in the sense that for any $ i 1, n$, $$ does not factor through $$\\mathcal {E}\\otimes \\Omega ^1(\\mathrm {log}(D-D_i))\\hookrightarrow \\mathcal {E}\\otimes \\Omega ^1_X(\\mathrm {log}\\, D).$$Such a logarithmic connection $ (E, )$ is called \\emph {flat} if its curvature $ 2$ is zero.$ We are particularly interested in the case where $X$ is a smooth projective curve (a compact Riemann surface).",
"Since then $X$ is of complex dimension one, any logarithmic connection over $X$ is automatically flat.",
"Moreover, since then $X$ is projective, any analytic logarithmic connection over $X$ is isomorphic to the analytification of a unique algebraic logarithmic connection over $X$ by one of Serre's GAGA theorems .",
"Monodromy.",
"The notion of the monodromy representation of a flat connection varies slightly in the literature.",
"For introductory and technical purposes, let us give the definition we are going to use.",
"This definition can only be formulated in the analytic category; in the algebraic case the monodromy representation is defined via analytification.",
"Let $X$ and $D$ be as above ($X$ has arbitrary dimension).",
"Denote $X^0:=X\\setminus D$ .",
"Let $(E,\\nabla )$ be an analytic logarithmic connection over $X$ with polar divisor $D$ .",
"Assume moreover that this analytic connection is flat, which is equivalent to it being integrable, i.e.",
"$\\mathcal {S}:=\\ker (\\nabla |_{X^0})$ is a locally constant sheaf of rank $r$ over $X^0$ .",
"Let $\\Sigma $ and $Y\\subset \\Sigma $ be topological spaces such that there is a homeomorphism $\\Phi : (\\Sigma ,Y)\\stackrel{\\sim }{\\rightarrow } (X,D)\\, .$ Fix such a homeomorphism and fix a point $\\tilde{y}_0\\in \\Sigma \\setminus Y$ .",
"Denote $\\tilde{x}_0:=\\Phi (\\tilde{y}_0)$ .",
"For any path $\\gamma : [0,1] \\rightarrow \\Sigma \\setminus Y$ , the pull back $(\\Phi \\circ \\gamma )^*\\mathcal {S}$ is locally constant and thus isomorphic to a constant sheaf.",
"Hence $\\gamma $ defines an isomorphism $\\gamma (\\mathcal {S}) : \\mathcal {S}_{\\gamma (1)}\\rightarrow \\mathcal {S}_{\\gamma (0)}$ .",
"This isomorphism is invariant by homotopy relative to $\\lbrace \\gamma (0)\\, , \\gamma (1)\\rbrace $ and satisfies $\\gamma _1\\, .\\, \\gamma _2(\\mathcal {S})=\\gamma _1(\\mathcal {S})\\circ \\gamma _2(\\mathcal {S})$ for any pair of paths $(\\gamma _1,\\gamma _2)$ .",
"We obtain a representation $\\pi _1(\\Sigma \\setminus Y, \\tilde{y}_0) \\rightarrow \\mathrm {GL}(\\mathcal {S}_{\\tilde{x}_0})$ .",
"Via an isomorphism $\\mathcal {S}_{\\tilde{x}_0}\\rightarrow r$ , one deduces a (non canonical) representation $\\rho _{\\nabla } \\in \\operatorname{Hom}(\\pi _1(\\Sigma , \\tilde{y}_0), \\mathrm {GL}_r $ and a canonical conjugacy class of representation $[\\rho _{\\nabla }]\\in {\\raisebox {.0em}{\\operatorname{Hom}(\\pi _1(\\Sigma \\setminus Y, \\tilde{y}_0), \\mathrm {GL}_r}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ \\mathrm {GL}_r}}\\, .",
"$ We refer to $\\rho _{\\nabla }$ as the monodromy representation and to $[\\rho _{\\nabla }]$ as the monodromy of $(E, \\nabla )$ with respect to $\\Phi $ .",
"Conversely, given $\\Phi $ , given a conjugacy class of representation $[\\rho ]\\in \\operatorname{Hom}(\\pi _1(\\Sigma \\setminus Y, \\tilde{y}_0), \\mathrm {GL}_r/{\\mathrm {GL}_r and a compatible choice of \\emph {mild transversal models} (see Section \\ref {Sec mild trans}), there is a flat logarithmic analytic connection (E, \\nabla ) over X, unique up to isomorphism, inducing these transversal models and such that [\\rho _\\nabla ]=[\\rho ] (see \\cite [Th.",
"3.3]{cousinisom}, adapted from \\cite [Prop.",
"5.4]{MR0417174}).",
"In our work, the use of the marking \\Phi is essential, as we wish to compare the monodromies of connections over various homeomorphic curves.\\vspace{8.5359pt}}\\textbf {Isomonodromy.",
"}Let $ C$ be a smooth projective curve of genus $ g$, let $ DC$ be a reduced divisor of degree $ n$ on $ C$.",
"Let $ (E0, 0)$ be a logarithmic connection over $ C$ with polar divisor~$ DC$.$ A $$ -isomonodromic deformation of $(C, E_0,\\nabla _0)$ consists in the following data: a $$ -family $(\\kappa : X\\rightarrow T, D)$ of $n$ -pointed smooth curves of genus $g$ (see Section REF ); a flat logarithmic $$ -connection $(E,\\nabla )$ over $X$ with polar divisor $D$ ; a point $t_0$ in $T$ ; we denote $X_{t_0}:=\\kappa ^{-1}(\\lbrace t_0\\rbrace )$ ; and an isomorphism of pointed curves with logarithmic connections $(\\psi ,\\Psi ): ((C, D_C),(E_0,\\nabla _0))\\stackrel{\\sim }{\\rightarrow }((X_{t_0}, D|_{X_{t_0}}),(E,\\nabla )|_{X_{t_0}})\\, .$ Why are such deformations called isomonodromic?",
"Again we have to work in the analytic category.",
"Up to shrinking $T$ to a sufficiently small polydisc $\\Delta $ containing $t_0$ , the family $\\kappa : (X,D)\\rightarrow \\Delta $ is topologically trivial.",
"Hence there is a homeomorphism $\\Phi : (\\Sigma _g,Y^n)\\times \\Delta \\stackrel{\\sim }{\\rightarrow } (X,D)$ commuting with the natural projections to $\\Delta $ .",
"Now for any $t\\in \\Delta $ , the morphism $\\pi _1(\\Sigma _g\\setminus Y^n, y_0) \\longrightarrow \\pi _1( (\\Sigma _g\\setminus Y^n)\\times \\Delta , (y_0,t) )\\, , $ induced by the inclusion of the fiber at $t$ , is an isomorphism.",
"On the other hand, $(E,\\nabla )|_{X_{t}}$ is a logarithmic connection over $X_t$ with polar divisor $D|_{X_t}$ .",
"By flatness of $\\nabla $ , its monodromy representation with respect to $\\Phi |_{t}$ and the base point $y_0$ can be identified with the monodromy representation of $(E,\\nabla )$ with respect to $\\Phi $ and the base point $(y_0,t)$ .",
"For $t=t_0$ , this means we can identify the monodromy representation of $(E,\\nabla )$ over $X$ with respect to $\\Phi $ with the monodromy representation of $(E_0,\\nabla _0)$ over $C$ with respect to $\\psi ^{-1}\\circ \\Phi |_{t_0}$ .",
"In that sense, we may say that with respect to some continuous “base point section” $t\\mapsto (y_0,t)$ , the monodromy representation along a germ of isomonodromic deformation is constant and given by the monodromy representation of $(E_0,\\nabla _0)$ .",
"More generally, one can say that an isomonodromic deformation induces a topologically locally trivial family of monodromy representations, leading to a phenomenon of monodromy of the monodromy representation.",
"The latter will become tangible in Section .",
"Statement of Theorem REF .",
"Following (see also , ), any triple $(C, E_0,\\nabla _0)$ as before admits a universal analytic isomonodromic deformation, which is unique up to unique isomorphism, and whose parameter space $T$ is the Teichmüller space ${g,n}$ .",
"This universal analytic isomonodromic deformation satisfies a universal property with respect to germs of analytic isomonodromic deformations of $(C, E_0,\\nabla _0)$ .",
"A universal algebraic isomonodromic deformation of $(C, E_0,\\nabla _0)$ , if it exists, would be an algebraic isomonodromic deformation whose analytic germification is isomorphic to the germification of the universal analytic isomonodromic deformation of $(C, E_0,\\nabla _0)$ .",
"In Section REF , we give an alternative definition and state a universal property of universal algebraic isomonodromic deformations.",
"Our main result is the following.",
"Theorem A Let $C$ be a smooth complex projective curve of genus $g$ .",
"Let $D_C$ be a set of $n$ distinct points in $C$ and let $\\Phi : (\\Sigma _g , Y^n)\\rightarrow (C, D_C)$ be an orientation preserving homeomorphism.",
"Let $(E_0,\\nabla _0)$ be an algebraic logarithmic connection of rank $r$ over $C$ with polar divisor $D_C$ and monodromy $[\\rho ]\\in \\chi _{g,n}(\\mathrm {GL}_r$ with respect to $\\Phi $ .",
"Assume that $2g-2+n>0$ and that $\\nabla _0$ is mild.",
"If $r>2$ , then assume further that $\\rho $ is semisimple.",
"The following are equivalent: There is a universal algebraic isomonodromic deformation of $(C, E_0, \\nabla _0)$ .",
"The orbit $\\Gamma _{g,n}\\cdot [\\rho ]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"Remark 1.A.1 Note that the orbit $\\Gamma _{g,n}\\cdot [\\rho ]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ does not depend on the choice of $\\Phi $ .",
"We prove this theorem by adapting the proof for the special case of genus $g=0$ , which has been established in .",
"The main ingredients of the proof of Theorem REF are: the logarithmic Riemann-Hilbert correspondence (see Section REF ); the introduction of a base point section for a family of punctured curves and the splitting of the fundamental group of the total space of the family (see Section REF ), together with its relation to the mapping class group (see Section REF ).",
"Both implications to be proven appear as special cases of stronger results: Theorem A1 and Theorem A2, respectively.",
"We give their statements and proofs in Section REF .",
"The statement of Theorem REF is natural in the following sense.",
"As we recall in Section REF , the (algebraic) moduli space $\\mathcal {M}_{g,n}$ of stable smooth $n$ -pointed genus-$g$ curves is the quotient of the (analytic) Teichmüller space $\\mathcal {T}_{g,n}$ by the natural action of $\\Gamma _{g,n}$ .",
"Intuitively, a universal algebraic isomonodromic deformation should be the quotient of the universal analytic isomonodromic deformation with respect to a sufficiently large subgroup of $\\Gamma _{g,n}$ that fixes $[\\rho ]$ ."
],
[
"Dynamical study of finite orbits in the reducible rank 2 case",
"Since the pure mapping class group is a finite index subgroup of the full mapping class group, for any representation $\\rho \\in \\operatorname{Hom}(\\Lambda _{g,n}, G)$ , the conjugacy class $[\\rho ]\\in \\chi _{g,n}(G)$ has finite orbit under $\\Gamma _{g,n}$ if and only if it has finite orbit under $\\hat{\\Gamma }_{g,n}$ .",
"Note that the size of $\\hat{\\Gamma }_{g,n}\\cdot [\\rho ]$ equals the size of the set of conjugacy classes of $m$ -tuples ${\\raisebox {.0em}{{\\left\\lbrace (\\rho ^{\\prime }(s_1), \\ldots , \\rho ^{\\prime }(s_m))~|~\\rho ^{\\prime }\\in \\operatorname{Hom}(\\Lambda _{g,n}, G)\\textrm { and }{[}\\rho ^{\\prime }{]}\\in \\hat{\\Gamma }_{g,n}\\cdot [\\rho ] \\right\\rbrace }}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ G }} \\, , $ where $\\lbrace s_1, \\ldots , s_m\\rbrace $ is a set of generators of $\\Lambda _{g,n}$ .",
"We introduce a specific presentation $\\Lambda _{g,n}= \\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g,\\gamma _1,\\ldots ,\\gamma _n ~\\vert ~ [\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]\\gamma _1\\cdots \\gamma _n=1\\right\\rangle \\, $ and a subgroup $\\hat{\\Gamma }_{g,n}^\\circ =\\left\\langle \\tau _{1}, \\ldots , \\tau _{3g+n-2}, \\sigma _1, \\ldots , \\sigma _{n-1}\\right\\rangle \\, $ of $\\hat{\\Gamma }_{g,n}^\\bullet $ which, as such, acts on $\\operatorname{Hom}(\\Lambda _{g,n}, G)$ , and which is sufficiently large in the sense that the $\\hat{\\Gamma }_{g,n}^\\circ $ -orbit of $[\\rho ]\\in \\chi _{g,n}(G)$ equals its $\\hat{\\Gamma }_{g,n}$ -orbit.",
"Moreover, the action of $\\hat{\\Gamma }_{g,n}^\\circ $ on $\\Lambda _{g,n}$ can be explicitely described (see Section ).",
"Table REF summarizes the explicit action of the generators $ \\tau _{1}, \\ldots , \\tau _{3g+n-2}, \\sigma _1, \\ldots , \\sigma _{n-1}$ of $\\hat{\\Gamma }_{g,n}^\\circ $ on the generators $ \\alpha _1, \\beta _1, \\ldots ,$ $ \\alpha _g, \\beta _g,\\gamma _1,\\ldots ,\\gamma _n$ of $\\Lambda _{g,n}$ .",
"Here we only indicate the action on those of our generators of $\\Lambda _{g,n}$ that are not fixed by the action of the generator of $\\hat{\\Gamma }_{g,n}^\\circ $ under consideration.",
"Table: Action of Γ ^ g,n ∘ \\hat{\\Gamma }_{g,n}^\\circ on Λ g,n \\Lambda _{g,n}.We then apply this explicit description of the mapping class group action to the specific study of finite $\\Gamma _{g,n}$ -orbits on $\\chi _{g,n}(\\mathrm {GL}_2\\mathbb {C})$ that correspond to reducible representations.",
"For $g=0$ , this study has been completely carried out in .",
"In this special case, the study can be reduced to linear dynamics.",
"More precisely, any reducible representation $\\rho \\in \\mathrm {Hom}(\\Lambda _{g,n}, \\mathrm {GL}_2\\mathbb {C})$ is conjugated to the tensor product of a character $\\rho _{*}\\in \\operatorname{Hom}(\\Lambda _{g,n}, *)$ and an affine representation $\\rho _{\\mathrm {Aff}}\\in \\operatorname{Hom}(\\Lambda _{g,n}, \\mathrm {Aff}(\\mathbb {C}))$ : $[\\rho ]=[ \\rho _{*}\\otimes \\rho _{\\mathrm {Aff}}]\\, .$ Moreover, $[\\rho ]$ has finite orbit under $\\Gamma _{g,n}$ in $\\chi _{g,n}(\\mathrm {GL}_2\\mathbb {C})$ if and only if $[\\rho _{*}]$ and $[\\rho _{\\mathrm {Aff}}]$ have finite orbit under $\\Gamma _{g,n}$ in $\\chi _{g,n}(*)$ and $\\chi _{g,n}(\\mathrm {Aff}()$ respectively.",
"On the other hand, for $g=0$ , the pure mapping class group acts trivially on $\\chi _{g,n}(*)$ and on the linear part of $\\rho _{\\mathrm {Aff}}$ .",
"Hence in the special case $g=0$ , the study of finite orbits reduces to the study of a certain linear action on the translation part of $\\rho _{\\mathrm {Aff}}$ .",
"For $g>0$ , the study of finite orbits of conjugacy classes of reducible $\\mathrm {GL}_2\\mathbb {C}$ -representations also reduces to the case of scalar and affine representations, but the linear part of $\\rho _{\\mathrm {Aff}}$ is no longer invariant and there is no effective means to reduce the study to linear dynamics.",
"However, Table REF allows to study the orbits explicitely.",
"In the case $g=1$ and $n>0$ , we find a particular type of representations whose conjugacy classes have finite orbit under $\\Gamma _{g,n}$ , namely the representations $\\rho _{\\mu , \\mathbf {c}} \\in \\operatorname{Hom}(\\Lambda _{g,n},\\mathrm {GL}_2\\mathbb {C})$ defined by $\\rho _{\\mu , \\mathbf {c}} (\\alpha _1):=\\begin{pmatrix} \\mu & 0 \\\\ 0 & 1\\end{pmatrix} \\quad \\rho _{\\mu , \\mathbf {c}}(\\beta _1):=\\begin{pmatrix} 1 & -\\frac{1}{\\mu -1} \\\\ 0 & 1\\end{pmatrix} \\quad \\rho _{\\mu , \\mathbf {c}}(\\gamma _i):=\\begin{pmatrix} 1 & c_i \\\\ 0 & 1\\end{pmatrix} \\quad \\forall i\\in \\llbracket 1, n\\rrbracket \\, $ where $\\mu \\in *\\setminus \\lbrace 1\\rbrace $ is a root of unity and $\\mathbf {c}=(c_1, \\ldots , c_{n})\\in {n}$ with $\\sum _{i=1}^{n}c_i=1$ .",
"Note that the condition $\\sum _{i=1}^{n}c_i=1$ is necessary for $\\rho _{\\mu , \\mathbf {c}}$ to be well defined.",
"The complete classification, for every $g>0$ and $n\\ge 0$ , of reducible rank-2 representations with finite $\\Gamma _{g,n}$ -orbit is the following.",
"Theorem B Let $g>0, n\\ge 0$ .",
"Let $\\rho \\in \\operatorname{Hom}(\\Lambda _{g,n},\\mathrm {GL}_2\\mathbb {C})$ be a reducible representation.",
"Consider its conjugacy class $[\\rho ] \\in \\chi _{g,n}(\\mathrm {GL}_2\\mathbb {C})$ .",
"Then the orbit $\\Gamma _{g,n}\\cdot [\\rho ]$ is finite if and only if one of the following conditions is satisfied.",
"The representation $\\rho $ is a direct sum of scalar representations with finite images.",
"We have $g=1$ , $n>0$ , there are a root of unity $\\mu \\in *\\setminus \\lbrace 1\\rbrace $ , $\\mathbf {c}=(c_1, \\ldots , c_{n})\\in {n}$ with $\\sum _{i=1}^{n}c_i=1$ and a scalar representation $\\lambda $ with finite image such that $[\\rho ] \\in \\Gamma _{g,n}\\cdot [\\lambda \\otimes \\rho _{\\mu , \\mathbf {c}}]\\, .\\vspace{5.69046pt}$ Moreover, if the orbit $\\Gamma _{g,n}\\cdot [\\rho ]$ is finite, we can give an estimate for its cardinality, which for $\\rho =\\lambda _1\\oplus \\lambda _2$ and $\\rho =\\lambda \\otimes \\rho _{\\mu , \\mathbf {c}}$ in the cases $(\\ref {case1})$ and $(\\ref {case2})$ respectively is $\\hspace{5.12128pt} \\frac{1}{2}\\cdot \\max \\left\\lbrace \\operatorname{\\mathrm {card}}(\\mathrm {Im}(\\lambda _i))^{2g-1}~|~i\\in \\lbrace 1,2\\rbrace \\right\\rbrace ~\\le ~\\operatorname{\\mathrm {card}}(\\Gamma _{g,n}\\cdot [\\rho ])~\\le ~\\operatorname{\\mathrm {card}}(\\mathrm {Im}(\\rho ))^{2g}$ and $\\hspace{12.80365pt}\\max \\left\\lbrace N_2\\, , \\, \\phi (N)(2N-\\phi (N))N^{n^{\\prime }-1}\\right\\rbrace ~\\le ~\\operatorname{\\mathrm {card}}(\\Gamma _{g,n}\\, \\cdot \\, [\\rho ])~\\le ~(N^2-1)N^{n^{\\prime }-1}N_2^{2}\\, ,$ where $\\phi $ denotes the Euler totient function, $n^{\\prime }:=\\operatorname{\\mathrm {card}}\\left\\lbrace i\\in \\llbracket 1, n\\rrbracket ~|~\\rho (\\gamma _i)\\notin *I_2\\right\\rbrace $ , $N:=\\mathrm {order}(\\mu )$ and $N_2:=\\operatorname{\\mathrm {card}}(\\mathrm {Im}(\\lambda ))$ .",
"The heart of the proof of Theorem REF is the complete classification of finite $\\hat{\\Gamma }_{g,n}$ -orbits in $\\chi _{g,n}(\\mathrm {Aff}(\\mathbb {C}))$ under the full mapping class group (see the beginning of Section for details on how we proceed).",
"In Section REF , we deduce an explicit description of the finite $\\Gamma _{g,n}$ -orbits for scalar and affine representations.",
"The decomposition of reducible representations into a tensor product of such representations then yields the result (see Sections REF and REF )."
],
[
"Universal isomonodromic deformations",
"In this section, we will recall some well known results about moduli spaces and universal families of curves.",
"For a more detailed exposition, see for example and .",
"Then we turn to the existence of analytic and algebraic universal isomonodromic deformations of connections over curves, and their respective universal properties.",
"The main purpose of this section is the precise setup of our notation and definitions.",
"The reader might want to skip this section at first and come back to it when needed (references will be given)."
],
[
"Moduli spaces of curves\n",
"We define a curve of genus $g$ to be a smooth projective complex curve $C$ with $H^1(C,\\mathbb {Z})=\\mathbb {Z}^{2g}$ .",
"From now on, we will assume $ 2g-2+n>0\\, .$ As a set, the Teichmüller space $\\mathcal {T}_{g,n}$ of $n$ -pointed genus-$g$ curves is the set of isomorphism classes $[C,D_C,\\varphi ]$ of triples $(C,D_C,\\varphi )$ , where $C$ is a genus $g$ curve, $D_C=\\lbrace x_1, \\ldots , x_n\\rbrace $ is a set of $n$ distinct points in $C$ and $\\varphi $ is a Teichmüller structure, i.e.",
"an orientation-preserving homeomorphism $\\varphi :(\\Sigma _g ,Y^n)\\rightarrow (C,D_C)$ .",
"Two $n$ -pointed genus-$g$ curves with Teichmüller structure $(C,D_C,\\varphi )$ and $(C^{\\prime },D^{\\prime }_{C^{\\prime }},\\varphi ^{\\prime })$ are said to be isomorphic if there exists an isomorphism of pointed curves $\\psi : (C^{\\prime },D^{\\prime }_{C^{\\prime }})\\rightarrow (C,D_C) $ such that $[\\varphi ^{\\prime }]=[\\psi \\circ \\varphi ]$ , where $[\\varphi ]$ denotes the isotopy class of $\\varphi $ .",
"We have a natural action of $\\Gamma _{g,n}$ on $\\mathcal {T}_{g,n}$ given by $[h]\\cdot [C,D_C,\\varphi ]:= [C,D_C, \\varphi \\circ h^{-1}]\\, ; ~~~~~ [h] \\in \\Gamma _{g,n}\\, ,\\, [C,D_C,\\varphi ]\\in \\mathcal {T}_{g,n}\\, .$ The kernel of this action is finite.",
"More precisely, we have (see ): Lemma 2.1.1 If the natural morphism $\\Gamma _{g,n}\\rightarrow \\mathrm {Aut}(\\mathcal {T}_{g,n})$ has nontrivial kernel $K_{g,n}$ , then $K_{g,n}\\simeq \\mathbb {Z}/2\\mathbb {Z}$ and one of the following holds.",
"$(g,n)=(2,0)$ and the non-trivial element of $K_{g,n}$ is the hyperelliptic involution of $\\Sigma _2$ .",
"$(g,n)=(1,1)$ and the non-trivial element of $K_{g,n}$ is the order 2 symmetry about the puncture, given, for $(\\Sigma _1,y_1)=( \\mathbb {Z}^2,0)$ , by $z\\mapsto -z$ .",
"As a set, the moduli space $\\mathcal {M}_{g,n}$ of curves of genus $g$ with $n$ (labeled) punctures is the set of isomorphism classes $[C,\\mathbf {x}]$ of pairs $(C,\\mathbf {x})$ , where $C$ is a genus $g$ curve and $\\mathbf {x}=(x_1,\\ldots , x_n)$ is a tuple of $n$ distinct points in $C$ .",
"The isomorphisms are isomorphisms of pointed curves that respect the labellings of the $n$ -tuples.",
"Notice that a Teichmüller structure $(C,D_C,\\varphi )$ defines such a pair $(C,\\mathbf {x})$ , by setting $\\mathbf {x}:=(\\varphi (y_i))_{i\\in \\llbracket 1, n\\rrbracket }$ .",
"In this way, we obtain a forgetful map $ \\pi _{g,n}: \\mathcal {T}_{g,n}\\rightarrow \\mathcal {M}_{g,n}\\,$ whose fibers are globally fixed by the action of $\\Gamma _{g,n}/K_{g,n}$ on $\\mathcal {T}_{g,n}$ .",
"Denote by $ \\mathcal {R}_{g,n}\\subset \\mathcal {T}_{g,n}$ the set consisting in points with non-trivial stabilizer for the action of $\\Gamma _{g,n}/K_{g,n}$ .",
"The subset $ \\mathcal {B}_{g,n}:=\\pi _{g,n}(\\mathcal {R}_{g,n})$ of $\\mathcal {M}_{g,n}$ characterizes pointed curves with automorphism groups not isomorphic to $K_{g,n}$ .",
"We say that these curves have exceptional automorphisms.",
"Recall that $\\mathcal {T}_{g,n}$ has a natural structure of a complex analytic manifold, and $\\mathcal {M}_{g,n}$ has a natural structure of a complex quasi-projective variety (see ).",
"The set $\\mathcal {B}_{g,n}$ of curves with exceptional automorphisms is a Zariski closed subset of $\\mathcal {M}_{g,n}$ (see ) which is a proper subset (see , , , ).",
"Moreover, the map $\\pi _{g,n}|_{\\mathcal {T}_{g,n}\\setminus \\mathcal {R}_{g,n}} :\\mathcal {T}_{g,n}\\setminus \\mathcal {R}_{g,n}\\rightarrow \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ is a non branched analytic cover, with Galois group $\\Gamma _{g,n}/K_{g,n}$ .",
"We thereby obtain a tautological morphism $ \\mathrm {taut}_{g,n}: \\pi _1(\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n},\\star )\\twoheadrightarrow {\\raisebox {.0em}{\\Gamma _{g,n}}\\raisebox {-.1em}{/}\\raisebox {-.2em}{ K_{g,n} }};$ such that any lift $\\tilde{\\gamma }$ in $\\mathcal {T}_{g,n}$ for a loop $\\gamma $ of $\\pi _1(\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n},\\star )$ satisfies $\\tilde{\\gamma }(1)=\\mathrm {taut}_{g,n}(\\gamma )\\cdot \\tilde{\\gamma }(0).$"
],
[
"Families of pointed curves",
"Let $C$ be a curve of genus $g$ and $D_C$ a reduced divisor of degree $n$ on $C$ .",
"Recall that we always assume (REF ); i.e.",
"the pointed curve $(C,D_C)$ is stable.",
"A $$ -family of $n$ -pointed genus-$g$ curves with central fiber $(C,D_C)$ is a datum $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi )\\, ,$ where $\\kappa : X\\rightarrow T$ is a proper surjective smooth morphism of $$ -manifolds; $D=\\sum _{i=1}^nD_i$ is a reduced divisor on $X$ such that there are pairwise disjoint sections $\\sigma _1, \\ldots , \\sigma _n$ of $\\kappa $ with $\\sigma _i(T)=D_i$ ; $t_0\\in T$ is a point and $X_{t_0}$ denotes $X_{t_0}:=\\kappa ^{-1}(\\lbrace t_0\\rbrace )$ ; $\\psi :(C,D_C) \\stackrel{\\sim }{\\rightarrow } (X_{t_0}, D|_{X_{t_0}}) $ is an isomorphism of $$ -manifolds.",
"We shall always denote by $X_t:=\\kappa ^{-1}(\\lbrace t\\rbrace )$ the fiber at a parameter $t\\in T$ .",
"When there is a smooth connected $$ -neighborhood $\\Delta $ of $t_0$ such that $\\mathcal {F}_{(C,D_C)}|_{\\Delta }$ satisfies a certain property, we may say that $\\mathcal {F}_{(C,D_C)}$ satisfies this property “up to shrinking”.",
"A morphism $\\mathfrak {f}: \\mathcal {F}^{\\prime }_{(C,D_C)}\\rightarrow \\mathcal {F}_{(C,D_C)}$ is a pair $\\mathfrak {f}=(\\mathfrak {f}^a, \\mathfrak {f}^b)$ , where $\\mathfrak {f}^a : X^{\\prime }\\rightarrow X$ and $\\mathfrak {f}^b : T^{\\prime }\\rightarrow T$ are morphisms of $$ -varieties such that the following diagram commutes (and in particular $\\mathfrak {f}^b(t_0^{\\prime })=t_0$ ).",
"${(C,D_C)@{=}[r]^{\\mathrm {id}}@{^{(}->}[d]_{\\psi ^{\\prime }}&(C,D_C)@{_{(}->}[d]^{\\psi }\\\\(X^{\\prime },D^{\\prime })[d]_{\\kappa ^{\\prime }} [r]^{\\mathfrak {f}^a} &(X,D) [d]^{\\kappa } \\\\T^{\\prime }[r]^{\\mathfrak {f}^{b}}& T}$ Remark 2.2.1 Note that this definition implies that $(X^{\\prime },D^{\\prime })$ is isomorphic to the pullback $\\mathfrak {f}^{b *}(X,D)$ (the fibered product with respect to $\\mathfrak {f}^b$ and $\\kappa $ ).",
"Suppose now that we have a Teichmüller structure for $(C,D_C)$ , given by an orientation preserving homeomorphism $\\varphi : (\\Sigma _g,Y^n)\\rightarrow (C,D_C)$ .",
"A $$ -family of $n$ -pointed genus-$g$ curves with Teichmüller structure with central fiber $(C,D_C,\\varphi )$ is a datum $\\mathcal {F}_{(C,D_C,\\varphi )}^+=(\\mathcal {F}_{(C,D_C)}, \\Phi )$ , where $\\mathcal {F}_{(C,D_C)}$ is as above and $\\Phi : (\\Sigma _g,Y^n)\\times T \\stackrel{\\sim }{\\rightarrow } (X,D)$ is a homeomorphism such that the following diagram commutes, where $\\mathrm {pr}$ denotes the projection to the second factor.",
"${ (\\Sigma _g,Y^n)\\times \\lbrace t_0\\rbrace [r]^{\\hspace{10.0pt}\\varphi }@{^{(}->}[d] & (C,D_C)@{_{(}->}[d]^{\\psi }\\\\(\\Sigma _g,Y^n)\\times T [dr]_{\\mathrm {pr}} [r]^{\\hspace{10.0pt}\\Phi } &(X,D) [d]^{\\kappa } \\\\& T}$ In particular, if we denote $\\Phi _t:=\\Phi |_{(\\Sigma _g,Y^n)\\times \\lbrace t\\rbrace }\\, , $ then $\\Phi _{t_0}=\\psi \\circ \\varphi $ .",
"Notice that by definition, a $$ -family with Teichmüller structure is topologically trivial.",
"For a given $\\varphi $ as above, up to shrinking, any analytic family $\\mathcal {F}_{(C,D_C)}$ lifts to a family $\\mathcal {F}_{(C,D_C, \\varphi )}^{+}$ with Teichmüller structure.",
"Let $\\mathcal {F}_{(C,D_C, \\varphi )}^+=(\\mathcal {F}_{(C,D_C)}, \\Phi )$ and ${\\mathcal {F}^{\\prime }}_{(C,D_C, \\varphi ^{\\prime })}^{+}=(\\mathcal {F}_{(C,D_C)}^{\\prime }, \\Phi ^{\\prime })$ be two $$ -families with Teichmüller structures.",
"A morphism ${\\mathcal {F}^{\\prime }}_{(C,D_C, \\varphi ^{\\prime })}^+\\rightarrow \\mathcal {F}_{(C,D_C, \\varphi )}^+$ is a datum $\\mathfrak {f}^+=(\\mathfrak {f}, \\mathfrak {f}^{\\mathrm {top}})$ , where $\\mathfrak {f}=(\\mathfrak {f}^a, \\mathfrak {f}^b)$ is as before and $\\mathfrak {f}^{\\mathrm {top}}$ is a continous map, such that the following diagram commutes.",
"${(\\Sigma _g,Y^n)\\times T^{\\prime }[d]_{\\Phi ^{\\prime }}[r]^{\\mathfrak {f}^{\\mathrm {top}}}& (\\Sigma _g,Y^n)\\times T \\\\(X^{\\prime },D^{\\prime }) [r]^{\\mathfrak {f}^a} &(X,D) [u]^{\\Phi ^{-1}} \\\\}$ Notice that the central fiber forces the map $\\mathfrak {f}^{\\mathrm {top}}$ to be $(\\varphi ^{-1}\\circ \\varphi ^{\\prime }) \\times \\mathfrak {f}^{b}$ , up to a fiberwise isotopy.",
"To a $$ -family $\\mathcal {F}_{(C,D_C)}$ [resp.",
"$$ -family with Teichmüller structure $\\mathcal {F}_{(C,D_C, \\varphi )}^+$ ] as before, one can associate a so-called $$ -family $\\mathcal {F}$ [resp.",
"$$ -family with Teichmüller structure $\\mathcal {F}^+$ ] with non specified central fiber, by forgetting $(C,D_C)$ [resp.",
"$(C,D_C,\\varphi )$ ] and the marking $t_0, \\psi $ .",
"A morphism of $$ -families with non specified central fiber is a datum $\\mathfrak {f}$ [respectively $\\mathfrak {f}^+$ ] as above for a convenient choice of a marked central fiber."
],
[
"Universal families of pointed curves",
"Let $\\mathcal {F}^+_{(C,D_C, \\varphi )}=(\\mathcal {F}_{(C,D_C)}, \\Phi )$ be a $$ -family with Teichmüller structure.",
"Then the classifying map $ {\\mathrm {class}^+}(\\mathcal {F}^+): \\left\\lbrace \\begin{array}{ccc} T &\\rightarrow &\\mathcal {T}_{g,n}\\\\ t & \\mapsto & [X_t,D|_{X_{t}},\\Phi _t] \\end{array}\\right.\\, $ is holomorphic with respect to the natural analytic manifold structure of $\\mathcal {T}_{g,n}$ .",
"The Teichmüller space $\\mathcal {T}_{g,n}$ carries a universal family (see for example ), which is an analytic family with Teichmüller structure $\\mathcal {F}_{g,n}^+=(\\mathcal {F}_{g,n}, \\Phi _{g,n})$ and non specified central fiber, satisfying ${\\mathrm {class}^+}(\\mathcal {F}_{g,n}^+)=\\mathrm {id}_{\\mathcal {T}_{g,n}}\\, .$ The universal Teichmüller curve enjoys the following universal property: If $\\mathcal {F}^+=(\\mathcal {F}, \\Phi )$ is an analytic family with Teichmüller structure and non specified central fiber, then there is a unique isomorphism $\\mathfrak {f}^+ : \\mathcal {F}^+ \\stackrel{\\sim }{\\longrightarrow } {\\mathrm {class}^+}(\\mathcal {F}^+)^*(\\mathcal {F}_{g,n})$ with $\\mathfrak {f}^b=\\mathrm {id}_T$ .",
"Let $\\mathcal {F}_{(C,D_C)}$ be a $$ -family.",
"Assume we have a labelling $\\mathbf {x}$ of $D_C$ , i.e.",
"$\\mathbf {x}=(x_i)_{i\\in \\llbracket 1, n\\rrbracket }\\in C^n$ and $D_C =\\sum _{i=1}^nx_i$ .",
"Then there is a well defined labelling $\\mathbf {D}=(D_i)_{i\\in \\llbracket 1, n\\rrbracket }$ of $D$ defined by $D=\\sum _{i=1}^n D_i$ and $\\psi (x_i)\\in D_i$ for all $i\\in \\llbracket 1, n\\rrbracket $ .",
"We then have a well defined classifying map $\\mathrm {class}(\\mathcal {F}): \\left\\lbrace \\begin{array}{ccc} T &\\rightarrow &\\mathcal {M}_{g,n}\\\\ t & \\mapsto & [X_t,\\mathbf {D}|_{X_{t}}] \\end{array}\\right.",
"\\, , $ which is a morphism of $$ -varieties with respect to the natural structure of $$ -variety on $\\mathcal {M}_{g,n}$ .",
"We say that the fiber $(X_t, D|_{X_t})$ of $\\mathcal {F}$ at $t\\in T$ has exceptional automorphisms if $\\mathrm {class}(\\mathcal {F})(t)\\in \\mathcal {B}_{g,n}$ .",
"This notion does not depend on the choice of a labelling.",
"Although there is no universal family of curves over $\\mathcal {M}_{g,n}$ in the strict sense, we can consider algebraic Kuranishi families.",
"Let $\\mathcal {F}$ be a $$ -family and let $t\\in T$ be a parameter.",
"Denote $\\mathcal {F}|_{\\Delta ^{\\mathrm {an}}}$ the analytic germification of $\\mathcal {F}$ at $t$ , which can be endowed with a Teichmüller structure $\\Phi _{\\Delta ^{\\mathrm {an}}}$ .",
"We say that $\\mathcal {F}$ is Kuranishi at $t$ if ${\\mathrm {class}^+}(\\mathcal {F}|_{\\Delta ^{\\mathrm {an}}},\\Phi _{\\Delta ^{\\mathrm {an}}})$ is an isomorphism.",
"The notion of being Kuranishi at $t$ does not depend on the choice of $\\Phi _{\\Delta ^{\\mathrm {an}}}$ .",
"We say that $\\mathcal {F}$ is Kuranishi if it is Kuranishi at each $t\\in T$ .",
"Notice that if $\\mathcal {F}^{\\mathrm {Kur}}$ is an algebraic Kuranishi family, then for any labelling, the classifying map $\\mathrm {class}(\\mathcal {F}^{\\mathrm {Kur}})$ is dominant and has finite fibers.",
"For any stable $n$ -pointed genus-$g$ curve $(C,D_C)$ , there exists an algebraic Kuranishi family $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ with central fiber $(C,D_C)$ .",
"Moreover, we have (see ): Proposition 2.3.1 (Universal property of Kuranishi families) Let $(C,D_C)$ and $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ be as above.",
"Let $\\mathcal {F}^{\\prime }_{(C,D_C)}$ be an algebraic family with central fiber $(C,D_C)$ .",
"Then there are an étale base change $p:(T^{\\prime \\prime },t_0^{\\prime \\prime })\\rightarrow (T^{\\prime },t_0^{\\prime })$ ; denote $\\mathcal {F}^{\\prime \\prime }_{(C,D_C)}:=p^*\\mathcal {F}^{\\prime }_{(C,D_C)}$ ; a morphism $q : (T^{\\prime \\prime },t_0^{\\prime \\prime })\\rightarrow (T,t_0)$ and an isomorphism $\\mathfrak {f}: \\mathcal {F}^{\\prime \\prime }_{(C,D_C)}\\stackrel{\\sim }{\\longrightarrow } q^*\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ with $\\mathfrak {f}^b=id_{\\Delta ^{\\prime \\prime }}$ ."
],
[
"Universal isomonodromic deformations",
"Let again $(C,D_C)$ be a stable $n$ -pointed genus-$g$ curve.",
"Let $(E_0, \\nabla _0)$ be logarithmic $$ -connection over $C$ with polar divisor $D_C$ .",
"Isomonodromic deformations.",
"A $$ -isomonodromic deformation of $(C, E_0, \\nabla _0)$ is a tuple $\\mathcal {I}_{(C, E_0, \\nabla _0)}=(\\mathcal {F}_{(C, D_C)}, E, \\nabla , \\Psi )$ , where $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi )$ is a $$ -family with central fiber $(C,D_C)$ , $(E,\\nabla )$ is a flat logarithmic $$ -connection over $X$ with polar divisor $D$ and $(\\psi ,\\Psi ) : (E_0\\rightarrow C, \\nabla _0)\\rightarrow (E\\rightarrow X, \\nabla )|_{X_{t_0}}$ is an isomorphism of $$ -logarithmic connections, i.e.",
"$\\Psi : E_0\\rightarrow E|_{X_{t_0}}$ is a $$ -vector bundle isomorphism over $\\psi : C\\rightarrow X_{t_0}$ satisfying $\\Psi ^*\\left(\\nabla |_{X_{t_0}}\\right)=\\nabla ^0$ .",
"Let $\\mathcal {I}_{(C, E_0, \\nabla _0)}$ and $\\mathcal {I}_{(C, E_0, \\nabla _0)}^{\\prime }$ be two $$ -isomonodromic deformations of $(C, E_0, \\nabla _0)$ .",
"A morphism $\\mathfrak {f}^{}: \\mathcal {I}_{(C, E_0, \\nabla _0)}\\rightarrow \\mathcal {I}_{(C, E_0, \\nabla _0)}^{\\prime }$ is a datum $\\mathfrak {f}^{ }=(\\mathfrak {f}^a, \\mathfrak {f}^b, \\mathfrak {f}^{\\mathrm {vb}})$ , where $(\\mathfrak {f}^a, \\mathfrak {f}^b)$ is a morphism $\\mathcal {F}_{(C,D_C)}^{\\prime }\\rightarrow \\mathcal {F}_{(C,D_C)}$ as in Section REF , and $ \\mathfrak {f}^{\\mathrm {vb}}$ is a morphism of $$ -vector bundles over $\\mathfrak {f}^a$ with ${\\nabla =\\mathfrak {f}^{vb}}^*\\nabla ^{\\prime }$ .",
"An algebraic isomonodromic deformation $\\mathcal {I}_{(C, E_0, \\nabla _0)}$ of $(C, E_0, \\nabla _0)$ as above is called regular if moreover $(E, \\nabla )$ is regular (with respect to a suitable meromorphic structure at infinity).",
"The definition of regularity can be found in .",
"Putting this regularity condition on $\\mathcal {I}_{(C, E_0, \\nabla _0)}$ may be seen as a way of standardizing algebraic isomonodromic deformations, as illustrated by the following statement.",
"Lemma 2.4.1 If $(E_0, \\nabla _0)$ is mild and $\\mathcal {I}_{(C, E_0, \\nabla _0)}$ is an algebraic isomonodromic deformation of $(C, E_0, \\nabla _0)$ , then the analytification of $\\mathcal {I}_{(C, E_0, \\nabla _0)}$ is isomorphic to the analytification of a regular algebraic isomonodromic deformation $\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}$ of $(C, E_0, \\nabla _0)$ .",
"This Lemma will be proven in Section REF , where we will also recall the notion of mildness, which is a minor technical condition.",
"Analytic universal isomonodromic deformations.",
"Let $\\varphi :(\\Sigma _g,Y^n) \\stackrel{\\sim }{\\rightarrow } (C,D_C)$ be an orientation preserving homeomorphism.",
"Consider the universal Teichmüller family $\\mathcal {F}_{g,n}^+=(\\mathcal {F}_{g,n}, \\Phi _{g,n})$ .",
"We shall denote $ \\mathcal {F}_{g,n}=(\\kappa _{g,n} : \\mathcal {X}\\rightarrow \\mathcal {T}_{g,n}, \\mathcal {D}) \\, ; \\quad \\Phi _{g,n} :(\\Sigma _g,Y^n)\\times \\mathcal {T}_{g,n}\\stackrel{\\sim }{\\rightarrow } (\\mathcal {X}, \\mathcal {D})\\, ; \\quad t_0:=[C,D_C, \\varphi ]\\in \\mathcal {T}_{g,n}\\, .$ By the definition of $\\mathcal {F}_{g,n}^+$ , we then have an isomorphism $\\psi : (C,D_C)\\stackrel{\\sim }{\\rightarrow } ({\\mathcal {X}_{t_0}} , \\mathcal {D}|_{\\mathcal {X}_{t_0}})$ .",
"In particular, $\\mathcal {F}^{\\mathrm {Teich}}_{(C, D_C)}:=( \\mathcal {F}_{g,n} , t_0, \\psi )$ is an analytic family with central fiber $(C,D_C)$ , which moreover is topologically trivial and has simply connected parameter space.",
"The inclusion $\\Phi _{g,n}^{-1}\\circ \\psi \\circ \\varphi $ of the topological fiber at $t_0$ then defines an isomorphism $\\Lambda _{g,n}=\\pi _1(\\Sigma _g\\setminus Y^n, y_0) \\stackrel{\\sim }{\\rightarrow } \\pi _1((\\Sigma _g\\setminus Y^n)\\times \\mathcal {T}_{g,n}, (y_0,t_0))\\, .$ Now let $[\\rho _{\\nabla _0}]$ be the monodromy of $(E_0, \\nabla _0)$ with respect to $\\varphi $ .",
"The representation $\\rho _{\\nabla _0}$ can then be trivially extended to a representation $\\rho $ of $\\pi _1((\\Sigma _g\\setminus Y^n)\\times \\mathcal {T}_{g,n}, (y_0,t_0))$ .",
"It turns out that the conjugacy class of this “extended representation” is the monodromy representation, with respect to $\\Phi _{g,n}$ , of a certain flat logarithmic connection $(E, \\nabla )$ over $ \\mathcal {X}$ with polar divisor $\\mathcal {D}$ such that the pullback $\\psi ^*(E,\\nabla )$ restricted to $\\mathcal {X}_{t_0}$ is canonically isomorphic to $(E_0, \\nabla _0)$ .",
"We obtain the universal analytic isomonodromic deformation $\\mathcal {I}^{\\mathrm {univ,\\, an}}_{(C, E_0, \\nabla _0)}:=(\\mathcal {F}^{\\mathrm {Teich}}_{(C, D_C)}, E, \\nabla , \\Psi ^\\mathrm {can})\\, .$ Its construction has been carried out in , using Malgrange's Lemma (see ) and the fact that $\\mathcal {T}_{g,n}$ is contractible by Fricke's Theorem.",
"It satisfies the following universal property: if $\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}=(\\mathcal {F}_{(C, D_C)}^{\\prime }, E^{\\prime }, \\nabla ^{\\prime } , \\Psi ^{\\prime })$ is an analytic isomonodromic deformation of $(C,E_0,\\nabla _0)$ , and if $\\Delta ^{\\prime }$ is a sufficiently small neighborhood of its central parameter $t_0^{\\prime }$ , then there is a morphism $q : (\\Delta ^{\\prime },t_0^{\\prime })\\rightarrow (\\mathcal {T}_{g,n}, t_0)$ and a canonical isomorphism $\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}|_{\\Delta ^{\\prime }} \\simeq q^{ *}\\mathcal {I}_{(C, E_0, \\nabla _0)}^{\\mathrm {univ,\\, an}}\\, .$ The construction of this analytic universal isomonodromic deformation and the proof of its universal property rely on the fact that up to shrinking the parameter space, analytic families of curves are topologically trivial and have simply connected parameter space.",
"This is of course no longer the case in the algebraic category, the “extension of the monodromy representation” of $(C, E_0, \\nabla _0)$ being the main challenge.",
"Algebraic universal isomonodromic deformations.",
"An algebraic universal isomonodromic deformation of $(C, E_0, \\nabla _0)$ is an algebraic isomonodromic deformation $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C, E_0, \\nabla _0)}=(\\mathcal {F}_{(C, D_C)}, E, \\nabla , \\Psi )$ , where $\\mathcal {F}_{(C, D_C)}=\\mathcal {F}^{\\mathrm {Kur}}_{(C, D_C)}$ is an algebraic Kuranishi family with central fiber $(C,D_C)$ .",
"Note that an algebraic universal isomonodromic deformation of $(C, E_0, \\nabla _0)$ does not need to exist; its existence is precisely the subject of Theorem REF .",
"When it does exist, it satisfies the following universal property, which will be proven in Section REF .",
"Proposition 2.4.2 (Universal property of universal algebraic isomonodromic deformations) Let $(C, E_0, \\nabla _0)$ and $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C, E_0, \\nabla _0)}$ be as above.",
"Let $\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}$ be another algebraic isomonodromic deformation of $(C, E_0, \\nabla _0)$ .",
"Assume that $(E_0, \\nabla _0)$ is mild; the monodromy representation of $(E_0, \\nabla _0)$ is irreducible; $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C, E_0, \\nabla _0)}$ and $\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}$ are both regular.",
"Then there are an étale base change $p:(T^{\\prime \\prime },t_0^{\\prime \\prime })\\rightarrow (T^{\\prime },t_0^{\\prime })$ ; denote $(\\mathcal {F}^{\\prime \\prime }_{(C, D_C)}, E^{\\prime \\prime }, \\nabla ^{\\prime \\prime } , \\Psi ^{\\prime \\prime }):=p^*\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}$ ; a flat algebraic connection $(L, \\xi )$ of rank 1 over $T^{\\prime \\prime }$ with empty polar divisor; a morphism $q : (T^{\\prime \\prime },t_0^{\\prime \\prime })\\rightarrow (T,t_0)$ and an isomorphism $\\mathfrak {f}: (\\mathcal {F}_{(C, D_C)}^{\\prime \\prime }, (E^{\\prime \\prime }, \\nabla ^{\\prime \\prime })\\otimes \\kappa ^{\\prime \\prime *}(L,\\xi ) , \\Psi ^{\\prime \\prime })\\stackrel{\\sim }{\\longrightarrow } q^*\\mathcal {I}^{\\mathrm {univ, alg}}_{(C, E_0, \\nabla _0)}$ with $\\mathfrak {f}^b=id_{\\Delta ^{\\prime \\prime }}$ .",
"Remark 2.4.3 It is not possible without further assumptions to prove a similar statement for initial connections $(E_0, \\nabla _0)$ with merely semisimple monodromy representations."
],
[
"Fundamental groups and the Riemann-Hilbert correspondence",
"In this section, we shall see that up to an étale base change, any algebraic family of pointed curves can be endowed with a section avoiding the punctures.",
"The existence of such a base point section allows us to decompose the fundamental group of the total space of the family of curves into an semi-direct product of the fundamental groups of the central fiber and the parameter space.",
"Together with the logarithmic Riemann-Hilbert correspondence, this will be used to prove the universal property of universal algebraic isomonodromic deformations."
],
[
"Splitting of the fundamental group",
"Lemma 3.1.1 (Existence of a base point section) Let $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi )$ be an algebraic family of pointed curves with central fiber $(C,D_C)$ as in Section REF .",
"Let $x_0$ be a point in $C\\setminus D_C$ .",
"Then there are a Zariski open neighborhood $\\Delta $ of $t_0$ in $T$ and a finite étale cover $p:(\\Delta ^{\\prime },t_0^{\\prime })\\rightarrow (\\Delta ,t_0)$ such that for $\\mathcal {F}^{\\prime }_{(C,D_C)}=(\\kappa ^{\\prime } : X^{\\prime }\\rightarrow \\Delta ^{\\prime },D^{\\prime }, t_0^{\\prime },\\psi ^{\\prime })$ , defined by $\\mathcal {F}^{\\prime }_{(C,D_C)}:=p^*\\mathcal {F}_{(C,D_C)}$ , there exists a section $\\sigma $ of $\\kappa ^{\\prime }$ with values in $X^{\\prime }\\setminus D^{\\prime }$ such that $\\sigma (t_0^{\\prime })=\\psi ^{\\prime }(x_0)$ .",
"Since $X$ is embedded in some projective space $\\mathbb {P}^N$ , by Bertini's Theorem, there exists a hyperplane $H$ of $\\mathbb {P}^N$ which intersects $X_{t_0}$ transversely, is disjoint from $D|_{X_{t_0}}$ and satisfies $\\psi (x_0)\\in H$ .",
"Since $H$ is ample, we have $\\mathrm {deg}(X_t\\cap H)>0$ for each $t\\in T$ .",
"In particular, $H\\cap X_t \\ne \\varnothing $ for each parameter $t\\in T$ .",
"By irreducibility of $T$ , there exists an irreducible component $T^{\\prime }$ of $X\\cap H$ such that $\\kappa (T^{\\prime })=T$ and $\\psi (x_0)\\in T^{\\prime }$ .",
"Now $\\kappa |_{T^{\\prime }}: T^{\\prime }\\rightarrow T$ is a connected finite ramified covering.",
"Denote by $Z_1\\subset T$ its branching locus.",
"Further, denote by $Z_2$ the adherence of $\\kappa (T^{\\prime }\\cap D)$ .",
"By construction, $Z:=Z_1\\cup Z_2$ is a Zariski closed proper subset of $T$ not containing $t_0$ .",
"Denote $ \\Delta :=T\\setminus Z$ and $\\Delta ^{\\prime }:=\\kappa ^{-1}(\\Delta )\\cap T^{\\prime }\\, .$ We now have $t_0^{\\prime }:=\\psi (x_0)\\in \\Delta ^{\\prime }$ and $p:=\\kappa |_{\\Delta ^{\\prime }} : (\\Delta ^{\\prime },t_0^{\\prime })\\rightarrow (\\Delta ,t_0)$ is a connected finite étale cover.",
"Consider the algebraic family $\\mathcal {F}^{\\prime }_{(C,D_C)} :=p^*\\mathcal {F}_{(C,D_C)}$ .",
"By definition of the pullback, its total space $X^{\\prime }$ is given by a fibered product $X^{\\prime }=\\lbrace (x,t^{\\prime })\\in X|_{\\kappa ^{-1}(\\Delta )}\\times \\Delta ^{\\prime } ~|~ \\kappa (x)=p(t^{\\prime })\\rbrace $ and we have $\\kappa ^{\\prime }:X^{\\prime }\\rightarrow \\Delta ^{\\prime }\\, ;~(x,t^{\\prime })\\mapsto t^{\\prime }\\, .$ On the other hand, $\\Delta ^{\\prime }$ is a subset of $X|_{\\kappa ^{-1}(\\Delta )}$ by construction and we can define a section $\\sigma $ of $\\kappa ^{\\prime }$ by $\\sigma :\\Delta ^{\\prime }\\rightarrow X^{\\prime }\\, ;~t^{\\prime }\\mapsto (t^{\\prime },t^{\\prime })\\, .$ Since moreover $\\Delta ^{\\prime }\\cap D=\\varnothing $ by the choice of $Z_2$ , we have $\\sigma (\\Delta ^{\\prime })\\cap D^{\\prime }=\\varnothing $ .",
"We conclude by noticing $\\sigma (t_0^{\\prime })=(\\psi (x_0),t_0^{\\prime })=\\psi ^{\\prime }(x_0)$ .",
"To fix notations, let us recall the definition of (inner) semi-direct products.",
"Let $G$ be a group and $A$ a subgroup.",
"Assume we have a group $\\widetilde{B}$ fitting into a split short exact sequence of groups, as follows.",
"$ {\\lbrace 1\\rbrace [r]& A [r] & G[r] & \\widetilde{B} @/_1pc/[l]_{\\sigma } [r]&\\lbrace 1\\rbrace \\,} $ Assume further that the map $A\\rightarrow G$ in that sequence is defined by the inclusion map.",
"Then $A$ is a normal subgroup of $G$ ; for $B:=\\sigma (\\widetilde{B})$ we have a natural morphism $\\eta \\in \\operatorname{Hom}(B, \\mathrm {Aut}(A))$ defined by $\\eta (b)(a)=b\\cdot a\\cdot b^{-1}$ for all $a\\in A\\, , b\\in B;$ we have a group $A\\rtimes _{\\eta }B$ defined as the set $A\\times B$ endowed with the group law $(a\\, ,b)\\cdot (a^{\\prime }\\, , b^{\\prime }) = (a\\cdot \\eta (b)(a^{\\prime }) \\, , b\\cdot b^{\\prime })\\, , $ and the natural morphism $A\\rtimes _{\\eta }B\\rightarrow G$ defined by $(a,b)\\mapsto a\\cdot b$ is bijective, allowing us to identify $G=A\\rtimes _{\\eta }B.$ Lemma 3.1.2 (Splitting) Let $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi )$ be an algebraic family as in Section REF .",
"Let $\\sigma : T\\rightarrow X$ be a section of $\\kappa $ such that $\\sigma (T)\\subset X^0=X\\setminus D$ .",
"Denote $C^0:=C\\setminus D_C$ and $x_0:=\\psi ^{-1}(\\sigma (t_0))$ .",
"Then $ \\pi _1(X^0,\\sigma (t_0)) = \\psi _*\\pi _1(C^0,x_0)\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0)\\, ,$ where for all $\\gamma \\in \\pi _1(C^0,x_0)$ and $ \\beta \\in \\pi _1(T,t_0)$ we have $\\begin{array}{rcl}\\eta (\\sigma _*\\beta ) (\\psi _*\\gamma ) &=& \\sigma _*\\beta \\cdot \\psi _*\\gamma \\cdot \\sigma _*\\beta ^{-1}\\, .",
"\\end{array}$ Since $\\sigma $ takes values in $X^0$ , we have a morphism of fundamental groups $\\sigma _* : \\pi _1(T,t_0)\\rightarrow \\pi _1(X^0,\\sigma (t_0))$ .",
"From the embedding of the central fiber, we get the morphism $\\psi _*: \\pi _1(C^0,x_0) \\rightarrow \\pi _1(X^0,\\sigma (t_0))\\,.$ Consider now the family of $n$ -punctured curves given by $\\kappa : X^0\\rightarrow T$ .",
"This family is a topologically locally trivial fibration and the fiber over $t_0$ identifies, via $\\psi $ , with $C^0$ .",
"Hence we have a long homotopy exact sequence $\\cdots \\longrightarrow \\pi _2(X^0 ,\\sigma (t_0))\\stackrel{\\kappa ^*}{\\longrightarrow } \\pi _2(T ,t_0) \\longrightarrow \\pi _1(C^0 ,\\sigma (t_0)) \\stackrel{\\psi ^*}{\\longrightarrow } \\pi _1(X^0 ,\\sigma (t_0))\\stackrel{\\kappa ^*}{\\longrightarrow } \\pi _1(T ,t_0)\\longrightarrow \\lbrace 1\\rbrace .",
"$ The maps ${\\sigma }_* : \\pi _j(T,t_0)\\rightarrow \\pi _j(X^0,\\tau (t_0))$ are sections for the corresponding $\\kappa _*$ and we may derive the following split short exact sequence: $ {\\textrm {~}\\hspace{19.91684pt} &\\lbrace 1\\rbrace [r]&\\pi _1(C^0,x_0) [r]^{\\hspace{-5.0pt}\\psi _*}& \\pi _1(X^0,\\tau (t_0)) [r]_-{\\kappa _*}& \\pi _1(T,t_0) @/^-1.2pc/[]!<-1ex,-2ex>;[l]!<+4ex,-1ex>_-{{\\sigma }_*} [r]&\\lbrace 1\\rbrace \\, .}",
"$ Given a decomposition (REF ), the monodromy representation of the flat connection underlying an isomonodromic deformation can be seen as an extension of the monodromy representation of the initial connection.",
"When does such an extension exist, and is it somehow unique?",
"Again we need a little group theory.",
"Lemma 3.1.3 (Extension of representations) Let $G=A\\rtimes _{\\eta }B$ be as before and let $\\rho _A \\in \\operatorname{Hom}(A, \\mathrm {GL}_r$ be a representation.",
"There exists a representation $\\rho \\in \\operatorname{Hom}(G, \\mathrm {GL}_r$ such that $\\rho |_A=\\rho _A$ if and only if there exists a representation $\\rho _B \\in \\operatorname{Hom}(B, \\mathrm {GL}_r$ such that for all $(a,b)\\in A\\times B$ we have $\\rho _A(b\\cdot a \\cdot b^{-1})=\\rho _B(b)\\cdot \\rho _A(a)\\cdot \\rho _B(b^{-1})\\, .",
"$ Let $\\rho , \\rho ^{\\prime } \\in \\operatorname{Hom}(G, \\mathrm {GL}_r$ be representations such that $\\rho |_A=\\rho ^{\\prime }|_A=\\rho _A$ .",
"Assume that $\\rho _A$ is irreducible.",
"Then there is $\\lambda \\in \\operatorname{Hom}(B,*)$ such that $\\rho =\\lambda \\otimes \\rho ^{\\prime }\\, .$ The proof of this Lemma is elementary and will be left to the reader.",
"A similar statement can be found in ."
],
[
"Logarithmic Riemann-Hilbert correspondence",
"Let us briefly recall some notions and results from , allowing to construct isomonodromic deformations from extensions of monodromy representations.",
"Denote by $\\mathbb {D}$ the unit disc around 0 in the complex line and denote by $\\mathbb {V}$ the trivial vector bundle of rank $r$ over $\\mathbb {D}$ .",
"Its sheaf of holomorphic sections shall be denoted by $\\mathcal {V}=\\oplus _{i=1}^r\\mathcal {O}_{\\mathbb {D}}$ .",
"A (logarithmic) transversal model is an analytic logarithmic connection $(\\mathbb {V}, \\xi )$ over $\\mathbb {D}$ with polar locus $\\lbrace 0\\rbrace $ .",
"It is called a mild transversal model if any automorphism of the locally constant sheaf $\\mathrm {ker}(\\xi |_{\\mathbb {D}\\setminus \\lbrace 0\\rbrace })$ is obtained by the restriction to $\\mathbb {D}\\setminus \\lbrace 0\\rbrace $ of an automorphism of the sheaf $\\mathcal {V}$ .",
"The isomorphism class of a transversal model is called a transversal type.",
"Accordingly, a mild transversal type is the transversal type of a mild transversal model.",
"Let $X$ be a $$ -manifold, and let $D\\subset X$ be a normal crossing hypersurface.",
"Denote $(D_i)_{i\\in I}$ the irreducible components of $D$ .",
"Let $\\rho \\in \\operatorname{Hom}(\\pi _1(X\\setminus D), \\mathrm {GL}_r$ be a representation and $\\mathcal {L}$ be a locally constant sheaf over $X\\setminus D$ with monodromy $\\rho $ .",
"For each $i\\in I$ , choose a holomorphic embedding $f_i : \\mathbb {D}\\hookrightarrow X$ such that $f_i (\\mathbb {D})$ intersects $D_i$ transversely exactly once, at $f_i(0)$ , a smooth point of $D$ .",
"We say that a transversal model $(\\mathbb {V}, \\xi _i)$ is compatible with $\\rho $ at $D_i$ if its monodromy is isomorphic to the one of $f_i^*\\mathcal {L}$ .",
"This is a well defined notion, independant of the choice of $f_i$ .",
"By isomorphism invariance, this adapts to a notion of compatible transversal type.",
"Compatible mild transversal models always exist, $e.g.$ one can choose non-resonant models.",
"Assume we have a flat $$ -logarithmic connection $\\nabla $ over $X$ , with polar locus in $D$ .",
"By , the transversal type defined by $f_i^*\\nabla $ is independant of the choice of $f_i$ , it depends only of $D_i$ and $\\nabla $ .",
"It is called the transversal type of $\\nabla $ at $D_i$.",
"The connection $\\nabla $ is said to be mild if for every component $D_i$ , the transversal type of $\\nabla $ at $D_i$ is mild.",
"Theorem 3.2.1 (Logarithmic Riemann-Hilbert) Let $X$ be a $\\omega $ -manifold, $D$ a normal crossing divisor on X and $\\rho : \\pi _1(X\\setminus D)\\rightarrow \\mathrm {GL}_r a representation.",
"For each $ iI$, let $ (V, i)$ be a mild transversal model compatible with $$.",
"Then up to isomorphism there is a unique flat $$-logarithmic connection $ (E, )$ over $ X$ with polar locus $ D$ such that\\begin{itemize}\\item the monodromy of (E, \\nabla ) is given by [\\rho ] and\\vspace{2.84544pt}\\item for each i\\in I, the transversal type of \\nabla at D_i is given by (V,\\xi _i);\\vspace{2.84544pt}\\item if ``= algebraic^{\\prime \\prime }, then (E, \\nabla ) is regular.\\end{itemize}$ The proof of this theorem in the analytic category can be found in .",
"We only need to check that it also holds in the algebraic category.",
"So assume now $X$ is a smooth irreducible quasiprojective variety.",
"By definition, there exists a smooth irreducible projective variety $\\widehat{X}$ containing $X$ as a Zariski open subset.",
"Denote by $\\widehat{D}_{j}, j\\in J$ , the irreducible components of $\\widehat{X} \\setminus X$ and by $\\widehat{D}_{i}$ the Zariski closure of $D_i$ in $\\widehat{X}$ for each $i\\in I$ .",
"By Hironaka's desingularization, we may suppose that $\\widehat{D}:=\\sum _{i\\in I\\cup J}\\widehat{D}$ is a normal crossing divisor.",
"Moreover, since $X\\setminus D=\\widehat{X}\\setminus \\widehat{D}$ , $\\rho $ defines $\\widehat{\\rho }=\\rho \\in \\operatorname{Hom}(\\pi _1(\\widehat{X}\\setminus \\widehat{D}), \\mathrm {GL}_r\\, .$ For each $j\\in J$ , choose an arbitrary mild transversal model $(V, \\xi _i)$ on $(\\mathbb {D},0)$ compatible with $\\widehat{\\rho }$ .",
"Then there exists an analytic logarithmic connection $(\\widehat{E}^\\mathrm {an},\\widehat{\\nabla }^\\mathrm {an})$ over $\\widehat{X}$ with polar divisor $\\widehat{D}$ by the analytic statement.",
"Since $\\widehat{X}$ is projective, this connection is however analytically isomorphic to an algebraic logarithmic connection $(\\widehat{E},\\widehat{\\nabla })$ on $\\widehat{X}$ by GAGA .",
"Since any logarithmic connection on $\\widehat{X}$ restricts to a regular connection on $X$ (see ), $(E,\\nabla ):=(\\widehat{E},\\widehat{\\nabla })|_X$ has the desired properties.",
"It remains to show uniqueness up to algebraic isomorphism.",
"By the analytic statement, we already know that the (analytic) isomorphism class of $(E^{\\mathrm {an}}, \\nabla ^{\\mathrm {an}})$ is unique.",
"Yet any analytic isomorphism between regular algebraic logarithmic connections $\\nabla _1, \\nabla _2$ over $X$ is algebraic, for the isomorphism can be seen as a horizontal section of $\\nabla _1\\otimes \\nabla _2^{\\vee }$ , which is regular by ."
],
[
"Proof of the universal property",
"Lemma REF , stated in Section REF , implying that under suitable generic conditions, algebraic universal isomonodromic deformations, if they exist, may be chosen to be regular is now an immediate consequence of the logarithmic Riemann-correspondence.",
"Moreover, we are now able to prove their universal property, also stated in Section REF .",
"Let $\\mathcal {I}_{(C, E_0, \\nabla _0)}=(\\mathcal {F}_{(C, D_C)}, E, \\nabla , \\Psi )$ with $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi )$ be an algebraic isomonodromic deformation of $(C, E_0, \\nabla _0)$ .",
"Let $\\rho \\in \\operatorname{Hom}(\\pi _1(X\\setminus D, x_0), \\mathrm {GL}_r$ be a representative of the monodromy $[\\rho ]$ of $(E, \\nabla )$ .",
"For $i\\in \\llbracket 1, n\\rrbracket $ , let $D_i$ be the component of $D$ passing through $\\psi (x_i)$ .",
"Since by assumption $(C, E_0, \\nabla _0)$ is mild, Theorem REF yields a regular algebraic connection $(E^{\\prime }, \\nabla ^{\\prime })$ over $X$ with polar divisor $D$ , monodromy $[\\rho ]$ and the same transversal types as $\\nabla $ at the components $(D_i)_{i\\in \\llbracket 1, n\\rrbracket }$ .",
"Moreover, also by Theorem REF , there is an isomorphism $\\widetilde{\\Psi } : (E, \\nabla )|_{X_{t_0}}\\stackrel{\\sim }{\\rightarrow }(E^{\\prime }, \\nabla ^{\\prime })|_{X_{t_0}}$ .",
"Then $\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}:=(\\mathcal {F}_{(C, D_C)}, E^{\\prime }, \\nabla ^{\\prime } ,\\widetilde{\\Psi }\\circ \\Psi )$ is a regular algebraic isomonodromic deformation of $(C, E_0, \\nabla _0)$ and there is an analytic isomorphism $(E^\\mathrm {an}, \\nabla ^\\mathrm {an})\\simeq ({E^{\\prime }}^\\mathrm {an}{\\nabla ^{\\prime }}^\\mathrm {an})$ .",
"In particular, the analytification of $\\mathcal {I}_{(C, E_0, \\nabla _0)}$ is isomorphic to the analytification of $\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}$ .",
"Let $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C, E_0, \\nabla _0)}=(\\mathcal {F}^{\\mathrm {Kur}}_{(C, D_C)}, E, \\nabla , \\Psi )$ be a regular algebraic universal isomonodromic deformation of $(C, E_0, \\nabla _0)$ with parameter space $(T,t_0)$ and let $\\mathcal {I}^{\\prime }_{(C, E_0, \\nabla _0)}=(\\mathcal {F}_{(C, D_C)}^{\\prime }, E^{\\prime }, \\nabla ^{\\prime } , \\Psi ^{\\prime })$ be a regular algebraic isomonodromic deformation of $(C, E_0, \\nabla _0)$ with parameter space $(T^{\\prime },t_0^{\\prime })$ .",
"By Lemma REF , there is an étale base change $\\tilde{p} : (\\widetilde{T} , \\tilde{t}_0) \\rightarrow (T, t_0)$ , such that for $\\widetilde{\\mathcal {F}}^{\\mathrm {Kur}}_{(C, D_C)} :=\\tilde{p}^*\\mathcal {F}^{\\mathrm {Kur}}_{(C, D_C)}, $ there is a section $\\sigma : \\widetilde{T}\\rightarrow \\widetilde{X}$ avoiding the marked points.",
"Since $\\widetilde{\\mathcal {F}}^{\\mathrm {Kur}}_{(C, D_C)}$ is still Kuranishi, by the universal property of Kuranishi families, we have an étale base change $ {p} : (T^{\\prime \\prime }, t_0^{\\prime \\prime })\\rightarrow (T^{\\prime } , {t}_0^{\\prime })$ , a morphism $\\tilde{q} : (T^{\\prime \\prime },t_0^{\\prime \\prime })\\rightarrow (\\widetilde{T}, \\tilde{t}_0)$ and an isomorphism $\\tilde{\\mathfrak {f}} : \\mathcal {F}^{\\prime \\prime }_{(C,D_C)}:={p}^* \\mathcal {F}^{\\prime }_{(C,D_C)} \\stackrel{\\sim }{\\longrightarrow } \\tilde{q}^*\\widetilde{\\mathcal {F}}^{\\mathrm {Kur}}_{(C,D_C)}\\, .$ In particular, $\\sigma $ lifts to a section $\\sigma ^{\\prime \\prime }:=\\tilde{f}^*\\tilde{q}^*\\sigma : T^{\\prime \\prime }\\rightarrow X^{\\prime \\prime }$ avoiding the marked points of $\\mathcal {F}_{(C, D_C)}^{\\prime \\prime }$ .",
"Denote by $\\rho ^{\\prime \\prime }\\, , \\tilde{\\rho } \\, \\in \\, \\operatorname{Hom}(\\pi _1(X^{\\prime \\prime }\\setminus D^{\\prime \\prime }\\, , \\sigma ^{\\prime \\prime }(t_0^{\\prime \\prime })), \\mathrm {GL}_r$ representatives of the conjugacy classes of the monodromy representations of $(E^{\\prime \\prime }; \\nabla ^{\\prime \\prime })$ and $\\tilde{f}^*\\tilde{q}^* \\tilde{p}^*(E,\\nabla ) $ respectively (with respect to the identity).",
"By the Splitting Lemma REF , we have $\\pi _1(X^{\\prime \\prime }\\setminus D^{\\prime \\prime } , \\sigma ^{\\prime \\prime }(t_0^{\\prime \\prime }))= \\psi ^{\\prime \\prime }_* \\pi _1(C\\setminus D_C, x_0) \\rtimes _{\\eta } \\sigma ^{\\prime \\prime }_*\\pi _1(T^{\\prime \\prime },t_0^{\\prime \\prime })\\, .$ Moreover, if $\\rho _{\\nabla _0}$ denotes a representative of the monodromy representation of $(E_0, \\nabla _0)$ (with respect to the identity), then $\\rho ^{\\prime \\prime }$ and $\\tilde{\\rho }$ could be chosen so that $\\rho ^{\\prime \\prime }|_{ \\psi ^{\\prime \\prime }_* \\pi _1(C\\setminus D_C, x_0) }= \\tilde{\\rho }|_{ \\psi ^{\\prime \\prime }_* \\pi _1(C\\setminus D_C, x_0) }=\\psi ^{\\prime \\prime }_*\\rho _{\\nabla _0}.$ Since $\\rho _{\\nabla _0}$ is irreducible, by Lemma REF there is a representation $\\lambda \\in \\operatorname{Hom}(\\pi _1(T^{\\prime \\prime }\\, ,\\, t_0^{\\prime \\prime })\\, , *)$ such that $\\lambda \\otimes (\\sigma ^{\\prime \\prime })^*\\rho ^{\\prime \\prime } = (\\sigma ^{\\prime \\prime })^*\\tilde{\\rho }$ .",
"By the Riemann-Hilbert correspondence, there is a regular flat algebraic connection $(L, \\xi )$ of rank 1 over $T^{\\prime \\prime }$ , without poles, whose monodromy representation is $\\lambda ^{-1}$ .",
"The monodromy representation of its lift ${\\kappa ^{\\prime \\prime }}^*(L, \\xi )$ is the trivial extension of ${\\sigma ^{\\prime \\prime }}^*\\lambda ^{-1}$ to a representation $\\psi ^{\\prime \\prime }_* \\pi _1(C\\setminus D_C, x_0) \\rtimes _{\\eta } \\sigma ^{\\prime \\prime }_*\\pi _1(T^{\\prime \\prime },t_0^{\\prime \\prime })\\, \\rightarrow *$ .",
"Now the monodromy representations of $(E^{\\prime \\prime }; \\nabla ^{\\prime \\prime })\\otimes {\\kappa ^{\\prime \\prime }}^*(L, \\xi )$ and $\\tilde{f}^*q^*p^*(E,\\nabla ) $ coincide.",
"Both connections are regular, have same monodromy representations and same transversal models, given by $(E_0, \\nabla _0)$ .",
"Hence they are isomorphic by the logarithmic Riemann-Hilbert correspondence."
],
[
"The monodromy of the monodromy",
"In this section, we introduce the so-called group of mapping classes of a $$ -family, which is the image of a canonical morphism from the fundamental group of the parameter space of the family to the fundamental group of the central fiber.",
"For an isomonodromic deformation, the action on the monodromy representation of the initial connection by the group of mapping classes of the underlying family of curves corresponds to the monodromy of the monodromy representation.",
"Under suitable conditions, this group can be canonically translated into a subgroup of $\\Gamma _{g,n}$ ."
],
[
"Mapping classes of the central fiber",
"As usual, let $(C,D_C)$ be a stable $n$ -pointed genus-$g$ curve.",
"Let $\\mathcal {F}_{(C,D_C)}$ be a $$ -family with parameter space $(T,t_0)$ .",
"Let $\\beta : [0,1]\\rightarrow T$ be a closed path with end point $t_0$ , i.e.",
"a continous map such that $\\beta (0)=\\beta (1)=t_0$ .",
"By , the pullback bundle $\\beta ^*(X,D) \\rightarrow [0,1]$ possesses a topological trivialization $\\Phi :(C, D_C)\\times [0,1] \\stackrel{\\sim }{\\rightarrow }\\beta ^*(X,D)$ .",
"For $s\\in [0,1]$ , we denote $\\Phi _s:=\\Phi |_{(C, D_C)\\times \\lbrace s\\rbrace } \\, $ and deduce a homeomorphism from the central fiber seen over $\\lbrace 1\\rbrace $ to the central fiber seen over $\\lbrace 0\\rbrace $ given by $\\psi ^{-1}\\circ \\Phi _0\\circ \\Phi _1^{-1} \\circ \\psi : (C, D_C) \\stackrel{\\sim }{\\rightarrow } (C, D_C)\\, .$ Its isotopy class shall be called the mapping class associated to $\\beta $ and $\\mathcal {F}_{(C,D_C)}$ and denoted $\\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )\\, .$ Lemma 4.1.1 The mapping class $\\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )$ is well defined, i.e.",
"it does not depend on the choice of a trivialization $\\Phi $ .",
"Moreover, $\\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )$ only depends on the homotopy class of $\\beta $ .",
"For fixed $\\beta $ , take two trivializations : $\\Phi ,\\widetilde{\\Phi }: (C, D_C)\\times [0,1] \\stackrel{\\sim }{\\rightarrow }\\beta ^*(X,D)$ .",
"The family $\\widetilde{\\Phi }_0 \\circ \\widetilde{\\Phi }_s^{-1}\\circ \\Phi _s\\circ \\Phi _1^{-1}$ gives an isotopy from $\\Phi _0\\circ \\Phi _1^{-1}$ to $\\widetilde{\\Phi }_0\\circ \\widetilde{\\Phi }_1^{-1}$ .",
"Consider now two paths $\\beta _1$ and $\\beta _2$ that are homotopic relative to their endpoints.",
"By definition, there exists a continuous map $\\theta : \\overline{ \\rightarrow T, where \\overline{ denotes the closed unit disc, such that \\beta _2(s)=\\theta (\\mathrm {e}^{i\\pi (1+s)}) and \\beta _1(s)=\\theta (\\mathrm {e}^{i\\pi (1-s)}).",
"Since \\overline{ is contractible, by \\cite [Cor.",
"10.3]{MR1249482}, there is a trivialization \\Phi of \\theta ^*(X,D).",
"It induces trivializations \\Phi ^i of \\beta _i^*X for i=1,2.",
"Since they are both induced by \\Phi , we have \\Phi ^1_0=\\Phi ^2_0=\\Phi _{-1} and \\Phi ^1_1=\\Phi ^2_1=\\Phi _{1}.",
"}}\\begin{prop} Let\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi ) be a -family as in Section \\ref {SecSetup}.",
"Assume that none of the fibers (X_t, D|_{X_t}) has exceptional automorphisms.",
"Let \\mathbf {x} be a labelling of D_C and denote\\mathrm {cl}: T\\rightarrow \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n} the corestriction of the induced classifying map \\mathrm {class}(\\mathcal {F}) (see Section \\ref {Sec univ fam}).",
"Then there exists an orientation preserving homeomorphism \\varphi : (\\Sigma _g,y^n)\\stackrel{\\sim }{\\rightarrow } (C,\\mathbf {x}) such that for all \\beta \\in \\pi _1(T,t_0), the following equation holds in \\Gamma _{g,n}/K_{g,n}:\\varphi ^{-1}\\circ \\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )\\circ \\varphi =\\mathrm {taut}_{g,n}(\\mathrm {cl}_*\\beta ) \\, ,where \\mathrm {taut}_{g,n} is the tautological morphism\\mathrm {taut}_{g,n}:\\pi _1( \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}, \\star )\\rightarrow \\Gamma _{g,n}/K_{g,n} (see (\\ref {def phi})) and \\star :=[C,\\mathbf {x}].\\end{prop}}{\\begin{xmlelement*}{proof}Fix an orientation preserving homeomorphism \\widetilde{\\varphi } : (\\Sigma _g,y^n)\\stackrel{\\sim }{\\rightarrow } (C, \\mathbf {x}) and denote \\hat{\\star }:=[C, D_C,\\widetilde{\\varphi }]\\in \\mathcal {T}_{g,n}\\, .\\end{xmlelement*}}$ As usual, denote $\\mathcal {F}^{+}_{g,n}=(\\mathcal {F}_{g,n}, \\Phi _{g,n})$ the universal Teichmüller curve $\\mathcal {F}_{g,n}=(\\kappa _{g,n} : \\mathcal {X}\\rightarrow \\mathcal {T}_{g,n}, \\mathcal {D})$ endowed with the Teichmüller structure $ \\Phi _{g,n}: (\\Sigma _g,Y^n)\\times \\mathcal {T}_{g,n}\\stackrel{\\sim }{\\rightarrow } (\\mathcal {X}, \\mathcal {D})$ .",
"For any point $t \\in \\mathcal {T}_{g,n}$ , we shall denote $\\Phi ^{g,n}_{t}:=\\Phi _{g,n}|_{(\\Sigma _g,Y^n)\\times \\lbrace t \\rbrace } : (\\Sigma _g,Y^n)\\times \\lbrace t \\rbrace \\stackrel{\\sim }{\\rightarrow } (\\mathcal {X}_{t}, \\mathcal {D}|_{\\mathcal {X}_{t }})\\, .$ Let $p : (\\widetilde{T}, \\tilde{t}_0) \\rightarrow (T,t_0)$ be a universal cover and consider the pulled-back family $\\widetilde{\\mathcal {F}}=(\\widetilde{\\kappa } : \\widetilde{X}\\rightarrow \\widetilde{T},\\widetilde{D}):=p^*(\\kappa : X\\rightarrow T,D)\\, .$ Now for any contractible analytic submanifold $\\widetilde{\\Delta } \\subset \\widetilde{T}$ containing $\\tilde{t}_0$ , there is a trivialization $\\widetilde{\\Phi }: (C,D_C)\\times \\widetilde{\\Delta }\\stackrel{\\sim }{\\longrightarrow } (\\widetilde{X} , \\widetilde{D})|_{\\widetilde{\\Delta }}$ of $\\widetilde{\\mathcal {F}}|_{\\widetilde{\\Delta }}$ , unique up to isotopy, such that $\\widetilde{\\Phi }_{\\tilde{t}_0}=\\psi $ with respect to the pullback identification $(\\widetilde{X}_{\\tilde{t}_0}, \\widetilde{D}|_{\\widetilde{X}_{\\tilde{t}_0}}){=}(X_{t_0},D|_{X_{t_0}})$ .",
"Setting $\\widetilde{\\mathcal {F}}^+:=(\\widetilde{\\mathcal {F}}|_{\\widetilde{\\Delta }}, \\widetilde{\\Phi }\\circ (\\widetilde{\\varphi }\\times \\mathrm {id}))$ defines an analytic family with Teichmüller structure.",
"By the universal property of the Teichmüller curve, we have an isomorphism $\\mathfrak {f}=(\\mathrm {id}_{\\widetilde{\\Delta }},\\mathfrak {f}^a, \\mathfrak {f}^{\\mathrm {top}})$ fitting into the following commutative diagram.",
"$ {(\\Sigma _g, Y^n)\\times \\widetilde{\\Delta } [rr]^{\\mathfrak {f}^{\\mathrm {top}}}_{\\sim }[d]_{\\widetilde{\\Phi }\\circ (\\widetilde{\\varphi }\\times \\mathrm {id})}&& (\\Sigma _g, Y^n)\\times \\widetilde{\\Delta }[d]^{{\\mathrm {class}^+}(\\widetilde{\\mathcal {F}}^+)^*\\Phi _{g,n}}\\\\(\\widetilde{X}, \\widetilde{D})|_{\\widetilde{\\Delta }}[rr]^{\\mathfrak {f}^a}_{\\sim }[d]_{\\widetilde{\\kappa }}&& {\\mathrm {class}^+}(\\widetilde{\\mathcal {F}}^+)^*(\\mathcal {X}, \\mathcal {D})[d]^{{\\mathrm {class}^+}(\\widetilde{\\mathcal {F}}^+)^*\\kappa _{g,n}}\\\\\\widetilde{\\Delta } @{=}[rr]&&\\widetilde{\\Delta }} $ By commutativity of the diagram, we have $\\mathfrak {f}^{\\mathrm {top}}=\\mathfrak {f}^{\\mathrm {top}}_{\\tilde{t}_0}\\times \\mathrm {id}_{\\widetilde{\\Delta }}$ up to fiberwise isotopy.",
"Define $\\varphi := \\widetilde{\\varphi }\\circ \\left(\\mathfrak {f}^{\\mathrm {top}}_{\\tilde{t}_0}\\right)^{-1} \\, .$ Now let $[\\beta ]\\in \\pi _1(T,t_0)\\setminus \\lbrace 1\\rbrace $ and consider $\\tilde{\\beta }: [0,1]\\rightarrow \\widetilde{T}$ , the lift of $\\beta $ with starting point $\\tilde{t}_0$ .",
"If the representative $\\beta $ of the homotopy class $[\\beta ]$ is well chosen, then $\\widetilde{\\beta }$ is a $\\mathcal {C}^{\\infty }$ -embedding.",
"By existence of tubular neighborhoods, there is a contractible neighborhood $\\widetilde{\\Delta }$ of $\\tilde{t}_0$ as above, containing $\\widetilde{\\beta }$ .",
"We claim that, up to isotopy, $ \\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )= \\widetilde{\\Phi }_{\\widetilde{\\beta }(1)}^{-1}\\circ \\widetilde{\\Phi }_{\\tilde{t}_0}\\, .$ Indeed, we have $\\beta ^*(X,D)=(p\\circ \\widetilde{\\beta })^*(X,D)=\\widetilde{\\beta }^*p^*(X,D)=\\widetilde{\\beta }^*(\\widetilde{X},\\widetilde{D})$ .",
"Since moreover $\\widetilde{\\beta }$ is an embedding, we have $\\beta ^*(X,D)=(\\widetilde{X},\\widetilde{D})|_{\\widetilde{\\beta }([0,1])}.$ The claim then follows from the fact that $ \\psi ^{-1}\\circ \\widetilde{\\Phi }_{\\tilde{t}_0}$ is the identity and from the definition of the mapping class.",
"Denote $\\widehat{\\beta }:={\\mathrm {class}^+}(\\widetilde{\\mathcal {F}}^+)_*\\widetilde{\\beta }$ , which is a path in $\\mathcal {T}_{g,n}$ with starting point $\\hat{\\star }$ .",
"By our definitions, the black part of the following diagram is commutative.",
"$ {(\\Sigma _g, Y^n)\\times \\lbrace \\tilde{t}_0\\rbrace [rr]^{\\widetilde{\\Phi }\\circ (\\widetilde{\\varphi }\\times \\lbrace \\tilde{t}_0\\rbrace )}_{\\sim }[d]_{\\mathfrak {f}^{\\mathrm {top}}_{\\tilde{t}_0}}^{\\hspace{3.0pt}\\text{\\begin{rotate}{90}\\sim \\end{rotate}}}&&(\\widetilde{X}_{\\tilde{t}_0}, \\widetilde{D}|_{\\widetilde{X}_{\\tilde{t}_0}})@{=}[r] [d]_{\\mathfrak {f}^{a}_{\\tilde{t}_0}}^{\\hspace{3.0pt}\\text{\\begin{rotate}{90}\\sim \\end{rotate}}}&(\\widetilde{X}_{\\widetilde{\\beta }(1)}, \\widetilde{D}|_{\\widetilde{X}_{\\widetilde{\\beta }(1)}}) [d]^{\\mathfrak {f}^{a}_{\\widetilde{\\beta }(1)}}_{\\hspace{3.0pt}\\text{\\begin{rotate}{90}\\sim \\end{rotate}}}&& (\\Sigma _g, Y^n)\\times \\lbrace \\widetilde{\\beta }(1)\\rbrace [ll]_{\\widetilde{\\Phi }\\circ (\\widetilde{\\varphi }\\times \\lbrace \\widetilde{\\beta }(1)\\rbrace )}^{\\sim } [d]^{\\mathfrak {f}^{\\mathrm {top}}_{\\widetilde{\\beta }(1)}}_{\\hspace{1.0pt}\\text{\\begin{rotate}{90}\\sim \\end{rotate}}}\\\\(\\Sigma _g, Y^n)\\times \\lbrace \\hat{\\star }\\rbrace [rr]_{\\Phi ^{g,n}_{\\hat{\\star }}}^{\\sim }&& (\\mathcal {X}_{\\hat{\\star }}, \\mathcal {D}|_{\\mathcal {X}_{\\hat{\\star }}}) @[gray][r]^{\\color {gray} \\hspace{-15.0pt}\\sim }_{\\color {gray} \\hspace{-15.0pt}\\widehat{\\psi }} & (\\mathcal {X}_{\\widehat{\\beta }(1)}, \\mathcal {D}|_{\\mathcal {X}_{\\widehat{\\beta }(1)}})&&(\\Sigma _g, Y^n)\\times \\lbrace \\widehat{\\beta }(1)\\rbrace [ll]^{\\Phi ^{g,n}_{\\widehat{\\beta }(1)}}_{\\sim }} $ We define $\\widehat{\\psi }=\\mathfrak {f}^a_{\\tilde{\\beta }(1)}\\circ (\\mathfrak {f}^a_{t_0})^{-1},$ so that adding the gray arrow maintains this commutativity.",
"Since $\\mathfrak {f}^{\\mathrm {top}}_{\\widetilde{\\beta }(1)}$ is isotopic to $\\mathfrak {f}^{\\mathrm {top}}_{\\tilde{t}_0}$ , the following equations hold up to isotopy: $\\left\\lbrace \\begin{array}{ccl}\\Phi ^{g,n}_{\\hat{\\star }} &=&\\mathfrak {f}^a_{\\tilde{t}_0}\\circ \\widetilde{\\Phi }_{\\tilde{t}_0}\\circ \\varphi \\vspace{2.84544pt}\\\\\\Phi ^{g,n}_{\\widehat{\\beta }(1)}&=&\\mathfrak {f}^a_{\\widetilde{\\beta }(1)}\\circ \\widetilde{\\Phi }_{\\widetilde{\\beta }(1)} \\circ \\varphi .",
"\\vspace{2.84544pt}\\\\\\end{array}\\right.$ On the other hand, $\\mathrm {cl}_*\\beta $ is a closed path in $\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ with end point $\\star $ .",
"By construction, it lifts, with respect to the forgetful map $\\pi _{g,n}$ , to $\\widehat{\\beta }$ , with $\\widehat{\\beta }(0)=\\hat{\\star }$ .",
"By definition of the tautological morphism $\\mathrm {taut}_{g,n}$ , we thus have, for $[h]:= \\mathrm {taut}_{g,n}(\\mathrm {cl}_*\\beta ) \\in \\Gamma _{g,n}/K_{g,n}$ : $[h]\\cdot \\left[\\mathcal {X}_{\\hat{\\star }}, \\mathcal {D}|_{\\mathcal {X}_{\\hat{\\star }}}, \\Phi ^{g,n}_{\\hat{\\star }}\\right]=\\left[\\mathcal {X}_{\\widehat{\\beta }(1)}, \\mathcal {D}|_{\\mathcal {X}_{\\widehat{\\beta }(1)}}, \\Phi ^{g,n}_{\\widehat{\\beta }(1)} \\right]\\, .$ Note that $\\widehat{\\psi }$ induces an isomorphism of pointed curves with Teichmüller structure.",
"By the definition of the action of the mapping class group on $\\mathcal {T}_{g,n}$ , we now have $[h]\\cdot \\left[\\mathcal {X}_{\\hat{\\star }}, \\mathcal {D}|_{\\mathcal {X}_{\\hat{\\star }}}, \\Phi ^{g,n}_{\\hat{\\star }}\\right] = [h]\\cdot \\left[\\mathcal {X}_{\\widehat{\\beta }(1)}, \\mathcal {D}|_{\\mathcal {X}_{\\widehat{\\beta }(1)}},\\widehat{\\psi }\\circ \\Phi ^{g,n}_{\\hat{\\star }} \\right]= \\left[\\mathcal {X}_{\\widehat{\\beta }(1)}, \\mathcal {D}|_{\\mathcal {X}_{\\widehat{\\beta }(1)}},\\widehat{\\psi }\\circ \\Phi ^{g,n}_{\\hat{\\star }} \\circ h^{-1}\\right]\\, .$ Hence there is an element $[k]\\in K_{g,n}$ such that, up to isotopy, $\\widehat{\\psi }\\circ \\Phi ^{g,n}_{\\hat{\\star }} = \\Phi ^{g,n}_{\\widehat{\\beta }(1)} \\circ h\\circ k\\, .$ Combined with (REF ) and (REF ), this implies, up to isotopy, $ \\widetilde{\\Phi }_{\\widetilde{\\beta }(1)}^{-1} \\circ \\widetilde{\\Phi }_{\\tilde{t}_0} = \\varphi \\circ h \\circ k\\circ \\varphi ^{-1}\\, ,$ which by (REF ) and the definitions of $h$ and $k$ yields the desired result.",
"Splitting and the mapping class group Let $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi )$ be a $$ -family of $n$ -pointed genus-$g$ curves as in Section REF .",
"Assume there is a section $\\sigma : T\\rightarrow X^0:=X\\setminus D$ of $\\kappa $ .",
"Then we can define a $$ -family of $n+1$ -pointed genus-$g$ curves $\\mathcal {F}^{\\bullet }_{(C,D_C^\\bullet )}:=(\\kappa : X\\rightarrow T,D^\\bullet , t_0, \\psi )$ by setting $D^\\bullet :=D+\\sigma (T)$ and $D_C^\\bullet :=D_C+x_0$ , where $x_0:=\\psi ^{-1}(\\sigma (t_0))$ .",
"To a labelling $\\mathbf {x} =(x_1, \\ldots , x_n)$ of $D_C$ we can associate a labelling $\\mathbf {x}^\\bullet :=(x_1, \\ldots , x_n,x_0)$ of $D_C^\\bullet $ .",
"Note that if a fiber of $\\mathcal {F}^\\bullet $ has exceptional automorphisms, then the corresponding fiber of $\\mathcal {F}$ also has exceptional automorphisms.",
"Proposition 4.2.1 Let $(C,D_C)$ be a stable $n$ -pointed genus-$g$ curve.",
"Let $\\mathcal {F}^{\\bullet }_{(C,D_C^\\bullet )}=(\\kappa : X\\rightarrow T,D^\\bullet , t_0, \\psi )$ be a $$ -family of $n+1$ -pointed genus-$g$ curves as above.",
"Assume that none of the fibers of $\\mathcal {F}^\\bullet $ has exceptional automorphisms.",
"Let $\\mathbf {x}^\\bullet $ be a labelling of $D_C^\\bullet $ as above.",
"Denote $\\mathrm {cl}^{\\bullet }: T\\rightarrow \\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}$ the corestriction of the induced classifying map $\\mathrm {class}(\\mathcal {F}^\\bullet )$ .",
"For a suitable homeomorphism $\\varphi : (\\Sigma _g,y^n, y_0)\\stackrel{\\sim }{\\rightarrow } (C,\\mathbf {x}, x_0)$ , we have $ \\pi _1(X^0,\\sigma (t_0)) =(\\psi \\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0)\\, ,$ where for all $\\alpha \\in \\Lambda _{g,n}$ and $ \\beta \\in \\pi _1(T,t_0)$ , we have $\\begin{array}{rcl}\\eta (\\sigma _*\\beta ) ((\\psi \\circ \\varphi )_*\\alpha ) &=& \\sigma _*\\beta \\cdot (\\psi \\circ \\varphi )_*\\alpha \\cdot \\sigma _*\\beta ^{-1}\\vspace{2.84544pt}\\\\&=& (\\psi \\circ \\varphi )_* \\, \\mathfrak {a}\\left( \\mathrm {taut}_{g,n+1}({\\mathrm {cl}^{\\bullet }}_*\\, \\beta )\\right)(\\alpha ) \\, , \\end{array}$ where, as we recall from the introduction, $\\mathfrak {a}(h)(\\alpha )=h_*\\alpha $ for all $h\\in \\Gamma _{g,n+1}$ and $\\alpha \\in \\Lambda _{g,n}$ .",
"Since $2g-2+n>0$ by assumption, we have $K_{g,n+1}=\\lbrace 1\\rbrace $ according to Lemma REF .",
"Then by Proposition , for a convenient choice of $\\varphi $ , the following equation holds in $\\Gamma _{g,n+1}$ for every $\\beta \\in \\pi _1(T,t_0)$ : $\\varphi ^{-1}\\circ \\mathrm {map}_{\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}}(\\beta )\\circ \\varphi =\\mathrm {taut}_{g,n+1}({\\mathrm {cl}^\\bullet }_*\\beta )\\, .$ Denote $C^0:=C\\setminus D_C$ .",
"We claim that for any $\\gamma \\in \\pi _1(C^0,x_0) $ and any $\\beta \\in \\pi _1(T,t_0)$ , the following equation holds in $\\pi _1(X^0,\\sigma (t_0))$ : $ \\psi _*\\mathrm {map}_{\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}}(\\beta )_*\\gamma = \\sigma _* \\beta \\cdot \\psi _*\\gamma \\cdot \\sigma _* \\beta ^{-1} \\, .$ Indeed, let $\\gamma : [0,1]\\rightarrow C^0$ be a closed path with end point $x_0$ .",
"For any $s_0\\in [0,1]$ , we have a closed path $\\gamma _{s_0}:=\\gamma \\times \\lbrace s_0\\rbrace $ in the product space $C^0\\times [0,1]$ .",
"We also have a path $\\theta : [0,1] \\rightarrow C^0\\times [0,1]\\, ; \\, s\\mapsto (x_0,s)$ .",
"The path $\\theta \\cdot \\gamma _1 \\cdot \\theta ^{-1}$ is closed and homotopic to $\\gamma _0$ .",
"Now let $\\beta \\in \\pi _1(T,t_0)$ and let $\\Phi : (C^0, x_0)\\times [0,1] \\stackrel{\\sim }{\\rightarrow } \\beta ^*(X^0, \\sigma (T))$ be a trivialization commuting with the natural projections to $[0,1]$ .",
"Define the homeomorphism $\\widetilde{\\Phi }:=\\Phi \\circ ((\\Phi _{1}^{-1}\\circ \\psi ) \\times \\mathrm {id}_{[0,1]}): (C^0,x_0)\\times [0,1] \\stackrel{\\sim }{\\rightarrow } \\beta ^*(X^0, \\sigma (T))\\, , $ which is another trivialization, satisfying $\\widetilde{\\Phi }_1=\\psi $ and $ \\widetilde{\\Phi }_0=\\psi _*\\mathrm {map}_{\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}}(\\beta ).$ Since $\\widetilde{\\Phi }$ is continuous, the closed paths $\\widetilde{\\Phi }_*\\gamma _0$ and $\\widetilde{\\Phi }_*\\theta \\cdot \\widetilde{\\Phi }_*\\gamma _1 \\cdot \\widetilde{\\Phi }_*\\theta ^{-1}$ are homotopic in $ \\beta ^*(X^0, \\sigma (T))$ .",
"Considering the natural projection $\\kappa : \\beta ^*(X^0, \\sigma (T)) \\rightarrow (X^0, \\sigma (T))$ , we have $\\kappa _*\\widetilde{\\Phi }_{*}\\gamma _0=\\widetilde{\\Phi }_{0*}\\gamma $ and $\\kappa _*\\widetilde{\\Phi }_*\\gamma _1=\\widetilde{\\Phi }_{1*}\\gamma $ .",
"Since moreover $\\kappa _*\\widetilde{\\Phi }_*\\theta =\\sigma _*\\beta $ , we have (REF ).",
"Since $\\varphi $ is a homeomorphism, the induced map $\\varphi _*:\\Lambda _{g,n}\\rightarrow \\pi _1(C^0,x_0)$ is an isomorphism.",
"The statement then follows from (REF ), (REF ) and the Splitting Lemma REF .",
"Necessary and sufficient conditions for algebraizability We shall see in Section REF that Theorem REF is a corollary of the juxtaposition of Theorem A1, showing that our algebraizability criterion for germs of universal isomonodromic deformations is necessary, and Theorem A2, showing that it is also sufficient.",
"We have already established the main ingredients for the proofs of both theorems.",
"For Theorem A2, we moreover need a representation-theoretical result developed in Section REF .",
"Extensions of representations We shall now consider the problem of extending a representation of the fundamental group of a fiber of a family of pointed curves to a representation for the whole family in light of Lemma REF and Proposition REF .",
"We begin with the elementary case of non semisimple rank 2 representations.",
"Let $A,B$ be groups.",
"Consider a representation $\\rho \\in \\operatorname{Hom}(A ,\\mathrm {Upp})$ , where $\\mathrm {Upp}$ is the group of invertible upper triangular matrices of rank 2.",
"To such a representation, we may associate two other ones : the scalar part $\\rho _{*} :\\alpha \\mapsto \\rho (\\alpha )_{2,2}$ and the affine part $\\rho _{\\mathrm {Aff}}:=\\rho _{*}^{-1} \\otimes \\rho $ .",
"The latter takes values in $\\mathrm {Aff}(:=\\lbrace (a_{i,j})\\in \\mathrm {Upp}~|~a_{2,2}=1\\rbrace $ which is isomorphic to the affine group of the complex line.",
"Lemma 5.1.1 Let $\\rho =\\rho _{*}\\otimes \\rho _{\\mathrm {Aff}}$ and $\\rho ^{\\prime } =\\rho ^{\\prime }_{*}\\otimes \\rho ^{\\prime }_{\\mathrm {Aff}}$ as above, and assume that they are not semisimple.",
"We have $[\\rho ]=[\\rho ^{\\prime }] \\in \\operatorname{Hom}(A , \\mathrm {GL}_{2}/\\mathrm {GL}_{2} if and only if $ *='*$ and$ [Aff]=['Aff] Hom(A , Aff(C))/Aff(C)$.$ The ”if”-part is trivial.",
"Assume $[\\rho ]=[\\rho ^{\\prime }]$ .",
"Since they take values in $\\mathrm {Upp}$ , both representations $\\rho $ and $\\rho ^{\\prime }$ leave the line $\\mathrm {span}(e_1)$ of 2 invariant.",
"By non semisimplicity, for each of the representations, there is no other globally invariant line.",
"Let $M=(m_{i,j})\\in \\mathrm {GL}_2 conjugate both representations.",
"Then $ M$ must leave $ span(e1)$ invariant, \\textit {i.e.}",
"$ MUpp$.",
"As the scalars are central in $ GL2, the element $M/m_{2,2}\\in \\mathrm {Aff}(\\mathbb {C})$ conjugates both representations.",
"In particular $\\rho _{*}=\\rho ^{\\prime }_{*}$ and $M/m_{2,2}$ conjugates $\\rho _{\\mathrm {Aff}}$ and $\\rho ^{\\prime }_{\\mathrm {Aff}}$ .",
"Lemma 5.1.2 Let $\\rho _A \\in \\operatorname{Hom}(A,\\mathrm {GL}_{2}$ be non semisimple.",
"Let $\\theta \\in \\operatorname{Hom}(B, \\mathrm {Aut}(A))$ such that for all $h\\in \\mathrm {Im}(\\theta )$ , we have $[\\rho _A]=h\\cdot [\\rho _A]:=[\\rho _A\\circ h^{-1}]$ .",
"Then there exists a representation $\\rho _B \\in \\operatorname{Hom}(B,\\mathrm {GL}_{2}$ such that $\\rho _A(\\theta (\\beta )^{-1}(\\alpha ))=\\rho _B(\\beta )^{-1} \\rho _A(\\alpha ) \\rho _B(\\beta ) \\quad \\forall \\alpha \\in A\\, , \\beta \\in B\\, .$ We may assume that $\\rho _A$ takes values in $\\mathrm {Upp}$ .",
"By assumption, for each $\\beta \\in B$ , there exists a matrix $M_\\beta \\in \\mathrm {GL}_{2} such that $ A(()-1())=M-1 A()M$.",
"By Lemma \\ref {lemred}, we may assume $ MAff(C)$.",
"If $ Im(A)Upp$ is non abelian, then it has trivial centralizer and the matrices $ MAff(C)$ are uniquely defined.",
"Otherwise, we have $ Im(A){ ( 1 01) | * , }$ and the matrices $ M$ are uniquely defined if we impose$ M{ ( 0 01) | *}$.",
"It is now straightforward to check that given these choices, the well-defined map $ M$ is a morphism of groups.",
"$ For a similar result for semisimple representations $\\rho _A$ (of arbitrary rank), the group $B$ , which in our case will be the fundamental group of a parameter space, might have to be enlarged, in order to take into account the non-unicity of the matrices $M_\\beta $ due to possible permutations of irreducible components.",
"Proposition 5.1.3 Let $\\rho _A \\in \\operatorname{Hom}(A, \\mathrm {GL}_r$ be semisimple.",
"Let $ (T,t_0)$ be a smooth connected quasi-projective variety, and let $\\theta \\in \\operatorname{Hom}(\\pi _1(T,t_0),{\\mathrm {Aut}(A)})$ such that $H:=\\mathrm {Im}(\\theta )$ stabilizes $[\\rho _A]$ .",
"Then there is an étale base change $p : (T^{\\prime },t_0^{\\prime }) \\rightarrow (T,t_0)$ and a representation $\\rho _B \\in \\operatorname{Hom}(\\pi _1(T^{\\prime },t_0^{\\prime }), \\mathrm {GL}_r$ such that $ \\rho _A (\\theta (p_*\\beta )^{-1} (\\alpha ) )= \\rho _B(\\beta )^{-1} \\cdot \\rho _A(\\alpha ) \\cdot \\rho _B(\\beta ) \\quad \\forall \\alpha \\in A\\, , \\beta \\in \\pi _1(T^{\\prime },t_0^{\\prime })\\, .",
"$ Let $\\rho _A = \\bigoplus _{i\\in I} \\rho ^i_A $ be a decomposition such that each $\\rho ^i_A$ is irreducible.",
"The subgroup $\\bigcap _{i\\in I} \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho ^i_A] \\subset \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho _A]\\, , $ stabilizing the conjugacy class $[\\rho ^i_A]$ for each $i\\in I$ , is of finite index (see for example ).",
"Hence the subgroup $\\widetilde{H}:=H \\cap _{i\\in I} \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho ^i_A]$ is of finite index in $H$ .",
"Consider now the finite connected unramified covering $\\tilde{p}: (\\widetilde{T},\\tilde{t}_0)\\rightarrow (T,t_0)$ characterized by $\\tilde{p}_*\\pi _1(\\widetilde{T},\\tilde{t}_0)=\\theta ^{-1}(\\widetilde{H})$ .",
"Note that $\\tilde{p}$ induces a structure of smooth quasi-projective variety on $\\widetilde{T}$ .",
"Since $\\widetilde{H}$ stabilizes $[\\rho ^i_A]$ , for every $h\\in \\widetilde{H}$ and every $i\\in I$ , there is a matrix $M^i_{h}\\in \\mathrm {GL}_{r_i} such that\\begin{equation} (M^i_h)^{-1} \\cdot \\rho ^i_A \\cdot M_h^i= [h]\\cdot \\rho ^i_A\\, .\\end{equation}Given $ i$ and $ h$, the choice of $ Mhi$ is unique up to an element of the centralizer of $ iA$.",
"Since $ iA$ is irreducible, this centralizer is given by the set of scalar matrices.",
"Denote by $MhiPGLri the projectivization of $ {M_h^i}\\in \\mathrm {GL}_{r_i}.",
"Then $Bi: M*p*i$ is a well defined element of $ Hom(1(T,t0), PGLri$.",
"According to the Lifting Theorem \\cite [Th.",
"$ 3.1$]{MR3300949}, there exists a Zariski closed subset $ Z$ of $ T$ not containing $ t0$, a finite morphism of smooth quasi-projective varieties $$p^{\\prime } : (\\widetilde{T}^{\\prime },t_0^{\\prime }) \\rightarrow ( \\widetilde{T}\\setminus \\widetilde{Z}, \\tilde{t}_0)\\, , $$ étale in a neighborhood $ T'$ of $ t0'$, and a representation $ iB Hom(1(T',t0'), GLri$ whose projectivization is $ p'*Bi$.",
"For a convenient choice of $ p'$, this property is satisfied for all $ iI$ at once.",
"We obtain a representation $ B:= iI iB $ in $ Hom(1(T',t0'), GLri$ satisfying the required properties with respect to $ p:=pp'|T'$.$ Finiteness and algebraization Theorem A1 Let $(C,D_C)$ be a stable $n$ -pointed genus $g$ -curve as in Section REF .",
"Let $\\varphi : (\\Sigma _g, Y^n) \\stackrel{\\sim }{\\rightarrow } (C,D_C)$ be an orientation preserving homeomorphism.",
"Let $(E_0, \\nabla _0)$ be an algebraic logarithmic connection over $C$ with polar divisor $D_C$ and denote by $[\\rho _{\\nabla _0}]\\in \\chi _{g,n}(\\mathrm {GL}_r$ its monodromy with respect to $\\varphi $ .",
"Let $\\mathcal {I}_{(C,E_0, \\nabla _0)}=(\\mathcal {F}_{(C,D_C)}, E, \\nabla , \\Psi )$ be an algebraic isomonodromic deformation of $(C,E_0, \\nabla _0)$ with parameter space $T$ as in Section REF .",
"Assume that the classifying map $\\mathrm {class}(\\mathcal {F}) : T\\rightarrow \\mathcal {M}_{g,n}$ is dominant (see Section REF ).",
"Then the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"The orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ does not depend on the choice of $\\varphi $ .",
"Moreover, it is canonically identified, for any $t_1\\in T$ , with the orbit $\\Gamma _{g,n}\\cdot [\\rho _{t_1}]$ of the monodromy of the connection $(E, \\nabla )$ restricted to the fiber over $t_1$ of the family $\\mathcal {F}$ .",
"Since $\\mathrm {class}(\\mathcal {F} )$ is dominant we may assume, without loss of generality, that $\\star :=\\mathrm {class}(\\mathcal {F} )(t_0)\\in \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ .",
"Moreover, up to restricting $\\mathcal {I}_{(C,E_0, \\nabla _0)}$ to a Zariski open neighborhood $\\Delta $ of $t_0$ in $T$ , we may assume that $\\mathrm {class}(\\mathcal {F} )(T)\\, \\cap \\, \\mathcal {B}_{g,n}\\, = \\varnothing .$ Notice that this property, as well as the assumption of $\\mathrm {class}(\\mathcal {F} )$ being dominant is not altered by finite covers and further excision of strict subvarieties not containing $t_0$ .",
"According to Lemma REF , up to such a manipulation, we may assume that $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ admits a section $\\sigma : T\\rightarrow X$ of $\\kappa $ with values in $X^0:=X\\setminus D$ such that $\\sigma (t_0)=\\psi \\circ \\varphi (y_0)$ .",
"Denote by $\\rho $ a representative of the monodromy representation of $(E, \\nabla )$ with respect to the identity such that the restriction of $\\rho $ to the subgroup $(\\psi \\circ \\varphi )_*\\Lambda _{g,n}$ of $\\pi _1(X^0, \\sigma (t_0))$ , given by the inclusion of the central fiber, is identical to $(\\psi \\circ \\varphi )_*\\rho _{\\nabla _0}$ .",
"Such a representative exists, as implies for example Theorem REF .",
"Since the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ does not depend on the choice of $\\varphi $ , we may assume that $\\varphi $ is convenient in the sense of Proposition REF .",
"We then have a semi-direct product decomposition $ \\pi _1(X^0,\\sigma (t_0)) =(\\psi \\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0)\\, ,$ where we have two different expressions for its structure morphism $\\eta $ , proving that ${H}:= \\mathrm {taut}_{g,n+1}({\\mathrm {cl}^\\bullet }_*\\pi _1(T,t_0))\\subset \\Gamma _{g,n+1}\\, $ acts on $\\rho _{\\nabla _0}\\in \\operatorname{Hom}(\\Lambda _{g,n}, \\mathrm {GL}_r$ by conjugation.",
"More precisely, for all $\\alpha \\in \\Lambda _{g,n}$ and $[h]= \\mathrm {taut}_{g,n+1}({\\mathrm {cl}^\\bullet }_*\\, \\beta )\\, \\in {H}$ , we have $\\rho _{\\nabla _0}\\left(\\mathfrak {a}(h)(\\alpha )\\right)=\\rho (\\sigma _*\\beta )\\cdot \\rho _{\\nabla _0}(\\alpha )\\cdot \\rho (\\sigma _*\\beta ^{-1})$ and in particular $[h^{-1}]\\cdot [\\rho _{\\nabla _0} ]= [\\rho _{\\nabla _0} ].$ In other words, $H$ is a subgroup of the stabilizer of $[\\rho _{\\nabla _0}]$ in $\\Gamma _{g, n+1}$ .",
"By definition of the mapping class group action, we then have $\\pi (H)\\subset \\mathrm {Stab}_{\\Gamma _{g,n}}[\\rho _{\\nabla _0}]\\, ,$ where $\\pi :\\Gamma _{g, n+1}\\rightarrow \\Gamma _{g,n}$ is the projection forgetting the marking $y_0$ .",
"Since the size of the orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ equals the index of $ \\mathrm {Stab}_{\\Gamma _{g,n}}[\\rho _{\\nabla _0}]$ in $\\Gamma _{g,n}$ , it now suffices to prove that $\\pi (H)$ has finite index in $\\Gamma _{g,n}$ .",
"Denote by $q:\\Gamma _{g,n}\\rightarrow \\Gamma _{g,n}/K_{g,n}$ the quotient by the normal subgroup $K_{g,n}$ , which, by Lemma REF , has order at most 2.",
"Hence for the indices, we have $[\\Gamma _{g,n}: \\pi (H)] \\le 2\\cdot [\\Gamma _{g,n}/K_{g,n} : q(\\pi (H))]\\, .$ We have a commutative diagram ${\\Gamma _{g,n+1}[d]_{\\pi }&&&& \\pi _1(T,t_0)[llll]_{\\mathrm {taut}_{g,n+1}\\, \\circ \\, {\\mathrm {cl}^\\bullet }_*}[d]^{\\mathrm {taut}_{g,n}\\, \\circ \\, \\mathrm {cl}_*}\\\\\\Gamma _{g,n}[rrrr]^{q}&&&& \\Gamma _{g,n}/K_{g,n}\\, ,}$ where $\\mathrm {cl}: T\\rightarrow \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ denotes the corestriction of $\\mathrm {class}(\\mathcal {F})$ .",
"On the other hand, by the dominance assumption and , the subgroup $\\mathrm {cl}_*\\, \\pi _1(T,t_0)$ of $\\pi _1\\left(\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n},\\star \\right)$ is of finite index.",
"In particular, since the tautological morphism $\\mathrm {taut}_{g,n}: \\pi _1\\left(\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n},\\star \\right) \\twoheadrightarrow \\Gamma _{g,n}/K_{g,n}$ is onto, the subgroup $q(\\pi (H))=\\mathrm {taut}_{g,n}(\\mathrm {cl}_*\\pi _1(T,t_0))$ of $\\Gamma _{g,n}/K_{g,n}$ has finite index.",
"Theorem A2 Let $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ be an algebraic family of stable $n$ -pointed genus-$g$ curves with central fiber $(C,D_C)$ as in Section REF .",
"Let $(E_0, \\nabla _0)$ be an algebraic logarithmic connection over $C$ with polar divisor $D_C$ and denote by $[\\rho _{\\nabla _0}]\\in \\chi _{g,n}(\\mathrm {GL}_r$ its monodromy with respect to an orientation preserving homeomorphism $\\varphi : (\\Sigma _g, Y^n) \\stackrel{\\sim }{\\rightarrow } (C,D_C)$ .",
"Assume that $(E_0, \\nabla _0)$ is mild, $r=2$ or $\\rho _{\\nabla _0}$ is semisimple, and the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"Then there are an étale base change $p: (T^{\\prime },t_0^{\\prime })\\rightarrow (T,t_0)$ and a flat algebraic logarithmic connection $(E,\\nabla )$ over $X^{\\prime }:=p^*X$ with polar divisor $p^*D$ , such that $\\psi ^*(E,\\nabla )|_{X^{\\prime }_{t_0^{\\prime }}}$ is isomorphic to $(E_0, \\nabla _0)$ .",
"Since $(C,D_C)$ is stable by assumption, it only admits a finite number of automorphisms.",
"Let $x_0\\in C\\setminus D_C$ be a point fixed by no automorphism other than the identity.",
"Up to isotopy, we may assume $\\varphi (y_0)=x_0$ .",
"Let $\\mathbf {x}^\\bullet $ be the labelling of $D_C^\\bullet =D_C+x_0$ induced by $\\varphi $ .",
"By construction, we have $\\star :=[C,\\mathbf {x}^\\bullet ]\\in \\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}$ .",
"Up to an étale base change, we may assume, by Lemma REF , that there is a section $\\sigma : T\\rightarrow X$ of $\\kappa $ with values in $X^0:=X\\setminus D$ and such that $\\sigma (t_0)=\\psi (x_0)$ .",
"With the notation of Section REF , we may consider the family of $n+1$ -pointed genus-$g$ curves $\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}=(\\kappa : X\\rightarrow T, D+\\sigma (T), t_0, \\psi )$ .",
"Altering neither $D_C^\\bullet $ , nor the labelling, nor the orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ , we may assume that $\\varphi $ is conveniently chosen in the sense of Proposition REF .",
"We then have a semi-direct product decomposition $ \\pi _1(X\\setminus D,\\sigma (t_0)) =(\\psi \\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0),$ where $\\eta (\\sigma _*\\beta )((\\psi \\circ \\varphi )_*\\alpha ) =\\sigma _*\\beta \\cdot (\\psi \\circ \\varphi )_*\\alpha \\cdot \\sigma _*\\beta ^{-1} = (\\psi \\circ \\varphi )_*\\mathfrak {a}(\\theta _*\\beta )(\\alpha )\\, $ and $\\theta : = \\mathrm {taut}_{g,n+1}\\circ {\\mathrm {cl}^\\bullet }_* : \\pi _1(T,t_0) \\rightarrow \\Gamma _{g,n+1}.$ Since the $\\Gamma _{g,n+1}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite, the stabilizer $H:=\\mathrm {Stab}_{\\Gamma _{g,n+1}}[\\rho _{\\nabla _0}]$ of the conjugacy class of $\\rho _{\\nabla _0}$ under the action of $\\Gamma _{g,n+1}$ has finite index in $\\Gamma _{g,n+1}$ .",
"Since the tautological morphism is onto, the subgroup $\\mathrm {taut}_{g,n+1}^{-1}(H)$ of $\\pi _1(\\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}, \\star )$ then has also finite index.",
"In particular, there is a finite connected étale cover $q : (U,u_0)\\rightarrow (\\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}, \\star )$ such that $\\pi _1(U,u_0)= \\mathrm {taut}_{g,n+1}^{-1}(H)$ .",
"Now consider the fibered product ${(T^{\\prime },t_0^{\\prime })[r]^{p}[d]&(T,t_0)[d]^{\\mathrm {class}(\\mathcal {F}^\\bullet )}\\\\(U,u_0)[r]^{\\hspace{-15.0pt}q}&(\\mathcal {M}_{g,n+1},\\star ).",
"}$ We denote the pullback family of curves by $\\mathcal {F}_{(C,D_C^\\bullet )}^{\\prime }=(\\kappa ^{\\prime } : X^{\\prime }\\rightarrow T^{\\prime }, D^{\\prime } +\\sigma ^{\\prime }(T^{\\prime }), t^{\\prime }_0, \\psi ^{\\prime }):=p^*\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}$ .",
"We further denote $\\mathrm {cl}^{\\prime }=\\mathrm {cl}^\\bullet \\circ p$ , which is the corestriction of $\\mathrm {class}(\\mathcal {F}^{\\prime })$ .",
"By construction, the morphism $\\theta ^{\\prime } : =\\theta \\circ p= \\mathrm {taut}_{g,n+1}\\circ {\\mathrm {cl}^{\\prime }}_* : \\pi _1(T^{\\prime },t_0^{\\prime }) \\rightarrow \\Gamma _{g,n+1}$ takes values in $H$ .",
"Again up to an étale base change of $(T^{\\prime },t_0^{\\prime })$ , by Proposition REF and Lemma REF , there is a representation $\\rho _B\\in \\operatorname{Hom}(\\pi _1(T^{\\prime },t_0^{\\prime }),\\mathrm {GL}_r$ such that for all $\\beta \\in \\pi _1(T^{\\prime },t_0^{\\prime })$ , $\\alpha \\in \\Lambda _{g,n}$ , we have $\\left([\\theta ^{\\prime }_* \\beta ]^{-1}\\cdot \\rho _{\\nabla _0}\\right) (\\alpha ) = \\rho _B(\\beta ) \\cdot \\rho _{\\nabla _0}(\\alpha ) \\cdot \\rho _B(\\beta ^{-1}) \\, .",
"$ Since by definition $\\left([\\theta ^{\\prime }_*\\beta ]^{-1}\\cdot \\rho _{\\nabla _0}\\right) (\\alpha )= \\rho _{\\nabla _0}(\\mathfrak {a}(\\theta ^{\\prime }_*\\beta )(\\alpha ))$ , we obtain a well defined representation $\\rho : \\left\\lbrace \\begin{array}{rcl} \\pi _1(X^{\\prime }\\setminus D^{\\prime },\\sigma ^{\\prime }(t_0^{\\prime })) & \\rightarrow & \\mathrm {GL}_r{.1cm}\\\\(\\psi ^{\\prime }\\circ \\varphi )_*\\alpha \\cdot \\sigma ^{\\prime }_*\\beta & \\mapsto & \\rho _{\\nabla _0}(\\alpha ) \\cdot \\rho _B(\\beta )\\end{array} \\right.\\, $ (see Lemma REF ) with respect to the semi-direct product decomposition $ \\pi _1(X^{\\prime }\\setminus D^{\\prime },\\sigma ^{\\prime }(t_0^{\\prime })) =(\\psi ^{\\prime }\\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma ^{\\prime }_*\\pi _1(T^{\\prime },t_0^{\\prime })$ .",
"By construction, $\\rho $ extends $ \\rho _{\\nabla _0}$ .",
"We conclude by the logarithmic Riemann-Hilbert correspondence (see Theorem REF ).",
"Let us first prove the implication $(\\ref {algitem alg})\\Rightarrow (\\ref {algitem finite})$ .",
"Let $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C,E_0, \\nabla _0)}=$ $(\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}, E, $ $\\nabla , \\Psi )$ be an algebraic universal isomonodromic deformation of $(C,E_0, \\nabla _0)$ as in Section REF .",
"Then by definition, the family $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ is Kuranishi.",
"In particular, the classifying map $\\mathrm {class}(\\mathcal {F}^{\\mathrm {Kur}} ) : T\\rightarrow \\mathcal {M}_{g,n}$ is dominant.",
"Then by Theorem A1, the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"Let us now prove the implication $(\\ref {algitem finite})\\Rightarrow (\\ref {algitem alg})$ .",
"Let $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ be any algebraic Kuranishi family with central fiber $(C,D_C)$ as in Section REF .",
"Note that such a family exists since $(C,D_C)$ is stable, and that it remains Kuranishi after pullback via an étale base change.",
"Up to such a manipulation, according to Theorem A2, the family $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ can be endowed with a flat algebraic logarithmic connection $(E,\\nabla )$ over $X$ with polar divisor $D$ such that there is an isomorphism $\\Psi : (E_0,\\nabla _0)\\rightarrow (E, \\nabla )|_{X_{t_0}}$ commuting with $\\psi $ via the natural projections to $(C,D_C)$ and $(X_{t_0}, D|_{X_{t_0}})$ respectively.",
"Now $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C,E_0, \\nabla _0)}:=(\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}, E, \\nabla , \\Psi )$ defines an algebraic universal isomonodromic deformation of $(C,E_0,\\nabla _0)$ (see Section REF ).",
"Dynamics Effective description of the mapping class group action In this section we describe the action of $\\hat{\\Gamma }_{g,n}$ on $\\Lambda _{g,n}$ in terms of specified generators for both groups.",
"Presentation of the fundamental group To give an effective description of $\\Lambda _{g,n}$ and how $\\hat{\\Gamma }_{g,n}$ acts, we will assume that $\\Sigma _g$ is the subsurface of genus $g$ of $\\mathbb {R}^3$ depicted in Figure $\\ref {figgens 1}$ .",
"On this surface we also depicted, in gray, an embedded closed disk $\\bar{\\Delta }\\subset \\Sigma _g$ , we will denote $\\Delta $ its interior.",
"We fix $n$ and we consider a subset $Y^n=\\lbrace y_1, \\ldots , y_n\\rbrace \\subset \\Delta $ of cardinality $n$ , as well as a point $y_{0}\\in \\bar{\\Delta }\\setminus \\Delta $ .",
"We have $\\pi _1(\\Sigma _g\\setminus \\Delta , y_0)=\\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g, \\delta ~|~ [\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]=\\delta ^{-1} \\right\\rangle ,$ where the mentioned generators correspond to the loops in Figure $\\ref {figgens 1}$ .",
"Figure: Preferred elements of the fundamental group, IThe loops in Figure $\\ref {figgens 2}$ correspond to the following presentation.",
"$\\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})=\\left\\langle \\gamma _1, \\ldots , \\gamma _n, \\delta ~|~ \\gamma _1\\cdots \\gamma _n=\\delta \\right\\rangle .$ Figure: Preferred elements of the fundamental group, IIBy the Van Kampen theorem, we have $\\begin{array}{rcl}\\Lambda _{g,n}&=&\\pi _1(\\Sigma \\setminus \\Delta , y_{0})*_\\delta \\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})\\vspace{5.69046pt}\\\\&=&\\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g,\\gamma _1,\\ldots ,\\gamma _n ~\\vert ~ \\gamma _1\\cdots \\gamma _n=\\left([\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]\\right)^{-1}\\right\\rangle .\\end{array}$ In the sequel, writing “the generators\" of $\\Lambda _{g,n}$ , we will refer to the above $(\\alpha _i)_{i\\in \\llbracket 1, g\\rrbracket }\\, , \\, (\\beta _i)_{i\\in \\llbracket 1, g\\rrbracket }\\, , \\, (\\gamma _j)_{j\\in \\llbracket 1, n\\rrbracket }\\, .$ Mapping class group generators We define $\\Gamma _{g}^1$ to be the mapping class group of orientation preserving homeomorphisms of $\\Sigma \\setminus \\Delta $ that restrict to the identity on $\\partial \\Delta $ .",
"Continuating such homeomorphisms by the identity on $\\Delta $ , we get a morphism $\\varphi _{g}: \\Gamma _{g}^1\\rightarrow \\hat{\\Gamma }_{g,n}^{\\bullet }\\, .$ After Lickorish (see also ), the group $\\Gamma _{g}^1$ is generated by the (right) Dehn-twists along the loops $\\tau _1, \\ldots ,\\tau _{3g-1}$ represented in Figure REF .",
"Figure: Dehn-twistsA right Dehn twist acts on paths which cross the corresponding Dehn curve as depicted in Figure REF .",
"This action can be summarized as “a path crossing the Dehn curve has to turn right”.",
"A left Dehn twist is the inverse of a right Dehn twist.",
"Figure: Dehn-twist actionOne can now easily check the following.",
"Lemma 6.2.1 (Dehn-twists) The action of the Dehn twists above on the fundamental group $\\pi _1(\\Sigma _g\\setminus \\Delta , y_{0})$ is given in Table $\\ref {DehnAction1}$ , where we only indicate the non-trivial actions on the generators.",
"Here for $\\tau _{2k-1}$ we give the formula for the left Dehn twist.",
"The other generators all correspond to right Dehn twists.",
"Moreover, for $k\\in \\llbracket 1, g-1\\rrbracket $ , the element $\\Theta _k$ described in Table $\\ref {DehnAction1}$ is fixed by $\\tau _{2g+k}$ .",
"Table: NO_CAPTIONOn the other hand, one can define the mapping class group of orientation preserving homeomorphisms of $\\bar{\\Delta }$ that preserve the set $Y^n$ and restrict to the identity on $\\partial \\Delta $ .",
"It is classically called the braid group on $n$ strands and denoted $B_n$ .",
"Continuating such homeomorphisms by the identity on the complement of $\\Delta $ in $\\Sigma _g$ , we get a morphism $\\varphi _{0}: B_n\\rightarrow \\hat{\\Gamma }_{g,n}^{\\bullet }.$ After Artin , the group $B_n$ is generated by half-twists $\\sigma _1, \\ldots , \\sigma _{n-1}$ , whose action is depicted in Figure REF .",
"Figure: half-twistsLemma 6.2.2 (half-twists) The action of $B_n=\\langle \\sigma _1, \\ldots , \\sigma _{n-1}\\rangle $ on the fundamental group $\\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})$ is described in Table $\\ref {DehnAction2}$ , where we only indicate the non-trivial actions on the generators.",
"Moreover, Table $\\ref {DehnAction2}$ indicates the action of $\\sigma _{cycl}:=\\sigma _{n-1}\\circ \\cdots \\circ \\sigma _{1} \\in B_n$ and some of its powers.",
"Table: Proof of Theorem"
],
[
"The monodromy of the monodromy",
"In this section, we introduce the so-called group of mapping classes of a $$ -family, which is the image of a canonical morphism from the fundamental group of the parameter space of the family to the fundamental group of the central fiber.",
"For an isomonodromic deformation, the action on the monodromy representation of the initial connection by the group of mapping classes of the underlying family of curves corresponds to the monodromy of the monodromy representation.",
"Under suitable conditions, this group can be canonically translated into a subgroup of $\\Gamma _{g,n}$ ."
],
[
"Mapping classes of the central fiber",
"As usual, let $(C,D_C)$ be a stable $n$ -pointed genus-$g$ curve.",
"Let $\\mathcal {F}_{(C,D_C)}$ be a $$ -family with parameter space $(T,t_0)$ .",
"Let $\\beta : [0,1]\\rightarrow T$ be a closed path with end point $t_0$ , i.e.",
"a continous map such that $\\beta (0)=\\beta (1)=t_0$ .",
"By , the pullback bundle $\\beta ^*(X,D) \\rightarrow [0,1]$ possesses a topological trivialization $\\Phi :(C, D_C)\\times [0,1] \\stackrel{\\sim }{\\rightarrow }\\beta ^*(X,D)$ .",
"For $s\\in [0,1]$ , we denote $\\Phi _s:=\\Phi |_{(C, D_C)\\times \\lbrace s\\rbrace } \\, $ and deduce a homeomorphism from the central fiber seen over $\\lbrace 1\\rbrace $ to the central fiber seen over $\\lbrace 0\\rbrace $ given by $\\psi ^{-1}\\circ \\Phi _0\\circ \\Phi _1^{-1} \\circ \\psi : (C, D_C) \\stackrel{\\sim }{\\rightarrow } (C, D_C)\\, .$ Its isotopy class shall be called the mapping class associated to $\\beta $ and $\\mathcal {F}_{(C,D_C)}$ and denoted $\\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )\\, .$ Lemma 4.1.1 The mapping class $\\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )$ is well defined, i.e.",
"it does not depend on the choice of a trivialization $\\Phi $ .",
"Moreover, $\\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )$ only depends on the homotopy class of $\\beta $ .",
"For fixed $\\beta $ , take two trivializations : $\\Phi ,\\widetilde{\\Phi }: (C, D_C)\\times [0,1] \\stackrel{\\sim }{\\rightarrow }\\beta ^*(X,D)$ .",
"The family $\\widetilde{\\Phi }_0 \\circ \\widetilde{\\Phi }_s^{-1}\\circ \\Phi _s\\circ \\Phi _1^{-1}$ gives an isotopy from $\\Phi _0\\circ \\Phi _1^{-1}$ to $\\widetilde{\\Phi }_0\\circ \\widetilde{\\Phi }_1^{-1}$ .",
"Consider now two paths $\\beta _1$ and $\\beta _2$ that are homotopic relative to their endpoints.",
"By definition, there exists a continuous map $\\theta : \\overline{ \\rightarrow T, where \\overline{ denotes the closed unit disc, such that \\beta _2(s)=\\theta (\\mathrm {e}^{i\\pi (1+s)}) and \\beta _1(s)=\\theta (\\mathrm {e}^{i\\pi (1-s)}).",
"Since \\overline{ is contractible, by \\cite [Cor.",
"10.3]{MR1249482}, there is a trivialization \\Phi of \\theta ^*(X,D).",
"It induces trivializations \\Phi ^i of \\beta _i^*X for i=1,2.",
"Since they are both induced by \\Phi , we have \\Phi ^1_0=\\Phi ^2_0=\\Phi _{-1} and \\Phi ^1_1=\\Phi ^2_1=\\Phi _{1}.",
"}}\\begin{prop} Let\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi ) be a -family as in Section \\ref {SecSetup}.",
"Assume that none of the fibers (X_t, D|_{X_t}) has exceptional automorphisms.",
"Let \\mathbf {x} be a labelling of D_C and denote\\mathrm {cl}: T\\rightarrow \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n} the corestriction of the induced classifying map \\mathrm {class}(\\mathcal {F}) (see Section \\ref {Sec univ fam}).",
"Then there exists an orientation preserving homeomorphism \\varphi : (\\Sigma _g,y^n)\\stackrel{\\sim }{\\rightarrow } (C,\\mathbf {x}) such that for all \\beta \\in \\pi _1(T,t_0), the following equation holds in \\Gamma _{g,n}/K_{g,n}:\\varphi ^{-1}\\circ \\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )\\circ \\varphi =\\mathrm {taut}_{g,n}(\\mathrm {cl}_*\\beta ) \\, ,where \\mathrm {taut}_{g,n} is the tautological morphism\\mathrm {taut}_{g,n}:\\pi _1( \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}, \\star )\\rightarrow \\Gamma _{g,n}/K_{g,n} (see (\\ref {def phi})) and \\star :=[C,\\mathbf {x}].\\end{prop}}{\\begin{xmlelement*}{proof}Fix an orientation preserving homeomorphism \\widetilde{\\varphi } : (\\Sigma _g,y^n)\\stackrel{\\sim }{\\rightarrow } (C, \\mathbf {x}) and denote \\hat{\\star }:=[C, D_C,\\widetilde{\\varphi }]\\in \\mathcal {T}_{g,n}\\, .\\end{xmlelement*}}$ As usual, denote $\\mathcal {F}^{+}_{g,n}=(\\mathcal {F}_{g,n}, \\Phi _{g,n})$ the universal Teichmüller curve $\\mathcal {F}_{g,n}=(\\kappa _{g,n} : \\mathcal {X}\\rightarrow \\mathcal {T}_{g,n}, \\mathcal {D})$ endowed with the Teichmüller structure $ \\Phi _{g,n}: (\\Sigma _g,Y^n)\\times \\mathcal {T}_{g,n}\\stackrel{\\sim }{\\rightarrow } (\\mathcal {X}, \\mathcal {D})$ .",
"For any point $t \\in \\mathcal {T}_{g,n}$ , we shall denote $\\Phi ^{g,n}_{t}:=\\Phi _{g,n}|_{(\\Sigma _g,Y^n)\\times \\lbrace t \\rbrace } : (\\Sigma _g,Y^n)\\times \\lbrace t \\rbrace \\stackrel{\\sim }{\\rightarrow } (\\mathcal {X}_{t}, \\mathcal {D}|_{\\mathcal {X}_{t }})\\, .$ Let $p : (\\widetilde{T}, \\tilde{t}_0) \\rightarrow (T,t_0)$ be a universal cover and consider the pulled-back family $\\widetilde{\\mathcal {F}}=(\\widetilde{\\kappa } : \\widetilde{X}\\rightarrow \\widetilde{T},\\widetilde{D}):=p^*(\\kappa : X\\rightarrow T,D)\\, .$ Now for any contractible analytic submanifold $\\widetilde{\\Delta } \\subset \\widetilde{T}$ containing $\\tilde{t}_0$ , there is a trivialization $\\widetilde{\\Phi }: (C,D_C)\\times \\widetilde{\\Delta }\\stackrel{\\sim }{\\longrightarrow } (\\widetilde{X} , \\widetilde{D})|_{\\widetilde{\\Delta }}$ of $\\widetilde{\\mathcal {F}}|_{\\widetilde{\\Delta }}$ , unique up to isotopy, such that $\\widetilde{\\Phi }_{\\tilde{t}_0}=\\psi $ with respect to the pullback identification $(\\widetilde{X}_{\\tilde{t}_0}, \\widetilde{D}|_{\\widetilde{X}_{\\tilde{t}_0}}){=}(X_{t_0},D|_{X_{t_0}})$ .",
"Setting $\\widetilde{\\mathcal {F}}^+:=(\\widetilde{\\mathcal {F}}|_{\\widetilde{\\Delta }}, \\widetilde{\\Phi }\\circ (\\widetilde{\\varphi }\\times \\mathrm {id}))$ defines an analytic family with Teichmüller structure.",
"By the universal property of the Teichmüller curve, we have an isomorphism $\\mathfrak {f}=(\\mathrm {id}_{\\widetilde{\\Delta }},\\mathfrak {f}^a, \\mathfrak {f}^{\\mathrm {top}})$ fitting into the following commutative diagram.",
"$ {(\\Sigma _g, Y^n)\\times \\widetilde{\\Delta } [rr]^{\\mathfrak {f}^{\\mathrm {top}}}_{\\sim }[d]_{\\widetilde{\\Phi }\\circ (\\widetilde{\\varphi }\\times \\mathrm {id})}&& (\\Sigma _g, Y^n)\\times \\widetilde{\\Delta }[d]^{{\\mathrm {class}^+}(\\widetilde{\\mathcal {F}}^+)^*\\Phi _{g,n}}\\\\(\\widetilde{X}, \\widetilde{D})|_{\\widetilde{\\Delta }}[rr]^{\\mathfrak {f}^a}_{\\sim }[d]_{\\widetilde{\\kappa }}&& {\\mathrm {class}^+}(\\widetilde{\\mathcal {F}}^+)^*(\\mathcal {X}, \\mathcal {D})[d]^{{\\mathrm {class}^+}(\\widetilde{\\mathcal {F}}^+)^*\\kappa _{g,n}}\\\\\\widetilde{\\Delta } @{=}[rr]&&\\widetilde{\\Delta }} $ By commutativity of the diagram, we have $\\mathfrak {f}^{\\mathrm {top}}=\\mathfrak {f}^{\\mathrm {top}}_{\\tilde{t}_0}\\times \\mathrm {id}_{\\widetilde{\\Delta }}$ up to fiberwise isotopy.",
"Define $\\varphi := \\widetilde{\\varphi }\\circ \\left(\\mathfrak {f}^{\\mathrm {top}}_{\\tilde{t}_0}\\right)^{-1} \\, .$ Now let $[\\beta ]\\in \\pi _1(T,t_0)\\setminus \\lbrace 1\\rbrace $ and consider $\\tilde{\\beta }: [0,1]\\rightarrow \\widetilde{T}$ , the lift of $\\beta $ with starting point $\\tilde{t}_0$ .",
"If the representative $\\beta $ of the homotopy class $[\\beta ]$ is well chosen, then $\\widetilde{\\beta }$ is a $\\mathcal {C}^{\\infty }$ -embedding.",
"By existence of tubular neighborhoods, there is a contractible neighborhood $\\widetilde{\\Delta }$ of $\\tilde{t}_0$ as above, containing $\\widetilde{\\beta }$ .",
"We claim that, up to isotopy, $ \\mathrm {map}_{\\mathcal {F}_{(C,D_C)}}(\\beta )= \\widetilde{\\Phi }_{\\widetilde{\\beta }(1)}^{-1}\\circ \\widetilde{\\Phi }_{\\tilde{t}_0}\\, .$ Indeed, we have $\\beta ^*(X,D)=(p\\circ \\widetilde{\\beta })^*(X,D)=\\widetilde{\\beta }^*p^*(X,D)=\\widetilde{\\beta }^*(\\widetilde{X},\\widetilde{D})$ .",
"Since moreover $\\widetilde{\\beta }$ is an embedding, we have $\\beta ^*(X,D)=(\\widetilde{X},\\widetilde{D})|_{\\widetilde{\\beta }([0,1])}.$ The claim then follows from the fact that $ \\psi ^{-1}\\circ \\widetilde{\\Phi }_{\\tilde{t}_0}$ is the identity and from the definition of the mapping class.",
"Denote $\\widehat{\\beta }:={\\mathrm {class}^+}(\\widetilde{\\mathcal {F}}^+)_*\\widetilde{\\beta }$ , which is a path in $\\mathcal {T}_{g,n}$ with starting point $\\hat{\\star }$ .",
"By our definitions, the black part of the following diagram is commutative.",
"$ {(\\Sigma _g, Y^n)\\times \\lbrace \\tilde{t}_0\\rbrace [rr]^{\\widetilde{\\Phi }\\circ (\\widetilde{\\varphi }\\times \\lbrace \\tilde{t}_0\\rbrace )}_{\\sim }[d]_{\\mathfrak {f}^{\\mathrm {top}}_{\\tilde{t}_0}}^{\\hspace{3.0pt}\\text{\\begin{rotate}{90}\\sim \\end{rotate}}}&&(\\widetilde{X}_{\\tilde{t}_0}, \\widetilde{D}|_{\\widetilde{X}_{\\tilde{t}_0}})@{=}[r] [d]_{\\mathfrak {f}^{a}_{\\tilde{t}_0}}^{\\hspace{3.0pt}\\text{\\begin{rotate}{90}\\sim \\end{rotate}}}&(\\widetilde{X}_{\\widetilde{\\beta }(1)}, \\widetilde{D}|_{\\widetilde{X}_{\\widetilde{\\beta }(1)}}) [d]^{\\mathfrak {f}^{a}_{\\widetilde{\\beta }(1)}}_{\\hspace{3.0pt}\\text{\\begin{rotate}{90}\\sim \\end{rotate}}}&& (\\Sigma _g, Y^n)\\times \\lbrace \\widetilde{\\beta }(1)\\rbrace [ll]_{\\widetilde{\\Phi }\\circ (\\widetilde{\\varphi }\\times \\lbrace \\widetilde{\\beta }(1)\\rbrace )}^{\\sim } [d]^{\\mathfrak {f}^{\\mathrm {top}}_{\\widetilde{\\beta }(1)}}_{\\hspace{1.0pt}\\text{\\begin{rotate}{90}\\sim \\end{rotate}}}\\\\(\\Sigma _g, Y^n)\\times \\lbrace \\hat{\\star }\\rbrace [rr]_{\\Phi ^{g,n}_{\\hat{\\star }}}^{\\sim }&& (\\mathcal {X}_{\\hat{\\star }}, \\mathcal {D}|_{\\mathcal {X}_{\\hat{\\star }}}) @[gray][r]^{\\color {gray} \\hspace{-15.0pt}\\sim }_{\\color {gray} \\hspace{-15.0pt}\\widehat{\\psi }} & (\\mathcal {X}_{\\widehat{\\beta }(1)}, \\mathcal {D}|_{\\mathcal {X}_{\\widehat{\\beta }(1)}})&&(\\Sigma _g, Y^n)\\times \\lbrace \\widehat{\\beta }(1)\\rbrace [ll]^{\\Phi ^{g,n}_{\\widehat{\\beta }(1)}}_{\\sim }} $ We define $\\widehat{\\psi }=\\mathfrak {f}^a_{\\tilde{\\beta }(1)}\\circ (\\mathfrak {f}^a_{t_0})^{-1},$ so that adding the gray arrow maintains this commutativity.",
"Since $\\mathfrak {f}^{\\mathrm {top}}_{\\widetilde{\\beta }(1)}$ is isotopic to $\\mathfrak {f}^{\\mathrm {top}}_{\\tilde{t}_0}$ , the following equations hold up to isotopy: $\\left\\lbrace \\begin{array}{ccl}\\Phi ^{g,n}_{\\hat{\\star }} &=&\\mathfrak {f}^a_{\\tilde{t}_0}\\circ \\widetilde{\\Phi }_{\\tilde{t}_0}\\circ \\varphi \\vspace{2.84544pt}\\\\\\Phi ^{g,n}_{\\widehat{\\beta }(1)}&=&\\mathfrak {f}^a_{\\widetilde{\\beta }(1)}\\circ \\widetilde{\\Phi }_{\\widetilde{\\beta }(1)} \\circ \\varphi .",
"\\vspace{2.84544pt}\\\\\\end{array}\\right.$ On the other hand, $\\mathrm {cl}_*\\beta $ is a closed path in $\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ with end point $\\star $ .",
"By construction, it lifts, with respect to the forgetful map $\\pi _{g,n}$ , to $\\widehat{\\beta }$ , with $\\widehat{\\beta }(0)=\\hat{\\star }$ .",
"By definition of the tautological morphism $\\mathrm {taut}_{g,n}$ , we thus have, for $[h]:= \\mathrm {taut}_{g,n}(\\mathrm {cl}_*\\beta ) \\in \\Gamma _{g,n}/K_{g,n}$ : $[h]\\cdot \\left[\\mathcal {X}_{\\hat{\\star }}, \\mathcal {D}|_{\\mathcal {X}_{\\hat{\\star }}}, \\Phi ^{g,n}_{\\hat{\\star }}\\right]=\\left[\\mathcal {X}_{\\widehat{\\beta }(1)}, \\mathcal {D}|_{\\mathcal {X}_{\\widehat{\\beta }(1)}}, \\Phi ^{g,n}_{\\widehat{\\beta }(1)} \\right]\\, .$ Note that $\\widehat{\\psi }$ induces an isomorphism of pointed curves with Teichmüller structure.",
"By the definition of the action of the mapping class group on $\\mathcal {T}_{g,n}$ , we now have $[h]\\cdot \\left[\\mathcal {X}_{\\hat{\\star }}, \\mathcal {D}|_{\\mathcal {X}_{\\hat{\\star }}}, \\Phi ^{g,n}_{\\hat{\\star }}\\right] = [h]\\cdot \\left[\\mathcal {X}_{\\widehat{\\beta }(1)}, \\mathcal {D}|_{\\mathcal {X}_{\\widehat{\\beta }(1)}},\\widehat{\\psi }\\circ \\Phi ^{g,n}_{\\hat{\\star }} \\right]= \\left[\\mathcal {X}_{\\widehat{\\beta }(1)}, \\mathcal {D}|_{\\mathcal {X}_{\\widehat{\\beta }(1)}},\\widehat{\\psi }\\circ \\Phi ^{g,n}_{\\hat{\\star }} \\circ h^{-1}\\right]\\, .$ Hence there is an element $[k]\\in K_{g,n}$ such that, up to isotopy, $\\widehat{\\psi }\\circ \\Phi ^{g,n}_{\\hat{\\star }} = \\Phi ^{g,n}_{\\widehat{\\beta }(1)} \\circ h\\circ k\\, .$ Combined with (REF ) and (REF ), this implies, up to isotopy, $ \\widetilde{\\Phi }_{\\widetilde{\\beta }(1)}^{-1} \\circ \\widetilde{\\Phi }_{\\tilde{t}_0} = \\varphi \\circ h \\circ k\\circ \\varphi ^{-1}\\, ,$ which by (REF ) and the definitions of $h$ and $k$ yields the desired result.",
"Splitting and the mapping class group Let $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T,D, t_0, \\psi )$ be a $$ -family of $n$ -pointed genus-$g$ curves as in Section REF .",
"Assume there is a section $\\sigma : T\\rightarrow X^0:=X\\setminus D$ of $\\kappa $ .",
"Then we can define a $$ -family of $n+1$ -pointed genus-$g$ curves $\\mathcal {F}^{\\bullet }_{(C,D_C^\\bullet )}:=(\\kappa : X\\rightarrow T,D^\\bullet , t_0, \\psi )$ by setting $D^\\bullet :=D+\\sigma (T)$ and $D_C^\\bullet :=D_C+x_0$ , where $x_0:=\\psi ^{-1}(\\sigma (t_0))$ .",
"To a labelling $\\mathbf {x} =(x_1, \\ldots , x_n)$ of $D_C$ we can associate a labelling $\\mathbf {x}^\\bullet :=(x_1, \\ldots , x_n,x_0)$ of $D_C^\\bullet $ .",
"Note that if a fiber of $\\mathcal {F}^\\bullet $ has exceptional automorphisms, then the corresponding fiber of $\\mathcal {F}$ also has exceptional automorphisms.",
"Proposition 4.2.1 Let $(C,D_C)$ be a stable $n$ -pointed genus-$g$ curve.",
"Let $\\mathcal {F}^{\\bullet }_{(C,D_C^\\bullet )}=(\\kappa : X\\rightarrow T,D^\\bullet , t_0, \\psi )$ be a $$ -family of $n+1$ -pointed genus-$g$ curves as above.",
"Assume that none of the fibers of $\\mathcal {F}^\\bullet $ has exceptional automorphisms.",
"Let $\\mathbf {x}^\\bullet $ be a labelling of $D_C^\\bullet $ as above.",
"Denote $\\mathrm {cl}^{\\bullet }: T\\rightarrow \\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}$ the corestriction of the induced classifying map $\\mathrm {class}(\\mathcal {F}^\\bullet )$ .",
"For a suitable homeomorphism $\\varphi : (\\Sigma _g,y^n, y_0)\\stackrel{\\sim }{\\rightarrow } (C,\\mathbf {x}, x_0)$ , we have $ \\pi _1(X^0,\\sigma (t_0)) =(\\psi \\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0)\\, ,$ where for all $\\alpha \\in \\Lambda _{g,n}$ and $ \\beta \\in \\pi _1(T,t_0)$ , we have $\\begin{array}{rcl}\\eta (\\sigma _*\\beta ) ((\\psi \\circ \\varphi )_*\\alpha ) &=& \\sigma _*\\beta \\cdot (\\psi \\circ \\varphi )_*\\alpha \\cdot \\sigma _*\\beta ^{-1}\\vspace{2.84544pt}\\\\&=& (\\psi \\circ \\varphi )_* \\, \\mathfrak {a}\\left( \\mathrm {taut}_{g,n+1}({\\mathrm {cl}^{\\bullet }}_*\\, \\beta )\\right)(\\alpha ) \\, , \\end{array}$ where, as we recall from the introduction, $\\mathfrak {a}(h)(\\alpha )=h_*\\alpha $ for all $h\\in \\Gamma _{g,n+1}$ and $\\alpha \\in \\Lambda _{g,n}$ .",
"Since $2g-2+n>0$ by assumption, we have $K_{g,n+1}=\\lbrace 1\\rbrace $ according to Lemma REF .",
"Then by Proposition , for a convenient choice of $\\varphi $ , the following equation holds in $\\Gamma _{g,n+1}$ for every $\\beta \\in \\pi _1(T,t_0)$ : $\\varphi ^{-1}\\circ \\mathrm {map}_{\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}}(\\beta )\\circ \\varphi =\\mathrm {taut}_{g,n+1}({\\mathrm {cl}^\\bullet }_*\\beta )\\, .$ Denote $C^0:=C\\setminus D_C$ .",
"We claim that for any $\\gamma \\in \\pi _1(C^0,x_0) $ and any $\\beta \\in \\pi _1(T,t_0)$ , the following equation holds in $\\pi _1(X^0,\\sigma (t_0))$ : $ \\psi _*\\mathrm {map}_{\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}}(\\beta )_*\\gamma = \\sigma _* \\beta \\cdot \\psi _*\\gamma \\cdot \\sigma _* \\beta ^{-1} \\, .$ Indeed, let $\\gamma : [0,1]\\rightarrow C^0$ be a closed path with end point $x_0$ .",
"For any $s_0\\in [0,1]$ , we have a closed path $\\gamma _{s_0}:=\\gamma \\times \\lbrace s_0\\rbrace $ in the product space $C^0\\times [0,1]$ .",
"We also have a path $\\theta : [0,1] \\rightarrow C^0\\times [0,1]\\, ; \\, s\\mapsto (x_0,s)$ .",
"The path $\\theta \\cdot \\gamma _1 \\cdot \\theta ^{-1}$ is closed and homotopic to $\\gamma _0$ .",
"Now let $\\beta \\in \\pi _1(T,t_0)$ and let $\\Phi : (C^0, x_0)\\times [0,1] \\stackrel{\\sim }{\\rightarrow } \\beta ^*(X^0, \\sigma (T))$ be a trivialization commuting with the natural projections to $[0,1]$ .",
"Define the homeomorphism $\\widetilde{\\Phi }:=\\Phi \\circ ((\\Phi _{1}^{-1}\\circ \\psi ) \\times \\mathrm {id}_{[0,1]}): (C^0,x_0)\\times [0,1] \\stackrel{\\sim }{\\rightarrow } \\beta ^*(X^0, \\sigma (T))\\, , $ which is another trivialization, satisfying $\\widetilde{\\Phi }_1=\\psi $ and $ \\widetilde{\\Phi }_0=\\psi _*\\mathrm {map}_{\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}}(\\beta ).$ Since $\\widetilde{\\Phi }$ is continuous, the closed paths $\\widetilde{\\Phi }_*\\gamma _0$ and $\\widetilde{\\Phi }_*\\theta \\cdot \\widetilde{\\Phi }_*\\gamma _1 \\cdot \\widetilde{\\Phi }_*\\theta ^{-1}$ are homotopic in $ \\beta ^*(X^0, \\sigma (T))$ .",
"Considering the natural projection $\\kappa : \\beta ^*(X^0, \\sigma (T)) \\rightarrow (X^0, \\sigma (T))$ , we have $\\kappa _*\\widetilde{\\Phi }_{*}\\gamma _0=\\widetilde{\\Phi }_{0*}\\gamma $ and $\\kappa _*\\widetilde{\\Phi }_*\\gamma _1=\\widetilde{\\Phi }_{1*}\\gamma $ .",
"Since moreover $\\kappa _*\\widetilde{\\Phi }_*\\theta =\\sigma _*\\beta $ , we have (REF ).",
"Since $\\varphi $ is a homeomorphism, the induced map $\\varphi _*:\\Lambda _{g,n}\\rightarrow \\pi _1(C^0,x_0)$ is an isomorphism.",
"The statement then follows from (REF ), (REF ) and the Splitting Lemma REF .",
"Necessary and sufficient conditions for algebraizability We shall see in Section REF that Theorem REF is a corollary of the juxtaposition of Theorem A1, showing that our algebraizability criterion for germs of universal isomonodromic deformations is necessary, and Theorem A2, showing that it is also sufficient.",
"We have already established the main ingredients for the proofs of both theorems.",
"For Theorem A2, we moreover need a representation-theoretical result developed in Section REF .",
"Extensions of representations We shall now consider the problem of extending a representation of the fundamental group of a fiber of a family of pointed curves to a representation for the whole family in light of Lemma REF and Proposition REF .",
"We begin with the elementary case of non semisimple rank 2 representations.",
"Let $A,B$ be groups.",
"Consider a representation $\\rho \\in \\operatorname{Hom}(A ,\\mathrm {Upp})$ , where $\\mathrm {Upp}$ is the group of invertible upper triangular matrices of rank 2.",
"To such a representation, we may associate two other ones : the scalar part $\\rho _{*} :\\alpha \\mapsto \\rho (\\alpha )_{2,2}$ and the affine part $\\rho _{\\mathrm {Aff}}:=\\rho _{*}^{-1} \\otimes \\rho $ .",
"The latter takes values in $\\mathrm {Aff}(:=\\lbrace (a_{i,j})\\in \\mathrm {Upp}~|~a_{2,2}=1\\rbrace $ which is isomorphic to the affine group of the complex line.",
"Lemma 5.1.1 Let $\\rho =\\rho _{*}\\otimes \\rho _{\\mathrm {Aff}}$ and $\\rho ^{\\prime } =\\rho ^{\\prime }_{*}\\otimes \\rho ^{\\prime }_{\\mathrm {Aff}}$ as above, and assume that they are not semisimple.",
"We have $[\\rho ]=[\\rho ^{\\prime }] \\in \\operatorname{Hom}(A , \\mathrm {GL}_{2}/\\mathrm {GL}_{2} if and only if $ *='*$ and$ [Aff]=['Aff] Hom(A , Aff(C))/Aff(C)$.$ The ”if”-part is trivial.",
"Assume $[\\rho ]=[\\rho ^{\\prime }]$ .",
"Since they take values in $\\mathrm {Upp}$ , both representations $\\rho $ and $\\rho ^{\\prime }$ leave the line $\\mathrm {span}(e_1)$ of 2 invariant.",
"By non semisimplicity, for each of the representations, there is no other globally invariant line.",
"Let $M=(m_{i,j})\\in \\mathrm {GL}_2 conjugate both representations.",
"Then $ M$ must leave $ span(e1)$ invariant, \\textit {i.e.}",
"$ MUpp$.",
"As the scalars are central in $ GL2, the element $M/m_{2,2}\\in \\mathrm {Aff}(\\mathbb {C})$ conjugates both representations.",
"In particular $\\rho _{*}=\\rho ^{\\prime }_{*}$ and $M/m_{2,2}$ conjugates $\\rho _{\\mathrm {Aff}}$ and $\\rho ^{\\prime }_{\\mathrm {Aff}}$ .",
"Lemma 5.1.2 Let $\\rho _A \\in \\operatorname{Hom}(A,\\mathrm {GL}_{2}$ be non semisimple.",
"Let $\\theta \\in \\operatorname{Hom}(B, \\mathrm {Aut}(A))$ such that for all $h\\in \\mathrm {Im}(\\theta )$ , we have $[\\rho _A]=h\\cdot [\\rho _A]:=[\\rho _A\\circ h^{-1}]$ .",
"Then there exists a representation $\\rho _B \\in \\operatorname{Hom}(B,\\mathrm {GL}_{2}$ such that $\\rho _A(\\theta (\\beta )^{-1}(\\alpha ))=\\rho _B(\\beta )^{-1} \\rho _A(\\alpha ) \\rho _B(\\beta ) \\quad \\forall \\alpha \\in A\\, , \\beta \\in B\\, .$ We may assume that $\\rho _A$ takes values in $\\mathrm {Upp}$ .",
"By assumption, for each $\\beta \\in B$ , there exists a matrix $M_\\beta \\in \\mathrm {GL}_{2} such that $ A(()-1())=M-1 A()M$.",
"By Lemma \\ref {lemred}, we may assume $ MAff(C)$.",
"If $ Im(A)Upp$ is non abelian, then it has trivial centralizer and the matrices $ MAff(C)$ are uniquely defined.",
"Otherwise, we have $ Im(A){ ( 1 01) | * , }$ and the matrices $ M$ are uniquely defined if we impose$ M{ ( 0 01) | *}$.",
"It is now straightforward to check that given these choices, the well-defined map $ M$ is a morphism of groups.",
"$ For a similar result for semisimple representations $\\rho _A$ (of arbitrary rank), the group $B$ , which in our case will be the fundamental group of a parameter space, might have to be enlarged, in order to take into account the non-unicity of the matrices $M_\\beta $ due to possible permutations of irreducible components.",
"Proposition 5.1.3 Let $\\rho _A \\in \\operatorname{Hom}(A, \\mathrm {GL}_r$ be semisimple.",
"Let $ (T,t_0)$ be a smooth connected quasi-projective variety, and let $\\theta \\in \\operatorname{Hom}(\\pi _1(T,t_0),{\\mathrm {Aut}(A)})$ such that $H:=\\mathrm {Im}(\\theta )$ stabilizes $[\\rho _A]$ .",
"Then there is an étale base change $p : (T^{\\prime },t_0^{\\prime }) \\rightarrow (T,t_0)$ and a representation $\\rho _B \\in \\operatorname{Hom}(\\pi _1(T^{\\prime },t_0^{\\prime }), \\mathrm {GL}_r$ such that $ \\rho _A (\\theta (p_*\\beta )^{-1} (\\alpha ) )= \\rho _B(\\beta )^{-1} \\cdot \\rho _A(\\alpha ) \\cdot \\rho _B(\\beta ) \\quad \\forall \\alpha \\in A\\, , \\beta \\in \\pi _1(T^{\\prime },t_0^{\\prime })\\, .",
"$ Let $\\rho _A = \\bigoplus _{i\\in I} \\rho ^i_A $ be a decomposition such that each $\\rho ^i_A$ is irreducible.",
"The subgroup $\\bigcap _{i\\in I} \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho ^i_A] \\subset \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho _A]\\, , $ stabilizing the conjugacy class $[\\rho ^i_A]$ for each $i\\in I$ , is of finite index (see for example ).",
"Hence the subgroup $\\widetilde{H}:=H \\cap _{i\\in I} \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho ^i_A]$ is of finite index in $H$ .",
"Consider now the finite connected unramified covering $\\tilde{p}: (\\widetilde{T},\\tilde{t}_0)\\rightarrow (T,t_0)$ characterized by $\\tilde{p}_*\\pi _1(\\widetilde{T},\\tilde{t}_0)=\\theta ^{-1}(\\widetilde{H})$ .",
"Note that $\\tilde{p}$ induces a structure of smooth quasi-projective variety on $\\widetilde{T}$ .",
"Since $\\widetilde{H}$ stabilizes $[\\rho ^i_A]$ , for every $h\\in \\widetilde{H}$ and every $i\\in I$ , there is a matrix $M^i_{h}\\in \\mathrm {GL}_{r_i} such that\\begin{equation} (M^i_h)^{-1} \\cdot \\rho ^i_A \\cdot M_h^i= [h]\\cdot \\rho ^i_A\\, .\\end{equation}Given $ i$ and $ h$, the choice of $ Mhi$ is unique up to an element of the centralizer of $ iA$.",
"Since $ iA$ is irreducible, this centralizer is given by the set of scalar matrices.",
"Denote by $MhiPGLri the projectivization of $ {M_h^i}\\in \\mathrm {GL}_{r_i}.",
"Then $Bi: M*p*i$ is a well defined element of $ Hom(1(T,t0), PGLri$.",
"According to the Lifting Theorem \\cite [Th.",
"$ 3.1$]{MR3300949}, there exists a Zariski closed subset $ Z$ of $ T$ not containing $ t0$, a finite morphism of smooth quasi-projective varieties $$p^{\\prime } : (\\widetilde{T}^{\\prime },t_0^{\\prime }) \\rightarrow ( \\widetilde{T}\\setminus \\widetilde{Z}, \\tilde{t}_0)\\, , $$ étale in a neighborhood $ T'$ of $ t0'$, and a representation $ iB Hom(1(T',t0'), GLri$ whose projectivization is $ p'*Bi$.",
"For a convenient choice of $ p'$, this property is satisfied for all $ iI$ at once.",
"We obtain a representation $ B:= iI iB $ in $ Hom(1(T',t0'), GLri$ satisfying the required properties with respect to $ p:=pp'|T'$.$ Finiteness and algebraization Theorem A1 Let $(C,D_C)$ be a stable $n$ -pointed genus $g$ -curve as in Section REF .",
"Let $\\varphi : (\\Sigma _g, Y^n) \\stackrel{\\sim }{\\rightarrow } (C,D_C)$ be an orientation preserving homeomorphism.",
"Let $(E_0, \\nabla _0)$ be an algebraic logarithmic connection over $C$ with polar divisor $D_C$ and denote by $[\\rho _{\\nabla _0}]\\in \\chi _{g,n}(\\mathrm {GL}_r$ its monodromy with respect to $\\varphi $ .",
"Let $\\mathcal {I}_{(C,E_0, \\nabla _0)}=(\\mathcal {F}_{(C,D_C)}, E, \\nabla , \\Psi )$ be an algebraic isomonodromic deformation of $(C,E_0, \\nabla _0)$ with parameter space $T$ as in Section REF .",
"Assume that the classifying map $\\mathrm {class}(\\mathcal {F}) : T\\rightarrow \\mathcal {M}_{g,n}$ is dominant (see Section REF ).",
"Then the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"The orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ does not depend on the choice of $\\varphi $ .",
"Moreover, it is canonically identified, for any $t_1\\in T$ , with the orbit $\\Gamma _{g,n}\\cdot [\\rho _{t_1}]$ of the monodromy of the connection $(E, \\nabla )$ restricted to the fiber over $t_1$ of the family $\\mathcal {F}$ .",
"Since $\\mathrm {class}(\\mathcal {F} )$ is dominant we may assume, without loss of generality, that $\\star :=\\mathrm {class}(\\mathcal {F} )(t_0)\\in \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ .",
"Moreover, up to restricting $\\mathcal {I}_{(C,E_0, \\nabla _0)}$ to a Zariski open neighborhood $\\Delta $ of $t_0$ in $T$ , we may assume that $\\mathrm {class}(\\mathcal {F} )(T)\\, \\cap \\, \\mathcal {B}_{g,n}\\, = \\varnothing .$ Notice that this property, as well as the assumption of $\\mathrm {class}(\\mathcal {F} )$ being dominant is not altered by finite covers and further excision of strict subvarieties not containing $t_0$ .",
"According to Lemma REF , up to such a manipulation, we may assume that $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ admits a section $\\sigma : T\\rightarrow X$ of $\\kappa $ with values in $X^0:=X\\setminus D$ such that $\\sigma (t_0)=\\psi \\circ \\varphi (y_0)$ .",
"Denote by $\\rho $ a representative of the monodromy representation of $(E, \\nabla )$ with respect to the identity such that the restriction of $\\rho $ to the subgroup $(\\psi \\circ \\varphi )_*\\Lambda _{g,n}$ of $\\pi _1(X^0, \\sigma (t_0))$ , given by the inclusion of the central fiber, is identical to $(\\psi \\circ \\varphi )_*\\rho _{\\nabla _0}$ .",
"Such a representative exists, as implies for example Theorem REF .",
"Since the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ does not depend on the choice of $\\varphi $ , we may assume that $\\varphi $ is convenient in the sense of Proposition REF .",
"We then have a semi-direct product decomposition $ \\pi _1(X^0,\\sigma (t_0)) =(\\psi \\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0)\\, ,$ where we have two different expressions for its structure morphism $\\eta $ , proving that ${H}:= \\mathrm {taut}_{g,n+1}({\\mathrm {cl}^\\bullet }_*\\pi _1(T,t_0))\\subset \\Gamma _{g,n+1}\\, $ acts on $\\rho _{\\nabla _0}\\in \\operatorname{Hom}(\\Lambda _{g,n}, \\mathrm {GL}_r$ by conjugation.",
"More precisely, for all $\\alpha \\in \\Lambda _{g,n}$ and $[h]= \\mathrm {taut}_{g,n+1}({\\mathrm {cl}^\\bullet }_*\\, \\beta )\\, \\in {H}$ , we have $\\rho _{\\nabla _0}\\left(\\mathfrak {a}(h)(\\alpha )\\right)=\\rho (\\sigma _*\\beta )\\cdot \\rho _{\\nabla _0}(\\alpha )\\cdot \\rho (\\sigma _*\\beta ^{-1})$ and in particular $[h^{-1}]\\cdot [\\rho _{\\nabla _0} ]= [\\rho _{\\nabla _0} ].$ In other words, $H$ is a subgroup of the stabilizer of $[\\rho _{\\nabla _0}]$ in $\\Gamma _{g, n+1}$ .",
"By definition of the mapping class group action, we then have $\\pi (H)\\subset \\mathrm {Stab}_{\\Gamma _{g,n}}[\\rho _{\\nabla _0}]\\, ,$ where $\\pi :\\Gamma _{g, n+1}\\rightarrow \\Gamma _{g,n}$ is the projection forgetting the marking $y_0$ .",
"Since the size of the orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ equals the index of $ \\mathrm {Stab}_{\\Gamma _{g,n}}[\\rho _{\\nabla _0}]$ in $\\Gamma _{g,n}$ , it now suffices to prove that $\\pi (H)$ has finite index in $\\Gamma _{g,n}$ .",
"Denote by $q:\\Gamma _{g,n}\\rightarrow \\Gamma _{g,n}/K_{g,n}$ the quotient by the normal subgroup $K_{g,n}$ , which, by Lemma REF , has order at most 2.",
"Hence for the indices, we have $[\\Gamma _{g,n}: \\pi (H)] \\le 2\\cdot [\\Gamma _{g,n}/K_{g,n} : q(\\pi (H))]\\, .$ We have a commutative diagram ${\\Gamma _{g,n+1}[d]_{\\pi }&&&& \\pi _1(T,t_0)[llll]_{\\mathrm {taut}_{g,n+1}\\, \\circ \\, {\\mathrm {cl}^\\bullet }_*}[d]^{\\mathrm {taut}_{g,n}\\, \\circ \\, \\mathrm {cl}_*}\\\\\\Gamma _{g,n}[rrrr]^{q}&&&& \\Gamma _{g,n}/K_{g,n}\\, ,}$ where $\\mathrm {cl}: T\\rightarrow \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ denotes the corestriction of $\\mathrm {class}(\\mathcal {F})$ .",
"On the other hand, by the dominance assumption and , the subgroup $\\mathrm {cl}_*\\, \\pi _1(T,t_0)$ of $\\pi _1\\left(\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n},\\star \\right)$ is of finite index.",
"In particular, since the tautological morphism $\\mathrm {taut}_{g,n}: \\pi _1\\left(\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n},\\star \\right) \\twoheadrightarrow \\Gamma _{g,n}/K_{g,n}$ is onto, the subgroup $q(\\pi (H))=\\mathrm {taut}_{g,n}(\\mathrm {cl}_*\\pi _1(T,t_0))$ of $\\Gamma _{g,n}/K_{g,n}$ has finite index.",
"Theorem A2 Let $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ be an algebraic family of stable $n$ -pointed genus-$g$ curves with central fiber $(C,D_C)$ as in Section REF .",
"Let $(E_0, \\nabla _0)$ be an algebraic logarithmic connection over $C$ with polar divisor $D_C$ and denote by $[\\rho _{\\nabla _0}]\\in \\chi _{g,n}(\\mathrm {GL}_r$ its monodromy with respect to an orientation preserving homeomorphism $\\varphi : (\\Sigma _g, Y^n) \\stackrel{\\sim }{\\rightarrow } (C,D_C)$ .",
"Assume that $(E_0, \\nabla _0)$ is mild, $r=2$ or $\\rho _{\\nabla _0}$ is semisimple, and the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"Then there are an étale base change $p: (T^{\\prime },t_0^{\\prime })\\rightarrow (T,t_0)$ and a flat algebraic logarithmic connection $(E,\\nabla )$ over $X^{\\prime }:=p^*X$ with polar divisor $p^*D$ , such that $\\psi ^*(E,\\nabla )|_{X^{\\prime }_{t_0^{\\prime }}}$ is isomorphic to $(E_0, \\nabla _0)$ .",
"Since $(C,D_C)$ is stable by assumption, it only admits a finite number of automorphisms.",
"Let $x_0\\in C\\setminus D_C$ be a point fixed by no automorphism other than the identity.",
"Up to isotopy, we may assume $\\varphi (y_0)=x_0$ .",
"Let $\\mathbf {x}^\\bullet $ be the labelling of $D_C^\\bullet =D_C+x_0$ induced by $\\varphi $ .",
"By construction, we have $\\star :=[C,\\mathbf {x}^\\bullet ]\\in \\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}$ .",
"Up to an étale base change, we may assume, by Lemma REF , that there is a section $\\sigma : T\\rightarrow X$ of $\\kappa $ with values in $X^0:=X\\setminus D$ and such that $\\sigma (t_0)=\\psi (x_0)$ .",
"With the notation of Section REF , we may consider the family of $n+1$ -pointed genus-$g$ curves $\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}=(\\kappa : X\\rightarrow T, D+\\sigma (T), t_0, \\psi )$ .",
"Altering neither $D_C^\\bullet $ , nor the labelling, nor the orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ , we may assume that $\\varphi $ is conveniently chosen in the sense of Proposition REF .",
"We then have a semi-direct product decomposition $ \\pi _1(X\\setminus D,\\sigma (t_0)) =(\\psi \\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0),$ where $\\eta (\\sigma _*\\beta )((\\psi \\circ \\varphi )_*\\alpha ) =\\sigma _*\\beta \\cdot (\\psi \\circ \\varphi )_*\\alpha \\cdot \\sigma _*\\beta ^{-1} = (\\psi \\circ \\varphi )_*\\mathfrak {a}(\\theta _*\\beta )(\\alpha )\\, $ and $\\theta : = \\mathrm {taut}_{g,n+1}\\circ {\\mathrm {cl}^\\bullet }_* : \\pi _1(T,t_0) \\rightarrow \\Gamma _{g,n+1}.$ Since the $\\Gamma _{g,n+1}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite, the stabilizer $H:=\\mathrm {Stab}_{\\Gamma _{g,n+1}}[\\rho _{\\nabla _0}]$ of the conjugacy class of $\\rho _{\\nabla _0}$ under the action of $\\Gamma _{g,n+1}$ has finite index in $\\Gamma _{g,n+1}$ .",
"Since the tautological morphism is onto, the subgroup $\\mathrm {taut}_{g,n+1}^{-1}(H)$ of $\\pi _1(\\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}, \\star )$ then has also finite index.",
"In particular, there is a finite connected étale cover $q : (U,u_0)\\rightarrow (\\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}, \\star )$ such that $\\pi _1(U,u_0)= \\mathrm {taut}_{g,n+1}^{-1}(H)$ .",
"Now consider the fibered product ${(T^{\\prime },t_0^{\\prime })[r]^{p}[d]&(T,t_0)[d]^{\\mathrm {class}(\\mathcal {F}^\\bullet )}\\\\(U,u_0)[r]^{\\hspace{-15.0pt}q}&(\\mathcal {M}_{g,n+1},\\star ).",
"}$ We denote the pullback family of curves by $\\mathcal {F}_{(C,D_C^\\bullet )}^{\\prime }=(\\kappa ^{\\prime } : X^{\\prime }\\rightarrow T^{\\prime }, D^{\\prime } +\\sigma ^{\\prime }(T^{\\prime }), t^{\\prime }_0, \\psi ^{\\prime }):=p^*\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}$ .",
"We further denote $\\mathrm {cl}^{\\prime }=\\mathrm {cl}^\\bullet \\circ p$ , which is the corestriction of $\\mathrm {class}(\\mathcal {F}^{\\prime })$ .",
"By construction, the morphism $\\theta ^{\\prime } : =\\theta \\circ p= \\mathrm {taut}_{g,n+1}\\circ {\\mathrm {cl}^{\\prime }}_* : \\pi _1(T^{\\prime },t_0^{\\prime }) \\rightarrow \\Gamma _{g,n+1}$ takes values in $H$ .",
"Again up to an étale base change of $(T^{\\prime },t_0^{\\prime })$ , by Proposition REF and Lemma REF , there is a representation $\\rho _B\\in \\operatorname{Hom}(\\pi _1(T^{\\prime },t_0^{\\prime }),\\mathrm {GL}_r$ such that for all $\\beta \\in \\pi _1(T^{\\prime },t_0^{\\prime })$ , $\\alpha \\in \\Lambda _{g,n}$ , we have $\\left([\\theta ^{\\prime }_* \\beta ]^{-1}\\cdot \\rho _{\\nabla _0}\\right) (\\alpha ) = \\rho _B(\\beta ) \\cdot \\rho _{\\nabla _0}(\\alpha ) \\cdot \\rho _B(\\beta ^{-1}) \\, .",
"$ Since by definition $\\left([\\theta ^{\\prime }_*\\beta ]^{-1}\\cdot \\rho _{\\nabla _0}\\right) (\\alpha )= \\rho _{\\nabla _0}(\\mathfrak {a}(\\theta ^{\\prime }_*\\beta )(\\alpha ))$ , we obtain a well defined representation $\\rho : \\left\\lbrace \\begin{array}{rcl} \\pi _1(X^{\\prime }\\setminus D^{\\prime },\\sigma ^{\\prime }(t_0^{\\prime })) & \\rightarrow & \\mathrm {GL}_r{.1cm}\\\\(\\psi ^{\\prime }\\circ \\varphi )_*\\alpha \\cdot \\sigma ^{\\prime }_*\\beta & \\mapsto & \\rho _{\\nabla _0}(\\alpha ) \\cdot \\rho _B(\\beta )\\end{array} \\right.\\, $ (see Lemma REF ) with respect to the semi-direct product decomposition $ \\pi _1(X^{\\prime }\\setminus D^{\\prime },\\sigma ^{\\prime }(t_0^{\\prime })) =(\\psi ^{\\prime }\\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma ^{\\prime }_*\\pi _1(T^{\\prime },t_0^{\\prime })$ .",
"By construction, $\\rho $ extends $ \\rho _{\\nabla _0}$ .",
"We conclude by the logarithmic Riemann-Hilbert correspondence (see Theorem REF ).",
"Let us first prove the implication $(\\ref {algitem alg})\\Rightarrow (\\ref {algitem finite})$ .",
"Let $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C,E_0, \\nabla _0)}=$ $(\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}, E, $ $\\nabla , \\Psi )$ be an algebraic universal isomonodromic deformation of $(C,E_0, \\nabla _0)$ as in Section REF .",
"Then by definition, the family $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ is Kuranishi.",
"In particular, the classifying map $\\mathrm {class}(\\mathcal {F}^{\\mathrm {Kur}} ) : T\\rightarrow \\mathcal {M}_{g,n}$ is dominant.",
"Then by Theorem A1, the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"Let us now prove the implication $(\\ref {algitem finite})\\Rightarrow (\\ref {algitem alg})$ .",
"Let $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ be any algebraic Kuranishi family with central fiber $(C,D_C)$ as in Section REF .",
"Note that such a family exists since $(C,D_C)$ is stable, and that it remains Kuranishi after pullback via an étale base change.",
"Up to such a manipulation, according to Theorem A2, the family $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ can be endowed with a flat algebraic logarithmic connection $(E,\\nabla )$ over $X$ with polar divisor $D$ such that there is an isomorphism $\\Psi : (E_0,\\nabla _0)\\rightarrow (E, \\nabla )|_{X_{t_0}}$ commuting with $\\psi $ via the natural projections to $(C,D_C)$ and $(X_{t_0}, D|_{X_{t_0}})$ respectively.",
"Now $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C,E_0, \\nabla _0)}:=(\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}, E, \\nabla , \\Psi )$ defines an algebraic universal isomonodromic deformation of $(C,E_0,\\nabla _0)$ (see Section REF ).",
"Dynamics Effective description of the mapping class group action In this section we describe the action of $\\hat{\\Gamma }_{g,n}$ on $\\Lambda _{g,n}$ in terms of specified generators for both groups.",
"Presentation of the fundamental group To give an effective description of $\\Lambda _{g,n}$ and how $\\hat{\\Gamma }_{g,n}$ acts, we will assume that $\\Sigma _g$ is the subsurface of genus $g$ of $\\mathbb {R}^3$ depicted in Figure $\\ref {figgens 1}$ .",
"On this surface we also depicted, in gray, an embedded closed disk $\\bar{\\Delta }\\subset \\Sigma _g$ , we will denote $\\Delta $ its interior.",
"We fix $n$ and we consider a subset $Y^n=\\lbrace y_1, \\ldots , y_n\\rbrace \\subset \\Delta $ of cardinality $n$ , as well as a point $y_{0}\\in \\bar{\\Delta }\\setminus \\Delta $ .",
"We have $\\pi _1(\\Sigma _g\\setminus \\Delta , y_0)=\\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g, \\delta ~|~ [\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]=\\delta ^{-1} \\right\\rangle ,$ where the mentioned generators correspond to the loops in Figure $\\ref {figgens 1}$ .",
"Figure: Preferred elements of the fundamental group, IThe loops in Figure $\\ref {figgens 2}$ correspond to the following presentation.",
"$\\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})=\\left\\langle \\gamma _1, \\ldots , \\gamma _n, \\delta ~|~ \\gamma _1\\cdots \\gamma _n=\\delta \\right\\rangle .$ Figure: Preferred elements of the fundamental group, IIBy the Van Kampen theorem, we have $\\begin{array}{rcl}\\Lambda _{g,n}&=&\\pi _1(\\Sigma \\setminus \\Delta , y_{0})*_\\delta \\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})\\vspace{5.69046pt}\\\\&=&\\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g,\\gamma _1,\\ldots ,\\gamma _n ~\\vert ~ \\gamma _1\\cdots \\gamma _n=\\left([\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]\\right)^{-1}\\right\\rangle .\\end{array}$ In the sequel, writing “the generators\" of $\\Lambda _{g,n}$ , we will refer to the above $(\\alpha _i)_{i\\in \\llbracket 1, g\\rrbracket }\\, , \\, (\\beta _i)_{i\\in \\llbracket 1, g\\rrbracket }\\, , \\, (\\gamma _j)_{j\\in \\llbracket 1, n\\rrbracket }\\, .$ Mapping class group generators We define $\\Gamma _{g}^1$ to be the mapping class group of orientation preserving homeomorphisms of $\\Sigma \\setminus \\Delta $ that restrict to the identity on $\\partial \\Delta $ .",
"Continuating such homeomorphisms by the identity on $\\Delta $ , we get a morphism $\\varphi _{g}: \\Gamma _{g}^1\\rightarrow \\hat{\\Gamma }_{g,n}^{\\bullet }\\, .$ After Lickorish (see also ), the group $\\Gamma _{g}^1$ is generated by the (right) Dehn-twists along the loops $\\tau _1, \\ldots ,\\tau _{3g-1}$ represented in Figure REF .",
"Figure: Dehn-twistsA right Dehn twist acts on paths which cross the corresponding Dehn curve as depicted in Figure REF .",
"This action can be summarized as “a path crossing the Dehn curve has to turn right”.",
"A left Dehn twist is the inverse of a right Dehn twist.",
"Figure: Dehn-twist actionOne can now easily check the following.",
"Lemma 6.2.1 (Dehn-twists) The action of the Dehn twists above on the fundamental group $\\pi _1(\\Sigma _g\\setminus \\Delta , y_{0})$ is given in Table $\\ref {DehnAction1}$ , where we only indicate the non-trivial actions on the generators.",
"Here for $\\tau _{2k-1}$ we give the formula for the left Dehn twist.",
"The other generators all correspond to right Dehn twists.",
"Moreover, for $k\\in \\llbracket 1, g-1\\rrbracket $ , the element $\\Theta _k$ described in Table $\\ref {DehnAction1}$ is fixed by $\\tau _{2g+k}$ .",
"Table: NO_CAPTIONOn the other hand, one can define the mapping class group of orientation preserving homeomorphisms of $\\bar{\\Delta }$ that preserve the set $Y^n$ and restrict to the identity on $\\partial \\Delta $ .",
"It is classically called the braid group on $n$ strands and denoted $B_n$ .",
"Continuating such homeomorphisms by the identity on the complement of $\\Delta $ in $\\Sigma _g$ , we get a morphism $\\varphi _{0}: B_n\\rightarrow \\hat{\\Gamma }_{g,n}^{\\bullet }.$ After Artin , the group $B_n$ is generated by half-twists $\\sigma _1, \\ldots , \\sigma _{n-1}$ , whose action is depicted in Figure REF .",
"Figure: half-twistsLemma 6.2.2 (half-twists) The action of $B_n=\\langle \\sigma _1, \\ldots , \\sigma _{n-1}\\rangle $ on the fundamental group $\\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})$ is described in Table $\\ref {DehnAction2}$ , where we only indicate the non-trivial actions on the generators.",
"Moreover, Table $\\ref {DehnAction2}$ indicates the action of $\\sigma _{cycl}:=\\sigma _{n-1}\\circ \\cdots \\circ \\sigma _{1} \\in B_n$ and some of its powers.",
"Table: Proof of Theorem"
],
[
"Necessary and sufficient conditions for algebraizability",
"We shall see in Section REF that Theorem REF is a corollary of the juxtaposition of Theorem A1, showing that our algebraizability criterion for germs of universal isomonodromic deformations is necessary, and Theorem A2, showing that it is also sufficient.",
"We have already established the main ingredients for the proofs of both theorems.",
"For Theorem A2, we moreover need a representation-theoretical result developed in Section REF ."
],
[
"Extensions of representations",
"We shall now consider the problem of extending a representation of the fundamental group of a fiber of a family of pointed curves to a representation for the whole family in light of Lemma REF and Proposition REF .",
"We begin with the elementary case of non semisimple rank 2 representations.",
"Let $A,B$ be groups.",
"Consider a representation $\\rho \\in \\operatorname{Hom}(A ,\\mathrm {Upp})$ , where $\\mathrm {Upp}$ is the group of invertible upper triangular matrices of rank 2.",
"To such a representation, we may associate two other ones : the scalar part $\\rho _{*} :\\alpha \\mapsto \\rho (\\alpha )_{2,2}$ and the affine part $\\rho _{\\mathrm {Aff}}:=\\rho _{*}^{-1} \\otimes \\rho $ .",
"The latter takes values in $\\mathrm {Aff}(:=\\lbrace (a_{i,j})\\in \\mathrm {Upp}~|~a_{2,2}=1\\rbrace $ which is isomorphic to the affine group of the complex line.",
"Lemma 5.1.1 Let $\\rho =\\rho _{*}\\otimes \\rho _{\\mathrm {Aff}}$ and $\\rho ^{\\prime } =\\rho ^{\\prime }_{*}\\otimes \\rho ^{\\prime }_{\\mathrm {Aff}}$ as above, and assume that they are not semisimple.",
"We have $[\\rho ]=[\\rho ^{\\prime }] \\in \\operatorname{Hom}(A , \\mathrm {GL}_{2}/\\mathrm {GL}_{2} if and only if $ *='*$ and$ [Aff]=['Aff] Hom(A , Aff(C))/Aff(C)$.$ The ”if”-part is trivial.",
"Assume $[\\rho ]=[\\rho ^{\\prime }]$ .",
"Since they take values in $\\mathrm {Upp}$ , both representations $\\rho $ and $\\rho ^{\\prime }$ leave the line $\\mathrm {span}(e_1)$ of 2 invariant.",
"By non semisimplicity, for each of the representations, there is no other globally invariant line.",
"Let $M=(m_{i,j})\\in \\mathrm {GL}_2 conjugate both representations.",
"Then $ M$ must leave $ span(e1)$ invariant, \\textit {i.e.}",
"$ MUpp$.",
"As the scalars are central in $ GL2, the element $M/m_{2,2}\\in \\mathrm {Aff}(\\mathbb {C})$ conjugates both representations.",
"In particular $\\rho _{*}=\\rho ^{\\prime }_{*}$ and $M/m_{2,2}$ conjugates $\\rho _{\\mathrm {Aff}}$ and $\\rho ^{\\prime }_{\\mathrm {Aff}}$ .",
"Lemma 5.1.2 Let $\\rho _A \\in \\operatorname{Hom}(A,\\mathrm {GL}_{2}$ be non semisimple.",
"Let $\\theta \\in \\operatorname{Hom}(B, \\mathrm {Aut}(A))$ such that for all $h\\in \\mathrm {Im}(\\theta )$ , we have $[\\rho _A]=h\\cdot [\\rho _A]:=[\\rho _A\\circ h^{-1}]$ .",
"Then there exists a representation $\\rho _B \\in \\operatorname{Hom}(B,\\mathrm {GL}_{2}$ such that $\\rho _A(\\theta (\\beta )^{-1}(\\alpha ))=\\rho _B(\\beta )^{-1} \\rho _A(\\alpha ) \\rho _B(\\beta ) \\quad \\forall \\alpha \\in A\\, , \\beta \\in B\\, .$ We may assume that $\\rho _A$ takes values in $\\mathrm {Upp}$ .",
"By assumption, for each $\\beta \\in B$ , there exists a matrix $M_\\beta \\in \\mathrm {GL}_{2} such that $ A(()-1())=M-1 A()M$.",
"By Lemma \\ref {lemred}, we may assume $ MAff(C)$.",
"If $ Im(A)Upp$ is non abelian, then it has trivial centralizer and the matrices $ MAff(C)$ are uniquely defined.",
"Otherwise, we have $ Im(A){ ( 1 01) | * , }$ and the matrices $ M$ are uniquely defined if we impose$ M{ ( 0 01) | *}$.",
"It is now straightforward to check that given these choices, the well-defined map $ M$ is a morphism of groups.",
"$ For a similar result for semisimple representations $\\rho _A$ (of arbitrary rank), the group $B$ , which in our case will be the fundamental group of a parameter space, might have to be enlarged, in order to take into account the non-unicity of the matrices $M_\\beta $ due to possible permutations of irreducible components.",
"Proposition 5.1.3 Let $\\rho _A \\in \\operatorname{Hom}(A, \\mathrm {GL}_r$ be semisimple.",
"Let $ (T,t_0)$ be a smooth connected quasi-projective variety, and let $\\theta \\in \\operatorname{Hom}(\\pi _1(T,t_0),{\\mathrm {Aut}(A)})$ such that $H:=\\mathrm {Im}(\\theta )$ stabilizes $[\\rho _A]$ .",
"Then there is an étale base change $p : (T^{\\prime },t_0^{\\prime }) \\rightarrow (T,t_0)$ and a representation $\\rho _B \\in \\operatorname{Hom}(\\pi _1(T^{\\prime },t_0^{\\prime }), \\mathrm {GL}_r$ such that $ \\rho _A (\\theta (p_*\\beta )^{-1} (\\alpha ) )= \\rho _B(\\beta )^{-1} \\cdot \\rho _A(\\alpha ) \\cdot \\rho _B(\\beta ) \\quad \\forall \\alpha \\in A\\, , \\beta \\in \\pi _1(T^{\\prime },t_0^{\\prime })\\, .",
"$ Let $\\rho _A = \\bigoplus _{i\\in I} \\rho ^i_A $ be a decomposition such that each $\\rho ^i_A$ is irreducible.",
"The subgroup $\\bigcap _{i\\in I} \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho ^i_A] \\subset \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho _A]\\, , $ stabilizing the conjugacy class $[\\rho ^i_A]$ for each $i\\in I$ , is of finite index (see for example ).",
"Hence the subgroup $\\widetilde{H}:=H \\cap _{i\\in I} \\mathrm {Stab}_{{\\mathrm {Aut}(A)}}[\\rho ^i_A]$ is of finite index in $H$ .",
"Consider now the finite connected unramified covering $\\tilde{p}: (\\widetilde{T},\\tilde{t}_0)\\rightarrow (T,t_0)$ characterized by $\\tilde{p}_*\\pi _1(\\widetilde{T},\\tilde{t}_0)=\\theta ^{-1}(\\widetilde{H})$ .",
"Note that $\\tilde{p}$ induces a structure of smooth quasi-projective variety on $\\widetilde{T}$ .",
"Since $\\widetilde{H}$ stabilizes $[\\rho ^i_A]$ , for every $h\\in \\widetilde{H}$ and every $i\\in I$ , there is a matrix $M^i_{h}\\in \\mathrm {GL}_{r_i} such that\\begin{equation} (M^i_h)^{-1} \\cdot \\rho ^i_A \\cdot M_h^i= [h]\\cdot \\rho ^i_A\\, .\\end{equation}Given $ i$ and $ h$, the choice of $ Mhi$ is unique up to an element of the centralizer of $ iA$.",
"Since $ iA$ is irreducible, this centralizer is given by the set of scalar matrices.",
"Denote by $MhiPGLri the projectivization of $ {M_h^i}\\in \\mathrm {GL}_{r_i}.",
"Then $Bi: M*p*i$ is a well defined element of $ Hom(1(T,t0), PGLri$.",
"According to the Lifting Theorem \\cite [Th.",
"$ 3.1$]{MR3300949}, there exists a Zariski closed subset $ Z$ of $ T$ not containing $ t0$, a finite morphism of smooth quasi-projective varieties $$p^{\\prime } : (\\widetilde{T}^{\\prime },t_0^{\\prime }) \\rightarrow ( \\widetilde{T}\\setminus \\widetilde{Z}, \\tilde{t}_0)\\, , $$ étale in a neighborhood $ T'$ of $ t0'$, and a representation $ iB Hom(1(T',t0'), GLri$ whose projectivization is $ p'*Bi$.",
"For a convenient choice of $ p'$, this property is satisfied for all $ iI$ at once.",
"We obtain a representation $ B:= iI iB $ in $ Hom(1(T',t0'), GLri$ satisfying the required properties with respect to $ p:=pp'|T'$.$ Finiteness and algebraization Theorem A1 Let $(C,D_C)$ be a stable $n$ -pointed genus $g$ -curve as in Section REF .",
"Let $\\varphi : (\\Sigma _g, Y^n) \\stackrel{\\sim }{\\rightarrow } (C,D_C)$ be an orientation preserving homeomorphism.",
"Let $(E_0, \\nabla _0)$ be an algebraic logarithmic connection over $C$ with polar divisor $D_C$ and denote by $[\\rho _{\\nabla _0}]\\in \\chi _{g,n}(\\mathrm {GL}_r$ its monodromy with respect to $\\varphi $ .",
"Let $\\mathcal {I}_{(C,E_0, \\nabla _0)}=(\\mathcal {F}_{(C,D_C)}, E, \\nabla , \\Psi )$ be an algebraic isomonodromic deformation of $(C,E_0, \\nabla _0)$ with parameter space $T$ as in Section REF .",
"Assume that the classifying map $\\mathrm {class}(\\mathcal {F}) : T\\rightarrow \\mathcal {M}_{g,n}$ is dominant (see Section REF ).",
"Then the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"The orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ does not depend on the choice of $\\varphi $ .",
"Moreover, it is canonically identified, for any $t_1\\in T$ , with the orbit $\\Gamma _{g,n}\\cdot [\\rho _{t_1}]$ of the monodromy of the connection $(E, \\nabla )$ restricted to the fiber over $t_1$ of the family $\\mathcal {F}$ .",
"Since $\\mathrm {class}(\\mathcal {F} )$ is dominant we may assume, without loss of generality, that $\\star :=\\mathrm {class}(\\mathcal {F} )(t_0)\\in \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ .",
"Moreover, up to restricting $\\mathcal {I}_{(C,E_0, \\nabla _0)}$ to a Zariski open neighborhood $\\Delta $ of $t_0$ in $T$ , we may assume that $\\mathrm {class}(\\mathcal {F} )(T)\\, \\cap \\, \\mathcal {B}_{g,n}\\, = \\varnothing .$ Notice that this property, as well as the assumption of $\\mathrm {class}(\\mathcal {F} )$ being dominant is not altered by finite covers and further excision of strict subvarieties not containing $t_0$ .",
"According to Lemma REF , up to such a manipulation, we may assume that $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ admits a section $\\sigma : T\\rightarrow X$ of $\\kappa $ with values in $X^0:=X\\setminus D$ such that $\\sigma (t_0)=\\psi \\circ \\varphi (y_0)$ .",
"Denote by $\\rho $ a representative of the monodromy representation of $(E, \\nabla )$ with respect to the identity such that the restriction of $\\rho $ to the subgroup $(\\psi \\circ \\varphi )_*\\Lambda _{g,n}$ of $\\pi _1(X^0, \\sigma (t_0))$ , given by the inclusion of the central fiber, is identical to $(\\psi \\circ \\varphi )_*\\rho _{\\nabla _0}$ .",
"Such a representative exists, as implies for example Theorem REF .",
"Since the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ does not depend on the choice of $\\varphi $ , we may assume that $\\varphi $ is convenient in the sense of Proposition REF .",
"We then have a semi-direct product decomposition $ \\pi _1(X^0,\\sigma (t_0)) =(\\psi \\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0)\\, ,$ where we have two different expressions for its structure morphism $\\eta $ , proving that ${H}:= \\mathrm {taut}_{g,n+1}({\\mathrm {cl}^\\bullet }_*\\pi _1(T,t_0))\\subset \\Gamma _{g,n+1}\\, $ acts on $\\rho _{\\nabla _0}\\in \\operatorname{Hom}(\\Lambda _{g,n}, \\mathrm {GL}_r$ by conjugation.",
"More precisely, for all $\\alpha \\in \\Lambda _{g,n}$ and $[h]= \\mathrm {taut}_{g,n+1}({\\mathrm {cl}^\\bullet }_*\\, \\beta )\\, \\in {H}$ , we have $\\rho _{\\nabla _0}\\left(\\mathfrak {a}(h)(\\alpha )\\right)=\\rho (\\sigma _*\\beta )\\cdot \\rho _{\\nabla _0}(\\alpha )\\cdot \\rho (\\sigma _*\\beta ^{-1})$ and in particular $[h^{-1}]\\cdot [\\rho _{\\nabla _0} ]= [\\rho _{\\nabla _0} ].$ In other words, $H$ is a subgroup of the stabilizer of $[\\rho _{\\nabla _0}]$ in $\\Gamma _{g, n+1}$ .",
"By definition of the mapping class group action, we then have $\\pi (H)\\subset \\mathrm {Stab}_{\\Gamma _{g,n}}[\\rho _{\\nabla _0}]\\, ,$ where $\\pi :\\Gamma _{g, n+1}\\rightarrow \\Gamma _{g,n}$ is the projection forgetting the marking $y_0$ .",
"Since the size of the orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ equals the index of $ \\mathrm {Stab}_{\\Gamma _{g,n}}[\\rho _{\\nabla _0}]$ in $\\Gamma _{g,n}$ , it now suffices to prove that $\\pi (H)$ has finite index in $\\Gamma _{g,n}$ .",
"Denote by $q:\\Gamma _{g,n}\\rightarrow \\Gamma _{g,n}/K_{g,n}$ the quotient by the normal subgroup $K_{g,n}$ , which, by Lemma REF , has order at most 2.",
"Hence for the indices, we have $[\\Gamma _{g,n}: \\pi (H)] \\le 2\\cdot [\\Gamma _{g,n}/K_{g,n} : q(\\pi (H))]\\, .$ We have a commutative diagram ${\\Gamma _{g,n+1}[d]_{\\pi }&&&& \\pi _1(T,t_0)[llll]_{\\mathrm {taut}_{g,n+1}\\, \\circ \\, {\\mathrm {cl}^\\bullet }_*}[d]^{\\mathrm {taut}_{g,n}\\, \\circ \\, \\mathrm {cl}_*}\\\\\\Gamma _{g,n}[rrrr]^{q}&&&& \\Gamma _{g,n}/K_{g,n}\\, ,}$ where $\\mathrm {cl}: T\\rightarrow \\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n}$ denotes the corestriction of $\\mathrm {class}(\\mathcal {F})$ .",
"On the other hand, by the dominance assumption and , the subgroup $\\mathrm {cl}_*\\, \\pi _1(T,t_0)$ of $\\pi _1\\left(\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n},\\star \\right)$ is of finite index.",
"In particular, since the tautological morphism $\\mathrm {taut}_{g,n}: \\pi _1\\left(\\mathcal {M}_{g,n}\\setminus \\mathcal {B}_{g,n},\\star \\right) \\twoheadrightarrow \\Gamma _{g,n}/K_{g,n}$ is onto, the subgroup $q(\\pi (H))=\\mathrm {taut}_{g,n}(\\mathrm {cl}_*\\pi _1(T,t_0))$ of $\\Gamma _{g,n}/K_{g,n}$ has finite index.",
"Theorem A2 Let $\\mathcal {F}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ be an algebraic family of stable $n$ -pointed genus-$g$ curves with central fiber $(C,D_C)$ as in Section REF .",
"Let $(E_0, \\nabla _0)$ be an algebraic logarithmic connection over $C$ with polar divisor $D_C$ and denote by $[\\rho _{\\nabla _0}]\\in \\chi _{g,n}(\\mathrm {GL}_r$ its monodromy with respect to an orientation preserving homeomorphism $\\varphi : (\\Sigma _g, Y^n) \\stackrel{\\sim }{\\rightarrow } (C,D_C)$ .",
"Assume that $(E_0, \\nabla _0)$ is mild, $r=2$ or $\\rho _{\\nabla _0}$ is semisimple, and the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"Then there are an étale base change $p: (T^{\\prime },t_0^{\\prime })\\rightarrow (T,t_0)$ and a flat algebraic logarithmic connection $(E,\\nabla )$ over $X^{\\prime }:=p^*X$ with polar divisor $p^*D$ , such that $\\psi ^*(E,\\nabla )|_{X^{\\prime }_{t_0^{\\prime }}}$ is isomorphic to $(E_0, \\nabla _0)$ .",
"Since $(C,D_C)$ is stable by assumption, it only admits a finite number of automorphisms.",
"Let $x_0\\in C\\setminus D_C$ be a point fixed by no automorphism other than the identity.",
"Up to isotopy, we may assume $\\varphi (y_0)=x_0$ .",
"Let $\\mathbf {x}^\\bullet $ be the labelling of $D_C^\\bullet =D_C+x_0$ induced by $\\varphi $ .",
"By construction, we have $\\star :=[C,\\mathbf {x}^\\bullet ]\\in \\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}$ .",
"Up to an étale base change, we may assume, by Lemma REF , that there is a section $\\sigma : T\\rightarrow X$ of $\\kappa $ with values in $X^0:=X\\setminus D$ and such that $\\sigma (t_0)=\\psi (x_0)$ .",
"With the notation of Section REF , we may consider the family of $n+1$ -pointed genus-$g$ curves $\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}=(\\kappa : X\\rightarrow T, D+\\sigma (T), t_0, \\psi )$ .",
"Altering neither $D_C^\\bullet $ , nor the labelling, nor the orbit $\\Gamma _{g,n}\\cdot [\\rho _{\\nabla _0}]$ , we may assume that $\\varphi $ is conveniently chosen in the sense of Proposition REF .",
"We then have a semi-direct product decomposition $ \\pi _1(X\\setminus D,\\sigma (t_0)) =(\\psi \\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma _*\\pi _1(T,t_0),$ where $\\eta (\\sigma _*\\beta )((\\psi \\circ \\varphi )_*\\alpha ) =\\sigma _*\\beta \\cdot (\\psi \\circ \\varphi )_*\\alpha \\cdot \\sigma _*\\beta ^{-1} = (\\psi \\circ \\varphi )_*\\mathfrak {a}(\\theta _*\\beta )(\\alpha )\\, $ and $\\theta : = \\mathrm {taut}_{g,n+1}\\circ {\\mathrm {cl}^\\bullet }_* : \\pi _1(T,t_0) \\rightarrow \\Gamma _{g,n+1}.$ Since the $\\Gamma _{g,n+1}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite, the stabilizer $H:=\\mathrm {Stab}_{\\Gamma _{g,n+1}}[\\rho _{\\nabla _0}]$ of the conjugacy class of $\\rho _{\\nabla _0}$ under the action of $\\Gamma _{g,n+1}$ has finite index in $\\Gamma _{g,n+1}$ .",
"Since the tautological morphism is onto, the subgroup $\\mathrm {taut}_{g,n+1}^{-1}(H)$ of $\\pi _1(\\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}, \\star )$ then has also finite index.",
"In particular, there is a finite connected étale cover $q : (U,u_0)\\rightarrow (\\mathcal {M}_{g,n+1}\\setminus \\mathcal {B}_{g,n+1}, \\star )$ such that $\\pi _1(U,u_0)= \\mathrm {taut}_{g,n+1}^{-1}(H)$ .",
"Now consider the fibered product ${(T^{\\prime },t_0^{\\prime })[r]^{p}[d]&(T,t_0)[d]^{\\mathrm {class}(\\mathcal {F}^\\bullet )}\\\\(U,u_0)[r]^{\\hspace{-15.0pt}q}&(\\mathcal {M}_{g,n+1},\\star ).",
"}$ We denote the pullback family of curves by $\\mathcal {F}_{(C,D_C^\\bullet )}^{\\prime }=(\\kappa ^{\\prime } : X^{\\prime }\\rightarrow T^{\\prime }, D^{\\prime } +\\sigma ^{\\prime }(T^{\\prime }), t^{\\prime }_0, \\psi ^{\\prime }):=p^*\\mathcal {F}^\\bullet _{(C,D_C^\\bullet )}$ .",
"We further denote $\\mathrm {cl}^{\\prime }=\\mathrm {cl}^\\bullet \\circ p$ , which is the corestriction of $\\mathrm {class}(\\mathcal {F}^{\\prime })$ .",
"By construction, the morphism $\\theta ^{\\prime } : =\\theta \\circ p= \\mathrm {taut}_{g,n+1}\\circ {\\mathrm {cl}^{\\prime }}_* : \\pi _1(T^{\\prime },t_0^{\\prime }) \\rightarrow \\Gamma _{g,n+1}$ takes values in $H$ .",
"Again up to an étale base change of $(T^{\\prime },t_0^{\\prime })$ , by Proposition REF and Lemma REF , there is a representation $\\rho _B\\in \\operatorname{Hom}(\\pi _1(T^{\\prime },t_0^{\\prime }),\\mathrm {GL}_r$ such that for all $\\beta \\in \\pi _1(T^{\\prime },t_0^{\\prime })$ , $\\alpha \\in \\Lambda _{g,n}$ , we have $\\left([\\theta ^{\\prime }_* \\beta ]^{-1}\\cdot \\rho _{\\nabla _0}\\right) (\\alpha ) = \\rho _B(\\beta ) \\cdot \\rho _{\\nabla _0}(\\alpha ) \\cdot \\rho _B(\\beta ^{-1}) \\, .",
"$ Since by definition $\\left([\\theta ^{\\prime }_*\\beta ]^{-1}\\cdot \\rho _{\\nabla _0}\\right) (\\alpha )= \\rho _{\\nabla _0}(\\mathfrak {a}(\\theta ^{\\prime }_*\\beta )(\\alpha ))$ , we obtain a well defined representation $\\rho : \\left\\lbrace \\begin{array}{rcl} \\pi _1(X^{\\prime }\\setminus D^{\\prime },\\sigma ^{\\prime }(t_0^{\\prime })) & \\rightarrow & \\mathrm {GL}_r{.1cm}\\\\(\\psi ^{\\prime }\\circ \\varphi )_*\\alpha \\cdot \\sigma ^{\\prime }_*\\beta & \\mapsto & \\rho _{\\nabla _0}(\\alpha ) \\cdot \\rho _B(\\beta )\\end{array} \\right.\\, $ (see Lemma REF ) with respect to the semi-direct product decomposition $ \\pi _1(X^{\\prime }\\setminus D^{\\prime },\\sigma ^{\\prime }(t_0^{\\prime })) =(\\psi ^{\\prime }\\circ \\varphi )_*\\Lambda _{g,n}\\rtimes _{\\eta } \\sigma ^{\\prime }_*\\pi _1(T^{\\prime },t_0^{\\prime })$ .",
"By construction, $\\rho $ extends $ \\rho _{\\nabla _0}$ .",
"We conclude by the logarithmic Riemann-Hilbert correspondence (see Theorem REF ).",
"Let us first prove the implication $(\\ref {algitem alg})\\Rightarrow (\\ref {algitem finite})$ .",
"Let $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C,E_0, \\nabla _0)}=$ $(\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}, E, $ $\\nabla , \\Psi )$ be an algebraic universal isomonodromic deformation of $(C,E_0, \\nabla _0)$ as in Section REF .",
"Then by definition, the family $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ is Kuranishi.",
"In particular, the classifying map $\\mathrm {class}(\\mathcal {F}^{\\mathrm {Kur}} ) : T\\rightarrow \\mathcal {M}_{g,n}$ is dominant.",
"Then by Theorem A1, the $\\Gamma _{g,n}$ -orbit of $[\\rho _{\\nabla _0}]$ in $\\chi _{g,n}(\\mathrm {GL}_r$ is finite.",
"Let us now prove the implication $(\\ref {algitem finite})\\Rightarrow (\\ref {algitem alg})$ .",
"Let $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}=(\\kappa : X\\rightarrow T, D, t_0, \\psi )$ be any algebraic Kuranishi family with central fiber $(C,D_C)$ as in Section REF .",
"Note that such a family exists since $(C,D_C)$ is stable, and that it remains Kuranishi after pullback via an étale base change.",
"Up to such a manipulation, according to Theorem A2, the family $\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}$ can be endowed with a flat algebraic logarithmic connection $(E,\\nabla )$ over $X$ with polar divisor $D$ such that there is an isomorphism $\\Psi : (E_0,\\nabla _0)\\rightarrow (E, \\nabla )|_{X_{t_0}}$ commuting with $\\psi $ via the natural projections to $(C,D_C)$ and $(X_{t_0}, D|_{X_{t_0}})$ respectively.",
"Now $\\mathcal {I}^{\\mathrm {univ, alg}}_{(C,E_0, \\nabla _0)}:=(\\mathcal {F}^{\\mathrm {Kur}}_{(C,D_C)}, E, \\nabla , \\Psi )$ defines an algebraic universal isomonodromic deformation of $(C,E_0,\\nabla _0)$ (see Section REF ).",
"Dynamics Effective description of the mapping class group action In this section we describe the action of $\\hat{\\Gamma }_{g,n}$ on $\\Lambda _{g,n}$ in terms of specified generators for both groups.",
"Presentation of the fundamental group To give an effective description of $\\Lambda _{g,n}$ and how $\\hat{\\Gamma }_{g,n}$ acts, we will assume that $\\Sigma _g$ is the subsurface of genus $g$ of $\\mathbb {R}^3$ depicted in Figure $\\ref {figgens 1}$ .",
"On this surface we also depicted, in gray, an embedded closed disk $\\bar{\\Delta }\\subset \\Sigma _g$ , we will denote $\\Delta $ its interior.",
"We fix $n$ and we consider a subset $Y^n=\\lbrace y_1, \\ldots , y_n\\rbrace \\subset \\Delta $ of cardinality $n$ , as well as a point $y_{0}\\in \\bar{\\Delta }\\setminus \\Delta $ .",
"We have $\\pi _1(\\Sigma _g\\setminus \\Delta , y_0)=\\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g, \\delta ~|~ [\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]=\\delta ^{-1} \\right\\rangle ,$ where the mentioned generators correspond to the loops in Figure $\\ref {figgens 1}$ .",
"Figure: Preferred elements of the fundamental group, IThe loops in Figure $\\ref {figgens 2}$ correspond to the following presentation.",
"$\\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})=\\left\\langle \\gamma _1, \\ldots , \\gamma _n, \\delta ~|~ \\gamma _1\\cdots \\gamma _n=\\delta \\right\\rangle .$ Figure: Preferred elements of the fundamental group, IIBy the Van Kampen theorem, we have $\\begin{array}{rcl}\\Lambda _{g,n}&=&\\pi _1(\\Sigma \\setminus \\Delta , y_{0})*_\\delta \\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})\\vspace{5.69046pt}\\\\&=&\\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g,\\gamma _1,\\ldots ,\\gamma _n ~\\vert ~ \\gamma _1\\cdots \\gamma _n=\\left([\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]\\right)^{-1}\\right\\rangle .\\end{array}$ In the sequel, writing “the generators\" of $\\Lambda _{g,n}$ , we will refer to the above $(\\alpha _i)_{i\\in \\llbracket 1, g\\rrbracket }\\, , \\, (\\beta _i)_{i\\in \\llbracket 1, g\\rrbracket }\\, , \\, (\\gamma _j)_{j\\in \\llbracket 1, n\\rrbracket }\\, .$ Mapping class group generators We define $\\Gamma _{g}^1$ to be the mapping class group of orientation preserving homeomorphisms of $\\Sigma \\setminus \\Delta $ that restrict to the identity on $\\partial \\Delta $ .",
"Continuating such homeomorphisms by the identity on $\\Delta $ , we get a morphism $\\varphi _{g}: \\Gamma _{g}^1\\rightarrow \\hat{\\Gamma }_{g,n}^{\\bullet }\\, .$ After Lickorish (see also ), the group $\\Gamma _{g}^1$ is generated by the (right) Dehn-twists along the loops $\\tau _1, \\ldots ,\\tau _{3g-1}$ represented in Figure REF .",
"Figure: Dehn-twistsA right Dehn twist acts on paths which cross the corresponding Dehn curve as depicted in Figure REF .",
"This action can be summarized as “a path crossing the Dehn curve has to turn right”.",
"A left Dehn twist is the inverse of a right Dehn twist.",
"Figure: Dehn-twist actionOne can now easily check the following.",
"Lemma 6.2.1 (Dehn-twists) The action of the Dehn twists above on the fundamental group $\\pi _1(\\Sigma _g\\setminus \\Delta , y_{0})$ is given in Table $\\ref {DehnAction1}$ , where we only indicate the non-trivial actions on the generators.",
"Here for $\\tau _{2k-1}$ we give the formula for the left Dehn twist.",
"The other generators all correspond to right Dehn twists.",
"Moreover, for $k\\in \\llbracket 1, g-1\\rrbracket $ , the element $\\Theta _k$ described in Table $\\ref {DehnAction1}$ is fixed by $\\tau _{2g+k}$ .",
"Table: NO_CAPTIONOn the other hand, one can define the mapping class group of orientation preserving homeomorphisms of $\\bar{\\Delta }$ that preserve the set $Y^n$ and restrict to the identity on $\\partial \\Delta $ .",
"It is classically called the braid group on $n$ strands and denoted $B_n$ .",
"Continuating such homeomorphisms by the identity on the complement of $\\Delta $ in $\\Sigma _g$ , we get a morphism $\\varphi _{0}: B_n\\rightarrow \\hat{\\Gamma }_{g,n}^{\\bullet }.$ After Artin , the group $B_n$ is generated by half-twists $\\sigma _1, \\ldots , \\sigma _{n-1}$ , whose action is depicted in Figure REF .",
"Figure: half-twistsLemma 6.2.2 (half-twists) The action of $B_n=\\langle \\sigma _1, \\ldots , \\sigma _{n-1}\\rangle $ on the fundamental group $\\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})$ is described in Table $\\ref {DehnAction2}$ , where we only indicate the non-trivial actions on the generators.",
"Moreover, Table $\\ref {DehnAction2}$ indicates the action of $\\sigma _{cycl}:=\\sigma _{n-1}\\circ \\cdots \\circ \\sigma _{1} \\in B_n$ and some of its powers.",
"Table: Proof of Theorem"
],
[
"Effective description of the mapping class group action",
"In this section we describe the action of $\\hat{\\Gamma }_{g,n}$ on $\\Lambda _{g,n}$ in terms of specified generators for both groups."
],
[
"Presentation of the fundamental group",
"To give an effective description of $\\Lambda _{g,n}$ and how $\\hat{\\Gamma }_{g,n}$ acts, we will assume that $\\Sigma _g$ is the subsurface of genus $g$ of $\\mathbb {R}^3$ depicted in Figure $\\ref {figgens 1}$ .",
"On this surface we also depicted, in gray, an embedded closed disk $\\bar{\\Delta }\\subset \\Sigma _g$ , we will denote $\\Delta $ its interior.",
"We fix $n$ and we consider a subset $Y^n=\\lbrace y_1, \\ldots , y_n\\rbrace \\subset \\Delta $ of cardinality $n$ , as well as a point $y_{0}\\in \\bar{\\Delta }\\setminus \\Delta $ .",
"We have $\\pi _1(\\Sigma _g\\setminus \\Delta , y_0)=\\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g, \\delta ~|~ [\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]=\\delta ^{-1} \\right\\rangle ,$ where the mentioned generators correspond to the loops in Figure $\\ref {figgens 1}$ .",
"Figure: Preferred elements of the fundamental group, IThe loops in Figure $\\ref {figgens 2}$ correspond to the following presentation.",
"$\\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})=\\left\\langle \\gamma _1, \\ldots , \\gamma _n, \\delta ~|~ \\gamma _1\\cdots \\gamma _n=\\delta \\right\\rangle .$ Figure: Preferred elements of the fundamental group, IIBy the Van Kampen theorem, we have $\\begin{array}{rcl}\\Lambda _{g,n}&=&\\pi _1(\\Sigma \\setminus \\Delta , y_{0})*_\\delta \\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})\\vspace{5.69046pt}\\\\&=&\\left\\langle \\alpha _1, \\beta _1, \\ldots , \\alpha _g, \\beta _g,\\gamma _1,\\ldots ,\\gamma _n ~\\vert ~ \\gamma _1\\cdots \\gamma _n=\\left([\\alpha _1, \\beta _1]\\cdots [\\alpha _g, \\beta _g]\\right)^{-1}\\right\\rangle .\\end{array}$ In the sequel, writing “the generators\" of $\\Lambda _{g,n}$ , we will refer to the above $(\\alpha _i)_{i\\in \\llbracket 1, g\\rrbracket }\\, , \\, (\\beta _i)_{i\\in \\llbracket 1, g\\rrbracket }\\, , \\, (\\gamma _j)_{j\\in \\llbracket 1, n\\rrbracket }\\, .$"
],
[
"Mapping class group generators",
"We define $\\Gamma _{g}^1$ to be the mapping class group of orientation preserving homeomorphisms of $\\Sigma \\setminus \\Delta $ that restrict to the identity on $\\partial \\Delta $ .",
"Continuating such homeomorphisms by the identity on $\\Delta $ , we get a morphism $\\varphi _{g}: \\Gamma _{g}^1\\rightarrow \\hat{\\Gamma }_{g,n}^{\\bullet }\\, .$ After Lickorish (see also ), the group $\\Gamma _{g}^1$ is generated by the (right) Dehn-twists along the loops $\\tau _1, \\ldots ,\\tau _{3g-1}$ represented in Figure REF .",
"Figure: Dehn-twistsA right Dehn twist acts on paths which cross the corresponding Dehn curve as depicted in Figure REF .",
"This action can be summarized as “a path crossing the Dehn curve has to turn right”.",
"A left Dehn twist is the inverse of a right Dehn twist.",
"Figure: Dehn-twist actionOne can now easily check the following.",
"Lemma 6.2.1 (Dehn-twists) The action of the Dehn twists above on the fundamental group $\\pi _1(\\Sigma _g\\setminus \\Delta , y_{0})$ is given in Table $\\ref {DehnAction1}$ , where we only indicate the non-trivial actions on the generators.",
"Here for $\\tau _{2k-1}$ we give the formula for the left Dehn twist.",
"The other generators all correspond to right Dehn twists.",
"Moreover, for $k\\in \\llbracket 1, g-1\\rrbracket $ , the element $\\Theta _k$ described in Table $\\ref {DehnAction1}$ is fixed by $\\tau _{2g+k}$ .",
"Table: NO_CAPTIONOn the other hand, one can define the mapping class group of orientation preserving homeomorphisms of $\\bar{\\Delta }$ that preserve the set $Y^n$ and restrict to the identity on $\\partial \\Delta $ .",
"It is classically called the braid group on $n$ strands and denoted $B_n$ .",
"Continuating such homeomorphisms by the identity on the complement of $\\Delta $ in $\\Sigma _g$ , we get a morphism $\\varphi _{0}: B_n\\rightarrow \\hat{\\Gamma }_{g,n}^{\\bullet }.$ After Artin , the group $B_n$ is generated by half-twists $\\sigma _1, \\ldots , \\sigma _{n-1}$ , whose action is depicted in Figure REF .",
"Figure: half-twistsLemma 6.2.2 (half-twists) The action of $B_n=\\langle \\sigma _1, \\ldots , \\sigma _{n-1}\\rangle $ on the fundamental group $\\pi _1(\\bar{\\Delta }\\setminus Y^n, y_{0})$ is described in Table $\\ref {DehnAction2}$ , where we only indicate the non-trivial actions on the generators.",
"Moreover, Table $\\ref {DehnAction2}$ indicates the action of $\\sigma _{cycl}:=\\sigma _{n-1}\\circ \\cdots \\circ \\sigma _{1} \\in B_n$ and some of its powers.",
"Table: Proof of Theorem"
]
] | 1612.05779 |
[
[
"Colourings with Bounded Monochromatic Components in Graphs of Given\n Circumference"
],
[
"Abstract We prove that every graph with circumference at most $k$ is $O(\\log k)$-colourable such that every monochromatic component has size at most $O(k)$.",
"The $O(\\log k)$ bound on the number of colours is best possible, even in the setting of colourings with bounded monochromatic degree."
],
[
"Colourings with Bounded Monochromatic Components in Graphs of given Circumference Bojan Mohar[3] Bruce Reed[2] David R. Wood[4] 2022/12/09 20:57:39 [3]Department of Mathematics, Simon Fraser University, Burnaby, Canada, ([email protected]).",
"On leave from IMFM, Ljubljana, Slovenia.",
"Research supported by an NSERC Discovery Grant, CRC program, and in part by ARRS, Research Program P1-0297.",
"[2]CNRS, Projet COATI, I3S (CNRS and UNS) UMR7271 and INRIA, Sophia Antipolis, France, ([email protected]).",
"Instituto Nacional de Matemática Pura e Aplicada (IMPA), Brasil.",
"Visiting Research Professor, ERATO Kawarabayashi Large Graph Project, Japan.",
"[4]School of Mathematical Sciences, Monash University, Melbourne, Australia, [email protected].",
"Supported by the Australian Research Council.",
"Abstract.",
"We prove that every graph with circumference at most $k$ is $O(\\log k)$ -colourable such that every monochromatic component has size at most $k$ .",
"The $O(\\log k)$ bound on the number of colours is best possible, even in the setting of colourings with bounded monochromatic degree.",
"In a vertex-coloured graph, a monochromatic component is a connected component of the subgraph induced by all the vertices of one colour.",
"As a relaxation of proper colouring, recent work has focused on graph colourings with monochromatic components of bounded size [1], [13], [17], [8], [14], [3] or bounded monochromatic degree [19], [2], [12], [15], [11], [7], [4], [16], [9], [6], [5], [10].",
"The circumference of a graph $G$ is the length of the longest cycle if $G$ contains a cycle, and is 2 if $G$ is a forest.",
"This paper studies colourings of graphs of given circumference with monochromatic components of bounded size.",
"Our primary goal is to minimize the number of colours, while reducing the order of the monochromatic components is a secondary objective.",
"Let $g(k)$ be the minimum integer $c$ for which there exists an integer $d$ such that every graph with circumference at most $k$ has a $c$ -colouring in which every monochromatic component has order at most $d$ .",
"Our main result is that $g(k)=\\Theta (\\log k)$ .",
"First we prove the upper bound.",
"Theorem 1 For every integer $k \\geqslant 2$ , every graph $G$ with circumference at most $k$ is $(3 \\log _{2} k)$ -colourable such that every monochromatic component has order at most $k$ .",
"This result is implied by the following lemma with $C=\\emptyset $ .",
"A clique is a set of pairwise adjacent vertices.",
"Lemma 2 For every integer $k \\geqslant 2$ , for every graph $G$ with circumference at most $k$ and for every pre-coloured clique $C$ of size at most 2 in $G$ , there is a $\\lfloor {3\\log _2 k}\\rfloor $ -colouring of $G$ such that every monochromatic component has order at most $k$ and every monochromatic component that intersects $C$ is contained in $C$ .",
"We proceed by induction on $k+|V(G)|$ .",
"The result is trivial if $|V(G)|\\leqslant 2$ .",
"Now assume $|V(G)|\\geqslant 3$ .",
"First suppose that $k=2$ .",
"Then $G$ is a forest, which is properly 2-colourable.",
"If $|C|\\le 1$ or $|C|=2$ and two colours are used on $C$ , we obtain the desired colouring (with $2< \\lfloor {3\\log _2 k}\\rfloor $ colours).",
"Otherwise, $|C|=2$ with the same colour on the vertices in $C$ .",
"Contract the edge $C$ and 2-colour the resulting forest by induction, to obtain the desired colouring of $G$ .",
"Now assume that $k \\geqslant 3$ .",
"Suppose that $G$ is not 3-connected.",
"Then $G$ has a minimal separation $(G_1,G_2)$ with $S := V(G_1 \\cap G_2)$ of size at most 2.",
"If $|S|=2$ , then add the edge on $S$ if the edge is not already present.",
"Consider both $G_1$ and $G_2$ to contain this edge.",
"Observe that since the separation is minimal, there is a path in each $G_j$ ($j=1,2$ ) between the two vertices of $S$ .",
"Therefore, adding the edge does not increase the circumference of $G$ .",
"Also note that any valid colouring of the augmented graph will be valid for the original graph.",
"Since $C$ is a clique, we may assume that $C \\subseteq V(G_1)$ .",
"By induction, there is a $\\lfloor {3\\log _2 k}\\rfloor $ -colouring of $G_1$ , with $C$ precoloured, such that every monochromatic component of $G_1$ has order at most $k$ and every monochromatic component of $G_1$ that intersects $C$ is contained in $C$ .",
"This colours $S$ .",
"By induction, there is a $\\lfloor {3\\log _2 k}\\rfloor $ -colouring of $G_2$ , with $S$ precoloured, such that every monochromatic component of $G_2$ has order at most $k$ and every monochromatic component of $G_2$ that intersects $S$ is contained in $S$ .",
"By combining the two colourings, every monochromatic component of $G$ has order at most $k$ and every monochromatic component of $G$ that intersects $C$ is contained in $C$ , as required.",
"Now assume that $G$ is 3-connected.",
"Every 3-connected graph contains a cycle of length at least 4.",
"Thus $k\\geqslant 4$ .",
"If $G$ contains no cycle of length $k$ , then apply the induction hypothesis for $k-1$ ; thus we may assume that $G$ contains a cycle $Q$ of length $k$ .",
"Let $\\mathcal {A}$ be the set of cycles in $G$ of length at least $\\lceil {\\tfrac{1}{2} (k-5)}\\rceil $ .",
"Suppose that a cycle $A \\in \\mathcal {A}$ is disjoint from $Q$ .",
"Since $G$ is 3-connected, there are three disjoint paths between $A$ and $Q$ .",
"It follows that $G$ contains three cycles with total length at least $2(|A|+|Q|+3) > 3k$ .",
"Thus $G$ contains a cycle of length greater than $k$ , which is a contradiction.",
"Hence, every cycle in $\\mathcal {A}$ intersects $Q$ .",
"Let $S:=V(Q)\\cup C$ .",
"As shown above, $G^{\\prime }:=G-S$ contains no cycle of length at least $\\lceil {\\frac{1}{2}(k-5)}\\rceil $ .",
"Then $G^{\\prime }$ has circumference at most $\\lceil {\\frac{1}{2}(k-7)}\\rceil $ , which is at most $\\lfloor {\\tfrac{1}{2} k}\\rfloor $ , which is at least 2.",
"By induction (with no precoloured vertices), there is a $\\lfloor {3\\log _{2} \\lfloor {\\frac{1}{2} k}\\rfloor }\\rfloor $ -colouring of $G^{\\prime }$ such that every monochromatic component of $G^{\\prime }$ has order at most $\\lfloor {\\frac{1}{2} k}\\rfloor $ .",
"Use a set of colours for $G^{\\prime }$ disjoint from the (at most two) preassigned colours for $C$ .",
"Use one new colour for $S\\setminus C$ , which has size at most $k$ .",
"In total, there are at most $\\lfloor {3\\log _2 \\lfloor {\\frac{1}{2} k}\\rfloor }\\rfloor +3 \\leqslant \\lfloor {3\\log _2 k}\\rfloor $ colours.",
"Every monochromatic component of $G$ has order at most $k$ , and every monochromatic component of $G$ that intersects $C$ is contained in $C$ .",
"Note that if $h$ is the function defined by the recurrence, $h(k) :={\\left\\lbrace \\begin{array}{ll}2 & \\text{ if }k=2\\\\5 & \\text{ if } 3\\leqslant k\\leqslant 9\\\\h(\\lceil {\\frac{1}{2} (k-7)}\\rceil )+3 & \\text{ if } k\\geqslant 10,\\end{array}\\right.",
"}$ then $\\lfloor {3\\log _2 k}\\rfloor $ can be replaced by $h(k)$ in Circumference.",
"We now show that the $O(\\log k)$ bound in Circumference is within a constant factor of optimal even in the setting of colourings of bounded monochromatic degree.",
"The following result is implicit in [19].",
"We include the proof for completeness.",
"Proposition 3 For any integers $k,d\\geqslant 1$ there is a graph $G_{k,d}$ with circumference at most $2^k$ , such that every $k$ -colouring of $G_{k,d}$ contains a vertex of monochromatic degree at least $d$ .",
"We proceed by induction on $k\\geqslant 1$ with $d$ fixed (and thus write $G_k$ instead of $G_{k,d}$ ), and with the additional property that $G_k$ contains no path of order $2^{k+1}$ .",
"For the base case, $k=1$ , let $G_1$ be the star $K_{1,d}$ , which has circumference 2 and no path of order 4.",
"Every 1-colouring of $G_1$ contains a vertex of monochromatic degree $d$ .",
"Now assume that $k\\geqslant 2$ and there is a graph $G_{k-1}$ with circumference $2^{k-1}$ and no path of order $2^k$ , such that every $(k-1)$ -colouring of $G_{k-1}$ contains a vertex of monochromatic degree at least $d$ .",
"Let $G_k$ be obtained from $d$ copies of $G_{k-1}$ by adding one new dominant vertex $v$ .",
"If $C$ is a cycle in $G_k$ with length at least $2^k+1$ , then $C$ is contained in one copy of $G_{k-1}$ plus $v$ , and thus $G_{k-1}$ contains a path of order $2^k$ , which is a contradiction.",
"Thus $G_k$ has circumference at most $2^k$ .",
"If $G_k$ contains a path $P$ of order $2^{k+1}$ , then $v$ is in $P$ , otherwise $P$ is contained in some copy of $G_{k-1}$ .",
"Hence $P-v$ includes a path component of order $\\lceil { \\frac{1}{2} (2^{k+1}-1) }\\rceil = 2^k$ contained in a copy of $G_{k-1}$ , which is a contradiction.",
"Hence $G_k$ contains no path of order $2^{k+1}$ .",
"Finally, consider a $k$ -colouring of $G_k$ .",
"Say $v$ is blue.",
"If every copy of $G_{k-1}$ contains a blue vertex, then $v$ has monochromatic degree $d$ , and we are done.",
"Otherwise, some copy of $G_{k-1}$ contains no blue vertex, in which case $G_{k-1}$ is $(k-1)$ -coloured, and thus $G$ contains a monochromatic vertex of degree at least $d$ .",
"Let $f(k)$ be the minimum integer $c$ for which there exists an integer $d$ such that every graph with circumference at most $k$ has a $c$ -colouring in which every monochromatic component has maximum degree at most $d$ .",
"In the language of [19], $f(k)$ is the defective chromatic number of the class of graphs with circumference at most $k$ .",
"Obviously, bounded size implies bounded degree, so $f(k)\\leqslant g(k)$ .",
"Circumference,LowerBound imply that $\\lfloor {\\log _2 k}\\rfloor +1 \\leqslant f(k) \\leqslant g(k)\\leqslant \\lfloor {3\\log _2 k}\\rfloor .$ We conclude this paper by placing our results in the context of a conjecture of [19].",
"The closure of a rooted tree $T$ is the graph obtained from $T$ by adding an edge between each ancestor and descendant.",
"The tree-depth of a graph $H$ , denoted $\\operatorname{td}(H)$ , is the minimum depth of a rooted tree for which $H$ is a subgraph of the closure of $T$ , where the depth of a rooted tree $T$ is the maximum number of vertices in a root-to-leaf path.",
"For a graph $H$ , let $f(H)$ be the minimum integer $c$ such that there exists an integer $d$ such that every $H$ -minor-free graph has a $c$ -colouring in which every monochromatic component has maximum degree at most $d$ .",
"[19] proved that $f(H)\\geqslant \\operatorname{td}(H)-1$ for every connected graph $H$ , and conjectured that $f(H)=\\operatorname{td}(H)-1.$ A graph has circumference $k$ if and only if it contains no $C_{k+1}$ minor; thus $f(k)=f(C_{k+1})$ .",
"It is easily seen that $\\operatorname{td}(C_{k+1})=1+\\lceil { \\log _{2}(k+1)}\\rceil =2+\\lfloor {\\log _2 k}\\rfloor .$ Thus the lower bound $f(H)\\geqslant \\operatorname{td}(H)-1$ , in the case of cycles, is equivalent to the lower bound on $f(k)$ in (REF ).",
"And conjecture (REF ), in the case of cycles, asserts that equality holds.",
"That is, $f(k)=f(C_{k+1})=\\operatorname{td}(C_{k+1})-1=\\lceil { \\log _{2}(k+1)}\\rceil .$ Hence Circumference, which proves that $f(k)\\leqslant \\lfloor {3\\log _2k}\\rfloor $ , is within a factor 3 of conjecture (REF ) for excluded cycles.",
"The best previous upper bound was linear in $k$ .",
"We obtain similar results for graph classes excluding a fixed path, which were identified by [19] as a key case for which their bounds on $f$ were far apart.",
"Let $P_k$ be the path on $k$ vertices.",
"Then $\\operatorname{td}(P_k)=\\lceil { \\log _{2}(k+1)}\\rceil $ ; see [18].",
"Of course, a graph contains a $P_k$ minor if and only if it contains a $P_k$ subgraph.",
"Thus conjecture (REF ), in the case of paths, asserts that $f(P_{k+1})=\\operatorname{td}(P_{k+1})-1=\\lceil { \\log _{2}(k+2)}\\rceil -1.$ Every graph with no $P_{k+1}$ -minor has circumference at most $k$ .",
"Thus Circumference implies that $f(P_{k+1})\\leqslant \\lfloor {3 \\log _2 k}\\rfloor $ , which is within a factor of 3 of conjecture (REF ) for excluded paths.",
"The best previous upper bound was linear in $k$ .",
"paragraph41.5ex plus.5ex minus.2ex-1emAcknowledgement.",
"This research was completed at the Australasian Conference on Combinatorial Mathematical and Combinatorial Computing (40ACCMCC) held at The University of Newcastle, December 2016.",
"Thanks to the conference organisers."
]
] | 1612.05674 |
[
[
"From ultraluminous X-ray sources to ultraluminous supersoft sources: NGC\n 55 ULX, the missing link"
],
[
"Abstract In recent work with high-resolution grating spectrometers (RGS) aboard XMM-Newton Pinto et al.",
"(2016) have discovered that two bright and archetypal ultraluminous X-ray sources (ULXs) have strong relativistic winds in agreement with theoretical predictions of high accretion rates.",
"It has been proposed that such winds can become optically thick enough to block and reprocess the disk X-ray photons almost entirely, making the source appear as a soft thermal emitter or ultraluminous supersoft X-ray source (ULS).",
"To test this hypothesis we have studied a ULX where the wind is strong enough to cause significant absorption of the hard X-ray continuum: NGC 55 ULX.",
"The RGS spectrum of NGC 55 ULX shows a wealth of emission and absorption lines blueshifted by significant fractions of the light speed (0.01 - 0.20)c indicating the presence of a powerful wind.",
"The wind has a complex dynamical structure with the ionization state increasing with the outflow velocity, which may indicate launching from different regions of the accretion disk.",
"The comparison with other ULXs such as NGC 1313 X-1 and NGC 5408 X-1 suggests that NGC 55 ULX is being observed at higher inclination.",
"The wind partly absorbs the source flux above 1 keV, generating a spectral drop similar to that observed in ULSs.",
"The softening of the spectrum at lower (~ Eddington) luminosities and the detection of a soft lag agree with the scenario of wind clumps crossing the line of sight, partly absorbing and reprocessing the hard X-rays from the innermost region."
],
[
"Introduction",
"Ultraluminous X-ray sources (ULXs) are bright, point-like, off-nucleus, extragalactic sources with X-ray luminosities above $10^{39}$ erg/s that result from accretion onto a compact object.",
"Previous studies have shown evidence of accretion onto neutron stars with strong magnetic fields (e.g.",
"Bachetti et al.",
"2014, Israel et al.",
"2016a,b, Fuerst et al.",
"2016).",
"It is suggested that ULXs are also powered by accretion onto stellar-mass black holes $(<100 M_{\\odot })$ at or in excess of the Eddington limit (e.g.",
"King et al.",
"2001, Poutanen et al.",
"2007, Gladstone et al.",
"2009, Middleton et al.",
"2013, Liu et al.",
"2013) or accretion at more sedate Eddington ratios onto intermediate mass black holes ($10^{3-5} M_{\\odot }$ , e.g.",
"Greene and Ho 2007, Farrell et al.",
"2009, Webb at al.",
"2012, Mezcua et al.",
"2016).",
"Ultraluminous supersoft sources (ULSs) are defined by a thermal spectrum with colour temperature $\\sim $ 0.1 keV, bolometric luminosity $\\sim $ a few $10^{39}$ erg/s, and almost no emission above 1 keV (Kong & Di Stefano 2003).",
"Deep exposures of ULSs have shown the presence of a hard tail, possibly due to a disk-like emission similar to ULXs (see, e.g, Urquhart and Soria 2016).",
"Classical X-ray binaries and ULXs, either have broad-band emission over the 1–10 keV range or a peak disk-blackbody temperature of a few keV.",
"Alternative models have tried to describe the X-ray spectra of ULSs with accreting intermediate-mass and stellar-mass black holes like for ULXs or with extreme supersoft sources powered by surface-nuclear-burning on white dwarf accretors (see, e.g., Di Stefano & Kong 2004, Soria & Kong 2016, Feng et al.",
"2016, and references therein).",
"ULXs and ULSs were initially considered different physical types of systems.",
"However, on the one hand, we have seen strong evidence of winds in classical ULXs (Middleton et al.",
"2015b, Pinto et al.",
"2016).",
"On the other hand, some ULSs (e.g.",
"in M 101 and NGC 247) exhibit a harder (fast variable) X-ray tail (e.g., Urquhart and Soria 2016), suggesting that some of the X-ray photons occasionally get through a rapidly changing wind (e.g., Middleton et al.",
"2011 and Takeuchi et al.",
"2013).",
"It is therefore speculated that ULXs and ULSs are simply two types of super-Eddington accretors, with respectively geometrically thinner and thicker outflows along the line of sight due to different viewing angles or mass accretion rates (e.g.",
"Poutanen et al.",
"2007; Urquhart and Soria 2016; Feng et al.",
"2016 and reference therein).",
"To some extent, this scenario is similar to the unification scenario proposed for active galactic nuclei (e.g., Elvis 2000, and references therein).",
"The core of our work is to test this model through the study of a bright and well isolated transitional object, which looks like a ULX when the wind is not optically thick and then shows some ULS signatures when the wind thickens: NGC 55 ULX.",
"The study of winds in ULXs and ULSs is now possible after the recent discovery (Pinto et al.",
"2016, hereafter Paper I) of rest-frame emission and blue-shifted absorption lines in NGC 1313 X-1 and NGC 5408 X-1 ULXs, with the high-resolution RGS gratings aboard XMM-Newton.",
"Similar features were also found in NGC 6946 X-1, albeit at lower significance.",
"The detections confirm the presence of powerful, relativistic ($\\sim 0.2c$ ), winds in these sources and by extension in several other ULXs with similar spectral residuals (e.g., Stobbart et al.",
"2006, Sutton et al.",
"2015, Middleton et al.",
"2015b).",
"The high-ionization Fe K part of the ultrafast outflow in stacked EPIC and NuSTAR spectra of NGC 1313 X-1 was also detected (Walton et al.",
"2016a).",
"The timing properties of ULXs provide an independent diagnostic of the accretion process.",
"Heil & Vaughan (2010) first detected a linear correlation between variability amplitude and flux in the ultraluminous X-ray source NGC 5408 X-1, similar to that found in Galactic X-ray binaries and active galactic nuclei (e.g., Uttley & McHardy 2001 and Vaughan et al.",
"2003).",
"Heil & Vaughan (2010) also found evidence of time delays between soft and hard energy bands, with the softer bands delayed with respect to the hard at frequencies of $\\sim 20$ mHz.",
"These time delays were later confirmed by De Marco et al.",
"(2013) in several newer XMM-Newton observations.",
"Delays between spectral components may provide some further clues on the geometry of the system and the different physical emission mechanisms in ULX and ULS accretion disks.",
"They could indeed be related to the different location of the soft/hard X-ray emitting regions.",
"This paper is structured as follows.",
"In Sect.",
"we report some well known characteristics of NGC 55 ULX that motivated us to search for evidence of winds.",
"We present the data in Sect.",
"and a detailed spectral modeling in Sect. .",
"Covariance and lag spectra of the source are shown in Sect. .",
"We discuss the results and provide some insights on future missions in Sect.",
"and give our conclusions in Sect. .",
"More technical detail on our analysis is reported in Appendix ."
],
[
"The ULX$-$ ULS hybrid in NGC 55",
"NGC 55 ULX is the brightest X-ray source in the nearby Magellanic-type galaxy NGC 55 (NED average distance $1.94$ Mpchttps://ned.ipac.caltech.edu/) with an X-ray luminosity peak of $\\sim 2\\times 10^{39}$ erg/s (see Table REF and Fig.",
"REF ).",
"The X-ray light curve exhibits a variety of features including sharp drops and 100s seconds dips.",
"During the dips most of the source flux is quenched in the 2.0–4.5 keV band.",
"Stobbart et al.",
"(2004) proposed that the accretion disk is viewed close to edge-on and that, during dips, orbiting clumps of obscuring material enter the line of sight and cause significant blocking or scattering of the hard thermal X-rays emitted from the inner disk.",
"The EPIC CCD spectra of NGC 55 ULX can be modelled with a broad-band component from the innermost region (either as a disk blackbody or as Comptonization) plus the standard T $\\sim $ 0.15 keV blackbody, which we see in most ULSs and the softest ULXs and probably originates from reprocessed emission, such as Compton down-scattered emission, and intrinsic disk emission (e.g., Sutton et al.",
"2013 and Pintore et al.",
"2015).",
"The spectrum is very soft (slope $\\Gamma >4$ , if modelled with a powerlaw) similar to ULSs, but there is significant flux and a bright hard tail above 1 keV (see Sect.",
"REF ), which is stronger than typical ULSs (e.g., Urquhart and Soria 2016).",
"Figure: ULX-to-ULS spectral states sequence.",
"A dotted line connects the standard ULS stateswith the soft ULX states of NGC 247 and NGC 55 ULXs.",
"For more detail see text.In Fig.",
"REF we show the observed EPIC-pn spectrum of NGC 55 ULX (XMM-Newton observation 0655050101) with a sequence of ULX spectra, from harder to softer: Holmberg IX X-1 (0693851801), NGC 1313 X-1 (0693851201), NGC 5408 X-1 (0653380501), NGC 247 ULX (0728190101 and 0601010101), M 51 ULS (0303420201), M 81 ULS (Chandra observation 735), and M 101 ULS (0212480201).",
"NGC 247 ULX is a very interesting source which oscillate around the boundary between the ULS and the soft ULX regime (see also, e.g., Feng et al.",
"2016).",
"Unfortunately, no deep grating observations are available for this object and for ULSs in general (see Sect.",
"REF ).",
"However, the high-flux spectrum of NGC 247 has a similar shape to that of NGC 55 ULX with both sources showing a turnover above 1 keV and, more importantly, is superimposable with the spectrum of the low-flux state of NGC 55 ULX (see, e.g., Fig.",
"REF ).",
"NGC 55 ULX has never shown a classical ULS spectrum during the XMM-Newton observations because the fraction of the energy carried by the photons above 1.5 keV has been above 15%, which is higher than the 10% limit commonly used to identify supersoft sources (see, e.g., Di Stefano and Kong 2013), but its similarity to NGC 247 ULX, which sometimes looks like a ULS, and its soft spectral shape, which just fits in between the classical ULXs and ULSs, are crucial.",
"There is a whole on-axis XMM-Newton orbit ($\\sim 120$ ks) of NGC 55 ULX plus other shorter exposures which provide a unique workbench to study the transition between the two phases.",
"The search for a wind in this object is highly motivated by the detection of soft X-ray residuals in the EPIC spectra of NGC 55 ULX that are similar in both shape and energy ($\\sim 1$ keV) to those that were observed and resolved in other ULXs (see, e.g., Middleton et al.",
"2015b and Paper I).",
"Table: XMM-Newton observations used in this paper.Figure: NGC 55 ULX EPIC/pn-mos stacked image.The yellow lines show the extraction region for the RGS spectrum.The bigger larger white circle is the EPIC background region.The smallest circle in the right indicates the NGC 55 galactic center."
],
[
"The data",
"In this work we utilize data from the broadband ($0.3-10$ keV) EPIC-pn CCD spectrometer (Turner et al.",
"2001) and the high-resolution Reflection Grating Spectrometer (RGS, $0.35-1.77$ keV) aboard XMM-Newton (den Herder et al.",
"2001) in order to constrain the spectral shape of the source and to search for evidence of a wind.",
"The XMM-Newton observations used in this paper are reported in Table REF .",
"We decided not to use another available observation (ID=0028740101) because it had the ULX near the edge of the field of view with a PSF highly distorted and affected by the background.",
"The combination of low flux and short exposure with corresponding limited statistics makes it not useful for our analysis.",
"The exposure with ID 0655050101 is the only one to provide significant RGS science data because the other exposures are off-axis with the ULX outside the RGS ($\\pm 2.5^{\\prime }$ ) field of view.",
"We reduced the data with the latest XMM-Newton/ SAS v15.0.0 (CALDB available on September, 2016) and corrected for contamination from soft-proton flares.",
"We extracted EPIC MOS 1-2 and pn images in the 0.5–3 keV energy range, which includes the vast majority of NGC 55 ULX photons, and stacked them with the emosaic task (see Fig.",
"REF ).",
"We also extracted EPIC-pn spectra from within a circular region of 1 arcmin diameter centred on the emission peak.",
"The background spectra were extracted from within a slightly larger circle in a nearby region on the same chip, but away from bright sources and the readout direction (see Fig.",
"REF ).",
"We also make sure that the background region was outside the copper emission ring (Lumb et al.",
"2002).",
"We also tested three other background regions for the EPIC-pn spectrum to confirm that the background does not produce instrumental features.",
"There was very little solar flaring during the observations.",
"We extracted the first-order RGS spectra in a cross-dispersion region of 1 arcmin width, centred on the emission peak and the background spectra by selecting photons beyond the 98% of the source point-spread-function and check for consistency with blank field observations.",
"For more detail on the RGS background see Appendix REF .",
"We stacked the RGS 1 and 2 (for displaying purposes only, using the method described in Paper I).",
"The RGS spectrum of NGC 55 ULX shows very interesting emission-like features and a sharp drop near 11 Å very similar to that one observed in EPIC spectra of ultraluminous supersoft sources (e.g., Urquhart and Soria 2016)."
],
[
"Broadband EPIC spectrum",
"We fitted the EPIC-pn spectrum with the new SPEX codehttp://www.sron.nl/spex v3.02 to constrain the broad-band continuum.",
"Each of MOS 1 and 2 cameras has 3–4 times less effective area than pn, which alone contains the vast majority of the counts.",
"The pn spectrum was sampled in channels equal to 1/3 of the spectral resolution for optimal binning and to avoid over-sampling.",
"We grouped the pn spectra in bins with at least 25 counts and use $\\chi ^2$ statistics.",
"Throughout the paper we adopt 1 $\\sigma $ error bars.",
"We focused on the pn spectrum of the second (0655050101) exposure to determine the best fitting continuum model because it has twice as many counts as the first observation, shows less variability, and its continuum determination is crucial for following high-resolution analysis.",
"We tested two alternative models to describe the broadband EPIC-pn spectrum of NGC 55: a combination of blackbody (bb) plus powerlaw (po) and a combination of blackbody (bb) plus disk blackbody modified by coherent Compton scattering (mbb, see SPEX manual for more detail).",
"All emission components were corrected by absorption due to the foreground interstellar medium and circumstellar medium (hot model in SPEX with low temperature $0.5$ eV, see e.g.",
"Pinto et al.",
"2012).",
"Table: Constraints on the broadband continuum model.Figure: NGC 55 ULX EPIC-pn background-subtracted spectrum withtwo alternative continuum models (see Table ).The dotted lines show interesting features that do not dependon the adopted continuum model, some of which can be resolved in the soft X-rayhigh-resolution RGS spectrum(see Fig.",
").The dotted lines show the model components for Model 1 (BB+PL)and Model 2 (BB+MBB); for more detail see Table .The bottom panel show the EPIC-pn residuals adopting the wind modelfrom the RGS high-resolution spectrum (see Sect.",
").The double blackbody model (bb + mbb) provides a better fit than the power-law model (see Table REF ) in agreement with the two-component model consisting of a blackbody (for the soft component) and a multicolour accretion disk (for the hard component) of Pintore et al.",
"(2015) for Chandra and Swift observations of NGC 55 ULX.",
"Moreover, the power-law model would imply a very high luminosity since the deabsorbed model diverges at low energies.",
"Of course, we cannot exclude one or the other continuum model according to the value of the $\\chi ^2$ only.",
"A good exercise consists in determining the column density, $N_{\\rm H}$ in a different way or using other observations and compare it to that measured with the blackbody and the power-law models.",
"Pintore et al.",
"(2015) have shown that the $N_{\\rm H}$ of NGC 55 ULX varies over a large range, $1-6 \\times 10^{21} {\\rm cm}^{-2}$ , which is always an order of magnitude above the Galactic value ($1-2 \\times 10^{20} {\\rm cm}^{-2}$ ).",
"Interestingly, if we fit the EPIC data with the power-law model, but excluding the hard tail above 2 keV, we obtain $4.1\\times 10^{21} {\\rm cm}^{-2}$ .",
"If instead we only fit the EPIC spectrum between 0.4 and 1 keV we obtain $2.7\\times 10^{21} {\\rm cm}^{-2}$ , which may suggest that the broadband power-law fit overestimates the column density.",
"We have also tested a Comptonization continuum (comt model in SPEX with $T_{\\rm seed}=0.13\\pm 0.01, T_{\\rm e}=0.83\\pm 0.03, \\, {\\rm and} \\; \\tau =6.8\\pm 0.3$ ) over the 0.3–10 keV and obtained $2.1 \\pm 0.1\\times 10^{21} {\\rm cm}^{-2}$ .",
"Following the procedure of Pinto et al.",
"(2013), adopting Solar abundances and a power-law continuum, we have also measured the $N_{\\rm H}$ through the depth of the absorption edges of the most abundant interstellar neutral species in the RGS spectrum: neon (13.4 Å), iron (17.4 Å), and oxygen (23.0 Å), see Fig.",
"REF .",
"We obtained $N_{\\rm H}=1.5\\pm 0.5 \\times 10^{21} {\\rm cm}^{-2}$ , without significant differences whether we fit the whole RGS spectrum or edge-by-edge.",
"This also prefers the solution obtained with the blackbody model.",
"The harder multicolour component is seen in most ULXs, although its temperature is typically 1.5–2.5 keV (hence the origin of the characteristic downturn at $\\sim $ 5–6 keV).",
"In the softest states of ULXs (e.g.",
"in NGC 55 ULX) and in classical ULSs can be as low as 0.7 keV.",
"It is thought to consist of emission produced by the inner accretion disk highly distorted by interaction with hot electrons in the inner regions (up-scattering) and by down-scattering with cool electrons in the wind (e.g.",
"Gladstone et al.",
"2009, Middleton et al.",
"2011a, Middleton et al.",
"2015a).",
"The soft ($\\sim 0.2$ keV) component is likely to be intrinsic to regions above the outer disk, where a radiatively-driven wind is expected to be launched at accretion rates comparable or higher than the Eddington limit (Poutanen et al.",
"2007).",
"Our broadband spectral fits are consistent with NGC 55 ULX being a soft ultraluminous (SUL) X-ray source according to the classification of Sutton et al.",
"(2013).",
"The first, shorter, observation also favours a multi-blackbody emission model.",
"The EPIC spectrum in 2001 basically exhibits hotter blackbody components (see below and Table REF ).",
"Interestingly, we found that the emitting area of the soft ($\\sim 0.2$ keV) blackbody component increases by a factor 2 from the brighter (2001) to the fainter (2010) observation in agreement with the study of ULSs of Urquhart and Soria (2016) and Feng et al.",
"(2016).",
"This is expected if the expansion of the photosphere and the decrease of the temperature shift the peak of the spectrum from X-ray towards the far-UV.",
"As previously mentioned, ULSs show a faint hard X-ray tail ($>$ 1.5 keV) whose origin is not quite understood.",
"It is thought to be produced in the inner regions similar to ULXs and characterized by either Bremsstrahlung or Comptonization emission (see, e.g., Urquhart and Soria 2016).",
"As a test, we have re-fitted the NGC 55 ULX EPIC spectrum by substituting the mbb component with a rest-frame Bremsstrahlung or collisionally-ionized emission model ($cie$ model in SPEX).",
"The bb + brems fit is bad ($\\chi ^2_{\\nu } = 13.8$ ).",
"We also tried to get an upper limit on the emission of a putative Bremsstrahlung component adding it on top of the best-fit bb + mbb model.",
"The maximum flux a Bremsstrahlung component can have is 10 times smaller than the flux of the mbb component.",
"The Bremsstrahlung temperature was however highly unconstrained.",
"Therefore, if the hard X-ray tail seen in ULSs is similar to the one that is well constrained in NGC 55 ULX, then a disk origin would be favoured.",
"Although the double blackbody model provides a better fit, it is still statistically rejected.",
"The presence of strong residuals around 1 keV shows an important similarity with the residuals seen in several ULXs with long exposures (emission peak at 1 keV and absorption-like features on both sides).",
"Middleton et al.",
"(2014, 2015b) interpreted them as either due to emission by collisionally-ionized gas or to absorption by outflowing photoionized gas and were identified in Paper I.",
"This motivated our investigation with the RGS.",
"The knowledge of the exact structure of the spectral continuum (either a double black body or power-law or Comptonization) does not strongly affect the following analysis, which focuses on the search for narrow spectral features and broadband spectral variations.",
"Figure: RGS stacked spectrum with overplotted the EPIC best-fit continuummodel (see Table ).",
"The solid blue line shows the RGS fitted continuumwith free normalization as only free parameter.The wavelengths of expected absorption lines produced by the Galactic absorber(Pinto et al.",
"2013)and the ∼1\\sim 1 keV peak seen in the EPIC spectrum are shown by blue dashedlines.",
"Some clear emission- (EM) and absorption-like (Abs) spectral featuresare marked with solid magenta lines.Our analysis focuses on the $7-27$ Å first order RGS 1 and 2 spectra from the long, on-axis, observation 2 (0655050101).",
"Below 7 Å the effective area is not well calibrated and above 27 Å the count rate is too low.",
"For the RGS analysis we have to use C-statistics because of the Poisson statistics and the need for high spectral resolution.",
"The spectrum was binned in channels equal to 1/3 of the spectral resolution for the optimal binning (see Kaastra and Bleeker 2016).",
"We plot the EPIC continuum models on top of the RGS data in Fig.",
"REF .",
"As for EPIC, the double blackbody spectral model describes the RGS spectrum better than the blackbody+powerlaw combination, although RGS does not have the same sensitivity.",
"Therefore, the choice of the continuum will not have a significant effect on our results.",
"We fit the EPIC model to the RGS spectrum leaving only the overall normalization as a free parameter and obtained a comparable fit with the overall flux lower by about 10 percent compared to the EPIC continuum model (see the solid blue line in Fig.",
"REF ).",
"This RGS continuum model is used in the following analysis.",
"One of the major issues in modelling the RGS spectra of faint sources, such as ULXs, is the background.",
"However, NGC 55 ULX is significantly brighter than the background between 7–25 Å.",
"Moreover the source is reasonably isolated; the RGS slit does not encounter any other bright object (see Fig.",
"REF ) and the background spectrum is rather featureless.",
"For more detail see Appendix REF ."
],
[
"Search for spectral features",
"Following the approach used in Paper I, we searched for spectral features on top of the spectral continuum by fitting a Gaussian spanning the $7-27$ Å wavelength range with increments of 0.05 Å and calculated the $\\Delta $ C-statistics.",
"We tested a few different linewidths: 500, 1000, 5000, and 10,000 km s$^{-1}$ .",
"We show the results obtained with 500 km s$^{-1}$ ($\\sim $ RGS resolution) and 10,000 km s$^{-1}$ (a case of relativistic broadening) in Fig.",
"REF .",
"The black points refer to lines with negative normalizations, i.e.",
"absorption lines.",
"We found several narrow emission features and some evidence for absorption.",
"The emission-like features are near some of the strongest transitions commonly observed in X-ray plasmas and are the same that we have detected in our previous work on NGC 1313 ULX-1, NGC 5408 ULX-1, and NGC 6946 ULX-1.",
"The RGS spectra amplify and resolve the residuals previously detected in the EPIC spectra (Middleton et al.",
"2015b).",
"If we identify the emission features with some of the strongest transitions in this energy band – Mg xii (8.42 Å), Fe xxii-xxiii (11.75 Å), Ne x (12.135 Å), Ne ix (13.45 Å), O viii (18.97 Å) and O vii (21.6 Å, 22.1 Å) – then they would require blueshift, which seems to be larger for higher ionization states.",
"For instance, the spike around 7.5 Å is likely produced by blueshifted Mg xii; the one seen around 11.4 Å could be a blend of blueshifted Ne x and high Fe ions.",
"The 21.8 Å emission-like feature could be a blueshifted O vii 22.1 Å forbidden line.",
"An O vii forbidden line (22.1 Å) stronger than the resonant line (21.6 Å) is a clear indication of photoionization (e.g.",
"Porquet and Dubau 2000).",
"The high ionization ions, e.g.",
"Mg xii, Fe xxii-xxiii and Ne ix also show strong resonant lines suggesting possible collisional ionization and, therefore, a complex wind ionization/temperature structure.",
"In the identification process we assumed that the lines were produced by the strongest transitions in the X-ray band and within $\\pm 0.2c$ from their rest-frame wavelengths.",
"We notice that the blueshifts of the strongest O vii and O viii emission lines are consistent within 1 $\\sigma $ .",
"The same applies to the Mg xii and Ne x emission lines.",
"In Appendix REF we show that the features detected in the RGS spectrum of NGC 55 ULX are intrinsic to the sources and are not of intrumental origin by applying the same line-search technique to the spectra of five active galactic nuclei with comparable statistics.",
"Figure: Δ\\Delta C-statisticsobtained by fitting a Gaussian spanning the 7-277-27 Å wavelengthrange with increments of 0.05 Å.Δ\\Delta C = 2.71, 4.00, and 9.00 referto 1.64σ\\sigma (90%), 2σ\\sigma (95.45%),and 3σ\\sigma (99.73%) confidence levels.We tested the effects of using two different linewidths: 500 km s -1 ^{-1}(∼\\sim RGS resolution, top panel) and 10,000 km s -1 ^{-1} (medium panel).Red and black points refer to emission and absorption lines, respectively.The ratio between the normalization of the gaussian and its erroris shown in the bottom panel.The rest-frame wavelengths of some relevant transitions are labelled.More detail on the line significance is report in the text."
],
[
"Monte Carlo simulations",
"The $\\Delta $ C-statistics of each feature provides a crude estimate of the significance but does not take into account the look-elsewhere-effect, which is due to the size of the parameter space used such as the number of velocity bins used, and other eventual effects due to the data quality.",
"We therefore performed Monte Carlo simulations to estimate the significance of the strongest features in the RGS spectrum such as those found at 7.5 Å, 11.4 Å, 15.55 Å, 18.9 Å, and 21.85 Å.",
"Adopting our best-fit spectral continuum model as template (Table REF , second column), we simulated 10,000 RGS 1 and 2 spectra (accounting for the uncertainties on the continuum parameters).",
"We then added the line as best-fitted in our line grid to the continuum model and calculated the changes in the C-statistics (see Fig.",
"REF ).",
"For each line, we then computed the number of occurrences where its addition to the continuum-simulated spectra provided an improvement (or $\\Delta $ C-statistics, with the position of the line free to vary within $\\pm 0.2c$ and the continuum parameters also free to vary) larger than that one provided by the line as fitted to the data.",
"This is basically equivalent to create a mock spectrum from the continuum model and search for lines in this spectrum using the same technique adopted in Sect.",
"REF .",
"The $p$ -values and the corresponding confidence intervals (C.I.)",
"and sigmas are reported in Table REF .",
"This provides much more conservative limits on the detection.",
"As expected, the blueshifted O viii line ($\\lambda _0=$ 18.97 Å, $\\lambda _{\\rm obs}\\sim $ 18.90 Å) is detected well above 3 $\\sigma $ , but all the strongest features are still detected above the $99\\%$ C.I.",
"each, even accounting for the look-elsewhere effect.",
"Although this procedure does not follow exactly all the steps adopted during the line search performed in Sect.",
"REF , we acknowledge that it is the best that can be done with the current data.",
"The combined detection of two pairs of strong lines sharing the velocity is certainly encouraging, see also Mg xii–Ne x and O vii–O viii lines in Fig.",
"REF , but more data are required to place strong constraints.",
"Table: NGC 55 ULX XMM/RGS fit : Monte Carlo simulations.Figure: RGS spectrum with two alternativewind models on top of the best-fit EPIC continuum:the red line has one photoionized outflowing emission (0.01c0.01c) and oneabsorption (0.16c0.16c) components;the blue line has two photoionized emission components (0.01c0.01c and 0.08c0.08c)and two absorbers (0.06c0.06c and 0.20c0.20c, seeTable ).Rest-frame wavelengths of relevant transitions are labelled.Note the line shift.Table: NGC 55 ULX XMM/RGS fit : main wind components.Figure: Results of the spectral fits using a grid of 8 photoionized absorbers(see Sect.",
").The ΔC-\\Delta \\,C-statistics indicate the significance of each absorber(each with three degrees of freedom).The column densities, the ionization parameters,and the absolute values of the outflow velocities are also shown.The cyan solid lines in the bottom panel show a linear fit withthe corresponding 3 σ\\sigma limits.The red filled squares indicate the values of the photoionizedemission components (see Table ).The transmission of the four absorbers indicatedby the magenta open squaresis shown in Fig.",
".The two absorbers with highest ΔC\\Delta C-stat arealso used in the multiphase wind model shown inFig.",
"(blue line).Figure: Transmission of the four most relevant absorbersaccording to a grid of 8 photoionized absorbers(see the magenta open squares Fig.",
").The outflow velocities and the ionization states are labelled.The fastest absorber significantly reduces the flux below 12 Å."
],
[
"Simple wind model",
"We modelled the low-ionization emission lines with a photoionized emission model ($photemis$ code imported from XSPEC, which uses XSTAR tableshttps://heasarc.gsfc.nasa.gov/xstar/docs/html/node106.html).",
"This model provides a good description of the O viii resonant line, the O vii k $\\alpha $ triplet and the O vii k $\\beta $ line at 18.627 Å $(\\log \\xi = 1.2 \\pm 0.1)$ , but requires a blueshift of $(0.011\\pm 0.001)c$ , i.e.",
"$\\sim $ 3000 km s$^{-1}$ .",
"This model provide a significant improvement to the fit with $\\Delta \\,\\chi ^2$ /$\\Delta $ $C/dof$ = 28/27/3 (see also Table REF ).",
"We notice that SPEX calculates the (blue/red) shift of any spectral model according to $E^{\\prime } = E / ( 1 - v / c)$ , which does not account for the relativistic corrections.",
"This was not taken into account in Paper I.",
"Hence, we show all the velocities as defined in SPEX to allow the readers to reproduce our work, but we then use the relativistic formula $z = \\sqrt{ (1 + v/c) / (1 - v/c) } - 1$ each time we report the blueshift, $z$ , of a certain line or model.",
"As previously done in Paper I, we modelled the absorption features with the $xabs$ model in SPEX.",
"This model calculates the transmission of a slab of material, where all ionic column densities are linked through a photoionization model.",
"The relevant parameter is the ionization parameter $\\xi = L/nr^2$ , with $L$ the source luminosity, $n$ the hydrogen density and $r$ the distance from the ionizing source (see, e.g., Steenbrugge et al.",
"2003).",
"The equivalent of this model in XSPEC is $warmabs$https://heasarc.gsfc.nasa.gov/xstar/docs/html/node102.html.",
"Some absorption-like features can be well described with one photoionized $xabs$ absorber with a higher ionization parameter $(\\log \\xi = 3.5 \\pm 0.3)$ and a relativistic outflow velocity of $(0.16\\pm 0.01)c$ , very similar to the $\\sim 0.18c$ outflow in NGC 1313 X-1 and NGC 5408 X-1 ($0.2c$ for the non-relativistic correction, see Paper I), see red line in Fig.",
"REF .",
"The absorber reproduces part of the drop below 11 Å and the strong absorption at 16 Å.",
"Several features are however missed by this simple wind model such as the blueshifted Mg xii (8.42 Å) emission line, part of the Fe xxii-xxiii (11.75 Å) – Ne x (12.135 Å) blend and some other absorption like features such as the dip at 15 Å.",
"This suggests that the wind may be more structured.",
"Line widths of 100 km s$^{-1}$ were adopted for all absorption and emission components.",
"A different line width does not improve the fit."
],
[
"Structured wind model",
"Multiple emission components – A significant improvement ($\\Delta \\,\\chi ^2$ /$\\Delta $ $C/dof$ = 30/30/3) is obtained if we include another, faster, photoionized emission with $0.082\\pm 0.001c$ , which provides an excellent description of the emission features at 7.5 Å and 11–13 Å (see blue line in Fig.",
"REF and Table REF ).",
"A comparable, slightly worse, fit ($\\Delta \\,\\chi ^2$ /$\\Delta $ $C/dof$ = 25/25/3) is obtained if we instead use a fast collisionally-ionized emission component ($cie$ in SPEX, $apec$ in XSPEC).",
"Multiple absorption components – The absorption features are less strong and less significant than the emission lines, but we have seen that a single component cannot reproduce all the strongest features.",
"For instance, the troughs between 8.8–10 Å and 15–16 Å (or 1.2–1.4 keV and 0.77–0.83, both also seen in EPIC-pn, e.g.",
"Fig.",
"REF ).",
"However, adding blindly more absorption components may miss some important features.",
"First, we considered a wind with a continuous distribution of ionization parameters from 0.2 to 3.8 all at the same velocity.",
"This was done using the $warm$ model in SPEX which adopts a continuous distribution of 19 absorbers with a $\\Delta \\xi $ step of 0.2.",
"This only provided a very small improvement with $\\Delta \\chi ^2$ and $\\Delta C=26$ for 19 new degrees of freedom with respect to the model of continuum plus emission lines.",
"We notice however that the best-fit outflow velocity of such multi-ionization component is 66000 km s$^{-1}$ , i.e.",
"$0.2c$ similar to NGC 1313 and NGC 5408 ULX-1.",
"This motivates a different fitting tactic.",
"The presence of emission lines at different velocities suggests a structured wind and, therefore, the absorption lines may also exhibit some complex dynamics.",
"To search for a trend of the velocity with the ionization state as well as for some missing components, we created a grid of photoionized absorbers with outflow velocities spanning the $0-0.25c$ range (8 $xabs$ components in SPEX with velocity steps of about 10,000 km s$^{-1}$ each).",
"We fit again the RGS data with the EPIC spectral continuum and calculated the improvement in the C-statistics with the addition of each component.",
"There are three degrees of freedom for each absorber: the column density $N_{\\rm H}$ , the ionization parameter $\\xi $ , and the line-of-sight velocity $v$ .",
"The results are shown in Fig.",
"REF .",
"We found at least two $xabs$ components with high significance: $\\sim 3\\sigma $ ($\\Delta C$ / $\\Delta \\chi ^2$ / d.o.f.",
"= 14/14/3) and $\\sim 3.5\\sigma $ ($\\Delta C$ / $\\Delta \\chi ^2$ / d.o.f.",
"= 18/19/3) respectively, at low ($\\sim 0.5$ ) and high ($\\sim 3.3$ ) ionization parameters.",
"The low-$\\xi $ plasma has a low velocity $\\sim 0.06c$ , while the high-$\\xi $ has is relativistically outflowing $v\\sim 0.20c$ .",
"There are several tentative ($2\\sigma $ ) detections of material at intermediate velocities, which were broadly reproduced by the single absorber in the simple model used in Sect.",
"REF .",
"Interestingly, the ionization parameter of the absorbers seems to show a strong trend with the velocity (see Fig.",
"REF , bottom panel).",
"The effective area of the RGS detector drops dramatically at energies below 1.77 keV (or wavelengths above 7 Å) with a corresponding loss of sensitivity to ionization parameters $\\log \\xi \\gtrsim 4.0$ .",
"This explains the large error bars at the highest velocities.",
"In Fig.",
"REF we show the transmission of the four absorbers with the highest $\\Delta C$ (8, 8, 14, and 18), which total $\\Delta C = 48$ for 12 degrees of freedom).",
"This plot is a simple diagnostic tool that shows how the lines shift with the higher outflowing velocities and ionization parameters.",
"We check the effect of a multiphase wind model on the overall spectral fitting by fitting again the RGS spectrum with the two most significant photoionized absorbers ($v\\sim 0.06c$ and $\\sim 0.20c$ ) and the two photoionized emitters discussed above on top of the EPIC spectral continuum model.",
"This structured wind model is shown in Fig.",
"REF (blue line) and provides $\\Delta C = 32$ for 6 degrees of freedom.",
"The model fits the RGS spectrum reproducing most narrow emission and absorption lines such as the drops below 11.5 Å and between 15–16 Å.",
"In Table REF we report the improvements in the fit for these four most significant emission and absorption components.",
"In Appendix REF we briefly discuss some systematic effects due to the adopted ionization balance."
],
[
"Do RGS wind components improve EPIC data fits?",
"A simultaneous fit of RGS and EPIC spectra is not straightforward due to their different characteristics.",
"EPIC-pn has high count rate but low spectral resolution, whilst RGS has low count rate but high spectral resolution.",
"Moreover, there are some cross-calibration uncertainties between them and the energy band of EPIC (0.3–10 keV) is much wider than that of RGS ($\\sim $ 0.35–1.77 keV).",
"However, it is a good excercise to test the wind components from the RGS spectral modeling onto the EPIC data and confirm their existence.",
"Therefore, starting from the EPIC-pn continuum spectral model in Sect.",
"REF (ID 0655050101), we have added each of the RGS wind components shown in Table REF on top of the EPIC double blackbody continuum with only the continuum parameters free to vary.",
"We remind that we used $\\chi ^2$ in the EPIC fits.",
"The addition of the slow low-$\\xi $ emitter has a small effect on the fits ($\\Delta \\chi ^2 = 10$ ) because is degenerate with the column density at the low spectral resolution of EPIC.",
"The fast high-$\\xi $ emitter helds $\\Delta \\chi ^2 = 12$ .",
"Interestingly, if we use the collisionally-ionized RGS solution (see Sect.",
"REF ) they provide a $\\Delta \\chi ^2 = 15$ each.",
"The introduction of the relativistic ($0.2c$ ) photoionized absorber also improves the fits with $\\Delta \\chi ^2 = 27$ .",
"The introduction of these components smears out most of the residuals that were detected with a continuum-only model (Fig.",
"REF ).",
"The EPIC-pn spectrum provides further support to the results obtained with the RGS only."
],
[
"Flux-resolved spectroscopy",
"Previous work has suggested that the spectral shape of NGC 55 changes with the source flux: softer when fainter (Stobbart et al.",
"2004) with the temperatures of the two blackbody components (tentatively) found to be correlated (Pintore et al.",
"2015).",
"Here we further investigate this problem using flux-resolved X-ray broadband spectroscopy.",
"We split the two observations into five flux regimes.",
"A detailed description is provided in Appendix REF .",
"Briefly, we extracted EPIC-pn lightcurves in the 0.3–10 keV energy band for the two observations, then chose two flux ranges for the first (shorter) observation and three flux ranges for the second observation, and then calculated the good time interval for each flux range.",
"The flux ranges were chosen such that the five spectra extracted in these good time intervals have comparable statistics.",
"The five flux-selected spectra are shown in Fig.",
"REF .",
"A prominent knee appears at 1 keV similar to that seen in some ultraluminous supersoft X-ray sources (e.g.",
"M 101 and NGC 247, Urquhart and Soria 2016).",
"In a preliminary fit performed to each of the five spectra with the best-fit bb+mbb continuum model (Sect.",
"REF ) we found that the column density $N_{\\rm H}$ of the neutral absorber and the temperature of the cooler blackbody $T_{bb}$ did not significantly change between flux-resolved spectra selected from the same observation.",
"We therefore fit simultaneously the spectra of observation 1 coupling the $N_{\\rm H}$ and the $T_{bb}$ .",
"The same coupling was repeated afterwards when simultaneously fitting the spectra of the observation 2.",
"The flux-resolved spectral fits are shown in Fig.",
"REF .",
"In Table REF we report the detail of the spectral fits including the observed unabsorbed luminosities.",
"The column density decreases with the increasing luminosity, whilst the temperatures of both the cooler blackbody and the hotter multicolour blackbody increase.",
"The flux-resolved spectra confirm the residuals detected with a higher confidence in the deeper, stacked, spectrum of the observation 2.",
"Moreover, there is some tentative evidence of variability in the residuals (see Appendix REF ).",
"Deeper observations are necessary to study their variability in detail.",
"Figure: NGC 55 ULX EPIC flux resolved spectra with a continuum modelof blackbody plus modified disk blackbody(see Table ).The dotted lines show that the residuals could vary."
],
[
"Timing analysis",
"NGC 55 ULX has shown an interesting temporal evolution in the last decade.",
"Stobbart et al.",
"(2004) found evidence of dips in the lightcurve, which seem to have significantly smoothed in more recent Chandra and Swift observations (Pintore et al.",
"2015).",
"We therefore looked for short and long term variations.",
"We extracted the source and background lightcurves in the same regions used to extract spectra (see Sect. )",
"and obtained a source background-subtracted lightcurve with the epiclccorr task.",
"Figure: EPIC/pn lightcurves of the first (left) and second (right) observations,The sizes of the time bins are different between the exposure (500 s and 2 ks)due to the different fluxes.",
"The soft band refers to the 0.5–1 keVenergy range and the hard band refers to 1–5 keV.For more detail on the energy band selection see Sect.",
".Notice the lower flux and variability in 2010.The area shaded in grey in the left panel includesthe count rate throughout the whole 2010 observationwhich seems to match the dips seen in 2001.Figure: Lag-energy spectra for different frequency rangeswith the NGC 55 ULX EPIC-pn (observation 1).Note the trend towards lower energies at low frequenciesin the bottom panel.The dotted red lines show the magnitude expected from thePoisson noise.",
"The pointsshould be randomly distributed between these linesin absence of any lag.Figure: Covariance spectra for the same frequency rangesas Fig.",
"(observation 1).The covariance spectra follow the shapes of the lagsmeasured at the corresponding frequency.The middle panel shows the fractional variability calculated ineach frequency range and divided by the mean rate.The bottom panel shows the ratios betweenthe fractional variability atsome frequencies."
],
[
"Lightcurves",
"NGC 55 ULX lightcurves are shown in Fig.",
"REF .",
"The lightcurves were extracted in a soft (0.5-1.0 keV) and a hard (1–5 keV) energy band in order to study the behaviour of the source on the low-energy and high-energy side, respectively, of the 1 keV bump seen in its EPIC spectrum as well as in the spectra of several ultraluminous X-ray sources (see e.g.",
"Pintore et al.",
"2015, Urquhart and Soria 2016).",
"The source behaviour has dramatically changed in the last decade.",
"During the observation of 2001 the hard band was highly variable, exhibiting strong dips as previously shown by Stobbart et al.",
"(2004).",
"During the observation of 2010 the source was much less variable, with the hard and soft band showing the same variability pattern (see Fig.",
"REF ).",
"Interestingly, the count rate in 2010 is consistent with the dips observed during the first 2001 observation.",
"We investigated the variability amplitude in each observation as a function of temporal (Fourier) frequency using power density spectra.",
"We follow the standard method of computing a periodogram in 10 ks lightcurve segments before averaging over the segments at each frequency bin, and subsequent binning up over adjacent frequency bins (van der Klis 1989, Vaughan et al.",
"2003).",
"The resulting power density spectra are shown in Fig.",
"REF and confirm that the source has very limited variability in 2010 compared to 2001 (see also Heil et al.",
"2009).",
"We therefore focus on the 2001 observation only in the remainder of this Section."
],
[
"Time lags and covariance spectra",
"We studied the timing properties of NGC 55 by calculating the cross-spectrum between light curves in different energy bands.",
"A detailed description of the method can be found in Alston, Vaughan & Uttley (2013) and Uttley et al.",
"(2014, and references therein).",
"We calculated the cross-spectrum at each Fourier frequency in 10 ks segments and then averaged over the segments at each frequency, before binning in geometrically spaced frequency bins by a factor of 1.6.",
"In a given frequency band we calculated the time lags between a comparison energy band versus a broad (in energy) reference band, whilst subtracting the comparison band from the reference band if it fell within it.",
"In this way, the time lag in each energy bin is the average lag or lead of the band versus the reference band.",
"We used the $0.3 - 1.0$ keV band as reference where the rms is high.",
"In Fig.",
"REF we show the resulting lag-energy spectrum for three low frequencies.",
"The power density spectrum is very flat and the Poisson noise dominates above a mHz (see, e.g., Heil et al.",
"2009).",
"We selected three frequency intervals for which it is possible to measure any lag and covariance.",
"Above these frequencies the lags are consistent with zero and a lot of the covariance energy bins are negative as the noise dominates.",
"We also estimated the lag magnitude expected from the Poisson noise contribution to the phase difference using Eq.",
"30 of Vaughan et al 2003, see red dotted lines in Fig.",
"REF .",
"In the absence of any intrinsic lag we should expect the bins to be randomly distributed between these limits, which is not the case at least at high and low frequencies.",
"At low frequencies ($0.5-1.5\\times 10^{-4}$ Hz in Fig.",
"REF bottom panel) there is evidence for the softer bands lagging behind harder bands.",
"As the frequency increases (from bottom to top panel), there is a systematic change in the lag above 0.9 keV with the profile flattening and possibly the hard bands lagging the soft at higher frequencies ($2.5-4.5\\times 10^{-4}$ Hz).",
"There is a tentative a sharp change in the lag below $0.9$ keV.",
"To better understand the behaviour in the lag energy spectra around 1 keV, we computed the covariance spectra at each Fourier frequency (Wilkinson & Uttley 2009).",
"The covariance spectra show the correlated variability between two energy bands, or between a comparison band and a broad reference band (minus the comparison band).",
"We computed the covariance from the cross-spectrum following Uttley et al.",
"(2014).",
"The covariance spectra for the same three frequencies are shown in Fig.",
"REF (top).",
"The overall (mean) covariance spectrum mimics the shape of the energy spectrum (see Fig.",
"REF .",
"The covariance spectra extracted in different frequency ranges follow the trend of the energy lag spectra (see Fig.",
"REF ).",
"The covariance increases towards lower energies, but then flattens or possibly drops below 1 keV.",
"On average, the high-frequency covariance spectrum appears to be harder than the low-frequency one, with a possible drop below 1 keV, suggesting the presence of different variability processes (Fig.",
"REF middle panel).",
"The covariance spectra extracted at different frequencies are very similar with the current statistics, but more data would certainly help to distinguish some differences at low and high frequencies.",
"The same applies to the lag-energy spectra.",
"More work will follow up in a forthcoming paper to provide better constraints on these trends and to compare the results with other ULXs."
],
[
"Discussion",
"Urquhart and Soria (2016) modelled the soft X-ray spectral features and the $\\sim $ 1 keV ($\\sim 12$ Å) drop in several CCD spectra of ULSs with a model of thermal emission and an absorption edge.",
"They interpreted the features as a result of absorption and photon reprocessing by an optically-thick wind which obscures the innermost regions where most hard X-rays are produced.",
"This suggested that ULSs are actually ULXs viewed almost edge on or maybe their outflow is thicker, at the same viewing angle.",
"A denser, more optically thick outflow may be the result of higher accretion rates above Eddington, assuming that ṁ$_{\\rm wind}$ is proportional to ṁ$_{\\rm disk}$ (e.g., Shakura and Sunyaev 1973).",
"Moreover, at higher super-Eddington rates, the outflow launching region moves to larger radii on the disk, where the escape velocity and the Compton temperature are lower, which favours denser winds (e.g., Poutanen et al.",
"2007).",
"Motivated by the recent detection of relativistic winds in the NGC 1313 and NGC 5408 ULX-1 high-resolution RGS spectra taken with XMM-Newton (see Paper I), we searched for a deep RGS observation of an object that shows the properties of both the ULX and ULS states and found NGC 55 ULX."
],
[
"Discovery of a powerful wind",
"The primary goal of our study was the search for unambiguous signatures of a wind.",
"The detection of (unresolved) spectral residuals in previous work done on NGC 55 ULX was certainly encouraging (e.g.",
"Middleton et al.",
"2015b).",
"We confirmed the presence of these residuals particularly in the 2010 EPIC deep spectrum (see Figs.",
"REF and REF ).",
"The evidence hinting at a wind was confirmed when we looked at the high-resolution RGS spectrum of NGC 55 ULX, which shows a wealth of emission and absorption features located at energies that are blueshifted compared to the energies of the strongest atomic transitions in the soft X-ray energy band (see Figs.",
"REF and REF ).",
"Accurate spectral modelling shows the presence of an outflowing plasma with a complex ionization and velocity structure (see Figs.",
"REF and REF ).",
"The best-fitting models suggest photoionization, but we cannot exclude some contribution from collisional ionization, particularly for high-ionization species.",
"The gas responsible for the emission lines moves with velocities $(0.01-0.08)c$ lower than those exhibited by the absorbing gas $(0.06-0.20)c$ , see Table REF and Fig.",
"REF .",
"Both the emission and the absorption lines indicate that the ionization state increases with the outflow velocity suggesting that we are detecting hotter/faster phases coming from inner regions.",
"Our limited statistics prevents us from claiming the detection of more than two photoionized emitters and absorbers in the line of sight, but additional observations would certainly help to understand the ionization and velocity structure of the wind in NGC 55 ULX.",
"An interesting result is the spectral signature of the high-ionization ($\\log \\xi \\sim 3.3$ ) absorber with the largest column density.",
"This component absorbs a substantial amount of photons below 12 Å (or above 1 keV), which provides a natural explanation for the spectral shape of NGC 55 ULX and the drop seen in several ULSs (e.g.",
"Urquhart and Soria 2016).",
"The main difference is that such a turnover is more pronounced in ULSs because they are seen more edge on or because the photosphere of the wind is simply further out due to a larger accretion rate (see e.g.",
"Middleton et al.",
"2015a and Urquhart and Soria 2016) with resulting increased obscuration of the inner regions.",
"It is useful to compare the wind power to the bolometric radiative luminosity of the source.",
"The kinetic luminosity of the outflow or wind power $L_{\\rm kin} = 1/2 \\dot{M} v^2_{\\rm out}$ can be expressed as $L_{\\rm kin} = 2 \\pi L_{\\rm ion}\\,m_{\\rm p}\\,\\mu \\,v^3_{\\rm out}\\,C_V\\,\\Omega / \\xi $ , where $L_{\\rm ion}$ is the ionizing luminosity, $m_{\\rm p}$ is the proton mass, $\\mu $ is the mean atomic weight ($\\sim 1.2$ for solar abundances), $v_{\\rm out}$ is the outflow velocity, $C_V$ is the volume filling factor (or `clumpiness'), $\\Omega $ is the solid angle, and $\\xi $ is the ionization parameter of the wind.",
"We calculated the ratio between the wind power and the bolometric luminosity for the fast and most significant absorber (see Table REF ) and obtained: $L_{\\rm kin} / L_{\\rm bol} = 1300 \\pm 650\\,C_V\\,\\Omega \\,L_{\\rm ion}/L_{\\rm bol}$ .",
"According to the simulations of Takeuchi et al.",
"(2013), a wind driven by a strong radiation field in super-Eddington accretion has a typical clumpiness factor of $\\sim 0.3$ and is launched over wide angles of $10^{\\circ }-50^{\\circ }$ from the disk rotation axis.",
"A significant fraction of the source radiative luminosity ($L_{\\rm bol}$ ) must have been used to ionize the wind before some reprocessing occurred.",
"In our broadband EPIC fits we found that, on average, the 0.3–10 keV total source flux was equally partioned between the hard (disk) and soft (reprocessed) components.",
"Therefore, it is reasonable to adopt $L_{\\rm ion}/L_{\\rm bol}\\sim 0.5$ .",
"This provides $L_{\\rm kin} / L_{\\rm bol} \\sim 30-100$ , which is similar to that measured for NGC 1313 X-1 and NGC 5408 X-1 (see paper I and Walton at al.",
"2016a), but is an extreme value if compared to the strongest outflows seen from sub-Eddington systems, e.g.",
"active galactic nuclei (e.g., Fabian 2012, and references therein)."
],
[
"Long and short term spectral variability",
"We studied the timing and spectral properties of NGC 55 ULX to understand if they also suggest that the source is a transition between classical ULXs and ULSs.",
"If the thick, outflowing, wind scenario is correct then we should expect a correlation between the spectral hardness of the source and the X-ray flux (e.g.",
"within 0.3–10 keV).",
"For instance, if either the accretion rate or the inclination decrease then the optical depth decreases, which corresponds to a lower down-scattering of the hard photons produced in the inner regions.",
"We test this picture by splitting the observations in different flux regimes which provided five high-quality EPIC spectra (see Fig.",
"REF ).",
"We modelled the EPIC spectra with a spectral continuum consisting of a blackbody and a multicolour blackbody emission components in agreement with Pintore et al.",
"(2015) and with our search for the best-fitting continuum.",
"The temperatures of both components (particularly the hotter multicolour blackbody) significantly increase with the luminosity supporting the wind scenario (see Table REF ).",
"This is confirmed by the observed decrease in column density of the neutral absorber with luminosity (the Galactic interstellar medium account less than 10 % according to the H I maps and does not affect our result).",
"In support for this scenario we also found that the emitting area of the soft ($\\sim 0.2$ keV) blackbody component increased by a factor 2 from the brighter (2001) to the fainter (2010) observation in agreement with the study of ULSs of Urquhart and Soria (2016) and Feng et al.",
"2016, where the expansion of the photosphere and the decrease of the temperature shift the peak of the spectrum from the X-ray to the far-UV energy band.",
"According to our spectral fits, the radius of the photosphere is 2000–3000 km and the temperature well above 0.15 keV, which explains why NGC 55 still looks like ULX rather than a ULS according to the classifications of Sutton et al.",
"(2013) and Urquhart and Soria (2016).",
"This may suggest that either the inclination or the accretion rate in NGC 55 is smaller than in typical ULSs.",
"There are no striking dips in the 2010 exposure if compared to those in 2001, which is not surprising since they were not detected in most recent observations taken with Chandra and Swift (see Pintore et al.",
"2015).",
"Lightcurves extracted with the same binning are shown in Appendix REF .",
"We remark that the flux level in 2010 is consistent with the lowest level of 2001, suggesting that the source is in a semi-obscured state (see Fig.",
"REF ).",
"The lack of variability and the low flux in 2010 is consistent with a picture where the innermost regions, which produce the high energy hard X-rays and the high variability, are partly obscured by an intervening absorber in agreement with the hypothesis of Stobbart et al.",
"(2004), Middleton et al.",
"(2011) and Sutton et al.",
"(2013).",
"Either the accretion rate increased, further inflating the outer regions of the disk, or the line of sight changed (e.g.",
"due to precession) with further obscuration of the innermost “hotter” regions.",
"The tentative detection of a large soft lag, $\\sim $ 1000s, at low frequencies further strengthens our scenario (see Fig.",
"REF ) suggesting that the soft emitting gas in the wind and the upper disk photosphere are reprocessing the photons produced in the innermost regions.",
"The covariance spectra computed within different frequency ranges follow the shape of the lag-energy spectra (see Fig.",
"REF ), confirm this picture and show evidence for two possible main processes that produce variability: fluctuation within the inner disk (harder) and slow reprocessing through the wind (softer).",
"It is also possible that when the column density of the wind decreases the wind becomes optically thin, but the hard X-ray photons start penetrating the wind before the soft photons (see, e.g., Kara et al.",
"2015).",
"This may explain the source hardening when the luminosity increases.",
"In a future project we will investigate all possible scenarios by comparing more sources and lags at different time scales.",
"Our results agree with Middleton et al.",
"(2015a).",
"They computed the covariance spectra on a broad frequency band and showed a significant lack of variability in the 2010 observation, suggesting that the variable, hard, X-ray component has intercepted cooler/optically thicker material.",
"They also found that the shape of the covariance spectra in NGC 55 ULX and several other ULXs was consistent with the variability originating in the hard component only, which agrees with a model where the variability on short and long timescales at moderate inclinations is dominated by obscuration of the high energy emission.",
"The lags tentatively detected here appear at much lower frequencies than those shown by Heil et al.",
"(2010) and De Marco et al.",
"(2013) in NGC 5408 X-1.",
"Hernandez-Garcia et al.",
"(2015) also found evidence of a $\\sim $ 1 ks time lag in NGC 5408 X-1 and showed that the time delays are energy-dependent and that their origin is not related to reflection from an accretion disk (`reverberation').",
"They also argued that associating the soft lag with a quasi-periodic oscillation (QPO) in these ULXs, drawing an analogy between soft lags in ULXs and soft lags seen in some low-frequency QPOs of Galactic X-ray binaries, is premature.",
"The two orders of magnitude in time delays (from a few seconds to hundred seconds in NGC 55 and NGC 5408 ULXs) point towards a complex energy lag spectrum and we also caution from using lags to derive mass estimates without a comprehensive analysis in a large frequency range.",
"Although this is beyond the scope of this paper, we provide a simple argument to compare lags in AGN, X-ray binaries, and ULXs.",
"The lag magnitude in NGC 55 ULX as a function of the variability timescale is $\\sim $ 10% (1000s; 1/f=10,000s).",
"The hard lags in AGN are typically 1%, the AGN soft lags are $\\sim $ 2% (e.g., Alston et al.",
"2013), whilst in hard state X-ray binaries they are 0.5-1% (e.g., Uttley et al.",
"2011).",
"De Marco et al.",
"(2013) measured about $\\sim $ 5% for the higher frequency lags in NGC 5408 X-1.",
"Hence, it is possible that the processes causing these lags are different.",
"The long soft lags in NGC 55 ULX and NGC 5408 X-1 could be related to some phenomena occurring in the outer region.",
"In fact, the 1000s magnitude is likely due to the combination of light travel time plus thermalization in the wind and additional local scattering processes before the photons are re-emitted towards the observer.",
"Such phenomena imply a distance that can be large enough to damp the variability with a corresponding decreasing of the covariance at high frequencies (see Fig.",
"REF ).",
"The short (a few seconds) soft lags seen by Heil et al.",
"(2010) and De Marco et al.",
"(2013) can be different processes occurring in the inner regions."
],
[
"NGC 55 vs ULXs & ULSs: the big picture",
"A lot of work has already been done in order to classify ULXs and ULSs and to understand the effect of the inclination (viewing angle) and the accretion rate on their spectral and timing behaviour (see, e.g, Gladstone et al.",
"2009, Sutton et al.",
"2013, Middleton et al.",
"2015, Urquhart and Soria 2016, and references therein).",
"Therefore, we briefly highlight why NGC 55 ULX looks like a transitional form between ULXs and ULSs for the shape of its spectral continuum.",
"More attention will be given on the way the detection of winds fits in the framework of super-Eddington accretion."
],
[
"The spectral shape",
"The higher-energy broad-band component is an interesting link between ULXs and ULSs.",
"The sequence from hard ULXs to soft ULXs and then to ULSs shows a progressively lower temperature (whether modelled as Comptonization or a modified disk, e.g., Gladstone et al.",
"2009).",
"The temperature is therefore a function of ṁ$_{\\rm wind}$ and/or the viewing angle.",
"The optical depth $\\tau $ (for a Comptonization model) instead increases along the same sequence, from $\\tau \\sim $ few for hard ULXs to $\\tau \\gtrsim $ 10 for soft ULXs, to $\\tau $ $\\longrightarrow $ infinity for ULSs.",
"Alternatively, the photon index $\\Gamma $ increases from $\\sim $ 1.5 to $\\sim $ 2.5 and then $>$ 4 along the same sequence in analogy with the ($T$ ,$\\tau $ ) changes for a comptonizing region.",
"The hard component of NGC 55 can be well modelled as either a steep ($\\Gamma >4$ ) power law or as a modified disk, which essentially means a large $\\tau $ (see e.g.",
"Tables REF and REF ).",
"We have also shown that the shape of NGC 55 ULX spectrum matches that of the high-flux state of NGC 247 ULX, an object that shifts between the ULX and the ULS classical states (see Fig.",
"REF ).",
"This suggests that NGC 55 ULX is indeed a transitional object between the ULXs and ULSs."
],
[
"The spectral features",
"The emission and absorption features detected in NGC 55 ULX are produced by the same ionic species detected in NGC 1313 X-1 and NGC 5408 X-1 RGS spectra (see Paper I).",
"For instance, the properties of the most significant absorber detected in NGC 55 ULX (see e.g.",
"Table REF ) seem to match those of the fast ($0.2c$ ) component in the other two ULXs.",
"The detection of a multiphase structure in NGC 55 ULX would point towards a tighter analogy to NGC 5408 X-1, which also exhibits a wind more complex than NGC 1313 X-1, with different ionization states and outflow velocities.",
"We notice that evidence for an edge-like feature around 11 Å was also found in the RGS spectrum of NGC 6946 X-1 (see Paper I), which sometimes exhibits very soft spectra like NGC 55 and NGC 5408 (the soft ultraluminous regime, Sutton et al.",
"2013).",
"Unfortunately the interpretation of this feature in NGC 6946 X-1 was difficult due to its flux at 1 keV (OBS ID 0691570101) $\\sim 50$ % lower than NGC 55 ULX.",
"The presence of blueshifted emission lines in NGC 55 ULX suggests that we may be looking through a different line of sight compared to NGC 1313 and 5408 ULXs.",
"The continuum is for instance significantly lower than NGC 5408 X-1 (see Extended Data Fig.",
"4 in Paper I), but at 1 keV NGC 55 ULX is comparable if not brighter, with the emission lines at other energies also matching the flux of the lines in the other ULXs.",
"Remarkably, the line-emitting gas phases account for 10–20 % of the total source flux in the X-ray (0.2–10 keV) energy band in agreement with NGC 5408 X-1, while in NGC 1313 X-1 the lines contribute about 5 % of the source flux.",
"If the emission lines in these ULXs have the same origin then we should expect that NGC 55 ULX is seen through a preferential line of sight where the outflow velocity of the emission component is almost maximum (see Fig.",
"REF ), while the other three ULXs may be seen through a line of sight where the motion of the line-emitting gas is more tangential.",
"This would explain why the emission lines in NGC 5408 X-1 appear to be broader by an order of magnitude ($\\sigma _v\\sim 1000$ km s$^{-1}$ , see Paper I).",
"NGC 1313 X-1 shows a large range of spectral hardness with strong absorption features at $\\sim 0.2c$ , which suggests moderate inclination angles with the wind variability possibly due to precession (see Middleton et al.",
"2015b).",
"Holmberg IX X-1 shows very shallow residuals in EPIC spectra and a spectral hardness among the highest measured in ULXs which argues in favour of a nearly face on view (e.g.",
"Sutton et al.",
"2013, Middleton et al.",
"2015a, Luangtip et al.",
"2016, Walton et al.",
"2016c).",
"The emission features detected in NGC 55 ULX are produced by the same ionic species that give origin to the blueshifted emission detected in the relativistic jet of the Galactic super-Eddington accretor SS 433 (see, e.g., Marshall et al.",
"2002 and references therein).",
"Similar, powerful, outflows have been discovered in Galactic black holes X-ray binaries such as IGR J17091–3624 (King et al.",
"2012), MAXI J1305–705 (Miller et al.",
"2014), 4U 1630–47 (Diaz-Trigo et al.",
"2013) and during superburst of the neutron star X-ray binary SAX J1808.4–3658 (Pinto et al.",
"2014).",
"Figure: Simplified scheme of high massaccretion rate sources (see also Middleton et al.",
"2011).The light blue region shows the soft X-ray emission of the accretion disk,altered by a photosphere of a radiatively-driven optically-thickwind.",
"The dark blue region, closer to the compactobject is dominated by highly variable, optically-thinner,turbulent Comptonization emitting high-energy (>1>1 keV) X-rays.The dashed lines indicated some possible sightlines for some famous ULXsand for the ULS regime (see e.g.",
"Urquhart and Soria 2016).We believe that the line of sight of NGC 55 ULX is somewhere betweenthe classical ULX and ULS sources, but still at high inclinationwhere outflowing material is both emitting and absorbingphotons from the inner regions."
],
[
"Ultraluminous supersoft sources",
"A comprehensive analysis of the ULX phenomenology also needs high-resolution spectroscopy of ULSs in order to study the cases of extreme absorption and to ultimately confirm an unification scenario for ULXs and ULSs (see Fig.",
"REF and, e.g., Poutanen et al.",
"2007, Urquhart and Soria 2016, and references therein).",
"The ULS archive has XMM-Newton exposures that are just too short ($\\lesssim 50$ ks) to provide RGS spectra with statistics good enough to significantly detect sharp spectral features.",
"NGC 247 ULX is likely the best candidate among all ULSs because of its brightness, good isolation, and sharp drop at 1 keV (see e.g.",
"Urquhart and Soria 2016, Feng et al.",
"2016).",
"The RGS archive has two on-axis exposures with a modest 30 ks exposure time each.",
"These prevent us from doing an accurate analysis, particularly for the low-flux ULS spectrum, which would require more data or additional spectra of other ULSs and, therefore, we defer this work to future work.",
"Here we present a preliminary result.",
"Briefly, we repeated the RGS analysis done in this work on NGC 55 ULX with the high-flux RGS exposure of NGC 247 ULX (ID 0728190101, see Sect. ).",
"The spectrum also shows hints for absorption and emission features, fainter than in NGC 55 ULX due to the worse statistics (see Fig.",
"REF ).",
"In particular, RGS confirms the spectral bending below 12 Å (above 1 keV) as seen in CCD spectra (see Fig.",
"REF ).",
"It is encouraging that the spectral features can be modelled with a wind model similar to that used in NGC 55 ULX, i.e.",
"two photoionized emitters and one photoionized absorber, with the ionization increasing with the outflow velocity (see Fig.",
"REF ).",
"The velocity of the line-emitters in NGC 247 ULX seems to be even higher than those in NGC 55 ULX showing a further preferential location of the line of sight.",
"Jin et al.",
"(2011) have analyzed the ULS-state EPIC-pn spectrum of NGC 247 that we have shown in Fig.",
"1 and found similar residuals to those detected in NGC 55 ULX.",
"In particular, absorption features at 0.7 and above 1 keV as well as emission at or just below 1 keV.",
"As suggested by Feng et al.",
"(2016) and Urquhart and Soria (2016), the supersoft ultraluminous (SSUL or ULS) regime is likely an extension of the soft ultraluminous state toward higher high accretion rates with the blackbody emission arising from the photosphere of thick outflows and the hard X-rays being emission leaked from the embedded accretion disk via the central low-density funnel or advected through the wind.",
"The wind thickens throughout the ULX-SUL-SSUL sequence and imprints the spectral curvature and the absorption edge at 1 keV.",
"Figure: NGC 55 ULX RGS spectrum with overlaid the structured with model (top)compared to NGC 247 ULX with a similar windmodel on top of a blackbody emission continuum (bottom):low-ξ\\xi slow (v=0.05cv=0.05c) and high-ξ\\xi fast (v=0.13cv=0.13c)photoionized emission components andhigh-ξ\\xi fast (v=0.14cv=0.14c) photoionized absorption.The red-dashed lines mark some relevant blueshifted emission line detected in NGC 55 ULX.Some interesting features appear on their blue side in NGC 247 ULX."
],
[
"XMM-Newton in the next decade",
"Ideally, we would need a statistical sample to probe the scenario shown in Fig.",
"REF comparing the properties of the wind in sources at different angles of inclination to the observer and mass accretion rate.",
"Only a handful of sources have been observed with deep, on-axis, XMM-Newton observations with statistics good enough to enable RGS analysis and no on-axis full-orbit observation of any bright ULS has ever been taken.",
"The fact that the wind lines can be studied imply that with a long exposure of a ULS we will learn more about their physics.",
"Deeper observations of other ULXs and ULSs will provide a complete sample of ultrafast winds, while deeper observations of those sources which already have well-exposed RGS spectra (e.g.",
"NGC 1313 X-1 and NGC 55 ULX) will enable to study the wind in detail and its dependence on flux, spectral hardness, and source quasi-periodicities (e.g.",
"Walton et al.",
"2016b).",
"All this is crucial to understand the relation between the wind and both the accretion rate and the viewing angle, which is useful to understand the launching mechanism.",
"This research field is still rather new and the ESA's XMM-Newton satellite has unique capabilities to further develop it.",
"The gratings on board NASA's Chandra satellite (HETGS/LETGS, e.g.",
"Brinkman et al.",
"2000 and Canizares et al.",
"2005) have the highest spectral resolution available in the X-ray band, but their effective area is smaller than that of RGS in the soft X-ray band where most ULX wind features are shown and where the flux is highest.",
"HETGS and LETGS are clearly optimal to distinguish spectral features in the brightest objects, whilst the RGS has unique capabilities to detect lines in weak objects, which means that the continuation of the XMM-Newton and Chandra missions is crucial to study the ULXs and ULSs in detail.",
"ULXs and ULSs will be excellent targets for a $Hitomi$ replacement mission, for the proposed $Arcus$ gratings mission and, particularly, for the two-square-meter ESA's $Athena$ mission.",
"$Arcus$ can significantly improve the RGS results due its much higher spectral resolution in the soft X-ray energy band (e.g., Kaastra 2016).",
"$Hitomi$ has already proven that we can change the way we perform X-ray spectroscopy thanks to its combination of excellent spectral resolution and collective area in the high energy X-ray band (Hitomi Collaboration 2016).",
"$Athena$ will completely revolutionize X-ray astronomy due to its combination of high spatial and spectral resolution (Nandra et al.",
"2013).",
"We remark that the study of the accretion flow in ULXs and ULSs is important to understand the phenomenology of super-Eddington accretion, which may be required to occur during the early stages of the Universe to build up the supermassive black holes that have been discovered to power AGN at high redshifts (see e.g.",
"Volonteri et al.",
"2013)."
],
[
"Conclusions",
"In the last decade it has been proposed that a substantial fraction of the population of ultraluminous X-ray sources (ULXs) and ultraluminous supersoft sources (ULSs) are powered by super-Eddington accretion onto compact objects such as neutron stars and black holes (see, e.g., King et al.",
"2001, Roberts 2007, Urquhart and Soria 2016, Feng et al.",
"2016, and references therein).",
"In particular, ULSs could be a category of ULXs observed at high inclination angles, possibly edge on, where a thick layer of material is obscuring the innermost hard X-ray emitting regions (e.g., Kylafis & Xilouris 1993 and Poutanen et al.",
"2007).",
"We have studied the NGC 55 ULX which we believe is a hybrid source showing properties common to both ULX and ULS.",
"The presence of a spectral curvature at 1 keV and a high energy tail that can be described as a $\\gtrsim 0.7$ keV disk-like blackbody place it just in between these two categories of X-ray sources.",
"We have found a powerful wind characterized by emission and absorption lines blueshifted by significant fractions of the speed of light ($0.01-0.20c$ ) in the XMM-Newton/RGS spectrum of NGC 55 ULX.",
"The detection of such a wind is consistent with the predictions of super-Eddington accretion (Takeuchi et al.",
"2013).",
"The wind has a complex dynamical structure with an ionization state that increases with the outflow velocity, which indicates launching from different regions of the accretion disk.",
"The comparison of the wind in NGC 55 ULX with that detected in other ULXs (Pinto et al.",
"2016) suggests that the source is being observed at high inclinations, but not high enough to look exactly like a ULS, in agreement with the classification on the basis of its spectral shape (e.g., Sutton et al.",
"2013).",
"However, the strongest wind component partly absorbs the source flux below 1 keV, generating a drop similar to that observed in ULSs.",
"The long and short term spectral variability of the source shows a softening of the spectrum at lower luminosities, i.e.",
"around the Eddington luminosity (for a $10\\,M_{\\odot }$ black hole), which agrees with the proposed scenario of wind clumps crossing the line of sight and partly obscuring the innermost region where most hard X-rays come from (e.g.",
"Middleton et al.",
"2015a, and references therein).",
"We have found evidence for a long $\\sim $ 1000s soft lag at low frequencies, similar to NGC 5408 X-1 (Hernandez-Garcia et al.",
"2015), which may indicate that part of the emission coming from the inner regions has been reprocessed in the outer regions before being re-emitted towards the observer.",
"This provides further support to the wind scenario.",
"Deeper XMM-Newton observations of NGC 55 ULX and other ULXs and ULSs will enable a detailed study of the dependence of the wind on the accretion rate and the inclination angle, which can help us to understand the geometry and the accretion flow in these extraordinary astronomical objects and in super-Eddington accretors in general."
],
[
"Acknowledgments",
"This work is based on observations obtained with XMM-Newton, an ESA science mission funded by ESA Member States and USA (NASA).",
"We also acknowledge support from ERC Advanced Grant Feedback 340442.",
"HE acknowledge support from the STFC through studentship grant ST/K501979/1.",
"TPR acknowledges funding from STFC as part of the consolidated grant ST/L00075X/1.",
"DJW and MJM acknowledge support from STFC via an Ernest Rutherford advanced grant.",
"We acknowledge the anonymous referee for useful comments that improved the paper."
],
[
"Systematic effects",
"In this section we briefly discuss some systematic effects that might have limited our analysis, such as the contamination from the background (instrumental and/or astrophysical) as well as the approximation of the ionization balance.",
"We also provide some more technical detail on the selection of flux intervals used to extract flux-resolved spectra and the plots of the power density spectra.",
"All this was shifted here to speed up the paper reading."
],
[
"RGS background",
"ULXs are faint sources if compared to common nearby Galactic X-ray binaries or active galactic nuclei.",
"The EPIC-pn camera has a very large effective area, which provides a lot of counts and an energy band broad enough to constrain the spectral continuum of ULXs.",
"RGS spectrometers distribute the photons in much more energy bins and therefore are much less sensitive.",
"Searching for weak features in their spectra must therefore be a careful process, particularly when accounting for the background.",
"Luckily, we have two ways of extracting an RGS background spectrum.",
"The first method uses the photons outside a certain cross-dispersion slit, which according to the standard XMM-Newton/SAS routine is the 98% of the PSF (see Fig.",
"REF , middle panel).",
"Another method uses a model background estimated with ultradeep exposures of blank fields and scaled by the count rate in the RGS CCD 9 where hardly any emission from the source is expected (see Fig.",
"REF , bottom panel).",
"The two background spectra are very similar and significantly dominate below $\\sim $ 7 Å and above $\\sim $ 26 Å.",
"The RGS 1 and 2 raw source spectra (and the spectral features) are significantly stronger than the background in our wavelength range.",
"For the work presented in this paper we have used the background determined from the exposure, but we have repeated all the work using the model background and found no significant change in our results.",
"All this shows that the background has minor systematic effects in our analysis.",
"Figure: NGC 55 ULX XMM RGS 1-2 source raw spectra (top) with two alternativebackground spectra: exposure-extracted background (middle)and blank-field-model background spectra (bottom).The background is severe below ∼\\sim 7 Å and above ∼\\sim 26 Åand higher for RGS 1.",
"RGS 1 and RGS 2 miss two chips coveringthe 10.4-13.810.4-13.8 Å and 20-2420-24 Å range, respectively."
],
[
"SED and ionization balance",
"Another source of systematic uncertainties is the choice of the spectral energy distribution (SED) since it may affect the ionization balance and therefore the calculation of ionization parameters and column densities in the photoionization components (both in emission and absorption).",
"For instance, the standard $xabs$ model in SPEX adopts a Seyfert 1 type of SED and ionization balance, whose shape of course deviates from the soft SED of a ULX such as NGC 55 ULX.",
"To understand the effects of the SED choice, we test the photoionization $pion$ model in SPEX, which assumes the SED measured with the fitted spectral continuum model - the observed spectral shape of NGC 55 ULX - and instantaneously calculates the photoionization equilibrium using the plasma routines available in SPEX (see the SPEX manual for more detail).",
"At the moment the $pion$ emission model calculates the thermal emission of the photoionized layer by ignoring the photon-induced processes.",
"This is not realistic enough, so for the moment we only check the effects of the SED on the absorption component.",
"We have fitted again the RGS spectrum of NGC 55 ULX with the EPIC spectral continuum on top of which we have added a $pion$ model to substitute the $xabs$ component of the $0.2c$ absorber detected with highest significance.",
"The main result is shown in Table REF along with the improvement in the fits previously obtained with the $xabs$ model (see also Table REF ).",
"The results with the $xabs$ and the $pion$ models are consistent within the statistical uncertainties mainly due to the limited statistics.",
"However, there is no indication of major systematics apart from the fact that the $pion$ model has larger uncertainties due to the larger dependence of the parameters on the continuum shape.",
"The $pion$ model certainly provides a more physical description of the absorber(s), but it requires much more memory and CPU time than the $xabs$ model, of course.",
"The search for a multiple/continuous photoionization structure of the wind would have been prohibitive with as many $pion$ components as the $xabs$ components used in Sect.",
"REF and Fig.",
"REF .",
"Table: Systematic effects on the ionization balance."
],
[
"EPIC lightcurves for flux resolved spectra",
"In Sect.",
"REF we extracted EPIC spectra in different flux ranges in order to understand the dependence of its spectral shape with the luminosity.",
"Here we briefly describe the selection of the flux intervals.",
"At first, we extracted an EPIC/pn lightcurve in the 0.3–10 keV energy band for both the first (ID=0028740201) and the second exposure (ID=0655050101).",
"The source and background regions were the same as those used to extract the spectra (see Sect. ).",
"The background lightcurves were subtracted with the epiclccorr task.",
"The background-subtracted lightcurves are shown in Fig.",
"REF .",
"We then split the two observation into five slices with comparable statistics.",
"In the first observation we selected bins with count rates above (or below) 1.895 count/s.",
"This provided two slices with 27 thousand counts each.",
"The second observation was instead split into three intervals of fluxes between ($<0.855$ ), (0.855–0.980), and ($>0.98$ ) counts/s.",
"This provided three slices with about 32 thousand counts each.",
"The statistics of these five time intervals are comparable.",
"The five time intervals can be also seen in Fig.",
"REF .",
"We built the good time intervals using these intervals and extracted source and background spectra, response matrices, and effective area auxiliary files with the evselect task.",
"The flux-resolved spectra have been shown in Fig.",
"REF .",
"Figure: EPIC/pn 0.3-10 keV lightcurves of the first (left)and second (right) observationsboth with time bins of 100 s.Flux thresholds were chosen in order have comparable statistics.For more detail see text."
],
[
"EPIC flux resolved spectra: Gaussian fits",
"We tested the behaviour of the EPIC spectral residuals by fitting a Gaussian line for some of the strongest residuals (0.75, 1.0, 1.2, and 1.4 keV, see Fig.",
"REF ).",
"We adopted a FWHM of 100 eV similar to the EPIC spectral resolution.",
"We fit these four Gaussian lines with fix energies to the five flux-resolved spectra and show the results in Fig.",
"REF .",
"The lines are broadly consistent within their mean values, but there are a few deviations.",
"Altough there is hint for variability, we notice that deeper observations are needed to confirm and study their variability in detail.",
"Figure: Gaussian fits to the flux-resolved EPIC pn spectra.Normalizations are in units of 10 44 10^{44} photons/s.Negative values refer to absorption lines.Luminosities are same as in Table .We slightly shifted the points along the X-axis for plotting purposes."
],
[
"Power density spectra",
"We follow the standard method of computing a periodogram in 10 ks lightcurve segments before averaging over the segments at each frequency bin, and subsequent binning up over adjacent frequency bins (van der Klis 1989, Vaughan et al.",
"2003).",
"The resulting power density spectra are shown in Fig.",
"REF .",
"Taking into account the level of Poisson noise, given by the dashed lines, it shows that the source has very limited variability in 2010 compared to 2001.",
"Figure: Power density spectra for the first (black) andsecond (red) observations computed in the 0.3–1.0 keV energy range.The dashed lines show the Poisson noise level."
],
[
"Cross-correlation spectra",
"For completeness, we have also computed the cross-correlation (CCF) between the soft energy band (0.3–1.0 keV - same as reference band used in cross-spectrum, CS, throughout the paper) and several other bands.",
"In Fig.",
"REF we show the CCF computed with respect to the 1.0–2.0 keV (top) and the 3.5–6.0 keV (bottom) energy bands.",
"The harder band is always defined as having the positive lag on the x-axis.",
"The plot shows that there is clearly some lag structure in agreement with what we see in the CS.",
"For the soft band separations, there is a strong correlation at zero time delay, with a skew towards a softer lag.",
"As the energy band separation increases, the zero-lag correlation weakens, and we start to see a hard lag, as well as a negative lag.",
"The changes seem to occur when looking at bands above $\\sim $ 0.9-1.0 keV, similar to where we see the changes in the covariance.",
"The CCFs are however harder to interpret because they average over all the frequency dependent behavior.",
"Figure: Cross-correlation (CCF) for thesecond observation computed between the (0.3–1.0 vs 1.0–2.0 keV, top) and the(0.3–1.0 vs 3.5–6.0 keV, bottom) energy bands.",
"Notice the lag structure."
],
[
"RGS spectra of other sources",
"In order to confirm that the lines detected in NGC 55 ULX are not produced by instrumental features, we have searched for similar features in five extragalactic sources whose spectra are expected to be mainly featureless.",
"We chose these objects because the statistics quality of their RGS spectra is comparable to the RGS spectrum of NGC 55 ULX, once accounted for the net exposure time and the flux.",
"In Table REF we report the detail of the exposures used.",
"We repeat the RGS data reduction as done in Sect.",
"and the line search as described in Sect.",
"REF for these five sources.",
"In Fig.",
"REF (top) we show the results obtained with the line-search routine on each object adopting a power-law continuum corrected by redshift and Galactic absorption.",
"In order to smear out any intrinsic features and strengthen the instrumental ones, we have also simultaneously fitted the RGS spectra of the five objects, adopting the best-fit power-law continuum for each of them, whilst fitting the grid of Gaussian lines coupled between the models of the five spectra.",
"In Fig.",
"REF (bottom) we compare the results from the line search obtained combining the five sources (solid line) with that one obtained for NGC 55 ULX (dotted line).",
"At first, we notice that no emission feature is significantly detected at or above $3\\sigma $ , but more importantly the strongest features detected in NGC 55 ULX are absent in the other sources and in their combined analysis.",
"This shows that the lines detected in the RGS spectrum of NGC 55 ULX are significant and intrinsic to the source.",
"Table: XMM-Newton observations of non-ULX sources.Figure: Line search for five power-law-like objects (top) and their simultaneous fitscompared to that of NGC 55 ULX (bottom).",
"The strong features detected in NGC 55 ULXsignificantly differ from the weak instrumental features present in the other objects."
]
] | 1612.05569 |
[
[
"Interaction Between Two Closely-Spaced Waving Slender Elastic Cylinders\n Immersed in a Viscous Fluid"
],
[
"Abstract We study the hydrodynamic interaction between two closely-spaced waving elastic cylinders immersed within a viscous liquid, at the creeping flow regime.",
"The cylinders are actuated by a forced oscillation of the slope at their clamped end and are free at the opposite end.",
"We obtain an expression for the interaction force and apply an asymptotic expansion based on a small parameter representing the ratio between the elastic deflections and the distance between the cylinders.",
"The leading-order solution is an asymmetric oscillation pattern at the two frequencies ($\\omega_1,\\omega_2$) in which the cylinders are actuated.",
"Higher orders oscillate at frequencies which are combinations of the actuation frequencies, where the first-order includes the $2\\omega_1,2\\omega_2,\\omega_1+\\omega_2$, and $\\omega_1-\\omega_2$ harmonics.",
"For in-phase actuation with $\\omega_1= \\omega_2$, the deflection dynamics are identical to an isolated cylinder with a modified Sperm number.",
"For configurations with $\\omega_1\\approx \\omega_2$, the $\\omega_1-\\omega_2$ mode represents the dominant first-order interaction effect due to significantly smaller effective Sperm number.",
"Experiments are conducted to verify and illustrate the theoretical predictions."
],
[
"Introduction",
"We study the interaction between two closely-spaced oscillating elastic cylinders which are immersed within a viscous liquid.",
"The cylinders are actuated by a forced oscillation of the slope at their clamped ends, which may vary in frequency, amplitude, and phase.",
"We focus on configurations with negligible inertial effects and linear elasticity, where the dynamics are governed by a balance between viscous and elastic forces.",
"Various previous works examined the viscous-elastic dynamics of a single elastic cylinder actuated by a forced oscillation at its clamped end.",
"These include [9] who was the first to analytically obtain the deflection modes of such a passive elastic filament for the case of actuation of the slope at the fixed end.",
"Using a similar approach, [13] and [14] studied deflection modes and propulsion forces for forced oscillations and impulses of the position of the fixed end, combined with a requirement of zero torque.",
"An experimental study was conducted by [12], who measured both deflection and propulsion for a single elastic filament actuated by oscillation of the slope.",
"The experimental data showed good agreement with both linear and non-linear theoretical predictions.",
"Other relevant works include [3] who studied the dynamics of an elastic cylinder actuated by internal moments and [1] who experimentally studied oscillating flexible sheet as a novel pumping mechanism in the creeping flow regime.",
"Previous studies on interaction between two oscillating elastic cylinders focused mainly on forced deformations in the context of synchronization dynamics between closely-spaced waving flagella.",
"One of the first works on synchronization of flagella was conduced by [11] who studied the simplified model of two infinite sheets with prescribed waveforms and showed that energy dissipation is minimized when the sheets oscillate in-phase.",
"More recently, [4] analyzed a similar simplified configuration and concluded that synchronization may occur solely from hydrodynamics forces and requires front-back asymmetry of the deformation modes.",
"Other works focused on experiments in biological systems, including [2] who demonstrated that synchronization dynamics of the Volvox carteri may indeed occur due to the hydrodynamic effects alone.",
"The aim of this work is to analytically and experimentally study the deflection dynamics of two interacting passive elastic cylinders actuated by a forced oscillation of the slope at their clamped end.",
"This work is arranged as follows: In §2 we present the problem formulation, compute the interaction forces and apply asymptotic expansions.",
"In §3 we present the deflection modes, define the experimental methodology and compare the experimental data to the analytical results.",
"In §4 we give concluding remarks."
],
[
"Analysis",
"We examine the fluidic interaction between two closely-spaced slender elastic cylinders immersed in a viscous liquid and actuated due to a forced oscillation of the slope at their clamped end.",
"The coordinates and configuration are illustrated in Fig.",
"REF .",
"The Cartesian coordinate system is denoted by $(x,y,z)$ and time is denoted by $t$ .",
"The cylinders, at rest, are parallel to the $x$ -direction and their centers oscillate within the $x-y$ plane.",
"The fluid viscosity and density are denoted by $\\mu $ and $\\rho $ , respectively.",
"The cylinder flexural rigidity is $s$ , the beam mass per-unit-length is $m$ , the gap between the centers of the cylinders is $d$ and the gap at rest is $d_0$ .",
"The radius and length of the cylinders are $r_c$ and $l$ , respectively.",
"The forced oscillations of the slope of the cylinders at $x=0$ are at frequencies $\\omega _i$ and amplitudes $\\phi _i$ (where $i=1$ and $i=2$ denote cylinders 1 and 2, respectively).",
"The phase difference between the forced oscillations is $\\gamma $ .",
"The deflection of the cylinders is $w_i$ , where we define an auxiliary average deflection $w_a=(w_1+w_2)/2$ and an auxiliary relative deflection $w_d=(w_1-w_2)/2$ .",
"The perpendicular drag coefficient of the cylinders is $\\xi _\\perp $ .",
"Figure: Illustration of the examined configuration consisting of two oscillating elastic cylinders immersed in a viscous fluid.",
"The cylinders, at rest, are parallel to the xx axis and their centers oscillate within the x-yx-y plane.",
"The distance between the bases of the cylinders is d 0 d_0 and the deflections are denoted by w 1 w_1 and w 2 w_2 for cylinder 1 and 2, respectively.",
"The length of both cylinders is ll.Hereafter, asterisk superscripts denote characteristic values and Capital letters denote normalized variables.",
"The characteristic average deflection is $w_a^*$ , characteristic relative deflection is $w_d^*$ and characteristic frequency is $\\omega ^*$ .",
"We define the small parameters (where $w^*=\\max {(w^*_a,w^*_d)}$ ) $ \\frac{\\rho \\omega ^* (w^*)^2}{\\mu }\\ll 1,\\quad \\frac{m (\\omega ^* w^*)^2}{s}\\ll 1,\\quad \\frac{d_0}{l}\\ll 1, \\quad \\frac{r_c}{d_0}\\ll 1,\\quad \\frac{w_a^*}{l}\\ll 1, \\quad \\frac{w_d^*}{d_0}\\ll 1,$ corresponding to assumptions of negligible fluidic inertia (small Womersley number), negligible solid inertia, small gap to cylinder length ratio, small cylinder radius to gap ratio, small average deflection to length ratio and small relative deflection to gap ratio.",
"In addition, we apply the commonly used approximation [9], [6], [8], [13], [14], [10], [12], [5] of perpendicular viscous drag of the form $\\Delta w \\xi _{\\perp }$ , where $\\Delta w$ is the relative perpendicular speed between the cylinder and the surrounding fluid and the coefficient $\\xi _\\perp $ is approximately constant throughout the cylinder.",
"We define the function $\\Lambda (d)$ as the ratio of the induced fluid speed due to the adjacent cylinder and the velocity of the adjacent cylinder.",
"Thus, under the above assumptions, the deflection of the cylinders is governed by $s\\frac{\\partial ^4 w_i}{\\partial x^4}=-\\xi _\\perp \\left[\\frac{\\partial w_i}{\\partial t}-\\Lambda \\left(d=d_0+\\frac{w_i-w_j}{j-i}\\right)\\frac{\\partial w_j}{\\partial t} \\right],$ supplemented by the boundary conditions $\\frac{\\partial w_i(0,t)}{\\partial x}=\\phi _ie^{i2\\pi (\\omega _i t+(i-1)\\gamma )},\\quad w_i(0,t)=\\frac{\\partial ^2 w_i(l,t)}{\\partial x^2}=\\frac{\\partial ^3 w_i(l,t)}{\\partial x^3}= 0,\\quad $ where for cylinder 1, $(i,j)=(1,2)$ and for cylinder 2, $(i,j)=(2,1)$ .",
"The flow field due to a slender cylinder moving relative to a viscous fluid in the creeping flow regime may be approximated by a uniform distribution of Stokeslets and dipoles positioned along the centerline of the cylinder.",
"For motion perpendicular to the centerline, the magnitude of the Stokeslet distribution is $\\xi _{\\perp }\\Delta w$ , and the dipole magnitude is $r_c^2\\xi _{\\perp }\\Delta w/4\\mu $ , where $\\xi _\\perp \\approx 8\\pi \\mu /(0.386+\\ln (l^2/r_c^2))$ [8].",
"Thus, the induced speed may be approximated [8] as $\\Lambda \\approx \\frac{\\xi _\\perp }{4\\pi \\mu }\\left[1+\\ln \\left(\\frac{2l}{d_0+w_1-w_2}\\right)\\right].$ Eqs.",
"() may be decoupled by subtracting the equation governing cylinder 2 from the equation governing cylinder 1, and substituting relative deflection $w_d=(w_1-w_2)/2$ , thus isolating $w_d$ .",
"Similarly, by addition of the governing equations of both cylinders and substituting average deflection $w_a=(w_1+w_2)/2$ , the governing equation for $w_a$ may be obtained, which however does depends on $w_d$ .",
"We define the normalized axial coordinate $X=x/l$ , normalized time $T=t2\\pi \\omega ^*$ , normalized average deflection $W_d=w_d/w^*_d$ , normalized relative deflection $W_a=w_a/w^*_a$ , normalized angular speeds $(\\Omega _1,\\Omega _2)=(\\omega _1/\\omega ^*,\\omega _2/\\omega ^*)$ and normalized gap $D=d/d_0$ .",
"Substituting normalized variables, the equations governing $W_a$ and $W_d$ are $\\frac{\\partial ^4 W_d}{\\partial X^4}=-S_p^4 \\left[1+\\Lambda \\left(D=1+2\\varepsilon W_d\\right) \\right]\\frac{\\partial W_d}{\\partial T}$ $\\frac{\\partial ^4 W_a}{\\partial X^4}=-S_p^4 \\left[1-\\Lambda \\left(D=1+2\\varepsilon W_d\\right) \\right]\\frac{\\partial W_a}{\\partial T},$ where $S_p=(\\xi _\\perp l^4 2\\pi \\omega ^*/s)^{1/4}$ is the Sperm number.",
"Eqs.",
"() are supplemented by the normalized boundary conditions $W_d\\big |_{X=0}=0,\\quad \\frac{\\partial W_d}{\\partial X}\\Big |_{X=0}=\\frac{\\phi _1 l}{2w^*_d} e^{i\\Omega _1 T}-\\frac{\\phi _2 l}{2w^*_d}e^{i(\\Omega _2 T+2\\pi \\gamma )},\\\\ W_a\\big |_{X=0}=0,\\quad \\frac{\\partial W_a}{\\partial X}\\Big |_{X=0}=\\frac{\\phi _1 l}{2w^*_a} e^{i\\Omega _1 T}+\\frac{\\phi _2 l}{2w^*_a}e^{i(\\Omega _2 T+2\\pi \\gamma )},$ representing the hinge boundary condition and slope actuation at $X=0$ and $W_d\\big |_{X=0}=\\frac{\\partial ^2 W_d}{\\partial X^2}\\Big |_{X=1}=\\frac{\\partial ^3 W_d}{\\partial X^3}\\Big |_{X=1}=W_a\\big |_{X=0}=\\frac{\\partial ^2 W_a}{\\partial X^2}\\Big |_{X=1}=\\frac{\\partial ^3 W_a}{\\partial X^3}\\Big |_{X=1}=0$ representing the free end at $X=1$ .",
"From (), the characteristic values $w_a^*$ and $w_d^*$ may be defined as $w_d^*=\\max \\left[\\frac{\\phi _1 l}{2} \\cos {(\\Omega _1 T)}-\\frac{\\phi _2 l}{2}\\cos {(\\Omega _2 T+2\\pi \\gamma )}\\right]$ and $w_a^*=\\max \\left[\\frac{\\phi _1 l}{2} \\cos {(\\Omega _1 T)}+\\frac{\\phi _2 l}{2}\\cos {(\\Omega _2 T+2\\pi \\gamma )}\\right].$ Asymptotic expansion of the nonlinear () allows for approximation to a set of linear equations at the limit, $\\varepsilon =\\frac{w_d^*}{d_0}\\ll 1,$ by presenting $\\Lambda (D)$ as a Taylor series around $D=1$ $\\Lambda (D)\\sim \\Lambda (1)+\\varepsilon 2W_d \\frac{\\partial \\Lambda (D)}{\\partial D}+\\frac{(2\\varepsilon W_d)^2}{2}\\frac{\\partial ^2 \\Lambda (D)}{\\partial D^2},$ as well as asymptotically expanding $W_d$ and $W_a$ $W_d\\sim W_{d,0}+\\varepsilon W_{d,1}+\\varepsilon ^2 W_{d,2},\\quad W_a \\sim W_{a,0}+\\varepsilon W_{a,1}+\\varepsilon ^2 W_{a,2}.$ Substituting (REF ) and (REF ) into () and defining the differential operators $\\textit {$ L+$}=\\partial ^4/\\partial X^4+S_p^4(1+\\Lambda (1))\\partial /\\partial T$ and $\\textit {$ L-$}=\\partial ^4/\\partial X^4+S_p^4(1-\\Lambda (1))\\partial /\\partial T$ , yields the leading-order $O(1)$ of () $\\textit {L^+}W_{d,0}=0, \\quad \\textit {L^-}W_{a,0}=0,$ as well as order $O(\\varepsilon )$ , $\\textit {L^+}W_{d,1}=-2S_p^4W_{d,0}\\frac{\\partial W_{d,0}}{\\partial T}\\frac{\\partial \\Lambda (D)}{\\partial D}, \\quad \\textit {L^-}W_{a,1}=2S_p^4W_{d,0}\\frac{\\partial W_{a,0}}{\\partial T}\\frac{\\partial \\Lambda (D)}{\\partial D},$ and order $O(\\varepsilon ^2)$ , $\\begin{aligned}\\textit {L^+}W_{d,2}=-2S_p^4\\left[\\frac{\\partial W_{d,0}}{\\partial T}\\left(W_{d,1} \\frac{\\partial \\Lambda (D)}{\\partial D}+W_{d,0}^2\\frac{\\partial ^2\\Lambda (D)}{\\partial D^2}\\right)+\\frac{\\partial W_{d,1}}{\\partial T}W_{d,0} \\frac{\\partial \\Lambda (D)}{\\partial D}\\right],\\\\\\textit {L^-}W_{a,2}=2S_p^4\\left[\\frac{\\partial W_{a,0}}{\\partial T}\\left(W_{d,1} \\frac{\\partial \\Lambda (D)}{\\partial D}+W_{d,0}^2\\frac{\\partial ^2\\Lambda (D)}{\\partial D^2}\\right)+\\frac{\\partial W_{a,1}}{\\partial T}W_{d,0}\\frac{\\partial \\Lambda (D)}{\\partial D}\\right]\\end{aligned}$ and so forth.",
"The boundary conditions for the $O(1)$ equations (REF ) are identical to ().",
"For the $O(\\varepsilon )$ (REF ) and $O(\\varepsilon ^2)$ (REF ) equations, the boundary conditions () are modified so that $\\partial W_{d,1}/\\partial X=\\partial W_{a,1}/\\partial X=\\partial W_{d,2}/\\partial X=\\partial W_{a,2}/\\partial X=0$ at $X=0$ .",
"The leading-order $W_{d,0}$ solution can be presented by $\\begin{aligned}W_{d,0}&=\\mathbb {Re}\\left(e^{i\\Omega _1T}F_{\\Omega _1,d}(X)-e^{i\\left(\\Omega _2T+2\\pi \\gamma \\right)}F_{\\Omega _2,d}(X)\\right)\\\\&=\\frac{1}{2}\\bigg [|F_{\\Omega _1,d}(X)|\\Big (e^{i\\left(\\Omega _1T+\\angle F_{\\Omega _1,d}(X) \\right)}- e^{-i\\left(\\Omega _1T+\\angle F_{\\Omega _1,d}(X) \\right)}\\Big )\\\\&-|F_{\\Omega _2,d}(X)|\\Big (e^{i\\left(\\Omega _2T+2\\pi \\gamma +\\angle F_{\\Omega _2,d}(X) \\right)}-e^{-i\\left(\\Omega _2T+2\\pi \\gamma +\\angle F_{\\Omega _2,d}(X) \\right)}\\Big )\\bigg ]\\end{aligned}$ where the functions $F_{\\Omega _1,d}(X)$ , $F_{\\Omega _2,d}(X)$ are $\\begin{aligned}F_{\\Omega _1,d}(X)=\\frac{\\phi _1 l}{2w^*_d}\\frac{e^{i\\frac{\\pi }{8}}\\left(S_p\\@root 4 \\of {\\Omega _1(1+\\Lambda (1))}\\right)^{-1}}{(2+2\\cos {\\phi _{d}^1}\\cosh {\\phi _{d}^1})}(\\sin {\\theta _{d}^1}+\\sinh {\\theta _{d}^1}\\\\+\\sin {\\phi _{d}^1}\\cosh {(\\phi _{d}^1-\\theta _{d}^1)}-\\cos {\\phi _{d}^1}\\sinh {(\\phi _{d}^1-\\theta _{d}^1)}\\\\-\\cosh {\\phi _{d}^1}\\sin {(\\phi _{d}^1-\\theta _{d}^1)}+\\sinh {\\phi _{d}^1}\\cos {(\\phi _{d}^1-\\theta _{d}^1)})\\end{aligned}$ and $\\begin{aligned}F_{\\Omega _2,d}(X)=\\frac{\\phi _2 l}{2w^*_d}\\frac{e^{i\\frac{\\pi }{8}}\\left(S_p\\@root 4 \\of {\\Omega _2(1+\\Lambda (1))}\\right)^{-1}}{(2+2\\cos {\\phi _{d}^2}\\cosh {\\phi _{d}^2})}(\\sin {\\theta _{d}^2}+\\sinh {\\theta _{d}^2}\\\\+\\sin {\\phi _{d}^2}\\cosh {(\\phi _{d}^2-\\theta _{d}^2)}-\\cos {\\phi _{d}^2}\\sinh {(\\phi _{d}^2-\\theta _{d}^2)}\\\\-\\cosh {\\phi _{d}^2}\\sin {(\\phi _{d}^2-\\theta _{d}^2)}+\\sinh {\\phi _{d}^2}\\cos {(\\phi _{d}^2-\\theta _{d}^2)})\\end{aligned}$ and where $\\theta _{d}^i=XS_p\\@root 4 \\of {\\Omega _i(1+\\Lambda (1))}r_1$ , $\\phi _{d}^i=S_p\\@root 4 \\of {\\Omega _i(1+\\Lambda (1))}r_1$ and $r_1=\\@root 4 \\of {-i}=0.92-0.38i$ .",
"Applying the relevant homogeneous boundary conditions and substituting (REF ) into (REF ), we obtain that $W_{d,1}$ is of the form $\\begin{aligned}W_{d,1}&=\\mathbb {Re}\\bigg (F_{2\\Omega _1,d}(X)e^{i2\\Omega _1T}+F_{2\\Omega _2,d}(X)e^{i2\\Omega _2T}\\\\&+F_{(\\Omega _1+\\Omega _2),d}(X)e^{i(\\Omega _1+\\Omega _2)T}+F_{(\\Omega _1-\\Omega _2),d}(X)e^{i(\\Omega _1-\\Omega _2)T}\\bigg )\\end{aligned}$ where the functions $F_{2\\Omega _1,d}(X)$ , $F_{2\\Omega _2,d}(X)$ , $F_{(\\Omega _1+\\Omega _2),d}(X)$ and $F_{(\\Omega _1-2\\Omega _2),d}(X)$ may be readily obtained by substituting (REF ) into (REF ), isolating each harmonic and solving the corresponding ordinary differential equation.",
"Substituting (REF ) and (REF ) into (REF ), yields $W_{d,2}$ with the harmonics $\\Omega _1$ , $3\\Omega _1$ , $\\Omega _2$ , $3\\Omega _2$ , $2\\Omega _1+\\Omega _2$ , $\\Omega _1+2\\Omega _2$ , $2\\Omega _1-\\Omega _2$ and $2\\Omega _1-\\Omega _2$ .",
"The corresponding functions $W_{a,0}$ , $W_{a,1}$ and $W_{a,2}$ can be calculated by applying a similar approach and will have identical frequencies, but different mode functions, compared to $W_{d,0}$ , $W_{d,1}$ and $W_{d,2}$ (see Appendix )."
],
[
"Deflection modes and Experimental Illustration",
"Fig.",
"REF presents the amplitude and phase of the harmonics comprising $w_1$ vs. the coordinate $X$ , for various configurations where $\\varepsilon =0.1$ and $S_p=2.1$ .",
"In all cases the cylinders are actuated at amplitudes $\\phi _1=\\phi _2=15^0$ and the normalized frequency of cylinder 1 is $\\Omega _1=0.5$ .",
"Panels (a-d) present the effect of phase difference $\\gamma $ for cylinders actuated at identical frequencies $\\Omega _2=\\Omega _1$ .",
"For comparison, an isolated cylinder is presented by a grey smooth line.",
"The interaction with an adjacent cylinder with $\\gamma =0$ decreases the effective Sperm number and thus increases the deformation of the cylinder.",
"This yields deflection dynamics identical to an isolated cylinder with a modified Sperm number $Sp(1-\\Lambda (1))^{1/4}$ , where $\\Lambda (1)$ is the leading-order interaction term.",
"Similarly, the leading-order effect of an adjacent cylinder oscillating at anti-phase $\\gamma =\\pi $ is to increase the effective Sperm number to $Sp(1+\\Lambda (1))^{1/4}$ , thus decreasing the deflection of the cylinder.",
"For $\\gamma =0$ the first-order is identically zero, while for both $\\gamma =\\pi /2$ and $\\gamma =\\pi $ (see panels c and d) the first-order correction is nearly independent of $X$ and includes only small value of phase $\\approx 10^0$ .",
"Panels (e-f) examine the effect of an adjacent oscillating cylinder with $\\Omega _2=0.2\\Omega _1$ and no phase $\\gamma =0$ .",
"Similarly, panels (i-l) examine the opposite case of $\\Omega _2=5\\Omega _1$ and $\\gamma =0$ .",
"The leading-order effect of the adjacent cylinder is significant for $\\Omega _2=0.2\\Omega _1$ (where it has a similar effect to the leading-order direct actuation of the cylinder, see panel e).",
"However, a much smaller effect is evident for the case $\\Omega _2=5\\Omega _1$ (see panel i).",
"Panels (f) and (j) examine the phase of the leading-order modes, presenting an opposite effect where the phase is small and nearly uniform for $\\Omega _2=0.2\\Omega _1$ and significant for the case of $\\Omega _2=5\\Omega _1$ .",
"The first-order modes (see panels g and k) are dominated by the small frequencies, corresponding to smaller effective Sperm numbers of the interaction.",
"Thus, for configurations in which the minimal frequency (from the set $2\\Omega _1,2\\Omega _2,\\Omega _1+\\Omega _2,|\\Omega _1-\\Omega _2|$ ) is significantly smaller than all other frequencies, the first-order dynamics may be reasonably approximated by the minimal frequency mode alone (see panel k).",
"Figure: (a) Schematic description of the experimental setup consisting of two elastic cylinders deforming due to a prescribed oscillation of the slope at their bases.",
"(b) An illustrative frame obtained by the Canon EOS 60D DSLR camera during an experiment.Experiments were conducted to quantify the interaction between the two oscillating elastic cylinders and validate the results presented in §2.",
"The experimental setup is illustrated in Fig.",
"REF .",
"Actuation is achieved by a pair of Faulhaber 3257G024CR DC motors operating outside of the fluid, where motion is transferred to the cylinders through levers connected to elongated rotation axes.",
"The bases of the cylinders are fixed while the slopes at the bases are forced to oscillate at predetermined amplitude and frequency.",
"The motors controller is iPOS4808 BX-CAN drive and the elastic cylinders are composed of carbon-fibre with diameter of $1\\textrm {mm}$ and length of $150\\textrm {mm}$ .",
"The gap at rest between the cylinders is $d_0=14mm$ .",
"The immersing fluid is Xiameter®PMX-200 Silicone oil with viscosity $\\mu =59.2 Pa\\cdot s$ and density of $\\rho _l=987 Kg/m^3$ .",
"The container dimensions are $0.4m\\times 0.3m\\times 0.2m$ and the elastic cylinders are placed symmetrically to both sides of the container center plane (see figure REF ).",
"A Canon EOS 60D DSLR camera with Canon EF-S $17-85\\textrm {mm}$ f/4-5.6 IS USM lens was used to record the motion of both cylinders at 25 frames-per-second and resolution of $1920\\; \\times \\; 1080$ pixels per frame.",
"The recorded data was processed by open source code [7].",
"Figure: Deflection W 1 W_1 of cylinder 1 for Ω 1 =5\\Omega _1=5 and amplitude φ 1 =φ 2 =15 0 \\phi _1=\\phi _2=15^0 at six equally spaced times along the full oscillation period PP.",
"Dimensional values are w 1 =W 1 ×1.4mmw_1=W_1\\times 1.4mm, ω=Ω×0.1Hz\\omega =\\Omega \\times 0.1Hz and t=T×0.62st=T\\times 0.62 s. Smooth lines denote theoretical results and dashed lines denote experimental data.",
"The examine cases are (a) without an adjacent cylinder, (b) an adjacent cylinder with Ω 1 =Ω 2 \\Omega _1=\\Omega _2, (c) Ω 2 =0.8Ω 1 \\Omega _2=0.8\\Omega _1 and (d) Ω 2 =1.2Ω 1 \\Omega _2=1.2\\Omega _1.",
"Inserts present the deflection of the cylinder at its free end W 1 (X=1)W_1(X=1) vs. time.",
"See supplementary information - movies 1-4.Fig.",
"REF presents the experimental (dashed lines) and theoretical (smooth lines) deflection patterns $W_1$ of cylinder 1 oscillating at frequency $\\Omega =0.5$ and amplitude at $\\phi _1=15^0$ adjacent to cylinder 2.",
"The inserts present the deflection of the free end of the cylinder $W_1(X=1)$ for a full cycle period defined as $P$ .",
"Dimensional values are related to the normalized values by $w_1=W_1\\times 1.4mm$ , $\\omega =\\Omega \\times 0.1Hz$ and $t=T\\times 0.62 s$ .",
"For reference, panel (a) (supplementary information - movie 1) presents the deflection of an elastic cylinder oscillating without the presence of a second adjacent cylinder.",
"Panel (b) (supplementary information - movie 2) presents the deflection $W_1$ for the case of an adjacent cylinder oscillating at identical frequency $\\Omega _1=\\Omega _2$ , identical amplitude $\\phi _1=\\phi _2$ and without phase $\\gamma =0$ .",
"A significant increase in amplitude of the deflection is clearly evident and the deflection patterns remain symmetric in this case.",
"Panels (c) and (d) (supplementary information - movies 3 and 4) present the effect of an adjacent cylinder oscillating at a slightly smaller frequency ($\\Omega _2=0.8\\Omega _1$ in panel c) and a slightly higher frequency ($\\Omega _2=1.2\\Omega _1$ in panel d).",
"Due to the multiple frequencies characterising panels (c) and (d), the full period is defined by the $\\Omega _1-\\Omega _2=1$ mode as $P=2\\pi $ , which is an integer multiplication of all other modes.",
"The values of $\\Omega _2$ in panels (c) and (d) were chosen to be similar to $\\Omega _1$ in order to reduce the effective Sperm number of the $\\Omega _1-\\Omega _2$ mode, thus increasing the first-order deflection to be experimentally significant.",
"Figure: Fourier decomposition of the experimental deflection W 1 W_1 at X=1{X=1}, presenting amplitude A Ω A_\\Omega vs. frequency Ω\\Omega .",
"Panels (a-d) present the frequency decomposition of inserts of panels (a-d) in figure , respectively.",
"Filled circle markers denote experimental data and smooth blue lines denote the theoretical predictions.",
"Red circles denote harmonics expected from the theoretical results.",
"Panels (e-h) are magnifications of (a-d) in order to present the amplitudes of the 2Ω 1 ,2Ω 2 ,Ω 1 +Ω 2 2\\Omega _1,2\\Omega _2,\\Omega _1+\\Omega _2 first-order frequencies.Fig.",
"REF presents frequency decomposition (based on MATLAB®FFT function) of the experimental data presented in the inserts of Fig.",
"REF .",
"The blue smooth lines are the theoretical predictions, and the full circles are the experimental amplitudes (predicted frequencies are filled red circles and other frequencies are filled black circles).",
"As expected, since $\\Omega _1\\approx \\Omega _2$ in panels (c,d), the deflection is dominated by the frequencies of the actuation in leading-order $\\Omega _1$ , $\\Omega _2$ and the $\\Omega _1-\\Omega _2$ first-order harmonic.",
"All other frequencies are small compared with the experimental resolution.",
"Panels (e-f) are magnifications of panels (a-d), focusing on of the amplitudes of the $2\\Omega _1$ , $2\\Omega _1$ , $\\Omega _1+\\Omega _2$ first-order frequencies.",
"For all examined cases in figures and REF and REF , a reasonable agreement between the experimental data and the theoretical results is evident."
],
[
"Concluding Remarks",
"While microscopic swimmers do not create propulsion by oscillating the slope at the base of their flagella, the results may still provide insight for biological mechanisms which inherently involve elasticity.",
"The presented analysis yielded that the effect of an adjacent oscillating cylinder, with identical in-phase actuation frequency and amplitude, is to decrease the effective Sperm number.",
"Thus, the optimal propulsion oscillation frequency for an array of flagella may be expected to be greater compared with the optimal frequency of an isolated flagellum.",
"For $\\Omega _1\\approx \\Omega _2$ , the slowest $\\Omega _1-\\Omega _2$ mode dominates the first-order dynamics since the amplitude of deflection is inverse to the effective Sperm number and the mode's frequency.",
"Future work may examine non-linear effects, propulsion dynamics, internal actuation distributed along the cylinder, as well as a study of the dynamics of a lattice of oscillating cylinders."
],
[
"leading-order average deflection, $W_{a,0}$",
"and where $\\theta _{a}^i=XS_p\\@root 4 \\of {\\Omega _i(1-\\Lambda (1))}r_1$ , $\\phi _{a}^i=S_p\\@root 4 \\of {\\Omega _i(1-\\Lambda (1))}r_1$ and $r_1=\\@root 4 \\of {-i}$ ."
]
] | 1612.05808 |
[
[
"Massive Star Evolution: What we do (not) know"
],
[
"Abstract The modelling of massive star evolution is a complex task, and is very sensitive to the way physical processes (such as convection, rotation, mass loss, etc.)",
"are included in stellar evolution code.",
"Moreover, the very high observed fraction of binary systems among massive stars makes the comparison with observations difficult.",
"In this paper, we focus on discussing the uncertainties linked to the modelling of convection and rotation in single massive stars."
],
[
"Introduction",
"The modelling of massive star is a complex task, involving a variety of physical processes.",
"Among the required ingredients of all stellar evolution codes are the treatment of the heat transfer in convective and radiative zones, the nuclear reaction network, the equation of state, the computation of opacities, and the inclusion of mass loss [26], [32].",
"During the past two decades, a variety of other processes were progressively added, such as rotation [13], [62], [36], transport of angular momentum and chemical species by internal magnetic fields [51], [34] or by internal waves [27], [55], [54], [16].",
"Moreover, massive stars are often found in multiple systems [50], and their modelling requires in addition the treatment of tidal interactions, Roche-lobe overflow, common envelope evolution and merging [28].",
"Each of these processes suffers from uncertainties in the way they should be implemented in stellar evolution code.",
"It leads to major uncertainties in our understanding of the evolution of massive stars, particularly of the post-main-sequence evolution [38], [7], [17].",
"In this paper, we focus on discussing some of the uncertainties linked to the modelling of convection and rotation in single massive stars."
],
[
"The modelling of convection in the interior of massive stars",
"Convection is ubiquitous in massive star evolution (see Fig.",
"REF ): successive convective cores (“CC”) are often linked to one of the burning stages.",
"After the main-sequence (MS), nuclear burning may also occur in convective shells (“CS”), producing a complex structure during the very late stages of the evolution.",
"During the MS, a very tiny convective zone is present near the surface [33], [4].",
"Finally, for stars evolving in the red part of the Hertzsprung-Russell diagram (HRD, typically the stars having a red supergiant phase), a deep external convective envelope (“DCE”) develops.",
"This highlights the need for a correct modelling of convection in massive star models.",
"Figure: Convective structure of solar metallicity 15M ⊙ 15\\,M_\\odot (left) and 60M ⊙ 60\\,M_\\odot stellar models (right).",
"The horizontal axis shows the time left until the star collapses.",
"The vertical axis is the mass coordinate (so-called “Kippenhahn diagram”).",
"The shaded area represent the convective zones inside the star.",
"The labels are for sub-surface convection (“SSC”), convective core (“CC”, highlighted by the thick solid black line), deep convective envelope (“DCE”), and convective shell (“CS”).",
"The thick line indicates the surface of the star.In most of stellar evolution codes, convection is modelled in two steps: The position of the convective zone boundaries are determined according to the Schwarzschild or Ledoux criterion [26].",
"Inside the convective zone, the thermal structure is computed by computing a thermal gradient.",
"This can be done in several ways: by assuming that convection is purely adiabatic, by using the “Mixing-Length Theory” [2] or more sophisticated models [58].",
"The sizes of the convective cores obtained in this way are known to be too small with respect to observations for a long time [29], and need to be artificially extended by an arbitrary length: the “overshoot”.",
"This overshoot cannot be predicted by design in the framework of the MLT.",
"In stellar evolution codes, the overshoot is usually considered to be “penetrative” [61], or “diffusive”, with a diffusion coefficient calibrated on observations or numerical simulations [15], [23].",
"Both approaches contain free parameters, and thus need to be calibrated.",
"This can be done in different ways, for example by reproducing the observed width of the MS [12], or by fitting the drop of the surface velocities of stars when their surface gravity decreases [3].",
"On the other hand, the development of multi-dimension hydrodynamics codes and of computing power has allowed for simulations of convection in a variety of physical conditions from first principles: envelope of cool stars [14], [6], [57], [37], or deep convection during different evolutionary stages of star life [40], [8], [59], [9], [46], [25].",
"Simulations of deep convection show that, at least during the advanced stages, convective boundaries are moving [40].",
"Moreover, there is a significant mixing across the boundary, making it less stiff than usually accounted for in 1d stellar evolution modelling [9].",
"From observations, a similar result is obtained thanks to asteroseismology for the overshoot during the MS [45].",
"The modelling of variable blue supergiants seems to be a promising way of constraining convection is stellar models [17].",
"In any case, it is clear that MLT is not able to correctly reproduce the behaviour of convective flows as seen in multi-d simulations, and there is an urgent need for a new way of treating convection in stellar evolution models [1]."
],
[
"The modelling of rotation in the interior of massive stars",
"The inclusion of rotation in 1d stellar evolution codes is not straightforward.",
"Due to its (at least) 2d nature, several hypotheses and approximations are requested to treat rotation in 1d (process sometimes called “1.5d”).",
"First of all, the star is described in the framework of the Roche model, assuming a spherical-symmetry gravitational potential on top of which the effects of centrifugal acceleration are added.",
"Moreover, a strong horizontal turbulence is assumed inside the star, homogenising the angular velocity on an isobar [62], [36].",
"In this framework, rotation has two main effects: the centrifugal acceleration modifies the usual stellar structure equations by adding corrective terms in the momentum equation and radiative transfer equation [41].",
"thermal non-equilibrium produces large scale currents inside the star [52].",
"In turn, these currents transport angular momentum and chemical species, modifying the internal rotation of the star.",
"Differential rotation can occur, generating shear turbulence.",
"The transport of angular momentum is modelled by the following relation [62], [36]: $\\rho \\partial _t\\left(r^2\\bar{\\Omega }\\right) = \\frac{1}{5r^2}\\partial _r\\left(\\rho r^4\\bar{\\Omega }U_2\\right) + \\frac{1}{r^2}\\partial _r\\left(\\rho D_v r^4\\partial _r\\bar{\\Omega }\\right),$ where $\\bar{\\Omega }$ is the mean angular velocity on an isobar, $U_2$ is the radial component of the meridional circulation velocity, and $D_v$ is the vertical turbulence diffusion coefficient.",
"The expression of $U_2$ is complex and can be found in [32].",
"The transport of chemical species can be modelled by a purely diffusive approach [5]: $\\rho \\partial _t\\left(X_i\\right) = \\frac{1}{r^2}\\partial _r\\left(\\rho r^2\\left(D_v + D_\\mathrm {eff}\\right)\\partial _r\\left(X_i\\right)\\right),$ where $X_i$ is the abundance of the element $i$ , and $D_\\mathrm {eff}$ is the effective diffusion coefficient, accounting for the effect of meridional circulation: $D_\\mathrm {eff} = \\frac{\\left(rU_2\\right)^2}{30D_\\mathrm {h}}$ , where $D_\\mathrm {h}$ is the horizontal turbulence diffusion coefficient."
],
[
"The vertical turbulence",
"The diffusion coefficient $D_v$ should account for any kind of turbulence arising in the vertical direction.",
"Depending on which stellar evolution code is used, different effects are accounted for: secular shear instability [30], [56], dynamical shear instability [21], Solberg-Hoiland instability [21], Goldreich-Schubert-Fricke instability [21], [24], Tayler-Spruit dynamo induced mixing [51], [34].",
"Most of time, the corresponding diffusion coefficients are summed up.",
"However, [35] propose a way to consider the combined effects of these instabilities at once."
],
[
"The horizontal turbulence",
"As in the case for the vertical turbulence, several prescriptions can be found in the literature for the diffusion coefficient linked to the horizontal turbulence [62], [31], [39]."
],
[
"Advection and diffusion or diffusion only?",
"As of today, we can distinguish two big families among the stellar evolution codes that contains the treatment of stellar rotation.",
"The first of them solves the full equation for the transport of angular momentum (eq.",
"REF ): the Geneva stellar evolution code [11], STAREVOL [10], FRANEC [7], ROSE [49]This list is indicative only and has no ambition to be complete.",
"It mostly covers the codes used in the massive star community.. On the other hand, other codes uses an approximate form of eq.",
"REF , where the advective term is replaced by another diffusion coefficient, making this equation fully diffusive: MESA [47], STERN [48], or Kepler [22].",
"There is so far no consensus about which implementation should be used.",
"However, the reader should keep in mind that both implementations provide different results in terms of evolution of the surface velocities and chemical species [12], [19], [3], [7], [38].",
"Another caution is linked to the uncertainty of the choice of the vertical or horizontal diffusion coefficients.",
"Different choices can also lead to qualitatively different results [44]."
],
[
"Surface abundances.",
"One of the most important effect of rotation is the modification of the surface chemical composition as a function of time, due to the rotation-induced internal mixing.",
"For most of massive star models, it implies that chemical species produced in the core of the star during hydrogen-burning are progressively brought to the surface.",
"For example, it implies an increase of the N abundance, while C and O abundances decrease.",
"This effect is generally stronger for higher mass star and at lower metallicity [12], [19].",
"In the most extreme cases, internal mixing favours the evolution towards the Wolf-Rayet stage, making it occur earlier in the lifetime of a massive star, or for lower initial mass stars [18]."
],
[
"Tracks in the HRD.",
"On the Zero-Age Main-Sequence, a rotating model is cooler and less luminous than its non-rotating counterpart.",
"This is due to the support of the centrifugal acceleration, making the model behaves as a lower mass one.",
"After the ZAMS, internal mixing brings fresh hydrogen into the core, making its mass diminish more slowly, and thus keeping the star at higher luminosity.",
"At the same time, the change in the surface abundances (more helium, less hydrogen) makes the star evolve at higher effective temperature than in the non-rotating case [42].",
"In some extreme cases, for very rapidly rotating star, the mixing is so efficient that the star can evolve nearly homogeneously [60], [43], [53]."
],
[
"Lifetimes.",
"Due to the ingestion of fresh hydrogen by the core during the MS due to rotational mixing, the lifetime of the star are increased [19].",
"The increase of the lifetime can reach several tens of percent with respect to the non-rotating case.",
"This has consequence on the computation of isochrones [20]."
],
[
"Conclusions",
"In this paper, we have briefly discussed the implementation of convection and rotation in stellar evolution codes, in particular in the context of massive star evolution.",
"Current implementations of convection are described, and we highlight the shortcomings shown by recent multi-dimensional hydrodynamics simulations or observations.",
"The various ways of dealing with rotation are also explained.",
"Finally, we discussed some impacts of the inclusion of rotation on our understanding of stellar evolution.",
"CG and SE acknowledge support from the Swiss National Science Foundation (project number 200020-160119).",
"RH acknowledges support from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 306901 and WPI IPMU."
]
] | 1612.05451 |
[
[
"Some Remarks on Preconditioning Molecular Dynamics"
],
[
"Abstract We consider a Preconditioned Overdamped Langevin algorithm that does not alter the invariant distribution (up to controllable discretisation errors) and ask whether preconditioning improves the standard model in terms of reducing the asymptotic variance and of accelerating convergence to equilibrium.",
"We present a detailed study of the dependence of the asymptotic variance on preconditioning in some elementary toy models related to molecular simulation.",
"Our theoretical results are supported by numerical simulations."
],
[
"Introduction",
"The problem of convergence to equilibrium for diffusion processes has attracted considerable attention in recent years.",
"Due to the significant computational cost associated with MCMC type algorithms it is important to understand, and where possible accelerate, the convergence to equilibrium of systems in statistical physics, materials science, biochemistry, machine learning and many other areas.",
"In such applications it is typically necessary to compute expectations of the form $ \\mu (f)= \\int _{\\mathbb {R}^N} f(x)\\mu (dx)$ of an observable $f$ with respect to a target probability distribution $\\mu (dx)$ on $\\mathbb {R}^N$ , where $\\mu $ is of the form $\\mu (dx)=Z_{\\mu }^{-1}{\\rm e}^{-E_N}dx,$ where $E_N:\\mathbb {R}^N \\rightarrow \\mathbb {R}$ is e.g.",
"a potential energy and $Z_{\\mu }^{-1}$ the normalisation constant.",
"Note that we have absorbed temperature into $E_N$ .",
"The usual difficulty is that the integration (REF ) cannot be performed directly due to the high dimensionality of the problem and MCMC methods are instead employed.",
"Ill-conditioning of $E_N$ , which can be induced by a variety of mechanisms, but in particular high-dimensionality (large $N$ ), is a common challenge to overcome in order to construct an efficient sampling scheme.",
"An attractive approach is to precondition the MCMC algorithm.",
"The algorithm is transformed by a well-chosen operator (the preconditioner) in a way that does not alter the invariant measure but (hopefully) accelerates convergence.",
"The purpose of this paper is to explore to what extent (or, whether at all) preconditioning of a Langevin-type algorithm helps to accelerate the computation of expectations.",
"Our study is motivated by recent advances such as the Riemannian Manifold MALA [giro09], Stochastic Newton Methods [mart12], non-reversible diffusions [dunc,Lelievre], optimal scaling for Langevin algorithms [bes09,Pillai] and affine-invariant sampler [hou].",
"All of these references present different variants of preconditioning or related modifications of MCMC methods and result in an improved performance.",
"Reviewing all these different approaches would go beyond the scope of this introduction, however, we mention the Riemannian Manifold Monte Carlo method [giro09, xifa] as the main motivation for the present study.",
"In this method, preconditioning is understood as performing MCMC on a Riemannian manifold defined via a “metric” $P(x)$ , the preconditioner.",
"A range of preconditioners, ideally the hessian $\\nabla ^2E_N(x)$ or a positive definite matrix $P(x)$ closely approximating the Hessian, are tested.",
"In a broad range of examples (including, e.g., logistic regression, a stochastic volatility model, and an ODE inference example) it has been shown in numerical tests that a well-chosen preconditioner improves both the mixing time as well as the convergence of the probability density functions to the target measure, with speed-ups ranging from moderate $O(1)$ factors to orders of magnitude depending on the application.",
"Motivated by these promising results we applied analogous preconditioned sampling algorithms to some model molecular systems, but did not always (rarely in fact) observe the speed-ups we expected.",
"For instance, in Figure REF we show the convergence of reconstructed free energy profiles from metadynamics [Laio02] simulations comparing the unpreconditioned and preconditioned dynamics using a Hessian–based preconditioner with varying parameter sets.",
"This type of preconditioner usually yields at least an order of magnitude speed-up in typical geometry optimisation tasks, but fails to accelerate the assembly of the free energy surface (see Appendix REF for the precise setup of the test).",
"The question naturally arises whether this failure is due to a lack of fine-tuning or due to a more fundamental limitation.",
"Figure: RMS errors of the reconstructed free energy surface profiles from metadynamics simulations without (black dashed line) and with preconditioning (coloured solid lines).",
"Error bars represent one standard deviation based on 10 independent simulations.",
"See for the detailed setup.Thus, to understand better these observations we will study some elementary analytical and numerical examples, which capture some essential characteristics of typical molecular systems, but where explicit results can still be obtained.",
"The origin of the difficulty comes from the fact that it is highly dependent not only on $\\mu $ but also on the observable $f$ whether preconditioning can achieve a significant (or, any) speed-up.",
"We will demonstrate that for some typical observables $f$ , even moderate preconditioning can achieve significant speed-ups while for other, equally common, observables no speed-up should be at all expected even if $E_N$ is highly ill-conditioned.",
"Moreover, we will show that the dimensionality (1D, 2D, 3D) of the molecular structure plays a crucial role.",
"While our discussion is primarily motivated by applications in molecular simulations, it should straightforwardly adapt the arguments and findings to other application areas.",
"The most commonly employed algorithms in molecular simulation are based on discretising the Langevin equation, but for the sake of simplicity we will focus on the overdamped Langevin equation, $ dX_t= - \\nabla E_N(X_t) dt + \\sqrt{2} dW_t,$ where $W_t$ is a $N$ -dimensional standard Brownian motion.",
"Under mild technical conditions on $E_N$ and $X_t$ it is known that the dynamics $(X_t)_{t\\ge 0}$ is ergodic with respect to the measure $\\mu $ (see [rob96]).",
"Discretising in time, $ X_{m+1} = X_m - \\delta \\nabla E_N(X_m) + \\sqrt{2\\delta } R_m,$ where $R_m \\sim N(0, I_{N \\times N})$ and $\\delta $ is a parameter quantifying the size of the discrete time increment (time-step), we obtain a Markov chain with invariant measure $\\tilde{\\mu }$ , where typically $\\tilde{\\mu }-\\mu \\sim O(t^{-1/2}) + O(\\delta )$ .",
"Here, $O(t^{-1/2})$ represents the statistical error due to the finite length of the simulation, while $O(\\delta )$ represents the bias due to time discretisation.",
"Adding metropolisation to (REF ) leads to the Metropolis Adjusted Langevin Algorithm (MALA).",
"It is well known that the measure $\\mu $ is invariant for the MALA [rob04,rob96].",
"In molecular simulation it is common to assume that the bias in $\\tilde{\\mu }$ is negligible compared to modelling and statistical errors and therefore we will not consider metropolisation in the present paper.",
"The resulting algorithm is called the Unadjusted Langevin Algorithm (ULA).",
"As a matter of fact, we will argue in Section REF that the issues we are addressing are largely unrelated to the time-discretisation, hence we will focus on the continuous process (REF ) instead of the Markov chain (REF ).",
"Thus, given an observable $f\\in L^1(\\mu )$ , we are interested in quantifying the convergence $ \\epsilon _T(f) := \\frac{1}{T}\\int _0^T f(X_t) dt \\rightarrow \\mu (f)\\qquad \\text{for $\\mu $-a.e } X_0.$ Under additional assumptions on $\\mu $ and $f$ , this convergence result is accompanied by a central limit theorem which characterises the asymptotic distribution of the fluctuations, i.e.",
"$\\sqrt{t} \\left(\\epsilon _t(f)-\\mu (f)\\right)\\xrightarrow{} \\mathcal {N}(0,\\sigma _f^2),$ where $\\sigma _f^2$ is known as the asymptotic variance for the observable $f$ (see [catt,kipnis]).",
"In Section we will explicitly compute $\\sigma _f^2$ for some simple energy functionals and observables, which mimic typical objects of interest in molecular simulations and demonstrate that for some typical observables, $\\sigma _f$ may be strongly dependent on the conditioning of $E_N$ , in particular on system size $N$ , while for others the dependence on $N$ is negligible."
],
[
"Preconditioned Langevin Algorithm",
"For a (fixed) preconditioner $P \\in \\mathbb {R}^{N \\times N}$ , symmetric positive definite, we consider the preconditioned Overdamped Langevin dynamics ($P$ -Langevin), $dX_t^P = - P^{-1}\\nabla E_N(X_t^P)dt + \\sqrt{2} P^{-1/2} dW_t.$ The standard overdamped Langevin dynamics is recovered by taking $P=I$ .",
"It is in principle possible to allow $P = P(X)$ but for simplicity we will not consider this in the present work.",
"Note that, if $P$ is fixed then the coordinate transform $Z = P^{1/2} X$ allows us to easily lift results from (REF ) to (REF ); see Section REF for more details.",
"The preconditioner does not affect the invariance property of the diffusion process, i.e., the target measure $\\mu $ is still invariant for the P-Langevin process (REF ).",
"However, it can affect the convergence of the process to the invariant measure.",
"That is, for $X_t^P \\sim \\mu _t^P = \\varphi _t^P dx$ , we will characterise in Section the rate of convergence of $\\varphi _t^P$ to $\\varphi _\\infty $ ($\\varphi _\\infty $ denotes the density of $\\mu $ ) and demonstrate how preconditioning improves this rate.",
"Moreover, we also obtain $ \\epsilon _T^P(f) := \\frac{1}{T}\\int _0^T f(X_t^P)dt\\rightarrow \\mu (f), \\qquad \\text{for $\\mu $-a.e } X_0.$ and analogously to the standard Langevin dynamics, a central limit theorem characterizes the asymptotic distribution of the fluctuations, $\\sqrt{t} \\left(\\epsilon _t^P(f) - \\mu (f) \\right)\\xrightarrow{} \\mathcal {N}(0,\\sigma _{f,P}^2),$ where $\\sigma _{f,P}^2$ is the asymptotic variance of $f$ under $P$ -Langevin dynamics.",
"The main aim of our paper is to present several simplified but still realistic examples at which we can observe whether or not preconditioning accelerates sampling in the sense that it achieves a reduction in the asymptotic variance, i.e., $\\sigma _{f,P}^2\\ll \\sigma _f^2$ ."
],
[
"Outline",
"The rest of the paper is organised as follows.",
"In Section we present the model Hamiltonians, $E_N$ , to motivate some key assumptions that we make throughout our analysis.",
"We also recall the coordinate transform that we use to reduce $P$ -Langevin to standard Langevin, and we explain why the time-discretisation is a negligible component and can therefore be ignored for our analysis.",
"In Section we show a long time convergence result to the invariant measure in the $P$ -Langevin process.",
"For some quadratic model Hamiltonians we can then precisely quantify the speed-up afforded through preconditioning.",
"Section is devoted to describe how the central limit theorem (REF ) arises from the solution of the Poisson equation associated with the generator of the dynamics.",
"This is then followed by a detailed analysis of $\\sigma _f^2$ and $\\sigma _{f,P}^2$ for some quadratic model Hamiltonians and observables.",
"In Section we show a numerical application to illustrate the reduction in the asymptotic variance."
],
[
"Coordinate transformation",
"As mentioned above, a convenience afforded by our assumption that the preconditioner $P$ is constant, is that a simple coordinate transformation can transform the P-Langevin dynamics (REF ) into standard Langevin dynamics (REF ) by taking $Z_t = P^{1/2} X_t$ .",
"We will now briefly review this transformation.",
"First, let us introduce the new coordinates and associated energy $z := P^{1/2} x \\qquad \\text{and} \\qquad E_N^P(z) := E_N(x) = E_N(P^{-1/2} z).$ The standard overdamped Langevin dynamics for $z$ then reads $ \\begin{split}dZ_t &= - \\nabla _z E_N(P^{-1/2} Z_t) dt + \\sqrt{2} dW_t \\\\&= - P^{-1/2} \\nabla _x E_N(P^{-1/2} Z_t) + \\sqrt{2} dW_t.\\end{split}$ Upon multiplying the equation with $P^{-1/2}$ we clearly recover (REF ); that is, (REF ) and (REF ) are indeed equivalent.",
"In the new coordinate system the infinitesimal generator operator of the diffusion process $Z_t$ is given by: $\\mathcal {L}_P= - \\nabla _z E_N^P(z)\\cdot \\nabla _z + \\Delta _z.$ In terms of estimating observables, we obtain that $ \\epsilon _T^P(f) = \\frac{1}{T}\\int _0^T f(X_t^P)dt= \\frac{1}{T} \\int _0^T f(P^{-1/2} Z_t) dt.$ The key observation in analysing how preconditioning changes the properties of $E_N^P$ and hence of the Langevin dynamics is that $\\nabla _z^2 E_N^P(z) = P^{-1/2} \\nabla _x^2 E_N(x) P^{-1/2}.$ In the following, if $H$ and $P$ are symmetric positive definite, we will write $H_P := P^{-1/2} H P^{-1/2}$ .",
"We also observe, for future reference, that the spectrum of $H_P$ satisfies $\\inf \\sigma (H_P) = \\inf _{v^T P v = 1} v^T H v \\qquad \\text{and} \\qquad \\sup \\sigma (H_P) = \\sup _{v^T Pv = 1} v^T H v.$"
],
[
"Model Hamiltonians",
"Let $x \\in \\mathbb {R}^N$ describe a system of $N/d$ particles at positions $(y_\\alpha )_{\\alpha = 1}^{N/d} \\subset \\mathbb {R}^d$ , then a simple model for potential energy is given by $E_N(y) = \\sum _{\\alpha = 1}^{N/d} \\sum _{\\beta \\ne \\alpha }\\phi ( |y_\\alpha - y_\\beta | ),$ where $\\phi $ is e.g.",
"a Lennard-Jones type potential, $\\phi (r) = r^{-12} - 2 r^{-6}$ .",
"Such systems exhibit complex meta-stable behaviour, which is an issue to be entirely separated from the ill-conditioning due to high dimension.",
"A much simpler situation is a mass-spring model, where $u = (u_\\alpha ) \\in \\mathbb {R}^N$ , with $u_\\alpha $ e.g.",
"denoting out-of-plane displacement of a particle, and particle connectivity described by an equivalence relation $\\alpha \\sim \\beta $ .",
"Then the energy can be written as $ E_N(u) = \\sum _{\\alpha \\sim \\beta } \\phi _{\\alpha \\beta }( u_\\alpha - u_\\beta ),$ where $\\phi _{\\alpha \\beta }$ could be taken as strictly convex to avoid meta-stability.",
"We will use systems of this kind in our numerical experiments.",
"In order to admit explicit analytical calculations we simplify further $E_N$ by expanding it about an equilibrium (e.g., at $u = 0$ ) which then yields a quadratic energy $ E_N(u) = \\sum _{\\alpha \\sim \\beta } {\\textstyle \\frac{k_{\\alpha \\beta }}{2}}|u_\\alpha - u_\\beta |^2 =: {\\textstyle \\frac{1}{2}} u^T H u.$ The spring constants $k_{\\alpha \\beta }$ could model how the interaction between different atomic species / environments differs.",
"We assume throughout that there exist bounds $\\underline{k}, \\bar{k}$ on the spring constants and a bound $\\bar{n}$ on the number of neighbours, which are both independent of $N$ .",
"More precisely, we assume that $ 0 < \\underline{k} \\le k_{\\alpha \\beta } \\le \\bar{k} \\qquad \\text{and} \\qquad \\# \\lbrace \\beta : \\alpha \\sim \\beta \\rbrace \\le \\bar{n}.$ Note that we have chosen a scaling of the energies where increasing system size $N \\rightarrow \\infty $ does not yield a continuum limit but rather an infinite lattice system.",
"The consequence is that if we have a bound in the spring constants $0 < k_{\\alpha \\beta } \\le \\bar{k}$ , and each atom $\\alpha $ is connected with at most $\\bar{n}$ neighbours, then it follows readily that $ |E_N(u)| \\lesssim \\bar{n} \\bar{k} |u|^2 \\qquad \\text{or, equivalently}\\qquad \\Vert H \\Vert \\lesssim \\bar{n} \\bar{k},$ where $\\Vert H\\Vert $ denotes the $\\ell ^2 \\rightarrow \\ell ^2$ operator norm.",
"To be even more specific let us assume that $E_N$ is a $d$ -dimensional lattice model, i.e., each atom index $\\alpha $ corresponds to a coordinate $\\ell _\\alpha \\in \\lbrace 0, \\dots , M+1\\rbrace ^d$ and $\\alpha \\sim \\beta $ if and only if $|\\ell _\\alpha -\\ell _\\beta | = 1$ , that is $ E_N(u) := \\sum _{\\alpha _1=0}^M \\cdots \\sum _{\\alpha _d = 0}^M\\sum _{j = 1}^d k_{\\alpha ,\\alpha +e_j} \\big |u(\\alpha +e_j) - u(\\alpha )\\big |^2.$ By clamping the boundary sites at $u = 0$ , we obtain $N = M^d$ free lattice sites.",
"In this case, $H$ is a (possibly inhomogeneous) discrete elliptic operator and employing (REF ) and using the min-max characterisation of eigenvalues (see, e.g., Sect XIII.1 in [ReedSimon]) to compare $H$ to the discrete Laplacian for which the spectrum can be computed explicitly [iserles], we can readily show that there exist constants $c_0, c_1$ such that $ c_0 (j/N)^{2/d} \\le \\lambda _j \\le c_1 (j/N)^{2/d},$ where $\\sigma (H) = \\lbrace \\lambda _j\\,|\\,j = 1,\\dots ,N\\rbrace $ is the ordered spectrum of $H$ .",
"This dimension-dependence of the eigenvalue distribution will be important later on."
],
[
"A model preconditioner",
"Although for the simple model problems described in Section REF it is straightforward to compute Hessians and use those as preconditioners, this would remove us from the practice of molecular simulations where Hessians are not normally computable.",
"Instead, we will consider preconditioners that only roughly capture the structure of the energy functionals and their Hessians.",
"Following [packwood], we will use a preconditioner of the form $ v^T P v = c \\sum _{\\alpha \\sim \\beta } |v_\\alpha - v_\\beta |^2,$ where $c$ is a free parameter to be fitted to the model.",
"The idea is that it captures the connectivity information but not the fine details of the Hessian.",
"For optimisation and saddle search, preconditioners of this kind have been shown to yield considerable speed-ups even for much more complex electronic structure type models [packwood].",
"For example, considering $E_N$ given by (REF ) with hessian $H := \\nabla ^2 E_N$ and choosing $c:= \\frac{1}{2} (\\underline{k} + \\bar{k})$ , we observe that $ {\\textstyle \\frac{2 \\underline{k}}{\\bar{k}+\\underline{k}}} v^T P v\\le v^T H v\\le {\\textstyle \\frac{2 \\bar{k}}{\\bar{k}+\\underline{k}}} v^T P v,$ which in particular implies that $\\kappa \\big ( H_P \\big ) \\le \\frac{\\bar{k}}{\\underline{k}}.$ Here, $\\kappa (A) := \\Vert A\\Vert \\Vert A^{-1}\\Vert $ denotes the condition number of the matrix $A$ .",
"That is, provided the inhomogeneity $\\bar{k}/\\underline{k}$ is not too severe, then the preconditioned energy landscape has only very moderate conditioning, independent of $N$ , while typically $\\kappa (\\nabla ^2 E) \\rightarrow \\infty $ as $N \\rightarrow \\infty $ , with a rate depending on the connectivity.",
"For instance, for a lattice model (REF ) implies that $\\kappa (\\nabla ^2 E_N) \\sim N^{2/d}$ ."
],
[
"Time Discretisation",
"To measure the cost/accuracy ratio of a practical sampling algorithm based on the Langevin process we need to account also for the time-discretisation, $ X_{n+1} = X_n - \\delta P^{-1} \\nabla E_N(X_n)+ \\sqrt{2 \\delta } P^{-1/2} R_n,$ where $R_n \\sim N(0, I_{N \\times N})$ .",
"It is tempting to assume that one advantage afforded by preconditioning is to take larger time-steps.",
"As we show in the following, this is a matter of scale, thus justifying our choice to focus purely on the time-continuous P-Langevin dynamics.",
"In the scaling that we have chosen in the model problems of Section REF , both the Hessians $H$ and preconditioned Hessians $H_P$ are bounded operators (on $\\ell ^2$ ), independent of system size $N$ .",
"Thus, the time-steps have similar restrictions for the preconditioned and unpreconditioned Langevin processes [hairr].",
"More precisely, in the quadratic model problem (REF ) we may assume $\\bar{k} = 1$ without loss of generality.",
"Then, using the model preconditioner (REF ) with $c = \\bar{k} = 1$ , say, we have that $\\sup \\sigma (H) = \\sup \\sigma (H_P)$ .",
"Suppose that $E_N(x) = \\frac{1}{2} x^T H x$ and $P \\in \\mathbb {R}^{N \\times N}$ with $H$ and $P$ both positive definite.",
"Suppose further that $\\frac{\\delta }{2}|P^{-1}H| < 1$ .",
"Then the invariant measure for (REF ) is Gaussian with covariance matrix $C_\\delta ^P = \\big (I - {\\textstyle \\frac{\\delta }{2}}P^{-1}H\\big )^{-1}H^{-1}.$ See Section REF .",
"It follows that the error in the covariance operator is $O(\\delta )$ with constant independence of $N$ , and independent of the choice of preconditioner.",
"Indeed, even with an “optimal” preconditioner $P = H$ , one would not obtain an improvement in the bias: if there exist eigenmodes $Hv=\\lambda v$ with $\\lambda \\ll 1$ , then | CH v - C v | = |(1-2)-1 - 1| | C v |, whereas | CI v - Cv | = | (1-2)-1 - 1| |Cv|, that is, $\\frac{| C_\\delta ^H v - C v |}{| C_\\delta ^I v - Cv |} \\sim \\lambda ^{-1}\\qquad \\text{for $\\delta , \\lambda $ sufficiently small,}$ where $C=H^{-1}$ is the covariance operator of the unbiased measure.",
"Therefore, for the remainder of the paper, we will not consider the effect of preconditioning on time-discretisation but only focus on the speed of convergence to equilibrium in the undiscretised (P-)Langevin dynamics.",
"We only stress that this point of view is only valid as long as $\\Vert H\\Vert $ and $\\Vert H_P\\Vert $ are comparable."
],
[
"Exponential convergence to the invariant measure",
"In this section, we prove exponential convergence to the equilibrium.",
"For the sake of simplicity we represent the probability density functions $\\mu $ and $\\mu _t^P$ by their respective densities $\\varphi _\\infty $ and $\\varphi _t^P$ .",
"Under the transformation $z = P^{1/2} x$ , we obtain transformed probability densities $\\psi _\\infty (z) := (\\det P)^{-1/2} \\varphi _\\infty (P^{-1/2} z) \\quad \\text{and} \\quad \\psi _t^P(z) := (\\det P)^{-1/2} \\varphi _t^P(P^{-1/2} z).$ Their evolution is described by the Fokker–Planck equation $\\partial _t\\psi _t^P=\\mathcal {L}_P^{*}\\psi _t^P:= \\nabla _z \\cdot \\left(\\nabla _z E_N(P^{-1/2} z) \\psi _t^P+ \\nabla _z \\psi _t^P \\right),$ where $\\mathcal {L}_P^{*}$ is the dual operator of $\\mathcal {L}_P$ (for the $L^2(dz)-$ scalar product) defined in (REF ).",
"The proof of the following theorem is inspired from [Lelievre] or A.19 in [Villani].",
"Suppose that $E_N\\in C^2(\\mathbb {R}^N)$ , such that $\\frac{1}{2}|\\nabla E_N(x)|^2-\\Delta E_N(x) \\rightarrow +\\infty $ as $|x|\\rightarrow +\\infty $ .",
"Then there exists $\\lambda _P > 0$ such that for all initial conditions $\\psi _0 \\in L^2(\\varphi _{\\infty }^{-1})$ , and for all times $t\\ge 0$ tP-L2(-1)2 e-P t0-L2(-1)2, or, equivalently, tP-L2(-1)2 e-P t0-L2(-1)2, $||.||_{L^2(\\psi _{\\infty }^{-1})} $ denotes the norm in ${L^2(\\mathbb {R}^N,\\psi _{\\infty }^{-1}dz)}$ , namely $||f||_{L^2(\\psi _{\\infty }^{-1})}^2=\\int _{\\mathbb {R}^N}f^2(z)\\psi _\\infty ^{-1}dz $ .",
"The exponent $\\lambda _P$ is the spectral gap of the Fokker–Planck operator $\\mathcal {L}_P^*$ defined in (REF ) (i.e., the smallest non-zero eigenvalue of $-\\mathcal {L}_P^*$ ).",
"See Section REF .",
"We now consider the quadratic case, $E_N(x) = \\frac{1}{2} x^T H x$ with $H$ symmetric and positive definite.",
"In this case, we have the following characterisation of the spectrum of $\\mathcal {L}_P^*$ and hence of $\\lambda _P$ .",
"The proof is based on [hit,Lelievre,ott].",
"The spectrum of the operator $\\mathcal {L}_P^*$ is $\\sigma (\\mathcal {L}_P^{*})= \\sigma (\\mathcal {L}_P)= \\displaystyle \\Big \\lbrace - {\\textstyle \\sum _{\\lambda \\in \\sigma (H_P)}} k_\\lambda \\lambda \\,:\\, k_\\lambda \\in \\mathbb {N}\\Big \\rbrace .$ See Section REF .",
"An immediate consequence of Theorem is that $ \\lambda _P = \\inf \\sigma (H_P) \\setminus \\lbrace 0\\rbrace ,$ in other words, the smallest non-zero generalised eigenvalue of $H v = \\lambda P v.$ Returning to the quadratic model Hamiltonians and model preconditioners introduced in Sections REF and REF , assume that $\\min \\sigma (H) \\sim N^{-s}$ (which is consistent with (REF )), while $\\min \\sigma (H_P) \\sim 1$ , then we obtain that $\\lambda _I \\sim N^{-s}$ with $\\lambda _P \\sim 1$ .",
"This result should give us confidence in the value of preconditioning.",
"The preconditioning ideas and results presented in Sections and are similar to the Brascamp–Lieb inequality [brasc].",
"In some sense, this inequality claims that a good preconditioner is the Hessian.",
"Precisely [Lelievre:17], if $E_N$ is strictly convex, for any function $f \\in L^2({\\rm e}^{-E_N})$ , $\\int \\left[f - \\int f {\\rm e}^{-E_N} \\right]^2 {\\rm e}^{-E_N} \\le \\int \\nabla f (\\nabla ^2 E_N)^{-1} \\cdot \\nabla f {\\rm e}^{-E_N}, $ where we assume here that the normalization $\\int {\\rm e}^{-E_N} = 1$ .",
"This means that if one considers the Fokker–Planck equation $\\partial _t \\psi = {\\rm div} [ (\\nabla ^2 E_N)^{-1} {\\rm e}^{-E_N} \\nabla ( \\psi {\\rm e}^{E_N}) ],$ which is associated to the following overdamped Langevin dynamics [xifa]: $dX_t = - (\\nabla ^2 E_N)^{-1} \\nabla E_N (X_t) dt + {\\rm div} [ (\\nabla ^2 E_N)^{-1} ] (X_t) dt + \\sqrt{2 (\\nabla ^2 E_N)^{-1}}(X_t) dW_t, $ then, if $E_N$ is strictly convex, we have $\\begin{aligned}\\frac{1}{2} \\frac{d}{dt} \\int \\left(\\frac{\\psi }{{\\rm e}^{-E_N}} - 1\\right)^2 {\\rm e}^{-E_N} & = - \\int (\\nabla ^2 E_N)^{-1} \\nabla \\left( \\frac{\\psi }{{\\rm e}^{-E_N}} \\right) \\cdot \\nabla \\left( \\frac{\\psi }{{\\rm e}^{-E_N}}\\right) {\\rm e}^{-E_N}\\\\&\\le \\int \\left(\\frac{\\psi }{{\\rm e}^{-E_N}} - 1\\right)^2 {\\rm e}^{-E_N}.\\end{aligned}$ And thus $\\int \\left(\\frac{\\psi }{{\\rm e}^{-E_N}} - 1\\right)^2{\\rm e}^{-E_N} \\le \\left[ \\int \\left(\\frac{\\psi _0}{{\\rm e}^{-E_N}} - 1\\right)^2 {\\rm e}^{-E_N}\\right] {\\rm e}^{-2t}, $ whatever the potential $E_N$ is.",
"It is in some sense “universal”.",
"For example, it does not depend on the temperature: if we multiply $E_N$ by a constant $\\beta $ (inverse of the temperature) it remains the same, whereas understanding the dependency of the spectral gap on the temperature is tricky in general."
],
[
"Analysis of the asymptotic variance",
"In this section we present sufficient conditions under which the estimator $\\epsilon _T^P(f)= \\frac{1}{T} \\int _0^T f(X_t^P) dt= \\frac{1}{T}\\int _0^T f(P^{-1/2}Z_t)dt$ satisfies a central limit theorem of the form (REF ) and we characterise the associated asymptotic variance."
],
[
"Generalities",
"The fundamental requirements to prove the central limit theorem is establishing the well-posedness of the Poisson equation $-\\mathcal {L}_P\\phi (z)=f(P^{-1/2}z)-\\mu (f),\\,\\, \\mu (\\phi )=0,$ for all bounded and continuous functions $f: \\mathbb {R}^N\\rightarrow \\mathbb {R}$ , where $\\mathcal {L}_P $ is defined by (REF ), and obtaining estimates on the growth of the unique solution $\\phi $ .",
"Recall that we shall assume that $\\mu $ admits a smooth, strictly positive density denoted by $\\psi _\\infty (z)$ , such that $\\int _{\\mathbb {R}^N}\\psi _\\infty (z)dz = 1$ and the SDE (REF ) has a unique strong solution.",
"Referring to results in [glynn,meyn] we suppose that the process $Z_t$ admits a Lyapunov function (see the Definition REF in Section REF ), which is sufficient to ensure the geometric ergodicity of $Z_t$ (see [mattin,twe]).",
"In terms of the potential energy $E_N$ and the preconditioner $P$ , we require that there exists $\\beta \\in (0, 1)$ such that $ \\displaystyle \\lim _{|z|\\rightarrow +\\infty }\\inf \\left[(1-\\beta )|\\nabla E_N^P(z)|^2+\\Delta E_N^P(z)\\right]>0.$ It is straightforward to check that this condition holds whenever $E_N$ is strongly convex and in particular if it is of the form (REF ).",
"If condition (REF ) holds, then the process $Z_t$ will be geometrically ergodic.",
"More specifically, the law of the process $Z_t$ started from a point $z\\in \\mathbb {R}^N$ will converge exponentially fast in the total-variation norm to the equilibrium distribution $\\mu $ (cf.",
"(REF )).",
"Assuming (REF ), we also obtain the following well-posedness result for the Poisson equation (REF ).",
"Suppose that (REF ) holds, then there exists $c > 0 $ , such that for any measurable observable $f$ satisfying $|f|^2 \\le e^{-\\beta E_N^P(z)}$ , the Poisson equation (REF ) admits a unique strong solution satisfying the bound $|\\phi (z)|^2 \\le c e^{-\\beta E_N^P(z)}$ .",
"In particular, $\\phi (P^{1/2}x) \\in L^2(\\mu )$ .",
"See Section REF .",
"The technique of using a Poisson equation to obtain a central limit theorem for an additive functional of a Markov process is widely known (see e.g.",
"[bhatt]).",
"For linear and quadratic observables, we can in fact produce an analytic solution to this Poisson problem; see Section REF below.",
"Under the conditions of Theorem REF , there exists a constant $0< \\sigma _{f,P}^2< \\infty $ such that the asymptotic distribution of the fluctuations of $\\epsilon _t^P(f)$ about $\\mu (f)$ are given by the central limit theorem $\\displaystyle \\sqrt{t} \\left(\\epsilon ^P_t(f)-\\mu (f)\\right)\\xrightarrow{} \\mathcal {N}(0,\\sigma _{f,P}^2),\\text{ as } t\\rightarrow +\\infty ,$ where $\\sigma _{f,P}^2$ (the asymptotic variance for the observable $f$ ) is given by $\\begin{split}\\sigma _{f,P}^2&= 2 \\int \\big | \\nabla _z \\phi (z) \\big |^2 \\mu ^P(dz) \\\\&= 2\\int \\big |\\nabla _z \\phi (P^{1/2} x)\\big |^2 \\mu (dx),\\end{split}$ where $\\mu ^P(dz) = \\psi _\\infty (z) dz.$ See Section REF ."
],
[
"Explicit solution for linear observables",
"In this section we exhibit explicit solutions $\\phi $ of the Poisson equation (REF ) when $E_N$ is quadratic and $f$ is linear, and compute the associated asymptotic variance $\\sigma _{f,P}^2$ .",
"This simplest possible case is of course well-known but we summarise it nevertheless to prepare for more interesting cases.",
"Suppose, therefore, that $P = I, \\quad E_N(x) = \\frac{1}{2} x^T H x \\quad \\text{and} \\quad f(x) = v \\cdot x,$ where $v \\in \\mathbb {R}^N$ .",
"From symmetry it follows that $\\mu (f) = 0$ , hence the Poisson equation (REF ) becomes $Hx \\cdot \\nabla \\phi (x) - \\Delta \\phi (x)= v \\cdot x.$ Seeking a solution of the form $\\phi (x) = d \\cdot x$ with $d \\in \\mathbb {R}^N$ , we obtain $d = H^{-1} v$ , i.e., $\\phi (x) = x \\cdot H^{-1} v,$ and in particular, $\\sigma _f^2= \\sigma _{f,I}^2 = 2\\int |H^{-1} v|^2 \\mu (dx) = 2|H^{-1} v|^2.$ In particular, choosing $v$ to be a normalised eigenmode of $H$ with associated eigenvalue $\\lambda $ , we obtain $\\sigma _f^2= 2\\lambda ^{-2}$ .",
"Focusing specifically on a $d$ -dimensional lattice model, we know from (REF ) that $\\min \\sigma (H) \\sim N^{-2/d}$ while $\\max \\sigma (H) \\sim 1$ .",
"Hence, it follows that $\\sigma _f^2$ is moderate for the high-frequency eigenmodes, but large for the observables corresponding to low-frequency eigenmodes.",
"Next, we turn to the preconditioned dynamics.",
"In this case we effectively replace $H$ with $H_P = P^{-1/2} H P^{-1/2}$ and $v$ with $P^{-1/2} v$ and thus obtain $ \\sigma _{f,P}^2= 2\\int |H_P^{-1} P^{-1/2} v|^2 \\mu (dx)= 2| P^{1/2} H^{-1} v |^2 =: 2|H^{-1} v|_P^2.$ If we assume that $c_0 = \\min \\sigma (H_P), c_1 = \\max \\sigma (H_P)$ , then a simple rewrite yields $ 2c_0 v^T H^{-1} v \\le \\sigma _{f,P}^2\\le 2c_1 v^T H^{-1} v.$ Comparing with $\\sigma _f^2=2v^T H^{-2} v$ and recalling our standing assumption (REF ) that $c_0 \\sim 1, c_1 \\sim 1$ (the spectrum is bounded above and below independently of $N$ ), we conclude that preconditioning does not entirely remove ill-conditioning but it is (potentially) diminished.",
"More concretely, for a $d$ -dimensional lattice system, we obtain that $N^{-2/d} \\lesssim \\frac{\\sigma _{f,P}^2}{\\sigma _f^2} \\lesssim 1,$ and both bounds are attained for specific observables.",
"We conclude that preconditioned Langevin can be significantly more efficient than standard Langevin (low-frequency observables) but that it will be comparable in efficiency for high-frequency observables.",
"Intuitively, low-frequency observables are “macroscopic” in nature and include e.g.",
"energy, average bond-length etc., while high-frequency observables include in particular single bonds, bond angles and dihedral angles (in a large molecule) or a bond-length near a crack-tip.",
"In the next sections, we consider three toy models mimicking “realistic” observables of these kinds, occurring in real-world simulations, to further substantiate our remarks."
],
[
"Example 1: Energy per particle",
"We now consider $E_N(x) = \\frac{1}{2} x^T H x$ and $f(x) = N^{-1} E_N(x)$ .",
"A straightforward computation yields $\\langle x \\cdot Bx \\rangle _\\mu = {\\rm Tr}(H^{-1} B) \\qquad \\text{for } B \\in \\mathbb {R}^{N \\times N},$ which in particular implies that $\\mu (f) = \\frac{1}{N} \\frac{\\int E_N(x) e^{- E_N(x)}}{ \\int e^{-E_N(x)}}= \\frac{{\\rm Tr}I}{2N} = \\frac{1}{2}.$ Thus, the Poisson equation becomes $H x \\cdot \\nabla \\phi (x) - \\Delta \\phi (x)= {\\textstyle \\frac{1}{2N}} x^T H x - {\\textstyle \\frac{1}{2}}$ We seek a solution of the form $\\phi (x) = \\frac{1}{2} x^T B x + l \\cdot x -{\\rm Tr}B$ (to ensure that $\\mu (\\phi ) = 0$ ), then this yields the equation $Hx \\cdot (Bx + l) - {\\rm Tr}B = {\\textstyle \\frac{1}{2N}} x^T H x - {\\textstyle \\frac{1}{2}}.$ This is satisfied for $B = \\frac{1}{N} I, l = 0$ , hence $\\phi (x) = \\frac{1}{2N} |x|^2 -\\frac{1}{2}$ .",
"We can now compute the asymptotic variance as $\\sigma _f^2= \\sigma _{f,I}^2 =2 \\int \\big |{\\textstyle \\frac{1}{N}} x\\big |^2 \\mu (dx)= {\\textstyle \\frac{2}{N^2}} {\\rm Tr}H^{-1}.$ Repeating the same argument in transformed coordinates $z = P^{1/2} x$ , we also obtain the asymptotic variance of the energy for the preconditioned Langevin dynamics: $\\sigma _{f,P}^2= {\\textstyle \\frac{2}{N^2}} {\\rm Tr}H_P^{-1}= {\\textstyle \\frac{2}{N^2}} {\\rm Tr}\\big ( H^{-1} P \\big ).$ Let us now focus on a lattice model, where we have (REF ).",
"Then we obtain that $\\frac{1}{2} \\sigma _f^2\\approx N^{-2} \\sum _{j = 1}^N (j/N)^{-2/d}\\approx N^{-1} \\int _{1/N}^1 s^{-2/d} ds \\approx {\\left\\lbrace \\begin{array}{ll}1, & d = 1, \\\\N^{-1} \\log N, & d = 2, \\\\N^{-1}, & d = 3,\\end{array}\\right.",
"}$ while, clearly, $\\frac{1}{2}\\sigma _{f,P}^2\\approx N^{-1}$ .",
"In summary, $\\frac{\\sigma _f^2}{\\sigma _{f,P}^2} \\approx {\\left\\lbrace \\begin{array}{ll}N, & d = 1, \\\\\\log N, & d = 2, \\\\1, & d = 3;\\end{array}\\right.",
"}$ that is, preconditioning only gives a significant speed-up in one-dimensional lattices but not in two- or three-dimensional lattices."
],
[
"Example 2: Bond-length",
"In our second example we observe a single bond in the crystal or molecule.",
"That is, we still use $E_N(x) = \\frac{1}{2} x^T H x$ but the observable is now given by $f(x) = x_{i} - x_j \\qquad \\text{for some fixed bond $i \\sim j$}.$ This is a linear observable, hence a special case of the discussion in Section REF .",
"Hence, we obtain $\\sigma _f^2= 2|H^{-1} l|^2 \\qquad \\text{where} \\quad l_n ={\\left\\lbrace \\begin{array}{ll}1, & n = i \\\\-1, & n = j, \\\\0, & \\text{otherwise.}\\end{array}\\right.",
"}$ In order to estimate $\\sigma _f^2$ further we consider again the $d$ -dimensional lattice model (REF ) and $P$ given by (REF ).",
"For $d \\ge 2$ , since $P$ is a homogeneous discrete elliptic operator, we know from [olson] that $\\big | [P^{-1}]_{ni} - [P^{-1}]_{nj} \\big | \\le C (1 + |n-i|)^{-d},$ where we note that $i,j$ are now neighbouring lattice sites; i.e., $[P^{-1}]_{ni} - [P^{-1}]_{nj}$ denotes a discrete gradient of the lattice Green's function.",
"Therefore, we obtain that $ \\sigma _f^2= 2|H^{-1} l|^2 \\lesssim 2|P^{-1} l|^2\\lesssim 2\\sum _{n \\in \\mathbb {Z}^d} (1+|n-i|)^{-2d} < \\infty ;$ that is, $\\sigma _f^2$ has an upper bound that is independent of $N$ .",
"A lower bound follows simply from the fact that $\\Vert H\\Vert \\le 1$ and hence $v^T H^{-1} v \\ge |v|^2$ , which implies $\\sigma _f^2\\ge |l|^2 = 1.$ To obtain bounds on $\\sigma _{f,P}^2$ , we use (REF ) to estimate $2c_0 = 2c_0 |l|^2 \\le \\sigma _{f,P}^2\\le 2c_1 l^T H^{-1} \\le 2c_1 |l| |H^{-1} l|,$ and we have already shown in (REF ) that this is bounded above, independently of $N$ .",
"In summary, we obtain that $\\frac{\\sigma _f^2}{\\sigma _{f,P}^2} \\lesssim 1 \\qquad \\text{for } d \\ge 2,$ that is, we expect no substantial (if any) speed-up for the bond-length observable from preconditioning for $d \\ge 2$ .",
"By contrast, for $d = 1$ , the system $P^{-1} l$ can be solved explicitly, and in this case one obtains $\\sigma _f^2\\sim N$ as $N \\rightarrow \\infty $ (specifically, $|P^{-1} l|^2 = N/12$ ), that is, $\\frac{\\sigma _f^2}{\\sigma _{f,P}^2} \\sim N \\qquad \\text{for } d = 1.$ Thus, we conclude that preconditioning helps to accelerate the computation of the bond-length observable only for one-dimensional structures.",
"The intuitive explanation of this effect is that far-away regions of space have little influence on a single bond and hence only local equilibration matters.",
"The difference in 1D is that elastic interaction is naturally more long-ranged than in dimension $d > 1$ ."
],
[
"Example 3: Umbrella sampling",
"Our final example is inspired by a technique called umbrella sampling [umbrella].",
"Given a potential energy $E_N(x)$ and a reaction coordinate $\\xi (x)$ , we wish to compute $A(\\xi _0) := - \\log \\int e^{-E_N(x)} \\delta (\\xi (x)-\\xi _0) \\,dx.$ Umbrella sampling achieves this by placing a restraint on the potential energy, $E_{N,K}(x) := E_N(x) + {\\textstyle \\frac{K}{2}} \\big ( \\xi (x) - \\xi _0 \\big )^2,$ for some $K > 0$ , $\\xi _0 \\in \\mathbb {R}$ .",
"Let $\\mu _{K}$ denote the corresponding equilibrium measure, then it can be shown [umbrella] that, for $K$ large, $\\frac{\\partial A}{\\partial \\xi }\\Big |_{\\xi = \\bar{\\xi }_0}\\approx - K (\\bar{\\xi }_0 - \\xi _0) \\qquad \\text{where }\\bar{\\xi }_0 = \\langle \\xi \\rangle _{\\mu _K}.$ Thus, $\\partial _\\xi A$ and hence $A$ can be reconstructed in this way.",
"More sophisticated variations of the idea exist of course, but for the sake of simplicity of presentation we will focus on this particularly simple variant.",
"To construct an analytically accessible toy problem mimicking umbrella sampling we consider again a quadratic energy $E_N(x) = \\frac{1}{2} x^T H x$ and a linear reaction coordinate $\\xi (x) := l \\cdot x$ where, for simplicity, we assume that $|l| = 1$ (for $|l| = O(1)$ , the argument is analogous).",
"The restrained potential with penalty parameter $K > 0$ is then given by $E_{N,K}(x) = {\\textstyle \\frac{1}{2}} x^T H x+ {\\textstyle \\frac{K}{2}} (\\xi (x) - \\xi _0)^2,$ for some $\\xi _0 \\in \\mathbb {R}$ , while the observable from which we can reconstruct the mean force is simply $f(x) = K \\big (l \\cdot x - \\xi _0\\big ).$ We are again in the context of Section REF and therefore obtain $\\sigma _f^2= 2K^2 \\big | H_K^{-1} l \\big |^2,$ where $H_K = \\nabla ^2 E_{N,K} = H + K l l^T.$ The Sherman–Morrison formula yields $ H_K^{-1} l = \\bigg ( H^{-1} - \\frac{K H^{-1} l l^T H^{-1}}{1 + K l^T H^{-1} l} \\bigg ) l = \\frac{H^{-1} l}{1 + K l^T H^{-1} l},$ and hence, f2= 2K2 |HK-1l|2 = 2K2 |H-1 l|2(1 + K lT H-1 l)2.",
"Since our focus in the present example is the ill-conditioning induced by large $K$ rather than ill-conditioning induced by $H$ (e.g.",
"through system size $N$ ), let us assume that $\\max \\sigma (H) = 1$ (as always) while $\\min \\sigma (H) \\ge c_0$ , for some moderate constant $c_0$ .",
"This would, e.g., be the typical situation for a small molecule, or if we preconditioned $H$ but without accounting for the umbrella.",
"We then obtain c02 2K2 c02(1 + K)2 f22K2(1+ K c0)2 2c02; that is, $\\frac{1}{2}\\sigma _f^2\\sim 1$ as $K \\rightarrow \\infty $ .",
"By contrast, suppose now that we choose a preconditioner $P_K := P + K l l^T,$ where $P$ is a preconditioner for $H$ satisfying $c_0^P v^T P v \\le v^THv \\le v^TP v$ .",
"Then a straightforward calculation yields $c_0^P \\le \\frac{v^T H v + K (l \\cdot v)^2}{v^T P v + K (l \\cdot v)^2}= \\frac{v^T H_K v}{v^T P_K v} \\le 1.$ It follows from (REF ) that $\\sigma _{f,P}^2= 2K^2 \\big | H_K^{-1} l \\big |_{P_K}^2\\approx 2K^2 |H_K^{-1/2} l|^2,$ where $\\approx $ now indicates upper and lower bounds with constants that are independent of $K$ .",
"Using (REF ) we obtain $\\sigma _{f,P}^2\\approx 2K^2 \\frac{l^T H^{-1} l}{1 + K l^T H^{-1} l} \\sim K\\qquad \\text{as $K \\rightarrow \\infty $.",
"}$ We can therefore conclude that $\\frac{\\sigma _{f,P}^2}{\\sigma _f^2} \\approx \\frac{(l^T H^{-1} l)(1 + K l^T H^{-1} l)}{|H^{-1} l|^2} \\sim 1+K\\qquad \\text{as $K \\rightarrow \\infty $;}$ that is, preconditioning the umbrella actually achieves a significant deterioration of the asymptotic variance and thus the $P$ -Langevin dynamics actually becomes less efficient than the standard Langevin dynamics.",
"However note the following crucial remark: [Step-sizes revisited] The surprising result of the present section does not in fact fully fall within our starting assumptions.",
"While $\\Vert H\\Vert = 1$ , $\\Vert H_K\\Vert $ is in fact of order $O(1 + K)$ which means that the time-step for the discretisation of the Langevin equation should be of order $O(K^{-1})$ , which exactly balances the lower mixing of the preconditioned dynamics and make the two schemes again comparable.",
"Indeed, in our computational examples we will need to choose $\\Delta t = O(K^{-1})$ to prevent instability.",
"In practice, the restraint parameter $K$ is chosen of the same order of magnitude of the stiffest bond in a molecule, while the reaction coordinate will normally be a function of the softest bonds and hence it would create no additional time-step restriction.",
"In such a situation, it is indeed preferable to not precondition the restraint.",
"However, we emphasise again that the interaction between preconditioning and time-stepping is an issue that we do not properly address in the present work and which will require further attention in the future.",
"We conclude our discussion by demonstrating the extension of our explicit computations to a mildly non-linear lattice model.",
"As potential energy $E_{M^d} : \\mathbb {R}^{\\lbrace 1,\\dots ,M\\rbrace ^d} \\rightarrow \\mathbb {R}$ , we choose $E_N(u) := \\sum _{\\alpha _1=0}^M \\cdots \\sum _{\\alpha _d = 0}^M\\sum _{j = 1}^d \\phi \\Big ({\\textstyle \\frac{1}{\\sqrt{d}}} \\big (u(\\alpha +e_j) - u(\\alpha )\\big )\\Big ),$ where $u(\\alpha ) := 0$ if any $\\alpha _j \\in \\lbrace 0, M+1\\rbrace $ , and with convex nearest-neighbour pair potential $\\phi (r) = {\\textstyle \\frac{1}{8}} \\big ( r^2 + \\sin (r)^2 \\big ).$ Upon choosing an arbitrary linear labelling of indices $\\alpha \\in \\lbrace 1,\\dots ,M\\rbrace ^d$ , this is a special case of (REF ).",
"As preconditioner we choose (REF ), which we can also write as $\\langle Pv, v \\rangle = \\frac{1}{4 d} \\sum _{\\alpha _1=0}^M \\cdots \\sum _{\\alpha _d = 0}^M\\sum _{j = 1}^d |u(\\alpha +e_j) - u(\\alpha )|^2.$ The occurrence of $d$ in the definitions of $E_N$ and $P$ ensures that $\\Vert \\nabla E_N\\Vert \\approx \\Vert P \\Vert \\approx 1$ ; moreover, since $\\phi $ is strictly convex, we have that ${\\rm cond}((\\nabla E_N(x))_{P})$ is bounded above independently of $N$ ; cf.",
"(REF ).",
"For all simulations (with small modifications for the umbrella sampling example) we choose $\\delta = 0.1, \\quad N_{\\rm steps} = 10^5, \\quad N_{\\rm runs} = 400.$ We then use a Cholesky factorisation to compute $P$ , i.e.",
"$P = L L^T$ , followed by $X_{n+1}^P = X_n^P - \\delta P^{-1} \\nabla E_N(X_N^P) + \\sqrt{2 \\delta } L^{-T} R_n,\\quad \\text{for } n = 1, \\dots , N_{\\rm steps},$ where $R_n \\sim N(0, I)^N$ .",
"The estimate of the observable value is then given by $\\bar{f} = N_{\\rm steps}^{-1} \\sum _{n = 1}^{N_{\\rm steps}} f(X_n^P)$ .",
"We compute $N_{\\rm runs}$ trajectories in order to estimate the asymptotic variance from $N_{\\rm runs}$ independent samples of $\\bar{f}$ .",
"For the umbrella sampling test, we choose $\\xi (x) := l \\cdot x$ to be the bond-length observable again with $\\xi _0 = 0.33$ .",
"The modification to the energy and observable is then as described in Section REF .",
"With $\\delta = 0.1$ , the discretised Langevin dynamics turns out to be unstable, hence we had to choose $\\delta _K = 0.1 / K$ instead.",
"To account for this (see also Remark REF ), we study $K \\sigma _f^2$ instead of $\\sigma _f^2$ in our tests.",
"We perform the umbrella sampling test only for $d = 2, N = 8^2$ , since we focus here on the magnitude on the restraint parameter $K$ rather than the system size.",
"The results of the simulations are shown in table REF .",
"The numbers closely match the analytical predictions of Sections REF , REF and REF .",
"To conclude this discussion we only remark that we did not fine-tune step-sizes, which means one could likely make small improvements to both the preconditioned and unpreconditioned processes.",
"However, we believe that the trends across dimension, system size and restraint parameter are reliable.",
"In particular, we stress that even though the preconditioned variants often have a smaller asymptotic variance, often (in particular for $d = 3$ ) this improvement is only by a moderate constant factor.",
"Because only the trends are reliable indicators in these tests only a successive improvement with increasing $N$ or $K$ (e.g.",
"as in the $d = 1$ tests) can be considered a success for the preconditioned algorithm.",
"Table: Numerically estimated asymptotic variances of the energy observable(Section ), the bond-length observable (Section) and the restraint observable occurring in umbrella sampling(Section ).",
"The nonlinear potential energy used in these testsis described in Section .",
"All results match theanalytical predictions of Sections , and ."
],
[
"Conclusion",
"In this paper we strived to develop an intuition what the effect of preconditioning has on molecular simulations.",
"The results are very mixed: it is clear that preconditioning accelerates convergence of the probability density functions to equilibrium (see Theorem as well as the discussion in Section REF ), and this necessarily implies accelerated convergence for some observables.",
"However, for many concrete observables of practical importance little (if any) benefit can be gained.",
"This was a surprising outcome for us and indicates that alternative avenues need to be explored on how a priori information about the analytical structure of configuration space should be exploited in molecular simulation.",
"We emphasize again that our (partially negative) conclusion, contrary to much of the existing literature, is due to the fact that we test the convergence of specific observables.",
"Moreover, we stress that we have only performed a limited set of tests on highly simplified toy models and a limited set of observables, while more realistic models may exhibit many features that we neglected."
],
[
"Proof of Proposition ",
"The covariance of the invariant measure associated to the dynamics (REF ) is given by the following identity: $\\begin{aligned}C^P_\\delta &=(I- \\delta P^{-1} H)C^P_\\delta (I- \\delta HP^{-1})+2 \\delta P^{-1}\\\\C^P_\\delta &=C^P_\\delta - \\delta P^{-1} HC^P_\\delta - \\delta C^P_\\delta HP^{-1}+ \\delta ^2P^{-1}HC^P_\\delta HP^{-1}+2 \\delta P^{-1}\\\\P^{-1}HC^P_\\delta + C^P_\\delta HP^{-1}&= 2 P^{-1}+ \\delta P^{-1}HC^P_\\delta HP^{-1}.\\end{aligned}$ Expanding $C^P_\\delta $ , we have $C^P_\\delta \\sim C_0+ \\delta C_1+\\delta ^2C_2+ O( \\delta ^3) ,$ then one gets $\\begin{aligned}P^{-1}HC_0 + C_0HP^{-1}=2P^{-1}&\\quad \\Rightarrow C_0=H^{-1}\\\\P^{-1}HC_1+C_1HP^{-1}= P^{-1}HP^{-1}&\\quad \\Rightarrow C_1=\\frac{1}{2}P^{-1}\\\\P^{-1}HC_2+C_2HP^{-1}=\\frac{1}{2}P^{-1}HP^{-1}HP^{-1}&\\quad \\Rightarrow C_2=\\frac{1}{4}P^{-1}HP^{-1}.\\end{aligned}$ Proceeding by induction, one can therefore obtain: $\\forall k\\in \\mathbb {N},\\quad C_k=2^{-k}(P^{-1}H)^kH^{-1}.$ Therefore $C^P_\\delta $ can be rewritten as: $ C^P_\\delta =\\displaystyle \\sum _{k=0}^{+\\infty }\\left({\\textstyle \\frac{ \\delta }{2}}P^{-1}H \\right)^kH^{-1}.",
"$ Indeed, the sum $\\displaystyle \\sum _{k\\ge 0}C_k$ converges since $\\frac{\\delta }{2}|P^{-1}H|<1.$ The covariance operator be can rewritten as $\\begin{aligned}C^P_\\delta & = \\left( \\displaystyle \\sum _{k=0}^\\infty \\left({\\textstyle \\frac{ \\delta }{2}}P^{-1}H\\right)^k\\right)H^{-1}\\\\& = \\left(I-{\\textstyle \\frac{ \\delta }{2}}P^{-1}H\\right)^{-1}H^{-1}\\\\&= (H-{\\textstyle \\frac{ \\delta }{2}}HP^{-1}H)^{-1},\\end{aligned}$ which concludes the proof."
],
[
"Proof of Theorem ",
"Under the assumptions on the potential $E_N$ , see A.19 in [Villani], the density $\\psi _\\infty $ satisfies a Poincaré inequality: there exists $\\lambda _P>0$ such that for all probability density functions $\\phi $ , we have: $\\displaystyle \\int _{\\mathbb {R}^N}\\left| \\frac{\\phi }{\\psi _\\infty }-1 \\right|^2\\psi _\\infty dz\\le \\frac{1}{\\lambda _P}\\int _{\\mathbb {R}^N}\\left| \\nabla \\left(\\frac{\\phi }{\\psi _\\infty }\\right) \\right|^2\\psi _\\infty dz.$ The optimal parameter $\\lambda _P$ in (REF ) is the opposite of the smallest (in absolute value) non-zero eigenvalue of the Fokker-Planck operator $\\mathcal {L}_P^* $ , which is self-adjoint in $L^2(\\mathbb {R}^N,\\psi _{\\infty }^{-1}dx)$ .",
"Thus the exponent $\\lambda _P$ is the spectral gap of $\\mathcal {L}_P^* $ .",
"If $\\psi _t^P$ is a solution of (REF ), therefore for all initial condition $\\psi _0\\in L^2(\\psi _{\\infty }^{-1})$ , $\\forall t\\ge 0:$ $\\displaystyle \\frac{d}{dt}\\Vert \\psi _t^P-\\psi _\\infty \\Vert _{L^2(\\psi _{\\infty }^{-1})}^2=-2\\int _{\\mathbb {R}^N}\\left|\\nabla \\left(\\frac{\\psi _t^P}{\\psi _\\infty }\\right)\\right|^2\\psi _\\infty dz.$ Indeed, $\\begin{aligned}\\displaystyle \\frac{d}{dt}\\Vert \\psi _t^P-\\psi _\\infty \\Vert _{L^2(\\psi _{\\infty }^{-1})}^2&=\\displaystyle \\frac{d}{dt}\\int \\left|\\psi _t^P-\\psi _\\infty \\right|^2\\psi _{\\infty }^{-1}dz\\\\&=2\\int \\partial _t\\psi _t^P\\left(\\psi _t^P-\\psi _\\infty \\right)\\psi _{\\infty }^{-1}dz\\\\&=2\\int \\nabla \\cdot \\left(\\nabla E_N^P(z)\\psi _t^P+\\nabla \\psi _t^P\\right)\\left(\\frac{\\psi _t^P}{\\psi _\\infty }-1\\right)dz\\\\&=-2\\int \\left(\\nabla E_N^P(z)\\psi _t^P+\\nabla \\psi _t^P\\right)\\cdot \\nabla \\left(\\frac{\\psi _t^P}{\\psi _\\infty }\\right)dz.\\end{aligned}$ But we have $\\begin{aligned}\\nabla E_N^P(z)\\psi _t^P+\\nabla \\psi _t^P&=-\\nabla \\left(\\ln \\left(\\psi _\\infty \\right)\\right)\\psi _t^P+\\nabla \\psi _t^P\\\\&=-\\frac{\\nabla \\psi _\\infty \\psi _t^P}{\\psi _\\infty }+\\nabla \\psi _t^P\\\\&=\\nabla \\left(\\frac{\\psi _t^P}{\\psi _\\infty }\\right)\\psi _\\infty ,\\end{aligned}$ which yields (REF ).",
"Therefore, using (REF ), $\\begin{aligned}\\displaystyle \\frac{d}{dt}\\Vert \\psi _t^P-\\psi _\\infty \\Vert _{L^2(\\psi _{\\infty }^{-1})}^2&\\le -2\\lambda _P \\int \\left| \\frac{\\psi _t^P}{\\psi _\\infty }-1 \\right|^2\\psi _\\infty dz\\\\&= -2\\lambda _P \\int \\left| \\psi _t^P-\\psi _\\infty \\right|^2\\psi _\\infty ^{-1} dz,\\end{aligned}$ then $\\Vert \\psi _t^P-\\psi _\\infty \\Vert _{L^2(\\psi _{\\infty }^{-1})}^2\\le {\\rm e}^{-\\lambda _P }\\Vert \\psi _0-\\psi _\\infty \\Vert _{L^2(\\psi _{\\infty }^{-1})}^2.$"
],
[
"Preliminaries for spectral analysis",
"In the linear case, i.e $E_N^P(z)=\\frac{1}{2}z^TH_Pz$ , where $H$ and $P$ are symmetric positive definite and $H_P := P^{-1/2} H P^{-1/2}$ , the analysis will be carried out in a suitable system of coordinates which simplifies the calculations and the proofs of the main theorems.",
"For this reason, we will perform one conjugation and one additional change of variables .",
"From the partial differential equation point of view and in order to use standard techniques from the spectral analysis of partial differential equations, then it appears to be useful to work in $L^2(\\mathbb {R}^N,dz; $ instead of $L^2(\\mathbb {R}^N,\\psi _\\infty dz; $ .",
"The mapping $\\phi \\mapsto \\psi _\\infty ^{-1/2}\\phi $ maps unitarily $L^2(\\mathbb {R}^N,dz; $ into $L^2(\\mathbb {R}^N,\\psi _\\infty dz; $ with the associated transformation rules for the differential operators: ${\\rm e}^{-\\frac{1}{2}H_P}\\nabla _z {\\rm e}^{\\frac{1}{2}H_P}=\\nabla _z +\\frac{1}{2}\\nabla _z H_P.$ Thus, the operator $\\mathcal {L}_P= -\\nabla _z E_N^P(z)\\cdot \\nabla _z +\\Delta _z$ is transformed into LP=e-12HPLPe12HP = z-14|z ENP(z)|2+12z ENP(z) = z-14zT HP2z+12Tr(HP).",
"The kernel of $ \\overline{\\mathcal {L}}_P$ is $^{-\\frac{z^TH_Pz}{4}} $ and the operator $\\overline{\\mathcal {L}}_P$ is unitarily equivalent to the operator $\\mathcal {L}_P$ .",
"In the goal of modifying the kernel of the operator $ \\overline{\\mathcal {L}}_P$ into a centered Gaussian with identity covariance matrix, we perform a second change of variables.",
"In the following, we introduce the new coordinates $y = H_P^{1/2}z$ , so that $\\nabla _z=H_P^{1/2}\\nabla _y$ .",
"Then the operator $ \\overline{\\mathcal {L}}_P$ becomes: $\\displaystyle \\tilde{\\mathcal {L}}_P=\\nabla _y^TH_P\\nabla _y-\\frac{1}{4}y^TH_Py+\\frac{1}{2}{\\rm Tr}(H_P).$ The operator $\\tilde{\\mathcal {L}}_P$ is still acting in $L^2(\\mathbb {R}^N,dz; $ .",
"In the new coordinate system ($Y_t=H_P^{1/2}Z_t$ ), the corresponding stochastic process is: $ dY_t=-H_PY_tdt+\\sqrt{2}H_P^{1/2}dW_t, $ so that ${\\rm Ker}(\\tilde{\\mathcal {L}}_P)=\\frac{1}{(2\\pi )^{N/4}}{\\rm e}^{\\frac{|y|^2}{4}}.$ The last conjugation and change of variables are used to compute the spectrum of $\\mathcal {L}_P^{*}$ needed to proof Theorem (see Section REF ).",
"Let us now introduce some additional notations.",
"Recall the space of rapidly decaying complex valued $\\mathcal {C}^{\\infty }$ functions $\\mathcal {S}(\\mathbb {R}^N)=\\displaystyle \\left\\lbrace f\\in \\mathcal {C}^{\\infty }(\\mathbb {R}^N),\\forall \\alpha ,\\beta \\in \\mathbb {N}^{N},\\exists C_{\\alpha ,\\beta }\\in \\mathbb {R}_+, \\sup _{x\\in \\mathbb {R}^N} |x^\\alpha \\partial _x^\\beta f(x) |\\le C_{\\alpha ,\\beta } \\right\\rbrace , $ and its dual is denoted $\\mathcal {S}^{\\prime }(\\mathbb {R}^N)$ .",
"The Weyl-quantization $q^W(x,D_x) $ of a symbol $q(x,\\xi )\\in \\mathcal {S}^{\\prime }(\\mathbb {R}^N) $ is an operator defined by its Schwartz-kernel $\\displaystyle \\left[q^W(x,D_x)\\right](x,y)=\\int _{\\mathbb {R}^N}{\\rm e}^{i(x-y)\\cdot \\xi }q\\left( \\frac{x+y}{2},\\xi \\right)\\frac{d\\xi }{(2\\pi )^N}.",
"$ For instance, the Weyl symbol of of the operator $-\\tilde{\\mathcal {L}}_P+\\frac{1}{2}{\\rm Tr}(H_P)= -\\nabla _y^TH_P\\nabla _y+\\frac{1}{4}y^TH_Py$ is $q(y,\\xi )=\\xi ^TH_P\\xi +\\frac{y^TH_Py}{4}.$ Those tools are essential to proof Theorem (see Section REF ).",
"For more details on Weyl-quantization, one can refer to [ott]."
],
[
"Proof of Theorem ",
"Referring to Theorem 1.2.2 in [hit], the spectrum of the operator $ q^W(y,D_y)=-\\tilde{\\mathcal {L}}_P+\\frac{1}{2}{\\rm Tr}(H_P)$ associated with the elliptic quadratic Weyl symbol $q(y,\\xi ) $ defined by (REF ) is given by $\\sigma (q^W(y,D_y))=\\displaystyle \\left\\lbrace \\sum _{{\\lambda \\in \\sigma (G)\\\\ {\\rm Im}\\lambda \\ge 0}}-i\\lambda ( r_\\lambda +2k_\\lambda ), k_\\lambda \\in \\mathbb {N}\\right\\rbrace .$ where $G$ is the so-called Hamilton map associated with $q$ , and $r_\\lambda $ is the algebraic multiplicity of $\\lambda \\in \\sigma (G)$ (the dimension of the characteristic space).",
"The Hamilton map is the $-linear map $ G: 2N2N $ associated with the matrix$ $G=\\begin{bmatrix}0 & H_P \\\\-\\frac{1}{4}H_P& 0 \\\\\\end{bmatrix}\\in {2N\\times 2N}.",
"$ The matrix $G$ is similar to another matrix denoted $\\overline{G}$ and defined by $\\overline{G}=\\begin{bmatrix}\\frac{1}{\\sqrt{2}} & 0 \\\\0& \\sqrt{2} \\\\\\end{bmatrix}G\\begin{bmatrix}\\sqrt{2}& 0 \\\\0& \\frac{1}{\\sqrt{2}} \\\\\\end{bmatrix}=\\frac{1}{2} \\begin{bmatrix}0 & H_P \\\\-H_P& 0 \\\\\\end{bmatrix}.",
"$ Now, the characteristic polynomial of $G$ can be computed by $\\begin{aligned}\\rm det(G-\\lambda I)&=\\rm det({\\overline{G}-\\lambda I })=2^{-2N}\\left|\\begin{array}{cc}-2\\lambda I &H_P\\\\-H_P&-2\\lambda I\\end{array}\\right|\\\\&=2^{-2N}\\left|\\begin{array}{cc}-2\\lambda I &H_P\\\\-H_P-i2\\lambda I& i(H_P+i2\\lambda I)\\end{array}\\right|\\\\&=2^{-2N}\\left|\\begin{array}{cc}i(-H_P+i2\\lambda I ) &H_P\\\\0& i(H_P+i2\\lambda I)\\end{array}\\right|\\\\&=2^{-2N}\\rm det(H_P-i2\\lambda I)\\rm det(H_P+i2\\lambda I).\\end{aligned}$ Since ${\\rm Re}(\\sigma (H_P) )\\ge 0$ , one thus obtains that $\\sigma (G)\\cap \\lbrace \\lambda , {\\rm Im}\\lambda \\ge 0 \\rbrace = \\frac{i}{2}\\sigma (H_P).",
"$ In particular, $\\displaystyle \\sum _{{\\lambda \\in \\sigma (G)\\\\ {\\rm Im}\\lambda \\ge 0}}-i\\lambda 2k_\\lambda =\\sum _{\\mu \\in \\sigma (H_P)}k_{\\frac{i}{2}\\mu }\\mu $ and $\\sum _{{\\lambda \\in \\sigma (G)\\\\ {\\rm Im}\\lambda \\ge 0}}-i\\lambda r_\\lambda =\\frac{{\\rm Tr}(H_P)}{2}, $ which concludes the proof of the theorem."
],
[
"Proof of Theorem ",
"(Foster-Lyapunov criterion) We say that the Foster-Lyapunov criterion holds for (REF ) if there exists a function $U:\\mathbb {R}^N \\rightarrow \\mathbb {R}$ and constants $C>0$ and $b\\in \\mathbb {R}$ such that $\\mu (U)<\\infty $ , $\\mathcal {L}_PU(z) \\le -cU(z)+b\\mathbf {1}_{C}$ and $U(z)\\ge 1,\\,z\\in \\mathbb {R}^N$ , where $\\mathbf {1}_{C}$ is the indication function over a petite Borel subset $C$ of $\\mathbb {R}^N$ (refer to [twe] for more details).",
"For the generator $\\mathcal {L}_P$ corresponding to (REF ), compact sets are always petite.",
"In the following we prove that the Foster-Lyapunov criterion holds for (REF ).",
"But first we need an assumption on the potential $E_N^P$ .",
"Assumption A There exists $k>0$ such that $\\mu ^P(z)$ is bounded from above for all $|z|\\ge k$ and, for some $0<\\beta <1$ , $\\displaystyle \\lim _{|z|\\rightarrow +\\infty }\\inf \\left[(1-\\beta )|\\nabla E_N^P(z)|^2+\\Delta E_N^P(z)\\right]>0.",
"$ Under Assumption A, the Foster-Lyapunov criterion holds for (REF ) with: $U(z)={\\rm e}^{\\beta E_N^P(z)}, \\, 0<\\beta <1.$ Recall the generator of (REF ) $\\mathcal {L}_P=-\\nabla _z E_N^P(z)\\nabla + \\Delta .$ For $U(z)={\\rm e}^{\\beta E_N^P(z)}$ , one obtains: LPU(z)= -U(z)ENP(z) + U(z) =-eENP(z)ENP(z)+(eENP(z) ) = -|ENP(z)|2 eENP(z)+ENP(z)eENP(z) +2 |ENP(z)|2 eENP(z) = -[ (1-)|ENP(z)|2+ENP(z) ] U(z).",
"Therefore, by Assumption A, for $\\varepsilon >0 $ , $\\exists k>0$ such that $\\forall \\, |z|>k$ : $(1-\\beta )|\\nabla E_N^P(z)|^2+\\Delta E_N^P(z)>\\varepsilon , $ and so also, $\\mathcal {L}_PU(z)\\le -\\beta \\varepsilon U(z)+b\\mathbf {1}_{C_k}, $ where $C_k=\\lbrace z\\in \\mathbb {R}^N;|z|\\le k\\rbrace $ and $b>0$ .",
"Finally, since $\\psi _\\infty (z)$ is bounded, then $U(z)$ is bounded away from zero uniformly.",
"Then $U(z)$ can be rescaled to satisfy the condition $U(z)\\ge 1$ .",
"Thus, we valid the Foster-Lyapunov criterion for (REF ).",
"Using this lemma, the proof of the well-posedness result for the Poisson equation (REF ) come straightforward using Theorem 3.2 in [bhatt]."
],
[
"Proof of Theorem ",
"We start the proof by decomposing $\\mu _t^P(f)-\\mu (f) $ into a martingale and a remainder terms: Using (REF ), (REF ) and (REF ) tP(f)-(f) =1t0tf(P-1/2Zs)ds -(f) = 1t0t(f(P-1/2Zs) -(f))ds = 1t0t-LP(Zs)ds = (Z0)-(Zt)t+2t0t(Zs)dWs := Rt+Mt.",
"Consider now the rescaling $\\sqrt{t}(\\epsilon _t^P(f)-\\mu (f) ) $ .",
"Using the central limit theorem for the martingale term $\\sqrt{t}M_t $ (see [hella] , Theorem 5.3), one obtains the following convergence in distribution $\\displaystyle \\sqrt{t}M_t \\xrightarrow{} \\mathcal {N}(0,\\sigma _{f,P}^2), $ with $\\sigma _{f,P}^2= \\displaystyle 2\\int |\\nabla _z\\phi (z) |^2\\mu ^P(dz)=2\\int |\\nabla _z\\phi (z) |^2\\mu (dx).$ It remains to study the remainder term $\\sqrt{t}R_t $ .",
"We consider the two cases: If $Z_0\\sim \\mu $ , then since $\\phi \\in L^2(\\mu ) $ , we have that $\\displaystyle \\sqrt{t}R_t \\xrightarrow{} 0 \\text{ in } L^2(\\mu ).",
"$ In the more general case, we must refer to a \"propagation of chaos\" argument (see for example [catt], Section 8 and [dunc]), to obtain the same result."
],
[
"Acknowledgments",
"We thank Tony Lelièvre and Jonathan Weare for helpful discussions.",
"The work was supported by ERC Starting Grant 335120."
],
[
"Setup of the Metadynamics example",
"In the example shown in Figure REF , we investigated how preconditioning the second order Langevin dynamics affects the sampling of different observables when applied to adaptive potential of mean force (PMF) techniques such metadynamics (MTD) [Laio02].",
"Unlike the previously discussed umbrella sampling method, adaptive PMF techniques build their bias and mean force or potential of mean force estimates on-the-fly during the dynamics.",
"In a general form, the corresponding biased and preconditioned equations of motion with preconditioner $P = P(q)$ can be written as dqt = M-1 pt dt dpt = -EN(qt) dt + (qt) Fb((qt), t) - P M-1 pt dt + 2 P1/2 dWt where $M \\in \\mathbb {R}^{N \\times N}$ is the diagonal mass matrix, $\\beta $ is the inverse temperature, $\\xi $ is the collective variable (reaction coordinate) and $F_{\\mathrm {b}}(\\xi , t)$ is the biasing force.",
"In the case of metadynamics, $F_{\\mathrm {b}}^{\\mathrm {MTD}}(\\xi , t) = -\\nabla _{\\xi }E_{\\mathrm {b}}^{\\mathrm {MTD}}(\\xi , t)$ , where $E_{\\mathrm {b}}^{\\mathrm {MTD}}(\\xi , t)$ is some history dependent biasing potential composed by Gaussians regularly deposited in the collective variable space: $E_{\\mathrm {b}}^{\\mathrm {MTD}}(\\xi , t) = \\sum \\limits _{t^{\\prime } < t} \\delta _{t^{\\prime }} \\exp \\left(-\\frac{1}{2} \\left(\\xi - \\xi (q_{t^{\\prime }}) \\right)^{T} w^{-2} \\left(\\xi - \\xi (q_{t^{\\prime }}) \\right) \\right)$ where $\\delta _{t^{\\prime }}$ is the height of the corresponding Gaussian and $w$ is a diagonal matrix including the widths of the collective variable components.",
"The molecular system we chose for this test was the 2-(formylamino) propionaldehyde in gas phase using the Amber99SB [Hornak06] force field.",
"The molecule has a single slow degree of freedom, a dihedral angle, that was selected as the one-dimensional collective variable (we note that this variable is highly associated to one of the two dihedrals of alanine dipeptide, a test system widely used in the computational chemistry field).",
"We performed 200 ps long molecular dynamics simulations at 300 K using the BBK integrator scheme [Brunger84] with 0.5 fs timestep.",
"In the case of unpreconditioned dynamics $P = \\gamma M$ was used with $\\gamma = 5.0$ ps$^{-1}$ , while for the preconditioned dynamics we applied $P = \\gamma M + \\tau \\tilde{H}$ , where $\\tilde{H}$ is a Hessian-based preconditioner, whose positiveness is guaranteed by rebuilding the matrix using the spectral decomposition of the Hessian with the absolute values of the eigenvalues [MonesPRE].",
"We used $\\gamma M$ as diagonal a stabiliser and varied $\\gamma $ and $\\tau $ parameters.",
"A deposition frequency of 1/50 fs$^{-1}$ and a starting height of $\\delta = 0.004$ eV were used in a well-tempered variant of MTD [Barducci08] with $T_{\\mathrm {w}} = 10000$ K. Free energy profiles were reconstructed simply as the negative of the actual history dependent biasing potential.",
"The reference for computing the RMS error of the profiles was obtained from a 200 ns long unpreconditioned constrained dynamics simulation [Mones16].",
"The result of this test is shown in Figure REF , where we plotted the RMS error of the reconstructed profiles based on 10 independent MTD simulations for each parameter set.",
"We observe that in general, preconditioning does not improve the convergence.",
"As a matter of fact, we performed a variety of similar tests, e.g.",
"with different parameters, or different observables, all of which led to similar conclusions.",
"Indeed, depending on the choice of observable, preconditioning often has an even larger negative impact."
]
] | 1612.05435 |
[
[
"Backdoors to Tractable Valued CSP"
],
[
"Abstract We extend the notion of a strong backdoor from the CSP setting to the Valued CSP setting (VCSP, for short).",
"This provides a means for augmenting a class of tractable VCSP instances to instances that are outside the class but of small distance to the class, where the distance is measured in terms of the size of a smallest backdoor.",
"We establish that VCSP is fixed-parameter tractable when parameterized by the size of a smallest backdoor into every tractable class of VCSP instances characterized by a (possibly infinite) tractable valued constraint language of finite arity and finite domain.",
"We further extend this fixed-parameter tractability result to so-called \"scattered classes\" of VCSP instances where each connected component may belong to a different tractable class."
],
[
"Introduction",
"Valued CSP (or VCSP for short) is a powerful framework that entails among others The authors acknowledge support by the Austrian Science Fund (FWF, project P26696).",
"Robert Ganian is also affiliated with FI MU, Brno, Czech Republic.the problems CSP and MAX-CSP as special cases [26].",
"A VCSP instance consists of a finite set of cost functions over a finite set of variables which range over a domain $D$ , and the task is to find an instantiation of these variables that minimizes the sum of the cost functions.",
"The VCSP framework is robust and has been studied in different contexts in computer science.",
"In its full generality, VCSP considers cost functions that can take as values the rational numbers and positive infinity.",
"CSP (feasibility) and Max-CSP (optimisation) arise as special cases by limiting the values of cost functions to $\\lbrace 0,\\infty \\rbrace $ and $\\lbrace 0,1\\rbrace $ , respectively.",
"Clearly VCSP is in general intractable.",
"Over the last decades much research has been devoted into the identification of tractable VCSP subproblems.",
"An important line of this research (see, e.g., [17], [18], [25]) is the characterization of tractable VCSPs in terms of restrictions on the underlying valued constraint language $\\Gamma $ , i.e., a set $\\Gamma $ of cost functions that guarantees polynomial-time solvability of all VCSP instances that use only cost functions from $\\Gamma $ .",
"The VCSP restricted to instances with cost functions from $\\Gamma $ is denoted by $\\textsc {VCSP}[\\Gamma ]$ .",
"In this paper we provide algorithmic results which allow us to gradually augment a tractable VCSP based on the notion of a (strong) backdoor into a tractable class of instances, called the base class.",
"Backdoors where introduced by Williams et al.",
"[27], [28] for SAT and CSP and generalize in a natural way to VCSP.",
"Let $\\mathcal {C}$ denote a tractable class of VCSP instances over a finite domain $D$ .",
"A backdoor of a VCSP instance $\\mathcal {P}$ into $\\mathcal {C}$ is a (small) subset $B$ of the variables of $\\mathcal {P}$ such that for all partial assignments $\\alpha $ that instantiate $B$ , the restricted instance $\\mathcal {P}|_\\alpha $ belongs to the tractable class $\\mathcal {C}$ .",
"Once we know such a backdoor $B$ of size $k$ we can solve $\\mathcal {P}$ by solving at most $|D|^k$ tractable instances.",
"In other words, VCSP is then fixed parameter tractable parameterized by backdoor size.",
"This is highly desirable as it allows us to scale the tractability for $\\mathcal {C}$ to instances outside the class, paying for an increased “distance” from $\\mathcal {C}$ only by a larger constant factor.",
"In order to apply this backdoor approach to solving a VCSP instance, we first need to find a small backdoor.",
"This turns out to be an algorithmically challenging task.",
"The fixed-parameter tractability of backdoor detection has been subject of intensive research in the context of SAT (see, e.g., [16]) and CSP (see, e.g., [2]).",
"In this paper we extend this line of research to VCSP.",
"First we obtain some basic and fundamental results on backdoor detection when the base class is defined by a valued constraint language $\\Gamma $ .",
"We obtain fixed-parameter tractability for the detection of backdoors into $\\textsc {VCSP}[\\Gamma ]$ where $\\Gamma $ is a valued constraint language with cost functions of bounded arity.",
"In fact, we show the stronger result: fixed-parameter tractability also holds with respect to heterogeneous base classes of the form $\\textsc {VCSP}[\\Gamma _1] \\cup \\dots \\cup \\textsc {VCSP}[\\Gamma _\\ell ]$ where different assignments to the backdoor variables may result in instances that belong to different base classes $\\textsc {VCSP}[\\Gamma _i]$ .",
"A similar result holds for CSP, but the VCSP setting is slightly more complicated as a valued constraint language of finite arity over a finite domain is not necessarily finite.",
"Secondly, we extend the basic fixed-parameter tractability result to so-called scattered base classes of the form $\\textsc {VCSP}[\\Gamma _1]\\oplus \\dots \\oplus \\textsc {VCSP}[\\Gamma _\\ell ]$ which contain VCSP instances where each connected component belongs to a tractable class $\\textsc {VCSP}[\\Gamma _i]$ for some $1\\le i \\le \\ell $ —again in the heterogeneous sense that for different assignments to the backdoor variables a single component of the reduced instance may belong to different classes $\\textsc {VCSP}[\\Gamma _i]$ .",
"Backdoors into a scattered base class can be much smaller than backdoors into each single class it is composed of, hence the gain is huge if we can handle scattered classes.",
"This boost in scalability does not come for free.",
"Indeed, already the “crisp” case of CSP, which was the topic of a recent SODA paper [14], requires a sophisticated algorithm which makes use of advanced techniques from parameterized algorithm design.",
"This algorithm works under the requirement that the constraint languages contain all unary constraints (i.e., is conservative); this is a reasonable requirement as one needs these unary cost functions to express partial assignments (see also Section for further discussion).",
"Here we lift the crisp case to general VCSP, and this also represents our main technical contribution.",
"To achieve this, we proceed in two phases.",
"First we transform the backdoor detection problem from a general scattered class $\\textsc {VCSP}(\\Gamma _1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell )$ to a scattered class $\\textsc {VCSP}(\\Gamma _1^{\\prime })\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell ^{\\prime })$ over finite valued constraint languages $\\Gamma _i^{\\prime }$ .",
"In the subsequent second phase we transform the problem to a backdoor detection problem into a scattered class $\\textsc {VCSP}(\\Gamma _1^{\\prime \\prime })\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell ^{\\prime \\prime })$ where each $\\Gamma _i^{\\prime \\prime }$ is a finite crisp language; i.e., we reduce from the VCSP setting to the CSP setting.",
"We believe that this sheds light on an interesting link between backdoors in the VCSP and CSP settings.",
"The latter problem can now be solved using the known algorithm [14]."
],
[
"Related Work",
"Williams et al.",
"[27], [28] introduced backdoors for CSP or SAT as a theoretical tool to capture the overall combinatorics of instances.",
"The purpose was an analysis of the empirical behaviour of backtrack search algorithms.",
"Nishimura et.",
"al [22] started the investigation on the parameterized complexity of finding a small SAT backdoor and using it to solve the instance.",
"This lead to a number of follow-up work (see [16]).",
"Parameterized complexity provides here an appealing framework, as given a CSP instance with $n$ variables, one can trivially find a backdoor of size $\\le k$ into a fixed tractable class of instances by trying all subsets of the variable set containing $\\le k$ variables; but there are $\\Theta (n^k)$ such sets, and therefore the running time of this brute-force algorithm scales very poorly in $k$ .",
"Fixed-parameter tractability removes $k$ from the exponent providing running times of the form $f(k)n^{c}$ which yields a significantly better scalability in backdoor size.",
"Extensions to the basic notion of a backdoor have been proposed, including backdoors with empty clause detection [6], backdoors in the context of learning [7], heterogeneous backdoors where different instantiations of the backdoor variables may result in instances that belong to different base classes [15], and backdoors into scattered classes where each connected component of an instance may belong to a different tractable class [14].",
"Le Bras et al.",
"[20] used backdoors to speed-up the solution of hard problems in materials discovery, using a crowd sourcing approach to find small backdoors.",
"The research on the parameterized complexity of backdoor detection was also successfully extended to other problem areas including disjunctive answer set programming [11], [10], abstract argumentation [9], and integer linear programming [13].",
"There are also several papers that investigate the parameterized complexity of backdoor detection for CSP.",
"Bessière et al.",
"[1], considered “partition backdoors” which are sets of variables whose deletion partitions the CSP instance into two parts, one falls into a tractable class defined by a conservative polymorphism, and the other part is a collection of independent constraints.",
"They also performed an empirical evaluation of the backdoor approach which resulted in promising results.",
"Gaspers et al.",
"[15] considered heterogeneous backdoors into tractable CSP classes that are characterized by polymorphisms.",
"A similar approach was also undertaken by Carbonnel et al.",
"[3] who also considered base classes that are “$h$ -Helly” for a fixed integer $h$ under the additional assumption that the domain is a finite subset of the natural numbers and comes with a fixed ordering."
],
[
"Valued Constraint Satisfaction",
"For a tuple $t$ , we shall denote by $t[i]$ its $i$ -th component.",
"We shall denote by $\\cal Q$ the set of all rational numbers, by ${\\cal Q}_\\ge 0$ the set of all nonnegative rational numbers, and by ${\\cal \\overline{Q}}_{\\ge 0}$ the set of all nonnegative rational numbers together with positive infinity, $\\infty $ .",
"We define $\\alpha +\\infty =\\infty +\\alpha =\\infty $ for all $\\alpha \\in {\\cal \\overline{Q}}_{\\ge 0}$ , and $\\alpha \\cdot \\infty = \\infty $ for all $\\alpha \\in {\\cal Q}_{\\ge 0}$ .",
"The elements of ${\\cal \\overline{Q}}_{\\ge 0}$ are called costs.",
"For every fixed set $D$ and $m\\ge 0$ , a function $\\varphi $ from $D^m$ to ${\\cal \\overline{Q}}_{\\ge 0}$ will be called a cost function on $D$ of arity $m$ .",
"$D$ is called the domain, and here we will only deal with finite domains.",
"If the range of $\\varphi $ is $\\lbrace 0,\\infty \\rbrace $ , then $\\varphi $ is called a crisp cost function.",
"With every relation $R$ on $D$ , we can associate a crisp cost function $\\varphi _R$ on $D$ which maps tuples in $R$ to 0 and tuples not in $R$ to $\\infty $ .",
"On the other hand, with every $m$ -ary cost function $\\varphi $ we can associate a relation $R_\\varphi $ defined by $(x_1,\\dots ,x_m)\\in R_\\varphi \\Leftrightarrow \\varphi (x_1,\\dots ,x_m)< \\infty $ .",
"In the view of the close correspondence between crisp cost functions and relations we shall use these terms interchangeably in the rest of the paper.",
"A VCSP instance consists of a set of variables, a set of possible values, and a multiset of valued constraints.",
"Each valued constraint has an associated cost function which assigns a cost to every possible tuple of values for the variables in the scope of the valued constraint.",
"The goal is to find an assignment of values to all of the variables that has the minimum total cost.",
"A formal definition is provided below.",
"[VCSP] An instance $\\cal P$ of the Valued Constraint satisfaction Problem, or VCSP, is a triple $(V,D,\\cal C)$ where $V$ is a finite set of variables, which are to be assigned values from the set $D$ , and $\\cal C$ is a multiset of valued constraints.",
"Each $c\\in \\cal C$ is a pair $c=(\\vec{x},\\varphi )$ , where $\\vec{x}$ is a tuple of variables of length $m$ called the scope of $c$ , and $\\varphi : D^m\\rightarrow {\\cal \\overline{Q}}_{\\ge 0}$ is an $m$ -ary cost function.",
"An assignment for the instance $\\cal P$ is a mapping $\\tau $ from $V$ to $D$ .",
"We extend $\\tau $ to a mapping from $V^k$ to $D^k$ on tuples of variables by applying $\\tau $ componentwise.",
"The cost of an assignment $\\tau $ is defined as follows: $\\text{Cost}_{\\cal P}(\\tau )=\\sum _{(\\vec{x},\\varphi )\\in \\cal C}\\varphi (\\tau (\\vec{x})).$ The task for VCSP is the computation of an assignment with minimum cost, called a solution to $\\cal P$ .",
"For a constraint $c$ , we will use $\\text{\\normalfont {\\bfseries var}}(c)$ to denote the set of variables which occur in the scope of $c$ .",
"We will later also deal with the constraint satisfaction problem, or CSP.",
"Having already defined VCSP, it is advantageous to simply define CSP as the special case of VCSP where each valued constraint has a crisp cost function.",
"The following representation of a cost function will sometimes be useful for our purposes.",
"A cost table for an $m$ -ary cost function $\\varphi $ is a table with $D^m$ rows and $m+1$ columns with the following property: each row corresponds to a unique tuple $\\vec{a}=(a_1,\\dots ,a_m)\\in D^m$ , for each $i\\in [m]$ the position $i$ of this row contains $a_i$ , and position $m+1$ of this row contains $\\varphi (a_1,\\dots ,a_m)$ .",
"A partial assignment is a mapping from $V^{\\prime }\\subseteq V$ to $D$ .",
"Given a partial assignment $\\tau $ , the application of $\\tau $ on a valued constraint $c=(\\vec{x},\\varphi )$ results in a new valued constraint $c|_{\\tau }=(\\vec{x}^{\\prime },\\varphi ^{\\prime })$ defined as follows.",
"Let $\\vec{x}^{\\prime }=\\vec{x}\\setminus V^{\\prime }$ (i.e., $\\vec{x}^{\\prime }$ is obtained by removing all elements in $V\\cap \\vec{x}$ from $\\vec{x}$ ) and $m^{\\prime }=|\\vec{x^{\\prime }}|$ .",
"Then for each $\\vec{a^{\\prime }}\\in D^{m^{\\prime }}$ , we set $\\varphi ^{\\prime }(\\vec{a^{\\prime }})=\\varphi (\\vec{a})$ where for each $i\\in [m]$ $\\vec{a}[i]=\\left\\lbrace \\begin{array}{ll}\\tau (\\vec{x}[i])& \\mbox{\\ if } \\vec{x}[i]\\in V^{\\prime } \\\\\\vec{a^{\\prime }}[i-j] & \\mbox{\\ otherwise, where } j=|\\lbrace \\,\\vec{x}[p] \\;{|}\\;p\\in [i] \\,\\rbrace \\cap V^{\\prime }|.\\end{array}\\right.",
"$ Intuitively, the tuple $\\vec{a}$ defined above is obtained by taking the original tuple $\\vec{a}^{\\prime }$ and enriching it by the values of the assignment $\\tau $ applied on the “missing” variables from $\\vec{x}$ .",
"In the special case when $\\vec{x}^{\\prime }$ is empty, the valued constraint $c|_{\\tau }$ becomes a nullary constraint whose cost function $\\varphi ^{\\prime }$ will effectively be a constant.",
"The application of $\\tau $ on a VCSP instance $\\cal P$ then results in a new VCSP instance ${\\cal P}|_{\\tau }=(V\\setminus V^{\\prime }, D, \\cal C^{\\prime })$ where ${\\cal C^{\\prime }}=\\lbrace \\,c|_{\\tau } \\;{|}\\;c\\in \\cal C\\,\\rbrace $ .",
"It will be useful to observe that applying a partial assignment $\\tau $ can be done in time linear in $|\\cal P|$ (each valued constraint can be processed independently, and the processing of each such valued constraint consists of merely pruning the cost table)."
],
[
"Valued Constraint Languages",
"A valued constraint language (or language for short) is a set of cost functions.",
"The arity of a language $\\Gamma $ is the maximum arity of a cost function in $\\Gamma $ , or $\\infty $ if $\\Gamma $ contains cost functions of arbitrarily large arities.",
"Each language $\\Gamma $ defines a set $\\textsc {VCSP}[\\Gamma ]$ of VCSP instances which only use cost functions from $\\Gamma $ ; formally, $(V, D, {\\cal C})\\in \\textsc {VCSP}[\\Gamma ]$ iff each $(\\vec{x},\\varphi )\\in \\cal C$ satisfies $\\varphi \\in \\Gamma $ .",
"A language is crisp if it contains only crisp cost functions.",
"A language $\\Gamma $ is globally tractable if there exists a polynomial-time algorithm which solves $\\textsc {VCSP}[\\Gamma ].$The literature also defines the notion of tractability [17], [19], which we do not consider here.",
"We remark that, to the best of our knowledge, all known tractable constraint languages are also globally tractable [17], [19].",
"Similarly, a class $\\cal H$ of VCSP instances is called tractable if there exists a polynomial-time algorithm which solves $\\cal H$ .",
"For technical reasons, we will implicitly assume that every language contains all nullary cost functions (i.e., constants); it is easily seen that adding such cost functions into a language has no impact on its tractability.",
"There are a few other properties of languages that will be required to formally state our results.",
"A language $\\Gamma $ is efficiently recognizable if there exists a polynomial-time algorithm which takes as input a cost function $\\varphi $ and decides whether $\\varphi \\in \\Gamma $ .",
"We note that every finite language is efficiently recognizable.",
"A language $\\Gamma $ is closed under partial assignments if for every instance $\\mathcal {P}\\in \\textsc {VCSP}[\\Gamma ]$ and every partial assignment $\\tau $ on $\\mathcal {P}$ and every valued constraint $c=(\\vec{x},\\varphi )$ in $\\mathcal {P}$ , the valued constraint $c|_{\\tau }=(\\vec{x}^{\\prime },\\varphi ^{\\prime })$ satisfies $\\varphi ^{\\prime }\\in \\Gamma $ .",
"The closure of a language $\\Gamma $ under partial assignments, is the language $\\Gamma ^{\\prime }\\supseteq \\Gamma $ containing all cost functions that can be obtained from $\\Gamma $ via partial assignments; formally, $\\Gamma ^{\\prime }$ contains a cost function $\\varphi ^{\\prime }$ if and only if there exists a cost function $\\varphi \\in \\Gamma $ such that for a constraint $c=(\\vec{x},\\varphi )$ and an assignment $\\tau :X\\rightarrow D$ defined on a subset $X\\subseteq \\text{\\normalfont {\\bfseries var}}(c)$ we have $c|_{\\tau }=(\\vec{x}^{\\prime },\\varphi ^{\\prime })$ .",
"If a language $\\Gamma $ is closed under partial assignments, then also $\\textsc {VCSP}[\\Gamma ]$ is closed under partial assignments, which is a natural property and provides a certain robustness of the class.",
"This robustness is also useful when considering backdoors into $\\textsc {VCSP}[\\Gamma ]$ (see Section ), as then every superset of a backdoor remains a backdoor.",
"Incidentally, being closed under partial assignments is also a property of tractable classes defined in terms of a polynomial-time subsolver [27], [28] where the property is called self-reducibility.",
"A language is conservative if it contains all unary cost functions [18].",
"We note that being closed under partial assignments is closely related to the well-studied property of conservativeness.",
"Crucially, for every conservative globally tractable language $\\Gamma $ , its closure under partial assignments $\\Gamma ^{\\prime }$ will also be globally tractable; indeed, one can observe that every instance $\\mathcal {P}\\in \\textsc {VCSP}[\\Gamma ^{\\prime }]$ can be converted, in linear time, to a solution-equivalent instance $\\mathcal {P}^{\\prime }\\in \\textsc {VCSP}[\\Gamma ]$ by using infinity-valued (or even sufficiently high-valued) unary cost functions to model the effects of partial assignments."
],
[
"Parameterized Complexity",
"We give a brief and rather informal review of the most important concepts of parameterized complexity.",
"For an in-depth treatment of the subject we refer the reader to other sources [5], [8], [12], [21].",
"The instances of a parameterized problem can be considered as pairs $(I,k)$ where $I$ is the main part of the instance and $k$ is the parameter of the instance; the latter is usually a non-negative integer.",
"A parameterized problem is fixed-parameter tractable (FPT) if instances $(I,k)$ of size $n$ (with respect to some reasonable encoding) can be solved in time $\\mathcal {O}(f(k)n^c)$ where $f$ is a computable function and $c$ is a constant independent of $k$ .",
"The function $f$ is called the parameter dependence, and algorithms with running time in this form are called fixed-parameter algorithms.",
"Since the parameter dependence is usually superpolynomial, we will often give the running times of our algorithms in $\\mathcal {O}^*$ notation which suppresses polynomial factors.",
"Hence the running time of an FPT algorithm can be simply stated as $\\mathcal {O}^*(f(k))$ .",
"The exists a completeness theory which allows to obtain strong theoretical evidence that a parameterized problem is not fixed-parameter tractable.",
"This theory is based on a hierarchy of parameterized complexity classes $\\text{\\normalfont W}[1]\\subseteq \\text{\\normalfont W}[2] \\subseteq \\dots $ where all inclusions are believed to be proper.",
"If a parameterized problem is shown to be $\\text{\\normalfont W}[i]$ -hard for some $i\\ge 1$ , then the problem is unlikely to be fixed-parameter tractable, similarly to an NP-complete problem being solvable in polynomial time [5], [8], [12], [21].",
"This section is devoted to establishing the first general results for finding and exploiting backdoors for VCSP.",
"We first present the formal definition of backdoors in the context of VCSP and describe how such backdoors once found, can be used to solve the VCSP instance.",
"Subsequently, we show how to detect backdoors into a single tractable VCSP class with certain properties.",
"In fact, our proof shows something stronger.",
"That is, we show how to detect heterogeneous backdoors into a finite set of VCSP classes which satisfy these properties.",
"The notion of heterogeneous backdoors is based on that introduced by Gaspers et al. [15].",
"For now, we proceed with the definition of a backdoor.",
"Let $\\cal H$ be a fixed class of VCSP instances over a domain $D$ and let ${\\cal P}=(V,D,\\cal C)$ be a VCSP instance.",
"A backdoor into $\\cal H$ is a subset $X\\subseteq V$ such that for each assignment $\\tau :X\\rightarrow D$ , the reduced instance ${\\cal P}|_{\\tau }$ is in $\\cal H$ .",
"We note that this naturally corresponds to the notion of a strong backdoor in the context of Constraint Satisfaction and Satisfiability [27], [28]; here we drop the adjective “strong” because the other kind of backdoors studied on these structures (so-called weak backdoors) do not seem to be useful in the general VCSP setting.",
"Namely, in analogy to the CSP setting, one would define a weak backdoor of a VCSP instance ${\\cal P}=(V,D,\\cal C)$ into $\\cal H$ as a subset $X\\subseteq V$ such that for some assignment $\\tau :X\\rightarrow D$ (i) the reduced instance ${\\cal P}|_{\\tau }$ is in $\\cal H$ and (ii) $\\tau $ can be extended to an assignment to $V$ of minimum cost.",
"However, in order to ensure (ii) we need to compare the cost of $\\tau $ with the costs of all other assignments $\\tau ^{\\prime }$ to $V$ .",
"If $X$ is not a strong backdoor, then some of the reduced instances ${\\cal P}|_{\\text{$\\tau ^{\\prime }$ restrictedto $X$}}$ will be outside of ${\\cal H}$ , and so in general we have no efficient way of determining a minimum cost assignment for it.",
"We begin by showing that small backdoors for globally tractable languages can always be used to efficiently solve VCSP instances as long as the domain is finite (assuming such a backdoor is known).",
"Let $\\cal H$ be a tractable class of VCSP instances over a finite domain $D$ .",
"There exists an algorithm which takes as input a $\\textsc {VCSP}$ instance $\\cal P$ along with a backdoor $X$ of ${\\mathcal {P}}=(V,D,\\cal C)$ into $\\cal H$ , runs in time $\\mathcal {O}^*(|D|^{|X|})$ , and solves ${\\mathcal {P}}$ .",
"Let $\\mathcal {B}$ be a polynomial-time algorithm which solves every ${\\mathcal {P}}$ in $\\cal H$ , i.e., outputs a minimum-cost assignment in ${\\mathcal {P}}$ ; the existence of $\\mathcal {B}$ follows by the tractability of $\\cal H$ .",
"Consider the following algorithm ${\\mathcal {A}}$ .",
"First, ${\\mathcal {A}}$ branches on the at most $|D|^{|X|}$ -many partial assignments of $X$ .",
"In each branch, ${\\mathcal {A}}$ then applies the selected partial assignment $\\tau $ to obtain the instance ${\\mathcal {P}}|_{\\tau }$ in linear time.",
"In this branch, ${\\mathcal {A}}$ proceeds by calling $\\mathcal {B}$ on ${\\mathcal {P}}|_{\\tau }$ , and stores the produced assignment along with its cost.",
"After the branching is complete ${\\mathcal {A}}$ reads the table of all of the at most $|D|^{|X|}$ assignments and costs outputted by $\\mathcal {B}$ , and selects one assignment (say $\\alpha $ ) with a minimum value (cost) $a$ .",
"Let $\\tau $ be the particular partial assignment on $X$ which resulted in the branch leading to $\\alpha $ .",
"${\\mathcal {A}}$ then outputs the assignment $\\alpha \\cup \\tau $ along with the value (cost) $a$ .",
"Already for crisp languages it is known that having a small backdoor does not necessarily allow for efficient (i.e., fixed-parameter) algorithms when the domain is not bounded.",
"Specifically, the W[1]-hard $k$ -clique problem can be encoded into a CSP with only $k$ variables [23], which naturally contains a backdoor of size at most $k$ for every crisp language under the natural assumption that the language contains the empty constraint.",
"Hence the finiteness of the domain in Lemma is a necessary condition for the statement to hold.",
"Next, we show that it is possible to find a small backdoor into $\\textsc {VCSP}[\\Gamma ]$ efficiently (or correctly determine that no such small backdoor exists) as long as $\\Gamma $ has two properties.",
"First, $\\Gamma $ must be efficiently recognizable; it is easily seen that this condition is a necessary one, since detection of an empty backdoor is equivalent to determining whether the instance lies in $\\textsc {VCSP}[\\Gamma ]$ .",
"Second, the arity of $\\Gamma $ must be bounded.",
"This condition is also necessary since already in the more restricted CSP setting it was shown that backdoor detection for a wide range of natural crisp languages (of unbounded arity) is W[2]-hard [15].",
"Before we proceed, we introduce the notion of heterogeneous backdoors for VCSP which represent a generalization of backdoors into classes defined in terms of a single language.",
"For languages $\\Gamma _1,\\dots ,\\Gamma _\\ell $ , a heterogeneous backdoor is a backdoor into the class ${\\cal H}=\\textsc {VCSP}[\\Gamma _1]\\cup \\dots \\cup \\textsc {VCSP}[\\Gamma _\\ell ]$ ; in other words, after each assignment to the backdoor variables, all cost functions in the resulting instance must belong to a language from our set.",
"We now show that detecting small heterogenous backdoors is fixed-parameter tractable parameterized by the size of the backdoor.",
"*5mm Let $\\Gamma _1,\\dots , \\Gamma _\\ell $ be efficiently recognizable languages over a domain $D$ of size at most $d$ and let $q$ be a bound on the arity of $\\Gamma _i$ for every $i\\in [\\ell ]$ .",
"There exists an algorithm which takes as input a $\\textsc {VCSP}$ instance $\\cal P$ over $D$ and an integer $k$ , runs in time $O^*( (\\ell \\cdot d \\cdot (q+1))^{k})$ , and either outputs a backdoor $X$ of $\\cal P$ into $\\textsc {VCSP}[\\Gamma _1]\\cup \\dots \\cup \\textsc {VCSP}[\\Gamma _\\ell ]$ such that $|X|\\le k$ or correctly concludes that no such backdoor exists.",
"The algorithm is a standard branching algorithm (see also [15]).",
"Formally, the algorithm is called ${\\sf Detectbd}$ , takes as input an instance $\\mathcal {P}=(V,D,\\cal C)$ , integer $k$ , a set of variables $B$ of size at most $k$ and in time $O^*((\\ell \\cdot d \\cdot (q+1))^k)$ either correctly concludes that $\\cal P$ has no backdoor $Z\\supseteq B$ of size at most $k$ into $\\textsc {VCSP}[\\Gamma _1]\\cup \\dots \\cup \\textsc {VCSP}[\\Gamma _\\ell ]$ or returns a backdoor $Z$ of $\\cal P$ into $\\textsc {VCSP}[\\Gamma _1]\\cup \\dots \\cup \\textsc {VCSP}[\\Gamma _\\ell ]$ of size at most $k$ .",
"The algorithm is initialized with $B=\\emptyset $ .",
"In the base case, if $|B|=k$ , and $B$ is a backdoor of $\\mathcal {P}$ into $\\textsc {VCSP}[\\Gamma _1]\\cup \\dots \\cup \\textsc {VCSP}[\\Gamma _\\ell ]$ then we return the set $B$ .",
"Otherwise, we return No.",
"We now move to the description of the case when $|B|<k$ .",
"In this case, if for every $\\sigma :B\\rightarrow D$ there is an $i\\in [\\ell ]$ such that $\\mathcal {P}|_{\\sigma }\\in \\textsc {VCSP}[\\Gamma _i]$ , then it sets $Z=B$ and returns it.",
"That is, if $B$ is already found to be a backdoor of the required kind, then the algorithm returns $B$ .",
"Otherwise, it computes an assignment $\\sigma :B\\rightarrow D$ and valued constraints $c_1,\\dots , c_\\ell $ in $\\mathcal {P}|_{\\sigma }$ such that for every $i\\in [\\ell ]$ , the cost function of $c_i$ is not in $\\Gamma _i$ .",
"Observe that for some $\\sigma $ , such a set of constraints must exist.",
"Furthermore, since every $\\Gamma _i$ is efficiently recognizable and $B$ has size at most $k$ , the selection of these valued constraints takes time $\\mathcal {O}^*(d^k)$ .",
"The algorithm now constructs a set $Y$ as follows.",
"Initially, $Y=\\emptyset $ .",
"For each $i\\in [\\ell ]$ , if the scope of the constraint $c_i$ contains more than $q$ variables then it adds to $Y$ an arbitrary $q+1$ -sized subset of the scope of $c_i$ .",
"Otherwise, it adds to $Y$ all the variables in the scope of $c$ .",
"This completes the definition of $Y$ .",
"Observe that any backdoor set for the given instance which contains $B$ must also intersect $Y$ .",
"Hence the algorithm now branches on the set $Y$ .",
"Formally, for every $x\\in Y$ it executes the recursive calls ${\\sf Detectbd}(\\mathcal {P},k,B\\cup \\lbrace x\\rbrace )$ .",
"If for some $x\\in Y$ , the invoked call returned a set of variables, then it must be a backdoor set of the given instance and hence it is returned.",
"Otherwise, the algorithm returns No.",
"Since the branching factor of this algorithm is at most $\\ell \\cdot (q+1)$ and the set $B$ , whose size is upper bounded by $k$ , is enlarged with each recursive call, the number of nodes in the search tree is bounded by $\\mathcal {O}((\\ell \\cdot (q+1))^k)$ .",
"Since the time spent at each node is bounded by $\\mathcal {O}^*(d^k)$ , the running time of the algorithm ${\\sf Detectbd}$ is bounded by $\\mathcal {O}^*((\\ell \\cdot (q+1)\\cdot d)^k)$ .",
"Combining Lemmas and , we obtain the main result of this section.",
"*5mm Let $\\Gamma _1,\\dots , \\Gamma _\\ell $ be globally tractable and efficiently recognizable languages each of arity at most $q$ over a domain of size $d$ .",
"There exists an algorithm which solves VCSP in time $O^*( (\\ell \\cdot d\\cdot (q+1))^{k^2+k})$ , where $k$ is the size of a minimum backdoor of the given instance into $\\textsc {VCSP}[\\Gamma _1]\\cup \\dots \\cup \\textsc {VCSP}[\\Gamma _\\ell ]$ ."
],
[
"Backdoors into Scattered Classes",
"Having established Corollary and knowing that both the arity and domain restrictions of the language are necessary, it is natural to ask whether it is possible to push the frontiers of tractability for backdoors to more general classes of VCSP instances.",
"In particular, there is no natural reason why the instances we obtain after each assignment into the backdoor should necessary always belong to the same language $\\Gamma $ even if $\\Gamma $ itself is one among several globally tractable languages.",
"In fact, it is not difficult to show that as long as each “connected component” of the instance belongs to some tractable class after each assignment into the backdoor, then we can use the backdoor in a similar fashion as in Lemma .",
"Such a generalization of backdoors from single languages to collections of languages has recently been obtained in the CSP setting [14] for conservative constraint languages.",
"We proceed by formally defining these more general classes of VCSP instances, along with some other required notions."
],
[
"Scattered Classes",
"A VCSP instance $(V,D,\\cal C)$ is connected if for each partition of its variable set into nonempty sets $V_1$ and $V_2$ , there exists at least one constraint $c\\in \\cal C$ such that $\\text{\\normalfont {\\bfseries var}}(c)\\cap V_1\\ne \\emptyset $ and $\\text{\\normalfont {\\bfseries var}}(c)\\cap V_2\\ne \\emptyset $ .",
"A connected component of $(V,D,\\cal C)$ is a maximal connected subinstance $(V^{\\prime },D,{\\cal C}^{\\prime })$ for $V^{\\prime }\\subseteq V$ , ${\\cal C}^{\\prime }\\subseteq \\cal C$ .",
"These notions naturally correspond to the connectedness and connected components of standard graph representations of VCSP instances.",
"Let $\\Gamma _1,\\dots ,\\Gamma _d$ be languages.",
"Then the scattered class $\\textsc {VCSP}(\\Gamma _1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _d)$ is the class of all instances $(V,D,\\cal C)$ which may be partitioned into pairwise variable disjoint subinstances $(V_1,D,{\\cal C}_1),\\dots ,(V_d,D,{\\cal C}_d)$ such that $(V_i,D,{\\cal C}_i)\\in \\textsc {VCSP}[\\Gamma _i]$ for each $i\\in [d]$ .",
"Equivalently, an instance ${\\mathcal {P}}$ is in $\\textsc {VCSP}(\\Gamma _1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _d)$ iff each connected component in ${\\mathcal {P}}$ belongs to some $\\textsc {VCSP}[\\Gamma _i]$ , $i\\in [d]$ .",
"Let $\\Gamma _1,\\dots , \\Gamma _d$ be globally tractable languages.",
"Then there exists a polynomial-time algorithm solving VCSP for all instances $P\\in \\textsc {VCSP}(\\Gamma _1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _d)$ .",
"It is worth noting that while scattered classes on their own are a somewhat trivial extension of the tractable classes defined in terms of individual languages, backdoors into scattered classes can be much smaller than backdoors into each individual globally tractable language (or, more precisely, each individual class defined by a globally tractable language).",
"That is because a backdoor can not only simplify cost functions to ensure they belong to a specific language, but it can also disconnect the instance into several “parts”, each belonging to a different language, and furthermore the specific language each “part” belongs to can change for different assignments into the backdoor.",
"As a simple example of this behavior, consider the boolean domain, let $\\Gamma _1$ be the globally tractable crisp language corresponding to Horn constraints [24], and let $\\Gamma _2$ be a globally tractable language containing only submodular cost functions [4].",
"It is not difficult to construct an instance ${\\mathcal {P}}=(V_1\\cup V_2\\cup \\lbrace x\\rbrace ,\\lbrace 0,1\\rbrace ,\\cal C)$ such that (a) every assignment to $x$ disconnects $V_1$ from $V_2$ , (b) in ${\\mathcal {P}}|_{x\\mapsto 0}$ , all valued constraints over $V_1$ are crisp Horn constraints and all valued constraints over $V_2$ are submodular, and (c) in ${\\mathcal {P}}|_{x\\mapsto 1}$ , all valued constraints over $V_1$ are submodular and all valued constraints over $V_2$ are crisp Horn constraints.",
"In the hypothetical example above, it is easy to verify that $x$ is a backdoor into $\\textsc {VCSP}[\\Gamma _1]\\oplus \\textsc {VCSP}[\\Gamma _2]$ but the instance does not have a small backdoor into neither $\\textsc {VCSP}[\\Gamma _1]$ nor $\\textsc {VCSP}[\\Gamma _2]$ .",
"It is known that backdoors into scattered classes can be used to obtain fixed-parameter algorithms for CSP, i.e., both finding and using such backdoors is FPT when dealing with crisp languages of bounded arity and domain size [14].",
"Crucially, these previous results relied on the fact that every crisp language of bounded arity and domain size is finite (which is not true for valued constraint languages in general).",
"We formalize this below.",
"[[14]] Let $\\Gamma _1,\\dots ,\\Gamma _\\ell $ be globally tractable conservative crisp languages over a domain $D$ , with each language having arity at most $q$ and containing at most $p$ relations.",
"There exists a function $f$ and an algorithm solving VCSP in time $\\mathcal {O}^*(f(\\ell ,|D|,q,k,p))$ , where $k$ is the size of a minimum backdoor into $\\textsc {VCSP}[\\Gamma _1] \\oplus \\dots \\oplus \\textsc {VCSP}[\\Gamma _\\ell ]$ .",
"Observe that in the above theorem, when $q$ and $|D|$ are bounded, $p$ is immediately bounded.",
"However, it is important that we formulate the running time of the algorithm in this form because in the course of our application, these parameters have to be bounded separately.",
"Our goal for the remainder of this section is to extend Theorem REF in the VCSP setting to also cover infinite globally tractable languages (of bounded arity and domain size).",
"Before proceeding, it will be useful to observe that if each $\\Gamma _1,\\dots ,\\Gamma _\\ell $ is globally tractable, then the class $\\textsc {VCSP}[\\Gamma _1] \\oplus \\dots \\oplus \\textsc {VCSP}[\\Gamma _\\ell ]$ is also tractable (since each connected component can be resolved independently of the others)."
],
[
"Finding Backdoors to Scattered Classes",
"In this subsection, we prove that finding backdoors for $\\textsc {VCSP}$ into scattered classes is fixed-parameter tractable.",
"This will then allow us to give a proof of our main theorem, stated below.",
"theoremfinaltheorem Let $\\Delta _1,\\dots ,\\Delta _\\ell $ be conservative, globally tractable and efficiently recognizable languages over a finite domain and having constant arity.",
"Then $\\textsc {VCSP}$ is fixed-parameter tractable parameterized by the size of a smallest backdoor of the given instance into $\\textsc {VCSP}{(\\Delta _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Delta _\\ell )}$ .",
"Recall that the closure of a conservative and globally tractable language under partial assignments is also a globally tractable language.",
"Furthermore, every backdoor of the given instance into $\\textsc {VCSP}{(\\Delta _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Delta _\\ell )}$ is also a backdoor into $\\textsc {VCSP}{(\\Gamma _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Gamma _\\ell )}$ where $\\Gamma _i$ is the closure of $\\Delta _i$ under partial assignments.",
"Due to Lemma , it follows that it is sufficient to compute a backdoor of small size into the scattered class $\\textsc {VCSP}{(\\Gamma _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Gamma _\\ell )}$ where each $\\Gamma _i$ is closed under partial assignments.",
"Our strategy for finding backdoors to scattered classes defined in terms of (potentially infinite) globally tractable languages relies on a two-phase transformation of the input instance.",
"In the first phase (Lemma REF ), we show that for every choice of $\\Gamma _1,\\dots ,\\Gamma _d$ (each having bounded domain size and arity), we can construct a set of finite languages $\\Gamma ^{\\prime }_1,\\dots ,\\Gamma ^{\\prime }_d$ and a new instance ${\\mathcal {P}}^{\\prime }$ such that there is a one-to-one correspondence between backdoors of ${\\mathcal {P}}$ into $\\Gamma _1\\oplus \\dots \\oplus \\Gamma _d$ and backdoors of ${\\mathcal {P}}^{\\prime }$ into $\\Gamma ^{\\prime }_1\\oplus \\dots \\oplus \\Gamma ^{\\prime }_d$ .",
"This allows us to restrict ourselves to only the case of finite (but not necessarily crisp) languages as far as backdoor detection is concerned.",
"In the second phase (Lemma REF ), we transform the instance and languages one more time to obtain another instance ${\\mathcal {P}}^{\\prime \\prime }$ along with finite crisp languages $\\Gamma ^{\\prime \\prime }_1,\\dots ,\\Gamma ^{\\prime \\prime }_d$ such that there is a one-to-one correspondence between the backdoors of ${\\mathcal {P}}^{\\prime \\prime }$ and backdoors of ${\\mathcal {P}}^{\\prime }$ .",
"We crucially note that the newly constructed instances are equivalent only with respect to backdoor detection; there is no correspondence between the solutions of these instances.",
"Before proceeding, we introduce a natural notion of replacement of valued constraints which is used in our proofs.",
"Let $\\mathcal {P}=(V,D,\\mathcal {C})$ be a $\\textsc {VCSP}$ instance and let $c=(\\vec{x},\\varphi )\\in \\mathcal {C}$ .",
"Let $\\varphi ^{\\prime }$ be a cost function over $D$ with the same arity as $\\varphi $ .",
"Then the operation of replacing $\\varphi $ in $c$ with $\\varphi ^{\\prime }$ results in a new instance $\\mathcal {P}^{\\prime }=(V,D,({\\mathcal {C}}\\setminus \\lbrace c\\rbrace )\\cup \\lbrace (\\vec{x},\\varphi ^{\\prime })\\rbrace )$ .",
"Let $\\Gamma _1,\\dots ,\\Gamma _\\ell $ be efficiently recognizable languages closed under partial assignments, each of arity at most $q$ over a domain $D$ of size $d$ .",
"There exists an algorithm which takes as input a $\\textsc {VCSP}$ instance ${\\mathcal {P}}=(V,D,\\cal C)$ and an integer $k$ , runs in time $\\mathcal {O}^*(f(\\ell ,d,k,q))$ for some function $f$ and either correctly concludes that $\\mathcal {P}$ has no backdoor into $\\textsc {VCSP}(\\Gamma _1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell )$ of size at most $k$ or outputs a $\\textsc {VCSP}$ instance ${\\mathcal {P}}^{\\prime }=(V,D^{\\prime },\\cal C^{\\prime })$ and languages $\\Gamma ^{\\prime }_1,\\dots ,\\Gamma ^{\\prime }_{\\ell }$ with the following properties.",
"For each $i\\in [\\ell ]$ , the arity of $\\Gamma _i^{\\prime }$ is at most $q$ For each $i\\in [\\ell ]$ , $\\Gamma ^{\\prime }_i$ is over $D^{\\prime }$ and $D^{\\prime }\\subseteq D$ Each of the languages $\\Gamma _1^{\\prime },\\dots , \\Gamma _\\ell ^{\\prime }$ is closed under partial assignments and contains at most $g(\\ell ,d,k,q)$ cost functions for some function $g$ .",
"For each $X\\subseteq V$ , $X$ is a minimal backdoor of ${\\mathcal {P}}$ into $\\textsc {VCSP}(\\Gamma _1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell )$ of size at most $k$ if and only if $X$ is a minimal backdoor of ${\\mathcal {P}}^{\\prime }$ into $\\textsc {VCSP}(\\Gamma ^{\\prime }_1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma ^{\\prime }_{\\ell })$ of size at most $k$ .",
"We will first define a function mapping the valued constraints in $\\mathcal {C}$ to a finite set whose size depends only on $\\ell ,d,k$ and $q$ .",
"Subsequently, we will show that every pair of constraints in $\\mathcal {C}$ which are mapped to the same element of this set are, for our purposes (locating a backdoor), interchangeable.",
"We will then use this observation to define the new instance $\\mathcal {P}^{\\prime }$ and the languages $\\Gamma ^{\\prime }_1,\\dots ,\\Gamma ^{\\prime }_{\\ell ^{\\prime }}$ .",
"To begin with, observe that if the arity of a valued constraint in $\\mathcal {P}$ is at least $q+k+1$ , then $\\mathcal {P}$ has no backdoor of size at most $k$ into $\\textsc {VCSP}(\\Gamma _1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell )$ .",
"Hence, we may assume without loss of generality that the arity of every valued constraint in $\\mathcal {P}$ is at most $q+k$ .",
"Let $\\mathcal {F}$ be the set of all functions from $[q+k]\\times 2^{[q+k]}\\times D^{[q+k]} \\rightarrow 2^{[\\ell ]}\\cup \\lbrace \\bot \\rbrace $ , where $\\bot $ is used a special symbol expressing that $\\mathcal {F}$ is “out of bounds.” Observe that $|\\mathcal {F}|\\le \\eta (\\ell ,d,k,q)=(2^\\ell +1)^{(2d)^{(q+k)+\\log (q+k)}}$ .",
"We will now define a function $\\og :\\mathcal {C}\\rightarrow \\mathcal {F}$ as follows.",
"We assume without loss of generality that the variables in the scope of each constraint in $\\mathcal {C}$ are numbered from 1 to $|\\text{\\normalfont {\\bfseries var}}(c)|$ based on their occurrence in the tuple $\\vec{x}$ where $c=(\\vec{x},\\varphi )$ .",
"Furthermore, recall that $|\\text{\\normalfont {\\bfseries var}}(c)|\\le q+k$ .",
"For $c\\in \\mathcal {C}$ , we define $\\og (c)=\\delta \\in \\mathcal {F}$ where $\\delta $ is defined as follows.",
"Let $r\\le q+k$ , $Q\\subseteq [q+k]$ and $\\gamma :[q+k]\\rightarrow D$ .",
"Let $\\gamma [Q\\cap [r]]$ denote the restriction of $\\gamma $ to the set $Q\\cap [r]$ .",
"Furthermore, recall that $c|_{\\gamma [Q\\cap [r]]}$ denotes the valued constraint resulting from applying the partial assignment $\\gamma $ on the variables of $c$ corresponding to all those indices in $Q\\cap [r]$ .",
"Then, $\\delta (r,Q,\\gamma )=\\bot $ if $r\\ne |\\text{\\normalfont {\\bfseries var}}(c)|$ .",
"Otherwise, $\\delta (r,Q,\\gamma )=L\\subseteq [\\ell ]$ where $i\\in [\\ell ]$ is in $L$ if and only if $c|_{\\gamma [Q\\cap [r]]}\\in \\textsc {VCSP}({\\Gamma _i})$ .",
"This completes the description of the function $\\og $ ; observe that $\\og (c)$ can be computed in time which is upper-bounded by a function of $\\ell , d, k, q$ .",
"For every $\\delta \\in \\mathcal {F}$ , if there is a valued constraint $c\\in \\mathcal {C}$ such that $\\og (c)=\\delta $ , we pick and fix one arbitrary such valued constraint $c^*_\\delta =(\\vec{x^*_\\delta },\\varphi ^*_\\delta )$ .",
"We now proceed to the definition of the instance $\\mathcal {P}^{\\prime }$ and the languages $\\Gamma ^{\\prime }_1,\\dots , \\Gamma ^{\\prime }_{\\ell ^{\\prime }}$ .",
"Observe that for 2 constraints $c=(\\vec{x_1},\\varphi ),c^{\\prime }=(\\vec{x^{\\prime }},\\varphi ^{\\prime })\\in \\mathcal {C}$ , if $\\og (c)=\\og (c^{\\prime })$ then $|\\text{\\normalfont {\\bfseries var}}(c)|=|\\text{\\normalfont {\\bfseries var}}(c^{\\prime })|$ .",
"Hence, the notion of replacing $\\varphi $ in $c$ with $\\varphi ^{\\prime }$ is well-defined (see Definition REF ).",
"We define the instance $\\mathcal {P}^{\\prime }$ as the instance obtained from $\\mathcal {P}$ by replacing each $c=(\\vec{x},\\varphi )\\in \\mathcal {C}$ with the constraint $(\\vec{x},\\varphi ^*_\\delta )$ where $\\delta =\\og (c)$ .",
"For each $i\\in [\\ell ]$ and cost function $\\varphi \\in \\Gamma _i$ , we add $\\varphi $ to the language $\\Gamma _i^{\\prime }$ if and only if for some $\\delta \\in \\mathcal {F}$ and some set $Q\\subseteq \\text{\\normalfont {\\bfseries var}}(c^*_\\delta )$ and assignment $\\gamma :Q\\rightarrow D$ , the constraint $c|_{\\gamma [Q]}=(\\vec{x}\\setminus Q,\\varphi )$ .",
"Clearly, for every $i\\in [\\ell ]$ , $|\\Gamma _i^{\\prime }|\\le d^q\\cdot |\\mathcal {F}|\\le d^q\\cdot \\eta (\\ell ,d,k,q)$ .",
"Finally, for each $\\Gamma _i^{\\prime }$ , we compute the closure of $\\Gamma _i^{\\prime }$ under partial assignments and add each relation from this closure into $\\Gamma _i^{\\prime }$ .",
"Since the size of each $\\Gamma _i^{\\prime }$ is bounded initially in terms of $\\ell ,d,k,q$ , computing this closure can be done in time $\\mathcal {O}^*(\\lambda (\\ell ,d,k,q))$ for some function $\\lambda $ .",
"Since each cost function has arity $q$ and domain $D$ , the size of the final language $\\Gamma _i^{\\prime }$ obtained after this operation is blown up by a factor of at most $d^q$ , implying that in the end, $|\\Gamma _i^{\\prime }|\\le d^{2q}\\cdot |\\mathcal {F}|\\le d^{2q}\\cdot \\eta (\\ell ,d,k,q)$ .",
"Now, observe that the first two statements of the lemma follow from the definition of the languages $\\lbrace \\Gamma _i^{\\prime }\\rbrace _{i\\in [\\ell ]}$ .",
"Furthermore, the number of cost functions in each $\\Gamma _i^{\\prime }$ is bounded by $d^q\\cdot \\eta (\\ell ,d,k,q)$ , and so the third statement holds as well.",
"Therefore, it only remains to prove the final statement of the lemma.",
"Before we do so, we state a straightforward consequence of the definition of $\\mathcal {P}^{\\prime }$ .",
"For every $Y\\subseteq V$ , $\\gamma :Y\\rightarrow D$ and connected component $\\mathcal {H}^{\\prime }$ of $\\mathcal {P}^{\\prime }|_\\gamma $ , there is a connected component $\\mathcal {H}$ of $\\mathcal {P}|_\\gamma $ and a bijection $\\psi :\\mathcal {H}\\rightarrow \\mathcal {H}^{\\prime }$ such that for every $c\\in \\mathcal {H}$ , $\\og (c)=\\og (\\psi (c))$ .",
"Furthermore, for every $c=(\\vec{x},\\varphi )\\in \\mathcal {H}$ , the constraint $\\psi (c)$ is obtained by replacing $\\varphi $ in $c$ with $\\varphi ^*_{\\og (c)}$ .",
"We now return to the proof of Lemma REF .",
"Consider the forward direction and let $X$ be a backdoor of size at most $k$ for $\\mathcal {P}$ into $\\textsc {VCSP}(\\Gamma _1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell )$ and suppose that $X$ is not a backdoor for $\\mathcal {P}^{\\prime }$ into $\\textsc {VCSP}(\\Gamma ^{\\prime }_1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell ^{\\prime })$ .",
"Then, there is an assignment $\\gamma :X\\rightarrow D$ such that for some connected component $\\mathcal {H}$ of $\\mathcal {P}^{\\prime }|_\\gamma $ , there is no $i\\in \\ell $ such that all constraints in $\\mathcal {H}^{\\prime }$ lie in $\\textsc {VCSP}(\\Gamma ^{\\prime }_i)$ .",
"By Observation REF above, there is a connected component $\\mathcal {H}$ in $\\mathcal {P}|_\\gamma $ and a bijection $\\psi :\\mathcal {H}\\rightarrow \\mathcal {H}^{\\prime }$ such that for every $c\\in \\mathcal {H}$ , $\\og (c)=\\og (\\psi (c))$ .",
"Since $X$ is a backdoor for $\\mathcal {P}$ , there is a $j\\in \\ell $ such that all constraints in $\\mathcal {H}$ lie in $\\textsc {VCSP}(\\Gamma _j)$ .",
"Pick an arbitrary constraint $c=(\\vec{x},\\varphi )\\in \\mathcal {H}$ .",
"Let $c^{\\prime }=(\\vec{x},\\varphi ^*_{\\og (c)})$ be the constraint $\\psi (c)$ .",
"By definition of $\\varphi ^*_{\\og (c)}$ it follows that $c^{\\prime }|_\\gamma \\in \\textsc {VCSP}(\\Gamma _j)$ .",
"The fact that this holds for an arbitrary constraint in $\\mathcal {H}$ along with the fact that $\\psi $ is a bijection implies that every constraint in $\\mathcal {H}^{\\prime }$ is in fact in $\\textsc {VCSP}(\\Gamma _j^{\\prime })$ , a contradiction.",
"The argument in the converse direction is symmetric.",
"This completes the proof of the final statement of the lemma.",
"The time taken to compute $\\mathcal {P}^{\\prime }$ and the languages $\\Gamma _1^{\\prime },\\dots , \\Gamma _\\ell ^{\\prime }$ is dominated by the time required to compute the function $\\og $ .",
"Since the languages $\\Gamma _1,\\dots , \\Gamma _\\ell $ are efficiently recognizable, this time is bounded by $\\mathcal {O}^*(|\\mathcal {F}|)$ , completing the proof of the lemma.",
"Let $\\Gamma _1,\\dots ,\\Gamma _\\ell $ be efficiently recognizable languages closed under partial assignments, each of arity at most $q$ over a domain $D$ of size $d$ .",
"Let $\\mathcal {P}^{\\prime }=(V,D^{\\prime },\\mathcal {C}^{\\prime })$ be the $\\textsc {VCSP}$ instance and let $\\Gamma _1^{\\prime },\\dots ,\\Gamma _\\ell ^{\\prime }$ be languages returned by the algorithm of Lemma REF on input $\\mathcal {P}$ and $k$ .",
"There exists an algorithm which takes as input $\\mathcal {P}^{\\prime }$ , these languages and $k$ , runs in time $\\mathcal {O}^*(f(\\ell ,d,k,q))$ for some function $f$ and outputs a CSP instance ${\\mathcal {P}}^{\\prime \\prime }=(V^{\\prime \\prime }\\supseteq V,D^{\\prime \\prime },\\cal C^{\\prime \\prime })$ and crisp languages $\\Gamma _1^{\\prime \\prime },\\dots , \\Gamma _\\ell ^{\\prime \\prime }$ with the following properties.",
"For each $i\\in [\\ell ]$ , the arity of $\\Gamma _i^{\\prime \\prime }$ is at most $q+1$ $D^{\\prime \\prime }\\supseteq D$ and $|D^{\\prime \\prime }|\\le \\beta (q,d,k)$ for some function $\\beta $ .",
"The number of relations in each of the languages $\\Gamma _1^{\\prime \\prime },\\dots , \\Gamma _\\ell ^{\\prime \\prime }$ is at most $\\alpha (q,d,k)$ for some function $\\alpha $ .",
"if $X$ is a minimal backdoor of arity at most $k$ of ${\\mathcal {P}}^{\\prime \\prime }$ into $\\textsc {CSP}(\\Gamma ^{\\prime \\prime }_1)\\oplus \\dots \\oplus \\textsc {CSP}(\\Gamma ^{\\prime \\prime }_{\\ell })$ , then $X\\subseteq V$ .",
"For each $X\\subseteq V$ , $X$ is a minimal backdoor of ${\\mathcal {P}}^{\\prime }$ into $\\textsc {VCSP}(\\Gamma _1^{\\prime })\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma _\\ell ^{\\prime })$ if and only if $X$ is a minimal backdoor of ${\\mathcal {P}}^{\\prime \\prime }$ into $\\textsc {CSP}(\\Gamma ^{\\prime \\prime }_1)\\oplus \\dots \\oplus \\textsc {CSP}(\\Gamma ^{\\prime \\prime }_{\\ell })$ .",
"We propose a fixed-parameter algorithm ${\\mathcal {A}}$ , and show that it has the claimed properties.",
"It will be useful to recall that we do not distinguish between crisp cost functions and relations.",
"We also formally assume that $D^{\\prime }$ does not intersect the set of rationals $\\cal Q$ ; if this is not the case, then we simply rename elements of $D^{\\prime }$ to make sure that this holds.",
"Within the proof, we will use $\\vec{a}\\circ b$ to denote the concatenation of vector $\\vec{a}$ by element $b$ .",
"First, let $T_i$ be the set of all values which are returned by at least one cost function from $\\Gamma ^{\\prime }_i$ , $i\\in [\\ell ]$ , for at least one input.",
"Let $T=\\bigcup _{i\\in [\\ell ]} T_i$ .",
"Observe that $|T|$ is upper-bounded by the size, domain and arity of our languages.",
"Let us now set $D^{\\prime \\prime }=D^{\\prime }\\cup T\\cup \\epsilon $ .",
"Intuitively, our goal will be to represent the cost function in each valued constraint in $\\mathcal {P}^{\\prime }$ by a crisp cost function with one additional variable which ranges over $T$ , where $T$ corresponds to a specific value which occurs in one of our base languages.",
"Note that this satisfies Condition $2.$ of the lemma, and that $T$ can be computed in linear time from the cost tables of $\\Gamma ^{\\prime }_1,\\dots ,\\Gamma ^{\\prime }_\\ell $ .",
"We will later construct $k+1$ such representations (each with its own additional variable) to ensure that the additional variables are never selected by minimal backdoors.",
"Next, for each language $\\Gamma _i^{\\prime }$ , $i\\in [\\ell ]$ , we compute a new crisp language $\\Gamma _i^{\\prime \\prime }$ as follows.",
"For each $\\varphi \\in \\Gamma _i^{\\prime }$ of arity $t$ , we add a new relation $\\psi $ of arity $t+1$ into $\\Gamma _i^{\\prime \\prime }$ , and for each tuple $(x_1,\\dots ,x_t)$ of elements from $D^{\\prime }$ we add the tuple $(x_1,\\dots ,x_t,\\varphi (x_1,\\dots ,x_t))$ into $\\psi $ ; observe that this relation exactly corresponds to the cost table of $\\varphi $ .",
"We then compute the closure of $\\Gamma _i^{\\prime \\prime }$ under partial assignments and add each relation from this closure into $\\Gamma _i^{\\prime \\prime }$ .",
"Observe that the number of relations in $\\Gamma _i^{\\prime \\prime }$ is bounded by a function of $|T|$ and $|\\Gamma _i^{\\prime }|$ , and furthermore the number of tuples in each relation is upper-bounded by $q^{|D^{\\prime }|}$ , and so Conditions $1.$ and $3.$ of the lemma hold.",
"The construction of each $\\Gamma _i^{\\prime \\prime }$ from $\\Gamma _i^{\\prime }$ can also be done in linear time from the cost tables of $\\Gamma ^{\\prime }_1,\\dots ,\\Gamma ^{\\prime }_\\ell $ .",
"Finally, we construct a new instance ${\\mathcal {P}}^{\\prime \\prime }=(V^{\\prime \\prime },D^{\\prime \\prime },\\cal C^{\\prime \\prime })$ from ${\\mathcal {P}}^{\\prime }=(V,D^{\\prime },\\cal C^{\\prime })$ as follows.",
"At the beginning, we set $V^{\\prime \\prime }:=V$ .",
"For each $c^{\\prime }=(\\vec{x}^{\\prime },\\varphi ^{\\prime })\\in \\mathcal {C}^{\\prime }$ , we add $k+1$ unique new variables $v^1_{c^{\\prime }},\\dots ,v^{k+1}_{c^{\\prime }}$ into $V^{\\prime \\prime }$ and add $k+1$ constraints $c^{\\prime \\prime 1},\\dots ,c^{\\prime \\prime k+1}$ into $\\mathcal {C}^{\\prime \\prime }$ .",
"For $i\\in [k+1]$ , each $c^{\\prime \\prime i}=(\\vec{x}^{\\prime } \\circ v^i_{c^{\\prime }},\\psi ^{\\prime \\prime })$ where $\\psi ^{\\prime \\prime }$ is a relation that is constructed similarly as the relations in our new languages $\\Gamma _i^{\\prime \\prime }$ above.",
"Specifically, for each tuple $(x_1,\\dots ,x_t)$ of elements from $D^{\\prime }$ we add the tuple $(x_1,\\dots ,x_t,\\varphi ^{\\prime }(x_1,\\dots ,x_t))$ into $\\psi ^{\\prime \\prime }$ , modulo the following exception.",
"If $\\varphi ^{\\prime }(x_1,\\dots ,x_t)\\notin D^{\\prime \\prime }$ , then we instead add the tuple $(x_1,\\dots ,x_t,\\epsilon )$ into $\\psi ^{\\prime \\prime }$ .",
"Clearly, the construction of our new instance ${\\mathcal {P}}^{\\prime \\prime }$ takes time at most $\\mathcal {O}(|{\\cal C^{\\prime }}|+q^{|D^{\\prime }|})$ .",
"This concludes the description of ${\\mathcal {A}}$ .",
"It remains to argue that Conditions $4.$ and $5.$ of the lemma hold.",
"First, consider a minimal backdoor $X$ of size at most $k$ of ${\\mathcal {P}}^{\\prime \\prime }$ into $\\textsc {CSP}[\\Gamma ^{\\prime \\prime }_1]\\oplus \\dots \\oplus \\textsc {CSP}[\\Gamma ^{\\prime \\prime }_{\\ell }]$ , and assume for a contradiction that there exists some $c^{\\prime }=(\\vec{x}^{\\prime },\\varphi ^{\\prime })\\in \\mathcal {C}^{\\prime }$ and $i\\in [k+1]$ such that $v^i_{c^{\\prime }}\\in X$ .",
"First, observe that this cannot happen if the whole scope of $c^{\\prime \\prime i}$ lies in $X$ .",
"By the size bound on $X$ , there exists $j\\in [k+1]$ such that $v^j_{c^{\\prime }}\\notin X$ .",
"Then for each partial assignment $\\tau $ of $X$ , the relation $\\varphi ^{\\prime \\prime }$ in $c^{\\prime \\prime j}$ belongs to the same globally tractable language as the rest of the connected component of $\\mathcal {P}^{\\prime \\prime }$ containing the scope of $c^{\\prime \\prime }$ (after applying $\\tau $ ).",
"Since the relation $\\varphi ^{\\prime \\prime }$ in $c^{\\prime \\prime j}$ is precisely the same as in $c^{\\prime \\prime i}$ and the scope of $c^{\\prime \\prime i}$ must lie in the same connected component as that of $c^{\\prime \\prime j}$ , it follows that $X\\setminus \\lbrace v_{x^{\\prime }}^i\\rbrace $ is also a backdoor of ${\\mathcal {P}}^{\\prime \\prime }$ into $\\textsc {CSP}(\\Gamma ^{\\prime \\prime }_1)\\oplus \\dots \\oplus \\textsc {CSP}(\\Gamma ^{\\prime \\prime }_{\\ell })$ .",
"However, this contradicts the minimality of $X$ .",
"Finally, for Condition $5.$ , consider an arbitrary backdoor $X$ of ${\\mathcal {P}}^{\\prime }$ into $\\textsc {VCSP}(\\Gamma ^{\\prime }_1)\\oplus \\dots \\oplus \\textsc {VCSP}(\\Gamma ^{\\prime }_{\\ell })$ , and let us consider an arbitrary assignment from $X$ to $D^{\\prime \\prime }$ .",
"It will be useful to note that while the contents of relations and/or cost functions in individual (valued) constraints depend on the particular choice of the assignment to $X$ , which variables actually occur in individual components depends only on the choice of $X$ and remains the same for arbitrary assignments.",
"Now observe that each connected component ${\\mathcal {P}}^{\\text{CSP}}$ of ${\\mathcal {P}}^{\\prime \\prime }$ after the application of the (arbitrarily chosen) assignment will fall into one of the following two cases.",
"${\\mathcal {P}}^{\\text{CSP}}$ could contain a single variable $v_{c^{\\prime }}$ with a single constraint whose relation lies in every language $\\Gamma ^{\\prime \\prime }_i$ , $i\\in [\\ell ]$ ; this occurs precisely when the whole scope of a valued constraint $c^{\\prime }\\in \\mathcal {C}^{\\prime }$ lies in $X$ , and the relation will either contain a singleton element from $T$ or be the empty relation.",
"In this case, we immediately conclude that ${\\mathcal {P}}^{\\text{CSP}}\\in \\textsc {CSP}(\\Gamma _i)$ for each $i\\in [\\ell ]$ .",
"Alternatively, ${\\mathcal {P}}^{\\text{CSP}}$ contains at least one variable $v\\in V$ .",
"Let ${\\mathcal {P}}^{\\text{VCSP}}$ be the unique connected component of ${\\mathcal {P}}^{\\prime }$ obtained after the application of an arbitrary assignment from $X$ to $D^{\\prime }$ which contains $v$ .",
"Observe that the variable sets of ${\\mathcal {P}}^{\\text{CSP}}$ and ${\\mathcal {P}}^{\\text{VCSP}}$ only differ in the fact that ${\\mathcal {P}}^{\\text{CSP}}$ may contain some of the newly added variables $v_{c^{\\prime }}$ for various constraints $c^{\\prime }$ .",
"Now let us consider a concrete assignment $\\tau $ from $X$ to $D^{\\prime }$ along with an $i\\in [\\ell ]$ such that after the application of $\\tau $ , the resulting connected component ${\\mathcal {P}}^{\\text{VCSP}}$ belongs to $\\textsc {VCSP}(\\Gamma ^{\\prime }_i)$ .",
"It follows by our construction that applying the same assignment $\\tau $ in ${\\mathcal {P}}^{\\prime \\prime }$ will result in a connected component ${\\mathcal {P}}^{\\text{CSP}}$ corresponding to ${\\mathcal {P}}^{\\text{VCSP}}$ such that ${\\mathcal {P}}^{\\text{VCSP}}\\in \\textsc {CSP}(\\Gamma ^{\\prime \\prime }_i)$ ; indeed, whenever $\\Gamma ^{\\prime }_i$ contains an arbitrary cost function $\\varphi (\\vec{x})=\\beta $ , the language $\\Gamma ^{\\prime \\prime }_i$ will contain the relation $(\\vec{x}\\circ \\beta )$ .",
"By the above, the application of an assignment from $X$ to $D^{\\prime }$ in ${\\mathcal {P}}^{\\prime \\prime }$ will indeed result in an instance in $\\textsc {CSP}(\\Gamma _1^{\\prime \\prime }\\oplus \\dots \\oplus \\Gamma _\\ell ^{\\prime \\prime })$ .",
"But recall that the domain of ${\\mathcal {P}}^{\\prime \\prime }$ is $D^{\\prime \\prime }$ , which is a superset of $D^{\\prime }$ ; we need to argue that the above also holds for assignments $\\tau $ from $X$ to $D^{\\prime \\prime }$ .",
"To this end, consider an arbitrary such $\\tau $ and let $\\tau _0$ be an arbitrary assignment from $X$ to $D^{\\prime }$ which matches $\\tau $ on all mappings into $D^{\\prime }$ .",
"Let us compare the instances ${\\mathcal {P}}^{\\prime \\prime }_{\\tau _0}$ and ${\\mathcal {P}}^{\\prime \\prime }_{\\tau }$ .",
"By our construction of ${\\mathcal {P}}^{\\prime }$ , whenever $\\tau $ maps at least one variable from the scope of some constraint $c^{\\prime \\prime }$ to $D^{\\prime \\prime }\\setminus D^{\\prime }$ , the resulting relation will be the empty relation.",
"It follows that each constraint in ${\\mathcal {P}}^{\\prime \\prime }_{\\tau }$ will either be the same as in ${\\mathcal {P}}^{\\prime \\prime }_{\\tau }$ , or will contain the empty relation.",
"But since the empty relation is included in every language $\\Gamma _1^{\\prime \\prime },\\dots ,\\Gamma _\\ell ^{\\prime \\prime }$ , we conclude that each connected component of ${\\mathcal {P}}^{\\prime \\prime }_\\tau $ must belong to at least one language $\\Gamma _i^{\\prime \\prime }$ , $i\\in [\\ell ]$ .",
"This shows that $X$ must also be a backdoor of ${\\mathcal {P}}^{\\prime \\prime }$ into $\\textsc {CSP}[\\Gamma ^{\\prime \\prime }_1]\\oplus \\dots \\oplus \\textsc {CSP}[\\Gamma ^{\\prime \\prime }_\\ell ]$ .",
"For the converse direction, consider a minimal backdoor $X$ of ${\\mathcal {P}}^{\\prime \\prime }$ into $\\textsc {CSP}[\\Gamma ^{\\prime \\prime }_1]\\oplus \\dots \\oplus \\textsc {CSP}[\\Gamma ^{\\prime \\prime }_\\ell ]$ .",
"Since we already know that Condition $4.$ holds, $X$ must be a subset of $V$ .",
"The argument from the previous case can then simply be reversed to see that $X$ will also be a backdoor of ${\\mathcal {P}}^{\\prime }$ into $\\textsc {VCSP}[\\Gamma ^{\\prime }_1]\\oplus \\dots \\oplus \\textsc {VCSP}[\\Gamma ^{\\prime }_\\ell ]$ ; in fact, the situation in this case is much easier since only assignments into $D^{\\prime }$ need to be considered.",
"Summarizing, we gave a fixed-parameter algorithm and then showed that it satisfies each of the required conditions, and so the proof is complete.",
"We are now ready to prove Theorem REF , which we restate for the sake of convenience.",
"* For each $i\\in [\\ell ]$ , let $\\Gamma _i$ denote the closure of $\\Delta _i$ under partial assignments.",
"Observe that every backdoor of the given instance into $\\textsc {VCSP}{(\\Delta _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Delta _\\ell )}$ is also a backdoor into $\\textsc {VCSP}{(\\Gamma _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Gamma _\\ell )}$ .",
"Furthermore, each $\\textsc {VCSP}{(\\Gamma _i)}$ is tractable since $\\textsc {VCSP}{(\\Delta _1)}$ is tractable and conservative.",
"Hence, it is sufficient to compute and use a backdoor of size at most $k$ into $\\textsc {VCSP}{(\\Gamma _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Gamma _\\ell )}$ .",
"The claimed algorithm has two parts.",
"The first one is finding a backdoor into $\\textsc {VCSP}{(\\Gamma _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Gamma _\\ell )}$ and the second one is using the computed backdoor to solve $\\textsc {VCSP}$ .",
"Given an instance $\\mathcal {P}$ and $k$ , we first execute the algorithm of Lemma REF to compute the instance $\\mathcal {P}^{\\prime }$ , and the languages $\\Gamma _1^{\\prime },\\dots , \\Gamma _\\ell ^{\\prime }$ with the properties stated in the lemma.",
"We then execute the algorithm of Lemma REF with input $\\mathcal {P}^{\\prime }$ , $k$ , and $\\Gamma _1^{\\prime },\\dots , \\Gamma _\\ell ^{\\prime }$ to compute the CSP instance $\\mathcal {P}^{\\prime \\prime }$ and crisp languages $\\Gamma _1^{\\prime \\prime },\\dots , \\Gamma _\\ell ^{\\prime \\prime }$ with the stated properties.",
"Following this, we execute the algorithm of Theorem REF with input $\\mathcal {P}^{\\prime \\prime },k$ .",
"If this algorithm returns No then we return No as well.",
"Otherwise we return the set returned by this algorithm as a backdoor of size at most $k$ for the given instance $\\mathcal {P}$ .",
"Finally, we use the algorithm of Lemma with $\\cal H$ set to be the class $\\textsc {VCSP}{(\\Gamma _1)} \\oplus \\cdots \\oplus \\textsc {VCSP}{(\\Gamma _\\ell )}$ , to solve the given instance.",
"The correctness as well as running time bounds follow from those of Lemmas REF and REF , Theorem REF , and Lemma .",
"This completes the proof of the theorem."
],
[
"Concluding Remarks",
"We have introduced the notion of backdoors to the VCSP setting as a means for augmenting a class of globally tractable VCSP instances to instances that are outside the class but of small distance to the class.",
"We have presented fixed-parameter tractability results for solving VCSP instances parameterized by the size of a smallest backdoor into a (possibly scattered and heterogeneous) tractable class satisfying certain natural properties.",
"Our work opens up several avenues for future research.",
"Since our main objective was to establish the fixed-parameter tractability of this problem, we have not attempted to optimize the runtime bounds for finding backdoors to scattered classes.",
"As a result, it is quite likely that a more focussed study of scattered classes arising from specific constraint languages will yield a significantly better runtime.",
"A second interesting direction would be studying the parameterized complexity of detection of backdoors into tractable VCSP classes that are characterized by specific fractional polymorphisms."
]
] | 1612.05733 |
[
[
"CMB Lens Sample Covariance and Consistency Relations"
],
[
"Abstract Gravitational lensing information from the two and higher point statistics of the CMB temperature and polarization fields are intrinsically correlated because they are lensed by the same realization of structure between last scattering and observation.",
"Using an analytic model for lens sample covariance, we show that there is one mode, separately measurable in the lensed CMB power spectra and lensing reconstruction, that carries most of this correlation.",
"Once these measurements become lens sample variance dominated, this mode should provide a useful consistency check between the observables that is largely free of sampling and cosmological parameter errors.",
"Violations of consistency could indicate systematic errors in the data and lens reconstruction or new physics at last scattering, any of which could bias cosmological inferences and delensing for gravitational waves.",
"A second mode provides a weaker consistency check for a spatially flat universe.",
"Our analysis isolates the additional information supplied by lensing in a model independent manner but is also useful for understanding and forecasting CMB cosmological parameter errors in the extended $\\Lambda$CDM parameter space of dark energy, curvature and massive neutrinos.",
"We introduce and test a simple but accurate forecasting technique for this purpose that neither double counts lensing information nor neglects lensing in the observables."
],
[
"Introduction",
"Power spectra of the cosmic microwave background (CMB) anisotropies have been extremely valuable in helping to confirm predictions of the standard $\\Lambda $ CDM cosmological model and constrain values of cosmological parameters [1].",
"Only recently has gravitational lensing of the CMB been detected, first through cross-correlation with galaxy surveys [2], [3], [4], and then by internal correlations of the temperature ($T$ ) [5], [6], [7], and polarization ($E$ ,$B$ ) [8], [9], [10], [11] fields, adding a new source of cosmological information.",
"This secondary signal depends on growth of structure in the universe, which can be leveraged to break certain parameter degeneracies in the CMB data and used to better constrain sum of neutrino masses and other parameters in models beyond $\\Lambda $ CDM (see [12] for a review).",
"Information carried by the lensing potential $\\phi $ can be recovered either by measuring its effect on CMB power spectra, in particular the smoothing of the acoustic peaks [13] or by measuring four point functions of the temperature and polarization maps.",
"The latter is possible, because gravitational lensing generates a correlation between measured CMB fields and their gradients [14], [15], [16], modifying the simple Gaussian statistics of the unlensed CMB.",
"This non-Gaussian structure can be used to measure the lensing potential, for example using a quadratic reconstruction [17] or iterative delensing [18], [19].",
"The reconstructed potential then serves as a new cosmological observable.",
"The same non-Gaussianity that makes lensing reconstruction possible is responsible for correlating the CMB observables and complicates their analysis.",
"Gravitational lensing induces nontrivial covariances between the lensed temperature and polarization data [20], [21].",
"Neglecting these covariances can affect parameter forecasts of future experiments and analysis of their data.",
"In particular, future experiments are expected to have their lensing information limited by sample variance of the lenses: the fact that on the same patch of sky, the gravitational lensing of all CMB observables is due to the same realizations of a finite number of lens modes.",
"In this work we use an extension of the analytical model of Ref.",
"[20] to include covariances between power spectra $C^{XY}_\\ell $ of the lensed CMB temperature and polarizations with the power spectra of the reconstructed lensing potential, recently also discussed in [22], [23].",
"With this model we then investigate how these covariances affect parameter forecasts and construct sharp consistency relations between the two types of observables that can be used to test for foregrounds, systematics or new physics.",
"The outline of the paper is as follows.",
"In § we present the analytical model for lens sample covariances.",
"We analyze their impact on cosmological parameters in § and separate information on them into lensing and non-lensing based sources.",
"Based on this separation, we determine the modes that most strongly covary between CMB power spectra and lens reconstruction in §.",
"These provide consistency relations between observables that are largely immune to lens sample variance and cosmological parameter uncertainties.",
"We discuss these results in §.",
"In the Appendix we use these results to develop a new accurate but simple Fisher forecasting technique in the extended $\\Lambda $ CDM parameter space that avoids double counting lensing information, and compare it with other similar but less accurate approaches."
],
[
"Analytic lens covariance model",
"In this section, we present an analytical model describing non-Gaussian covariances between the $C_\\ell ^{xy}$ power and cross spectra observables induced by gravitational lensing through the same lenses on the sky.",
"Here these $xy$ spectra are the CMB temperature power spectra $TT$ , $E$ -mode polarization power $EE$ , temperature-polarization cross spectra $TE$ , $B$ -mode polarization power $BB$ and the power spectrum of the lens potential $\\phi \\phi $ .",
"As a notational short hand, we denote the subset of $xy$ that includes only the CMB power spectra with capital letters $XY$ : $xy \\in \\lbrace XY,\\phi \\phi \\rbrace $ , whereas $XY\\in \\lbrace TT,EE,TE,BB\\rbrace $ .",
"Note that although the $T\\phi $ and $E\\phi $ spectra are also observable we omit them as a source of information but include them in the covariance of other spectra.",
"We comment more on this choice in §.",
"Covariances predicted by this model have been tested against numerical simulations in [20] for the $XY$ power spectra; here we use the physical intuition gained in [20] to extend the same model to include their covariance with measurements of ${\\phi \\phi }$ .",
"A similar model has recently been also used in [22], [23].",
"Figure: Comparison of the lensing potential power spectra C ℓ φφ C^{\\phi \\phi }_\\ell (solid) with thereconstruction noise forecast in this work (dashed, see text for details).",
"The forecast is lens sample variancelimited for ℓ≲10 3 \\ell \\lesssim 10^3.In this model the correlation matrix is split into “Gaussian part” $\\mathcal {G}$ that is diagonal in multipole space and $\\mathcal {N}$ which describes non-Gaussian correlations between multipoles, $\\mathrm {Cov}^{xy,wz}_{\\ell \\ell ^{\\prime }} = \\mathcal {G}^{xy,wz}_{\\ell \\ell ^{\\prime }}+\\mathcal {N}^{xy,wz}_{\\ell \\ell ^{\\prime }} .$ The Gaussian part is modelled after the covariance of Gaussian random fields as $\\mathcal {G}^{xy, wz}_{\\ell \\ell ^{\\prime }} = \\frac{\\delta _{\\ell \\ell ^{\\prime }}}{2\\ell + 1}\\left[C_{\\mathrm {exp},\\ell }^{xw}C_{\\mathrm {exp},\\ell }^{yz} +C_{\\mathrm {exp},\\ell }^{xz} C_{\\mathrm {exp},\\ell }^{yw}\\right] ,$ where the expectation value of the experimentally measured lensed CMB power spectra $C^{xy}_\\mathrm {exp}$ includes the noise power spectrum $N_\\ell ^{xy}$ $C_{\\mathrm {exp},\\ell }^{xy} = C_\\ell ^{xy} + N_\\ell ^{xy}.$ For noise in temperature and polarizations, we assume a Gaussian noise spectra [24] $N_\\ell ^{XY} = \\Delta _{XY}^2 e^{\\ell (\\ell +1){\\theta _\\mathrm {FWHM}^2}/{8 \\log 2} },$ where $\\Delta _{XY}$ is the instrumental noise (in $\\mu $ K-radian) and $\\theta _\\mathrm {FWHM}$ is the beam size (in radians).",
"In this work we investigate a simplified experimental setup of a full sky experiment with specifications inspired by CMB Stage 4 [25].",
"We consider a $1^{\\prime }$ beam, $\\Delta _{TT}=1\\,\\mu $ K$^{\\prime }$ , $\\Delta _{EE}=\\Delta _{BB}=1.4\\,\\mu $ K$^{\\prime }$ , and $\\Delta _{TE}=\\Delta _{TB}=\\Delta _{EB}=0$ and use measurements from $\\ell = 2 - 3000$ .",
"CMB Stage 4 measurements at $\\ell >3000$ have negligible impact on our results (see § and ).",
"We also assume measurements of $C_\\ell ^{\\phi \\phi }$ from $\\ell = 2 - 5000$ with the reconstruction noise $N_\\ell ^{\\phi \\phi }$ of the minimal variance quadratic estimator [17], commonly known as $N^{(0)}$ noise bias, and ignore other noise biases and trispectrum terms [21] (see §).",
"Comparison of the $C_\\ell ^{\\phi \\phi }$ with the reconstruction noise for our experiment $N^{\\phi \\phi }_\\ell $ is plotted in Figure REF .",
"Notice that for these specifications, the lens reconstruction is sample variance dominated for $\\ell \\lesssim 10^{3}$ .",
"This is the fundamental assumption underlying this work: that lens sample variance will in the future dominate the measurements of the lens power spectrum at low multipoles.",
"The consistency check proposed in § can be viewed as an operational test of this assumption and we comment more on current simulation-based tests in §.",
"Figure: Correlation matrix R ℓ XY ,ℓ φφ XY,φφ R^{XY,{\\phi \\phi }}_{\\ell _{XY},\\ell _{{\\phi \\phi }}} ()between the C ℓ XY C_\\ell ^{XY} CMB power spectraand the power spectra of the reconstructed lensingpotential C ℓ φφ C_\\ell ^{{\\phi \\phi }}.",
"Barely visible features for ℓ XY =ℓ φφ ≲50\\ell _{XY}=\\ell _{{\\phi \\phi }} \\lesssim 50 in the first three panels represent contributionsfrom the Gaussian terms due to nonzero C ℓ Tφ ,C ℓ Eφ C_\\ell ^{T\\phi }, C_\\ell ^{E\\phi }.Even if we assume that the unlensed CMB fields $\\tilde{X}$ and $\\phi $ are Gaussian, the lensed CMB fields $X$ are not.",
"In our model, we take two non-Gaussian terms to compose the full covariance, $\\mathcal {N}^{xy,wz}_{\\ell \\ell ^{\\prime }} = \\mathcal {N}^{(\\phi )xy,wz}_{\\ell \\ell ^{\\prime }} +\\mathcal {N}^{(E)xy,wz}_{\\ell \\ell ^{\\prime }} ,$ which we now describe.",
"Gravitational lensing induces non-Gaussian covariances between the data because all power spectra are affected by the same realization of the lensing potential; sample variance fluctuations of the lensing power produce coherent changes in all the observed power spectra.",
"The effect accumulates over the whole multipole range of the lenses and is largest for those $C^{XY}_\\ell $ which are most strongly affected by lensing.",
"It is modeled by adding an extra term $\\mathcal {N}^{(\\phi )xy,wz}_{\\ell \\ell ^{\\prime }} = \\sum _{L}\\frac{\\partial {C_\\ell ^{xy}}}{\\partial C_{L}^{{\\phi \\phi }}}\\mathrm {Cov}^{{\\phi \\phi }}_{LL}\\frac{\\partial {C_{\\ell ^{\\prime }}^{wz}}}{\\partial C_{L}^{{\\phi \\phi }}}$ to the non-Gaussian covariance $\\mathcal {N}$ .",
"The power spectra derivatives are in practice calculated using a two point central difference scheme from results obtained using CAMB [26].",
"For the reconstructed potential we take $\\mathcal {N}^{(\\phi ){\\phi \\phi },{\\phi \\phi }}_{\\ell \\ell ^{\\prime }} = 0$ as the corresponding variance is part of the Gaussian term.",
"Sample variance of the unlensed $\\tilde{E} \\tilde{E}$ power spectrum and its coherent propagation into the lensed power spectra through gravitational lensing produces similar but typically weaker effects.",
"Following [20] we include this contribution only for $\\mathrm {Cov}^{XY,BB}_{\\ell \\ell ^{\\prime }}$ with $\\mathcal {N}^{(E)XY,BB}_{\\ell , \\ell ^{\\prime }} = \\sum _{L}\\frac{\\partial {C_\\ell ^{XY}}}{\\partial C_{L}^{\\tilde{X} \\tilde{Y}}}\\mathrm {Cov}^{\\tilde{X}\\tilde{Y}, \\tilde{E} \\tilde{E}}_{L,L}\\frac{\\partial {C_{\\ell ^{\\prime }}^{BB}}}{\\partial C_{L}^{\\tilde{E} \\tilde{E}}}.$ Other sample covariance effects from unlensed fields on $XY$ are negligible in comparison [20].",
"We also assume that the analogous terms involving the reconstruction noise, e.g.",
"$\\partial N_l^{\\phi \\phi }/\\partial C_L^{\\tilde{E}\\tilde{E}}$ and other non-Gaussian reconstruction terms are negligible.",
"This should be a good approximation in the lens sample dominated regime $\\ell \\lesssim 10^3$ (see §).",
"The covariances $\\mathrm {Cov}^{XY,WZ}$ we obtain for the CMB power spectra qualitatively agree with those plotted in Fig.",
"1 of [20] for the same analytical model for covariances but for a slightly different cosmological model.",
"The less well studied covariances $\\mathrm {Cov}^{XY,{\\phi \\phi }}$ are shown in Figure REF ; for illustrative purposes we plot the correlation coefficient $R^{XY,{\\phi \\phi }}_{\\ell \\ell ^{\\prime }} =\\frac{\\mathrm {Cov}^{XY,{\\phi \\phi }}_{\\ell \\ell ^{\\prime }}}{\\sqrt{\\mathrm {Cov}^{XY,XY}_{\\ell \\ell }\\mathrm {Cov}^{{\\phi \\phi },{\\phi \\phi }}_{\\ell ^{\\prime }\\ell ^{\\prime }}}} .$ In this plot we assume experimental and reconstruction noise for our reference experiment.",
"We see that the covariances peak for $\\ell ^{\\prime } =\\ell _{\\phi \\phi }\\sim 100-200$ which reflects the fact that most of the lensing is caused by lenses at these scales.",
"In covariances with $TT, TE$ and $EE$ there are alternating regions of positive and negative correlations, corresponding to smearing of the peaks and troughs; correlation with $BB$ also shows acoustic features due to oscillations in the unlensed $C^{\\tilde{E} \\tilde{E}}_\\ell $ on top of a positive definite contribution.",
"The broad band $BB$ power thus coherently covaries with the lens power [27].",
"These results also agree with Ref.",
"[22], [23]."
],
[
"Parameter forecasts",
"In this section we investigate the impact of lens sample covariances between measurements of CMB power spectra and the lensing potential on cosmological parameters.",
"This impact comes through the additional information that lensing supplies on parameters.",
"We show that to good approximation this information in the lensed CMB power spectra can be considered independently from that of the unlensed CMB power spectra, effectively as direct measurements of the lens power spectrum itself."
],
[
"Cosmological parameters",
"In this work we focus on extensions of the standard 6 parameter $\\Lambda $ CDM cosmological model which we allow to vary two at a time: the sum of masses of the neutrino species $\\sum m_\\nu $ , the dark energy equation of state $w$ , and the spatial curvature $\\Omega _K$ .",
"For the $\\Lambda $ CDM parameters we take $\\Omega _b h^2$ , the physical baryon density; $\\Omega _c h^2$ , the physical cold dark matter density; $n_s$ , the tilt of the scalar power spectrum; $A_s$ , its amplitude; and $\\tau $ , the optical depth to recombination.",
"We choose $\\theta $ , the angular scale of the sound horizon at recombination, as opposed to the Hubble constant $h$ , as the sixth independent parameter given the angular diameter distance degeneracy between $h$ and parameters such as $w$ and $\\Omega _K$ in the unlensed CMB.",
"This choice also improves the numerical stability of forecasts.",
"We also assume that tensor modes are negligible so that there is no unlensed $B$ mode.",
"We call a set of 8 cosmological parameters of the extended $\\Lambda $ CDM family $\\theta _A$ .",
"Values of the cosmological parameters for the fiducial model used in this work are summarized in Table REF .",
"Our assumptions about measurement noise and characterization of lens sample variance in the covariance matrix are summarized in the previous section.",
"In general, we forecast parameter errors given a covariance matrix of a set of observables $D_i$ using the Fisher matrix $F_{AB} = \\sum _{ij} \\frac{\\partial D_i}{\\partial \\theta _A} \\, \\mathrm {Cov}^{-1}_{ij} \\,\\frac{\\partial D_j}{\\partial \\theta _B} .$ The inverse Fisher matrix represents an estimate of the covariance matrix of the parameters $\\mathrm {Cov}_{\\theta _A,\\theta _B} = (F_{AB})^{-1} .$ Prior information is included by adding its Fisher matrix before inverting.",
"Table: Fiducial parameters used in the analysis with extensions to the standardΛ\\Lambda CDM parameters listed last.In Figure REF we compare how the Fisher forecasts on the two extensions of $\\Lambda $ CDM change when we neglect the effect of $\\mathrm {Cov}^{XY,{\\phi \\phi }}$ , the lens sample covariances between the CMB and lens power spectra.",
"In these plots, $\\Lambda $ CDM parameters are marginalized over and the third $\\Lambda $ CDM extension fixed.",
"While for $w$ and $m_\\nu $ , the effect is sizable and amounts to $\\sim 20\\%$ , for $\\Omega _K$ and $m_\\nu $ the effect is much smaller.",
"These differences reflect parameter degeneracies in the lensing observables.",
"Figure: Forecasts for 2 parameter extensions to Λ\\Lambda CDM: ww-∑m ν \\sum m_\\nu (left) and Ω K \\Omega _K-∑m ν \\sum m_\\nu (right).",
"Black curvesshow Δχ 2 =1\\Delta \\chi ^2 = 1 constraints considering the full covariance (solid)and with covariances Cov XY,φφ \\mathrm {Cov}^{XY,{\\phi \\phi }}neglected (dashed); Λ\\Lambda CDM parameters are marginalized over.",
"The blue curvesshow the same constraints with Λ\\Lambda CDM parameters fixed to their fiducial values.We also show in Figure REF the same constraints with the 6 $\\Lambda $ CDM parameters fixed.",
"It is clear that the best constrained direction is limited by parameter degeneracies, especially with $\\Omega _c h^2$ [28].",
"The worst constrained direction is limited instead by the ability of lensing or other constraints to separate the two additional parameters.",
"Conversely, in the $\\Lambda $ CDM model with only the 6 standard parameters varied, parameter errors change by less that 4% when neglecting $\\mathrm {Cov}^{XY,{\\phi \\phi }}$ .",
"This reflects the fact that these parameters are well-constrained even in the absence of lensing.",
"One of the motivations for the rest of the paper will be to understand these behaviors in terms of the additional versus redundant information that lensing observables supply.",
"From the redundant information we will construct sharp consistency tests whose violation would imply systematic errors or violations of fundamental physical assumptions.",
"Note also that constraints on cosmological parameters depend strongly on how well $\\tau $ is constrained whereas those on the lensing power spectrum $C_\\ell ^{\\phi \\phi }$ itself do not [28].",
"For the measurements to cleanly separate $C_\\ell ^{\\phi \\phi }$ information, we primarily need the unlensed CMB in the acoustic regime $C_\\ell ^{\\tilde{X}\\tilde{Y}}$ to be well-characterized.",
"On the other hand, in terms of cosmological parameters, the amplitude of these spectra in this regime is proportional to $A_s e^{-2\\tau }$ .",
"The leverage on cosmological parameters gained through comparing the initial amplitude $A_s$ to the growth-dependent lensing amplitude depends on how well $\\tau $ is measured.",
"In our experimental setup we assumed for simplicity that polarization information will be obtained for the full range of multipoles $\\ell = 2-3000$ , which results in a nearly cosmic variance limited constraint on $\\tau $ of $\\sigma (\\tau )\\approx 0.002$ .",
"This is about five times better than current best constraints from Planck [29] and furthermore assumes a fixed functional form for reionization [30], [31].",
"If the final Planck release does not improve these constraints to substantially below $\\sigma (\\tau )\\sim 0.01$ , this uncertainty will dominate the interpretation of lensing constraints for cosmological parameters [28], [32] since it will be difficult to improve using ground-based instruments.",
"More concretely, removing polarization data from $\\ell < 30$ from our forecasts and replacing it with a prior of $\\sigma _\\tau = 0.01$ , the errors in the worst constrained direction in Figure REF do not significantly change, while those in the best constrained direction degrade by roughly a factor of two.",
"On the other hand, characterizing the information on the power spectrum of the lenses does not depend strongly on the measurements of $\\tau $ and this will be the main focus of the remainder of this work."
],
[
"Lens and unlensed information",
"CMB information on a given cosmological parameter comes both from its effect on the unlensed CMB power spectra $C_\\ell ^{\\tilde{X}\\tilde{Y}}$ with $\\tilde{X}\\tilde{Y} \\in \\tilde{T}\\tilde{T},\\tilde{T}\\tilde{E},\\tilde{E}\\tilde{E}$ and on the lenses $C_\\ell ^{\\phi \\phi }$ .",
"It is conceptually useful to separate these sources of information.",
"Indeed, beyond the cosmological parameters considered in the previous section, the total information in the CMB observables is carried by all two point functions for $\\tilde{T},\\tilde{E},\\tilde{B},\\phi $ , assuming they obey Gaussian statistics; recovery of this complete set of information is the ultimate goal of CMB delensing efforts.",
"By first extracting the lensing information we can also further separate the information from lensed CMB power spectra and reconstruction.",
"The latter can be used to form consistency tests between the two sources of lensing information.",
"Indeed, the Planck satellite found a mild discrepancy between the amount of lensing present in the $TT$ power spectrum and the $TT$ reconstructed lensing potential [33].",
"While these sources of lensing information are still limited by noise rather than by lens sample variance, if such discrepancies persist in future experiments, they may indicate systematic errors in the experiment or the data analysis technique which could obstruct delensing efforts.",
"By checking for consistency at the power spectra level, one can provide proof against such problems before making incorrect cosmological inferences.",
"In principle the full implementation of this approach would be to consider every multipole in $C_\\ell ^{\\tilde{X} \\tilde{Y}}$ and $C_\\ell ^{\\phi \\phi }$ as a parameter in its own right.",
"However, since the high redshift universe is well described by a $\\Lambda $ CDM-like model, we choose to parameterize the unlensed power spectra $C_\\ell ^{\\tilde{X} \\tilde{Y}}$ in terms of a small number of parameters $\\tilde{\\theta }_A$ .",
"These $\\tilde{\\theta }_A$ change the unlensed power spectra in exactly the manner of the $\\Lambda $ CDM parameters $\\theta _A$ , but unlike those, they have no effect on $C_\\ell ^{\\phi \\phi }$ .",
"The lens power spectrum is instead described by a more complete set of parameters $p_\\alpha $ , reflecting the wider range of possibilities during the acceleration epoch.",
"For practical reasons, instead of considering each multipole $C_\\ell ^{\\phi \\phi }$ of the lensing potential as a parameter, we assume that the power spectrum is sufficiently smooth in $\\ell $ that we can approximate it with binned perturbations around the fiducial model.",
"We then define a set of parameters $p_\\alpha $ by $\\ln C^{\\phi \\phi }_\\ell \\approx \\ln C^{\\phi \\phi }_\\ell \\Big |_\\mathrm {fid} + \\sum _{\\alpha = 1}^{N_\\phi } p_\\alpha B_\\alpha ^{\\phi , \\ell } ,$ where $B_\\alpha ^{\\phi , \\ell }$ describes the binning and is defined as $B^{\\phi , \\ell }_\\alpha ={\\left\\lbrace \\begin{array}{ll}1 & \\ell _\\alpha \\le \\ell < \\ell _{\\alpha + 1}\\\\0 & \\mathrm {otherwise}\\end{array}\\right.}",
".$ Expansion in $\\ln C_\\ell ^{{\\phi \\phi }}$ is chosen to assure positivity of the power spectrum.",
"Any cosmological model which predicts a smooth variation of $\\ln C^{\\phi \\phi }_\\ell $ from the fiducial model can be captured in these parameters as $p_\\alpha = \\frac{1}{\\Delta \\ell _\\alpha } \\sum _\\ell B^{\\phi , \\ell }_\\alpha \\delta \\ln C^{\\phi \\phi }_\\ell ,$ where $\\Delta \\ell _\\alpha = \\ell _{\\alpha + 1} - \\ell _\\alpha $ is the width of bin $\\alpha $ .",
"We consider uniform binning with bins of width 5 in this paper; we do not expect binning to have any effect on our conclusions.",
"Changes to the lensing potential are allowed up to $\\ell = 5000$ , given by the $\\ell $ range in which we assume the reconstruction data are measured.",
"The full set of parameters which we will constrain with a Fisher analysis is then $P_\\mathrm {tot} = \\lbrace \\tilde{\\theta }_1, \\tilde{\\theta }_2, \\dots , p_1, p_2, \\dots \\rbrace ,$ where $\\tilde{\\theta }_A$ only affect the unlensed power spectra and $p_\\alpha $ only affect the lensing potential.",
"A given cosmological parameter $\\theta _A$ jointly changes $\\tilde{\\theta }_A$ and $p_\\alpha $ .",
"In principle to fully represent a cosmological parameter in this way we would have to account for the covariance between the lens power spectrum and the unlensed CMB spectra induced by $C_\\ell ^{\\tilde{T}\\phi },C_\\ell ^{\\tilde{E}\\phi }$ – the ISW-lens and reionization-lens correlations respectively.",
"We could in principle add these as parameters to form a complete description.",
"However, these appear only on the largest, severely cosmic variance limited scales which will also be difficult to extract due to foregrounds and systematics.",
"For this reason we completely neglect them from this section onwards by setting $C_\\ell ^{T\\phi }= C_\\ell ^{E\\phi }=0$ everywhere, which means also in the Gaussian covariance.",
"We checked that omitting these contributions to the covariance matrix has only a small effect on parameter constraints in Figure REF ."
],
[
"Independent approximation",
"We can take the lens vs. unlensed information split of the previous section one step further and assume that the data constrain parameters of this split independently so that the $\\tilde{\\theta }_A$ and $p_\\alpha $ errors do not covary.",
"To the extent that this approximation is true, we can consider the lens information as independent.",
"Physically, this approximation involves the assumption that changes in the unlensed CMB and lens power spectra do not produce degenerate effects in the lensed CMB.",
"We can test this approximation by comparing cosmological parameter constraints on $\\theta _A$ as constructed from $\\tilde{\\theta }_A$ and $p_\\alpha $ with the direct forecasts.",
"Under this approximation we first construct independent Fisher matrices in the $p_\\alpha $ space ${F_{\\alpha \\beta }^\\mathrm {lenses} =}&&\\sum _{\\begin{array}{c}\\text{$ \\ell ,\\ell ^{\\prime }$}\\\\\\text{$xy, wz$}\\end{array}}\\frac{\\partial C^{xy}_\\ell }{\\partial p_\\alpha }\\left(\\mathrm {Cov}^{xy,wz}_{\\ell \\ell ^{\\prime }}\\right)^{-1}\\frac{\\partial C^{wz}_{\\ell ^{\\prime }}}{\\partial p_\\beta }$ with the unlensed CMB spectra $C_\\ell ^{\\tilde{X}\\tilde{Y}}$ fixed to their fiducial values and the $\\tilde{\\theta }_A$ space $F_{AB}^\\mathrm {unl} &=& \\sum _{ \\begin{array}{c}\\text{$ \\ell ,\\ell ^{\\prime }$}\\\\\\text{$XY, WZ$}\\end{array}}\\frac{\\partial C^{XY}_\\ell }{\\partial \\tilde{\\theta }_A}\\left(\\mathrm {Cov}^{XY,WZ}_{\\ell \\ell ^{\\prime }}\\right)^{-1}\\frac{\\partial C^{WZ}_{\\ell ^{\\prime }}}{\\partial \\tilde{\\theta }_B}$ with $C_\\ell ^{\\phi \\phi }$ fixed to their fiducial values.",
"Note that $\\phi \\phi $ has no dependence on $\\tilde{\\theta }_A$ and so those spectra do not enter into the sum.",
"We can then obtain the total Fisher matrix of the cosmological parameters by the Jacobian transform $F_{AB} = F_{AB}^\\mathrm {unl} + \\sum _{\\alpha ,\\beta } \\frac{\\partial p_\\alpha }{\\partial \\theta _A}F_{\\alpha \\beta }^\\mathrm {lenses}\\frac{\\partial p_\\beta }{\\partial \\theta _B} .$ In Figure REF we compare constraints obtained from the independent model (REF ) with constraints from the full Fisher analysis.",
"We see that the model indeed works very well and the assumption about independent measurement of the unlensed power spectra and the lensing potential is justified in these examples.",
"As we discuss in §REF , spaces that involve $\\Omega _K$ provide an especially stringent test of the independent approximation.",
"Because to calculate $F_{AB}^\\mathrm {unl}, F_{\\alpha \\beta }^\\mathrm {lenses}$ one needs to know the full covariance matrix for the lensed observables, this split does not represent any practical simplification for calculation of the Fisher matrix unlike the related “additive\" approximation in Ref.",
"[28] that utilizes the unlensed spectra as observables.",
"Conversely, we do not incur errors from conflating unlensed power spectra with direct observables.",
"In Appendix we introduce a new forecasting approximation which combines the virtues of these two approaches: simplicity and accuracy.",
"Figure: Accuracy of the independent lensing information model of () (red dashed)compared with the full Fisher forecast for the cases from Fig.",
"(black solid)."
],
[
"Redundancy and Consistency",
"Given the technique for isolating information about the lens power spectrum introduced in the previous section, we can now assess the level of redundancy and consistency between the information coming from lensed CMB power spectra and lensing power spectrum.",
"This study both helps explain constraints on cosmological parameters and enables the construction of sharp consistency tests between these two aspects of lensing in the data that are nearly immune to sample variance."
],
[
"Consistency of covarying modes",
"We can use the Karhunen-Loève (KL) transformThe KL transform is often used in cosmology to define signal-to-noise eigenmodes for optimal data compression [34], [35], [36]; our use follows [28] in comparing information in two different covariance matrices.",
"to extract the modes or linear combinations of the lens parameters $p_\\alpha $ that are most impacted by the $\\mathrm {Cov}^{XY,{\\phi \\phi }}$ covariance between the measurements of the lensed CMB power spectra $C_\\ell ^{XY}$ and the lens power spectra $C_\\ell ^{\\phi \\phi }$ .",
"These modes carry redundant information between $XY$ and $\\phi \\phi $ that can be used as a consistency check on the data and analysis techniques.",
"To assess the impact of the $XY,{\\phi \\phi }$ covariance, we consider two versions of the inverse Fisher matrix for $p_\\alpha $ , $\\mathrm {Cov}_{\\alpha \\beta } = [F_{\\alpha \\beta }^{\\rm lenses}]^{-1},$ from Eq.",
"(REF ), and $\\mathrm {Cov}_{\\alpha \\beta }^- =[F_{\\alpha \\beta }^{\\rm lenses}]^{-1}\\Big |_{\\mathrm {Cov}^{XY,{\\phi \\phi }}_{\\ell \\ell ^{\\prime }}=0},$ the same construction but with the $XY,\\phi \\phi $ covariance artificially set to zero.",
"We can then perform a KL transformation by finding all solutions to the generalized eigenvalue problem $\\sum _\\beta \\mathrm {Cov}_{\\alpha \\beta } v^{(k)}_\\beta =\\sum _\\beta \\lambda ^{(k)} \\mathrm {Cov}_{\\alpha \\beta }^{-} v^{(k)}_\\beta .$ Here $v^{(k)}_\\beta $ and $\\lambda ^{(k)}$ are the KL eigenvectors and eigenvalues.",
"The KL transform of the measurements $\\Psi ^{(k)} = \\sum _\\alpha v^{(k)}_\\alpha p_\\alpha $ provides a representation that is uncorrelated, or statistically orthogonal, with respect to both covariance matrices since solutions to (REF ) are simultaneously orthogonal with respect to the metrics defined by the covariance matrices, $\\mathrm {Cov}_{\\Psi ^{(k)}\\Psi ^{(l)}}= \\sum _{\\alpha \\beta } v^{(l)}_\\alpha \\mathrm {Cov}_{\\alpha \\beta } v^{(k)}_\\beta &=& \\lambda ^{(k)} \\delta _{kl} , \\nonumber \\\\\\mathrm {Cov}_{\\Psi ^{(k)}\\Psi ^{(l)}}^{-} = \\sum _{\\alpha \\beta } v^{(l)}_\\alpha \\mathrm {Cov}_{\\alpha \\beta }^{-}v^{(k)}_\\beta &=& \\delta _{kl} .$ We order $\\lambda ^{(k)}$ to be decreasing with $k$ and hence in the ratio of the variances between the two, i.e.",
"the degradation in the constraints due to $\\mathrm {Cov}^{XY,{\\phi \\phi }}_{\\ell \\ell ^{\\prime }}$ .",
"The eigenvectors are not necessarily mutually orthonormal in the ordinary Euclidean sense, $\\sum _{\\alpha } v^{(l)}_\\alpha v^{(k)}_\\alpha \\ne \\delta _{kl},$ as they would be in an ordinary eigenvector or principal component representation (see §REF ).",
"Consequently, the forward and inverse KL transforms are distinct: $p_\\alpha = \\sum _k w^{(k)}_\\alpha \\Psi ^{(k)} ,$ where $w^{(k)}_\\alpha $ is the matrix inverse of $v^{(k)}_\\alpha $ rather than its transpose.",
"As a function of the $\\alpha $ index, $v^{(k)}_\\alpha $ represents how strongly individual $p_\\alpha $ contribute to the $k$ th KL mode whereas $w^{(k)}_\\alpha $ represents how the $k$ th KL mode is distributed onto the original modes.",
"They can have very different shapes in $\\alpha $ .",
"We always use the forward KL transform and $v^{(k)}_\\alpha $ in the following discussion to avoid confusion.",
"We find two strongly degraded modes with $\\lambda ^{(1)} &=& 1.86, \\\\\\lambda ^{(2)} &=& 1.39 \\nonumber .$ These modes would be better constrained if there were no $XY,{\\phi \\phi }$ covariances, which agrees with the intuitive expectation that neglecting mutual covariances would lead to double counting of the lensing information.",
"The corresponding eigenvectors $v^{(1,2)}_\\alpha $ are plotted in Figure REF .",
"All other modes are only mildly affected and have eigenvalues between 0.93 and 1.08.",
"Figure: KL components of the lensing potential most affected by the covariances Cov XY,φφ \\mathrm {Cov}^{XY,{\\phi \\phi }} of CMB fields with the reconstructed lensing potential.By neglecting these covariances, constraints on the corresponding amplitude Ψ (k) \\Psi ^{(k)}would be overly optimistic due to double counting of lensing information.We see that measurements of the amplitude of the first mode $\\Psi ^{(1)}$ are degraded by almost a factor of two.",
"This means that constraint on this mode obtained from the $XY$ lensed power spectra alone is comparable to a constraint from the reconstructed lensing potential alone but that these two different measurements are highly correlated.",
"This occurs because both these measurements have their variances dominated by the sample variance of the lenses.",
"This sample variance is common to both measurements, which explains why the two variances are comparable and strongly correlated.",
"Table REF summarizes how well we can constrain $\\Psi ^{(1)}$ under various assumptions and provides quantitative justification of these claims.",
"The first two lines summarize the KL results – neglecting $\\mathrm {Cov}^{XY,{\\phi \\phi }}$ leads to a double counting of the lensing information and overly tight constraints in the full dataset.",
"Instead, we can constrain this mode separately from ${\\phi \\phi }$ and $XY$ data with variances that are both comparable to those of the full dataset.",
"The $XY$ result is not a trivial consequence of the KL results since the KL modes are not specifically constructed to be statistically orthogonal with $XY$ measurements alone.",
"Because the $XY$ power spectra provide only integrated constraints on $C_\\ell ^{\\phi \\phi }$ , we impose a mild theoretical prior of $\\sigma _{p_\\alpha } = 1$ to forbid numerical problems and degeneracies induced by unphysically large features in $C_\\ell ^{\\phi \\phi }$ (see also §REF ).",
"The minimum variance unbiased linear estimators of $\\Psi ^{(1)}$ from the separate $\\phi \\phi $ and $XY$ datasets have a correlation coefficient of 0.77, in agreement with values in Table REF .",
"Note that even when considering $XY$ separately, we include all of the internal covariances induced by lens sample variance.",
"Without the non-Gaussian covariances $\\mathcal {N}$ , $\\sigma _{\\Psi ^{(1)}}^2$ decreases significantly and is unphysically smaller than the lens sample variance limit by more than a factor of 3.",
"Finally we show that removing all of the non-Gaussian covariances in the full dataset leads to an even more extreme violation of the lens sample variance limit.",
"Table: Variance of KL consistency mode Ψ (1) \\Psi ^{(1)} obtained from various combinations of lensed CMB spectra XYXY andlens power spectra φφ\\phi \\phi measurements and assumptions about their variances and covariance.Because $\\Psi ^{(1)}$ is constrained by two independent but strongly correlated measurements, these measurements in principle provide an excellent systematic check on the experimental data that is nearly immune to sample variance and cosmological parameter uncertainties.",
"This check could be very valuable in future experiments, which are likely to be foreground and systematics limited: comparing $\\Psi ^{(1)}$ measured from power spectra and reconstruction separately could serve as a simple check on data quality and reconstruction algorithms before performing the delensing operation.",
"Identical conclusions are to a lesser degree valid also for $\\Psi ^{(2)}$ , which could also serve as a weaker consistency check, but valuable in its own right for reasons we discuss below.",
"Next we test the robustness of these results to our assumptions.",
"The eigenvectors $v^{(k)}$ and corresponding eigenvalues do not change appreciably if we discard in temperature and polarization information for $\\ell < 30$ , discard reconstruction information for $\\ell > 3000$ , or include polarization information out to $\\ell < 5000$ .",
"Unlike cosmological parameter inferences that involve breaking parameter degeneracies involving the standard $\\Lambda $ CDM parameters, $A_s, \\tau , \\Omega _c h^2$ , this consistency test involves just the lensing information.",
"In principle, the development of more sophisticated lens reconstruction algorithms beyond the damping tail may in the future allow additional consistency tests with $XY$ power spectra at $\\ell >3000$ .",
"However, this information does not significantly impact the $\\Psi ^{(1)}$ consistency test since it involves lens power on comparably high $\\ell $ scales.",
"The impact of neglecting $C_\\ell ^{T\\phi }, C_\\ell ^{E\\phi }$ should also not be significant, because unlike $v^{(1,2)}$ they are only significant at the lowest multipoles.",
"The most important assumption in this construction is that we can independently consider the information about the unlensed CMB and the lens power spectra.",
"While this is a good assumption in the extended $\\Lambda $ CDM parameter space for the full data set as demonstrated in Fig.",
"REF , it is less true when considering the lensed CMB $XY$ spectra alone if spatial curvature is allowed to vary.",
"Increasing $\\Omega _K$ impacts the unlensed CMB through $\\tilde{\\Omega }_K$ in a manner similar to the smoothing of the acoustic peaks by lensing [28].",
"Moreover, its impact on lensing through $p_\\alpha (\\Omega _K)$ is to decrease the amplitude of power (see Fig.",
"REF below), and so the overall sensitivity to curvature is degraded from what is assumed in the independent approximation.",
"Furthermore, the total impact of curvature on the lensed power spectrum becomes nearly degenerate with effects of the neutrino mass [28].",
"On the other hand, $BB$ partially breaks the degeneracy as it is not generated by curvature.",
"To investigate how severe these degeneracies are in the $XY$ dataset, we compare forecasted errors on $\\Psi ^{(1,2)}$ with fixed vs. marginalized $\\tilde{\\theta }_A$ in Table REF .",
"As before, we assume a mild theoretical prior $\\sigma _{p_\\alpha } = 1$ .",
"When $\\tilde{\\Omega }_K$ is held fixed, the variances of both $\\Psi ^{(1)}$ and $\\Psi ^{(2)}$ are negligibly increased by marginalizing the remaining 8 extended $\\Lambda $ CDM parameters.",
"When $\\tilde{\\Omega }_K$ is also marginalized the variance of $\\Psi ^{(1)}$ changes only by $\\sim 10\\%$ but that of $\\Psi ^{(2)}$ is close to doubled.",
"This mirrors the fact that changing $\\Psi ^{(1)}$ changes $BB$ significantly more – relative to the rest of the observables – than $\\Psi ^{(2)}$ does and cannot be mimicked by curvature in the unlensed spectra.",
"We conclude that $\\Psi ^{(1)}$ provides a robust consistency test for lensing in the full $\\Lambda $ CDM+$w$ +$\\Omega _K$ +$\\sum m_\\nu $ context whereas inconsistencies in $\\Psi ^{(2)}$ between $XY$ and $\\phi \\phi $ measurements may indicate a finite spatial curvature.",
"Violations of consistency in $\\Psi ^{(1)}$ would indicate systematics and foregrounds in the measurement or new physics at recombination that mimics the effect of lensing.",
"Either of these possibilities would lead to incorrect cosmological inferences and complicate delensing of the CMB if not discovered beforehand.",
"This relationship between lensing and curvature effects in the unlensed spectrum also leads to the small difference between the full Fisher forecast and the independent lensing information model in Figure REF which we discuss further in Appendix .",
"Table: Variance of KL consistency modes Ψ (1,2) \\Psi ^{(1,2)} obtained from XYXY lensed CMB powerspectra alone with and without unlensed CMB parameters θ ˜ A \\tilde{\\theta }_A marginalized.With mild theoretical prior σ p α =1\\sigma _{p_\\alpha } = 1."
],
[
"Principal component implementation",
"The consistency check discussed in §REF involves measuring the KL consistency parameter $\\Psi ^{(1)}$ from the CMB $XY$ power spectra alone.",
"There are practical obstacles to implementing this measurement given the many ill-constrained modes that compose the full lensing power spectrum $C_\\ell ^{\\phi \\phi }$ through $p_\\alpha $ .",
"Furthermore, with just $XY$ measurements alone, curvature $\\Omega _K$ mildly violates the assumption that the unlensed CMB parameters can be independently extracted from the lensed CMB as discussed in the previous section.",
"A full assessment will require going beyond the Fisher approximation with validation on numerical simulations which we postpone to a future work.",
"In this section, we take the first steps toward this goal by re-examining the lensing principal component decomposition introduced in Ref. [28].",
"A small set of these parameters completely characterizes the lensing information in the $XY$ data and can be measured jointly with those controlling the unlensed parameters $\\tilde{\\theta }_A$ , with or without curvature.",
"The forecasted covariance matrix of the $p_\\alpha $ lensing parameters measured by $XY$ power spectra is given by the inverse Fisher matrix (REF ), omitting ${\\phi \\phi }$ in the sum.",
"The orthonormal eigenvectors $K^{(i)}_\\alpha $ of this matrix represent an alternate basis for the measurements $\\Theta ^{(i)} = \\sum K^{(i)}_\\alpha p_\\alpha ,$ that yield uncorrelated parameters, rank ordered by their variance, in principle.",
"By keeping only the eigenvectors that are predicted to have low variance, we can measure the relevant information with a much smaller set of principal components (PCs).",
"Note that this differs from the KL basis in that it rank orders modes by total variance from $XY$ rather than by whether the joint measurements are noise or lens sample variance dominated.",
"The efficiency of the PC approach depends on the number of components needed to completely characterize the relevant information.",
"In our case, we find eigenvalues $10^3 \\lambda &=& 1.0, 4.0, 12, 19, 93, \\dots ,$ which indeed shows that relative importance of the components decreases rapidly and hits the $\\sigma _{p_\\alpha } = 1$ prior shortly thereafter.",
"The five most important components are shown in Figure REF .",
"The low order modes peak where the lenses have their largest impact on $XY$ and the higher modes are increasingly oscillatory, because they have to be orthogonal to the more important eigenmodes.",
"Figure: Five principal components K ℓ (i) K^{(i)}_\\ell of the lensing potentialbest measured by the lensed power spectra.It is sufficient to keep only several principal components to characterize the impact of cosmological parameters or the KL consistency modes completely.",
"Specifically, the mode $\\Psi ^{(1)}$ can be faithfully constructed from $XY$ measurements of the 5 lowest order PC components with the dominant contributions from the first two.",
"We have explicitly checked that truncating the remaining components has no significant effect on the error analysis, for example as displayed in Table REF .",
"Because of the truncation, the $\\sigma _{p_\\alpha } = 1$ prior plays little role and may be omitted.",
"This construction therefore provides a practical means of measuring $\\Psi ^{(1)}$ in the presence of the many unconstrained but unphysical modes.",
"We can also measure these $\\Theta ^{(i)}$ modes with lensing reconstruction and check consistency between $XY$ and $\\phi \\phi $ directly in PC space.",
"The results are summarized in Table REF .",
"Although the first mode is equally well constrained by $XY$ and $\\phi \\phi $ measurements, it does not produce as sharp a consistency test as $\\Psi ^{(1)}$ .",
"The reason is that lens sample variance only contributes less than $\\sim 2/3$ of the variance of either measurement and their results can therefore differ due to the remaining noise variance.",
"Higher modes are even less sample variance limited in $XY$ .",
"This mainly reflects the higher $\\ell $ weight in the PC components compared with $\\Psi ^{(1)}$ .",
"We can interpret $\\Psi ^{(1)}$ as essentially the linear combination of $\\Theta ^{(1)}$ and $\\Theta ^{(2)}$ that best isolates the low $\\ell $ , lens sample variance limited information.",
"Finally, while in this work we mainly focused on lensing information which is redundant, these results imply that the lensed $XY$ CMB power spectra actually improve constraints on lensing potential above roughly $\\ell \\sim 500$ (see Tab.",
"REF ).",
"In cosmological parameter errors this improvement is hidden because of the degeneracies with $\\Lambda $ CDM parameters as we discuss next.",
"Table: Variance of Θ (i) \\Theta ^{(i)} obtained from various datasets under variousassumptions about covariances of the data and noise."
],
[
"Parameter constraints revisited",
"The KL analysis exposes the fact that there is one mode which is nearly equally well measured by CMB power spectra $XY$ and lensing reconstruction $\\phi \\phi $ that reflects a large portion of the nearly lens sample dominated information on $C_\\ell ^{\\phi \\phi }$ at low $\\ell $ .",
"Our PC analysis highlights the fact that the decrease in lens sample variance at higher $\\ell $ means that despite being the highest in signal to noise, this consistency mode carries only a portion of the total information from lensing on the overall amplitude of the lensing spectrum.",
"Furthermore, as shown in Fig.",
"REF , the constraints from the overall amplitude of the lensing power spectrum on the $\\Lambda $ CDM extensions is limited by degeneracies since the $\\Lambda $ CDM parameters $A_s$ (implicitly $\\tau $ ) and $\\Omega _c h^2$ also affect $p_\\alpha $ , further reducing the impact of lens sample covariance.",
"It is when the low $\\ell $ lensing information strongly breaks a parameter degeneracy that the impact of the $\\mathrm {Cov}^{XY,{\\phi \\phi }}$ covariance is seen.",
"In Figure REF we show how the parameter constraints would change if we neglect the information carried by the $\\Psi ^{(1)}$ consistency mode.",
"In both the $w-\\sum m_\\nu $ and $\\Omega _K -\\sum m_\\nu $ cases, the impact is mainly in the degenerate direction but is only dramatic in the former.",
"The impact of $\\mathrm {Cov}^{XY,{\\phi \\phi }}$ shown in Fig.",
"REF can be understood from this result since the information on $\\Psi ^{(1)}$ is essentially double counted if this covariance is neglected.",
"Figure: Impact of eliminating the lens information associated with the KL consistencymode Ψ (1) \\Psi ^{(1)} (dashed line).",
"Solid lines represent the full Fisherforecast from Fig. .",
"Theconsistency mode carries a substantial amount of the total information especially incases where low ℓ\\ell lens information breaks parameter degeneracies.We can further understand the different parameter behaviors by examining the impact of parameters on $p_\\alpha $ or $\\ln C_\\ell ^{\\phi \\phi }$ (see Fig.",
"REF ).",
"Although the measurements determine the amplitude of the $C_\\ell ^{\\phi \\phi }$ well at $\\ell \\gtrsim 500$ , they are unable to separate out the contributions from the various cosmological parameters.",
"In particular, linear combinations of $\\ln A_s$ and $\\Omega _c h^2$ can mimic the impact of the extended $\\Lambda $ CDM parameters [28].",
"Therefore, while the best constrained direction in the 2-dimensional extended spaces correspond to combinations of the parameters that coherently change $C_\\ell ^{\\phi \\phi }$ at $\\ell \\gtrsim 500$ , the constraint itself is limited by how well $\\ln A_s$ and $\\Omega _c h^2$ are measured not by how well $C_\\ell ^{\\phi \\phi }$ is measured (see Fig.",
"REF ).",
"The degenerate or worst constrained direction corresponds to when the parameter variations cancel in their effect.",
"At $\\ell \\lesssim 500$ the degeneracy between $w$ and $\\Omega _K$ or $\\sum m_\\nu $ observed at high $\\ell $ starts to break, which allows us to meaningfully constrain also the perpendicular direction in the parameter space.",
"For $\\Omega _K$ and $\\sum m_\\nu $ this degeneracy breaking is noticeably weaker, especially at $\\ell \\gtrsim 50$ .",
"Given the large sample variance associated with the lowest multipoles, the limiting source of information in the degenerate direction in the $\\Omega _K-\\sum m_\\nu $ plane comes from the unlensed CMB rather than the lensing information.",
"Hence, the effect of lens sample covariance is smaller in this case.",
"Finally for these issues that relate to parameter degeneracies, it is important to remember that external information from measurements beyond the CMB, for example from baryon acoustic oscillations, can break these degeneracies and allow more of the information on $C_\\ell ^{\\phi \\phi }$ that our analysis uncovers to be used for parameter constraints.",
"Figure: Derivatives of lnC ℓ φφ \\ln C_\\ell ^{\\phi \\phi } with respect to cosmological parametersw,∑m ν ,Ω K ,Ω c h 2 ,lnA s w, \\sum m_\\nu , \\Omega _K, \\Omega _c h^2, \\ln A_s normalized at ℓ=1000\\ell =1000 to highlight degeneracies.These derivatives are taken at fixed acoustic scale θ\\theta ."
],
[
"Discussion",
"The lensing observables from the two and higher point statistics of the temperature and polarization fields are intrinsically correlated because they are lensed by the same realization of structure between last scattering and the observer.",
"While currently these observables are noise variance limited, in the future they are expected to be lens sample variance limited.",
"When jointly analyzing these observables, it will then be important to take these correlations into account both to prevent double counting of information and because they provide important consistency checks that are immune to sample variance, the chance fluctuations in the lenses.",
"In this work, we study a simple analytical model that consistently incorporates the lens sample covariance between CMB power spectra and lens reconstruction from higher point information.",
"This covariance model can be employed for cosmological parameter estimation to build the lens sample variance piece of the likelihood function as well as Fisher forecasts for future experiments.",
"While there is only a small effect on parameter errors of the covariances between the reconstructed lensing potential and the lensed power spectra in the $\\Lambda $ CDM and even the extended $\\Lambda $ CDM context, parameter errors, degeneracies and non-lensing information mask the full impact of the covariance.",
"To better expose this impact, we work in an approximation where information in the unlensed CMB power spectrum and the lensing potential $C_\\ell ^{\\phi \\phi }$ are considered independently.",
"Using a Karhunen-Loève analysis, we identify one mode in $C_\\ell ^{\\phi \\phi }$ that in the future should be nearly lens sample variance limited using either lensed power spectra or lensing reconstruction and hence nearly perfectly covaries between the two.",
"If this covariance is not taken into account then information on this mode will be double counted.",
"This mode peaks at somewhat lower multipole than the bulk of the information on the lensing power spectrum due to the larger signal versus noise variance there.",
"This mode can be measured separately through lens reconstruction and lensed CMB power spectra with the help of a principal component decomposition of the latter.",
"Notably, inconsistency between the measurements cannot be explained by chance lens realizations or parameter variations, and is immune to ambiguities due to $\\tau $ , the optical depth to reionization.",
"Instead, violations could indicate systematics, lens reconstruction errors, foregrounds or new physics at recombination which changes the unlensed power spectra, including the $BB$ power spectrum, in ways degenerate with lensing.",
"They would then lead to incorrect cosmological inferences and delensing if not taken into account.",
"The identification of this mode also explains the impact of covariances between the reconstructed lensing potential and the lensed power spectra on parameter constraints.",
"There is only a small effect within the $\\Lambda $ CDM model as these parameters are well constrained even without lensing.",
"The impact of covariance is mainly seen when measurements of the low $\\ell $ lensing power spectrum are useful in breaking parameter degeneracies in interpreting the measurements at higher $\\ell $ .",
"Specifically, for $w$ and $\\sum m_\\nu $ , the consistency mode has a strong impact on parameters and hence its double counting would lead to constraints overly optimistic by $\\sim $ 20%.",
"There is a second combination of $C_\\ell ^{\\phi \\phi }$ with similar properties, however there the correlation is weaker.",
"Despite being weaker, statistically significant violations of consistency in this mode are interesting since they may indicate nonzero spatial curvature as it has similar effects on the unlensed CMB as lensing.",
"While this work was in preparation, a similar analytic approach to modelling covariances was compared against numerical simulations [23].",
"That model was found to work well after realization-dependent noise subtraction.",
"As can be seen from their Figs.",
"3 and 4, these subtractions affect mostly correlations with lensing power spectra above $\\ell \\sim 1000$ and would be hidden by reconstruction noise in our approach.",
"They also show that the other trispectrum terms to the covariance, which we neglect, are subdominant.",
"Potentially more troublesome is their finding that there are some differences between the analytical model and simulations, especially in $\\mathrm {Cov}^{BB,{\\phi \\phi }}$ at low $\\ell _{BB}$ which they claim appears to be statistically significant [37].",
"If confirmed, then our analysis implicitly assumes that such additional effects can be modeled without breaking our consistency relations – in essence that both lensed CMB and reconstruction can measure this consistency mode to nearly the lens sample variance limit.",
"More generally, this consistency mode can be used to search for unaccounted for systematics in lens reconstruction.",
"We intend to study these issues and quantify their impact in a future work.",
"We thank Chen He Heinrich, Alessandro Manzotti and Julien Peloton for useful discussions.",
"This work was supported by U.S. Dept.",
"of Energy contract DE-FG02-13ER41958 and in part by the Kavli Institute for Cosmological Physics at the University of Chicago through grant NSF PHY-1125897 and an endowment from the Kavli Foundation and its founder Fred Kavli.",
"WH was additionally supported by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142 and NASA ATP NNX15AK22G and thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, where part of this work was completed.",
"ABL thanks CNES for financial support through its postdoctoral programme and KICP, where this work was initiated, for its visitor program.",
"We acknowledge use of the CAMB software package.",
"This work was completed in part with resources provided by the University of Chicago Research Computing Center."
],
[
"Simple forecast methods",
"In this appendix we compare various Fisher matrix approaches of how to estimate parameter constraints, including the standard calculation which uses the full analytical covariance matrix (REF ).",
"We also introduce a new forecasting approach, which we call the Simple Lensing Approximation (SLA), that is very accurate in predicting parameter constraints from CMB data only and does not require calculation of the full covariance matrix.",
"A frequently used approach to avoid double counting of the lensing information is to derive parameter constraints from the unlensed $\\tilde{X} \\tilde{Y}$ CMB power spectra and the reconstruction of the lensing potential assuming Gaussian statistics in each.",
"These constraints are equivalent to assuming that complete delensing in the CMB maps is possible, that it does not alter their noise properties and that no extra information on the lensing beyond reconstruction can be recovered from the $XY$ power spectra.",
"In the main text we have seen that while the lensing information in $XY$ is substantial, it is largely redundant with reconstruction or limited by parameter degeneracies.",
"For this reason, this approximation works fairly well in the $w-\\sum m_\\nu $ plane.",
"However, as seen from Figure REF , this approximation noticeably underestimates the errors on curvature since its effect on the unlensed spectrum and lensing work in opposite directions in the smoothing of the peaks, degrading the overall curvature sensitivity in the lensed CMB power spectra.",
"This problem is largely fixed by our independent lensing information model of Eq.",
"(REF ) which is shown in Fig.",
"REF .",
"In this model, the information from the unlensed power spectra is still considered as separate from that of the lens spectrum but the observable is the lensed $XY$ spectrum and lens sample covariance is taken into account in the covariances of observables.",
"The drawback is that to make forecasts, the cumbersome lens sample covariance matrix must be carried through all pieces of the construction.",
"We can combine the virtues of these two approaches in a new simple forecasting method, dubbed SLA, if all that is desired are parameter forecasts in the extended $\\Lambda $ CDM space from the CMB alone.",
"Namely, we can avoid double counting of the lens information by dropping the lens information in the $XY$ power spectra and along with it the non-Gaussian covariances induced by lensing.",
"Importantly, we still use the lensed $XY$ power spectra and not the unlensed $\\tilde{X}\\tilde{Y}$ power spectra as the observables.",
"Specifically, $F^\\mathrm {SLA}_{AB} = F_{AB}^{\\mathrm {unl,SLA}} + \\sum _{\\alpha ,\\beta } \\frac{\\partial p_\\alpha }{\\partial \\theta _A}F_{\\alpha \\beta }^{\\mathrm {lenses,SLA}}\\frac{\\partial p_\\beta }{\\partial \\theta _B}$ where we continue to assume Gaussian lens reconstruction noise as in the main text, ${F_{\\alpha \\beta }^{\\mathrm {lenses,SLA}} =}&&\\sum _{ \\ell }\\frac{\\partial C^{{\\phi \\phi }}_\\ell }{\\partial p_\\alpha }\\left(\\mathcal {G}^{{\\phi \\phi },{\\phi \\phi }}_{\\ell \\ell }\\right)^{-1}\\frac{\\partial C^{{\\phi \\phi }}_{\\ell }}{\\partial p_\\beta }.$ and omit any lensing information in the $XY$ power spectra.",
"The conceptual difference from Eq.",
"(REF ) is that when evaluating the unlensed Fisher matrix we assume Gaussian statistics, $F_{AB}^{\\mathrm {unl,SLA}} &=& \\sum _{ \\begin{array}{c}\\text{$ \\ell $}\\\\\\text{$XY, WZ$}\\end{array}}\\frac{\\partial C^{XY}_\\ell }{\\partial \\tilde{\\theta }_A}\\left(\\mathcal {G}^{XY,WZ}_{\\ell \\ell }\\right)^{-1}\\frac{\\partial C^{WZ}_{\\ell }}{\\partial \\tilde{\\theta }_B} \\nonumber .",
"\\\\$ As before, derivatives in (REF ) should be evaluated at fixed unlensed power spectra while derivatives in (REF ) should be evaluated at fixed lensing potential.",
"We show in Fig.",
"REF that this approximation provides simple but highly accurate constraints even when curvature is involved.",
"In fact it performs slightly better than the independent approximation of the main text in that it allows lensing to recover information that would otherwise be lost to the non-Gaussian correlations between multipole moments in the $XY$ power spectra.",
"On the other hand, this simple forecast scheme ignores the fact that the $XY$ power spectra provide strong constraints on the lensing power spectra at low multipole that serve as consistency checks against reconstruction measurements and provide additional constraints at high lens multipole when parameter degeneracies are broken by external measurements.",
"This is especially true beyond the $\\ell <3000$ limit for polarization measurements tested here but there the astrophysical uncertainties in modeling lenses in the nonlinear regime also limit cosmological parameter information.",
"Figure: New SLA forecasting approximation (solid blue) compared with the full forecast (black solid)and the frequently usedapproach of using the unlensed spectra and φ\\phi lensing power spectrum (green dashed).",
"The unlensed approachis simple but errs in assuming the unlensed spectra are directly observable.",
"The SLA approach employs thelensed CMB spectra but omits both their lensing information and covariances making it both accurate and simple."
]
] | 1612.05637 |
[
[
"On stochastic calculus with respect to q-Brownian motion"
],
[
"Abstract Following the approach and the terminology introduced in [A. Deya and R. Schott, On the rough paths approach to non-commutative stochastic calculus, J. Funct.",
"Anal., 2013], we construct a product L{\\'e}vy area above the $q$-Brownian motion (for $q\\in [0,1)$) and use this object to study differential equations driven by the process.We also provide a detailled comparison between the resulting \"rough\" integral and the stochastic \"It{\\^o}\" integral exhibited by Donati-Martin in [C. Donati-Martin, Stochastic integration with respect to $q$ Brownian motion, Probab.",
"Theory Related Fields, 2003]."
],
[
"Introduction: the $q$ -Brownian motion",
"The $q$ -Gaussian processes (for $q\\in [0,1)$ ) stand for one of the most standard families of non-commutative random variables in the literature.",
"Their consideration can be traced back to a paper by Frisch and Bourret in the early 1970s [8]: the dynamics is therein suggested as a model to quantify some possible non-commutativity phenomenon between the creation and annihilator operators on the Fock space, the limit case $q= 1$ morally corresponding to the classical probability framework.",
"The mathematical construction and basic stochastic properties of the $q$ -Gaussian processes were then investigated in the 1990s, in a series of pathbreaking papers by Bożejko, Kümmerer and Speicher [2], [3], [4].",
"For the sake of clarity, let us briefly recall the framework of this analysis and introduce a few notations that will be used in the sequel (we refer the reader to the comprehensive survey [11] for more details on the subsequent definitions and assertions).",
"First, recall that the processes under consideration consist of paths with values in a non-commutative probability space, that is a von Neumann algebra $\\mathcal {A}$ equipped with a weakly continuous, positive and faithful trace $\\varphi $ .",
"The sole existence of such a trace $\\varphi $ on $\\mathcal {A}$ (to be compared with the expectation in this setting) is known to give the algebra a specific structure, with $L^p$ -norms $\\Vert X\\Vert _{L^p(\\varphi )}:= \\varphi ( |X|^p)^{1/p} \\quad \\ (\\, |X|:=\\sqrt{XX^\\ast }\\, )$ closely related to the operator norm $\\Vert .\\Vert $ : $\\Vert X\\Vert _{L^p(\\varphi )} \\le \\Vert X\\Vert \\quad , \\quad \\Vert X\\Vert =\\lim _{p\\rightarrow \\infty } \\Vert X\\Vert _{L^p(\\varphi )} \\ , \\ \\text{for all} \\ X\\in \\mathcal {A}\\ .$ Now recall that non-commutative probability theory is built upon the following fundamental spectral result: any element $X$ in the subset $\\mathcal {A}_\\ast $ of self-adjoint operators in $\\mathcal {A}$ can be associated with a law that shares the same moments.",
"To be more specific, there exists a unique compactly supported probability measure $\\mu $ on $\\mathbb {R}$ such that for any real polynomial $P$ , $\\int _{\\mathbb {R}} P(x) \\mathrm {d}\\mu (x) = \\varphi (P(X))\\ .$ Based on this property, elements in $\\mathcal {A}_\\ast $ are usually referred to as (non-commutative) random variables, and in the same vein, the law of a given family $\\lbrace X^{(i)}\\rbrace _{i\\in I}$ of random variables in $(\\mathcal {A},\\varphi )$ is defined as the set of all of its joint moments $\\varphi \\big (X^{(i_1)} \\cdots X^{(i_r)}\\big ) \\ , \\quad i_1,\\ldots ,i_r \\in I \\ , \\ r\\in \\mathbb {N}\\ .$ With this stochastic approach in mind and using the terminology of [2], the definition of a $q$ -Gaussian family can be introduced along the following combinatorial description: Definition 1.1 1.",
"Let $r$ be an even integer.",
"A pairing of $\\lbrace 1,\\ldots ,r\\rbrace $ is any partition of $\\lbrace 1,\\ldots ,r\\rbrace $ into $r/2$ disjoint subsets, each of cardinality 2.",
"We denote by $\\mathcal {P}_2(\\lbrace 1,\\ldots ,r\\rbrace )$ or $\\mathcal {P}_2(r)$ the set of all pairings of $\\lbrace 1,\\ldots ,r\\rbrace $ .",
"2.",
"When $\\pi \\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,r\\rbrace )$ , a crossing in $\\pi $ is any set of the form $\\lbrace \\lbrace x_1,y_1\\rbrace ,\\lbrace x_2,y_2\\rbrace \\rbrace $ with $\\lbrace x_i,y_i\\rbrace \\in \\pi $ and $x_1 < x_2 <y_1 <y_2$ .",
"The number of such crossings is denoted by $\\mathrm {Cr}(\\pi )$ .",
"Definition 1.2 For any fixed $q\\in [0,1)$ , we call a $q$ -Gaussian family in a non-commutative probability space $(\\mathcal {A},\\varphi )$ any collection $\\lbrace X_i\\rbrace _{i \\in I}$ of random variables in $(\\mathcal {A},\\varphi )$ such that, for every integer $r\\ge $ 1 and all $i_1,\\ldots ,i_r \\in I$ , one has $\\varphi \\big ( X_{i_1}\\cdots X_{i_r}\\big )=\\sum _{\\pi \\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,r\\rbrace )} q^{\\mathrm {Cr}(\\pi )} \\prod _{\\lbrace p,q\\rbrace \\in \\pi } \\varphi \\big ( X_{i_p}X_{i_q}\\big ) \\ .$ Therefore, just as with classical (commutative) Gaussian families, the law of a $q$ -Gaussian family $\\lbrace X_i\\rbrace _{i \\in I}$ is completely characterized by the set of its covariances $\\varphi (X_iX_j)$ , $i,j\\in I$ .",
"In fact, when $q\\rightarrow 1$ and $\\varphi $ is - at least morally - identified with the usual expectation, relation (REF ) is nothing but the classical Wick formula satisfied by the joint moments of Gaussian variables.",
"When $q=0$ , such a family of random variables is also called a semicircular family, in reference to its marginal distributions (see [11] for more details on semicircular families, in connection with the so-called free central limit theorem).",
"We are now in a position to introduce the family of processes at the core of our study: Definition 1.3 For any fixed $q\\in [0,1)$ , we call $q$ -Brownian motion ($q$ -Bm) in some non-commutative probability space $(\\mathcal {A},\\varphi )$ any $q$ -Gaussian family $\\lbrace X_t\\rbrace _{t\\ge 0}$ in $(\\mathcal {A},\\varphi )$ with covariance function given by the formula $\\varphi \\big ( X_{s}X_{t}\\big )=s\\wedge t \\ .$ The existence of such a non-commutative process (in some non-commutative space $(\\mathcal {A},\\varphi )$ ) has been established by Bożejko and Speicher in [3].",
"In the same spirit as above, the $q$ -Bm distribution can be regarded as a straightforward extension of two well-known processes: $\\bullet $ When $q\\rightarrow 1$ , one recovers the classical Brownian-motion dynamics, with independent, stationary and normally-distributed increments.",
"$\\bullet $ The 0-Brownian motion coincides with the celebrated free Brownian motion, whose freely-independent increments are known to be closely related to the asymptotic behaviour of large random matrices, following Voiculescu's breakthrough results [14].",
"Thus, we have here at our disposal a family of processes which, as far as distribution is concerned, provides a natural smooth interpolation between two of the most central objects in probability theory: the standard and the free Brownian motions.",
"It is then natural to wonder whether the classical stochastic properties satisfied by each of these two processes can be lifted on the level of this interpolation, or in other words if the properties known for $q=0$ and $q\\rightarrow 1$ can be extended to every $q\\in [0,1)$ .",
"Of course, any such extension potentially offers an additional piece of evidence in favor of this interpolation model, as a privileged link between the free and the commutative worlds.",
"Some first results in this direction, focusing on the stationarity property and the marginal-distribution issue, can be found in [2]: Proposition 1.4 For any fixed $q\\in [0,1)$ , let $\\lbrace X_t\\rbrace _{t\\ge 0}$ be a $q$ -Brownian motion in some non-commutative probability space $(\\mathcal {A},\\varphi )$ .",
"Then for all $0\\le s<t$ , the random variable $X_t-X_s$ has the same law as $\\sqrt{t-s}\\, X_1$ , in the sense of identity (REF ).",
"In particular, any $q$ -Brownian motion $\\lbrace X_t\\rbrace _{t\\ge 0}$ is a $\\frac{1}{2}$ -Hölder path in $\\mathcal {A}$ , i.e.",
"$\\sup _{s<t} \\frac{\\Vert X_t-X_s\\Vert }{\\left|t-s \\right|^{1/2}} \\le \\Vert X_1\\Vert \\ < \\ \\infty \\ .$ Moreover, the law $\\mu _q$ of $X_1$ is absolutely continuous with respect to the Lebesgue measure; its density is supported on $\\big [\\frac{-2}{\\sqrt{1-q}},\\frac{2}{\\sqrt{1-q}} \\big ]$ and is given, within this interval, by the formula $\\mu _q(\\mathrm {d}x)= \\frac{1}{\\pi } \\sqrt{1-q} \\sin \\theta \\prod _{n=1}^\\infty (1-q^n) |1-q^n e^{2i\\theta }|^2\\ , \\quad \\text{where} \\ x=\\frac{2\\cos \\theta }{\\sqrt{1-q}} \\,\\,\\mbox{ with } \\theta \\in [0,\\pi ]\\ .$ A next natural step is to examine the possible extension, to all $q\\in [0,1)$ , of the stochastic integration results associated with the free/classical Brownian motion.",
"Let us here recall that the foundations of stochastic calculus with respect to the free Brownian motion (that is, for $q=0$ ) have been laid in a remarkable paper by Biane and Speicher [1].",
"Among other results, the latter study involves the construction of a free Itô integral, as well as an analysis of the free Wiener chaoses generated by the multiple integrals of the free Brownian motion.",
"These lines of investigation have been followed by Donati-Martin in [7] to handle the general $q$ -Bm case, with the construction of a $q$ -Itô integral and a study of the $q$ -Wiener chaos.",
"Let us also mention the results of [5] related to the extension of the fourth-moment phenomenon that prevails in Wiener chaoses.",
"In this paper, we intend to go further with the analysis related to the $q$ -Brownian motion.",
"To be more specific, we propose, in the continuation of [6], to adapt some of the main rough-path principles to this setting.",
"The aim here is to derive a very robust integration theory allowing, in particular, to consider the study of differential equations driven by the $q$ -Bm, i.e.",
"sophisticated dynamics of the form $dY_t=f(Y_t) \\cdot dX_t \\cdot g(Y_t) \\ ,$ for smooth functions $f,g$ .",
"In fact, thanks to the general (non-commutative) rough-path results proved in [6] (and which we will recall in Section ), the objective essentially reduces to the exhibition of a so-called product Lévy area above the $q$ -Bm, that is a kind of iterated integral of the process involving the product structure of $\\mathcal {A}$ .",
"Let us briefly recall that the definition of such an object (which appears as quite natural in this algebra setting) has been introduced in [6] as a way to overcome the possible non-existence problems arising from the study of more general Lévy areas, in the classical Lyons' sense [10] (see [13] for a description of the non-existence issue in the free case).",
"At this point, we would like to draw the reader's attention to the fact that the construction in [7] of a $q$ -Itô integral as an element of $L^2(\\varphi )$ would not be not sufficient for our purpose.",
"Indeed, the rough-path techniques are based on Taylor-expansion procedures, which, for obvious stability reasons, forces us to consider an algebra norm in the computations.",
"As a result, any satisfying notion of product Lévy area requires some control with respect to the operator norm, that is in $L^\\infty (\\varphi )$ (along (REF )), and not only with respect to the $L^2(\\varphi )$ -norm (see Section and especially Definition REF for more details on the topology involved in this control).",
"In the particular case of the free Bm ($q=0$ ), the Burkholder-Gundy inequality established by Biane and Speicher in [1] immediately gives rise to operator-norm controls on the free Itô integral, which we could readily exploit in [6] to deal with rough paths in the free situation.",
"Unfortunately, and at least for the time being, no similar operator-norm control has been shown for the $q$ -Itô integral when $q\\in (0,1)$ .",
"With our rough-path objectives in mind, we will be able to overcome this difficulty though, by resorting to a straightforward $L^\\infty (\\varphi )$ -construction of a product Lévy area - the latter object being actually much more specific than a general Itô integral.",
"This is the purpose of the forthcoming Section , which leads to the main result of the paper.",
"Injecting this construction into the general rough-path theory will immediately answer our original issue, that is the derivation of a robust stochastic calculus for the $q$ -Bm.",
"It is then possible to compare, a posteriori, the resulting rough integral with more familiar $q$ -Itô or $q$ -Stratonovich integrals, through a standard $L^2(\\varphi )$ -analysis and the involvement of the so-called second-quantization operator.",
"This comparison will be the topic of Section .",
"Let us however insist, one more time, on the fact that this sole $L^2(\\varphi )$ -analysis would not have been sufficient for the rough-path theory (and the powerful rough-path results) to be applied in this situation.",
"Our construction of a product Lévy area will only rely on the consideration of the law of the $q$ -Bm, that is on the process as given by Definition REF .",
"In other words, no reference will be made to any particular representation of the process as a map with values in some specific algebra (just as classical probability theory builds upon the law of the Brownian motion and not upon its representation).",
"The only reference to some particular representation of the $q$ -Brownian motion (namely its standard representation on the $q$ -Fock space) will occur in Section , as a way to compare our rough objects with the constructions of [7], based on the Fock space.",
"Besides, we have chosen in this study to focus on the case where $q\\in [0,1)$ and introduce the $q$ -Brownian motion as a natural interpolation between the free and the standard Brownian motions.",
"We are aware that the definition of a $q$ -Bm can also be extended to every $q\\in (-1,0)$ , that is up to the anticommutative situation $q\\rightarrow -1$ .",
"In fact, we must here specify that the positivity assumption on $q$ will be used in an essential way for the construction of the product Lévy area (see for instance (REF )), and at this point, we do not know if such an object could also be exhibited in the case $q<0$ .",
"As we already sketched it in the above description of our results, the study is organized as follows.",
"In Section , we will recall the general non-commutative rough-path results obtained in [6] and at the core of the present analysis.",
"Section is devoted to the construction of the main object involved in the rough-path procedure, that is a product Lévy area above the $q$ -Bm.",
"Finally, Section focuses on the $L^2(\\varphi )$ -comparison of the rough constructions with more standard Itô/Stratonovich definitions.",
"Acknowledgements.",
"We are very grateful to two anonymous referees for their attentive reading and insightful suggestions."
],
[
"General rough-path results in $C^\\ast $ -algebras",
"Our strategy to develop a robust $L^\\infty (\\varphi )$ -stochastic calculus for the $q$ -Bm is based on the non-commutative rough-path considerations of [6].",
"Therefore, before we can turn to the $q$ -Bm situation, it is necessary for us to recall the main results of the theoretical analysis carried out in [6].",
"This requires first a few brief preliminaries on functional calculus in a $C^\\ast $ -algebra (along the framework of [1]), as well as precisions on the topologies involved in the study.",
"Special emphasis will be put on the cornerstone of the rough-path machinery, the product Lévy area, around which the whole integration procedure can be naturally expanded.",
"Note that the considerations of this section apply to a general $C^\\ast $ -algebra $\\mathcal {A}$ , that we fix from now on.",
"In particular, no additional trace operator will be required here.",
"As before, we denote by $\\Vert .\\Vert $ the operator norm on $\\mathcal {A}$ , and $\\mathcal {A}_\\ast $ will stand for the set of self-adjoint operators in $\\mathcal {A}$ .",
"We also fix an arbitrary time horizon $T>0$ for the whole section."
],
[
"Tensor product",
"Let $\\mathcal {A}\\otimes \\mathcal {A}$ be the algebraic tensor product generated by $\\mathcal {A}$ , and just as in [1], denote by $\\sharp $ the natural product interaction between $\\mathcal {A}$ and $\\mathcal {A}\\otimes \\mathcal {A}$ , that is the linear extension of the formula $(U_1 \\otimes U_2) \\sharp X=X\\sharp (U_1 \\otimes U_2):=U_1X U_2\\ , \\quad \\text{for all} \\ U_1,U_2,X \\in \\mathcal {A}\\ .$ In a similar way, set, for all $U_1,U_2,U_3,X \\in \\mathcal {A}$ , $X \\sharp (U_1 \\otimes U_2 \\otimes U_3) :=(U_1X U_2) \\otimes U_3 \\quad , \\quad (U_1 \\otimes U_2 \\otimes U_3) \\sharp X:=U_1 \\otimes (U_2XU_3) \\ .$ Our developments will actually involve the projective tensor product $\\mathcal {A}\\hat{\\otimes }\\mathcal {A}$ of $\\mathcal {A}$ , that is the completion of $\\mathcal {A}\\otimes \\mathcal {A}$ with respect to the norm $\\Vert \\mathbf {U}\\Vert =\\Vert \\mathbf {U}\\Vert _{\\mathcal {A}\\hat{\\otimes } \\mathcal {A}}:=\\inf \\sum _i \\Vert U_i\\Vert \\Vert V_i\\Vert \\ ,$ where the infimum is taken over all possible representation $\\mathbf {U}=\\sum _i U_i \\otimes V_i$ of $\\mathbf {U}$ .",
"It is readily checked that for all $\\mathbf {U}\\in \\mathcal {A}\\otimes \\mathcal {A}$ and $X\\in \\mathcal {A}$ , one has $\\Vert \\mathbf {U} \\sharp X\\Vert \\le \\Vert \\mathbf {U}\\Vert \\Vert X\\Vert $ , and so the above $\\sharp $ -product continuously extends to $\\mathcal {A}\\hat{\\otimes }\\mathcal {A}$ .",
"These considerations can, of course, be generalized to the $n$ -th projective tensor product $\\mathcal {A}^{\\hat{\\otimes }n}$ , $n\\ge 1$ , and we will still denote by $\\Vert .\\Vert $ the projective tensor norm on $\\mathcal {A}^{\\hat{\\otimes }n}$ .",
"Along the same terminology as in [1], we will call any process with values in $\\mathcal {A}\\hat{\\otimes } \\mathcal {A}$ , resp.",
"$\\mathcal {A}\\hat{\\otimes } \\mathcal {A}\\hat{\\otimes }\\mathcal {A}$ , a biprocess, resp.",
"a triprocess."
],
[
"Functional calculus in a $C^\\ast $ -algebra",
"Following again the presentation of [1], let us introduce the class of functions $f$ defined for every integer $k\\ge 0$ by $\\mathbf {F}_k:=\\lbrace f:\\mathbb {R}\\rightarrow \\ f(x)=\\int _{\\mathbb {R}} e^{\\imath \\xi x} \\mu _f(\\mathrm {d}\\xi ) \\ \\text{with} \\ \\int _{\\mathbb {R}} |\\xi |^i \\, \\mu _f(\\mathrm {d}\\xi ) < \\infty \\ \\text{for every} \\ i \\in \\lbrace 0,\\ldots ,k\\rbrace \\rbrace ,$ and set, if $f\\in \\mathbf {F}_k$ , $\\Vert f\\Vert _k:=\\sum _{i=0}^k \\int _{\\mathbb {R}} |\\xi |^i \\, \\mu _f(\\mathrm {d}\\xi )$ .",
"Then, with all $f\\in \\mathbf {F}_0$ and $X\\in \\mathcal {A}_\\ast $ , we associate the operator $f(X)$ along the formula $f(X):=\\int _{\\mathbb {R}} e^{\\imath \\xi X} \\mu _f(\\mathrm {d}\\xi ) \\ ,$ where the integral in the right-hand side is uniformly convergent in $\\mathcal {A}$ .",
"This straightforward operator extension of functional calculus happens to be compatible with Taylor expansions of $f$ , a central ingredient towards the application of rough-path techniques.",
"The following notion of tensor derivatives naturally arises in the procedure (see the subsequent Examples REF and REF ): Definition 2.1 For every $f\\in \\mathbf {F}_1$ , resp.",
"$f\\in \\mathbf {F}_2$ , we define the tensor derivative, resp.",
"second tensor derivative, of $f$ by the formula: for every $X\\in \\mathcal {A}_\\ast $ , $\\partial f(X):=\\int _0^1 \\mathrm {d}\\alpha \\int _{\\mathbb {R}} \\imath \\xi \\, [e^{\\imath \\alpha \\xi X} \\otimes e^{\\imath (1-\\alpha )\\xi X}]\\, \\mu _f(\\mathrm {d}\\xi ) \\quad \\in \\mathcal {A}\\hat{\\otimes } \\mathcal {A}\\ ,$ $\\text{resp.}",
"\\quad \\partial ^2 f(X):=-\\iint _{\\begin{array}{c}\\alpha ,\\beta \\ge 0\\\\ \\alpha +\\beta \\le 1\\end{array}} \\mathrm {d}\\alpha \\, \\mathrm {d}\\beta \\int _{\\mathbb {R}} \\xi ^2 \\, [e^{\\imath \\alpha \\xi X}\\otimes e^{\\imath \\beta \\xi X} \\otimes e^{\\imath (1-\\alpha -\\beta )\\xi X}]\\, \\mu _f(\\mathrm {d}\\xi ) \\quad \\in \\mathcal {A}\\hat{\\otimes } \\mathcal {A}\\hat{\\otimes } \\mathcal {A}\\ .$"
],
[
"Filtration and Hölder topologies",
"From now on and for the rest of Section , we fix a process $X:[0,T]\\rightarrow \\mathcal {A}_\\ast $ and assume that $X$ is $\\gamma $ -Hölder regular, that is $\\sup _{0\\le s<t\\le T}\\frac{\\Vert X_t-X_s\\Vert }{|t-s|^\\gamma } \\ < \\ \\infty \\ ,$ for some fixed coefficient $\\gamma \\in (1/3,1/2)$ .",
"With this process in hand, we denote by $\\lbrace \\mathcal {A}_t\\rbrace _{t\\in [0,T]}=\\lbrace \\mathcal {A}^X_t\\rbrace _{t\\in [0,T]}$ the filtration generated by $X$ , that is, for each $t\\in [0,T]$ , $\\mathcal {A}_t$ stands for the closure (with respect to the operator norm) of the unital subalgebra of $\\mathcal {A}$ generated by $\\lbrace X_s\\rbrace _{0\\le s\\le t}$ .",
"For any fixed interval $I\\subset [0,T]$ , a process $Y: I\\rightarrow \\mathcal {A}$ is said to be adapted if for each $t\\in I$ , $Y_t\\in \\mathcal {A}_t$ .",
"In the same way, a biprocess $\\mathbf {U}:[0,T]\\rightarrow \\mathcal {A}\\hat{\\otimes }\\mathcal {A}$ , resp.",
"a triprocess $\\mathcal {U}:[0,T]\\rightarrow \\mathcal {A}\\hat{\\otimes } \\mathcal {A}\\hat{\\otimes }\\mathcal {A}$ , is adapted if for each $t\\in [0,T]$ , $\\mathbf {U}_t\\in \\mathcal {A}_t \\hat{\\otimes }\\mathcal {A}_t$ , resp.",
"$\\mathcal {U}_t\\in \\mathcal {A}_t \\hat{\\otimes } \\mathcal {A}_t \\hat{\\otimes }\\mathcal {A}_t$ .",
"Let us now briefly recall the topologies involved in the rough-path procedure, as far as time-roughness is concerned (and following Gubinelli's approach [9]).",
"For $V:=\\mathcal {A}^{\\hat{\\otimes }n}$ ($n\\ge 1$ ), let $\\mathcal {C}_1(I;V)$ be the set of continuous $V$ -valued maps on $I$ , and $\\mathcal {C}_2(I;V)$ the set of continuous $V$ -valued maps on the simplex $\\mathcal {S}_2:=\\lbrace (s,t)\\in I^2: \\ s\\le t\\rbrace $ that vanish on the diagonal.",
"The increments of a path $g\\in \\mathcal {C}_1(I;V)$ will be denoted by $\\delta g_{st}:=g_t-g_s$ ($s\\le t$ ) and for every $\\alpha \\in (0,1)$ , we define the $\\alpha $ -Hölder spaces $\\mathcal {C}_1^\\alpha (I;V)$ , resp.",
"$\\mathcal {C}_2^\\alpha (I;V)$ , as $\\mathcal {C}_1^\\alpha (I;V):=\\big \\lbrace h \\in \\mathcal {C}_1(I;V): \\ \\mathcal {N}[h;\\mathcal {C}_1^\\alpha (I;V)]:= \\sup _{s<t \\in I} \\frac{\\Vert \\delta h_{st}\\Vert }{\\left|t-s \\right|^\\alpha } \\ < \\infty \\big \\rbrace \\ ,$ resp.",
"$\\mathcal {C}_2^\\alpha (I;V):=\\big \\lbrace h \\in \\mathcal {C}_2(I;V): \\ \\mathcal {N}[h;\\mathcal {C}_2^\\alpha (I;V)]:=\\sup _{s<t \\in I} \\frac{\\Vert h_{st}\\Vert }{\\left|t-s \\right|^\\alpha } \\ < \\infty \\big \\rbrace \\ .$"
],
[
"The product Lévy area",
"Consider the successive spaces $\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }):=\\lbrace L=(L_{st})_{0\\le s < t \\le T}: \\ L_{st}\\in \\mathcal {L}(\\mathcal {A}_s\\hat{\\otimes } \\mathcal {A}_s,\\mathcal {A})\\rbrace \\ ,$ $\\mathcal {L}_T(\\mathcal {A}_{\\rightarrow }):=\\lbrace L=(L_{st})_{0\\le s < t \\le T}: \\ L_{st}\\in \\mathcal {L}(\\mathcal {A}_s\\hat{\\otimes } \\mathcal {A}_s,\\mathcal {A}_t)\\rbrace \\ ,$ and for every $\\lambda \\in [0,1]$ , denote by $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))$ , resp.",
"$\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_\\rightarrow ))$ , the set of elements $L\\in \\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup })$ , resp.",
"$L\\in \\mathcal {L}_T(\\mathcal {A}_\\rightarrow )$ , for which the following quantity is finite: $\\mathcal {N}[L;\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))]:=\\sup _{\\begin{array}{c}s<t\\in [0,T]\\\\ \\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes } \\mathcal {A}_s,\\mathbf {U}\\ne 0\\end{array}}\\frac{\\Vert L_{st}[\\mathbf {U}]\\Vert }{\\left|t-s\\right|^\\lambda \\Vert \\mathbf {U}\\Vert } \\ .$ At this point, recall that we have fixed a $\\gamma $ -Hölder process $X:[0,T]\\rightarrow \\mathcal {A}_\\ast $ ($\\gamma \\in (1/3,1/2)$ ) for the whole section .",
"Definition 2.2 We call product Lévy area above $X$ any process $\\mathbb {X}^{2}$ such that: (i) ($2\\gamma $ -roughness) $\\mathbb {X}^2\\in \\mathcal {C}_2^{2\\gamma }(\\mathcal {L}_T(\\mathcal {A}_\\rightarrow ))$ , (ii) (Product Chen identity) For all $s<u<t$ and $\\mathbf {U}\\in \\mathcal {A}_s\\hat{\\otimes }\\mathcal {A}_s$ , $\\mathbb {X}^2_{st}[\\mathbf {U}]-\\mathbb {X}^2_{su}[\\mathbf {U}]-\\mathbb {X}^2_{ut}[\\mathbf {U}]=(\\mathbf {U} \\sharp \\delta X_{su}) \\, \\delta X_{ut}\\ .$ Remark 2.3 Recall that Definition REF is derived from the theoretical analysis performed in [6] with equation (REF ) in mind.",
"At some heuristic level, and following the classical rough-path approach, the notion of product Lévy area must be seen as some abstract version of the iterated integral $\\mathbb {X}^2_{st}[\\mathbf {U}]= \\int _s^t (\\mathbf {U} \\sharp \\delta X_{su}) \\, \\mathrm {d}X_u \\ ,$ noting that definition of this integral is not clear a priori for a non-differentiable process $X$ .",
"As pointed out in [6], the above notion of Lévy area is specifically designed to handle the non-commutative algebra dynamics of (REF ), and it offers a much more efficient approach than general rough-path theory based on tensor Lévy areas (the object considered in [10]).",
"In a commutative setting (i.e., if $\\mathcal {A}$ were a commutative algebra), the basic process $\\mathbb {A}_{st}(\\mathbf {U}):=\\frac{1}{2} (\\mathbf {U}\\sharp \\delta X_{st}) \\, \\delta X_{st}$ would immediately provide us with such a product Lévy area.",
"In the general (non-commutative) situation though, this path only satisfies $\\mathbb {A}_{st}[\\mathbf {U}]-\\mathbb {A}_{su}[\\mathbf {U}]-\\mathbb {A}_{ut}[\\mathbf {U}]=\\frac{1}{2} \\big [ (\\mathbf {U}\\sharp \\delta X_{su}) \\, \\delta X_{ut}+(\\mathbf {U}\\sharp \\delta X_{ut}) \\, \\delta X_{su} \\big ] \\ ,$ so that $\\mathbb {A}$ may not meet the product-Chen condition $(ii)$ , making Definition REF undoubtedly relevant."
],
[
"Controlled (bi)processes and integration",
"A second ingredient in the rough-path machinery (in addition to a Lévy area) consists in the identification of a suitable class of integrands for the future rough integral with respect to $X$ .",
"The following definition naturally arises in this setting: Definition 2.4 Given a time interval $I\\subset [0,T]$ , we call adapted controlled process, resp.",
"biprocess, on $I$ any adapted process $Y\\in \\mathcal {C}_1^\\gamma (I;\\mathcal {A})$ , resp.",
"biprocess $\\mathbf {U}\\in \\mathcal {C}_1^\\gamma (I;\\mathcal {A}\\hat{\\otimes } \\mathcal {A})$ , with increments of the form $(\\delta Y)_{st}=\\mathbf {Y}^X_s \\sharp (\\delta X)_{st}+Y^\\flat _{st}\\ , \\quad s<t\\in I \\ ,$ resp.",
"$(\\delta \\mathbf {U})_{st}=(\\delta X)_{st} \\sharp \\mathcal {U}_s^{X,1} +\\mathcal {U}_s^{X,2} \\sharp (\\delta X)_{st}+ \\mathbf {U}^\\flat _{st} \\ , \\quad s<t\\in I \\ ,$ for some adapted biprocess $\\mathbf {Y}^X \\in \\mathcal {C}_1^{\\gamma }(I;\\mathcal {A}\\hat{\\otimes } \\mathcal {A})$ , resp.",
"adapted triprocesses $\\mathcal {U}^{X,1},\\mathcal {U}^{X,2}\\in \\mathcal {C}_1^{\\gamma }(I;\\mathcal {A}^{\\hat{\\otimes }3})$ , and $Y^\\flat \\in \\mathcal {C}_2^{2\\gamma }(I;\\mathcal {A})$ , resp.",
"$\\mathbf {U}^\\flat \\in \\mathcal {C}_2^{2\\gamma }(I;\\mathcal {A}\\hat{\\otimes } \\mathcal {A})$ .",
"We denote by $\\mathcal {Q}_X(I)$ , resp.",
"$\\mathbf {Q}_X(I)$ , the space of adapted controlled processes, resp.",
"biprocesses, on $I$ , and finally we define $\\mathcal {Q}_X^\\ast (I)$ as the subspace of controlled processes $Y \\in \\mathcal {Q}_X(I)$ for which one has both $Y^\\ast _s=Y_s$ and $(\\mathbf {Y}^X_s)^\\ast =\\mathbf {Y}^X_s$ for every $s\\in I$ .",
"Example 2.5 If $f,g \\in \\mathbf {F}_2$ and $Y\\in \\mathcal {Q}_X^\\ast (I)$ with decomposition (REF ), then $\\mathbf {U}:=f(Y) \\otimes g(Y) \\in \\mathbf {Q}_X(I)$ with $\\mathcal {U}^{X,1}_s := [\\partial f(Y_s) \\, \\mathbf {Y}^X_s ] \\otimes g(Y_s) \\quad , \\quad \\mathcal {U}^{X,2}_s=f(Y_s) \\otimes [\\partial g(Y_s) \\, \\mathbf {Y}^X_s] \\ .$ Example 2.6 If $f\\in \\mathbf {F}_3$ , then $\\mathbf {U}:=\\partial f(X) \\in \\mathbf {Q}_X([0,T])$ with $\\mathcal {U}^{X,1}_s =\\mathcal {U}^{X,2}_s=\\partial ^2 f(X_s)$ .",
"We are finally in a position to recall the definition of the rough integral with respect to $X$ , which can be expressed (among other ways) as the limit of corrected Riemann sums: Proposition 2.7 [6] Assume that we are given a product Lévy area $\\mathbb {X}^2$ above $X$ , in the sense of Definition REF , as well as a time interval $I=[\\ell _1,\\ell _2]\\subset [0,T]$ .",
"Then for every $\\mathbf {U}\\in \\mathbf {Q}_X(I)$ with decomposition (REF ), all $s<t\\in I$ and every subdivision $D_{st} = \\lbrace t_0=s<t_1 <\\ldots <t_n=t\\rbrace $ of $[s,t]$ with mesh $|D_{st}|$ tending to 0, the corrected Riemann sum $\\sum _{t_i\\in D_{st}} \\Big \\lbrace \\mathbf {U}_{t_i} \\sharp (\\delta X)_{t_it_{i+1}}+[\\mathbb {X}^2_{t_it_{i+1}}\\times \\mbox{Id}](\\mathcal {U}^{X,1}_{t_i})+[\\mbox{Id}\\times \\mathbb {X}^{2,\\ast }_{t_it_{i+1}}](\\mathcal {U}^{X,2}_{t_i})\\Big \\rbrace $ converges in $\\mathcal {A}$ as $|D_{st}| \\rightarrow 0$ .",
"We call the limit the rough integral (from $s$ to $t$ ) of $\\mathbf {U}$ against $\\mathbb {X}:=(X,\\mathbb {X}^2)$ , and we denote it by $\\int _s^t \\mathbf {U}_u \\sharp \\mathrm {d}\\mathbb {X}_u$ .",
"This construction satisfies the two following properties: $\\bullet $ (Consistency) If $X$ is a differentiable process in $\\mathcal {A}$ and $\\mathbb {X}^2$ is understood in the classical Lebesgue sense (that is, as in (REF )), then $\\int _s^t \\mathbf {U}_u \\sharp \\mathrm {d}\\mathbb {X}_u$ coincides with the classical Lebesgue integral $\\int _s^t [ \\mathbf {U}_u \\sharp X^{\\prime }_u] \\, du$ ; $\\bullet $ (Stability) For every $A\\in \\mathcal {A}$ , there exists a unique process $Z\\in \\mathcal {Q}_X(I)$ such that $Z_{\\ell _1}=A$ and $(\\delta Z)_{st}=\\int _s^t \\mathbf {U}_u \\sharp \\mathrm {d}\\mathbb {X}_u$ for all $s<t\\in I$ .",
"Theorem 2.8 [6] Assume that we are given a product Lévy area $\\mathbb {X}^2$ above $X$ .",
"Let $f=(f_1,\\ldots ,f_m) \\in \\mathbf {F}_3^m$ , $g=(f_1^\\ast ,\\ldots ,f_m^\\ast )$ or $(f_m^\\ast ,\\ldots ,f_1^\\ast )$ , and fix $A \\in \\mathcal {A}_\\ast $ .",
"Then the equation $Y_{0}=A \\quad , \\quad (\\delta Y)_{st}=\\sum _{i=1}^m \\int _s^t f_i(Y_u) \\, \\mathrm {d}\\mathbb {X}_u \\, g_i(Y_u)\\ , \\quad s<t \\in [0,T]\\ ,$ interpreted with Proposition REF , admits a unique solution $Y\\in \\mathcal {Q}_X^\\ast ([0,T])$ ."
],
[
"Approximation results",
"Another advantage of the rough-path approach - beyond its consistency and stability properties - lies in the continuity of the constructions with respect to the driving (rough) path.",
"In this non-commutative setting, and following the approach of [6], the phenomenon can be illustrated through several Wong-Zakaï-type approximation results, which we propose to briefly review here.",
"To this end, for every sequence of partitions $(D^n)$ of $[0,T]$ with mesh tending to zero, denote by $\\lbrace X^n_t\\rbrace _{t\\in [0,T]}=\\lbrace X^{D^n}_t\\rbrace _{t\\in [0,T]}$ the sequence of linear interpolations of $X$ along $D^n$ , i.e., if $D^n:=\\lbrace 0=t_0< t_1<\\ldots <t_k=T\\rbrace $ , $X^n_t:=X_{t_i}+\\frac{t-t_i}{t_{i+1}-t_i} \\delta X_{t_i t_{i+1}} \\quad \\text{for} \\ t\\in [t_i,t_{i+1}]\\ .$ Then consider the sequence of approximated product Lévy areas defined for every $\\mathbf {U}\\in \\mathcal {A}\\hat{\\otimes } \\mathcal {A}$ as $\\mathbb {X}^{2,n}_{st}[\\mathbf {U}]=\\mathbb {X}^{2,D^n}_{st}[\\mathbf {U}]:=\\int _s^t (\\mathbf {U}\\sharp \\delta X^n_{su}) \\, \\mathrm {d}X^n_u \\ ,\\quad s<t\\in [0,T]\\ ,$ where the integral is understood in the classical Lebesgue sense.",
"In other words, if $t_k \\le s <t_{k+1}\\le t_\\ell \\le t < t_{\\ell +1}$ , $&\\mathbb {X}^{2,n}_{st}[\\mathbf {U}]=\\int _s^{t_{k+1}} \\frac{du}{t_{k+1}-t_k} \\big ( \\mathbf {U}\\sharp \\delta X^n_{su} \\big ) (\\delta X)_{t_k t_{k+1}}\\\\&\\hspace{28.45274pt}+\\sum _{i=k+1}^{\\ell -1}\\int _{t_i}^{t_{i+1}} \\frac{du}{t_{i+1}-t_i} \\big ( \\mathbf {U}\\sharp \\delta X^n_{su} \\big ) (\\delta X)_{t_i t_{i+1}}+\\int _{t_\\ell }^{t} \\frac{du}{t_{k+1}-t_k} \\big ( \\mathbf {U}\\sharp \\delta X^n_{su} \\big ) (\\delta X)_{t_k t_{k+1}} \\ .$ Proposition 2.9 [6] Assume that there exists a product Lévy area $\\mathbb {X}^2$ above $X$ such that, as $n$ tends to infinity, $\\mathcal {N}[X^n - X ;\\mathcal {C}_1^\\gamma ([0,T];\\mathcal {A})]\\rightarrow 0 \\quad \\text{and} \\quad \\mathcal {N}[\\mathbb {X}^{2,n} -\\mathbb {X}^2; \\mathcal {C}_2^{2\\gamma }(\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))]\\rightarrow 0 \\ .$ Then for all $f,g \\in \\mathbf {F}_3$ , it holds that $\\int _.^.",
"f(X^n_u) \\, \\mathrm {d}X^n_u \\, g(X^n_u) \\xrightarrow{} \\int _.^.",
"f(X_u) \\, \\mathrm {d}\\mathbb {X}_u \\, g(X_u) \\quad \\text{in} \\ \\ \\mathcal {C}_2^\\gamma ([0,T];\\mathcal {A})\\ ,$ where the integral in the limit is interpreted with Proposition REF .",
"Similarly, for all $f \\in \\mathbf {F}_3$ , one has $\\int _.^.",
"\\partial f(X^n_u) \\sharp \\mathrm {d}X^n_u \\xrightarrow{} \\int _.^.",
"\\partial f (X_u) \\sharp \\mathrm {d}\\mathbb {X}_u \\quad \\text{in} \\ \\ \\mathcal {C}_2^\\gamma ([0,T];\\mathcal {A})\\ ,$ which immediately yields Itô's formula: for all $s<t\\in [0,T]$ , $\\delta f(X)_{st}=\\int _s^t \\partial f (X_u) \\sharp \\mathrm {d}\\mathbb {X}_u \\ .$ Finally, for some fixed $f=(f_1,\\ldots ,f_m) \\in \\mathbf {F}_3^m$ and $g=(f_1^\\ast ,\\ldots ,f_m^\\ast )$ (or $g:=(f_m^\\ast ,\\ldots ,f_1^\\ast )$ ), let us denote by $Y^n=Y^{D^n}$ the solution of the classical Lebesgue equation on $[0,T]$ $Y^n_0=A\\in \\mathcal {A}_\\ast \\quad , \\quad dY^n_t =\\sum \\nolimits _{i=1}^m f_i(Y^n_t) \\, \\mathrm {d} X^n_t \\, g_i(Y^n_t)\\ .$ Theorem 2.10 [6] Under the assumptions of Proposition REF , one has $Y^n \\xrightarrow{} Y$ in $\\mathcal {C}_1^\\gamma ([0,T];\\mathcal {A})$ , where $Y$ is the solution of (REF ) given by Theorem REF .",
"As we pointed it out in the introduction, these convergence results are based on Taylor-expansion procedures and accordingly, the consideration of an algebra norm for the control of $\\mathbf {U}$ and $L_{st}[\\mathbf {U}]$ in the roughness assumption (REF ) is an essential ingredient."
],
[
"A product Lévy area above the $q$ -Brownian motion",
"We go back here to the $q$ -Bm setting described in Section .",
"Namely, we fix $q\\in [0,1)$ and consider a $q$ -Brownian motion $(X_t)_{t\\ge 0}$ in some non-commutative probability space $(\\mathcal {A},\\varphi )$ .",
"With the developments of the previous section in mind, the route towards an efficient operator-norm calculus for $X$ is now clear: we need to exhibit a product Lévy area above $X$ , in the sense of Definition REF .",
"Our main result thus reads as follows: Theorem 3.1 Denote by $\\lbrace X^n_t\\rbrace _{t\\ge 0}$ the linear interpolation of $X$ along the dyadic partition $D^n:=\\lbrace t_i^n \\, , \\, i\\ge 0\\rbrace $ , $t_i^n:=\\frac{i}{2^n}$ .",
"Then there exists a product Lévy area $\\mathbb {X}^{2,S}$ above $X$ , in the sense of Definition REF , such that for every $T>0$ and every $0<\\gamma <1/2$ , one has $X^n \\rightarrow X \\quad \\text{in} \\ \\ \\mathcal {C}_1^\\gamma ([0,T];\\mathcal {A}) \\quad \\text{and} \\quad \\mathbb {X}^{2,n} \\rightarrow \\mathbb {X}^{2,S} \\quad \\text{in} \\ \\ \\mathcal {C}_2^{2\\gamma }(\\mathcal {L}_T(\\mathcal {A}_\\rightarrow )) \\ ,$ where $\\mathbb {X}^{2,n}$ is defined by (REF ).",
"We call $\\mathbb {X}^{2,S}$ the Stratonovich product Lévy area above $X$ .",
"Based on this result, the conclusions of Proposition REF , Theorem REF , Proposition REF and Theorem REF can all be applied to the $q$ -Brownian motion, with limits understood as rough integrals with respect to the product rough path $\\mathbb {X}^S:=(X,\\mathbb {X}^{2,S})$ .",
"The Stratonovich terminology is here used as a reference to the classical commutative situation, where the (almost sure) limit of the sequence of approximated Lévy areas would indeed coincide with the Stratonovich iterated integral (see also Corollary REF for another justification of this terminology).",
"Before we turn to the proof of Theorem REF , let us recall that the whole difficulty in constructing a stochastic integral with respect to the general $q$ -Bm, in comparison with the free ($q=0$ ) or the commutative ($q\\rightarrow 1$ ) cases, lies in the absence of any satisfying $q$ -freeness property for the increments of the process when $q\\in (0,1)$ (as reported by Speicher in [12]).",
"For instance, if $s<u<t$ , $\\varphi \\big ((X_u-X_s)(X_t-X_u)(X_u-X_s)(X_t-X_u)\\big ) =q\\, \\varphi \\big ((X_u-X_s)^2\\big )\\varphi \\big ((X_t-X_u)^2\\big )=q\\, |u-s| |t-u| \\ ,$ which shows that, for $q\\ne 0$ , the disjoint increments of a $q$ -Brownian motion $\\lbrace X_t\\rbrace _{t\\ge 0}$ are indeed not freely independent (in the sense of [6]), making most of the arguments of [1] unexploitable in this situation.",
"This being said, we can still rely here on the basic fact that for all $q\\in [0,1)$ , $\\varphi \\big ((X_u-X_s)(X_t-X_u)\\big )=0$ .",
"Together with Formula (REF ), this very weak freeness property of the increments will somehow be sufficient for our purpose, the construction of a product Lévy area being much more specific than the construction of a general stochastic integral (along Itô's standard procedure).",
"The proof of Theorem REF will also appeal to the two following elementary lemmas.",
"The first one (whose proof follows immediately from (REF )) is related to the linear stability of $q$ -Gaussian families: Lemma 3.2 For any fixed $q\\in [0,1)$ , let $Y:=\\lbrace Y_1,\\ldots ,Y_d\\rbrace $ be a $q$ -Gaussian vector in some non-commutative probability space $(\\mathcal {A},\\varphi )$ , and consider a real-valued $(d\\times m)$ -matrix $\\Lambda $ .",
"Then $Z:=\\Lambda Y$ is also a $q$ -Gaussian vector in $(\\mathcal {A},\\varphi )$ .",
"We will also need the following general topology property on the space accommodating any Lévy area: Lemma 3.3 The space $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))$ , endowed with the norm (REF ), is complete.",
"Although the arguments are classical, let us provide a few details here, since the $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))$ -structure is not exactly standard.",
"Consider a Cauchy sequence $L^n$ in $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))$ .",
"For every fixed $s\\in [0,T]$ , the sequence $L^n_{s.}$ defines a Cauchy sequence in the space $L^\\infty ([s,T]; \\mathcal {L}(\\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s,\\mathcal {A}))$ of bounded functions on $[s,T]$ (with values in $\\mathcal {L}(\\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s,\\mathcal {A})$ ), endowed with the uniform norm.",
"Therefore it converges in the latter space to some function $L_{s.}$ .",
"The fact that the so-defined family $\\lbrace L_{st}\\rbrace _{s<t}$ belongs to $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))$ is an immediate consequence of the boundedness of $L^n$ in $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))$ .",
"Finally, given $\\varepsilon >0$ and for all fixed $s<t$ , we know that there exists $M_{\\varepsilon ,s,t}\\ge 0$ such that for all $m\\ge M_{\\varepsilon ,s,t}$ , $\\Vert L^m_{st}-L_{st}\\Vert _{\\mathcal {L}(\\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s,\\mathcal {A})}\\le \\frac{\\varepsilon }{2} |t-s|^\\lambda $ .",
"On the other hand, there exists $N_\\varepsilon \\ge 0$ such that for all $n,m\\ge N_\\varepsilon $ and all $s<t$ , $\\Vert L^n_{st}-L^m_{st}\\Vert _{\\mathcal {L}(\\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s,\\mathcal {A})}\\le \\frac{\\varepsilon }{2} |t-s|^\\lambda $ .",
"Therefore, for all $n\\ge N_\\varepsilon $ and all $s<t$ , we get that for $m:=\\max (N_\\varepsilon ,M_{\\varepsilon ,s,t})$ , $\\Vert L^n_{st}-L_{st}\\Vert _{\\mathcal {L}(\\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s,\\mathcal {A})}\\le \\Vert L^n_{st}-L^m_{st}\\Vert _{\\mathcal {L}(\\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s,\\mathcal {A})}+\\Vert L_{st}-L^m_{st}\\Vert _{\\mathcal {L}(\\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s,\\mathcal {A})}\\le \\varepsilon |t-s|^\\lambda \\ ,$ and so $L^n\\rightarrow L$ in $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))$ , which achieves to prove that the latter space is complete.",
"Throughout the proof, we will denote by $A\\lesssim B$ any bound of the form $A\\le c B$ , where $c$ is a universal constant independent from the parameters under consideration.",
"The first-order convergence statement in (REF ) is a straightforward consequence of the $1/2$ -Hölder regularity of $X$ .",
"In fact, using (REF ), it can be checked that for all $n\\ge 0$ and $s<t$ , $\\Vert \\delta X^n_{st}\\Vert \\lesssim \\Vert X_1\\Vert |t-s|^{1/2} \\quad \\text{and} \\quad \\Vert \\delta (X^n-X)_{st}\\Vert \\lesssim \\Vert X_1\\Vert |t-s|^{\\gamma } 2^{-n(1/2-\\gamma )}\\ .$ Let us turn to the second-order convergence and to this end, fix $n\\ge 0$ and $s<t$ such that $t_k^n \\le s<t_{k+1}^n$ , $t_\\ell ^n \\le t < t_{\\ell +1}$ , with $k\\le \\ell $ .",
"If $|\\ell -k|\\le 1$ , or in other words if $|t-s|\\le 2^{-n+1}$ , the expected bound can be readily derived from the first estimate in (REF ), that is for every $\\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s$ , we get from (REF ) $\\Vert \\mathbb {X}^{2,n+1}_{st}[\\mathbf {U}]-\\mathbb {X}^{2,n}_{st}[\\mathbf {U}]\\Vert \\le \\Vert \\mathbb {X}^{2,n+1}_{st}[\\mathbf {U}]\\Vert +\\Vert \\mathbb {X}^{2,n}_{st}[\\mathbf {U}]\\Vert \\lesssim \\Vert X_1\\Vert ^2 |t-s|^{2\\gamma } 2^{-n(1/2-\\gamma )} \\ .",
"$ Assume from now on that $\\ell \\ge k+2$ and in this case consider the decomposition, for every $\\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s$ , ${\\mathbb {X}^{2,n+1}_{st}[\\mathbf {U}]-\\mathbb {X}^{2,n}_{st}[\\mathbf {U}]}\\nonumber \\\\&=& \\bigg [ \\int _{t_{k+1}^n}^{t_\\ell ^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{t_{k+1}^n u}\\big ) \\, \\mathrm {d}X^{n+1}_u-\\int _{t_{k+1}^n}^{t_\\ell ^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n}_{t_{k+1}^nu}\\big ) \\, \\mathrm {d}X^{n}_u \\bigg ]\\nonumber \\\\& & + \\bigg [ \\int _s^{t_{k+1}^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{su}\\big ) \\, \\mathrm {d}X^{n+1}_u +\\int _{t_\\ell ^n}^{t} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{su}\\big ) \\, \\mathrm {d}X^{n+1}_u -\\int _s^{t_{k+1}^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n}_{su}\\big ) \\, \\mathrm {d}X^{n}_u\\nonumber \\\\& & \\hspace{256.0748pt}-\\int _{t_\\ell ^n}^{t} \\big ( \\mathbf {U}\\sharp \\delta X^{n}_{su}\\big ) \\, \\mathrm {d}X^{n}_u\\bigg ]\\nonumber \\\\& & +\\bigg [\\int _{t_{k+1}^n}^{t_\\ell ^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{st_{k+1}^n}\\big ) \\, \\mathrm {d}X^{n+1}_u-\\int _{t_{k+1}^n}^{t_\\ell ^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n}_{st_{k+1}^n}\\big ) \\, \\mathrm {d}X^{n}_u \\bigg ] \\ .$ The boundary integrals within the second and third brackets can again be bounded individually using the first estimate in (REF ) only.",
"For instance, ${\\Big \\Vert \\int _{t_\\ell ^n}^{t} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{su}\\big ) \\, \\mathrm {d}X^{n+1}_u\\Big \\Vert }\\\\&\\lesssim & \\Vert X_1\\Vert \\Vert \\mathbf {U}\\Vert \\bigg [ {\\bf 1}_{\\lbrace t_{2\\ell }^{n+1} \\le t< t_{2\\ell +1}^{n+1}\\rbrace } \\int _{t_\\ell ^n}^t |s-u|^{1/2} \\big ( 2^{n+1} \\Vert \\delta X_{t_{2\\ell }^{n+1}t_{2\\ell +1}^{n+1}}\\Vert \\big ) \\\\& &\\hspace{42.67912pt} +{\\bf 1}_{\\lbrace t_{2\\ell +1}^{n+1} \\le t< t_{2\\ell +2}^{n+1}\\rbrace }\\int _{t_{2\\ell }^{n+1}}^{t_{2\\ell +1}^{n+1}} |s-u|^{1/2} \\big ( 2^{n+1} \\Vert \\delta X_{t_{2\\ell }^{n+1}t_{2\\ell +1}^{n+1}}\\Vert \\big )\\\\& &\\hspace{42.67912pt} +{\\bf 1}_{\\lbrace t_{2\\ell +1}^{n+1} \\le t< t_{2\\ell +2}^{n+1}\\rbrace }\\int _{t_{2\\ell +1}^{n+1}}^t |s-u|^{1/2} \\big ( 2^{n+1} \\Vert \\delta X_{t_{2\\ell +1}^{n+1}t_{2\\ell +2}^{n+1}}\\Vert \\big )\\bigg ]\\\\&\\lesssim & \\Vert X_1\\Vert ^2 \\Vert \\mathbf {U}\\Vert |t-s|^{2\\gamma } 2^{-n(1/2-\\gamma )} \\ .$ Therefore, we only have to focus on the first bracket in decomposition (REF ).",
"In fact, noting that $\\int _{t_i^n}^{t_{i+1}^n}\\big ( \\mathbf {U}\\sharp \\delta X^{n}_{t_{i}^nu}\\big ) \\, \\mathrm {d}X^{n}_u =\\frac{1}{2} \\big ( \\mathbf {U}\\sharp \\delta X_{t_i^n t_{i+1}^n} \\big ) \\, \\delta X_{t_i^n t_{i+1}^n} \\ ,$ we get ${\\int _{t_{k+1}^n}^{t_\\ell ^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{t_{k+1}^n u}\\big ) \\, \\mathrm {d}X^{n+1}_u-\\int _{t_{k+1}^n}^{t_\\ell ^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n}_{t_{k+1}^nu}\\big ) \\, \\mathrm {d}X^{n}_u} \\nonumber \\\\&=& \\sum _{i=k+1}^{\\ell -1} \\bigg \\lbrace \\int _{t_{2i}^{n+1}}^{t_{2i+1}^{n+1}} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{t_{k+1}^n u}\\big ) \\, \\mathrm {d}X^{n+1}_u+\\int _{t_{2i+1}^{n+1}}^{t_{2i+2}^{n+1}} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{t_{k+1}^n u}\\big ) \\, \\mathrm {d}X^{n+1}_u \\nonumber \\\\& &\\hspace{227.62204pt}-\\int _{t_{i}^n}^{t_{i+1}^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n}_{t_{k+1}^nu}\\big ) \\, \\mathrm {d}X^{n}_u \\bigg \\rbrace \\nonumber \\\\&=& \\sum _{i=k+1}^{\\ell -1} \\bigg \\lbrace \\bigg [\\int _{t_{2i}^{n+1}}^{t_{2i+1}^{n+1}} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{t_{2i}^{n+1} u}\\big ) \\, \\mathrm {d}X^{n+1}_u+\\int _{t_{2i+1}^{n+1}}^{t_{2i+2}^{n+1}} \\big ( \\mathbf {U}\\sharp \\delta X^{n+1}_{t_{2i+1}^{n+1} u}\\big ) \\, \\mathrm {d}X^{n+1}_u \\nonumber \\\\& &\\hspace{227.62204pt}-\\int _{t_{i}^n}^{t_{i+1}^n} \\big ( \\mathbf {U}\\sharp \\delta X^{n}_{t_{i}^nu}\\big ) \\, \\mathrm {d}X^{n}_u\\bigg ]\\nonumber \\\\& &\\hspace{56.9055pt} +\\Big [ \\big ( \\mathbf {U}\\sharp \\delta X_{t_{k+1}^n t_{2i}^{n+1}} \\big ) \\, \\delta X_{t_{2i}^{n+1} t_{2i+1}^{n+1}}+\\big ( \\mathbf {U}\\sharp \\delta X_{t_{k+1}^n t_{2i+1}^{n+1}} \\big ) \\, \\delta X_{t_{2i+1}^{n+1} t_{2i+2}^{n+1}}\\nonumber \\\\& &\\hspace{227.62204pt}-\\big ( \\mathbf {U}\\sharp \\delta X_{t_{k+1}^n t_{2i}^{n+1}} \\big ) \\, \\delta X_{t_{2i}^{n+1} t_{2i+2}^{n+1}} \\Big ] \\bigg \\rbrace \\nonumber \\\\&=& \\sum _{i=k+1}^{\\ell -1} \\bigg \\lbrace \\frac{1}{2}\\bigg [\\big ( \\mathbf {U}\\sharp \\delta X_{t_{2i}^{n+1} t_{2i+1}^{n+1}} \\big ) \\, \\delta X_{t_{2i}^{n+1} t_{2i+1}^{n+1}}+\\big ( \\mathbf {U}\\sharp \\delta X_{t_{2i+1}^{n+1}t_{2i+2}^{n+1}} \\big )\\delta X_{t_{2i+1}^{n+1}t_{2i+2}^{n+1}} \\nonumber \\\\& &\\hspace{199.16928pt}-\\big ( \\mathbf {U}\\sharp \\delta X_{t_{2i}^{n+1}t_{2i+2}^{n+1}} \\big ) \\delta X_{t_{2i}^{n+1}t_{2i+2}^{n+1}} \\bigg ]\\nonumber \\\\& &\\hspace{71.13188pt} +\\Big [ -\\big ( \\mathbf {U}\\sharp \\delta X_{t_{k+1}^n t_{2i}^{n+1}} \\big ) \\, \\delta X_{t_{2i+1}^{n+1} t_{2i+2}^{n+1}}+\\big ( \\mathbf {U}\\sharp \\delta X_{t_{k+1}^n t_{2i+1}^{n+1}} \\big ) \\, \\delta X_{t_{2i+1}^{n+1} t_{2i+2}^{n+1}}\\Big ]\\bigg \\rbrace \\nonumber \\\\&=& \\sum _{i=k+1}^{\\ell -1} \\bigg \\lbrace \\frac{1}{2}\\bigg [-\\big ( \\mathbf {U}\\sharp \\delta X_{t_{2i}^{n+1} t_{2i+1}^{n+1}} \\big ) \\, \\delta X_{t_{2i+1}^{n+1} t_{2i+2}^{n+1}}-\\big ( \\mathbf {U}\\sharp \\delta X_{t_{2i+1}^{n+1}t_{2i+2}^{n+1}} \\big )\\delta X_{t_{2i}^{n+1}t_{2i+1}^{n+1}} \\bigg ]\\nonumber \\\\& &\\hspace{199.16928pt} +\\Big [ \\big ( \\mathbf {U}\\sharp \\delta X_{t_{2i}^{n+1}t_{2i+1}^{n+1}} \\big ) \\, \\delta X_{t_{2i+1}^{n+1} t_{2i+2}^{n+1}}\\Big ]\\bigg \\rbrace \\nonumber \\\\&=&\\frac{1}{2} \\sum _{i=k+1}^{\\ell -1} \\bigg [ \\Big ( \\mathbf {U}\\sharp (\\delta X)_{t_{2i}^{n+1}t_{2i+1}^{n+1}} \\Big ) \\, (\\delta X)_{t_{2i+1}^{n+1}t_{2i+2}^{n+1}}-\\Big ( \\mathbf {U}\\sharp (\\delta X)_{t_{2i+1}^{n+1}t_{2i+2}^{n+1}} \\Big ) \\, (\\delta X)_{t_{2i}^{n+1}t_{2i+1}^{n+1}} \\bigg ] .$ Let us bound the two sums $S^{1,n}_{st}[\\mathbf {U}]:= \\sum _{i=k+1}^{\\ell -1} \\Big ( \\mathbf {U}\\sharp (\\delta X)_{t_{2i}^{n+1}t_{2i+1}^{n+1}} \\Big ) \\, (\\delta X)_{t_{2i+1}^{n+1}t_{2i+2}^{n+1}}$ and $S^{2,n}_{st}[\\mathbf {U}]:=\\sum _{i=k+1}^{\\ell -1} \\Big ( \\mathbf {U}\\sharp (\\delta X)_{t_{2i+1}^{n+1}t_{2i+2}^{n+1}} \\Big ) \\, (\\delta X)_{t_{2i}^{n+1}t_{2i+1}^{n+1}}$ separately.",
"Consider first the case where $\\mathbf {U}=\\sum _{j=1}^o U_j \\otimes V_j$ , with $U_j:=X_{s^j_1} \\cdots X_{s^j_{m_j}} \\quad , \\quad V_j:=X_{s^j_{m_j+1}} \\cdots X_{s^j_{m_j+p_j}} \\ ,$ and $s^j_p \\le s$ for all $j,p$ .",
"Besides, let us set $Y_i=Y_{i,n}:=(\\delta X)_{t_{i}^{n+1}t_{i+1}^{n+1}}$ .",
"With these notations, and for every $r\\ge 1$ , we have ${\\varphi \\big ( |S^{1,n}_{st}[\\mathbf {U}]|^{2r} \\big ) \\ =\\ \\varphi \\bigg ( \\bigg [ \\bigg ( \\sum _{i_1}\\sum _{j_1} U_{j_1}Y_{2i_1}V_{j_1}Y_{2i_1+1}\\bigg ) \\bigg ( \\sum _{i_2} \\sum _{j_2} U_{j_2}Y_{2i_2}V_{j_2}Y_{2i_2+1}\\bigg )^\\ast \\bigg ]^r\\bigg )}\\nonumber \\\\&=& \\sum _{i_1,\\ldots ,i_{2r}}\\sum _{j_1,\\ldots ,j_{2r}} \\varphi \\Big ( \\big [ U_{j_1}Y_{2i_1}V_{j_1}Y_{2i_1+1}Y_{2i_2+1} V_{j_2}^\\ast Y_{2i_2}U_{j_2}^\\ast \\big ]\\cdots \\nonumber \\\\& & \\hspace{85.35826pt}\\big [ U_{j_{2r-1}}Y_{2i_{2r-1}}V_{j_{2r-1}}Y_{2i_{2r-1}+1}Y_{2i_{2r}+1} V_{j_{2r}}^\\ast Y_{2i_{2r}}U_{j_{2r}}^\\ast \\big ] \\Big ) \\ ,$ where each index $i$ runs over $\\lbrace k+1,\\ldots ,\\ell -1\\rbrace $ and each index $j$ runs overs $\\lbrace 1,\\ldots ,o\\rbrace $ .",
"At this point, observe that for all fixed $\\mathbf {i}:=(i_1,\\ldots ,i_{2r})$ and $\\mathbf {j}:=(j_1,\\ldots ,j_{2r})$ , the family $\\lbrace X_{s^{j}_1},\\dots ,X_{s^{j}_{m_j+p_j}},Y_{2i},Y_{2i+1}, \\ i\\in \\lbrace i_1,\\ldots ,i_{2r}\\rbrace , \\ j\\in \\lbrace j_1,\\ldots ,j_{2r}\\rbrace \\rbrace $ is a $q$ -Gaussian family (due to Lemma REF ) and accordingly the associated joint moments obey Formula (REF ).",
"Besides, we have trivially $\\varphi \\big ( Y_{2i_a}Y_{2i_b+1}\\big )=0 \\quad , \\quad \\varphi \\big ( Y_{2i_a}Y_{2i_b}\\big )=\\varphi \\big ( Y_{2i_a+1}Y_{2i_b+1}\\big )={\\bf 1}_{\\lbrace i_a=i_b\\rbrace }2^{-(n+1)} \\varphi \\big ( |X_1|^2)$ and $\\varphi \\big ( Y_{2i} X_{s_a^j})=\\varphi \\big ( Y_{2i+1} X_{s_a^j})=0 \\ .$ Using these basic observations and going back to (REF ), it is clear that, when applying Formula (REF ) to the expectation in (REF ), we can restrict the sum to the set of pairings $\\pi \\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,N_r\\rbrace )$ ($N_r:=2\\big [(m_{j_1}+p_{j_1})+\\ldots +(m_{j_{2r}}+p_{j_{2r}})]+8r$ ) that decompose - in a unique way - as a combination of three sub-pairings, namely: 1) a pairing $\\pi ^1\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace )$ that connects the random variables $\\lbrace Y_{2i}\\rbrace $ to each other; 2) a pairing $\\pi ^2\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace )$ that connects the random variables $\\lbrace Y_{2i+1}\\rbrace $ to each other; 3) a pairing $\\pi ^3\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,N^{\\prime }_r\\rbrace )$ ($N^{\\prime }_r:=2\\big [(m_{j_1}+p_{j_1})+\\ldots +(m_{j_{2r}}+p_{j_{2r}})]$ ) that connects the random variables $\\lbrace X_{s_i^j}\\rbrace $ to each other.",
"Moreover, with this decomposition in mind, one has clearly $\\text{Cr}(\\pi )\\ge \\text{Cr}(\\pi ^1)+\\text{Cr}(\\pi ^2)+\\text{Cr}(\\pi ^3) \\ .$ Consequently, it holds that for all fixed $\\mathbf {i}:=(i_1,\\ldots ,i_{2r})$ and $\\mathbf {j}:=(j_1,\\ldots ,j_{2r})$ , ${\\Big |\\varphi \\Big ( \\big [ U_{j_1}Y_{2i_1}V_{j_1}Y_{2i_1+1}Y_{2i_2+1} V_{j_2}^\\ast Y_{2i_2}U_{j_2}^\\ast \\big ]\\cdots }\\nonumber \\\\& &\\hspace{113.81102pt}\\big [ U_{j_{2r-1}}Y_{2i_{2r-1}}V_{j_{2r-1}}Y_{2i_{2r-1}+1}Y_{2i_{2r}+1} V_{j_{2r}}^\\ast Y_{2i_{2r}}U_{j_{2r}}^\\ast \\big ] \\Big )\\Big |\\nonumber \\\\&\\le &\\sum _{\\begin{array}{c}\\pi ^1,\\pi ^2\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace ) \\\\ \\pi ^3\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,N^{\\prime }_r\\rbrace )\\end{array}} q^{\\text{Cr}(\\pi ^1)+\\text{Cr}(\\pi ^2)+\\text{Cr}(\\pi ^3)} \\nonumber \\\\& &\\hspace{28.45274pt}\\prod _{\\lbrace a,b\\rbrace \\in \\pi ^1} \\varphi \\big (Y_{2i_a}Y_{2i_b}\\big ) {\\bf 1}_{\\lbrace i_a=i_b\\rbrace }\\prod _{\\lbrace c,d\\rbrace \\in \\pi ^2} \\varphi \\big ( Y_{2i_c+1}Y_{2i_d+1}\\big ){\\bf 1}_{\\lbrace i_c=i_d\\rbrace } \\prod _{\\lbrace e,f\\rbrace \\in \\pi ^3} \\varphi \\big ( Z^{\\mathbf {j}}_e Z^{\\mathbf {j}}_f \\big )\\\\&\\le & 2^{-2r(n+1)} \\varphi \\big ( |X_1|^2)^{2r}\\bigg ( \\sum _{\\pi ^1\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace )}q^{\\text{Cr}(\\pi ^1)}\\prod _{\\lbrace a,b\\rbrace \\in \\pi ^1} {\\bf 1}_{\\lbrace i_a=i_b\\rbrace }\\bigg )\\nonumber \\\\& &\\bigg ( \\sum _{\\pi ^2\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace )}q^{\\text{Cr}(\\pi ^2)}\\prod _{\\lbrace c,d\\rbrace \\in \\pi ^2}{\\bf 1}_{\\lbrace i_c=i_d\\rbrace }\\bigg )\\bigg (\\sum _{\\pi ^3\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,N^{\\prime }_r\\rbrace )}q^{\\text{Cr}(\\pi ^3)}\\prod _{\\lbrace e,f\\rbrace \\in \\pi ^3} \\varphi \\big ( Z^{\\mathbf {j}}_e Z^{\\mathbf {j}}_f \\big )\\bigg ) \\ ,\\nonumber $ where $Z^{\\mathbf {j}}$ stands for the natural reordering of the variables $\\lbrace X_{s_m^j}\\rbrace $ , namely for all $a\\in \\lbrace 1,\\ldots ,2r\\rbrace $ and $b\\in \\lbrace 1,\\ldots ,m_{j_{a}}+p_{j_{a}}\\rbrace $ , $Z^{\\mathbf {j}}_{2[(m_{j_1}+p_{j_1})+\\ldots +(m_{j_{a-1}}+p_{j_{a-1}})]+b}=Z^{\\mathbf {j}}_{2[(m_{j_1}+p_{j_1})+\\ldots +(m_{j_{a-1}}+p_{j_{a-1}})]+[2(m_{j_{a}}+p_{j_{a}})-b]}:=X_{s_b^{j_a}} \\ .$ As a result, the double sum in (REF ) is bounded by ${2^{-2rn} \\varphi \\big ( |X_1|^2)^{2r}\\bigg ( \\sum _{\\pi ^1\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace )}q^{\\text{Cr}(\\pi ^1)}\\sum _{i_1,\\ldots ,i_{2r}=k+1}^{\\ell -1}\\prod _{\\lbrace a,b\\rbrace \\in \\pi ^1} {\\bf 1}_{\\lbrace i_a=i_b\\rbrace }\\bigg )}\\nonumber \\\\& &\\bigg ( \\sum _{\\pi ^2\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace )}q^{\\text{Cr}(\\pi ^2)}\\bigg )\\sum _{j_1,\\ldots ,j_{2r}=1}^o\\bigg (\\sum _{\\pi ^3\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,N^{\\prime }_r\\rbrace )}q^{\\text{Cr}(\\pi ^3)}\\prod _{\\lbrace e,f\\rbrace \\in \\pi ^3} \\varphi \\big ( Z^{\\mathbf {j}}_e Z^{\\mathbf {j}}_f \\big )\\bigg ) \\ .$ Now observe that the last sum in (REF ) actually corresponds to $\\sum _{j_1,\\ldots ,j_{2r}=1}^o\\bigg (\\sum _{\\pi ^3\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,N^{\\prime }_r\\rbrace )}q^{\\text{Cr}(\\pi ^3)}\\prod _{\\lbrace e,f\\rbrace \\in \\pi ^3} \\varphi \\big ( Z^{\\mathbf {j}}_e Z^{\\mathbf {j}}_f \\big )\\bigg )=\\varphi \\Big ( \\Big | \\sum _{j=1}^o U_jV_j \\Big |^{2r}\\Big ) \\ ,$ and for every fixed $\\pi ^1\\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace )$ , ${\\sum _{i_1,\\ldots ,i_{2r}=k+1}^{\\ell -1}\\prod _{\\lbrace a,b\\rbrace \\in \\pi ^1} {\\bf 1}_{\\lbrace i_a=i_b\\rbrace }}\\nonumber \\\\&=&\\bigg (\\sum _{i_1,\\ldots ,i_{2r}=k+1}^{\\ell -1}\\prod _{\\lbrace a,b\\rbrace \\in \\pi ^1} {\\bf 1}_{\\lbrace i_a=i_b\\rbrace }\\bigg )^{2(1-2\\gamma )} \\bigg (\\sum _{i_1,\\ldots ,i_{2r}=k+1}^{\\ell -1}\\prod _{\\lbrace a,b\\rbrace \\in \\pi ^1} {\\bf 1}_{\\lbrace i_a=i_b\\rbrace }\\bigg )^{4\\gamma -1}\\nonumber \\\\&\\le & (\\ell -(k+1))^{2(1-2\\gamma )r}(\\ell -(k+1))^{2r(4\\gamma -1)}\\nonumber \\\\& \\le &|t_\\ell ^n-t_{k+1}^n|^{4r\\gamma } 2^{4r\\gamma n} \\, \\le \\, |t-s|^{4r\\gamma } 2^{4r\\gamma n} \\, .",
"$ By injecting (REF ) and (REF ) into (REF ), we end up with the estimate ${\\varphi \\big ( |S^{1,n}_{st}[\\mathbf {U}]|^{2r} \\big )}\\\\&\\le & |t-s|^{4r\\gamma }2^{-2r(1-2\\gamma )n} \\varphi \\big ( |X_1|^2)^{2r}\\bigg ( \\sum _{\\pi \\in \\mathcal {P}_2(\\lbrace 1,\\ldots ,2r\\rbrace )}q^{\\text{Cr}(\\pi )}\\bigg )^2\\varphi \\Big ( \\Big | \\sum _{j=1}^o U_jV_j \\Big |^{2r}\\Big )\\\\&\\le & |t-s|^{4r\\gamma }2^{-2r(1-2\\gamma )n} \\varphi \\big ( |X_1|^2)^{2r}\\varphi \\big ( |X_1|^{2r}\\big )^{2}\\varphi \\Big ( \\Big | \\sum _{j=1}^o U_jV_j \\Big |^{2r}\\Big ) \\ ,$ and so ${\\varphi \\big ( |S^{1,n}_{st}[\\mathbf {U}]|^{2r} \\big )^{1/2r}}\\nonumber \\\\&\\le & |t-s|^{2\\gamma }2^{-(1-2\\gamma )n} \\varphi \\big ( |X_1|^2)\\varphi \\big ( |X_1|^{2r}\\big )^{1/r}\\varphi \\Big ( \\Big | \\sum _{j=1}^o U_jV_j \\Big |^{2r}\\Big )^{1/2r} \\nonumber \\\\&\\le & |t-s|^{2\\gamma }2^{-(1-2\\gamma )n} \\Vert X_1\\Vert ^4 \\Big \\Vert \\sum _{j=1}^o U_jV_j \\Big \\Vert \\nonumber \\\\& \\le & |t-s|^{2\\gamma }2^{-(1-2\\gamma )n} \\Vert X_1\\Vert ^4 \\Big (\\sum _{j=1}^o \\Vert U_j\\Vert \\Vert V_j\\Vert \\Big ) \\ .$ It is easy to see that the above arguments could also be applied to the more general situation where $\\mathbf {U}:=\\sum _{j=1}^o U_j \\otimes V_j$ with $U_j:=\\sum _{k=0}^{K_j} \\alpha _{j,k} X_{s^{j,k}_1} \\cdots X_{s^{j,k}_{m_{j,k}}}$ and $V_j:=\\sum _{\\ell =0}^{L_j}\\beta _{j,\\ell }X_{u^{j,\\ell }_{1}} \\cdots X_{u^{j,\\ell }_{p_{j,\\ell }}} \\ , \\quad \\alpha _{j,k},\\beta _{j,\\ell }\\in , \\ s_a^{j,k},u_b^{j,\\ell }\\in [0,s] \\ ,$ leading in the end to the same bound (REF ).",
"Therefore, this bound (REF ) can actually be extended to any $U_j,V_j \\in \\mathcal {A}_s$ , which then entails that for every $\\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s$ , $\\varphi \\big ( |S^{1,n}_{st}[\\mathbf {U}]|^{2r} \\big )^{1/2r} \\le |t-s|^{2\\gamma }2^{-(1-2\\gamma )n} \\Vert X_1\\Vert ^4 \\Vert \\mathbf {U}\\Vert \\ ,$ and by letting $r$ tend to infinity, we get by (REF ) that $\\Vert S^{1,n}_{st}[\\mathbf {U}]\\Vert \\le |t-s|^{2\\gamma }2^{-(1-2\\gamma )n} \\Vert X_1\\Vert ^4 \\Vert \\mathbf {U}\\Vert \\ .$ The very same reasoning can of course be used in order to estimate $\\Vert S^{2,n}_{st}[\\mathbf {U}]\\Vert $ , with the same resulting bound.",
"Going back to (REF ) and (REF ), we have thus proved that $\\mathbb {X}^{2,n}$ is a Cauchy sequence in $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightharpoonup }))$ , and by Lemma REF , we can therefore assert that it converges in this space to some element $\\mathbb {X}^{2,S}$ .",
"The product Chen identity (REF ) for $\\mathbb {X}^{2,S}$ is readily obtained by passing to the limit (in a pointwise way) in the product Chen identity that is trivially satisfied by $\\mathbb {X}^{2,n}$ .",
"Finally, in order to show that $\\mathbb {X}^{2,S}$ actually belongs to $\\mathcal {C}_2^\\lambda (\\mathcal {L}_T(\\mathcal {A}_{\\rightarrow }))$ , fix $s<t$ , $\\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes }\\mathcal {A}_s$ , and set $W^n:=\\mathbb {X}^{2,n}_{st}[\\mathbf {U}] \\ , \\ W:=\\mathbb {X}^{2,S}_{st}[\\mathbf {U}] \\ , \\ \\bar{W}^n:=\\int _s^{t_\\ell ^n} (\\mathbf {U}\\sharp \\delta X^n_{su})\\, \\mathrm {d}X^n_u \\ ,$ where $t_\\ell ^n$ is such that $s<t_\\ell ^n\\le t <t_{\\ell +1}^n$ (considering $n$ large enough).",
"Using the first estimate in (REF ), it is easy to check that $\\Vert W^n-\\bar{W}^n \\Vert \\rightarrow 0$ , and so, since $\\Vert W^n-W\\Vert \\rightarrow 0$ , we get that $\\Vert \\bar{W}^n - W\\Vert \\rightarrow 0$ .",
"As $\\bar{W}^n \\in \\mathcal {A}_t$ , we can conclude that $W\\in \\mathcal {A}_t$ , as expected.",
"Remark 3.4 Observe that in a commutative setting, the sum (REF ) would simply vanish, leading to an almost trivial proof, which clearly points out the specificity of our non-commutative framework (as evoked in Remark REF )."
],
[
"Comparison with $L^2(\\varphi )$ -constructions",
"Our objective in this section is to compare the previous $L^\\infty (\\varphi )$ -constructions (i.e., constructions based on the operator norm) with the $L^2(\\varphi )$ -constructions exhibited by Donati-Martin in [7].",
"In brief, we shall see that, when studied in $L^2(\\varphi )$ , the previous rough constructions correspond to Stratonovich-type integrals, while the constructions in [7] are more of a Itô-type.",
"This comparison relies on an additional ingredient, the so-called second-quantization operator, whose central role in $q$ -integration theory was already pointed out in Donati-Martin's work.",
"Since we intend to make specific references to some of the results of [7], we assume for simplicity that we are exactly in the same setting as in the latter study.",
"Namely, for a fixed $q\\in [0,1)$ , we assume that the $q$ -Bm $\\lbrace X_t\\rbrace _{t\\ge 0}$ we will handle in this section is constructed as the canonical process on the $q$ -Fock space $(\\mathcal {A},\\varphi )$ (see [7] for details on these structures).",
"As in the previous sections, we denote by $\\mathcal {A}_t$ the closure, with respect to the operator norm, of the algebra generated by $\\lbrace X_s\\rbrace _{s\\le t}$ ."
],
[
"Second quantization",
"Recall that the space $L^2(\\varphi )$ is defined as the completion of $\\mathcal {A}$ as a Hilbert space through the product $\\langle U,V\\rangle :=\\varphi ( UV^\\ast ) \\ .$ We will denote by $\\Vert .\\Vert _{L^2(\\varphi )}$ the associated norm, to be distinguished from the operator norm $\\Vert .\\Vert $ .",
"For every $t\\ge 0$ , let $\\mathcal {B}_t$ be the von Neumann algebra generated by $\\lbrace X_s\\rbrace _{s\\le t}$ (observe in particular that $\\mathcal {A}_t\\subset \\mathcal {B}_t \\subset \\mathcal {A}$ ) and denote by $\\varphi ( \\cdot |\\mathcal {B}_t)$ the conditional expectation with respect to $\\mathcal {B}_t$ .",
"In other words, for every $U\\in \\mathcal {A}$ , $\\varphi (U|\\mathcal {B}_t)$ stands for the orthogonal projection of $U$ onto $\\mathcal {B}_t$ , with respect to the product (REF ): $Z=\\varphi (U|\\mathcal {B}_t)$ if and only if $Z\\in \\mathcal {B}_t$ and $\\varphi (ZW^\\ast )=\\varphi (UW^\\ast )$ for every $W\\in \\mathcal {B}_t$ .",
"A possible way to introduce the second-quantization operator goes through the following invariance result: Lemma 4.1 [7].",
"For all $s_0<t_0$ , $s_1<t_1$ , with $s_0\\le s_1$ , and $U\\in \\mathcal {A}_{s_0}\\subset \\mathcal {A}_{s_1}$ , it holds that $\\frac{\\varphi \\big ((\\delta X)_{s_0t_0} U (\\delta X)_{s_0t_0} \\big | \\mathcal {B}_{s_0}\\big )}{|t_0-s_0|}=\\frac{\\varphi \\big ((\\delta X)_{s_1t_1}U (\\delta X)_{s_1t_1}\\big | \\mathcal {B}_{s_1}\\big )}{|t_1-s_1|} \\ .$ Definition 4.2 We call second quantization of $X$ the operator $\\Gamma _q: \\cup _{t\\ge 0} \\mathcal {A}_t \\rightarrow \\mathcal {A}$ defined for all $s\\ge 0$ and $U\\in \\mathcal {A}_s$ by the formula $\\Gamma _q(U):=\\varphi \\big ((\\delta X)_{s,s+1} U (\\delta X)_{s,s+1} \\big | \\mathcal {B}_{s}\\big ) \\ .$ In particular, for all $s\\ge 0$ and $U\\in \\mathcal {A}_s$ , $\\Gamma _q(U)\\in \\mathcal {B}_s$ , $\\Gamma _q(U)^\\ast =\\Gamma _q(U^\\ast )$ and $\\Vert \\Gamma _q(U)\\Vert _{L^2(\\varphi )}\\le \\Vert (\\delta X)_{s,s+1} U (\\delta X)_{s,s+1}\\Vert _{L^2(\\varphi )}\\le \\Vert X_1\\Vert ^2\\Vert U\\Vert \\ .$ Remark 4.3 For $q=0$ , it is easy to check that, thanks to the freeness properties of $X$ , the second quantization reduces to $\\Gamma _0(U)=\\varphi (U)$ , while in the commutative situation, that is when $q\\rightarrow 1$ , one has (at least morally) $\\Gamma _1(U)=U$ .",
"In fact, we will essentially use the operator $\\Gamma _q$ through the following result, which offers a quite general tool to study Itô/Stratonovich correction terms (for the sake of clarity, we have postponed the proof of this proposition to Section REF ): Proposition 4.4 For every adapted triprocess $\\mathcal {U}\\in \\mathcal {C}_1^\\varepsilon ([s,t];\\mathcal {A}^{\\hat{\\otimes }3})$ ($s<t$ , $\\varepsilon >0$ ) and every subdivision $\\Delta $ of $[s,t]$ whose mesh $|\\Delta |$ tends to 0, it holds that $\\sum _{(t_i)\\in \\Delta } (\\delta X_{t_i t_{i+1}} \\sharp \\mathcal {U}_{t_i})\\sharp \\delta X_{t_i t_{i+1}} \\longrightarrow \\int _s^t \\big [\\mbox{Id}\\times \\Gamma _q\\times \\mbox{Id}\\big ](\\mathcal {U}_{u})\\, \\mathrm {d}u \\quad \\text{in} \\ L^2(\\varphi ) \\ ,$ where $\\mbox{Id}\\times \\Gamma _q \\times \\mbox{Id}$ stands for the continuous extension, as an operator from $\\cup _{u\\ge 0}\\mathcal {A}_u^{\\hat{\\otimes }3}$ to $L^2(\\varphi )$ , of the operator $(\\mbox{Id}\\times \\Gamma _q \\times \\mbox{Id})(U_1 \\otimes U_2 \\otimes U_3):=U_1 \\Gamma _q(U_2) U_3 \\quad , \\ U_1,U_2,U_2\\in \\mathcal {A}_u \\ .$"
],
[
"Non-commutative Itô integral",
"Let us here slightly rephrase the results of [7] regarding Itô's approach to stochastic integration with respect to $X$ .",
"Definition 4.5 Fix an interval $I\\subset \\mathbb {R}$ .",
"An adapted biprocess $\\mathbf {U}: I\\rightarrow \\mathcal {A}\\hat{\\otimes } \\mathcal {A}$ is said to be Itô integrable against $X$ if it is adapted and if for every partition $\\Delta $ of $I$ whose mesh $|\\Delta |$ tends to 0, the sequence of Riemann sums $S^\\Delta _X(\\mathbf {U}):=\\sum _{t_i \\in \\Delta } \\mathbf {U}_{t_i}\\sharp \\delta X_{t_it_{i+1}}$ converges in $L^2(\\varphi )$ (as $|\\Delta |\\rightarrow 0$ ).",
"In this case, we call the limit of $S^\\Delta _X(\\mathbf {U})$ the product Itô integral of $\\mathbf {U}$ against $X$ , and we denote it by $\\int _I \\mathbf {U}_s \\sharp \\mathrm {d}X_s \\in L^2(\\varphi ) \\ .$ Given a biprocess $\\mathbf {U}:I\\rightarrow \\mathcal {A}\\hat{\\otimes }\\mathcal {A}$ and a partition $\\Delta $ of $I$ , we denote by $\\mathbf {U}^\\Delta $ the step-approximation $\\mathbf {U}^\\Delta :=\\sum _{t_i\\in \\Delta } \\mathbf {U}_{t_i} {\\bf 1}_{[t_i,t_{i+1}[} \\ .$ The following isometry property, to be compared with the classical Brownian Itô isometry, is the key ingredient to identify Itô-integrable processes: Proposition 4.6 [7].",
"For every interval $I\\subset \\mathbb {R}$ , all adapted biprocesses $\\mathbf {U}:I \\rightarrow \\mathcal {A}\\otimes \\mathcal {A}\\quad , \\quad \\mathbf {V}:I\\rightarrow \\mathcal {A}\\otimes \\mathcal {A}\\ ,$ and all partitions $\\Delta _1,\\Delta _2$ of $I$ , it holds that $\\langle S^{\\Delta _1}_X(\\mathbf {U}) , S^{\\Delta _2}_X(\\mathbf {V}) \\rangle _{L^2(\\varphi )} =\\int _0^\\infty \\langle \\langle \\mathbf {U}^{\\Delta _1}_u,\\mathbf {V}^{\\Delta _2}_u \\rangle \\rangle _q\\, \\mathrm {d}u \\ ,$ where $\\langle \\langle .,.",
"\\rangle \\rangle _q$ is the bilinear extension of the application defined for all $U_1,U_2,V_1,V_2\\in \\cup _{t\\ge 0}\\mathcal {A}_t$ as $\\langle \\langle U_1 \\otimes U_2,V_1\\otimes V_2 \\rangle \\rangle _q:=\\varphi \\big (U_1 \\Gamma _q(U_2 V_{2}^\\ast ) V_1^{\\ast } \\big ) \\ .$ Corollary 4.7 Let $\\mathbf {U}:I\\rightarrow \\mathcal {A}\\hat{\\otimes }\\mathcal {A}$ be an adapted biprocess such that $\\int _I \\Vert \\mathbf {U}_u\\Vert _{\\mathcal {A}\\hat{\\otimes } \\mathcal {A}}^2 \\, \\mathrm {d}u <\\infty \\quad \\text{and} \\quad \\int _I \\Vert \\mathbf {U}^\\Delta _u-\\mathbf {U}_u\\Vert _{\\mathcal {A}\\hat{\\otimes }\\mathcal {A}}^2 \\, \\mathrm {d}u \\rightarrow 0 \\ \\ \\text{as}\\ |\\Delta |\\rightarrow 0 \\ ,$ for every partition $\\Delta $ of $I$ .",
"Then $\\mathbf {U}$ is Itô integrable against $X$ and $\\Big \\Vert \\int _I \\mathbf {U}_u \\sharp \\mathrm {d}X_u \\Big \\Vert _{L^2(\\varphi )}^2=\\int _I \\langle \\langle \\mathbf {U}_u, \\mathbf {U}_u \\rangle \\rangle _q \\, \\mathrm {d}u \\ .$ Let us just provide a few details, the procedure being essential standard.",
"Consider a sequence $\\mathbf {U}^n:I\\rightarrow \\mathcal {A}\\otimes \\mathcal {A}$ of adapted biprocesses such that for every $t\\in I$ , $\\Vert \\mathbf {U}^n_t-\\mathbf {U}_t\\Vert \\rightarrow 0$ .",
"Then, given two partitions $\\Delta _1,\\Delta _2$ of $I$ , one has by (REF ) $\\big \\Vert S^{\\Delta _1}_X(\\mathbf {U}^n)-S^{\\Delta _2}_X(\\mathbf {U}^n) \\big \\Vert _{L^2(\\varphi )}^2 =\\int _I \\langle \\langle \\mathbf {U}^{n,\\Delta _1}_u-\\mathbf {U}^{n,\\Delta _2}_u, \\mathbf {U}^{n,\\Delta _1}_u-\\mathbf {U}^{n,\\Delta _2}_u \\rangle \\rangle _q \\, \\mathrm {d}u \\ .$ By applying Cauchy-Schwarz inequality and then (REF ), it is readily checked that for all $\\mathbf {V}\\in \\mathcal {A}_s\\otimes \\mathcal {A}_s$ , $\\langle \\langle \\mathbf {V},\\mathbf {V}\\rangle \\rangle _q \\le \\Vert X_1\\Vert ^2 \\Vert \\mathbf {V}\\Vert ^2 \\ ,$ and so $\\big \\Vert S^{\\Delta _1}_X(\\mathbf {U}^n)-S^{\\Delta _2}_X(\\mathbf {U}^n) \\big \\Vert _{L^2(\\varphi )}^2 \\le \\Vert X_1\\Vert ^2 \\int _I \\big \\Vert \\mathbf {U}^{n,\\Delta _1}_u-\\mathbf {U}^{n,\\Delta _2}_u\\big \\Vert ^2 \\, \\mathrm {d}u \\ ,$ which, by letting $n$ tend to infinity, leads us to $\\big \\Vert S^{\\Delta _1}_X(\\mathbf {U})-S^{\\Delta _2}_X(\\mathbf {U}) \\big \\Vert _{L^2(\\varphi )}^2 \\le \\Vert X_1\\Vert ^2 \\int _I \\big \\Vert \\mathbf {U}^{\\Delta _1}_u-\\mathbf {U}^{\\Delta _2}_u\\big \\Vert ^2 \\, \\mathrm {d}u \\ .$ The conclusion easily follows."
],
[
"Comparison with the rough integral",
"We now have all the tools to identify, as elements in $L^2(\\varphi )$ , the rough constructions arising from Sections and .",
"Let us first consider the situation at the level of the product Lévy area provided by Theorem REF .",
"To this end, given $0\\le s<t$ and $\\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes } \\mathcal {A}_s$ , observe that, by Corollary REF , the biprocess $\\mathbf {V}_u:=(\\mathbf {U}\\sharp \\delta X_{su}) \\otimes 1$ is known to be Itô-integrable on $[s,t]$ , which allows us to consider the integral $\\int _s^t (\\mathbf {U}\\sharp \\delta X_{su}) \\, \\mathrm {d}X_u \\ \\in L^2(\\varphi )\\ .$ Proposition 4.8 For all $0\\le s<t$ and every $\\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes } \\mathcal {A}_s$ , it holds that $\\mathbb {X}^{2,S}_{st}[\\mathbf {U}]=\\int _s^t (\\mathbf {U}\\sharp \\delta X_{su}) \\, \\mathrm {d}X_u+\\frac{1}{2} (t-s)\\big ( \\mbox{Id}\\times \\Gamma _q\\big ) [\\mathbf {U}] \\quad \\text{in}\\ L^2(\\varphi ) \\ ,$ where $\\text{Id} \\times \\Gamma _q$ stands for the continuous extension, as an operator from $\\mathcal {A}_s \\hat{\\otimes } \\mathcal {A}_s$ to $L^2(\\varphi )$ , of the operator $\\big (\\text{Id} \\times \\Gamma _q\\big )[U\\otimes V]:=U \\Gamma _q(V) \\ .$ Fix $s<t$ , $\\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes } \\mathcal {A}_s$ , and let $\\tilde{D}^n$ be the subdivision obtained by adding the two times $s,t$ to the dyadic partition $D^n:=\\lbrace i /2^{n},\\, i\\ge 0\\rbrace $ .",
"Denote by $\\tilde{X}^n$ the linear interpolation of $X$ along $\\tilde{D}^n$ and set $\\hat{X}^n:=\\sum _{\\hat{t}_i} X_{\\hat{t}_i} 1_{[\\hat{t}_i,\\hat{t}_{i+1})}$ where $\\lbrace s=\\hat{t}_1 < \\ldots < \\hat{t}_n=t\\rbrace :=\\tilde{D}^n \\cap [s,t]$ .",
"Besides, we recall that the notation $\\mathbb {X}^{2,D^n}$ (or $\\mathbb {X}^{2,\\tilde{D}^n}$ ) has been introduced in (REF ).",
"Using only the $1/2$ -Hölder regularity of $X$ (see (REF )), it is easy to check that for every $\\mathbf {U}\\in \\mathcal {A}_s \\hat{\\otimes } \\mathcal {A}_s$ , $\\Vert \\mathbb {X}^{2,D^n}_{st}[\\mathbf {U}]-\\mathbb {X}^{2,\\tilde{D}^n}_{st}[\\mathbf {U}] \\Vert \\le c \\Vert X_1\\Vert ^2 \\Vert \\mathbf {U} \\Vert \\left|t-s \\right|^{2\\gamma } 2^{-n(1/2-\\gamma )} \\ ,$ for some universal constant $c$ and for every $\\gamma \\in (0,1/2)$ .",
"Thus, by Theorem REF , we can assert that $\\mathbb {X}^{2,\\tilde{D}^n}_{st}[\\mathbf {U}]$ converges to $\\mathbb {X}^{2,S}_{st}[\\mathbf {U}]$ for the operator norm (and accordingly in $L^2(\\varphi )$ ).",
"Now write $\\mathbb {X}^{2,\\tilde{D}^n}_{st}[\\mathbf {U}]&=&\\int _s^t (\\mathbf {U}\\sharp \\delta \\tilde{X}^n_{su} )\\, \\mathrm {d}\\tilde{X}^n_u\\\\&=& \\sum _{k=1}^{n-1} \\frac{1}{\\hat{t}_{k+1}-\\hat{t}_k} \\int _{\\hat{t}_k}^{\\hat{t}_{k+1}}\\mathbf {U}\\sharp \\big ( \\delta X_{s\\hat{t}_k}+\\frac{u-\\hat{t}_k}{\\hat{t}_{k+1}-\\hat{t}_k}(\\delta X)_{\\hat{t}_{k}\\hat{t}_{k+1}} \\big ) \\, \\mathrm {d}u \\, (\\delta X)_{\\hat{t}_{k}\\hat{t}_{k+1}}\\nonumber \\\\&=& \\sum _{k=1}^{n-1} (\\mathbf {U}\\sharp \\delta X_{s \\hat{t}_k} )\\, \\delta X_{\\hat{t}_{k}\\hat{t}_{k+1}}+\\frac{1}{2}\\sum _{k=1}^{n-1} \\mathbf {U}\\sharp (\\delta X)_{\\hat{t}_{k}\\hat{t}_{k+1}} \\, (\\delta X)_{\\hat{t}_{k}\\hat{t}_{k+1}}\\nonumber \\\\&=& \\int _s^t (\\mathbf {U}\\sharp \\delta \\hat{X}^n_{su}) \\, \\mathrm {d}X_u+\\frac{1}{2}\\sum _{k=1}^{n-1} \\mathbf {U}\\sharp (\\delta X)_{\\hat{t}_{k}\\hat{t}_{k+1}} \\, (\\delta X)_{\\hat{t}_{k}\\hat{t}_{k+1}}.$ Thanks to (REF ), it holds that ${\\big \\Vert \\int _s^t (\\mathbf {U}\\sharp \\delta \\hat{X}^n_{su}) \\, \\mathrm {d}X_u-\\int _s^t (\\mathbf {U}\\sharp \\delta X_{su}) \\, \\mathrm {d}X_u \\big \\Vert _{L^2(\\varphi )}^2}\\\\&=&\\int _s^t \\langle \\langle \\, (\\mathbf {U}\\sharp [ \\delta \\hat{X}^n_{su}-\\delta X_{su}])\\otimes 1,(\\mathbf {U}\\sharp [ \\delta \\hat{X}^n_{su}- \\delta X_{su}])\\otimes 1\\, \\rangle \\rangle _q \\, \\mathrm {d}u \\\\&=& \\int _s^t \\big \\Vert \\mathbf {U}\\sharp [ \\delta \\hat{X}^n_{su}-\\delta X_{su}] \\big \\Vert _{L^2(\\varphi )}^2 \\, \\mathrm {d}u \\\\&\\le & \\Vert \\mathbf {U}\\Vert ^2 \\sum _{k=1}^{n-1} \\int _{\\hat{t}_k}^{\\hat{t}_{k+1}} \\Vert X_{\\hat{t}_k}-X_u\\Vert ^2 \\, \\mathrm {d}u \\\\& \\le & \\Vert \\mathbf {U}\\Vert ^2 \\sum _{k=1}^{n-1} \\int _{\\hat{t}_k}^{\\hat{t}_{k+1}} (u-\\hat{t}_k) \\, \\mathrm {d}u \\ \\le \\ \\frac{1}{2} \\Vert \\mathbf {U}\\Vert ^2 2^{-n} \\left|t-s \\right|\\ \\rightarrow \\ 0 \\ .$ Observe finally that the limit of the second term in (REF ) is immediately provided by Proposition REF , which achieves the proof of (REF ).",
"Let us now extend the correction formula (REF ) to any adapted controlled biprocess, that is to the class of biprocesses introduced in Definition REF .",
"Using again Corollary REF , it is easy to check that, as an adapted Hölder path in $\\mathcal {A}\\hat{\\otimes } \\mathcal {A}$ , any such controlled biprocess is Itô-integrable when considered on an interval $I$ of finite Lebesgue measure.",
"This puts us in a position to state the formula: Corollary 4.9 For all $0\\le s<t$ and every adapted controlled biprocess $\\mathbf {U}\\in \\mathbf {Q}_X([s,t])$ with decomposition (REF ), it holds that $\\int _s^t \\mathbf {U}_u\\sharp \\mathrm {d}\\mathbb {X}^{S}_u=\\int _s^t \\mathbf {U}_u \\sharp \\mathrm {d}X_u+\\frac{1}{2} \\int _s^t (\\mbox{Id}\\times \\Gamma _q \\times \\mbox{Id})[\\mathcal {U}^{X,1}_u+\\mathcal {U}^{X,2}_u] \\, \\mathrm {d}u \\quad \\text{in} \\ L^2(\\varphi ) \\ .$ The transition from (REF ) to (REF ) follows from the very same Taylor-expansion argument as in the proof of [6] (related to the free case), and so, for the sake of conciseness, we do not repeat it here.",
"At this point, observe that the combination of Proposition REF and Corollary REF immediately yields the following $q$ -extension of Itô/Stratonovich formula: for all $f\\in \\mathbf {F}_3$ and $s<t$ , $\\delta ( f(X))_{st}=\\int _s^t \\partial f(X_u) \\sharp \\mathrm {d}\\mathbb {X}^S_u=\\int _s^t \\partial f(X_u) \\sharp \\mathrm {d}X_u+\\int _s^t [\\mbox{Id}\\times \\Gamma _q \\times \\mbox{Id}](\\partial ^2 f(X_u)) \\, \\mathrm {d}u \\ .$ As another spin-off of Formula (REF ), we can finally derive an expression of the rough Stratonovich integral $\\int _s^t \\mathbf {U}_u\\sharp \\mathrm {d}\\mathbb {X}^S_u$ as the $L^2(\\varphi )$ -limit of mean-value Riemann sums.",
"The result, which emphasizes the analogy between the rough construction and the classical (commutative) Stratonovich integral, can be stated as follows: Corollary 4.10 For all $0\\le s<t$ and every adapted controlled biprocess $\\mathbf {U}\\in \\mathbf {Q}_X([s,t])$ , it holds that $\\int _s^t \\mathbf {U}_u\\sharp \\mathrm {d}\\mathbb {X}^S_u = \\lim _{|\\Delta |\\rightarrow 0} \\sum _{(t_i)\\in \\Delta } \\frac{1}{2}\\big (\\mathbf {U}_{t_i}+\\mathbf {U}_{t_{i+1}}\\big ) \\sharp \\delta X_{t_it_{i+1}} \\quad \\text{in} \\ L^2(\\varphi ) \\ ,$ for any subdivision $\\Delta $ of $[s,t]$ whose mesh $|\\Delta |$ tends to 0.",
"For any subdivision $\\Delta =(t_i)$ of $[s,t]$ , write ${\\frac{1}{2}\\big (\\mathbf {U}_{t_i}+\\mathbf {U}_{t_{i+1}}\\big )\\sharp \\delta X_{t_it_{i+1}}}\\\\&=&\\mathbf {U}_{t_i}\\sharp \\delta X_{t_it_{i+1}}+\\frac{1}{2}\\delta \\mathbf {U}_{t_it_{i+1}}\\sharp \\delta X_{t_it_{i+1}}\\\\&=&\\mathbf {U}_{t_i}\\sharp \\delta X_{t_it_{i+1}}\\\\& &+\\frac{1}{2} \\big [(\\delta X_{t_i t_{i+1}} \\sharp \\mathcal {U}_{t_i}^{X,1})\\sharp \\delta X_{t_it_{i+1}} +(\\mathcal {U}_{t_i}^{X,2} \\sharp \\delta X_{t_it_{i+1}})\\sharp \\delta X_{t_it_{i+1}}+ \\mathbf {U}^\\flat _{t_it_{i+1}}\\sharp \\delta X_{t_it_{i+1}}\\big ]\\ ,$ and observe that, with the notations of Section REF , we have $\\Vert \\mathbf {U}^\\flat _{t_it_{i+1}}\\sharp \\delta X_{t_it_{i+1}}\\Vert \\le |t_{i+1}-t_i|^{2\\gamma +1/2}\\Vert X_1\\Vert \\, \\mathcal {N}[\\mathbf {U}^\\flat ;\\mathcal {C}_2^{2\\gamma }([s,t])]\\ .$ Taking the sum over $i$ and then letting $|\\Delta |$ tend to 0, we get by Proposition REF that the sum in (REF ) converges in $L^2(\\varphi )$ to the right-hand side of (REF ), which leads us to the conclusion."
],
[
"Proof of Proposition ",
"When $\\mathcal {U}_t=U_t \\otimes V_t \\otimes W_t$ , the convergence property (REF ) has been shown in the proof of [7].",
"However, since we want the formula to hold for general adapted triprocesses here, we need to exhibit additional controls.",
"Let $\\mathcal {U}^n:[s,t]\\rightarrow \\mathcal {A}^{\\otimes 3}$ be a sequence of adapted triprocesses such that $\\big \\Vert \\mathcal {U}^n_u-\\mathcal {U}_u\\big \\Vert \\rightarrow 0$ for every $u\\in [s,t]$ , and fix a subdivision $\\Delta =(t_i)$ of $[s,t]$ .",
"Then set successively $Y_i:=\\delta X_{t_it_{i+1}}$ , $S_{\\Delta }(\\mathcal {U}):=\\sum _{(t_i)\\in \\Delta }\\big \\lbrace ( Y_i\\sharp \\mathcal {U}_{t_i}) \\sharp Y_i - (t_{i+1}-t_i)\\big [\\mbox{Id}\\times \\Gamma _q\\times \\mbox{Id}\\big ](\\mathcal {U}_{t_i})\\big \\rbrace $ $\\text{and}\\quad S_{\\Delta }^n(\\mathcal {U}):=\\sum _{(t_i)\\in \\Delta }\\big \\lbrace ( Y_i\\sharp \\mathcal {U}^n_{t_i}) \\sharp Y_i- (t_{i+1}-t_i)\\big [\\mbox{Id}\\times \\Gamma _q\\times \\mbox{Id}\\big ](\\mathcal {U}^n_{t_i})\\big \\rbrace \\ .$ If $\\mathcal {U}^n_t:=\\sum _{\\ell \\le L^n_t} U^n_{t,\\ell } \\otimes V^n_{t,\\ell } \\otimes W^n_{t,\\ell } \\in \\mathcal {A}_t^{\\otimes 3}$ , $S_{\\Delta }^n(\\mathcal {U})$ thus corresponds to $S_{\\Delta }^n(\\mathcal {U})=\\sum _{(t_i)\\in \\Delta }\\sum _{\\ell \\le L^n_{t_i}} M^n_{i,\\ell } \\ ,$ with $M^n_{i,\\ell }:=U^n_{t_i,\\ell }Y_i V^n_{t_i,\\ell } Y_i W^n_{t_i,\\ell }-(t_{i+1}-t_i)U^n_{t_i,\\ell }\\Gamma _q(V^n_{t_i,\\ell })W^n_{t_i,\\ell } \\ ,$ so $\\Vert S_\\Delta ^n(\\mathcal {U})\\Vert _{L^2(\\varphi )}^2 = \\sum _{(t_{i_1})\\in \\Delta }\\sum _{(t_{i_2})\\in \\Delta }\\sum _{\\ell _1 \\le L^n_{t_{i_1}}}\\sum _{\\ell _2 \\le L^n_{t_{i_2}}}\\varphi \\big ( M^n_{i_1,\\ell _1}(M^n_{i_2,\\ell _2})^\\ast \\big ) \\ .$ For more clarity, let us set $U^n_{i,\\ell }:=U^n_{t_i,\\ell }$ , $V^n_{i,\\ell }:=V^n_{t_i,\\ell }$ , $W^n_{i,\\ell }:=W^n_{t_i,\\ell }$ , and consider then the expansion ${\\varphi \\big ( M^n_{i_1,\\ell _1}(M^n_{i_2,\\ell _2})^\\ast \\big )}\\nonumber \\\\&=& \\varphi \\big (U^n_{i_1,\\ell _1}Y_{i_1} V^n_{i_1,\\ell _1}Y_{i_1}W^n_{i_1,\\ell _1}W^{n,\\ast }_{i_2,\\ell _2}Y_{i_2} V^{n,\\ast }_{i_2,\\ell _2}Y_{i_2}U^{n,\\ast }_{i_2,\\ell _2}\\big )\\nonumber \\\\& &-(t_{i_2+1}-t_{i_2}) \\varphi \\big (U^n_{i_1,\\ell _1}Y_{i_1} V^n_{i_1,\\ell _1}Y_{i_1}W^n_{i_1,\\ell _1}W^{n,\\ast }_{i_2,\\ell _2}\\Gamma _q(V^{n,\\ast }_{i_2,\\ell _2})U^{n,\\ast }_{i_2,\\ell _2}\\big )\\nonumber \\\\& &-(t_{i_1+1}-t_{i_1})\\varphi \\big ( U^n_{i_1,\\ell _1}\\Gamma _q(V^n_{i_1,\\ell _1})W^n_{i_1,\\ell _1}W^{n,\\ast }_{i_2,\\ell _2}Y_{i_2} V^{n,\\ast }_{i_2,\\ell _2}Y_{i_2}U^{n,\\ast }_{i_2,\\ell _2} \\big )\\nonumber \\\\& &+(t_{i_1+1}-t_{i_1})(t_{i_2+1}-t_{i_2}) \\varphi \\big (U^n_{i_1,\\ell _1}\\Gamma _q(V^n_{i_1,\\ell _1})W^n_{i_1,\\ell _1} W^{n,\\ast }_{i_2,\\ell _2}\\Gamma _q(V^{n,\\ast }_{i_2,\\ell _2})U^{n,\\ast }_{i_2,\\ell _2} \\big ) \\ .$ Step 1: Non-diagonal terms ($i_1\\ne i_2$ ).",
"Observe first that if for instance $i_1 < i_2$ , we have, by combining Lemma REF and Definition REF , ${\\varphi \\big (U^n_{i_1,\\ell _1}Y_{i_1} V^n_{i_1,\\ell _1}Y_{i_1}W^n_{i_1,\\ell _1}W^{n,\\ast }_{i_2,\\ell _2}Y_{i_2} V^{n,\\ast }_{i_2,\\ell _2}Y_{i_2}U^{n,\\ast }_{i_2,\\ell _2}\\big )}\\\\&=&\\varphi \\big (U^n_{i_1,\\ell _1}Y_{i_1} V^n_{i_1,\\ell _1}Y_{i_1}W^n_{i_1,\\ell _1}W^{n,\\ast }_{i_2,\\ell _2}\\varphi \\big ( Y_{i_2} V^{n,\\ast }_{i_2,\\ell _2}Y_{i_2}\\big | \\mathcal {B}_{t_{i_2}}\\big ) U^{n,\\ast }_{i_2,\\ell _2}\\big )\\\\&=&(t_{i_2+1}-t_{i_2}) \\varphi \\big (U^n_{i_1,\\ell _1}Y_{i_1} V^n_{i_1,\\ell _1}Y_{i_1}W^n_{i_1,\\ell _1}W^{n,\\ast }_{i_2,\\ell _2}\\Gamma _q(V^{n,\\ast }_{i_2,\\ell _2}) U^{n,\\ast }_{i_2,\\ell _2}\\big ) \\ ,$ and with the same conditioning argument $&\\varphi \\big ( U^n_{i_1,\\ell _1}\\Gamma _q(V^n_{i_1,\\ell _1})W^n_{i_1,\\ell _1}W^{n,\\ast }_{i_2,\\ell _2}Y_{i_2} V^{n,\\ast }_{i_2,\\ell _2}Y_{i_2}U^{n,\\ast }_{i_2,\\ell _2} \\big ) \\\\&=(t_{i_2+1}-t_{i_2}) \\varphi \\big (U^n_{i_1,\\ell _1}\\Gamma _q(V^n_{i_1,\\ell _1})W^n_{i_1,\\ell _1} W^{n,\\ast }_{i_2,\\ell _2}\\Gamma _q(V^{n,\\ast }_{i_2,\\ell _2})U^{n,\\ast }_{i_2,\\ell _2} \\big ) \\ ,$ so that, going back to (REF ), one has $\\varphi \\big ( M^n_{i_1,\\ell _1}(M^n_{i_2,\\ell _2})^\\ast \\big )=0$ .",
"Similar arguments lead to the same conclusion when $i_2<i_1$ .",
"Step 2: Diagonal terms ($i_1=i_2=i$ ).",
"First, observe that with the same conditioning argument as above, decomposition (REF ) actually reduces to $\\varphi \\big ( M^n_{i,\\ell _1}(M^n_{i,\\ell _2})^\\ast \\big )&=&\\varphi \\big (U^n_{i,\\ell _1}Y_{i} V^n_{i,\\ell _1}Y_{i}W^n_{i,\\ell _1}W^{n,\\ast }_{i,\\ell _2}Y_{i} V^{n,\\ast }_{i,\\ell _2}Y_{i}U^{n,\\ast }_{i,\\ell _2}\\big )\\\\& &-(t_{i+1}-t_{i})^2 \\varphi \\big (U^n_{i,\\ell _1}\\Gamma _q(V^n_{i,\\ell _1})W^n_{i,\\ell _1} W^{n,\\ast }_{i,\\ell _2}\\Gamma _q(V^{n,\\ast }_{i,\\ell _2})U^{n,\\ast }_{i,\\ell _2} \\big ) \\ .$ Now, on the one hand, using (REF ) and the Cauchy-Schwarz inequality, ${\\big |\\varphi \\big (U^n_{i,\\ell _1}Y_{i} V^n_{i,\\ell _1}Y_{i}W^n_{i,\\ell _1}W^{n,\\ast }_{i,\\ell _2}Y_{i} V^{n,\\ast }_{i,\\ell _2}Y_{i}U^{n,\\ast }_{i,\\ell _2}\\big )\\big |}\\\\&\\le & \\Vert Y_i\\Vert ^4 \\Vert U^n_{i,\\ell _1}\\Vert \\Vert V^n_{i,\\ell _1}\\Vert \\Vert W^n_{i,\\ell _1}\\Vert \\Vert W^{n}_{i,\\ell _2}\\Vert \\Vert V^{n}_{i,\\ell _2}\\Vert \\Vert U^{n}_{i,\\ell _2}\\Vert \\\\&\\le & (t_{i+1}-t_i)^2\\Vert X_1\\Vert ^4 \\Vert U^n_{i,\\ell _1}\\Vert \\Vert V^n_{i,\\ell _1}\\Vert \\Vert W^n_{i,\\ell _1}\\Vert \\Vert W^{n}_{i,\\ell _2}\\Vert \\Vert V^{n}_{i,\\ell _2}\\Vert \\Vert U^{n}_{i,\\ell _2}\\Vert \\ .$ On the other hand, using the definition of $\\Gamma _q(V_{j_1})$ , ${\\big | \\varphi \\big (U^n_{i,\\ell _1}\\Gamma _q(V^n_{i,\\ell _1})W^n_{i,\\ell _1} W^{n,\\ast }_{i,\\ell _2}\\Gamma _q(V^{n,\\ast }_{i,\\ell _2})U^{n,\\ast }_{i,\\ell _2} \\big ) \\big |}\\\\&=&\\big | \\varphi \\big (U^n_{i,\\ell _1}(\\delta X)_{t_i, t_i+1}V^n_{i,\\ell _1}(\\delta X)_{t_i,t_i+1}W^n_{i,\\ell _1} W^{n,\\ast }_{i,\\ell _2}\\Gamma _q(V^{n,\\ast }_{i,\\ell _2})U^{n,\\ast }_{i,\\ell _2} \\big ) \\big |\\\\&\\le & \\big \\Vert U^{n,\\ast }_{i,\\ell _2} U^n_{i,\\ell _1}(\\delta X)_{t_i, t_i+1}V^n_{i,\\ell _1}(\\delta X)_{t_i,t_i+1}W^n_{i,\\ell _1} W^{n,\\ast }_{i,\\ell _2} \\big \\Vert _{L^2(\\varphi )} \\big \\Vert \\Gamma _q(V^{n,\\ast }_{i,\\ell _2}) \\big \\Vert _{L^2(\\varphi )} \\ ,$ which, by (REF ), entails that ${\\big | \\varphi \\big (U^n_{i,\\ell _1}\\Gamma _q(V^n_{i,\\ell _1})W^n_{i,\\ell _1} W^{n,\\ast }_{i,\\ell _2}\\Gamma _q(V^{n,\\ast }_{i,\\ell _2})U^{n,\\ast }_{i,\\ell _2} \\big ) \\big |}\\\\&\\le & \\Vert X_1\\Vert ^4 \\Vert U^n_{i,\\ell _1}\\Vert \\Vert V^n_{i,\\ell _1}\\Vert \\Vert W^n_{i,\\ell _1}\\Vert \\Vert W^{n}_{i,\\ell _2}\\Vert \\Vert V^{n}_{i,\\ell _2}\\Vert \\Vert U^{n}_{i,\\ell _2}\\Vert \\ .$ Going back to (REF ), we have thus shown that $\\Vert S_\\Delta ^n(\\mathcal {U})\\Vert _{L^2(\\varphi )}^2\\le \\Vert X_1\\Vert ^4 \\sum _{(t_{i})\\in \\Delta }(t_{i+1}-t_i)^2\\Big (\\sum _{\\ell \\le L^n_{t_{i}}}\\Vert U^n_{i,\\ell }\\Vert \\Vert V^n_{i,\\ell }\\Vert \\Vert W^n_{i,\\ell }\\Vert \\Big )^2 \\ ,\\\\$ and so we can assert that $\\Vert S_\\Delta ^n(\\mathcal {U})\\Vert _{L^2(\\varphi )}^2&\\le & \\Vert X_1\\Vert ^4 \\sum _{(t_{i})\\in \\Delta }(t_{i+1}-t_i)^2 \\Vert \\mathcal {U}^n_{t_i}\\Vert ^2 \\\\&\\le &2 \\Vert X_1\\Vert ^4\\bigg \\lbrace \\sum _{(t_{i})\\in \\Delta }(t_{i+1}-t_i)^2 \\Vert \\mathcal {U}^n_{t_i}-\\mathcal {U}_{t_i}\\Vert ^2+\\big (\\sup _{u\\in [s,t]}\\Vert \\mathcal {U}_{u}\\Vert ^2\\big ) |t-s| |\\Delta | \\bigg \\rbrace \\ .$ By letting $n$ tend to infinity first, we can conclude that $\\Vert S_{\\Delta }(\\mathcal {U})\\Vert _{L^2(\\varphi )}^2 \\rightarrow \\ 0$ as the mesh $|\\Delta |$ tends to 0.",
"The convergence $\\sum _{(t_i)\\in \\Delta }(t_{i+1}-t_i)\\big [\\mbox{Id}\\times \\Gamma _q\\times \\mbox{Id}\\big ](\\mathcal {U}_{t_i}) \\rightarrow \\int _s^t \\big [\\mbox{Id}\\times \\Gamma _q\\times \\mbox{Id}\\big ](\\mathcal {U}_{u}) \\, \\mathrm {d}u \\quad \\text{in} \\ L^2(\\varphi )$ follows easily from the regularity of $\\mathcal {U}$ , by noting that for every $u$ and every $\\mathcal {V}\\in \\mathcal {A}_u^{\\hat{\\otimes }3}$ , $\\big \\Vert \\big [\\mbox{Id}\\times \\Gamma _q\\times \\mbox{Id}\\big ](\\mathcal {V}) \\big \\Vert _{L^2(\\varphi )} \\le \\Vert \\mathcal {V}\\Vert \\ .$ This achieves the proof of our statement."
]
] | 1612.05757 |
[
[
"Searches for $t\\bar{t}H$ and $tH$ with $H\\to b\\bar b$"
],
[
"Abstract The associated production of a Higgs boson with a top quark-antiquark pair ($\\text{t}\\bar{\\text{t}}\\text{H}$ production) or with a single top quark ($\\text{tH}$ production) allows a direct measurement of the top-Higgs-Yukawa coupling with minimal model dependence.",
"In this article, recent results of searches for $\\text{t}\\bar{\\text{t}}\\text{H}$ and $\\text{tH}$ production in the $\\text{H}\\rightarrow\\text{b}\\bar{\\text{b}}$ channel performed by the ATLAS and CMS experiments are reviewed.",
"The analyses use pp collision data collected at a centre-of-mass energy of 13$\\,$TeV corresponding to an integrated luminosity of up to 13.2$\\,$fb${}^{-1}$."
],
[
"Introduction",
"The coupling of the Higgs (H) boson to the top (t) quark is of particular interest.",
"In the Standard Model (SM), it is of Yukawa type with a strength $y_{\\text{t}}$ proportional to the t-quark mass, and hence, it is exceptionally large.",
"Thus, $y_{\\text{t}}$ contributes dominantly to various loop processes both in the SM and also in models of new physics.",
"The current value of $y_{\\text{t}}$ is dominated by indirect constraints derived from measurements of the gluon-gluon fusion H-boson production and the $\\text{H}\\rightarrow \\gamma \\gamma $ decay rate and depends on the assumption of absence of new particle contributions to the loop amplitudes.",
"The associated production of a H boson with a t quark-antiquark pair ($\\text{t}\\overline{\\text{t}}\\text{H}$ production) or with a single t quark ($\\text{t}\\text{H}$ production), on the other hand, allows a direct measurement of $y_{\\text{t}}$ with minimal model dependence, cf.",
"Fig.",
"REF .",
"However, the SM cross section of both processes is relatively small with approximately 0.5$\\,\\text{pb}$ and 90$\\,\\text{fb}$ at 13$\\,\\text{Te}\\hspace{-0.66666pt}\\text{V}$ centre-of-mass energy, respectively, making this a difficult measurement.",
"The bottom (b) quark-antiquark final state of the H boson benefits from a large branching ratio.",
"At the same time, the relatively poor jet-energy resolution and the huge combinatorial uncertainty in the event reconstruction require to use multivariate analysis methods to discriminate signal from background processes, where the signal cross section is determined with a fit of the discriminant distributions to the data.",
"Figure: Example leading-order Feynman diagrams contributing to the lepton+jets- channel tt ¯H(bb ¯)\\text{t}\\overline{\\text{t}}\\text{H}(\\text{b}\\overline{\\text{b}}) (left) and the t-channel tH(bb ¯)\\text{t}\\text{H}(\\text{b}\\overline{\\text{b}}) production (centre, right) , ."
],
[
"Searches for $\\text{t}\\overline{\\text{t}}\\text{H}(\\text{b}\\overline{\\text{b}})$ production",
"Both CMS [3] and ATLAS [4] have performed searches for $\\text{t}\\overline{\\text{t}}\\text{H}$ , $\\text{H}\\rightarrow \\text{b}\\overline{\\text{b}}$ , production in the dilepton and lepton+jets final states of the $\\text{t}\\overline{\\text{t}}$ system at 13$\\,\\text{Te}\\hspace{-0.66666pt}\\text{V}$ centre-of-mass energy: CMS has published the first analysis at 13$\\,\\text{Te}\\hspace{-0.66666pt}\\text{V}$ using 2.7$\\,\\text{fb}^{-1}$ of data collected in 2015 [1], ATLAS has published an analysis using 13.2$\\,\\text{fb}^{-1}$ of 2015 and 2016 data [5].",
"Events are generally selected by requiring in the dilepton channel 2 isolated, oppositely-charged leptons (electrons or muons) and $\\ge 3$ jets, $\\ge 2$ of which are identified as coming from b quarks (b-tagged), and in the lepton+jets channel 1 isolated lepton and $\\ge 4$ jets, $\\ge 2$ of which are b-tagged.",
"Additional, channel-dependent criteria are applied, such as a Z-boson-mass veto in same-flavour dilepton events.",
"Subsequently, the selected events are further divided into mutually exclusive categories in jet and b-tag multiplicity.",
"Signal events are expected to contribute particularly in the categories with higher multiplicities, cf.",
"Fig.",
"REF (left).",
"The SM background consists almost entirely of $\\text{t}\\overline{\\text{t}}+\\text{jets}$ events.",
"The additional jets stem either from gluons and light-flavour quarks ($\\text{t}\\overline{\\text{t}}+\\text{LF}$ ) or from heavy-flavour quarks ($\\text{t}\\overline{\\text{t}}+\\text{HF}$ ), such as $\\text{t}\\overline{\\text{t}}+\\text{b}\\overline{\\text{b}}$ events, which represent an irreducible background.",
"The background composition varies between the categories, cf.",
"Fig.",
"REF (left) as an example.",
"This is exploited in the final fit, which is performed simultaneously across all categories to constrain the uncertainties of the different processes.",
"Still, especially the $\\text{t}\\overline{\\text{t}}+\\text{HF}$ processes are difficult to model, and the associated uncertainties limit the overall sensitivity.",
"Figure: Predicted and observed event yields per category in the ATLAS analysis (left) and BDT output in one category of the CMS analysis (right) in the lepton+jets channel, after the fit to the data.The tt ¯+jets\\text{t}\\overline{\\text{t}}+\\text{jets} background is divided by the flavour of the additional jets, and the tt ¯H\\text{t}\\overline{\\text{t}}\\text{H} contribution, normalised to the best-fit value (left) and 15 times the SM expectation (right), is superimposed.For the CMS analysis, the $\\text{t}\\overline{\\text{t}}+\\text{jets}$ background is modelled using Powheg, interfaced with Pythia8 with the CUETP8M1 tune for parton showering.",
"The cross section is normalised to the next-to-next-leading order (NNLO) calculation with resummation to next-to-next-to-leading-logarithmic (NNLL) accuracy using the NNPDF3.0 PDF set.",
"Events are further separated based on the flavour, defined by the hadron content, of the additional jets within acceptance that do not originate from the t-quark decays, considering also the case that two close-by B hadrons form a single jet.",
"A rate uncertainty of 50% is assigned to each of these $\\text{t}\\overline{\\text{t}}+\\text{HF}$ processes.",
"Figure: Best-fit values of the signal-strength modifier μ\\mu with their one standard deviation confidence intervalsof the CMS (left) and ATLAS (right) analyses.For the ATLAS analysis, the $\\text{t}\\overline{\\text{t}}+\\text{jets}$ background is modelled using Powheg and Pythia6 with the Perugia2012 tune, reweighted to match the t-quark and $\\text{t}\\overline{\\text{t}}$ -system $p_{\\text{T}}$ spectra of the NNLO calculation, including resummation at NNLL accuracy.",
"Events with $\\ge 1$ additional b jets are reweighted to a Sherpa+OpenLoops prediction at next-to-leading order, using the CT10 4-flavour-scheme PDF set.",
"Uncertainties on the $\\text{t}\\overline{\\text{t}}+\\text{HF}$ processes are assigned based on the differences observed to a Madgraph5 aMC@NLO prediction, and the overall normalisation of the contributions from $\\text{t}\\overline{\\text{t}}$ +$\\,\\ge 1$ b or c jets events is left freely floating in the final fit.",
"Both analyses employ multivariate methods to combine in each category the information of several variables, such as kinematic properties or invariant masses of combinations of jets and leptons, into a final discriminating variable, cf.",
"Fig.",
"REF (right).",
"The signal cross section $\\sigma $ is determined with a binned fit of the background and signal discriminant distributions to the data, where the uncertainties, which affect the rate and the shape of the distributions, are taken into account via nuisance parameters.",
"At CMS, boosted decision trees (BDTs) and, in several categories, also a matrix-element-method (MEM) classifier are used.",
"The latter is a likelihood of the event kinematics under the signal or background hypothesis, taking into account response and acceptance effects, which is constructed to separate against the important $\\text{t}\\overline{\\text{t}}+\\text{b}\\overline{\\text{b}}$ background.",
"Depending on the category, the MEM variable is an input to the BDT or events are further separated into two sub-categories with low and high BDT-output and the MEM is used as final discriminant in each sub-category.",
"In a novel approach, also dedicated techniques are applied to reconstruct events in which the H boson and the hadronically decaying t quark are fairly boosted, resulting in reduced combinatorics in the jet assignment and thus a better event reconstruction efficiency.",
"At ATLAS, a two-staged multivariate approach is used in the signal-enriched categories.",
"A BDT is trained to assign the jets to the partons from the H-boson and t-quark decays under the signal hypothesis.",
"Based on this event reconstruction, additional separating variables, such as the invariant mass of the b jets from the H-boson decay, are computed.",
"They are used together with reconstruction-independent variables as input to a classification BDT or an artificial neural network that separate signal from background events.",
"In the signal-depleted categories in the lepton+jets channel (dilepton channel) the scalar sum of the jet (and lepton) $p_{\\text{T}}$ is used, which aims at constraining the systematic uncertainties of the background model.",
"CMS obtains a signal strength $\\mu = \\sigma /\\sigma _{\\text{SM}}$ relative to the SM expectation of $\\mu = -2.0^{+1.8}_{-1.8}$ with 2.7$\\,\\text{fb}^{-1}$ of data and ATLAS of $\\mu = 2.1^{+1.0}_{-0.9}$ with 13.2$\\,\\text{fb}^{-1}$ of data, cf.",
"Fig.",
"REF , which are compatible with the SM expectation within 1.7 standard deviations.",
"These correspond to observed (expected) upper limits on $\\mu $ at the 95% confidence level (C.L.)",
"of 2.6 ($3.6^{+1.6}_{-1.1}$ ) for CMS and 4.0 ($1.9^{+0.9}_{-0.5}$ ) for ATLAS."
],
[
"Search for $\\text{t}\\text{H}(\\text{b}\\overline{\\text{b}})$ production",
"At leading order, $\\text{t}\\text{H}$ production occurs predominantly via t-channel and associated tW production.",
"In both cases, the H boson can be emitted either from the t quark or the intermediate W boson, cf.",
"Fig.",
"REF .",
"The amplitudes of both contributions interfere depending on the coupling of the H boson to the t quark and to the W boson, expressed hereafter as coupling strengths $\\kappa _{\\text{t}}$ and $\\kappa _{\\text{V}}$ relative to the SM expectation, respectively.",
"Hence, $\\text{t}\\text{H}$ production is sensitive to both the magnitude and the sign of $y_{\\text{t}}$ .",
"The interference is destructive in the SM, but can in general also be constructive resulting in an enhanced $\\text{t}\\text{H}$ production cross section, e.g.",
"by a factor ten for $\\kappa _{\\text{t}} =-1$ and $\\kappa _{\\text{V}} =+1$ (inverted top coupling scenario, ITC).",
"CMS has performed a search for $\\text{t}\\text{H}$ production in the $\\text{H}\\rightarrow \\text{b}\\overline{\\text{b}}$ final state with a leptonically decaying t quark, using 2.3$\\,\\text{fb}^{-1}$ of data at 13$\\,\\text{Te}\\hspace{-0.66666pt}\\text{V}$ .",
"Events are selected requiring 1 isolated lepton and $\\ge 3$ b-tagged jets, targeting the b quarks from the H boson and t quark decay.",
"Events are further divided into two exclusive signal regions, one with 3 and one with 4 b-tagged jets as well as 1 additional non b-tagged jet in each case, targeting the fourth b jet, which often falls out of acceptance, and the additional light-flavour jet, cf.",
"Fig.",
"REF .",
"The dominant SM background in both signal regions stems from $\\text{t}\\overline{\\text{t}}+\\text{jets}$ production, which is modelled as for the CMS $\\text{t}\\overline{\\text{t}}\\text{H}$ search.",
"Events are reconstructed under both the signal and the $\\text{t}\\overline{\\text{t}}+\\text{jets}$ hypothesis, assigning the jets to the final-state quarks depending on the output of two dedicated BDTs that take into account b-tagging and jet kinematic information.",
"Based on the specific event reconstruction, separating variables, such as the $p_{\\text{T}}$ of the H boson, are computed.",
"These, together with reconstruction-independent variables, are used as input to a classification BDT that separates signal and background, cf.",
"Fig.",
"REF (left).",
"Figure: Classification BDT output for the ITC case (left), after the fit to the data.The tt ¯+jets\\text{t}\\overline{\\text{t}}+\\text{jets} background is divided by the flavour of the additional jets, and the tH\\text{t}\\text{H} contributions in both considered production channels, normalised to 100 times the expectation, are superimposed.Also shown are the observed and expected upper limits on the signal production cross section as a function of κ t \\kappa _{\\text{t}} for κ V =+1\\kappa _{\\text{V}} =+1 (right) .Limits on the signal cross section are obtained for 51 different points in the $(\\kappa _{\\text{t}},\\kappa _{\\text{V}})$ parameter space by fitting the corresponding BDT output distributions to the data, simultaneously in the two signal regions.",
"Since the kinematic properties of the signal events depend on $\\kappa _{\\text{t}}$ and $\\kappa _{\\text{V}}$ , dedicated BDT trainings are performed for each point.",
"The results for $\\kappa _{\\text{V}} =+1$ are shown in Fig.",
"REF (right), further results for $\\kappa _{\\text{V}} =+0.5$ and +1.5 have been derived [2].",
"For the SM and the ITC scenario, observed (expected) upper limits at 95% C.L.",
"of 113.7 ($98.6^{+60.6}_{-34.6}$) times the SM expectation and 6.0 ($6.4^{+3.7}_{-2.2}$) times the ITC expectation, respectively, are obtained."
],
[
"Summary",
"First searches by CMS and ATLAS for $\\text{t}\\overline{\\text{t}}\\text{H}(\\text{b}\\overline{\\text{b}})$ production at 13$\\,\\text{Te}\\hspace{-0.66666pt}\\text{V}$ result in a signal strength of $\\mu = -2.0^{+1.8}_{-1.8}$ with 2.7$\\,\\text{fb}^{-1}$ and $\\mu = 2.1^{+1.0}_{-0.9}$ with 13.2$\\,\\text{fb}^{-1}$ of data.",
"A 95% C.L.",
"upper limit on $\\text{t}\\text{H}(\\text{b}\\overline{\\text{b}})$ production with inverted couplings $\\kappa _{\\text{t}} =-1$ of 6.0 times the expectation is observed by CMS using 2.3$\\,\\text{fb}^{-1}$ of data.",
"The achieved sensitivities are close to or surpass the ones at 8$\\,\\text{Te}\\hspace{-0.66666pt}\\text{V}$ with only a fraction of the data."
]
] | 1612.05490 |
[
[
"Radiative heat transfer between metallic gratings using adaptive spatial\n resolution"
],
[
"Abstract We calculate the radiative heat transfer between two identical metallic one-dimensional lamellar gratings.",
"To this aim we present and exploit a modification to the widely-used Fourier modal method, known as adaptive spatial resolution, based on a stretch of the coordinate associated to the periodicity of the grating.",
"We first show that this technique dramatically improves the rate of convergence when calculating the heat flux, allowing to explore smaller separations.",
"We then present a study of heat flux as a function of the grating height, highlighting a remarkable amplification of the exchanged energy, ascribed to the appearance of spoof-plasmon modes, whose behavior is also spectrally investigated.",
"Differently from previous works, our method allows us to explore a range of grating heights extending over several orders of magnitude.",
"By comparing our results to recent studies we find a consistent quantitative disagreement with some previously obtained results going up to 50\\%.",
"In some cases, this disagreement is explained in terms of an incorrect connection between the reflection operators of the two gratings."
],
[
"Introduction",
"Two bodies kept at different temperatures and separated by a vacuum gap experience a radiative heat transfer mediated by photons.",
"This energy exchange is limited by the well known Stefan-Boltzmann's law in the far field, i.e.",
"when the distance separating the bodies is large compared to the thermal wavelength $\\hbar c/k_B T$ , of the order of $8\\,\\mu $ m at ambient temperature.",
"The pioneering works of Rytov [1], Polder and van Hove [2] first showed that this limit can be surpassed in the near-field regime, as a result of the tunneling of evanescent waves.",
"In particular, the heat transfer can exceed the one between two blackbodies (i.e.",
"the ideal far-field scenario predicted by Stefan-Boltzmann's law) even of several orders of magnitue when the materials support surface resonances, such as plasmons in metals (typically lying in the ultraviolet range of frequencies) and phonon-polaritons in dielectrics (typically in the infrared) [3].",
"Since the contribution of each field mode to radiative heat transfer is weighted by the Planck thermal distribution, negligible in the ultraviolet range at ordinary temperatures, dielectrics supporting surface resonances are typically the best candidates to maximize the heat flux.",
"Stimulated by the theoretical developments, several applications have been proposed for radiative heat transfer, ranging from thermophotovoltaic [4], [5], [6], [7], [8], [9] or solar thermal [10], [11] energy conversion, to heat-assisted data storage [12], nanoscale cooling [13], and the recent emerging field of thermotronics [14].",
"On the experimental side, the theoretical predictions have been verified during the last decade both in the plane-plane and sphere-plane configuration, for a wide range of distances, going from some nanometers to several microns [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [29], [28].",
"Recently, the idea of manipulating the heat flux through the manipulation of some external parameters have attracted a remarkable attention.",
"In fact, both the control of the overall value of the flux and its spectral properties can be extremely relevant for several applications, and in particular for energy conversion.",
"In the spirit of a manipulation thourgh geometrical properties, structured surfaces have been the topic of many theoretical investigations.",
"More specifically, by considering both 1D and 2D periodic gratings, both the radiative heat transfer [30], [31], [32], [33], [34], [35], [36], [37] and the Casimir force at and out of thermal equilibrium [38], [39], [40], [41], [42], [43], [44], [45], [46], [47] have been studied by employing a variety of theoretical approaches.",
"It has been shown that gratings represent indeed a tool to modify, both by reducing and amplifying, radiative heat transfer, as well as to influence its spectral properties.",
"Concerning the Casimir force, it must be mentioned that the force between a sphere and a dielectric [48], [49] or a metallic [50], [51] grating has been recently measured and theoretically investigated [47].",
"In the domain of radiative heat transfer, metallic gratings deserve ideed a special attention.",
"In fact, although, as mentioned above, surface resonances for a metal typically have ultraviolet frequencies, the presence of a periodically structured pattern can result in the presence of new surface resonances, referred to a spoof plasmons [52], [53], whose frequency can be adjusted as a function of the grating parameters and can be brought, for realistic values of the grating lenghtscales, in the infrared region, thus contributing to the flux.",
"Based on this behavior, the authors of Ref.",
"GueroutPRB12 have recently theoretically predicted a high enhancement of the flux between two identical gold gratings.",
"A similar study has been performed for 1D and 2D metallic gratings in Refs.",
"DaiPRB2015,DaiPRB2016a,DaiPRB2016b, and more recently in Ref. YangPRL16.",
"In these papers the reflection upon each grating has been described by means of the Fourier Modal Method (FMM) [54], equipped with the factorization rule introduced in Ref. GranetJOptSocAmA96.",
"It is worth stressing that, since the radiative heat transfer is calculated as an integral with respect to both frequency and wavevector, the choice of the method used to derive the grating reflection operators can drastically affect the computational time as well as the results.",
"This paper is precisely devoted to the study of radiative heat transfer between two gold gratings.",
"We present and discuss a modification to the FMM technique, known as Adaptive Spatial Resolution [56] (we will in the following refer to this modified method as ASR), a technique specifically introduced to accelerate the convergence.",
"This method has also been shown to overcome the known instabilities appearing for metallic gratings [57].",
"With respect to previous works, we extend here this technique to deal with arbitrary conical incidence.",
"Based on this approach, we provide a detailed study of the influence of the grating depth on the overall value of the flux as well as on its spectral properties.",
"We also compare our results to part of the works mentioned above on heat transfer between metallic gratings.",
"We show that (1) the ASR technique produces a dramatic increase of the convergence rate (thus implying a drastic reduction of computational time) and (2) the numerical results for heat transfer considerably differ from results previously obtained using the standard FMM.",
"The paper is structured as follows.",
"In Sec.",
"we present our physical system, describing also our notation and main definitions.",
"In Sec.",
"we discuss in detail the adaptive spatial resolution.",
"Then, in Sec.",
"we present our numerical results, by studying the radiative heat transfer between two gold gratings as a function of the grating height.",
"We finally give in Sec.",
"some conclusive remarks."
],
[
"Physical system",
"The system we are going to address consists of two 1D lamellar gratings separated by vacuum, as shown in Fig.",
"REF .",
"The two gratings are structured along the $x$ axis, translationally invariant along the $y$ axis, and separated by a vacuum gap of thickness $d$ along the $z$ axis.",
"The two gratings share the same period $D$ , while they can have in general different filling fractions $f=l/D$ (see Fig.",
"REF ).",
"Finally, they have different grating heights $h$ , while the substrate below each structured region is assumed to be infinitely thick for both gratings.",
"Figure: (Color online) Geometry of the system: two gratings, labeled with 1 and 2, at a distance dd.",
"The gratings, in general made of different materials, are infinite in the xyxy plane, and periodic in the xx direction with the same period DD.",
"They have corrugation depths h i h_i (i=1,2i=1,2), infinite thicknesses below the grating region, and lengths of the elevated part of the grating l i l_i.",
"This defines the filling factors f i =l i /Df_i=l_i/D.The two gratings, labeled with indexes 1 and 2, are kept by some external heat source at constant temperatures $T_1$ and $T_2$ .",
"Among the numerous theoretical techniques developed to calculate the radiative heat transfer between them, we employ an approach based on the knowledge of the individual scattering operators of the bodies involved, recently introduced for systems involving two [58], [59] and three [60], [61] bodies.",
"This method is based on a plane-wave mode decomposition of the fields, each mode $(\\omega ,\\mathbf {k},p,\\phi )$ being identified by the direction of propagation $\\phi =+,-$ along the $z$ axis, the polarization index $p$ [assuming the values $p=1,2$ which respectively correspond to transverse electric (TE) and transverse magnetic (TM) modes], the frequency $\\omega $ and the transverse wavevector $\\mathbf {k}=(k_x,k_y)$ .",
"In this description, the $z$ component of the wavevector $k_z$ is a dependent variable defined as $k_z=\\sqrt{\\frac{\\omega ^2}{c^2}-\\mathbf {k}^2}.$ In virtue of this mode decomposition, we define the trace of an operator $\\mathcal {O}$ as $\\operatorname{Tr}\\mathcal {O}=\\sum _p\\int \\frac{d^2\\mathbf {k}}{(2\\pi )^2}\\int _0^{+\\infty }\\frac{d\\omega }{2\\pi }\\displaystyle {\\langle p,\\mathbf {k}|}\\mathcal {O}\\displaystyle {|p,\\mathbf {k}\\rangle }.$ Assuming that the external environment is thermalized with body 1, the energy this body receives per unit surface and time is given by [59] $\\varphi =\\hbar \\operatorname{Tr}\\Bigl [\\omega n_{21}U^{(2,1)}f_{-1}(\\mathcal {R}^{(2)-})U^{(2,1)\\dag }f_1(\\mathcal {R}^{(1)+})\\Bigr ],$ where $\\mathcal {R}^{(1)+}$ ($\\mathcal {R}^{(2)-}$ ) is the reflection operator of body 1 (2) for an incoming wave propagating in direction $\\phi =-$ ($\\phi =+$ ) and we have defined $f_\\alpha (\\mathcal {R})={\\left\\lbrace \\begin{array}{ll}\\mathcal {P}_{-1}^{\\text{(pw)}}-\\mathcal {R}\\mathcal {P}_{-1}^{\\text{(pw)}}\\mathcal {R}^\\dag +\\mathcal {R}\\mathcal {P}_{-1}^{\\text{(ew)}}-\\mathcal {P}_{-1}^{\\text{(ew)}}\\mathcal {R}^\\dag \\\\\\hspace{150.79968pt}\\alpha =-1,\\\\\\mathcal {P}_1^{\\text{(pw)}}-\\mathcal {R}^\\dag \\mathcal {P}_1^{\\text{(pw)}}\\mathcal {R}+\\mathcal {R}^\\dag \\mathcal {P}_1^{\\text{(ew)}} -\\mathcal {P}_1^{\\text{(ew)}}\\mathcal {R}\\\\\\hspace{150.79968pt}\\alpha =1.\\end{array}\\right.",
"}$ Moreover, in Eq.",
"(REF ) we have defined the population differences $n_{ij}=n(\\omega ,T_i)-n(\\omega ,T_j)$ , with $n(\\omega ,T)=\\frac{1}{e^{\\frac{\\hbar \\omega }{k_\\text{B}T}}-1},$ the intracavity operator $U^{(21)}=\\sum _{n=0}^{+\\infty }\\bigl (\\mathcal {R}^{(2)-}\\mathcal {R}^{(1)+}\\bigr )^n=(1-\\mathcal {R}^{(2)-}\\mathcal {R}^{(1)+})^{-1},$ and the projection operators $\\displaystyle {\\langle p,\\mathbf {k}|}\\mathcal {P}_n^\\text{(pw/ew)}\\displaystyle {|p^{\\prime },\\mathbf {k}^{\\prime }\\rangle }=k_z^n\\displaystyle {\\langle p,\\mathbf {k}|}\\Pi ^\\text{(pw/ew)}\\displaystyle {|p^{\\prime },\\mathbf {k}^{\\prime }\\rangle },$ where $\\delta _{\\phi \\phi ^{\\prime }}$ is the Kronecker delta and being $\\Pi ^\\text{(pw)}$ [$\\Pi ^\\text{(ew)}$ ] the projector on the propagative ($k<\\omega /c$ ) [evanescent ($k>\\omega /c$ )] sector.",
"We remark that, as discussed in Ref.",
"NotoPRA14, the periodicity on the $x$ axis makes it natural to replace the mode variable $k_x$ with $k_{x,n}=k_x+\\frac{2\\pi }{D}n,$ with $k_x$ taking values in the first Brillouin zone $[-\\pi /D,\\pi /D]$ and $n$ assuming all integer values.",
"Based on Eq.",
"(REF ), the calculation of the flux $\\varphi $ is now reduced to the problem of describing the field reflection upon each grating.",
"The details about the ASR technique employed in this paper are discussed in detail in the next Section."
],
[
"Methods: Fourier modal method with adaptive spatial resolution",
"The physical system we consider is shown in Fig.",
"REF : it is a lamellar grating of period $D$ , invariant along the $y$ axis, with relative dielectric permittivity $\\varepsilon (x)$ .",
"The grating is inlayed between two homogeneous media whose permittivities are $\\varepsilon _i$ (input medium) and $\\varepsilon _o$ (output medium).",
"A monochromatic plane wave with frequency $\\omega $ and parallel wavevector $\\mathbf {k}=(k_x,k_y)$ illuminates the structure.",
"Throughout the calculation, the vacuum wave number is denoted by $k_0 = \\omega /c$ and we assume a time dependence of the form $e^{-i\\omega t}$ .",
"The particularity of the ASR [56] is the use of a new coordinates system $x=F(u)$ in which the coordinate in the $x$ direction is stretched around the permittivity discontinuities.",
"This allows to better describe the permittivity jump, which is crucial in the case of metallic structures.",
"Below, we derive the metric tensor and write the Maxwell's equations associated to the coordinates change.",
"Then we solve them in the three regions of space: $z \\le 0$ , $0 < z < h$ and $z \\ge h$ .",
"After, we derive the boundary conditions and obtain the fields amplitudes.",
"Based on the geometry depicted in Fig.",
"REF , the reflection operator we are going to calculate is $\\mathcal {R}^-$ for a vacuum-grating inferface located at $z=0$ .",
"This operator will allow us to easily deduce $\\mathcal {R}^{(1)+}$ and $\\mathcal {R}^{(2)-}$ as described in detail below.",
"Figure: (Color online) Geometry of the grating.",
"Three regions are identified: z≤0z\\le 0 and z≥hz\\ge h having uniform permittivity (ε i \\varepsilon _i and ε o \\varepsilon _o, respectively) and 0<z<h0<z<h, having a periodic ε(x)\\varepsilon (x)."
],
[
"Solution of Maxwell's equations in the three regions",
"We start by observing that, in general, the presence of a lamellar grating naturally divides the period region $[0,D]$ by a set of discontinuity points $\\lbrace x_l\\rbrace $ with $l=0,\\dots ,N$ , with $x_0=0$ and $x_N=D$ .",
"In view of the change of coordinates from $x$ to $u$ , we analogously define a set of points $\\lbrace u_l\\rbrace $ given by $u_l=Dl/N$ , i.e.",
"uniformly distributed between $u_0=0$ and $u_N=D$ .",
"For the explicit definition of the coordinate transformation $\\left\\lbrace x=F(u), y, z\\right\\rbrace $ we follow Ref.",
"Vallius and write $F(u)=a_{1l}+a_{2l}u+\\frac{a_{3l}}{2\\pi }\\sin \\left\\lbrace 2\\pi \\dfrac{u-u_{l-1}}{u_{l}-u_{l-1}} \\right\\rbrace ,$ in each interval $u\\in [u_{l-1},u_l]$ , with $l=1,\\dots ,N$ , where $\\begin{split}a_{1l}&=(u_{l}x_{l-1}-u_{l-1}x_{l})/(u_{l}-u_{l-1}),\\\\a_{2l}&=(x_{l}-x_{l-1})/(u_{l}-u_{l-1}),\\\\a_{3l}&=G(u_{l}-u_{l-1})-(x_{l}-x_{l-1}).\\end{split}$ In these expressions $G$ being a control parameter, for which we take the value $10^{-3}$ through all the numerical calculations presented in this paper.",
"The corresponding metric tensor is diagonal and reads $g_{ij}=\\operatorname{diag}\\left( f^2, 1, 1 \\right)$ , with $f(u)=dx/du=dF(u)/du$ .",
"Thus Maxwell's equations $\\zeta ^{ijk} \\partial _jE_k=i\\omega \\mu _0 \\sqrt{g}g^{ij}H_j$ and $\\zeta ^{ijk}\\partial _jH_k=-i\\omega \\varepsilon _0 \\varepsilon \\sqrt{g}g^{ij}E_j$ become: ${\\left\\lbrace \\begin{array}{ll}\\partial _y E_z-\\partial _z E_y=ik_0\\dfrac{1}{f(u)}\\widetilde{H}_u \\\\ \\partial _z E_u-\\partial _u E_z=ik_0f(u) \\widetilde{H}_y \\\\ \\partial _u E_y-\\partial _y E_u=ik_0f(u) \\widetilde{H}_z \\end{array}\\right.",
"}$ $ {\\left\\lbrace \\begin{array}{ll}\\partial _y \\widetilde{H}_z-\\partial _z \\widetilde{H}_y=-ik_0 \\varepsilon (u)\\dfrac{1}{f(u)}E_u \\\\ \\partial _z \\widetilde{H}_u-\\partial _u \\widetilde{H}_z=-i k_0 \\varepsilon (u) f(u) E_y \\\\ \\partial _u \\widetilde{H}_y-\\partial _y \\widetilde{H}_u=-ik_0 \\varepsilon (u) f(u) E_z \\end{array}\\right.",
"}$ with $\\widetilde{H}_i=Z_0H_i$ , $Z_0$ being the impedance of vacuum.",
"We now derive from Eqs.",
"(REF ) and (REF ) $E_z$ and $\\tilde{H}_z$ and plug them into the other equations, obtained the following system for the tangential components of the fields: $\\begin{split}&\\partial _z \\begin{pmatrix}E_u \\\\ E_y \\end{pmatrix} \\\\&=\\dfrac{i}{k_0}\\begin{pmatrix} -\\partial _u\\dfrac{1}{a(u)}\\partial _y & k_0^2 f(u)+\\partial _u\\dfrac{1}{a(u)}\\partial _u \\\\ -\\dfrac{k_0^2}{f(u)}-\\partial _y\\dfrac{1}{a(u)}\\partial _y & \\partial _y\\dfrac{1}{a(u)}\\partial _u \\end{pmatrix}\\begin{pmatrix} \\widetilde{H}_u \\\\ \\widetilde{H}_y \\end{pmatrix},\\end{split}$ $ \\begin{split}&\\partial _z \\begin{pmatrix}\\widetilde{H}_u \\\\ \\widetilde{H}_y \\end{pmatrix}\\\\&=\\dfrac{i}{k_0}\\begin{pmatrix} \\partial _u\\dfrac{1}{f(u)}\\partial _y & -k_0^2 a(u)-\\partial _u\\dfrac{1}{f(u)}\\partial _u \\\\ \\dfrac{k_0^2}{b(u)}+\\partial _y\\dfrac{1}{f(u)}\\partial _y & -\\partial _y\\dfrac{1}{f(u)}\\partial _u \\end{pmatrix}\\begin{pmatrix} E_u \\\\ E_y \\end{pmatrix}, \\end{split}$ with $a(u)=f(u)\\varepsilon (u)$ and $b(u)=f(u)/\\varepsilon (u)$ .",
"Next, following the procedure discussed more in detail e.g.",
"in Ref.",
"NotoPRA14, we rewrite this systems of equations in the Fourier space and truncate the series up to the truncation order $N$ , obtaining $ \\partial _z \\begin{pmatrix}\\mathcal {E}_u \\\\ \\mathcal {E}_y \\end{pmatrix}={F}\\begin{pmatrix} \\mathcal {\\widetilde{H}}_u \\\\ \\mathcal {\\widetilde{H}}_y \\end{pmatrix},\\qquad \\partial _z \\begin{pmatrix}\\mathcal {\\widetilde{H}}_u \\\\ \\mathcal {\\widetilde{H}}_y \\end{pmatrix} = {G} \\begin{pmatrix} \\mathcal {E}_u \\\\ \\mathcal {E}_y \\end{pmatrix},$ $ \\begin{split}{F}&=\\dfrac{i}{k_0}\\begin{pmatrix} k_y\\alpha \\llbracket a \\rrbracket ^{-1} & k_0^2 \\llbracket f \\rrbracket -\\alpha \\llbracket a \\rrbracket ^{-1} \\alpha \\\\ -k_0^2 \\llbracket f \\rrbracket ^{-1}+k_y^2\\llbracket a \\rrbracket ^{-1} & -k_y\\llbracket a \\rrbracket ^{-1} \\alpha \\end{pmatrix},\\\\{G}&=\\dfrac{i}{k_0}\\begin{pmatrix} -k_y\\alpha \\llbracket f \\rrbracket ^{-1} & -k_0^2 \\llbracket a \\rrbracket +\\alpha \\llbracket f \\rrbracket ^{-1} \\alpha \\\\ k_0^2 \\llbracket b \\rrbracket ^{-1}-k_y^2\\llbracket f \\rrbracket ^{-1} & k_y\\llbracket f \\rrbracket ^{-1} \\alpha \\end{pmatrix},\\end{split}$ where $\\alpha =\\operatorname{diag}(k_x+2\\pi n/D), n\\in [-N,N]$ , $k_x$ being defined in the first Brillouin zone $[-\\pi /D,\\pi /D]$ .",
"In Eq.",
"(REF ) we have gathered in the column vectors $\\mathcal {E}_{u/y} $ and $\\mathcal {\\widetilde{H}}_{u/y}$ the $2N+1$ Fourier components of the fields $E_{u/y}$ and $\\widetilde{H}_{u/y}$ .",
"Moreover, we have introduced the Toeplitz matrices $\\llbracket f \\rrbracket $ (resp.",
"$\\llbracket a \\rrbracket $ and $\\llbracket b \\rrbracket $ ) defined in terms of the Fourier coefficients of the function $f(u)$ [resp.",
"$a(u)$ and $b(u)$ ].",
"We can further simiplify the notation by introducing the following definitions $\\mathcal {E}=\\begin{pmatrix}\\mathcal {E}_u \\\\ \\mathcal {E}_y \\end{pmatrix}, \\qquad \\mathcal {\\widetilde{H}}=\\begin{pmatrix}\\mathcal {\\widetilde{H}}_u \\\\ \\mathcal {\\widetilde{H}}_y \\end{pmatrix},$ which allow us to write $ \\dfrac{\\partial ^2 \\mathcal {E}(z)}{\\partial z^2} = {F}{G}\\mathcal {E}(z)={P}{D}^2{P}^{-1}\\mathcal {E}(z).$ In this equation, ${D}^2$ is a diagonal matrix containing the eigenvalues of ${F}{G}$ and ${P}$ is the matrix of associated eigenvectors.",
"Then the solution of Eq.",
"(REF ) can be expressed under the form $ \\mathcal {E}(z)={P} \\left( e^{{D}z}{P}^{-1}a + e^{-{D}z}{P}^{-1}b \\right),$ and consequently [from Eq.",
"(REF )]: $ \\begin{split}\\mathcal {\\widetilde{H}}(z)&={G}{P}{D}^{-1} \\left( e^{{D}z}{P}^{-1}a - e^{-{D}z}{P}^{-1}b \\right)\\\\&={P}^{\\prime }\\left( e^{{D}z}{P}^{-1}a - e^{-{D}z}{P}^{-1}b \\right),\\end{split}$ where $a$ and $b$ are the amplitudes of the different waves traveling in the $\\phi =+$ and $\\phi =-$ directions, respectively.",
"The calculation presented here for the grating region, characterized by a periodic $\\varepsilon (x)$ , can be directly applied to the homogeneous regions $z\\le 0$ and $z\\ge h$ as well.",
"In this case, the framework can be simplified by observing that $\\llbracket a \\rrbracket =\\varepsilon \\llbracket f \\rrbracket $ and $\\llbracket b \\rrbracket =\\llbracket f \\rrbracket /\\varepsilon $ , where for the input (output) region $\\varepsilon =\\varepsilon _i$ ($\\varepsilon =\\varepsilon _o$ ).",
"This directly leads to the introduction of the four additional matrices ${P}_i$ , ${P}^{\\prime } _i$ , ${P}_o$ , and ${P}^{\\prime } _o$ .",
"In order to simplify the boundary conditions, the solution in the output medium is modified as follows $\\begin{split}\\mathcal {E}_o(z)&={P}_o \\left( e^{{D}_o(z-h)}{P}^{-1}_oa_o + e^{-{D}_o(z-h)}{P}^{-1}_ob_o \\right),\\\\\\mathcal {\\widetilde{H}}_o(z)&={P}^{\\prime }_o\\left( e^{{D}_o(z-h)}{P}^{-1}_oa_o - e^{-{D}_o(z-h)}{P}^{-1}_ob_o \\right),\\end{split}$ by means of the introduction of a phase factor."
],
[
"Boundary conditions",
"Having solved Maxwell's equations in each region, we can now write down the boundary conditions at the interfaces $z=0$ and $z=h$ : $ z=0 \\Rightarrow {\\left\\lbrace \\begin{array}{ll}a_{i} + b_{i}=a + b\\\\ {P}^{\\prime }_{i} {P}_{i}^{-1}\\left( a_{i} - b_{i} \\right)={P}^{\\prime }{P}^ {-1}\\left( a - b \\right)\\end{array}\\right.",
"}$ and, denoting $\\Phi =e^{h{D}}$ , $ z=h \\Rightarrow {\\left\\lbrace \\begin{array}{ll}{P} \\left( \\Phi {P}^ {-1}a + \\Phi ^{-1}{P}^ {-1}b \\right)=a_{o} + b_{o}\\\\{P}^{\\prime } \\left(\\Phi {P}^ {-1} a - \\Phi ^{-1}{P}^ {-1}b \\right)={P}^{\\prime }_o{P}_{o}^{-1} \\left( a_o - b_o \\right)\\end{array}\\right.",
"}$ In terms of the ${S}$ -matrix algorithm, this gives $ \\begin{split}\\begin{pmatrix} b_i \\\\ {P}^ {-1}a \\end{pmatrix}&=\\begin{pmatrix} -\\mathbb {1} & {P} \\\\ {P}^{\\prime }_i{P}_i^{-1} & {P}^{\\prime }\\end{pmatrix}^{-1}\\\\&\\,\\times \\begin{pmatrix} \\mathbb {1} & -{P} \\\\ {P}^{\\prime }_i{P}_i^{-1} & {P}^{\\prime }\\end{pmatrix}\\begin{pmatrix} a_i \\\\ {P}^ {-1}b \\end{pmatrix}={S}_1 \\begin{pmatrix} a_i \\\\ {P}^ {-1}b \\end{pmatrix},\\end{split}$ and $ \\begin{split}\\begin{pmatrix} {P}^ {-1}b \\\\ a_o \\end{pmatrix}&=\\begin{pmatrix} \\Phi & \\mathbb {0} \\\\ \\mathbb {0} & \\mathbb {1} \\end{pmatrix}\\begin{pmatrix} -{P} & \\mathbb {1} \\\\ {P}^{\\prime } & {P}^{\\prime }_o{P}^{-1}_o \\end{pmatrix}^{-1}\\begin{pmatrix} {P} & -\\mathbb {1} \\\\ {P}^{\\prime } & {P}^{\\prime }_o{P}^{-1}_o \\end{pmatrix}\\\\&\\,\\times \\begin{pmatrix} \\Phi & \\mathbb {0} \\\\ \\mathbb {0} & \\mathbb {1} \\end{pmatrix}\\begin{pmatrix} {P}^ {-1}a \\\\ b_o \\end{pmatrix}={S}_2 \\begin{pmatrix} {P}^ {-1}a \\\\ b_o \\end{pmatrix}.\\end{split}$ In the last two equations the unknowns are the amplitudes $a_i$ and $b_o$ of the incoming waves, $b_i$ and $a_o$ of the reflected and transmitted waves, and the amplitudes $a$ and $b$ of the field in the grating region, in which we have absorbed the factor ${P}^{-1}$ .",
"This is irrelevant for our purposes, since we are only interested in the field amplitudes in the homogeneous media.",
"The final part of the calculation is straightforward (see e.g.",
"Ref. NotoPRA14).",
"We define a chained ${S}$ matrix as [63] S=S1S2, having introduced the associative operation ${A}={B}\\circledast {C}$ , which for three square matrices ${A}$ , ${B}$ and ${C}$ of dimension $4(2N+1)$ is defined as $\\begin{split}{A}_{11}&={B}_{11}+{B}_{12}(\\mathbb {1}-{C}_{11}{B}_{22})^{-1}{C}_{11}{B}_{21},\\\\{A}_{12}&={B}_{12}(\\mathbb {1}-{C}_{11}{B}_{22})^{-1}{C}_{12},\\\\{A}_{21}&={C}_{21}(\\mathbb {1}-{B}_{22}{C}_{11})^{-1}{B}_{21},\\\\{A}_{22}&={C}_{22}+{C}_{21}(\\mathbb {1}-{B}_{22}{C}_{11})^{-1}{B}_{22}{C}_{12},\\end{split}$ where each matrix have been decomposed in four square blocks of dimension $2(2N+1)$ .",
"This ${S}$ matrix satisfies the relation $\\begin{pmatrix} b_i \\\\ a_o \\end{pmatrix}={S} \\begin{pmatrix} a_i \\\\ b_o \\end{pmatrix},$ thus its upper-left $2(2N+1)$ block, relating the reflected field $b_i$ to the incident one $a_i$ in the upper region, can be identified as the $\\mathcal {R}^-_u$ operator we are looking for in the $(u,y)$ reference system.",
"Figure: (Color online) Heat flux between two identical gratings of height h=2μh=2\\,\\mu m, period D=1μD=1\\,\\mu m, filling fraction f=0.5f=0.5, infinite thickness below the grating region, and temperatures T 1 =310T_1=310\\,K and T 2 =290T_2=290\\,K.",
"The flux is calculated using both the FMM (red dashed line) and the ASR (black solid line), for different truncation orders NN.",
"The black dashed line corresponds to the asymptotic value obtained using the ASR for N=10N=10, while the gray dot-dashed line is associated to a 10% error with respect to this asymptotic value."
],
[
"Transformation matrices",
"The steps described in the last Section lead to the derivation of a reflection matrix $\\mathcal {R}^-_u$ , expressed in the transformed reference system $(u,y)$ .",
"Actually, the operator $\\mathcal {R}^-$ needed for the calculation of the flux given by Eq.",
"(REF ) has to be expressed not only in the standard $(x,y)$ Cartesian reference, but also with respect to the basis of two polarization unity vectors $\\begin{split}\\hat{\\mbox{$\\epsilon $}}_\\text{TE}^{\\phi }(\\mathbf {k}_n,\\omega )&=\\frac{1}{k_n}(-k_y\\hat{\\mathbf {x}}+k_{x,n}\\hat{\\mathbf {y}}),\\\\\\hat{\\mbox{$\\epsilon $}}_\\text{TM}^{\\phi }(\\mathbf {k}_n,\\omega )&=\\frac{c}{\\omega }(-k_n\\hat{\\mathbf {z}}+\\phi k_{z,n}\\hat{\\mathbf {k}}_n),\\end{split}$ where $\\hat{\\mathbf {a}}=\\mathbf {a}/|\\mathbf {a}|$ and we have defined $\\mathbf {k}_n=(k_{x,n},k_y)$ and $k_{z,n}=\\sqrt{\\omega ^2/c^2-\\mathbf {k}_n^2}$ .",
"Thus, the final reflection matrix $\\mathcal {R}$ takes the form $\\mathcal {R}^-=({B}^-)^{-1}{T}\\mathcal {R}^-_u{T}^{-1}{B}^+,$ where ${T}$ is the matrix associated with the transformation from $(u,y)$ to $(x,y)$ , whereas ${B}^\\phi $ accounts for the transition from the (TE,TM) basis to the canonical $(x,y)$ basis for fields propagating in the $\\phi $ direction (note that we have $\\phi =+$ for the incident field and $\\phi =-$ for the reflected one, as manifest from Fig.",
"REF ).",
"In order to derive ${T}$ , we start observing that we have, labeling throughout this section with a prime the fields in the $(u,y)$ coordinate system, $E^{\\prime }_u=f(u)E_x$ and $E^{\\prime }_y=E_y$ .",
"This means that the ${T}$ matrix will be block-diagonal and take the form ${T}=\\begin{pmatrix} {T}_x & \\mathbb {0} \\\\ \\mathbb {0} & {T}_y.\\end{pmatrix}$ Moreover, since the coordinate stretch leaves the period unchanged, one just needs to express the $2N+1$ vectors $\\mathcal {E}_x$ and $\\mathcal {E}_y$ as a function of $\\mathcal {E}^{\\prime }_u$ and $\\mathcal {E}^{\\prime }_y$ for a given value of $\\omega $ and $k_x$ in the first Brillouin zone, as $\\mathcal {E}_x={T}_x\\mathcal {E}^{\\prime }_u$ and $\\mathcal {E}_y={T}_y\\mathcal {E}^{\\prime }_y$ .",
"Starting from the Fourier decompositions $E_x=\\sum _ne^{ik_{x,n}x}E_{x,n},\\qquad E^{\\prime }_u=\\sum _me^{ik_{x,m}u}E^{\\prime }_{u,m},$ we easily get $[{T}_x]_{n,m}=\\frac{1}{D}\\int _0^Ddu\\,e^{i[k_{x,m}u-k_{x,n}x(u)]},$ and in an analogous way $[{T}_y]_{n,m}=\\frac{1}{D}\\int _0^Ddu\\,f(u)e^{i[k_{x,m}u-k_{x,n}x(u)]}.$ Concerning the second transformation matrix ${B}^\\phi $ , it represents a basic change of basis and its action is written as $\\begin{pmatrix} \\mathcal {E}_x \\\\ \\mathcal {E}_y \\end{pmatrix}={B}^\\phi \\begin{pmatrix} \\mathcal {E}^\\phi _\\text{TE} \\\\ \\mathcal {E}^\\phi _\\text{TM} \\end{pmatrix}.$ where the (TE,TM) basis (and thus the matrix ${B}^\\phi $ ) depends on the propagation direction $\\phi $ .",
"Using Eq.",
"(REF ), the transformation matrix reads ${B}^\\phi =\\begin{pmatrix} -\\operatorname{diag}\\Bigl (\\frac{k_y}{k_n}\\Bigr ) & \\operatorname{diag}\\Bigl (\\frac{c\\phi k_{x,n}k_{z,n}}{\\omega k_n}\\Bigr ) \\\\ \\operatorname{diag}\\Bigl (\\frac{k_{x,n}}{k_n}\\Bigr ) & \\operatorname{diag}\\Bigl (\\frac{c\\phi k_yk_{z,n}}{\\omega k_n}\\Bigr ) \\end{pmatrix}.$"
],
[
"Reflection operators of the two gratings",
"As stated at the beginning of our calculation, the $\\mathcal {R}^-$ operator we calculated is relative to a grating having its interface with vacuum at $z=0$ .",
"As a consequence, in order to deduce the $\\mathcal {R}^{(2)-}$ relative to grating 2, we have to include a phase shift taking into account the fact that its vacuum-grating inferface coincides with the plane $z=d$ .",
"As described e.g.",
"in Ref.",
"NotoPRA14, the matrix elements of this modified operator are given by $\\begin{split}&\\displaystyle {\\langle p,\\mathbf {k},n,\\omega |}\\mathcal {R}^{(2)-}\\displaystyle {|p^{\\prime },\\mathbf {k}^{\\prime },n^{\\prime },\\omega ^{\\prime }\\rangle }\\\\&=\\exp [i(k_{z,n}+k^{\\prime }_{z,n^{\\prime }})d]\\displaystyle {\\langle p,\\mathbf {k},n,\\omega |}\\mathcal {R}^-\\displaystyle {|p^{\\prime },\\mathbf {k}^{\\prime },n^{\\prime },\\omega ^{\\prime }\\rangle }.\\nonumber \\end{split}$ We now focus on the issue of deriving $\\mathcal {R}^{(1)+}$ from the known operator $\\mathcal {R}^-$ .",
"For simplicity, in the following we will always consider two identical gratings, i.e.",
"having the same height $h$ and filling factor $f$ .",
"In this case, when evaluating the flux given by Eq.",
"(REF ), it is convenient to exploit the symmetry of the configuration and calculate (for each frequency and wavevector) only once the grating reflection matrix.",
"Nevertheless, one must keep in mind that in our formalism the unit polarization vector associated to TE polarization defined in Eq.",
"(REF ) is independent of the propagation direction $\\phi $ along the $z$ axis, while the $x$ and $y$ components of the TM polarization vector are proportional to $\\phi $ .",
"Since the $x$ and $y$ components are the ones which are conserved in the scattering process, this implies that the reflection matrix $\\mathcal {R}^{(1)+}$ coincides with $\\mathcal {R}^-$ in the diagonal blocks (TE,TE) and (TM,TM), while the non-diagonal blocks (TE,TM) and (TM,TE) undergo an overall sign change.",
"This sign-change issue is irrelevant, of course, in the case of a planar slab, since this system does not mix the two polarizations, but it must indeed be taken into account when dealing with bodies producing a coupling between TE and TM modes.",
"We are now ready to discuss our first numerical results.",
"These concern a couple of gratings having the same features used in Ref. GueroutPRB12.",
"The two gratings have filling factor $f=0.5$ , period $D=1\\,\\mu $ m, temperatures $T_1=290\\,$ K and $T_2=310\\,$ K, and are placed at a distance $d=1\\,\\mu $ m. Both gratings are made of gold, for which we have used a Drude model $\\varepsilon (\\omega )=1-\\frac{\\omega _P^2}{\\omega (\\omega +i\\gamma )},$ where the plasma frequency and the dissipation rate are respectively equal to $\\omega _P=9\\,$ eV and $\\gamma =35\\,$ meV.",
"In order to highlight the features of the ASR technique, we start by calculating the flux for a given grating height $h=2\\,\\mu $ m. As discussed in detail in Ref.",
"NotoPRA14, a crucial point when using the FMM technique is the choice of the truncation order $N$ , i.e.",
"the number of diffraction orders taken into account in the Fourier decomposition of the field, going from $-N$ to $N$ .",
"The same issue applies of course to the ASR method as well.",
"We show in Fig.",
"REF the value of the flux $\\varphi $ as a function of the truncation order, both using the FMM (red dashed line) and the ASR (black solid line).",
"It is manifest that in both cases the first truncation orders give very different results, suggesting that $N$ has to be further increased.",
"Nevertheless, starting from $N$ of the order of 5, the two curves are dramatically different.",
"The curve corresponding to the ASR technique very quickly converges to a stable result, and the points corresponding to $N=8,9,10$ are basically undistinguishable.",
"The dashed black line in the plot corresponds to the value obtained with the ASR for $N=10$ , while the dot-dashed line is associated to a 10% error with respect to this result.",
"The results obtained with the FMM show a very different behavior.",
"First of all, up to $N\\simeq 15$ , the obtained flux is non-monotonic, and has a quasi-oscillatory behavior.",
"Moreovoer, even when becoming monotonic as a function of $N$ , the flux converges very slowly to its asymptotic value.",
"More specifically, for $N=20$ the error is of the order of 10%, while for $N=51$ (the value used in Ref.",
"GueroutPRB12) it is of the order of 3%.",
"Figure: (Color online) Spectral heat flux between two identical gratings of height h=4.7μh=4.7\\,\\mu m, period D=0.5μD=0.5\\,\\mu m, filling fraction f=0.6f=0.6, infinite thickness below the grating region, and temperatures T 1 =300T_1=300\\,K and T 2 =301T_2=301\\,K.",
"The black solid line corresponds to our calculation using the ASR, while the blue dashed line gives the result obtained without inverting the sign of non-diagonal blocks of the reflection matrix (see text for more details).",
"The red points are extracted from Ref.",
"DaiPhD.Based on this discussion of the convergence, we use, in the following, a truncation order $N=8$ for the ASR, with an associated relative error of the order of 1% on the integrated flux.",
"Using these parameters, as in Ref.",
"GueroutPRB12, we study the heat transfer $\\varphi (h)$ as a function of the grating height $h$ , and discuss the amplification factor $\\varphi (h)/\\varphi (h=0)$ with respect to the case $h=0$ , i.e.",
"to the case of a planar slab.",
"The results are shown in the main part of Fig.",
"REF for a very wide range of values of $h$ , going from $h=0$ to 1 mm, and compared with the ones (red points) calculated in Ref.",
"GueroutPRB12 from $h=0$ to 6$\\,\\mu $ m. While for $h=100\\,$ nm the amplification factor is close to 1, increasing $h$ produces a huge amplification of the heat flux.",
"In particular, the ratio $\\varphi (h)/\\varphi (h=0)$ grows monotonically with $h$ , and reaches a horizonal asymptotote around $h=500\\,\\mu $ m, with a value slightly above 34.",
"In the inset of Fig.",
"REF , we show the same curve in linear $h$ scale from 0 to $6\\,\\mu $ m. As manifest from the plot, in spite of a qualitative agreement, the two curves are quantitatively different, with a disagreement going up to around 50% for $h=5\\,\\mu $ m. For this value of the grating height we have calculated the flux using FMM with $N=51$ and obtained a result in agreement within around 1% with the one coming from the ASR with $N=8$ .",
"We conclude that the truncation order $N=51$ is large enough to assure a good convergence for the integrated flux using FMM.",
"Thus, the difference between our results and the ones of Ref.",
"GueroutPRB12 does not seem to originate from an insufficient truncation order.",
"It is now instructive to investigate the evolution of the spectral properties of the heat flux as a function of the grating depth $h$ .",
"This behavior is illustrated in Fig.",
"REF .",
"Panel (a) shows the spectral flux $\\varphi (\\omega )$ for $h=0$ , i.e.",
"in a slab-slab configuration.",
"It is well known that in presence of surface modes, while approaching the near-field regime the heat flux becomes more and more monochromatic at the resonance frequency of the mode [3].",
"Even if the distance $d=1\\,\\mu $ m does not fully lie in this distance regime, a signature of the existence of such surface modes is already present.",
"Nevertheless, this is not the case for two gold slabs, simply because the plasma frequency for gold is located well outside the window of frequencies contributing to the flux, defined by the Planck function $n(\\omega ,T)$ .",
"As a result, the only non-monotonic behavior for the flux is observed in the low-frequency region $\\omega \\in [10^{10},10^{13}]\\,$ rad/s, shown in the inset of Fig.",
"REF (a).",
"The spectral flux tends to increase at small frequencies as a result of the divergence of $\\varepsilon (\\omega )$ for $\\omega \\rightarrow 0$ , this growth being at some point compensated by the fact that each mode of the field carries an energy $\\hbar \\omega $ , tending to 0 for $\\omega \\rightarrow 0$ .",
"Let us now focus on Fig.",
"REF (b), where the grating heights $h=1\\,\\mu $ m and $1.5\\,\\mu $ m are taken into account.",
"We first remark that the non-monotonic behavior at low frequencies is still present, but with a decreased maximum value.",
"This feature remains basically unchanged for all the higher values of $h$ shown in Fig.",
"REF .",
"The main feature of Fig.",
"REF is anyway the appearance of two peaks in the spectral flux, sign of the existence of two surface modes, whose frequency depends on the grating height, coherently with the fact that these are indeed spoof plamons.",
"The participation of these new modes produces an overall flux amplification equal to 2 and 2.7, respectively, for $h=1\\,\\mu $ m and $1.5\\,\\mu $ m. An analysis of panel (c) of the same figure shows that increasing $h$ produces both a decrease of the frequency of spoof-plasmon modes and an amplification of the peak height.",
"As shown in Fig.",
"REF (c)-(d) the scenario becomes even more interesting for higher values of $h$ .",
"In this case more and more resonances enter the window of frequencies relevant for radiative heat transfer.",
"This feature has been already discussed in Ref.",
"GueroutPRB12 by analyzing the heat-flux transmission coefficient for a given value of the wavevector $\\mathbf {k}$ .",
"We basically observe two resonances for $h=4\\,\\mu $ m and $7\\,\\mu $ m, while for $h=10\\,\\mu $ m and $20\\,\\mu $ m not only we start observing a comb of resonance frequencies, but we clearly see that the fact that they approach each other produces a constructive interference between them, increasing the overall value of $\\varphi (\\omega )$ in the frequency range considered.",
"This is even more evident in Fig.",
"REF (f), where for $h=100\\,\\mu $ m the frequency-comb behavior is manifest, as well as how this eventually produces a smooth asymptotic $\\varphi (\\omega )$ (reached for $h=1\\,$ mm) which does not show any abrupt change with respect to $\\omega $ .",
"This is the profile giving the asymptotic flux amplification close to 34 discussed above and shown in Fig.",
"REF .",
"In order to compare further our results with previous works, we focus on a configuration studied in Ref.",
"DaiPhD, namely the heat transfer between two identical gratings having height $h=4.7\\,\\mu $ m, period $D=500\\,$ nm and filling factor $f=0.6$ , placed at distance $d=1\\,\\mu $ m. The two chosen temperatures are $T_1=300\\,$ K and $T_2=301\\,$ K. In Ref.",
"DaiPhD, the authors show the spectrum associated with this heat transfer (calculated using the standard FMM) from which we have extracted some points, shown in red in Fig.",
"REF .",
"We have applied our numerical scheme to this scenario and obtained the black curve shown in the same figure.",
"It is manifest that, while sharing the same qualitative behavior, the two curves considerably differ quantitatively, in particular in the region between the two main peaks.",
"In order to try to explain this disagreement we have noticed that in one of their recent papers on this topic (namely, Ref.",
"DaiPRB2016b), the authors clearly state that the fact that the two structures coincide implies that the two reflection operators $\\mathcal {R}^{(1)+}$ and $\\mathcal {R}^{(2)-}$ are equal.",
"As discussed above, this is actually not correct because of the fact that the unit polarization vector in TM polarization depends on the propagation direction $\\phi $ .",
"Nevertheless, we have performed a new calculation for the same structure, using the ASR with $N=8$ , but omitting the sign change we need to deduce $\\mathcal {R}^{(1)+}$ from $\\mathcal {R}_u^{-}$ .",
"The result so obtained is given by the blue dashed curve in Fig.",
"REF and clearly shows an impressively increased agreement with the red points.",
"Thus, we interpret the difference between the two results as due to this sign change, missing in Ref.",
"DaiPhD and related works.",
"Moreover, it must be stressed that the results presented in Ref.",
"DaiPhD also show an oscillatory behavior in frequency, in particular in the region between the two main peaks, which is completely absent in the results obtained by exploiting the ASR, both with and without the sign change.",
"In our opinion, this oscillations are a result of the well-known instabilities of the FMM when dealing with metals.",
"With the aim of further trying to explain the discrepancy with Ref.",
"GueroutPRB12 shown in Fig.",
"REF , we have recalculated using the ASR one of the points of Fig.",
"REF , the one having $h=4.5\\,\\mu $ m, without the mentioned sign change.",
"The result is the blue dot in Fig.",
"REF which, even if approaching the result of Ref.",
"GueroutPRB12, still does not show a good agreement."
],
[
"Conclusions",
"We have addressed the caclulation of the radiative heat flux between two gold gratings.",
"To this aim we have made use of the ASR, a modified version of the FMM introduced to deal with the high dielectric contrast typical of metals.",
"We have shown that this technique produces a striking increase of the convergence rate, allowing us to obtain the heat flux at a distance of 1 $\\mu $ m with a truncation order as low as $N=8$ .",
"This implies a remarkable gain in computational time: to give an idea, for the grating parameters relative to Fig.",
"REF , the calculation of the spectral flux at frequency $\\omega =10^{14}\\,$ rad/s takes approximately 1 minute using the ASR with $N=8$ on a parallel 3.4 GHz 16-core machine, while 1.5 hours are needed using the FMM with $N=51$ .",
"Since the required truncation order increases when decreasing the distance, the ASR would allow to explore smaller separations, which would be prohibitive using the FMM.",
"By using this improved numerical method, we have made a detailed study of the heat-flux amplification as a function of the grating depth.",
"By studying both the flux ratio and the spectral properties, we have proved for our structure an amplification factor going up to 34, explained in terms of the appearance of a comb of spoof-plasmon resonances.",
"We have then compared our results to some previous works, and highlighted a quantitative disagreement both in the integrated flux and in its spectral distribution.",
"In some cases, we have argued that this disagreement is due to the incorrect assumption that the reflection matrices of the two gratings coincide.",
"Our results show that the physics behind metallic gratings makes them ideal candidates for the manipuation of both the overall heat flux and its frequency components, both possibilities being very relevant for several applications such as thermophotovoltaic energy conversion.",
"On a more technical side, our discussion proves that the FMM can provide slowly-converging and unstable results in the presence of metals, while the ASR represents a much more reliable approach even for high dielectric contrasts.",
"The authors acknowledge J. Dai for useful discussions."
]
] | 1612.05516 |
[
[
"On the Question of a Possible Infrared Zero in the Beta Function of the\n Finite-$N$ Gross-Neveu Model"
],
[
"Abstract We investigate whether the beta function of the finite-$N$ Gross-Neveu model, as calculated up to the four-loop level, exhibits evidence for an infrared zero.",
"As part of our analysis, we calculate and analyze Pad\\'e approximants to this beta function and evaluate effects of scheme dependence.",
"From our study, we find that in the range of coupling where the perturbative calculation of the four-loop beta function is reliable, it does not exhibit robust evidence for an infrared zero."
],
[
"Introduction",
"The Gross-Neveu (GN) model [1] is a quantum field theory in $d=2$ spacetime dimensions with an $N$ -component massless fermion $\\psi _j$ , $j=1,...,N$ , defined by the path integral $Z = \\int \\prod _x [{\\cal D}\\psi ][{\\cal D}\\bar{\\psi }] \\,e^{i\\int d^2x \\, {\\cal L}} \\ ,$ with the Lagrangian density [2] ${\\cal L} = i\\bar{\\psi }\\partial \\hspace{-5.05942pt}\\slash \\psi + \\frac{g}{2} (\\bar{\\psi }\\psi )^2 \\ .$ This model is of interest because it exhibits, albeit in a lower-dimensional, non-gauge-theory context, some properties of quantum chromodynamics (QCD), namely asymptotic freedom, dynamical symmetry breaking of a certain chiral symmetry, and the formation of a massive bound state of fermions.",
"These properties were shown by an exact solution of the model in [1] in an $N\\rightarrow \\infty $ limit that enabled Gross and Neveu to obtain nonperturbative information about the theory.",
"A semiclassical calculation of the bound-state spectrum of the model was carried out in [3].",
"The Gross-Neveu model has also been studied at finite $N$ , where it is not, in general, exactly solvable.",
"In these studies, one again makes use of a property that the model shares with QCD, namely asymptotic freedom, which allows one to carry out reliable perturbative calculations at high Euclidean energy/momentum scales $\\mu $ in the deep ultraviolet (UV), where the running four-fermion coupling, $g(\\mu )$ , approaches zero.",
"In this context, there is an interesting and fundamental question: how does this running coupling $g(\\mu )$ change as the scale $\\mu $ decreases from the deep UV to the infrared (IR) limit at $\\mu = 0$ ?",
"This change of $g(\\mu )$ as a function of $\\mu $ is described by the renormalization group (RG) [4] and the associated beta function, $\\beta =dg/dt$ , where $dt = d\\ln \\mu $ .",
"The asymptotic freedom property is equivalent to the fact that $\\beta $ is negative in the vicinity of the origin, $g=0$ , so that this point is a UV fixed point (UVFP) of the renormalization group.",
"As $\\mu $ decreases from the UV toward the IR, several different types of behavior of a theory are, a priori, possible.",
"One is that the (perturbatively calculated) beta function has no IR zero, so that as $\\mu $ decreases, $g(\\mu )$ eventually increases beyond the range where perturbative methods can be used to study its RG evolution.",
"An alternative possibility is that $\\beta $ has an IR zero at sufficiently small coupling so that it can be studied using perturbative methods.",
"An exact IR zero of $\\beta $ would be an IR fixed point (IRFP) of the renormalization group.",
"In the $N \\rightarrow \\infty $ limit used in [1] to solve the model, the resultant beta function (given below in Eq.",
"(REF )) does not exhibit any IR zero.",
"Ref.",
"[5] calculated $1/N$ corrections to the $N \\rightarrow \\infty $ limit in the Gross-Neveu model and excluded the presence of an IR zero to this order.",
"However, to our knowledge, there has not been an analysis of the beta function of the GN model for finite $N$ to higher-loop order to address the question of whether it exhibits evidence for an infrared fixed point.",
"In this paper we shall carry out this analysis of the beta function of the finite-$N$ Gross-Neveu model to address and answer the question of whether this function exhibits an IR zero.",
"We shall investigate the beta function to the highest loop order to which it has been calculated, namely four loops, making use of a recent computation of the four-loop term in Ref.",
"[6].",
"This paper is organized as follows.",
"In Section we review some background information about the Gross-Neveu model.",
"In Section we carry out our analysis of the beta function of the finite-$N$ Gross-Neveu model up to the four-loop level.",
"In Section we extend this analysis using Padé approximants.",
"Section contains an analysis of the effect of scheme transformations on the beta function.",
"In Section we comment further on the large-$N$ limit.",
"Our conclusions are given in Section ."
],
[
"Some Relevant Background on the Gross-Neveu Model",
"Here we briefly review some relevant background concerning the Gross-Neveu model.",
"We first comment on some notation.",
"In Ref.",
"[1], the coefficient in front of the $(\\bar{\\psi }\\psi )^2$ operator was written as a squared coupling, which we denote as $(g_{GN}^2/2)$ , while many subsequent works have written it as $g/2$ , so one has $g \\equiv g_{GN}^2 \\ .$ The analysis of the model in [1] made use of a functional integral identity to express the path integral as the $m \\rightarrow \\infty $ limit of a path integral containing an auxiliary real scalar field $\\phi $ with a mass $m$ and a Yukawa interaction ${\\cal L}_Y = g_{GN}m[\\bar{\\psi }\\psi ]\\phi \\ .$ Since $\\phi $ is a real field, the hermiticity of ${\\cal L}_Y$ implies that $g_{GN}$ must be real, which, in conjunction with Eq.",
"(REF ), implies that $g$ must be non-negative: $g \\ge 0 \\ .$ For $d=2$ (as more generally, for any even spacetime dimension), one can define a product of Dirac gamma matrices, denoted $\\gamma _5$ , that satisfies the anticommutation relation $\\lbrace \\gamma _5,\\gamma _\\mu \\rbrace =0$ for all $\\gamma _\\mu $ .",
"This $\\gamma _5$ matrix also satisfies $\\gamma _5^2=1$ and $\\gamma _5^\\dagger =\\gamma _5$ .",
"(An explicit representation is $\\gamma _0 =\\sigma _1$ , $\\gamma _1 = \\sigma _2$ , with $\\gamma _0 \\gamma _1 = i \\gamma _5 =i\\sigma _3$ , where $\\sigma _j$ are the Pauli matrices.)",
"One can then define chiral projection operators $P_{L,R} = (1/2)(1 \\pm \\gamma _5)$ .",
"As usual, one then defines left and right chiral components of the fermion field as $\\psi _L = P_L \\psi $ and $\\psi _R = P_R \\psi $ .",
"The Gross-Neveu model is invariant under a discrete global ${\\mathbb {Z}}_2$ group generated by the identity and the chiral transformation $\\psi \\rightarrow \\gamma _5 \\psi \\ .$ This discrete chiral transformation (REF ) takes $\\bar{\\psi }\\psi \\rightarrow -\\bar{\\psi }\\psi $ , and hence this ${\\mathbb {Z}}_2$ symmetry forbids (i) a mass term in the Lagrangian (REF ) and (ii) the generation of a nonzero condensate $\\langle \\bar{\\psi }\\psi \\rangle $ .",
"This is true to all (finite) orders of perturbation theory.",
"The Gross-Neveu model is also invariant under the continuous global (cg) symmetry group $G_{cg} = {\\rm U}(N)$ defined by the transformation $\\psi \\rightarrow U \\psi \\ ,$ where $U \\in {\\rm U}(N)$ (so $\\bar{\\psi }\\rightarrow \\bar{\\psi }U^\\dagger $ ).",
"In terms of the chiral components of the fermion field, the continuous global symmetry transformation (REF ) is $\\psi _L \\rightarrow U \\psi _L$ , $\\psi _R \\rightarrow U \\psi _R$ .",
"In contrast to the discrete $\\gamma _5$ symmetry, the continuous symmetry $G_{cg}$ leaves the operator $\\bar{\\psi }\\psi $ invariant [7].",
"An exact solution of the theory was obtained in [1] in the limit $N \\rightarrow \\infty $ and $g_{GN} \\rightarrow 0$ with the product $\\lambda \\equiv g_{GN}^2 N \\equiv gN$ a fixed and finite function of $\\mu $ .",
"We shall denote this as the LN limit (i.e., the large-$N$ limit with the condition (REF ) imposed).",
"In this limit, there is a nonperturbative generation of a nonzero bilinear fermion condensate, $\\langle \\bar{\\psi }\\psi \\rangle $ , dynamically breaking the discrete ${\\mathbb {Z}}_2$ chiral symmetry.",
"In this limit, there is also the formation of a massive bound state of fermions.",
"The beta function for $g_{GN}$ is $\\beta _{GN} = \\frac{dg_{GN}}{dt} \\ ,$ where $dt = d\\ln \\mu $ .",
"(The $\\mu $ dependence of the coupling will often be suppressed in the notation.)",
"This beta function is [1], [8] $\\beta _{GN} = -\\frac{g_{GN}\\lambda }{2\\pi } \\ .$ The fact that this beta function is negative is an expression of the asymptotic freedom of the theory.",
"This beta function does not exhibit any zero away from the origin, i.e., any infrared zero.",
"However, since the calculation in [1] was performed in the LN limit, this leaves open the possibility that at finite $N$ , there could be an IR zero in the beta function that would disappear in the LN limit.",
"We discuss this LN limit further in Section below."
],
[
"Beta Function for General $N$",
"Although the Gross-Neveu model is not, in general, solvable away from the LN limit, there has also been interest over the years in analyzing it for finite $N$ .",
"In terms of the coupling $g$ , the beta function of the finite-$N$ GN model is $\\beta = \\frac{dg}{dt} \\ ,$ where, as before, $dt = d\\ln \\mu $ .",
"For our purposes, it will be convenient to introduce a variable $a$ that includes the factor $1/(2\\pi )$ resulting from Feynman integrals in $d=2$ dimensions, namely $a = \\frac{g}{2\\pi } = \\frac{g_{GN}^2}{2\\pi } \\ .$ The model defined by the Lagrangian of Eq.",
"(REF ) can be generalized with the addition of further four-fermion operators [1], [9].",
"The regularization and renormalization of the Gross-Neveu model has been carried out in this more general context [9]-[13], [6].",
"As was true of other theories, such as the nonlinear $\\sigma $ model [14], one may consider this model in spacetime dimension $d > 2$ .",
"At finite $N$ , the model is not renormalizable for $d > 2$ , since the Maxwellian dimension of a four-fermion operator is $2(d-1)$ , which is larger than $d$ if $d > 2$ .",
"As in the case of the nonlinear $\\sigma $ model [14], in the $N\\rightarrow \\infty $ limit, one can still solve the model and study its properties.",
"Alternatively, for finite $N$ , one can regard it as a low-energy effective field theory.",
"With this generalization and $d \\mathrel {\\raisebox {-.6ex}{\\stackrel{\\textstyle >}{\\sim }}}2$ , $\\beta $ has the form $\\beta & = & g\\Big [ d-2 + \\sum _{\\ell =1}^\\infty b_\\ell \\, \\Big ( \\frac{g}{2\\pi }\\Big )^\\ell \\ \\Big ] \\cr \\cr & = & 2\\pi a\\Big [d-2 + \\sum _{\\ell =1}^\\infty b_\\ell \\, a^\\ell \\ \\Big ] \\ ,$ where $b_\\ell a^\\ell $ is the $\\ell $ -loop term.",
"The $n$ -loop ($n\\ell $ ) beta function, denoted $\\beta _{n\\ell }$ , is obtained by the replacement of $\\ell =\\infty $ by $\\ell =n$ in Eq.",
"(REF ).",
"Early discussions of the GN model for $d > 2$ include [1] and [10]; for more recent work see, e.g., [6], [15], and, for condensed-matter applications, [16], and references therein.",
"In this paper, aside from some comments in Section , we will restrict ourselves to the Gross-Neveu model in $d=2$ , where $g$ is dimensionless.",
"The $\\ell =1$ and $\\ell =2$ loop terms in $\\beta $ are independent of the scheme used for regularization and renormalization, while the terms at loop order $\\ell \\ge 3$ are scheme-dependent.",
"The beta function was calculated up to two-loop level in [11], with the results $b_1 = -2(N-1)$ and $b_2 = 2(N-1) \\ .$ (See also [17] for a two-loop calculation in a related Thirring model.)",
"The fact that $b_1$ in Eq.",
"(REF ) is negative means that in $d=2$ , this theory is asymptotically free for any finite $N > 1$ as well as in the $N\\rightarrow \\infty $ limit considered in [1].",
"The three-loop coefficient, $b_3$ , was calculated in [12], [13] in the commonly used scheme with dimensional regularization and modified minimal subtraction, denoted $\\overline{\\rm MS}$ [18], yielding the result $b_3 = \\frac{(N-1)(2N-7)}{2} \\ .$ Recently, the four-loop coefficient, $b_4$ has been calculated, again in the $\\overline{\\rm MS}$ scheme, to be [6] $b_4 = \\frac{1}{3}(N-1) \\Big [ -2N^2 - 19N + 24 - 6(11N-17)\\zeta _3 \\Big ] \\ ,$ where $\\zeta _s = \\sum _{n=1}^\\infty n^{-s}$ is the Riemann zeta function.",
"We comment on the dependence of the beta function coefficients on $N$ .",
"The property that these coefficients all contain a factor of $(N-1)$ is a consequence of the fact that for $N=1$ the GN model is equivalent to the massless abelian Thirring model [19], which has an identically zero beta function [20], [21].",
"Note that this statement about the beta function of the Thirring model is scheme-independent; if a beta function vanishes in one scheme, then it vanishes in all other schemes reached by acceptable (nonsingular) scheme transformations [22].",
"It follows that all of the coefficients $b_\\ell $ contain a factor of $(N-1)$ .",
"Therefore, it is only necessary to analyze the beta function of the Gross-Neveu model for $N > 1$ , where it is nonvanishing, and we will thus restrict to the physical integral values $N \\ge 2$ henceforth.",
"We next discuss how the $b_\\ell $ depend on $N$ in the relevant range $N > 1$ .",
"For this discussion, we consider $N$ to be extended from the positive integers to the real numbers.",
"The three-loop coefficient $b_3$ is a monotonically increasing function of $N$ that is negative for $N < 7/2$ , vanishes for $N=7/2$ , and is positive for $N > 7/2$ .",
"Thus, for physical, integral values, $b_3 < 0$ if $N=2$ or $N=3$ and $b_3 > 0$ if $N \\ge 4$ .",
"The coefficient $b_4$ is negative for large $N$ and is positive for $N$ in the interval $N_{b4z,m} < N < N_{b4z,p} \\ ,$ where the subscript $b4z$ stands for “$b_4$ zero” and $N_{b4z,(p,m)} = \\frac{-19-66\\zeta _3 \\pm \\sqrt{553+3324\\zeta _3+4356\\zeta _3^2}}{4}$ with $(p,m)$ corresponding to the $\\pm $ sign.",
"These have the values $N_{b4z,m} = -50.616$ and $N_{b4z,p}=1.448$ to the given floating-point accuracy.",
"Thus, in the relevant range $N > 1$ under consideration here, $b_4$ is negative.",
"We proceed to investigate the question of whether the beta function for the Gross-Neveu model at finite $N$ exhibits evidence for an infrared zero.",
"We denote an IR zero of the $n$ -loop beta function $\\beta _{n\\ell }$ as $a_{IR,n\\ell }$ , and the corresponding value of $g$ as $g_{IR,n\\ell }=2\\pi a_{IR,n\\ell }$ .",
"This IR zero of beta is a zero for positive $a$ closest to the origin (if there is such a zero), which one would thus reach as $\\mu $ decreases from the deep UV at large $\\mu $ to the IR at small $\\mu $ and $a$ increases from 0.",
"At the two-loop level, $\\beta _{2\\ell }$ has an IR zero at $a_{IR,2\\ell } = -\\frac{b_1}{b_2} = 1 \\ ,$ i.e., $g_{IR,2\\ell }=2\\pi $ .",
"Note that this value is independent of $N$ .",
"To judge whether this constitutes convincing evidence of an IR zero in the beta function, it is necessary to determine if higher-loop calculations confirm it.",
"We next carry out this task.",
"At the three-loop level, the condition that $\\beta _{3\\ell }=0$ away from the origin is the quadratic equation $b_1+b_2a+b_3 a^2=0$ .",
"This has two solutions, $a=\\frac{2[-1 \\pm \\sqrt{2(N-3)} \\ ]}{2N-7} \\ .$ If $N < 3$ , then these solutions are complex and hence unphysical.",
"If $N=3$ , these roots coincide, so that $a_{IR,3\\ell }=2$ , i.e., $g_{IR,3\\ell }=4\\pi $ .",
"For $N \\ge 3$ , there is only one physical root, namely $a_{IR,3\\ell } = \\frac{2[-1+\\sqrt{2(N-3)} \\ ]}{2N-7} \\ .$ However, this is not, in general, close to the two-loop zero of the beta function at $a_{IR,2\\ell }=1$ .",
"Furthermore, while $a_{IR,2\\ell }=1$ is independent of $N$ , $a_{IR,3\\ell }$ has a completely different behavior as a function of $N$ ; it decreases monotonically with $N$ in the interval $N \\ge 3$ over which it is physical and approaches zero asymptotically like $a_{IR,3\\ell } \\sim \\sqrt{\\frac{}{}}{2}{N} - \\frac{1}{N} +O \\Big ( \\frac{1}{N^{3/2}} \\Big ) \\quad {\\rm as} \\ N \\rightarrow \\infty \\ .$ At the four-loop level, the condition that $\\beta _{4\\ell }=0$ away from the origin is the cubic equation $b_1+b_2a+b_3 a^2 + b_4 a^3=0 \\ .$ The nature of the roots of this equation is determined by the discriminant, $\\Delta _3 = b_2^2b_3^2-27b_1^2b_4^2 - 4(b_1b_3^3+b_4b_2^3) + 18b_1b_2b_3b_4 \\ .$ This discriminant is negative for the relevant range $N \\ge 2$ (indeed, it is negative for all real $N$ ).",
"This implies that Eq.",
"(REF ) has one real root and a pair of complex-conjugate roots.",
"The real root is negative and hence is unphysical, since it violates the positivity requirement (REF ).",
"Moreover, since it is negative, it is clearly incompatible with the values of $a_{IR,2\\ell }$ and $a_{IR,3\\ell }$ , which are positive (discarding the unphysical complex value of $a_{IR,3\\ell }$ at $N=2$ ).",
"We therefore do not label this root as $a_{IR,4\\ell }$ , but instead as $a_{rt,4\\ell }$ , where $rt$ stands simply for the real root of Eq.",
"(REF ).",
"We find that the magnitude of $a_{rt,4\\ell }$ decreases toward zero monotonically as $N$ increases in the relevant interval $N \\ge 2$ , with the asymptotic behavior $a_{rt,4\\ell } \\sim -\\frac{3^{1/3}}{N^{2/3}} + \\frac{1}{2N} +O \\Big ( \\frac{1}{N^{4/3}} \\Big ) \\quad {\\rm as} \\ N \\rightarrow \\infty \\ .$ We list the values of $a_{IR,2\\ell }$ , $a_{IR,3\\ell }$ , and $a_{rt,4\\ell }$ in Table REF for $N$ from 2 to 10 and for three representative larger values, $N=100$ , 300, and $10^3$ .",
"Table: Values of a IR,2ℓ a_{IR,2\\ell }, a IR,3ℓ a_{IR,3\\ell }, anda rt,4ℓ a_{rt,4\\ell } for the beta function of the Gross-Neveu model, as afunction of NN.",
"Here, the three-loop and four-loop coefficients b 3 b_3 andb 4 b_4 are calculated in the MS ¯\\overline{\\rm MS} scheme.",
"If N=2N=2, thenthe zeros of β 3ℓ \\beta _{3\\ell } at nonzero aa form an unphysical complex (cmplx)pair.",
"As indicated, all of the values of a rt,4ℓ a_{rt,4\\ell } are negative and henceunphysical.",
"See text for further details.In our discussion above, we had stated that in order to judge whether the result for $a_{IR,2\\ell }$ constitutes convincing evidence of an IR zero in the beta function, it is necessary to determine if higher-loop calculations confirm it.",
"A necessary condition for the reliability of a perturbative calculation is that if one calculates some quantity to a given loop order, then there should not be a large fractional change in this quantity if one computes it to one higher order in the loop expansion.",
"This condition applies, in particular, to the calculation of a putative zero of the beta function.",
"Quantitatively, in order for the perturbative calculation of the IR zero of a beta function to be reliable, it is necessary that the fractional difference $\\frac{|a_{IR,(n-1)\\ell } - a_{IR,n\\ell }|}{\\frac{1}{2}[a_{IR,(n-1)\\ell }+ a_{IR,n\\ell }]}$ should be reasonably small and should tend to decrease with increasing loop order, $n$ .",
"As is evident both from our analytic formulas and from the numerical results listed in Table REF , this necessary condition is not satisfied in the present case.",
"The reason for this is clear from a plot of the beta functions $\\beta _{n\\ell }$ at loop orders $n=2$ , $n=3$ , and $n=4$ .",
"This shows that the IR zero in the two-loop beta function occurs at a value of $a$ that is too large for the perturbative calculation to be reliable.",
"In Figs.",
"REF and REF we plot the two-loop, three-loop, and four-loop beta functions for the Gross-Neveu model as functions of $a$ for two illustrative values of $N$ , namely $N=3$ and $N=10$ .",
"As is evident from these plots, the beta function does not satisfy the necessary criterion for the reliability of a calculation of an IR zero.",
"For the IR zero of the two-loop beta function at $a_{IR,2\\ell }=1$ to be reliable, one requires that the curves for the three-loop and four-loop beta functions should agree approximately with the curve for the two-loop beta function for $a \\simeq 1$ , and that these higher-loop beta functions should thus have respective IR zeros that are close to the two-loop zero at $a_{IR,2\\ell }=1$ .",
"But this is not the case; for $N=3$ , $\\beta _{3\\ell }$ has a double zero at the larger value, $a_{IR,3\\ell }=2$ and then goes negative again, while $\\beta _{4\\ell }$ has no IR zero in the physical region, $a > 0$ .",
"For $N=10$ the three-loop beta function $\\beta _{3\\ell }$ vanishes at a smaller value of $a$ than $a=1$ (and this value, $a_{IR,3\\ell }$ decreases as $N$ increases), while the four-loop beta function $\\beta _{4\\ell }$ again has no IR zero in the physical region, $a > 0$ .",
"The behavior illustrated for $N=10$ is generic for other values of $N \\ge 4$ .",
"Indeed, the curves for these beta functions at loop order $n=2, \\ 3, \\ 4$ only agree with each other close to the origin, and deviate strongly from each other before one gets to values of $a$ where a zero occurs.",
"Specifically, for $N=3$ , $\\beta _{2\\ell }$ and $\\beta _{3\\ell }$ only agree with each other for $a$ up to about 0.5, while $\\beta _{4\\ell }$ deviates from these lower-loop beta functions as $a$ increases beyond approximately 0.2.",
"As $N$ increases, these deviations occur for smaller $a$ .",
"Thus, for $N=10$ , $\\beta _{2\\ell }$ and $\\beta _{3\\ell }$ only agree with each other for $a$ up to roughly 0.15, while $\\beta _{4\\ell }$ deviates from these lower-loop beta functions as $a$ increases beyond about 0.08.",
"Figure: Plot of the nn-loop β\\beta functionβ a,nℓ \\beta _{a,n\\ell } of the Gross-Neveu model as a function of aa for N=3N=3 and(i) n=2n=2 (red), (ii) n=3n=3 (green), and (iii) n=4n=4 (blue) (colors inonline version).",
"At a=0.16a=0.16, going from bottom to top, the curves areβ 4ℓ \\beta _{4\\ell }, β 2ℓ \\beta _{2\\ell }, and β 3ℓ \\beta _{3\\ell }.Figure: Plot of the nn-loop β\\beta functionβ a,nℓ \\beta _{a,n\\ell } of the Gross-Neveu model as a function of aa for N=10N=10 and(i) n=2n=2 (red), (ii) n=3n=3 (green), and (iii) n=4n=4 (blue) (colors inonline version).",
"At a=0.2a=0.2, going from bottom to top, the curves areβ 4ℓ \\beta _{4\\ell }, β 2ℓ \\beta _{2\\ell }, and β 3ℓ \\beta _{3\\ell }.These results are similar to what was found in a search for a UV zero in the beta function of an IR-free theory, namely the O($N$ ) $\\lambda |{\\vec{\\phi }}|^4$ scalar field theory in $d=4$ spacetime dimensions [23].",
"In that theory, although the two-loop beta function exhibits a UV zero, higher-loop calculations up to five-loop order for general $N$ and up to six-loop order for $N=1$ do not confirm the two-loop result, and the reason was found to be that the two-loop UV zero occurs at too large a value of the quartic coupling for the two-loop perturbative calculation to be applicable and reliable."
],
[
"Analysis with Padé Approximants",
"In this section we carry out a further investigation of a possible IR fixed point in the renormalization-group flow for the Gross-Neveu model by calculating and analyzing Padé approximants (PAs) to the beta function at three-loop and four-loop level.",
"Since we are interested in a possible zero of the beta function away from the origin, it will be convenient to deal with a reduced ($rd$ ) beta function, $\\beta _{rd} \\equiv \\frac{\\beta }{2\\pi b_1 a^2} =1 + \\frac{1}{b_1} \\sum _{\\ell =2}^\\infty b_\\ell a^{\\ell -1} \\ .$ The $n$ -loop reduced beta function with $n \\ge 2$ , denoted $\\beta _{rd,n\\ell }$ , is obtained from Eq.",
"(REF ) by replacing $\\ell =\\infty $ by $\\ell =n$ as the upper limit in the summand.",
"This $n$ -loop reduced beta function is thus a polynomial of degree $n-1$ in $a$ .",
"The $[p,q]$ Padé approximant to this polynomial is the rational function $[p,q]_{\\beta _{rd,n\\ell }} =\\frac{1+\\sum _{j=1}^p \\, n_j x^j}{1+\\sum _{k=1}^q d_k \\, x^k}$ with $p+q=n-1 \\ ,$ where the $n_j$ and $d_k$ are $a$ -independent coefficients of the respective polynomials in the numerator and denominator of $[p,q]_{\\beta _{rd,n\\ell }}$ .",
"(Our notation follows [24].)",
"Hence, at a given $n$ -loop order, there are $n$ Padé approximants that one can calculate, namely $\\lbrace \\ [n-k,k-1]_{\\beta _{rd,n\\ell }} \\ \\rbrace \\quad {\\rm with} \\ 1 \\le k \\le n \\ .$ These provide rational-function approximations of the series expansion for $\\beta _{rd,n\\ell }$ that fits this series to the loop order $n$ .",
"As in our earlier work, e.g., [25], [26], these provide an alternate approach to investigating zeros of a beta function.",
"We shall label one of the $p$ zeros of a $[p,q]_{\\beta _{rd,n\\ell }}$ Padé approximant as $[p,q]_{zero}$ and one of the $q$ poles of this approximant as $[p,q]_{pole}$ ; in each case, the value of $n$ is given by Eq.",
"(REF ) as $n=p+q+1$ .",
"At the $n$ -loop level, the Padé approximant $[n-1,0]_{\\beta _{rd,n\\ell }}$ is equal to the reduced $n$ -loop beta function $\\beta _{rd,n\\ell }$ itself, which we have already analyzed in the previous section, and the PA $[0,n-1]_{\\beta _{rd,n\\ell }}$ has no zeros, and hence is not useful for our study.",
"Hence, at the $n$ -loop level, we focus on the $n-2$ PAs $[p,q]_{\\beta _{rd,n\\ell }}$ with $[p,q]=[n-k,k-1]$ having $2 \\le k \\le n-1$ .",
"At the $n=3$ loop level, we thus consider the $[1,1]_{\\beta _{rd,3\\ell }}$ Padé approximant.",
"This is $[1,1]_{\\beta _{rd,3\\ell }}=\\frac{1+\\Big (\\frac{b_2}{b_1}-\\frac{b_3}{b_2} \\Big )a}{1-\\Big ( \\frac{b_3}{b_2}\\Big ) a} = \\frac{1- \\Big ( \\frac{2N-3}{4} \\Big )a}{1- \\Big ( \\frac{2N-7}{4} \\Big )a} \\ .$ where the coefficients $b_1$ , $b_2$ , and $b_3$ were given in Eqs.",
"(REF )-(REF ) above.",
"This [1,1] PA has a zero at $[1,1]_{zero} = \\frac{4}{2N-3}$ and a pole at $[1,1]_{pole} = \\frac{4}{2N-7} \\ .$ The $a=[1,1]_{pole}$ is not relevant, since if $N = 2$ or 3, it has the respective negative and hence unphysical values $-4/3$ and $-4$ , while for $N \\ge 4$ , it lies farther from the origin than the zero.",
"This is clear from the fact that the difference $[1,1]_{pole} -[1,1]_{zero} = \\frac{16}{(2N-3)(2N-7)}$ is positive for this range $N \\ge 4$ .",
"Since the $[1,1]_{pole}$ lies farther from the origin than $[1,1]_{zero}$ , the coupling $a=a(\\mu )$ never reaches the pole as $\\mu $ decreases from large values in the UV to $\\mu =0$ and thus $a(\\mu )$ increases from 0 to $[1,1]_{zero}$ .",
"We list the values of the zero of the $[1,1]_{\\beta _{rd,3\\ell }}$ Padé approximant in Table REF .",
"For $N\\ge 3$ , the value of $a=[1,1]_{zero}$ is smaller than $a_{IR,3\\ell }$ and decreases more rapidly to zero as $N \\rightarrow \\infty $ than $a_{IR,3\\ell }$ .",
"If $N=3$ , the comparison cannot be made, since $a_{IR,3\\ell }$ is complex.",
"Thus, this analysis of the [1,1] Padé approximant to the reduced three-loop beta function, $\\beta _{rd,3\\ell }$ yields further evidence against a (reliably calculable) IR zero in the beta function up to the three-loop level.",
"At the $n=4$ loop level, there are two Padé approximants to analyze, namely $[2,1]_{\\beta _{rd,4\\ell }}$ and $[1,2]_{\\beta _{rd,4\\ell }}$ .",
"We calculate $[2,1]_{\\beta _{rd,4\\ell }}=\\frac{1+\\Big ( \\frac{b_2}{b_1}-\\frac{b_4}{b_3}\\Big )a+ \\Big ( \\frac{b_3}{b_1}-\\frac{b_2b_4}{b_1b_3} \\Big )a^2}{1- \\frac{b_4}{b_3}a} \\ ,$ where the coefficients $b_n$ were given in Eqs.",
"(REF )-(REF ).",
"The zeros of the numerator occur at $a=[2,1]_{zero,(i,ii)}$ , where $& & [2,1]_{zero,(i,ii)} = \\cr \\cr & & \\frac{b_2b_3-b_1b_4 \\pm \\Big [ b_1^2b_4^2+b_2^2b_3^2 -4b_1b_3^3+2b_1b_2b_3b_4 \\Big ]^{1/2}}{2(b_2b_4-b_3^2)} \\ .",
"\\cr \\cr & &$ and the subscripts $i$ and $ii$ correspond to the $\\pm $ sign in front of the square root.",
"It is straightforward to substitute the explicit expressions for the coefficients $b_2$ , $b_3$ , and $b_4$ in Eq.",
"(REF ), but the resultant expressions for these quadratic roots in terms of the explicit coefficients $b_n$ , $1 \\le n \\le 4$ are somewhat lengthy, so we do not display them.",
"The pole of the $[2,1]_{\\beta _{rd,4\\ell }}$ PA occurs at $a=[2,1]_{pole}$ , where $& & [2,1]_{pole} = \\frac{b_3}{b_4} \\cr \\cr & = & -\\frac{3(2N-7)}{2[2N^2+19N-24+6(11N-17)\\zeta _3]} \\ .$ If one has a series expansion of a function that contains $n_{zero}$ zeros and $n_{pole}$ poles, and one calculates $[r,s]$ Padé approximants to this series with $r > n_{zeros}$ and $s > n_{poles}$ , the approximants typically exhibit sets of nearly coincident zero-pole pairs in addition to fitting the actual zeros and poles of the function (e.g., see [24], [26]).",
"These nearly coincident zero-pole pairs may thus be ignored.",
"This happens in the present case.",
"For example, for $N=3$ , the $[2,1]_{\\beta _{rd,4\\ell }}$ PA has a zero at $a=0.99773$ , a zero at $a=0.009015$ and a pole at $a=0.009015$ , and similarly for other values of $N$ .",
"In Table REF we list the first zero, denoted $[2,1]_{zero,i}$ , as a function of $N$ .",
"Table: Values of [1,1] zero [1,1]_{zero} from [1,1] Padé approximantto the reduced three-loop beta function, β rd,3ℓ \\beta _{rd,3\\ell }, and[2,1] zero,i [2,1]_{zero,i} from the [2,1] Padé approximant to the four-loop betafunction, β rd,4ℓ \\beta _{rd,4\\ell }.",
"See text for further details.We calculate the $[1,2]_{\\beta _{rd,4\\ell }}$ Padé approximant to be $[1,2]_{\\beta _{rd,4\\ell }}=\\frac{1+\\Big [\\frac{b_1^2b_4+b_2^3-2b_1b_2b_3}{b_1(b_2^2-b_1b_3)} \\Big ]a}{1 + \\Big ( \\frac{b_1b_4-b_2b_3}{b_2^2-b_1b_3} \\Big ) a+ \\Big ( \\frac{b_3^2-b_2b_4}{b_2^2-b_1b_3} \\Big ) a^2 } \\ .$ The two poles of the $[1,2]_{\\beta _{rd,4\\ell }}$ approximant occur at $a=[1,2]_{pole,(i,ii)}$ , where $[1,2]_{pole,(i,ii)}= \\frac{b_1b_4 - b_2b_3 \\pm \\Big [ b_1^2b_4^2-3b_2^2b_3^2+4b_1b_3^3+4b_2^3b_4-6b_1b_2b_3b_4 \\Big ]^{1/2}}{2(b_2b_4-b_3^2)} \\ .$ The zero of this approximant occurs at $a=[1,2]_{zero}$ , where $& & [1,2]_{zero} = \\frac{b_1(b_1 b_3-b_2^2)}{b_1^2b_4+b_2^3-2b_1b_2b_3} \\cr \\cr & = & -\\frac{3(2N-3)}{2[2N^2+13N-9+6(11N-17)\\zeta _3]} \\ .$ Both of the poles $[1,2]_{pole,i}$ and $[1,2]_{pole,ii}$ are negative.",
"Furthermore, we find that this approximant has nearly coincident zero-pole pairs, which thus can both be ignored.",
"For example, for $N=3$ , the zero occurs at $a=-0.027540$ while one of the poles occurs at the nearly equal value, $a=-0.027556$ , and the other pole is at $a=-0.97919$ .",
"Similar results hold for other values of $N$ , i.e., the $[1,2]_{\\beta _{rd,4\\ell }}$ PA has a nearly coincident zero-pole pair (at negative $a$ ) together with a second unphysical pole at negative $a$ .",
"As we have discussed, the four-loop beta function yields a negative real root, in strong disagreement with the two-loop and three-loop beta functions.",
"At this four-loop level, the [1,2] PA does not exhibit any true zero, but only a zero that is nearly coincident with a pole and hence can be identified as an artifact.",
"The [2,1] PA yields a zero, but it is at a completely different value than the only real root of the actual four-loop beta function, $a_{rt,4\\ell }$ .",
"Thus, our analysis of the [2,1] and [1,2] Padé approximants to the four-loop (reduced) beta function yield further evidence against a robust IR zero in this four-loop beta function."
],
[
"Analysis Using Scheme Transformations",
"Since the coefficients $b_\\ell $ with $\\ell \\ge 3$ in the beta function are scheme-dependent, it is necessary to check that the conclusions from our analysis of the beta function with $b_3$ and $b_4$ calculated in the $\\overline{\\rm MS}$ scheme are robust with respect to scheme transformations.",
"To begin, we study scheme transformations that are designed to remove higher-loop terms in the beta function.",
"We first review some relevant background.",
"In [22], formulas were derived for the coefficients $b_\\ell ^{\\prime }$ resulting from a general scheme transformation $f(a^{\\prime })$ of the form $a = a^{\\prime }f(a^{\\prime }) \\ .$ Since a scheme transformation has no effect in the case of a free field theory, $f(a^{\\prime })$ satisfies the condition that $f(0)=1$ .",
"Expressing $f(a^{\\prime })$ as a power series in $a^{\\prime }$ , one has $f(a^{\\prime }) = 1 + \\sum _{s=1}^{s_{max}} k_s (a^{\\prime })^s \\ ,$ where the $k_s$ are constants and $s_{max}$ may be finite or infinite.",
"It follows that the Jacobian of this transformation, $J=da/da^{\\prime }$ satisfies the condition $J(0)=1$ and has the expansion $J = 1 + \\sum _{s=1}^{s_{max}} (s+1)k_s(a^{\\prime })^s \\ .$ Then in the transformed scheme, the coefficients of the three-loop and four-loop terms in the beta function are [22] $b_3^{\\prime } = b_3 + k_1b_2+(k_1^2-k_2)b_1 \\ ,$ $b_4^{\\prime } = b_4 + 2k_1b_3+k_1^2b_2+(-2k_1^3+4k_1k_2-2k_3)b_1 \\ ,$ and so forth for higher $b^{\\prime }_\\ell $ .",
"In [22] a set of conditions was given that should be obeyed by a nonpathological scheme transformation.",
"Condition C$_1$ was that the scheme transformation must map a physical (real, positive) $a$ to a real positive $a^{\\prime }$ , since a map that yields a negative or complex value of $a^{\\prime }$ would violate the unitarity of the theory.",
"As condition C$_2$ , we required that the scheme transformation should preserve perturbativity, and hence should not map a small or moderate value of $a$ to an excessively large value of $a^{\\prime }$ or vice versa.",
"Condition C$_3$ stated that the Jacobian $J$ should not vanish or diverge, since otherwise the transformation would be singular.",
"More generally, if $J$ were to become too small or too large, it could lead to a violation of condition C$_2$ .",
"Finally, condition C$_4$ was that if a beta function exhibited a zero at a sufficiently small value as to be perturbatively reliable, then a scheme transformation should not alter this property.",
"Ref.",
"[22] also gave the first explicit scheme transformation to set $b_\\ell ^{\\prime }=0$ for $\\ell \\ge 3$ , at least in the local vicinity of the origin, but it also showed that this does not, in general, work to remove these higher-loop terms at a point located away from the origin, i.e., an IR zero in an asymptotically free theory or a UV zero in an IR-free theory.",
"The reason, as shown in [22] and [27], if one attempts to apply such a scheme transformation to remove these higher-loop terms at a point away from the origin, then the transformation violates one or more of the conditions C$_1$ -C$_4$ for acceptability.",
"As in [27], we denote the scheme transformation presented in [22] (with $s_{max}=m$ ) that removes the coefficients in the beta function up to loop order $\\ell =m+1$ , at least near the origin, as $S_{R,m}$ .",
"We proceed with our analysis with the $S_{R,m}$ scheme transformation.",
"The $S_{R,2}$ transformation has [22] $k_2 = \\frac{b_3}{b_1}$ and the $S_{R,3}$ transformation has this $k_2$ and $k_3 = \\frac{b_4}{2b_1} \\ .$ We begin by determining whether the scheme transformation $S_{R,2}$ can be applied in the relevant region of $a$ where we need to apply it to set $b_3^{\\prime }=0$ and thus remove the three-loop term in the beta function.",
"Since the (scheme-independent) two-loop value is $a_{IR,2\\ell }=a_{IR,2\\ell }^{\\prime } = 1$ , the relevant region is in the neighborhood of $a=1$ .",
"This $S_{R,2}$ transformation is defined by Eq.",
"(REF ) with $s_{max}=2$ and $k_2$ given by Eq.",
"(REF ).",
"If the application of this $S_{R,2}$ transformation in the vicinity of $a=$ were possible, then it would follow from Eq.",
"(REF ) that $b_4^{\\prime }=b_4$ .",
"For $S_{R,2}$ , Eq.",
"(REF ) is $S_{R,2} \\ \\Longrightarrow \\ a=a^{\\prime }[1+k_2 (a^{\\prime })^2] = a^{\\prime }\\Big [1+\\frac{b_3}{b_1}(a^{\\prime })^2 \\Big ] \\ .$ Solving Eq.",
"(REF ) for $a^{\\prime }$ , we obtain three roots, and we require that at least one of these should be a physical (real, positive) value for $a$ in the relevant range of values comparable to $a_{IR,2\\ell }=1$ .",
"We find that this necessary condition, C$_1$ , is not satisfied.",
"Instead, two of the solutions of Eq.",
"(REF ) for $a^{\\prime }$ form a complex-conjugate pair, while the third is negative.",
"For example, for $a=a_{IR,2\\ell }=1$ and $N=4$ , the three solutions for $a^{\\prime }$ are $1.191 \\pm 0.509i$ and $-2.383$ , while for $N=10$ , the three solutions for $a^{\\prime }$ are $0.4125 \\pm 0.450i$ and $-0.825$ .",
"The Jacobian also exhibits pathological behavior; $J$ is given by $S_{R,2} \\ \\Longrightarrow \\ J & = & 1 + 3k_2(a^{\\prime })^2= 1 + \\frac{3b_3}{b_1}(a^{\\prime })^2 \\cr \\cr & = & 1 - \\frac{3(2N-7)}{4} \\, (a^{\\prime })^2 \\ .$ For $a_{IR,2\\ell }=a_{IR,2\\ell }^{\\prime }=1$ , $J=(25-6N)/4$ , which decreases through zero as $N$ (continued to the real numbers) increases through the value $N=25/6$ , violating condition C$_3$ .",
"It is therefore not possible to use this scheme transformation to remove the three-loop term in the beta function in the region of $a$ where we are trying to do this, namely the neighborhood of the (scheme-independent) value $a=a_{IR,2\\ell }=1$ .",
"We can also investigate whether the scheme transformation $S_{R,3}$ is physically acceptable to be applied in the relevant range of values of $a$ , namely $a=a_{IR,2\\ell }=1$ .",
"This transformation is defined by Eq.",
"(REF ) with $s_{max}=3$ and $k_2$ and $k_3$ given by Eqs.",
"(REF ) and (REF ): $S_{R,3} \\ \\Longrightarrow \\ a & = & a^{\\prime }[1+k_2 (a^{\\prime })^2+k_3(a^{\\prime })^3] \\cr \\cr & = & a^{\\prime }\\Big [1+\\frac{b_3}{b_1}(a^{\\prime })^2 + \\frac{b_4}{2b_1} (a^{\\prime })^3 \\Big ] \\ .$ The Jacobian for this transformation is $S_{R,3} \\ \\Longrightarrow \\ J & = & 1 + 3k_2(a^{\\prime })^2 + 4k_3(a^{\\prime })^3 \\cr \\cr & = & 1 + \\frac{3b_3}{b_1}(a^{\\prime })^2 + \\frac{2b_4}{b_1}(a^{\\prime })^3 \\ .$ With this $S_{R,3}$ scheme transformation we find that for the relevant range of $a \\simeq 1$ , $J$ can deviate excessively far from unity, violating condition C$_1$ .",
"For example, for $a=1$ and $N=10$ , we find that $J=339.8$ , much larger than unity.",
"One can also apply the various scheme transformations that we have devised in [22]-[29] to the beta function calculated in the $\\overline{\\rm MS}$ scheme and compare the resulting value(s) of the zero(s) of the beta function with the value(s) obtained at the three-loop and four-loop level in the $\\overline{\\rm MS}$ scheme.",
"Our general analyses in [22]-[29] (see also [30]) have shown that, for moderate values of the parameters determining these scheme transformations, the resultant values of the zero(s) are similar to those obtained in the original $\\overline{\\rm MS}$ scheme.",
"In particular, the negative, unphysical value of $a_{rt,4\\ell }$ will still be present in the transformed scheme.",
"Summarizing this section, we have shown that our conclusion, that the beta function of the finite-$N$ Gross-Neveu model, calculated up to four-loop order, does not exhibit an IR zero, is robust with respect to scheme transformations."
],
[
"Comparison with Results in the LN Limit and Behavior for $d > 2$",
"In this section we discuss how the conventional perturbative beta function reduces in the LN limit, and we also comment on some properties of the theory for spacetime dimension $d > 2$ .",
"From Eq.",
"(REF ), the quantity that remains finite and nonzero in the LN limit is $\\lambda = gN$ , and hence the corresponding beta function that is finite in this limit is $\\beta _\\lambda = \\frac{d\\lambda }{dt} = \\lim _{LN} N \\frac{dg}{dt}= \\lim _{LN} N \\beta \\ .$ With the limit $N \\rightarrow \\infty $ having been taken, $\\beta _\\lambda $ has the series expansion, for $d \\mathrel {\\raisebox {-.6ex}{\\stackrel{\\textstyle >}{\\sim }}}2$ , with $\\epsilon _d = d-2$ , $\\beta _\\lambda = \\lambda \\Big [ \\epsilon _d + \\sum _{\\ell =1}^\\infty \\hat{b}_\\ell \\xi ^\\ell \\Big ] \\ ,$ where $\\xi = \\lim _{LN} Na = \\frac{\\lambda }{2\\pi }$ and $\\hat{b}_\\ell = \\lim _{LN} \\frac{b_\\ell }{N^\\ell } \\ .$ Here we have used the fact that $b_\\ell a^\\ell = \\hat{b}_\\ell \\xi ^\\ell $ .",
"We find $\\hat{b}_1 = -2$ and $\\hat{b}_\\ell = 0 \\quad {\\rm for} \\ \\ell \\ge 2 \\ .$ The latter result follows from the fact that the structure of the bubble graphs in the calculation of $b_\\ell $ in, e.g., the $\\overline{\\rm MS}$ scheme, means that, for $\\ell \\ge 2$ , $b_\\ell $ is a polynomial in $N$ of degree $\\ell -1$ .",
"Although the $b_\\ell $ with $\\ell \\ge 3$ are scheme-dependent, this property is maintained by scheme transformations that are finite in the LN limit [22].",
"Hence, for $\\ell \\ge 2$ , $\\lim _{LN} b_\\ell /N^\\ell = 0$ , which is the result given in Eq.",
"(REF ).",
"Similarly, although $\\hat{b}_\\ell $ with $\\ell \\ge 3$ are, in general, scheme-dependent, if they are zero in one scheme, such as the $\\overline{\\rm MS}$ scheme, then they are also zero in any other scheme reached by a scheme transformation function that is finite in the LN limit [22].",
"It follows that in the LN limit, with $d=2+\\epsilon \\mathrel {\\raisebox {-.6ex}{\\stackrel{\\textstyle >}{\\sim }}}2$ , $\\beta _\\lambda = \\lambda [ \\epsilon - 2\\xi ] = \\lambda \\Big [ \\epsilon -\\frac{\\lambda }{\\pi } \\Big ] \\ .$ Hence, $d=2 \\ \\Longrightarrow \\ \\beta _\\lambda = -\\frac{\\lambda ^2}{\\pi } \\ ,$ with only the UV zero in this beta function at $\\lambda =0$ , and no IR zero.",
"We can relate this to the beta function that was calculated in [1] in the LN limit.",
"From Eqs.",
"(REF ) and (REF ), we have $\\beta = \\frac{dg}{dt} = 2g_{GN} \\, \\frac{dg_{GN}}{dt} = 2g_{GN}\\beta _{GN} \\ .$ Explicitly, in the LN limit, from Eqs.",
"(REF ) and (REF ), $\\beta _\\lambda = -\\frac{\\lambda ^2}{\\pi } = -\\lim _{LN} \\frac{g_{GN}^4 N^2}{\\pi } \\ .$ Combining Eqs.",
"(REF ), (REF ), and (REF ) yields $\\beta _{GN} = -g_{GN}^3 N/(2\\pi ) = -g_{GN}\\lambda /(2\\pi )$ , in agreement with Eq.",
"(REF ) above, or equivalently, Eq.",
"(3.7) in Ref.",
"[1].",
"This agreement was guaranteed, since the LN limit is a special limit of the result for finite $N$ .",
"Accordingly, our finding that there is no robust evidence for an IR zero in the finite-$N$ beta function of the ($d=2$ ) Gross-Neveu model is, a fortiori, in agreement with the fact that in the LN limit, the beta function $\\beta _\\lambda $ in Eq.",
"(REF ) (equivalently, $\\beta _{GN}$ in Eq.",
"(REF ) above), does not exhibit an IR zero.",
"If $d > 2$ , then for small $\\lambda $ , the GN theory is IR-free, with an IR zero of $\\beta _\\lambda $ at the origin, $\\lambda =0$ , and a UV zero of $\\beta _\\lambda $ at $\\lambda _{UV}= \\pi \\epsilon \\quad {\\rm for} \\ d \\mathrel {\\raisebox {-.6ex}{\\stackrel{\\textstyle >}{\\sim }}}2,\\quad {\\rm LN \\ limit} \\ ,$ which is a UV fixed point of the renormalization group.",
"This is closely analogous to the result found from an exact solution of the O($N$ ) nonlinear $\\sigma $ model (NL$\\sigma $ M) in $d=2+\\epsilon $ dimensions in the $N \\rightarrow \\infty $ limit [14].",
"In that theory, denoting the analogous finite coupling in this limit as $x = \\lim _{N \\rightarrow \\infty } N \\lambda _{NL\\sigma M} \\ ,$ the exact solution yielded the beta function, for $d \\mathrel {\\raisebox {-.6ex}{\\stackrel{\\textstyle >}{\\sim }}}2$ , $\\beta _x = \\frac{dx}{dt} = x\\Big [ \\epsilon - \\frac{x}{2\\pi } \\Big ] \\ .$ Thus, this nonlinear sigma model is, like the GN model in $d \\mathrel {\\raisebox {-.6ex}{\\stackrel{\\textstyle >}{\\sim }}}2$ , IR-free with a UV fixed point at $x_{UV} = 2\\pi \\epsilon \\ .$"
],
[
"Conclusions",
"The Gross-Neveu model in $d=2$ spacetime dimensions has long been of value as an asymptotically free theory which is exactly solvable in the LN limit and, in that limit, exhibits nonperturbative fermion mass generation and associated dynamical chiral symmetry breaking.",
"In this paper we have considered the finite-$N$ Gross-Neveu model.",
"We have addressed and answered a fundamental question about the UV to IR evolution of this model, as embodied in the beta function, namely whether this beta function exhibits evidence for an IR zero.",
"For the purpose of our study, we have analyzed the beta function to the highest-loop order to which it has been calculated, namely the four-loop order.",
"Our study used a combination of three methods, namely a direct analysis of the three-loop and four-loop beta functions, a study of Padé approximants, and a study of the effect of scheme transformations.",
"We find that in the range of coupling where the perturbative calculation of the four-loop beta function is reliable, it does not exhibit robust evidence for an infrared zero.",
"This research was supported in part by the Danish National Research Foundation grant DNRF90 to CP$^3$ -Origins at SDU (T.A.R.)",
"and by the U.S. NSF Grant NSF-PHY-16-1620628 (R.S.)"
]
] | 1612.05580 |
[
[
"Deep Reinforcement Learning with Successor Features for Navigation\n across Similar Environments"
],
[
"Abstract In this paper we consider the problem of robot navigation in simple maze-like environments where the robot has to rely on its onboard sensors to perform the navigation task.",
"In particular, we are interested in solutions to this problem that do not require localization, mapping or planning.",
"Additionally, we require that our solution can quickly adapt to new situations (e.g., changing navigation goals and environments).",
"To meet these criteria we frame this problem as a sequence of related reinforcement learning tasks.",
"We propose a successor feature based deep reinforcement learning algorithm that can learn to transfer knowledge from previously mastered navigation tasks to new problem instances.",
"Our algorithm substantially decreases the required learning time after the first task instance has been solved, which makes it easily adaptable to changing environments.",
"We validate our method in both simulated and real robot experiments with a Robotino and compare it to a set of baseline methods including classical planning-based navigation."
],
[
"Introduction",
"Autonomous navigation is one of the core problems in mobile robotics.",
"It can roughly be characterized as the ability of a robot to get from its current position to a designated goal location solely based on the input it receives from its on-board sensors.",
"A popular approach to this problem relies on the successful combination of a series of different algorithms for the problems of simultaneous localization and mapping (SLAM), localization in a given map as well as path planning and control, all of which often depend on additional information given to the agent.",
"Although individually the problems of SLAM, localization, path planning and control are well understood [1], [2], [3], and a lot of progress has been made on learning control [4], they have mainly been treated as separable problems within robotics and some often require human assistance during setup-time.",
"For example, the majority of SLAM solutions are implemented as passive procedures relying on special exploration strategies or a human controlling the robot for sensory data acquisition.",
"In addition, they typically require an expert to check as to whether the obtained map is accurate enough for path planning and localization.",
"Our goal in this paper is to make first steps towards a solution for navigation tasks without explicit localization, mapping and path planning procedures.",
"To achieve this we adopt a reinforcement learning (RL) perspective, building on recent successes of deep RL algorithms for solving challenging control tasks [5], [6], [7], [8], [9].",
"For such an RL algorithm to be useful for robot navigation we desire that it can quickly adapt to new situations (e.g., changing navigation goals and environments) while still preserving the solutions to earlier problems: a prerequisite that is not fulfilled by current state-of-the-art RL-based methods.",
"To achieve this, we employ successor representation learning, which has recently also been combined with deep nets [10], [7].",
"As we show in this paper, this formulation can be extended to handle sequential task transfer naturally, with minimal additional computational costs; its ability of retaining a compact representation of the Q functions of all encountered tasks enables it to cope with the limited memory and processing capabilities on robotic platforms.",
"Figure: Exemplary maze-like environment considered in this paper (Map6) and the optimal path from a randomly chosen start position to the goal (orange traffic cone) taken by the Robotino robot (top) together with the sensory input captured by the robot's on-board kinect sensor (bottom).We validate our approach and its fast transfer learning capabilities in both simulated and real world experiments, on both visual and depth inputs, where the agent must navigate different maze-like environments.",
"We compare it to several baselines such as a conventional planner (assuming perfect localization), a supervised imitation learner (assuming perfect localization during training only), and transfer with DQN.",
"In addition, we validate that deep convolutional neural networks (CNNs) can be used to imitate conventional planners in our considered domain."
],
[
"Relations To Existing Work",
"Our work is related to an increasing amount of literature on deep reinforcement learning.",
"We here highlight the most apparent connections to recent trends with a focus on value based RL (which we use as a basis).",
"A more detailed review of the concepts we built upon is then given in Sec. .",
"As mentioned, a growing amount of success has been reported for value-based RL in combination with deep neural networks.",
"This idea was arguably popularized by the Deep Q-networks (DQN) [5] approach followed by a large body of work deriving extended variants (e.g., recent adaptations to continuous control [6], [9] and improvements stabilizing its performance [11], [12], [13]).",
"While the DQN inspired RL algorithms were shown to be surprisingly effective, they also come with some caveats that complicate transfer to novel tasks (one of the key attributes we are interested in).",
"More precisely, although a neural network trained using Q-learning on a specific task is expected to learn features that are informative about both: i) the dynamics induced by the policy of the agent in a given environment (we refer to this as the policy dynamics in the following text), and ii) the association of rewards to states; these two sources of information cannot be assumed to be clearly separated within the network.",
"As a consequence, while fine-tuning a learned Q-network on a related task might be possible, it is not immediately clear how the aforementioned learned knowledge could be transferred in a way that keeps the policy on the original task intact.",
"One attempt at clearly separating reward attribution for different tasks while learning a shared representation is the idea of learning a general (or universal) value function [14] over many (sub)-tasks that has recently also been combined with DQN-type methods [15].",
"Our method can be interpreted as a special parametrization of a general value function architecture that facilitates fast task transfer.",
"Task transfer is one of the long standing problems in RL.",
"Historically, most existing work in this direction relied on simple task models and explicitly known relations between tasks or known dynamics [16], [17], [18].",
"More recently, there have been several attempts at combining task transfer with Deep RL [19], [20], [21], [22], [23], [24].",
"E.g., Parisotto et al.",
"[19] and Rusu et al.",
"[20] performed multi-task learning (transferring useful features between different ATARI games) by fine-tuning a DQN network (trained on a single ATARI game) on multiple “related” games.",
"More directly related to our work, Rusu et al.",
"[21] developed the Progressive Networks approach which trains an RL agent to progressively solve a set of tasks, allowing it to re-use the feature representation learned on tasks it has already mastered.",
"Their derivation has the advantage that performance on all considered tasks is preserved but requires an ever growing set of learned representations.",
"In contrast to this, our approach for task transfer aims to more directly tie the learned representations between tasks.",
"To achieve this, we build on the idea of successor representation learning for RL first proposed by Dayan [25], and recently combined with deep neural networks [10], [7].",
"This line of work makes the observation that Q-learning can be partitioned into two sub-tasks: 1) learning features from which the reward can be predicted reliably and 2) estimating how these features evolve over time.",
"While it was previously noted how such a partitioning can be exploited to speed up learning for cases where the reward changes scale or meaning [10], [7], we here show how this view can be extended to allow general – fast – transfer across tasks, including changes to the environment, the reward function and thus also the optimal policy.",
"We also note that the objective we use for learning descriptive features involves training a deep auto-encoder.",
"Learning state representations for RL via auto-encoders has previously been considered several times in the literature [26], [27], [28].",
"Among these, utilizing the priors on learned representation for robotics from Jonschkowski et al.",
"[27] could potentially further improve our model."
],
[
"Background",
"In this section we will first review the concepts of reinforcement learning upon which we build our approach."
],
[
"Reinforcement learning",
"We formalize the navigation task as a Markov Decision Process (MDP).",
"In an MDP an agent interacts with the environment through a sequence of observations, actions and reward signals.",
"In each time-step $t \\in [0, T ]$ of the decision process the agent first receives an observation from the environment $\\mathbf {x}_t \\in \\mathcal {X}$ (in our case an image of its surrounding).",
"Together with a history of recent observations $\\lbrace \\mathbf {x}_{t-H}, \\dots \\mathbf {x}_{t-1} \\rbrace $ – with history length $H$ –, $\\mathbf {x}_t$ informs the agent about the true state of the environment $\\mathbf {s}_t \\in \\mathcal {S}$ .",
"In the following always define $\\mathbf {s}_t$ as $\\mathbf {s}_t = \\lbrace \\mathbf {x}_{t-H}, \\dots \\mathbf {x}_{t-1}, \\mathbf {x}_t \\rbrace $ .The agent then selects an action $\\mathbf {a}_t \\in \\mathcal {A}$ according to a policy $\\mathbf {a}_t = \\pi (\\mathbf {s}_t)$ We restrict the following presentation to deterministic policies with discrete actions to simplify notation.",
"A generalization can easily be obtained.",
"and transits to the next state $\\mathbf {s}_{t+1}$ following the dynamics of the environment: $p(\\mathbf {s}_{t+1}|\\mathbf {s}_{t}, \\mathbf {a}_t)$ , receiving reward $R(\\mathbf {s}_t) \\in \\mathbb {R}$ and obtaining a new observation $\\mathbf {x}_{t+1}$ .",
"The agent's goal is to maximize the cumulative expected future reward (with discount factor $\\gamma $ ).",
"This quantity uniquely assigns an expected value to each state-action pair.",
"The action-value function (referred to as the Q-value function) of executing action $\\mathbf {a}$ in state $\\mathbf {s}$ under a policy $\\pi $ thus can be defined as: $Q(\\mathbf {s}, \\mathbf {a}; \\pi ) = \\mathbb {E}\\left[\\sum _{t=0}^{\\infty } \\gamma ^t R(\\mathbf {s}_t) \\Big | \\mathbf {s}_0=\\mathbf {s}, \\mathbf {a}_0=\\mathbf {a}, \\pi \\right],$ where the expectation is taken over the policy dynamics: the transition dynamics under policy $\\pi $ .",
"Importantly, the Q-function can be computed using the Bellman equation $Q(\\mathbf {s}_{t}, \\mathbf {a}_{t}; \\pi ) = R(\\mathbf {s}_t) + \\gamma \\mathbb {E}\\left[ Q(\\mathbf {s}_{t+1}, \\mathbf {a}_{t+1}; \\pi ) \\right],$ which allows for recursive estimation procedures such as Q-learning and SARSA [29].",
"Furthermore, assuming the Q-function for a given policy is known, we can find an improved policy $\\hat{\\pi }$ by greedily choosing $\\mathbf {a}_t$ in each state: $\\hat{\\pi }(\\mathbf {s}_t) = \\operatornamewithlimits{\\arg \\!\\max }_{\\mathbf {a}_t} Q(\\mathbf {s}_t, \\mathbf {a}_t; \\pi )$ .",
"When combined with powerful function approximators such as deep neural networks these principles form the basis of many recent successes in RL for control."
],
[
"Successor feature reinforcement learning",
"While direct learning of the Q-value function from Eq.",
"(REF ) with function approximation is possible, it results in a black-box approximator which makes knowledge transfer between tasks challenging (we refer to Sec.",
"for a discussion).",
"We will thus base our algorithm upon a re-formulation of the RL problem first introduced by [25] called successor representation learning which has recently also been combined with deep neural networks [10], [7], that we will first review here and then extend to naturally handle task transfer.",
"To start, we assume that the reward function can be approximately represented as a linear combination of learned features $\\phi (\\mathbf {s}; \\theta _{\\phi })$ (in our case features extracted from a neural network) with parameters $\\theta _\\phi $ and a reward weight vector $\\omega $ as $R(\\mathbf {s}) \\approx \\phi (\\mathbf {s}; \\theta _{\\phi })^T \\omega $ .",
"Using this assumption we can rewrite Eq.",
"(REF ) as $Q(\\mathbf {s}, \\mathbf {a}; \\pi )&\\approx \\mathbb {E}\\left[ \\sum _{t=0}^{\\infty } \\gamma ^t \\phi (\\mathbf {s}_{t}; \\theta _\\phi ) \\cdot \\omega \\Big | \\mathbf {s}_0=\\mathbf {s}, \\mathbf {a}_0=\\mathbf {a}, \\pi \\right]\\\\&=\\mathbb {E}\\left[ \\sum _{t=0}^{\\infty } \\gamma ^t \\phi (\\mathbf {s}_{t}; \\theta _\\phi ) \\Big | \\mathbf {s}_0=\\mathbf {s}, \\mathbf {a}_0=\\mathbf {a}, \\pi \\right] \\cdot \\omega \\\\&=\\psi ^{\\pi }(\\mathbf {s}, \\mathbf {a})^T \\omega ,$ where, in line with [10] we refer to $\\psi ^{\\pi }(\\mathbf {s}, \\mathbf {a}) = \\mathbb {E}\\left[ \\sum _{t=0}^{\\infty } \\gamma ^t \\phi (\\mathbf {s}_{t} ; \\theta _\\phi ) | \\mathbf {s}_0=\\mathbf {s}, \\mathbf {a}_0=\\mathbf {a}, \\pi \\right]$ as the successor features.",
"Consequently we will refer to the whole reinforcement learning algorithm as successor feature reinforcement learning (SF-RL).",
"As a special case we will assume that the features $\\phi (\\mathbf {s}; \\theta _{\\phi })$ themselves are representative of the state $\\mathbf {s}$ (i.e., we can reconstruct the state $\\mathbf {s}$ from $\\phi (\\mathbf {s}; \\theta _{\\phi })$ alone) which allows us to explicitly turn $\\psi (\\cdot )$ into a function $\\psi ^{\\pi }(\\phi (\\mathbf {s}_t; \\theta _\\phi ), \\mathbf {a}_t)$ .",
"In the following we use the short-hand $\\phi _\\mathbf {s}= \\phi (\\mathbf {s}; \\theta _\\phi )$ – omitting the dependency on the parameters $\\theta _\\phi $ – and write $\\psi ^{\\pi }(\\phi _{\\mathbf {s}_t}, \\mathbf {a}_t)$ to avoid cluttering notation.",
"Interestingly, these successor features can again be computed via a Bellman equation in which the reward function is replaced with $\\phi _{\\mathbf {s}_t}$ ; that is we have: $\\psi ^{\\pi }(\\phi _{\\mathbf {s}_t}, \\mathbf {a}_t) = \\phi _{\\mathbf {s}_t} + \\gamma \\mathbb {E} \\left[ \\psi ^{\\pi }\\left(\\phi _{\\mathbf {s}_{t+1}}, \\mathbf {a}_{t+1} \\right) \\right].$ And we can thus learn approximate successor features using a deep Q-learning like procedure [10], [7].",
"Effectively, this re-formulation separates the learning of the Q-function into two problems: 1) estimating the expectation of descriptive features under the current policy dynamics and 2) estimating the reward obtainable in a given state.",
"To show how learning with successor feature RL works let us consider the case where we are only interested in recovering the Q-function of the optimal policy $\\pi ^*$ .",
"In this case we can simultaneously learn the parameters $\\theta _\\phi $ of the feature mapping $\\phi _\\mathbf {s}$ (a convolutional neural network), the reward weights $\\omega $ and an approximate successor features mapping $\\psi (\\phi _\\mathbf {s}, \\mathbf {a}; \\theta _\\psi ) \\approx \\psi ^{\\pi ^*}(\\phi _{\\mathbf {s}}, \\mathbf {a})$ (a fully connected network with parameters $\\theta _\\psi $ ) by alternating stochastic gradient descent steps on two objective functions: $&L\\underset{(s, a, s^{\\prime }) \\in \\mathcal {D}_T}{(\\theta _\\psi ) = \\ \\mathbb {E}} \\left[ \\left( \\phi _\\mathbf {s}+ \\gamma \\psi (\\phi _{\\mathbf {s}^{\\prime }}, \\mathbf {a}^*; \\theta ^-_\\psi ) - \\psi (\\phi _{\\mathbf {s}}, \\mathbf {a}; \\theta _\\psi )\\right)^2 \\right],$ $\\begin{aligned}L(&\\theta _\\phi , \\theta _d, \\omega ) \\\\&= \\underset{(s, R(s)) \\in \\mathcal {D}_R}{ \\mathbb {E}} \\Big [\\left( R(\\mathbf {s}) - \\phi _{\\mathbf {s}}^T \\omega \\right)^2 + \\big ( \\mathbf {s}- d(\\phi _\\mathbf {s}; \\theta _d) \\big )^2 \\Big ],\\end{aligned}$ where $\\mathcal {D}_T$ and $\\mathcal {D}_R$ denote collected experience data for transitions and rewards, respectively, ${\\mathbf {a}^*} = \\operatornamewithlimits{\\arg \\!\\max }_{a^{\\prime }} Q(\\mathbf {s}^{\\prime }, \\mathbf {a}^{\\prime }; \\pi ^*)$ – computed by inserting the approximate successor features $\\psi (\\phi _{\\mathbf {s}^{\\prime }}, \\mathbf {a}^*; \\theta ^-_\\psi )$ into Eq.",
"(REF ) – and where $\\theta ^{-}_\\psi $ denotes the parameters of the current target successor feature approximation.",
"To provide stable learning these are occasionally copied from $\\theta _\\psi $ (a full discussion of the intricacies of this approach is out of the scope of this paper and we refer to [5] and [7] for details); we replace the target successor feature parameters every 5000 training steps.",
"The objective function from Eq.",
"(REF ) corresponds to learning the successor features via online Q-learning (with rewards $\\phi (\\cdot )$ ).",
"The objective from Eq.",
"(REF ) corresponds to learning the reward weights and the CNN feature mapping and consists of two parts: the first part ensures that the reward is regressed; the second part ensures that the features are representative of the state $\\mathbf {s}$ by enforcing that an inverse mapping from $\\phi (\\mathbf {s}; \\theta _\\phi )$ to $\\mathbf {s}$ exists through a third convolutional network, a decoder $d(\\cdot )$ , whose parameters $\\theta _d$ are also learned.",
"After learning, actions can be chosen greedily from $Q(\\mathbf {s}, \\mathbf {a}; \\pi ^*)$ by inserting the approximated successor features into Eq.",
"(REF )."
],
[
"Transferring successor features to new goals and tasks",
"As described above, the successor representation naturally decouples task specific reward estimation and the estimation of the expected occurrence of the features $\\phi (\\cdot )$ under the specific policy dynamics.",
"This makes successor feature based RL a natural choice when aiming to transfer knowledge between related tasks.",
"To see this let us first define two notions of knowledge transfer.",
"In both cases we assume that the learning occurs in $K$ different stages during each of which the agent can interact with a distinct task $k \\in [1, K]$ .",
"The aim for the agent is to solve all tasks at the end of training, using minimal interaction time for each task.",
"From the perspective of reinforcement learning this setup corresponds to a sequence of $K$ RL problems which have shared structure.",
"Knowledge transfer between tasks can then occur for two different scenarios: The first, and simplest, notion of knowledge transfer occurs if all $K$ tasks use the same environment and transition dynamics and differ only in the reward function $R$ .",
"In a navigation task this would be equivalent to finding paths to $K$ different goal positions in one single maze.",
"The second, and more general, notion of knowledge transfer occurs if all $K$ tasks use different environments (and potentially different reward functions) which share some similarities within their state space.",
"In a navigation task this includes changing the maze structure or robot dynamics between different tasks.",
"We can observe that successor feature RL lends itself well to transfer learning in scenarios of the first kind: If the features $\\phi (\\cdot )$ are expressive enough to ensure that the rewards for all tasks can be linearly predicted from them then for all tasks following the first (i.e., for $k > 1$ ) one only has to learn a new reward weight vector $\\omega ^k$ (keeping both the learned $\\phi (\\cdot )$ and $\\psi (\\cdot )$ fixed), although care has to be taken if the expectation of the features under the different policy changes (in which case the successor features would have to be adapted also).",
"Learning for $k > 1$ then boils down to solving a simple regression problem (i.e., minimizing Eq.",
"(REF ) wrt.",
"$\\omega $ ) and requires only the storage of an additional weight vector per task.",
"This idea has recently been explored [10], [7].",
"Kulkarni et al.",
"[7] showed large learning speedups for a special case of this setting where they changed the scale of the final reward.",
"We here argue that successor feature RL can be easily extended to transfer learning of the second kind with minimal additional memory and computational requirements.",
"Specifically, to derive a learning algorithm that works for both transfer scenarios let us first define the action-value function for task $k$ using the successor feature notation as $Q^{k}(\\mathbf {s}, \\mathbf {a}; \\pi ^k) \\approx \\mathbb {E}\\left[ \\sum _{t=0}^{\\infty } \\gamma ^t \\phi ^k_{\\mathbf {s}_{t}} \\Big | \\mathbf {s}_0=\\mathbf {s}, \\mathbf {a}_0=\\mathbf {a}, \\pi ^k \\right] \\cdot \\omega ^k,$ where we used the superscript $k$ to refer to task specific features and policies respectively and where we have again introduced the short-hand notation $\\phi ^k_{\\mathbf {s}_{t}} = \\phi ^k(\\mathbf {s}_{t}; \\theta _{\\phi ^k})$ for notational brevity.",
"Additionally, let us assume that there exists a linear relation between the task features, that is there exists a mapping $\\phi ^{i}_\\mathbf {s}= \\mathcal {B}^i \\phi ^{k}_\\mathbf {s}$ for all $i \\le k$ and we have $B^k = \\mathbf {I}$ .",
"We note that such a linear dependency between features does not imply a linear dependency between the observations (since $\\phi (\\cdot )$ is a nonlinear function implemented by a neural network), and hence this assumption is not very restrictive.",
"Then – again using the fact that the expectation is a linear operator – we obtain for $i \\le k$ : $Q^{i}(\\mathbf {s}, \\mathbf {a}; \\pi ^{i}) &\\approx \\mathbb {E}\\left[ \\sum _{t=0}^{\\infty } \\mathcal {B}^i \\gamma ^t \\phi ^k_{\\mathbf {s}_{t}} \\Big | \\mathbf {s}_0=\\mathbf {s}, \\mathbf {a}_0=\\mathbf {a}, \\pi ^i \\right] \\cdot \\omega ^i \\nonumber \\\\&= \\mathcal {B}^i \\mathbb {E}\\left[ \\sum _{t=0}^{\\infty } \\gamma ^t \\phi ^k_{\\mathbf {s}_{t}} \\Big | \\mathbf {s}_0=\\mathbf {s}, \\mathbf {a}_0=\\mathbf {a}, \\pi ^i \\right] \\omega ^i \\nonumber \\\\&= \\mathcal {B}^i \\psi ^{\\pi ^i}\\left( \\phi ^k_{\\mathbf {s}_{t}}, \\mathbf {a}\\right)^T \\omega ^i \\\\&= \\psi ^{\\pi ^i}\\left( \\mathcal {B}^i \\phi ^k_{\\mathbf {s}_{t}}, \\mathbf {a}\\right)^T \\omega ^i.",
"$ These equivalences now give us a straight-forward way to transfer knowledge to new tasks while keeping the solution found for old tasks intact (as long as we have access to all feature mappings $\\phi ^k$ and policies $\\pi ^k$ ): When training on task $k > 1$ initialize the parameters $\\theta _{\\phi ^k}$ and $\\theta _{\\psi ^k}$ with $\\theta _{\\phi ^{k-1}}$ and $\\theta _{\\psi ^{k-1}}$ respectively (otherwise initialize randomly) and train $\\psi ^{\\pi ^k}$ and $\\phi ^k$ via stochastic gradient descent on Eqs.",
"(REF )-(REF ).",
"In addition, train all $\\mathcal {B}^i$ with $i < k$ to preserve the relation $\\phi ^{i}_\\mathbf {s}\\approx \\mathcal {B}^i \\phi ^{k}_\\mathbf {s}$ .",
"To obtain successor features for the previous tasks, estimate the expectation of the features for the current task $k$ under the old task policies to obtain $\\psi ^{\\pi ^i}\\left( \\phi ^k_\\mathbf {s}, \\mathbf {a}\\right)$ so that Eq.",
"(REF ) can be computed during evaluation.",
"Note that this means we have to estimate the expectation of the current task features under all old task dynamics and policiesIn principle, the expectations for all tasks $i < k$ need to be evaluated with samples from these tasks.",
"In our case, we however found that the shared structure between tasks was large enough to allow for estimating all expectations based on the current tasks samples only..",
"Since we expect significant overlap between tasks in our experiments this can be implemented memory efficiently by using one single neural network with multiple output layers to implement all task specific successor features.",
"Alternatively, if the successor feature networks are small, one can just preserve the old task successor feature networks and use Eq.",
"() for selecting actions for old tasks.",
"When – as in Sec.",
"REF – we are only interested in finding the optimal policy $\\pi ^{i^*}$ for each task these steps correspond to alternating stochastic gradient descent steps on two objective functions analogous to Eqs.",
"(REF ),(REF ), under the model architecture depicted in Fig.",
"REF .",
"More precisely, we write $\\psi ^i(\\phi ^k_\\mathbf {s}, \\mathbf {a}; \\theta _{\\psi ^i}) \\approx \\psi ^{\\pi ^{i^*}}(\\phi ^k_{\\mathbf {s}}, \\mathbf {a})$ and obtain the following objectives for task $k$ : $&\\begin{aligned}&L^k\\Big (\\lbrace \\theta _{\\psi ^1}, \\dots , \\theta _{\\psi ^{k}} \\rbrace \\Big ) =\\\\&\\sum _{i \\le k}\\underset{\\begin{array}{c}(s, a, s^{\\prime }) \\\\ \\in \\mathcal {D}^i_T\\end{array}}{\\mathbb {E}} \\left[ \\left( \\phi ^k_\\mathbf {s}+ \\gamma \\psi ^i(\\phi ^k_{\\mathbf {s}^{\\prime }}, \\mathbf {a}^{i*}; \\theta ^-_{\\psi ^i}) - \\psi ^i(\\phi ^k_{\\mathbf {s}}, \\mathbf {a}; \\theta _{\\psi ^i})\\right)^2 \\right],\\end{aligned} \\\\&L^k\\Big (\\theta _\\phi , \\theta _d, \\omega ^k, \\lbrace \\mathcal {B}^1, \\dots , \\mathcal {B}^{k-1} \\rbrace \\Big ) \\nonumber =\\\\&\\begin{aligned}&\\underset{(s, R(s)) \\in \\mathcal {D}^k_R}{\\mathbb {E}} &\\Big [\\left( R(\\mathbf {s}) - {\\phi _{\\mathbf {s}}^{k}}^{T} \\omega ^k \\right)^2+ \\big ( \\mathbf {s}- d(\\phi _{\\mathbf {s}}^{k}; \\theta _{{d}^{k}}) \\big )^2 \\Big ]\\\\& \\ &+ \\sum _{i < k} \\underset{(s, R(s)) \\in \\mathcal {D}^i_R}{\\mathbb {E}} \\Big [\\big (\\phi ^i_s - \\mathcal {B}^i \\phi ^k_s \\big )^2 \\Big ],\\end{aligned}$ where $a^{i*} = \\operatornamewithlimits{\\arg \\!\\max }_{a^{\\prime }} Q(\\mathbf {s}^{\\prime }, \\mathbf {a}^{\\prime }; \\pi ^{i^*})$ is the current greedy best action for task $i$ and in cases where we are willing to store the old $\\psi ^i(\\cdot )$ for $i < k$ Eq.",
"(REF ) only needs to be optimized with respect to $\\theta _{\\psi ^k}$ (dropping all other terms)In practice there is no noticeable performance difference.. Several interesting details can be noted about this formulation.",
"First, if we assume that all $\\psi ^i(\\cdot )$ are implemented using one neural network with $k$ output layers – or if the successor feature networks are small – then the overhead for learning $k$ tasks is small (we only have to store $k-1$ additional weight matrices plus one additional reward weight vector per task) this is in contrast to other successful transfer learning approaches for RL that have recently been proposed such as [21].",
"Second, the regression of the old task features via the transformation matrices $\\mathcal {B}^i$ forces the CNN that outputs $\\phi ^k_s$ to represent the features for all tasks May be seen as special case of the distillation technique [30].. As such we expect this approach to work well when tasks have shared structure; if they have no shared structure one would have to increase the number of parameters (and thus possibly the dimensionality of $\\phi ^k$ ).",
"To gain some intuition for the reasons why the above model should work we here want to give a – hypothetical – example: Let us assume the set of extracted features $\\phi (\\cdot )$ to be the relative distance to a set of objects from the current position of the agent.",
"Then, the successor features $\\psi (\\cdot )$ would estimate the discounted sum of those relative distances under the current policy dynamics.",
"When transferring to a new environment, the spatial relationship of the objects could, for example, change.",
"Then $\\phi (\\cdot )$ would need to adapt accordingly.",
"But since we assume the two environments to share structure (e.g., they contain the same objects), filters in the early layers of $\\phi (\\cdot )$ could be largely reused (or transferred).",
"The adapted features (e.g., the relative distances from the current pose to the changed object positions) now would differ from those of the previous environments, this change in scale could be directly captured by a linear mapping $\\mathcal {B}$ .",
"$\\psi (\\cdot )$ would also need to be adapted, but due to the shared structure between environments and their similarity in the successor features we would expect adapting them to be fast.",
"Similarly, the reward mapping can either be re-learned quickly or transferred directly (e.g., if we assume that the reward penalizes proximity to objects)."
],
[
"Experimental setup",
"We first test our algorithm using a simulation of different maze-like 3D environments.",
"The environment contains cubic objects and a target for the agent to reach (rendered as a green sphere) (cf. Fig.",
"REF ).",
"We model the legal actions as four discrete choices: {stand still, turn left ($90^\\circ $ ), turn right ($90^\\circ $ ), go straight ($1 m$ ) } to simplify the problem (we note that in simulation the agent moves in a continuous manner).",
"The agent is a simulated Pioneer-3dx robot moving under a differential drive model (with Gaussian control noise, thus the robot will have observations of the environment from a continuous viewing position and angle space).",
"The agent obtains a reward of $-0.04$ for each step it takes, $-0.96$ for colliding with obstacles, 1 for reaching the goal; this reward structure forces time-optimal behavior.",
"Each episode starts with the agent in a random location and ends when it reaches the goal (unless noted otherwise).",
"In each time-step the agent receives as an observation a frame captured from the forward facing camera (as shown in Fig.",
"REF , re-scaled to $64 \\times 64$ pixels).",
"The state in each time-step is then given by the 4 most recently obtained observations.",
"The top-down views of the four different mazes we consider are shown in Fig.",
"REF .",
"For training the model (Fig.",
"REF ) we employed stochastic gradient descent with the ADAM optimizer [31], a minibatch size of 64 and a learning rate of $2.5\\times 10^{-4}$ and $2.5\\times 10^{-5}$ for visual inputs for the supervised learner and the reinforcement learners respectively, $5.0\\times 10^{-5}$ for depth inputs.",
"We performed a coarse grid search for each learning algorithm to choose the optimizer hyper parameters (learning rate in range $[1\\times 10^{-6}, 1\\times 10^{-3}$ ]) and use the same minibatch size across all considered approaches.",
"Training was performed alongside exploration in the environment (one batch is considered every 4 steps)."
],
[
"Baseline method - supervised learning & DQN ",
"As a baseline for our experiments, we train a CNN by supervised learning to directly predict the actions computed by an $A^*$ planner from the same visual input that the SF-RL model receives.",
"The network structure is the same as the CNN from the SF-RL model ($\\theta _{\\phi }$ ) and differs only in that the output 512 units are fed into a final softmax layer.",
"As an additional baseline we also compare to the DQN approach [5].",
"To ensure a fair comparison we evaluate DQN by learning from scratch and in a transfer learning situation in which we finetune the DQN model trained on the base task; such a fine-tuning approach is known to perform better than simply transferring with fixed features [32], [21] (we also conduct transfer learning experiments with fixed features for DQN for completeness).",
"The training data for the supervised learner is generated beforehand, consisting of $1.6e5$ labeled samples.",
"To generate these samples, full localization is required, while for evaluating the learned network it is not required.",
"As such, this setup can be thought of as the best case scenario for training a CNN to imitate a planner in this domain.",
"To ensure a fair comparison between different methods in the following plots, we scale the number of steps taken by the supervised learner, so that the number of updates matches that of the SF-RL model and that of DQN (the two reinforcement learners start learning at $3e4$ iterations and makes an update every 4 steps after that)."
],
[
"Visual navigation in 3D mazes",
"For the first experiment we trained our deep successor feature reinforcement learner (SF-RL), DQN and the supervised learner on the base map: Map1 (Fig.",
"REF ).",
"To compare the algorithms we perform a testing phase every 10,000 steps consisting of evaluating the performance of the current policy for 5,000 testing steps.",
"Figure: Top-down schematic view of the four different maze environments we consider for the simulated experiments."
],
[
"Base environment",
"We first train on Map1 from scratch.",
"We observe that the supervised learning and reinforcement learning (DQN and SF-RL) models converge to performance comparable to the optimal $A^*$ planner.",
"We also observe that the supervised learner converges significantly faster in this experiment.",
"This is to be expected since it has access to the optimal paths – as computed via $A^*$ – for starting positions covering the whole environment right from the beginning of training.",
"In contrast to this, the reinforcement learners gradually have to build up a dataset of experience and can only make use of the sparsely distributed reward signal to evaluate the actions taken.",
"Figure: Average reward ±\\pm one standard deviation obtained by A * A^* using the true system model, the supervised learner, as well as DQN and SF-RL when learning from scratch and with task transfer from Map1 () and Map3 ()."
],
[
"Transfer to different environment",
"We then perform a transfer learning experiment (using the trained models from above) to a changed environment Map2 where more walls are added (Fig.",
"REF ).",
"In Fig.",
"REF we show a performance comparison between the supervised learner (Supervised), DQN learning from scratch (DQN) and using task transfer (with fixed CNN layers: DQN-FixFeature, and by finetuning the whole network: DQN-Finetune), SF-RL from scratch (SF-RL) and using task transfer (SF-RL-Transfer).",
"We observe that SF-RL-Transfer converges to performance comparable to the optimal policy much faster than training from scratch.",
"Furthermore, in Fig.",
"REF the learning speed of SF-RL-Transfer is even comparable to that of Supervised, who is learning directly from perfectly labeled actions.",
"We observe that when training from scratch, DQN is slightly faster than SF-RL (we attribute this to the fact that SF-RL optimizes a more complicated loss function including e.g.",
"an auto-encoder loss).",
"In the transfer learning setting SF-RL-Transfer is comparable to DQN-Finetune, and converges faster than DQN-FixFeature.",
"It is important to realize that our method preserves the ability to solve the old task after this transfer occurred, which DQN-Finetune is not capable of.",
"To verify this preservation of the old policies we re-evaluated DQN-Finetune and SF-RL-Transfer on all tasks and summarize the results in Tab.",
"REF (DQN-FixFeature keeps the network for the initial tasks completely unchanged thus it is unnecessary to evaluate its performance again).",
"We note that our agent is still able to perform well on the old task, while the DQN agent deviated significantly from the optimal policy (it is still able to solve most of the episodes in this case via a “random-walk”).",
"We also want to emphasize that in contrast to DQN-FixFeature, SF-RL-Transfer has the ability of continuously adapting its features to new tasks while keeping a mapping to all previous task features.",
"Additionally, DQN-FixFeature has to perform the same transfer procedure for all kinds of transfer scenarios due to its black-box property; while with the flexibility of the more structured representation of SF-RL-Transfer, we only need to retrain the successor feature network $\\theta _{\\psi }$ and keep the reward mapping $\\omega $ fixed when only the dynamics changes, or if the dynamics of the environment stay fixed or close to the already observed dynamics, SF-RL-Transfer can adapt quickly by either changing only $\\omega $ or in combination with $\\theta _{\\psi }$ ."
],
[
"More complicated transfer scenarios",
"We then experiment in a more complicated transfer scenario: transferring a base controller from Map3 (Fig.",
"REF ) to Map4 (Fig.",
"REF ).",
"As can be seen from the visualization, the objects change significantly from Map3 to Map4.",
"Also, the goal location moves from the center of an open area to a “hidden” corner.",
"The results for this experiment are depicted in Fig.",
"REF , revealing a similar trend as for the simpler mazes.",
"A re-evaluation of the DQN-Finetune and SF-RL-Transfer agent is shown in Tab.",
"REF .",
"We note that the DQN-Finetune agent loses the policy for Map3 after being transferred to Map4 as the locations of the target and objects changed dramatically, while our agent still is able to solve the old task after the transfer.",
"Furthermore, when transferring from Map1 to Map2 we move from a simpler to a more complicated environment, while Map4 is “simpler” than Map3.",
"Table: Final testing statistics for all considered environments, each evaluated from 50 random starting positions.",
"The maximum number of steps per episode was: 200 steps for Map1&2, 500 steps for Map3&4."
],
[
"Analysis of learned representation",
"As an additional test, we analyzed the representation $\\phi ^k_s$ learned by the SF-RL approach.",
"Specifically, since the reward is defined on the pose of the agent and optimal path finding clearly depends on the agent being able to localize itself we analyzed as to whether $\\phi ^k_s$ encodes the robot pose.",
"To answer this, we extract features $\\phi ^k_s$ for all states along collected optimal trajectories and regressed the ground truth poses of the robot (obtained from our simulator) using a neural network with two hidden layers (128 units each).",
"Fig.",
"REF shows the results from this experiment, overlaying the ground truth poses with the predicted poses from our regressor on a held out example.",
"From these we can conclude that indeed, the transition dynamics is encoded and the agent is able to localize itself, and this information can reliably be retrieved post-hoc (i.e., after training)."
],
[
"Real-World Experiments",
"In order to show the applicability of our method to more realistic scenarios, we conducted additional experiments using a real robot.",
"We start by swapping the RGB camera input for a simulated depth sensor in simulation and then perform a transfer learning experiment to a different, real, environment from which we collect real depth images."
],
[
"Rendered Depth Experiments",
"To obtain a scenario more similar to a real world scene we might encounter, we build a maze-like environment Map5 (Fig.",
"REF ) in our robot simulator that includes realistic walls and object models.",
"In this setting the robot has to navigate to the target (traffic cone in the center) and avoid colliding with objects and walls.",
"We then simulate the robot within this environment, providing rendered depth images from a simulated kinect camera as the input modality (as opposed to the artificial RGB images we used before).",
"Figure: 3D Model of Map5.",
"The robot should avoid thecolored regions containing objects and navigate to the trafficcone in the center.",
"The bottom part shows pairs of images comparingbetween the rendered depth images from our simulator, and realdepth images taken by a kinect camera at approximately the samepose in a real environment modeledafter the simulator."
],
[
"Real World Transfer Experiments",
"We then change to a real robot experiment in which the robot can explore the maze depicted in Map6 (Fig.",
"REF ) (note that the position of the objects and the target are changed from the simulated environment Map5 in Fig.",
"REF ).",
"We collect real depth images in the actual maze-world using the on-board kinect sensor of a Robotino.",
"To avoid training for long periods of time in the real environment we pre-recorded images at all locations that the robot can explore (taking 100 images per position and direction with randomly perturbed robot pose to model noise).",
"The results of training from scratch in this real environment as well as when transfer from the simulated environment is performed are depicted in Fig.",
"REF (the agent starts to learn here after $1e4$ steps, whereas in previous experiments this number is set to $3e4$ ).",
"Similar to the previous experiments we see a large speed-up when transferring knowledge even though the simulated depth images contain none of the characteristic noise patterns present in the real-world kinect data.",
"We note that the agent achieves satisfactory performance at around 60,000 iterations, which corresponds to approximately 8 hours of real experience (assuming data is collected at a rate of 2Hz).",
"After training with the pre-recorded images, the robot is tested in real world environments.",
"A video of the real experiments in two changed environments: Map6 & Map7 (Map7 is not discussed here due to space constraints) can be found at: https://youtu.be/WcCcdkhgjdY.",
"Figure: Comparison between SF-RL trained on the real world Map6, and SF-RL-Transfer with the base model trained on the simulated Map5 transferred to Map6."
],
[
"Conclusion",
"We presented a method for solving robot navigation tasks from raw sensory data, based on an extension of the theory behind successor feature reinforcement learning.",
"Our algorithm is able to naturally transfer knowledge between related tasks and yields substantial speedups over deep reinforcement learning from scratch in the experiments we performed.",
"Despite of these encouraging results, there are several opportunities for future work including testing our approach in more complicated scenarios and extending it to more naturally handle partial observability."
]
] | 1612.05533 |
[
[
"A new generalization of the beta distribution"
],
[
"Abstract The beta distribution is the best-known distribution for modelling doubly-bounded data, \\eg percentage data or probabilities.",
"A new generalization of the beta distribution is proposed, which uses a cubic transformation of the beta random variable.",
"The new distribution is label-invariant like the beta distribution and has rational expressions for the moments.",
"This facilitates its use in mean regression.",
"The properties are discussed, and two examples of fitting to data are given.",
"A modification is also explored in which the Jacobian of the transformation is omitted.",
"This gives rise to messier expressions for the moments but better modal behaviour.",
"In addition, the Jacobian alone gives rise to a general quadratic distribution that is of interest.",
"The new distributions allow good fitting of unimodal data that fit poorly to the beta distribution, and could also be useful as prior distributions."
],
[
"Keywords",
"Beta distribution, cubic transformation, doubly-bounded data, Jacobian, label invariance, regression"
],
[
"Introduction",
"Doubly-bounded data occur in many application areas, as for example percentage data, and also when the random variable is a probability.",
"In this case, the probabilities are not often directly measured, and the data are binary, with events such as death that either occur or do not.",
"However, a distribution of probability may then be used as a prior distribution, in a Bayes or Empirical Bayes analysis.",
"There are relatively few distributions available for modelling doubly-bounded data.",
"The 2-parameter beta distribution defined on $[0,1]$ has pride of place, and many attempts have been made to generalize it to allow more flexible behaviour.",
"These include the generalized beta distribution reviewed by Pham-Gia and Duong (1989).",
"Other generalizations are considered by Nadarajah and Kotz (2006).",
"There are also two very simple generalizations.",
"One is a mixture with a uniform distribution, to allow greater variance.",
"Its use in regression is described in Bayes, Bazán and Garcia (2012).",
"Another is zero inflation, a mixture with a delta-function at zero (Stewart, 2013).",
"Attempts have also been made to replace the beta distribution with a more flexible distribution.",
"The best-known alternative is probably the Kumaraswamy distribution (ibid, 1980).",
"Some other replacement distributions are described in the book `Beyond Beta' by Kotz and van Dorp (2004), and include 2-sided power distributions, generalized trapezoidal distributions, the Topp and Leone distribution, and Johnson's $S_B$ distribution (for which see Johnson, Kotz and Balakrishnan, 1995).",
"There is also the log-Lindley distribution (Gómez-Déniz et al, 2014).",
"Many of these distributions are difficult to use, with likelihoods and moments only expressible in terms of hypergeometric functions, or with cusps (the 2-sided power and trapezoidal distributions).",
"No useful distribution can be completely simple, and even the normal distribution requires a special function, the error function, to compute its distribution function.",
"However, there are several requirements for a practically useful distribution to generalize the beta distribution.",
"One is that the moments should be simple to compute, or at least the mean.",
"This is because we often wish to regress the mean on a covariate, e.g.",
"mean percentage body fat can be regressed on the Quetelet index (body mass index or BMI).",
"To do this using a likelihood-based method, we must compute the mean $\\mu $ for an observation, as a function of the covariates, and in order to compute the log-likelihood, we must then be able to compute the model parameters.",
"For example, with the beta distribution itself with parameters $\\alpha , \\beta $ , the mean is $\\mu =\\alpha /(\\alpha +\\beta )$ , and Mielke (1975) suggests reparameterising to use $\\mu $ and $s=\\alpha +\\beta $ .",
"Then we compute $\\alpha =s\\mu $ and $\\beta =s(1-\\mu )$ and can compute the log-likelihood.",
"Without a simple formula for the mean in terms of the model parameters, such a regression would be difficult.",
"Another requirement is label-invariance.",
"With a random variable $X$ , and standardising the interval to be be $[0,1]$ , we can look at the distribution of $Y=1-X$ , where the ends of the scale have been flipped.",
"We have interchanged the labels, for example `success' and `failure', or `no treatment effect' and `100% treatment effect'.",
"Label-invariance means that $Y$ follows a distribution from the same family as $X$ .",
"Thus if $X \\sim Beta(\\alpha ,\\beta )$ , then $Y \\sim Beta(\\beta ,\\alpha )$ .",
"The fitted model will be the same with the same likelihood value and the same fitted parameter values whichever choice is made.",
"Some distributions, such as the Kumaraswamy distribution, are not label-invariant.",
"However, label invariance in general is a common requirement e.g.",
"in medicine.",
"(e.g.",
"Senn, 1996).",
"For example, suppose we wish to model the distribution of a disease activity index measured on a scale from zero to unity.",
"Without a label-invariant model, we would get a different distribution if we considered the corresponding health index $Y=1-X$ .",
"Which model should we believe?",
"We must also require that the pdf and distribution function can be easily computed on most platforms, i.e.",
"they do not require special functions that may not be available; this is not a crucial requirement, because if a distribution proves useful, the necessary special functions will soon be produced.",
"The distribution function is needed for computing the likelihood when data are censored, an extreme case being the application of a doubly-bounded distribution to fitting grouped data.",
"Random numbers are also needed, for example in Markov-chain Monte Carlo methods, and their generation should preferably be straightforward.",
"The beta distribution itself has the virtues of having a simple expression for the mean and of label-invariance.",
"To compute the likelihood we require the beta function, and the distribution function requires the incomplete beta function.",
"This is also needed for the t-distribution, so is commonly available.",
"Random number generation is not particularly simple, but there exist efficient methods for doing this.",
"It was desired to construct a practically useful generalized beta distribution, as there does not currently appear to be a distribution for doubly-bounded data that has relatively simple expressions for the moments, is free of cusps, and is label-invariant.",
"The log-Lindley distribution, for example, has simple expressions for the moments but is not label-invariant.",
"Bearing the above considerations in mind, we took the random variable $X$ as $X=\\sum _{j=1}^m c_j P^j$ , where $P \\sim Beta(\\alpha ,\\beta )$ , and the $c_j$ are chosen so that $\\,\\mbox{d}x/\\,\\mbox{d}p > 0$ , i.e.",
"we have a monotonic transformation of the beta random variable.",
"This has been explored with $m=2$ and $m=3$ .",
"The case $m=1$ of course simply gives the beta distribution.",
"These new distributions generalize the beta distribution and allow more flexible behaviour, e.g.",
"the skewness for a given mean can change substantially.",
"They are also label-invariant, as we can write $Y=1-X=\\sum _{j=1}^m c_j^{\\prime }(1-P)^j$ , where the $c_j^{\\prime }$ are linear functions of the $c_j$ .",
"The mean can also be computed as a rational expression, thus facilitating its regression on covariates.",
"The difficult task is restricting the $c_j$ to require that $\\,\\mbox{d}x/\\,\\mbox{d}p > 0$ .",
"We shall see that this is straightforward for $m=2$ (quadratic or Q-beta distribution) and less so for $m=3$ (cubic or C-beta distribution).",
"We have not yet gone beyond $m=3$ , where we already have two extra parameters, which gives plenty of flexibility.",
"An unexpected problem with the $m=3$ distribution led to the creation of `Jacobian-less' distributions, and these are regarded as the most useful distributions resulting from this approach.",
"The next section briefly discusses transformations of distributions, then the following sections discuss the detailed properties of the new distributions, after which two examples of their use are given.",
"Finally, the quadratic distributions arising when the transformation is applied to the uniform distribution are described in more detail in appendices."
],
[
"Transformation of pdfs",
"Consider a distribution with pdf $f_p(p)$ defined on $[0,1]$ and a transformation to $x=x(p)$ with an inverse transformation $p=p(x)$ .",
"Here $f_p$ is the pdf of the beta distribution with parameters $\\alpha , \\beta $ , and $x(p)=ap+bp^2+cp^3$ .",
"Denote the respective random variables as $P$ and $X$ respectively.",
"We require that the Jacobian $J(x)=\\,\\mbox{d}x/\\,\\mbox{d}p > 0$ , so that the transformation is monotonic and one-to-one.",
"Since $\\text{Prob}(X < x)=\\text{Prob}(P < p)$ we have that $f_x(x(p))=f_p(p) \\,\\mbox{d}p/\\,\\mbox{d}x=f_p(p)/(\\,\\mbox{d}x/\\,\\mbox{d}p)$ .",
"If the Jacobian becomes very small over some interval of $X$ , $f_x(x)$ will become large, and the distribution might be multimodal.",
"In view of this, a `Jacobian-less' distribution was constructed with pdf $g$ , for which $g_x(x(p))=Cf_p(p)$ for some unknown constant $C$ .",
"This distribution clearly has the same modal structure as $f_p(p)$ , because $\\,\\mbox{d}g_x(x)/\\,\\mbox{d}x=C(\\,\\mbox{d}f_p(p)/\\,\\mbox{d}p)/(\\,\\mbox{d}x/\\,\\mbox{d}p)$ .",
"Since $\\,\\mbox{d}x/\\,\\mbox{d}p > 0$ , the mode of $g_x(x)$ occurs at $x_m=x(p_m)$ , where $p_m$ is the mode of $f_p$ .",
"If $C$ could not be easily determined, this type of pdf would be of little interest.",
"The `parent' distribution of $P$ which is transformed to yield $g_x(x)$ is $Cf_p(p)\\,\\mbox{d}x/\\,\\mbox{d}p$ , which here is simply $Cf_p(p)(a+2bp+3cp^2)$ .",
"This is a mixture of beta-distributions, although some of the weights in the mixture may be negative.",
"It is therefore straightforward to evaluate $C$ by requiring that the pdf integrates to unity.",
"Discarding the Jacobian ensures a unimodal distribution if $f_p(p)$ is unimodal.",
"However, it unavoidably makes results for distribution functions and moments more complicated, and the motto when considering discarding Jacobians should be `if it's not broken, don't fix it'."
],
[
"Computing issues and notation",
"To solve cubics, and even quadratics, use of the Newton-Raphson iteration (e.g.",
"Press et al, 2007) is often recommended here in preference to analytic solutions.",
"This is very quick and easy to program.",
"If it happens that the variable $x$ strays outside $[0,1]$ this probably would not cause a problem, because it would wander back into $[0,1]$ again before convergence, but it is faster and safer to set $x \\rightarrow \\text{min}(\\text{max}(x,0),1)$ after each step.",
"It is safer because $\\,\\mbox{d}x/\\,\\mbox{d}p$ is not guaranteed to be positive for $p < 0$ or $p > 1$ and so the iteration might oscillate or diverge.",
"Computation was done using purpose-written fortran programs and the NAG library.",
"To define some notation, let $P$ be a r.v.",
"that follows the beta distribution with parameters $\\alpha , \\beta $ , so that the pdf $f(p)$ is: $f_p(p)=p^{\\alpha -1}(1-p)^{\\beta -1}/B(\\alpha ,\\beta ),$ where $B$ denotes the beta function.",
"We sometimes write for brevity $\\eta =\\alpha +\\beta $ .",
"Distributions are taken as having support in $[0,1]$ , but it is trivial to change the interval to an arbitrary interval by adding two more parameters.",
"Note that quantiles must always be found from the distribution function using Newton-Raphson iteration; they are not discussed further.",
"It is also straightforward to compute inverse moments such as $\\text{E}(1/X)$ or $\\text{E}\\lbrace (1-X)/X\\rbrace $ ; these are also not discussed further.",
"Bivariate distributions could be constructed, but currently the best procedure would be to use a copula."
],
[
"Properties: the Q-beta (quadratic) distribution",
"It seems that the $m=3$ (cubic) distribution is a lot more flexible than the $m=2$ distribution, and the Jacobian-less distribution still better, but we consider the simpler cases first."
],
[
"Pdf",
"Define $X=2\\gamma P+(1-2\\gamma )P^2$ where $0 < \\gamma < 1$ , where $a=2\\gamma , b=1-2\\gamma , c=0$ .",
"We say that $X$ follows the Q-beta (quadratic-beta) distribution, i.e.",
"$X \\sim QB(\\alpha ,\\beta ,\\gamma )$ .",
"First, since $x(p)=2\\gamma p+(1-2\\gamma )p^2$ , we have that $x(0)=0, x(1)=1$ and $\\,\\mbox{d}x/\\,\\mbox{d}p=2\\gamma +2(1-2\\gamma )p$ .",
"This quickly leads to the requirement $0 \\le \\gamma \\le 1$ , and when $\\gamma =1/2$ we regain the beta distribution.",
"The pdf $f_x(x)=f(p)\\,\\mbox{d}p/\\,\\mbox{d}x$ .",
"Solving the quadratic $2\\gamma p+(1-2\\gamma )p^2-x=0$ gives $p(x)=\\frac{-\\gamma +\\Delta (x)}{1-2\\gamma }=\\frac{x}{\\gamma +\\Delta (x)}$ where $\\Delta (x)=(\\gamma ^2+(1-2\\gamma )x)^{1/2}$ , and $\\,\\mbox{d}x/\\,\\mbox{d}p=2\\Delta (x)$ .",
"Hence the pdf $f_x(x)$ is $f_x(x)= (\\gamma +\\Delta (x))^{2-\\alpha -\\beta }x^{\\alpha -1}(\\gamma +\\Delta (x)-x)^{\\beta -1}/2\\Delta (x) B(\\alpha ,\\beta ).$ The distribution, like the beta distribution, is label-invariant, so that if $X \\sim QB(\\alpha ,\\beta ,\\gamma ), 1-X \\sim QB(\\beta ,\\alpha ,1-\\gamma )$ .",
"We have $1-x=2(1-\\gamma )(1-p)+(1-2(1-\\gamma ))(1-p)^2$ , and so $1-p=\\frac{1-x}{1-\\gamma +\\Delta (x)}$ and we have the alternative form $f_x(x)=\\frac{x^{\\alpha -1}(1-x)^{\\beta -1}}{2\\Delta (x) B(\\alpha ,\\beta )(\\gamma +\\Delta (x))^{\\alpha -1}(1-\\gamma +\\Delta (x))^{\\beta -1}}.$ In practice to compute the pdf a less analytic approach, which works well also for the cubic distribution considered in the next section, can be used.",
"We find $p(x)$ by solving $2\\gamma p+(1-2\\gamma ) p^2-x=0$ either by solving the quadratic or by Newton-Raphson iteration from $p=x$ and the pdf is then $f(p)/(2(\\gamma +(1-2\\gamma )p)$ ."
],
[
"Distribution function",
"The distribution function is also simply related to the distribution function of the beta distribution: using $f_x(x)\\,\\mbox{d}x=f(p)\\,\\mbox{d}p$ yields $F(x)=I(\\alpha ,\\beta ;p(x))=I(\\alpha ,\\beta ;x/(\\gamma +\\Delta (x)))$ , where $I(\\alpha ,\\beta ;p(x))$ denotes the incomplete beta function ratio."
],
[
"Random numbers",
"Random numbers are generated by piggy-backing off the beta distribution; generate $P \\sim Be(\\alpha ,\\beta )$ and then $X=2\\gamma P+(1-2\\gamma )P^2$ ."
],
[
"Moments",
"The moments are also simple to calculate, although messy.",
"We have that $\\text{E}(X^n)=\\int _0^1 f_x(x) x^n \\,\\mbox{d}x=\\int _0^1 (2\\gamma p+(1-2\\gamma ))p^2)^n f(p) \\,\\mbox{d}p,$ from which the moments can be read off using $\\int _0^1 p^m f(p)\\,\\mbox{d}p=(\\alpha )_m/(\\alpha +\\beta )_m,$ where $(a)_m$ denotes the Pochhammer symbol, the ascending factorial, so that $(a)_m=a(a+1)\\cdots (a+m-1)$ .",
"Finally, $\\text{E}(X^n)=\\sum _{j=0}^n {n \\atopwithdelims ()j}(1-2\\gamma )^j(2\\gamma )^{n-j}\\frac{(\\alpha )_{n+j}}{(\\alpha +\\beta )_{n+j}}.$ Specifically, $\\text{E}(X)=\\frac{\\alpha (2\\gamma \\beta +\\alpha +1)}{(\\alpha +\\beta )(\\alpha +\\beta +1)}.$ $\\text{var}(X)=4\\gamma ^2(\\alpha )_2/(\\alpha +\\beta )_2+4\\gamma (1-2\\gamma )(\\alpha )_3/(\\alpha +\\beta )_3+(1-2\\gamma )^2(\\alpha )_4/(\\alpha +\\beta )_4-\\text{E}(X)^2.$"
],
[
"Mode",
"The mode is best found from the pdf expressed in terms of $p(x)$ as $f_x(x)=p^{\\alpha -1}(1-p)^{\\beta -1}/2(\\gamma +(1-2\\gamma )p)B(\\alpha ,\\beta )$ .",
"Taking logarithms and writing $\\,\\mbox{d}\\ln f/\\,\\mbox{d}x=(\\,\\mbox{d}\\ln f/\\,\\mbox{d}p) (\\,\\mbox{d}p/\\,\\mbox{d}x)$ the mode is at $x_m=2\\gamma p_m+(1-2\\gamma )p_m^2$ , where $\\frac{\\alpha -1}{p_m}-\\frac{\\beta -1}{1-p_m}-\\frac{1-2\\gamma }{\\gamma +(1-2\\gamma )p_m}=0$ under the same conditions as for the mode of the beta distribution.",
"One can find $p_m$ and hence $x_m$ by solving this equation by Newton-Raphson iteration or by solving the corresponding quadratic $(1-2\\gamma )p^2+\\lbrace (\\alpha -2)(1-2\\gamma )-(\\alpha +\\beta -2)\\gamma )p+(\\alpha -1)(1-\\gamma )=0.$ For $\\alpha >1, \\beta > 1$ the distribution is unimodal and in general has exactly the same modality as the beta distribution."
],
[
"Parameters",
"We can add two parameters and create the variable $x(p)=ap+bp^2+cp^3$ .",
"Since $x(1)=1$ , $a+b+c=1$ , and we focus on $c$ and $a$ , which will yield parameters $\\gamma $ and $\\delta $ .",
"We require $\\,\\mbox{d}x/\\,\\mbox{d}p \\equiv J(p)>0$ , so $a+2bp+3cp^2 > 0$ , whence $a+2b+3c > 0$ or $a < c+2$ .",
"Given $a > 0, a < c+2$ , then $J(p) > 0$ at $p=0$ and $p=1$ and $J(p)$ could only become negative if the equation for zero slope, $a+2bp+3cp^2=0$ , has at least one root in $[0,1]$ .",
"A sufficient condition for this not to happen is that the determinant $\\Delta ^2(x)=b^2-3ac < 0$ .",
"Substituting for $b=1-a-c$ this yields limits for $a$ of $1+(1/2)c\\pm \\sqrt{3c(4-c)}/2$ .",
"Note by the way that when $c=4$ , $a=3$ , and that we must have $-2 \\le c \\le 4$ .",
"This last can be seen more convincingly from $J(1/2) > 0$ , i.e.",
"$a+b+(3/4)c > 0$ or $c < 4$ .",
"From $J(2/3)=a+(4/3)b+(4/3)c > 0$ , i.e.",
"$3a+4b+4c > 0$ , we have $a < 4$ .",
"The requirement that $\\Delta ^2 < 0$ is not a necessary condition for $J(p) > 0$ in $[0,1]$ , as real roots may exist but be outside the range $[0,1]$ .",
"When $c \\ge 1$ , the lower limit is as previously given, and when $ c \\le 1$ , the upper limit for $a$ is $c+2$ and the lower limit zero.",
"This can be seen as follows: write $J(p)=a(1-p)^2+2(a+b)p(1-p)+(a+2b+3c)p^2$ , i.e.",
"an expansion in Bernstein polynomials.",
"Clearly all coefficients are positive when $a < c+2$ (so that $a+2b+3c > 0$ ) if $a+b > 0$ , i.e.",
"$c < 1$ .",
"Hence for $c < 1$ the upper limit for $a$ such that $J(p) > 0$ in $[0,1]$ is $c+2$ and the lower limit is zero.",
"To summarise: $-2 \\le c \\le 4$ ; if $c \\le 1$ , the range of $a$ is $(0, c+2)$ ; if $c \\ge 1$ , the range of $a$ is $1+c/2-\\sqrt{3c(4-c)}/2, 1+c/2+\\sqrt{3c(4-c)}/2)$ .",
"We now consider how $a$ and $c$ transform when $X \\rightarrow 1-X$ .",
"Writing $1-X=a^\\prime (1-P)+b^\\prime (1-P)^2+c^\\prime (1-P)^3$ we have that $a^\\prime =a+2b+3c$ , so that $a^\\prime =2+c-a$ , and $c^\\prime =c$ .",
"Hence $c$ is invariant under the label transformation, and we take the first parameter $\\delta =(c+2)/6$ , so that $0 \\le \\delta \\le 1$ .",
"Next, for $c < 1$ (or $\\delta < 1/2$ ), define $\\gamma =a/(c+2)$ , and for $c > 1$ define $\\gamma =\\frac{a-(1+c/2)}{\\sqrt{3c(4-c)}}+1/2$ , so that $0 \\le \\gamma \\le 1$ always.",
"Under the label transformation, $\\delta $ stays the same, while $\\gamma \\rightarrow 1-\\gamma $ .",
"Thus $\\text{C-beta}(\\alpha ,\\beta ,\\gamma ,\\delta )\\rightarrow \\text{C-beta}(\\beta ,\\alpha , 1-\\gamma , \\delta )$ .",
"Distributions with $\\gamma =1/2$ and $\\alpha =\\beta $ are therefore symmetric.",
"This parameterisation allows $\\gamma , \\delta $ to each vary in $[0,1]$ and exhibits the label symmetry, but has the drawback that the model with $c=0$ will not have a zero value of $\\delta $ ; it occurs at $\\delta =1/3$ .",
"To convert $\\gamma ,\\delta $ to $a, b, c$ , we write $c=6\\delta -2$ , then if $\\delta < 1/2$ , $a=(c+2)\\gamma $ , otherwise $a=(\\gamma -1/2)\\sqrt{3c(4-c)}+1+c/2$ , and finally $b=1-a-c$ .",
"When $\\delta =1/3$ so that $c=0$ we have $a=2\\gamma $ .",
"Hence $\\gamma $ has the same meaning as for the $\\text{Q-beta}$ distribution, which is now seen to be the $\\text{C-beta}$ distribution with $\\delta =1/3$ .",
"A sensible fitting sequence to ensure convergence would be: fit the beta distribution in the usual way; fit the Q-beta distribution starting from the fitted $\\alpha $ and $\\beta $ values, with $\\gamma =1/2$ (so that $a=1$ ), $\\delta =1/3$ (so that $c=0$ ).",
"in case of difficulty, float $\\gamma $ and then $\\delta $ , or vice versa."
],
[
"Pdf",
"To obtain $p$ from $x$ we solve the cubic $cp^3+bc^2+ap-x=0$ to obtain $p(x)$ .",
"This could be done analytically, e.g.",
"using Vieta's method, but for some parameter values this method is numerically unstable, and a Newton-Raphson iteration starting from $p=x$ is fast and always converges quickly with no numerical problems.",
"The pdf can then be computed as $f_x(x)=\\frac{p(x)^{\\alpha -1}(1-p(x))^{\\beta -1}}{B(\\alpha ,\\beta )(3cp(x)^2+2bp(x)+a)}.$"
],
[
"Distribution function",
"We have again $F(x)=I(\\alpha ,\\beta ;p(x))$ ."
],
[
"Moments",
"The moments are found using the identity $\\text{E}(X^n)=\\int _0^1f(p)(a+bp+cp^2)^n \\,\\mbox{d}p,$ where $a, b, c$ are found from $\\gamma , \\delta $ .",
"The mean is $\\text{E}(X)\\equiv \\mu =\\frac{\\alpha }{\\alpha +\\beta }\\lbrace a+\\frac{\\alpha +1}{\\alpha +\\beta +1}\\lbrace b+\\frac{\\alpha +2}{\\alpha +\\beta +2}c\\rbrace \\rbrace ,$ which has been arranged for fast computation.",
"When regressing the mean on covariates, one can proceed as follows.",
"take as parameters $\\mu , \\eta =\\alpha +\\beta , \\gamma $ and $\\delta $ .",
"Solve $\\frac{\\alpha }{\\eta }\\lbrace a+\\frac{\\alpha +1}{\\eta +1}\\lbrace b+\\frac{\\alpha +2}{\\eta +2}c\\rbrace \\rbrace -\\mu =0$ for $\\alpha $ , either by solving the cubic, or (better) using Newton-Raphson iteration starting from $\\alpha =\\eta /2$ .",
"find $\\beta =\\eta -\\alpha $ and compute the log-likelihood as usual."
],
[
"Mode",
"The mode can be found by differentiating (REF ) as for the quadratic distribution.",
"Then the mode $x_m=ap_m+bp_m^2+cp_m^3$ where $\\frac{\\alpha -1}{p_m}-\\frac{\\beta -1}{1-p_m}-\\frac{2b+6cp_m}{a+2bp_m+3cp_m^2}=0.$ One can solve by Newton-Raphson iteration or by solving the resulting cubic equation for $p_m$ .",
"Compared with the beta distribution, this distribution has a different modal structure, unlike the Q-beta distribution, which had the same structure as the beta.",
"For $c \\ne 0$ there is always one mode in $(0,1)$ , besides the fact that the pdf will be infinite at zero if $\\alpha < 1$ and at unity if $\\beta < 1$ .",
"This behaviour, giving a mode superimposed on a U or J-shaped distribution, while interesting, is probably not often wanted.",
"It was this behaviour, caused by the fact that the Jacobian can be small over a range of $x$ , that led to the creation of the `Jacobian-less distribution, described on the next section.",
"Modal regression could be done by taking parameters $x_m, \\eta =\\alpha +\\beta , \\gamma $ and $\\delta $ .",
"Given the mode $x_m$ and setting $\\beta =\\eta -\\alpha $ one finds $p_m$ and then solves (REF ) for $\\alpha $ , which only requires solving a linear equation.",
"Then the likelihood can be computed in terms of $\\alpha , \\beta , a, b, c$ by setting $\\beta =\\eta -\\alpha $ ."
],
[
"Distributions lacking the Jacobian: SQ-beta and SC-beta distributions",
"The form of this distribution, as $Cf_p(p)(a+2bp+3cp^2)$ , was derived earlier.",
"Making $g_x(x)$ integrate to unity gives $C^{-1}=a+2b \\frac{\\alpha }{\\alpha +\\beta }+3c\\frac{\\alpha (\\alpha +1)}{(\\alpha +\\beta )(\\alpha +\\beta +1)}.$ The properties of the SC-beta distribution only are described; the SQ-beta distribution is of course similar but simpler."
],
[
"Pdf",
"The pdf is $g_x(x)=\\frac{p(x)^{\\alpha -1}(1-p(x))^{\\beta -1}}{B(\\alpha ,\\beta )(a+2b \\frac{\\alpha }{\\alpha +\\beta }+3c\\frac{\\alpha (\\alpha +1)}{(\\alpha +\\beta )(\\alpha +\\beta +1)})}$ where $a, b, c$ are derived from $\\gamma , \\delta $ as before and $p(x)$ is defined as before."
],
[
"Distribution function",
"From integrating the pdf this is $G(x)=C\\lbrace aI(\\alpha ,\\beta ;p(x))+2b\\frac{\\alpha }{\\alpha +\\beta } I(\\alpha +1,\\beta ;p(x))+3c\\frac{\\alpha (\\alpha +1)}{(\\alpha +\\beta )(\\alpha +\\beta +1)}I(\\alpha +2,\\beta ;p(x))\\rbrace .$ This form requires three evaluations of the incomplete beta function.",
"However, the computation can be made quicker (but messier) using the identity $I(\\alpha +1,\\beta ; x)=I(\\alpha ,\\beta ;x)-\\frac{x^\\alpha (1-x)^\\beta }{\\alpha B(\\alpha ,\\beta )},$ which is well-known, and can be derived by integrating $J(\\alpha +1,\\beta )=\\int _0^x u^\\alpha (1-u)^{\\beta -1}\\,\\mbox{d}u$ by parts, differentiating the $u^\\alpha $ term.",
"We then have $G(x)=C\\lbrace a+2b\\frac{\\alpha }{\\eta }+3c\\frac{\\alpha (\\alpha +1)}{\\eta (\\eta +1)}\\rbrace I(\\alpha ,\\beta ;x)-\\frac{Cx^\\alpha (1-x)^\\beta }{B(\\alpha ,\\beta )}\\lbrace \\frac{2b}{\\eta }+3c(\\frac{\\alpha +1}{\\eta (\\eta +1)}+\\frac{x}{\\eta +1})\\rbrace ,$ which requires only one evaluation of an incomplete beta function."
],
[
"Random numbers",
"The pdf of the parent distribution is the $Beta(\\alpha ,\\beta )$ pdf multiplied by $M(p)=C(a+2bp+3cp^2)$ .",
"The rejection method can be used to generate random numbers, by generating random numbers $P$ from $Beta(\\alpha ,\\beta )$ and accepting them with probability $M(P)/M_\\text{max}$ .",
"The maximum value of $M$ is either at $p=0$ or $p=1$ or a stationary value in $[0,1]$ and so is $CM_{\\text{max}}$ , where $M_{\\text{max}}$ is either $\\text{max}(a,a+2b+3c)$ or its maximum with the stationary value at $-b/3c$ , if $0 < -b/3c < 1$ .",
"Hence the acceptance probability is $\\frac{a+2bP+3cP^2}{M_\\text{max}}$ Then, given $P$ , one forms $X=aP+bP^2+cP^3$ .",
"The efficiency (proportion of generated random numbers retained) is $\\int _0^1 f_p(p)\\frac{(a+2bp+3cp^2)}{M_\\text{max}}\\,\\mbox{d}p=C^{-1}/M_\\text{max}.$ This varies depending on the parameter values, but for the examples it was 33.8% and 8%.",
"This method of generating random numbers works, but a more efficient method would be desirable.",
"However, designing a more efficient method would be another research project; there are many ways to proceed."
],
[
"Moments",
"The $n$ th moment is $\\text{E}(X^n)=C\\int _0^1 f_p(p)(a+2bp+3cp^2)(ap+bp^2+cp^3)^n \\,\\mbox{d}p.$ The mean is then $\\text{E}(X)=\\frac{C\\alpha }{\\alpha +\\beta }\\lbrace a^2+\\frac{\\alpha +1}{\\alpha +\\beta +1}\\lbrace 3ab+\\frac{\\alpha +2}{\\alpha +\\beta +2}\\lbrace 4ac+2b^2+\\frac{\\alpha +3}{\\alpha +\\beta +3}\\lbrace 5bc+\\frac{\\alpha +4}{\\alpha +\\beta +4}(3c^2)\\rbrace \\rbrace \\rbrace \\rbrace .$"
],
[
"Mode",
"The mode is simply $p_m(x)=\\frac{\\alpha -1}{\\alpha +\\beta -2}$ if it exists, or in full $x_m=ap_m+bp_m^2+cp_m^3$ .",
"Modal regression would thus be straightforward.",
"The parameters would be $(x_m, \\eta ,\\gamma , \\delta )$ and the equation for $x_m$ would be solved for $\\alpha $ , after which $\\beta =\\eta -\\alpha $ and the pdf can be computed.",
"The transformation used gives rise to two simple distributions that generalise the uniform distribution and allow modal or U-shaped distributions.",
"They are mentioned in appendices A and B for completeness and because they are new.",
"They may find some use in modelling."
],
[
"Fitting to data",
"Two datasets were fitted.",
"The first comprises 252 observations of calculated percentage of body fat plus a variety of other body size measurements, downloaded from statlib and supplied by Dr. A. Fisher.",
"It is referenced in Penrose et al (1985).",
"The second dataset, also from statlib, is a sample of 349 observations of glycosylated hemoglobin (HBA1c) readings reported in DCCT percentages from diabetic patients.",
"This is referenced in Daramola (2012).",
"The results of fitting the beta distribution model, and the Q-beta (quadratic) and C-beta (cubic) models, are shown in table REF .",
"The Jacobian-less distributions SQ-beta and SC-beta were also fitted.",
"Figures REF and REF show histograms of the data, with fitted beta and C-beta distributions.",
"In both cases, the cubic distribution gives a very significant improvement in the log-likelihood.",
"We have in the first case $X^2[2]=10.66, p=0.0048$ and in the second $X^2[2]=33.36, p < 0.001$ , showing that the two added parameters significantly improve the fit.",
"In the first case, the distribution of percentage body fat is almost normal, whereas the beta distribution skews it to the right.",
"The cubic distribution can correct this and give a good fit.",
"In the second case, the data are more skewed to the right than the beta distribution would allow.",
"The cubic distribution corrects this opposite problem.",
"The Jacobian-less distributions in fact fit slightly better in both cases, as seen in table REF .",
"The fitted parameters are quite similar."
],
[
"Conclusions",
"The beta distribution has been generalized by allowing quadratic or cubic functions of the beta random variable.",
"These distributions, especially the cubic, can greatly improve model fit for doubly bounded data.",
"They are fairly tractable, with moments that are rational functions, which allows a straightforward regression of the mean on covariates, and are label invariant like the beta distribution.",
"Modes are computable either as solutions of a quadratic/cubic equation or by Newton-Raphson iteration, so that modal regression is also possible.",
"Distribution functions and random number generation `piggy-back' off that for the parent beta distribution.",
"An obvious modification is to omit the Jacobian in the transformed distribution, so that the parent distribution is now a mixture of beta distributions, where some of the weights can be negative.",
"The rationale is that for some parameter values, a small Jacobian can introduce an extraneous peak into the distribution.",
"The modified cubic distribution fitted the examples slightly better than the originals.",
"It has a much simpler expression for the mode, and is unimodal for $\\alpha >1, \\beta > 1$ but it has messier expressions for the moments and the distribution function.",
"It would of course also be useful if carrying out modal regression rather than mean regression.",
"These distributions could be useful in data fitting and as prior distributions.",
"The beta distribution is well-known as the conjugate prior of the binomial distribution, and a more flexible prior can be useful, e.g.",
"for sensitivity analysis.",
"Obvious future work would be to study the 5-parameter quartic distribution.",
"However, the 4-parameter cubic distribution can already reproduce a wide range of behaviour, so this is not an urgent task.",
"More efficient generation of random numbers for the SC-beta distribution would be useful.",
"The method of generalizing the beta distribution proposed here can be applied to any distribution for doubly-bounded data, thus generating a vast number of possibilities."
],
[
"Appendix A: the C-beta$(1,1,\\gamma ,\\delta )$ (2-parameter) distribution",
"First, the C-beta(1,1) distribution is a special case of the C-beta$(\\alpha ,\\beta )$ distribution discussed earlier.",
"With parameters $\\gamma ,\\delta $ , define $a, b, c$ as before.",
"Then the pdf is $f(x)=1/(a+2bp(x)+3cp(x)^2)$ , where as before $p(x)$ solves $x=ap+bp^2+cp^3$ .",
"Figure REF shows the pdf for various values of $\\gamma $ and $\\delta $ .",
"The distribution function is simply $F(x)=p(x)$ .",
"The moments are $\\text{E}(X)=a/2+b/3+c/4,$ $\\text{var}(X)=a^2/12+4b^2/45+9c^2/112+ab/6+bc/6+3ac/20.$ The mode (which may be an antimode) is at $-b/3c$ if this lies in $(0,1)$ .",
"The curvature at the mode is $-6c/J(p)^4$ , so if $c > 0$ $(\\delta > 1/3)$ the curvature is negative, and it is a mode not an antimode.",
"For $\\delta < 1/2$ the mode may not exist, but for $\\delta > 1/2$ it always does.",
"Random numbers are generated by $X=aU+bU^2+cU^3$ where $U$ is uniform on $[0,1]$ .",
"This distribution with $\\delta > 1/2$ can give narrow peaks with a flattish background, and would be suitable as a prior distribution with fat tails.",
"The Jacobian-less distribution SC-beta$(1,1,\\gamma ,\\delta )$ is simply the uniform distribution.",
"However, its parent, a quadratic distribution, is of interest.",
"This type of distribution was not considered in the general case, because it did not fit the data as well as the C-beta and SC-beta distributions.",
"When $\\alpha =\\beta =1$ however we have a new and potentially useful distribution.",
"The distribution has pdf $f(p)=a+2bp+3cp^2$ .",
"This is shown in figure REF for various values of $\\gamma $ and $\\delta $ .",
"This gives distribution function $F(p)=ap+bp^2+cp^3$ .",
"Random numbers can be generated in at least two ways.",
"One is to solve $aP+bP^2+cP^3-U=0$ , where $U$ is a uniformly-distributed random number, either analytically or using Newton-Raphson iteration.",
"The other method is rejection sampling, by generating $U$ and accepting it with probability $f(U)/f_{\\text{max}}$ , as described for the Jacobian-less distribution.",
"The moments are $\\text{E}(P)=a/2+2b/3+3c/4,$ $\\text{var}(P)=a/3+b/2+3c/5-(a/2+2b/3+3c/4)^2.$ The moment-generating function can be found from $\\text{E}(\\exp (tP))=\\int _0^1 (a+2bp+3cp^2)\\exp (tp)\\,\\mbox{d}p,$ as $\\text{E}(\\exp (tP))=a\\frac{(\\exp (t)-1)}{t}+2b\\lbrace \\frac{\\exp (t)}{t}-\\frac{(\\exp (t)-1)}{t^2}\\rbrace +3c\\lbrace \\frac{\\exp (t)}{t}-2\\frac{\\exp (t)}{t^2}+2\\frac{(\\exp (t)-1)}{t^3}\\rbrace .$ The mode is again at $p=-b/3c$ if this lies in $(0,1)$ .",
"This distribution generalizes the uniform and U-quadratic distributions."
]
] | 1612.05426 |
[
[
"Erd\\H{o}s-Gallai-type results for total monochromatic connection of\n graphs"
],
[
"Abstract A graph is said to be {\\it total-colored} if all the edges and the vertices of the graph are colored.",
"A total-coloring of a graph is a {\\it total monochromatically-connecting coloring} ({\\it TMC-coloring}, for short) if any two vertices of the graph are connected by a path whose edges and internal vertices have the same color.",
"For a connected graph $G$, the {\\it total monochromatic connection number}, denoted by $tmc(G)$, is defined as the maximum number of colors used in a TMC-coloring of $G$.",
"In this paper, we study two kinds of Erd\\H{o}s-Gallai-type problems for $tmc(G)$ and completely solve them."
],
[
"Introduction",
"In this paper, all graphs are simple, finite and undirected.",
"We refer to book [1] for undefined notation and terminology in graph theory.",
"Throughout this paper, let $n$ and $m$ denote the order (number of vertices) and size (number of edges) of a graph, respectively.",
"Moreover, a vertex of a connected graph is called a leaf if its degree is one; otherwise, it is an internal vertex.",
"Let $l(T)$ and $q(T)$ denote the number of leaves and the number of internal vertices of a tree $T$ , respectively, and let $l(G)=\\max \\lbrace l(T) | $ $T$ is a spanning tree of $G$ $\\rbrace $ and $q(G)=\\min \\lbrace q(T) | $ $T$ is a spanning tree of $G$ $\\rbrace $ for a connected graph $G$ .",
"Note that the sum of $l(G)$ and $q(G)$ is $n$ for any connected graph $G$ of order $n$ .",
"A path in an edge-colored graph is a monochromatic path if all the edges on the path have the same color.",
"An edge-coloring of a connected graph is a monochromatically-connecting coloring (MC-coloring, for short) if any two vertices of the graph are connected by a monochromatic path of the graph.",
"For a connected graph $G$ , the monochromatic connection number of $G$ , denoted by $mc(G)$ , is defined as the maximum number of colors used in an MC-coloring of $G$ .",
"An extremal MC-coloring is an MC-coloring that uses $mc(G)$ colors.",
"Note that $mc(G)=m$ if and only if $G$ is a complete graph.",
"The concept of $mc(G)$ was first introduced by Caro and Yuster [4] and has been well-studied recently.",
"We refer the reader to [2], [6] for more details.",
"In [8], we introduced the concept of total monochromatic connection of graphs.",
"A graph is said to be total-colored if all the edges and the vertices of the graph are colored.",
"A path in a total-colored graph is a total monochromatic path if all the edges and internal vertices on the path have the same color.",
"A total-coloring of a graph is a total monochromatically-connecting coloring (TMC-coloring, for short) if any two vertices of the graph are connected by a total monochromatic path of the graph.",
"For a connected graph $G$ , the total monochromatic connection number, denoted by $tmc(G)$ , is defined as the maximum number of colors used in a TMC-coloring of $G$ .",
"An extremal TMC-coloring is a TMC-coloring that uses $tmc(G)$ colors.",
"It is easy to check that $tmc(G)=m+n$ if and only if $G$ is a complete graph.",
"Moreover, in [7] we determined the threshold function for a random graph to have $tmc(G)\\ge f(n)$ , where $f(n)$ is a function satisfying $1\\le f(n)<\\frac{1}{2}n(n-1)+n$ .",
"Actually, these concepts are not only inspired by the concept of monochromatic connection number but also by the concepts of monochromatic vertex connection number and total rainbow connection number of connected graphs.",
"For details about them we refer to [3], [9], [10], [11].",
"From the definition of the total monochromatic connection number, the following results are immediate.",
"Proposition 1 [8] If $G$ is a connected graph and $H$ is a connected spanning subgraph of $G$ , then $tmc(G)\\ge m(G)-m(H)+tmc(H)$ .",
"Theorem 1 [8] For a connected graph $G$ , $tmc(G)\\ge m-n+2+l(G)$ .",
"In particular, $tmc(G)=m-n+2+l(G)$ if $G$ is a tree.",
"In [8] we also showed that there are dense graphs that still meet this lower bound.",
"Theorem 2 [8] Let $G$ be a connected graph of order $n>3$ .",
"If $G$ satisfies any of the following properties, then $tmc(G)=m-n+2+l(G)$ .",
"$(a)$ The complement $\\overline{G}$ of $G$ is 4-connected.",
"$(b)$ $G$ is $K_3$ -free.",
"$(c)$ $\\Delta (G)<n-\\frac{2m-3(n-1)}{n-3}$ .",
"$(d)$ $diam(G)\\ge 3$ .",
"$(e)$ $G$ has a cut vertex.",
"Moreover, we gave an example in [8] to show that the lower bound $m-n+2+l(G)$ is not always attained.",
"Lemma 1 [8] Let $G= K_{n_1,\\ldots ,n_r}$ be a complete multipartite graph with $n_1 \\ge \\ldots \\ge n_t\\ge 2$ and $n_{t+1}=\\ldots =n_r=1$ .",
"Then $tmc(G)=m+r-t$ .",
"Let $G$ be a connected graph and $f$ be an extremal TMC-coloring of $G$ that uses a given color $c$ .",
"Note that the subgraph $H$ formed by the edges and vertices with color $c$ is a tree where the color of each internal vertex is $c$ ; see [8].",
"Now we define the color tree as the tree formed by the edges and vertices with color $c$ , denoted by $T_c$ .",
"If $T_c$ has at least two edges, the color $c$ is called nontrivial; otherwise, $c$ is trivial.",
"We call an extremal TMC-coloring simple if for any two nontrivial colors $c$ and $d$ , the corresponding trees $T_c$ and $T_d$ intersect in at most one vertex.",
"If $f$ is simple, then the leaves of $T_c$ must have distinct colors different from color $c$ .",
"Moreover, a nontrivial color tree of $f$ with $m^{\\prime }$ edges and $q^{\\prime }$ internal vertices is said to waste $m^{\\prime }-1+q^{\\prime }$ colors since the edges and internal vertices of a nontrivial color tree must have the same color.",
"In fact, we can use at most $m+n$ colors to assign its edges and vertices with different colors.",
"Thus, if $f$ wastes $x$ colors, then $tmc(G)=m+n-x$ .",
"For the rest of this paper we will use these facts without further mentioning them.",
"In addition, we list a helpful lemma below.",
"Lemma 2 [8] Every connected graph $G$ has a simple extremal TMC-coloring.",
"Among many interesting problems in extremal graph theory is the Erdős-Gallai-type problem to determine the maximum or minimum value of a graph parameter with some given properties.",
"In [2], [3], the authors investigated two kinds of Erdős-Gallai-type problems for monochromatic connection number and monochromatic vertex connection number, respectively.",
"Motivated by these, we study two kinds of Erdős-Gallai-type problems for $tmc(G)$ in this paper.",
"$\\displaystyle $Problem A.",
"Given two positive integers $n$ and $k$ , compute the minimum integer $f(n,k)$ such that for any connected graph $G$ of order $n$ , if $|E(G)|\\ge f(n,k)$ then $tmc(G)\\ge k$ .",
"$\\displaystyle $Problem B.",
"Given two positive integers $n$ and $k$ , compute the maximum integer $g(n,k)$ such that for any connected graph $G$ of order $n$ , if $|E(G)|\\le g(n,k)$ then $tmc(G)\\le k$ .",
"Note that for a connected graph $G$ we have $3\\le tmc(G)\\le \\binom{n}{2}+n$ , and that $g(n,k)$ does not exist for $3\\le k\\le n-1$ since for a star $S_n$ on $n$ vertices we have $tmc(S_n)=n$ .",
"Thus we just need to determine the exact values of $f(n,k)$ for $3\\le k\\le \\binom{n}{2}+n$ and $g(n,k)$ for $n\\le k\\le \\binom{n}{2}+n$ in the following.",
"Theorem 3 Given two positive integers $n$ and $k$ with $3\\le k\\le \\binom{n}{2}+n$ , $f(n,k)={\\left\\lbrace \\begin{array}{ll}n-1 &if\\ k=3, \\cr n+k-t-2 &if\\ k=\\binom{t}{2}+t+2-s,\\ where\\ 0\\le s\\le t-1 \\ and\\ 2\\le t\\le n-2, \\cr k &if\\ \\binom{n}{2}-n+4\\le k\\le \\binom{n}{2}+n-3\\lfloor \\frac{n}{2}\\rfloor \\ except\\ for\\ n\\ is\\ odd\\cr \\ \\ &\\ \\ \\ \\ and\\ k=\\binom{n}{2}+n-3\\lfloor \\frac{n}{2}\\rfloor ,\\cr \\binom{n}{2}-r &if\\ \\binom{n}{2}+n-3(r+1)<k\\le \\binom{n}{2}+n-3r,\\ where\\ 0\\le r\\le \\lfloor \\frac{n}{2}\\rfloor -1\\cr \\ \\ &\\ \\ \\ \\ or\\ n\\ is\\ odd,\\ r=\\lfloor \\frac{n}{2}\\rfloor \\ and\\ k=\\binom{n}{2}+n-3\\lfloor \\frac{n}{2}\\rfloor .\\end{array}\\right.",
"}$ Theorem 4 Given two positive integers $n$ and $k$ with $n\\le k\\le \\binom{n}{2}+n$ , $g(n,k)={\\left\\lbrace \\begin{array}{ll}k-n+t &if\\ \\binom{n-t}{2}+t(n-t-1)+n\\le k\\le \\binom{n-t}{2}+t(n-t)+n-2, \\cr k-n+t-1 &if\\ k=\\binom{n-t}{2}+t(n-t)+n-1, \\cr \\binom{n}{2}-1 &if\\ k=\\binom{n}{2}+n-1, \\cr \\binom{n}{2} &if\\ k=\\binom{n}{2}+n,\\end{array}\\right.",
"}$ for $2\\le t\\le n-1$ .",
"In the next sections we will give the proofs of the two theorems."
],
[
"Proof of Theorem ",
"Firstly, we give some useful lemmas.",
"Lemma 3 [5] Let $G$ be a connected graph with $|E(G)|\\ge |V(G)|+\\binom{t}{2}$ and $t\\le |V(G)|-3$ .",
"Then $G$ has a spanning tree with at least $t+1$ leaves and this is best possible.",
"Given three nonnegative integers $n$ , $t$ and $s$ such that $2\\le t\\le n-2$ and $0\\le s\\le t-1$ .",
"We can find a graph $G_{t,s}$ on $n$ vertices with $m(G_{t,s})=n+\\binom{t}{2}-1-s$ and $l(G_{t,s})=t$ .",
"Construct $G_{t,s}$ as follows: first let $H$ be the graph obtained from a complete graph $K_{t+1}$ by replacing its one edge $uv$ by a path of $n-t$ edges between the ends of $uv$ ; second we delete $s$ edges between $u$ and the vertices of $V(K_{t+1})\\backslash \\lbrace u,v\\rbrace $ from $H$ .",
"It can be checked that $m(G_{t,s})=n+\\binom{t}{2}-1-s$ and $l(G_{t,s})=t$ .",
"Next we will show that $tmc(G_{t,s})=m(G_{t,s})-n+2+l(G_{t,s})$ .",
"Lemma 4 Let $G$ be the graph $G_{t,s}$ described above.",
"Then $tmc(G)=m-n+2+l(G).$ Proof.",
"Let $f$ be a simple extremal TMC-coloring of $G$ .",
"Suppose that $f$ consists of $k$ nontrivial color trees, denoted by $T_1,\\ldots ,T_k$ .",
"Observe that every vertex appears in at least one of the nontrivial color trees.",
"Suppose $k\\ge 2$ .",
"Let $T^{\\prime }$ and $T^{\\prime \\prime }$ be any two nontrivial color trees of $f$ .",
"Since $f$ is simple, there is at most one common vertex between $T^{\\prime }$ and $T^{\\prime \\prime }$ .",
"If $T^{\\prime }$ and $T^{\\prime \\prime }$ have no common vertex, then there is a total monochromatic path between each vertex of $V(T^{\\prime })$ and each vertex of $V(T^{\\prime \\prime })$ .",
"If $T^{\\prime }$ and $T^{\\prime \\prime }$ have a common vertex $w^{\\prime }$ , then there is a total monochromatic path between each vertex of $V(T^{\\prime })$ and each vertex of $V(T^{\\prime \\prime })\\backslash \\lbrace w^{\\prime }\\rbrace $ .",
"Moreover, $\\delta (G)\\le 2$ .",
"Hence, $k=2$ and there exists a common vertex $w$ between $T_1$ and $T_2$ , which is a leaf of $T_1$ and $T_2$ , respectively.",
"In addition, $w$ is the unique vertex of degree two in $G$ .",
"If $t<n-2$ , then there exist at least two vertices of degree two in $G$ , a contradiction.",
"If $t=n-2$ , then there exists an edge between the two neighbors of $w$ , a contradiction to the construction of $G$ .",
"Hence, $k=1$ and so $tmc(G)=m-n+2+l(G)$ .",
"$\\Box $ Given two positive integers $n$ and $p$ with $\\frac{n}{2}<p<n-2$ , let $t=2(p+1)-n$ and $G_n^{t}$ be the graph defined as follows: partition the vertex set of the complete graph $K_n$ into $n-p$ vertex-classes $V_1,V_2,...,V_{n-p}$ , where $|V_1|=|V_2|=...=|V_{n-p-1}|=2$ and $|V_{n-p}|=t$ ; for each $j\\in \\lbrace 1,...,n-p\\rbrace $ , select a vertex $v_j^*$ from $V_j$ , and delete all the edges joining $v_j^*$ to the other vertices in $V_j$ .",
"Next we will show that $tmc(G_n^t)=m(G_n^t)$ .",
"Lemma 5 Let $G$ be the graph $G_n^t$ described above.",
"Then $tmc(G)=m$ .",
"Proof.",
"Let $f$ be a simple extremal TMC-coloring of $G$ .",
"Suppose that $f$ consists of $k$ nontrivial color trees, denoted by $T_1,\\ldots ,T_k$ , where $t_i=|V(T_i)|$ and $q_i=q(T_i)$ for $1\\le i\\le k$ .",
"Observe that every vertex appears in at least one of the nontrivial color trees.",
"Note that $m-n+2+l(G)=m$ and $tmc(G)\\ge m$ by Theorem REF .",
"As $T_i$ has $t_i-1$ edges and $q_i$ internal vertices, it wastes $t_i-2+q_i$ colors.",
"To show $tmc(G)\\le m$ , we just need to show that $f$ wastes at least $n$ colors, i.e.",
"$\\sum _{i=1}^k(t_i-2+q_i)\\ge n$ .",
"In fact, consider the spanning subgraph $G^{\\prime }$ consisting of the union of the $T_i$ 's and let $C_1,\\ldots ,C_s$ denote its components.",
"We claim that for $1\\le i \\le n-p$ , the vertices of $V_i$ are in the same component.",
"Otherwise, there exist two nonadjacent vertices of $V_i$ which are not total-monochromatically connected, a contradiction.",
"Thus, the components $C_1,\\ldots ,C_s$ form a partition of the vertex classes of $G$ .",
"Let $C$ be a component of $C_1,\\ldots ,C_s$ .",
"If there is exactly one nontrivial color tree in $C$ , it can not be a star.",
"Otherwise, there exist two nonadjacent vertices of the vertex class containing the center, which are not total-monochromatically connected, a contradiction.",
"Hence, there exist at least two internal vertices.",
"Then the nontrivial color tree of $C$ wastes at least $|V(C)|-2+2=|V(C)|$ colors.",
"Suppose $C$ contains $k_c$ ($\\ge 2$ ) nontrivial color trees, denoted by $T_1,\\ldots ,T_{k_c}$ without loss of generality.",
"If $q_i=1$ for some $i\\in \\lbrace 1,2,\\ldots ,k_c\\rbrace $ , then $T_i$ is a star and the center of $T_i$ must be in at least one other nontrivial color tree of $C$ since the vertices of the vertex-class containing the center must be total-monochromatically connected.",
"So we have that $\\begin{array}{llllll}\\sum _{i=1}^{k_c}(t_i-2+q_i)&\\ge \\sum _{i=1,q_i\\ge 2}^{k_c}(t_i-2+2)+ \\sum _{i=1,q_i=1}^{k_c}(t_i-2+1) \\\\[4pt]&=\\sum _{i=1,q_i\\ge 2}^{k_c} t_i+\\sum _{i=1,q_i=1}^{k_c}(t_i-1) \\\\[4pt]&=\\sum _{i=1}^{k_c}t_i-\\sum _{i=1,q_i=1}^{k_c}1 \\\\[4pt]&\\ge |V(C)|+\\sum _{i=1,q_i=1}^{k_c}1-\\sum _{i=1,q_i=1}^{k_c}1 \\\\[4pt]&=|V(C)|.\\end{array}$ Then the nontrivial color trees of $C$ waste at least $|V(C)|$ colors.",
"Thus for $1\\le i\\le s$ the nontrivial color trees of $C_i$ waste at least $|V(C_i)|$ colors.",
"Then $f$ wastes at least $\\sum _{i=1}^{s}|V(C_i)|=n$ colors and so $tmc(G)\\le m$ .",
"The proof is thus complete.",
"$\\Box $ Lemma 6 Let $n$ and $p$ be two integers with $0\\le p\\le n-3$ .",
"Then every connected graph $G$ with $n$ vertices and $m=\\binom{n}{2}-p$ edges satisfies that $tmc(G)\\ge \\binom{n}{2}+n-3p$ if $0\\le p\\le \\frac{n}{2}$ and $tmc(G)\\ge \\binom{n}{2}-p$ if $\\frac{n}{2}<p\\le n-3$ .",
"Proof.",
"It is trivial for $p=0$ , and so assume $1\\le p\\le n-3$ .",
"Let $\\widetilde{G}$ be the graph obtained from $\\overline{G}$ by deleting all the isolated vertices.",
"If $n(\\widetilde{G})\\le p+1 \\ (\\le n-2)$ , then we can find at least two vertices $v_1,v_2$ of degree $n-1$ in $G$ .",
"Take a star $S$ with $E(S)=\\lbrace v_1v:v\\in V(\\widetilde{G})\\rbrace $ .",
"We give all the edges and the internal vertex in $S$ one color, and every other edge and vertex in $G$ a different fresh color.",
"Obviously, it is a TMC-coloring of $G$ , which wastes at most $n(\\widetilde{G})$ colors.",
"If $n(\\widetilde{G})\\ge p+2$ , say $n(\\widetilde{G})=p+t \\ (t\\ge 2)$ , then $\\widetilde{G}$ has at least $t$ components since $m(\\widetilde{G})=p$ .",
"Let $u$ and $v$ be two vertices of $\\widetilde{G}$ which are in two different components.",
"We obtain a double star $S^{\\prime }$ in $G$ by connecting $u$ to each vertex in the same component with $v$ of $\\widetilde{G}$ and $v$ to the other vertices of $\\widetilde{G}$ .",
"Assign all the edges and internal vertices in $S^{\\prime }$ one color, and all the other edges and vertices in $G$ different new colors.",
"Clearly, this is a TMC-coloring of $G$ , which wastes $n(\\widetilde{G})$ colors.",
"If $1\\le p\\le \\frac{n}{2}$ , then $n(\\widetilde{G})\\le 2p$ since $m(\\widetilde{G})=p$ , implying $tmc(G)\\ge m+n-2p=\\binom{n}{2}+n-3p$ .",
"If $\\frac{n}{2}<p\\le n-3$ , then we have that $tmc(G)\\ge \\binom{n}{2}-p$ since $n(\\widetilde{G})\\le n$ .",
"The proof is now complete.",
"$\\Box $ Now we are ready to prove Theorem REF .",
"Proof of Theorem 3: Clearly, $f(n,3)=n-1$ , so the assertion holds for $k=3$ .",
"Suppose that $k=\\binom{t}{2}+t+2-s$ where $0\\le s\\le t-1$ and $2\\le t\\le n-2$ , namely $4\\le k\\le \\binom{n}{2}-n+3$ .",
"If a connected graph $G$ with $n$ vertices satisfies $m(G)\\ge n+k-t-2$ , then $l(G)\\ge t$ by Lemma REF since $n+k-t-2=n+\\binom{t}{2}-s\\ge n+\\binom{t-1}{2}$ .",
"By Theorem REF , we have that $tmc(G)\\ge m-n+2+l(G)\\ge n+k-t-2-n+2+t=k$ .",
"Thus $f(n,k)\\le n+k-t-2$ .",
"To show $f(n,k)\\ge n+k-t-2$ , it suffices to find a connected graph $G_k$ on $n$ vertices such that $m(G_k)=n+k-t-3$ and $tmc(G_k)<k$ .",
"Take $G_k$ as the graph $G_{t,s}$ described in Lemma REF such that $m(G_k)=n+\\binom{t}{2}-1-s=n+k-t-3$ and $l(G_k)=t$ .",
"By Lemma REF , we have that $tmc(G_k)=m(G_k)-n+2+l(G_k)=n+k-t-3-n+2+t=k-1<k$ .",
"Assume that $\\binom{n}{2}-n+4\\le k=\\binom{n}{2}-q\\le \\binom{n}{2}+n-3\\lfloor \\frac{n}{2}\\rfloor $ except for $n$ is odd and $k=\\binom{n}{2}+n-3\\lfloor \\frac{n}{2}\\rfloor $ .",
"For a connected graph $G$ with $n$ vertices satisfies $m(G)=\\binom{n}{2}-q^{\\prime }\\ge k\\ (q^{\\prime }\\le q)$ , it follows from Lemma REF that $tmc(G)\\ge \\binom{n}{2}+n-3q^{\\prime }\\ge \\binom{n}{2}-q^{\\prime }\\ge k$ if $0\\le q^{\\prime }\\le \\frac{n}{2}$ and $tmc(G)\\ge \\binom{n}{2}-q^{\\prime }\\ge k$ if $\\frac{n}{2}<q^{\\prime }\\le q$ , implying $f(n,k)\\le k$ .",
"To show $f(n,k)\\ge k$ , it suffices to find a connected graph $G_k$ on $n$ vertices such that $m(G_k)=k-1=\\binom{n}{2}-q-1$ and $tmc(G_k)<k=\\binom{n}{2}-q$ .",
"Take $G_k$ as $G_n^t$ such that $t=2(p+1)-n$ ($p=q+1$ ) and $m(G_k)=\\binom{n}{2}-q-1$ .",
"By Lemma REF , we have that $tmc(G_k)=m(G_k)=k-1<k$ .",
"Suppose that $\\binom{n}{2}+n-3(r+1)<k\\le \\binom{n}{2}+n-3r\\ (0\\le r\\le \\lfloor \\frac{n}{2}\\rfloor -1$ ) or $n$ is odd, $r=\\lfloor \\frac{n}{2}\\rfloor $ and $k=\\binom{n}{2}+n-3\\lfloor \\frac{n}{2}\\rfloor $ .",
"If a connected graph $G$ on $n$ vertices satisfies $m(G)=\\binom{n}{2}-r^{\\prime }\\ge \\binom{n}{2}-r\\ (r^{\\prime }\\le r)$ , then $tmc(G)\\ge \\binom{n}{2}+n-3r^{\\prime }\\ge \\binom{n}{2}+n-3r\\ge k$ by Lemma REF .",
"Thus, $f(n,k)\\le \\binom{n}{2}-r$ .",
"To show $f(n,k)\\ge \\binom{n}{2}-r$ , it suffices to find a connected graph $G_k$ on $n$ vertices such that $m(G_k)=\\binom{n}{2}-r-1$ and $tmc(G_k)<k$ .",
"For the case that $\\binom{n}{2}+n-3(r+1)<k\\le \\binom{n}{2}+n-3r$ , where $0\\le r\\le \\lfloor \\frac{n}{2}\\rfloor -1$ , take $G_k$ as a complete multipartite graph $K_{n_1,\\ldots ,n_{n-(r+1)}}$ with $n_1 =\\ldots =n_{r+1}= 2$ and $n_{r+2}=\\ldots =n_{n-(r+1)}=1$ .",
"It can be checked that $m(G_k)=\\binom{n}{2}-r-1$ and $tmc(G_k)=m(G_k)+n-(r+1)-(r+1)=\\binom{n}{2}+n-3(r+1)<k$ by Lemma REF .",
"For the case that $n$ is odd, $r=\\lfloor \\frac{n}{2}\\rfloor $ and $k=\\binom{n}{2}+n-3\\lfloor \\frac{n}{2}\\rfloor $ , take $G_k$ as $G_n^3$ such that $m(G_k)=\\binom{n}{2}-\\lfloor \\frac{n}{2}\\rfloor -1$ .",
"By Lemma REF , we have that $tmc(G_k)=m(G_k)=\\binom{n}{2}-\\lfloor \\frac{n}{2}\\rfloor -1<k$ .",
"The proof is thus complete.$\\Box $"
],
[
"Proof of Theorem ",
"In order to prove Theorem REF , we need the following lemma.",
"Recall that $\\binom{1}{2}=0$ .",
"Lemma 7 Let $G$ be a connected graph with $n$ vertices and $m$ edges.",
"If $\\binom{n-t}{2}+t(n-t)\\le m\\le \\binom{n-t}{2}+t(n-t)+(t-2)$ for some $t\\in \\lbrace 2,...,n-1\\rbrace $ , then $tmc(G)\\le m+n-t$ .",
"Moreover, the bound is sharp.",
"Proof.",
"We are given a simple extremal TMC-coloring $f$ of $G$ .",
"Since $2\\le t\\le n-1$ , we have $m\\le \\binom{n}{2}-1$ .",
"Then $G$ is not a complete graph and so there is at least one nontrivial color tree.",
"Suppose that $f$ consists of $k$ nontrivial color trees, denoted by $T_1,...,T_k$ where $t_i=|V(T_i)|$ and $q_i=q(T_i)$ for $1\\le i\\le k$ .",
"Since $T_i$ has $t_i-1$ edges and $q_i$ internal vertices, it wastes $t_i-2+q_i$ colors.",
"In order to show $tmc(G)\\le m+n-t$ , we just need to show that $f$ wastes at least $t$ colors, i.e.",
"$\\sum _{i=1}^{k}(t_i-2+q_i)\\ge t$ .",
"Next it suffices to show that $\\sum _{i=1}^{k}(t_i-2)\\ge t-1$ since $\\sum _{i=1}^{k}q_i\\ge 1$ .",
"Note that each $T_i$ can total-monochromatically connect at most $\\binom{t_i-1}{2}$ pairs of nonadjacent vertices in $G$ .",
"Then we have $\\sum _{i=1}^{k}\\binom{t_i-1}{2}\\ge \\binom{n}{2}-m.$ Suppose $\\sum _{i=1}^{k}(t_i-2)<t-1$ , that is, $\\sum _{i=1}^{k}(t_i-1)<t-1+k$ .",
"Since $T_i$ is nontrivial, we have $t_i-1\\ge 2$ .",
"Then $1\\le k\\le t-2$ .",
"By straight forward convexity, the expression $\\sum _{i=1}^{k}\\binom{t_i-1}{2}$ , subject to $t_i-1\\ge 2$ , is maximized when $k-1$ of the $t_i^{\\prime }$ s equal 3 and one of the $t_i^{\\prime }$ s, say $t_k$ , is as large as it can be, namely, $t_k-1$ is the largest integer smaller than $(t-1+k)-2(k-1)=t-k+1$ .",
"Hence $t_k-1=t-k$ .",
"Even in this extremal case, we have that $\\sum _{i=1}^{k}\\binom{t_i-1}{2}\\le (k-1)+\\binom{t-k}{2}\\le \\binom{t-1}{2}.$ In fact, $\\binom{t-1}{2}+m\\le \\binom{t-1}{2}+\\binom{n-t}{2}+t(n-t)+(t-2)=\\binom{n}{2}-1.$ Hence, $\\sum _{i=1}^{k}\\binom{t_i-1}{2}\\le \\binom{n}{2}-m-1<\\binom{n}{2}-m$ , a contradiction.",
"Next we will show that the bound is sharp.",
"Let $G^{*}$ be the graph defined as follows: first take a complete $(n-t+1)$ -partite graph with vertex-classes $V_1,...,V_{n-t+1}$ such that $|V_j|=1$ for $1\\le j\\le n-t$ and $V_{n-t+1}=t$ ; then add the remaining (at most $t-2$ ) edges to $V_{n-t+1}$ randomly.",
"Clearly, $G^*$ has a spanning subgraph isomorphic to a complete $(n-t+1)$ -partite graph $K_{1,\\ldots ,1,t}$ .",
"By Proposition REF and Theorem REF , it follows that $tmc(G)\\ge m+n-t$ .",
"Hence, $tmc(G)=m+n-t$ .",
"$\\Box $ $\\displaystyle $Proof of Theorem 4.",
"It is trivial for the case that $k=\\binom{n}{2}+n$ .",
"If $k=\\binom{n}{2}+n-1$ , we have $g(n,k)\\le \\binom{n}{2}-1 $ since $tmc(G)=\\binom{n}{2}+n$ for a complete graph $G$ .",
"If a connected graph $G$ on $n$ vertices satisfies $m(G)\\le \\binom{n}{2}-1$ , then there exist two nonadjacent vertices which are total-monochromatically connected by a nontrivial color tree and so it wastes at least two colors.",
"implying $tmc(G)\\le \\binom{n}{2}+n-3<k$ .",
"Thus, $g(n,k)\\ge \\binom{n}{2}-1$ and so $g(n,k)=\\binom{n}{2}-1$ .",
"For $\\binom{n-t}{2}+t(n-t-1)+n\\le k\\le \\binom{n-t}{2}+t(n-t)+n-2$ where $2\\le t\\le n-1$ , if a connected graph $G$ on $n$ vertices satisfies $m(G)\\le k-n+t(\\le \\binom{n-t}{2}+t(n-t)+t-2)$ , then $tmc(G)\\le m(G)+n-t\\le k$ by Lemma REF .",
"Hence, $g(n,k)\\ge k-n+t$ .",
"To show $g(n,k)\\le k-n+t$ , it suffices to find a connected graph $G$ on $n$ vertices such that $m(G)=k-n+t+1$ and $tmc(G)>k$ .",
"If $t=2$ , then $k=\\binom{n}{2}+n-3$ and take $G$ as a complete graph $K_n$ .",
"Hence $tmc(G)=\\binom{n}{2}+n=k+3>k$ .",
"If $t\\ge 3$ , then take $G$ as the graph $G^{*}$ described in Lemma REF such that $m(G)=k-n+t+1$ .",
"It follows from Lemma REF that $tmc(G)=m(G)+n-t=k+1>k$ for $\\binom{n-t}{2}+t(n-t-1)+n\\le k\\le \\binom{n-t}{2}+t(n-t)+n-3$ , and $tmc(G)=m(G)+n-(t-1)=k+2>k$ for $k=\\binom{n-t}{2}+t(n-t)+n-2$ .",
"Thus $g(n,k)=k-n+t$ .",
"For $k=\\binom{n-t}{2}+t(n-t)+n-1$ where $2\\le t\\le n-1$ , if a connected graph $G$ on $n$ vertices satisfies $m(G)\\le k-n+t-1(=\\binom{n-t}{2}+t(n-t)+t-2)$ , then $tmc(G)\\le m(G)+n-t\\le k-1<k$ by Lemma REF .",
"Hence, $g(n,k)\\ge k-n+t-1$ .",
"To show $g(n,k)\\le k-n+t-1$ , it suffices to find a connected graph $G$ on $n$ vertices such that $m(G)=k-n+t$ and $tmc(G)>k$ .",
"If $t=2$ , take $G$ as the complete graph $K_n$ and then $tmc(G)=\\binom{n}{2}+n=k+2>k$ .",
"If $t\\ge 3$ , take $G$ as the graph $G^{*}$ described in Lemma REF such that $m(G)=k-n+t$ .",
"It follows from Lemma REF that $tmc(G)=m(G)+n-(t-1)=k+1>k$ .",
"Thus, $g(n,k)=k-n+t-1$ .",
"The proof is now complete.",
"$\\Box $"
]
] | 1612.05381 |
[
[
"Understanding magnetic focusing in graphene $p$-$n$ junctions through\n quantum modeling"
],
[
"Abstract We present a quantum model which provides enhanced understanding of recent transverse magnetic focusing experiments on graphene $p$-$n$ junctions.",
"Spatially resolved flow maps of local particle current density show quantum interference and $p$-$n$ junction filtering effects which are crucial to explaining the device operation.",
"The Landauer-B\\\"{u}ttiker formula is used alongside dephasing edge contacts to give exceptional agreement between simulated non-local resistance and the recent experiment by Chen $\\textit{et al}$ ($\\textit{Science}$, 2016).",
"The origin of positive and negative focusing resonances and off resonance characteristics are explained in terms of quantum transmission functions.",
"Our model also captures subtle features from experiment, such as the previously unexplained $p$-$p^-$ to $p$-$p^+$ transition and the second $p$-$n$ focusing resonance."
],
[
"Introduction",
"Traditionally, transverse magnetic focusing (TMF) experiments have been restricted to unipolar conduction, in mediums such as metals [1] and two-dimensional electron gasses (2DEG) [2].",
"The discovery of graphene [3], in which electrons behave as massless Dirac fermions [4], has provided an exciting new platform for studying TMF.",
"Graphene's gapless band structure, allowing ambipolar conduction, has enabled several recent TMF experiments.",
"TMF in graphene has been studied as a function of carrier density [5] and imaged with scanning gate microscopy [6].",
"In addition, a large number of TMF peaks have been observed in graphene/hexagonal boron nitride superlattices [7].",
"Recently, p-n junctions in graphene have been used in TMF experiments to steer the focused beam [8], opening the door to new electron optics.",
"The p-n junction is a fundamental device and has received a significant amount of attention from the graphene community.",
"Graphene p-n junctions have rich physical properties, exhibiting chiral tunneling [9], [10], angle dependent transmission [11], [12], [13], quantized conductance in high magnetic fields [14], [15], [16], and ballistic interference [17].",
"Figure: (a) Model of the graphene device depicting the first TMF resonance for p-p' and p-n junctions.",
"(b) Schematic of device with four terminal measurement configuration.",
"The device simulated has dimensions D C =W=200D_C=W=200 nm, D W =50D_W=50 nm, and L C =60L_C=60 nm.",
"The red rectangles indicate dephasing contacts used in the simulation.",
"(c) Real space energy band diagram of the device.Figure: Maps of local particle current density () for a p-p' (p-n) junction in (a) ((b)) the first TMF resonance and (c) ((d)) the second TMF resonance.",
"The p-p' junctions in (a) and (c) are configured as E 1 =-50E_1=-50 meV and E 2 =-75E_2= -75 meV.",
"For the first p-n TMF resonance in (b) the junction is configured as E 1 =-E 2 =-50E_1=-E_2=-50 meV.",
"The second p-n TMF resonance in (d) is configured as E 1 =-50E_1=-50 meV and E 2 =100E_2=100 meV.",
"The scale bars are all 60 nm.In this paper, we use quantum transport methods to model the graphene p-n junction TMF experiment of Chen et al [8].",
"Our calculations, implemented in the KWANT package [18], intrinsically capture quantum interference, tunneling, and angle dependent transmission [19], which enables us to explain the results of Chen et al [8] in a completely quantum mechanical framework, without any fitting parameters.",
"Previously, we have used the same basic model to understand quantum Hall measurements in graphene p-n junctions [20].",
"By including large dephasing edge contacts and performing multi-terminal Landauer-Büttiker analysis[21], we are able to capture both the in-resonance and off-resonance characteristics of the device.",
"We achieve exceptionally strong agreement between our simulation and experiment [8], as shown in Fig.",
"REF .",
"When a magnetic field is applied perpendicular to a graphene p-n junction, electrons transporting across the junction will form snake states, arcing between the p and n sides of the junction [22].",
"In graphene, the arcs are characterized by the cyclotron radius, given by $r_c=\\frac{\\hbar \\sqrt{\\pi n}}{e|B|}$ with $\\hbar $ the reduced Planck's constant, $n$ the carrier density, $e$ the electron charge, and $B$ the applied magnetic field.",
"Snake states have been observed along graphene p-n junctions in several experiments [23], [24], [25].",
"Additionally, transport of electrons in snake states has been modeled using quantum mechanical[24], [26], [27], [28] and semi-classical [29], [30], [31], [32], [33], [34] methods.",
"The TMF experiment performed on graphene p-n junctions by Chen et al [8] probes a special case of snake state transport, in a device similar to that depicted in Figures REF a and REF b.",
"The device studied by Chen et al [8] is special because the distance between contacts on each side of the junction, $D_C$ , is approximately equal to the width of the device, $W$ .",
"When $2r_c \\approx D_C$ , the applied magnetic field focuses electrons directly between the contacts.",
"In a unipolar system the carriers are directed back to the side from which they originate.",
"Conversely, in a $p-n$ junction, the carriers will be steered towards the opposite side of the device.",
"These two paths are depicted in Fig REF a."
],
[
"Transport Model",
"In this paper, we study a tight-binding Hamiltonian describing low energy electrons in graphene, given by $\\hat{H}= \\sum _{i}^N\\epsilon _i\\hat{c}_i^\\dag \\hat{c}_i+\\sum _{i,j}^Nt_{i,j}\\hat{c}_i^\\dag \\hat{c}_j,$ where the second summation only takes place for atoms which are first nearest-neighbors.",
"$\\hat{c}_i^\\dag / \\hat{c}_j$ are Fermionic creation/annihilation operators, $\\epsilon _i$ is the on-site energy at site i, and $t_{i,j}$ is the hopping energy between sites i and j.",
"The effect of an applied magnetic field is included using Peierl's substitution, $t_{i,j} = t_s\\exp \\left[i\\frac{e}{\\hbar }\\int ^{\\textbf {r}_j}_{\\textbf {r}_i} \\textbf {A}\\cdot \\textbf {dr}\\right]$ , where we adopt a circular gauge for the vector potential $\\textbf {A}$ [35].",
"We use a scaled tight-binding model[36] where $a=s_{f}a_{0} \\textrm { and } t_{s}=t_{0}/s_f$ .",
"The term $s_f=10$ scales the lattice constant, $a_0$ , and the atomistic hopping energy , $t_0 \\approx 2.7$ eV[37], to yield more efficient simulations.",
"We simulate a six terminal Hall bar, as depicted in Fig REF b, with four small contacts (labeled one, two, four, and five) and two large contacts (labeled zero and three).",
"The spacing between the inner edges of the small contacts is set equal to the width of the Hall bar, $D_C=W$ , which is the critical element of device design to observe the first p-n focusing peak.",
"To form p-n junctions, the on-site energy on each side of the device may be tuned independently to $E_1$ and $E_2$ .",
"We set the on-site energy to change linearly between $E_1$ and $E_2$ over a junction width, $D_W$ , as shown in the energy band diagram in Fig REF c. The two large contacts, zero and three, are included as dephasing contacts.",
"The voltages of these contacts are allowed to float in the simulation, accounting for any dephasing which occurs as the carrier wave skips along the left or right side of the device.",
"This type of virtual dephasing contact has been used in quantum transport calculations in the past [38][39] and is critical for tying our results to experiment.",
"Since most TMF measurements are performed at cryogenic temperatures under very small biases, we adopt a zero-bias, zero-temperature approximation.",
"In this regime, we utilize the Landauer-Büttiker equation[21] to express the current in each lead p [40], $I_p = \\frac{2e^2}{h}\\sum _q \\left[ T_{qp}V_p-T_{pq}V_q\\right],$ where the summation takes place over all leads in the system, including the dephasing contacts.",
"For our simulation, (REF ) generates a system of six linear equations with six unknowns.",
"The term $T_{qp}$ is the quantum mechanical transmission function from lead p to q, defined as $T_{qp}(E)=\\sum _{n\\in p, m\\in q} \\mid S_{nm}(E)\\mid ^2,$ where $S_{nm}$ is the scattering matrix element between the $n^{\\textrm {th}}$ and $m^{\\textrm {th}}$ mode in leads $p$ and $q$ , respectively.",
"The summation in (REF ) takes place over the available modes in each lead at energy $E$ .",
"To connect with the multi-terminal measurement of Chen et al [8], we simulate driving a current between contacts one and five and calculate the voltage acquired by contacts two and four.",
"Practically, this requires setting $I_1=-I_5$ , $I_0=I_2=I_3=I_4=0$ , and choosing a contact to be grounded, in this case $V_1=0$ .",
"The non-local resistance for this configuration is defined as $R_{15,24} = \\frac{V_2-V_4}{I_1-I_5}.$ The components of (REF ) are attained by solving the linear system, $\\mathbf {I}=\\frac{2e^2}{h}\\mathbf {T}\\mathbf {V}$ , defined by (REF ), where $\\mathbf {I}$ and $\\mathbf {V}$ are column vectors of lead currents and voltages, respectively, and $\\mathbf {T}$ is a matrix of transmission functions.",
"Making the substitutions above, (REF ) may be reduced to $R_{15,24}=\\frac{h}{2e^2}\\frac{1}{2}\\left(R_{45}-R_{25}\\right)$ .",
"$R_{45}$ and $R_{25}$ are elements of the $R-$ matrix, defined as $\\mathbf {R}=\\mathbf {T}^{-1}$ , and are entirely comprised of transmission functions between different leads, thus, the problem is reduced to calculating the permutations of (REF ).",
"To understand the terminal characteristics of our simulation, we generate spatially resolved particle current density maps using $J_{\\mathbf {r_i},\\mathbf {r_j}}(E) = -2 \\sum _{n\\in p} \\textrm {Im}\\left[\\psi _n(\\mathbf {r_i},E)^\\dagger \\hat{H}_{i,j} \\psi _n(\\mathbf {r_j},E) \\right]$ where $\\mathbf {r_{i}}$ is the position of the $i^{\\textrm {th}}$ lattice site, $\\psi _n(\\mathbf {r_i},E)$ is the wave function of the $n^{\\textrm {th}}$ conducting mode in lead p. The summation takes place over all conductive modes in lead p available at energy $E$ .",
"However, separately resolving each mode is informative.",
"Figure: Table of mode resolved particle current density for each panel of Fig.",
".",
"The modes of each column are summed to give the final result in Fig.",
".",
"By looking at each mode individually, the interplay between the semi-classical and quantum mechanical nature of the system is visible.Figure: Comparison of particle current density for the device configured as in Fig.",
"b both (a) with and (b) without dephasing edge contacts.",
"When the dephasing edge contacts are removed, in (b), carrier density which is not focused into contact four will skip around the edge of the device until it exits out one of the small contacts.",
"The carrier density, which is not dephased, will interfere with the incoming waves and destroy the resonance condition.",
"This results in the extremely chaotic pattern seen in (b), with no observable focusing resonances.Figure: Non-local resistance as a function of magnetic field in an (a) p-p' and (b) p-n junction.",
"The junctions are configured as in Fig .",
"To compare to the carrier densities shown in Fig , the left side of (a) is set to -0.18×10 12 cm -2 -0.18\\times 10^{12} \\textrm { cm}^{-2} and the right side is set to -0.41×10 12 cm -2 -0.41\\times 10^{12} \\textrm { cm}^{-2}.",
"The left side of (b) is configured the same as (a) but the right side is now set to +0.18×10 12 cm -2 +0.18\\times 10^{12} \\textrm { cm}^{-2}.",
"Important transmission () functions are plotted for each configuration, as explained in the text.Figure: Off-resonance condition particle current density for the symmetric p-n junction studied in Fig.",
"b.",
"The junction is configured as E 1 =-E 2 =-50E_1=-E_2=-50 meV.",
"In (a) B=0.11B=0.11 T and in (b) B=0.16B=0.16 T. When the magnetic field is not strong enough to focus the carriers into contact four, the transmitted wave collides with the top of the device, to the right of contact four, and skips into contact three.",
"When the magnetic field is too strong, as in (b), the transmitted wave is focused to the left of contact four and again skips into contact three."
],
[
"Results and discussion",
"In Fig.",
"REF we plot vector flow maps of the local particle current density (REF ) injected by contact one for p-p' and p-n junctions.",
"When current is focused into contact two/four we observe positive/negative peaks in the non-local resistance, respectively.",
"When the p-p' junction is in the first TMF resonance, in Fig.",
"REF a, carriers injected by contact one are focused directly into contact two.",
"The carriers are injected and take on a broad spread of angles in the channel, but are primarily focused into a bright caustic which enters contact two.",
"The junction redirects the carriers slightly, elongating the orbit.",
"Due to the small size of the contacts, not all carriers which are injected by contact one are collected at contact two.",
"Some hit the bottom edge of the device and skip into contact three, from which corresponding interference fringes may be seen, especially on the caustic.",
"For the first resonance of the p-n junction, in Fig.",
"REF b, current injected from contact one is focused directly into contact four.",
"The 50 nm junction width acts as a low pass filter, allowing only current flowing close to normal to the junction to transmit.",
"On the left side of the junction, wave interference patterns indicate the current density reflected off the junction, which then exits out the contacts on the left side of the device.",
"The transmitted current predominately focuses into a caustic which enters contact four.",
"At low magnetic fields, a significant portion of the current injected from contact one hits the top edge of the device before crossing the junction, as seen for p-n junction in Fig.",
"REF b.",
"This is a consequence of the device geometry studied by Chen et al [8] and increasing the device width to avoid hitting the top edge prohibits one from probing the first p-n TMF resonance.",
"Interestingly, a component of the current hitting the top edge is redirected and transmits across the junction.",
"This subtle detail, captured by our model, contributes to the device's terminal characteristics and is important in many of the different junction configurations.",
"When the magnetic field is increased to the second TMF resonance, in Fig.",
"REF c and d, the current density will skip along the edge of the junction (p-n case) or the edge of the device (p-p' case).",
"p-n junctions do not exhibit the second resonance until the n-doping is stronger than the p-doping, thus we configure the junction in Fig.",
"REF d as $E_1=-50$ meV and $E_2=100$ meV.",
"In the p-n configuration, on the p-side, the current forms a circular orbit which reflects near the bottom of the junction and again almost half way up.",
"At each of these points there is a significant portion of current which is incoming normal to the junction and transmits to the other side, focusing on contact 4.",
"Due to the filtering effect of a smooth p-n junction, the second TMF resonance is significantly weaker.",
"To further understand the local particle current density of the devices in Fig.",
"REF , in Fig.",
"REF we resolve the characteristic by each propagating mode.",
"By resolving each mode which contributes to the results in Fig.",
"REF , we observe a combination of features reminiscent of semi-classical skipping orbits and quantum mechanical interference patterns.",
"The lowest mode is injected straight into the device, perpendicular to the semi-infinite contact.",
"In the first resonance of the p-p' and p-n junction, shown in columns one and three of Fig.",
"REF , respectively, the lowest mode is bent so that the wave is propagating approximately normal to the junction when it crosses it.",
"Thus, the lowest mode is nearly perfectly transmitted, with very few reflections (indicated by interference fringes) visible.",
"Higher modes are injected into the device with non-zero angles and arrive at the junction traveling at oblique angles.",
"For the first resonance of the p-p', the beam is noticeably refracted as it crosses the junction.",
"In the p-n junction, the higher order modes have significant components which are reflected off the junction, due to the angle dependent transmission across the junction.",
"For the second resonance of the p-p' and p-n' configurations, the local particle current density patterns in Fig.",
"REF are more complex than the first resonance.",
"By resolving each mode, we are able to develop a better picture of the important transport mechanisms.",
"The higher order modes for the p-p' junction have a component which transports nearly parallel to the lower edge of the device.",
"This is particularly evident in the fourth and fifth modes.",
"Most of the carriers which transport in this manner will miss contact two and transmit out contact three, resulting in a weaker signal for the second focusing resonance.",
"The second focusing resonance of the p-n' displays the most complex characteristics of the device, with predominant quantum characteristics not present in the other configurations.",
"At the higher magnetic field, the first and second modes appear to begin to form Landau levels when they collide with the junction, similar to what we have studied in our previous work [20].",
"The higher order modes, however, instead show a more complex, swirling pattern.",
"The carriers transport in skipping orbits which partially reflect of the junction, interfering with themselves.",
"A portion of each orbit transmits across the junction, contributing to the second p-n' resonance.",
"As mentioned previously, the dephasing edge contacts (labeled contact zero and three) are critical to attaining the results presented in this paper.",
"To demonstrate this importance, in Fig.",
"REF we plot the local particle current density for the device configured as in Fig.",
"REF b both with and without the dephasing contacts.",
"When the dephasing contacts are removed, in Fig.",
"REF b, the portions of the wave which normally exit contacts zero and three, instead scatters around the edge of the device.",
"The wave will continue to scatter around the device, interfering with itself, until exiting out one of the small contacts.",
"This process occurs until the device reaches steady state, resulting in the extremely chaotic pattern shown and the destruction of any resonance characteristics.",
"Fig.",
"REF shows the non-local resistance (REF ) and selected transmission coefficients (REF ) as a function of applied magnetic field for an asymmetric p-p' junction and a symmetric p-n junction.",
"The two junction configurations are doped the same as in Fig.",
"REF a and b, respectively.",
"It is non-trivial to extract specific terms from (REF ), in terms of transmission functions, which result in the final form of the non-local resistance.",
"The final magnitude and shape of the curve consists of permutations of transmission functions between every contact combined together.",
"However, we are able to target specific transmission functions which are important in understanding the problem.",
"In the unipolar p-p' configuration, we observe three well defined TMF resonances.",
"We are able to match the first two TMF resonances to a peak in the transmission from contact one into two, $T_{21}$ .",
"When the junction is switched to the p-n configuration, when in resonance, carriers are now focused from contact one into contact four.",
"This results in a negative peak in resistance at $B=0.128$ T, shown in Fig.",
"REF b.",
"The important transmission function for understanding the resonance condition of the p-n junction is $T_{41}$ , which is peaked while the device is in resonance.",
"Each subsequent TMF resonance of the unipolar junction configuration decreases in magnitude.",
"For higher order TMF resonances, an increase in $T_{31}$ indicates that the focusing effect is diminished.",
"This is due to interference caused by the increased number of scattering events off the edges of the device.",
"When either configuration of junction is not in resonance, there is an increase in the transmission from contact one into contact three, $T_{31}$ .",
"In the off-resonance state of the p-p' junction, carriers which are not focused from contact one into two will hit the bottom edge of the device and skip into contact three.",
"To maintain current conservation, the carriers will be re-injected by the floating contact three and the magnetic field will direct the carriers towards contact four, resulting in the negative off-resonance resistance in Fig.",
"REF a. Conversely, in the p-n junction, carriers which miss contact four will skip along the top edge of the device.",
"Again, they will be dephased by contact three, except this time the re-injected carriers will be directed towards contact two, which results in the positive off-resonance resistance in Fig.",
"REF b.",
"In Fig.",
"REF we illustrate the off-resonance particle current density for a symmetric p-n junction.",
"Figure: (a) Non-local resistance map for a fixed p-type doping as a function of the carrier density of the right side of the junction and magnetic field.",
"Scatter points mark configurations where we have demonstrated spatially resolved particle current density in Fig.",
".",
"Our results show exceptional agreement with (b) experimental measurements of Chen et al , reproduced with copyright permission.The junction filtering effect, seen in Fig.",
"REF , results in significantly weaker and fewer TMF resonances when the device is in the p-n configuration.",
"For the symmetric p-n junction, in Fig.",
"REF , only a single well defined resonance is observed.",
"At higher magnetic fields the beam of carriers skips along the edge of the junction; each time the beam hits the junction only a very small amount will leak through.",
"Since, in our model, no dephasing happens along the junction, the reflected wave of carriers will interfere with itself, further disrupting any resonance from setting up.",
"In addition, the non-local resistance tends towards zero for each configuration at around $B=0.65$ T. This is due to the carriers being forced into edge states as the device enters the quantum Hall regime.",
"This effect is also the reason why we do not see a well defined peak in $T_{21}$ for the third TMF resonance of the p-p' junction configuration.",
"Capturing this feature highlights the power of quantum transport modeling, where our simulations smoothly transition between carriers occupying semi-classical skipping orbits and edge states.",
"Finally, in Fig.",
"REF , we compare our model with the recent experimental data of Chen et al [8], reproduced with copyright permission.",
"In Fig.",
"REF a, we fix the doping of the left side of the junction to $E_1=-50$ meV (p-type) and vary the doping of the right side of the junction and applied magnetic field simultaneously.",
"For each configuration we calculate the non-local resistance (REF ) as before.",
"We report the doping of the right side in terms of carrier density $n$ , which has a similar functional form to the gate voltage applied in experiment.",
"Our simulation results show a striking similarity to the experimental data, capturing all of the major features.",
"These include the four unipolar junction TMF resonances, the first ambipolar TMF resonance, and the negative/positive peaks in resistance when the unipolar/ambipolar configurations are not in resonance, respectively.",
"We also are able to explain a number of subtle features seen experimentally which are due to the transitions between different types of junctions.",
"The second negative peak in the unipolar junction configuration begins to disappear as the right side of the device is more strongly doped p-type.",
"This transition occurs when the doping of the right side of the junction exceeds the doping of the left side.",
"The second ambipolar junction TMF resonance in Fig.",
"REF is extremely weak until the n-type doping of the right side of the junction exceeds the fixed p-type doping.",
"This effect is enabled by the increased number of modes available to conduct on the right side of the junction as the doping is increased.",
"The filtering effect due to the large junction width present in our model and in experiment [8] prohibits the traditional picture of the carrier density snaking across the junction several times in the second TMF resonance.",
"Instead, the resonance has the characteristic of the flow map shown in Fig.",
"REF d. In our simulation a larger magnetic field must be used, since our simulated Hall bar is about a factor of ten smaller than the experimental device.",
"The difference in device size and contact dimensions also accounts for the difference in magnitude of our simulated resistance.",
"Using larger contacts will result in smaller values of resistance.",
"However, the concepts we have discussed may still be applied to understand the experimental measurements of Chen et al [8]."
],
[
"Conclusion",
"In conclusion, we demonstrate a quantum transport model for a TMF experiment on graphene p-n by Chen et al [8].",
"Spatially resolved particle current density flow maps reveal the behavior of carriers in the first and second resonances of p-p' and p-n junctions.",
"Our results demonstrate the importance of wave interference and junction filtering effect for understanding TMF experiments.",
"A combination of dephasing edge contacts and use of the Landauer-Büttiker formula supplementing the standard tight-binding model yield extremely close agreement with experiment.",
"Our non-local resistance simulations show well defined positive and negative peaks, which are due to enhanced transmission into contacts three or four, respectively.",
"Many of the features seen by Chen et al [8] have been explained, including the transition into the quantum Hall regime for high magnetic fields and the transitions between different p-p' and p-n doping regimes.",
"The authors acknolwedge financial support provided by the U.S.",
"Naval Research Laboratory (Grant Number: N00173-14-1-G017)."
]
] | 1612.05657 |
[
[
"Optimizing Stochastic Scheduling in Fork-Join Queueing Models: Bounds\n and Applications"
],
[
"Abstract Fork-Join (FJ) queueing models capture the dynamics of system parallelization under synchronization constraints, for example, for applications such as MapReduce, multipath transmission and RAID systems.",
"Arriving jobs are first split into tasks and mapped to servers for execution, such that a job can only leave the system when all of its tasks are executed.",
"In this paper, we provide computable stochastic bounds for the waiting and response time distributions for heterogeneous FJ systems under general parallelization benefit.",
"Our main contribution is a generalized mathematical framework for probabilistic server scheduling strategies that are essentially characterized by a probability distribution over the number of utilized servers, and the optimization thereof.",
"We highlight the trade-off between the scaling benefit due to parallelization and the FJ inherent synchronization penalty.",
"Further, we provide optimal scheduling strategies for arbitrary scaling regimes that map to different levels of parallelization benefit.",
"One notable insight obtained from our results is that different applications with varying parallelization benefits result in different optimal strategies.",
"Finally, we complement our analytical results by applying them to various applications showing the optimality of the proposed scheduling strategies."
],
[
"=1"
]
] | 1612.05486 |
[
[
"The Pan-STARRS1 Surveys"
],
[
"Abstract Pan-STARRS1 has carried out a set of distinct synoptic imaging sky surveys including the $3\\pi$ Steradian Survey and the Medium Deep Survey in 5 bands ($grizy_{P1}$).",
"The mean 5$\\sigma$ point source limiting sensitivities in the stacked 3$\\pi$ Steradian Survey in $grizy_{P1}$ are (23.3, 23.2, 23.1, 22.3, 21.4) respectively.",
"The upper bound on the systematic uncertainty in the photometric calibration across the sky is 7-12 millimag depending on the bandpass.",
"The systematic uncertainty of the astrometric calibration using the Gaia frame comes from a comparison of the results with Gaia: the standard deviation of the mean and median residuals ($ \\Delta ra, \\Delta dec $) are (2.3, 1.7) milliarcsec, and (3.1, 4.8) milliarcsec respectively.",
"The Pan-STARRS system and the design of the PS1 surveys are described and an overview of the resulting image and catalog data products and their basic characteristics are described together with a summary of important results.",
"The images, reduced data products, and derived data products from the Pan-STARRS1 surveys are available to the community from the Mikulski Archive for Space Telescopes (MAST) at STScI."
],
[
"Introduction",
"The Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) is an innovative wide-field astronomical imaging and data processing facility developed at the University of Hawaii's Institute for Astronomy [58], [59].",
"The Pan-STARRS1 Science Consortium (PS1SC) was formed to use and extend the Pan-STARRS System for a series of surveys to address a set of science goals and in the process the PS1SC continued the development of the Pan-STARRS System.",
"An original goal the PS1SC set for itself was to insure the data would eventually become public.",
"This is the first in a series of seven papers that describe the Pan-STARRS1 Surveys, the data reduction techniques, the photometric and astrometric calibration of the data set, and the resulting data products.",
"These papers are concurrent with and are intended to support the public release of the Pan-STARRS1 data products http://panstarrs.stsci.edu/ from the Barbara A. Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute.",
"There are two Data Releases funded: Data Release 1, (DR1) containing the stacked images and the supporting database of the $3\\pi $ Steradian Survey, and Data Release 2 (DR2) containing all of the individual epoch data of the $3\\pi $ Survey.",
"Further Data Releases will depend on the availability of resources to support them.",
"This Paper (Paper I) provides an overview of the fully implemented Pan-STARRS System, the design and execution of the Pan-STARRS1 Surveys, the image and catalog data products, a discussion of the overall data quality and basic characteristics, and a summary of scientific results from the Surveys.",
"[82] describes how the various data processing stages are organised and implemented in the Imaging Processing Pipeline (IPP), including details of the the processing database which is a critical element in the IPP infrastructure .",
"[129] describes the details of the pixel processing algorithms, including detrending, warping, and adding (to create stacked images) and subtracting (to create difference images) and resulting image products and their properties.",
"[81] describes the details of the source detection and photometry, including point-spread-function and extended source fitting models, and the techniques for “forced\" photometry measurements.",
"[83] describes the final calibration process, and the resulting photometric and astrometric quality.",
"[33] describes the details of the resulting catalog data and its organization in the Pan-STARRS database.",
"Huber et al.",
"2017 (in preparation - Paper VII) describes the Medium Deep Survey in detail, including the unique issues and data products specific to that survey.",
"The Medium Deep Survey is not part of DR1.",
"Note: These papers are being placed on arXiv.org to provide crucial support information at the time of the public release of Data Release 1 (DR1).",
"We expect the arXiv versions to be updated prior to submission to the Astrophysical Journal in January 2017.",
"Feedback and suggestions for additional information from early users of the data products are welcome during the submission and refereeing process.",
"The paper is laid out as follows.",
"In Section of this paper we begin with an overview of the completed Pan-STARRS1 System, and a brief description of its associated subsystems: the Pan-STARRS Telescope #1, (PS1), the Gigapixel Camera #1 (GPC1), the Image Processing Pipeline (IPP), hierarchical database or Pan-STARRS Products System (PSPS), and the Science Servers: the Moving Object Pipeline (MOPS), Transient Science Server (TSS), Photo-Classification Server (PCS).",
"Section describes the various Pan-STARRS1 Surveys and their characteristics; the details of the observing strategy and the resulting impact on the time sampling and survey depth as a function of position on the sky.",
"Section provides a summary of the Pan-STARRS1 data products.",
"Section summarizes the overall astrometric and photometric calibration of the surveys.",
"Section provides an overview of the features and characteristics of the 3$\\pi $ Survey.",
"Finally, a summary of the legacy science of the PS1 Science Consortium and a brief discussion of the future of Pan-STARRS is provided in Section .",
"The Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) is an innovative wide-field astronomical imaging and data processing facility developed at the University of Hawaii's Institute for Astronomy [58], [59].",
"Approximately 80 percent of the construction and development funds came from the US Air Force Research Labs (AFRL) in response to a Broad Agency Announcement “to develop the technology to survey the sky\".",
"The remainder of the development funds came from NASA, the PS1 Science Consortium (PS1SC), the State of Hawaii, and some private funds.",
"The project's goal was originally to construct 4 separate 1.8-meter telescope units each equipped with a 1.4 gigapixel camera, and operate them in union.",
"The ambitious nature and full scale cost of the project led to a decision to build a prototype system of a single 1.8-meter telescope unit.",
"This provided an opportunity not only to test the hardware, software and design but also to carry out a unique science mission.",
"This system, located on the island of Maui, was named Pan-STARRS1 (PS1)."
],
[
" The PS1 Science Consortium",
"In order to execute and deliver a competitive and scientifically interesting set of sky surveys, the Institute for Astronomy (IfA) of the University of Hawaii (UH) assembled the PS1 Science Consortium (PS1SC).",
"This group of interested academic institutions established a set of science goals [15], and a Mission Concept Statement [14] and funded the operations of PS1 for the purpose of executing the PS1 Science Mission [13], [16].",
"The founding institutions of the PS1SC defined 12 Key Projects to ensure that the definition of the surveys and their implementation were shaped by science drivers covering a range of topics from solar system objects to the highest redshift QSOs.",
"The Memorandum of Agreement of the PS1SC established that the funding for operations was provided in return for the proprietary use of the Pan-STARRS1 data for scientific purposes.",
"As the PS1 Mission went on, additional members were added to bring in additional resources.",
"The member institutions of the PS1 Science Consortium are provided in Table REF Table: PS1 Science Consortium"
],
[
" The PS1 Science Mission ",
"The PS1 Telescope began formal operations on 2010 May 13, with the start of the PS1 Science Mission, funded by the PS1SC and with K. Chambers as PI and Director of PS1.",
"At the beginning of the PS1 Mission, the Image Processing Pipeline (IPP) - the software and hardware for managing and processing the data - was not at an advanced stage of development, nor were the characteristics of the unusual OTA devices well understood.",
"Furthermore, because of the AFRL funding, the imaging data was initially required to be censored.",
"The AFRL “Magic\" software was devised so that the pixels surrounding any feature in an individual image that could be interpreted as a potential satellite streak were masked.",
"This meant removal of pixels in a broad streak, or elongated box, which was large enough to prevent the determination of any orbital element of the artificial satellite before the images left the IfA servers to the consortium scientists.",
"This requirement hindered analysis of the very features that were triggering the censor, nearly all of which were not satellite streaks, but were inherent detector characteristics.",
"This effectively delayed the full and rapid analysis of the pixel data by consortium scientists until the ARFL finally dropped the requirement on 2011 Dec 12.",
"From that date, all Pan-STARRS1 images, including prior data taken during commissioning and from the start of survey operations, were no longer subject to any such masking software.",
"Earlier data was re-processed from the untouched original raw data without the streak removal.",
"There is no real time nor archival censorship of any Pan-STARRS data.",
"None of the data now being released in DR1 and DR2 suffer from any application of the “Magic\" streak removal software either in the individual or in the stack images.",
"At the start of the PS1 Mission the development of the IPP (software and hardware), and eventually the development of the PSPS, shifted from the Pan-STARRS Project Office (2003-2014) to the PS1 Science Consortium funded PS1 Operations team.",
"The Project Office went on to develop the second Pan-STARRS facility, Pan-STARRS2.",
"In August of 2014 the Pan-STARRS Office closed, and the Operations team also took over responsibility for the completion and commissioning of PS2 with the support of the NASA NEO Program, the State of Hawaii, and private funding.",
"No further involvement with the AFRL is expected."
],
[
" The STScI Mast Archive and Data Releases",
"To fully exploit the scientific potential of the PS1 survey data the PS1SC committed to make all PS1SC data public and accessible as soon as possible, but not before one year after the end of PS1SC survey operations.",
"The science consortium made this commitment in principle in order that the data reach as wide a usage as possible, however the original founding members of the PS1SC did not contain the resources or expertise to provide a public interface server.",
"To enable such a public release, the PS1SC joined forces with the Space Telescope Science Institute (STScI) and the Barbara A. Mikulski Archive for Space Telescopes (MAST).",
"The STScI joined the PS1 Science Consortium through a Memorandum of Agreement to contributing resources to create an archive of the PS1 Data Products that will serve the entire astronomical community.",
"DR1 contains the static sky and mean data (See Section ).",
"The individual detections and forced detections (See Section ) will come in a second release (DR2) in spring of 2017.",
"Our intention is to support multiple releases as resources allow."
],
[
"Flow of information in the Pan-STARRS System",
"An overview of the flow of information through the Pan-STARRS System is shown in Figure REF .",
"In brief: photons from astronomical objects are brought to a focus by the Telescope onto the focal plane of the Gigapixel Camera #1 (GPC1).",
"As discussed below, a feedback signal is generated from selected areas of GPC1 and fed back to the telescope through the Observatory, Telescope, and Instrument Software system or OTIS, see Section REF .",
"During the night, as new images are downloaded, they are processed by the IPP, see Section REF .",
"The results are passed to the Moving Object System (Section REF ) and the Transient Science Server (Section REF ) Near Earth Object (NEOs) candidates from MOPS are sent to the Minor Planet Center, and stationary transient objects are now posted on the IAU Transient Name Serverhttps://wis-tns.weizmann.ac.il/ for use by the community.",
"Offline from nightly processing, the IPP uses a variety of tools for calibration (Section REF ).",
"The catalog data products produced by IPP are passed on the PSPS database (Section REF ).",
"Both the PSPS database and all the image products from the IPP are then available to the community from the Barbara Mikulski Archive for Space Telescopes (MAST) at STScI.",
"Figure: Flow of information through the Pan-STARRS System.",
"The various subsystems are discussed in Section ."
],
[
"Site",
"The Pan-STARRS telescopes (both PS1 and PS2) are located at Haleakala Observatories (HO) on the island of Maui on the site of the Lunar Ranging Experiment (LURE) [12].",
"Measurements by the HO Differential Image Motion Monitor (DIMM) show the site has a median image quality of 0.83 arc-seconds (the mode is 0.66 arc-seconds).",
"On average 35% of the nights on Haleakala are photometric, with an additional 30% usable with very low extinction or more than 60% of the sky clear of clouds.",
"The wind pattern is predominately trade winds from the east-northeast, with occasional “Kona\" winds from west-southwest.",
"PS2 is due north of PS1, the center of the two telescope piers is separated by 20.05 meters.",
"The domes are situated in the wake of the flow from trade winds into the crater wall.",
"Detailed metrics of the site characteristics will be published elsewhere (Chambers, 2017 in prep).",
"More recently the Daniel K. Inouye Solar Telescope (DKIST)http://dkist.nso.edu/ has been erected to the south-south west of the Pan-STARRS facility.",
"The ultimate impact of DKIST operations on the Pan-STARRS environment is not yet fully known, their operational plan is to manufacture ice at night for use in the daytime cooling of DKIST, and subsequently dissipation of heat into the atmosphere at the summit.",
"The International Astronomical Union has determined that the acceptable level of Radio Frequency Interference outside an observatory doing optical and infrared observations should be less than $2\\mu W/m^2$ integrated over the radio spectrum.",
"This is exceeded at Haleakala and at the start of the PS1 Mission, radio frequency interference from various Federal and commercial transmission sites near the summit was an issue.",
"However with the relocation of TV broadcasters to the Ulukalapua site, this problem has been mitigated and we see no evidence of RFI in GPC1.",
"However cellphone transmission, wifi transmission, and microwave ovens have a noticeable effect and are not allowed at the Observatory."
],
[
"Telescope, optics, and control system ",
"The Pan-STARRS1 Telescope (PS1) is an alt-az telescope with an instrument rotator built by Electro Optic Systems Technologies Inc., Tucson, (EOST) with an enclosure by Electro Optic Systems Ltd. (EOS), Australia.",
"The PS1 Dome motion closely follows the telescope through a featherweight direct coupling.",
"The dome has four independently controllable vents for air flow through the dome.",
"The dome slit is covered by two independently controllable shutters that can be deployed over the top on to the back side of the dome.",
"When the moon is up the dome slit shutters are used to mitigate scattered light from the moon.",
"The Observatory, Instrument, Telescope, Software (OTIS) system controls all these aspects of the Observatory and collects and stores a wide variety of auxiliary and metadata on the conditions and all the functions of the Observatory.",
"The Pan-STARRS1 optical design [47], [48], [88] has a wide field Richey-Chretien configuration with a 1.8 meter diameter $f$ /4.44 primary mirror, and 0.9 m secondary.",
"The resulting converging beam then passes through two refractive correctors, one of six possible interference filters with a clear aperture diameter of 496 mm, and a final refractive corrector that is the cryostat window.",
"Note that the Pan-STARRS1 as-built optics are described by the Zemax model NOADC-3.0.",
"See Figure REF .",
"Table REF has summary of the Pan-STARRS1 telescope characteristics.",
"Figure: Pan-STARRS optical design .",
"The as-built design version was NOADC-M-3.0 shown here.",
"In the figure rays enter from left at the top of telescope where the spider legs that support the secondary introduce diffraction spikes in the images.",
"There are three baffles, one supported by the secondary support structure, a middle cone baffle that is suspended by cables aligned with the secondary support spiders, and a baffle supported from within the central hole of the primary.",
"The corrector lenses are labeled in order of passage as L1, L2, and L3 which is also the cryostat window.",
"Between L2 and L3 are the filter mechanism and the shutter.",
"The filter mechanism has 3 layers which can store 6 filters.The optical design has 4 aspheric surfaces; one each on the primary and secondary mirrors, one on the first corrector lens, L1, and a final aspheric on L3, the last corrector lens in the optical path and which also serves as the cryostat window.",
"The secondary mirror has a conic constant of $-20.43$ and a 6th order aspheric term of $4.5 \\times 10^{-19}$ , which made it a challenge to fabricate [88].",
"The Secondary Mirror is mounted on a hexapod and can be moved in five axes: x,y,z,tip, tilt.",
"The Primary Mirror is on a pneumatic support system and can be commanded in y,z, tip, and tilt.",
"The Primary Mirror can be moved in the x direction as well, but this is not on a powered actuator and must be done manually.",
"Furthermore the Primary Mirror has a 12 point astigmatic correction system.",
"Thus there are 22 independent mirror actuators that can be used to bring the optics into proper collimation and alignment with the optical axis as defined by the axis of the instrument rotator.",
"These actuators allow for modest amounts of Primary mirror deformation to remove trefoil, coma, and astigmatism.",
"The procedure for establishing the proper collimation and alignment is described in [88].",
"Given the system matrix, only minor adjustments are required to maintain collimation and alignment.",
"PS1 does have significant flexure, so empirical models have been determined to correct for that.",
"In practice the largest corrections are in the M2 tip and y (tangent to altitude) de-center.",
"The M1 figure correction also has an altitude dependent term.",
"The OTIS software applies these corrections for the destination of any commanded slew, corrections are disabled during exposures and the system tracks quiescently during the short exposures - generally not more than 2 minutes.",
"A focus offset is determined from each exposure based on the measured astigmatism, and this offset is applied to the empirically derived focus model.",
"The offset is calculated from an analysis of the ellipticity of the PSF across the focal plane calculated by the GPC1 software.",
"The calculation of the correction takes approximately one minute, and then can not be applied until the next pause between exposures while the camera is reading out.",
"Thus the telescope focus is maintained by the local focus model with an observationally based offset determined within a few minutes of a new exposure.",
"After large slews or starting a new chunk, a short exposure (10 seconds) is made to obtain a current focus correction.",
"This system maintains the correct M2 focus position to within $\\pm 5$ microns of true focus.",
"The collimation and alignment do drift occasionally, especially if there is maintenance performed on the telescope.",
"These drifts are corrected by a procedure of using above and below focus images of stars (donuts) [88] to make a correction.",
"The system to maintain the image quality is imperfect, and the results can be seen in some images.",
"Typically the impact is some combination of higher order aberrations that result in a asymmetric PSF.",
"The IPP fits only an elliptical PSF, so there is no systematic measure of this asymmetry or its effect on photometry, albeit it must be small.",
"The telescope illuminates a diameter of 3.3 degrees, with low distortion, and mild vignetting at the edge of this illuminated region.",
"The field of view is approximately 7 square degrees.",
"The 8 meter focal length at $f/4.4$ gives an approximate 10 micron pixel scale of 0.258 arcsec/pixel.",
"Table: Summary of PS1 Telescope Characteristics"
],
[
"GPC1 - the Gigapixel Camera #1 ",
"The Gigapixel Camera #1 (GPC1) uses Orthogonal Transfer Arrays devices, a concept developed by [120] and their development was key to the Pan-STARRS concept [57].",
"The detectors in GPC1 are CCID58 back-illuminated Orthogonal Transfer Arrays (OTAs), manufactured by Lincoln Laboratory [122], [123].",
"They have a novel pixel structure with 4 parallel phases per pixel [123] and required the development of a new type of controller ([99]).",
"GPC1 is actually populated with two different kinds of CCID58s, the CCID58a with a three phase serial register, and the CCID58b which has a two phase serial register [98].",
"Table REF has summary of GPC1 characteristics.",
"The intent of the OTA design was to allow charge to be moved in orthogonal directions providing an on-CCD tip-tilt image correction given a guide signal from a nearby cell being read at video rates, and Tonry's OPTIC camera did this successfully, [113].",
"However, with GPC1 when the Orthogonal Transfer mode of the detectors was turned on ,it produced an unacceptable level of non-uniform background noise [98].",
"The Pan-STARRS1 Surveys did not use the detectors in Orthogonal Transfer mode.",
"All Pan-STARRS1 Survey data was taken with the GPC1 devices operating as \"normal\" CCDs.",
"The detectors are read out using a StarGrasp CCD controller, with a total overhead of 10.3 seconds for a full unbinned image, see Table REF for a breakdown of the overhead.",
"Other performance characteristics of GPC1 are presented in [123], [98].",
"The focal plane of Pan-STARRS1 comprises a total of 60 CCID58 OTA devices [123].",
"Each of these devices consists of an $8\\times 8$ array of individual addressable CCDs called “cells”.",
"The overall format of a single OTA is a $4846\\times 4868$ pixel array with a pixel size of 10 $\\mu $ m which subtends 0.258 arcsec.",
"Each OTA device is made up of 64 cells where each cell is $590 \\times 598 $ pixels.",
"The cells are separated by a gap between columns, that is 18 “inactive\" pixels in size, and a gap between rows that is 12 inactive pixels in size.",
"Thus a single OTA device contains a single piece of silicon with 64 cells in an $8\\times 8$ array separated by a grid of $7\\times 7$ internal streets.",
"We will often refer to the OTA devices as \"chips\" in the data processing discussions.",
"Further more, there is a physical gap between the devices as mounted in GPC1.",
"The placement of the devices in the focalplane is shown in Figure REF .",
"The relative positions of each device, including rotation, were determined from a vast number of astrometric measurements on sky.",
"The separation between the OTA devices is 1400 microns (approximately 36 arcsec) in the $x$ direction and 2800 microns ( approximately 70 arcsec) in the $y$ direction.",
"In practice the devices are not perfectly spaced and can have some small rotation with respect to one another.",
"The astrometric solution for each device is solved independently without reference to one another, the only place where the determined relative position is used is telescope pointing and guiding.",
"Note there is a slight optical pin-cushion distortion of the sky on the focal plane, all of this is removed in the process of the astrometric registration (warping) by the IPP [81].",
"Table: Summary of GPC1 CharacteristicsFigure: Gigapixel Camera 1 focal plane layout and mask.",
"The non-functioning cells are blanked out in white.The telescope, detector devices, and control electronics each contribute a variety of artifacts to the GPC1 images.",
"Where possible these artifacts are identified and the pixels are masked or modified during processing and flags are set in the database.",
"These include optical ghosts from reflections in the optics, glints from scattered moon light, glints from structure in the camera, regions of poor charge transfer in the devices, persistence or “sticky charge\" from saturation leaving “burn-trails\" that persist for all successive images for tens of minutes, electronic ghosts from cross-talk in the electronics, and correlated read noise from the fiberflex that transmit the signal through the cryostat wall.",
"These are identified and masked where possible in the detrending procedure as part of the chip processing stage in the IPP.",
"There is a detailed discussion of the defects and how they are masked in [129].",
"These defects are visible on the focal plane in the single exposure frames, but the stacked images made from the multiple images taken over the survey duration are composed of dithered frames.",
"If we take the sky area covered by the GPC1 footprint to be the area of the inner blue circle in Figure REF (7 sq degrees) then the dead cells, pixel gaps and masking of defective pixels account for an overall loss of 20% of the focal plane in any one exposure.",
"There is an additional dynamic masking of around 2-3% per exposure, which mostly covers the “burn-trails\".",
"Therefore the overall fill factor of the camera is $76\\pm 1$ % per exposure and this is mitigated by the dither and stack techniques that were employed in the $3\\pi $ and Medium Deep Surveys.",
"The first data release (DR1) from the STScI MAST archive is the stack images only and hence the images will mostly look continuous, although there are areas where a combination of poor devices and fewer than 12 exposures mean masked regions creep through to the final product.",
"A subset of bright stars (mag$<12$ ) which fall on the focal plane are selected to be used as guide stars (suitably located across the camera), and a $100 \\times 100$ pixel box is defined, centered on the position where these stars are predicted to land based on the commanded telescope position.",
"This set of sub-arrays on different devices are read at video rates.",
"The centroid from these video frames are used to send a guide signal to the telescope control system.",
"Typically there are 4 to 10 stars chosen, which means these cells are then masked in the science exposures.",
"These additional masked cells are included in the “dynamic\" mask developed for each exposure that includes the masking due to the artifacts of that particular exposure and is added to the \"static\" mask as seen in Figure 3.",
"The shutter, built by the team at Bonn University, is a dual blade design.",
"The shutter aperture is approximately 40 cm across, and in closed position one blade covers the aperture and one is stored to the side.",
"When the shutter is opened, one side of the focal plane is exposed first.",
"At the conclusion of the exposure, the second blade traverses the aperture in the same direction, hence the total exposure time seen by each pixel is the same to the precision of the movement, or 10 milliseconds.",
"For the subsequent exposure the motion is in the opposite direction.",
"Short exposures are possible, where the blades follow each other trailing closely.",
"This does mean that the center time of the exposure is different by up to 0.5 seconds depending on placement in the focal plane.",
"The metadata exists to give every object a time of exposure depending on its position in the focal plane.",
"In principle a correction to the exact time of the center of the exposure can be calculated for every detection, such a correction is not currently made, and thus the UT of the exposure can be in error for any object by up to 0.5 seconds.",
"Even for moving asteroids this is not a serious limitation.",
"In practice the total overhead time between adjacent exposures, including the 1 second shutter movement time, is 10 seconds, see Table REF for details."
],
[
"Filter bandpasses and PS1 sensitivity",
"The Pan-STARRS1 observations are obtained through a set of five broadband filters, designated as $g_{\\rm P1}$ , $r_{\\rm P1}$ , $i_{\\rm P1}$ , $z_{\\rm P1}$ , and $y_{\\rm P1}$ .",
"Under certain circumstances Pan-STARRS1 observations are obtained with a sixth, “wide” filter designated as $w_{\\rm P1}$ that essentially spans $g_{\\rm P1}$ , $r_{\\rm P1}$ , and $i_{\\rm P1}$ .",
"There is full sky $3\\pi $ sky coverage in each of $grizy_{\\rm P1}$ but not in $w_{\\rm P1}$ , which was mostly used for near-earth object surveys.",
"Although the filter system for Pan-STARRS1 has much in common with that used in previous surveys, such as the Sloan Digital Sky Survey (SDSS, [132]), there are important differences, which is why the filters are labelled specifically with the P1 subscript.",
"The $g_{\\rm P1}$ filter extends 20 nm redward of $g_{SDSS}$ with the intention of providing greater sensitivity and lower systematics for photometric redshifts.",
"The strong [O I] 5577Å sky emission is on the filter edge but only at 1% transmission.",
"The $z_{\\rm P1}$ filter has a sharply defined cut-off at 922 nm, which is contrast to the SDSS $z-$ band which has no red cut off and the response is defined by the detector response.",
"The $r_{\\rm P1}$ and $i_{\\rm P1}$ filters are very similar to SDSS and colour differences between the two magnitude systems are small.",
"SDSS has no corresponding $y_{\\rm P1}$ filter.",
"The transmission of the Pan-STARRS1 filters, optics and total throughout were precisely measured with a calibrated photodiode and a tuneable laser, without use of celestial standards by [115] and this procedure was repeated in November 2016 (Stubbs et al.",
"in prep).",
"The definition of the photometric system has already been discussed in detail and published in [125].",
"Tabular data of the overall throughput of the PS1 system is available in the online data of [125] and the individual filter throughputs are in [115].",
"The PS1 total filter throughputs from [125] are reproduced here in Figure REF .",
"Figure: This figure is reproduced from for ease of reference.",
"The PS1 capture cross-section inm 2 ^{2}e -1 ^{-1}photon -1 ^{-1} to produce a detected e -1 ^{-1} for an incident photon for the six Pan-STARRS1 bandpasses,grizy P1 grizy_{\\rm P1} and w P1 w_{\\rm P1} for a standard airmass of 1.2.Photometry is in the “natural” Pan-STARRS1 system in “monochromatic AB magnitudes”[96] as described in [125] $m_{AB}(\\nu ) &= -2.5\\log (f_\\nu /3631~\\hbox{Jy})\\\\&= -48.600 - 2.5\\log (f_\\nu [\\hbox{erg/sec/cm$^2$/Hz])}$ Pan-STARRS1 magnitudes are interpreted as being at the top of the atmosphere, with 1.2 airmasses of atmospheric attenuation being included in the system response function.",
"No correction for Galactic extinction is applied to the Pan-STARRS1 magnitudes.",
"We stress that, like SDSS, Pan-STARRS1 uses the AB photometric system and there is no arbitrariness in the definition.",
"Flux representations are limited only by how accurately we know the system response function vs. wavelength.",
"The DR1 data has been calibrated with the updated values from [110], for details see [82]."
],
[
"IPP - Image Processing Pipeline",
"All images obtained by the Pan-STARRS1 system are processed through the Image Processing Pipeline (IPP) on a computer cluster at the Maui High Performance Computer Center.",
"The pipeline runs the images through a succession of stages, including de-trending or removing the instrumental signature, a flux-conserving warping to a sky-based image plane, masking and artifact removal, and object detection and photometry.",
"The IPP also performs image subtraction to allow for the prompt detection of moving objects, variables and transient phenomena.",
"Mask and variance arrays are carried forward at each stage of the IPP processing.",
"Photometric and astrometric measurements performed by the IPP system are published in a mysql relational database.",
"Below we give a brief summary of the Pan-STARRS image processing, full details are provided in the companion papers of [81], [83], [82], [33], [129].",
"Figure REF gives a simplified schematic of the processing stages.",
"Figure: Schematic of the image and analysis processing stages of the IPP (Magnier et al 2017b).",
"The images are read from the GPC1 to buffer storage at the summit.",
"The IPP polls this buffer and retrieves new images whenever they are created.",
"During the night the raw images are retrieved, and are processed individually through the single image analysis.",
"The nightly difference images for moving object or transient detection are created from a warp-stack image combination if the stack exists, or a warp-warp difference from a pair if there is no stack image.The post-processing stages work on the stacked images for the Static Sky Analysis and on the individual warp images for the Forced photometry."
],
[
"Chip Stage",
"In the “Chip Stage\" raw exposures are detrended [129] and sources in the images are detected and basic instrumental characterization is performed.",
"A PSF model is generated and all sources fitted with that model.",
"For sources above a minimal signal-to-noise limit (nominally 20), a simple galaxy model is fitted if the source appears to be extended.",
"The best model (PSF or galaxy) is subtracted and an additional source detection pass is made (down to S/N = 5).",
"This provides for some de-blending.",
"Reported values include instrumental positions, fluxes (PSF, seeing-matched aperture, Kron aperture), moments, and various quality flags are recorded for each source in the image.",
"The output from this stage consists of fits tables of detections and their properties called CMF files and detrended images and their associated variance, and mask pixel images."
],
[
"Camera Stage",
"In the “Camera Stage\" the instrumental measurements from all the chips in one exposure are gathered together for astrometric and photometric calibration by comparison with a reference catalog.",
"Initially a synthetic reference catalog was created based on 2MASS, USNO-B, and Tycho.",
"This was used for a photometric calibration as the survey proceeded.",
"In the re-processing and re-calibration that produced the data in DR1, the reference catalog uses Pan-STARRS itself, to create a precise and consistent internal calibration [83], [82] based on the “ubercal\" methods described in [105] and [32].",
"The primary data product from the Camera Stage is the collection of calibrated detection tables."
],
[
"Warp Stage",
"In the “Warp Stage\" the detrended pixel images generated by the chip stage are geometrically transformed to a predefined set of images which tessellate the relevant portion of the sky.",
"Specific examples are discussed in Section below.",
"A set of virtual rectilinear images with square pixels of 0.25 arcseconds size, on a local tangential projection center no bigger than about 4 degrees across are defined.",
"These virtual images are called “projection cells\" and one or more projection centers can be defined for specific areas of interest or arranged in some defined tessellation of the entire celestical sphere.",
"The total output from the warp stage is the collection of images that describe the signal, variance, and masking for each skycell."
],
[
"Stack Stage",
"Individual epoch skycell images (from the Warp Stage) are combined together to form deeper stack images of the sky, the details of the algorithms are in [129].",
"In the IPP analysis, stacks of different depths/quality may be made depending on the individual survey goals.",
"This is of particular application to the “Medium Deep\" survey.",
"The output from the stack stage consists of the signal, variance, and mask stack images."
],
[
"Difference Image Stage",
"The primary means for detecting a transient, moving, or variable object is through the process of subtracting a template image of a source from a single image to create a “difference\" or “diff\" image.",
"The IPP generates Alard-Lupton convolved difference images for skycells in various combinations depending on the survey goals.",
"The output from the diff stage is a collection of detections from the difference images, including both positive and negative difference detections."
],
[
"Static Sky Stage",
"The “Static Sky\" refers to a final stacked image.",
"The stack images from all filters are processed in a single analysis step to perform the deep source detection and characterization of objects detectable in the stacks.",
"This analysis step is similar to the source detection and characterization performed at the chip stage, with some important additions: First, 3 PSF-convolved galaxy models (Sersic, DeVaucouleurs, Exponential) are fitted to all objects with sufficient signal-to-noise and in regions outside the densest portions of the Galactic plane.",
"In addition, sources which are detected in only two of the 5 filters (or just in the $y_{\\rm P1}$ band, to allow for the presence of astrophysical objects which are dropouts in the bluer bands) are then used to force PSF photometry (and aperture and Kron flux measurements) at that same location in the other 3 (or 4) filters.",
"Finally, flux is measured for 7 radial aperture annuli, using apertures of the same radii in arc-seconds on the sky as used by SDSS.",
"These radial aperture fluxes are measured for the raw stack with its natural seeing as well as on a version of the stack convolved to match 1.5 and 2.0 arc-second seeing."
],
[
"Skycal Stage",
"The \"Skycal\" stage is similar to the chip stage, the staticsky stage analysis generates measurements in instrumental coordinates (X,Y,DN).",
"The skycal stage performs the calibration of the staticsky outputs relative to the reference catalog in an analogous fashion to the camera stage."
],
[
"Full Force Stage",
"Image quality variations between different exposures (and even within a single exposure) result in a stack PSF which can vary discontinuously on small scales.",
"PSF photometry and PSF-convolved galaxy model fitting on the stack cannot follow these variations.",
"The result is degraded performance in the stack photometry and morphology analysis.",
"To avoid this problem, we use the outputs from the “Static Sky\" stage analysis as the input to a “forced\" photometry analysis on each of the input warp images.",
"In this analysis, the positions of all objects detected in the stacks are used to measure the PSF photometry of those objects on each of the input warps images, using the appropriate PSF model determined for that position on that warp image.",
"The individual warp measurements are then combined in catalog space (in our photometry databasing system) to determine the mean photometry for each object.",
"In this step, input measurements with excessive masking are also excluded from this mean photometry calculation.",
"The result is a reliable photometry measurement for all objects down to the detection limits of the stack, as well as the data to study the variability and transient nature of the faintest sources.",
"In this stage, we also perform an analysis of the galaxy morphology using the “static sky\" galaxy model measurements as the seed ([83])."
],
[
"Post-Processing and DVO",
"After the pixel-level processing is performed, the catalogs of measurements extracted from the images are ingested into an instance of the Desktop Virtual Observatory or DVO .",
"DVO is a set of stand alone tools within the IPP system created to perform calibrations and provide further analysis of systematic effects.",
"In addition to the ingest into DVO at the IPP, the team of Eddie Schlafly (MPIA, LBL), Doug Finkbeiner (Harvard), and Greg Green (Stanford) also ingest the camera-stage data into a separate databasing system called LSD .",
"This system is similar in scope to DVO and allows similar calibration operations.",
"This team runs the “ubercal\" analysis on the detections from the chip and camera stage to measure zero points for photometric data.",
"In this analysis, relative photometry of overlapping images is used to constrain the zero points and airmass terms.",
"A rigid solution is determined by requiring a single zero point and airmass term for each night.",
"The resulting photometric system is shown to have a precision of 8, 7, 9, 11, 12 millimags for each of $grizy_{\\rm P1}$ respectively [105]."
],
[
"IPP-to-PSPS",
"Given the way the Pan-STARRS System evolved, it has been necessary to implement a translation layer to collate the catalog products produced by the IPP [81] in an optimal manner for ingest into the PSPS.",
"The IPP-to-PSPS produces batches of binary fits files containing catalog data.",
"There is a different kind of batch for each type of database table (e.g.",
"objects, stacks, detections, difference detections).",
"Each batch contains data from a localized region of the sky.",
"Some units are rationalized in the IPP-to-PSPS, so there is some manipulation of data values in this subsystem.",
"See [33] for a detailed discussion."
],
[
"PSPS - Published Science Products System",
"The Pan-STARRS Project teamed with the database development group at Johns Hopkins University to undertake the task of providing a hierarchical database for Pan-STARRS [44].",
"Since the JHU team was the major developer of the SDSS database [118], our goal was to reuse as much of the software developed for the SDSS as possible.",
"The Pan-STARRS database is commonly refered to as the “PSPS\".",
"The key to moving from the SDSS database to a system capable of dealing with Pan-STARRS data is the design of the Data Storage layers.",
"It was immediately clear that a single monolithic database design (like SDSS) would not work for the PS1 problem.",
"Our approach has been to use several features available within the Microsoft SQL Server product line to implement a system that would meet our requirements.",
"While SQL Server does not have (at present) a cluster implementation, this can be implemented by hand using a combination of distributed partition views and slices [44].",
"This allows us to partition data into smaller databases spread over multiple server machines and still treat the information as a unified table (from the users' perspective).",
"Further, by staying with SQL Server we are able to retain a wealth of software tools developed for SDSS, including the use of Hierarchical Triangular Mesh indexing for efficient spatial searches.",
"An overview of the PSPS system is shown in Figure REF ."
],
[
"Object Data Manager",
"The Object Data Manager is a collection of systems that are responsible for publishing data attributes measured by the IPP or other Pan-STARRS Science Servers to the end user (scientist).",
"The ODM manages the ingest of data products from the IPP (or other sources), integrates the new products with existing information in its data stores, and then makes the information available to the users in relational databases.",
"Catalog data from the IPP as prepared by the IPP-to-PSPS layer is contained in batches of binary fits tables.",
"These fits tables are read by a Data Transformation (or DX) Layer where data are grouped by declination zone and throttled in Right Ascension by the IPP-to-PSPS layer.",
"Then the data is loaded into `cold' or load slice machines by the DLP or data loading pipeline.",
"The slices are variable bands in declination, established to have nearly constant data density.",
"Once data are loaded on all declination slices through a given RA range, the data are merged, wherein they are stored and indexed on the slice machines so that data that are nearby in the sky are similarly nearby on disks and grouped by machine.",
"Once the data are successfully merged across the whole sky, the database is copied from the load/merge machines to the data storage machines where the user can access the database through the Query Manager (QM) and web-based Pan-STARRS Science Interface (PSI).",
"Figure: Data from the IPP enters the ODM through the DX layer as FITS binary tables.",
"The DX layer converts these into comma separated variable (CSV) files that are then passed to the loading pipeline.",
"The data flow is illustrated in the diagram above.",
"As illustrated in this figure, most of the processing inside the ODM takes place under the hood and is invisible to the users, who only see the data once it is loaded into the data stores that serve the hot and warm processing queues.The attributes are examined for basic validation (e.g., data in range checks).",
"The loading workflow process the input from the loading through to the merging of the new data records with the information already contained in the cold database.When a sufficient quantity of new input has been merged into the cold database we execute the copy-flip workflow.",
"In this stage the hot (fast queue) database is allowed to drain its query queue and is then taken off line.",
"The cold database is then copied in total to the hot database storage.",
"While the hot copy is taking place the warm (slow queue) is kept running to drain any remaining queries that have been staged for processing.",
"Once the hot copy has been completed we pause the slow queue, flip the active queue back to the fast queue and resume processing there, and execute the warm copy."
],
[
"The Data Retrieval Layer",
"The Data Retrieval Layer or DRL is the unseen hub of the PSPS system.",
"It sits between software that provides user access and the underlying data stores themselves.",
"The DRL provides the access to users and the databases through web browsers.",
"Only those users who want to write their own access clients will interact with the DRL directly.",
"A simple application programming interface (API) has been developed to allow one to develop such applications.",
"The DRL also provides the internal mechanisms for routing result sets from the PSPS databases back to the user.",
"The DRL API allows the PSPS to expand to incorporate the addition of new databases that can make science products created by PS1SC science servers available to the user community.",
"The API has been demonstrated to work with Microsoft SQL Server, MySQL, and PostgreSQL databases.",
"The DRL Layer is accessible through the CasJobs interface at the Pan-STARRS1 Archive at MAST."
],
[
"PSI Interface",
"The Pan-STARRS Science Interface (PSI) is a web application that has been developed by the PSPS development team.",
"It is designed provide users with easy access to the PSPS through a web browser.",
"PSI has tools to simplify the construction of querys and flags and a variety of useful features.",
"PSI is built on a improved version of CASJOBS, but it is not immediately backwards compatible with the version of CASJOBS at STScI.",
"Access to the Pan-STARRS1 archive at MAST at STScI is through the standard CASJOBS ([97], [117]) interface."
],
[
"Science Servers",
"The PS1 Science Servers were a project concept to add science value to the basic data products of object, position and flux.",
"The three projects that evolved to provide working code and data products are breifly described here."
],
[
"MOPS - Moving Object Processing System",
"The Pan-STARRS Moving Object Processing System (MOPS; [28]) is a modern software package that produces automatic asteroid discoveries and identifications from catalogs of transient detections from Pan-STARRS or any next-generation astronomical survey telescope.",
"As implemented as a subsytem in the Pan-STARRS System, it obtains difference detections from the IPP, performs linkages between detections, and makes initial orbit determinations.",
"Potential moving objects are evaluated by a human inspection system, and candidates are passed to the Minor Planet Center of the IAU.",
"Funded by the Pan-STARRS Project prior to the formation of the PS1SC, MOPS was the first integrated asteroid detector system able capable of automatically producing high-quality orbits from individual per-exposure transient catalogs.",
"MOPS is also able to search its own historical data for orphaned one-night detections after an orbit is generated.",
"As implemented as a subsytem in the Pan-STARRS System, it obtains difference detections from the IPP, performs linkages between detections, and makes initial orbit determinations.",
"Potential moving objects are evaluated by a human inspection system, and candiates are passed to the Minor Planet Center of the IAU.",
"MOPS has additional value as a research tool in survey design, able to simulate years of observations and detections given a catalog of synthetic asteroids and a hypothetical observation schedule.",
"The synthetic solar system model [41], containing $10^7$ objects representing populations of all major solar system bodies, remains the standard synthetic population for evaluating survey performance."
],
[
"TSS - Transient Science Server ",
"The vast majority detections in difference images requires a system for classifying real vs artifacts to manually select the most promising candidates.",
"The Queen's University Belfast group developed the Transient Science Server to systematically process difference detections from stationary transients from the IPP stream and apply machine learning techniques to classify them.",
"([131]).",
"This system continues to process transient events from Pan-STARRS and post discoveries on the IAU Transient Name Server.",
"In parallel the team at CfA, Harvard developed a custom version of the photpipe image subtraction and analysis pipeline and analyse the MDS data in real time [5], [101] The two teams cross-correlated transient discoveries and photometric measurements from both streams to improve efficiency and measurement precision of the IPP products.",
"Both were successful in different ways, and the QUB based TSS was the only one currently in operation for the 3$\\pi $ based searches and the ongoing Pan-STARRS Survey for Transients[51], [112]."
],
[
"PCS - Photometric Classification Server ",
"The Photometric Classification Server ([102]) is a set of software tools and hardware set up to compute photometric, color-based star/QSO/galaxy classification and best-fitting spectral energy distribution (SED) and photometric redshifts (photo-z) with errors for (reddish) galaxies.",
"The system can establish an interface to the PSPS database and results can be ingested back into the PSPS.",
"Results from the Photometric Classification Server will not be available in DR1."
],
[
"Pan-STARRS Operations",
"The observatories are operated remotely from the Pan-STARRS Remote Operations Center in the Institute for Astronomy (IfA) Advanced Technology Research Center (ATRC) in Pukalani, Maui.",
"There is no one at the summit at night or on weekends except in urgent or emergency situations.",
"The Observatory is approximately 45 minute drive from the ATRC.",
"A Pan-STARRS observer on a swing shift schedules the night's operations based on the overall science goals, state of the survey, and expected conditions.",
"The night Observer executes the plan prepared by the swing shift observer and modifies it in real time as circumstances demand.",
"The observing staff rotate through the swing and night shift and support the day crew at the summit.",
"The Staff at the ATRC also provides support for the telescope, scheduling software, and system administration for the IPP cluster in Kihei.",
"The IPP is a linux cluster that currently has 3100 cores and 5.5 Petabytes of storage.",
"The IPP cluster is currently located at the Maui Research and Technology Center in Kihei, Maui, about 20 minutes drive from the ATRC.",
"The computing facility (power, cooling, network connectivity to the outside world) is administered by the Maui High Performance Computing Center.",
"Additional computing resources were required for the PS1 Surveys including the Mustang Cluster (30,000 cores) at Los Alamos National Laboratory and the Cray cluster at the University of Hawaii (3600 cores).",
"Operationally the IPP and the PSPS are run remotely by the IPP team from IfA Manoa.",
"During night time operations, the raw exposures are immediately downloaded to the IPP cluster in Kihei, Maui.",
"Nighty data processing occurs automatically for exposures as they are obtained, with the analysis emphasis on the discovery of transient events, as well as data characterization for future re-processing.",
"The reprocessing versions and status are discussed in detail below and in the companion papers.",
"These data products have been loaded and merged in the PSPS database and transferred to STScI for exposure through the MAST archive."
],
[
"The PS1 Science Goals",
"The primary science design drivers for PS1 were originally put forth in the PS1 Science Goals Statement ([15]).",
"The top level goals were: Precision photometric and astrometric survey of stars in the Milky Way and the Local Group; Surveying our Solar System, including searching for Potentially Hazardous Objects amongst Near Earth Asteroids; New constraints on Dark Energy and Dark Matter; Exploration and categorization of the astrophysical time domain, including, but not limited to, explosive transients, microlensing events in M31, and a transit search for exo-planets.",
"Providing a development platform for prototyping PS4 components, subsystems, and survey strategy.",
"These goals drove the initial design and engineering requirements, and shaped real time development decisions.",
"On the last point, while the PS4 system has not yet been funded, PS1 did serve in this capacity for the development of PS2 [89].",
"The above outline goals do not begin to cover the vast array of solar system, galactic, extragalactic, and cosmological studies that can be done with the PS1 data products.",
"To refine this, the project and the PS1 Science Consortium Science Council generated the PS1 Mission Concept Statement [14] with a set of surveys as follows: (1) A 3$\\pi $ Steradian Survey; of 60 epochs in five passbands ($grizy_{\\rm P1}$ ) of the entire sky north of declination $\\delta =-30$ degrees, (2) A Medium Deep Survey with data in all of $grizy_{\\rm P1}$ of ten PS1 footprints on well studied fields totaling 70 square degrees at high galactic latitudes spaced around the sky, (3) A solar system ecliptic plane survey in the wide $w_{\\rm P1}$ passband with cadencing optimized for the discovery of Near Earth Objects and Kuiper Belt Objects, (4) a Stellar Transit Survey of 50 square degrees in the galactic bulge; and (5) a Deep Survey of M31 with an observing cadence designed to detect micro-lensing events and other transients.",
"In addition a special series of observations of spectro-photometric standards was carried out for calibration, and the Celestial North Pole was observed nightly for the last two years of the survey to track performance and measure atmospheric properties.",
"Table REF summarises these surveys and the approximate percentage time spent on each of the total operational science time.",
"The operational plan for execution of these surveys was articulated in the PS1 Design Reference Mission ([16]) or DRM, that served as a benchmark as the system transitioned from commissioning to operations.",
"This survey strategy evolved into a Modified Design Reference Mission as lessons learned were incorporated as the surveys progressed.",
"Table: The Pan-STARRS1 SurveysFigure: Top Left: One realization of the boresight tessellation.",
"Top Right: Schematic of seven field-of-views layed out in flat a hexagon pattern.",
"Most of the vertices of the boresight tessellation have six nearest neighbors, a few have five.",
"The figure shows the hexagon that is inscribed upon the field of view in Figure 3.",
"This shows the nominal overlap of adjacent field-of-views.",
"For a given exposure the (x,y) axes of the camera can have any orientation with respect to North, as the telescope is alt-az.",
"Lower Left: the RINGSV.3 tessellation of the sky with virtual rectangular images on tangential projection centers.",
"For the 3π\\pi Survey with a boresight southern declination limit of Dec >-3> -3 degrees, the RINGSV.3 has 2009 tangential projection centers, and a nominal 200,900 sky cells which extend to Dec=-31.81=-31.81 degrees.",
"However the southern edge of the set of images is ragged from the footprint extending down from the southern most possible boresight of Dec=-30=-30.",
"The number of skycells containing populated imaging data is then 200,684, see for more details.Furthermore there is a special tessellation for the north pole ().Bottom Right: a zoom showing several projection cells, each in a different color, and each divided up into overlapping sky cells.",
"This shows the overlap of sky cells and the overlap of projection cells.",
"Nearly all analysis of Pan-STARRS1 images is done on a sky cell basis."
],
[
"The 3$\\pi $ Steradian Survey ",
"The $3\\pi $ Steradian Survey covers the sky north of Dec $= -30$ degrees in five filters ($grizy_{\\rm P1}$ ) and includes data taken between 2009-06-02 and 2014-03-31.",
"This means that for a given sky tessellation, a field center was included in the survey only if it was above declination $\\delta =-30$ degrees.",
"For pointings with field centers that are close to $\\delta =-30^{\\circ }$ , close to half the field (up to 1.5 degrees) extended below the limit.",
"This means there is a ragged edge and an uneven declination limit to the survey between $-31.5\\mbox{$^\\circ $}< \\delta < -30\\mbox{$^\\circ $}$ .",
"The survey pattern and scheduling followed two different strategies over the course of the 3$\\pi $ survey: the initial pattern layed out in the Design Reference Mission (DRM) [16] followed by the Modified Design Reference Mission (MDRM).",
"We switched to MDRM on 2012-01-14.",
"All exposures in the DRM were taken in pairs, with each exposure separated by a Transient Time Interval or TTI of 12 to 24 minutes, for the purpose of detecting moving objects within the Solar System.",
"These were referred to as “TTI pairs\".",
"The original plan was then to take 2 TTI pairs over an observing season with $g_{\\rm P1}$ , $r_{\\rm P1}$ and $i_{\\rm P1}$ taken within the same lunation and separated by days to weeks.",
"The $z_{\\rm P1}$ and $y_{\\rm P1}$ were to be taken approximately 6 months apart to optimise stellar parallax and proper motion measurements (for low mass stars).",
"Over 3.5 years this would give (allowing for weather interruptions) 12 exposures in each band or 60 in total over all 5 filters.",
"In the MDRM, a series of 4 exposures, “quads\", all separated by approximately 15 minutes (therefore completed within about 1hr), were implemented for about half of the $g_{\\rm P1}$$r_{\\rm P1}$$i_{\\rm P1}$ exposures with the express purpose of increasing the recovery of Near Earth Objects (NEOs).",
"The relative exposure times in each were also chosen to make an asteroid of mean solar color (taken to be $(g_{\\rm P1}-r_{\\rm P1}) = 0.44, (r_{\\rm P1}-i_{\\rm P1})=0.14$ ) to have approximately the same signal-to-noise.",
"During data processing, the “Warp Stage\" takes the detrended pixel images generated by the “Chip stage\" and geometrically transforms (warps) and re-samples them onto a predefined set of images which tessellate the relevant portion of the sky (these processing stages are discussed in Section REF ).",
"For the $3\\pi $ survey, PS1 uses a modification (RINGS.V3) of the Budavari rings tessellation with tangential projection centers spaced $~4$ degrees apart.",
"A set of virtual images called “projection cells\" are defined to cover the sky about these projection centers without gaps.",
"These virtual projection cells are subdivided along cartesian pixel boundaries into “skycells\", the image regions onto which the native device pixels are warped.",
"All skycells have a pixel scale of 0.25 arcsec per pixel and are roughly 20 arcminutes on a side, which is comparable in size to the native device images (these chip images are the $4846\\times 4868$ pixel arrays which are 0.258 arscsec per pixel).",
"The main output from this stage is the collection of three separate pixel images each representing the signal, variance, and masking for the skycells.",
"The MD and similar surveys use special local projection cells centered on the fields of interest."
],
[
"Primary object resolution on the sky",
"The skycells and projection cells are defined to have an overlap of 60 arcseconds (120 pixels) on each edge in order to avoid objects being split between adjacent skycells.",
"Note that it is the same data which goes into the overlap regions - there is no new data involved here.",
"The problem of identifying a unique area, and thus assigning an object to a particular skycell, is called the primary resolution problem.",
"This is important, as data analysis is performed on each skycell independently, so an object near a boundary will have duplicate measurements.",
"IPP produces a tessellation tree file which contains RA and DEC limits for each projection cell, which can be used to define unique areas.",
"Objects landing within these limits are classed as primary objects and have the primary flag set in PSPS.",
"This flag should always be used to define a unique sample of objects on the sky.",
"Table: Properties of images in each filter.",
"The solar elongation indicates when twilight for that filter effectively starts."
],
[
"Scheduling of PS1 Surveys",
"The primary reason for a discussion of the scheduling of the PS1 Surveys is to explain why the time domain of the 3$\\pi $ Survey has the detailed structure that it has.",
"Prior to the formal start of the PS1 Mission on May 13, 2010, we used a contemporaneous version of the LSST scheduler to model the PS1 Mission as defined by the Design Reference Mission [16] and smaller in summer in accordance with the length of night.",
"We further tweaked the size of the $3\\pi $ slices to accommodate time for the smaller PAndromedra and Pan-Planets surveys, and assumed that the MD surveys, which are fairly evenly distributed in RA, could be fit into a constant nightly time allocation.",
"The observing pattern from the DRM (applicable from 2010-05-10 - 2012-01-14) is schematically shown in Figure REF .",
"An Observing Cycle (OC) is defined as one lunation.",
"The sky areas and filter coverage observed in an example OC are illustrated in this figure.",
"Clearly one needs to observe, on average, 1/12.37 of the sky per Observing Cycle per filter (12.37 is the number of lunations in one year).",
"This corresponds to a slice of sky from the pole to $\\delta =-30$ which is roughly 4 hrs in right ascension.",
"This mean value was expanded or compressed a-priori for the length of night and to adjust for the non-uniform impact of the smaller surveys in their RA distribution.",
"One aspect of the PS1 3$\\pi $ Survey is that the $z,y$ bands are observed out of phase with $g,r,i$ by months, whereas $g,r,i$ might be taken in the same night or be out of phase by days.",
"We defined two distinct kinds of slices, the Opposition slices, where the sky within about 2 hours RA of opposition was observed in $g,r$ and $i$ bands, and “Wing\" slices which were near the meridian at twilight.",
"There were several reasons for this “strategic\" approach: (i) because twilight (defined as the moment when the night sky reaches a constant sky brightness) occurs at increasing solar elongation as one proceeds through the filter set from red to blue $y,z,i,r,g$ , there is a period of time when the sky is as dark as it is going to get in $y$ band, but it is still in twilight in $z,i,r,g$ bands.",
"Thus it is most advantageous to use this time in $y$ band.",
"Once the sky becomes dark in $z$ band, the same is true.",
"The time differences in the other bands are more modest.",
"To illustrate this quantitatively Table REF gives details of the solar elongation angle at which the sky reaches its constant dark level (i.e.",
"end of twilight) in each filter.",
"(ii) We desire to measure as many stars with measurable parallaxes as possible.",
"These are the closest stars and are thus most likely to be brown dwarfs.",
"PS1 is a red sensitive instrument, and is already delivering on its goal of finding new populations of L and T dwarfs [22], [73].",
"It is therefore desirable to observe in $z,y$ bands at maximum parallax, i.e.",
"with a cadence of nearly six months.",
"As illustrated in Figure REF , the Wing slices here in the $z,y$ bands are separated by nearly six months: as the pattern marches to the left from “Month A\" to “Month B\".",
"This shows that in about six months time the same region of the sky will be observed again in the same filters.",
"(iii).",
"This approach also ensures that the sky areas surveyed in the $z,y$ bands are observed near the meridian, or close to optimum airmass.",
"During operations there typically was not quite enough time to get all of the $y$ and $z$ band observations in the twilight time of $g, r, i$ .",
"However near full moon, the sky is bright even in $i-$ band, and some $y$ band fields were observed closer to the middle of the night.",
"This required that the polar regions, which were beyond the 30 degree moon avoidance region, be shifted slightly closer to opposition and yet could still be observed at reasonable airmass.",
"During a night's observing the pattern from the above strategy was to observe “chunks\" where a chunk is simply a contiguous region of sky of approximately $4 \\times 4$ GPC1 footprints, with a pair of visits separated by a TTI in $y$ band, and then if possible the same chunk in $z$ band as the sky got darker.",
"Then the available Medium Deep Survey fields were inserted between the Wing Slices and the Opposition Slices.",
"Depending on the sky brightness (lunar illumination and distance) a chunk in $g,r$ or $i$ band would be observed near opposition.",
"Once opposition passed it would be back to the other programs and the morning Wing slice at twilight.",
"The prioritization of the chunks in declination was by image quality and transparency, so generally by airmass, followed by wind direction or partial clouds.",
"Figure: DRM twilight and opposition scheduling cadenceFigure: MDRM Opposition Cadence.",
"For the Modified Design Reference Mission the g P1 g_{\\rm P1}, r P1 r_{\\rm P1}, and i P1 i_{\\rm P1} observing cadence and pattern were changed, but the Wing cadence pairs of the DRM for z P1 z_{\\rm P1} and y P1 y_{\\rm P1} remained the same as in Figure Figure: Right Ascension in degrees vs Time in Unix seconds.It was eventually realized that a modification to the DRM was necessary.",
"The DRM was done entirely in pairs with the assumption that observations of NEOs in pairs separated by one or two nights could be linked.",
"As a matter of experience that turned out not to be the case.",
"However simply switching to quads, or four exposures per night separated by TTIs would put the years worth of exposures for a given field in a given filter all into one night.",
"This would have endangered the photometric survey, reducing the number of opportunities to have a photometric night, critical for ubercalibration [83].",
"The solution, called the Modified Design Reference Mission or MDRM, which was settled upon is shown in Figure REF .",
"The Wings pairs in $z,y$ remained the same in the MDRM as the DRM.",
"The sensitivity to discovering asteroids was low in $z,y$ although a few have been found by a pair in $y$ and a pair in $z$ .",
"In this compromise solution, one third of the data in a given filter is taken in a quad, while the remainder is taken in pairs on different nights.",
"The total number of exposures per field per year is still 4, but the cadence will be different in different parts of the sky.",
"This is roughly smoothed out with data over different years where the pattern is shifted by the position of opposition at new moon.",
"Furthermore the pattern is altered by the scheduling around weather and wind.",
"One way of visualizing the time domain is shown in Figure REF where the date (in Unix seconds Unix time is defined as the number of seconds that have elapsed since 00:00:00 Coordinated Universal Time (UTC), Thursday, 1 January 1970) of every exposure over the 3$\\pi $ Survey is plotted vs the right ascension.",
"The slanting bands are the yearly revisiting of the RA in opposition in $g,r,i$ bounded by two visits in $z,y$ per year.",
"For a given day, one can look along a row of constant time and see the range in RA ascension covered in a night.",
"This banding is by design in the “strategic\" approach, but because the constraints of airmass and sky brightness are generic to an all sky survey, one imagines that the results of the LSST scheduler should show the same pattern if the tension between the parallax cadence (six months) and the sky brightness (twilight) is balanced.",
"Compare Figure REF with Figure 1 of [14] in advance of the survey.",
"The effectiveness of the strategic scheduling approach and the patient efforts of the PS1 Observers who used the scheduling tools to solve the travelling salesman problem in multiple dimensions and responded interactively to the nightly conditions (clouds, wind speed, cadence, sky brightness, survey completeness) is demonstrated in Figure REF .",
"This shows the actual distribution of pointings in Dec vs Hour Angle for the entire $3\\pi $ Survey.",
"The hole in the middle is the keyhole characteristic of an Alt-Az telescope.",
"The hour angle distribution shows that 65% of the data is taken within $\\sim $ 1.5 hours of the meridian.",
"Figure: Distribution of telescope pointings for the 3π3\\pi Survey in Hour Angle vs Declination.",
"The hole in the middle is the keyhole of the Alt-Az PS1 telescope at latitude = XX.",
"The width of the HA distribution is XX."
],
[
"The Medium Deep Survey",
"The Medium Deep Survey consisted of 10 single PS1 footprints on well studied fields spaced approximately uniformly around the sky in Right Ascension.",
"The pointing centers of these 10 fields are listed in Table REF .",
"The table includes two additional fields of M31 which can be considered an MDS like field (see Section REF ) and a field at the north ecliptic pole (NEP).",
"The latter was not observed as extensively as the 10 main fields and was only observed over the period 2010-09-20 to 2011-06-17.",
"The individual exposure times per filter were considerably longer than those for the 3$\\pi $ survey (see Table REF ).",
"The Medium Deep Survey (MDS) component of the program regularly visited these 10 fields ( 7 sq.",
"deg.",
"each).",
"Each field was picked to have significant multi-wavelength overlap from previous and concurrent surveys by other teams and facilities (e.g.",
"DEEP2, ELIAS-N, CDFS, COSMOS, GALEX).",
"In total, 25% of the PS1 time was allocated to the MDS.",
"The cadence was generally composed of $g_{\\rm P1}$ and $r_{\\rm P1}$ together on one night, followed by $i_{\\rm P1}$ on the second and $z_{\\rm P1}$ on the third.",
"This pattern was repeated continually on a 3 day cycle over the 6-8 month observing season for the field, interrupted only by weather and the moon.",
"Around full moon, the $y_{\\rm P1}$ filter was primarily used and hence it does not have the same time cadence as the other 4.",
"Figure REF illustrates the cadence and observing seasons while Table REF [101] summarises the exposure times.",
"On each night the 8 separate exposures were dithered and the field was rotated.",
"The images were then combined into nightly stacks of 904 sec ($g_{\\rm P1}$ and $r_{\\rm P1}$ ) and 1902 sec ($i_{\\rm P1}$ , $z_{\\rm P1}$ and $y_{\\rm P1}$ ).",
"Roughly one a year these stacks are further combined to produce so-called reference stacks, which are then used as templates for difference imaging.",
"Finally, all the data are combined to produce very deep stacks, which contain several tens of hours worth of exposure.",
"The description of the MDS was initially presented in [124] and many papers on transients have already given an overview of the data products and survey [19], [38], [85], [103], [78].",
"Estimates of the typical 5$\\sigma $ depths of the MDS nightly stacks were given in [101] and are also listed in Table REF here.",
"Development work continued to improve the single exposure processing though to deep stacks during the transient event discovery and other science consortium programs over the course of the survey, the culmination of those improvements being applied in a more uniformly reprocessed dataset used for the public data release.",
"A full discussion of the Medium Deep Fields, including improved estimates of depths and their special processing will be presented inHuber et al.",
"2017 (in preparation - Paper VII).",
"No Medium Deep data will be released in DR1.",
"Figure: Medium Deep Field Survey: on the left the sky positions of the MDS fields are shown.",
"On the right, the cadence of the observing is illustrated.",
"With the grizy P1 grizy_{\\rm P1}filters labelled in blue, green yellow, brown and green respectivelyTable: The MDS cadence and exposure times.",
"This table was originally presented in and is reproduced here in identical form.Table: Pan-STARRS1 Medium-Deep Fields"
],
[
"Pan-Planets stellar transit survey",
"For Pan-Planets, seven slightly overlapping fields with overall 40 sq.",
"deg.",
"were observed with PS1, making up about 4% of the total survey time (see Table REF ).",
"Data were collected between 2009 and 2012 in the $i_{\\rm P1}$ -band.",
"Depending on seeing, exposure times were either 30 sec or 15 sec.",
"In the first two years of the survey, three fields were observed.",
"From 2011 on, four additional fields were added to the survey area, meaning that the previous three fields have a higher number of visits.",
"On each survey night, the exposures were cycled through the seven fields to minimize saturation effects.",
"We obtained at least 2000 exposures for each point in our FOV and up to 6000 in the overlapping areas between the fields.",
"The main goal of Pan-Planets is the search for transits from extrasolar planets, mainly hot gas giants close to their star with a special focus on M-dwarfs [1] There are up to 60000 M-dwarfs in the FOV with magnitudes between 13mag and 18mag in the i-band, which makes the survey one of the most comprehensive transit searches for M-dwarf exoplanets.",
"A description of the scientific results and analysis can be found in [95].",
"The Pan-Planets stellar transit data is not included in DR1.",
"Table: Pan-Planets Stellar Transit Survey Fields"
],
[
"PAndromeda, the M31 transient survey",
"PS1 had a special monitoring survey for M31 for 2% of the original PS1 survey time.",
"Data were taken from 2010 to 2012 (3 seasons), during the second half of each year when M31 was easily visible.",
"M31 was also covered in the regular 3$\\pi $ Steradian Survey.",
"As part of the separate survey, M31 was visited up to two times per night in the $r_{\\rm P1}$ and $i_{\\rm P1}$ filters.",
"Depending on the weather conditions, we obtained up to 14 times 60-second exposure in $r_{\\rm P1}$ and 10 times 60-second exposures in $i_{\\rm P1}$ .",
"These exposures were spread across the two visits per night to give some intra-night time resolution.",
"The survey strategy was optimized to detect short-term M31 microlensing events, but to also allow one to identify and analyze the variable star content in M31.",
"Observations were taken much more sparsely in the remaining filters ($g_{\\rm P1}$ , $z_{\\rm P1}$ and $y_{\\rm P1}$ ) in order to give multicolour maps of M31 in the full PS1 filter complement.",
"The first results and demonstration of data quality from the first 90 nights in 2010 were presented in [65].",
"M31 data will not be released in DR1."
],
[
"Spectro-photometric and Calspec Standard Stars",
"The AB magnitude system calibration of the Pan-STARRS1 photometric system by [10] used data from a single photometric night.",
"and special observations of the HST Calspec sample [10].",
"All standard stars were placed on OTA 34 and cell 33, so their integration was on the same silicon and used the same amplifier for read-out.",
"However, this position was very close to the center of the focal plane, where it has been noted that there is a strong gradient in the behavior of the chip (Rest et al.",
"2014), and thus these observations were not included in the subsequent study by [109], [110].",
"They analyzed a sample of faint Calspec standards observed over the course of the $3\\pi $ survey and re-determined the AB offsets for the $g_{\\rm P1}$ , $r_{\\rm P1}$ , $i_{\\rm P1}$ , $z_{\\rm P1}$ bands of the PS1 system.",
"The super-cal [110] AB offsets were used in the calibration of all the DR1 and DR2 data [82].",
"However [110] note that primary difference in the update arises from changes in the Calspec standards."
],
[
"The Celestial North Pole",
"The 3$\\pi $ Steradian Survey extends to the North Pole.",
"It was soon realized that a dedicated nightly pointing near the Celestial North Pole would provide continuous time coverage that could monitor the performance of the system as well as be of scientific interest for the unique cadence.",
"So a set of $grizy_{\\rm P1}$ exposures of 30 seconds was obtained each night on the meridian and at a declination of 89.5 degrees.",
"Observations were obtained every clear night between 2010-10-13 and 2014-02-13.",
"The net result is about a 4 square degree area with regular observations for 3.3 years.",
"This data is not included in DR1."
],
[
"Small Area Survey 2",
"In July 2011 a test area of the 3$\\pi $ survey, consisting of about 70 deg$^2$ centered on $(\\alpha ,\\delta ) = (334\\mbox{$^\\circ $}, 0\\mbox{$^\\circ $})$ (J2000), was observed to the expected final depth of the survey in $grizy_{\\rm P1}$ .",
"These data are described in depth in [87] where general issues of the data and the PS1 reduction software is subject to a rigorous investigation, with emphasis on the depth of the stacked survey.",
"A further paper [31] demonstrates how galaxy number counts and the angular two-point galaxy correlation function, w($\\theta $ ),can be reliably measured.",
"This data is not included in DR1.",
"Table: Image Processing Pipeline stages and data productsTable: Fundamental PSPS database tables"
],
[
"Overview of PS1 Data Products",
"The PS1 Data Products consist of images of various kinds, catalogs of attributes measured from the images organized in a hierarchical relational database, derived data products such as proper motions and photometric redshifts, and metadata for linking and tracking all of the above.",
"Here we provide a brief overview, see [33] for details.",
"We refer below to types of images and data files listed in Tables REF and REF .",
"The proper convention when reporting Pan-STARRS1 magnitudes is to use the nomenclature $grizy_{\\rm P1}$ (see Section REF ) and the convention for IAU names is $ {\\bf PSO JRRR.rrrr+DD.dddd} $ where the PSO identifier stands for Pan-STARRS Object, and the coordinates are in decimal degrees.",
"Another point of interest is the use of magnitudes and fluxes.",
"There are advantages and disadvantages to the use of these, and we employ both magnitude and fluxes where useful.",
"Luptitudes [79] also have some advantages, but we have made the decision not to use them in the PS1 data products.",
"One noteable advantage of fluxes, for example is that the magnitude of an aperture flux measurement can correctly be negative when measured on a sky subtracted image and, when the mean of a series of such measurements is computed, the result is well behaved.",
"All Pan-STARRS magnitudes are in AB magnitudes [125] and the fluxes are reported in the corresponding Janskys, where the absolute calibration is discussed in [109].",
"See also Section REF ."
],
[
"Image Data Products",
"In this section we discuss the specific kinds of images, their properties and location as produced by the IPP processing stages discussed above in Section REF .",
"A summary of the kinds of data files that exist including images is provided in Table REF .",
"The raw pixel data are archived in two geographically separate locations; one archival copy is retained on storage machines at the IfA's ATRC on Maui, and another is stored in the IPP cluster, which has moved from its inital location at the Maui High Performance Computing Center (MHPCC) to the Maui Research and Technology Center (MRTC-B) to its permanent location in the UH Information Technology Center on the Manoa campus.",
"Within the IPP all other files have at least two instances on separate Raid 10 machines.",
"Each Pan-STARRS image (an “exposure\" or “frame\") creates 60 fits image files, one for each device in the camera, and each fits file has 64 extensions, where each extension is the pixel data from one OTA cell (see Section REF ) Table REF lists the various image and binary fits table files produced by the IPP by each of its stages.",
"Some images are intermediate products and are not saved permanently, although they can be reproduced from the raw data.",
"The “chip\" images are the detrended images.",
"The signal image is now a float, and a matching mask and variance image are also produced.",
"The detection of objects and measurements of their positions and attributes (in detrended pixels) are stored binary fits tables (internally called CMF files).",
"These measurements are therefore in $(x,y)$ pixel coordinates and can have any orientation on the sky.",
"Together with the astrometric calibration from the “camera stage\" these measurements and their position in $(ra, dec)$ are the basis of the “Detection Table\" [33], These are also binary fits tables, (internally called SMF files).",
"The “warp\" images are astrometrically registered by a geometric transformation onto rectilinear North-South pixels in a tangential projection using the nearest projection center as defined by the RINGS3 tessellation The warp stage also produces warped mask and variance images, see [129] for complete discussion.",
"The “stack\" images are additions of accumulated warp images which should be precisely registered.",
"Variance and mask images for the stack are also created as well as a number image that shows the number of warps that contributed to the stack at any pixel.",
"Note, the different warps likely have different PSFs having been taken at different times and at different places in the focal plane, leading to what are essentially intractable problems in PSF measurements performed on the stacks.",
"This is the motivation for the “Forced Photometry\" stage [81].",
"The results of the analysis of the stacked images are stored in binary fits tables (again, internally labelled as CMF files) and are available in the Stack tables, [33].",
"Two convolved versions of the stack images are created by convolving with gaussians of width 6 or 8 pixels (precisely 1.5 and 2.0 arcseconds in 0.25 arcsecond sky cell pixels).",
"These are then used for aperture measurements [81].",
"The convolved images are intermediate products and are not saved.",
"The aperture measurments are stored in binary fits CMF files and are available in the Stack Aperture tables.",
"Difference images and their associated variance and mask images are created by the “Diff Stage\" and measured, with the results going in the Difference Tables.",
"The Difference images are not retained, but could in principle be regenerated from the stacks and warps.",
"Figure: Example of image types from a sky cell.",
"Upper Left:signal image.",
"Upper right: variance image.",
"Lower left: mask image.",
"Lower right: number image."
],
[
"Fundamental Data Products",
"The Pan-STARRS1 database schema ([33]) is organized into four sections: Fundamental Data Products.",
"These are attributes that are calculated from either detrended but untransformed pixels or warped pixels.",
"The instrumental fluxes or magnitudes have been re-calibrated, as have their positions.",
"Because of these calibrations, the catalog values are to be preferred to making a new measurement from the images.",
"See Table REF .",
"Derived Data Products.",
"These are higher order science products that have been calculated from the Fundamental data products, such as proper motions, photometric redshifts, associations of detections of moving objects by MOPS.",
"Observational Metadata.",
"This is metadata that provides detailed information about the individual exposures (e.g.",
"PSF model fit) or which exposures went into an image combination (stacks and diffs) of exposures, as well as information such as detection efficiencies.",
"System Metadata.",
"These tables have fixed information about the system and the database itself, including descriptions of various flags.",
"Various database \"Views\" or logical combinations of Tables are also constructed as an aide for common kinds of queries.",
"Note in the PSPS architecture, large tables (almost all except the \"head node\" ObjectThin, MeanObject, and GaiaFrameCoordinate) are actually \"views\" joining subsections (slices) of the data across different file partitions, but this structure is hidden from the casual user.",
"The classes of tables in the Fundamental Data Products include Detection, Object, Stack, Difference, and Forced (Table REF ).",
"We now consider each of these in turn."
],
[
"Detection Table",
"At the most basic level, an individual “Detection\" is a feature, likely a star or galaxy or artifact, detected above the noise in an individual exposure.",
"There are likely, but not always, multiple “detections\" of the same astronomical object from subsequent exposures.",
"The majority of single detections at the faint end are not real, but arise from systematic noise, primarily correlated read noise, in the GPC1 [129].",
"A wealth of flags are provided to help distinguish between real detections and artifacts [33].",
"Nonetheless there are detections that arise from systematic noise that are, by themselves, indistinguishable from real features For each single epoch detection, the Detection Table contains PSF magnitudes, total aperture based magnitudes, Kron magnitudes ([62]), assorted radial moments and combinations of moments, and circular radial aperture magnitudes in SDSS radii R3 through R11 [114].",
"See [83] for details."
],
[
"Object Tables",
"Individual “detections\" are associated into \"Objects\" by virtue of being approximately at the same location in $(ra, dec)$ .",
"The IPP makes this association of detections into objects in the “static sky stage\" if they are within 1.0 arcseconds and there are various complications for blended objects or single objects that become resolved in a subsequent, higher quality image [83].",
"It is possible that an astronomical object is only measured once even with multiple exposures with good pixels at the same location - for example a moving object, or a transient object, or an object that only rises above the noise in one image.",
"Systematic noise, especially the correlated read noise in the GPC1 detector [129], can also contribute a faint artifact that is interpreted as a single detection and becomes an Object.",
"Such single instances must also be elevated into \"Objects\" because at the time there is no independent way of knowing.",
"Thus the association of detections into objects is one-or-more to one.",
"Thus one-time-only false detections from artifacts are also promoted to Objects.",
"In the Pan-STARRS1 dataset, as a consequence of these features produced by the GPC1, such artifacts dominate the Detection and Object Tables.",
"One simple way to exclude them is to require an Object to have two or more detections; an occurrence which is decreasingly likely to happen if the feature isn't real.",
"This could also obviously exclude real moving and transient objects.",
"Sample queries to produce robust catalogs from the Pan-STARRS1 database are provided in [33], but one should always be aware of this aspect of Pan-STARRS1 data.",
"The Object Tables described in [33] include the ObjectThin Table, which contains the most minimal information set about an object, primarily its position and various indexes linking it to other tables.",
"There are two (ra,dec) positions provided, a “Mean\" position and a “Stack\" position.",
"Mean positions are the most accurate if available, as they come from a mean of all the individual epoch measurements, each of which have been calibrated on the Gaia [71] reference frame.",
"Objects that are only detected in the stack are fainter and their positions in DR1 have not been re-calibrated on the Gaia frame.",
"This is because their uncertainties are intrinsically larger and hence this is only an issue for the most demanding astrometry.",
"The MeanObject Table contains the mean photometric information for objects based on the single epoch data, calculated as described in [83].",
"To be included in this table, an object must be bright enough to have been detected at least once in an individual exposure.",
"PSF, Kron [62] and aperture magnitudes and statistics are provided for all filters.",
"The GaiaFrameCoordinate Table contains the re-calibration of the astrometric positions of all MeanObjects on the Gaia reference frame [82]."
],
[
"Stack Tables",
"Attributes measured on the stacked images are reported in the Stack Tables.",
"The StackObjectThin table contains the most minimal positional and photometric information for point-source photometry of stack detections.",
"The information for all filters are joined into a single row, with metadata indicating if this stack object represents the primary detection.",
"See Section REF .",
"The StackObjectAttributes table is analogous to the Detection table for single epoch images and it contains the PSF, Kron[62], and aperture fluxes for all filters in a single row, along with assorted radial moments and combinations of moments.",
"The StackApFlx Table contains the unconvolved fluxes within the SDSS R5, R6, R7 apertures [114] for all Stack Objects.",
"The StackApFlxExGal Unc, Conv6, Conv8, Tables contain the unconvolved fluxes within the SDSS R3 through R11 apertures [114] for objects in the extragalactic sky, i.e., they are not provided for objects in the Galactic plane because they are not useful in crowded areas.",
"For each aperture we report: flux (janskys), flux error, flux standard deviation (from the individual measurements), and the fill factor of the aperture (masked pixels could reduce this from 1.0).",
"The StackPetrosian and StackModelFit Exp, DeV, and Ser tables report the results of fits of extended sources to model PSF convolved surface brightness profiles.",
"The measurements include Petrosian magnitudes and radii, Exponential, de Vaucouleurs, and Sersic magnitudes and radii and elliptical aperture magnitudes and errors for a signal-to-noise ratio and galactic latitude limited sample.",
"See [83] for details."
],
[
"Difference Tables",
"The IPP generates Alard-Lupton convolved difference images for skycells in various combinations depending on the survey goals.",
"For the 3$\\pi $ Survey a difference image: $ diff = (warp - stack )$ is created for each epoch.",
"This difference image is then analyzed in the same fashion as an individual warp.",
"The DiffDetection table is analogous to the Detection table and has the same measurements.",
"If possible, DiffDetections are associated into DiffObjects, e.g.",
"different points on a light curve are associated into a Difference Object.",
"No attempt is made to associate Diff Objects with Objects.",
"While this might make sense for a variable object, a transient source, e.g.",
"a supernovae in a galaxy, could be undetectable in either an individual warp or stack, and yet be clear in the difference image.",
"In this case the closest Object would be the host galaxy, but that association would be incorrect.",
"Hence Difference Objects are a unique class and not contained in the Object Table.",
"On the other hand, a “good\" match between Objects and DiffObjects would provide a candidate for a variable object."
],
[
"Forced Photometry Tables",
"Forced photometry is carried out at the positions of all significant objects found in at least two bands.",
"This requirement keeps the number of forced objects to a practical number.",
"Single band detections, especially z-band dropouts or objects found only in y-band are a non-trivial subject of active research.",
"The forced detection measurements made on individual warp images are reported in the ForcedWarpMeasurement table.",
"Where the field-of-view of the exposure contains the position of the object, but its properties can not measured because the data happens to be masked at the position, the object's identify is stored in the ForcedWarpMasked Table.",
"The ForcedWarpExtended table contains the single epoch forced photometry fluxes within the SDSS R5, R6, and R7 apertures [114].",
"ForcedWarpLensing contains the contains the mean lensing parameters [56] of objects detected in stacked images measured on the individual single epoch data.",
"The individual epoch measurements are not reported, only their mean.",
"ForcedMeanObject has the mean properties of the individual forced measurements, including PSF, Kron, and aperture magnitudes, and R5, R6, and R7 apertures.",
"See [83]."
],
[
"Derived Data Products",
"Derived data products are results that can not be traced directly back to the pixels but are the result of systematic analysis of the Fundamental Data Products discussed above.",
"These include (i) measurements of proper motion and parallax (Magnier et al in prepation) made from an analyis of the minute changes in the positions of objects; (ii) Photometric redshifts deduced from aperture magnitudes using a variety of machine learning techniques ,(iii) the extinction and stellar parameters deduced from stellar photometry ; (iv) associating detections in the database with known or discovered moving objects in the MOPS database.",
"We have the tools to ingest these derived data products back into the PSPS and make them widely available to the community.",
"Our intention is to include these derived data products in future Data Releases."
],
[
"Astrometric",
"The Pan-STARRS1 astrometry has been re-calibrated[82] using Gaia [36] The Gaia DR1 [35] catalog [71] was used as the input reference catalog.",
"After recalibrating all individual epoch measurements to the Gaia Frame, and then re-constructing the mean PS1 positions we can estimate the astrometric error of the resulting PS1 postions.",
"The systematic uncertainty of the astrometric calibration using the Gaia frame comes from a comparison of the results with Gaia: the standard deviation of the mean and median residuals ($ \\Delta ra, \\Delta dec $ ) are (2.3, 1.7) milliarcsec, and (3.1, 4.8) milliarcsec.",
"The latter is a measurement of the bright end errors for average positions while the former is a measurement of the consistency of the PS1 and Gaia systems([82])."
],
[
"Photometric",
"The photometric accuracy of the PS1 data products has been demonstrated in the ubercal analysis [105] and relative photometric analysis [80].",
"Zero points for photometric data are determined with a reliability of 7-12 millimags.",
"Individual detections in the 3$\\pi $ survey have photometric accuracy limited at the bright end to $\\sim $ 12 millimags per epoch.",
"The current limits on the photometric precision are driven by our ability to model the 2D variations in the shape of the PSF.",
"The PSF shape in a given exposure changes on a variety of spatial scales due to 3 major effects: the atmosphere, the optics, and the detector.",
"To the extent that the PSF model is unable to follow the PSF variations, the PSF photometry is biased either high or low as the model PSF under or over predicts the size of the PSF.",
"The optics introduce image quality variations due to ripples in the focal surface.",
"These variations occur on spatial scales of 10 arcminutes and are relatively stable between exposures, introducing photometry errors of a few millimags.",
"The atmosphere introduces stochastic variations due to uncorrelated seeing across the focal plane, with a similar level of impact.",
"The detectors introduce PSF changes due to variable diffusion resulting from variations in the doping characteristics with spatial scales down to 10s of arcseconds.",
"With the density of PSF stars available in a typical PS1 exposure at high Galactic latitude, we are able to model the PSF variations on spatial scales of 3 arcmin, placing a limit on the accuracy of the PSF model on small scales.",
"See [82] for details.",
"Figure: Color image constructed from 16x16 binned g P1 g_{\\rm P1}, r P1 r_{\\rm P1}, and i P1 i_{\\rm P1}versions of the 3π\\pi stack images by Daniel Farrow.",
"The binned images were converted into HEALpix pixels and from this a color image was created using the software presented in .Figure: Left: FWHM cumulative probability distribution for all the observations in the 3π3\\pi Survey.",
"Right:Cumulative probability distribution for the sky brightness in mag/arc 2 ^2 for all the observations in the 3π3\\pi Survey.Table: 3π\\pi Steradian Survey Characteristics"
],
[
"3$\\pi $ Survey Characteristics",
"The $3\\pi $ Steradian Survey is comprised of 374,446 validated images taken between 2009-06-03 and 2015-02-25.",
"This includes some commissioning data taken before the start of the Mission and a modest number of images taken after the formal end of the survey, primarily in $z_{\\rm P1}$$y_{\\rm P1}$ during twilight to smooth out the spatial distribution."
],
[
"Summary of performance metrics",
"The image quality in the Pan-STARRS1 Surveys varies significantly, this is one reason for the forced photometry measurements.",
"The site, Haleakala Observatories (HO), is a well characterized site, and the lower limit to the seeing distribution is equivalent to Mauna Kea, but the distribution is broader and the median seeing as recorded by the HO Differential Image Motion Monitor (DIMM) is 0.84 arcsec, with a mode of 0.66 arcsec.",
"However, PS1 has a floor to its image quality, arising primarily from the wide field optics, so even the best images do not have a FWHM $< 06$ .",
"The image quality also depends on the filter, with the reddest bands displaying the best.",
"Figure REF shows the cumulative distribution of the image quality as characterized by a FWHM for each filter for the PS1 Surveys.",
"Haleakala is known for very low atmospheric scattering, that is why it is preferred to Mauna Kea for the Solar Telescope, the sky is even darker than Mauna Kea.",
"Solar astronomers assert this is due to the fact that the summit of Haleakala is primarily rock, whereas the summit of Mauna Kea is primarily cinder, and that summit of Mauna Kea is constantly surrounded by a halo of microscopic volcanic cinders.",
"Figure REF also shows the cumulative distribution of the sky brightness in each filter for the PS1 Surveys.",
"Table REF provides a summary of the characteristics of the 3$\\pi $ Survey."
],
[
"Simple Star/galaxy separation",
"For the DR1 and DR2 releases we recommend using a simple cut in (PSF - Kron) magnitude space to separate stars from galaxies.",
"Figure REF shows $i_{\\rm P1}$ v $i_{PSF}$ -$i_{Kron}$ for $i_{\\rm P1}$ -band data around the galactic pole region.",
"Unresolved objects form the tight sequence around PSF-Kron$ = 0.0$ .",
"A cut of (PSF-Kron)$<0.05$ does a reasonable job of selecting stars down to $i_{\\rm P1}$$\\sim 21$ .",
"Figure REF shows the star and galaxy counts resulting from such a cut.",
"Faintward of $i_{\\rm P1}$$\\sim 21$ the number of stars is over-predicted by a simple linear cut like this.",
"Also, at the brightest magnitudes, saturated stars tend to get classified as extended by this technique, resulting in a peak in the galaxy counts at $i_{\\rm P1}$$\\sim 13.5$ .",
"The use of the IPP flags or a more sophisticated non-linear cut can relieve this problem to some extent.",
"The distribution of stars and galaxies on the sky is visualised in Figure REF .",
"A more detailed discussion of this technique applied to PS1 data, including the behaviour of synthetic stars and galaxies, can be found in [31].",
"Figure: A demonstration of simple star galaxy separation using (PSF-Kron) magnitudes for a sample of i P1 i_{\\rm P1}-band chip detections around the galactic pole."
],
[
"Variation of $3\\pi $ Steradian Survey Depth",
"Although by the design of the survey each pixel on the sky notionally has 12 visits, in practice the coverage can be much more variable than this.",
"Figure REF shows the distribution of the number of exposures which contribute to each 16x16 binned pixel (4x4) over the whole of the 3$\\pi $ stacked survey.",
"The result of this is that the depth of the stacked survey varies significantly on quite small scales.",
"To estimate the depth, in the reduction of a skycell, artificial point-sources are added in magnitude bins and run through the process of being detected.",
"The numbers of these fake sources recovered and inserted as a function of magnitude is stored, for the stacked data, in the StackDetEffMeta table.",
"Maps of depth can be produced by finding at what magnitude a particular percentage of fake point-sources is recovered for each skycell, using linear interpolation between different magnitude bins when necessary.",
"To visualise the results across the whole survey, it is convenient to take the mean of these magnitudes for each skycell landing in a particular HEALpixhttp://healpix.sourceforge.net [40] pixel.",
"Figures REF and REF show the results of this procedure for recovery rates of 50% and 98%.",
"Not all the variation in limiting magnitude seen is due to the coverage.",
"For instance, in the galactic plane crowding can significantly reduce the number of recovered fakes.",
"It should also be noted that these limits are for point sources - [87] showed that the limits for extended sources are roughly $0.5$ mag brighter, although this is, of course, depends on the profile of the source.",
"Tests of how well these fake sources reproduce the true point-source recovery fractions, as well as a method of producing even higher resolution maps of depth will be presented in Farrow et al (in preparation).",
"Figure: Number counts of stars and galaxies from mean chip photometry (top) and stack photometry (bottom) for a region at the galactic pole.",
"A simple constant cut in (PSF-Kron) was used to separate stars from galaxies.Figure: The spatial distribution of (top) stars and (bottom) galaxies in the region |b|>60|b| > 60, selected using a simple cut in (PSF-Kron).Figure: The number of exposures contributing to each 4 binned pixel of the stacked 3π\\pi survey, for grizy P1 grizy_{\\rm P1}.Figure: left: The all-sky distribution of magnitude limits for 50% completeness on the 3π\\pi stacked data in the g P1 g_{\\rm P1}r P1 r_{\\rm P1}i P1 i_{\\rm P1}bands, based on the recovery of injected fake point-sources;right: 98% completeness for the same sample.Figure: As Figure but now showing the completeness for the z P1 z_{\\rm P1}and y P1 y_{\\rm P1}bands.Figure: i P1 i_{\\rm P1}v g P1 g_{\\rm P1}-i P1 i_{\\rm P1} color-magnitude diagrams for a variety of galactic star clusters and for the Local Group dwarf galaxy Leo I.",
"These data are taken from the meanObject table - see Section .Figure: Example r P1 r_{\\rm P1}v g P1 g_{\\rm P1}-r P1 r_{\\rm P1} color-magnitude diagram for a 1 degree square region around the Coma galaxy cluster (Abell 1656).",
"Galaxies are indicated in red, stars in blue.",
"This plot uses Kron magnitudes taken from the StackObjectThin table - see Section ."
],
[
"Examples of stellar and galactic photometry",
"In Figure REF we show examples of stellar color-magnitude diagrams for a variety of well-known galactic globular star clusters, as well as the Local Group dwarf galaxy Leo I.",
"These data were taken from the MeanObject table, and hence represent the mean of the measurements on individual exposures.",
"Despite the crowded nature of these fields, the stellar sequences are still quite tightly defined.",
"Figure REF demonstrates the use of Kron magnitudes for galaxies, taken from the stacked data.",
"Here we show the Coma galaxy cluster, with its prominent sequence at $g_{\\rm P1}$ -$r_{\\rm P1}$$\\sim 0.6$ .",
"Finally, in Figure REF we display the $g_{\\rm P1}$ -$r_{\\rm P1}$ v $r_{\\rm P1}$ -$i_{\\rm P1}$ , $r_{\\rm P1}$ -$i_{\\rm P1}$ v $i_{\\rm P1}$ -$z_{\\rm P1}$ and $i_{\\rm P1}$ -$z_{\\rm P1}$ v $y_{\\rm P1}$ -$z_{\\rm P1}$ color-color locii for stars and galaxies around the galactic pole region.",
"Figure: Color-Color plot for stars (left column) and galaxies (right column) with b >> 60."
],
[
"The PS1 Science Consortium Science Legacy",
"The Pan-STARRS1 Surveys of the PS1 Science Consortium have enabled science on topics ranging from Near Earth Objects to the most distant quasars.",
"New discoveries will be enabled by providing access to the community to the Pan-STARRS1 Archive at the Barbara A. Mikulski Archive for Space Telescopes.",
"While Pan-STARRS1 is not a space telescope, among other kinds of science these surveys will advance is a means to reprocess the astrometry of the Hubble Space Telescope Archive based on the Pan-STARRS1 extention of the Gaia Reference Frame [82].",
"Below we provide a brief summary of the legacy science from the PS1 Science Consortium as examples of the kind of science that can be done with the Pan-STARRS1 Surveys.",
"A primary goal of the PS1 mission was the Solar System Survey, designed to discover previously unknown Near-Earth Objects (NEOs) and provide additional orbital information on known bodies.",
"So far, PS1 has been responsible for discovering over 2900 NEOs, including potential targets for both robotic and manned space missions.",
"PS1 survey data has also led to the discovery of 129 comets and tens of thousands of new main-belt asteroids.",
"A major legacy of both the PS1SC Solar System Survey and the continuing NEO survey (PI Wainscoat) has been the reporting to date of $2.7\\times 10^7$ astrometric and photometric measurements of moving objects to the IAU Minor Planet Center.",
"A key science result from this treasure trove has been a determination of the luminosity distribution of NEOs down to diameters of just a few metres by [108] Additionally, the photometric properties of approximately a quarter of a million main-belt asteroids has been published by [127] Looking at rarer objects, PS1 has allowed characterisation of the Main-Belt Comet ([50]), and constrained the number of observable asteroid breakups ([29]).",
"Pre-discovery imaging of comet ISON out to Saturns' orbit demonstrated the value of deep solar system surveys in constraining the cometary activity of inbound long-period comets ([86]).",
"The rapid nightly processing of data to search for NEOs yielded objects with motions as slow as 0.05 deg/day, depending on seeing, and yielded numerous Centaurs, making Pan-STARRS1 one of the most prolific discovery telescopes for Centaurs.",
"The cadence of the observations was tuned for discovery of faster moving objects such as NEOs, making it not ideal for discovery of outer solar system objects.",
"Nevertheless, the Pan-STARRS1 dataset is rich in observations of the outer solar system.",
"[130] describe a search for distant solar system objects using the archival PS1 data for the period 2010 Feb 24 to 2015 July 31.",
"A total of 607 distant solar system objects were identified, 332 being new first observation discoveries, with an additional 24 significantly improving the astrometry of previously designated objects.",
"While a large number of new objects were found, no new extreme TNOs showing a clustering in their argument of perihelia were found, which, if present, could support the presence of a distant planetary sized perturber in the outer solar system.",
"[69] describe the discovery of five new Neptune Trojans in the PS1 data.",
"Four of these may be primordial, but the fifth is likely a recent capture.",
"[18] describe the discovery of a retrograde TNO in the PS1 dataset, and show that this object has similar orbital characteristics to other low semimajor axis high-inclination TNOs and Centaurs, hinting at a common orbital plane.",
"The scheduling and filter set of Pan-STARRS1 was designed from the beginning to enable the complete census and study of the ultracool dwarfs ($T_{\\rm eff}<2400$ K) of the Solar neighborhood, as well as white dwarfs.",
"The combination of proper motions and parallaxes allows for relaxed color selections to obtain complete, volume-limited samples [23].",
"Proper motions and colors also allow for the detection of substellar companions to main sequence stars, putting constraints on their age and mass [26], [24], [25]; peculiar brown dwarfs such as low-gravity, young brown dwarfs [74]; members of nearby stellar structures [2], [8], [39].",
"The depth of the stacked images provides the first colors in the visual and red part of the spectrum for a large number of ultra-cool stars and brown dwarfs (Best et al.",
"in prep).",
"The accuracy of both Pan-STARRS 1 astrometry and photometry also allowed the characterisation of Kepler target stars and the discovery of wide binary companions to planet hosts [27].",
"The panoptic Pan-STARRS1 $3\\pi $ survey provides a unique opportunity to map the distribution of stars (e.g.",
"[92]) in our own Milky Way and its outskirts and place it in a cosmological context.",
"In particular, it revealed the presence of multiple very faint Milky Way satellites (e.g.",
"[63]) and stellar streams likely stemming from the tidal disruption of globular clusters by our Galaxy [6].",
"Exploiting PS1 as a time-domain survey, the largest and deepest sample of RRLyrae candidate stars was identified [46], which provides for unprecedented 3-D mapping of the Milky Way's stellar halo out to $\\sim $ 120 kpc.",
"A major goal of Pan-STARRS1 Milky Way science was to map the interstellar dust in 3 dimensions using star colors.",
"For this purpose, PS1 has three main advantages over SDSS: it is deeper, goes one band redder, and covers more of the low-latitude sky.",
"The collaboration developed a method to infer the posterior on distance and reddening of each star, and then group stars into angular pixels and estimate the reddening as a function of distance in each [42].",
"They applied this method to PS1 photometry of 800 million stars (some with 2MASS photometry as well) and created a map with 2.4 million angular pixels and 31 distance bins, covering 3/4 of the sky [43].",
"They used a variant of the same technique to produce the largest catalog of molecular cloud distances [106].",
"More recently, PS1 photometry has served as the basis of a new parameterization of the reddening law, and study of its variation in 2 and 3 dimensions [107], [104].",
"The Pan-Planets survey was a dedicated exo-planet transit survey within the PS 1 project.",
"The survey covered an area of 42 sq degrees in the galactic disk for about 165 hr with the goal to constrain the occurrence rate of hot Jupiters around M dwarfs.",
"A combination of SED fiting, dust maps, and proper motion information allowed to identify more than 60 000 M dwarfs in the field.",
"This is the largest sample of low-mass stars observed in a transit survey.",
"With this large sample size, the Pan-Planets survey resulted in an occurrence rate of hot Jupiters of 0.11 (+0.37-0.02) % in case one of our candidates turns out to be a real detection.",
"If, however, none of our candidates turn out to be true planets, we are able to put an upper limit of 0.34% with a 95% confidence on the hot Jupiter occurrence rate of M dwarfs.",
"This is the best limit for the occurrence rate of hot Jupiters around M stars so far.",
"The major science goal for the Andromeda monitoring with Pan-STARRS1 (PAndromeda) originally was to identify a large number of gravitational microlensing events towards M31.",
"The final depths and image quality of the survey meant that early expectations of event rates were not met.",
"Nevertheless we found 6 events in a subfield of the first year of the survey [65] and the data are suitable for other PAndromeda stellar science.",
"Using only a subset of the PAndromeda data we identified and classified 1700 Cepheids and analysed their period luminosity/Wesenheit relations [60] and we found 300 eclipsing binaries in M31 [68] from which a handful of the brightest ones are suitable to derive an independent M31 distance.",
"We furthermore searched for rare variable stars which help to understand stellar structure (17 Beat Cepheids; [66]) and we also identified four new LBVs, i.e.",
"potential supernova progenitors [67].",
"The combination of the PAndromeda variability analysis and the HST-PHAT data [21] turned out to be very powerful.",
"In [61] we presented the largest M31 (HST) near infrared $J-$ band and $H-$ band sample at this time (371 Cepheids), studied their near infrared period luminosity relations and showed that the bright part of our sample is well suited for $H_0$ determination using M31 (having no metallicity issues compared to LMC/SMC) for the Cepheid distance ladder.",
"In summer 2016 we finished to completely (re)do the PAndromeda difference imaging for the full survey (time and area) by optimzing our pipeline for the data characteristics and by increasing the masking fraction.",
"We hence now have the final PAndromeda data products, i.e.",
"light curves for “all\" variable sources in M31.",
"The analysis of these light curves is ongoing, e.g., the final catalogue and analysis for 2700 PAndromeda Cepheids (750 with HST NIR photometry) will be made available to the community in 2017 (Kodric et al.",
"2017, in preparation).",
"The Medium Deep Survey (MDS, Section REF ) was designed for both deep fields and transient science with multi-colour temporal coverage of 70 square degrees in total.",
"In addition the multi-epoch aspect of the 3$\\pi $ survey also provided a new opportunity for time domain science.",
"The data for the MDS are not in DR1, but they will be released at a future date.",
"The cadence and filter coverage of the MDS fields was designed to both discover type Ia supernovae (SN Ia) before maximum light and to sample their lightcurves sufficiently for distance measurements.",
"We discovered $\\sim $ 3000 SNe Ia within $z \\approx 0.8$ , and obtained spectroscopic confirmation for $\\sim $ 500 SNe Ia.",
"[101], [109] published data from the first 1.5 years of the MDS, in which 146 SNe Ia were used to constrain the dark energy equation-of-state parameter, $w$ , to $\\sim $ 7% An analysis of the full sample is in preparation, and we have undertaken a first step in analyzing the photometric sample [55].",
"The MDS was also a rich source of exotic transients, with the discovery of high redshift superluminous supernovae.",
"These supernovae are 100 times brighter than normal core-collapse evens and peak at $M < -21$ mag.",
"They are UV bright and PS1 has discovered some of the most distant SLSNe, including one at $z = 1.566$ [5].",
"In a series of papers we studied their physical parameters [19], [75], [84], [78], and their host-galaxy environments [76], [77] and rates [85].",
"The MDS discovered two tidal disruption events, PS1-10jh [38] and PS1-11af [20], defined a class of fast-declining transients [30] and provided an extensive study of type II explosions[103].",
"In addition, the combination of the MDS and GALEX led to complementary UV data on some transients [11], [37].",
"The 3$\\pi $ survey provided discovery and critical lightcurve points for a number of interesting objects.",
"Low redshift super-luminous supernovae [100], [52], [93] were either discovered with the 3$\\pi $ survey or had lightcurve data at critical points.",
"The survey also provided the detection of a pre-supernova outburst of a type IIn explosion [34].",
"Initially we ran a transient search by catalogue matching with SDSS, which provided the discovery of slowly evolving blue transients at the centres of galaxies [64].",
"After the creation of an all sky stack, we progressed to routine difference imaging, leading to discovery of some super-luminous supernovae at lower redshift than in MDS [53], [94].",
"Pan-STARRS1 is now the world leading disoverer of low-redshift supernovae, according to the IAU statisticshttps://wis-tns.weizmann.ac.il/stats-maps.",
"The $3\\pi $ Steradian Survey is being used for a citizen-scientist enabled nearby galaxy survey based on the optical imaging from Pan-STARRS1, but also incorporating multi-wavelength data from the ultraviolet and infrared regimes (GALEX, WISE).",
"This project is called the PS1 Optical Galaxy Survey or POGShttp://pogs.theskynet.org [128].",
"Distributed computing resources contributed by tens of thousands of volunteers allow comprehensive pixel-by-pixel spectral energy distribution (SED) fitting for $>100,000$ galaxies, which in turn provides key physical parameters such as the local stellar mass surface density, star formation rate (SFR), and dust attenuation.",
"Sufficiently nearby galaxies are being processed using complete UV-optical-IR SED coverage, whereas distant (but more numerous) galaxies are analyzed with optical only data due to the resolution of ancillary observations.",
"With pixel SED fitting output, the POGS pipeline constrains parametric models of galaxy structure and measures non-parametric morphology indicators in a more meaningful way than ordinarily achieved, by operating on images of estimated physical parameters.",
"The depth, sky-coverage and time-domain capabilities of PS1 have also been leveraged to conduct various focused studies of galaxy properties, including host galaxy properties of variability selected AGN [45] the structure of outer galactic disks [133] and the influence of group environment on the SFR-stellar mass relation [70].",
"Early data from the 3$\\pi $ survey have been used to quantify galaxy angular clustering [31], confirm and determine redshifts for Planck cluster candidates [72] and detect a large void in front of the CMB Cold Spot [116].",
"Preliminary data from the MDS has been used to find galaxy groups and clusters and investigate the dependence of star formation on environment [54], [70].",
"One of the key science goals from the inception of PS1 was the discovery of quasars at the highest redshifts (z$\\sim $ 6), and to push the redshift barrier of z=6.4 (imposed at the time by the choice of the SDSS filters).",
"These high—redshift quasars are thought to be one of the most massive structures that exist in the first Gyr of the Universe.",
"The quasar host galaxies harbour accreting supermassive black holes, which allow detailed studies of key quasar properties, as well as the impact of the quasars on the surrounding intergalactic medium.",
"PS1 has now become the survey in which most of the z$\\sim $ 6 quasars have been discovered, and a number of papers have resulted from this effort (e.g., first papers by [90] and [3]).",
"The highest-redshift quasars found by PS1 are discussed in [126], and the most complete catalog of the high-z quasar population is published in [4].",
"The latter discusses the properties of 77 newly detected PS1 quasars (out of a total of 124 known) at z$>$ 5.6."
],
[
"Conclusions",
"We have presented an introduction to and overview of the Pan-STARRS1 surveys in preparation for the first data release (DR1) on 2016 December 19.",
"Under the auspices of the Pan-STARRS1 Science Consortium, PS1 observed the entire sky north of Dec$=-30$ deg (the 3$\\pi $ survey) in $grizy_{\\rm P1}$ (to $r_{\\rm P1}$$\\simeq 23.3$ ), additional ecliptic fields in $w_{\\rm P1}$ for a Solar System survey (to $w_{\\rm P1}$$\\simeq 22.5$ per visit CHECK!",
"), several extragalactic deep fields in $grizy_{\\rm P1}$ (to $r_{\\rm P1}\\simeq 23$ per visit), and some specialized fields (M31, transit survey fields).",
"These data have been calibrated to $\\sim 12$ mmag internal photometric precision and $\\sim 20$ mas internal astrometric precision.",
"DR1 consists of the image stacks and associated catalogs for the 3$\\pi $ survey, distributed through the MAST system at STScI.",
"The second data release (DR2, expected May 2017) will distribute the 3$\\pi $ time-domain data.",
"Future data releases will release the Medium Deep data, difference image data, and photometric redshifts from the Photo-Classification Server.",
"These data have already produced a variety of science, on subjects as varied as asteroids, Milky Way structure, galaxy formation, supernovae and cosmology, but we hope they will find further utility in achieving science goals beyond the scope of the Pan-STARRS1 Science Consortium.",
"The PS1 data are currently being used to provide targeting information for the SDSS-IV Time Domain Spectroscopic Survey [91], and to provide high resolution, deep, multi-colour reference images for transients in many transient surveys [111], [121], [49], [9].",
"Besides additional science investigations based directly on these data, the data have great legacy value in providing a high-quality network of calibration sources across the sky.",
"The data are already being used to calibrate the Hyper Suprime-Cam Survey [119], [17] and will also be useful in cross-checking the calibration of the northern areas imaged by the Dark Energy Survey and the Large Synoptic Survey Telescope.",
"But in addition to large-scale surveys, individual programmes with relatively small observing fields north of Dec$=-30$ deg will all have good PS1 calibration sources in the field, observed through exactly the same column as the target sources, allowing simple relative calibration for astrometry and photometry.",
"Finally, the legacy of the PS1 surveys extends beyond just the data.",
"The experience and lessons learned from designing and executing the PS1 surveys are assisting in the development of future survey projects.",
"Some algorithms and code from PS1 are being used in the development of LSST, and students and postdocs who built careers starting with PS1 are applying their experience to new and larger surveys.",
"We hope the PS1 surveys will be useful to the astronomical community for many years to come.",
"The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg, and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration Grants No.s NNX08AR22G, NNX12AR65G, and NNX14AM74G, the National Science Foundation under Grant No.",
"AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory and the Gordon and Betty Moore foundation.",
"This work has made use of data from the European Space Agency (ESA) mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium).",
"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.",
"SJS acknowledges funding from (FP7/2007-2013)/ERC Grant agreement n$^{\\rm o}$ [291222] and STFC grants ST/I001123/1 and ST/L000709/1.",
"DT acknowledges funding from the National Science Foundation under Grant No.",
"AST-1412596 for the POGS program.",
"The authors acknowledge the hard work and dedication of the University of Hawaii Institute for Astronomy staff who worked on Pan-STARRS1: Rick Anderson, Robert Calder, Greg Ching, Richard Harris, Haydn Huntley, Brooke Gibson, Jill Kajikara-Kent, Sifan Kahale, Chris Kaukali, Aaron Lee, Tom Melshiemer, Louis Robertson, Donna Roher, Freddie Ratuta, Gavin Seo, Diane Tokumura, Robin Uyeshiro, and Gail Yamada."
]
] | 1612.05560 |
[
[
"Implications of galaxy buildup for putative IMF variations in massive\n galaxies"
],
[
"Abstract Recent observational evidence for initial mass function (IMF) variations in massive quiescent galaxies at $z = 0$ challenges the long-established paradigm of a universal IMF.",
"While a few theoretical models relate the IMF to birth cloud conditions, the physical driver underlying these putative IMF variations is still largely unclear.",
"Here we use post-processing analysis of the Illustris cosmological hydrodynamical simulation to investigate possible physical origins of IMF variability with galactic properties.",
"We do so by tagging stellar particles in the simulation (each representing a stellar population of $\\approx10^{6}~\\mathrm{M}_{\\odot}$) with individual IMFs that depend on various physical conditions, such as velocity dispersion, metallicity, or SFR, at the time and place the stars are formed.",
"We then follow the assembly of these populations throughout cosmic time, and reconstruct the overall IMF of each $z=0$ galaxy from the many distinct IMFs it is comprised of.",
"Our main result is that applying the observed relations between IMF and galactic properties to the conditions at the star-formation sites does not result in strong enough IMF variations between $z = 0$ galaxies.",
"Steeper physical IMF relations are required for reproducing the observed IMF trends, and some stellar populations must form with more extreme IMFs than those observed.",
"The origin of this result is the hierarchical nature of massive galaxy assembly, and it has implications for the reliability of the strong observed trends, for the ability of cosmological simulations to capture certain physical conditions in galaxies, and for theories of star-formation aiming to explain the physical origin of a variable IMF."
],
[
"Introduction",
"The stellar initial mass function (IMF) has long been thought a universal feature of star formation.",
"Although observational support for a universal IMF has been mounting for more than half a century, the physical theory behind the IMF still remains an unsolved problem in star formation physics [2].",
"The elusive origin of the IMF stems from deficient knowledge of the physics governing how molecular clouds collapse and fragment to form protostars and the subsequent accretion of gas onto these protostars [65].",
"Observational measurements of the IMF have also proved to be particularly challenging, both for resolved stellar populations where individual stars can be counted, and for distant [10], unresolved stellar populations where the IMF must be inferred more indirectly [89].",
"Recent observations suggesting a variable IMF, as measured within massive elliptical galaxies, serve to further compound the elusive origin of the IMF [6], [20], [84], [28].",
"While understanding the shape of the IMF would certainly provide insight into the physical processes that control star formation, the IMF also has vast implications for the study of galaxy populations as the IMF influences nearly all observable galaxy properties including stellar mass, luminosity, metal content, and star formation history.",
"Observations of stellar populations within and near the Milky Way gave rise to the conception of a universal IMF.",
"The first determination of the IMF was by [74] who, using field stars in the solar neighborhood, found the IMF to follow a unimodal power law with a slope of $x = 2.35$ .",
"Since Salpeter's initial measurement the IMF has been extensively measured in nearby, resolved stellar populations.",
"These measurements have revealed the IMF to turn over at lower masses, following a [11] log-normal or a [44] segmented power law IMF below $\\sim $ 1 M$_{\\odot }$ and the original Salpeter IMF at stellar masses greater than $\\sim $ 1 M$_{\\odot }$ .",
"The shape of the IMF has been measured to be largely consistent across a variety of stellar populations within and near the Milky Way, including young clusters [38], [102], open clusters [55], [8], globular clusters [58], [45], the Large Magellanic Cloud (LMC) [42], [21] and Small Magellanic Cloud (SMC) [78], [73], and M31 and M32 [107].",
"For a detailed discussion of the nuances of local IMF observations we direct the reader to [10] and [2].",
"Surprisingly, observations over the past few years of more distant stellar populations suggest deviations from the universal IMF inferred in the Local Group.",
"Several independent methods have been implemented to study the IMF in unresolved stellar populations, including: (1) dynamical studies where stellar population synthesis (SPS) mass-to-light ratios are compared to dynamically derived mass-to-light ratios [25], [6], [7], [20], [15], [53], (2) absorption line studies where spectral features either sensitive or anti-sensitive to dwarf stars provide the constraints [9], [93], [26], [48], [84], [56], and (3) lensing studies where SPS masses are compared to masses derived from gravitational lensing [27], [28], [1], [91], [90], [70], [50].",
"These measurements predominately infer that the IMF of nearby early-type galaxies (ETGs) becomes more bottom-heavy, i.e.",
"having increasingly numerous low-mass stars with respect to high-mass stars, at higher values of galaxy properties such as velocity dispersion ($\\sigma $ ), metallicity ([M/H]), and metal abundance ratio ([$\\alpha $ /Fe]).",
"For example, [15] find IMF variations with velocity dispersion for a sample of compact ETGs by using mass-to-light ratio as a proxy for the fraction of low-mass stars, as these systems are believed to be stellar dominated at their centers.",
"They find low velocity dispersion galaxies ($\\sigma $ $\\sim $ 100 km s$^{-1}$ ) to be best described by a Milky Way-like IMF, galaxies with intermediate velocity dispersions ($\\sigma $ $\\sim $ 160 km s$^{-1}$ ) best fit by a Salpeter IMF, and galaxies with $\\sigma $ $\\sim $ 250 - 300 km s$^{-1}$ best described by an IMF even more bottom-heavy than the Salpeter IMF.",
"Similar IMF trends have also been observed to scale with metallicity.",
"For example, using a sample of ETGs from the CALIFA survey, [53] find the most metal-poor ETGs in their sample ([M/H] $\\sim $ -0.2) to be best described by an IMF slope of $x$ $\\sim $ 2 and the most metal-rich ETGs ([M/H] $\\sim $ 0.2) best described by an IMF slope of $x$ $\\sim $ 2.9.",
"IMF variations have also been observed within galaxies, highlighting the complications of systematics like aperture radius when comparing IMF measurements across studies , [49], (however, see [94] who find a constant IMF at all radii for two ETGs with $\\sigma $ = 410 and 260 km s$^{-1}$ ).",
"In particular, using deep spectroscopic data taken at various fractions of the effective radius ($R_{e}$ ), [52] find significant radial IMF trends for the highest velocity dispersion galaxies in their sample ($\\sigma $ $\\sim $ 300 km s$^{-1}$ ), starting with an IMF slope of $x$ $\\sim $ 3 at galaxy centers, down to an IMF slope of $x$ $\\sim $ 1.9 at $r = 0.7 ~\\mathrm {R}_{e}$ .",
"However, for the lower velocity dispersion ETG in their sample ($\\sigma $ $\\sim $ 100 km s$^{-1}$ ) the IMF is found to be constant with galactocentric distance.",
"These trends may be understood in the context of the `minor mergers' scenario [61], [66], [72], according to which the most massive ETGs accrete a large number of small galaxies in particular in their outer regions.",
"Similar IMF variations have also been recently reported for high redshift ETGs.",
"Comparing dynamical to SSP masses, [75], [76] find a Salpeter IMF, rather than the `universal' Chabrier, for their sample of massive (> 10$^{11}$ M$_{\\odot }$ ) ETGs at $z \\sim 0.8$ .",
"For a sample of ETGs at $z \\sim 1.4$ with both dynamical and photometric mass estimates, [33] report an IMF-$\\sigma $ relation consistent with the trends observed at $z \\sim 0$ .",
"They further posit that the IMF of dense (> 2500 M$_{\\odot }$ pc$^{-2}$ ) ETGs is independent of redshift over the past $\\sim $ 9 Gyr.",
"Additionally, based on the dwarf-sensitive TiO$_{2}$ feature, [54] find similar IMF variations for massive ETGs from 0.9 < $z$ < 1.5 that are consistent with a constant IMF over the past $\\sim $ 8 Gyr.",
"Though the evidence for IMF variations is mounting, a consensus has yet to be reached, as several studies report discrepant results.",
"For example, [20] find for their sample of ETGs the strongest IMF correlation to be with [Mg/Fe], with weaker IMF-$\\sigma $ and IMF-[M/H] correlations.",
"[47], on the other hand, report negligible IMF correlation with [Mg/Fe].",
"Additionally, based on studies of the low-mass X-ray binaries (LMXB) of ETGs, whose number is expected to scale with the IMF, [68] argue for an invariant IMF reporting a LMXB population per mass that is constant across a range of galaxy velocity dispersions.",
"Studies based on lensed galaxies have also yielded discrepant results.",
"[80] found that for two strong lens ETGs in the SINFONI Nearby Elliptical Lens Locator Survey (SNELLS) with $\\sigma $ $\\sim $ 300 km s$^{-1}$ a bottom-heavy IMF is ruled out in favor of a Kroupa IMF, but find their 1.14 Na I $\\mu $ m index strengths to suggest they have bottom-heavy IMFs [79].",
"Recently [64] compared lensing, dynamical, and SPS techniques for inferring the IMF of these SNELLS galaxies, finding that the SPS stellar mass-to-light ratios exceed the total lens mass-to-light ratio, and that there is even a significant discrepancy between the lensing and dynamical masses.",
"[64] discusses several possibilities for the origin of these tensions, but this study suggests there could be systematic errors in at least one of the techniques used to probe the IMF of ETGs.",
"These discrepancies highlight the importance of understanding the uncertainties in inferring IMF variations.",
"Emphasizing the difficulties of inferring the IMF from integrated light, [89] find that the degeneracy between a bottom-heavy IMF and decreasing AGB (asymptotic giant branch) strength is only confidently broken for old, metal-rich galaxies with a combination of accurate spectra and photometric observations at the .02 mag level.",
"[13] examines the influence of measurement error and selection bias on IMF variations using a sample of galaxies from the ATLAS-3D project.",
"They find that $\\sim $ 30% gaussian errors on kinematic measurements of mass-to-light ratios lead to similar IMF variations as reported in [6], emphasizing the importance of correctly modeling measurement errors.",
"[13] also find that galaxy selection can significantly influence the inferred IMF trend.",
"Placing a cut on star-formation (as most studies reporting IMF variations do) removes low velocity dispersion galaxies with IMFs comparable to quiescent, high velocity dispersion galaxies.",
"Additionally, excluding galaxies with kinematic masses below the ATLAS-3D mass completeness threshold (2$\\times 10^{10}$ M$_{\\odot }$ ) removes the IMF trend with velocity dispersion.",
"If however proven robust, a variable IMF will have significant implications for our current understanding of galaxy formation and evolution: an understanding that has largely been developed under the assumption of a universal IMF.",
"In particular, a variable IMF will affect the derived properties of galaxy populations.",
"For example, [14] quantifies the effect of the metallicity dependent IMF relation in [53] on the derived quantities of nearly $2\\times 10^{5}$ SDSS galaxies.",
"Inferred star formation rates increase by up to two orders of magnitude and stellar mass densities increase by a factor of 2.3 compared to what is inferred with the Chabrier IMF.",
"Adopting an IMF relation dependent on velocity dispersion, [57] find similar effects on the derived properties of SDSS galaxies, with the shape of the high-mass end of the galaxy stellar mass function shifting from the familiar exponential (using a Chabrier IMF) to a power-law.",
"From a more theoretical standpoint, [95] use a numerical chemical evolution code to quantify the effect of the IMF shape on metal yields of galaxies.",
"Exploring both the upper mass cut off and the slope of the IMF, they find that the metal yield can vary by up to a full order of magnitude.",
"Against this observational background, several star-formation models have been recently developed that predict IMF variations.",
"Some models find that the larger density fluctuations associated with higher Mach numbers in star-forming disks cause the low-mass turnover of the pre-stellar core mass function (CMF) to shift to lower masses, implying a more bottom-heavy IMF (e.g.",
"[40], [12], [37]; but see [4]).",
"These models are able to reconcile the universal IMF found across a range of Milky Way stellar populations with a bottom-heavy IMF in more extreme star-formation environments such as starbursts.",
"Also potentially able to account for differences between such environments is the [46] derivation of the low-mass turnover as a function of fundamental constants and a weak dependence on interstellar pressure and metallicity.",
"Another theory is IGIMF (Integrated Galaxy-wide stellar Initial Mass Function), which formulates the shape of the overall IMF of a galaxy based on the properties of its individual molecular clouds, which are in turn controlled by the total galaxy SFR [101].",
"In this theory, high SFR environments, such as starbursts, are predicted to undergo star-formation which follows a top-heavy IMF, so the excess mass inferred for massive elliptical galaxies is predicted to be in part due to stellar remnants.",
"Table: Simulation parametersSemi-analytical and numerical models have also recently been employed to study IMF variations.",
"The impact of IMF variations on the chemical abundances of galaxies has been quantified using both SAG [34] and GAEA [31].",
"[30] uses MORGANA to investigate the effect of IMF variations on both stellar mass and star-formation rate, reporting that top-heavy IMF in high star-formation rate environments produces the largest deviations from properties derived under a standard IMF.",
"A variable IMF has also been explored in the context of simulations of individual galaxies, finding that chemical evolution is dependent on the assumed IMF [3], [29].",
"Most recently, [81] developed a simple model of merger-driven galaxy evolution to predict the evolution of IMF trends with mass and velocity dispersion from $z=2$ to $z=0$ , predicting that the IMF slope of a galaxy is steeper at earlier times.",
"In this study, we use the Illustris cosmological hydrodynamical simulation to connect the physical conditions in which stars form to the global properties of galaxies at $z = 0$ .",
"We construct the IMF for a sample of $z=0$ Illustris galaxies by using prescribed IMF relations applied to the birth properties of the individual stellar populations that comprise each galaxy.",
"By attempting to reproduce observed relations between the overall IMF of a galaxy (or of its central parts) and its $z = 0$ properties, we are able to provide constrains on relations between IMF and physical conditions at the time of stellar birth.",
"This paper is organized as follows.",
"In Section we describe the Illustris simulation, our galaxy selection, and the method for constructing the IMF.",
"In Section we present the primary results: global IMF trends with galactic properties at $z=0$ .",
"In Section we explore additional constrains: radial trends, scatter, and redshift evolution.",
"In Section , we test the robustness of our results to both resolution and variations in the simulation.",
"Finally, in Section we discuss our findings, and summarize in Section .",
"To investigate possible physical origins of the observed IMF variations we primarily use cosmological simulations from the Illustris Project [99], [98], [35], and in particular the highest resolution hydrodynamical simulation in the suite, the Illustris simulation.",
"These simulations evolve down to $z=0$ a volume large enough to contain statistically significant galaxy populations, and incorporate crucial physics resulting in many realistic galaxy properties.",
"Illustris has been used to study a diverse range of topics in galaxy evolution including, in particular, topics directly relevant to this work, such as the formation of massive, compact ETGs [104], [105] and the stellar mass assembly of galaxies [72].",
"The Illustris simulation treats hydrodynamical calculations using the moving-mesh code AREPO [86], which has proven advantages over both adaptive mesh refinement (AMR) and smoothed particle hydrodynamics (SPH) techniques [77], [43], [97].",
"Gravitational forces are computed using a Tree-PM technique [106] that calculates short-range forces using the tree algorithm and long-range forces using the particle mesh (PM) method.",
"A $\\Lambda $ CDM cosmology with $\\Omega _{m} = 0.2726$ , $\\Omega _{\\Lambda } = 0.7274$ , $\\Omega _{b} = 0.0456$ , and $h = 0.704$ from WMAP9 [39] is adopted for all simulations used in this study.",
"The galaxy formation physics implemented in Illustris includes radiative cooling, star formation and evolution, including chemical enrichment, black hole seeding and accretion, stellar feedback in the forms of ISM pressure and galactic winds, as well as AGN feedback.",
"Since our study focuses on the stellar populations in Illustris, below we provide a short description of the stellar formation and evolution model.",
"For an in depth discussion of all of the physical models included in Illustris the reader is referred to [96].",
"Stellar particles form according to the Kennicutt-Schmidt relation [41] from dense ISM gas with a time scale of 2.2 Gyr at the density threshold of $n\\approx 0.13$ cm$^{-3}$ .",
"This gas is pressurized following an effective equation of state for a two-phase medium [87].",
"Each stellar particle represents a simple stellar population (SSP) consisting of stars formed at the same time with the same metallicity.",
"Stars in each stellar particle SSP return mass and metals to surrounding gas cells following their expected lifetimes with post-main sequence evolution occurring instantaneously, where low mass stars return mass through AGB winds and more massive stars return most of their mass to the ISM via supernovae.",
"Hence, the mass loss and metal production of each stellar particle as a function of its age are calculated using tables in accordance with the particle's initial mass, metallicity, and assumed IMF.",
"The SSP of each stellar particle in Illustris is assumed to have a stellar mass distribution described by a Chabrier IMF.",
"The IMF affects the mass and metal return via the evolution of high mass stars, as well as the energy available for galactic wind feedback, which however has a pre-factor that is a tunable parameter of the model.",
"In Section we explore simulations not included in the Illustris suite that are evolved with different IMFs, such as the Salpeter IMF as well as a variable IMF.",
"In addition, we study simulations that adopt different degrees of feedback.",
"In post-processing, structure is identified in each snapshot first using the FoF (friends-of-friends) algorithm [23] and then an updated version of the SUBFIND algorithm [88], [24].",
"The FoF algorithm identifies dark matter halos using a linking length of one-fifth the mean separation between dark matter particles, with the baryonic particles (gas, stars, and black holes) assigned to the FoF group of their closest dark matter particle if it is close enough by the same separation criterion.",
"The SUBFIND algorithm identifies gravitationally-bound substructure (subhalos) within each parent FoF group.",
"The dark matter and baryonic components of each subhalo constitute what we refer to as a galaxy.",
"The publicly released suite of hydrodynamical Illustris simulationshttp://www.illustris-project.org/data/ [63] includes three runs of the same volume at increasing resolution levels: Illustris-3,-2, and -1.",
"Illustris-1 includes 1820$^{3}$ dark matter particles with masses $m_{\\rm DM}$ = 6.26$\\times 10^{6}$ M$_{\\odot }$ and $\\sim $ 1820$^{3}$ baryonic resolution elements with an average mass of $\\overline{m_{b}}$ = 1.26$\\times 10^{6}$ M$_{\\odot }$ , evolved within a (106.5 Mpc)$^{3}$ cube.",
"More details about Illustris-1 and the other simulations used in this study are given in Table REF ."
],
[
"Galaxy selection",
"In selecting galaxies in Illustris, we aim to mimic the typical properties of the ETGs examined in the observational IMF studies.",
"We therefore first select galaxies with stellar masses greater than 10$^{10}$ M$_{\\odot }$ and specific star formation rates < 10$^{-11}$ yr$^{-1}$ .",
"The total stellar mass of each galaxy is calculated as the sum of the stellar particles assigned to it by SUBFIND and the specific star formation rate is calculated as the sum of the instantaneous SFRs of its gas cells, divided by the stellar mass.",
"Both the stellar mass and sSFR are calculated within two times the stellar half-mass radius of each galaxy.",
"For Illustris-1 at $z = 0$ these selection criteria result in a sample of 1160 galaxies.",
"The mean stellar mass of this sample is $M_{*}$ = 10$^{10.88}$ M$_{\\odot }$ and the mean specific star formation rate is sSFR = 2.45$\\times 10^{-12}$ yr$^{-1}$ .",
"Additionally, as done in [84], we limit our sample to galaxies with stellar velocity dispersions greater than 150 km s$^{-1}$ .The stellar velocity dispersion, $\\sigma _{*}$ , for each galaxy is calculated as the one-dimensional, $r$ -band luminosity-weighted velocity dispersion of the stellar particles falling within one-half the projectedThe projected half-mass radius $R^{p}_{1/2}$ is calculated as the radius containing half the total stellar mass of the galaxy, including stellar particles that fall within this projected radius as viewed from the $z$ -direction.",
"stellar half-mass radius (0.5$R^{p}_{1/2}$ ).",
"The velocity dispersion criterion reduces our sample from 1160 to 371 galaxies, with $\\overline{M_{*}}$ = 10$^{11.4}$ M$_{\\odot }$ , $\\overline{\\mathrm {sSFR}}$ = 1.22$\\times 10^{-12}$ yr$^{-1}$ , and $\\overline{\\sigma _{*}}$ = 198 km s$^{-1}$ ."
],
[
"IMF construction",
"Since the IMF is set at the time of stellar birth, we probe the birth conditions of the stellar particles belonging to our selection of $z = 0$ galaxies.",
"To do this we trace each of the 196335880 stellar particles belonging to the 371 $z = 0$ selected galaxies back to the snapshot in which it first appears and compute several physical quantities (discussed in Section ) that represent its birth conditions.",
"Following [84], [91] we assign an IMF mismatch parameter, $\\alpha _{\\rm IMF}$ , to each stellar particle: $ \\alpha _{\\rm IMF} = \\frac{(M_{*}/L)}{(M_{*}/L)_{\\rm Salp}},$ where $(M_{*}/L)_{\\rm Salp}$ is the mass-to-light ratio expected assuming a Salpeter IMF and $(M_{*}/L)$ is the actual mass-to-light ratio assumed for the particle.",
"For stellar populations with an IMF more `bottom-heavy' compared to the Salpeter IMF, $\\alpha _{\\rm IMF}$ > 1, while stellar populations `bottom-light' compared to the Salpeter IMF have $\\alpha _{\\rm IMF}$ < 1.",
"A Chabrier IMF is described by $\\alpha _{\\rm IMF}$ = 0.6.",
"To construct the overall $\\alpha _{\\rm IMF}$ of each galaxy described in Section REF we mass-weight $\\alpha _{\\rm IMF}$$^{-1}$ using the birth mass of all (or the innermost subset of) the stellar particles comprising a galaxy.",
"This is equivalent to summing up the light, $L/L_{x=2.35}$ , assigned to the stellar particles belonging to each galaxy.",
"Beginning our exploration, we are inspired by the observations when assigning an $\\alpha _{\\rm IMF}$ to the stellar particles.",
"For example, we use the $\\alpha _{\\rm IMF}$ -$\\sigma _{*}$ relation presented in [84] as an input relation applied to the local velocity dispersions of the stellar particles.",
"In general though, we have the freedom to construct input $\\alpha _{\\rm IMF}$ relations that scale with various physical quantities at the star-formation sites.",
"In Section we present the Illustris log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relations constructed using five different physical quantities associated with the birth conditions of each stellar particle: global stellar velocity dispersion ($\\sigma _{*}$ ), local dark matter velocity dispersion ($\\sigma _{\\mathrm {birth}}$ ), local metallicity ([M/H]), global star-forming gas velocity dispersion ($\\sigma _{\\mathrm {gas}}$ ), and global star-formation rate (SFR)."
],
[
"Global velocity dispersion",
"We begin our investigation by first constructing the overall log($\\alpha _{\\rm IMF}$ ) of each galaxy in the Illustris sample based on the global stellar velocity dispersion, $\\sigma _{*}$ .",
"We trace each stellar particle belonging to a $z=0$ galaxy back to the progenitor galaxy it was formed in, and compute that galaxy's stellar velocity dispersion in exactly the same way $\\sigma _{*}$ was computed for the $z=0$ galaxy sample.",
"With the global $\\sigma _{*}$ associated with each star particle, we construct the overall IMF mismatch parameter (log($\\alpha _{\\rm IMF}$ )) according to the prescription outlined in Section REF .",
"As a starting point, we apply the observed relation presented in [84], $ \\mathrm {log}(\\alpha _{\\rm IMF}) = (1.05\\pm .2) \\mathrm {log}(\\sigma _{*}) - (2.5\\pm .4),$ which is derived over a range of SDSS ETGs ($z \\le 0.05$ ) with velocity dispersions between $\\sigma _{*}$ = 150 km s$^{-1}$ and $\\sigma _{*}$ = 310 km s$^{-1}$ , by comparing spectral lines sensitive to low-mass stars to the corresponding index strengths in the [18] SSP models.",
"The mismatch value corresponding to a Chabrier IMF, log($\\alpha _{\\rm IMF}$ ) = -0.22, is adopted for all stellar particles with $\\sigma _{*}$ less than 150 km s$^{-1}$ .",
"Figure: Main panel: IMF mismatch parameter, log(α IMF \\alpha _{\\rm IMF}), as a function of z=0z = 0 global stellar velocity dispersion, σ * \\sigma _{*}.",
"Inset: the log(α IMF \\alpha _{\\rm IMF})-σ * \\sigma _{*} relations used as input physical laws that produce the curves in the main panel.",
"The solid black curve shows the observed relation from Spiniello et al.",
"2014 and the black dashed line shows an extrapolation of the relation.",
"The resulting z=0z=0 relations are always shallower than the input relations.",
"Red: input relation as in Spiniello et al.",
"2014, blue: an input relation that is 3.5×\\times steeper.Figure: Velocity dispersion distributions of the stars in four Illustris galaxies both today (right panels) and at birth, namely in the first snapshot they appear in (left panels).",
"Each pair of plots represents one galaxy, increasing in z=0z=0 stellar mass from top-left to bottom-right.",
"The grey curves correspond to all stars belonging to the z=0z=0 galaxy, while the light (dark) blue curve corresponds to just the stellar particles within one (half) projected stellar half-mass radius R 1/2 p R^{p}_{1/2} (.5R 1/2 p R^{p}_{1/2}) from the center of the z=0z=0 galaxy.",
"Evidently, the velocity dispersions of stars can change dramatically between their birth and z=0z=0, especially in massive galaxies, which present very broad distributions of σ birth \\sigma _{\\rm birth}.The red curve in Figure REF shows the resulting overall log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation for the $z=0$ Illustris galaxy sample.",
"This relation was constructed based on the innermost parts of each galaxy, using only the stellar particles residing within 0.5$R^{p}_{1/2}$ from the center of the galaxy, to approximately match [84].",
"The red curve in the inset shows the input relation used to construct log($\\alpha _{\\rm IMF}$ ), which is the same as the observed [84] relation.",
"The latter is repeated in the main panel as the black curve, to guide the eye.",
"As evident in Figure REF , the [84] relation applied at the time of stellar birth is not conserved through the assembly history of massive galaxies.",
"The overall log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation is $\\sim $ 2.5$\\times $ too shallow compared to the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation.",
"While the observed overall log($\\alpha _{\\rm IMF}$ ) is reproduced for the lowest velocity dispersion galaxies in the sample, the constructed log($\\alpha _{\\rm IMF}$ ) of the higher velocity dispersion galaxies becomes increasingly too low.",
"To reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation, we construct input relations steeper than Equation REF .",
"By minimizing the residuals between the resulting Illustris output relation and the [84] relation, we determine input relation that produces the best-fit.",
"The blue curves in Figure REF show that with an input relation 3.5$\\times $ steeper than the observed relation (shown in the inset), the resulting overall log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation (shown in the main panel) is able to reproduce both the slope and normalization of the observed trend within the reported errors.",
"This result suggests that the observed IMF variations with $z=0$ galactic velocity dispersion cannot be a correlation that exists in the galaxies in which the stars are actually born and where the IMF is set.",
"This observed relation is hence emergent rather than fundamental.",
"It is a manifestation of the complexities of $\\Lambda $ CDM galaxy formation through hierarchical assembly, where massive galaxies are composed of stellar populations that form in a plethora of progenitor galaxies with varied and evolving properties such as velocity dispersion.",
"Since galaxies with high $\\sigma _{*}$ contain stars that were formed inside galaxies with low $\\sigma _{*}$ and hence have relatively bottom-light IMFs, it is necessary that stars forming in-situ in galaxies with high $\\sigma _{*}$ have extremely bottom-heavy IMFs in order to combine together in their $z=0$ host galaxies and produce the observed relation.",
"In this section, we have used the same quantity on the horizontal axes of both the inset and the main panel, namely the assumed physical driver was the same as the independent variable of the $z=0$ relation.",
"We have shown that the input relation is not preserved through galaxy assembly.",
"For the remainder of this paper, we focus on connecting the global properties of $z=0$ galaxies to physical properties that are more closely associated with star-formation.",
"We examine the local velocity dispersion and metallicity of the stellar particles, as well global quantities, star-formation rate and gas velocity dispersion, which have been suggested as drivers of IMF variations on the star-formation scale."
],
[
"Local velocity dispersion",
"Still motivated by the observed IMF trends with velocity dispersion, but aiming for a more physically relevant IMF driver, we construct the overall log($\\alpha _{\\rm IMF}$ ) of each Illustris galaxy based on the local velocity dispersion around each individual stellar particle at the time of formation.",
"In particular, we use the one-dimensional velocity dispersion of the 64$\\pm $ 1 dark matter particles nearest to each stellar particle in the earliest snapshot where it exists.",
"This quantity, denoted as $\\sigma _{\\rm birth}$ , probes the local gravitational potential at the time of stellar birth.",
"Figure REF includes four example velocity dispersion distributions, comparing the local velocity dispersions the stars have in their $z=0$ host galaxy (right panels) to the velocity dispersions those same stars had at their individual formation times (left panels).",
"We also distinguish between the distributions of local velocity dispersions for stellar particles enclosed within different radii with respect to the center of the $z=0$ galaxy, which is defined as the position of the most bound particle belonging to the galaxy.",
"Figure REF shows four additional $\\sigma _{\\rm birth}$ distributions for comparison to the various other stellar birth properties that will be discussed in following sections.",
"Figure: Relations between the IMF mismatch parameter log(α IMF \\alpha _{\\rm IMF}) and various galaxy properties.",
"The insets show the input relations used to construct log(α IMF \\alpha _{\\rm IMF}), in each panel using a different physical quantity at star-formation time (Panel A: local velocity dispersion σ birth \\sigma _{\\rm birth}, Panel B: local metallicity [M/H], Panel C: global star-forming gas velocity dispersion σ gas \\sigma _{\\rm gas}, Panel D: global star-formation rate SFR).",
"The main panels show the resulting constructed relations between log(α IMF \\alpha _{\\rm IMF}) within 0.5R 1/2 p R^{p}_{1/2} and z=0z = 0 global stellar velocity dispersion, σ * \\sigma _{*}.",
"The red curves correspond to shallower input relations while the blue curves to steeper ones.",
"In each panel the solid black line shows the observed relation from Spiniello et al.",
"2014 and the black dashed line shows an extrapolation of the relation.",
"The dot-dashed lines at log(α IMF \\alpha _{\\rm IMF}) = 0 and log(α IMF \\alpha _{\\rm IMF}) = -0.22 indicate the Salpeter and Chabrier IMF mismatch parameters respectively.",
"All input relations and fits to the output relations are listed in Table A1.",
"Note: In Panel D, the blue log(α IMF \\alpha _{\\rm IMF})-σ * \\sigma _{*} relation has been shifted by -0.1 dex.As evident in Figure REF the velocity dispersions that the stellar particles have in their host galaxy at $z = 0$ can be substantially different from the velocity dispersions they had at their time of birth.",
"Qualitatively, the $\\sigma _{z=0}$ distributions for galaxies of all masses in our selected sample are singly peaked and best described as a gaussian or a gaussian with a low velocity dispersion tail.",
"In most cases the low velocity dispersion tail is largely built up by stellar particles residing outside the projected stellar half-mass radius of the galaxies.",
"Unlike the $z = 0$ distributions, the birth velocity dispersion distributions are quite varied.",
"Most lower mass galaxies in our sample ($M_{*}$ $\\lessapprox $ 10$^{11}$ $M_{\\odot }$ ) have near singly peaked birth distributions, spread out over a broader range of velocity dispersions than their $z = 0$ distributions.",
"Some of these lower mass galaxies end up with a higher mean velocity dispersion at $z = 0$ than a birth, and others vice versa.",
"Gas inflows, outflows, internal dynamical processes, as well as mergers, are all expected to play a role in shifting the velocity dispersion of a galaxy over time either to lower or higher values.",
"Generally, the range of birth velocity dispersions becomes larger for galaxies with a higher $z = 0$ stellar mass.",
"In particular, the birth velocity dispersion distributions of $M_{*}$ $\\gtrapprox $ 10$^{11}$ $M_{\\odot }$ galaxies in our sample are usually multi-peaked and spread across a large range of velocity dispersions.",
"This is reflective of the rich merger histories of these high mass galaxies, with their stellar particles being formed in numerous progenitor galaxies with varying masses and velocity dispersions.",
"For example, the $M_{*}$ = 10$^{11.69}$ $M_{\\odot }$ galaxy shown in the bottom left panel of Figure REF underwent 4 major mergers ($\\mu $ > 1/4), 7 minor mergers (1/4 > $\\mu $ > 1/10), and 529 very minor mergers ($\\mu $ < 1/10) throughout its history.",
"On the other hand, the lower mass galaxy shown in the top left panel, with $M_{*}$ = 10$^{10.95}$ $M_{\\odot }$ , only underwent 3 major mergers ($\\mu $ > 1/4), no minor mergers (1/4 > $\\mu $ > 1/10), and 96 very minor mergers ($\\mu $ < 1/10) throughout its history.",
"Radial trends are also present in the birth velocity dispersion distributions of massive galaxies, with the stellar particles closer to the center of each galaxy having, on average, higher birth velocity dispersions.",
"This is due to the spatial distribution of stellar particles inside galaxies set up by mergers.",
"As shown in [72], higher mass galaxies in Illustris consist of a larger fraction of stellar particles formed ex-situ, i.e.",
"not on the main progenitor branch.",
"Galaxies with stellar masses greater than 10$^{12}$ M$_{\\odot }$ can have up to 80% of their stellar particles formed ex-situ and later accreted onto the main galaxy via merging.",
"[72] finds that stellar particles formed in-situ tend to reside in the innermost regions of galaxies whereas stars formed ex-situ tend to lie in the outer regions at larger galactocentric distances.",
"With the birth velocity dispersion and mass of each stellar particle belonging to a galaxy, we construct the IMF mismatch parameter according to the prescription in Section REF .",
"To start, we shift the observed relation presented by [84] towards higher velocity dispersions, $ \\mathrm {log}(\\alpha _{\\rm IMF}) = 1.05\\mathrm {log}(\\sigma _{\\rm birth}) - 2.71,$ so that the minimum Chabrier IMF value of $\\alpha _{\\rm IMF}$ = 0.6 is adopted for all stellar particles with $\\sigma _{\\rm birth}$ less than 235 km s$^{-1}$ .",
"This shift in the input relation is applied because the $\\sigma _{\\rm birth}$ distributions, on average, cover higher values than the $\\sigma _{*}$ distributions.",
"A shift in the relation is needed to place the log($\\alpha _{\\rm IMF}$ ) of low velocity dispersion galaxies on the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation.Using the original [84] relation as input in this case is still unable to reproduce the slope of the observed relation.. Panel A of Figure REF shows the resulting log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation for $z = 0$ Illustris galaxies constructed based on $\\sigma _{\\rm birth}$ .",
"We show only the relations constructed using the stellar particles residing within 0.5$R^{p}_{1/2}$ of each $z = 0$ galaxy, as observations of IMF variations are mainly constructed using the innermost regions of galaxies, and the same holds for the other panels in Figure REF as well.",
"For the output relations constructed using all the stellar particles belonging to each galaxy, the reader is referred to Table A1.",
"The inset figure in each panel of Figure REF shows the input relations we used to construct log($\\alpha _{\\rm IMF}$ ) based on the indicated stellar particle property.",
"The red curve in Panel A shows the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ trend resulting from using Equation REF as the prescribed IMF relation applied to the birth velocity dispersions of each galaxy's stellar particles.",
"As evident in the figure, the output log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ is $\\sim $ 2.8$\\times $ too shallow compared to the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation (black curve).",
"This can be understood in terms of the birth velocity dispersion distributions shown in Figure REF .",
"Using Equation REF as the input relation, there is not enough of a differentiation created between the overall log($\\alpha _{\\rm IMF}$ ) of galaxies of different global $\\sigma _{*}$ values.",
"For example, the $\\sigma _{*}$ = 150 km s$^{-1}$ and $\\sigma _{*}$ = 195 km s$^{-1}$ galaxies shown in Figure REF have birth velocity dispersion distributions that cover a similar range of values.",
"So the log($\\alpha _{\\rm IMF}$ ) difference between the two galaxies, where the $\\sigma _{*}$ = 150 km s$^{-1}$ galaxy is found to have an overall log($\\alpha _{\\rm IMF}$ ) = -0.219 and the $\\sigma _{*}$ = 195 km s$^{-1}$ galaxy is found to have log($\\alpha _{\\rm IMF}$ ) = -0.210, is too small compared to the observed difference of $\\Delta $ log($\\alpha _{\\rm IMF}$ ) = .125.",
"To reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation, we construct various input log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm birth}$ relations with steeper slopes.",
"As before, we minimize the residuals between the resulting Illustris output relations and the [84] relation, and select the input relation that produces the best-fit.",
"The blue curve in Panel A of Figure REF shows the resulting log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation constructed using an input relation that is $\\sim $ 3.5$\\times $ steeper than Equation REF and a minimum Chabrier $\\alpha _{\\rm IMF}$ applied to stellar particles with $\\sigma _{\\rm birth}$ < 235 km s$^{-1}$ .",
"As seen in Panel A of Figure REF , this steeper input relation is able to reproduce both the slope and normalization of the observed trend with global velocity dispersion.",
"The increase in the slope of the input relation creates more of a differentiation between galaxies of different $\\sigma _*$ .",
"For example, the $\\sigma _{*}$ = 150 km s$^{-1}$ galaxy in Figure REF is found to have a similar overall log($\\alpha _{\\rm IMF}$ ) as before (log($\\alpha _{\\rm IMF}$ ) = -0.217), but the $\\sigma _{*}$ = 195 km s$^{-1}$ overall log($\\alpha _{\\rm IMF}$ ) increased by $\\Delta $ log($\\alpha _{\\rm IMF}$ ) = .025, placing it closer to the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation.",
"Figure: Birth properties of stellar particles belonging to four z=0z = 0 galaxies, showing how various stellar birth property distributions vary with galaxy mass.",
"Each row shows the properties of one galaxy, increasing in stellar mass from top to bottom where Column 1: local, birth velocity dispersion (σ birth \\sigma _{\\rm birth}), Column 2: birth metallicity ([M/H]), Column 3: gas velocity dispersion (σ gas \\sigma _{\\rm gas}), Column 4: star-formation rate.The grey distributions show the properties of all the stellar particles belonging to the z=0z = 0 galaxy while the the light blue and dark blue distributions show the properties of the stellar particles within one and one-half the projected stellar half-mass radius, respectively.In addition to increasing the slope of the output log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation, a steeper input relation acts to increase the scatter of the Illustris log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation.",
"As seen in Panel A of Figure REF , the scatter of the blue relation is nearly 2$\\times $ the scatter of the red relation.",
"The increase in scatter with the steeper input relation can also be understood with an example from Figure REF .",
"Although the 10$^{11.69}$ $M_{\\odot }$ and 10$^{11.85}$ $M_{\\odot }$ galaxies shown in Figure REF have similar masses and velocity dispersions, their birth velocity dispersion distributions are quite different.",
"Using Equation REF as the input relation, the difference between their overall log($\\alpha _{\\rm IMF}$ ) values is $\\Delta $ log($\\alpha _{\\rm IMF}$ ) = .025.",
"But, using the steeper input relation the difference becomes $\\Delta $ log($\\alpha _{\\rm IMF}$ ) = .10.",
"While a steeper input relation is able to create more of an overall differentiation between galaxies of different $\\sigma _{*}$ values, it also creates a larger scatter among galaxies of similar $\\sigma _{*}$ but different formation histories."
],
[
"Local metallicity",
"The second physical quantity of star-formation we examine is metallicity.",
"Aside from velocity dispersion, observational studies also report that the IMF scales with galaxy metallicity.",
"While both velocity dispersion and metallicity correlate with mass, the quantity that is more fundamentally associated with IMF variations is still unclear.",
"A metallicity-IMF correlation can be easily imagined through a reversed causal relationship.",
"Simply put, the IMF is expected to influence the metallicity because the number of high to low mass stars will directly affect the chemical evolution of a galaxy.",
"The more top-heavy the IMF, the more metals are injected into the ISM, which increases the metallicity of the stars born in subsequent star formation bursts.",
"So one might naively expect the overall metallicity of a galaxy by $z = 0$ to be higher with a more top-heavy its IMF.",
"This scenario is however in tension with observational IMF-metallicity relations, which infer a more bottom-heavy IMF for the most metal-rich galaxies [20], [53].",
"One way this tension may be reconciled is by invoking a time-dependent IMF: earlier star formation follows a flatter IMF to build up the metallicity of the ISM and star formation occurring later follows a bottom-heavy IMF to build up the population of low-mass stars [51], [100], [62].",
"Particularly, [100] propose that the ISM, enriched by episodes of high star-formation with a flat IMF, is exceptionally turbulent leading to increased fragmentation on lower mass scales.",
"Therefore, proceeding star-formation occurs with a steeper IMF slope.",
"In this scenario, a higher metallicity environment at the time of stellar birth is expected to correspond to a more bottom-heavy IMF.",
"Our current empirical approach for constructing log($\\alpha _{\\rm IMF}$ ) does not take into account how the IMF may influence metallicity.",
"Instead, motivated by [100], we assign an IMF based on the local metallicity of each stellar particle at the time of stellar birth with the idea that stellar particles born into higher metallicity environments form with a steeper IMF slope.",
"Thus, we construct the overall log($\\alpha _{\\rm IMF}$ ) of each Illustris galaxy based on the metallicity of each stellar particle at the time of formation.",
"As discussed in Section REF , the total mass in metals of a stellar particle in Illustris is inherited from the parent gas cell at the time of star formation.",
"The ratio of the total mass in metals heavier than helium to the total mass of the stellar particles at the time of birth, $Z$ , is output for each star in the snapshot files.",
"For the stellar particles in our sample, we convert the metallicity mass fraction to the metal abundance [M/H] by assuming each stellar particle to have a primordial hydrogen mass fraction of $X$ = 0.76 and scaling to solar units using $Z_{\\odot } = 0.02$ and $X_{\\odot } = 0.70$ .",
"The second column of Figure REF shows the distribution of stellar particle metallicities for four Illustris galaxies, increasing in $z = 0$ stellar mass from top to bottom.",
"The light grey histogram shows the distribution of metallicities of all stellar particles belonging to the galaxy while the light and dark blue distributions show the metallicities of the stellar particles within $R^{p}_{1/2}$ and 0.5$R^{p}_{1/2}$ , respectively.",
"The distributions are similarly shaped, mostly described as a gaussian with a significant low metallicity tail.",
"Similar to the velocity dispersion distributions in Figure REF , the stellar particles residing closer to the center of the galaxy have a higher average metallicity than the stellar particles residing in the outer reaches of the galaxy.",
"The global $z = 0$ metallicity of each galaxy, shown in the upper right of each panel, is calculated by taking the mass-weighted average of the metal abundance of the stellar particles within 0.5$R^{p}_{1/2}$ of each galaxy.",
"Figure: Global metallicity, [M/H], as a function of z=0z = 0 stellar velocity dispersion, σ * \\sigma _{*} for the 371 selected Illustris-1 galaxies.With the birth metallicities and mass of each stellar particle belonging to a galaxy, we construct the IMF mismatch parameter as before.",
"As a starting point, we are inspired by [52] to construct an input log($\\alpha _{\\rm IMF}$ )-[M/H] relation that is defined to have a Chabrier $\\alpha _{\\rm IMF}$ value at [M/H] = -0.29 and a Salpeter $\\alpha $ value at [M/H] = -0.07.",
"This results in the relation log($\\alpha _{\\rm IMF}$ ) = [M/H] + 0.07, where stellar particles with metallicities less than -0.29 are assigned $\\alpha _{\\rm IMF}$ = 0.6.",
"Panel B of Figure REF shows the resulting Illustris log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation (red curve) constructed based on the individual [M/H] values that each stellar particle belonging to a galaxy are formed with.",
"Again, we only show the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relations constructed using only the stellar particles residing within 0.5$R^{p}_{1/2}$ of each $z = 0$ galaxy.",
"Evidently, our initial input relation is unable to reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation, with the slope of the output Illustris relation being nearly flat and in fact slightly negative, showing the opposite trend to the observed one.",
"To try to reproduce the observed relation, we construct an input log($\\alpha _{\\rm IMF}$ )-[M/H] that is 4$\\times $ as steep and defined to have a minimum Chabrier $\\alpha $ at [M/H] = -0.05.",
"The increase in the Chabrier minimum is in attempt to decrease the overall log($\\alpha _{\\rm IMF}$ ) values of lower velocity dispersion galaxies.",
"The blue curve in Panel B of Figure REF shows the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation constructed using the steeper log($\\alpha _{\\rm IMF}$ )-[M/H] described.",
"The result is that the normalization of the output relation shifts to higher log($\\alpha _{\\rm IMF}$ ) values, but its slope is still negative.",
"Additionally, increasing the Chabrier minimum to [M/H] = -0.05 did not act to decrease the overall log($\\alpha _{\\rm IMF}$ ) of lower velocity dispersion galaxies, as increasing the Chabrier minimum in Section REF was able to do.",
"The inability to reproduce the observed IMF trend with $z = 0$ velocity dispersion using [M/H] as the physical driver of the IMF can be understood from the global [M/H]-$\\sigma _{*}$ relation.",
"As seen in Figure REF , this relation is almost flat for our sample comprised of massive $M_{*}$ > $\\sim $ 10$^{10}$ $M_{\\odot }$ galaxiesThe flatness of the [M/H]-$\\sigma _{*}$ relation is generally in agreement with observations of the stellar mass-metallicity relation, such as in [32], where the SDSS mass-metallicity becomes flat at high stellar masses with a scatter of $\\sim $ 0.3 dex.",
"Figure 5 is also in agreement with the observed velocity dispersion-mass relation over the appropriate velocity dispersion range [85].",
"In fact, where it is not completely flat, at $2.4<\\log {\\sigma _{*}}<2.6$ , the output log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ also has some slope.",
"The negative slope of the [M/H]-$\\sigma _{*}$ relation could either be due to intracluster light contamination or recycling of low metallicity gas.",
"However, even there, the mild slope of the [M/H]-$\\sigma _{*}$ relation combined with its large scatter result in galaxies of similar velocity dispersions having a wide range of global [M/H] values.",
"Since the global [M/H] of each galaxy is related to the distribution of the individual stellar particle [M/H] values (as seen in Figure REF ) and the widths of the [M/H] distributions are broad compared to the galaxy-to-galaxy differences in global [M/H], a differential effect in overall log($\\alpha _{\\rm IMF}$ ) between galaxies of different velocity dispersions is not produced using an input relation based on [M/H]."
],
[
"Global star-forming gas velocity dispersion",
"As mentioned in Section , a few analytical studies have focused on connecting the physics governing star formation to an environment dependent IMF.",
"In particular, [40] developed an analytical formulation where star-forming disks with higher Mach numbers cause the low-mass turnover of the pre-stellar core mass function (CMF) to be shifted to lower masses leading to a more bottom-heavy CMF which implies a more bottom-heavy IMF.",
"Physically, as discussed in [40], a higher star-forming disk Mach number leads to larger density fluctuations which causes more fragmentation on smaller mass scales.",
"Motivated by a Mach number dependent CMF, we construct the IMF of our Illustris galaxies using the one-dimensional, global star-forming gas velocity dispersion ($\\sigma _{\\rm gas}$ ) of the progenitor galaxies in which stellar particles are born.",
"We do not take into account differences in sound speed, but simply use $\\sigma _{\\rm gas}$ as a proxy for Mach number [12].",
"To calculate $\\sigma _{\\rm gas}$ we trace each stellar particle back to the progenitor galaxy in which it was born and consider only the gas cells in that galaxy with non-zero instantaneous star formation rates.",
"We remove net rotation by calculating the total angular momentum vector of each galaxy's star-forming gas component and calculate $\\sigma _{\\rm gas}$ as the mass-weighted standard deviation of the cell velocities parallel to that angular momentum vector.",
"The third column of Figure REF shows the $\\sigma _{\\rm gas}$ distributions for four galaxies in our sample.",
"The birth $\\sigma _{\\rm gas}$ distributions are similarly multi-peak and spread across a broad range of values as the $\\sigma _{\\rm birth}$ distributions shown in the first column of Figure REF .",
"However, there are a few qualitative differences between the two distributions.",
"First, the $\\sigma _{\\rm gas}$ distributions are less continuous than the corresponding $\\sigma _{\\rm birth}$ distributions, reflective of the fact that multiple stellar particles are often born in the same progenitor galaxy and therefore have the same $\\sigma _{\\rm gas}$ value.",
"The $\\sigma _{\\rm gas}$ distributions are also shifted to lower velocity dispersion values compared to their $\\sigma _{\\rm birth}$ counterparts due to the removal of rotation.",
"But similar to the $\\sigma _{\\rm birth}$ distributions, there is a radial trend in $\\sigma _{\\rm gas}$ especially for higher mass galaxies, with a larger fraction of stars born in high $\\sigma _{\\rm gas}$ galaxies residing closer to the center of their $z = 0$ host galaxy.",
"We construct the overall log($\\alpha _{\\rm IMF}$ ) of each galaxy based on the $\\sigma _{\\rm gas}$ distributions.",
"We first construct a log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm gas}$ relation inspired by Equation REF , but shift the Chabrier minimum of the relation to occur at $\\sigma _{\\rm gas}$ = 135 km s$^{-1}$ .",
"This results in the relation log($\\alpha _{\\rm IMF}$ ) = 1.05log($\\sigma _{\\rm gas}$ ) - 2.46, where stellar particles with $\\sigma _{\\rm gas}$ less than 135 km s$^{-1}$ are assigned $\\alpha _{\\rm IMF}$ = 0.6.",
"Panel C of Figure REF shows the resulting Illustris log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation (red curve) constructed based on the star-forming gas velocity dispersion of the progenitor galaxy that each stellar particle belonging to a $z = 0$ was formed in, showing just the relation constructed using the stellar particles residing within 0.5$R^{p}_{1/2}$ .",
"As with $\\sigma _{\\rm birth}$ , the initial input relation is unable to reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation, with the slope of the Illustris log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ $\\sim $ 2.6$\\times $ shallower than the observed relation.",
"To reproduce the observed relation, we construct a steeper log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm gas}$ relation.",
"The blue curve in Panel C of Figure REF shows the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation constructed using a log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm gas}$ input relation that is 4.1$\\times $ steeper than the [84] relation.",
"This steeper input relation does, within the reported uncertainty, reproduce both the slope and normalization of the observed trend with global velocity dispersion $\\sigma _{*}$ .",
"As with $\\sigma _{\\rm birth}$ as a physical driver, the increase in the slope of the input relation creates more of a differentiation between galaxies of different $\\sigma _{*}$ , which allows the observed relation to be reproduced.",
"Also similar to the Illustris log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation constructed based on $\\sigma _{\\rm birth}$ , increasing the slope of the input relation results in a larger scatter in the output relation.",
"This is because galaxies of similar $z = 0$ velocity dispersions can have a range of $\\sigma _{\\rm gas}$ distributions."
],
[
"Global star-formation rate",
"The last physical quantity we consider in constructing the overall log($\\alpha _{\\rm IMF}$ ) of each Illustris galaxy is star-formation rate.",
"Observationally, studies focusing on constraining the high mass end of the IMF suggest that the IMF correlates with galaxy SFR.",
"For example, [36] find that for a range of galaxies at $z$ < 0.35 with SFRs covering 10$^{-3}$ to 100 $M_{\\odot }$ yr$^{-1}$ , the most quiescent galaxies are best described by steeper IMF slopes ($x$$\\sim $ 2.4), whereas highly star-forming galaxies exhibit shallower IMFs ($x$$\\sim $ 1.8).",
"They translate their IMF-SFR to a relation between IMF and SFR surface density, finding that galaxies with higher SFR densities prefer flatter IMF slopes.",
"This result is consistent within the context of IGIMF theory [101], which connects the global SFR of a galaxy to the formation of stars within individual molecular clouds throughout the galaxy, where galaxies with higher SFR are expected to have a top-heavy galaxy-wide IMF.",
"On the other hand, [20] find their strongest IMF trend to be with [Mg/Fe], where galaxies with greater Mg enhancement have more bottom-heavy IMFs.",
"These galaxies with increased Mg abundances are interpreted as having shorter star-formation timescales.",
"Based on the inferred star-formation time scales of massive galaxies with enhanced Mg abundances, [20] infer that galaxies with high SFR surface densities are described by a more bottom-heavy IMF.",
"Physically, as pointed out by [20], in the context of the [40] analytical theory for IMF variations, high SFR surface densities promotes turbulence which leads to a more bottom-heavy IMF.",
"In regards to observations reporting that higher SFRs correspond to shallower IMFs and to IGIMF theory which predicts the same, it is suggested that the high $M_{*}/L$ ratio inferred for these massive elliptical galaxies is at least in part due to an excess of high-mass stellar remnants and not only an excess of low-mass stars.",
"In this scenario, high SFR starbursts induced by mergers form with a top-heavy IMF, and it is the remnants of these high mass stars that produce an excess of mass as measured by $z = 0$ .",
"For each stellar particle that comprises a $z = 0$ galaxy, we record the instantaneous star-formation rate of the progenitor galaxy in which the stellar particle is formed.",
"The fourth column of Figure REF shows the distribution of birth SFRs for the stellar particles comprising four galaxies of various masses.",
"As seen in the figure, the birth SFR distribution becomes multi-peaked and/or broader for galaxies with higher stellar mass, and includes more stellar particles with higher birth SFRs.",
"For lower mass galaxies, where a majority of their stellar populations are formed in-situ, the shape of the birth SFR distribution can be understood as the evolution of the star-formation rate of the main progenitor branch.",
"The low SFR tails of these distributions correspond to the formation of stellar particles before and after the period of peak star-formation where most of the stellar mass is formed.",
"Additionally, for these lower mass galaxies, there is little difference in the birth SFR distributions of all the stellar particles versus just the stellar particles residing within 0.5$R^{p}_{1/2}$ of each galaxy.",
"The higher mass galaxies shown in the figure ($M_{*}$ > 10$^{11.2}$ $M_{\\odot }$ ) have stellar particles that, on average, formed in progenitor galaxies with higher SFRs and also cover a broader range of SFRs.",
"The stellar particles formed in progenitor galaxies with SFRs $\\sim $ 100 $M_{\\odot }$ yr$^{-1}$ likely formed during merger-induced nuclear starbursts, whereas the stellar particles making up the lower SFR part of the distributions were formed either before or after the peak star-formation period of these merger events, or in lower SFR galaxies that are later accreted onto the main progenitor.",
"The SFR for the 10$^{11.88}$ $M_{\\odot }$ and 10$^{11.29}$ $M_{\\odot }$ galaxies particularly show that stellar particles residing closer to the center of the $z = 0$ galaxy are formed in galaxies with higher SFRs than stellar particles residing in the outer edges of the galaxy.",
"This is consistent with the $\\sigma _{\\rm birth}$ distribution of massive galaxies, where stellar particles closer to the center of a massive galaxy are formed in high velocity dispersion environments during periods of high SFR nuclear starbursts.",
"We construct the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation for our sample of Illustris galaxies based on the birth SFR distributions described above.",
"First, we use a log($\\alpha _{\\rm IMF}$ )-SFR relation that is defined to correspond to a Chabrier IMF at log(SFR) = 2.2 and a Salpeter IMF at log(SFR) = 0.7.",
"This starting point is inspired by [36] who for their sample of galaxies from GAMA find an IMF-SFR relation -x $\\approx $ 0.36 log(SFR) - 2.6.",
"Stellar particles born into galaxies with SFR = 0 $M_{\\odot }$ yr$^{-1}$ , where the SFR is an unresolved small value, are assigned zero light, i.e.",
"$\\alpha _{\\rm IMF}$$^{-1}$ = 0.",
"The red curve in Panel D of Figure REF shows the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation resulting from this initial input relation.",
"As seen in the figure, using SFR as the physical quantity of star-formation, where high SFR environments are expected to correspond to a shallower IMF, to build the Illustris log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ results in a trend opposite to that of the observations.",
"Inputting a $\\sim $ 4.7$\\times $ steeper log($\\alpha _{\\rm IMF}$ )-SFR relation, we are best able to reproduce the steepness of the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ trend, but in the opposite direction (blue curve).",
"This is because higher velocity dispersion galaxies in our sample have stellar particles that, on average, formed in progenitors with higher SFRs.",
"This is compared to the stellar particles belonging to low velocity dispersion galaxies, which generally form in progenitors with lower SFRs."
],
[
"Radial trends",
"As suggested by Figures REF and REF , the physical quantities that we investigate display various amounts of radial variation depending on the mass of the galaxy.",
"For the more massive galaxies in our sample, stellar particles residing closer to the centers of their $z = 0$ galaxy tend to have higher $\\sigma _{\\rm birth}$ , [M/H], $\\sigma _{\\rm gas}$ , or SFRs compared to the stellar particles residing in the outskirts of the galaxy.",
"Such a radial trend is weaker or non-existent for the lower mass galaxies we examine.",
"This is reflective of lower mass galaxies being composed of a smaller fraction of stellar particles formed ex-situ compared to high mass galaxies.",
"So in constructing the overall log($\\alpha _{\\rm IMF}$ ) using only stellar particles within 0.5$R^{p}_{1/2}$ , the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation is steeper than when constructing log($\\alpha _{\\rm IMF}$ ) based on all the stellar particles belonging to each galaxy.",
"Table REF lists the comparison between the output log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relations constructed using all stellar particles versus just the innermost stellar particles.",
"Figure: IMF slope as a function of galactocentric distance for 11 galaxies in our sample, using the steep log(α IMF \\alpha _{\\rm IMF})-σ birth \\sigma _{\\rm birth} relation (see text).",
"Each line represents a galaxy colored by its z=0z = 0 velocity dispersion.",
"The top panel shows the IMF slope constructed in cylinders, including all stars within each radius and the bottom panel shows the IMF slope constructed in hollow cylinders, including the stellar particles between two consecutive radii.",
"These constructed Illustris log(α IMF \\alpha _{\\rm IMF}) radial gradients are in qualitative agreement with observations of IMF gradients for galaxies of both low and high velocity dispersion.Here we examine in more detail radial trends of log($\\alpha _{\\rm IMF}$ ) for a sample of galaxies with varying global velocity dispersions.",
"To construct the overall IMF mismatch parameter we use the steep log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm birth}$ relation, log($\\alpha _{\\rm IMF}$ ) = 3.7log($\\sigma _{\\rm birth}$ ) - 8.99, for which we were able to reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ trend.",
"Figurfe REF shows log($\\alpha _{\\rm IMF}$ ) as a function of projected radius for 11 Illustris galaxies, starting at a radius of 1/7 the projected stellar half-mass radius, (1/7)$R^{p}_{1/2}$ , out to radius of 4 times the projected stellar half-mass radius, 4$R^{p}_{1/2}$ .",
"The top panel shows log($\\alpha _{\\rm IMF}$ ) constructed in cylinders (i.e.",
"spheres projected along the line of sight), including all stellar particles falling within the indicated radius.",
"The bottom panel shows log($\\alpha _{\\rm IMF}$ ) constructed in hollow cylinders, where each indicated radius shows log($\\alpha _{\\rm IMF}$ ) constructed with just the stellar particles falling within two consecutive radii.",
"The log($\\alpha _{\\rm IMF}$ ) value for (1/7)$R^{p}_{1/2}$ is constructed using stellar particles residing between (1/8)$R^{p}_{1/2}$ and (1/7)$R^{p}_{1/2}$ .",
"As evident in Figure REF , the highest velocity dispersion galaxies ($\\sigma $ $\\sim $ 250 - 350 km s$^{-1}$ ) exhibit the greatest decrement in log($\\alpha _{\\rm IMF}$ ) towards larger radii, whereas log($\\alpha _{\\rm IMF}$ ) for lower velocity dispersion galaxies ($\\sigma $ $\\sim $ 150 - 250 km s$^{-1}$ ) stays more constant with radius.",
"Qualitatively, this trend of higher $\\sigma _{*}$ galaxies displaying the largest radial IMF trends is in agreement with observations such as [52].",
"Though, it is difficult to directly compare our results to observations by radius due to the differences in how effective radius ($R_{e}$ ) is measured and how the stellar half-mass radius is calculated for Illustris galaxies.",
"The decrement in IMF mismatch parameter, $\\Delta $ log($\\alpha _{\\rm IMF}$ ), and the maximum log($\\alpha _{\\rm IMF}$ ) of our highest velocity dispersion Illustris galaxies is similar to what is reported in [52].",
"The decrement in log($\\alpha _{\\rm IMF}$ ) of the most bottom-heavy galaxy shown in Figure REF , with $\\sigma _{*}$ = 302 km s$^{-1}$ , is $\\Delta $ log($\\alpha _{\\rm IMF}$ ) = 0.36 in cylinders from a galactocentric radius of (1/7)$R^{p}_{1/2}$ to 4$R^{p}_{1/2}$ .",
"Considering hollow cylinders in which log($\\alpha _{\\rm IMF}$ ) is calculated, from (1/7)$R^{p}_{1/2}$ to 4$R^{p}_{1/2}$ there is a larger decrement of $\\Delta $ log($\\alpha _{\\rm IMF}$ ) = 0.57.",
"[52] reports for their high velocity dispersion galaxy ($\\sigma $ $\\sim $ 300 km s$^{-1}$ ) a decrement of $\\Delta $ x = 1.15, which roughly corresponds to $\\Delta $ log($\\alpha _{\\rm IMF}$ ) = 0.35, from the center of the galaxy (r = 0 $R_{e}$ ) to 0.7 $R_{e}$ ."
],
[
"Scatter",
"As mentioned, increasing the slope of the input $\\alpha _{\\rm IMF}$ relation increases the scatter of the resulting overall log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relations (Figure REF and Figure REF ).",
"This is consistent with observational studies that also show substantial scatter in the reported IMF-$\\sigma $ relations [70], [20], [6].",
"For example, based on dynamical modeling of ETGs in the ATLAS$^{\\rm 3D}$ project, [7] reports a 1$\\sigma $ scatter of $\\approx $ 0.12 dex (or 32%) in their derived relation between IMF mismatch parameter and velocity dispersion.",
"[7]'s reported scatter is comparable to the scatter seen in our Illustris log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relations.",
"Constructing the overall log($\\alpha _{\\rm IMF}$ ) of each galaxy based on the stellar velocity dispersion of each stellar particle's progenitor galaxy (Section REF ), we produce a 1$\\sigma $ scatter of 0.079 dex (20%) using the original [84] relation as input and a 1$\\sigma $ scatter of 0.123 dex (32.7%) using the 3.6$\\times $ steeper input relation.",
"Similarly, constructing the overall log($\\alpha _{\\rm IMF}$ ) using the local velocity dispersion of each stellar particle at the time of birth (Section REF ), we produce a 1$\\sigma $ scatter of 0.045 dex (11%) using the shallow input relation and a 1$\\sigma $ scatter of 0.125 dex (33%) using the steeper input relation that is able to reproduce the overall log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation.",
"As discussed in Section REF , the scatter in our Illustris IMF relations is due to galaxies of similar global $z=0$ velocity dispersions having varying stellar birth property distributions like $\\sigma _{\\rm birth}$ or $\\sigma _{*}$ .",
"These differences in the physical conditions of star-formation reflect an intrinsic scatter in the formation histories of galaxies with the same global $z=0$ properties."
],
[
"Redshift evolution",
"Finally, we examine the redshift evolution of the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation by repeating the analysis described in Section REF but now with a sample of galaxies at a higher redshift.",
"In Illustris-1 at $z = 2$ , we select the 311 galaxies with stellar masses greater than 10$^{10}$ M$_{\\odot }$ and stellar velocity dispersions greater than 150 km s$^{-1}$ .",
"We do not place a cut on star-formation as we did for the $z = 0$ sample since fewer galaxies meet the sSFR < 10$^{-11}$ yr$^{-1}$ criterion at $z = 2$ .",
"The average stellar mass, stellar velocity dispersion, and specific star formation rate of the $z = 2$ sample is $\\overline{M_{*}}$ = 10$^{10.89}$ , $\\overline{\\sigma _{*}}$ = 197 km s$^{-1}$ , and $\\overline{\\rm sSFR}$ = 7.60$\\times 10^{-10}$ yr$^{-1}$ .",
"Figure: IMF mismatch parameter versus global velocity dispersion for the z=2z = 2 massive galaxies.",
"The grey points show the z=2z=2 galaxies, while the blue relation shows the z=0z = 0 log(α IMF \\alpha _{\\rm IMF})-σ * \\sigma _{*} relation constructed using the steep input relation (shown in the inset as in Figure 3).",
"The red line shows the fit to the z=2z = 2 galaxies defined as quiescent with sSFR < 6.82×10 -11 \\times 10^{-11} yr -1 ^{-1} (outlined in black).",
"At fixed velocity dispersion, quiescent galaxies at z=2z=2 are more bottom-heavy than their z=0z=0 counterparts.We construct the overall log($\\alpha _{\\rm IMF}$ ) of each $z = 2$ galaxy based on the local, birth velocity dispersion of the stellar particles following the same procedure outlined in Section REF .",
"Figure REF shows the resulting $z = 2$ log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation using the same steep input relation that was able to reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation for the $z = 0$ galaxy sample.",
"As seen in the figure, the overall log($\\alpha _{\\rm IMF}$ ) values of the $z = 2$ galaxies are, on average, higher than the $z = 0$ relation.",
"Furthermore, the more quiescent galaxies generally have higher log($\\alpha _{\\rm IMF}$ ) values than the more star-forming galaxies.",
"To more directly compare to the $z = 0$ relation, which only includes quiescent galaxies, we fit the $z = 2$ log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation only for galaxies with sSFR < (3t$_{h}$ )$^{-1}$ yr$^{-1}$ where t$_{h}$ is the age of the Universe at a given redshift [22].",
"In Figure REF the red line shows the fit to the 29 quiescent $z = 2$ galaxies, which is $\\sim $ 1.4$\\times $ steeper and offset by $\\sim $ 0.17 dex compared to the $z = 0$ relation.",
"The offset of the $z = 2$ log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation towards higher log($\\alpha _{\\rm IMF}$ ) values compared to the $z = 0$ relation is due to the assembly history of massive galaxies.",
"In $\\Lambda $ CDM, massive galaxies are thought to first build up their in-situ stellar populations and then at later redshifts accrete smaller systems and build up their ex-situ stellar populations [60], [67].",
"In Illustris, quiescent galaxies at $z = 2$ have already formed a significant portion of their in-situ stellar particles, but have yet to accumulate a majority of their ex-situ stellar particles.",
"The higher log($\\alpha _{\\rm IMF}$ ) values of the $z = 2$ galaxies suggests the stellar particles already belonging to galaxies by $z = 2$ are formed in higher velocity dispersion environments than the stellar particles that will be added to the galaxies at later times.",
"To go from the $z = 2$ to the $z = 0$ log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation, stellar particles added to galaxies after $z = 2$ decrease the overall log($\\alpha _{\\rm IMF}$ ) values of the galaxies.",
"As will be discussed in Section , our $z=2$ log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation is seemingly in tension with IMF observations beyond $z=0$ , which suggest that the relation has remained constant over the past $\\sim $ 8 Gyrs.",
"Robust IMF determinations out to $z=2$ will be needed to fully assess the implications of our high redshift results."
],
[
"Convergence with resolution",
"First we confirm the convergence of our results to degradation in simulation resolution.",
"We repeat the same analysis as described in Section REF for Illustris-1 on the two lower resolution simulations, Illustris-2 and Illustris-3.",
"All three simulations have the same box size of (106.5 Mpc)$^{3}$ , but Illustris-1 contains $\\sim $ 2 $\\times $ 1820$^{3}$ resolution elements while Illustris-2 and Illustris-3 contain $\\sim $ 2 $\\times $ 910$^{3}$ and $\\sim $ 2 $\\times $ 455$^{3}$ resolution elements respectively.",
"In Illustris-2 the average baryonic particle mass is $\\overline{m_{b}}$ = 1.0$\\times 10^{7}$ M$_{\\odot }$ and in Illustris-3 it is $\\overline{m_{b}}$ = 8.05$\\times 10^{7}$ M$_{\\odot }$ .",
"Refer to Table REF for more Illustris-2 and Illustris-3 simulation parameters.",
"Implementing the same galaxy selection outlined in Section REF results in 229 galaxies in Illustris-2 and 103 galaxies in Illustris-3 that meet the three criteria at $z = 0$ of stellar mass, specific star formation rate, and velocity dispersion.",
"The average stellar mass, specific star formation rate, and stellar velocity dispersion of the Illustris-2 sample is $\\overline{M_{*}}$ = 10$^{11.44}$ , $\\overline{\\rm sSFR}$ = 2.52$\\times 10^{-12}$ yr$^{-1}$ , and $\\overline{\\sigma _{*}}$ = 199 km s$^{-1}$ while for the Illustris-3 sample they are $\\overline{M_{*}}$ = 10$^{11.48}$ , $\\overline{\\rm sSFR}$ = 2.64$\\times 10^{-12}$ yr$^{-1}$ , and $\\overline{\\sigma _{*}}$ = 204 km s$^{-1}$ .",
"To determine the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation we use both the observed relation as input (Equation REF ) and the steeper log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm birth}$ relation we constructed which was found to reproduce the observed trend with global velocity dispersion.",
"The leftmost panels of Figure REF show the resulting log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ trends for Illustris-2 and Illustris-3, constructed using just the birth velocity dispersions of the stellar particles within 0.5$R^{p}_{1/2}$ from the center of each galaxy.",
"As with Illustris-1, using the observed relation to set the IMF of stellar particles at their birth times results in a $z = 0$ log($\\alpha _{\\rm IMF}$ )-$\\sigma _*$ relation that is shallower than observed (red curve).",
"The Illustris-2 relation is $\\sim $ 2.5$\\times $ shallower than the observed relation while the Illustris-3 relation is $\\sim $ 2.8$\\times $ shallower than the observed relation.",
"We construct the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ using the same, steeper input relation we found to reproduce the global trend with Illustris-1, log($\\alpha _{\\rm IMF}$ ) = 3.7log($\\sigma _{\\rm birth}$ ) - 8.99.",
"The blue curve in the figure shows that the same steep input relation that was able to match the global $\\sigma _{*}$ trend in Illustris-1 is also able to reproduce the observed trend, within the uncertainties, in Illustris-2 and Illustris-3.",
"Thus, we conclude our main results to be robust to resolution degradation.",
"We also consider the effect of resolution on the radial IMF trends we explored in REF .",
"Figure REF shows the average IMF profile (measured in cylinders) in four velocity dispersion bins for Illustris-1, Illustris-2, and Illustris-3.",
"For the two lowest velocity dispersion bins, $\\sigma _{*}$ =150-200 km s$^{-1}$ and $\\sigma _{*}$ =200-250 km s$^{-1}$ , the radial profiles for the three resolution levels are similar, although in the $\\sigma _{*}$ =200-250 km s$^{-1}$ bin the Illustris-1 profile is slightly steeper at R$^{p}_{1/2}$ < 1 compared to Illustris-2 and -3.",
"For the $\\sigma _{*}$ =250-300 km s$^{-1}$ bin, the Illustris-2 and -3 average profiles are at higher log($\\alpha _{\\rm IMF}$ ) values compared to Illustris-1.",
"This is also seen comparing Figure REF to Panel A of Figure REF .",
"In this velocity dispersion bin, the Illustris-3 radial profile is significantly shallower than the Illustris-1 and -2 profiles, which is due to the larger smoothing length of the Illustris-3 simulation.",
"For the Illustris-2 and -3 simulations, the velocity dispersion bin with the highest log($\\alpha _{\\rm IMF}$ ) values is not the $\\sigma _{*}$ =300-350 km s$^{-1}$ bin, but the $\\sigma _{*}$ =250-300 km s$^{-1}$ bin.",
"This is likely due to the most massive galaxies in the lower resolution simulations being more affected by intracluster light, which acts to reduce the overall log($\\alpha _{\\rm IMF}$ ).",
"While the most relevant radial profile comparison would be between Illustris-1 and higher resolution simulations, as we discuss next, these high resolution simulations at the appropriate mass scale are not currently available.",
"Figure: The log(α IMF \\alpha _{\\rm IMF})-σ * \\sigma _{*} Illustris relations, constructed based on the birth velocity dispersion distributions of each galaxy, for the two lower resolution Illustris simulations (Illustris-2 and Illustris-3), the two simulations with varying feedback (no feedback and winds only), and the two simulations with varying IMFs (IMF-Salpeter and IMF-Spiniello).",
"In each panel, the red curve or points show the observed relation (Equation ) used as input to construct the overall log(α IMF \\alpha _{\\rm IMF}) of each galaxy and the blue curve shows the resulting relation using the steeper log(α IMF \\alpha _{\\rm IMF})-σ birth \\sigma _{\\rm birth} relation.",
"In each panel, the log(α IMF \\alpha _{\\rm IMF})-σ * \\sigma _{*} constructed using just the stellar particles within 0.5R 1/2 p R^{p}_{1/2} is shown.Additionally, we attempt to test the robustness of our results to higher resolutions.",
"This is motivated by [83], who present zoom-in simulations of Illustris galaxies with mass resolution up to 40 times better than that of Illustris-1.",
"While galaxies in Illustris-1 do undergo nuclear starbursts [104], in some of these zoom-in simulations the merger-driven nuclear starbursts are stronger.",
"This could potentially influence the constructed IMF for two reasons.",
"First, these starburst episodes produce more stellar mass.",
"Second, these extreme star-formation environments display larger velocity dispersions.",
"Hence they have the potential to be the sites where the bottom-heavy IMF of ETGs is built up.",
"These galaxies were selected by [83] based on their $z=0$ quiescence and that they undergo a major merger between $z=.5$ and $z=1$ .",
"Each is run at three resolution levels: 1 - $m_{\\rm dm}$ = 4.42$\\times 10^{6}$ $M_{\\odot }$ , 2 - $m_{\\rm dm}$ = 5.53$\\times 10^{5}$ $M_{\\odot }$ , 3 - $m_{\\rm dm}$ = 1.64$\\times 10^{5}$ $M_{\\odot }$ .",
"We focus on the two galaxies that exhibit the largest increase in star-formation rate at their respective merger times (galaxies 1349 and 1605).",
"For the IMF analysis, we construct the overall log($\\alpha _{\\rm IMF}$ ) based on the $\\sigma _{\\rm birth}$ distributions of the stellar particles residing within 0.5$R^{p}_{1/2}$ of each respective galaxy.",
"We find that the overall velocity dispersion of both galaxies increases with resolution level: $\\sigma _{*}$ = 115, 123, and 134 km s$^{-1}$ for galaxy 1349 and $\\sigma _{*}$ = 113, 123, and 139 km s$^{-1}$ for galaxy 1605.",
"However, using the steep input relation as discussed in Section REF , log($\\alpha _{\\rm IMF}$ ) hardly changes: -0.22, -0.22, -0.217 for galaxy 1349 and -0.22, -0.22, -0.22 for galaxy 1605, for zoom levels 1, 2, and 3, respectively.",
"The reason is that the corresponding $\\sigma _{\\rm birth}$ distributions lie mostly below the Chabrier minimum of log($\\alpha _{\\rm IMF}$ ) = -0.22 set at $\\sigma _{\\rm birth}$ = 235 km s$^{-1}$ .",
"Using a shifted input relation so that fewer stellar particles are assigned the minimum Chabrier value, we do see more significant increases in log($\\alpha _{\\rm IMF}$ ) with increasing resolution level.",
"This suggests the possibility that for zooms of higher velocity dispersion galaxies of at least $\\sigma _{*}$ = 300 km s$^{-1}$ a shallower input relation might suffice to reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation.",
"However, such simulations would require a very significant investment of computing time and are currently not available.",
"Hence, our zoom analysis is at this time inconclusive.",
"Lastly, we test the dependence of our results to time resolution by diluting the snapshots by a factor of 3 in our Illustris-2 analysis.",
"We find our main result to be unaffected by this decrease in time resolution, justifying our choice for the number of snapshots produced in modified physics runs which will be discussed in the following section."
],
[
"Variations in simulation physics",
"We also test the robustness of our results to variations in simulation physics, first considering variations in feedback.",
"We ran a box of 40 Mpc/h on a side with $2\\times 320^{3}$ resolution elements (with a Chabrier IMF) once with no feedback and once with galactic winds but no AGN feedback.",
"In each simulation we select galaxies with stellar masses greater than 10$^{10}$ M$_{\\odot }$ and stellar velocity dispersions greater than 150 km s$^{-1}$ .",
"Since no galaxies in the winds only simulation meet the sSFR criterion, we do not cut on sSFR.",
"This selection results in 178 $z = 0$ galaxies in the no feedback simulation and 41 $z = 0$ galaxies in the winds only simulation.",
"The average stellar mass, specific star formation rate, and stellar velocity dispersion of the no feedback sample is $\\overline{M_{*}}$ = 10$^{11.51}$ , $\\overline{\\rm sSFR}$ = 2.68$\\times 10^{-11}$ yr$^{-1}$ , and $\\overline{\\sigma _{*}}$ = 216 km s$^{-1}$ , while for the winds only sample $\\overline{M_{*}}$ = 10$^{11.43}$ , $\\overline{\\rm sSFR}$ = 1.74$\\times 10^{-10}$ yr$^{-1}$ , and $\\overline{\\sigma _{*}}$ = 223 km s$^{-1}$ .",
"The middle panels of Figure REF show the resulting log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relations constructed within 0.5$R^{p}_{1/2}$ for the no feedback simulation (top) and winds only simulation (bottom), where the overall log($\\alpha _{\\rm IMF}$ ) of each galaxy is calculated based on the local, birth velocity dispersions of the stellar particles.",
"For the no feedback simulation, the shallow input relation (as used in Section REF ) produces a log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation that is shallower than the observed relation, similar to the corresponding Illustris-1 relation.",
"We find that the same steep input relation that was necessary to reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ trend for Illustris-1 is also able to reproduce the observed trend in the no feedback simulation.",
"The winds only simulation, on the other hand, varies from the Illustris and no feedback results.",
"As seen in bottom, middle panel of Figure REF , the shallow input relation does reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation while the steeper input relation results in a log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation that is $\\sim $ 2$\\times $ too steep.",
"The impact of varying the simulation feedback on the overall log($\\alpha _{\\rm IMF}$ ) calculated for each galaxy can be understood by considering what star-formation is suppressed.",
"AGN feedback suppresses star-formation in massive galaxies through both quasar and radio mode, while galactic winds suppress star-formation in lower mass galaxies with shallower potentials.",
"In the no feedback simulation, without AGN feedback or galactic winds, star-formation is not suppressed either in low- or high-mass galaxies.",
"The fraction of high to low velocity dispersion stellar particles in the no feedback simulation ends up being similar to the fraction in the full physics Illustris simulations, and a steep input relation is needed to reproduce the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ trend.",
"In the winds only simulation, star-formation in low-mass galaxies is suppressed but not in high-mass galaxies.",
"The fraction of high to low velocity dispersion stellar particles in enhanced, leading to galaxies having higher log($\\alpha _{\\rm IMF}$ ) values.",
"Since more massive galaxies have an increasing fraction of stellar particles formed ex-situ in lower-mass galaxies, an increasing fraction of low velocity dispersion stellar particles are suppressed, allowing the shallow input relation to reproduce the slope of the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ trend.",
"Of the physical quantities and simulation variations explored in this paper, a stellar mass function tilted in favor of star-formation in high-mass galaxies is able preserve the overall IMF trend without requiring a steep, physical IMF relation.",
"Although, indicated by the lack of massive quiescent galaxies, the galaxy population in the winds only simulation is unrealistic.",
"As shown in [96], not including AGN feedback results in a $z=0$ stellar mass function and stellar mass to halo mass relation that are too high compared to observations and the fiducial Illustris model, as well as a star-formation rate density and a stellar mass density functions with redshift also being too high.",
"Figure: The average radial IMF profiles in four velocity dispersion bins for Illustris-1, -2, and -3.",
"Each curve represents the average IMF mismatch parameter as a function of galactocentric radius, colored by velocity dispersion bin.",
"For higher velocity dispersions, Illustris-3 exhibits shallower radial profiles at R 1/2 p ^{p}_{1/2} < 1 compared to Illustris-1 and -2.Lastly, we begin to explore the effect of varying the IMF with which the simulation is run.",
"As discussed in Section REF , a Chabrier IMF law is used to govern mass and metal return from stellar particles in Illustris, as well as to calculate the mass-loading factors of galactic winds.",
"Our empirical approach to constructing the IMF mismatch parameter of Illustris galaxies is not expected to depend directly on the IMF used to run the simulation.",
"But, different mass returns and feedback may affect the local velocity dispersions which stellar particles are born into.",
"To test if the IMF that the simulation is run with alters our main result, we run simulations with different IMF laws but otherwise with the same physics models as in Illustris, using smaller boxes at lower resolutions.",
"We run a box of 40 Mpc/h on a side with $2\\times 320^{3}$ resolution elements incorporating two IMF laws: 1) a pure Salpeter IMF law with a slope of $x = 2.35$ and 2) the variable IMF law presented by [84] dependent on the local dark matter velocity dispersion $\\sigma _{\\rm birth}$ .",
"More details about these additional simulations are listed in Table REF .",
"Using the same galaxy selection criteria outlined in Section REF , 23 galaxies in the Salpeter simulation at $z = 0$ and 31 galaxies in the Spiniello simulation at $z = 0$ meet the stellar mass, specific star formation rate, and velocity dispersion criteria.",
"The average stellar mass, specific star formation rate, and stellar velocity dispersion of the Salpeter sample is $\\overline{M_{*}}$ = 10$^{11.34}$ , $\\overline{\\rm sSFR}$ = 1.27$\\times 10^{-12}$ yr$^{-1}$ , and $\\overline{\\sigma _{*}}$ = 190 km s$^{-1}$ , while for the Spiniello sample $\\overline{M_{*}}$ = 10$^{11.38}$ , $\\overline{\\rm sSFR}$ = 3.94$\\times 10^{-13}$ yr$^{-1}$ , and $\\overline{\\sigma _{*}}$ = 189 km s$^{-1}$ .",
"Again, we determine the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation using both the observed relation as input (Equation REF ) and the steeper log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm birth}$ relation we constructed which was found to reproduce the observed trend with global $\\sigma _{*}$ in the Illustris-1 analysis.",
"The rightmost panels of Figure REF show the resulting log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relations constructed within 0.5$R^{p}_{1/2}$ for our Salpeter and Spiniello IMF runs.",
"Consistent with the results in Section REF , for both modified IMF simulations the output log($\\alpha _{\\rm IMF}$ ) values constructed using the steeper log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm birth}$ input relation lie closer to the observed relation than the log($\\alpha _{\\rm IMF}$ ) values constructed with the shallow input relation.",
"While there are a few outliers and the number of high velocity dispersion galaxies in each simulation is small, incorporating a different and even variable IMF in the Illustris galaxy formation model does not seem to significantly modify the results of this paper.",
"Since the formal fits of the Salpeter and Spiniello output log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relations are shallower than the observed relation, this serves to strengthen our claim that a steeper input relation is required to produce the observations of IMF variations.",
"Lastly, while the galaxy population in these modified IMF simulations undoubtedly differs from the galaxy population in Illustris, the investigation of these differences and the implications of simulations which self-consistently include a variable IMF is the topic of future work."
],
[
"Simulation limitations",
"Before we discuss the possible implications of this work, it is important to reiterate the extent to which the results are dependent on the simulation models and resolution.",
"While the Illustris galaxy formation and evolution models reproduce many key galaxy properties and scaling relations, there is certainly room for improvement [99], [98], [35].",
"For example, Illustris produced massive galaxies with too high of a stellar mass [98], and galaxy sizes that are too large [69].",
"In particular, relevant to studying IMF variations through the hierarchical build-up of galaxies, is the merger rate.",
"[71] show the Illustris merger rate to match some observations well, though there remain qualitative differences among the observations.",
"If the merger rate in Illustris is too high, this might explain why a steep physical IMF relation is needed to conserve the global IMF relation to $z=0$ .",
"Another simulation parameter that could be influencing our results is resolution.",
"Observational IMF studies are starting to reveal that IMF variations are confined to the most inner parts of galaxies, at radii typically < 0.3R$_{e}$ [52], [92].",
"The resolution of the Illustris simulation, with the highest resolution simulation having a baryonic smoothing length of 0.7 kpc, prevents us from gaining a realistic understanding of IMF gradients at small radii.",
"As considered in Section REF , higher resolution simulations could yield more powerful nuclear starbursts, which in turn could reduce the steepness of the physical IMF laws we currently find necessary.",
"At this time though, there are no simulations at the relevant mass scale to test this hypothesis.",
"Given the above dependence on simulation details, it is advised that the results of this work be interpreted in a more qualitative sense.",
"The exact input relation we find necessary to reproduce the observed IMF trends are certainly sensitive to the galaxy formation and evolution model used, as well as resolution.",
"While the quantitative results may change with new and improved simulation, the qualitative results are more robust."
],
[
"Implications for IMF observations",
"Observations suggest that with increasing galaxy velocity dispersion, galaxies also have increasingly bottom-heavy IMFs where the most massive $z = 0$ galaxies are characterized by super-Salpeter IMF slopes.",
"Since these massive galaxies are believed to be primarily composed of ex-situ stellar populations, which formed in smaller systems with lower velocity dispersions and only later accreted onto the main galaxy, then the physical explanation for the steepness of the correlation between $z = 0$ global galaxy properties and the IMF is unclear.",
"The result that even steeper physical IMF relations are needed to preserve the observed IMF variations through the assembly of massive galaxies has implications for observations of IMF variations and theoretical work predicting IMF variations.",
"One consequence of our analysis is that massive, quiescent galaxies at higher redshifts would have global IMFs even more bottom heavy than their $z = 0$ counterparts.",
"This is because massive galaxies in Illustris first build up their in-situ stellar populations, which are mainly formed in high velocity dispersion environments, and only later accrete smaller systems with stellar populations formed in low velocity dispersion environments.",
"For massive galaxies, these smaller systems reside even within an effective radius where global IMF measurements are typically made [72], reducing the overall IMF of the galaxy.",
"Based on studies of ETG populations at $z \\sim 1$ and $z \\sim 1.4$ , observations beyond the local Universe are beginning to suggest that the IMF-$\\sigma $ relation remains roughly constant over the last 8 Gyrs [54], [33].",
"Contrary to these studies, [82] find that the overall log($\\alpha _{\\rm IMF}$ ) at fixed velocity dispersion decreases from the $z \\sim 0$ value out to $z \\sim 0.8$ .",
"In [81], they postulate that a possible source of this apparent evolution could be their assumption of fixed dark matter density profile.",
"A robust determination of the evolution of IMF-$\\sigma $ relation from $z \\sim 0$ to $z \\sim 2$ will require careful determination of the dark matter fraction of galaxies to break the dark matter-IMF degeneracy.",
"To reconcile our prediction for $z \\sim 2$ with observations that suggest a constant IMF-$\\sigma $ evolution, one option is to invoke a time-dependent physical IMF `law'.",
"In the context of our analysis, this would mean applying a shallower log($\\alpha _{\\rm IMF}$ )-$\\sigma _{\\rm birth}$ relation to higher-redshift stellar population.",
"Figure: The satellite fraction of z=0z=0 Illustris-1 galaxies in 4 velocity dispersion bins, normalized by the total satellite fraction in the bin, for each log(α IMF \\alpha _{\\rm IMF}) quintile.",
"The bars are colored according to velocity dispersion, and each group of bars shows one quintile.",
"The errors are calculated assuming a binomial distribution, where we add in quadrature the error associated with the total satellite fraction and the error associated with the satellite fraction in each quintile.",
"In the 200-225 km s -1 ^{-1} bin, satellite galaxies make up 65% of the most extreme log(α IMF \\alpha _{\\rm IMF}) galaxies, which is ∼\\sim 1.8×\\times larger than their total fraction in the bin.A further prediction of our analysis is that satellite galaxies, at fixed velocity dispersion, should have more bottom-heavy IMFs than central galaxies.",
"Since satellites are expected to undergo fewer minor mergers than centrals, they should better preserve to $z=0$ their bottom-heavy IMFs that are in place at higher redshift before they become satellites.",
"To demonstrate this prediction, we examine the satellite fraction for $z=0$ Illustris-1 galaxies in four velocity dispersion bins, with widths of 25 km s$^{-1}$ except for the highest $\\sigma _{*}$ bin that includes all 78 galaxies with $\\sigma _{*}$ > 225 km s$^{-1}$ .",
"For each velocity dispersion bin, we determine log($\\alpha _{\\rm IMF}$ ) quintiles and calculate the fraction of satellite galaxies within each quintile.",
"Figure REF shows the satellite fraction, normalized by the total satellite fraction in the velocity dispersion bins, for each log($\\alpha _{\\rm IMF}$ ) quintile.",
"We note that the formal errors associated with the normalized satellite fractions are large, especially when comparing different velocity dispersion bins within each quintile.",
"However, some individual bins are above or below the total satellite fraction with statistical significance.",
"The fifth quintile represents the galaxies with the most bottom-heavy IMFs in each velocity dispersion bin.",
"For the lowest velocity dispersion bin, from 150 - 175 km s$^{-1}$ , the satellite fraction is the same as the total satellite fraction in the bin, indicating that for these low velocity dispersion galaxies, satellites do not have higher log($\\alpha _{\\rm IMF}$ ) values than centrals.",
"This can be understood as lower velocity dispersion galaxies are more dominated by in-situ evolution and less affected by minor mergers.",
"In higher velocity dispersion bins, the fraction of satellites composing the most extreme log($\\alpha _{\\rm IMF}$ ) galaxies is higher than the total satellite fraction.",
"In particular, in the 200-225 km s$^{-1}$ bin $\\sim $ 65% of the highest log($\\alpha _{\\rm IMF}$ ) galaxies are satellites, which is significantly enhanced with respect to their total fraction in the bin.",
"For these galaxies with 200-225 km s$^{-1}$ , the highest IMF values of individual galaxies are log($\\alpha _{\\rm IMF}$ ) $\\sim $ 0.32 (compared with the typical log($\\alpha _{\\rm IMF}$ ) $\\sim $ -0.05 for this $\\sigma _{*}$ bin), and a majority of these `extreme IMF' galaxies are satellites.",
"For galaxies with $\\sigma _{*}$ > 225 km s$^{-1}$ , of the 8 galaxies with log($\\alpha _{\\rm IMF}$ ) > 0.20, 2 are satellites and 6 are centrals.",
"The most extreme log($\\alpha _{\\rm IMF}$ ) value for a central in this bin is log($\\alpha _{\\rm IMF}$ ) = 0.32 and for a satellite it is log($\\alpha _{\\rm IMF}$ ) = 0.36."
],
[
"Implications for IMF theory",
"In our post-processing analysis of Illustris, we find that steep physical IMF relations, as applied to the birth properties of stellar particles, are required to reproduce the observed $z = 0$ IMF trend with global velocity dispersion.",
"Input relations more than 3$\\times $ steeper than the observed relation are needed, which means that some individual stellar populations must be formed with mass-to-light ratios up to $\\sim $ 20$\\times $ greater than the Salpeter mass-to-light ratio.",
"These required extreme mass-to-light ratios are $\\sim $ 10$\\times $ greater than the overall mass-to-light ratios measured in observations of massive galaxies.",
"To gain an idea of what IMF slope could give rise to an $M_{*}/L$ ratio excess this large, we calculate mass-to-light ratios with the FSPS (Flexible Stellar Population Synthesis) library [16], [17] and the Python FSPS packagehttp://dan.iel.fm/python-fsps/.",
"We model a single burst of star-formation with solar metallicity and an exponentially declining star-formation history truncated at 4 Gyrs, and calculate the $r$ -band $M_{*}/L$ ratio at an age of 10 Gyr for several IMF slopes.",
"We find that unimodal IMF slopes greater than $x=4$ are required to produce a $M_{*}/L$ ratio that is $\\sim $ 20$\\times $ greater than the Salpeter $M_{*}/L$ ratio, where an IMF slope of $x=4$ results in a $M_{*}/L$ excess of $\\sim $ 11.5 and an IMF slope of $x=4.5$ results in an $M_{*}/L$ excess of $\\sim $ 24.",
"While IMF slopes this steep have yet to be robustly observed, an IMF slope this extreme could have more of an immediate implication for analytical IMF variation theories.",
"For example, the functional form of the CMF could be mapped to a unimodal IMF slope to determine how high of a Mach number would be required to produce these extreme $M_{*}/L$ ratios in theories that predict that high Mach number environments promote a bottom-heavy IMF [40], [12], [37].",
"Additionally, in Section REF we determined the overall log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation with an input relation based on star-formation rate.",
"Motivated by IGIMF theory, we assigned stellar particles born into low SFR galaxies a higher log($\\alpha _{\\rm IMF}$ ) value and stellar particles born into high SFR lower log($\\alpha _{\\rm IMF}$ ) values.",
"With this input relation, we are able to reproduce the slope and normalization of the observed log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation, but in the opposite direction.",
"Combined with our ability to reproduce the slope of the observed relation and its sign with an input relation based on $\\sigma _{\\rm birth}$ , this suggests that stellar particles formed in high SFR environments also form in high velocity dispersion environments.",
"If this is the case, then there is an apparent tension between the analytical IMF theories which predict that high Mach number (and therefore high velocity dispersion) environments promote a bottom-heavy IMF [40], [12], [37], and IGIMF theory [101] which predicts that high SFR environments lead to a top-heavy IMF.",
"While this tension seems to exist, the main driver of IMF variations in analytical and simulation work is far from settled.",
"For example, recent SPH simulations of star-formation [5] show no correlation between Mach number and peak of the IMF in high density environments.",
"Moreover, in low density environments they actually find that a higher Mach number shifts the peak of the IMF to higher masses, implying a more top-heavy IMF – opposite of what [40] predicts.",
"In general, if observations prove to be robust, any theory predicting IMF variations will have to accommodate both the correlation of overall IMF with global $z=0$ velocity dispersion and with metallicity.",
"One possibility is an IMF `law' which depends on a combination of local velocity dispersion and metallicity.",
"Combined, this could be an additional probe of Mach number, with metallicity controlling cooling and therefore the sound speed.",
"As discussed in [101], to resolve the tension between IGIMF theory and the observed mass excess of ETGs it is suggested that at least part of the inferred mass is due to an increased number of stellar remnants.",
"IMF variation studies focused on measuring dwarf sensitive spectral lines do break the degeneracy between low-mass stars and stellar remnants , , suggesting that the inferred, excess mass is low-mass stars.",
"However, the exact parameterization of the IMF of massive ETGs, especially below 1 M$_{\\odot }$ , is still unconstrained.",
"Determining the physical driver of IMF variations will require both theoretical work predicting the shape of the IMF as a function of environment and the ability to observationally constrain the shape of the IMF.",
"Recent work to constrain the shape of low-mass end of the IMF from high-quality spectra appears promising [19]."
],
[
"Comparison to other work",
"The closest approach in the existing literature to the post-processing IMF analysis of Illustris presented in this paper is that of [81].",
"For a sample of galaxies at $z=2$ , [81] assume that all subsequent stellar mass growth occurs via dry mergers, based on a toy merging model.",
"By using empirical log($\\alpha _{\\rm IMF}$ )-$\\sigma $ or log($\\alpha _{\\rm IMF}$ )-$M_{*}$ relations, a log($\\alpha _{\\rm IMF}$ ) value is assigned to each central galaxy at $z = 2$ .",
"The overall log($\\alpha _{\\rm IMF}$ ) of each galaxy at later redshifts is determined by the addition of smaller systems, with their own IMFs assigned by the same empirical relations.",
"While [81] explore the mixing between IMF dependence on stellar mass and velocity dispersion, the most directly comparable result to this work is the log($\\alpha _{\\rm IMF}$ )-$\\sigma $ relation based on the velocity dispersion model.",
"Here they find that the slope and normalization of the log($\\alpha _{\\rm IMF}$ )-$\\sigma $ relation is preserved from $z=2$ to $z=0$ , such that galaxies at fixed velocity dispersion have the same log($\\alpha _{\\rm IMF}$ ) at different times.",
"As shown in Section REF , our analysis predicts that at fixed velocity dispersion log($\\alpha _{\\rm IMF}$ ) is higher at $z=2$ than at $z=0$ .",
"These qualitatively discrepant results could be due to a number of factors stemming from significant methodology differences.",
"First, we assign a log($\\alpha _{\\rm IMF}$ ) value to stellar particles based on the birth velocity dispersion, whereas in [81] log($\\alpha _{\\rm IMF}$ ) values are assigned based on the $z=2$ velocity dispersion for the central galaxies and based on later redshift values for the smaller systems.",
"This difference could be why our analysis required steeper input relations to reproduce the observed relation in the first place, as birth velocity dispersion values are generally lower than at $z=2$ and later.",
"Another difference between the works is that the [81] model does not take into account the addition of newly quenched galaxies to the population after $z=2$ .",
"This certainly affects the comparison to our $z=0$ IMF-$\\sigma $ relation, which does include more recently quenched galaxies.",
"Finally, as recognized by [81], the pure dry merger model does not correctly reproduce the redshift evolution of the velocity dispersion of galaxies, suggesting that a dissipational component is missing.",
"The addition of such a component to their model would also break the constancy of the log($\\alpha _{\\rm IMF}$ )-$\\sigma $ relation.",
"On the other hand, our analysis is based on the innermost region of each galaxy, following observational constraints, while [81] do not address radial gradients.",
"When we consider the full galaxy extent, we find an even stronger shallowing of the log($\\alpha _{\\rm IMF}$ )-$\\sigma $ relation with redshift, suggesting the differences between the two studies are even larger than at face value.",
"As mentioned in the Introduction, IMF variations have been studied in the context of semi-analytical models.",
"In particular, [31] incorporates IGIMF theory into the GAEA (GAlaxy Evolution and Assembly) model to study the implications of IMF variations on the chemical evolution and dynamical properties of galaxies.",
"There, a broken power-law IMF is employed where for stellar masses less than $\\sim $ 1$M_{\\odot }$ the standard Kroupa IMF is adopted, and for masses greater than that the slope of the IMF is determined by the instantaneous star-formation rate.",
"In accordance with IGIMF theory, higher star-formation rates correspond to shallower (more top-heavy) IMF slopes at the high-mass end.",
"For $z=0$ galaxies formed under the IGIMF model, [31] compares the stellar mass-to-light ratio ($M_{*}/L$ ) and stellar mass excesses to the corresponding Chabrier equivalents.",
"Overall, their results are in qualitative agreement with observations of IMF variations, where higher mass galaxies exhibit a greater mass (and $M_{*}/L$ ) excess compared to the mass (or $M_{*}/L$ ) derived assuming a Chabrier IMF.",
"But for 10$^{9}$ $M_{\\odot }$ < $M_{*}$ < 10$^{10.8}$ $M_{\\odot }$ , there is a negative slope to the `true' versus Chabrier stellar mass relation, which only turns over and becomes positive for stellar masses 10$^{10.8}$ $M_{\\odot }$ < $M_{*}$ < 10$^{12.2}$ $M_{\\odot }$ .",
"In contrast, the overall IMF-$\\sigma _{*}$ found in this study (Section REF ), assuming a SFR dependent IMF relation where higher SFRs correspond to a less bottom-heavy IMF, exhibits an overall negative slope from the lowest to highest velocity dispersion galaxies.",
"While this tension could possibly be highlighting an interesting difference between the Illustris and GAEA galaxy formation models, there are several factors that make the comparison difficult including the post-processing nature of our analysis and our different parameterization of the IMF-SFR relation.",
"Concerning the chemical enrichment of galaxies, [31] find that the IMF variations they include are actually able to reproduce the observed $\\alpha $ -enhancement of massive galaxies ($M_{*}$ > 10$^{11}$ $M_{\\odot }$ ), though they do note that IMF variations are not the only solution to produce the observed metallicity of galaxies.",
"Adopting a standard universal IMF, the [O/Fe] ratio of galaxies in GAEA with $M_{*}$ = 10$^{12}$ $M_{\\odot }$ is $\\sim $ 0.2 dex too low compared to observations of massive galaxies.",
"At the time, the post-processing nature of our analysis prevents us from making a meaningful comparison to this result.",
"In future work, as discussed in the next section, we will self-consistently incorporate different IMF laws into the Illustris galaxy formation models, and by tracking the build-up of individual metals we will be able to study how the chemical enrichment of galaxies is altered."
],
[
"Prospects for future work",
"Future work will include validating our results with both higher resolution simulations, and improved galaxy formation models [69], [103].",
"High-resolution simulations of individual galaxies, at the relevant mass scales, will allow for a more realistic study of radial gradients in the IMF and fully capture the impact of star-formation occurring in nuclear starbursts.",
"Additionally, improved galaxy formation models that can better match observed galaxy properties than the current fiducial Illustris model can, such as the $z = 0$ stellar mass function, will be a crucial test of the robustness of our main result.",
"In addition to validating our results with improved simulations, future work will include expanding on our analysis.",
"Inspired by observations and theoretical work, in this study we examined five physical quantities associated with star-formation and/or IMF variations.",
"There are other quantities not considered here that have also been advocated to influence the IMF, including pressure and redshift [46], [59].",
"Combinations of quantities can also be considered, such as metallicity and velocity dispersion.",
"Expanding on the analysis of the variable IMF simulations presented in Section REF , further work is also required for studying the impact of a variable IMF that is incorporated self-consistently on the evolution and properties of galaxy populations.",
"Since the number of low- to high-mass stars in a galaxy determines the amount of metals injected into the ISM, the energy available for supernovae feedback, and the mass of baryons trapped in low-mass stars, the inclusion of a variable IMF could significantly alter our current picture of galaxy evolution.",
"While a variable IMF remains controversial because measuring the IMF is observationally difficult, the numerous studies reporting IMF variations and the analytical work predicting IMF variations in extreme star-formation environments necessitates that the implications of a variable IMF be investigated."
],
[
"Summary & Conclusion",
"In this study, we present an investigation of the physical origin of IMF variations using the cosmological simulation Illustris in post-processing.",
"For a sample of massive ($M_{*}$ > 10$^{10}$ $M_{\\odot }$ ) and quiescent (sSFR < 10$^{-11}$ yr$^{-1}$ ) galaxies we connect the physical conditions in which stellar particles form to the properties of the galaxies they reside in at $z = 0$ .",
"We do this by constructing the overall IMF mismatch parameter, $\\alpha _{\\rm IMF}$ , of each galaxy based on various formation conditions associated with the individual stellar particles that comprise it.",
"By attempting to reproduce the observed relations between overall IMF and global velocity dispersion, we are able to gain insight into how galaxy-wide quantities at $z = 0$ are related to the IMF – a local property defined at the time of star-formation.",
"Our findings are summarized as follows: [leftmargin=7mm] A much steeper than observed physical IMF relation is needed to reproduce the reported IMF trends with global $z = 0$ velocity dispersion under the hierarchical assembly of massive galaxies.",
"This result ties observations of an IMF that varies with $z = 0$ galactic properties to a physical origin, but requires some individual stellar populations to be formed with super-Salpeter IMFs that are even steeper than the IMFs reported by observational studies.",
"These extreme IMFs are up to $\\sim $ 20$\\times $ in excess of the Salpeter $M_{*}/L$ , which could imply a unimodal IMF slope of $x$ > 4.",
"Of the five physical quantities we consider, we are able to reproduce the observed IMF trend with $z = 0$ velocity dispersion by constructing the overall $\\alpha _{\\rm IMF}$ of each galaxy based on the global and local birth velocity dispersion of the stellar particles, and the global star-forming gas velocity dispersion of the progenitor galaxy in which each stellar particle was formed.",
"All of these quantities are roughly related to Mach number, which in analytical models of IMF variations corresponds to increased fragmentation on lower mass scales in more extreme star-formation environments.",
"We are unable to reproduce the observed IMF trend with $z=0$ velocity dispersion when constructing the overall $\\alpha _{\\rm IMF}$ of each galaxy based on the metallicities of individual stellar particles.",
"The relations obtained in this way are too shallow.",
"This is due to the scatter and near flatness of the [M/H]-$\\sigma _{*}$ relation, and to the fact that the global metallicity of each galaxy is composed of a broad distribution of individual stellar particle metallicities.",
"Using the star-formation rate of the progenitor galaxy in which each individual stellar particle was formed to construct the overall $\\alpha _{\\rm IMF}$ , we obtain steep relations between IMF and $z=0$ velocity dispersion, but in the opposite direction to what is observed.",
"Inspired by IGIMF theory, we construct a log($\\alpha _{\\rm IMF}$ )-SFR relation in which a low SFR corresponds to a bottom-heavy IMF.",
"For the $z=0$ massive quiescent galaxies we focus on in this study, this simple relation does not reproduce the direction of the observed IMF trend.",
"If stellar particles which form in merger induced starbursts have both high velocity dispersions and high SFRs, there is tension between IMF theories as to whether these stellar populations are expected to form with a bottom-heavy or top-heavy IMF.",
"We find radial gradients in the constructed log($\\alpha _{\\rm IMF}$ ) for massive galaxies due to the different formation conditions of stellar particles that reside at the center of galaxies versus in the outer regions.",
"This result reinforces the need for consistent comparisons of IMF variations across observational studies that may be probing the IMF at different radii.",
"It also further supports the idea that the IMF is a local property of galaxies, which are composites of numerous stellar populations formed in a diverse range of physical conditions throughout cosmic time.",
"The scatter in the constructed Illustris IMF relations is reflective of the diverse formation histories of galaxies that have similar $z = 0$ velocity dispersions.",
"In other words, galaxies with similar galactic quantities can be comprised of stellar populations which formed in different physical environments.",
"This is in agreement with, and can provide a possible explanation for, the scatter in the observed IMF-$\\sigma {*}$ relations.",
"Based on our analysis we make two predictions for observations: (1) the log($\\alpha _{\\rm IMF}$ )-$\\sigma _{*}$ relation at high redshift would be more bottom-heavy than the $z=0$ relation, and (2) at high velocity dispersions galaxies with extreme bottom-heavy IMFs are preferentially satellite galaxies."
],
[
"Acknowledgements",
"We thank Richard Bower and Romain Teyssier for useful discussions, and Martin Sparre for sharing with us the outputs of his zoom-in simulations.",
"We also thank the anonymous referee for a helpful report.",
"KB is supported by the NSF Graduate Research Fellowship under grant number DGE 16-44869.",
"SG acknowledges support provided by NASA through Hubble Fellowship grant HST-HF2-51341.001-A awarded by the STScI, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.",
"GB acknowledges financial support from NASA grant NNX15AB20G and NSF grant AST-1615955.",
"The Flatiron Institute is supported by the Simons Foundation.",
"The Illustris simulation was run on the CURIE supercomputer at CEA/France as part of PRACE project RA0844, and the SuperMUC computer at the Leibniz Computing Centre, Germany, as part of project pr85je.",
"Modified simulations used in this study were run on Columbia University's High Performance Computing cluster Yeti and on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.",
"Post-processing analysis of simulation data was run on the Comet cluster hosted at SDSC, making use of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575."
],
[
"Input & output IMF relations",
"For each simulation studied, Table REF shows the input relations constructed based on the indicated star-formation quantity and the fit to the resulting log($\\alpha _{\\rm IMF}$ )-log($\\sigma _{*}$ ) output relation.",
"The output relations listed under 0.5$R^{p}_{1/2}$ correspond to the relations shown in Figure REF and Figure REF .",
"Table: IMF Relations"
]
] | 1612.05658 |
[
[
"A hadronic minute-scale GeV flare from quasar 3C 279?"
],
[
"Abstract The flat spectrum radio quasar 3C 279 is a known $\\gamma$-ray variable source that has recently exhibited minute-scale variability at energies $>100$ MeV.",
"One-zone leptonic models for blazar emission are severely constrained by the short timescale variability that implies a very compact emission region at a distance of hundreds of Schwarzschild radii from the central black hole.",
"Here, we investigate a hadronic scenario where GeV $\\gamma$-rays are produced via proton synchrotron radiation.",
"We also take into account the effects of the hadronically initiated electromagnetic cascades (EMC).",
"For a $\\gamma$-ray emitting region in rough equipartition between particles and kG magnetic fields, located within the broad-line region (BLR), the development of EMC redistributes the $\\gamma$-ray luminosity to softer energy bands and eventually leads to broad-band spectra that differ from the observed ones.",
"Suppression of EMC and energy equipartition are still possible, if the $\\gamma$-ray emitting region is located beyond the BLR, is fast moving with Doppler factor ($>70$), and contains strong magnetic fields ($>100$ G).",
"Yet, these conditions cannot be easily met in parsec-scale jets, thus disfavouring a proton synchrotron origin of the Fermi-LAT flare."
],
[
"Introduction",
"Variability ranging from few hours to several weeks has been commonly observed at various energy bands of the blazar spectrum.",
"What came as a surprise was the detection of short (few minutes) timescale variability at very high energy $\\gamma $ -rays (VHE, $E_{\\gamma }>100$ GeV) from several blazars, including BL Lac objects [Mrk 421 , Mrk 501 , and PKS 2155-304 ] and flat spectrum radio quasars (FSRQ) .",
"The latest addition to the above list is the minute-scale flare detected on June 2015 at GeV $\\gamma $ -rays with Fermi-LAT from FSRQ 3C 279 .",
"The 2015 Fermi-LAT flare was characterized by high apparent $\\gamma $ -ray luminosity ($L_{\\gamma }\\sim 10^{49}$ erg s$^{-1}$ ), short flux-doubling timescales ($\\lesssim 5$ minutes), and high values of the Compton dominance parameter ($\\kappa =100$ ).",
"The observed minute-scale variability suggests a very compact emission region at a distance of hundreds of Schwarzschild radii from the central black hole.",
"In leptonic external Compton models (ECS) , , the observed high Compton dominance requires a strongly matter-dominated emitting region .",
"Models invoking the presence of relativistic protons in the jet have been proposed as an alternative to the ECS scenario.",
"In these models, $\\gamma $ -rays may result from a hadronically initiated electromagnetic cascade (EMC) or from relativistic proton synchrotron radiation , .",
"The efficiencies of both hadronic processes are expected to be enhanced at sub-parsec dissipation distances .",
"In this letter, we investigate if a proton synchrotron origin of the GeV flare is possible.",
"The leptohadronic synchrotron model (LHS) for blazars is often criticized for its energetic requirements .",
"Recently, – henceforth PD16, showed that the LHS model of the VHE blazar radiation can have sub-Eddington absolute jet powers , whereas models of dominant $>100$ MeV radiation in FSRQ may still require excessive power .",
"We do not discuss a photomeson ($\\rm p\\gamma $ ) origin of the flare here, as this would have, in principle, higher energetic requirements than the LHS model , and would result in flatter X-ray-to-$\\gamma $ -ray spectra than observed; a detailed calculation will be presented elsewhere.",
"As a starting point, we seek for parameters that minimize the jet power and provide a successful fit to the time-averaged flare's spectrum from Orbit D .",
"To do so, we use the analytical expressions presented in PD16 and the main observables of the flare (variability timescale $t_{\\rm v}$ , $\\gamma $ -ray luminosity $L_{\\gamma }$ , and photon energy $E_{\\gamma }$ ).",
"It is possible that the produced $\\gamma $ -rays do not escape but they are absorbed by the softer photons in the source initiating EMC cascades which alter drastically the broad-band spectrum as they distribute the $\\gamma $ -ray energy to lower frequencies, thus destroying the attempted spectral fit.",
"To assess the role of EMC, we derive analytical expressions for the optical depth for internal photon-photon ($\\gamma \\gamma $ ) absorption and the efficiency for $\\rm p\\gamma $ interactions.",
"The parameter values derived in the first step are then used as an input to a numerical code that calculates the broad-band photon spectrum taking into account all relevant physical processes .",
"In our numerical investigation, we compare the broad-band photon spectra obtained with and without internal $\\gamma \\gamma $ absorption.",
"If, for parameters that minimize the jet power, the absorption of VHE radiation produced via $\\rm p\\gamma $ interactions initiates an EMC with a spectrum that does not describe well the data, we search for other parameters that may suppress the development of cascades.",
"Table: Acknowledgments"
]
] | 1612.05699 |
[
[
"Electromagnetic radiation and collectivity in small quark-gluon droplets"
],
[
"Abstract We study the multiplicity and rapidity dependence of thermal and prompt photon production in p+Pb collisions at 5.02 TeV, using a (3+1)D viscous hydrodynamic framework.",
"Direct photon anisotropic flow coefficients $v^\\gamma_{2,3}$ and nuclear modification factor $R^\\gamma_\\mathrm{pPb}(p_T)$ are presented in both the p-going (backward) and the Pb-going (forward) directions.",
"The interplay between initial state cold nuclear effect and final state thermal enhancement at different rapidity regions is discussed.",
"The proposed rapidity dependent thermal photon enhancement and direct photon anisotropic flow observables can elucidate non-trivial longitudinal dynamics of hot quark-gluon plasma droplets created in small collision systems."
],
[
"Introduction",
"Recent measurements of high energy proton-nucleus collisions at the Large Hadron Collider (LHC) show intriguing evidence of collective behaviour which suggest the creation of hot and strongly coupled quark-gluon plasma (QGP) [1], [2], [3].",
"Such collisions offer the opportunity to study QCD (Quantum Chromodynamics) under extreme conditions.",
"Theoretical analyses using viscous hydrodynamics can provide a reasonable description of the measured hadronic flow observables [4], [5], [6].",
"Understanding the origin of such a strong collectivity in small collision systems is currently a very active topic in the field.",
"In order to quantify the properties of the QCD matter produced in these collisions, one needs to study multiple aspects of experimental observables within a consistent framework.Thus, in addition to hadronic flow observables, we will study penetrating electromagnetic probes from these small collision systems.",
"Electromagnetic observables, such as direct photons, are regarded as clean probes to the produced matter in heavy-ion collisions [7], [8].",
"Owing to the small system size in p+Pb collisions, the produced photons can escape the medium without any further interaction.",
"Hence, they carry undistorted information from their local production points to the detectors.",
"This work studies the multiplicity and rapidity dependence of direct photon observables in p+Pb collisions at 5.02 TeV using event-by-event (3+1)D viscous hydrodynamic simulations.",
"The results presented here provide information complementary to that in our systematic study of Refs.",
"[6], [9]."
],
[
"Results and Discussion",
"The event-by-event dynamical evolution of p+Pb collision at 5.02 TeV is modelled using a hydrodynamics + hadronic cascade hybrid framework [6].",
"The Monte-Carlo Glauber model is employed to generate fluctuating initial conditions.",
"Collision-by-collision multiplicity fluctuations are included to reproduce the measured charged hadron multiplicity distributions [10].",
"The rapidity dependence of the initial energy density profile is parametrized with envelope functions [11], [6].",
"The individual energy density profile is then evolved using (3+1)D viscous hydrodynamics MUSIC [12] coupled with a lattice based equation of state s95p-v1.2 [13].",
"Individual fluid cells are converted to particles at an isothermal hyper-surface, where the switching temperature is $T_\\mathrm {sw} = 155$ MeV.",
"Particles are fed into a hadronic transport model (UrQMD) for further scatterings and decays.",
"We choose a specific shear viscosity of the QGP phase $\\eta /s = 0.1$ , to reproduce the hadronic flow observables (see Ref.",
"[6]).",
"For direct photon production, we consider contributions from prompt and thermal radiation.",
"Pre-equilibrium photon contributions from a saturated gluon phase [14] are not considered here.",
"The prompt photons are evaluated with perturbative QCD at next-to-leading order in $\\alpha _s$ [6].",
"The isospin effect is included for the Pb nucleus.",
"Cold nuclear effects are taken into account with the nCTEQ15 nuclear parton distribution functions [15].",
"Thermal radiation is computed by folding thermal photon emission rates with the evolving hot medium.",
"The details of the calculations can be found in Ref. [6].",
"Figure: Panel (a): Charged hadron multiplicity as a function of pseudorapdity compared with the ATLAS measurements in 0-1% p+Pb collisions at 5.02 TeV.",
"Positive rapidity is the Pb-going direction.",
"Panel (b): The rapidity dependence of thermal and prompt photon yields in 0-1% p+Pb collisions at 5.02 TeV.",
"Panel (c): The p T p_T-integrated direct photon v n v_n as a function of photon rapidity.",
"In Panels (b) and (c), the transverse momentum is integrated from 1 to 3 GeV.We perform numerical simulations of p+Pb collisions from central to 50% in collision centrality.",
"Fig.",
"REF a shows the prompt and thermal photon yields as a function of the final pion multiplicity.",
"Good power law scaling with $N_\\pi $ is found for the individual prompt and thermal component.",
"The prompt photon yield increases slower than $N_\\pi $ in p+Pb collisions, with a power exponent $\\sim 0.5$ .",
"It is smaller than its value in large symmetric nucleus-nucleus collisions, where $N^\\gamma _\\mathrm {prompt} \\propto N_\\pi ^{1.2}$ [16].",
"This slow growth of prompt photon yield helps the thermal photon radiation to shine out in the total direct photon signal in central p+Pb collisions.",
"The thermal photon yield goes like $N_\\pi ^{1.69}$ , much faster than the prompt source.",
"This can be understood as a superposition of the effects from an increase of the system's average temperature and of space-time volume as one goes to central collisions [6].",
"In Fig.",
"REF a, the difference between the open black circles and red filled points shows that the out-of-equilibrium correction to the photon rates coming from shear viscosity, usually denoted as $\\delta f$ , does not affect the $p_T$ -integrated photon yields.",
"The multiplicity dependence of direct photon anisotropic flow coefficients $v^\\gamma _{2,3}$ is presented in Fig.",
"REF b.",
"Compared to charged hadron $v^\\mathrm {ch}_n$ , direct photon $v^\\gamma _{2,3}$ shows a stronger multiplicity dependence.",
"In the low multiplicity events $N_\\pi < 10$ , prompt photons dominate the total direct photon signal.",
"Since they don't carry any momentum anisotropy in our framework, the direct photon $v^\\gamma _n$ is strongly suppressed.",
"As one goes to central p+Pb collisions, the thermal radiation wins over the prompt contribution in the direct photon signal.",
"The absolute values of direct photon $v^\\gamma _{2,3}$ are comparable to the charged hadron $v^\\mathrm {ch}_n$ .",
"Finally, by comparing the open symbols with filled points, we estimate that the $\\delta f$ corrections to the photon emission rates from shear viscosity suppress the direct photon $v^\\gamma _{2,3}$ by about 10-20%.",
"Because of the asymmetry of the colliding nuclei, p+Pb collisions break the longitudinal boost-invariance explicitly.",
"They provide an opportunity to study non-trivial longitudinal dynamics with direct photons.",
"Fig.",
"REF a shows the pseudo-rapidity distribution of the charged hadron multiplicity $dN^\\mathrm {ch}/d\\eta $ in 0-1% p+Pb collisions.",
"The longitudinal profile for the initial energy density in the MC-Glauber initial condition is adjusted to reproduce ATLAS measurements [17].",
"Since there are more participant nucleons from the Pb nucleus, more energy is deposited to the forward rapidity region compared to the backward p-going direction.",
"Thus, the fireball in the forward direction achieves a higher peak temperature at the initial time and lives longer compared the one in the backward rapidity region.",
"This longitudinal asymmetry of energy deposition directly imprints itself in the rapidity dependence of the thermal photon production $dN^\\gamma _\\mathrm {thermal}/dy$ , as shown in Fig.",
"REF b.",
"The thermal photon $dN^\\gamma _\\mathrm {thermal}/dy$ has a similar shape as that of the charged hadron $dN^\\mathrm {ch}/d\\eta $ .",
"The prompt photon yield is slightly suppressed in the Pb-going direction compared to its value at mid-rapidity.",
"This is caused by the nuclear shadowing effect from the Pb nuclear PDF.",
"This shadowing effect is reduced in the backward p-going direction.",
"So with such an interplay between thermal and prompt photons in the different rapidity regions, the thermal photon radiation shines over the prompt photon background in the Pb-going direction.",
"The maximum thermal to prompt ratio can be reached in the rapidity region $1 < y < 2$ .",
"Fig.",
"REF c further highlights the $p_T$ -integrated direct photon anisotropic flow coefficients as a function of rapidity.",
"The direct photon $v^\\gamma _{2,3}(y)$ also reaches a maximum in the rapidity region $1 < y < 2$ in the Pb-going direction where the thermal photon production peaks.",
"Because of the dilution by prompt photons, the direct photon $v^\\gamma _{2,3}$ has a stronger rapidity dependence than that of charged hadron $v^\\mathrm {ch}_n(\\eta )$ shown in Ref. [6].",
"Figure: The direct photon nuclear modification factor R pPb γ (p T )R^\\gamma _\\mathrm {pPb}(p_T) in 0-1% p+Pb collisions at three different rapidity regions.",
"Positive rapidity denotes the Pb-going side.Finally, the direct photon nuclear modification factor $R^\\gamma _\\mathrm {pPb}$ in different rapidity regions is summarized in Fig.",
"REF .",
"In central 0-1% p+Pb collisions, a thermal enhancement in $R^\\gamma _\\mathrm {pPb}$ can be observed in the region where $1 < p_T < 3$ GeV.",
"The enhancement is larger in the forward rapidity region owing to the hotter and longer lived fireball.",
"The maximum $R^\\gamma _\\mathrm {pPb}$ can reach up to $\\sim 2.5$ in the rapidity window $1 < y < 2$ ."
],
[
"Conclusion",
"We study the multiplicity and rapidity dependence of thermal photon radiation in p+Pb collisions at 5.02 TeV.",
"Both direct photon yield and its anisotropic flow coefficients shows a strong multiplicity dependence.",
"In central p+Pb collisions, thermal photons can leave a measurable signal in direct photon observables with $1 < p_T < 3$ GeV.",
"We found an interesting and opposite rapidity dependence for thermal and prompt photon production in p+Pb collisions.",
"Thermal enhancement in the total direct photon signal is expected to be more pronounced in the forward Pb-going region compared to the mid-rapidity and p-going direction.",
"The medium created in $1 < y < 2$ is hotter, and lives for a longer lifetime when compared with other rapidity windows.",
"Prompt photon production in this rapidity range is slightly suppressed by the larger nuclear shadowing effect in the Pb nucleus.",
"Future measurements of the rapidity dependence of direct photon observables can therefore help us to further understand the longitudinal dynamics of p+Pb collisions."
],
[
"Acknowledgement",
"This work was supported in part by the U.S. Department of Energy, Office of Science under contract No.",
"DE- SC0012704 and the Natural Sciences and Engineering Research Council of Canada.",
"C.S.",
"gratefully acknowledges a Goldhaber Distinguished Fellowship from Brookhaven Science Associates, and C. G. gratefully acknowledges support from the Canada Council for the Arts through its Killam Research Fellowship program.",
"J.-F.P.",
"was supported by the U.S. D.O.E.",
"Office of Science, under Award No.",
"DE-FG02-88ER40388.",
"Computations were made in part on the supercomputer Guillimin from McGill University, managed by Calcul Québec and Compute Canada.",
"The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), NanoQuébec, RMGA and the Fonds de recherche du Québec - Nature et technologies (FRQ-NT)."
]
] | 1612.05464 |
[
[
"Super-Walk Formulae for Even and Odd Laplacians in Finite Graphs"
],
[
"Abstract The number of walks from one vertex to another in a finite graph can be counted by the adjacency matrix.",
"In this paper, we prove two theorems that connect the graph Laplacian with two types of walks in a graph.",
"By defining two types of walks and giving orientation to a finite graph, one can easily count the number of the total signs of each kind of walk from one element to another of a fixed length."
],
[
"Introduction",
"In graph theory and combinatorics, an interesting problem is that to find the number of different ways a certain operation can be performed on a given graph.",
"For example, given two vertices $v_i$ and $v_j$ , a natural question is: how many different walks of length $k$ connect $v_i$ and $v_j$ .",
"The answer to this question has interesting application to random walks, spectral graph theory, and surprisingly, discrete versions of quantum mechanics, in which such number of paths appear in the partition function for graph quantum mechanics [3].",
"Such physical interpretation can be adapted to the case of super-symmetric quantum mechanics, for which both edges and vertices span the space of quantum states of the system.",
"The particular aim of this paper is to study generalized walks associated to a finite graph that we call of super-walks and edge super-walks.",
"Super-walks differ from the conventional walks between vertices by the fact that staying in a vertex after choosing an edge is allowed, and that such walks come equipped with a natural orientation.",
"Edge super-walks are in some way dual to the super-walks, in the sense that now the walks are considered to be between edges.",
"While powers of the adjacency matrix of a graph can be used to determine the number of conventional walks between vertices, we will show that another matrix associated with a graph, the even Laplacian $\\Delta ^+$ , provides a counting formula for super-walks (Theorem REF ).",
"Similar expressions were proved via Feynman-type expansion of the graph Laplacian in the case of regular graphs by Mnëv [3], [4], and for general finite graphs by del Vecchio [2].",
"We provide a combinatorial argument for such formulae, using induction on the length of the super-walks.",
"Edge super-walks appear naturally while studying the super-symmetric version of graph quantum mechanics [1], [2], [3], [4], for which the walks are considered to be between edges rather than vertices.",
"Theorem REF provides a counting formula for edge super-walks in terms of the odd Laplacian $\\Delta ^-$ .",
"It turns out that the number of such walks depends on the orientation of the graph, and in Section we will explain how.",
"This paper is organized as follows.",
"In Section , we introduce the different notions of walks and some examples.",
"In Section , super-walk is introduced for non-oriented graphs, and we prove the corresponding counting formula (Theorem REF ).",
"In Section , we introduce edge super-walks as a super-symmetric version of conventional walks, and we determine their counting formula (Theorem REF ).",
"In the last section, we give an outlook of further directions and generalizations."
],
[
"Walks",
"We start by recalling the usual notion of walk in a finite graph.",
"In this section, Theorem REF shows that the number of walks from vertices to vertices of fixed length can be counted by the corresponding adjacency matrix and its powers.",
"A graph $\\Gamma $ is an ordered pair $(V, E)$ comprising a set $V$ of vertices together with a set $E$ of edges, which are 2-element subsets of $V$ (that is an edge is associated with two vertices, and the association takes the form of the unordered pair of the vertices).",
"If the set $V$ and the set $E$ are finite sets, we say $\\Gamma $ is a finite graph.",
"A finite graph $\\Gamma $ is often indicated by the adjacency matrix which is defined by $A_{\\Gamma }:=A(i,j) = \\left\\lbrace \\begin{matrix}1 & \\text{if } i\\text{ is adjacent to }j\\\\0 & \\text{otherwise}\\end{matrix}\\right.$ .",
"The incidence matrix $I$ , the even graph Laplacian $\\Delta ^{+}_{\\Gamma }$ , and the odd graph Laplacian $\\Delta ^{-}_{\\Gamma }$ are also used to indicate a graph.",
"Let $\\Gamma $ be an oriented graph, with vertices $v_{1},v_{2},\\ldots ,v_{|V|}$ , and edges $e_{1},e_{2},\\ldots ,e_{|E|}$ .",
"The incidence matrix $I$ of $\\Gamma $ is a $|V|\\times |E|$ matrix given by $I(i,j)=\\left\\lbrace \\begin{matrix}1 & \\text{if } e_{j} \\text{ ends at } v_{i}\\\\-1 & \\text{if } e_{j} \\text{ starts at } v_{i}\\\\0 & \\text{otherwise}\\end{matrix}\\right.$ .",
"The even graph Laplacian is defined by $\\Delta ^{+}_{\\Gamma } :=II^{t}$ .",
"The odd graph Laplacian is defined by $\\Delta ^{-}_{\\Gamma } :=I^{t}I$ .",
"Definition 2.1 Given a graph $\\Gamma $ with vertices $v_{1}, v_{2}, \\ldots , v_{|V|}$ , and edges $e_{1}, e_{2}, \\ldots , e_{|E|}$ : (1) A walk of length 1 is a walk which starts at $v_{i}$ , goes to one of its neighbor $v_{j}$ .",
"(*) (2) A walk of length $k$ is a walk which starts at $v_{i}$ , repeats (*) $k$ times, and ends at $v_{j}$ .",
"Example 2.2 In Figure REF : (1) $v_{1}\\overset{e_{2}}{\\rightarrow }v_{4}$ is a walk from $v_{1}$ to $v_{4}$ of length 1; (2) $v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}\\overset{e_{3}}{\\rightarrow }v_{4}\\overset{e_{4}}{\\rightarrow }v_{5}\\overset{e_{7}}{\\rightarrow }v_{3}\\overset{e_{6}}{\\rightarrow }v_{6}$ and $v_{1}\\overset{e_{2}}{\\rightarrow }v_{4}\\overset{e_{4}}{\\rightarrow }v_{5}\\overset{e_{5}}{\\rightarrow }v_{6}\\overset{e_{6}}{\\rightarrow }v_{3}\\overset{e_{6}}{\\rightarrow }v_{6}$ are two walks from $v_{1}$ to $v_{6}$ of length 5.",
"Figure: The graph used in Example and Example Theorem 2.3 Let $A_\\Gamma $ be the adjacency matrix of a graph $\\Gamma $ .",
"$A_\\Gamma ^k(i,j)$ is the number of walks in $\\Gamma $ from $v_{i}$ to $v_{j}$ ."
],
[
"Super-Walks",
"Theorem REF gives the combinatorial interpretation of the entries of powers of the adjacency matrix.",
"It is also natural to consider the combinatorial interpretations of entries of graph Laplacians.",
"In this section, we formally define super-walk, and Theorem REF shows that the total signs of super-walks of fixed length can be calculated by the corresponding even graph Laplacian and its powers.",
"Definition 3.1 Given a graph $\\Gamma $ with vertices $v_{1}, v_{2}, \\ldots , v_{|V|}$ , and edges $e_{1}, e_{2}, \\ldots , e_{|E|}$ , we have: A super-walk of length 1 is a walk which starts at $v_{i_{1}}$ , and: (1) goes to one of its neighbor $v_{i_{2}}$ ; or (2) goes towards one of its neighbors $v_{i_{2}}$ , but does not reach it, and goes back to $v_{i_{1}}$ .",
"In case (1), we define the super-walk to have a negative sign, and in case (2), we define the super-walk to have a positive sign.",
"A super-walk of length $k$ from $v_{i}$ to $v_{j}$ is formed by repeating (1) and/or (2) $k$ times, starting at $v_{i}$ and ending at $v_{j}$ , where we determine the sign by multiplying together the signs of the steps.",
"Example 3.2 In Figure REF : (1) $v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}$ is a super-walk from $v_{1}$ to $v_{2}$ of length 1; (2) $v_{1}\\overset{e_{1}}{\\rightarrow }v_{1}$ and $v_{1}\\overset{e_{2}}{\\rightarrow }v_{1}$ are super-walks from $v_{1}$ to $v_{1}$ of length 1; (3) $v_{5}\\overset{e_{4}}{\\rightarrow }v_{5}$ , $v_{5}\\overset{e_{5}}{\\rightarrow }v_{5}$ , and $v_{5}\\overset{e_{7}}{\\rightarrow }v_{5}$ are super-walks from $v_{5}$ to $v_{5}$ of length 1.",
"Furthermore, we say (1) is a super-walk with negative sign, while (2) and (3) are super-walks with positive sign.",
"Moreover, (4) $v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}\\overset{e_{1}}{\\rightarrow }v_{2}$ , $v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}\\overset{e_{3}}{\\rightarrow }v_{2}$ , $v_{1}\\overset{e_{1}}{\\rightarrow }v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}$ , $v_{1}\\overset{e_{2}}{\\rightarrow }v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}$ , and $v_{1}\\overset{e_{2}}{\\rightarrow }v_{4}\\overset{e_{3}}{\\rightarrow }v_{2}$ are super-walks from $v_{1}$ to $v_{2}$ of length 2.",
"Furthermore, $v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}\\overset{e_{1}}{\\rightarrow }v_{2}$ , $v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}\\overset{e_{3}}{\\rightarrow }v_{2}$ , $v_{1}\\overset{e_{1}}{\\rightarrow }v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}$ , $v_{1}\\overset{e_{2}}{\\rightarrow }v_{1}\\overset{e_{1}}{\\rightarrow }v_{2}$ have one negative sign and one positive sign, thus we define each them with a negative sign.",
"While $v_{1}\\overset{e_{2}}{\\rightarrow }v_{4}\\overset{e_{3}}{\\rightarrow }v_{2}$ has two negative signs, thus we define it with a positive sign.",
"We now present one of our two main results, relating the number of super-walks with sign, to entries of power of the even Laplacian.",
"Theorem 3.3 Let $\\Delta ^+_\\Gamma $ be the even Laplacian of a graph $\\Gamma $ .",
"$(\\Delta {^{+}_{\\Gamma }})^{k}(i,j)=\\sum _{\\gamma ,i\\rightarrow j,k} sgn(\\gamma )$ where $\\gamma ,i\\rightarrow j,k$ is a super-walk $\\gamma $ that starts at vertex $i$ , ends at vertex $j$ , and has length $k$ .",
"Proof by induction on $k$ .",
"When $k=1$ , we have: $(\\Delta {^{+}_{\\Gamma }})^{k}=(\\Delta {^{+}_{\\Gamma }})^{1}$ .",
"Note that the even graph Laplacian can also be calculated by $\\Delta ^{+}_{\\Gamma } :=\\Delta ^{+}_{\\Gamma }(i,j)=\\left\\lbrace \\begin{matrix}\\text{val}(i) & \\text{if } i=j & (1)\\\\-1 & \\text{if } i\\text{ is adjacent to }j & (2)\\\\0 & \\text{otherwise} & (3)\\end{matrix}\\right.$ where val$(i)$ is the valence of a vertex $v$ , which equals to the number of neighbors of $v$ .",
"In case (1), $(\\Delta ^{+}_{\\Gamma })(i,i)=val(i)=$ the number of neighbors of $i=$ the number of super-walks of length 1 goes from vertex $i$ back to vertex $i$ .",
"Thus, the claim holds in case (1).",
"In case (2), $(\\Delta ^{+}_{\\Gamma })(i,j)=-1$ if $i$ is adjacent to $j$ .",
"In this case, there is only one super-walk from vertex $i$ to one of its neighbors vertex $j$ , and the sign is negative.",
"Thus, the claim holds in case (2).",
"In case (3), $(\\Delta ^{+}_{\\Gamma })(i,j)=0$ if vertex $i$ is not adjacent to vertex $j$ .",
"Since no super-walk of length 1 can go from vertex $i$ to vertex $j$ , the claim holds in case (3).",
"Therefore, the claim holds when $k=1$ .",
"Suppose the claim holds for some natural number $k$ .",
"Consider $k+1$ .",
"$(\\Delta {^{+}_{\\Gamma }})^{k+1}&=(\\Delta {^{+}_{\\Gamma }})^{k}\\cdot (\\Delta {^{+}_{\\Gamma }})^{1} \\\\ \\text{So, }(\\Delta {^{+}_{\\Gamma }})^{k+1}(i,j)&=\\sum _{q=1}^{n}(\\Delta {^{+}_{\\Gamma }})^{k}(i,q)(\\Delta ^{+}_{\\Gamma })(q,j) \\\\&=(\\Delta {^{+}_{\\Gamma }})^{k}(i,j)\\cdot (\\Delta ^{+} _{\\Gamma })(j,j)+\\overbrace{\\sum _{q=1}^{n}(\\Delta {^{+}_{\\Gamma }})^{k}(i,q)\\cdot (-1)}^{\\text{where } q \\text{ and } j \\text{ adjacent}}+\\overbrace{\\sum _{q=1}^{n}(\\Delta {^{+}_{\\Gamma }})^{k}(i,q)\\cdot 0}^{\\text{where } q \\text{ and } j \\text{ non-adjacent}} \\\\&=\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )\\cdot \\text{val}(j)+\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )\\cdot (-1)+0.$ Let $\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )\\cdot \\text{val}(j)=X$ .",
"In $X$ , we first count the super-walks from vertex $i$ to vertex $j$ of length $k$ by calculating $\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )$ .",
"Now we stand at vertex $j$ , and have to go from vertex $j$ to vertex $j$ of length 1.",
"Obviously, the number of the last step equals to val$(j)$ .",
"Since val$(j)$ is always non-negative, $X$ calculates $\\sum _{\\gamma } \\text{sgn}(\\gamma )$ where $\\gamma $ is one of the super-walks from vertex $i$ to vertex $j$ of length $k+1$ , and the last step going from vertex $j$ to vertex $j$ .",
"Let $\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )\\cdot (-1)=Y$ .",
"In $Y$ , we first count the super-walks from vertex $i$ to vertex $q$ of length $k$ by calculating $\\sum _{\\gamma ,i\\rightarrow q,k} \\text{sgn}(\\gamma )$ .",
"Now we stand at vertex $q$ , a neighbor of vertex $j$ , and have to go from vertex $q$ to vertex $j$ of length 1.",
"There is only one way to go from vertex $q$ to vertex $j$ of length 1, and its sign is negative.",
"That is $\\sum _{\\gamma ,q\\rightarrow j,1} \\text{sgn}(\\gamma )=-1$ .",
"Hence, $Y$ calculates $\\sum _{\\gamma } \\text{sgn}(\\gamma )$ where $\\gamma $ is one of the super-walks from vertex $i$ to vertex $j$ of length $k+1$ , and the last step going from vertex $q$ to vertex $j$ .",
"Therefore, $X+Y$ calculates $\\sum _{\\gamma ,i\\rightarrow j,k+1} \\text{sgn}(\\gamma )$ .",
"Hence, the claim holds when $k$ is replaced by $k+1$ .",
"This is the end of the proof."
],
[
"Edge Super-Walks",
"In this section, we define edge super-walk.",
"Similar to the super-walk, edge super-walk is a sign-sensitive edge-to-edge walk in an oriented graph.",
"Particularly, the sign of edge super-walk depends on not only the way each step goes, but also the orientation of the graph, which is defined in Definition REF in this section.",
"Theorem REF shows that the total signs of edge super-walks of fixed length can be calculated by the corresponding odd graph Laplacian and its powers.",
"Definition 4.1 Given an oriented graph $\\Gamma $ .",
"Let $i$ be an edge in $\\Gamma $ which starts at vertice $A$ and ends at vertice $B$ .",
"We say the sign of $A$ in $i$ is negative, and the sign of $B$ in $i$ is positive.",
"Example 4.2 In the figure REF , we say that the sign of $v_{5}$ in $e_{4}$ is positive; the sign of $v_{5}$ in $e_{5}$ in negative; the sign of $v_{5}$ in $e_{7}$ is positive.",
"Figure: The graph used in Example and Example Definition 4.3 Given an oriented graph $\\Gamma $ with vertices $v_{1}, v_{2}, \\ldots , v_{|V|}$ , and edges $e_{1}, e_{2}, \\ldots , e_{|E|}$ , we have: An edge super-walk of length 1 is a walk starts at $e_{i_{1}}$ , and: (1) goes to one of its neighbor edge $e_{i_{2}}$ through the vertex $v_{i}$ , where $e_{i_{1}}$ and $e_{i_{2}}$ intersect at $v_{i}$ ; or (2) goes through one of the end points and back to itself.",
"In case (1), if the sign of $v_{i}$ in $e_{i_{1}}$ and the sign of $v_{i}$ in $e_{i_{2}}$ are the same, we define this walk to have a positive sign; if not, we define this walk to have a negative sign.",
"In case (2), the walk is defined to have a positive sign.",
"An edge super-walk of length $k$ from $e_{i}$ to $e_{j}$ is formed by repeating (1) and/or (2) $k$ times, starting at $e_{i}$ and ending at $e_{j}$ , where we determine the sign by multiplying together the signs of the steps.",
"Example 4.4 In Figure REF : (1) $e_{1}\\overset{v_{1}}{\\rightarrow }e_{2}$ is an edge super-walk from $e_{1}$ to $e_{2}$ of length 1; (2) $e_{1}\\overset{v_{1}}{\\rightarrow }e_{1}$ and $e_{1}\\overset{v_{2}}{\\rightarrow }e_{1}$ are edge super-walks from $e_{1}$ to $e_{1}$ of length 1.",
"Furthermore, we say (1) is an edge super-walk with a negative sign, while both walks in (2) have a positive sign.",
"Moreover, $e_{1}\\overset{v_{1}}{\\rightarrow }e_{1}\\overset{v_{1}}{\\rightarrow }e_{2}$ , $e_{1}\\overset{v_{2}}{\\rightarrow }e_{1}\\overset{v_{1}}{\\rightarrow }e_{2}$ , $e_{1}\\overset{v_{1}}{\\rightarrow }e_{2}\\overset{v_{1}}{\\rightarrow }e_{2}$ , $e_{1}\\overset{v_{1}}{\\rightarrow }e_{2}\\overset{v_{4}}{\\rightarrow }e_{2}$ , and $e_{1}\\overset{v_{2}}{\\rightarrow }e_{3}\\overset{v_{4}}{\\rightarrow }e_{2}$ are edge super-walks from $e_{1}$ to $e_{2}$ of length 2.",
"Furthermore, all of the first four edge super-walks have a positive sign, while the last one has a negative sign.",
"We now present the other main result, relating the number of edge super-walks with sign, to entries of power of the odd Laplacian.",
"Theorem 4.5 Let $\\Delta ^-_\\Gamma $ be the odd Laplacian of a graph $\\Gamma $ .",
"$(\\Delta _{\\Gamma }^{-})^{k}=\\sum _{\\gamma ,i\\rightarrow j,k} sgn(\\gamma )$ where $\\gamma ,i\\rightarrow j,k$ is an edge super-walk $\\gamma $ that starts at edge $i$ , ends at edge $j$ , and has length $k$ .",
"Prove by induction on $k$ .",
"When $k=1$ , we have: $(\\Delta _{\\Gamma }^{-})^{k}=(\\Delta _{\\Gamma }^{-})^{1}=I^{t}I$ .",
"Therefore, $\\Delta _{\\Gamma }^{-}(i,j)=\\sum _{q=1}^{n}I_{qi}I_{qj}$ .",
"By the definition of incidence matrix, $I_{ij}=0 \\text{ or } 1 \\text{ or } -1$ .",
"Case (1), if $i=j$ , we have $\\Delta _{\\Gamma }^{-}(i,i)=\\sum _{q=1}^{n}I_{qi}I_{qi}=2$ .",
"Meanwhile, we have two edge super-walks of length 1 going from one edge to itself, and both of them are positive.",
"Hence, $\\Delta _{\\Gamma }^{-}(i,i)=2$ follows the claim.",
"The claim holds in case (1).",
"Case (2), if $i\\ne j$ , and edge $i$ is not incident to edge $j$ .",
"It follows that if $I_{qi}$ is non-zero, then $I_{qj}$ must be zero.",
"Hence, $\\Delta _{\\Gamma }^{-}(i,j)=\\sum _{q=1}^{n}I_{qi}I_{qj}=0$ .",
"Meanwhile, we cannot go from edge $i$ to edge $j$ of length 1 if two edges have no intersection.",
"Hence $\\Delta _{\\Gamma }^{-}(i,j)=0$ follows the claim.",
"The claim holds in case (2).",
"Case (3), if $i\\ne j$ , and edge $i$ intersects edge $j$ at vertex $A$ .",
"We have: $\\Delta _{\\Gamma }^{-}(i,j)=\\sum _{q=1}^{n}I_{qi}I_{qj}=\\pm 1$ .",
"Since only non-zero items contribute to the result, if the sign of $A$ in $i$ is the same as sign of $A$ in $j$ , then we get $(+1)$ ; if the sign of $A$ in $i$ is different from sign of $A$ in $j$ , then we get $(-1)$ .",
"Meanwhile, if the sign of $A$ in $i$ and the sign of $A$ in $j$ are the same, then both edges end at $A$ or start at $A$ .",
"We have one edge super-walk of length 1 going from edge $i$ to edge $j$ , and it is positive.",
"Hence, $\\Delta _{\\Gamma }^{-}(i,j)=1$ follows the claim.",
"Also, if the sign of $A$ in $i$ and the sign of $A$ in $j$ are different, then one of the edges ends at $A$ , and the other one starts at $A$ .",
"We have one edge super-walk of length 1 going from edge $i$ to edge $j$ , and it is negative.",
"Hence, $\\Delta _{\\Gamma }^{-}(i,j)=-1$ follows the claim.",
"The claim holds in case (3).",
"Therefore, the claim holds when $k=1$ .",
"Suppose the claim holds for some natural number $k$ .",
"Consider $k+1$ .",
"$(\\Delta {^{-}_{\\Gamma }})^{k+1}&=(\\Delta {^{-}_{\\Gamma }})^{k}\\cdot (\\Delta {^{-}_{\\Gamma }})^{1}\\\\\\text{So, }(\\Delta {^{-}_{\\Gamma }})^{k+1}(i,j)&=\\sum _{q=1}^{n}(\\Delta {^{-}_{\\Gamma }})^{k}(i,q)(\\Delta ^{-}_{\\Gamma })(q,j) \\\\&=(\\Delta _{\\Gamma }^{-})^{k}(i,j)\\cdot (\\Delta _{\\Gamma }^{-})(j,j)+\\overbrace{\\sum _{q=1}^{n}(\\Delta _{\\Gamma }^{-})^{k}(i,q)\\cdot (\\Delta _{\\Gamma }^{-})(q,j)}^{\\text{where } q \\text{ and } j \\text{ adjacent}}+\\overbrace{\\sum _{q=1}^{n}(\\Delta _{\\Gamma }^{-})^{k}(q,j)\\cdot 0}^{\\text{where } q \\text{ and } j \\text{ non-adjacent}} \\\\&=\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )\\cdot (\\Delta _{\\Gamma }^{-}) (j,j)+\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )\\cdot (\\Delta _{\\Gamma }^{-}) (q,j)+0.$ Let $\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )\\cdot (\\Delta _{\\Gamma }^{-}) (j,j)=X$ .",
"In $X$ , we first count the edge super-walks from edge $i$ to edge $j$ of length $k$ by calculating $\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )$ .",
"Now we stand at edge $j$ , and have to go from edge $j$ to edge $j$ of length 1.",
"The number of the last step is 2, because every edge has two ends.",
"Since $(\\Delta _{\\Gamma }^{-}) (j,j)$ is 2 for all $j$ , $X$ calculates $\\sum _{\\gamma } \\text{sgn}(\\gamma )$ where $\\gamma $ is the edge super-walks from edge $i$ to edge $j$ of length $k+1$ , and the last step going from edge $j$ to edge $j$ .",
"Let $\\sum _{\\gamma ,i\\rightarrow j,k} \\text{sgn}(\\gamma )\\cdot (\\Delta _{\\Gamma }^{-}) (q,j)=Y$ .",
"In $Y$ , we first count the edge super-walks from edge $i$ to edge $q$ of length $k$ by calculating $\\sum _{\\gamma ,i\\rightarrow q,k} \\text{sgn}(\\gamma )$ .",
"Now we stand at edge $q$ , an edge connected to the edge $j$ , and have to go from edge $q$ to edge $j$ of length 1.",
"There is only one way to go from edge $q$ to edge $j$ of length 1, and its sign can be negative or positive.",
"That is $\\sum _{\\gamma ,q\\rightarrow j,1} \\text{sgn}(\\gamma )=-1=(\\Delta _{\\Gamma }^{-}) (q,j)$ , or $\\sum _{\\gamma ,q\\rightarrow j,1} \\text{sgn}(\\gamma )=1=(\\Delta _{\\Gamma }^{-}) (q,j)$ .",
"Hence, $Y$ calculates $\\sum _{\\gamma } \\text{sgn}(\\gamma )$ where $\\gamma $ is the edge super-walks from edge $i$ to edge $j$ of length $k+1$ , and the last step going from edge $q$ to edge $j$ .",
"Therefore, $X+Y$ calculates $\\sum _{\\gamma ,i\\rightarrow j,k+1} \\text{sgn}(\\gamma )$ .",
"Hence, the claim holds when $k$ is replaced by $k+1$ .",
"This is the end of the proof."
],
[
"Conclusion and Perspectives",
"The combinatorial interpretation of the Laplacian has a physical counterpart.",
"In quantum mechanics, we have the evolution of quantum states $\\psi \\in \\mathcal {H} $ which is calculated via the so called Schrödinger equation: $\\frac{\\partial }{\\partial t}\\psi =\\mathbb {H}\\psi $ , where $\\mathbb {H}$ is the Schrödinger operator.",
"One can also calculate it in another way by looking at the wave equation.",
"$\\psi (y,t_{1})=\\int _{\\mathbb {R}}k(t_{1}-t_{0},x,y)\\psi (x,t_{0})dx$ .",
"This integral is the heat integral kernel, where $k$ is a propagator of the quantum system.",
"In order to compute such heat kernel, Feynman proposed the following description: $k(t_{1}-t_{0},x,y)=\\int _{\\mathbb {\\gamma }}\\mathcal {D}(\\Gamma )e^{i/\\hbar }S_{cl}(\\gamma )$ , where the integral is over paths $\\gamma $ in $\\mathbb {R}$ , and $\\mathcal {D}(\\Gamma )$ is a measure on the space of walks.",
"The two types of super-walks and their theorems given by this paper show the discrete form of the walk expansion for the partition function of the super-symmetric version of graph quantum mechanics.",
"In this combinatorial model of quantum mechanics, we know that $\\psi _{t_{1}}=exp((t_{0}-t_{1})\\Delta )\\psi _{t_{0}}$ , where $\\Delta $ is the super-Laplacian (the direct sum of the even and odd graph Laplacians).",
"Thus we can calculate $\\psi $ by summing all $\\psi _{t}$ 's: $\\psi = \\int _{\\mathbb {R}}\\sum _{\\gamma }\\frac{t^{\\left| \\gamma \\right|}}{\\left| \\gamma \\right|!",
"}e^{-kt}\\psi (x,t_{0})dx$ , where $\\gamma $ are all the super-walks defined in this paper.",
"The formulae in this paper also suggest a natural extension of walks to CW-complexes, with a given orientation, and such expressions would lead to a combinatorial interpretation of the partition function of super-symmetric quantum mechanics for CW-complexes.",
"In del Vecchio's work, walk sum formulae and gluing formulae are derived for the case of graph quantum mechanics , and it is conjectured that this approach is still useful in the more general setting of Laplacians for CW-complexes [2].",
"In this case the graph Laplacian is replaced by the Hodge Laplacian $\\left\\langle u|(\\Delta _{X})^{k}|v \\right\\rangle = \\left\\langle u|(d^{*}_{X}d_{X})^{k}|v \\right\\rangle +\\left\\langle u|(d_{X}d^{*}_{X})^{k}|v \\right\\rangle $ , where $X$ is the CW-complex such that $\\Delta _{X}:\\bigoplus _{i=0}^{n}Span_{\\mathbb {C}}(C_{i}(X))\\rightarrow Span_{\\mathbb {C}}(C_{i}(X))$ .",
"Since any graph is a 1-dimensional CW-complex, one can extend the two types of super-walks defined in this paper to CW-complexes so that $\\left\\langle u|(\\Delta _{X})^{k}|v \\right\\rangle = \\sum _{\\gamma }\\text{sgn}(\\gamma )$ , where $X$ is the CW-complex, $u$ and $v$ are cells, $\\gamma $ are the super-walks from $u$ to $v$ of length $k$ .",
"The super-walks for CW-complexes correspond to sequences of cells with alternating parity in dimension."
]
] | 1612.05505 |
[
[
"Trimer and Tetramer Bound States in Heteronuclear Systems"
],
[
"Abstract The Efimov effect in heteronuclear cold atomic systems is experimentally more easily accessible than the Efimov effect for identical atoms, because of the potentially smaller scaling factor.",
"We focus on the case of two or three heavy identical bosons and another atom.",
"The former case was recently observed in a mixture of 133Cs and 6Li atoms.",
"We employ the Gaussian Expansion Method as developed by Hiyama, Kino et al..",
"This is a variational method that uses Gaussians that are distributed geometrically over a chosen range.",
"Supplemental calculations are performed using the Skorniakov-Ter-Martirosian equation.",
"Blume et al.",
"previously investigated the scaling properties of heteronuclear systems in the unitary limit and at the three-body breakup threshold.",
"We have completed this picture by calculating the behaviour on the positive scattering length side of the Efimov plot, focussing on the dimer threshold."
],
[
"Introduction",
"The Efimov effect has been intensively investigated in recent years, both theoretically and experimentally.",
"It was predicted by Efimov [4] that in a system with a scattering length $a$ that is large compared to the typical interaction length of the system, a series of universal three-body bound states can be found with a fixed factor between them.",
"In the unitary limit, $\\frac{1}{a} = 0$ , the energies of consecutive states fulfil $\\frac{E_{n+1}}{E_n} = \\lambda ,$ where $\\lambda $ is a universal factor that does not depend on details of the interaction.",
"It does depend however on the number of interacting pairs and the mass imbalance, among other things.",
"The typical spectrum resulting from this is usually depicted in a so-called Efimov plot as presented in Fig.",
"REF .",
"Figure: Schematic Efimov plot of the mass imbalanced four-body system.",
"The H 2 LH_2L trimers arethe solid red lines, the H 3 LH_3L tetramers are the dashed green lines and the HLHL dimer thresholdis the dotted blue line.",
"Since we assume no interaction between the HH atoms, there is only onedimer.The most easily accessible systems experimentally are systems of two species of atoms (heavy bosons $H$ and one light atom $L$ ) with a large mass imbalance.",
"There the Efimov effect occurs as well and the large mass imbalance leads to a comparatively small factor $\\lambda $ between consecutive states [2].",
"In addition to the standard three-body effect, there has also been a lot of interest in expanding the universal picture into the $N$ -body sector.",
"In systems of identical bosons, for example, two universal tetramers attached to the lowest Efimov trimer have been found [5][8][3].",
"Bringing these two aspects together, Blume and Yan have studied four-body states in a heteronuclear system [1].",
"They focussed on the negative scattering length side of the Efimov spectrum, because this is the side most experiments use for their observations.",
"To complete this picture we have studied the side of positive scattering length.",
"This side can also be seen experimentally, although experiments are complicated by the existence of the shallow dimer.",
"Experiments typically measure atomic losses due to resonant recombination at the threshold, so we have focussed on the region near the dimer threshold using full four-body calculations with the Gaussian Expansion Method [7]."
],
[
"Method",
"The Gaussian Expansion Method employs the Rayleigh-Ritz Variational Principle with base functions that are selected via geometric progression between a minimum and a maximum range.",
"It uses various representations of the system in Jacobi Coordinates, which we call rearrangement channels, to cover different possible angular momentum couplings and arrangements.",
"The total wavefunction $\\Psi $ the Schrödinger Equation $(H-E)\\Psi = 0$ is projected onto is composed of a set of Gaussian functions for each Jacobi Coordinate, so in the three-body case we have $\\Psi ^{\\rm trimer}_{JM} = \\sum ^2_{c=1} \\sum _{n_c,N_c}\\sum _{\\ell _c,L_c} C^{(c)}_{n_c \\ell _c N_c L_c}[\\phi ^{(c)}_{n_c \\ell _c}({\\bf r}_c) \\psi ^{(c)}_{N_c L_c}({\\bf R}_c)]_{JM},$ with $\\phi _{nlm}({\\bf r}) = Y_{lm}({\\widehat{\\bf r}}) N_{nl}\\,r^l\\:e^{-(r/r_n)^2} \\qquad \\text{and}\\qquad \\psi _{NLM}({\\bf R}) = Y_{LM}({\\widehat{\\bf R}}) N_{NL}\\,R^L\\:e^{-(R/R_N)^2}.$ Figure: The two possible rearrangement channels for the H 2 LH_2L three-body system.",
"The big white circlesrepresent the heavy bosons HH, while the small black circlerepresents the light atom LL.The relevant rearrangement channels for the trimer are shown in Fig.",
"REF .",
"The $r_n$ are given by $r_n =r_{\\rm min}\\, \\left(\\frac{r_{\\mathrm {max}}}{r_{\\mathrm {min}}}\\right)^\\frac{n-1}{n_{\\rm max}-1},\\qquad n =1,..,n_{\\rm max}\\;,$ and analogously for $R_N$ , which means for each Jacobi coordinate we have 3 parameters, $r_{\\mathrm {max}}$ , $r_{\\mathrm {min}}$ and $n_{\\mathrm {max}}$ .",
"More details about this method can be found in [7].",
"Using several rearrangement channels and relative angular momentum configurations, the parameter space increases rapidly for higher $N$ systems.",
"Because of this, systematic sampling of base functions was impractical for the four-body system and we employed random sampling within relatively broad ranges to find optimized base functions."
],
[
"Interaction",
"Because there is no reason why the $HL$ interaction should be resonant at the same point as the $HH$ interaction, we can safely assume $a_{HH} \\ll a_{HL}$ and neglect $a_{HH}$ .",
"Adding an interaction between the heavy bosons only moves the whole Efimov spectrum up or down [9], which we can do independently by tuning our three-body force.",
"As we want to focus on universal behaviour we choose effective potentials, which have the added benefit of being numerically well behaved.",
"To keep the calculated energies in the range of validity of the effective theory we add a repulsive three-body force.",
"The inverse scattering length $\\frac{1}{a}$ is roughly linear in the two-body potential strength, which is how we tune it.",
"$V_{iN} = V_0 \\exp \\left({-\\frac{r_{iN}^2}{2r_0^2}}\\right),\\qquad W_{ijN} = W_0 \\exp \\left({-\\frac{r_{ij}^2 + r_{jN}^2 + r_{iN}^2}{16r_0^2}}\\right).$ Here $i$ and $j$ run over the heavy ($H$ ) bosons of mass $M$ from $1,..,N-1$ , while the $N$ th atom ($L$ ) is light and of mass $m$ .",
"The $r_{ij}$ are the distances between atoms $i$ and $j$ .",
"The natural energy scale of the problem is $E_s = \\frac{1}{2r_0^2}\\frac{m+M}{Mm},$ where $r_0$ is the effective range of the two-body potential.",
"We have ensured $E \\ll E_s$ throughout our calculations."
],
[
"Results",
"Our main result is that trimer and tetramer vanish at almost the same point for all systems we have looked at, which were $^7$ Li-$^6$ Li, $^{87}$ Rb-$^7$ Li and $^{133}$ Cs-$^6$ Li mixtures.",
"We have taken a closer look at the behaviour of trimer and tetramer near the dimer threshold.",
"Our results for the two most extreme cases, almost no mass imbalance ($^7$ Li-$^6$ Li) and very high mass imbalance ($^{133}$ Cs-$^6$ Li), are shown in Fig.",
"REF .",
"It can be seen that in the (7/6) case the states approach the threshold very slowly, whereas for the (133/6) case there is a slightly steeper slope.",
"Also, the difference $c_d$ between the point where the trimer vanishes, $c_3$ , and the point where the tetramer vanishes, $c_4$ , as shown in the right panel of Fig.",
"REF , is much larger for the (133/6) case than for the (7/6) case.",
"To quantify this, we have extracted these numbers for the three systems we have investigated and plotted them in Fig.",
"REF .",
"Figure: The difference c d c_d between trimer and tetramer crossing point in units of the trimer crossing point c 3 c_3as a function of the mass ratio M/mM/m.",
"The solid green (dashed red) lines are effective three-body calculations usingthe Skorniakov-Ter-Martirosian equationand fitted to reproduce the 133/6133/6 (87/787/7) point.To help interpret this picture, we carried out effective three-body calculations of the dimer-atom-atom system using the Skorniakov-Ter-Martirosian (STM) equation as described in [6].",
"This picture is valid near the dimer threshold where the trimer and tetramer are very shallow.",
"In this region the scattering length between $HL$ dimer and $H$ atom can be estimated by the inverse trimer binding energy and is very large ($a_{AD}/r_0 \\approx 10^7$ ) so an effective three-body treatment is justified.",
"The STM equation has one free parameter which we fitted to reproduce either the data point belonging to the $(133/6)$ -system, or the $(87/7)$ system.",
"In both cases, the data point of the $(7/6)$ -system is reproduced, but the $(133/6)$ and $(87/7)$ points cannot be fitted simultaneously.",
"This might mean that there is non-universal behaviour governing the $c_d/c_3$ value.",
"But it could also simply be the case that the effective three-body calculation does not capture enough of the four-body system to yield accurate predictions."
],
[
"Outlook",
"As a next step, we plan to investigate the dependence of $c_d$ on the potential shape and the strength of the three-body potential to rule out the possibility of non-universal behaviour.",
"Additionally, calculating more data points for Fig.",
"REF might deepen the understanding of the underlying functional dependence.",
"We thank Doerte Blume for useful discussions and comments.",
"Part of the calculations were carried out at YITP of Kyoto University.",
"C.H.",
"Schmickler thanks RIKEN for providing support under the IPA program."
]
] | 1612.05738 |
[
[
"Observation of spin-charge separation and boundary bound states via the\n local density of states"
],
[
"Abstract We numerically calculate the local density of states (LDOS) of a one-dimensional Mott insulator with open boundaries, which is modelled microscopically by a (extended) Hubbard chain at half filling.",
"In the Fourier transform of the LDOS we identify several dispersing features corresponding to propagating charge and spin degrees of freedom, thus providing a visualisation of the spin-charge separation in the system.",
"We also consider the effect of an additional boundary potential, which, if sufficiently strong, leads to the formation of a boundary bound state which is clearly visible in the LDOS as a non-dispersing feature inside the Mott gap."
],
[
"Introduction",
"One-dimensional systems remain a fascinating field in condensed-matter physics since they constitute prime examples for the breakdown of Fermi-liquid theory, which has to be replaced by the Luttinger-liquid paradigm.",
"[1] Arguably the most dramatic consequence of this is the absence of electron-like quasiparticles, manifesting itself in the separation of spin and charge degrees of freedom visible for example in angle-resolved photoemission,[2] transport,[3] scanning tunneling spectroscopy[4] or resonant inelastic X-ray scattering[5] experiments as well as analytical[6] and numerical studies of several one-dimensional models.",
"[7] The spectral properties of one-dimensional electron systems have been intensively investigated in the past.",
"These works considered the gapless Luttinger liquid,[1], [8] gapped systems like Mott insulators or charge-density wave states,[9] Luttinger liquids with impurities,[10] corrections to the Luttinger model due to non-linear dispersions[11] or the momentum dependence of the two-particle interaction,[12] as well as additional phonon degrees of freedom.",
"[13] These investigations uncovered universal power-law behaviour at low energies as well as deviations thereof, spin-charge separation visible in the propagation modes, and signatures of these features in various experimental probes.",
"In this article we consider another situation, namely the microscopic study of the boundary effects on one-dimensional Mott insulators.",
"Specifically we numerically study the local density of states (LDOS) of one-dimensional Hubbard models with open, ie, hard-wall, boundary conditions, where the system is at half filling and thus in its Mott phase.",
"A previous field-theoretical analysis[14] has shown that the Fourier transform of the LDOS[15] exhibits clear signatures of propagating spin and charge degrees of freedom, thus providing a way to detect spin-charge separation.",
"Furthermore, an additional boundary potential may lead to the formation of a boundary bound state, which manifests itself as a non-dispersing feature in the LDOS.",
"The aim of our work is to calculate the Fourier transform of the LDOS directly in the microscopic lattice model using a multi-target[16], [17] variant of the density matrix renormalisation group (DMRG) method [18] employing an expansion in Chebyshev polynomials.",
"We find our numerical results to be fully consistent with the analytical predictions both qualitatively, ie, concerning the number of dispersion modes and their basic properties, as well as quantitatively with respect to the numerical values of the effective parameters like the Mott gap and spin and charge velocities as compared to the exact results obtained from the Bethe ansatz.",
"[19] Thus our work provides a microscopic calculation of the Fourier transform of the LDOS in a gapped, strongly correlated electron system, showing spin-charge separation as well as the formation of a boundary bound state.",
"This paper is organised as follows: In Sec.",
"we present the microscopic models to be analysed and discuss the basic setup.",
"In Sec.",
"we give a brief summary of the numerical method we employ to calculate the single-particle Green function.",
"Our results for the LDOS of the Mott insulators with open boundary conditions are discussed in Sec. .",
"In Sec.",
"we study the effect of a boundary potential on the LDOS, in particular we analyse the properties of the boundary bound state existing for sufficiently strong boundary potentials.",
"In Sec.",
"we summarise our results."
],
[
"Model",
"In this work we analyse the LDOS of the one-dimensional Hubbard model [19] at half filling.",
"The Hamiltonian is given by $H=&-t \\sum _{\\sigma ,j=0}^{L-2}\\bigl ( c_{j,\\sigma }^{\\dagger }c_{j+1,\\sigma } + c_{j+1,\\sigma }^{\\dagger }c_{j,\\sigma }\\bigr )\\\\*&+U\\sum _{j=0}^{L-1} \\left(n_{j,\\uparrow }-\\frac{1}{2}\\right)\\left(n_{j,\\downarrow }-\\frac{1}{2}\\right)\\nonumber ,$ where $c_{j,\\sigma }$ and $c_{j,\\sigma }^\\dagger $ denote the annihilation and creation operators for electrons with spin $\\sigma =\\uparrow ,\\downarrow $ at lattice site $j$ and $n_{j,\\sigma }=c_{j,\\sigma }^{\\dagger }c_{j,\\sigma }$ the corresponding density operators.",
"The parameters $t$ and $U>0$ describe the hopping and repulsive on-site interaction respectively.",
"Furthermore we consider a chain with $L$ sites and open boundary conditions.",
"Since the system is assumed to be at half filling, the Fermi momentum is given by $k_\\text{F}=\\pi /2$ .",
"As is well known,[1], [19] in the Hubbard model at half filling, ie, when there are $L$ electrons in the system, the repulsive interaction opens a gap in the charge sector and the system becomes a Mott insulator.",
"Using bosonisation the low-energy behaviour of the system is described by the massive Thirring model;[20] the LDOS of which in the presence of boundaries has been analysed in Refs. SEJF08.",
"The main objective of our article is the comparison of the LDOS of the Hubbard model (REF ) with the field-theoretical results obtained in the Thirring model.",
"Hereby the effective parameters in the field theory, ie, the mass gap and velocities, can be obtained from the exact Bethe-ansatz solution of the Hubbard model.",
"This allows us to choose the microscopic parameters such that the expected features of the Fourier transformed LDOS can be easily resolved in the numerical results.",
"In addition to the standard Hubbard model (REF ) we also consider its extension including a nearest-neighbour interaction $V$ , ie, the Hamiltonian is given by[21] $H_\\text{ext}=H+V\\sum _{j=0}^{L-2}\\left(n_{j}-1\\right)\\left(n_{j+1}-1\\right)$ where $n_j=n_{j,\\uparrow }+n_{j,\\downarrow }$ is the total electron density.",
"The low-energy regime of the extended Hubbard model (REF ) is still described [20] by the massive Thirring model.",
"However, since (REF ) is no longer integrable, the explicit relation between the microscopic parameters $t$ , $U$ and $V$ and the field-theory ones is not known.",
"Thus the investigation of the phase diagram of the extended Hubbard model at half filling had to be performed by numerical means.",
"[22] Using these results we choose the microscopic parameters such that the system is well inside the Mott-insulating phase with an energy gap $\\Delta \\approx \\mathcal {O}(t)$ so that we are able to clearly resolve the interesting features in our numerical results."
],
[
"Green function",
"In order to determine the LDOS we calculate the retarded Green function in frequency space using an expansion of the occurring resolvent in Chebyshev polynomials.",
"[17] An alternative numerical approach consists in the expansion of the Lehmann representation of the spectral function in Chebyshev polynomials, the kernel polynomial method (KPM), see Refs. Weisse-06.",
"In contrast we specifically evaluate the complete (real and imaginary part) Green functions $G^\\text{R}(\\omega ,x)=G^+(\\omega ,x)-G^-(\\omega ,x)$ with $G^+(\\omega ,x)&=&\\left\\langle \\Psi _0\\right|c_{j,\\sigma }\\frac{1}{E_0-H+\\omega +\\text{i}\\eta }c_{j,\\sigma }^\\dagger \\left|\\Psi _0\\right\\rangle ,\\\\G^-(\\omega ,x)&=&\\left\\langle \\Psi _0\\right|c_{j,\\sigma }^\\dagger \\frac{1}{E_0-H-\\omega -\\text{i}\\eta }c_{j,\\sigma }\\left|\\Psi _0\\right\\rangle .$ Here $\\left|\\Psi _0\\right\\rangle $ denotes the ground state of the system with energy $E_0$ .",
"Note that since we are interested in the LDOS we have already taken the electron creation and annihilation operators to be at the same site $x=ja_0$ with $a_0$ denoting the lattice spacing.",
"Furthermore, since the systems we consider possess spin-rotation invariance we have suppressed the formal spin dependence of the Green functions.",
"In Eqs.",
"(REF ) and () we have included the convergence factor $\\eta $ , which in the continuum limit should be taken as $\\eta \\rightarrow 0^{+}$ .",
"In the numerical evaluations it has to be larger than the finite level splitting brought about by the finite system size.",
"At the same time $\\eta $ has to be smaller than any physically relevant energy scale in order to resolve the relevant features of the spectrum.",
"To attain a small value of $\\eta $ we employ a Chebyshev polynomial expansion approach for the resolvents in (REF ) and ().",
"More details on this approach can be found in Refs. Schmitteckert10,BraunSchmitteckert14.",
"The applied Chebyshev expansion is based on the representation of the functions $f^\\pm (\\omega ,z)=\\frac{1}{\\pm \\omega -z}$ in terms of Chebyshev polynomials $f^{\\pm }(\\omega ,z)=\\sum _{n=0}^{\\infty } \\alpha _n^{\\pm }(\\omega ) T_n(z),\\quad -1\\le z\\le 1.$ The expansion coefficients are given by $\\alpha _{n}^{\\pm }(\\omega ) &= \\frac{2}{\\pi (1+\\delta _{n,0})}\\int _{-1}^{1}\\text{d}z\\,\\frac{T_n(z)}{\\sqrt{1-z^2}}\\frac{1}{\\pm \\omega -z}\\\\\\nonumber &=\\frac{2-\\delta _{n,0}}{(\\pm \\omega )^{n+1}\\left(1+\\sqrt{\\omega ^2}\\frac{\\sqrt{\\omega ^2-1}}{\\omega ^2}\\right)^{n}\\sqrt{1-\\omega ^{-2}}},$ where $\\alpha _n^{\\pm }(\\omega )\\equiv \\alpha _n^{\\pm }(\\omega +\\text{i}\\eta )$ is a function of the artificial broadening $\\eta $ which would theoretically allow arbitrarily small $\\eta $ .",
"The Chebyshev polynomials $T_n(z)$ are defined by their recursion relation $T_0(z)&=1,\\\\T_1(z)&=z,\\\\T_{n+1}(z)&=2zT_n(z)-T_{n-1}(z),\\quad n\\ge 2,$ and fulfil $\\int _{-1}^{1} \\frac{\\text{d}z}{\\sqrt{1-z^2}}\\,T_n(z) T_m(z)=\\frac{\\pi }{2}\\delta _{n,m}(1+\\delta _{n,0})$ as well as $T_{2n}(z)&=2T_n(z)^2-T_0(z),\\\\T_{2n-1}(z)&=2T_{n-1}(z)\\,T_n(z)-T_1(z).$ In order to apply the expansion (REF ), which is only valid for $|z|\\le 1$ , to the resolvents appearing in the Green functions, we first have to rescale the energies.",
"To this end we run initial DMRG calculations to determine the ground-state energy $E_0$ as well as the smallest and the largest energies of the system with $L\\pm 1$ electrons.",
"This allows us to find the scaling factor $a$ and shift $b$ such that the operator $a(H-E_0)-b$ has a spectrum between $-1$ and 1 in the sectors with $L\\pm 1$ particles.",
"Then the Green function (REF ) can be expressed as $G^+(\\omega ,x)=a\\sum _{n=0}^\\infty \\alpha _n^+[a(\\omega +\\text{i}\\eta )-b]\\,\\mu _n^+(x),$ where the Chebyshev moments $\\mu _n^+(x)=\\left\\langle \\Psi _0\\right|c_{j,\\sigma }\\,T_n[a(H-E_0)-b]\\,c_{j,\\sigma }^\\dagger \\left|\\Psi _0\\right\\rangle $ (recall $x=ja_0$ ) can be evaluated recursively via $\\mu _{n}^{+} (x)= \\left\\langle \\Psi _0\\right|c_{\\sigma }(x)\\left|\\Phi _{n}^{+}\\right\\rangle $ with the recursion relations $\\vert \\Phi _{0}^{+} \\rangle &= c^{\\dagger }_{\\sigma }(x)\\vert \\Psi _0\\rangle ,\\\\\\vert \\Phi _{1}^{+} \\rangle &= [a(H-E_0)-b]\\vert \\Phi _0^{+}\\rangle ,\\\\\\vert \\Phi _{n+1}^{+} \\rangle &= 2[a(H-E_0)-b]\\vert \\Phi _{n}^{+}\\rangle -\\vert \\Phi _{n-1}^{+}\\rangle .$ Similarly, for the Green function () we obtain the expansion $G^-(\\omega ,x)=a\\sum _{n=0}^\\infty \\alpha _n^-[a(\\omega +\\text{i}\\eta )+b]\\,\\mu _n^-(x),$ where $\\mu _n^-(x)=\\left\\langle \\Psi _0\\right|c_{j,\\sigma }^\\dagger \\,T_n[a(H-E_0)-b]\\,c_{j,\\sigma }\\left|\\Psi _0\\right\\rangle .$ In the numerical evaluations the sums appearing in (REF ) and (REF ) are truncated at $N/2$ .",
"The moments $\\mu _n^\\pm $ are calculated iteratively from (REF )–() using DMRG.",
"During the DMRG finite-lattice sweeps we determine each state $\\left|\\Phi _0^\\pm \\right\\rangle , \\ldots , \\left|\\Phi _{N/2}^\\pm \\right\\rangle $ and include it into a modified density matrix.",
"By performing a singular-value decomposition of this modified density matrix we ensure that all the states $\\left|\\Phi _0^\\pm \\right\\rangle , \\ldots , \\left|\\Phi _{N/2}^\\pm \\right\\rangle $ are part of the Hilbert space after the DMRG truncation.",
"The moments for $n=N/2+1,\\ldots ,N$ are then obtained employing (REF ) and () as $\\mu _{2n}^\\pm =2\\left\\langle \\Phi _n^\\pm \\right|\\Phi _n^\\pm \\rangle -\\left\\langle \\Phi _0^\\pm \\right|\\Phi _0^\\pm \\rangle $ and $\\mu _{2n-1}^\\pm =2\\left\\langle \\Phi _{n-1}^\\pm \\right|\\Phi _n^\\pm \\rangle -\\left\\langle \\Phi _0^\\pm \\right|\\Phi _1^\\pm \\rangle $ .",
"Finally, we note that the Chebyshev moments $\\mu _n^\\pm $ are typically strongly oscillating with respect to the index $n$ .",
"Therefore, the final results oscillate slightly when changing the value of $N$ .",
"On the other hand, we find small oscillating parts in the spectral function if we choose $N$ too small.",
"Both effects can be avoided by implementing a smoothing window for the last $N_\\text{S}$ moments.",
"Throughout this article we use a $\\cos ^2$ -filter for the last $N_\\text{S}=N/5$ moments.",
"This way one can obtain a good approximation for the spectral function using a smaller number of moments $N$ .",
"Previously it was observed[17] that the number of required Chebyshev moments sufficient to approximate the Green function is inversely proportional to the width of the spectrum $a$ and the desired artificial broadening $\\eta $ , ie, $N\\simeq (a\\eta )^{-1}$ .",
"Throughout this work use $N\\ge 1000$ Chebyshev moments for the series expansion of the Green function.",
"Furthermore, $\\eta $ is chosen such that the resulting curves become smooth and artificial features are suppressed."
],
[
"LDOS",
"The LDOS is obtained from the retarded Green function (REF ) in the usual way.",
"As was noted by Kivelson et al.",
"[15] in the study of Luttinger liquids with boundaries, it is useful to consider the Fourier transform of the LDOS, as physical properties like the dispersions of propagating quasiparticles can be more easily identified.",
"Since we consider a finite chain of length $L$ we analyse $N(\\omega ,Q)=-\\frac{1}{\\pi }\\sqrt{\\frac{2}{L+1}}\\sum _{j=0}^{L-1}\\, \\text{Im}\\, G^\\text{R}(\\omega ,x)\\,\\sin [Q(j+1)],$ where the momenta $Q$ take the values $Q=\\pi k/(L+1)$ , $k=1,\\ldots , L$ .",
"We note that the LDOS is directly related to the tunneling current measured in scanning tunneling microscopy experiments, thus its Fourier transform (REF ) is experimentally accessible.",
"In the following we focus on the LDOS for positive energies; the LDOS for negative energies can be analysed analogously.",
"The LDOS of the low-energy effective field theory of the Hubbard models (REF ) and (REF ) has been analysed[25] in Refs. SEJF08.",
"In the field-theoretical description the momentum regimes $Q\\approx 0$ and $Q\\approx \\pm 2k_\\text{F}=\\pi $ are treated separately.",
"For small momenta $Q\\approx 0$ the main features of the Fourier transform (REF ) are a strong divergence at $Q=0$ as well as a propagating excitation in the gapped charge sector above the Mott gap.",
"In contrast, the behaviour at momenta $Q\\approx 2k_\\text{F}$ shows a divergence at $Q=2k_\\text{F}$ , a propagating excitation in the charge sector as well as a linearly dispersing excitation in the gapless spin sector.",
"Furthermore there exists a critical momentum above which a second linearly dispersing mode becomes visible.",
"In addition, it was shown that certain boundary conditions lead to the formation of boundary bound states which manifest themselves as non-propagating features in the LDOS.",
"The main aim of our article is the calculation of the Fourier transform of the LDOS (REF ) in the microscopic models (REF ) and (REF ) and its comparison to the field-theoretical predictions.",
"[14] We start with the standard Hubbard chain (REF ) before considering the extended version (REF ).",
"In Sec.",
"we then analyse the effect of additional boundary potentials which give rise to the existence of boundary bound states."
],
[
"Standard Hubbard model",
"We first consider the Fourier transform of the LDOS (REF ) in the standard Hubbard model (REF ).",
"The results in the vicinity of $Q=0$ and $Q=2k_\\text{F}=\\pi $ are shown in Figs.",
"REF and REF respectively, where we have chosen a repulsive interaction of $U=4.5\\, t$ corresponding to the dimensionless Hubbard parameter $u=U/(4t)=1.125$ and $L=90$ lattice sites.",
"Throughout our article we use the hopping parameter $t=1$ as our unit of energy.",
"As is well known, the Hubbard model (REF ) is exactly solvable by Bethe ansatz.",
"[19] In particular, the velocities of the spin and charge excitations $v_\\text{s}$ and $v_\\text{c}$ as well as the Mott gap $\\Delta $ can be determined analytically; the results in the thermodynamic limit read $&\\Delta = -2+2u+2\\int _{0}^{\\infty }\\frac{\\text{d}\\omega }{\\omega }\\,\\frac{J_1(\\omega )\\text{e}^{-u\\omega }}{\\cosh \\left(u\\omega \\right)},\\\\&v_\\text{c} =\\frac{2}{1-\\xi _{0,0}(u)}\\,\\sqrt{u-1+\\xi _{-1,1}(u)} \\sqrt{1-\\xi _{1,1}(u)},\\\\&v_\\text{s}=\\frac{2I_1\\left(\\frac{\\pi }{2u}\\right)}{I_0\\left(\\frac{\\pi }{2u}\\right)},\\\\&\\xi _{m,n}(u)=2\\int _0^{\\infty }\\frac{\\text{d}\\omega \\,\\omega ^m J_n(\\omega )}{1+\\exp (2\\omega u)},$ where $J_{n}(z)$ and $I_n(z)$ denote the Bessel functions and modified Bessel functions of the first kind respectively.",
"Our chosen parameters for the microscopic system correspond to[25] $v_\\text{c} > v_\\text{s}$ .",
"Figure: Fourier transform of the LDOS, N(ω,Q)N(\\omega ,Q), for interaction u=U/4=1.125u=U/4=1.125 (recall t=1t=1), L=90L=90 lattice sites, broadening η=0.2\\eta =0.2 and momenta Q=π/91,2π/91,...,18π/91Q=\\pi /91, 2\\pi /91,\\ldots , 18\\pi /91 (from bottom to top).",
"The curves are constant QQ-scans that have been offset along the y-axis by a constant with respect to one another.",
"N(ω,Q)N(\\omega ,Q) is dominated by a strong peak at Q=π/91≈0Q=\\pi /91\\approx 0 which is only partially displayed in the figure in order to improve visibility for the other cuts.",
"We clearly observe the Mott gap Δ\\Delta as well as a dispersing feature indicated by the arrow.",
"This feature corresponds to propagating charge excitations, it follows the dispersion relation E c (Q)E_\\text{c}(Q) given in () with v c ≃2.67v_\\text{c}\\simeq 2.67 obtained from ().In Fig.",
"REF we plot $N(\\omega ,Q)$ in the vicinity of small momenta $Q\\approx 0$ .",
"All features (except for a strong peak at $Q=\\pi /91$ ) appear at energies $\\omega \\ge \\Delta $ , clearly showing that the system is in a gapped phase.",
"The observed energy gap $\\Delta $ agrees perfectly with the value $\\Delta (u=1.125)\\simeq 0.83$ obtained from the Bethe ansatz (REF ) in the thermodynamic limit.",
"This suggests that the length of our chain is long enough to avoid significant finite-size effects in our results.",
"The Fourier transform of the LDOS for small momenta is dominated by a global maximum at $Q=\\pi /91\\approx 0$ .",
"This peak is attributed to a spin-density wave pinned at the boundary, it is also well visible in the field-theoretical results.",
"[14] At low energies above the energy gap we further observe a dispersing feature indicated by the arrow.",
"This again agrees well with the results from the field theory that predict a gapped, dispersing charge excitation with dispersion relation $E_\\mathrm {c}(q)=\\sqrt{\\left(\\frac{v_\\text{c}q}{2}\\right)^2+\\Delta ^2},$ where $q=Q$ and $v_\\text{c}$ is the velocity of the charge excitations.",
"The Bethe-ansatz solution () gives the value $v_\\text{c}(u=1.125)\\simeq 2.67$ , which is in excellent agreement with the velocity observed in the plot.",
"The physical origin of this dispersing feature is the decay of the electronic excitation into gapped charge and gapless spin excitations.",
"In the process giving rise to (REF ) the external momentum $q$ is taken by the charge excitation propagating through the system and eventually getting reflected at the boundary, while the spin excitation does not propagate and thus possesses zero momentum.",
"The appearance of $v_\\text{c}/2$ in (REF ) originates from the fact that the charge excitation has to propagate to the boundary and back, thus covering the distance $2x$ .",
"In addition, we find a second dispersing feature that seemingly follows the same dispersion relation albeit with a different value for the gap $\\Delta _2 \\simeq 5\\Delta /2$ .",
"This feature is not contained in the field-theory description of the LDOS, which solely focuses on the low-energy regime.",
"Furthermore, the field theory makes predictions about the power-law decay of $N(\\omega ,Q)$ at $Q=0$ which, however, cannot be resolved in our numerical data.",
"For the observation of such features we would require a significantly higher resolution, both in energy and momentum.",
"This can in turn only be achieved by turning to a significantly larger system size and a higher amount of calculated Chebyshev moments.",
"Figure: N(ω,2k F -q)N(\\omega ,2 k_\\text{F}-q) for momenta in the vicinity of Q=2k F =πQ=2 k_\\text{F}=\\pi with q=2k F -Q=π/91,2π/91,...,19π/Lq=2k_\\text{F}-Q=\\pi /91,2\\pi /91,\\ldots , 19\\pi /L (from top to bottom).",
"All other parameters are as in Fig. .",
"The curves are constant qq-scans that have been offset along the y-axis by a constant with respect to one another.",
"We observe two dispersing features (indicated by the arrows) at E c (q)E_\\text{c}(q) and E s (q)E_\\text{s}(q) originating from propagating charge and spin excitations respectively.We now turn our attention to momenta in the vicinity of $Q=2 k_\\text{F}=\\pi $ .",
"We first note that features in this momentum regime originate from umklapp processes coupling left- and right-moving modes which are absent in translationally invariant systems and thus constitute a particularly clean way to investigate the boundary effects.",
"In Fig.",
"REF we again observe the existence of the Mott gap as well as two dispersing features at $E_\\text{c}(q)$ as defined in (REF ) and $E_\\text{s}(q)=\\frac{v_\\text{s} \\vert q \\vert }{2}+\\Delta ,$ both indicated by the arrows.",
"The spin velocity observed in the plot is in excellent agreement with the Bethe-ansatz result () giving $v_\\text{s}(u=1.125)\\simeq 1.14$ .",
"While the feature adhering to (REF ) is again due to a propagating charge excitation, the feature following (REF ) originates from the propagation of spin excitations with the charge excitation possessing zero momentum.",
"Furthermore, we note that in contrast to the field-theoretical prediction we observe only one linearly dispersing mode.",
"In order to understand this we recall that the two linearly dispersing modes are energetically separated by[14] $\\Delta [1-\\sqrt{1-(v_\\text{s}/v_\\text{c})^2}]\\approx 0.1\\Delta \\approx 0.08$ , where in the last step we have put in the parameters used in Fig.",
"REF .",
"On the other hand, our resolution in energy is limited by finite-size effects to about $\\sim 2\\pi /L$ , implying that for the treatable system sizes the two linearly dispersing features cannot be separated.",
"Figure: Contour plot of the LDOS N(ω,Q)N(\\omega ,Q) for the parameters of Fig. .",
"The dominant, white peak at Q≈0Q\\approx 0 is due to the spin-density wave pinned at the boundary.",
"The solid and dashed lines indicate the holon dispersion () around Q=0Q=0 and Q=2k F Q=2k_\\text{F} respectively, the dashed-dotted line represents the spinon dispersion () around Q=2k F Q=2k_\\text{F}.",
"The parameters Δ\\Delta , v c v_\\text{c} and v s v_\\text{s} used in the plot were obtained from the Bethe ansatz for the bulk system ()–(), ie, there is no free fitting parameter.To summarise our results, in Fig.",
"REF we show a contour plot of the LDOS.",
"For comparison we plot the holon dispersion (REF ) around $Q=0$ and $Q=2k_\\text{F}$ as well as the spinon dispersion (REF ) around $Q=2k_\\text{F}$ , for which we used the parameters $\\Delta $ , $v_\\text{c}$ and $v_\\text{s}$ obtained from the Bethe ansatz for the bulk system.",
"In particular, we stress that there is no fitting parameter.",
"In conclusion, our results are in very good agreement with the features of the LDOS predicted by the field-theoretical investigations."
],
[
"Extended Hubbard model at half-filling",
"We have performed the analysis presented in the previous section for the extended Hubbard model (REF ) at half-filling and $L=88$ lattice sites.",
"Since the extended Hubbard model is not integrable, there exist no analytical results for the parameters $\\Delta $ , $v_\\text{c}$ and $v_\\text{s}$ .",
"Still, the field theory is expected to qualitatively describe the behaviour of the system in the low-energy limit.",
"The LDOS for momenta in the vicinity of $Q=0$ and $Q=2k_\\text{F}$ is shown in Figs.",
"REF and REF respectively.",
"In both plots we have renormalised the energy scale by the gap $\\Delta \\approx 2.1$ obtained from the data at $Q\\approx 0$ .",
"Figure: N(ω,2k F -q)N(\\omega ,2k_\\text{F}-q) for an extended Hubbard model in the vicinity of Q=2k F Q=2k_\\text{F} with q=2k F -Q=π/89,2π/89,...,21π/89q=2k_\\text{F}-Q=\\pi /89, 2\\pi /89, \\ldots , 21\\pi /89 (from top to bottom).",
"All other parameters are as in Fig. .",
"Similar to the standard Hubbard model, at low energies we observe two dispersing features at () and () respectively.At low energies the dispersing features are qualitatively identical to the ones seen for the standard Hubbard model, namely a propagating charge mode for $Q\\approx 0$ and both a propagating charge and spin mode around $Q=2k_\\text{F}$ .",
"The only difference is that the charge and spin velocities take the values $v_\\text{c}\\simeq 1.8\\Delta \\simeq 3.8$ and $v_\\text{s}\\simeq 0.35\\Delta \\simeq 0.7$ respectively, which were determined by comparison with the quasiparticle dispersions (REF ) and (REF ).",
"The energy gap $\\Delta $ and charge velocity $v_\\text{c}$ for the two different momentum regimes agree well.",
"We thus conclude that the low-energy sector is well described by the field theory.",
"Furthermore, for small momenta we again observe a second charge mode which now seems to have the gap $\\Delta _2\\simeq 3\\Delta /2$ ."
],
[
"Effect of a boundary potential",
"Having analysed the LDOS in the presence of open boundary conditions, we now turn to the investigation of the effect of a boundary chemical potential.",
"Specifically we consider the Hubbard model (REF ) with a boundary potential at site $j=0$ , $H_\\text{bp}=H+\\mu \\sum _{\\sigma } n_{j=0,\\sigma }.$ Using bosonisation such a boundary potential is translated into non-trivial boundary conditions for the bosonic degrees of freedom.",
"In particular, certain boundary conditions give rise to the existence of boundary bound states in the gapped charge sector[26] which manifest themselves[14] in the LDOS as non-propagating features inside the Mott gap.",
"The spectrum of the Hubbard chain with boundary potential (REF ) has been investigated by Bedürftig and Frahm[27] using the Bethe-ansatz solution.",
"In particular it was found that a boundary bound state corresponding to a charge bound at the first site exists for $\\mu <-1$ .",
"For even smaller boundary potentials, $\\mu <-2u-\\sqrt{1+4u^2}$ , two electrons in a spin singlet get bound to the surface.",
"Figure: N(ω,Q)N(\\omega ,Q) for interaction u=1.125u=1.125, boundary potential μ=-2\\mu =-2, L=90L=90 lattice sites and broadening η=0.4\\eta =0.4.",
"Besides the peak at Q=0Q=0 and the dispersing modes at ω≥Δ\\omega \\ge \\Delta we observe a non-dispersing feature inside the energy gap at ω=E bbs ≈Δ/2\\omega =E_{\\text{bbs}}\\approx \\Delta /2 which originates from the boundary bound state.The Fourier transform of the LDOS in the presence of a boundary chemical potential is shown in Fig.",
"REF .",
"Besides the peak at $Q=0$ due to the pinned charge-density wave and several dispersing modes above the Mott gap, we observe a clear, non-dispersing maximum inside the gap at $\\omega =E_\\text{bbs}\\approx \\Delta /2$ , which is a manifestation of the boundary bound state in the LDOS.",
"In the following we analyse this contribution in more detail by considering the LDOS $N(\\omega ,x)=-1/\\pi \\,\\text{Im}\\,G^\\text{R}(\\omega ,x)$ close to the boundary.",
"Figure: LDOS at the boundary, N(ω,x=0)N(\\omega ,x=0), for various values of μ\\mu and broadening η=0.1\\eta =0.1.",
"All other parameters are as in Fig. .",
"In the absence of a boundary potential (thick line) there is barely any spectral weight inside the energy gap.",
"For μ<-1\\mu <-1 the spectral density inside the gap grows continuously but its maximum is still located above the gap.",
"For μ≤-1.4\\mu \\le -1.4 the maximum is located inside the Mott gap, providing a clear manifestation of the boundary bound state.",
"Inset: Position E max E_{\\text{max}} of the maximum of N(ω,x=0)N(\\omega ,x=0) as a function of the boundary potential μ\\mu .",
"We observe that a potential μ≤-1.27\\mu \\le -1.27 is needed for E max <ΔE_{\\text{max}} < \\Delta .First we analyse the LDOS at the boundary site, $N(\\omega ,x=0)$ , which is shown in Fig.",
"REF for several values of the boundary potential $\\mu $ using an artificial broadening $\\eta =0.1$ .",
"One can clearly see that the maximum of the LDOS is shifted towards lower energies for decreasing $\\mu $ .",
"For $\\mu \\le -1$ we find a considerable spectral density inside the Mott gap $\\Delta $ ; for $\\mu \\lesssim -1.27$ the maximum of the LDOS is located inside the energy gap as well.",
"From this we deduce that for $\\mu \\lesssim -1.27$ there exists a clear boundary bound state contribution to the LDOS.",
"We attribute the deviation to the critical value $\\mu =-1$ obtained from the Bethe ansatz[27] to the finite system-size as well as the artificial broadening $\\eta $ introduced in our numerical calculations.",
"This is supported by the dependence of the energy of the maximum in the LDOS on the broadening presented in Fig.",
"REF , which shows that the energy of the maximum indeed decreases with decreasing $\\eta $ .",
"Extrapolating the results to $\\eta =0$ and keeping in mind the finite system size as well as the fact that for $\\mu \\rightarrow -1^-$ the contributions from the boundary bound state and the standard continuum at $\\omega \\ge \\Delta $ start to significantly overlap, we conclude that our results are consistent with the Bethe-ansatz solution.",
"This is further supported by the electron density at the boundary shown in the inset of Fig.",
"REF .",
"Figure: Maximum of N(ω,x=0)N(\\omega ,x=0) as a function of the artificial broadening η\\eta for u=1.125u=1.125 and L=90L=90.",
"Extrapolating to η=0\\eta =0 (indicated by lines) we find that that the energy of the maximum lies within the Mott gap for μ≲-1.15\\mu \\lesssim -1.15.",
"Inset: Electron density n 0 n_0 at the boundary showing very good agreement with the Bethe-ansatz result.",
"The dotted vertical lines indicate the positions μ=-2u-1+4u 2 \\mu =-2u-\\sqrt{1+4u^2} at which two electrons get bound to the boundary.Figure: Maximal value of the LDOS, N(ω=E max ,ja 0 )N(\\omega =E_{\\text{max}},ja_0), as a function of the distance to the boundary for u=1.125u=1.125, η=0.1\\eta =0.1 and L=90L=90.",
"For decreasing μ\\mu we observe that the spectral weight gets more and more localised at the boundary.Finally we consider the space dependence of the LDOS when going away from the boundary.",
"As is shown in Fig.",
"REF , lowering the boundary potential leads to an increase of the LDOS at the boundary, consistent with the formation of a boundary bound state localised at $j=0$ .",
"However, the system size and energy resolution is not sufficient to unveil an exponential space dependence of the LDOS as predicted by the field-theory analysis,[14] ie, $N(\\omega ,x)\\propto \\exp [-2x\\sqrt{\\Delta ^2-E_\\text{bbs}^2}/v_\\text{c}]$ ."
],
[
"Conclusion",
"In this work we have performed a numerical study of the LDOS of one-dimensional Mott insulators with an open boundary.",
"As microscopic realisations of the Mott insulator we have studied the (extended) Hubbard model at half filling.",
"The results for the Fourier transform of the LDOS revealed the existence of the Mott gap as well as several gapped and gapless dispersing modes.",
"These qualitative features were in perfect agreement with the results of field-theoretical calculations[14] of the LDOS in the Mott insulator.",
"Furthermore, we extracted quantitative values for the gap and velocities, which, in the case of the integrable Hubbard chain, were found to be in excellent agreement with the exact results.",
"[19] Besides open boundary conditions we have also considered the effect of a boundary potential.",
"For sufficiently strong potentials this results in the formation of a boundary bound state, which manifests itself in the LDOS as a non-dispersing feature inside the Mott gap.",
"In summary, our results show that spin-charge separation and the formation of boundary bound states can be observed in the Fourier transform of the LDOS amenable to numerical simulations or scanning tunneling spectroscopy experiments.",
"We thank Fabian Essler, Holger Frahm, Martin Hohenadler and Volker Meden for useful comments and discussions.",
"BS and DS were supported by the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO), under 14PR3168.",
"PS was supported by DFG-SFB 1170 and the ERC starting grant TOPOLECTRICS (ERC-StG-336012).",
"This work is part of the D-ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).",
"The authors acknowledge support by the state of Baden-Württemberg through bwHPC."
]
] | 1612.05597 |
[
[
"Parsec-scale Structure and Kinematics of Faint TeV HBLs"
],
[
"Abstract We present new multi-epoch Very Long Baseline Array (VLBA) observations of a set of TeV blazars drawn from our VLBA program to monitor all TeV-detected high-frequency peaked BL Lac objects (HBLs) at parsec scales.",
"Most of these sources are faint in the radio, so they have not been well observed with VLBI by other surveys.",
"Our previous measurements of apparent jet speeds in TeV HBLs showed apparent jet speeds that were subluminal or barely superluminal, suggesting jets with velocity structures at the parsec-scale.",
"Here we present apparent jet speed measurements for eight new TeV HBLs, which for the first time show a superluminal tail to the apparent speed distribution for the TeV HBLs."
],
[
"Introduction",
"At TeV energies (10$^{12}$ eV), three orders of magnitude higher than those studied by satellite-based detectors, gamma-ray astronomy is conducted with ground based telescopes such as H.E.S.S., VERITAS and MAGIC.",
"Over 175 TeV gamma-ray sources have now been catalogued (http://tevcat.uchicago.edu/), with over one third of these being extragalactic objects.",
"The majority of these (46 of 69) are classified as HBL (High-frequency–peaked BL Lac) objects, for which the synchrotron peak of the Spectral Energy Distribution (SED) lies at frequencies above 10$^{16.5}$ Hz.",
"Many of the well-studied HBL TeV sources have shown dramatic variability in their gamma-ray emission [1], [2].",
"The most rapid variations suggest extremely small emitting volumes and/or time compression by large relativistic Doppler factors of up to $\\sim $ 100 and challenge our understanding of relativistic jets [2], [3], [4].",
"The only way to directly obtain information on the parsec-scale structure of these blazar jets is by imaging the radio sources using the technique of Very Long Baseline Interferometry (VLBI).",
"However, many HBLs are fainter at radio wavelengths (typically tens of milli-janskys — see Table 1) than the more powerful quasars and BL Lac objects, and are not included in VLBI monitoring programs such as MOJAVE [5] and TANAMI [6].",
"Properties that can be measured from VLBI images — the apparent jet speed, radio core brightness temperature, core dominance, and jet–to–counter-jet brightness ratio — provide information on fundamental properties of the jet, such as the bulk Lorentz factor and viewing angle.",
"Yet despite the high Doppler factors inferred from TeV observations, our previous VLBA observations have established that TeV sources have only modest brightness temperatures, and jet component motions that are sub-luminal or only slightly superluminal [7], [8], [9], [10], [11], [12], [13], [14], [15].",
"This has been called the“doppler crisis” [16] or “bulk Lorentz factor crisis” [17].",
"Table: Current status of our VLBA monitoring program.",
"TeV sourcenames are those used by TeVCat, redshifts are those given in , where anasterisk denotes a tentative value or limit.",
"The NVSS flux density ismeasured at 1.4 GHz.",
"The number of epochs refers to the number ofVLBA images made in our monitoring program.",
"References are topapers presenting these image.Of the 46 HBLs detected at TeV energies: [leftmargin=*,labelsep=4mm] 11 have jet kinematics published previously by us.",
"(Some of these are also in MOJAVE.)",
"7 have, or soon will have, speeds determined by the MOJAVE program [5].",
"20 are included in the current phase of our program, with first epoch VLBA results for all 20 published [15].",
"We present here the kinematic results for the first portion of these sources.",
"4 are too far south to be studied with the VLBA, with several of these are part of the TANAMI monitoring program [6].",
"4 are recent detections which are yet to be monitored with VLBI.",
"The 39 TeV sources for which we have at least one VLBA image are listed in Table 1, together with their redshift, association in the NRAO VLA Sky Survey (NVSS) catalog [20], 1.4 GHz flux density, and details of our VLBA observations.",
"Images and data available at the project website: www2.whittier.edu/facultypages/gpiner/research/archive/archive.html .",
"Figure: VLBA images at 8 GHz of four Tev blazars in out VLBA monitoring program:(a) RGB J0152+017, (b) 1ES 0229+200, (c) RBS 0413 (0317+185), (d) 1ES 0347--121.TeV photons are attenuated by the infrared background [21] and, as is apparent in Table 1, the majority of extragalactic TeV sources are at relatively low redshift ($z <$ 0.2).",
"Conversely, studies of TeV gamma-ray spectra have offered a means of constraining the infrared background [21], [22]."
],
[
"Results",
"Images of four of the sources currently being monitored are shown in Figure 1.",
"All show parsec-scale morphologies typical of this class: a compact core (which hosts the supermassive black hole powering the source), and a weaker, one-sided jet that transitions to a decollimated structure with larger opening angle at a few tens of milli-arcseconds from the core (see image of 0229+200).",
"Multi-epoch studies of these jets over the course of several years allow the apparent speeds of the jet components to be determined.",
"Our previous VLBA studies indicated the absence of rapidly moving features in the jets of TeV HBLs; jet components were either nearly stationary or slowly moving ($< \\sim 1c$ ) [14].",
"With the addition of multi-epoch data from eight previously unpublished sources, including the four sources in Figure 1, the revised distribution of apparent jet speeds is shown in Figure 2.",
"This Figure incorporates the results of our previously published data and jet speeds for four sources that have been monitored as part of the MOJAVE project [5].",
"With the addition of new data, the tail of the distribution now extends to mildy superluminal apparent speeds for the first time; however, the majority of the TeV HBLs have peak apparent speeds of only about 1$c$ .",
"Combining these slow apparent speeds with the high Doppler factors ($\\delta $ ) implied by the TeV data to solve for the Lorentz factor ($\\Gamma $ ) and viewing angle ($\\theta $ ) results in unreasonbly small viewing angles ($\\theta <<$ 1$^\\circ $ ).",
"This would imply tiny jet opening angles, enormous linear sizes, and huge numbers of parent objects, and indicates that the combination of both high Doppler factor and slow apparent speed in the same jet region is unphysical.",
"If more realistic viewing angles of a few degrees are assumed, then the observed apparent speeds imply more modest Lorentz and Doppler factors for the radio jet.",
"The lack of detection of counterjets in the VLBI images for any TeV HBL [23], [15] requires that the Doppler factor cannot be arbitrarily low, and values of $\\delta $ and $\\Gamma $ of a few are most consistent with the combined VLBI data."
],
[
"Discussion",
"A variety of mechanisms have been proposed to try and reconcile the Doppler crisis [24], [25], [26], [27], [28], [29].",
"The most natural explanation is for a range of Doppler factors to coexist in the same jet on parsec scales through jet stratification.",
"One example is a jet that decelerates along its length [24].",
"In such a jet, the fast inner part sees blueshifted photons from the slower outer part, reducing the high Lorentz factor required in the fast portion.",
"This is a general feature of models with velocity structures; radiative interaction among the different regions allows the SED to be reproduced without the extremely high Lorentz factors and Doppler factors characterizing single-zone models.",
"Another alternative is a transverse velocity structure with a fast central spine and a slower outer sheath.",
"Radiative interaction between the spine and sheath naturally decelerates the spine, producing both radial and transverse velocity structures in the same jet [26].",
"Figure: A histogram of the peak jet speeds in TeV blazars.Blue denotes sources from our previously published data, red are from this work, and yellow are from MOJAVE.The highest apparent speed are observed inRBS 0413, (6.0±\\pm 1.2)cc, and RGB J0710+591, (5.8±\\pm 1.5)cc.If such spine–sheath jets are present in TeV HBLs, then the outer layer is expected to dominate the radio emission due to its SED shape, even with a lower Doppler factor than the spine [26].",
"An observational signature of this would be a limb-brightened transverse profile for the jet in VLBI images.",
"There is evidence for limb-brightening close to the core in a number of HBLs, e.g., Mkn 501 [11], 1ES 0502+675 [15], and H 1722+119 [15].",
"Other possible jet velocity structures have also been proposed, including multiple blobs [30] fast moving “needles” within the main jet [4], “minijets” powered by magnetic reconnection events [31], and turbulent subregions within the jet [29]."
],
[
"Conclusions",
"Our on-going VLBA monitoring of the growing number of TeV gamma-ray emitting HBLs has revealed that the distribution of peak apparent jet speeds in these sources extends to moderate superluminal speeds, $\\sim $ 6$c$ , but the majority display subluminal speeds, in contrast with the distribution for other classes of active galactic nuclei [5].",
"It has recently been proposed that jet kinematics may offer a better classification for blazars than the SED peak frequency [32], with HBLs tending to display quasi-stationary knots arising from recollimation shocks.",
"A possible physical explanation for this is based on TeV blazars having intrinsically weak jets that interact with the external medium forming a slow surrounding layer.",
"Radiative interaction between the spine and the sheath decelerates the spine, and eventually disrupts the jet.",
"Such jets are prominent in TeV-selected samples because selection favors rare high-synchrotron peak sources, which are drawn from the low end of the luminosity function where the source density is largest [33].",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.",
"This research has made use the TeVCat online source catalog (http://tevcat.uchicago.edu).",
"This research has made use of NASA's Astrophysics Data System.",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration."
]
] | 1612.05706 |
[
[
"Local Rigidity of Diophantine translations in higher dimensional tori"
],
[
"Abstract We prove a theorem asserting that, given a Diophantine rotation $\\alpha $ in a torus $\\T ^{d} \\equiv \\R ^{d} / \\Z ^{d}$, any perturbation, small enough in the $C^{\\infty}$ topology, that does not destroy all orbits with rotation vector $\\alpha$ is actually smoothly conjugate to the rigid rotation.",
"The proof relies on a K.A.M.",
"scheme (named after Kolmogorov-Arnol'd-Moser), where at each step the existence of an invariant measure with rotation vector $\\alpha$ assures that we can linearize the equations around the same rotation $\\alpha$.",
"The proof of the convergence of the scheme is carried out in the $C^{\\infty}$ category."
],
[
"Introduction",
"Let $R_{\\alpha } : x \\mapsto x+\\alpha \\mathbb {}\\mod {\\mathbb {Z}}^{d}$ be a translation on a torus ${d}$ with $d \\in \\mathbb {N}^{*}$ .",
"The search for conditions under which a diffeomorphism $f \\in \\mathrm {Diff}^{\\infty }({d})$ is guaranteed to be smoothly conjugate to $R_{\\alpha }$ is a very old subject in dynamical systems and the source of very deep and far-reaching studies, see for example [4] and [12] for the case $d=1$ .",
"To our best knowledge, the strongest rigidity result on perturbations of Diophantine rotations in higher dimensional tori in the literature is the one proved in [4].",
"This theorem, apart from the smallness assumptions, needs the preservation of a volume form, something that assures that every orbit rotates at the speed of the Diophantine rotation, so that the analogy with the one-dimensional theory is direct.",
"Our goal in the present article is to relax the condition of preservation of a (harmonic) volume form to a considerably weaker one, which seems partly optimal.",
"The closeness-to-rotations condition is, a priori at least, not indispensable, while the Diophantine property is known to be thus, since in the Liouvillean world rigid rotations tend to be fragile.",
"The present rigidity theorem, whose precise statement is given in thm REF , is in fact an instance of the strength of the Diophantine condition and of the K.A.M.",
"machinery.",
"Theorem A Let $\\alpha \\in {d}$ , $d \\in \\mathbb {N}^{*}$ , be a Diophantine rotation and $f \\in \\mathrm {Diff}^{\\infty } ({d}) $ be a small enough perturbation.",
"Then, if $\\alpha $ is in the convex hull of the rotation set of $f$ , the diffeomorphism $f$ is smoothly conjugate to the translation by $\\alpha $ .",
"The motivation for this theorem comes from a conjecture concerning diffeomorphisms of tori of dimension higher than 1.",
"In the one-dimensional case, the celebrated Denjoy theorem and examples establish a break in dynamical behaviour at the regularity threshold $C^{1+\\mathrm {BV}}({1})$ .By $C^{1+\\mathrm {BV}}({1})$ we denote the space of circle diffeomorphisms of the circle whose first derivative has bounded variation.",
"A circle diffeomorphism with irrational rotation number and regularity lower than $C^{1+\\mathrm {BV}}$ may have wandering intervals, while a diffeomorphism of regularity $C^{1+\\mathrm {BV}}$ cannot (see, e.g.",
"[5]).",
"In the one-dimensional case (see e.g.",
"[9] or [4]), a homeomorphism is assigned a unique rotation number, and, as soon as it is irrational, a continuous semi-conjugation to the rigid rotation can be readily constructed.",
"Denjoy's theorem is a rigidity theorem, stating that if the homeomorphism is sufficiently regular, the semi-conjugacy is in fact automatically a continuous conjugacy.",
"Arnold's theorem and subsequently the Herman-Yoccoz theory ([4] and [12]) is a further rigidity result in this setting, under additional regularity and arithmetical assumptions.",
"It is not known whether Denjoy's theory admits a reasonable generalization when the dimension of the torus is higher than one, but it is certainly not directly generalizable (due to the fact that the Denjoy-Koksma inequality fails, see [12]).",
"A homeomorphism of a higher dimensional torus does not, in general, have a unique rotation vector (see again [4]), and even if this is the case, the minimality of the corresponding translation does not imply the existence of a continuous semi-conjugacy to it (e.g.",
"[3]).",
"It is conjectured in [7], however, that for diffeomorphisms $f$ of ${d}$ , with $d\\ge 2$ , who satisfy the additional assumption that there exists $\\phi :{d} \\rightarrow {d}$ , continuous and surjective and such that $\\phi \\circ f = R_{\\alpha } \\circ \\phi $ with $R_{\\alpha }$ minimal, a similar break should appear.",
"That is, it is possible to construct such diffeomorphisms with wandering domains as long as the diffeomorphism is in, say $C^{d+1-\\varepsilon }$ , but not if it is more regular than, say, $C^{d+1}$ .",
"To our best knowledge, some examples of particular nature and of regularity lower than the conjectured threshold have been constructed by McSwiggen (see [7] and [8]) and by Sambarino and Passeggi (see [11]).",
"McSwiggen's examples on 2 are based on a Derived Anosov technique.",
"A linear Anosov diffeomorphism of 3 is deformed in $C^{\\infty } $ in order to turn the saddle around the fixed point into a repeller.",
"A diffeomorphism of 2 is then constructed as the holonomy map along the unstable foliation that is proved to survive the deformation.",
"The radical loss of regularity from $C^{\\infty } $ to $C^{3-\\varepsilon }$ comes from an inequality that has to be satisfied by an algebraic function of the eigenvalues of the original Anosov system and the need for contraction in the functional space of some bundle sections, and therefore in a quite indirect manner.",
"The diffeomorphism thus constructed is proved to have wandering domains, while it is semi-conjugate to the unstable holonomy map of the original Anosov diffeomorphism by collapsing the repelling basin of the origin to a point.",
"Moreover, even though it is not actually mentioned in the paper, the rotation vector is in fact Diophantine, since both the direction and the modulus of the translation vector are given by algebraic functions.",
"We think that, when compared to our result, this construction represents an instance of the fact that in low regularity arithmetic properties are irrelevant, while they become crucially so above some finite (and hopefully universal and explicit) threshold.",
"Acknowledgement: This work was supported by the ERC AdG grant no 339523 RGDD.",
"The author would like to thank Sebastian van Strien and Abed Bounemoura for some useful discussions during the preparation of the article."
],
[
"General notation",
"By $F : \\mathbb {R}^{d}\\rightarrow \\mathbb {R}^{d}$ we denote a lift of $f$ (and in general the corresponding capital letter will denote a lift whenever the small one denotes a diffeomorphism of the torus).",
"By a tilde we denote the lift of a point $x \\in {d} \\equiv \\mathbb {R}^{d} / \\mathbb {Z}^{d} $ to a representative in the covering space $\\tilde{x} \\in \\mathbb {R}^{d}$ .",
"A special case of diffeomorphisms of the torus is that of translations.",
"For $\\alpha \\in {d}$ , we define $R_{\\alpha } : x \\mapsto x+\\alpha \\mathbb {}\\mod {\\mathbb {Z}}^{d}$ We will denote the space of $C^{s}$ -smooth diffeomorphisms that are isotopic to the identity by $\\mathrm {Diff}^{s}_{0}({d})$ , and the distance in $\\mathrm {Diff}^{s}_{0}$ between two diffeomorphisms $f$ and $g$ by $d_{s} (f,g) = \\max _{0\\le \\sigma \\le s} \\Vert D^{\\sigma } F - D^{\\sigma } G \\Vert _{L^{\\infty }}$ The space of $C^{\\infty } $ diffeomorphisms will be furnished with the corresponding topology.",
"If $\\varphi : {d} \\rightarrow \\mathbb {R}$ , $\\hat{\\varphi }(k), k \\in \\mathbb {Z}^{d}$ are its Fourier coefficients, and $N \\in \\mathbb {N}^{*}$ , we denote by $T_{N}\\varphi (\\cdot )&=& \\sum _{|k| \\le N} \\hat{\\varphi }(k)e^{2i\\pi \\cdot } \\\\\\dot{T}_{N}\\varphi (\\cdot )&=& \\sum _{0<|k| \\le N} \\hat{\\varphi }(k)e^{2i\\pi \\cdot } \\\\R_{N}\\varphi (\\cdot )&=& \\sum _{|k| > N} \\hat{\\varphi }(k)e^{2i\\pi \\cdot }$ the inhomogeneous and homogeneous truncations and the rest, respectively, where $\\mathbb {Z}^{d}$ is equipped with the $\\ell ^{1}$ norm.",
"The estimates $\\Vert T_{N}\\varphi (\\cdot ) \\Vert _{s} &\\le & C_{s} N^{s+d/2} \\Vert \\varphi \\Vert _{0} \\\\\\Vert \\dot{T}_{N}\\varphi (\\cdot ) \\Vert _{s} &\\le & C_{s} N^{s+d/2} \\Vert \\varphi \\Vert _{0} \\\\\\Vert R_{N}\\varphi (\\cdot ) \\Vert _{s} &\\le & C_{s,s^{\\prime }} N^{-s^{\\prime } + s+d} \\Vert \\varphi \\Vert _{s^{\\prime }},$ are well known, where $0\\le s \\le s^{\\prime }$ .",
"If $f,g,u \\in \\mathrm {Diff}^{\\infty }_{0}({d})$ , then, see [6], $ \\Vert g \\circ f \\Vert _{s} &\\le & C_{s} \\Vert g \\Vert _{s} (1+\\Vert f \\Vert _{s})(1+ \\Vert f \\Vert _{0})^{s}\\\\\\left\\Vert g \\circ (f+u)-\\psi \\circ f\\right\\Vert _{s} &\\le &C_{s}\\left\\Vert gñ \\right\\Vert _{s+1}(1+\\left\\Vert f\\right\\Vert _{0})^{s}(1+\\left\\Vert f\\right\\Vert _{s})\\left\\Vert u\\right\\Vert _{s}$ Finally, the vector $ \\alpha \\in {d} $ is said to satisfy a Diophantine condition of type $\\gamma , \\tau $ , if the following holds: $\\alpha \\in DC(\\gamma ,\\tau ) \\Leftrightarrow \\forall k \\in \\mathbb {Z}^{d} \\setminus \\lbrace 0\\rbrace ,|k \\cdot \\alpha |_{\\mathbb {Z}^{d}} \\ge \\frac{\\gamma ^{-1}}{|k|^{-1}}$"
],
[
"Rotation vectors and sets",
"For this paragraph, see [10] or [2].",
"If $f \\in \\mathrm {Homeo}_{0}({d}) \\equiv \\mathrm {Diff}_{0}^{0}({d})$ and $x \\in {d}$ , we define $\\rho (x,f)$ as the following limit, provided that it exists: $\\frac{F^{n}(\\tilde{x})-\\tilde{x}}{n}$ It is defined $\\mathbb {}\\mod {\\mathbb {Z}}^{d}$ , due to the arbitrary choice of a lift for $f$ .",
"We also define $\\rho (f)$ , the rotation set of $f$ , as the accumulation points of $\\frac{F^{n_{i}}(\\tilde{x_{i}})-\\tilde{x_{i}}}{n_{i}}$ where $n_{i} \\rightarrow \\infty $ and the $x_{i} \\in {d}$ .",
"It can be shown that $\\rho (f)$ is the convex hull of $\\cup _{x \\in {d}} \\rho (x,f)$ .",
"If $\\nu \\in \\mathcal {M}(f)$ (i.e.",
"a probability measure on ${d}$ , invariant under $f$ ) then the quantity $\\int _{{d}} (F(\\tilde{x}) - \\tilde{x}) d\\nu $ is well defined and denoted by $\\rho (\\nu , f)$ .",
"The set $\\cup \\rho (\\nu ,f)$ , where the union is over $\\mathcal {M}(f)$ , is denoted by $\\rho _{meas}(f)$ .",
"M. Herman defines the rotation set of a homeomorphism precisely as $\\rho _{meas}(f)$ (his notation is different), and his conditions on $f$ and the volume that it preserves are needed in order to assure that $\\rho _{meas}(f) \\equiv \\rho (\\nu , f)\\equiv \\rho (\\mu , f) \\equiv \\rho (x,f)$ for every invariant measure $\\nu $ , and for every point $x\\in {d}$ , as is automatically the case in the circle (see [4]).",
"The condition imposed in thm.",
"REF can be written in the formBy $\\mathrm {Conv}$ we denote the convex hull of a set, i.e.",
"the smallest closed convex set containing the given one.",
"$\\alpha \\in \\mathrm {Conv}(\\rho (f)) = \\rho _{meas}(f)$ When $d =2$ , it can be shown that $\\rho (f)$ is convex, see [10], so that $\\rho (f) = \\rho _{meas}(f) $ ."
],
[
"Statement of the theorem",
"We can now restate our main theorem in a more precise way.",
"Theorem 3.1 Let $d \\in \\mathbb {N}^{*}$ , $\\gamma >0 $ and $\\tau > d$ .",
"Then, there exist $\\varepsilon >0$ and $s_{0} >0$ such that if $\\alpha \\in {d}$ and $f \\in \\mathrm {Diff}^{\\infty }_{0}({d})$ satisfy $\\alpha \\in DC(\\gamma , \\tau )$ $d_{0}(f(\\cdot ) , R_{\\alpha } ) < \\varepsilon $ and $d_{s_{0} } (f(\\cdot ) , R_{\\alpha } ) < 1$ $\\alpha \\in \\rho _{meas}(f) $ then $f $ is $C^{\\infty } $ conjugate to $R_{\\alpha }$ .",
"Moreover, the conjugation can be chosen close to the $Id$ .",
"Since such results tend to generalize to finite differentiability, we expect the following conjectural theorem to be true.",
"Theorem 3.2 (Conjectural) Let $d \\in \\mathbb {N}^{*}$ , $\\gamma >0 $ and $\\tau > d$ .",
"Then, there exist $\\varepsilon >0$ and $\\kappa , s_{0} >0$ with $0<\\kappa <s_{0}$ such that, if $\\alpha $ and $f \\in \\mathrm {Diff}^{s}({d})$ with $s>s_{0}$ satisfy the conditions of theorem REF in items $1-3$ , then $f$ is $C^{s-\\kappa }$ conjugate to $R_{\\alpha }$ .",
"The conjugation can be chosen close to the $Id$ .",
"This conjectural theorem is implied, for instance, by the proof in [4], which is carried out by approximation of finitely differentiable mappings by analytic ones.",
"The proof is valid as we point out in the answer to the following question.",
"Question 3.1 Does thm REF hold true in the real analytic category?",
"The is of course yes and an easy but non-optimal argument is as follows.",
"If the perturbation is small enough in some analytic norm, then it is small enough in $C^{\\infty } $ .",
"Therefore, thm REF applies and the diffeomorphism is $C^{\\infty } $ conjugate to the rigid rotation $R_{\\alpha }$ .",
"As a consequence, its rotation set is reduced to $\\lbrace \\alpha \\rbrace $ , and M. Herman's proof can be applied by just dropping the volume preservation assumption."
],
[
"Proof of theorem ",
"The proof relies on two lemmas.",
"The first one is a K.A.M.",
"lemma of very classical flavour and estimates, and represents one step of the K.A.M.",
"scheme that constructs successive conjugations reducing the diffeomorphism $f$ to the rigid rotation $R_{\\alpha }$ .",
"The second lemma is a geometric one and relates the size of the perturbation with $\\rho _{meas}$ .",
"For the scheme to produce a converging product of conjugations, two conditions are needed.",
"The first, more standard one, is a closeness to a rotation condition in an appropriate topology, and its general form is like the one of item REF of the statement of thm REF .",
"The second one is used in making sure that the perturbed diffemorphism $f$ does not drift away from $R_{\\alpha }$ : clearly, if $\\beta $ is a vector with rational coordinates, very close to $\\alpha $ , the two corresponding rotations are not conjugate.We remark that the rotation set of a diffeomorphism of ${d}$ is only preserved by conjugations that are isotopic to the $Id$ .",
"Such a condition can be imposed on the rotation set of $f$ , the perturbed diffeomorphism, and a possible condition would then be $\\rho (f) \\equiv \\lbrace \\alpha \\rbrace $ where $\\rho (f) $ is defined in paragraph REF .",
"In fact, the following weaker condition would be sufficient: $\\exists x \\in {d}, \\rho (x,f) = \\lbrace \\alpha \\rbrace $ However, it is enough that $\\alpha \\in \\mathrm {Conv} \\rho (f) = \\rho _{meas}(f)$ , which is exactly condition in item REF of the theorem.",
"All three conditions are weaker than $\\exists \\Phi :\\mathbb {R}^{d} \\rightarrow \\mathbb {R}^{d}, \\Vert \\Phi - Id \\Vert _{L^{\\infty }} < \\infty $ a measurable mapping, such that $\\Phi \\circ F = R _{\\alpha } \\circ \\Phi $ This last condition implies that for $a.e.$ $x \\in {d}$ $\\rho (x,f) = \\lbrace \\alpha \\rbrace $ If $\\Phi $ is assumed to be continuous, this holds for every $x \\in {d}$ and in fact $\\rho (f) = \\lbrace \\alpha \\rbrace $ .",
"Finally, this last condition is weaker than the existence of $ \\phi :{d} \\rightarrow {d}$ , surjective and respectively measurable or continuous, such that $\\phi \\circ f = \\mathbb {R}_{\\alpha } \\circ \\phi $ In the one-dimensional case, the existence of such a semi-conjugation is automatic as soon as the rotation number of the homeomorphism is irrational.",
"In the higher-dimesnional case, however, the existence of a semi-conjugation to a minimal rotation is an additional and restrictive hypothesis, and brings us back to the context of the conjecture mentioned in the introduction."
],
[
"Inductive lemma",
"We now state and recall the proof of the inductive conjugation lemma.",
"Lemma 4.1 Let $\\alpha \\in DC(\\gamma , \\tau ) \\subset {d}$ and $f \\in \\mathrm {Diff}^{\\infty }({d})$ , and call $\\Vert f - R_{\\alpha } \\Vert _{C^{s}} = \\varepsilon _{s}$ .",
"Then, for some absolute constant $C>0$ and for every $N \\in \\mathbb {N}^{*}$ such that $C \\gamma N^{2\\tau +d+2 }\\varepsilon _{0}<1$ there exists $\\phi \\in \\mathrm {Diff}^{\\infty }({d})$ such that $\\phi \\circ f \\circ \\phi ^{-1} = f^{\\prime }$ and the following hold true for the diffeomorphism $f^{\\prime } \\in \\mathrm {Diff}^{\\infty } $ thus defined.",
"There exists $\\beta \\in {d}$ such that $\\Vert f^{\\prime } - R_{\\beta } \\Vert _{C^{s}} = \\varepsilon ^{\\prime } _{s}$ satisfies $\\varepsilon ^{\\prime } _{s} \\le C_{s,s^{\\prime }}\\left(N^{s+2\\tau +d+2}\\varepsilon _{0}^{2}+N^{\\tau +d/2}\\varepsilon _{0}\\varepsilon _{s}+N^{s-s^{\\prime }+d} (1+ N^{s+ \\tau +d/2}\\varepsilon _{0} )\\varepsilon _{s^{\\prime }}\\right)$ for every $0\\le s \\le s^{\\prime } < \\infty $ .",
"Moreover, the conjugation $\\phi $ satisfies $\\Vert \\phi \\Vert _{s} \\le C_{s}\\gamma N^{s+\\tau + d/2}\\varepsilon _{0}$ Naturally, $\\beta \\simeq \\alpha + \\int (f - R_{\\alpha } )d\\mu $ and it represents a drift of the perturbed diffeomorphism with respect to $R_{\\alpha }$ .",
"The constants appearing in the statement depend on $ \\tau $ and $ d$ , but not on $N$ .",
"The proof is classical, but we sketch it for the sake of completeness.",
"Let $d=2$ in order to simplify notation, without any loss of generality.",
"Then, $f (\\cdot ) = R_{\\alpha } +\\begin{pmatrix}f _{1} (\\cdot ) \\\\f _{2} (\\cdot )\\end{pmatrix}$ where $f _{i} (\\cdot ) : {2} \\rightarrow \\mathbb {R}$ for $i = 1,2$ , are small in the $C^{\\infty } $ topology.",
"If we call $\\phi (\\cdot ) = Id +\\begin{pmatrix}\\phi _{1} (\\cdot ) \\\\\\phi _{2} (\\cdot )\\end{pmatrix}$ where $\\phi _{i} (\\cdot ) : {2} \\rightarrow \\mathbb {R}$ for $i = 1,2$ , then, for the conclusion of the lemma to be true, they need only satisfy the equation $\\phi _{i} (\\cdot ) \\circ R_{\\alpha } - \\phi _{i} (\\cdot )+ \\dot{T}_{N} f_{i} (\\cdot ) = 0$ Such functions $\\phi _{i}$ exist and are uniquely defined in $C^{\\infty } _{0} ({2} )$ .",
"They satisfy the estimate $\\Vert \\phi _{i} (\\cdot ) \\Vert _{s} \\le C_{s} \\gamma N^{s+\\tau + d/2} \\Vert f_{i}\\Vert _{0}$ Then, we can calculate $\\phi \\circ f \\circ \\phi ^{-1} &=&\\left(Id +\\begin{pmatrix}\\phi _{1} (\\cdot ) \\\\\\phi _{2} (\\cdot )\\end{pmatrix}\\right)\\circ \\left(R_{\\alpha } +\\begin{pmatrix}f _{1} (\\cdot ) \\\\f _{2} (\\cdot )\\end{pmatrix}\\right) \\circ \\left(Id -\\begin{pmatrix}\\phi _{1} (\\cdot ) \\\\\\phi _{2} (\\cdot )\\end{pmatrix}+ O(\\phi _{i}^{2})\\right) \\\\&=& \\left(Id +\\begin{pmatrix}\\phi _{1} (\\cdot ) \\\\\\phi _{2} (\\cdot )\\end{pmatrix}\\right)\\circ \\left(R_{\\alpha }-\\begin{pmatrix}\\phi _{1} (\\cdot ) \\\\\\phi _{2} (\\cdot )\\end{pmatrix}+ T_{N}\\begin{pmatrix}f _{1} (\\cdot ) \\\\f _{2} (\\cdot )\\end{pmatrix}+ O(\\cdot )\\right) \\\\&=&R_{\\alpha } +\\begin{pmatrix}\\hat{f} _{1} (0) \\\\\\hat{f} _{2} (0)\\end{pmatrix} +\\begin{pmatrix}\\phi _{1} (\\cdot ) \\\\\\phi _{2} (\\cdot )\\end{pmatrix}\\circ R_{\\alpha }-\\begin{pmatrix}\\phi _{1} (\\cdot ) \\\\\\phi _{2} (\\cdot )\\end{pmatrix}+ \\dot{T}_{N}\\begin{pmatrix}f _{1} (\\cdot ) \\\\f _{2} (\\cdot )\\end{pmatrix}+ O(\\cdot ) \\\\&=&R_{\\alpha } +\\begin{pmatrix}\\hat{f} _{1} (0) \\\\\\hat{f} _{2} (0)\\end{pmatrix} + O(\\phi _{i}^{2}, \\partial (\\dot{T}_{N} f _{i}).",
"\\phi _{i}, \\partial (\\phi _{i}) .",
"\\phi _{i},R_{N} f _{i} \\circ (Id - \\phi _{i}))$ The $O(\\cdot )$ term in the last line, which, anticipating the next section we call $\\begin{pmatrix}f ^{\\prime } _{1} (\\cdot ) \\\\f ^{\\prime } _{2} (\\cdot )\\end{pmatrix}$ so that $\\phi \\circ f \\circ \\phi ^{-1} =R_{\\alpha } +\\begin{pmatrix}\\hat{f} _{1} (0) \\\\\\hat{f} _{2} (0)\\end{pmatrix}+\\begin{pmatrix}f ^{\\prime } _{1} (\\cdot ) \\\\f ^{\\prime } _{2} (\\cdot )\\end{pmatrix}$ can be estimated in the $C^{s}$ -norm by $C_{s,s^{\\prime }}\\left(N^{s+2\\tau +d+2}\\varepsilon _{0}^{2}+N^{\\tau +d/2}\\varepsilon _{0}\\varepsilon _{s}+N^{s-s^{\\prime }+d} (1+ N^{s+ \\tau +d/2}\\varepsilon _{0} )\\varepsilon _{s^{\\prime }}\\right)$ This concludes the proof of the lemma."
],
[
"A posteriori estimate on the obstruction",
"The following elementary and well known observation establishes a relation between the displacement of points in the torus ${d}$ under a diffeomorphism $g$ with its rotation set $\\rho (g)$ .",
"Lemma 4.2 Let $g \\in \\mathrm {Diff}^{\\infty }_{0} ({d})$ and $\\beta \\in {d} $ .",
"If there exists $x \\in {d}$ such that $ \\rho (x,g) = \\lbrace \\beta \\rbrace $ , then $\\beta \\in \\mathrm {Conv} \\lbrace G (\\tilde{x}) -\\tilde{x} ,x \\in {d} \\rbrace $ .",
"Inspection of the proof shows that the condition on the existence of an orbit rotating at speed $\\beta $ can be relaxed to $\\beta \\in \\rho (g)$ .",
"Let $x \\in {d}$ be such that $ \\rho (x,g) = \\lbrace \\beta \\rbrace $ .",
"Then, $\\frac{G^{n} (\\tilde{x}) -\\tilde{x}}{n} =\\frac{G (G^{n-1} (\\tilde{x})) - G^{n-1} (\\tilde{x})+ G (G^{n-2} (\\tilde{x})) -G^{n-2} (\\tilde{x}) + \\cdots + G( \\tilde{x}) - \\tilde{x}}{n}$ converges to $\\beta $ .",
"Since the right-hand side is an element of $\\mathrm {Conv} \\lbrace G (\\tilde{x}) -\\tilde{x} ,x \\in {d} \\rbrace $ and the latter set is closed, the lemma is proved.",
"In the context of lemma REF , we obtain the following corollary.",
"Corollary 4.3 There exists an absolute constant $C>0$ depending only on $d$ such that, under the hypotheses of lem.",
"REF , and assuming additionally that $\\alpha \\in \\mathrm {Conv} ( \\rho (f) )$ , $\\left\\Vert \\begin{pmatrix}\\hat{f} _{1} (0) \\\\\\hat{f} _{2} (0)\\end{pmatrix}\\mathbb {}\\mod {\\mathbb {Z}}^{2}\\right\\Vert \\le C \\varepsilon ^{\\prime }_{0}$ In the proof we assume for simplicity that $ \\rho (x,f) = \\lbrace \\alpha \\rbrace $ for some $x \\in {2}$ .",
"The proof of the corollary as it is stated follows easily.",
"If $ \\rho (x,f) = \\lbrace \\alpha \\rbrace $ , then $\\rho (\\varphi (x),f^{\\prime }) = \\lbrace \\alpha \\rbrace $ .",
"Then, by lemma REF , $\\alpha \\in \\mathrm {Conv}(F^{\\prime }(\\cdot ) - Id)$ Consequently, $0 \\in \\mathrm {Conv}(F^{\\prime }(\\cdot ) - \\alpha ) =\\mathrm {Conv}\\left(\\begin{pmatrix}\\hat{f} _{1} (0) \\\\\\hat{f} _{2} (0)\\end{pmatrix}+\\begin{pmatrix}f ^{\\prime } _{1} (\\cdot ) \\\\f ^{\\prime } _{2} (\\cdot )\\end{pmatrix}\\right)$ The corollary follows directly.",
"Since invariant measures are accumulation points of Dirac measures uniformly distributed on finite segments of orbits, we can relax the condition $\\alpha \\in \\rho (f) $ to $\\alpha \\in \\rho _{meas}(f) $ : We must have $0 = \\int _{{d}}\\left(\\begin{pmatrix}\\hat{f} _{1} (0) \\\\\\hat{f} _{2} (0)\\end{pmatrix}+\\begin{pmatrix}f ^{\\prime } _{1} (\\cdot ) \\\\f ^{\\prime } _{2} (\\cdot )\\end{pmatrix}\\right)d(\\phi _{*}\\nu )$ for every (fixed) $\\nu \\in \\mathcal {M}(f)$ such that $\\rho (\\nu ,f)= \\alpha $ .",
"We immediately get the same estimate as in cor.",
"REF ."
],
[
"KAM scheme and convergence",
"The estimates provided by lemma REF are sufficient for the convergence of the corresponding scheme, provided that some smallness conditions are satisfied, and that we can linearize around the same rotation $\\alpha $ throughout the scheme, so that no \"counter-term\" is needed (as in the normal form version of the theorem in [4]), and the Diophantine condition can be kept constant throughout the scheme.",
"This second condition is assured by the existence of an orbit rotating like $R_{\\alpha }$ , and by corollary REF .",
"Let us state this formally in the following proposition.",
"Proposition 4.4 Let $\\alpha \\in DC(\\gamma , \\tau ) \\subset {d}$ and $f = f_{1} \\in \\mathrm {Diff}^{\\infty }({d})$ , and call $\\Vert f_{1} - R_{\\alpha } \\Vert _{C^{s}} = \\varepsilon _{s,1}$ .",
"Then, there exist $\\epsilon >0 $ and $s_{0} \\in \\mathbb {N}^{*} $ such that if $\\varepsilon _{0,1}<\\epsilon \\text{ and } \\varepsilon _{s_{0},1} <1$ and if $\\alpha \\in \\rho (f_{1})$ then there exist inductively defined sequences $\\phi _{n} \\in \\mathrm {Diff}^{\\infty }_{0}({d})$ and $f _{n} \\in \\mathrm {Diff}^{\\infty }_{0}({d})$ such that $\\phi _{n} \\circ f _{n} \\circ \\phi ^{-1}_{n} = f_{n+1}$ with $ \\Vert f_{n} - R_{\\alpha } \\Vert _{C^{s}} = \\varepsilon _{s,n} \\xrightarrow[n\\rightarrow \\infty ]{} 0 , \\forall s \\in \\mathbb {N}$ Moreover, $\\prod _{k=1}^{n} \\phi _{k} \\xrightarrow[n\\rightarrow \\infty ]{} \\phi \\in \\mathrm {Diff}^{\\infty }({d})$ Clearly, this proposition implies thm.",
"REF .",
"The proposition is proved by iteratively applying lemma REF and then corollary REF in the following, now classical, way.",
"Let $N = N_{1} \\in \\mathbb {N}^{*}$ be large enough, chose $\\sigma >0$ , and define inductively $N_{n} = N_{n-1}^{1+\\sigma } = N^{(1+\\sigma )^{n-1}}$ to be the order of truncation at the $n$ -th step, as in the proof of lem.",
"REF .",
"Assume, now, that $\\phi _{n-1}$ has already been constructed, so that $f_{n}$ is well defined.",
"Suppose, additionally, that $f_{n}$ satisfies the hypotheses of lem.",
"REF for $\\varepsilon _{s} = \\varepsilon _{s,n}$ and $N=N_{n}$ .",
"Then, application of lem.",
"REF and then of cor.",
"REF grants the existence of $\\phi _{n}$ such that $f_{n+1} = \\phi _{n} \\circ f _{n} \\circ \\phi _{n}^{-1}$ satisfies, for all $0 \\le s \\le s^{\\prime } < \\infty $ , the inequality s,n+1 Cs,s'( Nns+2+d+20,n2+ Nn+d/20,ns,n+ Nns-s'+d (1+ Nns++d/20,n )s',n ) We remind that $\\varepsilon _{n,s} $ , defined after eq.",
"REF , represents the distance of the diffeomorphism $f_{n}$ from the fixed rotation $R_{\\alpha }$ , thanks to the a posteriori estimate of cor.",
"REF .",
"Since the term in $\\varepsilon _{0,n} \\varepsilon _{s^{\\prime },n}$ is not present in thm.",
"10 of [1], we partially reprove the convergence of the scheme.",
"In fact, only the two main steps, lemmata 11 and 14 of the reference, have to be proved for the kind of estimates that we have hereinThe discrepancy is due to the fact that in the reference the dynamical system considered is a cocycle, and the last term herein comes from composition of mappings, whereas in the context of cocycles mappings are composed uniquely with the $\\exp $ and only products need to be considered.. We therefore need the following lemma, which is proved in the appendix, section .",
"Lemma 4.5 Let $\\varepsilon _{s,n}$ satisfy the inductive estimates of eq.",
"REF .",
"If, moreover $\\varepsilon _{0,1} &<& N_{1}^{-\\gamma _{0} } \\\\\\varepsilon _{s_{0},1} &<& N_{1}^{b}$ for some appropriately chosen $\\gamma _{0},b>0$ , then the double sequence $\\varepsilon _{s,n}$ is well defined and for all $n$ $\\varepsilon _{0,n} &<& N_{n}^{-\\gamma _{0}} \\\\\\varepsilon _{s_{0},n} &<& N_{n}^{b}$ We note, en passant, that this lemma implies that, under the relevant smallness conditions on $\\varepsilon _{s,0}$ , the smallness conditions of lemma REF are satisfied for all $n$ .",
"Therefore, the double sequence $\\varepsilon _{s,n}$ is well defined and we only need to establish its convergence.",
"We then show that, given the decay and growth rates granted by the previous lemma, we can actually do slightly better.",
"The following lemma allows us to bootstrap exactly like in [1] and conclude the convergence.",
"Lemma 4.6 Let $\\varepsilon _{s,n}$ satisfy the inductive estimates of eq.",
"REF .",
"If, moreover $\\varepsilon _{0,n} &=& O(N_{n}^{-\\gamma _{0}}) \\\\\\varepsilon _{s_{0},n} &=& O(N_{n}^{b})$ with $\\gamma _{0}$ , $b$ and $s_{0} $ as in the previous lemma then, there exist $\\omega _{0} , \\omega >0$ such that $\\varepsilon _{0,n} &=& O(N_{n}^{-(1+\\omega _{0} )\\gamma _{0}}) \\\\\\varepsilon _{(1+\\omega ) s_{0},n} &=& O(N_{n}^{b})$ Thus, we have successively proved the following assertions: at each step, the inductive smallness hypothesis for lem.",
"REF to be applicable is satisfied by $f_{n}$ and $N_{n}$ .",
"Therefore, the double sequence $\\varepsilon _{s,n}$ is well defined for all $n\\in \\mathbb {N}^{*}$ and $s\\in \\mathbb {N}$ .",
"for every fixed $s\\in \\mathbb {N}$ and $\\lambda \\ge 0$ $N^{\\lambda }_{n}\\varepsilon _{s,n} \\xrightarrow[n\\rightarrow \\infty ]{} 0$ The shorthand $f _{n} - R_{\\alpha } = O _{C^{\\infty }} (N_{n}^{-\\infty })$ is common.",
"This is obtained by the fast convergence of $\\varepsilon _{0,n}$ to 0, faster than any power of $N_{n}$ , and the slow growth of $\\varepsilon _{s,n}$ (as a fixed power of $N_{n}$ ) for every $s$ fixed.",
"Convexity estimates allow us to conclude that for every $0<s^{\\prime }<s$ , $\\varepsilon _{s^{\\prime },n}\\rightarrow 0$ .",
"the fast convergence of $\\varepsilon _{s,n}$ to 0 and the fact that $\\Vert \\Phi _{n} - Id \\Vert _{s} \\le C_{s} \\gamma N_{n}^{s+\\tau + d/2}\\varepsilon _{0,n}$ imply, together with eq.",
"REF that the product of successive conjugations $\\prod _{k=1}^{n} \\phi _{k} \\in \\mathrm {Diff}^{\\infty } ({d})$ converges in the $C^{\\infty } $ topology to a well defined diffeomorphism $\\phi $ .",
"This concludes the proof of the proposition, and thus of thm REF ."
],
[
"A remark on the proof",
"In the one-dimensional case, the theory of the rotation number for an orientation preserving homeomorphism of 1 is considerably stronger, thanks to the existence of a natural cyclic order (or of a total order in the covering space $\\mathbb {R}^{1}$ ).",
"The analogue of our argument in the one-dimensional case would be the following.",
"Consider a Diophantine rotation $\\alpha $ and perturb it.",
"Suppose that one orbit with rotation number $\\alpha $ survives under perturbation.",
"Solve the linear cohomological equation and observe that if the obstruction (the mean value of the perturbation) is not of \"second order\", a contradiction would be established, e.g.",
"by fitting a rational number between $\\mathrm {Conv} \\lbrace F (\\tilde{x}) -\\tilde{x} ,x \\in {1} \\rbrace =\\lbrace F (\\tilde{x}) -\\tilde{x} ,x \\in {1} \\rbrace $ and $\\alpha $ , see [5].",
"Of course, and this is a particularity of the one-dimensional theory, if one orbit has rotation number $\\alpha $ , then all orbits do.",
"This, however, is not an essential part of the proof of the existence of a smooth conjugacy to the rigid rotation, since the proof of the uniqueness of the rotation number is formally independent of the construction of the K.A.M.",
"scheme and of the proof of its convergence.",
"M. Herman in his thesis, [4], defines the rotation number for circle diffeomorphisms using an invariant measure instead of a combinatorial definition as in [5] or [9].",
"Accordingly to the one-dimensional case, his definition of the rotation set for diffeomorphisms preserving a volume form assures that the rotation set is thus reduced to a point, and this hypothesis is, in fact, needlessly strong."
],
[
"Appendix",
"We now provide the missing proofs of lemmata REF and REF .",
"We note that, when eq.",
"REF is compared with the corresponding eq.",
"$7.2$ of [1], which in our notation reads s,n+1 Cs,s'( Nna+Ms0,n1+0+ Nna+ms0,ns,n+ Nna - (s'-s) (s',n+ Nn s'0,n ) ) the agreeing terms correspond to the admissible choice of parameters (in the notation of the reference) $\\sigma _{0} = 1 &,& a = 2\\tau +d+2 \\\\M = 1 &,& m = 0 \\\\\\mu = 2 &,& \\bar{\\mu } = 0$ However, there do not seem to exist admissible values of the parameters $g $ , $\\sigma $ and $\\kappa $ of the reference for which either our estimates can be brought to the form of those of the reference, or for which the proof found therein produces convergence of the scheme for our type of estimates.",
"Consequently, we take up the proof of convergence for our type of estimates and we remark that the broader values of parameters (or the broader scope of types of estimates) are obtained thanks to the fact that we consider $a$ not as an \"affine\" parameter, but as a \"homogeneous\" one, when compared to $\\gamma _{0}$ , $b$ and $s_{0}$ .",
"The additional term appearing in eq.",
"REF , namely $N_{n}^{2s-s^{\\prime }+a}\\varepsilon _{0,n} \\varepsilon _{s^{\\prime },n}$ , is due to the composition of the conjugation with the perturbation (and is not present in the context of cocycles).",
"By defining $\\gamma _{0}$ , $b$ and $s_{0}$ as multiples of $a$ we manage to absorb the additional growth by $N_{n}^{s}$ into $\\varepsilon _{0,n}$ , which is shown to decay fast enough.",
"Let us now proceed to the actual proofs of the lemmata.",
"[Proof of lem.",
"REF ] In view of the estimate of eq.",
"REF by setting $s=0, s^{\\prime } =s_{0}$ and $s=s^{\\prime }=s_{0}$ , we only need to show that if $\\varepsilon _{0,n}<N_{n}^{-\\gamma _{0}}$ and $\\varepsilon _{s_{0},n}<N_{n}^{b}$ , then 0,n+1 C0,s0( Nna0,n2+ Nna/20,n2+ Nn-s0+a/2 (1+ Nna/20,n )s0,n ) and s0,n+1 Cs0,s0( Nns0+a0,n2+ Nna/20,ns0,n+ Nna/2 (1+ Nns0+a/20,n )s0,n ) satisfy $\\varepsilon _{0,n+1}<N_{n+1}^{-\\gamma _{0}}$ and $\\varepsilon _{s_{0},n+1}<N_{n+1}^{b}$ .",
"The inequalities that need to be verified in the limit of $N_{1}$ large enough read $a- 2\\gamma _{0} < -(1+\\sigma ) \\gamma _{0} &,& a/2 - 2\\gamma _{0} < -(1+\\sigma ) \\gamma _{0} \\\\-s_{0}+a/2 + b < -(1+\\sigma ) \\gamma _{0} &,& -s_{0}+a - \\gamma _{0} + b < -(1+\\sigma ) \\gamma _{0} \\\\s_{0}+a - 2\\gamma _{0} < (1+\\sigma )b &,& a/2 - \\gamma _{0} + b < (1+\\sigma )b \\\\a/2 +b < (1+\\sigma )b &,& s_{0} +a - \\gamma _{0} +b < (1+\\sigma )b$ Substitution by $\\gamma _{0} = \\lambda a$ , $s_{0} = \\mu a$ and $b = \\nu a$ where $\\lambda , \\mu , \\nu $ are positive reals, gives $(1-\\sigma ) \\lambda &>&1 \\\\\\mu - (1+\\sigma ) \\lambda - \\nu &>& 1/2 \\\\\\mu - \\nu -\\sigma \\lambda &>&1 \\\\2\\lambda + (1+\\sigma )\\nu -\\mu &>&1 \\\\\\sigma \\nu + \\lambda &>&1 \\\\\\sigma \\nu &>& 1/2 \\\\\\mu - \\sigma \\nu - \\lambda &>& 1 $ Inequality REF implies that $\\lambda >1$ , so that ineq.",
"is redundant.",
"For ineq.",
", , and to be compatible, we need $(1+\\sigma ) \\lambda + \\nu + 1/2 &<& 2\\lambda + (1+\\sigma )\\nu -1 \\\\1+ \\sigma \\lambda + \\nu &<& 2\\lambda + (1+\\sigma )\\nu -1 \\\\1+\\sigma \\nu +\\lambda &<& 2\\lambda + (1+\\sigma )\\nu -1$ or equivalently $3/2 &<& (1-\\sigma )\\lambda +\\sigma \\nu \\\\2 &<& (2-\\sigma )\\lambda + \\sigma \\nu \\\\2 &<& \\lambda + \\nu $ The first two ineq.",
"are implied by REF , and .",
"We thus need to impose $1 &>& \\sigma \\\\\\lambda + \\nu &>&2 \\\\(1-\\sigma ) \\lambda &>&1\\\\\\sigma \\nu &>& 1/2 $ and choose $\\mu $ such that $\\max \\lbrace (1+\\sigma ) \\lambda + \\nu + 1/2 ,1+ \\sigma \\lambda + \\nu , 1+\\sigma \\nu +\\lambda \\rbrace < \\mu < 2\\lambda + (1+\\sigma )\\nu -1$ The conditions are equivalent to $\\frac{\\lambda -1}{\\lambda } > \\sigma &>& \\frac{1}{2\\nu } \\\\\\lambda &>&\\frac{2\\nu }{2\\nu -1} \\\\\\nu &>&1/2$ At this point, we encourage the reader to check lemmata 12 and 13 in [1], as they prepare the following proof of the lemma corresponding to lem.",
"14 of the reference.",
"[Proof of lem.",
"REF ] Let $\\varepsilon _{0,n} < \\bar{C} N_{n}^{-\\gamma _{0}}$ and $\\varepsilon _{s_{0},n} < \\bar{C} N_{n}^{b}$ .",
"We first prove that there exists $\\omega _{0}$ such that $\\varepsilon _{0,n} = O(N_{n}^{-(1+\\omega _{0})\\gamma _{0}})$ We calculate directly $\\varepsilon _{0,n+1} &\\le & \\bar{C} C_{0,s_{0}}\\left(N_{n}^{a - \\gamma _{0}}\\varepsilon _{0,n}+N_{n}^{-s_{0}+a/2 + b} \\right) \\\\&\\le & \\bar{C} C_{0,s_{0}} ^{2} \\left(N_{n}^{a - \\gamma _{0}}\\left(N_{n-1}^{a - \\gamma _{0}}\\varepsilon _{0,n-1}+N_{n-1}^{-s_{0}+a/2 + b} \\right)+N_{n}^{-s_{0}+a/2 + b} \\right) \\\\&\\le & ( \\bar{C} C_{0,s_{0}} )^{2} \\left(N_{n}^{a - \\gamma _{0}}\\left(N_{n-1}^{a - 2\\gamma _{0}}+N_{n-1}^{-s_{0}+a/2 + b} \\right)+N_{n}^{-s_{0}+a/2 + b} \\right) \\\\&\\le & ( \\bar{C} C_{0,s_{0}} )^{2} \\left(N_{n}^{\\frac{2+\\sigma }{1+\\sigma } a - \\frac{3+\\sigma }{1+\\sigma }\\gamma _{0} } +N_{n}^{\\frac{3+2\\sigma }{2+2\\sigma }a - \\gamma _{0} - \\frac{s_{0} - b}{1+\\sigma }} +N_{n}^{-s_{0}+a/2 + b} \\right)$ Again in the limit $n \\rightarrow \\infty $ we only need to verify that $\\frac{2+\\sigma }{1+\\sigma } a - \\frac{3+\\sigma }{1+\\sigma }\\gamma _{0} &<& -(1+\\sigma )(1+\\omega _{0} )\\gamma _{0} \\\\\\frac{3+2\\sigma }{2+2\\sigma }a - \\gamma _{0} - \\frac{s_{0} - b}{1+\\sigma } &<& -(1+\\sigma )(1+\\omega _{0} )\\gamma _{0} \\\\-s_{0}+a/2 + b &<& -(1+\\sigma )(1+\\omega _{0} )\\gamma _{0}$ The first inequality holds true for $0<\\omega _{0} < \\frac{2\\lambda - \\sigma - 2 - \\lambda \\sigma (1+\\sigma )}{\\lambda (1+\\sigma )^{2}}$ The second holds true as long as $\\mu > \\frac{3}{2} + \\sigma + \\nu + (1+ \\sigma )((1+\\sigma )(1+\\omega _{0} )-1)\\lambda $ This inequality is verified for $\\omega _{0} = 0$ , by REF and and by the choice of $\\mu $ , and therefore also verified for $\\omega _{0}$ small enough.",
"The third inequality is equivalent to $\\mu > \\frac{1}{2} + \\nu + (1+\\sigma )(1+\\omega _{0})\\lambda $ which is verified provided that $\\omega _{0}$ is small enough, by the choice of $\\mu $ .",
"The second assertion of the lemma follows directly from gain in the speed of convergence for $\\varepsilon _{0,n}$ (i.e.",
"from the fact that we can replace $\\gamma _{0}$ by $(1 + \\omega _{0}) \\gamma _{0}$ for some fixed $\\omega _{0} >0 )$ ), and the fact that in the inequalities $s_{0}$ , , and $\\mu $ and $\\lambda $ appear with opposite signs, so that increase in $\\mu $ can be compensated by the increase in $\\lambda \\rightarrow (1+\\omega _{0})\\lambda $ without the inequality being violated for the same choice of $\\nu $ .",
"The eventual multiplicative constants can be absorbed in the exponents in the limit of $n $ large enough.",
"This last lemma can be used in order to boostrap and obtain prop.",
"REF ."
]
] | 1612.05564 |
[
[
"Comments on: \"Echoes from the abyss: Evidence for Planck-scale structure\n at black hole horizons\""
],
[
"Abstract Recently, Abedi, Dykaar and Afshordi claimed evidence for a repeating damped echo signal following the binary black hole merger gravitational-wave events recorded in the first observational period of the Advanced LIGO interferometers.",
"We discuss the methods of data analysis and significance estimation leading to this claim, and identify several important shortcomings.",
"We conclude that their analysis does not provide significant observational evidence for the existence of Planck-scale structure at black hole horizons, and suggest renewed analysis correcting for these shortcomings."
],
[
"colorlinks, linkcolor=red!50!black, citecolor=green!50!black, urlcolor=blue!80!black Comments on: “Echoes from the abyss: Evidence for Planck-scale structure at black hole horizons\" Gregory Ashton Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Ofek Birnholtz [email protected].",
"Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Miriam Cabero Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Collin Capano Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Thomas Dent Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Badri Krishnan Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Grant David Meadors Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Max-Planck-Institut für Gravitationsphysik, D-14476 Potsdam-Golm, Germany Alex B. Nielsen Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Alex Nitz Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Julian Westerweck Max-Planck-Institut für Gravitationsphysik, D-30167 Hannover, Germany Leibniz Universität Hannover, D-30167 Hannover, Germany Recently, Abedi, Dykaar and Afshordi claimed evidence for a repeating damped echo signal following the binary black hole merger gravitational-wave events recorded in the first observational period of the Advanced LIGO interferometers.",
"We discuss the methods of data analysis and significance estimation leading to this claim, and identify several important shortcomings.",
"We conclude that their analysis does not provide significant observational evidence for the existence of Planck-scale structure at black hole horizons, and suggest renewed analysis correcting for these shortcomings.",
"The detections by the Advanced LIGO detectors of gravitational wave signals from binary black hole mergers [1], [2], [3] has opened up the possibility of new tests of the nature of these objects [4], [5], [3].",
"A recent work [6] has claimed to find evidence of near-horizon Planck-scale structure using data[7] from the three Advanced LIGO events GW150914, LVT151012 and GW151226.",
"In the model of [6] this near-horizon structure gives rise to so-called echoes [8], [9], [10].",
"Their inferred amplitude parameters suggest that a lot of gravitational wave energy was emitted in the echoes: a very rough calculation implies that the amount of energy emitted in the echoes was approximately 0.1 solar masses (for GW150914) and 0.2 solar masses (for LVT151012).",
"This should be compared to the total estimated energy emitted by the original signal of 3 solar masses (for GW150914) and 1.5 solar masses (for LVT151012).",
"The data used is from the LIGO Open Science Center (LOSC) [7] which contains a total of 4096 seconds of strain data from both Advanced LIGO detectors around the three events.",
"Of these data the authors use only 32 seconds centered around each event for their analysis.",
"The authors claim to find such echoes in data following the three events with combined significance of 2.9$\\sigma $ (p-value $3.7\\times 10^{-3}$ ; with the one-sided significance convention used in [1], [2], [3], this value corresponds to 2.7$\\sigma $ ).",
"If this claim were true, it would force a major re-evaluation of the standard picture of black holes in vacuum Einstein gravity.",
"Besides the marginal claimed significance, there are a number of aspects of the analysis of [6] that lead us to suspect that the true significance of their detection may be considerably weaker.",
"Here we will not examine the theoretical motivations for the existence of such near-horizon Planck-scale structure, nor the model templates the authors have chosen to search for.",
"Instead we will focus on the data analysis methods as reported and the significance estimates assigned to them.",
"Regarding these we highlight some major data analysis drawbacks, which cast doubt on this aspect of their result.",
"The first problem arises at how strong the relative signal should be for the three events.",
"The two binary black hole events GW150914 and GW151226 were detected by the Advanced LIGO detectors with significance levels $>5.3\\sigma $ and signal-to-noise ratios of $23.7$ and $13.0$ respectively[3].",
"The other event, LVT151012, had a reported significance of only $1.7\\sigma $ and a signal-to-noise ratio of $9.7$ combined between the two Advanced LIGO detectors.",
"However, in Table II of [6] we see that the signal-to-noise ratio of the claimed echo signal is actually largest for LVT151012.",
"The higher SNR cannot be due to the projected number of echoes for LVT151012 over 32 seconds of data ($\\sim \\!180$ ) being greater than the number of echoes for GW150914 over that duration ($\\sim \\!110$ ), because late echoes are strongly damped, decreasing to a factor of 10 over $\\sim \\!22$ echoes.",
"Thus in order for the echoes of LVT15012 to have a higher SNR than the echoes of GW150914, their amplitude must be very high.",
"In fact to account for the reported SNRs, the initial amplitude for the first echo of LVT151012 would have to be about $10\\%$ higher than that of GW150914$\\frac{\\rho _{LVT151012}}{\\rho _{GW150914}} = \\frac{A_{LVT151012} \\sqrt{\\sum _{p=1}^{180} \\gamma ^{2p}}}{ A_{GW150914} \\sqrt{\\sum _{p=1}^{100} \\gamma ^{2p}}} = \\frac{4.52}{4.13} \\sim 1.1$ , where we have used the nomenclature of [6], and $\\gamma =0.9$ ., while the original event's peak is about 2-3 times lower for LVT151012 in comparison to GW150914's.",
"This would require their parameter $A$ to be about 2-3 times larger for LVT151012 than for GW150914.",
"It would therefore be interesting to see plots and estimated parameters for LVT151012 (and GW151226), similar to those presented in Table I of [6] for GW150914.",
"A second worrying aspect is the determination of the values for their echo waveform model, Equation 9.",
"The model depends on six parameters: a phase factor, three time parameters $\\Delta t_{\\rm echo}$ , $t_{\\rm echo}$ and $t_{0}$ , and two amplitude parameters $A$ and $\\gamma $ .",
"The phase is modeled as a simple sign flip at each reflectionignoring the phase accumulated over the travel between the light ring and the near-horizon Planck-scale boundary., $A$ is maximized over analytically, and $\\Delta t_{\\rm echo}$ is determined by the parameters of the final black hole given in [3] as given in Equation 6.",
"The three parameters $\\gamma $ , $t_0$ and $t_{\\rm echo}$ are determined by maximization, with $\\gamma $ and $t_0$ kept fixed between the different events.",
"This maximization is done over a prior range, as displayed in their Table I, and the values resulting from this maximization for $\\gamma $ and $t_{0}$ are found to lie very close to the boundary of this prior range, $0.9$ and $-0.1$ respectively.",
"This suggests that there may be support for values of these parameters that lie outside of this range (no error ranges are given).",
"This would be particularly worrisome in the case of $\\gamma $ since a value greater than unity means that each successive echo would have an amplitude greater than the previous echo.",
"Such a result would seem unphysical, and if supported by the analysis method, would cast considerable doubts on the method's validity.",
"Even if values $\\gamma \\ge 1$ are not supported, the railing of the reported parameter values against their prior range is a sign that these values may not be the best fits to the data; if these values are in fact arbitrary, reflecting the priors rather than the data, they cannot be reliably considered as evidence for a detection claim.",
"It would be both helpful and prudent to show results of the analysis for wider prior ranges.",
"The third problem relates to how the background is estimated for their result, as displayed in their Fig. 5.",
"For each time $t_{\\rm echo}$ in a window covering offsets up to $\\pm 5\\%$ of $\\Delta t_{\\rm echo}$ after the merger, the matched filter SNR [11], [12] is maximized over the remaining parameters $\\Delta t_{\\rm echo}$ , $t_{0}$ , $A$ and $\\gamma $ , either for GW150914 or for the combined events.",
"In both cases the resulting peak of SNR is found to actually lie within $0.54\\%$ of $\\Delta t_{\\rm echo}$ .",
"They then estimate in each case how likely this peak would be in random noise by finding how often such a high peak occurs in data away from the merger.",
"However, since they originally allowed the time offset to range over $\\pm 5\\% \\Delta t_{\\rm echo}$ , they should account for possible, comparable background peaks occurring over that full range not only the restricted range $(0-0.54\\%)\\Delta t_{\\rm echo}$ .",
"A naïve accounting for this post-hoc reduction in the extent of the parameter range would apply a trials factor of about 20 to the number of higher-SNR background samples, which would reduce the significance below $2\\sigma $ .",
"A more sophisticated treatment of false positives over the reduced parameter range [13] indicates a trials factor of $\\mathcal {O}(10)$ , weakly dependent on the number of independent samples in the SNR time series.",
"It is unclear why this background estimation was performed using a range of $t_{\\rm echo}$ values that is only 9 to 38 echo periods away from the merger.",
"If there is indeed an echo signal in the data then this region will not be entirely free of the signal being searched for.",
"At the beginning of the region the amplitude of the echoes would only have dropped by a factor $0.9^9 \\!\\sim \\!",
"0.4$ .",
"One therefore expects a contaminated background estimation.",
"Each of the data sets released at [7] consist of 4096 seconds of data.",
"Both GW150914 and LVT151012 are located 2048 seconds into this data, thus for large stretches of the data such contamination would be negligibly small.",
"We expect that use of this relatively uncontaminated data would give a more self-consistent background estimate.",
"A full analysis of the data is outside the scope of this comment.",
"Without a full analysis it is not possible to say whether the signals contain any true evidence of an echo signal, but as discussed here there are sufficient problems with the data analysis methodology of [6] to cast grave doubt on their claimed significance of a $2.9\\sigma $ effect.",
"It would be interesting to see the results of the analysis with these problems addressed, regarding both estimated parameters and estimated significance.",
"In conclusion, we find that the evidence as presented in [6] is lacking in several key aspects, such that their current methodology cannot provide observational evidence for or against the existence of near-horizon Planck-scale structure in black holes.",
"The authors thank Karl Wette, Francesco Salemi, Marco Drago, Andrew Lundgren and Vitor Cardoso for useful discussions, and the authors of [6] for helpful communications."
]
] | 1612.05625 |
[
[
"Polaron-induced phonon localization and stiffening in rutile TiO$_2$"
],
[
"Abstract Small polaron formation in transition metal oxides, like the prototypical material rutile TiO$_2$, remains a puzzle and a challenge to simple theoretical treatment.",
"In our combined experimental and theoretical study, we examine this problem using Raman spectroscopy of photo-excited samples and real-time time-dependent density functional theory (RT-TDDFT), which employs Ehrenfest dynamics to couple the electronic and ionic subsystems.",
"We observe experimentally the unexpected stiffening of the $A_{1g}$ phonon mode under UV illumination and provide a theoretical explanation for this effect.",
"Our analysis also reveals a possible reason for the observed anomalous temperature-dependence of the Hall mobility.",
"Small polaron formation in rutile TiO$_2$ is a strongly non-adiabatic process and is adequately described by Ehrenfest dynamics at time scales of polaron formation."
],
[
"Introduction.",
"A fundamental unsolved problem in polar materials is how electric charge carriers are generated and how they move through the solid.",
"In these materials electrons deform the highly polarizable crystal lattice and form quasi-particles (polarons) consisting of the electron and a lattice deformation associated with it.",
"In the case of so-called small polarons the strong and complicated electron-lattice interaction renders the usual electronic band-structure description of the charge carriers insufficient.",
"Polarons have also proved important in surface electron transfer processes and photocatalysis on titania where, due to their high binding energy, they act as electron scavengers (Refs.",
"hendersonsurface,kenji2016,kolesov2015anatomy and references therein).",
"Since the 1960s, the availability of high-quality samples and the absence of complex magnetic effects have made rutile titania the prototypical polaron-forming transition metal oxide (TMO).",
"Still, the basic properties of the polaron remain controversial.",
"For instance, reported effective mass values range from 2 – 150 $m_{e}$ and room-temperature drift mobility ($\\mu _{\\perp }$ ) values from 0.03 – 1.4 cm2$\\cdot $ V-1$\\cdot $ s-1 [4], [5], [6], [7], [8].",
"The temperature dependence of mobility also remains subject of a debate: the drift mobility increases with rising temperature for $T\\gtrsim $ 300 K, while the Hall mobility decreases.",
"Non-adiabatic polaron theory predicts that both mobilities should rise with temperature due to thermally-activated hopping mechanisms, while adiabatic theory predicts the opposite because of higher rate of scattering events [5], [9], [10], [11], [12], [13].",
"As a possible explanation it was suggested[14] that activation temperature for non-adiabatic hopping is higher in the case of Hall motion, and increase of the Hall mobility should be expected at higher temperatures.",
"However earlier experiments in Ref.",
"bransky1969hall where the Hall mobility was shown to decrease within all measured temperature range up to $T=1250$ K do not agree with this model.",
"Here we report yet another puzzling experimental observation, a phonon-stiffening effect, which is not captured by existing polaron theories that typically predict softening of the phonon modes [16], [17], [18], [19], [11].",
"The formation of small polarons in titania was first reported by Bogomolov et al.",
"[4], [20], who estimated the lower bound of the polaron binding energy to be $\\sim $ 0.4 eV.",
"Recent computations support these findings[21], [22], namely that formation of small polarons in rutile titania is energetically favored with a binding energy estimated at $\\gtrsim $ 0.5 eV.",
"The small polaron view has been challenged [6] while other computational studies reported smaller polaron binding energies of $\\le $ 0.15 eV [23], [24].",
"Here we study the properties of polarons in rutile TiO$_{2}$ both experimentally and computationally.",
"We use Raman spectroscopy on undoped samples excited with UV laser light.",
"We model the polaron using density functional theory (DFT) and probe the dynamics of its formation with real-time time-dependent density functional theory (RT-TDDFT) which employs mean-field classical-ion (Ehrenfest) dynamics to couple the electronic and ionic subsystems[25].",
"We show that this approach is adequate to capture the dynamics and time scales of formation of small polarons by photo-generated or injected carriers in rutile TiO$_{2}$ .",
"In both experiment and theory we observe stiffening of the A1g phonon mode in titania upon UV illumination and upon injection of electrons.",
"The key elements that lead to this effect are the strong electron interactions in d-shells of the TM atoms and, closely related to it, the large anharmonic effects of their coupling to the lattice.",
"Based on our results we propose a qualitative model that explains the temperature dependence of the Hall mobility."
],
[
"Experimental.",
"For the experimental study, we obtained an undoped monocrystalline rutile TiO$_{2}$ sample from SurfaceNet GmbH.",
"We measured the Raman spectra using a LabRAM HR Evolution (Horiba) spectrometer with the excitation induced by the 632.8 nm line of a He-Ne laser and the 325.03 nm line of a He-Cd laser.",
"The spectra at room temperature were obtained in the backscattering geometry using a Raman microscope.",
"The diameter of the laser beam spot on the sample surface was around 1-2 $\\mu $ m. The measurements were performed with a spectral resolution of 0.7 cm-1 with the 632.8 nm line and 4 cm-1 with the 325.03 nm line.",
"For the theoretical modeling, in both the adiabatic and non-adiabatic (Ehrenfest dynamics) calculations we use our code TDAP-2.0[25], which is based on the SIESTA package[26], [27], an efficient DFT code employing numerical atomic orbitals (NAO) as the basis set.",
"Following Refs.",
"[28], [21], [22] in this work we use the LDA$+U$ approach in its spherically-averaged form[29], for computational efficiency.",
"In the non-adiabatic part we use TD-DFT and mean-field (Ehrenfest) propagation of classical ions: $M_{J}\\frac{\\partial ^{2}\\mathbf {R}_{J}}{\\partial t^{2}}=-\\mathbf {\\nabla }_{\\mathbf {R}_{J}}E_{KS}$ where $J$ is the ion index and $E_{KS}$ is the expectation value of electronic energy.",
"The electronic wavefunctions are propagated with effective single-particle TD-DFT equations[30]: $i\\frac{\\partial \\phi _{n}(t)}{\\partial t}=\\hat{H}_{KS}[\\rho ](t)\\phi _{n}(t),$ with $\\phi _{n}$ being the single-particle Kohn-Sham orbitals, $\\hat{H}_{KS}$ the Kohn-Sham effective Hamiltonian operator and $\\rho \\left(\\mathbf {r},t\\right)=\\sum _{n=1}^{N}|\\phi _{n}(\\mathbf {r},t)|^{2}$ is the electronic density of the system containing $N$ electrons.",
"To calculate phonon modes and frequencies we used finite atom displacements to construct the dynamical matrix (frozen-phonon method).",
"In all cases described below phonon modes were calculated after complete lattice relaxation with force tolerance 0.02 eV/Å.",
"Simulation setup.",
"We used a $4\\times 4\\times 4$ rutile titania supercell in all our calculations.",
"This cell contains 384 atoms and was found by converging difference between the energies of neutral and charged ($-|q_{e}|$ ) supercells to within 0.1 eV (see Table S1).",
"Brillouin zone was sampled at $\\Gamma $ -point, except for smaller supercells used in convergence tests where we used Monkhorst-Pack $k$ -point grids starting from $8\\times 8\\times 12$ for a single unit cell.",
"As in the works prior to this [31], [3], [25], we used PBE$+U$ functional with $U=4.2$ eV in all our calculations.",
"In this study we used standard SIESTA pseudopotentials and SZP basis set.",
"In all simulations that involved an extra electron or electron-hole pair, we employed spin-polarized version.",
"In Ehrenfest RT-TDDFT simulations we used a time step $\\Delta t=1$ a.u.",
"($\\approx 24$ as).",
"We thermalized the system for 1 ps ($T=300$ K) with standard BOMD using 1 fs time steps."
],
[
"Results",
"Rutile TiO2 has four Raman-active modes with symmetries $A_{1g}$ , $E_g$ , $B_{1g}$ and $B_{2g}$ [32].",
"In Fig.",
"REF we present Raman spectra obtained in the normal state of the sample (red line) and after it was photo-excited with UV laser (blue line).",
"The peaks at 440 and 607 cm-1 correspond to the $E_{g}$ and $A_{1g}$ modes respectively[32], [33], shown schematically in Fig.",
"REF a.",
"The peak at 143 cm-1 ($B_{1g}$ phonon) is obscured by the combination peak at 235 cm-1 [32] and can not be seen in the blue line because it is below the resolution range of the spectrometer used in the UV experiment.",
"The peak at 823 cm-1 which corresponds to the $B_{2g}$ mode is known to be exceptionally weak and difficult to resolve[32], [33] and is not seen in the spectra presented here.",
"The most interesting feature of the two spectra is a shift of the $A_{1g}$ 607 cm-1 peak to the higher wavenumber of 617 cm-1 in the blue spectrum.",
"Typically at higher temperatures of the sample a shift of the peaks to lower wavenumber is expected[33].",
"Thus, the shift of the $A_{1g}$ mode to higher wavenumber cannot be explained by heating of the sample due to carrier relaxation to the band edges, but could be caused by polaron-induced structural changes.",
"Figure: Raman spectra of rutile titaniataken with the 632.8 nm laser (red line) and the 325 nm UV-laser (blue line)with ℏω=\\hbar \\omega =3.8 eV >E gap >E_{gap}.",
"Peak positions in experimentwere assigned by deconvolution in Lorentzian functions (see Fig.",
"S1).",
"The insetshows a magnified view of the spectra around the 607 cm-1and 617 cm-1peaks.Our calculation of the lowest charge-neutral excited state within the DFT $\\Delta $ SCF method [25] yielded excitation energy $E_{g}^{\\Delta SCF}=3.7$ eV in the bulk at $T$ =0 K. After correcting this energy for the spin contamination, $E_{g}\\approx 2 E_{g}^{\\Delta SCF}-E_{triplet}$ [34], [35], we obtained $E_{g}=3.4$ eV.",
"This agrees very well with the value obtained from BSE calculations (3.3 eV) [36], thus confirming our choice of the value $U=4.2$ eV, but overestimates the experimental band gap of 3.03 eV.",
"We calculate the polaron binding energy $E_{p}$ from the difference of total electronic energies of the charged system (with excess charge of $-e$ ) of the ideal bulk $4\\times 4\\times 4$ supercell and the relaxed supercell with the same charge.",
"Because of the high dielectric constant of titania and large supercell size, we disregarded energy corrections arising from charge interaction between replicas and the compensating uniform positive background charge that is usually introduced to obtain converged energies of charged systems in periodic-cell conditions.",
"We find $E_{p}\\approx 0.9$ eV.",
"The lower bound for $E_{p}$ in rutile titania was previously estimated from experimental data in the seminal work by Bogomolov et al.",
"[4] to be $\\sim 0.4$ eV, and by Austin and Mott [5] to be $\\sim 0.6$ eV, with authors suggesting that this value is likely to be underestimated.",
"Our value of $0.9$ eV is in reasonable agreement with these estimates, as well as with values obtained in other $+U$ and hybrid-functional DFT computations [22], [21].",
"To elucidate the origin of the $A_{1g}$ mode stiffening, we calculated vibrational eigenmodes and frequencies using the frozen phonon method.",
"Before performing frozen-phonon calculations in the periodic cell containing an extra-electron, (to which we refer as the `E' simulation) and an electron-hole excitation (to which we refer as the `E-H' simulation), we relax the geometry using a conjugate gradient algorithm, starting from the last point of the corresponding thermalized non-adiabatic trajectories described below.",
"The results for the $A_{1g}$ and $E_{g}$ modes are presented in Table REF .",
"Table: Frequencies of A 1g A_{1g} and E g E_{g} phonons obtained by theoryand experiment.",
"GS refers to the ground state of titania, whileE and E-H to the extra-electron and photo-excited electron-hole systems.The agreement between experiment and theory is surprisingly good.",
"Note that for the $E_{g}$ phonon the experimental peak is expected to red-shift with increasing temperature [33], while in the simulation the effect of heating is not included.",
"Given the limitations of the experimental measurements and of the simulations it could be argued that this surprising level of agreement is due to fortuitous cancellation of errors.",
"Because of the agreement in the trend, that is, the stiffening of the $A_{1g}$ mode only, as well as systematic stiffening of this mode with increasing value of $U$ (see Fig.",
"REF and text below), we suggest that the simulation captures, at least qualitatively, the mechanism of the stiffening, prompting us to analyze further the vibrational and electronic structure of the exciton-polaron system.",
"Figure: (a)Stereographic representation of atomicdisplacements corresponding to the A 1g A_{1g} and E g E_{g} phononsin the ground state (defect-free) bulk titania.",
"(b) The A 1g A_{1g} 607 cm-1phonon displacement in orthographic projection, in the ground state (GS titania.",
"(c) The localized A 1g * A_{1g}^{*} phonon at 616 cm-1in the E-H simulations (with the corresponding phonon at611 cm-1in the E-simulationbeing nearly identical, see Fig.",
"S2).In Fig.",
"REF c we show the atomic displacement corresponding to the $A_{1g}$ phonon in the E-H simulation (the phonon mode in the E simulation is nearly identical, see Fig.",
"S2).",
"Due to breaking of the crystal symmetry (space group $P\\frac{4_{2}}{m}\\frac{2_{1}}{n}\\frac{2}{m}$ ) by the lattice distortion associated with the polaron, the 616 cm-1 phonon in the E-H simulation (and the 611 cm-1 in the E simulation) is completely localized around one of the Ti atoms, which we will denote as Ti*.",
"This new phonon is a symmetric oxygen breathing mode which belongs to the $A_{1g}$ representation of the Ti$^*$ point group symmetry $\\frac{2}{m}\\mbox{$\\frac{2}{m}\\frac{2}{m}$}$ ($D_{2h}$ ).",
"We denote this localized oxygen breathing mode as $A_{1g}^{*}$ for clarity.",
"One of two degenerate $E_{g}$ modes (the one involving apical O atoms bonded to Ti$^*$ as shown in Fig.",
"REF a preserves its character upon formation of the polaron, while the other (at 444 cm-1) wraps around the polaron and does not involve involve O atoms bonded to Ti$^*$ (Fig.",
"S3).",
"We provide an explanation for this effect below.",
"The frequency of both $E_{g}$ modes is unchanged.",
"Figure: Electronic structure of the exciton-polaron.",
"(a) Themagnetization density: blue isosurface represents spin-up (electron) and orangespin-down (hole) magnetization densities.",
"(b) Charge-densitydifference between the electronic excited and ground states at the same atomicgeometry: magenta isosurface represents charge difference (excess of “hole”)and cyan negative (excess of “electron”) charge difference .",
"We also showthe gradient of the density difference on the (11 ¯01\\bar{1}0) cut-through plane,with blue-shades depicting excess electron density and red-shades depictingexcess hole density (see Fig.",
"S4 for a similar picture on the (110) cut-throughplane); the cut-through planes are labeled according to crystallographicdirections.In Fig.",
"REF a we show the magnetization density associated with the excited electron (spin-up) and the hole (spin-down).",
"The excited electron is primarily localized on the dxy orbital of the Ti$^*$ atom, one of the three non-bonding $t_{2g}$ orbitals Here for simplicity we use usual notation applied to octahedral environment.",
"Because in rutile the octahedron is distorted the actual orbital structure and bonding is more complex[38], [39].",
"It is partially screened by the hole which is weakly localized on the next-neighbor O atoms.",
"Most of the hole occupies delocalized p-type lone-pair orbitals of the O atoms in the system.",
"The presence of the electron in anti-bonding states and the hole in bonding states of the system, as well as the mobility of both carriers, should result in an overall softening of the potential.",
"This simple picture ignores the rearrangement of the total electron density caused by the localized electron.",
"In Fig.",
"REF b we present the electron density difference between the exciton-polaron excited state and the ground state at the same geometry.",
"This plot shows that the localized electron causes further polarization of the Ti$^*$ -O bonds with the electron on the $e_g$ -like orbital displaced towards the O atoms.",
"This increase of valence electron density on O atoms compensates for the presence of the hole; indeed, the net Hirshfeld atomic charge on the oxygen atom is the same as in the ground state at the relaxed geometry.",
"Figure: Total electronic energy as a function of displacement of the Oatom along the component of the A 1g A_{1g} eigenvector.",
"Green, orange and purplegraphs corresponds to the exciton-polaron obtained for different values of UUand dashed blue line corresponds to the ground state at the ground stategeometry (positive displacement indicates diminished O-Ti * ^*-O angle).We consider next the displacement of O atoms bonded to Ti$^*$ in the $\\pm x,\\mp y$ direction on the ($1\\bar{1}0$ ) plane, see Fig.",
"REF b.",
"This corresponds to a scissor-like motion of four in-plane O atoms in the direction of the $A_{1g}$ mode in Fig.REF a-b.",
"The inward motion ($-x,+y$ direction for the O atoms in front and $+x,-y$ for the atoms in back) in the excited system should experience a harder potential than in the ground state due to Coulomb repulsion between valence electrons surrounding the O atoms and the excited electron in the lobes of the dxy orbital of the Ti$^*$ atom.",
"Alternatively, this can be viewed as hardening due to the action of the Hubbard Uwith increasing overlap between the bonding p-states of O atoms and the dxy orbital of Ti$^*$ and the increasing population of the latter state.",
"In rutile titania the oxygen octahedron is distorted and d-orbitals belonging to the $T_{2g}$ (reducible) representation are not degenerate; thus the potential can not be softened by moving the excited electron to another $t_{2g}$ orbital.",
"Outward motion of the O atoms ($+x,-y$ direction for the O atoms in front) is on the contrary softened by the interaction with the excited electron as the atoms move away from it and because of additional Ti-O bond rearrangement.",
"This change in potential leads to almost complete suppression of the scissor-like motion of the four O atoms in the $xy$ -plane for the $A_{1g}^{*}$ phonon, see Fig.",
"REF b.",
"The effect of Coulomb repulsion is confirmed on Fig.",
"REF , where the potential for the O atom is mapped for different values of $U$ and displays systematic stiffening with increasing $U$ (graphs for $U=1$ and $U=6$ on Fig.REF , which conform to the same trend, were left out for clarity).",
"A similar description can be applied to the apical O atoms vibrating along the Ti$^*$ -O bond (along the $z$ -axis in Fig.",
"REF ), with the reduced effect of Coulomb repulsion due to the absence of lobes of the occupied dxy-orbital in the $z$ -direction (see Fig.",
"S4 and Fig.",
"S5b).",
"The overall effect is stiffening of the potential.",
"Similarly, displacement in the direction of one of the Eg modes, Fig.",
"REF a, does not change significantly the overlap of O p-states and the occupied dxy-state of Ti$^*$ and thus has a smaller effect on this phonon mode.",
"Figure: Scheme of polaron motion in Hall experiment.",
"(a) The polaron is hopping from a Ti * ^* atom on the left to a Ti * ^* atom onthe right in the xy-plane, with the electron occupying their overlapping in-planelowest-energy d || d_{||} (d xy d_{xy}) orbitals.",
"The electric field is inthe [001 ¯00\\bar{1}] direction.",
"Thermal ellipsoids around the oxygens depict theirrelative vibrational amplitude.",
"(b) Same as (a) withaddition of magnetic field; the Lorentz force acts normal to the plane.",
"TheHall current is directed toward orbital d || ' d^{\\prime }_{||} on a Ti ' ^{\\prime } atom through theintermediate orbitals d ⊥ d_\\perp on Ti * ^* atoms.",
"The vibrational displacement of apical O-atoms shown by ellipsoids elevates the energies of either d || ' d^{\\prime }_{||} or d ⊥ d_{\\perp } leading to reduced Hall current.The asymmetry of the potential in relation to O motion and Ti$^*$ $t_{2g}$ orbitals may have interesting consequences for polaron transport in the presence of a magnetic field.",
"In Fig.",
"REF a we show the polaron moving in the c-direction.",
"The electron is hopping from the lowest-energy in-plane $d_{||}$ ($d_{xy}$ ) orbital of Ti$^*$ on the left to the same orbital of the next Ti$^*$ on the right.",
"The higher the population of phonon modes that involve in-plane O-atoms the higher the rate of nonadiabatic hopping.",
"High population of the phonon modes causes larger displacement for the apical O-atoms than for in-plane O-atoms, because the potential for the apical O-atoms is softer.",
"These large displacements decrease the rate of hopping in c$_{\\perp }$ direction, because the large displacement of negatively charged apical O-atoms leads to increased Coulomb repulsion either by destination orbital $d^{\\prime }_{||}$ ($d_{x^{\\prime }y^{\\prime }}$ -orbital in the symmetry-transformed $x,y,z$ -coordinates) or by relevant intermediate orbitals belonging to Ti$^*$ octahedra, $d_{\\perp }$ (e.g.",
"$d_{xz}+d_{yz}$ ), through which such hopping must proceed.",
"This effect opposes the Lorentz force that drives the Hall current, see Fig.",
"REF b.",
"The same argument applies to the motion of the polaron in the c$_\\perp $ -direction, with the Lorentz force acting in the c-direction.",
"The higher the temperature, the higher the population of the polaronic phonon modes, and therefore the higher the drift velocity, the stronger this effect should be.",
"This explains the observed decrease of the Hall mobility with increasing temperature, while the drift mobility (hopping rate) increases."
],
[
"Formation of the polaron and exciton-polaron",
"We examined how the polaron is formed in two different simulations.",
"In the first case we introduced a single extra electron into the thermalized ($T=$ 300 K) system (`E' simulation).",
"In the second case we introduced an electron-hole (e-h, `E-H' simulation) pair by finding the lowest excited state with the $\\Delta $ SCF method.",
"The E-H trajectory is closer to the experimental situation, where e-h pairs are generated by UV light.",
"This choice implicitly assumes that at experimental time scales the majority of the excited electrons relax to the bottom of the conduction band and the holes to the top of the valence band.",
"Prior to introducing the e-h pair we thermalized the system at $T=300$ K using standard Born-Oppenheimer molecular dynamics (BOMD).",
"To make the comparison between the two simulations more informative, we start both at the same point of the BOMD trajectory, that is, with the same ionic coordinates and velocities.",
"These simulations were performed with Ehrenfest dynamics within RT-TDDFT.",
"As control samples, we used two BOMD trajectories branching from the same original configuration, of which the first corresponded to a neutral ground-state trajectory (GS trajectory) and the second to an adiabatic trajectory with an added electron (`ad-E').",
"In both non-adiabatic simulations we observed the formation of a polaron (Fig.",
"REF , Fig.",
"S6 and movie S1).",
"In the ad-E simulation, even though we are using the same small time-step as in the non-adiabatic simulations ($\\approx $ 24 as), we observed large deviations from energy-conservation which lead to divergence of the trajectory in less than 5 fs.",
"This was caused by violation of the conditions of the Born-Oppenheimer approximation: we assume that before the full polaron is formed, there are states with the electron localized on different Ti atoms that are close in energy, while interaction between them is small in comparison to the potential barrier separating them.",
"This confirms that, at least in this case, the formation of the polaron is indeed a non-adiabatic process.",
"A polaron consists of a localized charge trapped in a self-induced lattice deformation.",
"To quantify the deformation, for each nearest-neighbor $J$ of Ti$^*$ we compute the quantity $\\kappa _{J}$ , defined as $\\kappa _{J}(t_{n})=\\sum _{k=1}^{n}\\left[d_{J}\\left(t_{k}\\right)-d_{J}^{GS}(t_{k})\\right]/n,$ where $t_{n}$ is the time at time step $n$ , $d_{J}$ is the distance between Ti$^*$ and atom $J$ in the non-adiabatic trajectories (E and E-H) and $d_{J}^{GS}$ is the same distance in the GS BOMD trajectory.",
"We present in Fig.",
"REF the values of $\\kappa _{J}(t)$ for different species in the neighborhood of Ti$^*$ .",
"In both the E and E-H simulations, the deformation converges at roughly 50-60 fs, which matches nicely the 55 fs period of the $A_{1g}$ mode obtained in Raman experiments here and in Ref.",
"tio2raman.",
"This result is expected from small polaron theory [10], [13] and demonstrates that the Ehrenfest approximation captures well the dynamics of small polaron formation.",
"As expected, negatively-charged O atoms are repelled from the Ti$^*$ atom in both trajectories and the O-Ti$^*$ bond length increases substantially (by $\\sim 0.07$ Å)."
],
[
"Conclusion",
"Most theoretical polaron models employ a single-particle picture for the electrons and linear electron-phonon interaction terms in order to understand carrier behavior and to interpret experimental results.",
"From such models, softening of the polaron-renormalized phonon frequency $\\tilde{\\omega }$ is expected, for any strength of electron-phonon interaction [18], [19], [12], [17], [16], [11].",
"In contrast to this, we observe stiffening of the $A_{1g}$ mode, both in the experimental Raman spectra and in our DFT simulations.",
"Our analysis of the simulations reveals strong anharmonic effects: upon breaking of the crystal symmetry by the polaron, the original crystalline $A_{1g}$ mode is transformed into a new oxygen breathing mode which is localized on the polaron.",
"The formation of the polaron is a strongly nonlinear process that involves significant change of the valence electron density around the localized electron.",
"In the simulation, the localization and rearrangement is caused by the Coulomb interaction correction introduced by the Hubbard $U$ term in the DFT$+U$ method.",
"These effects cause overall stiffening and localization of the Raman-active $A_{1g}$ phonon mode.",
"The resulting potential is asymmetric with regard to motion of in-plane and apical oxygens, as well as to $t_{2g}$ and other orbitals in the distorted octahedron.",
"This observation led us to propose a qualitative explanation of the anomalous temperature dependence of the Hall mobility, which decreases with temperature, while the drift mobility increases (above an activation temperature) as expected from non-adiabatic small-polaron theory.",
"The asymmetry of the potential leads, upon excitation of the phonon modes required for polaron motion in one direction, to higher-amplitude motion of the out-of-plane oxygens, which, through Coulomb repulsion, increase the energy of the orbitals involved in hopping in the direction perpendicular to the drift current, thus reducing the Hall current.",
"As a first step in quantitative modeling of these problems we applied Ehrenfest dynamics with RT-TDDFT to study the formation of the polaron in TiO2 and demonstrated the usefulness and applicability of this quantum-classical method in describing the dynamics and the timescale of this process.",
"These findings may be relevant to other $d$ - and $f$ - materials which display small-polaron properties."
],
[
"Acknowledgments",
"This work was supported by an NSF grant EFRI 2-DARE: Quantum Optoelectronics, Magnetolectronics and Plasmonics in 2-Dimensional Materials Heterostructures, Award 1542807.",
"Computational resources were provided by XSEDE [40] (Grant No.",
"TG-DMR120073), which is supported by National Science Foundation Grant No.",
"ACI-1053575, NERSC (ERCAP request number 88881) and the Odyssey cluster, supported by the FAS Research Computing Group at Harvard University."
]
] | 1612.05679 |
[
[
"Exploiting non-trivial spatio-temporal correlations of thermal radiation\n for sunlight harvesting"
],
[
"Abstract The promise of any small improvement in the performance of light-harvesting devices, is sufficient to drive enormous experimental efforts.",
"However these efforts are almost exclusively focused on enhancing the power conversion efficiency with specific material properties and harvesting layers thickness, without exploiting the correlations present in sunlight -- in part because such correlations are assumed to have negligible effect.",
"Here we show, by contrast, that these spatio-temporal correlations are sufficiently relevant that the use of specific detector geometries would significantly improve the performance of harvesting devices.",
"The resulting increase in the absorption efficiency, as the primary step of energy conversion, may also act as a potential driving mechanism for artificial photosynthetic systems.",
"Our analysis presents design guidelines for optimal detector geometries with realistic incident intensities based on current technological capabilities"
],
[
"Thermal light statistics",
"Thermal light is partially coherent and statistically ranges between Poisson and Bose-Einstein distributions.",
"Poissonian detection is the is the limit where light fields have no correlation in space and time, and which correspond to reception processes where the events are perfectly independent.",
"On the other hand, the Bose-Einstein distribution describes the upper correlations limit which occurs for one mode thermal light.",
"The Bose-Einstein probability distribution models the boson nature of photons, expressing their bunching tendency in such a way that arrival events become strongly correlated.",
"Below, we review the statistical behavior of the light with any degree of coherence by using the factorial moments generating functional $G(\\lbrace s_i\\rbrace )$ and afterwards we show how the limits are obtained therein.",
"A generating functional is defined as an infinite series whose coefficients contain all the statistical information of a stochastic process.",
"The probability distribution and the statistical cumulants of a given stochastic process are obtained by differentiation of the factorial moments generating functional with respect to the set of expansion parameters $\\left\\lbrace s_i\\right\\rbrace $ , as we shall see.",
"The factorial moments generating functional is defined [1]: $G(\\left\\lbrace s_i\\right\\rbrace )\\equiv \\exp \\lbrace \\sum _{m=1}^{\\infty } \\frac{(-1)^m(s_1s_2\\ldots s_m)}{m!",
"}k_{[m]}\\rbrace ,$ where $k_{[m]}$ is the m-th order factorial cumulant containing the m-th order correlation.",
"The formula for $k_{[m]}$ can be seen as: $k_{[m]}=(m-1)!",
"(\\alpha _1 \\alpha _2\\ldots \\alpha _m) \\left\\lbrace \\Gamma ^{(m)}\\right\\rbrace .$ In this expression, $\\alpha _i$ is the efficiency or the effective cross section for absorption of the i-th detector and $ \\left\\lbrace \\Gamma ^{(m)}\\right\\rbrace \\equiv \\sum ^{N}_{l=1} \\int ^T_{0} dt \\Gamma ^{(m)}_{l,l}(t,t)$ is the trace operation, summing for all the $N$ detectors and integrating over detection time $T$The observation time $T$ in real scenarios is constrained by the time during which the detectors remain open to the incoming field..",
"The $m$ -th order correlation function is defined as [2], [3]: $\\hspace{-28.45274pt} \\Gamma ^{(m)}_{l_1,\\ldots ,l_m}(t_1,\\ldots ,t_m)=\\langle \\hat{\\vec{E}}^-(\\vec{r}_{l_1},t_1)\\ldots \\hat{\\vec{E}}^-(\\vec{r}_{l_m},t_m)\\hat{\\vec{E}}^+(\\vec{r}_{l_m},t_m)\\ldots \\hat{\\vec{E}}^+(\\vec{r}_{l_1},t_1) \\rangle ;$ where we assign an index $l_i$ for each detector, placed at the positon $\\vec{r}_{l_i}$ , and $t_i$ is the time at which the detector $l_i$ register the photon arrival.",
"The $m$ -th order correlation can be writen as [4], [5]: $\\Gamma ^{(m)}_{k,l}(t_1,t_2)&=\\Gamma _{k,l}(t_1-t_2) \\hspace{128.0374pt} \\text{$m=1$, }\\\\&=\\sum ^{N}_{i=1}\\int ^T_0 dt \\Gamma ^{(m-1)}_{k,i}(t_1,t)\\Gamma _{i,l}(t,t_2) \\hspace{34.14322pt} \\text{$m\\ge 2$ }$ The definition presented in Eq.REF is general since all the correlation orders of the stochastic process are included.",
"However in light detection there is a set of $N$ detectors and only $N$ -order correlations can be detected.",
"Once the generating functional is defined, we can apply the formalism for thermal light detection which is Gaussian distributed and therefore just the second cumulant exists.",
"After applying the equation in REF with $m=2$ , the thermal light generating functional in equation REF turns into the double product $G(\\left\\lbrace s_i\\right\\rbrace )=\\prod _{j=1}^T \\prod ^{N}_{k=1} (1+\\varpi _jb_ks_i)^{-1}$ presented in the main text (see Eq.1)[6], [7].",
"Here $\\varpi _j$ and $b_k$ accounts for the spatio-temporal correlations since they correspond to the eigenvalues of the temporal and spatial Fredholm equations presented thereAssuming cross spectral purity which separates the spatial from temporal correlations $\\Gamma _{k,l}(t_1-t_2)=\\sqrt{\\left\\langle n_k\\right\\rangle }\\sqrt{\\left\\langle n_l\\right\\rangle }\\gamma _{k,l}\\gamma (t_1-t_2)$.",
"The probability density function for the stochastic process is obtained by differentiation of the generating functional.",
"For light detection, the photo-counting probability to jointly detect $n_1,n_2 \\ldots n_N$ photons in the detectors labeled 1,2,...,$N$ is given by: $P(n_1,n_2,...,n_N;T)=\\left\\lbrace \\prod _{i=1}^N \\frac{(-1)^{n_i}}{n_i!",
"}\\frac{\\partial ^{n_i}}{\\partial s_i^{n_i}} \\right\\rbrace G(\\lbrace s_i\\rbrace ,T)|_{\\lbrace s_i=1\\rbrace },$ and the probability of detection of $n$ photons in the total set, regardless of the specific counting record of any individual detector is $P(n)=\\sum _{\\lbrace n_i \\rbrace }\\delta \\left( n-\\sum _{i=1}^Nn_i\\right)P(n_1,n_2,...,n_N)$ It is possible to map the photo-counting distribution $P(n)$ to the time domain in order to find the probability density function for the times between consecutive detections $f(t)$ ; it is obtained by time differentiation of $P(n=0)$ (probability to register zero arrivals when a time $T$ has passed) [1], [7], [8]: $f(t) =-\\left.\\frac{dP(n=0,T)}{dT}\\right|_{T=t};$ Now we are going to apply the presented formalism to achieve the maximal and minimal correlation limits.",
"The Poissonian limit is reached when detection times and areas are so large that there are no spatial or temporal correlations.",
"This limit is obtained from Equation REF with $m=1$ because only the first factorial cumulant exists for a non-correlated process and $\\Gamma ^{(1)}_{l,l}(t,t)=\\left\\langle n_l(t)\\right\\rangle $ [4].",
"Therefore, the probability to detect $n$ photons is given by the Poisson distribution [2], [3], [9]: $P(n)=\\frac{e^{-\\left\\langle n\\right\\rangle }\\left\\langle n\\right\\rangle ^n}{n!",
"},$ where $\\left\\langle n\\right\\rangle =\\sum ^{N}_{l=1} \\alpha _l \\int ^{T}_{0} n_{l}(t)=NAT(\\alpha \\left\\langle I^{\\prime }\\right\\rangle )$ is the average number of total detected photons in the detector's area $A$ , during the counting time $T$ .",
"The net average intensity is $\\alpha \\left\\langle I^{\\prime }\\right\\rangle $ for every detector with efficiency $\\alpha $ such that both quantities ($\\langle I^{\\prime }\\rangle $ and $\\alpha $ ) are position-independent.",
"Thus, the Poisson limit is only reached when two conditions are fulfilled: first, no correlations exist because the area, time, and number of detectors are large ($NAT\\rightarrow \\infty $ ) and second, the intensity received by each one is very small ($\\left\\langle I^{\\prime }\\right\\rangle \\rightarrow 0$ ) such that the total number of detections $NAT\\left\\langle I^{\\prime }\\right\\rangle $ is finite.",
"For instance, even in the case of no spatial correlations, if the detection time is short enough, the statistics will not reach the Poissonian behavior, remaining correlated despite the distance among detectors.",
"In the same way, a low intensity situation is not enough to guarantee Poissonian behavior and discard correlations.",
"The interplay of these quantities is responsible of the finite correlations in the statistics associated to the photo-detected fields.",
"The Bose-Einstein limit is obtained whenever the area covered by the set of detectors is much smaller than the coherence area ($A \\ll A_C$ ) and the detection time is also smaller than the coherence time ($T \\ll \\tau _C$ ).",
"In this case, $\\Gamma ^{(m)}_{k,l}(t_1,t_2)=(NT)^{m-1}\\left\\langle I^{\\prime }\\right\\rangle ^m$ no matter the positions of the detectors and the detection times.",
"The differentiation of the obtained generating functional reproduces the Bose-Einstein distribution (eq.",
"REF ), where the average detections at the $i$ th detector $\\left\\langle n_i\\right\\rangle =\\left\\langle n^{\\prime }\\right\\rangle $ is also independent of position because of the assumption of a very small area.",
"Again $\\left\\langle n\\right\\rangle =NAT(\\alpha \\left\\langle I^{\\prime }\\right\\rangle )$ is the average number of total detected photons in the detector's area $A$ .",
"$P(n)=\\frac{1}{\\left\\langle n\\right\\rangle +1}\\left(\\frac{\\left\\langle n\\right\\rangle }{\\left\\langle n\\right\\rangle +1}\\right)^n.$"
],
[
"Configuration sensitivity",
"Sensitive and no-sensitive regimes are explained by the second order correlation function plotted in Figure REF (a).",
"For very short separations (about $\\nu < 0.1$ ), the top of the second order coherence function is flat and changes of $\\nu $ about these value do not alter the spatial coherence, therefore configuration changes produce no effect on the photocounting probabilities.",
"In the other extreme ($\\nu > \\pi $ ) the function $\\gamma _{ij}$ has fade out enough such that spatial correlations are not relevant at those distances.",
"In between of these values, ie.",
"just before of the first zero of the Bessel function, the function's slop is steep enough to affect significantly $P(n)$ .",
"Figure: Second order spatial-coherence function for the radiation field.",
"The red arrow points the top of the function where first derivative tends to be constant.",
"Scheme for the bunched photon-arrivals in time in (b) and for the clustering on the detection screen in (c) (instantaneous detection).Figures REF and REF display the $P(n)$ curves for $N=3$ and $N=4$ detectors respectively, showing that configuration plays a role by changing photocounting probability distributions.",
"In figure REF (a) and (b), two configurations of three detectors (linear and triangular) are calculated and compared in (c).",
"The $P(n)$ curves for $\\nu \\le 0.1$ are all overlapped and curves for $\\nu \\ge 2\\pi $ as well, confirming that there is sensibility to changes in distances only when there is a competition between the detectors separation and the coherence length ($\\nu _{ij}\\approx 1$ ).",
"The comparison in Fig.REF (c) shows that only for values of $\\nu \\approx 1$ the distributions are different for different configurations.",
"Similarly Fig.",
"REF presents $P(n)$ probability distributions for three configurations of four detectors (linear, square and triangular) and compares them; leading to the same conclusion.",
"Figure: Photocounting probability distribution P(n)P(n) for N=3N=3 detectors and different spatial correlation degrees ν=ν k,l =a r2πd k,l λ\\nu =\\nu _{k,l}=\\frac{a}{r} \\frac{2 \\pi d_{k,l}}{\\lambda }.",
"(a) Linear and (b) triangular configuration.",
"(c) displays the comparison between the probability distributions of both configurations at ν=1\\nu =1.",
"〈n〉=10\\langle n \\rangle =10 and T/τ c ≪1T/\\tau _c\\ll 1.Figure: Photocounting probability distribution P(n)P(n) for N=4N=4 detectors and different spatial correlation degrees ν=ν k,l =a r2πd k,l λ\\nu =\\nu _{k,l}=\\frac{a}{r} \\frac{2 \\pi d_{k,l}}{\\lambda }.",
"(a) Linear configuration.",
"(b) Square configuration.",
"(c) Triangular configuration with a detector in the middle, covering the same area as in (b).",
"(d) Displays the comparison between the probability distributions of configurations presented in (a) to (c) at ν=1\\nu =1.",
"P(n)P(n) curves for ν≤0.1\\nu \\le 0.1 are all overlapped and curves for ν≥2π\\nu \\ge 2\\pi as well, confirming that only for values of ν≈1\\nu \\approx 1 one has a sensitivity to configuration.",
"〈n〉=10\\langle n \\rangle =10 and T/τ c ≪1T/\\tau _c\\ll 1.To figure out the reception behavior of the individual detectors, Figure REF displays the probability to detect 2 or 3 photons respectively in a set of $N=3$ detectors.",
"In this example the detectors labeled 1 and 3 are at fixed positions separated a distance $\\nu $ and the probability is calculated as a function of the position $d$ of the detector in the middle (labeled 2).",
"It is corroborated that in presence of correlations (plots for $\\nu =0.01$ and 1), the detection is more probable when photons are distributed among the detectors.",
"When spatial correlations are not present ($\\nu =100$ ) there are no preferred configurations The curves are not flat in the extremes because when the mobile receptor is close to one of the other two, the correlations play a role again..",
"Specifically at $\\nu =1$ and around, the probability to detect in one extreme increases when the mobile detectors is in the opposite side, confirming the result subsequently shown in Figs.4 and 5 at high intensity values.",
"Figure: Individual probability distributions (obtained from Eq.)",
"in a set of N=3N=3 detectors.",
"The positions of the extreme ones are fixed, separated by a distance ν\\nu and the middle one is mobile.",
"Now, the distance between detectors labeled 1 and 2 is changed and the curves show detection probabilities in all the cases: ν=0.01\\nu =0.01, ν=1\\nu =1 and ν=100\\nu =100.",
"In all the calculations 〈n〉=1\\langle n \\rangle =1 and T/τ c ≪1T/\\tau _c \\ll 1."
],
[
"What happen when detection time increases?",
"As done for the short detection time limit, Figure REF explores the behavior of simple $P(n)$ curves for $N=2$ detectors when the detection time $T$ is increased.",
"The plots show the approach to a Poissonian detection when $T\\rightarrow \\tau _c$ .",
"At this $T$ value, the sensitivity to configuration is almost lost and the $P(n)$ correlated curves practically overlap with the Poisson distribution.",
"This sensitivity is found for the same $\\nu $ range as in the short detection time case [$0.1 \\le \\nu \\le \\pi $ ] but curves get closer and closer during the transition from maximally correlated to the Poisson distribution, when $T$ is increased.",
"Figure: P(n)P(n) for N=2 detectors changing the detection time.",
"(The average number of detections n=1\\left\\langle n\\right\\rangle =1 is fixed).To better explore the photo-counting sensitivity to configurations around $\\nu =1$ , Figures REF and REF present contour and 3D probability distributions for sets of $N=3$ and 4 detectors respectively, for the detection time $T$ increased up to the coherence time.",
"We found that the probability function displays the same behavior as for $T\\ll \\tau _c$ , but for slightly higher $\\left\\langle n \\right\\rangle $ , the sensitivity to configuration is still conserved.",
"Figure: P(n=N)P(n=N) Contour and 3D plots for configurations of N=3N=3.",
"The detection time T/τ c =1T/\\tau _c=1.",
"Two detectors are fixed separated a distance ν=1\\nu =1 and the position of the third one is scanned.",
"The (x 3 ,y 3 )(x_3,y_3) coordinates are in dimensionless ν=ν k,l =2πr k,l /l c \\nu =\\nu _{k,l}=2\\pi r_{k,l}/l_c units.Figure: Contour and 3D plots of P(n=4)P(n=4) for configurations of N=4N=4 detectors.",
"The detection time T/τ c =1T/\\tau _c=1.",
"(a) Three detectors are fixed in the vertices of an equilateral triangle and the position of the fourth one is scanned (The triangle's side is ν=1\\nu =1).",
"(b) Three detectors are fixed in the corners of square of side ν=1\\nu =1 and the fourth one is mobile.",
"The (x 4 ,y 4 )(x_4,y_4) coordinates are in dimensionless ν=ν k,l =2πr k,l /l c \\nu =\\nu _{k,l}=2\\pi r_{k,l}/l_c units.Calculations of $f(t)$ for arbitrary detection times where also performed and are shown in figure REF .",
"Here the detection set is composed by 4 detectors in the configuration which maximizes the detection probabilities in Figures REF and REF .",
"It can be seen how the correlation degree and therefore, the burstiness measure, decrease when detection time increases.",
"This reduction confirms that for larger temporal counting window, the resolution of the measurement is less capable to detect the temporal correlations leading to more Poissonian behaviors.",
"Oppositely, the curves corroborates that for smaller detection times, $f(t)$ distributions become closer to the Pareto distribution (which corresponds to a Bose-Einstein process).",
"Figure: (a) f(t)f(t) probability functions for N=4N=4 detectors in the shown configuration.",
"〈n〉/N=0.01\\langle n\\rangle /N=0.01 in a coherence time.",
"(b) Simulations of detection events, for some of the detection times in (a)."
]
] | 1612.05705 |
[
[
"Dynamical system modeling fermionic limit"
],
[
"Abstract The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate values of mass parameters.",
"It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero."
],
[
"Introduction and motivation",
"Consider the following elliptic boundary value problem $\\Delta \\phi (u) = H_\\eta ^{-1}(c-\\phi (u))$ where $\\phi $ plays the role of the the gravitational potential generated by the cloud of diffusive particles with the self–agreed density $H_\\eta ^{-1}(\\phi (u)+c)$ distributed over $u\\in B(0,1)\\subset {\\mathbb {R}}^d$ and the constant $c$ satisfying, for given $M>0$ , the mass constraint $\\int _{B(0,1)} H_\\eta ^{-1}(c-\\phi (u))\\,du \\,=\\,M\\,.$ The origins of the function $H_\\eta $ stems from the statistical mechanics approach.",
"The function $H_\\eta $ is given and depends on the parameter $\\eta \\ge 0$ .",
"The form of $H_\\eta $ encompasses the models arising from the Maxwell–Boltzmann and the Fermi–Dirac statistics.",
"We shall prove the multiplicity results for the above nonlocal BVP for the intermediate values of the mass parameter $M>0$ while the parameter $\\eta >0$ is taken sufficiently close to zero.",
"Thus it can be seen as a singular perturbation of the Maxwell–Boltzmann statistics with $\\eta =0$ .",
"The problem can be reduced, by appropriate substitution, to some dynamical system stated in (REF ) for given $H_\\eta $ by defining the new nonlinearity $R_\\eta $ as $H^{\\prime }_\\eta (z)R_\\eta (z)=1\\,.$ We consider the following functions originating from the statistical mechanics: $R_0(z)=z$ in the Maxwell–Boltzmann model with $H_0(z)=\\log (z)$ , $R_\\eta (z)=(1/z+\\eta /z^{1/d})^{-1}$ in the simplified Fermi–Dirac model with $H_\\eta (z)=\\log (z)+\\eta z^{1-1/d}$ , $R_\\eta (z)=\\frac{\\mu (d-2)}{4}f_{d/2-2}(f_{d/2-1}^{-1}(2z/\\mu ))$ in the Fermi–Dirac model with $\\eta \\mu ^{2/d}=2d^{2/d-1}$ and the Fermi functions $f_\\alpha $ defined as $f_\\alpha (z)=\\int _0^{\\infty }\\frac{x^\\alpha }{1+\\exp (x-z)} dx\\,.",
"$ The Fermi–Dirac model was introduced to describe in a better way the existence of the galaxies or the gaseous stars than the Maxwell–Boltzmann model.",
"In the Maxwell–Boltzmann model the existence of blowing–up solutions for (REF – the so called the gravo-thermal castastrophe was proved.",
"It was accompanied by the lack of steady states for massive clouds but was not supported by observations of evolving galaxies or stars towards stable steady states, cf.",
"[24].",
"The motivation for considering such form of equations comes from the models of self–gravitating diffusive particles introduced by Chavanis et al.",
"in [11] and developed further in [8], [9].",
"Relating the potential $\\phi $ to the new variables $x$ reduced mass and $y$ the energy leads to the possibly nonautonomous system $\\left\\lbrace \\begin{array}{l}x^{\\prime }(s)=(2-d)\\,x(s)+\\,y(s)\\,,\\\\[6pt]y^{\\prime }(s)=2\\,y(s)-\\,x(s)\\,e^{2s}\\,R_\\eta (\\,e^{-2s}\\,y(s))\\,,\\end{array}\\right.$ with parameters $d\\in {\\mathbb {N}}\\cap [3,9], \\eta \\ge 0$ that reduces for $R_0={\\rm I}$ to the autonomous one $\\left\\lbrace \\begin{array}{l}x^{\\prime }(s)=(2-d)\\,x(s)+\\,y(s)\\,,\\\\[6pt]y^{\\prime }(s)=(2-\\,x(s))\\,y(s)\\,.\\end{array}\\right.$ Indeed the system (REF ) can be derived from the elliptic equation, up to constant studied in [26], [28], by considering $-\\,Q^{\\prime \\prime }+(d-1)\\,r^{-1}\\,\\,Q^{\\prime }=Q\\,R_\\eta (r^{1-d}\\,Q^{\\prime })$ with $Q(0)=0, Q(1)=\\sigma _d^{-1}M$ using the substitution relating $s, x, y$ to $r, Q, Q^{\\prime }$ given by $Q(e^s)=x(s)\\,e^{(d-2)s}, Q^{\\prime }(e^s)=y(s)\\,e^{(d-3)s}\\,.$ The latter equation (REF ) describes $Q(r)=\\sigma _d^{-1}\\int _{B(0,r)}\\rho (u) du$ the averaged (differing thus by a constant $\\sigma _d$ measure of the unit sphere from notation adopted in [26] and [28]), i.e.",
"integrated over the ball $B(0,r)$ , the radial density $\\rho (u)=H_\\eta ^{-1}(c-\\phi (u))$ of the particles preserving mass $\\int _{B(0,1)}\\rho (u) du=M$ .",
"The $x$ variable is related to the rescaled mass parameter, while $y$ can be vaguely referred to the energy of the system.",
"The precise reference is stated in the sequel.",
"One should note that while the system (REF ) referred to as Maxwell–Boltzmann case is well understood as been thoroughly examined in many papers, cf.",
"[1], [22] and references therein, the so called Fermi–Dirac like system (REF ) is less studied and not many results are available, cf.",
"[26], [28].",
"This difficulty is generated by the nonlinear nature of the $R_\\eta $ function causing some additional problems and posing some extra difficulties.",
"The problem can be also studied in slightly more general framework allowing $R_\\eta $ satisfying some condition cf.",
"Theorem 3.1 encompassing also the Fermi–Dirac case.",
"It should be noted that the results obtained for both models differ significantly for $d=3$ and large values of mass parameter, while for small and intermediate values of mass parameter they share the common features provided the parameter $\\eta $ related to the Planck constant is small enough.",
"The main result of this paper is the convergence of properly chosen solutions of the system (REF ) towards the solutions to (REF ) as $\\eta \\rightarrow 0$ and the mass parameter $M$ attains some intermediate values.",
"This results in the existence of multiple solutions for the Fermi–Dirac model for $\\eta $ small enough and properly chosen mass parameter $M>0$ with intermediate values as in the Maxwell–Boltzmann case.",
"This can be depicted in the phase diagram on Figure 1 illustrating the main Theorem 3.2 and Corollary 1 of the manuscript.",
"The results presented in this paper can be seen as continuous dependence of the solutions to the dynamical system on the parameter $\\eta \\ge 0$ but only for sufficiently small values of the parameter.",
"It should be underlined that solutions for the dynamical systems are defined on the non-compact interval.",
"Moreover, we choose some special family of the solutions characterized by the limit at minus infinity, not the whole set of possible solutions.",
"The continuous dependence on parameters of the whole set of solutions for elliptic equations was established among others in [4], [5], [6], [7].",
"One should point out that the passage to the singular limit was rigorously verified both for the related Navier–Stokes–Fourier–Poisson system by Laurençot and Feireisl in [15] while Golse and Saint–Raymond in [18] dealt with celebrated Navier–Stokes and Boltzmann equations.",
"The solutions of the BVP with elliptic equation considered above REF can be seen as steady states for the evolutions of the potential of particles with the density $\\rho $ and with no flux boundary condition evolving by $\\rho _t=\\nabla \\cdot N\\left( \\theta P_\\eta ^{\\prime } \\nabla \\rho + \\rho \\nabla \\Delta ^{-1} \\rho \\right)\\,,$ with some positive coefficient $N$ possibly depending on other variables, where $P_\\eta ^{\\prime }(z)=H^{\\prime }_\\eta (z)z\\,.$"
],
[
"Derivation of the dynamical systems",
"The results are the extension of the results obtained for the case $d=3$ in [12] to higher dimension $3\\le d\\le 9$ and more general pressure formulae $P_\\eta $ generating via $P_\\eta ^{\\prime }(z)=zH_\\eta ^{\\prime }(z),\\, H_\\eta ^{\\prime }(z)R_\\eta (z)=1$ with the function $R_\\eta $ appearing in the system (REF ) while $z=\\rho \\theta ^{-d/2}$ where $\\theta $ is the temperature of the system and $\\eta $ is the parameter related to the Planck constant.",
"Let us analyze the limit system (REF ) for which the point $(0,0)$ is a saddle, while the other stationary point $(2,2(d-2))$ can change character if any $d$ is considered but if $3\\le d \\le 9$ then it is a sink and a Lyapunov function $L(x,y)=\\frac{1}{2}(x-2)^2+y-2(d-2)-2(d-2)\\log (y/(2d-4))\\,$ governs convergence towards this point as was established in [1] and started in [22].",
"Indeed multiplying the equations (REF ) for $x^{\\prime }$ by $x-2$ and $y^{\\prime }/y$ by $2(2-d)$ and summing them with added $y^{\\prime }$ one obtains $\\frac{d}{dt}L(x(t),y(t))=x^{\\prime }(t)(x(t)-2)+y^{\\prime }(t)-2(d-2)y^{\\prime }(t)/y(t)=-(x(t)-2)^2\\le 0\\,.$ Moreover, using Taylor expansion in the neighborhood of $(x,y)\\sim (2,2(d-2))$ we can see that $L(x,y)\\sim \\frac{1}{2} (x-2)^2+\\frac{1}{4(d-2)}(y-2(d-2))^2\\,.$ Furthermore, note that the condition $Q(0)=0$ can be translated to $\\lim _{s\\rightarrow -\\infty } x(s)e^{(d-2)s} = 0\\,,$ while assuming $\\rho \\in L^{\\infty }$ guarantees $Q(r)$ be of order $r^{d}$ at zero thus assuring $x(s)e^{-2s}$ to be bounded.",
"Moreover, if $\\rho $ is continuous then the following limit exists and is finite $\\lim _{s\\rightarrow -\\infty } x(s)e^{-2s}<\\infty \\,.$ Additionally, $\\rho (0)=|\\rho |_\\infty =\\lim _{s\\rightarrow -\\infty } y(s)e^{-2s}<\\infty \\,.$ One assumes $R_\\eta $ to be continuous on $[0,\\infty )$ to claim the following lemma in the first, positive quadrant.",
"Lemma 2.1 For any solution $(x,y)$ to (REF ), finite $\\rho _0=\\lim _{s\\rightarrow -\\infty } y(s)e^{-2s}$ implies $\\lim _{s\\rightarrow -\\infty }\\frac{x(s)}{y(s)}=\\frac{1}{d}\\,.$ Proof.",
"Using de l'Hospital rule together with the system (REF ) one gets the claim by $M=\\lim _{s\\rightarrow -\\infty }\\frac{x(s)}{y(s)}=\\lim _{s\\rightarrow -\\infty }\\frac{x^{\\prime }(s)}{y^{\\prime }(s)}=\\lim _{s\\rightarrow -\\infty }\\frac{1+(2-d)\\frac{x(s)}{y(s)}}{2-e^{2s}R_\\eta (e^{-2s}y(s))\\frac{x(s)}{y(s)}}=\\frac{1+(2-d)M}{2}$"
],
[
"Convergence and multiplicity results",
"Consider the system describing the evolution of the difference $\\left\\lbrace \\begin{array}{l}w_\\eta =x_\\eta -x_0\\\\[6pt]v_\\eta =y_\\eta -y_0\\end{array}\\right.$ of solutions $(x_\\eta ,y_\\eta )$ to (REF ) and $(x_0,y_0)$ to (REF ) of the form $\\left\\lbrace \\begin{array}{l}w_\\eta ^{\\prime }=(2-d)\\,w_\\eta +\\,v_\\eta \\\\[6pt]v_\\eta ^{\\prime }=(2-x_0)\\,v_\\eta -y_\\eta w_\\eta -\\,x_\\eta \\,e^{2s}\\,S_\\eta (\\,e^{-2s}\\,y_\\eta )\\end{array}\\right.$ where $S_\\eta (z)=z-R_\\eta (z)\\,.$ Now we shall prove crucial a priori bound for the term $x_\\eta \\,e^{2s}\\,S_\\eta (\\,e^{-2s}\\,y_\\eta )$ appearing in (REF ).",
"Set $\\rho _0>0$ and take for any $\\rho \\le \\rho _0$ the solution $y$ such that $\\rho =\\lim _{s\\rightarrow -\\infty } y(s)e^{-2s}\\,.$ Then by Lemma REF we have that $y_\\eta e^{-2s}\\nearrow \\rho $ and $x_\\eta e^{-2s}\\nearrow \\frac{1}{d}\\rho $ as $s\\rightarrow -\\infty $ hence $y_\\eta \\le \\rho _0e^{2s}$ and $x_\\eta \\le \\frac{1}{d}\\rho _0 e^{2s}$ whence $dx_\\eta \\,e^{2s}\\,S_\\eta (\\,e^{-2s}\\,y_\\eta ) \\le \\rho _0 e^{4s} \\max _{[0,\\rho _0]} S_\\eta =\\rho _0 e^{4s} \\overline{S_{\\eta ,\\rho _0}}= \\rho _0 \\overline{S_{\\eta ,\\rho _0}}\\,,$ where $\\overline{S_{\\eta ,\\rho _0}}$ is increasing in $\\rho _0$ and decreasing to zero as $\\eta $ tends to 0.",
"Multiplying $w^{\\prime }_\\eta $ by $w_\\eta $ and $v^{\\prime }_\\eta $ by $v_\\eta $ respectively one obtains $w^{\\prime }_\\eta w_\\eta =(2-d)w_\\eta ^2+v_\\eta w_\\eta $ and $v^{\\prime }_\\eta v_\\eta =(2-x_0)v_\\eta ^2-y_\\eta w_\\eta v_\\eta - x_\\eta e^{2s}S_\\eta (e^{-2s}y_\\eta ) v_\\eta \\,.$ Next setting $\\chi = w_\\eta ^2 + v_\\eta ^2$ one obtains $ \\chi ^{\\prime } \\le \\alpha \\chi + \\beta $ where $\\alpha $ and $\\beta $ are the terms bounded with respect to $\\eta $ .",
"Indeed one can estimate $|(1-y_\\eta )v_\\eta w_\\eta | \\le \\frac{1}{2} (w_\\eta ^2+v_\\eta ^2) \\max \\lbrace 1,\\max y_\\eta -1\\rbrace $ and $|2-x_0|\\le \\max \\lbrace 2,\\max x_0-2 \\rbrace $ while $y_\\eta \\le \\rho _0, x_\\eta \\le \\frac{1}{d}\\rho _0, |v_\\eta | \\le \\frac{1}{2} (1+v_\\eta ^2)\\,.$ Thus coming back to the estimate on $\\chi $ and by the Gronwall lemma $\\chi \\le \\beta e^\\alpha \\,.$ But it should be noted that $\\beta \\le \\frac{1}{2} x_\\eta e^{2s}S_\\eta \\le \\frac{1}{2d} \\rho _0 S_\\eta \\le \\frac{1}{2d} \\rho _0 \\overline{S_{\\eta ,\\rho _0}}$ while $S_\\eta (z) \\le C(\\eta )D(z)$ where $C(\\eta )\\rightarrow 0$ as $\\eta \\rightarrow 0$ .",
"One can alternatively proceed with the same conclusion as in [12] defining $A_\\eta =\\sup _{s\\le t}e^{-2s}|x_\\eta (s)-x_0(s)|\\,, B_\\eta =\\sup _{s\\le t}e^{-2s}|y_\\eta (s)-y_0(s)|$ getting almost everywhere $ \\frac{d}{dt}(e^{dt}A_\\eta ) \\le e^{dt} B_\\eta $ and with some constant $\\kappa $ one gets $ \\frac{d}{dt} B_\\eta \\le (\\rho _0A_\\eta +\\frac{1}{d}\\rho _0B_\\eta +\\eta \\kappa )\\,.$ Then integration over $(-\\infty ,t)$ yields $dA_\\eta \\le B_\\eta \\le \\frac{1}{2} (\\rho _0A_\\eta +\\frac{1}{d}\\rho _0 B_\\eta + \\eta \\kappa )e^{2t}.$ Finally, as before, using a Gronwall estimate, one gets as required $0 \\le d A_\\eta \\le B_\\eta \\le \\frac{1}{2} \\eta \\kappa e^{\\rho _0/d}\\,.$ Thus we have proved the following convergence theorem Theorem 3.1 Fix any natural number $3\\le d\\le 9$ and $\\rho _0>0$ and take any $\\rho \\in (0,\\rho _0]$ such that $\\lim _{s\\rightarrow \\-\\infty } y(s)e^{-2s}=\\rho $ for some solution to the system.",
"Assume that the continuous function $R_\\eta $ satisfies $0\\le z-R_\\eta (z)\\le C(\\eta )D(z)$ where $C(\\eta )\\rightarrow 0$ as $\\eta \\rightarrow 0$ and $D:[0,\\infty )\\rightarrow [0,\\infty )$ is a continuous function such that $D(0)=0$ .",
"Then the solution $(x_\\eta , y_\\eta )$ converges uniformly to $(x_0,y_0)$ on $(-\\infty ,0]$ and in particular $x_\\eta (0)$ converges to $x_0(0)$ .",
"Recall from [12] in the case $d=3$ the generic for $3\\le d \\le 9$ phase portrait for the Maxwell–Boltzmann case with $R_0= I$ identity function and $(2,2(d-2))=(2,2)$ .",
"Figure: Left: the heteroclinic orbit joining the points (0,0)(0,0) and (2,2)(2,2) in the Maxwell–Boltzmann case.",
"Right: the mass–density diagram.An easy corollary, due to the fact that the mass of the system is related to $x_\\eta (0)$ of the Theorem REF , can be formulated as follows.",
"Theorem 3.2 For any mass parameter $M$ in the corresponding, intermediate range for Fermi–Dirac like models modeled by $R_\\eta $ , with $\\eta >0$ small enough, satisfying the condition from the Theorem REF there exists as many solutions as for the Maxwell–Boltzmann case with $R_0=I$ depicted in the Figure 1 and depending on the intersection of the vertical line (setting thus the mass $M>0$ ) with the bifurcation curves.",
"The details of the proof are the same as in [12] and are omitted herein but they focus on the continuous dependence of the mass $M$ on the density $\\rho $ or in other words $x_\\eta (0)$ on $\\lim _{s\\rightarrow -\\infty }e^{-2s}y(s)$ expressed in the language of the dynamical system variables.",
"Corollary 1 For the intermediate values of the mass parameter $M$ there exists multiple solutions to the Fermi–Dirac $R_\\eta $ and generalizations obeying the condition from Theorem REF provided the $\\eta $ parameter is sufficiently small.",
"We show that the phenomena appearing in Fermi–Dirac model for large values of the mass (cf.",
"[25]) parameter differentiating between dimensions $d=3$ (solution for any mass), $d\\ge 5$ (solution only up to some mass parameter) are not present for intermediate value of mass parameter, where the behavior is generic for any dimension $3\\le d\\le 9$ .",
"This is accompanied by existence of multiple solutions for any dimension if we are close enough to the Maxwell–Boltzmann case with $\\eta =0$ .",
"One should note also that for small values of mass parameter the uniqueness holds as was noted in [13]."
],
[
"Appendix",
"Lemma 4.1 For simplified Fermi–Dirac model we have straightforward estimate $z^{-1-2/d}(z-R_\\eta (z))\\le 1\\,.$ Lemma 4.2 For the Fermi–Dirac model we have $z^{-1-2/d}(z-R_\\eta (z))\\le \\left(\\frac{2}{\\mu }\\right)^{2/d}C\\,,$ where $C=\\max _{w\\in [0,\\infty )} w^{-1-2/d}(w-\\frac{d-2}{2}\\zeta (w))$ and $\\zeta (w) =f_{d/2-2}(f_{d/2-1}^{-1}(w))\\,.$ Proof.",
"Notice that due to the asymptotics of the Fermi functions [3] $\\frac{z-R_\\eta (z)}{z^{1+2/d}}\\cdot \\left(\\frac{\\mu }{2}\\right)^{1+2/d}=\\frac{\\frac{\\mu }{2}\\cdot \\frac{2z}{\\mu }-\\frac{\\mu (d-2)}{4}\\zeta (2z/\\mu )}{\\left(\\frac{2z}{\\mu }\\right)^{1+2/d}}=\\frac{\\mu }{2}\\frac{w-\\frac{d-2}{2}\\zeta (w)}{w^{1+2/d}}\\le \\frac{\\mu }{2}C\\,.$ Recall the relation between constants that appear above to agree behavior of the functions at $\\infty $ $\\eta \\mu ^{2/d}=2d^{2/d-1}\\,.$ Hence $\\mu \\rightarrow \\infty $ when $\\eta \\rightarrow 0^{+}\\,.$ Indeed using the estimates from [3] or [27] we have that $f_\\alpha (w) \\sim \\frac{1}{\\alpha +1} w^{\\alpha +1}, w\\sim \\infty $ while $f_\\alpha (w) \\sim \\Gamma (\\alpha +1) \\exp (w) , w\\sim 0^{+}\\,.$ The motivation for considering the pressure $p$ in the model equation $\\rho _t=\\nabla \\cdot N\\left( \\nabla p + \\rho \\nabla \\Delta ^{-1} \\rho \\right)\\,,$ in the form (REF ) with the specific dependence on the temperature $\\theta $ , the density $\\rho $ and the dimension of the ambient space $d$ reading $p(\\theta ,\\rho )=\\theta ^{d/2+1}P(\\rho \\theta ^{-d/2})$ with some given $P$ function (we drop dependence on $\\eta $ ) is threefold.",
"First of all one can for $zH^{\\prime }(z)=P^{\\prime }(z)$ with $z=\\rho \\theta ^{-d/2}$ establish the entropy formula ${\\mathcal {W}}=\\int _{B(0,1)} \\left(\\rho H(\\rho \\theta ^{-d/2}) - \\left(\\frac{d}{2}+1\\right)\\theta ^{d/2}P(\\rho \\theta ^{-d/2})\\right)$ due to this assumption on the pressure form, cf.",
"[2].",
"Then the number of astrophysically motivated examples can be found as: Maxwell–Boltzmann, Fermi–Dirac, Bose–Einstein or polytropic statistics modeling clouds of particles, galaxies or stars.",
"Finally, some monoatomic gases require this assumption which can be found in [14], [16], [17], [23].",
"To this end we recall for $d=3$ Maxwell's equation with kinetic internal energy per molecule $e$ $\\rho ^2e_\\rho =p-\\theta p_\\theta \\,.$ While for monoatomic gas the relation holds $3p=2\\rho e\\,.$ Hence $3p_\\rho =2e+2\\rho e_\\rho $ plugged into Maxwell's equation derived from the Gibb's relation (cf.",
"[16]) yields $2p-2\\theta p_\\theta =2\\rho ^2e_\\rho =3\\rho p_\\rho -2e\\rho =3\\rho p_\\rho -3p\\,.$ This gives the linear first order partial differential equation $5p=3\\rho p_\\rho +2\\theta p_\\theta $ that can be solved with characteristics i.e.",
"the system of equations $\\rho ^{\\prime }=3\\rho , \\theta ^{\\prime }=2\\theta , p^{\\prime }=5p$ with two first integrals of the form $p\\theta ^{-5/2}, \\rho \\theta ^{-3/2}\\,.$ This yields the solution in the implicit form $\\Phi (p\\theta ^{-5/2},\\rho \\theta ^{-3/2})=0$ or explicit form $p\\theta ^{-5/2}=P(\\rho \\theta ^{-3/2})$ giving $p=\\theta ^{5/2}P(\\rho \\theta ^{-3/2}).$ In higher dimension replacing 3 with $d$ would yield the corresponding formula $p=\\theta ^{d/2+1}P(\\rho \\theta ^{-d/2}).$"
],
[
"Open problems and possible extensions",
"One can consider the Dirichlet boundary value problem with elliptic equation $\\Delta \\phi (u)= \\rho (u) = H_\\eta ^{-1}(c-\\phi (u))$ where the constant $c$ is chosen so that the mass constraint holds $\\int _{B(0,1)} \\rho (u) du = \\int _{B(0,1)} H_\\eta ^{-1}(c-\\phi (u))du = M\\,.$ The entropy can be used in dimensions $d=3$ with any mass $M>0$ or $d=4$ and the mass $M$ sufficiently small to obtain the minimizer solving the related Euler-Lagrange equation.",
"To be more specific the dual approach, cf.",
"[25], uses the neg-entropy functional ${\\mathcal {V}}=\\int _{B(0,1)} \\left(\\rho H_\\eta (\\rho \\theta ^{-d/2}) - \\theta ^{d/2}P_\\eta (\\rho \\theta ^{-d/2})+\\frac{1}{2\\theta }\\rho \\Delta ^{-1}\\rho \\right)$ over the space of integrable functions $\\rho \\in L^{1+2/d}$ .",
"The functional is coercive and can be decomposed into compact and continuous part and lower–semicontinuous and convex part thus making the direct approach feasible to yield the existence of minimizer.",
"It seems that the results of [4], [5], [6], [7] can be used to get the continuity of the set of minimizers at least for sufficiently small mass $M>0$ .",
"The only obstacle is that the limiting functional is defined over the space of $\\rho \\log \\rho $ integrable functions as $\\eta \\rightarrow 0^{+}$ .",
"Moreover, one can consider with necessary modifications the following nonlinearities $R_\\eta (z)=\\frac{\\mu (d-2)}{4}g_{d/2-2}(g_{d/2-1}^{-1}(2z/\\mu ))$ in the Bose–Einstein model with Bose functions $g_\\alpha $ defined by $g_\\alpha (z)=\\int _0^{\\infty }\\frac{x^\\alpha }{1-\\exp (x-z)} dx $ requiring some limits for the density, or rather the ratio $\\rho /\\theta ^{d/2}$ , $R$ in classical King's model, cf.",
"[10], being the intermediate between Maxwell–Botlzmann and Fermi–Dirac cases ."
],
[
"Acknowledgements.",
"The remark on monoatomic gases that can be found in the Appendix was pointed out by Eduard Feireisl and we would like to thank him for this comment.",
"The impact of Jean Dolbeault who contributed to the three–dimensional case [12] could not be overestimated so the due credit is paid to him herein."
]
] | 1612.05442 |
[
[
"Systematic procedure for analyzing cumulants at any order"
],
[
"Abstract We present a systematic procedure for analyzing cumulants to arbitrary order in the context of heavy-ion collisions.",
"It generalizes and improves existing procedures in many respects.",
"In particular, particles which are correlated are allowed to belong to different phase-space windows, which may overlap.",
"It also allows for the analysis of cumulants at any order, using a simple algorithm rather than complicated expressions to be derived and coded by hand.",
"In the case of azimuthal correlations, it automatically corrects to leading order for detector non-uniformity, and it is useful for numerous other applications as well.",
"We discuss several of these applications: anisotropic flow, event-plane correlations, symmetric cumulants, net baryon and net charge fluctuations."
],
[
"Introduction",
"A nucleus-nucleus collision at ultrarelativistic energies typically emits thousands of particles [1].",
"The large multiplicity enables one to accurately measure various high-order correlations and cumulants.",
"Cumulants are connected correlations, and high-order cumulants are typically used to probe the emergence of collective effects.",
"More specifically, cumulants of azimuthal correlations have been measured [2], [3], [4], [5], [6] in order to study the collective expansion of the quark-gluon plasma in the direction of the impact parameter, or elliptic flow [7], [8].",
"Cumulants of the net proton [9] and net charge [10], [11] distribution are analyzed in order to search for the critical fluctuations associated with the QCD phase transition at finite baryon density [12], [13].",
"Recent years have witnessed the emergence of a great variety of new correlation observables: Event-plane correlations [14], [15], correlations between transverse momentum and anisotropic flow [16], or between two different Fourier harmonics of anisotropic flow [17], [18], which all involve cumulants of multiparticle correlations.",
"We propose a new framework for the cumulant analysis, which is more flexible than existing frameworks and allows to fully exploit the potential of multiparticle correlation analyses in high-energy collisions.",
"Our understanding of the collision dynamics has considerably evolved since cumulants were introduced in this context [19].",
"The importance of event-to-event fluctuations in the flow pattern [20], [21] was only recognized later [22], [23].",
"More specifically, it was recently shown that the rapidity dependence of anisotropic flow fluctuates event to event [24], [25].",
"This longitudinal decorrelation [26] induces a sizable variation of azimuthal correlations with the relative rapidity [27].",
"Existing analysis frameworks [28], [29], where all particles but oneThe analysis of differential flow with cumulants correlates one particle from a restricted phase space window with reference particles which are all in the same window.",
"are taken from the same rapidity window, do not allow to study this effect.",
"Precision studies of high-order cumulants are also needed.",
"They have been argued to be a crucial probe of collective behavior in proton-nucleus collisions [30], [31], [32], [33] and proton-proton collisions [34], [35] and first analyses of order 6 and 8 cumulants are promising [36].",
"High-order cumulants also provide insight into non-Gaussian fluctuations in nucleus-nucleus collisions [37], [38], [39], where they have also been measured up to order 8 [6], [33].",
"The maximum value of 8 is dictated by existing analysis frameworks, but higher orders are feasible experimentally.",
"The oldest framework [40] extracts cumulants by numerically tabulating the generating function and using a finite-difference approximation to compute its successive derivatives.",
"The numerical errors resulting from this procedure are hard to evaluate.",
"A new framework by Bilandzic et al.",
"[29] uses explicit expressions of cumulants in terms of moments of the flow vector, which is a more robust approach.",
"However, only a finite set of cumulants are provided, and azimuthal symmetry is assumed in order to simplify the algebraic complexity of these expressions.",
"Azimuthal asymmetries in the detector acceptance and efficiency must therefore be corrected beforehand.",
"On the other hand, the cumulant analysis automatically corrects for such asymmetries [19], so it is tempting to use cumulants for this purpose as well.",
"Our new framework generalizes the approach of Bilandzic [29] in several respects: It applies to arbitrary observables, not only to Fourier coefficients of the azimuthal distribution.",
"One can correlate particles from different bins in rapidity and transverse momentum.",
"No assumption is made regarding the detector acceptance and efficiency.",
"Cumulants can be evaluated to arbitrarily high order.",
"In Sec.",
", we illustrate the procedure on the simple example of azimuthal pair correlations.",
"We then generalize results to higher-order correlations in Sec.",
", where we show how to remove self-correlations to all orders [17], and in Sec.",
", where we give the inversion formulas for cumulants as a function of moments to all orders [41].",
"Specific implementations are discussed in Sec.",
"."
],
[
"Example: pair correlation",
"We illustrate the steps of the calculation on the simple example of azimuthal pair correlations [42].",
"One typically wants to compute an average value of $\\cos n\\Delta \\Phi $ , where $n$ is the harmonic order and $\\Delta \\Phi $ is the relative azimuthal angle between two particles in the same event [43].",
"Specifically, one takes the first particle from a region of momentum space $A$ and the second from region $B$ , and one evaluates $\\langle \\cos n\\Delta \\Phi \\rangle \\equiv \\frac{\\left\\langle \\sum _{\\rm pairs} e^{in(\\phi _j-\\phi _k)}\\right\\rangle }{\\left\\langle \\sum _{\\rm pairs} \\phantom{mm}1\\phantom{mm}\\right\\rangle },$ where $\\phi _j$ and $\\phi _k$ are the azimuthal angles of particles belonging to the same event, “pairs” is a shorthand notation for “$j\\in A$ , $k\\in B$ and $j\\ne k$ ”, and angular brackets in the right-hand side denote an average over events.The imaginary part of the right-hand side vanishes if parity is conserved, up to statistical fluctuations and asymmetries in the detector efficiency.",
"One could evaluate the sums over pairs as nested loops over $j$ and $k$ , but it is more efficient to instead factorize the sums [29].",
"For example, if $A$ and $B$ are disjoint: $\\sum _{\\rm pairs} e^{in(\\phi _j-\\phi _k)}=\\sum _{j\\in A}e^{in\\phi _j} \\sum _{k\\in B}e^{-in\\phi _k}.$ In the case where regions $A$ and $B$ overlap, such that they share some of the same particles, one must exclude the extra terms with $j=k$ , corresponding to a trivial correlation of a particle with itself (self-correlation): $\\sum _{\\rm pairs} e^{in(\\phi _j-\\phi _k)}=\\sum _{j\\in A}e^{in\\phi _j} \\sum _{k\\in B}e^{-in\\phi _k}-\\sum _{j\\in A\\cap B} 1,$ where the final sum is over the intersection of sets $A$ and $B$ , $A\\cap B$ , and each term has unit contribution since $e^{in(\\phi _j - \\phi _j)} = 1$ The next step is to average over a large number of collision events.",
"We first assume that particles are independent, in the sense that the number of particles in two disjoint momentum bins ${\\bf p}_1$ and ${\\bf p}_2$ are independent variables.",
"Then, the average number of pairs factorizes as a product of single averages, and $\\left\\langle \\sum _{\\rm pairs} e^{in(\\phi _j-\\phi _k)}\\right\\rangle =\\left\\langle \\sum _{j\\in A} e^{in\\phi _j}\\right\\rangle \\left\\langle \\sum _{k\\in B} e^{-in\\phi _k}\\right\\rangle .$ For an ideal (isotropic) detector and azimuthal symmetric regions $A$ and $B$ , the right-hand side vanishes identically, since every collision event has an arbitrary azimuthal orientation.",
"In a more realistic experimental situation, the detector efficiency has azimuthal asymmetries and the right-hand side is non-zero.",
"However, Eq.",
"(REF ) is still valid when particles are independent.",
"In the more general case where particles are not independent, we define the connected correlation as the difference between the two sides of this equation.",
"It thus isolates the physical correlation, and naturally corrects for asymmetries in the detector: $\\left\\langle \\sum _{\\rm pairs}e^{in(\\phi _j-\\phi _k)}\\right\\rangle _c&\\equiv &\\left\\langle \\sum _{\\rm pairs} e^{in(\\phi _j-\\phi _k)}\\right\\rangle \\cr &&-\\left\\langle \\sum _{j\\in A} e^{in\\phi _j}\\right\\rangle \\left\\langle \\sum _{k\\in B} e^{-in\\phi _k}\\right\\rangle , \\cr &&$ where the subscript $c$ in the left-hand side denotes the connected part of the correlation, i.e., the cumulant [19].",
"Note, however, that a non-uniform efficiency introduces a “cross-harmonic bias” [44] and the cumulant involves in general several harmonics of the particle distribution, not just harmonic $n$ [45].",
"Note that our definition of independent particles is not exactly equivalent to assuming that for a given pair of particles, azimuthal angles $\\phi _j$ and $\\phi _k$ are independent [29].",
"With this alternative definition, one can write an equation similar to Eq.",
"(REF ), where the left-hand side is divided by the average number of pairs, and the single-particle averages in the right-hand side are divided by the average number of particles in $A$ and $B$ .",
"The definition of the cumulant is then modified accordingly: $\\langle \\cos n\\Delta \\Phi \\rangle &\\equiv &\\frac{\\left\\langle \\sum _{\\rm pairs} e^{in(\\phi _j-\\phi _k)}\\right\\rangle }{\\left\\langle \\sum _{\\rm pairs}\\phantom{mm} 1\\phantom{mm}\\right\\rangle }\\cr &&-\\frac{\\left\\langle \\sum _{j\\in A} e^{in\\phi _j}\\right\\rangle }{\\left\\langle \\sum _{j\\in A}\\phantom{m} 1\\phantom{m}\\right\\rangle }\\frac{\\left\\langle \\sum _{k\\in B} e^{-in\\phi _k}\\right\\rangle }{\\left\\langle \\sum _{k\\in B} \\phantom{m}1\\phantom{m}\\right\\rangle }.",
"\\cr &&$ The method in this paper can be applied with either choice, as explained at the end of Sec. .",
"The advantage of our definition is that it simpler, and allows to treat multiplicity fluctuations and correlations on the same footing as anisotropic flow.",
"Its apparent drawback is that a correlation can be induced by a large fluctuation of the global multiplicity, which is of little physical interest.",
"However, one easily gets rid of this effect by using a fine centrality binning, which is recommended anyway for correlation analyses [46].Note that early cumulant analyses, which used wide centrality bins due to limited statistics, used a fixed subset of the multiplicity [3] in order to avoid the effects of multiplicity fluctuations in conjunction with detector asymmetries and nonflow correlations.",
"Instead of explicitly considering the connected correlation Eq.",
"(REF ) (or its variant Eq.",
"(REF )) to correct for detector anisotropy, one can do the correction in other ways.",
"Bilandzic et al.",
"[29] weight each particle with $1/p$ , where $p$ is the efficiency (probability of detection) at the point where the particle hits the detector.",
"After the correction, azimuthal asymmetry can be assumed, and the first term alone in the right-hand side of Eq.",
"(REF ) is equivalent to the connected part.",
"This “inverse weighting” method can still be used here, but is no longer necessary.",
"In particular, note that inverse weighting cannot be applied when there are holes in the detector, in which case the efficiency $p$ is 0.",
"Eliminating the nested sums gives the final expression for the numerator and denominator of the measurement $\\left\\langle \\sum _{\\rm pairs} e^{in(\\phi _j-\\phi _k)}\\right\\rangle _c\\equiv &\\left\\langle \\sum _{j\\in A}e^{in\\phi _j} \\sum _{k\\in B}e^{-in\\phi _k}\\right\\rangle \\nonumber \\\\&-\\left\\langle \\sum _{j\\in A\\cap B} 1\\right\\rangle \\nonumber \\\\&-\\left\\langle \\sum _{j\\in A} e^{in\\phi _j}\\right\\rangle \\left\\langle \\sum _{k\\in B} e^{-in\\phi _k}\\right\\rangle .$ and $\\left\\langle \\sum _{\\rm pairs} 1\\right\\rangle &=\\left\\langle \\sum _{j\\in A} 1 \\sum _{k\\in B} 1\\right\\rangle -\\left\\langle \\sum _{j\\in A\\cap B} 1\\right\\rangle .$ In the following Sections, we generalize the above discussion to higher-order correlations and arbitrary observables."
],
[
"Subtracting self-correlations",
"Cumulants can be constructed from moments, which are correlations that count $n$ -tuples in a collision event, where $n$ now denotes the order of the correlation.",
"Depending on the observable, one may weight every particle differently depending on its momentum, as in Eq.",
"(REF ).",
"In general, one evaluates in each event multiple sums of the type $Q(A_1,\\ldots ,A_n)\\equiv \\sum _{j_1\\in A_1,\\ldots ,j_n\\in A_n}q_1(j_1)\\ldots q_n(j_n),$ where $j_1,\\cdots ,j_n$ label particles chosen from $n$ sets labelled $A_1,\\cdots ,A_n$ (representing, e.g., specific regions in $p_T$ and $\\eta $ , or particular particle species), all indices in the sum are different, and $q_i(j)$ are functions of the particle momentum.",
"In the numerator of Eq.",
"(REF ), for example, we have $q_1(j) = e^{in\\phi _j}$ and $q_2(j) = e^{-in\\phi _j}$ .",
"The sum runs over all possible $n$ -tuples in the same event.",
"If one uses nested loops, the time needed to evaluate such sums grows with the multiplicity $M$ like $M^n$ , which can be computationally prohibitive.",
"In this section, we explain how to express them as a function of simple sums such as: $Q(A_i)\\equiv \\sum _{j_i\\in A_i} q_i(j_i).$ This reduces the number of operations from $M^n$ down to $M$ for any order $n$ .",
"In the case of the analysis of anisotropic flow, $Q(A_i)$ is the usual flow vector for particle set $A_i$ [47], [48].Recursive algorithms for subtracting self-correlations in this context are given by Bilandzic et al.",
"in Sec.",
"III A of Ref. [17].",
"The idea is to factorize the sum, as in Eq.",
"(REF ).",
"In the case when there is some overlap in the sets of particles $A_i$ , however, one must subtract terms in the sum where the same particle appears more than once, as in Eq.",
"(REF ).",
"In the case of pair correlations, the notation (REF ) allows to recast this subtraction in compact form: $Q(A_1,A_2)=Q(A_1)Q(A_2)-Q(A_1\\cap A_2).$ We have introduced the auxiliary notation $Q(A_1\\cap A_2)\\equiv \\sum _{j\\in A_1\\cap A_2} q_1(j)q_2(j),$ where the sum runs over all particles belonging to both $A_1$ and $A_2$ .",
"Consider now a 3-particle correlation.",
"The product $Q(A_1)Q(A_2)Q(A_3)$ contains all possible triplets, plus the self-correlations which must be removed.",
"One separates the sum into different contributions, depending on which particles are identical: (1) $j_1$ , $j_2$ , $j_3$ all different, (2) $j_1=j_2\\ne j_3$ , (3) $j_2=j_3\\ne j_1$ , (4) $j_1=j_3\\ne j_2$ , (5) $j_1=j_2= j_3$ .",
"This decomposition can be represented with Young diagrams: $(1,2,3)+(12,3)+(23,1)+(13,2)+(123),$ where each box is associated with a set $A_i$ .",
"In each diagram, different rows correspond to different values of the indices, and the values of the indices are identical for boxes belonging to the same horizontal row.",
"The first term in Eq.",
"(REF ) is the term we want to isolate and others must be subtracted.",
"The subtraction, which generalizes Eq.",
"(REF ) to third order, is derived in Appendix .",
"One obtains [17]: $Q(A_1,A_2,A_3)&=&Q(A_1)Q(A_2)Q(A_3)\\cr && -Q(A_1\\cap A_2)Q(A_3)\\cr &&-Q(A_2\\cap A_3)Q(A_1)\\cr &&-Q(A_1\\cap A_3)Q(A_2)\\cr &&+2 Q(A_1\\cap A_2\\cap A_3),$ which we represent diagrammatically as $Q(A_1,A_2,A_3)=(1,1,1)-(2,1)+2 \\,(3),$ where we have omitted the labels since the weights are identical for all permutations of the indices.",
"The right-hand side of Eq.",
"(REF ) is a sum over all partitions of the set $\\lbrace A_1,A_2,A_3\\rbrace $ .",
"This can be generalized to arbitrary $n$ , as shown in Appendix .",
"The contribution of a partition is the product of contributions of its subsets, called blocks.",
"Each row in the Young diagram corresponds to a block of the partition.",
"The contribution of a block of $k$ elements $\\lbrace A_{i_1},\\cdots ,A_{i_k}\\rbrace $ is the product of the integer weight $(-1)^{k-1} (k-1)!$ and $Q(A_{i_1}\\cap \\cdots \\cap A_{i_k})$ , which is defined by a straightforward generalization of Eq.",
"(REF ): $Q(A_{i_1}\\cap \\cdots \\cap A_{i_k})\\equiv \\sum _{j\\in A_{i_1}\\cap \\cdots \\cap A_{i_k}}q_{i_1}(j)\\cdots q_{i_k}(j).$ Blocks of $k=1$ , 2, 3 elements get respective weights of 1, $-1$ , 2, which explains the factors in front of each term in Eq.",
"(REF ).",
"We write explicitly, for sake of illustration, the corresponding formula for the 4-particle correlation [17]: $Q(A_1,A_2,A_3,A_4)&=&Q(A_1)Q(A_2)Q(A_3)Q(A_4)\\cr &&-Q(A_1\\cap A_2)Q(A_3)Q(A_4)\\cr &&-Q(A_1\\cap A_3)Q(A_2)Q(A_4)\\cr &&-Q(A_2\\cap A_3)Q(A_1)Q(A_4)\\cr &&-Q(A_1\\cap A_4)Q(A_2)Q(A_3)\\cr &&-Q(A_2\\cap A_4)Q(A_1)Q(A_3)\\cr &&-Q(A_3\\cap A_4)Q(A_1)Q(A_2)\\cr &&+Q(A_1\\cap A_2)Q(A_3\\cap A_4)\\cr &&+Q(A_1\\cap A_3)Q(A_2\\cap A_4)\\cr &&+Q(A_1\\cap A_4)Q(A_2\\cap A_3)\\cr &&+2 Q(A_1\\cap A_2\\cap A_3)Q(A_4)\\cr &&+2 Q(A_2\\cap A_3\\cap A_4)Q(A_1)\\cr &&+2 Q(A_1\\cap A_3\\cap A_4)Q(A_2)\\cr &&+2 Q(A_1\\cap A_2\\cap A_4)Q(A_3)\\cr &&-6 Q(A_1\\cap A_2\\cap A_3\\cap A_4),$ which we represent diagrammatically as: $(1,1,1,1)-(2,1,1)+(2,2)+2\\,(3,1)-6\\,(4).$ The weight of a given partition can be read directly from the Young diagram.",
"It is the product over all rows (all blocks) of $(-1)^{k-1} (k-1)!$ , where $k$ is the number of boxes in the row (number of elements in block).",
"In order to implement the subtraction in the most general case, one must generate all partitions of the set $\\lbrace A_1,\\cdots ,A_n\\rbrace $ .",
"An efficient algorithm has been described by Orlov, with a public domain C++ implementation available [49].",
"A sum can then be taken, with coefficient for each term given by the above formula."
],
[
"From moments to cumulants",
"We now define the cumulants of $n$ -particle correlations.",
"Let $f(p_1,\\cdots ,p_n)dp_1\\cdots dp_n$ denote the probability of finding a $n$ -tuple in $dp_1\\cdots dp_n$ .",
"We call $f(p_1,\\cdots ,p_n)$ the $n$ -point function.",
"It is normalized to the average number of $n$ -tuples: $\\int _p f(p_1,\\cdots ,p_n) = \\langle M(M-1)\\cdots (M-n+1)\\rangle ,$ where $\\int _p$ is a shorthand notation for $\\int dp_1\\cdots dp_n$ , and $M$ denotes the total multiplicity of an event.",
"The connected $n$ -point function $f_c(p_1,\\cdots ,p_n)$ , or cumulant, is the contribution of the $n$ -particle correlation.",
"For any order $n$ , it is defined recursively by isolating $n$ -particle clusters through an order-by-order decomposition of $f(p_1,\\cdots ,p_n)$ [41].",
"The 1-point functions are equal by definition: $f(p_1)=f_c(p_1).$ The two-point function is the sum of an uncorrelated part and a correlated part $f_c(p_1,p_2)$ : $f(p_1,p_2)=f_c(p_1,p_2)+f_c(p_1)f_c(p_2).$ Similarly, one defines $f_c(p_1,p_2,p_3)$ as the part of $f(p_1,p_2,p_3)$ which is not due to lower-order correlations [50]: $f(p_1,p_2,p_3)&=&f_c(p_1,p_2,p_3)\\cr &&+f_c(p_1,p_2)f_c(p_3)\\cr &&+f_c(p_2,p_3)f_c(p_1)\\cr &&+f_c(p_1,p_3)f_c(p_2)\\cr &&+f_c(p_1)f_c(p_2)f_c(p_3).$ The right-hand side of this equation is again a sum over partitions of the set $\\lbrace p_1,p_2,p_3\\rbrace $ , where each cluster corresponds to a block of the partition.",
"It can be represented diagrammatically as: $(123)+(12,3)+(23,1)+(13,2)+(1,2,3).$ Generalization to arbitrary order $n$ is straightforward.",
"The average of $Q(A_1,\\cdots ,A_n)$ over many collision events is a weighted integral of $f(p_1,\\cdots ,p_n)$ : $\\langle Q(A_1,\\cdots ,A_n)\\rangle =\\int _p q_1(p_1)\\cdots q_n(p_n) f(p_1,\\cdots ,p_n),$ We refer to such averages as moments.",
"The cumulant decomposition applies to moments after multiplying equations (REF ), (REF ) by $q_i(p_i)$ and integrating over $p_i$ .",
"The cumulant of order 2 is thus given by the inversion formula $\\langle Q(A_1,A_2)\\rangle _c\\equiv \\langle Q(A_1,A_2)\\rangle -\\langle Q(A_1)\\rangle \\langle Q(A_2)\\rangle ,$ which generalizes Eq.",
"(REF ), and which we rewrite synthetically as $\\langle Q_1Q_2\\rangle _c=\\langle Q_1Q_2\\rangle -\\langle Q_1\\rangle \\langle Q_2\\rangle .$ Note that the cumulant is unchanged if one shifts $Q_i$ by a constant value.",
"This property of translational invariance [51], which is true to all orders, explains why cumulants are remarkably stable with respect to detector imperfections.",
"We write for sake of illustration the inversion formulas giving cumulants of order 3 and 4 as a function of the corresponding moments: $\\langle Q_1Q_2Q_3\\rangle _c&=&\\langle Q_1Q_2Q_3\\rangle \\cr &&-\\langle Q_1Q_2\\rangle \\langle Q_3\\rangle \\cr &&-\\langle Q_1Q_3\\rangle \\langle Q_2\\rangle \\cr &&-\\langle Q_2Q_3\\rangle \\langle Q_1\\rangle \\cr &&+2\\langle Q_1\\rangle \\langle Q_2\\rangle \\langle Q_3\\rangle ,$ and $\\langle Q_1Q_2Q_3Q_4\\rangle _c&=&\\langle Q_1Q_2Q_3Q_4\\rangle \\cr &&-\\langle Q_1Q_2Q_3\\rangle \\langle Q_4\\rangle \\cr &&-\\langle Q_2Q_3Q_4\\rangle \\langle Q_1\\rangle \\cr &&-\\langle Q_1Q_3Q_4\\rangle \\langle Q_2\\rangle \\cr &&-\\langle Q_1Q_2Q_4\\rangle \\langle Q_3\\rangle \\cr &&-\\langle Q_1Q_2\\rangle \\langle Q_3Q_4\\rangle \\cr &&-\\langle Q_1Q_3\\rangle \\langle Q_2Q_4\\rangle \\cr &&-\\langle Q_1Q_4\\rangle \\langle Q_2Q_3\\rangle \\cr &&+2\\langle Q_1Q_2\\rangle \\langle Q_3\\rangle \\langle Q_4\\rangle \\cr &&+2\\langle Q_1Q_3\\rangle \\langle Q_2\\rangle \\langle Q_4\\rangle \\cr &&+2\\langle Q_2Q_3\\rangle \\langle Q_1\\rangle \\langle Q_4\\rangle \\cr &&+2\\langle Q_1Q_4\\rangle \\langle Q_2\\rangle \\langle Q_3\\rangle \\cr &&+2\\langle Q_2Q_4\\rangle \\langle Q_1\\rangle \\langle Q_3\\rangle \\cr &&+2\\langle Q_3Q_4\\rangle \\langle Q_1\\rangle \\langle Q_2\\rangle \\cr &&-6\\langle Q_1\\rangle \\langle Q_2\\rangle \\langle Q_3\\rangle \\langle Q_4\\rangle .$ The right-hand sides of these equations are again sums over partitions of the sets $\\lbrace A_1,A_2,\\cdots ,A_n\\rbrace $ for $n=2,3,4$ .",
"We represent them diagrammatically as $\\langle Q_1Q_2\\rangle _c&=&(2)-(1,1)\\cr \\langle Q_1Q_2 Q_3\\rangle _c&=&(3)-(2,1)+2\\,(1,1,1)\\cr \\langle Q_1Q_2Q_3Q_4\\rangle _c&=&(4)-\\,(3,1)\\cr &&-\\,(2,2)+2\\,(2,1,1)-6\\,(1,1,1,1)$ The weight of each term is given by a classic [41] Möbius inversion formula (see Appendix ).",
"It is equal to $(-1)^{n-1} (n-1)!$ , where $n$ is the number of blocks of the partition, i.e., the number of rows of the diagram.",
"Note the striking formal analogy between the subtraction of self-correlations and the cumulant expansion.",
"Both involve a summation over set partitions (so both can be generated by the same algorithm [49]).",
"The only difference is that the weight associated with each Young diagram involves the number of boxes in each row for self-correlations, and the number of rows for the cumulant expansion.",
"Finally, we point out that the corrections for self-correlations cancel to some extent in the cumulant.",
"Take as an example the order 3 cumulant, given by Eq.",
"(REF ).",
"Moments of order 2 and 3 in the right-hand side must be corrected for self-correlations using Eqs.",
"(REF ) and (REF ), and averaged over events.",
"After summing all terms, the correction to the cumulant from self-correlation can be written as $&&\\langle Q(A_1\\cap A_2)\\rangle \\langle Q(A_3)\\rangle - \\langle Q(A_1\\cap A_2) Q(A_3)\\rangle \\cr &+&\\langle Q(A_1\\cap A_3)\\rangle \\langle Q(A_2)\\rangle - \\langle Q(A_1\\cap A_3) Q(A_2)\\rangle \\cr &+&\\langle Q(A_2\\cap A_3)\\rangle \\langle Q(A_1)\\rangle - \\langle Q(A_2\\cap A_3) Q(A_1)\\rangle \\cr &+&2 \\langle Q(A_1\\cap A_2\\cap A_3)\\rangle .$ The first line is, up to a sign, the connected correlation between $Q(A_1\\cap A_2)$ and $Q(A_3)$ , which is usually much smaller than their respective magnitudes.",
"This is true to all orders [19], and the contribution to cumulants from self-correlations becomes smaller and smaller as the order increases when collective effects are present.In the context of the analysis of anisotropic flow, this is consistent with the expectation that higher-order cumulants are less sensitive to nonflow effects, since nonflow effects are of the same order as self-correlations [45].",
"It has been checked experimentally [52], [5] that one obtains compatible results with and without self-correlations in the limit of large cumulant order, through an analysis of Lee-Yang zeros [53], [54].",
"Self-correlations must be subtracted on an event-by-event basis, and the number of terms increases strongly with the order $n$ (it is the Bell number), therefore they are a limiting factor for the calculation.",
"However, it is expected that they are negligible beyond a certain order (which should explicitly be checked by doing the calculation with and without self correlations).",
"The cumulant expansion can then be pushed to arbitrarily high order, limited only by statistical uncertainty.",
"Our definition of cumulants in this section follows from our choice of random variables, which are the numbers of $n$ -tuples in a momentum bin $dp_1\\cdots dp_n$ .",
"As discussed in Sec.",
", an alternative choice is to choose as random variables the momenta $p_1,\\cdots ,p_n$ themselves [29].",
"Our method also accommodates this definition, at the expense of a slight complication: One must normalize each moment by the average number of $n$ -tuples before taking the cumulant (as in Eqs.",
"(7) and (8) of Ref. [29]).",
"This number is obtained by repeating the calculations of Sec.",
"with $q_k(j_k)=1$ in Eq.",
"(REF ).",
"However, this complication does not appear to offer any advantage.",
"Our formulation has the advantage that it provides a unified framework for all analyses of correlations and fluctuations, as we now explain."
],
[
"Applications",
"The interest of cumulants is that they subtract the effect of lower correlations, and isolate $n$ -particle correlations.",
"If a nucleus-nucleus collision is a superposition of a fixed number $N$ of independent nucleon-nucleon collisions, a moment of order $n$ scales like $N^n$ , while the corresponding cumulant is proportional to $N$ .",
"If, on the other hand, there are collective effects in the system, cumulants are typically of the same order as moments, so that large cumulants are a clear signature of collective effects.",
"By collective effect, one typically means an all-particle correlation.",
"Collective effects often arise from global fluctuations, which affect the whole system.",
"The fluctuation of the total multiplicity, already mentioned in Sec.",
", is a mundane collective effect which can be eliminated by a fine centrality binning [46].",
"On the other hand, fixing the total multiplicity $M$ in each event also generates cumulants to all orders, but they are proportional to $M$ at any order (see Appendix ).",
"As a result, working at a fixed multiplicity doesn't generate fake collective effects.",
"We now describe how usual cumulant analyses can be implemented within our framework."
],
[
"Net charge fluctuations and related studies",
"Our framework, where the random variables are numbers of particles in momentum bins (as opposed to momenta of given particles [29]), naturally encompasses correlation studies involving these numbers themselves, such as net charge and net baryon fluctuations.",
"We first illustrate the formulas derived in previous sections by discussing the simplest case, where one takes all particles from the same set, $A_1=\\cdots =A_n$ , and all the functions $q_i$ in Eq.",
"(REF ) are equal to 1.",
"In this case, the summation in this equation just counts the number of $n$ -tuples of $M$ particles (cf.",
"Eq.",
"(REF )): $Q(A_1,\\cdots ,A_n)=M(M-1)\\cdots (M-n+1).$ This result can be used to check the validity of Eqs.",
"(REF ), (REF ) and (REF ).",
"Each factor $Q(...)$ in these equations is equal to $M$ , therefore, they reduce to: $Q(A_1,A_2)&=&M^2-M\\cr Q(A_1,A_2,A_3)&=&M^3-3 M^2+2 M\\cr Q(A_1,A_2,A_3,A_4)&=&M^4-6M^3+11M^2-6M,$ which agree with the general result (REF ) after expanding in powers of $M$ .",
"Averaging $Q(A_1,\\cdots ,A_n)$ over events, one obtains the (unnormalized) factorial moment $F_n$ , which counts the average number of $n$ -tuples: $F_n\\equiv \\langle M(M-1)\\cdots (M-n+1)\\rangle $ If $M$ is distributed according to a Poisson distribution, $F_n=\\langle M\\rangle ^n$ .",
"The cumulants as defined in Sec.",
"are the factorial cumulants $K_n$ [55] of the distribution of $M$ ; unlike traditional multiplicity cumulants, they vanish for $n\\ge 2$ for a Poisson distribution, and are therefore automatically corrected for trivial statistical fluctuations [56].Notations are not yet standardized.",
"Bzdak et al.",
"use the notation $K_n$ for traditional cumulants, and $C_n$ for factorial cumulants [56], while the STAR Collaboration [57] uses $C_n$ for traditional cumulants.",
"In the same way, one can study correlations between multiplicities in two different rapidity windows, such as forward-backward correlations [58], [59] which have been proposed as a probe of longitudinal fluctuations [60].",
"Analyses could easily be extended to higher-order cumulants [61].",
"Consider next the case where $q_i$ in Eq.",
"(REF ) is the baryon number or the electric charge.",
"Assuming that $q_i=\\pm 1$ for all particles, the moment of order 2, as defined by Eq.",
"(REF ), is $Q(A_1,A_2)=\\Delta N_{\\rm ch}^2-M$ where $=\\Delta N_{\\rm ch}=\\sum _i q_i$ is the net charge and $M=\\sum _i (q_i)^2$ the charged multiplicity.",
"Thus trivial fluctuations are again subtracted, and such observables give more direct access to interesting quantities than the raw distribution of $\\Delta N_{\\rm ch}$ [11] or traditional cumulants [62]."
],
[
"Anisotropic flow",
"The analysis of anisotropic flow is one the most important practical applications of cumulants in high-energy physics.",
"We recall the definition of the relevant observables, and how they are obtained in our framework.",
"The flow paradigm [23], [63] states that the bulk of particle production is well described by independent-particle emission from an underlying probability distribution.",
"The classic picture is that of a relativistic fluid — near freeze-out, the system is a fluid, composed of well-defined particles that are uncorrelated with each other (e.g., as in a Boltzmann description).",
"In other words, if every event had the same hydrodynamic initial conditions (and a fixed orientation), the connected correlations defined by Eqs.",
"(REF ), (REF ) would vanish.",
"In a single event, then, all information is contained in the single-particle distribution $F(p)$ .",
"Note that this single particle distribution is different from $f(p)$ in Eq.",
"(REF ), which is $F(p)$ averaged over events (and azimuthal orientation).",
"Anisotropic flow is the particular observation that the single-particle distribution depends on azimuthal angle $\\phi $ : $F(p) = \\frac{N}{2\\pi } \\sum _{n=-\\infty }^{\\infty } V_n e^{in\\phi },$ with $V_0=1$ , $V_{-n}=V_n^*$ , and $N$ is the average number of particles in the event.",
"One denotes by $v_n\\equiv |V_n|$ the anisotropic flow coefficient in harmonic $n$ [64] in a particular event.",
"One can instead absorb the factor of $N$ into the coefficients [65] $F(p) = \\frac{1}{2\\pi } \\sum _{n=-\\infty }^{\\infty } {\\cal V}_n e^{in\\phi },$ where now ${\\cal V}_n$ = $N V_n$ .",
"While less standard, these are convenient quantities since they are additive with respect to rebinning in momentum space, and are more natural quantities with respect to the cumulant analysis.",
"The single-event distribution $F(p)$ fluctuates from one event to the next.",
"Upon averaging over events (and fluctuations), this generates correlations to all orders.",
"Moments and cumulants of multiparticle correlations are moments and cumulants of $F(p)$ .",
"There are non-trivial mathematical consequences to having uncorrelated particles in each event, e.g., for pair correlations [66], [67].",
"Even if the bulk of particle production can be described according to this single-particle distribution, it is expected for there to be small correlations that cannot be captured by $F(p)$ .",
"Such effects are typically referred to as “non-flow”, and can arise from sources such as Bose-Einstein correlations, resonance decays, unthermalized jets, momentum conservation, etc.",
"Observables involving higher-order cumulants are expected to be less sensitive to such nonflow effects [19].",
"The measure of $v_n$ from the cumulant of order $2k$ is usually noted $v_n\\lbrace 2k\\rbrace $ , and it can easily be obtained with the procedure outlined above.",
"This generalizes the discussion of pair correlations outlined in Sec. .",
"One now takes $2k$ -tuples, again taken from the same region of phase space (“integrated” flow).",
"The factor $q_i$ in Eq.",
"(REF ) is equal to $e^{in\\phi }$ for the first $k$ particles and to $e^{-in\\phi }$ for the next $k$ particles.",
"The subtraction of self correlations carried out in Sec.",
"generalizes that of Ref. [29].",
"For instance, one easily checks that the order 4 result Eq.",
"(REF ) reproduces the numerator of Eq.",
"(18) of Ref.",
"[29].The weight of each event is the same as in Ref.",
"[29], in the sense that all $n$ -tuples are treated on an equal footing (and as a result, each event can have significantly different contribution, if multiplicity fluctuates).",
"A variety of prescriptions are found in the literature.",
"The scalar-product analysis [2] and the Lee-Yang zero analysis [45] use a prescription equivalent to ours.",
"The cumulant analysis of Ref.",
"[40] weights each event by $1/M$ , and the analysis of event-plane correlations [15] implements a similar weight $1/E_T$ , where $E_T$ is the energy deposited in the calorimeter.",
"Finally, factors of $1/\\sqrt{M}$ were used in the cumulant analysis of Ref.",
"[19] and in studies of the distribution of the flow vector [2]).",
"We denote by $\\langle |Q_n|^{2k}\\rangle _c$ , the cumulant obtained after subtracting self-correlations, averaging over events and subtracting lower-order moments (Sec. ).",
"As explained above, it is a cumulant of the distribution of ${\\cal V}_n$ .",
"We thus define: ${\\cal V}_n\\lbrace 2k\\rbrace ^{2k}\\equiv \\frac{1}{a_{2k}} \\langle |Q_n|^{2k}\\rangle _c,$ where $a_{2k}$ is the ratio of the coefficients of the expansions of $\\ln I_0(x)$ (cumulants) and $I_0(x)$ (moments) to order $x^{2k}$ [19]: $\\ln I_0(x)=\\sum _{k=1}^{\\infty } \\frac{a_{2k} x^{2 k}}{2^{2k} (k!",
")^2},$ which gives $a_2=1$ , $a_4=-1$ , $a_6=4$ , $a_8=-33$ , $a_{10}=456$ , $a_{12}=-9460$ , etc.",
"This normalization ensures that ${\\cal V}_n\\lbrace 2k\\rbrace =|{\\cal V}_n|$ for all $k$ if $|{\\cal V}_n|$ were the same for all events.",
"In order to match the usual normalization convention of the flow coefficients $v_n$ , one normalizes this cumulant by the average number of $2k$ -tuples.",
"The standard azimuthal cumulants are thus given by $v_n\\lbrace 2k\\rbrace ^{2k}\\equiv \\frac{1}{a_{2k}}\\frac{\\langle |Q_n|^{2k}\\rangle _c}{F_{2k}}.$ This normalization ensures that again $v_n\\lbrace 2k\\rbrace =v_n$ when there are no fluctuations.",
"For the traditional measurement of “differential” flow [19], the only difference with the previous case is that the first particle is taken from a different (restricted) phase space window $B$ , while the $2k-1$ remaining particles are taken from the same set of particles $A$ .",
"One then scales the resulting cumulant with the corresponding number of $2k$ -tuples (with 1 particle in $B$ and $2k-1$ particles in $A$ ).",
"The flow in $B$ , traditionally denoted by $v^{\\prime }_n\\lbrace 2k\\rbrace $ , is again given by an equation similar to (REF ), where one replaces in the left-hand side $v_n\\lbrace 2k\\rbrace ^{2k}$ with $v^{\\prime }_n\\lbrace 2k\\rbrace v_n\\lbrace 2k\\rbrace ^{2k-1}$ .",
"Note that the coefficients $a_k$ are the same as for integrated flow.",
"In principle, each particle can be taken from an arbitrary region in momentum space, and many other differential analyses are possible [68], [69].",
"With our normalization conventions, all differential cumulants are additively related to integrated measurements."
],
[
"Symmetric cumulants, event-plane correlations",
"The normalized symmetric cumulant ${\\rm NSC}(m,n)$ with $n\\ne m$ can be defined by [18]As with previous cumulants, the ALICE Collaboration chose to normalize each individual moment by $F_2$ or $F_4$ .",
"See the discussion at the end of Sec. .",
"We choose to have no normalization factors in the normalized measurement.",
"${\\rm NSC}(m,n)&=\\frac{\\left\\langle Q_n Q_m Q_{-n} Q_{-m} \\right\\rangle _c}{\\left\\langle |Q_n|^2 \\right\\rangle _c \\left\\langle |Q_m|^2 \\right\\rangle _c} \\\\&\\stackrel{\\rm {(flow)}}{=}\\frac{\\langle |{\\cal V}_m|^2 |{\\cal V}_n|^2\\rangle -\\langle |{\\cal V}_m|^2\\rangle \\langle |{\\cal V}_n|^2\\rangle }{\\langle |{\\cal V}_m|^2\\rangle \\langle |{\\cal V}_n|^2\\rangle }.$ The numerator is a 4-particle cumulant [17] with $q_1=e^{im\\phi }$ , $q_2=e^{in\\phi }$ , $q_3=e^{-im\\phi }$ , $q_4=e^{-in\\phi }$ .",
"The fact that it is measured through a 4-particle cumulant guarantees that nonflow effects are small.",
"Equation () is the value assuming the flow paradigm of independent particles in each event.",
"The mean square values $\\langle |{\\cal V}_n|^2\\rangle =F_2 v_n\\lbrace 2\\rbrace ^2$ in the denominator are 2-particle cumulants obtained as described in Sec.",
"REF , which may be biased by short-range correlations (especially nonflow effects) unless a rapidity gap is applied.",
"The existing analysis [18] implements a gap in the denominator, but not in the numerator, hence neglecting the effect of longitudinal fluctuations.",
"it would be interesting to redo the analysis by implementing the same gap in the numerator and the denominator, and studing how ${\\rm NSC}(m,n)$ varies with the gap.",
"Event-plane correlations [15] are Pearson correlation coefficients between different complex flow hamonics [70], [71].",
"For instance, the two-plane correlation between ${\\cal V}_2$ and ${\\cal V}_4$ is defined as (we use the notation of ATLAS): $\\langle \\cos (4(\\Phi _2-\\Phi _4))\\rangle _w&\\equiv \\frac{\\left\\langle Q_4 Q_{-2} Q_{-2} \\right\\rangle _c}{\\sqrt{\\left\\langle |Q_4|^2\\right\\rangle _c\\left\\langle |Q_2|^4\\right\\rangle }} \\\\&\\stackrel{\\rm {(flow)}}{=}\\frac{\\langle {\\cal V}_4({\\cal V}_2^*)^2\\rangle }{\\sqrt{\\langle |{\\cal V}_4|^2\\rangle \\langle |{\\cal V}_2|^4\\rangle }}.$ The numerator is a 3-particle cumulant obtained with $q_1=e^{4i\\phi }$ , $q_2=q_3=e^{-2i\\phi }$ .",
"The ATLAS analysis uses two sets of particles separated with a rapidity gap, where particles 2 and 3 belong to the same bin.",
"Thus self-correlations between particles 2 and 3 are not removed.",
"It has been argued that they are small [72], but it will be interesting to check experimentally.",
"The denominator of Eq.",
"(REF ) involves moments which, as in the case of symmetric cumulants, may be biased by nonflow effect unless a rapidity gap is applied (note that the 4-particle $v_2$ factor is a moment, not a cumulant).",
"It has been pointed out [73] that the value of ${\\rm NSC}(4,2)$ measured by ALICE seems large compared to what one would expect based on the corresponding event-plane correlation measured by ATLAS [15], where a large rapidity gap is implemented.",
"It would be interesting to measure both the symmetric cumulant and the event-plane correlation with the same kinematic cuts."
],
[
"Correlation between transverse momentum and anisotropic flow",
"The correlation between transverse momentum and anisotropic flow recently proposed by Bozek as a further test of hydrodynamic behavior [16] is a straightforward application of our formalism.",
"It is a 3-particle cumulant with $q_1=p_t$ , $q_2=e^{in\\phi }$ , $q_3=e^{-in\\phi }$ .",
"Bozek recommends to use particles from three different rapidity intervals $A,B,C$ separated with gaps in order to avoid nonflow correlations but predicts that results should be identical if $A=B=C$ provided that self-correlations are subtracted."
],
[
"Conclusion",
"We have proposed a new, unified framework for cumulant analyses which is more systematic and flexible than existing frameworks, and discussed its practical implementation for the analysis of factorial cumulants and anisotropic flow.",
"A major improvement is that one can correlate particles in arbitrary regions of phase space.",
"Application to proton-nucleus and nucleus-nucleus at RHIC and the LHC should shed light on longitudinal fluctuations.",
"Our procedure is systematic and can be carried out to arbitrarily large orders, which is important in order to probe collective behavior.",
"In particular, we have argued that the subtraction of self-correlations, which is the limiting factor when going to higher orders, becomes a negligible correction at large orders.",
"This can be checked explicitly, and the cumulant expansion can then be extended to higher orders.",
"Finally, our new framework also extends beyond the analysis of anisotropic flow, and we anticipate a rich program of generalized cumulant analyses on this basis in the near future."
],
[
"Acknowledgments",
"This work is funded under the USP-COFECUB project Uc Ph 160-16 (2015/13) and under the FAPESP-CNRS project 2015/50438-8.",
"PDF is supported by the NSF Grant DMS-1301636 and the Morris and Gertrude Fine endowment.",
"MG is supported under DOE Contract No.",
"DE-SC0012185 and the Welch Foundation (Grant No.",
"C-1845), and thanks Wei Li and Hubert Hansen for discussions.",
"JYO thanks the Department of Theoretical Physics, Mumbai, where this paper was written, for hospitality and financial support.",
"We thank Giuliano Giacalone for careful reading of the manuscript."
],
[
"Möbius inversion",
"In this Appendix, we recall some known facts [74] on Möbius inversion applied to functions of set partitions.",
"We show in particular how it implies the moment/cumulant relations of Sec.",
", as well as the self-correlation subtraction of Section ."
],
[
"Set partitions and Möbius inversion",
"Let $[n]\\equiv \\lbrace 1,2,...,n\\rbrace $ and ${\\mathcal {P}}(n)$ the set of its partitions.",
"We denote by ${\\mathbf {I}}=\\lbrace I_1,...,I_k\\rbrace $ a partition, where $I_j$ are subsets of $[n]$ , called blocks of the partition.",
"The number of blocks $k$ is called the length of the partition ${\\mathbf {I}}$ and denoted by $|{\\mathbf {I}}|$ .",
"There is a natural partial order relation on partitions which we denote by $\\le $ and is defined as follows.",
"If ${\\mathbf {I}}$ and ${\\mathbf {J}}$ are two partitions of $[n]$ , ${\\mathbf {I}}\\le {\\mathbf {J}}$ if the partition ${\\mathbf {I}}$ is a refinement of the partition ${\\mathbf {J}}$ , that is, if each block of ${\\mathbf {I}}$ is included in a block of ${\\mathbf {J}}$ .",
"For instance we have $\\lbrace \\lbrace 1,2\\rbrace ,\\lbrace 3\\rbrace ,\\lbrace 4\\rbrace \\rbrace \\le \\lbrace \\lbrace 1,2\\rbrace ,\\lbrace 3,4\\rbrace \\rbrace $ but the order relation does not relate $\\lbrace \\lbrace 1,2\\rbrace ,\\lbrace 3,4\\rbrace \\rbrace $ to $\\lbrace \\lbrace 1,3\\rbrace ,\\lbrace 2,4\\rbrace \\rbrace $ .",
"The finest partition is denoted by $\\hat{\\mathbf {0}}\\equiv \\lbrace \\lbrace 1\\rbrace ,\\lbrace 2\\rbrace ,...,\\lbrace n\\rbrace \\rbrace $ and the coarsest partition by $\\hat{\\mathbf {1}}\\equiv {[n]}$ .",
"For all ${\\mathbf {I}}\\in {\\mathcal {P}}(n)$ , $\\hat{\\mathbf {0}}\\le {\\mathbf {I}}\\le \\hat{\\mathbf {1}}$ .",
"The Möbius inversion formula goes as follows [75].",
"Assume we have two real functions $f,g$ defined on ${\\mathcal {P}}(n)$ , such that for all ${\\mathbf {I}}\\in {\\mathcal {P}}(n)$ : $f({\\mathbf {J}})=\\sum _{\\hat{\\mathbf {0}}\\le {\\mathbf {I}}\\le {\\mathbf {J}}}g({\\mathbf {I}}).$ then we have the inverse relation: $g({\\mathbf {J}})=\\sum _{\\hat{\\mathbf {0}}\\le {\\mathbf {I}}\\le {\\mathbf {J}}}\\mu ({\\mathbf {I}},{\\mathbf {J}})\\, f({\\mathbf {I}}),$ where the Möbius function $\\mu ({\\mathbf {I}},{\\mathbf {J}})$ is given by $\\mu ({\\mathbf {I}},{\\mathbf {J}})=(-1)^{|{\\mathbf {I}}|-|{\\mathbf {J}}|}\\prod _{i=1}^n ((i-1)!",
")^{r_i({\\mathbf {I}},{\\mathbf {J}})},$ where $r_i({\\mathbf {I}},{\\mathbf {J}})$ denotes the number of blocks of $\\mathbf {J}$ containing exactly $i$ blocks of $\\mathbf {I}$ .",
"Likewise, there is a dual Möbius inversion formula: assuming $f,g$ are related via $ f({\\mathbf {I}})=\\sum _{{\\mathbf {I}}\\le {\\mathbf {J}}\\le \\hat{\\mathbf {1}}} g({\\mathbf {J}})$ for all ${\\mathbf {I}}\\in {\\mathcal {P}}(n)$ then we have the inverse relations: $g({\\mathbf {I}})=\\sum _{{\\mathbf {I}}\\le {\\mathbf {J}}\\le \\hat{\\mathbf {1}}} \\mu ({\\mathbf {I}},{\\mathbf {J}})\\, f({\\mathbf {J}})$"
],
[
"Applications",
"An example of the decomposition (REF ) is the decomposition of moments into cumulants (Section ).",
"One defines $f({\\mathbf {I}})$ as the product of moments over all blocks, and $g({\\mathbf {I}})$ as the product of cumulants over all blocks, that is; $f({\\mathbf {I}})&\\equiv &\\prod _{j=1}^{|{\\mathbf {I}}|}\\left\\langle \\prod _{l\\in I_j} Q_l\\right\\rangle \\cr g({\\mathbf {I}})&\\equiv &\\prod _{j=1}^{|{\\mathbf {I}}|}\\left\\langle \\prod _{l\\in I_j} Q_l\\right\\rangle _c.$ Then the Möbius inversion formula (REF ) specialized to ${\\mathbf {J}}=\\hat{\\mathbf {1}}$ gives the connected correlation function $\\langle Q_1\\cdots Q_n\\rangle _c$ in terms of moments as a sum over partitions, with coefficients $\\mu ({\\mathbf {I}},\\hat{\\mathbf {1}})=(-1)^{|{\\mathbf {I}}|-1}\\, (|{\\mathbf {I}}|-1)!$ Formulas (REF ), (REF ), (REF ) follow as well as the general case.",
"Similarly, the dual Möbius inversion formula applies directly to the problem of subtracting self-correlations (Section ).",
"We now define $f([n])\\equiv Q(A_1)...Q(A_n)$ (sum over all indices, including self correlations) $g([n])\\equiv Q(A_1,\\cdots ,A_n)$ (sum over different indices), and more generally: $f({\\mathbf {I}})&\\equiv & Q(\\cap _{i\\in I_1}A_i)Q(\\cap _{i\\in I_2}A_i)...Q(\\cap _{i\\in I_n}A_i)\\cr g({\\mathbf {I}})&\\equiv & Q(\\cap _{i\\in I_1}A_i, \\cap _{i\\in I_2}A_i,...,\\cap _{i\\in I_n}A_i).$ The relations (REF ) are clearly satisfied, as shown by splitting uncontrained sums into sums of constrained ones, for instance: $Q(A_1)Q(A_2)&=&\\sum _{i,j} q_1(i)q_2(j)\\cr &=&\\sum _{i\\ne j}q_1(i)q_2(j)+\\sum _{i=j} q_1(i)q_2(i)\\cr &=&Q(A_1, A_2)+Q(A_1\\cap A_2).$ The dual Möbius inversion formula (REF ) for ${\\mathbf {I}}=\\hat{\\mathbf {0}}$ gives the expression for $g(\\hat{\\mathbf {0}})=Q(A_1,A_2,...,A_n)$ in terms of the $f({\\mathbf {J}})$ , with coefficients $\\mu (\\hat{\\mathbf {0}},{\\mathbf {J}})=\\prod _{i=1}^{|{\\mathbf {J}}|}(-1)^{|J_i|-1}\\, (|J_i|-1)!$ where $|J_i|$ is the cardinality of the block $J_i$ , equal to the length of the $i$ -th row of the Young diagram of ${\\mathbf {J}}$ .",
"Formulas (REF ), (REF ), (REF ) and their generalization follow."
],
[
"Recursion relation",
"We derive a relation for generating the self-correlation corrections order by order.",
"It is simple to implement and easy to understand, but less efficient numerically than the general method exposed in Sec. .",
"Similar relations have been previously derived in Ref. [17].",
"We want to evaluate sums of the type $Q(A_1,\\ldots ,A_n)\\equiv \\sum _{j_1\\in A_1,\\ldots ,j_n\\in A_n}q_1(j_1)\\ldots q_n(j_n),$ where all indices in the sum are different.",
"Self correlations can be subtracted order by order.",
"Once one has a formula that works for $n-1$ , then to order $n$ one has $Q(A_1,\\ldots ,A_n)&=&Q(A_1,\\ldots ,A_{n-1})Q( A_n)\\cr &&-\\sum _{k=1}^{n-1}Q(A_1,...,A_k\\cap A_n,...,A_{n-1}).\\cr $ The first term in the right-hand side takes into account the conditions that the indices $i_1$ to $i_{n-1}$ are all different, and the last term subtracts the contributions from $i_k=i_n$ .",
"It is understood that the product $q_iq_j$ should be used together with the intersection $A_i\\cap A_j$ .",
"For $n=2$ , Eq.",
"(REF ) gives Eq.",
"(REF ).",
"For $n=3$ , it gives $Q(A_1,A_2,A_3)&=&Q(A_1,A_2)Q(A_3)-Q(A_1\\cap A_3,A_2)\\cr &&-Q(A_1,A_2\\cap A_3).$ Substituting Eq.",
"(REF ) into Eq.",
"(REF ), one obtains Eq.",
"(REF ).",
"For $n=4$ , Eq.",
"(REF ) gives $Q(A_1,A_2,A_3, A_4)&=&Q(A_1,A_2,A_3)Q(A_4)\\cr &&-Q(A_1\\cap A_4,A_2,A_3)\\cr &&-Q(A_1,A_2\\cap A_4,A_3)\\cr &&-Q(A_1,A_2,A_3\\cap A_4)$ Substituting Eq.",
"(REF ) into Eq.",
"(REF ), one obtains Eq.",
"(REF ).",
"Generating the subtraction to order $n$ with this method requires $n!$ operations so that it is less efficient in practice than generating partitions, since the number of partitions is the Bell number which grows more slowly with $n$ than $n!$ ."
],
[
"Generating function",
"Moments and cumulants to all orders are conveniently expressed in terms of generating functions.",
"The moment defined by Eq.",
"(REF ) can be obtained by expanding the generating function: $G(z_1,\\cdots ,z_n)\\equiv \\left\\langle \\prod _{j=1}^M\\left(1+z_1 q_1(j)+\\cdots +z_n q_n(j)\\right)\\right\\rangle ,$ where the product runs over all particles in the event.",
"If one expands the product, the moment defined by (the average over events of) Eq.",
"(REF ) is the coefficient in front of $z_1\\cdots z_n$ .",
"The corresponding cumulant is given by the expansion of $\\ln G$ to the same order: $\\langle Q_1\\cdots Q_n\\rangle _c=\\left.\\frac{\\partial ^n}{\\partial z_1\\cdots \\partial z_n}\\ln G(z_1,\\cdots ,z_n)\\right|_{z_1=\\cdots =z_n=0}.$ Writing $G$ as a product over all particles [40] guarantees that self-correlations do not appear at any order in the expansion.",
"If one does not subtract self correlations, then the generating function takes the usual exponential form [19]: $G(z_1,\\cdots ,z_n)\\equiv \\left\\langle \\exp \\left( z_1 Q(A_1)+\\cdots +z_n Q(A_n)\\right)\\right\\rangle ,$ with $Q(A_i)=\\sum _j q_i(j)$ .",
"Note that Eq.",
"(REF ) can also be written in an exponential form analogous to (REF ) by introducing Grassmann variables [76].",
"This formal analogy shows that the algebraic relations linking moments to cumulants, derived in Sec.",
", are identical irrespective of whether or not self-correlations are subtracted.",
"Both forms of the generating function have been used [52], [5] in the context of the analysis of elliptic flow with Lee-Yang zeros [45].",
"As an application of the formalism, we evaluate the generating function (REF ) in the simple case of independent particles, where all terms in the product are independent.",
"First, consider the case where the multiplicity $M$ is fixed.",
"Then, Eq.",
"(REF ) gives $G(z_1,\\cdots ,z_n)=\\left(1+z_1 \\langle q_1\\rangle +\\cdots +z_n \\langle q_n\\rangle \\right)^M.$ Therefore, $\\ln G$ is proportional to $M$ and cumulants of arbitrary order scale like $M$ , as stated in the first paragraph of Sec. .",
"If, on the other hand, all the connected $n$ -point functions $f(p_1,\\cdots , p_n)$ vanish for $n\\ge 2$ , then, the multiplicity $M$ follows a Poisson distribution: $p_M=\\frac{\\langle M\\rangle ^M}{M!}",
"e^{-\\langle M\\rangle }.$ Inserting into Eq.",
"(REF ) and summing the series, one obtains $G(z_1,\\cdots ,z_n)=\\exp \\left( \\langle M\\rangle (z_1 \\langle q_1\\rangle +\\cdots +z_n \\langle q_n\\rangle )\\right).$ Therefore, $\\ln G(z_1,\\cdots ,z_n)$ is linear in all the variables $z_1$ and cumulants of order $\\ge 2$ vanish identically, as expected."
]
] | 1612.05634 |
[
[
"ALPINE: A Bayesian System for Cloud Performance Diagnosis and Prediction"
],
[
"Abstract Cloud performance diagnosis and prediction is a challenging problem due to the stochastic nature of the cloud systems.",
"Cloud performance is affected by a large set of factors including (but not limited to) virtual machine types, regions, workloads, wide area network delay and bandwidth.",
"Therefore, necessitating the determination of complex relationships between these factors.",
"The current research in this area does not address the challenge of building models that capture the uncertain and complex relationships between these factors.",
"Further, the challenge of cloud performance prediction under uncertainty has not garnered sufficient attention.",
"This paper proposes develops and validates ALPINE, a Bayesian system for cloud performance diagnosis and prediction.",
"ALPINE incorporates Bayesian networks to model uncertain and complex relationships between several factors mentioned above.",
"It handles missing, scarce and sparse data to diagnose and predict stochastic cloud performance efficiently.",
"We validate our proposed system using extensive real data and trace-driven analysis and show that it predicts cloud performance with high accuracy of 91.93%."
],
[
"Introduction",
"Cloud computing through virtualization provides elastic, scalable, secure, on-demand and cheaper access to computing, network, and storage resources as-as-service [6].",
"The cloud system hides the complexity of managing these virtualized resources to provide an easy way for the end users to deploy their applications on the cloud.",
"The rapid surge in demand for cloud computing in the recent years has led to the emergence of several cloud providers such as Amazon Elastic Compute Cloud (EC2) and Google Compute Engine (GCE).",
"CloudHarmony [2], a major cloud provider comparison website lists ninety-six such cloud providers.",
"Most cloud providers offer relatively similar functionality, albeit at different prices and with different service level agreements.",
"Although each cloud provider aims to maximise their revenue by providing a broad range of applications and services to the end users, the quality of service (QoS) offered by them can differ substantially.",
"The multi-tenant model inherent in cloud systems, and the limitations posed by global Internet bandwidth may cause differences in QoS provided by the cloud providers that can hamper applications hosted on the clouds [9].",
"Cloud performance benchmarking (regarding QoS), diagnosis and prediction is a highly challenging problem [9], [17].",
"Each cloud provider may provide a complex combination of cloud service configurations at various geographically distributed regions all over the globe (in a cloud datacenter).",
"These service configurations include a plethora of virtual machine instance types, and network and storage services.",
"Zhang et al.",
"[18] note that Amazon Web Service alone offers six hundred and seventy-four such combinations differentiated by price, geographical region, and QoS.",
"Each combination of these services provided over the Internet may lead to QoS variations.",
"Therefore, it is imperative for the end users to monitor the QoS offered by the cloud providers during and after selection of a particular cloud provider for hosting their applications.",
"Cloud performance monitoring and benchmarking is a widely studied problem [17], [5].",
"Recent research in this area (e.g., [11], [4], [14]) has developed tools and platforms to monitor cloud resources across all cloud layers, i.e., Infrastrastrucure-as-a-Service (IaaS), Platform-as-a-Service (PaaS), and Software-as-a-Service (SaaS).",
"Further, recent research (e.g., [9]) has also widely studied the performance of several cloud platforms based on various applications, constraints, and experimental setups [9].",
"However, the challenge of performing root-cause diagnosis of cloud performance by critically studying the effect of multiple influencing factors taken together has not garnered sufficient attention.",
"Further, the current research does not deal with the challenge of handling uncertainty caused due to the uncontrollable (hidden) factors prevalent in the stochastic cloud environment.",
"Lastly, the current research does not aim to build a unifying model for cloud performance diagnosis and prediction.",
"Our contribution: This paper proposes, develops and validates ALPINE, a systematic and a unifying system for cloud performance diagnosis and prediction.",
"ALPINE incorporates Bayesian networks to model uncertain and complex relationships between several factors such as CPU type, geographical regions, time-of-the-day, day-of-the-week, cloud type, and the benchmark-type.",
"Using Bayesian networks and the Expectation Maximization algorithm, ALPINE handles missing, scarce and sparse data to diagnose and predict stochastic cloud performance efficiently.",
"We validate ALPINE using extensive real data and trace-driven analysis and show that it predicts cloud performance with high accuracy of 91.93%.",
"The rest of the paper is organised as follows: Section 2 presents the related work.",
"Section 3 presents ALPINE.",
"Section 4 presents the results analysis.",
"Finally, section 5 presents the conclusion and future work."
],
[
"Related Work ",
"The problem of cloud performance monitoring, benchmarking and prediction has got significant interest from both industry and academia [1], [2], [17], [16], [5].",
"There are already commercial and academic cloud monitoring and benchmarking systems available in the cloud domain.",
"For example, CloudHarmony [2] provides cloud benchmarking, reporting and selection service based on several parameters such as regions, latency and throughput.",
"Amazon EC2 provides CloudWatch [1], a cloud monitoring service for monitoring virtual machine instances running on Amazon EC2 clouds.",
"CloudWorkbench [14] provides a Web-based system for benchmarking IaaS clouds.",
"However, these systems and methods simply provide raw aggregated measurements and do not provide any analysis and recommendations.",
"The research work presented in this paper is motivated by [9] where the authors present an in-depth analysis of the results regarding performance variability in major cloud providers such as Amazon EC2 and Google AppEngine.",
"Most importantly, the authors studied performance variability and predictability of cloud resources by performing experimentation for several days and by collecting real data traces.",
"We used these data traces in this paper.",
"The work presented by [9] was limited based on several factors.",
"For instance, the authors did not critically determine the influence of multiple factors taken together to ascertain the degree of change that occurs when the values of these factors are varied.",
"Further, the authors did not develop a model that can be used to predict cloud performance under uncertainty and missing data values.",
"Compared to the work presented in [9], this paper presents a systematic and unifying model based on Bayesian networks (BNs) to model complex relationships between several factors for efficient cloud performance diagnosis and prediction.",
"Recently, BNs were applied in the area of cloud computing (e.g., [8], [7], [15]).",
"Bashar [7] use BNs for autoscaling of cloud datacenter resources by balancing the desired QoS and service level agreement targets.",
"The author using preliminary studies show the BNs can be utilised efficiently to model workloads, and QoS factors like CPU usage and response time.",
"However, they did not discuss in detail how BNs can be created and validated by the stakeholders.",
"Further, their work was limited to simpler simulation studies and did not consider realistic user workloads.",
"Compared to the work presented by [7], in this paper, we consider the challenge of efficient cloud performance diagnosis and prediction considering major public Cloud providers such as Amazon EC2 and Google AppEngine.",
"Compared to the state-of-the-art research in the area [8], [7], [15], [9], [16], the main aim of this paper is to develop a system for critical diagnosis and prediction of cloud performance under uncertainty.",
"Our system, ALPINE, considers several factors such as time-of-the-day, day-of-the-week, virtual machine-type, regions and different types of benchmarks and efficiently models complex relationships between these parameters for cloud performance diagnosis and prediction.",
"Using realistic data provided by Leitner and Cito [9], in this paper, we show how the stakeholders can develop BNs to perform probabilistic cloud performance diagnosis and prediction, and to determine the best combination of cloud resources for a given QoS level."
],
[
"ALPINE: Bayesian Cloud Performance Diagnosis and Prediction",
"This section presents ALPINE - a Bayesian system for cloud QoS diagnosis and prediction.",
"Fig.",
"1. shows our high-level approach.",
"As can be observed from this figure, first, benchmark data is collected by the stakeholders through experimentation or via third-party services such as Cloud Workbench [14] and CloudHarmony [2].",
"Second, this data is pre-processed and is stored in a database.",
"Third, a Bayesian Network (BN) is learned using the pre-processed data or is manually created by the domain expert.",
"In the case of manual BN creation, the model is created using domain expert's knowledge/experience; or it is learned using the pre-processed data which is then carefully calibrated by the domain expert.",
"Fourth, the modelled BN is then used for probabilistic diagnosis by entering the evidence in the form of probability assignment, i.e., a likelihood of a random variable (or facrtor) taking a particular value is determined by introducing evidence into the BN (discussed later in detail).",
"Fifth, if the diagnostic results are deemed to be sufficient, this BN can be used by the stakeholders for both diagnosis and prediction, and for actual usage; else, steps one to three are repeated to develop the best BN."
],
[
"Modelling Bayesian Networks for Cloud QoS Diagnosis and Prediction",
"We consider Bayesian Networks (BNs) for cloud QoS diagnosis and prediction.",
"We selected BNs over Fuzzy Logic, Neural Networks and Decision Trees as a method based on its several advantages.",
"These include: BNs learn efficiently from scarce and sparse data.",
"BNs deal effectively with uncertainty in stochastic environments (such as clouds and networks).",
"BNs handle both numerical and categorical data.",
"BNs can incorporate domain knowledge.",
"BNs do not require explicit rules to reason about factors.",
"BNs can be extended to dynamic Bayesian networks to reason about several hypotheses over time.",
"Finally, they can be used with utility theory to make decisions under uncertainty [13], [12].",
"We now show how BNs can be used to model several parameters for efficient for cloud performance diagnosis and prediction.",
"A BN can be defined as follows: Figure: A Bayesian Network for cloud QoS diagnosis and prediction.Definition 1.",
"A Bayesian network (BN) is a directed acyclic graph (DAG) where, random variables form the nodes of a network.",
"The directed links between nodes form the causal relationships.",
"The direction of a link from X to Y means that X is the parent of Y.",
"Any entry in the Bayesian network can be calculated using the joint probability distribution (JPD) denoted as: $P(x_{1},...,x_{m})=\\prod _{i=1}^{m}P(x_{i}|Parents(X_{i}))\\;\\blacksquare $ where, $parents(X_{i})$ , denotes the specific values of $Parents(X_{i})$ .",
"Each entry in the joint distribution is represented by the product of the elements of the conditional probability tables (CPTs) in a BN [13].",
"BNs provide a natural and a complete description of the problem domain; it provides a rich description of the causal relationships between several nodes (representing factors) in the BN model [13].",
"Fig.",
"2 shows example BNs for cloud QoS diagnosis and prediction.",
"In these BNs, the oval nodes represent the random variables that are modelled together to determine their effect on each other probabilistically.",
"In a BN, the direction of an arc from one node(s) to another node(s) denotes a parent-child relationship, where the parent node directly affects the child node probabilistically.",
"For example in Fig.",
"2 (d), the arcs from the nodes “Regions” and “Virtual Machine Size” towards “CPU” denote that these nodes are the parents of the child node “CPU”; and will be used to determine the effect of regions and virtual machine size on the types of CPU used.",
"Figure: Bayesian Networks for cloud QoS diagnosis and prediction.A BN can be created in a number of ways (see Fig.",
"2 (a) to Fig.",
"2 (d)) such as a Naive Bayes' Network, Noisy-Or Network, Tree-Augmented Naive Bayes' Network, or a more complex model (such as in Fig.",
"2 (d) where most of the nodes are connected to each other) based on the principle of causality.",
"Manual BN creation can be challenging as the causal relationships can be hard to define by the stakeholders/domain experts.",
"To test the causal relationship between two factors or random variables, consider the nodes A and B.",
"Assume that the domain expert fixes (assign probabilities) one of the state of node A ($s\\,\\in \\,S$ where $S$ is a set of states), to infer the states of node B.",
"Upon inference, if the states of node B do not change (degree of belief or probability of a state $s\\,\\in \\,S$ where $S$ is a set of states belonging to B), then the node A is not a cause of node B; otherwise it is.",
"For the sake of brevity, in the paper, we do not discuss various methods for manual BN creation.",
"The interested readers may refer to [13].",
"Each node in a BN represents a random variable (RV or factor in our case).",
"This RV can be discretized into a number of states $s\\,\\in \\,S$ .",
"The $S$ is then assigned probabilities that are represented via the conditional probability Table (CPT).",
"In the case of a continuous RVs, conditional probability distribution (CPD) is defined that can take any distribution; for example, Gaussian distribution.",
"The CPT for each RV can be learned using a dataset or can be set by the domain expert.",
"As mentioned previously, setting the CPTs can be quite challenging even if robust statistical methods are used [12].",
"In such cases, the methods that consider maximum entropy can be used.",
"To create a BN automatically, stakeholders can also consider BN structural learning algorithms such as structural expectation maximization and Markov Chain Monte Carlo [13].",
"For simplicity, let's assume a BN shown in Fig.",
"2 (d).",
"In this paper, we show that even simpler BNs can be used efficiently to model, diagnose and predict cloud QoS.",
"Cloud QoS is stochastic and can be influenced by $N$ number of factors.",
"Further, each $n\\,\\in \\,N$ can have $m\\,\\in \\,M$ number of states.",
"In a BN, all the states can be inferred together by entering the evidence $e\\,\\in \\,E$ in the network which is not possible in other methods such as regression analysis, decision trees, and neural networks.",
"By entering the evidence in a BN, we mean assigning a degree of belief (associating probability) to a particular state $s\\,\\in \\,S$ belonging to an RV.",
"For example, consider a BN as shown in Fig.",
"2 (d).",
"To determine the cloud QoS or “QoS Value” using the RV “Cloud”, the stakeholder can enter evidence into “Cloud” RV such as $P(``Cloud=aws^{\\prime \\prime }=1)\\wedge P(``Cloud=gce^{\\prime \\prime }=0)$ to depict the degree of belief that for a particular “QoS Value”,“aws” “Cloud” should be considered.",
"Similarly, the probability of occurrence of each $s\\;\\in S$ for all RVs can be entered as evidences $e\\,\\in \\,E$ to determine the probability $\\forall \\,S$ for “QoS Value” RV.",
"Once a BN is created via structural learning algorithms or by the domain experts, they need to be validated.",
"Usually, cross-validation is performed to check the correctness and accuracy of the BN [13].",
"In cross-validation, a part of the training data is is to train/learn the BN.",
"The rest of the data or the test data is used to check model's prediction accuracy.",
"For BN model parameter learning, we consider the most widely used Expectation-Maximization algorithm [13].",
"Once the stakeholders or domain experts are satisfied by BNs prediction accuracy, these BNs can be utilised in the real-world use cases."
],
[
"Results Analysis",
"This section presents the results related to ALPINE.",
"We validate ALPINE using GeNie Bayesian Network development environment [3] as well as a realistic cloud benchmark dataset recently collected by Leitner and Cito [9].",
"We chose this dataset based on the fact that it is recent, comprehensive and covers a broad range of factors that may affect the performance of clouds regarding communication, I/O and storage."
],
[
"Dataset",
"The cloud benchmark dataset [9] contains 30,140 unique records based on the data collected for one month regarding Amazon EC2 (AWS) and Google Compute Engine (GCE) in the United States and Europe regions.",
"In particular, this dataset contains records related to five benchmarks, namely, CPU, MEM, Compile, I/O, and OLTP.",
"The CPU benchmark was used to benchmark the compute capacity of the instance (running in Amazon of Google data centers) by computing the total time taken (in seconds (secs)) to check 20,000 natural numbers for primeness.",
"The MEM benchmark was used to measure the read-write memory speed in MB/s by allocating 64 MB arrays in memory and copy one array to the other fifty times.",
"The Compile benchmark was used to measure total cloning (from Github) and compilation time (in seconds) of the jCloudScale Java program using the OpenJDK 7.0 toolkit.",
"The I/O benchmark was used measure (in Mb/s) the average disk read/write speed, computed by reading and writing a 5 GB file for three minutes.",
"Finally, OLTP benchmark was used to measure the average number of queries per second (queries/sec).",
"Table 1 shows the statistics related to all QoS values.",
"We note that this dataset does not contain MEM QoS values for GCE.",
"Further, nearly all QoS values are widely distributed.",
"We now show that even with variability in this dataset, ALPINE can efficiently diagnose and predict cloud QoS.",
"For cloud QoS diagnosis, we considered several BNs, such as a simple Naive Bayes Network (NBN), Tree-augmented Naive Bayes Network (TAN), Noisy-Or network (NOR), and a complex BN (CBN) as shown in Fig.",
"2.",
"We created the first two BNs automatically from the dataset.",
"The latter two BNs were created using expert's knowledge (by the authors).",
"These BNs comprise six random variables or BN nodes depicting eight different factors present in the dataset.",
"These include CPU, VM size, regions, cloud providers, type of benchmark, time-of-the-day, day-of-the-week, and QoS values.",
"Except QoS value factor, all other factors were categorical, ranging from two to eleven states $(s\\in S)$ .",
"Table: Statistics related to all QoS values present in the dataset (Θ)(\\text{$\\Theta $})."
],
[
"CPU performance diagnosis",
"The CPU benchmark aims to study the performance of hardware-dominated applications hosted on the clouds.",
"In particular, it seeks to examine the effect of instance processing speed of cloud providers on the hosted applications (task completion time in seconds).",
"For this, we studied several hypotheses using ALPINE.",
"For instance, using a BN, we studied the impact of several factors including the instance type, time-of-the-day, day-of-the-week, region and CPU type on the applications' task completion time.",
"Using the same BN, we can not only determine the impact these factors on the QoS value, but also each other.",
"For example, we can easily answer the question that “for a certain QoS value, what is the most likely instance type, CPU type and the region?” i.e., using a single factor (CPU_type), we can infer the states of other factors (VM_size, CPU_type and the region).",
"Using a BN, we can infer the hidden truth (phenomena that cannot easily be explained by statistical tests) that may be masked by traditional statistical tests.",
"Most importantly, using probabilistic analysis, experts can also use their intuition (i.e., they can assign probabilities to particular states in a BN.",
"For example, a state region can be “us” and “eu”) to reach several conclusions by studying several hypotheses.",
"Traditional statistical methods and the methods presented in [11], [18], [16], [9] lack this capability.",
"The CPU dataset $(\\theta _{(cpu)})$ contains 6894 data points for both “aws” and “gce” clouds.",
"We discretised the QoS values into a ten states using hierarchical discretisation and by manual fine tuning as shown in Table II.",
"To study the impact of several factors on the QoS value, we first selected “us” region, “aws” as the cloud provider (cloud), and varied the VM_size as “micro”, “small”, “large”.",
"These selections were entered as evidence ($e\\in E$ ) in a BN.",
"For probabilistic inference, this can be written as: P (QoS value) = P (QoS Value | region = “us”, “cloud” = “aws”, VM_size= “micro”).",
"Through Bayesian analysis, we found clear differences offered by different VM sizes.",
"For instance, we found that for VM_size= “small”, there is 87% chance (probability) that the task will be completed between 82 and 103 seconds (state 9).",
"Further, there is 86% chance that cpu = “Intel Xeon 2650v0 2.0 GHz” will be used.",
"As expected, the “large” VM_size provided the best performance.",
"We concluded that for the “large” VM_size, there is 100% chance that the task will be completed between 11 and 20 seconds (state 2), offering up to five times better performance than “small” VM_size.",
"Further, we note that “aws” cloud only uses the Intel Xeon 2760v2 2.50GHz CPU for providing predictable performance.",
"To our surprise, we found out that in the case of “aws” the “micro” VM_size provided significantly better CPU performance than the “small” vm_size.",
"In that, there is more than 84% chance that the task will be completed between 39 to 54 seconds (state 5), leading us to believe that a “micro” vm_size offers two times better compute performance than the “small” vm_size.",
"Fig.",
"3 shows the screenshot of this case implemented in the GeNIe platform.",
"It is worth noting that for both “small” and “micro” vm_size mostly (84.5% chance) use an “Intel Xeon 2650v0 2 GHz” CPU in the case of “aws” cloud in the “us” region.",
"We then tested this hypothesis for the EU datacenter and found similar results.",
"The $\\theta _{cpu}$ also contains values for “ioopt” and “cpuopt” specialised instances for providing CPU and I/O optimised performance for “aws” cloud, respectively.",
"After BN diagnosis, we found out that the “ioopt” VM_size provides the best performance regarding QoS_value and with higher degree of certainty.",
"In this case, all the QoS_value lie below 11 seconds.",
"On the other hand, and to our surprise, the “cpuio” VM_size provides nearly the same performance as the “large” VM_size.",
"Finally, we studied the impact of several parameters on the QoS value for “gce”.",
"We found that “gce” provides highly predictable results compared to “aws”, and offers easily distinguishable performance with different VM_size.",
"Considering the “micro” VM_size, we found that there was greater than 94% chance that the task completion time was more than 103 seconds for both “eu” and “us” region.",
"This result shows that “aws” “micro” VM_size provides significantly better performance than “gce” “micro” VM_size.",
"On the other hand, we found that GCE’s “small” VM_size performs at least three times better than “aws” “small” VM_size with 100% chance that the task completion time would be between 20 to 32 seconds, compared to “aws” task completion time of 82 to 103 seconds with approx.",
"87% chance.",
"In THE case of the “large” VM_size, “gce” and “aws” performs similarly, offering task completion times between 11 to 20 seconds.",
"It’s also worth noting that “gce” always selects the same processors for similar VM_size in “eu” and “us” region leading to extremely high predictable CPU performance compared to AWS.",
"For example, “gce” always selects the “Intel Xeon 2.60 GHz” processor for predicable performance in both “us” and “eu” data centers or large VMs.",
"We also studied the impact of time and day_of_the_week on QoS_Value and found that these parameters do not significantly affect the CPU performances.",
"Table: QoS value states representation using hierarchal discretization forθ cpu \\theta _{cpu}.Figure: Screenshot of ALPINE implemented in GeNIe platform."
],
[
"Compile Diagnosis",
"The aim of the compile benchmark is to study application's performance on the clouds.",
"Therefore, using Bayesian diagnosis, we studied the impact of several factors mentioned above on the applications’ compile time.",
"As can be observed from Table I, the Compile dataset $(\\theta _{compile})$ contains a total of 7319 data points, representing the QoS values for both “aws” and “gce” clouds.",
"We discretised the QoS values into a fifteen states using hierarchical discretisation and by manual fine tuning as shown in Table III.",
"We first analyzed the performance of the “aws” cloud by varying the aforementioned parameters.",
"For example, by selecting the “micro” “VM_size” in both “eu” and “us” regions, we found the QoS values to the less predictable in the “us” region.",
"In that, we found that there is approx.",
"70% chance that the QoS values will lie between 41 and 233 secs.",
"; around 6% chance that these values will lie between 233 and 405 secs; and 19% chance that these values will lie between 405 and 701 secs.",
"However, the “micro” “vm_size” provides more predictable performance in the “eu” “region” where there is approx.",
"85% chance that the QoS values will lie in the range of 4 and 233 seconds, and there is 8% and 6% chance that these values will lie in the range of 233 to 405 seconds, and between 405 and 701 seconds, respectively.",
"The variation in the performance predicability can be attributed to the fact that in both “regions”, “aws” employs several different “cpu types” with varying probability.",
"However, in the “eu” region, “aws” selects one of the CPU (“Intel 2650 2 Ghz” processor) in majority of the cases (with 84% probability) compared to the “us” “region” where there is 72% chance that the same “cpu” will be used.",
"We also studied the performance of other VM types.",
"When we selected the “small” “VM_size”, the performance decreased slightly but it becomes highly predictable (compared to “micro” VM_size) with a 92% chance that the QoS values will lie between 233 and 405 secs.",
"We observed the similar behavior for both the regions.",
"We then selected the “large” “vm_size” and found that it performed better than both “micro” and “small” instances.",
"In particular, we found that there was 97% chance that the values will lie between 41 and 233 secs for both the regions.",
"For a thorough diagnosis, we also studied the impact of optimised “vm_size” such as, “ioopt” and “cpuopt” on the applications' performance.",
"As mentioned previously, these instances are optimised for I/O and CPU operations and should offer better and more predicable performance than the “micro”, “small” and “large” “vm_size”.",
"For instance, we diagnosed that the “ioopt” “VM_size” offers better QoS values (with compile time lower than 112 seconds) with 92% probability.",
"Further, the “cpuopt” VM_size also provides high QoS values with compile times in the range of 41 and 233 secs with 97% probability for the “eu” region.",
"There were no QoS values present in the $\\theta _{compile}$ dataset for the “us” region.",
"We also found similar performance for the “cpuopt” instance as well.",
"From our diagnosis we found it interesting to note that the performance of the “cpuopt” and “ioopt” “vm_size” is similar to the “large” VM_size.",
"This leads us to believe that instead of paying for “cpuopt” and “ioopt” VM_size, “large” instance can be selected at lower costs.",
"We also studied the performance of all the VM_size by also varying factors “day-of-the-week” and “time-of-the-day” and found no evidence that these factors” significantly affect the QoS values for this benchmark for “aws” cloud.",
"Finally, we also diagnosed the performance offered by the “gce” cloud present in both “us” and “eu” regions.",
"In the case of “micro” VM_size, there is approx.",
"91% chance that the QoS values will lie in the range of 405 and 701 secs.",
"(state 4) in both “eu” and “us” regions.",
"Further, there is approx.",
"99% chance that the QoS values will lie in the range of 41 to 233 secs (state 2) for “small” VM_size in both “us” and “eu” regions.",
"It is also interesting to note that “gce” always selects the same CPU for similar VM_size compared to “aws” cloud where different CPU types can be selected by the “aws” for same VM_size.",
"In this dataset, there were no data points for “cpuopt” and “ioopt” VM_size therefore, we could not study the optimised instances provided by “gce”.",
"However it is worth mentioning that the “gce” “large” VM_size performs similarly to the “aws” “large”,”cpuopt” and “ioopt” VM_size.",
"Overall “gce” provides more predictable performance than the “aws” cloud.",
"Finally, as in the “aws” case, we could not find any evidence that “day-of-the-week” and “time_of_the_day” affects the QoS for “gce” and “aws” clouds.",
"Table: QoS value states representation using hierarchal discretization forθ compile \\theta _{compile}."
],
[
"Memory Performance Diagnosis",
"Hardware dominated applications not only depends on CPU but also on memory.",
"The memory dataset $(\\theta _{memory})$ contains values related to “aws” cloud and has 4581 rows in total.",
"We again used hierarchical discretisation method with manual fine tuning to discretize the QoS values.",
"In all, we created thirteen states for this dataset as shown in Table IV.",
"The aim for memory diagnosis is to determine the effect of various factors on the memory dominated applications.",
"Therefore, in this case, we varied the states of all factors mentioned in Table I.",
"We started by selecting the “micro” VM_size in “us” region.",
"We found the performance of “micro” instance to be reasonably predictable where there was 78% chance that the values will lie in the range of 3612 and 3872 MB/sec (state 8).",
"We then varied the region and selected “eu” and found an increase in the performance not only in terns of bandwidth but also regarding certainty.",
"In particular, in this case, we found that most of the QoS values lie in the range between 4116 and 4539 MB/sec (state 10) with the probability of 87%.",
"We also found out that in this case, “aws” mostly employed the “Intel Xeon E5_2650 2GHz” CPU with the probability of more than 80% in both “us” and “eu” regions.",
"Table: QoS value states representation using hierarchal discretization forθ memory \\theta _{memory}.We then studied the performance of “small” VM_size and its effects on the QoS value.",
"As in the previous cases, this instance provided lower performance compared to the “micro” instance in both the regions.",
"In the case of the “eu” region, most of QoS values (93% probability) lie in the range of 1909 and 2318 MB/sec (state 4).",
"In the “us” region, nearly 79% of the QoS values lie in the range of 1425 to 1909 MB/sec (state 3).",
"The rest lie in lower ranges, i.e., between 1 and 1425 MB/sec (states 1 and 2).",
"The lower performance of “aws” VM_size in both the regions is attributed to the fact that “aws” consistently deploys VMs on one of the better-performing CPUs in “eu”; whereas, in the “us” region, other CPU types are also considered with a higher probability.",
"We also studied the performance of the “large” VM_size and their effects on QoS value.",
"We found out that even in this case (as with CPU and OLTP), these instance provides better and more predictable performance.",
"For instance, “large” VM_size in the “us” region can support QoS values in the range of 5101 to 5651 MB/sec (state 12) with 93% probability.",
"Further the same instance, in the “eu” region supports even higher QoS values that lie in the range of 5651 and 6316.1 MB/sec.",
"It is worth noting that “aws” employs the same CPU (“Intel E5_2670 2.50 GHz) in both the regions for “large” instances, leading to higher performance.",
"The $\\theta _{memory}$ dataset also contains values for “ioopt” and “cpuopt” specialised instances for the “eu” region.",
"We diagnosed the performance for both the instances and found that none of these instances match the performance of the “large” VM_size.",
"For example, for the “ioopt” case, there is greater than 74% chance that the QoS values will lie above 5101 MB/sec (state 11), and there is 21% chance that the QoS values will lie in the range of 3872 and 4116 MB/sec (state 9).",
"Similarly, for the “cpuopt” case, there is appox.",
"81% probability that the QoS values will lie above 5101 MB/sec (state 12), where there is approx.",
"79% chance that these values will lie above 5651 MB/sec (states 12); the rest of the QoS values mainly lie in the range of 4539 and 5101 MB/sec.",
"Finally, as in the previous cases, we did not find any evidence that “day-of-the-week” and “time-of-the-day” has any impact on any other parameter in a BN."
],
[
"OLTP Performance Diagnosis",
"The OLTP benchmark aims to study the performance related to multi-tenancy in cloud systems.",
"From Table 1, we note that in this dataset, there are 3969 entries for this dataset $(\\theta _{OLTP})$ .",
"The low number of values corresponds to the data regarding to EC2 cloud.",
"This data set does not contain values related to the GCE.",
"As can be observed from Table 1, for this benchmark, the QoS values are widely distributed with 95% of the data lying in the range of 0 queries/sec to 1000 queries/sec, and with the standard deviation of 281.74 queries/sec.",
"This variation in the QoS values can be attributed to the fact that multi-tenancy leads to low performance and leads to unpredictable behaviour [9].",
"As in the CPU diagnosis case mentioned above, for OLTP diagnosis, we created and tested several BNs.",
"Our aim was to study the effect of several factors on each other and most importantly, on the OLTP QoS values.",
"As QoS values were continuous, we discretized them into finite states of different sizes.",
"We used hierarchical discretization method and discretized the OLTP QoS values into three states with different counts as shown in Table V. As can be observed from the Table, most of the QoS values lie in the range of 0 to 196 queries/sec.",
"This followed by the range of 196 to 561 queries/sec, and lastly, the range of 561 to 1130 where only 33 values exist.",
"To study the impact of several factors on the QoS Value, we first selected the “us” region, “aws” as the cloud, and varied the VM_size as “micro”, “small”, “large”.",
"As discussed previously, these selections were entered as evidence $(e\\in \\,E)$ in a BN.",
"We studied several hypotheses such as “large VMs provide better QoS values”.",
"In this case, the larger VM should increase the throughput in queries/sec.",
"Firstly, we tested this hypothesis with “micro” VM_size and “us” region to determine the QoS value and CPU.",
"After performing the inference, we found out that nearly 98% of the QoS values lie in state 1, i.e., between the range of 0 to 196 queries/sec.",
"We also inferred that the “micro” VM_size in the “aws” “us” cloud mainly (82% probability) uses the “Intel Xeon 2650 cpu with 2 GHz” CPU.",
"We then tested the same hypothesis by only changing the evidence as “small” for the factor VM_size.",
"We noticed no change in the QoS value compared to the “micro” VM_size, leading us to believe that in the case of OLTP benchmark, “micro” and “small” VM_size perform rather similarly; with 78% probability Intel Xeon 2650 CPU with 2 GHz processor was used for the “small” VM_size as well.",
"In this case, our diagnosis is not absolute, rather based on the limited dataset and the variability of data, we reached this conclusion.",
"We assert that this OLTP based benchmarking should be done for a longer duration to build a larger dataset to retest this hypothesis.",
"We again tested the same hypothesis but now by keeping all the evidences fixed and by only varying the state of the factor VM_size to “large”.",
"From this test, we inferred that QoS value increases and lies mostly in the range of 196 to 561 queries/sec (state 2) validating the hypothesis that larger VM_size provide better QoS performance.",
"The VM_size also contains two other states namely “cpuopt” and “ioopt” representing CPU and IO optimised VMs in the dataset.",
"To verify whether I/O optimised VM_size leads to further QoS performance improvement, we kept all the evidences fixed but varied the state of the VM_size to “ioopt”.",
"After inference, we concluded that “ioopt” instance provided the best QoS values with most of values (with 93% probability) lying in the range of 561 queries to 1130 queries/sec (state 3).",
"We also found out that the “ioopt” VM_size employs a more powerful “Intel Xeon E5_2670 2.50 Ghz” CPU.",
"To study the impact of region on the OLTP QoS values, we studied the same hypothesis by changing the state of region from “us” to “eu”.",
"We then performed inference one by one by selecting the state of VM_size from “micro”, to “ioopt”, our analyses led us to conclude that OLTP performance remain rather stable across both regions for “micro”, “small”, and “large” VM_size.",
"We found that this dataset do not contain values related to “ioopt” VM_size for “us” regions.",
"Interestingly, we also concluded that in the “eu” region, more expensive “cpuopt” VM_size performs similarly to “large” VM_size.",
"Lastly, through Bayesian diagnosis, we inferred that “time-of-the-day” and “day-of-the-week” do not affect any other RV significantly.",
"Table: QoS value states representation using hierarchal discretization forθ oltp \\theta _{oltp}."
],
[
"I/O Performance Diagnosis",
"The I/O benchmark also aims to study the performance related to multi-tenancy in cloud systems.",
"From Table 1, we note that there were 7377 data points present in the dataset ($\\theta _{IO}$ ) representing the values for “aws” and “gce” clouds.",
"We did not find any significant variation in the QoS values.",
"As in the previous cases, we discretized the I/O QoS values which were continuous, into finite states of different sizes (see Table 5) We first analysed the performance of “aws” cloud by varying parameters listed above.",
"Initially, we selected the “micro” VM_size' in the “us” region and found that most of the QoS values (77% chance) lie in the range of 0 and 2 Mb/sec (state 1).",
"We then varied the region to “eu” and found similar results albeit with less predictability, where there is with only 66% chance that the values will lie in this range.",
"We then varied the VM_size to “small” and found nearly no change in the result.",
"Rather the QoS values become less predictable in the “us” region with close to half of the values lie in with states 1 and 2.",
"In the “eu” region, the value were widely distributed with 53% chance that QoS will lie in state 2, followed by 28% chance in state 1 and 18% chance that they will lie in state 3, respectively.",
"Again, in this case, we found that the “gce” cloud provides significantly high predictable values compared to the “aws” cloud.",
"In that, we concluded that “gce” and “micro” VM_size will lead to state 1 with 99.5% chance in both “us” and “eu” regions.",
"Similarly, in the case of “small” VM_size in the “eu” region, there is 100% chance that the QoS values will lie in state 2.",
"The performance for “gce” cloud in the “us” region was less predictable with only 71% chance that the QoS values will lie in state 2 and rest in state 1, respectively.",
"In the case of the “large” VM_size, “aws” cloud provided more predictable results in this case where there was an average 80.5% chance that the QoS values will lie in state 3, and the rest of the values will lie in state 2.",
"In the case of “gce”, there was only 67% chance that the QoS values will lie in state 3 and rest of the values will lie in state 2.",
"This dataset also contains QoS values for “oopt” and “cpuopt” VM_size for “aws” cloud.",
"The “oopt” VM_size performs very well with 100% chance that the values will lie in state 3.",
"The “cpuopt” VM_size performed rather poorly with only 55% chance that the QoS values will lie in state 3 and rest of the values will lie in state 2.",
"Again even in this case, we did not find any conclusive evidence that “time-of-the-day” and “day-of-the-week” factors have any significant impact on the QoS values for all the clouds.",
"Table: QoS value states representation using hierarchal discretization forθ IO \\theta _{IO}."
],
[
"Cloud QoS Prediction",
"The previous section validated ALPINE's cloud performance diagnosis capability under uncertainty.",
"This section presents the results related to cloud QoS prediction.",
"As referred to in section 2, a BN can be modelled in many ways.",
"It can be a simple Naive Bayes Model (NBN) (see Fig 2(a) where all the factors are conditionally independent given an outcome, i.e., QoS value.",
"Alternatively, it can be a more complex BN (CBN) (See Fig.",
"2 (d)) where more arcs between the factors are connected to determine more complex relationships between them.",
"Fig.",
"2 (c) shows another simple model; this is a Noisy-Or model (NOR) where all the factors directly affect the QoS value.",
"Finally, Fig.",
"2 (b) presents a Tree-augmented Naive Bayes Model (TAN); this model is similar to NBN.",
"However, in this model, more arcs are connected to determine more complex relationships between the factors.",
"All of these models were learned after we performed discretization on the raw QoS values.",
"To validate BNs prediction accuracy, we used 10-fold cross-validation which is a widely accepted method to determine the accuracy and correctness of a model [13], [12].",
"For training the model, we again used the EM algorithm [9].",
"Table VII shows the prediction accuracy of all BNs.",
"We conclude that BNs can predict QoS efficiently with an overall prediction accuracy of approximately 91.93%, which is an excellent result.",
"To our surprise, we found that even the simplest BNs could achieve high prediction accuracy (compared to CBN) using the dataset [10], [9] utilised in this paper.",
"The low prediction accuracy in the case of I/O dataset $(\\theta _{IO})$ was because of a very narrow distribution of I/O QoS values.",
"We assert that these results can be beneficial for the stakeholders for not only the best cloud selection but also to predict the QoS that their application might perceive by using a combination of factors mentioned above.",
"Table: Cloud QoS Prediction accuracy (%) for different type of BayesianNetworks."
],
[
"Conclusion and Future Work",
"This paper proposed, developed and validated ALPINE - a Bayesian system for cloud performance diagnosis and prediction.",
"The results presented in the paper clearly demonstrate that ALPINE can be used for efficiently diagnose cloud performance even in the case of limited data.",
"The major highlight of ALPINE is that it can consider several factors simultaneously (CPU, VM size, regions, cloud providers, type of benchmark, time-of-the-day, day-of-the-week, and QoS values) for the root-cause diagnosis of cloud performance.",
"In particular, a stakeholder can enter the evidence regarding multiple factors to determine their impact on other factors.",
"The state-of-the-art methods lack this capability.",
"ALPINE can model complex and uncertain relationships between these factors probabilistically to reason about several hypotheses regarding cloud performance.",
"We also validated ALIPNE's prediction performance and showed that it achieves an overall prediction accuracy of 91.93%.",
"Therefore, we assert that stakeholders can use ALPINE for efficient cloud ranking, selection, and orchestration.",
"As a future work, we will collect more data for several other cloud providers."
]
] | 1612.05477 |
[
[
"Canonical Weierstrass representations for minimal space-like surfaces in\n $\\RR^4_1$"
],
[
"Abstract A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero.",
"Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters.",
"For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representations - canonical Weierstrass representations via two holomorphic functions.",
"We find the expressions of the Gauss curvature and the normal curvature of the surface with respect to this pair of holomorphic functions.",
"We find the relation between two pairs of holomorphic functions generating one and the same minimal space-like surface of general type.",
"The canonical Weierstrass formulas allow us to establish geometric correspondence between minimal space-like surfaces of general type and classes of pairs of holomorphic functions in the Gauss plane."
],
[
"Introduction",
"A two-dimensional surface $\\mathcal {M}$ in the four-dimensional Minkowski space-time $\\mathbb {R}^4_1$ is said to be space-like if the induced metric on the tangential space at any point of $\\mathcal {M}$ is positive definite.",
"If $\\mathcal {M}$ is a space-like surface in $\\mathbb {R}^4_1$ , we denote by $T_p(\\mathcal {M})$ and $N_p(\\mathcal {M})$ the tangential space and the normal space at a point $p \\in \\mathcal {M}$ , respectively.",
"The flat Levi-Civita connection on $\\mathbb {R}^4_1$ is denoted by $\\nabla $ .",
"Then the second fundamental tensor $\\sigma $ of $\\mathcal {M}$ is given by $\\sigma (X,Y) = (\\nabla _X Y)^\\bot ;\\quad X,Y \\; \\text{tangent vectors to}\\; \\mathcal {M}\\;\\text{at a point} \\; p\\in \\mathcal {M}.$ The space-like surface $\\mathcal {M}$ is minimal if its mean curvature vector field $H=\\frac{1}{2}\\mathop {\\mathrm {trace}}\\nolimits \\sigma $ is zero, i.e.",
"$H =0$ .",
"A general approach to Weierstrass representations of minimal space-like surfaces in $\\mathbb {R}^4_1$ was given in [9] and [4].",
"In [1] minimal space-like surfaces in $\\mathbb {R}^4_1$ were studied with respect to special isothermal parameters and a fundamental theorem of Bonnet type in terms of the Gauss curvature $K$ and the normal curvature $\\varkappa $ was proved.",
"The question when a complete minimal space-like surface is a plane was studied in [5].",
"In this paper we consider canonical Weierstrass representations for minimal space-like surfaces in $\\mathbb {R}^4_1$ .",
"A point $p\\in \\mathcal {M}$ is said to be degenerate, if the set $\\lbrace \\sigma (\\mathrm {X},\\mathrm {Y});\\ \\mathrm {X}\\in T_p(\\mathcal {M}),\\mathrm {Y}\\in T_p(\\mathcal {M}) \\rbrace $ , is contained in one of the two light-like one-dimensional subspaces of $N_p(\\mathcal {M})$ .",
"We call a minimal space-like surface, free of degenerate points, a minimal space-like surface of general type.",
"Let $(\\mathcal {M},\\text{x}(u, v))$ be a space-like surface in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates $(u, v)$ .",
"In isothermal coordinates the space-like surface $\\mathcal {M}$ is minimal if and only if the position vector function $\\text{x}(u,v)$ is harmonic.",
"We describe the properties of minimal surfaces in terms of the complex vector function $\\Phi (t)=\\mathrm {x}_u-\\mathrm {i}\\mathrm {x}_v , \\;t=u+\\mathrm {i}v $ .",
"The standard Weierstrass representations for minimal space-like surfaces are in terms of three holomorphic functions.",
"Using special isothermal parameters (canonical parameters) on a minimal space-like surface of general type, we obtain canonical Weierstrass representations in terms of two holomorphic functions.",
"We call the isothermal parameters canonical of the first type $($ the second type$)$ if ${\\Phi ^\\prime }^2=+1 \\; ({\\Phi ^\\prime }^2=-1)$ .",
"The special parameters, used in [1] occur to be canonical of the first type.",
"In Theorem REF we prove that: Any minimal space-like surface in $\\mathbb {R}^4_1$ , free of degenerate points, admits locally canonical coordinates of both types.",
"In Theorem REF we prove the following statement.",
"Any minimal space-like surface ${\\mathcal {M}}$ of general type, parameterized by canonical coordinates of the first type, has the following Weierstrass representation: $\\Phi : \\quad \\begin{array}{rll}\\phi _1 &=& \\displaystyle \\frac{\\mathrm {i}}{2}\\; \\displaystyle \\frac{g_1 g_2+1}{\\sqrt{g^{\\prime }_1 g^{\\prime }_2}}\\,,\\\\[6mm]\\phi _2 &=& \\displaystyle \\frac{1}{2}\\; \\displaystyle \\frac{g_1 g_2-1}{\\sqrt{g^{\\prime }_1 g^{\\prime }_2}}\\,,\\\\[6mm]\\phi _3 &=& \\displaystyle \\frac{1}{2}\\; \\displaystyle \\frac{g_1 + g_2}{\\sqrt{g^{\\prime }_1 g^{\\prime }_2}}\\,,\\\\[6mm]\\phi _4 &=& \\displaystyle \\frac{1}{2}\\; \\displaystyle \\frac{g_1 - g_2}{\\sqrt{g^{\\prime }_1 g^{\\prime }_2}}\\,,\\\\\\end{array}$ where $(g_1,g_2)$ is a pair of holomorphic functions satisfying the conditions: $g^{\\prime }_1 g^{\\prime }_2 \\ne 0; \\quad g_1 \\bar{g}_2 \\ne -1.$ Conversely, if $(g_1,g_2)$ is a pair of holomorphic functions satisfying the conditions (REF ), then the formulas (REF ) generate a minimal space-like surface of general type, parameterized by canonical coordinates of the first type.",
"We call the representation of $\\Phi $ in Theorem REF canonical Weierstrass representation.",
"In terms of the above canonical representation the coefficients of the first fundamental form are given by $E = G =\\displaystyle \\frac{|1+g_1\\bar{g}_2|^2}{4|g^{\\prime }_1 g^{\\prime }_2|}.$ The Gauss curvature $K$ and the curvature of the normal connection $\\varkappa $ (the normal curvature) are given by $K = \\mathop {\\rm Re}\\nolimits \\frac{-16|g^{\\prime }_1 g^{\\prime }_2|\\, g^{\\prime }_1 \\bar{g^{\\prime }_2}}{|1 + g_1 \\bar{g}_2|^2\\; (1 + g_1 \\bar{g}_2)^2}\\,,\\qquad \\varkappa = \\mathop {\\rm Im}\\nolimits \\frac{-16|g^{\\prime }_1 g^{\\prime }_2|\\, g^{\\prime }_1 \\bar{g^{\\prime }_2}}{|1 + g_1 \\bar{g}_2|^2\\; (1 + g_1 \\bar{g}_2)^2}\\;.$ Theorem REF gives a representation of any minimal space-like surface of general type in terms of two holomorphic functions.",
"The following question arises naturally: If $(g_1, g_2)$ and $(\\hat{g}_1, \\hat{g}_2)$ are two pairs of holomorphic functions generating one and the same minimal space-like surface of general type, what is the relation between them?",
"We answer to this question in Theorem REF .",
"Let $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ be two minimal space-like surfaces of general type, given by the canonical Weierstrass representation of the type (REF ).",
"The following conditions are equivalent: $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ are related by a transformation in $\\mathbb {R}^4_1$ of the type: $\\hat{\\mathrm {x}}(t)=A\\mathrm {x}(t)+\\mathrm {b}$ , where $A \\in \\mathbf {SO}(3,1,\\mathbb {R})$ and $\\mathrm {b}\\in \\mathbb {R}^4_1$ .",
"The functions in the Weierstrass representations of $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ are related by the following equalities: $\\hat{g}_1 = \\displaystyle \\frac{ag_1+b}{cg_1+d}\\,; \\quad \\hat{g}_2 = \\displaystyle \\frac{\\phantom{-}\\bar{d} g_2 - \\bar{c}}{-\\bar{b} g_2 + \\bar{a}}\\;,\\qquad a,b,c,d \\in \\mathbb {C}, \\; ad-bc\\ne 0.$"
],
[
"Preliminaries",
"Let $\\mathbb {R}^4_1$ denote the standard Minkowski space-time.",
"This is a four-dimensional space endowed with the indefinite dot product: $\\mathrm {a}\\cdot \\mathrm {b}=a_1b_1+a_2b_2+a_3b_3-a_4b_4\\; .$ If $\\mathcal {M}$ is a two-dimensional manifold and $\\mathrm {x}: \\mathcal {M}\\rightarrow \\mathbb {R}^4_1$ is an immersion of $\\mathcal {M}$ into $\\mathbb {R}^4_1$ , then we say that $\\mathcal {M}$ is a (regular) surface in $\\mathbb {R}^4_1$ .",
"We denote by $T_p(\\mathcal {M})$ the tangential space to $\\mathcal {M}$ at a point $p$ identifying $T_p(\\mathcal {M})$ with the corresponding plane in $\\mathbb {R}^4_1$ .",
"$N_p(\\mathcal {M})$ will stand for the normal space to $\\mathcal {M}$ at the point $p$ , which is the orthogonal complement to $T_p(\\mathcal {M})$ in $\\mathbb {R}^4_1$ .",
"If the induced metric onto $T_p(\\mathcal {M})$ is positive definite, the surface $\\mathcal {M}$ is said to be space-like.",
"Then the induced metric onto the normal space $N_p(\\mathcal {M})$ is of signature (1,1).",
"The surface $\\mathcal {M}$ with the induced metric becomes a two-dimensional Riemannian space.",
"Let $E=\\mathrm {x}_u^2$ , $F=\\mathrm {x}_u\\cdot \\mathrm {x}_v$ and $G=\\mathrm {x}_v^2$ be the coefficients of the first fundamental form on $\\mathcal {M}$ .",
"The surface $\\mathcal {M}$ admits locally around any point $p \\in \\mathcal {M}$ isothermal coordinates (parameters) $(u,v)$ , which means that $E=G$ and $F=0$ .",
"Together with the real coordinates $(u,v) \\in we alsoconsider the complex coordinate $ t=u+iv$, identifying $ R2$with the complex plane $ C$.",
"Thus all functions defined around$ p$ can be considered as functions of the complex variable $ t$.Throughout this paper we consider isothermal coordinates $ (u,v)$ on$ M$.$ We also consider the complexified tangential space $T_{p,C}(\\mathcal {M})$ and the complexified normal space $N_{p,C}(\\mathcal {M})$ at a point $p$ in $\\mathcal {M}$ as the corresponding 2-planes in $\\mathbb {C}^4$ .",
"If $\\mathrm {a}$ and $\\mathrm {b}$ are two vectors in $\\mathbb {C}^4$ , then by $\\mathrm {a}\\cdot \\mathrm {b}$ (or $\\mathrm {a}\\mathrm {b}$ ) we denote the bilinear product in $\\mathbb {C}^4$ , which is the natural extension of the product in $\\mathbb {R}^4_1$ given by (REF ).",
"Together with the bilinear product in $\\mathbb {C}^4$ we also consider the indefinite Hermitian product of $\\mathrm {a}$ and $\\mathrm {b}$ , given by $\\mathrm {a}\\cdot \\bar{\\mathrm {b}}= a_1\\bar{b}_1+a_2\\bar{b}_2+a_3\\bar{b}_3-a_4\\bar{b}_4\\; .$ The square of $\\mathrm {a}$ with respect to the bilinear product is $\\mathrm {a}^2=\\mathrm {a}\\cdot \\mathrm {a}= a_1^2+a_2^2+a_3^2-a_4^2\\;$ and the norm of $\\mathrm {a}$ with respect to the Hermitian product is $\\Vert \\mathrm {a}\\Vert ^2=\\mathrm {a}\\cdot \\bar{\\mathrm {a}}= |a_1|^2+|a_2|^2+|a_3|^2-|a_4|^2\\; .$ The spaces $T_{p,C}(\\mathcal {M})$ and $N_{p,C}(\\mathcal {M})$ are closed with respect to the complex conjugation and are orthogonal with respect to both: bilinear and Hermitian product.",
"Therefore we have the following orthogonal decomposition $\\mathbb {C}^4 = T_{p,C}(\\mathcal {M}) \\oplus N_{p,C}(\\mathcal {M})\\; .$ For a given vector $\\mathrm {a}\\in \\mathbb {C}^4$ we denote by $\\mathrm {a}^\\top $ and $\\mathrm {a}^\\bot $ the orthogonal projections of $\\mathrm {a}$ into $T_{p,C}(\\mathcal {M})$ and $N_{p,C}(\\mathcal {M})$ , respectively, i.e.",
"$\\mathrm {a}=\\mathrm {a}^\\top + \\mathrm {a}^\\bot \\; .$ The above decomposition does not depend on the bilinear or the Hermitian dot product in $\\mathbb {C}^4$ .",
"The second fundamental form on $\\mathcal {M}$ is denoted by $\\sigma $ .",
"By definition we have: $\\sigma (\\mathrm {X},\\mathrm {Y}) = (\\nabla _\\mathrm {X}\\mathrm {Y})^\\bot \\; , $ where $\\mathrm {X}, \\mathrm {Y}\\in T(\\mathcal {M})$ , and $\\nabla $ is the canonical flat connection in $\\mathbb {R}^4_1$ .",
"Let $\\mathrm {X}_1$ and $\\mathrm {X}_2$ denote the unit tangent vectors to $\\mathcal {M}$ at a point $p$ having the same directions as the coordinate vectors $\\mathrm {x}_u$ and $\\mathrm {x}_v$ , respectively, i.e.",
"$\\mathrm {X}_1=\\frac{\\mathrm {x}_u}{\\Vert \\mathrm {x}_u\\Vert }=\\frac{\\mathrm {x}_u}{\\sqrt{E}};\\quad \\mathrm {X}_2=\\frac{\\mathrm {x}_v}{\\Vert \\mathrm {x}_v\\Vert }=\\frac{\\mathrm {x}_v}{\\sqrt{G}}=\\frac{\\mathrm {x}_v}{\\sqrt{E}}\\; .$ The mean curvature $\\mathrm {H}$ of $\\mathcal {M}$ is the vector function $\\mathrm {H}= \\frac{1}{2}\\mathop {\\mathrm {trace}}\\nolimits \\sigma = \\frac{1}{2}(\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)+\\sigma (\\mathrm {X}_2,\\mathrm {X}_2)) \\; .",
"$ A space-like surface $\\mathcal {M}$ in $\\mathbb {R}^4_1$ is said to be minimal if $\\mathrm {H}=0$ at any point of $\\mathcal {M}$ ."
],
[
"The complex function $\\Phi (t)$ . ",
"Let $\\mathcal {M}$ be a space-like surface in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates.",
"The complex-valued vector function $\\Phi (t)$ on $\\mathcal {M}$ with values in $\\mathbb {C}^4$ is defined by $\\Phi (t)=2\\frac{\\partial \\mathrm {x}}{\\partial t}=\\mathrm {x}_u-\\mathrm {i}\\mathrm {x}_v \\; .$ The defining equality (REF ) implies that: $ \\Phi ^2=(\\mathrm {x}_u - \\mathrm {i}\\mathrm {x}_v)^2=\\mathrm {x}_u^2-\\mathrm {x}_v^2-2\\mathrm {x}_u \\mathrm {x}_v \\mathrm {i}\\; .$ Then the following equalities are equivalent: $\\Phi ^2=0 \\ \\Leftrightarrow \\ \\begin{array}{l} \\mathrm {x}_u^2-\\mathrm {x}_v^2=0\\\\ \\mathrm {x}_u \\mathrm {x}_v=0 \\end{array} \\ \\Leftrightarrow \\ \\begin{array}{l} E=\\mathrm {x}_u^2=\\mathrm {x}_v^2=G\\\\ F=0\\end{array}.$ Hence, the parameters $(u,v)$ are isothermal if and only if: $\\Phi ^2=0\\; .$ For the norm of $\\Phi $ we find $\\Vert \\Phi \\Vert ^2=\\Phi \\bar{\\Phi }=\\mathrm {x}_u^2+\\mathrm {x}_v^2=2E=2G.$ Therefore $E=G=\\frac{1}{2}\\Vert \\Phi \\Vert ^2,\\quad F=0$ and $\\mathbf {I}=\\frac{1}{2}\\Vert \\Phi \\Vert ^2 (du^2 +dv^2)=\\frac{1}{2}\\Vert \\Phi \\Vert ^2|dt|^2 \\; .$ From the above it follows that $\\Phi $ satisfies the condition: $\\Vert \\Phi \\Vert ^2> 0\\; .$ Differentiating equality (REF ) and using that $\\frac{\\partial }{\\partial \\bar{t}} \\frac{\\partial }{\\partial t} =\\frac{1}{4}\\Delta $ , we find $\\frac{\\partial \\Phi }{\\partial \\bar{t}}= \\frac{\\partial }{\\partial \\bar{t}}\\, \\left(2\\, \\frac{\\partial \\mathrm {x}}{\\partial t} \\right)=\\frac{1}{2}\\Delta \\mathrm {x}\\; ,$ where $\\Delta $ denotes the Laplace operator.",
"The last formula implies that $\\displaystyle \\frac{\\partial \\Phi }{\\partial \\bar{t}}$ is a real vector function, i.e.",
"$\\frac{\\partial \\Phi }{\\partial \\bar{t}}=\\frac{\\partial \\bar{\\Phi }}{\\partial t} \\; .$ Thus, any space-like surface in $\\mathbb {R}^4_1$ parameterized by isothermal coordinates, determines a function $\\Phi $ given by (REF ), which satisfies the conditions: $\\Phi ^2=0, \\quad \\Vert \\Phi \\Vert ^2> 0, \\quad \\frac{\\partial \\Phi }{\\partial \\bar{t}}=\\frac{\\partial \\bar{\\Phi }}{\\partial t} \\; .$ Conversely, any function $\\Phi $ satisfying these three conditions determines locally a space-like surface in isothermal coordinates up to a translation.",
"The last assertion follows immediately from the fact that (REF ) is the integrability condition for the system $\\begin{array}{ll}\\mathrm {x}_u=\\ \\ \\,\\mathop {\\rm Re}\\nolimits (\\Phi ),\\\\[2mm]\\mathrm {x}_v=-\\mathop {\\rm Im}\\nolimits (\\Phi ).\\end{array}$ Next we express the vectors $\\mathrm {x}_u$ , $\\mathrm {x}_v$ and the second fundamental form $\\sigma $ of $\\mathcal {M}$ by means of $\\Phi $ .",
"Taking into account (REF ) we have: $\\begin{array}{ll}\\mathrm {x}_u=\\ \\ \\,\\mathop {\\rm Re}\\nolimits (\\Phi )=\\displaystyle \\frac{1}{2}(\\Phi +\\bar{\\Phi }),\\\\[3mm]\\mathrm {x}_v=-\\mathop {\\rm Im}\\nolimits (\\Phi )=\\displaystyle \\frac{\\mathrm {i}}{2}(\\Phi -\\bar{\\Phi }).\\end{array}$ Equality (REF ) implies that: $\\left(\\frac{\\partial \\Phi }{\\partial \\bar{t}}\\right)^\\bot =\\left(\\frac{1}{2}\\Delta \\mathrm {x}\\right)^\\bot = \\frac{1}{2}(\\mathrm {x}_{uu}^\\bot + \\mathrm {x}_{vv}^\\bot )= \\frac{1}{2}({\\nabla _{\\mathrm {x}_u}^\\bot \\mathrm {x}_u} +{\\nabla _{\\mathrm {x}_v}^\\bot \\mathrm {x}_v })= \\frac{1}{2}(\\sigma (\\mathrm {x}_u,\\mathrm {x}_u)+\\sigma (\\mathrm {x}_v,\\mathrm {x}_v)) .$ Differentiating (REF ) with respect to $t$ we find $\\frac{\\partial \\Phi }{\\partial t}= \\frac{1}{2}(\\mathrm {x}_{uu} - \\mathrm {x}_{vv}) -\\mathrm {i}\\mathrm {x}_{uv} .$ and $\\left(\\frac{\\partial \\Phi }{\\partial t}\\right)^\\bot =\\frac{1}{2}(\\sigma (\\mathrm {x}_u,\\mathrm {x}_u)-\\sigma (\\mathrm {x}_v,\\mathrm {x}_v)) - \\mathrm {i}\\sigma (\\mathrm {x}_u,\\mathrm {x}_v) .$ Therefore $\\begin{array}{l}\\displaystyle \\sigma (\\mathrm {x}_u,\\mathrm {x}_u) = \\mathop {\\rm Re}\\nolimits \\left(\\frac{\\partial \\Phi }{\\partial \\bar{t}}\\right)^\\bot +\\mathop {\\rm Re}\\nolimits \\left(\\frac{\\partial \\Phi }{\\partial t}\\right)^\\bot ;\\\\\\displaystyle \\sigma (\\mathrm {x}_v,\\mathrm {x}_v) = \\mathop {\\rm Re}\\nolimits \\left(\\frac{\\partial \\Phi }{\\partial \\bar{t}}\\right)^\\bot -\\mathop {\\rm Re}\\nolimits \\left(\\frac{\\partial \\Phi }{\\partial t}\\right)^\\bot ;\\\\\\displaystyle \\sigma (\\mathrm {x}_u,\\mathrm {x}_v) = -\\mathop {\\rm Im}\\nolimits \\left(\\frac{\\partial \\Phi }{\\partial t}\\right)^\\bot .\\end{array}$ Finally we give transformation formulas for the function $\\Phi $ under a change of the isothermal coordinates and under a motion in $\\mathbb {R}^4_1$ .",
"Let us consider the change of the isothermal coordinates given by $t=t(s)$ .",
"Since the transformation of the isothermal coordinates is conformal in $\\mathbb {C}$ , then the function $t(s)$ is either holomorphic or antiholomorphic.",
"Denote by $\\tilde{\\Phi }(s)$ the corresponding function in the new coordinates.",
"First, let us consider the holomorphic case.",
"Taking into account (REF ) we have: $\\tilde{\\Phi }(s)=2\\frac{\\partial \\mathrm {x}}{\\partial s}=2\\frac{\\partial \\mathrm {x}}{\\partial t}\\frac{\\partial t}{\\partial s}+2\\frac{\\partial \\mathrm {x}}{\\partial \\bar{t}}\\frac{\\partial \\bar{t}}{\\partial s}= 2\\frac{\\partial \\mathrm {x}}{\\partial t}\\frac{\\partial t}{\\partial s} \\;.$ Therefore, if the change $t=t(s)$ is holomorphic, then $\\tilde{\\Phi }(s)=\\Phi (t(s)) \\frac{\\partial t}{\\partial s} \\; .$ In the antiholomorphic case we have similarly $\\tilde{\\Phi }(s)=2\\frac{\\partial \\mathrm {x}}{\\partial s}=2\\frac{\\partial \\mathrm {x}}{\\partial t}\\frac{\\partial t}{\\partial s}+2\\frac{\\partial \\mathrm {x}}{\\partial \\bar{t}}\\frac{\\partial \\bar{t}}{\\partial s}= 2\\frac{\\partial \\mathrm {x}}{\\partial \\bar{t}}\\frac{\\partial \\bar{t}}{\\partial s},$ i.e.",
"$\\tilde{\\Phi }(s)=\\bar{\\Phi }(t(s)) \\frac{\\partial \\bar{t}}{\\partial s} \\;.$ Now let $(\\mathcal {M},\\mathrm {x})$ and $(\\hat{\\mathcal {M}},\\hat{\\mathrm {x}})$ be two surfaces in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates $t=u+\\mathrm {i}v$ in one and the same domain $\\mathbb {C}$ .",
"Suppose that $(\\hat{\\mathcal {M}},\\hat{\\mathrm {x}})$ is obtained by $(\\mathcal {M},\\mathrm {x})$ by means of a motion in $\\mathbb {R}^4_1$ : $\\hat{\\mathrm {x}}(t)=A\\mathrm {x}(t)+\\mathrm {b}; \\qquad A \\in \\mathbf {O}(3,1,\\mathbb {R}), \\ \\mathrm {b}\\in \\mathbb {R}^4_1 \\; .$ Differentiating (REF ) we find the relation between the corresponding functions $\\Phi $ and $\\hat{\\Phi }$ : $\\hat{\\Phi }(t)=A\\Phi (t); \\qquad A \\in \\mathbf {O}(3,1,\\mathbb {R}) \\; .$ Conversely, if $\\Phi $ and $\\hat{\\Phi }$ are connected by (REF ), then we have $\\hat{\\mathrm {x}}_u=A\\mathrm {x}_u$ and $\\hat{\\mathrm {x}}_v=A\\mathrm {x}_v$ , which implies (REF ).",
"Hence, the relations (REF ) and (REF ) are equivalent."
],
[
"Characterizing of minimal space-like surfaces in $\\mathbb {R}^4_1$ by means of {{formula:03a323ad-4e1e-45cd-bb3d-6c6917d0da5f}} ",
"Let $\\mathcal {M}$ be a surface in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates and let $\\Phi $ be the function defined by (REF ).",
"Differentiating (REF ), we find: $\\Phi \\cdot \\frac{\\partial \\Phi }{\\partial \\bar{t}}=0 \\; .$ In view of (REF ) the function $\\displaystyle \\frac{\\partial \\Phi }{\\partial \\bar{t}}$ is real and (REF ) after a complex conjugation implies that: $\\bar{\\Phi }\\cdot \\frac{\\partial \\Phi }{\\partial \\bar{t}}=0 \\; .$ Since $\\Phi $ and $\\bar{\\Phi }$ form a basis for $T_{C}(M)$ , then equalities (REF ) and (REF ) mean that $\\displaystyle \\frac{\\partial \\Phi }{\\partial \\bar{t}}$ is orthogonal to $T(\\mathcal {M})$ and therefore $\\frac{\\partial \\Phi }{\\partial \\bar{t}} \\in N(\\mathcal {M}) \\; .$ In view of (REF ) we find successively: $\\begin{array}{rl}\\displaystyle \\frac{\\partial \\Phi }{\\partial \\bar{t}}\\!\\!",
"&=\\displaystyle \\left(\\frac{\\partial \\Phi }{\\partial \\bar{t}}\\right)^\\bot =\\frac{1}{2}(\\Delta \\mathrm {x})^\\bot = \\frac{1}{2}(\\mathrm {x}_{uu}+\\mathrm {x}_{vv} )^\\bot =\\frac{1}{2}(\\nabla _{\\mathrm {x}_u} \\mathrm {x}_u + \\nabla _{\\mathrm {x}_v} \\mathrm {x}_v )^\\bot \\\\[2.5ex]&=\\displaystyle \\frac{1}{2}(\\sigma (\\mathrm {x}_u,\\mathrm {x}_u) + \\sigma (\\mathrm {x}_v,\\mathrm {x}_v) )= E\\;\\frac{1}{2}(\\sigma (\\mathrm {X}_1,\\mathrm {X}_1) + \\sigma (\\mathrm {X}_2,\\mathrm {X}_2) ) = E\\mathrm {H}\\; .\\end{array}$ Finally we have: $\\frac{\\partial \\Phi }{\\partial \\bar{t}}=\\frac{1}{2}\\Delta \\mathrm {x}= E\\mathrm {H}\\;.$ Equality (REF ) implies immediately the following statement.",
"Theorem 4.1 Let $(\\mathcal {M},\\mathrm {x})$ be a space-like surface in $\\mathbb {R}^4_1$ parameterized by isothermal coordinates $(u,v)\\in and $ (t)$ be thecomplex-valued vector function in $ defined by: $\\Phi (t)=2\\frac{\\partial \\mathrm {x}}{\\partial t}=\\mathrm {x}_u-\\mathrm {i}\\mathrm {x}_v, \\quad t=u+\\mathrm {i}.",
"v,$ Then the following conditions are equivalent: The function $\\Phi (t)$ is holomorphic $\\left( \\displaystyle \\frac{\\partial \\Phi }{\\partial \\bar{t}}= 0 \\right)$ ; The function $\\mathrm {x}(u,v)$ is harmonic $(\\Delta \\mathrm {x}= 0)$ ; $(\\mathcal {M},\\mathrm {x})$ is minimal space-like surface in $\\mathbb {R}^4_1$ $(\\mathrm {H}=0)$ .",
"Let $(\\mathcal {M},\\mathrm {x})$ be a minimal space-like surface.",
"Then the harmonic conjugate function $\\mathrm {y}$ to the function $\\mathrm {x}$ is determined by the Cauchy-Riemann equations $ \\mathrm {y}_u=-\\mathrm {x}_v; \\quad \\mathrm {y}_v=\\mathrm {x}_u \\ .$ Let us introduce the function $\\Psi $ by the equality $\\Psi =\\mathrm {x}+\\mathrm {i}\\mathrm {y}.$ The function $\\Psi $ is holomorphic and $\\mathrm {x}$ , $\\Phi $ are expressed by $\\Psi $ in the following way: $\\mathrm {x}=\\mathop {\\rm Re}\\nolimits \\Psi ; \\qquad \\Phi =\\mathrm {x}_u-\\mathrm {i}\\mathrm {x}_v=\\mathrm {x}_u+\\mathrm {i}\\mathrm {y}_u=\\frac{\\partial \\Psi }{\\partial u}=\\Psi ^{\\prime } .$ Since $\\displaystyle \\frac{\\partial \\Phi }{\\partial \\bar{t}}=0$ , then $\\displaystyle \\frac{\\partial \\Phi }{\\partial t}=\\Phi ^{\\prime }$ and $\\sigma (\\mathrm {X}_2,\\mathrm {X}_2)=-\\sigma (\\mathrm {X}_1,\\mathrm {X}_1).$ Therefore $\\sigma (\\mathrm {x}_v,\\mathrm {x}_v)=E\\sigma (\\mathrm {X}_2,\\mathrm {X}_2)=-E\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)=-\\sigma (\\mathrm {x}_u,\\mathrm {x}_u).$ Then formulas (REF ) and (REF ) get the following form: $\\Phi ^\\prime =\\frac{\\partial \\Phi }{\\partial u}=\\mathrm {x}_{uu}-\\mathrm {i}\\mathrm {x}_{uv};\\quad \\Phi ^{\\prime \\bot }=\\mathrm {x}_{uu}^\\bot -\\mathrm {i}x_{uv}^\\bot =\\sigma (\\mathrm {x}_u,\\mathrm {x}_u)-\\mathrm {i}\\sigma (\\mathrm {x}_u,\\mathrm {x}_v).$ Formulas (REF ) become correspondingly $\\begin{array}{l}\\sigma (\\mathrm {x}_u,\\mathrm {x}_u)=\\ \\ \\,\\mathop {\\rm Re}\\nolimits (\\Phi ^{\\prime \\bot })=\\ \\ \\,\\displaystyle \\frac{1}{2}(\\Phi ^{\\prime \\bot }+\\overline{\\Phi ^{\\prime \\bot }})=\\ \\ \\,\\displaystyle \\frac{1}{2}(\\Phi ^{\\prime \\bot }+{\\overline{\\Phi ^\\prime }}^\\bot )\\\\[4mm]\\end{array}\\sigma (\\mathrm {x}_v,\\mathrm {x}_v)= -\\mathop {\\rm Re}\\nolimits (\\Phi ^{\\prime \\bot })=-\\displaystyle \\frac{1}{2}(\\Phi ^{\\prime \\bot }+\\overline{\\Phi ^{\\prime \\bot }})=-\\displaystyle \\frac{1}{2}(\\Phi ^{\\prime \\bot }+{\\overline{\\Phi ^\\prime }}^\\bot )\\\\[4mm]$ (xu,xv)=-Im()= -12i(-)= i2(-)."
],
[
"Expressions for $K$ and {{formula:4e4254c2-2cb6-401f-8b7c-c27142c2fe1d}} of a minimal space-like surface by means of {{formula:2e6db202-e86b-4bdf-aa62-d5452eae6a55}}",
"Let $(\\mathcal {M},\\mathrm {x})$ be a minimal space-like surface in $\\mathbb {R}^4_1$ parameterized by isothermal parameters.",
"Choose a pair $\\mathrm {n}_1$ and $\\mathrm {n}_2$ of orthonormal vector functions in $N(\\mathcal {M})$ of $\\mathcal {M}$ , such that $\\mathrm {n}^2_1=1$ , $\\mathrm {n}^2_2=-1$ and the quadruple $(\\mathrm {X}_1,\\mathrm {X}_2,\\mathrm {n}_1,\\mathrm {n}_2)$ is right oriented in $\\mathbb {R}^4_1$ .",
"For a given normal vector $\\mathrm {n}$ we denote by $A_{\\mathrm {n}}$ the Weingarten operator in $T(\\mathcal {M})$ .",
"This operator and $\\sigma $ are related by the equality $A_{\\mathrm {n}}\\mathrm {X}\\cdot \\mathrm {Y}=\\sigma (\\mathrm {X},\\mathrm {Y})\\cdot \\mathrm {n}$ .",
"The condition $\\mathrm {H}=0$ implies that for any $\\mathrm {n}$ $\\mathop {\\mathrm {trace}}\\nolimits A_{\\mathrm {n}}=0 $ .",
"Then the operators $A_{\\mathrm {n}_1}$ and $A_{\\mathrm {n}_2}$ have the following representation $A_{\\mathrm {n}_1}= \\left(\\begin{array}{rr}\\nu & \\lambda \\\\\\lambda & -\\nu \\end{array}\\right); \\qquad A_{\\mathrm {n}_2}= \\left(\\begin{array}{rr}\\rho & \\mu \\\\\\mu & -\\rho \\end{array}\\right)$ and the components of $\\sigma $ are as follows: $\\begin{array}{l}\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)=(\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\cdot \\mathrm {n}_1)\\mathrm {n}_1-(\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\cdot \\mathrm {n}_2)\\mathrm {n}_2=\\nu \\mathrm {n}_1 - \\rho \\mathrm {n}_2\\\\\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)=(\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)\\cdot \\mathrm {n}_1)\\mathrm {n}_1-(\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)\\cdot \\mathrm {n}_2)\\mathrm {n}_2=\\lambda \\mathrm {n}_1 - \\mu \\mathrm {n}_2\\\\\\sigma (\\mathrm {X}_2,\\mathrm {X}_2)=-\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)=-\\nu \\mathrm {n}_1 + \\rho \\mathrm {n}_2.\\end{array}$ Denote by $R$ the curvature tensor of ${\\mathcal {M}}$ .",
"Then the Gauss equation and (REF ) give: $\\begin{array}{rl}K &=R(\\mathrm {X}_1,\\mathrm {X}_2)\\mathrm {X}_2\\cdot \\mathrm {X}_1\\\\&=\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\sigma (\\mathrm {X}_2,\\mathrm {X}_2)-\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_2)\\\\&=-\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)-\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2)\\;.\\end{array}$ Now (REF ) and (REF ) imply $K=-(\\nu ^2-\\rho ^2)-(\\lambda ^2-\\mu ^2)=\\det (A_{\\mathrm {n}_1})-\\det (A_{\\mathrm {n}_2})\\;.$ On the other hand we get from (REF ): $\\Phi ^{\\prime \\bot }=\\sigma ({\\sqrt{E}}{\\mathrm {X}_1},{\\sqrt{E}}{\\mathrm {X}_1})-\\mathrm {i}\\sigma ({\\sqrt{E}}{\\mathrm {X}_1},{\\sqrt{E}}{\\mathrm {X}_2})= E(\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)-\\mathrm {i}\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)) \\; .$ Calculating the norm $\\Vert \\Phi ^{\\prime \\bot }\\Vert $ , we find $\\begin{array}{rl}{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2=\\Phi ^{\\prime \\bot }\\cdot \\overline{\\Phi ^{\\prime \\bot }}&=E(\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)-\\mathrm {i}\\sigma (\\mathrm {X}_1,\\mathrm {X}_2))E(\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)+\\mathrm {i}\\sigma (\\mathrm {X}_1,\\mathrm {X}_2))\\\\[2mm]&=E^2(\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)+\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2)).\\end{array}$ Taking into account the last equality and (REF ) we have: $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)+\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2)=\\frac{{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2}{E^2}=\\frac{4{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2}{\\Vert \\Phi \\Vert ^4}\\;.$ Now (REF ) and (REF ) imply that $K= \\displaystyle \\frac{-4{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2}{\\Vert \\Phi \\Vert ^4}.$ In the last formula we can represent $\\Vert \\Phi ^{\\prime \\bot }\\Vert ^2$ in a different way.",
"Note that $\\Phi ^2=0$ means that $\\Phi $ and $\\bar{\\Phi }$ are orthogonal with respect to the Hermitian dot product in ${\\mathbb {C}}^4$ .",
"In view of (REF ) and (REF ) it follows that they form an orthogonal basis of the complexified tangential plane of ${\\mathcal {M}}$ .",
"Therefore the tangential projection of $\\Phi ^\\prime $ is as follows: $\\Phi ^{\\prime \\top }=\\displaystyle \\frac{\\Phi ^{\\prime \\top }\\cdot \\bar{\\Phi }}{\\Vert \\Phi \\Vert ^2}\\Phi +\\displaystyle \\frac{\\Phi ^{\\prime \\top }\\cdot \\Phi }{\\Vert \\bar{\\Phi }\\Vert ^2}\\bar{\\Phi }=\\displaystyle \\frac{\\Phi ^{\\prime } \\cdot \\bar{\\Phi }}{\\Vert \\Phi \\Vert ^2}\\Phi + \\displaystyle \\frac{\\Phi ^{\\prime } \\cdot \\Phi }{\\Vert \\bar{\\Phi }\\Vert ^2}\\bar{\\Phi }.",
"$ Differentiating $\\Phi ^2=0$ , we find $\\Phi \\cdot \\Phi ^\\prime =0$ .",
"Thus the projection of $\\Phi ^{\\prime }$ has the form: $\\Phi ^{\\prime \\top }= \\displaystyle \\frac{\\Phi ^{\\prime } \\cdot \\bar{\\Phi }}{\\Vert \\Phi \\Vert ^2}\\Phi ; \\quad \\quad \\Phi ^{\\prime \\bot }=\\Phi ^{\\prime }-\\Phi ^{\\prime \\top }= \\Phi ^{\\prime }-\\displaystyle \\frac{\\Phi ^{\\prime }\\cdot \\bar{\\Phi }}{\\Vert \\Phi \\Vert ^2}\\Phi .$ Using the second equality of (REF ) by means of complex conjugation we get: $\\begin{array}{rl}{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2&=\\Phi ^{\\prime \\bot }\\cdot \\overline{\\Phi ^{\\prime \\bot }}=\\left(\\Phi ^{\\prime }-\\displaystyle \\frac{\\Phi ^{\\prime } \\cdot \\bar{\\Phi }}{\\Vert \\Phi \\Vert ^2}\\Phi \\right)\\left(\\bar{\\Phi ^{\\prime }}-\\displaystyle \\frac{\\bar{\\Phi ^{\\prime }} \\cdot \\Phi }{\\Vert \\Phi \\Vert ^2}\\bar{\\Phi }\\right)\\\\[4mm]&=\\Phi ^{\\prime }\\cdot \\bar{\\Phi ^{\\prime }}-\\displaystyle \\frac{\\bar{\\Phi ^{\\prime }} \\cdot \\Phi }{\\Vert \\Phi \\Vert ^2}\\Phi ^{\\prime }\\cdot \\bar{\\Phi }-\\displaystyle \\frac{\\Phi ^{\\prime } \\cdot \\bar{\\Phi }}{\\Vert \\Phi \\Vert ^2}\\Phi \\cdot \\bar{\\Phi ^{\\prime }} +\\displaystyle \\frac{(\\Phi ^{\\prime } \\cdot \\bar{\\Phi })(\\bar{\\Phi ^{\\prime }} \\cdot \\Phi )}{\\Vert \\Phi \\Vert ^4}\\Phi \\cdot \\bar{\\Phi }\\\\[4mm]&=\\Vert \\Phi ^{\\prime }\\Vert ^2 - \\displaystyle \\frac{|\\bar{\\Phi ^{\\prime }} \\cdot \\Phi |^2}{\\Vert \\Phi \\Vert ^2} - \\displaystyle \\frac{|\\Phi ^{\\prime } \\cdot \\bar{\\Phi }|^2}{\\Vert \\Phi \\Vert ^2}+\\displaystyle \\frac{|\\Phi ^{\\prime } \\cdot \\bar{\\Phi }|^2}{\\Vert \\Phi \\Vert ^4}\\Vert \\Phi \\Vert ^2 = \\Vert \\Phi ^{\\prime }\\Vert ^2 - \\displaystyle \\frac{|\\bar{\\Phi ^{\\prime }} \\cdot \\Phi |^2}{\\Vert \\Phi \\Vert ^2}\\\\[4mm]&=\\displaystyle \\frac{\\Vert \\Phi \\Vert ^2\\Vert \\Phi ^{\\prime }\\Vert ^2-|\\bar{\\Phi }\\cdot \\Phi ^{\\prime }|^2}{\\Vert \\Phi \\Vert ^2}\\ .\\end{array}$ Since the norm of the bi-vector $\\Phi \\wedge \\Phi ^{\\prime }$ is given by: $\\Vert \\Phi \\wedge \\Phi ^{\\prime }\\Vert ^2=\\Vert \\Phi \\Vert ^2\\Vert \\Phi ^{\\prime }\\Vert ^2-|\\bar{\\Phi }\\cdot \\Phi ^{\\prime }|^2,$ then we have $\\Vert \\Phi ^{\\prime \\bot }\\Vert ^2=\\displaystyle \\frac{\\Vert \\Phi \\Vert ^2\\Vert \\Phi ^{\\prime }\\Vert ^2-|\\bar{\\Phi }\\cdot \\Phi ^{\\prime }|^2}{\\Vert \\Phi \\Vert ^2}=\\displaystyle \\frac{\\Vert \\Phi \\wedge \\Phi ^{\\prime }\\Vert ^2}{\\Vert \\Phi \\Vert ^2}\\ $ and $K= \\displaystyle \\frac{-4{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2}{\\Vert \\Phi \\Vert ^4}=\\displaystyle \\frac{-4\\Vert \\Phi \\wedge \\Phi ^{\\prime }\\Vert ^2}{\\Vert \\Phi \\Vert ^6}\\ .$ In order to obtain formulas for the normal curvature $\\varkappa $ , let us denote by $R^N$ the curvature tensor of the normal connection of ${\\mathcal {M}}$ .",
"The Ricci equation and (REF ) imply $\\begin{array}{rl}\\varkappa &= R^N(\\mathrm {X}_1,\\mathrm {X}_2,\\mathrm {n}_1,\\mathrm {n}_2)=R^N(\\mathrm {X}_1,\\mathrm {X}_2)\\mathrm {n}_2\\cdot \\mathrm {n}_1 \\\\ \\;&=[A_{\\mathrm {n}_2},A_{\\mathrm {n}_1}] \\mathrm {X}_1\\cdot \\mathrm {X}_2=A_{\\mathrm {n}_1}\\mathrm {X}_1\\cdot A_{\\mathrm {n}_2}\\mathrm {X}_2-A_{\\mathrm {n}_2}\\mathrm {X}_1\\cdot A_{\\mathrm {n}_1}\\mathrm {X}_2\\\\&=(\\nu \\mathrm {X}_1 + \\lambda \\mathrm {X}_2)\\cdot (\\mu \\mathrm {X}_1 - \\rho \\mathrm {X}_2)-(\\rho \\mathrm {X}_1 + \\mu \\mathrm {X}_2)\\cdot (\\lambda \\mathrm {X}_1 - \\nu \\mathrm {X}_2)\\\\&=\\nu \\mu - \\nu \\rho + \\lambda \\mu - \\lambda \\rho - (\\rho \\lambda - \\rho \\nu + \\mu \\lambda - \\mu \\nu ) \\\\&=2\\nu \\mu - 2\\rho \\lambda \\; .\\end{array}$ We denote by $\\det (\\mathrm {a},\\mathrm {b},\\mathrm {c},\\mathrm {d})$ the determinant of the vectors $\\mathrm {a}$ , $\\mathrm {b}$ , $\\mathrm {c}$ and $\\mathrm {d}$ , with respect to the standard basis in ${\\mathbb {C}}^4$ .",
"Taking into account (REF ), we have $\\begin{array}{rl}\\det (\\mathrm {x}_u,\\mathrm {x}_v,\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))&=\\det (\\sqrt{E} \\mathrm {X}_1,\\sqrt{E} \\mathrm {X}_2,\\sigma (\\sqrt{E} \\mathrm {X}_1,\\sqrt{E} \\mathrm {X}_1),\\sigma (\\sqrt{E} \\mathrm {X}_1,\\sqrt{E} \\mathrm {X}_2))\\\\&=E^3 \\det (\\mathrm {X}_1,\\mathrm {X}_2,\\sigma (\\mathrm {X}_1,\\mathrm {X}_1),\\sigma (\\mathrm {X}_1,\\mathrm {X}_2))\\\\&=E^3 \\det (\\mathrm {X}_1,\\mathrm {X}_2,\\nu \\mathrm {n}_1 - \\rho \\mathrm {n}_2,\\lambda \\mathrm {n}_1 - \\mu \\mathrm {n}_2)\\\\&=-E^3 \\det (\\mathrm {X}_1,\\mathrm {X}_2,\\nu \\mathrm {n}_1,\\mu \\mathrm {n}_2)-E^3 \\det (\\mathrm {X}_1,\\mathrm {X}_2,\\rho \\mathrm {n}_2,\\lambda \\mathrm {n}_1)\\\\&=E^3(-\\nu \\mu + \\rho \\lambda )\\det (\\mathrm {X}_1,\\mathrm {X}_2,\\mathrm {n}_1,\\mathrm {n}_2)=E^3(-\\nu \\mu + \\rho \\lambda ).\\end{array}$ From the last equation it follows that $-\\nu \\mu + \\rho \\lambda = \\displaystyle \\frac{1}{E^3}\\det (\\mathrm {x}_u,\\mathrm {x}_v,\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v)).$ Replacing $\\mathrm {x}_u$ and $\\mathrm {x}_v$ by (REF ) we find $\\begin{array}{l}\\displaystyle \\det (\\mathrm {x}_u,\\mathrm {x}_v,\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))=\\frac{\\mathrm {i}}{4}\\det (\\Phi +\\bar{\\Phi },\\Phi -\\bar{\\Phi },\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))\\\\\\displaystyle =\\frac{\\mathrm {i}}{4}\\det (\\Phi ,-\\bar{\\Phi },\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))+ \\frac{\\mathrm {i}}{4}\\det (\\bar{\\Phi },\\Phi ,\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))\\\\\\displaystyle =-\\frac{\\mathrm {i}}{2}\\det (\\Phi ,\\bar{\\Phi },\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v)).\\end{array}$ Similarly, using (REF ), we get: $\\begin{array}{l}\\displaystyle \\det (\\Phi ,\\bar{\\Phi },\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))=-\\frac{\\mathrm {i}}{2}\\det (\\Phi ,\\bar{\\Phi },\\Phi ^{\\prime \\bot },{\\overline{\\Phi ^\\prime }}^\\bot ).\\end{array}$ Now (REF ) and (REF ) give $\\begin{array}{l}\\displaystyle \\det (\\mathrm {x}_u,\\mathrm {x}_v,\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))=-\\frac{1}{4}\\det (\\Phi ,\\bar{\\Phi },\\Phi ^{\\prime \\bot },{\\overline{\\Phi ^\\prime }}^\\bot )\\\\\\displaystyle =-\\frac{1}{4}\\det (\\Phi ,\\bar{\\Phi },\\Phi ^\\prime -\\Phi ^{\\prime \\top },\\overline{\\Phi ^\\prime }-{\\overline{\\Phi ^\\prime }}^\\top ).\\end{array}$ Hence $\\begin{array}{l}\\displaystyle \\det (\\mathrm {x}_u,\\mathrm {x}_v,\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))=-\\frac{1}{4}\\det (\\Phi ,\\bar{\\Phi },\\Phi ^\\prime ,\\overline{\\Phi ^\\prime }).\\end{array}$ In view of (REF ), (REF ) and (REF ) we have $\\begin{array}{l}\\displaystyle \\varkappa = 2\\nu \\mu - 2\\rho \\lambda = \\frac{-2}{E^3}\\det (\\mathrm {x}_u,\\mathrm {x}_v,\\sigma (\\mathrm {x}_u,\\mathrm {x}_u),\\sigma (\\mathrm {x}_u,\\mathrm {x}_v))=\\frac{1}{2E^3}\\det (\\Phi ,\\bar{\\Phi },\\Phi ^\\prime ,\\overline{\\Phi ^\\prime }).\\end{array}$ Using (REF ) we find: $\\begin{array}{l}\\displaystyle \\varkappa = \\frac{4}{\\Vert \\Phi \\Vert ^6}\\det (\\Phi ,\\bar{\\Phi },\\Phi ^\\prime ,\\overline{\\Phi ^\\prime }).\\end{array}$ For any minimal space-like surface $(\\mathcal {M},\\mathrm {x})$ in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates the Gauss curvature $K$ and the normal curvature $\\varkappa $ are given by the formulas: $K= -\\nu ^2-\\lambda ^2+\\rho ^2+\\mu ^2, \\quad \\quad \\varkappa = 2\\nu \\mu -2\\rho \\lambda \\; ;$ $K= \\displaystyle \\frac{-4{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2}{\\Vert \\Phi \\Vert ^4}=\\displaystyle \\frac{-4\\Vert \\Phi \\wedge \\Phi ^{\\prime }\\Vert ^2}{\\Vert \\Phi \\Vert ^6}, \\quad \\quad \\varkappa = \\displaystyle \\frac{4}{\\Vert \\Phi \\Vert ^6}\\det (\\Phi ,\\bar{\\Phi },\\Phi ^\\prime ,\\overline{\\Phi ^\\prime }).$"
],
[
"Existence of canonical coordinates on a minimal space-like surface",
"Let $\\mathcal {M}$ be a minimal space-like surface in $\\mathbb {R}^4_1$ .",
"Definition 6.1 A point $p\\in \\mathcal {M}$ is said to be degenerate if the set $\\lbrace \\sigma (\\mathrm {X},\\mathrm {Y});\\ \\mathrm {X}\\in T_p(\\mathcal {M}),\\mathrm {Y}\\in T_p(\\mathcal {M}) \\rbrace $ , is contained into one of the light-like one-dimensional subspaces of $N_p(\\mathcal {M})$ .",
"Let $(\\mathcal {M},\\ \\mathrm {x}=\\mathop {\\rm Re}\\nolimits \\Psi )$ be a minimal space-like surface in $\\mathbb {R}^4_1$ parameterized by isothermal coordinates $(u,v)$ .",
"Theorem 6.2 A point $p\\in \\mathcal {M}$ is degenerate if and only if ${\\Phi ^{\\prime \\bot }}^2=0$ .",
"Proof.",
"Let us consider again equality (REF ).",
"Squaring both sides of the equality, we find $\\begin{array}{rl}{\\Phi ^{\\prime \\bot }}^2 \\!\\!\\!",
"&=E^2(\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)-\\mathrm {i}\\, 2\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)-\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2))\\\\&=E^2(\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)-\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2))-\\mathrm {i}\\, 2E^2 \\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\sigma (\\mathrm {X}_1,\\mathrm {X}_2).\\end{array}$ The last equality implies the following equivalence: $\\begin{array}{l}\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\bot \\: \\sigma (\\mathrm {X}_1,\\mathrm {X}_2)\\\\[2mm]\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)=\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2)\\end{array}\\quad \\Leftrightarrow \\quad {\\Phi ^{\\prime \\bot }}^2=0.$ Assuming that the point into consideration is degenerate, then it follows that $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)=0$ , $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_2)=0$ and $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)=0$ .",
"Now (REF ) implies that ${\\Phi ^{\\prime \\bot }}^2=0$ .",
"Let ${\\Phi ^{\\prime \\bot }}^2=0$ .",
"We have to prove that the vectors $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)$ and $\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ lie in one and the same light-like one-dimensional subspace of $N_p(\\mathcal {M})$ .",
"If we assume that $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)>0$ , then it follows that $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_2)>0$ and $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\bot \\: \\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ , which is a contradiction.",
"Similarly, assuming that $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)<0$ , we obtain the metric on $N_p(\\mathcal {M})$ is negative definite, which is a contradiction.",
"Thus $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)=0$ , $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_2)=0$ and $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\bot \\: \\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ .",
"Hence the vectors $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)$ and $\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ are light-like and lie in one and the same one-dimensional subspace of $N_p(\\mathcal {M})$ .",
"$\\blacksquare $ Next we prove that ${\\Phi ^{\\prime \\bot }}^2$ is a holomorphic function of $t$ .",
"In general it doesn't follow that the projection $\\Phi ^{\\prime \\bot }$ is a holomorphic function, but we shall prove that ${\\Phi ^{\\prime \\bot }}^2={\\Phi ^\\prime }^2$ .",
"In order to prove the last equality, we square the second equality in (REF ) and get: ${\\Phi ^{\\prime \\bot }}^2 = {\\Phi ^{\\prime }}^2-2\\Phi ^{\\prime }\\displaystyle \\frac{\\Phi ^{\\prime } \\cdot \\bar{\\Phi }}{\\Vert \\Phi \\Vert ^2}\\Phi +\\left(\\displaystyle \\frac{\\Phi ^{\\prime } \\cdot \\bar{\\Phi }}{\\Vert \\Phi \\Vert ^2}\\right)^2 \\Phi ^2.$ Taking into account equalities $\\Phi ^2=0$ and $\\Phi \\cdot \\Phi ^\\prime =0$ , we find: ${\\Phi ^{\\prime \\bot }}^2={\\Phi ^\\prime }^2 \\; .$ Thus we obtained that any degenerate point of $\\mathcal {M}$ is a zero of the holomorphic function ${\\Phi ^\\prime }^2$ .",
"This implies immediately the following characterization of the set of degenerate points of a minimal space-like surface: Theorem 6.3 If $\\mathcal {M}$ is a connected minimal space-like surface in $\\mathbb {R}^4_1$ , then: either it consists of degenerate points or the set of the degenerate points is countable without any limit points.",
"Further in this section we consider minimal space-like surfaces in $\\mathbb {R}^4_1$ without degenerate points.",
"We give the following definitions: Definition 6.4 The isothermal coordinates $(u, v)$ on a minimal space-like surface are said to be canonical of the first type if $\\begin{array}{l}\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\bot \\: \\sigma (\\mathrm {X}_1,\\mathrm {X}_2),\\\\[2mm]E^2(\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)-\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2))=1.\\end{array}$ Because of (REF ) the isothermal parameters $(u,v)$ are canonical of the first type if and only if ${\\Phi ^\\prime }^2={\\Phi ^{\\prime \\bot }}^2=1.$ Definition 6.5 The isothermal coordinates $(u, v)$ on a minimal space-like surface are said to be canonical of the second type if $\\begin{array}{l}\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\bot \\: \\sigma (\\mathrm {X}_1,\\mathrm {X}_2),\\\\[2mm]E^2(\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)-\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2))=-1.\\end{array}$ The isothermal coordinates $(u,v)$ are canonical of the second type if and only if: ${\\Phi ^\\prime }^2={\\Phi ^{\\prime \\bot }}^2=-1.$ Theorem 6.6 Any minimal space-like surface in $\\mathbb {R}^4_1$ , free of degenerate points, admits locally canonical coordinates of both types.",
"Proof.",
"For arbitrary isothermal coordinates $(u,v)$ on the surface, denote $t=u+v\\mathrm {i}$ .",
"Let us consider the change $t=t(\\tilde{t}\\:)$ of the complex variable $t$ by the new complex variable $\\tilde{t}\\:$ .",
"We shall find the conditions under which the new variable determines canonical coordinates.",
"First, the new coordinates have to be isothermal, i.e.",
"$t=t(\\tilde{t}\\,)$ is a conformal map in ${\\mathbb {C}}$ .",
"Therefore the function $t(\\tilde{t}\\,)$ is either holomorphic or antiholomorphic.",
"The case of an antiholomorphic function is reduced to the case of a holomorphic function by means of the additional change ${\\tilde{t}}=\\bar{s}$ .",
"It is enough to consider only the case of a holomorphic function $t(\\tilde{t}\\,)$ .",
"Let $\\tilde{\\Psi }$ be the holomorphic function representing ${\\mathcal {M}}$ with respect to the new coordinates and $\\tilde{\\Phi }$ be its derivative.",
"Then we have $\\tilde{\\Phi }= \\tilde{\\Psi }^{\\prime }_{\\tilde{t}} = \\Psi ^{\\prime }_t t^{\\prime } = \\Phi t^{\\prime }.$ The derivative of $\\tilde{\\Phi }$ with respect to $\\tilde{t}$ is given by $\\tilde{\\Phi }^{\\prime }_{\\tilde{t}}=\\Phi ^{\\prime }_t t^{\\prime 2}+\\Phi t^{\\prime \\prime }$ .",
"Since $\\Phi $ is tangent to the surface ${\\mathcal {M}}$ , then $\\Phi ^\\bot =0$ and consequently $\\begin{array}{lll}\\tilde{\\Phi }_{\\tilde{t}}^{\\prime \\bot }&=& (\\Phi ^{\\prime }_t t^{\\prime \\, 2}+\\Phi t^{\\prime \\prime })^\\bot = \\Phi _t^{\\prime \\bot }t^{\\prime \\, 2},\\\\\\left.\\tilde{\\Phi }_{\\tilde{t}}^{\\prime \\bot }\\right.^2 &=&{\\Phi _t^{\\prime \\bot }}^2 t^{\\prime 4}.\\end{array}$ According to (REF ) and (REF ) the new complex variable $\\tilde{t}$ determines canonical coordinates if $\\left.\\tilde{\\Phi }_{\\tilde{t}}^{\\prime \\bot }\\right.^2=\\pm 1$ .",
"If ${\\Phi _t^{\\prime \\bot }}^2=0$ , then by virtue of (REF ) it follows that $\\left.\\tilde{\\Phi }_{\\tilde{t}}^{\\prime \\bot }\\right.^2=0$ .",
"The last condition means that the point is degenerate, which is impossible.",
"Hence ${\\Phi ^{\\prime \\bot }}^2 \\ne 0$ .",
"Then the function $\\tilde{t}$ determines canonical coordinates if and only if ${\\Phi _t^{\\prime \\bot }}^2t^{\\prime 4}=\\pm 1$ , i.e.",
"$t(\\tilde{t}\\,)$ satisfies the following ordinary complex first order differential equation: $\\@root 4 \\of {\\pm {\\Phi _t^{\\prime \\bot }}^2}\\:dt = d{\\tilde{t}}.$ Integrating (REF ) and taking into account that the left side of the equality is holomorphic, we obtain $\\tilde{t}$ as a holomorphic function of $t$ .",
"The condition ${\\Phi _t^{\\prime \\bot }}^2 \\ne 0$ means that ${\\tilde{t}}^{\\prime }\\ne 0$ and the correspondence between ${\\tilde{t}}$ and $t$ is one-to-one.",
"Consequently ${\\tilde{t}}$ determines isothermal coordinates satisfying the condition $\\left.\\tilde{\\Phi }_{\\tilde{t}}^{\\prime \\bot }\\right.^2=\\pm 1$ , which implies that they are canonical.",
"$\\blacksquare $ Next we consider the question of uniqueness of canonical coordinates.",
"Suppose that $t$ and $\\tilde{t}$ are canonical of one and the same type.",
"Then $t = t(\\tilde{t}\\,)$ is either holomorphic or antiholomorphic.",
"According to (REF ) and (REF ) equality (REF ) implies that $\\pm 1 = \\left.\\tilde{\\Phi }_{\\tilde{t}}^{\\prime \\bot }\\right.^2={\\Phi _t^{\\prime \\bot }}^2 t^{\\prime 4} = \\pm 1t^{\\prime 4} = \\pm t^{\\prime 4}.$ Therefore $t^{\\prime 4} = 1$ and $t^{\\prime } = \\pm 1;\\ \\pm \\mathrm {i}$ .",
"We get from here that $t$ and $\\tilde{t}$ are related by one of the following equalities: $t=\\pm \\tilde{t}+c;\\ \\pm \\mathrm {i}\\tilde{t}+c$ , where $c=\\text{const}$ .",
"The anti-holomorphic case is reduced to the holomorphic one by the change ${\\tilde{t}}=\\bar{s}$ and we get: $t=\\pm \\bar{\\tilde{t}}+c;\\ \\pm \\mathrm {i}\\bar{\\tilde{t}}+c$ .",
"Thus we obtain eight possible relations between $t$ and $\\tilde{t}$ .",
"Under the natural initial condition $c=0$ , these relations mean that: The canonical coordinates of one and the same type are unique up to a direction and numbering of the coordinate lines.",
"Finally, we consider the relations between canonical coordinates of different type.",
"Let $t=u+v\\mathrm {i}$ be canonical coordinates of the first type and introduce new coordinates by means of the formula $t = e^{\\frac{\\pi \\mathrm {i}}{4}}\\tilde{t}$ .",
"Then $t^{\\prime 4}=-1$ , $\\left.\\tilde{\\Phi }_{\\tilde{t}}^{\\prime \\bot }\\right.^2=-1$ , and consequently $\\tilde{t}$ determines canonical coordinates of the second type.",
"This construction shows that the canonical coordinates of both types are obtained from each other by a rotation of the angle $\\frac{\\pi }{4}$ in the coordinate plane $(u,v)$ .",
"Let $(\\mathcal {M},\\mathrm {x})$ be a minimal space-like surface in $\\mathbb {R}^4_1$ free of degenerate points, parameterized by canonical coordinates of the first type.",
"We can precise the choice of the orthonormal pair $\\mathrm {n}_1, \\mathrm {n}_2$ in $N(\\mathcal {M})$ .",
"Since $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\bot \\:\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ , then we can choose $\\mathrm {n}_1$ and $\\mathrm {n}_2$ to be collinear with $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)$ and $\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ .",
"More precisely, if at a point we have $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)\\ne 0$ , then we choose $\\mathrm {n}_1$ with the same direction as $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)$ , and $\\mathrm {n}_2$ so that the quadruple $(\\mathrm {X}_1,\\mathrm {X}_2,\\mathrm {n}_1,\\mathrm {n}_2)$ is a positive oriented basis in $\\mathbb {R}^4_1$ .",
"Then $\\mathrm {n}_2$ is collinear with $\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ .",
"Under these conditions formulas (REF ) get the form: $\\begin{array}{l}\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)=\\phantom{-} \\nu \\, \\mathrm {n}_1, \\\\\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)= - \\mu \\, \\mathrm {n}_2, \\\\\\sigma (\\mathrm {X}_2,\\mathrm {X}_2)= - \\nu \\, \\mathrm {n}_1;\\end{array}\\qquad \\nu >0 \\; .$ Therefore we have $\\lambda =0$ and $\\rho =0$ and formulas (REF ) become as follows: $A_{\\mathrm {n}_1}= \\left(\\begin{array}{rr}\\nu & 0\\\\0 & -\\nu \\end{array}\\right); \\qquad A_{\\mathrm {n}_2}= \\left(\\begin{array}{rr}0 & \\mu \\\\\\mu & 0\\end{array}\\right).$ If at a given non-degenerate point $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)=0$ , then $\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)\\ne 0$ .",
"In this case we can choose first $\\mathrm {n}_2$ collinear with the same direction with $-\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ , and then $\\mathrm {n}_1$ so that the quadruple $(\\mathrm {X}_1,\\mathrm {X}_2,\\mathrm {n}_1,\\mathrm {n}_2)$ forms a positive oriented basis in $\\mathbb {R}^4_1$ .",
"The functions $\\nu $ and $\\mu $ also satisfy the following relations: $\\begin{array}{lr}\\nu ^2 \\!\\!",
"&= \\phantom{-} \\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_1),\\\\\\mu ^2 \\!\\!",
"&= -\\sigma ^2 (\\mathrm {X}_1,\\mathrm {X}_2).\\end{array}$ The functions $\\nu $ and $\\mu $ are a pair of scalar invariants of a minimal space-like surface, free of degenerate points.",
"These invariants completely determine the second fundamental form via (REF ).",
"The second condition in (REF ) implies that the first fundamental form is completely determined by the formula: $E=\\frac{1}{\\sqrt{\\nu ^2+\\mu ^2}} \\; .$ The relations between the pairs $(\\nu ,\\mu )$ and $(K ,\\varkappa )$ are as follows: $K= -\\nu ^2+\\mu ^2\\; , \\quad \\quad \\varkappa = 2\\nu \\mu \\; .$ $\\mu ^2=\\frac{\\sqrt{K^2+\\varkappa ^2}+K}{2}\\,, \\qquad \\nu ^2=\\frac{\\sqrt{K^2+\\varkappa ^2}-K}{2}\\,.$ Using the above formulas we can characterize the degenerate points of ${\\mathcal {M}}$ in terms of $K$ and $\\varkappa $ .",
"Theorem 6.7 Let $\\mathcal {M}$ be a minimal space-like surface with Gaussian curvature $K$ and normal curvature $\\varkappa $ .",
"A point $p\\in \\mathcal {M}$ is degenerate if and only if $K=0$ and $\\varkappa =0$ .",
"Proof.",
"If $p$ is not a degenerate point in $\\mathcal {M}$ , then we can introduce canonical coordinates of the first type in a neighborhood of $p$ .",
"Formulas (REF ) imply that at least one of $(\\nu ,\\mu )$ is different from 0.",
"Applying (REF ) we obtain that at least one of $(K,\\varkappa )$ is also different from 0.",
"If $p$ is a degenerate point, then $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)$ and $\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ are lightlike.",
"Then (REF ) implies that $K=0$ .",
"Further it follows that $\\sigma (\\mathrm {X}_1,\\mathrm {X}_1)$ and $\\sigma (\\mathrm {X}_1,\\mathrm {X}_2)$ are collinear.",
"Therefore the determinant of the four vectors $\\mathrm {x}_u$ ,$\\mathrm {x}_v$ ,$\\sigma (\\mathrm {x}_u,\\mathrm {x}_u)$ and $\\sigma (\\mathrm {x}_u,\\mathrm {x}_v)$ is zero.",
"Hence, in view of (REF ) and (REF ) it follows that $\\varkappa =0$ .",
"$\\blacksquare $ Finally we add some formulas for $\\nu $ , $\\mu $ and $\\varkappa $ in canonical coordinates of the first type.",
"Equalities (REF ) and (REF ) imply that $\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_1)-\\sigma ^2(\\mathrm {X}_1,\\mathrm {X}_2)=\\frac{1}{E^2}=\\frac{4}{\\Vert \\Phi \\Vert ^4}\\;.$ By virtue of (REF ), (REF ) and (REF ) we find $\\nu ^2 \\!\\!",
"= \\displaystyle \\frac{2(1+{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2)}{\\Vert \\Phi \\Vert ^4}\\,, \\qquad \\mu ^2 \\!\\!",
"= \\displaystyle \\frac{2(1-{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2)}{\\Vert \\Phi \\Vert ^4}\\,.$ Hence $|\\varkappa \\:|=|2\\nu \\mu |= \\displaystyle \\frac{4\\sqrt{{1-\\Vert \\Phi ^{\\prime \\bot }\\Vert }^4}}{\\Vert \\Phi \\Vert ^4}.$"
],
[
"General Weierstrass representations for minimal space-like surfaces.",
"In this section we give several types of general Weierstrass representations for minimal space-like surfaces in $\\mathbb {R}^4_1$ .",
"In $\\mathbb {R}^4$ such formulas were considered in [8], [7].",
"In $\\mathbb {R}^4_1$ general Weierstrass representations were used in [2], [3].",
"Let $(\\mathcal {M},\\mathrm {x})$ : $\\mathrm {x}=\\mathop {\\rm Re}\\nolimits \\Psi $ be a minimal space-like surface in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates, and let $\\Phi =\\Psi ^\\prime $ .",
"If $\\Phi =(\\phi _1,\\phi _2,\\phi _3,\\phi _4)$ , then the condition $\\Phi ^2=0$ is equivalent to that the coordinates are $\\phi _1^2+\\phi _2^2+\\phi _3^2-\\phi _4^2=0$ The relation (REF ) ca be 'parameterized' in different ways by means of three holomorphic functions.",
"First we shall find a representation of $\\Phi $ by means of trigonometric functions.",
"We write (REF ) in the following forms: $\\phi _1^2+\\phi _2^2=-\\phi _3^2+\\phi _4^2, \\quad \\phi _1^2+\\phi _3^2=-\\phi _2^2+\\phi _4^2, \\quad \\phi _2^2+\\phi _3^2=-\\phi _1^2+\\phi _4^2.$ At least one of these three quantities $\\phi _1^2+\\phi _2^2$ , $\\phi _1^2+\\phi _3^2$ and $\\phi _2^2+\\phi _3^2$ is different from zero.",
"(The opposite leads to $\\phi _1^2+\\phi _2^2+\\phi _3^2=0$ and $\\phi _1^2=\\phi _2^2=\\phi _3^2=\\phi _4^2=0$ , which contradicts to the condition ${\\mathcal {M}}$ is regular.)",
"Without loss of generality we can assume that $\\phi _1^2+\\phi _2^2\\ne 0$ , which means that there exists a holomorphic function $f \\ne 0$ , such that: $f^2=\\phi _1^2+\\phi _2^2=-\\phi _3^2+\\phi _4^2.$ The last equality is equivalent to the next one: $\\left(\\frac{\\phi _1}{f}\\right)^2+\\left(\\frac{\\phi _2}{f}\\right)^2=\\left(\\frac{\\phi _3}{\\mathrm {i}f}\\right)^2+\\left(\\frac{\\phi _4}{f}\\right)^2=1.$ Hence, there exist holomorphic functions $h_1$ and $h_2$ , such that: $\\frac{\\phi _1}{f}=\\cos h_1, \\quad \\frac{\\phi _2}{f}=\\sin h_1, \\quad \\frac{\\phi _3}{\\mathrm {i}f}=\\cos h_2, \\quad \\frac{\\phi _4}{f}=\\sin h_2.$ Thus we found the following representation of the function $\\Phi $ : $\\Phi : \\quad \\begin{array}{rlr}\\phi _1 &=& f\\cos h_1,\\\\\\phi _2 &=& f\\,\\sin h_1,\\\\\\phi _3 &=& \\mathrm {i}f\\cos h_2,\\\\\\phi _4 &=& f\\,\\sin h_2;\\\\\\end{array}\\qquad f \\ne 0.$ Next we have to express the condition $\\Vert \\Phi \\Vert ^2>0$ in terms of the triple $(f,h_1,h_2)$ .",
"Equality (REF ) implies that $\\begin{array}{rl}\\Vert \\Phi \\Vert ^2=\\Phi \\bar{\\Phi }&= |f|^2(\\cos h_1 \\cos \\bar{h}_1 + \\sin h_1 \\sin \\bar{h}_1\\\\&+\\cos h_2 \\cos \\bar{h}_2 - \\sin h_2 \\sin \\bar{h}_2)\\\\&= |f|^2(\\cos (h_1 - \\bar{h}_1) + \\cos (h_2 + \\bar{h}_2))\\\\&= |f|^2(\\cos (2\\mathrm {i}\\mathop {\\rm Im}\\nolimits h_1) + \\cos (2\\mathop {\\rm Re}\\nolimits h_2))\\\\&= |f|^2(\\cosh (2\\mathop {\\rm Im}\\nolimits h_1 ) + \\cos (2\\mathop {\\rm Re}\\nolimits h_2 )).\\end{array}$ Since $\\cosh (2\\mathop {\\rm Im}\\nolimits h_1 )\\ge 1 \\ge |\\cos (2\\mathop {\\rm Re}\\nolimits h_2 )|$ , then it follows from (REF ) that $\\Vert \\Phi \\Vert ^2\\ge 0$ .",
"The equality is equivalent to $\\cosh (2\\mathop {\\rm Im}\\nolimits h_1 )=1$ and $\\cos (2\\mathop {\\rm Re}\\nolimits h_2 )=-1$ , i.e.",
"$\\mathop {\\rm Im}\\nolimits h_1=0$ and $\\mathop {\\rm Re}\\nolimits h_2=\\frac{\\pi }{2}+k\\pi ;\\ k\\in {\\mathbb {Z}}$ .",
"Thus we obtained that the triple $(f,h_1,h_2)$ in the representation (REF ) satisfies the conditions $f\\ne 0; \\quad \\mathop {\\rm Im}\\nolimits h_1\\ne 0\\ \\text{or}\\ \\mathop {\\rm Re}\\nolimits h_2\\ne \\frac{\\pi }{2}+k\\pi ;\\ k\\in {\\mathbb {Z}}.$ Hence, any minimal space-like surface ${\\mathcal {M}}$ in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates, admits Weierstrass representation of the type (REF ), where the triple $(f,h_1,h_2)$ satisfies the conditions (REF ).",
"Conversely, any triple $(f,h_1,h_2)$ of holomorphic functions, defined in a domain in ${\\mathbb {C}}$ and satisfying the conditions (REF ), determines by (REF ) a holomorphic ${\\mathbb {C}}^4$ -valued function $\\Phi $ .",
"It follows from (REF ) that $\\Vert \\Phi \\Vert ^2>0$ .",
"By direct computations we get $\\Phi ^2=0$ .",
"Then the surface ${\\mathcal {M}}:\\; \\mathrm {x}=\\mathop {\\rm Re}\\nolimits (\\Psi )$ , where $\\Psi $ is determined by the equality $\\Psi ^\\prime = \\Phi $ , is a minimal space-like surface in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates.",
"So, we proved the following statement.",
"Any triple of holomorphic functions $(f,h_1,h_2)$ satisfying (REF ), generates by means of formulas (REF ) a minimal space-like surface in $\\mathbb {R}^4_1$ .",
"Finally let us establish to what extent the function $\\Phi $ determines the functions $(f,h_1,h_2)$ .",
"Suppose that one and the same function $\\Phi $ is represented by two different triples $(f,h_1,h_2)$ and $(\\hat{f},\\hat{h}_1,\\hat{h}_2)$ .",
"Then (REF ) and (REF ) imply the following relations between both triples: $\\begin{array}{ll}\\hat{f} \\!\\!",
"&=\\: f\\\\\\hat{h}_1 \\!\\!",
"&=\\: h_1 + 2k_1\\pi \\\\\\hat{h}_2 \\!\\!",
"&=\\: h_2 + 2k_2\\pi \\end{array}\\quad \\text{or}\\quad \\begin{array}{ll}\\hat{f} \\!\\!",
"&=\\: -f\\\\\\hat{h}_1 \\!\\!",
"&=\\: h_1 + (2k_1+1)\\pi \\\\\\hat{h}_2 \\!\\!",
"&=\\: h_2 + (2k_2+1)\\pi \\end{array};\\quad \\ \\ \\begin{array}{l}k_1\\in {\\mathbb {Z}}\\\\k_2\\in {\\mathbb {Z}}\\end{array}$ With the aid of different substitutions in (REF ) we can obtain other Weierstrass representations for minimal space-like surfaces in $\\mathbb {R}^4_1$ .",
"In order to obtain Weierstrass representation by means of hyperbolic functions, we make the following substitution in (REF ): $f\\rightarrow \\mathrm {i}f; \\quad h_1 \\rightarrow -\\mathrm {i}h_1; \\quad h_2\\rightarrow \\pi +\\mathrm {i}h_2.$ Thus we obtain the following Weierstrass representation: $\\Phi : \\quad \\begin{array}{rlr}\\phi _1 &=& \\mathrm {i}f\\cosh h_1,\\\\\\phi _2 &=& f\\,\\sinh h_1,\\\\\\phi _3 &=& f\\cosh h_2,\\\\\\phi _4 &=& f\\,\\sinh h_2.\\\\\\end{array}$ Taking into account (REF ), it follows that the functions $(f,h_1,h_2)$ satisfy the conditions: $f\\ne 0; \\quad \\mathop {\\rm Re}\\nolimits h_1\\ne 0\\ \\text{or}\\ \\mathop {\\rm Im}\\nolimits h_2\\ne \\frac{\\pi }{2}+k\\pi ;\\ k\\in {\\mathbb {Z}}.$ Further, let us change the functions $h_1$ and $h_2$ in $(\\ref {W2})$ by $w_1$ and $w_2$ in the following way: $\\begin{array}{l}w_1=h_1+h_2\\\\w_2=h_1-h_2\\end{array}$ Then we obtain the following representation of the surface: $\\Phi : \\quad \\begin{array}{rlr}\\phi _1 &=& \\mathrm {i}f \\cosh \\displaystyle \\frac{w_1+w_2}{2}\\\\[4mm]\\phi _2 &=& f \\sinh \\displaystyle \\frac{w_1+w_2}{2}\\\\[4mm]\\phi _3 &=& f \\cosh \\displaystyle \\frac{w_1-w_2}{2}\\\\[4mm]\\phi _4 &=& f \\sinh \\displaystyle \\frac{w_1-w_2}{2}\\\\\\end{array}$ It follows from (REF ) that $(f,w_1,w_2)$ satisfy the conditions: $f\\ne 0; \\quad \\mathop {\\rm Re}\\nolimits (w_1 + w_2) \\ne 0 \\ \\text{or}\\ \\mathop {\\rm Im}\\nolimits (w_1 - w_2) \\ne (2k+1)\\pi ;\\ k\\in {\\mathbb {Z}}.$ The last conditions can be written in the form: $f\\ne 0; \\quad \\mathop {\\rm Re}\\nolimits (w_1 + \\bar{w}_2) \\ne 0 \\ \\text{or}\\ \\mathop {\\rm Im}\\nolimits (w_1 + \\bar{w}_2) \\ne (2k+1)\\pi ;\\ k\\in {\\mathbb {Z}}.$ Thus we obtained the following more simple form for the conditions (REF ): $f\\ne 0; \\quad w_1 + \\bar{w}_2 \\ne (2k+1)\\pi \\mathrm {i};\\ k\\in {\\mathbb {Z}}.$ Next we introduce the functions $g_1$ and $g_2$ by the equalities: $g_1=e^{w_1}; \\quad g_2=e^{w_2}.$ Using these functions, we obtain from (REF ) the Weierstrass representation, which is the analogue of the classical Weierstrass representation for minimal surfaces in $\\mathbb {R}^3$ .",
"Consequently we calculate the coordinate functions: $\\phi _1 =\\frac{\\mathrm {i}f}{2} (e^{\\frac{w_1+w_2}{2}}+e^{-\\frac{w_1+w_2}{2}})=\\frac{\\mathrm {i}f}{2\\sqrt{g_1 g_2}}(g_1 g_2+1),$ $\\phi _2 = \\frac{f}{2} (e^{\\frac{w_1+w_2}{2}}-e^{-\\frac{w_1+w_2}{2}})=\\frac{f}{2\\sqrt{g_1 g_2}}(g_1 g_2-1).$ $\\phi _3 = \\frac{f}{2} (e^{\\frac{w_1-w_2}{2}}+e^{-\\frac{w_1-w_2}{2}})=\\frac{f}{2\\sqrt{g_1 g_2}}(g_1+g_2).$ $\\phi _4 = \\displaystyle \\frac{f}{2} (e^{\\frac{w_1-w_2}{2}}-e^{-\\frac{w_1-w_2}{2}})=\\displaystyle \\frac{f}{2\\sqrt{g_1 g_2}}(g_1 - g_2).$ In the last equalities we make the substitution $f \\rightarrow f2\\sqrt{g_1 g_2}$ and obtain the following 'polynomial' Weierstrass representation: $\\Phi : \\quad \\begin{array}{rll}\\phi _1 &=& \\mathrm {i}f(g_1 g_2+1),\\\\\\phi _2 &=& \\ f(g_1 g_2-1),\\\\\\phi _3 &=& \\ f(g_1+g_2),\\\\\\phi _4 &=& \\ f(g_1-g_2).\\\\\\end{array}$ Now we shall determine the conditions which satisfy the functions $(f, g_1, g_2)$ .",
"It follows from (REF ) that the condition $w_1 + \\bar{w}_2 \\ne (2k+1)\\pi \\mathrm {i};\\ k\\in {\\mathbb {Z}}$ is equivalent to the condition $e^{w_1 + \\bar{w}_2} \\ne e^{(2k+1)\\pi \\mathrm {i}}=-1;\\ k\\in {\\mathbb {Z}}$ , which gives $g_1 \\bar{g}_2 \\ne -1$ .",
"Therefore we obtained from (REF ) the following conditions: $f\\ne 0; \\quad g_1 \\bar{g}_2 \\ne -1.$ Conversely, if $(f,g_1,g_2)$ are three holomorphic functions defined in a domain in ${\\mathbb {C}}$ and satisfying (REF ), then formulas (REF ) determine a holomorphic function $\\Phi $ with values in ${\\mathbb {C}}^4$ .",
"Equalities (REF ) imply that $\\Vert \\Phi \\Vert ^2>0$ .",
"By direct computations we get from (REF ) equality (REF ), which is $\\Phi ^2=0$ .",
"If we determine the function $\\Psi $ by the equality $\\Psi ^\\prime = \\Phi $ and define ${\\mathcal {M}}:\\; \\mathrm {x}=\\mathop {\\rm Re}\\nolimits (\\Psi )$ , then ${\\mathcal {M}}$ is a minimal space-like surface in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates.",
"Thus we obtained: Any three holomorphic functions $(f,g_1,g_2)$ satisfying (REF ), generates via (REF ) a minimal space-like surface in $\\mathbb {R}^4_1$ .",
"Remark 7.1 We obtained the representation (REF ) using (REF ), which implies that the functions $g_1$ and $g_2$ are different from zero at any point.",
"This follows from the fact that we chose $\\phi _1^2+\\phi _2^2\\ne 0$ .",
"If any of the functions $(g_1, g_2)$ is zero at a fixed point, then it follows directly from (REF ) $\\Phi ^2=0$ and $\\Vert \\Phi \\Vert ^2=2|f|$ .",
"This means that (REF ) again determine a minimal space-like surface in $\\mathbb {R}^4_1$ .",
"Therefore there is no need to add new conditions for $g_1$ and $g_2$ other than these from (REF ).",
"In the end we show that the functions $(f,g_1,g_2)$ can be expressed by the components of the vector function $\\Phi $ .",
"Directly from (REF ) we get: $\\begin{array}{l}\\mathrm {i}\\phi _1+\\phi _2=-f(g_1 g_2+1)+f(g_1 g_2-1)=-2f,\\\\\\phi _3+\\phi _4=f(g_1+g_2)+f(g_1-g_2)=2fg_1,\\\\\\phi _3-\\phi _4=f(g_1+g_2)-f(g_1-g_2)=2fg_2.\\end{array}$ Hence, the functions $f$ , $g_1$ and $g_2$ are expressed as follows: $f=-\\displaystyle \\frac{1}{2}(\\mathrm {i}\\phi _1+\\phi _2), \\quad g_1=-\\displaystyle \\frac{\\phi _3+\\phi _4}{\\mathrm {i}\\phi _1+\\phi _2}\\,, \\quad g_2=-\\displaystyle \\frac{\\phi _3-\\phi _4}{\\mathrm {i}\\phi _1+\\phi _2}\\,.$"
],
[
"Some formulas, related to Weierstrass representations",
"In this section we use the Weierstrass representation (REF ) for minimal space-like surfaces by means of hyperbolic functions.",
"Using the functions $f, h_1, h_2$ , respectively $f, w_1, w_2$ , we obtain some formulas, which we use further.",
"First we introduce some subsidiary functions and denotations.",
"The holomorphic vector function $\\mathrm {a}$ is defined by the equality: $\\mathrm {a}=\\displaystyle \\frac{\\Phi }{f}.$ Next we introduce the following denotations: $\\alpha =\\mathop {\\rm Re}\\nolimits (h_1), \\quad \\beta =\\mathop {\\rm Im}\\nolimits (h_2).$ These functions determine the function $\\theta $ , given by: $\\theta = \\mathop {\\rm Re}\\nolimits h_1 + \\mathrm {i}\\mathop {\\rm Im}\\nolimits h_2 = \\alpha + \\mathrm {i}\\beta .$ The function $\\theta $ is a complex harmonic function, which in general is not holomorphic.",
"Under these denotations applying the Cauchy-Riemann equations, we have: $\\begin{array}{l}h^{\\prime }_1=\\mathop {\\rm Re}\\nolimits (h_1)^{\\prime }_u+\\mathrm {i}\\mathop {\\rm Im}\\nolimits (h_1)^{\\prime }_u=\\mathop {\\rm Re}\\nolimits (h_1)^{\\prime }_u-\\mathrm {i}\\mathop {\\rm Re}\\nolimits (h_1)^{\\prime }_v=\\alpha ^{\\prime }_u-\\mathrm {i}\\alpha ^{\\prime }_v\\;,\\\\h^{\\prime }_2=\\mathop {\\rm Re}\\nolimits (h_2)^{\\prime }_u+\\mathrm {i}\\mathop {\\rm Im}\\nolimits (h_2)^{\\prime }_u=\\mathop {\\rm Im}\\nolimits (h_2)^{\\prime }_v+\\mathrm {i}\\mathop {\\rm Im}\\nolimits (h_2)^{\\prime }_u=\\beta ^{\\prime }_v+\\mathrm {i}\\beta ^{\\prime }_u\\;.\\end{array}$ For $w^{\\prime }_1$ and $w^{\\prime }_2$ we find, respectively: $\\begin{array}{l}w^{\\prime }_1=(\\alpha ^{\\prime }_u+\\beta ^{\\prime }_v)-\\mathrm {i}(\\alpha ^{\\prime }_v-\\beta ^{\\prime }_u)\\;,\\\\w^{\\prime }_2=(\\alpha ^{\\prime }_u-\\beta ^{\\prime }_v)-\\mathrm {i}(\\alpha ^{\\prime }_v+\\beta ^{\\prime }_u)\\;.\\end{array}$ Using (REF ) and (REF ), we get the following formulas for $\\mathrm {a}$ , $\\bar{\\mathrm {a}}$ , $\\mathrm {a}^{\\prime }$ and $\\bar{\\mathrm {a}^{\\prime }}$ : $\\begin{array}{l}\\mathrm {a}=(\\ \\ \\, \\mathrm {i}\\cosh h_1,\\sinh h_1,\\cosh h_2,\\sinh h_2)\\\\\\bar{\\mathrm {a}}=(-\\mathrm {i}\\cosh \\bar{h}_1,\\sinh \\bar{h}_1,\\cosh \\bar{h}_2,\\sinh \\bar{h}_2)\\\\\\mathrm {a}^{\\prime }\\!",
"=(\\ \\ \\, \\mathrm {i}h^{\\prime }_1 \\sinh h_1,h^{\\prime }_1 \\cosh h_1,h^{\\prime }_2 \\sinh h_2,h^{\\prime }_2 \\cosh h_2)\\\\\\bar{\\mathrm {a}^{\\prime }}\\!",
"=(-\\mathrm {i}\\bar{h^{\\prime }_1} \\sinh \\bar{h}_1,\\bar{h^{\\prime }_1} \\cosh \\bar{h}_1,\\bar{h^{\\prime }_2} \\sinh \\bar{h}_2,\\bar{h^{\\prime }_2} \\cosh \\bar{h}_2)\\end{array}$ Further we find the scalar products between the functions $\\mathrm {a}$ , $\\bar{\\mathrm {a}}$ , $\\mathrm {a}^{\\prime }$ and $\\bar{\\mathrm {a}^{\\prime }}$ .",
"Differentiating the equality $\\mathrm {a}^2=0$ , we have: $\\mathrm {a}^2=\\mathrm {a}\\mathrm {a}^{\\prime }=\\bar{\\mathrm {a}}^2=\\bar{\\mathrm {a}}\\bar{\\mathrm {a}^{\\prime }}=0$ Taking scalar multiplications in (REF ), we also obtain: $\\begin{array}{rl}\\Vert \\mathrm {a}\\Vert ^2=\\mathrm {a}\\bar{\\mathrm {a}}&= \\cosh h_1 \\cosh \\bar{h}_1 + \\sinh h_1 \\sinh \\bar{h}_1 + \\cosh h_2 \\cosh \\bar{h}_2 - \\sinh h_2 \\sinh \\bar{h}_2\\\\&= \\cosh (h_1 + \\bar{h}_1) + \\cosh (h_2 - \\bar{h}_2)\\\\&= \\cosh (2\\mathop {\\rm Re}\\nolimits h_1 ) + \\cosh (2\\mathrm {i}\\mathop {\\rm Im}\\nolimits h_2 )\\\\&= 2\\cosh (\\mathop {\\rm Re}\\nolimits h_1+ \\mathrm {i}\\mathop {\\rm Im}\\nolimits h_2)\\cosh ( \\mathop {\\rm Re}\\nolimits h_1- \\mathrm {i}\\mathop {\\rm Im}\\nolimits h_2)\\\\&= 2\\cosh (\\theta )\\cosh (\\bar{\\theta })=2|\\cosh (\\theta )|^2;\\end{array}$ $\\begin{array}{rl}\\mathrm {a}\\bar{\\mathrm {a}^{\\prime }} &= \\bar{h^{\\prime }_1}\\cosh h_1 \\sinh \\bar{h}_1 +\\bar{h^{\\prime }_1}\\sinh h_1 \\cosh \\bar{h}_1 +\\bar{h^{\\prime }_2}\\cosh h_2 \\sinh \\bar{h}_2 - \\bar{h^{\\prime }_2}\\sinh h_2 \\cosh \\bar{h}_2\\\\&= \\bar{h^{\\prime }_1}\\sinh (h_1 + \\bar{h}_1) - \\bar{h^{\\prime }_2}\\sinh (h_2 - \\bar{h}_2)\\\\&= \\bar{h^{\\prime }_1}\\sinh (2\\mathop {\\rm Re}\\nolimits h_1 ) - \\bar{h^{\\prime }_2}\\sinh (2\\mathrm {i}\\mathop {\\rm Im}\\nolimits h_2 );\\end{array}$ $\\bar{\\mathrm {a}}\\mathrm {a}^{\\prime } = \\overline{\\mathrm {a}\\bar{\\mathrm {a}^{\\prime }}} = h^{\\prime }_1 \\sinh (2\\mathop {\\rm Re}\\nolimits h_1 ) + h^{\\prime }_2 \\sinh (2\\mathrm {i}\\mathop {\\rm Im}\\nolimits h_2 );$ $\\begin{array}{rl}\\mathrm {a}^{\\prime 2} &= -h^{\\prime 2}_1\\sinh ^2 h_1 + h^{\\prime 2}_1\\cosh ^2 h_1 + h^{\\prime 2}_2\\sinh ^2 h_2 - h^{\\prime 2}_2\\cosh ^2 h_2\\\\&= h^{\\prime 2}_1-h^{\\prime 2}_2 = w^{\\prime }_1 w^{\\prime }_2;\\end{array}$ $\\begin{array}{rl}\\Vert \\mathrm {a}^{\\prime }\\Vert ^2 = \\mathrm {a}^{\\prime }\\bar{\\mathrm {a}^{\\prime }} &= |h^{\\prime }_1|^2 \\sinh h_1 \\sinh \\bar{h}_1 + |h^{\\prime }_1|^2 \\cosh h_1 \\cosh \\bar{h}_1\\\\&+\\ |h^{\\prime }_2|^2 \\sinh h_2 \\sinh \\bar{h}_2 - |h^{\\prime }_2|^2 \\cosh h_2 \\cosh \\bar{h}_2\\\\&= |h^{\\prime }_1|^2\\cosh (h_1 + \\bar{h}_1) - |h^{\\prime }_2|^2\\cosh (h_2 - \\bar{h}_2)\\\\&= |h^{\\prime }_1|^2\\cosh (2\\mathop {\\rm Re}\\nolimits h_1 ) - |h^{\\prime }_2|^2\\cosh (2\\mathrm {i}\\mathop {\\rm Im}\\nolimits h_2 ).\\end{array}$ Further we obtain formulas for $\\mathrm {a}^{\\prime \\bot }$ , ${\\mathrm {a}^{\\prime \\bot }}^2$ and ${\\Vert \\mathrm {a}^{\\prime \\bot }\\Vert }^2$ expressed by means of $h_1, h_2$ and $w_1, w_2$ , respectively.",
"For $\\mathrm {a}^{\\prime \\bot }$ we have $\\mathrm {a}^{\\prime \\bot }=\\mathrm {a}^{\\prime }-\\mathrm {a}^{\\prime \\top }$ .",
"The equality $\\mathrm {a}^2=0$ means that the vectors $\\mathrm {a}$ and $\\bar{\\mathrm {a}}$ are orthogonal with respect to the Hermitian dot product in ${\\mathbb {C}}^4$ .",
"Therefore the tangential vector $\\mathrm {a}^{\\prime \\top }$ is decomposed as follows: $\\mathrm {a}^{\\prime \\top }=\\displaystyle \\frac{\\mathrm {a}^{\\prime \\top }\\cdot \\bar{\\mathrm {a}}}{\\Vert \\mathrm {a}\\Vert ^2}\\mathrm {a}+\\displaystyle \\frac{\\mathrm {a}^{\\prime \\top }\\cdot \\mathrm {a}}{\\Vert \\bar{\\mathrm {a}}\\Vert ^2}\\bar{\\mathrm {a}}=\\displaystyle \\frac{\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}}{\\Vert \\mathrm {a}\\Vert ^2}\\mathrm {a}+ \\displaystyle \\frac{\\mathrm {a}^{\\prime } \\cdot \\mathrm {a}}{\\Vert \\bar{\\mathrm {a}}\\Vert ^2}\\bar{\\mathrm {a}}.",
"$ Equality (REF ) implies that $\\mathrm {a}^{\\prime } \\cdot \\mathrm {a}= 0$ .",
"Thus we obtained: $\\mathrm {a}^{\\prime \\top }= \\displaystyle \\frac{\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}}{\\Vert \\mathrm {a}\\Vert ^2}\\mathrm {a};\\quad \\quad \\mathrm {a}^{\\prime \\bot }=\\mathrm {a}^{\\prime }-\\mathrm {a}^{\\prime \\top }=\\mathrm {a}^{\\prime }-\\displaystyle \\frac{\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}}{\\Vert \\mathrm {a}\\Vert ^2}\\mathrm {a}.$ Taking square in both sides of (REF ), we get: ${\\mathrm {a}^{\\prime \\bot }}^2 = {\\mathrm {a}^{\\prime }}^2-2\\mathrm {a}^{\\prime }\\displaystyle \\frac{\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}}{\\Vert \\mathrm {a}\\Vert ^2}\\mathrm {a}+\\left(\\displaystyle \\frac{\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}}{\\Vert \\mathrm {a}\\Vert ^2}\\right)^2 \\mathrm {a}^2.$ Taking again into account (REF ), we have $\\mathrm {a}^{\\prime } \\cdot \\mathrm {a}= 0$ and $\\mathrm {a}^2=0$ .",
"Consequently ${\\mathrm {a}^{\\prime \\bot }}^2 = {\\mathrm {a}^{\\prime }}^2$ .",
"Now by virtue of (REF ) we find ${\\mathrm {a}^{\\prime \\bot }}^2 = {\\mathrm {a}^{\\prime }}^2 = {h^{\\prime }_1}^2-{h^{\\prime }_2}^2 = w^{\\prime }_1 w^{\\prime }_2$ Using (REF ) and applying complex conjugation, we calculate ${\\Vert \\mathrm {a}^{\\prime \\bot }\\Vert }^2$ : $\\begin{array}{rl}{\\Vert \\mathrm {a}^{\\prime \\bot }\\Vert }^2&=\\mathrm {a}^{\\prime \\bot }\\cdot \\overline{\\mathrm {a}^{\\prime \\bot }}=\\left(\\mathrm {a}^{\\prime }-\\displaystyle \\frac{\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}}{\\Vert \\mathrm {a}\\Vert ^2}\\mathrm {a}\\right)\\left(\\bar{\\mathrm {a}^{\\prime }}-\\displaystyle \\frac{\\bar{\\mathrm {a}^{\\prime }} \\cdot \\mathrm {a}}{\\Vert \\mathrm {a}\\Vert ^2}\\bar{\\mathrm {a}}\\right)\\\\[4mm]&=\\mathrm {a}^{\\prime }\\cdot \\bar{\\mathrm {a}^{\\prime }}-\\displaystyle \\frac{\\bar{\\mathrm {a}^{\\prime }} \\cdot \\mathrm {a}}{\\Vert \\mathrm {a}\\Vert ^2}\\mathrm {a}^{\\prime }\\cdot \\bar{\\mathrm {a}}-\\displaystyle \\frac{\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}}{\\Vert \\mathrm {a}\\Vert ^2}\\mathrm {a}\\cdot \\bar{\\mathrm {a}^{\\prime }} +\\displaystyle \\frac{(\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}})(\\bar{\\mathrm {a}^{\\prime }} \\cdot \\mathrm {a})}{\\Vert \\mathrm {a}\\Vert ^4}\\mathrm {a}\\cdot \\bar{\\mathrm {a}}\\\\[4mm]&=\\Vert \\mathrm {a}^{\\prime }\\Vert ^2 - \\displaystyle \\frac{|\\bar{\\mathrm {a}^{\\prime }} \\cdot \\mathrm {a}|^2}{\\Vert \\mathrm {a}\\Vert ^2} - \\displaystyle \\frac{|\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}|^2}{\\Vert \\mathrm {a}\\Vert ^2}+\\displaystyle \\frac{|\\mathrm {a}^{\\prime } \\cdot \\bar{\\mathrm {a}}|^2}{\\Vert \\mathrm {a}\\Vert ^4}\\Vert \\mathrm {a}\\Vert ^2 = \\Vert \\mathrm {a}^{\\prime }\\Vert ^2 - \\displaystyle \\frac{|\\bar{\\mathrm {a}^{\\prime }} \\cdot \\mathrm {a}|^2}{\\Vert \\mathrm {a}\\Vert ^2}\\\\[4mm]&=\\displaystyle \\frac{\\Vert \\mathrm {a}\\Vert ^2\\Vert \\mathrm {a}^{\\prime }\\Vert ^2-|\\bar{\\mathrm {a}}\\cdot \\mathrm {a}^{\\prime }|^2}{\\Vert \\mathrm {a}\\Vert ^2}\\end{array}$ Let us denote the numerator in (REF ) by $k_1$ .",
"Applying equalities (REF ), (REF ) and (REF ) we find: $\\begin{array}{rl}k_1 &= \\Vert \\mathrm {a}\\Vert ^2\\Vert \\mathrm {a}^{\\prime }\\Vert ^2-|\\bar{\\mathrm {a}}\\cdot \\mathrm {a}^{\\prime }|^2\\\\&= (|h^{\\prime }_1|^2-|h^{\\prime }_2|^2)(1+\\cosh (2\\mathop {\\rm Re}\\nolimits h_1 )\\cos (2\\mathop {\\rm Im}\\nolimits h_2 ))\\\\&+ 2\\mathop {\\rm Im}\\nolimits (\\bar{h}^{\\prime }_1 h^{\\prime }_2)\\sinh (2\\mathop {\\rm Re}\\nolimits h_1 )\\sin (2\\mathop {\\rm Im}\\nolimits h_2 )\\\\&= ({\\alpha ^{\\prime }_u}^2+{\\alpha ^{\\prime }_v}^2-{\\beta ^{\\prime }_u}^2-{\\beta ^{\\prime }_v}^2)(1+\\cosh (2\\alpha )\\cos (2\\beta ))\\\\&+ 2(\\alpha ^{\\prime }_u\\beta ^{\\prime }_u+\\alpha ^{\\prime }_v\\beta ^{\\prime }_v)\\sinh (2\\alpha )\\sin (2\\beta )\\;.\\end{array}$ Denote the determinant of the vectors $\\mathrm {a}$ , $\\bar{\\mathrm {a}}$ , $\\mathrm {a}^{\\prime }$ and $\\bar{\\mathrm {a}^{\\prime }}$ by $k_2$ .",
"Applying formulas (REF ), we find: $\\begin{array}{rl}k_2 &= \\det (\\mathrm {a},\\bar{\\mathrm {a}}, \\mathrm {a}^{\\prime } , \\bar{\\mathrm {a}^{\\prime }})\\\\&= -2\\mathop {\\rm Im}\\nolimits (\\bar{h}^{\\prime }_1 h^{\\prime }_2)(1+\\cosh (2\\mathop {\\rm Re}\\nolimits h_1 )\\cos (2\\mathop {\\rm Im}\\nolimits h_2 ))\\\\&+ (|h^{\\prime }_1|^2-|h^{\\prime }_2|^2)\\sinh (2\\mathop {\\rm Re}\\nolimits h_1 )\\sin (2\\mathop {\\rm Im}\\nolimits h_2 )\\\\&= -2(\\alpha ^{\\prime }_u\\beta ^{\\prime }_u+\\alpha ^{\\prime }_v\\beta ^{\\prime }_v)(1+\\cosh (2\\alpha )\\cos (2\\beta ))\\\\&+ ({\\alpha ^{\\prime }_u}^2+{\\alpha ^{\\prime }_v}^2-{\\beta ^{\\prime }_u}^2-{\\beta ^{\\prime }_v}^2)\\sinh (2\\alpha )\\sin (2\\beta )\\;.\\end{array}$ Next we simplify the expressions for $k_1$ and $k_2$ calculating the complex quantity $-k_1+\\mathrm {i}k_2$ : $\\begin{array}{rl}-k_1+\\mathrm {i}k_2 &=-\\,({\\alpha ^{\\prime }_u}^2+{\\alpha ^{\\prime }_v}^2-{\\beta ^{\\prime }_u}^2-{\\beta ^{\\prime }_v}^2+2\\mathrm {i}(\\alpha ^{\\prime }_u\\beta ^{\\prime }_u+\\alpha ^{\\prime }_v\\beta ^{\\prime }_v))(1+\\cosh (2\\alpha )\\cos (2\\beta ))\\\\&\\phantom{=}+(2\\mathrm {i}(\\alpha ^{\\prime }_u\\beta ^{\\prime }_u+\\alpha ^{\\prime }_v\\beta ^{\\prime }_v)+{\\alpha ^{\\prime }_u}^2+{\\alpha ^{\\prime }_v}^2-{\\beta ^{\\prime }_u}^2-{\\beta ^{\\prime }_v}^2)\\sinh (2\\alpha )\\sinh (2\\mathrm {i}\\beta )\\\\&= -\\,((\\alpha ^{\\prime }_u+\\mathrm {i}\\beta ^{\\prime }_u)^2+(\\alpha ^{\\prime }_v+\\mathrm {i}\\beta ^{\\prime }_v)^2)\\\\&\\phantom{=} \\phantom{-}\\ \\, (1+\\cosh (2\\alpha )\\cosh (2\\mathrm {i}\\beta )-\\sinh (2\\alpha )\\sinh (2\\mathrm {i}\\beta ))\\\\&= -2((\\alpha ^{\\prime }_u+\\mathrm {i}\\beta ^{\\prime }_u)^2+(\\alpha ^{\\prime }_v+\\mathrm {i}\\beta ^{\\prime }_v)^2)\\cosh ^2(\\alpha -\\mathrm {i}\\beta )\\;.\\end{array}$ Using the function $\\theta $ , defined by (REF ), we obtain another form of $-k_1+\\mathrm {i}k_2$ : $-k_1+\\mathrm {i}k_2 =-2({\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2)\\cosh ^2(\\bar{\\theta })\\;.$ Further we express $-k_1+\\mathrm {i}k_2$ in terms of $w_1$ and $w_2$ .",
"For the first factor in (REF ) we have: $\\begin{array}{rl}{\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2&=(\\alpha ^{\\prime }_u+\\mathrm {i}\\beta ^{\\prime }_u)^2+(\\alpha ^{\\prime }_v+\\mathrm {i}\\beta ^{\\prime }_v)^2\\\\&=(\\alpha ^{\\prime }_u+\\mathrm {i}\\beta ^{\\prime }_u+\\mathrm {i}(\\alpha ^{\\prime }_v+\\mathrm {i}\\beta ^{\\prime }_v))(\\alpha ^{\\prime }_u+\\mathrm {i}\\beta ^{\\prime }_u-\\mathrm {i}(\\alpha ^{\\prime }_v+\\mathrm {i}\\beta ^{\\prime }_v))\\\\&=(\\alpha ^{\\prime }_u-\\beta ^{\\prime }_v+\\mathrm {i}(\\alpha ^{\\prime }_v+\\beta ^{\\prime }_u))(\\alpha ^{\\prime }_u+\\beta ^{\\prime }_v-\\mathrm {i}(\\alpha ^{\\prime }_v-\\beta ^{\\prime }_u))\\;.\\end{array}$ Comparing the last formula with (REF ), we get: ${\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2=(\\mathop {\\rm Re}\\nolimits w^{\\prime }_2 + \\mathrm {i}(-\\mathop {\\rm Im}\\nolimits w^{\\prime }_2))(\\mathop {\\rm Re}\\nolimits w^{\\prime }_1+ \\mathrm {i}\\mathop {\\rm Im}\\nolimits w^{\\prime }_1)\\;.$ The above formulas imply that: ${\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2=w^{\\prime }_1 \\bar{w^{\\prime }_2}\\;.$ In order to find the second factor in (REF ), first we find $\\theta $ : $\\begin{array}{rl}\\theta &= \\alpha +\\mathrm {i}\\beta = \\mathop {\\rm Re}\\nolimits h_1 + \\mathrm {i}\\mathop {\\rm Im}\\nolimits h_2\\\\&= \\frac{1}{2}(h_1 + \\bar{h}_1) + \\mathrm {i}\\frac{1}{2\\mathrm {i}}(h_2 - \\bar{h}_2)\\\\[0.4ex]&= \\frac{1}{2}(h_1 + h_2) + \\frac{1}{2}(\\bar{h}_1 - \\bar{h}_2)\\;.\\end{array}$ Taking into account the above equality and (REF ), we find: $\\theta =\\frac{w_1 + \\bar{w}_2}{2}\\;.$ Consequently $\\begin{array}{rl}\\cosh (\\theta ) &=\\frac{1}{2}(e^{\\frac{w_1 + \\bar{w}_2}{2}}+e^{-\\frac{w_1 + \\bar{w}_2}{2}})=\\frac{1}{2}e^{-\\frac{w_1 + \\bar{w}_2}{2}}(e^{w_1 + \\bar{w}_2}+1)\\\\&=\\frac{1}{2}e^{-\\frac{w_1}{2}}e^{-\\frac{\\bar{w}_2}{2}}(1+e^{w_1}e^{\\bar{w}_2})\\;.\\end{array}$ Finally we have $\\cosh ^2\\theta =\\frac{1}{4}e^{- w_1}e^{- \\bar{w}_2}(1+e^{w_1}e^{\\bar{w}_2})^2\\;.$ Now we replace (REF ) and (REF ) into (REF ) and obtain: $-k_1+\\mathrm {i}k_2 = -\\frac{1}{2}w^{\\prime }_1 \\bar{w^{\\prime }_2}e^{-\\bar{w}_1}e^{-w_2}(1+e^{\\bar{w}_1}e^{w_2})^2\\;.$"
],
[
"Canonical Weierstrass representation for minimal space-like surfaces of general type",
"In this section we introduce canonical Weierstrass representations for minimal space-like surfaces of general type in $\\mathbb {R}^4_1$ .",
"Weierstrass representations with respect to canonical coordinates were obtained in [6] for $\\mathbb {R}^3_1$ and in [7] for $\\mathbb {R}^4$ .",
"Definition 9.1 A minimal space-like surface in $\\mathbb {R}^4_1$ is said to be of general type if it is free of degenerate points in the sense of REF .",
"Let the minimal space-like surface ${\\mathcal {M}}$ of general type be parameterized by canonical coordinates of the first type.",
"Consider the Weierstrass representation (REF ) by means of hyperbolic functions.",
"The condition (REF ) leads to a relation between the three functions $f$ , $h_1$ and $h_2$ .",
"In order to obtain this relation, we express the condition ${\\Phi ^{\\prime \\bot }}^2=1$ via $f$ , $h_1$ and $h_2$ .",
"By virtue of (REF ) we have $\\Phi =f\\mathrm {a}$ and therefore $\\Phi ^{\\prime }=f^{\\prime }\\mathrm {a}+f\\mathrm {a}^{\\prime }$ .",
"Since the vector $\\mathrm {a}$ is tangential to ${\\mathcal {M}}$ , then we get: $\\Phi ^{\\prime \\bot }=(f^{\\prime }\\mathrm {a}+f\\mathrm {a}^{\\prime })^\\bot = f\\mathrm {a}^{\\prime \\bot }; \\quad \\quad {\\Phi ^{\\prime \\bot }}^2=f^2 {\\mathrm {a}^{\\prime \\bot }}^2.$ Because of (REF ) we have ${\\mathrm {a}^{\\prime \\bot }}^2 ={h^{\\prime }_1}^2-{h^{\\prime }_2}^2$ and consequently ${\\Phi ^{\\prime \\bot }}^2=f^2({h^{\\prime }_1}^2-{h^{\\prime }_2}^2)$ .",
"Taking into account the last equality and (REF ), we obtain that the minimal space-like surface ${\\mathcal {M}}$ given by (REF ) is parameterized by canonical coordinates of the first type if and only if: ${\\Phi ^{\\prime }}^2=f^2 ({h^{\\prime }_1}^2-{h^{\\prime }_2}^2)=1\\;.$ The last formula and (REF ) imply the following statement.",
"Theorem 9.2 Any minimal space-like surface ${\\mathcal {M}}$ of general type, parameterized by canonical coordinates of the first type, has the following Weierstrass representation: $\\Phi : \\quad \\begin{array}{rlr}\\phi _1 &=& \\mathrm {i}\\displaystyle \\frac{\\cosh h_1}{\\sqrt{{h^{\\prime }_1}^2 - {h^{\\prime }_2}^2}}\\,,\\\\[8mm]\\phi _2 &=& \\displaystyle \\frac{\\sinh h_1}{\\sqrt{{h^{\\prime }_1}^2 - {h^{\\prime }_2}^2}}\\,,\\\\[8mm]\\phi _3 &=& \\displaystyle \\frac{\\cosh h_2}{\\sqrt{{h^{\\prime }_1}^2 - {h^{\\prime }_2}^2}}\\,,\\\\[8mm]\\phi _4 &=& \\displaystyle \\frac{\\sinh h_2}{\\sqrt{{h^{\\prime }_1}^2 - {h^{\\prime }_2}^2}}\\,,\\\\\\end{array}$ where $(h_1,h_2)$ are holomorphic functions satisfying the conditions: ${h^{\\prime }_1}^2 \\ne {h^{\\prime }_2}^2; \\quad \\mathop {\\rm Re}\\nolimits h_1\\ne 0\\ \\text{or}\\ \\mathop {\\rm Im}\\nolimits h_2\\ne \\frac{\\pi }{2}+k\\pi ;\\ k\\in {\\mathbb {Z}}.$ Conversely, if $(h_1,h_2)$ is a pair of holomorphic functions satisfying the conditions (REF ), then formulas (REF ) give a minimal space-like surface of general type, parameterized by canonical coordinates of the first type.",
"We call the representation of $\\Phi $ in Theorem 9.2 canonical Weierstrass representation.",
"Using the functions $w_1$ and $w_2$ , given by (REF ), then the condition (REF ) gets the form: ${\\Phi ^{\\prime }}^2=f^2 w^{\\prime }_1 w^{\\prime }_2 = 1$ If we replace $h_1$ and $h_2$ with $w_1$ and $w_2$ into (REF ), then we obtain the following canonical Weierstrass representation for ${\\mathcal {M}}$ : $\\Phi : \\quad \\begin{array}{rlr}\\phi _1 &=& \\displaystyle \\frac{\\mathrm {i}}{\\sqrt{w^{\\prime }_1 w^{\\prime }_2}} \\cosh \\displaystyle \\frac{w_1+w_2}{2}\\,,\\\\[6mm]\\phi _2 &=& \\displaystyle \\frac{1}{\\sqrt{w^{\\prime }_1 w^{\\prime }_2}} \\sinh \\displaystyle \\frac{w_1+w_2}{2}\\,,\\\\[6mm]\\phi _3 &=& \\displaystyle \\frac{1}{\\sqrt{w^{\\prime }_1 w^{\\prime }_2}} \\cosh \\displaystyle \\frac{w_1-w_2}{2}\\,,\\\\[6mm]\\phi _4 &=& \\displaystyle \\frac{1}{\\sqrt{w^{\\prime }_1 w^{\\prime }_2}} \\sinh \\displaystyle \\frac{w_1-w_2}{2}\\,.\\\\\\end{array}$ According to (REF ), the functions $(w_1,w_2)$ satisfy the conditions: $w^{\\prime }_1 w^{\\prime }_2 \\ne 0; \\quad w_1 + \\bar{w}_2 \\ne (2k+1)\\pi \\mathrm {i};\\ k\\in {\\mathbb {Z}}.$ Conversely, if $(w_1,w_2)$ is a pair of holomorphic functions, satisfying the conditions (REF ), then the formulas (REF ) generate a minimal space-like surface of general type, parameterized by canonical coordinates of the first type.",
"Finally, using the functions $g_1$ and $g_2$ , given by (REF ), we obtain a canonical Weierstrass representation of the type (REF ).",
"Differentiating (REF ), we get: $g^{\\prime }_1=e^{w_1}w^{\\prime }_1=g_1 w^{\\prime }_1; \\quad g^{\\prime }_2=e^{w_2}w^{\\prime }_2=g_2 w^{\\prime }_2.$ From here we have: $w^{\\prime }_1=\\frac{g^{\\prime }_1}{g_1}\\,, \\quad w^{\\prime }_2=\\frac{g^{\\prime }_2}{g_2}.$ Applying (REF ) and (REF ) to the condition (REF ), we get $(f2\\sqrt{g_1 g_2})^2 \\displaystyle \\frac{g^{\\prime }_1}{g_1}\\displaystyle \\frac{g^{\\prime }_2}{g_2}=1$ .",
"Consequently the isothermal coordinates are canonical of the first type if and only if ${\\Phi ^{\\prime }}^2=4f^2 g^{\\prime }_1 g^{\\prime }_2 = 1.$ Next we express $f$ from the last equality of (REF ) and replace it into (REF ).",
"Thus we obtain the following statement.",
"Theorem 9.3 Any minimal space-like surface ${\\mathcal {M}}$ of general type, parameterized by canonical coordinates of the first type, has the following Weierstrass representation: $\\Phi : \\quad \\begin{array}{rll}\\phi _1 &=& \\displaystyle \\frac{\\mathrm {i}}{2}\\; \\displaystyle \\frac{g_1 g_2+1}{\\sqrt{g^{\\prime }_1 g^{\\prime }_2}}\\,,\\\\[6mm]\\phi _2 &=& \\displaystyle \\frac{1}{2}\\; \\displaystyle \\frac{g_1 g_2-1}{\\sqrt{g^{\\prime }_1 g^{\\prime }_2}}\\,,\\\\[6mm]\\phi _3 &=& \\displaystyle \\frac{1}{2}\\; \\displaystyle \\frac{g_1 + g_2}{\\sqrt{g^{\\prime }_1 g^{\\prime }_2}}\\,,\\\\[6mm]\\phi _4 &=& \\displaystyle \\frac{1}{2}\\; \\displaystyle \\frac{g_1 - g_2}{\\sqrt{g^{\\prime }_1 g^{\\prime }_2}}\\,.\\\\\\end{array}$ According to (REF ) the functions $(g_1,g_2)$ in this representation satisfy the conditions: $g^{\\prime }_1 g^{\\prime }_2 \\ne 0; \\quad g_1 \\bar{g}_2 \\ne -1.$ Conversely, if $(g_1,g_2)$ is a pair of holomorphic functions satisfying the conditions (REF ), then formulas (REF ) generate a minimal space-like surface of general type, parameterized by canonical coordinates of the first type.",
"The above canonical Weierstrass representation seems to be the most useful and applicable representation."
],
[
"The first fundamental form and the curvatures $K, \\varkappa $ in a general Weierstrass representation",
"Let ${\\mathcal {M}}$ be a minimal space-like surface in $\\mathbb {R}^4_1$ , parameterized by isothermal coordinates.",
"First we consider the case, when ${\\mathcal {M}}$ is given by (REF ).",
"In order to obtain a formula for $E$ , we use equalities (REF ), (REF ) and (REF ).",
"Thus we get: $E=\\frac{1}{2}\\Vert \\Phi \\Vert ^2=\\frac{1}{2}\\Vert f\\mathrm {a}\\Vert ^2=|f|^2|\\cosh \\theta |^2\\;,$ where $\\theta $ is the function (REF ).",
"Taking into account (REF ), we find the following formula for $E$ with respect to the representation (REF ): $E=|f|^2\\left|\\cosh \\frac{w_1 + \\bar{w}_2}{2}\\right|^2.$ In order to obtain a formula with respect to the representation (REF ), we express $\\cosh (\\theta )$ by means of $g_j$ , $(j=1;2)$ , given by (REF ).",
"Then (REF ) gives that $\\cosh ^2(\\theta ) =\\frac{(1+g_1\\bar{g}_2)^2}{4g_1\\bar{g}_2}\\;.$ Passing from the representation (REF ) by means of $w_j$ to the representation (REF ) by means of $g_j$ , $(j=1;2)$ , as a consequence of (REF ) $|f|^2$ has to be replaced by: $|f|^2 \\rightarrow 4|f|^2|g_1 g_2|\\;.$ Now applying (REF ) and (REF ) to (REF ), we find the following formula for the coefficient $E$ in the representation (REF ): $E=|f|^2|1+g_1\\bar{g}_2|^2\\;.$ Further we find the corresponding formulas for $K$ and $\\varkappa $ .",
"In the formula (REF ) we replace $\\Phi ^{\\prime \\bot }$ by (REF ) and get: $K = \\displaystyle \\frac{-4{\\Vert \\Phi ^{\\prime \\bot }\\Vert }^2}{\\Vert \\Phi \\Vert ^4} =\\displaystyle \\frac{-4{\\Vert f\\mathrm {a}^{\\prime \\bot }\\Vert }^2}{\\Vert f\\mathrm {a}\\Vert ^4} =\\displaystyle \\frac{-4{|f|^2\\Vert \\mathrm {a}^{\\prime \\bot }\\Vert }^2}{|f|^4\\Vert \\mathrm {a}\\Vert ^4} =\\displaystyle \\frac{-4{\\Vert \\mathrm {a}^{\\prime \\bot }\\Vert }^2}{|f|^2\\Vert \\mathrm {a}\\Vert ^4}.$ Using (REF ) and taking into account (REF ), we find: $K = \\displaystyle \\frac{-4(\\Vert \\mathrm {a}\\Vert ^2\\Vert \\mathrm {a}^{\\prime }\\Vert ^2-|\\bar{\\mathrm {a}}\\cdot \\mathrm {a}^{\\prime }|^2)}{|f|^2\\Vert \\mathrm {a}\\Vert ^6} = \\displaystyle \\frac{-4k_1}{|f|^2\\Vert \\mathrm {a}\\Vert ^6}.$ A similar formula for $\\varkappa $ can be derived using the second equality in (REF ).",
"We find consecutively: $\\displaystyle \\varkappa = \\displaystyle \\frac{4}{\\Vert f\\mathrm {a}\\Vert ^6}\\det (f\\mathrm {a},\\bar{f} \\bar{\\mathrm {a}},f^{\\prime }\\mathrm {a}+f\\mathrm {a}^{\\prime },\\bar{f}^{\\prime } \\bar{\\mathrm {a}}+ \\bar{f} \\bar{\\mathrm {a}^{\\prime }})= \\displaystyle \\frac{4|f|^4}{|f|^6\\Vert \\mathrm {a}\\Vert ^6}\\det (\\mathrm {a},\\bar{\\mathrm {a}},\\mathrm {a}^{\\prime },\\bar{\\mathrm {a}^{\\prime }}) = \\displaystyle \\frac{4k_2}{|f|^2\\Vert \\mathrm {a}\\Vert ^6}.$ Thus we have: $K = \\displaystyle \\frac{-4k_1}{|f|^2\\Vert \\mathrm {a}\\Vert ^6}; \\quad \\varkappa =\\displaystyle \\frac{4k_2}{|f|^2\\Vert \\mathrm {a}\\Vert ^6}\\;.$ It is useful to unite $K$ and $\\varkappa $ in one formula by the complex quantity $K+\\mathrm {i}\\varkappa $ .",
"It follows from (REF ) that: $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{4(-k_1+\\mathrm {i}k_2)}{|f|^2\\Vert \\mathrm {a}\\Vert ^6}\\;.$ Replacing $\\Vert \\mathrm {a}\\Vert ^2$ and $-k_1+\\mathrm {i}k_2$ respectively by (REF ) and (REF ) we get: $K+\\mathrm {i}\\varkappa =\\displaystyle \\frac{4(-2({\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2)\\cosh ^2(\\bar{\\theta }))}{|f|^28|\\cosh (\\theta )|^6}\\;.$ Finally we obtained: $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{-({\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2)}{|f|^2\\;|\\cosh (\\theta )|^2\\; \\cosh ^2(\\theta )}\\;,$ where $\\theta $ is the function (REF ).",
"Thus the formulas for $K$ and $\\varkappa $ related to the representation (REF ) are: $\\begin{array}{lll}K &=& \\mathop {\\rm Re}\\nolimits \\displaystyle \\frac{-({\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2)}{|f|^2\\; |\\cosh (\\theta )|^2\\; \\cosh ^2(\\theta )}\\\\[4ex]\\varkappa &=& \\mathop {\\rm Im}\\nolimits \\displaystyle \\frac{-({\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2)}{|f|^2\\;|\\cosh (\\theta )|^2\\; \\cosh ^2(\\theta )}\\;.\\end{array}$ In order to express $K$ and $\\varkappa $ by means of the functions $w_j$ , $(j=1;2)$ in the representation (REF ), we use (REF ) and (REF ).",
"Applying them to (REF ) we find: $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{- w^{\\prime }_1 \\bar{w^{\\prime }_2}}{|f|^2\\;\\left|\\cosh \\frac{w_1 + \\bar{w}_2}{2}\\right|^2\\;\\cosh ^2\\frac{w_1 + \\bar{w}_2}{2}}\\;.$ The corresponding formulas in terms of $g_j$ , $(j=1;2)$ in the representation (REF ) follow by using (REF ), (REF ) and (REF ): $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{- w^{\\prime }_1 \\bar{w^{\\prime }_2}}{4|f|^2|g_1 g_2|\\;|\\frac{1}{4}e^{-w_1}e^{-\\bar{w}_2}(1+e^{w_1}e^{\\bar{w}_2})^2|\\;\\frac{1}{4}e^{-w_1}e^{-\\bar{w}_2}(1+e^{w_1}e^{\\bar{w}_2})^2}\\;.$ By virtue of (REF ) и (REF ), the last formula takes the form: $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{-4 g^{\\prime }_1 \\bar{g^{\\prime }_2}}{|f|^2\\;|1 + g_1 \\bar{g}_2|^2\\;(1 + g_1 \\bar{g}_2)^2}\\;.$ Applying (REF ), we get: $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{-4 g^{\\prime }_1 \\bar{g^{\\prime }_2}}{E\\; (1 + g_1 \\bar{g}_2)^2}\\;.$ The corresponding formulas for $K$ and $\\varkappa $ , related to the representation (REF ) are: $\\begin{array}{lll}K &=& \\mathop {\\rm Re}\\nolimits \\displaystyle \\frac{-4 g^{\\prime }_1 \\bar{g^{\\prime }_2}}{|f|^2\\;|1 + g_1 \\bar{g}_2|^2\\; (1 + g_1 \\bar{g}_2)^2}=\\mathop {\\rm Re}\\nolimits \\displaystyle \\frac{-4 g^{\\prime }_1 \\bar{g^{\\prime }_2}}{E\\; (1 + g_1 \\bar{g}_2)^2}\\\\[3ex]\\varkappa &=& \\mathop {\\rm Im}\\nolimits \\displaystyle \\frac{-4 g^{\\prime }_1 \\bar{g^{\\prime }_2}}{|f|^2\\;|1 + g_1 \\bar{g}_2|^2\\; (1 + g_1 \\bar{g}_2)^2}=\\mathop {\\rm Im}\\nolimits \\displaystyle \\frac{-4 g^{\\prime }_1 \\bar{g^{\\prime }_2}}{E\\; (1 + g_1 \\bar{g}_2)^2}\\;.\\end{array}$ The above formulas have been found by Asperti A. and Vilhena J. in [2]."
],
[
"The first fundamental form and the curvatures $K$ , {{formula:209de643-8fca-4c8b-8e7f-c810e7eb3b0c}} , with respect to a canonical\nWeierstrass representation.",
"Let ${\\mathcal {M}}$ be a minimal space-like surface of general type, parameterized by canonical coordinates of the first type.",
"First we obtain a formula for the coefficient $E$ with respect to the canonical Weierstrass representation (REF ).",
"Applying (REF ) and (REF ), we find the following formula for $|f|^2$ : $|f|^2 = \\frac{1}{|w^{\\prime }_1 w^{\\prime }_2|} = \\frac{1}{|{\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2|}\\;.$ Replacing into the general formula (REF ), we get: $E = \\displaystyle \\frac{|\\cosh (\\theta )|^2}{|{\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2|}\\;.$ To obtain a formula for $E$ with respect to the canonical Weierstrass representation (REF ), we use (REF ) and (REF ).",
"Thus we find a formula for $E$ in terms of $w_1$ and $w_2$ : $E=\\displaystyle \\frac{\\left|\\cosh \\frac{w_1 + \\bar{w}_2}{2}\\right|^2}{|w^{\\prime }_1 w^{\\prime }_2|}.$ In a similar way, if ${\\mathcal {M}}$ is given by (REF ), we replace $f$ into the general formula (REF ) by the help of (REF ) and obtain: $E=\\displaystyle \\frac{|1+g_1\\bar{g}_2|^2}{4|g^{\\prime }_1 g^{\\prime }_2|}.$ Next we find formulas for the curvatures $K$ and $\\varkappa $ .",
"Using the representation (REF ), we replace $f$ into the general formula (REF ) by means of (REF ), and get: $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{-|{\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2|\\, ({\\theta ^{\\prime }_u}^2+{\\theta ^{\\prime }_v}^2)}{|\\cosh (\\theta )|^2\\; \\cosh ^2(\\theta )}\\;.$ To obtain a formula for $K+\\mathrm {i}\\varkappa $ , when ${\\mathcal {M}}$ is given by (REF ), we replace into the general formula (REF ) the function $f$ by means of (REF ).",
"Thus we have: $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{- |w^{\\prime }_1 w^{\\prime }_2|\\, w^{\\prime }_1 \\bar{w^{\\prime }_2}}{\\left|\\cosh \\frac{w_1 + \\bar{w}_2}{2}\\right|^2\\; \\cosh ^2\\frac{w_1 + \\bar{w}_2}{2}}\\;.$ To obtain a formula for $K+\\mathrm {i}\\varkappa $ , when ${\\mathcal {M}}$ is represented by (REF ), we replace into the general formula (REF ) the function $f$ by the help of (REF ).",
"Hence, we have: $K+\\mathrm {i}\\varkappa = \\displaystyle \\frac{-16|g^{\\prime }_1 g^{\\prime }_2|\\, g^{\\prime }_1 \\bar{g^{\\prime }_2}}{|1 + g_1 \\bar{g}_2|^2\\; (1 + g_1 \\bar{g}_2)^2}\\;.$ The formulas for the curvatures $K$ and $\\varkappa $ with respect to the representation (REF ) are as follows: $\\begin{array}{lll}K &=& \\mathop {\\rm Re}\\nolimits \\displaystyle \\frac{-16|g^{\\prime }_1 g^{\\prime }_2|\\, g^{\\prime }_1 \\bar{g^{\\prime }_2}}{|1 + g_1 \\bar{g}_2|^2\\; (1 + g_1 \\bar{g}_2)^2}\\,,\\\\[4ex]\\varkappa &=& \\mathop {\\rm Im}\\nolimits \\displaystyle \\frac{-16|g^{\\prime }_1 g^{\\prime }_2|\\, g^{\\prime }_1 \\bar{g^{\\prime }_2}}{|1 + g_1 \\bar{g}_2|^2\\; (1 + g_1 \\bar{g}_2)^2}\\;.\\end{array}$"
],
[
"Change of the functions $(g_1,g_2)$ under some basic geometric transformations of the minimal\nspace-like surface",
"Let ${\\mathcal {M}}$ be a minimal space-like surface of general type, parameterized by canonical coordi-nates $(u, v)$ of the first type.",
"The complex variable $t$ is given by $t=u+\\mathrm {i}v$ .",
"We suppose that ${\\mathcal {M}}$ is given by the canonical representation (REF ) by means of the pair $(g_1(t),g_2(t))$ of holomorphic functions.",
"The aim of this section is to study the changes of the pair $(g_1,g_2)$ under geometric transformations of the surface .",
"First we consider the case of a motion of the surface ${\\mathcal {M}}$ in $\\mathbb {R}^4_1$ .",
"We shall use some basic formulas and facts about the spinors in $\\mathbb {R}^4_1$ .",
"Let us recall some of these formulas in a form useful for an application to the theory of minimal space-like surfaces.",
"To any vector $\\mathrm {x}$ in $\\mathbb {R}^4_1$ we associate a Hermitian $2\\times 2$ -matrix $S$ as follows: $S=\\left(\\begin{array}{rr}x_3 +x_4 & \\mathrm {i}x_1 +x_2\\\\-\\mathrm {i}x_1 +x_2 & -x_3+x_4\\end{array}\\right) \\ \\leftrightarrow \\ \\mathrm {x}=(x_1,x_2,x_3,x_4)\\;.$ This correspondence is a linear isomorphism between $\\mathbb {R}^4_1$ and the space of Hermitian $2\\times 2$ -matrices.",
"This correspondence has the following property: $\\det S = -\\mathrm {x}^2$ .",
"The last property means that from any linear operator acting in the space of Hermitian $2\\times 2$ -matrices and preserving the determinant, can be obtained an orthogonal operator in $\\mathbb {R}^4_1$ .",
"If $\\tilde{A}$ is a complex $2\\times 2$ -matrix, then $\\tilde{A} S \\tilde{A}^*$ is a Hermitian matrix, where $\\tilde{A}^*$ is the Hermitian conjugate of $\\tilde{A}$ .",
"What is more, if $\\det \\tilde{A} = 1$ , then $\\det \\tilde{A} S \\tilde{A}^* = \\det S$ .",
"It follows from the above that to any matrix $\\tilde{A}$ in $\\mathbf {SL}(2,\\mathbb {C})$ corresponds an orthogonal matrix $A$ in $\\mathbf {O}(3,1,\\mathbb {R})$ .",
"Therefore we have a group homomorphism $\\tilde{A} \\rightarrow A$ , which can be written as follows: $\\hat{S} = \\tilde{A} S \\tilde{A}^* \\ \\rightarrow \\ \\hat{\\mathrm {x}}= A\\mathrm {x}\\;.$ The so obtained homomorphism from $\\mathbf {SL}(2,\\mathbb {C})$ into $\\mathbf {O}(3,1,\\mathbb {R})$ is called spinor map.",
"It is proved in the theory of spinors that the kernel of the spinor map consists of two elements: $\\pm I$ , where $I$ is the unitary matrix.",
"Further, the image of this map is the connected component of the unity element in $\\mathbf {O}(3,1,\\mathbb {R})$ , which is denoted in a standard way by $\\mathbf {SO}^+(3,1,\\mathbb {R})$ .",
"Briefly speaking, this is the group of the matrices determining those transformations in $\\mathbb {R}^4_1$ , preserving not only the orientation of $\\mathbb {R}^4_1$ , but also preserve both: the direction of time and the orientation of the three-dimensional Euclidean subspace of $\\mathbb {R}^4_1$ .",
"These transformations of $\\mathbb {R}^4_1$ are called orthochronous transformations.",
"The type of the kernel and the image of the spinor map (REF ) implies that the spinor map induces the following group isomorphism: $\\mathbf {SL}(2,\\mathbb {C})/\\lbrace \\pm I\\rbrace \\ \\cong \\ \\mathbf {SO}^+(3,1,\\mathbb {R})\\;.$ This means that $\\mathbf {SL}(2,\\mathbb {C})$ appears to be a two-sheeted covering of $\\mathbf {SO}^+(3,1,\\mathbb {R})$ and hence we can identify it with the spin group $\\mathbf {Spin}(3,1)$ of $\\mathbf {SO}^+(3,1,\\mathbb {R})$ .",
"In other words, (REF ) gives a representation of $\\mathbf {Spin}(3,1)$ as $\\mathbf {SL}(2,\\mathbb {C})$ , which is called the spinor representation.",
"Note that the group $\\mathbf {SL}(2,\\mathbb {C})$ is connected and simply connected, and consequently it follows from the isomorphisms (REF ) and (REF ) that $\\mathbf {SL}(2,\\mathbb {C})$ also appears to be universal covering group for $\\mathbf {SO}^+(3,1,\\mathbb {R})$ .",
"Now, let $\\mathrm {x}$ be an arbitrary complex vector in $\\mathbb {C}^4$ .",
"Up to now, considering different correspondences, we restricted $\\mathrm {x}$ to be a real vector in $\\mathbb {R}^4_1$ .",
"It is an easy verification that the relations (REF ) and (REF ) are linear with respect to $\\mathrm {x}$ .",
"Therefore, they are also valid when $\\mathrm {x}$ is an arbitrary complex vector in $\\mathbb {C}^4$ .",
"The only difference is that $S$ can be an arbitrary (not necessarily Hermitian) complex matrix.",
"Under a motion of the complex vector $\\mathrm {x}$ with a matrix in $\\mathbf {SO}^+(3,1,\\mathbb {R})$ , the matrix $S$ is transformed in the same way, as it is described in (REF ).",
"Let us return to minimal space-like surfaces.",
"With the help of the above formulas we shall find how the functions giving the Weierstrass representation of a minimal space-like surface are transformed under a motion of the surface in $\\mathbb {R}^4_1$ .",
"First, let the minimal space-like surface $({\\mathcal {M}},\\mathrm {x})$ be parameterized by arbitrary isothermal coordinates.",
"If the surface $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ is obtained from $({\\mathcal {M}},\\mathrm {x})$ by means of orthochronous transformation in $\\mathbb {R}^4_1$ , then we have $\\hat{\\mathrm {x}}(t)=A\\mathrm {x}(t)+\\mathrm {b}$ , where $A \\in \\mathbf {SO}^+(3,1,\\mathbb {R})$ and $\\mathrm {b}\\in \\mathbb {R}^4_1$ .",
"The function $\\Phi $ defined by (REF ), as we noted by the formula (REF ), is transformed by: $\\hat{\\Phi }=A\\Phi $ .",
"Next we introduce the complex matrix $S_{\\Phi }$ , which is obtained by $\\Phi $ according to the rule (REF ): $S_{\\Phi }=\\left(\\begin{array}{rr}\\phi _3 + \\phi _4 & \\mathrm {i}\\phi _1 + \\phi _2\\\\-\\mathrm {i}\\phi _1 + \\phi _2 & -\\phi _3 + \\phi _4\\end{array}\\right)\\;.$ We denote by $\\tilde{A}$ any of the two matrices in $\\mathbf {SL}(2,\\mathbb {C})$ , corresponding to $A$ by means of the homomorphism (REF ).",
"If $S_{\\hat{\\Phi }}$ is the matrix obtained from $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ , according to (REF ) it is related to $S_{\\Phi }$ as follows: $S_{\\hat{\\Phi }} = \\tilde{A} S_{\\Phi } \\tilde{A}^* \\;.$ Now, suppose that $\\mathcal {M}$ is given by a Weierstrass representation of the type (REF ).",
"By direct calculations we find: $\\begin{array}{l}\\phantom{-}\\mathrm {i}\\phi _1+\\phi _2= -f(g_1 g_2+1)+f(g_1 g_2-1)= -2f,\\\\-\\mathrm {i}\\phi _1+\\phi _2=\\phantom{-}f(g_1 g_2+1)+f(g_1 g_2-1)=\\phantom{-}2fg_1g_2,\\\\\\phantom{-}\\phi _3+\\phi _4=\\phantom{-}f(g_1+g_2)+f(g_1-g_2)=\\phantom{-}2fg_1,\\\\-\\phi _3+\\phi _4= -f(g_1+g_2)+f(g_1-g_2)= -2fg_2\\;.\\end{array}$ Consequently, the matrix $S_{\\Phi }$ is represented by means of $f$ , $g_1$ and $g_2$ as follows: $S_{\\Phi }=\\left(\\begin{array}{ll}2fg_1 & -2f \\\\2fg_1g_2 & -2fg_2\\end{array}\\right)\\;.$ Denoting the elements of $S_{\\Phi }$ by $s_{ij}$ , then we have the following expressions for $f$ , $g_1$ and $g_2$ : $f=-\\frac{1}{2}s_{12}, \\quad g_1=-\\frac{s_{11}}{s_{12}}, \\quad g_2=\\frac{s_{22}}{s_{12}}\\;.$ Since $S_{\\Phi }$ is transformed according the rule (REF ), then via (REF ) we shall find the transfor-mation formulas for the functions $f$ , $g_1$ and $g_2$ .",
"For that purpose we denote the elements of $\\tilde{A}$ in the following way: $\\tilde{A}=\\left(\\begin{array}{rr}\\bar{a} & -\\bar{b} \\\\-\\bar{c} & \\bar{d}\\end{array}\\right)\\,;\\quad a,b,c,d\\in \\mathbb {C}\\,;\\quad ad-bc=1\\;.$ After multiplying the matrices in (REF ) and simplifying, we get: $S_{\\hat{\\Phi }}=\\left(\\begin{array}{ll}2f(a g_1+b)( - \\bar{b} g_2 + \\bar{a}) & 2f(c g_1+d)(\\phantom{-}\\bar{b} g_2 - \\bar{a}) \\\\2f(a g_1+b)(\\phantom{-}\\bar{d} g_2 - \\bar{c}) & 2f(c g_1+d)( - \\bar{d} g_2 + \\bar{c})\\end{array}\\right)\\;.$ Applying (REF ) to $\\hat{f}$ , $\\hat{g}_1$ and $\\hat{g}_2$ , we find the transformation formulas of the functions in the Weierstrass representation of the type (REF ) under an orthochronous transformation of $\\mathcal {M}$ in $\\mathbb {R}^4_1$ : $\\begin{array}{l}\\hat{f} = f(cg_1+d)(-\\bar{b} g_2 + \\bar{a})\\,;\\\\[0.7ex]\\hat{g}_1 = \\displaystyle \\frac{ag_1+b}{cg_1+d}\\,; \\quad \\hat{g}_2 = \\displaystyle \\frac{\\phantom{-}\\bar{d} g_2 - \\bar{c}}{-\\bar{b} g_2 + \\bar{a}}\\;.\\end{array}$ Now, let us consider the inverse statement.",
"Suppose that $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ are two minimal space-like surfaces, given by the Weierstrass representation of the type (REF ) related by means of (REF ).",
"We shall show that they can be obtained one from the other by an orthochronous transformation in $\\mathbb {R}^4_1$ .",
"For that purpose, we introduce $\\tilde{A}$ by means of (REF ).",
"Let $A$ be the corresponding to $\\tilde{A}$ matrix under the homomorphism (REF ).",
"With the help of $A$ we obtain a third surface $({\\hat{\\hat{\\mathcal {M}}}},\\hat{\\hat{\\mathrm {x}}})$ given by the formula: $\\hat{\\hat{\\mathrm {x}}} = A\\mathrm {x}$ .",
"We proved that $\\hat{\\hat{\\mathcal {M}}}$ has a Weierstrass representation with functions also satisfying (REF ).",
"Therefore, $\\hat{\\mathcal {M}}$ and $\\hat{\\hat{\\mathcal {M}}}$ are generated by one and the same functions by means of formulas (REF ) and consequently they are obtained one from the other by a translation in $\\mathbb {R}^4_1$ .",
"Since $\\hat{\\hat{\\mathcal {M}}}$ is obtained from $\\mathcal {M}$ by an orthochronous transformation, then $\\hat{\\mathcal {M}}$ is also obtained from $\\mathcal {M}$ by orthochronous transformation.",
"Summarizing we obtain the following statement: Theorem 12.1 Let $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ be two minimal space-like surfaces in $\\mathbb {R}^4_1$ , given by Weierstrass representations of the type (REF ).",
"The following conditions are equivalent: $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ are related by an orthochronous transformation in $\\mathbb {R}^4_1$ of the type: $\\hat{\\mathrm {x}}(t)=A\\mathrm {x}(t)+\\mathrm {b}$ , where $A \\in \\mathbf {SO}^+(3,1,\\mathbb {R})$ and $\\mathrm {b}\\in \\mathbb {R}^4_1$ .",
"The functions in the Weierstrass representations of $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ are related by equalities (REF ), where $a,b,c,d\\in \\mathbb {C}$ , $ad-bc=1$ .",
"Up to now we considered only the case of a motion from the connected component of the identity in $\\mathbf {O}(3,1,\\mathbb {R})$ .",
"Next we show that any of the three remaining cases can be reduced to the considered one.",
"Let us consider the case of a transformation, which is not orthochronous.",
"Such a concrete transformation can be obtained by a change of the signs of the four coordinates: $\\hat{\\mathrm {x}}(t)=-\\mathrm {x}(t)$ .",
"This implies the change of the sign of the function $f$ in the Weierstrass representation (REF ), while the functions $g_1$ and $g_2$ remain the same.",
"Any non-orthochronous transformation can be obtained as a composition of this concrete transformation and an orthochronous transformation in $\\mathbb {R}^4_1$ .",
"Therefore, if two minimal space-like surfaces are obtained one from the other by a non-orthochronous transformation, then the functions in the Weierstrass representation are related by formulas, which are similar to (REF ) with the only difference in the sign of the formula for $\\hat{f}$ .",
"Further, we consider the case of a non-orthochronous improper transformation.",
"An example of such a transformation is the symmetry with respect to the hyperplane $x_4=0$ which is given by the change of the sign of $x_4$ .",
"This implies a change of the places of both functions $g_1$ and $g_2$ in the Weierstrass representation, while the function $f$ remains the same.",
"Any non-orthochronous improper transformation can be obtained as a composition of this symmetry and an orthochronous transformation.",
"Therefore the functions in the Weierstrass representation are changed similarly to (REF ), but this time the formulas for $\\hat{g}_1$ and $\\hat{g}_2$ change their places, while the formula for $\\hat{f}$ is the same.",
"Finally, we consider the case of an orthochronous improper transformation.",
"Such a transformation can be obtained as a combination of the last two cases.",
"Therefore, the transformation formulas for the functions in the Weierstrass representation are obtained from (REF ), by the change of the sign of $\\hat{f}$ and the change of the places of the formulas for $\\hat{g}_1$ and $\\hat{g}_2$ .",
"Now, let $\\hat{\\mathcal {M}}$ and $\\mathcal {M}$ be two minimal space-like surfaces, parameterized by canonical coordinates and the surface $\\hat{\\mathcal {M}}$ is obtained from $\\mathcal {M}$ by a motion in $\\mathbb {R}^4_1$ .",
"Suppose that $\\mathcal {M}$ is given by a canonical Weierstrass representation of the type (REF ).",
"Since $\\hat{\\mathrm {x}}=A\\mathrm {x}+\\mathrm {b}$ implies that $\\hat{\\Phi }^{\\prime }=A\\Phi ^{\\prime }$ , then we have $\\hat{\\Phi }^{\\prime }{}^2=\\Phi ^{\\prime 2}=1$ .",
"Consequently the canonical coordinates of $\\mathcal {M}$ appear to be also canonical coordinates for $\\hat{\\mathcal {M}}$ .",
"Taking into account that the canonical Weierstrass representation (REF ) is a special case of the representation (REF ), then the pair $(g_1,g_2)$ is transformed by the formulas (REF ).",
"Note that these formulas can be applied in the cases of an orthochronous or a non-orthochronous transformation.",
"This is so because the two cases differ from each other only the formula for the function $f$ .",
"Further, we see that it only remain the linear fractional functions from (REF ), which allowas us to replace the condition $ad-bc=1$ with the more general condition $ad-bc\\ne 0$ .",
"This is possible, because the linear fractional function does not change if its matrix is multiplied by a non zero factor.",
"Summarizing the above remarks, in view of Theorem REF we obtain the following statement.",
"Theorem 12.2 Let $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ be two minimal space-like surfaces of general type, given by the canonical Weierstrass representation of the type (REF ).",
"The following conditions are equivalent: $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ are related by a transformation in $\\mathbb {R}^4_1$ of the type: $\\hat{\\mathrm {x}}(t)=A\\mathrm {x}(t)+\\mathrm {b}$ , where $A \\in \\mathbf {SO}(3,1,\\mathbb {R})$ and $\\mathrm {b}\\in \\mathbb {R}^4_1$ .",
"The functions in the Weierstrass representations of $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ are related by the following equalities: $\\hat{g}_1 = \\displaystyle \\frac{ag_1+b}{cg_1+d}\\,; \\quad \\hat{g}_2 = \\displaystyle \\frac{\\phantom{-}\\bar{d} g_2 - \\bar{c}}{-\\bar{b} g_2 + \\bar{a}}\\;,$ where $a,b,c,d\\in \\mathbb {C}$ , $ad-bc\\ne 0$ .",
"If $({\\hat{\\mathcal {M}}},\\hat{\\mathrm {x}})$ and $({\\mathcal {M}},\\mathrm {x})$ are related by an improper transformation in $\\mathbb {R}^4_1$ , then in the formulas (REF ) one has to change the places of $\\hat{g}_1$ and $\\hat{g}_2$ .",
"In the end, we write down (REF ) in a form, which is useful for applications.",
"For that purpose, let us denote by $Gz$ , where $G\\in \\mathbf {GL}(2,\\mathbb {C})$ and $z\\in \\mathbb {C}$ , the standard action of the group $\\mathbf {GL}(2,\\mathbb {C})$ in the complex plane by means of linear fractional transformations.",
"Denoting by $B$ the matrix of the linear fractional function for $\\hat{g}_1$ in (REF ), by direct computations we see that the matrix of $\\hat{g}_2$ is up to a factor the matrix ${B^*}^{-1}$ .",
"Hence, the formulas (REF ) can be written briefly as follows: $\\hat{g}_1 = Bg_1\\,; \\quad \\hat{g}_2 = {B^*}^{-1} g_2\\,; \\quad B\\in \\mathbf {GL}(2,\\mathbb {C})\\;.$ Finally we give a natural approach to the family of the minimal space-like surfaces of general type, associated with a given one.",
"Let $(g_1(t),g_2(t))$ be a pair of holomorphic functions defined in a disc $\\mathcal {D}$ , centered at $(0, 0)$ in the parametric plane $\\mathbb {C}$ .",
"Consider the minimal space-like surface $({\\mathcal {M}},\\mathrm {x})$ , generated by the pair $(g_1(t),g_2(t))$ by means of (REF ).",
"For any complex number $a, \\, |a|=1$ we introduce the pair of holomorphic functions $(\\tilde{g}_1(s),\\tilde{g}_2(s))=(g_1(as),g_2(as)); \\quad s \\in \\mathcal {D}$ and denote by ${\\tilde{\\mathcal {M}}}$ the minimal space-like surface, generated by the pair $(\\tilde{g}_1,\\tilde{g}_2)$ by means of (REF ).",
"Further we denote by $\\Phi $ , $\\Psi $ and $\\tilde{\\Phi }$ , $\\tilde{\\Psi }$ the corresponding vector holomorphic functions on ${\\mathcal {M}}$ and ${\\tilde{\\mathcal {M}}}$ .",
"Replacing (REF ) into the representation (REF ), we get the following relation between $\\tilde{\\Phi }$ and $\\Phi $ : $\\tilde{\\Phi }(s)=\\displaystyle \\frac{1}{a}\\:\\Phi (as).$ Since $\\tilde{\\Phi }^{\\prime 2}=1$ , then $s=\\frac{t}{a}$ determines canonical coordinates on $\\tilde{\\mathcal {M}}$ .",
"After an integration we obtain the corresponding formula for $\\tilde{\\Psi }$ : $\\tilde{\\Psi }(s)=\\displaystyle \\frac{1}{a^2}\\:\\Psi (as).$ Denoting $a=e^{\\mathrm {i}\\frac{\\varphi }{2}}$ and $\\tilde{\\mathcal {M}}=\\mathcal {M}_{\\varphi }$ , we have: Any minimal space-like surface of general type $\\mathcal {M}: \\mathrm {x}=\\mathrm {x}(t); \\; t\\in \\mathcal {D}$ generates a one-parameter family $\\lbrace \\mathcal {M}_{\\varphi }\\rbrace $ of minimal space-like surfaces, given by the formula $\\mathcal {M}_\\varphi : \\; \\mathrm {x}_\\varphi (s) = \\mathop {\\rm Re}\\nolimits (e^{-\\mathrm {i}\\varphi }\\:\\Psi (e^{\\mathrm {i}\\frac{\\varphi }{2}}\\, s)); \\quad \\varphi \\in [0, \\frac{\\pi }{2}], \\quad s \\in \\mathcal {D}, $ where $s=e^{-\\mathrm {i}\\frac{\\varphi }{2}}t$ determines canonical coordinates on $\\mathcal {M}_\\varphi $ .",
"The surfaces of the family $\\lbrace \\mathcal {M}_\\varphi \\rbrace $ are said to be associated with the given surface $\\mathcal {M}$ .",
"Since the generating holomorphic functions of the family of the associated surfaces are given by (REF ), taking into account formulas (REF ) and (REF ), we observe that the transformation $\\mathcal {M}\\, \\rightarrow \\,\\mathcal {M}_{\\varphi }$ given by $s \\, \\rightarrow \\, e^{\\mathrm {i}\\frac{\\varphi }{2}}s$ preserves $E$ , $K$ and $\\varkappa $ , i.e.",
"it is a special isometry between $\\mathcal {M}$ and $\\mathcal {M}_{\\varphi }$ preserving the normal curvature $\\varkappa $.",
"Denote by ${\\bar{\\mathcal {M}}}$ the minimal space-like surface conjugate to ${\\mathcal {M}}$ , which is given by the formula $\\mathrm {y}=\\mathop {\\rm Im}\\nolimits (\\Psi ) = \\mathop {\\rm Re}\\nolimits (-\\mathrm {i}\\Psi )$ .",
"Then ${\\bar{\\mathcal {M}}}$ is the associated with $\\mathcal {M}$ surface $\\mathcal {M}_{\\varphi },\\;\\varphi = \\frac{\\pi }{2}$ .",
"Thus we have: If the minimal space-like surface $\\mathcal {M}$ , parameterized by canonical coordinates, is generated by the pair $(g_1(t),g_2(t))$ , then the minimal space-like surface ${\\bar{\\mathcal {M}}}$ , conjugate to $\\mathcal {M}$ , is generated by the pair $(g_1(e^{\\mathrm {i}\\frac{\\pi }{4}}\\, s),g_2(e^{\\mathrm {i}\\frac{\\pi }{4}}\\, s))$ with canonical parameter $s=e^{-\\mathrm {i}\\frac{\\pi }{4}}\\, t$ .",
"If $t$ determines canonical coordinates of the first type on ${\\mathcal {M}}$ , then $e^{-\\mathrm {i}\\frac{\\pi }{4}}\\, t$ gives canonical coordinates of the second type on $\\mathcal {M}$ and vice versa.",
"Hence: The canonical coordinates of the second type on $\\mathcal {M}$ are canonical coordinates of the first type on $\\bar{\\mathcal {M}}$ and the canonical coordinates of the first type on ${\\mathcal {M}}$ are canonical coordinates of the second type on ${\\bar{\\mathcal {M}}}$ .",
"Acknowledgments: The first author is partially supported by the National Science Fund, Ministry of Education and Science of Bulgaria under contract DFNI-I 02/14."
]
] | 1612.05504 |
[
[
"Design and Construction of the MicroBooNE Detector"
],
[
"Abstract This paper describes the design and construction of the MicroBooNE liquid argon time projection chamber and associated systems.",
"MicroBooNE is the first phase of the Short Baseline Neutrino program, located at Fermilab, and will utilize the capabilities of liquid argon detectors to examine a rich assortment of physics topics.",
"In this document details of design specifications, assembly procedures, and acceptance tests are reported."
],
[
"Introduction and Physics Motivation",
"The Micro Booster Neutrino Experiment (MicroBooNE) employes a large ($\\sim $ 100 tonnes) Liquid Argon Time Projection Chamber (LArTPC) detector designed for precision neutrino physics measurements.",
"MicroBooNE is the latest among a family of detectors that exploit the potential of liquified noble gases as the detection medium for neutrino interactions.",
"These detectors combine the advantages of high spatial resolution and calorimetry for excellent particle identification with the potential to scale to very large volumes.",
"Large calorimeters using cryogenic noble liquids combined with active components were recognized in the 1970s as having use for particle physics applications [1].",
"Specifically, much of the liquid argon based technology was developed within the ICARUS program [2], [3], [4] culminating in the realization of the ICARUS T600 detector [5].",
"On a much smaller scale than the ICARUS detector, the ArgoNeuT (Argon Neutrino Test) experiment operated a $\\sim $ 0.25 tonne LArTPC from 2009-2010 in the NuMI neutrino beam at Fermilab.",
"ArgoNeuT performed a series of detailed studies on the interaction of medium-energy neutrinos [6] producing the first published neutrino cross section measurements on argon [7], [8], [9].",
"Next generation LArTPCs for the Short Baseline Neutrino Detector (SBND) experiment and the Deep Underground Neutrino Experiment (DUNE) are now being designed and constructed.",
"MicroBooNE 's principal physics goal is to address short baseline neutrino oscillations, primarily the MiniBooNE observation of an excess of electron-like events at low energy [10], at the Fermi National Accelerator Laboratory (Fermilab).",
"MicroBooNE will be exposed to the 0.5-2 GeV on-axis Booster Neutrino Beam (BNB) at a $\\sim $ 500 m baseline, the same as was employed for MiniBoonE.",
"The MicroBooNE experiment is exploiting the LArTPC technology because of its superior capability for separation of signal electrons from the background of photon conversions.",
"While the mass of MicroBooNE is significantly less than the mass of MiniBooNE, this superior discrimination is expected to address the MiniBooNE result at the 5$\\sigma $ level.",
"In addition to MicroBooNE's signature oscillation analyses, a suite of precision cross-section measurements will be performed, critical both for future LArTPC oscillation experiments and for understanding neutrino interactions in general.",
"In the BNB, multiple interaction processes (quasi-elastic, resonances, deep inelastic scattering) are possible, and complicated nuclear effects in neutrino interactions on argon result in a variety of final states.",
"These can range from the emission of several nucleons to more complex topologies with multiple pions, all in addition to the leading lepton in charged-current events.",
"The LArTPC technology employed by MicroBooNE is particularly well suited for complicated topologies because of its excellent particle identification capability and calorimetric energy reconstruction down to very low detection thresholds.",
"MicroBooNE's physics program also encompasses searches for supernova and proton decay.",
"The detector is capable of recording neutrinos from a galactic supernova which would result in $\\sim $ 30 charged current neutrino interactions in MicroBooNE's active volume.",
"The detector will measure proton decay-like signatures and backgrounds and develop the analysis for this search; though its target mass is insufficiently large to enable a competitive sensitivity, the analysis will provide an important proof-of-principle for future searches in more massive detectors.",
"MicroBooNE began operations in late 2015 for an initial anticipated $\\sim $ 3 year run.",
"In 2018, MicroBooNE will continue operations as part of an expanded Short Baseline Neutrino (SBN) program [11] at Fermilab that includes continued operation of MicroBooNE (at 470 m) along with the SBND (at 110 m) and ICARUS (at 600 m) detectors.",
"The SBND and ICARUS experimental halls and detectors are presently under construction.",
"MicroBooNE will definitively address whether or not the MiniBooNE low energy excess in neutrino mode is due to electrons or photons in its initial run.",
"SBND will look for this low energy excess at the near location and ICARUS, with its larger mass, will enable the three detector program to cover the entire LSND-allowed region in neutrino parameter space with 5$\\sigma $ sensitivity in the $\\nu _e$ appearance channel.",
"This document describes the design, construction, and technical details of the MicroBooNE detector.",
"Section gives a brief review of the LArTPC technique and its implementation in MicroBooNE.",
"Section describes the cryogenic and purification systems which are required for maintaining a stable volume of highly purified liquid argon.",
"The LArTPC described in section is the centerpiece of the experiment, providing fine-grained images of neutrino interactions.",
"A light collection system, described in section , provides timing information, used primarily for triggering beam events, from the prompt scintillation light that is produced in the detector volume.",
"Signals from the light collection system and from the LArTPC are amplified, sampled, and recorded by a custom-designed electronic and readout system, as described in section .",
"Section describes the auxiliary instrumentation that monitor and control the detector and all of its associated systems, as well as provide an electrically quiet environment for the experiment to operate.",
"Finally, one of the main calibration sources for the experiment is an ultraviolet laser system, described in section , that provides the capability to map out geometric track distortions, as induced, for example, by space charge.",
"A cosmic ray tagger system, under construction at the time of the writing of this paper, will surround the detector to improve cosmic ray identification and rejection.",
"This system will be described in a subsequent publication.",
"More information on the LArTPC technology can be found in existing reviews (see, e.g., [12] and references therein)."
],
[
"Experiment Overview",
"The MicroBooNE detector at Fermilab in Batavia, Illinois is sited in the Liquid Argon Test Facility (LArTF) on axis in the BNB, 470 m downstream from the neutrino production target.",
"The BNB delivers a beam of predominantly muon neutrinos produced primarily from pion decays, with energies peaking at 700 MeV [13].",
"MicroBooNE is also exposed to an off-axis component of the NuMI beam [14] produced from pion and kaon decays with average neutrino energies of about 0.25 GeV and 2 GeV respectively.",
"MicroBooNE is located about 600 m downstream from the NuMI neutrino production target.",
"The characteristics of the BNB beamline are well measured and understood from many years of data taking and analysis from the MiniBooNE experiment [13], which operated directly downstream of the MicroBooNE location.",
"Figure REF shows the arrangement of MicroBooNE with respect to the BNB beamline at Fermilab.",
"The physics program of MicroBooNE will utilize both BNB and NuMI samples.",
"MicroBooNE will also collect data that is out-of-time with either beam, which will be useful for developing non-accelerator neutrino-based analyses (e.g.",
"proton decay and supernovae burst neutrino searches) relevant for next-generation detectors.",
"Figure: Aerial diagram showing location of MicroBooNE along the BNB (orange dashed line) at Fermilab."
],
[
"The MicroBooNE LArTPC ",
"Charged particles traversing a volume of liquid argon leave trails of ionization electrons in their wake and also create prompt vacuum ultraviolet (VUV) scintillation photons.",
"In a LArTPC, the liquid argon is highly purified so that the ionization trails can be transported with minimal attenuation over distances of the order of meters [15] under the influence of a uniform electric field in the detector volume, until they reach sense planes located along one side of the active volume.",
"The electric field is created by introducing voltage onto a cathode plane and gradually stepping that voltage down in magnitude across a field cage, which is formed from a series of equipotential rings surrounding the drift volume.",
"Non-uniformities in the electric field, diffusion, recombination, and space charge effects modify the tracks as they are transported.",
"Calibration of these effects is critical to reconstruction of the initial ionization trails.",
"The anode plane is arranged parallel to the cathode plane, and in MicroBooNE, parallel to the beam direction.",
"There are three planes comprised of sense wires with a characteristic pitch, held at a predetermined bias voltage, that continuously sense the signals induced by the ionization electrons drifting towards them [16].",
"The electrostatic potentials of the sequence of anode planes allow ionization electrons to pass undisturbed by the first two planes before ultimately ending their trajectory on a wire in the last plane.",
"The drifting ionization thus induces signals on the first planes (referred to as induction planes) and directly contributes to the signals in the final plane (referred to as the collection plane).",
"Figure REF depicts the arrangement of the MicroBooNE LArTPC and its operational principle.",
"Figure: Operational principle of the MicroBooNE LArTPC.The charged particle trajectory is reconstructed using the known positions of the anode plane wires and the recorded drift time of the ionization.",
"The drift time is the difference between the arrival times of ionization signals on the wires and the time the interaction took place in the detector ($t_0$ ) which is provided by an accelerator clock synchronized to the beam (BNB or NuMI) or from a trigger provided by the light collection system.",
"The characteristics of the waveforms observed by each wire provide a measure of the energy deposition of the traversing particles near that wire, and, when taken as a whole for each contained particle's trajectory, allow for determination of momentum and particle identity.",
"The scintillation photons are detected by a light collection system that is immersed in the liquid argon and faces into the detector volume.",
"This system provides signals that can establish the event $t_0$ and supplies trigger information to an electronic readout system.",
"The light collection system signals are vital in distinguishing detector activity that is in-time with the beam (and therefore possibly originating from beam interactions) from activity which is out-of-time (and therefore probably not associated with the beam), benefiting triggering and event reconstruction.",
"Information on the $z$ and $y$ position of an interaction can also be inferred from the light system, further aiding in the reconstruction.",
"Liquid argon as a target for neutrinos is attractive due to its density, allowing a more compact detector with a substantial boost in event rate over a comparable detector using less dense media.",
"A tradeoff to this aspect is the fact that the complicated structure of the argon nucleus (relative to hydrogen or helium, for example) will introduce nuclear effects that the data analysis must take into account.",
"The cryogenic temperatures at which the noble elements are in the liquid phase also introduces the need for additional design considerations to ensure stable and safe operations.",
"Table REF lists some of the properties of liquid argon that are salient for LArTPC design.",
"The noble liquids produce copious numbers of UV photons for every traversing charged particle, as well as large amounts of ionization.",
"The electrons from this ionization can be drifted for distances of meters under a modest electric field ($\\sim $ 500 V/cm).",
"Finally, building LArTPCs on increasingly large scales for neutrino detection becomes economically possible, given the abundance (1% of atmosphere) and low cost of argon and the convenient feature that it can be maintained as a liquid through refrigeration using liquid nitrogen which is plentiful and cheap.",
"Table: Selected properties of liquid argon.The successful implementation of the LArTPC technique depends critically on a number of factors.",
"The liquid argon must be purified of any electronegative contaminants, such as water or oxygen, to accommodate the very long drift path of ionization through a MicroBooNE-sized LArTPC without significant charge loss.",
"The signals that the ionization electrons create on the anode wires are very small, requiring low-noise electronics to discriminate between signal pulses and background noise.",
"The MicroBooNE collaboration has designed and constructed an experiment that addresses these considerations, providing critical technological development upon which the next generation of LArTPCs may capitalize."
],
[
"MicroBooNE LArTPC Implementation",
"MicroBooNE's LArTPC active volume, which is defined as the volume immediately within the confines of the LArTPC field cage, is a rectangular liquid argon volume with dimensions as given in table REF .",
"This is the maximum volume that can be used for physics analyses.",
"The cathode and the anode planes define the beam-left and beam-right sides of the active volume.",
"The end of the LArTPC that the beam first encounters is referred to as the “upstream” end, while the opposite end is referred to as “downstream.” Anode plane-to-plane spacing is 3 mm, and each plane has 3 mm wire pitch.",
"The induction plane wires are oriented at $\\pm 60^{\\circ }$ relative to vertical, and the collection plane wires are oriented vertically.",
"Field cage loops are employed to maintain uniformity of the electric field across the entire width of the detector, and these loops also act to define the top, bottom, upstream, and downstream sides of the active volume.",
"MicroBooNE uses a right-handed Cartesian coordinate system, with the origin defined to be located on the upstream face of the LArTPC, centered halfway up the vertical height of the active volume and horizontally centered on the anode plane closest to the cathode (the innermost anode plane).",
"In this system, $x$ ranges from 0.0 m at the innermost anode plane to $+2.6$ m at the cathode, $y$ ranges from $-1.15$ m on the bottom of the active volume to $+1.15$ m at the top of the active volume, and $z$ ranges from 0.0 m at the upstream end of the active volume to $+10.4$ m at the downstream end.",
"The light collection system, an array of photomultiplier tubes (PMTs) and lightguide paddles, is located directly behind the anode planes on beam-right, facing the detector volume through the anode planes.",
"The LArTPC and light collection system are immersed in liquid argon contained within a single-walled cryostat with a 170 tonne capacity.",
"Analog front-end electronics mounted directly on the LArTPC amplify the signals on the wires; these signals are then passed out of the cryostat for further processing and storage on disk.",
"Table REF lists the primary detector design parameters of MicroBooNE, and figure REF shows a schematic of the cross section of the detector.",
"Details of these design parameters and construction of all detector systems will be provided in the subsequent sections.",
"Figure: Schematic of the cross section of the MicroBooNE LArTPC.",
"In this view, the beam would be directed out of the page (in the zz direction).Table: Primary detector design parameters for MicroBooNE."
],
[
"Cryogenic System",
"The use of large quantities of highly-purified liquid argon as a detector medium in MicroBooNE requires a sophisticated cryogenic infrastructure that can maintain stable operations for many years with minimal downtime.",
"Not only must the purity of the liquid argon be maintained, but the pressure and temperature gradients within the LArTPC active volume must be tightly controlled as the drift velocity of electrons is dependent on these quantities.",
"A customized cryogenic system that serves these purposes has been built, and the requirements for this system are shown in table REF .",
"Table: Primary design requirements for MicroBooNE cryogenic and purification systems.The MicroBooNE cryogenic system is represented in figure REF .",
"The central component of the system is a cryostat that houses the complete LArTPC and light-collection detector systems.",
"The cryostat is supported by three major subsystems: the argon purification system, the nitrogen refrigeration system, and the controls and monitoring system.",
"These systems each represent the next generation of LArTPC cryogenic system after the Liquid Argon Purity Demonstrator (LAPD) [25] and make considerable use of the expertise gained during the design and implementation of that apparatus.",
"Figure: A rendering showing the MicroBooNE cryostat and cryogenic system, and the platform for the electronics racks as installed in LArTF."
],
[
"Cryostat Design Overview",
"Three major components make up the MicroBooNE cryostat: a type 304 stainless steel vessel to contain the liquid argon and all the active detector elements, front and rear supports to carry the weight of the fully loaded cryostat, and foam insulation covering the cryostat outer surfaces.",
"The foam insulation serves to reduce heat input from the ambient environment to a sufficiently low level to prevent large temperature gradients and boiling of the liquid argon.",
"The cryostat and the cryogenic systems are designed to achieve the requisite high purity of liquid argon needed to allow ionization electrons to drift to the anode wires with low probability of capture, and the high degree of thermal homogeneity needed to avoid introducing non-constant drift velocities for the ionization electrons.",
"Finally, the outer diameter of the vessel is designed to be the maximum standard size for highway transport.",
"The cryostat is constructed to the American Society of Mechanical Engineers (ASME) boiler code requirements [26] and features a single-walled construction, cylindrical shape, and domed caps closing each end, as shown in figure REF .",
"The cryostat is 12.2 m in overall length, with an inner diameter of 3.81 m, and a wall thickness of 11.1 mm.",
"When empty the cryostat weighs $\\sim $ 17,000 kg.",
"One end was removed for installation of the active detectors, welded back in place upon completion of that task, and then re-certified to the ASME code requirements.",
"Ionization electrons must not be significantly attenuated, via attachment to electronegative contaminants in the liquid argon while they drift up to 2.5 m across the active volume.",
"This dictates that the argon be kept free of electronegative contaminants to the level of 100 parts-per-trillion (ppt) oxygen-equivalent (O$_2$ equivalent), where the choice of O$_2$ equivalent units implies other polar molecules besides oxygen may be present, but the attachment rate constant for oxygen is used in calculations .",
"The cryostat is designed to minimize outgassing (desorption) and to avoid leakage and diffusion of air into the system.",
"This requirement imposes strict quality assurance demands on all welds for penetrations into the cryostat and on cleaning and handling procedures for the finished vessel.",
"Achieving the required level of purity is accomplished with a purification system, described in section REF , that removes electronegative contaminants from the argon during the initial fill and those introduced over time by leaks and outgassing of system components.",
"The electron drift velocity ($v_{d} = 1600$ m/s at an electric field of 500 V/cm, with a liquid argon temperature dependence $\\Delta v_{d}/v_{d} = -0.019\\Delta T$ ) must remain constant in magnitude and direction throughout the active liquid argon volume to avoid distortion of the mapping of drift time into the position along the drift ($\\hat{x}$ ) direction.",
"This requirement limits the allowable temperature variations of the liquid argon to less than 0.1 K and the laminar and turbulent flow rate of liquid argon to less than 1 m/s.",
"These requirements limit fractional errors in velocity, and therefore in the drift-coordinate determination, to be less than 0.1%.",
"The constraints on constancy of drift velocity affect the design by imposing limits on the acceptable heat flux through the insulation.",
"Figure: MicroBooNE cryostat with nozzle penetrations labeled.Upon installation of the sealed cryostat in its final location at the LArTF, 41 cm of spray-on, closed cell, polyurethane insulation was applied to the exterior of the cryostat, as shown in figure REF .",
"At the LArTF, to avoid ground loops that could interfere with the LArTPC signals, the cryostat vessel is grounded in only one place, allowing it to act as a Faraday cage.",
"This grounding scheme is explained further in section REF .",
"Figure: Photographs of the cryostat after application of exterior foam insulation.The vessel surface has 34 nozzle penetrations for cryogenic and electrical services, detailed in table REF .",
"All nozzles are sealed with feedthroughs, flanges, or pipes that are suitable for operation at the nominal pressure and temperature of the cryostat.",
"Table: List of nozzle penetrations in the MicroBooNE cryostat, their function, and their flange/pipe type and outer-diameter dimension.",
"CF=ConFlat flanges, RFWN=raised face weld neck flanges, SS = stainless steel pipe."
],
[
"Liquid Argon Purification Subsystem",
"The heart of the cryogenic system is the liquid argon purification subsystem.",
"The primary requirement of this subsystem is to keep the level of electronegative contamination to below 100 ppt of O$_2$ equivalent contaminants.",
"This requirement is determined by the physics needs of the experiment, namely the need to be able to reconstruct events at the longest drift distances in the LArTPC.",
"In addition to the requirement on the electronegative contamination, the system must maintain the level of nitrogen contamination in the argon, by minimizing the leak rate from the atmosphere, at less than 2 parts per million (ppm) [27] to keep the quenching and attenuation of the scintillation photons in the argon to a minimum.",
"The MicroBooNE argon purification subsystem consists of liquid argon pumps and filters that serve to circulate the argon and remove impurities (e.g.",
"O$_2$ and H$_2$ O) that degrade the quality of the data collected by the active detectors.",
"It should be noted that the filters do not remove $N_2$ and so the ultimate $N_2$ contamination is set by the quality of the delivered argon.",
"There are two pumps in the system arranged in parallel in order to allow for continuous recirculation while one pump is being serviced.",
"Similarly, there are two sets of filters arranged in parallel in the system.",
"Figure REF schematically depicts the flow of liquid and gaseous argon in the MicroBooNE cryogenic system.",
"Figure: Flow diagram of argon in MicroBooNE, showing direction of liquid and gaseous argon in the cryogenic system.",
"Dashed lines represent gas lines, solid lines represent liquid lines, and yellow lines are for the filter regeneration.",
"Gaseous argon from the cryostat is condensed and directed through the purification subsystem.",
"Liquid argon drawn from the cryostat volume is directed into the purification subsystem.The recirculation pumps are Barber-Nichols [28] BNCP-32B-000 magnetically-driven partial-emission centrifugal pumps.",
"Each pump isolates the liquid argon from the electric motor by a magnetic coupling of the impeller to the motor.",
"The impeller, inducer, and driving section of the magnetic coupling each have their own bearings that are lubricated by the liquid argon at the impeller.",
"The motor is controlled by a variable frequency drive (VFD) that allows adjustment of the pump speed to produce the desired head pressure and flow within the available power range of the motor.",
"Each filter skid contains two filters, as depicted in figure REF , each having identically-sized filtration beds of 77 liters.",
"The first filter that the argon stream enters contains a 4A molecular sieve supplied by Sigma-Aldrich [29] that primarily removes water contamination but can also remove small amounts of nitrogen and oxygen.",
"The second filter contains BASF CU-0226 S, a pelletized material of copper impregnated on a high-surface-area alumina, which removes oxygen [30] and to a lesser extent water.",
"Because the oxygen filter will absorb water and thereby reduce its capacity for removing oxygen, it is placed after the molecular sieve.",
"The oxygen-filtering material must be reduced to copper with the procedure described below before it can remove oxygen from the liquid argon.",
"The filters are insulated with vacuum jackets and aluminum radiation shields.",
"The metallic radiation shields were chosen because the filter regeneration temperatures, described below, would damage traditional aluminized mylar insulation.",
"Pipe supplying the filter regeneration gas is insulated both inside the filter vacuum-insulation space and outside the filter with Pyrogel XT, which is an aerogel-based insulation [31] that can withstand temperatures up to 920 K. Figure: Three-dimensional rendering of a MicroBooNE filter skid.",
"The left drawing shows the full skid, while the right drawing shows a cut-away of the vessels.The filters are initially activated and then regenerated as needed in situ using heated gas, by a procedure developed for the LAPD.",
"The filters are regenerated using a flow of argon gas that is heated to 473 K, supplied by a commercial 500 liter liquid argon dewar.",
"Once the argon gas reaches 473 K, a small flow of hydrogen is mixed into the primary argon flow and exothermically combined with oxygen captured by the filter to create water.",
"Too much hydrogen mixed in with the primary argon flow would induce temperatures that are sufficiently high to damage the copper-based filter media.",
"The damage is induced by sintering of the copper, which reduces the available filter surface area.",
"To avoid such damage, precautions are taken to maintain a hydrogen fraction below 2.5% of the heated gas mixture.",
"During the heated gas regeneration, five filter bed temperature sensors monitor the filter material temperature and the water content of the regeneration exhaust gas is measured.",
"To remove any remaining trace amounts of water, the filters are then evacuated using turbomolecular vacuum pumps while they cool.",
"A particulate filter with an effective filtration of 10 microns, positioned between the cryostat and the filter skids, prevents any debris in the piping from being introduced into the cryostat.",
"The particulate filter consists of a commercial stainless steel sintered-metal cylinder mounted in a custom cryogenic housing and vacuum jacket.",
"Filtration is accomplished by flowing liquid argon to the interior, then outward through the walls, of the sintered-metal cylinder.",
"Flanges on the argon piping, along with flanges and edge-welded bellows on the vacuum jacket, allow removal of the particulate filter.",
"The argon purification piping is 2.54 cm diameter stainless steel that was pre-insulated by the manufacturer with 10.2 cm of polyurethane foam.",
"During the fabrication process, all piping was washed with distilled water and detergent to remove oil and grease, then cleaned with ethanol.",
"All valves associated with the argon-purification piping utilize a metal seal with respect to ambient air, either through a bellows or a diaphragm, to prevent the diffusion of oxygen and water contamination.",
"The exhaust side of each relief valve is continuously purged with argon gas to prevent diffusion of oxygen and water from ambient air across the O-ring seal.",
"Where possible, ConFlat flanges with copper seals are used on both cryogenic and room-temperature argon piping.",
"Pipe flanges in the system are sealed using spiral-wound graphite gaskets.",
"Smaller connections are made with VCR fittings with stainless steel gaskets."
],
[
"Nitrogen Refrigeration",
"The cryostat and purification systems that contain the liquid argon are subject to heat load from the environment, as well as from the active detectors that have electrical power enabled.",
"To keep these systems operating at a stable temperature and pressure, a liquid nitrogen refrigeration system is present to provide the necessary cooling power.",
"The liquid nitrogen system contains two condensers that are arranged in parallel.",
"One of these is utilized for normal operations and one serves as a backup on standby.",
"Each condenser contains two liquid nitrogen coils, an inner and an outer, with the gas argon on the shell side.",
"Typically only one coil is actively running and the second can be manually activated during situations where the system heat load is higher than usual.",
"Each condenser is sized to handle a heat load of approximately 9.5 kW.",
"With the vessel full of liquid argon and no pump or liquid argon circulation running, the condenser uses $\\sim $ 2350 liters of liquid nitrogen per day, which equates to a 3.9 kW system heat load.",
"With the recirculation pumps and the electronics in operation, the usage rate is about 3400 liters/day corresponding to a total heat load from the cryostat system of about 6 kW."
],
[
"Controls and Purity Monitoring",
"MicroBooNE makes use of resistive thermal devices (RTDs) to measure temperatures throughout the experimental infrastructure.",
"Twelve RTDs are located along the walls of the cryostat, and another ten RTDs are mounted inside screws attached to the structure of the LArTPC.",
"Each of the filter vessels in the purification system contain nine RTDs.",
"An interlock based on the RTDs within the filter vessels prevents overheating that could potentially occur during filter regeneration with heated argon-hydrogen gas.",
"Liquid argon contaminations ranging between 300 and 50 ppt O$_2$ equivalent can be measured using double-gridded ion chambers, henceforth referred to as purity monitors, immersed in liquid argon.",
"The design of the purity monitors is based on the design presented by Carugno et al. [32].",
"A description of the purity monitors, the data-acquisition hardware and software used in LAPD can be found in [25].",
"MicroBooNE uses the same type of purity monitors, and the same data-acquisition hardware and software.",
"An estimate of the electron drift lifetime is made by measuring the fraction of electrons generated at the purity monitor cathode that subsequently arrive at the purity monitor anode $(Q_A/Q_C)$ after a drift time $t$ .",
"The ratio of $(Q_A/Q_C)$ is related to electron lifetime, $\\tau $ , such that $Q_A/Q_C = e^{-t/\\tau }.$ Measurement of liquid argon purity in the MicroBooNE cryogenic system are provided by three purity monitors of various lengths.",
"One purity monitor with a drift distance of 50 cm sits in a vessel just downstream of the filters and is used to monitor filter effectiveness.",
"Two purity monitors, one with a drift distance of 19 cm and the other with a drift distance of 50 cm, sit within the primary MicroBooNE vessel at each end of the LArTPC.",
"They are installed at different heights to allow purity measurements at different depths of the argon."
],
[
"Initial Purification",
"The MicroBooNE cryogenic system was designed to allow the cryostat to go from containing atmosphere (and the detector) to containing high purity liquid argon (and the detector) without ever evacuating the cryostat.",
"Such a process is considered essential to the development of multi-kiloton experiments where the cost of an evacuable cryostat would be prohibitive.",
"While a successful test had been carried out in the LAPD [25], before MicroBooNE, this process had never been performed in a cryostat with a fully instrumented large detector.",
"A more complete description of the initial purification process with operational details is available in [33].",
"The process for preparing the cryostat for filling with liquid argon involves three stages.",
"The first is the “piston-purge” stage where argon gas flows into the cryostat at its lowest point and the argon (being denser than the atmosphere) pushes the atmosphere out of the cryostat.",
"The second stage is a recirculation phase where the argon-gas loop is closed and the argon flows through a water-removal filter to reduce outgassing of water from detector materials.",
"The final stage is a cool down phase where the loop remains closed and the argon gas is cooled in a heat-exchanger and cools the detector to the point where the insertion of liquid argon will not damage the detector.",
"The so-called “piston-purge” was achieved using a single pipe for gas input at the bottom of the cryostat and an identical pipe for the output at the top; the pipes (referred to as “sparger” pipes) each have 4.76 mm diameter holes every 12.7 cm on both sides.",
"Calculations of the mutual diffusion of argon and nitrogen suggested that a rate of 770.4 Nm$^3$ /hr.",
"was adequate in avoiding turbulence and minimizing back-diffusion of the air [34].",
"A total of about 13 volume exchanges took one week and resulted in contamination levels of $<$ 10 ppm $H_2 O$ , $<$ 1 ppm $O_2$ and $<$ 1 ppm $N_2$ .",
"The recirculation phase used the same piping system as the piston-purge but at a significantly higher flow rate (3 cryostat changes/hr).",
"As mentioned, the recirculation passed the gas through the water removal filter and over a period of three weeks, the water concentration was reduced to $<$ 1 ppm.",
"To counteract the increase in $N_2$ and $O_2$ levels from outgassing observed during this phase, (typically a rise of a few ppb/hr), a small fraction of the gas was vented and replaced with fresh argon.",
"The final stage involved cooling the cryostat and detector to prepare for filling with liquid argon.",
"The TPC design imposed two requirements on the cooldown.",
"One was that the input gas be no more than 20 K colder than the frame of the TPC to avoid wire-breakage since the massive TPC frame takes a long time to cool while the wires immediately adopt the input gas temperature and shrink, thereby increasing the stress on the wires.",
"The second requirement was that the temperature difference between the top and bottom of the TPC be less than 20 K at any time to avoid warpage of the frame.",
"A set of RTDs screwed into the TPC frame, RTDs in the inlet gas piping, and a set of RTDs attached to the anode side of the cryostat were used to measure and monitor the process.",
"For the cooldown, the argon-gas was cooled using a three-pass counter-flow nitrogen heat-exchanger, and recirculated using the sparger pipes.",
"The cooldown was declared complete after three weeks when the average temperature had reached 105 K and the temperature difference between the top and the bottom of the detector was $<$ 10 K. Upon reaching this state, the O$_2$ level had been reduced to 18 ppb and the H$_2$ O concentration had fallen to 22 ppb.",
"These levels are reduced by a factor of $>$ 800 in the liquid and presented an excellent environment for the start of filling."
],
[
"Liquid Argon Time Projection Chamber",
"The MicroBooNE LArTPC drifts and collects charge to produce fine-grained images of the ionization that is liberated by charged particles traversing a volume of highly-purified liquid argon.",
"This section describes the design and implementation of the LArTPC in the experiment.",
"The LArTPC is composed of three major structures: the cathode, the field cage, and the anode.",
"A negative voltage is introduced via a feedthrough passing through nozzle N2 on the cryostat and applied at the cathode, which defines an equipotential surface.",
"A uniform electric field between the cathode and the anode planes is established by a series of field rings connected by a voltage divider chain starting at the cathode and ending at the anode plane.",
"Facing the cathode planes are the sense wire planes: two induction planes (referred to as the “U” and “V” planes) with wires oriented at $\\pm 60^{\\circ }$ from vertical, followed by one collection plane (referred to as the “Y” plane) with vertically-oriented wires.",
"The wires of the anode planes are the sensing elements that detect the ionization created by charged particles traveling through the LArTPC.",
"Figure REF depicts the assembled MicroBooNE LArTPC after insertion into the cryostat, showing details of the cathode, field cage, and anode plane.",
"Table REF lists the main parameters of the MicroBooNE LArTPC, which will be described in detail in this section.",
"Table: MicroBooNE LArTPC design parameters and nominal operating conditions.Figure: Schematic diagram of the MicroBooNE LArTPC , depicted as it is arranged inside the cryostat."
],
[
"Cathode",
"The cathode is assembled from 9 individual stainless steel sheets (Type 304, 2.3 mm thick) that are fastened to a supporting frame by hex-head stainless-steel button-screws.",
"The outer edge of the cathode frame consists of round stainless steel tubes of 5.08 cm outer diameter and 3.18 mm wall thickness.",
"Within this outer edge, square tubes with 5.08 cm $\\times $ 5.08 cm cross-sectional area, and 3.18 mm wall thickness, are fastened together with hex-head button-screws, forming a support structure upon which the cathode sheets are attached.",
"The individual components of the support structure are further welded together to eliminate sharp features from this high-potential surface.",
"The exterior frame and support structure of the cathode, and also an interior view, are shown in figure REF .",
"The cathode plane sheets are shimmed according to survey data to make the cathode as flat and as parallel to the anode frame as possible, resulting in the two surfaces being parallel to within 0.0413$^{\\circ }$ .",
"Flatness of the cathode is evaluated relative to a best fit plane of survey data (more than 10000 survey points recorded with a laser tracker).",
"The largest deviations of the cathode from the best fit plane are +6.6 mm and -6.5 mm.",
"Approximately 55$\\%$ of the measured survey points fall within +/-3 mm of the best fit plane, and more than 90$\\%$ of the points fall within $\\pm $ 5 mm.",
"Figure REF shows the results of the survey, with deviations from flat represented as color-coded data extending away from the nominal plane of an ideal cathode.",
"Figure: Top: Exterior view showing the cathode frame and structural supports to which cathode sheets are fastened.",
"Bottom: Interior view of cathode plane as viewed from the upstream end of the LArTPC , showing cathode sheets.",
"Note that the cathode sheets are polished, so a reflection is is clearly present in this photograph.Figure: Survey results showing the flatness of the cathode, as viewed from the interior (top) and exterior (bottom) sides, after shimming.",
"Color scale extends from -6.498 mm (red) to +6.636 mm (blue)."
],
[
"Field Cage",
"The field cage encloses the volume between the cathode plane and the anode wire planes, and creates a region with a uniform electric field.",
"The volume defined by the interior of the field cage, bounded by the anode and cathode planes, is referred to as the “active” volume.",
"The field cage structure consists of 64 individual sets of thin-walled stainless steel tubes (2.54 cm OD, 0.51 mm wall thickness), each shaped into a rectangular loop framing the perimeter of the active volume.",
"These 64 loops are mounted parallel to the cathode and anode planes, as shown in figure REF , and are held in place by a G-10 rib support structure.",
"Each field cage loop is electrically connected to its neighbors via a resistor divider chain (described in the following section), causing each loop to operate at a different electrical potential, which in turn maintains a uniform electric field between the cathode and anode planes.",
"For a nominal -128 kV cathode voltage, the difference in potential between adjacent field cage loops is 2 kV, ramping down the total potential in equidistant steps from cathode to anode.",
"The distance from center-to-center of adjacent field cage loops is 4.0 cm.",
"Each field cage loop is assembled from 2.07 m long vertical pipes on the upstream and downstream ends of the LArTPC , and on the top and bottom from two 5.18 m long horizontal pipes connected by a stainless steel coupling in the center.",
"Each tube has venting holes approximately every 15 cm to allow for effective purging from atmosphere and to avoid any trapped volumes.",
"The four corners of each field cage loop are curved with a radius of 5.24 cm.",
"Each corner is formed by three parts: two couplings and an elbow, shown in figure REF .",
"The couplings make the connections between the pipes and the elbow.",
"The thin-walled tubes and elbows slip-fit over the ends of the couplings with a 2.2 cm overlap.",
"Each coupling has two 6-32 NC tapped holes and the connections to the adjoining pieces are made by hex-head button-screws and split-ring lock washers with no teeth.",
"Figure: Photograph of the field cage during construction, with loops 0 (cathode) through 5 labeled.",
"Field cage loops closest to (and including) the cathode are modified to reduce sharp edges that would result in higher electric fields.In order to avoid electrical breakdown between the inner cryostat surface and field cage parts at high potential on or near the cathode, the electric field strength is minimized at the corners and edges of the field cage.",
"Loops 0, 1, and 2 are each designed differently than the other field cage loops.",
"Loop 0 is a special case in that it is made from larger diameter piping of 5.08 cm OD, and frames the cathode and also acts as its mechanical support.",
"It operates at the same electrical potential as the cathode plane sheets attached to it.",
"Loop 0 has a slightly smaller area than the other field cage loops, as shown in figure REF .",
"Loop 1 is the first of the 64 loops in the field cage with 2.54 cm pipe OD.",
"It surrounds the cathode plane and operates at cathode potential.",
"The elbow of loop 1 has a specially designed geometry in order to minimize the electric field potential.",
"The elbow of loop 2 has a larger radius of curvature than the standard elbows, also for the purpose of minimizing the electric field potential.",
"For all three of these loops (loop 0, 1, and 2), connections at corners and joints are made by welding instead of screws to avoid sharp edges that would result in higher electric fields and greater chance of electrical breakdown.",
"Another precaution to minimize the electrical field between the loops and the cryostat surface is the positioning of the coupling screws: for the first 20 loops, the screws are positioned on the sides facing the screws of the neighboring loops instead of facing inward to the LArTPC active volume and outward toward the grounded cryostat surface.",
"Hex-head button-screws and lock washers are also used here in order to minimize sharp metal edges.",
"Figure REF shows the simulated electric field values inside the MicroBooNE LArTPC , and some of the surrounding volume inside the cryostat, when the cathode is set to an operating voltage of -128 kV.",
"Figure: Cross-section view showing the electric field simulation inside the field cage when the cathode is set to a voltage of -128 kV.",
"Loop 0 is represented by the larger diameter circle on the left of the image.",
"The legend shows the absolute values of electric field modulus in units of V/m."
],
[
"Resistor Divider Chain",
"A resistor divider chain installed across the field cage loops steps the voltage down in magnitude from the cathode plane to the anode wire plane in equal steps.",
"For a nominal value of -128 kV on the cathode, this results in a potential difference of 2 kV between each pair of loops.",
"The value of the equivalent resistance between loops within the divider chain was chosen to be low enough such that the current flow through the divider circuit is much greater than the signal current flowing through the LArTPC .",
"The signal current in our case is dominated by the free ionization produced by the cosmic ray flux, and is estimated to be <50 nA.",
"An equivalent resistance of 250 M$\\Omega $ between each pair of field cage loops, corresponding to a current flow of 8 $\\mu $ A, was chosen.",
"The voltage divider chain is mounted on the inside of the field cage at the upstream end of the detector.",
"The couplings at the top corner of each field cage loop have additional holes facing the inside of the field cage, where the resistors are mounted.",
"On the first 16 field cage loops, pairs of Metallux HVR 969.23 499 M$\\Omega $ resistors (rated to 23 W, 48 kV in air) are mounted electrically in parallel to establish the beginning of the voltage divider chain.",
"On the remaining loops, four thick-film Ohmite Slim-Mox 104E metal-oxide epoxy-coated resistors with a lower power and voltage rating (1.5 W, 10 kV in air) are mounted in parallel, per loop.",
"Extensive testing was done on these two types of resistors [35].",
"For the loops with the Slim-Mox resistors, printed circuit boards span across eight field cage gaps and therefore have eight 250 M$\\Omega $ resistances in series, shown in figure REF .",
"The electrical connection between the boards and each field cage loop is made by metal contact pads on the back side of the boards, held in electrical contact with the field cage tube by a hex-head button-screw and lock washer.",
"Figure: The Ohmite Slim-Mox 104E resistors arranged in parallel sets of four on printed circuit boards that span eight field cage tubes.While the designed operating voltage difference across each resistor in the detector is 2 kV with a power flow of 4 mW, there is a slight possibility that the voltage drop and power could temporarily exceed the rating of the resistors in the case of discharge between the cathode plane or field cage loops and the cryostat wall, through the bulk liquid argon.",
"Recent studies [36] have shown that the value of the minimum breakdown electrical field decreases with the increasing argon purity; for purities as high as that required in the MicroBooNE detector, breakdown has been observed at electric fields as low as 40 kV/cm.",
"The field cage behaves like a capacitance network at high frequencies.",
"Based on measurements and simulations, the total energy stored inside the field cage when fully charged is estimated to be approximately 24 J.",
"In the case of a discharge between the cryostat and the cathode or one of the field cage loops close to the cathode, simulations show that voltages of up to 80 kV peak, with a discharge time constant of a few seconds, can develop across the resistors.",
"The observed peak voltages in such discharge scenarios decrease the further the breakdown occurs from the cathode, such that discharges occurring between the cryostat and field cage loops 32 through 63 do not exceed the 10 kV rating of the resistors.",
"Two strategies have been implemented to protect the resistors nearest the cathode from damage due to discharge.",
"The first is the use of the Mettalux resistors on the first 16 loops given their higher voltage rating of 48 kV as compared to the Slim-Mox resistors on the remaining loops.",
"Since the Metallux HVR 969.23 resistors are significantly larger physically than the loop-to-loop distance, they are mounted diagonally between each pair of field cage loops.",
"They are held by copper brackets, which are attached to studs welded onto the field cage tubes, shown in figure REF .",
"Figure: The Metallux HVR 969.23 resistors mounted on the 16 field cage loops closest to the cathode.The second protective measure is installation of surge protection circuits on field cage loops 1 through 32.",
"The chosen surge protection devices are designed to short the circuit in the case of a voltage spike, which protects any other electrical components installed in parallel.",
"Below their clamping voltage, they exhibit a very high resistance and do not influence the circuit.",
"The behavior of Gas Discharge Tubes (GDTs) and varistors in liquid argon has been studied extensively for application in the MicroBooNE field cage [37].",
"The surge protection device chosen is a Panasonic ERZ-V14D182 varistor with a clamping voltage of 1700 V. In order to obtain a very high resistance in normal operation and a clamping voltage above the 2 kV in normal operation mode, three of these devices are mounted in series across a block of four Slim-Mox 104E or two Metallux 969.23 resistors.",
"These additional varistor boards make electrical contact with the field cage via brass mounting brackets that are fastened to the field cage with button-head screws, as shown in figure REF .",
"Figure: Surge-protecting varistors (small black disks in the photograph) are installed in parallel with the voltage divider resistors for the first 32 field cage loops.",
"Here, they are shown mounted on small boards in sets of 3, and attached to the field cage by means of 6-32 hex-head button-screws."
],
[
"Anode Planes",
"The anode frame holds the induction and collection plane sense wires at tension and provides overall structural support for the beam-right side of the LArTPC.",
"Individual sense wires for all anode planes are held in place by wire carrier boards, which are printed circuit board assemblies that position the wires as well as provide the electrical connection to the electronic readout system of the experiment."
],
[
"Mechanical structure",
"The anode frame is comprised of a stainless steel C-channel hosting adjustable tensioning bars to which the wire carrier boards are attached.",
"The C-channel and tensioning bar assembly is depicted for one corner of the anode frame in figure REF .",
"Wire carrier boards attach to precision alignment pins distributed along the length of the tensioning bars.",
"Figure: Rendering of the anode frame assembly.",
"The C-channel is depicted in gray, and the adjustable tensioning bar assembly is shown in orange."
],
[
"Wire winding and quality assurance",
"The three anode planes are constructed from wire carrier boards that have individually-prepared wires attached to them in groups of 16 (for the U- and V- angled planes) or 32 (for the vertical Y-plane).",
"Consistent quality in wire preparation was achieved by a semi-automated winding machine, which terminated the ends of each wire via wrapping around 3 mm diameter brass ferrules as shown in figure REF .",
"The wire termination method via wrapping around a brass ferrule similar to that used in the ICARUS T600 detector.",
"Figure: Photograph of the wire termination on the brass ferrules.",
"Each ferrule is 3 mm in diameter, and 1.5 mm thick.Each wire was tested for strength on a tensioning stand where a load of 2.5 kg (more than 3 times the nominal load of 0.7 kg) was applied for 10 minutes, ensuring that the wire preparation did not leave any weaknesses that could result in a breakage.",
"Upon successful completion of the quality assurance testing, each wire was placed onto a wire carrier board, shown in figure REF .",
"Figure: Photograph of a collection plane wire carrier board that has been filled with wires, but has not yet had the cover plate installed.When installed on the wire carrier boards, the wires make contact with gold pins which are connected to a trace that routes to the cold electronics.",
"Once the wire-carrier board was filled with wires, a cover plate was installed and press-fit rivets were installed to hold the assembly together.",
"The assembled wire carrier board was then placed onto a tension stand, to reapply a 2.5 kg tension/wire to the whole board for 10 minutes.",
"This is to ensure that the wires were not weakened during the board assembly process.",
"The tension stand is depicted in figure REF .",
"A comprehensive description of the MicroBooNE wire preparation and associated quality assurance studies can be found in [38].",
"Figure: Photograph of a collection plane wire carrier board on the tension stand."
],
[
"Parts Preparation",
"The majority of the parts that make up the LArTPC are either stainless steel or G-10.",
"These two material types, as well as any others used in the LArTPC, were tested in the Fermilab Materials Test Stand (MTS) [39], whose purpose was to investigate the suitability of materials for use in LArTPCs.",
"The MTS confirmed that none of the materials used in the LArTPC assembly would contaminate the liquid argon.",
"Before assembly, all LArTPC parts were cleaned according to the procedures described in the following sections."
],
[
"Cleaning stainless steel",
"The delivered stainless steel parts were often greasy due to machining, and those with holes or interior cavities generally had a significant amount of trapped metal shavings due to the machining processes.",
"Many of the pieces also had markings from permanent ink pens, dirt smears, rust spots, and/or dried oil from machining.",
"These pieces were scrubbed with ScotchBrite 7447 general-purpose hand pads before cleaning.",
"Parts that were small enough to fit in an ultrasonic bath were prepared according to the following prescription.",
"A pre-rinse with tap water was performed to remove particulate matter, followed by deburring of sharp edges.",
"The first ultrasonic wash was 15 minutes in heated distilled water with a 3% solution of Citranox acid detergent [40].",
"After a first rinse in distilled water, a second ultrasonic wash was performed, again for 15 minutes, but using heated distilled water with a weak solution of Simple Green detergent [41].",
"A second rinse in distilled water was performed, followed by a final rinse in a fresh bath of distilled water.",
"The parts were then wiped dry with lint-free cloths, air dried completely, and wrapped in plastic film for storage.",
"Stainless steel parts that were too large to fit in the ultrasonic bath were prepared by a simpler prescription out of necessity.",
"A pre-rinse with tap water was followed by deburring to remove sharp edges.",
"Both the first and second washes were done with tap water and a weak solution of Simple Green detergent, scrubbing with brushes and lint-free sponges.",
"Two tap water rinses were done, and the parts were then wiped dry with lint-free cloths.",
"As a final additional step, each part was wiped with 200-proof ethyl alcohol, and then air dried completely.",
"These parts were also wrapped in plastic film for storage."
],
[
"Cleaning G-10",
"G-10 is known to absorb large quantities of water, which would outgas in the argon and could inhibit reaching the required argon purity in the detector.",
"For this reason all G-10 parts were cleaned and then baked to remove moisture.",
"The largest G-10 parts on the detector are beams that span the distance between the cathode and anode.",
"These were washed in 1900-liter ultrasonic baths that are overseen by Fermilab Accelerator Division, typically used for cleaning large sections of accelerator beam pipes.",
"An initial pre-wash was done with tap water to remove as much particulate matter as possible, since the machining process left a large amount of dust on the machined edges.",
"Pieces were then placed in the ultrasonic bath with heated deionized water and a 2% solution of Elma Clean 65 (EC 65) neutral cleanser [42].",
"Two ultrasonic bath rinses were performed, and the pieces were then sealed in plastic bags with clean dry nitrogen gas.",
"In order to remove the absorbed water, the large G-10 parts were then transported to the Fermilab Technical Division where they underwent an outgassing procedure to remove any remaining absorbed moisture.",
"They were baked in a large oven under vacuum until a plateau in the outgassing rate was reached, as reported by a monitor inside the oven.",
"Upon completion of the outgassing procedure, the parts were resealed in plastic bags for storage."
],
[
"Assembly",
"The full mechanical structure of the LArTPC is shown in figure REF , with the left image depicting the cathode frame as semi-transparent to show the support structures on which the cathode sheets are attached.",
"The anode frame is on the right of this image, with I-beams configured in a crossed pattern to maintain the shape and rigidity of the outer C-channel structure.",
"Ribs of G-10 connect the anode and cathode, electrically isolating them from each other while also providing mounting holes to hold in place each of 64 field cage loops that define the active volume of the LArTPC.",
"The field cage loops are visible in the photograph on the right of figure REF .",
"Figure: Left: Rendering of the full LArTPC frame assembly.",
"Right: Assembled LArTPC after wire and electronics installation.Assembly was done inside of a clean tent, shown on the top left in figure REF , on a flat surface made up of adjustable-height metal platforms that were installed on the assembly room floor.",
"These platforms were leveled to better than 0.5 mm before beginning assembly.",
"The anode frame was the first part of the detector to be assembled on this surface, shown on the top right of figure REF .",
"It was temporarily placed aside, and the cathode frame was assembled on the same set of platforms along with the G-10 ribs, which stood vertically with the help of temporary unistrut support pieces.",
"The combined cathode and G-10 frame was then lifted and rotated to the proper orientation, with G-10 ribs extending horizontally from the cathode to the anode, as shown on the bottom in figure REF .",
"Finally, the anode frame was brought back over and attached to the G-10 ribs, and the stainless steel tubes that make up the field cage loops were fed through the holes in the G-10 ribs to complete the mechanical structure of the LArTPC.",
"Figure: Top Left: Clean tent where MicroBooNE LArTPC assembly was conducted.",
"Top Right: Anode frame in the process of being moved from the metal assembly platforms.",
"Bottom: Cathode frame and G-10 ribs on metal assembly platforms."
],
[
"Wire installation and tension measurements",
"During detector assembly, the completed wire carrier assemblies (consisting of wires and supporting carrier boards on either end) were manually installed onto the adjustable tensioning bars residing in the C-channel of the supporting anode frame.",
"A team of two people installed each assembly onto the anode frame.",
"The collection plane was the first installed, followed by the middle induction plane, and then finally the inner induction plane.",
"Once all three anode planes were completely installed, the tensioning bars were adjusted and a survey was taken of the tension of all anode wires.",
"Tension was set according to the design criteria that it be small enough to prevent wire breakage during cool down and large enough to limit the maximum wire sag due to gravity to under 0.5 cm for any 5 m long U or V wire.",
"Tension was recorded through measurement of the resonant frequency of a laser beam reflected from a plucked wire and incident on a photodiode connected to a spectrum analyzer program [43].",
"The tension measuring equipment was developed and produced by the University of Wisconsin Physical Sciences Laboratory.",
"The tensioning bars were adjusted iteratively until the surveyed tension of all wires was within a range, approximately $\\pm $ 1.0 N of the nominal value of 6.9 N, where no single wire was too taught or loose to create detector performance issues.",
"Figure REF shows the final surveyed tension of the wires for each plane.",
"Figure: Final survey results for wire tension of the MicroBooNE LArTPC."
],
[
"High Voltage System",
"To create the drift field, negative voltage (referred to as the “high-voltage” or “HV”) is supplied to the LArTPC cathode.",
"This potential is generated outside of the cryostat by a Glassman LX150N12 power supply.",
"Before entering the cryostat, the output of the power supply is passed through a current-limiting resistor chain that serves as both a low-pass filter for the power supply ripple, and a partition for the stored energy in the system.",
"The resistor chain is a set of eight 10 M$\\Omega $ resistors connected in series and submerged in a transformer oil in an aluminum container.",
"This assembly was successfully tested to -200 kV in a dedicated test dewar.",
"The potential is introduced into the cryostat by a custom-designed HV feedthrough.",
"The feedthrough is based on an ICARUS design [5]; a 2.54 cm diameter stainless steel inner conductor is surrounded by a 5.08 cm outer diameter ultra-high molecular weight polyethylene (UHMW PE) tube that is encased in an outer ground tube.",
"A photograph and drawing of the production feedthrough are shown in figure REF .",
"Figure: Photograph and drawing of the production HV feedthrough.",
"The spherical probe tip is attached to the end of the inner conductor, on the right side of these figures.Figure REF shows the HV feedthrough extending into its receptacle cup attached to the MicroBooNE LArTPC cathode.",
"The lower termination of the outer ground tube is a torus chosen to reduce the electric field between both the feedthrough and the cathode plane, and along the feedthrough itself.",
"The electrical connection to the cathode is made with a hemispherical spring-loaded tip attached to the inner conductor of the feedthrough.",
"The UHMW PE tube is machined with grooves and extends into the receptacle cup attached to the cathode.",
"Figure: The production HV feedthrough inserted into the cathode receptacle cup inside the cryostat."
],
[
"Light Collection System",
"Liquid argon is a bright scintillator, and sampling the light from interactions in the argon can bring powerful new capabilities and information to complement the charge information from the LArTPC .",
"The light collection system in MicroBooNE is designed to meet MicroBooNE's physics goals for light collection, which are twofold.",
"First, for accelerator-induced events, the light collection system is designed to enable the experiment to trigger on $\\ge $ 40 MeV kinetic energy protons produced in beam neutrino interactions.",
"Second, for non-beam events, the system is designed for efficient observation of 5 to 10 MeV electrons from supernova neutrino interactions.",
"The light produced by neutrino interactions in MicroBooNE is an important input for both event selection and reconstruction.",
"One of the critical capabilities the light collection system provides is the ability to form a beam-event trigger when a pulse of light is observed in coincidence with the beam spill.",
"Because typically only one beam spill in 600 will produce a neutrino interaction in the detector, such a trigger will substantially reduce the overall data output rate.",
"For non-beam physics studies, the light system provides triggering and an event $t_0$ for the LArTPC system.",
"For accelerator-induced events, the time of the beam spill (1.6 microsecond duration) provides an adequate $t_0$ for tracks from the event.",
"However, because of the long window over which the ionization electrons of an event drift (about 1.6 ms maximum drift time at 500 V/cm field), almost every ($>$ 99%) accelerator-induced event will include one or more cosmic ray muons crossing the detector during the TPC livetime.",
"Utilizing the distribution of hits across the photodetectors, one can better reject cosmic ray muon tracks.",
"The light can also be used to select and trigger on specific types of cosmic-ray calibration events (Michel electrons, straight-through muons, etc.)",
"and non-beam events (supernova neutrinos, cosmic background events to proton decay studies, etc.).",
"The light collection system consists of primary and secondary sub-systems.",
"The primary light collection system is made up of “optical units,” each one consisting of a PMT located behind a wavelength-shifting plate.",
"In total, 32 optical units were installed, yielding 0.9% photocathode coverage.",
"The secondary system consists of four light guide paddles.",
"These paddles were introduced for R&D studies for future LArTPCs, and are placed near the primary optical units to allow a comparison of their performances.",
"A flasher system, used for calibration, consists of optical fibers bringing visible light from an LED to each PMT face.",
"The light collection detectors are located in the $y$ -$z$ plane behind the anode planes of the LArTPC, as shown in figure REF .",
"The combined transparency of the three anode planes is 86% for light at normal incidence.",
"This transparency value assumes 100$\\%$ of VUV photons impinging on the wires are absorbed.",
"The detectors were placed so as not to be obscured by the LArTPC structural cross-bars, shown in figure REF .",
"Locating the light detectors behind the anode plane places them in a very weak electric field due to the +440 V bias of the collection plane.",
"To test for an effect from weak electric fields, the response of a PMT placed between a +700 V mesh and ground, separated by 50 cm, was studied.",
"The PMT zenith was 25 cm from the voltage source.",
"No effects on the signal were observed.",
"This was expected, as the photocathode is held at ground, effectively acting as a Faraday cage.",
"Figure: The MicroBooNE light collection system consists of a primary system of 32 optical units and a secondary optical system of four lightguide paddles .",
"These are mounted behind the anode wire planes such that the view is not obscured by structural cross bars of the LArTPC.Throughout the design and construction of the light collection system, substantial R&D was performed.",
"The reader should refer to [27], [45], [46], [47], [48], [49], [50], [44], [51], [52], [53] for detailed results of these studies.",
"A useful overall review is available in [54].",
"Figure REF shows the light observed in two sequential events in the argon, consistent with a muon entering the detector followed by a Michel electron from the decay.",
"One can see that the light is relatively well localized.",
"This allows the light to be correlated with specific tracks in the detector.",
"This “flash-track matching” is used to identify and reconstruct the tracks that are in time with the beam spill–an important goal of the light collection system.",
"Figure: Displays showing two sequential events in the argon as seen by the light collection system.",
"The sequence is consistent with a muon that stops (top) and decays (bottom).",
"The circles correspond to the optical units.",
"Red circles indicate those units with hits within the time range indicated by the vertical bars drawn on the PMT waveforms, which are shown at the bottom of each display."
],
[
"Light Production in Argon ",
"Light produced in liquid argon arises from two processes: scintillation and Cherenkov radiation.",
"Scintillation light is produced by the formation and eventual radiative decay of excited argon dimers (or eximers) and is emitted in an isotropic distribution.",
"Liquid argon is an excellent scintillator: it produces a large amount of light per unit energy deposited (about 24,000 photons per MeV at 500 V/cm drift field) and is transparent to its own scintillation.",
"The scintillation light has a prompt and slow component with decay times of about 6 ns and 1.6 $\\mu $ s, respectively.",
"The two lifetimes correspond to the two lowest-lying eximer states with the prompt component coming from the decay of a singlet state and the slow from the decay of a triplet state.",
"The prompt to slow ratio is about 1:3 for minimum ionizing particles and varies with ionization density and particle type.",
"Both components consist of photons with a wavelength of 128 nm.",
"The effective lifetime of the triplet component may be modified by quenching (non-radiative dissociation of excimers by impurities) [55].",
"Other factors that can affect the arrival of the light include Rayleigh scattering, absorption by impurities, and obstructions.",
"For detailed discussion of the physics of scintillation light production and propagation in MicroBooNE, see [54].",
"Table REF summarizes information about the scintillation light.",
"Table: Important properties of scintillation light in liquid argon that affect detection in MicroBooNE.Because scintillation photons have a wavelength of 128 nm, they are are very difficult to detect using conventional photodetectors.",
"Figure REF summarizes the challenges involved in detection of the 128 nm scintillation light.",
"In order to detect the scintillation light (red distribution shown in figure REF ), the VUV photons must be shifted into the visible region.",
"MicroBooNE employs tetraphenyl-butadiene (TPB).",
"This organic fluor absorbs in the UV (green line) and emits in the visible with a peak at $425\\pm 20$ nm (green hatched region), the peak wavelength having a slight dependence on the micro-environment of the fluors.",
"This is a favorable wavelength for detection by the PMTs employed by MicroBooNE.",
"The efficiency for transmission through borosilicate PMT glass (black) and the quantum efficiency of the cryogenic tubes used in MicroBooNE (blue) are overlaid on the TPB spectrum.",
"Figure: Scintillation light emission spectrum (red) and TPB re-emission spectrum (green), in arbitary units.",
"Superimposed are relevant efficiencies (plotted in %\\% on the yy axis): Dark green line – absorption of VUV light by TPB; Black line – transmission of borosilicate glass; Blue line – efficiency of a R5912-02mod cryogenic PMT ."
],
[
"The Primary Light Collection System",
"Each of the 32 optical units of the primary light collection system consist of a cryogenic Hamamatsu 5912-02MOD PMT seated behind an acrylic plate coated with a TPB-rich layer and surrounded by a mu-metal shield.",
"Figure REF shows a diagram of one unit (left) and a photograph of the installed units (right).",
"Past experiments have directly coated PMTs with wavelength shifter [5].",
"However, the MicroBooNE design separates the PMT from the wavelength-shifting plate for simplicity of quality control and installation.",
"This proved important, as R&D indicated that TPB is particularly vulnerable to environmental degradation (see section REF ).",
"In this section, a description is provided for each component of the optical unit, as well as for the overall assembly.",
"Figure: Left: diagram of the optical unit; Right: units mounted in MicroBooNE, immediately prior to LArTPC installation."
],
[
"Photomultiplier Tubes, Bases, and Initial Tests",
"Reference [49] provides detailed information on the selection and testing of the 200 mm (8.0 in) diameter Hamamatsu R5912-02mod cryogenic PMTs employed in MicroBooNE.",
"In this section a brief summary of the findings from this testing is presented.",
"The R5912-02mod employs a bi-alkali photocathode.",
"Because the PMT is designed for cryogenic use, the R5912-02mod also features a thin platinum layer between the photocathode and the borosilicate glass envelope to preserve the conductance at low temperatures.",
"While this allows the PMT to function below 150 K, absorption in the platinum reduces the efficiency of the PMT by 20%.",
"Figure REF provides quantum efficiency curves for these PMTs.",
"The manufacturer's specifications do not include the effects of the platinum photocathode coating, but a wavelength dependent quantum efficiency was provided by Hamamatsu for 4 of the 32 installed PMTs.",
"The mean and standard deviation of these curves are shown in figure REF .",
"Figure: The specification for the non-platinum undercoated PMT from .",
"The blue band shows the mean and standard deviation of the four quantum efficiency curves provided by Hamamatsu for the installed PMTs.",
"Also shown is the measured emission spectra of the MicroBooNE wavelength-shifting coatings, discussed in section .",
"The R5912-02mod is a 14-stage PMT.",
"The high gain at room temperature ($10^9$ at $\\sim $ 1700 V), compensates for known reduced gain at 87 K in the liquid argon.",
"The high gain also has the additional advantage of allowing operation at lower than nominal voltage which reduces heat-loss in the PMT bases and the potential for high voltage breakdown at the feedthroughs.",
"The PMT base is designed such that the photocathode is grounded and the anode is held at large, positive voltage.",
"Thus the PMT bulb, which is closest to the LArTPC anode plane, is at ground and does not disturb the electric field on the wires.",
"The result is that the high voltage (HV) can be provided and the signal can be extracted from the PMT using a single cable, reducing the cable volume in the vapor region and removing a possible source of out-gassing water impurity.",
"The flat PC-board base was made of Rogers RO4000-series woven glass-reinforced laminate, which is the same material as the LArTPC cryogenic front end boards.",
"Use of Rogers 4000 series as the base material avoids the potential risk of contamination of the liquid argon.",
"The Rogers material has a similar temperature expansion coefficient as the surface mounted components which enhances the reliability of the electronics assembly when operated at cryogenic temperatures.",
"The base is attached $\\sim $ 1 cm from the bottom of the PMT and is the closest possible distance of safe approach to the PMT vacuum seal tip.",
"A schematic and photos of the PMT base are provided in reference [49].",
"The passive components include only metal film resistors and C0G/NP0 capacitors, which have the minimum temperature coefficients, and the performance of all components at cryogenic temperatures was tested.",
"Because the supplied HV and return signal share a single cable, the signal must be split from the HV through an AC-coupling capacitor, as is discussed in section REF .",
"All installed PMTs were tested in a PMT test stand, both at room temperature and in liquid nitrogen at 77 K. The details are described in reference [49].",
"In brief, the test stand consisted of a light-tight 346 liter, liquid nitrogen filled dewar into which up to four PMTs could be installed.",
"The PMTs were immersed and maintained in the dark environment for up to three days before most measurements of the dark rate and gain were performed.",
"A fiber brought in light from a pulsed blue LED, which was tested for linearity with bias voltage.",
"Among the important results from cryogenic testing were the following [49]: After the required period in the dark and cold, the PMTs could be ramped to voltage quickly in the cold environment, and after 30 minutes, the gains were found to be stable.",
"If the room temperature PMTs were immersed in liquid nitrogen, dramatic changes in the gain were observed after initial turn-on.",
"The PMT gain remained high in the first $\\sim $ 5 hours after immersion, and then suddenly dropped by more than a factor of two, afterwards reaching a stable value with a small drift.",
"The PMT response showed good linearity up to 100 photo-electrons (PE), which was the maximum attainable by the PMT test stand LED.",
"The HV for each PMT was selected to produce a gain of $3\\times 10^7$ in liquid nitrogen, and was typically chosen to be $\\sim $ 1300 V. The dark current plateaus extended up to 1800 V in liquid nitrogen, and the dark current is higher in liquid nitrogen than at room temperature.",
"The PMT performance depended on rate of the pulsed LED (see later in this section).",
"No PMTs were rejected on the basis of the testing.",
"However, there were three unexpected results to note here.",
"The first unexpected behavior was that, at room temperature, most of the PMTs showed gains that were 10 to 30% higher than manufacturer's specifications.",
"As expected at cryogenic temperatures, the gain is reduced by $\\sim $ 10% to 50%.",
"To measure the PMT gain, the LED was set to produce one to two PEs.",
"The gain was found from the separation of the single PE peak from the pedestal, where the single PE response was fit using the procedure described in [61].",
"Second, it was found that the PMTs are noisier in the cryogenic environment.",
"It would typically be expected that thermal emission is suppressed at cryogenic temperatures and one would expect a lower dark current for PMTs operating in this regime.",
"However, the dark current measured in the liquid nitrogen was higher than at room temperature.",
"Although the cause is unknown, this phenomenon has previously been observed [62].",
"A proposed explanation is provided in [63].",
"Third, an LED pulse-rate-dependent gain shift was found during testing, as shown in figure 9 of reference [49].",
"This behavior is described qualitatively in [64].",
"With 10 kHz LED pulsing, the gains of cold and dark-adapted PMTs were shown to steadily increase, requiring nearly 24 hours from turn-on to stabilize.",
"The effect was not observable at 10 Hz.",
"This is relevant to MicroBooNE because the cosmic muon rate in the MicroBooNE detector is $\\sim 5$ kHz.",
"Therefore a similar effect is expected in the MicroBooNE detector.",
"Preliminary results from measurements of the PMTs installed in the LAr-filled MicroBooNE cryostat shows the expected effect."
],
[
"Wavelength-Shifting Plates",
"In order to be sensitive to 128 nm VUV liquid argon scintillation photons, the optical assemblies use a wavelength-shifting coating to convert this VUV light to visible wavelengths that are detectable by PMTs.",
"In MicroBooNE, the active ingredient of this coating is TPB, an organic fluor which absorbs efficiently in the vacuum ultraviolet with an emission spectrum peaked around 425 nm [65] as shown in figure REF .",
"The optical unit PMTs observe light transmitted through TPB-coated, 305 mm diameter acrylic plates (see figure REF ).",
"The coating consists of a 1:1 TPB-to-polystyrene ratio, with 1 g of each dissolved in 50 ml of toluene.",
"A small amount of ethyl alcohol is added as a surfactant.",
"The coating is applied to the acrylic plate in three layers by brush-coating.",
"The solution dries in air at room temperature.",
"This leads to a final layer which is oversaturated with TPB, and white crystals form on the surface as the coating dries.",
"The presence of surface crystallization gives the MicroBooNE plates a white, opaque finish.",
"Details of the process are described in reference [66].",
"Figure: Left : Illustration of transmission mode, used by the optical units.",
"Right: photograph of a coated plate.The emission spectrum in figure REF was measured using a Hitachi F-4500 fluorescence spectrophotometer with an incident beam of wavelength 270 nm, selected by a diffraction grating from a xenon lamp.",
"A standard rhodamine dye calibration sample was used to correct for drift in the lamp and spectrometer.",
"The measurement was made in transmission mode, and an artifact peak at a harmonic of twice the incident wavelength was observed between 525 and 600 nm.",
"This region is omitted from the reported spectrum of figure REF .",
"As shown in figure REF , although the absolute quantum efficiency for the platinum-undercoated cryogenic PMT is lower than the non-cryogenic version, the wavelength dependence is similar, and overlap between the TPB emission spectrum and the sensitive wavelength range of the PMT remains high.",
"Using the measured TPB emission spectrum for plate coatings and the PMT quantum efficiency curve provided, the spectrum-averaged PMT quantum efficiency is 15.3% $\\pm $ 0.8% per visible photon incident on the photocathode.",
"Figure: Measured efficiencies of various wavelength-shifting coatings, from .",
"In this plot, 50% TPB-PS is the MicroBooNE plate coating used in the optical units.",
"The 33% TPB-UVT is the light guide coating for the secondary light collection system described below.",
"Connected points were measured in a vacuum monochromator at room temperature, and non-connected points were measured in liquid argon with 128 nm scintillation light.",
"All points are normalized to the performance of an evaporatively coated plate.The wavelength-shifting performance of the coating was measured at 128 nm relative to evaporative coatings of the type studied in [67].",
"Coating efficiencies were measured as a function of wavelength between 128 and 250 nm using a vacuum monochromator at room temperature.",
"They were also measured in liquid argon using 128 nm scintillation light, relative to the same evaporatively coated plate.",
"These data are shown in figure REF , and more information about these measurements can be found in [66].",
"The absolute efficiency of the MicroBooNE coatings can be obtained by multiplying the relative efficiencies of figure REF by the measured absolute efficiencies from [67], and accounting for the temperature dependence in the wavelength-shifting efficiency of pure TPB reported in [68].",
"The expected number of visible photons per incident 128 nm photon is $0.98 \\pm 0.17$ for the MicroBooNE plate coating."
],
[
"UV Light Protection for the Wavelength-Shifting Plates ",
"TPB coatings have been shown to degrade under exposure to ultraviolet light [47] through a radical-mediated photo-oxidation to the UV-blocker and photo-initiator benzophenone [69].",
"Several measures were taken to ensure that degradation was minimized during the construction of the experiment.",
"The TPB powder and coated elements were stored in the dark at all times, with coated plates and light guides kept wrapped in foil and stored in a dark container before installation.",
"The detector construction area was covered with a UV blocking plastic [70], and test plates were placed at various positions in the clean tent to check for degradation from stray light.",
"After several weeks of exposure, one test plate with a clear line of sight to the tent entrance demonstrated a few percent degradation, and all others showed no observable loss of efficiency.",
"The open end of the MicroBooNE cryostat was shielded from light by a black curtain after installation, and the feedthroughs of the cryostat were blocked when not in use to prevent stray light from entering.",
"The coated plates were the final component of the optical system to be installed into the detector to give the minimum possible light exposure during the detector construction process."
],
[
"Cryogenic Mu Metal Shields",
"The trajectories of electrons within the PMT, particularly those between the photocathode and the first dynode, can be deflected by the Earth's magnetic field thereby reducing the PMT response.",
"This effect can be reduced or removed by surrounding the PMT with mu metal, a metal of high magnetic permeability.",
"Commonly used mu-metal fails to provide shielding at cryogenic temperatures.",
"Two types of cryogenic mu metal, Cryoperm 10 and A4K, both products of Amuneal, were identified that did provide shielding at cryogenic temperature.",
"The mu metal shields were tested in the apparatus shown in figure REF .",
"The system allowed for the PMT to be positioned at an angle relative to the vertical axis, with the rotator set to 30 positions from 0$^{\\circ }$ to 348$^\\circ $ .",
"The set-up was on a dolly that allowed for rotation about the vertical axis.",
"PMT tests were performed in air and in liquid nitrogen.",
"PMTs were dark adapted for 5 hours before testing.",
"A blue LED provided 1 to 2 single PEs of light through an optical fiber.",
"The fiber was fixed to the PMT mount such that the endpoint was fixed with respect to the PMT as the system was rotated.",
"Figure: Schematic of the system used to study the mu-metal shields.",
"The design allows rotation along all three axes.The mean charge from the PMT as a function of angle was recorded.",
"The error was primarily systematic.",
"The 10$^6$ LED pulses per data point give $<1\\%$ statistical error on the mean.",
"However, 24-hour studies of a single point showed $\\sim $ 5% variations in collected charge.",
"Results showed that A4K and Cryoperm function essentially identically, to within the measurement error, and so MicroBooNE chose A4K, based on significant cost savings.",
"A plot of the relative PMT gain variation versus angle from vertical, figure REF , shows that adding the shield significantly improves PMT performance.",
"This plot shows the effect of the A4K shield with the top aligned with the equator of of the PMT (black) and aligned with the zenith of the bulb (blue), compared to no shield (red) as a function of PMT azimuthal angle.",
"Given the 5% systematic error, the two shield positions are indistinguishable.",
"Figure: Red: Angular dependence of a PMT response with no shield; Black: for a shield that reaches the tube equator; Blue: for a shield that fully covers the tube to the zenith.As a result of these tests, the A4K shields were designed to extend just past the equator of the PMT.",
"The shield has small holes in the backplate that allow the PMT cables to exit the shield.",
"MicroBooNE is the first LArTPC to use cryogenic mu metal shields in its light collection system."
],
[
"Implementation of the Primary System ",
"The light collection system is composed of optical units assembled as shown in the top picture of figure REF .",
"The PMT is seated within the mu metal shield on three teflon pads attached to an equatorial support ring.",
"The neck of the tube slides inside a loose wire guide-loop that prevents the PMT from tipping.",
"The PMT is held within this assembly using teflon-encased wires that extend across the bulb and connect to wire hooks attached to the equatorial ring with stainless steel springs.",
"Legs extending from the support at the equator are screwed into a backplate for final mounting.",
"Concern about differences in contraction of the materials led to this design which holds the PMT in place, but with only moderate rigidity.",
"The units were tested to ensure the PMTs would not be displaced during installation and filling.",
"Three posts extend upward from the equatorial ring to hold the plate $\\sim $ 3 mm above the apex of the bulb.",
"The optical units slide into a cylindrical mu-metal shield, which screws into the equatorial ring.",
"The unit is then mounted on stainless steel back-plates affixed to a support rack, as shown in the bottom picture of figure REF .",
"Figure: Top: The optical unit mount internal to the shield, with components labeled; Bottom: Unit mounted on rails.",
"The clear plates were replaced with TPB-coated plates immediately before LArTPC installation, as discussed in the text , .The support rack consists of five stainless steel components, or modules, for ease of installation.",
"Each module has vertical height 1.83 m and horizontal length 2.07 m, resulting in a total horizontal length of 10.36 m. Unlubricated Thomson bearings fitted to the lower edge allow each module to slide into the cryostat on rails mounted in the vessel.",
"The system was designed to allow the light detection system to slide into the vessel after the LArTPC was installed.",
"However, in the end, scheduling permitted installation of the system before the LArTPC installation.",
"This had the advantage of making installation and surveying easier, but the drawback that the system would be exposed to UV light for a longer period.",
"Therefore, the units were installed with dummy clear acrylic plates, and the TPB coated plates were installed only just before the LArTPC was moved in and the detector could be easily protected from light.",
"During optical unit installation, each rack module was supported by a temporary mounting rail.",
"The optical units were then mounted in positions chosen to avoid obstruction by the LArTPC cross-bars, as shown in figure REF .",
"As the units were mounted and slid into the cryostat, the cables were loosely tied to the bars of the rack for support and constraint.",
"The “splitter” circuit, located outside of the cryostat, is shown in figure REF .",
"The splitter separates the HV of the PMT from its output signal which is subsequently split into a high-gain (HG) and a low-gain (LG) channel.",
"The HG and LG channels respectively carry 18$\\%$ and 1.8$\\%$ of the output signal.",
"This allows a wide dynamic range for ADC readout of the PMT pulses.",
"The capacitance was chosen to minimize reflections, since the bases are not back-terminated.",
"The HV is supplied to the splitter using BiRa Corporation, Model 4877PS modules.",
"Figure: The “splitter” circuit.",
"The circuit connects the HV source to the PMT.",
"It also provides a pathway for signal pulses from the PMT to reach the readout electronics via an AC-coupling capaciter (C2).",
"The signal is split into two copies, one provided with an attenuation factor of 0.18 and another at 0.018.",
"Both signal sources are recorded by the readout electronics in order to provide two dynamic ranges.The PMT cable system delivers HV and returns signals between the external splitter and the optical unit in the cryostat.",
"A single cable runs from an external connector, through a feedthrough that is filled with epoxy, into the cryostat and to the PMT base.",
"The RG316/U coaxial cable has 50 $\\Omega $ impedance.",
"Cables were terminated with Pasternack PE4498 SHV to accommodate that the cable carries HV to the tube as well as signals from the tube.",
"The cable carries an AC voltage rating of 1100 V; however tests showed the DC rating to be at least three times higher and so suitable for this use.",
"The cables were routed through feedthroughs consisting of a pipe filled with solidified epoxy mounted on a conflat disk.",
"On the warm side of the feedthrough, the cables were terminated at a patch panel with SHV connectors.",
"The SHV connector impedance has a negligible effect on the 20-30 ns PMT signals.",
"SHV cables connect the patch panels to the splitters.",
"The impedance of every channel was tested at the feedthrough patch panel for a stable and correct value for the base resistance, which was 4.04$\\pm $ 0.02 M$\\Omega $ ."
],
[
"PMT Testing and Quality Assurance",
"A vertical slice test (VST) of the MicroBooNE optical system was performed in the Bo Cryostat at Fermilab.",
"The Bo cryostat was a 250-liter vacuum-insulated vessel with an inner diameter 56 cm and a depth of 102 cm used for R&D studies, and, relevant to this paper, a vertical slice test of the MicroBooNE optical units.",
"The system is described in detail in reference [54].",
"The cryostat can be filled with purified LAr with oxygen and water levels below 1 ppb and a typical nitrogen contamination below 1ppm.",
"The light collection system was tested with light from visible (420 nm) and UV (250 nm) LEDs piped in via fiber, as well as scintillation light from $^{210}$ Po alpha sources and cosmic rays.",
"The slice consisted of two PMTs with base electronics, mu-metal shield, TPB plates, cable feed throughs, splitters, the HV power supply and the interlock system.",
"Tests were performed without and with the mu-metal shield.",
"The test made use of the data acquisition components described in section , including the shaper, FEM, trigger card, control card, and server.",
"The MicroBooNE trace impurity monitors were also used.",
"The VST informed the final design of many components, as well as producing results relevant to understanding the running conditions and performance expectations.",
"For example, during studies of the response of the slice to 128 nm scintillation light from the alpha source, valuable information was gathered on the single photo-electron dark rate and cosmic ray rate that could be applied to the MicroBooNE detector expectations.",
"As a second example, these runs allowed characterization of the pulse shape nonlinearities of the optical units, as seen in figure REF .",
"These were shown to be significant at $\\sim $ 300 PE in pulse amplitude.",
"Full amplitude saturation occurred at $\\sim $ 670 PE.",
"Thus, it was concluded that for pulses of more than 300 PE, pulse shape cannot be described by a linear superposition of single photo-electron pulses.",
"Figure: As shown by the VST, linearity of cosmic-ray induced PMT pulses is maintained up to amplitudes of around -1.7 V (300 PE), and amplitude saturation occurs at -3.7 V (670 PE)"
],
[
"Secondary System: Acrylic Light Guides for R&D",
"A secondary light collection system consisting of four lightguide paddles was also installed.",
"Their design has several advantages for future large detectors such as DUNE.",
"First, the collection area per channel is larger than the optical units, providing more coverage for the same number of electronics channels, cables, and feedthroughs.",
"Second, the detectors have a narrow profile so they can be slid between chambers in a multi-LArTPC detector, minimizing space requirements of the light collection system.",
"In the case of MicroBooNE, the design gradually guides light in bent acrylic bars to a PMT.",
"This design was an early alternative to a perfectly flat design that guides the light to SiPMs.",
"Running this system will provide long-term information on performance of lightguide based systems.",
"It also enhances the MicroBooNE dynamic range, since the lightguide detectors saturate at a much higher light level than the optical units.",
"In the case of the lightguides, the 128 nm light is absorbed and shifted by a clear wavelength-shifting coating, and the re-emitted light is guided to a 5.08 cm (2.0 in) Hamamatsu R7725-MOD PMT, as illustrated in figure REF , left.",
"The installed paddles consist of six bars.",
"A photograph of one coated paddle with eight bars is shown in figure REF , right.",
"The active length of each bar is 50.8 cm.",
"This system was added for R&D purposes and made use of 4 of the 8 spare channels available of HV, cables, feedthroughs, and electronics.",
"As shown in figure REF , each paddle is installed next to an optical unit for direct comparison of performance.",
"Figure: Left : Illustration of guiding mode, used by the paddles.",
"Right: photograph of a coated paddle.The coating requirements for plate assemblies and light guides are different, and so the composition and coating methods for each were separately optimized.",
"In the case of the light guide coatings, the figure of merit is the light emitted in guided mode.",
"Guided mode light is the light that is detected at one of the ends of a test sample, which is orthogonal to the illuminated face of the sample.",
"In addition to the wavelength-shifting efficiency of the active layer, the detected light yield is affected by the reabsorption and scattering losses in the coating as visible light propagates along the bar.",
"The light guide assemblies have a TPB coating of 33% TPB to 67% UVT acrylic by mass, also with ethanol surfactant.",
"The coating is applied as a single layer and the TPB remains suspended in the acrylic matrix as the coating dries, leading to a smooth, visibly transparent surface.",
"The performance and attenuation behavior of similar light guides to those installed in MicroBooNE were studied experimentally in [48].",
"The reported non-exponential attenuation suggests that surface losses dominate over bulk losses as the attenuation mechanism, and that the fractional loss per reflection within the light guide is of order 2-3% [69].",
"In the light guide coatings, the TPB is suspended in an acrylic matrix which leads to a slight broadening of the emission spectrum compared to the spectrum from the plates.",
"This is an expected effect–TPB fluorescence has been shown to have dependence upon its microenvironment [71], [72], [68], [73], [74], and reference [68] demonstrated spectral broadening in the presence of a polystyrene substrate.",
"Using the monochromator described for the plate spectrum studies, the light guide coating spectrum was measured in guided mode.",
"A 10 cm section of light guide, with the incident beam perpendicular to the TPB coated surface, was used.",
"Figure REF shows the results of this measurement.",
"Based on reference [68] it is expected that the emission spectrum for the light guide coating, with TPB embedded in the substrate, will not change significantly as it cools to 87 K. The expected efficiency for the light guide coating is found to be $0.25 \\pm 0.05$ emitted visible photons per incident 128 nm photon.",
"Figure: Measured emission spectra of the light guide coating in guided mode, and the R7725 quantum efficiency.",
"Only the quantum efficiency of the non-undercoated PMT model is shown, from ."
],
[
"Calibration",
"The flasher system for the optical units and the light guides is described in reference [52].",
"This system was developed to check the timing of the installed optical units, exercise the optical units during construction and commissioning, and to calibrate them during detector operations.",
"The reference provides engineering drawings and details.",
"The system is briefly summarized here.",
"A control board pulses an array of 400 nm LEDs, each of which is coupled through an optical feedthrough to 10 m optical fibers within the cryostat.",
"The custom feedthrough/patch panel design encases the fibers in Arathane CW 5620 blue with HY 5610 hardener.",
"The internal fibers are Molex FVP polymide fibers with diameter of 600 $\\mu $ m, cladding of 30 $\\mu $ m, and an additional buffer layer of 25 $\\mu $ m. Each PMT has an individual calibration fiber.",
"Each fiber is routed along the rack and attached to the PMTs by a fiber holder constructed of an aluminum standoff with a nylon-tipped set screw at a distance of about 5 cm from the PMT glass.",
"Figure REF shows the results of flasher tests on all 32 optical units and four PMTs on the paddles.",
"Time is along the $x$ axis and PMT channel number is along the $y$ axis.",
"The white region separating the optical unit PMTs from the paddle PMTs represents unused channels.",
"The colored bars indicate charge detected in the PMTs during flashing.",
"One can see that all tubes respond properly.",
"Figure: Charge detected in each PMT due to flasher tests, shown as a function of time.",
"The y axis is the PMT channel number, the x axis follows the sequencing of signals to the individual PMTs, an the response is given by the color scale.",
"Each PMT is flashed for a short period.",
"The white band indicates unused channels."
],
[
"Coupling of PMT Signals to the Anode Wires",
"ICARUS observed an unexpected cross-talk between their PMTs and collection wire plane [76].",
"The Argontube test stand at the University of Bern also observed this effect [77].",
"Therefore, while the LArTPC was under final testing, before being rolled into the cryostat, an experiment was devised in order to determine the implication of this cross-talk for MicroBooNE.",
"One of the production 200 mm (8.0 in) Hamamatsu R5912-02mod PMTs was placed with its face 12.7 cm from the LArTPC collection wire plane.",
"This PMT was encased in a dark box with an optical fiber delivering light from a LED flasher.",
"The collection wire plane was read out using a test-stand that reflects the final data acquisition design.",
"A clear signal was observed on the collection plane when the PMT produced a signal.",
"The magnitude of the signal was characterized as a function of photo-electron count (figure REF ) and separation between the PMT and wire-plane.",
"The signal induced on the collection plane was estimated to be no more than $\\sim $ 10 ADC counts for a typical cosmic ray under normal operating conditions.",
"This was reduced to $\\sim $ 2 ADC counts when the PMT was encased in the $\\mu $ -metal shields.",
"This was deemed an acceptably small amount that further shielding was not required.",
"Figure: Area of signal pulses recorded on collection plane (in arbitrary units) as a function of wire number (arb.",
"offset).",
"The legend indicates the calibrated photo-electron count.",
"The signal pulses are averaged over 50,000 repetitions.",
"The plots are pinned together at wire 300, as the distribution baseline fluctuated over the course of the experiment due to intermittent noise.The effect appears to saturate with photo-electron count, and is reduced when shorting the resistors between the anode and last dynode.",
"A later test with Argontube showed that the signal was drastically amplified when using capacitors not able to withstand cryogenic temperatures in the PMT base [77].",
"This suggests that the effect is electrostatic in nature, and probably due to capacitive coupling between the PMT and wire-plane.",
"This is also suggested by the similarity between the signals observed and those produced by anode-coupled readout of a light collection system [53]."
],
[
"Initial Performance of the MicroBooNE Light Collection System",
"The system was first powered on after the cryostat had been filled with liquid argon.",
"All the PMTs in both the primary and secondary system were found to be operational.",
"Figure REF shows the waveforms for all the PMT channels around the time that a large pulse, potentially from a cosmic ray muon, is observed.",
"After initial checks of the system's health, the one photo-electron response of each PMT was set such that a one photo-electron pulse has an amplitude of 20 ADC counts as seen by the PMT readout system.",
"The flasher LED system, described previously, is used to set this response.",
"Figure REF shows a candidate single photo-electron pulse following arrival of a TTL logic pulse driving the LED flasher system.",
"The LED light level is set so that the majority (about 80%) of waveforms see no response in the region where pulses from the LED are expected to occur.",
"This ensures that for windows with pulses, the pulses are of single photo-electrons.",
"Figure REF shows an example of the area vs. maximum amplitude of such pulses seen by a PMT during the flasher runs.",
"Figure: Top: Example waveforms from all 32 PMTs of the primary light collection system over a 31.25 microsecond readout window.",
"The waveforms from the PMTs are in white.",
"The blue waveforms are from the secondary lightguide PMTs, which are off in this picture.",
"Waveforms from channels reserved for logic inputs are shown in green.",
"In this image, the PMTs see two successive flashes of light at different parts of the detector.",
"Middle and Bottom: Magenta highlights the early and late pulse, with the insets showing the PMTs which fired.",
"Each pulse is likely from two cosmic ray muons traveling through the detector.Figure: Example waveform captured by the PMT readout electronics during single photo-electron calibrations.",
"The black waveforms are logic pulses that mark the time at which an LED in the flasher system is driven.",
"After some delay coming from PMT cable lengths and the flasher system, a candidate single photo-electron pulse is seen (at ∼\\sim 230 ticks).Figure: Distribution of the maximum amplitude and charge of pulses collected during an LED flasher calibration run.",
"The central distribution of events is due to single photo-electron pulses."
],
[
"Electronics and Readout Systems",
"The analog signals that develop on a LArTPC during its operation must be amplified, digitized, and written to disk for use in analysis.",
"Custom low-noise electronics that are capable of operating in the liquid argon environment have been developed for this purpose in MicroBooNE.",
"The data from these LArTPC electronic channels, as well as from the PMTs, is sent to a readout system that digitizes and organizes the information before passing it along to a data acquisition (DAQ) system that stores it on disk.",
"The stages of signal processing are illustrated in figure REF .",
"The following subsections describe the LArTPC cold electronics, the LArTPC and PMT readout electronics systems, and the DAQ system in more detail.",
"Details of the trigger capabilities available in MicroBooNE are also provided in this section.",
"Figure: MicroBooNE LArTPC and PMT signal processing and readout stages."
],
[
"Cryogenic Low-Noise Electronics",
"To obtain optimum detector performance, MicroBooNE uses cryogenic low-noise front-end electronics for readout of the LArTPC.",
"To reduce electronic noise, the interconnection length between the LArTPC wires and preamplifier should be as short as possible thus minimizing the total capacitance seen at the preamplifier input.",
"To accomplish this, the analog front-end ASICs, which include a preamplifier, shaper, and signal driver are located inside the cryostat in addition to the wire bias voltage distribution system, decoupling capacitors, and calibration networks.",
"The front-end ASIC and associated circuits are implemented on a cold mother board which is directly attached to wire carrier boards on the LArTPC itself.",
"Cold cables are used to transmit output signals from cold motherboards to warm interface electronics installed on the top of the signal feed-through flanges."
],
[
"CMOS ASIC",
"The analog front end ASIC is designed in 180 nm CMOS technology, which integrates both the preamplifier and shaper on a single chip.",
"Each chip has 16 channels to read out signals from 16 wires.",
"Each channel also has a charge injection capacitor for precision calibration.",
"In MicroBooNE, the shaper has four programmable gain settings (4.7, 7.8, 14 and 25 mV/fC) and four programmable peaking time settings (0.5, 1.0, 2.0 and 3.0 $\\mu $ s) that provide increased flexibility to the readout system.",
"The ASIC also has programmable baseline settings (200 or 900 mV) to accommodate different detection wire configurations: either collection or induction plane.",
"It has a selectable AC/DC coupling mode with a 100 $\\mu $ s time constant for the AC coupling mode, which can be used to reduce low frequency noise.",
"The ASIC also has built-in band-gap reference and temperature sensors to facilitate biasing and monitoring.",
"The CMOS ASICs consume only 6 mW/channel in their default configuration.",
"The front end ASICs of the entire detector generate 50 W of heat load that is easily handled by the cryogenics system.",
"Design guidelines that constrain the electric field and the current density to address the lifetime of CMOS devices operated at cryogenic temperatures have been applied to every single transistor (total $\\sim $ 15,000 transistors) in the ASIC design.",
"A picture of the layout of the CMOS ASIC is shown in figure REF .",
"Test results agree well with simulations and indicate that the analog and the digital circuits (including the digital interface) operate as expected in the cryogenic environment.",
"Figure: Layout of the CMOS analog front end ASICThe MicroBooNE LArTPC has 8,256 readout channels and a total of 516 CMOS ASICs are required to fully instrument the detector.",
"The production testing of the CMOS ASICs required two steps: both a warm and cold test.",
"The warm test was performed with a dedicated test board housed in a Faraday box containing a socket to house the ASIC for ease in chip exchange.",
"All programmable parameters (gain, peaking time, baseline, AC/DC coupling etc.)",
"were exercised with the warm test setup for careful screening at room temperature.",
"The yield of the warm testing of the ASICs was 89%.",
"ASICs must have passed the warm test before going through cold testing.",
"The cold test was performed with a dedicated test board containing 6 sockets to facilitate testing of multiple chips in liquid nitrogen at the same time.",
"A total of 201 ASICs went through cold testing with a yield of 97%.",
"Based on this high yield, it was decided not to continue the cold screening test on the rest of the production chips, as they were tested cold after being installed on the motherboard.",
"After enough ($\\sim $ 600) ASICs passed the production screening test, they were sent to an assembly house to equip the cold motherboard."
],
[
"Cold Motherboards",
"A cold motherboard was designed to house the MicroBooNE CMOS ASICs.",
"In this capacity, the motherboard provides signal interconnections both between the detector wires and preamplifier inputs as well as between the driver outputs and cold cables to the signal feed-through.",
"The cold motherboard design provides sufficient protection of the ASICs against electrostatic discharge during installation.",
"It also provides a calibration network and bias voltage distribution for the wire planes.",
"Specifically, a calibration signal enters the cryostat via a feed-through and reaches the preamplifiers through the motherboard.",
"Each preamplifier channel in the ASIC has a built-in switch to individually cycle the calibration injection.",
"The bias voltage reaches the LArTPC wires via a two-fold redundant path on the motherboard that allows the detector to operate normally even if one bias voltage channel fails.",
"As with the PMT bases, the cold motherboards use Rogers 4000 series as the PC-board material due to the cleanliness and thermal properties previously described in section REF .",
"The different positions of the wire attachments along the top and sides of the LArTPC requires 2 types of cold motherboard.",
"The top version of the motherboard has 192 readout channels that includes 96 Y channels, 48 U channels, and 48 V channels.",
"The side version of the motherboard has 96 readout channels that are either U or V channels.",
"A picture of the top version of a cold motherboard with 12 mounted ASIC chips is shown in figure REF .",
"Figure: Top version of cold motherboard with 12 ASIC chips, including 6 chips mounted on the top layer and 6 chips on the bottom layer.The MicroBooNE LArTPC required a total of 36 top version motherboards and 14 side version motherboards to instrument the full detector.",
"A test stand was built for testing of the front end electronics.",
"This test stand included a full readout chain from the cold motherboard, cold cable, signal feed-through, warm interface electronics, warm cable and receiver ADC board to a DAQ board based on a Xilinx ML605 FPGA evaluation board which sends acquired data to a PC over a Gigabit Ethernet.",
"The production test of each motherboard also involved both a warm and cold test.",
"Both tests used the same test stand, except the motherboard was placed in a Faraday box for the warm test and submerged in a liquid nitrogen dewar for the cold test.",
"Noise, gain, peaking time, and linearity parameters were measured in both warm and cold to screen the motherboard.",
"Motherboards had to pass both tests before being installed on the detector."
],
[
"Cold Cables",
"Cold cables transmit the detector signals from the cold motherboard to an intermediate amplifier on top of the signal feed-through and distribute power to the CMOS ASICs.",
"The cold cable is a custom-built 32-pair twisted pair flat ribbon cable with Teflon FEP insulation and 100 $\\Omega ~(\\pm 10\\%)$ impedance, using AWG 26 stranded wire with silver-plated copper.",
"Custom designed shells with jack screws used in the cable assembly ensured proper alignment of the insertion on the signal feed-through pin carriers.",
"Cold cables of two different formats were assembled by an assembly house: signal cables and service cables.",
"Signal cables are used to transmit amplified detector signals while the service cable is used to transmit calibration pulses and slow control/monitoring signals.",
"Signal cables were produced in three different lengths: 203 cm, 254 cm, and 457 cm to accommodate the different lengths between the cold motherboards and the signal feedthroughs, while the service cables were produced in two different lengths: 254 cm and 457 cm.",
"All of the cold cable assemblies were tested with a cable tester in the assembly house before they were shipped out.",
"In addition, 10% of the cold cables were tested in the test stand at BNL to confirm the quality of the cable assembly."
],
[
"Electronic Calibration",
"The MicroBooNE cold electronics include a precision charge calibration system.",
"Through the cold cable and calibration network on the motherboard, a calibration signal enters the cryostat via a feed-through and reaches the preamplifiers.",
"A built-in switch in the ASIC makes it possible to power cycle the calibration injection for every channel individually.",
"The electronics calibration is based on charge injection through known capacitances (180 fF) in the ASIC.",
"This system enables gain (charge sensitivity) calibration, verification of sense wire integrity and noise measurements.",
"The built-in electronics calibration capability is an important tool in testing and characterizing the overall performance of the detector readout system.",
"It was extensively used in the cold electronics production testing and the electronics checkout during installation, commissioning, and data taking."
],
[
"Performance Tests",
"The development of the analog front end ASICs was initiated using 180 nm CMOS technology and 300 K models, though the performance parameters are extracted at 77 K. CMOS was found to function at cryogenic temperatures with increased gain and lower noise.",
"The noise, gain, and pulse shaping were found to be as expected in evaluation tests of the ASICs.",
"Extensive testing of the ASICs mounted on the motherboards was performed; these tests were done in liquid nitrogen rather than at room temperature, since noise levels and characteristics of the ASIC performance in liquid nitrogen are similar to the performance in liquid argon.",
"Thus, cold tests were performed on all production cold motherboards fully populated with 12 chips.",
"A total of $\\sim $ 2,200 chip-immersions were accumulated in liquid nitrogen without any failures due to thermal contraction or expansion.",
"Figure: Plot of noise vs. channel number for 192 channels (12 ASICs) and at two different temperatures.",
"Noise is ∼\\sim 1,200 e - e^{-} at 293 K, and ∼\\sim 550 e - e^{-} at 77 K with 150 pF C d C_{d}Figure: Plot of gain uniformity of 12 ASICs, total 192 channels, at 77 K with two different gain settings.The test results show the noise of the front end readout electronics system decreasing uniformly for all 768 channels from $\\sim $ 1,200 $e^{-}$ at 293 K to less than 600 $e^{-}$ at 77 K with 150 pF detector (sense wire) capacitance.",
"A plot of noise versus temperature of 12 ASICs for a total of 192 channels is shown in figure REF .",
"The response of the front end electronics exhibits excellent uniformity at cryogenic temperatures.",
"As shown in figure REF , the gain variation of a cold motherboard with 12 ASICs is only 7% peak-to-peak across 192 channels.",
"The spread of the gain variation is only 1% of the gain setting."
],
[
"Warm Electronic Amplification",
"Signals from the cold electronics are carried over the cold cables to dedicated feedthroughs mounted on the cryostat.",
"The cold cables are connected to pin-carriers located on 356 mm outer-diameter CF signal feedthrough flanges that are mounted on nozzles N1A-N1K of the cryostat (see figure REF ).",
"The signal feedthrough design must accommodate 100$\\%$ hermeticity and high signal density.",
"A design based on the ATLAS pin carrier style was developed for this purpose.",
"Two 8-row pin carriers and two 7-row pin carriers are welded onto the CF flange, as shown in figure REF , and create a vacuum-tight seal.",
"Nine of the 11 signal feedthroughs receive signals from the three LArTPC anode planes (384 Y-plane, 192 U-plane, 192 V-plane), while the remaining two on the extreme ends of the cryostat only receive signals from one of the angled induction planes (672 U-plane on one feedthrough, 672 V-plane on the other).",
"Figure: Signal feedthrough flange consisting of a 356 mm CF flange with two 8-row and two 7-row pin carriers welded in place.A Faraday cage is mounted on the external, warm, side of the signal feedthroughs to provide shielding for the intermediate amplifiers located inside.",
"The bias voltage feedthrough, which supplies anode plane bias voltages into the cryostat, is built onto a small 70 mm CF flange welded onto the signal feedthrough flange.",
"A filter board mounted on the bias voltage flange filters noise and ensures a good ground connection.",
"Figure REF shows details of the signal feedthrough assembly with electronics boards, bias voltage feedthrough, and Faraday cage.",
"Figure: Left: Diagram of the signal feedthrough assembly, which includes intermediate amplifiers, Faraday cage, bias-voltage feedthrough and filtering circuit.",
"Right: Photograph of one of the feedthrough assemblies, partially constructed and being tested.The intermediate amplifiers provide $\\sim $ 12 dB gain to the LArTPC signals to make them suitable for transmission over a 20 m long cable to the readout electronics (see section REF ).",
"Each intermediate amplifier has 32 channels installed on the signal feedthrough flange and housed inside the Faraday cage to provide noise isolation.",
"Figure REF shows a picture of a prototype intermediate amplifier plugged on the signal feedthrough pin carrier.",
"The intermediate amplifier uses a 68-pin SCSI-3 connector to drive the 32 channels of signal differentially for better noise immunity.",
"The layout and connector position have been carefully designed to ensure the card can be plugged on the pin carrier in either direction.",
"This efficiently utilizes the limited available space on the top of the feedthrough, which also makes the design of the Faraday cage easier.",
"In addition to the intermediate amplifiers, there are two service boards mounted on the top of each signal feedthrough.",
"The service board provides regulated low voltage, control and monitoring signals to the analog front end ASICs.",
"It also provides pulse injection to the preamplifiers for precision calibration.",
"The control, monitoring, and calibration signals are provided to the front end electronics with two-fold redundancy.",
"Should one set of signals become defective the detector can still operate normally with the redundant set.",
"Each service board plugs onto a 64-pin carrier row.",
"Figure REF shows a picture of a prototype service board.",
"Figure: Left: Photograph of one of the intermediate amplifier boards plugged into a pin carrier during testing.",
"Right: Photograph of a prototype service board."
],
[
"LArTPC Readout Electronics",
"The MicroBooNE readout electronics system consists of two subsystems: the LArTPC and PMT readout electronics.",
"The LArTPC readout electronics are responsible for the digitization, processing and readout of the induction and collection wire signals after amplification.",
"The PMT readout electronics are responsible for the amplification, shaping, digitization, and handling of PMT (and lightguide paddle) signals, and can be used to provide a trigger signal for the readout and Data Acquisition (DAQ) systems.",
"While the LArTPC and PMT readout systems share the same back-end design that organizes and packages the data for delivery to the DAQ system, they employ different analog front-end and digitization designs, which are described in this and the following subsection.",
"The LArTPC readout electronics are responsible for processing the signals from the 8,256 wires in MicroBooNE after pre-amplification and shaping in the cold electronics, as described in section REF .",
"The pre-amplified and shaped analog signals from the cold electronics are transmitted to the warm electronics outside the cryostat, as described in section REF , and then passed to 130 custom-designed ADC and Front End readout Modules (ADC/FEMs) distributed roughly evenly over nine readout crates, as shown in figure REF .",
"The readout modules digitize the analog signals and then process and prepare them for shipping to designated DAQ machines (one DAQ machine per readout crate).",
"These are part of the back-end DAQ system, described in section REF .",
"Figure: Rack containing two of the nine MicroBooNE TPC readout crates.The LArTPC readout crates communicate with the DAQ machines via three duplex 3.125 Gb/s optical links that connect to a crate controller module and data transmitter (XMIT) module on the crate end, and to three PCI Express boards on the DAQ machine end.",
"The controller is responsible for configuration, trigger and run control command distribution as well as the slow monitoring of each readout crate.",
"The controller occupies one of the optical links while the XMIT is responsible for sending two separate streams of readout data to the DAQ machines via the two other optical links.",
"The first XMIT stream contains losslessly compressed LArTPC data associated with event triggers received by the LArTPC readout crates, such as the BNB trigger, and is referred to as the “NU” data stream.",
"The second stream is a continuous LArTPC data stream which is compressed with some data loss.",
"The continuous data stream is used for beam-unrelated physics analyses, such as the study of potential supernova neutrino events, and is referred to as the “SN” data stream.",
"The compression schemes used in the NU and SN streams are described in section REF .",
"All readout crates are synchronized to a common 16 MHz clock.",
"The clock sync is provided by a clock fanout board and is sent via coaxial cables to a distribution board which is mounted on each crate backplane.",
"The frame size is set to 1.6 ms, which is equivalent to the time it takes for charge produced on the far end of the LArTPC to drift to the wire planes at the design cathode voltage of -128 kV."
],
[
" Data Digitization",
"The amplified and shaped analog LArTPC signals are differentially received and digitized in the first section of the ADC/FEM readout modules.",
"Each ADC module holds 8 AD9222 octal-channel 12-bit ADCs and handles signals from 64 wires.",
"The wire signals are grouped in two sets of 32 consecutive wire channels: either 32 induction wires plus 32 collection wires or two sets of 32 induction wires.",
"The induction channel sequence alternates wires between the two induction planes.",
"The ADC module digitizes the signals continuously at 16 MHz.",
"Each channel has a configurable baseline, which is either set low (450 ADC counts) for collection channels or at the middle of the dynamic range (2055 ADC counts) for induction channels, thus ensuring that both the collection plane unipolar differential signals and the induction plane bipolar differential signals can make use of the full ADC analog input range.",
"The requirement to observe a MIP produced at the far end of the LArTPC in the induction plane determines the lower end of the dynamic range, while the requirement to observe a highly-ionizing stopping proton at the close end of the LArTPC without saturation sets the upper end.",
"The digitized outputs from the ADC module are passed directly to a Front End Module (FEM) in the second section of the LArTPC readout module.",
"The FEM houses an FPGA for data processing, data reduction, and preparation for readout by the DAQ system as described in the following section."
],
[
" Data Handling",
"The FEM board consists of a 14-layer printed circuit board which is mechanically integrated with the ADC board as illustrated in figure REF .",
"The choice of a smaller board allows for short trace lengths which is beneficial for high speed signals.",
"The full assembly comes together as a standard VME 9U card in height, with a 280 mm depth.",
"Differential outputs from the ADCs connect to the FPGA through HM-Zd connectors that have individual ground shielding on each differential pair.",
"Figure: Photograph of a MicroBooNE ADC+FEM board.The digitized data stream moves from the ADCs to a Stratix III Altera FPGA, which reduces the sampling rate of the ADC from 16 MHz to 2 MHz.",
"The 2 MHz sampling rate is optimized by taking into consideration the expected pulse shape provided by the convolution of the cold electronics, the expected LArTPC field responses, and the O(1$\\mu $ s) diffusion effects which govern charge drift within the liquid argon.",
"The FPGA stores the data from all 64 wires per board sequentially in time in a 1M $\\times $ 36 bit 128 MHz SRAM, grouping two ADC words together in each 36 bit memory word.",
"This requires a data storage rate of (64/2) $\\times $ 2 MHz $=$ 64 MHz.",
"The SRAM chip size and memory access speed allow for continuous readout of the LArTPC data.",
"Since data reduction and compaction algorithms rely on the sequential time information of a given wire, the data readout out from the SRAM takes place in wire order in alternate clock cycles, again at a rate of 64 MHz.",
"This read in/out sequence is illustrated in figure REF .",
"Figure: MicroBooNE readout sequence in the TPC FEM.Separate DRAM multi-event buffers on the FEM store the NU and SN data streams.",
"The data divert into the NU readout stream when a trigger is issued and received (as shown in figure REF ) signaling for example, an accelerator neutrino-induced event.",
"When an event trigger is received, 4.8 ms worth of data, relevant to that event, are packeted per channel and sent to the DAQ through the NU data stream.",
"The 4.8 ms readout size is governed by the maximum drift time and spans three or four frames.",
"In order to reduce the amount of data being transmitted, the FPGA trims the three or four frames to span the exact 4.8 ms required, 1.6 ms before the trigger plus 3.2 ms after the trigger.",
"In parallel, the data is continually sent out through the SN data stream, frame by frame.",
"The compression and data reduction algorithms applied to each of the two streams are described in the following section.",
"Figure: MicroBooNE trigger readout.After processing by the FPGA, the data passes to the crate backplane dataway on connectors shown in figure REF .",
"A token-passing scheme is utilized to transfer data from each FEM board to the data transmitter module (XMIT) in a controlled way, whereby each FEM, in the order of closest to furthest away from the XMIT module, receives a token, transmits its data to the XMIT, and passes the token on to the next FEM in the sequence.",
"For the NU stream, each FEM sends all data associated with a particular trigger number; while for the SN stream, each FEM sends all data associated with a particular frame number.",
"This data transfer is relayed via the otherwise passive crate backplane, and is limited to 512 MB/s.",
"In the XMIT module, the data is buffered temporarily and sent to the DAQ machine through the two streams, SN and NU, which proceed effectively in parallel."
],
[
"Compression Schemes",
"In the case of the NU data stream, a lossless Huffman coding scheme implemented in the FEM FPGA compresses the data by approximately a factor of 4.5 (dependent on ASIC gain and shaping settings).",
"Further reduction in the overall data rate processed by the DAQ system is achieved by exploiting a PMT trigger in coincidence with the BNB trigger, as described in section REF .",
"Huffman coding provides for lossless data compression by taking advantage of the slow variation of the waveform TPC data in any given channel.",
"In particular, this compression scheme relies on the fact that successive data samples on any given wire vary relatively slowly in time.",
"As such, when noise levels are low, any two adjacent data samples either coincide or differ by 1 ADC count.",
"The most frequent values for the difference in ADCs between successive data samples are assigned pre-specified bit patterns with the lowest number of bits possible.",
"Those bit patterns are encoded in the 16-bit data words that would otherwise be used for a single 12-bit ADC sample value.",
"As such, data reduction of up to a factor of 14 is theoretically possible (the uppermost two bits are always reserved for header information, in each 16-bit word).",
"In practice, the data reduction is sensitive to noise levels and LArTPC activity, and is also dependent on the gain and shaping time setting.",
"The compression factor achieved during commissioning by MicroBooNE is shown in figure REF .",
"Figure: Compression factors achieved on ADC data with Huffman compression.",
"The default shaping time in MicroBooNE is 2 μ\\mu s.Because of the low trigger rate (the BNB trigger dictates an upper bound on the trigger rate of 15 Hz), lossless Huffman coding compression proves sufficient for the NU data stream.",
"However, for the continuous SN stream, further compression becomes necessary, resulting in unavoidable data loss.",
"A method called “dynamic decimation” (DD) handles this case.",
"The DD scheme relies on recognizing regions of interest (ROI) in the data stream that contain waveforms corresponding to drift ionization charges.",
"Portions of the data stream not containing ROI contribute to pedestal determination, and ROI are identified as deviations from the continually-updated pedestal, buffered, and read out to disk.",
"At the time of this writing, the MicroBooNE SN stream compression scheme is being finalized and the SN readout stream is being commissioned."
],
[
"PMT Readout Electronics",
"The PMT readout electronics are responsible for processing signals from the 32 PMTs described in section and identifying light signatures coincident with the BNB and NuMI beam spills.",
"The coincidences generate PMT triggers that can be later mixed with other triggers in the Trigger Board (TB).",
"Signals from the four lightguide paddle guides installed in the LArTPC are also recorded by the PMT readout electronics, but these signals do not participate in the PMT trigger generation.",
"The stages of signal processing are illustrated in figure REF .",
"First, each PMT signal (with the exception of the light paddle guide signals) is split into two different gains, as described in section REF , with the HG channel carrying 18$\\%$ of the PMT signal and the LG channel carrying 1.8$\\%$ of the PMT signal.",
"Each gain is split once again into HG1 and HG2, and LG1 and LG2, in order to allow more flexibility in separately processing beam-related and beam-unrelated PMT signals.",
"All 32$\\times $ 2$\\times $ 2 (PMT) plus 4 (light guide paddle) signals are pre-amplified and shaped in 16-channel pre-amp/shaper boards (section REF ).",
"Three PMT readout modules receive the analog shaped signals differentially and digitize them (section REF ) at 64 MHz.",
"The PMT readout modules then process the signals in order to prepare them for shipping to a designated DAQ machine and to form a possible PMT trigger (section REF ).",
"Each one of the three PMT ADC+FEM readout boards used in the PMT readout system handles one of the following: Readout of and PMT trigger generation using the HG1 PMT signals associated with neutrino beam events.",
"The paddle signals are also readout by this board but they do not participate in the trigger generation.",
"Readout of and PMT trigger generation using the HG2 PMT signals that are out of beam time (i.e.",
"cosmic rays and other cosmogenic backgrounds).",
"Readout of the LG1 PMT signals associated with neutrino beam events and signals that are out of beam time.",
"Figure: PMT signal processing stages, and digital signal processing in the PMT FEM.",
"Each PMT signal is discriminated and gated (in the presence of an external gate, such as a beam gate), and primitives such as hit amplitude and hit multiplicity are used to construct a PMT-based trigger.",
"All discriminated/gated data is transferred continuously to a DRAM for the SN stream.",
"When there is a Level-1 trigger, data corresponding to four frames, including the trigger frame and one/two frames preceding/following the trigger frame, are transferred to a DRAM for the NU data stream for readout by the DAQ.After signal processing, the data is sent to a designated DAQ machine via a transmitter (XMIT) module in the same way as is done for LArTPC data.",
"Two data streams are provided: a NU data stream associated with event triggers and a SN data stream which a continuous version of the NU stream readout.",
"The PMT readout and trigger electronics share the same 16 MHz clock as the LArTPC readout electronics, and likewise keep track of time since run-start in 1.6 ms-long frames.",
"All PMT readout electronics are housed in a single VME 6U crate."
],
[
"PMT Signal Amplification and Shaping",
"The preamp/shaper boards read raw PMT signals from the PMT HV/signal splitters and shape them into unipolar signals with a 60 ns rise time.",
"The shaped signals are sent to the LArTPC readout boards differentially via short front-panel cables, in order to minimize noise, where they are digitized at 64 MHz.",
"The 60 ns peaking time allows digitization of two or three samples on the rising edge.",
"This in turn enables an accurate determination of the event $t_0$ needed to determine the $x$ coordinates of ionization signals along the drift direction.",
"An accurate time measurement also helps reject other tracks, such as cosmic rays, that cross the detector during the drift time."
],
[
"PMT Data Digitization",
"The ADC (Texas Instruments, ADS5272) module part of the PMT readout board (figure REF ) is responsible for digitization of up to 48 differentially-driven input signals.",
"The differential signals are digitized at 64 MHz.",
"The 64 MHz clock used by the PMT readout is generated starting from the 16 MHz clock that is common to all readout crates (LArTPC and PMT).",
"Figure: The MicroBooNE PMT readout board which digitizes 48 input signals.In addition to HG and LG waveforms from the PMTs, each readout board also receives, digitizes, and processes beam gate signal markers which arrive 4 $\\mu $ s before the BNB (1.6 $\\mu $ s) and NuMI (10 $\\mu $ s) beam gates.",
"These gates are used to (a) specially mark regions of interest where PMT data are read out continuously with no compression and (b) look for coincident PMT light signatures for trigger generation."
],
[
"PMT Data Handling and PMT Trigger Generation",
"PMT information is recorded in the NU data stream for four 1.6 ms frames associated with an event trigger: the frame containing the (asynchronous) trigger, the frame preceding the trigger frame, and two frames following the trigger frame.",
"To avoid the inordinate amount of data that would be generated at a 64 MHz sampling rate, the FEM applies a zero-suppression immediately after digitization, retaining only samples above a given threshold as well as enough information before and after this useful data to establish a local baseline value; this collection of information is referred to as a PMT readout ROI.",
"An exception is formed for beam-related or other likewise-triggered data where, for example, the 4 $\\mu $ s-early BNB and NuMI beam gates mentioned in section REF instruct readout of 1500 consecutive samples (23.4 $\\mu $ s) surrounding and including the neutrino beam spill period regardless of signal activity.",
"Two different discriminators are used: one that is active inside the beam spill period(s), and one that is also active outside the beam-spill-surrounding 23.4 $\\mu $ s. The latter discriminator governs the readout activity due to cosmic rays and other non-beam related activity.",
"The former discriminator enables PMT channels with pulse heights above a configurable threshold (e.g.",
"corresponding to 1 photoelectron) to participate in trigger multiplicity and pulse height sum conditions, as described in the following section.",
"The thresholds for those two discriminators are set to different levels and configured with different dead times for the HG and LG signals."
],
[
"Level-1 Trigger Generation",
"The MicroBooNE Trigger Board (TB), which physically resides in the PMT readout crate, issues a “Level-1” trigger in order to flag frames that must be treated differently; in the case of the LArTPC readout, the TB flags the 4 frames that must be trimmed and readout through the NU data stream and, in the case of the PMT readout, it flags the 4 frames that must be readout in full through the NU data stream.",
"The inputs to the TB include a BNB trigger input (maximum rate of 15 Hz), a NuMI trigger input (1.25 Hz), a Fake Beam trigger input (configurable frequency), a PMT trigger input, and two calibration trigger inputs, provided by the laser calibration system and a cosmic ray muon tracker, respectively.",
"The TB also has the ability to receive, via the crate controller, DAQ-issued (via software) calibration triggers, which are used explicitly for cold electronics and PMT calibration.",
"The various input triggers can be independently pre-scaled, masked, and mixed together (OR or AND) to generate an event trigger, referred to as a Level-1 trigger.",
"The FPGA firmware in the PMT FEM can generate two different types of PMT triggers based on the PMT signals: a cosmic PMT trigger and a beam gate PMT trigger.",
"Beam gate PMT triggers are configured in the same way for the BNB, NuMI, and Fake Beam.",
"The nominal criteria for these triggers are (1) PMT multiplicity $\\ge $ 1 and (2) summed PMT pulse-height $\\ge $ 2 PE summed over all 32 HG1 PMT channels.",
"Both criteria must be met during any 100 ns time interval coincident with the beam spill duration (1.6 $\\mu $ s in the case of the BNB and Fake Beam gates and 10 $\\mu $ s in the case of the NuMI gate), and only channels enabled by the beam gate discriminator can participate in the active pulse-height and multiplicity sums.",
"The criteria for a cosmic PMT trigger are (1) PMT multiplicity $\\ge $ 1 and (2) summed PMT pulse-height $\\ge $ 40 PE summed over any one of 28 preset groups of 5 HG2 PMT channels that are grouped based on their spatial correlation.",
"Again, only channels enabled by the cosmic discriminators can participate in the trigger generation.",
"In addition, a software-based algorithm has been written to mimic the capabilities of the beam gate PMT trigger performed in the FPGA and provide more flexibility in trigger criteria settings.",
"Details of this higher-level software trigger are described in section REF .",
"Figure REF diagrams the PMT readout and trigger logic.",
"Activation or masking of each of the trigger inputs and outputs is DAQ-controlled.",
"The trigger condition and explicit PMT trigger type, if applicable, is available for every event in the NU data stream at both the event-building stage and offline; this information is read out via a dedicated optical data stream, directly from the TB.",
"The trigger number and trigger time are also propagated and available in the NU data streams sent independently from each LArTPC and PMT crate and “sub-event” DAQ machine to an “event assembler” DAQ machine.",
"They can therefore be used to correctly associate data from the same event.",
"Figure: PMT readout and trigger logic.The overall readout control sequence is illustrated in figure REF .",
"When a trigger is generated by the TB it is passed to a fan-out module on a single cable and from there it is distributed to all crate controllers (LArTPC and PMT).",
"Through the crate backplane, the trigger gets propagated to each FEM.",
"An FEM that receives a trigger temporarily inhibits the SN stream with its associated decimation and initiates the loss-less readout scheme to direct the data to the appropriate readout path.",
"SN readout resumes once the XMIT is done sending all NU data associated with an event to the DAQ.",
"Figure: Readout control sequence.",
"The MicroBooNE Level-1 trigger is a hardware trigger which consists of the OR between a BNB, NuMI, and Fake Beam (strobe) trigger.",
"Once received, the Level-1 trigger is propagated to all readout crates and instructs PMT and TPC data readout into ten dedicated sub-event buffer DAQ machines.",
"The data across different DAQ machines is correlated at event building stage by the trigger number and corresponding trigger frame and sample numbers recorded in each data stream, per event.",
"All readout crates are synchronized and correlated to the same 16 MHz clock."
],
[
"DAQ Design",
"The MicroBooNE DAQ system acquires data from the readout electronics, writes data to local disk before transferring it to long-term storage, configures and controls the readout electronics during data-taking periods, and monitors the data flow and detector conditions.",
"These tasks are performed on a network of commodity servers running both custom and open-source software.",
"The data from each crate of the backend electronics is sent to a dedicated server (called the sub-event buffer, or SEB) via optical fibers, arriving in dedicated cards on the SEB's PCIe bus.",
"A real-time application places these data in an internal buffer, collects all segments belonging to an event, and creates a sub-event fragment that may be routed to a specified destination.",
"For the NU data readout stream, in which the data arrives with every Level-1 trigger, these fragments are sent to a single event-building machine (EVB) over an internal network.",
"Full events are checked for consistency and written to local disk on the EVB before being sent offline for further processing.",
"A high-level software trigger, described in section REF , is applied to the data to determine whether events should be written locally or ignored.",
"For the SN stream the data remains on the SEB's where it is written to disk and only sent for offline analysis on explicit requests, e.g.",
"on receiving a Super Nova Early Warning System (SNEWS) alert [78].",
"Data writing to either triggered or SN streams is limited by the RAID6 disk write speeds which are roughly 300 MB/s.",
"This is much less than the network bandwidth bottleneck, which is 10 Gb/s.",
"The 300 MB/s disk write speed therefore sets the maximum aggregate rate at which all SEB fragments can ship data to the EVB without loss of data.",
"With Huffman compression, which gives a data reduction of approximately a factor of five (figure REF ) and the PMT trigger, which reduces the data rates by another factor of $>70$ , this is more than sufficient for MicroBooNE's maximum 15 Hz beam spill rate.",
"MicroBooNE expects a total triggered write rate of around 12 MB/s.",
"The SN stream circular buffers, which are aggressively (non-losslessly) compressed beyond what the NU stream experiences, fill each SEB's 14 TB in on the order of one day, which is ample time to respond to a SNEWS alert.",
"After data is written to disk, it is then copied to another server on the internal DAQ network, where the raw data is further compressed, shipped, and queued to be stored on tape and disk cache using the Fermilab central data management system known as SAM.",
"Offline applications then begin processing the raw data, converting the binary data format into a LArSoft [79] ROOT-based format which can be used as input for reconstruction algorithms.",
"LArSoft is a common framework of software tools utilized by many LArTPC experiments at Fermilab and elsewhere.",
"A separate process collects beam data and, during binary to LArSoft conversion, inserts that data into the built events.",
"A duplicate copy of the data is also stored offsite at Pacific Northwest National Laboratory (PNNL).",
"This collection of approximately 15 “projects” and the database which holds and monitors the state of the data flow is known as the Python/Postgres for MicroBooNE Scripting system (PUBS), and is patterned after a similar database state machine that the Double Chooz experiment used for data management.",
"As PUBS pushes the data through this process, the progress of each project is monitored and viewable via GUI.",
"PUBS can also monitor the state of the SN stream data, held locally on the SEBs.",
"A separate offline PUBS instance controls the processing of the data, including applying newly calculated calibration constants as part of data quality management.",
"Figure REF schematically depicts the flow of data throughout the MicroBooNE DAQ system.",
"Figure: MicroBooNE DAQ data flow from raw to processed.Additional software components handle the management of the main DAQ processes moving the data.",
"A run control application issues configuration and state-progressing commands to the SEBs and EVB.",
"Configuration states are stored in a dedicated run configuration database, which allows for the setting and preserving of configuration information for the DAQ, readout, and additional components.",
"This database not only allows for creating the large ($\\sim $ 200 parameters) intricate DAQ run configuration files, but also enforces certain conditions which must hold for consistency.",
"For example, the configuration that initiates the ASICS charge-injection calibration also dials and captures the settings on the external pulser that drives the calibration signal and assures that ASICS gains and peaking time parameters are enforced and recorded.",
"Another important aspect of the system is monitoring the health of the DAQ.",
"Monitoring of DAQ components is accomplished through Ganglia which monitors basic system states (such as CPU, memory, and network usage) as well as allows use of custom metrics to monitor the data flow and status of the readout electronics [80].",
"These metrics are sampled and collected by the Experimental Physics and Industrial Control System (EPICS) slow monitoring and control processes, which archives desired quantities and provides alarms when pre-defined thresholds are exceeded [81].",
"Some examples of Ganglia metrics that are monitored and alarmed in EPICS are the rates of growth of the SEB data buffers, the fragment rates leaving each SEB, and fragment arrival rates at the EVB.",
"Figure REF shows examples of Ganglia metrics.",
"Figure: Ganglia metrics showing data flow on the EVB machine during a 5 Hz test run.",
"Shown as a function of time are: the external trigger rate (top-left), the received fragment rate (top-right), the disk data write rate (bottom-left), and the number of assembled queued-up events waiting to be written (bottom-right).",
"(11 fragments constitute a complete event in the MicroBooNE DAQ.",
")Additional online monitoring exists to check data quality in more detail, through both programmed checks and visual checks including a real-time event display.",
"The online monitoring takes snapshots into shared memory segments on the SEBs and the EVB, and thus provides the desired low latency checks of newly-arriving data.",
"It continually walks through these $\\sim 150$ MB snap-shotted events and outputs histograms of occupancies and rates which are saved in ROOT files [82].",
"The histograms are then displayed in a web-based monitoring system that is easily accessible by the MicroBooNE shift crew.",
"Channels are aggregated in a variety of formats, including the order in which they appear in crates or across the wires and PMTs themselves.",
"In this way, potential problems across connectors or crates, for example, may be more readily identified.",
"Noisy, quiet, and unresponsive channels are easily marked and displayed to the shift crew.",
"Figure REF shows an example of available online monitoring information.",
"Figure: The online monitoring GUI (Lizard).",
"The pedestal-subtracted ADC values for all 8256 wires in one minimum bias event are shown.",
"Each box is a channel.",
"Not all crates are fully populated with electronics."
],
[
"High-Level Software Trigger",
"After events are collected on the DAQ event-builder server, a suite of trigger algorithms are applied to the data.",
"Currently, these algorithms mimic the PMT readout electronics' FPGA-based beam gate trigger algorithms, described in section REF .",
"A search over the PMT digitized waveforms from the beam-spill period is performed, and if there is a significant amount of light in coincidence with the expected arrival time of the neutrinos, the event is marked and saved.",
"This selection is performed on data that passes the Level-1 trigger, which typically includes data from the BNB and NuMI beams, and randomly selected off-beam data from an “external” (or fake beam) trigger.",
"A fraction of the data from each of these Level-1 trigger input streams is also retained via a random prescale, which provides a selection of data that has not been biased by the trigger.",
"The high-level trigger algorithms take approximately 10 ms to return a result, a latency that is well-below the event-taking rate and so does not impact data-taking performance.",
"The average pass rate for data-events in the PMT beam gate trigger algorithm is roughly 5%."
],
[
"Infrastructure and Monitoring Systems",
"MicroBooNE is housed at LArTF, which is located in the BNB at Fermilab.",
"Complete knowledge of the electrical and cryogenic systems housed within LArTF is necessary to maintain acceptable operating conditions for the experiment.",
"Continuous monitoring of the beam being delivered to LArTF is also necessary for subsequent physics analyses.",
"This section describes the details of infrastructure within LArTF, as well as monitoring of the experiment and beam conditions."
],
[
"Electronics Infrastructure at LArTF",
"Figure REF shows a diagram of the monitoring system at LArTF, depicting racks located on a platform directly above the cryostat that house electronics for: LArTPC control and readout, light collection system, drift high voltage, purity monitors, calibration laser, trigger, and cryogenic control systems.",
"Additional server racks containing the DAQ servers, beam timing, and external network electronics are located in a separate computer room above and adjacent to the above-cryostat platform.",
"The distribution of power, data, and network connections to and from all of these racks is also presented in figure REF , and is described in more detail in the following sections, along with electronics safety systems and interlocks.",
"Figure: Top: Diagram illustrating location of deployed electronics racks and separation of detector and building grounds and clean and building power for differing racks.",
"Middle: Photograph of installed electronics racks on the LArTF platform.",
"Bottom: A diagram illustrating the general scheme of signal, network, and timing signal cabling in the LArTF computer room and platform."
],
[
"AC Power Distribution and Grounding for Low-Noise LArTF Data-taking",
"As with any large detector operating with a high dynamic range, prevention of electromagnetic interference and its attendant effects on MicroBooNE data is an essential aspect of detector design.",
"MicroBooNE's strategy for producing a low-noise environment for the LArTPC and associated readout electronics can be largely summarized in a few key points.",
"AC power distribution-related items will be described here, while cabling, connections, and shielding will be described in a following section.",
"“Clean power,” or AC power electrically isolated from AC power for the rest of LArTF (“building power”), is supplied to all sensitive electronics via an isolation transformer.",
"Highly sensitive electronics are housed inside the Faraday cage provided by the detector cryostat or inside Faraday cages directly grounded to the cryostat.",
"The detector cryostat is grounded to a “detector ground,” which is physically and electrically isolated from the ground provided to all other LArTF power circuits, or “building ground.” No direct electrical connections are present between detector ground and building ground.",
"This is accomplished through the use of insulating platform and cryostat saddle materials, insulating cable trays and cables, and by inserting insulating “breaks” (i.e.",
"fiber data links or insulating cryo pipe sections) when connections between sensitive and potentially noisy detector components are necessary.",
"Indirect pickup on clean signals through capacitive coupling to adjacent noise sources is minimized through use of detector-grounded shielding and electrically-insulating cable trays.",
"Ground loops on detector ground are avoided wherever possible by connecting all electronics racks directly to the cryostat and by minimizing direct electrical connections between racks.",
"Direct or capacitive couplings between building and detector ground are constantly monitored during installation and operation with a custom-designed impedance monitor.",
"A line drawing describing the production of clean power and clean ground is shown in figure REF .",
"A pair of 200 A clean power circuits produced at isolation transformers are used to power all sensitive racks, which are indicated in figure REF .",
"All racks containing LArTPC readout electronics are placed on one circuit, while all other sensitive equipment is placed on the alternate circuit.",
"On the platform, all racks utilize clean power with the exception of the calibration laser and in-line purity monitor racks, which either contain noise-producing elements or support building-grounded components.",
"All racks in the LArTF computer room utilize building power.",
"Figure: Line drawing of clean power generation and distribution and connections to detector ground.The 208 volt, 3-phase power is distributed to each individual electronics rack.",
"For racks with significant power requirements or a large number of components, this power is delivered to a Fermilab-designed “AC switch box,” which distributes power to an Eaton Power Distribution Unit (PDU) only upon receiving an interlock signal from a smoke detection system in each rack, which will be described in more detail below.",
"Rack components then receive power from one of the three phases on this PDU.",
"For racks with fewer requirements, power is supplied to components directly from an interlocked simplified AC switch box or SurgeX SX-1120-RT PDU.",
"Racks with sensitive electronics are grounded to the cryostat via copper sheeting running throughout insulated cable trays above the cryostat.",
"Sensitive components within each rack are connected to a tin-plated copper grounding bar electrically connected to the rack bottom and running the height of the rack.",
"Mechanical attachments to the rack provide grounding for less sensitive rack components.",
"As mentioned before, any unintentional direct connection between building and detector ground is immediately alerted by the impedance monitor located on the LArTF platform.",
"Figure REF shows photographs of this equipment.",
"Figure: Photographs of the installed saturable inductor (top), AC switch box (top middle), detector ground strap and connection (bottom middle), and impedance monitor (bottom)."
],
[
"DC Power Distribution to the MicroBooNE Detector",
"DC power is provided to the LArTPC and readout electronics by power supplies in clean-powered, detector-grounded racks for a variety of purposes including Holding the LArTPC cathode plane at voltage to produce the desired ionization electron drift speed.",
"Holding the two ungrounded anode planes at the proper constant voltage to ensure the planes are transparent to drifting electrons.",
"Operating the light collection system PMTs.",
"Powering the cold electronics located inside the LArTPC .",
"Powering the warm electronics located in the LArTPC and PMT readout electronics racks Powering auxiliary systems, such as purity monitors.",
"Table REF summarizes the required voltages or currents for each of these purposes as well as the power supply make and model utilized in each case.",
"Power supplies are located in the relevant subsystem's electronics rack, with power and grounding connections as dictated by that rack.",
"Table: Overview of MicroBooNE DC power distribution.",
"Delivered voltages or currents are listed, along with power supply makes and models and whether each supply utilizes clean or building power.DC supply power consumption is minimal in most cases, with the exception of the warm LArTPC electronics PL-508 supplies.",
"Care was taken to distribute the AC power load by limiting the number of high-draw PC supplies per electronics rack."
],
[
"Network, Timing, and Data Distribution for Low-Noise LArTF Data-taking",
"Network, timing, and data connections must be made between the detector, building-ground, and detector-ground racks to properly read out MicroBooNE data.",
"However, as described above, these connections must be made while maintaining strict detector-building electrical isolation.",
"Deployed interconnections meeting both of these requirements are displayed in figure REF .",
"Timing and LArTF-external network signals are brought into LArTF via electronics in the computer room, where all racks are building grounded and powered.",
"These signals are distributed and processed in the computer room via copper cable, while network and processed timing signals to be sent to the platform are converted onto fiber cables and aggregated into a central fiber termination box.",
"A fiber trunk line then delivers these signals to the platform, where another fiber termination box on a detector-ground rack is used to fan out these signals.",
"Network connections are fanned out via fiber to a network switch in each platform rack, while timing signals are re-converted to copper and further processed for use by the trigger system on a different detector-grounded rack.",
"All rack-to-rack cables are run in insulating cable trays beneath the platform.",
"PMT and LArTPC data are transferred from each detector feedthrough to readout crates in detector-ground racks via insulated copper cable whose shield is tied to detector ground.",
"Digitized crate output is then sent to the aforementioned platform fiber termination box, where these signals are sent via fiber trunk line to the computer room.",
"In the computer room, these fibers are then fanned out to the appropriate DAQ computer.",
"Readout crate and cold electronics control commands are transmitted in the opposite direction utilizing a similar scheme, with crate controls delivered directly via fiber, and cold electronics commands delivered via fiber to a copper fanout in a detector-ground rack.",
"Clock and trigger signals must also be sent from a central trigger rack to all detector-ground LArTPC/PMT readout racks.",
"These signals are transmitted via copper connections, and represent the only source of ground loops on detector ground.",
"To further reduce the possible impact of induced noise in these and all copper cables mentioned above, all insulating cable trays beneath the platform are lined with copper sheeting grounded to the detector.",
"As an additional precaution, all LArTPC signal copper cables are run in separate cable trays from power and auxiliary cabling beneath the platform as well as inside every rack.",
"All cables between all detector components have been uniquely labelled with serial number, source, and destination to allow for ease of replacement and reconnection.",
"Ample fiber and copper spares for every major cable type are also installed along with the production cables to allow for quick replacement of any failed cable."
],
[
"Interlocks and Safety Systems",
"All electronics racks contain smoke-sensing and temperature-monitoring systems, which, when interlocked with AC and DC power transmission in each rack, constitute a rack protection system (RPS) designed to meet Fermilab safety requirements and reduce the risk of fire and related damage in LArTF and to individual rack components.",
"The RPS principally consists of a smoke sensor connected to a Fermilab-designed rack protection box.",
"This box produces and outputs a 12 V interlock signal when the rack protection box is on and receiving a “no-smoke” signal from the smoke sensor.",
"This 12 V signal can be sent to the AC distribution box located in each rack, as described above, to allow AC transmission to all rack components only if the RPS is on and not detecting smoke.",
"A similar 12 V “RPS Status” signal is also produced by the RPS box for input into the MicroBooNE slow control box, which will be described in following sections.",
"Alternate contacts are available on the rear of the RPS box for coupling the status of additional subsystems, such as the DAQ and calibration laser uninterruptible power supply (UPS), to smoke sensor or rack power status.",
"Temperature sensors deployed in two or three locations in each electronics rack sample air temperature within each rack.",
"Temperatures at each sensor are read out and recorded in the slow-control database by the slow-control monitoring box.",
"In addition, the box also produces a 5 V interlock signal if all sampled temperatures are within pre-programmed thresholds.",
"In electronics racks distributing PMT- or LArTPC-related DC power these temperature interlock signals are input into each relevant power supply, allowing DC power distribution only when this interlock signal is present, for safety purposes.",
"Additional hardware interlocks ensure the non-simultaneous operation of particular systems.",
"In particular, the PMT system is disabled when cryogenic system liquid level sensors detect a level below that of the highest PMT bases, or when the UV laser system is active.",
"The former requirement is enforced with a dry-contact hardware interlock, while the latter is enforced with a software interlock in the MicroBooNE online software.",
"The UV laser system is also dry-contact hardware interlocked.",
"Finally, the HV drift power supply is directly interlocked with the cryogenic system controls liquid-level sensor via a dry-contact hardware interlock."
],
[
"Performance Measurements",
"The proper operation of each production electronics rack's AC and DC distribution and RPS systems has been tested prior to installation at LArTF.",
"Furthermore, test stands exercising functionality of DAQ, PMT and LArTPC electronics, trigger, and drift HV systems have successfully incorporated and tested various aspects of these same AC and DC distribution and RPS systems.",
"Impedances between detector and building grounds were recorded throughout the installation of the rack infrastructure at LArTF using the impedance monitor located on the LArTF platform."
],
[
"Slow Monitoring and Control System",
"MicroBooNE uses EPICS for controlling and monitoring most devices and conditions important to the experiment.",
"These include power supply controls, temperatures, fan speeds, rack protection interlock status, and various environmental conditions.",
"The DAQ, cryogenics systems, and beam data collection systems operate independently of the EPICS slow monitoring, but export data which are imported into EPICS for archiving and status displays.",
"Applications from the Control System Studio software collection [83] are used for providing displays, alarm notifications, and data archiving.",
"Figure REF is a screenshot of the slow monitoring and controls display.",
"This system is responsible for monitoring approximately 4500 different variables for MicroBooNE.",
"Figure: Standard detector controls and monitoring operator page.",
"The overview panel (top, middle) provides a “one-click-away” access to any sub-system panel of the MicroBooNE detector.",
"The alarm area (top, right), alarm tree (bottom, left) and the alarm table (bottom) panels alert the operator and provide alarm related information at various levels.",
"Theheartbeats (top, right) show whether programs for important subsystems such as cryogenics, drift high voltage and beam are running.",
"The green LEDs (bottom, right) show the status of important power supplies used in the detector."
],
[
"EPICS architecture",
"An EPICS system consists of any number of server programs implementing the EPICS Channel Access (CA) protocol [84] to provide client programs access to any number of process variables, where each process variable represents a quantity being controlled (an output) or measured (an input).",
"The EPICS base distribution provides a standard type of channel access server called an Input/Output Controller (IOC), which can be extended to support specific hardware as desired."
],
[
"Power supply controls",
"Most power supplies are controllable over the network through the NetSNMP protocol [85].",
"Several EPICS driver modules are available for SNMP, and MicroBooNE utilizes one written at NSCL [86].",
"An IOC with this SNMP module runs on a central computer and contacts the power supplies over a private network for monitoring and control.",
"The photomultiplier power supplies are reused from the DØ experiment and have custom IOCs running in their own controllers.",
"The main high voltage power supply has only a simple RS-232 serial interface; control and monitoring for it is provided by a nearby computer running an IOC with the EPICS asynDriver [87] and StreamDevice [88] modules."
],
[
"Slow controls box",
"MicroBooNE has a number of racks in various positions above the detector and in an adjacent server room.",
"Each is equipped with a rack-protection system and multiple digital temperature sensors, and most contain one or two fan packs, each containing 6 fans.",
"To monitor and control these devices, each rack has an 1U rack-mount enclosure containing an ARM-based single-board computer (SBC) running Debian GNU/Linux 7 and a custom interface board, collectively known as a “slow controls box”.",
"An off-the-shelf GESBC-9G20 from Glomation Inc. [89] is utilized for the SBC.",
"The custom interface board [90] connects the SBC to front panel LEDs, temperature probes, fan packs, and rack-protection-status input.",
"The temperature sensors are DS1621 chips, controlled and read out over an I2C bus by the SBC's I2C controller.",
"The DS1621 also has a thermostat output with programmable trip and reset temperatures, which are connected via the interface board to outputs that can be used to interlock devices in the racks, such as power supplies.",
"The fans provide pulse-per-rotation outputs, which are monitored by a 12-channel tachometer implemented via a PIC16F887 microcontroller, and also read out by the I2C bus.",
"An EPICS IOC runs in each SBC, with custom device drivers for reading all status information and controlling the heartbeat LED and temperature sensor trip and reset points."
],
[
"External data sources",
"Data are imported into EPICS channels from a number of external sources.",
"The primary reason for duplicating these data in EPICS is to integrate displays and warnings into one system for the experiment operators, and to provide integrated archiving for sampled data in the archived database.",
"An IOC running on a central computer provides “soft” process-variables channels for these data.",
"The data acquisition system provides many metrics describing its operation via the Ganglia system[80], [91], which makes the data available in an XML format easily read by a Python script, which in turn writes to EPICS using the PyEPICS module [92].",
"The hardware and system status of the DAQ computers is monitored through the industry standard Intelligent Platform Management Interface (IPMI); rather than writing a script to import data from IPMI directly into EPICS, a IPMI-to-Ganglia interface provided by the FreeIPMI's “ipmi-sensors” package [93] is used, allowing data to be imported via the same mechanism used for the DAQ metrics.",
"Separate Python scripts periodically retrieve data about outside weather conditions from various sources, cryogenics system data from a file retrieved non-intrusively from the IFIX cryogenics control system, and beam data from Fermilab's Intensity Frontier Beam Database (IFDB) [94]."
],
[
"Beam Monitoring",
"The primary source of neutrinos for the MicroBooNE experiment is the BNB.",
"NuMI beam data is also recorded on a spill-by-spill basis.",
"The primary beamline is lined with instrumentation including toroids which indicate beam intensity, “multiwires” showing beam profile in the horizontal and vertical planes, and beam position monitors measuring the mean beam position.",
"Data from these monitors are stored on a spill-by-spill basis in the IFDB.",
"Many of MicroBooNE's physics analyses require that beam data are recorded for each spill and matched to detector events.",
"Primary beam monitoring in MicroBooNE is done using a “dashboard” interface to IFDB.",
"By using the IFBD instead of the accelerator control system, the experiment can also verify that data are being acquired by the IFBD.",
"The dashboard is accessible over the network using a web browser.",
"The final monitoring step includes a post-data-merge check, ensuring that beam data are successfully matched with detector data for all beam spills.",
"This is done once the detector DAQ binary data file is closed.",
"The dashboard presents a graphic representation of the data, allowing for easy error identification, as shown in figure REF .",
"The experiment monitors two toroids, which indicate beam intensity; three multiwires, each of which shows beam profile in each plane; and beam position monitors along the beamline, which show the vertical and the horizontal position.",
"Parameters pertaining to the target and horn, such as cooling air temperature and horn current, can also be monitored.",
"The dashboard allows the experiment to easily add additional devices if experience demonstrates the need for their monitoring.",
"Data are monitored in near real-time.",
"A reasonable history is also kept so that changes are easily identified.",
"The accelerator control system provides detailed diagnostics tools to experts and can be used in case of any problems.",
"Figure: The BNB dashboard, showing graphical representation of beam instrumentation data, is used to monitor the beam.",
"The top box shows the timestamp of the beam spill and indicates if data is stale by changing the color.",
"The two top plots show the primary proton beam position along the BNB.",
"The bottom left plot shows the recent beam spill intensity and rate.",
"The bottom right plot shows the beam as projected onto the Beryllium target (the grey circle in the middle with radius of 0.5 cm).",
"The dashboard is accessible via web page providing both real time updates and the review of past data.",
"The page can be easily extended to monitor additional beam devices."
],
[
"UV Laser System",
"Knowledge of the electric field inside the drift volume of a LArTPC is a necessary aspect for performing subsequent event reconstruction.",
"Distortions of particle tracks due to field non-uniformities affect the accuracy of the particle momentum reconstruction based on multiple scattering.",
"Deviations from a uniform drift field may arise mainly due to accumulation of positive argon ions in the drift volume.",
"These ions are produced by ionizing particles from neutrino interactions, as well as by cosmic rays.",
"While electrons produced by ionizing particles are quickly (within few milliseconds) swept towards the readout system, ions have significantly lower mobility.",
"Their drift velocity in the MicroBooNE detector at nominal drift field is of the order of 0.8 cm/s.",
"The rate of cosmic muons in the LArTPC volume is calculated to be 11,000 muons/s within the active volume (assuming no overburden, and a cosmic rate of 200 muons/m$^2$ /s through a horizontal plane at the earth's surface and 63 muons/m$^2$ /s through a vertical plane), traversing a summed total length of 1.9$\\times $ 10$^4$ m through the liquid argon [95], [96].",
"Assuming that cosmic muons are minimum-ionizing (2.1 MeV/cm) and produce 23.6 eV per ion pair, positive ion charge is produced at a rate of 2.8$\\times $ 10$^{-8}$ C/s in the MicroBooNE LArTPC.",
"These ions are continuously neutralized at the cathode.",
"The resulting positive space charge distribution distorts the nominal LArTPC electric field potential, as shown in figure REF .",
"Such accumulated space charge leads to noticeable distortion of the drift field and, consequently, to deviations of reconstructed track coordinates by up to 10 cm (see figure REF ).",
"The ion drift velocity is comparable to local argon flow velocities, produced by global argon recirculation flow and thermal convection.",
"Therefore the distribution of positive space charge inside the drift volume may be not only nonuniform (figure REF ), but also time varying.",
"Figure: Simulation of distorting potential distribution due to positive space charge in equilibrium in the MicroBooNE detector.Figure: Deviation of a crossing track from its true coordinates due to a simulated positive ion space charge in MicroBooNE.Figure: Simulation of distribution of the positive space charge in presence of argon circulation.A nonuniform drift field in the LArTPC leads to the apparent bending of truly straight tracks of high-momentum ionizing particles.",
"In principle, a set of events from such particles allows for the reconstruction of the field in any small region of the LArTPC drift volume, using the systematic apparent curvature of tracks at different angles passing through that region.",
"In practice, the rate of such events from cosmic muons is too low to acquire sufficient statistics in reasonable time.",
"A method to generate straight ionization tracks at a defined location in liquid argon is described in [97].",
"A collimated photon beam from a pulsed UV laser with $\\lambda $ =266 nm can ionize liquid argon via multi-photon absorption.",
"The resulting ionization track is straight, characterized by low electron density, therefore featuring little charge recombination loss, unlike cosmic muon tracks.",
"Laser tracks are also free from $\\delta $ -electrons, which complicate track reconstruction in the case of muons.",
"The method was successfully exploited in the Argontube long drift LArTPC [98], [99], [100] to derive the non-uniformity of the electric field along its 5 m long drift volume [101].",
"The MicroBooNE LArTPC requires a set of such tracks in order to cover the whole sensitive volume to reconstruct field distortions.",
"This is the purpose of the laser calibration system, which features two UV lasers, one on either end of the LArTPC.",
"Tracks are generated by steering a pulsed laser beam, introduced to the cryostat via a custom-designed opto-mechanical feedthrough [102], through the LArTPC active volume."
],
[
"UV Laser Calibration",
"In order to unambiguously reconstruct a drift field vector at any point within the detector fiducial volume, a minimum of two ionization tracks are required to cross in the region of interest.",
"The total number of crossing points is determined by the required reconstruction granularity.",
"In MicroBooNE the initial scenario is to acquire 100 tracks from each direction, producing a reconstructed 3-D map with voxels that are approximately 10x10x50 cm$^3$ in volume.",
"This map provides a rough picture of the space charge distribution.",
"Depending on the results of this measurement, areas of interest can be studied in more detail.",
"Repetitive study of small volumes may reveal dynamics in the space charge distribution due to turbulent circulation, and should further inform an optimized scenario for a standard UV laser calibration procedure.",
"An algorithm of drift field calibration utilizes an input array of detector events with one straight ionization track in each element of the array.",
"The result of the calibration is a coordinate correction map, to be applied to each track, which converts apparently curved track images back to the true coordinate system where they are straight.",
"The algorithm is iterative with an optimizable iteration step and required accuracy.",
"An example of simulated reconstruction in 2-D space is depicted in figure REF , showing that the magnitude of the field distortions can potentially be reduced from 10 cm down to several millimeters in 99% of the detector volume.",
"Figure: Top: Simulated true laser beam trajectories in the detector; Middle: A map of the discrepancy between reconstructed and true track Y coordinate under influence of non-uniform electric field, as a function of the true Z coordinate of the track; Bottom: Map of the residual (true - reconstructed) track Y-coordinates, as a function of the true Z coordinate of the track, after applying the UV laser electric field calibration methods."
],
[
"Laser Source and Optics",
"A Nd:YAG laser emitting light at a wavelength of 1024 nm is used as the primary light source [103].",
"Inside the laser head, nonlinear crystals are installed in the beam line for frequency doubling and summing, resulting in a wavelength of 266 nm needed for ionization of liquid argon.",
"For this wavelength, the Nd:YAG laser is specified to produce an output energy of 60 mJ for each 4 to 6 ns long pulse and a horizontal polarization.",
"The maximal repetition rate is 10 Hz; the beam has a divergence of 0.5 mrad.",
"Figure REF depicts the arrangement of the laser system in MicroBooNE.",
"An optical table, mounted on the cryostat, was developed to accommodate the necessary parts in a stable and compact environment.",
"The components were chosen to be accessible remotely where necessary.",
"The emitted laser beam contains not only ultraviolet light but also all other harmonics generated in the crystal and the primary light of 1024 nm.",
"Dichroic mirrors optimized to reflect only wavelengths in the UV region are used to filter higher wavelengths out.",
"To absorb the transmitted wavelength behind the mirrors, glass-ceramic plates are installed.",
"The beam leaving the laser head is reflected by the first $45^{\\circ }$ -mirror into an attenuator.",
"For optical adjustment and verification of the non-visible UV-beam, a green alignment laser is placed behind this mirror and adjusted such that its path is coincident with the UV-laser beam.",
"In the attenuator (Altechna Wattpilot) a turnable $\\lambda /2$ -plate enables rotation of the orientation of the laser beam polarization.",
"Behind the attenuator two parallel plates are installed such that the angle of the incident beam matches the Brewster angle of the reflector.",
"Modulating the polarization of the beam adjusts the intensity of the reflected beam.",
"An aperture is placed in the optical path of the beam after the attenuator to control the beam diameter.",
"The last part in the beam line is a remotely-controllable mirror mount (Zaber T-OMG), which directs the beam to the laser feedthrough on the cryostat.",
"A photodiode (Thorlabs DET10A/M), which is sensitive in the ultraviolet region, detects the scattered light when a laser pulse is fired.",
"Its signal is then used as a trigger for data taking.",
"Both the UV-laser head and the optical table are mounted on a 15 mm thick aluminum plate.",
"Figure: Left: A schematic drawing (not to scale) of the components used for laser beam configuration.",
"An alignment laser (visible light) is introduced along the UV-laser path at the first dichroic mirror (M1), such that the paths overlap.",
"In the attenuator the UV-laser beam intensity can be adjusted to the desired level, and the diameter of the beam is controlled by an aperture.",
"A motorized mirror (M2) deflects the light into the direction of the feedthrough.",
"Beam dumps (BD) are installed behind all mirrors to absorb the non-reflected laser light.",
"Right: Side view of the cryostat indicating the mirror support structure with respect to the LArTPC."
],
[
"Steering System",
"One of the main challenges of the laser calibration system is the introduction of a steerable laser beam into the detector.",
"Earlier, an evacuated quartz-glass [104] was utilized to introduce a laser beam into liquid argon, however this beam had a fixed path through the detector.",
"For the purpose of scanning the full detector a fully steerable mirror in liquid argon is necessary.",
"In the MicroBooNE detector, this is achieved by mounting a mirror on a horizontally-rotatable support structure.",
"A rack and pinion construction, where the mirror is mounted on the frontside of a half gear (pinion), provides the necessary freedom for the vertical movement (see figure REF right).",
"To steer the horizontal movement from outside the cryostat, a commercial differentially-pumped rotational feedthrough is deployed (see figure REF left).",
"The rack and pinion construction is attached to a linear feedthrough.",
"Both feedthroughs are motorized to allow for remote control and automation of the mirror movement.",
"The mirror support structure was fabricated out of polyamide-imide (Duratron T4301 PAI), which has a very low outgassing rate and low thermal expansion coefficient, and is certified for operation at 87 K. To minimize the probability of discharges due to the close location of the feedthrough to the field cage structure in MicroBooNE, no conductive parts were used in the support structure.",
"The support structure has a total length of 2.5 m in MicroBooNE.",
"Both feedthroughs are equipped with high precision position encoders from Heidenhain.",
"The accuracy of the encoders is chosen such that a position accuracy of 2 mm for the laser beam spot over 10 m distance is achieved.",
"An external interface box controls the encoders and records a position reading upon receiving a trigger signal from the photodiode.",
"The DAQ computer accesses the position information over an ethernet connection.",
"The same computer is also used for steering the two motors via a motor driver system (over a RS232 interface).",
"Figure: Left: CAD cutaway drawing of the feedthrough construction is shown.",
"The yellow line indicates the laser path.",
"Right, CAD drawing of the cold mirror including the support structure."
],
[
"Performance Tests and Initial Operation",
"A full performance test of the laser calibration system identical to the one installed in the MicroBooNE LArTPC was performed prior to the final installation.",
"Apart from the general proof-of-principle of the laser calibration system, several operationally relevant parameters were identified.",
"These include scanning speed, positioning accuracy, positioning limits, optimal laser beam intensity, beam diameter, and the minimal achievable field distortion which can be resolved.",
"The test system consists of a LArTPC equipped with 64 readout channels and an active area of about 400 cm$^2$ , with a drift distance of 40 cm (see [105] for further details).",
"Several tests of the motorized feedthrough were performed under warm conditions before cold tests were conducted.",
"One crucial parameter for the quality of the electric field calibration is the resolution at which laser tracks can be aimed in the detector.",
"For the rotational axis this angle is directly measured on a circular scale.",
"For the vertical movement the linear displacement of the bellow is translated into a rotation inside the cryostat, as can be seen in the CAD drawing in figure REF .",
"This construction introduces uncertainties to the measurement position and backlash.",
"The backlash can be compensated by always approaching positions from the same direction.",
"For the translation of the linear movement $\\Delta L$ into a rotation $\\Delta \\phi $ the translation ratio $s$ according to $\\Delta L = s \\cdot \\Delta \\phi $ was measured with a Bosch GPL3 laser alignment device.",
"The obtained ratio is $s=$ 0.3499$\\pm $ 0.0002 mm/$^{\\circ }$ .",
"The dominant uncertainty in the vertical position measurement is the accuracy of the encoder $\\sigma _{\\mathrm {linear}} = \\pm 1~\\mu m$ , which translates into a vertical rotation measurement accuracy $\\sigma _{\\mathrm {vertical}} =$ 0.050 mrad.",
"Horizontal movement limitations arise from the construction of the feedthrough system, namely the warm mirror support structure.",
"This limitation originates in the manner the laser table is mounted relative to the feedthrough on the MicroBooNE cryostat.",
"Vertically the mirror can be rotated more than 45$^{\\circ }$ relative to the horizon in both directions.",
"In an upward looking configuration, no limitations arise which would affect the coverage of the detector with the beam.",
"When the mirror faces the opposite downward direction and the laser is properly aligned onto the center of the mirror, only the laser diameter and the size of the mirror limit the achievable coverage.",
"However slight misalignment will affect this, since the beam will not be in the optimal spot anymore.",
"In warm tests a maximal downward angle of the beam of 52.5$^{\\circ }$ with respect to the horizon was achieved.",
"During the cold tests the horizontal and vertical movement speeds were set to 2.6 $^{\\circ }$ /s and 1 $^{\\circ }$ /s, respectively, and horizontal and vertical angles of 81$^{\\circ }$ and 22$^{\\circ }$ , respectively, were covered.",
"Warm tests of the fully expanded setup showed vibrations if the chosen movement speed was too large.",
"The vibrations are expected to be dampened with a more stable installation on the detector, and with the immersion of the setup in liquid argon.",
"Modulation of the beam energy with respect to the vertical alignment of the cold mirror was found to be crucial for obtaining sufficient ionization in the detector.",
"Investigations showed that the reflectivity of the selected dielectric mirrors, which were optimized for 45$^{\\circ }$ in air, are very sensitive to the angle of incidence in liquid argon.",
"Therefore during a calibration run, the beam energy has to be controlled.",
"The emitted UV-laser beam has a diameter of 6 mm and will spatially diverge during propagation.",
"A beam with this diameter will produce an ionization signal larger than the wire spacing, which will limit the capabilities of the full system.",
"With the aperture a small as possible diameter of the laser was selected to enter the detector.",
"Measurements of the diameter were performed with thermal paper (used for thermal printing) on which the selected beam spot burns in.",
"The minimal achieved diameter was 1 mm."
],
[
"Conclusion",
"The MicroBooNE detector is the culmination of several years of development and construction.",
"Innovations such as custom cold (in-liquid) electronics, non-evacuated cryostat, 2.5 m drift, and a UV laser calibration system represent major technological advances that future experiments will build upon.",
"Operations of MicroBooNE began in the summer of 2015, and a cosmic ray tagger system was added in the fall of 2016.",
"Future publications will be dedicated to describing the performance of the detector systems introduced in this paper.",
"This material is based upon work supported by the following: the United States Department of Energy, Office of Science, Offices of High Energy Physics (OHEP) and Nuclear Science, Intensity Frontier Fellowships and Neutrino Physics Center Fellowships through OHEP, the U.S. National Science Foundation, the Swiss National Science Foundation, the Science and Technology Facilities Council of the United Kingdom, and The Royal Society (United Kingdom).",
"Additional support for the laser calibration system and cosmic ray tagger was provided by the Albert Einstein Center for Fundamental Physics.",
"Fermilab is operated by Fermi Research Alliance, LLC under Contract No.",
"DE-AC02-07CH11359 with the United States Department of Energy."
]
] | 1612.05824 |
[
[
"Probing Bulk Superconducting Order Parameter in Ba(K)Fe$_2$As$_2$ by\n Four Complementary Techniques"
],
[
"Abstract Using four different experimental techniques, we performed comprehensive studies of the bulk superconductive properties of single crystals of the nearly optimally doped $Ba_{1-x}K_xFe_2As_2$ ($T_{c} \\approx 36\\,K$), a typical representative of the 122 family.",
"We investigated temperature dependencies of the (i) specific heat $C_{el}(T)$, (ii) first critical magnetic field $H_{c1}(T)$, (iii) intrinsic multiple Andreev reflection effect (IMARE), and (iv) infrared reflectivity spectra.",
"All data clearly show the presence of (at least) two superconducting nodeless gaps.",
"The quantitative data on the superconducting spectrum obtained by four different techniques are consistent with each other: (a) the small energy gap $\\Delta_S(0) \\approx 1.8 - 2.5\\,meV$, and the large gap energy $\\Delta_L(0) \\approx 9.5 - 11.3\\,meV$ that demonstrates the signature of an extended s-wave symmetry ($\\sim~33 \\%$ in-plane anisotropy), (b) the characteristic ratio $2\\Delta_L/k_BT_C$ noticeably exceeds the BCS value."
],
[
"Introduction",
"The symmetry structure of Cooper pairs is thought to be the key to the understanding of the pairing mechanism of their superconductivity.",
"It is well-known that in conventional superconductors the electron-phonon interaction gives rise to the attraction between two electrons, thus forming Cooper pairs.",
"However, Superconductors, whose averaged order parameter over the entire Fermi surface yields zero, are called unconventional.",
"In Iron-based superconductors, the popular opinion is that the electron-phonon is not strong enough to overcome Coulomb repulsion and form Cooper pairs.",
"The nature of the pairing state in iron-based superconductors is the subject of much debate [1], [2], [3], [4], [5], [6].",
"The ternary iron arsenide BaFe$_2$ As$_2$ shows superconductivity at about $37-38$ K by hole doping [7].",
"Among various known Fe-based superconductors (FeBS), these 122 type family compounds may be grown as high quality and large size single crystals with easily variable doping.",
"Band structure calculations show that the low energy bands are dominated by the Fe 3$d$ orbitals forming multiple band metallic state: hole-like Fermi surfaces (FS) around the $\\Gamma $ (0,0) point and electron-like Fermi sheets around the M $(\\pi ,\\pi )$ point in the Brillouin zone (BZ).",
"The electron and hole-like FS sheets in the normal state of (Ba$_{1-x}$ K$_x$ )Fe$_2$ As$_2$ observed in angle-resolved photoemission spectroscopy (ARPES) are gapped by either the spin density wave (SDW) [8], [9] or superconducting [10], [11] order in the parent ($x=0$ ) or superconducting ($x> 0.15$ ) compound, respectively.",
"It is well experimentally established that potassium doping leads to suppression of the SDW ordering in the parent BaFe$_2$ As$_2$ compound and induces superconducting (SC) state.",
"The Hall coefficient and thermoelectric power (TEP) measurements for the parent BaFe$_2$ As$_2$ indicate $n$ -type carriers, whereas potassium doping leads to change of the sign in Hall and TEP coefficients, thus indicating $p$ -type carriers in superconducting Ba$_{1-x}$ K$_x$ Fe$_2$ As$_2$ [10].",
"For the optimal doping $x \\approx 0.4$ the superconducting critical temperature reaches $T_c\\approx 38$ K [7], [12].",
"In the normal state, the electron and hole sheets of the FS are of a comparable size [13], [15], [16], [14].",
"In the superconducting state, several energy bands at the Fermi energy give rise to multiple energy gaps in the respective superconducting condensates [1].",
"Recent specific heat, magnetization, muon spin rotation ($\\mu $ SR), tunneling spectroscopy, Andreev reflection spectroscopy, and ARPES measurements provide clear evidence of multiple gap structures in 122-type FeBS.",
"The available quantitative experimental data on the key superconducting parameters probed by distinct techniques as well as in various experiments are far of being consistent.",
"Also, identification of the superconducting gaps with the relevant FS bands is hampered by the fact that each particular probe is sensitive only in a limited energy range.",
"Thus far, thermodynamic specific heat measurements with optimally doped BKFA crystals revealed either two nodeless superconducting gaps, $\\Delta _1 = 11$ meV and $\\Delta _2 =3.5$ meV [12], or one gap: $\\Delta =6$ meV [17], or 6.6 meV [18].",
"By fitting temperature dependence of the lower critical magnetic field $H_{c1}(T)$ extracted from low field magnetization measurements, two superconducting gaps were found in Ref.",
"[19], $\\Delta _1(0)=8.9 \\pm 0.4$ meV, and $\\Delta _2(0)=2.0 \\pm 0.3$ meV.",
"Penetration depth extracted from $\\mu $ SR leads to $\\Delta _1=9.1$ meV, and $\\Delta _2=1.5$ meV [20].",
"The specific heat (SH) measurements [21], [22], [12] are known to suffer of several evident problems with data treatment.",
"The SH data contains contribution from the lattice, that is subtracted to some extent in order to determine the electronic contribution.",
"The lattice contribution to the SH is typically estimated by suppressing the SC transition in high magnetic fields or by measuring SH for the parent non-SC compound.",
"Therefore, the lattice SH cannot be accurately obtained in FeBS because of the very high upper critical field and because of magnetic/structural phase transitions at higher temperature in the parent compound.",
"The majority of the earlier SH data suffer from a residual low-temperature non-superconducting electronic contribution and show Schottky anomalies [22], [23].",
"Moreover, superconductivity-induced electronic SH is very sensitive to the sample quality and phase purity [12].",
"Also, in the earlier SH data analysis, the data are commonly fitted to the phenomenological multiband $\\alpha $ -model [24], [25], that assumes a BCS temperature dependence of the gaps.",
"However, our direct measurements by means of multiple Andreev reflections effect (MARE) spectroscopy [26], [29], [28], [27], [30], [31] do not support this assumption and clearly show that the $\\Delta (T)$ dependences for the multiband superconductors (such as MgB$_{2}$ , and FeBS) deviate from the BCS-type because of the interband coupling.",
"Finally, fitting the SH data with the multiband model requires several adjustable parameters.",
"The amount of the SC gaps detected in ARPES measurements varies, depending, apparently, on the instrument resolution, crystal and its surface quality: initial experiments [15] reported large gap $\\Delta _1=12$ meV on both small hole-like and electron-like FS sheets, and a small gap $\\Delta _2=6$ meV on the large hole-like FS; similar results were reported in Ref.",
"[20]: $\\Delta _1=9.1$ meV, and $\\Delta _2 <4$ meV.",
"It should be noted that the small gaps developed on the inner hole and inner electron FS are difficult to resolve experimentally in ARPES measurements.",
"Later, some more SC nodeless gaps were observed; particularly, in Ref.",
"[32] the inner FS sheet around $\\Gamma $ point was found to show large ($10 - 12$ meV) and slightly momentum-dependent gap while the outer FS sheet has nearly isotropic small gap ($7 - 8$ meV).",
"In Ref.",
"[16] three hole condensates ($\\alpha , \\beta ,\\gamma $ ) were found around $\\Gamma $ point , and one electron condensate ($\\eta $ ) around M-point of the BZ, all with nodeless SC gaps.",
"$\\Delta _\\alpha $ was found warped along $k_z$ : $\\Delta _\\alpha =6-11.5 $ meV as $11.5\\cos (k_xa)\\cos (k_yb)+2.1\\cos (k_zc)$ , whereas $\\Delta _\\beta $ and $\\Delta _\\gamma $ were isotropic.",
"The $\\eta $ SC-gap is also almost isotropic along $z$ and a rhomb-like anisotropic in the $ab$ -plane.",
"Finally, in STS tunneling measurements two nodeless gaps $\\Delta _2=7.6$ , and $\\Delta _1 =3.3$ meV were found in Ref.",
"[33].",
"Substantial efforts have been made in order to understand the physics of the pairing mechanism.",
"On the theory side, for the Fe-based superconductors, which have both electron-like and hole-like pockets, there is general agreement among theoretical approaches [39], [37], [38], [35], [34], [36] that the starting point to the gap symmetry is the $s^{\\pm }$ type with opposite sign of the gap on the electron and hole pockets.",
"This symmetry, however, may change as FS sheets size changes [10], [40], or as nonmagnetic impurities are introduced [41].",
"The majority of experimental data, cited above reported nodeless $s$ -type symmetry gaps.",
"However, thermodynamic probes are sensitive to the line nodes with sufficiently high spectral weight, whose existence in BKFA they rule out.",
"ARPES measurements with optimally doped (Ba$_{1-x}$ K$_x$ )Fe$_2$ As$_2$ [34], [15], [42] and Ba(Fe$_{1-x}$ Co$_x$ )$_2$ As$_2$ [43], have identified nodeless gaps on the hole pockets.",
"We recall that thermodynamic measurements on these same materials [20], [44], [45], [46] also show nodeless behavior consistent with $s$ -wave gap symmetry.",
"Recent data of the $T_c$ dependence on nonmagnetic impurities in Ba(Fe$_{1-x}$ Co$_x$ )$_2$ As$_2$ disordered films [47] initially seemed to be inconsistent with $s^\\pm $ type theory predictions [39], however, a more detailed subsequent analysis of the same data [48] lead the authors to the conclusion on the $s_\\pm $ gap symmetry.",
"Finally, the phase sensitive SIS tunneling measurements [49] reported the $s^{\\pm }$ symmetry for current injected in the $ab$ -plane (and $s^{++}$ -wave – for current injected along $c$ ).",
"We conclude that the existing experimental data on the gap structure and anisotropy in $k$ -space are contradictory enough.",
"In this context, it is highly important to probe the superconducting properties with a set of independent experimental techniques.",
"Each of the experimental probes has its own limits of applicability and requires particular model assumptions for extracting the quantitative data from the observables.",
"Comparing the results obtained by several independent techniques one may test the validity of model assumption and obtain most reliable information.",
"In Ref.",
"[46] this approach has been implemented by applying two independent bulk probes, i.e.",
"by measuring the London penetration depth and MARE.",
"Despite the fact that well consistent data have been obtained in Ref.",
"[46] on the gap magnitude, these measurements did not fully address the problem since were performed with similar, though not identical samples and even of the nominally different composition, Ca$_{0.32}$ Na$_{0.68}$ Fe$_{2}$ As$_{2}$ and Ba$_{0.65}$ K$_{0.35}$ Fe$_{2}$ As$_{2}$ .",
"This drawback is improved in the current study, where we have succeeded in performing four types of measurements with one and the same large size single crystal of nearly optimally doped (Ba$_{1-x}$ K$_{x})$ Fe$_{2}$ As$_{2}$ (with $x=0.33-0.35$ ).",
"In particular, we have measured temperature dependences of the specific heat, lower critical field, $H_{c1}$ , multiple Andreev reflections effect, and infrared reflectance spectra.",
"We obtained self-consistent data that clearly shows the presence of two or more superconducting condensates with nodeless order parameters.",
"The quantitative data on the superconducting properties obtained by four complementary techniques may be summarized as follows: (a) the superconducting state has two (or more) nodeless gaps: the large gap, $\\Delta _L = 9.5 - 11.3$ meV with extended $s$ -wave symmetry, and the small gap, $\\Delta _S = 1.8-2.5$ meV; (b) both energy gaps fall with temperature in the way different from the single-band BCS-like behavior, (c) the characteristic ratio $2\\Delta _L/k_BT_C$ noticeably exceeds the BCS limit and indicates rather strong electron-boson coupling in the driving bands."
],
[
"Experimental details",
"The large size single crystal of Ba$_{1-x}$ K$_{x}$ Fe$_{2}$ As$_{2}$ was synthesized by self-flux technique using FeAs as the flux, for details see [50], [51].",
"For Ba-122 FeBS, optimal level corresponds to $x \\approx 0.4$ for K doping ($T_c ˜= 38.5$ K) [12], [52].",
"The chemical composition of our sample was verified by energy dispersive X-ray (EDX) spectroscopy probe.",
"According to the magnetic susceptibility measurements in zero field (see upper inset of Fig.",
"REF ) and specific heat measurements the critical temperature of the superconducting transition $T_c= 36.5 \\pm 0.2$ K. If one relies on the known phase diagram [52] the average bulk doping level of the studied samples may be concluded to correspond to $x=0.33$ .",
"The high quality of the crystals is confirmed by various physical characterizations: (i) a sharp superconducting transition observed in susceptibility and specific heat measurements at $T\\approx 37$ K [21] (see inset of Fig.REF ) confirming the good quality of the single crystal [21]; (ii) the chemical composition, crystal structure and lattice parameters tested by X-ray diffraction (Pan Analytical X'Pert Pro MRD).",
"The critical temperature $T_{c} \\approx $ 36.5-37 K, is evidenced by magnetization, DC transport measurements, and also by Andreev reflection spectra flattening measured at various points of the bulk crystal.",
"Low field magnetization measurements were performed by using a SQUID magnetometer MPMS-XL7, and specific heat measurements - with PPMS-9 system, both from Quantum Design.",
"Infrared reflectance (IR) spectra were measured with IFS-125HR Fourier transform infrared spectrometer from Bruker, and Andreev reflection spectra were obtained by the break-junction technique [30], [53]."
],
[
"DC magnetization",
"The London penetration depth $\\lambda $ is a fundamental parameter that carries signatures of the pairing mechanism, and therefore is a powerful tool for probing the superconducting state [54].",
"The London penetration depth is related to lower critical field $H_{c1}$ , that pinpoints the vortices penetration into the sample.",
"In this section we report measurements of the first critical field $H_{\\rm c1}$ for Ba$_{1-x}$ K$_x$ Fe$_2$ As$_2$ sample.",
"Our analysis of temperature dependence of the lower critical field $H_{c1}(T)$ for the $B|| c$ direction support the presence of two $s$ -wave-like gaps with strongly different magnitudes and slightly different contributions.",
"By analyzing the $H_{\\rm c1}$ temperature dependence we reveal the presence of the two SC condensates with $s-$ type symmetry of the order parameter.",
"The two SC gap values extracted from the $H_{\\rm c1}(T)$ analysis correspond to $2\\Delta _1(0)/k_BT_c =1.2 \\pm 0.2$ (or $\\Delta _1(0)= 2 \\pm 0.3$ meV), and $2\\Delta _2(0)/k_BT_c = 6.9 \\pm 0.3$ (or $\\Delta _2(0)= 11 \\pm 0.5$ meV); their weights extracted by fitting with the $\\alpha -$ model are $\\varphi _1=0.54 \\pm 0.02$ for the small gap and $\\varphi _2=0.46 \\pm 0.02$ for the large gap."
],
[
"Experimental",
"The DC magnetization measurements were performed with a rectangular slab, $3\\times 4 \\times 0.05$ mm$^3$ , cleft from the same large crystal used for all other measurements.",
"The approach used for extracting the first critical magnetic field is based on measuring the magnetic field value, for which the vortexes start penetrating into superconducting bulk destroying the ideal Meissner effect.",
"In other words, we determined such field value $H_{\\rm c1}$ which corresponds to the onset of nonlinear $M$ versus $H$ dependence.",
"Measurements were performed with MPMS-XL7 (Quantum Design) in the temperature range $2-36$ K with a step size of 1 K. Magnetic field direction was aligned with the crystal $c$ -axis.",
"Figure: Magnetic field dependence of the magnetization in the temperature range 2–36 K. The dashed straight line extrapolates the linear M(H)M(H) dependence observed in weak fields.",
"Upper inset shows magnetic susceptibility of the Ba 0.67 _{0.67}K 0.33 _{0.33}Fe 2 _2As 2 _2 sample, measured in zero field.",
"Lower inset: magnetization hysteresis loops measured at 20 and 30 K."
],
[
"Results",
"At first step we checked pinning properties by measuring of magnetic hysteresis loops at several temperatures (lower inset to Fig.",
"REF ).",
"Magnitization curve $M(H)$ is symmetric about the axis M=0 that indicates a strong bulk pinning and the absence of Bean-Levingston barier.",
"Also $M(H)$ shows no magnetic background.",
"The raw experimental data for $M(H)$ dependences in low fields are presented in Fig.",
"REF .",
"In fields above $H_{c1}$ the superconductor captures magnetic flux, that leads to departure of $M(H)$ dependence from the linear one.",
"Exact finding of the $H_{c1}$ values from the measured nonlinear $M(H)$ dependence is a hard task, taking into account a finite width of the linear-to-nonlinear crossover of the magnetization curves, and data scattering.",
"In our measurements the noise level corresponded to $\\approx (3-5)\\times 10^{-5}$ emu.",
"By modeling the $M(H)$ dependence with such noise level we found that the frequently used algorithm for the $H_{c1}$ determination based on the correlation parameter, (see e.g., [46], [55]) leads to artificially overestimated $H_{c1}$ data and excessive $H_{c1}(T)$ data scattering.",
"Correspondingly, instead of the above algorithm [46] based on regression calculation, we have developed a modified algorithm where the experimental $M(H)$ data are fitted with both, linear (for $H<H_{c1}$ ) and the second power polynomials (for $H>H_{c1}$ ).",
"The protocol of Refs.",
"[46], [55] and its shortcomings in the case of a large noise level, as well as the modified algorithm are described in detail in Appendix 1.",
"This approach minimizes the impact of a variable number of points on the correlation coefficient calculation and thus improves the accuracy of the $H_{c1}$ determination.",
"The $H_{c1}(M)$ dependence determined with the modified algorithm for the studied BKFA sample is shown in Fig.",
"REF It should be mentioned that the determined $H_{c1}$ value represents a critical field for the given sample.",
"In order to characterize the material parameter, one has also to take the demagnetization factor $N$ into account: $H_{c1}= \\frac{H_{c1}^{\\rm measured}}{1-N},$ where $N=\\frac{q\\frac{a}{b}}{q\\frac{a}{b}+1}.$ For the disk-shape sample [56]: $q_{\\rm disk}=\\frac{4}{3\\pi } +\\frac{2}{3\\pi }\\tanh \\left[1.27\\frac{b}{a} \\left( \\ln \\left(1+\\frac{a}{b}\\right)\\right) \\right].$ With the sample diameter $a=3\\pm 0.5$ mm and thickness $b=50\\pm 20$ $\\mu $ m the demagnetization factor for our sample $N = 0.96 \\pm 0.02$ , and the ratio of the material $H_{c1}$ and the measured $H_{c1}$ value amounts to $\\sim 16-50$ .",
"Correspondingly, the $H_{c1}$ value for Ba$_{0.67}$ K$_{0.33}$ Fe$_2$ As$_2$ falls into a range of fields $400-1250$ Oe.",
"Figure: H c1 H_{c1} temperature dependence for the BKFA sample compared with the single band BCS model and the two band α\\alpha -modelFor describing the lower critical field of a superconductor, it is convenient to introduce a normalized superfluid density [19]: $\\tilde{\\rho _s}(T)\\equiv \\lambda (0)_{ab}^2/\\lambda (T)_{ab}^2=H_{c1}(T)/H_{c1}(0).$ In the framework of the BCS theory for a single-band superconductor with an isotropic gap, the latter may be represented as [57], [58]: $\\tilde{\\rho _s}(T) = 1+2\\int _{\\Delta (T)}^\\infty \\frac{\\partial f}{\\partial E} \\frac{EdE}{\\sqrt{E^2-\\Delta ^2(T)}}.$ Here $f(T)$ is the Fermi distribution function, $\\Delta (T)=\\Delta _0\\tanh \\left[1.82 \\left(1.018 \\left(\\frac{T_c}{T}-1\\right)\\right)^{0.51} \\right]$ is the gap temperature dependence.",
"$E^2=\\varepsilon ^2+\\Delta ^2 (T)$ , $E$ is the total energy, and $\\varepsilon $ - single particle excitation energy counted from the Fermi energy.",
"The normalized superfluid density may be re-written in a more convenient for integration way as follows: $\\tilde{\\rho _s}(T) = 1-2 \\int _0^\\infty \\frac{e^y}{(e^y+1)^2}\\frac{1}{t}d\\xi ,$ with $t=T/T_c$ , $y=\\sqrt{\\xi ^2+\\left(\\frac{\\alpha \\delta (T)}{2}\\right)^2}/t$ , $\\xi =\\varepsilon /\\left(k_B T_c\\right) $ , and $\\alpha = 2\\Delta _0/\\left(k_B T_c \\right)$ .",
"The latter is the parameter in the given model.",
"Figure REF shows the least square fitting of the measured $H_{c1}(T)$ data within the above single-band BCS model.",
"The fitting parameter here $\\alpha =3.4$ .",
"One can see that the model fails to reproduce the experimental data.",
"Clearly, the single band model can not describe the curved $H_{c1}(T)$ dependence, especially in the interval $10-25$ K. The physical meaning of this failure is transparent: to fit the data successfully one needs to use a multiband model.",
"Correspondingly, at the next step for describing the experimental data we apply the so called two-band $\\alpha $ -model [57]: $\\tilde{\\rho _s} (T)= \\varphi _1 \\tilde{\\rho }_{s1}(T)+\\varphi _2 \\tilde{\\rho }_{s2} (T)$ This model considers a normalized superfluid density for the superconductor having two independent condensates with a normalized superfluid densities $\\rho _{s1}$ and $\\rho _{s2}$ in the first and second band respectively, taken with weighting factors $\\varphi _1$ and $\\varphi _2=1-\\varphi _1$ .",
"The result of fitting the $H_{c1}(T)$ data with $\\alpha $ model is shown in Fig.",
"REF .",
"This approach leads to a good agreement between the model and experimental data.",
"The fitting parameters are as follows: $\\alpha _1 = 1.2 \\pm 0.2$ ($\\Delta _1(0)= 2 \\pm 0.3$ meV), weight factor $\\varphi _1=0.54 \\pm 0.02$ ; and $\\alpha _2= 6.9 \\pm 0.3$ ($\\Delta _2(0)= 11 \\pm 0.5$ meV), weight factor $\\varphi _2=0.46 \\pm 0.02$ ; $H_{c1}(0) = 25.5$ Oe."
],
[
"Specific heat",
"The specific heat measurements are a powerful thermodynamic bulk probe [21], [22], [12], though there are several known problems with SH data treatment.",
"The SH data contains contribution from the lattice, that is to be subtracted in order to determine the electronic SH.",
"The lattice contribution to the SH is usually estimated by suppressing the superconducting transition in high magnetic fields.",
"For FeBS, the lattice SH cannot be suppressed because of the very high upper critical field.",
"The majority of the earlier SH data suffer from a residual low-temperature non-superconducting electronic contribution and show a Schottky anomaly [22], [23].",
"In this section we report our SH data and their analysis which evidence for the two-band superconducting condensate with $s-$ type order parameter symmetry.",
"The extracted superconducting gap values correspond to the characteristic ratios $2\\Delta _1(0)/k_BT_c =1.6 \\pm 0.1$ ($\\Delta _1(0)= 2.5 \\pm 0.2$ meV, weight factor $\\varphi _1=0.58 \\pm 0.02$ ) and $2\\Delta _2(0)/k_BT_c = 7.2 \\pm 0.2 (\\Delta _2(0)= 11.3 \\pm 0.3$ meV, weighting factor $\\varphi _2=0.42 \\pm 0.02$ ).",
"These parameters are consistent with those determined from magnetization measurements, IR reflection, and Andreev reflection spectra, descried in the corresponding sections.",
"Figure: (a)Temperature dependence of the specific heat for Ba 0.67 _{0.67}K 0.33 _{0.33}Fe 2 _2As 2 _2 sample, normalized by temperature at zero field and comparison with two models, using the common states approximation: for the 6-modes Einstein model and for the lattice SH of Ba(Fe 0.88 _{0.88}Mn 0.12 _{0.12}) 2 _2As 2 _2.",
"Insert compares the data for B=0B=0 and B=9B=9 T. (b) Low-temperature behavior of the normalized specific heat versus T 2 T^2.",
"Dots are the experimental data, black curve -- their fit with the Debye law.",
"For quantitative extraction of the parameters we used only the data in the range 2-42-4 K. (c) SH anomaly at the superconducting transition and determination of T c T_c."
],
[
"Experimental",
"The specific heat measurements were taken with a 1.93 mg-piece of Ba$_{1-x}$ K$_x$ Fe$_2$ As$_2$ ($x=0.33$ ) single crystal cleft from the same large crystal that was used for all other measurements; the sample had superconducting critical temperature $T_c=36.5$ K. Measurements were done using the thermal relaxation technique with PPMS-9, in the temperature range $2 - 200$ K. Temperature was swept with a stepsize of 0.2 K for the interval 2 - 50 K, 0.5 K for 50 - 100 K and 1 K for 100 - 200 K. For each temperature point the data have been averaged within 3 seconds."
],
[
"Results",
"The raw experimental SH data are shown in Fig.",
"REF at zero field; the insert shows results obtained with another piece of the same crystal (m=0.8 mg) in fields $B=0T$ and $9T$ .",
"The $C(T)/T$ data shows no features in the low temperature range (such as, e.g., growth towards the lowest $T$ ), thus evidencing for the absence of Schottky anomaly.",
"In the low-$T$ limit (for $T < 6-8$ K) the $C(T)/T$ data may be represented by the Debye law: $C(T)/T=\\gamma (0)+\\beta T^2$ , where $\\gamma (0)$ is the residual contribution of the non-superconducting phase, $\\beta T^2$ - is the lattice contribution.",
"The two parameters $\\gamma (0)$ and $\\beta $ may be easily found from fitting the model to the experimental data (see Fig.",
"REF b).",
"For $B= 0$ we found $\\gamma = 0.3 - 0.5$ mJ/mol K$^2$ and $\\beta = 0.74 - 0.71$ mJ/mol K$^4$ The negligeably low value of the residual electronic specific heat evidences for high quality of the sample.",
"It is worth noting that the above approach is rather approximate because beyond the linear approximation the electronic SH for superconducting materials depends on temperature, and because the lattice contribution includes higher order terms.",
"For this reason this approach is appropriate only for qualitative estimates, whereas for quantitative analysis more complex approach is needed, which is described below.",
"In the temperature interval 36-37 K the $C(T)$ data shows a sharp peak, related with the SC transition (see Fig.",
"REF ).",
"The peak width is about 1 K, and the jump in the $C/T$ data at the transition $\\Delta C/T = 119$ mJ/mol K$^2$ .",
"Due to the entropy conservation at the SC transition, the following equality must be fulfilled: $\\int _{T_c-t}^{T_c+t}\\frac{C_{exp}}{T} dT=\\int _{T_c-t}^{T_c}\\frac{C_{\\rm extrap}}{T}dT+\\int _{T_c}^{T_c+t}\\frac{C_{\\rm extrap}}{T} dT,$ where $C_{exp}$ - the measured SH data, $C_{\\rm extrap}$ - the data extrapolated to the region of the SC transition, and $2t$ is the superconducting transition width.",
"By implementing this implicit equation to the data in Fig.",
"REF c we determine the true $T_c$ value of 36.5 K, nicely consistent with that extracted from magnetic measurements."
],
[
"Separation of the lattice and electronic contributions to SH",
"Further experimental investigations of the structure and magnitude of the SC gaps by means of bulk specific heat data are of great interest.",
"In order to determine the specific heat related to the SC phase transition, we need to estimate the phonon (lattice) and electronic contributions to specific heat in the normal state.",
"These contributions are additive: $C_p(T)=C_e(T)+C_{lat}(T),$ where $C_e$ is the contribution related to electronic subsystem, and $C_{lat}$ is the lattice contribution.",
"The lattice term $C_{lat}$ however cannot be determined by direct measurements.",
"This problem may be solved by using the so called common states approximation [59], that consists in using, as a reference, of the lattice SH for a non-superconducting compound of a relative's composition.",
"For Ba$_{0.67}$ K$_{0.33}$ Fe$_2$ As$_2$ one of such compounds is the parent BaFe$_2$ As$_2$ that is non-superconducting though exhibits a magnetic phase transition at $\\approx 140$ K. Varying doping level or doping element leads to changes in the lattice spacings by a few percents.",
"In order to take account of these insignificant change one can use scaling factors proximate to unity.",
"Other possible reference materials for our Ba$_{0.67}$ K$_{0.33}$ Fe$_2$ As$_2$ sample are the nonsuperconducting compounds Ba(Fe$_{0.85}$ Co$_{0.15}$ )$_2$ As$_2$ [60], Ba(Fe$_{0.88}$ Mn$_{0.12}$ )$_2$ As$_2$ [12], and BaFe$_{1.75}$ Ni$_{0.25}$ As$_2$ [61].",
"Mathematically, the common states approximation for the specific heat may be written as follows: $C_{\\rm tot}^{SC}(T)=C_{el}^{SC}(T) + A C_{\\rm lat}^{nSC}(BT).$ Here $C_{\\rm tot}^{SC}(T)$ is the total calculated SH corresponding to the experimental data $C_{\\rm exp}(T)$ , $C_{el}^{SC}(T)$ is the electronic contribution to SH, $C_{\\rm lat}^{nSC}(T)$ - lattice SH for the nonmagnetic reference compound, $A$ and $B$ are the scaling factors.",
"For temperatures above $T_c$ the electronic SH may be written as $C_{el}^{SC}(T) = \\gamma _n T$ .",
"The factors $A$ and $B$ are selected based on least square fitting under the constraint of the entropy conservation: $\\int _0^{T_c}\\frac{C_{el}(T)}{T} dT = \\int _0^{T_c}\\gamma _n dT.$ By now, the lattice specific heat for Ba$_{0.68}$ K$_{0.32}$ Fe$_2$ As$_2$ was well described using the Debye-Einstein model [62].",
"In order to test whether or not we can apply the data of Refs.",
"[12], [62], [61] to the data processing for our Ba$_{0.67}$ K$_{0.33}$ Fe$_2$ As$_2$ sample, we have tested the results of Ref.",
"[12], [62], [61] for the lattice SH of non-superconducting and non-magnetic materials, and found that these data may be scaled to each other by using the common state approximation with factors $A$ and $B$ chosen rather close to unity, $0.95 - 1.05$ .",
"Correspondingly, for the analysis of our experimental data we used the model described in Ref.",
"[62] containing 6 Einstein modes.",
"We also used the lattice SH data for Ba(Fe$_{0.88}$ Mn$_{0.12}$ )$_2$ As$_2$ [12], since these measurements were done in the most wide temperature range.",
"Figure REF a shows that both models describe the experimental $C(T)$ data rather well.",
"The resulting electronic SH contribution $C_{el}/T$ obtained in this fit using the common state approximation is shown on Fig.",
"REF a, the inset to Fig.",
"REF b demonstrates the entropy conservation constraint for this calculations.",
"In the two respective fittings we obtained the two sets of factors: (i) $A= 0.998$ , $B= 0.974$ and $\\gamma _n= 63.6$ mJ/mol K$^2$ for the 1st scaling based on the lattice SH of Ba$_{0.68}$ K$_{0.32}$ Fe$_2$ As$_2$ , and (ii) $A= 0.954$ , $B= 0.996$ , $\\gamma _n= 58.0$ mJ/mol Ê$^2$ for the 2nd scaling based on the lattice SH of Ba(Fe$_{0.88}$ Mn$_{0.12}$ )$_2$ As$_2$ .",
"Figure: (a) Electronic SH calculated using the 6-mode Einstein model and the lattice SH of Ba(Fe 0.88 _{0.88}Mn 0.12 _{0.12}) 2 _2As 2 _2 .",
"Inset: comparison of the normalized electronic SH defined by the 6-mode Einstein model and the lattice SH of Ba(Fe 0.88 _{0.88}Mn 0.12 _{0.12}) 2 _2As 2 _2; (b) Normalized electronic SH of the superconducting condensate C es /Tγ n C_{es}/T \\gamma _n compared with the single band BCS model and the two band BCS α\\alpha -model.",
"Inset: Electronic entropy in normal and SC state in case of calculations with 6-mode Einstein model and the lattice SH of Ba(Fe 0.88 _{0.88}Mn 0.12 _{0.12}) 2 _2As 2 _2.In order to improve the accuracy of the data analysis we extrapolated the $C_{el}(T)$ data to $T= 0$ for both models.",
"From this extrapolation we obtained also the two estimates for normal state residual contribution, $\\gamma _r=2.3$ mJ/mol K$^2$ - in the analysis based on the lattice model [62] and $\\gamma _r = 1.1$ mJ/mol K$^2$ in the scaling based on the Ba(Fe$_{0.88}$ Mn$_{0.12}$ )$_2$ As$_2$ lattice.",
"We conclude that the non-superconducting residual contribution to SH is of the order of $2-4$ % ($\\gamma _r/\\gamma _n=0.019- 0.035$ ), that is comparable to the values reported for other superconducting FeBS [60].",
"Analyzing behavior of the superconducting condensate it is appropriate to consider the normalized electronic SH $C_{es}/T\\gamma _n$ versus normalized temperature $(T/T_c)$ , where $C_{es}$ is the SH of superconducting condensate [60], which may be obtained as: $\\frac{C_{es}}{\\gamma _nT}(T)=\\frac{C_e/T-\\gamma _r}{\\gamma _n-\\gamma _r}$ In Fig.",
"REF a, the $C_e/T$ data obtained by two approaches is consistent with each other.",
"Although there is a minor difference (much less than the peak height at $T_c$ ) between them in Fig.",
"REF a, the difference becomes almost invisible on the plot of the normalized SH of the superconducting condensate $C_{es}$ , Fig.",
"REF b.",
"For high temperatures, $T > 100$ K, the data description based on the Ba(Fe$_{0.88}$ Mn$_{0.12}$ )$_2$ As$_2$ lattice SH is somewhat worse: the difference between $C_e/T$ and è $\\gamma _n$ increases with temperature."
],
[
"Analysis of the normalized electronic SH",
"The normalized SH of the superconducting condensate may be calculated within the BCS theory as follows [25]: $\\frac{C(T)}{\\gamma _nT} &=&\\frac{d(S/\\gamma _nT_c)}{dt}, \\nonumber \\\\\\frac{S(T)}{\\gamma _nT_c} &=& \\frac{6}{\\pi ^2} \\int _0^\\infty \\left[f\\ln f +(1-f)\\ln (1-f) \\right]d\\varepsilon , \\nonumber \\\\f &=& \\left[\\exp \\frac{\\left(e^2+\\alpha ^2\\delta ^2(t)/4\\right)^{1/2}}{t} +1 \\right]^{-1}, \\nonumber \\\\\\delta (T) &=& \\tanh \\left[ 1.82\\left( 1.018\\left(\\frac{T}{T_c}-1 \\right) \\right)^{0.51}\\right].$ where $t=T/T_c$ , $\\alpha = 2\\Delta (0)/k_BT_c$ , $\\delta (T) =\\Delta (T)/\\Delta (0)$ is the temperature dependence of the gap, and $\\Delta (0)$ is the energy gap at $T= 0$ .",
"The above phenomenological formulae [46] generalizes calculations of [63] within the BCS model.",
"At the first step, for fitting the $C_{es}(T)$ data we applied the single-band BCS model using Eqs.",
"REF .",
"The model implies an isotropic $s$ -type order parameter $\\Delta $ .",
"Figure REF b shows the result of the mean square fitting with $2\\Delta /k_BT_c= 3.7$ .",
"Obviously, the single-band approach does not fit the experimental SH data and, particularly, does not reproduce the remarkable hump in $ C_{es}/T\\gamma _n$ clearly seen at $T/T_c \\sim 0.3-0.5$ .",
"At the second step we apply the phenomenological $\\alpha -$ model [25], [24] for the two-band superconductor, which sums up contributions of each band, calculated within the BCS model, Eq.",
"REF , with the corresponding weight factors $\\varphi _1$ and $\\varphi _2= 1-\\varphi _1$ : $C(T)= \\varphi _1 C_1(T) + \\varphi _2 C_2(T)$ This model has three adjustable parameters, $\\alpha _1=2\\Delta _1/k_BT_c$ , $\\alpha _2=2\\Delta _2/k_BT_c$ and $\\varphi _1$ , which may be found from least square fitting of the model to the experimental data.",
"$\\varphi _1$ and $\\varphi _2$ describe the relative share of each condensate in the total SH: $\\varphi _i= \\gamma _i/\\gamma _n$ , where $\\gamma _i$ is the specific heat of the $i$ -th condensate in the normal state.",
"The result of data fitting with the two-band model is shown in Fig.",
"REF b.",
"One can see that the two-band approach provides rather good fitting to the experimental data.",
"The difference between the model dependence and the experimental data does not exceed 5% of $C_{es}/T\\gamma _n$ , that corresponds to 4 mJ/mol K$^2$ .",
"The deviation is within the measurements uncertainty and in relative units does not exceed 1% of the total measured $C_{\\rm exp}$ .",
"With the two band model we find the following set of parameters: $\\alpha _1=2\\Delta _1/k_BT_c = 1.6 \\pm 0.1$ ($\\Delta _1= 2.5 \\pm 0.2$ meV), $\\alpha _2=2\\Delta _2/k_BT_c = 7.2 \\pm 0.2$ ($\\Delta _2 = 11.3 \\pm 0.3$ meV), and $\\varphi _1=0.58 \\pm 0.02$ ."
],
[
"Infrared reflection spectroscopy",
"Infrared (IR) spectroscopy is a powerful technique to investigate the electronic gap structure of superconductors.",
"Its large probe depth ensures the bulk nature of the measured quantities and its high-energy resolution and powerful sum rules enable a reliable determination of important physical parameters, such as the gap magnitude and the plasma frequency of the SC condensate [64].",
"In a simple one-band system, the standard Drude model with parameters plasma frequency $\\Omega $ and scattering rate $\\gamma $ describes the frequency-dependent complex conductivity $\\tilde{\\sigma }_N$ in the normal (N) state [65].",
"In the superconducting (S) state, the standard BCS model (Mattis-Bardeen equations [66], with parameters $\\sigma _0$ and superconducting gap $\\Delta $ ) can describe the complex conductivity $\\tilde{\\sigma }_S$ [67].",
"On this basis, far-infrared measurements can be of particular importance since a signature of the superconducting gap $\\Delta $ ) can be observed at $\\hbar \\omega \\sim 2\\Delta $ (optical gap) for an anisotropic $s$ -wave BCS superconductor.",
"The electromagnetic radiation below the gap energy $2\\Delta $ could not be absorbed.",
"For a bulk sample, in particular, a maximum at the optical gap is expected in the ratio $R_S/R_N$ , where $R_S$ and $R_N$ are the frequency-dependent reflectances in the superconducting and normal state, respectively [67]."
],
[
"Experimental",
"IR reflectance spectra $R(\\omega )$ were measured with Bruker IFS 125HR spectrometer with a spectral resolution of 2 cm$^{-1}$ over a wave number range of 400-50 cm$^{-1}$ (25-200 $\\mu $ m).",
"For measurements in FIR region we used a mylar beam splitters of various thickness.",
"Liquid-helium cooled Si bolometer was used to detect IR spectra.",
"For low-temperature measurements the sample was placed into the helium cryostat Optistat CF-V from Oxford Instruments with the wedged windows made of TPX plastic.",
"The reflectance measurements were carried out at near-normal incidence on the freshly cleaved surfaces.",
"The goal of IR measurements is to determine the frequency-dependent complex conductivity $\\tilde{\\sigma }(\\omega )=\\sigma _1(\\omega ) + i\\sigma _2(\\omega )$ which usually appears in discussion of the low-frequency electrodynamics of the system [64] and can describe its optical response.",
"The complex conductivity of the ideal single-band conducting system can be described using the Drude model in the normal state and the Bardeen-Mattis BCS model [66] generalized for an isotropic $s$ -wave BCS superconductor using the Zimmermann relations [68].",
"In this case, in the “dirty” limit the dissipative part of the optical conductivity $\\sigma _1(\\omega )$ at $T\\ll T_c$ vanishes abruptly below a frequency corresponding to doubled superconducting gap $2\\Delta $ .",
"Thus, in the vicinity of the frequency corresponding to $2\\Delta $ (optical gap) one should observe a peculiarity in the optical response of the system.",
"The relatively small size of the sample for IR measurements and irregular cleavage surface resulted in rather low accuracy of measurement of the absolute value of reflection coefficient; the latter hampered calculating the optical conductivity by using the Kramers-Kronig analysis.",
"For this reason we apply the technique described in [69] to determine the superconducting gaps.",
"It consists in the relative measurements of $R(T\\ll T_c)/R_N$ with no reference measurements, while sweeping the temperature within a narrow temperature range.",
"Here, $R_N$ is the reflectance in the normal state at temperature slightly above $T_c$ .",
"The measurements are performed in one cycle with the same detector and set of optical elements (beam splitter and cryostat windows).",
"In this way the sample position and orientation as well as the optical system were not changed during measurements.",
"This technique enables to minimize possible temperature-driven distortions of the optical set-up, which may yield frequency-dependent systematic errors in $R(\\omega )$ .",
"It should be noted that for the bulk superconductor of the $s-$ type symmetry the normalized reflectivity $R(T\\ll T_c)/R_N$ forms a maximum, whose energy corresponds to the superconducting gap $2\\Delta $ .",
"For the two gap superconductor, the maximum is expected to appear between the two SC gaps, closer to the one having a major contribution.",
"This enables one to estimate the value of the dominant gap."
],
[
"Results",
"Figure REF shows the $R(T)/R(T=40$ K) dependences for Ba$_{0.67}$ K$_{0.33}$ Fe$_2$ As$_2$ measured at $T=5-50$ K. One can see that the normalized reflectivity $R(T)/R(40$ K) maximum starts increasing as temperature $T$ decreases below $T_c$ .",
"This is because for $s$ -wave superconductor at temperatures below $T_c$ the reflectance approaches unity at energies $\\hbar \\omega < 2\\Delta $ .",
"As a result, a peak is formed with a maximum in the range of $\\sim 160$ cm$^{-1}$ (19.8 meV).",
"The peak position correlates with the magnitude of the greater of the superconducting gaps [12], [70], [71], [33], [72].",
"The smaller gap is beyond the frequency range of our IR measurements.",
"The kink in the normalized reflectivity at $\\sim 250$ cm$^{-1}$ is probably due to the IR active phonon mode $E_u$ related to the Fe($ab$ )-As($-ab$ ) vibrations [73].",
"This mode manifests itself in many AFe$_2$ As$_2$ materials including A = Ca, Sr, Eu and Ba."
],
[
"Intrinsic multiple Andreev reflection effect (IMARE) spectroscopy",
"In ballistic mode, superconductor - normal metal - superconductor (SnS) contact (whose diameter $2a$ is less than the carrier mean free path $l$ [74]) demonstrates multiple Andreev reflection effect (MARE) [75], [76], [78], [77].",
"MARE manifests itself in an excess current at low bias voltages in current-voltage characteristic (CVC) of SnS contact (so called foot area).",
"A series of dynamic conductance features called subharmonic gap structure (SGS) appears at bias voltages $V_{n} = 2\\Delta /en$ (where $n$ is a natural number) [75], [76], [78], [77], [79].",
"This simple formula enables to directly determine the superconducting gap value at any temperatures up to $T_c$ [75], [77].",
"For the high-transparency SnS-Andreev regime (typical for our break-junction contacts), SGS exhibits a series of dips for both nodeless and nodal gap [79], [81], [80].",
"The coexistence of two independent superconducting gaps would cause, obviously, two SGS's in the $dI(V)/dV$ -spectrum.",
"The $k$ -space angular distribution of the gap value strongly affects the SGS lineshape.",
"In case of an isotropic gap, the SGS minima are high-intensive and symmetrical, whereas a nodal gap (such as $d$ -wave) leads to strongly suppressed and asymmetric $dI(V)/dV$ minima [79], [81], [80].",
"For extended $s$ -wave nodeless symmetry, the SGS demonstrates doublet minima corresponding to the gap extremes in the $k$ -space [28], [80]."
],
[
"Experimental",
"For Andreev spectroscopy studies, we used a break-junction technique (for details, see [82], [80]) in order to create symmetric SnS contacts.",
"The studied sample is precisely cracked in cryogenic environment.",
"We cut from the single crystal a thin plate, $3 \\times 1.5 \\times 0.1$ mm$^{3}$ .",
"The crystal was attached to a springy holder by four In-Ga pads which insured true 4-probe connection and helped aligning the $ab$ -plane parallel to the sample holder.",
"After cooling down to 4.2 K, the sample holder was precisely bent, which caused cracking of the single crystal.",
"Its deformation generates a microcrack that represents the superconductor - constriction - superconductor contact (ScS), where the constriction formally acts as insulator or normal metal.",
"In our setup, the superconducting banks are kept touching each other and not separated to a valuable distance [80].",
"Taking in mind the metallic-type Ba spacers between superconducting Fe-As blocks of crystal structure, a formation of a metallic-type constriction is feasibly.",
"The observed $I(V)$ and $dI(V)/dV$ of the break junctions are typical for high-transparent SnS-Andreev mode [75], [76], [78], [77].",
"Obviously, a current flows through the break junction along the $c$ -direction (for the details see [80]), therefore, a gap anisotropy could be barely resolved in $k_xk_y$ plane [80].",
"Since in our setup the microcrack is located deep in the bulk of the sample and away from current leads, the cryogenic clefts are free of Joule overheating, and adverse surface influence such as possible degradation or impurity diffusing.",
"In layered sample, the break-junction probe often shows also array of the SnSn-...-S-type realized in natural steps and terraces onto cryogenic clefts of layered crystal.",
"In such arrays, an intrinsic multiple Andreev reflections effect occurs.",
"This effect is similar to the intrinsic Josephson effect [83], [84] and was first observed in Bi cuprates [85], further in all layered superconductors ([30], for a review, see [80]).",
"Since Andreev array consists of a sequence of $m$ identical SnS-junctions, the SGS dips appear at positions: $V_{n} = \\frac{m \\times 2\\Delta _i}{en}, m = 1, 2, \\dots $ In case of stack contacts, positions of other peculiarities caused by bulk properties of material also scale by a factor of $m$ [30], [80].",
"In our experiment, we were able to probe tens of arrays (containing various number of junctions $m$ ) by precisely readjusting the microcrack.",
"The latter opportunity helps one to collect a large amount of data and to check reproducibility of the bulk gap values and other peculiarities caused by bulk properties of material.",
"The number of junctions $m$ can be determined by normalizing the spectrum of array to that of the single SnS-contact; after such scaling, positions of each SGS should coincide.",
"Probing such $natural$ stack contacts, one obtains information about the true bulk properties of the sample (almost unaffected by surface states which seem to be significant in Ba-122 [86] locally, i.e.",
"within the contact size $a \\approx 7-50$ nm.",
"This feature favors accuracy increasing in the superconducting gap measurements [30]."
],
[
"Results",
"Figure REF a shows a typical current-voltage characteristic (blue line, $T=4.2$ K) for a break-junction in nearly optimal Ba$_{0.65}$ K$_{0.35}$ Fe$_2$ As$_2$ with critical temperature $T_c^{\\rm local} \\approx 36$ K. The excess current at low bias voltages (foot area) manifests a formally metallic-type constriction with ballistic $c$ -axis transport [75], [76], [78], [77].",
"Taking the contact resistance $R \\approx 15$ $\\Omega $ , the bulk in-plane resistivity of the studied crystal $\\rho ^{ab} \\approx 0.4 \\cdot 10^{-5}$ $\\rm \\Omega \\cdot cm$ , and using the value $\\rho ^{ab}l \\approx 0.45 \\cdot 10^{-9}$ $\\rm \\Omega \\cdot cm^2$ [87], we estimate the elastic mean free path of carriers $l^{ab} \\approx 1.1$ $\\rm \\mu m$ , and the contact radius $a = \\sqrt{\\frac{4}{3\\pi } \\cdot \\frac{\\rho ^{ab}l}{R}} \\approx 36$ nm.",
"This rough estimation gives the contact dimension $2a << l$ , which satisfies the conditions of MARE observation.",
"Figure: a) Normalized to a single junction current-voltage characteristic (left axis) and dynamic conductance spectrum(right axis) for Andreev array in Ba 0.65 _{0.65}K 0.35 _{0.35}Fe 2 _2As 2 _2 with T c local ≈36T_c^{local} \\approx 36 K measured at T=4.2T=4.2 K.Black vertical lines depict the positions of subharmonic gap structure (SGS) dips, corresponding to Δ L ≈6.4–9.5\\Delta _L \\approx 6.4 –9.5 meV (33%~ 33\\%anisotropy in k-space angle distribution).",
"b) Low-bias fragments of dynamic conductance spectra demonstrating SGS (marked with black vertical lines) of the small gap Δ S ≈1.8\\Delta _S \\approx 1.8 meV.",
"The dI(V)/dVdI(V)/dV of m=6m=6 and m=2m=2 contacts (lower curves) were normalized to the single SnS-junction spectrum (upper curve).",
"Monotonic background was suppressed for clarity.",
"c) The positions of Δ L \\Delta _L (vertical bars depict the gap anisotropy) and Δ S \\Delta _S (open symbols) subharmonics in the dynamic conductance spectra shown in a,b panels.",
"Black lines are guidelines.The corresponding $dI(V)/dV$ spectrum (red line in Fig.",
"REF a) shows a set of dynamic conductance dips typical for clean classical SnS-Andreev array of 2 junctions (a natural SnSnS structure).",
"In order to normalize $I(V)$ and $dI(V)/dV$ in Fig.",
"REF a to those for a single SnS-junction, the voltage axis was scaled by a factor of $m=2$ .",
"The large gap SGS starts with the clear dips at $\\approx \\pm 18$ mV corresponding, in accordance with the SGS expression, to $2\\Delta _L/e$ .",
"The next features at $\\approx \\pm 11.5$ mV do not match the expected position ($\\approx \\pm 9$ mV) of the second subharmonic of the large gap, therefore these two dips could be interpreted as a doublet $n =1$ feature caused by a $~ 33~\\%$ gap anisotropy.",
"The positions of the next pair of dynamic conductance features, $\\approx \\pm 9.6$ and $\\approx \\pm 6.4$ mV, corresponds well with those of the second subharmonic of the large gap.",
"Note that the $n=2$ doublet is right twice narrower than the $n=1$ one, agreeing with the subharmonic set.",
"To say, whether the $dI(V)/dV$ doublets are caused by the in-plane gap anisotropy in the $k$ -space, or the order parameter fine splitting, one needs a further study of the dynamic conductance lineshape.",
"In Fig.REF a, the real shape of $\\Delta _L$ subharmonics is rather ambiguous since overlapped by the pronounced excess conductance.",
"Surely, the intensity and the shape of the $\\Delta _L$ dips is inconsistent with that expected for $d$ -wave or fully anisotropic (nodal) $s$ -wave symmetry [81], [79]; we conclude therefore that the large gap is nodeless.",
"On the other hand, comparing the current data with those obtained earlier with Ba(K)-122 single crystals with a bit lower $T_c \\approx 34$ K [46], the extended $s$ -wave symmetry of the large gap is more likely.",
"Using Eq.",
"(15), we directly determine the large gap edges $\\Delta _L^{min} \\approx 6.4$ meV, $\\Delta _L^{max} \\approx 9.5$ meV, and corresponding BCS ratios $2\\Delta _L/k_BT_c \\approx 4.1 - 6.1$ .",
"When trying to regard this array as corresponding to a single SnS-junction, we get the twice BCS-ratio up to 12.2 seemed too large for Ba-122; on the other hand, given $m \\ge 3$ , we get $2\\Delta _L/k_BT_c \\approx 2 - 3$ which is impossible for driving gap since lies below the weak-coupling limit 3.5.",
"This simple check demonstrates a way for correct determination of the number of junctions in the array; in the case, the 2-junction structure is identified unambiguously.",
"Figure REF b shows low-bias fragments of dynamic conductance spectra of $m=6$ and $m=2$ Andreev arrays, and a single SnS junction (upper curve).",
"The width and the outlook of the pronounced foot near zero bias is reproducible in all the curves.",
"The monotonic background was suppressed in order to clarify the small gap SGS.",
"Black vertical lines in Fig.",
"REF b mark the first feature at $V_{S1} \\approx \\pm 3.5$ mV and the second feature at $V_{S2} \\approx \\pm 1.8$ mV.",
"These subharmonics, obviously, do not belong to the large gap SGS (see Fig.",
"REF a), rather, they originate from a small gap $\\Delta _S \\approx 1.8$ meV.",
"Unlike the $\\Delta _L$ dips, the small gap peculiarities are not split and are rather symmetric, thus pointing to nearly isotropic $\\Delta _S$ in $k-$ space.",
"Despite the fact that the three dynamic conductance spectra shown in Fig.",
"REF b are obtained with different Ba(K)-122 samples (with the same $T_c$ ), the positions of $\\Delta _S$ SGS's are reproducible.",
"The sharpening of Andreev features with the $m$ increasing is a representative for IMARE spectroscopy [30], [80] and evidences the bulk nature of the $\\Delta _S$ order parameter.",
"The dependence of SGS positions $V_n$ $versus$ their inverse number $1/n$ shown in Fig.",
"REF c agrees with Eq.",
"REF and represents straight lines crossing the origin.",
"Two independent SGS observed in $dI(V)/dV$ spectra are caused by a presence of at least two distinct condensates with $\\Delta _L$ and $\\Delta _S$ order parameters.",
"The temperature dependences $\\Delta _L(T)$ (corresponding to the positions of the outer dip of doublet-like SGS) and $\\Delta _S(T)$ obtained directly are shown in Fig.",
"REF .",
"The dependence of the inner $\\Delta _L$ extremum is an issue of further studies.",
"The local critical temperature (corresponding to the contact area of $~ 1$ $\\mu $ m size transition to the normal state) $T_c^{local} \\approx 36$ K is a bit lower than the bulk $T_c$ determined with a bulk probe (see the resistive transition in Fig.",
"REF ).",
"A single-band model (dash-dot line), obviously, is inconsistent to describe the experimental temperature dependences of the large and the small gaps.",
"$\\Delta _L(T)$ passes below the single-band BCS-like curve, whereas $\\Delta _S(T)$ bends down significantly.",
"These deviation from the single-band type are caused by a moderate interband interaction.",
"As a result, both gaps turn to zero at common critical temperature $T_c^{local}$ .",
"Figure: Temperature dependences of the large gap (blue solid circles) and the small gap (open circles) in Ba 0.65 _{0.65}K 0.35 _{0.35}Fe 2 _2As 2 _2.",
"Single-band BCS-like curve (dash-dot line) and bulk resistive transition (rhombs) are shown for comparison.To approximate the experimental $\\Delta _{L,S}(T)$ , we used a two effective bands model based on Moskalenko and Suhl gap equations [88], [89] with a renormalized BCS-integral.",
"The shape of gap temperature behavior depends on a set of electron-boson coupling constants $\\lambda _{ij} = V_{ij}N_j$ , where $i,j = L,S$ , $V_{ij}$ are matrix interaction elements, $N_{i,j}$ —normal density of states (DOS) in the corresponding bands at the Fermi level.",
"We took the Debye energy $\\hbar \\omega _D = 20$ meV [90]; as fitting parameters, we used $\\alpha = N_S / N_L$ ratio (hereafter “L” index is linked with the driving bands), and the relation between intra- and interband coupling $\\beta = \\sqrt{V_LV_S}/V_{LS}$ , the fitting is detailed in [27], [91].",
"Theoretical $\\Delta _{L,S}(T)$ shown by solid lines agree well with the experimental dependences, therefore, the simple two effective bands model is applicable to describe the IMARE data.",
"The observed $\\Delta _{L,S}(T)$ are typical for a strong intraband coupling in the driving bands.",
"The large gap BCS-ratio far exceeding the weak-coupling BCS limit, also favors the latter statement.",
"In contrast, the Moskalenko-Suhl fit proves a weak-pairing superconductivity in the driven bands solely.",
"In a hypothetical case of zero interband interaction ($V_{LS} = 0$ ), we estimate $2\\Delta _S/k_BT_c^S \\approx 3.5$ ($T_c^S$ is the eigen critical temperature of the bands where the small gap is developed).",
"Taking zero Coulomb pseudopotentials $\\mu ^{\\ast }=0$ suggested, for example, in [92], [93], we get $\\lambda _{LL} = 0.59$ , $\\lambda _{SS} = 0.28$ , $\\lambda _{LS} = 0.24$ , $\\lambda _{SL} = 0.02$ leading to extremely large DOS ratio $\\alpha \\sim $ 12, and intra- to interband coupling ratio $\\beta \\sim 6$ , which is impossible for the so-called s$^{\\pm }$ scenario proposed in [92], [93].",
"When accepting a moderate nonzero Coulomb repulsion $\\mu ^{\\ast }=0.2$ , we roughly estimate $\\lambda _{LL} = 0.73$ , $\\lambda _{SS} = 0.43$ , $\\lambda _{LS} = 0.4$ , $\\lambda _{SL} = 0.13$ , seemed more realistic.",
"In the latter case, $\\alpha \\approx 3$ , whereas the intraband coupling is $~ 2.4$ times stronger than interband one."
],
[
"Discussion",
"The gap values obtained using the four complementary techniques are summarized in the Table 1.",
"$H_{c1}$ and $C(T)$ probe bulk properties, IR spectroscopy provides information about crystal subsurface layer, whereas IMARE is a direct local probe of the bulk order parameter.",
"Our experimental data $C_{el}(T)$ and $H_{c1}(T)$ may be well fitted with the two isotropic nodeless gaps.",
"The Andreev spectroscopy data points at two distinct gaps, the anisotropic large gap and isotropic small gap.",
"All the data converge on the absence of nodes for both gaps.",
"For the large gap, we report the BCS ratio $2\\Delta _L/k_BT_c = 6.1 - 7.2$ exceeding the BCS-limit.",
"This slight variation could be caused by several reasons, such as (a) out-of-plane anisotropy of the order parameter discussed in [94], (b) a complex and nontrivial in-plane angle distribution of the large gap in the $k$ -space, (c) a possible presence of a large gap splitting, (c) a surface sensitivity of superconducting properties, (d) a significant contribution of high-energy ($\\omega > \\Delta $ ) pairs with $Re[\\Delta (\\omega )] > \\Delta _{exp}$ (where $\\Delta _{exp}$ is a gap edge of the Eliashberg function) accounted in bulk probes.",
"As for the small gap, the determined values give $2\\Delta _S/k_BT_c = 1.2 - 1.6$ which lies well below the 3.5 limit and point to a nonzero interaction between the condensates.",
"It is noteworthy that our extracted gap values are comparable with the two-band $s$ -wave fit, $\\Delta _{1,2}(0)$ = 2 and 8.9 meV, reported for Ba$_{0.6}$ Ka$_{0.4}$ Fe$_{2}$ As$_{2}$ in [19] and $\\Delta _{1,2}(0)$ = 3.5 and 11 meV in [12].",
"The value of the gap amplitudes obtained for this material scales relatively well with its $T_{c}$ in light of the recent results for the FeBS [46], [95].",
"It is important to note that ARPES studies also report two $s$ -wave nodeless gaps of 2.3 and 7.8 meV for the outer and the inner Fermi surface sheets, respectively [96].",
"In fact, ARPES results hint towards the conclusion about strong dependence of the gap value on orbital character of the bands forming the corresponding Fermi surfaces: the larger gap appears on d$_{xz}$ /d$_{yz}$ bands [97].",
"Very recently, and based on a multi-band Eliashberg analysis, for Ca$_{0.32}$ Na$_{0.68}$ Fe$_{2}$ As$_{2}$ the superconducting electronic specific heat was shown to be described by a three-band model with an unconventional $s_{\\pm }$ pairing symmetry with gap magnitudes of approximately 2.35, 7.48, and formally -7.50 meV [21].",
"It has been well demonstrated that the model based on Eliashberg equations is a simplified model of the real four bands model taking into account the similarities between the two 3D Fermi sheets and between the two 2D Fermi sheets.",
"Based on them for the determination of $T_{c}$ and for the gap functions there can be considered only a distinct gap for every 2D, and respectively 3D sets of bands [98].",
"In fact, the Eliashberg equations may be solved in two ways.",
"The first way is to solve the equations which contain dependences of real frequency, and the second one – to solve this equations on the imaginary axis, summing on Matsubara frequencies [99].",
"Thus, the uncertainty in the number of SC condensates to be involved into the data processing affects the parameters extracted from the experiment.",
"In this work we used the simple $\\alpha $ -model that is not self-consistent, but is often used by experimentalists for fitting their thermodynamic data that deviate from the BCS predictions and for quantifying those deviations [100].",
"From the temperature dependence of the lower critical field data or specific heat data alone it is difficult to be sure whether one, two or three bands can describe well our investigated system, since in the case of multiband superconductivity low-energy quasiparticle excitations can be always explained by the contribution from an electron group with a small gap.",
"By complementing presented data as well as the data on BaFe$_{1.9}$ Ni$_{0.1}$ As$_2$ single crystals ($T_c \\approx $ 19 K) [101] obtained with MARE spectroscopy with the existing ARPES results [15], [96], [72], [20], one could make conclusion on ab-plane anisotropy of the large order parameter $\\Delta _L$ .",
"Comparing the $H_{c1}$ , $C(T)$ , and IMARE data, the two-band model seems to be sufficient to describe the experimental temperature dependences of superconducting parameters."
],
[
"Conclusions",
"Using four complementary experimental techniques, we studied single crystals of the 122 family, nearly optimally doped Ba$_{1-x}$ K$_x$ Fe$_{2}$ As$_{2}$ , and obtained consistent data on the structure of the superconducting order parameter.",
"Our data extracted from (i) temperature dependence of lower critical field, and (ii) temperature dependence of the specific heat, are inconsistent with a single $s$ -wave order parameter but is rather in favor of the presence of two gaps without nodes.",
"Our infrared reflection spectra supports the magnitude of the large gap, obtained from SH and lower critical field data, and its nodeless character.",
"The IMARE spectroscopy data, obtained on SnS-Andreev arrays, refine the conclusions on the two nodeless gaps: the large gap, $\\Delta _{L} = 6.4 - 9.5$ meV with extended $s$ -wave symmetry and anisotropy in the $k$ -space not less than $\\approx $ 30%, and the small gap, $\\Delta _S = 1.8 \\pm 0.3$ meV.",
"The BCS-ratio for the the upper extremum of the large gap is $2\\Delta _L/k_BT_c \\approx 6.1 - 7.2$ .",
"All our data clearly show that the superconducting energy gaps in nearly optimally doped Ba$_{1-x}$ K$_x$ Fe$_{2}$ As$_{2}$ are nodeless.",
"In addition, the obtained gaps are consistent with those determined from ARPES measurements."
],
[
"Acknowledgements",
"This work is supported by the Russian Science Foundation (16-12-10507).",
"Magnetic measurements were carried out with the support of the Russian Foundation for Basic Research (16-32-00663).",
"M.A.",
"acknowledges funding by DFG in the project MO 3014/1-1.",
"YuAA acknowledges the support of the Competitiveness Program of NRNU MEPhI.",
"Authors also acknowledge the Shared Facility Center at LPI for using their equipment."
],
[
"Data processing protocol",
"As mentioned in the main text, the noise level in our $M(H)$ measurements was rather high, $3-5 \\times 10^{-5}$ emu with the signal of the order of $10^{-2}$ emu.",
"We firstly investigated the possibility of determination $H_{c1}$ using the conventional method [55], [46].",
"For this purpose we model the typical $M(H)$ dependence using a piecewise analytical formulae.",
"We break the range of measurements in two regions.",
"In the low-field region we model the $M(H)$ data with a linear model dependence, whereas above a certain field $H_0$ – with a parabolic one.",
"The parameters for both linear and non-linear parts are fitted to the measured $M(H)$ data; the step size for the model dependences was chosen 0.5 Oe.",
"We further add a random signal within a chosen noise level to each data point of the model $M(H)$ dependence, and consider how this noise affects the correlation coefficient $R$ , calculated by the conventional method [55], [46], and also the extracted $H_{c1}$ value.",
"It appears that the correlation coefficient calculated for the noise level about $10^{-6}$ emu coincides with that calculated in Ref. [46].",
"Particularly, it exhibits a plateau below $H_{c1}$ .",
"By taking the field value where $R$ starts sharply decreasing we obtain the $H_{c1}$ value that also coincides with the $H_{0}$ parameter of the model.",
"The inset to Fig.",
"REF a shows a typical correlation coefficient $R(H)$ calculated for the noise level $10^{-6}$ emu.",
"This dependence has a weak maximum at $H=16$ Oe, which is only by 6% higher than the parameter $H_0=15$ Oe included in the model.",
"However, as noise increases, the $R(H)$ dependence changes drastically: the maximum becomes more clearly pronounced and its departure from $H_0$ increases.",
"Figure REF a shows the correlation coefficient $R$ calculated for the model $M(H)$ dependence with a $50\\times $ bigger noise level, $5\\times 10^{-5}$ (typical for the experiment), and for $H_0=15$ Oe.",
"Instead of plateau, $R(H)$ here exhibits a maximum at $\\approx 25$ Oe which is essentially higher than the given $H_0$ value.",
"Figure: (a) Correlation coefficient RR calculated for model data with noise level about 5×10 -5 5 \\times 10^{-5} emu by method used in , .Inset: correlation coefficient RR calculated for model data with noise level 10 -6 10^{-6} emu.",
"(b) Deviation of the correlation index from unity δR ' =1-R ' \\delta R^{\\prime }=1-R^\\prime versus HH calculated by the modified method for the same model data.In order to overcome the problem of extraction the $H_{c1}$ value in the presence of noise, we have modified the above algorithm of Refs.",
"[55], [46].",
"In the modified method we expand the trapped magnetization $M$ as $M (H) \\propto (H -H_{c1})^2$ in the vicinity of $H_{c1}$ .",
"Correspondingly, the magnetization may be written as follows: $M(H)&=& aH+b \\nonumber \\qquad \\qquad \\qquad \\quad \\text{\\rm for } H<H^*\\\\M(H)&=& aH+b +c (H-H^*)^2 \\quad \\text{\\rm for } H>H^*.$ For every running data point $H_i$ we take $H^*=H_i$ and find the best fitting of the experimental data with the model curve Eq.",
"(REF ), using $a, b$ and $c$ as fitting parameters (the parameter $b$ corresponds to an insignificant, an order of $10^{-4}$ emu, possible residual zero field magnetization $M (H = 0)$ ).",
"For every $H^*=H_i$ we calculate the correlation index (coefficient of determination) as follows: $R^{\\prime }(H)=\\sqrt{1-\\frac{\\sum {(M_{exper}(H)-M_{fit}(H))^2}}{\\sum {(M_{exper}(H)-\\overline{M})^2}}}.$ Here $M_{fit}$ is the magnetization calculated within the model Eq.",
"(REF ) for the given set of parameters $a_i, b_i$ and $c_i$ which are determined at point $H_i$ , and $\\overline{M}$ is the averaged magnetization value.",
"The model Eq.",
"(REF ) is expected to give the best fit of the experimental data at the $H_i=H_{c1}$ , therefore we interpret the $R^{\\prime }$ maximum point as $H_{c1}$ .",
"Figure REF b shows the deviation from unity of correlation index calculated using the modified method for the same model function $M(H)$ as that used above for calculations of $R$ in Fig.",
"REF a, and for the same noise level $5\\times 10^{-5}$ emu.",
"This dependence has a maximum at $H=15.5$ Oe that agrees with $H_0=15$ Oe used in the model $M(H)$ .",
"By random varying the magnetization within the the same noise level we found that the maximum of $R^{\\prime }$ (and therefore $H_{c1}$ ) varies within 2 Oe; we consider this as the estimate of the uncertainty of $H_{c1}$ .",
"Figure REF shows the deviation from unity of correlation index $\\delta R^{\\prime }=1-R^{\\prime }$ , versus $H$ calculated from our experimental $M(H)$ data measured at $T=15$ K. The upper inset shows this dependence near it's minimum.",
"This minimum is taken as the best estimate of $H_{c1}$ .",
"The lower inset shows deviation of the experimental $M(H)$ data from the best linear fit calculated with parameters $a_i$ and $b_i$ defined for the point of maximum $R^{\\prime }(H_i)$ .",
"The high fit quality demonstrates the applicability of the model Eq.",
"(REF ) to the experimental data.",
"Figure: Deviation from unity of the correlation index δR ' =1-R ' (H)\\delta R^{\\prime }=1-R^\\prime (H) calculated by the modified method for the data taken at T=15T = 15 K. The upper inset shows the δR ' (H)\\delta R^{\\prime }(H) dependence near its minimum.",
"The lower inset shows deviation of the measured M(H)M(H) data from the linear M(H)M(H) dependence."
]
] | 1612.05540 |
[
[
"Event-by-event picture for the medium-induced jet evolution"
],
[
"Abstract We discuss the evolution of an energetic jet which propagates through a dense quark-gluon plasma and radiates gluons due to its interactions with the medium.",
"Within perturbative QCD, this evolution can be described as a stochastic branching process, that we have managed to solve exactly.",
"We present exact, analytic, results for the gluon spectrum (the average gluon distribution) and for the higher n-point functions, which describe correlations and fluctuations.",
"Using these results, we construct the event-by-event picture of the gluon distribution produced via medium-induced gluon branching.",
"In contrast to what happens in a usual QCD cascade in vacuum, the medium-induced branchings are quasi-democratic, with offspring gluons carrying sizable fractions of the energy of their parent parton.",
"We find large fluctuations in the energy loss and in the multiplicity of soft gluons.",
"The multiplicity distribution is predicted to exhibit KNO (Koba-Nielsen-Olesen) scaling.",
"These predictions can be tested in Pb+Pb collisions at the LHC, via event-by-event measurements of the di-jet asymmetry.",
"Based on JHEP 1605 (2016) 008 and arXiv:1609.06104"
],
[
"Introduction",
"One of the observables in which the formation of a collective medium in heavy-ion collisions manifests in a very clear way is the dijet asymmetry, the energy difference between two approximately back-to-back jets [3], [4].",
"The usual interpretation of this observation is the following: The two energetic jets are initially created in a hard process, due to momentum conservation the two jets will have back-to-back directions and approximately the same energy.",
"For simplicity we consider central collisions in which the interaction plane has rotational symmetry.",
"The point of the collision region in which the hard process takes place is not always the center, a deviation from this point will have as a consequence that the effective size of the medium seen by each jet will be different.",
"The two jets will lose energy inside of the medium, however the amount of energy loss will depend on the size of the medium that they traverse.",
"In summary, the fact that the formation of the two jets does not happen at the center of the fireball translates in an asymmetry in the effective length of the medium seen by each jet that at the same time translates into an asymmetry in the energy loss.",
"However, this might not be the whole story.",
"In the previous discussion we were assuming that the energy loss is always the same at fixed medium size, in other words, we were neglecting fluctuations.",
"The question is then, how big are these fluctuations?",
"This is one of the main problems we are going to address in this proceedings and the answer we are going to find is that the typical deviation in the energy loss is of the order of the average value and therefore fluctuations can not be neglected.",
"We are going to arrive to this conclusion by performing an analytical computation based on the results obtained in [5], [6].",
"A similar result was obtained recently by a Monte Carlo computation in [7].",
"Another issue we want to discuss in this proceedings is what are the event-by-event properties of gluons produced by the energy loss mechanism."
],
[
"Jet quenching formalism",
"We are going to perform the computation using the BDMPS-Z theory [8], [9].",
"In this formalism all the information that we need from the medium is encoded in its length $L$ and a parameter called $\\hat{q}$ that controls the amount of jet broadening induced by the medium.",
"There are two time-scales that have a very important role in this problem.",
"The first one we are going to discuss is the formation time.",
"This is given by the uncertainty relation $\\tau _f\\sim \\frac{2\\omega }{k_\\perp ^2}$ where $\\omega $ is the energy of the gluon that is being emitted.",
"In a medium $k_\\perp \\sim \\sqrt{\\hat{q}\\tau _f}$ , this gives a self consistent equation that results in $\\tau _f\\sim \\sqrt{\\frac{2\\omega }{\\hat{q}}}$ .",
"Another important time-scale is the branching time.",
"In the BDMPS-Z theory the probability to emit a gluon during a small time $\\Delta t$ is $P(\\omega ,\\Delta t)\\propto \\frac{N_c\\alpha _s}{\\pi }\\sqrt{\\frac{\\hat{q}}{\\omega }}\\Delta t\\,,$ the branching time $\\tau _{br}$ is the period after which we are almost sure that a gluon with a given energy will be emitted, looking at the previous equation we can see that $\\tau _{br}(\\omega )=\\frac{\\pi }{N_c \\alpha _s}\\tau _f(\\omega )$ , this shows that in perturbation theory the formation time is much smaller than the branching time and therefore, at first approximation, the branching process can be thought as an almost classical process in which gluons are formed instantaneously.",
"The branching time allows to divide the gluons in two different types: Soft gluons have an energy such that $\\tau _{br}\\ll L$ therefore they will be emitted abundantly.",
"The harder gluons which are likely to be emitted are those with $\\tau _{br}\\sim L$ , this implies that they will have an energy of order $\\omega _{br}\\sim \\alpha _s^2\\hat{q}L^2$ .",
"Their emission by the leading particle will dominate the energy loss.",
"However, this gluons with energy $\\omega _{br}$ will subsequently branch and at the end of the day what will be found is a lot of soft gluons emitted at large angles.",
"The equations and the consequences of the multiple branching obtained with the previous assumptions were discussed in [5], [6], there it was observed the importance of the so-called democratic branching, the process in which a parton branches in a way such that the resulting partons have a similar energy.",
"This will be a rare event for the leading particle because their energy is much bigger than $\\omega _{br}$ , however for the gluons emitted by the leading particle, that will typically have an energy of the order of $\\omega _{br}$ or smaller, this will be a very common process and a very efficient way to transfer energy into low energy gluons emitted at large angles."
],
[
"The gluon spectrum and the average energy loss",
"The main focus of this section is going to be the gluon spectrum that we define as $D(x,t)=x\\langle \\sum _i\\delta (x_i-x)\\rangle \\,,$ where $x$ is the energy fraction carried by the gluon.",
"This quantity evolves with time following the equation [10] $\\frac{\\partial }{\\partial \\tau } D(x,\\tau )=\\int {\\rm d}z \\,{\\cal K}(z)\\left[\\sqrt{\\frac{z}{x}}D\\left(\\frac{x}{z},\\tau \\right)-\\frac{z}{\\sqrt{x}}D(x,\\tau )\\right]\\,,$ where $\\tau =\\frac{\\alpha _s N_c}{\\pi }\\sqrt{\\frac{\\hat{q}}{E}}t=\\frac{t}{\\tau _{br}(E)}$ .",
"$E$ in this case is the energy of the leading particle.",
"The case interesting for jet quenching at LHC is therefore the one in which $\\tau \\ll 1$ , however the case $\\tau \\sim 1$ is also interesting in order to understand how jets are absorbed by the medium.",
"The kernel ${\\cal K}(z)$ in eq.",
"(REF ) has the form ${\\cal K}(z)=\\frac{[1-z(1-z)]^{5/2}}{[z(1-z)]^{3/2}}\\,,$ however eq.",
"(REF ) has not been analytically solved so far with this kernel.",
"In [6] it was solved with the approximate kernel ${\\cal K}_0(z)=\\frac{1}{[z(1-z)]^{3/2}}$ and the initial condition $D(x,0)=\\delta (1-x)$ $D(x,\\tau )=\\frac{\\tau }{\\sqrt{x}(1-x)^{3/2}}\\exp \\lbrace -\\frac{\\pi \\tau ^2}{1-x}\\rbrace \\,.$ The previous formula implies that the energy decreases with time, in fact $\\langle X(\\tau )\\rangle =\\int _0^1\\,dxD(x,\\tau )=e^{-\\pi \\tau ^2}\\,.$ This leaves the question of where this missing energy goes.",
"Eq.",
"(REF ) is only valid for particles with a momentum much bigger than that of the particles in the medium.",
"This can be quantified by an infrared cut-off in momentum fraction $x_0$ , remarkably $D(x,\\tau )$ can be accurately computed setting $x_0=0$ .",
"The energy that is not captured inside of $D(x,\\tau )$ goes to degrees of freedom with energy fraction smaller than $x_0$ , in other words, to the medium.",
"In summary, the energy loss into the medium is $\\mathcal {E}(\\tau )=E\\left(1-e^{-\\pi \\tau ^2}\\right)\\,.$"
],
[
"The 2-point function and the fluctuations of the energy loss",
"In order to quantify the importance of the energy loss fluctuations we will compute the variance (more details on the computation are given in [1]) $\\sigma ^2_\\mathcal {E}=E^2(\\langle X^2\\rangle -\\langle X\\rangle ^2)\\,.$ We already computed the value of $\\langle X(\\tau )\\rangle $ .",
"In order to compute $\\langle X^2(\\tau )\\rangle $ apart from $D(x,\\tau )$ we also need the 2-point function defined as $D^{(2)}(x,x^{\\prime },t)=xx^{\\prime }\\langle \\sum _{i\\ne j}\\delta (x_i-x)\\delta (x_j-x^{\\prime })\\rangle \\,,$ which gives information about the pairs of partons with different energy found inside the jet.",
"Knowing this $\\langle X^ 2\\rangle $ is determined as $\\langle X^2(t)\\rangle =\\int _0^1\\,dx xD(x,t)+\\int _0^1\\,dx\\int _0^1\\,dx^{\\prime }D^{(2)}(x,x^{\\prime },t)\\,.$ $D^{(2)}$ fulfills an evolution equation similar to the one in eq.",
"(REF ) , this has the solution $D^{(2)}(x,x^{\\prime },\\tau )\\,=\\,\\frac{1}{2\\pi }\\frac{1}{\\sqrt{x x^{\\prime } (1-x-x^{\\prime })}}\\left[{\\rm e}^{-\\frac{\\pi \\tau ^2}{1-x-x^{\\prime }}}-{\\rm e}^{-\\frac{4\\pi \\tau ^2}{1-x-x^{\\prime }}}\\right].$ At small times this equation has a peak corresponding to pairs whose sum of energy correspond approximately to the original energy of the leading particle $x+x^{\\prime }\\sim 1$ .",
"However this peak will disappear quite quickly as time passes.",
"There is another peak that indicates that there will be a large number of pairs formed by soft particles (both $x$ and $x^{\\prime }$ much smaller than 1).",
"With this result we can already compute $\\sigma ^2_\\mathcal {E}$ .",
"In the limit $\\tau \\ll 1$ , which is the one interesting for LHC physics $\\sigma ^2_{\\mathcal {E}}(\\tau )=E^2\\left(\\frac{1}{3}\\pi ^2\\tau ^4-\\,\\frac{11}{15}\\pi ^3\\tau ^6\\right)+\\mathcal {O}(E^2\\tau ^8)\\,,$ this result means that the typical deviation will go like $E\\tau ^2\\sim \\omega _{br}$ .",
"This means that both the average and the typical deviation are of the same order of magnitude and that both are of the size of $\\omega _{br}$ .",
"Let us now discuss the phenomenological consequences of this result.",
"We focus in back-to-back pairs from which we assume that initially both of them have an energy $E$ but they will typically see different path length.",
"On top of that we will have the fluctuations of the energy loss mechanism itself that we have just computed, in this situation $\\langle E_1-E_2\\rangle ^2=(N_c\\alpha _s\\hat{q})^2(\\langle L_1^2\\rangle -\\langle L_2^2\\rangle )^2\\,,$ where the symbol $\\langle \\cdot \\rangle $ applied on $L_x$ means average over the geometry of the fireball in the different events.",
"This equation tells us that the observation of a $\\langle E_1-E_2\\rangle ^2$ different from 0 indicates an asymmetry in the path length seen by the jets.",
"However, what is observed experimentally is $\\langle |E_1-E_2|\\rangle $ rather than $\\langle E_1-E_2\\rangle $ .",
"Therefore the following quantity might give a more precise picture of what is actually observed in experiments $&&\\sigma ^2_{E_1-E_2}=\\langle (E_1-E_2)^2\\rangle -\\langle E_1-E_2\\rangle ^2=\\nonumber \\\\&&(N_c\\alpha _s\\hat{q})^2\\left[\\frac{1}{3}(\\langle L_1^4\\rangle +\\langle L_2^4\\rangle )+\\sigma ^2_{L_1^2}+\\sigma ^2_{L^2_2}\\right]\\,,$ looking at this equation we see that indeed the asymmetry on the path length contributes but we also see that even in the case $L_1=L_2$ there will be a non-zero contribution.",
"We also see, looking at eq.",
"(REF ), that both effects are of the same order of magnitude."
],
[
"The n-point functions and KNO scaling",
"In order to compute the average energy loss $\\langle \\mathcal {E}\\rangle $ and the average number of particles inside the jet $\\langle N\\rangle $ we need to know the gluon spectrum $D$ .",
"If we want to compute $\\langle \\mathcal {E}^2\\rangle $ and $\\langle N^2\\rangle $ we also need to know $D^{(2)}$ .",
"If we want to have more detailed information on the energy loss and the distribution of particles we need to compute higher order n-point functions $D^{(n)}$ .",
"They fulfill an evolution equation similar to the one of $D^{(2)}$ , they can be analytically solved using the same approximations [2] $D^{(n)}(x_1,\\cdots ,x_n|\\tau )=\\frac{(n!",
")^2}{2^{n-1}n}\\frac{(1-\\sum _{i=1}^nx_i)^{\\frac{n-3}{2}}}{\\sqrt{x_1\\cdots x_n}}h_n\\left(\\frac{\\tau }{\\sqrt{1-\\sum _{j=1}^nx_j}}\\right)\\,,$ where $h_n(l)=\\int _0^l\\,dl_{n-1}\\cdots \\int _0^{l_2}\\,dl_1(nl-\\sum _{i=1}^{n-1}l_i){\\rm e}^{-\\pi (nl-\\sum _{j=1}^{n-1}l_j)^2}\\,.$ As was already mentioned, the interesting limit for LHC is $\\tau \\ll 1$ .",
"If we are also in the limit $x_0\\ll \\tau ^2$ (very small resolution scale) the number of particles will be completely dominated by soft gluons and we can compute the leading order contribution to $\\langle N\\rangle $ analytically.",
"In the more restrictive case in which $x_0\\ll \\tau ^2$ and also $\\pi n^2\\tau ^2\\ll 1$ we can, using eq.",
"(REF ), do the same for $\\langle N^n\\rangle $ .",
"All the moments of the number of particles will diverge as $x_0\\rightarrow 0$ , however the ratio $C_p=\\frac{\\langle N^p\\rangle }{\\langle N\\rangle ^p}=\\frac{(p+1)!",
"}{2^p}\\,,$ will be a constant that does only depend of $p$ .",
"This property is called KNO scaling [11] and appears in several processes in heavy-ion as well as in collider physics.",
"In fact, eq.",
"(REF ) corresponds to a negative binomial distribution with parameter $k=2$ .",
"This distribution gives the probability of having $n$ successful attempts in a Bernoulli trial before having $k$ failures, in this case 2.",
"Similar properties were also found in the vacuum [12], there it was seen that KNO scaling is also fulfilled and that the distribution of emitted gluons was approximately described by a negative binomial distribution but this time with $k=3$ .",
"In conclusion we can see that the distribution of gluons produced by a jet, either in a medium or in the vacuum, can be approximately described by a negative binomial distribution and therefore they approximately fulfill KNO scaling.",
"The difference is that in a medium fluctuations and correlations are much more important."
],
[
"Conclusions",
"In this proceedings we have reviewed the computation of the fluctuations of the energy loss.",
"We have seen that they large, of the order of the average value.",
"This means that they can not be neglected when interpreting experimental results.",
"This is particularly important for the dijet asymmetry, our result shows that such an asymmetry can be generated even if the medium path length that each jet traverses is the same.",
"This is in contradiction with the usual picture.",
"We have also shown that the gluons emitted during the process in which the jet loses energy fulfill KNO scaling and can be approximately described by a negative binomial distribution.",
"Remarkably this is similar to what is found in the vacuum where the physics is very different.",
"Comparing the two cases we see that in the medium correlations and fluctuations are much bigger."
],
[
"Acknowledgments",
"The work of M.A.E.",
"has been supported, during the preparation of the talk and the proceedings, in part by the European Research Council under the Advanced Investigator Grant ERC-AD-267258 and in part by the Academy of Finland, project 303756."
]
] | 1612.05485 |
[
[
"Monogamy inequalities for entanglement using continuous variable\n measurements"
],
[
"Abstract We consider three modes $A$, $B$ and $C$ and derive continuous variable monogamy inequalities that constrain the distribution of bipartite entanglement amongst the three modes.",
"The inequalities hold for all such tripartite states, without the assumption of Gaussian states, and are based on measurements of two conjugate quadrature phase amplitudes $X_{i}$ and $P_{i}$ at each mode $i=A,B$.",
"The first monogamy inequality is $D_{BA}+D_{BC}\\geq1$ where $D_{BA}<1$ is the widely used symmetric entanglement criterion, for which $D_{BA}$ is the sum of the variances of $(X_{A}-X_{B})/2$ and $(P_{A}+P_{B})/2$.",
"A second monogamy inequality is $Ent_{BA}Ent_{BC}\\geq\\frac{1}{\\left(1+(g_{BA}^{(sym)})^{2}\\right)\\left(1+(g_{BC}^{(sym)})^{2}\\right)}$ where $Ent_{BA}<1$ is the EPR variance product criterion for entanglement.",
"Here $Ent_{BA}$ is a normalised product of variances of $X_{B}-g_{BA}^{(sym)}X_{A}$ and $P_{B}+g_{BA}^{(sym)}P_{A}$, and $g_{BA}^{(sym)}$ is a parameter that gives a measure of the symmetry between the moments of $A$ and $B$.",
"We also show that the monogamy bounds are increased if a standard steering criterion for the steering of $B$ is not satisfied.",
"We illustrate the monogamy for continuous variable tripartite entangled states including the effects of losses and noise, and identify regimes of saturation of the inequalities.",
"The monogamy relations explain the experimentally observed saturation at $D_{AB}=0.5$ for the entanglement between $A$ and $B$ when both modes have 50\\% losses, and may be useful to establish rigorous bounds of correlation for the purpose of quantum key distribution protocols."
],
[
"Introduction",
"Entanglement is the major resource for many applications in quantum information processing.",
"Measurable quantifiers exist to determine the amount of entanglement shared between two separated parties, or subsystems, that we denote $A$ and $B$ .",
"According to quantum mechanics, the amount of entanglement that exists between two parties $A$ and $B$ puts a constraint on the amount of entanglement that exists between one of those parties ($B$ say) and a third party, $C$ .",
"This fundamental result is called monogamy of entanglement.",
"If the entanglement between two parties $A$ and $B$ can be quantified, it is useful to be able to place a numerical bound on the quantifiable entanglement between the parties $B$ and $C$ .",
"As an example, such relations have application to quantum key distribution, where the amount of bipartite entanglement between two parties gives a measure of the correlation between the bit sequences (and hence the key) that each party possesses.",
"The monogamy relations can thus quantify the security of the information shared between two parties.",
"Monogamy relations may also be useful to understand how the bipartite entanglement can be distributed for various types of multipartite entangled states.",
"A quantifiable monogamy relation involving the concurrence measure of bipartite entanglement was originally derived for three qubit systems by Coffman, Kundu and Wootters [1].",
"Since then, the interest in understanding and quantifying monogamy of entanglement has expanded [6], [5], [4], [7], [11], [8], [13], [2], [10], [12], [15], [9], [14], [3].",
"Work by Adesso et al [6], [5], [4] formulated monogamy relations for systems involving Gaussian states [16] and continuous variable measurements.",
"Barrett et al, Masanes et al and Toner et alred investigated the monogamy of Bell nonlocality [12], [11].",
"Monogamy relations for the Einstein-Podolsky-Rosen (EPR) paradox and EPR steering, which are directional forms of nonlocality [19], [18], [22], [21], [20], have been studied and derived in Refs.",
"[13], [15], [14].",
"In this paper, we derive entanglement monogamy relations for continuous variable entanglement quantifiers based on Einstein-Podolsky-Rosen (EPR) correlations [20].",
"The amount of EPR correlation between the field quadrature phase amplitudes $X_{A,B}$ and $P_{A,B}$ of two modes $A$ , $B$ can be defined using variances [21], [23], [24], [26].",
"Often the EPR correlations are as for a two-mode squeezed state, where the correlation is between $X_{A}$ and $X_{B}$ , and $P_{A}$ and $-P_{B}$ , so that the variances of $X_{A}-X_{B}$ and $P_{A}+P_{B}$ vanish in a limit of perfect entanglement [21].",
"This limit is thus also a limit of infinite two-mode squeezing.",
"Entanglement can be inferred if the sum of these two variances drops below a critical level [26], [24], [23].",
"Considering the sum of the two variances $D_{AB}=\\lbrace (\\Delta (X_{A}-X_{B}))^{2}+(\\Delta (P_{A}+P_{B}))^{2}\\rbrace /4$ , the Tan-Duan-Giedke-Cirac-Zoller (TDGCZ) criterion for entanglement is $D_{AB}<1$ [23], [24].",
"Here we use a scaling of the quadratures so that the uncertainty principle gives $\\Delta X_{A}\\Delta P_{A}\\ge 1$ and $\\Delta X_{B}\\Delta P_{B}\\ge 1$ .",
"This criterion has been used in numerous experiments to detect entanglement [22].",
"Importantly, it is a symmetric criterion, in that the criterion is unchanged if the labels $A$ and $B$ are exchanged.",
"In this paper, we consider three modes $A$ , $B$ and $C$ and derive the monogamy inequality $D_{BA}+D_{BC}\\ge \\max \\lbrace 1,S_{B|{\\lbrace AC\\rbrace }}\\rbrace $ that holds to describe the distribution of the bipartite entanglement for all states, without the assumption of Gaussianity.",
"Here $S_{B|\\lbrace AC\\rbrace }$ is an EPR steering parameter that certifies steering of mode $B$ by measurements on the combined system $AC$ if $S_{B|\\lbrace AC\\rbrace }<1$ .",
"This steering parameter was used in the experiments described in Ref.",
"[22] that detected the continuous variable EPR paradox.",
"The relation of Eq.",
"(REF ) may be useful to establish rigorous bounds of correlation for the purpose of quantum key distribution protocols and also explains the experimentally observed saturation at $D_{AB}=0.5$ by Bowen et al.",
"[25] for the entanglement between $A$ and $B$ when both modes undergo 50% attenuation of intensity .",
"Figure: Entanglement monogamy: The entanglement is shared in a waythat ensures D BA +D BC ≥1D_{BA}+D_{BC}\\ge 1.",
"Here, D AB D_{AB} is a quantifierof symmetric bipartite entanglement between AA and BB.",
"Bipartiteentanglement is depicted by the dashed lines.",
"A stricter conditiongiven by Eq.",
"() places bounds on the valueof the more general entanglement bipartite quantifiers Ent AB Ent_{AB}and Ent BC Ent_{BC}, in terms of bipartite symmetry parameters g BA (sym) g_{BA}^{(sym)}and g BC (sym) g_{BC}^{(sym)}.",
"The main text shows how stricter conditionsapply if steering of system BB by the combined systems ACAC cannotbe demonstrated using a standard steering criterion.It was shown by blackDuan et al [24] and Giovannetti et al [26] that a more sensitive entanglement criterion is possible if one considers the variances of $X_{A}+gX_{B}$ and $P_{A}-gP_{B}$ , where $g$ are real constants.",
"This leads to a criterion for entanglement of $Ent_{AB}<1$ , where $Ent_{AB}$ is defined as a normalised product of the variances of $X_{A}+gX_{B}$ and $P_{A}-gP_{B}$ , the $g$ being optimally chosen [27], [28].",
"For an important class of two-mode Gaussian systems, such criteria have been shown to be necessary and sufficient and equivalent to the Peres-Simon Positive-Partial-Transpose (PPT) criterion [27], [24], [29].",
"In Section IV of this paper, we derive monogamy relations for this PPT-EPR variance entanglement quantifier.",
"Specifically, we show that $Ent_{BA}Ent_{BC}\\geqslant \\frac{\\max {\\lbrace 1,S_{B|\\lbrace AC\\rbrace }^{2}\\rbrace }}{\\left(1+(g_{BA}^{(sym)})^{2}\\right)\\left(1+(g_{BC}^{(sym)})^{2}\\right)}$ Here, $g_{BA}^{(sym)}$ and $g_{BC}^{(sym)}$ are symmetry parameters introduced in Refs.",
"[27], [28], which quantify the amount of symmetry between the moments of $B$ and $A$ , and $B$ and $C$ , respectively.",
"These parameters take the value $g^{(sym)}=1$ when the moments are perfectly symmetrical.",
"It has been explained in the Refs.",
"[27], [28] how the effect of thermal noise and dissipation on the different modes can alter the values of the symmetry parameters.",
"In Sections III and IV, we illustrate the application of the monogamy relations Eqs.",
"(REF ) and (REF ) to the tripartite entangled system created using a two-mode squeezed state and a beam splitter [30].",
"Such tripartite entangled states (or very similar states) have been realised experimentally [34], [36], [35], [31], [33], [32].",
"We show that the second inequality (Eq.",
"(2)) is more sensitive and saturates for this tripartite system for all beam splitter couplings, in regimes corresponding to a high squeeze parameter $r$ where the tripartite entanglement is maximised in terms of the underlying EPR correlations.",
"The first relation (Eq.",
"(1)) is useful however for identifying bounds on the symmetric entanglement quantified by $D_{AB}$ , which has been shown useful for specific teleportation protocols [28].",
"We also study the monogamy relations where coupling of the modes to the environment will create additional losses and thermal noise.",
"Such couplings can be asymmetric (hence altering the symmetry parameters) and also lead to a decrease in the amount of EPR steering possible.",
"We are thus able to verify the decrease in overall entanglement when the steering identified by the parameter $S_{A|\\lbrace BC\\rbrace }$ is diminished.",
"Importantly, we identify regimes of saturation of the monogamy inequality (2), for almost all values of attenuation of the shared mode (depicted by $B$ in Figure 1), if mode $C$ has been created by an eavesdropper using a 50/50 beam splitter to tap mode $A$ .",
"This gives a fundamental explanation of the observed optimal value of $D_{BA}=0.5$ measured in the experiment of Bowen et al, for the symmetric case when the mode $B$ has 50% attentuation [25].",
"Our derivations are general for three-mode tripartite states $A$ , $B$ and $C$ and do not depend on the assumption of Gaussian states [16].",
"The derivations are based on a previous monogamy result for EPR-steering given in reference [13] and in fact we find that the steering plays an important role in the monogamy relations.",
"Only if the steering between $B$ and $AC$ is preserved in a specific directional sense is the monogamy bound limited by the quantum noise level.",
"This gives an indication that directional properties of entanglement, such as steering for which the parties are not interchangeable, play an important role in quantum information applications."
],
[
"Monogamy of Entanglement using the symmetric TDGCZ criterion",
"The symmetric Tan-Duan-Giedke-Cirac-Zoller (TDGCZ) criterion for certifying entanglement between two modes is defined in terms of the sum of the Einstein-Podolsky-Rosen variances $D_{AB}=\\frac{1}{4}\\Bigl (\\mathrm {Var}(X_{A}-X_{B})+\\mathrm {Var}(P_{A}+P_{B})\\Bigr )$ and is given as [23], [24]: $D_{AB}<1$ Here $X_{A},$ $P_{A}$ and $X_{B}$ , $P_{B}$ are the quadrature phase amplitudes for modes symbolised by $A$ and $B$ respectively, and $\\mathrm {Var}(X)=(\\Delta X)^{2}=\\langle X^{2}\\rangle -\\langle X\\rangle ^{2}$ denotes the variance of $X$ .",
"Denoting the boson annihilation operators of each mode by $\\hat{a}$ and $\\hat{b}$ , we have selected $X_{A}=\\hat{a}+\\hat{a}^{\\dagger }$ , $P_{A}=(\\hat{a}-\\hat{a}^{\\dagger })/i,$ $X_{B}=\\hat{b}+\\hat{b}^{\\dagger }$ and $P_{B}=(\\hat{b}-\\hat{b}^{\\dagger })/i$ for which the uncertainty relation is $\\Delta X\\Delta P\\ge 1$ .",
"The criterion $D_{AB}<1$ is sufficient (though not necessary) to detect entanglement for all two-mode states, regardless of assumptions about the nature of the two-mode state.",
"Nonetheless, for two-mode Gaussian symmetric fields where the moments of fields $A$ and $B$ are equal, the entanglement criterion can be shown necessary and sufficient for two-mode entanglement for some choice of quadrature phase amplitudes $X$ and $P$ (defined by a phase angle $\\theta $ ) [24], [29].",
"Result (1): The first main result of the paper is that for any three modes $A$ , $B$ and $C$ , the following monogamy relation holds: $D_{BA}+D_{BC}\\ge 1$ The proof is given in the Appendix and is based on an earlier result for monogamy of steering [13].",
"The relation holds for all three-mode quantum states.",
"In particular, the relation does not rely on the assumption of Gaussian states.",
"The monogamy relation (REF ) has an inherent asymmetry with respect to $B$ and the remaining two systems $A$ and $C$ .",
"This asymmetry is depicted in the Fig.",
"(REF ).",
"In fact, we notice that it is possible to prove a result relating the entanglement sharing to a steering parameter $S_{B|\\lbrace AC\\rbrace }$ for the steering of the system $B$ by the composite system $AC$ .",
"We introduce the steering parameter as $S_{A|B}$ as follows.",
"A sufficient condition to demonstrate steering of $B$ (by measurements made on $A$ ) is [21] $S_{B|A}<1$ where the steering parameter $S_{B|A}$ is defined as $S_{B|A}=\\Delta (X_{B}-g_{x}X_{A})\\Delta (P_{B}+g_{p}P_{A})<1$ Here $g_{x}$ and $g_{p}$ are real constants chosen to minimise the value of $S_{B|A}$ .",
"The condition becomes necessary and sufficient for two-mode Gaussian states, if $g_{x}$ and $g_{p}$ and the choice of quadrature phase amplitudes $X$ , $P$ are optimised [18], [17], [43].",
"blackThe steering parameter can be more generally defined as [22] $S_{B|A}=\\Delta _{inf}X_{B\\vert A}\\Delta _{inf}P_{B\\vert A}$ where $[\\Delta _{inf}(X_{B}|X_{A})]^{2}=\\sum _{x_{A}}P(x_{A})\\sum _{x_{B}}P(x_{B}|x_{A})(x_{B}-\\mu _{B|x_{A}})^{2}$ is the average conditional variance for $X_{B}$ given the measurement $X_{A}$ at $A$ .",
"The $\\lbrace x_{A}\\rbrace $ is the set of all possible outcomes for $X_{A}$ and $\\mu _{B|x_{A}}$ is the mean of $P(X_{B}|X_{A}=x_{A})$ .",
"blackThe $\\Delta _{inf}X_{B|A}$ is taken as the minimum value of $\\Delta _{inf}(X_{B}|X_{A})$ over all possible choices of measurement $X_{A}$ that can be made at $A$ .",
"The results of this paper hold for both definitions of $S_{B|A}$ , as is apparent from the proofs given in the Appendix.",
"For the example of two-mode Gaussian states, the definitions become equivalent [22].",
"blackThe next result indicates that the distribution of the entanglement as detected by the $D_{AB}$ parameter in accordance with inequality (REF ) can only be optimised if steering exists between the system $B$ and the composite system $AC.$ Result (2): The following inequality holds: $D_{BA}+D_{BC}\\ge \\max {\\lbrace 1,S_{B|\\lbrace AC\\rbrace }\\rbrace }$ blackThe proof is given in the Appendix.",
"For this inequality, it is necessary to ensure that the measured steering parameter is the optimal one, obtained by optimising the values of $g_{x}$ and $g_{p}$ and the choice of quadrature phase angle for the inference of $B$ .",
"For this reason, the second definition (REF ) involving the conditional variances is generally more useful, where care is taken to ensure the conditional variances are defined for the choice of quadratures at both $A$ and $C$ that minimise the conditional variance.",
"The monogamy inequality (REF ) relates the TDGCZ entanglement to the value of the steering parameter $S_{B|\\lbrace AC\\rbrace }$ .",
"If the noise levels are such that there is no steering of $B$ by the composite system $AC$ detectable by $S_{B|\\lbrace AC\\rbrace }<1$ , then the amount of entanglement is reduced.",
"blackThe monogamy relation states that the lower bound $D_{BA}+D_{BC}=1$ can only be reached if there is steering of $B$ by the composite system $AC$ .",
"Otherwise the monogamy is restricted by the value of the steering parameter."
],
[
"Illustration of Monogamy for tripartite CV entangled systems",
"blackThe relations given by Results (1) and (2) can be verified experimentally and are useful to explain past experimental observations.",
"We consider the continuous variable (CV) tripartite system generated by placing squeezed vacuum inputs through a series of beam splitters,black or else via nondegenerate down conversion followed by a beam splitter on one mode, as shown in the diagram of Fig.",
"2black [30].",
"At the output of the device are three modes, that we label $A$ , $B$ and $C$ .",
"blackThe essential feature is that a two-mode squeezed state is first generated for two modes that we label $A$ and $F$ .",
"The two-mode squeezing corresponds to an EPR entanglement between modes $B$ and $F$ , which can be generated as the outputs of a beam splitter $BS1$ with squeezed vacuum inputs, or else via a parametric down conversion (PDC) process [37].",
"The amount of entanglement (two-mode squeeing) between the two modes $B$ and $F$ is determined by the two-mode squeezing parameter .",
"The amount of entanglement increases as $r$ increases [21].",
"The mode $F$ is then coupled to a second beam splitter $BS2$ which has two output modes, $A$ and $C$ .",
"The transmission efficiency for $A$ is given by $\\eta _{0}$ ; that for $C$ is therefore $1-\\eta _{0}$ .",
"The second input to the beam splitter $BS2$ is a vacuum state [30].",
"Figure: Configuration for the generation of a tripartite entangled system.",
"Here, a two-mode squeezed state is generated using a parametricdown conversion (PDC) process.",
"Similar two-mode entanglement can becreated using single mode squeezed vacuum states that are input toa beam splitter BS1BS1.",
"Either way, entangled modes BB and FFare created at the outputs of the first device.",
"The entanglement betweenFF and BB is determined by a squeeze parameter rr.",
"Final modesAA and CC are created at the output of the second beam splitterBS2BS2 which has a transmission efficiency η 0 \\eta _{0} and secondvacuum input.",
"The resulting three modes BB, AA and CC are genuinelytripartite-entangled.The two-mode entanglement between modes $B$ and $F$ can be modelled as that of a two-mode squeezed state, given as the output of a parametric down conversion [21].",
"The nonzero covariance matrix elements in this case are denoted $n=\\langle X_{B}^{2}\\rangle $ , $m=\\langle X_{F}^{2}\\rangle $ and $c=\\langle X_{B}X_{F}\\rangle $ where here $\\langle X_{F}\\rangle =\\langle P_{F}\\rangle =..=0$ and the solutions for the two-mode squeezed state are $n=\\cosh 2r$ , $m=\\cosh 2r$ and $c=\\sinh (2r)$ .",
"We see that $[\\Delta (X_{B}-X_{F})]^{2}=[\\Delta (P_{B}+P_{F})]^{2}=n-2c+m$ .",
"The mode $F$ is then input to a beam splitter with transmission efficiency $\\eta _{0}$ (Fig.",
"2) and we see that the two-mode entanglement between $B$ and $A$ is calculated by defining A as the beam $F$ transmitted with efficiency $\\eta _{0}$ .",
"We now test the relation of Result (1) for the three output modes, specified in the diagram of Figure 2 by $A$ , $B$ and $C$ .",
"The beam splitter coupling is given by a unitary transformation, and we evaluate the correlation between $B$ and $A$ in terms of $\\eta _{0}$ and $r$ by tracing over the mode $C$ where the beam splitter transmission efficiency for $A$ is given by $\\eta _{0}$ .",
"Similarly, for the calculation of $D_{BC}$ the mode $C$ is evaluated by tracing over the mode $A$ .",
"The beam splitter efficiency for the transmission of the field $C$ is $1-\\eta _{0}$ .",
"The covariances become $n_{BA} & = & \\cosh 2r\\nonumber \\\\m_{BA} & = & \\eta _{0}\\cosh 2r+\\left(1-\\eta _{0}\\right)\\nonumber \\\\c_{BA} & = & \\sqrt{\\eta _{0}}\\sinh 2r$ blackand those for modes $B$ and $C$ are obtained by replacing $\\eta _{0}$ with $1-\\eta _{0}$ .",
"Hence $D_{BA} & = & (n_{BA}-2c_{BA}+m_{BA})/2\\nonumber \\\\& = & \\Bigl (\\eta _{0}\\cosh 2r+\\left(1-\\eta _{0}\\right)+\\cosh 2r\\nonumber \\\\& & -2\\sqrt{\\eta _{0}}\\sinh 2r\\Bigr )/2$ blackand $D_{BC} & = & \\Bigl ((1-\\eta _{0})\\cosh 2r+\\eta _{0}+\\cosh 2r\\nonumber \\\\& & -2\\sqrt{1-\\eta _{0}}\\sinh 2r\\Bigr )/2$ blackWe notice from the expression of $D_{BA}$ that $D_{BA}\\le 1$ for all $r$ when $\\eta _{0}=1$ .",
"For larger $r$ , $D_{BA}$ exceeds 1 for smaller $\\eta _{0}$ values.blue The monogamy relation of Result (1) is illustrated in Figure 3 for the configuration of Figure 2 where the modes $B$ and $F$ are generated as a two-mode squeezed state.",
"Figure 3 plots the values of $D_{BA}$ , $D_{BC}$ and $D_{BA}+D_{BC}$ for various $\\eta _{0}$ .",
"We note the relation is verified, but that saturation (achieved when the equality $D_{BA}+D_{BC}=1$ is reached) does not occur.",
"Figure: Monogamy for the tripartite state of Figure 2 with no extralosses or noise present.",
"The D BA D_{BA}, D BC D_{BC} and D BA +D BC D_{BA}+D_{BC}versus η 0 \\eta _{0} for the state of Figure 2.",
"Here r=0.5r=0.5 (a) andr=2r=2 (b).",
"The monogamy bound of 1 is indicated by the gray dottedline.blackTo test the relation of Result (2), we need to consider the steering of $B$ .",
"The value of the steering parameter $S_{B|F}$ is minimized to $S_{B|F} & = & \\left(n_{BF}-c_{BF}^{2}/m_{BF}\\right)$ using the optimal factorsred $g_{x}=c_{BF}/m_{BF}$ , $g_{p}=c_{BF}/m_{BF}$ .",
"blueThus, the steering parameter is [21] $S_{B|\\lbrace AC\\rbrace }=1/\\cosh (2r)$ whichred blackcannot exceed 1 for any $r$ .green The smallness of the steering parameter gives a measure of the degree of the steering.",
"The steering parameter $S_{B|\\lbrace AC\\rbrace }$ is evaluated as 0 for large $r$ by noting that the full knowledge of quadrature phase amplitudes of both $A$ and $C$ enables prediction of the amplitudes at $B$ .",
"This type of collective steering was examined in Ref.",
"[38].",
"Hence the sum $D_{BA}+D_{BC}$ is bounded below by the quantum noise level given by black1.",
"The Result (2) will be verified in Sections III.B and C for different scenarios.red"
],
[
"Extra losses for the shared mode $B$",
"We can model the effects of extra loss for mode $B$ , by coupling the mode $B$ to an imaginary beam splitter ($BS3$ ).",
"This widely-used method [39] models loss that occurs after the interaction that creates the two-mode squeezing.",
"The final detected mode at the site $B$ is modelled as the output transmitted mode from the imaginary beam splitter $BS3$ (which for simplicity we also denote by $B$ ).",
"We model the overall loss for the mode $B$ by the transmission efficiency $\\eta _{B}$ .",
"Thblackus, we find the new covariances describing the two-mode entanglement between $F$ and $B$ to be $n_{BF} & = & \\eta _{B}\\cosh 2r+\\left(1-\\eta _{B}\\right)\\nonumber \\\\m_{BF} & = & \\cosh 2r\\nonumber \\\\c_{BF} & = & \\sqrt{\\eta _{B}}\\sinh 2r$ blackThe modes $A$ and $C$ are created by the beam splitter $BS2$ coupled to mode $F$ as in the diagram of Figure 2.",
"Here, extra losses created for the output modes $A$ and $C$ are ignored, for the sake of simplicity.",
"We then solve for the final covariances.",
"Denoting $n_{BA}=\\langle X_{B}^{2}\\rangle $ , $m_{BA}=\\langle X_{A}^{2}\\rangle $ and $c_{BA}=\\langle X_{B}X_{A}\\rangle $ where here $\\langle X_{A}\\rangle =\\langle P_{A}\\rangle =..=0$ , we find $n_{BA} & = & \\eta _{B}\\cosh 2r+\\left(1-\\eta _{B}\\right)\\nonumber \\\\m_{BA} & = & \\eta _{0}\\cosh 2r+\\left(1-\\eta _{0}\\right)\\nonumber \\\\c_{BA} & = & \\sqrt{\\eta _{0}}\\sqrt{\\eta _{B}}\\sinh 2r$ Hence $D_{BA} & = & (n_{BA}-2c_{BA}+m_{BA})/2\\nonumber \\\\& = & \\Bigl (\\eta _{B}\\cosh 2r+\\left(1-\\eta _{B}\\right)+\\eta _{0}\\cosh 2r\\nonumber \\\\& & +(1-\\eta _{0})-2\\sqrt{\\eta _{B}}\\sqrt{\\eta _{0}}\\sinh 2r\\Bigr )/2$ Similarly, evaluating the entanglement between modes $B$ and the second output $C$ of the $BS2$ , we replace the transmission $\\eta _{0}$ by $1-\\eta _{0}$ , to obtaingreen $D_{BC} & = & \\Bigl (\\eta _{B}\\cosh 2r+1-\\eta _{B}+(1-\\eta _{0})\\cosh 2r\\nonumber \\\\& & +\\eta _{0}-2\\sqrt{\\eta _{B}}\\sqrt{1-\\eta _{0}}\\sinh 2r\\Bigr )/2$ We note that if $\\eta _{0}=0.5$ , then $D_{BA}=D_{BC} & = & \\Bigl (\\eta _{B}\\cosh 2r+1-\\eta _{B}+0.5+0.5\\cosh 2r\\nonumber \\\\& & -2\\sqrt{\\eta _{B}}\\sqrt{0.5}\\sinh 2r\\Bigr )/2$ bluewhich when $\\eta _{B}=0.5$ becomes $D_{BA}=D_{BC}=0.5\\left(1+e^{-2r}\\right)$ blackWe see that $D_{BA}=D_{BC}$ for $\\eta _{0}=0.5$ , independent of the value of $r$ and $\\eta _{B}$ .",
"For $\\eta _{0}=\\eta _{B}=0.5$ , $D_{BA}=D_{BC}\\approx 0.5$ in the highly entangled (or squeezed) limit, $r\\rightarrow \\infty $ [25].",
"blackTo test the relation of Result (2), we would need to consider where the steering of $B$ is not possible, so that $S_{B|\\lbrace AC\\rbrace }>1$ .",
"The value of the steering parameter $S_{B|F}$ is minimized to $S_{B|F}=\\left(n_{BF}-c_{BF}^{2}/m_{BF}\\right)$ using the optimal factorsred $g_{x}=c_{BF}/m_{BF}$ , $g_{p}=c_{BF}/m_{BF}$ .",
"blueThus, the steering parameter is [22] $S_{B|\\lbrace AC\\rbrace }=\\eta _{B}\\cosh 2r+(1-\\eta _{B})-\\eta _{B}\\sinh ^{2}(2r)/\\cosh (2r)$ whichred blackcannot exceed 1 for any $r$ .green blackWe next analyse the special case of $\\eta _{B}=\\eta _{0}$ .",
"This is the situation where modes $A$ and $B$ are known to have an equal amount of attenuation.",
"This situation is what two observers (one at each mode) may typically assume after transmission of an entangled state so that the modes $A$ and $B$ are spatially separated.",
"The aim is to understand limitations imposed on the entanglement between $B$ and a third mode $C$ , based on the motivation that mode $C$ might have been created, or be accessible, by an eavesdropper (Eve).",
"The value of $D_{BA}$ can be measured by observers at modes $A$ and $B$ , and that value gives the restriction on $D_{BC}$ based on the monogamy relation $D_{BA}+D_{BC}\\ge 1$ .",
"Figure: Monogamy for the tripartite state of Figure 2 with equal observedlosses for modes BB and AA (η B =η 0 \\eta _{B}=\\eta _{0}).",
"The D BA D_{BA},D BC D_{BC} and D BA +D BC D_{BA}+D_{BC} versus η 0 \\eta _{0} where mode BBhas loss η B =η 0 \\eta _{B}=\\eta _{0}.teal Here r=0.5r=0.5 (left)and r=2r=2 (right).purple blackThe plot(b) shows the saturation (D BA +D BC =1D_{BA}+D_{BC}=1)black ofthe monogamy relation () blackthatoccurs for large rr at η 0 =0.5\\eta _{0}=0.5 in this case.",
"See text forexplanation.",
"The monogamy bound of 1 is indicated by the graydotted line.black The steering parameter S B|{AC} S_{B|\\lbrace AC\\rbrace }is plotted for r=0.5r=0.5 (c), and for r=2r=2 (d).",
"Steering of BBoccurs when S B|{AC} <1S_{B|\\lbrace AC\\rbrace }<1.blackLetting $\\eta _{B}=\\eta _{0}$ , we obtain the actual solutions for this case where the state is generated as in Figure 2:black $D_{BA} & = & 1+\\eta _{B}\\left(e^{-2r}-1\\right)\\nonumber \\\\D_{BC} & = & \\left(\\cosh 2r+1-2\\sqrt{\\eta _{B}(1-\\eta _{B})}\\sinh 2r\\right)/2$ Figure 4 plots the values of $D_{BA}$ , $D_{BC}$ and $D_{BA}+D_{BC}$ for various $\\eta _{B}=\\eta _{0}$ .",
"blackFrom the expression $D_{BA}$ , we see that $D_{BA}\\le 1$ for all $r$ , implying that entanglement is preserved between $A$ and $B$ for all attenuation values $\\eta $ .",
"However, the value of entanglement between $B$ and $C$ as measurable by $D_{BC}$ is limited by the monogamy result $D_{BC}+D_{BA}\\ge 1$ , as verified by the Figure 4 which gives the specific values for this particular scenario.",
"The plot Figure 4b show the saturation of the inequality (REF ) at large entanglement ($r\\rightarrow \\infty $ ) to obtain $D_{BA}+D_{BC}=1$ for the tripartite configuration when $\\eta _{B}=\\eta _{0}=0.5$ .",
"This occurs where the modes have symmetric moments, each being subject to an equal attenuation.",
"We note that the monogamy result explains the experimental observation by Bowen et al of a 50% reduction in the value of $D_{BA}$ for the two-mode system where the modes $B$ and $A$ each have a 50% attenuation.black"
],
[
"blackSymmetric tripartite states and asymmetrical attenuation\n$\\eta _{B}\\rightarrow 0$",
"The plots of Figure 5 illustrate the case where there is symmetry between modes $A$ and $C$ so that $\\eta _{0}=0.5$ , but where the attenuation for mode $B$ is varied.",
"This implies a variable transmission efficiency $\\eta _{B}$ .",
"In this case, the steering parameter satisfies $S_{B|\\lbrace AC\\rbrace }<1$ , as shown by Eq.",
"(REF ).",
"Also, $D_{BA}=D_{BC}$ .",
"The value of $D_{BA}=D_{BC}$ reduces below 1 only for a regime where $\\eta _{B}\\sim \\eta _{0}$ .",
"This does not imply that there is no bipartite entanglement however, as will be evident in Section IV where a more sensitive entanglement criterion is used.",
"Figure: Monogamy for the symmetric tripartite state of Figure 2 whereη 0 =0.5\\eta _{0}=0.5 and with extra losses for mode BB.",
"The D BA D_{BA},D BC D_{BC} and D BA +D BC D_{BA}+D_{BC} versus η B \\eta _{B}, the detection efficiencyfor mode BB.",
"Here r=0.5r=0.5 (left) and r=2r=2 (right).",
"Always, forthese parameters, D BA =D BC D_{BA}=D_{BC}.",
"The monogamy bound of 1 isindicated by the gray dotted line.blackTo test the relation of Result (2), we need to consider scenarios where steering of $B$ as detected by $S_{B|\\lbrace AC\\rbrace }$ is not possible, so that $S_{B|\\lbrace AC\\rbrace }>1$ .",
"blackThis can be done by placing thermal noise on the mode $B$ [40], [41], or else by adding additional losses to the modes $A$ and $C$ [42], [22].",
"Without loss or extra thermal noise, $S_{B|\\lbrace AC\\rbrace }$ becomes zero in the limit of large $r$ .black The smallness of the steering parameter gives a measure of the degree of the steering.",
"In this Section, we test the monogamy relation of Result (2) by adding losses to modes $A$ and $C$ .",
"The extra loss on mode $A$ is modelled by a beam splitter $BS4$ wth transmission (or detection) efficiency $\\eta _{A}$ .",
"The beam splitter has two inputs, mode $A$ and a second mode that is in a vacuum state (denoted by boson operator $a_{vac}$ ).",
"The relevant detected moments after loss are then modelled by those of the transmitted mode, with boson operator $a_{det}=\\sqrt{\\eta _{A}}a+\\sqrt{1-\\eta _{A}}a_{vac}=\\sqrt{\\eta _{A}}(\\sqrt{\\eta _{0}}a+\\sqrt{1-\\eta _{0}}a_{0,vac})+\\sqrt{1-\\eta _{A}}a_{vac}$ .",
"The covariances become: $n_{BA} & = & \\cosh 2r\\nonumber \\\\m_{BA} & = & \\eta _{0}\\eta _{A}\\cosh 2r+1-\\eta _{0}\\eta _{A}\\nonumber \\\\c_{BA} & = & \\sqrt{\\eta _{A}\\eta _{0}}\\sinh 2r$ Similarly, if the extra losses for mode $C$ are modelled similarly by a transmission efficiency $\\eta _{C}$ , the covariances become (replacing $\\eta _{0}$ with $1-\\eta _{0}$ ) $n_{BC} & = & \\cosh 2r\\nonumber \\\\m_{BC} & = & (1-\\eta _{0})\\eta _{C}\\cosh 2r+1-(1-\\eta _{0})\\eta _{C}\\nonumber \\\\c_{BC} & = & \\sqrt{\\eta _{C}(1-\\eta _{0})}\\sinh 2r$ blackHence $D_{BA} & = & (n_{BA}-2c_{BA}+m_{BA})/2\\nonumber \\\\& = & \\Bigl (\\eta _{0}\\eta _{A}\\cosh 2r+\\left(1-\\eta _{0}\\eta _{A}\\right)+\\cosh 2r\\nonumber \\\\& & -2\\sqrt{\\eta _{0}\\eta _{A}}\\sinh 2r\\Bigr )/2$ blackand $D_{BC} & = & \\Bigl ((1-\\eta _{0})\\eta _{C}\\cosh 2r+1-\\eta _{C}+\\eta _{0}\\eta _{C}+\\cosh 2r\\nonumber \\\\& & -2\\sqrt{\\eta _{C}(1-\\eta _{0})}\\sinh 2r\\Bigr )/2$ Figure: Monogamy for the tripartite state of Figure 2 with extra lossesfor modes AA and CC.black The D BA D_{BA}, D BC D_{BC}and D BA +D BC D_{BA}+D_{BC} versus η A \\eta _{A} (η B =1\\eta _{B}=1, r=2r=2).",
"Themonogamy bound is shown by the grayblack dotted line.Plots (a) and (b) assume η A =η C \\eta _{A}=\\eta _{C} and η 0 =0.5\\eta _{0}=0.5(a) and η 0 =0.8\\eta _{0}=0.8 (b).",
"Plots (c) and (d) shows values for η C =1\\eta _{C}=1.Plot (e) gives the steering parameter S B|{AC} S_{B|\\lbrace AC\\rbrace } (Eq.",
"())in each case.",
"The S B|{AC} S_{B|\\lbrace AC\\rbrace } is independent of η 0 \\eta _{0}.The steering parameter is changed by the attenuation of modes $A$ and $C$ .",
"The inference of the quadrature phase amplitudes of $B$ by amplitudes $A$ and $C$ cannot be better than the inference made by measurements of amplitudes of $F$ .",
"The quadrature amplitudes of $F$ can be determined from those of $A$ and $C$ in a lossless situation as described above.",
"The total effective intensity of mode $F$ can be summed as the intensity of modes $A$ and $C$ .",
"The total transmitted intensity (in units of photon number) with the loss present is given by $\\eta _{0}\\eta _{A}+(1-\\eta _{0})\\eta _{C}$ .",
"The lowest possible value for the steering $S_{B|\\lbrace AC\\rbrace }$ in the presence of loss for modes $A$ and $C$ is thus given as the steering parameter $S_{B|F}$ where mode $F$ is attenuated by the transmission factor $\\eta _{F}=\\eta _{0}\\eta _{A}+(1-\\eta _{0})\\eta _{C}$ The solution is given in Refs.",
"[22], [42].",
"We find $S_{B|F} & = & 1-\\eta _{B}\\frac{\\left[\\cosh \\left(2r\\right)-1\\right]\\left[2\\eta _{F}-1\\right]}{\\left[1-\\eta _{F}+\\eta _{F}\\cosh \\left(2r\\right)\\right]}\\,$ where $\\eta _{F}$ is the transmission efficiency for mode F and $\\eta _{B}$ is that for mode $B$ .",
"Here we take $\\eta _{B}=1$ so that $S_{B|F} & = & 1-\\frac{\\left[\\cosh \\left(2r\\right)-1\\right]\\left[2\\eta _{F}-1\\right]}{\\left[1-\\eta _{F}+\\eta _{F}\\cosh \\left(2r\\right)\\right]}\\,$ As summarised in Ref.",
"[22], $S_{B|F}<1$ for all $\\eta _{F}>0.5$ , given that $\\eta _{B},r\\ne 0$ .",
"For $\\eta _{F}\\le 0.5$ , it is possible to obtain $S_{B|F}\\ge 1$ .",
"Figure 6 demonstrates the monogamy relation for both regimes, where we assume $\\eta _{A}=\\eta _{C}$ .",
"In Figures 6 (a) and (b), the extra losses for modes $A$ and $C$ are assumed equal: $\\eta _{A}=\\eta _{C}$ .",
"The steering parameter exceeds 1 in that case when $\\eta _{F}=\\eta _{A}<0.5$ (Figure 6 (e)).",
"In Figure 6 (c) and (d) we assume no extra loss on the mode $C$ , modelling a best possible scenario for an eavesdropper who has access to mode $C$ .",
"blackWe see that the eavesdropper does not gain access to the symmetric form of entanglement that is indicated by $D_{BC}<1$ ."
],
[
"Squeezed thermal two-mode state",
"blackIn this Section, we test the monogamy relation by adding thermal noise $n_{B}$ on the mode $B$ .",
"This can be done in several ways, depending on what model is used for the creation of the thermal noise.",
"The simplest procedure is to assume that the modes $B$ and $F$ are initially thermally excited states, and then coupled to the interaction that generates the two-mode squeezing [44] .",
"The entanglement that is formed between the modes $B$ and $F$ is then modified by the inclusion of the two thermal reservoirs, with thermal excitation numbers given by $n_{a}$ and $n_{b}$ respectively.",
"This might also serve as a simple model for mixed states in optical systems (where thermal noise is negligible at room temperature).",
"The covariance elements of the entangled outputs modes $B$ and $F$ of a squeezed thermal state are [44]: green $n_{BF} & = & (n_{F}+n_{B}+1)\\cosh (2r)+(n_{B}-n_{F})\\nonumber \\\\m_{BF} & = & (n_{F}+n_{B}+1)\\cosh (2r)-(n_{B}-n_{F})\\nonumber \\\\c_{BF} & = & (n_{F}+n_{B}+1)\\sinh (2r)$ We will take $n_{th}=n_{F}=n_{B}$ .",
"The steering parameter becomes in that case $S_{B|\\lbrace AC\\rbrace } & = & n_{BF}-c_{BF}^{2}/m_{BF}\\nonumber \\\\& = & \\left(2n_{th}+1\\right)/\\cosh 2r$ and can be shown to exceed 1 for any given $r,$ for sufficient thermal noise.",
"Figure: Monogamy relations with thermal noise present: Plots of D BA D_{BA}given by Eq.",
"(), D BC D_{BC} given by Eq.",
"(),and D BA +D BC D_{BA}+D_{BC} as for Figure 2 with η B =1\\eta _{B}=1 and r=1r=1.Graphs give curves versus η th \\eta _{th} for various η 0 \\eta _{0} wherewe take n th =n F =n B n_{th}=n_{F}=n_{B}.",
"The monogamy bound given by max{1,S B|{AC} }\\max {\\lbrace 1,S_{B|\\lbrace AC\\rbrace }\\rbrace }is given by the gray dotted curve.",
"Here S B|{AC} S_{B|\\lbrace AC\\rbrace } is given byEq.",
"() as shown in (e).",
"The value of η 0 \\eta _{0}is 0.20.2 (a), 0.50.5 (b), 0.80.8 (c).",
"blueredThe modes $A$ and $C$ are created by the beam splitter $BS2$ with transmission $\\eta _{0}$ .",
"We find that $\\langle X_{B}^{2}\\rangle $ is unchanged, and thus $n_{BA}=n_{BF}$ .",
"However, $\\langle X_{B}X_{A}\\rangle =\\langle X_{B}(\\sqrt{\\eta _{0}}X_{F}+\\sqrt{(1-\\eta _{0})}X_{vac})\\rangle $ where $X_{vac}$ is the quadrature operator from the uncorrelated vacuum input.",
"Also, $\\langle X_{A}^{2}\\rangle =\\langle (\\sqrt{\\eta _{0}}X_{F}+\\sqrt{(1-\\eta _{0})}X_{vac})^{2}\\rangle $ and thus $\\langle X_{A}^{2}\\rangle =\\eta _{0}\\langle X_{F}^{2}\\rangle +1-\\eta _{0}$ .",
"Hence $n_{BA} & = & (n_{F}+n_{B}+1)\\cosh (2r)+(n_{B}-n_{F})\\nonumber \\\\m_{BA} & = & \\eta _{0}(n_{F}+n_{B}+1)\\cosh (2r)\\nonumber \\\\& & \\ \\ \\ \\ \\ -\\eta _{0}(n_{B}-n_{F})+1-\\eta _{0}\\nonumber \\\\c_{BA} & = & \\sqrt{\\eta _{0}}(n_{F}+n_{B}+1)\\sinh (2r)$ and the covariances $n_{BC}$ , $m_{BC}$ and $c_{BC}$ are obtained from those for $BA$ by replacing $\\eta _{0}$ with $1-\\eta _{0}$ .",
"The solutions give black $D_{BA} & = & \\frac{1}{2}\\left(n_{BA}+m_{BA}-2c_{BA}\\right)\\nonumber \\\\& = & \\frac{1}{2}\\left[\\left(n_{B}+n_{F}+1\\right)\\cosh 2r+\\left(n_{B}-n_{F}\\right)\\right.\\nonumber \\\\& & +\\eta _{0}\\left(n_{B}+n_{F}+1\\right)\\cosh 2r-\\eta _{0}\\left(n_{B}-n_{F}\\right)\\nonumber \\\\& & \\left.+\\left(1-\\eta _{0}\\right)-2\\sqrt{\\eta _{0}}\\left(n_{F}+n_{B}+1\\right)\\sinh 2r\\right]\\nonumber \\\\$ The covariance elements for mode $B$ and $C$ are $n_{BC}=\\left(n_{B}+n_{F}+1\\right)\\cosh 2r+\\left(n_{B}-n_{F}\\right)$ , $m_{BC}=\\left(1-\\eta _{0}\\right)\\left(n_{F}+n_{B}+1\\right)\\cosh 2r-\\left(1-\\eta _{0}\\right)\\left(n_{B}-n_{F}\\right)+\\eta _{0}$ and $c_{BC}=\\sqrt{1-\\eta _{0}}\\left(n_{F}+n_{B}+1\\right)\\sinh 2r$ .",
"$D_{BC}$ is then black $D_{BC} & = & \\frac{1}{2}\\left(n_{BC}+m_{BC}-2c_{BC}\\right)\\nonumber \\\\& = & \\frac{1}{2}\\left[\\left(n_{B}+n_{F}+1\\right)\\cosh 2r+\\left(n_{B}-n_{F}\\right)\\right.\\nonumber \\\\& & +\\left(1-\\eta _{0}\\right)\\left(n_{F}+n_{B}+1\\right)\\cosh 2r\\nonumber \\\\& & -\\left(1-\\eta _{0}\\right)\\left(n_{B}-n_{F}\\right)+\\eta _{0}\\nonumber \\\\& & -2\\sqrt{1-\\eta _{0}}\\left(n_{F}+n_{B}+1\\right)\\sinh 2r\\nonumber \\\\$ The Figure 7 gives plots of the $D_{BA}$ , $D_{BC}$ and the steering parameter $S_{A|\\lbrace BC\\rbrace }$ , given in Eq.",
"(REF ), with different noise values to validate the monogamy relation of Result (2)."
],
[
"Monogamy relations for a more general entanglement quantifier",
"The monogamy relations for the symmetric criterion $D_{AB}$ are useful, since resources satisfying $D_{AB}<1$ are often required for certain protocols [28].",
"However, with the motivation to obtain more sensitive monogamy relations, we next derive relations for the more general entanglement quantifier that has been shown necessary and sufficient for detecting the two-mode entanglement of Gaussian resources.",
"An entanglement criterion considered by blackGiovannetti et al is [26] $Ent_{AB}(\\mathbf {g_{AB}})<1$ where we define $Ent_{AB}(g_{AB})=\\frac{\\Delta (X_{A}-g_{AB,x}X_{B})\\Delta (P_{A}+g_{AB,p}P_{B})}{(1+g_{AB,x}g_{AB,p})}$ blueThe $g_{AB}=(g_{AB,x},g_{AB,p})$ where $g_{AB,x}$ , $g_{AB,p}$ are real constants that can be optimally chosen to minimize the value of $Ent_{AB}(\\mathbf {g_{AB}})$ .",
"This minimum value is denoted $Ent_{AB}$ and it has been shown previously that $Ent_{AB}=Ent_{BA}$ [27].",
"This is seen by noting that $Ent_{AB}(g_{AB})=Ent_{BA}(1/g_{AB})$ and we will see below that the optimal $g_{AB}$ written as $g_{AB}^{(sym)}$ can be shown to be $1/g_{BA}^{(sym)}$ , the reciprocal of the optimal $g_{BA}$ .",
"We note that the order $AB$ in the suffix of $Ent_{AB}$ does have a real meaning, since the coefficients appear before the $X_{B}$ and $P_{B}$ (but not the $X_{A}$ and $P_{A}$ ).",
"The entanglement criterion (REF ) holds as a valid criterion to detect entanglement, for any choice of constants $g_{AB,x}$ , $g_{AB,p}$ .",
"For the restricted subclass of Gaussian EPR resources where there is symmetry between the $X$ and $P$ moments (we call this class $X-P$ symmetric), a single $g_{AB}=g_{AB,x}=g_{AB,p}$ is optimal.",
"The optimal choice is $g_{AB}=g_{AB}^{(sym)}$ where [27], [28] $g_{AB}^{(sym)}\\equiv \\frac{1}{2c_{AB}}\\left(n_{AB}-m_{AB}+\\sqrt{(n_{AB}-m_{AB})^{2}+4c_{AB}^{2}}\\right)$ blackWe note that here (as in Section III) we define the covariances so that $n_{IJ}=\\langle X_{I},X_{I}\\rangle $ and $m_{IJ}=\\langle X_{J},X_{J}\\rangle $ .",
"Hence $n_{IJ}=m_{JI}$ and $n_{IJ}\\ne n_{JI}$ .blue It has been shown that $g_{AB}^{(sym)}=1/g_{BA}^{(sym)}$ [28], [27].",
"It has also been shown that the condition (REF ) reduces to the Simon-Peres positive partial transpose (PPT) condition for entanglement in this case, provided the choices of $X$ and $P$ 's are optimal [27].",
"For two-mode Gaussian states, the PPT condition is necessary and sufficient for entanglement [29].",
"Where the moments of $A$ and $B$ are identical, the value of the parameter is $g_{AB}=1$ .",
"The optimal value $g_{AB}$ is then an indicator of the “symmetry” of the entanglement with respect to the modes $A$ and $B$ .",
"We refer to $g_{AB}$ as the symmetry parameter.",
"In the fully symmetric case where $g_{AB}^{(sym)}=1$ , the condition $Ent_{AB}<1$ becomes equivalent to $D_{AB}<1$ .",
"The entanglement criterion has been applied to asymmetric systems in Refs.",
"[45].",
"The next result gives the entanglement monogamy relations in terms of the entanglement parameter $Ent_{AB}(g_{AB})$ .",
"Result (3): We select $g_{AB,x}=g_{AB,p}=g_{AB}$ so that the entanglement criterion (REF ) reduces to $Ent_{AB}(g_{AB})=\\frac{\\Delta \\left(X_{B}-g_{AB}X_{A}\\right)\\Delta \\left(P_{B}+g_{AB}P_{A}\\right)}{\\left(1+g_{AB}^{2}\\right)}<1$ bluefor any real $g_{AB}$ .",
"We noted above that $Ent_{AB}(g_{AB})=Ent_{BA}(1/g_{AB})$ .",
"The following monogamy inequality holds $Ent_{BA}(g_{BA})Ent_{BC}(g_{BC})\\geqslant \\frac{\\max {\\lbrace 1,S_{B|\\lbrace AC\\rbrace }^{2}\\rbrace }}{\\left(1+g_{BA}^{2}\\right)\\left(1+g_{BC}^{2}\\right)}$ for any real values $g_{BA}$ , $g_{BC}$ .",
"The following inequality also holds: $Ent_{BA}(g_{BA})+Ent_{BC}(g_{BC}) & \\geqslant & \\frac{S_{B\\vert \\lbrace AC\\rbrace }(2+g_{BA}^{2}+g_{BC}^{2})}{\\left(1+g_{BA}^{2}\\right)\\left(1+g_{BC}^{2}\\right)}\\nonumber \\\\$ The proofs are given in the Appendix.",
"The monogamy relation (REF ) reduces to that of (REF ) when we select $g_{BA}=g_{BC}=1$ and use that $2xy\\le x^{2}+y^{2}$ for any $x,y\\in \\Re $ .red bluered We consider a physical scenario where the fields are $X-P$ symmetric so that the entanglement criterion (REF ) is equivalent to the PPT criterion.",
"The next Result follows from the previous one.",
"We select the values of $g_{BA}$ , $g_{BC}$ to be given by (REF ), in which case we can write the monogamy relation in terms of the PPT entanglement: Result (4): For any three systems $A$ , $B$ and $C$ , it follows that $Ent_{BA}Ent_{BC}\\geqslant \\frac{\\max {\\lbrace 1,S_{B|\\lbrace AC\\rbrace }^{2}\\rbrace }}{\\left(1+(g_{BA}^{(sym)})^{2}\\right)\\left(1+(g_{BC}^{(sym)})^{2}\\right)}$ We rewrite this relation as $Ent_{BA}Ent_{BC}\\ge M_{B}$ where we define the monogamy bound as $M_{B}=\\frac{\\max {\\lbrace 1,S_{B|\\lbrace AC\\rbrace }^{2}\\rbrace }}{\\left(1+(g_{BA}^{(sym)})^{2}\\right)\\left(1+(g_{BC}^{(sym)})^{2}\\right)}$ The Result (4) tells us that the bound for entanglement distribution is determined by the symmetry parameters $g_{BA}^{(sym)}$ and $g_{BC}^{(sym)}$ .",
"These symmetry parameters are fixed for a given field pair.",
"A consequence that is immediately evident is that where the entanglement between modes $A$ and $B$ is maximum (so that $Ent_{BA}\\rightarrow 0$ ), the value of $Ent_{BC}\\rightarrow \\infty $ .",
"This is a stronger result than the sum relation $D_{BA}+D_{BC}\\ge 1$ , which is not so useful where a high degree of entanglement is present ($D_{BA}\\rightarrow 0$ ).",
"The collective steering of the system $B$ (by the composite system $AC$ ) determines the lower bound on the monogamy relation.",
"If there is no steering of this type, then the overall bipartite entanglement as determined by the smallness of the product $Ent_{BA}Ent_{BC}$ is reduced.",
"The sensitivity however depends on the value of the symmetry parameters, since if $g_{BC}^{(sym)}\\gg 1$ , it might be possible for both pairs $BA$ and $BC$ to share a large degree of bipartite entanglement.",
"If $A$ and $B$ are sites for observers that want to use their shared entanglement $Ent_{BA}$ , then the observers $A$ and $B$ may want to ensure that the entanglement $Ent_{BC}$ is reduced (meaning a large value of $Ent_{BC}$ ).",
"Knowledge of the symmetry parameter $g_{BC}^{(sym)}$ , in particular factors that would make $g_{BC}^{(sym)}$ large without decreasing the steering of $B$ , would be useful.",
"Figure: Monogamy of the bipartite PPT entanglement quantifier EntEntfor the tripartite state of Figure 2, assuming no extra loss or noise.The Ent BA Ent_{BA}, Ent BC Ent_{BC}, Ent BA Ent BC Ent_{BA}Ent_{BC} and the monogamybound M B M_{B} (gray dotted line) given by equation ()are plotted versus η 0 \\eta _{0}.",
"In this case, there is steering (S B|{AC} <1S_{B|\\lbrace AC\\rbrace }<1)for all η 0 \\eta _{0}.",
"Here r=0.5r=0.5 (a) and r=2r=2 (b).",
"For thissystem, the symmetry parameters (Fig.",
"(c) and (d)) satisfy g BA (sym) ,g BC (sym >1g_{BA}^{(sym)},g_{BC}^{(sym}>1.redblackSaturation of the monogamy relation() is observed for all η 0 \\eta _{0} for largerrr (Fig.",
"(b)), where the grayblack dotted and greendashed lines coincide.In Figure 8 we illustrate the monogamy relation with respect to the idealised tripartite system depicted in Figure 2 and Figure 3.",
"For this system there are no additional losses or noise and the covariances are given by Eq.",
"(REF ).",
"The expression for $Ent_{BA}$ is $Ent_{BA}=\\frac{n_{BA}-2g_{BA}^{(sym)}c_{BA}+\\left(g_{BA}^{(sym)}\\right)^{2}m_{BA}}{1+\\left(g_{BA}^{(sym)}\\right)^{2}}$ where $n_{BA},m_{BA}$ and $c_{BA}$ are given by (REF ).",
"The $Ent_{BC}$ is given similarly, blackreplacing $\\eta _{0}$ with $1-\\eta _{0}$ .",
"It can be verified that the symmetry parameters satisfy $g_{BA}^{(sym)},\\ g_{BC}^{(sym}>1$ implying that the monogamy bound $M_{B}$ reduces below 1.",
"In fact for this case, we find black $g_{BA}^{(sym)} & = & \\frac{1}{2\\sqrt{\\eta _{0}}\\sinh 2r}\\biggl (\\cosh 2r(1-\\eta _{0})-\\left(1-\\eta _{0}\\right)\\nonumber \\\\& & +\\sqrt{\\left(\\cosh 2r(1-\\eta _{0})-1+\\eta _{0}\\right)^{2}+4\\eta _{0}\\sinh ^{2}2r}\\Biggr )\\nonumber \\\\$ and $g_{BC}^{(sym)}$ is obtained by replacing $\\eta _{0}$ with $1-\\eta _{0}$ .",
"These parameters are plotted in Figure 8. blackWe see that $g_{BA}^{(sym)}=g_{BC}^{(sym)}$ for $\\eta _{0}=0.5$ .blue redblackThe results for the monogamy of entanglement as measured by the quantifier $Ent$ indeed show a greater sensitivity than those for $D$ .",
"Where the squeeze parameter $r$ is higher, there is a greater bipartite entanglement created between modes $B$ and $F$ and the collective steering is greater.",
"The higher value of $r$ also indicates a greater degree of genuine tripartite entanglement between the three modes, as measured by inequalities derived in Refs.",
"[34], [30].",
"We see from the Figure 8(b) that the monogamy relation is saturated for blackallblack values of $\\eta _{0}$ in the high $r$ regime.",
"This contrasts with the result of Figure 4 for $D$ where the saturation is only at $\\eta _{0}=0.5$ ."
],
[
"Extra loss in the shared mode $B$",
"This case is discussed in the Section III.",
"A, where it was shown that steering exists (such that $S_{B|\\lbrace AC\\rbrace }<1$ ) over all values of the attenuated efficiency $\\eta _{B}$ for mode $B$ .",
"In Figure 9 we plot the entanglement monogamy relations and the symmetry parameters for the case of Figure 4 where the value of loss on mode $B$ matches that of $\\eta _{0}$ ($\\eta _{B}=\\eta _{0}$ ).",
"This implies symmetry of entanglement between $A$ and $B$ so that $g_{BA}^{(sym)}=1$ .",
"As $\\eta _{B}$ is varied from 1 (no loss) to zero (high loss), the symmetry parameter $g_{BC}^{(sym)}$ varies from above to below 1, being equal to 1 when $\\eta _{b}=\\eta _{0}=0.5$ (Figures 9 (c) and (d)).",
"At that point, the monogamy relation for $Ent_{BA}$ and $Ent_{BC}$ is then equivalent to that for $D_{BA}$ and $D_{BC}$ , and there is saturation of the monogamy inequality.",
"As $\\eta _{B}\\rightarrow 1$ , $g_{BC}^{(sym)}$ becomes large and the monogamy bound $M_{B}$ becomes small.",
"Indeed the entanglement product is small in this regime.",
"We note however that with $A$ and $B$ sharing excellent symmetric entanglement, the amount of entanglement shared between $B$ and $C$ is reduced and, as seen from the monogamy relation for the symmetric entanglement $D_{BC}$ (Figure 4), is necessarily highly asymmetric.",
"In Figure 10 we plot the monogamy relation and the symmetry parameters for the situation of Figure 5 where the loss in mode $B$ is varied across all values for fixed $\\eta _{0}$ .",
"The value of $g_{BA}^{(sym)}$ becomes small when there is considerable loss at the mode $B$ , so that $\\eta _{B}\\ll \\eta _{0}$ .",
"Similarly, the value of $g_{BC}^{(sym)}$ is small if $\\eta _{B}\\ll 1-\\eta _{0}$ .",
"This implies an increased lower bound $M_{B}$ for the monogamy relation.",
"We note from Figure 10 (a) a second regime of saturation of the monogamy relation, where $\\eta _{0}=0.5$ and $\\eta _{B}$ varies from 0 to 1.",
"This regime corresponds to collective steering where $S_{B|\\lbrace AC\\rbrace }<1$ , but we note that (unlike the saturation case of Fig.",
"8 (b)), the steering parameter is not optimal ($S_{B|\\lbrace AC\\rbrace }>0$ )."
],
[
"Extra loss for modes $A$ and {{formula:398ca658-aec0-4eff-a851-51ca5a383d13}} ",
"blackTo test the relation of Result (2), we need to consider where the steering of $B$ as detected by $S_{B|\\lbrace AC\\rbrace }<1$ is not possible, so that $S_{B|\\lbrace AC\\rbrace }>1$ .",
"blackWe test the relation by adding losses to the “steering modes”, $A$ and $C$ .",
"The covariances are given in the Section III.B.",
"The steering parameter is given by Eq.",
"(REF ) and can exceed 1 when $\\eta _{F}<0.5$ .",
"In Figure 11, we plot the values for the entanglement quantifiers and demonstrate the entanglement monogamy relation.",
"Both the relevant symmetry parameters become large as the extra loss increases ($\\eta _{A}$ , $\\eta _{C}$ becoming small).",
"The steering reduces ($S_{B|\\lbrace AC\\rbrace }>1$ ) and overall the monogamy bound $M_{B}$ also becomes small, despite the lack of collective steering (Fig.",
"6 (e)).",
"We note there is entanglement maintained between both parties ($Ent_{BA}<1$ , $Ent_{BC}<1$ ) over the full parameter range.",
"The entanglement shows high asymmetry however, as indicated by the symmetry parameters plotted in Fig.",
"11(e) and by the contrasting results for the symmetric entanglement ($D_{BA}$ , $D_{BC}$ ) given in Figure 6."
],
[
"Squeezed thermal two-mode state",
"blackAs for the Section III.C, we can also test the full monogamy relation by including thermal noise.",
"This is achieved by considering the squeezed thermal two-mode state, as described in Section III.",
"C. In Figure 12 we plot for various thermal noise values the monogamy product against the lower bound $M_{B}$ , allowing for when there is no steering so that $S_{B|\\lbrace AC\\rbrace }^{2}\\ge 1$ .",
"We note the values of the symmetry parameters are above 1, but the steering parameter can also exceed 1.",
"The overall monogamy bound is plotted in Figure 11 and shows significant increase as the thermal noise increases and the steering indicated by $S_{B|\\lbrace AC\\rbrace }<1$ is lost.",
"The entanglement product therefore must similarly increase, making entanglement between both parties impossible.",
"This contrasts with the results of Figure 11 where, although the steering is lost, the monogamy bound is low being better balanced by the symmetry parameters.",
"In that case, the entanglement product goes below 1 and we obtain bipartite entanglement between both pairs."
],
[
"Conclusion",
"We have derived monogamy relations for the bipartite entanglement distribution of three systems $A$ , $B$ and $C$ modelled as modes.",
"The relations hold for three modes regardless of the tripartite state involved, and may therefore have application to quantum information protocols where two observers $A$ and $B$ have knowledge of the entanglement between them and desire to place a lower bound on the entanglement between one of their parties $B$ and a third observer, Eve, at site $C$ .",
"In Section IV, we present Result (4) where we use as a bipartite entanglement quantifier the Einstein-Podolsky-Rosen variance product involving continuous variable (CV) measurements, that we call $Ent_{AB}$ .",
"This quantifier has been shown to give an entanglement condition $Ent_{AB}<1$ equivalent to the Peres-Simon necessary and sufficient condition for highly useful continuous variable Gaussian state resources.",
"Ideal entanglement is achieved when $Ent_{AB}\\rightarrow 0$ .",
"We show that the lower bound for the entanglement product $Ent_{BA}Ent_{BC}$ depends on the quantum noise level, and also the size of a conventional steering parameter $S_{B|\\lbrace AC\\rbrace }$ (that for $S_{B|\\lbrace AC\\rbrace }<1$ certifies a steering of system $B$ from the combined system $A$ and $C$ ).",
"When there is steering $S_{B|\\lbrace AC\\rbrace }<1$ , the monogamy lower bound is determined by the vacuum quantum noise level.",
"Otherwise, the bound is higher and is constrained by the steering parameter.",
"The lower bound also depends on symmetry parameters $g_{BA}^{(sym)}$ and $g_{BC}^{(sym)}$ which quantify the amount of symmetry between the moments of $B$ and $A$ , and $B$ and $C$ , respectively.",
"These parameters take the value $g^{(sym)}=1$ when the moments are perfectly symmetrical.",
"In the Section IV, we illustrate the application of this monogamy relation for the tripartite system created using a two-mode squeezed state and a beam splitter.",
"We show that when the two-mode squeezing is high, the monogamy relation is always saturated (reaching the lowest possible monogamy bound).",
"We also study the case of a thermal two-mode squeezed state and where there is extra dissipation.",
"In Section IV, we also obtain more general monogamy relations (Result (3)) that constrain the shared entanglement as measured by other entanglement certifiers.",
"An example is a relation for the well-known TDGCZ bipartite certifier $D_{BA}$ that gives a necessary and sufficient entanglement condition for symmetric two-mode gaussian states (where $g_{BA}^{(sym)}=1$ ), which we study in Section II.",
"Although not as sensitive as the more general relation, this can be useful in establishing rigorous bounds on the value of $D_{BA}$ when knowledge of the symmetry parameters is absent, or where only the symmetric form of entanglement is required.",
"We are able to replicate the saturation result of Section IV where the modes have complete symmetry with respect to bipartite moments, hence giving insight into the experiment of Bowen et al.",
"For convenience, in the proofs below we may abbreviate the notation for the variance to denote $(\\Delta X)^{2}=\\Delta ^{2}X$ .",
"First we note that $\\Delta (X_{B}-X_{A})\\ge \\Delta (X_{B}|X_{A})$ where $(\\Delta (X_{B}|X_{A}))^{2}$ denotes the average variance defined as $\\Delta ^{2}(X_{B}|X_{A})=\\sum _{x_{A}}P(x_{A})\\sum _{x_{B}}P(x_{B}|x_{A})(x_{B}-\\mu _{B|x_{A}})^{2}$ where $\\lbrace x_{A}\\rbrace $ is the set of all possible outcomes for $X_{A}$ and denoting $\\mu _{B|x_{A}}$ as the mean of $P(X_{B}|X_{A}=x_{A})$ .",
"The result (REF ) is proved as follows: We write $\\Delta ^{2}(X_{B}-X_{A}) & = & \\sum _{x_{B},x_{A}}P(x_{B},x_{A})\\left[x_{B}-x_{A}-\\langle x_{B}-x_{A}\\rangle \\right]^{2}\\nonumber \\\\& = & \\sum _{x_{A}}P(x_{A})\\sum _{x_{B}}P(x_{B}|x_{A})(x_{B}-g_{x_{A}})^{2}\\nonumber \\\\& \\ge & \\sum _{x_{A}}P(x_{A})\\sum _{x_{B}}P(x_{B}|x_{A})(x_{B}-\\mu _{B|x_{A}})^{2}\\nonumber \\\\$ where we introduce $g_{x_{A}}=x_{A}-\\langle x_{A}\\rangle +\\langle x_{B}\\rangle $ .",
"We have used that for any distribution, $\\sum _{x}P(x)(x-g)^{2}$ where $g$ is a constant is minimised by the choice $g=\\langle x\\rangle =\\sum _{x}P(x)x$ .green We can introduce the notation $\\Delta _{inf}^{2}X_{B\\vert A}=\\Delta ^{2}(X_{B}|X_{A})$ .",
"The notation $\\Delta _{inf}^{2}X_{B\\vert A}$ is also often taken to mean the minimum conditional variance average where the measurement $X_{A}$ , that we might call $X_{\\theta }$ , has been chosen optimally to give the best average inference of $X_{B}$ .",
"Regardless of the definitions, $\\Delta ^{2}\\left(X_{B}\\vert X_{A}\\right)\\geqslant \\Delta _{inf}^{2}X_{B\\vert A},$ and it follows that $\\Delta ^{2}(X_{B}-X_{A})\\geqslant \\Delta _{inf}^{2}X_{B\\vert A}$ .",
"The main proof will now be made by contradiction.",
"Let us consider that $D_{BA}+D_{BC}<1$ .",
"Then it follows that: $\\Delta _{inf}^{2}X_{B|A}+\\Delta _{inf}^{2}P_{B|A}+\\Delta _{inf}^{2}X_{B|C}+\\Delta _{inf}^{2}P_{B|C} & < & 4\\nonumber \\\\$ We use the identity $2xy\\leqslant x^{2}+y^{2},$ to get that $2\\Delta _{inf}X_{B\\vert A}\\Delta _{inf}P_{B\\vert A}\\leqslant \\Delta _{inf}^{2}X_{B\\vert A}+\\Delta _{inf}^{2}P_{B\\vert A},$ and $2\\Delta _{inf}X_{B\\vert C}\\Delta _{inf}P_{B\\vert C}\\leqslant \\Delta _{inf}^{2}X_{B\\vert C}+\\Delta _{inf}^{2}P_{B\\vert C}.$ This implies $\\Delta _{inf}X_{B\\vert A}\\Delta _{inf}P_{B\\vert A}+\\Delta _{inf}X_{B\\vert C}\\Delta _{inf}P_{B\\vert C}<2$ Next we notice that we can write the above inequality in terms of the steering parameter $S_{B\\vert A}$ which is defined as[21]: $S_{B\\vert A}=\\Delta _{inf}X_{B\\vert A}\\Delta _{inf}P_{B\\vert A}$ It follows that $S_{B\\vert A}+S_{B\\vert C}<2,$ and this implies that $S_{B|A}S_{B|C}<1$ (using the identity $2xy\\leqslant x^{2}+y^{2}$ again).",
"Thus, $D_{BA}+D_{BC}<1$ implies $S_{B|A}S_{B|C}<1$ , which gives a contradiction, since it has been proved in [13] that $S_{B|A}S_{B|C}\\ge 1$ , which is a monogamy inequality for steering.",
"The steering monogamy result was proved valid in Ref.",
"[13] for all three mode quantum states.",
"Details of the proof were also given in the Supplementary Materials of Ref.",
"[36].",
"Thus, by contradiction, we have proved $D_{BA}+D_{BC}\\ge 1$ as required.",
"Since we have already proved $D_{BA}+D_{BC}\\geqslant 1,$ we only require to prove $D_{BA}+D_{BC}\\geqslant S_{B\\vert \\lbrace AC\\rbrace }$ .",
"We prove by contradiction.",
"Let us assume that $D_{BA}+D_{BC}<S_{B|\\lbrace AC\\rbrace }$ .",
"In analogy to the proof for (REF ) we obtain: $& & \\Delta _{inf}^{2}X_{B|A}+\\Delta _{inf}^{2}P_{B|A}+\\Delta _{inf}^{2}X_{B|C}+\\Delta _{inf}^{2}P_{B|C}\\nonumber \\\\& & <4S_{B|\\lbrace AC\\rbrace }$ Next using the identity $2xy\\leqslant x^{2}+y^{2},$ we get that $\\Delta _{inf}X_{B|A}\\Delta _{inf}P_{B|A}+\\Delta _{inf}X_{B|C}\\Delta _{inf}P_{B|C}<2S_{B|\\lbrace AC\\rbrace }$ .",
"Using the definition of the steering parameter defined in Eq.",
"(REF ) we obtain that $S_{B\\vert A}+S_{B\\vert C}<2S_{B\\vert \\left\\lbrace AC\\right\\rbrace }$ , which gives a contradiction, since it has been proved in [13] that $S_{B\\vert A}+S_{B\\vert C}\\geqslant 2S_{B\\vert \\left\\lbrace AC\\right\\rbrace }$ blackbased on the fact the accuracy to give an inference of $X_{B}$ cannot be decreased if the extra system $C$ is included with $A$ , so that $S_{B|A}\\ge S_{B|\\lbrace AC\\rbrace }$ .",
"Straightforward extension of the proof given in lines (REF -REF ) blackleads to the following result: By definition, $\\Delta ^{2}(X_{B}-g_{x}X_{A})=\\sum _{x_{B},x_{A}}P(x_{B},x_{A})\\left[x_{B}-g_{x}x_{A}-\\langle x_{B}-g_{x}x_{A}\\rangle \\right]^{2}$ .",
"Hence we can rewrite $\\Delta ^{2}(X_{B}-g_{x}X_{A}) & = & \\sum _{x_{A}}P(x_{A})\\sum _{x_{B}}P(x_{B}|x_{A})(x_{B}-f\\left(x_{A}\\right))^{2}\\\\& \\ge & \\sum _{x_{A}}P(x_{A})\\sum _{x_{B}}P(x_{B}|x_{A})(x_{B}-\\mu _{B|x_{A}})^{2}\\\\& = & \\Delta _{inf}^{2}X_{B|A}$ where $f\\left(x_{A}\\right)=g_{x}x_{A}+\\langle x_{B}\\rangle -g_{x}\\langle x_{A}\\rangle $ .",
"Here, $f\\left(x_{A}\\right)$ minimises the expression when $f\\left(x_{A}\\right)=\\sum _{x_{B}}P(x_{B}|x_{A})x_{B}=\\mu _{B|x_{A}}$ .",
"This is true for any real constant $g_{x}$ .",
"Thus $\\Delta ^{2}(X_{B}-g_{x}X_{A})\\geqslant \\Delta _{inf}^{2}X_{B\\vert A}$ and similarly $\\Delta ^{2}(P_{B}-g_{p}P_{A})\\geqslant \\Delta _{inf}^{2}P_{B\\vert A}$ where $g_{x}$ , $g_{p}$ are any real constants.",
"Using the definition given in Eq.",
"(REF ) and with similar identities for $Ent_{BC}$ , we obtain: $& & Ent_{BA}Ent_{BC}\\nonumber \\\\& & \\geqslant \\frac{\\Delta _{inf}X_{B\\vert A}\\Delta _{inf}P_{B\\vert A}}{1+g_{BA}^{2}}\\frac{\\Delta _{inf}X_{B\\vert C}\\Delta _{inf}P_{B\\vert C}}{1+g_{BC}^{2}}\\nonumber \\\\& & =\\frac{S_{B\\vert A}S_{B\\vert C}}{\\left(1+g_{BA}^{2}\\right)\\left(1+g_{BC}^{2}\\right)}\\nonumber \\\\& & \\ge \\frac{1}{\\left(1+g_{BA}^{2}\\right)\\left(1+g_{BC}^{2}\\right)}$ as required.",
"Here we have used the steering monogamy inequality $S_{B\\vert A}S_{B\\vert C}\\ge 1$ of Ref.[13].",
"We next prove the second inequality.",
"Since $S_{B\\vert \\left\\lbrace AC\\right\\rbrace }\\leqslant S_{B\\vert A}$ and $S_{B\\vert \\left\\lbrace AC\\right\\rbrace }\\leqslant S_{B\\vert C}$ , we can write the following identities: $Ent_{BA}\\geqslant \\frac{\\Delta _{inf}X_{B\\vert A}\\Delta _{inf}P_{B\\vert A}}{1+g_{BA}^{2}}=\\frac{S_{B\\vert A}}{1+g_{BA}^{2}}\\geqslant \\frac{S_{B\\vert \\left\\lbrace AC\\right\\rbrace }}{1+g_{BA}^{2}}$ with a similar relation for $Ent_{B\\vert C}$ : $Ent_{BC}\\geqslant \\frac{S_{B\\vert \\left\\lbrace AC\\right\\rbrace }}{1+g_{BC}^{2}}$ From the inequalities given in Eq.",
"(REF ) and Eq.",
"(REF ), we can derive the following monogamy relations: $Ent_{BA}+Ent_{BC}\\geqslant S_{B\\vert \\lbrace AC\\rbrace }\\left(\\frac{1}{1+g_{BA}^{2}}+\\frac{1}{1+g_{BC}^{2}}\\right)$ and $Ent_{BA}Ent_{BC}\\geqslant \\frac{S_{B\\vert \\left\\lbrace AC\\right\\rbrace }^{2}}{\\left(1+g_{BA}^{2}\\right)\\left(1+g_{BC}^{2}\\right)}$ Here, we outline the derivation of the steering monogamy result $S_{A|C}S_{A|B}\\ge 1$ where $S_{A|B}=\\Delta _{inf}X_{A|B}\\Delta _{inf}P_{A|B}$ that has been used in the above proofs.",
"The monogamy result is proven in Ref.",
"[13], but for the sake of completeness is given here in the more detailed form previously presented in the Supplementary Materials of Ref.",
"[36].",
"The average conditional “inference” variances are defined in Section II as: $[\\Delta _{inf}X_{A|B}]^{2}=\\sum _{x_{B}}P(x{}_{B})[\\Delta (X_{A}|x{}_{B})]^{2}$ and $[\\Delta _{inf}P_{A|B}]^{2}=\\sum _{p_{B}}P(p{}_{B})[\\Delta (P_{A}|p{}_{B})]^{2}$ where $x_{B}$ ($p_{B}$ ) are the possible results of a measurement performed on system $B$ and on comparing with the Eq.",
"(REF ) we see that $[\\Delta (X_{A}|x_{B})]^{2}=\\sum _{x_{A}}P(x_{A}|x_{B})(x_{A}-\\mu _{A|x_{B}})^{2}$ and similarly for $\\Delta (P_{A}|p_{B})$ .",
"The best choice of measurement for $x_{B}$ is that which optimizes the inference of $X_{A}$ though this is not essential to the validity of the monogamy result.",
"Similarly, $p_{B}$ are the possible results for a second measurement performed at $B$ , usually chosen to optimize the inference of $P_{A}$ .",
"To derive the relation, we note that the observer (Bob) at $B$ can make a local measurement $O_{B}$ to infer a result for an outcome of $X_{A}$ at $A$ .",
"The set of values denoted by $x_{B}$ are the results for the measurement $O_{B}$ , and $P(x_{B})$ is the probability for the outcome $x_{B}$ .",
"The conditional distribution $P(X_{A}|x_{B})$ has a variance which we denote by $[\\Delta (X_{A}|x{}_{B})]^{2}$ .",
"The $(\\Delta _{inf}X_{A|B})^{2}$ is thus the average conditional variance.",
"Similarly, the observer can make another measurement, denoted $Q_{B}$ , to infer a result for the outcome of $P_{A}$ at $A$ .",
"Denoting the results of this measurement by the set $p_{B}$ , we define the conditional variances as for $X_{A}$ .",
"A third observer $C$ (“Charlie”) can also make such inference measurements, with uncertainty $\\Delta _{inf}X_{B|C}$ and $\\Delta _{inf}P_{B|C}$ .",
"Let us denote the outcomes of Charlie's measurements, for inferring Alice's $X_{A}$ or $P_{A}$ , by $x_{C}$ and $p_{C}$ respectively.",
"Since Bob and Charlie can make the measurements simultaneously, a conditional quantum density operator $\\rho _{A|\\lbrace x_{B},p_{C}\\rbrace }$ for system $A$ , given the outcomes $x_{B}$ and $p_{C}$ for Bob and Charlie's measurements, can be defined.",
"The $P(x_{B},p_{C})$ is the joint probability for these outcomes.",
"The moments predicted by this conditional quantum state must satisfy the Heisenberg uncertainty relation.",
"That is, we can define the variance of $X_{A}$ conditional on the joint measurements as $\\Delta (X_{A}|x{}_{B},p_{C})$ and $\\Delta (P_{A}|x{}_{B},p_{C})$ and these must satisfy $\\Delta (X_{A}|x{}_{B},p_{C})\\Delta (P_{A}|x_{B},p_{C})\\ge 1$ .",
"We also note that $ $$[\\Delta _{inf}X_{A|B}]^{2}\\ge \\sum _{x_{B},p_{C}}P(x_{B},p_{C})[\\Delta (X_{A}|x{}_{B},p_{C})]^{2}$ and $[\\Delta _{inf}P_{A|C}]^{2}\\ge \\sum _{x_{B},p_{C}}P(x_{B},p_{C})[\\Delta (P_{A}|x{}_{B},p_{C})]^{2}$ (proved in the Result L below).",
"We see that $[\\Delta _{inf}X_{A|B}\\Delta _{inf}P_{A|C}]^{2} & = & \\sum _{x_{B},p_{c}}P(x_{B},p_{C})[\\Delta (X_{A}|x{}_{B},p_{C})]^{2}\\nonumber \\\\& & \\times \\sum _{x_{B},p_{c}}P(x_{B},p_{C})[\\Delta (P_{A}|x{}_{B},p_{C})]^{2}\\nonumber \\\\$ Then using the Cauchy-Schwarz inequality and taking the square root, we obtain $\\Delta _{inf}X_{A|B}\\Delta _{inf}P_{A|C} & = & \\sum _{x_{B},p_{C}}P(x_{B},p_{C})\\nonumber \\\\& & \\times \\Bigl (\\Delta (X_{A}|x{}_{B},p_{C})\\Delta (P_{A}|x_{B},p_{C})\\Bigr )\\nonumber \\\\& \\ge & 1$ Similarly, Bob can measure to infer $P_{A}$ and Charlie can measure to infer $X_{A}$ , and it must also be true that $\\Delta _{inf}P_{A|B}\\Delta _{inf}X_{A|C}\\ge 1.$ Hence, it must be true that $S_{A|B}S_{A|C}\\ge 1$ .",
"Result L: Step by step we show the following: $[\\Delta (X_{A}|x_{B})]^{2} & = & \\sum _{X_{A}}P(X_{A}|x_{B})(X_{A}-\\mu _{x_{B}})^{2}\\\\& = & \\frac{1}{P(x_{B})}\\sum _{X_{A}}P(X_{A},x_{B})(X_{A}-\\mu _{x_{B}})^{2}\\\\& = & \\frac{1}{P(x_{B})}\\sum _{X_{A}}\\sum _{p_{C}}P(X_{A},x_{B},p_{C})(X_{A}-\\mu _{x_{B}})^{2}\\\\& \\ge & \\frac{1}{P(x_{B})}\\sum _{p_{C}}p(x_{B},p_{C})\\\\& & \\times \\sum _{X_{A}}P(X_{A}|x_{B},p_{C})(X_{A}-\\mu _{x_{B},p_{C}})^{2}\\\\& = & \\frac{1}{P(x_{B})}\\sum _{p_{C}}p(x_{B},p_{C})[\\Delta (X_{A}|x_{B},p_{C})]^{2}$ Here $\\mu _{x_{B}}$ is the mean of $P(X_{A}|x_{B})$ and $\\mu _{x_{B},p_{C}}$ is the mean of $P(X_{A}|x_{B},p_{C})$ .",
"We note that the value of the constant $\\mu $ that minimizes $\\langle (x-\\mu )^{2}\\rangle $ will be the mean of the associated probability distribution.",
"$\\square $ In many papers, and in the calculations given in Sections III of this paper, the values for the inference variances as defined in (REF -REF ) are determined by linear optimization.",
"It is explained in Ref.",
"[22] that the value determined this way cannot be less than the (smallest) value given by the definition (REF -REF ).",
"Furthermore, for Gaussian states, the values according to the two definitions become equal."
]
] | 1612.05727 |
[
[
"Approximation of the Partition Number After Hardy and Ramanujan: An\n Application of Data Fitting Method in Combinatorics"
],
[
"Abstract Sometimes we need the approximate value of the partition number in a simple and efficient way.",
"There are already several formulae to calculate the partition number p(n).",
"But they are either inconvenient for most people (not majored in math) who do not want do write programs, or unsatisfying in accuracy.",
"By bringing in two parameters in the Hardy-Ramanujan's Asymptotic formula and fitting the data of the two parameters by least square method, iteration method and some other special designed methods, several revised elementary estimation formulae with high accuracy for p(n) are obtained.",
"With these estimation formulae, the approximate value of p(n) can be calculated by a pocket calculator without programming function.",
"The main difficulty is that the usual methods to fit the data of the two parameters by an elementary function is defective here.",
"These method could be used in finding the fitting functions of some other complex data."
],
[
"Introduction",
"The partition number $p(n)$ is an interesting topic which attracts many attention.",
"There are already a lot of literatures on many aspects of $p(n)$ .",
"Many mathematicians, such as Euler, Hardy, Ramanujan, Rademacher, Newman, Erdős, Andrews, Berndt and Ono, have made important contribution to this topic.",
"Some important literatures may be found in [1], or in the references of [21], [5], [4] and [17].",
"In recent years, a very important result dues to Ken Ono and his team who connected the partition function with the modular form and found the principles of the congruence property of $p(n)$ that may even be considered as the revealing of the nature of numbers (refer [2], [9], [6] and [7]).",
"For a positive integer $n$ , an integer solution of the equation $s_{1}+s_{2}+\\cdots +s_{q}=n\\ \\ (1\\leqslant s_{1}\\leqslant s_{2}\\leqslant \\cdots \\leqslant s_{q},\\,q\\geqslant 1),$ for all the possible integer $q$ (where $s_{1}$ , $s_{2}$ , $\\cdots $ , $s_{q}$ are unknowns) is called a partition of $n$ .",
"The number of all the partitions of $n$ is denoted by $p(n)$ , which is also called the partition number or the partition function.",
"In a lot of occasions, we need the value of $p(n)$ .",
"There are already several formulae to calculate $p(n)$ .",
"In reference [10] (p.53, p.57) or [15], we may find the generation function of $p(n)$ obtained by Euler: $F(x) & =\\sum \\limits _{n=0}^{\\infty }p(n)x^{n}=\\dfrac{1}{1-x}\\dfrac{1}{1-x^{2}}\\dfrac{1}{1-x^{3}}\\cdots \\nonumber \\\\& \\dfrac{1}{1-x^{i}}\\cdots \\cdots =\\prod \\limits _{i=1}^{\\infty }\\left(1-x^{i}\\right)^{-1},$ and a formula $p(n)=\\dfrac{1}{2\\pi i}{\\oint }_{C}\\dfrac{F(x)}{x^{n+1}}\\mbox{d}x,$ where $C$ is a contour around the original point.",
"Of course, we seldom use (REF ) to compute the value of $p(n)$ in practical.",
"There is a recursion for $p(n)$ ([10], p.55), $p(n) & =p(n-1)+p(n-2)-p(n-5)\\nonumber \\\\& \\quad \\quad -p(n-7)+\\cdots \\nonumber \\\\& \\quad \\quad +(-1)^{k-1}p\\left(n-\\dfrac{3k^{2}\\pm k}{2}\\right)+\\cdots \\cdots \\nonumber \\\\& =\\sum \\limits _{k=1}^{k_{1}}(-1)^{k-1}p\\left(n-\\dfrac{3k^{2}+k}{2}\\right)\\nonumber \\\\& \\quad \\quad +\\sum \\limits _{k=1}^{k_{2}}(-1)^{k-1}p\\left(n-\\dfrac{3k^{2}-k}{2}\\right),$ where $k_{1}=\\left\\lfloor \\dfrac{\\sqrt{24n+1}-1}{6}\\right\\rfloor ,\\ k_{2}=\\left\\lfloor \\dfrac{\\sqrt{24n+1}+1}{6}\\right\\rfloor ,$ and assume that $p(0)=1$ .",
"Here $\\left\\lfloor x\\right\\rfloor $ stands for the maximum integer that will not exceed the real number $x$ .",
"Equation (REF ) is much better for computing $p(n)$ .",
"We can obtain the exact value of $p(n)$ efficiently with a program based on it.",
"But it is not convenient for many people who do not want to write programs.",
"Further more, if we want to calculate $p(n)$ by (REF ) by a small program written in C or some other general computer language, it is usually necessary to decide the size of the space in memory to store the results beforehand, which means we should know the approximate value of $p(n)$ before the calculation started, (actually, here it is sufficient to know $\\left\\lceil \\frac{\\log _{2}p(n)+1}{8}\\right\\rceil $ , where $\\left\\lceil x\\right\\rceil $ stands for the minimum integer that is greater than or equal to the real number $x$ ) otherwise we have to do some extra work for overflow handling and consequently change the size of the space in memory to store the value of the variable that stands for $p(n)$ .",
"Obviously, the datatypes already defined in the C language itself are not suitable.",
"If we use the Dynamic Memory Allocation method, this problem is solved at the price of the program being a little more complicated.",
"Actually, in a lot of cases, we can not decide the approximate size of the result, it is the best choice available.",
"If we can use maple, maximal, axiom or some other computer algebra systems, there is no need to consider this problem.",
"But it is not always an option, especially when the function to do this job is part of a big program written in a compile language while mixing programming of an interpretative language and a compile language is nearly unavailable in most cases (with very few exceptions, such as mixing programming C and matlab).",
"$\\ $ The analysis of $p(n)$ by contour integral with (REF ) (refer [10], p. 57) resulted a very good estimation of $p(n)$ , $p(n)=\\sum \\limits _{q=1}^{\\left\\lfloor \\alpha \\sqrt{n}\\right\\rfloor }A_{q}(n)\\cdot \\phi _{q}(n)+O(n^{-{1}{2}}),$ called the Hardy-Ramanujan formula (refer [11] and [16]), that 6 terms of this formula contain an error of 0.004 when $n$ = 100, while 8 terms of this formula contain an error of 0.004 when $n$ = 200.",
"Here $\\alpha $ is an arbitrary constant, $\\phi _{q}(n)=\\dfrac{\\sqrt{q}}{2\\pi \\sqrt{2}}\\cdot \\dfrac{\\mathrm {d}}{\\mathrm {d}n}\\left(\\dfrac{\\exp \\left(\\tfrac{\\pi }{q}\\sqrt{\\tfrac{2}{3}\\left(n-\\tfrac{1}{24}\\right)}\\right)}{\\sqrt{n-\\tfrac{1}{24}}}\\right),$ $A_{q}(n)=\\underset{\\tiny \\begin{array}{c}0<p<q\\\\(p,q)=1\\end{array}}{{\\Sigma }}\\omega _{p,q}\\cdot \\exp \\left(\\dfrac{-2np\\pi i}{q}\\right)$ (while $p$ runs through the non-negative integers that are prime to $q$ and less than $q$ ), $\\omega _{p,q}$ is a certain 24$q$ -th root of unity, $\\left(\\dfrac{a}{b}\\right)$ is the Legendre symbol.",
"$b$ is an odd prime, and $p^{\\prime }$ is any positive integer such that $q\\,|\\,(1+pp^{\\prime })$ .",
"When $n$ is very large, $p(n)$ is the integer nearest to $\\sum \\limits _{q=1}^{\\left\\lfloor \\alpha \\sqrt{n}\\right\\rfloor }A_{q}(n)\\cdot \\phi _{q}(n)$ .",
"In [10] or [16], a convergent series for $p(n)$ modified from (REF ) by Rademacher in 1937 is presented, $p(n)=\\sum \\limits _{q=1}^{\\infty }A_{q}(n)\\cdot \\psi _{q}(n),$ where $A_{q}(n)$ is the same as mentioned above and $\\psi _{q}(n)=\\dfrac{\\sqrt{q}}{\\pi \\sqrt{2}}\\dfrac{\\mathrm {d}}{\\mathrm {d}n}\\left(\\dfrac{\\sinh \\left(\\tfrac{\\pi }{q}\\sqrt{\\frac{2}{3}\\left(n-\\tfrac{1}{24}\\right)}\\right)}{\\sqrt{n-\\tfrac{1}{24}}}\\right).$ Figure: The Relative Error of R h (n)R_{\\mathrm {h}}(n) to p(n)p(n) when 1K ⩽n⩽\\leqslant n\\leqslant 10KEquations (REF ) or (REF ) are valuable in theory and can be used to calculate the value of $p(n)$ with very high accuracy.",
"But they are not convenient for practical usage especially when $n$ is small, since it is very difficult for programmers, engineers or other ordinary people (not familiar with any computer algebra system softwares) since they are too complicated and they contain some special functions that most people (not majored in mathematics) are not familiar with.",
"It is very difficult for them to use these two formulae to calculate $p(n)$ on a pocket science calculator without programming function.",
"In references [21] or [3], we may find the famous asymptotic formula for $p(n)$ , [Rh(n)]$R_{\\mathrm {h}}(n)$ The Hardy-Ramanujan's asymptotic formula.",
"$p(n)\\sim \\dfrac{1}{4n\\sqrt{3}}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi n^{{1}{2}}\\right),$ obtained by Godfrey Harold Hardy and Srinivasa Ramanujan in 1918 in the famous paper [11].",
"(Two different proofs can be found in [8] and [14].",
"The evaluation of the constants was shown in [13].)",
"This formula will be called the Hardy-Ramanujan's asymptotic formula in this paper.",
"This asymptotic formula is with great importance in theory.",
"Equation (REF ) is much more convenient than formulae (REF ) and (REF ) for ordinary people not majored in mathematics.",
"Let $R_{\\mathrm {h}}(n)=\\dfrac{1}{4n\\sqrt{3}}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right).$ be the asymptotic function by Hardy and Ramanujan.",
"By the figure in reference [18], this asymptotic formula fits $p(n)$ very well when $n$ is huge.",
"But when $n$ is small, the relative error of $R_{\\mathrm {h}}(n)$ to $p(n)$ is not so satisfying as shown in Table REF (when $n$ $\\leqslant $ 1000) on page REF .",
"When $n$ $\\leqslant $ 25, the relative error is greater than 9%; when 25 < $n$ $\\leqslant $ 220, the relative error is greater than 3%; when $n$ $\\leqslant $ 1000, the relative error is greater than 1.4%.",
"From Figure 1.REF , we will find out that the relative error is greater than 0.44% when 1000 $\\leqslant n\\leqslant $ 10000.",
"Considering that $p(n)$ is an integer and $R_{\\mathrm {h}}(n)$ is definitely not, the round approximation of $R_{\\mathrm {h}}(n)$ may be a little more accurate, but that does not help.",
"Although (REF ) is not so accurate when $n$ is small, it provides some important clue for a more accurate formula for small $n$ .",
"Table: The relative error of R h (n)R_{\\mathrm {h}}(n) to p(n)p(n) when n⩽1000n\\leqslant 1000.By revising (REF ), some other estimation formulae with high accuracy is obtained here.",
"In sec:main-idea, the main idea is introduced, two parameters $C_{1}$ and $C_{2}$ are brought in the Hardy-Ramanujan's asymptotic formula, they will be fitted in sections and , respectively.",
"Sections and will show some other methods to obtain estimation formulae.",
"Section displays an estimation formula with more accuracy when $n\\leqslant 100$ .",
"The main difficulty is that it is too hard to obtain the appropriate functions to fit the data of $C_{1}$ (or $C_{2}$ or some others) generated here since we know very little about them and the usual methods to find fitting functions are invalid here.",
"If we fit the data directly, the results are far from satisfactory, at least the accuracy is not as good as that of (REF )."
],
[
"Main idea ",
"There are many different ways to modify $R_{\\mathrm {h}}(n)$ , e.g.",
"we could also construct a function $p_{1}(n)$ to estimate $R_{\\mathrm {h}}(n)-p(n)$ , then $R_{\\mathrm {h}}(n)-p_{1}(n)$ may reach a better accuracy when estimating $p(n)$ , or we can estimate the value of $\\frac{R_{\\mathrm {h}}(n)}{p(n)}$ by a function $f_{1}(n)$ then estimate $p(n)$ by $\\frac{R_{\\mathrm {h}}(n)}{f_{1}(n)}$ , etc.",
"The problem is that the accuracy of $R_{\\mathrm {h}}(n)-p_{1}(n)$ is not so satisfying if we do not use the idea shown in (REF ) in sec:main-idea, because the shape of the figure of $\\ln \\left(R_{\\mathrm {h}}(n)-p(n)\\right)$ is nearly the same as the shape of the figure of $\\ln \\left(p(n)\\right)$ , at least we can not tell the difference by our eyes as shown on Figure REF and Figure REF (on page REF ), though they are different in theory.",
"$\\ $ Since $p(n)$ $\\sim $ $R_{\\mathrm {h}}(n)$ , we believe that an approximate formula with better accuracy may be in this form $p(n)\\approx \\dfrac{1}{4\\sqrt{3}(n+C_{2})}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}}\\right).$ Figure: The graph of the data n,C 1 n\\left(n,\\,C_{1}\\left(n\\right)\\right) (n⩾120n\\geqslant 120).Where $C_{1}$ (or $C_{2}$ ) may be a constant or a function of $n$ that increases slowly than $n$ , so as to have $\\underset{n\\rightarrow \\infty }{\\lim }\\frac{\\frac{1}{4\\sqrt{3}(n+C_{2})}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}}\\right)}{R_{\\mathrm {h}}(n)}$ = 1, or $\\underset{n\\rightarrow \\infty }{\\lim }\\frac{\\frac{1}{4\\sqrt{3}(n+C_{2})}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}}\\right)}{p(n)}$ = 1.",
"There are some other ways to modify $R_{\\mathrm {h}}(n)$ , we will discuss the details in section .",
"As we can not determine $C_{1}$ and $C_{2}$ at the same time because of technique problems, $\\ $ Usually, we will get the value of $C_{1}$ and/or $C_{2}$ from a number of pairs of $\\left(n,p(n)\\right)$ by the least square method, not from two pairs of $\\left(n,p(n)\\right)$ only.",
"Many software can get efficiently the undetermined coefficients (by the least square method) by solving a system of (incompatible) linear equations, while it is very difficult to “solve” a system of tens or hundreds of transcendental equations that are incompatible.",
"we may decide $C_{1}$ first then determine $C_{2}$ , the main reason is that $\\frac{1}{(n+C_{2})}$ and $\\frac{1}{n}$ differs very little when $n$ is very huge, at least we believe that the difference is much less that the difference of $\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}}\\right)$ and $\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)$ .",
"$\\ $ It is not difficult to know that $\\frac{1}{(n+\\delta )}$ $\\approx $ $\\frac{1}{n}\\left(1-\\frac{\\delta }{n}\\right)$ , $\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+\\delta }\\right)$ $\\approx $ $\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)\\left(1+\\frac{\\pi }{\\sqrt{6}}\\frac{\\delta }{\\sqrt{n}}\\right)$ , when $\\delta \\ll n$ .",
"Obviously, $\\frac{\\delta }{n}\\ll \\frac{\\pi }{\\sqrt{6}}\\frac{\\delta }{\\sqrt{n}}$ (when $\\max \\lbrace \\delta ,1\\rbrace \\ll n$ ).",
"Figure: The graph of the data n,C 1 n\\left(n,\\,C_{1}\\left(n\\right)\\right) (80⩽n⩽20080\\leqslant n\\leqslant 200).So, when $n$ $\\gg $ 1, we believe $p(n)\\doteq \\dfrac{1}{4\\sqrt{3}n}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}}\\right),$ hence $4\\sqrt{3}n\\times p(n)\\doteq \\exp \\left(\\pi \\sqrt{\\frac{2}{3}(n+C_{1})}\\right)$ , then $C_{1}\\left(n\\right)\\doteq \\dfrac{3}{2}\\cdot \\dfrac{\\left(\\ln \\left(4n\\sqrt{3}p(n)\\right)\\right)^{2}}{\\pi ^{2}}-n.$ If we point the data $\\left(n,\\ \\frac{3}{2}\\cdot \\frac{\\left(\\ln \\left(4n\\sqrt{3}p(n)\\right)\\right)^{2}}{\\pi ^{2}}\\right)$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) in the coordinate system, we will find that they lie in a straight line, as shown in the Figure REF on page REF , which means that the Hardy-Ramanujan's asymptotic formula is close to perfect.",
"Here every tiny cycle stands for a data point."
],
[
"Fit the Exponent ",
"If we point the data $\\left(n,\\,C_{1}(n)\\right)$ , i.e., $\\left(n,\\ \\frac{3}{2}\\cdot \\frac{\\left(\\ln \\left(4n\\sqrt{3}p(n)\\right)\\right)^{2}}{\\pi ^{2}}-n\\right)$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) in the coordinate system, we will get the Figure REF on page REF .",
"Here the points when $n\\leqslant 120$ are not shown on Figure REF , partly because the deduction above is based on $n\\gg 1$ , the main reason is that the points obviously do not lie in a curve when $n\\leqslant 120$ , as shown on Figure REF and Figure REF (on page REF ).",
"Figure REF looks like a logarithmic curve or a hyperbola.",
"The author has tried many functions (by a small program written in MAPLE) like $a\\cdot \\left(\\ln (x^{e_{1}}+c_{1})\\right)^{e_{2}}+b,$ where $e_{1}$ , $e_{2}$ and $c_{1}$ are given constants while $a$ and $b$ are undermined coefficients to be decided.",
"But none of them fits the data very well.",
"A function $y=a\\cdot \\left(\\ln \\left(\\left(\\dfrac{7}{20}\\cdot x-16\\right)^{29/32}+2.5\\right)\\right)^{1/32}+b,$ where $a$ = 0.06656839293 and $b$ = -0.4166945066, may fit the data better, but it is not as good as we expect, as shown on Figure REF on page REF .",
"A hyperbola like $y=\\dfrac{a}{x}+b$ does not fit the data very well, either.",
"Then we consider this type of functions $y=\\dfrac{a}{(x+c_{2})^{e_{2}}}+b,$ where $a$ , $b$ , $c_{2}$ and $e_{2}$ are undetermined constants.",
"This seems much better.",
"For technique reason, we can not decide all the undetermined coefficients $a$ , $b$ , $c_{2}$ , $e_{2}$ at the same time.",
"$\\ $ Because most computer algebra system (CAS) could not solve the system of many incompatible nonlinear equations by the least square method, or the time-consumption is unacceptable.",
"These undetermined coefficients may be obtained in this way: A1.",
"Give $c_{2}$ and $e_{2}$ initial values; A2.",
"Fit the data $\\left(n,\\,C_{1}(n)\\right)$ by the least square method with Equation (REF ) and obtain the values of $a$ and $b$ , then get the average error of the fitting function for the values of $c_{2}$ , $e_{2}$ , $a$ , $b$ ; $\\ $ Here we use the square root of the mean square deviation $s=\\sqrt{\\dfrac{1}{m}\\sum _{i=1}^{m}\\left(y_{i}-f(x_{i})\\right)^{2}}$ to measure the average error of the fitting function $y=f(x)$ to the original data $\\left(x_{i},\\,y_{i}\\right)$ ($i$ = 1, 2, $\\cdots $ , $m$ ).",
"A3.",
"Reevaluate $e_{2}$ and $a$ .",
"Plot the points of the data $\\Bigl (\\ln \\left(n+c_{2}\\right),\\,\\ln \\left(b-C_{1}(n)\\right)\\Bigr )$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) in the coordinate system with the values of $b$ and $c_{2}$ just found, $\\ $ Such as shown in Figure REF on page REF when $c_{2}$ = 2.5 and $b$ = $-$ 0.3456348045.",
"$\\ $ The purpose of this step is to obtain more accurate values of $e_{2}$ and $a$ .",
"Since $C_{1}(n)=\\frac{a}{(n+c_{2})^{e_{2}}}+b$ , then $b-C_{1}(n)=\\frac{-a}{(n+c_{2})^{e_{2}}}$ , (considering that $a<0$ ), $\\ln \\left(b-C_{1}(n)\\right)$ = $\\ln \\left(-a\\right)$ + $e_{2}\\cdot \\ln \\left(n+c_{2}\\right)$ , so the figure of data $\\Bigl (\\ln \\left(n+c_{2}\\right),\\,\\ln \\left(b-C_{1}(n)\\right)\\Bigr )$ will be some points on a straight line if the previous assumption is correct and meanwhile the values of $b$ and $c_{2}$ are proper.",
"fit the data by the least square method with $y=e_{1}\\cdot x+a_{1}$ and find the values of $a_{1}$ and $e_{1}$ , then reevaluate $e_{2}$ and $a$ by $e_{2}=-e_{1},\\ a=-\\exp (a_{1});$ A4.",
"Reevaluate $c_{2}$ .",
"Plot the points of the data $\\left(n,\\,\\left(\\frac{a}{C_{1}(n)-b}\\right)^{1/e_{2}}\\right)$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) in the coordinate system with the value of $b$ and the new values of $a$ and $e_{2}$ , $\\ $ Such as shown on Figure REF on page REF when $b$ = $-$ 0.3456365954, $e_{2}$ = 0.5012314726 and $a$ = $-$ 0.02661232627.",
"$\\ $ The main idea of this step: since $C_{1}(n)=\\frac{a}{(n+c_{2})^{e_{2}}}+b$ , then $n+c_{2}=\\left(\\frac{a}{C_{1}(n)-b}\\right)^{1/e_{2}}$ , hence the figure of data $\\left(n,\\,\\cdot \\left(\\frac{a}{C_{1}(n)-b}\\right)^{1/e_{2}}\\right)$ will be some points on a straight line.",
"fit the data by the least square method with $y=x+c_{1}$ and find the value of $c_{1}$ , then reevaluate $c_{2}$ by $c_{2}=c_{1}$ .",
"A5.",
"goto step 2 until a fitting function with the least average error is obtained.",
"For example, in step A1, the initial value could be set by $c_{2}$ = 2.5, $e_{2}$ = 0.5 (or some other values).",
"In step A2, if $c_{2}$ = 2.5, $e_{2}$ = 0.5, then $a$ = $-$ 0.02635983935, $b$ = $-$ 0.3456348045.",
"$\\ $ If we plot the figure of (REF ) with the value of $c_{2}$ , $e_{2}$ , $a$ , $b$ , and compare the figure with Figure REF on page REF , we will get a graph nearly the same as Figure REF (although there should be a little different, but we can not distinguish the difference by our eyes).",
"The average error of the fitting function for the values of $c_{2}$ , $e_{2}$ , $a$ , $b$ mentioned above is 1.074574171$\\times 10^{-5}$ , which seems to be very tiny.",
"In Step A3, $\\ $ if $c_{2}$ = 2.5, $b$ = $-$ 0.3456365954, then $a_{1}$ = $-$ 3.626380777, $e_{1}$ = $-$ 0.5012314726.",
"After reevaluation, $e_{2}$ = 0.5012314726, $a$ = $-$ 0.02661232627.",
"In Step A4, for the values of $b$ , $e_{2}$ and $a$ mentioned before, after reevaluation $c_{2}$ = 4.871833842.",
"Figure: The graph of the data ln n+c 2 , ln b-C 1 (n)\\bigl (\\ln \\left(n+c_{2}\\right),\\,\\ln \\left(b-C_{1}(n)\\right)\\bigr )Actually, only a few times of repeating the steps form A2 to A4, we will obtain a very good fitting function, as shown on Figure REF on page REF .",
"$\\ $ For the initial value $c_{2}$ = 2.5, $e_{2}$ = 0.5, after repeating 41 times of the steps from A2 to A4, we will find a fitting function $y=\\dfrac{-0.02594609078}{(x+3.320623832)^{0.4963284361}}-0.3456286995,$ with a minimal average error $9.010349470\\times 10^{-8}$ .",
"After a few times more of iteration, a result with similar coefficients will be found but with a little more error.",
"There are some explanations about the steps above: (1).",
"In step A4, we did not plot the points of the data $\\left(n,\\,\\left(\\frac{a}{C_{1}(n)-b}\\right)^{1/e_{2}}-n\\right)$ because the shape of the figure is not a horizontal line as shown on Figure REF on page REF (the points in the right hand side are not so smooth because only 10 significance digits are kept in the process, if more significance digits are calculated, it will be better).",
"Actually, it is a little complicated.",
"But it will not help us to obtain better values of the undetermined in (REF ) if we fit the data $\\left(n,\\,\\left(\\frac{a}{C_{1}(n)-b}\\right)^{1/e_{2}}-n\\right)$ with a more accurate fitting function.",
"(2).",
"In step A3, if we do not reevaluate $a$ , the fitting parameters will not converge in general (even if we computing more significant figures in the process), or we can not continue the iterations steps at all since imaginary numbers appear.",
"(3).",
"If we started with a different initial value of $c_{2}$ and keep the initial value of $e_{2}$ , such as $c_{2}$ = 15, after repeating 78 times of the steps from A2 to A4, we will find a fitting function $y=\\dfrac{-0.02593608938}{(x+3.272445238)^{0.4962730054}}-0.3456286681,$ with a minimal average error $9.109686836\\times 10^{-8}$ .",
"If we started with some different initial values for both $c_{2}$ and $e_{2}$ , such as $c_{2}$ = 15 and $e_{2}$ = 0.7, (from Figure REF on page REF , we will find that $e_{2}$ should be less that 1.0), we will get a similar result.",
"After repeating 125 times of the steps from A2 to A4, we will find a fitting function $y=\\dfrac{-0.02593617719}{(x+3.273513225)^{0.4962727258}}-0.3456286655,$ with a minimal average error $9.105941452\\times 10^{-8}$ .",
"After that, $e_{2}$ and $c_{2}$ will decrease slowly and slowly, and the average error will increase little by little if we continue the steps from A2 to A4.",
"As concerned to the errors in computing, the valid value of the undermined $a$ , $b$ , $c_{2}$ and $e_{2}$ should be $-0.0259361$ , $-0.34562866$ , $3.273$ , $0.49627$ , the average absolute error of the fitting function of $C_{1}(n)$ is about $9.1\\times 10^{-8}$ .",
"Figure: The graph of the data n,·a C 1 (n)-b 1/e 2 -n\\left(n,\\,\\cdot \\left(\\frac{a}{C_{1}(n)-b}\\right)^{1/e_{2}}-n\\right)Considering that (REF ) is an approximate formula, we may believe that the best value of $e_{2}$ is $0.5$ , as we prefer a simple exponent.",
"Then it will be more convenient to obtain $a$ , $b$ and $c_{2}$ .",
"Below $e_{2}$ is supposed to be $1/2$ , which means that the fitting function of $C_{1}(n)$ is $y=\\dfrac{a}{\\sqrt{x+c_{2}}}+b.$ When $e_{2}$ is fixed to be $1/2$ , if we use the iteration method described above but keep the value of $e_{2}$ in step A3, i.e., substitute step A3 by If if if A3'.",
"Reevaluate $a$ by $\\ $ or equivalently, Plot the points of the data $\\Bigl (\\ln \\left(n+c_{2}\\right),\\,\\ln \\left(b-C_{1}(n)\\right)\\Bigr )$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) in the coordinate system with the values of $b$ , $e_{2}$ and $c_{2}$ just found, fit the data by the least square method with $y=e_{2}\\cdot x+a_{1}$ and find the values of $a_{1}$ , then reevaluate $a$ by $a=-\\exp (a_{1});$ $a & =-\\exp \\biggl (\\dfrac{1}{395}\\sum \\limits _{k=1}^{395}\\Bigl (\\ln \\left(b-C_{1}(20k+100)\\right)-\\\\& \\quad \\quad e_{2}\\cdot \\ln \\left(20k+100+c_{2}\\right)\\Bigr )\\biggr );$ (that means we evaluate $a$ twice in every loop) the sequence of fitting functions of $C_{1}(n)$ will diverge.",
"But we will obtain a converged sequence of the determinants if $n$ ranges from 120 to 6000, (i.e., consider only the data $\\left(n,\\,p(n)\\right)$ when $n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 295).",
"The fitting function of $C_{1}(n)$ obtained in this way (when $n$ ranges from 120 to 6000, step 20) is $y=\\dfrac{-0.02650620466}{\\sqrt{x+4.855479108}}-0.3456326154,$ with the minimal average error 2.374935895$\\times 10^{-7}$ .",
"$\\ $ If we use the value of $c_{1}$ already found above, such as $c_{2}$ = 3.273513225 in (REF ), the fitting function is $y=\\dfrac{-0.02640970103}{\\sqrt{x+3.273513225}}-0.3456340228,$ with an average error $7.404647856\\times 10^{-7}$ , which is about 3 times than that above.",
"If we choose $c_{2}$ = 3.320623832 in eq:C1(n)1-A, the fitting function is $y=\\dfrac{-0.02641281526}{\\sqrt{x+3.320623832}}-0.3456339736,$ with an average error $7.205944166\\times 10^{-7}$ .",
"For the fixed value $1/2$ of $e_{2}$ , if we continue use the iteration method described above but ignore step 3, which means we reevaluate $a$ only once in every loop, we will meet the same situation.",
"The sequence of fitting functions of $C_{1}(n)$ will diverge if $n$ ranges from 120 to 8000 (or 6000) even if we calculate more significance digits (such as 18 significance digits) in the process, but it will converge if $n$ ranges from 120 to 4000.",
"The fitting function of $C_{1}(n)$ obtained in this way (when $n$ ranges from 120 to 4000, step 20) is $y=\\dfrac{-0.02647712648}{\\sqrt{x+4.55083607}}-0.345633305,$ with the minimal average error 1.993012726$\\times 10^{-7}$ when the initial value of $c_{2}$ is 10 (iterated 4 times).",
"But after more times of iteration, for several initial values of $c_{2}$ (such as 5, 10, 15, etc), the fitting functions converge to $y=\\dfrac{-0.0268\\cdots }{\\sqrt{x+4.888\\cdots }}-0.345632760\\cdots ,$ with the average error 2.68$\\cdots \\times 10^{-7}$ .",
"$\\ $ Unlike the previous method, by the results mentioned above and some other results not mentioned here, the sequence of fitting functions of $C_{1}(n)$ usually converges to a function which is obviously different from the one with the minimal average error.",
"In order to get a fitting function with errors as tiny as possible, we can design another algorithm.",
"By the results described above, we known that $c_{2}$ is probably between 3 and 5, so we can find the fitting function of $C_{1}(n)$ and the corresponding average error for many values of $c_{2}$ in the possible range, then choose the one with minimal average error.",
"To be cautious, we test the value of $c_{2}$ in the interval $[0.5,\\ 15]$ .",
"The main steps are as below: (1) Initial $c_{\\mathrm {a}}$ , $c_{\\mathrm {b}}$ , $c_{0}$ , $s_{0}$ , $D_{\\mathrm {t}}$ , $a_{0}$ , $b_{0}$ .",
"Let $c_{\\mathrm {a}}$ $:=$ 0.5, $c_{\\mathrm {b}}$ $:=$ 15, $c_{0}$ $:=$ 0, $s_{0}$ $:=$ 1, $a_{0}$ $:=$ 0, $b_{0}$ $:=$ 0, $D_{\\mathrm {t}}$ $:=$ 8, $s_{\\mathrm {t}}$ $:=$ 0.1,.",
"(2) for $c_{2}$ from $c_{\\mathrm {a}}$ to $c_{\\mathrm {b}}$ by $s_{\\mathrm {t}}$ do WW Fit the data $\\left(n,\\,C_{1}(n)\\right)$ by the least square method with (REF ) and get the values of $a$ and $b$ , then get the average error $s_{1}$ of the fitting function for the values of $c_{2}$ , $a$ , $b$ ; if $s_{1}$ < $s_{0}$ , then let $c_{0}$ $:=$ $c_{2}$ , $s_{0}$ $:=$ $s_{1}$ , $a_{0}$ $:=$ $a$ , $b_{0}$ $:=$ $b$ ; end if; end do (3) If $D_{\\mathrm {t}}$ > 1, then set $D_{\\mathrm {t}}$ $:=$ $D_{\\mathrm {t}}-1$ , $c_{\\mathrm {a}}$ $:=$ $c_{0}-5s_{\\mathrm {t}}$ , $c_{\\mathrm {b}}$ $:=$ $c_{0}+5s_{\\mathrm {t}}$ ; WW set $s_{\\mathrm {t}}$ $:=$ $s_{\\mathrm {t}}/10$ ; goto step (2); else, terminate the process.",
"end if; Here the symbol “$x$ $:=$ $y$ ” means that the variable $x$ is evaluated by the value of the variable $y$ ; in step (1), $D_{\\mathrm {t}}$ $:=$ 8 means that we will get 8 significance digits of the value of $c_{2}$ .",
"In the algorithm above, we have assumed implicitly that the average error is a smooth and continuous function of $a$ , $b$ , $c_{2}$ for the values of $x_{k}=20k+100$ , ($k$ = 1, 2, $\\cdots $ , 395).",
"For every $c_{2}$ , we can get the value of $a$ and $b$ , then obtain the the average error $s_{1}$ , so $s_{1}$ could be believed as a convex and smooth function of $c_{2}$ (hence it will have only one minimum point) in the interval we are considering.",
"This could be verified by plotting the figure of the curve $s_{1}$ = $s_{1}(c_{2})$ in the given interval (although this work is not easy in practice).",
"If $n$ ranges from 120 to 8000 (step 20), we can get a fitting function of $C_{1}(n)$ , $y=\\dfrac{-0.02651010067}{\\sqrt{x+4.8444724}}-0.3456324524,$ with a minimal average error $2.446731760\\times 10^{-7}$ .",
"If $n$ ranges from 120 to 6000 (step 20), the fitting function of $C_{1}(n)$ is, $y=\\dfrac{-0.02649625326}{\\sqrt{x+4.7152127}}-0.3456327903,$ with a minimal average error $2.279396699\\times 10^{-7}$ .",
"In the next section, (REF ) will be used to estimate $C_{1}(n)$ , i.e., $C_{1}(n)\\doteq \\dfrac{-0.02651010067}{\\sqrt{n+4.8444724}}-0.3456324524.$"
],
[
"Fit the Denominator ",
"By (REF ) and (REF ), we have $C_{2}(n)\\doteq \\dfrac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}(n)}\\right)}{4\\sqrt{3}p(n)}-n.$ Figure: Fit n,C 2 (n)\\left(n,\\,C_{2}(n)\\right), the odd, Part BIf we point out the data $\\left(n,\\,C_{2}(n)\\right)$ (1 $\\leqslant $ $n$ $\\leqslant $ 80) on the coordinate system as shown on Figure REF on page REF , we will immediately know than $C_{2}(n)$ can not be fit by a simple function.",
"From the Figure REF (or the value of $C_{2}(n)$ calculated by a small program), it is clear that $C_{2}(n)$ is very small when $n$ $>$ 40, at least much less than $n$ , so there is no need to fit $C_{2}(n)$ when $n$ $>$ 40.",
"When $n$ is odd, the points of $\\left(n,\\,C_{2}(n)\\right)$ in Figure REF are above the horizontal-axis, it is not difficult to separate them into two parts and fit them by two cubic curves, as shown on Figure REF and Figure REF .",
"The two fitting functions are $y= & -1.548835311\\times 10^{-6}\\times x^{3}+\\\\& \\quad 1.880663805\\times 10^{-4}\\times x^{2}-\\\\& \\quad 0.008334098201\\times x+0.1399798428,\\\\y= & -5.416501948\\times 10^{-6}\\times x^{3}+\\\\& \\quad 5.728510889\\times 10^{-4}\\times x^{2}-\\\\& \\quad 0.02125835759\\times x+0.2882706948.$ Figure: The Relative Error of R h1 (n)R_{\\mathrm {h1}}(n) when 1000 ⩽\\leqslant nn ⩽\\leqslant 10000For the points of $\\left(n,\\,C_{2}(n)\\right)$ under the horizontal-axis (when $n$ is even) in Figure REF , we have to separate them into at least 4 parts so as to fit them smoothly, two or three parts are not convenient.",
"As a result, we have to fit $C_{2}(n)$ by a hybrid function with at least 6 pieces, or fit $p(n)$ by a piecewise-defined function with 7 pieces, which is very complicated.",
"This seems to contradict with our purpose at the beginning of this paper.",
"From Figure REF on page REF we found that the value of $C_{2}(n)$ are much less than $n$ when $n$ $\\geqslant $ 15, so the error will be very tiny if we omit $C_{2}(n)$ .",
"Hence we can calculate $p(n)$ directly by $R_{\\mathrm {h1}}(n)=\\dfrac{1}{4\\sqrt{3}n}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+\\dfrac{a_{1}}{\\sqrt{n+c_{1}}}+b_{1}}\\right),$ where $a_{1}=-0.02651010067$ , $b_{1}=-0.3456324524$ and $c_{1}=4.8444724$ .",
"The error of (REF ) to $p(n)$ (when $n$ $\\leqslant $ 1000) is shown on Table REF on page REF .",
"The accuracy is much better than (REF ).",
"Although this fitting function is obtained when $n$ $\\geqslant $ 120, the relative error is less than $6\\times 10^{-7}$ when $n$ $\\geqslant $ 100, less than 1 when $n$ $\\geqslant $ 26, less than $1\\%$ when $n$ $\\geqslant $ 11.",
"When 1000 $\\leqslant $ $n$ $\\leqslant $ 3000, the relative error is less than $1\\times 10^{-8}$ .",
"When 3000 $\\leqslant $ $n$ $\\leqslant $ 10000, the relative error is less than $5.3\\times 10^{-9}$ , as shown on Figure REF on page REF .",
"But the relative error is not so satisfying when $n$ $\\leqslant $ 7, especially when $n=1$ .",
"Consider that $p(n)$ is an integer, if we take the round approximation of (REF ), [Rh1(n)]$R^{\\prime }_{\\mathrm {h1}}(n)$ The Hardy-Ramanujan's revised estimation formula 1.",
"$R^{\\prime }_{\\mathrm {h1}}(n)=\\left\\lfloor \\dfrac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+\\dfrac{a_{1}}{\\sqrt{n+c_{1}}}+b_{1}}\\right)}{4\\sqrt{3}n}+\\dfrac{1}{2}\\right\\rfloor ,$ (we may call it Hardy-Ramanujan's revised estimation formula 1), it will solve perfectly the relative error problem when $n$ $<$ 11, as shown on Table REF on page REF , although the relative error will increase very little for some $n$ , which is negligible.",
"(The average relative error is less than $2\\times 10^{-8}$ when $n$ $\\geqslant $ 200.)",
"Take an example, when $n=100$ , $R_{\\mathrm {h2}}(100)$ = 190569177, $p(100)$ = 190569292, the difference is 115; when $n=200$ , $R_{\\mathrm {h2}}(200)$ = 3972999059745, $p(200)$ = 3972999029388, the difference is 30357.",
"Although the errors are much greater than the error 0.004 of Hardy-Ramanujan formula with 6 terms ($n=100$ ) or 8 terms ($n$ = 200) (refer [11] or [16]), it contains only one term of elementary functions, and is convenient for a junior middle school student to calculate the value of $p(n)$ with high accuracy.",
"Table: The relative error of R h1 ' (n)R^{\\prime }_{\\mathrm {h1}}(n) to p(n)p(n) when n⩽80n\\leqslant 80.When n⩾70n\\geqslant 70, the relative error differs very little."
],
[
"Some Other Methods ",
"In the previous sections, we assume that $C_{1}\\left(n\\right)\\doteq \\frac{3\\left(\\ln \\left(4n\\sqrt{3}p(n)\\right)\\right)^{2}}{2\\pi ^{2}}-n$ , then fit the data $\\left(n,\\ \\frac{3\\left(\\ln \\left(4n\\sqrt{3}p(n)\\right)\\right)^{2}}{2\\pi ^{2}}\\right)$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395), and estimate $p(n)$ by $R_{\\mathrm {h2}}(n)=\\left\\lfloor \\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}\\left(n\\right)}\\right)}{4\\sqrt{3}n}+\\dfrac{1}{2}\\right\\rfloor $ ."
],
[
"Modify the Denominator only",
"If we assume that $p(n)\\doteq \\frac{\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)}{4\\sqrt{3}(n+C_{2})}$ , then $C_{2}(n)\\doteq \\dfrac{1}{4\\sqrt{3}p(n)}\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)-n,$ Figure: The graph of the data n,expπ2 3n 43p(n)-n\\left(n,\\ \\frac{\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)}{4\\sqrt{3}p(n)}-n\\right)we wonder whether we can fit the data $\\left(n,\\ \\frac{\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)}{4\\sqrt{3}p(n)}-n\\right)$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) by a function $C_{2}$ and estimate $p(n)$ by $\\left\\lfloor \\frac{\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)}{4\\sqrt{3}(n+C_{2})}+\\dfrac{1}{2}\\right\\rfloor $ ?",
"The data $\\left(n,\\ \\frac{\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)}{4\\sqrt{3}p(n)}-n\\right)$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) are shown on Figure REF on page REF (together with the figure of a fitting function).",
"It is not difficult to know that a function in this form $y=a_{1}\\times (x+c_{1})^{e_{1}}+b_{1}$ will fit the points very well, and $e_{1}=0.5$ will be very satisfying.",
"By the same method to fit $C_{1}(n)$ , we can obtain a fitting function $y=0.4432884566\\times \\sqrt{x+0.274078}+0.1325096085$ to fit $C_{2}(n)$ with an average error $3.65\\times 10^{-6}$ .",
"Hence we can calculate $p(n)$ by $R_{\\mathrm {h2}}(n)=\\dfrac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{4\\sqrt{3}\\left(n+a_{2}\\sqrt{n+c_{2}}+b_{2}\\right)},$ where $a_{2}=0.4432884566$ , $b_{2}=0.1325096085$ and $c_{2}=0.274078$ , when $n$ is not so small.",
"Table: The relative error of R h2 ' (n)R^{\\prime }_{\\mathrm {h2}}(n) to p(n)p(n) when n⩽80n\\leqslant 80.When n⩾40n\\geqslant 40, the relative error differs very little.The error of (REF ) to $p(n)$ is shown on Table REF on page REF when $n$ $\\leqslant $ 1000.",
"The accuracy is much better than (REF ).",
"Compared with Table REF (page REF ), the accuracy are almost the same when $n$ $\\leqslant $ 1000.",
"When 1500 $\\leqslant $ $n$ $\\leqslant $ 10000, the relative error is obviously less than that of (REF ), as shown on Figure REF on page REF (compared with Figure REF on page REF ).",
"Which means that $R_{\\mathrm {h2}}(n)$ is more accurate than $R_{\\mathrm {h1}}(n)$ .",
"(If we change the range of $n$ of the data points, the accuracy of the fitting function obtained may not be so good.)",
"Consider that $p(n)$ is an integer, we can take the round approximation of (REF ), [Rh2(n)]$R^{\\prime }_{\\mathrm {h2}}(n)$ The Hardy-Ramanujan's revised estimation formula 2.",
"REF $R^{\\prime }_{\\mathrm {h2}}(n)=\\left\\lfloor \\dfrac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{4\\sqrt{3}\\left(n+a_{2}\\sqrt{n+c_{2}}+b_{2}\\right)}+\\dfrac{1}{2}\\right\\rfloor ,$ for small values of $n$ .",
"We may call it Hardy-Ramanujan's revised estimation formula 2.",
"The error of (REF ) to $p(n)$ is shown on Table REF (on page REF ) when $n$ $\\leqslant $ 1000."
],
[
"Fit ${R_{\\mathrm {h}}(n)}{p(n)}$",
"At the beginning of section , some other methods to estimate $p(n)$ are mentioned, such as estimating the value of $\\frac{R_{\\mathrm {h}}(n)}{p(n)}$ by a function $f_{1}(n)$ , then estimate $p(n)$ by $\\frac{R_{\\mathrm {h}}(n)}{f_{1}(n)}$ .",
"The data $\\left(n,\\ \\frac{R_{\\mathrm {h}}(n)}{f_{1}(n)}\\right)$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) are shown on Figure REF on page REF (together with the figure of a fitting function).",
"It is not difficult to find out that a function $y=1+\\dfrac{1}{\\sqrt{a_{3}x+b_{3}}},$ where $a_{3}=5.062307637$ and $b_{3}=-75.65700620$ , will fit the data very well, as shown on the figure, with an average error $1.41\\times 10^{-4}$ .",
"(because the data $\\left(n,\\ \\left(\\frac{R_{\\mathrm {h}}(n)}{f_{1}(n)}-1\\right)^{-2}\\right)$ lies exactly on a straight line $y=a_{3}x+b_{3}$ , as shown on Figure REF on page REF ) Figure: The Relative Error of R d3 (n)R_{d\\mathrm {3}}(n) when 1000 ⩽\\leqslant nn ⩽\\leqslant 10000So we have another fitting function for $p(n)$ , $R_{d3}(n)=\\dfrac{R_{\\mathrm {h}}(n)}{1+\\dfrac{1}{\\sqrt{a_{3}n+b_{3}}}}.$ However, this formula does not fit $p(n)$ very well when $n$ is small.",
"When $n\\leqslant 14$ , the value of $R_{d3}(n)$ is an imaginary number.",
"Unfortunately, when $n$ > 1000, the error of $R_{d3}(n)$ to $p(n)$ is about 1000 times of the error of $R_{\\mathrm {h2}}(n)$ , as shown on Figure REF on page REF .",
"Actually, $R_{\\mathrm {h2}}(n)$ is in the form $\\frac{R_{\\mathrm {h}}(n)}{f_{1}(n)}$ , since $\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{4\\sqrt{3}\\left(n+a_{2}\\sqrt{n+c_{2}}+b_{2}\\right)}$ = $\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{4\\sqrt{3}n}$ $\\frac{n}{n+a_{2}\\sqrt{n+c_{2}}+b_{2}}$ = $\\frac{R_{\\mathrm {h}}(n)}{1+\\frac{a_{2}}{n}\\sqrt{n+c_{2}}+\\frac{b_{2}}{n}}$ .",
"As $1+\\frac{a_{2}\\sqrt{n+c_{2}}}{n}+\\frac{b_{2}}{n}$ fits $\\frac{R_{\\mathrm {h}}(n)}{p(n)}$ with very little error, $1+\\frac{1}{\\sqrt{a_{3}n+b3}}$ will not reach that accuracy.",
"Figure: The data n,R h (n) f 1 (n)-1 -2 \\left(n,\\ \\left(\\frac{R_{\\mathrm {h}}(n)}{f_{1}(n)}-1\\right)^{-2}\\right)and the fitting function"
],
[
"Result 1",
"It is not difficult to verify that $R_{\\mathrm {h}}(n)-R_{\\mathrm {h}}(n-1)\\sim \\dfrac{\\pi }{12\\sqrt{2n^{3}}}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right).$ (refer Sec.",
"3.1 of [12]).",
"As $R_{\\mathrm {h}}(n)$ is obviously greater than $p(n)$ , we wander whether we can fit $R_{\\mathrm {h}}(n)-p(n)$ by an expression similar like the right part of REF , such as $\\frac{\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{12\\sqrt{2C_{3}(n)}}$ , where $C_{3}(n)$ is a cubic function, or equivalently, fit $\\left(\\frac{\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{12\\sqrt{2}\\left(R_{\\mathrm {h}}(n)-p(n)\\right)}\\right)^{2}$ by a cubic function $C_{3}(n)$ , from the data with the data $(n,\\,p(n))$ ($n=20k+60$ , $k$ = 1, 2, $\\cdots $ , 397).",
"The result is $C_{3}(n)=a_{1}n^{3}+b_{1}n^{2}+c_{1}n+d_{1},$ where $a_{1} & =8.383485427,\\\\b_{1} & =130.0792015,\\\\c_{1} & =-1.197477259\\times 10^{5},\\\\d_{1} & =4.188653689\\times 10^{7}.$ Figure: The graph of the data n , ln R h (n)-pn\\bigl (n,\\,\\ln \\left(R_{\\mathrm {h}}(n)-p\\left(n\\right)\\right)\\bigr )Here $c_{1}$ and $d_{1}$ are very huge, which suggests that this result may not be so satisfying.",
"As a sequence, if we estimate $p(n)$ by $F_{3}(n)=R_{\\mathrm {h}}(n)-\\dfrac{\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{12\\sqrt{2C_{3}(n)}},$ the relative error differs very little with the relative error of $R_{\\mathrm {h}}(n)$ to $p(n)$ when $n<50$ , but the relative error is not satisfying when $n<280$ , as shown in Table REF on page REF .",
"If we fit $\\left(\\frac{\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{12\\sqrt{2}\\left(R_{\\mathrm {h}}(n)-p(n)\\right)}\\right)^{2}$ by a function like $C_{3}(n)=a_{2}n^{3}+b_{2}n^{2.5}+c_{2}n^{2}+d_{2}n^{1.5}+e_{2}n+f_{2}n^{0.5}+g_{2},$ the result are even worse, since imaginary number appeared (as concerned to the data mentioned in this section.",
"If we fit less data, the imaginary problem might be avoid).",
"So we have to consider a different method."
],
[
"Result 2",
"In the previous sub-subsection, we obtained the asymptotic order of $p(n)-p(n-1)$ , and revised it to fit $R_{\\mathrm {h}}(n)-p(n)$ .",
"Since $R_{\\mathrm {h}}(n)$ is always a little greater than $p(n)$ , we may guess that there is a $t_{0}$ such that $R_{\\mathrm {h}}(n-t_{0})$ is closer to $p(n)$ than $R_{\\mathrm {h}}(n)$ .",
"Then we can find the asymptotic order of $R_{\\mathrm {h}}(n)-R_{\\mathrm {h}}(n-t_{0})$ and use the new asymptotic order to fit $R_{\\mathrm {h}}(n)-p(n)$ .",
"Table: The relative error of F 3 (n)F_{3}(n) to p(n)p(n) when n⩽1000n\\leqslant 1000.By the same idea described in the algorithm mentioned on page REF , we can obtain the value $t_{0}\\doteq 0.3594143172$ .",
"When $n\\gg 1$ and $n\\gg t$ , $\\ $ a $r(n)$ $=$ $R_{\\mathrm {h}}(n)-R_{\\mathrm {h}}(n-t)$ $=$ $\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{4\\sqrt{3}n}-$ $\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t}\\right)}{4\\sqrt{3}(n-t)}$ = $\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{4\\sqrt{3}n}-\\left(\\frac{\\exp \\left(\\frac{t\\pi \\sqrt{2/3}}{\\sqrt{n}+\\sqrt{n-t}}\\right)}{n}-\\frac{1}{(n-t)}\\right)$ $\\sim $ $\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t}\\right)}{4\\sqrt{3}n}\\left(\\frac{\\exp \\left(\\frac{t\\pi \\sqrt{2/3}}{2\\sqrt{n-t/2}}\\right)}{n}-\\frac{1}{(n-t)}\\right)$ $\\sim $ $\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t}\\right)}{4\\sqrt{3}n}\\left(\\frac{1+\\frac{t\\pi }{\\sqrt{6(n-t/2)}}}{n}-\\frac{1}{(n-t)}\\right)$ $\\sim $ $\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t}\\right)}{4\\sqrt{3}n}\\left(\\frac{t\\pi \\sqrt{n+t/2}}{\\sqrt{6}n(n-t)}\\right)$ = $\\frac{t\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t}\\right)}{12\\sqrt{2}(n-t)\\sqrt{(n-t/2)}}$ $\\sim $ $\\frac{t\\pi }{12\\sqrt{2}\\sqrt{n^{3}}}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right).$ As $r(n)\\sim \\dfrac{t\\pi }{12\\sqrt{2}\\sqrt{n^{3}}}\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right),$ so we may consider fitting $R_{\\mathrm {h}}(n)-p(n)$ by $\\frac{\\sqrt{2}t_{0}\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t_{0}}\\right)}{24C_{4}(n)}$ , where $C_{4}(n)=a_{2}(n-t_{0})^{1.5}+b_{2}(n-t_{0})+c_{2}(n-t_{0})^{0.5}+d_{2}.$ Table: The relative error of R h3 ' (n)R^{\\prime }_{\\mathrm {h3}}(n) to p(n)p(n) when n⩽30n\\leqslant 30.When n⩾21n\\geqslant 21, the relative error differs very little.When $t_{0}\\doteq 0.3594143172$ , $\\ $ In [19] (or [20]) or some other papers, there is a theoretic value $\\dfrac{1}{24}$ .",
"it is not difficult to find out that $a_{2} & =1.039888529,\\\\b_{2} & =-0.3305606395,\\\\c_{2} & =0.6134039843,\\\\d_{2} & =-0.8582793693,$ from the data $(n,\\,p(n))$ ($n=20k+60$ , $k$ = 1, 2, $\\cdots $ , 397).",
"Here none of the coefficients is very huge, which seems better than the previous result mentioned in this section.",
"As a matter of fact, if we estimate $p(n)$ by $R_{\\mathrm {h3}}(n)=R_{\\mathrm {h}}(n)-\\dfrac{\\sqrt{2}t_{0}\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t_{0}}\\right)}{24C_{4}(n)},$ the relative error is very small even when $n<10$ (except the cases when $n$ = 1 or 2) as shown on Table REF on page REF .",
"This is the first time to have an estimation formula of $p(n)$ which can reach a good accuracy without taking round approximation even when $n<10$ .",
"Further more, if we take the round value of $R_{\\mathrm {h3}}(n)$ , [Rh3(n)]$R^{\\prime }_{\\mathrm {h3}}(n)$ The Hardy-Ramanujan's revised estimation formula 3.",
"REF $R^{\\prime }_{\\mathrm {h3}}(n)=\\left\\lfloor R_{\\mathrm {h}}(n)-\\dfrac{\\sqrt{2}t_{0}\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t_{0}}\\right)}{24C_{4}(n)}+\\dfrac{1}{2}\\right\\rfloor ,$ the relative error to error is even less, especially when $n=15$ or $1<n<12$ (it reaches 0), as shown on Table REF on page REF .",
"The relative error is less than $3\\times 10^{-9}$ when $2500<n<10000$ , as shown on Figure REF on page REF .",
"This formula will be called Hardy-Ramanujan's revised estimation formula 3.",
"Figure: The Relative Error of R h4 (n)R_{\\mathrm {\\mathrm {h}4}}(n) when 1000 ⩽\\leqslant nn ⩽\\leqslant 10000"
],
[
"Result 3",
"Now that we can fit $R_{\\mathrm {h}}(n)-p(n)$ by $\\frac{\\sqrt{2}t_{0}\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n-t_{0}}\\right)}{24C_{4}(n)}$ , where $C_{4}(n)=a_{2}(n-t_{0})^{1.5}+b_{2}(n-t_{0})+c_{2}(n-t_{0})^{0.5}+d_{2}$ , maybe we can also fit $R_{\\mathrm {h}}(n)-p(n)$ by $\\frac{\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{12\\sqrt{2}C_{5}(n)}$ directly, where $C_{5}(n)=a_{3}n^{1.5}+b_{3}n+c_{3}n^{0.5}+d_{3},$ or equivalently, to fit $\\frac{\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{12\\sqrt{2}\\left(R_{\\mathrm {h}}(n)-p(n)\\right)}$ by a function $C_{5}(n)$ in the form mentioned above.",
"We can easily obtain the value of the unknown coefficients in the equation above by the least square method.",
"$a_{3} & =2.893270736,\\\\b_{3} & =0.4164546941,\\\\c_{3} & =-0.08501098214,\\\\d_{3} & =-0.4621004962.$ Again, none of the coefficients is very huge.",
"As a result, the relative error of $R_{\\mathrm {h4}}(n)=R_{\\mathrm {h}}(n)-\\dfrac{\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{12\\sqrt{2}C_{5}(n)},$ to $p(n)$ is also very small when $n<10$ (even in the cases when $n$ = 1 or 2) as shown on Table REF on page REF .",
"This is the first time to obtain an estimation formula of $p(n)$ which can reach a very good accuracy even when $n<10$ .",
"Further more, if we get the round value of $R_{\\mathrm {h4}}(n)$ , [Rh4(n)]$R^{\\prime }_{\\mathrm {h4}}(n)$ The Hardy-Ramanujan's revised estimation formula 4.",
"REF $R^{\\prime }_{\\mathrm {h4}}(n)=\\left\\lfloor R_{\\mathrm {h4}}(n)-\\dfrac{\\pi \\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{12\\sqrt{2}C_{5}(n)}+\\dfrac{1}{2}\\right\\rfloor ,$ the relative error to error is even less, especially when $n=15$ or $1<n<12$ it reaches 0, as shown on Table REF on page REF .",
"The relative error is less than $1\\times 10^{-9}$ when $2500<n<10000$ , as shown on Figure REF on page REF .",
"That is much better than $R_{\\mathrm {h3}}(n)$ and $R^{\\prime }_{\\mathrm {h3}}(n)$ , besides, it is more simple.",
"This formula will be called Hardy-Ramanujan's revised estimation formula 4.",
"Table: The relative error of R h4 ' (n)R^{\\prime }_{\\mathrm {h4}}(n) to p(n)p(n) when n⩽30n\\leqslant 30.When n⩾22n\\geqslant 22, the relative error differs very little."
],
[
"Estimate $p(n)$ When {{formula:21910f81-39c2-4db5-9eaa-74868471490c}} ",
"Until now, all the estimation function generated for $p(n)$ are with very good accuracy when $n$ is greater than 100, but they are not so accurate when $n<50$ .",
"Although $R^{\\prime }_{\\mathrm {h2}}(n)$ and $R^{\\prime }_{\\mathrm {h4}}(n)$ are better than others, the relative error are still greater than 1 for some values of $n$ .",
"On the other hand, in sections and , when $n<100$ , it is nearly impossible to fit $C_{1}\\left(n\\right)\\doteq \\dfrac{3}{2}\\cdot \\dfrac{\\left(\\ln \\left(4n\\sqrt{3}p(n)\\right)\\right)^{2}}{\\pi ^{2}}-n$ or $C_{2}(n)\\doteq \\dfrac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}(n)}\\right)}{4\\sqrt{3}p(n)}-n$ by a simple piecewise function with less than 4 pieces with high accuracy, as shown on Figure REF , Figure REF (on page REF ) and Figure REF (on page REF ), since the points do not lie on less than 4 smooth simple curves.",
"Can we reach a better accuracy when estimating $p(n)$ by a formula not too complicated?",
"In subsection REF , we fit the data $\\left(n,\\ \\frac{\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)}{4\\sqrt{3}p(n)}-n\\right)$ ($n=20k+100$ , $k$ = 1, 2, $\\cdots $ , 395) by a function $C_{2}(n)$ and obtained a very good estimation of $p(n)$ when $n$ > 50.",
"Figure: The graph of the data n,exp2 3πn+C 1 (n) 43p(n)-n\\left(n,\\,\\frac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n+C_{1}(n)}\\right)}{4\\sqrt{3}p(n)}-n\\right)when nn ⩽\\leqslant 100So we wander whether we can fit the data $\\left(n,\\ \\frac{\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)}{4\\sqrt{3}p(n)}-n\\right)$ ($n$ = 3, 4, $\\cdots $ , 100) by a piecewise function (with 2 pieces) so as to get a better estimation of $p(n)$ when $n$ $\\leqslant $ 100?",
"The figure of the points of the data $\\left(n,\\ \\frac{\\exp \\left(\\pi \\sqrt{\\frac{2}{3}n}\\right)}{4\\sqrt{3}p(n)}-n\\right)$ ($n$ = 3, 4, $\\cdots $ , 100) are shown on Figure REF (on page REF ).",
"It is not difficult to find that the even points (where $n$ is even) lie roughly on a smooth curve, so are the odd points.",
"If we try to fit them respectively, we will have the fitting function below: $C^{\\prime }_{2}(n)={\\left\\lbrace \\begin{array}{ll}0.4527092482\\times \\sqrt{n+4.35278}-\\\\\\quad \\quad 0.05498719946,\\\\\\quad \\quad \\quad \\quad \\quad \\quad \\quad n=3,5,7,\\cdots ,99;\\\\0.4412187317\\times \\sqrt{n-2.01699}+\\\\\\quad \\quad 0.2102618735,\\\\\\quad \\quad \\quad \\quad \\quad \\quad \\quad n=4,6,8\\cdots ,100.\\end{array}\\right.",
"}$ Hence we can calculate $p(n)$ by $R_{\\mathrm {h0}}(n)=\\dfrac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{4\\sqrt{3}\\left(n+C^{\\prime }_{2}(n)\\right)},\\quad 1\\leqslant n\\leqslant 100.$ Consider that $p(n)$ is an integer, we can take the round approximation of (REF ), [Rh0(n)]$R^{\\prime }_{\\mathrm {h0}}(n)$ The Hardy-Ramanujan's revised estimation formula when $1 \\leqslant n \\leqslant 100$ .",
"$R^{\\prime }_{\\mathrm {h0}}(n)=\\left\\lfloor \\dfrac{\\exp \\left(\\sqrt{\\frac{2}{3}}\\pi \\sqrt{n}\\right)}{4\\sqrt{3}\\left(n+C^{\\prime }_{2}(n)\\right)}+\\dfrac{1}{2}\\right\\rfloor ,\\quad 1\\leqslant n\\leqslant 100.$ The relative error of $R_{\\mathrm {h0}}(n)$ (or $R^{\\prime }_{\\mathrm {h0}}(n)$ ) to $p(n)$ are shown on Table REF (or Table REF ) on page REF .",
"Compared with Table REF on page REF , we will find that when $n\\geqslant 80$ , $R^{\\prime }_{\\mathrm {h2}}(n)$ is more accurate than $R^{\\prime }_{\\mathrm {h0}}(n)$ ; when $n\\leqslant 50$ , $R^{\\prime }_{\\mathrm {h0}}(n)$ is obviously better.",
"Table: The relative error of R h0 ' (n)R^{\\prime }_{\\mathrm {h0}}(n) to p(n)p(n) when n⩽30n\\leqslant 30.When n⩾26n\\geqslant 26, the relative error differs very little."
],
[
"Conclusions",
"In this paper, we have presented several elementary estimation formulae with high accuracy to calculated $p(n)$ , that can be operated on a pocket science calculator without programming function.",
"When $n\\leqslant 80$ , we can use $R^{\\prime }_{\\mathrm {h0}}(n)$ (Equation (REF )) , with a relative error less than 0.004%; when $n>80$ , we can use $R^{\\prime }_{\\mathrm {h2}}(n)$ (Equation (REF )).",
"Equations (REF ), (REF ) and (REF ) are also very accurate although they are not as good as (REF ).",
"By the construction of these estimation formulae, when $n$ $\\rightarrow $ $\\infty $ , the relative error will approaches 0.",
"(But the absolute error may approaches infinity).",
"If we can find the accurate expression $\\ $ such as in the form $\\pi ^{a/b}$ ($a,b\\in \\mathbb {Z}$ , $ab\\ne 0$ ) or $\\pi ^{a}\\mathrm {e}^{b}$ ($a,b\\in \\mathbb {Z}$ ).",
"of the coefficients $a_{2}\\doteq 1.039888529$ in (REF ), $t_{0}\\doteq 0.3594143172$ in (REF ) and $a_{3}\\doteq 2.893270736$ in (REF ), and can find the explanation in theory, we may gain better results.",
"The ideas described here could be used to acquire elementary estimation formulae in some other cases when approximate values are frequently wanted while the asymptotic formulae are less accurate than expectation and the methods to calculate the exact values are inconvenient, such as the computation of some kinds of restricted partition numbers if we have ( or can deduce) the asymptotic formulae beforehand.",
"These methods to fitting $C_{1}(n)$ and $C_{2}(n)$ could also be used in searching for the fitting functions of some classes of data obtain in experiments if we want more accuracy."
],
[
"Acknowledgements",
"tocsectionAcknowledgements The author would like to express the gratitude to his supervisor Prof. LI Shangzhi from Beihang University (in China) for his help and encouragement.",
"Special thanks must be given to Prof. Ian M. Wanless from Monash University (in Australia), Prof. Jack H. Koolen and Prof. WANG Xinmao from USTC (in China), Dr. ZHANG Zhe from Xidian University and Dr. WANG Qihan from Anhui University (in China) for their valuable advice.",
"tocsectionBibliography$\\ $"
]
] | 1612.05526 |
[
[
"A modulation technique for the blow-up profile of the vector-valued\n semilinear wave equation"
],
[
"Abstract We consider a vector-valued blow-up solution with values in $\\mathbb{R}^m$ for the semilinear wave equation with power nonlinearity in one space dimension (this is a system of PDEs).",
"We first characterize all the solutions of the associated stationary problem as an m-parameter family.",
"Then, we show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic cases.",
"Our analysis is not just a simple adaptation of the already handled real or complex case.",
"In particular, there is a new structure of the set a stationary solutions."
],
[
"Introduction",
"We consider the vector-valued semilinear wave equation $ \\left\\lbrace \\begin{array}{l}\\displaystyle \\partial ^2_{t} u = \\partial ^2_{x} u+|u|^{p-1}u, \\\\u(0)=u_{0} \\mbox{ and } u_{t}(0) = u_{1},\\end{array}\\right.",
"$ where here and all over the paper $|.|$ is the euclidian norm in $\\mathbb { R}^m$ , $u(t): x\\in \\mathbb {R}\\rightarrow u(x,t) \\in \\mathbb {R}^m,\\, m\\ge 2,\\,p>1$ , $u_0 \\in H^1_{loc,u}$ and $ u_1\\in L^2_{loc,u}$ with $||v||^2_{L^2_{loc,u}}=\\displaystyle \\sup \\limits _{a\\in \\mathbb { R}}\\int _{|x-a|<1}|v(x)|^2 dx $ and $|| v||^2_{H^1_{loc,u}}=||v||^2_{L^2_{loc,u}}+|| \\nabla v||^2_{L^2_{loc,u}}\\cdot $ The Cauchy problem for equation (REF ) in the space $H^1_{loc,u}\\times L^2_{loc,u}$ follows from the finite speed of propagation and the wellposedness in $H^1\\times L^2$ .",
"See for instance Ginibre, Soffer and Velo [9], Ginibre and Velo [10], Lindblad and Sogge [14] (for the local in time wellposedness in $H^1\\times L^2$ ).",
"Existence of blow-up solutions follows from ODE techniques or the energy-based blow-up criterion of [13].",
"More blow-up results can be found in Caffarelli and Friedman [6], Alinhac [1] and [2], Kichenassamy and Littman [12], [11] Shatah and Struwe [25]).",
"The real case (in one space dimension) has been understood completely, in a series of papers by Merle and Zaag [18], [19], [21] and [22] and in Côte and Zaag [7] (see also the note [20]).",
"Recently, the authors give an extension to higher dimensions in [24] and [23], where the blow-up behavior is given, together with some stability results.",
"For other types of nonlinearities, we mention the recent contribution of Azaiez, Masmoudi and Zaag in [5], where we study the semilinear wave equation with exponential nonlinearity, in particular we give the blow-up rate with some estimations.",
"In [4], we consider the complex-valued solution of (REF ) (or $\\mathbb {R}^2$ -valued solution), characterize all stationary solutions and give a trapping result.",
"The main obstruction in extending those results to the vector case $m\\ge 3$ was the question of classification of all self similar solutions of (REF ) in the energy space.",
"In this paper we solve that problem and show that the real valued and complex valued classification also hold in the vector-valued case $m\\ge 3$ (see Proposition REF below), with an adequate choice in $S^{m-1}$ .",
"This is in fact our main contribution in this paper, and it allows us to generalize the results of the complex case to the vector valued case $m\\ge 3$ .",
"In this paper, we aim at proving similar results for the general case $u(x,t) \\in \\mathbb {R}^m,$ for $m\\ge 3$ .",
"Let us first introduce some notations before stating our results.",
"If u is a blow-up solution of (REF ), we define (see for example Alinhac [1]) a continuous curve $\\Gamma $ as the graph of a function ${x \\rightarrow T(x)}$ such that the domain of definition of $u$ (or the maximal influence domain of $u$ ) is $D_u=\\lbrace (x,t)| t<T(x)\\rbrace .$ From the finite speed of propagation, $T$ is a 1-Lipschitz function.",
"The time $\\bar{T}=\\inf _{x \\in R}T(x)$ and the graph $\\Gamma $ are called (respectively) the blow-up time and the blow-up graph of $u$ .",
"Let us introduce the following non-degeneracy condition for $\\Gamma $ .",
"If we introduce for all $x \\in {R},$ $t\\le T(x)$ and $\\delta >0$ , the cone $\\mathcal {C}_{x,t, \\delta }=\\lbrace (\\xi ,\\tau )\\ne (x,t)\\,| 0\\le \\tau \\le t-\\delta |\\xi -x|\\rbrace ,$ then our non-degeneracy condition is the following: $x_0$ is a non-characteristic point if $\\exists \\delta = \\delta (x_0) \\in (0,1) \\mbox{ such that } u \\mbox{ is defined on }\\mathcal {C}_{x_0,T(x_0), \\delta _0}.$ If condition (REF ) is not true, then we call $x_0$ a characteristic point.",
"Already when $u$ is real-valued, we know from [21] and [7] that there exist blow-up solutions with characteristic points.",
"Given some $x_0 \\in R,$ we introduce the following self-similar change of variables: $w_{x_0}(y,s) =(T(x_0)-t)^\\frac{2}{p-1}u(x,t), \\quad y=\\frac{x-x_0}{T(x_0)-t}, \\quad s=-\\log (T(x_0)-t).$ This change of variables transforms the backward light cone with vertex $(x_0, T(x_0))$ into the infinite cylinder $(y,s)\\in (-1,1) \\times [-\\log T(x_0),+\\infty ).$ The function $w_{x_0}$ (we write $w$ for simplicity) satisfies the following equation for all $|y|<1$ and $s\\ge -\\log T(x_0)$ : $ \\partial ^2_{s} w=\\mathcal {L}w-\\frac{2(p+1)}{(p-1)^2}w+|w|^{p-1}w-\\frac{p+3}{p-1} \\partial _s w- 2 y \\partial _{ys} w$ $\\mbox{where }\\mathcal {L} w=\\frac{1}{\\rho }\\partial _y (\\rho (1-y^2)\\partial _y w)\\, \\mbox{ and }\\, \\rho (y)= (1-y^2)^\\frac{2}{p-1}.$ This equation will be studied in the space $\\mathcal { H}=\\lbrace q \\in H_{loc}^1 \\times L_{loc}^2 ((-1,1),\\mathbb {R}^m) \\, \\Big | \\parallel q \\parallel _{\\mathcal { H}}^2 \\equiv \\int _{-1}^{1}(|q_1|^2+|q^{\\prime }_1|^2(1-y^2)+|q_2|^2)\\rho \\;dy< +\\infty \\rbrace ,$ which is the energy space for $w$ .",
"Note that $\\mathcal { H}=\\mathcal { H}_0\\times L_{\\rho }^2$ where $\\mathcal { H}_0=\\lbrace r \\in H_{loc}^1 ((-1,1),\\mathbb {R}^m) \\,\\Big | \\parallel r \\parallel _{\\mathcal { H}_0}^2 \\equiv \\int _{-1}^{1}(|r^{\\prime }|^2(1-y^2)+|r|^2)\\rho \\;dy< +\\infty \\rbrace .$ In some places in our proof and when this is natural, the notation $\\mathcal { H}$ , $\\mathcal { H}_0$ and $L_{\\rho }^2$ may stand for real-valued spaces.",
"Let us define $E(w,\\partial _s w)=\\int _{-1}^{1} \\left( \\frac{1}{2} |\\partial _s w|^2+\\frac{1}{2} |\\partial _y w|^2 (1-y^2)+\\frac{p+1}{(p-1)^2}|w|^2-\\frac{1}{p+1}|w|^{p+1}\\right) \\rho dy.$ By the argument of Antonini and Merle [3], which works straightforwardly in the vector-valued case, we see that $E$ is a Lyapunov functional for equation (REF )."
],
[
"Blow-up rate",
"Only in this subsection, the space dimension will be extended to any $N \\ge 1$ .",
"We assume in addition that $p$ is conformal or sub-conformal: $1<p\\le p_c \\equiv 1+\\frac{4}{N-1}.$ We recall that for the real case of equation (REF ), Merle and Zaag determined in [15] and [16] the blow-up rate for (REF ) in the region $\\lbrace (x,t)\\,|\\, t<\\bar{T}\\rbrace $ in a first step.",
"Then in [17], they extended their result to the whole domain of definition $\\lbrace (x,t)\\,|\\, t< T(x)\\rbrace $ .",
"In fact, the proof of [15], [16] and [17] is valid for vector-valued solutions, since the energy structure (see(REF )), which is the main ingredient of the proof, is preserved.",
"This is the growth estimate near the blow-up surface for solutions of equation (REF ).",
"Proposition 1 (Growth estimate near the blow-up surface for solutions of equation (REF )) If $u$ is a solution of (REF ) with blow-up surface $\\Gamma \\,:\\, \\lbrace x\\rightarrow T(x)\\rbrace ,$ and if $x_0\\in \\mathbb {R}^N$ is non-characteristic (in the sense (REF )) then, (i) (Uniform bounds on $w$ ) For all $s\\ge -\\log \\frac{T(x_0)}{4}$ : $||w_{x_0}(s)||_{H^1(B)}+||\\partial _s w_{x_0}(s)||_{L^2(B)}\\le K.$ (ii) (Uniform bounds on $u$ ) For all $t\\in [\\frac{3}{4}T(x_0),T(x_0))$ : $&(T(x_0)-t)^\\frac{2}{p-1}\\frac{||u(t)||_{L^2(B(x_0,T(x_0)-t))}}{T(x_0)-t)}\\\\&+(T(x_0)-t)^{\\frac{2}{p-1}+1}\\left( \\frac{||\\partial _t u(t)||_{L^2(B(x_0,T(x_0)-t))}}{(T(x_0)-t)^{N/2}}+\\frac{||\\nabla u(t)||_{L^2(B(x_0,T(x_0)-t))}}{(T(x_0)-t)^{N/2}}\\right) \\le K,$ where the constant $K$ depends only on $N,\\, p,$ and on an upper bound on $T(x_0),1/T(x_0)$ , $\\delta _0 (x_0)$ and the initial data in $H^1_{loc,u}\\times L^2_{loc, u}$ .",
"Blow-up profile This result is our main novelty.",
"In the following, we characterize the set of stationary solutions for vector-valued solutions.",
"Proposition 2 (Characterization of all stationary solutions of equation (REF ) in $\\mathcal { H}_0$ ).",
"(i) Consider $w \\in \\mathcal {H}_0$ a stationary solution of (REF ).",
"Then, either $w\\equiv 0$ or there exist $d\\in (-1,1)$ and $\\Omega \\in \\mathbb {S}^{m-1}$ such that $w(y)=\\Omega \\kappa (d,y)$ where $\\forall (d, y) \\in (-1,1)^2,\\;\\kappa (d,y)= \\kappa _0 \\frac{(1-d^2)^\\frac{1}{p-1}}{(1+dy)^\\frac{2}{p-1}} \\mbox{ and }\\kappa _0=\\left(\\frac{2(p+1)}{(p-1)^2}\\right)^\\frac{1}{p-1}.$ (ii) It holds that $E(0,0)=0 \\mbox{ and }\\forall d \\in (-1,1),\\,\\forall \\Omega \\in \\mathbb {S}^{m-1} ,\\, E( \\kappa (d,.",
")\\Omega ,0)=E(\\kappa _0,0)>0$ where $E$ is given by (REF ).",
"Thanks to the existence of the Lyapunov functional $E(w,\\partial _s w)$ defined in (REF ), we show that when $x_0$ is non-characteristic, then $w_{x_0}$ approaches the set of non-zero stationary solutions: Proposition 3 (Approaching the set of non-zero stationary solutions near a non-characteristic point) Consider $u$ a solution of (REF ) with blow-up curve $\\Gamma :\\lbrace x \\rightarrow T(x)\\rbrace .$ If $x_0 \\in R$ is non-characteristic, then: (A.i) $\\inf _{\\lbrace \\Omega \\in {S}^{m-1},\\; |d|<1\\rbrace }||w_{x_0}(.,s)- \\kappa (d,.",
")\\Omega ||_{H^1(-1,1)}+||\\partial _s w_{x_0}||_{L^2(-1,1)} \\rightarrow 0$ as $s \\rightarrow \\infty $ .",
"(A.ii) $E(w_{x_0}(s),\\partial _s w_{x_0}(s))\\rightarrow E(\\kappa _0,0)$ as $s \\rightarrow \\infty .$ We write the fundamental theorem of our paper: Theorem 4 (Trapping near the set of non-zero stationary solutions of (REF )) There exist positive $\\epsilon _0$ , $\\mu _0$ and $C_0$ such that if $w\\in C([s^*,\\infty ),\\mathcal {H})$ for some $s^*\\in \\mathbb {R}$ is a solution of equation (REF ) such that $ \\forall s \\ge s^*, E(w(s),\\partial _s w(s)) \\ge E(\\kappa _0,0),$ and $\\Big |\\Big |\\begin{pmatrix} w(s^*)\\\\\\partial _s w(s^*) \\end{pmatrix} -\\begin{pmatrix} \\kappa (d^*,.",
")\\Omega ^*\\\\0\\end{pmatrix} \\Big |\\Big |_{\\mathcal { H}}\\le \\epsilon ^*$ for some $d^* \\in (-1,1), \\Omega ^* \\in \\mathbb {S}^{m-1}$ and $\\epsilon ^*\\in (0,\\epsilon _0]$ , then there exists $d_{\\infty } \\in (-1,1)$ and $\\Omega ^\\infty \\in \\mathbb {S}^{m-1}$ such that $|\\arg \\tanh d_{\\infty }-\\arg \\tanh d^*|+|\\Omega _\\infty -\\Omega ^*| \\le C_0 \\epsilon ^* $ and for all $s\\ge s^*$ : $\\Big |\\Big |\\begin{pmatrix} w(s)\\\\\\partial _s w(s) \\end{pmatrix} -\\begin{pmatrix} \\kappa (d_\\infty ,.",
")\\Omega _{\\infty }\\\\0\\end{pmatrix} \\Big |\\Big |_{\\mathcal { H}}\\le C_0\\epsilon ^* e^{-\\mu _0(s-s^*)}.$ Combining Proposition REF and Theorem REF , we derive the existence of a blow-up profile near non-characteristic points in the following: Theorem 5 (Blow-up profile near a non-characteristic point) If $u$ a solution of (REF ) with blow-up curve $\\Gamma :\\lbrace x \\rightarrow T(x)\\rbrace $ and $x_0 \\in R$ is non-characteristic (in the sense (REF )), then there exist $d_\\infty (x_0) \\in (-1,1),\\, \\Omega ^\\infty (x_0)\\in \\mathbb {S}^{m-1}$ and $s^*(x_0) \\ge -\\log T(x_0)$ such that for all $s\\ge s^*(x_0),$ (REF ) holds with $\\epsilon ^*=\\epsilon _0$ , where $C_0$ and $\\epsilon _0$ are given in Theorem REF .",
"Moreover, $||w_{x_0}(s)- \\kappa (d_\\infty (x_0))\\Omega ^\\infty (x_0)||_{H^1(-1,1)}+||\\partial _s w_{x_0}(s)||_{L^2(-1,1)} \\rightarrow 0 \\mbox{ as } s \\rightarrow \\infty .$ Remark: From the Sobolev embedding, we know that the convergence takes place also in $L^\\infty $ , in the sense that $||w_{x_0}(s)- \\kappa (d_\\infty (x_0))\\Omega ^\\infty (x_0)||_{L^\\infty (-1,1)} \\rightarrow 0 \\mbox{ as } s \\rightarrow \\infty .$ In this paper, we give the proofs of Proposition REF and Theorem REF , which present the novelties of this work comparing with the handled real and complex cases, since Propositions REF , REF and Theorem REF , can be generalized from the real case treated in [18] without any difficulty.",
"Let us remark that our paper is not a simple adaptation of the complex case.",
"In fact, the vector-valued structure of our solution implies a new characterization of the set of stationary solutions in ${R}^m$ (see Proposition REF above).",
"In addition, in order to apply the modulation theory, we need more parameters, and for that, a suitable $ m\\times m$ rotation matrix will be defined (see (REF ) and (REF ) below; see the beginning of the proof of Proposition REF page REF below), and we have to treat delicately the terms coming from the rotation matrix.",
"This paper is organized as follows: - In Section 2, we give the proof of Proposition REF .",
"- In Section 3, we give the proof of Theorem REF .",
"Characterization of the set of stationary solutions In this section, we prove Proposition REF which characterizes all $\\mathcal {H}_0$ solutions of $\\frac{1}{\\rho } (\\rho (1-y^2) w^{\\prime })^{\\prime }-\\frac{2(p+1)}{(p-1)^2}w+|w|^{p-1}w=0,$ the stationary version of (REF ).",
"Note that since 0 and $\\kappa _0 \\Omega $ are trivial solutions to equation (REF ) for any $\\Omega \\in S^{m-1}$ , we see from a Lorentz transformation (see Lemma 2.6 page 54 in [18]) that $\\mathcal {T}_d e^{i\\theta }\\kappa _0=\\kappa (d,y)$ is also a stationary solution to (REF ).",
"Let us introduce the set $S\\equiv \\lbrace 0, \\kappa (d,.",
")\\Omega , |d|<1, \\Omega \\in S^{m-1}\\rbrace .$ Now, we prove Proposition REF which states that there are no more solutions of (REF ) in $\\mathcal {H}_0$ outside the set $S$ .",
"We first prove $(ii)$ , since its proof is short.",
"(ii) Since we clearly have from the definition (REF ) that $E(0,0)=0$ , we will compute $E(\\Omega \\kappa (d,.",
"),0)$ .",
"From (REF ) and the proof of the real case treated in page 59 in [18], we see that $ E(\\kappa (d,.",
")\\Omega ,0)= E(\\kappa (d,.",
"),0)=E(\\kappa _0,0)>0.$ Thus, (REF ) follows.",
"(i) Consider $w \\in \\mathcal {H}_0$ an $ \\mathbb {R}^m$ non-zero solution of (REF ).",
"Let us prove that there are some $d\\in (-1,1)$ and $\\Omega \\in \\mathbb {S}^{m-1}$ such that $w=\\kappa (d,.",
")\\Omega .$ For this purpose, define $\\xi =\\frac{1}{2} \\log \\left(\\frac{1+y}{1-y}\\right) (\\mbox{that is } y=\\tanh \\xi )\\mbox{ and } \\bar{w}( \\xi )=w(y) (1-y^2)^\\frac{1}{p-1}.$ As in the real case, we see from straightforward calculations that $\\bar{w}\\lnot \\equiv 0$ is a $H^1({R})$ solution to $\\partial _\\xi ^2\\bar{w} +|\\bar{w}|^{p-1}\\bar{w}-\\frac{4}{(p-1)^2}\\bar{w}=0,\\,\\forall \\xi \\in {R}.$ Our aim is to prove the existence of $\\Omega \\in \\mathbb {S}^{m-1}$ and $\\xi _0 \\in \\mathbb {R}$ such that $\\bar{w}(\\xi )=\\Omega \\bar{k} (\\xi +\\xi _0)$ where $\\bar{k}(\\xi )=\\frac{\\kappa _0}{\\cosh ^\\frac{2}{p-1}(\\xi )}.$ Since $\\bar{w} \\in H^1(R) \\subset C^\\frac{1}{2}(R),$ we see that $\\bar{w}$ is a strong $C^2$ solution of equation (REF ).",
"Since $\\bar{w}\\lnot \\equiv 0$ , there exists $\\xi _0 \\in \\mathbb {R}$ such that $\\bar{w}(\\xi _0)\\ne 0$ .",
"By invariance of (REF ) by translation, we may suppose that $\\xi _0=0$ .",
"Let $G^*=\\left\\lbrace \\xi \\in \\mathbb {R} \\,|\\, \\bar{w}(\\xi )\\ne 0 \\right\\rbrace ,$ a nonempty open set by continuity.",
"Note that $G^*$ contains some non empty interval $I$ containing 0.",
"We introduce $\\rho $ and $\\Omega $ by $\\rho =|\\bar{w}|,\\,\\Omega =\\frac{\\bar{w}}{|\\bar{w}|},\\mbox{ whenever }\\xi \\in G^*.$ From equation (REF ), we see that $\\rho ^{\\prime \\prime }\\Omega +2 \\rho ^{\\prime } \\Omega ^{\\prime }+ \\rho \\Omega ^{\\prime \\prime }+\\rho ^p\\Omega - \\frac{4}{(p-1)^2} \\rho \\Omega =0.$ Now, since $|\\Omega |=1$ , we immediately see that $ \\Omega ^{\\prime }.",
"\\Omega =0$ and $ \\Omega ^{\\prime \\prime }.",
"\\Omega +|\\Omega ^{\\prime }|^2=0$ .",
"Let $H(\\xi )=|\\Omega ^{\\prime }|^2$ .",
"Projecting equation (REF ) according to $\\Omega $ and $\\Omega ^{\\prime }$ we see that $\\forall \\xi \\in G^*,\\,\\left\\lbrace \\begin{array}{l}\\rho ^{\\prime \\prime }(\\xi )-\\rho (\\xi ) H(\\xi ) -c_0\\rho (\\xi )+\\rho (\\xi )^p=0,\\; c_0= \\frac{4}{(p-1)^2}\\\\4\\rho ^{\\prime }(\\xi )H(\\xi )+\\rho (\\xi ) H^{\\prime }(\\xi )=0\\end{array}\\right.$ Integrating the second equation on the interval $I\\subset G^*$ , we see that for all $\\xi \\in I$ , $H(\\xi )=\\frac{H(0)(\\rho (0))^4}{(\\rho (\\xi ))^4}$ .",
"Plugging this in the first equation, we get $\\forall \\xi \\in I,\\,\\rho ^{\\prime \\prime }(\\xi )-\\frac{\\mu }{(\\rho (\\xi ))^3}-c_0\\rho (\\xi )+\\rho ^p(\\xi )=0 \\mbox{ where }\\mu =H(0)(\\rho (0))^4.$ Now let $G= \\left\\lbrace \\xi \\in G^*, \\forall \\xi ^{\\prime }\\in I_\\xi ,\\, H(\\xi ^{\\prime })=\\frac{H(0)\\rho (0)^4}{\\rho (\\xi ^{\\prime })^4} \\right\\rbrace ,$ where $I_\\xi =[0,\\xi )$ if $\\xi \\ge 0$ or $I_\\xi =(\\xi ,0]$ if $\\xi \\le 0$ .",
"Note that $I \\subset G$ .",
"Now, we give the following: Lemma 2.1 There exists $\\epsilon _0 >0$ such that $\\forall \\xi \\in G, \\,\\forall \\xi ^{\\prime }\\in I_\\xi ,\\, 0< \\epsilon _0 \\le |\\bar{w}(\\xi ^{\\prime })|\\le \\frac{1}{\\epsilon _0}.$ The proof is the same as in the complex-case, see page 5898 in [4].",
"But for the reader's convenience and for the sake of self-containedness, we recall it here.",
"Take $\\xi \\in G$ .",
"By definition (REF ) of $G$ , we see that equation (REF ) is satisfied for all $\\xi ^{\\prime }\\in I_\\xi $ .",
"Multiplying $\\rho ^{\\prime \\prime }(\\xi )-\\frac{\\mu }{(\\rho (\\xi ))^3}-c_0\\rho (\\xi )+\\rho ^p(\\xi )=0 $ by $\\rho ^{\\prime }$ and integrating between 0 and $\\xi $ , we get: $\\forall \\xi \\in I_\\xi ,\\, \\mathcal {E}(\\xi ^{\\prime })=\\mathcal {E}(0), \\mbox{ where }\\mathcal {E}(\\xi ^{\\prime })=\\frac{1}{2}(\\rho ^{\\prime }(\\xi ^{\\prime }))^2+\\frac{\\mu }{2( (\\rho (\\xi ^{\\prime }))^2}-\\frac{c_0}{2}\\rho ^2(\\xi ^{\\prime })+\\frac{\\rho ^{p+1}(\\xi ^{\\prime })}{p+1},$ or equivalently, $\\forall \\xi ^{\\prime } \\in I_\\xi ,\\, F(\\rho (\\xi ^{\\prime }))=\\frac{1}{2} \\rho ^{\\prime }(\\xi ^{\\prime })^2\\ge 0 \\mbox{ where }F(r)=\\frac{\\mu }{2r^2}+\\frac{c_0}{2} r^2-\\frac{r^{p+1}}{p+1}+\\mathcal {E}(0).$ Since $F(r)\\rightarrow -\\infty $ as $r \\rightarrow 0$ or $r \\rightarrow \\infty $ , there exists $\\epsilon _0=\\epsilon _0(\\mu , E(0)) >0$ such that $\\epsilon _0\\le \\rho (\\xi ^{\\prime }) \\le \\frac{1}{\\epsilon _0}$ , which yields to the conclusion of the Claim REF .",
"We claim the following: Lemma 2.2 It holds that $G=\\mathbb {R}$ .",
"Note first that by construction, $G$ is a nonempty interval (note that $0\\in I \\subset G$ where $I$ is defined right before (REF )).",
"We have only to prove that $\\sup G=+\\infty $ , since the fact that $\\inf G=-\\infty $ can be deduced by replacing $\\bar{w}(\\xi )$ by $\\bar{w}(-\\xi )$ .",
"By contradiction, suppose that $\\sup G=a<+\\infty $ .",
"First of all, by Lemma REF , we have for all $ \\xi ^{\\prime } \\in [0,a), 0< \\epsilon _0 \\le |\\bar{w}(\\xi ^{\\prime })|\\le \\frac{1}{\\epsilon _0}$ .",
"By continuity, this holds also for $\\xi ^{\\prime }=a$ , hence, $\\bar{w}(a)\\ne 0$ , and $a\\in G^*$ .",
"Furthermore, by definition of $G$ and continuity, we see that $\\forall \\xi \\in [0, a], H(\\xi )=\\frac{H(0)\\rho (0)^4}{\\rho (\\xi )^4}.$ Therefore, we see that $a\\in G$ .",
"By continuity, we can write for all $\\xi \\in (a-\\delta ,a+\\delta ),$ where $\\delta >0 $ is small enough, $ \\left\\lbrace \\begin{array}{l}\\rho ^{\\prime \\prime }(\\xi )-\\rho (\\xi )H(\\xi )-c_0\\rho (\\xi )+\\rho (\\xi )^p=0,\\; c_0= \\frac{4}{(p-1)^2}\\\\4\\rho ^{\\prime }(\\xi )H(\\xi )+\\rho (\\xi ) H^{\\prime }(\\xi )=0.\\end{array}\\right.$ From the second equation and (REF ) applied with $\\xi =a$ , we see that $H(\\xi )=\\frac{H(a)(\\rho (a))^4}{(\\rho (\\xi ))^4}=\\frac{H(0)(\\rho (0))^4}{(\\rho (\\xi ))^4}$ .",
"Therefore, it follows that $(a, a+\\delta )\\in G$ , which contradicts the fact that $a=\\sup G$ .",
"Note from Lemma REF that (REF ) and (REF ) holds for all $\\xi \\in {R}$ .",
"We claim that $H(0)=0$ .",
"Indeed, if not, then by (REF ), we have $\\mu \\ne 0$ , and since $G={R}$ , we see from Lemma REF that for all $\\xi \\in {R}$ , $|\\bar{w}(\\xi )|\\ge \\epsilon _0$ , therefore $w \\notin L^2({R})$ , which contradicts the fact that $\\bar{w}\\in H^1(\\mathbb {R})$ .",
"Thus, $H(0)=0$ , and $\\mu =0$ .",
"By uniqueness of solutions to the second equation of (REF ), we see that $H(\\xi )=0$ for all $\\xi \\in {R}$ , so $\\Omega (\\xi )=\\Omega (0)$ , and $ \\left\\lbrace \\begin{array}{l}\\bar{w}(0)=\\rho (0) \\Omega (0)\\\\\\bar{w}^{\\prime }(0)=\\rho ^{\\prime }(0) \\Omega (0).\\end{array}\\right.$ Let $W$ be the maximal real-valued solution of $ \\left\\lbrace \\begin{array}{l}W^{\\prime \\prime }-c_0 W+|W|^{p-1}W=0\\\\W(0)=\\rho (0)\\\\W^{\\prime }(0)=\\rho ^{\\prime }(0).\\end{array}\\right.$ By uniqueness of the Cauchy problem of equation (REF ), we have for all $\\xi \\in \\mathbb {R}, \\bar{w}(\\xi )=W(\\xi ) \\Omega (0)$ , and as $\\bar{w}\\in H^1(\\mathbb {R})$ , $W$ is also in $H^1(\\mathbb {R})$ .",
"It is then classical that there exists $\\xi _0$ such that for all $\\xi \\in \\mathbb {R}$ , $W(\\xi )=\\bar{k }(\\xi +\\xi _0)$ (remember that $\\rho (0)>0$ , hence we only select positive solutions here).",
"In addition, for $ \\Omega _0= \\Omega (0)$ , $\\bar{w}(\\xi )=\\bar{k }(\\xi +\\xi _0)\\Omega _0$ .",
"Thus, for $d=\\tanh \\xi _0\\,\\in \\, (-1,1)$ and $y=\\tanh \\xi $ , we get $&\\bar{w}(\\xi )=\\kappa _0 \\left[1-\\tanh (\\xi +\\xi _0)^2\\right]^\\frac{1}{p-1}\\Omega _0=\\kappa _0 \\left[1-\\left(\\frac{\\tanh \\xi +\\tanh \\xi _0}{1+\\tanh \\xi \\tanh \\xi _0}\\right)^2\\right]^\\frac{1}{p-1}\\Omega _0\\\\&= \\kappa _0 \\left[1-\\left(\\frac{y+d}{1+dy}\\right)^2\\right]^\\frac{1}{p-1}\\Omega _0= \\kappa _0 \\left[\\frac{(1-d^2)(1-y^2)}{(1+dy)}^2\\right]^\\frac{1}{p-1}\\Omega _0= \\kappa (d,y) (1-y^2)^\\frac{1}{p-1}\\Omega _0.$ By (REF ), we see that $w(y)= \\kappa (d,y)\\Omega _0$ .",
"This concludes the proof of Proposition REF .",
"Outline of the proof of Theorem REF The proof of Theorem REF is not a simple adaptation of the complex-case to the vector-valued case, in fact, it involves a delicate modulation.",
"In this section, we will outline the proof, insisting on the novelties, and only recalling the features which are the same as in the real-valued complex-valued cases.",
"This section is organized as follows: - In Subsection 3.1, we linearize equation (REF ) around $\\kappa (d,y)e_1$ where $e_1=(1,0,...,0)$ and figure-out that, with respect to the complex-valued case, our linear operator is just a superposition of one copy of the real part operator, with $(m-1)$ copies of the imaginary part operator.",
"- In Subsection 3.2, we recall from [18] the spectral properties of the real-part operator.",
"- In Subsection 3.3, we recall from [4] the spectral properties of the imaginary-part operator.",
"- In Subsection 3.4, assuming that $\\Omega ^*=e_1$ (possible thanks to rotation invariance of (REF )), we introduce a modulation technique adapted to the vector-valued case.",
"This part makes the originality of our work with respect to the complex-valued case.",
"- In Subsection 3.5, we write down the equations satsified by the modulation parameters along with the PDE satisfied by $q(y,s)$ and its components.",
"- In Subsection 3.6, we conclude the proof of Theorem REF .",
"The linearized operator around a non-zero stationary solution We study the properties of the linearized operator of equation (REF ) around the stationary solution $\\kappa (d,y)$ $(\\ref {defk})$ .",
"Let us introduce $q=(q_1,q_2)\\in \\mathbb {R}^m\\times \\mathbb {R}^m$ for all $ s\\in [s_0, \\infty )$ , for a given $s_0 \\in {R}$ , by $\\begin{pmatrix} w(y,s)\\\\\\partial _s w(y,s) \\end{pmatrix} =\\begin{pmatrix} \\kappa (d,y)e_1\\\\0\\end{pmatrix} +\\begin{pmatrix} q_1(y,s)\\\\q_2(y,s)\\end{pmatrix}.$ Let us introduce the coordinates of $q_1$ and $q_2$ by $q_1=(q_{1,1},q_{1,2},...,q_{1,m})$ , $q_2=(q_{2,1},q_{2,2},...,q_{2,m})$ .",
"We see from equation (REF ), that $q$ satisfies the following equation for all $s\\ge s_0$ : $ \\frac{\\partial }{\\partial s} \\begin{pmatrix} q_1\\\\q_2 \\end{pmatrix} = L_d\\begin{pmatrix}q_1\\\\q_2\\end{pmatrix}+\\begin{pmatrix}0\\\\ f_d(q_1)\\end{pmatrix},$ where $ L_d\\begin{pmatrix}q_1\\\\q_2\\end{pmatrix}=\\begin{pmatrix}q_2 \\\\\\mathcal {L}q_1+\\bar{\\psi }(d, y)q_{1,1}e_1+\\sum _{j=2}^{m} \\tilde{\\psi }(d, y)q_{1,j}e_j-\\frac{p+3}{p-1} q_2- 2 y \\partial _y q_2 \\end{pmatrix}, $ $\\bar{\\psi }(d,y) =p \\kappa (d, y)^{p-1}-\\frac{2(p+1)}{(p-1)^2}$ $\\tilde{\\psi }(d,y) = \\kappa (d, y)^{p-1}-\\frac{2(p+1)}{(p-1)^2}$ $f_d(q_1)={f_{d,1}}(q_1)e_1+ \\sum _{j=2}^{m} f_{d,j}(q_1)e_j,$ where $f_{d,1}(q_1)=|\\kappa (d, y)e_1+q_1|^{p-1}(\\kappa (d, y)+q_{1,1})-\\kappa (d, y)^{p}-p\\kappa ^{p-1}(d, y)q_{1,1}.$ $f_{d,j}( q_1)=|\\kappa (d, y)e_1+q_1|^{p-1}q_{1,j}-\\kappa ^{p-1}(d,y)q_{1,j}.$ Projecting (REF ) on the first coordinate, we get for all $s\\ge s_0$ : $\\frac{\\partial }{\\partial s}\\begin{pmatrix} q_{1,1}\\\\q_{2,1}\\end{pmatrix}= \\bar{L}_d \\begin{pmatrix} q_{1,1} \\\\q_{2,1} \\end{pmatrix}+\\begin{pmatrix}0\\\\ f_{d,1}(q_1)\\end{pmatrix},$ where $\\bar{L}_d $ is given by: $\\bar{L}_d \\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix}=\\begin{pmatrix}q_{2,1} \\\\\\mathcal {L} q_{1,1}+ \\bar{\\psi }(d,y) q_{1,1}-\\frac{p+3}{p-1} q_{2,1} - 2 y \\partial _y q_{2,1}\\end{pmatrix},$ Now, projecting equation (REF ) on the j-th coordinate with $j=2,..,m,$ we see that $\\frac{\\partial }{\\partial s}\\begin{pmatrix} q_{1,j}\\\\q_{2,j}\\end{pmatrix}= \\tilde{L}_d\\begin{pmatrix} q_{1,j} \\\\q_{2,j} \\end{pmatrix}+\\begin{pmatrix}0\\\\ f_{d,j}( q_1)\\end{pmatrix},$ where $\\tilde{L}_d \\begin{pmatrix} q_{1,j}\\\\q_{2,j} \\end{pmatrix}=\\begin{pmatrix}q_{2,j} \\\\\\mathcal {L} q_{1,j}+ \\tilde{\\psi }(d,y) q_{1,j}-\\frac{p+3}{p-1} q_{2,j} - 2 y \\partial _y q_{2,j}\\end{pmatrix},$ Remark: Our linearized operator $L_d$ is in fact diagonal in the sens that $ L_d\\begin{pmatrix}q_1\\\\q_2\\end{pmatrix}=\\bar{L}_d \\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix}e_1+ \\sum _{j=2}^{m} \\tilde{L}_d \\begin{pmatrix} q_{1,j}\\\\q_{2,j} \\end{pmatrix}e_j.$ We mention that for $j=1$ , equation (REF ) is the same as the equation satisfied by the real part of the solution in the complex case (see Section 3 page 5899 in [4]), whereas for $j=2,..,m$ , equation (REF ) is the same as the equation satisfied by the imaginary part of the solution operator in the complex case.",
"Thus, the reader will have no difficulty in adapting the remaining part of the proof to the vector-valued case.",
"Thus, the dynamical system formulation we performed when $m=2$ can be adapted straightforwardly to the case $m\\ge 3$ .",
"Note from (REF ) that we have $||q||_\\mathcal {H}=[\\phi (q, q)]^\\frac{1}{2}<+\\infty ,$ where the inner product $\\phi $ is defined by $\\phi (q, r)=\\phi \\left(\\begin{pmatrix} q_1\\\\q_2 \\end{pmatrix}, \\begin{pmatrix} r_1\\\\r_2 \\end{pmatrix}\\right)=\\int _{-1}^{1}(q_1.r_1+q^{\\prime }_1.",
"r^{\\prime }_1(1-y^2)+q_2.r_2)\\rho \\;dy.$ where $q_1.r_1=\\sum _{j=1}^m q_{1,j}.r_{1,j}$ is the standard inner product in $\\mathbb {R}^m$ , with similar expressions for $q^{\\prime }_1.r^{\\prime }_1$ and $q_2.r_2$ .",
"Using integration by parts and the definition of $\\mathcal {L}$ (REF ), we have the following: $\\phi (q, r)=\\int _{-1}^{1}(q_1\\cdot (-\\mathcal {L} r_1+ r_1)+q_2 \\cdot r_2)\\rho \\,dy.$ In the following two sections, we recall from [18] and [4] the spectral properties of $\\bar{L}_d$ and $\\tilde{L}_d$ .",
"Spectral theory of the operator $\\bar{L}_d$ From Section 4 in [18], we know that $\\bar{L}_d$ has two nonnegative eigenvalues $\\lambda =1$ and $\\lambda =0$ with eigenfunctions $\\bar{F}_1^d (y)= (1-d^2)^{\\frac{p}{p-1}}\\begin{pmatrix}(1+dy)^{-\\frac{p+1}{p-1}}\\\\(1+dy)^{-\\frac{p+1}{p-1}}\\end{pmatrix}\\mbox{and }\\;\\bar{F}_0^d (y)= (1-d^2)^{\\frac{1}{p-1}}\\begin{pmatrix}\\frac{y+d}{(1+dy)^\\frac{p+1}{p-1}}\\\\0\\end{pmatrix}.",
"$ Note that for some $C_0>0$ and any $\\lambda \\in \\lbrace 0,1\\rbrace $ , we have $\\forall |d|<1,\\;\\; \\frac{1}{C_0}\\le ||\\bar{F}_\\lambda ^d||_{\\mathcal {H}} \\le C_0 \\;\\mbox{ and }\\; ||\\partial _d \\bar{F}_\\lambda ^d||_{\\mathcal {H}} \\le \\frac{C_0}{1-d^2}.$ Also, we know that $\\bar{L}_d^*$ the conjugate operator of $\\bar{L}_d$ with respect to $\\phi $ is given by $\\bar{L}_d^*\\begin{pmatrix}r_1\\\\r_2\\end{pmatrix}= \\begin{pmatrix}\\bar{R}_d(r_2)\\\\-\\mathcal {L} r_1+r_1+\\frac{p+3}{p-1}r_2+2y r_2^{\\prime }-\\frac{8}{(p-1)}\\frac{r_2}{(1-y^2)}\\end{pmatrix}$ for any $(r_1, r_2)\\in (\\mathcal {D}(\\mathcal {L}))^2$ , where $r=\\bar{R}_d (r_2)$ is the unique solution of $-\\mathcal {L} r+r=\\mathcal {L} r_2+\\bar{\\psi } (d, y)r_2.$ Here, the domain $\\mathcal {D}(\\mathcal {L})$ of $\\mathcal {L}$ defined in (REF ) is the set of all $r \\in L_{\\rho }^2$ such that $\\mathcal {L} r \\in L_{\\rho }^2.$ Furthermore, $\\bar{L}_d^*$ has two nonnegative eigenvalues $\\lambda =0$ and $\\lambda =1$ with eigenfunctions $\\bar{W}_{\\lambda }^d$ such that $\\bar{W}_{1, 2}^d (y)= \\bar{c}_1 \\frac{(1-y^2)(1-d)^\\frac{1}{p-1}}{(1+dy)^\\frac{p+1}{p-1}},\\,\\bar{W}_{0, 2}^d (y)= \\bar{c}_0 \\frac{(y+d)(1-d)^\\frac{1}{p-1}}{(1+dy)^\\frac{p+1}{p-1}},$ withIn section 4 of [18], we had non explicit normalizing constants $\\bar{c}_\\lambda =\\bar{c}_\\lambda (d)$ .",
"In Lemma 2.4 in [24], the authors compute the explicit dependence of $\\bar{c}_\\lambda (d)$ .",
"$\\frac{1}{\\bar{c}_\\lambda }=2(\\frac{2}{p-1}+\\lambda )\\int _{-1}^1 (\\frac{y^2}{1-y^2} )^{1-\\lambda }\\rho (y) \\,dy,$ and $\\bar{W}_{\\lambda , 1}^d $ is the unique solution of the equation $-\\mathcal {L} r+ r =\\left(\\lambda -\\frac{p+3}{p-1}\\right) r_2- 2 y r^{\\prime }_2 + \\frac{8}{p-1} \\frac{r_2}{1-y^2}$ with $r_2= \\bar{W}_{\\lambda , 2}^d$ .",
"We also have for $\\lambda =0,1$ $||\\bar{W}_\\lambda ^d ||_{\\mathcal {H}}+ (1-d^2)||\\partial _d \\bar{W}_\\lambda ^d ||_{\\mathcal {H}} \\le C, \\forall |d|<1.$ Note that we have the following relations for $\\lambda =0$ or $\\lambda =1$ $\\phi (\\bar{W}_\\lambda ^d,\\bar{F_\\lambda ^d})=1\\mbox{ and }\\phi (\\bar{W}_\\lambda ^d,\\bar{F_{1-\\lambda } ^d})=0.$ Let us introduce for $\\lambda \\in \\lbrace 0, 1\\rbrace $ the projectors $\\bar{\\pi }_\\lambda (r)$ , and $\\bar{\\pi }_-^d(r)$ for any $r\\in \\mathcal {H}$ by $ \\bar{\\pi }_\\lambda ^d (r)=\\phi (\\bar{W}_\\lambda ^d, r),$ $ r=\\bar{\\pi }_0^d (r) \\bar{F}_0^d (y)+\\bar{\\pi }_1^d (r) \\bar{F}_1^d (y)+ \\bar{\\pi }_{-}^d (r),$ and the space $\\bar{\\mathcal {H}}_{-}^d \\equiv \\lbrace r \\in \\mathcal {H} \\,|\\, \\bar{\\pi }_1^d (r)=\\bar{\\pi }_0^d(r)=0\\rbrace .$ Introducing the bilinear form $\\bar{\\varphi }_{d} (q,r)&=&\\int _{-1}^{1} (-\\bar{\\psi }(d,y) q_1r_1+q_1^{\\prime } r_1^{\\prime }(1-y^2)+q_2 r_2 ) \\rho dy,$ where $\\bar{\\psi }(d,y)$ is defined in (REF ), we recall from Proposition 4.7 page 90 in [18] that there exists $C_0>0$ such that for all $|d|<1$ , for all $r\\in \\bar{\\mathcal {H}}^d_{-},$ $\\frac{1}{C_0} ||r||_{\\mathcal {H} }^2\\le \\bar{\\varphi }_d (r,r)\\le C_0 ||r||_\\mathcal {H} ^2.$ Furthermore, if $r\\in \\mathcal {H},$ then $\\frac{1}{C_0} ||r||_{\\mathcal {H} }\\le \\left(|\\bar{\\pi }_{0}^d (r)| +|\\bar{\\pi }_{1}^d (r)| +\\sqrt{ \\bar{\\varphi }_d (r_{-},r_{-})}\\right)\\le C_0 ||r||_\\mathcal {H}\\mbox{ where }r_{-}=\\bar{\\pi }_{-}^d (r) .$ In the following section we recall from [4] the spectral properties of $\\tilde{L}_d$ .",
"Spectral theory of the operator $\\tilde{L}_d$ From Section 3 in [4], we know that $\\tilde{L}_d$ has one nonnegative eigenvalue $\\lambda =0$ with eigenfunction $\\tilde{F}_0^d(y)= \\begin{pmatrix}\\kappa (d,y)\\\\0\\end{pmatrix}.$ Note that for some $C_0>0$ we have $\\forall |d|<1,\\;\\; \\frac{1}{C_0}\\le ||\\tilde{F}_0^d||_{\\mathcal {H}} \\le C_0 \\;\\mbox{ and }\\; ||\\partial _d \\tilde{F}_0^d||_{\\mathcal {H}} \\le \\frac{C_0}{1-d^2}.$ We know also that the operator $\\tilde{L}_d^*$ conjugate of $\\tilde{L}_d$ with respect to $\\phi $ is given by $\\tilde{L}_d^*\\begin{pmatrix}r_1\\\\r_2\\end{pmatrix}= \\begin{pmatrix}\\tilde{R}_d(r_2)\\\\-\\mathcal {L} r_1+r_1+\\frac{p+3}{p-1}r_2+2y r_2^{\\prime }-\\frac{8}{(p-1)}\\frac{r_2}{(1-y^2)}\\end{pmatrix}$ for any $(r_1, r_2)\\in (\\mathcal {D}(\\mathcal {L}))^2$ , where $r=\\tilde{R}_d (r_2)$ is the unique solution of $-\\mathcal {L} r+r=\\mathcal {L} r_2+\\tilde{\\psi } (d, y)r_2.",
"$ Furthermore, $\\tilde{L}_d^*$ have one nonnegative eigenvalue $\\lambda =0$ with eigenfunction $\\tilde{W}_0^d$ such that $\\tilde{W}_{0, 2}^d (y)= \\tilde{c}_0 \\kappa (d, y) \\mbox{ and }\\frac{1}{\\tilde{c}_0}=\\frac{4\\kappa _0^2}{p-1} \\int _{-1}^1 \\frac{\\rho (y)}{1-y^2}dy$ and $\\tilde{W}_{0, 1}^d $ is the unique solution of the equation $-\\mathcal {L} r+ r =-\\frac{p+3}{p-1} r_2- 2 y r^{\\prime }_2 + \\frac{8}{p-1} \\frac{r_2}{1-y^2}$ with $r_2= \\tilde{W}_{0, 2}^d$ .",
"We also have for $\\lambda =0,1$ $||\\tilde{W}_0 ^d ||_{\\mathcal {H}}+ (1-d^2)||\\partial _d \\tilde{W}_\\lambda ^d ||_{\\mathcal {H}} \\le C,\\; \\forall |d|<1.$ Moreover, we have $\\phi (\\tilde{W}_0^d,\\tilde{F_0^d})=1.$ Let us introduce the projectors $\\tilde{\\pi }_0^d (r)$ and $ \\tilde{\\pi }_{-}^d (r))$ for any $r \\in \\mathcal {H}$ by $ \\tilde{\\pi }_0^d (r)=\\phi (\\tilde{W}_0^d, r),$ $ r=\\tilde{\\pi }_0^d (r) \\tilde{F}_0^d (y)+ \\tilde{\\pi }_{-}^d (r).$ and the space $\\tilde{\\mathcal {H}}_{-}^d \\equiv \\lbrace r \\in \\mathcal {H} \\,|\\, \\tilde{\\pi }_{0}^d (r)=0\\rbrace .$ Introducing the bilinear form $\\tilde{\\varphi }_{d} (q,r)&=&\\int _{-1}^{1} (-\\tilde{\\psi }(d,y) q_1r_1+q_1^{\\prime } r_1^{\\prime }(1-y^2)+q_2 r_2 ) \\rho dy,\\\\$ where $\\tilde{\\psi }(d,y) $ is defined in (REF ), we recall from Proposition $3.7$ page 5906 in [4] that there exists $C_0 >0$ such that for all $|d|<1,$ for all $r\\in \\tilde{\\mathcal {H}}^d_{-},$ $\\frac{1}{C_0} ||r||_{\\mathcal {H} }^2\\le \\tilde{\\varphi }_d (r,r)\\le C_0 ||r||_\\mathcal {H} ^2.$ A modulation technique We start the proof of Theorem REF here.",
"Let us consider $w\\in C ([s^*,\\infty ), \\mathcal {H})$ for some $s^*\\in \\mathbb {R}$ a solution of equation (REF ) such that $\\forall s\\ge s^*, E(w(s),\\partial _s w(s))\\ge E(\\kappa _0,0)$ and $\\Big |\\Big |\\begin{pmatrix} w(s^*)\\\\\\partial _s w(s^*) \\end{pmatrix} -\\begin{pmatrix} \\kappa (d^*,.",
")\\Omega ^*\\\\0\\end{pmatrix} \\Big |\\Big |_{\\mathcal { H}}\\le \\epsilon ^*$ for some $d^*\\in (-1,1)$ , $\\Omega ^*\\in \\mathbb {S}^{m-1}$ and $\\epsilon ^*>0$ to be chosen small enough.",
"Our aim is to show the convergence of $(w(s),\\partial _s w(s))$ as $s\\rightarrow \\infty $ to some $(\\kappa (d_\\infty ,0)\\Omega _\\infty ,0)$ , for some $(d_\\infty ,\\Omega _\\infty )$ close to $(d^*,\\Omega ^*)$ .",
"As one can see from (REF ), $(w,\\partial _s w)$ is close to a one representative of the family of the non-zero stationary solution $S^*\\equiv \\lbrace (\\kappa (d,y),0)\\Omega , |d|<1, \\Omega \\in S^{m-1}\\rbrace .$ From the continuity of $(w,\\partial _s w)$ from $[s^*,\\infty )$ to $\\mathcal { H}$ , $(w(s),\\partial _s w(s))$ will stay close to a soliton from $ S^*$ , at least for a short time after $s^*$ .",
"In fact, we can do better, and impose some orthogonality conditions, killing the zero directions of the linearized operator of equation (REF ) (see the operator $L_d$ defined in (REF )).",
"From the invariance of equation (REF ) under rotations in $\\mathbb {R}^{m}$ , we may assume that $\\Omega ^*=e_1.$ We recall that at this level of the study in the complex case (i.e.",
"for $m=2$ ), we were able to modulate $(w,\\partial _s w)$ as follows $\\begin{pmatrix} w(y,s)\\\\\\partial _s w(y,s) \\end{pmatrix} =e^{i \\theta (s)}\\left[\\begin{pmatrix} \\kappa (d(s),y)\\\\0\\end{pmatrix} +\\begin{pmatrix} q_1(y,s)\\\\q_2(y,s)\\end{pmatrix}\\right].$ for some well chosen $d(s)\\in (-1,1)$ and $\\theta (s)\\in \\mathbb {R}$ , such that $\\bar{\\pi }_0^{d(s)}\\begin{pmatrix} q_{1,1}(s)\\\\q_{2,1}(s)\\end{pmatrix} =\\tilde{\\pi }_0^{d(s)}\\begin{pmatrix} q_{1,2}(s)\\\\q_{2,2}(s)\\end{pmatrix}=0$ where $\\bar{\\pi }_0^{d}$ and $\\tilde{\\pi }_0^{d}$ are defined in (REF ) and (REF ) and $q=(q_1, q_2)$ is small in $\\mathcal { H}$ .",
"From (REF ), we see that we have a rotation in the complex plane, which has to be generalized to the vector-valued case.",
"In order to do so, we introduce for $i=2,...,m$ $R_i\\equiv \\begin{pmatrix}\\cos \\theta _i&0&\\cdots &-\\sin \\theta _i &\\cdots &0\\\\0 &1&\\cdots &0 &\\cdots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\\\sin \\theta _i&0&\\cdots &\\cos \\theta _i &\\cdots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\0 &0&\\cdots &0&\\cdots &1\\\\\\end{pmatrix}.$ Note that $R_i$ is an $m\\times m$ orthonormal matrix which rotates the $(e_1,e_i)$ -plane by an angle $\\theta _i$ and leaves all other directions invariant.",
"We introduce $ R_\\theta $ by $R_\\theta \\equiv R_2 R_3\\cdots R_m,$ where $\\theta =(\\theta _2,\\theta _3,\\cdots , \\theta _m)$ .",
"Clearly, $R_\\theta $ is an $m\\times m$ orthonormal matrix.",
"We also define $A_j$ by $A_j=R_{\\theta }^{-1} \\frac{\\partial R_\\theta }{\\partial \\theta _j}.$ In the appendix, we show a different expression for $A_j$ : $A_j=\\frac{\\partial R_\\theta ^{-1} }{\\partial \\theta _j} R_{\\theta }.$ In fact, this formalism is borrowed from Filippas and Merle [8] who introduced the modulation technique for the vector-valued heat equation $\\partial _t u=\\Delta u+|u|^{p-1}u.$ We are ready to give our modulation technique result well adapted to the vector-valued case: Proposition 3.1 (Modulation of $w$ with respect to $\\kappa (d, .",
")\\Omega $ , where $\\Omega \\in \\mathbb {R}^{m-1}$ ) There exists $\\epsilon _0 > 0$ and $K_1 >0$ such that for all $\\epsilon \\le \\epsilon _0$ if $v\\in \\mathcal {H}$ , $ d\\in (-1,1) $ and $\\hat{\\theta }=(\\hat{\\theta }_2,...,\\hat{\\theta }_{m})\\in \\mathbb {R}^{m-1}$ are such that $\\forall i=2,...,m, \\cos {\\hat{\\theta }}_i\\ge \\frac{3}{4}\\mbox{ and } ||\\hat{q}||_{\\mathcal {H}}\\le \\epsilon \\mbox{ where }v =R_{\\hat{\\theta }}\\left[\\begin{pmatrix} \\kappa (\\hat{d},.",
")e_1\\\\0\\end{pmatrix} +\\hat{q}\\right],$ then, there exist $d\\in (-1,1),$ $\\hat{\\theta }=(\\hat{\\theta }_2,...,\\hat{\\theta }_{m})\\in \\mathbb {R}^{m-1}$ such that $\\bar{\\pi }_{0}^{d}\\begin{pmatrix} q_{1,1}\\\\ q_{2,1}\\end{pmatrix} =0,\\mbox{ and }\\tilde{\\pi }_{0}^{d}\\begin{pmatrix} q_{1,j}\\\\ q_{2,j}\\end{pmatrix} =0,\\; \\forall j=2,..m,$ where $q=(q_1, q_2)$ is defined by: $\\Big | \\log \\left(\\frac{1+d}{1-d}\\right)-\\log \\left(\\frac{1+\\hat{d}}{1-\\hat{d}}\\right)\\Big |+ |\\theta -\\hat{\\theta }|\\le C_0 ||\\hat{q}||_{\\mathcal {H}}\\le K_1 \\epsilon ,$ $\\forall i=2,...,m,\\; \\cos \\theta _i\\ge \\frac{1}{2}\\mbox{ and }||q||_{\\mathcal {H}}\\le K_1 \\epsilon .$ In order to prove this proposition, we need the following estimates on the matrix $A_j$ given in $(\\ref {A_i})$ and $(\\ref {61,5})$ : Lemma 3.2 (Orthogonality and continuity results related to the matrix $A_i$ (REF )) $ $ i) For any $i\\in \\lbrace 2,...,m\\rbrace $ , $A_i e_1=( \\prod \\limits _{j =i+1}^{m} \\cos \\theta _j ) e_i$ ii) For any $i\\in \\lbrace 2,...,m\\rbrace $ , $z\\in \\mathbb {R}^m$ , we have $ |A_i (z) |\\le |z| .$ The proof is straightforward though a bit technical.",
"For that reason, we give it in Appendix Now, we are ready to prove Proposition REF .",
"[Proof of Proposition REF ] The proof is similar to the complex-valued case.",
"However, since our notations are somehow complicated, we give details for the reader's convenience.",
"First, we recall that $\\theta =(\\theta _2,\\theta _3,...,\\theta _m)\\in \\mathbb {R}^{m-1}$ .",
"From (REF ) and (REF ), we see that the condition (REF ) becomes $\\Phi (v, d, \\theta )=0 $ where $ \\Phi \\in C(\\mathcal {H}\\times (-1,1)\\times \\mathbb {R}^{m-1}, \\mathbb {R}^m) $ is defined by $\\begin{array}{c}\\Phi (v, d, \\theta )=\\begin{pmatrix}\\bar{\\Phi }(v, d, \\theta )\\\\\\tilde{\\Phi }_2 (v, d, \\theta )\\\\\\vdots \\\\\\tilde{\\Phi }_m (v, d, \\theta )\\end{pmatrix}=\\begin{pmatrix}\\phi \\left(\\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}-\\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix},\\bar{W}_0^d\\right)\\\\\\phi \\left(\\begin{pmatrix}V_{1,j}\\\\V_{2,j} \\end{pmatrix},\\tilde{W}_0^d\\right)_{j=2...m}\\end{pmatrix}\\end{array}$ where $V=\\begin{pmatrix}V_1\\\\V_2\\end{pmatrix}\\in \\mathbb {R}^m\\times \\mathbb {R}^m$ is given by $V=R^{-1}_\\theta v$ .",
"We claim that we can apply the implicit function theorem to $\\Phi $ near the point $(\\hat{v},\\hat{d},\\hat{\\theta })$ with $\\hat{v}=R_{\\hat{\\theta }}(\\kappa (\\hat{d},.",
")e_1,0)$ .",
"Three facts have to be checked: 1-First, note that $\\hat{v}=R_{\\hat{\\theta }}^{-1}(\\hat{v})$ , hence $\\Phi (R_{\\hat{\\theta }}(\\kappa (\\hat{d},.",
")e_1,0),\\hat{d},\\hat{\\theta })=0.$ 2-Then, we compute from (REF ), for all $u\\in \\mathcal {H}$ , $D_v \\bar{\\Phi }(v,d,\\theta )(u)=\\phi (\\begin{pmatrix}U_{1,1}\\\\U_{2,1} \\end{pmatrix},\\bar{W}_0^d),$ and for all $ j=2...m$ , we have $D_v \\tilde{\\Phi }_j (v,d,\\theta )(u)=\\phi (\\begin{pmatrix}U_{1,j}\\\\U_{2,j} \\end{pmatrix},\\tilde{W}_0^d) ,$ so we have from (REF ) and (REF ) $||D_v \\bar{\\Phi }(v,d,\\theta )||\\le C_0 \\mbox{ and } ||D_v \\tilde{\\Phi }_j (v,d,\\theta )||\\le C_0.$ 3-Let $J(\\bar{\\Phi },\\tilde{\\Phi }_{j,j=2..m})$ the jacobian matrix of $\\Phi $ with respect to $(d,\\theta )$ , and $D$ its determinant so $J\\equiv \\begin{pmatrix}\\partial _d \\bar{\\phi }&\\partial _{\\theta _2} \\bar{\\phi }&\\cdots &\\partial _{\\theta _m} \\bar{\\phi }\\\\\\partial _d \\tilde{\\phi }_2&\\partial _{\\theta _2} \\tilde{\\phi }_2&\\cdots &\\partial _{\\theta _m} \\tilde{\\phi }_2\\\\\\vdots &\\vdots &\\vdots &\\vdots \\\\\\partial _d \\tilde{\\phi }_m&\\partial _{\\theta _2} \\tilde{\\phi }_m&\\cdots &\\partial _{\\theta _m} \\tilde{\\phi }_m\\\\\\end{pmatrix}.$ Then, we compute from (REF ): $ \\partial _d \\bar{\\Phi }&=&-\\phi ((\\partial _d \\kappa (d,.",
"),0),\\bar{W}_0^d)+\\phi (\\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}-\\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix}, \\partial _d \\bar{W}_0^d), $ and for $i,j=2,..m$ $\\partial _d \\tilde{\\Phi }_j&=&\\phi (\\begin{pmatrix}V_{1,j}\\\\V_{2,j} \\end{pmatrix},\\partial _d \\tilde{W}_0^d),\\\\\\partial _{\\theta _i} \\bar{\\Phi }&=&\\phi ( \\partial _{\\theta _i} \\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix},\\bar{W}_0^d)=\\phi ( \\begin{pmatrix} <e_1, \\frac{\\partial R_\\theta ^{-1}}{ \\partial {\\theta _i} }v_{1}>\\\\ <e_1, \\frac{\\partial R_\\theta ^{-1}}{ \\partial {\\theta _i} }v_{2}> \\end{pmatrix},\\bar{W}_0^d),\\\\\\partial _{\\theta _i} \\tilde{\\Phi }_j&=&\\phi (\\begin{pmatrix} <e_j, \\frac{\\partial R_\\theta ^{-1}}{ \\partial {\\theta _i} }v_{1}>\\\\ <e_j, \\frac{\\partial R_\\theta ^{-1}}{ \\partial {\\theta _i} }v_{2}> \\end{pmatrix},\\partial _d \\tilde{W}_0^d).\\\\$ Now, we assume that $|\\theta |+\\big |\\log \\left(\\frac{1+d}{1-d}\\right)- \\log \\left(\\frac{1+\\hat{d}}{1-\\hat{d}}\\right)\\big |+ \\big |\\big | v -R_{\\hat{\\theta }}\\begin{pmatrix} \\kappa (\\hat{d},.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\epsilon _1$ for some small $\\epsilon _1>0$ .",
"In the following, we estimate each of the derivatives whose expressions where given above.",
"- Since $\\begin{pmatrix} \\partial _d \\kappa (d,y)\\\\0\\end{pmatrix}=\\frac{-2\\kappa _0}{(p-1)(1-d^2)}\\bar{F}_0^d,$ by definiftion (REF ) and (REF ), it follows from the orthogonality condition (REF ) that $\\phi ((\\partial _d \\kappa (d,.",
"),0),\\bar{W}_0^d)=\\frac{-2\\kappa _0}{(p-1)(1-d^2)}.$ Therefore, from (REF ), we write $ \\partial _d \\bar{\\Phi }=\\frac{2\\kappa _0}{(p-1)(1-d^2)}+\\phi (\\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}-\\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix}), \\partial _d \\bar{W}_0^d).$ Since $\\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}=\\begin{pmatrix} <e_1, R_\\theta ^{-1}v_{1}>\\\\ <e_1, R_\\theta ^{-1}v_{2}> \\end{pmatrix},$ we write $ &\\;&\\big |\\big | \\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}- \\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\big |\\big | R_\\theta ^{-1}v-\\begin{pmatrix}\\kappa (d,.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\\\&\\le &\\big |\\big | ( R_\\theta ^{-1}- R_{\\hat{\\theta }}^{-1}) v \\big |\\big |_{\\mathcal {H} }+\\big |\\big | R_{\\hat{\\theta }}^{-1} v-\\begin{pmatrix}\\kappa (\\hat{d},.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }+ \\big |\\big | \\kappa (\\hat{d},.",
")-\\kappa (d,.)",
"\\big |\\big |_{\\mathcal {H} }.$ Since, $\\forall \\theta , \\theta ^{\\prime } \\in \\mathbb {R}, \\big | R_{\\theta }-R_{\\theta ^{\\prime }}\\big |+\\big | R_{\\theta }^{-1}-R_{\\theta ^{\\prime }}^{-1}\\big | \\le C \\big | \\theta -\\theta ^{\\prime } \\big |, $ (see (REF ) below for $R_{\\theta }$ , and use an adhoc change of variables for $R_{\\theta }^{-1}$ ), recalling the following continuity result from estimate $(174)$ page 101 in [18]: $\\big |\\big | \\kappa (d_1,.",
")- \\kappa (d_2,.)",
"\\big |\\big |_{\\mathcal {H}_0 }\\le C\\big | \\left(\\frac{1+d_1}{1-d_1}\\right)- \\left(\\frac{1+d_2}{1-d_2}\\right)\\big | ,$ we see from the Cauchy-Schwartz inequality, (REF ), (REF ) and (REF ) that $ \\big |\\big | \\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}- \\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\big |\\big | V-\\begin{pmatrix}\\kappa (d,.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\ \\le C \\epsilon _1.$ $ \\big | \\partial _d \\bar{\\Phi }-\\frac{2\\kappa _0}{(p-1)(1-d^2)}\\big | \\le C \\epsilon _1.$ - Since $ \\begin{pmatrix}V_{1,j}\\\\V_{2,j} \\end{pmatrix}=\\begin{pmatrix} <e_j, R_\\theta ^{-1}v_{1}>\\\\ <e_j, R_\\theta ^{-1}v_{2}> \\end{pmatrix},$ we write $\\big |\\big | \\begin{pmatrix}V_{1,j}\\\\V_{2,j} \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\big |\\big | R_{\\theta }^{-1} v-\\begin{pmatrix}\\kappa (d,.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le C \\epsilon _1$ by the same argument as for (REF ).",
"Using the Cauchy-Schwarz inequality together with (REF ), we see from (REF ) that $ \\big | \\partial _d \\tilde{\\Phi }_j\\big | \\le \\frac{C \\epsilon _1}{1-d^2}.$ From (REF ), we see that $\\frac{\\partial R_\\theta ^{-1}}{\\partial \\theta _i}= A_i R_\\theta ^{-1}$ .",
"Therefore using $ii)$ of Lemma REF and the fact that the rotation $R_\\theta $ does not change the norm in $\\mathcal {H}$ , we write $ \\big | \\big | \\frac{\\partial R_{\\theta }^{-1}}{\\partial \\theta _i}(v)- \\begin{pmatrix}\\kappa (d,.",
")A_i (e_1)\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\big | \\big | v- R_{\\theta } \\begin{pmatrix}\\kappa (d,.",
")e_1\\\\0 \\end{pmatrix} \\big | \\big | _{\\mathcal {H} }\\le C \\epsilon _1,$ by the same argument as for (REF ).",
"Using the Cauchy-Schwarz identity together with (REF ), we see from () that $ \\big | \\partial _{\\theta _i} \\bar{\\Phi }\\big | \\le C \\epsilon _1.$ - By the same argument as for (REF ), we obtain from () $ \\big | \\partial _{\\theta _i} \\tilde{\\Phi }_j\\big | \\le C \\epsilon _1 \\mbox{ if }i\\ne j.$ Now, if $i= j$ , noting from (REF ) that $\\frac{\\partial R_\\theta ^{-1}}{\\partial \\theta _i}v=A_i R_\\theta ^{-1}(v)=A_i V,$ applying the operator $A_i$ to (REF ), then taking the scalar product with $e_i$ , we see from Lemma REF that $ \\big | \\big | \\begin{pmatrix} <e_i, \\frac{\\partial R^{-1}_{\\theta }}{\\partial \\theta _i} v_{1}> \\\\ <e_i, \\frac{\\partial R^{-1}_{\\theta }}{\\partial \\theta _i} v_{2}>\\end{pmatrix} - \\begin{pmatrix}\\kappa (d_i,.",
")\\prod \\limits _{k=i+1}^{m}\\cos \\theta _k\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le C \\epsilon _1.$ Since we know from (REF ) and (REF ) that $\\phi (\\kappa (d,.",
"), \\tilde{W}_0^d)=1,$ it follows from () that $ \\big | \\partial _{\\theta _i} \\tilde{\\Phi }_i -\\prod \\limits _{k=i+1}^{m}\\cos \\theta _k\\big | \\le C \\epsilon _1.$ Collecting (REF ), (REF ), (REF ), (REF ) and (REF ) we see that $ \\big | D-\\frac{2\\kappa _0}{(p-1)(1-d^2)} -\\cos \\theta _3 (\\cos \\theta _4)^2... (\\cos \\theta _m)^{m-2}\\big | \\le \\frac{C \\epsilon _1}{1-d^2}.$ Since $\\cos \\theta _i \\ge \\frac{3}{4}$ by hypothesis, we have the non-degeneracy of $\\Phi $ (voir (REF )) near the point $(\\hat{v},\\hat{d},\\hat{\\theta })$ with $\\hat{v}= R_{\\hat{\\theta }}(\\kappa (\\hat{d},.",
")e_1,0)$ .",
"Applying the implicit function theorem, we conclude the proof of Proposition REF .",
"Dynamics of $q$ , $d$ and $\\theta $ Let us apply Proposition REF with $v=(w,\\partial _s w) (s^*)$ , $\\hat{d}=d^*$ and $\\hat{\\theta }=0$ .",
"Clearly, from (REF ) and (REF ), we have $ ||\\hat{q}||_{\\mathcal {H}}\\le \\epsilon ^*.$ Assuming that $\\epsilon ^*\\le \\epsilon _0$ defined in Proposition REF , we see that the proposition applies, and from the continuity of $(w,\\partial _s w)$ from $[s^*,\\infty )$ to $\\mathcal {H}$ , we have a maximal $\\bar{s}> s^*$ , such that $(w(s),\\partial _s w(s))$ can be modulated in the sense that $\\begin{pmatrix} w(y,s)\\\\\\partial _s w(y,s) \\end{pmatrix} =R_{\\theta (s)}\\left[\\begin{pmatrix} \\kappa (d(s),y)e_1\\\\0\\end{pmatrix} +\\begin{pmatrix} q_1(y,s)\\\\q_2(y,s)\\end{pmatrix}\\right],$ where the parameters $d(s)\\in (-1,1)$ and $\\theta (s)=(\\theta _2(s),..., \\theta _m(s))$ are such that for all $s\\in [s^*, \\bar{s})$ $\\bar{\\pi }_{0}^{d(s)}\\begin{pmatrix} q_{1,1}(s)\\\\ q_{2,1}(s)\\end{pmatrix} =0,\\mbox{ and }\\tilde{\\pi }_{0}^{d(s)}\\begin{pmatrix} q_{1,j}(s)\\\\ q_{2,j}(s)\\end{pmatrix} =0,\\; \\forall j=2,..m,$ and $\\forall i=2,...,m\\; \\cos \\theta _i(s)\\ge \\frac{1}{2} \\mbox{ and } ||q(s)||_{\\mathcal {H}}\\le \\epsilon \\equiv 2K_0 K_1 \\epsilon ^*,$ where $K_1>0$ is defined in Proposition REF and $K_1>1$ is a constant that will be fixed below in (REF ).",
"Two cases then arise: - Case 1: $\\bar{s}=+\\infty ;$ - Case 2: $\\bar{s}<+\\infty ;$ in this case, we have an equality case in (REF ), i.e.",
"$cos \\theta _i(\\bar{s})= \\frac{1}{2} $ for some $i=2,...,m$ , or $ ||q(\\bar{s})||_{\\mathcal {H}}= 2K_0 K_1 \\epsilon ^*$ .",
"At this stage, we see that controlling the solution $(w(s),\\partial _s w (s))\\in \\mathcal {H} $ is equivalent to controlling $q\\in \\mathcal {H}$ , $d\\in (-1,1)$ and $\\theta (s)\\in \\mathbb {R}^{m-1}$ .",
"Before giving the dynamics of this parameters, we need to introduce some notations.",
"From (REF ), we will expand $\\bar{q}$ and $\\tilde{q}$ respectively according to the spectrum of the linear operators $\\bar{L}_d$ and $\\tilde{L}_d$ as in (REF ) and (REF ): $\\begin{pmatrix} q_{1,1}(y,s)\\\\ q_{2,1}(y,s) \\end{pmatrix}&=\\alpha _{1,1} \\bar{F}_1^d (y)+\\begin{pmatrix} q_{-,1,1}(y,s)\\\\ q_{-,2,1} (y,s)\\end{pmatrix} \\\\\\forall j \\in \\lbrace 1,...,m\\rbrace ,\\; \\begin{pmatrix} q_{1,j}(y,s)\\\\ q_{2,j} (y,s)\\end{pmatrix}&=\\begin{pmatrix} q_{-,1,j}(y,s)\\\\ q_{-,2,j}(y,s) \\end{pmatrix}$ where $\\alpha _{1,1} = \\bar{\\pi }_1^{d(s)} \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix},\\;\\alpha _{0,1} = \\bar{\\pi }_0^{d(s)} \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}=0,\\;\\alpha _{-,1}(s)=\\sqrt{\\bar{\\varphi }_d (\\begin{pmatrix} q_{-,1,1}\\\\ q_{-,2,1} \\end{pmatrix},\\begin{pmatrix} q_{-,1,1}\\\\ q_{-,2,1} \\end{pmatrix})}$ $\\alpha _{0,j}= \\tilde{\\pi }_0^{d(s)}\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix}=0,\\;\\alpha _{-,j}(s)=\\sqrt{\\tilde{\\varphi }_d (\\begin{pmatrix} q_{-,1,j}\\\\ q_{-,2,j} \\end{pmatrix},\\begin{pmatrix} q_{-,1,j}\\\\ q_{-,2,j} \\end{pmatrix})}$ and $\\begin{pmatrix} q_{-,1,1}\\\\ q_{-,2,1} \\end{pmatrix}=\\bar{\\pi }_{-}^{d}\\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix}$ $\\forall j \\in \\lbrace 1,...,m\\rbrace ,\\;\\begin{pmatrix} q_{-,1,j}\\\\ q_{-,2,j} \\end{pmatrix}=\\tilde{\\pi }_{-}^{d}\\begin{pmatrix} q_{1,j}\\\\q_{2,j} \\end{pmatrix}$ From (REF ), (), (REF ) (REF ) and (REF ), we see that for all $s \\ge s_0$ , $\\frac{1}{C_0} \\alpha _{-,1}(s) &\\le & ||\\begin{pmatrix} q_{-,1,1}\\\\ q_{-,2,1} \\end{pmatrix}||_{\\mathcal {H} } \\le C_0 \\alpha _{-,1}(s)\\\\\\frac{1}{C_0}(| \\alpha _{1,1}(s)|+ \\alpha _{-,1}(s)) &\\le & || \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}||_{\\mathcal {H} } \\le C_0(| \\alpha _{1,1}(s)|+ \\alpha _{-,1}(s)) \\\\\\frac{1}{C_0} \\alpha _{-,j}(s) &\\le & || \\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix}||_{\\mathcal {H} } \\le C_0 \\alpha _{-,j}(s) $ for some $C_0 > 0$ .",
"In the following proposition, we derive from (REF ) and () differential inequalities satisfied by $ \\alpha _{1,1}(s)$ , $ \\alpha _{-,1}(s)$ , $ \\alpha _{-,j}(s)$ , $\\theta _i(s)$ and $d(s)$ .",
"Introducing $R_{-}(s)=-\\int _{-1}^{1} \\mathcal { F}_{d }(q_1) \\rho dy,$ where $\\mathcal {F}_{d(s)} (q_1 (y,s))=\\frac{|\\kappa (d ,\\cdot )e_1+q_1|^{p+1}}{p+1}-\\frac{\\kappa (d ,\\cdot )^{p+1}}{p+1}-\\kappa (d ,\\cdot )^p q_{1,1}-\\frac{p}{2} \\kappa (d ,\\cdot )^{p-1} q_{1,1}^2-\\frac{\\kappa (d ,\\cdot )^{p-1}}{2} \\sum _{j=2}^{m} q_{1,j}^2,$ we claim the following Proposition 3.3 (Dynamics of the parameters) For $\\epsilon ^*$ small enough and for all $s\\in [s^*,\\bar{s})$ , we have: (i) (Control of the modulation parameter) $\\sum _{i=2}^{m}|\\theta _i^{\\prime }|+\\frac{|d^{\\prime }|}{1-d^2}\\le C_0 || q||_{\\mathcal {H}}^2.$ (ii) (Projection of equation (REF ) on the different eigenspaces of $\\bar{L}_d$ and $\\tilde{L}_d$ ) $| \\alpha _{1,1}^{\\prime }(s) - \\alpha _{1,1}(s)|&\\le C_0 || q||_{H}^2.\\\\\\left( R_{-}+\\frac{1}{2}( \\alpha _{1,1}^2+ \\alpha _{-,j}^2)\\right)^{\\prime }&\\le -\\frac{4}{p-1}\\int _{-1}^{1}(q_{-,2,1}^2+ q_{-,2,j}^2)\\frac{\\rho }{1-y^2}dy+ C_0 ||q(s)||_{\\mathcal {H}}^{3},$ for $j\\in \\lbrace 2,...,m\\rbrace $ and $R_{-}(s)$ defined in $(\\ref {209})$ , satisfying $|R_{-}(s)|\\le C_0 ||q(s)||_{\\mathcal {H}}^{1+\\bar{p}} \\mbox{ where } \\bar{p}=\\min (p,2)>1.$ (iii) (An additional relation) $\\frac{d}{ds}\\int _{-1}^{1} q_{1,1} q_{2,1} \\rho \\le -\\frac{4}{5}\\bar{\\alpha }_{-,1}^2+ C_0\\int _{-1}^{1} q_{-,2,1}^2\\frac{\\rho }{1-y^2}+ C_0 ||q(s)||_{\\mathcal {H}}^{2}.$ For $j\\in \\lbrace 2,...,m\\rbrace ,$ we have: $\\frac{d}{ds}\\int _{-1}^{1} q_{1,j} q_{1,j} \\rho \\le -\\frac{4}{5}\\tilde{\\alpha }_{-,j}^2+ C_0\\int _{-1}^{1} q_{2,j}^2\\frac{\\rho }{1-y^2}+ C_0 ||q(s)||_{\\mathcal {H}}^{2}.$ (iv) (Energy barrier) $\\alpha _{1,1} (s)\\le C_0\\alpha _{-,1}(s)+ C_1 \\sum _{j=2}^{m} \\alpha _{-,j}(s).$ The proof follows the general framework developed by Merle and Zaag in the real case (see Proposition $5.2$ in [18]), then adapted to the complex-valued case in [4] 5(see Proposition $4.2$ page 5915 in [4]).",
"However, new ideas are needed, mainly because we have $(m-1)$ rotation parameters in the modulation technique (see Proposition REF above), rather than only one in the complex-valued case.",
"For that reason, in the following, we give details only for the \"new\" terms, referring the reader to the earlier literature for the \"old\" terms.",
"Let us first write an equation satisfied by $q$ defined in (REF ).",
"We put the equation (REF ) satisfied by $w$ in vectorial form: $\\begin{array}{l}\\partial _s w_1=w_2\\\\\\partial _s w_2=\\mathcal {L}w_1-\\frac{2(p+1)}{(p-1)^2}w_1+|w_1|^{p-1}w_1-\\frac{p+3}{p-1} w_2- 2 y \\partial _y w_2.\\end{array}$ We replace all the terms of (REF ) by their expressions from (REF ).",
"Precisely, for the terms of the right hand side of (REF ) we have: $\\begin{array}{l}\\partial _s w_1=R_{\\theta }\\left(d^{\\prime }\\partial _d \\kappa e_1+\\partial _s q_1\\right)+\\sum _{i=1}^{m}\\theta _i^{\\prime }\\frac{\\partial R_\\theta }{\\partial \\theta _j}\\left( \\kappa _d e_1+q_1\\right),\\\\\\partial _s w_2=R_{\\theta }\\left(\\partial _s q_1\\right)+\\sum _{i=1}^{m}\\theta _i^{\\prime }\\frac{\\partial R_\\theta }{\\partial \\theta _j}\\left(q_2\\right).",
"\\end{array}$ For the terms on the left hand side of (REF ) we have: $w_2=R_{\\theta }(q_2), \\,\\mathcal {L}w_1=R_{\\theta }( \\mathcal {L}(\\kappa _d e_1)+\\mathcal {L} q_1),\\,|w_1|=|\\kappa _d e_1+ q_1|,\\,\\partial _y w_2=R_{\\theta }\\partial _y q_2.$ Then, multiplying by $R_{\\theta }^{-1}$ , using the fact that $(\\kappa (d ,\\cdot ),0)$ is a stationnary solution and dissociating the first and $j$ th component of these equations, we get for all $s\\in [s^*,\\bar{s})$ , for all $j \\in \\lbrace 2,...,m\\rbrace $ : $\\frac{\\partial }{\\partial s}\\begin{pmatrix} q_{1,1}\\\\ q_{2,1}\\end{pmatrix}&=\\bar{L}_{d(s)} \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}+\\begin{pmatrix}0\\\\ {f}_{d(s),1}(q_1)\\end{pmatrix}-d^{\\prime }(s)\\begin{pmatrix}\\partial _d \\kappa (d,y)\\\\ 0\\end{pmatrix}-\\sum _{i=2}^{m}\\theta _i^{\\prime }(s)\\begin{pmatrix} a_{i,1,1} \\\\ a_{i,2,1} \\end{pmatrix},\\\\\\frac{\\partial }{\\partial s} \\begin{pmatrix} q_{1,j}\\\\q_{2,j}\\end{pmatrix}&=\\tilde{L}_{d(s)}\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix}+\\begin{pmatrix} 0\\\\{f}_{d(s),j}(q_1) \\end{pmatrix}-\\sum _{i=2}^{m}\\theta _i^{\\prime }(s)\\begin{pmatrix} a_{i,1,j} \\\\ a_{i,2,j} \\end{pmatrix},$ where $\\bar{L}_{d(s)} ,\\tilde{L}_{d(s)}, f_{d(s),1}$ and $f_{d(s),j}$ are defined in (REF ), (REF ), (REF ) and (REF ), and $a_i$ by $\\begin{pmatrix} a_{i,1}\\\\a_{i,2}\\end{pmatrix}=\\begin{pmatrix}A_i (\\kappa _d e_1+q_1)\\\\A_i (q_2) \\end{pmatrix},$ with $a_{i,1}=(a_{i,1,1},a_{i,1,2},...,a_{i,1,m})\\in \\mathbb {R}^{m}$ and $a_{i,2}=(a_{i,2,1},a_{i,2,2},...,a_{i,2,m})\\in \\mathbb {R}^{m}$ .",
"Let $i\\in \\lbrace 2,\\cdots ,m\\rbrace $ , Projecting equation (REF ) with the projector $\\bar{\\pi }_\\lambda ^d$ (REF ) for $\\lambda =0$ and $\\lambda =1$ , we write $\\bar{\\pi }_\\lambda ^d (\\partial _s \\begin{pmatrix} q_{1,1}\\\\ q_{2,1}\\end{pmatrix})&=&\\bar{\\pi }_\\lambda ^d (\\bar{L}_{d(s)} \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix})+\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}0\\\\ {f}_{d(s),1}(q_1) \\end{pmatrix}-d^{\\prime }(s)\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}\\partial _d \\kappa (d,y)\\\\ 0\\end{pmatrix}\\\\&-&\\sum _{i=2}^{m}\\theta _i^{\\prime }(s)\\bar{\\pi }_\\lambda ^d\\begin{pmatrix} a_{i,1,1} \\\\ a_{i,2,1} \\end{pmatrix}.$ Note that, expect the last term, all the terms of (REF ) can be controled exactly like the real case using (REF ) (for details see page 105 in [18]).",
"So, we recall that we have: $|\\bar{\\pi }_\\lambda ^d( \\partial _s \\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix})-\\alpha _{\\lambda ,1}^{\\prime }|\\le \\frac{C_0}{1-d^2} |d^{\\prime }||| q||_{\\mathcal { H}},$ $\\bar{\\pi }_\\lambda ^d( L_d \\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix})=\\lambda \\alpha _{\\lambda ,1},$ $|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}0\\\\ {f}_{d(s),1}(q_1) \\end{pmatrix}|\\le C_0|| q||_{\\mathcal { H}}^2,$ $|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}\\partial _d \\kappa (d,y)\\\\ 0\\end{pmatrix}|=-\\frac{2\\kappa _0}{(p-1)(1-d^2)} \\bar{\\pi }_\\lambda ^d (F_0^d) =-\\frac{2\\kappa _0}{(p-1)(1-d^2)}\\delta _{\\lambda ,0}.$ Now, we focus on the study of the last term of (REF ).",
"From the definition of $a_i$ (REF ) and $i)$ of Lemma REF , we have: $\\begin{pmatrix} a_{i,1,1}\\\\a_{i,2,1}\\end{pmatrix}=\\kappa _d \\begin{pmatrix}<e_1,A_i (e_1)>\\\\0 \\end{pmatrix}+\\begin{pmatrix}<e_1,A_i (q_1)>\\\\<e_1,A_i (q_2)> \\end{pmatrix}=\\begin{pmatrix}<e_1,A_i (q_1)>\\\\<e_1,A_i (q_2)> \\end{pmatrix}.$ Applying the projector $\\bar{\\pi }_\\lambda ^d$ (REF ), we get $|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix} a_{i,1,1}\\\\a_{i,2,1} \\end{pmatrix}|&=&|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}<e_1,A_i (q_1)>\\\\<e_1,A_i (q_2)> \\end{pmatrix}|\\le C || \\begin{pmatrix}<e_1,A_i (q_1)>\\\\<e_1,A_i (q_2)> \\end{pmatrix} ||_{\\mathcal { H}}\\\\&\\le & C (|| <e_1,A_i (q_1)>||_{\\mathcal { H}_0}+|| <e_1,A_i (q_2)>||_{ L_\\rho ^2}).$ Using $ii)$ of Lemma REF , we have: $|| <e_1,A_i (q_2)>||^2_{ L_\\rho ^2}=\\int _{-1}^{1} <e_1,A_i (q_2)>^2\\rho dy\\le \\int _{-1}^{1} |A_i (q_2)|^2\\rho dy\\le \\int _{-1}^{1} |q_2|^2\\rho dy,$ and by the same way, using $ii)$ of Lemma REF and the definition of $\\mathcal { H}_0$ (REF ), we have $|| <e_1,A_i (q_1)>||^2_{\\mathcal { H}_0}&=&\\int _{-1}^{1} <e_1,A_i (q_1)>^2\\rho dy+\\int _{-1}^{1} (<e_1,A_i (\\partial _y q_1)>)^2(1-y^2)\\rho dy\\\\&\\le &\\int _{-1}^{1}( |q_1|^2+(1-y^2)|\\partial _y q_1|^2)\\rho dy.$ From (REF ), (REF ) and (REF ), we have $|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix} a_{i,1,1}\\\\a_{i,2,1} \\end{pmatrix}|\\le C_0 || q||_{\\mathcal { H}}.$ Using (REF ), (REF ), (REF ), (REF ), (REF ), (REF ), and the fact that $\\alpha _{0,1}\\equiv \\alpha _{0,1}^{\\prime }\\equiv 0$ (see (REF )), we get for $\\lambda =0,1$ : $\\frac{2 \\kappa _0}{(p-1)(1-d^2)} |d^{\\prime }|&\\le \\frac{C_0}{1-d^2}|d^{\\prime }| || q||_{\\mathcal { H}}+C_0 || q||_{\\mathcal { H}}^2+C_0 || q||_{\\mathcal { H}}\\sum _{j=2}^{m}|\\theta _i^{\\prime }|\\\\|\\bar{\\alpha }_1^{\\prime }(s)-\\bar{\\alpha }_1(s)|&\\le \\frac{C_0}{1-d^2}|d^{\\prime }| || q||_{\\mathcal { H}} +C_0 || q||_{\\mathcal { H}}^2+C_0|| q||_{\\mathcal { H}}\\sum _{j=2}^{m}|\\theta _i^{\\prime }|.$ Now, projecting equation () with the projector $\\tilde{\\pi }_0^d$ (REF ), where $j\\in \\lbrace 2,\\cdots ,m\\rbrace $ , we get: $\\tilde{\\pi }_0^d(\\partial _s\\begin{pmatrix} q_{1,j}\\\\q_{2,j}\\end{pmatrix} )=\\tilde{\\pi }_0^d( \\tilde{L}_{d(s)}\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})+\\tilde{\\pi }_0^d \\begin{pmatrix} 0\\\\f_{d(s),j} (q_1) \\end{pmatrix}-\\sum _{i=2}^{m}\\theta _i^{\\prime }(s) \\tilde{\\pi }_0^d\\begin{pmatrix} a_{i,1,j} \\\\ a_{i,2,j} \\end{pmatrix}.$ From the complex-valued case we recall that we have (for details see page 5917 in [4], together with Lemma REF ): $|\\tilde{\\pi }_0^d (\\partial _s \\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})|\\le \\frac{C_0}{1-d^2} |d ^{\\prime }| || q||_{\\mathcal { H}},$ $\\tilde{\\pi }_0^d (\\tilde{L}_{d }\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})=0,$ $\\Big {|}\\tilde{\\pi }_0^d \\begin{pmatrix} 0\\\\\\tilde{f}_{d(s)}(q_1) \\end{pmatrix}\\Big {|} \\le C_0 || q||_{\\mathcal { H}}^2,$ $\\tilde{\\pi }_0^d\\begin{pmatrix} \\kappa _d \\\\0 \\end{pmatrix} =1.$ Thus, only the last term in (REF ) remains to be treated in the following.",
"From the definition of $a_i$ (REF ), we recall that $\\begin{pmatrix} a_{i,1,j}\\\\a_{i,2,j}\\end{pmatrix}= \\begin{pmatrix}<e_j,A_i (e_1)>\\kappa _d\\\\0 \\end{pmatrix}+\\begin{pmatrix}<e_j,A_i (q_1)>\\\\<e_j,A_i (q_2)> \\end{pmatrix}.$ By $i)$ of Lemma REF : $\\sum _{i=2}^{m}\\theta _i^{\\prime }(s) \\begin{pmatrix} a_{i,1,j}\\\\a_{i,2,j}\\end{pmatrix}=\\theta _j^{\\prime }(s)\\begin{pmatrix} (\\prod \\limits _{l =j+1}^{m} \\cos \\theta _l ) \\kappa _d \\\\0 \\end{pmatrix}+\\sum _{i=2}^{m}\\theta _i^{\\prime }(s) \\begin{pmatrix}<e_j,A_i (q_1)>\\\\<e_j,A_i (q_2)> \\end{pmatrix},$ where by convention $\\prod \\limits _{l =m+1}^{m} \\cos \\theta _l =1$ if $j=m$ .",
"Applying the projection $\\tilde{\\pi }_0^d$ to (REF ) and using (REF ), we see that $\\Big {|} \\sum _{i=2}^{m}\\theta _i^{\\prime }(s) \\tilde{\\pi }_0^d \\begin{pmatrix} a_{i,1,j}\\\\a_{i,2,j}\\end{pmatrix}-\\theta _j^{\\prime }(s) \\prod \\limits _{l =j+1}^{m} \\cos \\theta _l \\Big {|} &\\le & \\sum _{i=2}^{m}|\\theta _i^{\\prime }(s)| \\Big {|} \\tilde{\\pi }_0^d \\begin{pmatrix} <e_j,A_i (q_1)>\\\\<e_j,A_i (q_2)> \\end{pmatrix} \\Big {|}\\\\&\\le & C_0 || q||_{\\mathcal { H}}\\sum _{i=2}^{m}|\\theta _i^{\\prime }(s)|,$ where, we use the fact that $|\\tilde{\\pi }_0^d\\begin{pmatrix} <e_j,A_i (q_1)>\\\\<e_j,A_i (q_2)> \\end{pmatrix}|\\le C_0 || q||_{\\mathcal { H}},$ which follows by the same techniques as in (REF ,) (REF ) and (REF ).",
"Using (REF ),(REF ), (REF ), (REF ) and (REF ), and recalling from (REF ) that $\\prod \\limits _{l =j+1}^{m} \\cos \\theta _l \\ge (\\frac{1}{2})^{m-j},$ we get for any $j\\in \\lbrace 2,...,m\\rbrace $ : $\\Big {|} \\theta _j^{\\prime }(s) \\Big {|} \\le \\frac{C_0}{1-d^2}|d^{\\prime }| || q||_{\\mathcal { H}}+C_0 || q||_{\\mathcal { H}}^2+C_0|| q||_{\\mathcal { H}}\\sum _{i=2}^{m}| \\theta _i^{\\prime }| .$ Using (REF ) together with (REF ), we see that $\\sum _{j=2}^{m} \\Big {|} \\theta _j^{\\prime }(s) \\Big {|}+ \\frac{|d^{\\prime }|}{1-d^2} \\le C_0 \\frac{|d^{\\prime }|}{1-d^2} || q||_{\\mathcal { H}}+C_0 || q||_{\\mathcal { H}}^2+C_0| || q||_{\\mathcal { H}}\\sum _{i=2}^{m}| \\theta _i^{\\prime }| ,$ Thus, using again (REF ) and taking $\\epsilon $ small enough, we get $\\sum _{j=2}^{m} \\Big {|} \\theta _j^{\\prime }(s) \\Big {|}+ \\frac{|d^{\\prime }|}{1-d^2} \\le C_0 || q||_{\\mathcal { H}}^2,$ which yields (REF ).",
"Then, using () together with (REF ) gives (REF ).",
"For estimations (REF ) (REF ) (REF ) (REF ), the study in the complex case (Subsection $4.3$ page 5914 in [4]) can be adapted without any difficulty to the vector-valued case.",
"For the reader convenience, we detail for example the energy barrier (REF ): Using the definition of $q(y,s)$ (REF ), we can make an expansion of $E(w(s),\\partial _s w(s))$ (REF ) for $q\\rightarrow 0$ in $\\mathcal {H}$ and get after from straightforward computations: $E(w(s),\\partial _s w(s))= E(\\kappa _0,0)+\\frac{1}{2}\\left[\\bar{\\varphi }_d(\\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}, \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix})+\\sum _{i=2}^{m}\\tilde{\\varphi }_d(\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix},\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})\\right]-\\int _{-1}^{1}\\mathcal {F}_d (q_1) \\rho dy $ where $ \\bar{\\varphi }_d$ , $ \\tilde{\\varphi }_d$ and $\\mathcal {F}_d (q_1)$ are defined in (REF ), (REF ) and (REF ).",
"Using the argument in the real case (see page 113 in [18]) we see that for some $C_0,C_1 >0$ we have: $\\bar{\\varphi }_d(\\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}, \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}) \\le C_0 \\alpha _{1,1}^2- C_1\\alpha _{-,1}^2.$ From (), (REF ) and (REF ), we see by definition that $0\\le \\tilde{\\varphi }_d(\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix}, \\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})=\\alpha _{-,j}^2.$ Since we have from (REF ), (REF ), (REF ), (REF ) and (REF ): $\\left| \\int _{-1}^{1}\\mathcal {F}_d (q_1) \\rho dy \\right| \\le C ||q(s)||_{\\mathcal {H}}^{\\bar{p}+1}\\le C \\epsilon ^{\\bar{p}-1} ( \\alpha _{1,1}^2+ \\alpha _{-,1}^2+\\sum _{i=2}^{m} \\alpha _{-,j}^2),$ Using (REF ), (REF ), (REF ) and (REF ), we see that taking $\\epsilon $ small enough so that $C \\epsilon ^{\\bar{p}-1}\\le \\frac{C_1}{4}$ , we get $0\\le E(w(s),\\partial _s w(s))-E(\\kappa _0,0)\\le \\left( \\frac{C_0}{2}+\\frac{C_1}{4}\\right) \\alpha _{-,1}^2-\\frac{C_1}{4}\\alpha _{1,1}^2+\\left(\\frac{1}{2}+\\frac{C_1}{4}\\right)\\sum _{i=2}^{m} \\alpha _{-,j}^2.$ which yields (REF ).",
"Exponential decay of the different components Our aim is to show that $||q(s)||_{\\mathcal {H}}\\rightarrow 0$ and that both $\\theta $ and $d$ converge as $s\\rightarrow \\infty $ .",
"An important issue will be to show that the unstable mode $\\alpha _{1,1}$ , which satisfies equation (REF ) never dominates.",
"This is true thanks to item $(iv)$ in Proposition REF .",
"If we introduce $\\lambda (s)=\\frac{1}{2} \\log \\left(\\frac{1+d(s)}{1-d(s)}\\right), a(s)= \\alpha _{1,1}(s)^2\\, \\mbox{and}\\, b(s)=\\alpha _{-,1}(s)^2+\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2+R_-(s)$ (note that $d(s)=\\tanh (\\lambda (s))$ ), then we see from (REF ), (REF ) and (REF ) that for all $s\\in [s^*, \\bar{s})$ $|R_-(s)|=|b(s)-(\\alpha _{-,1}(s)^2+\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2)|\\le C_0 \\epsilon ^{\\bar{p}-1}(\\alpha _{1,1}(s)^2+\\alpha _{-,1}(s)^2+\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2)$ , hence $\\frac{99}{100}\\alpha _{-,1}(s)^2+ \\frac{99}{100}\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2-\\frac{1}{100} a\\le b\\le \\frac{101}{100}\\alpha _{-,1}(s)^2+ \\frac{101}{100}\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2+\\frac{1}{100} a$ for $\\epsilon $ small enough.",
"Therefore, using Proposition REF , estimate (REF ), (REF ) and the fact that $\\lambda ^{\\prime }(s)=\\frac{d^{\\prime }(s)}{1-d(s)^2}$ , we derive the following: Claim 3.4 (Relations between $a$ , $b$ , $\\lambda $ , $\\theta $ , $\\int _{-1}^1 q_{1,1} q_{2,1} \\rho $ and $\\int _{-1}^1 q_{1,j}q_{2,j} \\rho $ ) There exist positive $\\epsilon _4$ , $K_4$ and $K_5$ such that if $\\epsilon ^*\\le \\epsilon _4$ , then we have for all $s\\in [s^*,\\bar{s}]$ and $j=2,...,m$ : (i) (Size of the solution) $\\frac{1}{K_4}(a(s)+b(s))\\le || q(s)||_\\mathcal {H}^2&\\le K_4 (a(s)+b(s))\\le K_4^2 \\epsilon ^2,\\\\|\\theta ^{\\prime }(s)|+|\\lambda ^{\\prime }(s)|&\\le K_4 (a(s)+b(s))\\le K_4^2 ||q(s)||_\\mathcal {H}^2 ,\\\\\\left| \\int _{-1}^{1}q_{1,1}q_{1,1}\\rho \\right|&\\le K_4 (a(s)+b(s)),\\\\\\left| \\int _{-1}^{1}q_{1,j}q_{1,j}\\rho \\right|&\\le K_4 b(s),$ and (REF ) holds.",
"(ii) (Equations) $\\frac{3}{2} a-K_4 \\epsilon b&\\le a^{\\prime } \\le \\frac{5}{2} a-K_4 \\epsilon b,\\\\b^{\\prime }&\\le -\\frac{8}{p-1}\\int _{-1}^{1}(q_{-,2,1}^2+q_{-,2,j}^2)\\frac{\\rho }{1-y^2}dy+ K_4\\epsilon (a+b),\\\\\\frac{d}{ds}\\int _{-1}^1 (q_{1,1} q_{2,1}+ q_{1,j} q_{2,j}) \\rho &\\le -\\frac{3}{5} b+K_4 \\int _{-1}^1 (q_{-,2,1}^2+ q_{2,j}^2)\\frac{\\rho }{1-y^2}+K_4 a.$ (iii) (Energy barrier) we have $a(s)\\le K_5 b(s).$ [End of the Proof of Theorem REF ] Now, we are ready to finish the proof of Theorem REF just started at the beginning of Section REF .",
"Let us define $s_2^* \\in [s^*,\\bar{s}]$ as the first $s \\in [s^*,\\bar{s}]$ such that $a(s)\\ge \\frac{b(s)}{5 K_4}$ where $K_4$ is introduced in Corollary REF , or $s^*_2=\\bar{s}$ if (REF ) is never satisfied on $[s^*,\\bar{s}]$ .",
"We claim the following: Claim 3.5 There exist positive $\\epsilon _6$ , $\\mu _6$ , $K_6$ and $f\\in C^1([s^*, s^*_2]$ such that if $\\epsilon \\le \\epsilon _6$ , then for all $s\\in [s^*,s_2^*]$ : (i) $\\frac{1}{2}f(s)\\le b(s)\\le 2 f(s)\\mbox{ and }f^{\\prime }(s)\\le -2\\mu _6f(s),$ (ii) $|| q(s)||_\\mathcal {H}\\le K_6 || q(s^*)||_\\mathcal {H}e^{-\\mu _6(s-s^*)}\\le K_6 K_1 \\epsilon ^*e^{-\\mu _6(s-s^*)}.$ The proof of Claim 5.6 page 115 in [18] remains valid where $f(s)$ is given by $f(s)=b(s)+\\eta _6\\int _{-1}^1 (q_{1,1} q_{2,1}+ \\sum _{j=2}^{m}q_{1,j} q_{2,j}) \\rho ,$ where $\\eta _6 >0$ is fixed small independent of $\\epsilon $ .",
"Claim 3.6 (i) There exists $\\epsilon _7>0$ such that for all $\\sigma >0$ , there exists $K_7(\\sigma )>0$ such that if $\\epsilon \\le \\epsilon _7$ , then $\\forall s \\in [s_2^*, \\min (s_2^*+\\sigma , \\bar{s})],\\,|| q(s)||_\\mathcal {H}\\le K_7 || q(s^*)||_\\mathcal {H}e^{-\\mu _6(s-s^*)}\\le K_7 K_1 \\epsilon ^*e^{-\\mu _6(s-s^*)}$ and $|\\theta _i(s)|\\le C\\frac{(K_7 K_1 \\epsilon ^*)^2}{2\\mu _6}$ where $\\mu _6$ has been introduced in Claim REF .",
"(ii) There exists $\\epsilon _8>0$ such that if $\\epsilon \\le \\epsilon _8$ , then $\\forall s \\in (s_2^*, \\bar{s}],\\; b(s)\\le a(s) \\left( 5 K_4 e^{-\\frac{(s-s_2^*)}{2}}+\\frac{1}{4 K_5}\\right)$ where $K_4$ and $K_5$ have been introduced in Corollary REF .",
"The proof is the same as the proof of Claim 5.7 page 117 in [18].",
"Now, in order to conclude the proof of Theorem REF , we fix $\\sigma _0>0$ such that $5K_4^{-\\frac{\\sigma _0}{2}}+\\frac{1}{4K_5}\\le \\frac{1}{2K_5},$ where $K_4$ and $K_5$ are introduced in Claim REF .",
"Then, we fix the value of $K_0=\\max (2,K_6,K_7(\\sigma _0)),$ and the constants are defined in Claims REF and REF .",
"Then, we fix $\\epsilon _0=\\min \\left(1,\\epsilon _1,\\frac{\\epsilon _i}{2K_0K_1}\\mbox{ for }i\\in \\lbrace 4,6,7,8\\rbrace \\right)$ and the constants are defined in Claims REF , REF and REF .",
"Now, if $\\epsilon ^*\\le \\epsilon _0$ , then Claim REF , Claim REF and Claim REF apply.",
"We claim that for all $s\\in [s^*,\\bar{s}]$ , $|| q(s)||_\\mathcal {H}\\le K_0 || q(s^*)||_\\mathcal {H}e^{-\\mu _6(s-s^*)}\\le K_0 K_1 \\epsilon ^*e^{-\\mu _6(s-s^*)}=\\frac{\\epsilon }{2}e^{-\\mu _6(s-s^*)}.$ Indeed, if $s\\in [s^*,\\min (s_2^*+\\sigma _0,\\bar{s})]$ , then, this comes from $(ii)$ of Claim REF or $(i)$ of Claim REF and the definition of $K_0$ (REF ).",
"Now, if $s_2^*+\\sigma _0<\\bar{s}$ and $s\\in [s_2^*+\\sigma _0,\\bar{s}]$ , then we have from (REF ) and the definition of $\\sigma _0$ , $b(s)\\le \\frac{a(s)}{2K_5 }$ on the one hand.",
"On the other hand, from $(iii)$ in Claim REF , we have $a(s)\\le K_5 b(s)$ , hence, $a(s)=b(s)=0$ and from (REF ), $q(y,s)\\equiv 0$ , hence (REF ) is satisfied trivially.",
"In particular, we have for all $s\\in [s~^*,\\bar{s}],\\, ||q||_\\mathcal {H}\\le \\frac{\\epsilon }{2}$ and $ \\cos \\theta _i\\ge 1-C \\frac{\\epsilon ^2}{\\mu _6^2}\\ge \\frac{3}{4}$ , hence, by definition of $\\bar{s}$ given right before (REF ), this means that $\\bar{s}=\\infty $ .",
"From $(i)$ of Claim REF and (), we have $\\forall s \\ge s^*,||q(s)||_\\mathcal {H}\\le \\frac{\\epsilon }{2} e^{-\\mu _6(s-s^*)}\\mbox{ and }|\\theta ^{\\prime }(s)|+|\\lambda ^{\\prime }(s)|\\le K_4^2 \\frac{\\epsilon ^2}{4} e^{-2\\mu _6(s-s^*)},$ where, $\\theta (s)=(\\theta _2 (s),...,\\theta _m (s))$ .",
"Hence, there is $\\theta _\\infty \\in \\mathbb {R}^{m-1}$ , $\\lambda _\\infty $ in $\\mathbb {R}$ such that $\\theta (s) \\rightarrow \\theta _\\infty $ , $\\lambda (s) \\rightarrow \\lambda _\\infty $ as $s\\rightarrow \\infty $ and $\\forall s \\ge s^*,| \\lambda _\\infty -\\lambda (s) |\\le C_1\\epsilon ^{*2} e^{-2\\mu _6(s-s^*)}=C_2\\epsilon ^{2} e^{-2\\mu _6(s-s^*)}$ $\\forall s \\ge s^*,| \\theta _\\infty -\\theta (s) |\\le C_1\\epsilon ^{*2} e^{-2\\mu _6(s-s^*)}=C_2\\epsilon ^{2} e^{-2\\mu _6(s-s^*)}$ for some positive $C_1$ and $C_2$ .",
"Taking $s=s^*$ here, we see that $| \\lambda _\\infty -\\lambda ^* |+|\\theta _\\infty |\\le C_0\\epsilon ^* ,$ where $\\Omega =R_{\\theta _\\infty }(e_1)$ .",
"If $d_\\infty =\\tanh \\lambda _\\infty ,$ then we see that $|d_\\infty -d^*|\\le C_3 (1-d^{*2})\\epsilon ^*.$ Using the definition of $q$ (REF ), (REF ), (REF ) and (REF ) we write $&\\,&\\Bigg |\\Bigg |\\begin{pmatrix}w(s)\\\\\\partial _s w(s)\\end{pmatrix}-\\begin{pmatrix}\\kappa (d _\\infty ,\\cdot )\\Omega _\\infty \\\\0\\end{pmatrix}\\Bigg |\\Bigg |_\\mathcal {H}\\\\&\\le &\\Bigg |\\Bigg |\\begin{pmatrix}w(s)\\\\\\partial _s w(s)\\end{pmatrix}-\\begin{pmatrix}\\kappa (d (s),\\cdot )\\Omega _\\infty \\\\0\\end{pmatrix}\\Bigg |\\Bigg |_\\mathcal {H}+||(\\kappa (d (s),\\cdot )-\\kappa (d _\\infty ,\\cdot )) \\Omega _\\infty ||_{\\mathcal {H}_0}\\\\&+&||\\kappa (d_\\infty ,\\cdot )||_{\\mathcal {H}_0} |R_{\\theta (s)}(e_1)-R_{\\theta _{\\infty }}(e_1)|\\\\&\\le & ||q(s)||_\\mathcal {H}+C|\\lambda _\\infty -\\lambda (s)|+C|\\theta _\\infty -\\theta (s)|\\le C_4 \\epsilon ^*e^{-\\mu _6(s-s^*)},$ where, we used the fact that $\\theta \\in \\mathbb {R}^{m-1}\\mapsto \\mathcal {O}^m$ is a Lipschitz function (see (REF ) to be convinced) and $\\lambda \\in \\mathbb {R}\\mapsto \\kappa (d,\\cdot )\\in \\mathcal {H}_0$ is also Lipschitz, where $d=\\tanh \\lambda $ (see REF ).This concludes the proof of Theorem REF in the case where $\\Omega ^*=e_1$ (see (REF )).",
"From rotataion invariance of equation (REF ), this yields the conclusion of Theorem REF in the general case.",
"A some technical estimates In this section, we give the proof of estimate (REF ) and Lemma REF .",
"Proof of estimate (REF ): Using (REF ), we see that $\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= R_2 ... R_{j-1}\\frac{\\partial R_j}{\\partial \\theta _j}R_{j+1}...R_m.$ From (REF ), we see that $\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= \\Pi _j\\circ R_{j}(\\theta _j+\\frac{\\pi }{2})= R_{j}(\\theta _j+\\frac{\\pi }{2})\\circ \\Pi _j,$ where $\\Pi _j$ is the orthogonal projection on the plane spanned by $e_1$ and $e_j$ , and the rotation $ R_{j}(\\alpha )$ is given by considering the matrix of $R_j$ defined in (REF ), and changing $\\theta _j$ into $\\alpha $ .",
"Since $\\partial R_{j}^{-1 }\\partial R_{j}(\\theta _j+\\frac{\\pi }{2})=\\partial R_{j}(\\frac{\\pi }{2}),$ it follows from $(\\ref {R_theta})$ and $(\\ref {60,5})$ that $R_{\\theta }^{-1}\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= R_m^{-1}...R_{j+1}^{-1} R_{j}(\\frac{\\pi }{2}) \\Pi _j R_{j+1}...R_m.$ By the same argument, we drive that $\\frac{\\partial R_{\\theta }^{-1}}{\\partial \\theta _j}\\partial R_{\\theta }$ has the same expression, thus, (REF ) holds from (REF ) and (REF ).",
"Now, we give the proof of Lemma REF .",
"Proof of Lemma REF: i) We first give the expression of the $m\\times m$ matrix $R_\\theta $ defined (REF ).",
"Indeed, using (REF ) and (REF ), we have: $R_\\theta \\equiv \\begin{pmatrix}\\varphi _{2,m}&-\\sin \\theta _2&\\cdots &-\\sin \\theta _j \\varphi _{2,l-1}&\\cdots &-\\sin \\theta _m \\varphi _{2,m-1}\\\\ \\\\\\end{pmatrix}\\sin \\theta _2 \\varphi _{3,m}&\\cos \\theta _2& & & &.\\\\$ $\\vdots $ R, k,l $\\vdots $ $\\vdots $ j $\\vdots $ k k+1,m 0 $\\vdots $ $\\vdots $ $\\vdots $ mm+1,m0 0m where for $k\\ge 1$ , $l\\ge 2$ : $ R_{\\theta , k, l}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _l \\varphi _{2,l-1}\\; \\mbox{ if } k=1\\\\-\\sin \\theta _k \\sin \\theta _l \\varphi _{k+1,l-1}\\; \\mbox{ if } 2\\le k\\le l-1\\\\\\cos \\theta _k\\; \\mbox{ if } k=l\\\\\\; 0\\; \\; \\mbox{ if } k\\ge l+1\\end{array}\\right.",
"$ with $ \\varphi _{ k, l}=\\left\\lbrace \\begin{array}{l}\\prod \\limits _{n=k}^{l}\\cos \\theta _n \\mbox{ if } k \\le l\\\\1\\mbox{ if } k\\ge l+1.\\end{array}\\right.$ In fact, we will prove the following identities, which imply item $i)$ : $(A)$ For all $i$ ,$j$ $\\in \\lbrace 2,...,m\\rbrace $ , such that $ i\\ne j$ , we have $<e_j,A_i (e_1)>=0.$ $(B)$ For all $i\\in \\lbrace 2,...,m\\rbrace $ $< e_1,A_i e_1>=0.$ $(C)$ For all $i\\in \\lbrace 2,...,m\\rbrace $ , we have $< e_i,A_i e_1>=\\varphi _{ i+1, m},$ where $A_i$ and $\\varphi _{i+1,m}$ are given in (REF ) and (REF ) .",
"$\\blacktriangleright $Proof of $(A)$.",
"Let $i$ ,$j$ $\\in \\lbrace 2,...,m\\rbrace $ , such that $ i\\ne j$ .",
"The idea is to compute $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>$ instead of $<e_j,A_i e_1 >$ .",
"In fact, using the conservation of the inner product after a rotation and the fact that $A_i=R_{\\theta }^{-1} \\frac{\\partial R_\\theta }{\\partial \\theta _i}$ (by (REF )), we have: $<e_j,A_i e_1 >=<R_\\theta e_j,R_\\theta A_i e_1>=< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ In the following, we distinguish two cases: - Case 1: $i\\le j-1$ , - Case 2: $i\\ge j+1$ .",
"We first handle Case 1.",
"Case 1: $i\\le j-1$ .",
"Using (REF ) and its derivative with respect to $\\theta _i$ , we write: $ R_\\theta e_j=(R_\\theta e_j)_{k=1,...,m}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _j \\varphi _{2,j-1}, \\mbox{ if } k=1\\\\-\\sin \\theta _k \\sin \\theta _j \\varphi _{k+1,j-1}\\; \\mbox{ if } 2\\le k\\le j-1\\\\\\cos \\theta _j\\; \\mbox{ if } k=j\\\\\\; 0\\; \\; \\mbox{ if } k\\ge j+1\\end{array}\\right.",
"$ and $ \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1= (\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1)_{ k=1,...., m}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _i \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\mbox{ if } k =1\\\\-\\sin \\theta _i \\sin \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\mbox{ if } 2\\le k\\le i-1\\\\\\cos \\theta _i \\varphi _{i+1,m}, \\mbox{ if } k=i\\\\0\\mbox{ if } k\\ge i+1.\\end{array}\\right.",
"$ Therefore, $&&< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sum _{k=1}^{m}R_{\\theta ,k,j}{\\frac{\\partial R_\\theta }{\\partial \\theta _i}}_{k,1}\\\\&=&\\sin \\theta _i \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\sin \\theta _j \\varphi _{2,j-1}+\\sum _{k=2}^{i-1}(\\sin \\theta _k \\sin \\theta _i \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\sin \\theta _k \\sin \\theta _j \\varphi _{k+1,j-1})\\\\&-& \\cos \\theta _i \\varphi _{i+1,m} \\sin \\theta _i \\sin \\theta _j \\varphi _{i+1,j-1}\\\\&=& \\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}+ \\sum _{k=2}^{i-1}(\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}) - \\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} \\right).$ In order to transform the sum term in the previous identity, we make in the following a finite induction where the parameter $q$ decreases from $i-1$ to 1: Lemma A.1 We have: $ \\forall q \\in \\lbrace 1,...,i-1\\rbrace ,\\;\\sum _{k=2}^{i-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} &\\times \\varphi _{k+1,j-1}-\\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} =\\\\\\sum _{k=2}^{q}&\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\frac{ \\varphi _{q+1,m} \\varphi _{q+1,j-1} }{\\cos \\theta _i}.",
"$ Remark: If $q=1$ , the sum in the right hand side is naturally zero.",
"See below.",
"Applying this Lemma, we conclude the proof of $(A)$ in Case 1 (i. e. when $i\\le j-1$ ).",
"Indeed, from (REF ) and Lemma REF with $q=1$ we write $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1} - \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}\\right)=0.$ It remains now to prove Lemma REF .",
"[Proof of Lemma REF ] First, we give the following: Claim A.2 We have $\\varphi _{i,j-1}=\\cos \\theta _i \\varphi _{i+1,j-1}.$ Since $i\\le j-1$ , we have two cases: - If $i\\le j-2$ : trivial.",
"- If $i=j-1$ : $\\varphi _{i,j-1}=\\varphi _{i,i} =\\cos \\theta _i $ and $\\varphi _{i+1,j-1} = \\varphi _{i+1,i}=1 $ , and the result follows.",
"Now, we are ready to start the proof of Lemma REF .",
"Let us prove the result using an induction with a decreasing index.",
"For $q=i-1$ , (REF ) is satisfied using Claim REF .",
"Assume now that (REF ) is true for $q=i-1,...,2$ and let us prove it for $q-1$ .",
"Using (REF ) with $q$ , we write $&&\\sum _{k=2}^{i-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1} - \\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} =\\\\&& \\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}+ \\sin \\theta _q^2 \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1} -\\frac{ \\varphi _{q+1,m} \\varphi _{q+1,j-1} }{\\cos \\theta _i} =\\\\&& \\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}- \\cos \\theta _q^2 \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1}= \\\\&&\\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1} -\\frac{ \\varphi _{q,m} \\varphi _{q,j-1} }{\\cos \\theta _i}.$ Thus, (REF ) is satisfied for $q-1$ .",
"This concludes the proof of Lemma REF and identity $(A)$ when $i\\le j-1$ .",
"Now, we handle Case 2.",
"- Case 2: $i\\ge j+1$ .",
"Using (REF ) and (REF ), we write: $&&< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sum _{k=1}^{m}R_{\\theta ,k,j}{\\frac{\\partial R_\\theta }{\\partial \\theta _i}}_{k,1}\\\\&=& \\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}+ \\sum _{k=2}^{j-1}(\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}) - \\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} \\right).$ In order to transform the sum term in the previous identity, we make in the following a finite induction where the parameter $q$ decreases from $j-1$ to 1: Lemma A.3 We have: $ \\forall q \\in \\lbrace 1,...,j-1\\rbrace ,\\;\\sum _{k=2}^{j-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} &\\times \\varphi _{k+1,j-1}-\\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} =\\\\\\sum _{k=2}^{q}&\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i} .",
"$ Remark: If $q=1$ , the sum in the right hand side is naturally zero.",
"See below.",
"Applying this Lemma, we conclude the proof of $(A)$ in Case 2 (i. e. when $i\\ge j+1$ ).",
"Indeed, from (REF ) and Lemma REF with $q=1$ we write $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1} - \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}\\right)=0.$ It remains now to prove Lemma REF .",
"[Proof of Lemma REF ] We prove the result using an induction with a decreasing index.",
"For $q=j-1$ , (REF ) is satisfied.",
"Assume now that (REF ) is true for $q=j-1,...,2$ and let us prove it for $q-1$ .",
"Using (REF ) with $q$ , we write $&&\\sum _{k=2}^{j-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} =\\\\&&\\sum _{k=2}^{q}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i}= \\\\&&\\sum _{k=2}^{q-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}+\\sin ^2 \\theta _q \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i} =\\\\&&\\sum _{k=2}^{q-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}- \\frac{ \\varphi _{q,m}}{\\cos \\theta _i}\\varphi _{q,j-1}.$ Thus, (REF ) is satisfied for $q-1$ .",
"This concludes the proof of Lemma REF .",
"$\\blacktriangleright $Proof of $(B)$: As for (REF ) we have: $<e_1,A_i e_1 >=<R_\\theta e_1, \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ From (REF ), we have: $R_\\theta e_1=( \\varphi _{2,m},\\sin \\theta _2\\varphi _{3,m}\\,\\cdots , \\sin \\theta _{i} \\varphi _{i+1,m} ,\\cdots ,\\sin \\theta _m).$ Therefore, using (REF ), we have: $< R_\\theta e_1,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=-\\cos \\theta _i\\sin \\theta _i\\left((\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 +\\sum _{k=2}^{i-1}\\sin ^2\\theta _k (\\frac{\\displaystyle \\varphi _{k+1,m}}{\\cos \\theta _i})^2 \\right)+\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2.$ In order to transform the sum term in the previous identity, we make in the following a finite induction: Lemma A.4 We have: $\\forall q\\in \\lbrace 2,..., i-1\\rbrace ,\\;\\displaystyle (\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 +\\sum _{l=2}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2=(\\frac{\\displaystyle \\varphi _{q,m}}{\\cos \\theta _i})^2 +\\sum _{l=q}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2.$ Remark: If $q=i$ , the sum in the right hand side is naturally zero.",
"Using Lemma REF with $q=i$ we get $< R_\\theta e_1,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=-\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2 +\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2=0,$ which yields the result.",
"In order to conclude $(B)$ we give the proof of Lemma REF .",
"[Proof of Lemma REF ] We proceed by induction for $q\\in \\lbrace 2,..., i-1\\rbrace $ .",
"For $q=2$ , (REF ) is satisfied.",
"Assume that (REF ) is true for $q=2,...,i-1$ and prove it for $q+1$ .",
"Using (REF ) with $q$ , we write $\\displaystyle (\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 &+&\\sum _{l=2}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2=(\\frac{\\displaystyle \\varphi _{q,m}}{\\cos \\theta _i})^2 +\\sum _{l=q}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2\\\\&=&\\cos ^2\\theta _q (\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2 +\\sin ^2\\theta _q (\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2\\\\&=&(\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2 +\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2.$ Thus (REF ) is satisfied for $q+1$ .",
"This concludes the proof of Lemma REF and identity $(B)$ .",
"$\\blacktriangleright $Proof of $(C)$: Consider $i\\in \\lbrace 2,...,m\\rbrace $ .",
"As for (REF ) we have: $<e_i,A_i e_1 >=<R_\\theta e_i, \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ Using (REF ) and (REF ) $<e_i,A_i e_1 >=\\sin ^2\\theta _i \\varphi _{i+1,m} \\left( \\varphi _{2,i-1}^2+\\sum _{k=2}^{i-1}\\sin ^2\\theta _k \\varphi _{k+1,i-1}^2\\right)+\\cos ^2 \\theta _i \\varphi _{i+1,m}.$ In order to transform the sum term in the previous identity, we make in the following a finite induction: Lemma A.5 We have: $\\forall q\\in \\lbrace 2,..., i\\rbrace $ , $\\varphi _{2,i-1}^2+\\sum _{l=2}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2=\\varphi _{q,i-1}^2+\\sum _{l=q}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2.$ Remark: If $q=i$ , the sum in the right hand side is naturally zero.",
"From (REF ) and (REF ) with $q=i$ we get $< R_\\theta e_i,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin ^2\\theta _i \\varphi _{i+1,m}+\\cos ^2\\theta _i \\varphi _{i+1,m}= \\varphi _{i+1,m}.$ which yields the result.",
"In order to conclude $(C)$ we give the proof of Lemma REF .",
"[Proof of Lemma REF ] We proceed by induction for $q\\in \\lbrace 2,..., i\\rbrace $ .",
"For $q=2$ , (REF ) is satisfied.",
"Assume now that (REF ) is true for $q=2,...,i-1$ and prove it for $q+1$ .",
"Using (REF ) with $q$ , we write $\\varphi _{2,i-1}^2+\\sum _{l=2}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2&=&\\varphi _{q,i-1}^2+\\sum _{l=q}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2\\\\&=&\\cos ^2\\theta _q \\varphi _{q+1,i-1}^2+\\sin ^2\\theta _q \\varphi _{q+1,i-1}^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2\\\\&=&\\varphi _{q+1,i-1}^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2.$ Thus (REF ) is satisfied for $q+1$ .",
"This concludes the proof of Lemma REF .",
"ii)We recall from (REF ) that we have $\\frac{\\partial R_\\theta }{\\partial \\theta _j}=R_2 \\cdots R_{j-1} \\frac{\\partial R_j}{\\partial \\theta _j} R_{j+1} \\cdots R_m,$ so by (REF ), $A_j$ is given explicitly by $A_j=R_m^{-1} R_{m-1}^{-1} \\cdots R_{j}^{-1} \\frac{\\partial R_j}{\\partial \\theta _j} R_{j+1} \\cdots R_m.$ From a straightforward geometrical observation, we can see that the rotation conserves the euclidien norm in $\\mathbb {R}^m$ .",
"For $\\frac{\\partial R_j}{\\partial \\theta _j} $ , it can be seen as a composition of a projection on the plane $(e_1,e_j)$ and a rotation with angle $\\theta _j+\\frac{\\pi }{2}$ , which decreases the norm.",
"This concludes the proof of Lemma REF .",
"Address: Université de Cergy-Pontoise, Laboratoire Analyse Géometrie Modélisation, CNRS-UMR 8088, 2 avenue Adolphe Chauvin 95302, Cergy-Pontoise, France.",
"e-mail: [email protected] Université Paris 13, Institut Galilée, Laboratoire Analyse Géometrie et Applications, CNRS-UMR 7539, 99 avenue J.B. Clément 93430, Villetaneuse, France.",
"e-mail: [email protected]"
],
[
"Characterization of the set of stationary solutions",
"In this section, we prove Proposition REF which characterizes all $\\mathcal {H}_0$ solutions of $\\frac{1}{\\rho } (\\rho (1-y^2) w^{\\prime })^{\\prime }-\\frac{2(p+1)}{(p-1)^2}w+|w|^{p-1}w=0,$ the stationary version of (REF ).",
"Note that since 0 and $\\kappa _0 \\Omega $ are trivial solutions to equation (REF ) for any $\\Omega \\in S^{m-1}$ , we see from a Lorentz transformation (see Lemma 2.6 page 54 in [18]) that $\\mathcal {T}_d e^{i\\theta }\\kappa _0=\\kappa (d,y)$ is also a stationary solution to (REF ).",
"Let us introduce the set $S\\equiv \\lbrace 0, \\kappa (d,.",
")\\Omega , |d|<1, \\Omega \\in S^{m-1}\\rbrace .$ Now, we prove Proposition REF which states that there are no more solutions of (REF ) in $\\mathcal {H}_0$ outside the set $S$ .",
"We first prove $(ii)$ , since its proof is short.",
"(ii) Since we clearly have from the definition (REF ) that $E(0,0)=0$ , we will compute $E(\\Omega \\kappa (d,.",
"),0)$ .",
"From (REF ) and the proof of the real case treated in page 59 in [18], we see that $ E(\\kappa (d,.",
")\\Omega ,0)= E(\\kappa (d,.",
"),0)=E(\\kappa _0,0)>0.$ Thus, (REF ) follows.",
"(i) Consider $w \\in \\mathcal {H}_0$ an $ \\mathbb {R}^m$ non-zero solution of (REF ).",
"Let us prove that there are some $d\\in (-1,1)$ and $\\Omega \\in \\mathbb {S}^{m-1}$ such that $w=\\kappa (d,.",
")\\Omega .$ For this purpose, define $\\xi =\\frac{1}{2} \\log \\left(\\frac{1+y}{1-y}\\right) (\\mbox{that is } y=\\tanh \\xi )\\mbox{ and } \\bar{w}( \\xi )=w(y) (1-y^2)^\\frac{1}{p-1}.$ As in the real case, we see from straightforward calculations that $\\bar{w}\\lnot \\equiv 0$ is a $H^1({R})$ solution to $\\partial _\\xi ^2\\bar{w} +|\\bar{w}|^{p-1}\\bar{w}-\\frac{4}{(p-1)^2}\\bar{w}=0,\\,\\forall \\xi \\in {R}.$ Our aim is to prove the existence of $\\Omega \\in \\mathbb {S}^{m-1}$ and $\\xi _0 \\in \\mathbb {R}$ such that $\\bar{w}(\\xi )=\\Omega \\bar{k} (\\xi +\\xi _0)$ where $\\bar{k}(\\xi )=\\frac{\\kappa _0}{\\cosh ^\\frac{2}{p-1}(\\xi )}.$ Since $\\bar{w} \\in H^1(R) \\subset C^\\frac{1}{2}(R),$ we see that $\\bar{w}$ is a strong $C^2$ solution of equation (REF ).",
"Since $\\bar{w}\\lnot \\equiv 0$ , there exists $\\xi _0 \\in \\mathbb {R}$ such that $\\bar{w}(\\xi _0)\\ne 0$ .",
"By invariance of (REF ) by translation, we may suppose that $\\xi _0=0$ .",
"Let $G^*=\\left\\lbrace \\xi \\in \\mathbb {R} \\,|\\, \\bar{w}(\\xi )\\ne 0 \\right\\rbrace ,$ a nonempty open set by continuity.",
"Note that $G^*$ contains some non empty interval $I$ containing 0.",
"We introduce $\\rho $ and $\\Omega $ by $\\rho =|\\bar{w}|,\\,\\Omega =\\frac{\\bar{w}}{|\\bar{w}|},\\mbox{ whenever }\\xi \\in G^*.$ From equation (REF ), we see that $\\rho ^{\\prime \\prime }\\Omega +2 \\rho ^{\\prime } \\Omega ^{\\prime }+ \\rho \\Omega ^{\\prime \\prime }+\\rho ^p\\Omega - \\frac{4}{(p-1)^2} \\rho \\Omega =0.$ Now, since $|\\Omega |=1$ , we immediately see that $ \\Omega ^{\\prime }.",
"\\Omega =0$ and $ \\Omega ^{\\prime \\prime }.",
"\\Omega +|\\Omega ^{\\prime }|^2=0$ .",
"Let $H(\\xi )=|\\Omega ^{\\prime }|^2$ .",
"Projecting equation (REF ) according to $\\Omega $ and $\\Omega ^{\\prime }$ we see that $\\forall \\xi \\in G^*,\\,\\left\\lbrace \\begin{array}{l}\\rho ^{\\prime \\prime }(\\xi )-\\rho (\\xi ) H(\\xi ) -c_0\\rho (\\xi )+\\rho (\\xi )^p=0,\\; c_0= \\frac{4}{(p-1)^2}\\\\4\\rho ^{\\prime }(\\xi )H(\\xi )+\\rho (\\xi ) H^{\\prime }(\\xi )=0\\end{array}\\right.$ Integrating the second equation on the interval $I\\subset G^*$ , we see that for all $\\xi \\in I$ , $H(\\xi )=\\frac{H(0)(\\rho (0))^4}{(\\rho (\\xi ))^4}$ .",
"Plugging this in the first equation, we get $\\forall \\xi \\in I,\\,\\rho ^{\\prime \\prime }(\\xi )-\\frac{\\mu }{(\\rho (\\xi ))^3}-c_0\\rho (\\xi )+\\rho ^p(\\xi )=0 \\mbox{ where }\\mu =H(0)(\\rho (0))^4.$ Now let $G= \\left\\lbrace \\xi \\in G^*, \\forall \\xi ^{\\prime }\\in I_\\xi ,\\, H(\\xi ^{\\prime })=\\frac{H(0)\\rho (0)^4}{\\rho (\\xi ^{\\prime })^4} \\right\\rbrace ,$ where $I_\\xi =[0,\\xi )$ if $\\xi \\ge 0$ or $I_\\xi =(\\xi ,0]$ if $\\xi \\le 0$ .",
"Note that $I \\subset G$ .",
"Now, we give the following: Lemma 2.1 There exists $\\epsilon _0 >0$ such that $\\forall \\xi \\in G, \\,\\forall \\xi ^{\\prime }\\in I_\\xi ,\\, 0< \\epsilon _0 \\le |\\bar{w}(\\xi ^{\\prime })|\\le \\frac{1}{\\epsilon _0}.$ The proof is the same as in the complex-case, see page 5898 in [4].",
"But for the reader's convenience and for the sake of self-containedness, we recall it here.",
"Take $\\xi \\in G$ .",
"By definition (REF ) of $G$ , we see that equation (REF ) is satisfied for all $\\xi ^{\\prime }\\in I_\\xi $ .",
"Multiplying $\\rho ^{\\prime \\prime }(\\xi )-\\frac{\\mu }{(\\rho (\\xi ))^3}-c_0\\rho (\\xi )+\\rho ^p(\\xi )=0 $ by $\\rho ^{\\prime }$ and integrating between 0 and $\\xi $ , we get: $\\forall \\xi \\in I_\\xi ,\\, \\mathcal {E}(\\xi ^{\\prime })=\\mathcal {E}(0), \\mbox{ where }\\mathcal {E}(\\xi ^{\\prime })=\\frac{1}{2}(\\rho ^{\\prime }(\\xi ^{\\prime }))^2+\\frac{\\mu }{2( (\\rho (\\xi ^{\\prime }))^2}-\\frac{c_0}{2}\\rho ^2(\\xi ^{\\prime })+\\frac{\\rho ^{p+1}(\\xi ^{\\prime })}{p+1},$ or equivalently, $\\forall \\xi ^{\\prime } \\in I_\\xi ,\\, F(\\rho (\\xi ^{\\prime }))=\\frac{1}{2} \\rho ^{\\prime }(\\xi ^{\\prime })^2\\ge 0 \\mbox{ where }F(r)=\\frac{\\mu }{2r^2}+\\frac{c_0}{2} r^2-\\frac{r^{p+1}}{p+1}+\\mathcal {E}(0).$ Since $F(r)\\rightarrow -\\infty $ as $r \\rightarrow 0$ or $r \\rightarrow \\infty $ , there exists $\\epsilon _0=\\epsilon _0(\\mu , E(0)) >0$ such that $\\epsilon _0\\le \\rho (\\xi ^{\\prime }) \\le \\frac{1}{\\epsilon _0}$ , which yields to the conclusion of the Claim REF .",
"We claim the following: Lemma 2.2 It holds that $G=\\mathbb {R}$ .",
"Note first that by construction, $G$ is a nonempty interval (note that $0\\in I \\subset G$ where $I$ is defined right before (REF )).",
"We have only to prove that $\\sup G=+\\infty $ , since the fact that $\\inf G=-\\infty $ can be deduced by replacing $\\bar{w}(\\xi )$ by $\\bar{w}(-\\xi )$ .",
"By contradiction, suppose that $\\sup G=a<+\\infty $ .",
"First of all, by Lemma REF , we have for all $ \\xi ^{\\prime } \\in [0,a), 0< \\epsilon _0 \\le |\\bar{w}(\\xi ^{\\prime })|\\le \\frac{1}{\\epsilon _0}$ .",
"By continuity, this holds also for $\\xi ^{\\prime }=a$ , hence, $\\bar{w}(a)\\ne 0$ , and $a\\in G^*$ .",
"Furthermore, by definition of $G$ and continuity, we see that $\\forall \\xi \\in [0, a], H(\\xi )=\\frac{H(0)\\rho (0)^4}{\\rho (\\xi )^4}.$ Therefore, we see that $a\\in G$ .",
"By continuity, we can write for all $\\xi \\in (a-\\delta ,a+\\delta ),$ where $\\delta >0 $ is small enough, $ \\left\\lbrace \\begin{array}{l}\\rho ^{\\prime \\prime }(\\xi )-\\rho (\\xi )H(\\xi )-c_0\\rho (\\xi )+\\rho (\\xi )^p=0,\\; c_0= \\frac{4}{(p-1)^2}\\\\4\\rho ^{\\prime }(\\xi )H(\\xi )+\\rho (\\xi ) H^{\\prime }(\\xi )=0.\\end{array}\\right.$ From the second equation and (REF ) applied with $\\xi =a$ , we see that $H(\\xi )=\\frac{H(a)(\\rho (a))^4}{(\\rho (\\xi ))^4}=\\frac{H(0)(\\rho (0))^4}{(\\rho (\\xi ))^4}$ .",
"Therefore, it follows that $(a, a+\\delta )\\in G$ , which contradicts the fact that $a=\\sup G$ .",
"Note from Lemma REF that (REF ) and (REF ) holds for all $\\xi \\in {R}$ .",
"We claim that $H(0)=0$ .",
"Indeed, if not, then by (REF ), we have $\\mu \\ne 0$ , and since $G={R}$ , we see from Lemma REF that for all $\\xi \\in {R}$ , $|\\bar{w}(\\xi )|\\ge \\epsilon _0$ , therefore $w \\notin L^2({R})$ , which contradicts the fact that $\\bar{w}\\in H^1(\\mathbb {R})$ .",
"Thus, $H(0)=0$ , and $\\mu =0$ .",
"By uniqueness of solutions to the second equation of (REF ), we see that $H(\\xi )=0$ for all $\\xi \\in {R}$ , so $\\Omega (\\xi )=\\Omega (0)$ , and $ \\left\\lbrace \\begin{array}{l}\\bar{w}(0)=\\rho (0) \\Omega (0)\\\\\\bar{w}^{\\prime }(0)=\\rho ^{\\prime }(0) \\Omega (0).\\end{array}\\right.$ Let $W$ be the maximal real-valued solution of $ \\left\\lbrace \\begin{array}{l}W^{\\prime \\prime }-c_0 W+|W|^{p-1}W=0\\\\W(0)=\\rho (0)\\\\W^{\\prime }(0)=\\rho ^{\\prime }(0).\\end{array}\\right.$ By uniqueness of the Cauchy problem of equation (REF ), we have for all $\\xi \\in \\mathbb {R}, \\bar{w}(\\xi )=W(\\xi ) \\Omega (0)$ , and as $\\bar{w}\\in H^1(\\mathbb {R})$ , $W$ is also in $H^1(\\mathbb {R})$ .",
"It is then classical that there exists $\\xi _0$ such that for all $\\xi \\in \\mathbb {R}$ , $W(\\xi )=\\bar{k }(\\xi +\\xi _0)$ (remember that $\\rho (0)>0$ , hence we only select positive solutions here).",
"In addition, for $ \\Omega _0= \\Omega (0)$ , $\\bar{w}(\\xi )=\\bar{k }(\\xi +\\xi _0)\\Omega _0$ .",
"Thus, for $d=\\tanh \\xi _0\\,\\in \\, (-1,1)$ and $y=\\tanh \\xi $ , we get $&\\bar{w}(\\xi )=\\kappa _0 \\left[1-\\tanh (\\xi +\\xi _0)^2\\right]^\\frac{1}{p-1}\\Omega _0=\\kappa _0 \\left[1-\\left(\\frac{\\tanh \\xi +\\tanh \\xi _0}{1+\\tanh \\xi \\tanh \\xi _0}\\right)^2\\right]^\\frac{1}{p-1}\\Omega _0\\\\&= \\kappa _0 \\left[1-\\left(\\frac{y+d}{1+dy}\\right)^2\\right]^\\frac{1}{p-1}\\Omega _0= \\kappa _0 \\left[\\frac{(1-d^2)(1-y^2)}{(1+dy)}^2\\right]^\\frac{1}{p-1}\\Omega _0= \\kappa (d,y) (1-y^2)^\\frac{1}{p-1}\\Omega _0.$ By (REF ), we see that $w(y)= \\kappa (d,y)\\Omega _0$ .",
"This concludes the proof of Proposition REF ."
],
[
"Outline of the proof of Theorem ",
"The proof of Theorem REF is not a simple adaptation of the complex-case to the vector-valued case, in fact, it involves a delicate modulation.",
"In this section, we will outline the proof, insisting on the novelties, and only recalling the features which are the same as in the real-valued complex-valued cases.",
"This section is organized as follows: - In Subsection 3.1, we linearize equation (REF ) around $\\kappa (d,y)e_1$ where $e_1=(1,0,...,0)$ and figure-out that, with respect to the complex-valued case, our linear operator is just a superposition of one copy of the real part operator, with $(m-1)$ copies of the imaginary part operator.",
"- In Subsection 3.2, we recall from [18] the spectral properties of the real-part operator.",
"- In Subsection 3.3, we recall from [4] the spectral properties of the imaginary-part operator.",
"- In Subsection 3.4, assuming that $\\Omega ^*=e_1$ (possible thanks to rotation invariance of (REF )), we introduce a modulation technique adapted to the vector-valued case.",
"This part makes the originality of our work with respect to the complex-valued case.",
"- In Subsection 3.5, we write down the equations satsified by the modulation parameters along with the PDE satisfied by $q(y,s)$ and its components.",
"- In Subsection 3.6, we conclude the proof of Theorem REF ."
],
[
" The linearized operator around a non-zero stationary solution",
"We study the properties of the linearized operator of equation (REF ) around the stationary solution $\\kappa (d,y)$ $(\\ref {defk})$ .",
"Let us introduce $q=(q_1,q_2)\\in \\mathbb {R}^m\\times \\mathbb {R}^m$ for all $ s\\in [s_0, \\infty )$ , for a given $s_0 \\in {R}$ , by $\\begin{pmatrix} w(y,s)\\\\\\partial _s w(y,s) \\end{pmatrix} =\\begin{pmatrix} \\kappa (d,y)e_1\\\\0\\end{pmatrix} +\\begin{pmatrix} q_1(y,s)\\\\q_2(y,s)\\end{pmatrix}.$ Let us introduce the coordinates of $q_1$ and $q_2$ by $q_1=(q_{1,1},q_{1,2},...,q_{1,m})$ , $q_2=(q_{2,1},q_{2,2},...,q_{2,m})$ .",
"We see from equation (REF ), that $q$ satisfies the following equation for all $s\\ge s_0$ : $ \\frac{\\partial }{\\partial s} \\begin{pmatrix} q_1\\\\q_2 \\end{pmatrix} = L_d\\begin{pmatrix}q_1\\\\q_2\\end{pmatrix}+\\begin{pmatrix}0\\\\ f_d(q_1)\\end{pmatrix},$ where $ L_d\\begin{pmatrix}q_1\\\\q_2\\end{pmatrix}=\\begin{pmatrix}q_2 \\\\\\mathcal {L}q_1+\\bar{\\psi }(d, y)q_{1,1}e_1+\\sum _{j=2}^{m} \\tilde{\\psi }(d, y)q_{1,j}e_j-\\frac{p+3}{p-1} q_2- 2 y \\partial _y q_2 \\end{pmatrix}, $ $\\bar{\\psi }(d,y) =p \\kappa (d, y)^{p-1}-\\frac{2(p+1)}{(p-1)^2}$ $\\tilde{\\psi }(d,y) = \\kappa (d, y)^{p-1}-\\frac{2(p+1)}{(p-1)^2}$ $f_d(q_1)={f_{d,1}}(q_1)e_1+ \\sum _{j=2}^{m} f_{d,j}(q_1)e_j,$ where $f_{d,1}(q_1)=|\\kappa (d, y)e_1+q_1|^{p-1}(\\kappa (d, y)+q_{1,1})-\\kappa (d, y)^{p}-p\\kappa ^{p-1}(d, y)q_{1,1}.$ $f_{d,j}( q_1)=|\\kappa (d, y)e_1+q_1|^{p-1}q_{1,j}-\\kappa ^{p-1}(d,y)q_{1,j}.$ Projecting (REF ) on the first coordinate, we get for all $s\\ge s_0$ : $\\frac{\\partial }{\\partial s}\\begin{pmatrix} q_{1,1}\\\\q_{2,1}\\end{pmatrix}= \\bar{L}_d \\begin{pmatrix} q_{1,1} \\\\q_{2,1} \\end{pmatrix}+\\begin{pmatrix}0\\\\ f_{d,1}(q_1)\\end{pmatrix},$ where $\\bar{L}_d $ is given by: $\\bar{L}_d \\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix}=\\begin{pmatrix}q_{2,1} \\\\\\mathcal {L} q_{1,1}+ \\bar{\\psi }(d,y) q_{1,1}-\\frac{p+3}{p-1} q_{2,1} - 2 y \\partial _y q_{2,1}\\end{pmatrix},$ Now, projecting equation (REF ) on the j-th coordinate with $j=2,..,m,$ we see that $\\frac{\\partial }{\\partial s}\\begin{pmatrix} q_{1,j}\\\\q_{2,j}\\end{pmatrix}= \\tilde{L}_d\\begin{pmatrix} q_{1,j} \\\\q_{2,j} \\end{pmatrix}+\\begin{pmatrix}0\\\\ f_{d,j}( q_1)\\end{pmatrix},$ where $\\tilde{L}_d \\begin{pmatrix} q_{1,j}\\\\q_{2,j} \\end{pmatrix}=\\begin{pmatrix}q_{2,j} \\\\\\mathcal {L} q_{1,j}+ \\tilde{\\psi }(d,y) q_{1,j}-\\frac{p+3}{p-1} q_{2,j} - 2 y \\partial _y q_{2,j}\\end{pmatrix},$ Remark: Our linearized operator $L_d$ is in fact diagonal in the sens that $ L_d\\begin{pmatrix}q_1\\\\q_2\\end{pmatrix}=\\bar{L}_d \\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix}e_1+ \\sum _{j=2}^{m} \\tilde{L}_d \\begin{pmatrix} q_{1,j}\\\\q_{2,j} \\end{pmatrix}e_j.$ We mention that for $j=1$ , equation (REF ) is the same as the equation satisfied by the real part of the solution in the complex case (see Section 3 page 5899 in [4]), whereas for $j=2,..,m$ , equation (REF ) is the same as the equation satisfied by the imaginary part of the solution operator in the complex case.",
"Thus, the reader will have no difficulty in adapting the remaining part of the proof to the vector-valued case.",
"Thus, the dynamical system formulation we performed when $m=2$ can be adapted straightforwardly to the case $m\\ge 3$ .",
"Note from (REF ) that we have $||q||_\\mathcal {H}=[\\phi (q, q)]^\\frac{1}{2}<+\\infty ,$ where the inner product $\\phi $ is defined by $\\phi (q, r)=\\phi \\left(\\begin{pmatrix} q_1\\\\q_2 \\end{pmatrix}, \\begin{pmatrix} r_1\\\\r_2 \\end{pmatrix}\\right)=\\int _{-1}^{1}(q_1.r_1+q^{\\prime }_1.",
"r^{\\prime }_1(1-y^2)+q_2.r_2)\\rho \\;dy.$ where $q_1.r_1=\\sum _{j=1}^m q_{1,j}.r_{1,j}$ is the standard inner product in $\\mathbb {R}^m$ , with similar expressions for $q^{\\prime }_1.r^{\\prime }_1$ and $q_2.r_2$ .",
"Using integration by parts and the definition of $\\mathcal {L}$ (REF ), we have the following: $\\phi (q, r)=\\int _{-1}^{1}(q_1\\cdot (-\\mathcal {L} r_1+ r_1)+q_2 \\cdot r_2)\\rho \\,dy.$ In the following two sections, we recall from [18] and [4] the spectral properties of $\\bar{L}_d$ and $\\tilde{L}_d$ ."
],
[
"Spectral theory of the operator $\\bar{L}_d$",
"From Section 4 in [18], we know that $\\bar{L}_d$ has two nonnegative eigenvalues $\\lambda =1$ and $\\lambda =0$ with eigenfunctions $\\bar{F}_1^d (y)= (1-d^2)^{\\frac{p}{p-1}}\\begin{pmatrix}(1+dy)^{-\\frac{p+1}{p-1}}\\\\(1+dy)^{-\\frac{p+1}{p-1}}\\end{pmatrix}\\mbox{and }\\;\\bar{F}_0^d (y)= (1-d^2)^{\\frac{1}{p-1}}\\begin{pmatrix}\\frac{y+d}{(1+dy)^\\frac{p+1}{p-1}}\\\\0\\end{pmatrix}.",
"$ Note that for some $C_0>0$ and any $\\lambda \\in \\lbrace 0,1\\rbrace $ , we have $\\forall |d|<1,\\;\\; \\frac{1}{C_0}\\le ||\\bar{F}_\\lambda ^d||_{\\mathcal {H}} \\le C_0 \\;\\mbox{ and }\\; ||\\partial _d \\bar{F}_\\lambda ^d||_{\\mathcal {H}} \\le \\frac{C_0}{1-d^2}.$ Also, we know that $\\bar{L}_d^*$ the conjugate operator of $\\bar{L}_d$ with respect to $\\phi $ is given by $\\bar{L}_d^*\\begin{pmatrix}r_1\\\\r_2\\end{pmatrix}= \\begin{pmatrix}\\bar{R}_d(r_2)\\\\-\\mathcal {L} r_1+r_1+\\frac{p+3}{p-1}r_2+2y r_2^{\\prime }-\\frac{8}{(p-1)}\\frac{r_2}{(1-y^2)}\\end{pmatrix}$ for any $(r_1, r_2)\\in (\\mathcal {D}(\\mathcal {L}))^2$ , where $r=\\bar{R}_d (r_2)$ is the unique solution of $-\\mathcal {L} r+r=\\mathcal {L} r_2+\\bar{\\psi } (d, y)r_2.$ Here, the domain $\\mathcal {D}(\\mathcal {L})$ of $\\mathcal {L}$ defined in (REF ) is the set of all $r \\in L_{\\rho }^2$ such that $\\mathcal {L} r \\in L_{\\rho }^2.$ Furthermore, $\\bar{L}_d^*$ has two nonnegative eigenvalues $\\lambda =0$ and $\\lambda =1$ with eigenfunctions $\\bar{W}_{\\lambda }^d$ such that $\\bar{W}_{1, 2}^d (y)= \\bar{c}_1 \\frac{(1-y^2)(1-d)^\\frac{1}{p-1}}{(1+dy)^\\frac{p+1}{p-1}},\\,\\bar{W}_{0, 2}^d (y)= \\bar{c}_0 \\frac{(y+d)(1-d)^\\frac{1}{p-1}}{(1+dy)^\\frac{p+1}{p-1}},$ withIn section 4 of [18], we had non explicit normalizing constants $\\bar{c}_\\lambda =\\bar{c}_\\lambda (d)$ .",
"In Lemma 2.4 in [24], the authors compute the explicit dependence of $\\bar{c}_\\lambda (d)$ .",
"$\\frac{1}{\\bar{c}_\\lambda }=2(\\frac{2}{p-1}+\\lambda )\\int _{-1}^1 (\\frac{y^2}{1-y^2} )^{1-\\lambda }\\rho (y) \\,dy,$ and $\\bar{W}_{\\lambda , 1}^d $ is the unique solution of the equation $-\\mathcal {L} r+ r =\\left(\\lambda -\\frac{p+3}{p-1}\\right) r_2- 2 y r^{\\prime }_2 + \\frac{8}{p-1} \\frac{r_2}{1-y^2}$ with $r_2= \\bar{W}_{\\lambda , 2}^d$ .",
"We also have for $\\lambda =0,1$ $||\\bar{W}_\\lambda ^d ||_{\\mathcal {H}}+ (1-d^2)||\\partial _d \\bar{W}_\\lambda ^d ||_{\\mathcal {H}} \\le C, \\forall |d|<1.$ Note that we have the following relations for $\\lambda =0$ or $\\lambda =1$ $\\phi (\\bar{W}_\\lambda ^d,\\bar{F_\\lambda ^d})=1\\mbox{ and }\\phi (\\bar{W}_\\lambda ^d,\\bar{F_{1-\\lambda } ^d})=0.$ Let us introduce for $\\lambda \\in \\lbrace 0, 1\\rbrace $ the projectors $\\bar{\\pi }_\\lambda (r)$ , and $\\bar{\\pi }_-^d(r)$ for any $r\\in \\mathcal {H}$ by $ \\bar{\\pi }_\\lambda ^d (r)=\\phi (\\bar{W}_\\lambda ^d, r),$ $ r=\\bar{\\pi }_0^d (r) \\bar{F}_0^d (y)+\\bar{\\pi }_1^d (r) \\bar{F}_1^d (y)+ \\bar{\\pi }_{-}^d (r),$ and the space $\\bar{\\mathcal {H}}_{-}^d \\equiv \\lbrace r \\in \\mathcal {H} \\,|\\, \\bar{\\pi }_1^d (r)=\\bar{\\pi }_0^d(r)=0\\rbrace .$ Introducing the bilinear form $\\bar{\\varphi }_{d} (q,r)&=&\\int _{-1}^{1} (-\\bar{\\psi }(d,y) q_1r_1+q_1^{\\prime } r_1^{\\prime }(1-y^2)+q_2 r_2 ) \\rho dy,$ where $\\bar{\\psi }(d,y)$ is defined in (REF ), we recall from Proposition 4.7 page 90 in [18] that there exists $C_0>0$ such that for all $|d|<1$ , for all $r\\in \\bar{\\mathcal {H}}^d_{-},$ $\\frac{1}{C_0} ||r||_{\\mathcal {H} }^2\\le \\bar{\\varphi }_d (r,r)\\le C_0 ||r||_\\mathcal {H} ^2.$ Furthermore, if $r\\in \\mathcal {H},$ then $\\frac{1}{C_0} ||r||_{\\mathcal {H} }\\le \\left(|\\bar{\\pi }_{0}^d (r)| +|\\bar{\\pi }_{1}^d (r)| +\\sqrt{ \\bar{\\varphi }_d (r_{-},r_{-})}\\right)\\le C_0 ||r||_\\mathcal {H}\\mbox{ where }r_{-}=\\bar{\\pi }_{-}^d (r) .$ In the following section we recall from [4] the spectral properties of $\\tilde{L}_d$ ."
],
[
"Spectral theory of the operator $\\tilde{L}_d$",
"From Section 3 in [4], we know that $\\tilde{L}_d$ has one nonnegative eigenvalue $\\lambda =0$ with eigenfunction $\\tilde{F}_0^d(y)= \\begin{pmatrix}\\kappa (d,y)\\\\0\\end{pmatrix}.$ Note that for some $C_0>0$ we have $\\forall |d|<1,\\;\\; \\frac{1}{C_0}\\le ||\\tilde{F}_0^d||_{\\mathcal {H}} \\le C_0 \\;\\mbox{ and }\\; ||\\partial _d \\tilde{F}_0^d||_{\\mathcal {H}} \\le \\frac{C_0}{1-d^2}.$ We know also that the operator $\\tilde{L}_d^*$ conjugate of $\\tilde{L}_d$ with respect to $\\phi $ is given by $\\tilde{L}_d^*\\begin{pmatrix}r_1\\\\r_2\\end{pmatrix}= \\begin{pmatrix}\\tilde{R}_d(r_2)\\\\-\\mathcal {L} r_1+r_1+\\frac{p+3}{p-1}r_2+2y r_2^{\\prime }-\\frac{8}{(p-1)}\\frac{r_2}{(1-y^2)}\\end{pmatrix}$ for any $(r_1, r_2)\\in (\\mathcal {D}(\\mathcal {L}))^2$ , where $r=\\tilde{R}_d (r_2)$ is the unique solution of $-\\mathcal {L} r+r=\\mathcal {L} r_2+\\tilde{\\psi } (d, y)r_2.",
"$ Furthermore, $\\tilde{L}_d^*$ have one nonnegative eigenvalue $\\lambda =0$ with eigenfunction $\\tilde{W}_0^d$ such that $\\tilde{W}_{0, 2}^d (y)= \\tilde{c}_0 \\kappa (d, y) \\mbox{ and }\\frac{1}{\\tilde{c}_0}=\\frac{4\\kappa _0^2}{p-1} \\int _{-1}^1 \\frac{\\rho (y)}{1-y^2}dy$ and $\\tilde{W}_{0, 1}^d $ is the unique solution of the equation $-\\mathcal {L} r+ r =-\\frac{p+3}{p-1} r_2- 2 y r^{\\prime }_2 + \\frac{8}{p-1} \\frac{r_2}{1-y^2}$ with $r_2= \\tilde{W}_{0, 2}^d$ .",
"We also have for $\\lambda =0,1$ $||\\tilde{W}_0 ^d ||_{\\mathcal {H}}+ (1-d^2)||\\partial _d \\tilde{W}_\\lambda ^d ||_{\\mathcal {H}} \\le C,\\; \\forall |d|<1.$ Moreover, we have $\\phi (\\tilde{W}_0^d,\\tilde{F_0^d})=1.$ Let us introduce the projectors $\\tilde{\\pi }_0^d (r)$ and $ \\tilde{\\pi }_{-}^d (r))$ for any $r \\in \\mathcal {H}$ by $ \\tilde{\\pi }_0^d (r)=\\phi (\\tilde{W}_0^d, r),$ $ r=\\tilde{\\pi }_0^d (r) \\tilde{F}_0^d (y)+ \\tilde{\\pi }_{-}^d (r).$ and the space $\\tilde{\\mathcal {H}}_{-}^d \\equiv \\lbrace r \\in \\mathcal {H} \\,|\\, \\tilde{\\pi }_{0}^d (r)=0\\rbrace .$ Introducing the bilinear form $\\tilde{\\varphi }_{d} (q,r)&=&\\int _{-1}^{1} (-\\tilde{\\psi }(d,y) q_1r_1+q_1^{\\prime } r_1^{\\prime }(1-y^2)+q_2 r_2 ) \\rho dy,\\\\$ where $\\tilde{\\psi }(d,y) $ is defined in (REF ), we recall from Proposition $3.7$ page 5906 in [4] that there exists $C_0 >0$ such that for all $|d|<1,$ for all $r\\in \\tilde{\\mathcal {H}}^d_{-},$ $\\frac{1}{C_0} ||r||_{\\mathcal {H} }^2\\le \\tilde{\\varphi }_d (r,r)\\le C_0 ||r||_\\mathcal {H} ^2.$"
],
[
" A modulation technique",
"We start the proof of Theorem REF here.",
"Let us consider $w\\in C ([s^*,\\infty ), \\mathcal {H})$ for some $s^*\\in \\mathbb {R}$ a solution of equation (REF ) such that $\\forall s\\ge s^*, E(w(s),\\partial _s w(s))\\ge E(\\kappa _0,0)$ and $\\Big |\\Big |\\begin{pmatrix} w(s^*)\\\\\\partial _s w(s^*) \\end{pmatrix} -\\begin{pmatrix} \\kappa (d^*,.",
")\\Omega ^*\\\\0\\end{pmatrix} \\Big |\\Big |_{\\mathcal { H}}\\le \\epsilon ^*$ for some $d^*\\in (-1,1)$ , $\\Omega ^*\\in \\mathbb {S}^{m-1}$ and $\\epsilon ^*>0$ to be chosen small enough.",
"Our aim is to show the convergence of $(w(s),\\partial _s w(s))$ as $s\\rightarrow \\infty $ to some $(\\kappa (d_\\infty ,0)\\Omega _\\infty ,0)$ , for some $(d_\\infty ,\\Omega _\\infty )$ close to $(d^*,\\Omega ^*)$ .",
"As one can see from (REF ), $(w,\\partial _s w)$ is close to a one representative of the family of the non-zero stationary solution $S^*\\equiv \\lbrace (\\kappa (d,y),0)\\Omega , |d|<1, \\Omega \\in S^{m-1}\\rbrace .$ From the continuity of $(w,\\partial _s w)$ from $[s^*,\\infty )$ to $\\mathcal { H}$ , $(w(s),\\partial _s w(s))$ will stay close to a soliton from $ S^*$ , at least for a short time after $s^*$ .",
"In fact, we can do better, and impose some orthogonality conditions, killing the zero directions of the linearized operator of equation (REF ) (see the operator $L_d$ defined in (REF )).",
"From the invariance of equation (REF ) under rotations in $\\mathbb {R}^{m}$ , we may assume that $\\Omega ^*=e_1.$ We recall that at this level of the study in the complex case (i.e.",
"for $m=2$ ), we were able to modulate $(w,\\partial _s w)$ as follows $\\begin{pmatrix} w(y,s)\\\\\\partial _s w(y,s) \\end{pmatrix} =e^{i \\theta (s)}\\left[\\begin{pmatrix} \\kappa (d(s),y)\\\\0\\end{pmatrix} +\\begin{pmatrix} q_1(y,s)\\\\q_2(y,s)\\end{pmatrix}\\right].$ for some well chosen $d(s)\\in (-1,1)$ and $\\theta (s)\\in \\mathbb {R}$ , such that $\\bar{\\pi }_0^{d(s)}\\begin{pmatrix} q_{1,1}(s)\\\\q_{2,1}(s)\\end{pmatrix} =\\tilde{\\pi }_0^{d(s)}\\begin{pmatrix} q_{1,2}(s)\\\\q_{2,2}(s)\\end{pmatrix}=0$ where $\\bar{\\pi }_0^{d}$ and $\\tilde{\\pi }_0^{d}$ are defined in (REF ) and (REF ) and $q=(q_1, q_2)$ is small in $\\mathcal { H}$ .",
"From (REF ), we see that we have a rotation in the complex plane, which has to be generalized to the vector-valued case.",
"In order to do so, we introduce for $i=2,...,m$ $R_i\\equiv \\begin{pmatrix}\\cos \\theta _i&0&\\cdots &-\\sin \\theta _i &\\cdots &0\\\\0 &1&\\cdots &0 &\\cdots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\\\sin \\theta _i&0&\\cdots &\\cos \\theta _i &\\cdots &0\\\\\\vdots &\\vdots &\\ddots &\\vdots &\\ddots &\\vdots \\\\0 &0&\\cdots &0&\\cdots &1\\\\\\end{pmatrix}.$ Note that $R_i$ is an $m\\times m$ orthonormal matrix which rotates the $(e_1,e_i)$ -plane by an angle $\\theta _i$ and leaves all other directions invariant.",
"We introduce $ R_\\theta $ by $R_\\theta \\equiv R_2 R_3\\cdots R_m,$ where $\\theta =(\\theta _2,\\theta _3,\\cdots , \\theta _m)$ .",
"Clearly, $R_\\theta $ is an $m\\times m$ orthonormal matrix.",
"We also define $A_j$ by $A_j=R_{\\theta }^{-1} \\frac{\\partial R_\\theta }{\\partial \\theta _j}.$ In the appendix, we show a different expression for $A_j$ : $A_j=\\frac{\\partial R_\\theta ^{-1} }{\\partial \\theta _j} R_{\\theta }.$ In fact, this formalism is borrowed from Filippas and Merle [8] who introduced the modulation technique for the vector-valued heat equation $\\partial _t u=\\Delta u+|u|^{p-1}u.$ We are ready to give our modulation technique result well adapted to the vector-valued case: Proposition 3.1 (Modulation of $w$ with respect to $\\kappa (d, .",
")\\Omega $ , where $\\Omega \\in \\mathbb {R}^{m-1}$ ) There exists $\\epsilon _0 > 0$ and $K_1 >0$ such that for all $\\epsilon \\le \\epsilon _0$ if $v\\in \\mathcal {H}$ , $ d\\in (-1,1) $ and $\\hat{\\theta }=(\\hat{\\theta }_2,...,\\hat{\\theta }_{m})\\in \\mathbb {R}^{m-1}$ are such that $\\forall i=2,...,m, \\cos {\\hat{\\theta }}_i\\ge \\frac{3}{4}\\mbox{ and } ||\\hat{q}||_{\\mathcal {H}}\\le \\epsilon \\mbox{ where }v =R_{\\hat{\\theta }}\\left[\\begin{pmatrix} \\kappa (\\hat{d},.",
")e_1\\\\0\\end{pmatrix} +\\hat{q}\\right],$ then, there exist $d\\in (-1,1),$ $\\hat{\\theta }=(\\hat{\\theta }_2,...,\\hat{\\theta }_{m})\\in \\mathbb {R}^{m-1}$ such that $\\bar{\\pi }_{0}^{d}\\begin{pmatrix} q_{1,1}\\\\ q_{2,1}\\end{pmatrix} =0,\\mbox{ and }\\tilde{\\pi }_{0}^{d}\\begin{pmatrix} q_{1,j}\\\\ q_{2,j}\\end{pmatrix} =0,\\; \\forall j=2,..m,$ where $q=(q_1, q_2)$ is defined by: $\\Big | \\log \\left(\\frac{1+d}{1-d}\\right)-\\log \\left(\\frac{1+\\hat{d}}{1-\\hat{d}}\\right)\\Big |+ |\\theta -\\hat{\\theta }|\\le C_0 ||\\hat{q}||_{\\mathcal {H}}\\le K_1 \\epsilon ,$ $\\forall i=2,...,m,\\; \\cos \\theta _i\\ge \\frac{1}{2}\\mbox{ and }||q||_{\\mathcal {H}}\\le K_1 \\epsilon .$ In order to prove this proposition, we need the following estimates on the matrix $A_j$ given in $(\\ref {A_i})$ and $(\\ref {61,5})$ : Lemma 3.2 (Orthogonality and continuity results related to the matrix $A_i$ (REF )) $ $ i) For any $i\\in \\lbrace 2,...,m\\rbrace $ , $A_i e_1=( \\prod \\limits _{j =i+1}^{m} \\cos \\theta _j ) e_i$ ii) For any $i\\in \\lbrace 2,...,m\\rbrace $ , $z\\in \\mathbb {R}^m$ , we have $ |A_i (z) |\\le |z| .$ The proof is straightforward though a bit technical.",
"For that reason, we give it in Appendix Now, we are ready to prove Proposition REF .",
"[Proof of Proposition REF ] The proof is similar to the complex-valued case.",
"However, since our notations are somehow complicated, we give details for the reader's convenience.",
"First, we recall that $\\theta =(\\theta _2,\\theta _3,...,\\theta _m)\\in \\mathbb {R}^{m-1}$ .",
"From (REF ) and (REF ), we see that the condition (REF ) becomes $\\Phi (v, d, \\theta )=0 $ where $ \\Phi \\in C(\\mathcal {H}\\times (-1,1)\\times \\mathbb {R}^{m-1}, \\mathbb {R}^m) $ is defined by $\\begin{array}{c}\\Phi (v, d, \\theta )=\\begin{pmatrix}\\bar{\\Phi }(v, d, \\theta )\\\\\\tilde{\\Phi }_2 (v, d, \\theta )\\\\\\vdots \\\\\\tilde{\\Phi }_m (v, d, \\theta )\\end{pmatrix}=\\begin{pmatrix}\\phi \\left(\\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}-\\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix},\\bar{W}_0^d\\right)\\\\\\phi \\left(\\begin{pmatrix}V_{1,j}\\\\V_{2,j} \\end{pmatrix},\\tilde{W}_0^d\\right)_{j=2...m}\\end{pmatrix}\\end{array}$ where $V=\\begin{pmatrix}V_1\\\\V_2\\end{pmatrix}\\in \\mathbb {R}^m\\times \\mathbb {R}^m$ is given by $V=R^{-1}_\\theta v$ .",
"We claim that we can apply the implicit function theorem to $\\Phi $ near the point $(\\hat{v},\\hat{d},\\hat{\\theta })$ with $\\hat{v}=R_{\\hat{\\theta }}(\\kappa (\\hat{d},.",
")e_1,0)$ .",
"Three facts have to be checked: 1-First, note that $\\hat{v}=R_{\\hat{\\theta }}^{-1}(\\hat{v})$ , hence $\\Phi (R_{\\hat{\\theta }}(\\kappa (\\hat{d},.",
")e_1,0),\\hat{d},\\hat{\\theta })=0.$ 2-Then, we compute from (REF ), for all $u\\in \\mathcal {H}$ , $D_v \\bar{\\Phi }(v,d,\\theta )(u)=\\phi (\\begin{pmatrix}U_{1,1}\\\\U_{2,1} \\end{pmatrix},\\bar{W}_0^d),$ and for all $ j=2...m$ , we have $D_v \\tilde{\\Phi }_j (v,d,\\theta )(u)=\\phi (\\begin{pmatrix}U_{1,j}\\\\U_{2,j} \\end{pmatrix},\\tilde{W}_0^d) ,$ so we have from (REF ) and (REF ) $||D_v \\bar{\\Phi }(v,d,\\theta )||\\le C_0 \\mbox{ and } ||D_v \\tilde{\\Phi }_j (v,d,\\theta )||\\le C_0.$ 3-Let $J(\\bar{\\Phi },\\tilde{\\Phi }_{j,j=2..m})$ the jacobian matrix of $\\Phi $ with respect to $(d,\\theta )$ , and $D$ its determinant so $J\\equiv \\begin{pmatrix}\\partial _d \\bar{\\phi }&\\partial _{\\theta _2} \\bar{\\phi }&\\cdots &\\partial _{\\theta _m} \\bar{\\phi }\\\\\\partial _d \\tilde{\\phi }_2&\\partial _{\\theta _2} \\tilde{\\phi }_2&\\cdots &\\partial _{\\theta _m} \\tilde{\\phi }_2\\\\\\vdots &\\vdots &\\vdots &\\vdots \\\\\\partial _d \\tilde{\\phi }_m&\\partial _{\\theta _2} \\tilde{\\phi }_m&\\cdots &\\partial _{\\theta _m} \\tilde{\\phi }_m\\\\\\end{pmatrix}.$ Then, we compute from (REF ): $ \\partial _d \\bar{\\Phi }&=&-\\phi ((\\partial _d \\kappa (d,.",
"),0),\\bar{W}_0^d)+\\phi (\\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}-\\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix}, \\partial _d \\bar{W}_0^d), $ and for $i,j=2,..m$ $\\partial _d \\tilde{\\Phi }_j&=&\\phi (\\begin{pmatrix}V_{1,j}\\\\V_{2,j} \\end{pmatrix},\\partial _d \\tilde{W}_0^d),\\\\\\partial _{\\theta _i} \\bar{\\Phi }&=&\\phi ( \\partial _{\\theta _i} \\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix},\\bar{W}_0^d)=\\phi ( \\begin{pmatrix} <e_1, \\frac{\\partial R_\\theta ^{-1}}{ \\partial {\\theta _i} }v_{1}>\\\\ <e_1, \\frac{\\partial R_\\theta ^{-1}}{ \\partial {\\theta _i} }v_{2}> \\end{pmatrix},\\bar{W}_0^d),\\\\\\partial _{\\theta _i} \\tilde{\\Phi }_j&=&\\phi (\\begin{pmatrix} <e_j, \\frac{\\partial R_\\theta ^{-1}}{ \\partial {\\theta _i} }v_{1}>\\\\ <e_j, \\frac{\\partial R_\\theta ^{-1}}{ \\partial {\\theta _i} }v_{2}> \\end{pmatrix},\\partial _d \\tilde{W}_0^d).\\\\$ Now, we assume that $|\\theta |+\\big |\\log \\left(\\frac{1+d}{1-d}\\right)- \\log \\left(\\frac{1+\\hat{d}}{1-\\hat{d}}\\right)\\big |+ \\big |\\big | v -R_{\\hat{\\theta }}\\begin{pmatrix} \\kappa (\\hat{d},.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\epsilon _1$ for some small $\\epsilon _1>0$ .",
"In the following, we estimate each of the derivatives whose expressions where given above.",
"- Since $\\begin{pmatrix} \\partial _d \\kappa (d,y)\\\\0\\end{pmatrix}=\\frac{-2\\kappa _0}{(p-1)(1-d^2)}\\bar{F}_0^d,$ by definiftion (REF ) and (REF ), it follows from the orthogonality condition (REF ) that $\\phi ((\\partial _d \\kappa (d,.",
"),0),\\bar{W}_0^d)=\\frac{-2\\kappa _0}{(p-1)(1-d^2)}.$ Therefore, from (REF ), we write $ \\partial _d \\bar{\\Phi }=\\frac{2\\kappa _0}{(p-1)(1-d^2)}+\\phi (\\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}-\\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix}), \\partial _d \\bar{W}_0^d).$ Since $\\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}=\\begin{pmatrix} <e_1, R_\\theta ^{-1}v_{1}>\\\\ <e_1, R_\\theta ^{-1}v_{2}> \\end{pmatrix},$ we write $ &\\;&\\big |\\big | \\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}- \\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\big |\\big | R_\\theta ^{-1}v-\\begin{pmatrix}\\kappa (d,.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\\\&\\le &\\big |\\big | ( R_\\theta ^{-1}- R_{\\hat{\\theta }}^{-1}) v \\big |\\big |_{\\mathcal {H} }+\\big |\\big | R_{\\hat{\\theta }}^{-1} v-\\begin{pmatrix}\\kappa (\\hat{d},.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }+ \\big |\\big | \\kappa (\\hat{d},.",
")-\\kappa (d,.)",
"\\big |\\big |_{\\mathcal {H} }.$ Since, $\\forall \\theta , \\theta ^{\\prime } \\in \\mathbb {R}, \\big | R_{\\theta }-R_{\\theta ^{\\prime }}\\big |+\\big | R_{\\theta }^{-1}-R_{\\theta ^{\\prime }}^{-1}\\big | \\le C \\big | \\theta -\\theta ^{\\prime } \\big |, $ (see (REF ) below for $R_{\\theta }$ , and use an adhoc change of variables for $R_{\\theta }^{-1}$ ), recalling the following continuity result from estimate $(174)$ page 101 in [18]: $\\big |\\big | \\kappa (d_1,.",
")- \\kappa (d_2,.)",
"\\big |\\big |_{\\mathcal {H}_0 }\\le C\\big | \\left(\\frac{1+d_1}{1-d_1}\\right)- \\left(\\frac{1+d_2}{1-d_2}\\right)\\big | ,$ we see from the Cauchy-Schwartz inequality, (REF ), (REF ) and (REF ) that $ \\big |\\big | \\begin{pmatrix}V_{1,1}\\\\V_{2,1} \\end{pmatrix}- \\begin{pmatrix}\\kappa (d,.",
")\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\big |\\big | V-\\begin{pmatrix}\\kappa (d,.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\ \\le C \\epsilon _1.$ $ \\big | \\partial _d \\bar{\\Phi }-\\frac{2\\kappa _0}{(p-1)(1-d^2)}\\big | \\le C \\epsilon _1.$ - Since $ \\begin{pmatrix}V_{1,j}\\\\V_{2,j} \\end{pmatrix}=\\begin{pmatrix} <e_j, R_\\theta ^{-1}v_{1}>\\\\ <e_j, R_\\theta ^{-1}v_{2}> \\end{pmatrix},$ we write $\\big |\\big | \\begin{pmatrix}V_{1,j}\\\\V_{2,j} \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\big |\\big | R_{\\theta }^{-1} v-\\begin{pmatrix}\\kappa (d,.",
")e_1\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le C \\epsilon _1$ by the same argument as for (REF ).",
"Using the Cauchy-Schwarz inequality together with (REF ), we see from (REF ) that $ \\big | \\partial _d \\tilde{\\Phi }_j\\big | \\le \\frac{C \\epsilon _1}{1-d^2}.$ From (REF ), we see that $\\frac{\\partial R_\\theta ^{-1}}{\\partial \\theta _i}= A_i R_\\theta ^{-1}$ .",
"Therefore using $ii)$ of Lemma REF and the fact that the rotation $R_\\theta $ does not change the norm in $\\mathcal {H}$ , we write $ \\big | \\big | \\frac{\\partial R_{\\theta }^{-1}}{\\partial \\theta _i}(v)- \\begin{pmatrix}\\kappa (d,.",
")A_i (e_1)\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le \\big | \\big | v- R_{\\theta } \\begin{pmatrix}\\kappa (d,.",
")e_1\\\\0 \\end{pmatrix} \\big | \\big | _{\\mathcal {H} }\\le C \\epsilon _1,$ by the same argument as for (REF ).",
"Using the Cauchy-Schwarz identity together with (REF ), we see from () that $ \\big | \\partial _{\\theta _i} \\bar{\\Phi }\\big | \\le C \\epsilon _1.$ - By the same argument as for (REF ), we obtain from () $ \\big | \\partial _{\\theta _i} \\tilde{\\Phi }_j\\big | \\le C \\epsilon _1 \\mbox{ if }i\\ne j.$ Now, if $i= j$ , noting from (REF ) that $\\frac{\\partial R_\\theta ^{-1}}{\\partial \\theta _i}v=A_i R_\\theta ^{-1}(v)=A_i V,$ applying the operator $A_i$ to (REF ), then taking the scalar product with $e_i$ , we see from Lemma REF that $ \\big | \\big | \\begin{pmatrix} <e_i, \\frac{\\partial R^{-1}_{\\theta }}{\\partial \\theta _i} v_{1}> \\\\ <e_i, \\frac{\\partial R^{-1}_{\\theta }}{\\partial \\theta _i} v_{2}>\\end{pmatrix} - \\begin{pmatrix}\\kappa (d_i,.",
")\\prod \\limits _{k=i+1}^{m}\\cos \\theta _k\\\\0 \\end{pmatrix} \\big |\\big |_{\\mathcal {H} }\\le C \\epsilon _1.$ Since we know from (REF ) and (REF ) that $\\phi (\\kappa (d,.",
"), \\tilde{W}_0^d)=1,$ it follows from () that $ \\big | \\partial _{\\theta _i} \\tilde{\\Phi }_i -\\prod \\limits _{k=i+1}^{m}\\cos \\theta _k\\big | \\le C \\epsilon _1.$ Collecting (REF ), (REF ), (REF ), (REF ) and (REF ) we see that $ \\big | D-\\frac{2\\kappa _0}{(p-1)(1-d^2)} -\\cos \\theta _3 (\\cos \\theta _4)^2... (\\cos \\theta _m)^{m-2}\\big | \\le \\frac{C \\epsilon _1}{1-d^2}.$ Since $\\cos \\theta _i \\ge \\frac{3}{4}$ by hypothesis, we have the non-degeneracy of $\\Phi $ (voir (REF )) near the point $(\\hat{v},\\hat{d},\\hat{\\theta })$ with $\\hat{v}= R_{\\hat{\\theta }}(\\kappa (\\hat{d},.",
")e_1,0)$ .",
"Applying the implicit function theorem, we conclude the proof of Proposition REF ."
],
[
"Dynamics of $q$ , {{formula:2b5265e7-5707-4af2-9f03-bc5d16b5bb87}} and {{formula:953dad1d-59c8-4ee0-8eb3-f2ecf20c4884}}",
"Let us apply Proposition REF with $v=(w,\\partial _s w) (s^*)$ , $\\hat{d}=d^*$ and $\\hat{\\theta }=0$ .",
"Clearly, from (REF ) and (REF ), we have $ ||\\hat{q}||_{\\mathcal {H}}\\le \\epsilon ^*.$ Assuming that $\\epsilon ^*\\le \\epsilon _0$ defined in Proposition REF , we see that the proposition applies, and from the continuity of $(w,\\partial _s w)$ from $[s^*,\\infty )$ to $\\mathcal {H}$ , we have a maximal $\\bar{s}> s^*$ , such that $(w(s),\\partial _s w(s))$ can be modulated in the sense that $\\begin{pmatrix} w(y,s)\\\\\\partial _s w(y,s) \\end{pmatrix} =R_{\\theta (s)}\\left[\\begin{pmatrix} \\kappa (d(s),y)e_1\\\\0\\end{pmatrix} +\\begin{pmatrix} q_1(y,s)\\\\q_2(y,s)\\end{pmatrix}\\right],$ where the parameters $d(s)\\in (-1,1)$ and $\\theta (s)=(\\theta _2(s),..., \\theta _m(s))$ are such that for all $s\\in [s^*, \\bar{s})$ $\\bar{\\pi }_{0}^{d(s)}\\begin{pmatrix} q_{1,1}(s)\\\\ q_{2,1}(s)\\end{pmatrix} =0,\\mbox{ and }\\tilde{\\pi }_{0}^{d(s)}\\begin{pmatrix} q_{1,j}(s)\\\\ q_{2,j}(s)\\end{pmatrix} =0,\\; \\forall j=2,..m,$ and $\\forall i=2,...,m\\; \\cos \\theta _i(s)\\ge \\frac{1}{2} \\mbox{ and } ||q(s)||_{\\mathcal {H}}\\le \\epsilon \\equiv 2K_0 K_1 \\epsilon ^*,$ where $K_1>0$ is defined in Proposition REF and $K_1>1$ is a constant that will be fixed below in (REF ).",
"Two cases then arise: - Case 1: $\\bar{s}=+\\infty ;$ - Case 2: $\\bar{s}<+\\infty ;$ in this case, we have an equality case in (REF ), i.e.",
"$cos \\theta _i(\\bar{s})= \\frac{1}{2} $ for some $i=2,...,m$ , or $ ||q(\\bar{s})||_{\\mathcal {H}}= 2K_0 K_1 \\epsilon ^*$ .",
"At this stage, we see that controlling the solution $(w(s),\\partial _s w (s))\\in \\mathcal {H} $ is equivalent to controlling $q\\in \\mathcal {H}$ , $d\\in (-1,1)$ and $\\theta (s)\\in \\mathbb {R}^{m-1}$ .",
"Before giving the dynamics of this parameters, we need to introduce some notations.",
"From (REF ), we will expand $\\bar{q}$ and $\\tilde{q}$ respectively according to the spectrum of the linear operators $\\bar{L}_d$ and $\\tilde{L}_d$ as in (REF ) and (REF ): $\\begin{pmatrix} q_{1,1}(y,s)\\\\ q_{2,1}(y,s) \\end{pmatrix}&=\\alpha _{1,1} \\bar{F}_1^d (y)+\\begin{pmatrix} q_{-,1,1}(y,s)\\\\ q_{-,2,1} (y,s)\\end{pmatrix} \\\\\\forall j \\in \\lbrace 1,...,m\\rbrace ,\\; \\begin{pmatrix} q_{1,j}(y,s)\\\\ q_{2,j} (y,s)\\end{pmatrix}&=\\begin{pmatrix} q_{-,1,j}(y,s)\\\\ q_{-,2,j}(y,s) \\end{pmatrix}$ where $\\alpha _{1,1} = \\bar{\\pi }_1^{d(s)} \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix},\\;\\alpha _{0,1} = \\bar{\\pi }_0^{d(s)} \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}=0,\\;\\alpha _{-,1}(s)=\\sqrt{\\bar{\\varphi }_d (\\begin{pmatrix} q_{-,1,1}\\\\ q_{-,2,1} \\end{pmatrix},\\begin{pmatrix} q_{-,1,1}\\\\ q_{-,2,1} \\end{pmatrix})}$ $\\alpha _{0,j}= \\tilde{\\pi }_0^{d(s)}\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix}=0,\\;\\alpha _{-,j}(s)=\\sqrt{\\tilde{\\varphi }_d (\\begin{pmatrix} q_{-,1,j}\\\\ q_{-,2,j} \\end{pmatrix},\\begin{pmatrix} q_{-,1,j}\\\\ q_{-,2,j} \\end{pmatrix})}$ and $\\begin{pmatrix} q_{-,1,1}\\\\ q_{-,2,1} \\end{pmatrix}=\\bar{\\pi }_{-}^{d}\\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix}$ $\\forall j \\in \\lbrace 1,...,m\\rbrace ,\\;\\begin{pmatrix} q_{-,1,j}\\\\ q_{-,2,j} \\end{pmatrix}=\\tilde{\\pi }_{-}^{d}\\begin{pmatrix} q_{1,j}\\\\q_{2,j} \\end{pmatrix}$ From (REF ), (), (REF ) (REF ) and (REF ), we see that for all $s \\ge s_0$ , $\\frac{1}{C_0} \\alpha _{-,1}(s) &\\le & ||\\begin{pmatrix} q_{-,1,1}\\\\ q_{-,2,1} \\end{pmatrix}||_{\\mathcal {H} } \\le C_0 \\alpha _{-,1}(s)\\\\\\frac{1}{C_0}(| \\alpha _{1,1}(s)|+ \\alpha _{-,1}(s)) &\\le & || \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}||_{\\mathcal {H} } \\le C_0(| \\alpha _{1,1}(s)|+ \\alpha _{-,1}(s)) \\\\\\frac{1}{C_0} \\alpha _{-,j}(s) &\\le & || \\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix}||_{\\mathcal {H} } \\le C_0 \\alpha _{-,j}(s) $ for some $C_0 > 0$ .",
"In the following proposition, we derive from (REF ) and () differential inequalities satisfied by $ \\alpha _{1,1}(s)$ , $ \\alpha _{-,1}(s)$ , $ \\alpha _{-,j}(s)$ , $\\theta _i(s)$ and $d(s)$ .",
"Introducing $R_{-}(s)=-\\int _{-1}^{1} \\mathcal { F}_{d }(q_1) \\rho dy,$ where $\\mathcal {F}_{d(s)} (q_1 (y,s))=\\frac{|\\kappa (d ,\\cdot )e_1+q_1|^{p+1}}{p+1}-\\frac{\\kappa (d ,\\cdot )^{p+1}}{p+1}-\\kappa (d ,\\cdot )^p q_{1,1}-\\frac{p}{2} \\kappa (d ,\\cdot )^{p-1} q_{1,1}^2-\\frac{\\kappa (d ,\\cdot )^{p-1}}{2} \\sum _{j=2}^{m} q_{1,j}^2,$ we claim the following Proposition 3.3 (Dynamics of the parameters) For $\\epsilon ^*$ small enough and for all $s\\in [s^*,\\bar{s})$ , we have: (i) (Control of the modulation parameter) $\\sum _{i=2}^{m}|\\theta _i^{\\prime }|+\\frac{|d^{\\prime }|}{1-d^2}\\le C_0 || q||_{\\mathcal {H}}^2.$ (ii) (Projection of equation (REF ) on the different eigenspaces of $\\bar{L}_d$ and $\\tilde{L}_d$ ) $| \\alpha _{1,1}^{\\prime }(s) - \\alpha _{1,1}(s)|&\\le C_0 || q||_{H}^2.\\\\\\left( R_{-}+\\frac{1}{2}( \\alpha _{1,1}^2+ \\alpha _{-,j}^2)\\right)^{\\prime }&\\le -\\frac{4}{p-1}\\int _{-1}^{1}(q_{-,2,1}^2+ q_{-,2,j}^2)\\frac{\\rho }{1-y^2}dy+ C_0 ||q(s)||_{\\mathcal {H}}^{3},$ for $j\\in \\lbrace 2,...,m\\rbrace $ and $R_{-}(s)$ defined in $(\\ref {209})$ , satisfying $|R_{-}(s)|\\le C_0 ||q(s)||_{\\mathcal {H}}^{1+\\bar{p}} \\mbox{ where } \\bar{p}=\\min (p,2)>1.$ (iii) (An additional relation) $\\frac{d}{ds}\\int _{-1}^{1} q_{1,1} q_{2,1} \\rho \\le -\\frac{4}{5}\\bar{\\alpha }_{-,1}^2+ C_0\\int _{-1}^{1} q_{-,2,1}^2\\frac{\\rho }{1-y^2}+ C_0 ||q(s)||_{\\mathcal {H}}^{2}.$ For $j\\in \\lbrace 2,...,m\\rbrace ,$ we have: $\\frac{d}{ds}\\int _{-1}^{1} q_{1,j} q_{1,j} \\rho \\le -\\frac{4}{5}\\tilde{\\alpha }_{-,j}^2+ C_0\\int _{-1}^{1} q_{2,j}^2\\frac{\\rho }{1-y^2}+ C_0 ||q(s)||_{\\mathcal {H}}^{2}.$ (iv) (Energy barrier) $\\alpha _{1,1} (s)\\le C_0\\alpha _{-,1}(s)+ C_1 \\sum _{j=2}^{m} \\alpha _{-,j}(s).$ The proof follows the general framework developed by Merle and Zaag in the real case (see Proposition $5.2$ in [18]), then adapted to the complex-valued case in [4] 5(see Proposition $4.2$ page 5915 in [4]).",
"However, new ideas are needed, mainly because we have $(m-1)$ rotation parameters in the modulation technique (see Proposition REF above), rather than only one in the complex-valued case.",
"For that reason, in the following, we give details only for the \"new\" terms, referring the reader to the earlier literature for the \"old\" terms.",
"Let us first write an equation satisfied by $q$ defined in (REF ).",
"We put the equation (REF ) satisfied by $w$ in vectorial form: $\\begin{array}{l}\\partial _s w_1=w_2\\\\\\partial _s w_2=\\mathcal {L}w_1-\\frac{2(p+1)}{(p-1)^2}w_1+|w_1|^{p-1}w_1-\\frac{p+3}{p-1} w_2- 2 y \\partial _y w_2.\\end{array}$ We replace all the terms of (REF ) by their expressions from (REF ).",
"Precisely, for the terms of the right hand side of (REF ) we have: $\\begin{array}{l}\\partial _s w_1=R_{\\theta }\\left(d^{\\prime }\\partial _d \\kappa e_1+\\partial _s q_1\\right)+\\sum _{i=1}^{m}\\theta _i^{\\prime }\\frac{\\partial R_\\theta }{\\partial \\theta _j}\\left( \\kappa _d e_1+q_1\\right),\\\\\\partial _s w_2=R_{\\theta }\\left(\\partial _s q_1\\right)+\\sum _{i=1}^{m}\\theta _i^{\\prime }\\frac{\\partial R_\\theta }{\\partial \\theta _j}\\left(q_2\\right).",
"\\end{array}$ For the terms on the left hand side of (REF ) we have: $w_2=R_{\\theta }(q_2), \\,\\mathcal {L}w_1=R_{\\theta }( \\mathcal {L}(\\kappa _d e_1)+\\mathcal {L} q_1),\\,|w_1|=|\\kappa _d e_1+ q_1|,\\,\\partial _y w_2=R_{\\theta }\\partial _y q_2.$ Then, multiplying by $R_{\\theta }^{-1}$ , using the fact that $(\\kappa (d ,\\cdot ),0)$ is a stationnary solution and dissociating the first and $j$ th component of these equations, we get for all $s\\in [s^*,\\bar{s})$ , for all $j \\in \\lbrace 2,...,m\\rbrace $ : $\\frac{\\partial }{\\partial s}\\begin{pmatrix} q_{1,1}\\\\ q_{2,1}\\end{pmatrix}&=\\bar{L}_{d(s)} \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}+\\begin{pmatrix}0\\\\ {f}_{d(s),1}(q_1)\\end{pmatrix}-d^{\\prime }(s)\\begin{pmatrix}\\partial _d \\kappa (d,y)\\\\ 0\\end{pmatrix}-\\sum _{i=2}^{m}\\theta _i^{\\prime }(s)\\begin{pmatrix} a_{i,1,1} \\\\ a_{i,2,1} \\end{pmatrix},\\\\\\frac{\\partial }{\\partial s} \\begin{pmatrix} q_{1,j}\\\\q_{2,j}\\end{pmatrix}&=\\tilde{L}_{d(s)}\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix}+\\begin{pmatrix} 0\\\\{f}_{d(s),j}(q_1) \\end{pmatrix}-\\sum _{i=2}^{m}\\theta _i^{\\prime }(s)\\begin{pmatrix} a_{i,1,j} \\\\ a_{i,2,j} \\end{pmatrix},$ where $\\bar{L}_{d(s)} ,\\tilde{L}_{d(s)}, f_{d(s),1}$ and $f_{d(s),j}$ are defined in (REF ), (REF ), (REF ) and (REF ), and $a_i$ by $\\begin{pmatrix} a_{i,1}\\\\a_{i,2}\\end{pmatrix}=\\begin{pmatrix}A_i (\\kappa _d e_1+q_1)\\\\A_i (q_2) \\end{pmatrix},$ with $a_{i,1}=(a_{i,1,1},a_{i,1,2},...,a_{i,1,m})\\in \\mathbb {R}^{m}$ and $a_{i,2}=(a_{i,2,1},a_{i,2,2},...,a_{i,2,m})\\in \\mathbb {R}^{m}$ .",
"Let $i\\in \\lbrace 2,\\cdots ,m\\rbrace $ , Projecting equation (REF ) with the projector $\\bar{\\pi }_\\lambda ^d$ (REF ) for $\\lambda =0$ and $\\lambda =1$ , we write $\\bar{\\pi }_\\lambda ^d (\\partial _s \\begin{pmatrix} q_{1,1}\\\\ q_{2,1}\\end{pmatrix})&=&\\bar{\\pi }_\\lambda ^d (\\bar{L}_{d(s)} \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix})+\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}0\\\\ {f}_{d(s),1}(q_1) \\end{pmatrix}-d^{\\prime }(s)\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}\\partial _d \\kappa (d,y)\\\\ 0\\end{pmatrix}\\\\&-&\\sum _{i=2}^{m}\\theta _i^{\\prime }(s)\\bar{\\pi }_\\lambda ^d\\begin{pmatrix} a_{i,1,1} \\\\ a_{i,2,1} \\end{pmatrix}.$ Note that, expect the last term, all the terms of (REF ) can be controled exactly like the real case using (REF ) (for details see page 105 in [18]).",
"So, we recall that we have: $|\\bar{\\pi }_\\lambda ^d( \\partial _s \\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix})-\\alpha _{\\lambda ,1}^{\\prime }|\\le \\frac{C_0}{1-d^2} |d^{\\prime }||| q||_{\\mathcal { H}},$ $\\bar{\\pi }_\\lambda ^d( L_d \\begin{pmatrix} q_{1,1}\\\\q_{2,1} \\end{pmatrix})=\\lambda \\alpha _{\\lambda ,1},$ $|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}0\\\\ {f}_{d(s),1}(q_1) \\end{pmatrix}|\\le C_0|| q||_{\\mathcal { H}}^2,$ $|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}\\partial _d \\kappa (d,y)\\\\ 0\\end{pmatrix}|=-\\frac{2\\kappa _0}{(p-1)(1-d^2)} \\bar{\\pi }_\\lambda ^d (F_0^d) =-\\frac{2\\kappa _0}{(p-1)(1-d^2)}\\delta _{\\lambda ,0}.$ Now, we focus on the study of the last term of (REF ).",
"From the definition of $a_i$ (REF ) and $i)$ of Lemma REF , we have: $\\begin{pmatrix} a_{i,1,1}\\\\a_{i,2,1}\\end{pmatrix}=\\kappa _d \\begin{pmatrix}<e_1,A_i (e_1)>\\\\0 \\end{pmatrix}+\\begin{pmatrix}<e_1,A_i (q_1)>\\\\<e_1,A_i (q_2)> \\end{pmatrix}=\\begin{pmatrix}<e_1,A_i (q_1)>\\\\<e_1,A_i (q_2)> \\end{pmatrix}.$ Applying the projector $\\bar{\\pi }_\\lambda ^d$ (REF ), we get $|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix} a_{i,1,1}\\\\a_{i,2,1} \\end{pmatrix}|&=&|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix}<e_1,A_i (q_1)>\\\\<e_1,A_i (q_2)> \\end{pmatrix}|\\le C || \\begin{pmatrix}<e_1,A_i (q_1)>\\\\<e_1,A_i (q_2)> \\end{pmatrix} ||_{\\mathcal { H}}\\\\&\\le & C (|| <e_1,A_i (q_1)>||_{\\mathcal { H}_0}+|| <e_1,A_i (q_2)>||_{ L_\\rho ^2}).$ Using $ii)$ of Lemma REF , we have: $|| <e_1,A_i (q_2)>||^2_{ L_\\rho ^2}=\\int _{-1}^{1} <e_1,A_i (q_2)>^2\\rho dy\\le \\int _{-1}^{1} |A_i (q_2)|^2\\rho dy\\le \\int _{-1}^{1} |q_2|^2\\rho dy,$ and by the same way, using $ii)$ of Lemma REF and the definition of $\\mathcal { H}_0$ (REF ), we have $|| <e_1,A_i (q_1)>||^2_{\\mathcal { H}_0}&=&\\int _{-1}^{1} <e_1,A_i (q_1)>^2\\rho dy+\\int _{-1}^{1} (<e_1,A_i (\\partial _y q_1)>)^2(1-y^2)\\rho dy\\\\&\\le &\\int _{-1}^{1}( |q_1|^2+(1-y^2)|\\partial _y q_1|^2)\\rho dy.$ From (REF ), (REF ) and (REF ), we have $|\\bar{\\pi }_\\lambda ^d \\begin{pmatrix} a_{i,1,1}\\\\a_{i,2,1} \\end{pmatrix}|\\le C_0 || q||_{\\mathcal { H}}.$ Using (REF ), (REF ), (REF ), (REF ), (REF ), (REF ), and the fact that $\\alpha _{0,1}\\equiv \\alpha _{0,1}^{\\prime }\\equiv 0$ (see (REF )), we get for $\\lambda =0,1$ : $\\frac{2 \\kappa _0}{(p-1)(1-d^2)} |d^{\\prime }|&\\le \\frac{C_0}{1-d^2}|d^{\\prime }| || q||_{\\mathcal { H}}+C_0 || q||_{\\mathcal { H}}^2+C_0 || q||_{\\mathcal { H}}\\sum _{j=2}^{m}|\\theta _i^{\\prime }|\\\\|\\bar{\\alpha }_1^{\\prime }(s)-\\bar{\\alpha }_1(s)|&\\le \\frac{C_0}{1-d^2}|d^{\\prime }| || q||_{\\mathcal { H}} +C_0 || q||_{\\mathcal { H}}^2+C_0|| q||_{\\mathcal { H}}\\sum _{j=2}^{m}|\\theta _i^{\\prime }|.$ Now, projecting equation () with the projector $\\tilde{\\pi }_0^d$ (REF ), where $j\\in \\lbrace 2,\\cdots ,m\\rbrace $ , we get: $\\tilde{\\pi }_0^d(\\partial _s\\begin{pmatrix} q_{1,j}\\\\q_{2,j}\\end{pmatrix} )=\\tilde{\\pi }_0^d( \\tilde{L}_{d(s)}\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})+\\tilde{\\pi }_0^d \\begin{pmatrix} 0\\\\f_{d(s),j} (q_1) \\end{pmatrix}-\\sum _{i=2}^{m}\\theta _i^{\\prime }(s) \\tilde{\\pi }_0^d\\begin{pmatrix} a_{i,1,j} \\\\ a_{i,2,j} \\end{pmatrix}.$ From the complex-valued case we recall that we have (for details see page 5917 in [4], together with Lemma REF ): $|\\tilde{\\pi }_0^d (\\partial _s \\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})|\\le \\frac{C_0}{1-d^2} |d ^{\\prime }| || q||_{\\mathcal { H}},$ $\\tilde{\\pi }_0^d (\\tilde{L}_{d }\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})=0,$ $\\Big {|}\\tilde{\\pi }_0^d \\begin{pmatrix} 0\\\\\\tilde{f}_{d(s)}(q_1) \\end{pmatrix}\\Big {|} \\le C_0 || q||_{\\mathcal { H}}^2,$ $\\tilde{\\pi }_0^d\\begin{pmatrix} \\kappa _d \\\\0 \\end{pmatrix} =1.$ Thus, only the last term in (REF ) remains to be treated in the following.",
"From the definition of $a_i$ (REF ), we recall that $\\begin{pmatrix} a_{i,1,j}\\\\a_{i,2,j}\\end{pmatrix}= \\begin{pmatrix}<e_j,A_i (e_1)>\\kappa _d\\\\0 \\end{pmatrix}+\\begin{pmatrix}<e_j,A_i (q_1)>\\\\<e_j,A_i (q_2)> \\end{pmatrix}.$ By $i)$ of Lemma REF : $\\sum _{i=2}^{m}\\theta _i^{\\prime }(s) \\begin{pmatrix} a_{i,1,j}\\\\a_{i,2,j}\\end{pmatrix}=\\theta _j^{\\prime }(s)\\begin{pmatrix} (\\prod \\limits _{l =j+1}^{m} \\cos \\theta _l ) \\kappa _d \\\\0 \\end{pmatrix}+\\sum _{i=2}^{m}\\theta _i^{\\prime }(s) \\begin{pmatrix}<e_j,A_i (q_1)>\\\\<e_j,A_i (q_2)> \\end{pmatrix},$ where by convention $\\prod \\limits _{l =m+1}^{m} \\cos \\theta _l =1$ if $j=m$ .",
"Applying the projection $\\tilde{\\pi }_0^d$ to (REF ) and using (REF ), we see that $\\Big {|} \\sum _{i=2}^{m}\\theta _i^{\\prime }(s) \\tilde{\\pi }_0^d \\begin{pmatrix} a_{i,1,j}\\\\a_{i,2,j}\\end{pmatrix}-\\theta _j^{\\prime }(s) \\prod \\limits _{l =j+1}^{m} \\cos \\theta _l \\Big {|} &\\le & \\sum _{i=2}^{m}|\\theta _i^{\\prime }(s)| \\Big {|} \\tilde{\\pi }_0^d \\begin{pmatrix} <e_j,A_i (q_1)>\\\\<e_j,A_i (q_2)> \\end{pmatrix} \\Big {|}\\\\&\\le & C_0 || q||_{\\mathcal { H}}\\sum _{i=2}^{m}|\\theta _i^{\\prime }(s)|,$ where, we use the fact that $|\\tilde{\\pi }_0^d\\begin{pmatrix} <e_j,A_i (q_1)>\\\\<e_j,A_i (q_2)> \\end{pmatrix}|\\le C_0 || q||_{\\mathcal { H}},$ which follows by the same techniques as in (REF ,) (REF ) and (REF ).",
"Using (REF ),(REF ), (REF ), (REF ) and (REF ), and recalling from (REF ) that $\\prod \\limits _{l =j+1}^{m} \\cos \\theta _l \\ge (\\frac{1}{2})^{m-j},$ we get for any $j\\in \\lbrace 2,...,m\\rbrace $ : $\\Big {|} \\theta _j^{\\prime }(s) \\Big {|} \\le \\frac{C_0}{1-d^2}|d^{\\prime }| || q||_{\\mathcal { H}}+C_0 || q||_{\\mathcal { H}}^2+C_0|| q||_{\\mathcal { H}}\\sum _{i=2}^{m}| \\theta _i^{\\prime }| .$ Using (REF ) together with (REF ), we see that $\\sum _{j=2}^{m} \\Big {|} \\theta _j^{\\prime }(s) \\Big {|}+ \\frac{|d^{\\prime }|}{1-d^2} \\le C_0 \\frac{|d^{\\prime }|}{1-d^2} || q||_{\\mathcal { H}}+C_0 || q||_{\\mathcal { H}}^2+C_0| || q||_{\\mathcal { H}}\\sum _{i=2}^{m}| \\theta _i^{\\prime }| ,$ Thus, using again (REF ) and taking $\\epsilon $ small enough, we get $\\sum _{j=2}^{m} \\Big {|} \\theta _j^{\\prime }(s) \\Big {|}+ \\frac{|d^{\\prime }|}{1-d^2} \\le C_0 || q||_{\\mathcal { H}}^2,$ which yields (REF ).",
"Then, using () together with (REF ) gives (REF ).",
"For estimations (REF ) (REF ) (REF ) (REF ), the study in the complex case (Subsection $4.3$ page 5914 in [4]) can be adapted without any difficulty to the vector-valued case.",
"For the reader convenience, we detail for example the energy barrier (REF ): Using the definition of $q(y,s)$ (REF ), we can make an expansion of $E(w(s),\\partial _s w(s))$ (REF ) for $q\\rightarrow 0$ in $\\mathcal {H}$ and get after from straightforward computations: $E(w(s),\\partial _s w(s))= E(\\kappa _0,0)+\\frac{1}{2}\\left[\\bar{\\varphi }_d(\\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}, \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix})+\\sum _{i=2}^{m}\\tilde{\\varphi }_d(\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix},\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})\\right]-\\int _{-1}^{1}\\mathcal {F}_d (q_1) \\rho dy $ where $ \\bar{\\varphi }_d$ , $ \\tilde{\\varphi }_d$ and $\\mathcal {F}_d (q_1)$ are defined in (REF ), (REF ) and (REF ).",
"Using the argument in the real case (see page 113 in [18]) we see that for some $C_0,C_1 >0$ we have: $\\bar{\\varphi }_d(\\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}, \\begin{pmatrix} q_{1,1}\\\\ q_{2,1} \\end{pmatrix}) \\le C_0 \\alpha _{1,1}^2- C_1\\alpha _{-,1}^2.$ From (), (REF ) and (REF ), we see by definition that $0\\le \\tilde{\\varphi }_d(\\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix}, \\begin{pmatrix} q_{1,j}\\\\ q_{2,j} \\end{pmatrix})=\\alpha _{-,j}^2.$ Since we have from (REF ), (REF ), (REF ), (REF ) and (REF ): $\\left| \\int _{-1}^{1}\\mathcal {F}_d (q_1) \\rho dy \\right| \\le C ||q(s)||_{\\mathcal {H}}^{\\bar{p}+1}\\le C \\epsilon ^{\\bar{p}-1} ( \\alpha _{1,1}^2+ \\alpha _{-,1}^2+\\sum _{i=2}^{m} \\alpha _{-,j}^2),$ Using (REF ), (REF ), (REF ) and (REF ), we see that taking $\\epsilon $ small enough so that $C \\epsilon ^{\\bar{p}-1}\\le \\frac{C_1}{4}$ , we get $0\\le E(w(s),\\partial _s w(s))-E(\\kappa _0,0)\\le \\left( \\frac{C_0}{2}+\\frac{C_1}{4}\\right) \\alpha _{-,1}^2-\\frac{C_1}{4}\\alpha _{1,1}^2+\\left(\\frac{1}{2}+\\frac{C_1}{4}\\right)\\sum _{i=2}^{m} \\alpha _{-,j}^2.$ which yields (REF ).",
"Exponential decay of the different components Our aim is to show that $||q(s)||_{\\mathcal {H}}\\rightarrow 0$ and that both $\\theta $ and $d$ converge as $s\\rightarrow \\infty $ .",
"An important issue will be to show that the unstable mode $\\alpha _{1,1}$ , which satisfies equation (REF ) never dominates.",
"This is true thanks to item $(iv)$ in Proposition REF .",
"If we introduce $\\lambda (s)=\\frac{1}{2} \\log \\left(\\frac{1+d(s)}{1-d(s)}\\right), a(s)= \\alpha _{1,1}(s)^2\\, \\mbox{and}\\, b(s)=\\alpha _{-,1}(s)^2+\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2+R_-(s)$ (note that $d(s)=\\tanh (\\lambda (s))$ ), then we see from (REF ), (REF ) and (REF ) that for all $s\\in [s^*, \\bar{s})$ $|R_-(s)|=|b(s)-(\\alpha _{-,1}(s)^2+\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2)|\\le C_0 \\epsilon ^{\\bar{p}-1}(\\alpha _{1,1}(s)^2+\\alpha _{-,1}(s)^2+\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2)$ , hence $\\frac{99}{100}\\alpha _{-,1}(s)^2+ \\frac{99}{100}\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2-\\frac{1}{100} a\\le b\\le \\frac{101}{100}\\alpha _{-,1}(s)^2+ \\frac{101}{100}\\sum _{j=2}^{m}\\alpha _{-,j}(s)^2+\\frac{1}{100} a$ for $\\epsilon $ small enough.",
"Therefore, using Proposition REF , estimate (REF ), (REF ) and the fact that $\\lambda ^{\\prime }(s)=\\frac{d^{\\prime }(s)}{1-d(s)^2}$ , we derive the following: Claim 3.4 (Relations between $a$ , $b$ , $\\lambda $ , $\\theta $ , $\\int _{-1}^1 q_{1,1} q_{2,1} \\rho $ and $\\int _{-1}^1 q_{1,j}q_{2,j} \\rho $ ) There exist positive $\\epsilon _4$ , $K_4$ and $K_5$ such that if $\\epsilon ^*\\le \\epsilon _4$ , then we have for all $s\\in [s^*,\\bar{s}]$ and $j=2,...,m$ : (i) (Size of the solution) $\\frac{1}{K_4}(a(s)+b(s))\\le || q(s)||_\\mathcal {H}^2&\\le K_4 (a(s)+b(s))\\le K_4^2 \\epsilon ^2,\\\\|\\theta ^{\\prime }(s)|+|\\lambda ^{\\prime }(s)|&\\le K_4 (a(s)+b(s))\\le K_4^2 ||q(s)||_\\mathcal {H}^2 ,\\\\\\left| \\int _{-1}^{1}q_{1,1}q_{1,1}\\rho \\right|&\\le K_4 (a(s)+b(s)),\\\\\\left| \\int _{-1}^{1}q_{1,j}q_{1,j}\\rho \\right|&\\le K_4 b(s),$ and (REF ) holds.",
"(ii) (Equations) $\\frac{3}{2} a-K_4 \\epsilon b&\\le a^{\\prime } \\le \\frac{5}{2} a-K_4 \\epsilon b,\\\\b^{\\prime }&\\le -\\frac{8}{p-1}\\int _{-1}^{1}(q_{-,2,1}^2+q_{-,2,j}^2)\\frac{\\rho }{1-y^2}dy+ K_4\\epsilon (a+b),\\\\\\frac{d}{ds}\\int _{-1}^1 (q_{1,1} q_{2,1}+ q_{1,j} q_{2,j}) \\rho &\\le -\\frac{3}{5} b+K_4 \\int _{-1}^1 (q_{-,2,1}^2+ q_{2,j}^2)\\frac{\\rho }{1-y^2}+K_4 a.$ (iii) (Energy barrier) we have $a(s)\\le K_5 b(s).$ [End of the Proof of Theorem REF ] Now, we are ready to finish the proof of Theorem REF just started at the beginning of Section REF .",
"Let us define $s_2^* \\in [s^*,\\bar{s}]$ as the first $s \\in [s^*,\\bar{s}]$ such that $a(s)\\ge \\frac{b(s)}{5 K_4}$ where $K_4$ is introduced in Corollary REF , or $s^*_2=\\bar{s}$ if (REF ) is never satisfied on $[s^*,\\bar{s}]$ .",
"We claim the following: Claim 3.5 There exist positive $\\epsilon _6$ , $\\mu _6$ , $K_6$ and $f\\in C^1([s^*, s^*_2]$ such that if $\\epsilon \\le \\epsilon _6$ , then for all $s\\in [s^*,s_2^*]$ : (i) $\\frac{1}{2}f(s)\\le b(s)\\le 2 f(s)\\mbox{ and }f^{\\prime }(s)\\le -2\\mu _6f(s),$ (ii) $|| q(s)||_\\mathcal {H}\\le K_6 || q(s^*)||_\\mathcal {H}e^{-\\mu _6(s-s^*)}\\le K_6 K_1 \\epsilon ^*e^{-\\mu _6(s-s^*)}.$ The proof of Claim 5.6 page 115 in [18] remains valid where $f(s)$ is given by $f(s)=b(s)+\\eta _6\\int _{-1}^1 (q_{1,1} q_{2,1}+ \\sum _{j=2}^{m}q_{1,j} q_{2,j}) \\rho ,$ where $\\eta _6 >0$ is fixed small independent of $\\epsilon $ .",
"Claim 3.6 (i) There exists $\\epsilon _7>0$ such that for all $\\sigma >0$ , there exists $K_7(\\sigma )>0$ such that if $\\epsilon \\le \\epsilon _7$ , then $\\forall s \\in [s_2^*, \\min (s_2^*+\\sigma , \\bar{s})],\\,|| q(s)||_\\mathcal {H}\\le K_7 || q(s^*)||_\\mathcal {H}e^{-\\mu _6(s-s^*)}\\le K_7 K_1 \\epsilon ^*e^{-\\mu _6(s-s^*)}$ and $|\\theta _i(s)|\\le C\\frac{(K_7 K_1 \\epsilon ^*)^2}{2\\mu _6}$ where $\\mu _6$ has been introduced in Claim REF .",
"(ii) There exists $\\epsilon _8>0$ such that if $\\epsilon \\le \\epsilon _8$ , then $\\forall s \\in (s_2^*, \\bar{s}],\\; b(s)\\le a(s) \\left( 5 K_4 e^{-\\frac{(s-s_2^*)}{2}}+\\frac{1}{4 K_5}\\right)$ where $K_4$ and $K_5$ have been introduced in Corollary REF .",
"The proof is the same as the proof of Claim 5.7 page 117 in [18].",
"Now, in order to conclude the proof of Theorem REF , we fix $\\sigma _0>0$ such that $5K_4^{-\\frac{\\sigma _0}{2}}+\\frac{1}{4K_5}\\le \\frac{1}{2K_5},$ where $K_4$ and $K_5$ are introduced in Claim REF .",
"Then, we fix the value of $K_0=\\max (2,K_6,K_7(\\sigma _0)),$ and the constants are defined in Claims REF and REF .",
"Then, we fix $\\epsilon _0=\\min \\left(1,\\epsilon _1,\\frac{\\epsilon _i}{2K_0K_1}\\mbox{ for }i\\in \\lbrace 4,6,7,8\\rbrace \\right)$ and the constants are defined in Claims REF , REF and REF .",
"Now, if $\\epsilon ^*\\le \\epsilon _0$ , then Claim REF , Claim REF and Claim REF apply.",
"We claim that for all $s\\in [s^*,\\bar{s}]$ , $|| q(s)||_\\mathcal {H}\\le K_0 || q(s^*)||_\\mathcal {H}e^{-\\mu _6(s-s^*)}\\le K_0 K_1 \\epsilon ^*e^{-\\mu _6(s-s^*)}=\\frac{\\epsilon }{2}e^{-\\mu _6(s-s^*)}.$ Indeed, if $s\\in [s^*,\\min (s_2^*+\\sigma _0,\\bar{s})]$ , then, this comes from $(ii)$ of Claim REF or $(i)$ of Claim REF and the definition of $K_0$ (REF ).",
"Now, if $s_2^*+\\sigma _0<\\bar{s}$ and $s\\in [s_2^*+\\sigma _0,\\bar{s}]$ , then we have from (REF ) and the definition of $\\sigma _0$ , $b(s)\\le \\frac{a(s)}{2K_5 }$ on the one hand.",
"On the other hand, from $(iii)$ in Claim REF , we have $a(s)\\le K_5 b(s)$ , hence, $a(s)=b(s)=0$ and from (REF ), $q(y,s)\\equiv 0$ , hence (REF ) is satisfied trivially.",
"In particular, we have for all $s\\in [s~^*,\\bar{s}],\\, ||q||_\\mathcal {H}\\le \\frac{\\epsilon }{2}$ and $ \\cos \\theta _i\\ge 1-C \\frac{\\epsilon ^2}{\\mu _6^2}\\ge \\frac{3}{4}$ , hence, by definition of $\\bar{s}$ given right before (REF ), this means that $\\bar{s}=\\infty $ .",
"From $(i)$ of Claim REF and (), we have $\\forall s \\ge s^*,||q(s)||_\\mathcal {H}\\le \\frac{\\epsilon }{2} e^{-\\mu _6(s-s^*)}\\mbox{ and }|\\theta ^{\\prime }(s)|+|\\lambda ^{\\prime }(s)|\\le K_4^2 \\frac{\\epsilon ^2}{4} e^{-2\\mu _6(s-s^*)},$ where, $\\theta (s)=(\\theta _2 (s),...,\\theta _m (s))$ .",
"Hence, there is $\\theta _\\infty \\in \\mathbb {R}^{m-1}$ , $\\lambda _\\infty $ in $\\mathbb {R}$ such that $\\theta (s) \\rightarrow \\theta _\\infty $ , $\\lambda (s) \\rightarrow \\lambda _\\infty $ as $s\\rightarrow \\infty $ and $\\forall s \\ge s^*,| \\lambda _\\infty -\\lambda (s) |\\le C_1\\epsilon ^{*2} e^{-2\\mu _6(s-s^*)}=C_2\\epsilon ^{2} e^{-2\\mu _6(s-s^*)}$ $\\forall s \\ge s^*,| \\theta _\\infty -\\theta (s) |\\le C_1\\epsilon ^{*2} e^{-2\\mu _6(s-s^*)}=C_2\\epsilon ^{2} e^{-2\\mu _6(s-s^*)}$ for some positive $C_1$ and $C_2$ .",
"Taking $s=s^*$ here, we see that $| \\lambda _\\infty -\\lambda ^* |+|\\theta _\\infty |\\le C_0\\epsilon ^* ,$ where $\\Omega =R_{\\theta _\\infty }(e_1)$ .",
"If $d_\\infty =\\tanh \\lambda _\\infty ,$ then we see that $|d_\\infty -d^*|\\le C_3 (1-d^{*2})\\epsilon ^*.$ Using the definition of $q$ (REF ), (REF ), (REF ) and (REF ) we write $&\\,&\\Bigg |\\Bigg |\\begin{pmatrix}w(s)\\\\\\partial _s w(s)\\end{pmatrix}-\\begin{pmatrix}\\kappa (d _\\infty ,\\cdot )\\Omega _\\infty \\\\0\\end{pmatrix}\\Bigg |\\Bigg |_\\mathcal {H}\\\\&\\le &\\Bigg |\\Bigg |\\begin{pmatrix}w(s)\\\\\\partial _s w(s)\\end{pmatrix}-\\begin{pmatrix}\\kappa (d (s),\\cdot )\\Omega _\\infty \\\\0\\end{pmatrix}\\Bigg |\\Bigg |_\\mathcal {H}+||(\\kappa (d (s),\\cdot )-\\kappa (d _\\infty ,\\cdot )) \\Omega _\\infty ||_{\\mathcal {H}_0}\\\\&+&||\\kappa (d_\\infty ,\\cdot )||_{\\mathcal {H}_0} |R_{\\theta (s)}(e_1)-R_{\\theta _{\\infty }}(e_1)|\\\\&\\le & ||q(s)||_\\mathcal {H}+C|\\lambda _\\infty -\\lambda (s)|+C|\\theta _\\infty -\\theta (s)|\\le C_4 \\epsilon ^*e^{-\\mu _6(s-s^*)},$ where, we used the fact that $\\theta \\in \\mathbb {R}^{m-1}\\mapsto \\mathcal {O}^m$ is a Lipschitz function (see (REF ) to be convinced) and $\\lambda \\in \\mathbb {R}\\mapsto \\kappa (d,\\cdot )\\in \\mathcal {H}_0$ is also Lipschitz, where $d=\\tanh \\lambda $ (see REF ).This concludes the proof of Theorem REF in the case where $\\Omega ^*=e_1$ (see (REF )).",
"From rotataion invariance of equation (REF ), this yields the conclusion of Theorem REF in the general case.",
"A some technical estimates In this section, we give the proof of estimate (REF ) and Lemma REF .",
"Proof of estimate (REF ): Using (REF ), we see that $\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= R_2 ... R_{j-1}\\frac{\\partial R_j}{\\partial \\theta _j}R_{j+1}...R_m.$ From (REF ), we see that $\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= \\Pi _j\\circ R_{j}(\\theta _j+\\frac{\\pi }{2})= R_{j}(\\theta _j+\\frac{\\pi }{2})\\circ \\Pi _j,$ where $\\Pi _j$ is the orthogonal projection on the plane spanned by $e_1$ and $e_j$ , and the rotation $ R_{j}(\\alpha )$ is given by considering the matrix of $R_j$ defined in (REF ), and changing $\\theta _j$ into $\\alpha $ .",
"Since $\\partial R_{j}^{-1 }\\partial R_{j}(\\theta _j+\\frac{\\pi }{2})=\\partial R_{j}(\\frac{\\pi }{2}),$ it follows from $(\\ref {R_theta})$ and $(\\ref {60,5})$ that $R_{\\theta }^{-1}\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= R_m^{-1}...R_{j+1}^{-1} R_{j}(\\frac{\\pi }{2}) \\Pi _j R_{j+1}...R_m.$ By the same argument, we drive that $\\frac{\\partial R_{\\theta }^{-1}}{\\partial \\theta _j}\\partial R_{\\theta }$ has the same expression, thus, (REF ) holds from (REF ) and (REF ).",
"Now, we give the proof of Lemma REF .",
"Proof of Lemma REF: i) We first give the expression of the $m\\times m$ matrix $R_\\theta $ defined (REF ).",
"Indeed, using (REF ) and (REF ), we have: $R_\\theta \\equiv \\begin{pmatrix}\\varphi _{2,m}&-\\sin \\theta _2&\\cdots &-\\sin \\theta _j \\varphi _{2,l-1}&\\cdots &-\\sin \\theta _m \\varphi _{2,m-1}\\\\ \\\\\\end{pmatrix}\\sin \\theta _2 \\varphi _{3,m}&\\cos \\theta _2& & & &.\\\\$ $\\vdots $ R, k,l $\\vdots $ $\\vdots $ j $\\vdots $ k k+1,m 0 $\\vdots $ $\\vdots $ $\\vdots $ mm+1,m0 0m where for $k\\ge 1$ , $l\\ge 2$ : $ R_{\\theta , k, l}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _l \\varphi _{2,l-1}\\; \\mbox{ if } k=1\\\\-\\sin \\theta _k \\sin \\theta _l \\varphi _{k+1,l-1}\\; \\mbox{ if } 2\\le k\\le l-1\\\\\\cos \\theta _k\\; \\mbox{ if } k=l\\\\\\; 0\\; \\; \\mbox{ if } k\\ge l+1\\end{array}\\right.",
"$ with $ \\varphi _{ k, l}=\\left\\lbrace \\begin{array}{l}\\prod \\limits _{n=k}^{l}\\cos \\theta _n \\mbox{ if } k \\le l\\\\1\\mbox{ if } k\\ge l+1.\\end{array}\\right.$ In fact, we will prove the following identities, which imply item $i)$ : $(A)$ For all $i$ ,$j$ $\\in \\lbrace 2,...,m\\rbrace $ , such that $ i\\ne j$ , we have $<e_j,A_i (e_1)>=0.$ $(B)$ For all $i\\in \\lbrace 2,...,m\\rbrace $ $< e_1,A_i e_1>=0.$ $(C)$ For all $i\\in \\lbrace 2,...,m\\rbrace $ , we have $< e_i,A_i e_1>=\\varphi _{ i+1, m},$ where $A_i$ and $\\varphi _{i+1,m}$ are given in (REF ) and (REF ) .",
"$\\blacktriangleright $Proof of $(A)$.",
"Let $i$ ,$j$ $\\in \\lbrace 2,...,m\\rbrace $ , such that $ i\\ne j$ .",
"The idea is to compute $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>$ instead of $<e_j,A_i e_1 >$ .",
"In fact, using the conservation of the inner product after a rotation and the fact that $A_i=R_{\\theta }^{-1} \\frac{\\partial R_\\theta }{\\partial \\theta _i}$ (by (REF )), we have: $<e_j,A_i e_1 >=<R_\\theta e_j,R_\\theta A_i e_1>=< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ In the following, we distinguish two cases: - Case 1: $i\\le j-1$ , - Case 2: $i\\ge j+1$ .",
"We first handle Case 1.",
"Case 1: $i\\le j-1$ .",
"Using (REF ) and its derivative with respect to $\\theta _i$ , we write: $ R_\\theta e_j=(R_\\theta e_j)_{k=1,...,m}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _j \\varphi _{2,j-1}, \\mbox{ if } k=1\\\\-\\sin \\theta _k \\sin \\theta _j \\varphi _{k+1,j-1}\\; \\mbox{ if } 2\\le k\\le j-1\\\\\\cos \\theta _j\\; \\mbox{ if } k=j\\\\\\; 0\\; \\; \\mbox{ if } k\\ge j+1\\end{array}\\right.",
"$ and $ \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1= (\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1)_{ k=1,...., m}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _i \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\mbox{ if } k =1\\\\-\\sin \\theta _i \\sin \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\mbox{ if } 2\\le k\\le i-1\\\\\\cos \\theta _i \\varphi _{i+1,m}, \\mbox{ if } k=i\\\\0\\mbox{ if } k\\ge i+1.\\end{array}\\right.",
"$ Therefore, $&&< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sum _{k=1}^{m}R_{\\theta ,k,j}{\\frac{\\partial R_\\theta }{\\partial \\theta _i}}_{k,1}\\\\&=&\\sin \\theta _i \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\sin \\theta _j \\varphi _{2,j-1}+\\sum _{k=2}^{i-1}(\\sin \\theta _k \\sin \\theta _i \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\sin \\theta _k \\sin \\theta _j \\varphi _{k+1,j-1})\\\\&-& \\cos \\theta _i \\varphi _{i+1,m} \\sin \\theta _i \\sin \\theta _j \\varphi _{i+1,j-1}\\\\&=& \\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}+ \\sum _{k=2}^{i-1}(\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}) - \\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} \\right).$ In order to transform the sum term in the previous identity, we make in the following a finite induction where the parameter $q$ decreases from $i-1$ to 1: Lemma A.1 We have: $ \\forall q \\in \\lbrace 1,...,i-1\\rbrace ,\\;\\sum _{k=2}^{i-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} &\\times \\varphi _{k+1,j-1}-\\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} =\\\\\\sum _{k=2}^{q}&\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\frac{ \\varphi _{q+1,m} \\varphi _{q+1,j-1} }{\\cos \\theta _i}.",
"$ Remark: If $q=1$ , the sum in the right hand side is naturally zero.",
"See below.",
"Applying this Lemma, we conclude the proof of $(A)$ in Case 1 (i. e. when $i\\le j-1$ ).",
"Indeed, from (REF ) and Lemma REF with $q=1$ we write $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1} - \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}\\right)=0.$ It remains now to prove Lemma REF .",
"[Proof of Lemma REF ] First, we give the following: Claim A.2 We have $\\varphi _{i,j-1}=\\cos \\theta _i \\varphi _{i+1,j-1}.$ Since $i\\le j-1$ , we have two cases: - If $i\\le j-2$ : trivial.",
"- If $i=j-1$ : $\\varphi _{i,j-1}=\\varphi _{i,i} =\\cos \\theta _i $ and $\\varphi _{i+1,j-1} = \\varphi _{i+1,i}=1 $ , and the result follows.",
"Now, we are ready to start the proof of Lemma REF .",
"Let us prove the result using an induction with a decreasing index.",
"For $q=i-1$ , (REF ) is satisfied using Claim REF .",
"Assume now that (REF ) is true for $q=i-1,...,2$ and let us prove it for $q-1$ .",
"Using (REF ) with $q$ , we write $&&\\sum _{k=2}^{i-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1} - \\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} =\\\\&& \\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}+ \\sin \\theta _q^2 \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1} -\\frac{ \\varphi _{q+1,m} \\varphi _{q+1,j-1} }{\\cos \\theta _i} =\\\\&& \\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}- \\cos \\theta _q^2 \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1}= \\\\&&\\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1} -\\frac{ \\varphi _{q,m} \\varphi _{q,j-1} }{\\cos \\theta _i}.$ Thus, (REF ) is satisfied for $q-1$ .",
"This concludes the proof of Lemma REF and identity $(A)$ when $i\\le j-1$ .",
"Now, we handle Case 2.",
"- Case 2: $i\\ge j+1$ .",
"Using (REF ) and (REF ), we write: $&&< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sum _{k=1}^{m}R_{\\theta ,k,j}{\\frac{\\partial R_\\theta }{\\partial \\theta _i}}_{k,1}\\\\&=& \\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}+ \\sum _{k=2}^{j-1}(\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}) - \\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} \\right).$ In order to transform the sum term in the previous identity, we make in the following a finite induction where the parameter $q$ decreases from $j-1$ to 1: Lemma A.3 We have: $ \\forall q \\in \\lbrace 1,...,j-1\\rbrace ,\\;\\sum _{k=2}^{j-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} &\\times \\varphi _{k+1,j-1}-\\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} =\\\\\\sum _{k=2}^{q}&\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i} .",
"$ Remark: If $q=1$ , the sum in the right hand side is naturally zero.",
"See below.",
"Applying this Lemma, we conclude the proof of $(A)$ in Case 2 (i. e. when $i\\ge j+1$ ).",
"Indeed, from (REF ) and Lemma REF with $q=1$ we write $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1} - \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}\\right)=0.$ It remains now to prove Lemma REF .",
"[Proof of Lemma REF ] We prove the result using an induction with a decreasing index.",
"For $q=j-1$ , (REF ) is satisfied.",
"Assume now that (REF ) is true for $q=j-1,...,2$ and let us prove it for $q-1$ .",
"Using (REF ) with $q$ , we write $&&\\sum _{k=2}^{j-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} =\\\\&&\\sum _{k=2}^{q}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i}= \\\\&&\\sum _{k=2}^{q-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}+\\sin ^2 \\theta _q \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i} =\\\\&&\\sum _{k=2}^{q-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}- \\frac{ \\varphi _{q,m}}{\\cos \\theta _i}\\varphi _{q,j-1}.$ Thus, (REF ) is satisfied for $q-1$ .",
"This concludes the proof of Lemma REF .",
"$\\blacktriangleright $Proof of $(B)$: As for (REF ) we have: $<e_1,A_i e_1 >=<R_\\theta e_1, \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ From (REF ), we have: $R_\\theta e_1=( \\varphi _{2,m},\\sin \\theta _2\\varphi _{3,m}\\,\\cdots , \\sin \\theta _{i} \\varphi _{i+1,m} ,\\cdots ,\\sin \\theta _m).$ Therefore, using (REF ), we have: $< R_\\theta e_1,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=-\\cos \\theta _i\\sin \\theta _i\\left((\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 +\\sum _{k=2}^{i-1}\\sin ^2\\theta _k (\\frac{\\displaystyle \\varphi _{k+1,m}}{\\cos \\theta _i})^2 \\right)+\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2.$ In order to transform the sum term in the previous identity, we make in the following a finite induction: Lemma A.4 We have: $\\forall q\\in \\lbrace 2,..., i-1\\rbrace ,\\;\\displaystyle (\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 +\\sum _{l=2}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2=(\\frac{\\displaystyle \\varphi _{q,m}}{\\cos \\theta _i})^2 +\\sum _{l=q}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2.$ Remark: If $q=i$ , the sum in the right hand side is naturally zero.",
"Using Lemma REF with $q=i$ we get $< R_\\theta e_1,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=-\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2 +\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2=0,$ which yields the result.",
"In order to conclude $(B)$ we give the proof of Lemma REF .",
"[Proof of Lemma REF ] We proceed by induction for $q\\in \\lbrace 2,..., i-1\\rbrace $ .",
"For $q=2$ , (REF ) is satisfied.",
"Assume that (REF ) is true for $q=2,...,i-1$ and prove it for $q+1$ .",
"Using (REF ) with $q$ , we write $\\displaystyle (\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 &+&\\sum _{l=2}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2=(\\frac{\\displaystyle \\varphi _{q,m}}{\\cos \\theta _i})^2 +\\sum _{l=q}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2\\\\&=&\\cos ^2\\theta _q (\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2 +\\sin ^2\\theta _q (\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2\\\\&=&(\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2 +\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2.$ Thus (REF ) is satisfied for $q+1$ .",
"This concludes the proof of Lemma REF and identity $(B)$ .",
"$\\blacktriangleright $Proof of $(C)$: Consider $i\\in \\lbrace 2,...,m\\rbrace $ .",
"As for (REF ) we have: $<e_i,A_i e_1 >=<R_\\theta e_i, \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ Using (REF ) and (REF ) $<e_i,A_i e_1 >=\\sin ^2\\theta _i \\varphi _{i+1,m} \\left( \\varphi _{2,i-1}^2+\\sum _{k=2}^{i-1}\\sin ^2\\theta _k \\varphi _{k+1,i-1}^2\\right)+\\cos ^2 \\theta _i \\varphi _{i+1,m}.$ In order to transform the sum term in the previous identity, we make in the following a finite induction: Lemma A.5 We have: $\\forall q\\in \\lbrace 2,..., i\\rbrace $ , $\\varphi _{2,i-1}^2+\\sum _{l=2}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2=\\varphi _{q,i-1}^2+\\sum _{l=q}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2.$ Remark: If $q=i$ , the sum in the right hand side is naturally zero.",
"From (REF ) and (REF ) with $q=i$ we get $< R_\\theta e_i,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin ^2\\theta _i \\varphi _{i+1,m}+\\cos ^2\\theta _i \\varphi _{i+1,m}= \\varphi _{i+1,m}.$ which yields the result.",
"In order to conclude $(C)$ we give the proof of Lemma REF .",
"[Proof of Lemma REF ] We proceed by induction for $q\\in \\lbrace 2,..., i\\rbrace $ .",
"For $q=2$ , (REF ) is satisfied.",
"Assume now that (REF ) is true for $q=2,...,i-1$ and prove it for $q+1$ .",
"Using (REF ) with $q$ , we write $\\varphi _{2,i-1}^2+\\sum _{l=2}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2&=&\\varphi _{q,i-1}^2+\\sum _{l=q}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2\\\\&=&\\cos ^2\\theta _q \\varphi _{q+1,i-1}^2+\\sin ^2\\theta _q \\varphi _{q+1,i-1}^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2\\\\&=&\\varphi _{q+1,i-1}^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2.$ Thus (REF ) is satisfied for $q+1$ .",
"This concludes the proof of Lemma REF .",
"ii)We recall from (REF ) that we have $\\frac{\\partial R_\\theta }{\\partial \\theta _j}=R_2 \\cdots R_{j-1} \\frac{\\partial R_j}{\\partial \\theta _j} R_{j+1} \\cdots R_m,$ so by (REF ), $A_j$ is given explicitly by $A_j=R_m^{-1} R_{m-1}^{-1} \\cdots R_{j}^{-1} \\frac{\\partial R_j}{\\partial \\theta _j} R_{j+1} \\cdots R_m.$ From a straightforward geometrical observation, we can see that the rotation conserves the euclidien norm in $\\mathbb {R}^m$ .",
"For $\\frac{\\partial R_j}{\\partial \\theta _j} $ , it can be seen as a composition of a projection on the plane $(e_1,e_j)$ and a rotation with angle $\\theta _j+\\frac{\\pi }{2}$ , which decreases the norm.",
"This concludes the proof of Lemma REF .",
"Address: Université de Cergy-Pontoise, Laboratoire Analyse Géometrie Modélisation, CNRS-UMR 8088, 2 avenue Adolphe Chauvin 95302, Cergy-Pontoise, France.",
"e-mail: [email protected] Université Paris 13, Institut Galilée, Laboratoire Analyse Géometrie et Applications, CNRS-UMR 7539, 99 avenue J.B. Clément 93430, Villetaneuse, France.",
"e-mail: [email protected]"
],
[
"A some technical estimates",
"In this section, we give the proof of estimate (REF ) and Lemma REF .",
"Proof of estimate (REF ): Using (REF ), we see that $\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= R_2 ... R_{j-1}\\frac{\\partial R_j}{\\partial \\theta _j}R_{j+1}...R_m.$ From (REF ), we see that $\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= \\Pi _j\\circ R_{j}(\\theta _j+\\frac{\\pi }{2})= R_{j}(\\theta _j+\\frac{\\pi }{2})\\circ \\Pi _j,$ where $\\Pi _j$ is the orthogonal projection on the plane spanned by $e_1$ and $e_j$ , and the rotation $ R_{j}(\\alpha )$ is given by considering the matrix of $R_j$ defined in (REF ), and changing $\\theta _j$ into $\\alpha $ .",
"Since $\\partial R_{j}^{-1 }\\partial R_{j}(\\theta _j+\\frac{\\pi }{2})=\\partial R_{j}(\\frac{\\pi }{2}),$ it follows from $(\\ref {R_theta})$ and $(\\ref {60,5})$ that $R_{\\theta }^{-1}\\frac{\\partial R_{\\theta }}{\\partial \\theta _j}= R_m^{-1}...R_{j+1}^{-1} R_{j}(\\frac{\\pi }{2}) \\Pi _j R_{j+1}...R_m.$ By the same argument, we drive that $\\frac{\\partial R_{\\theta }^{-1}}{\\partial \\theta _j}\\partial R_{\\theta }$ has the same expression, thus, (REF ) holds from (REF ) and (REF ).",
"Now, we give the proof of Lemma REF .",
"Proof of Lemma REF: i) We first give the expression of the $m\\times m$ matrix $R_\\theta $ defined (REF ).",
"Indeed, using (REF ) and (REF ), we have: $R_\\theta \\equiv \\begin{pmatrix}\\varphi _{2,m}&-\\sin \\theta _2&\\cdots &-\\sin \\theta _j \\varphi _{2,l-1}&\\cdots &-\\sin \\theta _m \\varphi _{2,m-1}\\\\ \\\\\\end{pmatrix}\\sin \\theta _2 \\varphi _{3,m}&\\cos \\theta _2& & & &.\\\\$ $\\vdots $ R, k,l $\\vdots $ $\\vdots $ j $\\vdots $ k k+1,m 0 $\\vdots $ $\\vdots $ $\\vdots $ mm+1,m0 0m where for $k\\ge 1$ , $l\\ge 2$ : $ R_{\\theta , k, l}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _l \\varphi _{2,l-1}\\; \\mbox{ if } k=1\\\\-\\sin \\theta _k \\sin \\theta _l \\varphi _{k+1,l-1}\\; \\mbox{ if } 2\\le k\\le l-1\\\\\\cos \\theta _k\\; \\mbox{ if } k=l\\\\\\; 0\\; \\; \\mbox{ if } k\\ge l+1\\end{array}\\right.",
"$ with $ \\varphi _{ k, l}=\\left\\lbrace \\begin{array}{l}\\prod \\limits _{n=k}^{l}\\cos \\theta _n \\mbox{ if } k \\le l\\\\1\\mbox{ if } k\\ge l+1.\\end{array}\\right.$ In fact, we will prove the following identities, which imply item $i)$ : $(A)$ For all $i$ ,$j$ $\\in \\lbrace 2,...,m\\rbrace $ , such that $ i\\ne j$ , we have $<e_j,A_i (e_1)>=0.$ $(B)$ For all $i\\in \\lbrace 2,...,m\\rbrace $ $< e_1,A_i e_1>=0.$ $(C)$ For all $i\\in \\lbrace 2,...,m\\rbrace $ , we have $< e_i,A_i e_1>=\\varphi _{ i+1, m},$ where $A_i$ and $\\varphi _{i+1,m}$ are given in (REF ) and (REF ) .",
"$\\blacktriangleright $Proof of $(A)$.",
"Let $i$ ,$j$ $\\in \\lbrace 2,...,m\\rbrace $ , such that $ i\\ne j$ .",
"The idea is to compute $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>$ instead of $<e_j,A_i e_1 >$ .",
"In fact, using the conservation of the inner product after a rotation and the fact that $A_i=R_{\\theta }^{-1} \\frac{\\partial R_\\theta }{\\partial \\theta _i}$ (by (REF )), we have: $<e_j,A_i e_1 >=<R_\\theta e_j,R_\\theta A_i e_1>=< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ In the following, we distinguish two cases: - Case 1: $i\\le j-1$ , - Case 2: $i\\ge j+1$ .",
"We first handle Case 1.",
"Case 1: $i\\le j-1$ .",
"Using (REF ) and its derivative with respect to $\\theta _i$ , we write: $ R_\\theta e_j=(R_\\theta e_j)_{k=1,...,m}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _j \\varphi _{2,j-1}, \\mbox{ if } k=1\\\\-\\sin \\theta _k \\sin \\theta _j \\varphi _{k+1,j-1}\\; \\mbox{ if } 2\\le k\\le j-1\\\\\\cos \\theta _j\\; \\mbox{ if } k=j\\\\\\; 0\\; \\; \\mbox{ if } k\\ge j+1\\end{array}\\right.",
"$ and $ \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1= (\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1)_{ k=1,...., m}=\\left\\lbrace \\begin{array}{l}-\\sin \\theta _i \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\mbox{ if } k =1\\\\-\\sin \\theta _i \\sin \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\mbox{ if } 2\\le k\\le i-1\\\\\\cos \\theta _i \\varphi _{i+1,m}, \\mbox{ if } k=i\\\\0\\mbox{ if } k\\ge i+1.\\end{array}\\right.",
"$ Therefore, $&&< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sum _{k=1}^{m}R_{\\theta ,k,j}{\\frac{\\partial R_\\theta }{\\partial \\theta _i}}_{k,1}\\\\&=&\\sin \\theta _i \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\sin \\theta _j \\varphi _{2,j-1}+\\sum _{k=2}^{i-1}(\\sin \\theta _k \\sin \\theta _i \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\sin \\theta _k \\sin \\theta _j \\varphi _{k+1,j-1})\\\\&-& \\cos \\theta _i \\varphi _{i+1,m} \\sin \\theta _i \\sin \\theta _j \\varphi _{i+1,j-1}\\\\&=& \\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}+ \\sum _{k=2}^{i-1}(\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}) - \\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} \\right).$ In order to transform the sum term in the previous identity, we make in the following a finite induction where the parameter $q$ decreases from $i-1$ to 1: Lemma A.1 We have: $ \\forall q \\in \\lbrace 1,...,i-1\\rbrace ,\\;\\sum _{k=2}^{i-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} &\\times \\varphi _{k+1,j-1}-\\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} =\\\\\\sum _{k=2}^{q}&\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\frac{ \\varphi _{q+1,m} \\varphi _{q+1,j-1} }{\\cos \\theta _i}.",
"$ Remark: If $q=1$ , the sum in the right hand side is naturally zero.",
"See below.",
"Applying this Lemma, we conclude the proof of $(A)$ in Case 1 (i. e. when $i\\le j-1$ ).",
"Indeed, from (REF ) and Lemma REF with $q=1$ we write $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1} - \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}\\right)=0.$ It remains now to prove Lemma REF .",
"[Proof of Lemma REF ] First, we give the following: Claim A.2 We have $\\varphi _{i,j-1}=\\cos \\theta _i \\varphi _{i+1,j-1}.$ Since $i\\le j-1$ , we have two cases: - If $i\\le j-2$ : trivial.",
"- If $i=j-1$ : $\\varphi _{i,j-1}=\\varphi _{i,i} =\\cos \\theta _i $ and $\\varphi _{i+1,j-1} = \\varphi _{i+1,i}=1 $ , and the result follows.",
"Now, we are ready to start the proof of Lemma REF .",
"Let us prove the result using an induction with a decreasing index.",
"For $q=i-1$ , (REF ) is satisfied using Claim REF .",
"Assume now that (REF ) is true for $q=i-1,...,2$ and let us prove it for $q-1$ .",
"Using (REF ) with $q$ , we write $&&\\sum _{k=2}^{i-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1} - \\cos \\theta _i \\varphi _{i+1,m} \\varphi _{i+1,j-1} =\\\\&& \\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}+ \\sin \\theta _q^2 \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1} -\\frac{ \\varphi _{q+1,m} \\varphi _{q+1,j-1} }{\\cos \\theta _i} =\\\\&& \\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}- \\cos \\theta _q^2 \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1}= \\\\&&\\sum _{k=2}^{q-1}\\sin \\theta _k^2 \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1} -\\frac{ \\varphi _{q,m} \\varphi _{q,j-1} }{\\cos \\theta _i}.$ Thus, (REF ) is satisfied for $q-1$ .",
"This concludes the proof of Lemma REF and identity $(A)$ when $i\\le j-1$ .",
"Now, we handle Case 2.",
"- Case 2: $i\\ge j+1$ .",
"Using (REF ) and (REF ), we write: $&&< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sum _{k=1}^{m}R_{\\theta ,k,j}{\\frac{\\partial R_\\theta }{\\partial \\theta _i}}_{k,1}\\\\&=& \\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}+ \\sum _{k=2}^{j-1}(\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}) - \\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} \\right).$ In order to transform the sum term in the previous identity, we make in the following a finite induction where the parameter $q$ decreases from $j-1$ to 1: Lemma A.3 We have: $ \\forall q \\in \\lbrace 1,...,j-1\\rbrace ,\\;\\sum _{k=2}^{j-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} &\\times \\varphi _{k+1,j-1}-\\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} =\\\\\\sum _{k=2}^{q}&\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i} .",
"$ Remark: If $q=1$ , the sum in the right hand side is naturally zero.",
"See below.",
"Applying this Lemma, we conclude the proof of $(A)$ in Case 2 (i. e. when $i\\ge j+1$ ).",
"Indeed, from (REF ) and Lemma REF with $q=1$ we write $< R_\\theta e_j,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin \\theta _i \\sin \\theta _j \\left( \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1} - \\frac{\\varphi _{2,m}}{\\cos \\theta _i} \\varphi _{2,j-1}\\right)=0.$ It remains now to prove Lemma REF .",
"[Proof of Lemma REF ] We prove the result using an induction with a decreasing index.",
"For $q=j-1$ , (REF ) is satisfied.",
"Assume now that (REF ) is true for $q=j-1,...,2$ and let us prove it for $q-1$ .",
"Using (REF ) with $q$ , we write $&&\\sum _{k=2}^{j-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\cos \\theta _j \\frac{ \\varphi _{j+1,m}}{\\cos \\theta _i} =\\\\&&\\sum _{k=2}^{q}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i}= \\\\&&\\sum _{k=2}^{q-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}+\\sin ^2 \\theta _q \\frac{\\varphi _{q+1,m}}{\\cos \\theta _i} \\times \\varphi _{q+1,j-1}-\\varphi _{q+1,j-1} \\frac{ \\varphi _{q+1,m}}{\\cos \\theta _i} =\\\\&&\\sum _{k=2}^{q-1}\\sin ^2 \\theta _k \\frac{\\varphi _{k+1,m}}{\\cos \\theta _i} \\times \\varphi _{k+1,j-1}- \\frac{ \\varphi _{q,m}}{\\cos \\theta _i}\\varphi _{q,j-1}.$ Thus, (REF ) is satisfied for $q-1$ .",
"This concludes the proof of Lemma REF .",
"$\\blacktriangleright $Proof of $(B)$: As for (REF ) we have: $<e_1,A_i e_1 >=<R_\\theta e_1, \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ From (REF ), we have: $R_\\theta e_1=( \\varphi _{2,m},\\sin \\theta _2\\varphi _{3,m}\\,\\cdots , \\sin \\theta _{i} \\varphi _{i+1,m} ,\\cdots ,\\sin \\theta _m).$ Therefore, using (REF ), we have: $< R_\\theta e_1,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=-\\cos \\theta _i\\sin \\theta _i\\left((\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 +\\sum _{k=2}^{i-1}\\sin ^2\\theta _k (\\frac{\\displaystyle \\varphi _{k+1,m}}{\\cos \\theta _i})^2 \\right)+\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2.$ In order to transform the sum term in the previous identity, we make in the following a finite induction: Lemma A.4 We have: $\\forall q\\in \\lbrace 2,..., i-1\\rbrace ,\\;\\displaystyle (\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 +\\sum _{l=2}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2=(\\frac{\\displaystyle \\varphi _{q,m}}{\\cos \\theta _i})^2 +\\sum _{l=q}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2.$ Remark: If $q=i$ , the sum in the right hand side is naturally zero.",
"Using Lemma REF with $q=i$ we get $< R_\\theta e_1,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=-\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2 +\\cos \\theta _i\\sin \\theta _i (\\varphi _{i+1,m})^2=0,$ which yields the result.",
"In order to conclude $(B)$ we give the proof of Lemma REF .",
"[Proof of Lemma REF ] We proceed by induction for $q\\in \\lbrace 2,..., i-1\\rbrace $ .",
"For $q=2$ , (REF ) is satisfied.",
"Assume that (REF ) is true for $q=2,...,i-1$ and prove it for $q+1$ .",
"Using (REF ) with $q$ , we write $\\displaystyle (\\frac{\\displaystyle \\varphi _{2,m}}{\\cos \\theta _i})^2 &+&\\sum _{l=2}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2=(\\frac{\\displaystyle \\varphi _{q,m}}{\\cos \\theta _i})^2 +\\sum _{l=q}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2\\\\&=&\\cos ^2\\theta _q (\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2 +\\sin ^2\\theta _q (\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2\\\\&=&(\\frac{\\displaystyle \\varphi _{q+1,m}}{\\cos \\theta _i})^2 +\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l (\\frac{\\displaystyle \\varphi _{l+1,m}}{\\cos \\theta _i})^2.$ Thus (REF ) is satisfied for $q+1$ .",
"This concludes the proof of Lemma REF and identity $(B)$ .",
"$\\blacktriangleright $Proof of $(C)$: Consider $i\\in \\lbrace 2,...,m\\rbrace $ .",
"As for (REF ) we have: $<e_i,A_i e_1 >=<R_\\theta e_i, \\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>.$ Using (REF ) and (REF ) $<e_i,A_i e_1 >=\\sin ^2\\theta _i \\varphi _{i+1,m} \\left( \\varphi _{2,i-1}^2+\\sum _{k=2}^{i-1}\\sin ^2\\theta _k \\varphi _{k+1,i-1}^2\\right)+\\cos ^2 \\theta _i \\varphi _{i+1,m}.$ In order to transform the sum term in the previous identity, we make in the following a finite induction: Lemma A.5 We have: $\\forall q\\in \\lbrace 2,..., i\\rbrace $ , $\\varphi _{2,i-1}^2+\\sum _{l=2}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2=\\varphi _{q,i-1}^2+\\sum _{l=q}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2.$ Remark: If $q=i$ , the sum in the right hand side is naturally zero.",
"From (REF ) and (REF ) with $q=i$ we get $< R_\\theta e_i,\\frac{\\partial R_\\theta }{\\partial \\theta _i} e_1>=\\sin ^2\\theta _i \\varphi _{i+1,m}+\\cos ^2\\theta _i \\varphi _{i+1,m}= \\varphi _{i+1,m}.$ which yields the result.",
"In order to conclude $(C)$ we give the proof of Lemma REF .",
"[Proof of Lemma REF ] We proceed by induction for $q\\in \\lbrace 2,..., i\\rbrace $ .",
"For $q=2$ , (REF ) is satisfied.",
"Assume now that (REF ) is true for $q=2,...,i-1$ and prove it for $q+1$ .",
"Using (REF ) with $q$ , we write $\\varphi _{2,i-1}^2+\\sum _{l=2}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2&=&\\varphi _{q,i-1}^2+\\sum _{l=q}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2\\\\&=&\\cos ^2\\theta _q \\varphi _{q+1,i-1}^2+\\sin ^2\\theta _q \\varphi _{q+1,i-1}^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2\\\\&=&\\varphi _{q+1,i-1}^2+\\sum _{l=q+1}^{i-1}\\sin ^2\\theta _l \\varphi _{l+1,i-1}^2.$ Thus (REF ) is satisfied for $q+1$ .",
"This concludes the proof of Lemma REF .",
"ii)We recall from (REF ) that we have $\\frac{\\partial R_\\theta }{\\partial \\theta _j}=R_2 \\cdots R_{j-1} \\frac{\\partial R_j}{\\partial \\theta _j} R_{j+1} \\cdots R_m,$ so by (REF ), $A_j$ is given explicitly by $A_j=R_m^{-1} R_{m-1}^{-1} \\cdots R_{j}^{-1} \\frac{\\partial R_j}{\\partial \\theta _j} R_{j+1} \\cdots R_m.$ From a straightforward geometrical observation, we can see that the rotation conserves the euclidien norm in $\\mathbb {R}^m$ .",
"For $\\frac{\\partial R_j}{\\partial \\theta _j} $ , it can be seen as a composition of a projection on the plane $(e_1,e_j)$ and a rotation with angle $\\theta _j+\\frac{\\pi }{2}$ , which decreases the norm.",
"This concludes the proof of Lemma REF .",
"Address: Université de Cergy-Pontoise, Laboratoire Analyse Géometrie Modélisation, CNRS-UMR 8088, 2 avenue Adolphe Chauvin 95302, Cergy-Pontoise, France.",
"e-mail: [email protected] Université Paris 13, Institut Galilée, Laboratoire Analyse Géometrie et Applications, CNRS-UMR 7539, 99 avenue J.B. Clément 93430, Villetaneuse, France.",
"e-mail: [email protected]"
]
] | 1612.05427 |
[
[
"Robust Improvement of the Age of Information by Adaptive Packet Coding"
],
[
"Abstract We consider a wireless communication network with an adaptive scheme to select the number of packets to be admitted and encoded for each transmission, and characterize the information timeliness.",
"For a network of erasure channels and discrete time, we provide closed form expressions for the Average and Peak Age of Information (AoI) as functions of admission control and adaptive coding parameters, the feedback delay, and the maximum feasible end-to-end rate that depends on channel conditions and network topology.",
"These new results guide the system design for robust improvements of the AoI when transmitting time sensitive information in the presence of topology and channel changes.",
"We illustrate the benefits of using adaptive packet coding to improve information timeliness by characterizing the network performance with respect to the AoI along with its relationship to throughput (rate of successfully decoded packets at the destination) and per-packet delay.",
"We show that significant AoI performance gains can be obtained in comparison to the uncoded case, and that these gains are robust to network variations as channel conditions and network topology change."
],
[
"Introduction",
"As wireless networks become more ubiquitous and diverse, different quality of service (QoS) requirements arise to serve different applications.",
"In particular, many sensing, monitoring, decision, and control applications require the wireless network to be optimized to deliver timely information.",
"Examples of such applications include surveillance systems, autonomous vehicles, healthcare monitoring, Internet of Things (IoT), and Ultra-Reliable Low-Latency Communication (URLLC) services in 5G.",
"On the other hand, long-distance communications such as satellite communications and beyond line of sight (BLoS) high frequency (HF) communications impose long propagation delays that make the delivery of information in a timely manner even a more challenging task.",
"To address the need to design and optimize a wireless network to deliver fresh information about an observed process (such as a status update), metrics related to the Age of Information (AoI) have received much attention recently [1].",
"AoI provides means to quantify information timeliness (or freshness) when multiple observations are to be sequentially transmitted through a network to a destination that is interested in the most up-to-date observation, using metrics to characterize the average AoI [2] or the peak AoI [3].",
"Prior to this special attention paid to alternative performance metrics such as AoI, most work on network optimization has focused on the traditional metrics of throughput and delay.",
"The benefits of packet level coding such as network coding to improve the network throughput are well known [4], [5].",
"Capacity achieving schemes are available, promoting resiliency to channel erasures and reliability with more efficient feedback [6].",
"Network coding has been applied to wireless settings by accounting for channel effects and interactions with other network layers [7], [8].",
"The importance of network coding has been highlighted in [9] for systems that require information timeliness, and the AoI has been characterized in a multicast network with packet-level coding.",
"The implication of timeliness for network coding design in terms of meeting hard delay deadlines has been studied in [10].",
"While promoting improved throughput, network coding may incur other transmission and processing delay costs [11], [12], [13].",
"The trade-off between delay sensitivity and throughput can be formally captured using $\\mathcal {(}l)_p$ norms of the packet arrival times, as shown in [14].",
"This class of delay metrics captures the average delay, hence the rate of transmission, at one extreme, and the maximum ordered inter-arrival delay at the other extreme.",
"An adaptive coding scheme has been proposed in [15], extending the generation-based random linear network coding [16] with a coding bucket of variable size, which takes the role of head-of-the-line, containing the packets that will be encoded together and sent through the wireless channel.",
"The point-to-point scheme proposed in [15] has been extended in [14] to multi-hop line networks with several feedback schemes.",
"For time-sensitive applications, we are mostly interested in the use of recoding at intermediate nodes and end-to-end acknowledgement (ACK).",
"In this case, the intermediate nodes do not attempt to decode packets or send link-level ACKs, but simply encode the received packets together and send them towards the destination.",
"When the destination node receives sufficient degrees of freedom to decode all the original packets in a bucket, it sends an ACK message back to the source node.",
"The proposed metrics based on $\\mathcal {(}l)_p$ norms provide a framework to optimize the coding and scheduling depending on the desired level of delay sensitivity.",
"However, when timeliness of information is the objective, optimizing the network with respect to throughput or delay may not be equivalent to optimizing the network with respect to AoI metrics.",
"We highlight that the AoI encompasses the effects of throughput and delay, in addition to the effect of admission control in regulating the rate at which information is injected into the network.",
"For example, a node may have outdated information about a process of interest if update messages are not available to be injected in the network.",
"In this case, even if the delays are small, the AoI at the destination node would be large.",
"Hence, characterizing the AoI is important to optimize the system when the application requires timely information.",
"In this paper, we study the intricate relationships between AoI, per-packet delay, and throughput metrics, and demonstrate that adaptive network coding promotes a significant AoI performance improvement that is robust with respect to variations of channel conditions and network topology.",
"In a network of erasure channels, packets are combined using random linear network coding (RLNC) at the source and sent towards a destination that is interested in fresh information of decoded packets and sends ACKs reaching back the source after a feedback delay.",
"The source can control the packet admission and the number of packets to be encoded in each transmission for timely information delivery.",
"In this setting, we provide closed form expressions for the Average and Peak AoI as functions of the design parameters for the admission control and adaptive coding schemes, the feedback delay, and the maximum rate that can be achieved from the source to the destination through different networks, ranging from single-hop to multi-hop and multi-path.",
"We demonstrate that adaptive coding sustains a robust improvement in the presence of changes in channel conditions and network topology.",
"The rest of the paper is organized as follows.",
"Section II describes the adaptive coding scheme that we consider in this paper.",
"Section III presents the timeliness performance metrics.",
"Section IV analyzes performance trade-offs for both point-to-point communications and multi-hop line networks.",
"Section V presents numerical results.",
"Section VI concludes the paper."
],
[
"Adaptive Coding",
"We consider the adaptive coding scheme proposed in [15] to transmit status update packets through a network of erasure channels.",
"Adaptive coding is illustrated in Fig.REF .",
"The source may apply admission control to regulate the packet arrival rate $\\lambda $ .",
"An example of a simple admission control is to sample from the arriving process, keeping packets with probability $\\beta $ and discarding packets with probability $1-\\beta $ .",
"The admitted packets are kept in a buffer, in order of arrival.",
"The source node keeps a coding bucket with up to $K$ packets, corresponding to the head of the line used in generation-based RLNC [16].",
"The packets in the bucket are coded using RLNC.",
"A coded packet is a linear combination of all the packets in the bucket, with a vector of coefficients drawn uniformly at random from the space ${F}_q^K$ over a finite field ${F}_q$ of size $q$ .",
"We assume that the field is sufficiently large, so that coded packets are linearly independent, and every packet successfully received at the destination is informative.",
"Coded packets are generated and transmitted, one per time slot, as long as there is at least one packet in the bucket.",
"The maximum feasible rate $r$ that can be achieved from the source to the destination depends on erasure probabilities and network topology.",
"For a single link, $r=1-\\epsilon $ for erasure probability $\\epsilon $ .",
"We determine $r$ for multi-hop and multi-path networks in Section REF .",
"An arriving packet will be included in the next coded packet if upon arrival it finds the bucket with less than $K$ packets, otherwise it will wait in the queue.",
"The source node transmits linear combinations of the packets in the coding bucket, together with the coding coefficients (note that this overhead is negligible compared to the packet size), until the destination node receives enough packets to decode them all using Gaussian elimination.",
"The destination sends an ACK message to the source, and we assume this feedback arrives successfully after the delay $D$ .",
"Once the feedback is received, the source node empties the coding bucket by discarding those packets, and moves new packets that may be waiting in the queue.",
"The maximum number of packets included in the coding bucket, $K$ can be determined depending on timeliness and throughput requirements.",
"The throughput, denoted with $\\mu $ , is defined as the rate of packets that are successfully decoded at the destination, and is a function of $r$ , $D$ and $K$ , as well as the packet size $L$ , as we discuss in more detail in Section .",
"We characterize the trade-off between performance metrics related to timeliness and the more traditional metrics of throughput and delay.",
"Figure: Model: adaptive network coding is performed at source node, coded packets are transmitted through an erasure channel, feedback is sent from destination when packets are decoded."
],
[
"Timeliness Performance Metrics",
"We use the AoI metrics to measure information timeliness.",
"AoI is a time-evolving process evaluated at the destination that tracks the time since the last received update was generated.",
"Let $U(t)$ be the time at which the most up-to-date message was generated.",
"For discrete (slotted) time, the AoI evolves as $A(t+1) ={\\left\\lbrace \\begin{array}{ll}A(t)+1 & \\text{if no update received,}\\\\\\min \\lbrace t-U(t),A(t)\\rbrace & \\text{if update received.}\\end{array}\\right.",
"}$ The commonly used metrics to characterize the AoI are the Average AoI (AAoI), which we denote with $A_A$ , and the Peak AoI (PAoI), which we denote with $A_P$ .",
"For the case of discrete time, these metrics can be formally defined as $A_A \\triangleq \\limsup _{T\\rightarrow \\infty }\\frac{1}{T}\\sum _{t=1}^{T}A(t),$ $A_P \\triangleq \\limsup _{T\\rightarrow \\infty }\\frac{\\sum _{t=1}^{T}A(t){1}_{\\lbrace A(t+1)\\le A(t)\\rbrace }}{\\sum _{t=1}^{T}{1}_{\\lbrace A(t+1)\\le A(t)\\rbrace }}.$ Assuming a Bernoulli arrival process of rate $\\lambda $ at the source node, a general service time distribution, and a single server (or a single path from source to destination), we have a Ber/G/1 queue.",
"Let $S$ denote the service time, with $\\mathbb {E}[S]=1/\\mu $ and $\\rho =\\lambda \\mathbb {E}[S]$ .",
"The waiting time in the Ber/G/1 queue [17] is given by $\\mathbb {E}[W]=\\frac{\\lambda \\mathbb {E}[S^2]-\\rho }{2(1-\\rho )}.$ The AAoI and PAoI for the Ber/G/1 queue have been presented in [18] and are given by $A_A = 1+\\frac{1}{\\mu }+\\frac{(1-\\lambda )(1-\\rho )}{\\lambda \\mathcal {L}_S(1-\\lambda )}+\\frac{\\lambda \\mathbb {E}[S^2]-\\rho }{2(1-\\rho )}, \\;\\text{and}$ $A_P = \\frac{1}{\\lambda }+\\frac{1}{\\mu }+\\frac{\\lambda \\mathbb {E}[S^2]-\\rho }{2(1-\\rho )},$ where $ \\mathcal {L}_S(\\cdot )$ represents the probability generating function (PGF) of the service time $S$ ."
],
[
"Point-to-Point Communication",
"We denote with $L$ the packet length and with $N$ the number of original packets that should be delivered.",
"We also let $\\Delta T_i$ represent the ordered inter-delivery time for the $i$ th packet.",
"That is, if $T_i$ represents the in-order delivery time of the $i$ th packet, then $\\Delta T_i = T_i - T_{i-1}, \\; i\\in \\lbrace 1,\\ldots ,N\\rbrace $ , where we define $T_0\\triangleq -D$ to account for the feedback delay $D$ associated to each bucket of at least one packet.",
"The delay sensitivity of the receiver is modeled using the $\\mathcal {L}_p$ -norm of the sequence of variables $(\\Delta T_1, \\Delta T_2,\\ldots ,\\Delta T_N)$ with a delay cost function defined by [15]: $d(p)=\\frac{1}{L} \\left[\\frac{1}{N} \\sum _{i=1}^{N} \\mathbb {E}[\\Delta T_i]^p \\right]^{\\frac{1}{p}} ,\\;\\;\\;p\\in [1,\\infty ).$ When $p=1$ , we have the average delay per packet, normalized by the total size of the received data.",
"Hence, minimizing $d(1)$ is tantamount to maximizing the throughput $\\mu = d(1)^{-1}$ .",
"When $p=\\infty $ , we have $d(\\infty )=\\frac{1}{L}\\max _{i=1\\ldots N} \\Delta T_i$ , which is the maximum expected inter-arrival time between two successive packets, or per-packet delay.",
"Hence, minimizing $d(\\infty )$ is equivalent to minimizing the per-packet delay.",
"Consider the transmission of $N$ packets using coding buckets with $K$ packets.",
"We identify the packets in the $i$ th bucket as $\\lbrace P_{i1}, \\ldots ,P_{iK}\\rbrace $ , and denote with $\\Delta T_{ij}$ the inter-delivery time of the $j$ th packet within the $i$ th bucket.",
"The cost function in (REF ) can be rewritten as $d(p)=\\frac{1}{L} \\left[\\frac{1}{N}\\frac{N}{K}\\sum _{j=1}^{K} \\mathbb {E}[\\Delta T_{ij}]^p \\right]^{\\frac{1}{p}} ,\\;\\;\\;p\\in [1,\\infty ).$ Note that, within the same bucket, the packets are assumed to have the same delivery time, which is the time when all the packets can be decoded at the destination node.",
"Let $X$ represent the time to deliver the entire bucket, while $r$ denotes the maximum feasible end-to-end rate.",
"Then, $\\Delta T_{ij}=X+D$ for $j\\equiv 1 $ (mod $K$ ), and zero otherwise.",
"Under the assumption of an erasure channel, the number of time slots $X$ needed to successfully deliver $K$ linearly independent coded packets follows a negative binomial distribution $NegBin(r,K)$ , with $\\mathbb {E}[X]=K/r$ .",
"Using these properties in (REF ), yields $d(p)=\\frac{1}{LK^{\\frac{1}{p}}} \\left(\\frac{K}{r}+D\\right).$ If we look at a single packet, as opposed to the entire bucket, the service rate is $S=\\frac{X+D}{LK}$ , where $X$ is the random variable representing the time to deliver all the $K$ packets in a bucket, $D$ is a constant feedback delay, $L$ is the packet length, and $K$ is the bucket size.",
"As a result, we have $\\mathbb {E}[S] & = & \\frac{1}{rL}+\\frac{D}{LK},\\\\Var(S) & = & \\frac{1-r}{L^2K r^2},\\\\\\mathbb {E}[S^2]&=& \\frac{(1-r)K+(K+Dr)^2}{(LrK)^2},\\\\\\mathcal {L}_S(z)&=&z^{\\frac{D}{LK}}\\mathcal {L}_X(z^{\\frac{1}{LK}}).$ Note that $\\mathbb {E}[S]=d(1)$ .",
"Also, for a negative binomial random variable $X$ representing the number of attempts until $K$ successful transmissions, we have $\\mathcal {L}_X(z)=\\left(\\frac{rz}{1-(1-r)z}\\right)^K,$ hence for the service time of each packet, the PGF is $\\mathcal {L}_S(z)=z^{\\frac{D}{LK}}\\left(\\frac{rz^{\\frac{1}{LK}}}{1-(1-r)z^{\\frac{1}{LK}}}\\right)^K.$ Using (REF )–(REF ) together with (REF ) and (REF ) yields closed form expressions for AAoI and PAoI as a function of $\\lambda $ , $L$ , $D$ , $r$ , and $K$ .",
"For the PAoI, we obtain rCl AP = 1+K+DrKLr+ [K(1-r)-(KLr)(K+Dr)+(K+Dr)2]2[(KLr)2-(KLr)(K+Dr)], which can be used together with (REF ) to obtain the expression for the AAoI as rCl AA =AP+(1-1) + (1-)(1-(K+Dr)KLr)(1-)DLK(r(1-)1LK1-(1-r)(1-)1LK)K .",
"Note that the variables that can be controlled are the arrival rate $\\lambda $ , which can be modified through admission control policies, and the bucket size $K$ , which can be adapted according to the sensitivity to the timeliness, as well as the delay and throughput constraints.",
"From [15], we have the mean maximum inter-delivery time $d (\\infty ) = \\frac{K}{rL} + \\frac{D}{L}$ , which increases with $K$ , and the throughput given by $d(1)^{-1}$ benefits from larger $K$ .",
"On the other hand, $d(1)= \\frac{1}{rL}+\\frac{D}{LK}$ decreases with $K$ .",
"The metrics related to timeliness capture this trade-off between throughput and delay, and provide guidance to select the optimal values of $K$ .",
"We evaluate these trade-offs numerically in Section ."
],
[
"Robustness to Different Network Topologies",
"The previous results hold for different network topologies, under the assumption that the propagation delay is negligible compared to the delays caused by network congestion and by the need to wait for sufficient degrees of freedom in order to decode the packets.",
"We assume that the number of packets in a bucket remains the same at each intermediate node.",
"At each hop, the node mixes the packets it has together and sends it.",
"The feedback is an end-to-end feedback as described in Fig.",
"REF .",
"To extend to a multi-hop network, we consider a network consisting of $H$ links in tandem, each link with erasure probability $\\epsilon _h$ , $h = 1,\\ldots ,H$ , as described in [14].",
"Given our focus on systems where the timeliness of information is relevant, we consider that intermediate nodes do not attempt to decode packets, and perform a recode-and-forward scheme.",
"In this case, the end-to-end rate at which the encoded packets are received at the destination is obtained through the min-cut max-flow theorem as $r_{\\text{multihop}} = \\min _{h=1,\\ldots ,H}(1-\\epsilon _h).$ For the multi-path case, the main challenge is to determine the allocation of the new coded packets of information to be transmitted over the available paths.",
"It is necessary to balance between the number of packets, sent to maximize throughput, and the number of retransmissions due to channel erasures, sent to minimize delay.",
"This problem was addressed in [19], where the authors proposed a bit-filling scheme to allocate transmitted packets to the paths at each time slot.",
"Consider the availability of $Z$ paths, with independent erasure probabilities given by $\\epsilon _j$ , where $j\\in \\lbrace 1,\\ldots ,Z\\rbrace $ .",
"Let the number of packets allocated to path $j$ be denoted with $k_j$ , such that all packets in the coding bucket are allocated to a path, satisfying the sum constraint $\\sum _{j=1}^{Z}z_j=K$ .",
"The delivery time is determined by the path with maximum number of transmissions, so we write $T_{\\text{multipath}} = \\max _{j=1, \\ldots Z} \\left(k_j\\frac{1}{1-\\epsilon _j}\\right).$ The minimization of $T_{\\text{multipath}}$ subject to the sum constraint is solved using discrete water filling, and the resulting end-to-end rate is $r_{\\text{multipath}}=K/T_{\\text{multipath}}$ .",
"For illustration purposes, consider the simple topology in Fig.",
"REF .",
"In this case, there is a multi-hop path that we identify as the relayed path, and there is a direct path to destination.",
"The discrete water filling will allocate $k_{\\text{relayed}}$ and $k_{\\text{direct}}$ to each path, such that $k_{\\text{relayed}}+k_{\\text{direct}}=K$ .",
"The resulting end-to-end rate in this particular scenario becomes $r_{\\text{multipath}}=\\frac{K}{\\max \\left(\\frac{k_{\\text{relayed}}}{\\min ((1-\\epsilon _1),(1-\\epsilon _2))},\\frac{k_{\\text{direct}}}{(1-\\epsilon _3)}\\right)}.$ Figure: Simple network topology for multi-hop multi-path illustration.Effectively, the multi-hop and multi-path scenarios change the maximum feasible end-to-end rate we denoted with $r$ , and our previous results hold when replacing $r$ with the correct representation as discussed above.",
"Processing times in each hop would simply shift the results.",
"Hence, the results we present hold for different network topologies (refer to Fig.",
"REF and Table REF for numerical examples)."
],
[
"Performance Evaluation",
"Unless specified otherwise, we set the packet length $L=1$ , the feedback delay $D=1$ , the channel utilization factor $\\rho =0.6$ , and the rate $r=0.8$ .",
"The arrival rate is limited by the service rate to guarantee the stability of the queue, which is tantamount to limiting the system utilization factor $\\rho \\triangleq \\lambda /\\mu <1$ .",
"Note that $\\mu =d(1)^{-1}$ in this case.",
"In general, varying the arrival rate $\\lambda $ produces two effects in AoI metrics.",
"While increasing the arrival rate may result in delivering messages more often to the destination, reducing the AoI, it may also result in network congestion and larger delays, increasing the AoI.",
"The two effects are observed when plotting the AoI versus $\\rho $ , which also describes the arrival rate as a fraction of the service rate, as shown in Fig.",
"REF .",
"We note that departing from the uncoded case $K=1$ provides the most significant gains with respect to the AoI metrics.",
"Figure: AAoI versus system utilization factor ρ=λ/μ\\rho =\\lambda /\\mu .Fig.",
"REF shows the effect of the coding bucket size $K$ , and presents curves for different values of the system utilization factor $\\rho $ , illustrating the trade-off between the metrics of AoI, delay, and throughput.",
"Clearly, the AoI is decreasing with $K$ and, once more, we note that departing from the uncoded case $K=1$ provides significant performance gains with respect to timeliness.",
"While the scenarios with more congestion (larger $\\rho $ ) can benefit from further increasing $K$ , we note that, in general, it is sufficient to combine a very small number (such as 10) of original packets into a coded packet in order to obtain significant improvement in timeliness for all levels of congestion.",
"We note that, for any given value of $K$ , there is significant gain in increasing the utilization factor from $\\rho =0.2$ to $\\rho =0.6$ .",
"In this range, we can increase throughput, and the destination can receive updated messages more often.",
"Further increasing the utilization factor from $\\rho =0.6$ to $\\rho =0.9$ creates congestion, increasing the waiting times in queue, and increasing the AoI.",
"For larger $\\rho $ the PAoI metric also becomes more distinct from the AAoI and can be used as an upper bound on AoI, given that it is simpler to calculate.",
"Figure: AoI versus coding bucket size KK for different levels of congestion represented by ρ\\rho .Fig.",
"REF depicts, on the left, the AoI versus throughput $\\mu = d(1)^{-1}$ .",
"We obtain the curves by changing the bucket size $K\\in [1,\\infty )$ .",
"Increasing $K$ yields larger throughput and reduces the AoI, even though for very large $K$ the maximum expected delay for a packet may increase significantly, as it takes longer to decode all the packets in the bucket.",
"This trade-off is illustrated with the curves associated to the axis on the right, which shows $d(\\infty )$ as a function of the throughput.",
"Under the conditions assumed in this case, the AoI is dominated by $d(1)$ .",
"The effect of the feedback delay is also represented in Fig.",
"REF .",
"It determines the range of feasible pairs $(d(1),d(\\infty ))$ and also impacts the acceptable range of arrival rates such that the system is stable.",
"Larger $D$ requires smaller arrival rates, and also results in larger average delays $d(1)$ , which result in significant increase in AoI.",
"Figure: AAoI A A A_A and per-packet delay d(∞)d(\\infty ) versus throughput μ=d(1) -1 \\mu =d(1)^{-1}.",
"Channel utilization is fixed to ρ=0.6\\rho =0.6.We illustrate the effect of the feedback delay $D$ in Fig.",
"REF .",
"Both the AoI and the delay increase with $D$ , as expected.",
"However, the impact of $D$ on the AoI decreases with $K$ , and a larger $D$ can be tolerated by using a larger $K$ .",
"Meanwhile, the per-packet delay $d(\\infty )$ presents similar slope, i.e.",
"increases with $D$ at similar rate, for different values of $K$ .",
"Figure: AAoI A A A_A and per-packet delay d(∞)d(\\infty ) versus feedback delay DD.",
"Channel utilization is fixed to ρ=0.6\\rho =0.6.Fig.",
"REF shows the AoI and throughput as a function of $r$ .",
"For single hop, $r=1-\\epsilon $ , while for multi-hop and multi-path we can use (REF ) and (REF ) to obtain a modified rate as exemplified by (REF ).",
"By modeling a multi-hop multi-path network using a modified erasure channel, we note that the performance improvement with respect to timeliness is robust to the network topology.",
"Nonetheless, the performance is affected by the maximum end-to-end rate that can be achieved in the network.",
"In the case of AoI, the rate has more impact for $r<0.2$ .",
"In that range, an increase in rate results in significant reduction of the AoI.",
"In other words, except in the case of very poor channel conditions, the performance with respect to AoI is very stable to variations in the rate.",
"In the case of throughput, the performance improves more steadily with the increasing rate.",
"For both AoI and throughput we observe significant gains in departing from the uncoded case ($K=1$ ).",
"We highlight that the gain of using $K>1$ is observed under any channel conditions.",
"The reduction in AoI values with respect to the uncoded case increases as the channel conditions deteriorate, and it is significantly larger for larger feedback delay, as shown in Table REF , indicating that the adaptive coding scheme provides a robust improvement of timeliness, even under unfavorable channel or network conditions.",
"In summary, we observed that the adaptive coding scheme is very robust to variations in network topology, channel conditions, and size of coded packets.",
"It requires only a small code to produce significant performance improvement with respect to timeliness metrics.",
"The system is robust to the choice of a coding bucket size, so the number of packets combined in an encoded packet can vary around $K=10$ , but any value $K>1$ results in better performance than the uncoded case $K=1$ .",
"These gains are observed for a wide range of channel conditions, with stable performance as long as the erasures are not extreme.",
"The AoI takes small values as long as the system is kept away from extreme cases of low or high utilization, so there is a wide range, say $0.2<\\rho <0.8$ where the system is forgiving to variations in arrival and service rates.",
"The gains extend to general network topology.",
"In fact, our results hold for a multi-hop multi-path scenario with the proper adjustment of the maximum feasible end-to-end rate such that adaptive coding provides significant and robust gains with respect to timeliness in wireless networks.",
"Figure: AAoI A A A_A and throughput μ=d(1) -1 \\mu =d(1)^{-1} versus maximum feasible end-to-end rate rr.",
"Channel utilization ρ=06\\rho =06.",
"and feedback delay D=5D=5.Table: Reduction in AAoI w.r.t.",
"uncoded case (K=1K=1)"
],
[
"Conclusion",
"We studied the performance of adaptive coding of packet traffic with respect to timeliness metrics associated with the Age of Information (AoI).",
"For a communication network modeled with an erasure channel and discrete time, we presented closed form expressions for the Average and Peak AoI as functions of a tunable parameter $K$ that defines the number of original packets to be coded together.",
"While the benefits of network coding with respect to throughput and delay are well known and documented, this work has shown that AoI metrics may also be significantly improved by transmitting linear combinations of a few original packets.",
"We observed that the AoI is decreasing for a large range of values for $K$ , and noted that the biggest gain is obtained when departing from the (uncoded) case of $K=1$ , so coding a small number ($K\\le 10$ ) packets together was demonstrated to greatly improve performance in systems that are sensitive to information timeliness.",
"We showed that these AoI gains are robust to variations in $K$ , feedback delay, and end-to-end rate that encapsulates channel and network topology effects."
],
[
"ACKNOWLEDGMENTS",
"(a) Contractor acknowledges Government's support in the publication of this paper.",
"This material is based upon work funded by AFRL, under AFRL Contract No.",
"# FA8750-18-C-0190.",
"(b) Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of AFRL."
]
] | 2012.05172 |
[
[
"First measurement using a nuclear emulsion detector of the $\\nu_{\\mu}$\n charged-current cross section on iron around the 1$\\,$GeV energy region"
],
[
"Abstract We have carried out $\\nu_{\\mu}$ charged-current interaction measurement on iron using an emulsion detector exposed to the T2K neutrino beam in the J-PARC neutrino facility.",
"The data samples correspond to 4.0$\\times$10$^{19}$ protons on target, and the neutrino mean energy is 1.49$\\,$GeV.",
"The emulsion detector is suitable for precision measurements of charged particles produced in neutrino-iron interactions with a low momentum threshold thanks to thin-layered structure and sub-$\\mu$m spatial resolution.",
"The charged particles are successfully detected, and their multiplicities are measured using the emulsion detector.",
"The cross section was measured to be $\\sigma^{\\mathrm{Fe}}_{\\mathrm{CC}} = (1.28 \\pm 0.11({\\mathrm{stat.}})^{+0.12}_{-0.11}({\\mathrm{syst.}}))",
"\\times 10^{-38} \\, {\\mathrm{cm}}^{2}/{\\mathrm{nucleon}}$.",
"The cross section in a limited kinematic phase space of induced muons, $\\theta_{\\mu} < 45^{\\circ}$ and $p_{\\mu} > 400 \\, {\\rm MeV}/c$, on iron was $\\sigma^{\\mathrm{Fe}}_{\\mathrm{CC \\hspace{1mm} phase \\hspace{0.5mm} space}} = (0.84 \\pm 0.07({\\mathrm{stat.}})^{+0.07}_{-0.06}({\\mathrm{syst.}}))",
"\\times 10^{-38} \\, {\\mathrm{cm}}^{2}/{\\mathrm{nucleon}}$.",
"The cross-section results are consistent with previous values obtained via different techniques using the same beamline, and they are well reproduced by current neutrino interaction models.",
"These results demonstrate the capability of the detector towards the detailed measurements of the neutrino-nucleus interactions around the 1$\\,$GeV energy region."
],
[
"Introduction",
"Long-baseline neutrino oscillation experiments, such as T2K [1], are typically performed in the vicinity of 1 GeV.",
"It is essential to understand neutrino-nucleus interactions for future neutrino oscillation experiments because the experimental precision will be limited by uncertainties of neutrino interaction models.",
"In this energy region, the dominant modes of neutrino charged-current interactions are quasi-elastic scattering and resonant pion production.",
"In addition, the existence of two-particle-two-hole excitations has been posited.",
"Measuring the multiplicity and kinematics of protons and pions from neutrino interactions is important for constructing reliable neutrino interaction models, but it is a difficult task because the produced hadrons have low energies.",
"It is especially difficult to observe all these protons in detectors that use scintillators because their momentum thresholds are below the required sensitivity.",
"The NINJA experiment uses emulsion-based detectors to study the interactions between neutrinos and nuclei for energies ranging from hundreds of MeV to several GeV.",
"Nuclear emulsion is well suited for performing high precision measurements of the positions and angles of charged particles emitted from neutrino interactions because it provides sub-micron spatial resolution.",
"Since 2014, a series of pilot experiments [2], [3], [4] has been run using the emulsion-based detectors.",
"The emulsion detector is capable of detecting slow protons with momenta as low as 200 MeV/$c$ , representing a distinct advantage compared to other detectors with higher proton momentum thresholds of approximately 400–700 MeV/$c$ [5], [6], [7].",
"This paper reports a measurement of the flux-averaged cross section of $\\nu _{\\mu }$ charged-current interaction using an emulsion detector combined with a 65-kg iron target.",
"This represents a significant advance for the precise studies of neutrino-nucleus interactions using an emulsion detector."
],
[
"Detector configuration and data samples",
"The detector is located in the near detector hall of the T2K experiment at J-PARC.",
"Figure REF shows a schematic view of the detector.",
"The detector is a hybrid apparatus composed of an iron target emulsion cloud chamber (ECC), an emulsion multi-stage shifter (Shifter) [8], [9], [10], and interactive neutrino grid (INGRID) detector [11].",
"The ECC consists of 12 basic units called bricks, which are made of emulsion films interleaved with iron plates.",
"The ECC bricks and the Shifter are installed upstream of the INGRID module, which is one of the near detectors in the T2K experiment.",
"The Shifter adds timing information to each track observed in an ECC brick, which helps to match the tracks with a corresponding muon track in INGRID.",
"The ECC bricks and the Shifter are enclosed in a cooling shelter to maintain a temperature of approximately 10 $^{\\circ }$ C and protect the emulsion films from sensitivity degradation and fading.",
"In this study, INGRID is used as a muon range detector, from which $\\nu _{\\mu }$ charged-current interactions are selected.",
"In the following, the X- and Y-axes are defined as the horizontal and vertical directions perpendicular to the beam direction (Z-axis), respectively.",
"Figure: Schematic view of the detector.",
"The ECC bricks and the Shifter are enclosed in a cooling shelter, which is placed in front of an INGRID module."
],
[
"J-PARC neutrino beamline",
"The J-PARC accelerator provides a high-intensity 30 GeV proton beam.",
"Each proton beam spill consists of eight bunches.",
"The width of each bunch is approximately 58 ns, while the interval between the bunches is approximately 581 ns.",
"These spills are delivered to a graphite target every 2.48 s. Hadrons, mainly pions, are produced by the interaction of protons with the target.",
"The charged pions are parallel focused by three magnetic horns.",
"During their flight in the decay volume, they decay primarily into muons and muon-neutrinos.",
"By changing the polarity of the magnetic horns, the neutrino and anti-neutrino beam modes can be switched.",
"Thus, an almost pure $\\nu _{\\mu }$ beam is delivered to the neutrino near detector hall.",
"The neutrino beam has energies ranging from hundreds of MeV to a few GeV at the detector location, with a peak at approximately 1 GeV.",
"Further details of the J-PARC neutrino beamline can be found in Ref.",
"[12]."
],
[
"INGRID",
"INGRID is the on-axis near detector for the T2K experiment located at 280 m downstream from the proton target [1], [11].",
"Figure REF top shows the position of the 14 INGRID modules arranged in a cross shape.",
"One of the horizontal modules next to the central module serves as a muon range detector in this measurement.",
"Each INGRID module comprises 11 scintillator planes interleaved with nine iron plates, as shown in the bottom half of Fig.",
"REF .",
"Each iron plate measures 124 cm $\\times $ 124 cm $\\times $ 6.5 cm.",
"Each of the 11 scintillator planes features 24 $\\times $ 2 plastic scintillator bars, with alternated X- and Y- directions.",
"This structure makes it possible to reconstruct three-dimensional muon tracks.",
"The dimensions of each scintillator bar are 5 cm $\\times $ 1 cm $\\times $ 120 cm, and the scintillation light is collected by a wavelength-shifting fiber, which is inserted into a hole in the center of the scintillator strip.",
"One end of the fiber is attached to a multi-pixel photon counter (MPPC) with an optical connector, and the light is read out using the MPPC.",
"Figure: (Top) Projected view of the INGRID modules and (bottom) exploded view of an INGRID module."
],
[
"ECC",
"The nuclear emulsion comprises AgBr crystals embedded in gelatin.",
"The crystal volume occupancy of the emulsion used in this experiment is 45%.",
"Properties of this emulsion are described in Refs.",
"[13], [14].",
"The area and thickness of the emulsion film are 25 cm $\\times $ 25 cm and 300 $\\mu $ m, respectively.",
"Each emulsion film is a 180 $\\mu $ m polystyrene sheet with a 60-$\\mu $ m thick nuclear emulsion layer on each face.",
"Charged particle trajectories are leaving latent images in the emulsion which are transformed into visible rows of grains during development.",
"The rows of grains are measured using an optical microscope.",
"Figure REF shows an image of charged particles emitted from a neutrino-iron interaction in an emulsion layer, which was acquired using a microscope system called fine track selector (FTS) [15], [16].",
"The black lines in Fig.",
"REF represent the charged particle tracks.",
"As shown in Fig.",
"REF , the ECC brick is composed of 23 emulsion films interleaved with 22 iron plates, each of which measures 25 cm $\\times $ 25 cm $\\times $ 0.05 cm.",
"The iron plates are made of stainless steel (SUS304) instead of pure iron which causes chemical reactions in contact with emulsion films.",
"It consists of iron (72.3%), chromium (18.1%), nickel (8.0%), and other contaminations, including manganese, silicon, phosphorus, and sulfur (1.6% in total).",
"Numbers of neutrons and protons in nuclei composing the stainless steel are close to those in iron.",
"Furthermore, the neutron to proton number ratio in the stainless steel is 1.149, compared to 1.150 for iron.",
"Therefore, in the following analysis, the target material in the ECC bricks is treated as iron.",
"For this pilot experiment, the ECC comprised 12 bricks, with the iron plates having a total mass of 65 kg.",
"As shown in Fig.",
"REF , four ECC bricks were placed in the XY-plane in a square configuration, with another two bricks placed behind them along the Z-direction, i.e., the four-brick square was three bricks deep.",
"In addition, a subsidiary emulsion film was placed between the ECC bricks as well as on the brick face farthest downstream to facilitate track connection, as shown in the right-hand schematic of Fig.",
"REF .",
"One of the bricks was taken out from the detector about one month after the beginning of the exposure and developed to check the emulsion quality.",
"This brick was not used in the following analysis.",
"Figure: FTS image of charged particles emitted from a neutrino-iron interaction in an emulsion layer.",
"The black lines are the charged particle tracks, with the white arrows showing the direction of each track.Figure: Structure of the iron ECC brick.",
"Each ECC brick consists of 23 emulsion films interleaved with 22 iron plates which are made of stainless steel (SUS304).Figure: (Left) Front view and (right) side view of the ECC bricks.",
"Four ECC bricks were placed in the XY-plane, with this configuration repeated to produce a structure that was three bricks deep along the Z-direction."
],
[
"Shifter",
"The emulsion shifter technique was originally developed for a balloon experiment to study cosmic-ray electrons [17] and was introduced into the GRAINE experiment [9], [18].",
"The Shifter is composed of three stages (S1, S2, S3), as shown in Fig.",
"REF .",
"Seven films, each with an area of 25 cm $\\times $ 30 cm, are mounted on the three stages in a 2:3:2 ratio.",
"Stages S1 and S2 are separated by a gap of 3 mm, while S2 and S3 are separated by a gap of 2 mm.",
"Each stage is driven at a different speed along the Y-direction in order to add timing information to ECC tracks.",
"Figure REF shows the respective positions of each stage over time: S1 has a cyclical motion pattern, moving at a speed of 0.553 $\\rm \\mu $ m/s and a stroke of 3000 $\\rm \\mu $ m; S2 shifts at each instant when S1 changes direction, and is driven by a stepping motor with a step size of 150 $\\rm \\mu $ m and a stroke of 7500 $\\rm \\mu $ m (it has a repetition time of 1.51 h); S3 shifts when S2 changes direction, and is driven by a stepping motor with a step of 150 $\\rm \\mu $ m (it has a repetition time of 3.1 days).",
"A full cycle of the Shifter operation lasts about 155 days.",
"The Shifter allows the addition of time information to the ECC tracks up to 155 days.",
"Figure: Elevational structure of the Shifter.",
"Two emulsion films are mounted on both S1 and S3, with three mounted on S2.Figure: Respective positions of each Shifter stage over time.",
"Each stage is driven at a different speed to add timing information to the ECC tracks."
],
[
"Data samples",
"The ECC bricks and the Shifter were exposed to the neutrino beam between February and May 2016.",
"The neutrino beamline was operated in both neutrino and anti-neutrino beam modes.",
"There were two periods of exposure in the neutrino beam mode: the first was from February 1–3, and the second was from May 19–27.",
"After live-time correction of the detectors, we analyzed data samples in the neutrino beam mode, corresponding to $4.0 \\times 10^{19}$ protons on target (POT)."
],
[
"Monte Carlo simulation",
"The signal and background events, neutrino flux, and detection efficiency were estimated using Monte Carlo (MC) simulations.",
"These MC simulations consisted of three parts: (i) JNUBEAM [12] to predict the neutrino flux, (ii) NEUT [19], [20] to model the interactions between neutrinos and nuclei, and (iii) a GEANT4 [21], [22], [23]-based framework to simulate the detector response.",
"The MC predictions were normalized with respect to the POT value and the target mass."
],
[
"Neutrino beam",
"The neutrino flux at the NINJA detector is estimated using JNUBEAM.",
"JNUBEAM was developed to predict the flux and spectrum of neutrinos at the T2K detectors and is based on a GEANT3 framework [24].",
"We used JNUBEAM version 13av6.1.",
"In addition, FLUKA2011.2 [25], [26] was used to simulate the hadronic interactions of primary protons on the graphite target.",
"Then, JNUBEAM takes the secondary particle information simulated using FLUKA, and models their propagation, interaction, and decay events.",
"The hadronic interaction simulation was tuned using hadron production data from experiments such as CERN NA61/SHINE [27], where a combination of measurements with a replica of the T2K target [28], [29] and a thin graphite target [30], [31], [32], [33] was used.",
"In the neutrino beam mode, the mean energy and fraction of $\\nu _{\\mu }$ components are 1.49 GeV and 94.9%.",
"The fraction of $\\bar{\\nu }_{\\mu }$ components is 4.3%, with $\\nu _{e}$ and $\\bar{\\nu }_{e}$ components representing the remaining 0.8%.",
"Figure REF shows the neutrino energy spectrum of each beam component at the NINJA detector in the neutrino beam mode.",
"Figure: Neutrino energy spectrum of each beam component at the NINJA detector in the neutrino beam mode.",
"These spectra were predicted using JNUBEAM."
],
[
"Event generation",
"The Super-Kamiokande [34] and T2K experiments use the NEUT neutrino event generator.",
"In addition to simulating primary neutrino interactions, NEUT also simulates final state interactions (FSIs), such as scattering, absorption, particle production, and charge-exchange of hadrons produced by neutrino interactions in the nuclear medium prior to escape.",
"We used version 5.4.0 of NEUT.",
"To predict the signal and background events in the ECC bricks, $\\nu _{\\mu }$ , $\\bar{\\nu }_{\\mu }$ , $\\nu _{e}$ , and $\\bar{\\nu }_{e}$ interactions on iron were generated using NEUT.",
"The MC prediction for an iron target was adapted to the stainless steel target using the difference in the fractions of protons and neutrons between iron and the stainless steel.",
"In addition, neutrino interactions in the upstream wall of the detector hall and the INGRID modules were generated as background sources.",
"The neutrino interaction models and nominal parameters used in NEUT are listed in Table REF .",
"Charged-current (CC) quasi-elastic (QE) and neutral-current (NC) elastic scatterings, two-particle-two-hole (2p2h) excitations, CC and NC resonant interactions (RES), coherent pion productions (COH $\\pi $ ), and deep inelastic scatterings (DIS) were simulated.",
"The one-particle-one-hole (1p1h) model by Nieves $\\textit {et al.",
"}$ [35], [36] was used to simulate the CCQE.",
"In this model, a local Fermi gas (LFG) model with random phase approximation (RPA) corrections is used for the nuclear model, and the axial mass $M_{\\rm A}^{\\rm QE}$ is set to 1.05 GeV/$c^{2}$ .",
"Nieves $\\textit {et al.",
"}$ have also modeled the 2p2h interaction [37].",
"The RES was simulated using the Rein-Sehgal model [38], and the axial mass $M_{\\rm A}^{\\rm RES}$ is set to 0.95 GeV/$c^{2}$ .",
"In addition, we used the COH $\\pi $ model described by Rein-Sehgal model in Refs.",
"[39], [40].",
"To describe DIS, we applied parton distribution function (PDF) GRV98 with Bodek and Yang correction [41], [42], [43].",
"NEUT models the FSI for hadrons using a semi-classical intra-nuclear cascade model [20], [44], [45].",
"Figure REF shows the neutrino-nucleus cross sections per nucleon of an iron nucleus predicted by NEUT.",
"Table: Neutrino interaction models used in the nominal MC simulation.Figure: Neutrino-nucleus cross sections per nucleon of an iron nucleus predicted by NEUT."
],
[
"Detector response",
"The detector simulation for the particles was developed with a GEANT4 framework.",
"We used GEANT4 version 9.2.1 and the QGSP BERT physics list [46].",
"The geometry of the ECC bricks, Shifter, INGRID modules, and wall of the detector hall were modeled for the detector simulation.",
"The base track (described in Sec.",
"REF ) was reconstructed using the positions that charged particles pass through on both faces of the polystyrene sheet.",
"Therefore, the detection efficiency of the base tracks evaluated by the data can be incorporated in the MC simulation.",
"A volume pulse height (VPH) [47] corresponding to an energy deposit in the emulsion film is reconstructed on the basis of the correlation between a given slope, momentum, and VPH in the data.",
"This process is repeated for the MC simulation of the track connections between the films in individual ECC bricks, between the ECC bricks, and between the ECC bricks and INGRID.",
"For the track connections between the ECC bricks and the Shifter, we used the connection efficiency based on the data.",
"The background events produced by cosmic rays and misconnected events between the ECC bricks, the Shifter, and INGRID were estimated using the data rather than the MC simulation."
],
[
"Track reconstruction in the ECC bricks",
"The track pieces recorded in an emulsion layer are called “micro tracks.” The positions $(x, y)$ and slopes (tan$\\theta _{x}$ , tan$\\theta _{y}$ ) of the micro tracks were measured using the hyper-track selector (HTS) [48].",
"The HTS recognizes a series of grains on a straight line as a micro track by taking 16 tomographic images in the emulsion layer.",
"The slope acceptance of the HTS was set to $|$ tan$\\theta _{x(y)}$$|$$<$ 1.7 ($|\\theta _{x(y)}|\\lesssim 60^{\\circ }$ ).",
"The track angle $\\theta $ is defined as the angle with respect to the Z-direction.",
"The pulse height (PH) [49] and VPH were measured, revealing a strong correlation with the blackness of the track.",
"The PH is defined as the number of tomographic images that have pixels associated with the track, while the VPH is the total number of pixels associated with the track in all 16 tomographic images.",
"A single scan covers an area of 130 mm $\\times $ 90 mm, with each emulsion film covered by six scans.",
"After scanning, the tracks are reconstructed via a NETSCAN [50], [51]-based procedure.",
"The tracks connecting the positions of micro tracks on both sides of the polystyrene sheet are called “base tracks,” and are used as track segments in this analysis.",
"The base track detection efficiency ranges between 95 and 99%, with variation caused by individual film differences.",
"After reconstructing the base tracks, the rotation, slant, parallel translation, and gap between emulsion films are adjusted.",
"This alignment process determines the relative positions of the films during the beam exposure.",
"Following the alignment process, the ECC tracks are reconstructed by connecting the base tracks both in adjacent films and in films separated by one or two films.",
"The slope- and position-related tolerances associated with the base track connections are defined as functions of the track slope.",
"The connection efficiency exceeds 99%.",
"In this analysis, the ECC tracks are required to pass through at least one iron plate and two emulsion films."
],
[
"Time-stamping to the ECC tracks",
"As shown in Sec.",
"REF , the Shifter comprises seven emulsion films mounted on three stages.",
"The scanning area and the slope acceptance of the films in the Shifter are equivalent to those of the ECC brick films.",
"Each Shifter film is covered by eight scanning areas.",
"The Shifter tracks were also reconstructed using the NETSCAN software package.",
"Next, the tracks in the ECC bricks and the Shifter are connected in order to add timing information to the ECC tracks.",
"First, the ECC bricks are aligned with each of the Shifter stages using as reference the fixed position of the stages for one week after the Shifter has completed its operation.",
"After this alignment, the tracks between the ECC bricks and each Shifter stage are connected in the following order: first, the tracks between the most downstream film of the ECC brick and S3 are connected; then, the tracks between S3 and S2 are connected; finally, the tracks between S2 and S1 are connected.",
"The tracks are connected using slope- and position-based matching, with slope- and position-related tolerances set to 0.025 and 75 $\\mu $ m in the XZ- and YZ-planes, respectively.",
"Thus, the Shifter combines the ECC track data and time information."
],
[
"Track reconstruction in INGRID",
"The track reconstruction and selection processes used in this analysis for the INGRID detector are similar to those used in the T2K experiment [52], [53].",
"Each track is composed of a series of hits, where a hit is defined as an MPPC channel that exceeds a signal equivalent to 2.5 photoelectrons.",
"The hits are clustered within $\\pm $ 50 ns from the average hit time.",
"A tracking plane that contains at least one hit in both X- (horizontal) and Y- (vertical) layers is defined as an active plane.",
"Events are required to have at least three active planes, corresponding to a muon momentum threshold of approximately 300 MeV/$c$ .",
"Two-dimensional tracks in the XZ- and YZ-planes are reconstructed independently using a track reconstruction algorithm based on cellular automaton [54], while three-dimensional tracks are reconstructed by merging track pairs in the XZ- and YZ-planes.",
"Events are selected within $\\pm $ 100 ns from the event timing, which is defined as the hit timing of the channel with the largest number of photoelectrons.",
"In this analysis, the INGRID tracks are required to start at the most upstream plane."
],
[
"Track matching",
"After connecting the tracks between the ECC bricks and the Shifter, track matching between the ECC bricks and the INGRID module is performed.",
"The ECC tracks are extrapolated to the most upstream plane of INGRID and matched with an INGRID track using the slope, position, and timing information.",
"Each ECC event is required to have a time residual within 200 s from the INGRID event timing.",
"The slope- and position-related tolerances are set to 0.100 and 5.0 cm in the XZ- and YZ-planes, respectively.",
"Figure.",
"REF shows the time residuals between the ECC and INGRID events.",
"The standard deviation of the time residuals distribution represents the Shifter time resolution, which was approximately 50 s in this study.",
"The connection efficiencies of muon tracks among the ECC bricks, the Shifter, and INGRID are shown in Fig.",
"REF .",
"Furthermore, events with more than one possible connections, representing approximately 4% of the total, were not used.",
"Figure: Time residuals between the ECC events and INGRID events.",
"The data samples are all events that occurred during the Shifter operation.",
"The standard deviation of the time residuals distribution (approximately 50 s) corresponds to the estimated time resolution of the Shifter.",
"The black lines and arrows represent the time tolerance for the ECC–INGRID track matching.Figure: Connection efficiencies of the muon tracks among the ECC bricks, the Shifter, and INGRID as a function of tanθ\\theta .",
"The black and red markers represent the connection efficiencies, and their vertical errors represent statistical errors.",
"The gray histogram represents with an arbitrary normalization the expected tanθ\\theta distribution of muons emitted from ν μ \\nu _{\\mu } CC interactions in the ECC fiducial volume described in Sec.",
"."
],
[
"Event selection",
"In this analysis, the $\\nu _{\\mu }$ CC interactions in the iron target are defined as signal events.",
"Each step of the event selection process is described below.",
"ECC–Shifter–INGRID track matching The ECC tracks that are matched between the ECC bricks, Shifter, and INGRID were selected as muon candidates from $\\nu _{\\mu }$ CC interactions.",
"A total of 9 397 such events were selected.",
"Scanback The muon candidates were traced back from INGRID to the neutrino interaction vertices in the ECC bricks.",
"This procedure is known as the scanback method [50], [55], [56], [57], [58].",
"In this method, if no track satisfying slope- and position-related tolerances is found in three consecutive films, the most upstream track segment of a muon candidate is defined as its starting segment.",
"The iron plate on the upstream side of the starting segment is defined as the interaction plate.",
"Fiducial volume cut Most of the muon candidates are so-called sand muons, which are produced by neutrino interactions in the upstream wall of the detector hall.",
"An escaping edge originates from the escape of sand muons and cosmic rays across the scanning area, and is located at a position 1–2 mm from the border of the scanning area.",
"An area 5 mm inside from the escaping edges is excluded from the fiducial scanning areas, with the average fiducial scanning area of each film measuring 116 mm $\\times $ 78 mm.",
"In the Z-axis direction, the fiducial volume (FV) is defined as the volume between the fourth film from the upstream face and the second film from the downstream face of each ECC brick.",
"As a result, the target mass in the FV is 42 kg.",
"The muon candidate tracks were extrapolated from the starting segment positions to the positions in the three upstream films and defined as outbound FV tracks if they are escaping from the fiducial area.",
"If a muon candidate started from a film that is damaged by scratches or within three films downstream of the damaged film, it was excluded from the neutrino interaction candidates in the FV.",
"Each film features four 3-mm diameter holes to facilitate film development.",
"If the muon candidates passed through holes when the tracks were extrapolated from the starting segments to the three upstream films, they were excluded from the interaction candidates in the FV.",
"After these cuts, 236 events remained as interaction candidates occurring in the FV of the ECC bricks.",
"Manual microscope check A process called “manual check” consists in a careful examination with a microscope of the region around the starting segment of the muon candidates.",
"First, the film just upstream of the starting segment is examined.",
"If a base track is found that can be connected to the starting segment, it is defined as the new starting segment.",
"The FV cuts are applied to this new starting segment and some additional events are rejected as possible sand muons.",
"The other role of the manual check is to determine in which material the neutrino interaction took place.",
"If the track is starting inside the emulsion layer, the interaction is considered as occurring in the emulsion.",
"If the track is observed only in the emulsion layer on the downstream side of the polystyrene sheet, the interaction is considered as occurring in the polystyrene.",
"The other events are classified as interactions in the iron target.",
"As a result of the manual check, 203 events were defined as interactions in the iron target, 13 events as interactions in the emulsion and 14 events as interactions in the polystyrene.",
"In addition, 6 events were excluded as possible sand muon tracks.",
"The relative rates of interactions in the different materials are consistent with their mass ratios within the statistical uncertainty.",
"In the MC simulations, the efficiency of the manual check is assumed to be 100%.",
"Partner track search The tracks attached to the muon candidates are charged hadrons from the neutrino interactions and are called “partner tracks.” In this study, we defined partner tracks with VPH $<$ 150 as thin tracks, whereas tracks with VPH $\\ge $ 150 were defined as black tracks.",
"Our search for partner tracks was performed under the following conditions: for thin tracks, the minimum distance between the muon candidate and its partner track had to be less than 50 $\\mu $ m. For black tracks, this minimum distance had to be less than 60 $\\mu $ m. In both cases, the distance along the Z-direction between the starting segment and the position of closest approach had to be less than 800 $\\mu $ m. Moreover, we required the thin tracks to have at least three track segments, while the black tracks were required to have at least two track segments.",
"If multiple tracks for a particular event were connected to INGRID, the track with the highest momentum (see sub-section 7) was assumed to be the muon candidate.",
"For duplicate events, whereby two tracks were connected to INGRID, one of the events was discarded.",
"Kink event cut Two track events for which the opening angle $\\alpha $ of the track pair is almost 180$^{\\circ }$ typically represent background events from sand muons or cosmic rays.",
"If a charged particle coming from the wall exceeds the track connection tolerance, the resulting event looks like a two-track event, consisting of a forward track and a backward track connected at a vertex.",
"Such events are called “kink events.” Kink events are characterized by their large opening angle: in the region cos$\\alpha $$<$$-$ 0.96, the background fraction is 98.2% according to our MC study.",
"Next, we conducted a particle identification process [59], [4] based on the momentum and the VPH of the partner track, which allows to separate pion-like and proton-like tracks.",
"Two-track events consisting of a muon candidate and a pion-like track with an opening angle in the region of cos$\\alpha $$<$$-$ 0.96 were assumed as kink events, and were discarded.",
"In contrast, two-track events consisting of a muon candidate and a proton-like track were kept.",
"As a result of the kink cut process, 7 events were rejected.",
"Momentum consistency check There are two methods for estimating the momentum of a muon.",
"One involves measuring its multiple Coulomb scattering in the ECC bricks, while the other is to measure its energy from the track range in the ECC bricks and the INGRID detector.",
"The values measured by the two methods can be compared to exclude misconnected backgrounds.",
"Muons were considered to exhibit momentum inconsistencies if they met the following criteria: if the momentum estimated by an angular scattering measurement [60] was greater (smaller) than 218% (17%) of that measured by the range; if the momentum estimated by a positional scattering measurement [61] was greater (smaller) than 352% (45%) of that measured by the range.",
"The maximum and minimum limits were based on the two-sigma confidence interval of the momentum measurement accuracy.",
"In the case of muons passing through or side-escaping INGRID, only the minimum limits were considered.",
"The momentum consistency check led to the exclusion of 12 events.",
"These events can be mainly due to misconnection of the ECC track to INGRID or to cosmic muon coming from downstream and stopping in an ECC brick.",
"Figure REF shows the display of a selected neutrino-iron interaction candidate.",
"The event contains a muon and a proton-like track.",
"Figure: Event display of a neutrino-iron CC interaction candidate.",
"The left-hand side of the figure shows the event display in the ECC brick, while the right-hand side shows the event display both in the ECC bricks and in INGRID.",
"On the left-hand side, with their colors representing the track segment and their width indicating the VPH.",
"On the right-hand side, the blue and pink lines are the ECC tracks extrapolated to INGRID, with the blue line representing a muon candidate and the pink line representing a proton-like track.",
"The width of the blue and pink lines indicates the VPH.",
"The red markers represent hits and their size represent deposited photoelectrons, and the black line represents the reconstructed track in INGRID.The number of selected events remaining after each selection step is summarized in Table REF .",
"The purity is defined as the fraction of $\\nu _{\\mu }$ CC interactions on iron in the MC sample.",
"Finally, a total of 183 events were confirmed as $\\nu _{\\mu }$ CC interactions on the iron target, corresponding to 188.8 events in the MC sample, in good agreement within the statistical uncertainty.",
"The events predicted by the MC simulation were categorized as follows: 88.2% of the events were signals produced by $\\nu _{\\mu }$ CC interactions, 4.8% were misconnected backgrounds, 3.4% were hadron interactions caused by neutrons, protons and charged pions emitted from the neutrino interactions in the upstream wall, 2.7% were events arising from $\\bar{\\nu }_{\\mu }$ interactions, and 0.8% were events caused by $\\nu _{\\mu }$ NC interactions.",
"The other sources of background contribute for less than 0.1%.",
"Table: Number of selected events remaining after each selection check described in Sec. .",
"The purity indicates the fraction of ν μ \\nu _{\\mu } CC interactions in the MC prediction.",
"Comparisons of the data with the MC prediction are only possible for numbers of neutrino-iron interactions which occurred inside the ECC bricks.Figure REF shows the selection efficiency of the $\\nu _{\\mu }$ CC interactions as a function of the neutrino energy as estimated by the MC simulation.",
"The selection efficiency was evaluated using the number of $\\nu _{\\mu }$ CC interactions in the FV as the denominator and the number of selected events as the numerator.",
"The profile of selection efficiency curve is determined by the following factors: the connection efficiency between the ECC bricks and the Shifter, and between the ECC bricks and INGRID; the muon momentum threshold of the ECC–INGRID track matching.",
"The mean selection efficiency is 25.3%.",
"Figure: Selection efficiency of the ν μ \\nu _{\\mu } CC interactions as a function of the neutrino energy.",
"The selection efficiency was evaluated using the number of ν μ \\nu _{\\mu } CC interactions in the FV as the denominator and the number of selected events as the numerator.",
"The neutrino energy spectrum of the CC interactions in the FV is shown in gray."
],
[
"Cross-section measurement",
"The flux-averaged $\\nu _{\\mu }$ CC inclusive cross section is measured from the number of selected events after background subtraction and efficiency correction.",
"The flux-averaged cross section is expressed as $\\sigma _{\\rm CC} = \\frac{N_{\\rm sel} - N_{\\rm bkg}}{\\phi T \\varepsilon },$ where $N_{\\rm sel}$ is the number of events selected from the data, $N_{\\rm bkg}$ is the number of background events predicted by the MC simulation, $T$ is the number of target nucleons in the FV, $\\phi $ is the integrated $\\nu _{\\mu }$ flux for the ECC bricks, and $\\varepsilon $ is the selection efficiency predicted by the MC simulation.",
"The values used in the cross-section measurement are summarized in Table REF .",
"Table: Cross-section measurement inputs."
],
[
"Systematic uncertainties",
"The sources of systematic uncertainties in the cross-section measurement can be categorized into four groups: neutrino flux, neutrino interaction models, background estimation and detector response."
],
[
"Neutrino flux",
"Uncertainties in the neutrino flux are due to uncertainties in hadron production and neutrino beam line optics.",
"A covariance matrix at the detector position was prepared following the same procedure as for the T2K experiment [12], with the relative errors in each energy bin shown in Fig.",
"REF .",
"In the cross-section measurement, the neutrino flux is made to fluctuate according to the covariance matrix and the $\\pm $ 1$\\sigma $ change of the cross-section result is taken as the systematic uncertainty.",
"The cross-section uncertainty resulting from the flux uncertainties is found to be $-$ 5.8%/$+$ 6.6%.",
"Figure: Neutrino flux uncertainties at the location of the detector arising from the hadron production uncertainties and the T2K beamline uncertainties."
],
[
"Neutrino interaction",
"Table REF summarizes the parameters [62], [63] used for modeling the neutrino interactions and the FSI in NEUT.",
"It lists the nominal values and the $1 \\sigma $ uncertainties for all the parameters.",
"These uncertainties affect the number of background events and the selection efficiency predicted by the MC simulation.",
"The systematic uncertainties on the cross-section arising from the neutrino interaction models are listed in Table REF .",
"The combined uncertainty is $-$ 4.1%/$+$ 4.6%.",
"Table: Nominal parameter values and their uncertainties in the neutrino interaction models.",
"Detailed descriptions of the parameters are given in Refs.",
", ."
],
[
"Background estimation",
"For the background estimation, the uncertainties associated with the wall backgrounds and the misconnected backgrounds are considered.",
"The number of sand muons in the MC simulation was found to be 30% smaller than in the data.",
"To estimate the background, we used the observed number of sand muons and considered the 30% difference as a systematic uncertainty which reflects our lack of understanding of the flux and the materials surrounding the detector hall.",
"The uncertainty attributed to misconnected events was evaluated using mock data, which are the combination of the nominal and fake data in which the time information of the ECC tracks is shifted.",
"This uncertainty is asymmetric because the misconnection rate of the beam-induced tracks and that of the cosmic-ray tracks are different.",
"The positive and negative uncertainties corresponding to the number of misconnected events were $+$ 24% and $-$ 39%, respectively.",
"The corresponding uncertainties on the cross-section result are listed in Table REF .",
"The combined uncertainty is $-$ 1.8%/$+$ 2.4%, which is smaller than the other uncertainties."
],
[
"Detector response",
"The uncertainties associated with the detector response were estimated using the data and the MC simulation.",
"We considered the contributions of the following uncertainties: the detection efficiency of the base tracks, the track reconstruction in the ECC brick, the track connection between the ECC bricks, the track connection between the ECC bricks and the Shifter, the track matching between the ECC bricks and INGRID, the track reconstruction in the INGRID module, the kink cut, the momentum consistency check, the target mass, and the difference between iron and the stainless steel.",
"To evaluate the effect of the detector response uncertainties on the cross section, the MC simulations were run using each of the detector responses with their $1 \\sigma $ uncertainty applied.",
"The difference between the cross-section result and its nominal result was defined as the systematic uncertainty for each detector response.",
"These uncertainties are collected in Table REF .",
"The effect on the cross-section measurement due to using the stainless steel plates instead of iron plates was estimated to be 0.3%.",
"The main uncertainty components are the track matching between the ECC bricks and the Shifter, and the ECC bricks and INGRID, both of which were found to be 2–3%.",
"In total, the uncertainty associated with the detector response on the cross-section result is $-$ 4.2%/$+$ 4.4%."
],
[
"Summary of the systematic uncertainties",
"Table REF summarizes the uncertainties involved in the cross-section measurement.",
"The total systematic uncertainty involved in the cross-section measurement was $-$ 8.5%/$+$ 9.4%.",
"Table: Summary of the systematic uncertainties involved in the cross-section measurement."
],
[
"Results and discussion",
"The measured flux-averaged $\\nu _{\\mu }$ CC inclusive cross section on iron is $\\sigma ^{\\mathrm {Fe}}_{\\mathrm {CC}} = (1.28 \\pm 0.11({\\mathrm {stat.",
"}})^{+0.12}_{-0.11}({\\mathrm {syst.}}))",
"\\times 10^{-38} \\, {\\mathrm {cm}}^{2}/{\\mathrm {nucleon}}, $ at a mean neutrino energy of 1.49 GeV.",
"This is the cross section per nucleon for an iron nucleus.",
"Figure REF shows the cross-section result obtained in this study as well as those reported by other experiments.",
"The result measured by the emulsion-based detector is consistent with the T2K measurements using INGRID on the same beamline[52], [53].",
"The measured cross section agrees well with the MC prediction of 1.30$\\times $ 10$^{-38}$ cm$^{2}$ .",
"Furthermore, the cross section for a restricted phase space of induced muons, $\\theta _{\\mu } < 45^{\\circ }$ and $p_{\\mu } > 400 \\, {\\rm MeV}/c$ , on iron is $\\sigma ^{\\mathrm {Fe}}_{\\mathrm {CC \\hspace{2.84526pt} phase \\hspace{1.42262pt} space}} = (0.84 \\pm 0.07({\\mathrm {stat.",
"}})^{+0.07}_{-0.06}({\\mathrm {syst.}}))",
"\\times 10^{-38} \\, {\\mathrm {cm}}^{2}/{\\mathrm {nucleon}}.",
"$ This result is also consistent with the T2K measurement using INGRID with the same phase space[63] as well as the MC prediction of 0.87$\\times $ 10$^{-38}$ cm$^{2}$ .",
"These results demonstrate the reliability of our detector and our data analysis.",
"Figure: Flux-averaged ν μ \\nu _{\\mu } CC inclusive cross section on iron.",
"Our data point is plotted at the mean flux energy.",
"The vertical error bar represents the total (statistical and systematic) uncertainty and the horizontal bar represents 68% of the flux at each side of the mean energy.",
"The MINOS and T2K results are also plotted.",
"The MINOS results are the ν μ \\nu _{\\mu } CC inclusive cross sections on iron.",
"The T2K results are the flux-averaged ν μ \\nu _{\\mu } CC inclusive cross section and the ν μ \\nu _{\\mu } CC inclusive cross sections on iron.",
"The neutrino flux at the detector position is shown in gray."
],
[
"Conclusions",
"In this study, we report the first $\\nu _{\\mu }$ cross-section measurement using an iron-target emulsion detector in the NINJA pilot experiment.",
"The measurement was performed by exposing a 65-kg iron target to the neutrino beam at J-PARC.",
"From the data acquired during the 4.0$\\times $ 10$^{19}$ POT exposure, the flux-averaged $\\nu _{\\mu }$ CC inclusive cross sections on iron at a mean neutrino energy of 1.49 GeV were measured.",
"We have reported the measurement of cross sections in the full phase space and a limited phase space for the kinematics of the induced muons with $\\theta _{\\mu } < 45^{\\circ }$ and $p_{\\mu } > 400 \\, {\\rm MeV}/c$ .",
"The results of the cross-section measurement agree closely with the T2K measurements as well as current neutrino interaction models, demonstrating that we have a good understanding of the events occurring in the detector.",
"The emulsion detector provides sub-micron spatial resolution and a low momentum threshold for the protons and charged pions produced by neutrino interactions.",
"These results represent an important step for the precise measurement of these particles.",
"Our future work will conduct a detailed study of the neutrino interactions in the 1 GeV energy region using emulsion detectors.",
"This study will be important for future long-baseline neutrino oscillation experiments."
],
[
"Acknowledgments",
"We would like to acknowledge the support of the T2K Collaboration in performing the experiment.",
"Furthermore, we appreciate the assistance of the T2K neutrino beam group in providing a high-quality beam and the MC simulation.",
"We thank the T2K INGRID group for providing access to their data.",
"We acknowledge the work of the J-PARC staff in facilitating superb accelerator performance.",
"We would like to thank P. Vilain (Brussels University, Belgium) for his careful reading of the manuscript and for his valuable comments.",
"This work was financially supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers JP25105001, JP25105006, JP26105516, JP26287049, JP25707019, JP20244031, JP26800138, JP16H00873, JP18K03680, JP17H02888, JP18H03701, JP18H05537, and JP18H05541."
]
] | 2012.05221 |
[
[
"Theory of competing excitonic orders in insulating WTe$_2$ monolayers"
],
[
"Abstract We develop a theory of the excitonic phase recently proposed as the zero-field insulating state observed near charge neutrality in monolayer WTe$_2$.",
"Using a Hartree-Fock approximation, we numerically identify two distinct gapped excitonic phases: a spin density wave state for weak non-zero interaction strength and spin spiral order at stronger interactions, separated by a narrow window of non-excitonic quantum spin Hall insulator.",
"We introduce a simplified model capturing key features of the WTe$_2$ band structure, in which these phases appear as distinct valley ferromagnetic orders.",
"We link the competition between the excitonic phases to the orbital structure of electronic wavefunctions at the Fermi surface and hence its proximity to the underlying gapped Dirac point in WTe$_2$.",
"We briefly discuss collective modes of the two excitonic states, and comment on implications for experiments."
],
[
"$k\\cdot p$ Model and Hartree-Fock (HF)",
"The setup here closely follows the supplement of Ref. [16].",
"In the basis $\\lbrace \\mathinner {|{d\\uparrow }\\rangle },\\mathinner {|{d\\downarrow }\\rangle },\\mathinner {|{p\\uparrow }\\rangle },\\mathinner {|{p\\downarrow }\\rangle }\\rbrace $ , the $k\\cdot p $ Hamiltonian of monolayer WTe2 is $H_0=\\left(a\\mathbf {k}^2+b\\mathbf {k}^4+\\frac{\\delta }{2}\\right)\\begin{pmatrix}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0\\end{pmatrix}+\\left(-\\frac{\\mathbf {k}^2}{2m}-\\frac{\\delta }{2}\\right)\\begin{pmatrix}0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & 1\\end{pmatrix}+v_xk_x\\tau _xs_y+v_yk_y\\tau _ys_0\\\\a=-3,\\quad b=18,\\quad m=0.03,\\quad \\delta =-0.9,\\quad v_x=0.5,\\quad v_y=3$ where energies are measured in eV and lengths in Å.",
"The symmetries are inversion $\\hat{P}=\\tau _z$ and time-reversal $\\hat{T}=is_y\\hat{K}$ , leading to a two-fold degeneracy of the bands under $\\hat{P}\\hat{T}$ .",
"With the above parameters, the bandstructure at charge neutrality consists of a hole pocket at the zone centre, and two electron pockets with minima at $\\mathbf {q_c}=\\pm 0.3144\\hat{x}$ .",
"The undistorted lattice has reciprocal lattice vector lengths $G_x=1.81$ and $G_y=1.01$ .",
"The Fermi energy is $E_F\\simeq -0.493$ .",
"Without the SOC term, the bandstructure contains two overtilted Dirac cones at $\\mathbf {q}_D=\\pm 0.2469\\hat{x}$ .",
"The $U(1)_s$ symmetric SOC term gaps the Dirac point, leading to an indirect negative band gap.",
"The interaction Hamiltonian is taken to be density-density in spin and orbital space $ H_{\\text{int}}=\\frac{1}{2N\\Omega }\\sum _{\\mathbf {k},\\mathbf {p},\\mathbf {q}}\\sum _{\\alpha ,\\beta }V(\\mathbf {q})c^\\dagger _{\\mathbf {k}+\\mathbf {q},\\alpha }c^{\\dagger }_{\\mathbf {p}-\\mathbf {q},\\beta }c^{\\phantom{\\dagger }}_{\\mathbf {p},\\beta }c^{\\phantom{\\dagger }}_{\\mathbf {k},\\alpha }\\\\V(q)=\\frac{e^2}{2\\epsilon \\epsilon _0 q}\\tanh \\frac{q\\xi }{2}$ where $N$ is the total number of unit cells in the system, $\\Omega $ is the real-space unit cell area, and $\\alpha ,\\beta $ are combined orbital/spin indices.",
"The interaction potential is of dual-gate screened form, with $\\xi $ the gate distance and $\\epsilon $ the relative permittivity of the encapsulating hBN.",
"Since we are working with a $k\\cdot p$ model, our calculations require a momentum cutoff, which is taken to be $|k_x|<\\frac{3q_c}{2},|k_y|<\\frac{G_y}{4}$ .",
"The prefactor in Eq REF is set by the density of momentum points, which would be $A_{BZ}/N$ .",
"However in our calculations our momentum cutoff has area $A_{kp}$ with $N_{kp}$ points, so we should replace $N\\Omega \\rightarrow N_{kp}\\Omega \\frac{A_{BZ}}{A_{kp}}$ .",
"Therefore the interaction Hamiltonian can be rewritten $ H_{\\text{int}}=\\frac{1}{2N_{kp}}\\sum _{\\mathbf {k},\\mathbf {p},\\mathbf {q}}\\sum _{\\alpha ,\\beta }U(\\mathbf {q})c^\\dagger _{\\mathbf {k}+\\mathbf {q},\\alpha }c^{\\dagger }_{\\mathbf {p}-\\mathbf {q},\\beta }c^{\\phantom{\\dagger }}_{\\mathbf {p},\\beta }c^{\\phantom{\\dagger }}_{\\mathbf {k},\\alpha }\\\\U(q)\\equiv \\frac{2U_0}{q\\xi }\\tanh \\frac{q\\xi }{2}.$ For parameters $\\epsilon =3.5$ and $\\xi =250$ relevant to experiments, we obtain $U_0\\approx 37$ .",
"Anticipating excitonic pairing at $\\pm \\mathbf {q_c}$ , we perform self-consistent HF calculations allowing for coherence by multiples of $\\mathbf {q_c}$ , i.e.",
"$\\langle c^\\dagger _{\\mathbf {k}\\alpha }c^{\\phantom{\\dagger }}_{\\mathbf {k}+n\\mathbf {q_c}\\beta }\\rangle $ can take non-zero values for integer $n$ .",
"Representative (folded) band structures are shown in Figure REF .",
"The presence of excitonic condensation is diagnosed by the integrated order parameter $\\Delta _\\text{exc}\\equiv \\sqrt{\\sum _{\\alpha ,\\beta } |\\langle c^\\dagger _{\\mathbf {k}\\alpha }c^{\\phantom{\\dagger }}_{\\mathbf {k}+\\mathbf {q_c}\\beta } \\rangle |^2}$ .",
"The spin-valley nature of the ordering is diagnosed by computing charge/spin densities $\\rho ^\\mu _{\\mathbf {Q}}\\propto \\sum _{\\sigma \\sigma ^{\\prime }\\mathbf {k}a}\\sigma ^\\mu _{\\sigma \\sigma ^{\\prime }}\\langle c^\\dagger _{\\mathbf {k}-\\mathbf {Q}\\sigma a}c^{\\phantom{\\dagger }}_{\\mathbf {k}\\sigma ^{\\prime } a}\\rangle $ where $a$ runs over the orbital degree of freedom.",
"HF calculations were also performed without allowing for excitonic coherence, in order to obtain the `parent' self-consistent states appropriate for an analytic weak-coupling treatment of excitonic pairing.",
"Representative results are shown in Figure REF .",
"Figure: Folded HF band structures for different interaction strengths—in order of increasing U 0 U_0, the phases are semimetal, SDW, trivial insulator, spin spiral, spin spiral (at experimentally relevant U 0 U_0), trivial insulator.",
"Labeled momentum points are M ' =(-q c /2,G y /4),Y ' =(0,G y /4),X=(-q c /2,0)M^{\\prime }=(-q_c/2,G_y/4),Y^{\\prime }=(0,G_y/4),X=(-q_c/2,0).",
"Given the momentum cutoff of the k·pk\\cdot p theory, there are 12 bands per momentum in the folded BZ.",
"All phases except the spin spiral have doubly-degenerate bands due to P ^T ^\\hat{P}\\hat{T} symmetry.",
"Calculations were done on a 75×2575\\times 25 momentum grid.Figure: HF band structures along the k x k_x axis, when the HF is restricted to forbid translation symmetry breaking.",
"Calculations were done on a 75×2575\\times 25 momentum grid."
],
[
"Trial Excitonic Insulator States",
"For generality, consider the situation with one valence pocket at $\\mathbf {\\Gamma }$ , and $N_\\lambda $ equivalent conduction valleys at $\\mathbf {Q}(\\lambda )$ .",
"The insulating excitonic trial states considered in the main text are of the following form $\\mathinner {|{\\Phi }\\rangle } = \\prod _{\\mathbf {k},\\sigma } \\alpha _{\\mathbf {k}\\sigma }^\\dagger \\mathinner {|{0}\\rangle },\\,\\alpha _{\\mathbf {k}\\sigma } = u_{\\mathbf {k}}a_{\\mathbf {k}\\sigma } + v_{\\mathbf {k}}\\sum _{s\\lambda } w^{\\sigma }_{s\\lambda } b_{\\mathbf {k}s\\lambda },$ where $u_k,v_k$ are real and even, $u_k^2+v_k^2=1$ , $\\sum _{s\\lambda }w^{\\sigma }_{s\\lambda }\\bar{w}^{\\sigma ^{\\prime }}_{s\\lambda }=\\delta _{\\sigma \\sigma ^{\\prime }}$ , and overbar denotes complex conjugation.",
"$u_k,v_k$ parameterizes the (small)-momentum structure of exciton coherence, while $w_{s\\lambda }^\\sigma $ parameterizes the spin-valley structure of pairing.",
"Note that, for a fixed choice of gauge for the Bloch operators, the choice of orthonormal complex vectors $w^\\uparrow ,w^\\downarrow $ in $\\mathbb {C}^{2N_\\lambda }$ uniquely specifies the trial state without any redundancy—there is no gauge redundancy corresponding to unitary rotation within occupied orbitals.",
"Now specialize to the case of two valleys $\\lambda =\\pm $ , so that $\\mathbf {Q}(\\lambda )=\\lambda \\mathbf {q_c}$ .",
"The trial states considered by Ref.",
"[2] are a strict subset of Eqn.",
"REF , and can be parameterized by $w^\\sigma _{s\\lambda }=lM^{(+)}_{\\sigma s}\\delta _{\\lambda +}+mM^{(-)}_{\\sigma s}\\delta _{\\lambda -}$ where $l^2+m^2=1$ , and the $M$ matrices are unitary.",
"This can describe the SDW, but can only describe the spin spiral if $l$ and $m$ are allowed to be $\\sigma $ -dependent.",
"It can be shown that global spin-rotation $\\hat{U}^s$ and valley-rotation $\\hat{U}^v$ act as $\\hat{U}^s:&\\quad w^\\sigma _{s\\lambda }\\rightarrow \\sum _{\\sigma ^{\\prime }s^{\\prime }}U^\\dagger _{\\sigma \\sigma ^{\\prime }}w^{\\sigma ^{\\prime }}_{s^{\\prime } \\lambda }U^{\\phantom{\\dagger }}_{s^{\\prime }s}\\\\\\hat{U}^v:&\\quad w^\\sigma _{s\\lambda }\\rightarrow \\sum _{\\lambda ^{\\prime }}w^\\sigma _{s\\lambda ^{\\prime } }U_{\\lambda ^{\\prime }\\lambda }$ where $U=\\exp \\left(\\frac{i\\theta }{2}\\hat{n}\\cdot \\mathbf {\\sigma }\\right)$ is a $SU(2)$ unitary.",
"$U(1)_{eh}$ rotations corresponding to separate conservation of conduction and valence populations act as $w^\\sigma _{s\\lambda }\\rightarrow w^\\sigma _{s\\lambda }e^{i\\theta }$ ."
],
[
"Dominant Term Approximation (DTA) Equations",
"The DTA equations [5], [2] determine the internal momentum structure of excitonic coherence (i.e.",
"the coefficients $u_k,v_k$ ).",
"The starting point is an effective model that describes the band extrema of a self-consistent non-excitonic band structure, which can be semimetallic or insulating.",
"For simplicity consider the `two-pocket' case (one conduction minimum $b^\\dagger _{k\\sigma }$ and valence maximum $a^\\dagger _{ k\\sigma }$ )—the multi-valley case can be treated analogously.",
"In the DTA, only intra-pocket interactions are included, and the gauge is chosen smooth so that $F^{nn;\\sigma }_{k,k+q}\\simeq 1$ for small momentum transfer $q$ , leading to the Hamiltonian $\\hat{H}_\\text{DTA}=\\sum _{kn\\sigma }\\epsilon ^n_{k}d^\\dagger _{nk\\sigma }d_{nk\\sigma }+\\frac{1}{2}\\sum _{kk^{\\prime }qnn^{\\prime }\\sigma \\sigma ^{\\prime }}U(q)d^\\dagger _{n,k+q,\\sigma }d^\\dagger _{n^{\\prime },k^{\\prime }-q,\\sigma ^{\\prime }}d^{\\phantom{\\dagger }}_{n^{\\prime },k^{\\prime },\\sigma ^{\\prime }}d^{\\phantom{\\dagger }}_{n,k,\\sigma }.$ Therefore there is an emergent $U(1)$ symmetry corresponding to separate conservation of conduction and valence band electrons.",
"There is also now global $SU(2)_s$ spin rotation symmetry, as well as $SU(2)_v$ valley rotation symmetry.",
"We consider an insulating excitonic ansatz $\\mathinner {|{\\Phi (w)}\\rangle }$ described by the operator for the filled bands $\\alpha _{k\\sigma }=u_ka_{k\\sigma }+v_k\\sum _{s}w^\\sigma _{s}b_{ks},\\quad \\sum _{s}w^{\\sigma }_{s}w^{\\sigma ^{\\prime *}}_{s }=\\delta _{\\sigma \\sigma ^{\\prime }}.$ where $u_k,v_k$ are real and even, and $u_k^2+v_k^2=1$ .",
"We now recall that the parameters of the model are extracted from a self-consistent band structure.",
"Therefore when counting the interactions of any distorted state, we need to measure the density relative to the reference self-consistent state $\\Phi _0$ : $E_\\text{DTA}[\\Phi (w)]=\\text{const}+2\\sum _{k}v_k^2(\\epsilon ^b_{k}-\\epsilon ^a_{k})-\\frac{1}{2}\\sum _{kk^{\\prime }nn^{\\prime }\\sigma \\sigma ^{\\prime }}U(k-k^{\\prime })\\langle d^\\dagger _{nk^{\\prime }\\sigma }d^{\\phantom{\\dagger }}_{n^{\\prime }k^{\\prime }\\sigma ^{\\prime }}\\rangle ^{\\prime }\\langle d^\\dagger _{n^{\\prime }k\\sigma ^{\\prime }}d^{\\phantom{\\dagger }}_{nk\\sigma }\\rangle ^{\\prime }\\\\\\langle a^\\dagger _{k\\sigma }a^{\\phantom{\\dagger }}_{k\\sigma ^{\\prime }}\\rangle ^{\\prime }=(u_k^2-N^0_{ak})\\delta _{\\sigma \\sigma ^{\\prime }}\\\\\\langle b^\\dagger _{k\\sigma }a^{\\phantom{\\dagger }}_{k\\sigma ^{\\prime }}\\rangle ^{\\prime }=g_k w^{\\sigma ^{\\prime }}_{\\sigma }\\\\\\langle b^\\dagger _{k\\sigma }b^{\\phantom{\\dagger }}_{k\\sigma ^{\\prime }}\\rangle ^{\\prime }=(v_k^2-N^0_{bk})\\delta _{\\sigma \\sigma ^{\\prime }}.$ where $g_k=u_kv_k$ , and $N^0_{nk}$ is the filling of $\\Phi _0$ (for the insulating parent state in the main text, we have $N^0_{ak}=1$ ).",
"The direct contributions with $q=0$ are canceled by the neutralizing background.",
"Evaluating the interaction term, we obtain the DTA energy $E_\\text{DTA}=\\text{const}+ 2\\sum _{k}v_k^2(\\epsilon ^b_{k}-\\epsilon ^a_{k})-\\sum _{kk^{\\prime }}U(k-k^{\\prime })\\left[(u_k^2-N^0_{ak})(u_{k^{\\prime }}^2-N^0_{ak^{\\prime }})+(v_k^2-N^0_{bk})(v_{k^{\\prime }}^2-N^0_{bk^{\\prime }})+2g_kg_{k^{\\prime }}\\right]$ which is independent of $w$ , leading to a huge degeneracy at DTA level.",
"We minimize this energy with respect to $v_k$ $0=\\partial _{v_p}E_{\\text{DTA}}=4(\\epsilon ^b_{p}-\\epsilon ^a_{p})v_p-4\\sum _{k}U(k-p)\\left[(2v_k^2-1+N^0_{ak}-N^0_{bk})v_p+g_k\\frac{1-2v_p^2}{\\sqrt{1-v_p^2}}\\right]\\\\\\rightarrow \\left[\\epsilon _{pb}-\\sum _k U(k-p)\\left(v_k^2-N^0_{bk}\\right)-\\epsilon _{pa}+\\sum _k U(k-p)\\left(u_k^2-N^0_{ak}\\right)\\right]v_p=\\frac{1-2v_p^2}{\\sqrt{1-v_p^2}}\\sum _k U(k-p)g_k\\\\\\rightarrow 2\\xi _pv_p=\\frac{1-2v_p^2}{\\sqrt{1-v_p^2}}\\Delta _p$ where we have defined $\\bar{\\epsilon }^a_{k}=\\epsilon _{ka}-\\sum _{k^{\\prime }}U(k-k^{\\prime })(u_{k^{\\prime }}^2-N^0_{ak^{\\prime }})\\\\\\bar{\\epsilon }^b_{k}=\\epsilon _{kb}-\\sum _{k^{\\prime }}U(k-k^{\\prime })(v_{k^{\\prime }}^2-N^0_{bk^{\\prime }})\\\\\\xi _k=\\frac{1}{2}(\\bar{\\epsilon }^b_{k}-\\bar{\\epsilon }^a_{k})\\\\\\Delta _k=\\sum _{k^{\\prime }}U(k-k^{\\prime })g_{k^{\\prime }}.$ The minimization condition can be recast as the coupled integral equations $v_k=\\sqrt{\\frac{1}{2}\\left(1-\\frac{\\xi _k}{\\sqrt{\\xi _k^2+\\Delta _k^2}}\\right)}$ which are solved by iteration.",
"The energy bands of the excitonic state are given by $E_{k\\alpha }=\\frac{\\bar{\\epsilon }^a_{k}+\\bar{\\epsilon }^b_{k}}{2}-\\sqrt{\\xi _k^2+\\Delta _k^2}\\\\E_{k\\beta }=\\frac{\\bar{\\epsilon }^a_{k}+\\bar{\\epsilon }^b_{k}}{2}+\\sqrt{\\xi _k^2+\\Delta _k^2}.$ If we have two valleys, we will have an additional energy band $E_{k\\gamma }=\\epsilon ^b_k$ which remains unaltered.",
"In this case it is possible that the excitonic state remains semimetallic if the parent state is semimetallic."
],
[
"Beyond Dominant Term Approximation (bDTA) Splitting Terms",
"While the DTA equations determine $u_k,v_k$ , the choice of $w$ can only be resolved by considering the neglected inter-pocket interactions [2].",
"Assuming a good DTA/bDTA separation of scales, we can use first-order perturbation theory to evaluate the neglected terms of $\\mathinner {\\langle {\\Phi (w)|\\hat{H}|\\Phi (w)}\\rangle }$ .",
"In the two-valley case, we obtain $\\delta E[w]&=&\\sum _{\\sigma \\sigma ^{\\prime }}\\left(B_{\\sigma \\sigma ^{\\prime }}(w^\\sigma _{\\sigma +}+\\bar{w}^{\\bar{\\sigma }}_{\\bar{\\sigma }-})(\\bar{w}^{\\sigma ^{\\prime }}_{\\sigma ^{\\prime }+}+w^{\\bar{\\sigma ^{\\prime }}}_{\\bar{\\sigma ^{\\prime }}-})-2\\text{Re}\\,C_{\\sigma \\sigma ^{\\prime }}w^{\\sigma ^{\\prime }}_{\\sigma +}w^{\\sigma }_{\\sigma ^{\\prime }-}\\right)\\\\&+&\\sum _{\\sigma \\sigma ^{\\prime }ss^{\\prime }}\\left(Dw^\\sigma _{s+}\\bar{w}^{\\sigma }_{s-}\\bar{w}^{\\sigma ^{\\prime }}_{s^{\\prime }+}w^{\\sigma ^{\\prime }}_{s^{\\prime }-}-J_{ss^{\\prime }}w^\\sigma _{s+}\\bar{w}^{\\sigma }_{s^{\\prime }+}w^{\\sigma ^{\\prime }}_{s^{\\prime }-}\\bar{w}^{\\sigma ^{\\prime }}_{s-}\\right)$ $B_{\\sigma \\sigma ^{\\prime }}&=&U(q_c)\\sum _{k}g_kF^{ab;\\sigma ^*}_{k,k+q_c}\\sum _{k^{\\prime }}g_{k^{\\prime }}F^{ab;\\sigma ^{\\prime }}_{k^{\\prime },k^{\\prime }+q_c}\\\\C_{\\sigma \\sigma ^{\\prime }}&=&\\sum _{kk^{\\prime }}g_kg_{k^{\\prime }}U(k-k^{\\prime }+q_c)F^{ab;\\sigma ^*}_{k^{\\prime },k+q_c}F^{ba;\\sigma ^{\\prime }}_{k^{\\prime }-q_c,k}\\\\D&=&U(2q_c)|\\sum _{k}v_k^2F^{bb;\\sigma }_{k-q_c,k+q_c}|^2\\\\J_{ss^{\\prime }}&=&\\frac{1}{2}\\sum _{kk^{\\prime }}v_k^2v_{k^{\\prime }}^2U(k-k^{\\prime }+2q_c)[F^{bb;s^*}_{k^{\\prime }-q_c,k+q_c}F^{bb;s^{\\prime }}_{k^{\\prime }-q_c,k+q_c}+\\text{c.c.",
"}].$ where $B$ is Hermitian, $J,C$ are symmetric, and the spin-quantization axis is chosen along the preserved direction (SOC is $U(1)_s$ preserving).",
"The $F$ refer to the form factors of the effective model, and the momentum labels are absolute momenta measured from the zone center.",
"For example, $F^{bb;\\sigma }_{k-q_c,k^{\\prime }+q_c}$ is an intervalley form factor because the $k,k^{\\prime }$ always represent `small' momenta.",
"The $B$ -term and $D$ -term are Hartree terms that penalize charge density wave modulations at wavevector $q_c$ and $2q_c$ respectively.",
"The $C$ -term and $J$ -term are exchange terms at momentum transfer $q\\sim q_c$ and $2q_c$ respectively.",
"In the limit of vanishing SOC, and specializing to a restricted class of states (that includes the SDW but not the spin spiral), we recover the bDTA expression of Ref. [2].",
"Using the transformations in Eqns REF ,, it can be shown that $\\delta E[w]$ is invariant under $U(1)_s$ and $U(1)_v$ symmetries.",
"$U(1)_s$ is present because the starting model was already assumed to have this symmetry.",
"$U(1)_v$ can be seen by investigating the possible inter-pocket interaction terms which conserve momentum.",
"This symmetry ceases to be sensible once excitonic coherence remains strong out to momenta $k\\sim q_c/2$ in the folded BZ, since then the division of the relevant low-energy Bloch states into small `pockets' fails, and the weak-coupling perspective is no longer useful.",
"There is no $U(1)_{eh}$ symmetry corresponding to separate conservation of valence/conduction electrons, because bDTA contains interaction terms $\\sim a^\\dagger a^\\dagger b^{\\phantom{\\dagger }}_+b^{\\phantom{\\dagger }}_-$ .",
"These $U(1)_{eh}$ -violating terms are reflected in the $B$ - and $C$ -terms of the bDTA energy functional."
],
[
"Spiral/SDW Competition for Two Excitons",
"In this section we argue that the spin spiral vs SDW competition outlined in the main text is invisible to a single exciton, and is a selection mechanism at the many-exciton level.",
"Let $\\mathinner {|{\\Phi _0}\\rangle }=\\prod _{k\\sigma }a^\\dagger _{k\\sigma }\\mathinner {|{\\text{vac}}\\rangle }$ be the parent insulating state of the effective model.",
"An exciton creation operator (with net momentum $q=0$ in the folded BZ) can be parameterized as a linear combination of single particle-hole operators: $B^\\dagger _\\sigma (w)=\\sum _{ks\\lambda }f_k w^*_{s\\lambda }b^\\dagger _{ks\\lambda }a^{\\phantom{\\dagger }}_{k\\sigma }$ where $f_k$ , satisfying $\\sum _k f_k^2=1$ , is real and even, and parameterizes the exciton momentum structure (predominantly determined by $q\\sim 0$ interactions), while $w$ indicates the valley/spin structure of the electron.",
"We focus on the $q\\sim 2q_c$ components of the interaction Hamiltonian, since these were found to mediate the SDW/spiral competition.",
"Consider a single exciton state $\\mathinner {|{\\sigma w}\\rangle }=B_\\sigma ^\\dagger (w)\\mathinner {|{\\Phi _0}\\rangle }.$ This vanishes under the action of $q\\sim 2q_c$ interaction terms, since $b^\\dagger b^\\dagger bb$ will always annihilate the above state.",
"Hence a single exciton is not sensitive to the competition described in the main text.",
"Now we consider the interaction energy of two-exciton states.",
"For simplicity we assume the quasi-boson approximation and consider the following two-exciton states $\\mathinner {|{\\sigma w;\\sigma ^{\\prime } w^{\\prime }}\\rangle }=B^\\dagger _\\sigma (w)B^\\dagger _{\\sigma ^{\\prime }}(w^{\\prime })\\mathinner {|{0}\\rangle }$ where we neglect the normalization.",
"We will be interested in cases where the $w,w^{\\prime }$ describe excitons with spin/valley structures corresponding to spiral or SDW phases.",
"Focusing on the $q\\sim 2q_c$ contributions again, we obtain after some algebra $\\mathinner {\\langle {\\sigma w;\\sigma w^{\\prime }|\\hat{H}_{q\\sim 2q_c}|\\sigma w;\\sigma w^{\\prime }}\\rangle }&=\\frac{2}{N}\\bigg [U(2q_c)|\\sum _k f_k^2\\mathcal {F}^{\\uparrow }_{k,k}|^2\\sum _{ss^{\\prime }}(w_{s+}\\bar{w}_{s-}\\bar{w}^{\\prime }_{s^{\\prime }+}w^{\\prime }_{s^{\\prime }-}+\\text{c.c.",
"})\\\\&-\\frac{1}{2}\\sum _{kk^{\\prime }ss^{\\prime }}f_k^2f_{k^{\\prime }}^2U(k-k^{\\prime }+2q_c)(\\mathcal {F}^{s^*}_{k^{\\prime }k}\\mathcal {F}^{s^{\\prime }}_{k^{\\prime }k}w_{s+}\\bar{w}_{s^{\\prime }+}w^{\\prime }_{s^{\\prime }-}\\bar{w}^{\\prime }_{s-}+\\text{c.c.",
"})\\bigg ].$ For $w,w^{\\prime }$ corresponding to the spin spiral (Eqn 10 in main text), the above contributions vanish as expected since there is no $2q_c$ coherence.",
"For $w,w^{\\prime }$ corresponding to the SDW (Eqn 9 in main text), we recover the competition between the direct and exchange terms, which take analogous forms to the $D$ - and $J$ -terms in the bDTA.",
"We note that these calculations (involving $q\\sim 0$ and $q\\sim q_c$ terms as well) can be generalized to derive the interaction terms of an effective quasi-boson Hamiltonian."
],
[
"Elliptical Spin Spirals and Charge Order",
"In this section we consider the more general class of elliptical spin spirals, which encompasses the limiting cases of SDW and (circular) spin spiral discussed in the main text.",
"The spin/valley structure of these states can be parameterized using the language of Eq REF $w^{\\uparrow }_{\\downarrow +}&= e^{-i(\\alpha +\\phi )}\\sin (\\chi -\\frac{\\pi }{4})\\\\w^{\\uparrow }_{\\downarrow -}&= e^{i(\\alpha -\\phi )}\\cos (\\chi -\\frac{\\pi }{4})\\\\w^{\\downarrow }_{\\uparrow +}&= e^{i(-\\alpha +\\phi )}\\cos (\\chi -\\frac{\\pi }{4})\\\\w^{\\downarrow }_{\\uparrow -}&= e^{i(\\alpha +\\phi )}\\sin (\\chi -\\frac{\\pi }{4}).$ The spin spiral is recovered for $\\chi =\\pm \\pi /4$ (corresponding to the two senses of rotation), while $\\chi =0$ corresponds to the SDW.",
"The spin and charge densities for these states are $\\mathbf {\\rho }^s(\\mathbf {r})\\sim \\begin{bmatrix}\\sin \\chi \\cos \\phi \\cos \\big [q_cx+\\alpha \\big ]+\\cos \\chi \\sin \\phi \\sin \\big [q_cx+\\alpha \\big ]\\\\0\\\\\\sin \\chi \\sin \\phi \\cos \\big [q_cx+\\alpha \\big ]+\\cos \\chi \\cos \\phi \\sin \\big [q_cx+\\alpha \\big ]\\end{bmatrix}\\\\\\rho ^c(\\mathbf {r})\\sim \\cos 2\\chi \\cos \\big [2(q_cx+\\alpha )\\big ].$ Hence the principal axes of the elliptical spiral are controlled by $\\phi $ and lie along $[\\cos \\phi ,0,\\sin \\phi ]$ and $[-\\sin \\phi ,0,\\cos \\phi ]$ , while $\\alpha $ controls the position along $x$ .",
"$\\chi $ is related to the ellipticity of the spin order, and also controls the strength $\\sim \\cos ^2 2\\chi $ of the associated $2q_c$ charge density wave.",
"To further understand the relation between the ellipticity of the spiral and the charge order, we can analyze their coupling within Landau theory [26].",
"With the constraints given by TRS, $U(1)_s$ about $s^y$ , and translation (there are no Umklapp processes since $\\mathbf {q_c}$ is not at a high-symmetry point), the lowest order coupling between charge density $\\rho ^c$ and $x-z$ spin density $\\mathbf {\\rho }^{s,\\perp }$ is $F\\sim \\int dx \\rho ^c(x)\\big [\\mathbf {\\rho }^{s,\\perp }(x)\\big ]^2\\sim \\sum _{p,p^{\\prime }}\\rho ^c_{-p-p^{\\prime }}\\mathbf {\\rho }^{s,\\perp }_p\\cdot \\mathbf {\\rho }^{s,\\perp }_{p^{\\prime }}$ where we have used the fact that there is no order along the $y$ -direction.",
"Since the spin order has non-trivial contributions for momenta $p=\\pm q_c$ , we focus on the case $p=p^{\\prime }$ which couples to the $2q_c$ charge order.",
"With appropriate choice of coordinate and spin axes, the spin order parameter of the elliptical spiral can be chosen as $\\mathbf {\\rho }^{s,\\perp }(x)\\sim [\\cos \\chi \\sin (q_c x),\\sin \\chi \\cos (q_c x)]$ , with Fourier components $\\mathbf {\\rho }^{s,\\perp }_{\\pm q_c}\\sim [\\pm i\\cos \\chi ,\\sin \\chi ]$ , leading to $\\mathbf {\\rho }^{s,\\perp }_{\\pm q_c}\\cdot \\mathbf {\\rho }^{s,\\perp }_{\\pm q_c}\\sim \\cos 2\\chi $ .",
"Hence the coupling between $2q_c$ charge order and $q_c$ spin order contains a multiplicative factor of $\\cos 2\\chi $ .",
"This vanishes for the circular spiral, which can be intuited from the fact that $\\big [\\mathbf {\\rho }^{s,\\perp }(x)\\big ]^2$ is spatially uniform.",
"This argument holds for higher order terms in Landau theory, since in-plane spin must enter as $\\big [\\mathbf {\\rho }^{s,\\perp }(x)\\big ]^2$ due to $U(1)_s$ .",
"In the bDTA, the energy of the elliptical spiral is $E_\\text{bDTA}=-2\\text{Re}\\,C_{\\uparrow \\downarrow }+\\cos ^22\\chi \\left(D-\\frac{J}{2}\\right)$ .",
"Hence the energetics mean that we have $\\chi \\rightarrow \\pm \\pi /4,0$ depending on whether $D-\\frac{J}{2}$ is positive or negative.",
"$U(1)_s$ symmetry breaking and CDW modulation in a circular spiral phase: Note that when $U(1)_s$ symmetry in the $xz$ spin plane is broken (e.g., by a magnetic field perpendicular to the monolayer), the reduction in symmetry admits additional terms such as $({\\rho }^{s,x}{\\rho }^{s,x} - {\\rho }^{s,z}{\\rho }^{s,z})\\rho _s$ , allowing even a circular spiral to generate a CDW modulation."
]
] | 2012.05255 |
[
[
"INetCEP: In-Network Complex Event Processing for Information-Centric\n Networking"
],
[
"Abstract Emerging network architectures like Information-centric Networking (ICN) offer simplicity in the data plane by addressing named data.",
"Such flexibility opens up the possibility to move data processing inside network elements for high-performance computation, known as in-network processing.",
"However, existing ICN architectures are limited in terms of data plane programmability due to the lack of (i) in-network processing and (ii) data plane programming abstractions.",
"Such architectures can benefit from Complex Event Processing (CEP), an in-network processing paradigm to efficiently process data inside the data plane.",
"Yet, it is extremely challenging to integrate CEP because the current communication model of ICN is limited to consumer-initiated interaction that comes with significant overhead in a number of requests to process continuous data streams.",
"In contrast, a change to producer-initiated interaction, as favored by CEP, imposes severe limitations for request-reply interactions.",
"In this paper, we propose an in-network CEP architecture, INetCEP that supports unified interaction patterns (consumer- and producer-initiated).",
"In addition, we provide a CEP query language and facilitate CEP operations while increasing the range of applications that can be supported by ICN.",
"We provide an open-source implementation and evaluation of INetCEP over an ICN architecture, Named Function Networking, and two applications: energy forecasting in smart homes and a disaster scenario."
],
[
"Introduction",
"Emerging network architectures like Information-Centric Networking (ICN) simplify the data plane of the current Internet by changing its addressing scheme from named hosts to named data.",
"ICN has evolved as a key paradigm towards a content-centric Internet, as currently adopted by academia and industry, e.g., by Internet2, Cisco, and Intel [6] for real-world deployment [37].",
"The data plane abstractions of ICN are particularly useful since users can define what data they need instead of identifying where to get it from.",
"Additionally, exploiting data plane programmability on in-network elements of ICN can offer high throughput by processing packets at line rate, while delivering them at low latency, typically known as in-network processing.",
"However, existing ICN architectures like NDN and NFN are restricted in terms of data plane programmability due to lack of (i) in-network processing and (ii) data plane programming abstractions.",
"Figure: Taxonomy of ICN architectures based on the supported interaction patterns.This makes Complex Event Processing (CEP) a paradigm of choice for an ICN architecture.",
"CEP is a powerful in-network processing paradigm that takes a query as an input to describe the correlations over a set of incoming data streams in order to deliver data notifications in response to the query.",
"For instance, in a disaster scenario, a heat map query can describe correlations over a set of data streams, e.g., location updates from victims to deliver a heat map distribution of survivors to better coordinate the activities of rescue workers.",
"However, employing CEP on top of an ICN architecture is extremely challenging.",
"An important challenge is that the communication model of current ICN architectures has strong limitations in supporting the processing of periodic data streams.",
"For instance, NDN uses a consumer-initiated interaction pattern where a consumer pulls data by sending an Interest (request) to the network.",
"The NDN network forwards this request to one or more producers that satisfy the request and then forward the Data (reply) back to the consumer.",
"For continuous data streams, consumer-initiated interaction poses significant overhead in terms of number of request messages and in the delay until fresh data becomes available.",
"On the other hand, changing to a pure producer-initiated interaction as favored by CEP is problematic for many IoT applications that build on the request-reply interactions.",
"For example, applications like Amazon Alexa [5] need personalized request-reply interaction.",
"In fact, ICN ideally should offer efficient support for both interaction patterns as part of a unified communication model as illustrated in fig:taxonomy.",
"Initial work in the context of content routing has shown the potential of a unified communication model [19].",
"However, performing CEP operations inside the network while efficiently realizing a unified communication model in ICN is a challenge that we aim to address in this paper.",
"Thus, we present a novel INetCEP architecture [8] with the following contributions: a unified communication layer to provide the functionality of CEP at the network level, a general CEP query language that specifies patterns for meaningful event detection over the ICN substrate, in the form of query interests, a query processing algorithm to resolve query interests, and an open source implementation and evaluation of the proposed approach on a state-of-the-art ICN architecture, NFN, with two IoT case studies.",
"The paper is organized as follows.",
"In Section , we present preliminaries required to understand our approach.",
"In Section , we present two motivational IoT use cases.",
"In Section , we describe the problem space and our system model.",
"In Section , we present the INetCEP architecture, and in Section , we provide an evaluation.",
"In Section , we provide a comparison with related work.",
"Finally, in Section , we discuss possible extensions to our architecture before concluding in Section ."
],
[
"Background",
"In this section, we briefly explain the building blocks of our work: CCN that evolved into Named Data Networking (NDN) and Named Function Networking (NFN), and Complex Event Processing (CEP).",
"Content-Centric Networking.",
"Jacobson et al.",
"[26] proposed CCNIn the remainder of the paper, we will use the terms ICN and CCN interchangeably., where communication is consumer-initiated, consisting of two packets: Interest and Data.",
"A data object (payload of a Data packet) satisfies an interest if the name in the Interest packet is a prefix of the name in the Data packet.",
"Thus, when a packet arrives on a faceface stands for interface in CCN terminology.",
"(identified by $face\\_id$ ) of a CCN node, the longest prefix match is performed on the name and the data is returned based on a lookup.",
"CCN data plane: Each CCN node maintains three major data structures: FIB, CS (also known as in-network cache), and PIT.",
"Once an Interest arrives on a face, the node first checks its Content Store for a matching Data packet by name.",
"Upon a match, the Data packet is sent via the same face it arrived from.",
"Otherwise, the node continues its search in the PIT that stores all the Interest packets (along with its incoming and outgoing face) that are not satisfied.",
"If an entry exists in the PIT, the face is updated and the Interest is discarded, because an Interest packet has already been sent upstream.",
"Otherwise, the node looks for a matching FIB entry and forwards the Interest to the potential source(s) of the data.",
"NDN and NFN: NDN [45] emerged as a prominent architecture that builds on the principles of CCN's named data.",
"NFN is another emerging architecture that focuses on addressing named functions in addition to named data by extending the principles of NDN.",
"NFN blends [42] data computations with network forwarding, by performing computational tasks across the CCN network.",
"It represents named functions on the data as Church's $\\lambda -$ calculus expressions that are the basis of functional programming.",
"We aim to encapsulate CEP operators (cf.",
"next section) as NFN named functions and hence resolve them in the network.",
"Yet, the proof-of-concept design of NFN focuses mainly on resolving functions on top of the CCN substrate.",
"In contrast, we focus on continuous and discrete computations (push and pull), expressive representation of the computation tasks and their efficient distribution (cf.",
"Section ).",
"Complex Event Processing.",
"CEP can process multiple online and unbounded data streams using compute units called operators to deliver meaningful events to the consumers.",
"The consumers specify interest in the form of a query comprising of multiple operators.",
"Some of the commonly used operators are defined in the following.",
"Filter ($\\sigma $ ) checks a condition on the attribute of an event tuple and forwards the event if the condition is satisfied.",
"Aggregate applies an aggregation function such as $max$ , $min$ , $count$ , $sum$ , $avg$ , etc., on one or more event tuples.",
"Hence, the data stream must be bounded to apply these operations.",
"For this purpose, window can be used.",
"Window limits the unbounded data stream to a window based on time or tuple size, such that operators like Aggregate can be applied on the selected set of tuples.",
"Join ($\\bowtie $ ) combines two data streams to one output stream based on a filter condition applied on a window of limited tuples.",
"Sequence ($\\rightarrow $ ) detects causal or temporal relationship among two events applied on a window of selected event tuples from a data stream, e.g., if event $a$ caused event $b$ and event $a$ happened before event $b$ .",
"Figure: Limitations when standard consumer initiated communication is used in CCN to support CEP.These operators can be stateless or stateful.",
"Filter and Aggregate are stateless operators, while the other operators are stateful and maintain the state of input tuples before emitting the complex event and therefore depend on multiple input tuples to be accumulated before actual emission."
],
[
"Motivating Use Cases",
"Use case I: Disaster Scenario.",
"A natural disaster scenario is a prominent use case of ICN architecture research.",
"It is drafted as one of the baseline scenarios in an active Internet Research Task Force (IRTF) draft of ICN working groups [36].",
"A typical disaster management application is to generate a heat map showing live distributions of survivors in a disaster area [29].",
"An important property of such an application is that the information must be updated continuously to delegate rescue workers to the hot spots and to monitor their operations, and hence is producer-initiated.",
"However, there are limitations of developing such an application using current ICN architectures such as NDN alone, due to the absence of support for producer-initiated communication.",
"Use case II: Internet of Things.",
"We consider an IoT application as the second use case for our approach since: it is one of the baseline scenarios for ICN architectures [36] and IoT traffic is among the most used type of traffic in the current Internet [9].",
"An intrinsic property of such applications is that IoT devices produce continuous data streams, e.g., sensor data that needs to be analysed, filtered, and derived to retrieve meaningful information for the end consumers, and hence it is oriented towards producer-initiated interaction.",
"One such application in the context of smart homes is short term load forecasting of energy consumed by smart plugs, which is useful, e.g., for energy providers.",
"The DEBS grand challenge 2014 focuses on this application.",
"Although we target the above use cases in the following sections, our solution is not limited to these use cases, but the presented scenarios are representative to cover the design space of our solution.",
"Problem Space In this section, we first discuss the limitations of using straightforward solutions to motivate the need of our architecture (cf.",
"subsec:decisions) and then we explain the system model of INetCEP (cf.",
"sec:systemmodel).",
"Design Challenges We discuss the limitations of using straightforward solutions, e.g., standard consumer-initiated, producer-initiated communication, or long lived Interest packets for the purpose of supporting a wide variety of applications as also pointed here [19].",
"Then, we study limitations of using Interest packets to represent queries.",
"Finally, we present limitations in performing operator graph processing at the consumer end.",
"A straightforward solution to support CEP is to use the standard consumer initiated communication of the CCN architecture.",
"We illustrate the problems using this naive solution in Figure REF .",
"One way is that the consumers continuously issue [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; a query at regular intervals, and [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; the producer replies with the event of interest in a data (notification) packet.",
"However, there are multiple problems with this solution, as indicated in fig:lim1.",
"Figure: Limitation 3: Three way message and two kind of Interest packets required.Limitation 1: The continuous polling of a query by consumers generates a lot of overhead traffic and network state in the form of pending interests for only a few meaningful data packets.",
"Each time a data packet is received, the pending interest is removed from PIT since it is satisfied (represented as strikethrough in the figure).",
"However, depending on the query interval a new entry is again created in the PIT for each query.",
"Also, the interval length of issuing query might determine the maximum latency at which the notification is delivered to the consumer, which might not be acceptable for latency sensitive applications, e.g., autonomous cars.",
"Limitation 2 is to deal with the stale data in the cache or CS, as represented in fig:lim2.",
"The consumers in a CEP application often need real-time updates on the latest data.",
"For this reason, the query needs to be updated each time, otherwise it will retrieve the last cached Data packet which is stale or obsolete in time.",
"For instance, in fig:lim2, the broker still sends the old data to the consumer while the producer has generated a new data item for the query.",
"In addition, there should be a mechanism to expire the Data packet at the right time, perhaps, immediately for the real-time updates.",
"Solutions like appending sequence numbers (similar to TCP) to the Data packet can be applied.",
"However, this will require additional synchronization mechanisms.",
"Alternatively, another possibility is to support just producer-initiated transmission while using CCN primitives (cf.",
"fig:lim3).",
"Although this is a viable option for some applications [20], it results in a three-way message exchange of what amounts to a one-way message.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; The producer sends an asynchronous Interest packet that is not intended to fetch a Data packet from the network but to announce the data name and the callback from the consumer.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The consumer then shows interest in the data name, which is [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 3; fulfilled by a Data packet from the producer.",
"Figure: INetCEP communication model supports pull-based communication and fetches latest Data packet.Limitation 3: Besides the overhead generated by a 3-way message, this design has other major issues.",
"The interests leave an in-network state in the PIT of CCN nodes such that data can be fetched by the same path.",
"CCN typically performs a unicast of the Interest packet, so that the Data packet can follow the same path to the consumer.",
"Such an application has to support two kinds of Interest packets: a packet that is not supposed to fetch data, and a packet that is supposed to fetch data.",
"Limitation 4: Long-lived Interest packets can be used in place of a query, but this also has multiple side-effects.",
"Similar to multiple interests, long-lived Interest packets will also result in large in-network state (cf.",
"fig:lim1).",
"In addition, the long-lived Interest packets will have to deal with stale data as explained earlier in Limitation 2 (cf.",
"fig:lim2).",
"To solve the aforementioned issues, we propose to have both consumer-initiated and producer-initiated interaction patterns coexisting under a unified CCN communication layer.",
"A CCN architecture is unable to achieve this using existing packets and data structures as we saw above.",
"Hence, we propose additional packets as a part of the communication model and handle them while processing CEP queries in the network as defined in Section REF .",
"Limitation 5: CCN/NDN assumes a hierarchical naming scheme to address named data, e.g., $/node/nodeA/temperature$ , in order to fetch data objects e.g., $35^\\circ C$ , from the producers.",
"A simple way to specify CEP operations over data would be to represent this using the standard naming scheme, e.g., a $min$ operator as $/node/nodeA/min/temperature$ .",
"However, there are problems with this approach: the name cannot be used to correlate data from multiple producers, this would mean the processing is performed always at the consumer, which is inefficient and this is not extensible and not expressive, since adding more operators would mean appending them in the naming scheme, which reduces readability.",
"Hence, we need more than just CCN Interest packets that encapsulate name prefixes as stated above to represent CEP queries over a CCN network.",
"We propose an expressive query language that can correlate data from multiple producers and an efficient query parser to execute queries in the network (cf.",
"Section REF ).",
"Limitation 6: The query specified by the consumers must be processed within the CCN network.",
"The CCN resolution engine can resolve only Interest packets to retrieve Data packets based on the matching name prefix, but it cannot express query.",
"A naive way to deal with this is to process the query at the consumer.",
"However, this would overload the network with all the unnecessary data that could have been filtered on the way to the consumer and overload the consumer with all the processing.",
"This might result in a single node of failure, when the data becomes very big.",
"Thus, the processing needs to be performed in the CCN network, e.g., at the broker while being transmitted to the consumer.",
"We provide this in two ways: (i) centralized query processing, where the entire query is processed at a single broker and (ii) distributed query processing, where the query operators are assigned to in-network nodes for processing (cf.",
"Section REF ).",
"Figure: INetCEP communication model supports push-based communication without creating endless PIT entries.INetCEP System Model Every CCN node can act either as a producer, a consumer or a broker.",
"Here, a broker is an in-network element, i.e., an INetCEP aware CCN router, while a producer or consumer is an end device, e.g., a sensor or a mobile device.",
"On the one hand, [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; consumers can request a specific data item using an Interest packet, where broker(s) forward(s) the request received by consumers to support anonymous request-reply communication, as illustrated in fig:inetceppull.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The producer replies with a data object contained in a Data packet.",
"On the other hand, broker(s) process(es) the unbounded and ordered data streams generated by producers to provide event-based communication, which happens as illustrated in fig:inetceppush and is explained below.",
"Table: Description of differences in traditional ICN vs INetCEP architecture (\"-\" means no support).A producer multicasts the data stream (Data Stream packet) towards the broker network, which disseminates the stream all over the network (push).",
"The Data Stream packet is forwarded to further brokers in the network if there are consumers downstream for query interest ($qi$ ).",
"An efficient event dissemination can be achieved by using routing algorithms, e.g., defined in this work [13], by looking at the similarity score of the $qi$ .",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; A consumer issues a query by sending an Add Query Interest packet comprising $qi$ (top fig:inetceppush).",
"Each $qi$ encapsulates a CEP query $q$ that is processed by interconnected brokers in $B$ forming a broker network.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The $qi$ is stored in the PIT of the receiving broker until a [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 3; Remove Query Interest packet is received that triggers the removal of $qi$ from the PIT (bottom fig:inetceppush).",
"Unlike a conventional CEP system, the event-based communication in the INetCEP happens in the underlay CCN network.",
"The query $q$ induces a directed acyclic operator graph $G$ , where a vertex is an operator $\\omega \\in \\Omega $ and an edge represents the data flow of the data stream $D$ .",
"Each operator $\\omega $ dictates a processing logic $f_\\omega $ .",
"We explain the constituents: the communication model, the query model, and the operator graph model below.",
"Communication Model.",
"We provide five types of packets to support both kinds of interaction patterns.",
"The Interest (request) packet is equivalent to CCN's Interest packet that is used by the consumer to specify interest in any named data or named function.",
"The Data (reply) packet is a CCN data packet that satisfies an Interest.",
"It also encapsulates the complex event ($ce$ ) as described later.",
"The Data Stream packet represents a data stream of the form $<ts, a_1, \\ldots , a_m>$ .",
"Here, $ts$ is the time at which a tuple is generated and $a_i$ are the attributes of the tuple.",
"The Add Query Interest packet represents the event of interest in the form of a CEP query $q$ .",
"The Remove Query Interest packet represents the CEP query that must to be removed for the respective consumer (so that it no longer receives complex events).",
"The CCN forwarding engine (data plane) is enhanced to handle these packets, as below.",
"Query Model.",
"The INetCEP query language is based on two main design goals: it should deal with both pull (data from relations) and push (time series data streams) kind of traffic, and support standard CEP operators (as identified in Section ) over the CCN data plane.",
"Thus, a query ($q$ ) must be able to capture time series data streams as well as relations of the form $<ts, a_1, .., a_m>$ and define an operator $\\omega $ with processing logic $f_\\omega $ in a way that it is extensible.",
"Operator Graph Model.",
"The operator graph $G$ is a directed acyclic graph of plan nodes.",
"The vertex of the graph is a plan node that encapsulates a single operator ($\\omega $ ), while the links between plan nodes represents the data flow from the bottom of the graph to the top.",
"The operator graph can be processed centrally or collaboratively in a distributed manner by mapping it to the underlay CCN network.",
"In distributed CEP, typically, an operator placement mechanism defines a mapping of an operator graph $G$ onto a set of brokers, to collaboratively process the query.",
"The placement needs to be coordinated with the forwarding decisions for efficient processing over the CCN data plane.",
"INetCEP architecture We identify the following three broad requirements for the INetCEP architecture from our discussion in the previous section.",
"R1 A unified communication layer supporting both producer and consumer initiated communication (cf.",
"Section REF ).",
"R2 An expressive CEP query language for specifying the event of interest (cf.",
"Section REF ).",
"R3 Resolution of CEP queries by efficient and scalable in-network query processing (cf.",
"Section REF ).",
"We address each of these requirements below.",
"Unified Communication Layer In this section, we explain the extension of the CCN data plane to enable CEP.",
"In our approach, each CCN node $n \\in N$ maintains a Content Store or cache (CS), a Pending Interest Table (PIT), a Forwarding Information Base (FIB) and a CEP engine.",
"In the following, we explain the function of these main building blocks and the data plane handling.",
"Subsequently, we detail on the newly introduced packets Add Query Interest, Remove Query Interest and Data Stream and simultaneously the solution to the limitations identified in subsec:decisions.",
"Figure: High level view of packet handling in INetCEP architecture.Node Components The CS stores all the data objects associated with the CCN Interest, additionally, the time-stamped data objects associated with the query interests ($qi$ ).",
"The data object returns a value associated with the $qi$ , e.g., if the $qi$ is the sum of victims in a disaster location, then the data object contains value 20.",
"Hence, if multiple consumers are interested in the same $qi$ , the query is not reprocessed but the data is fetched directly from the CS.",
"To deal with Limitation 2 of stale cache entries for $qi$ (cf.",
"Section REF ), we store logical timestamps along with cache entries while storing $qi$ in the CS.",
"Hence, always up to-date data objects are stored, while old entries are discarded (cf.",
"fig:inetceppull).",
"The PIT stores the pending $qi$ so that the complex event ($ce$ ) could follow the path, i.e., the in-network state created in PIT to the consumer.",
"The $\\mathit {face}$ information is also stored in the PIT entry to keep track of $qi$ 's consumers.",
"In contrast to consumer-initiated interaction, the $ce$ must be notified to the interested consumers as and when detected in real-time.",
"Thus, as new data is received, the query interests in the PIT are re-evaluated as explained later in Algorithm REF .",
"In the former, we are referring to only continuous Data Stream packet since we also support fetching data from relations (this is handled conventionally using a CCN Interest packet as explained in Section ).",
"The reasons why we distinguish between Add Query Interest ($qi$ ) and CCN Interest packets are: (i) $qi$ is invoked on receipt of Add Query Interest as well as the Data Stream packet, (ii) removal of the PIT entry is not based on a Data packet retrieval but on the reception of the Remove Query Interest packet and (iii) $qi$ retrieves Data packets asynchronously.",
"In summary, we deal with Limitation 3 and Limitation 4 by asynchronously handling the $qi$ instead by 3-way message exchange and efficiently managing PIT entries, respectively.",
"We deal with Limitation 1 by storing $qi$ in PIT and asynchronously delivering event notifications to the consumers.",
"The FIB table gets populated as the producer multicasts to the broker network leaving a trail to the data source.",
"In this way the data processing is performed efficiently along the path from producer and consumer.",
"Finally, the CEP engine holds the processing logic $f_\\omega $ for each operator $\\omega $ and is responsible for parsing, processing, and returning the result to the next node towards consumer (cf.",
"Sections REF and REF ).",
"1em $\\mathit {CS} \\leftarrow \\text{content store of current node}$ $\\mathit {PIT} \\leftarrow \\text{pending interest table of current node}$ $\\mathit {FIB} \\leftarrow \\text{forwarding information base}$ $\\mathit {qi} \\leftarrow \\text{requested query interest}$ $\\mathit {result} \\leftarrow \\text{query result}$ $\\mathit {facelist} \\leftarrow \\text{list of all faces in PIT}$ $\\texttt {DataStream} \\leftarrow \\text{\\texttt {Data Stream} packet}$ $\\mathit {data} \\leftarrow \\text{data that resolves the qi}$ $\\textsc {AddQueryInterest}(\\mathit {qi})$ $qi$ is found in $\\mathit {CS.\\textsc {lookup}(qi)}$ $ \\mathit {data} \\leftarrow \\mathit {CS}.\\textsc {fetchContent}(\\mathit {qi}) $ $ \\text{return } \\mathit {data}$ (Discard AddQueryInterest) $qi \\text{ found in } \\mathit {PIT.\\textsc {lookup}(qi)} $ $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\texttt {AddQueryInterest})$ $qi \\text{ found in } \\mathit {FIB.\\textsc {lookup}(qi)} $ $\\mathit {\\textsc {CreateOperatorGraph}(\\mathit {qi})}$ (Refer Algorithm REF ) Forward AddQueryInterest (Discard AddQueryInterest) $\\textsc {ProcessDataStream}(\\texttt {DataStream})$ $\\mathit {qi} \\in PIT$ DataStream satisfies $qi$ $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\texttt {DataStream})$ Forward DataStream $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\mathit {packet})$ $\\mathit {packet} == \\texttt {DataStream} \\text{ and } \\mathit {packet}.\\mathit {ts} > \\mathit {qi.ts}$ $\\mathit {\\textsc {CreateOperatorGraph}(\\mathit {qi})}$ (Refer Algorithm REF ) Forward $\\mathit {packet}$ $\\mathit {facelist} \\leftarrow \\mathit {PIT}.\\textsc {getFaces}(\\mathit {qi}) $ $qi.\\mathit {face}$ is not found in $\\mathit {facelist}$ $ \\mathit {PIT}.\\textsc {addFace}(\\mathit {qi}) $ Discard $\\mathit {packet}$ -15pt Add Query Interest and Data Stream packet handling.",
"Data Plane Handling In Algorithm REF (lines REF -REF ) and fig:overall, we define the handling of Add Query Interest and Data Stream packets at the broker end in a CCN network.",
"The processing of $qi$ stored in PIT is triggered based on the receipt of these two packets as follows: when an Add Query Interest packet is received at a broker (line REF ) and due to the continuous arrival of new Data Stream packets (line REF ).",
"This is in contrast to the PIT entry of CCN Interest, which is checked only on the receipt of a new Interest packet.",
"When an Add Query Interest arrives, the broker checks if the (up to-date) data object corresponding to the $qi$ exists in the CS.",
"If this is true, the broker forwards the data object to the consumer and discards the $qi$ (lines REF -REF ).",
"This is because the $qi$ is already processed at one or more brokers and a matching Data packet (with latest timestamp) is found in the CS or cache.",
"The resolution to the Data packet is explained later in Section REF .",
"If the cache entry is not found, the broker continues its search in the PIT table (lines REF -REF ).",
"If $qi$ is found in PIT and the face corresponding to the query interest does not exist (lines REF -REF ), a new ${face\\_id}$ (from which the interest is received) is added.",
"Conversely, if the face entry is found in PIT, this means the $qi$ is being processed and hence the packet is discarded (lines REF -REF ).",
"However, if no entry in PIT exists, this means that the consumer's interest reaches first time at the broker network.",
"Therefore, a new entry for $qi$ is created and the $qi$ is processed by first generating an operator graph (cf.",
"Section REF ) and then processing it (cf.",
"Section REF ) (lines REF -REF ).",
"A Data Stream packet is also handled similarly to the Add Query Interest packet (lines REF -REF ), except for the fact that the query processing is triggered if Data Stream satisfies a query interest in PIT and it is a new packet (lines REF -REF ).",
"This means that the $qi$ performs an operation on the data object contained in the Data Stream packet.",
"In this case, query processing is triggered because it may contribute to the generation of a new $ce$ .",
"In addition, if the broker does not have a matching $qi$ in PIT entry, this means it is not allocated to operator graph processing and hence it is forwarded to the next broker (line REF ).",
"The Data Stream is forwarded if there are consumers downstream by looking at the FIB entry.",
"When a Remove Query Interest packet is received at a broker, the node looks up its PIT table for an entry of the $qi$ .",
"If found, it removes the PIT entry for $qi$ and the Remove Query Interest packet is forwarded to the next node.",
"It is done in a similar way as the PIT entry corresponding to a CCN Interest packet is removed when a matching Data packet is found.",
"To summarize, in tab:differences, we show the differences of the INetCEP architecture in comparison to the standard CCN architecture in terms of the packet types, the data plane, and the processing engine.",
"We show that with minimum changes in the data plane, we support both consumer- and producer-initiated traffic.",
"A General CEP Query Language In this section, we present a general CEP language to resolve Limitation 5.",
"By doing this, we provide a means to resolve CEP queries expressed as $qi$ (query interests) on the data plane of CCN.",
"The grammar definition of the query language can be found in apndix:grammar.",
"We aim for three main design goals for the query language and the parser: distinguishing between pull and push based traffic, translating a query to an equivalent name prefix of the CCN architecture, and supporting conventional relational algebraic operators and being extensible such that additional operators can be integrated with minimum changes.",
"This is to ensure easy integration of existing and new IoT applications.",
"We provide the definition of INetCEP query language in Section REF and the parser in Section REF .",
"Query Language Each operator in a query behaves differently based on the input source type, i.e., consumer- and producer-initiated interaction, which is done based on the reception of a Data packet or a Data Stream packet, respectively.",
"The Data packet is processed and returned as a data object, as conventionally done in the CCN architecture.",
"For instance, a Join operator placed on broker $C$ can join two data objects, $<lat1, long1>$ with name prefix $/node/A$ and $<lat2, long2>$ with $/node/B$ to produce $<lat1, long1, lat2, long2>$ with $/node/C$ .",
"Figure: Join of two data stream packets with window size of 2s in a CCN network.In contrast, a Data Stream is processed and transformed either into an output stream (another Data Stream packet), or can be transformed to derive a Data packet, containing, e.g., a boolean variable depending on the CEP query.",
"For instance, a join of two continuous data streams expressing location attributes of producer A with location attributes of producer B leads to the generation of a new data stream, as illustrated by broker C in fig:icnjoin.",
"We express the standard CEP operators as explained in sec:preliminaries using the INetCEP language belowWe have implemented all standard CEP operators defined in Section including SEQUENCE operator but only present the representative operators relevant for our use cases.. call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries WINDOW(GPS_S1, 4s) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries FILTER(WINDOW(GPS_S1, 4s),'latitude'<50) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries JOIN( FILTER(WINDOW(GPS_S1, 4s), 'latitude'<50), FILTER(WINDOW(GPS_S2, 4s), 'latitude'<50), GPS_S1.'ts' = GPS_S2.",
"'ts' ) The stateful operators e.g., Window and JOIN must store the accumulated tuples in some form of a readily available storage.",
"For this, we make use of in-network cache, the CS, that readily provides data for the window operator.",
"This can be highly beneficial, e.g., in a dynamic environment where state migration is necessary.",
"The INetCEP query language provides an abstract, simple, and expressive external DSL that translates the CEP query to equivalent NFN lambda ($\\lambda $ ) expressions, as explained later in subsubsec:parser.",
"The INetCEP language abstracts over the complexity of lambda expressions (as seen in equation below), so that CEP developers can easily perform data plane query processing on an ICN network.",
"For instance, a general $\\lambda $ expression of a JOIN query defined in Query is given as follows.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries (call <no_of_params> /node/nodeQuery/nfn_service_Join (call <no_of_params> /node/nodeQuery/nfn_service_Filter (call <no_of_params> /node/nodeQuery/nfn_service_Window 4s), 'latitude' < 50) (call <no_of_params> /node/nodeQuery/nfn_service_Filter (call <no_of_params> /node/nodeQuery/nfn_service_Window 4s), 'latitude' < 50) GPS_S1.'ts' = GPS_S2.",
"'ts') 1em [t] $query \\leftarrow $ the input CEP query $\\tau \\mathit {curList} \\leftarrow $ top down list of 3 $\\omega $ of tuple $\\tau $ $\\omega _{cur} \\leftarrow $ current operator $\\textsc {createOperatorGraph} (query)$ $\\tau \\mathit {curList} \\leftarrow $ getCurList($query$ ) parseQuery($\\tau \\mathit {curList})$ $\\textsc {parseQuery}(\\tau \\mathit {curList})$ $\\omega _{cur} \\leftarrow $getOperator$(\\tau \\mathit {curList})$ $\\mathit {nfnExp} \\leftarrow \\textsc {constructNFNQuery}(\\omega _{cur})$ $node \\leftarrow \\text{new } \\textsc {Node}(\\mathit {nfnExp})$ $size(\\tau \\mathit {curList}) == 1$ return $node$ $size(\\tau \\mathit {curList}) > 1$ parseQuery($\\tau \\mathit {curList}.left)$ parseQuery($\\tau \\mathit {curList}.right)$ return $node$ Recursively generating the operator graph 1em Here, <no_of_params> is the number of parameters in the $\\lambda $ expression, nfn_service_Join is the name of the operator (join operator) in the query, $4s$ is the window size and the remaining are the filter and join conditions, respectively.",
"Each operator is preceded by /node/nodeQuery/.. which represents the name of the node that is used to place the operator e.g., nodeA.",
"This is done at runtime by placing nodeQuery on the node name selected by the operator placement algorithm (cf.",
"Section REF ) to process the operator in a centralized or a distributed manner.",
"The translation of a CEP query to the above $\\lambda $ expression is discussed in the next section.",
"Query Parser In Algorithm REF , we express the INetCEP query parser as a recursive algorithm to map the query (e.g., Query ) to generate an equivalent NFN's $\\lambda $ expression (e.g., given above).",
"A CEP query is transformed into an operator graph $G$ (lines REF -REF ), which is a binary graph tree defined as a tuple $\\tau = (L, S, R)$ .",
"Here, $L$ and $R$ are binary trees or an empty set and $S$ is a singleton set, e.g., a single operator ($\\omega $ ).",
"The query parser starts parsing the query in a specific order, i.e., in a top-down fashion that marks the dependency of operators as well.",
"This implies each leaf operator is dependent on its parent.",
"Thus, the parser starts by iterating top down the binary tree starting from the root operator $\\omega _{cur}$ (line REF ), where $cur = root$ in the first step.",
"The traversal is performed in a depth-first pre-order manner (visit parent first, then left (L) and then right (R) children) (lines REF -REF ).",
"The workflow of the query parser algorithm for an example query of the form of Query , is illustrated in Fig.",
"REF and explained in the following.",
"We start by extracting the operator name ($\\omega $ ) by separating the parameters into a list.",
"We create a logical operator graph for each query by instantiating the operators and their data flow.",
"An operator is created only after the semantic checks on the operator are verified, e.g., if the $\\omega $ is valid, and/or it has valid parameters.",
"Once all the semantic checks are verified, we continue processing the query recursively as in Algorithm REF , by generating the corresponding NFN query and creating logical plan nodes for operators that are assigned to the broker network for processing the operator graph.",
"Figure: Query parser workflow based on Algorithm .",
"Operator Graph Processing This module processes the query either centrally or in a distributed way by mapping the operator graph plan nodes to the brokers ($B$ ) or INetCEP aware CCN routers after operator graph construction (cf.",
"subsec:language).",
"This is because central processing might not be sufficient for all the use cases, e.g., when the amount of resources required to process the queries increases with the number of operators and/or queries.",
"It is, therefore, necessary to distribute operators on multiple brokers.",
"In this way, the network is also not unreasonably loaded by the queries, while the network forwarding is not disturbed.",
"In fig:ceptimeline, we show the timeline of distributed query processing.",
"In case of central processing, only parsing and deployment is required.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; The broker that first receives the $qi$ from the consumer parses the CEP query and forms an operator graph (as described in subsubsec:parser).",
"This broker becomes the placement coordinator and coordinates the further actions taken for operator graph processing.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The coordinator builds the path where the operator graph is processed based on a criteria, e.g., minimum latency and selects other broker nodes for operator placement.",
"This is along the path from producer towards the consumer.",
"It is important because the Data packets are forwarded as well as processed along this path (in-network processing).",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 3; The coordinator recursively traverses the operator graph, while assigning the CEP operators or translated named functions (cf.",
"subsec:language) to the CCN routers.",
"The resulting $ce$ is encapsulated in a Data Stream or a Data packet, which is received at the root node of the operator graph and forwarded to the consumer.",
"In the following, we describe the collection of the monitoring information related to the CCN nodes by the placement coordinator (cf.",
"subsubsec:nwdiscovery) and the assignment of operators based on this information by the placement module (cf. subsubsec:placement).",
"In principle, the role of the coordinator is decided by the concrete operator placement algorithm.",
"INetCEP supports different means of coordination and hence operator placement algorithms.",
"Yet, for the algorithm defined below, the node where the first (or root) operator is deployed is the coordinator.",
"Network Discovery Service The placement coordinator fetches and maintains the monitoring information related to the node or network to place the operators on the right set of brokers or a single broker.",
"Since different CEP applications might be interested in optimizing distinct QoS metrics, the network discovery service can be updated accordingly to monitor the respective metric(s).",
"At the moment, we provide monitoring for the end-to-end delay which is important for our representative use cases.",
"The end-to-end delay is defined as the complete timeline as illustrated in fig:ceptimeline from query parsing to the delivery of the complex event.",
"Figure: Timeline of distributed query processing.The node and network information is retrieved as a Data packet with name prefix, e.g., /node/node_id/delay only on fetch basis (whenever required).",
"The placement coordinator subscribes for this information and hence maintains the global (centralized) or local (decentralized) knowledge on the network.",
"The cluster coordinators can be elected for decentralized placement as dictated in the placement literature [33].",
"By looking at the node and network characteristics, e.g., average delay, the placement coordinator selects one or more nodes for operator placement (defined next).",
"Operator Placement Module The operator placement module handles distributed query processing in case the processing requests, e.g., in terms of query interests, exceed the network or node capacity.",
"It works in conjunction with the placement coordinator, which is a primary component to provide operator placement decisions.",
"This module is responsible for building a path for operator placement while optimizing one or more QoS metrics (based on the knowledge from network discovery service), placing the plan node with CEP queries on the selected physical brokers and collaboratively processing the deployed query and delivering the complex event.",
"In principle, this module can be extended to support different QoS metrics, design characteristics and hence placement decisions.",
"Evaluation We evaluate the INetCEP architecture by answering two questions: EQ1 Is the INetCEP system extensible and expressive?",
"EQ2 How is the performance of INetCEP system?",
"To this end, we explain the evaluation setup in Section REF , EQ1 in Section REF and EQ2 in Section REF .",
"Evaluation Environment We selected the NFN architecture [42] to implement our solution, due to its built-in support of resolving named functions as so-called $\\lambda $ expressions on top of the ICN substrate.",
"However, a major difference to our architecture is that the communication plane in NFN is only consumer-initiated.",
"In contrast, we provide unified communication layer for co-existing consumer- and producer-initiated interactions, while doing CEP operations in the network.",
"As a consequence, we embedded CEP operators as named functions while leveraging NFN's abstract machine to resolve them.",
"NFN works together with CCN-lite [7], which is a lightweight implementation of CCNx and NDN protocol.",
"We have developed unified interfaces of our design on top of NFN (v0.1.0) and CCN-lite (v0.3.0) for the Linux platform [8].",
"We have enhanced the NDN protocol implementation in the CCN-lite and the NFN architecture by: including the additional packet types and their handling, as described in subsec:unified, implementing the extensible general CEP query language, parser, and CEP operators as NFN services, as described in Section REF and implementing a network discovery service with modifications in both CCN and NFN, and operator placement as an NFN service (cf.",
"Section REF ).",
"We evaluated our implementation using the CCN-lite emulator on two topologies: centralized (cf.",
"fig:topoa) and distributed (cf. fig:topob).",
"Each node in our topology is an Ubuntu 16.04 virtual machine (VM) with 8 GiB of memory.",
"Here, each VM (node) is a CCN-NFN relay, which hosts a NFN compute server encapsulating the CEP operator logic.",
"For running the experiments, we first created a CCN network topology as illustrated in fig:topo.",
"Second, we deployed the INetCEP architecture that works on the NFN compute server, the CCN-NFN relay, as well as on the links.",
"Here, as intended, the nodes communicate using the NDN protocol instead of IP.",
"In the centralized topology, we have two producers, a single broker that processes the query and one consumer.",
"In the distributed topology, we have one consumer, two producers and six brokers, as shown in the figure.",
"The data structures CS and PIT are utilized as explained in the previous sections (cf. subsec:unified).",
"We use Queries - (cf.",
"subsec:language) for our evaluation with the DEBS grand challenge 2014 smart home dataset and the disaster field dataset.",
"The dataset is explained in subsec:eq1.",
"Figure: Topology for evaluation.Evaluation Question I: Extensibility To show the extensibility and expressiveness of our approach, we extended the INetCEP architecture for the two representative IoT use cases that we introduced in Section , with a heat map query and a load prediction query.",
"We extended the INetCEP query language and CEP operators to include the heat map [29] and prediction operators [34] by making a few additions to our implementation in our extensible query language and parser.",
"We used real world datasets to evaluate the queries: the 2014 DEBS grand challenge and a disaster field dataset.",
"Dataset 1.",
"For the heat map query, we use a dataset [12] of a field test mimicking a post-disaster situation.",
"The field test mimics two fictive events, a lightning strike and a hazardous substance release from a chemical plant, which resulted in a stressful situation.",
"The collected dataset consists of sensor data, e.g., location coordinates.",
"It was collected from smartphones provided to the participants.",
"Each sensor data stream has a schema specifying the name of the attributes, e.g., the GPS data stream has the following schema: $<ts, s\\_id, latitude, longitude, altitude, accuracy, distance, speed>$ Query.",
"We use the $latitude$ and $longitude$ attributes of this schema to generate the heat map distribution of the survivors from the disaster field test.",
"A typical heat map application joins the GPS data stream from a given set of survivors, derives the area by finding minimum and maximum latitude and longitude values, and visualizes the heat map distribution of the location of the survivors in this area.",
"For simplicity, we consider a data stream from two survivors, as shown in the operator graph in fig:bothqueriesa.",
"Here, $p_1$ and $p_2$ are the producers or GPS sensors, $\\omega _{\\bowtie }$ is the join operator, and $\\omega _{h}$ is the heat map generation operator algorithm [29].",
"This is easily possible using the INetCEP language and parser implementation that follows an Abstract Factory design pattern.",
"First, we included the algorithm for heat map generation, which is $\\tilde{20}$ LOC.",
"Second, we extended the language implementation to include the user defined operator by adding $\\tilde{20}$ LOC.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries HEATMAP( 'cell_size', 'area', JOIN(WINDOW(GPS_S1, 1m), WINDOW(GPS_S2, 1m)) GPS_S1.'ts' = GPS_S2.",
"'ts') ) Figure: Two applications for evaluation (a) a heat map query for post-disaster relief and (b) an energy load forecasting query for smart homes.Dataset 2.",
"The second dataset comes from the 2014 DEBS grand challenge [4] scenario focused on solving a short-term load forecasting problem in a smart grid.",
"The data for the challenge is based on real-world profiles collected from smart home installations.",
"The dataset captured load measurements from unique smart plugs with the following schema: $<ts, id, value, property, plug\\_id, household\\_id, house\\_id>$ Query.",
"We apply an existing solution [34] to perform prediction by extending the INetCEP architecture for two smart plugs.",
"The corresponding operator graph for such a prediction is illustrated in fig:bothqueriesb and listed below.",
"Here, $p_3$ and $p_4$ are the producers or smart plugs, $\\omega _{\\bowtie }$ is a join operator and $\\omega _{pr}$ is a prediction operator based on the algorithm [34].",
"In the first query, we notify the consumer about the predictions for five minutes into the future, while in the second query we notify only if the predictions of load are above a threshold.",
"Similarly to the heat map application, we implemented a prediction algorithm by adding $\\tilde{50}$ LOC and the language implementation with $\\tilde{20}$ LOC.",
"The detailed description of the algorithms for the respective use cases in order to achieve the extensibility is presented in apndix:extensibility.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries JOIN( PREDICT(5m, WINDOW(PLUG_S1, 1m)), PREDICT(5m, WINDOW(PLUG_S2, 1m)) PLUG_S1.'ts' = PLUG_S2.",
"'ts' ) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries FILTER(JOIN( PREDICT(5m, WINDOW(PLUG_S1, 1m)), PREDICT(5m, WINDOW(PLUG_S2, 1m)) PLUG_S1.'ts' = PLUG_S2.",
"'ts' ), 'load'>20) Evaluation Question II: Performance We evaluated the performance of the INetCEP architecture on standard CEP queries including the Queries -.",
"Additionally, we evaluated the heat map and prediction queries (Queries -).",
"Our aim is to understand the performance of: the query parser on increasing the number of nested operators in operator graph, the operator graph module for different kind of queries, and the operator placement module on increasing number of queries.",
"Figure: Performance of the query parser depending on increasing the number of nested operators.Query Parsing In the query parser design (cf.",
"Section REF ), we performed a complexity analysis of the Algorithm REF .",
"In this section, we verify the analysis experimentally by increasing the number of operators in the operator graph while using the centralized topology shown in fig:topoa.",
"Figure: End-to-end and communication delay observed in Queries 1 to 6.In fig:parser, we show the performance of query parser algorithm in terms of runtime (in ms) for Query or Filter operators and Query or Join operators.",
"We show in a line plot with a confidence interval of 95% for 20 runs of the emulation that the two queries scales reasonably and can be processed in a few milliseconds.",
"Centralized Query Processing We measured the end-to-end delay in processing the six queries defined above using the operator graph module (cf.",
"§ REF ).",
"In fig:evaldelay, we show the results as box plots with a confidence interval of 95% for 10 runs of the emulation.",
"For Query -, the total delay perceived is less than a few milliseconds.",
"It increased for the new queries Query -, where we introduced prediction and heat map operators, primarily due to increased consumption of data and the computational complexity of the algorithms for prediction [34] and heat map [29], respectively.",
"We further show the distribution of the mean end-to-end delay in tab:latency to understand the primary reason for the delay.",
"We listed the time spent in each of the modules of the INetCEP architecture, namely, operator graph creation, placement of operators, and communication of events.",
"The values shown in the table are the mean of the values observed for 10 executions.",
"For basic CEP queries -, the major portion of time ($\\tilde{98}\\%$ ) is spent in communication.",
"This can be explained, as also confirmed in other works [20], by the limitations of the CCNx implementation of the NDN protocol.",
"Hence, we observe the communication delays for all the queries in fig:commq1toq6, where it takes up to 500 ms for delivering results of complex queries like prediction and heatmap.",
"Table: The division of mean end-to-end delay in ms for of operator graph creation, placement of operators, and (communication) delay for centralized placement (see fig:q1toq6).Figure: End-to-end delay observed for Query 3 on incrementally increasing the query workload.",
"Operator Placement To understand the behavior of the operator placement module, we utilize the distributed topology of seven VMs, as illustrated in fig:topob, to place operators based on the information collected by the network discovery service.",
"To take full advantage of distributed CEP, we increased the query load of Query starting from node 1 to 5.",
"For example, the first 10 queries were initialized at the first broker node followed by the next 10 queries at the second node, and so on.",
"Hence, the number of consumers also increased in the network with the query load.",
"In fig:multipleq3, we see that the total delay in retrieving the complex event increased incrementally with the query load, which is reasonable given the static network size (7 nodes) and the resources.",
"However, the queries were distributed evenly using the operator placement algorithm at distinct nodes.",
"To summarize, we evaluated the performance of INetCEP on two topologies: centralized and distributed, using two IoT applications and six different queries.",
"Our evaluation shows that IoT applications can be integrated seamlessly using the INetCEP architecture, CEP queries can be formulated and can be extended for more use cases, and simple CEP queries can be processed in milliseconds.",
"Related Work We now review the state-of-the-art ICN architectures in terms of their support for consumer- and producer-initiated interaction patterns and existing INP architectures in subsec:icnarch, and CEP and networking architectures in subsec:eparch.",
"ICN Architectures Interaction Patterns in ICNs: In Figure REF , we highlight the main ICN architectures NDN [45], NFN [42], DONA [28], PURSUIT [2], PSIRP [1], in terms of their support of a unified communication layer as presented in our work.",
"However, only those appearing in the green box, namely CONVERGENCE [35], GreenICN [3], and Carzaniga et al.",
"[19], provide support for both kind of invocation mechanisms.",
"The CONVERGENCE system combines the publish/subscribe interaction paradigm on top of an information-centric network layer.",
"In contrast, we provide a unified interface such that pull and push based interaction patterns could co-exist in a single network layer while performing in-network computations.",
"GreenICN is an ICN architecture for post-disaster scenarios by combining NDN (pull-based) with COPSS [20] (push-based).",
"However, GreenICN introduces additional data structures, e.g., a subscription table (ST), while we provide this combination using the existing data structures of ICN.",
"Furthermore, it is not clear if GreenICN can function as a whole in a single ICN architecture [41].",
"Carzaniga et al.",
"propose a unified network interface similar to our work, however, the authors only propose a preliminary design of their approach [19] without implementing it in an ICN architecture, and subsequently focus on routing decisions [18] rather than on distributed processing.",
"INP and IoT Architectures in ICNs: Authors in work [43], [38] propose an approach to distribute computational tasks in the network by extending the NFN architecture similar to our work.",
"However, the authors do not deal with the lack of abstractions required for processing continuous data stream.",
"In contrast, we propose a unified communication layer to support CEP over ICN and an extensible query grammar and parser that opens a wide range of operators.",
"Krol et al.",
"[31] propose NFaaS based on unikernels, which is a container based virtualization approach to encapsulate named functions placed on NDN nodes.",
"However, they do not provide support for stateful functions, while IoT functions can be stateful, e.g., involving time windows, which is supported by our architecture.",
"Ahmed et al.",
"[10] propose a smart home approach using NDN and support both push and pull interaction patterns similar to our work.",
"However, in their architecture they only support retrieving raw data, e.g., humidity sensor readings, but not meaningful events as we do.",
"Shang et al.",
"[39] propose a publish/subscribe based approach for modern building management systems (BMS) in NDN.",
"However, the authors build on standard consumer-initiated interaction, as described in Limitation 1 (cf. subsec:decisions).",
"Publish-subscribe deployment for NDN in the IoT scenarios has been discussed in previous works [23], [24].",
"These works confirm the need of integrating producer-initiated interaction in NDN, however, do not provide a unified layer for both interaction patterns as we do.",
"CEP and Networking Architectures CEP Architectures: Several event processing architectures exist, ranging from, e.g., the open source Apache Flink [16] to Twitter's Heron [32] and Google's Millwheel [11].",
"One possibility is to interface one of them with an ICN architecture.",
"Initial work implemented Hadoop on NDN [22] for datacenter applications.",
"However, this requires changing the network model to push in contrast to our work, which would limit the support for a wide range of applications, as discussed above.",
"Networking Architectures: Another emerging network architecture is Software-Defined Networking (SDN) [30], which is gradually being deployed, e.g., in Google's data centers.",
"It allows network managers to program the control plane to support efficient traffic monitoring and engineering.",
"The SDN architecture is complementary to our work, since SDN empowers the control plane, while ICN upgrades the data plane of the current Internet architecture.",
"Data Plane and Query Languages: The literature discusses many CEP query languages [16].",
"The novelty of the proposed query language is to allow for a mapping of operations to ICN's data plane.",
"Alternative designs builds on P4 [14] in the context of SDN.",
"Initial work on programming ICN with P4 [40] faced several difficulties due to lack of key language features and the strong coupling of the language to SDN's data plane model.",
"Discussion In this section, we discuss important future challenges that could be interesting to provide more sophisticated networking, reliability and optimization mechanisms in the INetCEP architecture.",
"Sophisticated Flow and Congestion Control: In CCN, the PIT table ensures the flow balance since one Data packet is sent for each Interest packet.",
"The Data Stream packets of INetCEP could disturb the rule of flow balance since the producer could overflow the buffer on the broker side.",
"For this, INetCEP implements a simple flow control mechanism where we restrict the receiver (consumer/broker) to specify maximum outstanding messages at a time.",
"However, since the forwarding logic of Data Stream packets is similar to IP multicast, existing sophisticated multicast congestion control solutions like TCP-Friendly Multicast Congestion Control [44] and similar can provide sophisticated flow and congestion control.",
"Reliability: The brokers or consumers could miss packets when the available bandwidth and resources at their end is lower than the sending rate.",
"This makes the presence of a module of reliability relevant, which can be catered by extending our work with existing reliable CEP solutions [27] or by looking into equivalent IP solutions such as Scalable Reliable Multicast [21].",
"Query Optimization: In INetCEP we provide a placement module that maps the operator graph to in-network elements of CCN.",
"Another complementary direction could be to generate an optimal operator graph, e.g., based on operator selectivity or even partition operator graph by performing query optimization [15].",
"Optimizing QoS: In INetCEP we provide a programming abstraction for the developers to write CEP queries over ICN data plane substrate.",
"In addition, the placement module can be extended to look into further decentralized solutions and even other QoS metrics [17] like throughput, availability, etc.",
"Conclusion In this paper, we proposed the INetCEP architecture that implements a unified communication layer for co-existing consumer-initiated and producer-initiated interaction patterns.",
"We studied important design challenges to come up with our design of a unified communication layer.",
"In the unified layer, both consumer- and producer-initiated interaction patterns can co-exist in a single ICN architecture.",
"In this way, a wide range of IoT applications are supported.",
"With the proposed query language, we can express interest in aggregated data that is resolved and processed in a distributed manner in the network.",
"In our evaluation, we demonstrated in the context of two IoT case studies that our approach is highly extensible.",
"The performance evaluation showed that queries are efficiently parsed and deployed, which yields - thanks to the in-network deployment - a low end-to-end delay, e.g., simple queries induce only few milliseconds of overall delay.",
"Interesting research directions for future work are: (i) enhancing the performance of query processing by using parallelization, (ii) porting CCNx implementation on real hardware to accomplish low communication delays, and (iii) developing CEP compliant caching strategies.",
"Acknowledgements This work has been co-funded by the German Research Foundation (DFG) as part of the project C2, A3 and C5 within the Collaborative Research Center (CRC) SFB 1053 – MAKI.",
"Appendix In this section, we define our query grammar in apndix:grammar, the implementation details of our extensible design in apndix:extensibility and the algorithms used for extending the system with load forecasting and a heat map application in apndix:applications.",
"Query Grammar Definition 10.1 A grammar consists of four components: A set of terminals or tokens.",
"Terminals are the symbols that occur in a language.",
"A set of non-terminals or syntactic variables.",
"Each of them represent a set of strings.",
"We define them the way we want to use them.",
"A set of production rules that define which non-terminals can be replaced by which terminals, or non-terminals or a combination of both.",
"Here, the terminal is the head of the left side of the production, and the replacement is the body or right side of the production.",
"For example, head $\\rightarrow $ body One of the non-terminals is designated as start symbol for each production.",
"Following the above definition REF and Chomsky-Hierarchy [25], we selected a type 2 grammar or a context-free grammar for CEP over ICN language since a query may consist of multiple subqueries (or operators), which can be expressed (out of many possible ways) using parenthesis \"$()$ \".",
"Table: NO_CAPTIONThe context-free grammar allows us to combine the production rules.",
"The head of each production consist only of one non-terminal and the bodies are not limited by only one terminal and/or one non-terminal.",
"This is because the language needs to embed operators in parenthesis and with a regular grammar we cannot have an arbitrary number of parentheses.",
"We need a way to memorize each parenthesis and this ability is given by context-free grammars.",
"We define an initial INetCEP language grammar and represent it using BNF (Backus-Naur form) in tab:grammar, considering the aforementioned design decisions.",
"We use regular expressions represented as $\\mathit {REG} (\\ldots )$ , where $(\\ldots )$ can be literals $\\mathit {[a-z]}$ and $\\mathit {[A-Z]}$ in lower and upper case, respectively, and numbers $\\mathit {[0-9]}$ .",
"The plus ($+$ ) sign in $\\mathit {REG ([a-z]+)}$ means that at least one lowercase letter has to appear, while a 1 in $\\mathit {REG ([a-z]\\lbrace 1\\rbrace )}$ means that exactly one lowercase letter has to appear.",
"The relational operators given by $\\mathit {comparison}$ define a binary relation between two entities, e.g., two column names of a schema, a column name to a number or a column index to a number.",
"Extensibility To make our query language extensible, we follow a well-known Abstract Factory design pattern from object-oriented programming for our operator definition as illustrated in fig:parserUml.",
"Algorithm REF is the starting point of our operator graph creation.",
"This is implemented in the OperatorTree class.",
"Each operator inherits the abstract class OperatorA which defines the interpret (parseQuery in Algorithm) function as seen in the figure.",
"The checkParameters verifies the correctness of the parameters.",
"Figure: Operator Definition in UMLIf a new operator is to be included, it will override the existing methods of the abstract class and the parameters correctness has to be defined.",
"This allows minimal changes in the implementation for each new application developed using our system.",
"Applications Short term Load Forecasting For the DEBS Grand Challenge in 2014, Martin et al.",
"derived requirements for predicting future energy consumption of a plug [34].",
"We formulate our requirements with respect to our INetCEP system: To meet the goal of making an estimation on future energy consumption, it is necessary to use historical data as a reference.",
"In order to run on a machine with limited resources, the prediction algorithm needs to be lightweight in computation power and storage.",
"Figure: Flow chart explaining the prediction algorithmThe formula for predicting future load is given by the publishers of the DEBS Grand Challenge and is as follows: $ predicted\\_load\\left( s_{i+2} \\right) = \\left( avgLoad\\left( s_{ i } \\right) + median\\left( \\left\\lbrace avgLoads\\left( s_{ j } \\right) \\right\\rbrace \\right) \\right) $ , Here, $i$ is the current timestamp, $s_{i}$ the currently recorded values at time $i$ and $s_{j}$ the past values at a corresponding time $j=i+2$ .",
"The load two steps in the future is therefore made up of the current average electricity consumption and the average electricity consumption from the past.",
"In fig:prediction1operatorflow, we represent the flow of the prediction algorithm as per the requirements defined above and Query .",
"For each time window of 1 minute, we first determine if it is the time for next prediction, which is provided as an input in the query.",
"If the time has not come yet, a value for prediction (average load) is calculated and stored so that it can be used for the equation defined above.",
"Inversely, if it is the time to make a prediction, a prediction tuple of the following form is emitted.",
"$<ts, plug\\_id; household\\_id; house\\_id; predicted\\_load>$ Here, $ts$ is the timestamp of the prediction, $plug\\_id$ identifies a socket in a household, $household\\_id$ identifies a household within a house and $house\\_id$ identifies a house and $predicted\\_load$ is the prediction as specified in the equation above.",
"Heat Map Algorithm REF describes the heat map creation and visualization for the location updates from survivors of the disaster field test used in this work based on [29].",
"[h] $loc$ : Window of location $<lat, long>$ tuple of the survivor $Lat_{min}$ : The minimum latitude value $Lat_{max}$ : The maximum latitude value $Long_{min}$ : The minimum longitude value $Long_{min}$ : The maximum longitude value $HC$ : The number of horizontal cells needed to map the values $VC$ : The number of vertical cells needed to map the values $cell\\_size$ : The granularity $Grid$ : A two dimensional array $HC$ = $\\lfloor {\\frac{Long_{max}- Long_{min}}{cell\\_size}}\\rfloor $ $VC$ = $\\lfloor {\\frac{Lat_{max}- Lat_{min}}{cell\\_size}}\\rfloor $ line in $S_D$ absLatVal = $loc$ [lat] - $Lat_{min}$ absLongVal = $loc$ [long] - $Long_{min}$ $Grid$ [$ \\lfloor {\\frac{absLatVal}{cell\\_size}}\\rfloor $ ][$ \\lfloor {\\frac{absLongVal}{cell\\_size}}\\rfloor $ ] += 1 $Grid$ Algorithm for the heat map operator In line 1, we calculate the number of horizontal cells required for the desired heat map.",
"For this we divide the difference between the maximum and minimum longitude by the desired cell_size, which indicates how large and finely meshed the resulting heat map should be and then round this value down to the next smaller number (given by floor function).",
"In line 2, similarly we compute the vertical cells.",
"For each of these location tuples in the current window, the first absolute latitude and longitude values are computed in line 4 and 5, respectively.",
"By dividing these values by the cell_size, we obtain the corresponding position in the heat map in line 6."
],
[
"Problem Space",
"In this section, we first discuss the limitations of using straightforward solutions to motivate the need of our architecture (cf.",
"subsec:decisions) and then we explain the system model of INetCEP (cf.",
"sec:systemmodel)."
],
[
"Design Challenges",
"We discuss the limitations of using straightforward solutions, e.g., standard consumer-initiated, producer-initiated communication, or long lived Interest packets for the purpose of supporting a wide variety of applications as also pointed here [19].",
"Then, we study limitations of using Interest packets to represent queries.",
"Finally, we present limitations in performing operator graph processing at the consumer end.",
"A straightforward solution to support CEP is to use the standard consumer initiated communication of the CCN architecture.",
"We illustrate the problems using this naive solution in Figure REF .",
"One way is that the consumers continuously issue [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; a query at regular intervals, and [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; the producer replies with the event of interest in a data (notification) packet.",
"However, there are multiple problems with this solution, as indicated in fig:lim1.",
"Figure: Limitation 3: Three way message and two kind of Interest packets required.Limitation 1: The continuous polling of a query by consumers generates a lot of overhead traffic and network state in the form of pending interests for only a few meaningful data packets.",
"Each time a data packet is received, the pending interest is removed from PIT since it is satisfied (represented as strikethrough in the figure).",
"However, depending on the query interval a new entry is again created in the PIT for each query.",
"Also, the interval length of issuing query might determine the maximum latency at which the notification is delivered to the consumer, which might not be acceptable for latency sensitive applications, e.g., autonomous cars.",
"Limitation 2 is to deal with the stale data in the cache or CS, as represented in fig:lim2.",
"The consumers in a CEP application often need real-time updates on the latest data.",
"For this reason, the query needs to be updated each time, otherwise it will retrieve the last cached Data packet which is stale or obsolete in time.",
"For instance, in fig:lim2, the broker still sends the old data to the consumer while the producer has generated a new data item for the query.",
"In addition, there should be a mechanism to expire the Data packet at the right time, perhaps, immediately for the real-time updates.",
"Solutions like appending sequence numbers (similar to TCP) to the Data packet can be applied.",
"However, this will require additional synchronization mechanisms.",
"Alternatively, another possibility is to support just producer-initiated transmission while using CCN primitives (cf.",
"fig:lim3).",
"Although this is a viable option for some applications [20], it results in a three-way message exchange of what amounts to a one-way message.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; The producer sends an asynchronous Interest packet that is not intended to fetch a Data packet from the network but to announce the data name and the callback from the consumer.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The consumer then shows interest in the data name, which is [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 3; fulfilled by a Data packet from the producer.",
"Figure: INetCEP communication model supports pull-based communication and fetches latest Data packet.Limitation 3: Besides the overhead generated by a 3-way message, this design has other major issues.",
"The interests leave an in-network state in the PIT of CCN nodes such that data can be fetched by the same path.",
"CCN typically performs a unicast of the Interest packet, so that the Data packet can follow the same path to the consumer.",
"Such an application has to support two kinds of Interest packets: a packet that is not supposed to fetch data, and a packet that is supposed to fetch data.",
"Limitation 4: Long-lived Interest packets can be used in place of a query, but this also has multiple side-effects.",
"Similar to multiple interests, long-lived Interest packets will also result in large in-network state (cf.",
"fig:lim1).",
"In addition, the long-lived Interest packets will have to deal with stale data as explained earlier in Limitation 2 (cf.",
"fig:lim2).",
"To solve the aforementioned issues, we propose to have both consumer-initiated and producer-initiated interaction patterns coexisting under a unified CCN communication layer.",
"A CCN architecture is unable to achieve this using existing packets and data structures as we saw above.",
"Hence, we propose additional packets as a part of the communication model and handle them while processing CEP queries in the network as defined in Section REF .",
"Limitation 5: CCN/NDN assumes a hierarchical naming scheme to address named data, e.g., $/node/nodeA/temperature$ , in order to fetch data objects e.g., $35^\\circ C$ , from the producers.",
"A simple way to specify CEP operations over data would be to represent this using the standard naming scheme, e.g., a $min$ operator as $/node/nodeA/min/temperature$ .",
"However, there are problems with this approach: the name cannot be used to correlate data from multiple producers, this would mean the processing is performed always at the consumer, which is inefficient and this is not extensible and not expressive, since adding more operators would mean appending them in the naming scheme, which reduces readability.",
"Hence, we need more than just CCN Interest packets that encapsulate name prefixes as stated above to represent CEP queries over a CCN network.",
"We propose an expressive query language that can correlate data from multiple producers and an efficient query parser to execute queries in the network (cf.",
"Section REF ).",
"Limitation 6: The query specified by the consumers must be processed within the CCN network.",
"The CCN resolution engine can resolve only Interest packets to retrieve Data packets based on the matching name prefix, but it cannot express query.",
"A naive way to deal with this is to process the query at the consumer.",
"However, this would overload the network with all the unnecessary data that could have been filtered on the way to the consumer and overload the consumer with all the processing.",
"This might result in a single node of failure, when the data becomes very big.",
"Thus, the processing needs to be performed in the CCN network, e.g., at the broker while being transmitted to the consumer.",
"We provide this in two ways: (i) centralized query processing, where the entire query is processed at a single broker and (ii) distributed query processing, where the query operators are assigned to in-network nodes for processing (cf.",
"Section REF ).",
"Figure: INetCEP communication model supports push-based communication without creating endless PIT entries.INetCEP System Model Every CCN node can act either as a producer, a consumer or a broker.",
"Here, a broker is an in-network element, i.e., an INetCEP aware CCN router, while a producer or consumer is an end device, e.g., a sensor or a mobile device.",
"On the one hand, [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; consumers can request a specific data item using an Interest packet, where broker(s) forward(s) the request received by consumers to support anonymous request-reply communication, as illustrated in fig:inetceppull.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The producer replies with a data object contained in a Data packet.",
"On the other hand, broker(s) process(es) the unbounded and ordered data streams generated by producers to provide event-based communication, which happens as illustrated in fig:inetceppush and is explained below.",
"Table: Description of differences in traditional ICN vs INetCEP architecture (\"-\" means no support).A producer multicasts the data stream (Data Stream packet) towards the broker network, which disseminates the stream all over the network (push).",
"The Data Stream packet is forwarded to further brokers in the network if there are consumers downstream for query interest ($qi$ ).",
"An efficient event dissemination can be achieved by using routing algorithms, e.g., defined in this work [13], by looking at the similarity score of the $qi$ .",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; A consumer issues a query by sending an Add Query Interest packet comprising $qi$ (top fig:inetceppush).",
"Each $qi$ encapsulates a CEP query $q$ that is processed by interconnected brokers in $B$ forming a broker network.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The $qi$ is stored in the PIT of the receiving broker until a [baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 3; Remove Query Interest packet is received that triggers the removal of $qi$ from the PIT (bottom fig:inetceppush).",
"Unlike a conventional CEP system, the event-based communication in the INetCEP happens in the underlay CCN network.",
"The query $q$ induces a directed acyclic operator graph $G$ , where a vertex is an operator $\\omega \\in \\Omega $ and an edge represents the data flow of the data stream $D$ .",
"Each operator $\\omega $ dictates a processing logic $f_\\omega $ .",
"We explain the constituents: the communication model, the query model, and the operator graph model below.",
"Communication Model.",
"We provide five types of packets to support both kinds of interaction patterns.",
"The Interest (request) packet is equivalent to CCN's Interest packet that is used by the consumer to specify interest in any named data or named function.",
"The Data (reply) packet is a CCN data packet that satisfies an Interest.",
"It also encapsulates the complex event ($ce$ ) as described later.",
"The Data Stream packet represents a data stream of the form $<ts, a_1, \\ldots , a_m>$ .",
"Here, $ts$ is the time at which a tuple is generated and $a_i$ are the attributes of the tuple.",
"The Add Query Interest packet represents the event of interest in the form of a CEP query $q$ .",
"The Remove Query Interest packet represents the CEP query that must to be removed for the respective consumer (so that it no longer receives complex events).",
"The CCN forwarding engine (data plane) is enhanced to handle these packets, as below.",
"Query Model.",
"The INetCEP query language is based on two main design goals: it should deal with both pull (data from relations) and push (time series data streams) kind of traffic, and support standard CEP operators (as identified in Section ) over the CCN data plane.",
"Thus, a query ($q$ ) must be able to capture time series data streams as well as relations of the form $<ts, a_1, .., a_m>$ and define an operator $\\omega $ with processing logic $f_\\omega $ in a way that it is extensible.",
"Operator Graph Model.",
"The operator graph $G$ is a directed acyclic graph of plan nodes.",
"The vertex of the graph is a plan node that encapsulates a single operator ($\\omega $ ), while the links between plan nodes represents the data flow from the bottom of the graph to the top.",
"The operator graph can be processed centrally or collaboratively in a distributed manner by mapping it to the underlay CCN network.",
"In distributed CEP, typically, an operator placement mechanism defines a mapping of an operator graph $G$ onto a set of brokers, to collaboratively process the query.",
"The placement needs to be coordinated with the forwarding decisions for efficient processing over the CCN data plane.",
"INetCEP architecture We identify the following three broad requirements for the INetCEP architecture from our discussion in the previous section.",
"R1 A unified communication layer supporting both producer and consumer initiated communication (cf.",
"Section REF ).",
"R2 An expressive CEP query language for specifying the event of interest (cf.",
"Section REF ).",
"R3 Resolution of CEP queries by efficient and scalable in-network query processing (cf.",
"Section REF ).",
"We address each of these requirements below.",
"Unified Communication Layer In this section, we explain the extension of the CCN data plane to enable CEP.",
"In our approach, each CCN node $n \\in N$ maintains a Content Store or cache (CS), a Pending Interest Table (PIT), a Forwarding Information Base (FIB) and a CEP engine.",
"In the following, we explain the function of these main building blocks and the data plane handling.",
"Subsequently, we detail on the newly introduced packets Add Query Interest, Remove Query Interest and Data Stream and simultaneously the solution to the limitations identified in subsec:decisions.",
"Figure: High level view of packet handling in INetCEP architecture.Node Components The CS stores all the data objects associated with the CCN Interest, additionally, the time-stamped data objects associated with the query interests ($qi$ ).",
"The data object returns a value associated with the $qi$ , e.g., if the $qi$ is the sum of victims in a disaster location, then the data object contains value 20.",
"Hence, if multiple consumers are interested in the same $qi$ , the query is not reprocessed but the data is fetched directly from the CS.",
"To deal with Limitation 2 of stale cache entries for $qi$ (cf.",
"Section REF ), we store logical timestamps along with cache entries while storing $qi$ in the CS.",
"Hence, always up to-date data objects are stored, while old entries are discarded (cf.",
"fig:inetceppull).",
"The PIT stores the pending $qi$ so that the complex event ($ce$ ) could follow the path, i.e., the in-network state created in PIT to the consumer.",
"The $\\mathit {face}$ information is also stored in the PIT entry to keep track of $qi$ 's consumers.",
"In contrast to consumer-initiated interaction, the $ce$ must be notified to the interested consumers as and when detected in real-time.",
"Thus, as new data is received, the query interests in the PIT are re-evaluated as explained later in Algorithm REF .",
"In the former, we are referring to only continuous Data Stream packet since we also support fetching data from relations (this is handled conventionally using a CCN Interest packet as explained in Section ).",
"The reasons why we distinguish between Add Query Interest ($qi$ ) and CCN Interest packets are: (i) $qi$ is invoked on receipt of Add Query Interest as well as the Data Stream packet, (ii) removal of the PIT entry is not based on a Data packet retrieval but on the reception of the Remove Query Interest packet and (iii) $qi$ retrieves Data packets asynchronously.",
"In summary, we deal with Limitation 3 and Limitation 4 by asynchronously handling the $qi$ instead by 3-way message exchange and efficiently managing PIT entries, respectively.",
"We deal with Limitation 1 by storing $qi$ in PIT and asynchronously delivering event notifications to the consumers.",
"The FIB table gets populated as the producer multicasts to the broker network leaving a trail to the data source.",
"In this way the data processing is performed efficiently along the path from producer and consumer.",
"Finally, the CEP engine holds the processing logic $f_\\omega $ for each operator $\\omega $ and is responsible for parsing, processing, and returning the result to the next node towards consumer (cf.",
"Sections REF and REF ).",
"1em $\\mathit {CS} \\leftarrow \\text{content store of current node}$ $\\mathit {PIT} \\leftarrow \\text{pending interest table of current node}$ $\\mathit {FIB} \\leftarrow \\text{forwarding information base}$ $\\mathit {qi} \\leftarrow \\text{requested query interest}$ $\\mathit {result} \\leftarrow \\text{query result}$ $\\mathit {facelist} \\leftarrow \\text{list of all faces in PIT}$ $\\texttt {DataStream} \\leftarrow \\text{\\texttt {Data Stream} packet}$ $\\mathit {data} \\leftarrow \\text{data that resolves the qi}$ $\\textsc {AddQueryInterest}(\\mathit {qi})$ $qi$ is found in $\\mathit {CS.\\textsc {lookup}(qi)}$ $ \\mathit {data} \\leftarrow \\mathit {CS}.\\textsc {fetchContent}(\\mathit {qi}) $ $ \\text{return } \\mathit {data}$ (Discard AddQueryInterest) $qi \\text{ found in } \\mathit {PIT.\\textsc {lookup}(qi)} $ $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\texttt {AddQueryInterest})$ $qi \\text{ found in } \\mathit {FIB.\\textsc {lookup}(qi)} $ $\\mathit {\\textsc {CreateOperatorGraph}(\\mathit {qi})}$ (Refer Algorithm REF ) Forward AddQueryInterest (Discard AddQueryInterest) $\\textsc {ProcessDataStream}(\\texttt {DataStream})$ $\\mathit {qi} \\in PIT$ DataStream satisfies $qi$ $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\texttt {DataStream})$ Forward DataStream $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\mathit {packet})$ $\\mathit {packet} == \\texttt {DataStream} \\text{ and } \\mathit {packet}.\\mathit {ts} > \\mathit {qi.ts}$ $\\mathit {\\textsc {CreateOperatorGraph}(\\mathit {qi})}$ (Refer Algorithm REF ) Forward $\\mathit {packet}$ $\\mathit {facelist} \\leftarrow \\mathit {PIT}.\\textsc {getFaces}(\\mathit {qi}) $ $qi.\\mathit {face}$ is not found in $\\mathit {facelist}$ $ \\mathit {PIT}.\\textsc {addFace}(\\mathit {qi}) $ Discard $\\mathit {packet}$ -15pt Add Query Interest and Data Stream packet handling.",
"Data Plane Handling In Algorithm REF (lines REF -REF ) and fig:overall, we define the handling of Add Query Interest and Data Stream packets at the broker end in a CCN network.",
"The processing of $qi$ stored in PIT is triggered based on the receipt of these two packets as follows: when an Add Query Interest packet is received at a broker (line REF ) and due to the continuous arrival of new Data Stream packets (line REF ).",
"This is in contrast to the PIT entry of CCN Interest, which is checked only on the receipt of a new Interest packet.",
"When an Add Query Interest arrives, the broker checks if the (up to-date) data object corresponding to the $qi$ exists in the CS.",
"If this is true, the broker forwards the data object to the consumer and discards the $qi$ (lines REF -REF ).",
"This is because the $qi$ is already processed at one or more brokers and a matching Data packet (with latest timestamp) is found in the CS or cache.",
"The resolution to the Data packet is explained later in Section REF .",
"If the cache entry is not found, the broker continues its search in the PIT table (lines REF -REF ).",
"If $qi$ is found in PIT and the face corresponding to the query interest does not exist (lines REF -REF ), a new ${face\\_id}$ (from which the interest is received) is added.",
"Conversely, if the face entry is found in PIT, this means the $qi$ is being processed and hence the packet is discarded (lines REF -REF ).",
"However, if no entry in PIT exists, this means that the consumer's interest reaches first time at the broker network.",
"Therefore, a new entry for $qi$ is created and the $qi$ is processed by first generating an operator graph (cf.",
"Section REF ) and then processing it (cf.",
"Section REF ) (lines REF -REF ).",
"A Data Stream packet is also handled similarly to the Add Query Interest packet (lines REF -REF ), except for the fact that the query processing is triggered if Data Stream satisfies a query interest in PIT and it is a new packet (lines REF -REF ).",
"This means that the $qi$ performs an operation on the data object contained in the Data Stream packet.",
"In this case, query processing is triggered because it may contribute to the generation of a new $ce$ .",
"In addition, if the broker does not have a matching $qi$ in PIT entry, this means it is not allocated to operator graph processing and hence it is forwarded to the next broker (line REF ).",
"The Data Stream is forwarded if there are consumers downstream by looking at the FIB entry.",
"When a Remove Query Interest packet is received at a broker, the node looks up its PIT table for an entry of the $qi$ .",
"If found, it removes the PIT entry for $qi$ and the Remove Query Interest packet is forwarded to the next node.",
"It is done in a similar way as the PIT entry corresponding to a CCN Interest packet is removed when a matching Data packet is found.",
"To summarize, in tab:differences, we show the differences of the INetCEP architecture in comparison to the standard CCN architecture in terms of the packet types, the data plane, and the processing engine.",
"We show that with minimum changes in the data plane, we support both consumer- and producer-initiated traffic.",
"A General CEP Query Language In this section, we present a general CEP language to resolve Limitation 5.",
"By doing this, we provide a means to resolve CEP queries expressed as $qi$ (query interests) on the data plane of CCN.",
"The grammar definition of the query language can be found in apndix:grammar.",
"We aim for three main design goals for the query language and the parser: distinguishing between pull and push based traffic, translating a query to an equivalent name prefix of the CCN architecture, and supporting conventional relational algebraic operators and being extensible such that additional operators can be integrated with minimum changes.",
"This is to ensure easy integration of existing and new IoT applications.",
"We provide the definition of INetCEP query language in Section REF and the parser in Section REF .",
"Query Language Each operator in a query behaves differently based on the input source type, i.e., consumer- and producer-initiated interaction, which is done based on the reception of a Data packet or a Data Stream packet, respectively.",
"The Data packet is processed and returned as a data object, as conventionally done in the CCN architecture.",
"For instance, a Join operator placed on broker $C$ can join two data objects, $<lat1, long1>$ with name prefix $/node/A$ and $<lat2, long2>$ with $/node/B$ to produce $<lat1, long1, lat2, long2>$ with $/node/C$ .",
"Figure: Join of two data stream packets with window size of 2s in a CCN network.In contrast, a Data Stream is processed and transformed either into an output stream (another Data Stream packet), or can be transformed to derive a Data packet, containing, e.g., a boolean variable depending on the CEP query.",
"For instance, a join of two continuous data streams expressing location attributes of producer A with location attributes of producer B leads to the generation of a new data stream, as illustrated by broker C in fig:icnjoin.",
"We express the standard CEP operators as explained in sec:preliminaries using the INetCEP language belowWe have implemented all standard CEP operators defined in Section including SEQUENCE operator but only present the representative operators relevant for our use cases.. call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries WINDOW(GPS_S1, 4s) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries FILTER(WINDOW(GPS_S1, 4s),'latitude'<50) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries JOIN( FILTER(WINDOW(GPS_S1, 4s), 'latitude'<50), FILTER(WINDOW(GPS_S2, 4s), 'latitude'<50), GPS_S1.'ts' = GPS_S2.",
"'ts' ) The stateful operators e.g., Window and JOIN must store the accumulated tuples in some form of a readily available storage.",
"For this, we make use of in-network cache, the CS, that readily provides data for the window operator.",
"This can be highly beneficial, e.g., in a dynamic environment where state migration is necessary.",
"The INetCEP query language provides an abstract, simple, and expressive external DSL that translates the CEP query to equivalent NFN lambda ($\\lambda $ ) expressions, as explained later in subsubsec:parser.",
"The INetCEP language abstracts over the complexity of lambda expressions (as seen in equation below), so that CEP developers can easily perform data plane query processing on an ICN network.",
"For instance, a general $\\lambda $ expression of a JOIN query defined in Query is given as follows.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries (call <no_of_params> /node/nodeQuery/nfn_service_Join (call <no_of_params> /node/nodeQuery/nfn_service_Filter (call <no_of_params> /node/nodeQuery/nfn_service_Window 4s), 'latitude' < 50) (call <no_of_params> /node/nodeQuery/nfn_service_Filter (call <no_of_params> /node/nodeQuery/nfn_service_Window 4s), 'latitude' < 50) GPS_S1.'ts' = GPS_S2.",
"'ts') 1em [t] $query \\leftarrow $ the input CEP query $\\tau \\mathit {curList} \\leftarrow $ top down list of 3 $\\omega $ of tuple $\\tau $ $\\omega _{cur} \\leftarrow $ current operator $\\textsc {createOperatorGraph} (query)$ $\\tau \\mathit {curList} \\leftarrow $ getCurList($query$ ) parseQuery($\\tau \\mathit {curList})$ $\\textsc {parseQuery}(\\tau \\mathit {curList})$ $\\omega _{cur} \\leftarrow $getOperator$(\\tau \\mathit {curList})$ $\\mathit {nfnExp} \\leftarrow \\textsc {constructNFNQuery}(\\omega _{cur})$ $node \\leftarrow \\text{new } \\textsc {Node}(\\mathit {nfnExp})$ $size(\\tau \\mathit {curList}) == 1$ return $node$ $size(\\tau \\mathit {curList}) > 1$ parseQuery($\\tau \\mathit {curList}.left)$ parseQuery($\\tau \\mathit {curList}.right)$ return $node$ Recursively generating the operator graph 1em Here, <no_of_params> is the number of parameters in the $\\lambda $ expression, nfn_service_Join is the name of the operator (join operator) in the query, $4s$ is the window size and the remaining are the filter and join conditions, respectively.",
"Each operator is preceded by /node/nodeQuery/.. which represents the name of the node that is used to place the operator e.g., nodeA.",
"This is done at runtime by placing nodeQuery on the node name selected by the operator placement algorithm (cf.",
"Section REF ) to process the operator in a centralized or a distributed manner.",
"The translation of a CEP query to the above $\\lambda $ expression is discussed in the next section.",
"Query Parser In Algorithm REF , we express the INetCEP query parser as a recursive algorithm to map the query (e.g., Query ) to generate an equivalent NFN's $\\lambda $ expression (e.g., given above).",
"A CEP query is transformed into an operator graph $G$ (lines REF -REF ), which is a binary graph tree defined as a tuple $\\tau = (L, S, R)$ .",
"Here, $L$ and $R$ are binary trees or an empty set and $S$ is a singleton set, e.g., a single operator ($\\omega $ ).",
"The query parser starts parsing the query in a specific order, i.e., in a top-down fashion that marks the dependency of operators as well.",
"This implies each leaf operator is dependent on its parent.",
"Thus, the parser starts by iterating top down the binary tree starting from the root operator $\\omega _{cur}$ (line REF ), where $cur = root$ in the first step.",
"The traversal is performed in a depth-first pre-order manner (visit parent first, then left (L) and then right (R) children) (lines REF -REF ).",
"The workflow of the query parser algorithm for an example query of the form of Query , is illustrated in Fig.",
"REF and explained in the following.",
"We start by extracting the operator name ($\\omega $ ) by separating the parameters into a list.",
"We create a logical operator graph for each query by instantiating the operators and their data flow.",
"An operator is created only after the semantic checks on the operator are verified, e.g., if the $\\omega $ is valid, and/or it has valid parameters.",
"Once all the semantic checks are verified, we continue processing the query recursively as in Algorithm REF , by generating the corresponding NFN query and creating logical plan nodes for operators that are assigned to the broker network for processing the operator graph.",
"Figure: Query parser workflow based on Algorithm .",
"Operator Graph Processing This module processes the query either centrally or in a distributed way by mapping the operator graph plan nodes to the brokers ($B$ ) or INetCEP aware CCN routers after operator graph construction (cf.",
"subsec:language).",
"This is because central processing might not be sufficient for all the use cases, e.g., when the amount of resources required to process the queries increases with the number of operators and/or queries.",
"It is, therefore, necessary to distribute operators on multiple brokers.",
"In this way, the network is also not unreasonably loaded by the queries, while the network forwarding is not disturbed.",
"In fig:ceptimeline, we show the timeline of distributed query processing.",
"In case of central processing, only parsing and deployment is required.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; The broker that first receives the $qi$ from the consumer parses the CEP query and forms an operator graph (as described in subsubsec:parser).",
"This broker becomes the placement coordinator and coordinates the further actions taken for operator graph processing.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The coordinator builds the path where the operator graph is processed based on a criteria, e.g., minimum latency and selects other broker nodes for operator placement.",
"This is along the path from producer towards the consumer.",
"It is important because the Data packets are forwarded as well as processed along this path (in-network processing).",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 3; The coordinator recursively traverses the operator graph, while assigning the CEP operators or translated named functions (cf.",
"subsec:language) to the CCN routers.",
"The resulting $ce$ is encapsulated in a Data Stream or a Data packet, which is received at the root node of the operator graph and forwarded to the consumer.",
"In the following, we describe the collection of the monitoring information related to the CCN nodes by the placement coordinator (cf.",
"subsubsec:nwdiscovery) and the assignment of operators based on this information by the placement module (cf. subsubsec:placement).",
"In principle, the role of the coordinator is decided by the concrete operator placement algorithm.",
"INetCEP supports different means of coordination and hence operator placement algorithms.",
"Yet, for the algorithm defined below, the node where the first (or root) operator is deployed is the coordinator.",
"Network Discovery Service The placement coordinator fetches and maintains the monitoring information related to the node or network to place the operators on the right set of brokers or a single broker.",
"Since different CEP applications might be interested in optimizing distinct QoS metrics, the network discovery service can be updated accordingly to monitor the respective metric(s).",
"At the moment, we provide monitoring for the end-to-end delay which is important for our representative use cases.",
"The end-to-end delay is defined as the complete timeline as illustrated in fig:ceptimeline from query parsing to the delivery of the complex event.",
"Figure: Timeline of distributed query processing.The node and network information is retrieved as a Data packet with name prefix, e.g., /node/node_id/delay only on fetch basis (whenever required).",
"The placement coordinator subscribes for this information and hence maintains the global (centralized) or local (decentralized) knowledge on the network.",
"The cluster coordinators can be elected for decentralized placement as dictated in the placement literature [33].",
"By looking at the node and network characteristics, e.g., average delay, the placement coordinator selects one or more nodes for operator placement (defined next).",
"Operator Placement Module The operator placement module handles distributed query processing in case the processing requests, e.g., in terms of query interests, exceed the network or node capacity.",
"It works in conjunction with the placement coordinator, which is a primary component to provide operator placement decisions.",
"This module is responsible for building a path for operator placement while optimizing one or more QoS metrics (based on the knowledge from network discovery service), placing the plan node with CEP queries on the selected physical brokers and collaboratively processing the deployed query and delivering the complex event.",
"In principle, this module can be extended to support different QoS metrics, design characteristics and hence placement decisions.",
"Evaluation We evaluate the INetCEP architecture by answering two questions: EQ1 Is the INetCEP system extensible and expressive?",
"EQ2 How is the performance of INetCEP system?",
"To this end, we explain the evaluation setup in Section REF , EQ1 in Section REF and EQ2 in Section REF .",
"Evaluation Environment We selected the NFN architecture [42] to implement our solution, due to its built-in support of resolving named functions as so-called $\\lambda $ expressions on top of the ICN substrate.",
"However, a major difference to our architecture is that the communication plane in NFN is only consumer-initiated.",
"In contrast, we provide unified communication layer for co-existing consumer- and producer-initiated interactions, while doing CEP operations in the network.",
"As a consequence, we embedded CEP operators as named functions while leveraging NFN's abstract machine to resolve them.",
"NFN works together with CCN-lite [7], which is a lightweight implementation of CCNx and NDN protocol.",
"We have developed unified interfaces of our design on top of NFN (v0.1.0) and CCN-lite (v0.3.0) for the Linux platform [8].",
"We have enhanced the NDN protocol implementation in the CCN-lite and the NFN architecture by: including the additional packet types and their handling, as described in subsec:unified, implementing the extensible general CEP query language, parser, and CEP operators as NFN services, as described in Section REF and implementing a network discovery service with modifications in both CCN and NFN, and operator placement as an NFN service (cf.",
"Section REF ).",
"We evaluated our implementation using the CCN-lite emulator on two topologies: centralized (cf.",
"fig:topoa) and distributed (cf. fig:topob).",
"Each node in our topology is an Ubuntu 16.04 virtual machine (VM) with 8 GiB of memory.",
"Here, each VM (node) is a CCN-NFN relay, which hosts a NFN compute server encapsulating the CEP operator logic.",
"For running the experiments, we first created a CCN network topology as illustrated in fig:topo.",
"Second, we deployed the INetCEP architecture that works on the NFN compute server, the CCN-NFN relay, as well as on the links.",
"Here, as intended, the nodes communicate using the NDN protocol instead of IP.",
"In the centralized topology, we have two producers, a single broker that processes the query and one consumer.",
"In the distributed topology, we have one consumer, two producers and six brokers, as shown in the figure.",
"The data structures CS and PIT are utilized as explained in the previous sections (cf. subsec:unified).",
"We use Queries - (cf.",
"subsec:language) for our evaluation with the DEBS grand challenge 2014 smart home dataset and the disaster field dataset.",
"The dataset is explained in subsec:eq1.",
"Figure: Topology for evaluation.Evaluation Question I: Extensibility To show the extensibility and expressiveness of our approach, we extended the INetCEP architecture for the two representative IoT use cases that we introduced in Section , with a heat map query and a load prediction query.",
"We extended the INetCEP query language and CEP operators to include the heat map [29] and prediction operators [34] by making a few additions to our implementation in our extensible query language and parser.",
"We used real world datasets to evaluate the queries: the 2014 DEBS grand challenge and a disaster field dataset.",
"Dataset 1.",
"For the heat map query, we use a dataset [12] of a field test mimicking a post-disaster situation.",
"The field test mimics two fictive events, a lightning strike and a hazardous substance release from a chemical plant, which resulted in a stressful situation.",
"The collected dataset consists of sensor data, e.g., location coordinates.",
"It was collected from smartphones provided to the participants.",
"Each sensor data stream has a schema specifying the name of the attributes, e.g., the GPS data stream has the following schema: $<ts, s\\_id, latitude, longitude, altitude, accuracy, distance, speed>$ Query.",
"We use the $latitude$ and $longitude$ attributes of this schema to generate the heat map distribution of the survivors from the disaster field test.",
"A typical heat map application joins the GPS data stream from a given set of survivors, derives the area by finding minimum and maximum latitude and longitude values, and visualizes the heat map distribution of the location of the survivors in this area.",
"For simplicity, we consider a data stream from two survivors, as shown in the operator graph in fig:bothqueriesa.",
"Here, $p_1$ and $p_2$ are the producers or GPS sensors, $\\omega _{\\bowtie }$ is the join operator, and $\\omega _{h}$ is the heat map generation operator algorithm [29].",
"This is easily possible using the INetCEP language and parser implementation that follows an Abstract Factory design pattern.",
"First, we included the algorithm for heat map generation, which is $\\tilde{20}$ LOC.",
"Second, we extended the language implementation to include the user defined operator by adding $\\tilde{20}$ LOC.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries HEATMAP( 'cell_size', 'area', JOIN(WINDOW(GPS_S1, 1m), WINDOW(GPS_S2, 1m)) GPS_S1.'ts' = GPS_S2.",
"'ts') ) Figure: Two applications for evaluation (a) a heat map query for post-disaster relief and (b) an energy load forecasting query for smart homes.Dataset 2.",
"The second dataset comes from the 2014 DEBS grand challenge [4] scenario focused on solving a short-term load forecasting problem in a smart grid.",
"The data for the challenge is based on real-world profiles collected from smart home installations.",
"The dataset captured load measurements from unique smart plugs with the following schema: $<ts, id, value, property, plug\\_id, household\\_id, house\\_id>$ Query.",
"We apply an existing solution [34] to perform prediction by extending the INetCEP architecture for two smart plugs.",
"The corresponding operator graph for such a prediction is illustrated in fig:bothqueriesb and listed below.",
"Here, $p_3$ and $p_4$ are the producers or smart plugs, $\\omega _{\\bowtie }$ is a join operator and $\\omega _{pr}$ is a prediction operator based on the algorithm [34].",
"In the first query, we notify the consumer about the predictions for five minutes into the future, while in the second query we notify only if the predictions of load are above a threshold.",
"Similarly to the heat map application, we implemented a prediction algorithm by adding $\\tilde{50}$ LOC and the language implementation with $\\tilde{20}$ LOC.",
"The detailed description of the algorithms for the respective use cases in order to achieve the extensibility is presented in apndix:extensibility.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries JOIN( PREDICT(5m, WINDOW(PLUG_S1, 1m)), PREDICT(5m, WINDOW(PLUG_S2, 1m)) PLUG_S1.'ts' = PLUG_S2.",
"'ts' ) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries FILTER(JOIN( PREDICT(5m, WINDOW(PLUG_S1, 1m)), PREDICT(5m, WINDOW(PLUG_S2, 1m)) PLUG_S1.'ts' = PLUG_S2.",
"'ts' ), 'load'>20) Evaluation Question II: Performance We evaluated the performance of the INetCEP architecture on standard CEP queries including the Queries -.",
"Additionally, we evaluated the heat map and prediction queries (Queries -).",
"Our aim is to understand the performance of: the query parser on increasing the number of nested operators in operator graph, the operator graph module for different kind of queries, and the operator placement module on increasing number of queries.",
"Figure: Performance of the query parser depending on increasing the number of nested operators.Query Parsing In the query parser design (cf.",
"Section REF ), we performed a complexity analysis of the Algorithm REF .",
"In this section, we verify the analysis experimentally by increasing the number of operators in the operator graph while using the centralized topology shown in fig:topoa.",
"Figure: End-to-end and communication delay observed in Queries 1 to 6.In fig:parser, we show the performance of query parser algorithm in terms of runtime (in ms) for Query or Filter operators and Query or Join operators.",
"We show in a line plot with a confidence interval of 95% for 20 runs of the emulation that the two queries scales reasonably and can be processed in a few milliseconds.",
"Centralized Query Processing We measured the end-to-end delay in processing the six queries defined above using the operator graph module (cf.",
"§ REF ).",
"In fig:evaldelay, we show the results as box plots with a confidence interval of 95% for 10 runs of the emulation.",
"For Query -, the total delay perceived is less than a few milliseconds.",
"It increased for the new queries Query -, where we introduced prediction and heat map operators, primarily due to increased consumption of data and the computational complexity of the algorithms for prediction [34] and heat map [29], respectively.",
"We further show the distribution of the mean end-to-end delay in tab:latency to understand the primary reason for the delay.",
"We listed the time spent in each of the modules of the INetCEP architecture, namely, operator graph creation, placement of operators, and communication of events.",
"The values shown in the table are the mean of the values observed for 10 executions.",
"For basic CEP queries -, the major portion of time ($\\tilde{98}\\%$ ) is spent in communication.",
"This can be explained, as also confirmed in other works [20], by the limitations of the CCNx implementation of the NDN protocol.",
"Hence, we observe the communication delays for all the queries in fig:commq1toq6, where it takes up to 500 ms for delivering results of complex queries like prediction and heatmap.",
"Table: The division of mean end-to-end delay in ms for of operator graph creation, placement of operators, and (communication) delay for centralized placement (see fig:q1toq6).Figure: End-to-end delay observed for Query 3 on incrementally increasing the query workload.",
"Operator Placement To understand the behavior of the operator placement module, we utilize the distributed topology of seven VMs, as illustrated in fig:topob, to place operators based on the information collected by the network discovery service.",
"To take full advantage of distributed CEP, we increased the query load of Query starting from node 1 to 5.",
"For example, the first 10 queries were initialized at the first broker node followed by the next 10 queries at the second node, and so on.",
"Hence, the number of consumers also increased in the network with the query load.",
"In fig:multipleq3, we see that the total delay in retrieving the complex event increased incrementally with the query load, which is reasonable given the static network size (7 nodes) and the resources.",
"However, the queries were distributed evenly using the operator placement algorithm at distinct nodes.",
"To summarize, we evaluated the performance of INetCEP on two topologies: centralized and distributed, using two IoT applications and six different queries.",
"Our evaluation shows that IoT applications can be integrated seamlessly using the INetCEP architecture, CEP queries can be formulated and can be extended for more use cases, and simple CEP queries can be processed in milliseconds.",
"Related Work We now review the state-of-the-art ICN architectures in terms of their support for consumer- and producer-initiated interaction patterns and existing INP architectures in subsec:icnarch, and CEP and networking architectures in subsec:eparch.",
"ICN Architectures Interaction Patterns in ICNs: In Figure REF , we highlight the main ICN architectures NDN [45], NFN [42], DONA [28], PURSUIT [2], PSIRP [1], in terms of their support of a unified communication layer as presented in our work.",
"However, only those appearing in the green box, namely CONVERGENCE [35], GreenICN [3], and Carzaniga et al.",
"[19], provide support for both kind of invocation mechanisms.",
"The CONVERGENCE system combines the publish/subscribe interaction paradigm on top of an information-centric network layer.",
"In contrast, we provide a unified interface such that pull and push based interaction patterns could co-exist in a single network layer while performing in-network computations.",
"GreenICN is an ICN architecture for post-disaster scenarios by combining NDN (pull-based) with COPSS [20] (push-based).",
"However, GreenICN introduces additional data structures, e.g., a subscription table (ST), while we provide this combination using the existing data structures of ICN.",
"Furthermore, it is not clear if GreenICN can function as a whole in a single ICN architecture [41].",
"Carzaniga et al.",
"propose a unified network interface similar to our work, however, the authors only propose a preliminary design of their approach [19] without implementing it in an ICN architecture, and subsequently focus on routing decisions [18] rather than on distributed processing.",
"INP and IoT Architectures in ICNs: Authors in work [43], [38] propose an approach to distribute computational tasks in the network by extending the NFN architecture similar to our work.",
"However, the authors do not deal with the lack of abstractions required for processing continuous data stream.",
"In contrast, we propose a unified communication layer to support CEP over ICN and an extensible query grammar and parser that opens a wide range of operators.",
"Krol et al.",
"[31] propose NFaaS based on unikernels, which is a container based virtualization approach to encapsulate named functions placed on NDN nodes.",
"However, they do not provide support for stateful functions, while IoT functions can be stateful, e.g., involving time windows, which is supported by our architecture.",
"Ahmed et al.",
"[10] propose a smart home approach using NDN and support both push and pull interaction patterns similar to our work.",
"However, in their architecture they only support retrieving raw data, e.g., humidity sensor readings, but not meaningful events as we do.",
"Shang et al.",
"[39] propose a publish/subscribe based approach for modern building management systems (BMS) in NDN.",
"However, the authors build on standard consumer-initiated interaction, as described in Limitation 1 (cf. subsec:decisions).",
"Publish-subscribe deployment for NDN in the IoT scenarios has been discussed in previous works [23], [24].",
"These works confirm the need of integrating producer-initiated interaction in NDN, however, do not provide a unified layer for both interaction patterns as we do.",
"CEP and Networking Architectures CEP Architectures: Several event processing architectures exist, ranging from, e.g., the open source Apache Flink [16] to Twitter's Heron [32] and Google's Millwheel [11].",
"One possibility is to interface one of them with an ICN architecture.",
"Initial work implemented Hadoop on NDN [22] for datacenter applications.",
"However, this requires changing the network model to push in contrast to our work, which would limit the support for a wide range of applications, as discussed above.",
"Networking Architectures: Another emerging network architecture is Software-Defined Networking (SDN) [30], which is gradually being deployed, e.g., in Google's data centers.",
"It allows network managers to program the control plane to support efficient traffic monitoring and engineering.",
"The SDN architecture is complementary to our work, since SDN empowers the control plane, while ICN upgrades the data plane of the current Internet architecture.",
"Data Plane and Query Languages: The literature discusses many CEP query languages [16].",
"The novelty of the proposed query language is to allow for a mapping of operations to ICN's data plane.",
"Alternative designs builds on P4 [14] in the context of SDN.",
"Initial work on programming ICN with P4 [40] faced several difficulties due to lack of key language features and the strong coupling of the language to SDN's data plane model.",
"Discussion In this section, we discuss important future challenges that could be interesting to provide more sophisticated networking, reliability and optimization mechanisms in the INetCEP architecture.",
"Sophisticated Flow and Congestion Control: In CCN, the PIT table ensures the flow balance since one Data packet is sent for each Interest packet.",
"The Data Stream packets of INetCEP could disturb the rule of flow balance since the producer could overflow the buffer on the broker side.",
"For this, INetCEP implements a simple flow control mechanism where we restrict the receiver (consumer/broker) to specify maximum outstanding messages at a time.",
"However, since the forwarding logic of Data Stream packets is similar to IP multicast, existing sophisticated multicast congestion control solutions like TCP-Friendly Multicast Congestion Control [44] and similar can provide sophisticated flow and congestion control.",
"Reliability: The brokers or consumers could miss packets when the available bandwidth and resources at their end is lower than the sending rate.",
"This makes the presence of a module of reliability relevant, which can be catered by extending our work with existing reliable CEP solutions [27] or by looking into equivalent IP solutions such as Scalable Reliable Multicast [21].",
"Query Optimization: In INetCEP we provide a placement module that maps the operator graph to in-network elements of CCN.",
"Another complementary direction could be to generate an optimal operator graph, e.g., based on operator selectivity or even partition operator graph by performing query optimization [15].",
"Optimizing QoS: In INetCEP we provide a programming abstraction for the developers to write CEP queries over ICN data plane substrate.",
"In addition, the placement module can be extended to look into further decentralized solutions and even other QoS metrics [17] like throughput, availability, etc.",
"Conclusion In this paper, we proposed the INetCEP architecture that implements a unified communication layer for co-existing consumer-initiated and producer-initiated interaction patterns.",
"We studied important design challenges to come up with our design of a unified communication layer.",
"In the unified layer, both consumer- and producer-initiated interaction patterns can co-exist in a single ICN architecture.",
"In this way, a wide range of IoT applications are supported.",
"With the proposed query language, we can express interest in aggregated data that is resolved and processed in a distributed manner in the network.",
"In our evaluation, we demonstrated in the context of two IoT case studies that our approach is highly extensible.",
"The performance evaluation showed that queries are efficiently parsed and deployed, which yields - thanks to the in-network deployment - a low end-to-end delay, e.g., simple queries induce only few milliseconds of overall delay.",
"Interesting research directions for future work are: (i) enhancing the performance of query processing by using parallelization, (ii) porting CCNx implementation on real hardware to accomplish low communication delays, and (iii) developing CEP compliant caching strategies.",
"Acknowledgements This work has been co-funded by the German Research Foundation (DFG) as part of the project C2, A3 and C5 within the Collaborative Research Center (CRC) SFB 1053 – MAKI.",
"Appendix In this section, we define our query grammar in apndix:grammar, the implementation details of our extensible design in apndix:extensibility and the algorithms used for extending the system with load forecasting and a heat map application in apndix:applications.",
"Query Grammar Definition 10.1 A grammar consists of four components: A set of terminals or tokens.",
"Terminals are the symbols that occur in a language.",
"A set of non-terminals or syntactic variables.",
"Each of them represent a set of strings.",
"We define them the way we want to use them.",
"A set of production rules that define which non-terminals can be replaced by which terminals, or non-terminals or a combination of both.",
"Here, the terminal is the head of the left side of the production, and the replacement is the body or right side of the production.",
"For example, head $\\rightarrow $ body One of the non-terminals is designated as start symbol for each production.",
"Following the above definition REF and Chomsky-Hierarchy [25], we selected a type 2 grammar or a context-free grammar for CEP over ICN language since a query may consist of multiple subqueries (or operators), which can be expressed (out of many possible ways) using parenthesis \"$()$ \".",
"Table: NO_CAPTIONThe context-free grammar allows us to combine the production rules.",
"The head of each production consist only of one non-terminal and the bodies are not limited by only one terminal and/or one non-terminal.",
"This is because the language needs to embed operators in parenthesis and with a regular grammar we cannot have an arbitrary number of parentheses.",
"We need a way to memorize each parenthesis and this ability is given by context-free grammars.",
"We define an initial INetCEP language grammar and represent it using BNF (Backus-Naur form) in tab:grammar, considering the aforementioned design decisions.",
"We use regular expressions represented as $\\mathit {REG} (\\ldots )$ , where $(\\ldots )$ can be literals $\\mathit {[a-z]}$ and $\\mathit {[A-Z]}$ in lower and upper case, respectively, and numbers $\\mathit {[0-9]}$ .",
"The plus ($+$ ) sign in $\\mathit {REG ([a-z]+)}$ means that at least one lowercase letter has to appear, while a 1 in $\\mathit {REG ([a-z]\\lbrace 1\\rbrace )}$ means that exactly one lowercase letter has to appear.",
"The relational operators given by $\\mathit {comparison}$ define a binary relation between two entities, e.g., two column names of a schema, a column name to a number or a column index to a number.",
"Extensibility To make our query language extensible, we follow a well-known Abstract Factory design pattern from object-oriented programming for our operator definition as illustrated in fig:parserUml.",
"Algorithm REF is the starting point of our operator graph creation.",
"This is implemented in the OperatorTree class.",
"Each operator inherits the abstract class OperatorA which defines the interpret (parseQuery in Algorithm) function as seen in the figure.",
"The checkParameters verifies the correctness of the parameters.",
"Figure: Operator Definition in UMLIf a new operator is to be included, it will override the existing methods of the abstract class and the parameters correctness has to be defined.",
"This allows minimal changes in the implementation for each new application developed using our system.",
"Applications Short term Load Forecasting For the DEBS Grand Challenge in 2014, Martin et al.",
"derived requirements for predicting future energy consumption of a plug [34].",
"We formulate our requirements with respect to our INetCEP system: To meet the goal of making an estimation on future energy consumption, it is necessary to use historical data as a reference.",
"In order to run on a machine with limited resources, the prediction algorithm needs to be lightweight in computation power and storage.",
"Figure: Flow chart explaining the prediction algorithmThe formula for predicting future load is given by the publishers of the DEBS Grand Challenge and is as follows: $ predicted\\_load\\left( s_{i+2} \\right) = \\left( avgLoad\\left( s_{ i } \\right) + median\\left( \\left\\lbrace avgLoads\\left( s_{ j } \\right) \\right\\rbrace \\right) \\right) $ , Here, $i$ is the current timestamp, $s_{i}$ the currently recorded values at time $i$ and $s_{j}$ the past values at a corresponding time $j=i+2$ .",
"The load two steps in the future is therefore made up of the current average electricity consumption and the average electricity consumption from the past.",
"In fig:prediction1operatorflow, we represent the flow of the prediction algorithm as per the requirements defined above and Query .",
"For each time window of 1 minute, we first determine if it is the time for next prediction, which is provided as an input in the query.",
"If the time has not come yet, a value for prediction (average load) is calculated and stored so that it can be used for the equation defined above.",
"Inversely, if it is the time to make a prediction, a prediction tuple of the following form is emitted.",
"$<ts, plug\\_id; household\\_id; house\\_id; predicted\\_load>$ Here, $ts$ is the timestamp of the prediction, $plug\\_id$ identifies a socket in a household, $household\\_id$ identifies a household within a house and $house\\_id$ identifies a house and $predicted\\_load$ is the prediction as specified in the equation above.",
"Heat Map Algorithm REF describes the heat map creation and visualization for the location updates from survivors of the disaster field test used in this work based on [29].",
"[h] $loc$ : Window of location $<lat, long>$ tuple of the survivor $Lat_{min}$ : The minimum latitude value $Lat_{max}$ : The maximum latitude value $Long_{min}$ : The minimum longitude value $Long_{min}$ : The maximum longitude value $HC$ : The number of horizontal cells needed to map the values $VC$ : The number of vertical cells needed to map the values $cell\\_size$ : The granularity $Grid$ : A two dimensional array $HC$ = $\\lfloor {\\frac{Long_{max}- Long_{min}}{cell\\_size}}\\rfloor $ $VC$ = $\\lfloor {\\frac{Lat_{max}- Lat_{min}}{cell\\_size}}\\rfloor $ line in $S_D$ absLatVal = $loc$ [lat] - $Lat_{min}$ absLongVal = $loc$ [long] - $Long_{min}$ $Grid$ [$ \\lfloor {\\frac{absLatVal}{cell\\_size}}\\rfloor $ ][$ \\lfloor {\\frac{absLongVal}{cell\\_size}}\\rfloor $ ] += 1 $Grid$ Algorithm for the heat map operator In line 1, we calculate the number of horizontal cells required for the desired heat map.",
"For this we divide the difference between the maximum and minimum longitude by the desired cell_size, which indicates how large and finely meshed the resulting heat map should be and then round this value down to the next smaller number (given by floor function).",
"In line 2, similarly we compute the vertical cells.",
"For each of these location tuples in the current window, the first absolute latitude and longitude values are computed in line 4 and 5, respectively.",
"By dividing these values by the cell_size, we obtain the corresponding position in the heat map in line 6."
],
[
"We identify the following three broad requirements for the INetCEP architecture from our discussion in the previous section.",
"R1 A unified communication layer supporting both producer and consumer initiated communication (cf.",
"Section REF ).",
"R2 An expressive CEP query language for specifying the event of interest (cf.",
"Section REF ).",
"R3 Resolution of CEP queries by efficient and scalable in-network query processing (cf.",
"Section REF ).",
"We address each of these requirements below."
],
[
"Unified Communication Layer",
"In this section, we explain the extension of the CCN data plane to enable CEP.",
"In our approach, each CCN node $n \\in N$ maintains a Content Store or cache (CS), a Pending Interest Table (PIT), a Forwarding Information Base (FIB) and a CEP engine.",
"In the following, we explain the function of these main building blocks and the data plane handling.",
"Subsequently, we detail on the newly introduced packets Add Query Interest, Remove Query Interest and Data Stream and simultaneously the solution to the limitations identified in subsec:decisions.",
"Figure: High level view of packet handling in INetCEP architecture.",
"The CS stores all the data objects associated with the CCN Interest, additionally, the time-stamped data objects associated with the query interests ($qi$ ).",
"The data object returns a value associated with the $qi$ , e.g., if the $qi$ is the sum of victims in a disaster location, then the data object contains value 20.",
"Hence, if multiple consumers are interested in the same $qi$ , the query is not reprocessed but the data is fetched directly from the CS.",
"To deal with Limitation 2 of stale cache entries for $qi$ (cf.",
"Section REF ), we store logical timestamps along with cache entries while storing $qi$ in the CS.",
"Hence, always up to-date data objects are stored, while old entries are discarded (cf.",
"fig:inetceppull).",
"The PIT stores the pending $qi$ so that the complex event ($ce$ ) could follow the path, i.e., the in-network state created in PIT to the consumer.",
"The $\\mathit {face}$ information is also stored in the PIT entry to keep track of $qi$ 's consumers.",
"In contrast to consumer-initiated interaction, the $ce$ must be notified to the interested consumers as and when detected in real-time.",
"Thus, as new data is received, the query interests in the PIT are re-evaluated as explained later in Algorithm REF .",
"In the former, we are referring to only continuous Data Stream packet since we also support fetching data from relations (this is handled conventionally using a CCN Interest packet as explained in Section ).",
"The reasons why we distinguish between Add Query Interest ($qi$ ) and CCN Interest packets are: (i) $qi$ is invoked on receipt of Add Query Interest as well as the Data Stream packet, (ii) removal of the PIT entry is not based on a Data packet retrieval but on the reception of the Remove Query Interest packet and (iii) $qi$ retrieves Data packets asynchronously.",
"In summary, we deal with Limitation 3 and Limitation 4 by asynchronously handling the $qi$ instead by 3-way message exchange and efficiently managing PIT entries, respectively.",
"We deal with Limitation 1 by storing $qi$ in PIT and asynchronously delivering event notifications to the consumers.",
"The FIB table gets populated as the producer multicasts to the broker network leaving a trail to the data source.",
"In this way the data processing is performed efficiently along the path from producer and consumer.",
"Finally, the CEP engine holds the processing logic $f_\\omega $ for each operator $\\omega $ and is responsible for parsing, processing, and returning the result to the next node towards consumer (cf.",
"Sections REF and REF ).",
"1em $\\mathit {CS} \\leftarrow \\text{content store of current node}$ $\\mathit {PIT} \\leftarrow \\text{pending interest table of current node}$ $\\mathit {FIB} \\leftarrow \\text{forwarding information base}$ $\\mathit {qi} \\leftarrow \\text{requested query interest}$ $\\mathit {result} \\leftarrow \\text{query result}$ $\\mathit {facelist} \\leftarrow \\text{list of all faces in PIT}$ $\\texttt {DataStream} \\leftarrow \\text{\\texttt {Data Stream} packet}$ $\\mathit {data} \\leftarrow \\text{data that resolves the qi}$ $\\textsc {AddQueryInterest}(\\mathit {qi})$ $qi$ is found in $\\mathit {CS.\\textsc {lookup}(qi)}$ $ \\mathit {data} \\leftarrow \\mathit {CS}.\\textsc {fetchContent}(\\mathit {qi}) $ $ \\text{return } \\mathit {data}$ (Discard AddQueryInterest) $qi \\text{ found in } \\mathit {PIT.\\textsc {lookup}(qi)} $ $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\texttt {AddQueryInterest})$ $qi \\text{ found in } \\mathit {FIB.\\textsc {lookup}(qi)} $ $\\mathit {\\textsc {CreateOperatorGraph}(\\mathit {qi})}$ (Refer Algorithm REF ) Forward AddQueryInterest (Discard AddQueryInterest) $\\textsc {ProcessDataStream}(\\texttt {DataStream})$ $\\mathit {qi} \\in PIT$ DataStream satisfies $qi$ $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\texttt {DataStream})$ Forward DataStream $\\textsc {ProcessQiInPIT}(\\mathit {qi}, \\mathit {packet})$ $\\mathit {packet} == \\texttt {DataStream} \\text{ and } \\mathit {packet}.\\mathit {ts} > \\mathit {qi.ts}$ $\\mathit {\\textsc {CreateOperatorGraph}(\\mathit {qi})}$ (Refer Algorithm REF ) Forward $\\mathit {packet}$ $\\mathit {facelist} \\leftarrow \\mathit {PIT}.\\textsc {getFaces}(\\mathit {qi}) $ $qi.\\mathit {face}$ is not found in $\\mathit {facelist}$ $ \\mathit {PIT}.\\textsc {addFace}(\\mathit {qi}) $ Discard $\\mathit {packet}$ -15pt Add Query Interest and Data Stream packet handling."
],
[
"Data Plane Handling",
"In Algorithm REF (lines REF -REF ) and fig:overall, we define the handling of Add Query Interest and Data Stream packets at the broker end in a CCN network.",
"The processing of $qi$ stored in PIT is triggered based on the receipt of these two packets as follows: when an Add Query Interest packet is received at a broker (line REF ) and due to the continuous arrival of new Data Stream packets (line REF ).",
"This is in contrast to the PIT entry of CCN Interest, which is checked only on the receipt of a new Interest packet.",
"When an Add Query Interest arrives, the broker checks if the (up to-date) data object corresponding to the $qi$ exists in the CS.",
"If this is true, the broker forwards the data object to the consumer and discards the $qi$ (lines REF -REF ).",
"This is because the $qi$ is already processed at one or more brokers and a matching Data packet (with latest timestamp) is found in the CS or cache.",
"The resolution to the Data packet is explained later in Section REF .",
"If the cache entry is not found, the broker continues its search in the PIT table (lines REF -REF ).",
"If $qi$ is found in PIT and the face corresponding to the query interest does not exist (lines REF -REF ), a new ${face\\_id}$ (from which the interest is received) is added.",
"Conversely, if the face entry is found in PIT, this means the $qi$ is being processed and hence the packet is discarded (lines REF -REF ).",
"However, if no entry in PIT exists, this means that the consumer's interest reaches first time at the broker network.",
"Therefore, a new entry for $qi$ is created and the $qi$ is processed by first generating an operator graph (cf.",
"Section REF ) and then processing it (cf.",
"Section REF ) (lines REF -REF ).",
"A Data Stream packet is also handled similarly to the Add Query Interest packet (lines REF -REF ), except for the fact that the query processing is triggered if Data Stream satisfies a query interest in PIT and it is a new packet (lines REF -REF ).",
"This means that the $qi$ performs an operation on the data object contained in the Data Stream packet.",
"In this case, query processing is triggered because it may contribute to the generation of a new $ce$ .",
"In addition, if the broker does not have a matching $qi$ in PIT entry, this means it is not allocated to operator graph processing and hence it is forwarded to the next broker (line REF ).",
"The Data Stream is forwarded if there are consumers downstream by looking at the FIB entry.",
"When a Remove Query Interest packet is received at a broker, the node looks up its PIT table for an entry of the $qi$ .",
"If found, it removes the PIT entry for $qi$ and the Remove Query Interest packet is forwarded to the next node.",
"It is done in a similar way as the PIT entry corresponding to a CCN Interest packet is removed when a matching Data packet is found.",
"To summarize, in tab:differences, we show the differences of the INetCEP architecture in comparison to the standard CCN architecture in terms of the packet types, the data plane, and the processing engine.",
"We show that with minimum changes in the data plane, we support both consumer- and producer-initiated traffic.",
"A General CEP Query Language In this section, we present a general CEP language to resolve Limitation 5.",
"By doing this, we provide a means to resolve CEP queries expressed as $qi$ (query interests) on the data plane of CCN.",
"The grammar definition of the query language can be found in apndix:grammar.",
"We aim for three main design goals for the query language and the parser: distinguishing between pull and push based traffic, translating a query to an equivalent name prefix of the CCN architecture, and supporting conventional relational algebraic operators and being extensible such that additional operators can be integrated with minimum changes.",
"This is to ensure easy integration of existing and new IoT applications.",
"We provide the definition of INetCEP query language in Section REF and the parser in Section REF .",
"Query Language Each operator in a query behaves differently based on the input source type, i.e., consumer- and producer-initiated interaction, which is done based on the reception of a Data packet or a Data Stream packet, respectively.",
"The Data packet is processed and returned as a data object, as conventionally done in the CCN architecture.",
"For instance, a Join operator placed on broker $C$ can join two data objects, $<lat1, long1>$ with name prefix $/node/A$ and $<lat2, long2>$ with $/node/B$ to produce $<lat1, long1, lat2, long2>$ with $/node/C$ .",
"Figure: Join of two data stream packets with window size of 2s in a CCN network.In contrast, a Data Stream is processed and transformed either into an output stream (another Data Stream packet), or can be transformed to derive a Data packet, containing, e.g., a boolean variable depending on the CEP query.",
"For instance, a join of two continuous data streams expressing location attributes of producer A with location attributes of producer B leads to the generation of a new data stream, as illustrated by broker C in fig:icnjoin.",
"We express the standard CEP operators as explained in sec:preliminaries using the INetCEP language belowWe have implemented all standard CEP operators defined in Section including SEQUENCE operator but only present the representative operators relevant for our use cases.. call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries WINDOW(GPS_S1, 4s) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries FILTER(WINDOW(GPS_S1, 4s),'latitude'<50) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries JOIN( FILTER(WINDOW(GPS_S1, 4s), 'latitude'<50), FILTER(WINDOW(GPS_S2, 4s), 'latitude'<50), GPS_S1.'ts' = GPS_S2.",
"'ts' ) The stateful operators e.g., Window and JOIN must store the accumulated tuples in some form of a readily available storage.",
"For this, we make use of in-network cache, the CS, that readily provides data for the window operator.",
"This can be highly beneficial, e.g., in a dynamic environment where state migration is necessary.",
"The INetCEP query language provides an abstract, simple, and expressive external DSL that translates the CEP query to equivalent NFN lambda ($\\lambda $ ) expressions, as explained later in subsubsec:parser.",
"The INetCEP language abstracts over the complexity of lambda expressions (as seen in equation below), so that CEP developers can easily perform data plane query processing on an ICN network.",
"For instance, a general $\\lambda $ expression of a JOIN query defined in Query is given as follows.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries (call <no_of_params> /node/nodeQuery/nfn_service_Join (call <no_of_params> /node/nodeQuery/nfn_service_Filter (call <no_of_params> /node/nodeQuery/nfn_service_Window 4s), 'latitude' < 50) (call <no_of_params> /node/nodeQuery/nfn_service_Filter (call <no_of_params> /node/nodeQuery/nfn_service_Window 4s), 'latitude' < 50) GPS_S1.'ts' = GPS_S2.",
"'ts') 1em [t] $query \\leftarrow $ the input CEP query $\\tau \\mathit {curList} \\leftarrow $ top down list of 3 $\\omega $ of tuple $\\tau $ $\\omega _{cur} \\leftarrow $ current operator $\\textsc {createOperatorGraph} (query)$ $\\tau \\mathit {curList} \\leftarrow $ getCurList($query$ ) parseQuery($\\tau \\mathit {curList})$ $\\textsc {parseQuery}(\\tau \\mathit {curList})$ $\\omega _{cur} \\leftarrow $getOperator$(\\tau \\mathit {curList})$ $\\mathit {nfnExp} \\leftarrow \\textsc {constructNFNQuery}(\\omega _{cur})$ $node \\leftarrow \\text{new } \\textsc {Node}(\\mathit {nfnExp})$ $size(\\tau \\mathit {curList}) == 1$ return $node$ $size(\\tau \\mathit {curList}) > 1$ parseQuery($\\tau \\mathit {curList}.left)$ parseQuery($\\tau \\mathit {curList}.right)$ return $node$ Recursively generating the operator graph 1em Here, <no_of_params> is the number of parameters in the $\\lambda $ expression, nfn_service_Join is the name of the operator (join operator) in the query, $4s$ is the window size and the remaining are the filter and join conditions, respectively.",
"Each operator is preceded by /node/nodeQuery/.. which represents the name of the node that is used to place the operator e.g., nodeA.",
"This is done at runtime by placing nodeQuery on the node name selected by the operator placement algorithm (cf.",
"Section REF ) to process the operator in a centralized or a distributed manner.",
"The translation of a CEP query to the above $\\lambda $ expression is discussed in the next section.",
"Query Parser In Algorithm REF , we express the INetCEP query parser as a recursive algorithm to map the query (e.g., Query ) to generate an equivalent NFN's $\\lambda $ expression (e.g., given above).",
"A CEP query is transformed into an operator graph $G$ (lines REF -REF ), which is a binary graph tree defined as a tuple $\\tau = (L, S, R)$ .",
"Here, $L$ and $R$ are binary trees or an empty set and $S$ is a singleton set, e.g., a single operator ($\\omega $ ).",
"The query parser starts parsing the query in a specific order, i.e., in a top-down fashion that marks the dependency of operators as well.",
"This implies each leaf operator is dependent on its parent.",
"Thus, the parser starts by iterating top down the binary tree starting from the root operator $\\omega _{cur}$ (line REF ), where $cur = root$ in the first step.",
"The traversal is performed in a depth-first pre-order manner (visit parent first, then left (L) and then right (R) children) (lines REF -REF ).",
"The workflow of the query parser algorithm for an example query of the form of Query , is illustrated in Fig.",
"REF and explained in the following.",
"We start by extracting the operator name ($\\omega $ ) by separating the parameters into a list.",
"We create a logical operator graph for each query by instantiating the operators and their data flow.",
"An operator is created only after the semantic checks on the operator are verified, e.g., if the $\\omega $ is valid, and/or it has valid parameters.",
"Once all the semantic checks are verified, we continue processing the query recursively as in Algorithm REF , by generating the corresponding NFN query and creating logical plan nodes for operators that are assigned to the broker network for processing the operator graph.",
"Figure: Query parser workflow based on Algorithm .",
"Operator Graph Processing This module processes the query either centrally or in a distributed way by mapping the operator graph plan nodes to the brokers ($B$ ) or INetCEP aware CCN routers after operator graph construction (cf.",
"subsec:language).",
"This is because central processing might not be sufficient for all the use cases, e.g., when the amount of resources required to process the queries increases with the number of operators and/or queries.",
"It is, therefore, necessary to distribute operators on multiple brokers.",
"In this way, the network is also not unreasonably loaded by the queries, while the network forwarding is not disturbed.",
"In fig:ceptimeline, we show the timeline of distributed query processing.",
"In case of central processing, only parsing and deployment is required.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 1; The broker that first receives the $qi$ from the consumer parses the CEP query and forms an operator graph (as described in subsubsec:parser).",
"This broker becomes the placement coordinator and coordinates the further actions taken for operator graph processing.",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 2; The coordinator builds the path where the operator graph is processed based on a criteria, e.g., minimum latency and selects other broker nodes for operator placement.",
"This is along the path from producer towards the consumer.",
"It is important because the Data packets are forwarded as well as processed along this path (in-network processing).",
"[baseline=(char.base)] shape=circle,draw,inner sep=0.3pt,color=white,fill=black] (char) 3; The coordinator recursively traverses the operator graph, while assigning the CEP operators or translated named functions (cf.",
"subsec:language) to the CCN routers.",
"The resulting $ce$ is encapsulated in a Data Stream or a Data packet, which is received at the root node of the operator graph and forwarded to the consumer.",
"In the following, we describe the collection of the monitoring information related to the CCN nodes by the placement coordinator (cf.",
"subsubsec:nwdiscovery) and the assignment of operators based on this information by the placement module (cf. subsubsec:placement).",
"In principle, the role of the coordinator is decided by the concrete operator placement algorithm.",
"INetCEP supports different means of coordination and hence operator placement algorithms.",
"Yet, for the algorithm defined below, the node where the first (or root) operator is deployed is the coordinator.",
"Network Discovery Service The placement coordinator fetches and maintains the monitoring information related to the node or network to place the operators on the right set of brokers or a single broker.",
"Since different CEP applications might be interested in optimizing distinct QoS metrics, the network discovery service can be updated accordingly to monitor the respective metric(s).",
"At the moment, we provide monitoring for the end-to-end delay which is important for our representative use cases.",
"The end-to-end delay is defined as the complete timeline as illustrated in fig:ceptimeline from query parsing to the delivery of the complex event.",
"Figure: Timeline of distributed query processing.The node and network information is retrieved as a Data packet with name prefix, e.g., /node/node_id/delay only on fetch basis (whenever required).",
"The placement coordinator subscribes for this information and hence maintains the global (centralized) or local (decentralized) knowledge on the network.",
"The cluster coordinators can be elected for decentralized placement as dictated in the placement literature [33].",
"By looking at the node and network characteristics, e.g., average delay, the placement coordinator selects one or more nodes for operator placement (defined next).",
"Operator Placement Module The operator placement module handles distributed query processing in case the processing requests, e.g., in terms of query interests, exceed the network or node capacity.",
"It works in conjunction with the placement coordinator, which is a primary component to provide operator placement decisions.",
"This module is responsible for building a path for operator placement while optimizing one or more QoS metrics (based on the knowledge from network discovery service), placing the plan node with CEP queries on the selected physical brokers and collaboratively processing the deployed query and delivering the complex event.",
"In principle, this module can be extended to support different QoS metrics, design characteristics and hence placement decisions.",
"Evaluation We evaluate the INetCEP architecture by answering two questions: EQ1 Is the INetCEP system extensible and expressive?",
"EQ2 How is the performance of INetCEP system?",
"To this end, we explain the evaluation setup in Section REF , EQ1 in Section REF and EQ2 in Section REF .",
"Evaluation Environment We selected the NFN architecture [42] to implement our solution, due to its built-in support of resolving named functions as so-called $\\lambda $ expressions on top of the ICN substrate.",
"However, a major difference to our architecture is that the communication plane in NFN is only consumer-initiated.",
"In contrast, we provide unified communication layer for co-existing consumer- and producer-initiated interactions, while doing CEP operations in the network.",
"As a consequence, we embedded CEP operators as named functions while leveraging NFN's abstract machine to resolve them.",
"NFN works together with CCN-lite [7], which is a lightweight implementation of CCNx and NDN protocol.",
"We have developed unified interfaces of our design on top of NFN (v0.1.0) and CCN-lite (v0.3.0) for the Linux platform [8].",
"We have enhanced the NDN protocol implementation in the CCN-lite and the NFN architecture by: including the additional packet types and their handling, as described in subsec:unified, implementing the extensible general CEP query language, parser, and CEP operators as NFN services, as described in Section REF and implementing a network discovery service with modifications in both CCN and NFN, and operator placement as an NFN service (cf.",
"Section REF ).",
"We evaluated our implementation using the CCN-lite emulator on two topologies: centralized (cf.",
"fig:topoa) and distributed (cf. fig:topob).",
"Each node in our topology is an Ubuntu 16.04 virtual machine (VM) with 8 GiB of memory.",
"Here, each VM (node) is a CCN-NFN relay, which hosts a NFN compute server encapsulating the CEP operator logic.",
"For running the experiments, we first created a CCN network topology as illustrated in fig:topo.",
"Second, we deployed the INetCEP architecture that works on the NFN compute server, the CCN-NFN relay, as well as on the links.",
"Here, as intended, the nodes communicate using the NDN protocol instead of IP.",
"In the centralized topology, we have two producers, a single broker that processes the query and one consumer.",
"In the distributed topology, we have one consumer, two producers and six brokers, as shown in the figure.",
"The data structures CS and PIT are utilized as explained in the previous sections (cf. subsec:unified).",
"We use Queries - (cf.",
"subsec:language) for our evaluation with the DEBS grand challenge 2014 smart home dataset and the disaster field dataset.",
"The dataset is explained in subsec:eq1.",
"Figure: Topology for evaluation.Evaluation Question I: Extensibility To show the extensibility and expressiveness of our approach, we extended the INetCEP architecture for the two representative IoT use cases that we introduced in Section , with a heat map query and a load prediction query.",
"We extended the INetCEP query language and CEP operators to include the heat map [29] and prediction operators [34] by making a few additions to our implementation in our extensible query language and parser.",
"We used real world datasets to evaluate the queries: the 2014 DEBS grand challenge and a disaster field dataset.",
"Dataset 1.",
"For the heat map query, we use a dataset [12] of a field test mimicking a post-disaster situation.",
"The field test mimics two fictive events, a lightning strike and a hazardous substance release from a chemical plant, which resulted in a stressful situation.",
"The collected dataset consists of sensor data, e.g., location coordinates.",
"It was collected from smartphones provided to the participants.",
"Each sensor data stream has a schema specifying the name of the attributes, e.g., the GPS data stream has the following schema: $<ts, s\\_id, latitude, longitude, altitude, accuracy, distance, speed>$ Query.",
"We use the $latitude$ and $longitude$ attributes of this schema to generate the heat map distribution of the survivors from the disaster field test.",
"A typical heat map application joins the GPS data stream from a given set of survivors, derives the area by finding minimum and maximum latitude and longitude values, and visualizes the heat map distribution of the location of the survivors in this area.",
"For simplicity, we consider a data stream from two survivors, as shown in the operator graph in fig:bothqueriesa.",
"Here, $p_1$ and $p_2$ are the producers or GPS sensors, $\\omega _{\\bowtie }$ is the join operator, and $\\omega _{h}$ is the heat map generation operator algorithm [29].",
"This is easily possible using the INetCEP language and parser implementation that follows an Abstract Factory design pattern.",
"First, we included the algorithm for heat map generation, which is $\\tilde{20}$ LOC.",
"Second, we extended the language implementation to include the user defined operator by adding $\\tilde{20}$ LOC.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries HEATMAP( 'cell_size', 'area', JOIN(WINDOW(GPS_S1, 1m), WINDOW(GPS_S2, 1m)) GPS_S1.'ts' = GPS_S2.",
"'ts') ) Figure: Two applications for evaluation (a) a heat map query for post-disaster relief and (b) an energy load forecasting query for smart homes.Dataset 2.",
"The second dataset comes from the 2014 DEBS grand challenge [4] scenario focused on solving a short-term load forecasting problem in a smart grid.",
"The data for the challenge is based on real-world profiles collected from smart home installations.",
"The dataset captured load measurements from unique smart plugs with the following schema: $<ts, id, value, property, plug\\_id, household\\_id, house\\_id>$ Query.",
"We apply an existing solution [34] to perform prediction by extending the INetCEP architecture for two smart plugs.",
"The corresponding operator graph for such a prediction is illustrated in fig:bothqueriesb and listed below.",
"Here, $p_3$ and $p_4$ are the producers or smart plugs, $\\omega _{\\bowtie }$ is a join operator and $\\omega _{pr}$ is a prediction operator based on the algorithm [34].",
"In the first query, we notify the consumer about the predictions for five minutes into the future, while in the second query we notify only if the predictions of load are above a threshold.",
"Similarly to the heat map application, we implemented a prediction algorithm by adding $\\tilde{50}$ LOC and the language implementation with $\\tilde{20}$ LOC.",
"The detailed description of the algorithms for the respective use cases in order to achieve the extensibility is presented in apndix:extensibility.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries JOIN( PREDICT(5m, WINDOW(PLUG_S1, 1m)), PREDICT(5m, WINDOW(PLUG_S2, 1m)) PLUG_S1.'ts' = PLUG_S2.",
"'ts' ) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries FILTER(JOIN( PREDICT(5m, WINDOW(PLUG_S1, 1m)), PREDICT(5m, WINDOW(PLUG_S2, 1m)) PLUG_S1.'ts' = PLUG_S2.",
"'ts' ), 'load'>20) Evaluation Question II: Performance We evaluated the performance of the INetCEP architecture on standard CEP queries including the Queries -.",
"Additionally, we evaluated the heat map and prediction queries (Queries -).",
"Our aim is to understand the performance of: the query parser on increasing the number of nested operators in operator graph, the operator graph module for different kind of queries, and the operator placement module on increasing number of queries.",
"Figure: Performance of the query parser depending on increasing the number of nested operators.Query Parsing In the query parser design (cf.",
"Section REF ), we performed a complexity analysis of the Algorithm REF .",
"In this section, we verify the analysis experimentally by increasing the number of operators in the operator graph while using the centralized topology shown in fig:topoa.",
"Figure: End-to-end and communication delay observed in Queries 1 to 6.In fig:parser, we show the performance of query parser algorithm in terms of runtime (in ms) for Query or Filter operators and Query or Join operators.",
"We show in a line plot with a confidence interval of 95% for 20 runs of the emulation that the two queries scales reasonably and can be processed in a few milliseconds.",
"Centralized Query Processing We measured the end-to-end delay in processing the six queries defined above using the operator graph module (cf.",
"§ REF ).",
"In fig:evaldelay, we show the results as box plots with a confidence interval of 95% for 10 runs of the emulation.",
"For Query -, the total delay perceived is less than a few milliseconds.",
"It increased for the new queries Query -, where we introduced prediction and heat map operators, primarily due to increased consumption of data and the computational complexity of the algorithms for prediction [34] and heat map [29], respectively.",
"We further show the distribution of the mean end-to-end delay in tab:latency to understand the primary reason for the delay.",
"We listed the time spent in each of the modules of the INetCEP architecture, namely, operator graph creation, placement of operators, and communication of events.",
"The values shown in the table are the mean of the values observed for 10 executions.",
"For basic CEP queries -, the major portion of time ($\\tilde{98}\\%$ ) is spent in communication.",
"This can be explained, as also confirmed in other works [20], by the limitations of the CCNx implementation of the NDN protocol.",
"Hence, we observe the communication delays for all the queries in fig:commq1toq6, where it takes up to 500 ms for delivering results of complex queries like prediction and heatmap.",
"Table: The division of mean end-to-end delay in ms for of operator graph creation, placement of operators, and (communication) delay for centralized placement (see fig:q1toq6).Figure: End-to-end delay observed for Query 3 on incrementally increasing the query workload.",
"Operator Placement To understand the behavior of the operator placement module, we utilize the distributed topology of seven VMs, as illustrated in fig:topob, to place operators based on the information collected by the network discovery service.",
"To take full advantage of distributed CEP, we increased the query load of Query starting from node 1 to 5.",
"For example, the first 10 queries were initialized at the first broker node followed by the next 10 queries at the second node, and so on.",
"Hence, the number of consumers also increased in the network with the query load.",
"In fig:multipleq3, we see that the total delay in retrieving the complex event increased incrementally with the query load, which is reasonable given the static network size (7 nodes) and the resources.",
"However, the queries were distributed evenly using the operator placement algorithm at distinct nodes.",
"To summarize, we evaluated the performance of INetCEP on two topologies: centralized and distributed, using two IoT applications and six different queries.",
"Our evaluation shows that IoT applications can be integrated seamlessly using the INetCEP architecture, CEP queries can be formulated and can be extended for more use cases, and simple CEP queries can be processed in milliseconds.",
"Related Work We now review the state-of-the-art ICN architectures in terms of their support for consumer- and producer-initiated interaction patterns and existing INP architectures in subsec:icnarch, and CEP and networking architectures in subsec:eparch.",
"ICN Architectures Interaction Patterns in ICNs: In Figure REF , we highlight the main ICN architectures NDN [45], NFN [42], DONA [28], PURSUIT [2], PSIRP [1], in terms of their support of a unified communication layer as presented in our work.",
"However, only those appearing in the green box, namely CONVERGENCE [35], GreenICN [3], and Carzaniga et al.",
"[19], provide support for both kind of invocation mechanisms.",
"The CONVERGENCE system combines the publish/subscribe interaction paradigm on top of an information-centric network layer.",
"In contrast, we provide a unified interface such that pull and push based interaction patterns could co-exist in a single network layer while performing in-network computations.",
"GreenICN is an ICN architecture for post-disaster scenarios by combining NDN (pull-based) with COPSS [20] (push-based).",
"However, GreenICN introduces additional data structures, e.g., a subscription table (ST), while we provide this combination using the existing data structures of ICN.",
"Furthermore, it is not clear if GreenICN can function as a whole in a single ICN architecture [41].",
"Carzaniga et al.",
"propose a unified network interface similar to our work, however, the authors only propose a preliminary design of their approach [19] without implementing it in an ICN architecture, and subsequently focus on routing decisions [18] rather than on distributed processing.",
"INP and IoT Architectures in ICNs: Authors in work [43], [38] propose an approach to distribute computational tasks in the network by extending the NFN architecture similar to our work.",
"However, the authors do not deal with the lack of abstractions required for processing continuous data stream.",
"In contrast, we propose a unified communication layer to support CEP over ICN and an extensible query grammar and parser that opens a wide range of operators.",
"Krol et al.",
"[31] propose NFaaS based on unikernels, which is a container based virtualization approach to encapsulate named functions placed on NDN nodes.",
"However, they do not provide support for stateful functions, while IoT functions can be stateful, e.g., involving time windows, which is supported by our architecture.",
"Ahmed et al.",
"[10] propose a smart home approach using NDN and support both push and pull interaction patterns similar to our work.",
"However, in their architecture they only support retrieving raw data, e.g., humidity sensor readings, but not meaningful events as we do.",
"Shang et al.",
"[39] propose a publish/subscribe based approach for modern building management systems (BMS) in NDN.",
"However, the authors build on standard consumer-initiated interaction, as described in Limitation 1 (cf. subsec:decisions).",
"Publish-subscribe deployment for NDN in the IoT scenarios has been discussed in previous works [23], [24].",
"These works confirm the need of integrating producer-initiated interaction in NDN, however, do not provide a unified layer for both interaction patterns as we do.",
"CEP and Networking Architectures CEP Architectures: Several event processing architectures exist, ranging from, e.g., the open source Apache Flink [16] to Twitter's Heron [32] and Google's Millwheel [11].",
"One possibility is to interface one of them with an ICN architecture.",
"Initial work implemented Hadoop on NDN [22] for datacenter applications.",
"However, this requires changing the network model to push in contrast to our work, which would limit the support for a wide range of applications, as discussed above.",
"Networking Architectures: Another emerging network architecture is Software-Defined Networking (SDN) [30], which is gradually being deployed, e.g., in Google's data centers.",
"It allows network managers to program the control plane to support efficient traffic monitoring and engineering.",
"The SDN architecture is complementary to our work, since SDN empowers the control plane, while ICN upgrades the data plane of the current Internet architecture.",
"Data Plane and Query Languages: The literature discusses many CEP query languages [16].",
"The novelty of the proposed query language is to allow for a mapping of operations to ICN's data plane.",
"Alternative designs builds on P4 [14] in the context of SDN.",
"Initial work on programming ICN with P4 [40] faced several difficulties due to lack of key language features and the strong coupling of the language to SDN's data plane model.",
"Discussion In this section, we discuss important future challenges that could be interesting to provide more sophisticated networking, reliability and optimization mechanisms in the INetCEP architecture.",
"Sophisticated Flow and Congestion Control: In CCN, the PIT table ensures the flow balance since one Data packet is sent for each Interest packet.",
"The Data Stream packets of INetCEP could disturb the rule of flow balance since the producer could overflow the buffer on the broker side.",
"For this, INetCEP implements a simple flow control mechanism where we restrict the receiver (consumer/broker) to specify maximum outstanding messages at a time.",
"However, since the forwarding logic of Data Stream packets is similar to IP multicast, existing sophisticated multicast congestion control solutions like TCP-Friendly Multicast Congestion Control [44] and similar can provide sophisticated flow and congestion control.",
"Reliability: The brokers or consumers could miss packets when the available bandwidth and resources at their end is lower than the sending rate.",
"This makes the presence of a module of reliability relevant, which can be catered by extending our work with existing reliable CEP solutions [27] or by looking into equivalent IP solutions such as Scalable Reliable Multicast [21].",
"Query Optimization: In INetCEP we provide a placement module that maps the operator graph to in-network elements of CCN.",
"Another complementary direction could be to generate an optimal operator graph, e.g., based on operator selectivity or even partition operator graph by performing query optimization [15].",
"Optimizing QoS: In INetCEP we provide a programming abstraction for the developers to write CEP queries over ICN data plane substrate.",
"In addition, the placement module can be extended to look into further decentralized solutions and even other QoS metrics [17] like throughput, availability, etc.",
"Conclusion In this paper, we proposed the INetCEP architecture that implements a unified communication layer for co-existing consumer-initiated and producer-initiated interaction patterns.",
"We studied important design challenges to come up with our design of a unified communication layer.",
"In the unified layer, both consumer- and producer-initiated interaction patterns can co-exist in a single ICN architecture.",
"In this way, a wide range of IoT applications are supported.",
"With the proposed query language, we can express interest in aggregated data that is resolved and processed in a distributed manner in the network.",
"In our evaluation, we demonstrated in the context of two IoT case studies that our approach is highly extensible.",
"The performance evaluation showed that queries are efficiently parsed and deployed, which yields - thanks to the in-network deployment - a low end-to-end delay, e.g., simple queries induce only few milliseconds of overall delay.",
"Interesting research directions for future work are: (i) enhancing the performance of query processing by using parallelization, (ii) porting CCNx implementation on real hardware to accomplish low communication delays, and (iii) developing CEP compliant caching strategies.",
"Acknowledgements This work has been co-funded by the German Research Foundation (DFG) as part of the project C2, A3 and C5 within the Collaborative Research Center (CRC) SFB 1053 – MAKI.",
"Appendix In this section, we define our query grammar in apndix:grammar, the implementation details of our extensible design in apndix:extensibility and the algorithms used for extending the system with load forecasting and a heat map application in apndix:applications.",
"Query Grammar Definition 10.1 A grammar consists of four components: A set of terminals or tokens.",
"Terminals are the symbols that occur in a language.",
"A set of non-terminals or syntactic variables.",
"Each of them represent a set of strings.",
"We define them the way we want to use them.",
"A set of production rules that define which non-terminals can be replaced by which terminals, or non-terminals or a combination of both.",
"Here, the terminal is the head of the left side of the production, and the replacement is the body or right side of the production.",
"For example, head $\\rightarrow $ body One of the non-terminals is designated as start symbol for each production.",
"Following the above definition REF and Chomsky-Hierarchy [25], we selected a type 2 grammar or a context-free grammar for CEP over ICN language since a query may consist of multiple subqueries (or operators), which can be expressed (out of many possible ways) using parenthesis \"$()$ \".",
"Table: NO_CAPTIONThe context-free grammar allows us to combine the production rules.",
"The head of each production consist only of one non-terminal and the bodies are not limited by only one terminal and/or one non-terminal.",
"This is because the language needs to embed operators in parenthesis and with a regular grammar we cannot have an arbitrary number of parentheses.",
"We need a way to memorize each parenthesis and this ability is given by context-free grammars.",
"We define an initial INetCEP language grammar and represent it using BNF (Backus-Naur form) in tab:grammar, considering the aforementioned design decisions.",
"We use regular expressions represented as $\\mathit {REG} (\\ldots )$ , where $(\\ldots )$ can be literals $\\mathit {[a-z]}$ and $\\mathit {[A-Z]}$ in lower and upper case, respectively, and numbers $\\mathit {[0-9]}$ .",
"The plus ($+$ ) sign in $\\mathit {REG ([a-z]+)}$ means that at least one lowercase letter has to appear, while a 1 in $\\mathit {REG ([a-z]\\lbrace 1\\rbrace )}$ means that exactly one lowercase letter has to appear.",
"The relational operators given by $\\mathit {comparison}$ define a binary relation between two entities, e.g., two column names of a schema, a column name to a number or a column index to a number.",
"Extensibility To make our query language extensible, we follow a well-known Abstract Factory design pattern from object-oriented programming for our operator definition as illustrated in fig:parserUml.",
"Algorithm REF is the starting point of our operator graph creation.",
"This is implemented in the OperatorTree class.",
"Each operator inherits the abstract class OperatorA which defines the interpret (parseQuery in Algorithm) function as seen in the figure.",
"The checkParameters verifies the correctness of the parameters.",
"Figure: Operator Definition in UMLIf a new operator is to be included, it will override the existing methods of the abstract class and the parameters correctness has to be defined.",
"This allows minimal changes in the implementation for each new application developed using our system.",
"Applications Short term Load Forecasting For the DEBS Grand Challenge in 2014, Martin et al.",
"derived requirements for predicting future energy consumption of a plug [34].",
"We formulate our requirements with respect to our INetCEP system: To meet the goal of making an estimation on future energy consumption, it is necessary to use historical data as a reference.",
"In order to run on a machine with limited resources, the prediction algorithm needs to be lightweight in computation power and storage.",
"Figure: Flow chart explaining the prediction algorithmThe formula for predicting future load is given by the publishers of the DEBS Grand Challenge and is as follows: $ predicted\\_load\\left( s_{i+2} \\right) = \\left( avgLoad\\left( s_{ i } \\right) + median\\left( \\left\\lbrace avgLoads\\left( s_{ j } \\right) \\right\\rbrace \\right) \\right) $ , Here, $i$ is the current timestamp, $s_{i}$ the currently recorded values at time $i$ and $s_{j}$ the past values at a corresponding time $j=i+2$ .",
"The load two steps in the future is therefore made up of the current average electricity consumption and the average electricity consumption from the past.",
"In fig:prediction1operatorflow, we represent the flow of the prediction algorithm as per the requirements defined above and Query .",
"For each time window of 1 minute, we first determine if it is the time for next prediction, which is provided as an input in the query.",
"If the time has not come yet, a value for prediction (average load) is calculated and stored so that it can be used for the equation defined above.",
"Inversely, if it is the time to make a prediction, a prediction tuple of the following form is emitted.",
"$<ts, plug\\_id; household\\_id; house\\_id; predicted\\_load>$ Here, $ts$ is the timestamp of the prediction, $plug\\_id$ identifies a socket in a household, $household\\_id$ identifies a household within a house and $house\\_id$ identifies a house and $predicted\\_load$ is the prediction as specified in the equation above.",
"Heat Map Algorithm REF describes the heat map creation and visualization for the location updates from survivors of the disaster field test used in this work based on [29].",
"[h] $loc$ : Window of location $<lat, long>$ tuple of the survivor $Lat_{min}$ : The minimum latitude value $Lat_{max}$ : The maximum latitude value $Long_{min}$ : The minimum longitude value $Long_{min}$ : The maximum longitude value $HC$ : The number of horizontal cells needed to map the values $VC$ : The number of vertical cells needed to map the values $cell\\_size$ : The granularity $Grid$ : A two dimensional array $HC$ = $\\lfloor {\\frac{Long_{max}- Long_{min}}{cell\\_size}}\\rfloor $ $VC$ = $\\lfloor {\\frac{Lat_{max}- Lat_{min}}{cell\\_size}}\\rfloor $ line in $S_D$ absLatVal = $loc$ [lat] - $Lat_{min}$ absLongVal = $loc$ [long] - $Long_{min}$ $Grid$ [$ \\lfloor {\\frac{absLatVal}{cell\\_size}}\\rfloor $ ][$ \\lfloor {\\frac{absLongVal}{cell\\_size}}\\rfloor $ ] += 1 $Grid$ Algorithm for the heat map operator In line 1, we calculate the number of horizontal cells required for the desired heat map.",
"For this we divide the difference between the maximum and minimum longitude by the desired cell_size, which indicates how large and finely meshed the resulting heat map should be and then round this value down to the next smaller number (given by floor function).",
"In line 2, similarly we compute the vertical cells.",
"For each of these location tuples in the current window, the first absolute latitude and longitude values are computed in line 4 and 5, respectively.",
"By dividing these values by the cell_size, we obtain the corresponding position in the heat map in line 6."
],
[
"Evaluation",
"We evaluate the INetCEP architecture by answering two questions: EQ1 Is the INetCEP system extensible and expressive?",
"EQ2 How is the performance of INetCEP system?",
"To this end, we explain the evaluation setup in Section REF , EQ1 in Section REF and EQ2 in Section REF ."
],
[
"Evaluation Environment",
"We selected the NFN architecture [42] to implement our solution, due to its built-in support of resolving named functions as so-called $\\lambda $ expressions on top of the ICN substrate.",
"However, a major difference to our architecture is that the communication plane in NFN is only consumer-initiated.",
"In contrast, we provide unified communication layer for co-existing consumer- and producer-initiated interactions, while doing CEP operations in the network.",
"As a consequence, we embedded CEP operators as named functions while leveraging NFN's abstract machine to resolve them.",
"NFN works together with CCN-lite [7], which is a lightweight implementation of CCNx and NDN protocol.",
"We have developed unified interfaces of our design on top of NFN (v0.1.0) and CCN-lite (v0.3.0) for the Linux platform [8].",
"We have enhanced the NDN protocol implementation in the CCN-lite and the NFN architecture by: including the additional packet types and their handling, as described in subsec:unified, implementing the extensible general CEP query language, parser, and CEP operators as NFN services, as described in Section REF and implementing a network discovery service with modifications in both CCN and NFN, and operator placement as an NFN service (cf.",
"Section REF ).",
"We evaluated our implementation using the CCN-lite emulator on two topologies: centralized (cf.",
"fig:topoa) and distributed (cf. fig:topob).",
"Each node in our topology is an Ubuntu 16.04 virtual machine (VM) with 8 GiB of memory.",
"Here, each VM (node) is a CCN-NFN relay, which hosts a NFN compute server encapsulating the CEP operator logic.",
"For running the experiments, we first created a CCN network topology as illustrated in fig:topo.",
"Second, we deployed the INetCEP architecture that works on the NFN compute server, the CCN-NFN relay, as well as on the links.",
"Here, as intended, the nodes communicate using the NDN protocol instead of IP.",
"In the centralized topology, we have two producers, a single broker that processes the query and one consumer.",
"In the distributed topology, we have one consumer, two producers and six brokers, as shown in the figure.",
"The data structures CS and PIT are utilized as explained in the previous sections (cf. subsec:unified).",
"We use Queries - (cf.",
"subsec:language) for our evaluation with the DEBS grand challenge 2014 smart home dataset and the disaster field dataset.",
"The dataset is explained in subsec:eq1.",
"Figure: Topology for evaluation.Evaluation Question I: Extensibility To show the extensibility and expressiveness of our approach, we extended the INetCEP architecture for the two representative IoT use cases that we introduced in Section , with a heat map query and a load prediction query.",
"We extended the INetCEP query language and CEP operators to include the heat map [29] and prediction operators [34] by making a few additions to our implementation in our extensible query language and parser.",
"We used real world datasets to evaluate the queries: the 2014 DEBS grand challenge and a disaster field dataset.",
"Dataset 1.",
"For the heat map query, we use a dataset [12] of a field test mimicking a post-disaster situation.",
"The field test mimics two fictive events, a lightning strike and a hazardous substance release from a chemical plant, which resulted in a stressful situation.",
"The collected dataset consists of sensor data, e.g., location coordinates.",
"It was collected from smartphones provided to the participants.",
"Each sensor data stream has a schema specifying the name of the attributes, e.g., the GPS data stream has the following schema: $<ts, s\\_id, latitude, longitude, altitude, accuracy, distance, speed>$ Query.",
"We use the $latitude$ and $longitude$ attributes of this schema to generate the heat map distribution of the survivors from the disaster field test.",
"A typical heat map application joins the GPS data stream from a given set of survivors, derives the area by finding minimum and maximum latitude and longitude values, and visualizes the heat map distribution of the location of the survivors in this area.",
"For simplicity, we consider a data stream from two survivors, as shown in the operator graph in fig:bothqueriesa.",
"Here, $p_1$ and $p_2$ are the producers or GPS sensors, $\\omega _{\\bowtie }$ is the join operator, and $\\omega _{h}$ is the heat map generation operator algorithm [29].",
"This is easily possible using the INetCEP language and parser implementation that follows an Abstract Factory design pattern.",
"First, we included the algorithm for heat map generation, which is $\\tilde{20}$ LOC.",
"Second, we extended the language implementation to include the user defined operator by adding $\\tilde{20}$ LOC.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries HEATMAP( 'cell_size', 'area', JOIN(WINDOW(GPS_S1, 1m), WINDOW(GPS_S2, 1m)) GPS_S1.'ts' = GPS_S2.",
"'ts') ) Figure: Two applications for evaluation (a) a heat map query for post-disaster relief and (b) an energy load forecasting query for smart homes.Dataset 2.",
"The second dataset comes from the 2014 DEBS grand challenge [4] scenario focused on solving a short-term load forecasting problem in a smart grid.",
"The data for the challenge is based on real-world profiles collected from smart home installations.",
"The dataset captured load measurements from unique smart plugs with the following schema: $<ts, id, value, property, plug\\_id, household\\_id, house\\_id>$ Query.",
"We apply an existing solution [34] to perform prediction by extending the INetCEP architecture for two smart plugs.",
"The corresponding operator graph for such a prediction is illustrated in fig:bothqueriesb and listed below.",
"Here, $p_3$ and $p_4$ are the producers or smart plugs, $\\omega _{\\bowtie }$ is a join operator and $\\omega _{pr}$ is a prediction operator based on the algorithm [34].",
"In the first query, we notify the consumer about the predictions for five minutes into the future, while in the second query we notify only if the predictions of load are above a threshold.",
"Similarly to the heat map application, we implemented a prediction algorithm by adding $\\tilde{50}$ LOC and the language implementation with $\\tilde{20}$ LOC.",
"The detailed description of the algorithms for the respective use cases in order to achieve the extensibility is presented in apndix:extensibility.",
"call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries JOIN( PREDICT(5m, WINDOW(PLUG_S1, 1m)), PREDICT(5m, WINDOW(PLUG_S2, 1m)) PLUG_S1.'ts' = PLUG_S2.",
"'ts' ) call, JOIN, WINDOW, FILTER, SUM, MIN, MAX, AVG, COUNT, SEQUENCE, PREDICT, HEATMAP blackbfseries FILTER(JOIN( PREDICT(5m, WINDOW(PLUG_S1, 1m)), PREDICT(5m, WINDOW(PLUG_S2, 1m)) PLUG_S1.'ts' = PLUG_S2.",
"'ts' ), 'load'>20) Evaluation Question II: Performance We evaluated the performance of the INetCEP architecture on standard CEP queries including the Queries -.",
"Additionally, we evaluated the heat map and prediction queries (Queries -).",
"Our aim is to understand the performance of: the query parser on increasing the number of nested operators in operator graph, the operator graph module for different kind of queries, and the operator placement module on increasing number of queries.",
"Figure: Performance of the query parser depending on increasing the number of nested operators.Query Parsing In the query parser design (cf.",
"Section REF ), we performed a complexity analysis of the Algorithm REF .",
"In this section, we verify the analysis experimentally by increasing the number of operators in the operator graph while using the centralized topology shown in fig:topoa.",
"Figure: End-to-end and communication delay observed in Queries 1 to 6.In fig:parser, we show the performance of query parser algorithm in terms of runtime (in ms) for Query or Filter operators and Query or Join operators.",
"We show in a line plot with a confidence interval of 95% for 20 runs of the emulation that the two queries scales reasonably and can be processed in a few milliseconds.",
"Centralized Query Processing We measured the end-to-end delay in processing the six queries defined above using the operator graph module (cf.",
"§ REF ).",
"In fig:evaldelay, we show the results as box plots with a confidence interval of 95% for 10 runs of the emulation.",
"For Query -, the total delay perceived is less than a few milliseconds.",
"It increased for the new queries Query -, where we introduced prediction and heat map operators, primarily due to increased consumption of data and the computational complexity of the algorithms for prediction [34] and heat map [29], respectively.",
"We further show the distribution of the mean end-to-end delay in tab:latency to understand the primary reason for the delay.",
"We listed the time spent in each of the modules of the INetCEP architecture, namely, operator graph creation, placement of operators, and communication of events.",
"The values shown in the table are the mean of the values observed for 10 executions.",
"For basic CEP queries -, the major portion of time ($\\tilde{98}\\%$ ) is spent in communication.",
"This can be explained, as also confirmed in other works [20], by the limitations of the CCNx implementation of the NDN protocol.",
"Hence, we observe the communication delays for all the queries in fig:commq1toq6, where it takes up to 500 ms for delivering results of complex queries like prediction and heatmap.",
"Table: The division of mean end-to-end delay in ms for of operator graph creation, placement of operators, and (communication) delay for centralized placement (see fig:q1toq6).Figure: End-to-end delay observed for Query 3 on incrementally increasing the query workload.",
"Operator Placement To understand the behavior of the operator placement module, we utilize the distributed topology of seven VMs, as illustrated in fig:topob, to place operators based on the information collected by the network discovery service.",
"To take full advantage of distributed CEP, we increased the query load of Query starting from node 1 to 5.",
"For example, the first 10 queries were initialized at the first broker node followed by the next 10 queries at the second node, and so on.",
"Hence, the number of consumers also increased in the network with the query load.",
"In fig:multipleq3, we see that the total delay in retrieving the complex event increased incrementally with the query load, which is reasonable given the static network size (7 nodes) and the resources.",
"However, the queries were distributed evenly using the operator placement algorithm at distinct nodes.",
"To summarize, we evaluated the performance of INetCEP on two topologies: centralized and distributed, using two IoT applications and six different queries.",
"Our evaluation shows that IoT applications can be integrated seamlessly using the INetCEP architecture, CEP queries can be formulated and can be extended for more use cases, and simple CEP queries can be processed in milliseconds.",
"Related Work We now review the state-of-the-art ICN architectures in terms of their support for consumer- and producer-initiated interaction patterns and existing INP architectures in subsec:icnarch, and CEP and networking architectures in subsec:eparch.",
"ICN Architectures Interaction Patterns in ICNs: In Figure REF , we highlight the main ICN architectures NDN [45], NFN [42], DONA [28], PURSUIT [2], PSIRP [1], in terms of their support of a unified communication layer as presented in our work.",
"However, only those appearing in the green box, namely CONVERGENCE [35], GreenICN [3], and Carzaniga et al.",
"[19], provide support for both kind of invocation mechanisms.",
"The CONVERGENCE system combines the publish/subscribe interaction paradigm on top of an information-centric network layer.",
"In contrast, we provide a unified interface such that pull and push based interaction patterns could co-exist in a single network layer while performing in-network computations.",
"GreenICN is an ICN architecture for post-disaster scenarios by combining NDN (pull-based) with COPSS [20] (push-based).",
"However, GreenICN introduces additional data structures, e.g., a subscription table (ST), while we provide this combination using the existing data structures of ICN.",
"Furthermore, it is not clear if GreenICN can function as a whole in a single ICN architecture [41].",
"Carzaniga et al.",
"propose a unified network interface similar to our work, however, the authors only propose a preliminary design of their approach [19] without implementing it in an ICN architecture, and subsequently focus on routing decisions [18] rather than on distributed processing.",
"INP and IoT Architectures in ICNs: Authors in work [43], [38] propose an approach to distribute computational tasks in the network by extending the NFN architecture similar to our work.",
"However, the authors do not deal with the lack of abstractions required for processing continuous data stream.",
"In contrast, we propose a unified communication layer to support CEP over ICN and an extensible query grammar and parser that opens a wide range of operators.",
"Krol et al.",
"[31] propose NFaaS based on unikernels, which is a container based virtualization approach to encapsulate named functions placed on NDN nodes.",
"However, they do not provide support for stateful functions, while IoT functions can be stateful, e.g., involving time windows, which is supported by our architecture.",
"Ahmed et al.",
"[10] propose a smart home approach using NDN and support both push and pull interaction patterns similar to our work.",
"However, in their architecture they only support retrieving raw data, e.g., humidity sensor readings, but not meaningful events as we do.",
"Shang et al.",
"[39] propose a publish/subscribe based approach for modern building management systems (BMS) in NDN.",
"However, the authors build on standard consumer-initiated interaction, as described in Limitation 1 (cf. subsec:decisions).",
"Publish-subscribe deployment for NDN in the IoT scenarios has been discussed in previous works [23], [24].",
"These works confirm the need of integrating producer-initiated interaction in NDN, however, do not provide a unified layer for both interaction patterns as we do.",
"CEP and Networking Architectures CEP Architectures: Several event processing architectures exist, ranging from, e.g., the open source Apache Flink [16] to Twitter's Heron [32] and Google's Millwheel [11].",
"One possibility is to interface one of them with an ICN architecture.",
"Initial work implemented Hadoop on NDN [22] for datacenter applications.",
"However, this requires changing the network model to push in contrast to our work, which would limit the support for a wide range of applications, as discussed above.",
"Networking Architectures: Another emerging network architecture is Software-Defined Networking (SDN) [30], which is gradually being deployed, e.g., in Google's data centers.",
"It allows network managers to program the control plane to support efficient traffic monitoring and engineering.",
"The SDN architecture is complementary to our work, since SDN empowers the control plane, while ICN upgrades the data plane of the current Internet architecture.",
"Data Plane and Query Languages: The literature discusses many CEP query languages [16].",
"The novelty of the proposed query language is to allow for a mapping of operations to ICN's data plane.",
"Alternative designs builds on P4 [14] in the context of SDN.",
"Initial work on programming ICN with P4 [40] faced several difficulties due to lack of key language features and the strong coupling of the language to SDN's data plane model.",
"Discussion In this section, we discuss important future challenges that could be interesting to provide more sophisticated networking, reliability and optimization mechanisms in the INetCEP architecture.",
"Sophisticated Flow and Congestion Control: In CCN, the PIT table ensures the flow balance since one Data packet is sent for each Interest packet.",
"The Data Stream packets of INetCEP could disturb the rule of flow balance since the producer could overflow the buffer on the broker side.",
"For this, INetCEP implements a simple flow control mechanism where we restrict the receiver (consumer/broker) to specify maximum outstanding messages at a time.",
"However, since the forwarding logic of Data Stream packets is similar to IP multicast, existing sophisticated multicast congestion control solutions like TCP-Friendly Multicast Congestion Control [44] and similar can provide sophisticated flow and congestion control.",
"Reliability: The brokers or consumers could miss packets when the available bandwidth and resources at their end is lower than the sending rate.",
"This makes the presence of a module of reliability relevant, which can be catered by extending our work with existing reliable CEP solutions [27] or by looking into equivalent IP solutions such as Scalable Reliable Multicast [21].",
"Query Optimization: In INetCEP we provide a placement module that maps the operator graph to in-network elements of CCN.",
"Another complementary direction could be to generate an optimal operator graph, e.g., based on operator selectivity or even partition operator graph by performing query optimization [15].",
"Optimizing QoS: In INetCEP we provide a programming abstraction for the developers to write CEP queries over ICN data plane substrate.",
"In addition, the placement module can be extended to look into further decentralized solutions and even other QoS metrics [17] like throughput, availability, etc.",
"Conclusion In this paper, we proposed the INetCEP architecture that implements a unified communication layer for co-existing consumer-initiated and producer-initiated interaction patterns.",
"We studied important design challenges to come up with our design of a unified communication layer.",
"In the unified layer, both consumer- and producer-initiated interaction patterns can co-exist in a single ICN architecture.",
"In this way, a wide range of IoT applications are supported.",
"With the proposed query language, we can express interest in aggregated data that is resolved and processed in a distributed manner in the network.",
"In our evaluation, we demonstrated in the context of two IoT case studies that our approach is highly extensible.",
"The performance evaluation showed that queries are efficiently parsed and deployed, which yields - thanks to the in-network deployment - a low end-to-end delay, e.g., simple queries induce only few milliseconds of overall delay.",
"Interesting research directions for future work are: (i) enhancing the performance of query processing by using parallelization, (ii) porting CCNx implementation on real hardware to accomplish low communication delays, and (iii) developing CEP compliant caching strategies.",
"Acknowledgements This work has been co-funded by the German Research Foundation (DFG) as part of the project C2, A3 and C5 within the Collaborative Research Center (CRC) SFB 1053 – MAKI.",
"Appendix In this section, we define our query grammar in apndix:grammar, the implementation details of our extensible design in apndix:extensibility and the algorithms used for extending the system with load forecasting and a heat map application in apndix:applications.",
"Query Grammar Definition 10.1 A grammar consists of four components: A set of terminals or tokens.",
"Terminals are the symbols that occur in a language.",
"A set of non-terminals or syntactic variables.",
"Each of them represent a set of strings.",
"We define them the way we want to use them.",
"A set of production rules that define which non-terminals can be replaced by which terminals, or non-terminals or a combination of both.",
"Here, the terminal is the head of the left side of the production, and the replacement is the body or right side of the production.",
"For example, head $\\rightarrow $ body One of the non-terminals is designated as start symbol for each production.",
"Following the above definition REF and Chomsky-Hierarchy [25], we selected a type 2 grammar or a context-free grammar for CEP over ICN language since a query may consist of multiple subqueries (or operators), which can be expressed (out of many possible ways) using parenthesis \"$()$ \".",
"Table: NO_CAPTIONThe context-free grammar allows us to combine the production rules.",
"The head of each production consist only of one non-terminal and the bodies are not limited by only one terminal and/or one non-terminal.",
"This is because the language needs to embed operators in parenthesis and with a regular grammar we cannot have an arbitrary number of parentheses.",
"We need a way to memorize each parenthesis and this ability is given by context-free grammars.",
"We define an initial INetCEP language grammar and represent it using BNF (Backus-Naur form) in tab:grammar, considering the aforementioned design decisions.",
"We use regular expressions represented as $\\mathit {REG} (\\ldots )$ , where $(\\ldots )$ can be literals $\\mathit {[a-z]}$ and $\\mathit {[A-Z]}$ in lower and upper case, respectively, and numbers $\\mathit {[0-9]}$ .",
"The plus ($+$ ) sign in $\\mathit {REG ([a-z]+)}$ means that at least one lowercase letter has to appear, while a 1 in $\\mathit {REG ([a-z]\\lbrace 1\\rbrace )}$ means that exactly one lowercase letter has to appear.",
"The relational operators given by $\\mathit {comparison}$ define a binary relation between two entities, e.g., two column names of a schema, a column name to a number or a column index to a number.",
"Extensibility To make our query language extensible, we follow a well-known Abstract Factory design pattern from object-oriented programming for our operator definition as illustrated in fig:parserUml.",
"Algorithm REF is the starting point of our operator graph creation.",
"This is implemented in the OperatorTree class.",
"Each operator inherits the abstract class OperatorA which defines the interpret (parseQuery in Algorithm) function as seen in the figure.",
"The checkParameters verifies the correctness of the parameters.",
"Figure: Operator Definition in UMLIf a new operator is to be included, it will override the existing methods of the abstract class and the parameters correctness has to be defined.",
"This allows minimal changes in the implementation for each new application developed using our system.",
"Applications Short term Load Forecasting For the DEBS Grand Challenge in 2014, Martin et al.",
"derived requirements for predicting future energy consumption of a plug [34].",
"We formulate our requirements with respect to our INetCEP system: To meet the goal of making an estimation on future energy consumption, it is necessary to use historical data as a reference.",
"In order to run on a machine with limited resources, the prediction algorithm needs to be lightweight in computation power and storage.",
"Figure: Flow chart explaining the prediction algorithmThe formula for predicting future load is given by the publishers of the DEBS Grand Challenge and is as follows: $ predicted\\_load\\left( s_{i+2} \\right) = \\left( avgLoad\\left( s_{ i } \\right) + median\\left( \\left\\lbrace avgLoads\\left( s_{ j } \\right) \\right\\rbrace \\right) \\right) $ , Here, $i$ is the current timestamp, $s_{i}$ the currently recorded values at time $i$ and $s_{j}$ the past values at a corresponding time $j=i+2$ .",
"The load two steps in the future is therefore made up of the current average electricity consumption and the average electricity consumption from the past.",
"In fig:prediction1operatorflow, we represent the flow of the prediction algorithm as per the requirements defined above and Query .",
"For each time window of 1 minute, we first determine if it is the time for next prediction, which is provided as an input in the query.",
"If the time has not come yet, a value for prediction (average load) is calculated and stored so that it can be used for the equation defined above.",
"Inversely, if it is the time to make a prediction, a prediction tuple of the following form is emitted.",
"$<ts, plug\\_id; household\\_id; house\\_id; predicted\\_load>$ Here, $ts$ is the timestamp of the prediction, $plug\\_id$ identifies a socket in a household, $household\\_id$ identifies a household within a house and $house\\_id$ identifies a house and $predicted\\_load$ is the prediction as specified in the equation above.",
"Heat Map Algorithm REF describes the heat map creation and visualization for the location updates from survivors of the disaster field test used in this work based on [29].",
"[h] $loc$ : Window of location $<lat, long>$ tuple of the survivor $Lat_{min}$ : The minimum latitude value $Lat_{max}$ : The maximum latitude value $Long_{min}$ : The minimum longitude value $Long_{min}$ : The maximum longitude value $HC$ : The number of horizontal cells needed to map the values $VC$ : The number of vertical cells needed to map the values $cell\\_size$ : The granularity $Grid$ : A two dimensional array $HC$ = $\\lfloor {\\frac{Long_{max}- Long_{min}}{cell\\_size}}\\rfloor $ $VC$ = $\\lfloor {\\frac{Lat_{max}- Lat_{min}}{cell\\_size}}\\rfloor $ line in $S_D$ absLatVal = $loc$ [lat] - $Lat_{min}$ absLongVal = $loc$ [long] - $Long_{min}$ $Grid$ [$ \\lfloor {\\frac{absLatVal}{cell\\_size}}\\rfloor $ ][$ \\lfloor {\\frac{absLongVal}{cell\\_size}}\\rfloor $ ] += 1 $Grid$ Algorithm for the heat map operator In line 1, we calculate the number of horizontal cells required for the desired heat map.",
"For this we divide the difference between the maximum and minimum longitude by the desired cell_size, which indicates how large and finely meshed the resulting heat map should be and then round this value down to the next smaller number (given by floor function).",
"In line 2, similarly we compute the vertical cells.",
"For each of these location tuples in the current window, the first absolute latitude and longitude values are computed in line 4 and 5, respectively.",
"By dividing these values by the cell_size, we obtain the corresponding position in the heat map in line 6."
],
[
"Related Work",
"We now review the state-of-the-art ICN architectures in terms of their support for consumer- and producer-initiated interaction patterns and existing INP architectures in subsec:icnarch, and CEP and networking architectures in subsec:eparch."
],
[
"ICN Architectures",
" Interaction Patterns in ICNs: In Figure REF , we highlight the main ICN architectures NDN [45], NFN [42], DONA [28], PURSUIT [2], PSIRP [1], in terms of their support of a unified communication layer as presented in our work.",
"However, only those appearing in the green box, namely CONVERGENCE [35], GreenICN [3], and Carzaniga et al.",
"[19], provide support for both kind of invocation mechanisms.",
"The CONVERGENCE system combines the publish/subscribe interaction paradigm on top of an information-centric network layer.",
"In contrast, we provide a unified interface such that pull and push based interaction patterns could co-exist in a single network layer while performing in-network computations.",
"GreenICN is an ICN architecture for post-disaster scenarios by combining NDN (pull-based) with COPSS [20] (push-based).",
"However, GreenICN introduces additional data structures, e.g., a subscription table (ST), while we provide this combination using the existing data structures of ICN.",
"Furthermore, it is not clear if GreenICN can function as a whole in a single ICN architecture [41].",
"Carzaniga et al.",
"propose a unified network interface similar to our work, however, the authors only propose a preliminary design of their approach [19] without implementing it in an ICN architecture, and subsequently focus on routing decisions [18] rather than on distributed processing.",
"INP and IoT Architectures in ICNs: Authors in work [43], [38] propose an approach to distribute computational tasks in the network by extending the NFN architecture similar to our work.",
"However, the authors do not deal with the lack of abstractions required for processing continuous data stream.",
"In contrast, we propose a unified communication layer to support CEP over ICN and an extensible query grammar and parser that opens a wide range of operators.",
"Krol et al.",
"[31] propose NFaaS based on unikernels, which is a container based virtualization approach to encapsulate named functions placed on NDN nodes.",
"However, they do not provide support for stateful functions, while IoT functions can be stateful, e.g., involving time windows, which is supported by our architecture.",
"Ahmed et al.",
"[10] propose a smart home approach using NDN and support both push and pull interaction patterns similar to our work.",
"However, in their architecture they only support retrieving raw data, e.g., humidity sensor readings, but not meaningful events as we do.",
"Shang et al.",
"[39] propose a publish/subscribe based approach for modern building management systems (BMS) in NDN.",
"However, the authors build on standard consumer-initiated interaction, as described in Limitation 1 (cf. subsec:decisions).",
"Publish-subscribe deployment for NDN in the IoT scenarios has been discussed in previous works [23], [24].",
"These works confirm the need of integrating producer-initiated interaction in NDN, however, do not provide a unified layer for both interaction patterns as we do."
],
[
"CEP and Networking Architectures",
" CEP Architectures: Several event processing architectures exist, ranging from, e.g., the open source Apache Flink [16] to Twitter's Heron [32] and Google's Millwheel [11].",
"One possibility is to interface one of them with an ICN architecture.",
"Initial work implemented Hadoop on NDN [22] for datacenter applications.",
"However, this requires changing the network model to push in contrast to our work, which would limit the support for a wide range of applications, as discussed above.",
"Networking Architectures: Another emerging network architecture is Software-Defined Networking (SDN) [30], which is gradually being deployed, e.g., in Google's data centers.",
"It allows network managers to program the control plane to support efficient traffic monitoring and engineering.",
"The SDN architecture is complementary to our work, since SDN empowers the control plane, while ICN upgrades the data plane of the current Internet architecture.",
"Data Plane and Query Languages: The literature discusses many CEP query languages [16].",
"The novelty of the proposed query language is to allow for a mapping of operations to ICN's data plane.",
"Alternative designs builds on P4 [14] in the context of SDN.",
"Initial work on programming ICN with P4 [40] faced several difficulties due to lack of key language features and the strong coupling of the language to SDN's data plane model."
],
[
"Discussion",
"In this section, we discuss important future challenges that could be interesting to provide more sophisticated networking, reliability and optimization mechanisms in the INetCEP architecture.",
"Sophisticated Flow and Congestion Control: In CCN, the PIT table ensures the flow balance since one Data packet is sent for each Interest packet.",
"The Data Stream packets of INetCEP could disturb the rule of flow balance since the producer could overflow the buffer on the broker side.",
"For this, INetCEP implements a simple flow control mechanism where we restrict the receiver (consumer/broker) to specify maximum outstanding messages at a time.",
"However, since the forwarding logic of Data Stream packets is similar to IP multicast, existing sophisticated multicast congestion control solutions like TCP-Friendly Multicast Congestion Control [44] and similar can provide sophisticated flow and congestion control.",
"Reliability: The brokers or consumers could miss packets when the available bandwidth and resources at their end is lower than the sending rate.",
"This makes the presence of a module of reliability relevant, which can be catered by extending our work with existing reliable CEP solutions [27] or by looking into equivalent IP solutions such as Scalable Reliable Multicast [21].",
"Query Optimization: In INetCEP we provide a placement module that maps the operator graph to in-network elements of CCN.",
"Another complementary direction could be to generate an optimal operator graph, e.g., based on operator selectivity or even partition operator graph by performing query optimization [15].",
"Optimizing QoS: In INetCEP we provide a programming abstraction for the developers to write CEP queries over ICN data plane substrate.",
"In addition, the placement module can be extended to look into further decentralized solutions and even other QoS metrics [17] like throughput, availability, etc."
],
[
"Conclusion",
"In this paper, we proposed the INetCEP architecture that implements a unified communication layer for co-existing consumer-initiated and producer-initiated interaction patterns.",
"We studied important design challenges to come up with our design of a unified communication layer.",
"In the unified layer, both consumer- and producer-initiated interaction patterns can co-exist in a single ICN architecture.",
"In this way, a wide range of IoT applications are supported.",
"With the proposed query language, we can express interest in aggregated data that is resolved and processed in a distributed manner in the network.",
"In our evaluation, we demonstrated in the context of two IoT case studies that our approach is highly extensible.",
"The performance evaluation showed that queries are efficiently parsed and deployed, which yields - thanks to the in-network deployment - a low end-to-end delay, e.g., simple queries induce only few milliseconds of overall delay.",
"Interesting research directions for future work are: (i) enhancing the performance of query processing by using parallelization, (ii) porting CCNx implementation on real hardware to accomplish low communication delays, and (iii) developing CEP compliant caching strategies."
],
[
"Acknowledgements",
"This work has been co-funded by the German Research Foundation (DFG) as part of the project C2, A3 and C5 within the Collaborative Research Center (CRC) SFB 1053 – MAKI."
],
[
"Appendix",
"In this section, we define our query grammar in apndix:grammar, the implementation details of our extensible design in apndix:extensibility and the algorithms used for extending the system with load forecasting and a heat map application in apndix:applications."
],
[
"Query Grammar",
"Definition 10.1 A grammar consists of four components: A set of terminals or tokens.",
"Terminals are the symbols that occur in a language.",
"A set of non-terminals or syntactic variables.",
"Each of them represent a set of strings.",
"We define them the way we want to use them.",
"A set of production rules that define which non-terminals can be replaced by which terminals, or non-terminals or a combination of both.",
"Here, the terminal is the head of the left side of the production, and the replacement is the body or right side of the production.",
"For example, head $\\rightarrow $ body One of the non-terminals is designated as start symbol for each production.",
"Following the above definition REF and Chomsky-Hierarchy [25], we selected a type 2 grammar or a context-free grammar for CEP over ICN language since a query may consist of multiple subqueries (or operators), which can be expressed (out of many possible ways) using parenthesis \"$()$ \".",
"Table: NO_CAPTIONThe context-free grammar allows us to combine the production rules.",
"The head of each production consist only of one non-terminal and the bodies are not limited by only one terminal and/or one non-terminal.",
"This is because the language needs to embed operators in parenthesis and with a regular grammar we cannot have an arbitrary number of parentheses.",
"We need a way to memorize each parenthesis and this ability is given by context-free grammars.",
"We define an initial INetCEP language grammar and represent it using BNF (Backus-Naur form) in tab:grammar, considering the aforementioned design decisions.",
"We use regular expressions represented as $\\mathit {REG} (\\ldots )$ , where $(\\ldots )$ can be literals $\\mathit {[a-z]}$ and $\\mathit {[A-Z]}$ in lower and upper case, respectively, and numbers $\\mathit {[0-9]}$ .",
"The plus ($+$ ) sign in $\\mathit {REG ([a-z]+)}$ means that at least one lowercase letter has to appear, while a 1 in $\\mathit {REG ([a-z]\\lbrace 1\\rbrace )}$ means that exactly one lowercase letter has to appear.",
"The relational operators given by $\\mathit {comparison}$ define a binary relation between two entities, e.g., two column names of a schema, a column name to a number or a column index to a number."
],
[
"Extensibility",
" To make our query language extensible, we follow a well-known Abstract Factory design pattern from object-oriented programming for our operator definition as illustrated in fig:parserUml.",
"Algorithm REF is the starting point of our operator graph creation.",
"This is implemented in the OperatorTree class.",
"Each operator inherits the abstract class OperatorA which defines the interpret (parseQuery in Algorithm) function as seen in the figure.",
"The checkParameters verifies the correctness of the parameters.",
"Figure: Operator Definition in UMLIf a new operator is to be included, it will override the existing methods of the abstract class and the parameters correctness has to be defined.",
"This allows minimal changes in the implementation for each new application developed using our system."
],
[
"Short term Load Forecasting",
"For the DEBS Grand Challenge in 2014, Martin et al.",
"derived requirements for predicting future energy consumption of a plug [34].",
"We formulate our requirements with respect to our INetCEP system: To meet the goal of making an estimation on future energy consumption, it is necessary to use historical data as a reference.",
"In order to run on a machine with limited resources, the prediction algorithm needs to be lightweight in computation power and storage.",
"Figure: Flow chart explaining the prediction algorithmThe formula for predicting future load is given by the publishers of the DEBS Grand Challenge and is as follows: $ predicted\\_load\\left( s_{i+2} \\right) = \\left( avgLoad\\left( s_{ i } \\right) + median\\left( \\left\\lbrace avgLoads\\left( s_{ j } \\right) \\right\\rbrace \\right) \\right) $ , Here, $i$ is the current timestamp, $s_{i}$ the currently recorded values at time $i$ and $s_{j}$ the past values at a corresponding time $j=i+2$ .",
"The load two steps in the future is therefore made up of the current average electricity consumption and the average electricity consumption from the past.",
"In fig:prediction1operatorflow, we represent the flow of the prediction algorithm as per the requirements defined above and Query .",
"For each time window of 1 minute, we first determine if it is the time for next prediction, which is provided as an input in the query.",
"If the time has not come yet, a value for prediction (average load) is calculated and stored so that it can be used for the equation defined above.",
"Inversely, if it is the time to make a prediction, a prediction tuple of the following form is emitted.",
"$<ts, plug\\_id; household\\_id; house\\_id; predicted\\_load>$ Here, $ts$ is the timestamp of the prediction, $plug\\_id$ identifies a socket in a household, $household\\_id$ identifies a household within a house and $house\\_id$ identifies a house and $predicted\\_load$ is the prediction as specified in the equation above."
],
[
"Heat Map",
"Algorithm REF describes the heat map creation and visualization for the location updates from survivors of the disaster field test used in this work based on [29].",
"[h] $loc$ : Window of location $<lat, long>$ tuple of the survivor $Lat_{min}$ : The minimum latitude value $Lat_{max}$ : The maximum latitude value $Long_{min}$ : The minimum longitude value $Long_{min}$ : The maximum longitude value $HC$ : The number of horizontal cells needed to map the values $VC$ : The number of vertical cells needed to map the values $cell\\_size$ : The granularity $Grid$ : A two dimensional array $HC$ = $\\lfloor {\\frac{Long_{max}- Long_{min}}{cell\\_size}}\\rfloor $ $VC$ = $\\lfloor {\\frac{Lat_{max}- Lat_{min}}{cell\\_size}}\\rfloor $ line in $S_D$ absLatVal = $loc$ [lat] - $Lat_{min}$ absLongVal = $loc$ [long] - $Long_{min}$ $Grid$ [$ \\lfloor {\\frac{absLatVal}{cell\\_size}}\\rfloor $ ][$ \\lfloor {\\frac{absLongVal}{cell\\_size}}\\rfloor $ ] += 1 $Grid$ Algorithm for the heat map operator In line 1, we calculate the number of horizontal cells required for the desired heat map.",
"For this we divide the difference between the maximum and minimum longitude by the desired cell_size, which indicates how large and finely meshed the resulting heat map should be and then round this value down to the next smaller number (given by floor function).",
"In line 2, similarly we compute the vertical cells.",
"For each of these location tuples in the current window, the first absolute latitude and longitude values are computed in line 4 and 5, respectively.",
"By dividing these values by the cell_size, we obtain the corresponding position in the heat map in line 6."
]
] | 2012.05239 |
[
[
"Subdivision schemes on a dyadic half-line"
],
[
"Abstract In this paper subdivision schemes, which are used for functions approximation and curves generation, are considered.",
"In classical case, for the functions defined on the real line, the theory of subdivision schemes is widely known due to multiple applications in constructive approximation theory, signal processing as well as for generating fractal curves and surfaces.",
"Subdivision schemes on a dyadic half-line, which is a positive half-line, equipped with the standard Lebesgue measure and the digitwise binary addition operation, where the Walsh functions play the role of exponents, are defined and studied.",
"Necessary and sufficient convergence conditions of the subdivision schemes in terms of spectral properties of matrices and in terms of the smoothness of the solution of the corresponding refinement equation are proved.",
"The problem of the convergence of subdivision schemes with non-negative coefficients is also investigated.",
"Explicit convergence criterion of the subdivision schemes with four coefficients is obtained.",
"As an auxiliary result fractal curves on a dyadic half-line are defined and the formula of their smoothness is proved.",
"The paper contains various illustrations and numerical results."
],
[
"a4paper, left=30mm, right=15mm, top=24mm, bottom=24mm 3 Karapetyants M.A.",
"Subdivision schemes on a dyadic half-line Abstract In this paper subdivision schemes, which are used for functions approximation and curves generation, are considered.",
"In classical case, for the functions defined on the real line, the theory of subdivision schemes is widely known due to multiple applications in constructive approximation theory, signal processing as well as for generating fractal curves and surfaces.",
"Subdivision schemes on a dyadic half-line – positive half-line, equipped with the standard Lebesgue measure and the digitwise binary addition operation, where the Walsh functions play the role of exponents, are defined and studied.",
"Necessary and sufficient convergence conditions of the subdivision schemes in terms of spectral properties of matrices and in terms of the smoothness of the solution of the corresponding refinement equation are proved.",
"The problem of the convergence of subdivision schemes with non-negative coefficients is also investigated.",
"Explicit convergence criterion of the subdivision schemes with four coefficients is obtained.",
"As an auxiliary result fractal curves on a dyadic half-line are defined and the formula of their smoothness is proved.",
"The paper contains various illustrations and numerical results.",
"Bibliography: 26 items.",
"Keywords: Subdivision schemes, dyadic half-line, fractal curves, smoothness of fractal curves, spectral properties of matrices.",
"We begin with the definition of the dyadic half-line [1], [20].",
"We consider sequences $x = \\lbrace \\ldots , x_{j-1}, x_j, x_{j+1}, \\ldots \\rbrace ,$ where $ x_j \\in \\lbrace 0, 1\\rbrace $ and $ x_j = 0 $ , $ j > k $ , for sufficiently large $ k > 0 $ .",
"Each element $x$ corresponds to a convergent series $\\sum _{n\\in {{\\mathbb {Z}}}}x_n 2^n$ , which is the binary decomposition of a number in $ \\mathbb {R}_+ = [0; \\infty )$ .",
"Define algebraic summation operation as follows: sum of two sequences $ x $ and $ y $ is sequence $ z $ , such that $z \\ = \\ x \\oplus y \\ = \\ \\lbrace x_i \\oplus y_i \\rbrace $ for each $i$ ; $x_i \\oplus y_i = \\left\\lbrace \\begin{array}{cc}0, & x_i + y_i \\in \\lbrace 0, 2\\rbrace , \\\\1, & x_i + y_i = 1.\\end{array}\\right.$ It follows, that $x \\oplus x \\ = \\ 0 \\ $ for each $x \\in {\\mathbb {R}}_+$ , which implicates the coincidence of $\\oplus $ operation and inverse operation $\\ominus $ .",
"For instance, $3 = 11_{2}, \\ 6 = 110_{2}, \\ 3 \\oplus 6 = 5$ The continuity of a dyadic function on ${\\mathbb {R}}_+$ in literature is called $W$ -continuity (after Walsh, it seems).",
"A function $f$ is $W$ -continuous at the point $x$ , if $\\lim _{h \\rightarrow 0} |f(x \\oplus h) - f(x)| = 0$ Basically, $W$ -continuous function $f$ is continuous from the right at dyadic rational points of the half-line and continuous in usual sense at all others.",
"Half-line ${\\mathbb {R}}_+$ equipped with $\\oplus $ operation and $W$ -continuity we call dyadic half-line.",
"Unit shift of the dyadic half-line is represented below: linear function alters as it is shown on the figure 1.",
"Figure: Unit shift of the dyadic half-line.The first examples of subdivision schemes (in the classical case, that is, on a straight line with regular addition) appeared in the early fifties in the works of G. de Rham (the famous algorithm of ”cutting corners” [5], [14]).",
"Then this idea was developed in the works of G. Chaikin [9], G. Deslauriers and S. Dubuc [15].",
"The general theory of subdivision algorithms was represented by N. Dyn, A. Levin, C. Micchelli, V. Dahmen and A. Kavarette, K. Conti, P. Osvald, and other researchers.",
"(eg.",
"[8], [16], [18], [19] and references in these papers).",
"All these papers are devoted to the approximation of functions, or the construction of fractal curves on the usual (classical) line, or generalizations to functions of several variables.",
"We restrict ourselves to the functions of one variable.",
"Let us recall the basic concepts and facts about the subdivision schemes on the real line.",
"Consider the space $\\ell _\\infty $ and a linear operator $ \\tilde{S}: \\ell _\\infty \\rightarrow {\\ell }_\\infty $ , defined by a finite set of coefficients called mask: $ { \\lbrace c_j \\rbrace }_{j \\in \\mathbb {J}} $ .",
"This operator acts according to the following rule: for any sequence $a \\in {\\ell }_{\\infty }$ $(\\tilde{S} a)(\\alpha ) \\ = \\ \\sum _{\\beta \\in {{\\mathbb {Z}}}} c_{\\alpha - 2\\beta } \\ a_\\beta $ One iteration of subdivision scheme consists in the application of the operator $ \\tilde{S} $ to some sequence $a$ .",
"We match sequence $a$ piecewise linear function with nodes at integer points, with the value of the function at the node $k$ equal to $a_k$ .",
"So we do with the sequence $ \\tilde{S} a $ , but with nodes in half-integer points (on $\\frac{1}{2} \\mathbb {Z}$ ).",
"We say, that subdivision scheme converges, if for any bounded sequence there is a continuous function $f = f_a$ such, that $\\lim _{n \\rightarrow \\infty } \\Vert \\ f(2^{-n} \\cdot ) - (\\tilde{S}^na)(\\cdot ) \\Vert _{\\infty } = 0$ Thus, if the scheme converges, then for each sequence $a$ there is a continuous function $f_a$ .",
"Dyadic subdivision operator matches sequence $a \\in {\\ell }_{\\infty }$ sequence $Sa$ , defined as follows: $(Sa)(\\alpha ) \\ = \\ \\sum _{\\beta \\in {{\\mathbb {Z}}}} c_{\\alpha \\ominus 2\\beta } \\ a_\\beta $ By dyadic subdivision scheme we mean the sequential application of the operator $S$ to some bounded sequence $a$ .",
"The convergence of the dyadic subdivision scheme is determined similarly to the classical one, but the limit function will be defined on $ \\mathbb {R}_+ $ , and will be $W$ -continuous.",
"Definition 1 A dyadic subdivision scheme converges if, for any bounded sequence, there exists a $W$ -continuous function $f$ such that $\\lim _{n \\rightarrow \\infty } \\Vert \\ f(2^{-n} \\cdot ) - (S^n a)(\\cdot ) \\Vert _{\\infty } = 0$ This paper is devoted to the study of various properties of dyadic subdivision operators and schemes, as well as the associated dyadic fractals.",
"In Section 2 the necessary and sufficient conditions for the convergence of the dyadic subdivision schemes are formulated.",
"In Section 3 the conditions under which two affine operators generate a dyadic fractal curve are investigated.",
"This result is used in the construction of limit functions of subdivision schemes.",
"Section 4 examines some spectral properties of subdivision schemes.",
"Sections 5 and 6 are devoted to dyadic subdivision schemes with positive and non-negative coefficients, respectively.",
"Section 7 discusses in detail the simplest special case of the dyadic scheme defined by four coefficients, and its explicit convergence conditions are obtained.",
"Section 8 consists entirely of examples and illustrations of various dyadic subdivision schemes.",
"In order to apply a subdivision scheme to interpolation problems, it is necessary for this scheme to converge.",
"First of all, it should be established under what conditions the subdivision scheme is guaranteed to converge for any initial sequence.",
"To do this, we need several auxiliary assertions.",
"Define $\\delta $ -sequence as follows: $\\delta (\\cdot ) = (1, 0,0, \\ldots )$ .",
"Lemma 1 The subdivision scheme converges if and only if it converges on $\\delta $ -sequence.",
"If the subdivision scheme converges, then, obviously, it converges on $\\delta $ -sequence.",
"Now, note that any bounded sequence can be represented as a linear combination of shifts of $\\delta $ -sequence: $a \\ = \\ \\sum _{k \\in {{\\mathbb {Z}}}} \\delta (\\cdot \\ominus k) a_k$ Therefore, if the subdivision scheme converges on $\\delta $ -sequence, then it converges on any bounded sequence due to the linearity of the subdivision operator.",
"This fact means that it is sufficient to verify the convergence on $\\delta $ -sequence, and if it is possible to establish the fact of convergence on $ \\delta $ , then it is transferred to all other sequences in view of the lemma.",
"Lemma 2 Let $\\varphi $ be limit function of the subdivision scheme on $\\delta $ -sequence.",
"If there is a limit function $f_a$ on any sequence $a \\in \\ell _\\infty $ , then it is given as follows: $f \\ = \\ \\sum _{k \\in {{\\mathbb {Z}}}}\\, {a}_k\\, \\varphi (x \\ominus k)$ As mentioned above, any limited sequence can be represented as (REF ).",
"We apply $k$ times to the both parts of (REF ) operator $S$ and tend $k$ to infinity.",
"For each term on the right side of the equality, there is a limit function, therefore, the limit function for the left side of the equality also exists and is given by (REF ).",
"So, let $\\varphi $ be the limit function of the subdivision scheme on the delta sequence (provided the scheme converges).",
"Is it possible to find this function without using a subdivision scheme?",
"The answer is affirmative: this function is the solution of a certain difference equation with the compression of the argument.",
"Lemma 3 Let $\\varphi $ be a limit function of a subdivision scheme.",
"Then $\\varphi $ is the solution of the following equation $\\varphi (x)\\ = \\ \\sum _{k\\in {{\\mathbb {Z}}}} c_k \\varphi (2x \\ominus k)$ Remark 1 Equation (REF ) is called refinement regarding $\\varphi $ .",
"Let the subdivision scheme with the coefficients $ \\lbrace c_k\\rbrace _{k = 0}^N $ converge and $\\varphi $ be its limit function on $\\delta $ -sequence.",
"Since after one iteration $S$ on $\\delta $ we get a sequence consisting entirely of coefficients that define subdivision operator, then $S^n(\\delta ) = \\sum _{k\\in {{\\mathbb {Z}}}} c_k S^{n-1}(\\delta (\\cdot \\ominus k)).$ On each step $n$ we match $\\ S^n(\\delta )$ with the sequence on $R_+$ with interval $2^{-n}$ , beginning from zero, and connect the obtained points piecewise linearly.",
"The functions thus obtained $f_n$ will converge to $\\varphi $ due to the convergence of the scheme: $\\varphi = \\lim _{n\\rightarrow \\infty }f_n,$ and, similarly $f_n(\\cdot ) = \\sum _{k\\in {{\\mathbb {Z}}}} c_k f_{n-1}(\\cdot \\ominus k).$ It only remains to note that multiplying the initial sequence by a constant is equivalent to multiplying the limit function by a constant, while shifting the argument of the initial sequence shifts the argument of the limiting function.",
"Passing to the limit as $n \\rightarrow \\infty $ in equality (REF ) and given that we have done one iteration \"manually\", we obtain $\\varphi (x) = \\sum _{k\\in {{\\mathbb {Z}}}} c_k \\varphi (2x \\ominus k).$ Not any subdivision scheme converges.",
"Our immediate goal is to obtain convergence conditions.",
"We start with the necessary conditions.",
"They repeat exactly the same conditions for the classical subdivision schemes on the line.",
"Function $f$ is called uniformly continuous on $\\mathbb {R}_+$ , if for each $ \\varepsilon > 0$ there exists $ \\delta > 0 $ such, that $ \\Vert f(y) - f(x) \\Vert < \\varepsilon $ , as soon as $ \\Vert y \\ominus x \\Vert < \\delta , \\ \\ x, y \\in \\mathbb {R}_+ $ .",
"Theorem 4 Given a subdivision scheme with mask $ \\lbrace c_k\\rbrace _{k = 0}^N $ .",
"If the scheme converges, then $\\sum _{k \\in {{\\mathbb {N}}}} c_{2k} = \\sum _{k \\in {{\\mathbb {N}}}} c_{2k+1} = 1.$ Let the initial sequence be given: $a \\ = \\ \\lbrace \\ldots , a_{-1}, a_0, a_1, \\ldots \\rbrace .$ It can be associated with the function $ f_0(x) $ defined on a uniform lattice on $ R_+ $ with a step one.",
"Similarly the sequence $Sa$ could be associated with the function $f_1(x)$ , but with a different step: $\\frac{1}{2}$ , etc.",
"Consider the sum $\\sum _{k \\in {{\\mathbb {N}}}} c_{2k} f_n(x_{2k}),$ where $f_n(x_{2k})$ is value of function $f_n(x)$ , associated with coefficient $c_{2k}$ .",
"Since the scheme converges to some function f(x), then for any $ \\varepsilon > 0 $ there is a number $N$ such, that for each $n > N$ the inequality $ |f_n(x)-f(x)| < \\varepsilon $ holds.",
"For each n $\\ge $ N on the iteration n intervals between function values $f_n(x)$ are small and equal to $2^{-n}$ , consequently, the values of function $f_n(x)$ do not differ much, because for $f_n(x)$ there is a uniform continuity on each of the segments whose ends are binary rational points.",
"Let for $ {\\varepsilon }_n = 1/n $ be a number $ n_0 > N $ such, that for each $ p \\in \\mathbb {N} $ the inequality $ |f_{n_0+p}(x)-f(x)| < {\\varepsilon }_n $ holds.",
"Consider one of these segments and denote ${\\hat{f}}_n$ as a value of function $f_n(x)$ at the left end of this segment.",
"We suppose that on $n+1$ step the values of the function $f_{n+1}(x)$ differ in the corresponding points from ${\\hat{f}}_n$ no more than $2{\\varepsilon }_n$ (otherwise, we will select a new $ n_0 $ to satisfy this condition).",
"On the other hand, $\\sum _{k \\in {{\\mathbb {N}}}} c_{2k} ( {\\hat{f}}_n - 2{\\varepsilon }_n ) \\ \\le \\ f_{n+1}(x) \\ = \\ \\sum _{k \\in {{\\mathbb {N}}}} c_{2k} f_n(x_{2k}) \\ \\le \\ \\sum _{k \\in {{\\mathbb {N}}}} c_{2k} ( {\\hat{f}}_n + 2{\\varepsilon }_n ),$ where ${\\varepsilon }_n \\rightarrow 0$ as $n \\rightarrow \\infty $ .",
"But $|f_{n+1}(x) - {\\hat{f}}_n| < 2{\\varepsilon }_n$ , from which and the last inequality it follows that $\\sum _{k \\in {{\\mathbb {N}}}} c_{2k} = 1.$ Similarly, it is proved, that $ \\sum _{k \\in {{\\mathbb {N}}}} c_{2k+1} = 1 $ .",
"Theorem 4 is a necessary condition for the convergence of a subdivision scheme in terms of its mask.",
"What is sufficient for the convergence of the subdivision scheme?",
"To answer this question we need to state a few auxiliary definitions.",
"In future, unless otherwise specified, we will omit the symbol $ W $ in the designation of $W$ -continuity, i.e.",
"by continuity we mean $W$ -continuity.",
"By $C(\\mathbb {R}_+)$ we define a space of $W$ -continuous functions.",
"Let $f$ be a compactly supported continuous function.",
"Definition 2 We say, that function $f$ possesses linearly independent integer shifts if for any sequence $a$ $\\sum _{k\\in {{\\mathbb {Z}}}} a_k f(x \\ominus k) \\ \\lnot \\equiv \\ 0$ The function $f$ is called stable if (REF ) is satisfied for any $a \\in {\\ell }_\\infty $ .",
"We define now the linear operator $T:C(\\mathbb {R}_+) \\rightarrow C(\\mathbb {R}_+)$ , acting according to the rule $T(f(x)) \\ = \\ \\sum _{k=0}^N p_k f(2x \\ominus k),$ where $ \\lbrace p_k \\rbrace _{k = 0}^N $ is a finite set of coefficients.",
"In literature (in the classical case) this operator is called the transition operator.",
"We now establish a connection between the operators $S$ and $T$ .",
"Lemma 5 Let $f(x)$ be continuous and compactly supported.",
"There is an equality: $(T^nf)(x) = \\sum _{k \\in {{\\mathbb {Z}}}} (S^n \\delta )(k) f(2^nx \\ominus k)$ We prove this formula by induction.",
"For $n \\ = \\ 1$ we have $Tf(x) = \\sum _{k=0}^{N} p_k f(2x \\ominus k).$ On the other side, $\\sum _{k=0}^N p_k f(2x \\ominus k) \\ = \\ \\sum _{k=0}^N (S(\\delta ))(k) f(2x \\ominus k).$ Let the formula be true for $n-1$ : $(T^{n-1}f)(x)\\ = \\ \\sum _{k=0}^{N} (S^{n-1}(\\delta ))(k) f(2^{n-1}x \\ominus k).$ Consider $T(T^{n-1}f)(x) = T^{n}(x)$ : $T(\\sum _{k=0}^N p_k f(2^{n-1}x \\ominus k)) \\ = \\ T(\\sum _{k=0}^N (S^{n-1}(\\delta ))(k) f(2^{n-1}x \\ominus k) \\ = \\ $ $= \\ \\sum _{j=0}^N \\sum _{k=0}^N (S^{n-1}(\\delta ))(k) p_j f(2^nx \\ominus j \\ominus 2k) \\ = \\ \\sum _{j=0}^N \\sum _{k=0}^N (S^{n-1}(\\delta ))(k) p_{j \\ominus 2k} f(2^nx \\ominus j) \\ = \\ $ $= \\ \\sum _{j=0}^N (S^n(\\delta ))(j) f(2^nx \\ominus j).$ Lemma is proved.",
"Now we formulate a theorem providing conditions sufficient for the convergence of the scheme.",
"Its proof practically repeats the classical case.",
"Theorem 6 If the refinement equation $\\varphi (x) \\ = \\ \\sum _{k \\in {{\\mathbb {Z}}}} c_k \\varphi (2x \\ominus k)$ has a continuous and stable solution $f(x)$ , then the corresponding subdivision scheme converges.",
"Due to the lemma REF we only have to prove this theorem for $\\delta $ -sequence.",
"By lemma REF : $(T^n f)(x) = \\sum _{k \\in {{\\mathbb {Z}}}} (S^n(\\delta ))(k) f(2^n x \\ominus k).$ If the scheme converges, then it converges to a continuous function.",
"$f(x)$ and $f(x) = \\sum _{k \\in {{\\mathbb {Z}}}} (S^n(\\delta ))(k) f(2^n x \\ominus k),$ because $f(x)$ is an eigenfunction of $T$ .",
"Consider 1-periodical function $F(x) = \\sum _{k \\in {{\\mathbb {Z}}}} f(x \\ominus k)$ .",
"We show that it is constant.",
"Indeed, $\\sum _{i \\in {{\\mathbb {Z}}}} f(\\frac{x}{2} \\ominus i) \\ = \\ \\sum _{i \\in {{\\mathbb {Z}}}} \\sum _{j \\in {{\\mathbb {Z}}}} {c}_j f(x \\ominus 2i \\ominus j) \\ = \\ $ $\\ = \\ \\sum _{i \\in {{\\mathbb {Z}}}} \\sum _{j \\in {{\\mathbb {Z}}}} {c}_{j-2i} f(x \\ominus j) \\ = \\ \\sum _{i \\in {{\\mathbb {Z}}}} f(x \\ominus i),$ since in order for the scheme to converge, it is necessary that the sum of all even (as well as odd) coefficients are equal to one.",
"We obtained, that $F(\\frac{x}{2}) = F(x)$ - the function is constant at half-integer points.",
"Similarly, we obtain that it is constant at all binary rational points, and since this set is everywhere dense on ${\\mathbb {R}}_+$ , then $F(x)$ is constant on ${\\mathbb {R}}_+$ due to its continuity.",
"By condition the function $f(x)$ is stable on $\\ell _\\infty $ , therefore, function $F(x)$ differs from zero and we may put $F(x) = \\sum _{k \\in {{\\mathbb {Z}}}} f(x \\ominus k) = 1$ .",
"As function $f(x)$ is continuous and compactly supported, then for each $ \\varepsilon > 0 $ $|f(x) - \\sum _{k \\in {{\\mathbb {Z}}}} f(2^{-n}k) f(2^nx \\ominus k)| < \\varepsilon \\ \\ \\forall x \\in {{{\\mathbb {R}}}}_+$ for large enough $n$ .",
"Futhermore, $|\\sum _{k \\in {{\\mathbb {Z}}}} \\lbrace f(2^{-n}k) - (S^n(\\delta ))(k)\\rbrace f(2^n x \\ominus k)| < \\varepsilon \\ \\ \\forall x \\in {{{\\mathbb {R}}}}_+,$ and from the condition of stability of the function $f(x)$ it follows, that $|f(2^{-n}k) - (S^n(\\delta ))(k)| < const \\cdot \\varepsilon \\ \\ \\forall k \\in {{\\mathbb {Z}}}$ and for large enough $n$ , that is, we have established the convergence of the subdivision scheme on $\\delta $ -sequence, and therefore on all sequences from ${\\ell }_\\infty $ .",
"Even in the classical case, studying the refinement equation for the existence of a continuous solution is not easy.",
"It is known [2], that in the classical case when the necessary conditions are met (REF ), the refinement equation always has a generalized compactly supported solution.",
"Moreover, this solution is unique up to normalization.",
"[2].",
"However, it may not be continuous.",
"The criterion of continuity of the solution was first obtained in the work [11], based on the developed in [13] matrix method.",
"In work [6] This method was generalized to dyadic refinement equations.",
"The idea of the method is as follows: instead of studying a functional equation, one can go to an equation for a vector function, that is, from a refinement equation to the so-called self-similarity equation, which will be described below.",
"The self-similarity equation is a special case of the equation for fractal curves.",
"We will prove this result in the most general form, for arbitrary fractal curves.",
"The next chapter is fully devoted to the question of generating fractal curves on a dyadic half-line.",
"For an arbitrary pair of affine operators in $ \\mathbb {R}^N$ , consider a functional equation for a vector function ${v}: [0,1] \\rightarrow \\mathbb {R}^N$ ${v}(t) = \\left\\lbrace \\begin{array}{cc}A_0 {v}(2t), & t \\in [0, \\frac{1}{2}), \\\\A_1 {v}(2t \\ominus 1), & t \\in [\\frac{1}{2}, 1].\\end{array}\\right.$ This equation with binary compression of an argument in literature is called self-similarity equation, by analogy with the same equation on the segment [0, 1] with the usual addition [2].",
"In the classical case, the name is justified by the fact that many famous fractal curves, such as the de Rham curve, the Koch curve, etc.",
"are solutions of such equations.",
"Indeed, it follows from the equation that the arc of a curve ${v}(t)$ between $t = 0$ and $t = \\frac{1}{2}$ affine-like (by operator $A_0$ ) the whole curve, and so is (by operator $A_1$ ) the second arc of a curve, from $t = \\frac{1}{2}$ to $t =1$ .",
"So, the point $ {v}(\\frac{1}{2}) $ divides the curve $\\lbrace {v}(t), t \\in [0, 1] \\rbrace $ into two arcs, each of which is affine-like throughout the curve.",
"A fractal curve is a continuous solution of the equation (REF ) (with the usual addition).",
"On the dyadic half-line we mean the W-continuous solution.",
"We formulate the conditions under which two affine operators generate a fractal curve.",
"For this we need an auxiliary Theorem 7 For any two operators $A_0$ , $A_1$ in $ \\mathbb {R}^N $ and for each $\\varepsilon > 0$ there exists a norm in $\\mathbb {R}^N$ such, that $(\\rho - \\varepsilon ) \\Vert x\\Vert \\le \\max (\\Vert {A}_0 x \\Vert , \\Vert {A}_1 x \\Vert ) \\le (\\rho + \\varepsilon ) \\Vert x\\Vert $ for any $ x \\in \\mathbb {R}^N $ , where $\\rho $ is a joint spectral radius of operators, namely: $\\rho \\ = \\ \\lim _{r \\rightarrow \\infty } \\max _{(d_1, ..., d_r) \\in { \\lbrace 0, 1 \\rbrace }^r } { \\Vert A_{d_1} \\cdot \\ldots \\cdot A_{d_r} \\Vert }^{ \\frac{1}{r} }.$ The proof of this theorem could be found in [23].",
"Dyadic modulus of continuity of a function $f$ is a number $ {w}(f, \\delta ) $ such, that $\\forall \\delta > 0 \\ \\ {w}(f, \\delta ) = \\displaystyle { {\\sup }_{x, y \\in [0, 2^{n-1}), 0 \\le y < \\delta } } \\lbrace \\Vert f(x \\oplus y) - f(x) \\Vert \\rbrace $ If $ {w}(f, 2^{-n}) \\le const \\cdot 2^{- \\alpha n} $ for some $\\alpha > 0$ , then $ {w}(f, \\delta ) \\le const(f, \\alpha ) \\cdot \\delta ^{\\alpha }$ Dyadic Hölder exponent ${\\alpha }_{f}$ of a function $f$ is: ${\\alpha }_{f} = \\sup _{\\alpha > 0} \\lbrace \\alpha : {w}(f, \\delta ) \\le const(f, \\alpha ) \\cdot \\delta ^\\alpha \\rbrace .$ Let $A_0$ , $A_1$ $\\in Aff({{\\mathbb {R}}}^N)$ и $\\Vert A_0\\Vert \\le \\rho $ , $\\Vert A_1\\Vert \\le \\rho $ , $\\rho \\in (0,1)$ .",
"Then Theorem 8 If $\\rho (A_0, A_1) <1$ , then equation (REF ) has a continuous solution ${v}(t)$ .",
"Wherein, ${\\alpha }_{{v}} \\ge - \\log _2 \\rho $ .",
"If there are no common affine subspaces of operators $ A_0, A_1 $ , then ${\\alpha }_{{v}} = - \\log _2 \\rho $ .",
"Consider binary rational $x$ and $y$ and without loss of generality we assume, that $x, y \\in (0, 1)$ и $x < y$ .",
"Let us evaluate the norm $\\Vert {v}(x) - {v}(y)\\Vert $ .",
"Consider the set $P \\subset {{\\mathbb {N}}}$ of indices $p \\in {{\\mathbb {N}}}$ such, that $p$ -th digits in the binary decomposition $x$ and $y$ are different.",
"The set $P$ is finite due to $x$ and $y$ being binary rational.",
"Let ${p}_0$ be the smallest element of $P$ ; it is obvious, that $|y \\ominus x| \\ge 2^{-{p}_0}$ .",
"Then $\\Vert {v}(y) - {v}(x)\\Vert \\ = \\ \\Vert \\sum _{p \\in {{\\mathbb {P}}}} {v}(0.",
"{d}_1 \\ldots {d}_{p-1}1) - {v}(0.",
"{d}_1 \\ldots {d}_{p-1}0)\\Vert \\le $ $\\le \\sum _{p \\in {{\\mathbb {P}}}} \\Vert {v}(0.",
"{d}_1 \\ldots {d}_{p-1}1) - {v}(0.",
"{d}_1 \\ldots {d}_{p-1}0) \\Vert \\ =\\sum _{p \\in {{\\mathbb {P}}}} \\Vert {A}_{{d}_1} \\ldots {A}_{{d}_{p-1}} ({A}_1 {{v}}_0 - {A}_0 {{v}}_0)\\Vert \\le $ $\\le \\ \\sum _{p \\in {{\\mathbb {P}}}} \\Vert {A}_{{d}_1} \\ldots {A}_{{d}_{p-1}} \\Vert \\cdot \\Vert {A}_1 {{v}}_0 - {A}_0 {{v}}_0 \\Vert \\ \\le \\ \\sum _{p \\in {{\\mathbb {P}}}} {( \\rho ({A}_0, {A}_1) + \\varepsilon )}^{p-1} \\cdot \\Vert {A}_1 {{v}}_0 - {A}_0 {{v}}_0 \\Vert ,$ where ${\\rho ({A}_0, {A}_1)}$ is a joint spectral radius of operators ${A}_0$ и ${A}_1$ .",
"Let $ \\sigma $ be a sum of a finite number series $\\sum _{p \\in {{\\mathbb {P}}}} {( \\rho ({A}_0, {A}_1) + \\varepsilon )}^{p-1}$ .",
"Then there is a constant ${C}_1$ such, that $ {C}_1 = \\frac{\\sigma }{ {( \\rho ({A}_0, {A}_1) + \\varepsilon )}^{ {p}_0 - 1} } $ .",
"Let us denote $ {C}_2 = \\Vert {A}_1 {{v}}_0 - {A}_0 {{v}}_0 \\Vert = \\Vert {A}_1 {{v}}_0 - {{v}}_0 \\Vert $ .",
"Let $ C = {C}_1 \\cdot {C}_2 $ .",
"We have: $\\sum _{p \\in {{\\mathbb {P}}}} {( \\rho ({A}_0, {A}_1) + \\varepsilon )}^{p-1} \\cdot \\Vert {A}_1 {{v}}_0 - {A}_0 {{v}}_0 \\Vert = C \\cdot \\frac{{( \\rho ({A}_0, {A}_1) + \\varepsilon )}^{ {p}_0 }}{\\rho ({A}_0, {A}_1) + \\varepsilon }$ Consequently, $\\Vert {v}(y) - {v}(x)\\Vert \\le \\tilde{C} {|y \\ominus x|}^{-\\log _2{\\rho (A_0, A_1) + \\varepsilon }},$ where $ \\tilde{C} = \\frac{C}{\\rho ({A}_0, {A}_1) + \\varepsilon } $ and function ${v}(t)$ is uniformly continuous at binary-rational points of the interval (0,1), and, therefore, on the segment [0,1].",
"From the evaluation above it also follows, that ${\\alpha }_{{v}} \\ge -\\log _2{\\rho ({A}_0, {A}_1)}.$ If for the operators $ A_0, \\ A_1 $ there are no common affine subspaces, then the reverse inequality is proved in a similar way (the proof almost literally coincides with similar arguments in the proof of theorem 5.1.4 of [2]).",
"Due to the fact that $ \\varepsilon $ can be arbitrarily small, we obtain ${\\alpha }_{{v}} = -\\log _2 \\rho (A_0, A_1).$ Note that in the classical case the Theorem REF is false.",
"For the existence of a continuous solution on the classical real line, an additional condition is necessary, sometimes called cross condition or Barnsley condition: $A_0 {{v}}_1 = A_1 {{v}}_0$ , where ${{v}}_0, {{v}}_1$ are the fixed points of operators $A_0, A_1$ from (REF ) respectively.",
"This condition eliminates the possibility of the function ${v}(t)$ to be discontinuous at the point $t = \\frac{1}{2}$ .",
"In the dyadic case, $W$ -continuity admits the existence of discontinuities at binary rational points, so that in the dyadic case this condition is not required.",
"Now we apply the results of Section 3 on fractal curves to study the convergence of refinement schemes.",
"Operator $S$ possesses two $ N \\times N $ matrices, where $ N = 2^{n-1}, $ ${T_0}_{ij} = c_{2(i-1)\\oplus (j-1)}\\, , \\qquad {T_1}_{ij} = c_{(2i-1)\\oplus (j-1)}\\, , \\qquad 1 \\le i, j \\le N = 2^{n-1}$ We represent the explicit form of these matrices for $n = 3$ : $T_0 \\ = \\ \\begin{pmatrix}c_0 & c_1 & c_2 & c_3 \\\\c_2 & c_3 & c_0 & c_1 \\\\c_4 & c_5 & c_6 & c_7 \\\\c_6 & c_7 & c_4 & c_5\\end{pmatrix}$ $T_1 \\ = \\ \\begin{pmatrix}c_1 & c_0 & c_3 & c_2 \\\\c_3 & c_2 & c_1 & c_0 \\\\c_5 & c_4 & c_7 & c_6 \\\\c_7 & c_6 & c_5 & c_4\\end{pmatrix}$ By conditions (REF ) for ${T}_0$ , ${T}_1$ there is a common invariant affine subspace $W = \\lbrace x \\in {{{\\mathbb {R}}}}^N: \\sum _{i=1}^{N} x_i = 1 \\rbrace $ .",
"On this subspace of the matrices $T_0, \\, T_1$ define affine operators.",
"We denote $ V \\subset W $ as a smallest common affine subspace containing eigenvector $T_0$ , corresponding to eigenvalue 1, and ${T}_0{|}_V = {A}_0$ , ${T}_1{|}_V = {A}_1$ .",
"Now we return to the refinement equation and apply the method described in [13].",
"So, for the refinement equation $\\varphi (t) = \\sum _{k=0}^N c_k \\varphi (2t \\ominus k)$ we define a vector-function $ {v}(t) = ( \\varphi (t), \\ \\varphi (t+1), \\ ..., \\ \\varphi (t-N+1) ) \\in \\mathbb {R}^N$ at arbitrary $ t $ , besides, ${v}(t) = \\left\\lbrace \\begin{array}{cc}T_0 {v}(2t), & t \\in [0, \\frac{1}{2}], \\\\T_1 {v}(2t \\ominus 1), & t \\in [\\frac{1}{2}, 1].\\end{array}\\right.$ Since $W$ is invariant with respect to operators $ T_0, T_1 $ , the function $ {v}$ also satisfies ${v}(t) = \\left\\lbrace \\begin{array}{cc}A_0 {v}(2t), & t \\in [0, \\frac{1}{2}], \\\\A_1 {v}(2t \\ominus 1), & t \\in [\\frac{1}{2}, 1],\\end{array}\\right.$ which combined with the Theorem REF leads to the following result: Theorem 9 The refinement equation $\\varphi (t) = \\sum _{k=0}^{N} {c}_k \\varphi (2t - k)$ possesses a continuous solution if and only if $\\rho (A_0,A_1) < 1$ .",
"Moreover, $\\alpha _{\\varphi } = - \\log _2 \\rho (A_0, A_1)$ .",
"Corollary 1 If $\\rho (T_0|_{W},T_1|_{W} ) < 1$ , then the solution of the refinement equation is continuous and $\\alpha _{\\varphi } \\ge - \\log _2 \\rho (T_0|_{W},T_1|_{W} )$ .",
"Remark 2 The corollary 1 has been proven also in[6].",
"Theorem REF is the consequence of Theorem REF and it is a criterion that the refinement equation has a continuous solution in terms of operators ${A}_0$ , ${A}_1$ .",
"It allows us to establish whether the limit function of the subdivision scheme is continuous without finding its explicit form.",
"And, accordingly, if the limit function is discontinuous, then there is no point in speaking about the convergence of the scheme.",
"So we first switched from the functional equation (REF ) to the equation for the vector function(REF ), figured out under what conditions it has a continuous solution (Theorem REF ), then went back to the refinement equation (Theorem REF ).",
"Now it is easy to verify whether the solution is stable and, applying the Theorem REF , find out if the corresponding subdivision scheme converges.",
"We ended up with a convergence theorem for subdivision schemes, which is fairly easy to use.",
"Next, we consider some special cases of subdivision schemes.",
"We start with the subdivision schemes defined by positive coefficients.",
"To begin with we will study some of their combinatorial properties.",
"A matrix is called stochastic in columns (rows) if it is non-negative and the sum of all elements of each column (row) equals one.",
"The following auxiliary result is well known in the theory of Markov chains.",
"We give his proof for the convenience of the reader.",
"Recall that $ W \\ = \\ \\lbrace x: \\displaystyle {\\sum _{k \\in \\mathbb {Z}} x_k = 1} \\rbrace $ , $ \\Delta \\ = \\ \\lbrace x \\in W: x \\ge 0 \\rbrace $ is a unit simplex.",
"Theorem 10 If $A$ is column-stochastic and possesses at least one positive row, and $x$ , $y$ are the elements of $ \\Delta $ , then there exists $ q \\in (0,1) $ such, that ${\\Vert Ax - Ay\\Vert }_{{\\ell }_1} \\le q{\\Vert x - y\\Vert }_{{\\ell }_1}.$ Consider the function $\\Vert x - y\\Vert _{\\ell _1} = f(x,y)$ : on a compact set $ \\Delta $ its maximum is reached at extreme points (because $ f(x, y) $ is a convex function), that is, points that are not the midpoints of any segments in this set.",
"Consequently, to prove that the distance between two points decreases under the action of a certain matrix, it is sufficient to prove that it decreases for the extreme points.",
"Without loss of generality, let $x$ be the first basis vector in $W$ (with a 1 in the first place and zeros on all others), $y$ be the second one (with a 1 in second place and zeros on all others), and $\\Vert x - y\\Vert _{\\ell _1} = 2$ .",
"Multiplying the matrix $ A $ by each of them, we get the first and second columns of it, respectively.",
"It is necessary to prove that $\\Vert Ax - Ay\\Vert _{\\ell _1} < 2$ .",
"Let $i$ be a number of the positive row of $A$ , and $m < 1$ is the smallest element in the row.",
"Element $Ax$ has at least $m$ on the $i$ -th place.",
"So does $Ay$ .",
"Consider $Ax$ and represent it as $Ax \\ = \\ {x}_i + \\sum _{j \\ne i} {x}_j, \\ {x}_i \\ge m,$ where elements $ x_j $ have a non-zero component only on $ j $ -th place.",
"Similarly for $Ay$ .",
"For each $j \\ne i$ it holds that $|({Ax})_j - ({Ay})_j| \\le {x}_j - {y}_j.$ In the $i$ -th row: $|({Ax})_i - ({Ay})_i| \\le {x}_i - {y}_i - 2m.$ Consequently, ${\\Vert Ax - Ay \\Vert }_{\\ell _1} \\le \\sum _{i \\in {{\\mathbb {N}}}} |{x}_i - {y}_i| - 2m \\ = \\ 2 - 2m,$ that is, the norm has changed $ 1-m $ times, $1 > 1-m > 0$ .",
"This fact allows us to prove the convergence of dyadic subdivision schemes with a positive mask with the help of the following property of convergence of schemes, the proof of which the reader can find in [17] (in the case of the dyadic half-line, it completely repeats the classical analogue).",
"Theorem 11 If there is a norm in which the operators $T_0, \\ T_1$ are contractions, then the subdivision scheme converges.",
"Theorem 12 The subdivision scheme with a strictly positive mask converges.",
"It is enough to establish that the matrices ${T}_0$ , ${T}_1$ always have at least one positive row.",
"Then by the Theorem REF they are contractions in $ \\ell _1 $ norm, which implies the convergence of the scheme according to the Theorem REF .",
"Note that both ${T}_0$ , ${T}_1$ are column stochastic.",
"Let $I \\ = \\ \\lbrace i: {c}_i > 0\\rbrace $ be the set of indices, corresponding to positive mask coefficients of a particular subdivision scheme.",
"Let also $N = 2^{n-1}$ be the dimension of matrices ${T}_0$ , ${T}_1$ , and $2^n$ is the number of coefficients.",
"In order for each of the matrices to have at least one positive line, it is necessary that for a fixed $ i $ the following inequalities are satisfied: $0 \\le 2(i-1) \\oplus (j-1) \\le 2^n - 1,$ $0 \\le (2i-1) \\oplus (j-1) \\le 2^n - 1.$ That is, all indices of the elements of each matrix in some row belong to the set $I$ .",
"Let $i = 1$ , consider ${T}_0$ .",
"It is remained to investigate whether the minimum and maximum of $i$ -th row of ${T}_0$ are in $I$ or not.",
"The inequality above will take the form of $0 \\le j - 1 \\le 2^n - 1.$ The left part of the inequality is always satisfied.",
"$j - 1$ will be maximum when $j = N$ ($j$ takes the highest possible even value).",
"Therefore, we have $2^{n-1} - 1 \\le 2^n - 1,$ which obviously also holds.",
"We obtain that when $i = 1$ all the coefficients of the matrix ${T}_0$ are positive, which means the matrix has a positive row.",
"Consider now ${T}_1$ and likewise we set $i = 1$ .",
"We have $0 \\le j \\oplus 2 \\le 2^n - 1.$ The left side of the inequality, again, is always fulfilled.",
"Let the number $N = 2^{n-1}$ contain in its binary decomposition the rank of two.",
"$j - 1$ will be maximum when $j = N \\oplus 3$ ($j$ takes the highest possible even value).",
"Then we have $2^{n-1} - 1 \\le 2^n - 1,$ which is obviously fulfilled.",
"If the number $N = 2^{n-1}$ do not contain in its binary decomposition the rank of two, then $j - 1$ will be maximum when $j = N$ ($j$ takes the maximum possible value).",
"Then the inequality takes the following form: $2^{n-1} \\oplus 2 \\le 2^n - 1 = 2 \\cdot 2^{n-1} - 1,$ which is true under these conditions.",
"Therefore, the matrix $T_1$ also has a positive row and the theorem is proved.",
"We now consider subdivision schemes with a mask of non-negative elements.",
"Let $I \\ = \\ \\lbrace i: {c}_i > 0\\rbrace $ , ${I}_N \\ = \\ I \\oplus 2I \\oplus \\ldots \\oplus 2^{N-1}I$ .",
"We present a criterion for the convergence of dyadic refinement schemes with nonnegative coefficients (A similar result for classical refinement schemes can be found in [21], its proof is completely transferred to the dyadic case): Theorem 13 Subdivision scheme with non-negative coefficients converges if and only if there exists $N$ such, that for each $i$ there is $j$ such, that $i \\oplus l \\ominus 2^N j$ in ${I}_N$ , l = 0, ..., n - 1, $N = 2^{n-1}$ That is, set ${I}_N$ should be dense enough on the segment $ [\\min \\lbrace i: i \\in {I}_N \\rbrace , \\max \\lbrace i: i \\in {I}_N \\rbrace ].",
"$ Based on the criterion, we hypothesize: Conjecture 1 Subdivision scheme with nonnegative coefficients converges unless greatest common divisor of $ I \\ne 1 $ .",
"Remark 3 Note that the classical analogue of the hypothesis above includes another case where the schemes obviously do not converge, namely: $I$ has exactly one odd element and it is the last one (or $I$ has exactly one even element and it is the first one).",
"If this condition is satisfied, then the solution of the refinement equation will be discontinuous, at least at zero, but in the case of dyadic subdivision schemes, such discontinuities are quite acceptable.",
"Moreover, many numerical examples (section 8, example 9) show that when executed (REF ), such dyadic subdivision schemes converge.",
"We give an example of a scheme, which GCD($I$ ) $ \\ne $ 1, and show that it diverges.",
"Let the set of indices of non-negative coefficients be as follows: $I = \\lbrace 0, 6, 9, 15\\rbrace $ .",
"Consider the sum $I \\oplus 2I \\ = \\ \\lbrace 0, 6, 9, 15; 12, 10, 5, 3; 18, 20, 27, 29; 30, 24, 23, 17\\rbrace \\ = \\ \\lbrace I, {I}_1, {I}_2, {I}_3\\rbrace $ It can be seen that there are no numbers $\\lbrace 1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28\\rbrace $ .",
"That is, by Theorem REF such a scheme does not converge.",
"Indeed, the set $I \\oplus 2I \\oplus 4I$ does not fill these gaps, but only add new ones, for example: $I \\oplus 2I \\oplus 24 \\ = \\ I \\oplus 2I.$ A whole series of numbers falls out, and this is repeated for the set ${I}_n$ at least once for each $n$ .",
"That is, the conditions of the criterion are obviously not fulfilled.",
"Hypothesis 1 states that the convergence of non-negative subdivision schemes can be simply verified.",
"For arbitrary schemes, this is not so: the convergence of a subdivision scheme depends on the magnitude of the joint spectral radius of the matrices.",
"$A_0, \\, A_1$ (section 3).",
"Another class of schemes whose convergence is relatively easy to verify is schemes with a small number of coefficients.",
"In the next section, we show that the convergence of schemes with four coefficients is verified elementary.",
"Note that in the classical theory of subdivision schemes this is not the case!",
"Consider arbitrary dyadic subdivision scheme with coefficients $ \\lbrace {c}_0, {c}_1, {c}_2, {c}_3 \\rbrace $ .",
"If it converges, then by the Theorem REF the sequence of its coefficients can be rewritten as $ \\lbrace c_0, c_1, 1 - c_0, 1 - c_1 \\rbrace $ .",
"Matrices $ T_0, T_1 $ have the form, respectively: $\\begin{pmatrix}c_0 & c_1 \\\\1 - c_0 & 1 - c_1\\end{pmatrix}$ $\\begin{pmatrix}c_1 & c_0 \\\\1 - c_1 & 1 - c_0\\end{pmatrix}$ Consider the restriction of these matrices to a common linear invariant subspace $X \\ = \\ \\lbrace {x} : {x}_1 + {x}_2 = 0 \\rbrace $ and represent them in basis $ {(1; -1)}^T $ We obtain: $T_0 \\cdot {(1; -1)}^T = (c_0 - c_1) \\cdot {(1; -1)}^T$ $T_1 \\cdot {(1; -1)}^T = (c_1 - c_0) \\cdot {(1; -1)}^T$ It is known [8], that the subdivision scheme converges, if the joint spectral radius of $T_0, T_1$ , which here equals to $ \\max \\lbrace | c_0 - c_1 | , | c_1 - c_0 | \\rbrace $ , is less than one.",
"We obtain: $ \\max \\lbrace | c_0 - c_1 | , | c_1 - c_0 | \\rbrace < 1$ and, therefore, ${\\left\\lbrace \\begin{array}{ll}c_0 - c_1 < 1, \\\\c_1 - c_0 < 1.\\end{array}\\right.",
"}$ The system above can be reduced to a double inequality: $ c_0 - 1 < c_1 < c_0 + 1 $ .",
"We depict its solution in the figure below: Figure: The connection between c 0 c_0 and c 1 c_1.Thus, all the schemes of the four coefficients, of which the first two satisfy the strict inequality above, obviously converge.",
"Remark 4 The classical theory of subdivision schemes does not provide a complete description of all cases in which the scheme of four coefficients converges.",
"(e.g.",
"[2], [4]).",
"We present some illustrations of dyadic subdivision schemes and compare them with their classical counterparts.",
"For simplicity, we represent only the refinable functions of the subdivision schemes (the remaining sequences are obtained by various shifts, classical and dyadic, of these functions).",
"All the images below were taken after ten iterations of the subdivision schemes.",
"Example 6 illustrates the case when in the classical case the subdivision scheme diverges, and in the dyadic it converges.",
"Let the scheme be given by a sequence of coefficients $c = \\lbrace 0.3, 0.1, 0.7, 0.9 \\rbrace $ .",
"We construct the refinable function of this scheme in the classical and dyadic case.",
"Figure: NO_CAPTIONIn both cases the refinable function is continuous.",
"In the classical case it is supported by a segment $ [0; 3] $ .",
"At the point $ x = 3 $ the function is zero, despite it is quite unclear on the figure 3 (the function rapidly decreases to zero).",
"In the dyadic case the refinable function is supported by the segment $ [0; 2] $ and is zero at the point $ x = 2 $ correspondingly.",
"Wherein at the point $ x = 2 $ there is a discontinuity, which, as we know, does not contradict the definition of $W$ -continuity.",
"Next, we present a scheme with coefficients $c = \\lbrace 0.6, 0.9, 0.4, 0.1 \\rbrace $ .",
"Figure: NO_CAPTION Now the coefficients are equal to $c = \\lbrace 0.6, 1.1, 0.4, -0.1 \\rbrace $ .",
"Figure: NO_CAPTION Now $c = \\lbrace \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{4} \\rbrace $ .",
"Figure: NO_CAPTION Let us give an example of divergent scheme: $c = \\lbrace 2.6, 0.7, -1.6, 0.3 \\rbrace $ .",
"Figure: NO_CAPTION Let us give an example, when in the classical case the subdivision scheme diverges, and in the dyadic case it converges: $c = \\lbrace 0.4, -0.1, 0.6, 1.1 \\rbrace $ .",
"Figure: NO_CAPTIONIn the classical case, it cannot converge, since the last coefficient is greater than one (see [8]).",
"In the dyadic case, it converges for reasons from the section 7: the conditions REF for the first two coefficients are fulfilled: $c_0 - 1 < c_1 < c_0 + 1,$ $0.4 - 1 < -0.1 < 0.4 + 1,$ and, therefore, such a scheme converges.",
"Consider the schemes given by the eight coefficients.",
"Put all eight coefficients equal to $ \\frac{1}{4} $ : Figure: NO_CAPTION Now $c = \\lbrace \\frac{1}{8}, \\frac{3}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{1}{8} \\rbrace $ .",
"Figure: NO_CAPTION Let us give an example of a dyadic interpolating ( $ c_0 \\ = \\ 1 $ , the rest of the even coefficients are zeros) subdivision scheme defined by sixteen coefficients: $c = \\lbrace 1, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8} \\rbrace $ .",
"Figure: NO_CAPTION Let the scheme be given by a sequence of coefficients $c = \\lbrace 0.3, 0.1, 0.7, 0.9 \\rbrace $ .",
"We construct the refinable function of this scheme in the classical and dyadic case.",
"Figure: NO_CAPTIONIn both cases the refinable function is continuous.",
"In the classical case it is supported by a segment $ [0; 3] $ .",
"At the point $ x = 3 $ the function is zero, despite it is quite unclear on the figure 3 (the function rapidly decreases to zero).",
"In the dyadic case the refinable function is supported by the segment $ [0; 2] $ and is zero at the point $ x = 2 $ correspondingly.",
"Wherein at the point $ x = 2 $ there is a discontinuity, which, as we know, does not contradict the definition of $W$ -continuity.",
"Next, we present a scheme with coefficients $c = \\lbrace 0.6, 0.9, 0.4, 0.1 \\rbrace $ .",
"Figure: NO_CAPTIONNow the coefficients are equal to $c = \\lbrace 0.6, 1.1, 0.4, -0.1 \\rbrace $ .",
"Figure: NO_CAPTIONNow $c = \\lbrace \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{4} \\rbrace $ .",
"Figure: NO_CAPTIONLet us give an example of divergent scheme: $c = \\lbrace 2.6, 0.7, -1.6, 0.3 \\rbrace $ .",
"Figure: NO_CAPTIONLet us give an example, when in the classical case the subdivision scheme diverges, and in the dyadic case it converges: $c = \\lbrace 0.4, -0.1, 0.6, 1.1 \\rbrace $ .",
"Figure: NO_CAPTIONIn the classical case, it cannot converge, since the last coefficient is greater than one (see [8]).",
"In the dyadic case, it converges for reasons from the section 7: the conditions REF for the first two coefficients are fulfilled: $c_0 - 1 < c_1 < c_0 + 1,$ $0.4 - 1 < -0.1 < 0.4 + 1,$ and, therefore, such a scheme converges.",
"Consider the schemes given by the eight coefficients.",
"Put all eight coefficients equal to $ \\frac{1}{4} $ : Figure: NO_CAPTIONNow $c = \\lbrace \\frac{1}{8}, \\frac{3}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{1}{8} \\rbrace $ .",
"Figure: NO_CAPTIONLet us give an example of a dyadic interpolating ( $ c_0 \\ = \\ 1 $ , the rest of the even coefficients are zeros) subdivision scheme defined by sixteen coefficients: $c = \\lbrace 1, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8}, 0, \\frac{1}{8} \\rbrace $ .",
"Figure: NO_CAPTIONThe author is grateful to Professor V. Yu.",
"Protasov for valuable advice and remarks."
]
] | 2012.05204 |
[
[
"Electron-beam interaction with emission-line clouds in blazars"
],
[
"Abstract Context: An electron-positron beam escaping from the magnetospheric vacuum gap of an accreting black hole interacts with recombination-line photons from surrounding gas clouds.",
"Inverse-Compton scattering and subsequent pair production initiate unsaturated electromagnetic cascades exhibiting a characteristic spectral energy distribution.",
"Aims: By modelling the interactions of beam electrons (positrons) with hydrogen and helium recombination-line photons, we seek to describe the spectral signature of beam-driven cascades in the broad emission-line region of blazar jets.",
"Methods: Employing coupled kinetic equations for electrons (positrons) and photons including an escape term, we numerically obtain their steady-state distributions, and the escaping photon spectrum.",
"Results: We find that cascade emission resulting from beam interactions can produce a narrow spectral feature at TeV energies.",
"Indications of such an intermittent feature, which defies an explanation in the standard shock-in-jet scenario, have been found at $\\approx\\,4\\,\\sigma$ confidence level at an energy of $\\approx$ 3 TeV in the spectrum of the blazar Mrk 501.",
"Conclusions: The energetic requirements for explaining the intermittent 3 TeV bump with the beam-interaction model are plausible: Gap discharges that lead to multi-TeV beam electrons (positrons) carrying $\\approx$ 0.1 % of the Blandford-Znajek luminosity, which interact with recombination-line photons from gas clouds that reprocess $\\approx$ 1 % of the similar accretion luminosity are required."
],
[
"Introduction",
"The launching of astrophysical jets from accreting black holes (BHs) involves fundamental gravitomagnetic processes.",
"The frame-dragging effect on the plasma of the inner accretion flow and its advected magnetic field creates a rotating magnetosphere that can lead to a vacuum voltage drop of $10^{20} \\, \\rm {eV}$ for a typical spinning supermassive BH in the centres of galaxies [43].",
"It is commonly thought that thermal pair production (PP) in photon-photon collisions provides charge carriers in sufficient numbers to supply the Goldreich-Julian density, which allows electric currents to remove the space charges resulting from the electric potential and to reshape the magnetosphere into a force-free configuration.",
"Any imbalance in the supply of charged particles, however, results in the immediate creation of vacuum (electrostatic) gaps.",
"In vacuum gaps the actual charge density, provided by photon-photon PP [56], is different from the Goldreich-Julian charge density.",
"Because of this deviation, the force-free condition is not satisfied and the electric field component parallel to the magnetic field lines is non-vanishing and able to accelerate intruding charge carriers to relativistic velocities, extracting energy from the central rotating BH [17], [15], [32], [41].",
"The interaction of a particle beam with a low-energy radiation field results in electromagnetic cascades.",
"Inverse-Compton (IC) scattering of the primary electrons off low-energy photons produces gamma rays that then undergo PP by interacting with the same photon field leading to a secondary generation of electrons.",
"This sequence repeats until the gamma rays can escape freely [47], [9], [62], [68], [6].",
"Cascades have been studied in a variety of astrophysical environments.",
"In outer magnetospheric vacuum gaps of pulsars, curvature radiation is thought to initiate the pair cascade, limiting the gap growth by supplying charged particles [22], [30], [63].",
"The gamma rays produced by the cascade and leaving the pulsar magnetosphere in a confined beam are made responsible for the observed high-energy (HE) and very high energy (VHE) pulsed emission.",
"In the galactic centre, vacuum gaps are hypothesised to drive cascades that are responsible for PeV cosmic rays and for the VHE gamma-ray point source coincident with Sagittarius A* [43], [39].",
"In active galactic nuclei (AGN) with high accretion rates such as Seyfert galaxies, IC pair cascades in the coronal plasma of the accretion disk are thought to shape their X-ray spectrum up to MeV energies [62], [68].",
"In AGN with jets such as blazars, unsaturated IC pair cascades can develop from which gamma rays at energies well above MeV can escape.",
"The cascades can be initiated by relativistic particles accelerated in vacuum gaps in the BH magnetosphere when the accretion rate is too low to supply the Goldreich-Julian density but their magnetic field is strong enough to launch a jet [54], [55], [43], [35], [38], [31].",
"The gaps are thought to be located at the poles of the spinning BH magnetosphere near to the null surface where the Goldreich-Julian charge density is equal to zero [15], [32], [59], [34], [44], [25], [20], [40].",
"It was also found that gaps form around the divide of the magnetohydrodynamic (MHD) flow, called the stagnation surface [65], [18], [33], [5], or near to the inner light surface of the magnetosphere [23].",
"Time-dependent one-dimensional, general relativistic MHD simulations by [44], [42], [21], and [20] and two-dimensional simulations by [23] have recently found inherently non-stationary gap solutions.",
"[42] and [40] indicate that the gap activity relaxes to low-amplitude quasi-steady oscillations after an initial spark event, while [21] and [20] find cyclically enduring gap activity.",
"Gap-accelerated particles then interact with ambient photon fields, that is, photons from the accretion flow or from the broad line region (BLR), initiating the onset of a cascade [16], [18], which gives rise to an additional component of IC up-scattered photons that are emitted along a beam in the same direction as the original electrons were moving.",
"When these cascaded photons can escape from the AGN gamma sphere without being absorbed by PP, and when the line of sight coincides with the beam, the emission could cause an observable imprint on the spectral energy distribution (SED) of the AGN.",
"Intermittency of the particle flux injected from the gap or of the target photon field results in time variability of the emerging radiation and in transient features of the total SED.",
"This could serve as an explanation of the observed minute- and hour-scale variability of HE and VHE emission in blazars [4], [10], [27], [58], [45].",
"Similar short-term variability has been found in AGN with jets at inclination angles larger than a few degrees, which supports a similar interpretation [11], [49].",
"Alternative scenarios to explain the short-time variability are the interaction of the jet with the radiation field from a stellar envelope, for instance, or a relativistically propagating emission zone in a relativistically moving jet [5], [60].",
"In this paper, we examine IC pair cascades in blazars emphasising the interaction of beam electrons from (short-lived) vacuum gaps with recombination-line photons.",
"Blazars are subdivided into two subclasses, flat-spectrum radio quasars (FSRQs) and BL Lacertae (BL Lac) objects [26].",
"These two types were originally distinguished by the presence of broad optical emission lines in the case of FSRQs, with equivalent width $|W_{\\lambda }| > 0.5 \\, \\rm {nm}$ , and the absence of broad optical emission lines for BL Lac objects [64].",
"However, even in BL Lac objects, in which by definition $|W_{\\lambda }| < 0.5 \\, \\rm {nm}$ , optical and ultraviolet recombination lines from the BLR are often present [61], [13], and might imprint a diagnostically important signature on the gamma-ray spectra emerging from the innermost regions.",
"This paper is organised as follows: In Sect.",
"we describe the physical model for the beam-photon interaction scenario and its underlying assumptions, including the set of coupled kinetic equations that were employed to numerically evaluate the model.",
"We outline the numerical procedure for solving them to obtain the steady-state particle and photon distributions in Sect.",
".",
"In Sect.",
"we apply this numerical scheme to fit the observational SED of the BL Lac object Markarian 501 (Mrk 501), which exhibited a peculiar spectral feature at $\\approx 3$ TeV during a strong flaring activity in 2014 July [48], and discuss the implications of the proposed scenario for the jet formation in Mrk 501.",
"Finally, Sect.",
"summarises our findings.",
"In the following $m_{\\rm e}$ , $c$ , $e$ , $\\sigma _{\\rm Th}$ , $k_{\\rm B}$ , and $h$ denote the electron rest mass, the velocity of light, the elementary charge, the Thomson cross section, the Boltzmann constant, and Planck's constant, respectively.",
"SI units are employed unless noted otherwise."
],
[
"Unsaturated inverse-Compton pair cascade model",
"The electron beam from an active spark gap region close to the accreting BH can initiate IC pair cascades [18], [59], [25], [42], [20].",
"When the beam electrons propagate further along the jet axis, they can encounter the intense photon field of an emission-line cloud, as depicted in Fig.",
"REF , and initiate IC pair cascades.",
"Because the threshold energy for PP in photon-photon collisions with the optical and ultraviolet photons characteristic for recombination-line spectra is high, the cascade spectra do not saturate at the electron rest mass energy delivering X-ray spectra, but instead produce spectra peaking at VHE gamma rays.",
"Such cascades are also linear because the reservoir of low-energy target photons is not modified by the cascade in a major way.",
"For their SEDs, it is essential to consider an escape term accounting for the energy-dependent gamma sphere resulting from the pair creation optical depth, which decreases towards the edge of the emission region."
],
[
"Unsaturated inverse-Compton pair cascade with escape",
"For the model of the emission region, we treat electrons and positrons identically and assume three reservoirs of particle species to be present: Relativistic electrons with spectral number density $N(\\gamma )$ , where $\\gamma $ denotes the Lorentz factor of the electron.",
"Highly energetic photonsIn what follows, the term \"highly energetic photon\" does not only denote photons with energies in the established HE range from 100 MeV to 100 GeV, but more generally photons with energies well above $m_{\\rm e} c^2$ , in contrast to low-energy photons with energies well below $m_{\\rm e} c^2$ .",
"with spectral number density $n_\\gamma (x_\\gamma )$ , where $x_\\gamma $ denotes the photon energy in units of the electron rest mass energy $m_{\\rm e}c^2$ .",
"External low-energy photons with spectral number density $n_0(x)$ with $x_1 \\le x \\le x_0 \\ll 1$ .",
"We assume that all number densities are time-independent, isotropic, and homogeneous throughout the interaction region, and that interactions take place only inside of it.",
"The interactions are assumed to be dominated by IC scattering and photon-photon PP.",
"The cascade is then driven by PP from collisions in which the low-energy target photons turn the highly energetic photons into electron-positron pairs that then IC scatter off the same low-energy target photons to become the next generation of highly energetic photons, and so on (cf.",
"Fig.",
"REF ).",
"We consider four additional mechanisms acting as sinks or sources for the highly energetic photons or the electrons: The injection of highly energetic photons with spectral rate $\\dot{n}_{\\gamma ,\\,\\rm i}(x_\\gamma )$ and $x_\\gamma \\gg 1$$\\dot{n}_{\\gamma ,\\,\\rm i}$ and $\\dot{N}_{\\rm i}$ are called spectral injection rate henceforth, and are defined as the number of particles, injected per unit space volume interval, per unit time interval, and per unit energy interval.. Additionally, we demand that the threshold condition for PP in collisions with the isotropic target radiation field with $x_\\gamma \\, x > 1$ is fulfilled.",
"The injection of electrons with spectral rate $\\dot{N}_{\\rm i}(\\gamma )$ REF.",
"We require that $\\dot{N}_{\\rm i}$ is non-vanishing only for $\\gamma \\gg 1$ , hence the electrons have to be highly relativistic.",
"Furthermore, we require that $\\gamma \\, x > 1$ is satisfied for the injected electrons.",
"This is equivalent with primary IC scattering events being in the Klein-Nishina (KN) regime.",
"According to [68], this type of cascade is denoted IC-Klein-Nishina pair cascade.",
"The escape of highly energetic photons from the interaction region with timescale $T_{\\rm {ph\\,esc}}(x_\\gamma )$ .",
"The electron escape with timescale $T_{\\rm {e\\,esc}}(\\gamma )$ .",
"This is the main difference of our treatment to that by [68] and [66], who considered saturated cascades, and makes our scenario more realistic.",
"A finite escape timescale is equivalent to a bounded interaction region and is thus closer to reality than an infinite escape timescale, which is equivalent to an infinitely large interaction volume.",
"The cascades considered here are therefore not necessarily saturated.",
"We assume that the reservoir of target photons is not modified by the cascade.",
"In other words, $n_0(x)$ is kept constant.",
"This assumption is equivalent to assuming that a cascade is linear [68].",
"Linearity of a cascade means that the low-energy photons alone serve as targets.",
"Self-interactions of the highly energetic photons and of the electrons are neglected.",
"[68] showed by comparing cross sections that when the energy density of low-energy photons is at least similar to that of the highly energetic photons, the condition for a cascade to be linear is met whose interactions occur in the KN regime.",
"Repetition of IC scattering events successively decreases the electron energy, until IC scattering events take place in the Thomson regime ($\\gamma \\, x < 1$ ).",
"Here, cascades are always non-linear because $x_\\gamma \\, x < 1$ implies that PP is not possible and thus PP is only possible through highly energetic photon self-interactions.",
"Figure: Sketch of the considered linear IC pair cascade with escape terms."
],
[
"Description through kinetic equations",
"To determine the change rates of the spectral number densities of photons and electrons, we use the spectral interaction rates of IC scattering and of PP given by [68].",
"For IC events with incident electrons of energy $\\gamma \\gg 1$ scattering off photons with energy $x \\ll 1$ and spectral number density $n_0(x)$ , which result in down-scattered electrons with energy $\\gamma ^{\\prime }$ and highly energetic photons with energy $x_\\gamma $ , the spectral IC scattering interaction rate is denoted by $C(\\gamma ,\\gamma ^{\\prime })$ and determined with Eq.",
"A1 by [68], cf.",
"Appendix .",
"For such events, the threshold $\\gamma _{\\rm {IC,\\,th}}(x_\\gamma ,x)$ as defined in Appendix is the minimum required value of $\\gamma $ .",
"The maximum possible $\\gamma $ is $\\gamma _{\\rm {IC,\\,max}}(\\gamma ^{\\prime },x)$ , as outlined in Appendix .",
"Furthermore, we define $\\gamma ^{\\prime }_{\\rm {IC,\\,min}}(\\gamma ,x)$ as the minimum possible $\\gamma ^{\\prime }$ and $x_{\\gamma ,\\,\\mathrm {max}}(\\gamma ,x)$ as the maximum energy of the up-scattered photon.",
"Given the interaction of highly energetic photons of energy $x_\\gamma $ with low-energy photons of energy $x$ and spectral number density $n_0(x)$ , resulting in the PP of electrons of energy $\\gamma $ , the spectral PP interaction rate is called $P(x_{\\gamma },\\gamma )$ and is computed with Eq.",
"B1 of [68], cf.",
"Appendix .",
"The PP threshold, that is, the minimum required $x_\\gamma $ , is defined as $x_{\\gamma ,\\,\\rm {PP,\\,th}}(\\gamma ,x)$ in Appendix .",
"The values of $\\gamma $ that can be reached are limited by $\\gamma _{\\rm {PP,\\,min}}(x_\\gamma ,x)$ and $\\gamma _{\\rm {PP,\\,max}}(x_\\gamma ,x)$ , as described in Appendix .",
"The kinetic equation of the electrons and of the photons ensues after all the relevant sinks and sources are specified.",
"The change rate of the spectral number density of the electrons (electron energy distribution) is affected by the electron spectral injection rate $\\dot{N}_{\\rm i}$ , the spectral loss rate (as the sum of the escape rate with timescale $T_{\\rm {e\\,esc}}$ and of the IC down-scattering rate to lower energies), and by the production rate (as the sum of the IC down-scattering rate from higher energies and of the PP rate).",
"Similarly, the change rate of the spectral number density of the photons is determined from the photon spectral injection rate $\\dot{n}_{\\gamma ,\\,\\rm i}$ , the loss rate (which is the sum of the escape rate with timescale $T_{\\rm {ph\\,esc}}$ and of the pair absorption rate), and the production rate $\\dot{n}_{\\gamma , \\, \\rm {IC}}$ .",
"Assuming steady state, we obtain, as shown in Appendix , $n_\\gamma (x_\\gamma ) =\\underbrace{\\frac{\\dot{n}_{\\gamma ,\\,\\mathrm {i}}(x_\\gamma )}{\\frac{1}{T_{\\rm {ph\\,esc}}(x_\\gamma )} + \\int ^{\\gamma _{\\mathrm {PP,\\,max}}(x_\\gamma ,x_0)}_{\\gamma _{\\mathrm {PP,\\,min}}(x_\\gamma ,x_0)} P(x_\\gamma ,\\gamma ) \\, \\mathrm {d}\\gamma }}_{:= \\;n_{\\gamma ,\\,\\mathrm {i}}(x_\\gamma )} +\\underbrace{\\frac{\\int _{\\gamma _{\\mathrm {IC,\\,th}}(x_\\gamma ,x_0)}^{\\infty } N(\\gamma ) C(\\gamma ,\\gamma -x_\\gamma ) \\, \\mathrm {d}\\gamma }{\\frac{1}{T_{\\rm {ph\\,esc}}(x_\\gamma )} + \\int ^{\\gamma _{\\mathrm {PP,\\,max}}(x_\\gamma ,x_0)}_{\\gamma _{\\mathrm {PP,\\,min}}(x_\\gamma ,x_0)} P(x_\\gamma ,\\gamma ) \\, \\mathrm {d}\\gamma }}_{:= \\; n_{\\gamma ,\\,\\mathrm {IC}}(x_\\gamma )},$ where we have defined two contributions to the photon spectral number density.",
"$n_{\\gamma ,\\,\\mathrm {i}}$ describes the contribution that arises from the interplay of injection and escape and absorption losses, while $n_{\\gamma ,\\,\\mathrm {IC}}$ describes the contribution that is achieved from the interplay of production through IC up-scattering and both losses.",
"Exchanging $\\gamma $ and $\\gamma ^{\\prime }$ in Eq.",
"REF and plugging this into Eq.",
"REF , we can eliminate $n_\\gamma $ , N() = Ni() + 'IC, max(,x0) N(') C(',) d' + x, PP, th(,x0) ( n, i(x) + 'IC, th(x,x0) N(') C(','-x) d' ) 2 P(x,)1Tph esc(x) + 'PP, max(x,x0)'PP, min(x,x0) P(x,') d' dx1Te esc() + 'IC, min(,x0) C(,') d'"
],
[
"Numerical solution procedure",
"In this section, we outline how explicit solutions for $N(\\gamma )$ and $n_\\gamma (x_\\gamma )$ are found.",
"The main task is to find solutions for $N$ .",
"This is done by solving Eq.",
"REF iteratively.",
"The right-hand side of Eq.",
"REF directly incorporates $\\gamma $ , $T_{\\rm {ph\\,esc}}$ , $T_{\\rm {e\\,esc}}$ , $N$ , $\\dot{N}_{\\rm {i}}$ , $\\dot{n}_{\\gamma ,\\,\\rm i}$ , $C$ , $P$ , and the functions in the integration borders, which have been defined in the previous section.",
"$C$ and $P$ are known as soon as $n_0$ has been specified.",
"Hence, Eq.",
"REF can briefly be written as $N(\\gamma ) = \\mathcal {F}^{\\prime }(n_0, \\dot{N}_{\\rm i}, \\dot{n}_{\\gamma ,\\,\\rm i}, T_{\\rm {ph\\,esc}}, T_{\\rm {e\\,esc}}, N, \\gamma )$ .",
"As described in Sect.",
"REF , the first five arguments of $\\mathcal {F}^{\\prime }$ are quantities that describe the physical setting.",
"Hence, they should be understood as input functions and have to be prescribed to specify the physical problem.",
"When this is done, we obtain the equation $N(\\gamma ) = \\mathcal {F}(N, \\gamma )$ , where $\\mathcal {F}$ was defined accordingly.",
"Now, we choose a sequence $\\left( \\gamma _k \\right)_{k = 1,\\dots ,\\kappa }$ of $\\kappa $ discrete values $\\gamma _k$ along which the kinetic equation is to be solved, in other words, along which $N(\\gamma )$ is to be determined.",
"We write $N(\\gamma _k) = N_k$ for convenience.",
"As a starting point of the iteration, we guess an initial course $N_{\\rm {init}}(\\gamma )$ of the function $N(\\gamma )$ and determine the corresponding sequence of values $\\left( N_k \\right)_{k = 1,\\dots ,\\kappa ;\\,j_{\\rm {init}}}$ .",
"In Appendix , we discuss how the converged sequence is determined numerically.",
"We perform the iteration using the python3 language.",
"To numerically compute the integrals with an infinite integration range, we restrict the integration range we used to a finite range.",
"This is possible when $n_0(x)$ , $\\dot{n}_{\\gamma ,\\,\\rm i}(x_\\gamma )$ , and $\\dot{N}_{\\rm i}(\\gamma )$ are equal to zero everywhere above an upper cut-off, which we denote by $x_0$ , $x_{\\gamma ,\\,0}$ , and by $\\gamma _{{\\mathrm {i}},\\,0}$ , respectively.",
"To summarise, the physical setting is specified as soon as the following functions are defined: The low-energy photon spectral number density $n_0$ The highly energetic photon spectral injection rate $\\dot{n}_{\\gamma ,\\,\\rm i}$ The electron spectral injection rate $\\dot{N}_{\\rm i}$ The highly energetic photon escape time $T_{\\rm {ph\\,esc}}$ The electron escape time $T_{\\rm {e\\,esc}}$ By iterating, our code then determines the steady state $N$ .",
"When $N$ is known, we use Eq.",
"REF to determine $n_\\gamma $ , and from this, we obtain the spectral flux density $F_{\\rm {casc}}$ .",
"This approach differs from that of [66].",
"Therein, although $T_{\\rm {ph\\,esc}}=T_{\\rm {e\\,esc}}=\\infty $ (no escape) was used in the kinetic equation, it was assumed that the IC up-scattered photons leave the interaction region from its shell-like boundary.",
"Accordingly, [66] directly converted the photon spectral production rate (last term in Eq.",
"REF ) into the observed spectral flux density.",
"This appears to be a very crude approach because first it neglects that the injected highly energetic photons can also leave the interaction region (which we take into account by $n_{\\gamma ,\\,\\mathrm {i}}$ in Eq.",
"REF ), and second it assumes that all IC up-scattered photons escape from the boundary of the interaction region and that no IC up-scattered photons escape from its interior.",
"On the one hand, this second assumption contradicts the $T_{\\rm {ph\\,esc}}=T_{\\rm {e\\,esc}}=\\infty $ assumption and on the other hand it neglects the energy-dependent behaviour of photon attenuation (which we take into account through the division of the photon spectral injection and production rate (numerator of Eq.",
"REF ) by the spectral loss rate (denominator of Eq.",
"REF )).",
"Based on $n_\\gamma $ , we determine the spectral flux density $F_{\\rm {casc}}$ of highly energetic photons (i.e.",
"the number of photons per unit energy, per unit area, and per unit time) through $F_{\\mathrm {casc}}(x_\\gamma ) = n_{\\gamma }(x_\\gamma ) \\cdot \\frac{4 \\pi \\, c \\, R_{\\rm {ph\\,esc}}(x_\\gamma )^2}{\\Omega (\\phi ) \\, D^2 \\ m_{\\rm {e}} c^2}.$ We assumed that the photons leave the interaction region, whose approximate radial size is $R_{\\rm {ph\\,esc}}(x_\\gamma )$ , only through a conical beam of opening angle $\\phi $ and solid angle $\\Omega = 4 \\pi \\cdot \\sin ^2(\\phi /4)$ , and that $D$ is the distance to the observer.",
"The size of the interaction region is linked with the escape time of the photons and electrons.",
"A natural choice for this would be $R_{\\rm {ph\\,esc}}(x_\\gamma ) = T_{\\rm {ph\\,esc}}(x_\\gamma ) \\, c$ .",
"The division by $m_{\\rm {e}} c^2$ arises because $F_{\\rm {casc}}$ is measured in $\\rm {m}^{-2} \\, \\rm {s}^{-1} \\, \\rm {J}^{-1}$ , while $n_{\\gamma }$ is in units of $\\rm {m}^{-3}$ ."
],
[
"Indication of gap activity in Markarian 501",
"The numerical procedure described above is used below to explain a peculiar feature in the VHE spectrum of the blazar Mrk 501 during a flaring activity on 2014 July 19.",
"The broadband emission from blazars, and in particular from the high-synchrotron-peaked BL Lac object Mrk 501, is characterised by a two-bump structure.",
"This emission is usually described within the one-zone synchrotron-self-Compton (SSC) framework [51], [24].",
"In this model, a blob of relativistic electrons is assumed to move along the jet that points towards the observer.",
"The blob is permeated by a randomly distributed magnetic field, producing synchrotron radiation from the electrons.",
"This radiation is responsible for the low-energy SED bump in blazars.",
"The origin of the HE bump is typically ascribed to the IC scattering of the relativistic electrons with the synchrotron photons.",
"In addition to the SSC scenario, other theoretical models have been proposed to explain the HE and VHE emission in blazars, involving hadronic components and IC pair cascading.",
"Minute-scale variability spectral components mean that the standard one-zone SSC scenario may be insufficient and suggest that an additional sporadic emission component is emitted from the innermost regions of the jet, see [8].",
"Recently, the [48] has presented a two-week flare from Mrk 501, mainly detected in X-rays with the Swift satellite and in VHE gamma rays observed with the major atmospheric gamma-ray imaging Cherenkov (MAGIC) telescopes.",
"During this flaring state, the source displayed hard X-ray spectra as well as the highest X-ray count rate detected by the Swift X-Ray Telescope during its 15 years of operation.",
"In coincidence with the peak of the X-ray count rate, which took place on 2014 July 19 (MJD 56857.98), a narrow feature in the TeV spectrum from the MAGIC telescopes was detected with a significance of $3 \\, \\sigma - 4 \\, \\sigma $ .",
"This feature, located around $3 \\, \\rm {TeV}$ , is inconsistent with the standard functions to describe the VHE spectra from blazars (power law, log-parabola, and log-parabola with exponential cut-off) at more than $3\\,\\sigma $ confidence level.",
"The VHE spectrum on 2014 July 19 is better explained assuming a narrow component (a narrow log-parabola or a Gaussian function) in addition to a broad log-parabola.",
"Such a double function fit is preferred with respect to a single log-parabola at $3.2 \\, \\sigma - 4.5\\, \\sigma $ , depending on the a priori assumptions [48].",
"This was the first time that a narrow feature (not consistent with the usual smooth spectral functions) was found with a significance higher than $3\\,\\sigma $ in the VHE spectrum of Mrk 501, and any TeV blazar in general.",
"Indications for similar (but statistically insignificant) spectral features at $\\approx 2\\,\\rm {TeV}$ can be found by inspection of flare spectra reported by the very energetic radiation imaging telescope array system (VERITAS) [1] and the MAGIC collaboration [7].",
"Under the assumption that the spectral feature described above is real, three different theoretical scenarios were proposed by the [48] to reproduce the narrow spectral feature: a) a pileup in the electron energy distribution produced by stochastic acceleration, b) a two-zone SSC model, and c) a narrow emission from an IC pair cascade scenario induced by electrons accelerated in a magnetospheric vacuum gap.",
"Details for the model describing scenario c) are provided in this work.",
"In the following, we evaluate the model we fitted to the observational data to infer its implications and consequences."
],
[
"Specification of the modelled setting",
"We assume a vacuum gap near to the poles of the Mrk 501 BH magnetosphere.",
"The potential drop in the gap accelerates seed electrons to ultra-relativistic energies.",
"The seed electrons enter the gap both through PP of gamma rays from the hot advection-dominated accretion flow (ADAF), and possibly through direct leakage from it [55].",
"Seed electrons that have been accelerated in the gap, leave it in the direction of the electric field (i.e.",
"in the direction of the magnetic field), and propagate along a beam that is well collimated to an opening angle $\\approx \\phi $ .",
"During the propagation away from the gap, the seed electrons still encounter the ADAF photon field, and a post-gap cascade [18], [20] occurs and increases the number of electrons by repeated curvature-radiation and IC emission and PP events.",
"This post-gap cascade ceases when curvature-radiation emission becomes negligible and when IC scattering and PP can no longer be sustained by ADAF photons.",
"Several tens or hundreds of Schwarzschild radii away from the gap, the beam of electrons is assumed to enter a region in which background photons from ionised gas clouds are present.",
"This induces a cascade in this interaction region.",
"This model is depicted in Fig.",
"REF .",
"For this cascade, the model of Sect.",
"is applied.",
"The electron beam from the gap serves as electron injection mechanism in our model.",
"We describe the electron spectral injection rate with the following cut-off Gaussian: $\\dot{N}_{\\rm i}(\\gamma ) = \\left\\lbrace \\begin{array}{ll}\\frac{K_{\\rm {G}}}{\\varsigma \\sqrt{(}2 \\pi )} \\cdot \\exp \\left( -\\frac{(\\gamma -\\gamma _{\\mathrm {mean}})^2}{2 \\, \\varsigma ^2} \\right) & \\mathrm {if} \\; \\gamma _{{\\mathrm {i}},\\,1} \\le \\gamma \\le \\gamma _{{\\mathrm {i}},\\,0} \\mathrm {,} \\\\0 & \\mathrm {otherwise}\\end{array}\\right.$ Here, $\\gamma _{\\rm {mean}}$ denotes the mean Lorentz factor of the injected electrons, and $\\varsigma $ parametrises the width of the distribution.",
"For the cut-off values we choose $\\gamma _{{\\rm {i}},\\,1} = \\gamma _{\\rm {mean}} - 3.0 \\, \\varsigma $ and $\\gamma _{{\\rm {i}},\\,0} = \\gamma _{\\rm {mean}} + 3.0 \\, \\varsigma $ , satisfying the condition $\\gamma \\cdot x > 1$ (cf.",
"Sect.",
"REF ).",
"The normalisation of the Gaussian, which describes the total number of electrons that are injected per unit time and per unit space volume, is called $K_{\\rm {G}}$ .",
"Furthermore, we assume that the external low-energy photons of the cascade are represented by emission-line photons from photo-ionised gas clouds in the surroundings of the Mrk 501 BH magnetosphere.",
"Although Mrk 501, as a BL Lac object, does not have a prominent BLR, its BLR has clearly been confirmed observationally, possibly due to outshining by the boosted jet emission, due to a weak accretion rate, or due to the lack of ambient gas [52], [61].",
"After they are triggered by minor galaxy mergers, interstellar gas clouds, consisting mainly of hydrogen and helium [67], can migrate from the host galaxy into its centre and thus into the central part of Mrk 501.",
"Radiation from the accretion flow or from OB-type stars in the dense galactic centre will inevitably ionise passing clouds.",
"As an effect of recombination, emission-line photons will be abundant in the proximity of the clouds.",
"The external low-energetic photon field can also stem from the envelope of a post-main-sequence star with low metallicity [14], [12].",
"In any case, we assume that the interaction region is both penetrated by the electron beam and pervaded by emission-line photons, and consequently, that an IC pair cascade will develop there.",
"It is natural to describe the spectral number density by a sum of Dirac delta distributions, $n_{0}(x) = K_{\\rm {lines}} \\cdot \\sum _{i=1}^4 \\frac{K_{{\\rm {line}},\\,i}}{x_{0,\\,i}} \\cdot \\delta _{\\mathrm {Dirac}} \\left( x - x_{0,\\,i} \\right)$ Here, $x_{0,\\,i} = h / (\\lambda _{0,\\,i} \\, m_{\\rm {e}} \\, c)$ , and $\\lambda _{0,\\,i}$ are the dimensionless energy, and the wavelength of the $i$ -th line, respectively.",
"$K_{{\\rm {line}},\\,i}$ gives the relative flux density contribution of the $i$ -th line$K_{{\\rm {line}},\\,i}$ gives the flux density of the emission line $i$ with respect to the flux density of a hypothetical hydrogen Balmer-$\\beta $ line..",
"Accordingly, $K_{{\\rm {line}},\\,i}/x_{0,\\,i}$ describes the relative number density contribution of the $i$ -th line.",
"The parameter $K_{\\rm {lines}}$ is a measure for the photon total number density.",
"We restrict the sum to the three typically most important hydrogen lines plus the most important helium line.",
"We compiled these lines and their respective $K_{{\\rm {line}},\\,i}$ based on the ultraviolet spectroscopic blazar observations by [57] and on the synthetic photo-ionisation spectra by [2].",
"We list them in Table REF .",
"From an energetic point of view, the emission-line radiation is just a reprocessed fraction of the more luminous ADAF radiation.",
"Nevertheless, we do not include the ADAF radiation in the target photon field.",
"This is justified by the following geometrical argument.",
"The emission of line photons in the ionised cloud is approximately isotropic.",
"The interaction angles of injected electrons and photons are therefore equally distributed; the interaction angle is on average 90.",
"This also justifies the usage of the interaction rates $C$ and $P$ .",
"In contrast, for interactions of electrons with ADAF photons the collision angle would be at best 90, and on average smaller than 90.",
"Thus, glancing collisions occur, and for such interactions, the interaction rates are lower than for right-angled collisions.",
"This results in increased interaction rates for the line photons in comparison to ADAF photons, and thus we can neglect the latter.",
"Moreover, except for the reprocessed photons from the ionised cloud, reprocessed ADAF photons might be scarce in the surroundings of the beam due to low ambient gas density in the BL Lac object, which means that they drop out as target photons as well.",
"We have to mention furthermore, that ADAF spectra usually extend to higher energies and $x \\ll 1$ , which was assumed in the derivation of $C$ and $P$ , is satisfied only very critically.",
"We do not have injection of highly energetic photons in this setting.",
"Therefore we set the photon spectral injection rate $\\dot{n}_{\\gamma ,\\,\\rm i}(x_\\gamma ) = 0.$ Concerning the choice of the escape timescales, we choose $T_{\\rm {ph\\,esc}}(x_\\gamma ) = T_{\\rm {e\\,esc}}(\\gamma ) = \\frac{R}{c} := T_{\\rm {esc}}.$ In other words, we approximate the interaction volume as a spherical region with radius $R$ .",
"Electrons and photons have equal, and energy independent, that is, constant, escape timescales.",
"The function $R_{\\rm {ph\\,esc}}(x_\\gamma )$ in Eq.",
"REF is therefore also set equal to the parameter $R$ ."
],
[
"Modelling results",
"According to the procedure outlined in Sect.",
", we determine $N(\\gamma )$ , $n_\\gamma (x_\\gamma )$ , and $F_{\\rm {casc}}(x_\\gamma )$ of the cascade that evolves through interaction of the electrons originally stemming from the gap with the emission-line photons, such that $F_{\\rm {casc}}(x_\\gamma )$ is fitted to the narrow SED feature (MAGIC telescopes' data) and a background SSC emission is fitted to the multi-wavelength SED.",
"We understand $K_{\\rm {G}}$ , $K_{\\rm {lines}}$ , $\\gamma _{\\mathrm {mean}}$ , $\\varsigma $ , $R$ , and $\\phi $ as fitting parameters, while we use the luminosity distance $D = 149.4 \\, \\rm {Mpc}$ (corresponding to redshift $z=0.034$ and a cosmologyHere, $\\Omega _{\\rm {m}}$ , $\\Omega _{\\Lambda }$ , and $H_0$ are the dimensionless density parameter of matter, the dimensionless density parameter of dark energy, and the Hubble constant, respectively.",
"$\\Omega _{\\rm {m}} = 0.3$ , $\\Omega _{\\Lambda } = 0.7$ , $H_0 = 70 \\, \\rm {km} \\, \\rm {s}^{-1} \\, \\rm {Mpc}^{-1}$ ) and the line contributions $K_{{\\rm {line}},\\,i}$ given in Table REF .",
"Because the electron beam penetrates into the interaction region from one direction, the kinematics of IC scattering and PP ensure that the highly energetic photons leave the interaction region in this same direction, hence along a beam of opening angle $\\phi $ .",
"Interim results of the numerical procedure are shown in Figs.",
"REF , REF , and REF and are discussed now.",
"Figure: Various contributions (cf.",
"Eq. )",
"to the photon spectral loss rate on the left-hand side ordinate in dependence on the product of the dimensionless highly energetic photon energy with the dimensionless energy of the highest energetic line.",
"On the right-hand side ordinate, we plot the optical depth that corresponds to the respective contribution to the spectral loss rate.The results of our iterative determination of $N(\\gamma )$ are shown in Fig.",
"REF .",
"The initial function of $N$ was chosen to be $N_{\\rm {init}}(\\gamma ) = \\frac{2 \\cdot \\dot{N}_{\\rm i}(\\gamma )}{\\int ^{\\gamma _{{\\mathrm {i}},\\,1}}_{\\gamma ^{\\prime }_{\\rm {IC,\\,min}}(\\gamma _{{\\mathrm {i}},\\,1},x_0)} C(\\gamma _{{\\mathrm {i}},\\,1},\\gamma ^{\\prime }) \\, \\mathrm {d}\\gamma ^{\\prime } + \\int ^{\\gamma _{{\\mathrm {i}},\\,0}}_{\\gamma ^{\\prime }_{\\rm {IC,\\,min}}(\\gamma _{{\\mathrm {i}},\\,0},x_0)} C(\\gamma _{{\\mathrm {i}},\\,0},\\gamma ^{\\prime }) \\, \\mathrm {d}\\gamma ^{\\prime }},$ cf.",
"Eq.",
"REF .",
"This choice is a compromise between a function that can be evaluated fast and is accurate.",
"It is a good approximation for the final $N$ in the highest energetic regime because the electron spectral production rate is negligible here.",
"For decreasing $\\gamma $ , a higher number of iteration steps is necessary because the energy transfer in one IC scattering event decreases.",
"We choose the convergence criterion as follows: One iteration is considered as finished, as soon as the relative change $\\left| N_{k,\\,j-1} - N_{k,\\,j} \\right| / N_{k,\\,j}$ between two successive values is smaller than $0.001$ at the points $\\gamma _k > 10 \\, x_{0,\\,1}^{-1}$ and smaller than $0.001 \\cdot (\\gamma _k x_{0,\\,1} / 10)^{2}$ at the points $\\gamma _k \\le 10 \\, x_{0,\\,1}^{-1}$ .",
"In other words, the demanded accuracy increases quadratically the deeper we iterate into the Thomson regime.",
"There, about $j_{\\rm {final}}=100$ iteration steps are necessary to meet the convergence criterion, while only two or three steps are necessary in the KN regime, cf.",
"Fig.",
"REF .",
"The shallow ridge of $N$ at $\\gamma \\approx 2 \\, x_{0,\\,1}^{-1}$ is caused by PP of highly energetic photons with hydrogen Lyman photons (mainly hydrogen Lyman-$\\alpha $ photons), while the very shallow ridge at $\\gamma \\approx 0.5 \\, x_{0,\\,1}^{-1}$ is caused by PP of highly energetic photons with the helium II Lyman-$\\alpha $ photons.",
"In other words, these ridges in $N$ are the effect of peaks of the spectral PP rate (fourth summand in Eq.",
"REF ).",
"In the Thomson regime, with decreasing $\\gamma $ the course of $N$ deviates more and more from the $\\sim \\gamma ^{-2}$ dependence of the saturated cascade case considered by [68] because particles escape in our scenario.",
"The finite value of the escape time prevents the electrons from penetrating deep into the Thomson regime.",
"From Fig.",
"REF it is obvious that the photon spectral loss rate is dominated by pair absorption in the approximate range $3\\,x_{0,\\,1}^{-1}<x_\\gamma <800\\,x_{0,\\,1}^{-1}$ corresponding to a photon energy between $19 \\, \\rm {GeV}$ and $5.1 \\, \\rm {TeV}$ , and by escape outside this energy range.",
"The maximum of the photon spectral loss rate occurs at $x_\\gamma \\approx 12 \\, x_{0,\\,1}^{-1}$ , corresponding to the maximum of the hydrogen Lyman-$\\alpha $ absorption.",
"The optical depths, which correspond to the respective spectral loss rate contributions, are shown in Fig.",
"REF on the right-hand side ordinate and were determined by multiplication of the respective spectral loss rate contribution with the escape time $T_{\\rm {esc}}$ .",
"Figure: High-energy bump of the SED of Mrk 501 from 2014 July 19 (MJD 56857.98).",
"The dashed grey line depicts the IC bump of a one-zone SSC model (contribution F SSC F_{\\rm {SSC}}).",
"The dot-dashed red line depicts the emission of the cascade (contribution F casc F_{\\rm {casc}}, cf.",
"green line in Fig. ).",
"The sum FF of both contributions is depicted by the solid black line.",
"The spectral data from the MAGIC telescopes are shown as red circles, while measurements and upper limits by the Fermi Large Area Telescope are drawn by black and yellow triangles.",
"The details on the data analysis and the SSC modelling can be found in We show the photon spectral production rate $\\dot{n}_{\\gamma , \\, \\rm {IC}}$ in Fig.",
"REF .",
"From $N$ , it inherits the peak around $500 \\, x_{0,\\,1}^{-1}$ as well as the deviation from a power law in the Thomson regime due to escape [68].",
"In this figure we also show the spectral number density $n_\\gamma $ of highly energetic photons.",
"The contribution $n_{\\gamma ,\\,\\mathrm {i}}$ vanishes because of the choice Eq.",
"REF .",
"The contribution $n_{\\gamma ,\\,\\mathrm {IC}}$ is non-vanishing and ensues as a consequence of IC scattering, pair absorption, and escape.",
"The pronounced trough above $x_{\\gamma } \\approx 4 \\, x_{0,\\,1}^{-1}$ is due to absorption by PP with photons from the hydrogen Lyman-$\\alpha $ lineBecause of the smallness of $K_{{\\rm {line}},\\,2}$ and $K_{{\\rm {line}},\\,3}$ , absorption features of the hydrogen Lyman-$\\beta $ line and series are almost invisible.",
"(cf.",
"Fig.",
"REF ).",
"The shallow dip of the spectral number density above $x_{\\gamma } \\approx x_{0,\\,1}^{-1}$ is due to absorption on the helium II Lyman-$\\alpha $ line.",
"Thus, the slight bump at $x_{\\gamma } \\approx x_{0,\\,1}^{-1}$ results from cascaded photons that are immediately below the threshold for PP and hence are not absorbed.",
"The peaky feature around $500 \\, x_{0,\\,1}^{-1}$ is more pronounced in $n_{\\gamma }$ than in $\\dot{n}_{\\gamma , \\, \\rm {IC}}$ because the spectral loss rate (i.e.",
"the denominator of $n_{\\gamma ,\\,\\rm {IC}}$ ) is higher below $500 \\, x_{0,\\,1}^{-1}$ than at $500 \\, x_{0,\\,1}^{-1}$ .",
"If $n_0$ were continuous and distributed along a wider range of $x$ instead of consisting of a few strong lines, the peak around $500 \\, x_{0,\\,1}^{-1}$ would be wider and extend to lower energies.",
"The total energy density of highly energetic photons is obtained by integrating of the spectral energy density $x_{\\gamma } \\, n_{\\gamma }(x_{\\gamma })$ : $\\int _0^{x_{\\gamma ,\\,\\mathrm {max}}(\\gamma _{{\\mathrm {i}},\\,0},x_{0,\\,1})} x_{\\gamma } \\, n_{\\gamma }(x_{\\gamma }) \\, \\mathrm {d}x_{\\gamma } \\cdot m_{\\rm e} c^2 \\approx 4.5 \\, \\mathrm {J}/\\mathrm {m}^3$ The total energy density of the low-energy photons is $K_{\\rm {lines}} \\cdot \\sum _{i=1}^4 K_{{\\rm {line}},\\,i} \\cdot m_{\\rm e} c^2 \\approx 6.5 \\, \\mathrm {J}/\\mathrm {m}^3$ and thus of similar magnitude.",
"Our cascade is therefore indeed linear (cf.",
"Sect.",
"REF ).",
"The cascaded spectral flux density $F_{\\rm {casc}}$ of highly energetic photons adds to the spectral flux density $F_{\\rm {SSC}}$ of photons that is produced farther downstream the jet in a common one-zone SSC scenario.",
"Both these components contribute to the observed spectral flux density $F$ .",
"We fit both components, $F_{\\rm {SSC}}$ and $F_{\\rm {casc}}$ , to the multi-wavelength data of Mrk 501 of the night of MJD 56857.98 as presented by the [48].",
"As shown in Fig.",
"10 of [48] and in Fig.",
"REF in the present paper, we can describe the narrow SED feature without the need of extreme SSC parameters.",
"The parameters found by the fit are given for the cascaded emission component in Table REF of the present paper and for the SSC emission component in Table 6 and Fig.",
"7 in [48].",
"The emission from a cascade initiated by electrons originating from a vacuum gap can thus explain the observed narrow peak-like feature."
],
[
"Physical inferences",
"In this subsection we discuss the results of modelling Mrk 501.",
"We assess the findings in a broader context."
],
[
"Number and size of emission-line clouds",
"We compare the assumed hydrogen Lyman-$\\alpha $ photon abundance with the hydrogen Lyman-$\\alpha $ luminosity of $L_{\\rm {Ly} \\,\\alpha ,\\,\\rm {obs}} = 5.2 \\cdot 10^{33} \\, \\rm {W}$ that was detected from Mrk 501 by [61].",
"The total number density of all low-energy photons is given by $K_{\\rm {lines}} \\cdot \\sum _{i=1}^4 K_{{\\rm {line}},\\,i} / x_{0,\\,i} \\approx 3.2 \\cdot 10^{18} \\, \\rm {m}^{-3}$ .",
"This number density describes the photon field of the ionised cloud assumed to be interacting with the electron beam from the gap.",
"The total number density of hydrogen Lyman-$\\alpha $ photons is determined by $K_{\\rm {lines}} \\cdot K_{{\\rm {line}},\\,4} / x_{0,\\,4} \\approx 2.6 \\cdot 10^{18} \\, \\rm {m}^{-3}$ .",
"We further assume that there are $N_{\\rm {cl}}$ clouds of approximate radial size $R_{\\rm {cl}}$ in the BLR.",
"The hydrogen Lyman-$\\alpha $ luminosity $L_{\\rm {Ly} \\,\\alpha ,\\,\\rm {cl}}$ of all these clouds is responsible for the observed hydrogen Lyman-$\\alpha $ luminosity.",
"Hence, we can set $L_{\\rm {Ly} \\,\\alpha ,\\,\\rm {obs}} \\overset{!",
"}{=} L_{\\rm {Ly} \\,\\alpha ,\\,\\rm {cl}} = N_{\\rm {cl}} \\cdot 4 \\pi R_{\\rm cl}^2 \\: K_{\\rm {lines}} \\frac{K_{{\\rm {line}},\\,4}}{x_{0,\\,4}} \\, c \\: x_{0,\\,4} \\, m_{\\rm e} c^2.$ As BL Lac objects are considered to be evolved objects running short of cold gas supplies [46], [26], we can assume that $N_{\\rm {cl}}$ is quite small.",
"On the other hand, to obtain a Gaussian line profile for the emission lines resulting from the Doppler broadening, the number of clouds must not be too small.",
"Then, solving Eq.",
"REF for the radial size yields an estimate of $R_{\\rm cl}$ .",
"Choosing $N_{\\rm {cl}} = 10$ yields $R_{\\rm cl} \\approx 1.8 \\cdot 10^{11} \\, {\\rm m}$ , which roughly agrees with our value used for the radial size of the interaction region $R$ (cf.",
"Table REF ).",
"This size of the cloud corresponds to the size of a red giant, which is suggestive of cloud formation by photospheric abrasion through beam interaction.",
"In terms of Schwarzschild radii $r_{\\rm S}$ for a central BH of mass $M = 10^9 \\, M_{\\odot }$ , it is $R_{\\rm cl} = 0.061 \\, r_{\\rm S}$ .",
"From the modelling we infer that in Mrk 501 only few major gas clouds are present in the BLR.",
"The encounter of a major cloud with electrons from the gap-beam must therefore be a rare event.",
"Presuming that the gas clouds revolve around the BH on unperturbed Keplerian orbits, an encounter of a cloud with the beam of electrons will at best repeat with the Keplerian orbital period of the cloud.",
"These conclusions are in line with the fact that the peak-like SED feature has not been found elsewhere in Mrk 501 data."
],
[
"Reprocessing of the accretion flow luminosity",
"The total line luminosity of these $N_{\\rm {cl}}$ clouds of radius $R_{\\rm cl}$ is determined as $L_{\\rm {lines,\\,tot}} = N_{\\rm {cl}} \\cdot 4 \\pi R_{\\rm cl}^2 \\: K_{\\rm {lines}} \\cdot \\sum _{i=1}^4 K_{{\\rm {line}},\\,i} \\, c \\, m_{\\rm e} c^2.$ For the above chosen $N_{\\rm {cl}} = 10$ and resulting $R_{\\rm cl} = 1.8 \\cdot 10^{11} \\, {\\rm m}$ , it is $L_{\\rm {lines,\\,tot}} \\approx 7.8 \\cdot 10^{33} \\, \\rm {W}$ , a factor 5 lower than the upper limit of the Mrk 501 BLR luminosity found by [19].",
"We recall that this finding is obtained from our modelling and from the observed hydrogen Lyman-$\\alpha $ luminosity.",
"In canonical BLR models, the BLR luminosity is assumed to be about $10 \\, \\%$ of the accretion flow luminosity [28].",
"As Mrk 501 is a gas-depleted object, its BLR reprocesses even less of the incident accretion flow radiation.",
"Thus, the reprocessing fraction $\\xi = L_{\\rm {lines,\\,tot}}/L_{\\rm {ADAF,\\,tot}}$ might be lower than $10 \\, \\%$ , say $1 \\, \\%$ .",
"Here, $L_{\\rm {ADAF,\\,tot}}$ is the total luminosity of the ADAF.",
"To compare the emission-line luminosity with the accretion flow luminosity, we apply the synthetic ADAF spectrum brought forth by [50].",
"There, the spectral luminosity of an ADAF as a function of frequency was parametrised by the viscosity parameter $\\alpha $ , the pressure ratio $\\beta $ , the inner boundary $R_{\\rm in}$ of the hot accretion flow, its outer boundary $R_{\\rm out}$ , the central BH mass $M$ , the electrons' temperature $T_{\\rm e}$ , and by the dimensionless mass accretion rate $\\dot{m}$ , which is defined as the ratio of the mass accretion rate to the Eddington accretion rate.",
"We integrate numerically, and for the purpose of cross checking also analytically, along the photon energy $\\epsilon $ over the spectral luminosity $L_{\\rm {ADAF}}(\\epsilon )$ We integrate essentially over the superposition of the cyclosynchrotron emission, its Comptonised part, and of the bremsstrahlung contribution.",
"to obtain $L_{\\rm {ADAF,\\,tot}}$ .",
"We use the canonical choices $\\alpha = 0.3$ , $\\beta = 0.5$ , $R_{\\rm in} = 3 \\, r_{\\rm S}$ , and $R_{\\rm out} = 1000 \\, r_{\\rm S}$ as well as $M = 10^9 \\, M_{\\odot }$ .",
"Furthermore, we choose the energy $\\epsilon _{\\rm {ADAF},\\,0} = 300 \\, k_{\\rm B} \\, T_{\\rm e}$ as an upper integration border because the spectrum essentially vanishes above this energy.",
"We let $\\dot{m}$ and $T_{\\rm e}$ be free parameters.",
"The chosen requirement $L_{\\rm {lines,\\,tot} \\approx 0.01 \\cdot L_{\\rm {ADAF,\\,tot}}(\\dot{m}, T_{\\rm e})}$ can be met with the exemplary $(\\dot{m}, T_{\\rm e})$ pairs listed in row 1 of Table REF .",
"This shows that the present scenario fits to the general picture of accretion flows as illuminators of BLRs.",
"Moreover, it gives evidence that Mrk 501 is indeed fed by a low-rate, hot ADAF.",
"When the reprocessing fraction $\\xi $ is increased from $1 \\, \\%$ to the canonical $10 \\, \\%$ , the values of $T_{\\rm e}$ decrease by about $25 \\, \\%$ (cf.",
"Table REF )."
],
[
"Constraints on the in-gap and post-gap multiplication",
"The parameter $K_{\\rm {G}}$ expresses the number density of electrons that intrude into the interaction region per time interval.",
"In our model, these electrons enter the interaction region after their number was increased in the gap and during the post-gap cascade.",
"The origin of the electron beam along which the multiplication takes place is the vacuum gap (cf.",
"Fig.",
"REF ), where in turn the origin of the seed electrons is PP through self-interaction of ADAF radiation.",
"In other words, photons that are emitted by the ADAF can collide in the BH magnetosphere and pair-produce seed electrons.",
"We can determine the number density of ADAF photons capable of PP with the help of the above used ADAF model by [50].",
"The spectral number density of ADAF photons is given by $n_{\\rm {ADAF}}(\\epsilon ) = L_{\\rm {ADAF}}(\\epsilon ) / (c \\cdot \\epsilon \\cdot A_{\\rm {ADAF}})$ , where $A_{\\rm {ADAF}} = 2 \\, \\pi \\, R_{\\rm out}^2 + 2 \\, \\pi \\, R_{\\rm in}^2 + 2 \\, \\pi \\cdot \\left( R_{\\rm out} \\, \\sqrt{(}1.25 \\, R_{\\rm out}^2) - R_{\\rm in} \\, \\sqrt{(}1.25 \\, R_{\\rm in}^2) \\right)$ is a rough estimate of the surface areaWe determine the total surface area as a sum of the outer cylindrical lateral area, the inner cylindrical lateral area, and twice the outer conical lateral area subtracted by the inner conical lateral area.",
"Additionally, we assume the height of the ADAF to be $r$ at radius $r$ .",
"of the ADAF.",
"As mentioned above, the ADAF spectrum is non-vanishing up to the energy $\\epsilon _{\\rm {ADAF},\\,0}$ .",
"Because PP is a threshold process, the lowest possible ADAF photons capable of PP have the energy $\\epsilon _{\\rm {ADAF,\\,PP\\,th}} = m_{\\rm e}^2 c^4 /\\epsilon _{\\rm {ADAF},\\,0}$ .",
"Consequently, the total number density of ADAF photons capable of PP is obtained by $n_{{\\rm PP}, \\, 1} \\approx \\int _{\\epsilon _{\\rm {ADAF,\\,PP\\,th}}}^{\\epsilon _{\\rm {ADAF},\\,0}} n_{\\rm {ADAF}}(\\epsilon ) \\, \\mathrm {d}\\epsilon .$ As a comparison, we determine the total number density of pair-producing MeV photons based on the approximation $n_{{\\rm PP}, \\, 2} \\approx 1.4 \\cdot 10^{17} \\, \\dot{m}^2 \\,\\frac{10^9 M_{\\odot }}{M} \\, \\rm {m}^{-3}$ from [43], which is motivated by findings for the bremsstrahlung luminosity of an ADAF by [53].",
"For the ADAF parameters we used above, the assumed $\\xi = 1 \\, \\%$ , and the constrained pairs of $\\dot{m}$ and $T_{\\rm e}$ , we give the correspondingly values of $n_{{\\rm PP}, \\, 1}$ and $n_{{\\rm PP}, \\, 2}$ in the second and third row of Table REF .",
"Except for the last pair, we have an order-of-magnitude agreement.",
"In contrast, for the canonical $\\xi = 10 \\, \\%$ and the corresponding lower $T_{\\rm e}$ , we obtain values for $n_{{\\rm PP}, \\, 1}$ that are about two orders of magnitude lower (cf.",
"Table REF ) than in the $1 \\, \\%$ reprocessing fraction case.",
"From the number density of pair-producing ADAF photons, we estimate the materialisation rate in the gap.",
"With $0.2 \\, \\sigma _{\\rm Th}$ as an approximation for the PP cross section, the number density of pairs that are produced in the gap region per unit time is $K_{\\rm {PP,\\,gap}} \\approx 0.2 \\, \\sigma _{\\rm Th} \\, n_{{\\rm PP}, \\, 1}^2 \\, c$ .",
"We show the values of $K_{\\rm {PP,\\,gap}}$ for the $(\\dot{m}, T_{\\rm e})$ pairs in the fourth row of Table REF .",
"All four estimates in the $\\xi = 1 \\, \\%$ case are about six orders of magnitude smaller than our determined $K_{\\rm {G}}$ and all four estimates in the $\\xi = 10 \\, \\%$ case are about ten orders smaller.",
"This does not necessarily cast doubt on our model.",
"These pair-produced seed electrons are accelerated in the gap and multiplied in the gap and during the propagation to the interaction with the emission-line photons.",
"The multiplication is mediated by the emission of synchro-curvature radiation in the magnetospheric magnetic field, by the emission of IC up-scattered radiation in the ADAF photon field, and by the subsequent materialisation of this radiation through PP.",
"In net effect, this multiplication increases the number of electrons by the number $\\mathcal {N}$ and also lowers the energy per electron by $\\mathcal {N}$ .",
"Because we neither include the magnetic field and the emission of synchro-curvature radiation nor the exact spatial distribution of the ADAF photon field into our model, a detailed analysis of the in-gap and post-gap multiplication is beyond the scope of this work.",
"We can, however, estimate from comparison of $K_{\\rm {PP,\\,gap}}$ with $K_{\\rm {G}}$ that it has to be $\\mathcal {N} \\approx 10^6$ in the $1 \\, \\%$ reprocessing fraction case and $\\mathcal {N} \\approx 10^{10}$ in the $10 \\, \\%$ case.",
"The finding $\\mathcal {N} \\approx 10^6$ is three orders of magnitude larger than the multiplication found by [18].",
"These authors, however, determined the multiplication only in the post-gap cascade.",
"Considering that up to $10^7$ curvature photons can be emitted by one electron in the initial spark event of the gap [42], $\\mathcal {N} \\approx 10^6$ after the post-gap cascade is plausible and does not inhibit gap stability.",
"Furthermore, to assess the magnitude of $\\mathcal {N}$ , we have to consider the magnetic field structure (especially the divergence of the magnetic field), which can dilute the number density of the electron bunch.",
"The maximum potential drop in the gap is connected with the maximum Lorentz factor of the electrons.",
"Knowing from our modelling (cf.",
"Eq.",
"REF and Table REF ) that the maximum energy per electron after the multiplication is $\\gamma _{{\\rm {i}},\\,0} \\, m_{\\rm {e}} c^2 = (\\gamma _{\\rm {mean}} + 3 \\, \\varsigma ) \\, m_{\\rm {e}} c^2 = 5.7 \\cdot 10^{12} \\, \\rm {eV}$ , we can estimate that the maximum energy that one original seed electron has attained in the gap voltage is $\\approx \\mathcal {N} \\cdot \\gamma _{{\\rm {i}},\\,0} \\, m_{\\rm {e}} c^2$ .",
"In the $\\xi = 1 \\, \\%$ case, this gives $\\approx 5.7 \\cdot 10^{18} \\, \\rm {eV}$ , which agrees with estimated [43] and simulated [33] values for the potential drop in a vacuum gap.",
"In the $\\xi = 10 \\, \\%$ case, this gives $\\approx 5.7 \\cdot 10^{22} \\, \\rm {eV}$ , which seems hard to reach even for extreme magnetospheric gap conditions.",
"Thus, the canonical choice $\\xi = 10 \\, \\%$ for the reprocessing fraction does not fit into a consistent picture of Mrk 501.",
"[44], [42], [20], [23], and [40] have recently cast doubt on the prior attitude of vacuum gaps being steady-state phenomena.",
"Instead, they might be of intermittent or cyclic character.",
"Concretely, [42] found that charge-starved gaps, after ignition by injection of gamma rays or electrons abruptly discharge in a gamma-ray flaring event on a timescale $r_{\\rm S}/(2 c)$ , followed by relaxation towards a state of weak-amplitude oscillations of the electric field and plasma.",
"In this case, rule-of-thumb estimates for the maximum potential drop would not be adequate for conclusions about the attained electron Lorentz factor.",
"We therefore leave detailed inferences from our input parameters about conditions in the vacuum gap to future studies.",
"If the electron beam is emitted from an oscillating gap, one might expect that this also leads to cyclic behaviour of the cascade with the emission-line photons.",
"However, [18] found that if the gap width exceeds its height, then the gap consists of a number of causally independent tubes.",
"The cyclic behaviour after the initial spark can therefore hardly be traced because the emitted electron beam is the transverse average of many independently fluctuating gap tubes.",
"Table: Fitting parameters that were used to describe the narrow SED feature by emission from an IC pair cascade."
],
[
"Comparison of energy budgets",
"The total power that is injected into the interaction volume is determined to be $P_{\\rm i} = 4 \\pi R^3/3 \\cdot \\int _{\\gamma _{{\\mathrm {i}},\\,1}} ^{\\gamma _{{\\mathrm {i}},\\,0}} \\gamma \\dot{N}_{\\rm i}(\\gamma ) {\\rm d}\\gamma \\cdot m_{\\rm e} c^2 = 1.9 \\cdot 10^{33} \\, {\\rm W}$ , where $R$ is the radial size of the interaction region.",
"This power is due to the electrons that have inherited their energy from gap acceleration.",
"Hence, $P_{\\rm i}$ has to be smaller than the maximum power $P_{\\rm gap}$ that can be extracted from the gap, which itself cannot exceed the corresponding Blandford-Znajek power $P_{\\rm BZ}$ .",
"Moreover, we determine the power, that is radiated by the electrons in the SSC emitting zone to be $P_{\\rm SSC} = 4.4 \\cdot 10^{36} \\, {\\rm W}$ .",
"The magnetic flux density at the BH horizon upon assuming an open gap was estimated by [5] to be $B_{\\rm BH} = 0.19 \\cdot (\\beta _{\\rm m}/M_9)^{4/7} \\, {\\rm T}$ , where $\\beta _{\\rm m}$ is the magnetisation of the ADAF.",
"Assuming the maximum possible electric field strength, gap extension, BH spin, and charge density, [5] obtained $P_{\\rm gap} = 4.8 \\cdot 10^{37} \\, \\beta _{\\rm m}^{8/7} \\, \\kappa \\, h \\, M_9^{6/7} \\ \\sin ^2\\theta \\,\\, {\\rm W}$ , where we have substituted for the variability timescale by the height $h$ of the gap in units of Schwarzschild radii, and where $\\theta $ is the polar angle, and $\\kappa $ is the multiplicity in the gap as defined by [5].",
"For the values $\\theta =\\pi /4$ , $\\kappa =0.1$ (weakly screened gap), $h=0.25$ (thickness is half a gravitational radius), and $\\beta _{\\rm m}=0.5$ (magnetic pressure is half the gas pressure), we find $P_{\\rm gap}=2.7 \\cdot 10^{35} \\, {\\rm W}$ .",
"For the same parameters and for the dimensionless BH spin parameter $a_{\\ast } = 0.99$ , we find $P_{\\rm BZ}= 10^{38} \\, a_{\\ast }^2 \\, M_9 \\, (B_{\\rm BH}/{\\rm T})^2 \\, {\\rm W}= 1.6 \\cdot 10^{36} \\, {\\rm W}$ [34], which is twice the accretion flow luminosity (cf.",
"Sect.",
"REF .",
"To conclude, we find $P_{\\rm i} / P_{\\rm BZ} \\approx 0.001$ .",
"This agrees with estimates by [20] and [40] and especially with the finding of [42], that a fraction $\\approx 0.001$ of $P_{\\rm BZ}$ is dissipated in a time span $\\approx 7 \\, r_{\\rm S}/(2 \\, c)$ after the initial discharge of a gap.",
"When we again assume $M_9 = 1$ , this time span is $\\approx 9.6 \\, {\\rm h}$ , the approximate duration of the night of observation of the narrow feature.",
"The comparison $P_{\\rm gap} / P_{\\rm BZ} \\approx 0.2$ is due to the upper limit nature of the estimates by [5].",
"The ratio $P_{\\rm SSC} / P_{\\rm BZ}\\approx 3$ is acceptable within the uncertainty of $M$ and changes of the magnetic field strength.",
"Table: Four combinations of physical quantities connecting ADAF properties to the materialisation rate in the vacuum gap.",
"The first row shows pairs of values for the dimensionless accretion rate and for the electron temperature of an ADAF that are permitted by reconciliation of the ADAF luminosity with the luminosity of the line emitting clouds.",
"The corresponding values of the total number density of pair producing ADAF photons according to the model by , of the total number density of pair producing ADAF photons according to the estimation by , and of the materialisation rate of pair-produced electrons in the gap region are shown in the second, third, and fourth line, respectively.",
"In each field, the upper item was obtained with the assumption of ξ=1%\\xi = 1 \\, \\% and the lower item was obtained with ξ=10%\\xi = 10 \\, \\%.",
"As n PP ,2 n_{{\\rm PP}, \\, 2} is independent of T e T_{\\rm e}, it has the same value for both cases."
],
[
"Summary",
"We have modelled the interaction of particle beams from vacuum gaps in the magnetospheres of spinning BHs in blazars with recombination-line photons from surrounding gas clouds.",
"Such clouds are commonly found in the BLR of quasars, but could also arise in early-type galaxies as a result of red giants that cross the particle beam.",
"Numerically solving the coupled kinetic equations corresponding to linear IC pair cascades with escape terms, we obtained the steady-state solution of the SED of the photons escaping from the interaction zone.",
"Compared to previous studies of cascades in AGN, our approach considered three main ramifications: The inclusion of an escape term corresponds to a source that is finitely extended and not necessarily optically thick.",
"Thus, in contrast to the work of [68], which dealt with saturated and hence optically thick cascades, our scenario can be applied to astrophysical objects that are optically thin to the emission of HE gamma rays.",
"This is only a first step towards the self-consistent treatment of a spherically symmetric emission region with a radially varying emission and absorption coefficients.",
"For the SSC mechanism without pair cascades, such a treatment was developed by [29].",
"In our framework, the photon density reacts back upon itself through the electron density.",
"Hence, we did not treat the emission coefficient as prescribed.",
"[66] determined the photon spectral number density $n_\\gamma $ essentially as the accumulation of all the IC up-scattered photons.",
"In contrast, we determined $n_\\gamma $ by considering both IC up-scattering and losses through escape and pair absorption (cf.",
"Eq.",
"REF ).",
"We used an efficient numerical iteration scheme (cf.",
"Sect. )",
"that saves computational cost by iterating from values $\\gamma _k$ with high $k$ to lower $k$ , and so using the optimal course of $N(\\gamma )$ in each iteration step.",
"Applying the model to the blazar Mrk 501, we showed that the recently observed peculiar narrow spectral feature at 3 TeV can readily be explained when rather plausible physical parameters for the beam and recombination-line clouds are adopted.",
"In the scenario supported by this observation, photons from a weakly accreting hot ADAF materialise as pairs in the central BH magnetosphere.",
"These seed electrons are accelerated in a vacuum gap and are multiplied by post-gap cascades while travelling away from the central region and carry about $0.1 \\, \\%$ of the Blandford-Znajek luminosity.",
"The ADAF also irradiates and ionises ambient gas clouds.",
"In agreement with measurements of the hydrogen Lyman-$\\alpha $ luminosity in Mrk 501, these clouds reprocess about $1 \\, \\%$ of the original ADAF luminosity as emission-line photons.",
"They act as a target colliding with the electron beam.",
"The resulting cascade photons are emitted in the direction of the primary beam electrons, shaping the intermittent narrow bump at about 3 TeV in the SED.",
"In combination with the SSC spectrum produced by particles presumably accelerated at shock waves further downstream, the model explains the broadband SED of Mrk 501 at MJD 56857.98, assuming no further pair attenuation within the host galaxy to occur.",
"Predictions about the duty cycle of this feature are difficult considering the intermittency of gap formation and randomness of nearby cloud passages.",
"We thank Dorit Glawion, and Amit Shukla for fruitful discussions on the topic.",
"We are also thankful to the anonymous referee for constructive criticism on the manuscript.",
"Moreover, we thank Astrid Peter and Julian Sitarek for comments on this work as well as J. D. Hunter and co-workers for the development of matplotlib [36].",
"C. W. gratefully acknowledges support by the project \"Promotion inklusive\" of the Universität zu Köln and the German Bundesministerium für Arbeit und Soziales.",
"J.",
"B. G. acknowledges the support of the Viera y Clavijo program funded by ACIISI and ULL."
],
[
"Derivation of the kinetic equations",
"As an attachment to Subsect.",
"REF , here we give a precise derivation of the kinetic equations.",
"To quantify the change rates of the spectral number densities of the photons and of the electrons, it is necessary to introduce the spectral interaction rates of IC scattering and of PP.",
"We do this in line with [68].",
"Given an IC scattering event of an incident electron with original energy $\\gamma \\gg 1$ off a photon with energy $x \\ll 1$ that results in a down-scattered electron with final energy $\\gamma ^{\\prime }$ and a highly energetic photon with energy $x_\\gamma $ .",
"Energy conservation yields $\\gamma \\approx \\gamma + x = \\gamma ^{\\prime } + x_\\gamma $ .",
"From IC scattering kinematics, it follows $\\gamma x > E_{\\ast }$ with the abbreviation $E_{\\ast } = (\\gamma / \\gamma ^{\\prime } - 1)/4$ [68].",
"This ensures that the photon energy has a maximum possible value $x_{\\gamma ,\\,\\mathrm {max}}(\\gamma ,x) := (4 \\gamma ^2 x) / (1 + 4 \\gamma x)$ .",
"This function is $< \\gamma $ for all realistic values of $\\gamma $ , which reflects that the electron cannot transfer all its energy to the photon.",
"Moreover, we remark that $x_{\\gamma ,\\,\\mathrm {max}}(\\gamma ,x)$ increases with increasing $\\gamma $ as well as with increasing $x$ .",
"Hence, if $N$ has an upper cut-off at $\\gamma _{\\rm {i},\\,0}$ and $n_0$ has an upper cut-off at $x_0$ , then the distribution of highly energetic photons vanishes above $x_{\\gamma ,\\,\\mathrm {max}}(\\gamma _{\\rm {i},\\,0},x_0)$ and is non-vanishing below this.",
"The spectral IC scattering interaction rate $C$ for such events on the field of low-energy photons with spectral number density $n_0$ is given by [68] as an approximation of the exact one found by [37], $C(\\gamma ,\\gamma ^{\\prime }) = \\int _{\\max (x_1, \\, E_{\\ast }/\\gamma )}^{x_0} n_0(x) \\frac{3 \\sigma _{\\rm {Th}} c}{4 \\gamma ^2 x} \\left[ r + (2-r) \\frac{E_{\\ast }}{\\gamma x} - 2 \\left( \\frac{E_{\\ast }}{\\gamma x} \\right)^2 - 2 \\frac{E_{\\ast }}{\\gamma x} \\ln {\\frac{\\gamma x}{E_{\\ast }}} \\right] \\, \\mathrm {d}x$ Here it is $r = (\\gamma / \\gamma ^{\\prime } + \\gamma ^{\\prime } / \\gamma )/2$ and the second argument in the lower integration border is due to Ineq.",
"REF .",
"$N(\\gamma ) C(\\gamma ,\\gamma ^{\\prime })$ is the number of IC scattering events with original energy $\\gamma $ and final energy $\\gamma ^{\\prime }$ per unit time, per unit space volume, per unit original energy, and per unit final energy.",
"Substituting $\\gamma ^{\\prime }$ with $\\gamma - x_\\gamma $ in $C$ , we realise that $C(\\gamma ,\\gamma - x_\\gamma )$ describes the probability for an IC scattering event with original electron energy $\\gamma $ to produce a photon with energy $x_\\gamma $ , per unit time, and per unit $x_\\gamma $ -interval.",
"$N(\\gamma ) C(\\gamma ,\\gamma - x_\\gamma )$ stands for the number of IC scattering events with original electron energy $\\gamma $ that result in a photon with energy $x_\\gamma $ , per unit time, per unit space volume, per unit $\\gamma $ -interval, and per unit $x_\\gamma $ -interval.",
"Now, we consider a collision of a highly energetic photon of energy $x_\\gamma $ with a photon of energy $x$ , resulting in the PP of an electron of energy $\\gamma $ and a positron with energy $x_\\gamma + x - \\gamma \\approx x_\\gamma - \\gamma $ .",
"Based on [3], [68] gave the corresponding kinematic relationship $x_{\\gamma } x > E_{\\ast } > 1$ where $E_{\\ast }=x_{\\gamma }^2/(4 \\gamma (x_{\\gamma }-\\gamma ))$ as well as the spectral PP interaction rate $P$ , again for a photon field $n_0$ , $P(x_{\\gamma },\\gamma ) = \\int _{\\max (x_1, \\, E_{\\ast }/\\gamma )}^{x_0} n_0(x) \\frac{3 \\sigma _{\\rm {Th}} c}{4 x_{\\gamma }^2 x} \\left[ r - (2+r) \\frac{E_{\\ast }}{x_{\\gamma } x} + 2 \\left( \\frac{E_{\\ast }}{x_{\\gamma } x} \\right)^2 + 2 \\frac{E_{\\ast }}{x_{\\gamma } x} \\ln {\\frac{x_{\\gamma } x}{E_{\\ast }}} \\right] \\, \\mathrm {d}x$ Here it is $r = (\\gamma / (x_{\\gamma }-\\gamma ) + (x_{\\gamma }-\\gamma ) / \\gamma )/2$ and the second argument in the lower integration border is due to the first part of Ineq.",
"REF .",
"$n_\\gamma (x_\\gamma ) P(x_\\gamma ,\\gamma )$ is the number of PP events with photon energy $x_\\gamma $ that create an electron with energy $\\gamma $ per unit time, per unit space volume, per unit photon energy, and per unit electron energy.",
"To infer the kinetic equation of the electrons, we have to specify all relevant sinks and sources of the electron energy distribution in terms of change rates of the electron spectral number density $N$ : The spectral injection rate $\\dot{N}_{\\rm i}(\\gamma )$ of electrons directly enters the kinetic equation.",
"Electrons leave the interaction region on the escape timescale $T_{\\rm {e\\,esc}}$ .",
"This is expressed with the term ${N(\\gamma )}/{T_{\\rm {e\\,esc}}(\\gamma )}$ .",
"An electron of original energy $\\gamma $ loses energy by becoming IC down-scattered to a lower energy $\\gamma ^{\\prime } < \\gamma $ , resulting in a sink at $\\gamma $ .",
"We remark, however, that for given $x$ and $\\gamma $ , the minimum possible value of $\\gamma ^{\\prime }$ is not equal to 1 but has a lower limit at $\\gamma ^{\\prime }_{\\rm {IC,\\,min}}(\\gamma ,x) := \\gamma /(1 + 4 x \\gamma )$ as a result of scattering kinematics.",
"From Ineq.",
"REF the limitation $\\gamma ^{\\prime } > \\gamma ^{\\prime }_{\\rm {IC,\\,min}}$ follows.",
"It always is $\\gamma ^{\\prime }_{\\rm {IC,\\,min}} > 1$ , reflecting the fact that the electron retains an amount of kinetic energy in every case.",
"$\\gamma ^{\\prime }_{\\rm {IC,\\,min}}$ decreases with increasing $x$ .",
"Now, if the low-energy photons are not mono-energetic, but are distributed in energy space with an upper cut-off of $n_0(x)$ at some $x_0$ , then the value of $\\gamma ^{\\prime }$ that can at least be reached for an electron with original energy $\\gamma $ , is $\\gamma ^{\\prime }_{\\rm {IC,\\,min}}(\\gamma ,x_0)$ .",
"Therefore we describe the number of IC scattering events with original energy $\\gamma $ and any allowed final energy $\\gamma ^{\\prime }$ per unit time, per unit space volume, and per unit original energy by $\\int ^{\\gamma }_{\\gamma ^{\\prime }_{\\rm {IC,\\,min}}(\\gamma ,x_0)} N(\\gamma ) C(\\gamma ,\\gamma ^{\\prime }) \\, \\rm {d}\\gamma ^{\\prime }$ The lower integration border is in contrast to the one mistakenly used by [68] in the second term on the right-hand side of his Eq.",
"1 as well as in the first integral in A21.. a source at $\\gamma ^{\\prime }$ .",
"Though, we can infer from Ineq.",
"REF that a final electron energy $\\gamma ^{\\prime }$ cannot be reached from any original electron energy $\\gamma $ .",
"For $\\gamma ^{\\prime } \\ge 1/(4 x)$ (scattering into the KN regime), the original electron energy is indeed unbounded.",
"However, for $\\gamma ^{\\prime } < 1/(4 x)$ (scattering into the Thomson regime), the original electron energy has to be lower than $\\gamma ^{\\prime }/(1 - 4 x \\gamma ^{\\prime })$ , which increases with increasing $x$ .",
"Again, if the low-energy photon spectral number density is extended with an upper cut-off at $x_0$ , then the upper boundary of $\\gamma $ is expressed by $\\gamma ^{\\prime }/(1 - 4 x_0 \\gamma ^{\\prime })$ .",
"In what follows we define $\\gamma _{\\mathrm {IC,\\,max}}(\\gamma ^{\\prime },x) := \\left\\lbrace \\begin{array}{ll}\\gamma ^{\\prime }/(1 - 4 x \\gamma ^{\\prime }) & \\mathrm {for} \\; \\; \\gamma ^{\\prime } < 1/(4 x) \\mathrm {,} \\\\+ \\infty & \\mathrm {for} \\; \\; \\gamma ^{\\prime } \\ge 1/(4 x)\\mathrm {.",
"}\\end{array}\\right.$ With this, we can quantify the number of IC scattering events with final energy $\\gamma ^{\\prime }$ and any permitted original energy $\\gamma $ per unit time, per unit space volume, and per unit final energy by $\\int _{\\gamma ^{\\prime }}^{\\gamma _{\\rm {IC,\\,max}}(\\gamma ^{\\prime },x_0)} N(\\gamma ) C(\\gamma ,\\gamma ^{\\prime }) \\, \\rm {d}\\gamma $ The upper integration border is in contrast to the one erroneously used by [68] in the third term on the right-hand side of his Eq. 1..",
"When this term is plugged into the kinetic equation, we have to exchange $\\gamma $ with $\\gamma ^{\\prime }$ in this term because the kinetic equation refers to the variable $\\gamma $ .",
"PP of an electron with energy $\\gamma $ is a source at this energy.",
"However, PP on a photon with energy $x$ that produces an electron with energy $\\gamma $ is possible only if the photon energy $x_\\gamma $ exceeds the PP threshold $x_{\\gamma ,\\,\\rm {PP,\\,th}}(\\gamma ,x) := \\gamma /(1 - (1/(4 \\gamma x)))$ .",
"This threshold can be inferred from Ineq.",
"REF .",
"$x_{\\gamma ,\\,\\rm {PP,\\,th}}$ adopts finite and positive values only for $\\gamma > 1/(4 x)$ , expressing that electrons cannot be pair-produced below $1/(4 x)$ .",
"In this range, $x_{\\gamma ,\\,\\rm {PP,\\,th}}$ has a minimum at $\\gamma = 1/(2 x)$ .",
"This minimum corresponds to $x_{\\gamma ,\\,\\rm {PP,\\,th}}(1/(2 x),x) = 1/x$ , which is the well-known set-in of PP.",
"For $\\gamma > 1/(4 x)$ , it is of course $x_{\\gamma ,\\,\\rm {PP,\\,th}} > \\gamma $ .",
"Furthermore, we can show that $x_{\\gamma ,\\,\\rm {PP,\\,th}}$ decreases with increasing $x$ .",
"Hence, if the low-energy photons are not mono-energetic, but are distributed with $n_0(x)$ , which vanishes above an upper cut-off $x_0$ , then PP is possible as soon as $x_\\gamma > x_{\\gamma ,\\,\\rm {PP,\\,th}}(\\gamma ,x_0)$ .",
"The number of kinematically allowed PP events that create an electron with energy $\\gamma $ per unit time, per unit space volume, and per unit electron energy, is accordingly expressed by $\\int _{x_{\\gamma ,\\,\\rm {PP,\\,th}}(\\gamma ,x_0)}^{\\infty } n_\\gamma (x_\\gamma ) P(x_\\gamma ,\\gamma ) \\, {\\rm {d}}x_\\gamma $ .",
"As $P$ is normalised to the production of only one electron, this term has to be multiplied by 2 in the kinetic equation.",
"The electron kinetic equation is obtained by setting the rate of change $\\dot{N}(\\gamma )$ of the electron spectral number density equal to the sum of all source terms subtracted by all sinks, $\\dot{N}(\\gamma ) = \\dot{N}_{\\mathrm {i}}(\\gamma ) - N(\\gamma ) \\left( \\frac{1}{T_{\\rm {e\\,esc}}(\\gamma )} + \\int ^{\\gamma }_{\\gamma ^{\\prime }_{\\rm {IC,\\,min}}(\\gamma ,x_0)} C(\\gamma ,\\gamma ^{\\prime }) \\, \\mathrm {d}\\gamma ^{\\prime } \\right) + \\int _{\\gamma }^{\\gamma ^{\\prime }_{\\mathrm {IC,\\,max}}(\\gamma ,x_0)} N(\\gamma ^{\\prime }) C(\\gamma ^{\\prime },\\gamma ) \\, \\mathrm {d}\\gamma ^{\\prime } + 2 \\cdot \\int _{x_{\\gamma ,\\,\\mathrm {PP,\\,th}}(\\gamma ,x_0)}^{\\infty } n_\\gamma (x_\\gamma ) P(x_\\gamma ,\\gamma ) \\, \\mathrm {d}x_\\gamma $ In what follows the first term of this equation is called electron spectral injection rate, the part within round brackets is called electron spectral loss rate (which is the sum of the spectral escape rate and of the spectral IC down-scattering rate to lower energies), and the sum of the third and fourth term are called electron spectral production rate (which is the sum of the spectral IC down-scattering rate from higher energies and of the spectral PP rate)We note that the dimension of the electron spectral injection rate and of the electron spectral production rate is $[\\gamma ]^{-1} [t]^{-1} [l]^{-3}$ , respectively, whereas that of the electron spectral loss rate is $[\\gamma ]^{-1} [t]^{-1}$ .. Analogously, we identify all source and sink terms that determine the rate of change $\\dot{n}_\\gamma (x_\\gamma )$ of the highly energetic photon spectral number density: The injection term $\\dot{n}_{\\gamma ,\\,\\rm i}(x_\\gamma )$ has to be included.",
"The spectral loss rate due to escape from the interaction volume is described by ${n_\\gamma (x_\\gamma )}/{T_{\\rm {ph\\,esc}}(x_\\gamma )}$ .",
"PP is a sink of the highly energetic photon distribution at the energy $x_\\gamma $ of the incident photons.",
"For given $x$ and $x_\\gamma $ , the energies $\\gamma $ that can be reached by the created electrons, are limited due to the interaction kinematics Ineq.",
"REF .",
"The electron energy $\\gamma $ that can be obtained obeys $\\gamma _{\\rm {PP,\\,min}} < \\gamma < \\gamma _{\\rm {PP,\\,max}}$ with the limits $\\gamma _{\\rm {PP,\\,min}}(x_\\gamma ,x) := x_\\gamma (1 - (1 - 1 /(x_\\gamma x))^{1/2})/2$ and $\\gamma _{\\rm {PP,\\,max}}(x_\\gamma ,x) :=\\\\ x_\\gamma (1 + (1 - 1 /(x_\\gamma x))^{1/2})/2$ .",
"With increasing $x$ , the limit $\\gamma _{\\rm {PP,\\,min}}$ decreases and $\\gamma _{\\rm {PP,\\,max}}$ increases.",
"Consequently, if $n_0$ is extended and non-vanishing only below and at an upper cut-off $x_0$ , then the electron energies that are kinematically allowed satisfy $\\gamma _{\\rm {PP,\\,min}}(x_\\gamma ,x_0) < \\gamma < \\gamma _{\\rm {PP,\\,max}}(x_\\gamma ,x_0)$ .",
"With this in mind, the number of PP events with incident photon energy $x_\\gamma $ and any possible electron energy per unit time, per unit space volume, and per unit $x_\\gamma $ -interval can be written as $\\int ^{\\gamma _{\\rm {PP,\\,max}}(x_\\gamma ,x_0)}_{\\gamma _{\\rm {PP,\\,min}}(x_\\gamma ,x_0)} n_\\gamma (x_\\gamma ) P(x_\\gamma ,\\gamma ) \\, \\rm {d}\\gamma $ .",
"The IC up-scattering of photons of energy $x$ to energy $x_\\gamma $ is a source of photons.",
"The energy $x_\\gamma $ can, however, only be reached, when the original electron energy $\\gamma $ exceeds a threshold $\\gamma _{\\rm {IC,\\,th}}$ .",
"This threshold is defined by $\\gamma _{\\rm {IC,\\,th}}(x_\\gamma ,x) := x_\\gamma (1 + (1 + 1 /(x_\\gamma x))^{1/2})/2$ , as can be inferred from Ineq.",
"REF , and is $> x_\\gamma $ for all realistic $x_\\gamma $ .",
"The constraint by this threshold is again the result of the electron not being able to transfer all its energy to the photon, and hence the original electron energy has to exceed $x_\\gamma $ by $\\gamma ^{\\prime }$ , which cannot fall below the value $\\gamma _{\\rm {IC,\\,th}} - x_\\gamma $ .",
"We note that $\\gamma _{\\rm {IC,\\,th}}$ decreases with increasing $x$ .",
"Thus, if $n_0$ is extended over a range of energies and has an upper cut-off at $x_0$ , then IC scattering to the energy $x_\\gamma $ sets in as soon as $\\gamma > \\gamma _{\\rm {IC,\\,th}}(x_\\gamma ,x_0)$ .",
"Then, $\\int _{\\gamma _{\\rm {IC,\\,th}}(x_\\gamma ,x_0)}^{\\infty } N(\\gamma ) C(\\gamma ,\\gamma -x_\\gamma ) \\, \\rm {d}\\gamma $ quantifies the number of IC scattering events with any kinematically allowed original electron energy that result in a photon with energy $x_\\gamma $ per unit time, per unit space volume, and per unit energy.",
"The highly energetic photon kinetic equation is obtained by setting the rate of change $\\dot{n}_\\gamma (x_\\gamma )$ of the photon spectral number density equal to the sum of all source terms subtracted by all sinks, $\\dot{n}_\\gamma (x_\\gamma ) = \\dot{n}_{\\gamma ,\\,\\mathrm {i}}(x_\\gamma ) - n_\\gamma (x_\\gamma ) \\left( \\frac{1}{T_{\\rm {ph\\,esc}}(x_\\gamma )} + \\int ^{\\gamma _{\\mathrm {PP,\\,max}}(x_\\gamma ,x_0)}_{\\gamma _{\\mathrm {PP,\\,min}}(x_\\gamma ,x_0)} P(x_\\gamma ,\\gamma ) \\, \\mathrm {d}\\gamma \\right) + \\underbrace{\\int _{\\gamma _{\\mathrm {IC,\\,th}}(x_\\gamma ,x_0)}^{\\infty } N(\\gamma ) C(\\gamma ,\\gamma -x_\\gamma ) \\, \\mathrm {d}\\gamma }_{:= \\, \\dot{n}_{\\gamma , \\, \\rm {IC}}(x_\\gamma )}$ In what follows the first term of this equation is called photon spectral injection rate, the part within round brackets is called photon spectral loss rate (which is the sum of the spectral escape rate and of the spectral pair absorption rate), and the last term $\\dot{n}_{\\gamma , \\, \\rm {IC}}$ is called photon spectral production rateWe remark that the dimension of the photon spectral injection rate and of the photon spectral production rate is $[x_{\\gamma }]^{-1} [t]^{-1} [l]^{-3}$ , respectively, whereas that of the photon spectral loss rate is $[x_{\\gamma }]^{-1} [t]^{-1}$ .. We assume the spectral number densities to be in steady state.",
"Thus, the respective rates of change vanish in Eq.",
"REF and REF .",
"Then, we can solve the respective equation for the respective spectral number density in the second term on the right-hand side of the respective equation.",
"This yields $N(\\gamma ) = \\frac{\\dot{N}_{\\mathrm {i}}(\\gamma ) + \\int _{\\gamma }^{\\gamma ^{\\prime }_{\\mathrm {IC,\\,max}}(\\gamma ,x_0)} N(\\gamma ^{\\prime }) C(\\gamma ^{\\prime },\\gamma ) \\, \\mathrm {d}\\gamma ^{\\prime } + 2 \\cdot \\int _{x_{\\gamma ,\\,\\mathrm {PP,\\,th}}(\\gamma ,x_0)}^{\\infty } n_\\gamma (x_\\gamma ) P(x_\\gamma ,\\gamma ) \\, \\mathrm {d}x_\\gamma }{\\frac{1}{T_{\\rm {e\\,esc}}(\\gamma )} + \\int ^{\\gamma }_{\\gamma ^{\\prime }_{\\rm {IC,\\,min}}(\\gamma ,x_0)} C(\\gamma ,\\gamma ^{\\prime }) \\, \\mathrm {d}\\gamma ^{\\prime }}$ and Eq.",
"REF ."
],
[
"Discussion of numerical solution procedure",
"As an extension to Sect.",
", in this Appendix we concisely explain how explicit converging solutions for $N$ are found from iteratively solving $N(\\gamma ) = \\mathcal {F}(N, \\gamma )$ along the sequence of points $\\left( \\gamma _k \\right)_{k = 1,\\dots ,\\kappa }$ after prescribing the initialisation $N_{\\rm {init}}(\\gamma )$ .",
"One could iterate the following way: One could determine $\\mathcal {F}$ at all the points $\\gamma _k$ based on the sequence $\\left( N_k \\right)_{j_{\\rm {init}}}$ .",
"Then, one would assign the attained values to a new sequence $\\left( N_k \\right)_0$ .",
"In short, the initialisation step of the iteration would be described by $N_{k,\\,0} = \\mathcal {F}(\\left( N_k \\right)_{j_{\\rm {init}}}, \\gamma _k)$ .",
"The same technique would be pursued in the subsequent iteration steps.",
"In each iteration step, one would compute a complete new sequence $\\left( N_k \\right)_j$ (with integer $j$ running along $j_{\\rm {init}},\\,0,\\,1,\\dots ,\\,j_{\\rm {final}}$ and denoting the number of the iteration step, or equivalently, the number of the sequence of values $N_{k,\\,j}$ ).",
"In other words, in the $j$ -th iteration step, one would determine the $\\kappa $ values $N_{k,\\,j} = \\mathcal {F}(\\left( N_k \\right)_{j-1}, \\gamma _k)$ based on the sequence $\\left( N_k \\right)_{j-1}$ of the previous iteration.",
"Iterating would have to proceed until all points $N_{k,\\,j}$ converge with a value called $N_{k,\\,j_{\\rm {final}}}$ (and hence the function $N(\\gamma )$ converges), which is always the case according to the Banach fixed-point theorem.",
"[66] used this iteration scheme.",
"However, this scheme is not favoured because it is computationally inefficient.",
"This can be seen as follows.",
"By inspection of Eq.",
"REF we realise that a computation of $\\mathcal {F}$ at a given point $\\gamma _{k^{\\prime }}$ evaluates (and thus needs to know) $N$ only at points $\\gamma _k \\ge \\gamma _{k^{\\prime }}$ .This observation is obvious in the case of the second term in the numerator of Eq.",
"REF because the integration range stretches from $\\gamma $ upwards.",
"In the case of the third term, we have to consider the values of the lower integration borders of the nested integrals.",
"First, we make use of the fact that the lower border $x_{\\gamma ,\\,\\rm {PP,\\,th}}(\\gamma ,x_0)$ of the outer integral is the PP threshold and thus $> \\gamma $ (for all realistic values of $\\gamma $ ).",
"Second, as mentioned in Sect.",
"REF , the lower border $\\gamma _{\\mathrm {IC,\\,th}}(x_\\gamma ,x_0)$ of the inner integral is always $> x_\\gamma $ .",
"As an effect, the lowest value, at which $N$ has to be evaluated during the computation of the third term, is $\\gamma _{\\mathrm {IC,\\,th}}(x_{\\gamma ,\\,\\rm {PP,\\,th}}(\\gamma ,x_0),x_0)$ , which is always $> \\gamma $ .",
"We consider the computation of a certain value $N_{k^{\\prime },\\,j}$ , in other words, the $j$ -th computation of the value of $N$ at $\\gamma _{k^{\\prime }}$ .",
"This computation is based on all the values $N_{k,\\,j-1}$ with $k \\ge k^{\\prime }$ , in other words, on the sequence $\\left( N_k \\right)_{k = k^{\\prime }+1,\\dots ,\\kappa ;\\,j-1}$ .",
"As long as the values with $k>k^{\\prime }$ have not yet converged, the computation of $N_{k^{\\prime },\\,j}$ is based on inaccurate prerequisites and thus becomes inaccurate by itself.",
"It is therefore reasonable to implement the following iteration scheme: In contrast to above, where the complete sequence $\\left( N_k \\right)_j$ of $\\kappa $ values was determined in one iteration step, we iterate point-wise.",
"In the first step, only the value $N_{\\kappa ,\\,j}$ at the largest $\\gamma $ is iterated.",
"Briefly written, we calculate $N_{\\kappa ,\\,0} = \\mathcal {F}(\\left( N_k \\right)_{j_{\\rm {init}}}, \\gamma _\\kappa )$ and $N_{\\kappa ,\\,j} = \\mathcal {F}(\\left( N_k \\right)_{j-1}, \\gamma _\\kappa )$ .",
"Not until convergence of $N_{\\kappa ,\\,j}$ was achieved with the final value being $N_{\\kappa ,\\,j_{\\rm {final}}}$ , did we iterate at the next lower point $\\gamma _{\\kappa -1}$ .",
"In short, we compute $N_{\\kappa -1,\\,0} = \\mathcal {F}(\\left( N_k \\right)_{j_{\\rm {init}}}, \\gamma _{\\kappa -1})$ and $N_{\\kappa -1,\\,j} = \\mathcal {F}(\\left( N_k \\right)_{j-1}, \\gamma _{\\kappa -1})$ until $N_{\\kappa -1,\\,j}$ converges with the final value being $N_{\\kappa -1,\\,j_{\\rm {final}}}$ .",
"Then, we switch to $\\gamma _{\\kappa -2}$ and iterate again until convergence.",
"This is done successively at all points until convergence was reached at the point $k=1$ .",
"This procedure is advantageous because during the iteration at a point $\\gamma _{k^{\\prime }}$ , the final converged sequence $\\left( N_k \\right)_{k = k^{\\prime }+1,\\dots ,\\kappa ;\\,j_{\\rm {final}}}$ of values is already known and can be used, which reduces inaccuracies at $\\gamma _{k^{\\prime }}$ .",
"In the range $\\gamma > 1/(4 x_0)$ pairs can be produced.",
"This is reflected in the third term in the numerator of Eq.",
"REF being non-zero.",
"This makes the computation of $\\mathcal {F}$ expensive in this regime.",
"At $\\gamma \\le 1/(4 x_0)$ , PP does not happen and the third term vanishes, which speeds up the computations.",
"On the other side, more iteration steps are necessary for lower values of $\\gamma $ due to the decreasing energy transfer in one IC scattering event.",
"In the deep Thomson regime (i.e.",
"for $\\gamma \\ll 1/(4 x_0)$ ) Eq.",
"REF in the steady state can be expressed as a continuity equation and could be solved by a differential equation solver (e.g.",
"a Runge-Kutta method, cf.",
"[68]).",
"However, as we are mainly interested in the course of $N$ in the KN regime, we use the iteration procedure over the complete range."
]
] | 2012.05215 |
[
[
"Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset\n Superspaces"
],
[
"Abstract We study Chevalley-Eilenberg cohomology of physically relevant Lie superalgebras related to supersymmetric theories, providing explicit expressions for their cocycles in terms of their Maurer-Cartan forms.",
"We then include integral forms in the picture by defining a notion of integral forms related to a Lie superalgebra.",
"We develop a suitable generalization of Chevalley-Eilenberg cohomology extended to integral forms and we prove that it is isomorphic to the ordinary Chevalley-Eilenberg cohomology of the Lie superalgebra.",
"Next we study equivariant Chevalley-Eilenberg cohomology for coset superspaces, which plays a crucial role in supergravity and superstring models.",
"Again, we treat explicitly several examples, providing cocycles' expressions and revealing a characteristic infinite dimensional cohomology."
],
[
"Introduction",
"The mathematical development of cohomology of Lie algebras [19] [29] has been prompted and characterized by a twofold reason in relation to the theory of Lie groups.",
"On one hand, in a diverging direction with respect to Lie groups, Lie algebra cohomology unties the representation theory of Lie algebras from the corresponding representation theory of Lie groups, by allowing a completely algebraic proof of the Weyl theorem [43], which was originally of analytic nature.",
"On the other hand, in a converging direction with respect to Lie groups, in many important instances Lie algebra cohomology makes computations of the de Rham cohomology of the corresponding Lie groups easier.",
"Nowadays, applications of Lie algebra cohomology range from representation theory in pure mathematics to modern physics - let us just recall that Kac-Moody and Virasoro algebras, which play central role in string theory, are central extensions of the polynomial loop-algebra and the Witt algebra respectively, and, as such, they are related to Lie algebra's 2-cohomology group.",
"While it is quite natural to generalize a cohomology theory from Lie algebra to Lie superalgebra [30] [31] (more recent reviews and computations can be found in [32], [33], [34], [35]).",
"both from a derived-functorial point of view and, more concretely, via cochain complexes, it can be seen that the two directions sketched above are meant to breakdown as one moves to the super setting.",
"Indeed, in the representation theoretic direction, there is no Weyl theorem for Lie superalgebras, initially leading to the opinion that the cohomology theory is rather empty and meaningless.",
"Further, in the topological direction, when working with Lie supergroups and their related Lie superalgebras, Cartan theorem resists to a naive “super” generalization, as it only encodes topological informations.",
"On the other hand a different point of view is possible, namely one can look at the the failure of Weyl theorem in the supersymmetric setting as an opportunity, rather than a pathological feature of the theory, for it suggests that the cohomology groups of Lie superalgebras might have a much richer structure than the one that can be guessed by analogy with the ordinary theory.",
"Remarkably, physics is paving this way: cocycles arising from cohomology of Lie superalgebras - in particular, Poincaré superalgebras - are getting related to higher Wess-Zumino-Witten (WZW) terms in supersymmetric Lagrangians (the so-called brane scan and its recent higher-version, the brane bouquet, which promotes Lie superalgebras to $L_{\\infty }$ -superalgebras and consider their cohomology) [1] [2] [20] [21].",
"It is fair to observe, thought, that even the cohomology of a finite dimensional Lie superalgebra does not vanish in general for degree greater than the dimension of the algebra - as it happens in the ordinary case instead -: this makes the actual computation of the cohomology of Lie superalgebras into a very difficult task in general.",
"Accordingly, results can be found in literature for specific choices of superalgebras - in particular in low-degree [41] -, but only very few results encompassing the whole framework are available [23], even just for the Betti numbers of Lie superalgebras.",
"Even less is known regarding the cohomology and the structure of cocycles of coset or homogeneous superspaces, which play a fundamental role in many superstring and supergravity models.",
"If on one hand it is likely that a detailed knowledge of this equivariant cohomologies would help understanding the geometric nature and invariant structure of convoluted supergravity Langrangians [27] [28], it is also fair to notice that - once again - computations are difficult even in the most basic examples.",
"On a different note, getting back to the relations between algebras and groups, as mentioned above, it is a well-known fact that the de Rham cohomology of a Lie group can be formulated in terms of its underlying Lie algebra, thus making feasible computations otherwise very difficult.",
"Whereas one tries to generalize this to Lie supergroups, (s)he would run into an issue, which is deeply ingrained in the theory of forms and the related integration theory in supergeometry.",
"Indeed, in order to formulate a coherent notion of geometric integration on supermanifolds [36], besides differential forms, one also need to take into account integral forms, a notion which is crucial, thought not widely known and understood: for example, a supergeometric analogue of Stokes' theorem [37] [44] is proved using integral forms.",
"On the other hand, it needs to be remarked that Lie superalgebra cohomology is nothing but a “$\\mathbb {Z}_2$ -graded generalization” of the ordinary Lie algebra cohomology, and, as such, it is not capable to account for objects other than differential forms on supermanifolds, such as in particular, integral forms, which simply do not enter the picture [45].",
"It is natural to ask if it is possible to provide a formulation of Lie superalgebra cohomology capable of capturing properties of integral forms as well, and, in turn, what are the relations between the ordinary Lie superalgebra cohomology and this newly defined cohomology.",
"In the present work, after a brief review of Chevalley-Eilenberg cohomology of Lie algebras and superalgebras and a basic introduction to integral forms - which aims at making the paper as self-consistent as possible -, we extend the notion of integral forms to a Lie superalgebraic context and we define a related notion of Chevalley-Eilenberg cohomology.",
"We establish an isomorphism between the Chevalley-Eilenberg cohomology of integral forms of a superalgebra and the ordinary Chevalley-Eilenberg cohomology of the superalgebra in question.",
"We then proceed to explicit computations of these cohomologies in several cases of physical interest, by looking at the Lie superalgebra of symmetries of relevant superspaces.",
"However, it is fair to remark that, even if Lie supergroups - or supergroup manifolds, as they are called in the physics community - and their associated Lie superalgebras appear in several physical applications and have allowed to establish important results, coset supermanifolds actually open up to the most interesting and rich scenarios, offering several ways to take into account different amount of symmetries.",
"For this reason, the last part of the paper is dedicated to the computations of equivariant Chevalley-Eilenberg cohomology for coset superspaces: several examples are discussed and typical phenomenology is pointed out."
],
[
"Lie Algebras and Lie Superalgebras",
"We start providing the basic definitions, first in the usual setting, then in the super one.",
"Let $\\mathfrak {g}$ be an ordinary finite dimensional Lie algebra defined over the field $k$ , and let $V$ be a $\\mathfrak {g}$ -module or a representation space for $\\mathfrak {g}.$ We define the (Chevalley-Eilenberg) $p$ -cochains of $\\mathfrak {g}$ valued in $V$ to be alternating $k$ -linear maps from $\\mathfrak {g}$ to $V$ [19], $C_{CE}^p (\\mathfrak {g}, V) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =Homk (p g, V ), where we note in particular that $C_{CE}^0 (\\mathfrak {g}, V) = Hom_k (k, V) \\cong V$ for $p=0$ and where, in taking the exterior power $\\mathfrak {g}$ is looked at as a vector space.",
"Further, notice that if we take trivial coefficient, i.e.",
"$V = k$ , as we will do in the rest of the paper, we simply have $C_{CE}^p (\\mathfrak {g}, k) = \\bigwedge ^p \\mathfrak {g}^\\ast $ .",
"The above (REF ) can be lifted into a complex by introducing the (Chevalley-Eilenberg) differential $d_{\\mathfrak {g}}^p : C_{CE}^p (\\mathfrak {g}, V) \\rightarrow C_{CE}^{p+1} (\\mathfrak {g}, V),$ defined as $ d_{\\mathfrak {g}}^p f (x_1 \\wedge \\ldots \\wedge x_{p+1}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ = 1 i < j p+1 (-1)i+j f ([xi, xj] x1...xi ...xj ...xp+1) + + i = 1p+1 (-1)i+1 xi f (x1 ...xi ...xp+1), for $f \\in Hom_k (\\wedge ^p \\mathfrak {g}, V)$ and where the hatted entry is omitted.",
"Once again notice that if $V$ is a trivial $\\mathfrak {g}$ -module, as in the case $V = k$ , the second summand vanishes identically, so that one has $d_{\\mathfrak {g}}^p f (x_1 \\wedge \\ldots \\wedge x_{p+1}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ = 1 i < j p+1 (-1)i+j f ([xi, xj] x1...xi ...xj ...xp+1) It is not too hard to prove that $d^{p+1} \\circ d^{p} = 0$ , so that one can define the Chevalley-Eilenberg complex of $\\mathfrak {g}$ valued in $V$ as the pair $(C_{CE}^\\bullet (\\mathfrak {g}, V), d^\\bullet )$ .",
"Given this definition, the cohomology is defined in the usual way: we call Chevalley-Eilenberg cocycles the elements of the vector space $Z_{CE}^p (\\mathfrak {g}, V) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ ={ f CCEp (g, V): dp f =0 }, and Chevalley-Eilenberg coboundaries the elements in the vector space $B_{CE}^{p} (\\mathfrak {g}, V) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ ={f CCEp (g, V) : g CCEp-1 (g, V) : f = dp-1 g }, and we define the Chevalley-Eilenberg $p$ -cohomology group of $\\mathfrak {g}$ valued in $V$ as the quotient vector space $H_{CE}^p (\\mathfrak {g}, V) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =$Z_{CE}^p (\\mathfrak {g}, V)$/$B_{CE}^p (\\mathfrak {g}, V)$.. Denoting now $\\mathfrak {g}$ a Lie superalgebra with $\\mathfrak {g} = \\mathfrak {g}_0 \\oplus \\mathfrak {g}_1$ its even and odd components in the $\\mathbb {Z}_2$ -grading, one can easily generalize the above construction just by taking care of the signs related to the $\\mathbb {Z}_2$ -grading (parity).",
"In particular, the definition of cochains and cohomology groups is unchanged and the previous differential in (REF ) modifies to [31] $ d^p f (x_1 \\wedge \\ldots \\wedge x_{p+1}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ = 1 i < j p+1 (-1)i+ j + i, j + i-1, j f ([xi, xj] x1...xi ...xj ...xp+1) + + i = 1p+1 (-1)i+1 + r-1, r xi f (x1 ...xi ...xp+1), where $\\delta _{i, j} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =|xi| (|f| + k=0i |xk|) $ for any $ f p gV$, $ xi g$, in order to take into account the parity, \\emph {i.e.",
"}\\ the $ Z2$-grading of the elements.",
"Also, notice that as soon as the odd dimension of the Lie superalgebra is greater than zero, \\emph {i.e.}",
"if $ g$ is a true Lie superalgebra and not just a Lie algebra, the Chevalley-Eilenberg cochain complex is not bounded from above, in pretty much the same fashion of the de Rham complex of a supermanifold, \\emph {i.e.",
"}\\begin{eqnarray}C^{\\bullet }_{CE} (\\mathfrak {g}, V) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=\\bigoplus _{p \\in \\mathbb {Z}} C^p_{CE}(\\mathfrak {g}, V) \\quad \\mbox{with} \\quad C^{p}_{CE} (\\mathfrak {g}, V) \\ne 0 \\quad \\forall p \\ge 0.$ As previously mentioned, Chevalley-Eilenberg cohomology has made its entrance years ago in physics, in particular in the context of supergravity and more specifically in the “FDA” (Free Differential Algebra) approach to supergravity due to D'Auria and Fre [4].",
"Construction of (semi) Free Differential Algebras, or $n/\\infty $ -Lie (super)algebras were indeed given iteratively in terms of Chevalley-Eilenberg cocycles of a given Lie (super)algebra.",
"In some sense, because of its supergravity origin, this approach is closer to Cartan geometry than the previous one, which has more algebraic taste.",
"One starts with a Lie group $G$ - or group manifold in the supergravity literature - and a $G$ -module $V$ , i.e.",
"a $k$ -vector space endowed with an action $\\rho : G \\times V \\rightarrow V$ of $G$ on $V$ , such that $\\rho _g \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =(g, ) Autk (V)$ for any $ g G.$ Now, an $ n$-form on the Lie group $ G$ valued in $ V$, \\emph {i.e.",
"}\\ an element of the vector space $ n (G, V) =n (G) k V$, is said to be $ G$-\\emph {equivariant} if $ g = g $ for any $ g G$ and where $ g : G G$ is the left translation by $ g$.",
"We call $ n (G, V)eq$ the space of equivariant $ n$-form valued in the $ G$-module $ V$.",
"It is clear that a $ G$-equivariant form is determined by its value at the origin on $ G$, and in particular it can be proved that $ p (G, V)eq CCEp (g, V)$.",
"Further, $ (deq)e = dg$, where $ dg$ is the Chevalley-Eilenberg differential (\\ref {differential}) and $ d$ is the de Rham differential.",
"This shows that the Lie algebra cohomology can be described in terms of the de Rham cohomology of (equivariant) differential forms on the Lie group whose the Lie algebra is associated, \\emph {i.e.}",
"$ Hp (g, V) Hp (p (G, V)eq, d)$, thus making contact between two seemingly different cohomologies and making possible to compute Lie algebra cohomology via \\emph {forms}, see for example \\cite {Knapp}.",
"\\\\$ The above remarks are completely general.",
"In order to make contact with the notation employed and results in the following sections, we will now look at the description of the Chevalley-Eilenberg cohomology in terms of forms in some more details in the case we will be concerned with, that of the trivial $\\mathfrak {g}$ -module $V=k$ , where $k$ is the ground field.",
"In this case we will simply write $C^p_{CE} (\\mathfrak {g}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =CpCE (g, k)$ for the Chevalley-Eilenberg cochains defined above, and we recall that $ CpCE (g) = p g$, see the definition (\\ref {cochains}).",
"Likewise, equivariance of forms becomes simply left-invariance, \\emph {i.e.}",
"the requirement $ g = $.",
"This means that denoting $ pL (G)$ the vector space of left-invariant forms one has that the above isomorphism becomes $ CpCE (g) pL (G)$.",
"Let us consider a $ k$-basis of left-invariant forms $ i 1L (G)$ together with its dual basis of left-invariant vector field $ Xi (1L(G))$, with $ ig (Xj, g) = ij$ for any $ g G$.",
"Then, the $ i 1L (G)$ satisfy the \\emph {Maurer-Cartan structure equation}\\begin{eqnarray}d \\omega ^i = -\\frac{1}{2} C_{~jk}^i \\, \\omega ^j \\wedge \\omega ^k,\\end{eqnarray}where the $ C jki $ are the \\emph {structure constants} relative to the basis $ i$.The sums over repeated indices are understood.",
"These equations are equivalent to the Lie braket relations for the basis $ Xi$ of the algebra of left-invariant vector fields, $ [Xj, Xk ] = Cjki Xi$.",
"Also it can be easily checked that $ dd = 0$ is equivalent to \\emph {Jacobi identity}, as{\\begin{@align}{1}{-1}d(d\\omega ^k) & = - \\frac{1}{2} C^{k}_{~ij} \\, d\\omega ^i \\wedge \\omega ^j + \\frac{1}{2} C^{k}_{~ij} \\, \\omega ^i \\wedge d \\omega ^j = \\frac{1}{2} C^{k}_{~i[j} C^{i}_{~l m]}\\, \\omega ^{l} \\wedge \\omega ^m \\wedge \\omega ^j = 0,\\end{@align}}where $ i L (G)$ and where $ Ck i[j Ci l m] = 0$ is indeed the Jacobi identity.",
"These will be the fundamental ingredients to actually compute cohomologies (notice that the differential is a derivation, so that it extends to higher forms).\\\\$ In the present paper we will deal only with matrix Lie groups, i.e.",
"Lie groups which admit an embedding into some $GL$ -group: in this case, the above is equivalent to take a basis of forms $\\mathpzc {V} = dg g^{-1}$ , where $g \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =(gij)$ is matrix-valued, which we call Maurer-Cartan forms, as they satisfy Maurer-Cartan equations (\\ref {MCeq1}) by construction.",
"In turns, we will take the cochains to be generated starting from the basis of Maurer-Cartan forms $ {Vi}$, \\emph {i.e.}",
"the \\emph {vielbeins} in the physics literature, so that\\begin{eqnarray}C^p (\\mathfrak {g}) = \\Omega ^{p}_L (G) = \\bigg \\lbrace c_{i_1, \\ldots , i_p} \\mathpzc {V}^{i_1} \\wedge \\ldots \\wedge \\mathpzc {V}^{i_p} \\bigg \\rbrace \\quad \\mbox{for} \\quad c_{i_1, \\ldots , i_p} \\in k.\\end{eqnarray}Notice that the above discussion is readily generalizable to the $ Z2$-graded super-setting of a Lie \\emph {super}group $ G$ and its Lie \\emph {super}algebra $ g$, but a remark about the \\emph {parity} is in order: indeed, instead of considering the vector bundle of forms, we will consider its parity reversed version $ 1(G) =T(G)$, as it is customary in supergeometry: notice that in this convention the de Rham differential $ d$ is an \\emph {odd} derivation.",
"This leads to consider \\emph {even} and \\emph {odd} vielbeins $ {| Vi }$ generating the $ Z2$-graded vector space $ 1L (G)$, where the even $$^{\\prime }s arise from odd coordinates and the odd $ Vi$^{\\prime }s arise from even coordinates.",
"What it is crucial to observe is that, accordingly, this should be related to the \\emph {parity changed dual} of the Lie superalgebra $ g$, that is at the level of the cochains one has \\begin{eqnarray}C^\\bullet (\\Pi \\mathfrak {g})\\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=S^\\bullet \\Pi \\mathfrak {g}^\\ast \\cong \\Omega ^\\bullet _L (\\mathpzc {G}),$ where $S^\\bullet $ is the supersymmetric product functor (symmetrization operation) [37].",
"Likewise, at the level of the differentials, the de Rham differential is extended to both even and odd coordinates.",
"The commutators characterizing the algebra or, dually, the Maurer-Cartan equations, become supercommutators.",
"In particular, on the parity reversed algebra $\\Pi \\mathfrak {g}$ , if $\\pi X$ and $\\pi Y \\in \\Pi \\mathfrak {g}$ we put $[ \\pi X , \\pi Y \\rbrace \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =[X, Y} $ for $ X$ and $ Y$ in $ g.$$"
],
[
"A Primer of Integral Forms on Supermanifolds",
"Given a supermanifold $\\mathpzc {M}$ , say of dimension $n|m$ , differential forms in $\\Omega ^\\bullet (\\mathpzc {M}) $ are not enough to define a coherent notion of integration on $\\mathpzc {M}$ .",
"This leads to the introduction of integral forms, which are geometrically as important as differential forms, see [37] and the recent papers [8], [6], [7], [25], [39], [9], [10], [11], [15], [16], [17], [18], [12], [3].",
"Loosely speaking, whereas differential forms lead to a consistent geometric integration on ordinary bosonic submanifolds (i.e.",
"sub-manifolds of codimension $p|m$ ) in $\\mathpzc {M}$ , integral forms plays the same role on sub-supermanifolds of codimension $p|0$ in $\\mathpzc {M}$ , and in particular, they control integration on $\\mathpzc {M}$ itself.",
"Notice that, even if it is often left understood or not stated, integral forms are ubiquitous in theoretical high energy physics: for example, the Lagrangian density of a supersymmetric theory in superspace is indeed a top integral form.",
"There are (at least) two ways to introduce integral forms, which we now briefly recall.",
"The first approach is to define integral forms as generalized functions on $\\mbox{Tot}\\, \\Pi \\mathcal {T} (\\mathpzc {M})$ [44], that is elements $\\omega (x^1, \\ldots , x^n, d\\theta ^1, \\ldots , d\\theta ^m | \\theta ^1, \\ldots , \\theta ^m, dx^1, \\ldots dx^n) \\in \\Pi \\mathcal {T} (\\mathpzc {M})$ , where $x^i|\\theta ^\\alpha $ are local coordinate for $\\mathpzc {M}$ , which only allows a distributional dependence supported in $d\\theta ^1 = \\ldots = d\\theta ^m = 0$ .",
"Algebraically, integral forms can be (roughly) described as $\\Omega ^\\bullet (\\mathpzc {M})$ -modules generated over the set (of Dirac delta distributions and their derivatives) $\\lbrace \\delta ^{(r_1)} (d\\theta ^1)\\wedge \\ldots \\wedge \\delta ^{(r_m)} (d\\theta ^m) \\rbrace $ , for $r_i \\ge 0$ , together with the defining relations $d\\theta ^{\\alpha }\\delta ^{(k)}(d\\theta ^{\\alpha })=-k\\delta ^{(k-1)}(d\\theta _{\\alpha }) \\quad \\mbox{for} \\quad k\\ge 0$ for any $\\alpha = 1, \\ldots , m$ and any $k \\ge 0$ , which are deduced analytically by integration by parts.",
"Notice that the case $k=0$ tells that the expressions $d\\theta ^\\alpha \\delta ^{(0)} (d\\theta ^\\alpha ) $ vanishes, so that the presence of the delta's can be seen as a localization in the locus $d\\theta ^\\alpha = 0$ in $\\mbox{Tot}\\, \\Pi \\mathcal {T} (\\mathpzc {M})$ .",
"Locally, an integral form $\\omega _{int}$ is written as a (generalized) tensor $ \\omega _{int} (x, & d\\theta | \\theta , dx) = \\nonumber \\\\& = \\sum _{i=1}^n \\sum _{j =1}^m\\sum _{a_i \\in \\lbrace 0,1\\rbrace , r_j \\ge 0} \\omega _{[a_{1}\\dots a_{m}r_{1}\\dots r_{m}]} (x |\\theta ) (dx^1)^{a_{1}}\\dots (dx^n)^{a_{m}}\\delta ^{(r_{1})}(d\\theta ^1 )\\dots \\delta ^{(r_{m})}(d\\theta ^m),$ where all indices are antisymmetric (recalling that two delta's anticommute with each other), and where we note that there cannot be $d\\theta $ 's thanks to the above relations (REF ).",
"In what follows we will say that an integral forms has picture $m$ , to mean that we are considering expressions that admits only a distributional dependence on all of the $m$ coordinates $d\\theta ^1, \\ldots , d\\theta ^m$ on $\\mbox{Tot}\\, \\Pi \\mathcal {T} (\\mathpzc {M})$ .",
"Further, with reference to the previous expression REF , we assign a degree to an integral form according to the definition $\\deg (\\omega _{int}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =i = 1n aj - j=1m rj, so that we will say that an integral form has picture $m$ and degree $p \\le n.$ In particular, a top integral form is an integral form of degree $n$ , $\\omega ^{top}_{int} = \\omega (x | \\theta ) dx^1 \\ldots dx^n \\delta (d\\theta ^1) \\ldots \\delta (d\\theta ^m),$ and it can be checked that this expression has the transformation properties of a section of the Berezinian line bundle $\\mathcal {B}er (\\mathpzc {M}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =Ber(T($\\mathpzc {M}$ ))$ of the supermanifold $$\\mathpzc {M}$$.",
"Notice that all of the integral forms as in {intf} can be generated from the above \\ref {berdel} by repeatedly acting with \\emph {contractions} along (coordinate) vector fields, \\emph {i.e.",
"}\\begin{eqnarray}\\omega _{int}^{n-\\ell } = \\iota _{X_1}\\ldots \\iota _{X_\\ell } \\, \\omega ^{top}_{int},\\end{eqnarray}where we recall that in particular, for the coordinate vector fields $ xi | $ one has that $ |xi| = 1$ and $ || = 0$.",
"The modules of integrals forms are then structured into a complex letting $ d$ operate as the usual de Rham differential on $ ($\\mathpzc {M}$ )$ and declaring that its action on the delta^{\\prime }s, is trivial \\emph {i.e.}",
"posing $ d ((d) ) = 0$ for any $$.$ In the second approach one defines integral forms of degree $p$ as sections of the vector bundle on $\\mathpzc {M}$ $\\Sigma ^p (\\mathpzc {M}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =Ber ($\\mathpzc {M}$ ) O$\\mathpzc {M}$ (n-p ($\\mathpzc {M}$ ))= Ber ($\\mathpzc {M}$ ) O$\\mathpzc {M}$ Sn-p (T($\\mathpzc {M}$ )).",
"where $\\mathcal {B}er (\\mathpzc {M})$ is the Berezinian line bundle of $\\mathpzc {M}$ and $\\Pi \\mathcal {T} (\\mathpzc {M})$ the parity-reversed tangent bundle.",
"The correspondence between integral forms in the different representations reads $\\omega ^{(n-\\ell )} = \\mathpzc {D} \\otimes \\left( \\pi X_1 {\\odot } \\ldots \\odot \\pi X_\\ell \\right) \\ \\leftrightsquigarrow \\ \\omega ^{(n-\\ell )} = \\iota _{X^1} \\ldots \\iota _{X^\\ell } \\omega ^{top}_{int}$ where $\\mathpzc {D}$ is a section of $\\mathcal {B}er(\\mathpzc {M})$ and $\\pi X_1 {\\odot } \\ldots \\odot \\pi X_\\ell $ is a section of $S^\\ell \\Pi \\mathcal {T} (\\mathpzc {M})$ , together with the correspondence of sections of Berezinian line bundle, or integral top forms, mentioned above, i.e.",
"$\\omega ^{top}_{int} \\leftrightsquigarrow \\mathpzc {D}$ .",
"Clearly, given the above tensor product structure, defining a nilpotent differential acting as $\\delta ^p : \\Sigma ^p (\\mathpzc {M}) \\rightarrow \\Sigma ^{p+1} (\\mathpzc {M})$ is not at all trivial matter, as originally discussed in [37] and recently realized in [3], but this can be done as getting a complex which will in general be unbounded from below ${\\ldots [r] & \\mathcal {B}er (\\mathpzc {M}) \\otimes S^{n-p} (\\Pi \\mathcal {T}(\\mathpzc {M})) [r] & \\ldots [r] & \\mathcal {B}er (\\mathpzc {M}) \\otimes \\Pi \\mathcal {T}(\\mathpzc {M}) [r] & \\mathcal {B}er (\\mathpzc {M}) [r] & 0.",
"}$ Remarkably, these different approaches, which agree in terms of general results, complement each others.",
"If on one hand this second approach is probably more suitable when it comes to deal with mathematical and foundational issues where well-definiteness is crucial, on the other hand the first approach proves quite more effective when it comes to actual computations, and for this reason is favoured in applications to theoretical physics.",
"The different nature of these two approaches is mirrored, for example, in the proof of which is probably the most important result in the theory, i.e.",
"the (natural) isomorphism between the cohomology of differential form $H^{p}_{d} (\\Omega ^{\\bullet } (\\mathpzc {M}))$ and integral forms $H^p_{\\delta } (\\Sigma ^\\bullet (\\mathpzc {M}) )$ on supermanifolds, namely introducing in the first approach the crucial notion of Picture Changing Operators (see, e.g., [11]), which maps differential to integral forms and vice-versa, and via a spectral sequence argument in the second approach [3]."
],
[
"Defining Chevalley-Eilenberg Cohomology of Integral Forms",
"In this section we investigate to what extent, in the case the supermanifold $\\mathpzc {M}$ is a Lie supergroup $\\mathpzc {G}$ with Lie superalgebra $\\mathfrak {g}$ , it is possible to define a notion of “integral form” and in particular a “Chevalley-Eilenberg cohomology” of integral forms related to $\\mathfrak {g}$ .",
"Notice that, as explained above, the Chevalley-Eilenberg cohomology can be analogously introduced as the cohomology of the vector (super)space of the left-invariant differential forms for a certain Lie group: since over a supermanifold differential forms need to be supplemented by integral forms, it can be expected that there must exist an analogous notion of cohomology of left-invariant integral, better than differential, forms.",
"The first of the two approaches presented above is probably more straightforward in this respect.",
"One takes a basis of Maurer-Cartan forms $\\lbrace \\psi ^\\alpha | \\mathpzc {V}^i \\rbrace $ , or supervielbein, with even $\\psi $ 's and odd $\\mathpzc {V}$ 's and restrict to consider only integral forms written in terms of them.",
"More precisely, if $\\mathpzc {Y}_{i | \\alpha } \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ ={ Pi |Q}$ is the basis of generators of the Lie superalgebra $ g$ which is dual (up to a parity shift) to the basis of the Maurer-Cartan forms above, so that $ (Q) = $ and $ Vi (Pj) = ij$, then the most general integral form on $ g$ of degree $ n-$, see (\\ref {mgit}), will be written as\\begin{eqnarray}\\omega ^{n-\\ell }_{\\mathfrak {g}} = \\omega ^{i_{1} \\ldots i_{\\ell } } \\iota _{\\mathpzc {Y}^{i_1}} \\ldots \\iota _{\\mathpzc {Y}^{i_\\ell }} \\omega ^{top}_{\\mathfrak {g}},\\end{eqnarray}for $ Y$ spanning both even and odd dimensions of $ g$ and the indices of the tensor $ i1 ...i $ symmetrized or anti-symmetrized according to the parity of the related contraction (the sum over repeated indices is understood).",
"In the above expression one fixes the integral top form up to a multiplicative constant to be\\begin{eqnarray}\\omega ^{top}_{\\mathfrak {g}} = \\mathpzc {V}^1\\ldots \\mathpzc {V}^n \\delta (\\psi ^1) \\ldots \\delta ( \\psi ^m ),\\end{eqnarray}that is $ topg$ is again expressed only in terms of of the Maurer-Cartan forms, which makes it formally left-invariant.",
"Having set this stage and in the light of the discussion in the previous subsection, one can therefore generalize the Maurer-Cartan differential as to act on integral forms in the following way\\begin{eqnarray}\\omega ^{n-\\ell }_{\\mathfrak {g}} \\longmapsto d (\\omega ^{n-\\ell }_{\\mathfrak {g}}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=\\frac{1}{2} C^{A}_{\\; \\; BC} (\\pi \\mathpzc {Y}^\\ast )^B (\\pi \\mathpzc {Y}^\\ast )^{C} \\iota _{\\mathpzc {Y}^A} \\bigg ( \\omega ^{i_{1} \\ldots i_{\\ell } } \\iota _{\\mathpzc {Y}^{i_1}} \\ldots \\iota _{\\mathpzc {Y}^{i_\\ell }} \\omega ^{top}_{\\mathfrak {g}} \\bigg ),$ where $C^{A}_{\\; \\; BC}$ are the structure constants of the Lie superalgebra $\\mathfrak {g}$ and $A,B$ and $C$ are the cumulative indices for $i|\\alpha $ .",
"Notice that the right-hand ride of the () defines indeed an integral form of degree $n-\\ell + 1$ and, once again, that the differential is indeed nilpotent thanks to Jacobi identity for the Lie superalgebra $\\mathfrak {g}.$ Defining integral forms for $\\mathfrak {g}$ in the second approach requires some further explanations: the discussion is somewhat formal, therefore the reader can skip to the next section at the first reading.",
"One can proceed specializing the definition (REF ) to a Lie supergroup $\\mathpzc {G}$ , but first the question of how to intrinsically define a left-invariant Berezinian needs to be addressed.",
"One might start with an analogy with the ordinary case, where the Haar determinant - which, integrated, gives the volume of a compact Lie group -, is constructed by taking the top exterior power of the left-invariant 1-forms $\\mbox{Span}_{\\mathbb {R}} \\lbrace \\omega ^1, \\ldots , \\omega ^n\\rbrace = \\Omega ^1_L (G)$ over the $n$ -dimensional ordinary Lie group $G$ , i.e.",
"$\\det (G) = \\mathbb {R} \\cdot \\omega ^1 \\wedge \\ldots \\wedge \\omega ^n.$ This construction cannot be generalized in a straighforward manner, mainly because the Berezinian of a vector space is not a top-exterior form.",
"On the other hand, there exists a less known construction of the Berezinian of a vector superspace via the cohomology of a suitable generalization of the Koszul complex (see the quite recent papers [40] and [3]): this should not surprise, as also the determinant appears in the same way from the Koszul complex.",
"More precisely, given a vector $\\mathbb {R}$ -superspace $V$ of dimension $n|m$ , one finds that the cohomology of the (dual of the) Koszul complex is concentrated in degree $n$ , i.e.",
"$Ext^n_{S^\\bullet V^\\ast } (\\mathbb {R}, S^\\bullet V^\\ast ) \\cong \\Pi ^{n+m} \\mathbb {R}$ and an automorphism $\\phi \\in Aut (V)$ induces an automorphism on $Ext^n_{S^\\bullet V^\\ast } (\\mathbb {R}, S^\\bullet V^\\ast )$ which is just the multiplication by the Berezinian of the automorphism $\\mbox{Ber} (\\phi )$ , so that one rightfully defines $\\mbox{Ber} (V ) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =ExtnSV (R, SV)$ \\cite {NR}.",
"The computation of this cohomology is particularly useful also because it gives the generator of the Berezinian of $ V$ in terms of the generators of the vector space $ V$.",
"Shifting from algebra to geometry, one defines the Berezinian of the supermanifold to be the Berezinian of the tangent bundle $ T($\\mathpzc {M}$ )$, or analogously the dual of the Berezinian of the parity-reversed cotangent bundle $ 1 ($\\mathpzc {M}$ )$ as above.",
"One finds that the Berezinian line bundle is (locally) generated by the class\\begin{eqnarray}\\mathcal {B}er (\\mathpzc {M}) \\cong \\mathcal {O}_\\mathpzc {M}\\cdot [dx^1 \\wedge \\ldots \\wedge dx^n \\otimes \\partial _{\\theta ^1} \\wedge \\ldots \\wedge \\partial _{\\theta ^m}]\\end{eqnarray}in the corresponding $ Ext$-sheaf, where $ xi | $ for $ i=1,...,n$ and $ = 1, ..., m$ are local coordinates for the supermanifold $$\\mathpzc {M}$$.",
"\\\\This is what is needed in order to write a corresponding \\emph {left-invariant Berezinian}, or \\emph {Haar Berezinian} for a Lie supergroup: it is enough to consider the left-invariant \\emph {odd} vector fields, call them $ { ()1, ..., ()m }$, generating $ g1$ and the left-invariant \\emph {odd} 1-forms, call them $ { ()1, ..., ()n }$, generating $ g1$: then the Haar Berezinian is generated over $ R$ by the expression\\begin{eqnarray}\\mbox{Ber}^{\\mathpzc {H}} (\\mathfrak {g}) \\cong \\mathbb {R} \\cdot [\\omega ^{(\\ell )1} \\wedge \\ldots \\wedge \\omega ^{(\\ell )n} \\otimes \\Psi ^{(\\ell )}_1 \\wedge \\ldots \\wedge \\Psi ^{(\\ell )}_m ].\\end{eqnarray}Notice that this corresponds to the vector superspace of densities of the vector space underlying the Lie superalgebra $ g.$ ``Dually^{\\prime \\prime } to ordinary Chevally-Eilenberg cochains for a Lie superalgebra, integral forms cochains can then be introduced into this Lie-algebraic framework by looking at the definition (\\ref {intform2}) as\\begin{eqnarray}C_{CE, \\mathpzc {int}}^p (\\mathfrak {g}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=\\mbox{Ber}^{\\mathpzc {H}} (\\mathfrak {g}) \\otimes S^{n-p} \\Pi \\mathfrak {g},$ where we are exploiting the usual isomorphism between left invariant vector fields on a Lie supergroup $\\mathpzc {G}$ and elements of its Lie algebra $\\mathfrak {g}$ .",
"In order to distinguish between them we henceforth call differential Chevalley-Eilenberg $p$ -cochains the elements in the vector superspace $C^p_{CE, \\mathpzc {dif}} (\\mathfrak {g}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =Sp g$ and \\emph {integral} Chevalley-Eilenberg $ p$-cochains the elements in the vector superspace $ CpCE, int (g) =BerH (g) Sn-p g,$ as above.\\\\In order to structure this into a true cochain complex, one has to introduce a nilpotent differential acting as $ p : CpCE, int (g) Cp+1CE, int (g)$.",
"We first extend the notion of Lie derivative, or \\emph {supercommutator}, to the whole supersymmetric product $ Sn g$, this can be done recursively as follows.Given $ X g$, having already defined $ LX : Sh g Sh g$ for $ h<p$ we uniquely define the action of $ LX $ on $ Sp g$ via the relation\\begin{eqnarray}\\mathcal {L}_{\\mathpzc {X}} (\\langle \\omega , \\tau \\rangle ) = \\langle \\mathcal {L}_\\mathpzc {X} (\\omega ) , \\tau \\rangle + (-1)^{|\\omega | |\\mathpzc {X}|} \\langle \\omega , \\mathcal {L}_\\mathpzc {X} (\\tau ) \\rangle \\end{eqnarray}for any $ Si>0g$ and $ Sp g$, and where $ , $ is the duality pairing between $ g$ and $ g$, extended to higher tensor powers.",
"Notice that from (\\ref {recursive}) it follows that \\begin{eqnarray}\\mathcal {L}_X (Y)= \\pi [X , \\pi Y]\\end{eqnarray}for any $ Y g$, \\emph {i.e.",
"}\\ the Lie derivative of a parity-reversed field is a commutator, as it should.",
"We now use this to introduce a differential, namely we define the following odd operator\\begin{eqnarray}@R=1.5pt{\\delta ^p : C^p_{CE, \\mathpzc {int}} (\\mathfrak {g}) [r] & C^{p+1}_{CE, \\mathpzc {int}} \\\\\\mathpzc {D} \\otimes \\tau @{|->}[r] & \\delta ^p (\\mathpzc {D} \\otimes \\tau ) = \\mathpzc {D} \\otimes \\sum _{A} \\iota _{\\pi \\mathpzc {X}^\\ast _A} \\mathcal {L}_{\\mathpzc {X}_A} (\\tau )}\\end{eqnarray}where the index $ A$ runs over both even and odd coordinates and where $ D$ is a Haar Berezinian tensor density in $ BerH (g)$ and $ { XA }$ are left-invariant vector fields generating $ g$, so that hence $ { XA }$ are generators for $ g$.",
"Here $ XA$ is the contraction with the form $ XA$, so that the above can be re-written as\\begin{eqnarray}\\delta ^p (\\mathpzc {D} \\otimes \\tau ) = \\mathpzc {D} \\otimes \\sum _{A} \\langle \\pi \\mathpzc {X}_A^\\ast , \\mathcal {L}_{\\mathpzc {X}_A} (\\tau ) \\rangle .\\end{eqnarray}Nilpotency can be checked formally as{\\begin{@align}{1}{-1}\\frac{1}{2}\\lbrace \\delta , \\delta \\rbrace & = \\sum _{A,B} (\\iota _{\\pi \\mathpzc {X}^\\ast _A} \\mathcal {L}_{\\mathpzc {X}_A} \\iota _{\\pi \\mathpzc {X}^\\ast _B} \\mathcal {L}_{\\mathpzc {X}_B} + \\iota _{\\pi \\mathpzc {X}^\\ast _B} \\mathcal {L}_{\\mathpzc {X}_B}\\iota _{\\pi \\mathpzc {X}^\\ast _A} \\mathcal {L}_{\\mathpzc {X}_A}) \\nonumber \\\\& = \\sum _{A,B}\\left( (-1)^{|\\mathpzc {X}_A||\\mathpzc {X}_B| + |\\mathpzc {X}_A|} + (-1)^{|\\mathpzc {X}_A||\\mathpzc {X}_B| + |\\mathpzc {X}_A| +1}\\right) \\iota _{\\pi \\mathpzc {X}^\\ast _A} \\iota _{\\pi \\mathpzc {X}^\\ast _B} \\mathcal {L}_{\\mathpzc {X}_A} \\mathcal {L}_{\\mathpzc {X}_B}=0.\\end{@align}}We thus introduce the cochain complex $ (CpCE, int (g), p )$ and we define the corresponding \\emph {integral} Chevalley-Eilenberg cohomology of the Lie superalgebra $ g$ in the usual way\\begin{eqnarray}H^p_{CE, \\mathpzc {int}} (\\mathfrak {g}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=\\frac{\\ker \\big ( \\delta ^p : C^p_{CE, \\mathpzc {int}} (\\mathfrak {g}) \\rightarrow C^{p+1}_{CE, \\mathpzc {int}} (\\mathfrak {g}) \\big )}{\\mbox{im} \\big ( \\delta ^{p-1} : C^{p-1}_{CE, \\mathpzc {int}} (\\mathfrak {g}) \\rightarrow C^p_{CE, \\mathpzc {int}} (\\mathfrak {g}) \\big )}.$ Notice that the differential only acts on $S^\\bullet \\Pi \\mathfrak {g}$ , as can be seen in ().",
"One can therefore alternatively define the above cohomology () starting from the cochains $\\widehat{C}^p_{S} (\\mathfrak {g}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =Sp g$ on which $$.",
"The related cohomology is then defined as\\begin{eqnarray}H^p_{S} (\\mathfrak {g}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=\\frac{\\ker \\big ( \\delta ^p : \\widehat{C}^p_{S} (\\mathfrak {g}) \\rightarrow \\widehat{C}^{p+1}_{S} (\\mathfrak {g}) \\big )}{\\mbox{im} \\big ( \\delta ^{p-1} : \\widehat{C}^{p-1}_{S} (\\mathfrak {g}) \\rightarrow \\widehat{C}^p_{S} (\\mathfrak {g}) \\big )},$ and in turn the integral Chevalley-Eilenberg cohomology $H^p_{CE, \\mathpzc {int}} (\\mathfrak {g})$ becomes a twist of $H^p_{S} (\\mathfrak {g})$ by $\\mbox{Ber}^{\\mathpzc {H}} (\\mathfrak {g})$ , namely $H^{\\bullet }_{CE, \\mathpzc {int}} (\\mathfrak {g}) \\cong \\mbox{Ber}^{\\mathpzc {H}}(\\mathfrak {g}) \\otimes H^\\bullet _S (\\mathfrak {g}).$ Observe that the Haar Berezinian can be seen as a shift by degree $n$ in cohomology and that the cochains are really dual one another, as $C^p_{CE, dif} = S^p \\Pi \\mathfrak {g}^\\ast $ and $C^p_{CE, int} (\\mathfrak {g}) = S^p \\Pi \\mathfrak {g}$ .",
"So, the question is: if $\\omega \\in C^p_{dif} (\\mathfrak {g})$ is closed, then is $\\omega ^\\ast \\in C^p_{int} (\\mathfrak {g})$ closed?",
"And viceversa.",
"This is proved in the following.",
"Let us first consider some calculations in the two formalisms, showing that they are equivalent: consider the case of a $(n-1)$ -integral form $\\omega ^{(n-1)} = \\mathpzc {D} \\otimes \\sum _{A= 1}^{m+n} T^{A} \\left( \\pi \\mathpzc {Y}_{A} \\right) \\equiv T^{A} \\iota _{\\mathpzc {Y}_{A}} \\omega ^{top} \\ .$ We can apply the operator $\\delta ^{(1)} \\equiv d$ to $\\omega ^{(n-1)}$ thus obtaining $\\delta ^{(1)} \\omega ^{(n-1)} = \\mathpzc {D} \\otimes \\sum _{B} \\sum _A \\iota _{(\\pi \\mathpzc {Y}^\\ast _B)} T^{A} \\mathcal {L}_{\\mathpzc {Y}_B} \\left( \\pi \\mathpzc {Y}_{A} \\right) = = \\mathpzc {D} \\otimes \\sum _{B} \\sum _{A,C} \\iota _{(\\pi \\mathpzc {Y}^\\ast _B)} T^{A} f^C_{B A} \\left( \\pi \\mathpzc {Y}_{C} \\right) = \\mathpzc {D} \\otimes \\sum _{B} \\sum _{A,C} T^{A} f^C_{B A} \\delta _{B C} = 0 \\ ,$ where we have used the () for the Lie derivative, $\\displaystyle \\iota _{(\\pi \\mathpzc {Y}^\\ast _A)} \\left( \\pi \\mathpzc {Y}_{B} \\right) = \\delta ^A_{B}$ and the properties of the structure constants.",
"On the other hand we have $d \\omega ^{(n-1)} = \\frac{1}{2} f^A_{B C} \\left( \\pi \\mathpzc {Y}^{*} \\right)^B \\left( \\pi \\mathpzc {Y}^{*} \\right)^C \\iota _{\\mathpzc {Y}_A} T^D \\iota _{\\mathpzc {Y}_D} \\omega ^{top} = f^A_{BC } \\delta ^{~B}_A \\delta _D^{~C}T^D \\omega ^{top} = 0 \\ .$ Notice that actually we can use the isomorphism $H^\\bullet _{CE} \\left( \\mathfrak {g} , \\mathbb {R} \\right) \\cong H^\\bullet _{dR} \\left( G \\right)^{G}$ (i.e.",
"that the Chevalley-Eilenberg cohomology of the superalgebra $\\mathfrak {g}$ is isomorphic to the de Rham cohomology of the supergroup $G$ restricted to the left-invariant forms) to obtain (REF ) in a different way: $d \\omega ^{(n-1)} = d T^D \\iota _{\\mathpzc {Y}_D} \\omega ^{top} = T^D \\mathcal {L}_{\\mathpzc {Y}_D} \\omega ^{top} + \\left( -1 \\right)^{|\\pi \\mathpzc {Y}_D |} T^D \\iota _{\\mathpzc {Y}_D} d \\omega ^{top} = 0 \\ ,$ where we have used the fact that $\\omega ^{top}$ is the Haar Berezinian tensor density in $\\mbox{Ber}^\\mathpzc {H} (\\mathfrak {g})$ , hence the (left) invariant top form.",
"The previous example is two-folded: first it is an example of calculation in both realisations with a check of equivalence, second it shows that the Haar Berezinian $\\mathpzc {D} \\equiv \\omega ^{top}$ , which is obviously closed with respect to $\\delta ^{(\\bullet )} \\equiv d$ , is not exact, thus showing that it is always a cohomology representative."
],
[
"Isomorphism Between Superform and Integral Form Cohomologies.",
"In this section we show that the cohomology of superforms is isomorphic to the cohomology of integral forms.",
"In order to do so, we will use the formalism where the Haar Berezinian is treated as a differential form as in () and the nilpotent operator is actually the Cartan differential.",
"The proof for integral forms written as in () with respect to the differential () follows from the “dictionary\" between the two established formalisms.",
"Let us start by considering a superform $\\omega ^{(1)}$ , such that $d \\omega ^{(1)} =0$ .",
"We define its “Berezinian complement\" $\\star \\omega ^{(1)}$ as $@R=1.5pt{\\star : \\Omega ^1_{CE, \\mathpzc {dif}} (\\mathfrak {g}) [r] & \\Omega ^{n-1}_{CE, \\mathpzc {int}} (\\mathfrak {g}) \\\\\\omega ^{(1)} @{|->}[r] & \\star \\omega ^{(1)} = \\left( \\star \\omega \\right)^{(n-1)} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}}=\\iota _\\mathpzc {Y} \\omega ^{top}_\\mathfrak {g} \\ ,$ where $\\displaystyle \\omega ^{(1)} (\\mathpzc {Y}) = 1$ , i.e.",
"$\\pi \\mathpzc {Y}$ is the vector field dual to $\\omega ^{(1)}$ .",
"Then we have $\\displaystyle d \\star \\omega ^{(1)} = d \\iota _\\mathpzc {Y} \\omega ^{top}_\\mathfrak {g} = 0$ , as we have shown in (REF ).",
"For a generic $p$ -superform the generalization follows by extending (REF ) as $@R=1.5pt{\\star : \\Omega ^p_{CE, \\mathpzc {dif}} (\\mathfrak {g}) [r] & \\Omega ^{n-p}_{CE, \\mathpzc {int}} (\\mathfrak {g})\\\\\\omega ^{(p)} @{|->}[r] & \\star \\omega ^{(p)} = \\left( \\star \\omega \\right)^{(n-p)} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}}=\\iota _{\\mathpzc {Y}_1} \\ldots \\iota _{\\mathpzc {Y}_p} \\omega ^{top}_\\mathfrak {g} \\ ,$ where $\\displaystyle \\omega ^{(p)} (\\mathpzc {Y}_1 , \\ldots , \\mathpzc {Y}_p) = 1 $ .",
"Given $\\omega ^{(p)}\\in H^{p}_{CE, \\mathpzc {dif}} (\\mathfrak {g})$ , we have $d \\left( \\omega _{A_1 \\ldots A_p} \\left( \\pi \\mathpzc {Y}^* \\right)^{A_1} \\wedge \\ldots \\wedge \\left( \\pi \\mathpzc {Y}^* \\right)^{A_p} \\right) = p \\omega _{A_1 \\ldots A_p} f^{A_1}_{~~R S} \\left( \\pi \\mathpzc {Y}^* \\right)^R \\left( \\pi \\mathpzc {Y}^* \\right)^S \\left( \\pi \\mathpzc {Y}^* \\right)^{A_2} \\wedge \\ldots \\wedge \\left( \\pi \\mathpzc {Y}^* \\right)^{A_p} = 0 \\iff \\ \\omega _{A_1 \\ldots A_p} f^{A_1}_{~~R S} = 0 \\ .$ We now show that this condition implies $d \\star \\omega ^{(p)} = 0$ .",
"First of all, we observe that the integral form dual to $\\omega ^{(p)}$ reads $\\star \\omega ^{(p)} = T^{A_1 \\ldots A_p} \\iota _{\\mathpzc {Y}^{A_1}} \\ldots \\iota _{\\mathpzc {Y}^{A_p}} \\omega ^{top}_\\mathfrak {g} \\ , \\ \\mbox{ such that } \\ T^{A_1 \\ldots A_p} \\omega _{A_1 \\ldots A_p} = 1 \\ .$ It is easy to see that $d \\star \\omega ^{(p)}=0 \\ \\iff \\ T^{A_1 A_2 \\ldots A_{p}} f_{~~A_1 A_2}^R = 0 \\ .$ Recalling that every basic classical Lie superalgebra admits a non-degenerate bilinear form, see e.g.",
"[22], we can use the (non-degenerate) bilinear form $g_{AB}$ in order to write the coefficients $T$ of the integral form in terms of the coefficients $\\omega $ of the superform as $T^{A_1 A_2 \\ldots A_{p}} = \\frac{1}{||\\omega ||^2} g^{A_1 B_1} \\ldots g^{A_p B_p} \\omega _{B_1 \\ldots B_p} \\ , \\ \\text{where} \\ ||\\omega ||^2 = \\omega _{A_1 \\ldots A_p} g^{A_1 B_1} \\ldots g^{A_p B_p} \\omega _{B_1 \\ldots B_p} \\ .$ By substituting (REF ) in the left hand side of (REF ), we obtain $\\frac{1}{||\\omega ||^2}g^{A_1 B_1} \\ldots g^{A_p B_p} \\omega _{B_1 \\ldots B_p} f_{~~A_1 A_2}^R = \\frac{1}{||\\omega ||^2} g^{A_3 B_3} \\ldots g^{A_p B_p} \\omega _{B_1 \\ldots B_p} f^{R B_1 B_2} = 0 \\ ,$ as a consequence of (REF ).",
"Hence the result $d \\omega ^{(p)} = 0 \\ \\Rightarrow \\ d \\star \\omega ^{(p)} = 0$ .",
"The converse can be shown in a similar way.",
"From the previous argument we can now infer the isomorphism between the cohomologies of super and integral forms.",
"In particular, if $\\displaystyle \\omega ^{(p)} \\in H^{p}_{CE, \\mathpzc {dif}} (\\mathfrak {g})$ , we have $\\omega ^{(p)} \\wedge \\star \\omega ^{(p)} = \\omega ^{top}_\\mathfrak {g} \\in \\mbox{Ber}^{\\mathpzc {H}}(\\mathfrak {g}) \\ .$ By contradiction, let us assume $\\left( \\star \\omega \\right)^{(n-p)} = d \\Lambda ^{(n-p-1)}$ , we get $\\omega ^{top}_\\mathfrak {g} = d \\left( \\omega ^{(p)} \\wedge \\Lambda ^{(n-p-1)} \\right) \\ ,$ contradicting that $\\omega ^{top}_\\mathfrak {g}$ is a cohomology representative as shown in the previous section.",
"This argument shows that the operator $\\star $ is indeed an isomorphism: $\\star : H^{\\bullet }_{CE, \\mathpzc {dif}} (\\mathfrak {g}) \\overset{\\cong }{\\underset{}{\\longrightarrow }} H^{n-\\bullet }_{CE, \\mathpzc {int}} (\\mathfrak {g}) \\ .$"
],
[
"Poincaré Polynomials and Betti Numbers",
"Before we move to compute examples of Chevalley-Eilenberg cohomologies, we review the definition of Poincaré series and Poincaré polynomials.",
"For $X$ a graded $k$ -vector space with direct decomposition into $p$ -degree homogeneous subspaces given by $X = \\bigoplus _{p \\in \\mathbb {Z}} X_p$ we call the formal series $\\mathpzc {P}_X(t) = \\sum _p ({\\rm dim}_k \\, X_p) (-t)^p$ the Poincaré series of $X$ .",
"Notice that we have implicitly assumed that $X$ is a of finite type, i.e.",
"its homogeneous subspaces $X_p$ are finite dimensional for every $p.$ The unconventional sign in $(-t)^p$ takes into account the parity of $X_p$ , which takes values in $\\mathbb {Z}_2$ and it is given by $p \\, \\mbox{mod}\\, 2$ : this will be particularly useful in the super setting.",
"If also $\\dim _k X$ is finite, then $\\mathpzc {P}_X(t)$ becomes a polynomial $\\mathpzc {P}_X [t]$ , called Poincaré polynomial of $X$ .",
"The evaluation of the Poincaré polynomial at $t=1$ yields the so-called Euler characteristics $\\chi _{_X} = \\mathpzc {P}_X[t=1] = \\sum _p (-1)^p \\dim _k X_p$ of $X$ .",
"If we assume that the pair $(X, \\delta )$ is a differential complex for $X$ a graded vector space and $\\delta : X_p \\rightarrow X_{p+1}$ for any $p$ , then the cohomology $H_{\\delta }^\\bullet (X) = \\bigoplus _{p\\in \\mathbb {Z}} H_{\\delta }^p(X)$ is a graded space.",
"Here we are interested into the case of the de Rham cohomology, where $X = \\bigwedge ^\\bullet \\mathcal {T}^{\\ast } M$ , i.e.",
"the exterior bundle of a certain differentiable manifold $M$ and the differential $\\delta = d : \\bigwedge ^p \\mathcal {T}^\\ast M \\rightarrow \\bigwedge ^{p+1} \\mathcal {T} M$ is the de Rham differential: then $H^\\bullet _{dR}(M)$ is a graded vector space and we call $b_p (M) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =k HpdR (M)$ the $ p$-th Betti number of $ M$.",
"The Poincaré polynomial of $ M$, defined as (Euler-Poincaré formula)\\begin{eqnarray}\\mathpzc {P}_{M} [t] \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=\\mathpzc {P}_{H_{dR} (M)}[t] = \\sum _{p} b_p (M) (-t)^p$ is the generating function of the Betti numbers of $M$ .",
"This property is known as telescopic nesting which implies that from the easy computation of $P_X(t)$ , one deduces $P_{H(X)}(t)$ .",
"From the latter one can read the cohomology classes by their gradings and the parity.",
"Even if the notion of Betti numbers is originally related to the topology of a certain manifold or topological space, by extension, in this paper we will call Betti numbers the dimensions of any cohomology space valued in a field, in particular, we will call $p$ -th Betti numbers of a certain Lie (super)algebra the dimension of its Chevalley-Eilenberg $p$ -cohomology group $b_p (\\mathfrak {g}) = \\dim _k H^p_{CE} (\\mathfrak {g}),$ so that the Poincaré series of the Lie (super)algebra $\\mathfrak {g}$ is the generating function of its Betti number $\\mathpzc {P}_{\\mathfrak {g}} (t) = \\sum _p b_p (\\mathfrak {g}) (-t)^p.$ Notice that we used the notation $\\mathpzc {P} (t)$ on purpose: indeed, as we shall see, $H^\\bullet _{CE} (\\mathfrak {g})$ is not in general finite dimensional for a generic Lie superalgebra $\\mathfrak {g}$ .",
"In this context, we can retrieve some useful results using the Poincaré series.",
"For example, Künneth theorem, which computes the cohomology of products of spaces, can simply be written as $\\mathpzc {P}_{X \\otimes Y} (t) = \\mathpzc {P}_X(t) \\cdot \\mathpzc {P}_{Y}(t).$ Sometimes, it is useful to introduce a second grading.",
"In that case the space is said to be bigraded vector space $X = \\sum _{p,q \\in \\mathbb {Z}} X^{p,q}$ , then the gradation $X = \\sum _r X^{r}$ given by $X^r = \\sum _{p+q=r} X^{p,q}$ is called the induced total gradation.",
"One can write a double Poincaré series $\\mathpzc {P}_X(t,s) = \\sum _{p,q} (-t)^p s^q {\\rm dim}X^{p,q}$ which, in any case, allows an easier identification of cohomological classes (see, e.g., [24] where double Poincaré series have been used to select different type cohomologies)."
],
[
"Dimension 1: Example of Infinite Cohomology",
"In order to get familiar with cohomology computations of Lie superalgebras we start from a “simple model”, that is the Lie superalgebra of the supertranslations of the superspace $\\mathbb {R}^{1|2}$ , which we will denote $\\mathpzc {susy} (\\mathbb {R}^{1|2})$ , and we spell out all the details.",
"Starting from the supermanifold structure, here - and in the following examples - the superspace $\\mathbb {R}^{1|2}$ is actually not to be looked at as just the “bare” flat superspace $\\mathbb {R}^{1|2}$ , characterized by the pair $(\\mathbb {R}, \\mathcal {O}_{\\mathbb {R}} \\otimes \\wedge ^\\bullet [ \\theta _1, \\theta _2 ])$ as a ringed space, where the first entry is just the ordinary manifold $\\mathbb {R}$ and the second entry is a sheaf of exterior algebras generated over two anti-commuting variables $\\theta _1 $ and $\\theta _2$ , i.e.",
"the structure sheaf $\\mathcal {O}_{\\mathbb {R}^{1|2}}$ of the supermanifold $\\mathbb {R}^{1|2}$ .",
"Instead, $\\mathbb {R}^{1|2}$ carries some additional data, namely an odd distribution of the tangent bundle of $\\mathbb {R}^{1|2}$ , we denote it as $\\mathpzc {Susy} \\subset \\mathcal {T} (\\mathbb {R}^{1|2})$ , which is generated by the fields $\\mathcal {Q}_1 \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =1 - 2 x, Q2 =2 - 1 x, and which satisfies the commutation relations $\\lbrace \\mathcal {Q}_\\alpha , \\mathcal {Q}_\\beta \\rbrace = 2\\delta _{\\alpha \\beta } \\mathcal {P},$ for $\\alpha , \\beta = 1, 2$ , where we have defined $\\mathcal {P} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =-x$.",
"This means that the distribution $ Susy$ generated by $ { Q1, Q2}$ is \\emph {non-integrable} and the triple $ { Q1, Q2, P }$ generates the tangent bundle at any point.Adding to the previous relations (\\ref {susy1}) also the obvious commutation relations $ [P, P] = 0$ and $ [P, Qi] = 0$ for any $ i=1,2$ one gets the \\emph {supersymmetry translation algebra}, or supertranslation algebra for short, which we denote $ susy (R1|2)$.\\\\Switching from fields to forms, in order to write the cochains $ CpCE (susy (R1|2) ) = Sp susy (R1|2),$ we have to find the dual vielbeins (up to a parity shift) to the above fields.",
"These are\\begin{eqnarray}\\mathpzc {V} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=dx - \\theta ^1 d\\theta ^2 - \\theta ^2 d\\theta ^1, \\quad \\psi ^\\alpha = d\\theta ^\\alpha ,$ for $\\alpha = 1, 2,$ and it can be easily checked that $\\mathpzc {V} (\\pi \\mathcal {P}) = 1 $ and $ \\psi ^\\alpha (\\pi \\mathcal {Q}_\\beta ) = \\delta ^\\alpha _\\beta $ .",
"The Maurer-Cartan equations for the vielbeins $C^1_{CE} ( \\mathpzc {susy}( \\mathbb {R}^{1|2})) = \\lbrace \\psi ^\\alpha | \\mathpzc {V} \\rbrace $ are easily computed to be $d \\mathpzc {V} = -2\\psi ^1 \\psi ^2, \\qquad d \\psi ^\\alpha = 0,$ for $\\alpha = 1, 2.$ Now the cohomology is readily computed observing that terms involving $\\mathpzc {V}$ will never be closed and terms involving the product $\\psi ^1 \\psi ^2$ will always be exact.",
"This leads to the following differential Chevalley-Eilenberg cohomology $H^p_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1|2})) \\cong \\mathbb {R}\\cdot \\lbrace (\\psi ^\\alpha )^p \\rbrace ,$ for any $p \\ge 0$ and $\\alpha = 1, 2.$ Assigning the weights to Maurer-Cartan forms according to $\\mathpzc {W} (\\mathpzc {V}) = 1$ and $\\mathpzc {W} (\\psi ^\\alpha ) = 1/2$ for any $\\alpha = 1, 2$ , one finds for the Poincaré series $\\mathpzc {P}^{\\mathpzc {dif}}_{\\mathpzc {susy} (\\mathbb {R}^{1|2})} (t) = \\frac{1-t}{\\left( 1- \\sqrt{t} \\right)^2} = \\frac{1+ \\sqrt{t}}{1-\\sqrt{t}} = 1 + 2 \\sum _{n=1}^\\infty t^{{n}/{2}} \\ ,$ where the denominator has been expanded around 0.",
"This is in agreement with the previous computation, which indeed says that the Betti numbers of the superalgebra are $b_1 (\\mathpzc {susy} (\\mathbb {R}^{1|2})= 1, \\quad b_{p\\ge 1} (\\mathpzc {susy} (\\mathbb {R}^{1|2}) = 2.$ Let us now look at the integral Chevalley-Eilenberg cohomology.",
"Repeating the above analysis, posing $\\mathpzc {D}_{\\mathpzc {susy}(\\mathbb {R}^{1|2})} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =V ()() BerH (susy (R1|2)), one finds that $H^{1-p}_{CE, \\mathpzc {int}} (\\mathpzc {susy} (\\mathbb {R}^{1|2})) = \\mathbb {R} \\cdot \\left\\lbrace (\\iota _{\\pi \\mathcal {Q}_1 } )^p \\mathpzc {D}_{\\mathpzc {susy} (\\mathbb {R}^{1|2})} , \\; (\\iota _{\\pi \\mathcal {Q}_2 } )^p \\mathpzc {D}_{\\mathpzc {susy}( \\mathbb {R}^{1|2})}\\right\\rbrace ,$ where we notice in particular that the (Haar) Berezinian $(\\mathpzc {D}_{\\mathpzc {susy} (\\mathbb {R}^{1|2})})$ generates the integral 1-cohomology $H^1_{CE, \\mathpzc {int}} (\\mathpzc {susy} (\\mathbb {R}^{1|2})).$ Mirroring what above, terms coming from a double contraction $\\iota _{\\pi \\mathcal {Q}_1} \\iota _{\\pi \\mathcal {Q}_2} $ are not closed, while terms that do not contain $\\mathpzc {V}$ are exact.",
"Just like above, this matches the Poincaré series computation, namely $\\mathpzc {P}^{\\mathpzc {int}}_{\\mathpzc {susy} (\\mathbb {R}^{1|2})} (t) = \\frac{1-t}{\\left( 1- {1}/{\\sqrt{t}} \\right)^2} \\left( \\frac{-1}{\\sqrt{t}} \\right)^2 = \\frac{1+ \\sqrt{t}}{1-\\sqrt{t}} = -1 - 2 \\sum _{n=1}^\\infty t^{-{n}/{2}},$ where now we expanded the denominator around infinity, as to represent the cohomology spaces with negative form degree.",
"Before we move to higher dimensional examples, some remarks are in order.",
"First of all, already in this example it has to be noted an obvious yet striking difference between the ordinary and the super Chevalley-Eilenberg cohomology, namely the fact that even the cohomology of finite dimensional Lie superalgebras can be infinite, whereas clearly every finite-dimensional Lie algebras has a finite-dimensional Chevalley-Eilenberg cohomology.",
"This is allowed by the very structure of the cochain complex, which is not bounded from above for a true Lie superalgebra, i.e.",
"a Lie superalgebra whose odd dimension is different from zero, and by the structure of the commutators, which can leave “unconstrained” an even form, such as in the case of $\\psi ^\\alpha $ above."
],
[
"Dimension 2: “Flat” and “Curved” Cases",
"We now pass to study some more interesting cases of cohomology of Lie superalgebras, which both have two bosonic dimensions.",
"Namely we study the Lie superalgebra of supertranslations related to the superspace $\\mathbb {R}^{1,1 | 2}$ , which we will call flat superspace as it is constructed over the Minkowski space $\\mathbb {R}^{1,1}$ , and the Lie superalgebra $\\mathfrak {u} (1|1)$ .",
"For the sake of completeness and readability of the paper, the general mathematical structure of the Lie superalgebra $\\mathfrak {u} (n|m)$ is described in Appendix ."
],
[
"Flat Case: Supertranslations of the ${D=2}$ , {{formula:26ba2d9c-10ab-49ce-8e52-242575761b5a}} Superspace",
"Repeating the above discussion for the 1-dimensional case, one is lead to consider the algebra of supertranslations generated by the following vector fields $\\mathcal {Q}_\\alpha \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ = - ( )i xi, Pi =- xi, where $x^i | \\theta ^\\alpha $ for $i = 0,1$ and $\\alpha = 1, 2$ are coordinate for $\\mathbb {R}^{1,1 |2}$ and the gamma matrices $\\Gamma _{\\alpha \\beta }^i$ generating the spin representation of $\\mathfrak {so}(1,1)$ in which the odd coordinates transform, are given by $\\Gamma ^1_{\\alpha \\beta } = \\left( \\begin{array}{cc}1 & 0 \\\\0 & -1\\end{array} \\right), \\qquad \\Gamma ^2_{\\alpha \\beta } = \\left( \\begin{array}{cc}0 & 1 \\\\1 & 0\\end{array} \\right).$ The non-trivial commutation relations (supersymmetries) characterizing the Lie superalgebra $\\mathpzc {susy} (\\mathbb {R}^{1,1|2})$ read $\\lbrace \\mathcal {Q}_\\alpha , \\mathcal {Q}_{\\beta } \\rbrace = - 2 \\Gamma ^i_{\\alpha \\beta } \\mathcal {P}_{i}.$ Dually, we introduce the vielbeins, which will generate the cochains for the Lie superalgebra.",
"These are given by $\\mathpzc {V}^i = dx^a - \\theta ^\\alpha \\Gamma ^a_{\\alpha \\beta } d \\theta ^\\beta , \\quad \\psi ^\\alpha = d \\theta ^\\alpha $ again for $i = 0,1$ and $\\alpha = 1,2$ .",
"The Maurer-Cartan equations reads $d\\mathpzc {V}^i = \\psi ^\\alpha \\Gamma ^a_{\\alpha \\beta } \\psi ^\\beta , \\qquad d \\psi ^\\alpha = 0,$ leading to the following cohomology: $& H^0_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot 1, \\qquad H^1_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot \\lbrace \\psi ^1, \\psi ^2 \\rbrace \\nonumber \\\\& H^2_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot \\bigg \\lbrace \\sum _{\\alpha = 1}^2(\\psi ^\\alpha )^2 \\bigg \\rbrace , \\qquad H^{p > 2}_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong 0.$ This result is in agreement with the Poincaré polynomial, as indeed $\\mathpzc {P}^{\\mathpzc {dif}}_{\\mathpzc {susy} (\\mathbb {R}^{1,1|2})} [\\sqrt{t}] = \\frac{\\left( 1-t \\right)^2}{\\left( 1- \\sqrt{t} \\right)^2} = 1 + 2 \\sqrt{t} + t.$ Switching to integral forms, posing as above $\\mathpzc {D}_{\\mathpzc {susy}(\\mathbb {R}^{1,1|2})} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =V1 V2 (1)(2) BerH (susy (R1,1|2), and repeating the above computations one gets accordingly that $& H^2_{CE, \\mathpzc {int}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot \\mathpzc {D}_{\\mathpzc {susy}(\\mathbb {R}^{1,1|2})}, \\qquad H^1_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot \\lbrace \\iota _{\\pi \\mathcal {Q}_\\alpha } \\mathpzc {D}_{\\mathpzc {susy}(\\mathbb {R}^{1,1|2})} \\rbrace \\nonumber \\\\& H^0_{CE, \\mathpzc {int}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot \\bigg \\lbrace \\sum _{\\alpha = 1}^2 (\\iota _{\\pi \\mathcal {Q}_\\alpha })^2 \\mathpzc {D}_{\\mathpzc {susy}(\\mathbb {R}^{1,1|2})} \\bigg \\rbrace , \\qquad H^{p < 0}_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong 0.$ The Poincaré polynomial reads $\\mathpzc {P}^{\\mathpzc {int}}_{\\mathpzc {susy} (\\mathbb {R}^{1,1|2})} [t]= \\frac{\\left( 1-t \\right)^2}{\\left( 1- {1}/{\\sqrt{t}} \\right)^2} \\left( \\frac{-1}{\\sqrt{t}} \\right)^2 = 1 + 2 \\sqrt{t} + t.$"
],
[
"Curved Case: Lie Superalgebra $\\mathfrak {u} (1|1)$",
"We now aim at computing the cohomology of the $2|2$ -dimensional Lie superalgebra $\\mathfrak {u} (1|1)$ .",
"Further, later on, we briefly comment on Cartan's theorem on the cohomology of compact and connected Lie groups in the supersetting.",
"Before we start, we recall that, for the sake of the readability of the paper, the construction of the Lie superalgebra $\\mathfrak {u}(n|m)$ for arbitrary values of $n$ and $m$ is given in Appendix .",
"In the easiest case $\\mathfrak {u} (1|1)$ , one has a $2|2$ -dimensional Lie superalgebra, whose general element can be given in the following form $X = \\left( \\begin{array}{c|c}i a & \\theta + i \\psi \\\\\\hline - \\psi - i \\theta & i b\\end{array}\\right),$ for $a, b \\in \\mathbb {R}$ and $\\theta , \\psi \\in \\Pi \\mathbb {R},$ so that the even and odd generators can be chosen to be the matrices $X_1 = \\left( \\begin{array}{c|c}i & \\\\\\hline & 0\\end{array}\\right), \\quad X_2 = \\left( \\begin{array}{c|c}0 & \\\\\\hline & i\\end{array}\\right), \\quad \\Psi _1 = \\left( \\begin{array}{c|c}& 1 \\\\\\hline -i &\\end{array}\\right), \\quad \\Psi _2 = \\left( \\begin{array}{c|c}& i \\\\\\hline -1 &\\end{array}\\right),$ together with the commutation relations $ [X_i, X_j] = 0, \\quad [X_1, \\Psi _1 ] = \\Psi _2, \\quad [X_1, \\Psi _2] = - \\Psi _1, \\quad [X_2, \\Psi _1] = - \\Psi _2, \\quad [X_2, \\Psi _2] = \\Psi _1 \\\\\\lbrace \\Psi _1, \\Psi _1 \\rbrace = -2 X_1 - 2X_2, \\quad \\lbrace \\Psi _2, \\Psi _2 \\rbrace = - 2 X_1 - 2X_2, \\quad \\lbrace \\Psi _1, \\Psi _2 \\rbrace = 0 .$ Introducing the dual (up to parity) basis of Maurer-Cartan forms of $\\Pi \\mathfrak {u} (1|1)^\\ast $ , defined so that $\\Pi \\mathfrak {u} (1|1)^\\ast = \\mbox{Span}_\\mathbb {R} \\lbrace \\mathpzc {V}^i | \\psi ^\\alpha \\rbrace $ for $i = 1, 2$ and $\\alpha = 1, 2$ , with $V^i (\\pi X_j) = \\delta ^i_j$ and $\\psi ^\\alpha (\\pi \\Psi _\\beta ) = \\delta ^\\alpha _\\beta $ , one sees from (REF ) and () that the Maurer-Cartan equations read $dV^1 = dV^2 = - \\sum _{\\alpha =1}^2 (\\psi ^\\alpha )^2, \\quad d\\psi ^1 = \\psi ^2 \\frac{( V^1 - V^2 )}{2}, \\quad d \\psi ^2 = \\psi ^1 \\frac{(- V^1 + V^2 )}{2}.$ Changing the basis to $U \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =V1 - V22$ and $ W =V1 + V22$, the Maurer-Cartan equations simplify to\\begin{eqnarray}dU = 0, \\qquad dW = - \\sum _{\\alpha =1}^2 (\\psi ^\\alpha )^2, \\qquad d\\psi ^1 = U \\psi ^2, \\qquad d \\psi ^2 = -U \\psi ^1.\\end{eqnarray}Starting from the $ p$-cochains $ Cp (u (1|1)) = Sp (u (1|1))$ and using the above Maurer-Cartan equations (\\ref {mcUW}), it is not hard to compute the related Chevalley-Eilenberg cohomology:\\begin{eqnarray}H^0_{CE, \\mathpzc {dif}} (\\mathfrak {u} (1|1)) \\cong \\mathbb {R} \\cdot 1, \\qquad H^1_{CE, \\mathpzc {dif}} (\\mathfrak {u} (1|1)) \\cong \\mathbb {R} \\cdot \\lbrace U \\rbrace , \\quad H^{p > 1}_{CE, \\mathpzc {dif}} (\\mathfrak {u} (1|1)) = 0.\\end{eqnarray}This is in agreement with the computation of the Poincaré polynomial, which is the one of the ordinary 1-dimensional unitary Lie algebra $ u (1)$ as shown also by Fuks:\\begin{eqnarray}\\mathpzc {P}^{\\mathpzc {dif}}_{\\mathfrak {u} (1|1)} [t] = \\mathpzc {P}_{\\mathfrak {u} (1)}^{\\mathpzc {dif}} [t] = 1-t.\\end{eqnarray}In the case of integral Chevalley-Eilenberg cohomology one has the following cohomology{\\begin{@align}{1}{-1}H^2_{CE, \\mathpzc {int}} (\\mathfrak {u} (1|1)) \\cong \\mathbb {R} \\cdot \\mathpzc {D}_{\\mathfrak {u}(1|1)}, & \\qquad H^1_{CE, \\mathpzc {int}} (\\mathfrak {u} (1|1)) \\cong \\mathbb {R} \\cdot \\lbrace \\iota _{\\pi U^\\ast } \\mathpzc {D}_{\\mathfrak {u} (1|1)} \\rbrace , \\nonumber \\\\ & H^{p < 1}_{CE, \\mathpzc {int}} (\\mathfrak {u} (1|1)) = 0.\\end{@align}}where we have posed again $ Du (1|1) = UW (1) (2)$ so that for example the representative of the 1-cohomology group is given by $ W ( ) ()$, and accordingly the Poincaré polynomial is computed to be\\begin{eqnarray}\\mathpzc {P}^{\\mathpzc {int}}_{\\mathfrak {u} (1|1)} [t] = t^2 - t.\\end{eqnarray}$"
],
[
"A Remark on Cartan Theorem on Compact Lie Groups",
"A crucial result in Lie algebra cohomology theory is a theorem due to Cartan, which states that under the topological assumptions of compactness and connectedness, the de Rham cohomology of a Lie group $G$ is isomorphic to the cohomology of its Lie algebra (valued in the real numbers), i.e.",
"$H^p_{dR} (G) \\cong H^p_{CE} (\\mathfrak {g})$ ; clearly, the result is remarkable not only from a conceptual point of view, but also from a computational point of view, for it allows to get topological informations on large interesting classes of Lie groups via linear algebra.",
"The above result on $\\mathfrak {u} (1|1)$ shows that the result does not hold true in the supersetting, whereas one naively substitutes the ordinary compact Lie group $G$ with a compact Lie supergroup $\\mathpzc {G}$ and the Lie algebra $\\mathfrak {g}$ with its Lie superalgebra.",
"Let us look indeed at the Lie supergroup $U (1|1) $ related to $\\mathfrak {u}(1|1)$ .",
"Especially in this context, it is convenient to introduce the unitary supergroup $U(1|1)$ as the super Harish-Chandra pair $(U(1) \\times U(1), \\mathfrak {u} (1|1)),$ since the categories of Lie supergroups and super Harish-Chandra pairs are indeed equivalent [5].",
"As it is well-known [8], the de Rham cohomology of a supermanifold only depends on its underlying topological space, and as such it is completely determined by the first entry, i.e.",
"the ordinary Lie group, of the super Harish-Chandra pair.",
"In our case, we obtain the cohomology of a 2-torus $S^1 \\times S^1 \\cong U(1) \\times U(1)$ : $H^p_{dR} (U(1|1)) \\cong \\left\\lbrace \\begin{array}{lc}\\mathbb {R} & p = 0\\\\\\Pi \\mathbb {R}^2 & p=1\\\\\\mathbb {R} & p=2.\\end{array}\\right.$ This shows that the de Rham cohomology of compact Lie supergroups, such as for example $U(1|1)$ which is topologically a 2-torus, is not isomorphic to the Chevalley-Eilenberg cohomology of superforms of their related Lie superalgebras.",
"Notice by the way, that the isomorphism is restored once one reduces to deal with the even - or topological - part of a Lie superalgebra.",
"In other words, if, as a vector space, a Lie superalgebra is such that $\\mathfrak {g} = \\mathfrak {g}_0 \\oplus \\mathfrak {g}_1$ , and its related (e.g.",
"via Harish-Chandra pair) Lie supergroup $\\mathpzc {G}$ is topologically compact as a (super)manifold, then one finds that for any $p$ $H^p_{dR} (\\mathpzc {G}) \\cong H^p_{CE, {\\mathpzc {dif}}} (\\mathfrak {g}_0 ).$ This is readily seen in the above case for the Lie superalgebra $\\mathfrak {u} (1|1)$ , where modding out the odd part of the underlying vector space, one is left with Maurer-Cartan equations of the form $dU=0$ and $dW=0$ , which indeed lead to the same cohomology of the 2-torus.",
"Once again, it has therefore to be stressed that whilst fermions play really no role when computing de Rham cohomology of a supermanifold as nilpotents do not modify topology, in the case of Chevalley-Eilenberg cohomology of a Lie superalgebra, which is ultimately determined by the structure of commutators or, equivalently, by the Maurer-Cartan equations, fermions play a crucial role and they do indeed determine the cohomology structure, which might be very different - either richer or poorer - from the cohomology of the topological even part of the superalgebra."
],
[
"Dimension 3: “Flat” and “Curved” Cases",
"In this section we study two examples of superalgebras that have 3 bosonic dimensions.",
"In particular, we study the cohomology of the Lie superalgebra of superstranslations of flat superspace $\\mathbb {R}^{1,2|2}$ and, after having reviewed (in appendix B) the construction of the (simple) Lie superalgebra $\\mathfrak {osp} (n|2m)$ for generic values of $n$ and $m$ we study the cohomology of its simplest case, namely $\\mathfrak {osp}(1|2),$ corresponding to the classical simple Lie superalgebra $B(0,1)$ in Kac's classification."
],
[
"Flat Case: Superstranslations of $D=3$ , {{formula:ce83b396-1462-4feb-818a-dd283d48b231}} Superspace",
"We describe the supermanifold $\\mathbb {R}^{1,2|2}$ , based on the Minkowski space $\\mathbb {R}^{1,2}$ by a set of two coordinates $(x^{a},\\theta ^{\\alpha })$ .",
"In terms of these coordinates, we have the following supersymmetry generators $\\mathcal {Q}_\\alpha \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ = - ( )a xa.",
"Notice that in the above we are using real and symmetric gamma matrices $\\gamma ^a_{\\alpha \\beta }$ , which are defined via charge conjugation, given by the Pauli matrix $C \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =- i 2 = $, so that we have\\begin{eqnarray}&&\\gamma ^0_{\\alpha \\beta } \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=(C \\Gamma ^0)_{\\alpha \\beta } = - {\\mathbf {1}}, \\qquad \\gamma ^1_{\\alpha \\beta } \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =(C 1) = 3, 2 =(C 2) = - 1 , where $\\Gamma ^0 \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =i 2 = ( cc 0 1 -1 0 ), 1 =i 1 = ( cc 0 1 1 0 ), 2 =3 = ( cc 1 0 0 -1 ), which satisfies the the Clifford algebra relation $\\lbrace \\Gamma ^a, \\Gamma ^b \\rbrace = 2 \\eta ^{ab} \\mathbf {1}$ , with $\\eta ^{ab}$ the Minkowski metric and they must be looked at as $(\\Gamma ^a)^{\\alpha }_{\\; \\beta }$ from the point of view of spinor indices.",
"Defining as above $\\mathcal {P}_a = - \\frac{\\partial }{\\partial x^a}$ , we have the commutation relations of the algebra $\\mathpzc {susy} (\\mathbb {R}^{1,2|2})$ $\\lbrace \\mathcal {Q}_\\alpha , \\mathcal {Q}_{\\beta } \\rbrace = 2 \\gamma ^a_{\\alpha \\beta } \\mathcal {P}_a.$ Switching to forms, we have the following dual (up to parity) basis of Maurer-Cartan 1-forms $C^1_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,2|2})) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ ={ | Va } $ for $ a = 0,..., 2$ and $ = 1, 2$ with\\begin{eqnarray}\\mathpzc {V}^a \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=dx^a - \\theta ^\\alpha \\gamma ^a_{\\alpha \\beta } d\\theta ^\\beta , \\qquad \\psi ^\\alpha \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =d.",
"From the commutation relation above one reads the Maurer-Cartan equations $d \\mathpzc {V}^a = \\psi ^\\alpha \\gamma ^a_{\\alpha \\beta } \\psi ^\\beta , \\qquad d \\psi ^\\alpha = 0\\,,$ which in turns leads to the following (differential) Chevalley-Eilenberg cohomologySome results have been already appeared in the liturature [47].",
"$&H^0_{CE, \\mathpzc {dif}}(\\mathpzc {susy} (\\mathbb {R}^{1,2|2} )) \\cong \\mathbb {R} \\cdot 1, \\nonumber \\\\&H^1_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,2|2}) ) \\cong \\mathbb {R} \\cdot \\lbrace \\psi ^\\alpha \\rbrace , \\nonumber \\\\& H^2_{CE, \\mathpzc {dif}} (\\mathpzc {susy} ( \\mathbb {R}^{1,2|2} ) ) \\cong \\mathbb {R} \\cdot \\bigg \\lbrace \\mathpzc {V}^a \\gamma _{a, \\alpha \\beta } \\psi ^\\beta \\bigg \\rbrace , \\nonumber \\\\& H^3_{CE, \\mathpzc {dif}} (\\mathpzc {susy} ( \\mathbb {R}^{1,2|2} ) ) \\cong \\mathbb {R} \\cdot \\bigg \\lbrace \\mathpzc {V}^a \\psi ^\\alpha \\gamma _{a, \\alpha \\beta } \\psi ^\\beta \\bigg \\rbrace ,$ and $H^{p>3}_{CE, \\mathpzc {dif}} (\\mathpzc {susy} ( \\mathbb {R}^{1,2|2} ) ) = 0.$ The closure of the above 2-forms and 3-form is easily seen by using Fierz identities.",
"Accordingly, the computations of the Poincaré polynomial in the present case gives $\\mathpzc {P}_{\\mathpzc {susy} (\\mathbb {R}^{1,2|2})}^{\\mathpzc {dif}} [\\sqrt{t}] = \\frac{(1-t)^3}{(1-\\sqrt{t})^2}= (1-\\sqrt{t}) (1+ \\sqrt{t})^3 = 1 + 2 \\sqrt{t} - 2 t \\sqrt{t} - t^2,$ where we observe that the signs indeed match the parity of the representatives.",
"Passing to the integral Chevalley-Eilenberg cohomology we find, keeping explicit the structure of the generators and leaving the wedge product understood, $& H^3_{CE, \\mathpzc {int}} (\\mathpzc {susy} (\\mathbb {R}^{1,2|2})) \\cong \\mathbb {R} \\cdot \\lbrace \\mathpzc {V}^0 \\mathpzc {V}^1 \\mathpzc {V}^2 \\delta (\\psi ^1)\\delta (\\psi ^2) \\mathpzc {}\\rbrace , \\nonumber \\\\& H^2_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot \\lbrace \\mathpzc {V}^0 \\mathpzc {V}^1 \\mathpzc {V}^2 \\iota _{\\pi \\mathcal {Q}_{\\alpha }} \\delta (\\psi ^1) \\delta (\\psi ^2) \\rbrace \\nonumber \\\\& H^1_{CE, \\mathpzc {int}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot \\bigg \\lbrace \\mathpzc {V}^a \\mathpzc {V}^b \\gamma _{ab, \\alpha \\beta } \\iota _{\\pi \\mathcal {Q}_\\beta } \\delta (\\psi ^1) \\delta (\\psi ^2) \\bigg \\rbrace , \\nonumber \\\\& H^{0}_{CE, \\mathpzc {int}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong \\mathbb {R} \\cdot \\bigg \\lbrace \\mathpzc {V}^a \\mathpzc {V}^b \\iota _{\\pi \\mathcal {Q}_{\\alpha } }\\gamma _{ab,\\alpha \\beta } \\iota _{\\pi \\mathcal {Q}_\\beta } \\delta (\\psi ^1) \\delta (\\psi ^2) \\bigg \\rbrace \\nonumber \\\\& H^{p < 0}_{CE, \\mathpzc {dif}} (\\mathpzc {susy} (\\mathbb {R}^{1,1|2})) \\cong 0,$ where we have defined $\\gamma ^{ab} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =12[a, b]$ and it can be seen that $ ab = abc c$.",
"Also notice that, as above, the highest integral cohomology group is indeed generated by the (Haar) Berezinian, namely $ R Dsusy(R1,1|2) = abcVa Vb Vc ()().$ The Poincaré polynomial reads\\begin{eqnarray}\\mathpzc {P}^{\\mathpzc {int}}_{\\mathpzc {susy} (\\mathbb {R}^{1,2|2})} [\\sqrt{t}]= \\frac{\\left( 1-t \\right)^3}{\\left( 1- {1}/{\\sqrt{t}} \\right)^2} \\left( \\frac{-1}{\\sqrt{t}} \\right)^2 = 1 + 2 \\sqrt{t} - 2 t \\sqrt{t} - t^2.\\end{eqnarray}where we have used the assignement of the charges as in the previous sections.",
"The factor $ (1-t)3$ is due to$ Va$^{\\prime }s, the factor $ (1- 1/t)2$ in the denominator is due to the contractions$ Q$ (being a contraction w.r.t.",
"an odd vector a commuting object).",
"The factor $ (-1/t)2$is due to the term $ ()()$.",
"Notice that the Poincaré polynomialis exactly the same as in (\\ref {PPB}).",
"\\\\$ As it is known, beside integral and differential forms, there are also forms with non-maximal and non-zero picture number, which are usually called pseudoforms [44], [10], [11], [12], [15], [16].",
"Just as a hint, in the present case, pseudoforms have picture number 1 and form other two complexes unbounded both from above and from below.",
"The prototype for these forms is $\\mathpzc {V} \\dots \\mathpzc {V} (\\psi ^1)^a (\\iota _2)^b \\delta (\\psi ^2)$ where $a, b \\ge 0$ (by exchanging $\\psi ^1$ with $\\psi ^2$ , we get the other complex).",
"Counting again the scaling dimensions we have $\\mathpzc {P}_{\\rm pseudo}[\\sqrt{t}] = \\frac{(1-t)^3}{\\left(1-\\frac{1}{\\sqrt{t}}\\right) (1 - \\sqrt{t})} \\left(\\frac{-1}{\\sqrt{t}}\\right)=1 + 2 \\sqrt{t} - 2 t \\sqrt{t} - t^2$ where the factor $(1-t)^3$ is due to the $\\mathpzc {V}$ 's, $1/(1-\\frac{1}{\\sqrt{t}})$ takes into account the powers of $\\iota _2$ , $1/(1-\\sqrt{t})$ takes into account the powers of $\\psi ^1$ .",
"Finally, $ \\frac{-1}{\\sqrt{t}}$ represents $\\delta (\\psi ^2)$ which scales as $1/\\sqrt{t}$ and the minus sign takes into account the fermion nature of a single delta.",
"We do not explore any further this “sector” of the cohomology, but it will turn out to be crucial for a complete understanding of the Chevalley-Eilenberg cohomology in this extended framework [13]."
],
[
"Curved Case: Lie Superalgebra $\\mathfrak {osp} (1|2)$ and its İnönü-Wigner Contraction to {{formula:49fb5695-13fb-45ac-aa6d-8aca32a654d0}}",
"For the sake of readability of the paper, we review in Appendix the construction of the orthosymplectic Lie superalgebra $\\mathfrak {osp}(n|2m)$ for generic values of $n$ and $m$ .",
"Here we restrict to the case $\\mathfrak {osp}(1|2) = B(0,1)$ and compute its cohomology.",
"Last, we relate the computation with the case of the previous “flat” case of the Lie superalgebra $\\mathpzc {susy} (\\mathbb {R}^{1,2|2})$ considered above.",
"The choice of a basis for $\\mathfrak {osp}(1|2)$ using the relations (REF ) is reflected into the commutation relations.",
"However, a neat and convenient choice is provided as follows: $\\mathcal {P}_0 = \\frac{1}{2}\\left(\\begin{array}{c|cc}0 & 0 & 0 \\\\\\hline 0 & 0 & 1 \\\\0 & - 1 & 0\\end{array}\\right), \\qquad \\mathcal {P}_1 = \\frac{1}{2} \\left(\\begin{array}{c|cc}0 & 0 & 0 \\\\\\hline 0 & 1 & 0 \\\\0 & 0 & -1\\end{array}\\right), \\qquad \\mathcal {P}_2 = \\frac{1}{2}\\left(\\begin{array}{c|cc}0 & 0 & 0 \\\\\\hline 0 & 0 & 1 \\\\0 & 1 & 0\\end{array}\\right), \\quad $ $\\mathcal {Q}_1 = \\left(\\begin{array}{c|cc}0 & 1 & 1 \\\\\\hline 1 & 0 & 0 \\\\-1 & 0 & 0\\end{array}\\right), \\qquad \\mathcal {Q}_2 = \\left(\\begin{array}{c|cc}0 & 1 & -1 \\\\\\hline -1 & 0 & 0 \\\\-1 & 0 & 0\\end{array}\\right).$ Making use of the previously introduced (real and symmetric) gamma matrices $\\gamma ^i_{\\alpha \\beta }$ the commutation relations can be written in the following very convenient way $& [\\mathcal {P}_a, \\mathcal {P}_b] = - \\epsilon _{abc}\\mathcal {P}_c, \\qquad \\lbrace \\mathcal {Q}_\\alpha , \\mathcal {Q}_\\beta \\rbrace = -2 \\gamma ^a_{\\alpha \\beta } \\mathcal {P}_a, \\qquad [\\mathcal {Q}_\\alpha , \\mathcal {P}_a] = - \\gamma ^{\\; \\; a}_{\\alpha \\; \\beta } \\mathcal {Q}_\\beta $ where $\\epsilon _{abc}$ is the Levi-Civita symbol and where we observe that the first commutation relation follows by the isomorphism $\\mathfrak {sp} (2, \\mathbb {R}) \\cong \\mathfrak {so}(2,1, \\mathbb {R}) \\cong \\mathfrak {su} (1,1, \\mathbb {C})$ .",
"We now introduce the Maurer-Cartan forms which are dual to the above generators of the Lie superalgebra $\\mathfrak {osp} (1|2)$ up to parity.",
"More precisely we introduce a basis of forms such that $C^1_{CE, \\mathpzc {dif}} (\\mathfrak {osp} (1|2)) = \\Pi \\mathfrak {osp}(1|2)^\\ast = \\mbox{Span}_{\\mathbb {R}} \\lbrace \\psi ^\\alpha | \\mathpzc {V}^a \\rbrace $ for $a = 0, 1, 2$ and $\\alpha = 1,2$ with $\\mathpzc {V}^a (\\pi \\mathcal {P}_b) = \\delta ^a_b$ and $\\psi ^\\alpha (\\pi \\mathcal {Q}_\\beta ) = \\delta ^\\alpha _\\beta $ .",
"The above commutation relations lead to the following set of Maurer Cartan equations (up to a sign redefinition): $d \\mathpzc {V}^a = \\epsilon _{bc}^{\\; \\; \\; a} \\mathpzc {V}^b \\mathpzc {V}^c + \\psi ^\\alpha \\gamma ^a_{\\alpha \\beta } \\psi ^\\beta , \\qquad d \\psi ^\\alpha = \\mathpzc {V}^a \\gamma _{a, \\alpha \\beta } \\psi ^\\beta $ The cohomology reads $&H^0_{CE, \\mathpzc {dif}} (\\mathfrak {osp}(1|2)) \\cong \\mathbb {R} \\cdot 1, \\nonumber \\\\& H^1_{CE, \\mathpzc {dif}} (\\mathfrak {osp}(1|2)) \\cong 0, \\nonumber \\\\& H^2_{CE, \\mathpzc {dif}} (\\mathfrak {osp}(1|2)) \\cong 0 \\nonumber \\\\& H^3_{CE, \\mathpzc {dif}} (\\mathfrak {osp} (1|2)) \\cong \\mathbb {R} \\cdot \\bigg \\lbrace \\frac{1}{2} \\mathpzc {V}^a (\\psi \\gamma _a \\psi ) - \\frac{1}{6} \\epsilon _{abc} \\mathpzc {V}^a \\mathpzc {V}^b \\mathpzc {V}^c\\ \\bigg \\rbrace $ and $H^{p>3}_{CE, \\mathpzc {dif}} (\\mathfrak {osp} (1|2)) \\cong 0.$ Notice that this result is confirmed by the theorem of Fuks, which states that the cohomology of $\\mathfrak {osp} (1|2)$ is isomorphic to that of its bosonic subalgebra $\\mathfrak {sp}(2, \\mathbb {R})$ , thus leading to the Poincaré polynomial $\\mathpzc {P}_{\\mathfrak {osp} (1|2)}^{\\mathpzc {dif}} [t] = \\mathpzc {P}_{\\mathfrak {sp} (2, \\mathbb {R})} = 1 - t^3.$ Notice, though, that with respect to the bosonic Lie algebra $\\mathfrak {sp}(2, \\mathbb {R})$ the representative of the 3-cohomology of the Lie superalgebra $\\mathfrak {osp}(1|2)$ is shifted in the fermionic directions as can be seen directly by the above expression.",
"Quite similarly, the integral Chevalley-Eilenberg cohomology reads $&H^3_{CE, \\mathpzc {int}} (\\mathfrak {osp}(1|2)) \\cong \\mathbb {R} \\cdot \\epsilon _{abc} \\mathpzc {V}^a \\mathpzc {V}^b \\mathpzc {V}^c \\epsilon _{\\alpha \\beta } \\delta (\\psi ^\\alpha ) \\delta (\\psi ^\\beta ), \\nonumber \\\\& H^2_{CE, \\mathpzc {int}} (\\mathfrak {osp}(1|2)) \\cong 0, \\nonumber \\\\& H^1_{CE, \\mathpzc {int}} (\\mathfrak {osp}(1|2)) \\cong 0, \\nonumber \\\\& H^0_{CE, \\mathpzc {int}} (\\mathfrak {osp} (1|2)) \\cong \\mathbb {R} \\cdot \\bigg \\lbrace \\frac{1}{2}\\mathpzc {V}^a \\mathpzc {V}^b (\\iota _{\\pi \\mathcal {Q}_\\alpha } \\gamma _{[ab], \\alpha \\beta }\\iota _{\\pi \\mathcal {Q}_\\beta }) \\epsilon _{\\alpha \\beta } \\delta (\\psi ^\\alpha ) \\delta (\\psi ^\\beta ) - \\frac{1}{6} \\epsilon _{\\alpha \\beta } \\delta (\\psi ^\\alpha ) \\delta (\\psi ^\\beta ) \\bigg \\rbrace \\ .$ It is worth to observe the relation between the “curved” and “flat” 3-dimensional case.",
"Indeed, simply redefining the generators of the superalgebra $\\mathfrak {osp}(1|2)$ by a constant parameter $\\lambda $ as follows, $\\mathcal {Q}^{\\lambda }_\\alpha \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =1 Q, Pa =1 Pa, one finds that the new Maurer-Cartan equations for $\\mathpzc {V}_{\\lambda }^a$ and $\\psi ^\\alpha _\\lambda $ read $d \\mathpzc {V}^a_{\\lambda } = \\lambda \\epsilon _{bc}^{\\; \\; \\; a} \\mathpzc {V}^b_{\\lambda } \\mathpzc {V}^c_{\\lambda } + \\psi ^\\alpha _{\\lambda } \\gamma ^a_{\\alpha \\beta } \\psi ^\\beta _{\\lambda }, \\qquad d \\psi _\\lambda ^\\alpha = \\lambda \\mathpzc {V}^a \\gamma _{a, \\alpha \\beta }.$ The limit $\\lambda \\rightarrow 0$ is called İnönü-Wigner contraction and it is immediate to see that it gives back the Maurer-Cartan equations for the superalgebra $\\mathpzc {susy} (\\mathbb {R}^{1,2|2}):$ in this sense $\\mathpzc {susy} (\\mathbb {R}^{1,2|2})$ can be seen as the “flat” limit of the orthosymplectic superalgebra $\\mathfrak {osp}(1|2)$ ."
],
[
"Flat Case: Supertranslations of the $D=4$ , {{formula:6bc0ac5f-45e5-4dbb-b010-55de3622b63d}} Superspace {{formula:5c7ba707-8b79-4b63-aad4-8a5ad5602c19}}",
"Let us now move to a 4 dimensional example.",
"Here we consider the physically relevant superspace $\\mathbb {R}^{1,3|4}$ based upon the 4-dimensional Minkowski space $\\mathbb {R}^{1,3}$ .",
"This is the usual superspace for rigid supersymmetry models $\\mathcal {N}=1$ and therefore the first step toward supergravity models.",
"Some of results of the present section have been also discussed in [46].",
"Again, we describe this flat supermanifold via the coordinates $(x^a | \\theta ^\\alpha )$ for $a=0,\\ldots 3$ and $\\alpha = 1, \\ldots 4$ .",
"The supersymmetry generators read exactly as in equation (REF ), but clearly now the gamma's are 4-dimensional Dirac matrices, instead of 2-dimensional.",
"Accordingly, passing to the Maurer-Cartan forms and defining $\\mathpzc {V}^a = dx^a + \\theta ^\\alpha \\gamma ^a_{\\alpha \\beta } d\\theta ^\\beta $ and $\\psi ^\\alpha = d\\theta ^\\alpha $ one has that the generators of the 1-cochains of the Lie superalgebra satisfies the Maurer-Cartan equations $d\\mathpzc {V}^a = \\psi ^\\alpha \\gamma ^a_{\\alpha \\beta } \\psi ^\\beta ,\\qquad d\\psi ^\\alpha = 0\\,.$ Notice, by the way, that it is convenient switching the reducible Dirac representation $\\psi \\in (1/2, 0) \\oplus (0, 1/2)$ to its irreducible components, the (left) Weyl spinors $\\chi ^\\alpha \\in (1/2,0)$ and (right) anti-Weyl spinors $\\bar{\\lambda }^{\\dot{\\alpha }} \\in (0, 1/2)$ for $\\alpha , \\dot{\\alpha }= 1, 2 $ so that $\\psi = (\\chi ^\\alpha , \\bar{\\lambda }^{\\dot{\\alpha }})$ .",
"The above Maurer-Cartan modifies to $d\\mathpzc {V}^{\\alpha \\dot{\\alpha }} = \\chi ^\\alpha \\bar{\\lambda }^{\\dot{\\alpha }}, \\qquad d\\chi ^\\alpha = 0, \\qquad d \\bar{\\lambda }^{\\dot{\\alpha }} = 0.$ Here we are using the spin structure to represent the odd 1-forms $\\mathpzc {V}^a$ as bispinors: $\\mathpzc {V}^{\\alpha \\dot{\\alpha }} = \\bar{\\sigma }_{a}^{\\alpha \\dot{\\alpha }} \\mathpzc {V}^a $ , where we have used the matrices $\\bar{\\sigma }$ of the $(0, 1/2)$ irreducible component.",
"Instead of giving the cohomology classes, let us first look at the Poincaré polynomial.",
"Assigning weights $1/2$ to the Maurer-Cartan forms $(\\chi ^\\alpha , \\bar{\\lambda }^{\\dot{\\alpha }})$ and 1 to the Maurer-Cartan form $\\mathpzc {V}^i$ respectively, as already done early on, one considers $\\hspace{-11.38092pt}\\mathpzc {P}_{\\mathpzc {susy} (\\mathbb {R}^{1,3|4})}^{\\mathpzc {dif}} [\\sqrt{t}] = \\frac{(1-t)^4}{(1-\\sqrt{t})^4} = \\frac{(1-\\sqrt{t})^4 (1+ \\sqrt{t})^4}{(1-\\sqrt{t})^4} =1 + 4 \\sqrt{t} + 6 t + 4 t \\sqrt{t} + t^2.$ Here the numerator corresponds to product of the $\\mathpzc {V}^{\\alpha \\dot{\\alpha }}$ 's - these are odd forms, thus they appear in the numerators - and the denominator corresponds to product of the $\\chi ^\\alpha $ 's and $\\bar{\\lambda }^{\\dot{\\alpha }}$ 's - these are even forms, thus they appear in the numerator.",
"Let us now see explicitly the first cohomology groups: $& H^0_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\cong \\mathbb {R} \\cdot 1, \\nonumber \\\\& H^1_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\cong \\mathbb {R} \\cdot \\lbrace \\chi ^\\alpha , \\; \\bar{\\lambda }^{\\dot{\\alpha }}\\rbrace \\nonumber \\\\& H^2_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\cong \\mathbb {R} \\cdot \\lbrace \\chi ^\\alpha \\chi ^\\beta , \\; \\bar{\\lambda }^{\\dot{\\alpha }} \\bar{\\lambda }^{\\dot{\\beta }}, \\; \\chi ^\\alpha \\epsilon _{\\alpha \\beta } V^{\\beta \\dot{\\beta }}, \\; \\bar{\\lambda }^{\\dot{\\alpha }}\\epsilon _{\\dot{\\alpha }\\dot{\\beta }} V^{\\beta \\dot{\\beta }} \\rbrace , \\nonumber \\\\& H^3_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\cong \\mathbb {R} \\cdot \\lbrace \\chi ^\\alpha \\chi ^\\beta \\chi ^\\gamma , \\,\\bar{\\lambda }^{\\dot{\\alpha }} \\bar{\\lambda }^{\\dot{\\beta }} \\bar{\\lambda }^{\\dot{\\gamma }}, \\,\\chi ^\\gamma \\chi ^\\alpha \\epsilon _{\\alpha \\beta } V^{\\beta \\dot{\\beta }}, \\,\\bar{\\lambda }^{\\dot{\\gamma }}\\bar{\\lambda }^{\\dot{\\alpha }}\\epsilon _{\\dot{\\alpha }\\dot{\\beta }} V^{\\beta \\dot{\\beta }}, \\,\\chi ^\\alpha \\bar{\\lambda }^{\\dot{\\alpha }} \\epsilon _{\\alpha \\beta } \\epsilon _{\\dot{\\alpha }\\dot{\\beta }} V^{\\beta \\dot{\\beta }}\\rbrace , \\nonumber \\\\& H^4_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\cong \\mathbb {R} \\cdot \\lbrace \\chi ^\\alpha \\chi ^\\beta \\chi ^\\gamma \\chi ^\\delta , \\,\\bar{\\lambda }^{\\dot{\\alpha }} \\bar{\\lambda }^{\\dot{\\beta }} \\bar{\\lambda }^{\\dot{\\gamma }}\\bar{\\lambda }^{\\dot{\\delta }}, \\dots \\rbrace $ where the sum over repeated indices is understood, and the ellipses in the 4-cohomology group stays for other cohomology representatives which we have not written.",
"Notice that the cohomology is again infinite dimensional, for example one has that $\\chi ^{\\alpha _1} \\ldots \\chi ^{\\alpha _p} \\in H^p_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}}))$ for any $p\\ge 1$ .",
"With reference to the assigned weights, one sees that the Poincaré series (in $\\sqrt{t}$ ) is reconstructed as follows: $& \\dim H^0_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\rightsquigarrow 1, \\nonumber \\\\& \\dim H^1_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\rightsquigarrow 2\\sqrt{t} + 2\\sqrt{t}, \\nonumber \\\\& \\dim H^2_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\rightsquigarrow - 2t \\sqrt{t} - 2t \\sqrt{t} + 3 t + 3 t \\nonumber \\\\& \\dim H^3_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\rightsquigarrow 4t \\sqrt{t} + 4t \\sqrt{t} - 4 t^2 - 4 t^2 - t^2 \\nonumber \\\\& \\dim H^4_{CE, \\mathpzc {dif}} (\\mathpzc {susy}(\\mathbb {R}^{1,3|{4}})) \\rightsquigarrow 5 {t}^2 + 5 {t}^2 + \\ldots .$ Summing up the above terms, this leads to $1 + 4 \\sqrt{t} + 6 t + 4 t \\sqrt{t} + t^2 = \\mathpzc {P}_{\\mathpzc {susy} (\\mathbb {R}^{1,3|4})}^{\\mathpzc {dif}} $ .",
"Comparing explicit computations with the above Poincaré polynomial one indeed sees that the contributions for weights higher than 2 vanishes, or better, they sum up to zero, even if there is cohomology at any degree higher than 4.",
"However, the Maurer-Cartan equations allow to take different weights, namely distinguish between the left spinors and the right spinors, and associating to the $\\chi $ 's the weight $\\sqrt{t}$ and to $\\bar{\\lambda }$ 's the weight $\\sqrt{\\bar{t}}$ , so that $\\mathpzc {V}$ is associated with $\\sqrt{t} \\sqrt{\\bar{t}}$ .",
"This choice leads indeed to the series: $\\hspace{-14.22636pt}\\mathpzc {P}_{\\mathpzc {susy} (\\mathbb {R}^{1,3|4})}^{\\mathpzc {dif}} ( \\sqrt{t}, \\sqrt{\\bar{t}} ) = 1 + 2 (\\sqrt{t} + \\sqrt{\\bar{t}}) + 3 (t + \\bar{t}) + 4(t \\sqrt{t} + t \\sqrt{t}) - 2 (\\bar{t} \\sqrt{t} + t \\sqrt{\\bar{t}}) + \\ldots \\nonumber \\\\$ where each monomial of the series is in a one-to-one correspondence with a cohomology representative and the signs stand for even and odd parity."
],
[
"Curved Case: Lie Superalgebra $\\mathfrak {osp}({2|2})$",
"Before discussing coset superspaces - which appear more suitable to provide useful example of supergravity backgrounds - we consider a “curved” 4 dimensional case, studying the cohomology of the Lie superalgebra $\\mathfrak {osp}({2|2}) = C(2)$ .",
"The setting is given exactly as above and we refer to Appendix .",
"Before we start, though, it is useful to stress that the related Lie supergroup $OSp(2|2)$ cannot be given an interpretation from Minkowskian point of view, since it breaks the $SO(1,3)$ -invariance to the subgroup $SO(2) \\times Sp(2)$ - and indeed fermionic coordinates transforms under this subgroup.",
"However, the example provides a useful comparison with the remarkable “flat superspace” case above.",
"The Maurer-Cartan forms are $\\mathpzc {V}^{a} = \\gamma ^a_{\\alpha \\beta } \\mathpzc {V}^{\\alpha \\beta }$ , $\\mathpzc {V}^0$ and $\\psi ^\\alpha _I$ , having separated a “time” direction.",
"They satisfy the Maurer-Cartan equations $d \\mathpzc {V}^{\\alpha \\beta } &=& (\\mathpzc {V}\\wedge \\mathpzc {V})^{\\alpha \\beta } + \\psi ^\\alpha _I \\eta ^{IJ} \\psi ^\\beta _J\\,, \\nonumber \\\\d \\mathpzc {V}^{0} &=&- \\epsilon _{\\alpha \\beta } \\psi ^\\alpha _I \\epsilon ^{IJ} \\psi ^\\beta _J\\,, \\nonumber \\\\d \\psi ^\\alpha _I &=& (V\\wedge \\psi )^\\alpha _I + \\epsilon _{I}^J \\mathpzc {V}^0 \\psi ^\\alpha _J\\,.$ Notice that in the suitable “flat” limit, one retrieves the flat model discussed in the previous section.",
"According to Fuks, the cohomology should match with the one of the $\\mathfrak {sp}({2}, \\mathbb {R})$ subalgebra, and therefore we expect the Poincaré polynomial to be of the form $\\mathpzc {P}_{ \\mathfrak {osp}({2|2})}^{\\mathpzc {dif}}[t] = (1-t^3).$ The cohomology generators are indeed found to read $H^{0}_{CE, \\mathpzc {dif}}(\\mathfrak {osp}({2|2})) &=& \\mathbb {R} \\cdot 1 \\,, \\nonumber \\\\H^{3}_{CE, \\mathpzc {dif}}(\\mathfrak {osp}({2|2})) &=& \\mathbb {R} \\cdot \\left\\lbrace \\psi ^\\alpha _I \\eta ^{IJ} \\psi ^\\beta _J \\mathpzc {V}_{\\alpha \\beta } +\\psi ^\\alpha _I \\epsilon ^{IJ} \\psi ^\\beta _J \\epsilon _{\\alpha \\beta } \\mathpzc {V}^0 + \\mathpzc {V}\\wedge \\mathpzc {V}\\wedge \\mathpzc {V}\\right\\rbrace $ On the other hand, cohomology classes in the integral form sector are explicitly given by $H^{1}_{CE, \\mathpzc {int}}(\\mathfrak {osp}({2|2}) &=& \\mathbb {R} \\cdot \\left\\lbrace \\iota _\\alpha ^I \\eta _{IJ} \\iota _\\beta ^J \\mathpzc {V}^0 ( \\mathpzc {V} \\wedge \\mathpzc {V} )^{\\alpha \\beta } \\delta ^4(\\psi ) +\\iota _\\alpha ^I \\epsilon _{IJ} \\iota _\\beta ^J \\epsilon ^{\\alpha \\beta } (\\mathpzc {V} \\wedge \\mathpzc {V} \\wedge \\mathpzc {V}) \\delta ^4(\\psi ) + \\mathpzc {V}^0 \\delta ^4(\\psi ) \\right\\rbrace \\nonumber \\\\H^{4}_{CE, \\mathpzc {int}}(\\mathfrak {osp}({2|2}) &=& \\mathbb {R} \\cdot \\left\\lbrace \\mathpzc {V}^0 \\mathpzc {V}\\wedge \\mathpzc {V}\\wedge \\mathpzc {V} \\delta ^4(\\psi ) \\right\\rbrace .$"
],
[
"Coset Superspaces and Equivariant Chevalley-Eilenberg Cohomology",
"In this section we briefly introduce equivariant Chevalley-Eilenberg cohomology, a crucial tool to study the cohomology of coset or homogeneous superspaces $\\mathcal {G} / \\mathcal {H}$ where $\\mathcal {G}$ is a Lie supergroup and $\\mathcal {H}$ is a Lie sub-supergroup of $\\mathcal {G}$ .",
"Very few examples of Lie supergroup, or group supermanifolds, are indeed solutions of supergravity/string equations of motion, for example $AdS_3$ in the case of non-critical strings and few others.",
"Nonetheless, the space of geometric backgrounds modelled on coset spaces is much richer, in particular the case of supersymmetric background built on coset supermanifolds.",
"In this context, the most important instance is that of a coset supermanifold realized by modding out a certain bosonic subgroup: the infamous examples of $AdS_5 \\times S^5$ and $AdS_4 \\times \\mathbb {CP}^3$ belong this category [38] [42].",
"Furthermore, a less explored instance it that obtained by modding out a true Lie sub-supergroup.",
"In any of each cases, it is interesting to compute their (equivariant) cohomology, as it can uncover insights in the physics related to the model.",
"Given a Lie supergroup $\\mathcal {G}$ and a Lie sub-supergroup $\\mathcal {H}$ of $\\mathcal {G}$ we define the related Lie superalgebras by $\\mathfrak {g}$ and $\\mathfrak {h}$ .",
"Then, attached to the coset superspace $\\mathcal {G}/ \\mathcal {H}$ we will have, correspondingly, the quotient $\\mathfrak {g} / \\mathfrak {h}$ , whose elements are equivalence classes $ g\\, \\mbox{mod} \\, \\mathfrak {h}$ .",
"As a vector superspace, there always exists a direct linear decomposition of $\\mathfrak {g}$ such that $\\mathfrak {g} = \\mathfrak {h} \\oplus \\mathfrak {C},$ but the choice of $\\mathfrak {C}$ is ambiguous and different compatibility conditions between this direct linear decomposition and the Lie algebra structures can be imposed.",
"More in details, the coset superspace $\\mathcal {G} / \\mathcal {H}$ is said to be reductive if there exists an $\\mbox{ad}(\\mathfrak {h})$ -invariant choice of $\\mathfrak {C}$ , i.e.",
"$\\mbox{ad} (\\mathfrak {h}) \\cdot \\mathfrak {C} = [\\mathfrak {h}, \\mathfrak {C}] \\subset \\mathfrak {C}.$ Further, imposing that $[\\mathfrak {C}, \\mathfrak {C}] \\subset \\mathfrak {h}$ we get that the coset $\\mathcal {G} / \\mathcal {H}$ is a symmetric superspace, but in the following we will consider the more general relation $[\\mathfrak {C}, \\mathfrak {C}] \\subset \\mathfrak {g}.$ As in the ordinary setting, left-translation in the coset superspace induces a map $(\\ell _{[g^{-1}]})_\\ast : \\mathcal {T}_{[g]} \\mathcal {G} / \\mathcal {H} \\rightarrow \\mathcal {T}_{[e]} \\mathcal {G} / \\mathcal {H} \\cong \\mathfrak {g}/ \\mathfrak {h}$ which can be seen as $\\mathfrak {g} / \\mathfrak {h}$ -valued 1-forms, the so-called Maurer-Cartan forms.",
"As above, we will always deal with matrix Lie superalgebras.",
"In this case the Maurer-Cartan is usually written starting from the coset superspace elements as $\\omega _{MC}^g = [g^{-1} dg].$ Notice that, choosing another representative $gh $ for $h \\in \\mathcal {H}$ instead of $g$ , we get $\\omega ^{gh}_{MC} = [\\mbox{ad} (h) (g^{-1} dg)] = \\mbox{ad} (h)\\cdot \\omega ^g_{MC},$ since $[h^{-1} dh] = 0 $ in the quotient $\\mathfrak {g}/\\mathfrak {h}$ .",
"Passing from the above coordinate-invariant formalism to a particular choice of coordinates, in line with the general philosophy of the paper of finding explicit expressions, we choose a certain direct linear decomposition of $\\mathfrak {g}$ as above and, in turn, a basis $\\lbrace h_i \\rbrace $ for $i = 1, \\ldots , \\dim \\mathfrak {h}$ of generators for $\\mathfrak {h}$ and a basis $\\lbrace {k}_J \\rbrace $ for $J = 1, \\ldots , \\dim \\mathfrak {C}$ of generators for $\\mathfrak {C}$ .",
"Notice that the parametrization of the elements of the coset superspace $[g] \\in \\mathcal {G} /\\mathcal {H}$ is far from being unique.",
"The Maurer-Cartan form related to this decomposition and choice of basis can be computed as to get $\\omega _{MC} = \\mathpzc {V}^J k_J + \\omega ^i h_i,$ where the $\\mathpzc {V}^i$ 's are the supervielbein forms and the $\\omega ^j$ 's are interpreted as the connection forms associated with the action of the sub-superalgebra $\\mathfrak {h}.$ The vielbein and connection forms satisfy the following Maurer-Cartan equations that can be read off the (REF ) and (REF ) $d \\mathpzc {V}^I = f^I_{JK} \\mathpzc {V}^J \\wedge \\mathpzc {V}^K + f^I_{i J} \\, \\omega ^i\\wedge \\mathpzc {V}^J, \\nonumber \\\\d\\omega ^i = f^i_{jk} \\omega ^j \\wedge \\omega ^k + f^i_{IJ} \\mathpzc {V}^I \\wedge \\mathpzc {V}^K.$ The second one can be re-written as $\\mathpzc {R}^i \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =di - fijk j k = fiIJ VI VK.",
"Here the structure constants are written with respect to the above decomposition of the Lie superalgebra $\\mathfrak {g} = \\mathfrak {h} \\oplus \\mathfrak {C}$ and $\\mathpzc {R}^i$ is referred to as the “curvature” of the gauge connection $\\omega ^i$ related to the sub-algebra $\\mathfrak {h}$ .",
"The form of the first Maurer-Cartan equation in (REF ) in turn makes convenient to introduce a covariant differential defined as $\\mathcal {D} \\mathpzc {V}^I \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =d VI - fIi J iVJ.",
"Notice that this differential is not nilpotent, indeed one has $\\mathcal {D}^2 \\mathpzc {V}^I = - \\mathpzc {R}^i f_{iJ}^{I} \\mathpzc {V}^J$ using Jacobi identity.",
"This can be re-written as $\\mathcal {D}^2 \\mathpzc {V}^I = - {\\mathcal {L}}_\\mathpzc {R} V^I$ , where we have denoted $\\mathcal {L}_{\\mathpzc {R}}$ the action of the Lie derivative on the vielbeins $\\mathpzc {V}^I$ along the (vertical) vector $\\mathpzc {R}^i h_i$ .",
"The above expression makes clear that in order to have a well-defined differential cochain complex for coset superspaces, we need to impose further conditions on the forms to take into account.",
"Namely, we need the Maurer-Cartan forms, call them $\\Omega $ 's, to be basic, this means that we require $\\mathcal {L}_{H} \\Omega = 0, \\qquad \\iota _{H} \\Omega = 0,$ for any vector $H$ coming from the sub-algebra $\\mathfrak {h}$ .",
"Roughly speaking, one can visualize these requirements thinking about a principal $\\mathcal {H}$ -bundle $\\pi : \\mathcal {P} \\rightarrow \\mathcal {G} / \\mathcal {H}$ : in this respect a basic form $\\Omega $ is a form defined on the principal bundle $\\Omega = \\pi ^\\ast (\\mathpzc {V})$ such that it has no vertical components (it is horizontal) and no vertical variation (it stays horizontal), i.e.",
"basic forms are in the intersection $\\ker (\\iota _{H} )\\cap \\ker (\\mathcal {L}_H)$ .",
"Calling $C^p_{\\mathpzc {Basic}} (\\mathfrak {g}/\\mathfrak {h})$ the (vector) superspace of the basic $p$ -forms, we accordingly define the equivariant (Chevalley-Eilenberg) cohomology to be the cohomology of the basic forms with respect to the differential $\\mathcal {D}$ introduced above.",
"${H}^{p}_{\\mathpzc {EQ}} ({\\mathfrak {g} / \\mathfrak {h}}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ ={ CpBasic (g / h) : D = 0 }{ CpBasic (g/ h) : Cp-1Basic (g / h) = D } ."
],
[
"Methods for Computations: Poincaré Polynomial Revised",
"In absence of encompassing “structure theorems”, different methods are possible in order to compute cohomology of coset superspaces.",
"Our strategy will be to supplement brute force computations with the indications coming from the Poincaré polynomial of coset superspaces.",
"This will tell, for example, when a cohomology space is expected to be infinite dimensional, as we shall see.",
"Following [26], if $\\mathfrak {g}$ is a Lie superalgebra with Poincaré series given by $\\mathpzc {P}_\\mathfrak {g} (t) = \\sum _i b_i^\\mathfrak {g} t^i$ , where the $b_i^\\mathfrak {g}$ 's are the Betti numbers of the Lie superalgebra $\\mathfrak {g}$ , i.e.",
"$b^\\mathfrak {g}_i \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =HiCE (g)$ and $ h$ is a Lie sub-superalgebra of $ g$, of the same rank (Cartan Pairs, see \\cite {greub}) having Poincaré series given by $ Ph (t) = j bih tj$, then the Poincaré series for the coset will be given by the following formula\\begin{eqnarray}\\mathpzc {P}_{\\mathfrak {g}/\\mathfrak {h}}(t) = \\frac{\\sum _{i} b^\\mathfrak {g}_i t^{i+1}}{\\sum _{j} b^\\mathfrak {h}_j t^{j+1}} =\\frac{\\prod _{l} (1-t^{c^\\mathfrak {g}_l+1})}{\\prod _{m} (1-t^{c^\\mathfrak {h}_m+1})}\\end{eqnarray}where $ cgl$ and $ chm$ are the usual exponents in the factorised form of the polynomial.",
"This product formula is very helpful since it provides some informations regarding thedifferent cohomology classes.$ A remark is in order: for ordinary coset spaces arising from purely bosonic finite dimensional Lie algebras $\\mathfrak {h} \\subset \\mathfrak {g}$ the above formula () always yields a polynomial: $\\mathpzc {P}_{\\mathfrak {h}} (t)$ divides $\\mathpzc {P}_\\mathfrak {g} (t)$ (which is a polynomial), or equivalently the $c^{\\mathfrak {h}}_m$ are a subset of the $c^{\\mathfrak {g}}_l$ .",
"For example, a $2n$ -dimensional sphere $S^{2n}$ can be seen as the coset manifold given by the quotient of Lie groups $SO(2n+1) / SO(2n)$ .",
"In the simplest case $n=1$ we obtain a 2-sphere $S^2$ : passing to the algebras we have the coset $\\mathfrak {so}(3)/\\mathfrak {so}(2)$ which is isomorphic to the coset $\\mathfrak {su}(2)/ \\mathfrak {u}(1)$ .",
"The vielbeins $\\lbrace \\mathpzc {V}^0, \\mathpzc {V}^{+}, \\mathpzc {V}^{-} \\rbrace $ for $\\mathfrak {so}(3)\\cong \\mathfrak {su}(2)$ satisfies the following Maurer-Cartan equations $d \\mathpzc {V}^0 = \\mathpzc {V}^+ \\wedge \\mathpzc {V}^-\\,, ~~~~~ d \\mathpzc {V}^+ = i \\mathpzc {V}^0 \\wedge \\mathpzc {V}^+\\,, ~~~~ d \\mathpzc {V}^- = - i \\mathpzc {V}^0 \\wedge V^+.$ These can be rewritten as $\\mathpzc {R} = d \\mathpzc {V}^0 = \\mathpzc {V}^+ \\wedge \\mathpzc {V}^-, ~~~~~ \\mathcal {D} \\mathpzc {V}^\\pm = 0\\,, ~~~~$ where $\\mathcal {D}$ is as above and the Bianchi identities imply that $\\mathcal {D} \\mathpzc {R} =0$ and $\\mathcal {D}^2 V^\\pm = \\pm i \\mathpzc {R} \\wedge \\mathpzc {V}^\\pm =0$ .",
"Notice that the nilpotent of $\\mathcal {D}$ holds true since since $\\mathpzc {R} = \\mathpzc {V}^+ \\wedge \\mathpzc {V}^-$ and $\\mathpzc {V}^\\pm \\wedge \\mathpzc {V}^\\pm =0$ .",
"In addition, we notice that ${\\mathcal {L}}_\\mathpzc {T} \\mathpzc {V}^\\pm = i \\mathpzc {V}^\\pm $ where $\\mathpzc {T}$ is the generator of $\\mathfrak {u}(1)$ subalgebra.",
"This implies that the only basic forms are given by $\\lbrace 1, \\mathpzc {V}^+ \\wedge \\mathpzc {V}^-\\rbrace $ , indeed notice also that $\\iota _\\mathpzc {T} \\mathpzc {V}^0 =1 \\ne 0$ , thus $\\mathpzc {V}^0$ is not a basic form.",
"The basic forms $\\lbrace 1, \\mathpzc {V}^+ \\wedge \\mathpzc {V}^-\\rbrace $ are closed and not exact.",
"The first statement is obvious.",
"For the latter we observe that $\\mathpzc {V}^+ \\wedge \\mathpzc {V}^- = \\mathcal {D} \\mathpzc {V}^0$ , with $\\mathcal {D}$ given in (REF ) (notice that $\\mathcal {D}\\mathpzc {V}^0 = d \\mathpzc {V}^0$ ) but since $\\mathpzc {V}^0$ is not basic.",
"It follows that $\\mathpzc {V}^+ \\wedge \\mathpzc {V}^-$ defines indeed an equivariant cohomology class.",
"The Poincaré polynomial for $\\mathfrak {so}(3)$ is given by $\\mathpzc {P}_{\\mathfrak {so}(3)} (t) = \\mathpzc {P}_{\\mathfrak {su}(2)} (t)= 1 - t^3$ , while the subalgebra $\\mathfrak {u}(1) \\cong \\mathfrak {so}(2)$ has Poincaré polynomial given by $\\mathpzc {P}_{\\mathfrak {u}(1)} (t) = 1-t$ , so that according to the above formula the coset has Poincaré polynomial given by $\\mathpzc {P}_{\\mathfrak {so}(3) / \\mathfrak {so}(2)} [t] = \\frac{1-t^4}{1-t^2} = 1 + t^2,$ thus matching the above calculation.",
"In the next section we generalize this easy example to the case of the supersphere."
],
[
"Lower Dimensional Cosets of $\\mathfrak {osp}(1|2)$ and {{formula:230980aa-cc74-49e4-b529-c0f304b31750}}",
"Let us now consider the Lie superalgebra $\\mathfrak {osp}(1|2)$ introduced above.",
"In agreement with an early result by Fuks, we have seen that $H^\\bullet _{CE} (\\mathfrak {osp}(1|2)) \\cong H^\\bullet _{CE} (\\mathfrak {sp} (2, \\mathbb {R}))$ and in particular, its Poincaré polynomial reads $\\mathpzc {P}_{\\mathfrak {osp} (1|2)} (t) = 1-t^3$ with the 3-cohomology class generated by $\\omega ^{(3)} = \\psi \\gamma _a \\psi \\mathpzc {V}^a + \\frac{1}{3} \\epsilon _{abc} \\mathpzc {V}^a \\mathpzc {V}^b \\mathpzc {V}^c$ , where the vielbeins $\\psi $ 's and $\\mathpzc {V}$ 's have been introduced above together with the gamma matrices $\\gamma ^i_{\\alpha \\beta }.$ Looking at the Lie supergroup $OSp(1|2)$ related to $\\mathfrak {osp}(1|2)$ it is natural to consider two coset manifolds.",
"The first one is the coset $OSp(1|2) / SO(1,1)$ , which is known as the supersphere $S^{2|2}$ .",
"The second one is a purely fermionic superspace, actually a “fat point”, given by the coset $OSp(1|2) / Sp(2, \\mathbb {R})$ , which is a $0|2$ -dimensional superspace.",
"Let us start from the supersphere: the Lie algebra coset Poincaré polynomial reads $\\mathpzc {P}_{\\mathfrak {osp}(1|2)/ \\mathfrak {so}(1,1)} [t] = \\frac{1-t^4}{1-t^2} = 1+t^2,$ upon using the so-called Weyl trick, in order to identify the Chevalley-Eilenberg cohomology of $\\mathfrak {so}(1,1)$ with that of $\\mathfrak {so}(2) \\cong \\mathfrak {u}(1).$ This suggests that besides the constants there is a single cohomology class at degree two.",
"Explicitely, introducing the Maurer-Cartan vielbeins $\\lbrace \\mathpzc {V}^0, \\mathpzc {V}^\\ddagger , \\mathpzc {V}^=| \\psi ^\\pm \\rbrace $ - notice that the the $\\mathpzc {V}$ 's are odd forms and the $\\psi $ 's are even forms - one gets the following Maurer-Cartan equations: $&\\mathcal {D} \\mathpzc {V}^0 = \\mathpzc {R} = i \\mathpzc {V}^\\ddagger \\wedge \\mathpzc {V}^= + \\psi ^+ \\wedge \\psi ^-\\,, ~~~~\\mathcal {D} \\mathpzc {V}^\\ddagger = i \\psi ^+\\wedge \\psi ^+\\,, ~~~~\\mathcal {D} V^= = - i \\psi ^-\\wedge \\psi ^-\\,, ~~~~\\nonumber \\\\&\\qquad \\qquad \\qquad \\qquad \\mathcal {D} \\psi ^+ = \\mathpzc {V}^\\ddagger \\wedge \\psi ^- \\,, ~~~~\\mathcal {D} \\psi ^- = \\mathpzc {V}^= \\wedge \\psi ^+ \\,.",
"~~~~$ The infinitesimal action of the subgroup is given by ${\\mathcal {L}}_\\mathpzc {T} V^\\ddagger = 2 i V^\\ddagger \\,, ~~~~{\\mathcal {L}}_\\mathpzc {T} V^= = -2 i V^=\\,, ~~~~{\\mathcal {L}}_\\mathpzc {T} \\psi ^\\pm = \\pm i \\psi ^\\pm .$ Note that the 2-form $\\mathpzc {R} = i \\mathpzc {V}^\\ddagger \\wedge \\mathpzc {V}^= + \\psi ^+ \\wedge \\psi ^-\\,$ is (real) basic and closed.",
"It is not exact because $\\mathpzc {R} = \\mathcal {D} \\mathpzc {V}^0$ , but $\\mathpzc {V}^0$ is not a basic form.",
"Notice in particular that the 3-cohomology class of $\\mathfrak {osp} (1|2)$ is no longer a cohomology class of the coset; it is invariant under the action of the subgroup, but it is not basic.",
"We have $H^p_\\mathpzc {EQ}(\\mathfrak {osp}(1|2) / \\mathfrak {osp}(1|1)) = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p =0, 2\\\\0 & \\quad & \\mbox{else}.\\end{array}\\right.$ In the case of the fermionic coset the Poincaré polynomial reads $\\mathpzc {P}_{\\mathfrak {osp}(1|2)/ \\mathfrak {sp}(2)} (t) = \\frac{1-t^4}{1-t^4} = 1.$ We expect therefore only the constants be in the cohomology, which is indeed the case since now $\\mathpzc {R}$ is not basic as now the forms $\\mathpzc {V}^{\\ddagger }$ and $\\mathpzc {V}^{=}$ are not vielbeins, but connections instead, coming from the subalgebra $\\mathfrak {sp}(2)$ : $H^p_\\mathpzc {EQ}(\\mathfrak {osp}(1|2) / \\mathfrak {sp}(2, \\mathbb {R})) = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p =0\\\\0 & \\quad & \\mbox{else}.\\end{array}\\right.$ We now consider the case of $\\mathfrak {u}(1|1)$ , whose Chevalley-Eilenberg cohomology has been discussed above.",
"Namely, we consider the coset $\\mathfrak {u}(1|1) / \\mathfrak {u}(1)$ of dimension $1|2$ and $\\mathfrak {u}(1|1) / \\mathfrak {u}(1)\\oplus \\mathfrak {u}(1)$ of dimension $0|2$ .",
"Let us start from the first coset space.",
"A subtle point is that we have to choose how to embed the abelian factor $\\mathfrak {u}(1)$ inside $\\mathfrak {u}(1|1)$ : indeed the automorphism of $\\mathfrak {u}(1|1)_{0} = \\mathfrak {u}(1) \\oplus \\mathfrak {u}(1)$ that exchange the factors does not lift to $\\mathfrak {u}(1|1)$ (see, e.g., [22]).",
"With reference to the previous section, we can embed $\\mathfrak {u} (1)$ in such a way that its Maurer-Cartan form (the connection, in view of the equivariant cohomology) is associated either to $U$ or to $W$ .",
"In the case it is associated to $U$ , then the cohomology trivializes as can be readily observed from the Maurer-Cartan equations: the only non-zero equivariant cohomology group is the zeroth-cohomology group: $H^p_\\mathpzc {EQ}(\\mathfrak {u}(1|1) / \\mathfrak {u}_U(1)) = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p =0\\\\0 & \\quad & \\mbox{else},\\end{array}\\right.$ having called $\\mathfrak {u}(1|1) / \\mathfrak {u}_U(1)$ the related coset.",
"Things changes drastically in case $\\mathfrak {u}(1)$ is embedded in a way such that its Maurer-Cartan forms correspond with $W$ .",
"In this case $U$ is the generator of a cohomology class, indeed it is closed and basic.",
"Moreover, also the bilinears $(\\psi ^1 \\psi ^2)^p$ for any $p$ are in the equivariant cohomology: indeed they can be seen to be exact with respect to a non-basic term.",
"The cohomology is therefore infinite-dimensional and generated by the elements $\\lbrace 1, U \\rbrace \\otimes \\lbrace (\\psi ^1 \\psi ^2)^p\\rbrace $ for any $p\\ge 0.$ $H^p_\\mathpzc {EQ}(\\mathfrak {u}(1|1) / \\mathfrak {u}_W(1)) = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p \\mbox{ even}\\\\\\Pi \\mathbb {R} & \\quad & p \\mbox{ odd},\\end{array}\\right.$ having called $\\mathfrak {u}(1|1) / \\mathfrak {u}_W(1)$ the related coset.",
"Finally, considering the coset $\\mathfrak {u}(1|1) / \\mathfrak {u}(1) \\oplus \\mathfrak {u}(1)$ we have that in this case both $U$ and $W$ correspond to Maurer-Cartan forms for the subalgebra.",
"From the Maurer-Cartan equations it follows that the cohomology is generated by the representatives given by the fermionic bilinears $\\lbrace (\\psi ^1 \\psi ^2)^p \\rbrace ,$ for any $p\\ge 0$ so that the equivariant cohomology is non-zero in any even degree.",
"$H^p_\\mathpzc {EQ}(\\mathfrak {u}(1|1) / \\mathfrak {u}{(1)} \\otimes \\mathfrak {u}{(1)}) = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p \\mbox{ even}\\\\0 & \\quad & p \\mbox{ odd},\\end{array}\\right.$ The corresponding Poincaré series read $\\mathpzc {P}_{\\mathfrak {u}(1|1) / \\mathfrak {u}_W(1)}(t) = \\frac{1-t}{1-t^2} = \\frac{1}{1+t}= 1 - t + t^2 - t^3\\dots .\\ .$ and $\\mathpzc {P}_{\\mathfrak {u}(1|1) / \\mathfrak {u}{(1)} \\otimes \\mathfrak {u}{(1)}}(t) = \\frac{1-t^2}{\\left( 1 - t^2 \\right)^2} = \\frac{1}{1- t^2} =1 + t^2 + t^4 + \\dots \\,.$ which match the computations above."
],
[
"Higher Dimensional Cosets of\n$\\mathfrak {osp}(2|2)$ , {{formula:6fd6286f-b383-4387-8179-e40c73d5481a}} and {{formula:21228560-3a17-4179-81e3-b29a775ea865}}",
"We now consider higher-dimensional cosets of $\\mathfrak {osp}(n|2)$ for $n=2, 3, 4$ .",
"We start with some general considerations, and then we specialize to the single cases together with their cosets.",
"On a general ground, the Maurer-Cartan equations for $\\mathfrak {osp}(n|2)$ $& d \\mathpzc {V}_{(\\alpha \\beta )} = \\psi ^I_\\alpha \\psi ^J_\\beta \\eta _{IJ} + (\\mathpzc {V} \\wedge \\mathpzc {V})_{(\\alpha \\beta )}\\,, \\nonumber \\\\& d \\mathpzc {T}^{[IJ]} = - \\psi ^I_\\alpha \\psi ^J_\\beta \\Omega ^{\\alpha \\beta } + (\\mathpzc {T}\\wedge \\mathpzc {T})^{[IJ]}\\,, \\nonumber \\\\& d \\psi ^I_\\alpha = \\mathpzc {V}_{\\alpha \\beta } \\Omega ^{\\beta \\gamma } \\psi _\\gamma ^I + \\mathpzc {T}^{[IJ]} \\eta _{JK} \\psi ^K_\\alpha $ where the Maurer-Cartan forms are given by $\\lbrace \\psi ^I_\\alpha | \\mathpzc {V}^{(\\alpha \\beta )}, \\mathpzc {T}^{[IJ]} \\rbrace $ for $\\alpha , \\beta =1,2$ and $I,J,K, \\ldots =1, \\dots , n$ .",
"We have $ (\\mathpzc {V} \\wedge \\mathpzc {V})_{(\\alpha \\beta )} = \\mathpzc {V}_{(\\alpha \\gamma )} \\Omega ^{\\gamma \\delta } \\mathpzc {V}_{(\\delta \\beta )}$ and $(\\mathpzc {T}\\wedge \\mathpzc {T})^{[IJ]} = \\mathpzc {T}^{[IK]} \\eta _{KL} \\mathpzc {T}^{[LJ]}$ , where $\\Omega ^{\\alpha \\beta }$ is the 2-symplectic invariant tensor (from $\\mathfrak {sp}(2)$ ) and $\\eta _{IJ}$ is the Euclidean rotation-invariant metric (from $\\mathfrak {so}(n)$ ).",
"For any $n$ the 3-form $\\hspace{-14.22636pt}\\omega ^{(3)} = \\psi ^I_\\alpha \\psi ^J_\\beta \\eta _{IJ} \\mathpzc {V}^{(\\alpha \\beta )} + \\psi ^I_\\alpha \\psi ^J_{\\beta } \\Omega ^{\\alpha \\beta } \\mathpzc {T}_{IJ} +(\\mathpzc {V}\\wedge \\mathpzc {V}\\wedge \\mathpzc {V})_{\\alpha \\beta } \\Omega ^{\\alpha \\beta } + (\\mathpzc {T}\\wedge \\mathpzc {T}\\wedge \\mathpzc {T})_{IJ} \\eta ^{IJ}$ gives an invariant which is a representative of the 3-cohomology group $H^3_{CE} (\\mathfrak {osp}(n|2))$ , shared by all of the $\\mathfrak {osp}(n|2)$ .",
"This is the unique cohomology class up to $n=3$ (besides the constants in the 0-cohomology), indeed we have that $\\mathpzc {P}_{\\mathfrak {osp} (2|2)} [t] = \\mathpzc {P}_{\\mathfrak {osp} (3|2)} [t]= \\mathpzc {P}_{\\mathfrak {osp} (1|2)} [t] = \\mathpzc {P}_{\\mathfrak {sp} (2, \\mathbb {R})} [t] = 1- t^3.$ Things start changing in the case of $\\mathfrak {osp}(4|2)$ , indeed in this case one has that $\\mathpzc {P}_{\\mathfrak {osp} (4|2)} [t] = \\mathpzc {P}_{\\mathfrak {so} (4) } [t] = (1-t^3)^2,$ where we recall that $D_2 \\cong A_1 \\otimes A_1$ for the complexified algebras and the Poincaré polynomial for $A_1$ is indeed $1-t^3$ .",
"We therefore expect a further 3-form in the cohomology $H^3_{CE}(\\mathfrak {osp}(4|2)).$ This is indeed the case and the extra cohomology representative is given by $\\widetilde{\\omega }^{(3)} = \\epsilon _{IJKL} \\psi ^I_\\alpha \\psi ^J_\\beta \\Omega ^{\\alpha \\beta } \\mathpzc {T}^{KL} + \\epsilon _{IJK[M} \\eta _{N]L} \\mathpzc {T}^{IJ} \\mathpzc {T}^{KL} \\mathpzc {T}^{MN}.$ Cohomology classes for higher dimensional $\\mathfrak {osp}(n|2)$ for $n>4$ can be constructed in similar way."
],
[
"$\\mathfrak {osp}(2|2)$",
"Let us now get back to the specific case of the Lie superalgebra $\\mathfrak {osp}(2|2)$ .",
"This is a Lie superalgebra of dimension $4|4$ , whose reduced algebra is given by $\\mathfrak {osp}(2|2)_0 = \\mathfrak {so}(2) \\oplus \\mathfrak {sp}(2, \\mathbb {R})$ .",
"We consider its cosets $\\mathfrak {osp}(2|2)/ \\mathfrak {so}(1,1)$ and $\\mathfrak {osp}(2|2)/ \\mathfrak {so}(2)\\oplus \\mathfrak {so}(1,1)$ , respectively of dimension $3|4$ and $2|4$ .",
"While the Poincaré series for the first coset can not be guessed by () (because the two superalgebras have different rank), it can be immediately written for the double coset: $& \\mathpzc {P}_{\\mathfrak {osp}(2|2)/\\left( \\mathfrak {so}(2)\\oplus \\mathfrak {so}(1,1) \\right)} (t) = \\frac{(1-t^4)}{(1-t^2)^2}= \\frac{1+ t^2}{1-t^2}.$ Let us calculate explicitly the cohomology of the two cosets: by looking at the Maurer-Cartan equations one finds $&d \\mathpzc {V}^{(\\alpha \\beta )} = (\\mathpzc {V} \\wedge \\mathpzc {V})^{(\\alpha \\beta )} + \\psi ^\\alpha _i \\wedge \\psi ^\\beta _j \\eta ^{ij} , \\nonumber \\\\&d \\mathpzc {W} = \\psi ^\\alpha _i \\wedge \\psi ^\\beta _j \\epsilon ^{ij} \\epsilon _{\\alpha \\beta }, \\nonumber \\\\& d \\psi ^\\alpha _i = \\mathpzc {V}^{(\\alpha \\gamma )} \\epsilon _{\\gamma \\beta }\\wedge \\psi ^\\beta _i - \\mathpzc {W} \\epsilon _{ij} \\wedge \\psi ^{\\alpha i}\\,.$ for $\\alpha = 1,2$ and $i = 1, 2$ , where $\\eta ^{ij}$ is the Minkowski metric.",
"In the case of the first coset $\\mathfrak {osp}(2|2) / \\mathfrak {so}(1,1)$ , there are two ways to embed $\\mathfrak {so}(1,1)$ in $\\mathfrak {osp}(2|2)$ : we can embed it in the $\\mathfrak {sp} (2)$ part or in the $\\mathfrak {so}(2)$ part (after a suitable signature redefinition via unitary trick, i.e.",
"we can identify $ \\mathfrak {so} (1,1) $ and $\\mathfrak {so} (2) $ ).",
"In the first case, one finds the cohomology class $\\mathpzc {R} = (\\gamma ^0)_{\\alpha \\beta } ((\\mathpzc {V} \\wedge \\mathpzc {V})^{(\\alpha \\beta )} + \\psi ^\\alpha _i \\wedge \\psi ^\\beta _j \\eta ^{ij}) \\ ,$ where $\\gamma ^0$ is the 0-th Dirac gamma matrix: it is easy to check that this is indeed a basic closed and not exact form.",
"To do this it is convenient to decompose the vielbeins as $\\mathpzc {V}^{(\\alpha \\beta )} = \\gamma _{a (\\alpha \\beta )} V^a, a \\in \\lbrace 0,\\pm \\rbrace $ (as in the previous section for $\\mathfrak {osp} (1|2)$ ), then we are doing the quotient w.r.t.",
"$\\mathpzc {V}^0$ .",
"Hence (REF ) represent a form which is closed by construction, basic since it does not depend on $\\mathpzc {V}^0$ and non-exact, being exact w.r.t.",
"a non-basic object.",
"In the second case, we have that the $\\mathfrak {so}$ algebra is embedded in the $\\mathfrak {so}$ subalgebra of $\\mathfrak {osp}$ , hence in this case we are doing the quotient w.r.t.",
"$\\mathpzc {W}$ .",
"In this case we immediately see from the MC equations (REF ) that the bilinear $(\\psi \\cdot \\psi ) = \\psi ^\\alpha _i \\wedge \\psi ^\\beta _j \\epsilon ^{ij} \\epsilon _{\\alpha \\beta } = \\mathcal {D} \\mathpzc {W} \\ ,$ together with its powers, is a basic, closed, non-exact form.",
"On the other hand, we can study the second coset $\\mathfrak {osp} (2|2) / \\left( \\mathfrak {so}(1,1) \\oplus \\mathfrak {so}(2)\\right)$ .",
"In this case, either $\\mathpzc {R}$ or the bilinear (and all its powers) $(\\psi \\cdot \\psi )$ are part of the equivariant cohomology, making the cohomology infinite dimensional, generated by $\\lbrace 1, \\mathpzc {R} \\rbrace \\otimes \\lbrace (\\psi \\cdot \\psi )^p\\rbrace $ for any $p$ .",
"The cohomologies that we have studied explicitly then read $& H^p_\\mathpzc {EQ}(\\mathfrak {osp}(2|2) / \\mathfrak {so}(1,1))_{\\mathpzc {V}^0} = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p =0, 2\\\\0 & \\quad & \\mbox{else},\\end{array}\\right.",
"\\\\& H^p_\\mathpzc {EQ}(\\mathfrak {osp}(2|2) / \\mathfrak {so}(1,1))_{\\mathpzc {W}} = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p \\mbox{ even}\\\\0 & \\quad & \\mbox{else},\\end{array}\\right.",
"\\\\& H^p_\\mathpzc {EQ}(\\mathfrak {osp}(2|2) / \\left( \\mathfrak {so}(1,1) \\oplus \\mathfrak {so}(2) \\right)) = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p = 0, \\\\\\mathbb {R}^2 & \\quad & p = 2, 4, \\ldots \\\\0 & \\quad & \\mbox{else}.\\end{array}\\right.$"
],
[
"$\\mathfrak {osp}(3|2)$",
"Let us look at the case of the cosets of $\\mathfrak {osp}(3|2)$ .",
"Cosets by $\\mathfrak {so}(2)$ or $\\mathfrak {so}(1,1)$ works in pretty the same way as the above case of $\\mathfrak {osp}(2|2)$ .",
"On the other hand, it is interesting to deal with the case $\\mathfrak {osp}(3|2) / \\mathfrak {so}(2) \\oplus \\mathfrak {so}(1,1)$ .",
"First, observe that the subgroup and the supergroup have the same rank, so by () the Poincaré series reads $\\mathpzc {P}_{\\mathfrak {osp}(3|2) / \\mathfrak {so}(2) \\oplus \\mathfrak {so}(1,1)} (t) = \\frac{1+t^2}{1-t^2},$ which is the same as in the case of the coset $\\mathfrak {osp}(2|2) / \\mathfrak {so}(2) \\oplus \\mathfrak {so}(1,1)$ .",
"However, in this case we run into a problem.",
"Indeed two equivariant cohomology classes can be singled out: $& \\mathcal {D} \\mathpzc {V}_{0} = \\psi ^I_\\alpha \\psi ^J_\\beta \\eta _{IJ} \\gamma _0^{\\alpha \\beta } + \\mathpzc {V}_+ \\wedge \\mathpzc {V}_-, \\nonumber \\\\& \\mathcal {D} \\mathpzc {T}^0 = - \\psi ^I_\\alpha \\psi ^J_\\beta \\Omega ^{\\alpha \\beta } s_{IJ } + \\mathpzc {T}_+\\wedge \\mathpzc {T}_-,$ where $s_{IJ} = - s_{JI}$ , which select “direction\" $\\mathfrak {so}(3)$ in denoted as $\\mathpzc {T}_0$ - in pretty much the same way as the $\\gamma ^0$ allows to select a “direction” in the Lie algebra $\\mathfrak {sp}(2, \\mathbb {R}).$ Notice that the above elements are not exact as $\\mathpzc {V}^0$ and $\\mathpzc {T}_0$ are not basic as they being the generators of the subgroup.",
"This result seems contradicting the above Poincaré series computation though.",
"Actually, to take into account the two independent cohomology classes above one needs to have a term of the kind $(1+t^2)^2$ in the numerator: we multiply and divide the above series by $1+t^2$ so that we get: $\\mathpzc {P}_{\\mathfrak {osp}(3|2) / \\mathfrak {so}(2) \\oplus \\mathfrak {so}(1,1)} (t) = \\frac{(1+t^2)^2}{1-t^4}.$ This suggests that the cohomology must be infinite, depending on a 4-form which can indeed be found to be $\\mathpzc {X}^{(4)} = \\psi ^I_{\\alpha } \\gamma _a^{\\alpha \\beta } \\psi ^J_\\beta \\eta ^{ab} \\epsilon _{R IJ} \\eta ^{RS} \\epsilon _{S KL}\\psi ^K_{\\alpha ^{\\prime }} \\gamma _b^{\\alpha ^{\\prime }\\beta ^{\\prime }} \\psi ^L_{\\beta ^{\\prime }},$ which is basic, closed and not exact.",
"The cohomology therefore reads $& H^p_\\mathpzc {EQ}(\\mathfrak {osp}(3|2) / \\mathfrak {so}(2)\\oplus \\mathfrak {so}(1,1)) = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p = 0 \\\\\\mathbb {R}^2 & \\quad & p \\mbox{ even} \\\\0 & \\quad & p \\mbox{ odd}.\\end{array}\\right.$"
],
[
"$\\mathfrak {osp}(4|2)$",
"Finally, let us take a brief look at an interesting coset of $\\mathfrak {osp}(4|2)$ , namely $\\mathfrak {osp}(4|2) / \\mathfrak {u}(2)$ .",
"In this case the problem can be studied by considering the spinor representation of $\\mathfrak {so}(1,3) \\sim \\mathfrak {so}(4) \\cong \\mathfrak {su}(2) \\times \\mathfrak {su}(2)$ .",
"In this formulation we have $ && \\mathpzc {T}^{[IJ]} \\rightarrow {\\left\\lbrace \\begin{array}{ll}T^{(AB)} = (\\sigma _{[IJ]})^{(AB)} \\mathpzc {T}^{[IJ]} \\\\\\tilde{T}^{(\\dot{A} \\dot{B})} = (\\sigma _{[IJ]})^{(\\dot{A} \\dot{B})} \\mathpzc {T}^{[IJ]}\\end{array}\\right.}",
"\\ , \\ A,B=1,2 \\ , \\\\ && \\psi ^I_\\alpha \\ \\rightarrow \\ \\psi _{\\alpha A \\dot{A}} = (\\sigma _I)_{A \\dot{A}} \\psi ^{I}_\\alpha \\ ,$ where $(\\sigma ^I)_{AB}$ and $(\\sigma ^I)_{\\dot{A}\\dot{B}}$ are the two copies of the Pauli matrices of $\\mathfrak {su}(2) \\times \\mathfrak {su}(2)$ , $\\displaystyle (\\sigma _{[IJ]})^{(AB)} = \\left[ (\\sigma _{I})^{A\\dot{A}} , (\\sigma _{J})_{\\dot{A}}^{\\ B} \\right]$ and $\\displaystyle (\\sigma _{[IJ]})^{(\\dot{A} \\dot{B})} = \\left[ (\\sigma _{I})^{A\\dot{A}} , (\\sigma _{J})_ {A}^{\\ \\dot{B}} \\right]$ .",
"The MC (REF ) then become $& d \\mathpzc {V}_{(\\alpha \\beta )} = \\left( \\psi \\cdot \\psi \\right)_{(\\alpha \\beta )} + (\\mathpzc {V} \\wedge \\mathpzc {V})_{(\\alpha \\beta )}\\,, \\nonumber \\\\& d \\mathpzc {T}^{(AB)} = - \\left( \\psi \\cdot \\psi \\right)^{(AB)} + (\\mathpzc {T}\\wedge \\mathpzc {T})^{(AB)}\\,, \\nonumber \\\\& d \\tilde{\\mathpzc {T}}^{(\\dot{A}\\dot{B})} = - \\left( \\psi \\cdot \\psi \\right)^{(\\dot{A}\\dot{B})} + (\\tilde{\\mathpzc {T}}\\wedge \\tilde{\\mathpzc {T}})^{(\\dot{A}\\dot{B})}\\,, \\\\& d \\psi _{\\alpha A \\dot{A}} = \\mathpzc {V}_{\\alpha \\beta } \\Omega ^{\\beta \\gamma } \\psi _{\\gamma A \\dot{A}} + \\sigma _{IA \\dot{A}} \\left( (\\sigma ^{[IJ]})_{(CD)}\\mathpzc {T}^{(CD)} + (\\sigma ^{[IJ]})_{(\\dot{C}\\dot{D})}\\mathpzc {T}^{(\\dot{C}\\dot{D})} \\right) \\eta _{JK} \\sigma ^{K A \\dot{A}} \\psi _{\\alpha A \\dot{A}} \\nonumber \\ .$ Let us consider the coset $\\displaystyle \\mathfrak {osp}(4|2)/\\mathfrak {su}(2)$ : we can quotient out one of the two $\\mathfrak {su}(2)$ , for example the one generated by $\\tilde{\\mathpzc {T}}$ .",
"We immediately see that the bilinears $\\left( \\psi \\cdot \\psi \\right)^{(\\dot{A}\\dot{B})} = - \\nabla \\tilde{\\mathpzc {T}}^{(\\dot{A}\\dot{B})}$ become exact, with respect to non basic objects, hence are cohomology representatives of the coset algebra.",
"The same holds for any power and product of these bilinears.",
"Moreover, we have another cohomology representative which is given by (REF ) but with just the non-modded out set of $\\mathpzc {T}$ 's: $\\omega ^{(3)} = \\left( \\psi \\cdot \\psi \\right)_{(\\alpha \\beta )} \\mathpzc {V}^{(\\alpha \\beta )} + \\left( \\psi \\cdot \\psi \\right)^{(AB)} \\mathpzc {T}_{(AB)} +(\\mathpzc {V}\\wedge \\mathpzc {V}\\wedge \\mathpzc {V})_{\\alpha \\beta } \\Omega ^{\\alpha \\beta } + (\\mathpzc {T}\\wedge \\mathpzc {T}\\wedge \\mathpzc {T})_{AB} \\eta ^{AB} \\ .$ Hence, the cohomology is generated by $\\displaystyle \\left\\lbrace 1 , \\omega ^{(3)} \\right\\rbrace \\otimes \\left\\lbrace \\left[ \\left( \\psi \\cdot \\psi \\right)^{(\\dot{A}\\dot{B})} \\right]^n \\right\\rbrace , \\forall n \\in \\mathbb {N}$ .",
"The computation of the dimensions of the cohomology spaces is not difficult, but tedious since it heavily relies on the Fierz identities, hence it is not written here.",
"Notice that the “finite part\" of the cohomology (to be precise, its bosonic restriction) is exactly what is left from the bosonic quotient $\\mathfrak {so}(4)/\\mathfrak {su}(2) \\cong \\mathfrak {su}(2)$ .",
"Keeping in mind this observation, we will comment further on this in the next paragraph.",
"We could now proceed further by quotienting by another $\\mathfrak {u}(1)$ , in order to study the coset space $\\displaystyle \\mathfrak {osp}(4|2)/\\mathfrak {u}(2)$ .",
"Given the previous results, the calculation is straightforward: in modding out with respect to $\\mathfrak {u}(1)$ , we can either embed it into the remaining $\\mathfrak {su}(2)$ , which is generated by the $\\mathpzc {T}^{(AB)}$ 's, or into $\\mathfrak {sp}(2)$ , which is generated by the $\\mathpzc {V}^{(\\alpha \\beta )}$ 's.",
"However, the two embeddings are equivalent since $\\mathfrak {sp}(2) \\cong \\mathfrak {su}(2)$ at the level of complex algebras.",
"Suppose then that we perform the quotient in the $\\mathfrak {sp}(2)$ part, hence we immediately see that (REF ) is no longer basic, hence it does not contribute to the cohomology.",
"However, a 2-form as the one in (REF ) appears, making contribution to the cohomology.",
"It follows that the cohomology of the coset $\\displaystyle \\mathfrak {osp}(4|2)/\\mathfrak {u}(2)$ is generated by $\\displaystyle \\left\\lbrace 1 , \\mathpzc {R}^{(2)} \\right\\rbrace \\otimes \\left\\lbrace \\left[ \\left( \\psi \\cdot \\psi \\right)^{(\\dot{A}\\dot{B})} \\right]^n \\right\\rbrace , \\forall n \\in \\mathbb {N}$ .",
"A comment is now mandatory: as we already noticed at the end of the previous paragraph, the “finite part\" of the cohomology (again, its bosonic restriction) corresponds to what is left from the bosonic quotient $\\mathfrak {so}(4)/\\mathfrak {u}(2) \\cong \\mathfrak {su}(2)/\\mathfrak {u}(1)$ .",
"We can interpret this result, and the previous one, as follows: Fuks' theorem states that $H^p_{CE, \\mathpzc {dif}} (\\mathfrak {osp}(4|2)) = H^p_{CE, \\mathpzc {dif}} (\\mathfrak {so}(4)) $ .",
"When modding out the sub-algebra, we have found that bosonic restriction of the finite part is actually given by the coset of the (purely bosonic) subalgebra which is selected by Fuks' theorem.",
"Notice that, not completely surprisingly, this holds true as long as we are embedding the sub-algebra in the part which actually contributes to the full cohomology of the algebra at the numerator.",
"Indeed, we have seen in the $\\mathfrak {u}(1|1)/\\mathfrak {u}(1)$ example that if we embed the divisor subalgebra in the subalgebra not contributing to the cohomology, we obtain a different finite part.",
"Hence, under the discussed assumption, it can be conjectured that, for example, if we consider the superalgebra $\\mathfrak {osp}(n|m)$ , given a subalgebra $\\mathfrak {h}$ , one will find that $\\left[ H^p_{CE} \\left( \\frac{\\mathfrak {osp}(n|m)}{\\mathfrak {h}} \\right) \\right]_{FP} \\cong {\\left\\lbrace \\begin{array}{ll}H^p_{CE} \\left( \\frac{\\mathfrak {so}(n)}{\\mathfrak {h}} \\right) \\ , \\ \\text{if} \\ n \\ge 2m \\\\H^p_{CE} \\left( \\frac{\\mathfrak {sp}(m)}{\\mathfrak {h}} \\right) \\ , \\ \\text{if} \\ n < 2m\\end{array}\\right.}",
"\\ ,$ where the subscript “FP\" denotes the finite part of the cohomology.",
"An evidence supporting this claim is provided by the Poincaré polynomials, which can be computed combining Fuks' results with [26], in the case of equal rank pairs as follows $\\mathpzc {P}_{{\\mathfrak {osp}(n|m)}/{\\mathfrak {h}}} (t) = \\frac{\\mathpzc {P}^{\\prime }_{{\\mathfrak {osp}(n|m)}} (t) }{\\mathpzc {P}^{\\prime }_{{\\mathfrak {h}}}(t)}$ where the prime denotes the augmented power by one for all powers in the Poincaré series.",
"It would be interesting to verify if this holds for cosets of the other basic Lie superalgebras as well as to improve general results comprising also quotient spaces, as in [26]."
],
[
"Cosets of $\\mathfrak {osp}(1|4)$ : {{formula:3a1e76f2-072f-4c9d-89b3-781c4c88da21}} , {{formula:6d932b49-4423-4437-8793-f5a128ee4dcf}} Anti de Sitter Superspace",
"It is well-known that the ordinary anti de Sitter spacetimes $AdS_D$ in $D$ -dimensions can be obtained starting from the Lie groups $SO(2, D-1)$ and $SO (1, D-1)$ as the coset manifold $SO(2, D-1) / SO(1,D-1)$ , for example the $AdS_4$ is obtained by taking the quotient of the Lie group $SO(2,3)$ by the Lorentz group $SO(1,3)$ (see [4] for a complete discussion on the present case in relation to supergravity and in particular in relation with Chevalley-Eilenberg cohomology.",
"In [4] the computation of the easiest CE cohomology has been performed) .",
"This construction can be generalized to a coset superspace as to obtain the superspace extension of the anti-de Sitter spacetimes.",
"Namely, in this section we are interested into computing the equivariant cohomology of one such construction, namely the $D=4$ , $\\mathcal {N}=1$ anti-de Sitter superspace $AdS_{4|4}$ realized as the quotient supermanifold $OSp(1|4)/SO(1,3)$ .",
"At the level of the Lie superalegbras one starts analyzing the $\\mathfrak {osp}(1|4),$ of dimension $10|4$ , whose reduced Lie algebra is $\\mathfrak {osp}(1|4)_0 = \\mathfrak {sp}(4, \\mathbb {R})$ .",
"Using that $\\mathfrak {sp} (4, \\mathbb {R}) \\cong \\mathfrak {so}(2,3)$ one can trace back the quotient yielding the anti de Sitter 4-space $AdS_4$ at the level of the groups.",
"Notice that the quotient manifold $OSp(1|4) / SO(1,3)$ has dimension $4|4$ , therefore it is $\\mathcal {N}=1$ (minimal) supersymmetric extension for the $AdS_4$ and we call it $AdS_{4|4}$ .",
"We will denote the corresponding coset at the Lie superalgebra level $\\mathfrak {ads}_{4|4} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =osp(1|4) / so(1,3)$.\\\\$ Let us start analyzing the the Chevalley-Eilenberg cohomology of $\\mathfrak {osp}(1|4)$ .",
"At the level of the Poincaré polynomial we have $\\mathpzc {P}_{\\mathfrak {osp}(1|4)} [t]= \\mathpzc {P}_{\\mathfrak {sp}(4, \\mathbb {R})} [t] = 1 - t^3 - t^7 + t^{10}.$ Introducing a set of gamma matrices $\\gamma ^a_{\\alpha \\beta }$ for $a = 0, \\ldots , 9$ and $\\alpha , \\beta = 1, \\ldots , 4$ we represent the Maurer-Cartan odd forms by bi-spinors as follows $\\mathpzc {V}^a = \\gamma ^a_{\\alpha \\beta } \\mathpzc {V}^{\\alpha \\beta } \\ , \\ a=1,\\ldots ,10, \\quad \\alpha , \\beta =1,\\ldots ,4.$ Notice that is consistent as long as the indices $\\alpha , \\beta $ are symmetrized, i.e.",
"$\\mathpzc {V}^{\\alpha \\beta } = \\mathpzc {V}^{(\\alpha \\beta )}$ , as to yield 10 components.",
"Further, we use the (standard) symplectic matrix $\\Omega _{\\alpha \\beta }$ and its inverse $\\Omega ^{\\alpha \\beta } $ to lower and raise indices.",
"This representation is particularly convenient, as the Maurer-Cartan equations read $\\nonumber & d \\mathpzc {V}_{\\alpha \\beta } = \\psi _\\alpha \\psi _\\beta + \\left( \\mathpzc {V} \\Omega \\mathpzc {V} \\right)_{\\alpha \\beta }, \\\\ & d \\psi _\\alpha = \\left( \\mathpzc {V} \\Omega \\psi \\right)_\\alpha ,$ having introduced the (even) vielbeins $\\psi ^\\alpha $ as well and where we have made use of the notation $\\displaystyle \\left( \\mathpzc {V} \\Omega \\mathpzc {V} \\right)_{\\alpha \\beta } = \\mathpzc {V}_{\\alpha \\gamma } \\Omega ^{\\gamma \\delta } \\mathpzc {V}_{\\delta \\beta } $ and $\\displaystyle \\left( \\mathpzc {V} \\Omega \\psi \\right)_\\alpha = \\mathpzc {V}_{\\alpha \\beta } \\Omega ^{\\beta \\gamma } \\psi _\\gamma .$ Let us look for the 3-form explicitely: the most general 3-form reads $\\omega ^{(3|0)} = c_1 \\left( \\mathpzc {V}^{\\alpha _1 \\beta _1} \\Omega _{\\beta _1 \\alpha _2} \\mathpzc {V}^{\\alpha _2 \\beta _2} \\Omega _{\\beta _2 \\alpha _3} \\mathpzc {V}^{\\alpha _3 \\beta _3} \\Omega _{\\beta _3 \\alpha _1} \\right) + c_2 \\mathpzc {V}^{\\alpha \\beta } \\psi _\\alpha \\psi _\\beta ,$ where $c_1$ and $c_2$ are constants coefficient.",
"We shorten the previous expression by $\\omega ^{(3)} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =c1 V(3) + c2 V (2).$ By compatibility with the cohomology of the reduced algebra $ sp(4, R)$ we conclude that $ c1 0$, and in particular, we put $ c1 = 1$.",
"Imposing the closure condition $ d(3) = 0$ we fix the second coefficient:\\begin{equation}0 = d \\omega ^{(3)} = 3 \\left[ \\left( \\psi ^{\\alpha _1} \\psi ^{\\beta _1} + \\left( \\mathpzc {V} \\Omega \\mathpzc {V} \\right)^{\\alpha _1 \\beta _1} \\right) \\Omega _{\\beta _1 \\alpha _2} V^{\\alpha _2 \\beta _2} \\Omega _{\\beta _2 \\alpha _3} \\mathpzc {V}^{\\alpha _3 \\beta _3} \\Omega _{\\beta _3 \\alpha _1} \\right] + + c_2 \\left[ \\psi ^{\\alpha } \\psi ^\\beta \\psi _\\alpha \\psi _\\beta - 2 \\mathpzc {V}^{\\alpha \\beta } \\mathpzc {V}_{\\alpha \\gamma } \\Omega ^{\\gamma \\delta } \\psi _\\delta \\psi _\\beta \\right].\\end{equation}Let us look at the terms in this expression: the second term, namely the one proportional to $ V4$ is zero by trace identity, indeed we can write $ V3 V = - V V3$, but on the other hand, by cyclicity we have $ V3 V = VV3$.",
"The third term, namely the one proportional to $ 4$, is zero since $ = = 0$, being the $$^{\\prime }s even and $$ antisymmetric.",
"This allows us to fix $ c2 = 3/2$ as to get that the first cancel the last term and obtaining a closed form.",
"Further, in order to show that $ (3)$ is not exact, we have to consider the most general even $ 2$-form and show that its Chevalley-Eilenberg differential cannot generate $ (3)$.",
"However a crucial observation simplifies the job: we cannot construct a (non-zero) $ 2$-form which is a \\emph {singlet}, \\emph {i.e.",
"}\\ having all of the indices contracted (the only case would be $ V V + $ which is equal to zero, as shown above).",
"Hence we have (after multiplying by an overall factor)\\begin{equation}H^{3}_{CE} \\left( \\mathfrak {osp} \\left( 1|4 \\right) \\right) = \\left\\lbrace \\frac{1}{3} \\left( V^{\\alpha _1 \\beta _1} \\Omega _{\\beta _1 \\alpha _2} V^{\\alpha _2 \\beta _2} \\Omega _{\\beta _2 \\alpha _3} V^{\\alpha _3 \\beta _3} \\Omega _{\\beta _3 \\alpha _1} \\right) + \\frac{1}{2} V^{\\alpha \\beta } \\psi _\\alpha \\psi _\\beta \\right\\rbrace \\ .\\end{equation}With completely analogous arguments we can construct the most general odd $ 7$-form as\\begin{eqnarray}\\nonumber \\omega ^{(7)} &=& c_1 \\left( \\mathpzc {V}^{\\alpha _1 \\beta _1} \\Omega _{\\beta _1 \\alpha _2} \\mathpzc {V}^{\\alpha _2 \\beta _2} \\Omega _{\\beta _2 \\alpha _3} \\mathpzc {V}^{\\alpha _3 \\beta _3} \\Omega _{\\beta _3 \\alpha _4} V^{\\alpha _4 \\beta _4} \\Omega _{\\beta _4 \\alpha _5} \\mathpzc {V}^{\\alpha _5 \\beta _5} \\Omega _{\\beta _5 \\alpha _6} \\mathpzc {V}^{\\alpha _6 \\beta _6} \\Omega _{\\beta _6 \\alpha _7} V^{\\alpha _7 \\beta _7} \\Omega _{\\beta _7 \\alpha _1} \\right) + \\\\\\nonumber &+& c_2 \\left( V^{\\alpha _1 \\beta _1} \\Omega _{\\beta _1 \\alpha _2} \\mathpzc {V}^{\\alpha _2 \\beta _2} \\Omega _{\\beta _2 \\alpha _3} \\mathpzc {V}^{\\alpha _3 \\beta _3} \\Omega _{\\beta _3 \\alpha _4} \\mathpzc {V}^{\\alpha _4 \\beta _4} \\Omega _{\\beta _4 \\alpha _5} \\mathpzc {V}^{\\alpha _5 \\beta _5} \\right) \\psi _{\\alpha _1} \\psi _{\\alpha _5} + \\\\\\nonumber &+& c_3 \\left( \\mathpzc {V}s^{\\alpha _1 \\beta _1} \\Omega _{\\beta _1 \\alpha _2} \\mathpzc {V}^{\\alpha _2 \\beta _2} \\Omega _{\\beta _2 \\alpha _3} \\mathpzc {V}^{\\alpha _3 \\beta _3} \\Omega _{\\beta _3 \\alpha _1} \\right) \\left( \\mathpzc {V}^{\\alpha _1 \\mu } \\Omega _{\\mu \\nu } \\mathpzc {V}^{\\nu \\alpha _2} \\right) \\psi _{\\alpha _1} \\psi _{\\alpha _2} + \\\\ &+& c_4 \\left( \\mathpzc {V}^{\\alpha _1 \\beta } \\Omega _{\\beta \\gamma } \\mathpzc {V}^{\\gamma \\alpha _2} \\right) \\mathpzc {V}^{\\alpha _3 \\alpha _4} \\psi _{\\alpha _1} \\psi _{\\alpha _2} \\psi _{\\alpha _3} \\psi _{\\alpha _4}.\\end{eqnarray}We note that we do not have a term of the form $ V 6$ since it would be trivially 0, as can be checked.",
"We can write $ (7)$ in a more compact way as\\begin{equation}\\omega ^{(7)} = c_1 \\mathpzc {V}^7 + c_ 2\\left( \\mathpzc {V}^5 \\right)^{\\alpha \\beta } \\psi _\\alpha \\psi _\\beta + c_3 \\mathpzc {V}^3 \\left( \\mathpzc {V}^2 \\right)^{\\alpha \\beta } \\psi _\\alpha \\psi _\\beta + c_4 \\left( \\mathpzc {V}^2 \\right)^{\\alpha \\beta } \\mathpzc {V}^{\\gamma \\delta } \\psi _\\alpha \\psi _\\beta \\psi _\\gamma \\psi _\\delta \\ ,\\end{equation}where the contractions are omitted.",
"Again by compatibility with the reduced algebra cohomology, we need to have $ c10$.",
"The remaining coefficients $ c2, c3, c4$ can be easily fixed imposing $ d (7) = 0$: again, as above, the resulting form will not be exact since it is not possible to have a non-trivial singlet represented by an even $ 6$-form.",
"\\\\Finally the top representative in the cohomology, the form $ (10)$ is simply given given by the multiplication\\begin{equation}\\omega ^{(10)} = \\omega ^{(3)} \\wedge \\omega ^{(7)},\\end{equation}exploiting the ring structure of the cohomology.",
"Notice that $ (10)$ is non-zero since, for example, the term of the form $ V3 V7$ is non-vanishing, and since either $ (3)$ or $ (7)$ are closed and non-exact it follows that $ (10)$ is closed and non-exact as well.\\\\$ We now pass to study the equivariant cohomology of the coset superspace $\\mathfrak {ads}_{4|4} = \\mathfrak {osp}(1|4) / \\mathfrak {so}(1,3)$ .",
"In order to do so, we have to “split” the Maurer-Cartan forms $\\mathpzc {V}^{\\alpha \\beta }$ coming from the $\\mathfrak {sp}(4, \\mathbb {R}) \\subset \\mathfrak {osp}(1|4)$ into the coset Maurer-Cartan forms (vielbeins) and those coming from $\\mathfrak {so}(1,3)$ (connections).",
"Again, making use of the gamma matrices, i.e.",
"of the spin structure, we can decompose the vielbeins as $\\mathpzc {V}_{(\\alpha \\beta )} = \\gamma ^a_{(\\alpha \\beta )} \\mathpzc {V}_a + \\gamma ^{[a b]}_{(\\alpha \\beta )} \\mathpzc {V}_{[ab]},$ for $a = 1,\\ldots ,4$ and $\\alpha =1,\\ldots ,4$ , where the $\\mathpzc {V}^a$ are the four vierbein of the coset space that lifts to $AdS_4$ and $\\mathpzc {V}_{[ab]}$ are the six vielbeins of $\\mathfrak {so}(1,3)$ .",
"The Poincaré polynomial can be computed using the result by [26] - notice that both the algebras involved have the same rank, actually 2 - and it reads $\\mathpzc {P}_{\\mathfrak {ads}_{4|4}} [t] = \\frac{\\left( 1 - t^4 \\right) \\left( 1 - t^8 \\right)}{\\left( 1 - t^4 \\right)^2} = 1 + t^4.$ We therefore expect a single equivariant cohomology class at degree 4, besides the constants.",
"In particular, we expect this to be related to the “volume form” $\\omega ^{(4)}_{\\mathfrak {ads}_4} $ coming from the $AdS_4$ space.",
"Using the above decomposition (REF ) and the previously obtained Maurer-Cartan equations (REF ) one gets the following Maurer-Cartan equations $&& \\mathcal {D} \\mathpzc {V}_a = \\psi \\gamma _a \\psi \\,, ~~~~\\nonumber \\\\&& \\mathcal {D} \\mathpzc {\\psi }_\\alpha = \\mathpzc {V}_a \\gamma ^a \\psi \\,, ~~~~\\nonumber \\\\&& R_{ab} \\equiv d \\mathpzc {V}_{[ab]} + ( \\mathpzc {V} \\wedge \\mathpzc {V})_{[ab]} = \\psi \\gamma _{[ab]} \\psi $ where the covariant derivative $\\mathcal {D}$ is with respect to the connection $ \\mathpzc {V}_{[ab]}$ of the subgroup $\\mathfrak {so}(1,3)$ .",
"Working as above, we have that the most general even 4-singlet reads $\\omega ^{(4|0)} = c_1 \\epsilon _{abcd} \\mathpzc {V}^a \\mathpzc {V}^b \\mathpzc {V}^c \\mathpzc {V}^d + c_2 \\mathpzc {V}^a \\mathpzc {V}^b \\left( \\psi \\gamma _{ab} \\psi \\right).$ Notice that there cannot be terms of the form $ \\psi ^4 = \\left( \\psi \\gamma ^{ab} \\psi \\right) \\left( \\psi \\gamma _{ab} \\psi \\right) $ , since they vanish because of the Fierz identities.",
"As above, we have that $c_1\\ne 0$ by compatibility with the cohomology of the reduced algebra $\\omega ^{(4)}_{\\mathfrak {ads}_4} = \\epsilon _{abcd} V^a V^b V^c V^d$ .",
"Hence we can fix $c_1=1$ without loss of generality.",
"The coefficient $c_2$ is fixed by imposing that $\\mathcal {D} \\omega ^{(4)} = 0$ : $0 = \\mathcal {D} \\omega ^{(4)} = 4 \\epsilon _{abcd} \\left( \\psi \\gamma ^a \\psi \\right) \\mathpzc {V}^b \\mathpzc {V}^c \\mathpzc {V}^d + 2 c_2 \\left[ \\psi \\gamma ^a \\psi \\mathpzc {V}^b \\left( \\psi \\gamma _{ab} \\psi \\right) + \\mathpzc {V}^a \\mathpzc {V}^b \\left( \\left( \\mathpzc {V}^c \\gamma _c \\psi \\right) \\gamma _{ab} \\psi \\right) \\right] \\ .$ The second term in the sum vanishes because of Fierz identities, while the last term, after using $\\gamma $ matrices properties, cancels the first one upon fixing $\\displaystyle c_2=-2$ .",
"Finally, we can conclude that $\\omega ^{(4)}$ is not exact, since, once again, it is not possible to write an odd 3-singlet that generates the term $\\mathpzc {V}^4$ .",
"Hence we have $H^{4}_{\\mathpzc {EQ}} \\left(\\mathfrak {ads}_{4|4} \\right) = \\mathbb {R}\\cdot \\left\\lbrace \\epsilon _{abcd} \\mathpzc {V}^a \\mathpzc {V}^b \\mathpzc {V}^c \\mathpzc {V}^d -2 \\mathpzc {V}^a \\mathpzc {V}^b \\left( \\psi \\gamma _{ab} \\psi \\right) \\right\\rbrace .$ All in all we have: $H^{p}_{\\mathpzc {EQ}} \\left(\\mathfrak {ads}_{4|4} \\right) = \\left\\lbrace \\begin{array}{lll}\\mathbb {R} & \\quad & p = 0, 4 \\\\0 & \\quad & \\mbox{else}.\\end{array}\\right.$ We conclude with the integral form Chevalley-Eilenberg cohomology.",
"As discussed in the previous section, by the isomorphism, we have two cohomology classes at picture four, the maximal picture degree.",
"They have the explicit expressions $H^{(0|4)} \\left(\\mathfrak {ads}_{4|4} \\right) &=& \\mathbb {R} \\cdot \\left\\lbrace 2 \\mathpzc {V}^a \\mathpzc {V}^b \\iota _{\\pi Q} \\gamma _{ab} \\iota _{\\pi Q} \\delta ^4(\\psi ) + \\delta ^4(\\psi )\\right\\rbrace \\nonumber \\\\H^{(4|4)} \\left(\\mathfrak {ads}_{4|4} \\right) &=& \\mathbb {R} \\cdot \\left\\lbrace \\epsilon _{abcd} \\mathpzc {V}^a \\mathpzc {V}^b \\mathpzc {V}^c \\mathpzc {V}^d \\delta ^4(\\psi )\\right\\rbrace $ matching again the cohomology for superforms."
],
[
"Conclusions and Outlook",
"The present work spawns from the observation that since Lie superalgebra cohomology is nothing but a straightforward generalization of ordinary Lie algebra cohomology, it is not capable to account for objects different than differential forms on the corresponding Lie supergroup.",
"On the other hard, it is well-know that in order to make a meaningful connection with integration theory, when working on supermanifolds, differential forms has to be supplemented by integral forms, whose geometry is not at all captured by Chevalley-Eilenberg cohomology.",
"To this end, after reviewing Chevalley-Eilenberg cohomology for ordinary Lie algebras and Lie superalgebras and its relations to forms on the corresponding Lie groups or Lie supergroups, we extend the notion of Chevalley-Eilenberg cochains as to include also integral forms and we define a corresponding cohomology.",
"We thus show a duality between the ordinary Chevalley-Eilenberg cohomology for a certain Lie superalgebra - which looks at forms on the corresponding Lie supergroup - and this newly defined (Chevalley-Eilenberg) cohomology accounting for integral forms instead.",
"We observe that, most notably - and differently from de Rham cohomology -, this cohomology always feature the true analog of a top-form, a Berezinian form appearing in the integral form complex.",
"Nonetheless, beside general results, a great deal of focus in this paper is on explicit direct computations: in particular, we provide explicit expressions for cocycles of Lie superalgebras of physical interest, namely supertranslations of flat superspaces and classical Lie superalgebras, up to dimension 4, in terms of their Maurer-Cartan forms.",
"The second part of the paper is devoted to equivariant Chevalley-Eilenberg cohomology, which is related to the (super)symmetries of coset supermanifolds, which provides very important backgrounds for supergravity and superstring theories.",
"Again, several example up to dimension 4 are studied and explicit expressions for their cocycles are provided, culminating with the case of super anti-de Sitter space $AdS_{4|4}$ .",
"Here, a mixture of techniques have been exploited, spanning from Poincaré polynomials computations for equal rank pairs to brute force computations.",
"We remark that our analysis have uncovered new cocycles spawning from fermionic generators - both in ordinary and equivariant Chevalley-Eilenberg cohomology - and several characteristic examples of infinite dimensional cohomology.",
"In hoping that the present results might come useful to understand the geometry of supergravity and string backgrounds and the mathematics behind it, we stress though, that this research scenario looking at relating Chevalley-Eilenberg cohomology and the extended geometry of forms on supermanifolds is far from being exhausted.",
"Indeed, just as an example, in the present paper we have only hinted at pseudoforms, which nonetheless plays an important role both in the integration theory on superspaces and in its applications: it is legit to ask it they can be fitted in the picture we have presented and which role they play.",
"We will address this problem in a forthcoming paper [13], arguing that pseudoforms are indeed crucial to understand the general structure of the cohomology."
],
[
"Acknowledgements",
"This work has been partially supported by Università del Piemonte Orientale research funds.",
"We thank L. Castellani and P. Aschieri for many useful discussions."
],
[
"The Unitary Lie Superalgebra $\\mathfrak {u} (n|m)$",
"Following [14], in order to introduce the Lie superalgebra $\\mathfrak {u} (n|m)$ one can start with the ordinary Hermitian vector space $(\\mathbb {C}^{n+m}, \\langle \\cdot , \\cdot \\rangle _{\\mathbb {C}^{n+m}})$ , where $\\langle \\cdot , \\cdot \\rangle _{\\mathbb {C}^{n|m}} $ is the standard Hermitian product: promoting $\\mathbb {C}^{n+m}$ to a vector superspace $\\mathbb {C}^{n|m}$ using the $\\mathbb {Z}_2$ -gradation, one then defines the super Hermitian forms on $\\mathbb {C}^{n|m}$ as $\\langle u , v \\rangle _{\\mathbb {C}^{n|m}} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =(-1)|u| |v| u, vCn+m,$ where $ u$ and $ v$ are $ Z2$-homogeneous vectors in $ Cn|m$.",
"Notice that $ u, vCn|m = (-1)|u| |v| v, u Cn|m$, so that this bilinear form is indeed Hermitian in the usual sense.",
"Using this, the \\emph {superadjoint} of an endomorphism $ A End (Cn|m)$ is naturally defined as $ A u , v Cn|m = (-1)|A||u|u, Av Cn|m$ and it is easy to see that $ T= i|T| T$, where the map $ T T$ does not involve the supertransposition, but just the ordinary transposition, \\emph {i.e.}",
"$ T$ is the usual adjoint with respect to standard Hermitian form on $ Cn+m$.",
"These definitions leads immediately to $ [A, B]= - [A, B]$, which spell out the relations between superadjoint and the supercommutator, which is what is needed in order to define a unitary representation of a Lie superalgebra: in particular if $ : g End (Cn|m)$ is a representation of $ g$, we will say that it is a \\emph {unitary} representation if $ (A)= - (A)$ for $ A g.$ For the case of supermatrices $ X gl (n|m, C)$, this leads to the definition\\begin{eqnarray}\\mathfrak {u} (n|m) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=\\left\\lbrace X \\in \\mathfrak {gl} (n|m, \\mathbb {C}): X^\\ast = -X \\right\\rbrace .$ Realizing the above conditions in terms of matrices $\\mathbb {C}^{n|m} \\times \\mathbb {C}^{n|m}$ , one finds $X = \\left( \\begin{array}{c|c}A & \\Theta \\\\\\hline - i \\Theta ^\\dagger & B\\end{array}\\right),$ for $A \\in \\mathfrak {u}(p)$ and $B \\in \\mathfrak {u} (q)$ , so that $A^\\dagger = -A$ and $B^\\dagger = -B$ , and $\\Theta \\in Hom (\\Pi \\mathbb {C}^{m} , \\mathbb {C}^{n}),$ i.e.",
"an odd matrix.",
"This easily yield $\\dim _{\\mathbb {R}} \\mathfrak {u} (n|m) = n^2 + m^2 | 2nm.$"
],
[
"The Orthosymplectic Lie Superalgebra $\\mathfrak {osp}(n|2m)$",
"Working in the most general setting following again [14], given a number field $k$ of characteristic 0 the natural representation of the general linear Lie superalgebra $\\mathfrak {gl} (n|m, k)$ on the vector superspace $k^{n|m}$ can be extended to a representation acting on the tensor algebra $\\mathpzc {T}ens (k^{n|m}) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =n0 (kn|m)n$, upon using the graded Leibniz rule, \\emph {i.e.",
"}\\ for $ A gl(n|m, k)$ and $ vi kn|m$ homogeneous vectors one has{\\begin{@align}{1}{-1}A \\cdot (v_1 \\otimes \\ldots \\otimes v_n \\otimes \\ldots ) = &(A \\cdot v_1) \\otimes v_2 \\otimes \\ldots \\otimes v_n \\otimes \\ldots + \\nonumber \\\\& + (-1)^{|A||v_1|} v_1 \\otimes (A\\cdot v_2) \\otimes v_3 \\otimes \\ldots \\otimes v_n \\otimes \\ldots + \\ldots + \\nonumber \\\\& + (-1)^{\\sum _{i=1}^{n-1} |A||v_i|} v_1 \\otimes \\ldots \\otimes v_{n-1} \\otimes (A\\cdot v_n) \\otimes v_{n+1} \\otimes \\ldots .\\end{@align}}Choosing $ k=R$, it is possible to introduce a bilinear form on $ G: Rn|2m Rn|2m R$, such that for the standard basis $ SpanR {ei } = Rn|2m$, one has\\begin{eqnarray}G (e_i \\otimes e_j ) = {g}_{ij} \\quad \\mbox{with} \\quad {g}_{ij} = \\left( \\begin{array}{c|cc}{1}_{n\\times n} & {0}_{m \\times n} & {0}_{m \\times n} \\\\\\hline {0}_{m \\times n} & {0}_{m \\times m} & {1}_{m \\times m} \\\\{0}_{m\\times n} & -{1}_{m \\times m} & {0}_{m \\times m}\\end{array}\\right).\\end{eqnarray}For short, we define\\begin{eqnarray}G \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}\\end{eqnarray}=\\left(\\begin{array}{c|c}1_n & \\\\\\hline & J_{2m}\\end{array}\\right) \\quad \\mbox{with} \\quad J_{2m} \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =( cc 0m 1m -1m 0m ).",
"Here $J$ is just the standard symplectic matrix, which has the property that $J = - J^t.$ The orthosymplectic Lie superalgebra can therefore be defined as $\\mathfrak {osp}(n|2m) \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ ={ X gl (n|2m, R) : G (X ( v1 v2)) = 0, v1, v2 Rn|2m }.",
"One sees that, unraveling the above definition, one gets the following condition on $X \\in \\mathfrak {gl}(n|2m, \\mathbb {R}):$ $X^tG + G X = 0.$ In turn, writing $X$ in block-form $X \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =( c|c A B ) for $A \\in Hom_{\\mathbb {R}} (\\mathbb {R}^{n|0}, \\mathbb {R}^{n|0})$ , $B \\in Hom_\\mathbb {R} (\\mathbb {R}^{0|2m}, \\mathbb {R}^{0|2m})$ even matrices and $\\Phi \\in Hom_{\\mathbb {R}} (\\mathbb {R}^{0|2m} , \\mathbb {R}^{n|0})$ and $\\Psi \\in Hom_{\\mathbb {R}} (\\mathbb {R}^{n|0} , \\mathbb {R}^{0|2m})$ odd matrices one finds the conditions $ A^{t} = - A , \\qquad B^t = JBJ, \\qquad \\Psi = J \\Phi ^t,$ having used that $-J^t = J$ in the relation for $B$ , so that the generic element of the superalgebra can be written as $\\mathfrak {osp} (n|2m) \\ni X \\mathrel {\\hbox{\\scriptsize .",
"}\\hbox{\\scriptsize .",
"}}$ =( c|c A J t B ), with $A \\in \\mathfrak {so} (n, \\mathbb {R})$ and $B \\in \\mathfrak {sp} (2n, \\mathbb {R})$ , which explains the denomination orthosymplectic.",
"Also, the above conditions makes it easy to count the dimensions of this Lie superalgebra, namely one finds $\\dim _\\mathbb {R} \\mathfrak {osp} (n|2m) = \\frac{1}{2} n(n-1) + m (2m + 1) | 2mn.$"
]
] | 2012.05246 |
[
[
"Topological Protection of Coherence in Noisy Open Quantum Systems"
],
[
"Abstract We consider topological protection mechanisms in dissipative quantum systems in the presence of quenched disorder, with the intent to prolong coherence times of qubits.",
"The physical setting is a network of qubits and dissipative cavities whose coupling parameters are tunable, such that topological edge states can be stabilized.",
"The evolution of a fiducial qubit is entirely determined by a non-Hermitian Hamiltonian which thus emerges from a bona-fide physical process.",
"It is shown how even in the presence of disorder winding numbers can be defined and evaluated in real space, as long as certain symmetries are preserved.",
"Hence we can construct the topological phase diagrams of noisy open quantum models, such as the non-Hermitian disordered Su-Schrieffer- Heeger dimer model and a trimer model that includes longer-range couplings.",
"In the presence of competing disorder parameters, interesting re-entrance phenomena of topologically non-trivial sectors are observed.",
"This means that in certain parameter regions, increasing disorder drastically increases the coherence time of the fiducial qubit."
],
[
"Introduction",
"Due to their characteristic protection against environmental noise, topological phases of matter [1], [2], [3] are considered to be promising candidates for the realization of noise-resilient quantum computers [4], [5], [6], [7], [8], [9], [10], [11].",
"Furthermore, it was shown in [12] that the presence of topological edge states can preserve quantum mechanical features, e.g.",
"coherence of a fiducial qubit, in the presence of dissipation (see also [13], [14], [15] for other works in a similar spirit).",
"In that work, dissipative one-dimensional (1D) quantum optical qubit-cavity architectures were analyzed, where effective non-Hermitian Hamiltonians of the form of a tight-binding chain with diagonal complex entries were derived.",
"The time evolution of the boundary-qubit coherence, driven by such non-Hermitian Hamiltonians, was extensively studied for choices of hopping parameters that admit symmetry-protected topological states localized at the edges of the system.",
"It was found that (quasi-)dark modes, i.e., boundary states with exponentially small (in system size) imaginary parts, protect the edge qubits from decoherence effects via photon leakage through cavities.",
"Moreover, disordered as well as non-Hermitian generalizations of 1D topological insulators, such as the Su-Schrieffer-Heeger (SSH) model [16], [17], have been studied theoretically [18], [19], where the real space winding number was analyzed for different disorder strengths on the hopping parameters.",
"Here, we build on the work presented in [12], addressing the role of quenched disorder in the qubit-cavity arrays.",
"We fully characterize the disordered, non-Hermitian system's topology by computing the winding number in real space in the parameter space spanned by the coupling amplitude and the disorder strength.",
"From this characterization, accurate predictions for the behavior of a fiducial qubit's coherence can be made for long times, therefore expanding the discussion of the quantum optical systems to a broader physical context, considering both quenched disorder and dissipation.",
"The remainder of this paper is organized as follows.",
"In Sec.",
", we derive the effective non-Hermitian Hamiltonian describing qubit-cavity arrays using the Lindblad formalism.",
"In Sec.",
", the topological characterization of dissipative, disordered systems is illustrated, which is then applied to non-Hermitian dimer and trimer models in Sec .",
"Special focus is put on the coherence of the qubit located at the boundary, whose faith can be accurately predicted from the phase diagrams.",
"We then briefly discuss possible applications in quantum computation via dark-state braiding in Sec.",
"and conclude in Sec.",
"."
],
[
"The Setup",
"We consider a network consisting of qubits coupled to dissipative cavities in a Jaynes-Cumming fashion.",
"Specifically, we study networks of $M$ qubits and $K$ dissipative cavities, as illustrated in Fig.",
"REF for $M=4$ and $K=5$ .",
"The Hamiltonian of the system has the following form $H & = & \\sum _{l,m=1}^{K}J_{l,m}(a_{l}^{\\dagger }a_{m}+\\text{h.c.})\\nonumber \\\\& + & \\sum _{i=1}^{M}\\sum _{l=1}^{K}\\tilde{J}_{l,i}(a_{l}^{\\dagger }\\sigma _{i}^{-}+\\text{h.c.}),$ where $a_{l}^{\\dagger }$ and $a_{l}$ are the bosonic creation and annihilation operators for cavity mode $l$ , and $\\sigma _{i}^{\\pm }$ are the ladder operators for qubit $i$ .",
"We consider a Lindblad master equation $\\dot{\\rho }=\\mathcal {L}[\\rho ]$ , where $\\mathcal {L}=\\mathcal {K}+\\mathcal {D}$ , the coherent part is $\\mathcal {K}=-i\\left[H,\\bullet \\right]$ , whereas the dissipative part is $\\mathcal {D}[\\rho ]=\\sum _{l=1}^{K}\\Gamma _{l}\\Big [2a_{l}\\rho a_{l}^{\\dagger }-\\lbrace a_{l}^{\\dagger }a_{l},\\rho \\rbrace \\Big ].$ Such Lindbladian description is accurate at sufficiently low temperature in particular in circuit QED experiments.",
"Figure: Example of a network of qubits (filled circles) interacting with lossycavities (hollow circles).",
"Wavy lines indicate coherent hopping matrixelements J ˜ i,j \\tilde{J}_{i,j}, connecting cavity ii with qubit jj,and J i,j J_{i,j}, connecting cavity ii with cavity jj.",
"Arrows indicateincoherent decay Γ i \\Gamma _{i} in cavity ii.The Lindbladian can also be written as $\\mathcal {L}=\\mathcal {K}^{\\prime }+\\mathcal {D}^{\\prime }$ , where $\\mathcal {K}^{\\prime }(\\rho )=-i(H^{\\prime }\\rho -\\rho {H^{\\prime }}^{\\dagger })$ defines the non-Hermitian Hamiltonian $H^{\\prime }$ : $H^{\\prime }=H-i\\sum _{l=1}^{K}\\Gamma _{l}a_{l}^{\\dagger }a_{l},$ and $\\mathcal {D}^{\\prime }(\\rho )=\\sum _{l=1}^{K}2\\Gamma _{l}a_{l}\\rho a_{l}^{\\dagger }.$ Consider the Fock space of the system $\\mathcal {F}=\\oplus _{n=0}^{\\infty }\\mathcal {H}_{n}$ where $\\mathcal {H}_{n}$ is the Hilbert space of $n$ particles (at this level the distinction between spins and bosons is unimportant).",
"In the space of operators on $\\mathcal {F}$ we define the Hilbert-Schmidt scalar product $\\langle \\!\\langle x|y\\rangle \\!\\rangle =\\operatorname{Tr}(x^{\\dagger }y)$ .",
"Using the isomorphism $\\mathcal {B}_{HS}(\\mathcal {F})\\simeq \\mathcal {F}\\otimes \\mathcal {F}^{*}$ ($\\mathcal {B}_{HS}(\\mathcal {F})$ is the space of bounded Hilbert-Schmidt operators on $\\mathcal {F}$ ), the space of operators $\\mathcal {B}_{HS}(\\mathcal {F})$ can be identified with $\\mathcal {B}_{HS}(\\mathcal {F})\\simeq \\bigoplus _{i,j=0}^{\\infty }\\mathcal {H}_{i}\\otimes \\mathcal {H}_{j}^{*}.$ In simpler terms, $\\mathcal {B}_{HS}(\\mathcal {F})$ has a block structure with two labels $(i,j)$ each label being a particle number.",
"The non-Hermitian Hamiltonian $H^{\\prime }$ preserves the number of particles and correspondigly $\\mathcal {K}^{\\prime }$ is block-diagonal in $(i,j)$ .",
"Instead, $\\mathcal {D}^{\\prime }$ connects the sector $(i,j)$ with the sector $(i-1,j-1)$ , i.e.",
"it decreases the number of particles by one.",
"In this paper we will be mostly interested in the coherence of a fiducial qubit, that, without loss of generality we place at site 1.",
"In the standard basis, the coherence of a qubit in state $\\rho $ , can be defined as $\\mathcal {C}=2\\vert \\rho _{\\downarrow ,\\uparrow }\\vert $ [20].",
"We initialize the system such that all cavities are empty and qubits are in the lowest state ($|\\downarrow \\rangle $ ), while on the fiducial qubit the state is $|\\psi \\rangle =\\alpha |\\downarrow \\rangle +\\beta |\\uparrow \\rangle $ .",
"We further fix $|\\alpha \\beta |=1/2$ which means that at the beginning the coherence assumes its maximal value one.",
"We are interested in the evolution of the coherence as a function of time.",
"Let $|0\\rangle $ be the vacuum state with no excitation on any qubit or cavity, while $|j\\rangle $ denotes a single excitation on the $j$ th site, describing either an excited qubit or a cavity hosting a photon.",
"We use the notation $|j\\rangle \\!\\rangle \\leftrightarrow |0\\rangle \\langle j|$ .",
"It can be shown [12] that the evolution of the coherence at later time is given by $\\mathcal {C}(t)=2|\\rho _{\\downarrow ,\\uparrow }(t)| & = & 2|\\langle 0|\\rho (t)|1\\rangle |=2|\\operatorname{Tr}|1\\rangle \\langle 0|\\rho (t)|\\nonumber \\\\& = & 2|\\langle \\!\\langle 1|\\rho (t)\\rangle \\!\\rangle |=2|\\langle \\!\\langle 1|e^{t\\mathcal {L}}|\\rho (0)\\rangle \\!\\rangle |.$ Because of the block structure of the Lindbladian, $\\langle \\!\\langle 1|e^{t\\mathcal {L}}|\\rho (0)\\rangle \\!\\rangle =\\langle \\!\\langle 1|e^{t\\tilde{\\mathcal {L}}}|\\tilde{\\rho }(0)\\rangle \\!\\rangle $ , where $\\tilde{X}$ is the operator $X$ restricted to the linear space $\\mathcal {V}_{0,1}=\\mathrm {Span}(|0\\rangle \\langle j|),j=1,2,\\ldots ,N$ .",
"In particular, $\\tilde{\\rho }(0)=\\overline{\\alpha }\\beta |0\\rangle \\langle 1|=(1/2)|0\\rangle \\langle 1|$ .",
"For what regard the restriction of the Lindbladian we have $\\tilde{\\mathcal {L}}=\\tilde{\\mathcal {K}^{\\prime }}$ .",
"Moreover, in $\\mathcal {V}_{0,1}$ , ${\\mathcal {K}^{\\prime }}_{l,m} & =\\operatorname{Tr}\\left(|l\\rangle \\langle 0|{\\mathcal {K}^{\\prime }}(|0\\rangle \\langle m|)\\right)\\\\& =i\\langle m|{H^{\\prime }}^{\\dagger }|l\\rangle \\\\& =i\\overline{\\langle l|H^{\\prime }|m\\rangle }.$ In other words, the evolution of the coherence of the fiducial qubit is entirely determined by the non-Hermitian Hamiltonian $-\\overline{H^{\\prime }}$ in the one-particle sector.",
"Calling $\\mathsf {H}:=-\\left.\\overline{H^{\\prime }}\\right|_{\\mathrm {one\\ particle}}$ we have finally $\\mathcal {C}(t)=\\left|\\langle \\!\\langle 1|e^{-it\\mathsf {H}}|1\\rangle \\!\\rangle \\right|.$ We would like to stress that, in this setting, a non-Hermitian Hamiltonian emerges from a genuine bona-fide quantum evolution whereas in most current proposals non-Hermitian Hamiltonians are simulated in classical dissipative wave-guides via the analogy between Helmoltz and Schrödinger equation (see e.g. [21]).",
"In order to prolong the coherence Eq.",
"(REF ), one seeks a (non-Hermitian) Hamiltonian $\\mathsf {H}$ that admits i) long-lived states, i.e.",
"eigenstates of $\\mathsf {H}$ with small (negative) imaginary part; and ii) that also have large amplitude on the site $|1\\rangle \\!\\rangle $ (conventionally placed at the beginning of the chain).",
"Interestingly, both these requirement are satisfied to a large degree in one dimensional topological systems which admit edge states with the required properties in the non-trivial phase.",
"The classification of such dissipative, non-Hermitian, topological chains has been done in [22] and utilized to prolong quantum coherence for the first time in [12].",
"Here we extend the analysis of [12] to disordered systems where translational invariance is broken.",
"The phase diagrams of topological dissipative chains will tell us which parameter regions and models can be used to prolong the quantum coherence of the fiducial qubit."
],
[
"Topological invariant of dissipative systems in real space",
"We now briefly recall the topological classification of non-Hermitian quantum systems provided by Rudner et al.",
"in Ref. [22].",
"For non-Hermitian quantum systems hosting dissipative sites, the topological invariant can be defined as the winding number around the dark-state manifold in the Hamiltonian parameter space.",
"A non-trivial phase in dissipative systems corresponds to long-lived edge modes with infinite or exponential large lifetimes.",
"In previous work [12], the topological classification of non-Hermitian models was formulated within the framework of Bloch theory, which we briefly outline here for comparison with the real-space approach to be introduced.",
"Consider a one-dimensional periodic non-Hermitian chain with $n$ sites per unit cell.",
"In the thermodynamic limit, the Hamiltonian is given by $\\mathsf {H}=\\oint dk/(2\\pi )\\sum _{\\alpha ,\\beta =1}^{n}H_{\\alpha ,\\beta }(k)|k,\\alpha \\rangle \\langle k,\\beta |$ .",
"We shall only focus on the cases with one leaky site per unit cell, as the topological characterization is trivial in all other cases if no additional constraints are imposed [22].",
"The Bloch Hamiltonian of any such system is an $n\\times n$ matrix, which can be written as $H(k)=\\left(\\begin{array}{cc}h(k) & v_{k}\\\\v_{k}^{\\dagger } & \\Delta (k)-i\\Gamma \\end{array}\\right),$ where $h(k)$ is an $(n-1)\\times (n-1)$ Hermitian matrix, $v_{k}$ is a $(n-1)$ -dimensional vector and $\\Delta (k)-i\\Gamma $ is a complex number.",
"The Hamiltonian can be further decomposed in the following manner $H(k)=\\left(\\begin{array}{cc}U(k) & 0\\\\0 & 1\\end{array}\\right)\\left(\\begin{array}{cc}\\tilde{h}(k) & \\tilde{v}_{k}\\\\\\tilde{v}_{k}^{\\dagger } & \\Delta (k)-i\\Gamma \\end{array}\\right)\\left(\\begin{array}{cc}U(k)^{\\dagger } & 0\\\\0 & 1\\end{array}\\right),$ where $U(k)$ is a $(n-1)\\times (n-1)$ unitary matrix whose columns are the eigenvectors of $h(k)$ , and $\\tilde{h}(k)$ is the $(n-1)\\times (n-1)$ diagonal matrix of the corresponding eigenvalues.",
"The phases of the eigenvectors are fixed by making all entries of the $(n-1)$ -dimensional vector $\\tilde{v}_{k}$ real and positive.",
"Any $U(k)$ satisfying the above criteria can be chosen without affecting the final result.",
"Since $U(k)$ is the only component parametrizing the Hamiltonian that can lead to non-trivial topology [22], the winding number of $H(k)$ reduces to the one of $U(k)$ , which is given by $W=\\oint \\frac{dk}{2\\pi i}\\partial _{k}\\ln \\operatorname{det}\\lbrace U(k)\\rbrace .$ We now construct a real-space representation of the winding number that remains well defined when translation invariance is destroyed by e.g.",
"the presence of disorder.",
"Consider a chain with $n$ sites in each cell and $M$ number of unit cells.",
"For what we said previously, we consider only one leaky site per unit cell, which, without loss of generality, we place at the final site of the cell.",
"The one-particle (non-Hermitian) Hamiltonian can be written as $\\mathsf {H}=\\sum _{i,j=1}^{M}\\sum _{\\alpha ,\\beta =1}^{n}H_{\\alpha ,\\beta }^{i,j}|i,\\alpha \\rangle \\langle j,\\beta |$ Generally one thinks of the chain as being made of $M$ cells with $n$ sites each, but one may as well think of $n$ sections with $M$ sites each.",
"In other words, we rearrange Eq.",
"(REF ) according to the following block structure $\\mathsf {H}=\\left(\\begin{array}{ccccc}H_{1,1} & H_{1,2} & H_{1,3} & \\dots & H_{1,n}\\\\H_{2,1} & H_{2,2} & H_{2,3} & \\dots & H_{2,n}\\\\\\vdots & \\vdots & \\vdots & & \\vdots \\\\H_{n,1} & H_{n,2} & H_{n,3} & \\dots & H_{n,n}\\end{array}\\right),$ where each $H_{\\alpha ,\\beta }$ is a $M\\times M$ matrix.",
"The matrices $H_{\\alpha ,\\alpha }$ $\\alpha =1,\\ldots ,(n-1)$ are diagonal with chemical potentials on the diagonal.",
"Since we put the leaky site at position $\\alpha =n$ , the matrix $H_{n,n}=\\epsilon _{n}-i\\Gamma \\leavevmode {\\rm 1\\hspace{-2.66663pt}I}$ , where $\\epsilon _{n}$ is a diagonal matrix of chemical potentials and for simplicity we set the leakage to have value $\\Gamma $ on each site.",
"Recalling the approach used in $k$ -space, we first write the real-space Hamiltonian as $H & =\\left(\\begin{array}{cc}\\Lambda & V\\\\V^{\\dagger } & \\epsilon _{n}-i\\Gamma \\leavevmode {\\rm 1\\hspace{-2.66663pt}I}\\end{array}\\right)\\nonumber \\\\& =\\left(\\begin{array}{cc}U & 0\\\\0 & \\leavevmode {\\rm 1\\hspace{-2.66663pt}I}\\end{array}\\right)\\left(\\begin{array}{cc}\\tilde{\\Lambda } & \\tilde{V}\\\\\\tilde{V}^{\\dagger } & \\epsilon _{n}-i\\Gamma \\leavevmode {\\rm 1\\hspace{-2.66663pt}I}\\end{array}\\right)\\left(\\begin{array}{cc}U^{\\dagger } & 0\\\\0 & \\leavevmode {\\rm 1\\hspace{-2.66663pt}I}\\end{array}\\right).$ $\\Lambda $ is a $(n-1)M\\times (n-1)M$ Hermitian matrix, while $V$ is a $(n-1)M\\times M$ real matrix describing the hopping between decaying and non-decaying sites.",
"$\\tilde{\\Lambda }$ is a $(n-1)L\\times (n-1)L$ diagonal matrix with real eigenvalues of $\\Lambda $ , and $U$ is a $(n-1)L\\times (n-1)L$ unitary matrix that diagonalizes $\\Lambda $ .",
"The degrees of freedom for the choice of $U$ are fixed by making each $L\\times L$ submatrix in $\\tilde{V}$ positive-definite, analogous to the procedure in reciprocal space.",
"With these preparations, the winding number of the unitary matrix $U$ in real space can be evaluated with the prescription of [23] and further elaborations of Refs.",
"[18], [24], [19].",
"In particular, $\\int _{0}^{2\\pi }(dk/2\\pi )\\times \\operatorname{tr}\\lbrace \\rbrace $ and $\\partial _{k}$ become trace per volume and the commutator $-i[X,]$ ($X$ being the position operator), respectively.",
"Note that $X$ is the $M$ -sized cell position operator, i.e.",
"$X=\\operatorname{diag}(1,2,\\ldots ,M,1,2,\\dots ,M,\\ldots ,M-1,M)$ .",
"Thus, Eq.",
"(REF ) in real space can be written as $W=\\frac{1}{L^{\\prime }}\\operatorname{tr}^{\\prime }(U^{\\dagger }[X,U]).$ Here, $\\operatorname{tr}^{\\prime }$ stands for trace with truncation.",
"Specifically, we take the trace over the middle interval of length $M^{\\prime }$ and leave out $\\ell $ sites on each side (total length $M=M^{\\prime }+2\\ell $ ).",
"With Eq.",
"(REF ), we can explore topological phases in presence of dissipation and disorder.",
"Note that in the model that we will consider, the matrix $\\Lambda $ is not noisy.",
"In general, the model supports a non-trivial topological phase as long as a certain (chiral) symmetry is preserved.",
"Disorder on the elements of $\\Lambda $ destroys the symmetry and consequently the system becomes topologically trivial."
],
[
"Disordered non-Hermitian Systems",
"We now apply the real space formalism to investigate topological features in two explicit network geometries, namely the disordered non-Hermitian SSH dimer model and a disordered non-Hermitian trimer model."
],
[
"Disordered non-Hermitian SSH Dimer Model",
"This model describes an open quantum system of coupled qubits and optical cavities which are arranged in an alternating manner, as shown in Fig.",
"REF .",
"In the super-one-particle sector, the corresponding restricted Hamiltonian $\\mathsf {H}$ in the presence of disorder is given by $\\mathsf {H} & = & \\sum _{j=1}^{M}\\epsilon _{A,j}|j,A\\rangle \\langle j,A|+(\\epsilon _{B,j}-i\\Gamma )|j,B\\rangle \\langle j,B|\\nonumber \\\\& + & \\sum _{j=1}^{M}(J_{1,j}|j,B\\rangle \\langle j,A|+\\mathrm {h.c.})\\nonumber \\\\& + & \\sum _{j=1}^{M-1}(J_{2,j}|j+1,A\\rangle \\langle j,B|+\\mathrm {h.c.}).$ Due to the chiral symmetry and the pseudo-anti-hermiticity of the non-dissipative and dissipative model, respectively [12], [25], the topological states are expected to be robust against the chiral symmetry preserving off-diagonal disorder, i.e., noise in the hopping parameters.",
"In contrast, disorder in the on-site potentials breaks the symmetries and is thus expected to quickly diminish topological features.",
"Indeed, diagonal disorder leads to a unit cell as large as the system, thus having more than one dissipative site per unit cell and hence preventing the existence of topological dark states according to the argument in [22].",
"We therefore restrict the randomness to act on the hopping parameters, i.e., $J_{1,j}\\equiv J_{1}+\\mu _{1}\\omega _{1,j}$ and $J_{2,j}\\equiv J_{2}+\\mu _{2}\\omega _{2,j}$ , where $\\omega _{\\alpha ,j}$ are independent random variables with uniform distribution in the range $[-1,+1]$ .",
"Figure: Complex density of states of the restricted Hamiltonian 𝖧\\mathsf {H}.",
"(a)&(b) Topologically trivial regime for the clean and disordered(μ=1\\mu =1) case, respectively.",
"(c)&(d) Topologically non-trivialphase for clean and disordered (μ=1\\mu =1) systems, respectively.",
"Resultsare averaged over 1000 diagonalizations.",
"Here, N=20N=20, Γ=0.5\\Gamma =0.5,J 2 =1J_{2}=1 and J 1 =1.5J_{1}=1.5 (J 1 =0.5J_{1}=0.5) for the topologically trivial(non-trivial) configurations.The effect of off-diagonal disorder on the spectrum of the restricted Hamiltonian $\\mathsf {H}$ is illustrated in Figure REF , where the density of states is plotted in the complex plane.",
"In the topologically trivial regime of the clean system, Fig.",
"REF (a), all eigenvalues have imaginary part $-\\Gamma /2$ .",
"When disorder is introduced, they mainly wash out on axis $\\mathrm {Im}(E)=-\\Gamma /2$ , as seen in Fig.",
"REF (b) .",
"There is, however, a notable non-vanishing density of states emerging in the vicinity of $\\operatorname{Re}(E)=0$ .",
"In the topologically non-trivial regime, Figs.",
"REF (c)&(d), a dark state with corresponding $\\operatorname{Im}(E)=0$ can be found.",
"Its topological protection against off-diagonal disorder manifests itself in its eigenvalue being left almost unchanged when disorder disturbs the system, while the bulk states featuring eigenvalues with imaginary part $-\\Gamma /2$ blur out.",
"The protected dark state corresponds to an edge state having support only on the non-dissipative sites, thus not decaying through the cavities.",
"Another state emerging in the non-trivial phase lives, on the contrary, only on the dissipative sites, with eigenvalue satisfying $\\mathrm {Im}(E)=-\\Gamma $ , as also seen in Fig.",
"REF (c).",
"The mentioned destructive character of on-site potential disorder is discussed in the Appendix, Sec.",
", where the density of states for diagonal disorder is analyzed, see Figs REF (a)-(d).",
"We now turn to the computation of the winding number.",
"In absence of disorder we can go to reciprocal space and realize that the unitary $U(k)$ in Eq.",
"(REF ) is simply given by the phase of $J_{1}+J_{2}e^{-ik}$ .",
"The winding number of the dissipative system is thus the same as the winding number of the closed, Hermitian SSH-chain, resulting in $W=\\Theta (|J_{2}|-|J_{1}|),$ where $\\Theta $ is the Heaviside function ($\\Theta (x)=1$ for $x>0$ and $\\Theta (x)=0$ for $x<0$ ).",
"In order to compute the winding number in real space for the non-Hermitian SSH model, we follow the steps described in Sec. .",
"First, the Hamiltonian is written in the order of sublattices and divided into four blocks, as in Eq.",
"(REF ).",
"In this case, $\\Lambda =H_{1,1}=\\epsilon _{A}\\leavevmode {\\rm 1\\hspace{-2.66663pt}I}$ , and $V=H_{1,2}$ .",
"From Eq.",
"(REF ), we get $U\\tilde{V}=V$ , where $U$ , $V$ and $\\tilde{V}$ are all of dimension $M\\times M$ .",
"To determine the unitary matrix $U$ , we need to fulfill two requirements: i) the columns of $U$ need to be eigenvectors of $\\Lambda $ and ii) $\\tilde{V}$ needs to be positive definite.",
"Since $\\Lambda \\propto \\leavevmode {\\rm 1\\hspace{-2.66663pt}I}$ , the first requirement is satisfied for any vector.",
"In order to satisfy the second requirement, we recall that the polar decomposition of an invertible square matrix $V$ is a factorization of the form $V=U\\tilde{V}$ , where $U$ is a unitary matrix and $\\tilde{V}$ is a positive-definite Hermitian matrix.",
"$\\tilde{V}$ is uniquely determined by $\\tilde{V}=(V^{\\dagger }V)^{1/2}$ .",
"As a result, $U$ can be written as $U=V(V^{\\dagger }V)^{-1/2}.$ Finally, the winding number $W$ can be calculated via Eq.",
"(REF ).",
"From here on, we set the on-site potentials to be zero, i.e., $\\epsilon _{A}=\\epsilon _{B}=0$ .",
"Figure: Phase diagram of the disordered, dissipative SSH model for N=1000N=1000, Γ=0.5\\Gamma =0.5 and J 2 =1J_2=1.",
"Results are averaged over 40 random realizations.",
"(a) isotropic disorder (μ 1 =μ 2 =μ\\mu _{1}=\\mu _{2}=\\mu ), (b) anisotropic disorder(μ 1 =2μ 2 =μ\\mu _{1}=2\\mu _{2}=\\mu ).",
"White lines indicate points of diverginglocalization length in the thermodynamic limit.Fig.",
"REF presents the phase diagrams of the disordered dissipative SSH model as a function of coupling and disorder strength.",
"In Fig.",
"REF (a), the disorder is isotropic, i.e.",
"$\\mu _{1}=\\mu _{2}=\\mu $ and $J_{2}=1$ , while in Fig.",
"REF (b), we consider anisotropic disorder with $\\mu _{1}=2\\mu _{2}=\\mu $ and $J_{2}=1$ .",
"The exact location of the phase transition, illustrated by the white lines in Fig.",
"REF , can be obtained analytically by studying loci of the divergences in the localization length of the edge modes [18], [26], as elucidated in more detail in the Appendix, Sec. .",
"In Fig.",
"REF (a), the phase transition occurs at $|J_{2}/J_{1}|=1$ for all disorder strengths as for the clean case.",
"Fig.",
"REF (b) shows a non-trivial topology by disorder effect.",
"Namely, for fixed value of $|J_{1}|>1$ close to one, one enters the topologically non-trivial region by increasing the disorder strength $\\mu $ , before transitioning into the topologically trivial regime after further increasing the noise.",
"This widening of the topological phase boundary is observed for any kind of anisotropic disorder $\\mu _{1}\\ne \\mu _{2}$ .",
"As already mentioned, the exact phase transition points can be evaluated from the divergence of the localization length.",
"In particular, the phase boundary of the disordered SSH model is given by the equation $\\mathbb {E}(\\log |J_{1,j}|)=\\mathbb {E}(\\log |J_{2,j}|)$ , where $\\mathbb {E}(\\bullet )$ denotes average over disorder (see Eq.",
"(REF )).",
"We first discuss the widening at small disorder strengths observed in Fig.",
"REF (b).",
"The second order Taylor expansion of $\\mathbb {E}(\\log |X|)$ in $\\mu _{i}/J_{i}$ reads $\\mathbb {E}[\\log |X|]\\simeq \\log (\\mathbb {E}[X])-\\frac{\\mathbb {V}[X]}{2\\mathbb {E}[X]^{2}}$ [27], resulting in the following approximation of the phase boundary equation, $\\log |J_{1}|-\\frac{\\mu _{1}^{2}}{6J_{1}^{2}}\\simeq \\log |J_{2}|-\\frac{\\mu _{2}^{2}}{6J_{2}^{2}}.$ Fixing $J_{2}$ and $\\mu _{2}$ such that the right hand side of Eq.",
"(REF ) is constant, we see that the function $\\log |J_{1}|-\\frac{\\mu _{1}^{2}}{6J_{1}^{2}}$ is monotonically increasing in $J_{1}$ and decreasing in $\\mu _{1}$ .",
"Hence, if $\\mu _{1}$ increases, $J_{1}$ needs to grow as well in order to compensate.",
"This corresponds to a widening of the topologically non-trivial region for small increasing noise.",
"In the opposite, strong disorder limit, we can expand $\\mathbb {E}(\\log |J_{\\alpha ,i}|)$ in $J_{\\alpha }/\\mu _{\\alpha }$ , obtaining $\\mathbb {E}(\\log |J_{\\alpha ,i}|)=\\log |\\mu _{\\alpha }|-1+O(J_{\\alpha }/\\mu _{\\alpha })$ .",
"The phase boundary equation in this regime becomes $\\log |\\mu _{1}|\\simeq \\log |\\mu _{2}|.$ Hence, for strong disorder, the phase boundary is roughly independent of $J_{1}$ accounting for the horizontal boundary in Fig.",
"REF (b).",
"Similar disorder-induced topological characteristics were also recently discussed in the context of other non-Hermitian models [19], [28].",
"Figure: Coherence of the first qubit in the disordered, dissipative SSH model for N=100N=100, Γ=0.5\\Gamma =0.5 and J 2 =1J_2=1.",
"Results are averaged over 40 random realizations.",
"(a) isotropic disorder (μ 1 =μ 2 =μ\\mu _{1}=\\mu _{2}=\\mu ) along the vertical line where J 1 =0J_1=0 with μ=0.1\\mu =0.1 (orange), μ=0.5\\mu =0.5 (purple), μ=1.0\\mu =1.0 (red), and in the topologically trivial regime J 1 =1.5J_1=1.5 with μ=0.5\\mu =0.5 (black).",
"(b) anisotropic disorder(μ 1 =2μ 2 =μ\\mu _{1}=2\\mu _{2}=\\mu ) along the vertical line where J 1 =1.2J_1=1.2 with μ=0.5\\mu =0.5 (purple), μ=1.5\\mu =1.5 (red), μ=2.5\\mu =2.5 (black), and for J 1 =1.5J_1=1.5 with μ=0.5\\mu =0.5 (orange).",
"Solid black lines indicate the asymptotic prediction 𝔼(1-x 2 )\\mathbb {E}(1-x^2) Eq.",
"() valid for small disorders.For each phase diagram, we now fix $J_2=1$ and choose four characteristic parameter configurations in order to get representative coherence time evolutions for the different topological sectors, depicted Fig.",
"REF .",
"For isotropic disorder, Fig.",
"REF (a), we choose three points along the vertical $J_1=0$ with $\\mu =0.1,0.5,1.0$ as well as the configuration $J_1=1.5, \\mu =0.5$ , representing the disordered topologically non-trivial and trivial regime, respectively.",
"The coherence decays to a non-zero (respectively zero) value at large times in the topologically non-trivial (respectively trivial) sector, thus matching the phase diagram Fig.",
"REF (a).",
"In the topologically non-trivial regime, increasing disorder leads to a smaller asymptotic value of the coherence.",
"Similarly, for anisotropic disorder, Fig.",
"REF (b), we choose three points along the vertical $J_1=1.2$ with $\\mu =0.5,1.5,2.5$ as well as $J_1=0,\\mu =0.5$ .",
"The former three parameter pairs lie on a vertical line cutting through the broadening of the topologically non-trivial regime, thus representing the reentrance phenomenon into a higher topological phase.",
"It can be seen that a finite coherence of the first qubit is present at large times only for $\\mu =1.5$ , being in consent with the corresponding phase diagram Fig.",
"REF (b).",
"For $J_1=0$ and $\\mu =0.5$ , a similar behavior as for the isotropic disordered chain can be observed, with a large asymptotic coherence value.",
"In previous work [12], it was shown that for large chains in the topologically non-trivial regime, the coherence saturates to approximately $\\mathcal {C}(t\\rightarrow \\infty )\\approx 1-x^2,$ where $x=J_1/J_2$ , with $|x|<1$ .",
"It is thus natural to assume that the expectation value of the asymptotic coherence including disorder is given by $\\mathbb {E}[\\mathcal {C}(t\\rightarrow \\infty )]\\approx \\mathbb {E}(1-x^2) = \\nonumber \\\\ \\frac{1}{4\\mu _1 \\mu _2} \\int _{-\\mu _2}^{\\mu _2}\\int _{-\\mu _1}^{\\mu _1} 1- \\Big ( \\frac{J_1+\\mu _1}{J_2+\\mu _2} \\Big )^2 d\\mu _1 d\\mu _2 = \\\\ 1-\\frac{3J_1^2 + \\mu _1^2}{3J_2^2-3\\mu _2^2} \\, .",
"\\nonumber $ Note that this only holds for weak to moderate disorder such that no change of topological phase can be generated randomly, i.e., $\\mu _1 + \\mu _2 < |J_2| - |J_1|$ .",
"In Fig.",
"REF , the prediction Eq.",
"(REF ) is illustrated by black solid lines for disorder strengths falling into the discussed regime.",
"For large disorder, random phase changes result in a decrease of the mean coherence in the simulation, and Eq.",
"(REF ) breaks down.",
"It is important to note that, even though the phase diagram is the same as those found in previous works [18], [19], the physical interpretation is different, as our models include dissipation.",
"Edge states do not correspond to actual electronic states located at one of the boundaries of the chain, but rather describe the physics of the projected density matrix introduced in Sec. .",
"A non-trivial topological phase, resulting in quasi-dark states of the restricted Hamiltonian, leads to having an exponentially long (in system size) coherence time of the edge qubit.",
"In the topologically trivial regime, the decoherence of the edge qubit is governed by dissipation, leading to a finite coherence time."
],
[
"Disordered non-Hermitian Trimer Model",
"Next, we consider a trimer chain with nearest-neighbor as well as next-nearest-neighbor couplings, as depicted in Fig.",
"REF .",
"The corresponding non-Hermitian Hamiltonian, derived from the restricted Lindbladian, is given by $\\mathsf {H} & = & \\sum _{j=1}^{M}(J_{1,j}|j,B\\rangle \\langle j,A|+\\text{h.c.})\\nonumber \\\\& + & \\sum _{j=1}^{M-1}(J_{2,j}|j,C\\rangle \\langle j,B|+\\text{h.c.})\\nonumber \\\\& + & \\sum _{j=1}^{M-1}(J_{j}|j,C\\rangle \\langle j,A|+\\text{h.c.}) \\\\& + & \\sum _{j=1}^{M-1}(J_{3,j}|j+1,A\\rangle \\langle j,C|+\\text{h.c.})\\nonumber \\\\& + & \\sum _{j=1}^{M}\\epsilon _{A}|j,A\\rangle \\langle j,A|+\\sum _{j=1}^{N}\\epsilon _{B}|j,B\\rangle \\langle j,B|\\nonumber \\\\& + & \\sum _{j=1}^{M}(\\epsilon _{C,j}-i\\Gamma )|j,C\\rangle \\langle j,C|.\\nonumber $ It has been demonstrated that robust chiral edge modes exist in non-dissipative trimer chains, even in the absence of inversion symmetry [29].",
"It has been argued that their topological character is inherited through a mapping of a higher-dimensional model, namely the commensurate off-diagonal Aubry-André-Harper model, which is topologically equivalent to a two dimensional tight-binding lattice pierced by a magnetic flux [30].",
"The topological classification by Rudner et al.",
"including dissipation, however, imposes only translational symmetry.",
"In fact, it turns out that the winding number in Eq.",
"(REF ) can be used as a reliable predictor for the number of (quasi)-dark states located on the edge of the trimer chain with open boundary conditions.",
"In previous work [12], it was found that in the clean case, the presence of next-nearest-neighbor couplings enable winding numbers $W=0,1,2$ .",
"Concretely, $W$ is given by $W & =\\Theta \\left(\\left|J_{3}\\right|-\\left|J+J_{2}\\tan (\\vartheta /2)\\right|\\right)\\nonumber \\\\& +\\Theta \\left(\\left|J_{3}\\right|-\\left|J-J_{2}\\cot (\\vartheta /2)\\right|\\right),$ where $\\vartheta =\\arccos \\left[\\left(\\epsilon _{A}-\\epsilon _{B}\\right)/\\sqrt{4J_{1}^{2}+\\left(\\epsilon _{A}-\\epsilon _{B}\\right)^{2}}\\right].$ We further verify the above equation in the Appendix, Sec.",
", by solving the system analytically for a convenient system size and counting the number of dark states localized on one edge of the chain.",
"Figure: Topology and coherence for the clean, non-disordered trimer model.",
"(a) Winding number and (b)-(d) time dependent coherence for threeparameter configurations corresponding to the three topological sectors.The dotted lines indicate the theoretically predicted asymptotic coherenceas t→∞t\\rightarrow \\infty .",
"Dissipation is set to Γ=0.5\\Gamma =0.5 anda chain with N=300N=300, J 1 =1,J 2 =2J_{1}=1,J_{2}=2 and J=1J=1 is considered.The three time evolutions of the coherence in the topological sectorsW=0,1,2W=0,1,2 correspond to parameter choices J 3 =0.5,2.0,3.5J_{3}=0.5,2.0,3.5, respectively.In order to calculate the winding number using the the real-space approach, we first rewrite the Hamiltonian with respect to its sublattices and decompose it as in Eq.",
"(REF ).",
"In this case, the matrices $\\Lambda $ and (respectively $V$ ) with dimensions $2M\\times 2M$ (respectively $2M\\times M$ ) are given by $\\Lambda =\\begin{pmatrix}\\epsilon _{A}\\mathbb {1} & H_{AB}\\\\H_{BA} & \\epsilon _{B}\\mathbb {1}\\end{pmatrix} ;\\qquad V=\\begin{pmatrix}H_{AC}\\\\H_{BC}\\end{pmatrix}.$ Here, $H_{AB}=J_{1}\\mathbb {1}$ .",
"Due to the symmetry of $\\Lambda $ , $U$ from Eq.",
"(REF ) can be written as $U=\\begin{pmatrix}-\\cos (\\vartheta /2)U_{-} & \\sin (\\vartheta /2)U_{+}\\\\\\sin (\\vartheta /2)U_{-} & \\cos (\\vartheta /2)U_{+}\\end{pmatrix},$ where $U_{\\pm }$ are two $M\\times M$ so far unspecified unitaries and $\\vartheta $ has been given above.",
"From Eq.",
"(REF ), we further get $U\\tilde{V}=V$ , which gives $\\begin{pmatrix}-\\cos (\\vartheta /2)U_{-} & \\sin (\\vartheta /2)U_{+}\\\\\\sin (\\vartheta /2)U_{-} & \\cos (\\vartheta /2)U_{+}\\end{pmatrix}\\begin{pmatrix}\\tilde{V}_{-}\\\\\\tilde{V}_{+}\\end{pmatrix}=\\begin{pmatrix}H_{AC}\\\\H_{BC}\\end{pmatrix},$ where $\\tilde{V}:=(\\tilde{V}_{-},\\tilde{V}_{+})^{T}$ .",
"From the above equation we find $U_{+}\\tilde{V}_{+} & =\\frac{1}{2}(\\cos (\\vartheta /2)H_{AC}+\\sin (\\vartheta /2)H_{BC}),\\nonumber \\\\U_{-}\\tilde{V}_{-} & =\\frac{1}{2}(-\\sin (\\vartheta /2)H_{AC}+\\cos (\\vartheta /2)H_{BC}).$ Recall that we must fix the gauge freedom in $U$ by requiring the submatrices $\\tilde{V}_{\\pm }$ to be positive definite.",
"Consequently, $U_{\\pm }$ can be determined by polar decomposition of the right hand side of Eq.",
"(REF ), after which the unitary matrix $U$ is obtained using Eq.",
"(REF ).",
"Finally, the winding number is computed via Eq.",
"(REF ).",
"Interestingly, it can be shown that the winding number of $U$ is nothing more than the sum of the winding numbers of $U_{+}$ and $U_{-}$ .",
"Figure: Phase diagram of the disordered, dissipative trimer model for N=1500N=1500, Γ=0.5\\Gamma =0.5, J 1 =2J_1=2, J 2 =2J_2=2, J 3 =3J_3=3.",
"Results are averaged over 40 random realizations.In (a), μ 2 =μ J =μ 3 =μ\\mu _{2}=\\mu _{J}=\\mu _{3}=\\mu , whereas (b) describes disorderwith 2μ J =2μ 2 =μ 3 =μ2\\mu _{J}=2\\mu _{2}=\\mu _{3}=\\mu .",
"White lines indicatethe loci of diverging localization lengths in the thermodynamic limit.For simplicity, we again limit our considerations to the case of vanishing the on-site chemical potentials, i.e., $\\epsilon _{A}=\\epsilon _{B}=\\epsilon _{C}=0$ .",
"Using the real-space winding number approach for the clean trimer model results in Fig.",
"REF (a), matching Eq.",
"(REF ).",
"Figs REF (b)-(d) show the typical behavior of the coherence in the three distinct topological sectors $W=0,1,2$ in the clean trimer model, respectively.",
"In the topologically trivial regime, no dark states are present, driving decoherence of the first qubit.",
"For $W=1$ , the dark state manifold is one-dimensional, leading to a saturation of the coherence at infinite times.",
"For $W=2$ , the existence of two dark states result in Rabi like oscillations of the first qubit's coherence.",
"The asymptotic solution, Eq.",
"(REF ), is also featured in Figs.",
"REF (b)-(d).",
"Because of the $J_{1}$ dependence of the dark states, disorder in $J_{1}$ is expected to quickly destroy the topological features of the system.",
"This is further suggested by the degree of freedom of the matrix $U$ , Eq.",
"(REF ), which collapses as soon as $J_{1}$ becomes disordered, leading to an immediate collapse of a well-defined winding number.",
"Therefore, we shall from now on focus on the analysis of the disordered regime where only $J_{2},J_{3},J$ are exposed to noise, which we control via additive random noise drawn from a uniform distribution.",
"Concretely, if $j$ labels the unit cell and $\\lbrace \\omega _{1}\\rbrace $ , $\\lbrace \\omega _{2}\\rbrace $ , $\\lbrace \\omega \\rbrace $ are sets of independent, uniformly distributed random variables $\\in [-1,1]$ , $J_{i,j}=J_{i}+\\mu _{i}\\omega _{i,j}$ for $i=2,3$ , $J_{j}=J+\\mu _{J}\\omega _{j}$ , and $J_{1,j}=J_{1}$ for all $j$ .",
"Looking at the density of states for the different disorder types, depicted in Figure REF , the selection rules for the type of disorder under which topological dark states are stable is further underlined.",
"As for the disordered non-Hermitian SSH model, the full phase diagram for different disorder strengths can be constructed, shown in Fig.",
"REF .",
"Again, the exact phase transition points in the thermodynamic limit are depicted by white lines, which are derived via the dark state localization length considering disorder in the Appendix, Sec. .",
"The phase diagram features rich structures, presenting widenings of topologically non-trivial phases for moderate (high) disorder strengths in the chain with equal (different) disorder amplitudes.",
"Note that the system with different distributions on the disordered parameters, $2\\mu _{2}=2\\mu _{J}=\\mu _{3}=\\mu $ , is more similar to what we called anisotropic disorder in the SSH model, being due to the competition between $|J_{2}\\pm J|$ and $J_{3}$ deciding the topological phase for the trimer model Eq.",
"(REF ).",
"When computing the localization length, the disorder amplitudes of $J_{2}$ and $J$ hence add up, as is explicitly seen in Eq.",
"(REF ).",
"Note, however, that the effective disorder on $|J_{2}\\pm J|$ is $\\mu /\\sqrt{2}<\\mu $ , which results in having a widening of the non-topological phases in the large disorder regime.",
"Analogously, the trimer system having equal disorder on all hopping parameters resembles the case $\\mu _{1}>\\mu _{2}$ of the SSH-model, featuring a widening of the topologically non-trivial regimes for small disorders.",
"Figure: Coherence of the first qubit in the disordered, dissipative trimer model for N=300N=300, Γ=0.5\\Gamma =0.5, J 1 =1J_1=1, J 2 =2J_2=2 and J 3 =3J_3=3.",
"Results are averaged over 40 random realizations.",
"(a) For μ J =μ 2 =μ 3 =μ\\mu _{J}=\\mu _{2}=\\mu _{3}=\\mu , we show the coherence for (J,μ)=(0,1)(J,\\mu )=(0,1) (orange), (J,μ)=(3,1)(J,\\mu )=(3,1) (purple), and (J,μ)=(0,7)(J,\\mu )=(0,7) (black), corresponding to W=2,1,0W=2,1,0, respectively.",
"(b) For 2μ J =2μ 2 =μ 3 =μ2\\mu _{J}=2\\mu _{2}=\\mu _{3}=\\mu , we highlight the reentrance into a higher topological phase along the vertical line J=1.2J=1.2, with μ=1\\mu =1 (purple) and μ=7\\mu =7 (orange), corresponding to W=1,2W=1,2, respectively.",
"We further show the trivial regime by evaluating the coherence for (J,μ)=(6,1)(J,\\mu )=(6,1) (black).",
"The observable broadening of the curves is due to the error of the mean, pictured by error bars for every data point.We shall again pick three points in each phase diagram and illustrate the corresponding time evolution of the first qubits coherence, seen in Fig.",
"REF .",
"For $\\mu _{J}=\\mu _{2}=\\mu _{3}=\\mu $ , Fig.",
"REF (a), we choose the parameter pairs $(J,\\mu ) = (0,1), (3,1), (0,7)$ , belonging to winding numbers $W=2,1,0$ , respectively (cf.",
"Fig.",
"REF ).",
"For all configurations, we find that the asymptotic behavior of the coherence the one of the clean case, namely a decrease to zero for $W=0$ , a convergence to a constant larger than zero for $W=1$ , and an oscillation for $W=2$ .",
"For different disorder strengths $2\\mu _{J}=2\\mu _{2}=\\mu _{3}=\\mu $ , we focus on the reentrance phenomenon $W=1\\rightarrow 2$ by computing the coherence for $(J,\\mu )=(1.2,1), (1.2,7)$ .",
"Indeed, we find that for large enough disorder, an oscillating behavior emerges, signaling the change of topological phase.",
"For completeness, we also include $(J,\\mu )=(6,1)$ representing the trivial sector, where a vanishing coherence can be observed at large times."
],
[
"Application to Quantum Computation",
"Ever since Kitaev's proposal [23] to braid anyons in order to realize non-trivial quantum gates, the field of topological quantum computation has been an exceptionally active field of research [4], [5], [6], [7], [8], [9], [10], [11].",
"This is mainly due to the promising protection against environmental noise governed by the non-locality of the state manifold used for braiding [31].",
"Spinless p-wave superconductor wires hosting non-Abelian Majorana fermions bound to topological defects have been of particular interest [32], as the intrinsic particle-hole symmetry of the BdG-Hamiltonian promises a realizable topological protection.",
"Recently, the SSH model has been analyzed in terms of its applicability to quantum computation [33], where it was found that the non-trivial braiding statistics of the topological edge modes can be used to build quantum gates via Y-junctions.",
"However, as for all quantum gates based on symmetry protected topological states, the set of quantum gates is not universal [31].",
"Nevertheless, studying the braiding statistics for our concrete open disordered models seems like an exciting and promising work for future projects."
],
[
"Conclusions",
"We have analyzed and topologically classified disordered dissipative qubit-cavity dimer and trimer architectures, with special focus on topological protection mechanisms of the coherence measure in a fiducial qubit.",
"The evolution of the coherence's qubit is exactly given by a non-Hermitian Hamiltonian which thus emerges from a bona-fide physical system.",
"We demonstrated the use of a real-space topological invariant $W$ , which accurately predicts the number of non-trivial (quasi-)dark modes in disordered, non-Hermitian models, as long as certain symmetries are preserved by the disorder operators.",
"We then computed the phase diagrams of dimer and trimer chains in the parameter space spanned by the tunneling amplitude and the disorder strength, predicting the faith of the fiducial qubit's coherence at long times, i.e., decay to zero, a constant value or oscillatory behavior for winding numbers $W=0,1,2$ , respectively.",
"For certain choices of disorder strengths or the hopping parameters, reentrance phenomena into topological phases with higher winding numbers were observed, leading to an increase of coherence times (exponentially large in system size) when introducing higher noise levels.",
"Possible applications in topological quantum computing via braiding of dark modes were briefly discussed, opening up interesting questions for future research.",
"Furthermore, generalizations of the classification to larger numbers of sites per unit cell and systems of higher dimension would be of great interest.",
"Acknowledgements: We would like to thank Hubert Saleur for useful discussions.",
"This work was supported by the US Department of Energy under grant number DE-FG03-01ER45908.",
"L.C.V.",
"acknowledges partial support from the Air Force Research Laboratory award no.",
"FA8750-18-1- 0041.",
"The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office contract W911NF-17-C-0050.",
"The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government.",
"The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon."
],
[
"Dark states in the dissipative trimer model",
"We here derive an exact form of the asymptotic coherence dynamics and the topological phase transition in the trimer model by studying the dark states, i.e., by finding all states that obey $\\mathsf {H}|\\psi \\rangle \\!\\rangle =E|\\psi \\rangle \\!\\rangle $ with $E\\in \\mathbb {R}$ .",
"For the sake of convenience, the following considerations assume chain lengths $N\\mod {3}=2$ , as the system then hosts exact dark states with vanishing imaginary part.",
"For all other system sizes the states are quasi-dark, as they have an imaginary part exponentially small in the system size.",
"Of course, in the thermodynamic limit, these differences vanish, and the dynamics is exactly described by the result below.",
"The ansatz is to look for possible dark states with energies $E=\\pm J_{1}$ , i.e., to find the kernel of the matrix $\\mathsf {H}\\mp \\mathbb {1}J_{1}=\\begin{pmatrix}\\mp J_{1} & J_{1} & J & 0 & 0 & 0\\\\J_{1} & \\mp J_{1} & J_{2} & 0 & 0 & 0\\\\J & J_{2} & \\mp J_{1}-i\\Gamma & J_{3} & 0 & 0\\\\0 & 0 & J_{3} & \\mp J_{1} & J_{1} & J\\\\0 & 0 & 0 & J_{1} & \\mp J_{1} & \\ddots \\\\0 & 0 & 0 & J & \\ddots & \\ddots \\end{pmatrix}.$ For $N\\mod {3}=2$ , solutions of (REF ) are of the form $v_{+} & = & \\big (1,1,0,-\\delta _{+},-\\delta _{+},0,(-\\delta _{+})^{2},(-\\delta _{+})^{2},0,...,(-\\delta _{+})^{\\frac{N-2}{3}},(-\\delta _{+})^{\\frac{N-2}{3}}\\big )^{T},\\nonumber \\\\v_{-} & = & \\big (1,-1,0,\\delta _{-},-\\delta _{-},0,\\delta _{-}^{2},-\\delta _{-}^{2},0,...,\\delta _{-}^{\\frac{N-2}{3}},-\\delta _{-}^{\\frac{N-2}{3}}\\big )^{T}.$ These solutions are intuitive and analogous to the open SSH model [12], in the sense that they disappear on all dissipative sites.",
"The condition $E=\\pm J_{1}$ signals the equivalence of the first two sites of each unit cell up to a sign factor.",
"Eq.",
"(REF ) leads to $\\delta _{\\pm }=\\frac{|J\\pm J_{2}|}{J_{3}},$ where the sign of the solution is fixed without loss of generality by assuming $\\delta _{\\pm }$ to be positive.",
"The winding number classification is illustrated in the corresponding vectors, as we find zero, one, or two dark states localized at the outer left qubit for different topological sectors, i.e., $W=\\Theta (J_{3}>|J-J_{2}|)+\\Theta (J_{3}>J+J_{2})$ .",
"Taking into account the normalization factor of the solutions, $A_{\\pm }^{-2}=2\\sum _{k=0}^{\\frac{N-2}{3}}\\delta _{\\pm }^{2k}=2\\frac{1-\\delta _{\\pm }^{\\frac{2N-4}{3}}}{1-\\delta _{\\pm }^{2}},$ the time dependent coherence can be approximated for large times $t\\gg 1/\\Gamma $ , $\\mathcal {C}(t) & =|\\langle \\langle 1|e^{-i\\mathsf {H}t}|1\\rangle \\rangle |\\approx |e^{-iJ_{1}t}A_{+}^{2}+e^{iJ_{1}t}A_{-}^{2}|\\nonumber \\\\& =|A_{+}^{4}+A_{-}^{4}+2A_{+}^{2}A_{-}^{2}\\cos 2J_{1}t|.$"
],
[
"Analytical Determination of Critical Phase Transition Contours",
"In [18], the critical phase transition surface was derived for the Hermitian SSH model, using the numerical transfer matrix method and level-spacing statistics analysis.",
"The analytical critical phase transition contour for non-Hermitian models can be calculated in a similar manner.",
"To see this, consider the non-Hermitian SSH model.",
"Here, the dark edge state is exactly at zero energy and only lives on the non-decaying sublattice.",
"We consider the critical phase transition in the thermodynamic limit, such that the results for the linear chain of odd length coincide with the results of even length.",
"Now recall that the edge state of the disordered Hermitian SSH model is also supported entirely by one sublattice or the other.",
"Its zero energy edge state on sublattice A ($\\psi _{n,B}=0$ ) can be written as $\\psi _{n,A}=i^{n-1}\\prod _{j=1}^{n}\\left|\\frac{J_{1j}}{J_{2j}}\\right|\\psi _{1,A},$ where $J_{1}j$ and $J_{2}j$ are the two perturbed hopping parameters in the $j$ th unit cell.",
"The edge states in the two systems share an identical distribution in the clean limit.",
"Consequently, in this case, the non-Hermitian problem follows the same localization length and phase transition as a one-dimensional Hermitian SSH model.",
"With the exact wave function distribution as in Eq.",
"(REF ), the inverse localization length of a edge mode can be obtained by $\\Lambda ^{-1} & = & -\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\left|\\psi _{n,A}\\right|\\nonumber \\\\& = & \\left|\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{j=1}^{n}\\left(\\ln \\left|J_{1j}\\right|-\\ln \\left|J_{2j}\\right|\\right)\\right|$ An analytical result can be obtained by taking the ensemble average of the last expression.",
"The limit of the sum turns into an integration for independent and identically distributed disorder, $\\Lambda ^{-1}=\\frac{1}{4}\\left|\\int _{-1}^{1}d\\omega \\int _{-1}^{1}d\\omega ^{\\prime }\\left(\\ln \\left|J_{1}+\\mu _{1}\\omega \\right|-\\ln \\left|J_{2}+\\mu _{2}\\omega ^{\\prime }\\right|\\right)\\right|,$ where $J_{1}$ and $J_{2}$ are the unperturbed hopping parameters.",
"$\\mu _{1}$ and $\\mu _{2}$ control the strength of disorder in $J_{1}$ and $J_{2}$ respectively.",
"The random variables $\\omega $ and $\\omega ^{\\prime }$ are both drawn from a uniform distribution in the range $[-1,1]$ , leading to a normalization prefactor $1/4$ .",
"The analytic solution to this integral has been obtained in [18], $\\Lambda ^{-1}=\\frac{1}{4\\mu _{1}}\\left[\\left(J_{1}+\\mu _{1}\\right)\\log \\left|J_{1}+\\mu _{1}\\right|-\\left(J_{1}-\\mu _{1}\\right)\\log \\left|J_{1}-\\mu _{1}\\right|\\right]\\\\-\\frac{1}{4\\mu _{2}}\\left[\\left(J_{2}+\\mu _{2}\\right)\\log \\left|J_{2}+\\mu _{2}\\right|-\\left(J_{2}-\\mu _{2}\\right)\\log \\left|J_{2}-\\mu _{2}\\right|\\right]$ For small disorder, $\\mu _{1},\\mu _{2}\\ll J_{2},J_{1}$ , the localization length Eq.",
"(REF ) can be approximated by $\\Lambda ^{-1} & \\propto & \\int _{-1}^{1}\\int _{-1}^{1}d\\omega _{1}d\\omega _{2}\\ln |J_{1}+\\omega _{1}\\mu _{1}|-\\ln |J_{2}+\\omega _{2}\\mu _{2}|\\nonumber \\\\& = & \\int _{-1}^{1}\\int _{-1}^{1}d\\omega _{1}d\\omega _{2}\\ln |J_{1}|+\\frac{\\omega _{1}\\mu _{1}}{J_{1}}-\\frac{1}{2}\\Big (\\frac{\\omega _{1}\\mu _{1}}{J_{1}}\\Big )^{2}-\\Big [\\ln |J_{2}|+\\frac{\\omega _{2}\\mu _{2}}{J_{2}}-\\frac{1}{2}\\Big (\\frac{\\omega _{2}\\mu _{2}}{J_{2}}\\Big )^{2}\\Big ]\\nonumber \\\\& + & \\mathcal {O}\\Big (\\Big (\\frac{\\mu _{1}}{J_{1}}\\Big )^{3}\\Big )+\\mathcal {O}\\Big (\\Big (\\frac{\\mu _{2}}{J_{2}}\\Big )^{3}\\Big ).$ Performing the integration up to order $\\mathcal {O}\\big (\\big (\\frac{\\mu _{1}}{J_{1}}\\big )^{3}\\big )$ and $\\mathcal {O}\\big (\\big (\\frac{\\mu _{2}}{J_{2}}\\big )^{3}\\big )$ , one finds that the localization length diverges for $|J_{1}|(\\mu _{1},\\mu _{2})=|J_{2}|\\exp \\Big (\\frac{J_{2}^{2}\\mu _{1}^{2}-J_{1}^{2}\\mu _{2}^{2}}{J_{1}^{2}J_{2}^{2}}\\Big ),$ which, up to leading order in the expansion of the exponential function, reduces to $|J_{1}|(\\mu _{1},\\mu _{2})=|J_{2}|\\exp \\Big (\\frac{\\mu _{1}^{2}-\\mu _{2}^{2}}{J_{2}^{2}}\\Big ).$ We thus arrive at the conclusion that the value of $J_{1}$ where the non-trivial$\\leftrightarrow $ trivial transition occurs increases (decreases) compared to the clean case for small disorder strengths if $\\mu _{2}<\\mu _{1}$ ($\\mu _{2}>\\mu _{1}$ ).",
"This corresponds to the topology by disorder effect discussed in the main text and can be nicely seen in Fig.",
"REF (b).",
"For $\\mu _{1}=\\mu _{2}$ , the phase transition always occurs at $J_{1}=J_{2}$ , as observed in Fig.",
"REF (a).",
"Now we continue to generalize the result to the non-Hermitian trimer model.",
"In Sec.",
", we have shown that the trimer model of length $N\\mod {3}$ can host two dark edge modes with energies $E=\\pm J_{1}$ .",
"These edge modes are supported purely by non-decaying sublattices.",
"The wave functions of the two dark states for disordered three-site model is given by $\\psi _{n,ds\\pm }=(-1)^{n-1}\\prod _{j=1}^{n}\\left|\\frac{J_{2j}\\pm J_{j}}{J_{3j}}\\right|\\psi _{1,ds\\pm },$ where $J_{j}$ , $J_{2j}$ and $J_{3j}$ are perturbed hopping parameters in the $j$ th unit cell.",
"Since there exist two different edge modes, we would expect two disjoint localization lengths, $\\Lambda _{ds\\pm }^{-1} & =-\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\left|\\psi _{n,ds\\pm }\\right|\\\\& =\\left|\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\sum _{j=1}^{n}\\left(\\ln \\left|J_{2j}\\pm J_{j}\\right|-\\ln \\left|J_{3j}\\right|\\right)\\right|.$ Again, we take the ensemble average, and the summation turns into an integration, which gives $\\Lambda _{ds\\pm }^{-1}=\\frac{1}{8}\\left|\\int _{-1}^{1}d\\omega \\int _{-1}^{1}d\\omega ^{\\prime }\\int _{-1}^{1}d\\omega ^{\\prime \\prime }\\left(\\ln \\left|(J+\\mu \\omega )\\pm (J_{2}+\\mu _{2}\\omega ^{\\prime })\\right|-\\ln \\left|J_{3}+\\mu _{3}\\omega ^{\\prime \\prime }\\right|\\right)\\right|.$ Here $J$ , $J_{2}$ and $J_{3}$ are unperturbed hopping parameters.",
"$\\mu $ $\\mu _{2}$ and $\\mu _{3}$ define the amplitudes of disorder.",
"$\\omega $ , $\\omega ^{\\prime }$ and $\\omega ^{\\prime \\prime }$ are three independent and identically distributed random variables in the range of $[-1,1]$ .",
"After performing the integration explicitly, we arrive at $\\Lambda _{ds\\pm }^{-1} & = & {2\\mu \\mu _{2}}\\Big \\lbrace \\left(J\\pm J_{2}-\\mu -\\mu _{2}\\right)^{2}\\log \\left(\\left|J\\pm J_{2}-\\mu -\\mu _{2}\\right|\\right)-\\left(J\\pm J_{2}+\\mu -\\mu _{2}\\right)^{2}\\log \\left(\\left|J\\pm J_{2}+\\mu -\\mu _{2}\\right|\\right)\\nonumber \\\\& - & \\left(J\\pm J_{2}-\\mu +\\mu _{2}\\right)^{2}\\log \\left(\\left|J\\pm J_{2}-\\mu +\\mu _{2}\\right|\\right)+\\left(J\\pm J_{2}+\\mu +\\mu _{2}\\right)^{2}\\log \\left|J\\pm J_{2}+\\mu +\\mu _{2}\\right|\\Big \\rbrace \\nonumber \\\\& - & \\frac{1}{\\mu _{3}}\\Big \\lbrace \\left(J_{3}+\\mu _{3}\\right)\\log \\left(\\left|J_{3}+\\mu _{3}\\right|\\right)-\\left(J_{3}-\\mu _{3}\\right)\\log \\left(\\left|J_{3}-\\mu _{3}\\right|\\right)\\Big \\rbrace -4.$ Eq.",
"(REF ) and Eq.",
"(REF ) allow us to trace the exact critical phase transition contours in the non-Hermitian SSH dimer and trimer models."
],
[
"Diagonal Disorder",
"As argued in the main text, diagonal disorder destroys the protective chiral symmetry of the SSH model, making it collapse to a topologically trivial phase.",
"This effect can be nicely seen when considering the eigenspectrum density of states of the restricted Hamiltonian, as already introduced in the main text for symmetry conserving disorder.",
"In analogy to off-diagonal disorder, the on-site potentials $\\epsilon _{A,i}$ and $\\epsilon _{B,i}$ are chosen to be uniformly distributed between $[-\\mu ,\\mu ]$ .",
"Fig.",
"REF (a)-(d) illustrates how the topological dark states appearing in the clean system quickly wash out, joining the non-topological bulk state manifold.",
"This is in in stark contrast to a finite symmetry conserving off-diagonal disorder, where the topological dark states were almost unaffected by the noise, cf.",
"Figure REF .",
"To underline the destructive effect further, the edgequbit's coherence is inspected.",
"As soon as disorder on the on-site potentials is introduced, the coherence time is not infinite anymore, but it is reduced to a finite value $\\tau $ .",
"By assuming an exponential decay in time, i.e., $\\mathcal {C}(t)=\\mathcal {C}(t_{0})e^{-(t-t_{0})/\\tau }$ for some $t_{0}\\gg 1/\\Gamma $ , we can extract $\\tau $ by integrating over the time evolution of the coherence, i.e., $I:=\\int _{t_{0}}^{t_{1}}\\mathcal {C}(t)dt=\\int _{t_{0}}^{t_{1}}\\mathcal {C}(t_{0})e^{-(t-t_{0})/\\tau }dt=\\tau (\\mathcal {C}(t_{1})-\\mathcal {C}(t_{0})).$ Numerical integration leads to the results depicted in Fig.",
"REF (e), where a sharp drop of the coherence time away from the fully dimerized, clean limit can be observed (notice the logarithmic scaling on the z-axis).",
"For the trimer model, very similar behavior is being observed, for disorder acting on either on-site potentials or the coupling parameter $J_{1}$ , see Fig.",
"REF for the DOS.",
"Figure: Density of states for the trimer model.",
"(a)-(c) Clean density of statesfor W=2,1,0W=2,1,0, respectively.",
"(d)-(f) Diagonal disorder μ=1\\mu =1.",
"(g)-(i) Off diagonal disorder on J 2 ,J 3 ,JJ_{2},J_{3},J with μ=1\\mu =1.",
"Here,N=21N=21, J 1 =J 2 =2J_{1}=J_{2}=2, J 3 =3J_{3}=3, and J=0,3,6J=0,3,6 for the topologicalphases W=2,1,0W=2,1,0, respectively.",
"It is seen how WW quasi-dark stateswith energies E=±J 1 E=\\pm J_{1} exist in the clean system, being unstable(stable) for the considered diagonal (off-diagonal) disorder."
]
] | 2012.05274 |
[
[
"Revisiting the Water Quality Sensor Placement Problem: Optimizing\n Network Observability and State Estimation Metrics"
],
[
"Abstract Real-time water quality (WQ) sensors in water distribution networks (WDN) have the potential to enable network-wide observability of water quality indicators, contamination event detection, and closed-loop feedback control of WQ dynamics.",
"To that end, prior research has investigated a wide range of methods that guide the geographic placement of WQ sensors.",
"These methods assign a metric for fixed sensor placement (SP) followed by \\textit{metric-optimization} to obtain optimal SP.",
"These metrics include minimizing intrusion detection time, minimizing the expected population and amount of contaminated water affected by an intrusion event.",
"In contrast to the literature, the objective of this paper is to provide a computational method that considers the overlooked metric of state estimation and network-wide observability of the WQ dynamics.",
"This metric finds the optimal WQ sensor placement that minimizes the state estimation error via the Kalman filter for noisy WQ dynamics -- a metric that quantifies WDN observability.",
"To that end, the state-space dynamics of WQ states for an entire WDN are given and the observability-driven sensor placement algorithm is presented.",
"The algorithm takes into account the time-varying nature of WQ dynamics due to changes in the hydraulic profile -- a collection of hydraulic states including heads (pressures) at nodes and flow rates in links which are caused by a demand profile over a certain period of time.",
"Thorough case studies are given, highlighting key findings, observations, and recommendations for WDN operators.",
"Github codes are included for reproducibility."
],
[
"Introduction and literature review",
"In dynamic infrastructure sciences, the sensor placement (SP) problem is concerned with the time-varying selection or one-time placement of sensors, while optimizing desired objective functions.",
"This problem exists widely in dynamic networks such as transportation systems, electric power systems, and water distribution networks (WDN).",
"The optimal placement of water quality (WQ) sensors is a crucial issue in WDN due to the dangers brought by accidental or intentional contamination, the expensiveness of sensors and their installation cost, and their potential in performing real-time feedback control of water quality—control that requires high-frequency WQ sensor data.",
"WQ sensor placement in WDN serves different purposes.",
"The high-level one is minimizing the potential public health impacts of a contamination incident given a limited number of sensors.",
"To quantify this high-level objective, the WQ literature considers various mathematical objectives and metrics.",
"Specifically, the SP problem has been studied in [1], [2], [3], [4], [5], [6], [7] considering different contamination risks, optimization objectives, optimization formulations, uncertainty, the solution methodology and its computational feasibility, and the use of mobile sensors.",
"Rathi and Gupta [8] classify methodologies from over forty studies into two categories as single- and multi-objective SP problem.",
"Two other comprehensive surveys focusing on optimization strategies are also conducted in [9], [10].",
"As we mentioned, the most common objective of sensor placement in WDN is to minimize the potential public health caused by contamination incident, and it can be formulated as maximizing the coverage of water with a minimum number of sensors.",
"Lee and Deininger introduce the concept of “Demand Coverage“ and solve the problem using a mixed integer programming (MIP) method [11].",
"Kumar et al.",
"[12] and Kansal et al.",
"[13] propose heuristic methods to find optimal sensor location one by one by selecting one optimal location first and then selecting the next location by modifying the coverage matrix.",
"To consider nodes with lower water quality, Woo et al.",
"modify the objective by placing weights for each term and normalizing the concentrations [14].",
"Alzahrani et al.",
"[15] and Afshar and Marino [16] use genetic algorithm (GA) and ant colony optimization (ACO) respectively to find optimal placement strategy to maximize the demand coverage.",
"Ghimire et al.",
"[17] and Rathi and Gupta also suggested heuristic methods to solve the problem.",
"We briefly summarize the more recent literature on this problem followed by identifying the key research gap.",
"Recently, He et al.",
"propose a multi-objective SP method to explicitly account for contamination probability variations.",
"Hooshmand et al.",
"address SP problem with the identification criterion assuming that a limited sensor budget is available, followed by minimizing the number of vulnerable nodes using mixed integer programming (MIP).",
"A combined management strategy for monitoring WDN is proposed in based on the application of water network partitioning and the installation of WQ sensors.",
"Winter et al.",
"investigate optimal sensor placements by introducing two greedy algorithms in which the imperfection of sensors and multiple objectives are taken into account.",
"Giudicianni et al.",
"present a method that relies on a priori clustering of the WDN and on the installation of WQ sensors at the most central nodes of each cluster—selected according to different topological centrality metrics.",
"Hu et al.",
"propose a customized genetic algorithm to solve multi-objective SP in WDN.",
"Based on graph spectral techniques that take advantage on spectrum properties of the adjacency matrix of WDN graph, a sensor placement strategy is discussed in Di Nardo et al. .",
"Different objective functions leads to different placement strategies, and Tinelli et al.",
"discuss the impact of objective function selection on optimal placement of sensors.",
"Zhang et al.",
"investigate the global resilience considering all likely sensor failures that have been rarely explored.",
"The research community thoroughly investigated water quality sensor placement strategies considering various socio-technical objectives (as briefly discussed above).",
"The objective of this paper is not to develop a computational method to solve such SP problems with the aforementioned metrics/objectives.",
"The objective herein is to find optimal SP of water quality sensors considering an overlooked, yet significant metric: the state observability and estimation metric jointly with Kalman filtering.$\\endcsname $In dynamic systems, the Kalman filter is a widely used algorithm that computes unmeasured state estimates of a system given a dynamic model and data from sensor measurements subject to noise.",
"In short, this metric maps sensor placements given a fixed hydraulic profile to a scalar value to be minimized.",
"This value quantifies the observability of unmeasured WQ states (i.e., concentrations of chlorine) in the entire water network.",
"The observability quantification metric is depicted as a state estimation error measuring the difference between the actual WQ states and their estimates.",
"Accordingly, this proposed metric finds the optimal WQ sensor placement that minimizes the state estimation error via the vintage Kalman filter for noisy WQ dynamics and measurement models.",
"To the best of our knowledge, this is the first attempt to find the optimal sensor placement jointly with optimizing Kalman filter performance for WQ dynamics.",
"The most related research is the ensemble Kalman filter-based techniques by Rajakumar et al.",
", where the authors explore the impact of sensor placement on the final state estimation performance.",
"However, the study (i) does not provide sensor placement strategy, (ii) mainly focuses on estimating water quality states and reaction parameters, and (iii) a dynamic model for WQ is not present to guide optimal SP.",
"To that end, the objective of this study is to provide a control- and network-theoretic method that determines the optimal geographic placements of water quality sensors while optimizing the Kalman filter performance.",
"The specific paper contributions are: The state-space, control-theoretic dynamics depicting the evolution of WQ states, i.e., concentrations of chlorine, are shown.",
"Specifically, we are modeling and tracking chlorine concentrations as a surrogate for contamination—this has been showcased in various studies depicting rapid depletion of chlorine upon the introduction of contaminants .",
"The dynamics of chlorine concentrations are represented as a time-varying state-space model.",
"This model is then utilized to formulate the water quality sensor placement (WQSP) problem that optimizes the Kalman filter state estimation performance.",
"This formulation (i) takes into account and builds a mapping between a WDN observability metric and the performance of Kalman filter and (ii) is a set function optimization (an optimization problem that takes sets as variables) that is difficult to solve for large networks.",
"To account for the time-varying nature of the dynamic WQ model (due to the changes in the hydraulic profiles that are caused by demand profiles), an algorithm that computes a sensor placement for the most common hydraulic profiles is presented.",
"Furthermore, scalability of this algorithm is investigated.",
"The algorithm is based on an important theoretical feature for set function optimization called submodularity.",
"This feature has been identified in recent control-theoretic studies for sensor placement strategies , .",
"In particular, the developed approach is based on a greedy algorithm which returns a suboptimal placement strategy with guarantees on the distance to optimality.",
"An efficient implementation of the algorithm is also presented.",
"Compared to , , the proposed algorithm takes into account the time-varying nature of WQ dynamics.",
"Thorough case studies on three water distribution networks under different conditions are presented.",
"The case studies consider varying scales of water networks, significant demand variability, different number of allocated sensors, and their impact on the state estimation performance and WQSP solution.",
"Important observations and recommendations for water system operators are given.",
"Github codes are included for reproducibility.",
"The rest of the paper is organized as follows.",
"Section introduces network-oriented water quality dynamic model by presenting the models of each component in detail.",
"An abstract, linear, state-space format for the water quality model is given first considering the first-order reaction model with known reaction rate coefficients.",
"WQ observability and its metric (observability Gramian) are introduced in Section , and then WQSP problem is formulated and solved by taking advantage of submodularity property of set function optimization in Section .",
"A scalable implementation of the problem is showcased.",
"Section presents case studies to support the computational algorithms.",
"Appendix outlines components of the scalable implementation of the presented computational methods.",
"The notation for this paper is introduced next.",
"Paper's Notation $\\;$ Italicized, boldface upper and lower case characters represent matrices and column vectors: $a$ is a scalar, $a$ is a vector, and $A$ is a matrix.",
"Matrix $I_n$ denotes a identity square matrix of dimension $n$ -by-$n$ , whereas $0_{m \\times n}$ denotes a zero matrix with size $m$ -by-$n$ .",
"The notations $\\mathbb {R}$ and $\\mathbb {R}_{++}$ denote the set of real and positive real numbers.",
"The notations $\\mathbb {R}^n$ and $\\mathbb {R}^{m\\times n}$ denote a column vector with $n$ elements and an $m$ -by-$n$ matrix in $\\mathbb {R}$ .",
"For any two matrices $A$ and $B$ with same number of columns, the notation $\\lbrace A, B\\rbrace $ denotes $[A^\\top \\ B^\\top ]^\\top $ .",
"For a random variable $x \\in \\mathbb {R}^n$ , $\\mathbb {E}(x)$ is its expected value, and its covariance is denoted by $\\mathbb {C}(x) = \\mathbb {E}\\left( (x - \\mathbb {E}(x))(x - \\mathbb {E}(x))^\\top \\right)$ ."
],
[
"State-Space Water Quality Dynamic Model",
" We model WDN by a directed graph $\\mathcal {G} = (\\mathcal {W},\\mathcal {L})$ .",
"Set $\\mathcal {W}$ defines the nodes and is partitioned as $\\mathcal {W} = \\mathcal {J} \\bigcup \\mathcal {T} \\bigcup \\mathcal {R}$ where $\\mathcal {J}$ , $\\mathcal {T}$ , and $\\mathcal {R}$ are collection of junctions, tanks, and reservoirs.",
"For the $i$ -th node, set $\\mathcal {N}_i$ collects its neighboring nodes (any two nodes connected by a link) and is partitioned as $\\mathcal {N}_i = \\mathcal {N}_i^\\mathrm {in} \\bigcup \\mathcal {N}_i^\\mathrm {out}$ , where $\\mathcal {N}_i^\\mathrm {in}$ and $\\mathcal {N}_i^\\mathrm {out}$ are collection of inflow and outflow nodes.",
"Let $\\mathcal {L} \\subseteq \\mathcal {W} \\times \\mathcal {W}$ be the set of links, and define the partition $\\mathcal {L} = \\mathcal {P} \\bigcup \\mathcal {M} \\bigcup \\mathcal {V}$ , where $\\mathcal {P}$ , $\\mathcal {M}$ , and $\\mathcal {V}$ represent the collection of pipes, pumps, and valves.",
"In this paper, we the use Lax-Wendroff scheme to space-discretize pipes and each pipe with length $L$ is split into $s_{L}$ segments.",
"The number of junctions, reservoirs, tanks, pipes, pumps and valves is denoted as $n_{\\mathrm {J}}$ , $n_{\\mathrm {R}}$ , $n_{\\mathrm {TK}}$ , $n_{\\mathrm {P}}$ , $n_{\\mathrm {M}}$ , and $n_{\\mathrm {V}}$ .",
"Hence, the number of nodes and links are $n_\\mathrm {N} = n_{\\mathrm {J}}+n_{\\mathrm {R}}+n_{\\mathrm {TK}}$ and $n_\\mathrm {L} = n_{\\mathrm {P}} \\cdot s_{L} +n_{\\mathrm {M}}+n_{\\mathrm {V}}$ .",
"The principal component of the presented state-space, control-theoretic water quality modeling is a state-vector defining the concentrations of the disinfectant (chlorine) in the network.",
"Concentrations at nodes such as junctions, reservoirs, and tanks are collected in vector $c_\\mathrm {N} \\triangleq \\lbrace c_\\mathrm {J}, c_\\mathrm {R}, c_\\mathrm {T} \\rbrace $ ; concentrations at links such as pipes and pumps are collected in $c_\\mathrm {L} \\triangleq \\lbrace c_\\mathrm {P}, c_\\mathrm {M}, c_\\mathrm {V} \\rbrace $ .",
"We define WQ state $x(t) \\triangleq x$ at time $t$ as: $x(t) = \\lbrace c_\\mathrm {N},c_\\mathrm {L} \\rbrace = \\lbrace c_\\mathrm {J}, c_\\mathrm {R}, c_\\mathrm {T}, c_\\mathrm {P}, c_\\mathrm {M}, c_\\mathrm {V}\\rbrace \\in \\mathbb {R}^{n_x}, n_x = n_\\mathrm {N} + n_\\mathrm {L}.",
"$ We also make two assumptions: (i) the mixing of the solute is complete and instantaneous at junctions and in tanks with a continuously stirred tank reactors (CSTR) model , and (ii) the first-order reaction for single-species that describes disinfectant decay both in the bulk flow and at the pipe wall are assumed herein.",
"The assumptions are widely used in the literature , , ."
],
[
"Conservation of mass",
"The water quality model represents the movement of all chemical and/or microbial species (contaminant, disinfectants, DBPs, metals, etc.)",
"within a WDN as they traverse various components of the network.",
"Specifically, we are considering the single-species interaction and dynamics of chlorine.",
"This movement or time-evolution is based on three principles: (i) mass balance in pipes, which is represented by chlorine transport in differential pipe lengths by advection in addition to its decay/growth due to reactions; (ii) mass balance at junctions, which is represented by complete and instantaneous mixing of all inflows, that is the concentration of chlorine in links flowing into this junction; and (iii) mass balance in tanks, which is represented by a continuously stirred tank reactors (CSTRs) model with complete and instantaneous mixing and growth/decay reactions.",
"The modeling of each component is introduced next.",
"The water quality modeling for pipes involves modeling the chlorine transport and reaction by 1-D advection-reaction (A-R) equation.",
"For any Pipe $i$ , the 1-D A-R model is given by a PDE: $ ~{\\partial _t c_\\mathrm {P}} = -v_{i}(t) {\\partial _x c_\\mathrm {P}} + r_{i} c_\\mathrm {P} ,$ where $v_{i}(t)$ is flow velocity, $r_{i}$ is the first-order reaction rate and remains constant, which is related with the bulk and wall reaction rate and mass transfer coefficient between the bulk flow and the pipe wall , .",
"Here, the Lax-Wendroff (L-W) scheme shown in Fig.",
"REF is used to approximate the solution of the PDE (REF ) in space and time; this model has been used and accepted in the literature , , .",
"Pipe $i$ with length $L_{i}$ is split into $s_{L_{i}}$ segments, and the discretized form for segment $s$ is given by $~\\hspace{-10.0pt} c_{i,s}(t\\hspace{-1.0pt}+\\hspace{-1.0pt}\\Delta t) = \\underline{\\alpha } c_{i,s-1}(t)+ (\\alpha + r_i)c_{i,s}(t) +\\bar{\\alpha } c_{i,s+1}(t),$ where L-W coefficients for previous, current, and next segment are $\\underline{\\alpha } = 0.5 \\beta (1+\\beta )$ , ${\\alpha } = 1- \\beta ^2 $ , and $\\bar{\\alpha } = -0.5 \\beta (1-\\beta )$ .",
"Note that $\\beta \\in \\left(0,1\\right]$ for Pipe $i$ at time $t$ is a constant related with stability condition of L-W scheme, and can be decided by ${v_{i}(t) \\Delta t}(\\Delta x_{i})^{-1}$ , where $\\Delta t$ and $\\Delta x_{i}$ are the time step and the space-discretization step in Fig.",
"REF .",
"Hence, to stabilize L-W scheme, the water quality time step $\\Delta t \\le \\min ({\\Delta x_{i}}/v_{i}(t))$ , for all $i \\in \\mathcal {P}$ .",
"The L-W scheme coefficients $\\underline{\\alpha }$ , $\\alpha $ , and $\\bar{\\alpha }$ are a function of time but vary much slower than $x(t)$ , and they only change when $v_i(t)$ changes after the $\\Delta t$ and $\\Delta x_i$ are fixed.",
"That is, they only update each hydraulic time step.",
"Equation (REF ) can be lumped in a matrix-vector form for all segments $s$ for all Pipes $i \\in \\mathcal {P}$ as: Figure: Time-space discretization of Pipe ii based on the L-W scheme.$~c_\\mathrm {P}(t+ \\Delta t) = A_\\mathrm {P}(t) c_\\mathrm {P}(t) + A_\\mathrm {N}(t) c_\\mathrm {N}(t),$ where matrices $A_\\mathrm {P}$ and $A_\\mathrm {N}$ map the scalar equation (REF ) into the vector form (REF ).",
"The Github codes of this paper shows how these matrices are computed."
],
[
"Chlorine mass balance at junctions",
" Mass conservation of the disinfectant (i.e., chlorine) for Junction $i$ at time $t$ can be described by $\\hspace{-15.0pt}\\textstyle \\sum _{k = 1}^{|\\mathcal {N}_i^\\mathrm {in}|} q_{ki}(t)c_{ki}(t)=d_i(t) c_{i}(t) + \\textstyle \\sum _{j = 1}^{|\\mathcal {N}_i^\\mathrm {out}|} q_{ij}(t) c_{ij}(t),~$ where $ \\lbrace ki : k \\in \\mathcal {N}_i^\\mathrm {in} \\rbrace $ and $ \\lbrace ij : j \\in \\mathcal {N}_i^\\mathrm {out} \\rbrace $ represent the sets of links with inflows and outflows of Junction $i$ ; $d_i$ is its demand; $q_{ki}(t)$ and $q_{ij}(t)$ are the flow rate in Links $ki$ and $ij$ ; $c_{ki}(t)$ and $c_{ij}(t)$ are the corresponding concentrations.",
"Specifically, when links are pipes, $c_{ki}(t)$ and $c_{ij}(t)$ should be the last and first segment of Pipes $ki$ and $ij$ .",
"The matrix form when considering all junctions is given as $ ~c_\\mathrm {J}(t+ \\Delta t) = A_\\mathrm {J}(t) c_\\mathrm {J}(t) + A_\\mathrm {L}(t) c_\\mathrm {L}(t).$"
],
[
"Chlorine mass balance at tanks",
"Akin to dealing with junctions, we can express the mass balance equations for each tank, the details are similar and omitted for brevity and ease of exposition.",
"With that in mind, the provided Github codes present all of the necessary details that are required to arrive at the high-fidelity state-space description.",
"We directly give the matrix form of all tanks as $ ~c_\\mathrm {T}(t+ \\Delta t) = A_\\mathrm {T}(t) c_\\mathrm {T}(t) + A^{\\prime }_\\mathrm {P}(t) c_\\mathrm {P}(t),$ where $A_\\mathrm {T}$ is in terms of tank volumes $V_\\mathrm {T}$ , time step $\\Delta t$ , and flow rates flowing in or out of tanks."
],
[
"Chlorine mass balance at reservoirs",
"Without loss of generality, it is assumed that the chlorine sources are only located at reservoirs, and the concentration at a reservoir is constant.",
"That is, $c_\\mathrm {R}(t + \\Delta t) = c_\\mathrm {R}(t) .$"
],
[
"Chlorine transport in pumps and valves",
" We consider that the lengths of pumps to be null, i.e., the distance between its upstream node and downstream node is zero, and hence they neither store any water nor are discretized into different segments.",
"Therefore, the concentrations at pumps or valves equal the concentration of upstream nodes (a reservoir) they are connecting.",
"That is $\\hspace{-10.0pt} c_{j}(t+\\Delta t) = c_i(t + \\Delta t) = c_i(t) = c_j(t), i \\in \\mathcal {R}, j \\in \\mathcal {M},~$ and the corresponding matrix form for pumps is $~c_\\mathrm {M}(t+\\Delta t) = c_\\mathrm {M}(t).$ As for valves installed on pipes, it is simply treated as a segment of that pipe.",
"In this case, the concentration in valves equals the segment concentrations in pipes.",
"We next show how these matrix forms can yield state-space formulation of water quality modeling."
],
[
"Water quality modeling in state-space form",
" The briefly summarized water quality model of each component from the previous section can be written as a state-space Linear Difference Equation (LDE) as in (REF ) where $I$ is an identity matrix of appropriate dimension.",
"Figure: NO_CAPTIONFor the ease of exposition, we consider that $\\Delta t = 1 \\sec $ and the time-index $t$ is replaced with another time-index $k$ .",
"The state-space form of the water quality model is presented as a linear time-variant (LTV) system: $~{x}(k+1) =A(k) x(k)+w(k), \\;\\; y (k)=C x(k)+ v(k),$ where $x(k) \\in \\mathbb {R}^{n_x}$ is the state vector defined above; $y(k) \\in \\mathbb {R}^{n_y}$ represents a vector of data from WQ sensors; $w(k) \\in \\mathbb {R}^{n_x}$ and $v(k) \\in \\mathbb {R}^{n_y}$ are the process and measurement noise; $C \\in \\mathbb {R}^{n_y \\times n_x}$ is a matrix depicting the location of the placed sensors where $n_y << n_x$ .",
"We note the following.",
"First, although the argument of the state-space matrices $A(k)$ is in terms of $k$ , this is somewhat of an abuse for the notation seeing that $A(k)$ encodes the hydraulic profile (heads and flow rates) that does not change with the same frequency as the water quality states $x(k)$ .",
"Hence, the state-space model (REF ) is time varying as system matrix $A(k)$ changes for different hydraulic simulation, but remains the same $A$ in a single simulation.",
"Second, and without loss of generality, the input vector from booster stations is implicitly embedded within the state-space matrix $A$ .",
"Third, for all $k \\ge 0$ , it is assumed that initial condition, process noise $w(k)$ and the measurement noise $v(k)$ are uncorrelated and the noise variance from each sensor is $\\sigma ^2$ .",
"Finally, we like to point out that extensive details for the above state-space model can be studied from our recent work on model predictive control of water quality dynamics ."
],
[
"Observability Metrics for WQ Dynamics",
" The objective of this section is two-fold.",
"First, to introduce water system engineers and researchers to control-theoretic approaches for ensuring or optimizing the observability of the water quality dynamics.",
"Herein, observability is defined as the ability to estimate water quality model states $x(k)$ from available measurements $y(k)$ via a state estimation routine.",
"This provides situational awareness for the operator given data from few water quality sensors.",
"Second, to define a simple observability metric that maps the number and location of fixed sensors to a scalar metric acting as a proxy for the state estimation."
],
[
"Metrics for observability and its interpretations",
" In dynamic systems theory, observability is a measure of how the system state vector $x(k) \\in \\mathbb {R}^{n_x}$ can be inferred from knowledge of its output $y(k) \\in \\mathbb {R}^{n_y}$ over either finite- or infinite-time horizons.",
"In particular, given sensor data $y(0), y(1), \\ldots , y(k_f-1)$ for finite $k_{f} = k_{final}$ time-steps, observability is concerned with reconstructing or estimating the initial unknown state vector $x(0)$ from the $k_f$ measurements, and subsequently computing $x(1), \\ldots , x(k_f)$ assuming noiseless system.",
"Accordingly, a linear dynamic system (such as the water quality model (REF )) is observable if and only if the observability matrix for $k_f$ time-steps $\\mathcal {O}(k_f) = \\lbrace C, C A, \\hdots , C A^{k_f-1}\\rbrace \\in \\mathbb {R}^{k_f n_y \\times n_x }~$ is full column rank , i.e., $\\operatorname{rank}(\\mathcal {O}(k_f))=n_x$ assuming that $k_fn_y > n_x$ .",
"In this section, and for brevity, we assume that the hydraulic variables are not changing during each hydraulic simulation period and hence $A(k)= A$ .",
"With that in mind, the proposed sensor placement formulations considers changing hydraulic simulations.",
"For the infinite-time horizon case with $k_f = \\infty $ (that is, data has been collected over a long period of time), a system is observable if and only if the observability matrix $\\mathcal {O}(k_f=n_x) \\in \\mathbb {R}^{n_x n_y \\times n_x }$ is full column rank .",
"However, observability is a binary metric—it cannot indicate how observable a dynamic system is.",
"Due to the complexity and dimension of the water quality model (REF ), this dynamic model is not observable, i.e., it fails the aforementioned rank condition for various water networks and hydraulic simulation profiles.",
"Specifically, it is virtually impossible to accurately reconstruct all chlorine concentrations (states $x(k)$ ) unless water quality sensors are ubiquitously available and widespread in the network, i.e., installed at each junction.",
"To that end, a more elaborate, non-binary quantitative metric for observability is needed for the water quality model and the sensor placement problem.",
"One metric is based on the observability Gramian defined as the $k_f$ sum of matrices $W(k_f)=\\sum _{\\tau =0}^{k_f}\\left(A^{\\top }\\right)^{\\tau } C^{\\top } C A^{\\tau }.~$ The system is observable at time-step $k_f$ if matrix $W(k_f)$ is nonsingular and is unobservable if $W(k_f)$ is singular.",
"Similarly, this definition extends for the infinite-horizon case with $k_f = \\infty $ .",
"However, $W$ is still a matrix and the aforementioned observability-singularity discussion is still binary.",
"As a result, various non-binary metrics have been explored in the literature , .",
"This includes: the minimum eigenvalue $\\lambda _{\\mathrm {min}}(W)$ , the log determinant $\\log \\operatorname{det} (W)$ , the $\\operatorname{trace} (W)$ , and the sums or products of the first $m$ eigenvalues $\\lambda _1,\\hdots ,\\lambda _m$ of $W$ .",
"These metrics differ in their practical application, interpretation, and theoretical properties; the reader is referred to for a thorough discussion.",
"In this paper, we utilize the $\\log \\operatorname{det} (W)$ metric due to various reasons outlined in the ensuing sections, but the formulations presented in the paper can be extend to other metrics."
],
[
"Metrics for water quality observability matrix",
" In this section, we provide a discussion on the utilized metric for observability for the sensor placement problem.",
"To do so, we consider the time-invariant state-space matrices for a single hydraulic simulation $k \\in [0,k_f]$ which is also a single instant of hydraulic simulation and demand profile.",
"That is, to ease the ensuing exposition we assume that the state-space matrix $A(k) = A$ is fixed rather than being time-varying (the actual methods consider time-varying demand pattern).",
"The objective of this section is to formulate a water quality observability metric that maps collection of water quality data $y(k)$ from a specific number of sensors $n_y$ to a scalar observability measure under the noise from water quality dynamics and measurement models.",
"First, consider the augmented measurement vector $\\bar{y}(k_f) \\triangleq \\lbrace y(0), \\ldots ,y(k_f) \\rbrace $ for $k_f+1$ time-steps.",
"Given (REF ), this yields: $~\\hspace{-14.70001pt}\\includegraphics [width=0.92valign=c]{Fig_3.pdf},$ where $z(k_f)$ lumps initial unknown state $x_0 = x(0)$ and process noise $w(k_f)$ , and $\\bar{v}(k_f)$ collects all measurement noise.",
"Note that the left-hand side of (REF ) is known, whereas vectors $z(k_f)$ and $\\bar{v}(k_f)$ are unknown vectors.",
"To that end, the problem of estimating $z(k_f) \\triangleq z \\in \\mathbb {R}^{n_z}$ , where $n_z= (k_f+1)n_x$ , is important to gain network-wide observability of water quality state $x$ which will guide the real-time estimation.",
"As a probabilistic surrogate to estimating $z$ , we utilize the minimum mean square estimate (MMSE) defined as $\\mathbb {E}(z - \\hat{z})$ , and its corresponding posterior error covariance matrix $\\Sigma _{z}$ .",
"These two quantities provide estimates of means and variances of the unknown variable $z(k_f)$ .",
"Interestingly, these can be written in terms of the sensor noise variance $\\sigma ^2$ , the collected sensor data $\\bar{y}(k_f)$ , the observability-like matrix $\\mathcal {O}(k_f)$ in (REF ), and the expectation and covariance of the unknown variable $z(k_f)$ given by $\\mathbb {E}(z(k_f)), \\; \\mathbb {C}(z(k_f)) = \\mathbb {E}\\left( (z - \\mathbb {E}(z))(z - \\mathbb {E}(z))^\\top \\right) ~$ Given these developments, and to guide the sensor placement problem formulation, a metric is needed to map the covariance matrix $\\Sigma _{z}$ to a scalar value.",
"In particular, the metric $\\log \\operatorname{det}\\left(\\Sigma _{z}\\right)$ , which maps an $n_z$ -by-$n_z$ matrix $\\Sigma _z$ to a scalar value, can be used to achieve that.",
"Fortunately, $\\log \\det (\\Sigma _{z})$ has a closed form expression given by: $~\\hspace{-10.0155pt}\\log \\det (\\Sigma _{z}) = 2 n_z \\log (\\sigma )\\hspace{-2.0pt}-\\hspace{-2.0pt}\\log \\det \\left(\\sigma ^{2} \\mathbb {C}^{-1}\\left(z\\right)+W_o \\right)$ where $W_o = \\mathcal {O}^{\\top }(k_f)\\mathcal {O}(k_f)$ .",
"The reader is referred to for the derivation of (REF ).",
"We note the following: (i) the closed-form expression of $\\log \\operatorname{det}\\left(\\Sigma _{z}\\right)$ in (REF ) assumes a fixed sensor placement while associating a scalar measure of water quality observability given a collection of sensor data and the system's parameters.",
"This closed form expression is rather too complex to be incorporated within a sensor placement formulation and does not allow for near real-time state estimation.",
"The next section discusses simple solutions to these issues.",
"(ii) We use the $\\log \\det (\\cdot )$ metric here as it is endowed with desirable theoretical properties (namely super/sub-modularity) that makes it amenable to large-scale networks, it exhibits a closed-form expression as in (REF ), and has been used in various sensor placement studies in the literature.",
"With that in mind, other metrics can be used including the $\\mathrm {trace}$ operator."
],
[
"Relationship with the Kalman filter",
"The above discussions yield a metric that can be used for quantifying observability of the water quality model (REF ), in addition to probabilistically estimating the unknown, initial state vector $x(0)$ .",
"A relevant problem is the real-time state estimation via the Kalman filter, which essentially reconstructs or estimates in real-time states $x(k)$ from output measurements $y(k)$ .",
"This is in contrast with the batch state estimation as in (REF ).",
"While the initial state estimation problem discussed in the previous section provides a starting point for reconstructing $x$ , the Kalman filter presents a more general approach to the estimation problem.",
"In fact, ignoring the process noise $w$ and setting variances of sensor data to $\\sigma ^2 = 1$ , the Kalman filter becomes equivalent to a real-time version of the above probabilistic estimator.",
"Most importantly, the metric $\\log \\operatorname{det} (\\cdot )$ degenerates to: $\\hspace{-11.38092pt} \\log \\operatorname{det}\\left(\\Sigma _{z}\\right) &= - \\log \\operatorname{det} ( I_{n_{x}} + W(k_f) ) ~ \\\\&= - \\log \\operatorname{det} \\left( I_{n_{x}} + \\sum _{\\tau =0}^{k_f}\\left(A^{\\top }\\right)^{\\tau } C^{\\top } C A^{\\tau } \\right) $ where $I_{n_{x}}$ is an identity matrix of size $n_x$ .",
"This is shown in the recent control theoretic literature , .",
"In short, this is a simple metric that maps the number of installed or placed sensors (i.e., number of rows of matrix $C$ ) to a metric that defines the quality of the state estimates.",
"When no sensor is installed or $C$ is a zero matrix, the observability Gramian $W(k_f)$ is also a zero matrix, and intuitively the $\\log \\det (\\cdot )$ metric defined above has the maximum error of 0.",
"When the network is fully sensed—that is $n_y = n_x$ , $C = I_{n_{x}}$ , and all states are measured—then $W(k_f) = I_{n_x} + A + \\hdots + A^{k_f}$ and the smallest error is achieved.",
"Building on that, the control theoretic literature thoroughly investigated bounds for the estimation error and the corresponding metric with respect to the number of sensors; see .",
"The objective of this paper is to build on these developments and investigate how such metric relates with the performance of the Kalman filter.",
"The next section formulates the water quality sensor problem using the introduced metric."
],
[
"Water Quality Sensor Placement Formulation",
" The objective of the presented water quality sensor placement (WQSP) formulation is to minimize the error covariance of the Kalman filter while using at most $r$ water quality sensors.",
"In WDN, water quality sensors are installed at nodes, that is, at most $r$ sensors are selected from the set $\\mathcal {W} = \\mathcal {J} \\bigcup \\mathcal {T} \\bigcup \\mathcal {R}$ where the cardinality of set $|\\mathcal {W}| = n_N$ , i.e., the set $\\mathcal {W}$ contains $n_N$ possible locations at various junctions, tanks, and reservoirs.",
"This forms a sensor set $\\mathcal {S} \\subset \\mathcal {W}$ where $|\\mathcal {S}| = n_{\\mathcal {S}} \\le r$ .",
"The specific geographic placement and locations of these $n_{\\mathcal {S}}$ sensors are encoded in matrix $C$ of (REF ) through binary indicators.",
"In short, the presented WQSP seeks to find the optimal set $\\mathcal {S}^*_{r}$ that optimizes the state estimation performance with at most $r$ WQ sensors.",
"The metric discussed in the previous section assumes that the state-space matrix $A$ (encoding network and hydraulic simulation parameters) is time-varying due to varying demand and flow/head profiles.",
"In short, the metric (REF ) yields a time-varying value and hence different state estimation performance for each hydraulic simulation reflected with a different $A(k)$ matrix.",
"As a result, considering a varying hydraulic simulation profile within the sensor placement problem is important, i.e., the sensor placement solution needs to be aware of the most probable demand and hydraulic scenarios.",
"Consequently, we define $D_i \\in \\mathbb {R}^{n_{\\mathrm {J}} \\times T_h k_f}, \\forall i \\in \\lbrace 1,\\ldots ,n_d\\rbrace $ for all $n_{\\mathrm {J}}$ junctions during $T_h$ distinct hydraulic simulations, each lasting $k_f \\, \\sec $ .",
"The notation $D_{i,k}$ defines the $k$ th column vector of matrix $D_i$ .",
"Parameter $n_d$ reflects the number of potential demand patterns; concrete examples are given in case study section.",
"Demand profiles $D_i \\in \\mathcal {D}$ essentially define the most common varying demand profiles experienced by the system operator from historical data.",
"Each demand profile results in a different hydraulic profile and hence a different state-space matrix$\\endcsname $We defined $A(k)$ earlier due to the change in the hydraulic and demand profiles.",
"The notation $A(D_{i,k})$ is equivalent to $A(k)$ but offers more clarity.",
"$A(D_{i,k})\\triangleq A(k)$ .",
"Given these definitions and for an a priori defined $D_{i,k} \\in \\mathcal {D}$ , one useful instant of the WQSP problem can be abstractly formulated as: $\\begin{split}\\mathrm {{WQSP:}} \\;\\;\\;\\; \\operatornamewithlimits{minimize}\\;\\; \\; & f(\\mathcal {S}; A(D_{i,k})) \\\\\\operatorname{subject\\ to}\\;\\;\\;&{\\mathcal {S} \\subset \\mathcal {W}, \\;\\; |\\mathcal {S}| = n_\\mathcal {S}} \\le r.\\end{split}$ The design variable in the optimization problem $\\mathrm {WQSP}$ is the location of the installed sensors reflected via set $\\mathcal {S}$ defined earlier.",
"The objective function $f(\\cdot ;\\cdot ): \\mathbb {R}^{n_{\\mathcal {S}}} \\times \\mathbb {R}^{n_x \\times n_x} \\rightarrow \\mathbb {R}$ maps the optimal sensor placement candidate $\\mathcal {S}$ and given hydraulic demand profile $D_{i,k}$ and its corresponding matrix $A(D_{i,k}) $ to the state estimation, Kalman filter performance.",
"We note that when the objective function has a set as the variable (i.e., $\\mathcal {S}$ in $f(\\cdot ;\\cdot )$ ), the objective function is often referred to as a set function.",
"We use these terms interchangeably.",
"In this paper, the set (objective) function takes the form of (REF ) which indeed takes explicitly the sensor placement set $\\mathcal {S}$ through matrix $C$ as well as the a priori known hydraulic profiles and the corresponding state-space matrices $A(D_{i,k})$ .",
"The constraint set of $\\mathrm {WQSP}$ represents the number of utilized sensors and their location in the network.",
"For small-scale water networks, one may solve the set function optimization (REF ) via brute force, but this is impossible for large-scale networks—such problems are known to be an NP-hard one, i.e., a computational problem that is suspected to have no polynomial-time algorithm to optimally solve.",
"To address this computational challenge, we resort to a widely-used approach in combinatorial optimization: exploiting special property of the set function $f(\\mathcal {S};A(D_{i,k}))$ via sub/super-modularity defined as follows.",
"A set function $f(\\cdot )$ is submodular if and only if $ f(\\mathcal {A} \\cup \\lbrace a\\rbrace )-f(\\mathcal {A}) \\ge f(\\mathcal {B} \\cup \\lbrace a\\rbrace )-f(\\mathcal {B})$ for any subsets $\\mathcal {A} \\subseteq \\mathcal {B} \\subseteq \\mathcal {V}$ and $\\lbrace a\\rbrace \\in \\mathcal {V} \\backslash \\mathcal {B}$ .",
"A set function $f(\\cdot )$ is supermodular if $-f(\\cdot )$ is submodular.",
"Intuitively, submodularity is a diminishing returns property where adding an element to a smaller set gives a larger gain than adding one to a larger set .",
"The computational framework of submodularity of set function optimization allows one to use greedy algorithms with desirable performance while being computationally tractable.",
"Although greedy algorithms are known to return suboptimal solutions, they are also known to return excellent performance when the set function is especially sub/super-modular.",
"Interestingly, the set function in $\\mathrm {WQSP}$ given in (REF ) is indeed supermodular .",
"Given this property, a vintage greedy algorithm—applied to solve the NP-hard problem $\\mathrm {WSQP}$ —can return a solution $\\mathcal {S}$ with objective function value $f(\\mathcal {S})$ at least 63% of the optimal solution $f(\\mathcal {S}^{*})$ .",
"Empirically, a large body of work , , shows that the solution provided by some greedy algorithms can be near-optimal, rather than being 63% optimal.",
"[t] Number of sensors $r$ , all demand profiles $\\mathcal {D}$ , water network parameters, $k = i = 1$ , $\\tilde{\\mathcal {S}} = \\emptyset $ Optimal sensor set $\\mathcal {S^{\\star }}$ Compute: $A(D_{i,k})=A, \\forall i, k \\in \\lbrace 1,\\ldots ,n_d\\rbrace , \\lbrace 1,\\ldots , T_h k_f\\rbrace $ $k\\le T_hk_f$ // For each single hydraulic simulation interval $k$ $i = 1$ , $\\bar{\\mathcal {S}}=\\emptyset $ $i \\le n_d$ // For each demand profile $j = 1, \\mathcal {S}_j = \\emptyset $ $j \\le r$ $e_{j} \\leftarrow \\mathrm {argmax} _{e \\in \\mathcal {W} \\backslash \\mathcal {S} }\\left[f(\\mathcal {S};A )-f(\\mathcal {S} \\cup \\lbrace e\\rbrace ;A)\\right]$ $\\mathcal {S}_j \\leftarrow \\mathcal {S}_j \\cup \\left\\lbrace e_{j}\\right\\rbrace $ $j \\leftarrow j+1$ $\\bar{\\mathcal {S}} \\leftarrow \\bar{\\mathcal {S}} \\bigcup {\\mathcal {S}}_{j}$ , $i \\leftarrow i+1$ $ {\\mathcal {S}}^{(k)} \\leftarrow \\arg \\max _{\\mathcal {S} \\in \\bar{\\mathcal {S}}} f(\\mathcal {S}; A)$ $\\tilde{\\mathcal {S}} \\leftarrow \\tilde{\\mathcal {S}} \\bigcup {\\mathcal {S}}^{(k)}$ $k \\leftarrow k+k_f$ $\\mathcal {S}^* \\leftarrow \\arg \\max _{\\mathcal {S} \\in \\tilde{\\mathcal {S}}} {T}(\\mathcal {S})$ // Greedy-optimal sensor placement Greedy algorithm to solve WQSP problem.",
"We apply a greedy algorithm to solve the WQSP for various hydraulic profiles.",
"The details of this algorithm are given in Algorithm .",
"The notation $\\mathcal {S}_j$ denotes the sensor set with $j$ placed sensors.",
"The notation $\\mathcal {S}^{(k)}$ denotes the sensor set at iteration $k$ .",
"The sets $\\tilde{\\mathcal {S}}$ and $\\bar{\\mathcal {S}}$ are super-sets that include various sets $\\mathcal {S}$ .",
"Variable $e \\in \\mathcal {S}$ defines an element (i.e., a junction) in the set $\\mathcal {S}$ .",
"The inputs for the algorithm are the number of sensors $r$ , all demand profiles $D_{i,k} \\in \\mathcal {D}$ , and WDN parameters.",
"The output of the algorithm is greedy-optimal sensor set $\\mathcal {S}^*$ .",
"The first step of the algorithm is to compute all state-space matrices $A(D_{i,k})$ for various demand profiles $D_{i,k}$ .",
"Then, given a fixed hydraulic simulation interval $k$ , a fixed demand profile $i$ , and fixed number of sensors $j$ , Step 9 computes the optimal element in the set $\\mathcal {W} \\backslash \\mathcal {S}_j$ that yields the best improvement in the set function optimization reflecting the Kalman filter performance—a core component of the greedy algorithm and supermodular optimization.",
"At each iteration inside the while loop, the algorithm finds the optimal element $e_j$ (i.e., the sensor through a junction ID) that results in the best improvement in the state estimation performance metric.",
"Then, the $n_d$ sets $\\mathcal {S}_j$ (that include the optimal sensor sets for all $n_d$ demand profiles) are stored in a master set $\\bar{\\mathcal {S}}$ .",
"This is then followed by finding the optimal sensor sets from $\\bar{\\mathcal {S}}$ for all $T_h$ hydraulic simulations; these are all included in another master set $\\tilde{\\mathcal {S}}$ .",
"Finally, the algorithm terminates by computing the final optimal sensor locations $\\mathcal {S}^*$ via picking the combination that maximizes the occupation time ${T}(\\mathcal {S})$ for all $\\mathcal {S} \\in \\tilde{\\mathcal {S}}$ , i.e., a metric that defines the frequency of a specific sensor activation.",
"Finally, we note that this algorithm returns the greedy-optimal solution.",
"This solution is not necessarily the optimal solution as discussed above with the 63% optimality guarantees.",
"Thorough case studies are given in the ensuing section.",
"Figure: (a) Three-node network, (b) Net1, and (c) Net3."
],
[
"Case Studies",
" We present three simulation examples (three-node network, Net1, and Net3 network , ) to illustrate the applicability of our approach.",
"The three-node network is designed to illustrate the details of proposed method and help readers understand the physical meaning of results intuitively.",
"Then, we test Net1 with looped network topology considering the impacts on final WQSP from choosing (i) the length of a single hydraulic simulation $t$ , (ii) L-W scheme time-step $\\Delta t$ (or equally dynamic number of segments), (iii) different base demands, and (iv) different patterns.",
"Net3 network is used to test scalability of proposed algorithm and verify our findings further.",
"Considering that the LDE model (REF ) produces accurate state evolution, we eliminate the process noise and set the sensor noise standard deviation to $\\sigma = 0.1$ .",
"The simulations are performed via EPANET Matlab Toolkit on Windows 10 Enterprise with an Intel(R) Xeon(R) CPU E5-1620 v3 @3.50 GHz.",
"All codes, parameters, tested networks, and results are available on Github which includes an efficient and scalable implementation of Algorithm .",
"The details of this implementation are included in Appendix ."
],
[
"Three-node network",
"The three-node network shown in Fig.",
"REF a includes one junction, one pipe, one pump, one tank, and one reservoir.",
"A chlorine source ($ c_\\mathrm {R1} = 0.8$ mg/L) is installed at Reservoir 1.",
"The initial chlorine concentrations at or in the other components are 0 mg/L.",
"Only Junction 2 consumes water, and its base demand is $d_{\\mathrm {base}} = 2000\\ \\mathrm {GPM}$ .",
"The corresponding pattern $\\mathrm {Pattern\\ I}$ (viewed as a row vector) for Junction 2 in 24 hours is presented in Fig.",
"REF .",
"Hence, only one demand profile for a day is computed as $D = d_{\\mathrm {base}} \\times \\mathrm {Pattern\\ I}$ .",
"The pipe is split into fixed as $s_{L_{23}} = 150$ segments, and the single hydraulic simulation interval is set to $k_f = 300 \\sec $ and $T_h = 24$ hydraulic simulations are considered.",
"To help the readers understand intuitively about the water quality modeling in state-space form and the observability (Gramian), an illustrative code including step by step comments for this small three-node network is available in our Github for the convenience of readers.",
"Figure: Pattern for Three-node and Net1 networks.Figure: Different base demands (a) and demand patterns (b) for nodes in Net1.Figure: Sensor placement results for the three-node network (a) and Net1 (b) in 24 hours with k f =300seck_f = 300 \\sec , Δt=5sec\\Delta t = 5\\sec , Pattern I, Base demand 1.For the three-node network, there are three possible sensor locations ($\\mathrm {R}1$ , $\\mathrm {J}2$ , and $\\mathrm {T}3$ ); therefore, $r$ is set to 1 or 2 in Algorithm .",
"The final sensor placement results are presented as Fig.",
"REF .",
"When $r = 1$ , $\\mathrm {J}2$ is the best location or the center of the network, and when $r = 2$ , locations $\\mathrm {J}2$ and $\\mathrm {T}3$ are selected.",
"To qualify the centrality or importance of a specific location during 24 hours, occupation time ${T}(\\mathcal {S}) = \\frac{\\mathrm {Selected\\ time}}{\\mathrm {Total\\ time}}$ is defined as a percentage of the selected time by Algorithm in a day.",
"This measure indicates the importance of the selected sensor locations.",
"If the sensor location does not change during 24 hours, the occupation time would be 100%; see Tab.",
"REF .",
"With that in mind, this 100% figure of sensor occupation time rarely happens for any junction in larger networks—its occurrence in the three-node network is due to its simple topology.",
"We show more interesting results with varying occupation time in the next sections."
],
[
"Looped Net1 network",
"Net1 network , shown in Fig.",
"REF b is composed of 9 junctions, 1 reservoir, 1 tank, 12 pipes, and 1 pump.",
"Beyond optimal sensor placements, here we investigate the impact of the length of a single hydraulic simulation length $k_f$ , L-W scheme time-step $\\Delta t$ , and the demand profile on the final sensor placement result.",
"This network is more complex than the three-node network because its flow direction changes and flow rates or velocities vary dramatically every hour.",
"To balance the performance of L-W scheme and computational burden, $s_{L_i}$ for each pipe is set to an integer which is the ceiling of $\\frac{L_i}{ v_i(t) \\Delta t}$ , and dynamic number of segments setting makes $\\Delta t = 5\\sec $ .",
"Furthermore, If the parameter $\\Delta t = 10\\sec $ is needed, and this can be achieved conveniently by reducing the $s_{L_i}$ for each pipe by half."
],
[
"Base case scenario and its result",
"The base case is considered with the following settings: $\\Delta t = 5\\sec $ , single hydraulic simulation $k_f = 300 \\sec $ , and demand profile for a single interval $D_k = \\mathrm {Base \\ demand \\ 1} \\times \\mathrm {Pattern\\ I}$ shown in Fig.",
"REF .",
"There are 11 possible sensor locations (see Fig.",
"REF b), and the number of sensor locations $r$ is chosen as $[1, 3, 5]$ in (REF ).",
"Similarly, we consider 24 hours in Algorithm .",
"The final result is presented in Fig.",
"REF , and the sensor placement results in terms of occupation time $T$ are presented in Tab.",
"REF .",
"From Fig.",
"REF and Tab.",
"REF , when $r = 1$ , $\\mathrm {J}10$ in Fig.",
"REF is the best sensor location most of the time ($T_{\\mathrm {J}10} = 66.4\\%$ ) and the best location switches to $\\mathrm {J}12$ or $\\mathrm {J}21$ occasionally ($T_{\\mathrm {J}12} = 18.6\\%$ , $T_{\\mathrm {J}21} = 14.8\\%$ ).",
"Hence, the solution of WQSP is $\\mathcal {S}_{r=1}^* = \\lbrace \\mathrm {J}10\\rbrace $ (marked as blue in Tab.",
"REF ).",
"Similarly, the locations with the largest $r$ occupation time are selected as the final results when $r = 3$ and 5.",
"These greedy-optimal placements are given by $\\mathcal {S}_{r=3}^* = \\lbrace \\mathrm {J}10, \\mathrm {J}12, \\mathrm {J}21\\rbrace $ and $\\mathcal {S}_{r=5}^* = \\mathcal {S}_{r=3}^* \\bigcup \\lbrace \\mathrm {T}2,\\mathrm {J}31\\rbrace $ .",
"This showcases supermodularity of the set function optimization, seeing that $\\mathcal {S}_{r=3}^* \\subset \\mathcal {S}_{r=5}^*$ .",
"Table: Sensor placement results with detailed occupation time (Base case of Net1: Δt=5sec\\Delta t = 5\\sec , k f =300seck_f= 300\\sec , Pattern I, Base demand 1).Table: Sensor placement results considering the impacts of L-W scheme time-step Δt\\Delta t and the length of the single observation time tt (Case A: Δt=10sec\\Delta t = 10\\sec , k f =300seck_f = 300\\sec ; Case B: Δt=5sec\\Delta t = 5\\sec , k f =60seck_f = 60\\sec )."
],
[
"The impacts of L-W scheme time-step and the length of single observation time",
"Here, we study the impact of L-W scheme time-step and the length of single observation time parameters on the final WQSP results—in comparison with the base case from the previous section.",
"At first, only $\\Delta t$ is increased from 5 (from the base case) to $10\\sec $ (Case A).",
"Accordingly, the number of segments of all pipes is reduced by $50\\%$ , while still maintaining the accuracy of LDE state-space model compared to the EPANET water quality simulation.",
"We also define Case B by reducing $k_f$ from $300\\sec $ (base case) to $60 \\sec $ .",
"The results for this experiment are shown in Tab.",
"REF .",
"We observe the following: (i) the final results are exactly the same as the ones under the base case for $r = 1,5$ , and the differences are materialized only in the slightly changed occupation time; (ii) the results under $r = 3$ are different from the base case as the solution changes from $\\mathcal {S} = \\lbrace \\mathrm {J}10, \\mathrm {J}12, \\mathrm {J}21\\rbrace $ (base case) to $\\mathcal {S} = \\lbrace \\mathrm {J}10, \\mathrm {T}2, \\mathrm {J}21\\rbrace $ (Cases A and B).",
"This is due to the fact that the base case did not produce a clear winner in terms of the sensor placement—the occupation times ($T_{\\leavevmode {\\color {blue}\\mathrm {J}12}} = 56.7\\%,$ $T_{\\mathrm {T}2} = 53.6\\%$ ) are similar.",
"We note that even if the sensor placement strategy is changed when $r = 3$ , the final performances of these three cases are comparable, and the relative error of Kalman filter performance in (REF ) reached between Base case and Case A (Case B) is $17.2\\%$ ($7.9\\%$ ) even though the difference in $\\Delta t$ is two times and the difference in $k_f$ is 5 times, which is acceptable.",
"Hence, one could make this preliminary conclusion: the impacts of L-W scheme time-step $\\Delta t$ and the length of hydraulic simulation $k_f$ on the final sensor placement results are negligible assuming that the number of pipe segments (the partial differential equation space discretization parameter) is large enough to ensure the accuracy of LDE model.",
"Figure: Sensor placement results for Net1 with k f =300seck_f = 300\\sec , Δt=5sec\\Delta t = 5\\sec , Pattern I under (a) Base demand 2, (b) Base demand 3."
],
[
"The impact of various demand patterns",
"In this section, the impact of demand profiles on the final sensor placement result is explored.",
"Note that the demand in 24 hours at a node is decided by its base demand and the corresponding patterns simultaneously.",
"Furthermore, other demand patterns could reflect other days of the weeks such as a weekend, rather than assuming a week-long demand curve.",
"First, the Pattern I is fixed as the stair-shape in Fig.",
"REF or the dotted line in Fig.",
"REF , and base demands 1, 2, and 3 in Fig.",
"REF are used.",
"That is, we have $n_d = 3$ different demand profiles which is an input for Algorithm .",
"Note that these base demands are generated for illustrative purposes.",
"Base demand 1 is designed to assign nearly identical base demand at each node.",
"Base demand 2 assigns more base demands to the nodes on the right half part of the network in Fig.",
"REF b, such as $\\lbrace \\mathrm {J}12, \\mathrm {J}13, \\mathrm {J}22,\\mathrm {J}23,\\mathrm {J}32\\rbrace $ .",
"Base demand 3 assigns larger base demands to the nodes on the left half part of the topology in Fig.",
"REF b, such as $\\lbrace \\mathrm {J}11, \\mathrm {J}21, \\mathrm {J}31\\rbrace $ .",
"Figure: Sensor placement results for Net1 with k f =300seck_f= 300\\sec , Δt=5sec\\Delta t = 5\\sec , and Base demand 1 under (a) Pattern II, (b) Pattern III.Figure: Final sensor placement results for Net1 with k f =300seck_f= 300\\sec , Δt=5sec\\Delta t = 5\\sec consider five different demand profiles (fusion of Fig.",
", Fig.",
", and Fig.",
").The final sensor placement strategies under the three base demands 1, 2, and 3 are shown as Fig.",
"REF , Fig.",
"REF , and Fig.",
"REF , and the corresponding detailed occupation time are not shown for brevity.",
"It can be observed that the greedy-optimal location switches from $\\mathcal {S}_{r=1}^* = \\lbrace \\mathrm {J}10 \\rbrace $ (under base demand 1) to $\\mathcal {S}_{r=1}^* = \\lbrace \\mathrm {J}12 \\rbrace $ (under base demand 3) along with changing base demand; when $r=3$ , it switches from $\\mathcal {S}_{r=3}^* = \\lbrace \\mathrm {J}10, \\mathrm {J}12, \\mathrm {J}21 \\rbrace $ (under base demand 1) to $\\mathcal {S}_{r=3}^* = \\lbrace \\mathrm {J}11, \\mathrm {J}12, \\mathrm {J}32 \\rbrace $ (under base demand 3); when $r=5$ , it switches from $\\mathcal {S}_{r=5}^* = \\lbrace \\mathrm {J}10, \\mathrm {J}12, \\mathrm {J}21, \\mathrm {J}31, \\mathrm {T}2\\rbrace $ (under base demand 1) to $\\mathcal {S}_{r=5}^* = \\lbrace \\mathrm {J}11, \\mathrm {J}12, \\mathrm {J}23, \\mathrm {J}31, \\mathrm {J}32 \\rbrace $ (under base demand 2).",
"This showcases changing base demands or different demand profiles indeed have an impact on the sensor placement, but Algorithm still returns the best placement according to the chosen metrics.",
"Second, to test the impact of patterns, Patterns I, II, and III in Fig.",
"REF are used when base demand 1 is fixed (see Fig.",
"REF ).",
"We have another $n_d = 3$ different group of demand profiles.",
"Again, these patterns are only used for illustrative purposes to test the algorithm's performance.",
"It can be seen that Pattern I is relatively flatter compared with the other patterns, while Patterns II and III vary dramatically and are complementary to each other.",
"The final sensor placement strategies under Patterns I, II, and III are shown as Fig.",
"REF , Fig.",
"REF , and Fig.",
"REF that can also be viewed as three corresponding matrices with only zeros and ones element (not selected or selected).",
"It can be observed that the greedy-optimal location switches from $\\mathcal {S}_{r=1}^* = \\lbrace \\mathrm {J}10 \\rbrace $ to $\\mathcal {S}_{r=1}^* = \\lbrace \\mathrm {J}21 \\rbrace $ and from $\\mathcal {S}_{r=3}^* = \\lbrace \\mathrm {J}10, \\mathrm {J}12, \\mathrm {J}21 \\rbrace $ to $\\mathcal {S}_{r=3}^* = \\lbrace \\mathrm {J}21, \\mathrm {J}22, \\mathrm {J}31 \\rbrace $ .",
"With the above comparisons, we claim that both base demands and patterns would have impacts on the final sensor placement solution in Net1.",
"In order to quantify the similarity between two sensor placement strategies $S_1^*$ and $S_2^*$ (viewed as matrices with only zeros and ones), we define a similarity metric as $-\\sum \\sum \\oplus (S_1^*,S_2^*)$ , where $\\oplus $ stands for element-wise logical operator xor.",
"Note that this similarity metric is always a negative value, and when two matrices are the same, the largest similarity value 0 is reached.",
"With applying this similarity metric, Fig.",
"REF is closer or more similar to Fig.",
"REF than Fig.",
"REF , That is, the pattern tends to cause less impacts than the base demand in Net1 case.",
"This conclusion may extend to the other networks, and it is always safe to claim that varying demand profiles at each node has significant impact on the sensor placement strategy.",
"If we consider all discussed $n_d = 5$ demand profiles $D_i \\in \\mathbb {R}^{5 \\times k_f}$ where $i = 1, \\ldots , n_d$ , and run Algorithm , the final sensor placement results considering Patterns I with Base demand 1,2, and 3, and Patterns II and III are shown as Fig.",
"REF , which is the fusion of Fig.",
"REF , Fig.",
"REF , and Fig.",
"REF .",
"The final solution $\\mathcal {S}_{r=1}^* = \\lbrace \\mathrm {J}10\\rbrace $ , $\\mathcal {S}_{r=3}^* = \\mathcal {S}_{r=1}^* \\bigcup \\lbrace \\mathrm {J}21, \\mathrm {T}2\\rbrace $ , and $\\mathcal {S}_{r=5}^* = \\mathcal {S}_{r=3}^* \\bigcup \\lbrace \\mathrm {J}12, \\mathrm {J}31\\rbrace $ , thereby showcasing the greedy-optimal solution for Algorithm that exploits supermodularity of the set function optimization.",
"In this section, the conclusions drawn from looped Net1 network in previous section are further corroborated via the Net3 water network shown in Fig.",
"REF c with 90 junctions, 2 reservoirs, 3 tanks, 117 pipes, and 2 pumps.",
"The base demands of all junctions are assumed as fixed, and a relative flatten pattern (varies slightly) are tested.",
"The results selecting $r = 2,8,14$ from 95 node locations are shown as Fig.",
"REF , the detailed locations are presented in Tab.",
"REF , and set $\\mathcal {S}^*_{r=2} \\subset \\mathcal {S}^*_{r=8} \\subset \\mathcal {S}^*_{r=14}$ indicates the supermodularity property of the solution for this Net3 network.",
"This showcases this property for even a larger network, further reaffirming the performance of the greedy algorithm.",
"Besides that, the motivations behind testing Net3 network from a practical point of view come in two aspects, that are (i) whether it is effective or not via adding extra sensors to reduce the Kalman filter estimation error?",
"and (ii) is the strategy from Algorithm better than random strategies?",
"Figure: Sensor placement results for Net3 with r=2,8,14r = 2,8,14 (95 node IDs are not shown for brevity).Table: Sensor placement results for Net 3."
],
[
"Estimation performance and comparing with random SP",
"This section computationally investigates two important issues closely related to the two motivations aforementioned: First, the performance of the state estimation and Kalman filter as the number of utilized sensors $r$ in the water network varies.",
"The second issue is whether a uniform (i.e., placing a sensor every other junction) or random sensor placement strategy yields a comparable performance—in terms of the state estimation metric—when comparison with the greedy-optimal presented in Algorithm .",
"Both issues are investigated for the larger network Net3.",
"First, the relationship between performance of the Kalman filter $f(\\mathcal {S}_r)$ (REF ) and the number of sensors is shown as Fig.",
"REF .",
"Interestingly, Kalman filter performance $f(\\mathcal {S}_r)$ decreases roughly linearly as the number of sensors $r$ increases from 1 to 14 for three different hydraulic simulations.",
"Specifically, Fig.",
"REF showcases the performance of the greedy-optimal solution when $r$ is fixed in Algorithm with fixed hydraulic profiles ($T_h = 0^\\mathrm {th}, 10^\\mathrm {th}, 20^\\mathrm {th}$ hour) i.e., the three figures in Fig.",
"REF show similar trend for three different hydraulic profiles.",
"The best performance or lower bounds under the corresponding cases are reached when all sensor locations are selected ($r = 95$ ).",
"This indicates that one would not expect a large improvement of Kalman filter performance via increasing the number of sensors even the locations of added sensors are all greedy-optimal.",
"Furthermore, the time-varying Kalman filter performance $f(\\mathcal {S}^*_{r = 14})$ for 24 hours is depicted via the blue line in Fig.",
"REF .",
"the performance value can easily reach $10^{5}$ level for this relatively large-scale network due to (i) the large dimension of $z$ ($n_z = 3.066 \\times 10^6$ ), (ii) covariance matrix $\\mathbb {C}$ with tiny diagonal element (i.e., $5 \\times 10^{-3}$ ), and (iii) the typical value of $k_f$ is 200 in Net 3 resulting in $W_o$ with huge value element in (REF ).",
"Moreover, the trend of the blue line is decided by the hydraulic profile such as the flow rates for 24 hours, the plot of flow rates are not shown for brevity.",
"To address the second issue, we showcase the performance of a random sensor placement with a fixed number of sensors $r=14$ .",
"Specifically, ten random sensor placements are generated for every hydraulic simulation.",
"To quantify the performance of the proposed optimal placement, we define the relative performance of a random placement strategy $\\hat{\\mathcal {S}}$ as $\\Delta f(\\hat{\\mathcal {S}}_{r=14}) = f(\\hat{\\mathcal {S}}_{r=14}) - f(\\mathcal {S}^*_{r=14})$ .",
"A smaller value of $\\Delta f(\\hat{\\mathcal {S}}_{r=14})$ implies a better optimal placement.",
"The red lines in Fig.",
"REF are the relative performance of ten different randomizations—all of them are greater than zero showcasing a much better state estimation performance through the greedy algorithm.",
"Even though the differences of performance are only 100-200 on average, the actual Kalman filter performance is orders of magnitude better due to fact that the $\\log \\det $ function is used to quantify the state estimation accuracy.",
"That is, the $\\mathcal {S}^*_{r=14}$ obtained from Algorithm performs significantly better than any random strategy $\\hat{\\mathcal {S}}^*_{r=14}$ .",
"Figure: Kalman filter performance f(𝒮 r * )f(\\mathcal {S}_{r}^*) with r={1,...,14,95}r = \\lbrace 1,\\ldots ,14,95\\rbrace when T h =0 th ,10 th ,20 th T_h = 0^\\mathrm {th}, 10^\\mathrm {th}, 20^\\mathrm {th} hour (a), performance f(𝒮 r=14 * )f(\\mathcal {S}^*_{r=14}) for 24 hours (blue line in (b)), and the relative performance of ten randomized sensor placements Δf(𝒮 ^ r=14 )\\Delta f(\\hat{\\mathcal {S}}_{r=14}) (red lines in (b))."
],
[
"Conclusions, Paper Limitations, and Future Directions",
"The paper presents a new computational method that results in sensor placements of WQ sensing devices in water networks.",
"The method exclusively focuses on the WDN observability in regards to the WQ dynamics.",
"After thoroughly testing three networks, we summarize the findings.",
"First, the impacts of choosing L-W scheme time-step $\\Delta t$ (or the number of segments $s_L$ ) and the length of a single hydraulic simulation $k_f$ on the sensor placement strategy is minor and can be neglected.",
"Second, our proposed method can be applied to practical networks to determine sensor placement strategy because in practice historical data for demand patterns are available, thereby furnishing the sensor placement with the most common demand patterns.",
"Hence, there is a possibility that the optimal sensor placement in terms of occupation time obtained would relatively be time-invariant.",
"Third, the algorithm verifies the supermodular nature of the advocated set function optimization as corroborated via different test cases on three different networks.",
"Fourth, and even if demand patterns change significantly, the algorithm can still be used to obtain a sensor placement that optimizes the sensor occupation time.",
"The paper does not advocate for only looking at state estimation metrics for water quality sensor placement.",
"As mentioned in Section , a plethora of social and engineering objectives are typically considered in the literature to solve the WQSP.",
"To that end, it is imperative that the proposed approach in this paper be studied in light of the other metrics and objectives discussed in the literature (such as minimizing the expected population and amount of contaminated water affected by an intrusion event).",
"Consequently, an investigation of balancing engineering and state estimation objectives with more social-driven ones is needed.",
"Hence, the objective of this paper is to provide a tool for the system operator that quantifies network observability vis-a-vis water quality sensor placements.",
"The water system operator can also balance the objective of obtaining network-wide observability with these other metrics.",
"Future work will focus on this limitation of the present work, in addition to considering multi-species dynamics that are nonlinear in the process model, which necessitate alternate approaches to quantify observability of nonlinear dynamic networks.",
"This will also allow examining the potential reaction between contaminants and chlorine residuals that the sensors are monitoring."
],
[
"Data Availability Statement",
"Some or all data, models, or code used during the study were provided by a third party.",
"Specifically, we provide a Github link that includes all the models, the data, and the results from the case study .",
"This material is based upon work supported by the National Science Foundation under Grants 1728629, 1728605, 2015671, and 2015603."
],
[
"Scalability and Efficient Algorithm Implementation",
"This section presents a brief discussion on an efficient implementation of Algorithm in light of the large-scale nature of the problem.",
"This nature is due to the size of water networks, but mainly due to the space discretization of the PDE.",
"This results in a large state-space dimension for the LDE model (REF ).",
"Considering Net3 for example with a single hydraulic simulation (i.e., $t = 5$ minutes or equally $k_f = 300$ time-steps when $\\Delta t$ = 1 second to reach a decent performance in L-W scheme), the number of segments of different pipes is set as $s_L = 1000$ .",
"Given these figures, the corresponding dimension of the state-space model (REF ) and (REF ) is $n_x = 117,099$ .",
"That is, $A \\in \\mathbb {R}^{117,099 \\times 117,099 }$ , $C \\in \\mathbb {R}^{95 \\times 117,099 }$ in (REF ), $\\mathcal {O}(k_f) \\in \\mathbb {R}^{95 k_f \\times 117,099 k_f}$ in (REF ), and $W_o \\in \\mathbb {R}^{117,099 k_f \\times 117,099 k_f}$ in (REF ) with $k_f = 300$ .",
"Next, we discuss the balance between the accuracy of our LDE and the computational burden.",
"From the above example, the problem dimension is determined via number of the water quality simulation time-steps $k_f$ , the number of pipes $n_{\\mathrm {P}} $ , and the number of segments of each single pipe $s_{L}$ .",
"Note that $k_f = t/\\Delta t$ , where $\\Delta t \\le \\min (\\frac{L_i}{s_{L_i} v_{i}(t)})$ for all $i \\in \\mathcal {P}$ .",
"Ideally, parameter $s_{L_i}$ should be as large as possible to ensure accuracy of LDE model.",
"After fixing the length of the interval $t$ , parameter $k_f$ ($\\Delta t$ ) should be as small (large) as possible to reduce the dimension of $\\mathcal {O}$ in (REF ).",
"This subsequently reduces the computational burden of finding the $\\log $ $\\operatorname{det}$ of a large-scale matrix in (REF )—and hence improves the computational tractability of Algorithm .",
"Evidently, there exists a conflict between increasing of the accuracy of the LDE model (via increasing $s_L$ for each pipe) and reducing computational burden (via decreasing $k_f$ , increasing $\\Delta t$ or, equally, decreasing $s_L$ ) in a single hydraulic simulation.",
"We next show simple approaches which can alleviate this conflict, significantly reduce the computational burden while maintaining the accuracy, and yield a scale algorithmic implementation.",
"First, the state-space matrices $A$ and $C$ , which are the major component of the dynamic water quality model and all subsequent matrices and Algorithm , are extremely sparse.",
"In fact, more than 99.9% of the entries of these matrices are zeros.",
"Thus, matrices $A$ and $C$ can be expressed in the sparse matrix form thereby reducing the computational burden and the needed memory by many orders of magnitude.",
"We use this in our Github codes.",
"Second, to reduce the number of time-steps $k_f$ or increase $\\Delta t$ , the dynamic number of segments of each pipe should be adopted due to the fact that the $\\Delta t$ is related with pipe length $L_i$ and its velocity $v_{i}(t)$ which depends on the hydraulic simulation and demand profile.",
"That is, for the short (long) Pipe $i$ with large (small) velocity for interval $t$ , the number of segments $s_{L_i}$ can be chosen as a relatively small (large) value that is still enough to ensure the accuracy of L-W scheme.",
"With varying number of segments, the dimension of $A$ (i.e., $n_x$ ) varies in different hydraulic simulations.",
"The interested reader is referred to our Github for the details of this implementation.",
"For example, if the velocities during interval $t_1$ are two times less than the ones during interval $t_2$ , then the size of $n_x$ can be reduced by half in $t_2$ .",
"Third, the WQSP problem parameters (REF ) in each single hydraulic simulation are independent on each other.",
"This is due to $A$ and $C$ in (REF ) are time-invariant in a single interval.",
"After obtaining initial conditions and matrices $A$ , $C$ offline, Algorithm can be implemented through parallel computing.",
"That is, multi-intervals can be calculated simultaneously on a multi-core computer.",
"Moreover, the bottleneck of Algorithm is located in calculating the $\\log \\operatorname{det}$ of large-scale matrix (REF ).",
"To that end, we adopt the LU and Cholesky decompositions to accelerate the evaluation of the $\\log \\operatorname{det}$ objective function for different sensor placements.",
"Fourth, it is clear that reducing the length of a single hydraulic simulation $t$ would be another effective trick, since it would reduce the dimension of $\\mathcal {O}(k)$ and $W_{O}(k)$ significantly according to (REF ).",
"Finally, it is worth mentioning that Algorithm is indeed offline seeing that it solves a placement problem of water quality sensors: sensors that cannot have varying geographic locations.",
"Powerful computational resources, at the disposal of water network operators, can hence be used to run the algorithm.",
"We present the tested computational time for running a horizon of 24 hours for the tested three networks.",
"For the three-node network, it takes $45.7 \\sec $ to terminate Algorithm ; for Net1 with larger discretization time-step $\\Delta t = 10 \\sec $ , the computational time is $129.6 \\sec $ , and the computational time becomes $188.1 \\sec $ when the time-step is reduced by half, that is, $\\Delta t = 5 \\sec $ .",
"When we change a new demand profile for the Net1 network, the computational time can reach $555.6 \\sec $ ; for the Net 3 network it takes almost four hours for the entire 24 hour simulation horizon.",
"The reasons of the varying computational time are: (i) special demand profiles may result in time-consuming LU or Cholesky decompositions, (ii) the discretization time-step $\\Delta t$ and length of single observation time $k_f$ have an impact on the size of $W_o$ and result in different computational time, (iii) and the tested computer have different numbers of cores and capacities of memory."
]
] | 2012.05268 |
[
[
"Reconstruction of a single-active-electron potential from electron\n momentum distribution produced by strong-field ionization using optimization\n technique"
],
[
"Abstract We present a method for retrieving of single-active electron potential in an atom or molecule from a given momentum distribution of photoelectrons ionized by a strong laser field.",
"In this method the potential varying within certain limits is found as the result of the optimization procedure aimed at reproducing the given momentum distribution.",
"The optimization using numerical solution of the time-dependent Schrodinger equation for ionization of a model one-dimensional atom shows the good accuracy of the potential reconstruction method.",
"This applies to different ways used for representing of the potential under reconstruction, including a parametrization and determination of the potential by specifying its values on a spatial grid."
],
[
"Introduction",
"Ionization of molecules by strong laser pulses has been attracting considerable attention both in experiment and theory, see [1], [2] for recent reviews.",
"The reason for this is that the accurate description of ionization step is necessary for understanding of a variety of phenomena arising from the interaction of strong laser pulses with molecules.",
"These phenomena include above-threshold ionization (ATI), formation of the high-energy plateau in the ATI spectrum (high-order ATI), generation of high-order harmonics, nonsequential double ionization, etc (see [3], [4], [5], [6] for reviews).",
"Among the main theoretical approaches used to describe ionization of atoms by strong laser pulses are the direct numerical solution of the time-dependent Schrödinger equation (TDSE) (see, e.g., [7], [8], [9], [10]), the strong-field approximation [11], [12], [13], and the semiclassical models using classical mechanics to describe the motion of an electron after it has been promoted to the continuum.",
"The two-step [14], [15], [16] and the three-step [17], [18] models are the widely known examples of semiclassical approaches.",
"All these theoretical methods have been generalized to the molecular case.",
"Obviously, it is more difficult to describe the strong-field ionization of a molecule than this process in an atom.",
"Indeed, due to the nuclear motion a molecule has extra degrees of freedom as compared to the atomic case.",
"These additional degrees of freedom lead to a necessity of taking into account the corresponding time scales.",
"Furthermore, electronic orbitals in molecules often have complicated shapes.",
"In principle, the direct numerical solution of the TDSE allows to fully take into account all the features of the molecular ionization.",
"However, this numerical solution of the three-dimensional (3D) TDSE for a molecule interacting with a strong laser pulse is a very complicated task.",
"It is possible to solve the 3D TDSE only for the simplest molecules selecting the most relevant degrees of freedom [19], [20].",
"Therefore, a number of approximations has to be used to solve the TDSE for strong-field ionization of a molecule.",
"Usually it is assumed that the positions of the atomic nuclei are fixed.",
"Simultaneously, the single-active electron (SAE) approximation [21], [22] is applied.",
"Within the SAE the strong-field ionization of an atom or molecule is described as an interaction of only one active electron with the laser radiation.",
"This active electron moves in the combined effective potential and the electric field of the laser pulse.",
"Therefore, it is important to obtain suitable SAE potentials for various molecules.",
"However, the calculation of the SAE potential is a non-trivial task.",
"The potential that describes the interaction of one active electron with the frozen core can be approximated as $V\\left(\\mathbf {r}\\right)=V_{en}\\left(\\mathbf {r}\\right)+V_{ee}^{cl}\\left(\\mathbf {r}\\right)+V_{xc}\\left(\\mathbf {r}\\right)$ , where $V_{en}\\left(\\mathbf {r}\\right)$ is the electron-electron nuclear interaction, $V_{ee}^{cl}\\left(\\mathbf {r}\\right)$ is the Hartree electron-electron repulsion, and $V_{xc}\\left(\\mathbf {r}\\right)$ is the exchange-correlation potential.",
"The occupied orbitals and therefore the Hartree term $V_{ee}^{cl}\\left(\\mathbf {r}\\right)$ can be obtained using the standard quantum chemistry packages: GAMESS [23], Gaussian [24], etc.",
"In exchange-only calculations, the exchange-correlation potential can be approximated by the local density approximation [25].",
"To ensure that the resulting SAE potential has Coulomb behavior at large distances, a gradient correction to the local density [26] can be applied.",
"The approach described here was used in [27], [28] to calculate the potentials for some selected simple molecules.",
"However, simpler model potentials are used in many studies applying the solution of the TDSE for molecules.",
"Some of these model potentials are constructed as simple combinations of a few Coulomb soft-core atomic potentials with the centers at the atomic nuclei, whereas the others rely on more complicated expressions (see, e.g., [29], [30], [31], [32]).",
"Usually the model potential depends on one or few parameters.",
"While one parameter of the model potential allows to reproduce only one bound-state energy, the presence of several parameters makes it possible to recover the predefined energies of a few different bound states.",
"Usually the parameters of a model potential are adjusted using an optimization technique.",
"This suggests to explore the possibility of obtaining the SAE potential as a result of some optimization procedure.",
"It is well known that the potential experienced by the ionized electron is encoded in the measured photoelectron momentum distribution (PMD).",
"Therefore, it should be possible to optimize the unknown SAE potential in order to reproduce a given PMD generated by strong-field ionization of an atom or molecule.",
"In this paper we develop such an approach to retrieval of the SAE potentials.",
"We assume that the given momentum distribution is measured in an experiment and refer to it as the target PMD (TPMD).",
"The TPMD is considered as a goal that is to be achieved by an optimization algorithm considering the unknown values of the potential (or expansion coefficients of the potential in a given basis) as parameters that are optimized.",
"It should be stressed that there is a substantial difference between the approach proposed here and the quantum optimal control theory (QOCT), see [33], [34] for reviews.",
"The latter provides a powerful theoretical approach to the optimization and control of various quantum phenomena, including those in strong laser fields (see, e.g., [35], [36], [37], [38]).",
"Indeed, the QOCT treats the electric field of the laser pulse as a control function, whereas the effective potential experienced by an electron is to be predefined and does not change in optimization.",
"Any optimization requires a specification of a measure that allows to identify whether the optimization target is achieved.",
"Since a PMD is a picture, it is natural to specify such a measure using tools employed in image analysis and pattern recognition.",
"Therefore, the measures applied to compare different digital images or videos (see, e.g.",
"[39], [40]) can be used to estimate the similarity of the two PMD's: the result of current iteration of an optimization algorithm and the target distribution.",
"However, a valid choice of the specific measure and its application to comparison of the PMD's require thorough studies.",
"For this reason, we leave the application of image recognition tools for future investigations.",
"In this paper we retrieve the SAE potential from momentum distribution produced in ionization of a one-dimensional (1D) model atom.",
"The 1D momentum distribution is a function of only one variable (momentum component along one spatial axis).",
"Therefore, the widely-known measures used in variation calculus and functional analysis (see, e.g., [41]) can be applied for comparison of different PMD's.",
"It was shown in the studies of the ATI process that the vast majority of the photoelectrons do not recollide with their parent ions.",
"These electrons are referred to as direct electrons.",
"They are detected with the energies below $2U_{p}$ , where $U_{p}=F_{0}^{2}/\\omega ^2$ is the ponderomotive energy.",
"Here $F_{0}$ is the field strength and $\\omega $ is the angular frequency (atomic units are used throughout the paper).",
"There are also rescattered electrons that due to oscillations of the laser field come back to their parent ions and rescatter on them by large angles.",
"The rescattered electrons form the high-energy plateau in the ATI spectrum, i.e., they are responsible for the high-order ATI.",
"In our optimization-based approach only momentum distributions of the direct electrons are used.",
"The optimization technique relying on the distributions of rescattered electrons will be the subject of further studies.",
"The paper is organized as follows.",
"In Sec.",
"II we briefly discuss our approach to solve the 1D TDSE, measures used to compare electron momentum distributions, and derivative-free optimization algorithms.",
"In Sec.",
"III we apply our method to retrieve the soft-core Coulomb potential from the PMD's produced by ionization of a 1D model atom.",
"We test our approach for two different ways of representing the unknown potential and discuss various optimization strategies.",
"The conclusions and outlook are given in Sec.",
"IV.",
"We define a few-cycle laser pulse linearly polarized along the $x$ -axis by specifying its vector-potential: $\\mathbf {A}\\left(t\\right)=\\left(-1\\right)^{n_p+1}\\frac{F_0}{\\omega }\\sin ^2\\left(\\frac{\\omega t}{2n_p}\\right)\\sin \\left(\\omega t +\\varphi \\right)\\mathbf {e}_{x},$ where $n_p$ is number of optical cycles within the pulse, $\\varphi $ is the carrier envelope phase, $\\mathbf {e}_x$ is a unit vector, and the laser pulse is present between $t=0$ and $T=\\left(2\\pi /\\omega \\right)n_{p}$ .",
"The electric field can be calculated from vector-potential (REF ) by $\\mathbf {F}=-d\\mathbf {A}/dt$ .",
"In the velocity gauge, the 1D TDSE for an electron interacting with the lase pulse is given by $i\\frac{\\partial }{\\partial t}\\Psi \\left(x,t\\right)=\\left[\\frac{1}{2}\\left(-i\\frac{\\partial }{\\partial x}+A_{x}\\left(t\\right)\\right)^2+V\\left(x\\right)\\right]\\Psi \\left(x,t\\right),$ where $\\Psi \\left(x,t\\right)$ is the time-dependent wave function in coordinate representation and $V\\left(x\\right)$ is the SAE potential.",
"In the absence of the electric field, the time-independent Schrodinger equation reads as $\\left[-\\frac{1}{2}\\frac{d^2}{dx^2}+V\\left(x\\right)\\right]\\Psi \\left(x\\right)=E\\Psi \\left(x\\right),$ where $\\Psi \\left(x\\right)$ and $E$ are the eigenfunction and the corresponding energy eigenvalue, respectively.",
"The eigenvalue problem (REF ) is solved on a grid using the three-step formula to approximate the second derivative.",
"Hence, the diagonalization routine developed for sparse matrices [42] can be used.",
"Alternatively, the ground state and the first few excited states can be found by imaginary time propagation (see, e.g., [43], [44]).",
"The computational box is centered at $x=0$ and extends to $\\pm x_{max}$ , i.e., $x\\in \\left[-x_{max},x_{max}\\right]$ .",
"Typically, we set $x_{max}=250.0$ a.u and use a grid consisting of $N=4096$ points, what corresponds to the grid spacing $dx=0.1225$ a.u.",
"The well-known split-operator method [45] is used to solve the TDSE (REF ).",
"The time step is $\\Delta t=0.055$ a.u.",
"We prevent unphysical reflections from the boundary of the grid by using absorbing boundaries, i.e., at every step of the time propagation we multiply the wave function in the region $\\left|x\\right|>x_{b}$ by the mask: $M\\left(x\\right)=\\cos ^{1/6}\\left[\\frac{\\pi \\left(\\left|x\\right|-x_{b}\\right)}{2\\left(x_{max}-x_{b}\\right)}\\right]$ with $x_b=3x_{max}/4$ .",
"Hence, $x=\\pm x_{b}$ correspond to the internal boundaries of the absorbing regions.",
"As the result, at every time step the part of the wave function in the mask region is absorbed without any effect on the $\\left|x\\right|<x_{b}$ domain.",
"We calculate the electron momentum distributions using the mask method [46]."
],
[
"Comparison of electron momentum distributions and optimization algorithms",
"The optimization procedure developed here is based on comparison of the 1D electron momentum distributions, which are functions of one variable.",
"The following metrics are widely used to calculate the distance between continuous functions $f\\left(x\\right)$ and $g\\left(x\\right)$ defined for $x\\in \\left[a, b\\right]$ : $\\rho _{1}\\left[f\\left(x\\right),g\\left(x\\right)\\right]=\\max _{x\\in \\left[a,b\\right]}\\left|f\\left(x\\right)-g\\left(x\\right)\\right|$ and $\\rho _{2}\\left[f\\left(x\\right),g\\left(x\\right)\\right]=\\left\\lbrace \\int _{a}^{b}dx\\left[f\\left(x\\right)-g\\left(x\\right)\\right]^{2}\\right\\rbrace ^{1/2},$ see, e.g., [41] for a textbook treatment.",
"We use metric (REF ) as a measure of difference between two PMD's.",
"We note that before calculating the distance (REF ) we normalize electron momentum distributions to the total ionization yield (i.e., the area under the graph of the PMD).",
"Therefore, our optimization procedure relies only on the shape of momentum distributions, but not on the ionization probabilities.",
"We believe that similar approach when applied to the 3D case will help to facilitate the retrieval of the unknown potential from experimental electron momentum distributions.",
"Since the derivatives of the similarity measure with respect to unknown potential values (or any other parameters used to represent the potential) can be calculated only numerically, it is not practical to use any gradient-based optimization method.",
"Instead, it is appropriate to use a derivative-free optimization technique, see [47], [48] for recent reviews.",
"We apply particle swarm optimization method [49], [50], surrogate optimization technique [51], and pattern search method [52], [53], [54].",
"The MATLAB system [55] is used for simulations."
],
[
"Reconstruction of SAE potential on a grid",
"In this work we reconstruct the soft-core Coulomb potential: $V\\left(x\\right)=\\frac{Z}{\\sqrt{x^2+a}}$ with $Z=1.0$ and $a=1.0$ , see [56].",
"At first, we do not use any parametrization to represent the potential (REF ).",
"Instead, we determine the potential that is to be retrieved by specifying its values in certain points of the $x$ -axis.",
"The potential values in any other points are found by interpolation.",
"Here we use cubic spline interpolation [57].",
"As many other single-active electron potentials the potential $V\\left(x\\right)$ changes more rapidly for small values of $x$ .",
"Therefore, it is natural to use a non-uniform grid to represent the potential (REF ).",
"Here we apply the following grid used for development of generalized pseudospectral methods: $x=\\gamma \\frac{1+x_{0}}{1-x_{0}+x_{0}^{m}},$ where $x_{0}^{m}=2\\gamma /x_{max}$ and $\\gamma $ is the mapping parameter [58], [59], [60].",
"Equation (REF ) transforms a uniform grid within the domain $x_{0}\\in \\left[-1.0, 1.0\\right]$ to a nonuniform grid in the domain $x\\in \\left[0, x_{max}\\right]$ .",
"The points of this grid are to be reflected with respect to $x=0$ and thus a nonuniform grid in the whole range $x\\in \\left[-x_{max}, x_{max}\\right]$ is obtained.",
"Then the question arises: How many points of the grid (REF ) are required to represent the potential with sufficient accuracy?",
"To answer this question, we choose different numbers of points for the uniform grid in the range $\\left[-1.0, 1.0\\right]$ , find the corresponding point of the nonuniform grid (REF ), and calculate the potential values at these points.",
"For each number of points of the nonuniform grid we interpolate the potential $V\\left(x\\right)$ at every point of the dense uniform grid with $N$ points.",
"For this interpolated potential we find energy eigenvalues and the corresponding eigenfunctions, solve the TDSE for a given laser pulse, and calculate electron momentum distributions.",
"We next compare these PMD's with the reference momentum distribution obtained in the case that the potential (REF ) is directly calculated on the dense uniform grid consisting of $N$ points, see Figure REF .",
"It is seen that about 20 points of the nonuniform grid (REF ) is sufficient to reproduce the reference PMD accurately enough.",
"Indeed, the difference between these distributions calculated in accord with the measure (REF ) is about $0.01$ .",
"Taking into account that the potential $V\\left(x\\right)$ is an even function, the number of the points can be reduced by a factor of two, i.e., only 10 points are needed.",
"Figure: The electron momentum distributions for ionization of a 1D atom by a laser pulse with a duration of n p =4n_p=4 cycles, wavelength of 800 nm, phase ϕ=0\\varphi =0, and intensity of 2.0×10 14 2.0\\times 10^{14} W/cm 2 ^2.",
"The distributions are obtained from the solution of the TDSE () with the potential () calculated on a uniform grid consisting of N=4096N=4096 points (thick light-blue curve), as well as with the same potential determined by its values on the non-uniform grid () with γ=20.5\\gamma =20.5 consisting of 10 (dashed green curve), 15 (dotted black curve), and 20 (thin magenta curve) points.Using the nonuniform grid we first attempt to reconstruct the unknown potential using the ground state and the first excited state energies only.",
"Such an attempt may raise questions.",
"Indeed, it is well-known that even in the 1D case the potential cannot be unambiguously determined from one or a few energy eigenvalues.",
"Only a symmetric reflectionless potential can be restored based on its complete set of the bound state energies [61].",
"However, it is evident that the optimization of a “black-box\" function that depends on 10-20 parameters is a difficult numerical problem.",
"An optimization algorithm used to solve this problem requires an initial approximation to the maximum (minimum).",
"The success of the optimization and the convergence speed critically depend on the quality of the initial approximation.",
"It turns out that satisfactory initial approximations can often be obtained as a result of optimization of only a few bound state energies.",
"The optimization methods also require the specification of the boundaries, within which the optimization parameters (in this case, the values of the potential in the grid points) can vary.",
"We specify these ranges by sandwiching the unknown potential between the two known potential functions.",
"As these functions we use $V_{1,2}\\left(x\\right)=\\frac{Z_{1,2}}{\\sqrt{x^2+a_{1,2}}}.$ At first, we chose $Z_{1,2}=1.0$ and $a_1$ and $a_2$ equal to $0.4$ and $1.5$ , respectively.",
"Therefore, $V_1\\left(x\\right)$ is the lower boundary for the potential $V\\left(x\\right)$ that is to be retrieved, whereas $V_2\\left(x\\right)$ is the upper boundary.",
"The SAE potential reconstructed by optimization the ground state energy is shown in Figure 2 (a) together with the boundaries $V_1\\left(x\\right)$ and $V_2\\left(x\\right)$ .",
"For the optimization we use the particle swarm method.",
"It is seen that the obtained potential is in a good quantitative agreement with the one that we wanted to reconstruct.",
"However, the situation changes dramatically, if the boundaries for the potential values under reconstruction are not as tight as in the example shown in Figure REF (a).",
"Figure: The values of the SAE potential on the non-uniform grid () (magenta circles) reconstructed by optimization of the bound state and first excited states energies only, boundaries for the potential values (dashed blue curves), the potential obtained by spline interpolation based on the reconstructed values (thin magenta curve), and the soft-core Coulomb potential (thick black curve).",
"Panel (a) shows the optimization result for the case where the optimized potential values are bounded by the potentials () with Z 1,2 =1.0Z_{1,2}=1.0 and a 1 a_1 and a 2 a_2 equal to 0.40.4 and 1.51.5, respectively.",
"Panel (b) displays the potential obtained for the optimization parameters restricted by V 1 xV_{1}\\left(x\\right) calculated from equation () with Z 1 =1.0Z_{1}=1.0 and a 1 =0.4a_1=0.4 and V 2 x=0V_{2}\\left(x\\right)=0.",
"The parameters are the same as in Figure .In Figure REF (b) we display one of the potentials that can be obtained by optimization of the bound state energy in the case where the lower boundary for the potential values is again given by $V_1\\left(x\\right)$ , but the upper boundary is chosen to be zero: $V_2\\left(x\\right)=0$ .",
"It should be stressed that the optimization result is not unique for the chosen boundaries.",
"The successive runs of the optimization algorithms lead to a whole family of the potentials with very close ground state energies.",
"This agrees with the conclusions of [61].",
"It is seen that there is even no qualitative agreement between the potential shown in Figure 2 (b) and the soft-core Coulomb potential (REF ) that is to be reconstructed.",
"We now turn to the retrieval of the potential from the electron momentum distributions.",
"At first, we use the same boundaries for the unknown potential values as in the example shown in Figure REF (a).",
"We minimize the difference as defined by the measure (REF ) between the PMD calculated for a potential defined by its values in the grid points and the PMD obtained for the potential (REF ) with the same laser pulse.",
"The minimum value of the metric $\\rho _2\\left[{\\rm PMD},{\\rm TPMD}\\right]$ obtained in optimization is $0.019$ .",
"The potential retrieved in the optimization procedure is shown in Figure REF (a).",
"It is seen that the obtained potential almost coincides with the soft-core Coulomb potential (REF ) we wanted to reconstruct.",
"The same is true for the distributions calculated for the retrieved potential and the potential (REF ), see Fig.",
"REF (b).",
"The question arises how sensitive is the optimization result to a change in the boundaries $V_1\\left(x\\right)$ and $V_2\\left(x\\right)$ .",
"To answer this question, we perform another optimization with the broader boundaries for the allowed potential values.",
"Specifically, we sandwich the potential that is to be retrieved by $V_1\\left(x\\right)$ and $V_2\\left(x\\right)$ with $a_{1,2}=1.0$ and $Z_1$ and $Z_2$ equal to $2.0$ and $0.5$ , respectively.",
"The optimization result for these broader boundaries quantitatively agrees with the potential (REF ), see Figure REF (c).",
"The same is also true for the electron momentum distributions, see Figure REF (d).",
"The minimum value of the metric (REF ) obtained in optimization with these broader boundaries is $0.03$ , which is only slightly higher than in the previous case.",
"Figure: (a),(c) The values of the SAE potential (magenta circles) reconstructed by optimization of the electron momentum distribution, boundaries for the potential values (dashed blue curves), the potential obtained by spline interpolation using the reconstructed values (thin magenta curve), and the soft-core Coulomb potential (thick black curve).",
"(b),(d) The PMD calculated using the optimized potential (thin magenta curve) and the TPMD (thick light-blue curve).",
"Panels (a) and (b) show the optimization results for the case where the allowed potential values are bounded by the potentials V 1,2 xV_{1,2}\\left(x\\right) [equation ()] with Z 1,2 =1.0Z_{1,2}=1.0 and a 1 a_1 and a 2 a_2 equal to 0.40.4 and 1.51.5, respectively.",
"Panels (c) and (d) correspond to the boundaries V 1,2 xV_{1,2}\\left(x\\right) for the potential values given by () with a 1,2 =1.0a_{1,2}=1.0 and Z 1 Z_1 and Z 2 Z_2 equal to 0.50.5 and 2.02.0, respectively.",
"The potential under reconstruction is determined by its values on the grid () with 20 points.",
"The laser parameters are the same as in Figs.",
"and .We are now able to address the more important questions to the optimization-based approach, namely, how vulnerable is it to intensity fluctuations that are inevitable in an experiment?",
"Can the actual laser intensity be restored by applying the optimization technique?",
"In order to answer the first question, we try to retrieve the potential at the intensity of $1.0\\times 10^{14}$ W/cm$^2$ using the TPMD calculated for the higher intensity of $2.0\\times 10^{14}$ W/cm$^2$ .",
"The potential obtained in such an optimization is shown in Figure REF (a).",
"It is seen that this potential substantially differs from the potential (REF ) we expected to reconstruct.",
"Indeed, the minimum obtained value of the metric (REF ) is $0.61$ , and the PMD calculated for the retrieved potential does not agree with the target one, see Figure REF (b).",
"Figure: (a),(c) The values of the SAE potential on the non-uniform grid () with 20 points (magenta circles) reconstructed by optimization of the PMD, boundaries for the potential values (dashed blue curves), the potential obtained by spline interpolation based on the reconstructed values (thin magenta curve), and the soft-core Coulomb potential (thick black curve).",
"(b),(d) The PMD calculated using the optimized potential (thin magenta curve) and the TPMD (thick light-blue curve).",
"Panels (a) and (b) correspond to the case where the TPMD is calculated for the intensity of 2.0×10 14 2.0\\times 10^{14} W/cm 2 ^2 and the optimization of the momentum distributions is performed at the intensity of 1.0×10 14 1.0\\times 10^{14} W/cm 2 ^2.",
"Panels (c) and (d) show the results obtained treating the field strength F 0 F_0 as an additional parameter that is to be optimized.",
"The boundaries for the optimized potential values are given by equations () with Z 1,2 =1.0Z_{1,2}=1.0 and a 1 a_1 and a 2 a_2 equal to 0.40.4 and 1.51.5, respectively.",
"The parameters are the same as in Figs.",
"-.To generalize our approach to the case where the TPMD is obtained at different laser intensity, we add the field strength to the parameter set that is to be optimized.",
"This allows us to reconstruct the actual value of the laser intensity, at which the TPMD that we want to reproduce in optimization is obtained.",
"In this case, the surrogate optimization turns out to be slightly more efficient than the particle swarm method.",
"The results of this modified approach are shown in Figures REF (c) and (d).",
"It is seen from Figure REF (c) that the retrieved potential is in a quantitative agreement with the soft-core Coulomb potential (REF ) used to calculate the TPMD.",
"The same is true for the resulting electron momentum distribution, see Figure REF (d).",
"The difference between the momentum distributions $\\rho _2\\left[{\\rm PMD},{\\rm TPMD}\\right]$ comes to only $0.032$ .",
"The optimized value of the field strength is $0.0763$ a.u., whereas the exact value of $F_0$ equals to $0.0755$ a.u.",
"Although these results are encouraging, it is clear that further studies are needed to completely tackle the question regarding the intensity fluctuations.",
"To mimic a real experimental situation, it is necessary to average the PMD over the intensity distribution within the focal volume at every iteration of the optimization process.",
"We leave this modification of the proposed method for further studies implying the generalization to the 3D case.",
"The results shown in Figures REF -REF were obtained for the non-uniform grid (REF ) in the range $\\left[-x_{max}, x_{max}\\right]$ .",
"However, the typical molecular potential we intend to reconstruct has a long-range Coulomb asymptotic at large distance: $V\\left(r\\right)\\rightarrow 1/r$ at $r\\rightarrow \\infty $ .",
"Therefore, there is no need to find the values of the potential at large $x$ , and an optimization of the potential values at $x\\rightarrow \\infty $ leads to a waste of computational resources.",
"At the same time, it is highly desirable to have a denser grid for small $x$ , where the potential can vary significantly.",
"To address both these issues, from this point on we use a smaller computational box $\\left[-x_{C}, x_{C}\\right]$ setting $V\\left(x\\right)=1/\\left|x\\right|$ for $\\left|x\\right|>x_{C}$ .",
"Typically, we chose $x_{C}=10.0$ a.u.",
"and use uniform grid within the range $\\left[-x_{C}, x_{C}\\right]$ .",
"The optimization results are shown in Figures REF (a) and (b).",
"It is seen from Figure REF (a) that retrieved potential agrees well with the soft-core Coulomb potential (REF ).",
"The optimization results in a measure (REF ) equal to $0.04$ what corresponds to a good agreement between the obtained and target electron momentum distributions, see Figure REF (b).",
"A different optimization method was used here.",
"At first, we performed an optimization of the bound state and first excited state energies.",
"As mentioned above, such optimization results in a family of different potentials, unless the boundaries for the potential values are close to each other.",
"Some of these potentials resemble the desired potential (REF ), whereas the others are very different from it and do not match the Coulomb asymptote at $x=\\pm x_{C}$ .",
"Then we use all the obtained potentials as an initial approximation for the second optimization procedure that minimize the difference between the corresponding PMD and the target one.",
"For the second optimization we use the pattern search algorithm (Hook-Jeeves method [52]), see, e.g., [53], [54] for reviews.",
"This combined two-step approach to optimization of the PMD turns out to be computationally more efficient than particle swarm optimization we used before and many other gradient-free optimization methods, including simulated annealing, genetic algorithms, and surrogate optimization, if these methods start from a random initial approximation.",
"Figure: Panels (a) and (b) show the same as Figures (a) and (b) for the optimization parameters varying in wider ranges and the potential determined by the values on the uniform grid consisting of 10 points between 0 and 10.010.0 a.u.",
"Panels (c) and (d) display the optimization results for the same grid consisting of 20 points.",
"The optimization parameters are bounded by the potential V 1 xV_1\\left(x\\right) calculated from equation () with Z 1 =1.0Z_1=1.0 and a 1 =0.4a_1=0.4 and V 2 x=-0.09V_{2}\\left(x\\right)=-0.09 a.u.",
"The parameters are the same as in Figures -.The optimization results presented in Figures REF (a) and (b) were obtained for 20 grid points in the range $\\left[-x_{C}, x_{C}\\right]$ (i.e, for 10 grid points for $x\\in \\left[0, x_{C}\\right]$ ).",
"This corresponds to the grid spacing $dx=1.0$ a.u.",
"Suppose that we need better resolution along the $x$ axis, e.g., $dx=0.5$ a.u., what corresponds to 20 grid points between 0 and $x_{C}$ .",
"The most efficient way to perform the optimization of the PMD on a denser grid is to use the results obtained for a sparser grid as an initial approximation.",
"In doing so the potential values in the points of a denser grid can be restored by interpolation.",
"The results of the application of this approach are shown in Figures REF (c) and (d).",
"Note that for $dx=0.5$ a.u.",
"we achieve a perfect agreement between the retrieved potential and the desired soft-core Coulomb potential (REF ).",
"The corresponding PMD's are also almost indistinguishable from each other, see Figure REF (d).",
"The optimization algorithm terminates when the distance $\\rho _2\\left[{\\rm PMD},{\\rm TPMD}\\right]$ reduces to $0.0042$ ."
],
[
"Reconstruction of parametrized potential",
"Up to this point we have not used any parametrization to represent the unknown SAE potential.",
"It is clear, however, that this parametrization-free approach being extended to the two-dimensional and especially the 3D case will become a very difficult computational task.",
"Indeed, such an extension will lead to a necessity of optimization of a function that depends on hundred of variables.",
"We note that this task is feasible with modern computational facilities and optimization algorithms.",
"Such problems arise, for example, in research on magnetically confined plasmas for fusion energy, see, e.g.",
"[62], [63].",
"It is nevertheless of interest to test the optimization-based algorithm in the case where the unknown potential is in some way parametrized.",
"It is natural to express the SAE potential as $V\\left(x\\right)=V_{0}\\left(x\\right)\\left[1+V_{1p}\\left(x\\right)\\right],$ where $V_{0}\\left(x\\right)$ is a known potential with correct asymptotic behavior and the potential $V_{1}\\left(x\\right)$ is to be parametrized and determined.",
"Here we choose $V_{0}\\left(x\\right)=Z_{0}/\\left(x^4+a_0\\right)^{\\frac{1}{4}}$ with $Z_{0}=1.0$ and $a_{0}=1000.0$ what corresponds to a very shallow and wide potential well, which is substantially different from the potential (REF ) that should be restored.",
"The question thus arises how to parametrize the potential $V_{1}\\left(x\\right)$ in the best possible way.",
"Since we know the potential $V\\left(x\\right)$ we want to reconstruct, this question is easy to answer.",
"Indeed, by trying different options and applying standard curve fitting routines [55], we find that the rational interpolation can be used to approximate the function $V_{1p}\\left(x\\right)=V\\left(x\\right)/V_{0}\\left(x\\right)-1$ with a good accuracy.",
"Therefore, the potential $V_{1p}\\left(x\\right)$ can be represented as a quotient of two polynomials $P_m\\left(x\\right)$ and $Q_{n}\\left(x\\right)$ : $V_{1p}\\left(x\\right)=\\frac{P_{m}\\left(x\\right)}{Q_{n}\\left(x\\right)}$ It is clear that since $V_{1p}\\left(x\\right)\\rightarrow 0$ at $x\\rightarrow 0$ , the inequality $m<n$ should be fulfilled.",
"It is easy to see that the measure $\\rho _2$ of the difference between the corresponding momentum distribution and the target one dramatically changes for different choices of the $m$ and $n$ .",
"If, for example, $m=0$ and $n=2$ , the minimum value of the $\\rho _2$ that can be achieved is equal to $0.29$ .",
"This corresponds to $V_{1p}\\left(x\\right)=p_{1}/\\left(x^{2}+q_1\\right)$ with $p_1=29.99$ and $q_1=3.62$ .",
"The linear function in the nominator $\\left(m=1\\right)$ and the quadratic function in the denominator $\\left(n=2\\right)$ , i.e., $V_{1p}\\left(x\\right)=\\left(p_{1}x+p_{2}\\right)/\\left(x^2+q_{1}x+q_{2}\\right)$ , allow to reduce the measure (REF ) to $0.047$ .",
"This value is achieved for $p_{1}=-1.15$ , $p_{2}=36.90$ , $q_{1}=0.92$ , and $q_{2}=23.95$ .",
"The minimum value of the $\\rho _2$ we have obtained using the equation (REF ) is equal to $0.01$ .",
"It corresponds to the constant in the numerator and the fourth-order polynomial in the denominator of the quotient (REF ): $V_{1p}\\left(x\\right)=\\frac{p_{1}}{x^4+q_{1}x^3+q_{2}x^2+q_{3}x+q_{4}},$ where $p_{1}=1261.0$ , $q_{1}=-9.747$ , $q_{2}=66.5$ , $q_{3}=10.4$ , and $q_{4}=138.7$ .",
"It should be noted that not all of the coefficients in the denominator of this formula are positive.",
"When using the optimization-based method in practice, a series of optimizations applying different ways to parametrize the unknown potential should be performed.",
"This will allow to compare the optimization results and to choose the best way of the parametrization similarly to what we do here with the fitting of the known potential.",
"We now try to recover the parameter values in the parametrization (REF ) by optimizing the PMD.",
"At first glance it would seem that optimization of function (REF ) depending on only 5 parameters is a simple task compared to the one performed in Sec. 3.1.",
"But this is not the case, since negative values of the parameters $q_{i} \\left(i=1,...,4\\right)$ may lead to $1+V_{1p}\\left(x\\right)<0$ and, therefore, $V\\left(x\\right)>0$ for certain ranges of $x$ .",
"To prevent this situation, we use constrained optimization.",
"Specifically, instead of minimizing the measure (REF ) alone, we now look for a minimum of $\\rho _2\\left[{\\rm PMD},{\\rm TPMD}\\right]+w \\cdot V_{m}^{2}\\left[1+{\\rm sgn}\\left(V_m\\right)\\right],$ where $V_m$ is the maximum value of $V\\left(x\\right)$ in the interval $\\left[0, x_0\\right]$ , and $w$ is some weight factor.",
"Typically, we use $w$ in the range between $5.0$ and $20.0$ .",
"We allow for the following ranges of the optimization parameters: $p_1\\in \\left[-2000.0, 2000.0\\right]$ and $q_{i}\\in \\left[-100.0, 100.0\\right] \\left(i=1,...,4\\right)$ .",
"To speed up the simulations, we again perform the optimization in two steps, i.e., we use the results obtained in optimization of the ground state, first, and second excited states energies as initial approximations for the optimization of the PMD.",
"The retrieved potential and the corresponding electron momentum distribution are shown in Figures REF (a) and (b), respectively.",
"The following parameter values were obtained: $p_{1}$ =403.44, $q_{1}=3.08$ , $q_{2}=17.03$ , $q_{3}=35.57$ , and $q_{4}=80.45$ .",
"The difference $\\rho _2$ between the resulting PMD and the TPMD is $0.0164$ , what is higher than the one corresponding to the parameters obtained by approximation of the known dunction $V_{1p}\\left(x\\right)$ .",
"Therefore, some local minimum, albeit quite close to the desired global one, is found in optimization.",
"It is seen that the retrieved potential agrees well with the exact result, see Figure (9) (a).",
"The same is also true for the electron momentum distributions [see Figure REF (b)].",
"Figure: Optimization results for parametrized SAE potential.",
"Panels (a) and (c) show the reconstructed potentials (thin magenta curve) and the soft-core Coulomb potential (thick black curve).",
"Panels (b) and (d) display the comparison of the PMD's obtained from the TDSE with the retrieved (thin magenta curve) and the exact (thick light-blue curve) SAE potentials.",
"Panels [(a), (b)] and [(c), (d)] correspond to the parametrization () and (), respectively.",
"The laser parameters are the same as in Figs.",
"-.As a next step in testing the method, we represent our potential as a sum of a few Gaussian functions: $V_{1p}\\left(x\\right)=\\sum _{k=1}^{k_{max}}a_{k}\\exp \\left[-\\frac{\\left(x-b_{k}\\right)^2}{c_{k}^{2}}\\right]$ The representation (REF ) is obviously more flexible, i.e., it allows for finer variations of the potential function, as compared to the one applying rational function.",
"Here we choose $k_{max}=5$ what corresponds to 15 parameters to be optimized.",
"It should be stressed that in contrast to all other examples shown in this paper, here we do not assume any symmetry of the potential $V\\left(x\\right)$ .",
"Indeed, the allowed values of the optimization parameters are: $a_{k}\\in \\left[-4.0, 4.0\\right]$ , $b_{k}\\in \\left[-20.0, 20.0\\right]$ , and $c_{k}\\in \\left[0, 100.0\\right]$ $\\left(k=1,...,5\\right)$ .",
"As an initial approximation for the optimization of the PMD's we use the potential shown in Figure 8 (a), i.e., the one obtained with parametrization (REF ).",
"The optimization results are presented in Figures REF (c) and (d).",
"As expected, a better agreement between the retrieved potential and the exact one can be achieved with parametrization (REF ).",
"However, the measure (REF ) remains practically unchanged: $\\rho _2\\left[{\\rm PMD},{\\rm TPMD}\\right]=0.0159$ .",
"The examples above demonstrate that in the 1D case the parametrization of the potential does not offer any decisive advantages compared to the direct representation on a grid.",
"Nevertheless, it is shown that the optimization-based method also works in the case, where the potential is determined by a number of parameters.",
"This is essential in view of the possible extension of the approach onto the 3D case, where parametrization of the unknown potential is expected to become particularly important."
],
[
"Conclusions and outlook",
"In conclusion, we have developed a method capable to retrieve the SAE potential in an atom or molecule from a given momentum distribution of photoelectrons ionized by a strong laser pulse.",
"In this method the potential is found by minimization of the difference between the given momentum distribution and the one obtained from the solution of the TDSE with the SAE potential that varies in the optimization process.",
"The unknown potential is either represented by a set of its values in points of a spatial grid, or by a set of parameters.",
"We have shown that the optimization can be performed using a number of different derivative-free techniques, including particle swarm method, surrogate optimization, and pattern search.",
"It is found that the most efficient approach is based on the use of potentials obtained in optimization of a few bound-state energies as initial approximations for the optimization of the PMD.",
"We have tested our method by reconstructing of the soft-core Coulomb potential from the corresponding PMD generated in ionization of a 1D atom by a strong few-cycle laser field.",
"It is shown that the retrieved SAE potential is in a quantitative agreement with the potential we aimed to reconstruct.",
"This is true for both ways used to represent the potential under reconstruction.",
"In the case where the potential is represented by its values on a grid the spatial resolution can be effectively improved by using the optimization results on a sparse grid as an initial approximation for optimization on the dense grid.",
"This allows to avoid severe computational costs when optimizing a function depending on a few dozens of variables.",
"It is clear that the measured electron momentum distributions are affected by focal averaging.",
"We have shown that the actual laser intensity can be restored together with the SAE potential in the optimization approach.",
"Nevertheless, further work is needed to fully explore the question how sensitive is the proposed method to the focal averaging.",
"Furthermore, extension of the method to the real 3D case require a reliable measure used to compare different momentum distributions.",
"To this end, the tools of image analysis and pattern recognition can be applied.",
"It remains to be studied which of these tools are the most appropriate for the problem at hand.",
"Finally, we have restricted ourselves by the optimization of only one part of the PMD created by the direct electrons.",
"However, our preliminary results show that the momentum distribution of the rescattered photoelectrons are more sensitive optimization target that can be used for the retrieval of the potential.",
"It is thus of interest to develop a method that optimizes the distributions of the rescattered electrons.",
"Therefore, future work is needed to address the issues listed here.",
"Developments in these directions have already begun.",
"We believe that the advent of the method for retrieval of the SAE potential from the electron momentum distribution will be an important step forward in the studies of strong-field ionization.",
"We are grateful to Professor Manfred Lein, Florian Oppermann, Simon Brennecke, and Shengjun Yue for valuable discussions and continued interest to this work.",
"This work was supported by the Deutsche Forschungsgemeinschaft (Grant No.",
"SH 1145/1-2)."
]
] | 2012.05179 |
[
[
"MLComp: A Methodology for Machine Learning-based Performance Estimation\n and Adaptive Selection of Pareto-Optimal Compiler Optimization Sequences"
],
[
"Abstract Embedded systems have proliferated in various consumer and industrial applications with the evolution of Cyber-Physical Systems and the Internet of Things.",
"These systems are subjected to stringent constraints so that embedded software must be optimized for multiple objectives simultaneously, namely reduced energy consumption, execution time, and code size.",
"Compilers offer optimization phases to improve these metrics.",
"However, proper selection and ordering of them depends on multiple factors and typically requires expert knowledge.",
"State-of-the-art optimizers facilitate different platforms and applications case by case, and they are limited by optimizing one metric at a time, as well as requiring a time-consuming adaptation for different targets through dynamic profiling.",
"To address these problems, we propose the novel MLComp methodology, in which optimization phases are sequenced by a Reinforcement Learning-based policy.",
"Training of the policy is supported by Machine Learning-based analytical models for quick performance estimation, thereby drastically reducing the time spent for dynamic profiling.",
"In our framework, different Machine Learning models are automatically tested to choose the best-fitting one.",
"The trained Performance Estimator model is leveraged to efficiently devise Reinforcement Learning-based multi-objective policies for creating quasi-optimal phase sequences.",
"Compared to state-of-the-art estimation models, our Performance Estimator model achieves lower relative error (<2%) with up to 50x faster training time over multiple platforms and application domains.",
"Our Phase Selection Policy improves execution time and energy consumption of a given code by up to 12% and 6%, respectively.",
"The Performance Estimator and the Phase Selection Policy can be trained efficiently for any target platform and application domain."
],
[
"Introduction",
"The number and complexity of embedded systems are constantly growing , .",
"Recent years saw an advent of the *iot and *cps, and their subsequent applications , , .",
"These systems are tightly resource-constrained, requiring latency-limited real-time operation with a very low power budget.",
"Software running on them must be optimized and tailored to the specific hardware.",
"Major optimizing compilers, like LLVM and GCC , provide an ever-increasing number of optimization phases to improve operational characteristics of embedded software.",
"The phases are applied during compilation in sequence.",
"Their optimal selection and ordering depend on the program to be compiled and on the target platform, as well as on the final optimization objective.",
"The value of the objective function must be estimated at compile-time to tune phase sequencing.",
"Standard phase selection and ordering policies in optimizing compilers , are fixed algorithms that have been tuned for the average case and do not exactly fit to actual use-cases.",
"State-of-the-art approaches for choosing the optimization phases are based on *ml, e.g.",
"*sl and *rl , , , , and other adaptive mechanisms , .",
"Some approaches ignore phase ordering and deal with phase selection only , , while the order is important for the quality of the generated code .",
"Most of the works optimize programs for one specific metric only, like execution time , , , or energy consumption .",
"The state-of-the-art solutions are typically not generic, i.e., they provide good results only in a limited environment and for one specific metric at a time.",
"Moreover, these methods gather the required metrics by profiling execution, which is super-expensive in time, and should be replaced by a fast-yet-accurate estimation method to reduce the total adaptation time.",
"State-of-the-art estimation models can be distinguished as *ml-based , , and formal , , ones.",
"*ml-based models tend to use accurate sensors and interfaces to estimate the power, and hence require external modifications.",
"However, these methods focus only on a single metric and employ only a small selection of models with at most 5% relative error.",
"Formal models estimate the power using formulas and accurate simulation of switching activity that guarantees high accuracy.",
"However, they require deep knowledge of the internal details of the target platform.",
"Table: Comparison of*ml-based state-of-the-artphase selection policies.Table: Comparison of state-of-the-art performance estimators.Our work aims at overcoming specific limitations of state-of-the-art solutions in compiler phase sequencing and performance estimation.",
"Tables REF and REF highlight the shortcomings of major state-of-the-art with respect to different properties/features, and show comprehensive and superior coverage of our MLComp methodology.",
"Our phase sequence selector possesses the following key attributes: [leftmargin=*] It utilizes *rl, which has been given little attention in the literature for obtaining optimal phase sequences.",
"It optimizes programs for multiple objectives in contrast to typical single-objective optimizations.",
"It supports fast adaptation for different application domains.",
"The latter is enabled by fast performance estimation in the adaptation phase, which requires proper performance modeling of target platforms.",
"To adapt to the different target platforms efficiently, potential models need to be evaluated and the best-fitting one selected.",
"This process, which is usually done by manual analysis and design , is automated in our solution.",
"Our efforts are focused on the following scientific challenges, which, to the best of our knowledge, have not been addressed in the literature before: [leftmargin=*] automatic evaluation of different *ml-based performance models to support adaptation to different platforms; efficient training of adaptive phase sequence selection policies for multi-objective optimization of programs.",
"Novel Contributions: To address the above challenges, we propose a novel methodology MLComp that employs: [leftmargin=*] adaptive analytical models for estimating energy consumption and execution time, which are trained on features of target applications executing on a given target platform; and adaptive phase sequence selection policies, which can be trained for quasi-Pareto-optimal code size, energy consumption, and execution time of target applications running on a target platform.",
"The paper makes the following additional novel contributions: [leftmargin=*] testing environment to collect static and dynamic features of target applications on a target platform; framework to adapt performance models by analyzing and modeling energy consumption and execution time based on the collected code features; and training framework for *rl-based adaptation of phase selection policies with respect to estimated dynamic features, and utilizing trained policies in LLVM.",
"After presenting a brief overview of the required background knowledge in Section , we explain our novel MLComp methodology in Section .",
"Experimental setup is explained in Section , which is followed by our results in Section .",
"Conclusion is drawn in Section ."
],
[
"Background",
"A compiler converts a given software code implemented in a high-level programming language into machine executable code.",
"The compilation flow is divided into 3 main parts: the front end transforms source code into the compiler's *ir; the middle end performs analyses and transformations in *ir to ensure quality and prepares for code generation; the back end generates a target-specific executable from *ir.",
"Optimizing transformations are performed in each stage.",
"Optimizations in the front end are specific to the programming language, while those in the back end tune hardware-specific low-level details.",
"We work with *ir-level optimizations in the middle end as depicted in Fig.",
"REF .",
"Those optimizations are general and applied independently to the source language and target platform.",
"However, they can affect the performance in different ways depending on the target, which calls for their adaptive selection and ordering.",
"Standard *ml methods are available to solve different kinds of algorithmic and modeling problems.",
"*sl is used to learn a model representation that can fit input data to output predictions.",
"The model is trained in iterative passes.",
"In each pass, the model predicts output for the given input data, and the results are used to update the model weights to return predictions closer to the correct ones.",
"*rl , on the other hand, uses a reward-based system to learn the operations which should be done from the current state of the system itself.",
"The reward reflects how an operation contributes to reaching the objective.",
"Programs are represented by their characterising features for *ml approaches.",
"The main types of program features are: [leftmargin=*] source code features that characterize application code in a programming language and *ir during compilation; graph-based features that provide information about data and control dependencies during compilation; dynamic features that describe operational aspects in architecture-dependent or architecture-independent ways.",
"Architecture-dependent dynamic features such as execution time and energy consumption are objectives for compiler optimization, while static features describe programs during compilation , , .",
"Dynamic features are time-expensive to determine because they require direct execution of compiled programs.",
"This problem can be circumvented through model-based prediction.",
"The performance of *ml models is improved by scaling and filtering the input features , , an example of which is *pca , ."
],
[
"The MLComp Methodology",
"The flow of our methodology is depicted in Fig.",
"REF and discussed hereafter.",
"The concept is based on two models: *pe is a fast and efficient way of estimating dynamic features for a given application domain, which is represented by a set of target applications, and for a given target platform.",
"It allows for accurate prediction adapted to the given domain faster than standard estimation methods.",
"*pss is based on a policy for selection and ordering of optimization phases, and supports *rl-based adaptation for different target applications and target platforms.",
"Deploying *pss reduces development cost and realizes faster time-to-market by relieving performance engineering from the details of phase selection and ordering."
],
[
"Data Extraction",
"Data Extraction is the first step of the methodology.",
"We collect a training dataset for the *pe model using the flow depicted in box [baseline=(C.base)] inner sep=0pt] (C) 1g1; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g; in Fig.",
"REF .",
"Data is extracted for a target platform from a set of target applications by exploring different permutations of optimization phases.",
"Permutations increase the number of data points as differently optimized variants of programs.",
"For each combination of permutations and applications, the corresponding optimized code is compiled and its features are collected.",
"We extract *ir features that are similar to those of Milepost GCC , such as *ir instruction counts, data and control dependencies, loop hierarchies, and call chains.",
"Our tool also extracts platform-specific instruction counts from generated code for *pe training.",
"Dynamic features such as execution time, energy consumption, and code size are obtained via profiling the compiled code.",
"All features are collected in a dataset that is used for training the *pe model.",
"The size of the dataset depends on the specific set of optimizations and the target applications."
],
[
"Performance Estimator (PE)",
"The next step is the *pe Model Training in box [baseline=(C.base)] inner sep=0pt] (C) 1g2; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g; in Fig.",
"REF .",
"We search for the preprocessing method and *ml model that fits the best to the profiling data based on the code features.",
"The list of methods and models to search is given as input.",
"The search process is detailed in Alg.",
"REF .",
"Tables REF and REF present the preprocessing methods and *ml models used by our *pe modeling in this paper.",
"The set of output metrics is completely customizable.",
"As a training dataset is collected for one target platform, the *pe model is to be trained for each target platform separately to achieve high accuracy.",
"The trained *pe model is used in later steps to predict a program's dynamic features from its *ir features."
],
[
"Phase Selection Policy",
"The trained *pe model is used for the Phase Selection Policy Training in box [baseline=(C.base)] inner sep=0pt] (C) 1g3; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g; in Fig.",
"REF .",
"We use *rl to train the policy that selects the best optimization phase to apply to a program characterized by its *ir features, and thereby enables an efficient phase sequence to be created iteratively.",
"The training is done in batches of episodes as listed in Alg.",
"REF .",
"The policy is optimized using the REINFORCE policy gradient method , .",
"The training algorithm creates a phase sequence for a randomly selected target application in each episode with the current policy as depicted in box [baseline=(C.base)] inner sep=0pt] (C) 1g3; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g; in Fig.",
"REF .",
"The reward in each iteration of an episode reflects how well the last phase changed the dynamic features.",
"Furthermore, the reward guides the training to Pareto-optimal outcomes by penalizing any degradation of the dynamic features.",
"Accumulating rewards over an episode gives a discounted reward, which indicates the overall fitness of the policy for creating a Pareto-optimal phase sequence with respect to final dynamic features.",
"At the end of each batch, the policy is updated according to the episodes' discounted rewards and corresponding phase sequences.",
"The policy is trained with a given *pe and a set of programs that represent a target platform and an application domain, respectively.",
"The training time is reduced compared to other methods by using *pe for fast estimation of dynamic features.",
"Phase Selection Policy is the model used in the *pss."
],
[
"Phase Sequence Selection (PSS)",
"The last step of MLComp is Deployment in box [baseline=(C.base)] inner sep=0pt] (C) 1g4; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g; in Fig.",
"REF , which is the *pss utilizing a trained Phase Selection Policy.",
"We apply the *pss model to drive a compiler's optimizer by selecting phases one after the other.",
"The policy predicts how probable it is that a phase improves dynamic features of the program and accordingly the phase with the highest probability is applied.",
"In case the selected phase did not change the program, which might happen because of the uncertainty of the selection, the best predicted phase remains the same for the next iteration.",
"*pss overcomes that situation by applying the second best, the third best, and so on until a predefined limit, which is “Max.",
"inactive subsequence length” in Table REF .",
"Phase selection ends when that limit is reached or when the total number of applied phases reaches a threshold.",
"Note, *pss does not require a *pe model because the policy learns the platform-specific knowledge.",
"Decoupling the *pe and *pss models allows their separate training so that a platform vendor might provide a trained *pe model for application developers, who can train a custom *pss model with a set of representative applications.",
"Although the *pss model is trained to reach the Pareto-front by selecting locally optimal phases, we observe that both the *pe and *pss models have approximation uncertainties and true Pareto-optimality can not be guaranteed.",
"The accuracy of *pss might be quantified by applying probabilistic dominance , which requires an in-depth empirical evaluation and statistical analysis beyond the scope of this paper.",
"Our evaluation in Section still shows quasi-Pareto-optimality of the results.",
"A detailed toolflow for our experimental setup is shown in Fig.",
"REF .",
"We worked with two different target platforms: profiling for an x86 target is done on an Intel Core i7 system using the RAPL interface to measure power consumption, and dynamic features for a RISC-V target are obtained by accurate simulation with the industrial-grade simulator HIPERSIM integrated with the open-source McPAT .",
"Programs are compiled with LLVM version 9.0.0, which is able to target both platforms.",
"The size of the collected dataset depends on the target applications and the set of optimization phases chosen at compile time: in this evaluation, we used between 200 and 600 data points for both the PARSEC benchmark on x86 target and the BEEBS benchmark on RISC-V target.",
"The training of the *pe model is implemented in Python using Optuna , scikit-learn , and pandas .",
"It covers preprocessing methods and *ml models listed in Tables REF and REF , respectively.",
"The set of output metrics has been chosen to analyze different patterns and distributions.",
"Even though power consumption has a slight correlation with execution time, as in the number of cycles and the number of instructions , increasing the complexity of the system reduces the correlation of these metrics.",
"Therefore, each of them is important for learning the dynamic behaviour of the system.",
"*pss training is implemented also in Python using PyTorch for realizing the model.",
"The 63 code features that our static analysis obtains are preprocessed by *pca with *mle before being passed to Deep *rl .",
"The *pss model is trained with the parameters listed in Table REF , applying optimization phases shown in Table REF .",
"The trained model is stored in TorchScript format, to be loaded into and utilized by our custom LLVM optimization with LibTorch, the PyTorch C++ API.",
"Note that *pe and *pss are independent of the target platform and the used application set.",
"The necessary adaptations when changing the target platform are limited to adjusting target-specific compiler flags and utilizing a tool with support for gathering dynamic features inside the Data Extraction block.",
"Furthermore, any application can be used with our training frameworks as long as it supports a build method using LLVM and allows controlling optimization phases via parameters or environment variables."
],
[
"Evaluation",
"To evaluate our MLComp methodology, we trained and tested both *pe and *pss models on different target platforms with different benchmarks as target applications."
],
[
"PARSEC Benchmark Evaluation on x86 Platform",
"Here we focus our analysis on the PARSEC benchmark , running on an x86 platform.",
"First, we gather the required dataset by profiling the execution of programs from the benchmark compiled with different optimization phases.",
"Then, we use our framework to train different *ml models and select the best one; the results are shown in Fig.",
"REF .",
"As we can see, the exact distributions and the ones generated by our *pe model are almost identical for all the 4 metrics.",
"Note that the blackscholes benchmark has a very tight distribution, while all the others have wider distributions.",
"Referring to [baseline=(C.base)] inner sep=0pt] (C) 1g1; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g;, we can see that the only visible difference resides in the facesim benchmark.",
"However, there is always a high fidelity, as the error between the correct and the predicted distributions always has the same bias.",
"This property is important for the training of the *pss model, giving the correct positive/negative reward to the current choice, even if a limited prediction error is present.",
"Figure: *pss validation results for PARSEC applications on x86 platform.",
"Values are relative to those of unoptimized code, the lower is the better.After validating the *pe model, we used it to train the corresponding *pss model.",
"In Fig.",
"REF , we can see the result of the validation executed after the training.",
"Specifically, distributions are pretty similar across standard state-of-the-art optimizations and MLComp.",
"However, in some cases, as shown by [baseline=(C.base)] inner sep=0pt] (C) 1g1; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g; and [baseline=(C.base)] inner sep=0pt] (C) 1g3; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g;, some standard phase usage can increase both the energy consumption and the execution time between 8x and 10x, respectively, while MLComp shows slight improvements.",
"Regarding memory size, as pointed by [baseline=(C.base)] inner sep=0pt] (C) 1g2; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g;, there are minimal gains, which could be related to the benchmarks being synthetic applications."
],
[
"BEEBS Benchmark Evaluation on RISC-V Platform",
"We performed a similar evaluation for BEEBS on the RISC-V platform.",
"In this case, the number of benchmarks is much higher compared to PARSEC, and since the *pe results are similar to those with PARSEC, we show only an overview of the distribution points in Fig.",
"REF .",
"Figure: *pss validation results for BEEBS applications on RISC-V platform.",
"Values are relative to those of unoptimized code, the lower is the better.This *pe model was then used to train a *pss model, obtaining the results shown in Fig.",
"REF .",
"Here, at pointer [baseline=(C.base)] inner sep=0pt] (C) 1g1; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g;, we can see that our MLComp performs better on average than standard state-of-the-art policies: reducing energy while also optimizing other objectives.",
"Also with BEEBS, we can see that the memory size does not improve or worsen much, as pointed by [baseline=(C.base)] inner sep=0pt] (C) 1g2; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g;.",
"In addition, MLComp results in similar patterns of execution time and energy consumption.",
"Focusing on pointer [baseline=(C.base)] inner sep=0pt] (C) 1g3; draw, circle, inner sep=0.8pt, yshift=1pt] at (C.center) 1g;, we can see how our MLComp obtain more balanced results compared to standard state-of-the-art policies."
],
[
"Discussion and Key Takeaways",
"Our *pe model has a maximum percentage error smaller than 2% across all four metrics, while that of the comparable state of the art is in the range of 2%-7% on a single metric , , , , , .",
"Moreover, the efficient setup for data extraction and the heuristic search of models help us to reach higher accuracy with less time spent for acquiring data and training the model.",
"Profiling the applications and training the models took only 2 days, compared to 15, 30 or 108 days , .",
"Drawing a straight comparison is more difficult for our *pss model, as related techniques optimize a single objective only.",
"The state-of-the-art results oscillate between 5% and 30% improvement in execution time , , which makes our results fall in their average with up to 12% improvement in that metric.",
"However, our *pss model considers additional objectives and reaches up to 6% reduction in energy consumption while not increasing code size.",
"There is actually a slight 0.1% improvement in the latter.",
"We can summarize the following key observations: [leftmargin=*] The *pe model realizes fast estimation with high accuracy, as it is capable of reproducing the profiled distributions.",
"The *pss model performs better than standard optimizations on average and also provides quasi-optimal results for multiple objectives.",
"Our MLComp methodology is fully automated and is usable with different target platforms and applications, enabling for fast estimation and optimization without manual analysis and modeling required."
],
[
"Conclusion",
"We propose the MLComp methodology to overcome limitations of current solutions in compiler optimization phase sequencing and performance modeling.",
"State-of-the-art optimizers can be applied for different target platforms and applications case by case, but their adaptation is expensive and they typically optimize one metric only.",
"MLComp supports adaptive selection of Pareto-optimal phase sequences with respect to execution time, power consumption, and code size by a *pss with an *rl-based policy.",
"Fast adaptation of the policy for different target platforms and application domains is enabled by an *ml-based *pe model, which provides fast-yet-accurate prediction of dynamic program features.",
"The *pe model is trained for a target platform by automatically selecting the most suitable data preprocessing method and *ml model for accurate prediction.",
"This is a novel contribution in performance modeling as current solutions require manual analysis and modeling.",
"Experiments with LLVM on the x86 and RISC-V platforms show that our methodology is efficiently reusable with different target platforms and applications.",
"The *pe model realizes fast estimation with very high accuracy, and the *pss model performs better than state-of-the-art optimizations with multiple objectives."
],
[
"Acknowledgment",
"The presented work has been conducted in the “Cost Efficient Smart System Software Synthesis - COGUTS II (Code Generation for Ultra-Thin Systems)” project, funded and supported by the Austrian Research Promotion Agency (FFG) under grant agreement 872663, and affiliated under the umbrella of the EU Eureka R&D&I ITEA3 “COMPACT” Cluster programme."
]
] | 2012.05270 |
[
[
"Noiseless attack and counterfactual security of quantum key distribution"
],
[
"Abstract Counterfactual quantum key distribution (QKD) enables two parties to share a secret key using an interaction-free measurement.",
"Here, we point out that the efficiency of counterfactual QKD protocols can be enhanced by including non-counterfactual bits.",
"This inclusion potentially gives rise to the possibility of noiseless attacks, in which Eve can gain knowledge of the key bits without introducing any errors in the quantum channel.",
"We show how this problem can be resolved in a simple way that naturally leads to the idea of \"counterfactual security\", whereby the non-counterfactual key bits are indicated to be secure by counterfactual detections.",
"This method of enhancing the key rate is shown to be applicable to various existing quantum counterfactual key distribution protocols, increasing their efficiency without weakening their security."
],
[
"Introduction",
"Quantum key distribution (QKD), which enables secure communication, is currently the most promising quantum technology and studied extensively in a practical context [1], [2].",
"The tradeoff between information gain and state disturbance [3], which is related to Heisenberg uncertainty and the no-cloning theorem, lies at the heart of the information-theoretic security of QKD, and is based on the laws of physics, whereas in classical cryptography, the security stems from certain computational assumptions.",
"Typically, at the end of the quantum stage of a QKD protocol, and after the sifting procedure, sender Alice and receiver Bob estimate the quantum bit error rate (QBER) $e$ in the communication channel.",
"If $e$ exceeds a threshold $e_0$ , they abort the protocol, assuming that too much information has been gained by the eavesdropper Eve.",
"If $e < e_0$ , they proceed with publicly reconciling their respective keys using error correction (EC), and then use privacy amplification (PA) to distill a smaller, final key $K$ over which Eve is guaranteed to have exponentially little knowledge.",
"If Alice and Bob find that $e=0$ , then obviously no EC is required, and furthermore, in light of the information-disturbance tradeoff, it seems natural to expect that no information has leaked to Eve, and that therefore no PA would be required.",
"In this work, surprisingly, we point out for certain QKD protocols, specifically a class of those based on the principle of interaction-free measurement (IFM) [4], that the condition $e=0$ does not imply that Eve has no information.",
"Such a noiseless attack (i.e., one where $e=0$ ) is not in contradiction with the information-disturbance tradeoff.",
"Rather, what happens is that the disturbance produced by Eve's attack does not show up in the QBER analysis.",
"IFM allows for certain information to be communicated or processed even without the interaction of the systems concerned.",
"As a result, IFM-based quantum information processing schemes are usually referred to as “counterfactual”.",
"IFM has been applied to cryptography [5], [6], [7], [8], [9], entanglement generation [10], quantum computation [11], [12] and to direct communication [13], [14].",
"The definition of counterfactuality, based on a weaker or more stringent criterion, has been debated in the context of direct communication [15], [16], [17].",
"Counterfactual QKD has been experimentally implemented [18], [19] as well.",
"This article is arranged as follows.",
"The prototypical counterfactual QKD scheme, the Noh protocol of 2009 [6] (usually called “Noh09”), is briefly reviewed in Section .",
"In Section , our main result, namely a noiseless attack on Noh09, whereby Eve obtains partial information of the sifted key, is proposed and shown to be the optimal, individual, noiseless attack.",
"The implication of the proposed attack to the realistic (noisy) case, where the protocol can potentially become insecure, is discussed in Section .",
"This attack adapted for other counterfactual QKD protocols is investigated in Section .",
"In Section , we explore the possibility of a noiseless attack on other types of QKD protocols besides those based on IFM.",
"Specifically, we show that an analogous attack on the ping-pong protocol [20], which is also two-way but not IFM-based, is not feasible.",
"Finally, we conclude in Section ."
],
[
"Counterfactual quantum cryptography ",
"Noh09 brought considerable attention to the idea of using IFM for key distribution.",
"The protocol works as follows: (1) Alice prepares a string of $n$ single photons in the horizontal/vertical ($H/V$ ) polarization basis.",
"(2) Each photon is incident on a beam-splitter (BS) of a Michelson interferometer, whose one arm (internal) is retained in Alice's station whilst the other arm (external) extends to Bob's station (Fig.",
"REF ).",
"The photonic modal state may be represented by: $\\mathinner {|{\\Psi }\\rangle }_{AB} = \\frac{1}{\\sqrt{2}}(\\mathinner {|{0,j}\\rangle }+ \\mathinner {|{j,0}\\rangle })_{AB},$ where $j \\in \\lbrace H, V\\rbrace $ and $\\mathinner {|{0}\\rangle }$ denotes the vacuum state.",
"(3) In each round, Bob may either reflect the photon of polarization $H$ whilst blocking one with polarization $V$ (action $R_H$ ), or vice versa (action $R_V$ ), with equal probability.",
"(4) If Bob's action $R_k$ ($k \\in \\lbrace H,V\\rbrace $ ) matches with the polarization $j$ of the photon (i.e., $j=k$ ), then it is reflected and detected deterministically at Alice's detector $D_2$ .",
"(5) In case of a mismatch (i.e., $j \\ne k$ ), then there are three possibilities of its detection: (i) at $D_B$ in Bob's station (with probability $\\frac{1}{2}$ ); (ii) at $D_2$ (with probability $\\frac{1}{4}$ ); (iii) at $D_1$ in Alice's station (with probability $\\frac{1}{4}$ ).",
"(6) Alice publicly announces the $D_1$ detection instances, which constitutes the sifting process.",
"The polarization of the corresponding photon forms the sifted secret bit.",
"(7) On a fraction of the sifted key, they announce their respective actions to estimate the QBER given by $e \\equiv \\frac{1}{2}\\big [ P(H, R_H|D_1) + P(V, R_V|D_1) \\big ],$ where $P(\\cdot |\\cdot )$ denotes the conditional probability.",
"If $e$ is too large, they abort the protocol.",
"Figure: Experimental setup for the Noh09 protocol.Although the encoding states in Eq.",
"(REF ) are orthogonal, the corresponding states $\\rho _j$ accessible to Eve are not.",
"In particular, $\\rho _j = \\frac{1}{2}\\left(\\mathinner {|{0}\\rangle }_B\\mathinner {\\langle {0}|} + \\mathinner {|{j}\\rangle }_B\\mathinner {\\langle {j}|}\\right),$ whereby the trace distance $D = |\\rho _H - \\rho _V| = \\frac{1}{2}$ .",
"Therefore, Eve's optimal probability [21] to guess the correct polarization is $p_{\\rm guess} = \\frac{1}{2}(1+D) = \\frac{3}{4}$ , so that $e^\\prime \\equiv 1-p_{\\rm guess}=\\frac{1}{4}$ is Eve's minimum error in distinguishing $\\rho _H$ and $\\rho _V$ ."
],
[
"Security considerations ",
"In step (7) of Noh09, if $e=0$ in Eq.",
"(REF ), conventionally one would assume that there can have been no attack from an eavesdropper.",
"However, the fact that in a two-way protocol such as Noh09, Eve can access the external particle twice, once before and then after the encoding action by Bob, allows her, as we show below, to obtain partial information by attacking the raw quantum transmission.",
"This turns out to provide a twist to the situation.",
"But, because $e=0$ , no EC is performed, and therefore there is no corresponding side-information leaked to Eve.",
"In the onward leg, Eve introduces a probe qubit prepared in state $\\mathinner {|{\\epsilon _0}\\rangle }$ , which interacts with particle $B$ in the external arm by the unitary $\\mathcal {U}$ whose action is as follows: |, 0BE |, BE ({0, H, V}) |0, jBE |0, jBE (j {H, V}), where the three probe states $\\mathinner {|{\\epsilon _\\alpha }\\rangle }$ are orthogonal to each other.",
"In the return leg, she performs the “unattack” $\\mathcal {U}^{-1}$ .",
"To see how the attack works, note that the joint state after the onward attack, and before Bob's action, is given by: $\\mathinner {|{\\Psi ^\\prime }\\rangle }_{ABE} = \\frac{1}{\\sqrt{2}}(\\mathinner {|{0,j, \\epsilon _j}\\rangle } + \\mathinner {|{j,0,\\epsilon _0}\\rangle })_{ABE}.$ After Bob's action of $R_k$ and then Eve's subsequent unattack, the above state evolves to: |ABE Rk { ll |ABE (Rj) 12(|0,0, j + |j,0,0)ABE (Rj) , .",
"U-1 { ll |AB|0E (Rj) 12(|0,0, j + |j,0,0)ABE (Rj) .",
".",
"The first case in Eq.",
"() leads to a detection at $D_2$ and hence no key generation.",
"In the second case, there remains a residual entanglement between modes $A$ and $E$ that disturbs the coherence, but the key point is that the detection at $D_1$ or $D_2$ arises only from the state $\\mathinner {|{j}\\rangle }_A$ in the second term in the r.h.s, and does not involve measuring in basis such as $\\mathinner {|{0}\\rangle }_A\\pm \\mathinner {|{j}\\rangle }_A$ requiring both the terms.",
"Thus the lack of coherence cannot be observed in the detection statistics– hence, the “noiseless\" attack.",
"It follows from Eq.",
"() that if Bob blocked a photon in polarization $j$ , then conditioned on Alice's announcing a $D_1$ detection, Eve's probe is left in one of the states: $\\rho _E^{(0)} = \\frac{1}{2}(\\mathinner {|{\\epsilon _0}\\rangle }_E\\mathinner {\\langle {\\epsilon _0}|} + \\mathinner {|{\\epsilon _H}\\rangle }_E\\mathinner {\\langle {\\epsilon _H}|}), \\nonumber \\\\\\rho _E^{(1)} = \\frac{1}{2}(\\mathinner {|{\\epsilon _0}\\rangle }_E\\mathinner {\\langle {\\epsilon _0}|} + \\mathinner {|{\\epsilon _V}\\rangle }_E\\mathinner {\\langle {\\epsilon _V}|}),$ corresponding to Alice's bit being 0 or 1, respectively.",
"The trace distance $D = |\\rho _E^{(0)} - \\rho _E^{(1)}| = \\frac{1}{2}$ , so that her error $e^\\prime =1-p_{\\rm guess} = \\frac{1}{4}$ in the eavesdropping channel, which reaches the bound given in Eq.",
"(REF ).",
"Therefore, the above proposed attack is the most powerful individual noiseless attack possible.",
"With regard to Eq.",
"(REF ), Eve's optimal strategy to measure her probe is in the basis $\\lbrace \\mathinner {|{\\epsilon _0}\\rangle }, \\mathinner {|{\\epsilon _H}\\rangle }, \\mathinner {|{\\epsilon _V}\\rangle }\\rbrace $ when a $D_1$ detection is announced.",
"She makes a deterministic deduction of $j$ on finding her probe in the state $\\mathinner {|{\\epsilon _j}\\rangle }$ ($j \\in \\lbrace H, V\\rbrace $ ), and assigns the polarization to be $H(X_i\\equiv 0)$ or $V(X_i\\equiv 1)$ with equal probability on finding her probe in the state $\\mathinner {|{\\epsilon _0}\\rangle }$ .",
"We then have for any $i$ that $P({X_i=0}|{Z_i=0}) = P({X_i=1}|{Z_i=1}) = \\frac{3}{4}$ , where $X_i$ and $Z_i$ denote the $i^{\\rm th}$ bit of strings $X$ and $Z$ , respectively.",
"It follows from the above that $I(A{:}B) = 1 > I(A{:}E) = 1 - h\\left(\\frac{1}{4}\\right),$ which shows Alice and Bob can asymptotically obtain a secure key.",
"On the other hand, since $I(A{:}E) > 0$ , it follows that Eve can eavesdrop to an extent without introducing any error.",
"We remark that there this does not contradict the information-disturbance tradeoff.",
"As noted above, Eve's attack indeed produces a disturbance, but this doesn't show up in the error checking step (7).",
"Therefore, Alice and Bob must employ PA [22] even when QBER vanishes; otherwise, Eve, knowing part of a message encoded using the key, could potentially gain full knowledge of the message using a known-plaintext attack [23].",
"That PA is required even for the noiseless situation flies in the face of the conventional quantum cryptographic expectation that a vanishing QBER indicates the absence of Eve, and thus does not require PA. We now consider the degree of compression required.",
"Eve's average collision probability [24] on each attacked qubit is found to be $p_c \\equiv \\sum _{X_i, Z_i} P^2(X_i|Z_i)P(Z_i)= \\frac{5}{8}.$ That $p_c > \\frac{1}{2}$ is another manifestation of the fact that Eve acquires non-zero knowledge through her attack.",
"The final key rate $r$ (after PA) is given by $r \\le -\\log _2(p_c) -s \\approx 0.678 - s,$ in view of Eq.",
"(REF ), where $s~ (\\in (0,-\\log _2(5/8)])$ is a security parameter chosen by Alice and Bob to determine the compression used for PA. Then, Eve's information about the final key $\\kappa $ is given by $I_E = \\mathcal {O}(2^{-s})$ [25].",
"That our attack is the optimal noiseless attack entails that any individual attack that is more powerful will necessarily be noisy.",
"More generally, Eve may launch a collective attack by measuring all her probes together.",
"From a purely communication perspective, an upper-bound on the accessible information is given by the Holevo bound [26], which in this case is $\\chi = S\\left(\\frac{1}{2}\\left[\\rho _E^{(0)}+\\rho _E^{(1)}\\right]\\right) - \\frac{1}{2}\\sum _{j=0,1} S(\\rho ^{(j)}_E) := \\frac{1}{2}$ per qubit.",
"Note that the quantity $\\chi $ provides an upper bound on the average classical information that can be block-coded with the symbols states given by Eq.",
"(REF ).",
"Thus, it is safe to say that even if Eve launches a collective attack by measuring all her probes collectively, she cannot gain on average more than 0.5 bits per qubit.",
"The corresponding error $e^\\prime _\\chi $ in the eavesdropping channel, assumed binary symmetric, can be estimated by setting $\\chi := 1 - h(e^\\prime _\\chi )$ .",
"Solving, we find $e^\\prime _\\chi \\approx 11\\%$ , from which corresponds to an average collision probability of $p_c \\approx 0.804$ , so that $r \\le -\\log _2(p_c)-s \\approx 0.314 - s$ .",
"This can be interpreted as the lower bound on the PA compression factor, for security against Eve's possible collective noiseless attack, assuming that Alice only emits single-photons."
],
[
"Significance for the realistic (noisy) case",
"The noiseless attack has a direct relevance for the realistic, noisy case, since an Eve, powerful enough to launch the above noiseless attack, can introduce noise into Bob's channel, to thereby render the protocol insecure.",
"Since the individual noiseless attack is optimal, therefore Eve's information in Eq.",
"(REF ) corresponds to the minimum error in her eavesdropping channel under the assumption of an individual attack, ideal sources of single-photons and no particle substitutions.",
"Therefore, it follows that in a noisy situation, under these assumptions, Alice and Bob must find $e < \\frac{1}{4}$ for the protocol to be secure.",
"As in the case of noiseless attack, one can consider a collective attack by Eve here as well.",
"Setting $I(A{:}B) = 1-h(e) > \\chi $ , we find $e < e^\\prime _\\chi \\approx 11\\%$ .",
"This provides a lower bound on the maximal tolerable error rate on Bob's communication channel for the protocol to be secure.",
"Note that the above estimate assumes that Eve can introduce errors into Bob's channel, without increasing the error rate in her own channel.",
"A specific attack that illustrates this scenario is the following.",
"Eve modifies the noiseless attack whereby she makes Bob's channel noisy, without introducing additional noise in her own eavesdropping channel.",
"On fraction $f$ of the qubits, Eve implements an attack/unattack similar to that of Eq.",
"(), except that $\\mathinner {|{j,\\epsilon _0}\\rangle }_{BE} \\longrightarrow \\mathinner {|{\\overline{j},\\epsilon _j}\\rangle }_{BE}$ , and the noiseless attack strategy on the remaining qubits.",
"Thereby, she introduces an error $e=f$ in Bob's channel, while her own error rate remains minimal.",
"She subsequently measures her probes individually or collectively.",
"The above error thresholds assume that Eve's probes interact with Alice's particle, but she does not substitute the latter.",
"Relaxing this assumption leads potentially to tighter error thresholds, but such attacks can usually be detected by Alice or Bob even without public discussion, and thus do not constitute good strategies for Eve.",
"As an illustration, in the context of individual attacks, suppose Eve performs a simple intercept-resend attack $\\mathcal {Q}$ on a fraction $f$ of the communicated photons, and the noiseless attack on the remaining photons.",
"In case of $\\mathcal {Q}$ , Eve intercepts the photon sent towards Bob.",
"If she detects it, she injects an identical one back towards Alice.",
"If not, she injects a photon of random polarization $j$ towards Alice, ideally leading to a double detection on Alice's side.",
"Alice and Bob can unilaterally detect the attack, since it will create a fraction $\\frac{f}{2}$ of double detections on Alice's side, and a corresponding dip in detections on Bob's side.",
"For the sake of argument, suppose the duo attribute these errors to imperfect state preparation and detection, and estimate QBER based only on the single detections at the $D_1$ detector.",
"On these detections, Eve has polarization information and therefore she gains full knowledge of the key.",
"Bob's average information is $I({A{:}B}) = 1 - h(e)$ , with QBER $e = \\frac{f}{1+f}$ .",
"Eve's information, averaged over the two attacks, is given by $I({A{:}E}) = (\\frac{1-f}{1+f})\\left(1 - h(\\frac{1}{4})\\right) + (\\frac{2f}{1+f})$ , which exceeds her information in case of the noiseless attack (Eq.",
"(REF )), for $f>0$ .",
"The protocol is insecure if $I(A{:}E) \\ge I(A{:}B)$ , or $f \\ge 0.16$ .",
"Accordingly, the maximum tolerable QBER is $e_{\\rm max} \\approx 13.79\\%$ and the protocol must be aborted for any higher error rate."
],
[
"Other counterfactual quantum QKD protocols",
"It turns out that the above possibility of a noiseless attack on Noh09 holds also for other similar counterfactual QKD protocols.",
"In the semi-counterfactual QKD (SC-QKD) protocol [8], unlike in Noh09, a fixed polarization state is used (here taken to be $H$ ), and hence the encoding is not polarization-based.",
"Furthermore, Alice may, like Bob, either apply a reflect or block operation on the pre-agreed polarization.",
"The counterfactual detections arise half the time when their actions are dissimilar, i.e., their joint action is (block, reflect) or (reflect, block), which correspond to $X_i=0$ or $X_i=1$ , respectively.",
"As in Noh09, the key generation probability per photon is $\\frac{1}{8}$ .",
"In the noiseless attack on SC-QKD, Eve's attack and unattack operations work similarly as on Noh09, except that Eve's probe state conditioned on a $D_1$ announcement would be, rather than Eq.",
"(REF ), one of E(0) = |0E0|, E(1) = 12(|0E0| + |HEH|), where $\\mathinner {\\langle {\\epsilon _0}\\rangle }{\\epsilon _H} = 0$ .",
"The trace distance $D = |\\rho _E^{(0)} - \\rho _E^{(1)}|~ {=}~ \\frac{1}{2}$ between the probe states, and hence Eve's error $e^\\prime \\equiv \\frac{1}{2}(1-D) ~{=}~\\frac{1}{4}$ , are identical with that between the reduced states of particle $B$ conditioned on a $D_1$ detection.",
"This entails that the proposed noiseless attack is optimal.",
"Eve's optimal strategy to measure her probe is in the $\\lbrace \\mathinner {|{\\epsilon _0}\\rangle }, \\mathinner {|{\\epsilon _H}\\rangle }\\rbrace $ basis and to infer $X_i=0$ if $Z_i=0$ and $X_i=1$ if $Z_i=1$ .",
"The latter follows immediately from Eq.",
"().",
"To see the former, suppose she instead chooses $X_i=1$ with probability $q$ .",
"Then her error conditioned on detecting her probe state $\\mathinner {|{\\epsilon _0}\\rangle }$ is $e^\\ast \\equiv \\frac{2}{3}(q) + \\frac{1}{3}(1-q) = \\frac{1+q}{3}$ , which is minimal if $q=0$ .",
"We thus find $e^\\ast ({q := 0})P(\\epsilon _0) = \\frac{1}{4}$ , which is the optimal error $e^\\prime $ for Eve.",
"It is assumed here that Alice and Bob perform the block/reflect actions with equal probability.",
"One finds that $P({X_i=1}|{Z_i=1}) = 1$ and $P({X_i=0}|{Z_i=0}) = \\frac{2}{3}$ , and this leads to the same situation as in Eq.",
"(REF ), but Eve's information being $I(A{:}E) = h\\left(\\frac{1}{4}\\right) - \\frac{1}{2}$ .",
"Thus evidently, Eve has more information here than in the case of Noh09.",
"Eve's average collision probability on each key bit is found to be $p_c^{\\rm sc} = \\frac{2}{3}.$ Accordingly, the final key rate is $r \\le -\\log _2(p_c^{\\rm sc}) - s \\approx 0.585 - s$ , showing that that the PA requirement is more stringent than in the case of the Noh09 (Eq.",
"(REF )).",
"The Guo-Shi protocol [5], which works like the SC-QKD scheme but with a Mach-Zehnder setup, has both the arms exposed to Eve during the phase after Alice's block/pass intervention and before Bob's corresponding intervention.",
"Eve's optimal noiseless attack turns out to be the one wherein she implements attack $\\mathcal {U}$ in Alice's arm, followed by unattack $\\mathcal {U}^{-1}$ on the other arm.",
"It can be shown that this leads to the same eavesdropping scenario as in SC-QKD, and a similar key rate.",
"This is interesting in that although both these protocols involve block/pass actions by both Alice and Bob, technically the Guo-Shi protocol is not two-way and furthermore Eve has access to both interferometric arms.",
"Therefore, careful time-control would be required to prevent Eve from gaining full information of the key.",
"In the cascaded version of Noh09 [7], a sequence of Michelson interferometers is used, with the succeeding beam-splitter placed in the transmission arm of the preceding one (Figure 2 of Ref.",
"[7]).",
"This setup ensures that the probability amplitude of the state leaving Alice's station and reaching Bob is exponentially low in $n$ , the number of beam splitters placed in sequence.",
"Through a similar attack-unattack strategy as discussed for Noh09 in Sec.",
", implemented on the transmission arm of the $n$ th beam-splitter, Eve after her attack on the onward leg generates entanglement between her probe and Alice's and Bob's systems (analogous to Eq.",
"(REF )), given by the $(n+1)$ -particle entangled state: $\\mathinner {|{\\Psi ^\\prime }\\rangle }_{ABE} =\\bigg (\\sum _{k=1}^{n} 2^{-k/2}\\mathinner {|{\\textbf {0}_k}\\rangle }\\bigg )\\mathinner {|{\\epsilon _0}\\rangle } + 2^{-n/2}\\mathinner {|{\\textbf {0}_n}\\rangle } \\otimes \\mathinner {|{\\epsilon _j}\\rangle },$ where $\\mathinner {|{\\epsilon _j}\\rangle }$ ($j \\in \\lbrace H, V\\rbrace )$ are Eve's probe states, and $\\mathinner {|{\\textbf {0}_k}\\rangle } \\equiv \\mathinner {|{0}\\rangle }^{\\otimes (k-1)} \\otimes \\mathinner {|{1}\\rangle } \\otimes \\mathinner {|{0}\\rangle }^{\\otimes (n-k-1)}$ , with the first $n-1$ registers being with Alice, the $n$ th one with Bob.",
"Therefore, in place of Eq.",
"(REF ), we have $\\rho _E^{(j)} = (1-2^{-n})\\mathinner {|{\\epsilon _0}\\rangle }_E\\mathinner {\\langle {\\epsilon _0}|} + 2^{-n}\\mathinner {|{\\epsilon _j}\\rangle }_E\\mathinner {\\langle {\\epsilon _j}|}).$ The trace distance $D = |\\rho _E^{(0)} - \\rho _E^{(1)}| = 2^{-n}$ , meaning that Eve learns exponentially little about the secret key.",
"Correspondingly, we find $p_c^{\\rm ca} = \\frac{1}{2}\\left(1+\\frac{1}{2^n}\\right)$ ."
],
[
"Contrast with attack on ping-pong protocol",
"Keeping in mind the issues raised in Section , we now address the question of what other types of QKD protocols are vulnerable to the noiseless attack.",
"It seems safe to suppose that any such protocol should allow Eve to access the channel twice, once before and then after a party's intervention, thereby allowing Eve to launch an attack and a subsequent unattack (or an equivalent operation) during these two instances.",
"We now ask whether the feature of IFM is also necessary in the protocol.",
"Our analysis in the remaining section suggests that the answer is in the affirmative.",
"However, it turns out that IFM is not sufficient for the vulnerability.",
"The ping-pong protocol [20], [27] is a two-way scheme similar to Noh09, except that it employs a two-qubit entanglement $\\mathinner {|{\\Phi }\\rangle }_{AB} = \\frac{1}{\\sqrt{2}}(\\mathinner {|{\\uparrow \\downarrow }\\rangle } + \\mathinner {|{\\downarrow \\uparrow }\\rangle })_{AB}$ in place of particle-vacuum entanglement of Eq.",
"(REF ).",
"As against the reflect/block choices for Bob in Noh09, here the choices are to apply the Pauli operation $I$ or $Z$ on qubit $B$ .",
"Accordingly, an attack-unattack strategy similar to the one proposed here can be implemented against the ping-pong protocol [28].",
"Even so, this strategy cannot be noiseless, as we point out below.",
"The ping-pong protocol involves a control mode, where the two qubits are locally measured in the $Z$ basis to monitor for QBER.",
"The two-way scheme mentioned above pertains to the message mode, which is used for direct communication and not monitored for QBER.",
"In the message mode, the final tripartite entanglement between particles $A$ , $B$ and Eve's probe $E$ , after Eve's unattack in the return leg, is given by $\\mathinner {|{\\Phi ^\\prime }\\rangle }_{ABE} = \\left\\lbrace \\begin{array}{ll}\\mathinner {|{\\Phi }\\rangle }_{AB} \\otimes \\mathinner {|{\\epsilon _0}\\rangle }_E & (j=0) \\\\\\frac{1}{\\sqrt{2}} \\big [\\mathinner {|{\\uparrow ,\\downarrow }\\rangle }_{AB}\\mathinner {|{\\epsilon _0}\\rangle }_E - \\mathinner {|{\\downarrow ,\\uparrow }\\rangle }_{AB}\\mathinner {|{\\epsilon _1}\\rangle }_E\\big ] & (j=1)\\end{array} \\right.$ where $j$ is the secret bit, $\\mathinner {|{\\epsilon _j}\\rangle }$ are Eve's probe states and Eq.",
"(REF ) is the analogue of Eq.",
"().",
"The residual entanglement in the ${j=1}$ case of Eq.",
"(REF ) leads to a QBER of 50%, meaning that Eve's attack cannot be noiseless.",
"Note that although in the original ping-pong protocol QBER is not monitored in the message mode, yet in principle it is detectable, and is indeed monitored in the adaptation for the ping-pong protocol for QKD [29], [30].",
"As discussed earlier, the $R_{\\overline{j}}$ case in Eq.",
"() also shows residual entanglement.",
"Even so, the reason why no QBER is generated is the following.",
"The state on Alice's side is the mixed state $\\sigma _1 = \\frac{1}{2}(\\mathinner {|{0}\\rangle }_A\\mathinner {\\langle {0}|} + \\mathinner {|{j}\\rangle }_A\\mathinner {\\langle {j}|})$ , whereas it is $\\sigma _2 = \\frac{1}{2}(\\mathinner {|{0}\\rangle } + \\mathinner {|{j}\\rangle })_A(\\mathinner {\\langle {0}|} + \\mathinner {\\langle {j}|})$ if Eve did not attack.",
"Thus, in principle, Eve would generate QBER if Alice were to measure in a diagonal basis such as $\\frac{1}{\\sqrt{2}}(\\mathinner {|{0}\\rangle } \\pm \\mathinner {|{j}\\rangle })$ .",
"However, we note that $\\mathinner {|{0}\\rangle }$ represents the vacuum state, and thus such an unusual sort of diagonal measurement is certainly not implemented through the $D_1$ and $D_2$ detections.",
"Instead, these two represent diagonal measurements in the basis spanned by the spatial modes of a given photon number and polarization.",
"It can be shown that if Noh09 allows non-orthogonal encoding, e.g., by including all the four BB84 states, then the noiseless attack is not possible.",
"In such a modified protocol, Bob could have the option of a reflect/block action in either basis, and correspondingly Eve has an attack-unattack strategy customized for each action.",
"Consider the case where Alice sends state $\\mathinner {|{+}\\rangle } = \\frac{\\mathinner {|{H}\\rangle }+\\mathinner {|{V}\\rangle }}{\\sqrt{2}}$ , and Bob applies $R_+$ .",
"Not knowing which basis was used, Eve cannot determine which attack strategy to perform.",
"If she implements the one in Eq.",
"(), then the state of the photon after Bob's action becomes |ABE = |0A22 [|HB|VE + |VB|HE + |HB|HE + |VB|VE] + 12|00AB(|H - |V)E + 12|+0AB|0E.",
"The first two terms of the r.h.s (appearing in the first line) here are such that Eve's unattack will not disentangle her probe, thus potentially giving rise to QBER.",
"In this case, if $e=0$ , then Eve's attack can indeed be ruled out à la BB84.",
"Therefore, the final key rate here for the noiseless case would be $r = 0.5$ .",
"Here, the factor half is due to a BB84-like sifting and the compression parameter $s$ is set to zero since the (pre-emptive) PA is not required.",
"Whilst the key rate in this case is greater than the key rate in Eq.",
"(REF ), the experimental implementation is more difficult as it involves a greater number of choices of quantum operations for Bob.",
"Moreover, this modified protocol is no longer an orthogonal-state-based protocol.",
"One can show that the inclusion of non-orthogonal states has the consequence for the other counterfactual QKD protocols discussed above.",
"This indicates that if non-orthogonal-state-based encoding is used in counterfactual QKD protocols, then the security loop-hole due to the above noiseless attack is eliminated."
],
[
"Discussion and conclusions",
"The security of QKD is ultimately rooted in the information-disturbance tradeoff, a fundamental principle that asserts that an eavesdropper cannot obtain information about an unknown quantum state without disturbing the system.",
"In this work, surprisingly, we have identified a noiseless attack on a class of QKD protocols, where the eavesdropper Eve can obtain partial information about the key noiselessly, i.e., without generating errors that are detectable during the protocol's error checking.",
"The possibility of the attack does not contradict the information-disturbance tradeoff, nor the no-cloning theorem, since Eve indeed disturbs the eavesdropped states, but this disturbance is not reflected in the error rate as estimated during the error checking step of the considered protocols.",
"An implication of this noiselessness is that, even if the estimated channel-error vanishes, though EC is not required, still PA is required.",
"On the flip side, Eve can of course make the protocol noisy to her own advantage.",
"Thus, the level of compression required in the noiseless case is directly related to the level of error rate that can be sustained while keeping the protocol secure.",
"The protocols vulnerable to this attack feature a double use of a quantum channel exposed to Eve, and furthermore involve IFM with orthogonal-state encoding.",
"The noiseless attack can be detected by modifying these counterfactual protocols to include non-orthogonal states in the encoding scheme, but this will make the practical implementation more difficult, and furthermore the protocols so modified would no longer be orthogonal-state-based.",
"VNR acknowledges the support and encouragement from AMEF.",
"RS thanks support from Interdisciplinary Cyber Physical Systems (ICPS) programme of the Department of Science and Technology (DST), India, Grant No.",
"DST/ICPS/QuST/Theme-1/2019/14."
]
] | 2012.05157 |
[
[
"The two-point correlation function in the six-vertex model"
],
[
"Abstract We study numerically the two-point correlation functions of height functions in the six-vertex model with domain wall boundary conditions.",
"The correlation functions and the height functions are computed by the Markov chain Monte-Carlo algorithm.",
"Particular attention is paid to the free fermionic point ($\\Delta=0$), for which the correlation functions are obtained analytically in the thermodynamic limit.",
"A good agreement of the exact and numerical results for the free fermionic point allows us to extend calculations to the disordered ($|\\Delta|<1$) phase and to monitor the logarithm-like behavior of correlation functions there.",
"For the antiferroelectric ($\\Delta<-1$) phase, the exponential decrease of correlation functions is observed."
],
[
"Introduction",
"The goal of this paper is the computation of correlation functions in the six-vertex model directly by Markov chain Monte-Carlo simulations.",
"This model was introduced by Pauling [35] who proposed it to describe the crystal where the oxygen groups form a square lattice with a hydrogen atom between each pair of lattice sites.",
"He proposed the ice rule: each lattice site has two hydrogen atoms close to it and two further apart; see a recent historical review [6].",
"Another crystal with such a structure is the potassium dihydrogen phosphate.",
"Slater was the first who suggested that the two dimensional case, known as the six-vertex model, is important to understand universal thermodynamic properties of these structures [41].",
"The states in this model are configurations of arrows on edges which satisfy the ice rules, see Fig.",
"REF .",
"An arrow indicates to which of two sites (the oxygen atoms) the hydrogen atom (which is approximately in the middle of an edge) is closer.",
"Equivalently, the configurations of arrows can be regarded as configurations of lattice paths such that paths may meet at a vertex, turn or pass, as it is shown in Fig.",
"REF .",
"Boltzmann weight of a configuration is the product of Boltzmann weights assigned to vertices.",
"The weight of a vertex depends on the configurations of paths on adjacent edges, see Fig.",
"REF .",
"The probability of state $\\sigma $ is $Prob(\\sigma )=\\frac{1}{Z} \\, w(\\sigma ),$ where $w(\\sigma )=\\prod _v w_v(\\sigma )$ is the weight of state $\\sigma $ , $w_v(\\sigma )$ is the weight of the vertex $v$ in the state $\\sigma $ , and $Z=\\sum _\\sigma w(\\sigma )$ is the partition function.",
"Locally, lattice paths of the six-vertex model on a planar lattice can be regarded as level curves of a step function defined on faces [37].",
"We assume it is increasing when we move to the right and up.",
"This integer valued function is called the height function $\\chi (n,m)$ .",
"It is a random variable with values in integers $\\mathbb {Z}$ with the probability distribution given by Boltzmann weights described above.",
"On a planar simply connected lattice domain there is a bijection between configurations of paths with fixed positions on the boundary and height functions with corresponding boundary valuesWe assume that the value of a height function is fixed at some reference point on the domain..",
"Figure: Local configurations and weights of the six-vertex model.",
"For the symmetric model, τ=τ 1 =τ 2 \\tau =\\tau _{1}=\\tau _{2}, where τ=a,b,c\\tau =a,b,c.The first breakthrough in the study of the model came in works of Lieb, Yang, Sutherland and others where the Bethe ansatz was used for finding the spectrum of transfer matrices with periodic boundary conditions [28], [44], [42].",
"Then, came works of Baxter where the role of commuting transfer matrices became clear and the partition function of the eight-vertex model was obtained [2], see [3] for an overview of these developments.",
"Then, many important algebraic structures came in the framework of the algebraic Bethe ansatz and the quantum inverse scattering method [12], for an overview see [25], [4], [36].",
"In the last decade, substantially better understanding of thermodynamic properties of the six-vertex model with domain wall boundary conditions (DW) on a square lattice has been achieved.",
"These boundary conditions correspond to paths coming through each edge on the top side of the square and leaving through edges on the right side only.",
"These particular boundary conditions are quite remarkable because the partition function in this case can be written as a determinant [17], [25], [9] as well as because of the relation to the alternating sign matrices [26].",
"In the large volume limit (the thermodynamic limit, $N\\rightarrow \\infty $ ), properly normalized height function converges, as a random variable, to a deterministic function $h_{0}(x,y): D=[0,1]\\times [0,1]\\rightarrow \\mathbb {R}$ known as the limit shape height function.",
"Such a behavior is known as the limit shape phenomenon, see [10], [33] for an overview.",
"This phenomenon is an analogue of the central limit theorem in probability theory.",
"It predicts the following behavior of the height function as $N\\rightarrow \\infty $This means the convergence of $\\chi (n,m)/N\\rightarrow h_0\\left(\\frac{n}{N}, \\frac{m}{N}\\right)$ and $\\chi (n,m)-Nh_0\\left(\\frac{n}{N}, \\frac{m}{N}\\right)\\rightarrow \\phi \\left(\\frac{n}{N}, \\frac{m}{N}\\right)$ in probability.",
": $\\chi \\left(n,m \\right) \\rightarrow N h_0\\left( \\frac{n}{N}, \\frac{m}{N} \\right)+\\phi \\left(\\frac{n}{N}, \\frac{m}{N} \\right).$ Here $h_0(x,y)$ is the limit shape height function which can be computed using the variational principle.",
"The variational principle was developed and proved for dimer models in [22], [33].",
"It was adopted to the six-vertex model in [46], [34].",
"The random variable $\\phi (x,y)$ is a free Gaussian quantum field in the Euclidean space time with the metric determined by the height function $h(x,y)$ , see for example [46], [16].",
"For generic values of parameters in the six-vertex model, mathematically, the variational principle is still a hypothesis [34], [15].",
"It is proven in some special cases of stochastic weights in [5] and it follows from [7] for the free fermionic case $\\Delta =0$ .",
"An important characteristic of the (symmetric) six-vertex model is the parameter [2] $\\Delta =\\frac{a^2+b^2-c^2}{2ab}.$ When $\\Delta =0$ , the model can be mapped to a dimer model and the partition function and correlation functions can be computed in terms of the determinant and the minors of the Kasteleyn matrix [19], respectively.",
"It has already been shown earlier [1], [20] how to use configurations generated by a Markov chain Monte-Carlo simulations for calculating the limit shape of the height function of the six-vertex model with DW boundary conditions.",
"In this paper, we show how, based on the generated configurations, to compute numerically the two-point correlation function.",
"When $\\Delta =0$ both the limit shape height function and the correlation functions are known from the exact solution because in this case the six-vertex can be mapped to a dimer model on a modified (decorated) square lattice, details can be found in [34], [38].",
"We demonstrate that in this case the usual averaging over time in Markov process gives an excellent agreement of numerical results with the exact ones.",
"After that, we apply the same algorithm for other values of $\\Delta $ .",
"When $|\\Delta |\\le 1$ the model is critical, i.e.",
"the Gaussian field $\\phi (x,y)$ is a massless field on the space time with the metric induced by the limit shape.",
"The numerics confirms that correlation functions are conformal at short distances.",
"Of course, we should not expect conformal invariance at all distances for $\\Delta $ other than zero.",
"When $\\Delta <-1$ an antiferroelectric diamond shape droplet forms in the middle of the limit shape.",
"Because the antiferroelectric ground state is double degenerate, the Markov process gets stuck in one of the ground states for exponentially long time.",
"Numerical estimations for this case are given in the last section.",
"The paper is organized as follows.",
"In section , we outline a derivation of the two-point correlation function from the exact solution of the six-vertex model for $\\Delta =0$ .",
"In section , we demonstrate the results of numerical Monte-Carlo simulations and comparisons with the exact solution.",
"In the appendices, we provide the technical details to specify the model to obtain the exact solution.",
"Acknowledgements.",
"We would like to thank A. G. Pronko for discussions and for sharing a draft of the manuscript [18], D. Keating and A. Sridhar for numerous discussions and the latest version of the Monte-Carlo code.",
"We also benefited from discussions with A. A. Nazarov.",
"The work of N. Yu.",
"Reshetikhin was partly supported by the NSF grant DMS-1902226 and the RSF grant 18-11-00297.",
"P. A. Belov is grateful to the Russian Science Foundation, grant no.",
"18-11-00297, for the financial support.",
"The calculations were carried out using the facilities of the SPbU Resource Center “Computational Center of SPbU”."
],
[
"The free fermionic point of the six-vertex model and mapping to dimers",
"In this paper, we focus on the symmetric six-vertex model with weights $a_{1}=a_{2}=a$ , $b_{1}=b_{2}=b$ , $c_{1}=c_{2}=c$ .",
"The parameter $\\Delta $ , Eq.",
"(REF ), is an important characteristic of the model.",
"It determines the phases of the model on the $M\\times N$ torus when $M,\\, N \\rightarrow \\infty $ .",
"For simplicity, in the following we assume that $M=N$ .",
"When $\\Delta >1$ the model develops a ferroelectric, totally ordered phase.",
"For $|\\Delta |<1$ it develops a disordered phase and for $\\Delta <-1$ it transitions to an antiferroelectric phase.",
"When $\\Delta =\\pm 1$ the model undergoes phase transitions (in parameter $\\Delta $ ).",
"When the weights of the six-vertex model satisfy the condition $\\Delta =0$ the six-vertex model can be mapped to the dimer model on a modified lattice, see for example [43], [38].",
"The partition function of the dimer model is the sum of Pfaffians (the number of terms is determined by the topology of the lattice) [19], [14], [31].",
"Each Pfaffian can be regarded as the Gaussian Grassmann integral.",
"Because of this and because it implies that the multipoint correlators can be expressed as Pfaffians of the two-point correlation functions, the case of $\\Delta =0$ is also known as a free fermionic point of the six-vertex model.",
"Because the weights can be multiplied by an overall constant factor without changing the probability measure, we can put $c=1$ .",
"Then, we can parametrize weights $a$ and $b$ as $a=\\cos {(u)}, b=\\sin {(u)}.$ This is a particular case of Baxter's parametrization of weights of the six-vertex model [3].",
"Note that the mapping $a\\mapsto b, \\ \\ b\\mapsto a$ is a symmetry of the probability measure, see Appendix for details.",
"This is why we can assume, without loosing generality, that $b/a\\le 1$ ."
],
[
"The variational principle",
"Here we will recall the variational principle for deriving the limit shape.",
"Let $\\sigma (s,t)$ be the free energy per site for the six-vertex model on a torus with $s$ and $t$ being fixed densities of edges occupied vertical and horizontal paths respectively.",
"The limit shape height function $h_0(x,y)$ for the six-vertex model with DW boundary conditions is a real valued function on $\\mathcal {D}=[0,1]\\times [0,1]$ which minimizes the large deviation rate functional $S[h]=-{\\int \\int }_{\\mathcal {D}} \\sigma (\\partial _xh,\\partial _yh) \\, dx \\, dy$ in the space of functions with boundary conditions $h(0,y)=h(x,0)=0, \\ \\ h(1,y)=y, \\ \\ h(x,1)=x$ and the constraints $|h(x,y)-h(x^{\\prime },y)|\\le |x-x^{\\prime }|, \\ \\ |h(x,y)-h(x,y^{\\prime })|\\le |y-y^{\\prime }|.$ For dimer models it follows from [7].",
"The critical value $S[h_{0}]$ is the minus free energy of the model.",
"If $Z_{N}$ is the partition function of the six-vertex model with DW boundary conditions, then $S[h_{0}] = \\lim _{N\\rightarrow \\infty } \\frac{1}{N^{2}} \\ln Z_{N}.$ The six-vertex model at the free fermionic point ($\\Delta =0$ ) can be mapped to the dimer model on a decorated square lattice, see for example [38] and references therein.",
"As it was already stated earlier, the corresponding dimer model can be solved by the Pfaffian method.",
"This method gives the formula for the partition function of the model as a Pfaffian (or a determinant) of certain $N\\times N$ matrix, called the Kasteleyn matrix [19].",
"In this case $\\sigma (s,t)$ can be computed explicitly as the Legendre transform of the free energy $f(H,V)$ of the six-vertex model on a torus (with $\\Delta =0$ ) in the presence of electric fields $H$ and $V$ : $\\sigma (s,t)=\\min _{H,V} \\left( Hs + Vt - f(H,V) \\right).$ The function $f(H,V)$ is given by the double integral $f(H,V)=\\frac{1}{(2\\pi i)^2}\\int _{|z|=\\exp (H)}\\int _{|w|=\\exp (V)} \\ln |P(z,w)| \\frac{dz}{z}\\frac{dw}{w},$ where $P(z,w)=a(wz-1)+b(z+w)$ is the spectral polynomial of the Kasteleyn matrix, see [19], [31], [22].",
"Note that $f(H,V)$ is convex, $\\text{det}(\\partial _{i}\\partial _{j} f)>0$ , and $\\sigma (s,t)$ is concave, $\\text{det}(\\partial _{i}\\partial _{j} \\sigma )<0$ .",
"Euler-Lagrange equations for the large deviation rate functional (REF ) can be written explicitly as follows (see [22], [23] for details).",
"Consider complex valued functions $z(x,y)$ and $w(x,y)$ such that $arg(z(x,y))=\\pi \\partial _x h(x,y), \\ \\ arg(w(x,y))=-\\pi \\partial _y h(x,y).$ Then the Euler-Lagrange equations for $h(x,y)$ can be written as a system of equations for $z(x,y)$ and $w(x,y)$ as $\\partial _y \\log (z)+\\partial _x \\log (w)=0, \\ \\ P(z,w)=0.$ Figure: The xyxy plane with the denoted domains.",
"The ellipse is the boundary of the limit shape.",
"The height function smoothinside the ellipse and linear outside.",
"Here, we also show the lines n x =0n_x=0, n y =0n_y=0 and areas AA, BB, CC, and DD."
],
[
"The limit shape",
"Here we describe the limit shape height function $h_0(x,y)$ for DW boundary conditions.",
"We use the result of [18] where the density of the horizontal edges occupied by the paths is derived.",
"Define the function $D(x,y)$ as $D(x,y) = \\alpha (1-\\alpha ) \\left[ \\frac{(y-x)^{2}}{\\alpha }+\\frac{(1-x-y)^{2}}{1-\\alpha }-1 \\right].$ Here, the parameter $\\alpha $ is determined by values of Boltzmann weights of the model as $\\alpha =\\frac{b}{a}.$ In Baxter's parametrization $ \\alpha =\\tan {(u)}$ .",
"Define the region $E= \\lbrace (x,y) | D(x,y) \\le 0 \\rbrace $ as the interior of the ellipse $\\partial E= \\lbrace (x,y) | D(x,y)= 0 \\rbrace $ which is inscribed in the square $0\\le x \\le 1, 0\\le y \\le 1$ as it is shown in Fig.",
"REF .",
"The ellipse is the boundary of the limit shape, or the “arctic curve” [8].",
"The following expression was derived in [18]: $\\partial _{y} h_0(x,y) = \\left\\lbrace \\begin{array}{cc}\\frac{1}{\\pi } \\text{arccot} \\left( \\frac{-n_{y}}{\\sqrt{-D(x,y)}} \\right), & (x,y) \\in E \\\\0, & (x,y) \\in A \\cup B, n_{y}<0 \\\\1, & (x,y) \\in C \\cup D, n_{y}>0 \\\\\\end{array}\\right.$ Regions $A,B,C,D$ are shown in Fig.",
"REF Here $n_{y}=(1-\\alpha )(y-x)+\\alpha (1-x-y)=x+(2\\alpha -1) y -\\alpha $ .",
"Integrating this expression, we obtain the following formula for the limit shape height function itself: $h_0(x,y) = \\left\\lbrace \\begin{array}{cc}\\frac{1}{\\pi } \\left( y \\, \\text{arccot} \\left[ \\frac{-n_{y}}{\\sqrt{-D(x,y)}} \\right] -\\frac{1}{2} \\, \\text{arctan}\\left[\\frac{-x^2+(y-\\alpha )(\\alpha -1)+x(1+y-2y\\alpha )}{(1-x-\\alpha )\\sqrt{-D(x,y)}} \\right] + \\right.",
"& \\\\\\left.",
"+(\\frac{1}{2}-x) \\, \\text{arctan} \\left[ \\frac{-n_{x}}{\\sqrt{-D(x,y)}} \\right] \\right) +\\frac{x}{2} & (x,y) \\in E \\mbox{ }\\& \\mbox{ } x<1-\\alpha \\\\\\frac{1}{\\pi } \\left( y \\, \\text{arccot} \\left[ \\frac{-n_{y}}{\\sqrt{-D(x,y)}} \\right] -\\frac{1}{2} \\, \\text{arctan}\\left[\\frac{-x^2+(y-\\alpha )(\\alpha -1)+x(1+y-2y\\alpha )}{(1-x-\\alpha )\\sqrt{-D(x,y)}} \\right] + \\right.",
"& \\\\\\left.",
"+(\\frac{1}{2}-x) \\, \\text{arctan} \\left[ \\frac{-n_{x}}{\\sqrt{-D(x,y)}} \\right] \\right) +\\frac{x}{2}-\\frac{1}{2} & (x,y) \\in E \\mbox{ }\\& \\mbox{ } x \\ge 1-\\alpha \\\\0 & (x,y) \\in A \\\\x & (x,y) \\in B \\\\x+y-1 & (x,y) \\in C \\\\y & (x,y) \\in D\\end{array}\\right.$ Here $n_{x}=(1-\\alpha )(y-x)+\\alpha (x+y-1)=y+(2\\alpha -1) x -\\alpha $ .",
"Figure: The density of horizontal edges occupied by paths, or ∂ y h 0 (x,y)\\partial _{y} h_0(x,y), for α=9/25\\alpha =9/25.Differentiating this expression in $x$ , we obtain the density of edges occupied with horizontal paths: $\\partial _{x} h_0(x,y) = \\left\\lbrace \\begin{array}{cc}-\\frac{1}{\\pi } \\text{arctan} \\left( \\frac{-n_{x}}{\\sqrt{-D(x,y)}} \\right)+\\frac{1}{2}, & (x,y) \\in E \\\\0, & (x,y) \\in A \\cup D, n_{x}<0 \\\\1, & (x,y) \\in B \\cup C, n_{x}>0 \\\\\\end{array}\\right.$ Here we use the branch of the function $\\text{arctan}$ which behaves as $-\\frac{1}{\\pi } \\text{arctan} \\left( \\frac{-n_{x}}{\\sqrt{-D(x,y)}} \\right) \\rightarrow \\left\\lbrace \\begin{array}{cc}-\\frac{1}{2}, & n_{x}<0, (x,y) \\in A \\cup D \\\\\\frac{1}{2}, & n_{x}>0, (x,y) \\in B \\cup C \\\\\\end{array}\\right.$ when $(x,y)$ approach to the boundary of $E$ .",
"As an example, in Fig.",
"REF we show the partial derivative of the height function (REF ) for $\\alpha =9/25$ .",
"Inside the arctic curve, it is given by the nontrivial part of Eq.",
"(REF ) and outside that one, it equals to zero or one.",
"The limit shape height function $h_{0}(x,y)$ , Eq.",
"(REF ), is shown in Fig.",
"REF .",
"Figure: The limit shape height function h 0 (x,y)h_{0}(x,y), Eq.",
"(), for α=9/25\\alpha =9/25."
],
[
"The function $z(x,y)$",
"An important property of functions $z(x,y)$ and $w(x,y)$ is that $z$ maps the inner part of the ellipse $D(x,y)=0$ (the arctic curve) to the upper half plane.",
"Indeed, as we saw in the previous section, the $x$ -derivative of the height function (REF ) is non-negative when $0\\le x\\le 1$ and, therefore, the imaginary part of the function $z(x,y)$ is also non-negative.",
"In our case we already know the height function, so in order to find functions $z$ and $w$ it is sufficient to solve the algebraic equation in (REF ).",
"Moreover, since we know the height function, we know the arguments of $z$ and $w$ , therefore, we just have to solve the equation $P(z,w)=0$ for absolute values of $z$ and $w$ .",
"This will give the conformal mapping $z$ from the interior $E$ of our ellipse to the upper half of the complex plane.",
"Solving the quadratic equation $P(z,w)=0$ for the absolute values of $z$ and $w$ and taking into account (REF ), we obtain: $\\nonumber |z|=\\frac{1}{2 \\, a \\, b \\, \\sin [-\\pi \\partial _yh_0]}\\Bigg (a^2 \\sin [ \\pi (\\partial _xh_0-\\partial _yh_0) ]+b^2 \\sin [ \\pi (\\partial _xh_0+\\partial _yh_0)] \\\\\\hspace{28.45274pt}\\mp \\sqrt{4 a^2 b^2 (\\sin [-\\pi \\partial _yh_0])^2+\\bigg (a^2 \\sin [\\pi (\\partial _xh_0-\\partial _yh_0)]+b^2 \\sin [\\pi (\\partial _xh_0+\\partial _yh_0)]\\bigg )^2}\\Bigg ), \\\\\\nonumber |w|=\\frac{1}{2 \\, a \\, b \\, \\sin [\\pi \\partial _xh_{0}]}\\Bigg (a^2 \\sin [\\pi (\\partial _xh_0-\\partial _yh_0)]+b^2 \\sin [\\pi (\\partial _xh_0+\\partial _yh_0)] \\\\\\hspace{28.45274pt}\\pm \\sqrt{4 a^2 b^2 (\\sin [-\\pi \\partial _yh_{0}])^2+\\bigg (a^2 \\sin [ \\pi (\\partial _xh_0-\\partial _yh_0)]+b^2 \\sin [ \\pi (\\partial _xh_0+\\partial _yh_0)])\\bigg )^2}\\Bigg ) .$ We almost constructed the mapping $z: E \\rightarrow H=\\lbrace z | Re(z) \\ge 0 \\rbrace , \\partial E \\rightarrow \\mathbb {R}$ .",
"The last step is to determine the signs in ().",
"In the Appendix , we determine the signs and the mapping.",
"It maps the boundary of $E$ bijectively to the real line in the following way: $z: A\\cap \\partial E\\rightarrow (0, \\frac{a}{b})$ , $A\\cap B\\mapsto 0$ , $A\\cap D\\mapsto \\frac{a}{b}$ $z: B\\cap \\partial E\\rightarrow (-\\frac{b}{a}, 0)$ , $B\\cap C\\mapsto -\\frac{b}{a}$ , $B\\cap A\\mapsto 0$ $z: C\\cap \\partial E\\rightarrow (-\\infty , -\\frac{b}{a})$ , $C\\cap D|_{C}\\mapsto -\\infty $ , $C\\cap B\\mapsto -\\frac{b}{a}$ $z: D\\cap \\partial E\\rightarrow (\\frac{a}{b},\\infty )$ , $D\\cap A\\mapsto \\frac{a}{b}$ , $D\\cap C|_{D}\\mapsto \\infty $"
],
[
"The two-point correlation function",
"In the continuum limit, the fluctuations of the height function are described by the massless Euclidean quantum Bose field in the interior of the arctic curve with the metric determined by the second variation $S^{(2)}$ of the large deviation rate functional (REF ) computed at the limit shape height function.",
"It reads $S^{(2)}[h_{0}]= \\frac{1}{2} \\iint _{\\mathcal {D}} \\left( \\partial _{1}^{2} \\sigma (\\vec{\\nabla } h_{0}) (\\partial _{x} \\phi )^{2}+2 \\partial _{1} \\partial _{2} \\sigma (\\vec{\\nabla } h_{0}) \\partial _{x} \\phi \\, \\partial _{y} \\phi +\\partial _{2}^{2} \\sigma (\\vec{\\nabla } h_{0}) (\\partial _{y} \\phi )^{2} \\right) \\, dx \\, dy.$ The mapping $z$ brings the functional $S^{(2)}$ with the kernel defined on functions on the interior of $E$ to the Dirichlet functional for the Laplace operator acting on functions on the upper half of the complex plane.",
"This defines the two-point correlation function for fluctuations of the height function on $E$ as the Green's function for the Laplace operator on the upper half plane with Dirichlet boundary conditions on the real line: $\\left\\langle \\phi (x_{1},y_{1}),\\phi (x_{2},y_{2}) \\right\\rangle = -\\frac{1}{2\\pi } \\ln \\Bigg | \\frac{z(x_{1},y_{1})-z(x_{2},y_{2})}{z(x_{1},y_{1})- \\overline{z(x_{2},y_{2})}} \\Bigg |.$ Here, $\\phi $ is the fluctuation field from (REF ).",
"The formula (REF ) means that the two-point correlation function at the free fermionic point ($\\Delta =0$ ) has a logarithmic dependence on the distance between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ , when this distance is small [24].",
"For free fermionic models, local correlation functions (multipoint correlation functions) $\\langle \\phi (\\vec{r}_{1}),\\ldots ,\\phi (\\vec{r}_{n}) \\rangle $ of fluctuations of the height function are determined by the two-point correlation functions through the Wick's formula.",
"Figure: The behavior of the normalized volume under the height function values as a function of a number of flips for three different lattice sizes and three values of Δ\\Delta : 0, 1/21/2, and -7/2-7/2 of the six-vertex model."
],
[
"Computation of observables and the thermalization.",
"As it was mentioned in the introduction, we use the Markov chain sampling algorithm to generate a sequence of random states of the six-vertex model [1].",
"This method is known as the Markov chain Monte-Carlo simulation.",
"It is based on the special choice of the transition probabilities to transfer from an arbitrary distribution to the desired one.",
"It is also known as the Metropolis algorithm [32].",
"An overview of these numerical methods and their applications to statistical mechanics can be found in [27].",
"See [47], [20], [21], [29], [30] for related numerical simulations.",
"The idea of Markov sampling is to create a random process that will follow the most likely states in the model.",
"This is guaranteed by the choice of the matrix of transition probabilities which is symmetrizable (detailed balanced condition) by the diagonal matrix with entries given by Boltzmann weights of the system.",
"This condition (plus an assumption of nondegeneracy of the largest eigenvalue) also guarantees the asymptotical convergence of the process to the Boltzmann distribution starting from any distribution.",
"Also, in this case the Boltzmann distribution is the Perron-Frobenius eigenvector of the matrix of transition probabilitiesThese are all standard facts about Markov processes, for details see for example [27], [40], [39]..",
"When a random process is constructed, the expectation values of observables with respect to the Boltzmann distribution can be computed by averaging along the random process.",
"This procedure is especially effective when the Boltzmann distribution is concentrated in a small neighborhood of the most likely state (the limit shape).",
"In probability theory this is known as large deviations, in non-equilibrium statistical physics this is known as a hydrodynamic limit.",
"In dimer models it was proven rigorously [7] that there exists a most probable state and the probability for any other state to be “macroscopically distant” from it is exponentially suppressed: $Prob(h)\\propto \\exp \\left[N^2(S[h_{0}]-S[h])\\right].$ Here, $h_0$ is the height function corresponding to the limit shape (REF ).",
"It minimizes the large deviation rate functional.",
"The minimal value $S[h_{0}]$ is exactly (minus) the free energy of the system.",
"The six-vertex model with $\\Delta =0$ is equivalent to a dimer model.",
"Therefore, in this case we can use the probability distribution and results from the corresponding dimer model.",
"For other values of $\\Delta $ in the six-vertex model the analysis is more complicated, but we expect a similar structure of the distribution, suggesting the formation of the limit shape $h_{0}$ .",
"The localization (concentration) of random states near the limit shape makes the numerical computation of observables easy once the Markov process is thermalized i.e.",
"when it moves along the states in a vicinity of the limit shape.",
"Thus, the main challenge for computing observables is to know when the process is thermalized.",
"Unfortunately, it is very hard to have an effective criterium for thermalization.",
"Instead, we use a simple empirical technique: we monitor the fluctuations of the normalized volume under the height function $vol(h)=\\frac{1}{N^3}\\sum _{(n,m)} h(n,m).$ As it is clear from Fig.",
"REF , the normalized volume “drifts”, when the process is not yet thermalized.",
"Then it starts to fluctuate around the normalized volume under the limit shape $h_{0}$ .",
"Thus, we can start measurements to compute observables using the Markov chain simulations.",
"For example, Fig.",
"REF shows that for the lattice of size $90\\times 90$ and $\\Delta =1/2$ it is safe to start averaging after about $10^{7}$ flips [1].",
"Figure: (left plot) The numerical height function for Δ=0\\Delta =0 and for the lattice size of 40×4040\\times 40.",
"The parameter α=9/25\\alpha =9/25.",
"The height function is the result of averaging over measurements.",
"(right plot) The difference between the theoretical limit shape height function and the one obtained from numerical simulation.Once the thermalization is achieved, we compute an observable by time averaging: $\\left\\langle O\\right\\rangle \\simeq \\frac{O(s_1)+\\dots +O(s_K)}{K}.$ Here $s_i$ is a random state at time $T_i$ counting from the first measurement, $K$ is the total number of measurements.",
"The right side depends on random states $s_i$ and is a random variable, but as $K\\rightarrow \\infty $ it converges to the Boltzmann expectation value.",
"Of course, numerically $K\\rightarrow \\infty $ simply means large values.",
"We will use this to compute the limit shape $h_{0}$ and correlation functions.",
"In particular, the two-point correlation function of points $(x_{i},y_{i})$ and $(x_{j},y_{j})$ is calculated as $\\left\\langle \\phi (x_{i},y_{i}),\\phi (x_{j},y_{j})\\right\\rangle = \\left\\langle \\chi (x_{i},y_{i}) \\chi (x_{j},y_{j}) \\right\\rangle -\\left\\langle \\chi (x_{i},y_{i})\\right\\rangle \\left\\langle \\chi (x_{j},y_{j}) \\right\\rangle ,$ where $\\left\\langle \\chi (x_{i},y_{i})\\right\\rangle = \\frac{1}{K}\\sum _{k=1}^K \\chi _k(x_{i},y_{i}),$ the height function $\\chi $ is from Eq.",
"(REF ), and indices $i,j=1,\\ldots ,N$ numerate the lattice sites.",
"Here $\\chi _{k}$ are random variables, but the sum represents a deterministic quantity as $K\\rightarrow \\infty $ .",
"Figure: The plots of (a) the exact limit shape two-point correlation function inside the arctic curve for Δ=0\\Delta =0, (b) the difference between the numerical correlation function on the square lattice of 60×6060\\times 60 and the exact one, the numerical ones on the square lattices of sizes (c) 40×4040\\times 40 and (d) 90×9090\\times 90.",
"The parameter α=9/25\\alpha =9/25."
],
[
"Numerical computation of the limit shape and correlation functions at the free fermionic point.",
"We start by comparison of the calculated height function with the exact one, the limit shape $h_{0}$ given by Eq.",
"(REF ), for $\\Delta =0$ and $\\alpha =9/25$ .",
"The difference between the exact height function and the numerical one is shown in Fig.",
"REF .",
"It should be noted that the numerical height function is smooth since it was averaged over a number of measurements, as described above.",
"The difference between $h_{0}$ and the numerical height function reveals the Airy asymptotic near the boundary of the limit shape.",
"The difference vanishes as the lattice size $N\\rightarrow \\infty $ .",
"Figure: The values of the two-point correlation function along the slice y=0.5y=0.5 for Δ=0\\Delta =0.",
"The curve shows the theoretical limit shape correlation function ().",
"The points are the calculated values for different lattice sizes.",
"The inset shows the values of numerically obtained coefficient against logarithm in Eq.",
"() with respect to the lattice size as well as a fit of these coefficients.The results of computations of two-point correlation functions $\\left\\langle \\phi (x_{i},y_{i}),\\phi (x_{j},y_{j})\\right\\rangle $ are presented in Fig.",
"REF .",
"We show plots of data (REF ) for one point $(x_i,y_i)=(0.5,0.5)$ and another point $(x_j,y_j)$ running through the square $[0,1]\\times [0,1]$ of the lattice domain with step $1/N$ .",
"The plot (a) shows the theoretical limit shape correlation function, given by Eq.",
"(REF ).",
"The plots (c),(d) represent the computed values itself obtained in a “single run” of the Markov process described above for different linear sizes $N$ of the system.",
"The measurements are taken after thermalization after each several hundred thousand iterations of the process.",
"The total number of measurements is $10^{5}$ .",
"The plot (b) shows the difference between the theoretical exact values and the numerical computation.",
"Again, the difference shows the Airy waves propagating from the boundary of the limit shape.",
"As in the case of the height function, one can see the Airy waves which decrease with increasing of $N$ .",
"The agreement of theoretical values of the correlation function and the corresponding numerical values can be seen qualitatively by comparing pictures in Fig.",
"REF .",
"When $\\Delta =0$ the six-vertex model maps to a dimer model and therefore in the limit $N\\rightarrow \\infty $ correlation functions converge to conformally invariant correlation functions (REF ).",
"One can see the logarithmic behavior in the two-point correlation function in the vicinity of $(x_i,y_i)$ , where $\\left\\langle \\phi (\\vec{r_{i}}),\\phi (\\vec{r}_{j})\\right\\rangle \\sim -1/(2\\pi )\\, \\ln {|\\vec{r}_{i}-\\vec{r}_{j}|}$ .",
"By plotting the results of numerics along the slice $y=0.5$ we can examine the logarithmic behavior carefully, see Fig.",
"REF .",
"There is a good agreement between the theoretical result and numerical data for different lattice sizes: the calculated values converge to the theoretical prediction as the lattice size increases.",
"The numerical values of the coefficient against the logarithm in Eq.",
"(REF ) have been obtained from the fit to data.",
"They are shown in the inset of Fig.",
"REF .",
"We see that, as the lattice size increases, $N\\rightarrow \\infty $ , the value of the coefficient against the logarithm tends to the exact one $-1/(2\\pi )\\approx -0.159155$ .",
"For example, a fit by logarithm for the lattice size $90\\times 90$ yields the coefficient $-0.167$ .",
"A fit of the values of the coefficient (red line in the inset), in turn, gives the approximate value for the infinite lattice to be $-0.1584\\pm 0.0082$ , which is close to the exact one."
],
[
"Numerical results for $\\Delta =1/2$",
"The agreement of theoretical and numerical results at the free fermionic point, $\\Delta =0$ , suggests that the numerics should work equally well for other values of $\\Delta $ , where the analytical results are still unknown.",
"Here, we present numerical results for $\\Delta =1/2$ .",
"We choose this value of $\\Delta $ randomly, but note that it is also known as the combinatorial point where the model has many extra interesting features [45].",
"The results of numerical computation of the two-point correlation function for $a/c=b/c=1$ are shown in Fig.",
"REF .",
"Three plots correspond to three lattice sizes: $40\\times 40$ , $60\\times 60$ , and $90\\times 90$ .",
"The behavior of the correlation function at short distances, as expected, is very similar to that for the free fermionic point in Fig.",
"REF .",
"Figure: The correlation function for the case Δ=1/2\\Delta =1/2, a/c=b/c=1a/c=b/c=1.",
"The shown data are for lattice sizes (a) 40×4040\\times 40, (b) 60×6060\\times 60, and (c) 90×9090\\times 90.The numerical values of the two-point correlation function along the slice $y=0.5$ for $\\Delta =1/2$ are shown in Fig.",
"REF .",
"The short distance asymptotic of the correlation function is again logarithmic.",
"This is in agreement with the fact that the model is in the disordered phase.",
"The difference with the free fermionic case is that the global correlation function is not given by a conformal mapping anymore, but is given by an effective Gaussian field theory, see for example the discussion in [15].",
"However, as in any disordered phase, at distances which are larger than the lattice step, but much smaller than the characteristic size of the lattice, the correlation functions are still given by an effective conformal field theory.",
"In the case of the six-vertex model, this is $c=1$ Gaussian CFT model with logarithmic correlators.",
"The fitted coefficient against logarithm in Eq.",
"(REF ) approaches the exact value $-1/(2\\pi )$ as the lattice size $N \\rightarrow \\infty $ in this case as well.",
"However, the numerical values of this coefficient when $\\Delta =1/2$ are notably worse than the same values for $\\Delta =0$ .",
"For $\\Delta =1/2$ , the values for smaller lattices are systematically smaller than those for $\\Delta =0$ .",
"They are naturally expected to converge to the exact value, but the rate of a convergence is less than for $\\Delta =0$ .",
"For example, when the lattice size is $90\\times 90$ , the fit by logarithm gives the value $-0.182$ for the coefficient against $\\log $ .",
"The convergence is shown in the inset of Fig.",
"REF with the extrapolated value for the infinite lattice being $-0.1588\\pm 0.0058$ , which is still close to the expected $-1/(2\\pi )$ .",
"Figure: The values of the two-point correlation function along the slice y=0.5y=0.5 for Δ=1/2\\Delta =1/2.",
"The calculated values for different lattice sizes are shown.",
"The inset shows the values of numerically obtained coefficient against logarithm in the logarithm-like fit with respect to the lattice size as well as a fit of these coefficients.As in the case $\\Delta =0$ one can see the Airy waves near the boundary of the limit shape.",
"They disappear when $N$ is increasing.",
"Figure: The values of the two-point correlation function for the antiferroelectric phase Δ=-7/2\\Delta =-7/2.",
"The lattice size is N=90N=90.",
"The sharp peak is given by the exponential fall which is fitted as 0.484+0.516exp(-99±4)|r → i -r → j |0.484+0.516 \\exp {\\left((-99\\pm 4) |\\vec{r}_{i}-\\vec{r}_{j}|\\right)}."
],
[
"Numerical results for $\\Delta =-7/2$",
"For $\\Delta <-1$ , the antiferroelectric phase of the six-vertex model is opened in the form of a diamond shape droplet.",
"This droplet has already been observed earlier [1], [47], [11], [29].",
"The Gaussian field $\\phi (x,y)$ in this region is massive which predicts the exponential decay of correlation functions at short distances.",
"The numerical observation of the exponential decay of correlation function in this phase is challenging.",
"In order to carry out such computations of correlation functions at distances deep in the antiferroelectric droplet, the characteristic length of the droplet should be much larger than the correlation length.",
"But for these values of $N$ and $\\Delta $ the thermalization is expected exponentially long [13].",
"We carried out calculations of two-point correlation functions for $\\Delta =-7/2$ and the lattice size $N=90$ .",
"The result is given in Fig.",
"REF , and the thermalization was presented in Fig.",
"REF .",
"The numerics are in qualitative agreement with the theoretical prediction that the correlation function should exponentially decrease.",
"One can see a sharp peak over a relatively flat background.",
"The sharp peak is given by the exponential fall and can be fitted as $0.484+0.516 \\exp {\\left((-99\\pm 4) |\\vec{r}_{i}-\\vec{r}_{j}|\\right)}$ .",
"The background “pillow” in Fig.",
"REF is expected to be a result of “mesoscopic” effects.",
"The lattice size $N=90$ is relatively small and correlation functions get affected by the Airy processes on the boundaries of the disordered region.",
"The parallel GPU computations on large lattices may resolve this issue [20].",
"More careful analysis of comparative values of the linear size of the droplet and of the correlation length will be given in a separate publication both numerically and from the exact solution."
],
[
"Conclusion",
"In this paper, we numerically calculated the two-point correlation functions for the six-vertex model with the domain wall boundary conditions.",
"The disordered ($|\\Delta |<1$ ) phase has mainly been studied.",
"Particular attention was paid to the free fermionic point ($\\Delta =0$ ), for which the correlation function has been also obtained analytically in the thermodynamic limit, $N\\rightarrow \\infty $ .",
"The logarithm-like behavior of correlation functions at the small scales has been confirmed.",
"For antiferroelectric phase, the exponential decrease of the correlator has been observed.",
"The numerics for $N=90$ and $\\Delta =-7/2$ show that it might be interesting to study correlation functions in the mesoscopic region where the size of the antiferroelectric droplet is comparable to the correlation length in the antiferroelectric phase.",
"We plan to continue studies of correlation functions and, in particular, their asymptotics in the limit $\\Delta \\rightarrow -1-0$ when the relatively small characteristic size of the droplet requires computations on large lattices.",
"For such lattices, the implementation of Markov sampling on GPU may be of great practical significance."
],
[
"The symmetry",
"Consider the following mapping of the weights and configurations of the six-vertex model.",
"On weights it acts as $a\\mapsto b, \\ \\ b\\mapsto a,\\ \\ c\\mapsto c$ .",
"On states, it replaces each horizontal edge which is not occupied by a path with an edge occupied by a path and each occupied edge by an empty edge.",
"It is clear that the probability measure is invariant with respect to this mapping: $Prob_{a,b,c}(\\gamma , \\beta )=Prob_{b,a,c}(\\overline{\\gamma }, \\overline{\\beta })$ Here $\\gamma $ is a state of the six-vertex model and $\\beta $ is the boundary configuration of paths, that is fixed.",
"For the DW boundary conditions we have $Prob_{a,b,c}(\\gamma )=Prob_{b,a,c}(\\overline{\\gamma })$ Note that the probability measure depends only on the ratios $a:b:c$ , therefore, when $c\\ne 0$ , we can set $c=1$ .",
"At the free fermionic point $\\Delta =0$ , this, together with the symmetry described above, implies that the probability measure with $\\alpha =b/a$ is equal to the one with $\\alpha ^{-1}$ .",
"Therefore, we can assume that $0<\\alpha <1$ ."
],
[
"Let us consider the asymptotical behavior of $|z|$ when $(x,y)\\in E$ approaches $\\partial E\\cap D$ .",
"We know that there $\\partial _x h_{0} \\rightarrow 0$ , $\\partial _yh_{0} \\rightarrow 1$ .",
"Introduce variables $\\delta _{x}=\\pi h_{x}$ and $\\delta _{y} = \\pi - \\pi h_{y}$ .",
"At generic points of $\\partial E$ the ratio $\\delta _{x}/\\delta _{y}$ is finite.",
"However, when $\\partial E$ is tangent to the boundary of the domain (see points $DC$ , $CB$ , $BA$ , $AD$ in Fig.",
"REF ) there is a change of the asymptotic of the height function at $\\partial E$ and, as a consequence, $\\delta _{x}/\\delta _{y} \\rightarrow 0$ or $\\infty $ .",
"Below, we analyze these asymptotics and check the signs in the formula for $|z|$ .",
"In the current case, we have $\\delta _{x},\\delta _{y} \\rightarrow +0$ as $(x,y)\\rightarrow \\partial E\\cap D$ , therefore $\\sin {(-\\pi h_{y})}=\\sin {(-\\pi +\\delta _{y})}\\rightarrow -\\delta _{y},$ $\\sin {(\\pi (h_{x}-h_{y}))}=\\sin {(-\\pi +\\delta _{x}+\\delta _{y})}\\rightarrow -(\\delta _{x}+\\delta _{y}),$ $\\sin {(\\pi (h_{x}+h_{y}))}=\\sin {(\\pi +\\delta _{x}-\\delta _{y})}\\rightarrow -(\\delta _{x}-\\delta _{y}).$ From here we derive the asymptotic of $|z|$ near the $D$ -part of the boundary $|z|=\\frac{1}{2 \\, a \\, b \\, \\delta _{y}} \\left( (a^{2}+b^{2}) \\delta _{x} + (a^{2}-b^{2}) \\delta _{y} \\pm \\sqrt{ (a^{2}+b^{2})^{2} (\\delta _{x}^{2}+\\delta _{y}^{2})+ 2 (a^{4}-b^{4}) \\delta _{x} \\delta _{y} } \\right).$ We have an obvious inequality $\\left( (a^{2}+b^{2}) \\delta _{x} + (a^{2}-b^{2}) \\delta _{y} \\right)^{2} < (a^{2}+b^{2})^{2} (\\delta _{x}^{2}+\\delta _{y}^{2})+ 2 (a^{4}-b^{4}) \\delta _{x} \\delta _{y}$ which means that one of the solutions is negative, the other is positive, corresponding to the plus sign in (REF ).",
"The positive solution is $|z|=\\frac{1}{2 \\, a \\, b \\, \\delta _{y}} \\left( (a^{2}+b^{2}) \\delta _{x} + (a^{2}-b^{2}) \\delta _{y} + \\sqrt{ (a^{2}+b^{2})^{2} (\\delta _{x}^{2}+\\delta _{y}^{2})+ 2 (a^{4}-b^{4}) \\delta _{x} \\delta _{y} } \\right)=$ $\\frac{a^2+b^2}{2ab} \\left( \\frac{\\delta _{x}}{\\delta _{y}} + \\frac{a^2-b^2}{a^2+b^2} + \\sqrt{1+ \\left( \\frac{\\delta _{x}}{\\delta _{y}} \\right)^2 + 2 \\frac{a^2-b^2}{a^2+b^2} \\frac{\\delta _{x}}{\\delta _{y}} } \\right).$ We have two boundary points of the segment $\\partial E\\cap D$ , one is $AD=A\\cap D=(0,\\alpha )$ , the other is $DC=C\\cap D=(1,1-\\alpha )$ .",
"As $(x,y)\\rightarrow AD$ near $\\partial E\\cap D$ , we have $\\frac{\\delta _{x}}{\\delta _{y}} \\rightarrow 0$ , and therefore $|z| \\rightarrow \\frac{a^2+b^2}{2ab} \\left(1 + \\frac{a^2-b^2}{a^2+b^2} \\right) = \\frac{a}{b}=\\frac{1}{\\alpha }.$ When $(x,y)\\rightarrow DC$ near $\\partial E\\cap D$ , we have $\\frac{\\delta _{x}}{\\delta _{y}} \\rightarrow \\infty $ and therefore $ |z| \\rightarrow \\infty $ .",
"Thus, on $D \\cap \\partial E$ , $z$ is real and $\\frac{a}{b} < z < \\infty .$"
],
[
"Let us consider the asymptotical behavior of $|z|$ when $(x,y)\\in E$ approaches $\\partial E\\cap C$ .",
"As in the previous case, we define $\\delta _x,\\delta _y$ as $\\delta _{x} = \\pi - \\pi h_{x}$ , $\\delta _{y}= \\pi - \\pi h_{y} $ .",
"Then, since near the boundary $\\delta _{x},\\delta _{y} \\rightarrow +0$ , $\\sin {(-\\pi h_{y})}=\\sin {(-\\pi +\\delta _{y})}\\rightarrow -\\delta _{y},$ $\\sin {(\\pi (h_{x}-h_{y}))}=\\sin {(\\delta _{y}-\\delta _{x})}\\rightarrow \\delta _{y}-\\delta _{x},$ $\\sin {(\\pi (h_{x}+h_{y}))}\\rightarrow -(\\delta _{x}+\\delta _{y}).$ For $|z|$ , we obtain $|z|=-\\frac{1}{2 \\, a \\, b \\, \\delta _{y}} \\left( a^{2} \\left(\\delta _{y}-\\delta _{x} \\right) - b^{2} \\left(\\delta _{x}+\\delta _{y} \\right) \\mp \\sqrt{ 4 a^2 b^2 \\delta _{y}^{2} + \\left( a^{2} \\left(\\delta _{x}-\\delta _{y} \\right) - b^{2} \\left(\\delta _{x}+\\delta _{y} \\right) \\right)^{2} } \\right)=$ $\\frac{1}{2 \\, a \\, b \\, \\delta _{y}} \\left( (a^{2}+b^{2}) \\delta _{x} - (a^{2}-b^{2}) \\delta _{y} \\pm \\sqrt{ (a^{2}+b^{2})^{2} (\\delta _{x}^{2}+\\delta _{y}^{2})- 4 (a^{4}-b^{4}) \\delta _{x} \\delta _{y} } \\right).$ The inequality $\\left( (a^{2}+b^{2}) \\delta _{x} - (a^{2}-b^{2}) \\delta _{y} \\right)^{2} < (a^{2}+b^{2})^{2} (\\delta _{x}^{2}+\\delta _{y}^{2})- 4 (a^{4}-b^{4}) \\delta _{x} \\delta _{y}$ implies that the positive solution corresponds to the minus sign in (REF ) and its asymptotic is $|z|=\\frac{a^2+b^2}{2ab} \\left( \\frac{\\delta _{x}}{\\delta _{y}} - \\frac{a^2-b^2}{a^2+b^2} + \\sqrt{\\left( \\frac{\\delta _{x}}{\\delta _{y}} \\right)^2 -4 (a^4-b^4) \\frac{\\delta _{x}}{\\delta _{y}} + 1 } \\right).$ When $(x,y)\\rightarrow CB=(1-\\alpha ,1)$ near $\\partial E\\cap C$ , we have $\\frac{\\delta _{x}}{\\delta _{y}} \\rightarrow 0$ and therefore $|z| \\rightarrow \\frac{a^2+b^2}{2ab} \\left( 1 - \\frac{a^2-b^2}{a^2+b^2} \\right)=\\frac{b}{a}.$ When $(x,y)\\rightarrow DC=(1, 1-\\alpha )$ near $\\partial E\\cap C$ , we have $\\frac{\\delta _{x}}{\\delta _{y}} \\rightarrow \\infty $ and therefore $|z| \\rightarrow \\infty $ Thus, on $C \\cap \\partial E$ , $z$ is real and $-\\infty < z < -\\frac{b}{a}.$"
],
[
"Let us consider the asymptotical behavior of $|z|$ when $(x,y)\\in E$ approaches $\\partial E\\cap B$ .",
"Now define $\\delta _x$ and $\\delta _y$ as $\\delta _{x}= \\pi - \\pi h_{x}$ and $\\delta _{y}=\\pi h_{y}$ .",
"Near the boundary $\\delta _{x}, \\delta _{y} \\rightarrow +0$ and $\\sin {(-\\pi h_{y})}\\rightarrow -\\delta _{y}$ , $\\sin {(\\pi (h_{x}-h_{y}))}=\\sin {(\\pi -\\delta _{x}-\\delta _{y})}\\rightarrow \\delta _{x}+\\delta _{y}$ , $\\sin {(\\pi (h_{x}+h_{y}))}=\\sin {(\\pi -\\delta _{x}+\\delta _{y})}\\rightarrow \\delta _{x}-\\delta _{y}$ .",
"Thus, for $z$ we obtain $|z|=-\\frac{1}{2 \\, a \\, b \\, \\delta _{y}} \\left( a^{2} (\\delta _{x}+\\delta _{y}) + b^{2} (\\delta _{x}-\\delta _{y}) \\mp \\sqrt{ 4 a^{2} b^{2} \\delta _{y}^{2}+ \\left( a^{2} \\left(\\delta _{x}+\\delta _{y} \\right) + b^{2} \\left(\\delta _{x}-\\delta _{y} \\right) \\right)^{2} } \\right).$ The inequality $(a^{2}+b^{2})^{2} \\left(\\delta _{x}^{2}+\\delta _{y}^{2} \\right) +2 (a^{4}-b^{4}) \\delta _{x} \\delta _{y}> \\left( (a^{2}+b^{2}) \\delta _{x} +(a^{2}-b^{2}) \\delta _{y} \\right)^2$ implies that the positive solution corresponds to the plus sign, and for the asymptotic we have $|z|=\\frac{a^2+b^2}{2ab} \\left( -\\frac{\\delta _{x}}{\\delta _{y}} - \\frac{a^2-b^2}{a^2+b^2} + \\sqrt{ 1 + \\left(\\frac{\\delta _{x}}{\\delta _{y}} \\right)^2 +2 \\frac{a^2-b^2}{a^2+b^2} \\frac{\\delta _{x}}{\\delta _{y}} } \\right).$ When $(x,y)\\rightarrow CB=(1-\\alpha ,1)$ , we have $\\frac{\\delta _{x}}{\\delta _{y}} \\rightarrow 0$ and $|z| \\rightarrow \\frac{a^2+b^2}{2ab} \\left( 1 - \\frac{a^2-b^2}{a^2+b^2} \\right)=\\frac{b}{a}.$ When $(x,y)\\rightarrow BA=(0,\\alpha )$ we have $\\frac{\\delta _{x}}{\\delta _{y}} \\rightarrow \\infty $ and $|z|=O \\left( \\frac{\\delta _y}{\\delta _x} \\right) \\rightarrow 0.$ Thus, on $B \\cap \\partial E$ $z$ is real and $-\\frac{b}{a} < z < 0.$"
],
[
"Let us consider the asymptotical behavior of $|z|$ when $(x,y)\\in E$ approaches $\\partial E\\cap A$ .",
"Define $\\delta _{x}=\\pi h_{x}$ and $\\delta _{y}=\\pi h_{y}$ .",
"Near the boundary $\\delta _{x}, \\delta _{y} \\rightarrow +0$ and $\\sin {(-\\pi h_{y})}\\rightarrow -\\delta _{y}$ , $\\sin {(\\pi (h_{x}-h_{y}))}=\\delta _{x}-\\delta _{y}$ , $\\sin {(\\pi (h_{x}+h_{y}))}=\\delta _{x}+\\delta _{y}$ .",
"For the asymptotic of $|z|$ in this region we obtain $|z|=-\\frac{1}{2 \\, a \\, b \\, \\delta _{y}} \\left( a^{2} (\\delta _{x}-\\delta _{y}) + b^{2} (\\delta _{x}+\\delta _{y}) \\mp \\sqrt{ 4 a^{2} b^{2} \\delta _{y}^{2}+ \\left( a^{2} \\left(\\delta _{x}-\\delta _{y} \\right) + b^{2} \\left(\\delta _{x}+\\delta _{y} \\right) \\right)^{2} } \\right)=$ $-\\frac{1}{2 \\, a \\, b \\, \\delta _{y}} \\left( (a^{2}+b^{2}) \\delta _{x} - (a^{2}-b^{2}) \\delta _{y} \\mp \\sqrt{ (a^{2}+b^{2})^{2} (\\delta _{x}^{2}+\\delta _{y}^{2}) -4 (a^{4}-b^{4}) \\delta _{x} \\delta _{y} } \\right).$ From the inequality $\\left( (a^{2}+b^{2}) \\delta _{x} -(a^{2}-b^{2}) \\delta _{y} \\right)^{2} < (a^{2}+b^{2})^{2} \\left(\\delta _{x}^{2}+\\delta _{y}^{2}\\right) -4 (a^{4}-b^{4}) \\delta _{x} \\delta _{y}$ we conclude that the positive solution corresponds to the plus sign and $|z|=\\frac{1}{2ab} \\left( a^2-b^2 - (a^2+b^2)\\frac{\\delta _{x}}{\\delta _{y}} + \\sqrt{ (a^2+b^2)^2 \\left(1 + \\left(\\frac{\\delta _{x}}{\\delta _{y}} \\right)^2 \\right) -4 (a^{4}-b^{4}) \\frac{\\delta _{x}}{\\delta _{y}}} \\right).$ When $(x,y)\\rightarrow BA=(0, \\alpha )$ near $\\partial E\\cap A$ we have $\\frac{\\delta _{x}}{\\delta _{y}} \\rightarrow 0$ and, therefore, $ |z| \\rightarrow \\frac{a}{b}$ When $(x,y)\\rightarrow AD=(\\alpha ,0)$ near $\\partial E\\cap A$ we have $\\frac{\\delta _{x}}{\\delta _{y}} \\rightarrow \\infty $ and $|z| \\rightarrow 0$ .",
"Thus, on $A \\cap \\partial E$ , $z$ is real and $0 < z < \\frac{a}{b}.$ The obtained mapping is depicted in Fig.",
"REF ."
]
] | 2012.05182 |
[
[
"The spaces of dyadic distributions"
],
[
"Abstract In this paper subdivision schemes, which are used for functions approximation and curves generation, are considered.",
"In classical case, for the functions defined on the real line, the theory of subdivision schemes is widely known due to multiple applications in constructive approximation theory, signal processing as well as for generating fractal curves and surfaces.",
"Subdivision schemes on a dyadic half-line, which is the positive half-line, equipped with the standard Lebesgue measure and the digitwise binary addition operation, where the Walsh functions play the role of exponents, are defined and studied.",
"Necessary and sufficient convergence conditions of the subdivision schemes in terms of spectral properties of matrices and in terms of the smoothness of the solution of the corresponding refinement equation are proved.",
"The problem of the convergence of subdivision schemes with non-negative coefficients is also investigated.",
"Explicit convergence criterion of the subdivision schemes with four coefficients is obtained.",
"As an auxiliary result fractal curves on a dyadic half-line are defined and the formula of their smoothness is proved.",
"The paper contains various illustrations and numerical results."
],
[
"In this paper the spaces of distributions on a dyadic half-line, which is the positive half-line equipped with the digitwise binary addition and Lebesgue measure, are studied.",
"We prove the non-existence of such a space of dyadic distributions which satisfies a number of natural requirements (for instance, the property of being invariant with respect to the Walsh-Fourier transform) and, in addition, is invariant with respect to multiplication by linear functions.",
"This, in particular, allows the space of dyadic distributions suggested by S. Volosivets in 2009 to be optimal.",
"We also show the applications of dyadic distributions to the theory of refinement equations as well as wavelets on a dyadic half-line.",
"Key words: dyadic half-line, distributions, Walsh functions, Walsh-Fourier transform, refinement equations, wavelets.",
"1.",
"Introduction The existence of more or less relevant space of dyadic distribution was discussed in the literature and in conferences since early 2000s.",
"The dyadic half-line is the positive half-line equipped with the specific addition operation, namely, if $x \\, = \\, \\sum _{k\\in {\\mathbb {Z}}} x_k2^{k}, \\, y \\, = \\, \\sum _{k\\in {\\mathbb {Z}}} y_k2^k$ are arbitrary positive numbers presented in their binary form (each series starts with the sequence of zeros), then their dyadic sum is defined as follows: $x \\oplus y\\, = \\, \\sum _{k\\in {\\mathbb {Z}}} 2^{k}\\,(x_k+y_k)_2$ , where $(x_k+y_k)_2$ is the sum modulo 2 of $k$ -th elements in the binary expansion of $x$ and $y$ respectively.",
"Herewith, $x \\ominus y = x \\oplus y$ .",
"The Lebesgue measure on the dyadic half-line ${\\mathbb {R}}_+$ coincides with its standard analogue, and so do all the spaces $L_p({\\mathbb {R}}_+)$ .",
"The Walsh functions play the role of exponents in the dyadic harmonic analysis.",
"The Walsh-Fourier transform is isometric in the space $L_2({\\mathbb {R}}_+)$ .",
"More about the properties of functions on a dyadic half-line and its applications can be found in [3], [14].",
"In 2007 B. Golubov [2] defined a space of distributions on a dyadic half-line ${\\mathbb {R}}_+$ .",
"It consists of continuous linear functionals on the space of test functions $ D_d(\\mathbb {R}_+)$ , which is the space of infinitely differentiable (in the dyadic sense) functions on $\\mathbb {R}_+$ such that 1)the supports of each function $ \\varphi $ alongside with all its dyadic derivatives ${\\varphi }^{\\alpha }, \\ \\alpha \\in \\mathbb {N}, $ are contained in some dyadic interval $ \\delta $ ; 2)for all $ \\alpha \\in \\mathbb {N} $ , the sequence $ \\psi \\ast ( { {\\Lambda }^{\\alpha } }_n )(x) $ uniformly converges to $ {\\psi }^{\\alpha } $ on $ \\mathbb {R}_+ $ as $ n \\rightarrow \\infty $ , where ${ {\\Lambda }^{\\alpha } }_n = \\int _{0}^{2^n} { (h(t)) }^{- \\alpha } \\psi (x, t) dt, \\ x \\in \\mathbb {R},$ $h(x) = 2^{-n}, \\ 2^n \\le x < 2^{n+1}, \\ n \\in \\mathbb {Z},$ $ \\psi (x, t) $ is the the generalized Walsh functions (see the corresponding definition in section 2).",
"The space of rapidly decreasing in the neighbourhood of infinity functions $ S_d(\\mathbb {R}_+) $ was also suggested.",
"It consists of infinitely smooth functions (in dyadic sense) such that for each $ \\alpha , \\ \\beta \\ \\in { \\mathbb {Z} }_+$ , we have $\\lim _{x \\rightarrow \\infty } {(h(x))}^{- \\beta } { \\varphi }^{ \\alpha }(x) \\ = \\ 0.$ In this case the space of distributions was also defined as the space of linear continuous functionals on $ S_d(\\mathbb {R}_+) $ .",
"The main problem of the spaces $ D_d$ and $ S_d$ is that they are not invariant with respect to the Walsh-Fourier transform.",
"Thus, the Walsh-Fourier transform cannot be well defined on the corresponding spaces of distributions $D_d^{\\prime }, S_d^{\\prime }$ .",
"This problem, however, was solved by S.S. Volosivets in 2009.",
"Note that similar constructions in the space $L_2$ on other groups appeared in [4], [5], [6].",
"In [1] he suggested another, more narrow than $D_d$ , space of test functions $H_d$ , which is the space of ”dyadic-analytic” functions $f$ which are the finite linear combinations of the indicator function $\\chi _{\\Delta _{j,k}}$ , where $\\Delta _{j,k} \\, = \\, [2^{-j}k, 2^{-j}(k+1))$ is the dyadic interval of the rank $j\\in \\mathbb {Z}$ .",
"For all such functions $f$ it holds that $f(\\cdot + h) - f(\\cdot ) \\equiv 0$ for an arbitrary $h \\in (0, 2^{-n})$ , where $n$ is the highest rank of the intervals in the linear combination.",
"The topology is defined by convergence to zero: $f_k \\rightarrow 0$ as $ k \\rightarrow \\infty $ if the ranks of the intervals of the sequence $\\lbrace f_k\\rbrace _{k \\in {\\mathbb {N}}}$ are bounded above and are contained in the fixed segment, and the sequence itself converges to zero pointwise.",
"It is easy to show that $H_d$ is a complete linear space invariant with the respect to the Walsh-Fourier transform.",
"Linearity and completeness can be checked directly and the last property follows from the fact that if ${\\rm supp} f\\, \\subset \\, [0, 2^m]$ and $n$ is the highest rank of the intervals in $f$ , then ${\\rm supp} \\widehat{f}\\, \\subset \\, [0, 2^n]$ and $m$ is the highest rank of the intervals in $\\widehat{f}$ .",
"Thus the Walsh-Fourier transform maps the space into itself.",
"The Walsh-Fourier transform on the corresponding space of distributions $H_d$ is defined as usual: for each $f\\in H_d^{\\prime }$ , we have $(\\widehat{f}, \\varphi ) \\, = \\, (f, \\check{\\varphi }), \\, \\varphi \\in H_d$ , where $\\check{\\varphi }$ means the inverse Walsh-Fourier transform.",
"2.",
"On the possibility of multiplication by smooth functions.",
"The main result.",
"The space of distributions suggested by S. Volosivets is very convenient due to its simplicity and variety of applications.",
"Since the space of the test functions is very narrow (it only contains functions generated by binary dilates and shifts of the function $\\chi _{[0,1)}$ ), the space of distributions is rather wide.",
"For instance, every locally summable on ${\\mathbb {R}}_+$ function $f$ belongs to $H_d^{\\prime }$ with its Walsh-Fourier transform.",
"The exponent also belongs to $H_d^{\\prime }$ which is not true for the classical Schwartz space.",
"The main disadvantage of $H_d^{\\prime }$ is that it is not invariant with respect to multiplication by smooth functions, in particular, by polynomials.",
"For example, not for each function $f \\in H_d^{\\prime }$ is true that also $xf \\in H_d^{\\prime }$ .",
"The natural question is whether there is a space of distributions on the dyadic half-line that is invariant with respect to both the Walsh-Fourier transform and the multiplication by smooth functions, e.g.",
"by polynomials?",
"Theorem REF gives a negative answer and establishes the non-improvability of the space $H_d^{\\prime }$ .",
"There is no extension of $H_d^{\\prime }$ that allows the multiplication even by linear functions.",
"First we need to introduce some notation.",
"For arbitrary $x, y \\in {\\mathbb {R}}_+$ , we denote $(y, x) \\, = \\, \\sum _{k \\in {\\mathbb {Z}}}^{\\infty } y_k x_{-1-k}$ , where $x_i, y_i$ are the digits in the binary expansion of $x$ and $y$ respectively.",
"This sum always contains a finite number of nonzero terms.",
"For an integer $k \\ge 0$ , the Walsh function is defined as ${\\mathbf {w}}_{k}(x) \\, = \\, (-1)^{(k, x)}$ and $ \\psi (x, y) \\ = \\ {\\mathbf {w}}_{[y]}(x) \\cdot {\\mathbf {w}}_{[x]}(y) $ , where $ [y] $ is the integer part of $ y $ .",
"The Walsh-Fourier transform of the function $f\\in L_1({\\mathbb {R}}_+)$ is $\\widehat{f} (y)\\, = \\, \\int _{{\\mathbb {R}}_+} \\psi (x, y) f(x) \\, dx $ , and it can be extended to $L_2({\\mathbb {R}}_+)$ in a usual way.",
"The Walsh-Fourier transform is an invertible orthogonal transform of $L_2({\\mathbb {R}}_+)$ [3], [14].",
"What can be the space of the test functions to define distributions on a dyadic half-line?",
"It is natural to require that this space contains the indicator function $\\chi _{[0,1)}$ and is invariant with respect to integer shifts of a function $f(x) \\, \\mapsto f(x\\oplus 1)$ as well as with respect to binary contraction and expansion $f(x) \\, \\mapsto f(2x), \\ f(x) \\, \\mapsto f(x/2)$ .",
"In this case it already contains $H_d$ .",
"Thus, $H_d$ is the smallest by inclusion functional space satisfying these requirements.",
"The property of being invariant with respect to the Walsh-Fourier transform is fulfilled automatically.",
"Indeed, $\\widehat{\\chi }_{[0,1)}\\, = \\, \\chi _{[0,1)}$ , so the Walsh-Fourier transform maps $H_d$ into itself.",
"The question arises whether it is possible to extend $H_d$ so that it would be also invariant with respect to the multiplication by algebraic polynomials?",
"If so, the corresponding space of distributions would also possess this property: if $f\\in H_d^{\\prime }$ , then $xf$ is defined as in the classical case: $(xf, \\varphi ) \\, = \\, (f, x\\varphi ), \\, \\varphi \\in H_d$ .",
"Theorem REF provides a negative answer to the question above under another natural condition: the space of distributions must contain the functional space $L_2({\\mathbb {R}}_+)$ , which means that each $g \\in L_2({\\mathbb {R}}_+)$ acts naturally on $H_d$ , e.g., the integral $\\int _{{\\mathbb {R}}_+} fg \\, dx$ is defined for each function $f \\in H_d$ .",
"Theorem 1 There is no space of measurable functions on a dyadic half-line that contains the indicator function $\\chi _{[0,1)}$ , is invariant with respect to both the Walsh-Fourier transform and the multiplication by linear functions, and on which every element of the space $ L_2({\\mathbb {R}}_+) $ acts by the formula of the inner product.",
"Proof.",
"Suppose such a space exists, we denote it by $\\tilde{H}_d$ .",
"Computing the Walsh-Fourier transform of the function $f(x) = x\\chi _{[0,1)}(x)$ , we obtain: $\\widehat{f}(y)\\ = \\ \\int _0^1 x \\cdot \\psi (x, y) dx \\ = \\ \\int _0^1 x \\cdot {\\mathbf {w}}_{[x]}(y)\\cdot {\\mathbf {w}}_{[y]}(x) dx \\ = \\ \\int _0^1 x \\cdot {\\mathbf {w}}_{[y]}(x) dx\\, ,$ since $ {\\mathbf {w}}_{[x]}(y) \\ = \\ {\\mathbf {w}}_{0}(y)\\, = \\, 1 $ .",
"This integral depends only on the integer part of $ y $ .",
"We calculate it for $y \\in [2^n, 2^{n}+1)$ , when $[y] = 2^n, \\,n \\in {\\mathbb {N}}$ .",
"So, $[y] = 10\\ldots 0$ ($n$ zeroes); then ${\\mathbf {w}}_{[y]}(x) \\, = \\, (-1)^{(2^n, x)} \\, = \\, (-1)^{x_{-n-1}}$ .",
"Thus, if $y\\in [2^n, 2^n+1)$ , we obtain $\\int _0^1 x \\cdot {\\mathbf {w}}_{[y]}(x) dx \\ = \\ \\int _0^1 x \\cdot x_{-n-1} \\, dx \\ = \\ \\sum _{k=0}^{2^{n+1}}(-1)^k\\int _{k2^{-n-1}}^{(k+ 1)2^{-n-1}}\\, x \\, dx\\ = \\ $ $\\ \\sum _{k=0}^{2^{n+1}}(-1)^k\\frac{(k+ 1)^2-k^2}{2^{2n+3}} \\ = \\ - 2^{-(n+1)}\\, .$ Finally, $\\widehat{f}(y)\\quad = \\quad - \\, 2^{-(n+1)} \\ , \\qquad y \\in {[2^n, 2^{n}+1)}\\, , \\quad n \\in {\\mathbb {N}}\\, .$ By the assumption, $\\widehat{f}(y) \\in \\tilde{H}_d$ , therefore $y\\widehat{f}(y) \\in \\tilde{H}_d$ .",
"On the other hand, each element from $L_2({\\mathbb {R}}_+)$ acts naturally on $\\tilde{H}_d$ .",
"We choose the following function $g\\in L_2({\\mathbb {R}}_+)$ : $g(y) \\ = \\ \\left\\lbrace \\begin{array}{cl}-\\, \\frac{1}{n+1}, & \\ y \\in [2^n; 2^n + 1), \\ n \\in \\mathbb {N} \\, , \\\\0, & \\ \\mbox{ else}.\\end{array}\\right.$ Then $\\Bigl (g\\, , \\, y \\, \\widehat{f}(y)\\Bigr ) \\ =\\ - \\, \\sum _{n = 1}^{\\infty }\\, \\frac{1}{n+1} \\, \\int _{2^n}^{2^n+1} \\, \\frac{y}{2^{n+1}}\\, dy\\ = \\ - \\, \\sum _{n = 1}^{\\infty }\\, \\frac{1}{n+1} \\, \\frac{y^2}{2^{n+2}}\\Bigl |_{2^n}^{2^n+1} \\ = \\ $ $\\ = \\ - \\, \\sum _{n = 1}^{\\infty }\\, \\frac{1}{n+1}\\, \\frac{2^{2n} \\, - \\, (2^n+1)^2}{2^{n+2}}\\ =\\ \\sum _{n = 1}^{\\infty }\\, \\frac{\\frac{1}{2} + 2^{-n-2}}{n+1}\\ = \\ \\infty .$ Consequently, $\\Bigl (g\\, , \\, y \\widehat{f}(y)\\Bigr )$ is not defined, which leads to a contradiction.",
"$\\Box $ Remark 1 One may also define a space of dyadic smooth rapidly decreasing functions $Q_d$ .",
"Dyadic smoothness means that for a function $f$ , the following holds: $\\Bigl \\Vert f(x \\oplus t)\\, - \\, f(x) \\Bigr \\Vert _{2} \\ \\le \\ C(\\alpha ) \\cdot { t }^\\alpha \\, \\qquad t > 0,\\ $ for each $ \\alpha >0$ , and $f$ is rapidly decreasing if $ |f(x)| \\le C{(x+1)}^{-n} $ , for each $ n \\in \\mathbb {N} $ , $ x \\in \\mathbb {R}_+ $ .",
"The space $ Q_d $ is invariant with respect to the Walsh-Fourier transform, but it is also invariant with respect to multiplication by dyadic smooth rapidly decreasing functions.",
"However, the dyadic smoothness is not the same as the smoothness in the classical sense, that is why $ Q_d $ is not invariant with respect to multiplication by linear functions.",
"3.",
"Applications to the wavelet theory The first examples of systems of wavelets on the dyadic half-line and its various generalizations can be found in the paper of Lang [9], more general constructions were presented in [15], [12], [13].",
"The wavelets on the Abelian groups were also studied [10].",
"To obtain a system of wavelets one needs to solve the refinement equation, which is a functional equation on a function $ \\varphi $ with binary expansion of the argument $x$ : $\\varphi \\ = \\ \\sum _{k=0}^{2^n} c_k \\varphi (2x \\ominus k), \\ x \\in {\\mathbb {R}}_+$ Such equations are also used when studying dyadic approximate algorithms [8].",
"The theory of refinement equations on the classical real line ${\\mathbb {R}}$ was developed by the end of 1980s [7], [11].",
"However, many questions remained unsolved on a dyadic half-line.",
"Does the refinement equation always have a solution and, if so, which class does the solution belong to?",
"Will the solution be unique up to multiplication by a constant?",
"If the solution is compactly supported, what is the length of its support?",
"The space of dyadic distributions $H_d$ allows us to fully answer these questions in Theorem REF .",
"First let us introduce some further notation.",
"The solution of the refinement equation is called a refinable function, which is a fixed point of the transition operator $T$: $Tf \\ = \\ \\sum _{k=0}^{2^n} c_k f(2x \\ominus k)$ .",
"Set $m(y) = \\frac{1}{2} \\, \\sum _{k=0}^{2^n} c_k {\\mathbf {w}}_k (y)$ .",
"This Walsh polynomial is called the mask of the refinement equation.",
"It is known [12] that the studying of the general refinement equations could be reduced to the case $\\sum _k c_k = 2$ , which is equivalent to $m(0)=1$ .",
"Theorem 2 For each sequence of complex coefficients $\\lbrace c_k\\rbrace _{k=0}^{2^n}$ , the sum of which equals 2, the refinement equation has a unique up to multiplication by a constant solution in terms of distributions $\\varphi \\in H_d^{\\prime }$ .",
"The support of this function $\\varphi $ lies in the segment $[0, 2^n]$ and the Walsh-Fourier transform of $\\varphi $ is given by the formula: $\\widehat{\\varphi }(y) \\ = \\ \\prod _{j=1}^{\\infty } m\\bigl (2^{-j}y \\bigr )$ Moreover, for each finite summable function $f\\in L_2({\\mathbb {R}}_+)$ , the sequence $T^kf$ converges in $H_d^{\\prime }$ to the solution of the refinement equation $c\\, \\varphi $ , where $c = \\int _{{\\mathbb {R}}_+}f(x)dx$ .",
"Proof.",
"Using the properties of the Walsh-Fourier transform, we obtain: $\\widehat{Tf}(y) \\, = \\, m\\bigl (\\frac{y}{2} \\bigr )\\widehat{f}\\bigl (\\frac{y}{2} \\bigr )$ .",
"Consequently, for each $k$ we have $\\widehat{T^kf} (y) \\ = \\ \\widehat{f}\\bigl (2^{-k} y\\bigr )\\, \\prod _{j=1}^{\\infty }m\\bigl (2^{-j}y \\bigr )\\, .$ We show that for each finite function $f\\in L_1({\\mathbb {R}}_+)$ , the product (REF ) converges uniformly on each segment $[0, 2^N]$ .",
"Note that the function $m(y)$ is a Walsh polynomial of the power $2^n$ , so it is constant on the dyadic intervals of rank $n$ .",
"Since $\\sum _{i}c_i = 2$ , we deduce $m(0) = 1$ .",
"Thus, $m(z) = 1$ for each $z \\in [0, 2^{-n})$ .",
"Therefore, as $k > n + N$ , we have $2^{-k} y \\in [0, 2^{-n})$ for each $y \\in [0, 2^N]$ , and hence $m\\bigl (2^{-k} y\\bigr ) \\, = \\, 1$ .",
"So, on a segment $[0, 2^N]$ each term in (REF ) with the number $k > n + N$ is identically equal to one on $[0, 2^{N}]$ , and the product converges on this segment.",
"Thus, the product (REF ) converges uniformly on each compact set in ${\\mathbb {R}}_+$ , consequently, it converges in the space of distributions in $H_d^{\\prime }$ .",
"Therefore $T^kf$ converges in $H_d^{\\prime }$ to a distribution $\\psi $ .",
"Then $T\\psi = \\psi $ , e.g., $\\psi $ is the solution of the refinement equation.",
"Since $f$ is finite, let its support be in $[0, 2^{\\ell })$ .",
"Then on the segment $[0, 2^{-\\ell })$ the function $\\widehat{f}$ is identically equal to $\\widehat{f}(0) = \\int _{{\\mathbb {R}}_+} f\\, dt\\, = \\, c$ .",
"As $k > \\ell + N$ we obtain $2^{-k} y \\in [0, 2^{-\\ell })$ for each $y \\in [0, 2^N]$ , hence $\\widehat{f}\\bigl (2^{-k} y\\bigr ) \\, = \\, c$ .",
"Thus, the product (REF ) converges as $k \\rightarrow \\infty $ to $\\, c\\, \\prod _{j=1}^{\\infty } m\\bigl (2^{-j}y \\bigr )$ .",
"We see that the inverse Walsh-Fourier transform of this product is nothing but a solution of the refinement equation.",
"Therefore, the inverse Walsh-Fourier transform of $\\prod _{j=1}^{\\infty } m\\bigl (2^{-j}y \\bigr )$ is also a refinable function, which is denoted by $\\varphi $ .",
"Finally, for each finite function $f$ , the sequence $T^kf$ converges to $ \\varphi $ .",
"Now let $\\varphi _0$ be an arbitrary finite solution of the refinement equation.",
"Since $T\\varphi _0 = \\varphi _0$ , it follows that $T^k\\varphi _0 = \\varphi _0$ for each $k$ .",
"Hence, the sequence $T^k\\varphi _0$ converges to $\\varphi _0$ , the function $\\varphi _0$ is proportional to $\\varphi $ (namely, $\\varphi _0 = \\varphi \\int _{{\\mathbb {R}}}\\varphi _0 dx$ ).",
"From this the uniqueness of the solution follows.",
"Finally, to prove that the support of the solution lies in $[0, 2^n]$ , it is enough to consider an arbitrary function $f$ supported by this segment and apply the operator $T$ .",
"The function $Tf$ is also supported on the same segment and so do all the functions $T^kf$ , therefore the limit (which is the solution of a refinement equation) of the sequence $T^kf$ is also supported on $[0, 2^n]$ .",
"$\\Box $ Theorem REF provides tools for studying the properties of many refinements equations.",
"For instance, the following fact is useful in probability theory, approximation theory and the theory of subdivision schemes.",
"Corollary 1 If all the coefficients $c_k$ of a refinement equation are non-negative, then its solution $\\varphi $ normalized by the condition $\\int _{{\\mathbb {R}}_+}\\varphi dx = 1$ , is a non-negative distribution.",
"Proof.",
"Take $f=\\chi _{[0,1)}$ and consider the sequence $T^kf$ .",
"Since the operator $T$ respects the non-negativity of functions, it follows that all the elements of that sequence are also non-negative.",
"Hence, its limit, which is the solution, is non-negative as well.",
"Acknowledgments.",
"The authors are grateful to S.S. Volosivets for valuable remarks and interesting discussions and to an anonymous referee for a thorough reading and many useful advises.",
"The second author was supported by the Program of fundamental research of Higher School of Economics (National Research University) and was financed within the framework of the state support for the leading universities of the Russian Federation ”5-100”.",
"$\\Box $"
]
] | 2012.05209 |
[
[
"Holographic Entanglement Entropy of the Coulomb Branch"
],
[
"Abstract We compute entanglement entropy (EE) of a spherical region in $(3+1)$-dimensional $\\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory in states described holographically by probe D3-branes in $AdS_5 \\times S^5$.",
"We do so by generalising methods for computing EE from a probe brane action without having to determine the probe's back-reaction.",
"On the Coulomb branch with $SU(N)$ broken to $SU(N-1)\\times U(1)$, we find the EE monotonically decreases as the sphere's radius increases, consistent with the $a$-theorem.",
"The EE of a symmetric-representation Wilson line screened in $SU(N-1)$ also monotonically decreases, although no known physical principle requires this.",
"A spherical soliton separating $SU(N)$ inside from $SU(N-1)\\times U(1)$ outside had been proposed to model an extremal black hole.",
"However, we find the EE of a sphere at the soliton's radius does not scale with the surface area.",
"For both the screened Wilson line and soliton, the EE at large radius is described by a position-dependent W-boson mass as a short-distance cutoff.",
"Our holographic results for EE and one-point functions of the Lagrangian and stress-energy tensor show that at large distance the soliton looks like a Wilson line in a direct product of fundamental representations."
],
[
"Introduction",
"Quantum entanglement is of fundamental importance, being a manifestation of purely quantum correlations.",
"For a bipartite system in a pure state, the amount of entanglement between the two complementary subspaces may be quantified by entanglement entropy (EE), defined as the von Neumann entropy of the reduced density matrix of one of the subspaces [1], [2].",
"EE plays a prominent role in disparate areas of physics.",
"For instance, in quantum field theory (QFT) the EE of a spatial subregion has ultraviolet (UV) divergences due to correlations across the subregion's boundary.",
"The leading divergence is proportional to the subregions' surface area [3].",
"This area law received much attention in attempts to understand the Bekenstein-Hawking entropy of black holes.",
"EE also plays a prominent role in condensed matter physics [4], [5], [6], for example as a probe of quantum phase transitions [7].",
"Besides the leading area law term, EE contains subleading divergences and finite contributions that can provide important information [8], [9], [10], [11], [12].",
"For example, in $(1+1)$ -dimensional critical systems the EE of an interval has a logarithmic violation of the area law driven by the central charge [9], [13], [14].",
"This feature makes the EE a good quantity from which to infer the universality class of spin chains from numerical computations [15], [16], [17].",
"More generally, a logarithmic divergence arises in the EE of any even-dimensional conformal field theory (CFT).",
"Its coefficient, which is universal and regularisation independent, turns out to be a linear combination of the CFT's central charges.",
"This opened a new perspective on monotonicity theorems, such as the well-known $c$ -theorem in $1+1$ dimensions [18].",
"Indeed, ref.",
"[19] proved an entropic version of the $c$ -theorem: in any Lorentz invariant QFT describing a renormalisation group (RG) flow from a UV CFT to an infrared (IR) CFT, a certain $c$ -function, defined from a spatial interval's EE, decreases monotonically with the interval's length, and agrees with the central charge in the limits of zero length (the UV) and infinite length (the IR).",
"In higher dimensions EE has been used to prove similar monotonicity theorems, such as the $(2+1)$ -dimensional $F$ -theorem [20], [21] and the $(3+1)$ -dimensional $a$ -theorem [22], [23], [24], [25], [26], [27].",
"While string theory has provided methods to compute black hole entropy [28], the anti-de Sitter/conformal field theory (AdS/CFT) correspondence [29], [30], [31], [32], otherwise known as holography, renewed interest in EE in the high-energy physics community.",
"For a gauge theory at large $N$ and strong coupling with a holographic dual, Ryu and Takayanagi (RT) proposed that the EE of a spatial region is proportional to the area of the minimal surface extending in the holographic direction and anchored to the entangling surface at the asymptotic boundary [33], [34].",
"Originally a conjecture, this prescription was later proved by Lewkowycz and Maldacena [35].",
"The RT formula and its covariant generalisation [36], [37] have turned out to be useful for many reasons, including two in particular.",
"First, holographic EE (HEE) provides a powerful tool to quantify entanglement in strongly-coupled QFTs.",
"In general EE is difficult to compute directly in QFT, even for free theories.An exception is $(1+1)$ -dimensional CFT, where the Virasoro symmetry provides a systematic approach to computing EE [9], [14].",
"On the other hand, HEE is relatively straightforward to calculate.",
"Second, the RT formula suggests a deep connection between entanglement and gravity.",
"For example, refs.",
"[38], [39], [40], [41] argued that the geometry of an asymptotically AdS space-time should be related to the entanglement structure of the dual QFT's quantum state.",
"In this paper we use HEE to study RG flows described holographically by probe branes in AdS space-time.",
"In the holographic framework, branes in the gravity theory can describe fields, states, and objects ranging from fields in the fundamental representation of the gauge group (i.e.",
"flavour fields) [42], to baryons [43], to Wilson lines [44], [45], [46], [47], [48], [49], [50], and more.",
"Broadly speaking, a brane that reaches the AdS boundary is dual to fields added to the QFT, while a brane that does not reach the AdS boundary describes a state of the QFT.",
"We employ a common simplification, namely the probe limit, in which the brane's back-reaction on the metric and other bulk fields is neglected.",
"In the dual field theory, the probe limit corresponds to a probe sector with a number of degrees of freedom parametrically smaller in $N$ than the adjoint-representation fields' order $N^2$ degrees of freedom.",
"How can we calculate HEE from a probe brane?",
"More precisely, how can we holographically compute the leading contribution in $N$ to the EE from the probe sector?",
"Given that the RT approach depends only on the metric, the obvious answer is to compute the brane's linearised back-reaction on the metric, and from that, the resulting change in the minimal surface's area.",
"Although streamlined methods for doing this have been developed in certain cases [51], [52], in general this remains a difficult problem, especially for branes that break symmetries and hence may make the back-reaction complicated.",
"We will instead use Karch and Uhlemann's method for computing the leading order contribution of a probe brane to HEE directly from a probe brane's action, without computing back-reaction [53].",
"Their method generalises Lewkowycz and Maldacena's method for proving the RT formula to probe branes (see ref.",
"[54] for precursor work in conformal cases).",
"Refs.",
"[55], [56] and [57] used the Karch-Uhlemann method to obtain the HEE of non-conformal D-brane and M-brane solutions, respectively.",
"Figure: Cartoons of the D3-brane solutions discussed in the paper.",
"The green surfaces depict the D3-branes, while the pale blue planes represent the conformal boundary.In this paper, we use the Karch-Uhlemann method to compute the contribution to HEE from various probe D3-branes in the $AdS_5\\times S^5$ background of type IIB supergravity.",
"The holographically dual CFT is $(3+1)$ -dimensional $\\mathcal {N}=4$ supersymmetric Yang-Mills (SYM) theory with gauge group $SU(N)$ , with large $N$ and large 't Hooft coupling, $\\lambda $ [29].",
"This theory can be realised as the worldvolume theory of a stack of $N$ D3-branes in $\\mathbb {R}^{1,9}$ .",
"The theory has a Coulomb branch of supersymmetric vacua in which the adjoint scalar superpartners of the gluons acquire non-zero vacuum expectation values (VEVs) and Higgs $SU(N)$ to a subgroup.",
"This corresponds to separating the D3-branes from each other in transverse directions.",
"Separations between D3-branes set the lengths of strings stretched between them, and hence the masses of the W-bosons and their superpartners.",
"We will focus on the situation with precisely one D3-brane separated from the stack, thus Higgsing $SU(N)$ to $SU(N-1)\\times U(1)$ .",
"In the holographic limits $N \\rightarrow \\infty $ and $\\lambda \\rightarrow \\infty $ we replace the stack of D3-branes with their near-horizon geometry, $AdS_5 \\times S^5$ , and treat the single D3-brane as a probe describing an RG flow from $SU(N)$ ${\\mathcal {N}}=4$ SYM to $SU(N-1)\\times U(1)$ ${\\mathcal {N}}=4$ SYM.",
"We will also consider D3-branes that describe supersymmetric Wilson lines and spherical solitons in this vacuum.",
"Cartoons of the relevant D3-brane solutions are shown in figure REF , with the $AdS_5$ boundary at the top and the Poincaré horizon at the bottom.",
"In the figure we depict the $N$ D3-branes generating $AdS_5 \\times S^5$ , but of course these D3-branes are not actually present in $AdS_5 \\times S^5$ , having “dissolved” into five-form flux.",
"We show them simply for intuition.",
"Figure REF a depicts the D3-brane holographically dual to a straight Wilson line in the $k$ -th rank symmetric representation of $SU(N)$ , where the D3-brane carries $k$ units of string charge [46], [58], [49], [50].",
"We show this solution at the origin of the Coulomb branch, where all $N$ D3-branes generating $AdS_5 \\times S^5$ remain coincident.",
"This solution is 1/2-BPS and preserves defect conformal symmetry, so we call it the (conformal) Wilson line D3-brane.",
"Figure REF b depicts the probe D3-brane dual to a point on the Coulomb branch where $SU(N) \\rightarrow SU(N-1) \\times U(1)$ [59].",
"This D3-brane is parallel to the other $N-1$ D3-branes, but is separated from them by a distance $Lv$ in the holographic direction, where $L$ is the $AdS_5$ radius and $v$ sets the VEV of a single adjoint scalar field.",
"If we descend into $AdS_5$ from the boundary, then, when we cross this D3-brane, the five-form flux drops from $N$ to $N-1$ .",
"Recalling that the holographic coordinate is dual to the RG scale, with the regions near the boundary and Poincaré horizon dual to the UV and IR, respectively, this probe D3-brane clearly describes an RG flow in which $SU(N) \\rightarrow SU(N-1) \\times U(1)$ .",
"We call this the Coulomb branch D3-brane.",
"Figure REF c depicts the D3-brane dual to the Wilson line on the Coulomb branch, now screened by the adjoint scalar VEV.",
"In particular, as we descend into $AdS_5$ this D3-brane initially resembles the Wilson line D3-brane of figure REF a, including the $k$ units of string charge, but then interpolates smoothly to the Coulomb branch D3-brane of figure REF b. Crucially, this D3-brane is not present below $Lv$ .",
"The dual CFT interpretation is that in this Coulomb branch vacuum the Wilson line present in the UV is absent in the IR, that is, the Wilson line has been screened by the adjoint scalar VEV [60], [56], [61].",
"This D3-brane is 1/2-BPS but not conformal.",
"We call it the screened Wilson line D3-brane.",
"Figure REF d depicts the D3-brane dual to an excited state on the Coulomb branch, namely a spherically symmetric soliton [62], [63], [64], [65], interpreted in refs.",
"[65], [66] as a phase bubble, or domain wall, separating $SU(N)$ inside from $SU(N-1) \\times U(1)$ outside.",
"Indeed, this D3-brane is essentially a Coulomb branch D3-brane with a cylindrical “spike” that reaches the Poincaré horizon with non-zero radius.",
"This D3-brane carries $k$ units of string charge, dual to $k$ units of $U(1)$ charge uniformly spread over the domain wall.",
"This D3-brane is 1/2-BPS but not conformal: the soliton's mass is $\\propto v$ .",
"We call it the spherical soliton D3-brane.",
"In this paper we calculate the contribution of each probe D3-brane mentioned above to the EE of a spherical region of radius $R$ , centred on the Wilson line or spherical soliton, using the Karch-Uhlemann method.",
"More precisely, we calculate the change in EE from the case with no probe D3-brane.",
"This difference has no UV divergences.",
"For all cases except the conformal Wilson line, our results are novel, and revealing.",
"Moreover, they provide a crucial lesson about the Karch-Uhlemann method: for non-conformal probe branes, this method requires a careful accounting for a boundary contribution on the brane worldvolume.",
"Although this boundary term vanishes for the conformal Wilson line, it is non-zero for all our non-conformal probe branes.",
"This boundary term has been neglected in all previous applications of the Karch-Uhlemann method [53], [55], [56], [57].",
"The contribution to EE from the conformal, symmetric-representation Wilson line, $S_\\mathrm {symm}$ , was computed using conformal symmetry and supersymmetric localisation in ref.",
"[67], and using the Karch-Uhlemann method in ref. [56].",
"We reproduce those results.",
"For the contribution to EE from the Coulomb branch D3-brane, $S_\\mathrm {Coul}$ , we find a new, analytical (i.e.",
"non-numerical) result.",
"As a non-trivial check, we reproduce our $S_\\mathrm {Coul}$ using the RT formula in the fully back-reacted solution for the Coulomb branch.",
"We further show that our $S_\\mathrm {Coul}$ obeys the entropic $a$ -theorem [23], [24], [25], [26], [27] and also the entropic “area law,” which states that the coefficient of the area law term must decrease along the RG flow [68], [27].",
"We compute the contribution to EE from the screened Wilson line D3-brane, $S_\\mathrm {screen}$ , numerically.",
"When $R \\rightarrow 0$ we find $S_\\mathrm {screen} \\rightarrow S_\\mathrm {symm}$ , as expected, since these two cases coincide in the UV.",
"As $R$ increases, we find that $S_\\mathrm {screen}$ decreases monotonically.",
"We know of no physical principle that requires such behaviour.",
"Indeed, this case involves an RG flow of the bulk QFT, triggered by the scalar VEV $\\propto v$ , which in turn triggers an RG flow on the Wilson line, whose degrees of freedom couple to the scalar such that the non-zero VEV acts as a mass term [44], [58].",
"No monotonicity theorem is known for such a situation.",
"We compute the contribution to EE from the spherical soliton D3-brane, $S_\\mathrm {soliton}$ , numerically.",
"the result is not monotonic in $R$ .",
"When $R \\rightarrow 0$ we find $S_\\mathrm {soliton}\\rightarrow 0$ , as expected.",
"As we increase $R$ we find a maximum near the domain wall, after which $S_\\mathrm {soliton}$ decreases.",
"Curiously, the spherical soliton's mass and charge are both proportional to its radius.",
"In refs.",
"[65], [66] Schwarz observed that an asymptotically flat extremal Reissner-Nordström black hole with sufficiently large charge shares this property, raising the question of whether the soliton reproduces any other features of extremal Reissner-Nordström.",
"Indeed, in ref.",
"[69], some of us showed that the spherical soliton supports a spectrum of quasi-normal modes with both qualitative and quantitative similarities to those of an extremal black hole.",
"In this paper we address one of Schwarz's key questions, namely whether at large charge the EE of a sphere coincident with the soliton scales with surface area (after suitable regularisation), similar to a black hole's Bekenstein-Hawking entropy.",
"We find numerically that this EE scales not with surface area, but with a power of the soliton's radius $\\approx 1.3$ In the IR limit $R \\rightarrow \\infty $ we expect both $S_\\mathrm {screen}$ and $S_\\mathrm {soliton}$ to approach $S_\\mathrm {Coul}$ , since in that limit all the dual D3-branes look like the Coulomb branch D3-brane.",
"In the large-$R$ asymptotics of $S_\\mathrm {screen}$ and $S_\\mathrm {soliton}$ we indeed find $S_\\mathrm {Coul}$ , however we also find other contributions, including a term that grows as $R$ , as well as $R$ -independent constants.",
"Remarkably, we found a simple and intuitive way to reproduce these terms, as follows.",
"In figures REF b, c and d, a W-boson is dual to a string stretched between the probe D3-brane and the Poincaré horizon.",
"In particular, the W-boson mass is the length of such a string times its tension.",
"In $SU(N-1)\\times U(1)$ ${\\mathcal {N}}=4$ SYM, the W-boson mass acts as a UV cutoff.",
"Indeed, our result for $S_\\mathrm {Coul}$ resembles that of a CFT with this UV cutoff.",
"In figures REF c and d the W-boson clearly acquires a position-dependent mass, hence this UV cutoff becomes position dependent.",
"Remarkably, for $S_\\mathrm {screen}$ and $S_\\mathrm {soliton}$ in the $R \\rightarrow \\infty $ limit, we find that taking $S_\\mathrm {Coul}$ and replacing the constant cutoff with the position-dependent cutoff reproduces our numerical results for $S_\\mathrm {screen}$ and $S_\\mathrm {soliton}$ up to and including order $1/R$ .",
"In particular, in $S_\\mathrm {soliton}$ this reproduces the term $\\propto R$ and an $R$ -independent term linear in $k$ .",
"In a large-$v$ , large-$k$ limit we show that this contribution arises from the part of the probe D3-brane near the Poincaré horizon, which behaves as a cylindrical shell of $k$ strings.",
"Specifically, this contribution is precisely that of a Wilson line in the direct product of $k$ fundamental representations.",
"We also compute the probe sector's contribution to the VEV of the Lagrangian, via the D3-branes' linearised back-reaction on the dilaton.",
"We obtain new, analytical results in various limits.",
"For example, for the screened Wilson line, when $R \\rightarrow \\infty $ our result resembles that of a point charge in Maxwell theory.",
"What begins in the UV as a Wilson line of $SU(N)$ appears in the IR as a point charge of the $U(1)$ and a singlet of $SU(N-1)$ .",
"In other words, it is screened, as expected.",
"For the spherical soliton we find a maximum near the soliton's radius.",
"In the large-$v$ and large-$k$ limits we compute the probe sector's contribution to the VEVs of the Lagrangian and stress-energy tensor analytically, and find results consistent with $S_\\mathrm {soliton}$ .",
"In particular, in each VEV, when $R \\rightarrow \\infty $ we find a term linear in $k$ , with the form expected of a Wilson line in the direct product of $k$ fundamental representations.",
"All of our results for the EE of non-conformal branes rely crucially on the boundary term in the Karch-Uhlemann method, as we discuss in detail in each case.",
"More generally, our results pave the way for pursuing many fundamental questions and applications of HEE directly in the probe limit, without computing backreaction.",
"This paper is organized as follows.",
"In section we review EE, the RT formula, and the Karch-Uhlemann method, and illustrate the method in a simple example, a fundamental representation Wilson line.",
"In section we review the probe D3-branes described above, and in section we compute their contributions to HEE.",
"In section we compute the Lagrangian VEVs, and for the soliton, also the stress-energy tensor.",
"Section is a summary and discussion of future research.",
"We collect many technical details in four appendices."
],
[
"Holographic Entanglement Entropy",
"In this section we review the calculation of EE in holography.",
"In section REF we review the definition of EE and its computation in holography as proposed by Ryu and Takayanagi (RT) [33], [34].",
"Our focus will be on AdS space-times containing one or more probe branes.",
"A convenient method for calculating the leading order contribution of probe branes to EE was given by Karch and Uhlemann in ref. [53].",
"In section REF we review their method, and highlight the presence of a contribution they neglected.",
"This contribution, which takes the form of a boundary term on the probe brane, is in fact non-zero in all our non-conformal examples.",
"Indeed, it can provide a crucial contribution to the EE as we will see in section ."
],
[
"Review: entanglement entropy and holography",
"Given a generic quantum state described by a density matrix $\\rho $ and a bipartition of the Hilbert space $H$ into a subspace $A$ and its complement $\\bar{A}$ , the reduced density matrix on $A$ is defined as $\\rho _A \\equiv {\\rm Tr}_{\\bar{A}} \\rho $ .",
"The EE of $A$ is defined as the von Neumann entropy of $\\rho _A$ , $S_{A} \\equiv - {\\rm Tr}_A \\rho _A \\log \\rho _A\\,.$ If the total state $\\rho $ is pure, which will be the case in all that follows, then $S_A = S_{\\bar{A}}$ and the EE is a good measure of the amount of quantum entanglement between $A$ and $\\bar{A}$ .",
"In QFT, a natural way to partition the Hilbert space is to do so geometrically, taking $A$ to be a set of states in a subregion of a Cauchy hypersurface.",
"The EE may then be computed through the replica trick [8], an approach which has been particularly successful for two-dimensional CFTs [9], [13], [14], [70], [71].",
"The first step of the replica trick is to construct the quantity ${\\rm Tr}_A \\rho _A^n$ with $n\\in \\mathbb {Z}^+$ , which is equal to the partition function $Z$ of $n$ copies of the original theory glued together along $A$ .",
"This is equivalent to the partition function of the original theory on a manifold $\\mathcal {M}_n$ with conical deficit $2\\pi /n$ at the entangling surface between $A$ and $\\bar{A}$ , namely ${\\rm Tr}_A \\rho _A^n = Z[\\mathcal {M}_n]/Z[\\mathcal {M}_1]^n$ .",
"Defining the $n$ -th Rényi entropy as $S_A^{(n)} \\equiv \\frac{1}{1-n} \\log {\\rm Tr}_A \\rho _A^n\\,,$ the EE is given by the limit $S_A = \\lim _{n\\rightarrow 1} S_A^{(n)} = - \\lim _{n\\rightarrow 1}\\partial _n {\\rm Tr}_A \\rho _A^n\\,.$ Defining a generating functional $W[\\mathcal {M}_n]\\equiv -\\log Z[\\mathcal {M}_n]$ , eq.",
"(REF ) may be rewritten as $S_A = \\lim _{n\\rightarrow 1} (\\partial _n-1)W[\\mathcal {M}_n]\\,.$ This form of the EE will be useful in the following.",
"For a QFT with a holographic dual, the EE of a time-independent state can be computed through the RT formula [33], [34] $S_{A} = \\frac{\\mathcal {A}\\left[\\gamma _A^{(\\mathrm {min})}\\right]}{4 G_N}\\,,$ where $\\gamma _A^{(\\mathrm {min})}$ is the minimal surface in the bulk space-time homologous to the region $A$ at the asymptotic boundary, $\\mathcal {A}$ is the area of $\\gamma _A^{(\\mathrm {min})}$ , and $G_N$ is Newton's constant of the bulk gravity theory.",
"This result was extended to time-dependent cases in refs.",
"[36], [37].",
"The RT formula was proved by Lewkowycz and Maldacena in ref.",
"[35] by defining a generalised gravitational entropy, in an extension of the usual Gibbons-Hawking thermodynamic interpretation of Euclidean gravity solutions [72] to cases with no $U(1)$ symmetry.See also ref.",
"[73] for an earlier proof of the RT formula for the special case of a spherical entangling region in a CFT.",
"Consider a semiclassical gravitational theory on a Euclidean manifold with a boundary.",
"We assume that the boundary has a direction which is topologically a circle, parametrised by a coordinate $\\tau $ with $\\tau \\sim \\tau + 2\\pi $ , but it need not have a $U(1)$ isometry.",
"The boundary conditions are assumed to respect $\\tau $ 's periodicity.",
"In the context of EE, $\\tau $ winds around the boundary of the subregion $A$ .",
"Let $I(n)$ be the on-shell action of the gravity theory with the period extended to $\\tau \\sim \\tau + 2\\pi n$ , with $n \\in \\mathbb {Z}^+$ , while maintaining boundary conditions invariant under $\\tau \\rightarrow \\tau + 2\\pi $ .",
"The generalised gravitational entropy is then defined as $S_\\mathrm {grav} = \\lim _{n\\rightarrow 1} (\\partial _n-1) I(n)\\,.$ For a theory on a static asymptotically AdS space-time, the on-shell action is equal to the dual QFT's generating functional $W$ .",
"The right-hand side of eq.",
"(REF ) is then identical to the right-hand side of eq.",
"(REF ), and so $S_\\mathrm {grav} = S_A$ .",
"Evaluating the right-hand side of eq.",
"(REF ) requires analytic continuation of $I(n)$ to non-integer $n$ .",
"A convenient prescription for this continuation is as follows [35].",
"For integer $n$ , assuming the bulk respects the symmetry of the boundary conditions under $\\tau \\rightarrow \\tau + 2\\pi $ , we have $I(n)= \\left.",
"n I(n)\\right|_{2\\pi }$ , where $\\left.I(n)\\right|_{2\\pi }$ denotes the on-shell action with $\\tau $ integrated only over the range $[0,2\\pi )$ .",
"With the crucial assumption that this result applies also at non-integer $n$ , the right-hand side of eq.",
"(REF ) becomes $S_\\mathrm {grav} = \\lim _{n\\rightarrow 1} \\left.",
"\\partial _n I(n) \\right|_{2\\pi }\\,.$ In ref.",
"[35], the authors showed that the right-hand sides of eqs.",
"(REF ) and (REF ) are equivalent for asymptotically AdS space-times, thus proving the RT formula.",
"We will henceforth denote the generalised gravitational entropy and EE by the same symbol $S_A$ ."
],
[
"Holographic entanglement entropy of probe branes",
"In this paper we compute EE in holographic duals of spacetimes containing branes.",
"We write the total action for the gravitational theory as $I = I_\\mathrm {bulk} + I_\\mathrm {brane}\\,,$ where $I_\\mathrm {bulk}$ denotes the bulk action of the $(d+1)$ -dimensional gravitational theory, while $I_\\mathrm {brane}$ denotes the action of a $(p+1)$ -dimensional brane, with $p\\le d$ .",
"We assume that $I_\\mathrm {brane}$ is proportional to a tension $T_p$ .",
"We work in the probe limit, defined as the limit in which $T_p$ is small in units of $G_N$ and the $AdS_{d+1}$ curvature radius, $L$ , so that the back-reaction of the brane on the bulk fields is negligible.",
"More precisely, the probe limit is an expansion in the dimensionless parameter $\\epsilon \\equiv T_p G_N L^{p+2-d} \\ll 1$ to order $\\epsilon $ .",
"Before taking the probe limit, the presence of the brane changes the metric of the bulk space-time.",
"This leads to a change in HEE due to a change in the area of the minimal surface in eq.",
"(REF ).",
"Our goal is to determine the EE in the probe limit, meaning to order $\\epsilon $ .",
"In ref.",
"[53], Karch and Uhlemann proposed a method for computing this without having to compute the full back-reaction of the brane, i.e.",
"directly in the probe limit, by extending Lewkowycz and Maldacena's arguments to probe branes.",
"We decompose the bulk and brane actions as $I_\\mathrm {bulk} = \\int _{\\zeta _h}^\\infty d\\zeta \\int d^d x \\, \\mathcal {L}_\\mathrm {bulk} + I_\\mathrm {ct,bulk}\\,,\\qquad I_\\mathrm {brane} = \\int _{\\zeta _h}^\\infty d\\zeta \\int d^p y \\, \\mathcal {L}_\\mathrm {brane} + I_\\mathrm {ct,brane}\\,.$ Here we have introduced a coordinate $\\zeta \\in [\\zeta _h,\\infty )$ , where $\\zeta = \\zeta _h$ is the locus where the $\\tau $ -circle degenerates, and the boundary of $AdS_{d+1}$ is at $\\zeta \\rightarrow \\infty $ .",
"We have denoted the remaining bulk coordinates by $x$ , and the remaining coordinates on the brane by $y$ .",
"The counterterm action $I_\\mathrm {ct,bulk}$ consists of boundary terms at a large-$\\zeta $ cut-off $\\zeta _c$ , and is needed to render the on-shell action finite and the variational problem well-defined [74].",
"The brane counterterms $I_\\mathrm {ct,brane}$ are also needed if the brane reaches the boundary of AdS [75].",
"We collectively denote the bulk fields as $\\Phi $ and the fields on the brane as $X$ .",
"In general, the bulk Lagrangian $\\mathcal {L}_\\mathrm {bulk}$ depends on $\\Phi $ and its derivatives $\\partial \\Phi $ , and similarly $\\mathcal {L}_\\mathrm {brane}$ depends on $X$ and $\\partial X$ .",
"The brane Lagrangian will also depend on $\\Phi $ , but we assume that it is independent of $\\partial \\Phi $ .",
"This is true generically for D-brane actions in string theory, including the D3-branes we consider below.",
"With this decomposition, we can write the generalised gravitational entropy in eq.",
"(REF ) as $S_A = \\lim _{n \\rightarrow 1} \\biggl [&\\int _{\\zeta _h}^\\infty d\\zeta \\int d^d x \\left(\\frac{\\delta \\mathcal {L}_\\mathrm {bulk}}{\\delta \\Phi } \\partial _n \\Phi + \\partial _\\mu \\Theta ^\\mu _\\mathrm {bulk} \\right)- \\int d^d x \\, \\left.\\mathcal {L}_\\mathrm {bulk} \\right|_{\\zeta =\\zeta _h}\\partial _n \\zeta _h+ \\partial _n I_\\mathrm {ct,bulk}\\nonumber \\\\&+\\int _{\\zeta _h}^\\infty d\\zeta \\int d^p y \\left(\\frac{\\delta \\mathcal {L}_\\mathrm {brane}}{\\delta \\Phi } \\partial _n \\Phi + \\frac{\\delta \\mathcal {L}_\\mathrm {brane}}{\\delta X} \\partial _n X + \\partial _\\mu \\Theta ^\\mu _\\mathrm {brane} \\right) \\\\& - \\int d^p y \\, \\left.",
"\\mathcal {L}_\\mathrm {brane} \\right|_{\\zeta = \\zeta _h}\\partial _n \\zeta _h+ \\partial _n I_\\mathrm {ct,brane} \\biggr ]_{2\\pi }\\,,\\nonumber $ where the total derivative terms $\\partial _\\mu \\Theta ^\\mu _\\mathrm {bulk}$ and $\\partial _\\mu \\Theta ^\\mu _\\mathrm {brane}$ arise due to the dependence of $\\mathcal {L}_\\mathrm {bulk}$ and $\\mathcal {L}_\\mathrm {brane}$ on derivatives of $\\Phi $ and $X$ , respectively.",
"It can be shown that eq.",
"(REF ) is equivalent to the RT formula (REF ) in the presence of the brane [35], [53].",
"To evaluate eq.",
"(REF ) in the probe limit, we imagine solving the equations of motion for $\\Phi $ and $X$ as power series in the small parameter $\\epsilon $ .",
"Concretely, we write $\\Phi = \\Phi ^{(0)} + \\Phi ^{(1)} + \\dots $ where $\\Phi ^{(n)} \\sim \\mathcal {O}(\\epsilon ^n)$ , and similarly for $X$ and $S_A$ .",
"Crucially, $\\Phi ^{(0)}$ is the solution of the bulk equations of motion in the absence of the brane, and $X^{(0)}$ is the solution of the brane equations of motion when $\\Phi = \\Phi ^{(0)}$ , so $\\left.",
"\\frac{\\delta \\mathcal {L}_\\mathrm {bulk}}{\\delta \\Phi } \\right|_{\\Phi = \\Phi ^{(0)}} &= 0\\,, &\\left.",
"\\frac{\\delta \\mathcal {L}_\\mathrm {brane}}{\\delta X} \\right|_{\\Phi = \\Phi ^{(0)},\\,X=X^{(0)}} &= 0\\,.$ In the small $\\epsilon $ expansion, the bulk contributions to $S_A$ (the terms on the first line of eq.",
"(REF )) are dominated by an $\\mathcal {O}(\\epsilon ^{0})$ term that depends only on $\\Phi ^{(0)}$ .",
"This is the gravitational entropy in the absence of the brane.",
"Since by definition $\\Phi ^{(0)}$ extremises the bulk action, there is no $\\mathcal {O}(\\epsilon )$ contribution to the first line of eq.",
"(REF ).",
"The $\\mathcal {O}(\\epsilon )$ piece of $S_A$ , which we denote $S_A^{(1)}$ , is thus obtained purely from the brane contribution, i.e.",
"the second and third lines of eq.",
"(REF ), evaluated on the leading order solutions $\\Phi ^{(0)}$ and $X^{(0)}$ .",
"We can simplify this contribution by noting that $\\delta \\mathcal {L}_\\mathrm {brane}/\\delta X$ vanishes by eq.",
"(REF ).",
"Furthermore, the total derivative term $\\partial _\\mu \\Theta ^\\mu $ may be integrated over $\\zeta $ , yielding boundary terms at $\\zeta =\\zeta _h$ and $\\zeta = \\zeta _c$ .",
"The $\\zeta _c$ boundary term must cancel part of the counterterm contribution.",
"Concretely, if we write $I_\\mathrm {ct,brane} = \\int d^p y \\left.",
"\\mathcal {L}_\\mathrm {ct,brane} \\right|_{\\zeta =\\zeta _c}$ , then $ \\partial _n I_\\mathrm {ct,brane} = \\int d^p y \\, \\left[ \\frac{\\delta \\mathcal {L}_\\mathrm {ct,brane}}{\\delta X} \\partial _n X + \\frac{\\delta \\mathcal {L}_\\mathrm {ct,brane}}{\\delta \\Phi } \\partial _n \\Phi \\right]_{\\zeta =\\zeta _c}\\,.$ The boundary term at $\\zeta =\\zeta _c$ from $\\partial _\\mu \\Theta ^\\mu $ must cancel the term containing $\\delta \\mathcal {L}_\\mathrm {ct,brane}/\\delta X$ in eq.",
"(REF ), since the variational problem demands that we choose boundary conditions for $X$ such that the whole action, including boundary terms, is stationary on a solution of the equations of motion.",
"In total, the $\\mathcal {O}(\\epsilon )$ contribution to the EE is then $S_A^{(1)} = \\lim _{n \\rightarrow 1} \\biggl [\\int _{\\zeta _h}^\\infty d\\zeta \\int d^p y & \\frac{\\delta \\mathcal {L}_\\mathrm {brane}}{\\delta \\Phi } \\partial _n \\Phi + \\int d^p y \\frac{\\delta \\mathcal {L}_\\mathrm {ct,brane}}{\\delta \\Phi } \\partial _n \\Phi \\nonumber \\\\&- \\int d^p y \\left.",
"\\mathcal {L}_\\mathrm {brane}\\right|_{\\zeta = \\zeta _h} \\partial _n \\zeta _h+ \\int d^p y \\left.",
"N_\\mu \\Theta ^\\mu _\\mathrm {brane} \\right|_{\\zeta = \\zeta _h}\\biggr ]_{2\\pi }\\,,$ where $N_\\mu $ is the unit normal vector to the surface at $\\zeta =\\zeta _h$ , and this expression is to be evaluated on the leading order solutions $\\Phi = \\Phi ^{(0)}$ and $X = X^{(0)}$ .",
"In ref.",
"[53] Karch and Uhlemann argued that the final term in eq.",
"(REF ) generically vanishes.",
"However, we find that this is not the case.",
"Indeed, we find that this term is non-zero for all the probe D3-branes we study that break conformal symmetry in the dual CFT.",
"We will see that this boundary term will be crucial to obtaining the correct expression for the EE of the Coulomb branch D3-brane in section REF , and a finite and continuous expression for the EE of the spherical soliton in section REF ."
],
[
"HEE of spherical entangling regions",
"A significant challenge when using eq.",
"(REF ) is that we need to know the solution for the metric $g_{\\mu \\nu }$ and other bulk fields when $n \\ne 1$ , in order to evaluate the derivatives $\\partial _n \\Phi $ and $\\partial _n \\zeta _h$ .",
"In general, this solution can be very difficult to find.",
"However, for a spherical entangling region $A$ of radius $R$ , and in the CFT vacuum state dual to $AdS_{d+1}$ space-time, the appropriate metric was found in refs.",
"[76], [73].",
"We will make extensive use of this solution, so we provide a brief review of it here.",
"To begin, consider the Euclidean $AdS_{d+1}$ metric in Poincaré coordinates, $ds^2 = \\frac{L^2}{r^2} \\, d r^2 + \\frac{r^2}{L^2} \\left( d t_E^2 + d\\rho ^2 + \\rho ^2 d \\Omega _{d-2}^2 \\right).$ Here $r$ is the $AdS_{d+1}$ radial coordinate, with the Poincaré horizon at $r=0$ and the boundary at $r \\rightarrow \\infty $ , $t_E$ is the Euclidean time, $\\rho $ denotes the CFT radial coordinate, and $d \\Omega _{d-2}^2$ is the round metric on $ S^{d-2} $ .",
"We now make a coordinate transformation to hyperbolic slicing, defining coordinates $(\\zeta ,u,\\tau )$ such that $r = \\frac{L^2}{R} \\left(\\zeta \\cosh u +\\sqrt{\\zeta ^2 -1} \\cos \\tau \\right) \\,, \\qquad t_E =R\\frac{\\sqrt{\\zeta ^2 -1} \\sin \\tau }{\\zeta \\cosh u + \\sqrt{\\zeta ^2 -1} \\cos \\tau }\\,, \\qquad \\nonumber \\\\\\rho =R\\frac{ \\zeta \\sinh u}{\\zeta \\cosh u + \\sqrt{\\zeta ^2 -1} \\cos \\tau }\\,.$ The new coordinates are dimensionless and take values in the following intervals: $u \\in [0, +\\infty )$ , $\\zeta \\in [1, + \\infty )$ and $\\tau \\in [0,2\\pi ) $ .",
"The metric in eq.",
"(REF ) now becomes $ds^2 &= L^2 \\left[ \\frac{d\\zeta ^2}{f(\\zeta )} + f(\\zeta ) d \\tau ^2 + \\zeta ^2 du^2 + \\zeta ^2 \\sinh ^2u \\, d \\Omega ^2_{d-2}\\right] \\,, & f(\\zeta ) &= \\zeta ^2 - 1\\,.$ At the $AdS_{d+1}$ boundary, the coordinate transformation in eq.",
"(REF ) implements a conformal transformation that maps the sphere's causal development to $S^1$ (parametrised by $\\tau $ ) times a hyperbolic plane [73].",
"Eq.",
"(REF ) thus maps the reduced density matrix $\\rho _A$ for spherical region $A$ to a thermal density matrix on the hyperbolic plane.",
"Indeed, the metric in eq.",
"(REF ) has a Euclidean horizon at $\\zeta _h = 1$ , where the radius of the circle parametrised by $\\tau $ shrinks to zero, with inverse Hawking temperature $2 \\pi $ .",
"From eq.",
"(REF ) we find that in Poincaré coordinates the horizon is located at $r^2 + \\rho ^2 =R^2$ , which is precisely the RT surface of region $A$ [33], [34].",
"The EE thus maps to the Bekenstein-Hawking entropy of the horizon, as expected.",
"Of course, we merely changed coordinates, so the space is still precisely $AdS_{d+1}$ , and the horizon is simply that due to the observer's acceleration [76], [73].",
"For the gravitational theories of interest in this paper, the period of the Euclidean time can be modified to $\\tau \\sim \\tau + 2\\pi n$ by replacing $f(\\zeta )$ in eq.",
"(REF ) with $f_n(\\zeta ) &= \\zeta ^2 - 1 - \\frac{\\left( \\zeta _h^d- \\zeta _h^{d-2}\\right)}{\\zeta ^{d-2}}\\,, &\\zeta _h &= \\frac{\\sqrt{1+n^2 d (d-2) }+1}{n d }\\,,$ where the horizon is now at $\\zeta = \\zeta _h$ .",
"The metric in eq.",
"(REF ) with $f(\\zeta )$ in eq.",
"(REF ) is the topological black hole in hyperbolic slicing studied in ref. [76].",
"Given a solution for the worldvolume fields on a probe brane in $AdS_{d+1}$ , we will compute $S_A^{(1)}$ by mapping the solution to hyperbolic coordinates via eq.",
"(REF ), plugging the result into eq.",
"(REF ), and then performing the integrals.",
"As mentioned in section , EE generically has UV divergences arising from correlations across $A$ 's boundary.",
"In eq.",
"(REF ), we reach $A$ 's boundary by fixing $\\zeta $ and sending $u \\rightarrow \\infty $ : these limits send $r \\rightarrow \\infty $ , taking us to the $AdS_{d+1}$ boundary, and $\\rho \\rightarrow R$ , taking us to the surface of $A$ .",
"The UV divergences of the EE would thus appear on a probe brane as large-$u$ divergences.",
"These will not be cancelled by the probe brane's counterterms, $I_\\mathrm {ct,brane}$ in eq.",
"(REF ), as those cancel divergences that are independent of the choice of $A$ .",
"To be explicit, $I_\\mathrm {ct,brane}$ cancels divergences that arise when we fix $u$ and send $\\zeta \\rightarrow \\infty $ : these limits send $r \\rightarrow \\infty $ but with $\\rho $ determined by $u$ and $\\tau $ , such that $\\rho $ need not be at $A$ 's surface.",
"These are the usual near-boundary divergences of $AdS_{d+1}$ , which could be regulated by a Fefferman-Graham cutoff $r = L^2/\\varepsilon $ with $\\varepsilon \\ll 1$ .",
"None of our probe D3-branes below will exhibit large-$u$ divergences, either because they do not reach the $AdS_5$ boundary at all, like the Coulomb branch or spherical soliton D3-branes in figures REF b and d, or because they reach the $AdS_5$ boundary only at a point, which does not produce divergences near $\\rho = R$ , like the conformal or screened Wilson line D3-branes in figures REF a and c. As a result, all of our results for $S_A^{(1)}$ below will be finite for any finite $R$ , i.e.",
"they will require no UV regulator."
],
[
"A simple example: the probe string",
"In order to illustrate the Karch-Uhlemann method of computing probe brane EE, we now apply it to a straight Wilson line in the fundamental representation of $\\mathcal {N}=4$ SYM theory with gauge group $SU(N)$ .",
"A result for the spherical EE of a Wilson line in the fundamental representation, valid in the large $N$ limit and for any value of the 't Hooft coupling $\\lambda $ , was obtained in ref.",
"[67] using only conformal symmetry and supersymmetric localisation.",
"We will reproduce the holographic computation of this EE using the Karch-Uhlemann method in ref.",
"[56], which agrees with ref.",
"[67] when $\\lambda \\gg 1$ .",
"We choose this example both for its simplicity and because the final result will be useful in section REF .",
"In the Maldacena limit of large-$N$ followed by $\\lambda \\gg 1$ , the holographic dual of the fundamental representation Wilson line is a string anchored to the boundary of $AdS_5 \\times S^5$ [44], [45], which can be treated as a probe.",
"The action for the probe string is $I_\\mathrm {string} = \\frac{1}{2\\pi \\alpha ^{\\prime }} \\int d t_\\mathrm {E} \\, dr \\sqrt{P[g]} + I_\\mathrm {ct,string}\\,,$ where $1/(2\\pi \\alpha ^{\\prime })$ is the string tension, $P[g]$ is the induced metric on the string, and we have chosen $(t_E,r)$ as coordinates on the string.",
"The solution dual to the Wilson line is located at $\\rho =0$ and at an arbitrary point on the $S^5$ .",
"Mapping to hyperbolic slicing using eq.",
"(REF ), we find that we can parametrise the string by $(\\zeta ,\\tau )$ , and the solution becomes $u=0$ .",
"The on-shell action for generic $n$ , with $\\tau $ restricted to the range $[0,2\\pi )$ , becomes $\\left.",
"I_\\mathrm {string}(n)\\right|_{2\\pi } = \\frac{L^2}{2\\pi \\alpha ^{\\prime }} \\int _0^{2\\pi } d\\tau \\int _{\\zeta _h}^{\\infty } d\\zeta + I_\\mathrm {ct,string}\\,,$ and the EE is $\\left.\\frac{\\partial }{\\partial n} I_\\mathrm {string}(n)\\right|_{2\\pi }$ evaluated at $n=1$ .",
"In QFT terms, the counterterm action $I_\\mathrm {ct,string}$ is only sensitive to UV physics, and so is independent of $n$ .",
"Denoting the EE contribution of the fundamental representation Wilson line as $S_A^{(1)} = S_{\\square }$ , we find $S_{\\square } = \\left.",
"\\partial _n \\left.",
"I_\\mathrm {string}(n)\\right|_{2\\pi } \\right|_{n=1} = - \\frac{L^2}{2\\pi \\alpha ^{\\prime }} \\left.",
"\\partial _n \\, \\left( 2\\pi \\, \\zeta _h \\right) \\right|_{n=1} = \\frac{\\sqrt{\\lambda }}{3}\\,,$ where we used $\\alpha ^{\\prime } = L^2 /\\sqrt{\\lambda }$ .",
"This result agrees with that of ref.",
"[67] when $\\lambda \\gg 1$ .",
"In this simple case, the probe string's contribution to the EE arises entirely from the variation of the lower endpoint of integration $\\zeta _h$ with respect to $n$ , i.e.",
"the third term in eq.",
"(REF ).",
"This will not be the case in the more complicated examples below."
],
[
"Probe D3-brane Solutions",
"In this section we briefly review the probe D3-brane solutions in $AdS_5 \\times S^5$ that we will consider in this paper.",
"We will use the $AdS_5$ coordinates of eq.",
"(REF ), but in Lorentzian signature, so our $AdS_5 \\times S^5$ metric is $ds^2 = \\frac{L^2}{r^2} \\, d r^2 + \\frac{r^2}{L^2} \\left[ -d t^2 + d\\rho ^2 + \\rho ^2 (d\\theta ^2 + \\sin ^2 \\theta \\, d\\phi ^2) \\right]+ L^2 d \\Omega _5^2\\,,$ with $\\theta \\in [0,\\pi )$ and $\\phi \\in [0,2\\pi )$ .",
"In these coordinates, we choose a gauge in which the Ramond-Ramond 4-form field takes the formIn eq.",
"(REF ) we neglect a contribution to $C_4$ with components only in the $S^5$ directions.",
"This term is necessary to ensure that $F_5 \\equiv d C_4$ is self-dual, but it will play no role in our calculations since it has vanishing pullback on all D3-brane solutions we consider.",
"$C_4 = \\frac{r^4}{L^4} \\,\\rho ^2 \\sin \\theta \\, dt \\wedge d\\rho \\wedge d\\theta \\wedge d\\phi \\, .$ The D3-brane's action is $I_{D3} = - T_{D3} \\int d^4 \\xi \\sqrt{- \\text{det}(P[g]_{ab}+ F_{ab})} + T_{D3} \\int P[C_4]+I_\\mathrm {ct}\\,,$ where $T_{D3} = N/(2\\pi ^2)$ is the D3-brane tension, $\\xi ^a$ with $a=1,2,3,4$ are the worldvolume coordinates, $P[g]_{ab}$ is the pullback of the metric to the brane, $F_{ab}$ is the worldvolume $U(1)$ field strength, and $P[C_4]$ is the pullback of $C_4$ to the brane.",
"We assume that the D3-brane is static and spherically symmetric, and spans the $t$ direction, wraps $S^2$ inside $AdS_5$ , and sits at a point on $S^5$ .",
"The D3-brane will then trace a curve in the $(r,\\rho )$ directions, which we parametrise as $r= r(\\rho )$ .",
"We also assume that the only non-trivial field strength component is $F_{t\\rho }(\\rho )$ .",
"Plugging this ansatz into eq.",
"(REF ) and integrating over the $S^2$ then gives $\\begin{split}I_{D3}= -4 \\pi T_{D3} \\int dt \\, d\\rho \\, \\frac{\\rho ^2\\,r^4}{L^4} \\left[ \\sqrt{ 1+ \\frac{L^4}{r^4} \\left[\\left(\\frac{\\partial r}{\\partial \\rho }\\right)^2 - F_{t\\rho }^2\\right]} - 1 \\right] + I_\\mathrm {ct}\\,.\\end{split}$ The counterterm action $I_\\mathrm {ct}$ is non-zero only when the D3-brane reaches the $AdS_5$ boundary, and can be split as $I_\\mathrm {ct}= I_{\\text{UV}}+I_{U(1)}$ .",
"Here, $I_{\\text{UV}}$ are the boundary terms needed to make the action finite and $I_{U(1)}$ is a finite Lagrange multiplier, $I_{U(1)} &= - \\kappa \\, (4 \\pi L \\, T_{D3}) \\int dt \\, d\\rho \\, F_{t\\rho }\\,, & \\kappa &\\equiv k \\, \\frac{\\sqrt{\\lambda }}{4 N}\\,,$ which enforces the condition that the D3-brane is endowed with $k>0$ units of string charge.",
"We will henceforth ignore $I_{\\text{UV}}$ , which is independent of $n$ and thus will not contribute to the probe brane EE in eq.",
"(REF ).",
"As shown in refs.",
"[65], [66] and references therein, the equations of motion coming from the action in eq.",
"(REF ) have BPS solutions with $F_{t\\rho } = \\partial r/\\partial \\rho $ and $r(\\rho ) = v\\,L \\pm \\frac{\\kappa \\,L^2 }{\\rho }\\,,$ where $v>0$ is an integration constant.",
"The solution is thus determined by the two integration constants, $\\kappa $ and $v$ , and by the sign in eq.",
"(REF ).",
"Different choices for $v$ , $\\kappa $ , and the sign lead to the solutions described in figure REF , with very different interpretations in the dual CFT, as we will now summarise."
],
[
"Conformal Wilson line",
"While a Wilson line in the fundamental representation corresponds to a single string ending at the $AdS_5$ boundary, higher-rank representations correspond to multiple coincident strings.",
"When the number of strings is of order $N$ , a convenient holographic description becomes D-branes carrying string charge [77], [78].",
"In this description, the type of brane depends on the representation of the Wilson line.",
"For example, the $k$ -th antisymmetric representation is described by a D5-brane along $AdS_2 \\times S^4$ endowed with $k$ units of string charge [48], [58].",
"The $AdS_2$ factor implies the dual Wilson line preserves $(0+1)$ -dimensional conformal symmetry.",
"Indeed, the Wilson line breaks ${\\mathcal {N}}=4$ SYM's $SO(4,2)$ conformal symmetry group to $SO(1,2) \\times SO(3)$ , where $SO(1,2)$ are conformal transformations preserving the line and $SO(3)$ is the $S^2$ isometry.",
"The solution in eq.",
"(REF ) with $v=0$ and $\\kappa \\ne 0$ with a plus sign describes a conformal Wilson line in the $k$ -th symmetric representation of $SU(N)$ [46].",
"In that case, $r(\\rho ) = \\frac{\\kappa \\, L^2}{\\rho }\\,,$ so that $r \\rightarrow \\infty $ as $\\rho \\rightarrow 0$ and the D3-brane reaches the $AdS_5$ boundary at a point.",
"The solution describes a D3-brane shaped like a cone with apex at the $AdS_5$ boundary as depicted in figure REF a.",
"Both the opening angle and $k$ are determined by $\\kappa $ , as the D3-brane carries $k=\\kappa \\,4 N/\\sqrt{\\lambda }$ units of string charge [69].",
"In this case the D3-brane's worldvolume is $AdS_2 \\times S^2$ , so like the D5-branes mentioned above, this D3-brane describes a conformal Wilson line preserving $SO(1,2) \\times SO(3) \\subset SO(4,2)$ .",
"The EE contribution to a spherical region in ${\\cal N}=4$ SYM from a Wilson line in a generic representation, at large $N$ and any $\\lambda $ , was related to the expectation value of the circular Wilson loop in ref.",
"[67] using conformal symmetry and supersymmetry.",
"Simple closed form expressions exist for the specific cases of Wilson lines in symmetric and antisymmetric tensor representations, as the corresponding circular Wilson loops can be computed explicitly using matrix model techniques [49].",
"In section REF we compute the conformal D3-brane's contribution to the EE of a spherical region, finding agreement with the result for the symmetric representation Wilson line in ref.",
"[67] when $\\lambda \\gg 1$ ."
],
[
"Coulomb branch",
"The simplest non-conformal solution coming from eq.",
"(REF ) is $v>0$ and $\\kappa =0$ , so that $r(\\rho ) = L v \\,.$ This describes a D3-brane sitting at fixed $r$ , as depicted in figure REF b.",
"We can imagine that such a solution comes from a stack of $N$ D3-branes at $r=0$ after one D3-brane has been pulled to $r=Lv$ .",
"The probe approximation is then clearly justified since $N \\gg 1$ .",
"The field theory interpretation of this solution is $\\mathcal {N}=4$ $SU(N)$ SYM at large $N$ and strong coupling at a point on the Coulomb branch where $SU(N)\\rightarrow SU(N-1) \\times U(1)$ .",
"More specifically, one of the six adjoint-valued scalar fields of ${\\mathcal {N}}=4$ SYM, $\\Psi $ , acquires a non-zero VEV, $\\langle \\Psi \\rangle \\propto v$ [59].",
"Such a state describes an RG flow from a UV CFT, $SU(N)$ ${\\mathcal {N}}=4$ SYM, to an IR CFT, $SU(N-1)$ ${\\mathcal {N}}=4$ SYM and a decoupled $U(1)$ ${\\mathcal {N}}=4$ SYM.",
"The two CFTs in the IR interact only via massive degrees of freedom, namely the W-boson supermultiplet, which is bi-fundamental under $SU(N-1) \\times U(1)$ .",
"A W-boson is dual to a string stretched between the Poincaré horizon and the probe D3-brane.",
"A solution of type IIB supergravity describing fully back-reacted Coulomb branch D3-branes is known [59], and has been used to calculate EE using the RT formula for example in refs.",
"[79], [80].",
"In appendix we use the RT formula to compute the EE at the point on the Coulomb branch with the breaking $SU(N) \\rightarrow SU(N-1) \\times U(1)$ , and then take a probe limit.",
"We find perfect agreement with our probe limit result below, obtained using the Karch-Uhlemann method, eq.",
"(REF ).",
"Crucially, the agreement occurs only if we include the last term in eq.",
"(REF ), which Karch and Uhlemann overlooked in ref.",
"[53]."
],
[
"Screened Wilson line",
"Taking $v>0$ , $\\kappa >0$ , and the plus sign in eq.",
"(REF ) results in a solution that interpolates between the symmetric-representation Wilson line and Coulomb branch D3-branes, $r(\\rho ) = v\\,L + \\frac{\\kappa \\, L^2}{\\rho }\\,.$ In particular, as $\\rho \\rightarrow 0$ the solution approaches the Wilson line solution of eq.",
"(REF ), $r = \\kappa \\, L^2/\\rho $ , and as $\\rho \\rightarrow \\infty $ the solution approaches the Coulomb branch solution of eq.",
"(REF ) $r = v\\,L$ .",
"The solution in eq.",
"(REF ) is depicted in figure REF c. Ref.",
"[61] argued that the solution in eq.",
"(REF ) describes a symmetric-representation Wilson line screened by the adjoint scalar $\\Psi $ that acquires a VEV.",
"In the language of condensed matter physics, the Wilson line is an “impurity.” Thinking of the adjoint of $SU(N)$ as the combination of fundamental and anti-fundamental, the scalar VEV $\\langle \\Psi \\rangle $ acts as a collection of colour dipoles.",
"In the presence of the impurity these dipoles are polarised, and form a spherically-symmetric screening cloud around the impurity.",
"Such screening is clear qualitatively in figure REF c: the impurity is present in the UV (near the $AdS_5$ boundary) but absent in the IR (near the Poincaré horizon).",
"Ref.",
"[61] provided quantitative evidence for screening by showing the worldvolume fields support quasi-normal modes dual to quasi-bound states localised at the impurity, a clear signature of screening.",
"In section REF we compute the EE of a spherical region centered on the screened Wilson line.",
"This case is not conformal, so the EE can have non-trivial dependence on the sphere's radius $R$ .",
"More precisely, since $R$ and the scalar VEV $\\propto v$ are the only scales available, and EE is dimensionless, the EE can have non-trivial dependence on the dimensionless combination $Rv/L$ .",
"We find that as $Rv/L$ increases the impurity's contribution to the EE decreases monotonically, although to our knowledge no physical principle requires such behaviour.",
"In particular, we know of no monotonicity theorem for a case like this, where the bulk RG flow, triggered by $\\langle \\Psi \\rangle \\propto v$ , in turn triggers an RG flow of Wilson line degrees of freedom, which couple to $\\Psi $ in such a way that $\\langle \\Psi \\rangle \\propto v$ gives them a mass [44], [58].",
"In section REF we find a more detailed description of the screening.",
"We compute the VEV of the probe sector's Lagrangian, which at large $R$ has the form of a point charge with a Coulomb potential, as in Maxwell theory.",
"We thus learn that what begins in the UV as a Wilson line of $SU(N)$ appears in the IR as a point charge in the $U(1)$ sector, but is absent in the $SU(N-1)$ sector, i.e.",
"it is screened in the $SU(N-1)$ sector."
],
[
"Spherical soliton",
"Finally, taking $v>0$ , $\\kappa >0$ , and the minus sign in eq.",
"(REF ) results in a solution discussed in detail by Schwarz in refs.",
"[65], [66], $r(\\rho ) = v\\,L - \\frac{\\kappa \\, L^2}{\\rho }\\,.$ As $\\rho \\rightarrow \\infty $ this solution reduces to the Coulomb branch solution eq.",
"(REF ) like the previous case, but now as $\\rho \\rightarrow 0$ the D3-brane bends towards the Poincaré horizon.",
"Indeed, when $\\rho = \\kappa L/v$ the D3-brane intersects the Poincaré horizon $r=0$ , as depicted in figure REF d. This behaviour implies very different physics compared to the screened Wilson line discussed above.",
"In particular, in refs.",
"[65], [66] Schwarz interpreted this solution as a spherically-symmetric soliton “phase bubble,” separating ${\\mathcal {N}}=4$ SYM with gauge group $SU(N)$ inside from ${\\mathcal {N}}=4$ SYM with gauge group $SU(N-1)\\times U(1)$ outside.",
"The soliton is a spherical shell charged under the $U(1)$ of $SU(N-1) \\times U(1)$ , and being BPS thus has non-zero mass: it is an excited state of the Coulomb branch.",
"The soliton radius is $R_0\\equiv \\kappa L/v$ , which is proportional to its total charge $k$ , and its mass is $4 \\pi T_{\\text{D3}} \\, L^3 \\, v \\, \\kappa = 4 \\pi T_{\\text{D3}} \\, L^2 \\, v^2\\, R_0\\,,$ where on the right-hand side we emphasised that the soliton's mass is proportional to its radius, just like an extremal black hole.",
"As discussed in section , based on this similarity, Schwarz asked in ref.",
"[66] whether the EE of a sphere of radius $R_0$ centred on the soliton scales with the soliton's surface area, i.e.",
"as $R_0^2$ , when $\\kappa $ is large.",
"In section REF we compute this EE, finding that it scales approximately as $R_0^{1.3}$ , in contrast to area-law scaling $R_0^2$ .",
"In sections REF and we show that in the limits $\\kappa \\rightarrow \\infty $ and $v \\rightarrow \\infty $ with $R_0$ fixed, a spherical region's EE, the Lagrangian's VEV, and the stress-energy tensor's VEV all diverge at $R_0$ , and as $\\rho \\rightarrow \\infty $ approach the result for a Wilson line in a direct product of $k$ fundamental representations.",
"The natural interpretation is that in these limits the spherical soliton is an infinitely thin shell at $R_0$ that at large distances looks like a Wilson line in a direct product of $k$ fundamental representations."
],
[
"Holographic Entanglement Entropy of D3-branes",
"In this section we compute the holographic EE contribution of the probe D3-branes reviewed in section .",
"In the first part of this section, we consider a generic probe D3-brane, finding all the terms we need to obtain their contribution to the EE.",
"Subsequently, we obtain explicit results for the conformal Wilson line in section REF , the Coulomb branch in section REF , the screened Wilson line in section REF , and finally the spherical soliton in section REF .",
"For the remainder of the paper, we use units in which the $AdS_5$ radius is unity, $L\\equiv 1$ ."
],
[
"General case",
"As discussed in section , we consider (Euclidean) $AdS_5$ in hyperbolic slicing whose metric was given in eq.",
"(REF ).",
"The D3-brane's worldvolume scalars are the $AdS_5$ coordinates whose profile determines the embedding.",
"Parametrising the embedding as $u(\\zeta ,\\tau )$ gives a D3-brane action $\\begin{aligned}I_{D3} =\\;& T_{D3} \\int d\\theta \\, d \\phi \\, d\\tau \\, d \\zeta \\, \\sin \\theta \\, \\zeta ^2 \\, \\sinh ^2 u\\sqrt{\\zeta ^2 (\\partial _\\zeta u)^2f_n(\\zeta )+\\frac{\\zeta ^2 ( \\partial _\\tau u)^2}{f_n(\\zeta )}-F_{\\tau \\zeta }^2+ 1} \\\\& - T_{D3} \\int P[C_4] + I_\\mathrm {ct}\\,.\\end{aligned}$ A delicate part of writing eq.",
"(REF ) is finding a convenient gauge for $C_4$ .",
"To this end, we point out an observation first made in ref. [46].",
"The gauge in eq.",
"(REF ) gives the correct result for the expectation value of a straight Wilson line.",
"However, after performing the mapping to hyperbolic slicing in eq.",
"(REF ), one does not obtain the correct value for a circular Wilson loop.",
"The reason is that the action in eq.",
"(REF ) in that case would need an additional boundary term to be gauge invariant.",
"To the best of our knowledge the form of such a boundary term has not been found.",
"We will therefore just pick a gauge that reproduces the correct circular Wilson loop expectation value, namely that used in ref.",
"[46], in whichNote that the minus sign on the $C_4$ is due to a different choice of orientation with respect to ref. [46].",
"$\\begin{aligned}C_4 =\\;& - \\zeta ^2 (\\zeta ^2-1) \\sinh ^2 u \\sin \\theta \\, du \\wedge d\\tau \\wedge d \\theta \\wedge d\\phi \\\\&+ \\frac{\\zeta \\sinh ^2 u \\sin \\theta (\\sinh u - \\cos \\theta \\cosh u)}{\\cosh u -\\cos \\theta \\sinh u} \\, d \\zeta \\wedge d\\tau \\wedge d\\theta \\wedge d\\phi \\\\&- \\frac{\\zeta \\sinh u \\sin ^2 \\theta }{\\cosh u -\\cos \\theta \\sinh u} \\, d \\zeta \\wedge d\\tau \\wedge du \\wedge d\\phi \\,.\\end{aligned}$ When $n=1$ , the first term in eq.",
"(REF ) vanishes at the horizon $\\zeta = \\zeta _h =1$ .",
"This is no accident: it is necessary to avoid a singularity at the horizon.",
"When $n \\ne 1$ , this condition is no longer satisfied because $\\zeta _h \\ne 1$ .",
"In appendix we find the requisite gauge transformation to make $C_4$ regular at the horizon, with the result $\\begin{aligned}C_4 =\\;& - \\zeta ^2 f_n(\\zeta ) \\sinh ^2 u \\sin \\theta \\, du \\wedge d\\tau \\wedge d \\theta \\wedge d\\phi \\\\&+ \\frac{\\zeta \\sinh ^2 u \\sin \\theta (\\sinh u - \\cos \\theta \\cosh u)}{\\cosh u -\\cos \\theta \\sinh u} \\, d \\zeta \\wedge d\\tau \\wedge d\\theta \\wedge d\\phi \\\\&- \\frac{\\zeta \\sinh u \\sin ^2 \\theta }{\\cosh u -\\cos \\theta \\sinh u} \\, d \\zeta \\wedge d\\tau \\wedge du \\wedge d\\phi \\,.\\end{aligned}$ The only change is in the first line, where $(\\zeta ^2-1)$ has been replaced by the function $f_n(\\zeta )$ defined in eq.",
"(REF ).",
"The pull-back of $C_4$ in eq.",
"(REF ) to the D3-brane worldvolume is then $P[C_4] =- \\left[\\zeta ^2 f_n(\\zeta ) (\\partial _\\zeta u) - \\zeta \\frac{ \\sinh u - \\cos \\theta \\cosh u}{\\cosh u -\\cos \\theta \\sinh u} \\right] \\; \\sinh ^2 u \\sin \\theta \\, d\\zeta \\wedge d\\tau \\wedge d \\theta \\wedge d\\phi \\,.$ In the following we will show that this choice reproduces the correct EE of both the conformal Wilson line and the Coulomb branch D3-branes.",
"If we also split $I_\\mathrm {ct} = I_\\mathrm {UV}+I_{U(1)}$ , with $I_{U(1)}$ defined in eq.",
"(REF ), then the complete action of the D3-brane in eq.",
"(REF ) is $\\begin{aligned}\\left.I_{D3}(n)\\right|_{2\\pi } =\\;&4\\pi T_{D3} \\int d \\zeta d\\tau \\Bigg \\lbrace \\zeta ^2 \\sinh ^2 u\\sqrt{\\zeta ^2 (\\partial _\\zeta u)^2f_n(\\zeta )+\\frac{\\zeta ^2 ( \\partial _\\tau u)^2}{f_n(\\zeta )}-F_{\\tau \\zeta }^2+ 1} \\\\& - \\kappa F_{\\tau \\zeta } +\\omega _\\pm (u) \\left( \\zeta ^2 f_n(\\zeta ) (\\partial _\\zeta u) \\sinh ^2 u + \\zeta \\left( u -\\sinh u \\cosh u \\right) \\right)\\Bigg \\rbrace \\\\&+ I_{\\text{UV}}\\,.\\end{aligned}$ where $\\omega _\\pm (u)= \\pm $ takes into account a possible change in the orientation of the D3-brane due to the explicit parametrization of the embedding $u(\\zeta , \\tau )$ .",
"In the explicit examples below this will be the case only for the soliton solution studied in section REF .",
"To compute the EE using eq.",
"(REF ), we need to evaluate the action in eq.",
"(REF ) on-shell.",
"We can easily solve $F_{\\tau \\zeta }$ equation of motion and plug the result back into the action, finding $\\begin{split}\\left.I_{D3}(n)\\right|_{2\\pi } =\\;& 4\\pi T_{D3} \\int d \\zeta d\\tau \\Bigg \\lbrace \\, \\sqrt{\\left(\\kappa ^2 + \\zeta ^4\\sinh ^4 u\\right) \\left(1+\\zeta ^2 (\\partial _\\zeta u )^2 f_n(\\zeta )+\\frac{\\zeta ^2 ( \\partial _\\tau u)^2}{f_n(\\zeta )}\\right)} \\\\& + \\omega _\\pm (u) \\left( \\zeta ^2 f_n(\\zeta ) (\\partial _\\zeta u) \\sinh ^2 u + \\zeta \\left( u -\\sinh u \\cosh u \\right) \\right) \\Bigg \\rbrace \\\\&+ I_{\\text{UV}}\\,.\\end{split}$ The EE is then given by a derivative with respect to $n$ of this expression.",
"In particular, we need to consider the four terms in $S_A^{(1)}$ in eq.",
"(REF ).",
"They respectively correspond to: Term 1: $\\partial /\\partial n$ of the integrand in eq.",
"(REF ).",
"The integral of this we call $\\mathcal {S}$ .",
"Term 2: $\\partial /\\partial n$ of the boundary term $I_{\\text{UV}}$ in eq.",
"(REF ).",
"As mentioned below eq.",
"(REF ), $I_{\\text{UV}}$ is independent of $n$ , so this term vanishes.",
"Term 3: $\\partial /\\partial n$ acting on the lower limit of the $\\zeta $ -integration, $\\zeta _h$ .",
"This is non-zero since $\\zeta _h$ depends on $n$ through eq.",
"(REF ).",
"We call this term $\\mathcal {S}^{(bdy)}_1$ .",
"Term 4: the boundary contribution from the equations of motion of $u(\\zeta ,\\tau )$ , which we denote by $\\mathcal {S}^{(bdy)}_2$ .",
"In summary, the EE has three distinct contributions: $S^{(1)}_A = \\mathcal {S} + \\mathcal {S}^{(bdy)}_1 + \\mathcal {S}^{(bdy)}_2\\,.$ For $\\mathcal {S}$ , we take $\\partial /\\partial n$ of the integrand of eq.",
"(REF ), evaluate the result at $n=1$ , and integrate over $\\tau $ and $\\zeta $ , with the result $\\mathcal {S} = \\frac{8\\pi }{3}T_{D3} \\int d \\zeta d\\tau \\left[\\frac{\\left( (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ f^2_1(\\zeta )}\\right)\\sqrt{ \\kappa ^2+\\zeta ^4 \\sinh ^4 u}}{2 \\sqrt{1+\\zeta ^2 (\\partial _\\zeta u)^2f_1(\\zeta )+ \\frac{\\zeta ^2 (\\partial _\\tau u)^2}{f_1(\\zeta )} }} + \\omega _\\pm (u)(\\partial _\\zeta u) \\sinh ^2 u\\right].$ For the contribution $\\mathcal {S}^{(bdy)}_1$ from the limit of integration, we find $\\begin{aligned}\\mathcal {S}^{(bdy)}_1 =\\;& \\frac{8\\pi ^2}{3} T_{D3} \\bigg \\lbrace \\sqrt{\\left(\\kappa ^2 + \\zeta ^4\\sinh ^4 u_h\\right) \\left(1+\\zeta _h^2 (\\partial _\\zeta u_h )^2 f_n(\\zeta )+\\frac{\\zeta ^2 ( \\partial _\\tau u_h)^2}{f_n(\\zeta _h )}\\right)} \\\\& + \\omega _\\pm (u_h)\\left[ \\zeta _h^2 f_n(\\zeta _h) (\\partial _\\zeta u_h) \\sinh ^2 u_h + \\zeta _h \\left( u_h -\\sinh u_h \\cosh u_h \\right) \\right] \\bigg \\rbrace \\,,\\end{aligned}$ with $u_h \\equiv u(\\zeta _h,\\tau )$ and $\\partial _\\alpha u_h \\equiv (\\partial _\\alpha u) (\\zeta _h,\\tau )$ , where $\\alpha =\\zeta , \\tau $ , and $\\zeta _h=1$ because $n=1$ .",
"For the contribution $\\mathcal {S}^{(bdy)}_2$ coming from the equations of motion's boundary term, we find $\\begin{aligned}\\mathcal {S}^{(bdy)}_2 = - 4\\pi T_{D3} \\int _0^{2\\pi } d\\tau \\Bigg \\lbrace \\, \\sqrt{\\frac{\\kappa ^2 + \\zeta _h ^4\\sinh ^4 u_h}{ 1+\\zeta _h^2 (\\partial _\\zeta u_h )^2 f_n(\\zeta )+\\frac{\\zeta _h^2 ( \\partial _\\tau u_h)^2}{f_n(\\zeta _h )}}} \\zeta _h^2 (\\partial _\\zeta u_h ) f_n(\\zeta _h) \\\\+ \\omega _\\pm (u_h) \\,\\zeta _h^2 f_n(\\zeta _h) \\sinh ^2 u_h \\Bigg \\rbrace \\partial _n u_h \\Bigg |_{n =1}&\\,.\\end{aligned}$ To evaluate eq.",
"(REF ), we need $\\partial _n u_h$ .",
"In general this can be very complicated to compute because in most cases we do not know the solution to the equations of motion when $n \\ne 1$ .",
"Luckily, we can show that whenever this term is non-vanishing, it is completely fixed by the on-shell solution at $n=1$ .",
"To do so, we start with the following ansatz for the behaviour of the embedding close to the hyperbolic horizon, $\\zeta \\sim \\zeta _h$ , $u (\\zeta ,\\tau ) = u_n^{(0)}(\\tau ) + u_n^{(1)}(\\tau ) \\sqrt{\\zeta -\\zeta _h} + u_n^{(2)}(\\tau ) (\\zeta -\\zeta _h) +\\mathcal {O}\\left[(\\zeta -\\zeta _h)^{3/2}\\right].$ The equation of motion at lowest order in a small $(\\zeta -\\zeta _h)$ expansion gives $ \\partial _\\tau u_n^{(0)}(\\tau ) = 0$ , which is solved by constant $u_n^{(0)}(\\tau ) = u_n^{(0)}$ .",
"At the next order, the equation of motion gives $\\zeta _ h^2 \\, \\partial _\\tau ^2 u_n^{(1)}(\\tau )+\\left(1-2 \\zeta _h^2\\right)^2 u_n^{(1)}(\\tau ) =0\\,,$ with general solution $u_n^{(1)}(\\tau ) = c_1(n) \\sin \\left( \\frac{\\tau }{n} \\right) + c_2 (n) \\cos \\left( \\frac{\\tau }{n} \\right)\\,.$ In principle, we should fix the integration constants $c_1(n)$ and $c_2(n)$ for all $n$ .",
"However, we will see that this is not necessary because the final result will depend only on the coefficients evaluated at $n=1$ , and not on their derivatives with respect to $n$ .",
"In other words, knowing the embedding at $n=1$ is enough.",
"Indeed, plugging our solutions for $u_n^{(0)}(\\tau )$ and $u_n^{(1)}(\\tau )$ into eq.",
"(REF ) gives $\\mathcal {S}^{(bdy)}_2 = -2 \\sqrt{2} \\pi T_{D3}\\int _0^{2\\pi } d\\tau \\left[ \\frac{\\frac{1}{\\sqrt{8 n^2+1}}+1}{4 n^2}\\frac{ \\zeta _h^3 \\left(2 \\zeta _h^2-1\\right) \\sqrt{\\kappa ^2 +\\sinh ^4\\bigl (u_n^{(0)}\\bigr )} \\;\\bigl (u_n^{(1)}(\\tau )\\bigr )^2}{\\sqrt{\\frac{\\zeta _h^3 (\\partial _\\tau u_n^{(1)}(\\tau ))^2}{2 \\zeta _h^2-1}+\\left(2 \\zeta _h^2-1\\right) \\zeta _ h\\bigl (u_n^{(1)}(\\tau )\\bigr )^2+2}}\\right]_{n =1}.$ The limit $n \\rightarrow 1$ , for which $\\zeta _h \\rightarrow 1$ , is non-singular, and gives $\\mathcal {S}^{(bdy)}_2 = -\\frac{2 \\sqrt{2} \\pi }{3} T_{D3}\\int _0^{2\\pi } d\\tau \\,\\sqrt{\\frac{\\kappa ^2 + \\sinh ^4u_1^{(0)}}{2+ \\big (\\partial _\\tau u_1^{(1)}(\\tau )\\bigr )^2+ \\bigl (u_1^{(1)}(\\tau )\\bigr )^2} } \\, \\bigl (u_1^{(1)}(\\tau )\\bigr )^2.$ Note that $\\mathcal {S}^{(bdy)}_2\\ne 0$ only if $u_1^{(1)}(\\tau )\\ne 0$ .",
"In the examples below $u_1^{(1)}(\\tau )\\ne 0$ in all cases except the conformal Wilson line.",
"In other words, $\\mathcal {S}^{(bdy)}_2\\ne 0$ in all of our non-conformal examples.",
"This is no surprise.",
"Any configuration that preserves conformal symmetry will map to a $\\tau $ -independent embedding.",
"The equation of motion eq.",
"(REF ) then forces $u_1^{(1)}(\\tau )=0$ and so $\\mathcal {S}^{(bdy)}_2=0$ .",
"For example, as mentioned in section REF , the Wilson line D3-brane has worldvolume $AdS_2 \\times S^2$ .",
"For a spherical region centred on the Wilson line, the coordinate transformation in eq.",
"(REF ) produces a D3-brane extended along $\\tau $ and wrapping the equator of the hyperbolic plane.",
"In particular, its embedding $u(\\zeta ,\\tau )$ is $\\tau $ -independent (see eq.",
"(REF ) below).",
"Displacing the sphere and/or introducing a non-conformal brane introduces at least one other scale besides $R$ , and hence breaks conformal symmetry.",
"Generically, the coordinate transformation in eq.",
"(REF ) will then produce a $\\tau $ -dependent embedding, and thus $u_1^{(1)}(\\tau )\\ne 0$ and $\\mathcal {S}^{(bdy)}_2\\ne 0$ , as we will see in our non-conformal examples below.",
"In writing $\\mathcal {S}_1^{(bdy)}$ and $\\mathcal {S}_2^{(bdy)}$ we implicitly assumed that the D3-brane has precisely one connected component at the horizon $\\zeta _h=1$ .",
"In Poincaré coordinates, this corresponds to the RT surface intersecting the brane only once.",
"This is not always the case.",
"If the D3-brane has multiple disconnected components at the hyperbolic horizon, we must sum over all of them.",
"As we will see, this happens for the spherical soliton in section REF .",
"Another possibility is that the D3-brane does not reach the hyperbolic horizon, i.e.",
"the RT surface does not intersect the D3-brane.",
"In that case, $\\mathcal {S}_1^{(bdy)}=0$ and $\\mathcal {S}_2^{(bdy)}=0$ identically.",
"In order to compute the $S^{(1)}_A$ we need specific solutions for the embedding $u(\\zeta ,\\tau )$ to plug into eqs.",
"(REF ), (REF ) and (REF ).",
"In the following subsections we compute $S^{(1)}_A$ for the four D3-brane embeddings reviewed in section ."
],
[
"Conformal Wilson line",
"The conformal Wilson line in the $k$ -th symmetric representation, reviewed in section REF , is the simplest case we consider.",
"As mentioned in sections REF and REF , an exact result for the EE appears in ref.",
"[67], and a holographic calculation using the Karch-Uhlemann method appears in ref. [56].",
"Here, we review the latter computation both for completeness and to highlight the salient steps that will be important for the following cases.",
"The D3-brane embedding corresponding to the conformal Wilson line was given in eq.",
"(REF ).",
"In the hyperbolic coordinates of eq.",
"(REF ) it becomes $u(\\zeta ,\\tau ) = \\sinh ^{-1} \\left(\\kappa /\\zeta \\right)\\,,$ which is manifestly independent of $\\tau $ , and $\\omega _\\pm (u) =+$ .",
"As discussed below eq.",
"(REF ), we thus have $\\mathcal {S}_2^{(bdy)}=0$ .",
"The other two contributions to $S^{(1)}_A$ in eq.",
"(REF ) are $\\mathcal {S} &= &-\\frac{8 \\pi }{3}\\, T_{D3} \\,\\int _0^{2\\pi } d\\tau \\int _1^{\\infty } d \\zeta \\, \\frac{\\kappa ^3}{2 \\zeta ^3 \\sqrt{\\kappa ^2+ \\zeta ^3}} = \\frac{4 \\pi ^2 }{3} T_{D3} \\left(\\sinh ^{-1}\\kappa - \\kappa \\sqrt{\\kappa ^2+1}\\right), \\nonumber \\\\\\mathcal {S}^{(bdy)}_1 &= &\\frac{8 \\pi ^2 }{3} \\, T_{D3} \\,\\sinh ^{-1}\\kappa \\,.$ Denoting the symmetric representation Wilson line EE as $S^{(1)}_A = S_\\mathrm {symm}$ , we thus have $S_\\mathrm {symm} = \\frac{4 \\pi ^2 }{3} \\, T_{D3} \\, \\left(3\\sinh ^{-1}\\kappa - \\kappa \\sqrt{\\kappa ^2+1}\\right) = \\frac{2}{3}\\, N\\, \\left(3\\sinh ^{-1}\\kappa - \\kappa \\sqrt{\\kappa ^2+1}\\right),$ where in the second equality we used $T_{D3} = N/(2\\pi ^2)$ .",
"As mentioned above, eq.",
"(REF ) agrees with the exact result of ref. [67].",
"Also note that since this case is conformal, the Wilson line's contribution to the EE is independent of the entangling region's radius $R$ ."
],
[
"Coulomb branch",
"The embedding of the Coulomb branch brane was given in eq.",
"(REF ), and has $\\kappa =0$ .",
"In the hyperbolic coordinates of eq.",
"(REF ) this becomes $u(\\zeta ,\\tau )= \\cosh ^{-1} \\left(\\frac{R v-\\sqrt{\\zeta ^2 -1} \\cos \\tau }{\\zeta }\\right)\\,.$ As we discussed below eq.",
"(REF ), for this non-conformal D3-brane $u(\\zeta ,\\tau )$ depends on $\\tau $ .",
"In this case, all three contributions to $S^{(1)}_A = \\mathcal {S} + \\mathcal {S}^{(bdy)}_1 + \\mathcal {S}^{(bdy)}_2$ in eq.",
"(REF ) are non-zero.",
"For the first contribution, we plug eq.",
"(REF ) into eq.",
"(REF ) to find $\\begin{split}\\mathcal {S} = \\frac{4\\pi }{3} T_{D3} \\int d \\zeta d\\tau \\left\\lbrace \\frac{1}{Rv}\\left[ (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ (\\zeta ^2-1)^2}\\right]\\zeta ^2\\sinh ^3 u +2(\\partial _\\zeta u) \\sinh ^2 u\\right\\rbrace \\,,\\end{split}$ where we used $\\omega _\\pm (u)=+$ .",
"In fact, we can show that this integral reduces to a boundary term as follows.",
"We use $\\partial _\\zeta u &=- \\frac{1}{\\zeta ^2 \\,\\sinh u} \\left( Rv + \\frac{\\cos \\tau }{\\sqrt{\\zeta ^2-1}} \\right), & \\partial _\\tau u &= \\frac{\\sqrt{\\zeta ^2-1}}{\\zeta \\sinh u} \\sin \\tau \\,,$ to show that the first term of the integrand in eq.",
"(REF ) can be written as $\\frac{1}{Rv}\\left[ (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ (\\zeta ^2-1)^2}\\right]\\zeta ^2\\sinh ^3 u= \\varepsilon _{ \\alpha \\beta } \\partial _\\alpha J_\\beta - (\\partial _\\zeta u) \\sinh ^2 u\\,,$ where $\\varepsilon _{\\alpha \\beta }$ is the 2d Levi-Civita symbol with $\\varepsilon _{\\zeta \\tau }=-\\varepsilon _{\\tau \\zeta }=1$ , and we defined $J_\\alpha \\equiv \\frac{\\sinh ^2 u \\, \\sin \\tau }{Rv \\sqrt{\\zeta ^2 -1}}\\, \\partial _\\alpha u\\,.$ The second term of the integrand in eq.",
"(REF ) is also a total derivative, namely $2(\\partial _\\zeta u) \\sinh ^2 u = - \\partial _\\zeta (u -\\sinh u \\cosh u)\\,.$ We can thus finally write $\\mathcal {S} = \\frac{4\\pi }{3}\\, T_{D3}\\, \\int d \\zeta d\\tau \\left[ \\varepsilon _{ \\alpha \\beta } \\partial _\\alpha J_\\beta - \\frac{1}{2}\\partial _\\zeta (u -\\sinh u \\cosh u) \\right].$ A boundary only exists if $Rv>1$ , i.e.",
"when the RT surface intersects the D3-brane.",
"This means that if $Rv<1$ then $\\mathcal {S}=0$ .This can be easily verified by explicit computation.",
"For $Rv>0$ , we can use Stokes' theorem to evaluate the first term in eq.",
"(REF ) and the divergence theorem for the second term.",
"We thus obtain $\\begin{split}\\mathcal {S} & = - \\frac{4\\pi }{3} \\, T_{D3}\\, \\int _0^{2\\pi } d \\tau \\left[ J_\\tau - \\frac{1}{2} (u-\\cosh u \\sinh u) \\right]_{\\zeta =1} \\\\& = \\frac{4\\pi ^2 }{3} \\, T_{D3}\\, \\left[ \\cosh ^{-1} ( Rv )- \\left( Rv + \\frac{1}{Rv} \\right) \\sqrt{(Rv)^2-1} \\right], \\qquad Rv>1\\,,\\end{split}$ where we used $u_h\\equiv u(1,\\tau ) = \\cosh ^{-1}(Rv)$ .",
"The second contribution is straightforward to compute from eq.",
"(REF ), and we find $\\mathcal {S}^{(bdy)}_1 = \\frac{8\\pi ^2}{3} T_{D3} \\cosh ^{-1} (Rv)\\,, \\qquad Rv >1\\,,$ where we used $f_n(\\zeta _h) (\\partial _\\zeta u_h) \\rightarrow 0$ when $\\zeta _h \\rightarrow 1$ .",
"The third and final contribution is $\\mathcal {S}_{2}^{(bdy)}$ in eq.",
"(REF ).",
"Expanding the embedding $u(\\zeta ,\\tau )$ in eq.",
"(REF ) near the horizon as in eq.",
"(REF ), extracting the coefficients $u_1^{(0)}$ and $u_1^{(1)}(\\tau )$ , and plugging them into eq.",
"(REF ), we find $\\mathcal {S}^{(bdy)}_2 =- \\frac{4\\pi }{3} \\, T_{D3}\\,\\int _0^{2\\pi } d\\tau \\, \\frac{\\sqrt{(Rv)^2-1}}{Rv}\\cos ^2 \\tau = - \\frac{4\\pi ^2 }{3}\\, T_{D3}\\, \\frac{\\sqrt{(Rv)^2-1}}{Rv} \\,.$ Figure: Left: Contribution of the Coulomb branch D3-brane to EE, S Coul S_\\mathrm {Coul} in eq.",
"(), divided by NN, as a function of RvRv.",
"Right: The quantity a Coul (R)a_\\mathrm {Coul}(R) defined in eq.",
"() as a function of RvRv.",
"The function is continuous but not analytic at Rv=1Rv=1.Summing all three contributions in eqs.",
"(REF ), (REF ) and (REF ), and denoting the contribution of the Coulomb branch D3-brane to the EE as $S^{(1)}_A = S_\\mathrm {Coul}$ , we find $S_\\mathrm {Coul}={\\left\\lbrace \\begin{array}{ll}0\\,, & Rv<1\\,, \\\\\\frac{ 2}{3} \\, N \\, \\left[3 \\cosh ^{-1}(Rv)-\\left( Rv + \\frac{2}{Rv} \\right) \\sqrt{(Rv)^2-1}\\right], & Rv > 1\\,.\\end{array}\\right.",
"}$ This is the first original result of this paper.",
"As mentioned in section REF , $S_\\mathrm {Coul}$ depends only on the dimensionless combination $Rv$ .",
"We plot eq.",
"(REF ) in figure REF on the left.",
"In appendix we show that eq.",
"(REF ) agrees with a calculation using the RT method in the fully back-reacted geometry describing the Coulomb branch, as discussed in section REF .",
"This agreement of course is only possible because $\\mathcal {S}^{(bdy)}_2$ in eq.",
"(REF ), i.e.",
"the boundary term coming from the D3-brane's equations of motion, is non-zero.",
"For a $(3+1)$ -dimensional Poincaré-invariant QFT describing an RG flow from a UV CFT with central charge $a_\\mathrm {UV}$ to an IR CFT with central charge $a_\\mathrm {IR}$ , the $a$ -theorem is the statement that unitarity requires $a_\\mathrm {UV} \\ge a_\\mathrm {IR}$ [22], [25].",
"As discussed in section REF , the Coulomb branch D3-brane describes an RG flow from a UV CFT, namely ${\\mathcal {N}}=4$ SYM with gauge group $SU(N)$ , to an IR CFT, namely ${\\mathcal {N}}=4$ SYM with gauge group $SU(N-1)\\times U(1)$ .",
"In the large-$N$ limit we thus have $a_\\mathrm {UV} \\approx N^2/4$ and $a_\\mathrm {IR} \\approx (N-1)^2/4 \\approx N^2/4 - N/2$ , where we dropped terms of $\\mathcal {O}(N^0)$ .",
"Hence, the $a$ -theorem is obeyed, as expected.",
"We can show that our $S_\\mathrm {Coul}$ in eq.",
"(REF ) obeys the $a$ -theorem, as follows.",
"Eq.",
"(REF ) vanishes in the UV limit $Rv \\ll 1$ .",
"In that case the only contribution to the EE comes from the RT calculation in $AdS_5 \\times S^5$ [33], [34], $S_{SU(N)} =N^2 \\left[ \\frac{R^2}{\\varepsilon ^2} - \\log \\left( \\frac{R}{\\varepsilon } \\right) + \\mathcal {O}(1) \\right],$ where $\\varepsilon $ is the UV cut-off.",
"Eq.",
"(REF ) has the form expected for a $(3+1)$ -dimensional CFT.",
"The first term on the right-hand side is the area-law term, which is $\\varepsilon $ -dependent and hence unphysical.",
"The logarithm in the second term also depends on $\\varepsilon $ , however its coefficient is independent of rescalings of $\\varepsilon $ and hence is physical.",
"Indeed, that coefficient is $-4\\,a_\\mathrm {UV}$ .",
"In the IR limit $Rv \\gg 1$ we still have the contribution from the $AdS_5 \\times S^5$ background in eq.",
"(REF ), but now $S_\\mathrm {Coul}$ in eq.",
"(REF ) also contributes.",
"When $Rv \\gg 1$ , eq.",
"(REF ) gives $S_\\mathrm {Coul} \\approx \\frac{2}{3} \\,N \\,\\left[ - (Rv)^2 + 3 \\log (Rv) + 3 \\log 2 +\\frac{3}{2} \\right] + \\mathcal {O}\\left((Rv)^{-2}\\right) \\,, \\qquad Rv \\gg 1\\,.$ At leading order, this has the CFT form of eq.",
"(REF ), including an area-law term and logarithm, but now with cutoff $1/v$ .",
"This is intuitive: in the IR we expect the cutoff to be the inverse W-boson mass, which is indeed $\\propto 1/v$ .",
"The probe D3-brane's contribution to the logarithm's coefficient is $2N$ , hence we reproduce $a_\\mathrm {IR} = N^2/4 - N/2$ , with the $N^2/4$ from the $AdS_5 \\times S^5$ background and the $-N/2$ from the probe D3-brane.",
"Our results thus obey the $a$ -theorem exactly as expected.",
"However, a stronger version of the $a$ -theorem exists, namely the entropic $a$ -theorem of ref. [27].",
"For the EE of a spherical region of radius $R$ , $S(R)$ , we define an effective position-dependent central charge, $a(R) = \\frac{1}{8} \\left[ R^2 \\frac{d^2}{d R^2}S(R) - R \\frac{d}{d R}S(R)\\right],$ which in the UV limit $R \\rightarrow 0$ obeys $a(R) \\rightarrow a_\\mathrm {UV}$ , and in the IR limit $R \\rightarrow \\infty $ obeys $a(R) \\rightarrow a_\\mathrm {IR}$ .",
"The entropic $a$ -theorem is then the statement that strong subadditivity of EE and the Markov property of a CFT vacuum require [27], $a(R) \\le a_\\mathrm {UV}\\,.$ The effective central charge $a(R)$ in eq.",
"(REF ) is an entropic $a$ -function, defined not only at the fixed points of the RG flow, but for all scales in between.",
"The entropic $a$ -theorem is thus a constraint for all $R$ , not just the UV and IR.",
"However, when evaluated in the IR limit $R \\rightarrow \\infty $ , eq.",
"(REF ) reproduces the original $a$ -theorem.",
"Notice the entropic $a$ -theorem does not require $a(R)$ to be monotonic in $R$ , but simply imposes an upper limit.",
"We can show that our $S_\\mathrm {Coul}$ obeys the entropic $a$ -theorem as follows.",
"Eq.",
"(REF ) gives $a_\\mathrm {Coul}(R) = {\\left\\lbrace \\begin{array}{ll} 0\\,, & Rv<1\\,, \\\\ - \\dfrac{N}{2} \\, \\dfrac{\\sqrt{(Rv)^2-1}}{Rv}\\,, & Rv>1\\,,\\end{array}\\right.",
"}$ where $a_\\mathrm {Coul}(R) \\equiv a(R) - a_\\mathrm {UV}$ , and hence for the total EE, $S(R) = S_{SU(N)} + S_\\mathrm {Coul}$ , $a(R) = {\\left\\lbrace \\begin{array}{ll}\\dfrac{N^2}{4}\\,, & Rv<1\\,,\\\\[1em]\\dfrac{N^2}{4} -\\dfrac{N}{2} \\dfrac{\\sqrt{(Rv)^2-1}}{Rv}\\,, & Rv > 1\\,,\\end{array}\\right.",
"}$ which is clearly always $\\le a_\\mathrm {UV} = N^2/4$ , hence the entropic $a$ -theorem is obeyed, as expected.",
"Figure REF on the left makes clear that our $S_\\mathrm {Coul}$ is continuous and decreases monotonically as $Rv$ increases.",
"However, although our entropic $a$ -function $a(R)$ in eq.",
"(REF ) is also continuous and decreases monotonically as $Rv$ increases, it is not analytic at $Rv=1$ , because $S_\\mathrm {Coul}$ in eq.",
"(REF ) is only twice differentiable.",
"The non-analyticity of our $a(R)$ at $Rv=1$ is clear in figure REF on the right, where we plot $a_\\mathrm {Coul}(R)/N = a(R)/N - N/4$ .",
"We can also show that $S_\\mathrm {Coul}$ obeys the four-dimensional “area theorem” of refs.",
"[68], [27], which in our notation states that $\\frac{1}{2R} \\partial _R S_\\mathrm {Coul}(R)$ monotonically decreases as $R$ increases.",
"From eq.",
"(REF ) we straightforwardly find $\\frac{1}{2R} \\partial _R S_\\mathrm {Coul}(R) = {\\left\\lbrace \\begin{array}{ll}0\\,, & R v < 1\\,,\\\\- \\dfrac{2}{3} N v^2 \\left[1 - \\dfrac{1}{(R v)^2} \\right]^{3/2}, & R v > 1\\,,\\end{array}\\right.",
"}$ which indeed decreases monotonically from zero at $R=0$ to $-\\frac{2}{3} N v^2$ at $R \\rightarrow \\infty $ , thus satisfying the area theorem.",
"The boundary term $\\mathcal {S}_{2}^{(bdy)}$ is crucial for this.",
"If we neglect the contribution of $\\mathcal {S}_2^{(bdy)}$ to the EE, then the right-hand side of eq.",
"(REF ) receives an additive contribution $\\frac{1}{3} N R^{-4}v^{-2} \\left[1 - (R v)^{-2}\\right]^{-1/2}$ for $R v > 1$ .",
"This diverges to $+\\infty $ as $Rv \\rightarrow 1^+$ .",
"In that case, as $Rv$ increases through $Rv=1$ , $\\frac{1}{2R} \\partial _R S_\\mathrm {Coul}(R)$ jumps from zero to $+\\infty $ .",
"Clearly that would not be a monotonic decrease, and the area theorem would be violated."
],
[
"Screened Wilson line",
"The embedding of the D3-brane describing a screened Wilson line was given in eq.",
"(REF ), which in the hyperbolic coordinates of eq.",
"(REF ) is given implicitly by $\\zeta \\sinh u=\\frac{\\kappa \\left(\\sqrt{\\zeta ^2-1} \\cos \\tau +\\zeta \\cosh u\\right)}{-Rv+\\sqrt{\\zeta ^2-1} \\cos \\tau +\\zeta \\cosh u}\\,.$ As we discussed below eq.",
"(REF ), for this non-conformal D3-brane $u(\\zeta ,\\tau )$ depends on $\\tau $ .",
"In this case, all three contributions to $S^{(1)}_A = \\mathcal {S} + \\mathcal {S}^{(bdy)}_1 + \\mathcal {S}^{(bdy)}_2$ in eq.",
"(REF ) are non-zero.",
"Furthermore, also in this case we have $\\omega _\\pm (u)=+$ .",
"We evaluate $\\mathcal {S}$ numerically.",
"Some details of our numerical evaluation appear in appendix REF .",
"The boundary term arising from the derivative of the limits of integration is $\\mathcal {S}^{(bdy)}_1 = \\frac{8 \\pi ^2}{3} T_{D3} \\left(\\frac{Rv \\left(\\kappa ^2+\\sinh ^4u_h \\right)}{Rv \\kappa \\cosh u_h+\\sinh u_h \\left(\\kappa -\\sinh u_h\\right)^2}+ u_h-\\cosh u_h \\sinh u_h\\right),$ where $u_h$ is a function of $\\kappa $ and $Rv$ defined from eq.",
"(REF ) by setting $\\zeta =1$ and solving the resulting equation, $\\sinh u_h=\\frac{\\kappa \\cosh u_h}{\\cosh u_h - Rv}\\,.$ Eq.",
"(REF ) has only one real and positive solution, which we will not write explicitly since the expression is rather cumbersome.",
"To compute the boundary term of the equations of motion, $\\mathcal {S}^{(bdy)}_2$ , we start from $u(\\zeta ,\\tau )$ in eq.",
"(REF ), perform the expansion in eq.",
"(REF ) with $n=1$ , extract $u_1^{(0)}(\\tau )=u_h$ and $u_1^{(1)}(\\tau )$ (both of which are non-zero), plug these into eq.",
"(REF ) for $\\mathcal {S}^{(bdy)}_2$ , and perform the integration over $\\tau $ , obtaining $\\mathcal {S}^{(bdy)}_2 = -\\frac{4 \\pi ^2}{3}T_{D3}\\frac{(Rv)^2 \\kappa ^2 \\sqrt{\\frac{\\kappa ^2+\\sinh ^4 u_h}{(Rv)^2 \\kappa ^2+\\left[Rv \\kappa \\sinh u_h+\\cosh u_h (Rv-\\cosh u_h)^2\\right]^2}}}{ Rv \\kappa \\sinh u_h+\\cosh u_h (Rv-\\cosh u_h)^2 }\\,.$ We denote this probe D3-brane contribution to the EE as $S^{(1)}_A = S_\\mathrm {screen}$ .",
"We show in figure REF our results for $S_\\mathrm {screen}/(2N/3)$ as a function of $Rv$ , for several values of $\\kappa $ .",
"In all cases this EE decreases monotonically as $Rv$ increases.",
"In contrast, the result for $S_\\mathrm {screen}$ in ref.",
"[56] had a maximum.",
"The discrepancy is due to the boundary term from the equations of motion, $\\mathcal {S}^{(bdy)}_2$ , which was neglected in ref. [56].",
"However, as mentioned in sections and REF , we know of no physical principle that requires $S_\\mathrm {screen}$ to be monotonic in $Rv$ .",
"Figure: Our numerical result for the EE of the screened Wilson line, S screen /(2N/3)S_\\mathrm {screen}/(2N/3), as a function of RvRv for κ=0.5\\kappa =0.5 (red), 1 (blue), and 1.51.5 (green).",
"The inset shows that S screen /(2N/3)S_\\mathrm {screen}/(2N/3) approaches a non-zero value in the limit Rv→0Rv \\rightarrow 0, namely the value for the conformal Wilson line in eq.",
"().Since the EE depends on $R$ and $v$ only through their product $Rv$ , the $R = 0$ and $v \\rightarrow 0$ limits are equivalent.",
"Hence, at small $Rv$ we find that $S_\\mathrm {screen}$ approaches the EE of the conformal Wilson line in eq.",
"(REF ), $S_\\mathrm {symm}$ , which had $v=0$ .",
"The inset in figure REF shows $S_\\mathrm {screen}$ near $Rv \\rightarrow 0$ , which indeed approaches the non-zero value in eq.",
"(REF ).",
"In the IR limit $Rv \\rightarrow \\infty $ , we expect $S_\\mathrm {screen}$ to approach the EE of the Coulomb branch in the same limit, $S_\\mathrm {Coul}$ in eq.",
"(REF ).",
"As we move towards smaller $Rv$ , we expect a different set of finite-$(Rv)$ corrections to those in $S_\\mathrm {Coul}$ .",
"As mentioned in section , we have found a simple and intuitive derivation of these corrections, up to order $1/(Rv)$ , as follows.",
"As mentioned below eq.",
"(REF ), on the Coulomb branch at large $Rv$ the UV cutoff is $1/v$ , which is proportional to the inverse W-boson mass.",
"In the holographic picture of the Coulomb branch in figure REF b, the W-boson is a string stretched from the probe D3-brane to the Poincaré horizon.",
"In the Maldacena limit the mass of such a string is $v/(2\\pi \\alpha ^{\\prime })$ (in our units where $L\\equiv 1$ ).",
"For the screened Wilson line of figure REF c, the mass of such a string clearly increases upon approaching the Wilson line at $\\rho =0$ .",
"More specifically, such a string's mass is determined by the D3-brane's embedding, $(v+\\kappa /\\rho )/(2\\pi \\alpha ^{\\prime })$ .",
"In CFT terms, the W-boson acquires a position-dependent mass, hence the UV cutoff becomes position-dependent.",
"We thus introduce the position-dependent effective cutoff, $v_\\mathrm {eff}(\\rho ) = v+\\frac{\\kappa }{\\rho }\\,.$ We then find numerically that if we start with the Coulomb branch result $S_\\mathrm {Coul}$ at large $Rv$ in eq.",
"(REF ), make the replacement $v \\rightarrow v_\\mathrm {eff}$ , and then expand in $Rv \\gg 1$ , the result agrees precisely with our $S_\\mathrm {screen}$ at large $Rv$ , up to order $1/(Rv)$ .",
"Explicitly, for $Rv \\gg 1 $ , $S_\\mathrm {screen}(Rv) & = & S_\\mathrm {Coul}(Rv_\\mathrm {eff}) + \\mathcal {O}\\left((Rv_\\mathrm {eff})^{-2}\\right)\\nonumber \\\\ & = & \\frac{2}{3} \\, N \\,\\left[ - (R\\,v_\\mathrm {eff}(R))^2 + 3 \\log (R\\,v_\\mathrm {eff}(R)) + 3 \\log 2 +\\frac{3}{2} \\right] + \\mathcal {O}\\left((R\\,v_\\mathrm {eff}(R))^{-2}\\right) \\nonumber \\\\& = & S_\\mathrm {Coul}(Rv) + \\frac{2}{3} \\, N\\, \\left(- 2 \\,\\kappa \\,Rv - \\kappa ^2 + \\frac{3\\kappa }{Rv}\\right)+ \\mathcal {O}\\left((Rv)^{-2}\\right)\\,$ agrees with our numerical results for $S_\\mathrm {screen}$ up to order $1/(Rv)$ .",
"Indeed, in figure REF on the left we show our numerical results for $\\Delta S_\\mathrm {screen} \\equiv S_\\mathrm {screen}-S_\\mathrm {Coul} + \\frac{2}{3} N (2 \\kappa \\,Rv)\\,,$ divided by $\\frac{2}{3} N$ , which clearly approaches $-\\kappa ^2$ as $Rv \\rightarrow \\infty $ , consistent with eq.",
"(REF ).",
"Furthermore, at large but finite $Rv$ we show in figure REF on the right that $\\frac{2}{3} N (-\\kappa ^2 + \\frac{3\\kappa }{Rv})$ provides an even better approximation to $\\Delta S_\\mathrm {screen}$ than $\\frac{2}{3} N (-\\kappa ^2)$ alone.",
"We have not been able to resolve numerically whether the replacement $v \\rightarrow v_\\mathrm {eff}$ reproduces corrections at $\\mathcal {O}\\left((Rv)^{-2}\\right)$ or higher.",
"However, $S_\\mathrm {Coul}$ with $v \\rightarrow v_\\mathrm {eff}(R)$ cannot reproduce $S_\\mathrm {screen}$ for all $Rv$ .",
"For example, as mentioned above, as $Rv \\rightarrow 0$ our $S_\\mathrm {screen}$ approaches the conformal Wilson line result $S_\\mathrm {symm}$ in (REF ), whereas $S_\\mathrm {Coul}$ with $v \\rightarrow v_\\mathrm {eff}(R)$ does not.",
"Nevertheless, the fact that such a simple and intuitive replacement works at large $Rv$ up to order $1/(Rv)$ is surprising.",
"In the next section we will see that a similar replacement works for the spherical soliton EE.",
"Figure: Left: The dots are our numerical results for ΔS screen \\Delta S_\\mathrm {screen} defined in eq.",
"(), divided by 2N/32N/3, as a function of the region's size RvRv for κ=0.5\\kappa =0.5 (red), 1 (blue), and 1.51.5 (green).",
"The solid lines are the expression in eq.",
"() without the 𝒪(Rv) -2 \\mathcal {O}\\left((Rv)^{-2}\\right) corrections, and the horizontal dashed lines are -κ 2 -\\kappa ^2.",
"Right: The dots are our numerical results for ΔS screen /(2N/3)\\Delta S_\\mathrm {screen}/(2N/3) for Rv=30Rv = 30, for various κ\\kappa .",
"The solid blue line is -κ 2 -\\kappa ^2 while the solid red line is -κ 2 +3κ/(Rv)-\\kappa ^2 + 3\\kappa /(Rv).",
"The inset is a close-up that clearly shows the latter is a better approximation to our numerical results than the former."
],
[
"Spherical soliton",
"In this section, we consider the spherical soliton discussed in section REF , whose embedding in the hyperbolic coordinates of eq.",
"(REF ) is given implicitly by $\\zeta \\sinh u = \\frac{\\kappa \\left(\\sqrt{\\zeta ^2-1} \\cos \\tau +\\zeta \\cosh u\\right)}{Rv -\\sqrt{\\zeta ^2-1} \\cos \\tau -\\zeta \\cosh u} \\,.$ As we discussed below eq.",
"(REF ), for this non-conformal D3-brane $u(\\zeta ,\\tau )$ depends on $\\tau $ .",
"We discuss several features of eq.",
"(REF ) in appendix REF .",
"Here we focus on what happens when the RT surface, $\\zeta =1$ , intersects the D3-brane.",
"Figure REF is a cross section of figure REF d showing the spherical soliton D3-brane as the green curve.",
"We expect that for sufficiently small $Rv$ the RT surface will not intersect the D3-brane, as depicted by the purple curve in figure REF .",
"As we increase $Rv$ we expect to find a critical value, $\\left(Rv\\right)_\\mathrm {crit}$ , at which the RT surface is tangent to the D3-brane, as depicted by the red curve in figure REF .",
"For $Rv > \\left(Rv\\right)_\\mathrm {crit}$ we expect the RT surface to intersect the D3-brane at two points, as depicted by the orange curve in figure REF .",
"To determine the intersection points, we set $\\zeta =1$ in eq.",
"(REF ), which gives $\\sinh u_h = \\frac{\\kappa \\cosh u_h}{Rv -\\cosh u_h}\\,.$ We indeed find a critical radius, $(Rv)_\\mathrm {crit} =\\left(\\kappa ^{2/3}+1\\right)^{3/2},$ where if $Rv < (Rv)_\\mathrm {crit}$ , then eq.",
"(REF ) has no real solutions, and so the RT surface does not intersect the D3-brane.",
"If $Rv >(Rv)_\\mathrm {crit}$ , then the equation has two real solutions, so the RT surface intersects the D3-brane twice, as expected.",
"We denote these two solutions as $u_1(\\kappa ,Rv)$ and $u_2(\\kappa ,Rv)$ such that $0< u_1(\\kappa ,Rv) < u_2(\\kappa ,Rv)$ .",
"These correspond to different orientations of the D3-brane, namely $\\omega _\\pm ( u_1) = -1$ and $\\omega _\\pm ( u_2) = +1$ .",
"For the Coulomb branch D3-brane, which has $\\kappa =0$ , we expect $(Rv)_\\mathrm {crit}=1$ , which is indeed the case when $\\kappa =0$ in eq.",
"(REF ).",
"Figure: Cross sections of RT surfaces of increasing RvRv (purple, red, orange) and of the D3-brane embedding (green) for κ=0.5\\kappa =0.5.",
"When Rv<(Rv) crit Rv<(Rv)_\\mathrm {crit}, with (Rv) crit (Rv)_\\mathrm {crit} in eq.",
"(), the two do not intersect (purple).",
"When Rv=(Rv) crit Rv = (Rv)_\\mathrm {crit}, they are tangent (red).",
"When Rv>(Rv) crit Rv > (Rv)_\\mathrm {crit}, they intersect twice (orange).As in the previous two cases, all three contributions to $S^{(1)}_A = \\mathcal {S} + \\mathcal {S}^{(bdy)}_1 + \\mathcal {S}^{(bdy)}_2$ in eq.",
"(REF ) are non-zero.",
"For the first contribution we find $\\mathcal {S} = \\frac{4\\pi }{3} T_{D3} \\int d \\zeta \\, d\\tau \\, \\Bigg [&\\dfrac{1}{ Rv }&\\left( (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ f^2_1(\\zeta )}\\right)\\left|\\kappa Rv \\cosh u- \\sinh u (\\kappa +\\zeta \\sinh u)^2\\right| \\nonumber \\\\&+& 2\\,\\omega _\\pm (u) (\\partial _\\zeta u) \\sinh ^2 u\\Bigg ].$ The integral in the first line of eq.",
"(REF ) is non-trivial for any value of $Rv$ , while the one in the second line, being a boundary term, is non-trivial only for $Rv >(Rv)_\\mathrm {crit}$ .",
"We evaluate the first line of eq.",
"(REF ) numerically.",
"In appendix REF we discuss in detail our numerical evaluation for $Rv >(Rv)_\\mathrm {crit}$ , which is the most subtle case.",
"In the appendix REF , we also show that the first line of eq.",
"(REF ) diverges in the limit $Rv \\rightarrow (Rv)^+_\\mathrm {crit}$ .In the limit from below, namely $Rv \\rightarrow (Rv)^-_\\mathrm {crit}$ , the integral goes to a finite value.",
"To characterise the divergence, we take $Rv = (Rv)_{\\text{crit}}+ \\delta $ and expand the first line of eq.",
"(REF ) around $\\delta = 0$ , obtaining $\\mathcal {S}=-\\pi \\sqrt{\\frac{2}{3}} \\frac{\\left(\\kappa ^{2/3}+1\\right)^{5/4} \\kappa ^{2/3}}{\\sqrt{\\delta }} + \\mathcal {O}(\\sqrt{\\delta })\\,, \\qquad Rv \\gtrsim (Rv)_\\mathrm {crit}\\,,$ which indeed diverges as $1/\\sqrt{\\delta }$ as $\\delta \\rightarrow 0$ .",
"This divergence is cancelled by $\\mathcal {S}^{(bdy)}_1$ and $\\mathcal {S}^{(bdy)}_2$ , as we discuss below.",
"The second line of eq.",
"(REF ) reduces to boundary terms, $2\\int d \\zeta \\, d\\tau \\, \\omega _\\pm (u) (\\partial _\\zeta u) \\sinh ^2 u= - 2\\pi \\sum _{i=1,2} \\omega _\\pm (u_i) \\left( u_i -\\sinh u_i \\cosh u_i \\right).$ For $\\mathcal {S}^{bdy}_1$ a straightforward calculation gives $\\begin{split}\\mathcal {S}^{(bdy)}_1 = \\frac{8\\pi ^2}{3} T_{D3} \\sum _{i=1,2} & \\Bigg \\lbrace Rv \\frac{\\left(\\kappa ^2 + \\sinh ^4 u_i\\right) }{ \\left|\\kappa Rv \\cosh u_i - \\sinh u_i (\\kappa +\\ \\sinh u_i)^2\\right|} \\\\& + \\omega _\\pm (u_i) \\left( u_i -\\sinh u_i \\cosh u_i \\right) \\Bigg \\rbrace , \\qquad Rv > (Rv)_{\\text{crit}} \\,.\\end{split}$ The denominator of the first term in eq.",
"(REF ) vanishes at $Rv = (Rv)_{\\text{crit}}$ , hence $\\mathcal {S}^{(bdy)}_1$ diverges there.",
"Again taking $Rv = (Rv)_{\\text{crit}}+ \\delta $ and expanding in $\\delta $ , we find $\\mathcal {S}^{(bdy)}_1 = 2 \\, \\pi \\,\\sqrt{\\frac{2}{3}} \\frac{ \\left(\\kappa ^{2/3}+1\\right)^{5/4} \\kappa ^{2/3}}{\\sqrt{\\delta }}+\\mathcal {O}\\left(\\sqrt{\\delta } \\right), \\qquad Rv \\gtrsim (Rv)_{\\text{crit}}\\,.$ For the boundary contribution from the equations of motion, $\\mathcal {S}^{(bdy)}_2$ , we start from $u(\\zeta ,\\tau )$ in eq.",
"(REF ), perform the expansion in eq.",
"(REF ) with $n=1$ , extract $u_1^{(0)}(\\tau )=u_h$ and $u_1^{(1)}(\\tau )$ (both of which are non-zero), plug these into eq.",
"(REF ) for $\\mathcal {S}^{(bdy)}_2$ , and perform the integration over $\\tau $ , obtaining for $Rv \\ge (Rv)_{\\text{crit}}$ $\\mathcal {S}^{(bdy)}_2 = -\\frac{4 \\pi ^2}{3}T_{D3} \\sum _i \\frac{ (Rv)^2 \\kappa ^2 \\sqrt{\\frac{\\kappa ^2+\\sinh ^4u_i }{(Rv)^2 \\kappa ^2+\\left[\\cosh u_i (Rv-\\cosh u_i)^2-Rv \\kappa \\sinh u_i\\right]^2}}}{\\left|\\cosh u_i (Rv-\\cosh u_i)^2-Rv \\kappa \\sinh u_i\\right|}\\,.$ This term also diverges at $Rv = (Rv)_{\\text{crit}}$ .",
"Once again taking $Rv = (Rv)_{\\text{crit}}+ \\delta $ and expanding around $\\delta = 0$ , we find $\\mathcal {S}^{(bdy)}_2 = -\\pi \\sqrt{\\frac{2}{3}} \\frac{\\left(\\kappa ^{2/3}+1\\right)^{5/4} \\kappa ^{2/3}}{\\sqrt{\\delta }} + \\mathcal {O}(\\sqrt{\\delta })\\,, \\qquad Rv \\gtrsim (Rv)_{\\text{crit}}\\, .$ Clearly, in $S^{(1)}_A = \\mathcal {S} + \\mathcal {S}^{(bdy)}_1 + \\mathcal {S}^{(bdy)}_2$ the divergences in eqs.",
"(REF ), (REF ), and (REF ) cancel, so that $S^{(1)}_A$ will be finite and continuous at $Rv = (Rv)_{\\text{crit}}$ .",
"Indeed, denoting the contribution of the spherical soliton D3-brane to the EE as $S^{(1)}_A = S_\\mathrm {soliton}$ , we find Figure: Our numerical results for S soliton /(2N/3)S_{\\text{soliton}}/(2N /3) in eq.",
"() as a function of RvRv for (from left to right) κ=1\\kappa =1 (blue), 2 (green), 3 (purple), 4 (orange), 5 (red).",
"The dashed vertical lines denote the location of the critical radius Rv=(Rv) crit Rv = (Rv)_{\\text{crit}} defined in eq. ().",
"Each curve also has a maximum at (Rv) max ≳(Rv) crit (Rv)_\\mathrm {max} \\gtrsim (Rv)_{\\text{crit}}.$S_\\mathrm {soliton}=\\;& \\frac{2N}{3 \\pi } \\int d\\tau d \\zeta \\, \\Bigg \\lbrace \\dfrac{1}{ Rv }\\left( (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ f^2_1(\\zeta )}\\right)\\left|\\kappa Rv \\cosh u- \\sinh u (\\kappa +\\zeta \\sinh u)^2\\right| \\Bigg \\rbrace \\nonumber \\\\&+ \\frac{2 N}{3} \\sum _{i=1,2} \\left\\lbrace \\vphantom{\\frac{ R_v^2 \\kappa ^2 \\sqrt{\\frac{\\kappa ^2+\\sinh ^4u_i }{R_v^2 \\kappa ^2+\\left(\\cosh u_i (R_v-\\cosh u_i)^2-R_v k \\sinh u_i\\right)^2}}}{\\left|\\cosh u_i (R_v-\\cosh u_i)^2-R_v \\kappa \\sinh u_i\\right|}} 2Rv \\frac{\\left(\\kappa ^2 + \\sinh ^4 u_i\\right) }{ \\left|\\kappa Rv \\cosh u_i - \\sinh u_i (\\kappa +\\ \\sinh u_i)^2\\right|} \\right.\\nonumber \\\\& + \\omega _\\pm (u_i) \\left( u_i -\\sinh u_i \\cosh u_i \\right) \\\\&\\left.- \\frac{ (Rv)^2 \\kappa ^2 \\sqrt{\\frac{\\kappa ^2+\\sinh ^4u_i }{(Rv)^2 \\kappa ^2+\\left[\\cosh u_i (Rv-\\cosh u_i)^2-Rv \\kappa \\sinh u_i\\right]^2}}}{\\left|\\cosh u_i (Rv-\\cosh u_i)^2-Rv \\kappa \\sinh u_i\\right|} \\right\\rbrace \\Theta (Rv-(Rv)_{\\text{crit}}) \\nonumber \\,,$ where $\\Theta $ is the Heaviside step function.",
"Figure REF shows our numerical result for $S_\\mathrm {soliton}/(2N/3)$ as a function of $R v$ for several $\\kappa $ values.",
"It is clearly finite and continuous at $(Rv)_\\mathrm {crit}$ , which we indicate for each $\\kappa $ by a dashed vertical line.",
"One prominent feature of $S_\\mathrm {soliton}$ is very different from the previous cases.",
"While $S_\\mathrm {Coul}$ in section REF and $S_\\mathrm {screen}$ in section REF were monotonic in $Rv$ , our $S_\\mathrm {soliton}$ has a maximum at an $Rv$ value $(Rv)_\\mathrm {max}$ slightly larger than $(Rv)_\\mathrm {crit}$ .",
"In ref.",
"[65], Schwarz interpreted the spherical soliton as an infinitely thin shell with $U(1)$ charge.",
"Although our result for $S_\\mathrm {soliton}$ does not show any “smoking gun” features characteristic of an infinitely thin interface, our results are consistent with a charge distribution peaked at $(Rv)_\\mathrm {max}$ : if charges are entangled with each other, then regions with a larger charge density will contribute more to the EE.",
"Furthermore, we will see below that in the limit $\\kappa , v \\rightarrow +\\infty $ our $S_\\mathrm {soliton}$ may be interpreted as that of an infinitely thin shell.",
"As mentioned in section , Schwarz in ref.",
"[66] asked whether the EE of a sphere coincident with the soliton might scale with surface area at large $\\kappa $ , similar to a black hole's Bekenstein-Hawking entropy.",
"Recalling from section REF that the spherical soliton's radius is $R_0\\equiv \\kappa /v$ (in our units with $L\\equiv 1$ ), Schwarz's question becomes whether at $R=R_0$ and large $\\kappa $ we find $S_\\mathrm {soliton}\\propto \\kappa ^2$ .",
"Our numerical results suggest this is not the case.",
"Figure REF on the left shows our numerical results for the value of $S_\\mathrm {soliton}/N$ at $(Rv)_\\mathrm {max}$ as a function of $(Rv)_\\mathrm {max}$ .",
"We also show a fit to a function $\\propto (Rv)_\\mathrm {max}^{1.2}$ , which is clearly better than a $(Rv)_\\mathrm {max}^2$ fit, suggesting that the scaling is not with surface area.",
"Figure REF on the right shows our result for $S_\\mathrm {soliton}/N$ at $R_0$ (with $v=1$ ) as a function of $\\kappa $ .",
"To answer Schwarz's question: at large $\\kappa $ our results fit a function $\\propto \\kappa ^{1.3}$ better than a function $\\propto \\kappa ^2$ , suggesting again that the scaling is not with surface area.",
"Figure: Left: The black dots are our numerical results for S soliton /NS_{\\text{soliton}}/N in eq.",
"() at the position of its maximum (Rv) max (Rv)_\\mathrm {max} as a function of (Rv) max (Rv)_\\mathrm {max}.",
"The solid red line is a fit to a function ∝(Rv) max 1.2 \\propto (Rv)_\\mathrm {max}^{1.2}.",
"Right: The black dots are our numerical results for S soliton /NS_{\\text{soliton}}/N at R=R 0 R=R_0, with R 0 =κ/vR_0 = \\kappa /v the spherical soliton's radius (and we set v=1v=1), as a function of κ\\kappa .",
"The solid red line is a fit to a function ∝κ 1.3 \\propto \\kappa ^{1.3}.In the IR limit $Rv \\rightarrow \\infty $ , we expect $S_\\mathrm {soliton}$ to approach the EE of the Coulomb branch in the same limit, $S_\\mathrm {Coul}$ in eq.",
"(REF ).",
"As we move towards smaller $Rv$ we expect a different set of finite-$(Rv)$ corrections to those in $S_\\mathrm {Coul}$ .",
"Numerically we find $S_\\mathrm {soliton} = S_\\mathrm {Coul} + \\frac{2}{3} N \\left(2\\kappa Rv - \\kappa ^2 - \\frac{3\\kappa }{Rv} \\right) + \\frac{4}{3} N \\kappa + \\mathcal {O}\\left((Rv)^{-2}\\right), \\qquad Rv \\gg 1 \\,.$ Indeed, figure REF shows our numerical results for $\\Delta S_\\mathrm {soliton} \\equiv S_\\mathrm {soliton}-S_\\mathrm {Coul} - \\frac{2}{3} N (2 \\kappa \\,Rv)\\,,$ divided by $2N/3$ , at large values of $R v$ and for several values of $\\kappa $ .",
"In the figure, the dots show our numerical results, while the curves show $(-\\kappa ^2 + 2 \\kappa -3 \\kappa /(R v))$ , from the $\\mathcal {O}\\left((Rv)^0\\right)$ and $\\mathcal {O}\\left((Rv)^{-1}\\right)$ terms in eq.",
"(REF ).",
"We find very good agreement between the two, showing that the behaviour of $S_\\mathrm {soliton}$ at large $Rv$ is indeed given by eq.",
"(REF ).",
"Figure: The dots are our numerical results for ΔS soliton \\Delta S_\\mathrm {soliton} in eq.",
"(), divided by 2N/32N/3, as a function of RvRv and for κ=3\\kappa =3 (orange), 4 (purple), 5 (green), 6 (blue), 7 (red).",
"The solid lines are the prediction from eq. ().",
"The horizontal dashed lines show the limit lim Rv→∞ ΔS soliton =-κ 2 +2κ\\lim _{Rv\\rightarrow \\infty }\\Delta S_\\mathrm {soliton} = - \\kappa ^2 + 2 \\kappa .The large-$(Rv)$ behaviour in eq.",
"(REF ) up to order $\\mathcal {O}((R v)^{-1})$ consists of two contributions.",
"The first contribution, in round brackets in eq (REF ), comes from the same “trick” we used in eq.",
"(REF ) to compute the leading $1/(Rv)$ corrections to $S_\\mathrm {screen}$ .",
"We introduce an effective position-dependent cutoff like that in eq.",
"(REF ), but with $\\kappa \\rightarrow -\\kappa $ , $v_\\mathrm {eff}(\\rho ) = v-\\frac{\\kappa }{\\rho }\\,.$ In the large-$(Rv)$ Coulomb branch result, $S_\\mathrm {Coul}$ in eq.",
"(REF ), we then make the replacement $v \\rightarrow v_\\mathrm {eff}$ , and expand in $Rv \\gg 1$ , obtaining the terms in the round brackets in eq.",
"(REF ).",
"The fact that this “trick” works for both $S_\\mathrm {screen}$ and $S_\\mathrm {soliton}$ is remarkable, and suggests this is neither a trick nor a coincidence, but a genuine physical effect: in these cases, at large $Rv$ , the UV cutoff is an inverse W-boson mass that has acquired position dependence given by $v_\\mathrm {eff}$ in eq.",
"(REF ) or (REF ).",
"The second contribution is $\\frac{4}{3}N \\kappa $ .",
"Recalling from below eq.",
"(REF ) that this D3-brane carries $k=\\kappa \\,4 N/\\sqrt{\\lambda }$ units of string charge, this term is $\\kappa \\, 4N/3 = k \\sqrt{\\lambda }/3$ , which is precisely $k \\, S_{\\square }$ with $S_{\\square }=\\sqrt{\\lambda }/3$ from eq.",
"(REF ).",
"In other words, this second contribution is that of a Wilson line in a direct product of $k$ fundamental representations.",
"To summarise, we have shown that the $Rv \\gg 1$ result in eq.",
"(REF ) is $S_\\mathrm {soliton} (Rv) = S_\\mathrm {Coul}(R\\,v_\\mathrm {eff}) + k \\, S_{\\square } + \\mathcal {O}\\left((Rv)^{-2}\\right), \\qquad Rv \\gg 1\\,.$ We can in fact show that the $k \\, S_{\\square }$ contribution comes from the part of the D3-brane that reaches the Poincaré horizon.",
"Consider the limits $\\kappa \\rightarrow \\infty $ and $v \\rightarrow \\infty $ with the soliton's radius $R_0 \\equiv \\kappa /v $ fixed.",
"In this limit, eq.",
"(REF ) for the embedding reduces to $\\sinh u = \\frac{R_0}{R} \\frac{ \\sqrt{\\zeta ^2-1} \\cos \\tau +\\zeta \\cosh u}{\\zeta }\\,,$ which in Poincaré coordinates is simply a cylinder $\\rho = R_0$ .",
"Starting from figure REF d, intuitively these limits correspond to sending the Coulomb branch part of the D3-brane to the $AdS_5$ boundary while simultaneously increasing the size of the spike down to the Poincaré horizon, until all that remains in the limit is a cylinder of radius $R_0$ extending from the $AdS_5$ boundary to the Poincaré horizon.",
"This cylinder carries $k$ units of string charge, hence we interpret this solution as a uniform cylindrical distribution of $k$ strings that have expanded into a D3-brane via the Myers effect [81], [82].",
"This limit drastically alters the EE.",
"In particular, we will show that the EE in this limit diverges at $R_0$ , and when $R \\rightarrow \\infty $ reproduces the $k \\, S_{\\square }$ term in eq.",
"(REF ) (and none of the other terms).",
"Such behaviour suggests this solution is dual to an infinitely thin spherical shell of charge at $R_0$ that at $R \\rightarrow \\infty $ produces the EE of a Wilson line in the direct product of $k$ fundamental representations.",
"A similar divergent behaviour in the EE can also be found in boundary conformal field theories when the entangling region approaches the boundary [83], [84], [85].",
"To demonstrate these features, we return to eq.",
"(REF ) for the D3-brane action in the hyperbolic black hole background with arbitrary $n$ , plug in the embedding given by eq.",
"(REF ), take large $\\kappa $ , and retain only the leading contribution, which is linear in $\\kappa $ .",
"The result is $\\begin{split}\\left.",
"I_{D3}(n)\\right|_{2\\pi } = 4\\pi \\, T_{D3} \\, \\kappa \\int d\\tau \\int d \\zeta \\, \\sqrt{ 1+\\zeta ^2 (\\partial _\\zeta u )^2 f_n(\\zeta )+\\frac{\\zeta ^2 ( \\partial _\\tau u)^2}{f_n(\\zeta )}} \\,.\\end{split}$ Crucially, the term involving $C_4$ is subleading when $\\kappa \\rightarrow \\infty $ , so we can safely ignore it.",
"Taking $\\partial /\\partial n$ and $n=1$ then gives the expected form, $S_A^{(1)} = \\mathcal {S} + \\mathcal {S}_1^{(bdy)}+\\mathcal {S}_2^{(bdy)}$ , with $\\mathcal {S} & = & \\frac{4\\pi }{3} T_{D3} \\, \\kappa \\,\\int d\\tau \\int d \\zeta \\frac{\\left( (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ f^2_1(\\zeta )}\\right)}{ \\sqrt{\\left(1+\\zeta ^2 (\\partial _\\zeta u)^2f_1(\\zeta )+\\frac{\\zeta ^2 (\\partial _\\tau u)^2}{f_1(\\zeta )}\\right) }} \\nonumber \\\\& = & \\frac{2\\pi ^2}{3}\\, T_{D3}\\, \\kappa \\left[\\frac{\\left(\\tilde{R}^2-3\\right) \\log \\left(\\frac{| \\tilde{R} -1| }{\\tilde{R}+1}\\right)-6\\tilde{R}}{ \\tilde{R}} + \\frac{2 \\left(4 \\tilde{R}^2-5\\right)}{\\tilde{R} \\sqrt{\\tilde{R}^2-1}} \\Theta \\left(\\tilde{R}-1\\right) \\right],$ where we introduced $\\tilde{R} \\equiv R/R_0$ .",
"We also find $\\begin{split}\\mathcal {S}_1^{(bdy)} &= \\frac{8\\pi ^2}{3} T_{D3} \\kappa \\frac{\\tilde{R}}{ \\left|\\tilde{R} \\cosh u_h- \\sinh u_h \\right|} = \\frac{8\\pi ^2}{3} T_{D3} \\kappa \\frac{\\tilde{R}}{ \\left|\\tilde{R}^2-1\\right|\\sinh u_h} \\\\& = \\frac{8\\pi ^2}{3} T_{D3} \\kappa \\frac{\\tilde{R}}{ \\sqrt{\\tilde{R}^2-1}}\\,, \\qquad \\tilde{R} > 1\\,,\\end{split}$ $\\mathcal {S}_2^{(bdy)} = -\\frac{4 \\pi }{3}T_{D3} \\int _0^{2\\pi } d\\tau \\,\\frac{\\cos ^2\\tau }{ \\tilde{R} \\sqrt{\\tilde{R}^2-1}}\\Theta \\left(\\tilde{R}-1\\right) = -\\frac{4 \\pi ^2 }{3} \\, T_{D3}\\frac{1}{\\tilde{R} \\sqrt{\\tilde{R}^2-1}}\\Theta \\left(\\tilde{R}-1\\right).$ Denoting the contribution of this D3-brane to the EE as $S_A^{(1)} = S_\\infty $ , we thus find $S_\\infty = \\frac{1}{3}\\,N\\kappa \\, \\left\\lbrace \\frac{\\left(\\tilde{R}^2-3\\right) \\log \\left(\\frac{| \\tilde{R} -1| }{\\tilde{R}+1}\\right)-6\\tilde{R}}{ \\tilde{R}} + \\frac{12 \\sqrt{\\tilde{R}^2-1}}{\\tilde{R}} \\Theta \\left(\\tilde{R}-1\\right) \\right\\rbrace ,$ where the prefactor is $N\\kappa /3 = k \\sqrt{\\lambda }/3$ .",
"Figure REF shows $S_\\infty /(k \\sqrt{\\lambda }/3)$ as a function of $\\tilde{R}$ .",
"Clearly $S_\\infty $ diverges at $\\tilde{R} = 1$ , and $S_\\infty \\rightarrow k \\sqrt{\\lambda }/3$ at large $\\tilde{R}$ , as advertised.",
"Figure: Our result for S ∞ /(kλ/3)S_\\infty /(k \\sqrt{\\lambda }/3) from eq.",
"() as a function of R ˜≡R/R 0 \\tilde{R} \\equiv R/R_0, with R 0 ≡κ/vR_0 \\equiv \\kappa /v fixed as κ→∞\\kappa \\rightarrow \\infty and v→∞v \\rightarrow \\infty .",
"Clearly S ∞ S_\\infty diverges at R ˜=1\\tilde{R}=1, and approaches kλ/3k\\sqrt{\\lambda }/3 as R ˜→∞\\tilde{R} \\rightarrow \\infty , as indicated by the horizontal dashed line.In the next section we compute the VEV of the Lagrangian for the screened Wilson line and spherical soliton.",
"For the latter we find large-$R$ behaviour that can also be interpreted as the contribution of a Wilson line in the direct product of $k$ fundamental representations.",
"Furthermore, for the spherical soliton in the limits $\\kappa \\rightarrow \\infty $ and $v \\rightarrow \\infty $ with $R_0$ fixed we find behaviour similar to that of $S_\\infty $ , namely a divergence at $R_0$ and the large-$(Rv)$ limit of a Wilson line in a direct product of $k$ fundamental representations."
],
[
"Lagrangian and Stress-Energy Tensor",
"In a gauge theory the one-point functions of single-trace gauge-invariant operators provide a natural way to characterise the spatial profiles of objects like a Wilson line, screened or not, or a spherical soliton.",
"In this section we will consider two such operators.",
"The first is an exactly marginal scalar operator, namely the ${\\mathcal {N}}=4$ SYM Lagrangian density, which we denote $\\mathcal {O}_{F^2} \\equiv \\frac{1}{N} \\text{Tr} F_{mn}F^{mn} + \\dots $ , where the $\\ldots $ represents supersymmetric completion.",
"This operator is holographically dual to the dilaton.",
"The second is the stress-energy tensor, $T_{mn}$ , which in a CFT can acquire a non-zero vacuum expectation value in the presence of a conformal defect of codimension two or higher.",
"This operator is holographically dual to the metric.",
"In each case we will consider a single probe D3-brane's effect on the dilaton or metric, so in the dual CFT, where the breaking $SU(N) \\rightarrow SU(N-1)\\times U(1)$ and large $N$ , we will compute the order $N$ contribution to $\\langle \\mathcal {O}_{F^2}\\rangle $ or $\\langle T_{mn} \\rangle $ , respectively.",
"Evaluating $\\left\\langle \\mathcal {O}_{F^2}\\right\\rangle $ is straightforward for all cases discussed in this paper.",
"However, for $\\langle T_{mn} \\rangle $ a subtlety arises: in general we need to take into account the D3-brane's back-reaction on $C_4$ .",
"As a simplification, we will only consider $\\langle T_{mn}\\rangle $ for the spherical soliton in the limit $\\kappa \\rightarrow \\infty $ and $v \\rightarrow \\infty $ with $R_0=\\kappa /v$ fixed, where the contribution of $C_4$ is negligible, as mentioned below eq.",
"(REF )."
],
[
"Expectation value of $\\mathcal {O}_{F^2}$",
"We follow refs.",
"[86], [87], [88] and compute $\\langle \\mathcal {O}_{F^2}\\rangle $ holographically by computing the linearised perturbation of the dilaton field generated by the D3-brane, and then reading $\\langle \\mathcal {O}_{F^2}\\rangle $ from its near-boundary behavior, per the standard AdS/CFT dictionary.",
"Let us first recall existing results for $\\langle \\mathcal {O}_{F^2}\\rangle $ for the fundamental-representation Wilson line, the symmetric-representation Wilson line, and the screened Wilson line.",
"For the fundamental-representation Wilson line, the result of refs.",
"[86], [87] is $\\langle \\mathcal {O}_{F^2} \\rangle _{\\square } = \\frac{\\sqrt{\\lambda }}{16 \\pi ^2} \\frac{1}{\\rho ^4}\\,.$ For conformal Wilson lines, the dependence on $\\rho $ is fixed simply by dimensional analysis to be $1/\\rho ^4$ , as in eq.",
"(REF ).",
"The nontrivial information is therefore the dimensionless coefficient, $\\frac{\\sqrt{\\lambda }}{16 \\pi ^2} $ .",
"For the conformal Wilson line in a symmetric representation the result of ref.",
"[88] is $\\langle \\mathcal {O}_{F^2} \\rangle _\\mathrm {symm} = \\frac{N }{4 \\pi ^2} \\frac{\\kappa \\sqrt{1+\\kappa ^2}}{\\rho ^4} =\\frac{k \\sqrt{\\lambda }}{16 \\pi ^2} \\sqrt{1+\\frac{k^2 \\lambda }{16 N^2}} \\frac{1}{\\rho ^4}\\,,$ which in the limit $k \\ll N$ reduces to $k \\langle \\mathcal {O}_{F^2} \\rangle _{\\square }$ .",
"For the screened Wilson line, ref.",
"[56] was able to reduce the result to the integral, with $z=1/r$ , $\\langle \\mathcal {O}_{F^2} \\rangle _{\\text{screen}} =\\frac{3 N}{16 \\pi ^2} \\frac{1}{\\rho ^4} \\int _0^{\\frac{1}{\\rho v}}& dz \\, \\left( 1- \\rho v z \\right) z \\\\&\\!",
"\\times \\left\\lbrace \\left[ z^2 + \\left(1 - \\frac{\\kappa z}{1- \\rho v z}\\right)^2\\right]^{-\\frac{5}{2}} - \\left[ z^2 + \\left(1 + \\frac{\\kappa z}{1 - \\rho v z}\\right)^2\\right]^{-\\frac{5}{2}} \\right\\rbrace .",
"\\nonumber \\nonumber $ At small and large $\\rho v$ we can perform this integral, with the results $\\langle \\mathcal {O}_{F^2} \\rangle _{\\text{screen}} \\approx {\\left\\lbrace \\begin{array}{ll} \\dfrac{N }{4 \\pi ^2} \\dfrac{\\kappa \\sqrt{1+\\kappa ^2}}{\\rho ^4}\\,, & \\quad \\rho v \\ll 1\\,, \\\\[1em] \\dfrac{ N }{4 \\pi ^2} \\dfrac{\\kappa ^2}{\\rho ^4}\\,, & \\quad \\rho v \\gg 1\\,.\\end{array}\\right.",
"}$ As expected, in both the UV and IR limits, $\\rho v \\ll 1$ and $\\rho v \\gg 1$ , respectively, we find $1/\\rho ^4$ , as required by scale invariance.",
"For the dimensionless coefficient, the UV limit reproduces the conformal Wilson line result in eq.",
"(REF ), while the IR limit produces a factor $\\propto \\kappa ^2$ .",
"For the screened Wilson line, ref.",
"[61] provided evidence for the screening in the form of a quasi-normal mode spectrum, a feature characteristic of any screened impurity.",
"Eq.",
"(REF ) provides more detailed information, as follows.",
"Consider $(3+1)$ -dimensional Maxwell theory, which is a CFT, and in which a point electric charge $Q$ produces an electric field $\\propto Q/\\rho ^2$ , and hence $F^2 \\propto Q^2/\\rho ^4$ .",
"Eq.",
"(REF ) has the same form, including in particular the IR limit $\\rho v \\gg 1$ , where $\\kappa $ plays a role analogous to $Q$ .",
"Clearly, in the $U(1)$ sector the Wilson line is not screened in the IR, but rather survives and appears as a point-like electric charge with strength $\\kappa $ .Recall that our IR includes ${\\mathcal {N}}=4$ SYM with gauge group $U(1)$ , so eq.",
"(REF ) includes contributions from both a $U(1)$ Maxwell field and its scalar superpartners.",
"We thus learn that what appears in the UV as a Wilson line of $SU(N)$ becomes in the IR, where $SU(N) \\rightarrow SU(N-1) \\times U(1)$ , a point charge of the $U(1)$ sector, and is completely screened in the $SU(N-1)$ sector.",
"Let us now consider the Coulomb branch spherical soliton.",
"This case has not been considered in the literature, so here our results are novel.",
"The embedding for the D3-brane dual to the spherical soliton in eq.",
"(REF ) is of the same form as the embedding for the D3-brane dual to the screened Wilson line in eq.",
"(REF ), but with $\\kappa \\rightarrow -\\kappa $ .",
"As a result, we can obtain an integral for $\\langle \\mathcal {O}_{F^2} \\rangle _\\mathrm {soliton}$ simply by sending $\\kappa \\rightarrow -\\kappa $ in eq.",
"(REF ) and taking the region of integration to be the complement, $\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}= \\frac{3 N}{16 \\pi ^2} \\frac{1}{\\rho ^4} \\int _{\\frac{1}{ \\rho v}}^\\infty & d z \\, \\left( 1-\\rho v z\\right)z \\\\& \\!",
"\\times \\left\\lbrace \\left[ z^2 + \\left( 1 + \\frac{\\kappa z}{ \\rho v z-1}\\right)^2 \\right]^{-\\frac{5}{2}}- \\left[z^2 + \\left( 1 - \\frac{\\kappa z}{ \\rho v z -1}\\right)^2 \\right]^{-\\frac{5}{2}} \\right\\rbrace .",
"\\nonumber $ In general we must evaluate this integral numerically.",
"Figure REF on the left shows our numerical results for $\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}/N$ as a function of $\\rho /R_0$ , for various values of $\\kappa $ and $v=1$ , and on the right shows our numerical results for $\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}/(N\\kappa )$ as a function of $\\rho /R_0$ , with $\\kappa =v$ , for various values of $v$ .",
"We can also obtain analytical results for $\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}$ in various limits, as follows.",
"Figure: Left: Our numerical results for 〈𝒪 F 2 〉 soliton /N\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}/N from eq.",
"() as a function of ρ/R 0 \\rho /R_0 with R 0 =κ/vR_0=\\kappa /v for v=1v=1 and κ=0.1\\kappa =0.1 (red), 0.50.5 (blue), 1 (green), 1.51.5 (purple), 3 (brown), and 5 (orange).",
"Right: Our numerical results for 〈𝒪 F 2 〉 soliton /(Nκ)\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}/(N\\kappa ) from eq.",
"() as a function of ρ/R 0 \\rho /R_0 with κ=v\\kappa =v and v=1v=1 (red), 5 (blue), 10 (green), and 20 (brown).",
"The black curve corresponds to the analytical solution in the v→∞v \\rightarrow \\infty limit given by eq.",
"().In the large-$(\\rho v)$ limit, following arguments similar to those in ref.",
"[56], we find $\\langle \\mathcal {O}_{F^2}\\rangle _{\\text{soliton}} = \\frac{ N }{4 \\pi ^2} \\frac{\\kappa ^2 + \\kappa }{\\rho ^4}\\,, \\qquad \\rho v \\rightarrow \\infty \\,.$ Similarly to $S_\\mathrm {soliton}$ in the large-$R v$ limit of eq.",
"(REF ), $\\langle \\mathcal {O}_{F^2}\\rangle _{\\text{soliton}}$ includes contributions of order $\\kappa ^2$ and $\\kappa $ .",
"The order $\\kappa ^2$ contribution to $\\langle \\mathcal {O}_{F^2}\\rangle _{\\text{soliton}}$ is identical to that of $\\langle \\mathcal {O}_{F^2}\\rangle _{\\text{screen}}$ at large $\\rho v$ in eq.",
"(REF ).",
"The order $\\kappa $ contribution is precisely that of a Wilson line in a direct product of $k$ fundamental representations [56].",
"We thus interpret the order $\\kappa ^2$ term as a contribution from the $U(1)$ sector, where the spherical soliton looks like a point charge, and the order $\\kappa $ term as a contribution from the $SU(N-1)$ sector, where the spherical soliton looks like a Wilson line in the direct product of $k$ fundamental representations.",
"When we approach the origin inside the spherical soliton, $\\rho v \\rightarrow 0$ , we find that $\\langle {\\cal O}_{F^2}\\rangle _\\mathrm {soliton}$ approaches a constant $\\propto v^4$ , unlike the screened Wilson line's $1/\\rho ^4$ behaviour in eq.",
"(REF ).",
"The power of $v$ is fixed by dimensional analysis, but comes with a coefficient that is a nontrivial function of $\\kappa $ , $\\langle {\\cal O}_{F^2}\\rangle _\\mathrm { soliton}=\\frac{N}{64\\pi ^2}\\frac{v^4 \\kappa }{(1+\\kappa ^2)^7} \\left[p_{10}(\\kappa )-\\frac{p_8(\\kappa )}{\\sqrt{1+\\kappa ^2}}\\left(\\sinh ^{-1}\\kappa +\\sinh ^{-1}\\tfrac{1}{\\kappa }\\right)\\right], \\qquad \\rho v \\rightarrow 0\\,,$ where $p_{8}(\\kappa )$ and $p_{10}(\\kappa )$ are degree eight and ten polynomials, respectively, $p_8(\\kappa ) &= 105 \\kappa ^2 (40 \\kappa ^6- 204 \\kappa ^4+ 165 \\kappa ^2-20)\\,,\\nonumber \\\\p_{10}(\\kappa ) &= \\kappa ^{10}- 120 \\kappa ^9+ 1072 \\kappa ^8+8790 \\kappa ^7 - 15624 \\kappa ^6- 25179 \\kappa ^5+23380 \\kappa ^4\\\\\\nonumber &\\phantom{=} + 10572 \\kappa ^3- 4955 \\kappa ^2- 384 \\kappa +30\\,.$ In the small and large $\\kappa $ limits, we find, $\\langle \\mathcal {O}_{F^2} \\rangle _{\\rm soliton}= {\\left\\lbrace \\begin{array}{ll} \\dfrac{15N}{32\\pi ^2} v^4 \\kappa \\,, & \\quad \\kappa \\rightarrow 0\\,, \\\\[1em] \\dfrac{N}{4\\pi ^2} \\dfrac{v^4}{\\kappa ^3}\\,,& \\quad \\kappa \\rightarrow \\infty \\,,\\end{array}\\right.}",
"\\qquad \\qquad \\rho v \\rightarrow 0\\,.$ As $\\kappa \\rightarrow 0$ the spherical soliton's charge and size vanish, and correspondingly $\\langle \\mathcal {O}_{F^2} \\rangle _{\\rm soliton}$ at $\\rho v =0$ in eq.",
"(REF ) vanishes as well.",
"As $\\kappa \\rightarrow \\infty $ , the spherical soliton's radius $R_0 = \\kappa /v \\rightarrow \\infty $ as we expect to recover the results of the $SU(N)$ ${\\mathcal {N}}=4$ SYM conformal vacuum, where $\\langle \\mathcal {O}_{F^2} \\rangle =0$ .",
"Indeed, in that limit $\\langle \\mathcal {O}_{F^2} \\rangle _{\\rm soliton}$ in eq.",
"(REF ) vanishes.",
"As discussed in section REF for the EE, our results for $\\langle \\mathcal {O}_{F^2} \\rangle _{\\rm soliton}$ in general do not exhibit “smoking gun” features characteristic of an infinitely thin interface.",
"However, figure REF clearly shows that in general $\\langle \\mathcal {O}_{F^2} \\rangle _{\\rm soliton}$ has a maximum, consistent with a charge distribution peaked near $R_0$ .",
"In particular, figure REF on the left shows that if $\\kappa $ is small then $\\langle {\\cal O}_{F^2}\\rangle _\\mathrm {soliton}$ is peaked at $\\rho =0$ , and as $\\kappa $ grows the peak moves to $\\rho \\ne 0$ .",
"Clearly, some critical value $\\kappa _\\mathrm {crit}$ exists where the peak first leaves $\\rho =0$ .",
"We can determine $\\kappa _\\mathrm {crit}$ by computing $\\partial ^2/\\partial \\rho ^2$ of $\\langle {\\cal O}_{F^2}\\rangle _\\mathrm {soliton}$ at $\\rho =0$ and setting it to zero.",
"This results in a transcendental equation for $\\kappa $ with numerical solution $\\kappa _\\mathrm {crit} \\approx 0.901$ .",
"A natural interpretation is that when $\\kappa <\\kappa _\\mathrm {crit}$ , the spherical soliton is a lump at $\\rho =0$ , and as we increase $\\kappa $ through $\\kappa _\\mathrm {crit}$ the lump turns into an interface or bubble at $\\rho \\ne 0$ .",
"A natural question is whether any other observables exhibit similar behaviour.",
"As discussed at the end of section REF , when $\\kappa \\rightarrow \\infty $ and $v \\rightarrow \\infty $ with $\\kappa /v \\equiv R_0$ fixed, the D3-brane becomes a cylinder of radius $R_0$ extending from the $AdS_5$ boundary to the Poincaré horizon.",
"We saw that $S_\\mathrm {soliton}$ then exhibited a divergence at $R = R_0$ and at $R \\rightarrow \\infty $ approached the result for a Wilson line in the direct product of $k$ fundamental representations.",
"The same occurs in $\\langle \\mathcal {O}_{F^2} \\rangle _{\\rm soliton}$ : in the same limits, we find $\\begin{split}\\langle \\mathcal {O}_{F^2}\\rangle _{\\text{soliton}} & = \\kappa \\, \\frac{3 N }{16 \\pi ^2} \\int _{0}^\\infty d z \\, \\frac{z^2}{\\rho R_0 } \\left\\lbrace \\left[ z^2 + \\left( \\rho - R_0 \\right)^2\\right]^{-\\frac{5}{2}} - \\left[ z^2 + \\left( \\rho + R_0\\right)^2 \\right]^{-\\frac{5}{2}}\\right\\rbrace \\\\& = \\kappa \\, \\frac{ N }{4 \\pi ^2} \\frac{1}{(\\rho ^2-R_0^2)^2} = k \\, \\frac{ \\sqrt{\\lambda } }{16 \\pi ^2} \\frac{1}{(\\rho ^2-R_0^2)^2} \\,,\\end{split}$ which we show in figure REF on the right as a black line.",
"Clearly $\\langle \\mathcal {O}_{F^2}\\rangle _{\\text{soliton}}$ in eq.",
"(REF ) diverges at $\\rho =R_0$ , and as $\\rho \\rightarrow \\infty $ approaches $k$ times $\\langle \\mathcal {O}_{F^2}\\rangle _{\\square }$ in eq.",
"(REF ), as expected."
],
[
"Stress-energy tensor",
"In this section we consider the one-point function of the stress-tensor, $\\langle T_{mn}\\rangle $ , for the spherical soliton.",
"To compute it we will need to find the linearised back-reaction of the D3-brane on the $AdS_5$ metric.",
"This can be done via the method of ref. [89].",
"One obtains the linearised correction to the metric by integrating the graviton propagator against the probe D3-brane's stress-energy tensor, and then extracts $\\langle T_{mn}\\rangle $ from the metric's near-boundary behaviour, per the usual AdS/CFT dictionary [74], [90].Note that we compute the D3-brane's linearised back-reaction on the metric only in a near-boundary limit, to extract $\\langle T_{mn}\\rangle $ .",
"Computing the D3-brane's contribution to EE using the RT formula requires computing the linearised back-reaction on the metric everywhere in the bulk, not just near the boundary.",
"A subtlety is that the D3-brane also sources $C_4$ .",
"In general, we would need solve for the linearised back-reaction of the D3-brane on both the metric and $C_4$ , since they satisfy coupled equations of motion (see e.g.",
"ref. [51]).",
"We will avoid this issue by considering only the limits introduced at the end of section REF , $\\kappa \\rightarrow \\infty $ and $v \\rightarrow \\infty $ with the spherical soliton's radius $R_0 \\equiv \\kappa /v$ fixed.",
"In these limits the D3-brane becomes a cylinder of radius $R_0$ extending from the $AdS_5$ boundary to the Poincaré horizon, and only sources components of $C_4$ orthogonal to the non-zero components in the background solution in eq.",
"(REF ).",
"The linearised equations of motion for the metric and $C_4$ then decouple, making it sufficient to consider the D3-brane's back-reaction on the metric only.",
"Similarly to $S_\\mathrm {soliton}$ and $\\langle \\mathcal {O}_{F^2}\\rangle _{\\text{soliton}}$ , in these limits we expect $\\langle T_{mn}\\rangle $ to diverge at $R_0$ , and as $\\rho \\rightarrow \\infty $ to look like the result for a Wilson line in a direct product of $k$ fundamental representations, as computed in refs.",
"[91], [67].",
"We will see that is indeed the case.",
"Details of our computation appear in appendix .",
"Here we merely present the result, $\\begin{split}\\left< T_{t_E t_E} \\right> &= \\kappa \\,\\frac{N}{2 \\pi ^2} \\frac{2}{3}\\frac{1}{ \\left(R_0^2-\\rho ^2\\right)^2}\\,, \\\\\\left<T_{\\rho \\rho } \\right> &= \\kappa \\,\\frac{N}{2 \\pi ^2} \\left(\\frac{\\log \\frac{\\left(R_0 + \\rho \\right)^2}{\\left(R_0-\\rho \\right)^2}}{12\\rho ^3 R_0}-\\frac{1}{3 \\rho ^2 \\left(R_0^2-\\rho ^2\\right)}\\right), \\\\\\left<T_{\\theta \\theta } \\right> &= \\kappa \\,\\frac{N}{2 \\pi ^2} \\left(\\frac{ \\left(R_0^2-3 \\rho ^2\\right)}{6 \\left(R_0^2-\\rho ^2\\right)^2}-\\frac{\\log \\frac{\\left(R_0 + \\rho \\right)^2}{\\left(R_0-\\rho \\right)^2}}{24 \\rho R_0}\\right), \\\\\\left<T_{\\phi \\phi }\\right> &= \\left<T_{\\theta \\theta }\\right> \\sin ^2 \\theta \\,,\\end{split}$ with all other components vanishing.",
"This one-point function is finite as $\\rho \\rightarrow 0$ , $\\begin{aligned}\\left< T_{t_E t_E}\\right> &= \\kappa T_{D3}\\left(\\frac{2}{3 R_0^4} + \\frac{4\\rho ^2}{3 R_0^6}\\right) + \\mathcal {O}(\\rho ^4)\\,, & \\left<T_{\\rho \\rho }\\right>& = -\\kappa T_{D3}\\left(\\frac{4 \\rho ^2}{15 R_0^6}+\\frac{2}{9 R_0^4}\\right) + \\mathcal {O}(\\rho ^4) \\,, \\\\\\left<T_{\\theta \\theta }\\right> &= -\\kappa T_{D3}\\frac{2 \\rho ^2 }{9 R_0^4} + \\mathcal {O}(\\rho ^4)\\,, & \\left<T_{\\phi \\phi }\\right> &= \\left<T_{\\theta \\theta }\\right>\\sin ^2 \\theta \\,.\\end{aligned}$ However, at $\\rho = R_0$ it diverges, as expected, $\\begin{aligned}\\left< T_{t_E t_E} \\right> &= \\frac{\\kappa T_{D3}}{6 R_0^2 (\\rho -R_0)^2} + \\mathcal {O}\\left( \\frac{1}{\\rho -R_0} \\right)\\,, & \\left< T_{\\rho \\rho } \\right> &= \\frac{\\kappa T_{D3}}{6 R_0^3 (\\rho -R_0)} + \\mathcal {O}\\left[\\log \\left( \\frac{1}{\\rho - R_0} \\right) \\right]\\,,\\\\\\left< T_{\\theta \\theta } \\right> &= - \\frac{\\kappa T_{D3}}{12 (\\rho -R_0)^2} + \\mathcal {O}\\left( \\frac{1}{\\rho -R_0} \\right)\\,,&\\left< T_{\\phi \\phi }\\right> &= \\left< T_{\\theta \\theta } \\right> \\sin ^2 \\theta \\, .\\end{aligned}$ In the large distance limit $\\rho \\rightarrow \\infty $ , we expect to recover the result for $\\langle T_{mn}\\rangle $ in the presence of a one-dimensional conformal defect, which takes the form [92] $\\left< T_{t_E t_E} \\right> &= \\frac{h}{\\rho ^4}\\,, & \\left< T_{i j} \\right> &= - h \\frac{\\delta _{ij}-2 n_i n_j}{\\rho ^4} \\,, & \\left< T_{t j} \\right> = 0\\,,$ where $h$ is a constant and $n_i = x_i / \\rho $ with $x_i$ , $i=1,2,3$ , the spatial Cartesian coordinates on the boundary.",
"Indeed, from eq.",
"(REF ) we find the leading order behaviour at large $\\rho $ $\\left< T_{t_E t_E} \\right> &= \\frac{2}{3} \\frac{\\kappa T_{D3} }{ \\rho ^4} \\,,& \\left< T_{\\rho \\rho } \\right> &= \\frac{2}{3} \\frac{\\kappa T_{D3} }{ \\rho ^4} \\,, & \\left< T_{\\theta \\theta } \\right> &= - \\frac{2}{3} \\frac{\\kappa T_{D3} }{ \\rho ^2}\\,, & \\left< T_{\\phi \\phi } \\right> &= \\left< T_{\\theta \\theta } \\right>\\sin ^2 \\theta \\,,$ which agrees with eq.",
"(REF ) after changing from spherical to Cartesian coordinates, with $h=2\\kappa T_\\mathrm {D3}/3$ .",
"Using $\\kappa = k \\sqrt{\\lambda }/4N$ and $T_\\mathrm {D3} = N/2\\pi ^2$ , this becomes $k \\frac{\\sqrt{\\lambda }}{12 \\pi ^2} = k\\, h_\\square \\,,$ where $h_\\square =\\sqrt{\\lambda }/12\\pi ^2$ is the value of $h$ for a fundamental representation Wilson line at large $N$ and strong coupling [91], [67].",
"We have thus shown that in the limits $\\kappa \\rightarrow \\infty $ and $v \\rightarrow \\infty $ with $R_0 \\equiv \\kappa /v$ fixed, all three of $S_\\mathrm {soliton}$ , $\\langle \\mathcal {O}_{F^2}\\rangle _{\\text{soliton}}$ , and $\\langle T_{mn}\\rangle $ diverge at $R_0$ and at $\\rho \\rightarrow \\infty $ approach the result for a Wilson line in the direct product of $k$ fundamental representations.",
"Our interpretation is that in these limits the spherical soliton becomes an infinitely thin shell of $U(1)$ charge at $R_0$ that at large distances looks like a point charge in the $U(1)$ sector and Wilson line in the direct product of $k$ fundamental representations in the $SU(N-1)$ sector."
],
[
"Summary and Outlook",
"We used holography to compute the EE of a spherical region of radius $R$ in large-$N$ , strongly coupled ${\\mathcal {N}}=4$ SYM theory on the Coulomb branch where an adjoint scalar VEV $\\propto v$ breaks the gauge group from $SU(N)$ to $SU(N-1) \\times U(1)$ .",
"We also considered cases with a screened symmetric-representation Wilson line or a spherical soliton separating $SU(N)$ inside from $SU(N-1) \\times U(1)$ outside.",
"Each case is supersymmetric, BPS, and non-conformal.",
"The objects are described holographically by probe D3-branes in $AdS_5 \\times S^5$ .",
"We used the Karch-Uhlemann method [53], i.e.",
"Lewkowycz and Maldacena's generalised gravitational entropy [35] applied to probe branes, to compute the probe's contribution to the EE directly from the probe D3-brane action, without ever computing the D3-brane's back-reaction on the bulk fields.",
"For the Coulomb branch vacuum, we found the closed-form result for the EE, $S_{\\rm Coul}$ in eq.",
"(REF ).",
"In this case, a fully back-reacted solution is available, so we could compare to a calculation using the RT minimal-area prescription, in a probe limit.",
"We found perfect agreement.",
"This agreement relied crucially on a certain boundary term in the probe D3-brane's action, our $\\mathcal {S}^{(bdy)}_2$ in eq.",
"(REF ), that was neglected by Karch and Uhlemann [53] and in previous calculations using their method [55], [56], [57].",
"Our result for $S_{\\rm Coul}$ vanishes when $Rv<1$ and is a monotonically decreasing function of $Rv$ for $Rv > 1$ .",
"Our $S_{\\rm Coul}$ is consistent with the $a$ -theorem [22], [25], its more stringent entropic cousin [27], and the entropic area theorem [68], [27].",
"Our $S_{\\rm Coul}$ also has a square root non-analyticity at $Rv = 1$ , being continuous and monotonic in $Rv$ but having a discontinuous third derivative with respect to $Rv$ at $Rv=1$ .",
"Such behaviour is strongly reminiscent of the potential experienced by a probe eigenvalue at a large-$N$ matrix model saddle point, where the eigenvalues are distributed along the branch cut of the matrix model's resolvent function.",
"On this branch cut, the force on a probe eigenvalue is vanishing (or the potential is flat).",
"Outside the branch cut the force is non-zero and dictated by the non-analytic resolvent function.",
"A natural question for future research is whether this similarity is more than coincidence and could be made precise, for example by generalising the calculation to include multiple branes, or groups of branes, including back-reaction where necessary.",
"More directly, in the field theory on hyperbolic space we could attempt to compute an effective action for Coulomb branch VEVs using perturbative methods and/or supersymmetric localisation.",
"Similar to our result for $S_{\\rm Coul}$ , our numerical result for the screened Wilson line EE, $S_{\\rm screen}$ , was a monotonically decreasing function of $Rv$ .",
"In contrast, a previous attempt to calculate the same EE in ref.",
"[56] claimed to find a maximum.",
"Crucially, ref.",
"[56] overlooked $\\mathcal {S}^{(bdy)}_2$ , the effect being the apparent, but false, maximum.",
"This once again highlights the importance of $\\mathcal {S}^{(bdy)}_2$ .",
"In this case, no known physical principle, such as a monotonicity theorem like the $c$ - or $a$ -theorems, requires the EE to be monotonic in $Rv$ .",
"Monotonicity theorems have been proven for RG flows on certain defects or impurities in CFTs [93], [94], [95], [96], [97].",
"However, explicit examples of RG flows on line defects in $(3+1)$ -dimensional CFTs show that the spherical EE does not have to be monotonic in $R$ [98].",
"Moreover, in our case a bulk RG flow, from $SU(N)$ to $SU(N-1) \\times U(1)$ , actually triggers the RG flow on our Wilson line.",
"Our result raises the question of whether a monotonicity theorem for EE could be proven in such cases.",
"At the very least, our results do not rule out the possibility of such a monotonicity theorem.",
"For the screened Wilson line we also computed the probe's contribution to the one-point function of the field theory's Lagrangian density, which we denoted $\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {screen}$ .",
"At large distances the result had the same form as a point charge in Maxwell theory.",
"This revealed more detailed information about the screening, namely that what appears in the UV as a Wilson line of $SU(N)$ appears in the IR as a point charge in the $U(1)$ sector, while being completely absent, i.e.",
"screened, in the $SU(N-1)$ sector.",
"Our numerical result for the EE of the spherical soliton, $S_{\\rm soliton}$ in eq.",
"(REF ), has a maximum just outside the soliton's radius, $R_0 = \\kappa /v$ , where $\\kappa $ determines the soliton's $U(1)$ charge.",
"Here again $\\mathcal {S}^{(bdy)}_2\\ne 0$ is crucial: without it, the EE appears to diverge as $R \\rightarrow R_0^+$ .",
"Schwarz in refs.",
"[65], [66] proposed that the spherical soliton is an infinitesimally thin shell, i.e.",
"a phase bubble or domain wall.",
"In general we did not find clear evidence for an infinitesimally thin shell.",
"At least, our results were consistent with a shell of finite thickness around $R_0$ .",
"In fact, we computed $\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}$ in this case, and found a critical value $\\kappa _\\mathrm {crit}\\approx 0.901$ such that when $\\kappa < \\kappa _\\mathrm {crit}$ the spherical soliton appears to be a lump localised at the origin, rather than a shell, bubble, or domain wall.",
"A natural question is whether VEVs of other single-trace operators display similar behaviour.",
"However, in the limits $v \\rightarrow \\infty $ and $\\kappa \\rightarrow \\infty $ with $R_0$ fixed, we found that all three of $S_{\\rm soliton}$ , $\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}$ , and the stress-energy tensor diverge at $R_0$ , consistent with an infinitesimally thin shell.",
"This is nicely consistent with the form of the D3-brane solution in these limits.",
"In particular, the D3-brane becomes a cylindrical shell with radius $R_0$ and $k$ units of string charge, stretching from the $AdS_5$ boundary to the Poincaré horizon.",
"Schwarz in refs.",
"[65], [66] also asked whether $S_{\\rm soliton}$ at $R_0$ scales for large $\\kappa $ with the surface area, i.e.",
"like $R_0^2$ , similar to a black hole's Bekenstein-Hawking entropy.",
"Our numerical results suggest this is not the case: we found that $S_{\\rm soliton}$ at $R_0$ scales for large $\\kappa $ as $R_0^{1.3}$ .",
"Nevertheless, the spherical soliton has many features similar to an asymptotically flat extremal black hole, such as a mass and radius proportional to its charge and a spectrum of quasi-normal modes [69].",
"A natural question is thus to what extent this soliton, and other similar QFT objects [99], [100], [101], [102], [103], can model aspects of black hole physics, and in particular whether any could indeed have EE that scales with their surface area.",
"A tantalizing fact is that in the limits $v \\rightarrow \\infty $ and $\\kappa \\rightarrow \\infty $ with $R_0$ fixed, the probe D3-brane becomes a cylindrical shell with radius $R_0$ and $k$ units of string charge, stretching from the $AdS_5$ boundary to the Poincaré horizon.",
"The D3-brane then resembles a supertube [104], a cylindrical D2-brane solution with non-zero entropy whose microstates have been counted explicitly [105], [106], and are simply BPS zero-mode deformations of the supertube's shape.",
"A natural question is whether our D3-brane cylinder has similar microstates, and if so, do they contribute to $S_{\\rm soliton}$ , or any other entropy?",
"Our result for $S_{\\rm Coul}$ at large $R$ has the form of a $(3+1)$ -dimensional CFT with cutoff $1/v$ .",
"This makes sense, since in the IR the cutoff should be the inverse W-boson mass, which is indeed $\\propto 1/v$ .",
"Remarkably, however, we find that the large-$R$ asymptotics of $S_{\\rm screen}$ and $S_{\\rm soliton}$ can be determined from $S_{\\rm Coul}$ as follows.",
"As mentioned in section , the shape of the D3-branes in figures REF c and d suggest that the W-boson mass, and hence the cutoff, become position-dependent.",
"We thus introduce an effective value of $v$ , $v_{\\rm eff}^\\pm (R)\\,=\\,v\\pm \\frac{\\kappa }{R}\\,,$ with the plus sign for the screened Wilson line and the minus sign for the spherical soliton.",
"We found that, for $Rv \\gg 1$ , $\\begin{split}&S_{\\rm screen}(Rv) = S_{\\rm Coul}( Rv_{\\rm eff}^+)+\\mathcal {O}\\left((Rv)^{-2}\\right)\\,,\\\\&S_{\\rm soliton}(Rv) = S_{\\rm Coul}( Rv_{\\rm eff}^-)+k \\,S_{\\Box }+\\mathcal {O}\\left((Rv)^{-2}\\right),\\end{split}$ with $S_{\\Box }$ the EE of a conformal Wilson line in the fundamental representation, eq.",
"(REF ).",
"We thus find that at large $Rv$ both $S_{\\rm screen}$ and $S_{\\rm soliton}$ approach $S_{\\rm Coul}$ .",
"Indeed, in figure REF all three D3-branes look the same at large $Rv$ .",
"However, the behaviour in eq.",
"(REF ) captures some of the $1/R$ corrections, some of which are difficult to explain otherwise.",
"For example, although $S_{\\rm Coul}$ at large $R$ has the form of a $(3+1)$ -dimensional CFT, including an area law term $\\propto (Rv)^2$ and a $\\log (Rv)$ term, the corrections include a term $\\propto Rv$ that does not look like a CFT or an impurity entropy, which would be independent of $R$ .",
"Eq.",
"(REF ) captures the term $\\propto (Rv)$ , and others, in a simple and intuitive way.",
"For $S_{\\rm soliton}$ , we showed that the term $k \\,S_{\\Box }$ in eq.",
"(REF ) comes from the part of the D3-brane that reaches the Poincaré horizon.",
"Indeed, when $v \\rightarrow \\infty $ and $\\kappa \\rightarrow \\infty $ with $R_0$ fixed, that part of the D3-brane becomes the cylindrical shell with charge $k$ and radius $R_0$ .",
"We found also that in these limits $\\langle \\mathcal {O}_{F^2}\\rangle _\\mathrm {soliton}$ and the stress tensor's one-point function at large distances take the form of $k$ charges of the $U(1)$ plus a Wilson line in a direct product of $k$ fundamental representations of $SU(N-1)$ , consistent with eq.",
"(REF ) in these limits.",
"Although we focused on probe D3-branes in $AdS_5 \\times S^5$ , the Karch-Uhlemann method, upgraded to include $\\mathcal {S}^{(bdy)}_2$ , is of course applicable to practically any probe object in any holographic space-time.",
"We thus have an enormous number of possibilities for future research.",
"An obvious starting point is to re-visit previous calculations using the Karch-Uhlemann method where $\\mathcal {S}^{(bdy)}_2$ was neglected [55], [56], [57].",
"More generally, RG flows and other forms of conformal symmetry breaking are typically easier to study using probe branes, whose equations of motion are usually easier to solve than Einstein's equation.",
"As a result, the Karch-Uhlemann method, with $\\mathcal {S}^{(bdy)}_2$ included, could potentially be used to address questions about EE, such as its behaviour with RG flows, defects, non-perturbative objects like baryons and solitons, and so on, and more broadly questions about its relation to monotonicity theorems, the emergence of probe actions from probe sectors of QFT, and much more.",
"We intend to pursue these and many related questions in the future, using this paper as a foundation."
],
[
"Acknowledgements",
"We would like to thank Zohar Komargodski, Christoph Uhlemann and Konstantin Zarembo for useful discussions.",
"A. C. is supported by the Royal Society award RGF/EA/180098.",
"S. P. K. acknowledges support from STFC grant ST/P00055X/1.",
"A. O’B.",
"is a Royal Society University Research Fellow.",
"A. P. is supported by SFI and the Royal Society award RGF/EA/180167.",
"The work of R. R. was supported by the D-ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).",
"J. S. is supported by the Royal Society award RGF/EA/181020."
],
[
"HEE of the Coulomb Branch from Back-Reaction",
"In this appendix we consider the Coulomb branch D3-brane introduced in section REF , and compute its contribution to the EE of a spherical region of radius $R$ , using the RT formula, eq.",
"(REF ).",
"More precisely, we consider the solution of type IIB supergravity describing the back-reaction of the D3-brane and calculate the EE using the RT formula in a probe limit.",
"We then compare to our result computed directly in the probe limit eq.",
"(REF ) (rather than including back-reaction and then taking the probe limit).",
"We find perfect agreement.",
"If we begin with $N$ coincident D3-branes, with low-energy worldvolume theory $SU(N)$ ${\\mathcal {N}}=4$ SYM, then a generic point on the Coulomb branch is described by multiple separate stacks of D3-branes, say with $N_I$ D3-branes in each stack, breaking $SU(N) \\rightarrow S\\left[\\prod _{I} U (N_I) \\right]$ with $\\sum _I N_I =N$ .",
"In this case, if we arrange the six adjoint scalars as a vector $\\vec{\\Psi }$ , then $\\langle \\vec{\\Psi }\\rangle \\propto \\left( \\begin{matrix}\\vec{d}_1 \\, \\mathbb {I}_{N_1} && 0 && \\dots && 0 \\\\0 && \\vec{d}_2 \\, \\mathbb {I}_{N_2} && 0 && \\dots \\\\0 && 0 &&\\ddots && 0\\end{matrix} \\right),$ with constants $\\vec{d}_1$ , $\\vec{d}_2$ , etc.",
"The fully back-reacted solution of type IIB supergravity describing such generic points on the Coulomb branch is known [59].",
"Its metric is $d s^2 = H(\\vec{y})^{-1/2} \\eta _{mn} d x^m d x^n + H(\\vec{y})^{1/2} \\delta _{ij} d y^i d y^j\\,,$ $H(\\vec{y}) \\equiv 1 + 4 \\pi g_s \\alpha ^{\\prime 2} \\sum _I \\frac{N_I}{|\\vec{y} - \\vec{d}_I|^4}\\,,$ where $m,n$ label the four field theory directions, $i,j=1,\\ldots ,6$ and $g_s$ is the string coupling.",
"We place $N-M$ D3-branes at $\\vec{y}=0$ and $M$ at $\\vec{y} = \\vec{y}_0$ .",
"In the decoupling limit $\\alpha ^{\\prime } \\rightarrow 0$ with $\\vec{y}/\\alpha ^{\\prime }$ fixed, the metric becomes $d s^2 &= L^{-2} \\left(\\frac{N-M}{N|\\vec{y}|^4} + \\frac{M}{N|\\vec{y} - \\vec{y}_0|^4}\\right)^{-1/2} \\eta _{mn} d x^m d x^n\\nonumber \\\\ &\\phantom{=}+ L^2 \\left( \\frac{N - M}{N|\\vec{y}|^4} + \\frac{M}{N|\\vec{y} - \\vec{y}_0|^4} \\right)^{1/2} \\delta _{ij} d y^i d y^j\\,,$ where we also used $L^2 = \\alpha ^{\\prime } \\sqrt{4 \\pi g_s N}$ .",
"Let us switch to polar coordinates in the $\\vec{y}$ directions, with radius $|\\vec{y}| = r$ and polar angle $\\chi $ , orienting the axes such that $\\vec{y}_0$ points in the $\\chi =0$ direction.",
"We also define $z = L^2/r$ and $|\\vec{y}_0| = Lv$ .",
"The metric in eq.",
"(REF ) then becomes $d s^2 = \\frac{L^2}{z^2} f(z,\\chi )^{-\\frac{1}{2}} \\left(-d t^2 + d \\rho ^2 + \\rho ^2 d \\Omega _2^2 \\right) + L^2 f(z,\\chi )^{\\frac{1}{2}} \\left[\\frac{ d z^2}{z^2} + d \\chi ^2 + \\sin ^2 \\chi \\, d \\Omega _4^2\\right],$ $f(z,\\chi ) \\equiv 1 - \\frac{M}{N} + \\frac{M}{N} \\left(1 + \\frac{v^2 z^2}{L^2} - \\frac{2 vz}{L} \\cos \\chi \\right)^{-2}\\,.$ The RT surface is extended along $z$ , $\\chi $ , and the $S^2$ and $S^4$ , and thus is given by $\\rho = \\rho (z,\\chi )$ .",
"After integration over the $S^2$ and $S^4$ , the area of this surface is $\\mathcal {A} = \\frac{32 L^8 \\pi ^3}{3} \\int _0^\\pi d\\chi \\int _\\epsilon ^{z_*} d z \\frac{\\rho ^2 \\sin ^4 \\chi }{z^4} \\sqrt{f(z,\\chi )}\\sqrt{(\\partial _\\chi \\rho )^2 + z^2 \\left[f(z,\\chi ) + (\\partial _z \\rho )^2 \\right]}\\,,$ where $\\epsilon $ is a small-$z$ cut-off, and $z_*$ denotes the maximal extension of the surface into the bulk, which in general will depend on $\\chi $ .",
"We will work in the probe limit $\\eta \\equiv M/N \\ll 1$ .",
"For $\\eta = 0$ , the metric (REF ) reduces to that of $AdS_5 \\times S^5$ .",
"The RT surface is given by $\\rho =\\sqrt{ R^2 - z^2}$ [33], [34], whose maximal extension into the bulk is $\\left.z_*\\right|_{\\eta =0}=R$ .",
"The $\\mathcal {O}(\\eta )$ term in a small $\\eta $ expansion of the area functional eq.",
"(REF ) is the sum of three contributions, $ \\mathcal {A}=\\mathcal {A}_1+ \\mathcal {A}_2 + \\mathcal {A}_3$ .",
"Their origins are, respectively: $\\mathcal {A}_1$ comes from the linearised change in the embedding $\\rho (z,\\chi )$ due to non-zero $\\eta $ .",
"The equation of motion for $\\rho (z,\\chi )$ reduces this contribution to a boundary term.",
"$\\mathcal {A}_2$ comes from the change in $z_*$ due to non-zero $\\eta $ .",
"$\\mathcal {A}_3$ comes from the explicit dependence of $f(z,\\chi )$ on $\\eta $ .",
"We will consider each contribution in turn.",
"Contribution 1.",
"If we write $\\rho (z,\\chi ) = \\sqrt{R^2 - z^2} + \\eta \\, \\delta \\rho (z,\\chi ) + \\mathcal {O}(\\eta ^2)$ , then we find the change in the area from $\\delta \\rho (z,\\chi )$ to be $\\begin{aligned}\\mathcal {A}_1 &= - \\frac{32 \\pi ^3 L^8}{3 R} \\int _0^\\pi d \\chi \\int _\\epsilon ^{z_*} dz \\, \\sin ^4 \\chi \\, \\partial _z \\left[ \\left(\\frac{R^2}{z^2} - 1 \\right) \\delta \\rho (z,\\chi ) \\right]\\\\&= - \\frac{32 \\pi ^3 L^8}{3 R} \\int _0^\\pi d \\chi \\sin ^4 \\chi \\, \\left[ \\left(\\frac{R^2}{z^2} - 1 \\right) \\delta \\rho (z,\\chi ) \\right]_{z=\\epsilon }^{z=R}\\,.\\end{aligned}$ The prefactor of $\\delta \\rho (z,\\chi )$ vanishes at $z=R$ .",
"If we require $\\delta \\rho (z,\\chi )$ to be regular, then the contribution to eq.",
"(REF ) from $z=R$ vanishes.",
"On the other hand, the prefactor diverges as $\\epsilon ^{-2}$ near $z=0$ , and so we could get a contribution from small $z$ , depending on the near-boundary behaviour of $\\delta \\rho (z,\\chi )$ .",
"However, we find that the leading-order term in a near-boundary expansion of $\\delta \\rho (z,\\chi )$ is $\\mathcal {O}(z^3)$ .",
"Concretely, $ \\delta \\rho (z,\\chi ) = - z^3 \\frac{v}{L R} \\cos \\chi + \\mathcal {O}(z^4) $ , so the small-$z$ contribution to (REF ) vanishes as well.",
"Thus, $\\mathcal {A}_1 = 0$ .",
"Contribution 2.",
"The fundamental theorem of calculus implies that the change in eq.",
"(REF ) due to a small change in $z_*$ is given by $\\eta \\frac{d z_*}{d \\eta }$ times the $\\eta =0$ integrand, evaluated at $z = z_*|_{\\eta =0} = R$ .",
"The $\\eta =0$ integrand is proportional to $\\sqrt{R^2 - z^2}$ , so $\\mathcal {A}_2 = 0$ .",
"It is perhaps unsurprising that this boundary term vanishes, since it arises purely from our choice of coordinates on the RT surface.",
"In particular, since the area functional is diffeomorphism invariant we could instead choose coordinates in which the only boundary is the physical one at $z=0$ , so we expect that all boundary contributions at $z_*$ must vanish.",
"Contribution 3.",
"Since the first two contributions vanish, the $\\mathcal {O}(\\eta )$ contribution to the entanglement entropy comes entirely from $\\mathcal {A}_3$ .",
"By expanding the integrand in eq.",
"(REF ) to linear order in $\\eta $ and evaluating it on the unperturbed solution $\\rho (z) = \\sqrt{R^2 - z^2}$ , we find $\\mathcal {A}_3 = \\frac{16 \\pi ^3 L^8 M v}{3 N R} \\int _0^\\pi d \\chi \\int _0^{R} dz &\\frac{(2 R^2 - z^2)\\sqrt{R^2 - z^2}}{z^2} \\frac{\\sin ^4 \\chi }{\\left( L^2 + v^2 z^2 - 2 L v z \\cos \\chi \\right)^2}\\\\&\\times \\Bigl \\lbrace 4 L (L^2 + v^2 z^2) \\cos \\chi - v z \\left[4 L^2 + v^2 z^2 + 2 L^2 \\cos (2\\chi ) \\right]\\Bigr \\rbrace \\,.\\nonumber $ The EE is then obtained by performing the integrals over $z$ and $\\chi $ .",
"The result is $S_{\\mathrm {Coul}}= \\frac{\\mathcal {A}_3}{4 G_N} ={\\left\\lbrace \\begin{array}{ll}0\\,, \\quad &R v < 1\\,,\\\\\\dfrac{2 }{3 } \\, N M \\, \\left[3 \\cosh ^{-1} \\left( R v \\right) - \\left(R v + \\dfrac{2 }{R v} \\right)\\sqrt{ \\left(R v \\right)^2 -1 }\\right], \\quad & R v > 1\\,,\\end{array}\\right.",
"}$ where we have chosen units in which $L\\equiv 1$ .",
"In these units, $1/G_N = 2N^2/\\pi ^4$ .",
"When $M=1$ , the result in eq.",
"(REF ) agrees perfectly with the one obtained by employing the Karch-Uhlemann method eq.",
"(REF ), namely eq.",
"(REF )."
],
[
"Modification of $C_4$ for {{formula:1bda9f10-5c2f-47e7-9a41-22e482cc090e}}",
"In this appendix we discuss the gauge choice that we require for the Ramond-Ramond 4-form $C_4$ to be non-singular at the hyperbolic horizon.",
"Consider $C_4$ as given by eq.",
"(REF ).",
"When $n=1$ , the hyperbolic horizon is located at $\\zeta =\\zeta _h=1$ , and the first term in $C_4$ vanishes at the horizon.",
"This behavior is a requirement for regularity at the horizon.",
"If we change $n$ , then the location of the horizon changes to $\\zeta _h \\ne 1$ .",
"This modification introduces a singularity in $C_4$ that must be cancelled.",
"To make the discussion above more precise, it is convenient to make the following coordinate transformation from $(\\zeta ,\\tau )$ to $(x,y)$ , $x = \\sqrt{\\zeta ^2 - \\zeta _h^2}\\cos \\tau \\,, \\qquad y = \\sqrt{\\zeta ^2 - \\zeta _h^2}\\sin \\tau \\, .$ Here we take $\\zeta _h$ to be arbitrary, i.e.",
"not necessarily equal to one.",
"The hyperbolic horizon $\\zeta =\\zeta _h$ now corresponds to $x=0$ and $y=0$ .",
"Using $d\\zeta = \\frac{x \\, dx}{\\sqrt{x^2 + y^2 + \\zeta ^2_h}} + \\frac{y \\, dy}{\\sqrt{x^2 + y^2 + \\zeta ^2_h}}\\,, \\qquad d\\tau = \\frac{x dy}{x^2 + y^2} - \\frac{y dx}{x^2 + y^2} \\,,$ eq.",
"(REF ) becomes $\\begin{split}C_4 =\\;& - x \\frac{(x^2 + y^2 + \\zeta ^2_h)(x^2 + y^2 + \\zeta ^2_h-1)}{x^2 + y^2}\\sinh ^2 u \\sin \\theta \\, du \\wedge dy \\wedge d\\theta \\wedge d\\phi \\\\&+ y \\frac{(x^2 + y^2 + \\zeta ^2_h)(x^2 + y^2 + \\zeta ^2_h-1)}{x^2 + y^2}\\sinh ^2 u \\sin \\theta \\, du \\wedge dx \\wedge d\\theta \\wedge d\\phi \\\\&+ \\frac{\\sinh ^2 u \\sin \\theta (\\sinh u - \\cos \\theta \\cosh u)}{\\cosh u -\\cos \\theta \\sinh u} \\, d x \\wedge d y \\wedge d\\theta \\wedge d\\phi \\\\&- \\frac{\\sinh u \\sin ^2 \\theta }{\\cosh u -\\cos \\theta \\sinh u} \\, dx \\wedge dy \\wedge du \\wedge d\\phi \\,.\\end{split}$ If $\\zeta _h\\ne 1$ , then the first two terms are singular at the horizon $x=0$ and $y=0$ .",
"Specifically, the singular terms are $\\begin{split}\\left.",
"C_4 \\right|_\\mathrm {sing} =\\;& - x \\frac{\\zeta ^2_h(\\zeta ^2_h-1)}{x^2 + y^2}\\sinh ^2 u \\sin \\theta \\, du \\wedge dy \\wedge d\\theta \\wedge d\\phi \\\\&+ y \\frac{ \\zeta ^2_h( \\zeta ^2_h-1)}{x^2 + y^2}\\sinh ^2 u \\sin \\theta \\, du \\wedge dx \\wedge d\\theta \\wedge d\\phi \\,,\\end{split}$ which in terms of $(\\zeta ,\\tau )$ become $\\begin{split}\\left.",
"C_4 \\right|_\\mathrm {sing} = -\\zeta _h^2(\\zeta ^2_h-1)\\sinh ^2 u \\sin \\theta \\, du \\wedge d\\tau \\wedge d\\theta \\wedge d\\phi \\,.\\end{split}$ Crucially, $\\left.",
"C_4 \\right|_\\mathrm {sing}$ is exact everywhere but at the horizon, so we can perform a (singular) gauge transformation to remove the singularity.",
"Starting from the singular $C_4$ in eq.",
"(REF ), we thus perform a gauge transformation that simply subtracts the $\\left.",
"C_4 \\right|_\\mathrm {sing}$ in eq.",
"(REF ).",
"The result is the non-singular $C_4$ in eq.",
"(REF ), which we use throughout our calculations."
],
[
"Details of the HEE Computation",
"In this section we discuss the integral $\\mathcal {S}$ in eq.",
"(REF ) in more detail.",
"It consists of the sum of two pieces.",
"The term with $\\text{sign}(\\partial _\\zeta u)$ is a total derivative and thus straightforward to compute.",
"This appendix is devoted to the other piece, $\\mathcal {S}_1 \\equiv \\frac{4\\pi }{3}T_{D3} \\int d \\zeta d\\tau \\, \\frac{\\left( (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ f^2_1(\\zeta )}\\right)\\sqrt{ \\kappa ^2+\\zeta ^4 \\sinh ^4 u}}{ \\sqrt{1+\\zeta ^2 (\\partial _\\zeta u)^2f_1(\\zeta ) + \\frac{\\zeta ^2 (\\partial _\\tau u)^2}{f_1(\\zeta )} }} \\,,$ which in most of the cases needs to be evaluated numerically.",
"The general strategy is to make a coordinate transformation that maps the embedding from hyperbolic coordinates back to Poincaré coordinates.",
"The embeddings are much simpler in the Poincaré patch, and the problem is more tractable.",
"However, a subtlety arises as we will see shortly.",
"For later convenience, we note the inverse transformation of eqs.",
"(REF ), $\\zeta &= \\frac{\\sqrt{R^4+2 R^2 \\left(-\\rho ^2+t^2+z^2\\right)+\\left(\\rho ^2+t^2+z^2\\right)^2}}{2 R z}\\,, \\\\[1em]\\sinh u &= \\frac{2 \\rho R}{\\sqrt{R^4+2 R^2 \\left(-\\rho ^2+t^2+z^2\\right)+\\left(\\rho ^2+t^2+z^2\\right)^2}}\\,,\\\\[1em]\\cos \\tau &= \\frac{R^2 -\\rho ^2-t^2-z^2}{\\sqrt{R^4+2 R^2 \\left(-\\rho ^2+t^2-z^2\\right)+\\left(\\rho ^2+t^2+z^2\\right)^2}}\\,,$ where we have set $L\\equiv 1$ , and for notational simplicity we drop the subscript $E$ on the Euclidean time in this appendix.",
"Let us begin by considering the integral $\\mathcal {S}_1$ in eq.",
"(REF ).",
"For all the non-conformal embeddings that we study, its integrand diverges at the horizon $\\zeta =1$ like $1/(\\zeta ^2-1)$ .",
"The divergence in the $\\zeta $ -integration can be easily regularised, e.g.",
"by introducing a cut-off.",
"Subsequently performing the integral over the Euclidean time $\\tau $ , one finds that the leading-order piece in $\\zeta ^2-1$ vanishes, and we can safely take the cutoff to zero.",
"Nonetheless, this singular behaviour makes the numerics unstable in the hyperbolic coordinates.",
"For this reason, our strategy will be to subtract and add the singular term.",
"The integrand minus the singular piece will be computed numerically in Poincaré coordinates, while the additional singular term will be treated in hyperbolic coordinates as outlined above.",
"Employing the ansatz eq.",
"(REF ), one can straightforwardly find the singular part of the integral $\\mathcal {S}_1$ in eq.",
"(REF ) for a generic non-conformal embedding.",
"The result is $\\mathcal {S}_1^{(\\mathrm {sing})} = \\frac{4 \\pi }{3} T_{\\text{D3}}\\int d\\tau d\\zeta \\, \\frac{\\sqrt{\\kappa ^2+\\sinh ^4 u^{(0)}_1} \\left(\\left(u^{(1)}_1\\right)^2-\\left(\\partial _\\tau u^{(1)}_1\\right)^2 \\right)}{ \\sqrt{2} \\sqrt{2+\\left(\\partial _\\tau u_1^{(1)}\\right)^2+\\left(u^{(1)}_1\\right)^2}}\\frac{1}{\\zeta ^2-1}+\\mathcal {O}\\left(\\frac{1}{\\sqrt{\\zeta ^2 -1}}\\right)\\,.$ We now illustrate our method of subtraction of this singular part for the Coulomb branch D3-brane.",
"The resulting numerical evaluation of the EE agrees perfectly with the analytical evaluation found in section REF .",
"We will then give some details of the subtraction and numerical evaluation for the screened Wilson line and the spherical soliton."
],
[
"Coulomb branch",
"Recall that the Coulomb branch D3-brane reaches the hyperbolic horizon if $R v > 1$ .",
"In this case, expanding the solution eq.",
"(REF ) near the horizon we find $u_1^{(0)} = \\cosh ^{-1}(Rv)$ and $u_1^{(1)}(\\tau ) = - \\sqrt{\\frac{2}{(Rv)^2-1}} \\cos \\tau $ .",
"The integral in eq.",
"(REF ) becomes $\\mathcal {S}_1^{(\\mathrm {sing})} = \\frac{4 \\pi }{3} T_\\text{D3}\\frac{\\sqrt{(R v)^2-1}}{R v} \\int _0^{2\\pi } d\\tau \\int _{1}^{\\tilde{\\zeta }(\\tau )} d \\zeta \\, \\frac{ \\cos 2 \\tau }{ \\zeta ^2 -1}\\,,$ where $\\tilde{\\zeta }(\\tau ) = \\frac{R v - \\cos \\tau \\sqrt{(R v)^2 -\\sin ^2 \\tau }}{\\sin ^2 \\tau } $ is the upper limit on the integration over $\\zeta $ , arising from the condition that $\\cosh u(\\zeta ,\\tau )$ in eq.",
"(REF ) is $\\ge 1$ .",
"In order to deal with the singularity at $\\zeta =1$ , we introduce a small cut-off such that the lower limit of integration is $\\zeta = 1+ \\epsilon $ .",
"Evaluating eq.",
"(REF ) for non-zero $\\epsilon $ , and then sending $\\epsilon \\rightarrow 0$ , we find $\\mathcal {S}_1^{(\\mathrm {sing})} = 0$ .",
"Now consider changing coordinates in $\\mathcal {S}_1$ from hyperbolic to Poincaré coordinates.",
"The integral picks up the following Jacobian $\\left| \\frac{\\partial \\tau }{\\partial \\rho } \\frac{\\partial \\zeta }{\\partial t} - \\frac{\\partial \\tau }{\\partial t} \\frac{\\partial \\zeta }{\\partial \\rho } \\right|=\\frac{\\sinh u}{R \\, z}\\,.$ We transform the divergent piece eq.",
"(REF ) to Poincaré coordinates and subtract it from the first integrand in eq.",
"(REF ).",
"The result is $\\mathcal {S}_1 = && \\dfrac{4 \\pi }{3} T_\\text{D3}\\int _0^{+\\infty } dt \\, d\\rho \\,\\, 16 R v \\rho \\left\\lbrace -\\frac{\\sqrt{\\frac{(R v)^2-1}{{\\cal F}}} \\left({\\cal F}-4 R^2 v^2 \\left(2 t^2+1\\right)\\right)}{\\left({\\cal F}-4 R^2 v^2\\right)^2} \\right.",
"\\\\ && \\left.",
"+ \\frac{2 R^3 v^3 \\rho \\left(\\frac{{\\cal F} \\left(-2 {\\cal F}+f_-^2+f_+^2\\right)}{4 R^2 v^2}+4 R^2 v^2 {\\cal F}+{\\cal F}^2-4 {\\cal F}f_-+2 (3 {\\cal F}-2 {\\cal F} f_+)+f_-^2+f_+^2\\right)}{{\\cal F}^2 \\left({\\cal F}-4(R v)^2\\right)^2}\\right\\rbrace , \\nonumber $ where we have defined $f_+ &\\equiv (R v+\\rho )^2+t^2+1\\,, & f_- &\\equiv (R v-\\rho )^2+t^2+1\\,, & {\\cal F} &\\equiv f_+ f_- \\,.$ Evaluating the $t$ and $\\rho $ integrals numerically, we obtain perfect agreement with the analytical result $\\mathcal {S}_1 = \\frac{4 \\pi ^2 T_{\\text{D3}}}{3}\\frac{ \\left(R^2 v^2-1\\right)^{3/2}-R v \\cosh ^{-1}(R v)}{R v}$ found in section REF .",
"Concretely, in figure REF we show the numerical result (blue dots) and the analytical answer (red curve).",
"Evidently they agree very well.",
"Figure: Comparison between the numerical integration of eq.",
"() (blue dots) and the analytical expression eq.",
"() (red curve).",
"The numerical values agree perfectly with the analytical result."
],
[
"Screened Wilson line",
"For the screened Wilson line we need to compute the following integral $\\begin{split}\\mathcal {S}_1 &= \\frac{4\\pi }{3} T_{\\text{D3}} \\int d\\tau \\int d \\zeta \\left[\\frac{\\left( (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ (\\zeta ^2-1)^2}\\right)\\left[\\kappa R v \\cosh u + \\sinh u (\\kappa - \\zeta \\sinh u)^2\\right]}{ R v } \\right],\\end{split}$ which is obtained by substituting the solution in eq.",
"(REF ) into eq.",
"(REF ).",
"In this case the integral in eq.",
"(REF ) becomes $\\mathcal {S}_1^{(\\mathrm {sing})} = \\frac{4 \\pi }{3}T_\\text{D3}\\mathbf {C}_{\\mathrm {screen}}(\\kappa ,R v) \\int _0^{2\\pi } d\\tau \\int _{1}^{\\infty } d \\zeta \\,\\frac{ \\cos 2 \\tau }{ \\zeta ^2 -1}\\,,$ where $\\mathbf {C}_{\\mathrm {screen}}(\\kappa ,R v) &\\equiv \\frac{c_0^2 \\sqrt{\\kappa ^2+\\sinh ^4 u_0}}{ \\sqrt{1+ c_0^2}}\\,, & c_0 &\\equiv \\frac{ R v \\, \\kappa }{R v \\kappa \\sinh u_0+\\cosh u_0 (R v-\\cosh u_0)^2}\\,.$ It is straightforward to show that again $\\mathcal {S}_1^{(\\mathrm {sing})}=0$ .",
"The Jacobian for the transformation to Poincaré coordinates takes the form $\\left| \\frac{\\partial \\tau }{\\partial \\rho } \\frac{\\partial \\zeta }{\\partial t} - \\frac{\\partial \\tau }{\\partial t} \\frac{\\partial \\zeta }{\\partial \\rho } \\right|= \\frac{(v z(\\rho )-1)^2 \\left| R v \\, \\kappa \\cosh u +\\sinh u (\\kappa -\\zeta \\sinh u)^2\\right|}{R v \\, \\kappa ^2 \\, z(\\rho )^2}\\,,$ where $z=z(\\rho )$ is the embedding as given by eq.",
"(REF ) (recall that $r=1/z$ for $L\\equiv 1$ ).",
"By transforming the integral in eq.",
"(REF ) to Poincaré coordinates and subtracting the singular part eq.",
"(REF ), we obtain $\\mathcal {S}_1 = \\frac{8 \\pi }{3} T_{\\text{D3}}\\int _0^{+\\infty } dt\\, d\\rho \\, \\bigg [ F_{\\kappa ,R v}(\\rho ,t) - F^{\\mathrm {sing}}_{\\kappa ,R v}(\\rho ,t) \\bigg ]\\,.$ The functions $F_{\\kappa ,R v}$ and $F^{\\mathrm {sing}}_{\\kappa ,R v}$ are $\\begin{split}F_{\\kappa ,R v} (\\rho ,t) \\equiv \\;& 16 R^3 v^3 \\kappa ^3 \\rho ^2 \\left[\\frac{\\rho ^4 \\left(\\sqrt{4 R^2v^2 \\kappa ^2 \\rho ^2+f_{\\mathrm {screen}}^2}-2 R^2 v^2\\right)^2}{f_{\\mathrm {screen}}^2 \\left(4 R^2 v^2 \\rho ^2-f_{\\mathrm {screen}}^2 (\\rho +1)^2\\right)^2}\\right.",
"\\\\&+\\frac{\\rho ^4}{f_{\\mathrm {screen}}^2 (\\rho +1)^2 \\left(4 R^2 v^2 \\rho ^2-f_{\\mathrm {screen}}^2 (\\rho +1)^2\\right)} \\\\ & \\left.+\\frac{R^2 v^2 \\left(f_{\\mathrm {screen}}^2 (\\rho +1)^3-2 \\rho ^2 \\left(\\rho \\sqrt{4 R^2 v^2 \\kappa ^2 \\rho ^2+f_{\\mathrm {screen}}^2}+2R^2 v^2\\right)\\right)^2}{f_{\\mathrm {screen}}^4 (\\rho +1)^2 \\left(4 R^2 v^2 \\rho ^2-f_{\\mathrm {screen}}^2 (\\rho +1)^2\\right)^2}\\right],\\end{split}$ $\\begin{split}F^{\\mathrm {sing}}_{\\kappa ,R v} (\\rho ,t) \\equiv \\;\\mathbf {C}_{\\mathrm {screen}}(\\kappa ,R v) \\frac{4 R^3 v^3 \\left[R^2 v^2 \\left(t^2+1\\right)+\\rho ^2 \\left(\\kappa ^2 (2 \\rho +1)+\\frac{1}{(\\rho +1)^2}\\right)\\right]}{ (\\rho +1)^2 \\left(f_{\\mathrm {screen}}^2-\\frac{4 R^2 v^2 \\rho ^2}{(\\rho +1)^2}\\right)^2 f_{\\mathrm {screen}}} & \\\\\\times \\left(f_{\\mathrm {screen}}^2-\\frac{4 R^2 v^2 \\left(2 R^2 v^2 (\\rho +1)^2 t^2+\\rho ^2\\right)}{(\\rho +1)^2}\\right)& \\,,\\end{split}$ where we defined $f_{\\mathrm {screen}} \\equiv \\sqrt{\\frac{2 R^2 v^2 \\rho ^2 \\left(\\kappa ^2 (\\rho +1)^2 \\left(t^2-1\\right)+t^2+1\\right)}{(\\rho +1)^2}+R^4 v^4 \\left(t^2+1\\right)^2+\\frac{\\rho ^4 \\left(\\kappa ^2 (\\rho +1)^2+1\\right)^2}{(\\rho +1)^4}} \\,.$ The results and plots in section REF were obtained by numerically evaluating the integral in eq.",
"(REF )."
],
[
"Spherical soliton",
"Finally, let us discuss the integral in eq.",
"(REF ) for the spherical soliton case.",
"When $R v < (Rv)_\\mathrm {crit}$ , where $(Rv)_\\mathrm {crit}$ is defined in eq.",
"(REF ), the D3-brane does not reach the horizon $\\zeta =1$ .",
"In this case, performing the integral numerically is straightforward.",
"Let us therefore focus on the more subtle case $R v > (Rv)_\\mathrm {crit}$ .",
"As anticipated in section REF , the D3-brane embedding has two disconnected boundaries located at the hyperbolic horizon $\\zeta =1$ , corresponding to the two loci where the D3-brane intersects the RT surface.",
"In figure REF , we plot the embedding for $\\kappa =1$ and $R v=3$ for two different values of $\\tau = \\pi /2 \\pm 0.2$ (in red and black, respectively).",
"The embedding reaches the horizon $\\zeta =1$ at two distinct points, $u_1$ and $u_2$ , which are independent of $\\tau $ .",
"Their values are given by the solution of eq.",
"(REF ).",
"Note also the qualitatively different behaviour of the two curves in the $(\\zeta , u)$ plane.",
"Finally, we observe that $u_1$ and $u_2$ approach the same point in the limit $R v \\rightarrow (Rv)_\\mathrm {crit}$ .",
"The reason is very intuitive: at the critical value the RT surface is exactly tangent to the D3-brane.",
"We now discuss the evaluation of the first integral in eq.",
"(REF ), which reads $\\begin{split}\\mathcal {S}_1 &= \\frac{4\\pi }{3} T_{\\text{D3}} \\int d\\tau \\int d \\zeta \\left[\\frac{\\left( (\\partial _\\zeta u)^2-\\frac{(\\partial _\\tau u)^2}{ (\\zeta ^2-1)^2}\\right)\\left|R v \\, \\kappa \\cosh u - \\sinh u (\\kappa + \\zeta \\sinh u)^2\\right|}{ R v } \\right].\\end{split}$ Applying the same procedure as above, we first find the singular part of the integrand.",
"Since there are two distinct points at the horizon, there are two singular parts, namely $\\mathcal {S}_i^{(\\mathrm {sing})} &= \\frac{4 \\pi }{3}T_\\text{D3}\\mathbf {C}^{(i)}_{\\mathrm {soliton}}(\\kappa ,R v) \\int _0^{2\\pi } d\\tau \\int _{1}^{\\zeta _{\\text{max}}(\\tau )} d \\zeta \\, \\frac{ \\cos 2 \\tau }{ \\zeta ^2 -1}\\,, & i&=1,2\\,.$ Here $\\zeta _{\\text{max}}(\\tau )$ is the largest value of $\\zeta $ that satisfies the embedding equation, as illustrated on the left side of figure REF .",
"It can be found numerically by solving $1/\\partial _\\zeta u =0$ .",
"The coefficients $\\mathbf {C}^{(i)}_{\\mathrm {soliton}}$ are given by $\\mathbf {C}^{(i)}_{\\mathrm {soliton}}(\\kappa ,R v)&\\equiv \\frac{c_i^2 \\sqrt{\\kappa ^2+\\sinh ^4 u_i}}{ \\sqrt{1+ c_i^2}} \\,, & c_i &\\equiv \\frac{ R v \\, \\kappa }{R v \\, \\kappa \\sinh u_i-\\cosh u_i (R v-\\cosh u_i)^2}\\,.$ Unlike the previous cases, the integrals $\\mathcal {S}_i^{(\\mathrm {sing})}$ are now non-zero.",
"We straightforwardly find $\\mathcal {S}_{\\mathrm {tot}}^{\\mathrm {sing}}\\equiv \\sum _{i=1,2}\\mathcal {S}_i^{(\\mathrm {sing})} =-\\frac{4\\pi }{3}T_{\\text{D3}} \\sum _{i=1,2}\\mathbf {C}^{(i)}_{\\mathrm {soliton}}(\\kappa ,R v) \\int _ {0}^{2\\pi } d\\tau \\,\\tanh ^{-1}\\left[\\zeta _ {\\text{max}}(\\tau )\\right] \\cos 2\\tau \\,.$ Finally, we apply our usual trick: we subtract the singular piece in the form of eq.",
"(REF ) and add it again in the form of eq.",
"(REF ).",
"Transforming the full integral and the subtracted piece to Poincaré coordinates, we find $\\begin{split}\\mathcal {S}_1 = \\frac{8\\pi L^4}{3}T_{D3} & \\Bigg \\lbrace \\int _{0}^{+\\infty } d t \\int _{0}^{\\rho _*(t)} d \\rho \\, \\bigg [ F_{\\kappa ,R v}(\\rho ,t) - F^{\\mathrm {sing},(1)}_{\\kappa ,R v}(\\rho ,t) \\bigg ] + \\\\& + \\int _{0}^{+\\infty } d t \\int _{\\rho _*(t)}^{+\\infty } d \\rho \\, \\bigg [ F_{\\kappa ,R v}(\\rho ,t) - F^{\\mathrm {sing},(2)}_{\\kappa ,R v}(\\rho ,t) \\bigg ] \\Bigg \\rbrace + \\mathcal {S}_{\\mathrm {tot}}^{\\mathrm {sing}}\\,.\\end{split}$ In the above, we defined $\\begin{split}F_{\\kappa ,R v}(\\rho ,t)=\\; &\\frac{4 R v \\kappa ^3 (\\rho +1)^2 \\left(-\\sqrt{4 R^2 v^2 \\kappa ^2 (\\rho +1)^2+f_{\\mathrm {soliton}}^2}+R^2 v^2+\\kappa ^2 (\\rho +1)^2+\\frac{(\\rho +1)^2}{\\rho ^2}\\right)}{f_{\\mathrm {soliton}}^2\\left(\\frac{f_{\\mathrm {soliton}}^2 \\rho ^2}{4 R^2 v^2 (\\rho +1)^2}-1\\right)^2} \\\\& + \\frac{R v \\kappa ^3 \\left(-2 (\\rho +1)^2 \\sqrt{4 R^2 v^2 \\kappa ^2 (\\rho +1)^2+f_{\\mathrm {soliton}}^2}+4 R^2 v^2 (\\rho +1)+\\frac{f_{\\mathrm {soliton}}^2 \\rho ^3}{\\rho +1}\\right)^2}{f_{\\mathrm {soliton}}^4 \\rho ^2\\left(\\frac{f_{\\mathrm {soliton}}^2 \\rho ^2}{4 R^2 v^2 (\\rho +1)^2}-1\\right)^2}\\,,\\end{split}$ $\\begin{split}F^{\\mathrm {sing},(i)}_{\\kappa ,R v}(\\rho ,t) \\equiv \\frac{ 2 R^2 v^2 \\rho ^2 \\left(2 \\kappa (\\rho +1) \\sqrt{\\frac{f_{\\mathrm {soliton}}^2}{4 R^2 v^2 \\kappa ^2 (\\rho +1)^2}+1}-2 R v\\right)^2+4 R^2 v^2 (\\rho +1)^2-f_{\\mathrm {soliton}}^2 \\rho ^2}{4 R v f_{\\mathrm {soliton}} (\\rho +1)^3 \\left(\\frac{f_{\\mathrm {soliton}}^2 \\rho ^2}{4 R^2 v^2 (\\rho +1)^2}-1\\right)^2}& \\\\\\times \\left|2 \\kappa ^2 (\\rho +1)^2-2 R v \\kappa \\sqrt{\\frac{f_{\\mathrm {soliton}}^2}{4 R^2 v^2 \\kappa ^2 (\\rho +1)^2}+1}\\right| \\mathbf {C}^{(i)}_{\\mathrm {soliton}}(\\kappa ,R v) & \\,,\\end{split}$ $f_{\\mathrm {soliton}} \\equiv \\sqrt{\\frac{2 R^2 v^2 (\\rho +1)^2 \\left(\\kappa ^2 \\rho ^2 \\left( t^2-1\\right)+ t^2+1\\right)}{\\rho ^2}+R^4 v^4 \\left( t^2+1\\right)^2+\\frac{(\\rho +1)^4 \\left(\\kappa ^2 \\rho ^2+1\\right)^2}{ \\rho ^4}}\\,.$ Note that the integral over $\\rho $ in eq.",
"(REF ) splits into two parts, $(0,\\rho _*(t))$ and $(\\rho _*(t),+\\infty )$ .",
"This is because we have two boundaries at the horizon with different $\\mathbf {C}^{(i)}_{\\mathrm {soliton}}(\\kappa ,R v)$ .",
"The value $\\rho _*(t)$ corresponds to $\\zeta _{\\text{max}}$ in Poincaré coordinates, and it is given by the unique positive root of the equation $R v \\sqrt{\\frac{f_{\\mathrm {soliton}}^2}{4 R^2 v^2 \\kappa ^2 (\\rho +1)^2}+1}- \\kappa (\\rho +1)^2=0 \\,.$ Before concluding this appendix, let us discuss the behaviour of the integral eq.",
"(REF ) near the critical point $(Rv)_\\mathrm {crit} $ .",
"The integral must be evaluated numerically, and we find that it diverges at $(Rv)_\\mathrm {crit}$ .",
"Numerically, we see that the divergent piece reads $\\mathcal {S}_1^{(div)}= -\\frac{\\sqrt{\\frac{2}{3}} \\pi \\left(\\kappa ^{2/3}+1\\right)^{5/4} \\kappa ^{2/3}}{\\sqrt{R-(Rv)_\\mathrm {crit}}} + \\dots \\, .$ On the right-hand side of figure REF we plot the numerical result of the integral eq.",
"(REF ) (blue dots) for $\\kappa =0.5$ close to $(Rv)_\\mathrm {crit}$ .",
"The red curve is given by eq.",
"(REF ).",
"We see that the numerics agrees very well with the expression for the divergent behaviour.",
"Eq.",
"(REF ) is the divergence we write in eq.",
"(REF ).",
"Notice that the divergence in eq.",
"(REF ) matches exactly the one coming from the boundary term $\\mathcal {S}^{(bdy)}_2$ in eq.",
"(REF ).",
"This demonstrates the cancellation of the divergences at $(Rv)_\\mathrm {crit}$ once all the contributions, i.e.",
"$\\mathcal {S}_1^{(bdy)}$ , $\\mathcal {S}_2^{(bdy)}$ and $\\mathcal {S}$ , are taken into account, as mentioned below eq.",
"(REF )."
],
[
"Stress-Energy Tensor One-Point Function",
"In this appendix we present details of the derivation of the one-point function of the stress tensor for the spherical soliton solution in the infinite $\\kappa $ and $v$ limit whose embedding is eq.",
"(REF ).",
"Our strategy will be to find the leading-order back-reaction of the probe D3-brane on the Euclidean $AdS_5$ metric $ds^2 = \\frac{1}{z^2} \\bigg (d t_E^2 + d\\rho ^2 + \\rho ^2 d\\theta ^2+ \\rho ^2 \\sin ^2 \\theta \\, d\\phi ^2 + dz^2 \\bigg )\\,.$ From the linearised back-reacted metric $g_{\\mu \\nu }=g_{\\mu \\nu }^{(0)}+h_{\\mu \\nu }$ we may then extract the one-point function of the stress tensor by employing the standard AdS/CFT dictionary as outlined in ref. [74].",
"To find the first-order correction we will use the following result of ref.",
"[89] $h_{\\mu \\nu }(z) = \\int d^{d+1}x^{\\prime } \\, \\sqrt{g} \\, G_{\\mu \\nu ; \\mu ^{\\prime } \\nu ^{\\prime }}(x, x^{\\prime }) T^{\\mu ^{\\prime } \\nu ^{\\prime }}(x^{\\prime })\\, .$ Here, $G_{\\mu \\nu ; \\mu ^{\\prime } \\nu ^{\\prime }}$ is the graviton propagator.",
"It takes the form $G_{\\mu \\nu ; \\mu ^{\\prime } \\nu ^{\\prime }} = \\left( \\partial _\\mu \\partial _{\\mu ^{\\prime }} u \\, \\partial _\\nu \\partial _{\\nu ^{\\prime }}u + \\partial _\\mu \\partial _{\\nu ^{\\prime }} u \\,\\partial _\\nu \\partial _{\\mu ^{\\prime }}u\\right) G(u) + g_{\\mu \\nu }g_{\\mu ^{\\prime }\\nu ^{\\prime }} H(u)\\,,$ where the variable $u$ (not to be confused with the hyperbolic coordinate) is defined as $u &\\equiv &\\frac{(x-x^{\\prime })^2}{2 z z^{\\prime }} \\\\& =&\\frac{(t_E-t_E^{\\prime })^2 + (z-z^{\\prime })^2+\\rho ^2+\\rho ^{\\prime 2}-2\\rho \\rho ^{\\prime } \\left[ \\cos (\\theta -\\theta ^{\\prime })+\\sin \\theta \\sin \\theta ^{\\prime } \\left( \\cos (\\phi -\\phi ^{\\prime })-1\\right) \\right]}{2 z z^{\\prime }}\\,.",
"\\nonumber $ The partial derivatives $\\partial _{\\mu }$ and $\\partial _{\\mu ^{\\prime }}$ are taken with respect to $x$ and $x^{\\prime }$ , respectively.",
"For $d=4$ , i.e.",
"the present case of interest, the functions $G(u)$ and $H(u)$ in eq.",
"(REF ) are $\\begin{aligned}G(u) &= -\\frac{1}{8 \\pi ^2} \\left\\lbrace \\frac{(1+u)\\left(2(1+u)^2-3\\right)}{\\left(u(2 +u)\\right)^\\frac{3}{2}} -2\\right\\rbrace ,\\\\H(u) &= \\frac{1}{12 \\pi ^2} \\left\\lbrace \\frac{(1+u)\\left(6(1+u)^4-9 (1+u)^2+2\\right)}{\\left(u(2 +u)\\right)^\\frac{3}{2}} -6(1+u)^2\\right\\rbrace .\\end{aligned}$ From the DBI part of the action eq.",
"(REF ), we find the on-shell D3-brane stress tensor in the Euclidean space-time.",
"In the $\\kappa \\rightarrow + \\infty $ limit it reads $\\sqrt{g}\\, T^{\\mu \\nu } = T_{D3}\\left(\\begin{array}{ccccc}\\kappa \\sin \\theta & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & \\kappa \\sin \\theta \\end{array} \\right) \\delta (\\rho - R_0)\\,,$ where $R_0 \\equiv \\kappa /v$ .",
"Since we are interested in the expansion for small $z$ , it is convenient to re-express the integral in eq.",
"(REF ) in terms of the variable $v \\equiv 1/u$ (not to be confused with the adjoint scalar VEV).",
"Moreover, we change integration variables from $t_E^{\\prime }$ to $v$ , which produces the following Jacobian $\\left|\\frac{\\partial t_E^{\\prime }}{ \\partial v}\\right|&=\\frac{\\sqrt{z z^{\\prime }}}{\\sqrt{2}v^{3/2}}\\frac{1}{\\sqrt{1-v/v_m}}\\,, & v_m &\\equiv \\frac{2 z z^{\\prime }}{(x -x^{\\prime })^2 + (z-z^{\\prime })^2}\\,,$ where $x$ collectively denotes the three spatial field theory directions.",
"Let us illustrate the computation by considering the simplest piece, which is the one that contains $H(u)$ .",
"Explicitly, $I^H_{\\mu \\nu } = 2 g_{\\mu \\nu }\\int d^4 x^{\\prime } \\int _0^{v_m} dv \\, \\frac{\\sqrt{z z^{\\prime }}}{\\sqrt{2}v^{3/2}}\\frac{ H(1/v) T}{\\sqrt{1-v/v_m}}\\,, \\qquad T \\equiv \\sqrt{g} T^{\\mu \\nu }g_{\\mu \\nu } = \\frac{2 T_{D3}\\kappa }{z^{\\prime 2}} \\delta (\\rho ^{\\prime }-R_0)\\,.$ We now change variables once more to $\\varpi = v/v_m $ , and expand the integrand to 5th order in $v_m$ .",
"This corresponds to an expansion in small $z$ .",
"Performing the $\\varpi $ -integral gives $ I^H_{\\mu \\nu } = \\kappa T_{D3}\\sqrt{z} \\int _0^\\pi d\\theta ^{\\prime }\\sin \\theta ^{\\prime } \\int _0^{2\\pi } d\\phi ^{\\prime }\\int _0^\\infty dz^{\\prime } \\, \\frac{\\tilde{v}_m^{3/2}}{z^{\\prime 3/2}} \\frac{ ( 35 \\tilde{v}^3_m-15 \\tilde{v}_m^2+6 \\tilde{v}_m-4)}{96 \\sqrt{2} \\pi }\\,,$ where now $\\begin{split}\\tilde{v}_m & = \\frac{2 z z^{\\prime }}{\\rho ^2+ R_0^2 -2 \\rho R_0 [\\cos (\\theta -\\theta ^{\\prime })+ \\sin \\theta \\sin \\theta ^{\\prime } (\\cos (\\phi -\\phi ^{\\prime })-1)] + (z-z^{\\prime })^2} \\\\& =\\frac{2 z z^{\\prime }}{\\rho ^2+ R_0^2 -2 \\rho R_0 \\cos \\theta ^{\\prime } + (z-z^{\\prime })^2}\\,.\\end{split}$ We used spherical symmetry to set $\\theta = 0$ in the second step.",
"Evaluating the integrals in eq.",
"(REF ), we find $I^H_{\\mu \\nu } = -\\kappa T_{D3} g_{\\mu \\nu }\\left[\\frac{ z^2 \\log \\left(\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right)}{12 \\rho R_0}+\\mathcal {O}\\left(z^6\\right)\\right].$ The contributions from the piece containing $G(u)$ in eq.",
"(REF ) can be found in a similar way.",
"The derivation is straightforward but tedious.We found the following identity useful for evaluating those terms $\\partial _\\mu \\partial _{\\nu ^{\\prime }} u = - \\frac{1}{z z^{\\prime }} \\left[ \\delta _{\\mu \\nu ^{\\prime }}+ \\frac{1}{z^{\\prime }} (x-x^{\\prime })_\\mu \\delta _{\\nu ^{\\prime } z }+ \\frac{1}{z}(x^{\\prime } - x)_{\\nu ^{\\prime }} \\delta _{\\mu z}- u \\delta _{\\mu z}\\delta _{\\nu ^{\\prime } z} \\right] .$ We will skip it and only present the final result for the correction $h_{\\mu \\nu }$ : $\\begin{aligned}h_{t_E t_E} &= -\\frac{\\kappa T_{D3} \\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{12 \\rho R_0}+ \\frac{\\kappa T_{D3} z^2}{\\left(R_0^2-\\rho ^2\\right)^2} + \\mathcal {O}(z^4)\\,,\\\\h_{\\theta \\theta } & = - \\kappa T_{D3} \\frac{ \\rho \\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{12 R_0} + \\kappa T_{D3} \\frac{ \\frac{4 \\rho R_0 \\left(\\rho ^2+R_0^2\\right)}{\\left(R_0^2-\\rho ^2\\right)^2}+\\log \\left[\\frac{(R_0-\\rho )^2}{(\\rho +R_0)^2}\\right]}{16 \\rho R_0}z^2+ \\mathcal {O}(z^4) \\,,\\\\h_{\\phi \\phi } & = h_{\\theta \\theta } \\sin ^2 \\theta \\,,\\\\h_{\\rho \\rho } & = -\\kappa T_{D3} \\frac{\\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{12 \\rho R_0} +\\kappa T_{D3} \\frac{ \\frac{\\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{R_0}-\\frac{4 \\rho \\left(R_0^2-3 \\rho ^2\\right)}{\\left(\\text{R0}^2-\\rho ^2\\right)^2}}{8 \\rho ^3}z^2 + \\mathcal {O}(z^4) \\,,\\\\h_{zz} & = \\kappa T_{D3} \\frac{ \\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{6 \\rho R_0}-\\kappa T_{D3} \\frac{8 }{3 \\left(R_0^2-\\rho ^2\\right)^2}z^2 + \\mathcal {O}(z^4) \\,,\\\\h_{\\rho z} & = \\kappa T_{D3} \\left(\\frac{2}{\\rho \\left(R_0^2-\\rho ^2\\right)}-\\frac{\\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{8 \\rho ^2 R_0}\\right)z + \\mathcal {O}(z^3)\\,.\\end{aligned}$ In order to straightforwardly obtain the stress tensor from results presented in ref.",
"[74], we would like to switch to Fefferman-Graham gauge $ds^2 = \\frac{1}{z^2} \\Bigg [ dz^2 + \\bigg (\\delta _{mn} + h_{mn} (z ,x)\\bigg ) dx^m dx^n \\Bigg ]\\,,$ where we assumed that the metric is time-independent.",
"To this end we perform an infinitesimal diffeomorphism $h_{\\mu \\nu } \\rightarrow h_{\\mu \\nu } + \\nabla _\\mu \\xi _\\nu + \\nabla _\\nu \\xi _\\mu $ to remove the off-diagonal terms $h_{\\rho z}=h_{z\\rho }$ and $h_{zz}$ .",
"Any $\\xi _\\mu $ with components $\\begin{aligned}\\xi _\\rho &= a_\\rho ^{(2)}(\\rho ) \\frac{z^2}{\\rho ^3} + \\mathcal {O}(z^3)\\,, \\\\\\xi _z &= a_z^{(1)}(\\rho ) \\frac{z}{\\rho ^2} + a_z^{(3)}(\\rho ) \\frac{z^3}{\\rho ^4} + \\mathcal {O}(z^4)\\,,\\end{aligned}$ and $\\begin{aligned}a_z^{(1)}(\\rho ) &=- \\kappa T_{D3} \\frac{\\rho \\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{24 R_0}\\,,\\\\a_\\rho ^{(2)}(\\rho ) &= \\frac{1}{48} \\kappa T_{D3} \\rho \\left(\\frac{4 \\rho }{\\rho ^2-R_0^2}+\\frac{\\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{R_0}\\right)\\,, \\\\a_z^{(3)}(\\rho ) &=\\frac{1}{3} \\kappa T_{D3} \\frac{\\rho ^4}{\\left(R_0^2-\\rho ^2\\right)^2}\\,,\\end{aligned}$ will do the job.",
"This transformation puts the metric in eq.",
"(REF ) in the form $\\begin{aligned}h_{t_E t_E} &= \\frac{\\kappa T_{D3} }{3 \\left(R_0^2-\\rho ^2\\right)^2} z^2 + \\mathcal {O}(z^4) \\,, \\\\h_{\\theta \\theta } &= \\kappa T_{D3} \\left(\\frac{ \\left(R_0^2-3 \\rho ^2\\right)}{12 \\left(R_0^2-\\rho ^2\\right)^2}-\\frac{\\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{48 \\rho R_0}\\right) z^2 + \\mathcal {O}(z^4)\\,, \\\\h_{\\phi \\phi } &= h_{\\theta \\theta }\\sin ^2 \\theta \\,, \\\\h_{\\rho \\rho } &= \\kappa T_{D3} \\left(\\frac{\\log \\left[\\frac{(\\rho +R_0)^2}{(R_0-\\rho )^2}\\right]}{24\\rho ^3 R_0}-\\frac{1}{6 \\rho ^2 \\left(R_0^2-\\rho ^2\\right)}\\right) z^2 + \\mathcal {O}(z^4)\\,.\\end{aligned}$ The field theory stress-energy tensor is then given by [74] $\\langle T_{mn}\\rangle = 2 \\, h_{mn}^{(4)}\\,,$ where we have written the metric as $h_{mn} (z, x)= h^{(0)}_{mn}(x)+ h^{(2)}_{mn}(x)z^2 + h^{(4)}_{mn}(x) z^4 + \\mathcal {O}(z^6)\\,.$ Substituting eq.",
"(REF ), we thus find the result quoted in the main text in eq.",
"(REF )."
]
] | 2012.05188 |
[
[
"Magnetic Field Transport in Propagating Thermonuclear Burn"
],
[
"Abstract High energy gain in inertial fusion schemes requires the propagation of a thermonuclear burn wave from hot to cold fuel.",
"We consider the problem of burn propagation when a magnetic field is orthogonal to the burn wave.",
"Using an extended-MHD model with a magnetized $\\alpha$ energy transport equation we find that the magnetic field can reduce the rate of burn propagation by suppressing electron thermal conduction and $\\alpha$ particle flux.",
"Magnetic field transport during burn propagation is subject to competing effects: field can be advected from cold to hot regions by ablation of cold fuel, while the Nernst and $\\alpha$ particle flux effects transport field from hot to cold fuel.",
"These effects, combined with the temperature increase due to burn, can cause the electron Hall parameter to grow rapidly at the burn front.",
"This results in the formation of a self-insulating layer between hot and cold fuel that reduces electron thermal conductivity and $\\alpha$ transport, increases the temperature gradient and reduces the rate of burn propagation."
],
[
"Introduction",
"The ultimate goal of inertial fusion energy is to produce high energy gain, in which the energy liberated in thermonuclear reactions is orders of magnitude larger than the energy of the driver used to compress the fusion fuel.",
"High energy gain can be achieved by using the driver energy to ignite a small fraction of the fuel, the hotspot, which then drives a propagating thermonuclear burn wave into a surrounding layer of dense, cold fuel.",
"[1], [2] The burn wave consists of significant energy transport due to $\\alpha $ particles, electron thermal conduction and radiative processes.",
"[3] Magnetic fields are a central feature of a broad class of inertial fusion schemes referred to as Magneto-Inertial Fusion (MIF)[4].",
"In MIF, a magnetic field is applied to the fuel during the compression phase.",
"This magnetic field provides magnetothermoinsulation[5] during the compression phase by reducing electron thermal conduction losses from hot plasma.",
"This reduces the implosion velocity required to reach the ignition temperature.",
"The magnetic field also confines $\\alpha $ particles within the fuel during thermonuclear burn but with a Larmor radius that is significantly larger than that of the thermal electrons.",
"A number of MIF schemes under current investigation, such as MagLIF at Sandia National Laboratories[6], [7], aim to achieve volumetric thermonuclear burn.",
"Volumetric burn schemes are designed to heat all of the fuel to fusion temperatures during the compression phase.",
"If the fuel ignites then a net energy gain can be achieved without the need for thermonuclear burn propagation (even though the theoretical maximum energy gain is smaller for volumetric burn compared with propagating burn).",
"For volumetric burn schemes it is desirable to have a magnetic field which maximizes the magnetothermoinsulation, as long as the magnetic pressure remains small relative to thermal pressure.",
"In the case of MIF schemes involving propagating burn, such as high gain MagLIF[2] and magnetized indirect-drive Inertial Confinement Fusion(ICF)[8], [9], the magnetothermoinsulation effect is required during the compression phase to aid the formation of an igniting hotspot.",
"However, the role of the magnetic field during the propagating burn phase is less clear.",
"The suppression of thermal conduction and $\\alpha $ particle transport reduces the rate of burn propagation into the cold fuel.",
"[2], [10], [11] This can limit the energy gain achieved before the target disassembles.",
"While some magnetic confinement of $\\alpha $ particles during burn propagation could be desirable, to allow for lower areal densities, it is clear that a magnetic field which is too large could be detrimental to the achievable yield.",
"This question of how a magnetic field affects thermonuclear burn propagation motivates the present work.",
"In addition to propagating burn MIF schemes, this question may also be relevant to conventional ICF since simulations have shown that the Biermann battery mechanism[12] can generate magnetic fields at the surface of the hotspot in perturbed ICF implosions.",
"[13] In this work we study magnetic field dynamics in a propagating burn wave and the effect of the magnetic field on the evolution of burn propagation.",
"The principal novelty of the work is our detailed treatment of the magnetic field transport.",
"We utilize an MHD model (described in detail in section ) in which the induction equation includes advection, resistive diffusion, electron temperature gradients (the Nernst effect) and the flux of $\\alpha $ particles.",
"This last term is typically not included in simulations of MIF schemes.",
"We find that the various terms in the induction equation result in competing dynamics of the magnetic field at a propagating burn front: field is advected from cold to hot regions as the burn wave causes ablation of cold fuel whilst temperature gradients and $\\alpha $ particle fluxes transport field in the opposite direction.",
"These dynamics have a significant effect on the shape and rate of propagation of the burn wave.",
"The magnetic field profile at the onset of burn in MIF is determined by the implosion phase and there has been a number of studies showing the complex field transport occurring in that phase.",
"[14], [15], [16], [17] For simplicity, we do not study the implosion phase in the current work.",
"Instead we take as our starting point a region of hot fuel that is undergoing significant $\\alpha $ heating adjacent to a cold fuel layer.",
"In order to elucidate the field dynamics during burn, we focus on three simple cases for the initial magnetic field profile: a spatially uniform field, a spatially uniform electron Hall parameter and a magnetic field that is non-zero only in a small region at the burn front.",
"Results for each of these cases are discussed in section .",
"This is not an exhaustive search of magnetic field strengths and profiles but the cases considered show that there is a complex interplay between the magnetic field dynamics and the evolution of the propagating burn wave.",
"The suppression of thermal conduction and $\\alpha $ transport by the magnetic field depends on both the magnitude of the magnetic field and the collisionality of the plasma, where the relevant metric is the Hall parameter.",
"As thermonuclear burn heats the plasma, collisionality decreases rapidly.",
"Combining this with the magnetic field transport effects leads to enhanced suppression of energy transport from hot to cold fuel.",
"In our case studies we identify scenarios in which a self-insulating magnetized layer can form between hot and cold fuel due to feedback between decreasing collisionality in the burn wave and the magnetic field transport.",
"This self-insulating layer significantly reduces thermal conductivity and $\\alpha $ transport across it and leads to the development of large temperature gradients.",
"In section we formulate an equation for the growth rate of the Hall parameter in order to better understand the conditions leading to formation of the self-insulating layer.",
"This equation illustrates, inter alia, the dependence of the growth rate on the Hall parameter value, $\\alpha $ heating, and the temperature and $\\alpha $ energy density profiles.",
"Finally, we present some conclusions in section .",
"While recent theoretical[15] and experimental[7] work has demonstrated the importance of magnetic field transport during the implosion phase, our results show that field transport effects during the burn phase are also of interest."
],
[
"Model Outline",
"Our model is a one-dimensional, planar, classical DT plasma consisting of semi-infinite regions of cold and hot fuel, separated by a smooth transition region with an orthogonal magnetic field.",
"This idealized geometry is not intended to model a specific MIF scheme, but instead allows us to study how thermonuclear burn propagates from hot to cold fuel.",
"The system is governed by an equation of continuity, the isobaric condition, an induction equation and energy equations for the fuel and the $\\alpha $ particles as follows: $\\frac{\\partial n}{\\partial \\hat{t}}+\\frac{\\partial }{\\partial \\hat{x}}\\left(n\\hat{u}\\right)&=&0,\\\\\\frac{\\partial }{\\partial \\hat{x}}\\left(2nT+\\frac{B^{2}}{2\\mu _{0}}\\right)&=&0,\\\\\\frac{\\partial B}{\\partial \\hat{t}}+\\underbrace{\\frac{\\partial }{\\partial \\hat{x}}\\left(\\hat{u}B\\right)}_\\text{advect.",
"}&=&\\frac{\\partial }{\\partial \\hat{x}}\\left[\\underbrace{\\hat{\\alpha }\\frac{\\partial B}{\\partial \\hat{x}}}_\\text{res.",
"diff.",
"}+\\left(\\underbrace{\\hat{\\beta }\\frac{\\partial \\ln T}{\\partial \\hat{x}}}_\\text{Nernst}+\\underbrace{\\hat{\\gamma }\\frac{\\partial \\ln \\mathcal {E}_{\\alpha }}{\\partial \\hat{x}}}_\\text{$\\alpha $-e coll.",
"}\\right)B\\right],\\\\3n\\frac{D T}{D \\hat{t}}+2nT\\frac{\\partial \\hat{u}}{\\partial \\hat{x}}&=& \\frac{\\partial }{\\partial \\hat{x}}\\left[\\underbrace{\\hat{\\kappa }\\frac{\\partial T}{\\partial \\hat{x}}}_\\text{cond.",
"}+\\underbrace{\\hat{\\beta }\\frac{B}{\\mu _{0}}\\frac{\\partial B}{\\partial \\hat{x}}}_\\text{Ettings.",
"}\\right]+\\underbrace{\\hat{q}_{\\alpha }}_\\text{$\\alpha $ heat.}-\\underbrace{\\hat{P}}_\\text{brem.",
"},\\\\\\frac{D \\mathcal {E}_{\\alpha }}{D \\hat{t}}+\\frac{5}{3}\\mathcal {E}_{\\alpha }\\frac{\\partial \\hat{u}}{\\partial \\hat{x}} &=& \\underbrace{\\frac{\\partial }{\\partial \\hat{x}}\\left(\\hat{\\delta }_\\mathcal {E}\\frac{\\partial \\mathcal {E}_{\\alpha }}{\\partial \\hat{x}}\\right)}_\\text{$\\alpha $ energy diff.",
"}-\\hat{q}_{\\alpha }+\\underbrace{\\hat{Q}}_\\text{reac.",
"},$ where $\\hat{t}$ , $\\hat{x}$ and $\\hat{u}$ represent dimensionless time, position and fluid velocity of the fuel and $\\frac{D }{D \\hat{t}} = \\frac{\\partial }{\\partial \\hat{t}}+\\hat{u}\\cdot \\frac{\\partial }{\\partial \\hat{x}},\\nonumber $ is the convective derivative.",
"The normalizing constants are $L_{T}$ , representing the width of the transition region between hot and cold fuel and $\\mathcal {T}_{h}$ , the stopping time for a $3.45\\,MeV$ $\\alpha $ particle in the initial hot fuel.",
"Other variables include fuel density, $n$ , magnetic field, $B$ , fuel temperature, $T$ , and $\\mathcal {E}_{\\alpha }$ is the energy density of $\\alpha $ particles.",
"SI units are used for quantities that have not been made dimensionless.",
"The induction, (), and fuel energy, (), equations contain classical magnetized transport coefficients for resistivity, thermoelectricity and thermal conductivity[18] $\\hat{\\alpha } &=& \\frac{\\mathcal {T}_{H}}{L_{T}^{2}}\\frac{m_{e}}{\\mu _{0}e^{2} n\\tau _{ei}}\\alpha _{\\bot }^{c},\\qquad \\hat{\\beta } = \\frac{\\mathcal {T}_{H}}{L_{T}^{2}}\\frac{\\tau _{ei}T}{m_{e}}\\frac{\\beta _{\\wedge }}{\\chi _{e}},\\nonumber \\\\\\hat{\\kappa } &=& \\frac{\\mathcal {T}_{H}}{L_{T}^{2}}\\frac{n T \\tau _{ei}}{m_{e}}\\left(\\kappa _{\\bot e}^{c}+ \\sqrt{\\frac{2m_{e}}{m_{i}}}\\kappa _{\\bot i}^{c}\\right),\\nonumber $ The coefficients $\\alpha _{\\bot }^{c}$ , $\\beta _{\\wedge }$ , $\\kappa _{\\bot e}^{c}$ and $\\kappa _{\\bot i}^{c}$ are functions of electron Hall parameter, $\\chi _{e} = eB\\tau _{ei}/m_{e}$ , where $\\tau _{ei} = 6\\varepsilon _{0}^{2}\\sqrt{2\\pi ^{3}m_{e}}T^{\\frac{3}{2}}/\\left(e^{4}\\ln \\Lambda _{ei} n\\right)$ .",
"We use the tabulated values of Epperlein and Haines[19] for these coefficients.",
"The chosen geometry of our problem means that the Hall term[20] and Biermann battery term[12] do not need to be considered in the induction equation.",
"The final term on the rhs of () represents the collisionally-induced current arising from the interaction of the $\\alpha $ particle flux with the thermal electrons.",
"Similar fast ions effects have previously been studied for collisionless plasmas in current drive in tokamaks[21], cosmic rays in astrophysical plasmas[22] and relativistic particles in laser-plasma interactions.",
"[23] Here we follow the procedure of Appelbe et al[24] to include this effect in a collisional, magnetized plasma.",
"An analogy may be drawn between the Nernst and the $\\alpha $ -e collisional terms in ().",
"The Nernst effect is a result of the effect of a magnetic field on the thermal force.",
"[18] Electrons being driven by a temperature gradient and experiencing friction due to the background ions are deflected by a magnetic field that is orthogonal to the temperature gradient, generating an electron current orthogonal to both.",
"In the case of the $\\alpha $ -e collisional term the electron current is instead driven by a flux of $\\alpha $ particles.",
"Both the electron temperature gradient and the $\\alpha $ flux are themselves dependent on the magnetic field.",
"The interplay between temperature gradient, magnetic field and driven electron current has been well studied[15] and is often included in simulations.",
"However, the equivalent interplay between the magnetic field, the $\\alpha $ flux and the driven electron current has not received such attention.",
"In our geometry the $\\alpha $ -e collisional effect can be expressed as $\\hat{\\gamma }&=&\\mathcal {E}_{\\alpha }\\left(\\frac{\\partial \\mathcal {E}_{\\alpha }}{\\partial \\hat{x}}\\right)^{-1}\\frac{Z_{\\alpha }}{n}\\left(\\gamma _{\\bot }F_{\\bot }+\\gamma _{\\wedge }F_{\\wedge }\\right),\\\\\\gamma _{\\bot } &=& -j_{\\bot }^{\\nu }\\left(1+\\frac{\\alpha _{\\wedge }^{c}}{\\chi _{e}}\\right)+j_{\\wedge }^{\\nu }\\frac{\\alpha _{\\bot }^{c}}{\\chi _{e}},\\\\\\gamma _{\\wedge } &=& j_{\\wedge }^{\\nu }\\left(1+\\frac{\\alpha _{\\wedge }^{c}}{\\chi _{e}}\\right)+j_{\\bot }^{\\nu }\\frac{\\alpha _{\\bot }^{c}}{\\chi _{e}},$ where $F_{\\bot }$ , $F_{\\wedge }$ are components of $\\alpha $ particle flux in the directions of burn propagation and orthogonal to burn propagation and $B$ field, respectively, and $j_{\\bot }^{\\nu }$ , $j_{\\wedge }^{\\nu }$ are functions of $\\chi _{e}$ .",
"[24] The $\\alpha $ particle fluxes can be estimated from the gradients of $\\mathcal {E}_{\\alpha }$ and $\\alpha $ particle diffusion coefficients as follows[25] $F_{\\bot ,\\wedge } &=& -\\frac{\\hat{\\delta }_{p\\bot ,\\wedge }}{\\langle E_{\\alpha }\\rangle }\\frac{\\partial \\mathcal {E}_{\\alpha }}{\\partial \\hat{x}},\\\\\\hat{\\delta }_{p\\bot } &=& \\frac{\\mathcal {T}_{h}}{L_{T}^{2}}\\frac{E_{\\alpha 0}\\tau _{\\alpha }}{3m_{\\alpha }\\left(1+\\chi _{\\alpha }^{2}\\right)},\\quad \\hat{\\delta }_{p\\wedge } = \\chi _{\\alpha } \\hat{\\delta }_{p\\bot }\\nonumber $ where $\\langle E_{\\alpha }\\rangle \\approx E_{\\alpha 0}/2$ is the mean $\\alpha $ particle energy.",
"The $\\alpha $ energy transport equation, (), represents a single group diffusion approximation.",
"[25] It contains terms for magnetized diffusion, slowing on electrons and a source due to DT reactions [26] $\\hat{\\delta }_{\\mathcal {E}} &=& \\frac{\\mathcal {T}_{h}}{L_{T}^{2}}\\frac{E_{\\alpha 0}\\tau _{\\alpha }}{m_{\\alpha }\\left(9+\\chi _{\\alpha }^{2}\\right)},\\quad \\hat{\\nu } = \\frac{\\mathcal {T}_{h}}{\\tau _{\\alpha }},\\quad \\hat{Q} = \\mathcal {T}_{h}\\frac{n^{2}}{4}\\langle \\sigma v\\rangle _{DT}E_{\\alpha 0},\\nonumber \\\\\\chi _{\\alpha } &=& \\frac{Z_{\\alpha }eB}{m_{\\alpha }}\\tau _{\\alpha },\\quad \\tau _{\\alpha } = 3\\sqrt{\\frac{\\pi ^{3}}{2m_{e}}}\\frac{m_{\\alpha }\\varepsilon _{0}^{2}}{e^{4}\\ln \\Lambda _{\\alpha e}}\\frac{T^{\\frac{3}{2}}}{n},\\nonumber $ where $\\chi _{\\alpha }$ is the $\\alpha $ Hall parameter, $\\tau _{\\alpha }$ is slowing time on electrons and $E_{\\alpha 0}=3.45\\,MeV$ .",
"Electron and $\\alpha $ Hall parameters are related by $\\chi _{\\alpha }/\\chi _{e}=Z_{\\alpha }\\ln \\Lambda _{ei}/\\left(4\\ln \\Lambda _{\\alpha e}\\right)$ .",
"Finally, $\\hat{q}_{\\alpha }=2\\hat{\\nu }\\mathcal {E}_{\\alpha }$ and bremsstrahlung losses[27] are $\\hat{P} = 1.69\\times 10^{-38}n^{2}\\sqrt{\\frac{T}{e}}\\mathcal {T}_{h}\\,J\\,m^{-3}$ .",
"Returning to the induction equation, (), we note that it has been written such that $\\hat{\\alpha }$ , $\\hat{\\beta }$ and $\\hat{\\gamma }$ can all be treated as diffusion coefficients.",
"In the burning plasma regime the magnetic Lewis number,[14] defined as $Lm = \\left.\\hat{\\kappa }/\\left(3n\\hat{\\alpha }\\right)\\right|_{\\chi _{e}=0},$ is large, meaning that thermal diffusivity is much larger than magnetic diffusivity and so resistive diffusion plays only a minor role in $B$ field transport.",
"Using (REF ) for $\\alpha $ particle fluxes we can write (REF ) more compactly as $\\hat{\\gamma } = \\frac{\\mathcal {T}_{h}}{L_{T}^{2}}\\frac{Z_{\\alpha }\\mathcal {E}_{\\alpha }E_{\\alpha 0}\\tau _{\\alpha }}{3m_{\\alpha }n\\langle E_{\\alpha }\\rangle }\\underbrace{\\left[\\frac{-\\gamma _{\\bot }-\\chi _{\\alpha }\\gamma _{\\wedge }}{1+\\chi _{\\alpha }^{2}}\\right]}_\\text{$\\Gamma _{\\chi }$},\\nonumber $ where the dependency on magnetization is now fully contained in the square brackets which we now denote as $\\Gamma _{\\chi }$ .",
"We note that even though () and () contain terms $\\propto 1/\\chi _{e}$ , the term $\\Gamma _{\\chi }$ does not diverge at $\\chi _{e}= 0$ since $\\alpha _{\\wedge }^{c},j_{\\wedge }^{\\nu },F_{\\wedge } \\rightarrow 0$ as $\\chi _{e}\\rightarrow 0$ .",
"The $\\Gamma _{\\chi }$ term is compared with the dependence of $\\hat{\\beta }$ (which determines the magnitude of the Nernst effect) on $\\chi _{e}$ in fig.",
"REF .",
"Comparing the unmagnetized components of $\\hat{\\beta }$ and $\\hat{\\gamma }$ then gives $\\frac{\\hat{\\gamma }}{\\hat{\\beta }} \\sim \\frac{E_{\\alpha 0}}{6 T}\\frac{n_{\\alpha }}{n}\\frac{\\ln \\Lambda _{ei}}{\\ln \\Lambda _{\\alpha e}}\\Gamma _{\\chi }\\left(\\frac{\\chi _{e}}{\\beta _{\\wedge }}\\right),$ where $n_{\\alpha }$ is the number density of non-thermalised $\\alpha $ particles.",
"We can assume that the ratio of Coulomb Logarithms is of order unity and $E_{\\alpha 0}/\\left(6 T\\right)\\sim 10^{2}$ for an igniting plasma.",
"Therefore, a relative density of $\\alpha $ particles of $n_{\\alpha }/n\\sim 10^{-2}$ could be sufficient for the $\\hat{\\gamma }$ term to dominate.",
"It is also worth noting that $\\hat{\\beta }\\ge 0$ for all values of $\\chi _{e}$ .",
"This is because in a plasma the friction force exerted by background ions is lower for faster electrons.",
"However, $\\Gamma _{\\chi }<0$ for very large and very small values of $\\chi _{e}$ .",
"This is due to $j_{\\bot }^{\\nu } < 0$ at small values of $\\chi _{e}$ , while for large values the $-\\chi _{\\alpha }\\gamma _{\\wedge }$ term dominates.",
"The system of equations (REF )-() is solved using an implicit Lagrangian model in order to accurately resolve the steep gradients that exist at the burn front.",
"[14], [15] We use the isobaric condition, (), in place of a momentum equation for simplicity as our principal interest is the transport processes occurring in ()-().",
"However, this limits our studies to burn propagation regimes driven by deflagration rather than detonation.",
"Figure: The variation of the diffusion coefficients β ^\\hat{\\beta } and γ ^\\hat{\\gamma } with χ e \\chi _{e}, where we assume χ α =χ e /2\\chi _{\\alpha }=\\chi _{e}/2.",
"Note that β ∧ /χ e ≥0∀χ e \\beta _{\\wedge }/\\chi _{e}\\ge 0\\,\\forall \\, \\chi _{e} while Γ χ <0\\Gamma _{\\chi }<0 for χ e ≪1\\chi _{e}\\ll 1 and χ e ≫1\\chi _{e}\\gg 1.",
"For χ e =0\\chi _{e}=0 we have Γ χ ≈-1.7\\Gamma _{\\chi }\\approx -1.7.",
"The scale of the horizontal axis of the inset plot represents log 10 χ e \\log _{10}\\chi _{e}."
],
[
"Burn propagation dynamics for various $B$ profiles",
"In this section we apply the model to a variety of different initial $B$ field profiles.",
"In each case the initial temperature profile is chosen to be sigmoidal $T_{0}\\left(\\hat{x}\\right) = T_{c}+\\left(T_{h}-T_{c}\\right)\\left(1+\\exp \\left(-10\\hat{x}\\right)\\right)^{-1},\\nonumber $ where $T_{c}$ ($T_{h}$ ) is the temperature of the cold (hot) fuel.",
"The factor of 10 ensures that 98% of the temperature change takes place within a region of unity length.",
"The normalizing constant, $L_{T}$ , is the burn front scale length.",
"Its value is chosen to be some multiple of the stopping distance of a $3.45\\,MeV$ $\\alpha $ particle at the conditions of the midpoint of the initial temperature profile."
],
[
"Uniform $B$ field",
"We begin with the case of a $B$ field whose strength is initially uniform across the burn front.",
"Since $B$ is constant, $\\chi _{e}\\propto \\tau _{ei}$ , and the initial value of $\\chi _{e}$ will vary significantly across the burn front with the largest value in the hot fuel.",
"Figure REF shows results for three different initial values of $B$ in which $L_{T}$ is $0.3$ times the $\\alpha $ stopping distance, corresponding to $\\sim 15\\,\\mu m$ .",
"The initial values of $\\chi _{e}$ in the hot fuel were $0.1,6,40$ , corresponding to $B_{0} \\approx 80, 500, 3500\\,T$ .",
"The corresponding values in the cold fuel were a factor of $\\sim 800$ smaller.",
"We note that at $\\chi _{e} =0.1,\\, 6,\\,40$ , the electron thermal conductivity is a factor of $\\sim 1,\\,10^{-1}\\,10^{-3}$ lower than its unmagnetized value.",
"The maximum mean free path of thermal electrons in this system is $\\lambda _{ei}\\approx 0.009$ , ensuring validity for our MHD model.",
"As burn propagates the evolution of the burn front is highly sensitive to the $B$ field.",
"The $\\chi _{e}=0.1$ case (which closely resembles an unmagnetized burn front) develops a temperature profile that is far less steep than the more magnetized cases and propagates furthest into the cold fuel.",
"However, this burn front also contains a localized region of steep temperature gradient near the cold fuel, which we refer to as the foot of the burn front.",
"This is caused by rapid $\\alpha $ heating of the cold, dense fuel.",
"In the more magnetized cases, the burn propagates less far due to suppression by the magnetic field of thermal conduction and $\\alpha $ transport from hot to cold fuel.",
"Transport of the $B$ field during burn is also dependent on $\\chi _{e}$ .",
"The middle diagram of fig.",
"REF illustrates competing transport effects.",
"The Nernst and $\\alpha $ flux effects transport $B$ field from hot to cold plasma.",
"This is most clearly evidenced by the spikes at the foot of the burn front where temperature gradients rapidly increase.",
"This is similar to the Nernst waves that have previously been studied in MagLIF and magnetized laser ablation.",
"[15], [28] Magnetic resistivity limits the sharpness of these spikes.",
"Meanwhile, ablation of cold fuel by the burn front leads to $B$ field being advected from cold to hot fuel.",
"Evidence of the ablative effect is shown in the $\\chi _{e}=6$ case, where a dip in $B$ field centered at $\\hat{x}=1.7$ coincides with a shallower temperature gradient.",
"The dynamics at an evolving burn front can cause both compression and rarefaction of a magnetic field.",
"Due to the isobaric condition and the increasing temperature of the plasma, advection does not cause any compression of $B$ field, only rarefaction.",
"However, the Nernst effect also contributes to rarefaction of the $B$ field in regions behind the compression wave.",
"This effect has previously been observed in the compression phase of MagLIF[6] and will be more clearly shown in fig.",
"REF .",
"The bottom diagram of fig.",
"REF shows profiles for $\\sigma _{\\chi }$ , the exponential growth rate for $\\chi _{e}$ at $\\hat{t}=5\\mathcal {T}_{h}$ .",
"The value of $\\chi _{e}$ can increase due to both heating of the plasma (which increases $\\tau _{ei}$ ) and $B$ field transport.",
"In the cases of $\\chi _{e} = 0.1$ and 6, peak values of $\\sigma _{\\chi }$ occur near the foot of the burn front and are significantly larger than the $\\sigma _{\\chi }$ values in the hot fuel.",
"In the $\\chi _{e}=40$ case, the temperature at the foot of the burn front is increasing at a slower rate and so the value of $\\sigma _{\\chi }$ is approximately the same as that in the hot fuel.",
"The effects of these transport phenomena on burn propagation are illustrated in fig.",
"REF .",
"There is a significant drop in the rate of burn propagation as the initial value of $\\chi _{e}$ increases.",
"The hot fuel is insulated from the cold fuel and, since $\\sigma _{\\chi }$ is large at the burn front, this insulating effect increases as burn evolves.",
"Similar behaviour is found for a wide range of initial parameters ($n_{h}\\sim \\left[10^{28},5\\times 10^{31}\\right]\\,m^{-3}$ and $T_{h}>\\sim 5\\,keV$ ).",
"We conclude this subsection with the results of an integrated simulation, see fig.",
"REF , carried out using the Chimera code[3] which includes radiation transport, extended-MHD and $\\alpha $ transport package.",
"However, the $\\alpha -e$ collisional term has not yet been included in this model.",
"These simulations used a similar set-up and initial conditions to those shown in fig.",
"REF but with an initial value of $\\chi _{e}\\approx 1$ in the hot fuel.",
"The Nernst term causes a significant spike in the $B$ field at the burn front, with a corresponding spike in $\\chi _{e}$ , by compressing the $B$ field.",
"A comparison of the $B$ field profiles with and without Nernst, shown in the middle panel, demonstrates how the Nernst also causes rarefaction of the $B$ field behind this compression wave.",
"The bottom panel of fig.",
"REF shows the velocity components of $B$ field transport with $v_{fluid}$ ($v_{Nernst}$ ) equivalent to $\\hat{u}$ ($\\hat{\\beta }\\partial \\ln T/\\partial \\hat{x}$ ) in ().",
"From this we can see that the spike in $B$ field corresponds to a region in which the absolute value of $v_{Nernst}$ has a local minimum.",
"Although the isobaric condition no longer applies, the evolution of the profiles of $\\rho $ , $T$ and $B$ is qualitatively similar to the results obtained from the model outlined in section and suggest that observations from this model are not unduly compromised by the isobaric condition."
],
[
"Uniform electron Hall parameter",
"We next consider the case of a burn front in which $\\chi _{e}$ is initially uniform.",
"Figure REF shows an example of burn propagation where initially $\\chi _{e}=0.1$ .",
"To achieve uniformity of $\\chi _{e}$ , the initial $B$ field needs to be significantly larger in the cold fuel and so we choose a parameter value of $n_{h} = 10^{30}\\,m^{-3}$ to ensure that the magnetic pressure is small compared with thermal pressure.",
"The top diagram of fig.",
"REF illustrates how $\\chi _{e}$ grows rapidly in the burn front region due to both increasing temperature and increasing magnetic field.",
"The bottom diagram shows different terms contributing to magnetic field growth in the burn front region.",
"The final $T$ profile has a distinctive shape with very steep temperature gradients developing at two locations.",
"First, at the foot of the burn pulse where $\\alpha $ heating is largest (see fig.",
"REF ) and secondly in the region where $\\chi _{e}$ reaches a maximum.",
"This second steep temperature gradient evolves to compensate for the drop in electron thermal conductivity caused by increasing $\\chi _{e}$ .",
"This temperature profile is reminiscent of double ablation front structures[29], [30] which occur in direct-drive ICF.",
"In that scenario radiation energy and electron heat flux are absorbed at different locations, leading to the formation of the double ablation front.",
"In our case the first front is formed by $\\alpha $ heating while the second front is due to the suppression of energy transport.",
"At the location of the second steep temperature gradient, advection is causing $B$ to grow while $\\alpha -e$ collisional and Nernst terms are preventing $B$ from penetrating further into the hot fuel, as can be seen from the bottom diagram of fig.",
"REF .",
"We describe the region where $\\chi _{e}$ grows rapidly as self-insulating since the growth is driven by the dynamics of burn propagation.",
"The top diagram of fig.",
"REF also shows the final profiles of $\\chi _{e}$ that are obtained if these two terms are neglected in the induction equation.",
"Clearly, these terms play a crucial role in establishing the self-insulating region.",
"Finally, fig.",
"REF shows profiles of quantities related to the $\\alpha $ particles.",
"It is found that $\\chi _{\\alpha }$ evolves in a similar way to $\\chi _{e}$ with a spike in $\\chi _{\\alpha }$ occurring at the burn front.",
"This causes a sharp drop in $F_{\\bot }$ and a spike in $F_{\\wedge } $ in the self-insulating region, while $\\alpha $ heating propagates less far into the cold fuel compared with the unmagnetized case."
],
[
"Local Magnetization",
"Integrated simulations of unmagentized ICF implosions have shown that self-generated $B$ fields can develop at the interface of hot and cold fuel.",
"[13] In this section, we consider how such a local $B$ field could evolve during burn by assuming that the initial $\\chi _{e}$ has a Gaussian profile that is narrower than $L_{T}$ .",
"An example of such a scenario is shown in fig.",
"REF where the FWHM of the initial $\\chi _{e}$ profile is $1/3L_{T}$ with a peak value of $\\chi _{e}=1$ .",
"The $B$ field is rapidly advected from cold to hot fuel by the propagating burn.",
"However, the Nernst and $\\alpha -e$ collision terms cause this advected $B$ field to maintain a steep front.",
"This coincides with the front of the rapidly growing $\\chi _{e}$ profile and a region of very steep temperature gradient.",
"As in the case of initially uniform $\\chi _{e}$ , the growth in $\\chi _{e}$ leads to the formation of a self-insulating layer between hot and cold fuel.",
"This behaviour is rather counter-intuitive.",
"Our usual expectation would be that the increasing temperature would increase the rate at which transport processes act to smooth out the steep gradients.",
"Instead we see that the increasing temperature leads to increasing $\\chi _{e}$ and a reduction in energy transport, such that the gradients become even steeper.",
"Figure REF also shows the reduction in burn rate of cold fuel as a function of time.",
"We can see from this that even though the majority of fuel is unmagnetized the growth of $\\chi _{e}$ during burn can significantly reduce the rate of burn propagation.",
"We note that the value of $\\kappa _{\\bot e}^{c}$ , the electron thermal conductivity coefficient, is about an order of magnitude smaller at $\\chi _{e}=7$ compared with $\\chi _{e}=1$ .",
"Theoretical models of burn in unmagnetized ICF[31], [32], [33] typically do not account for such rapid changes in conductivity but this may be necessary if the presence of self-generated $B$ fields is verified.",
"Such fields may also alter the ablative stabilization of Rayleigh-Taylor instabilities during burn.",
"[34]"
],
[
"Hall parameter growth rates",
"The formation of a self-insulating layer with two steep temperature gradients during burn propagation is observed across a broad range of simulation parameters and initial $B$ field profiles.",
"It is a result of the Hall parameter growing faster at the burn front than in the hot fuel.",
"We can investigate its formation by considering the factors on which $\\sigma _{\\chi }$ depends.",
"Starting with the definition $\\chi _{e} = eB\\tau _{ei}/m_{e}$ and assuming an isobaric, high-beta plasma with thermal pressure $P_{0} = 2nT$ we obtain $\\sigma _{\\chi }=\\frac{D \\ln \\chi _{e}}{D \\hat{t}} = \\frac{1}{B}\\frac{D B}{D \\hat{t}}+\\frac{g}{T}\\frac{D T}{D \\hat{t}},$ where $g = \\left(5-3\\left(\\ln \\Lambda _{ei}\\right)^{-1}\\right)/2\\approx 5/2$ .",
"If we assume that $\\chi _{e}$ is uniform, neglect bremsstrahlung losses ($\\hat{q}_{\\alpha }\\gg \\hat{P}$ ) and neglect resistive diffusion and Ettingshausen heat flow then using () and () results in $\\sigma _{\\chi } = &&\\frac{2\\left(g-1\\right)}{5P_{0}}\\hat{q}_{\\alpha }+\\hat{\\beta }\\left(\\frac{\\partial ^{2} \\ln T}{\\partial \\hat{x}^{2}}+\\left(\\frac{\\partial \\ln T}{\\partial \\hat{x}}\\right)^{2}\\right)+\\frac{2\\left(g-1\\right)}{5P_{0}}T\\hat{\\kappa }\\left(\\frac{\\partial ^{2} \\ln T}{\\partial \\hat{x}^{2}}+\\left(g+1\\right)\\left(\\frac{\\partial \\ln T}{\\partial \\hat{x}}\\right)^{2}\\right)\\nonumber \\\\&&+\\hat{\\gamma }\\left(\\frac{\\partial ^{2} \\ln \\mathcal {E}_{\\alpha }}{\\partial \\hat{x}^{2}}+\\left(\\frac{\\partial \\ln \\mathcal {E}_{\\alpha }}{\\partial \\hat{x}}\\right)^{2}+\\frac{\\partial \\ln T}{\\partial \\hat{x}}\\frac{\\partial \\ln \\mathcal {E}_{\\alpha }}{\\partial \\hat{x}}\\right),$ The first term on the right-hand-side (the $\\alpha $ heating term) will always be positive but the three remaining terms could be positive or negative, depending on the profiles of $T$ and $\\mathcal {E}_{\\alpha }$ .",
"This equation illustrates why the steep, opposing gradients in $\\chi _{e}$ and $T$ are observed in figs.",
"REF and REF : locations with $\\partial ^{2} \\ln T/\\partial \\hat{x}^{2} <0$ lead to $\\sigma _{\\chi }<0$ and vice versa.",
"Assuming that the gradient length scales of $T$ and $\\mathcal {E}_{\\alpha }$ are similar, we can obtain the following estimates of the strengths of the different terms $\\frac{2\\left(g-1\\right)}{5P_{0}}\\hat{q}_{\\alpha } &\\approx & \\mathcal {T}_{H}6.29\\times 10^{-5}\\ln \\Lambda _{\\alpha e}\\frac{\\mathcal {E}_{\\alpha }}{T_{keV}^{\\frac{5}{2}}},\\nonumber \\\\\\hat{\\beta } &\\approx & \\mathcal {T}_{H}\\frac{6.14\\times 10^{12}}{\\ln \\Lambda _{ei}}\\frac{T_{keV}^{\\frac{7}{2}}}{L_{T\\mu }^{2}P_{0G}}\\frac{\\beta _{\\wedge }}{\\chi _{e}},\\nonumber \\\\\\frac{2\\left(g-1\\right)}{5P_{0}}T\\hat{\\kappa } &\\approx & \\mathcal {T}_{H}\\frac{1.84\\times 10^{12}}{\\ln \\Lambda _{ei}}\\frac{T_{keV}^{\\frac{7}{2}}}{L_{T\\mu }^{2}P_{0G}}\\left(\\kappa _{\\bot e}^{c}+ \\sqrt{\\frac{2m_{e}}{m_{i}}}\\kappa _{\\bot i}^{c}\\right),\\nonumber \\\\\\hat{\\gamma } &\\approx & \\mathcal {T}_{H}\\frac{4.09\\times 10^{-2}}{\\ln \\Lambda _{\\alpha e}}\\mathcal {E}_{\\alpha }\\frac{T_{keV}^{\\frac{7}{2}}}{L_{T\\mu }^{2}P_{0G}^{2}}\\Gamma _{\\chi },\\nonumber $ where $T_{keV}$ is fuel temperature in $keV$ , $P_{0G}$ is fuel pressure in $Gbar$ and $L_{T\\mu }$ is temperature length scale in $\\mu m$ .",
"From these relations we can see that the Nernst and thermal conductivity terms are very similar in magnitude.",
"Their dependence on $\\chi _{e}$ is shown in fig.",
"REF .",
"This figure illustrates that we expect the largest magnetization growth rates to occur at small values of $\\chi _{e}$ .",
"It is likely that the cryogenic fuel layers of many MIF schemes will have initially low $\\chi _{e}$ values due to the high density and so $\\chi _{e}$ can increase rapidly in such regions as burn propagates.",
"Finally, we can take the ratio of the $\\alpha $ heating term to the $\\alpha $ -e collisional term to obtain a criterion (which is indepedent of $\\mathcal {E}_{\\alpha }$ ) for when the latter term is larger $T_{keV}^{6}\\left|\\Gamma _{\\chi }\\right|>\\frac{\\left(Ł_{T\\mu }P_{0G}\\ln \\Lambda _{\\alpha e}\\right)^{2}}{650}.$ From this formula we can, for given values of $\\chi _{e}$ , $L_{T\\mu }$ and $P_{0G}$ , calculate the temperature above which the $\\alpha $ -e collisional term is larger.",
"An example of this is shown in fig.",
"REF .",
"Figure: The variation of the dimensionless thermoelectricity and thermal conductivity coefficients with χ e \\chi _{e}.",
"For χ e >∼6\\chi _{e}> \\sim 6, ion thermal conductivity is larger than electron thermal conductivity.Figure: A plot of the minimum temperature for which the α\\alpha -e collisional term dominates the α\\alpha heating term in the growth rate of χ e \\chi _{e} for L Tμ =10L_{T\\mu }=10 and P 0G =100P_{0G}=100.",
"This result may be scaled according to () for other values of L Tμ L_{T\\mu } and P 0G P_{0G}.",
"The dashed vertical lines indicate the values of χ e \\chi _{e} for which Γ χ =0\\Gamma _{\\chi }=0."
],
[
"Conclusions",
"In this work we have studied the propagation of a subsonic thermonuclear burn wave across a magnetic field in a high beta plasma using an MHD model.",
"We can summarize our findings as follows: (i) Ablation of cold fuel due to thermonuclear burn advects $B$ field from cold to hot fuel at the burn front, while the Nernst and $\\alpha -e$ collision terms transport $B$ field from hot to cold fuel.",
"These combined transport effects confine $B$ field in the burn front region and cause both compression and rarefaction of $B$ .",
"(ii) The electron and $\\alpha $ Hall parameters, $\\chi _{e}$ and $\\chi _{\\alpha }$ , can grow rapidly at the burn front due to $\\alpha $ heating, magnetic field transport and thermal conduction processes.",
"The growth rate at the burn front can be significantly larger than in the hot fuel, causing a self-insulating layer to form at the burn front.",
"(iv) The temperature profile of a magnetized burn front is steeper than the unmagnetized case due to the suppression of thermal conduction.",
"Furthermore, the formation of a self-insulating layer can lead to very steep temperature gradients in localized regions of the burn front.",
"(iv) A magnetized burn front reduces the rate at which cold fuel is burnt due to the self-insulating effect.",
"These results demonstrate that careful modelling of the $B$ field transport during burn is required to accurately simulate the propagating burn wave and predict the energy gain achieved.",
"However, some of the assumptions in our idealized model prevent us from making quantitative predictions for how $B$ field transport could affect specific MIF schemes.",
"The isobaric condition restricts us to regimes of subsonic burn (deflagration).",
"Our results prompt the question of whether the formation of a self-insulating layer at the burn front could cause a transition to detonation of the hot fuel.",
"Furthermore, the planar geometry used here is suitable for idealized studies of burn front dynamics since the hot fuel at infinity is unaffected by the cold fuel.",
"It is likely that there will be a more complex interplay between hot fuel and burn front in finite sized targets.",
"Questions such as these will be addressed in future work using integrated simulations.",
"The results also provide further evidence[2], [11] that an optimum magnetic field strength and profile shape could exist for a given MIF scheme.",
"Such a field would need to provide the correct amount of magnothermoinsulation during compression and ignition, followed by sufficient confinement of $\\alpha $ particles during burn propagation without unnecessary slowing of the burn wave.",
"The strength and shape of this optimum magnetic field and how it can be generated remain open questions."
]
] | 2012.05280 |
[
[
"Refinements of Kool-Thomas Invariants via Equivariant $K$-theoretic\n invariants"
],
[
"Abstract In this article we are defining a refinement of Kool-Thomas invariants of local surfaces via the equivariant $K$-theoretic invariants proposed by Nekrasov and Okounkov.",
"Kool and Thomas defined the reduced obstruction theory for the moduli of stable pairs $\\mathcal{P}_{\\chi}(X,i_{*}\\beta)$ as the degree of the virtual class $\\left[\\mathcal{P}_{\\chi}(S,\\beta)\\right]^{red}$ after we apply $\\tau([pt])^{m}\\in H^{*}(\\mathcal{P}_{\\chi}(X,i_{*}\\beta),\\mathbb{Z})$.",
"$\\tau([pt])$ contain the information of the incidence of a point and a curve supporting a $(\\mathcal{F},s)$.",
"$\\textbf{Keywords: }$Kool-Thomas invariants, $K$-theoretic invariants, G\\\"ottsche Shende invariants"
],
[
"Introduction",
"Fix a nonsingular projective surface $S$ and a sufficently ample line bundle $\\mathcal {L}$ on $S$ .",
"A $\\delta $ -nodal curve $C$ on $S$ is a 1 dimensional subvariety of $S$ which has nodes at $\\delta $ points and is regular outside these singular points.",
"For any scheme $Y$ , let $Y^{[n]}$ be the Hilbert scheme of $n$ -points i.e.",
"$Y^{[n]}$ parametrizes subschemes $Z\\subset Y$ of length $n$ .",
"Given a family of curves $\\mathcal {C}\\rightarrow B$ over a base $B$ , we denote by $\\text{Hilb}^{n}(\\mathcal {C}/B)$ the relative Hilbert scheme of points.",
"Kool, Thomas and Shende showed that some linear combinations $n_{r,C}$ of the Euler characteristic of $C^{[n]}$ counts the number of curves of arithmetic genus $r$ mapping to $C$ .",
"Applying this to the family $\\mathcal {C}\\rightarrow \\mathbb {P}^{\\delta }$ where $\\mathbb {P}^{\\delta }\\subset |\\mathcal {L}|$ , the number of $\\delta $ -nodal curves is given by a coefficient of the generating function of the Euler characteristic of $\\text{Hilb}(\\mathcal {C}/\\mathbb {P}^{\\delta })$ after change of variable[11].",
"By replacing euler characteristic with Hirzebruch $\\chi _{y}$ -genus, Götsche and Shende give a refined counting of $\\delta $ -nodal curves.",
"Pandharipande and Thomas showed that a stable pair $(\\mathcal {\\mathcal {F}},s)$ on a surface $S$ is equivalent to the pair $(C,Z)$ of a curve $C$ on $S$ supporting the sheaf $\\mathcal {\\mathcal {F}}$ with $Z\\subset C$ a subscheme of finite length.",
"Thus the moduli space of stable pairs on a surface $S$ is a relative Hilbert scheme of points corresponding to a family of curves on $S$ .",
"The study of the moduli space of stable pairs on Calabi-Yau threefold $Y$ is an active area of research.",
"This moduli space gives a compactification of the moduli space of nonsingular curves in $Y$ .",
"To get an invariant of the moduli space Behrend and Fantechi introduce the notion of perfect obstruction theory.",
"With this notion we can construct a class in the Chow group of dimension 0 that is invariant under some deformations of $Y$[1].",
"The homological invariants of the stable pair moduli space $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ of the total space $X$ of $K_{S}$ of some smooth projective surface $S$ contain the information of the number of $\\delta $ -nodal curves in a hyperplane $\\mathbb {P}^{\\delta }\\subset |\\mathcal {L}|$ .",
"Notice that $X$ is Calabi-Yau.",
"There exist a morphism of schemes $\\text{div}:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\rightarrow |\\mathcal {L}|$ that maps a point $(\\mathcal {\\mathcal {F}},s)\\in \\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ to a divisor $\\text{div}\\left(\\pi _{*}\\mathcal {\\mathcal {F}}\\right)$ that support $\\pi _{*}\\mathcal {\\mathcal {F}}$ on $S$ where $\\pi :X\\rightarrow S$ is the structure morphism of $X$ as a vector bundle over $S$ .",
"Using descendents, Kool and Thomas translate the information of the incidence of a curve with a point into cutting down the moduli space by a hypersurface pulledback from $|\\mathcal {L}|$ so that after cutting down, we have a moduli space that parameterize Hilbert scheme of curves in $\\mathbb {P}^{\\delta }$[12].",
"The famous conjecture of Maulik, Nekrasov Okounkov and Pandharipande states that the invariants corresponding to the moduli space of stable pairs have the same information as the invariants defined from the moduli space of stable maps and the Hilbert schemes.",
"The next development in the theory of PT invariants is to give a refinement of the homological invariant.",
"The end product of this homological invariant is a number.",
"A refinement of this invariant would be a Laurent polynomial in a variable $t$ such that when we evaluate $t$ at 1 we get the homological invariant.",
"There are several methods that have been introduced to give a refinement for DT invariants, for example both motivic and $K$ -theoretic definitions.",
"In this thesis we use the $K$ -theoretic definition which has been proposed by Nekrasov and Okounkov in [15] where we compute the holomorphic Euler characteristic of the twisted virtual structure sheaf of the coresponding moduli space.",
"In the case when $S=\\mathbb {P}^{2}$ or $S=\\mathbb {P}^{1}\\times \\mathbb {P}^{1}$ Choi, Katz and Klemm have computed a $K$ -theoretic invariant of the moduli space of stable pairs in the paper [2].",
"Their computation does not include any information about the incidence of subschemes of $S$ ."
],
[
"Equivariant Chow Groups and $K$ -theory",
"In this section we will describe the notatioan and definition we use regarding equivariant Chow Groups and $K$ -theory."
],
[
"Equivariant Chow Groups",
"In this section we review the definition of equivariant Chow groups given in [3], [4].",
"We will use $g$ to denote the dimension of our group $G$ as a scheme over $\\mathbb {C}$ .",
"Given $i\\in \\mathbb {Z}.$ Let $X$ be a $G$ -scheme with $\\dim \\,X=d$ .",
"Let $V$ be $G$ -vector space of dimension $l$ .",
"Assume that there exists an open subscheme $U\\subset V$ and a principal $G$ -bundle $\\pi :U\\rightarrow U_{G}.$ By giving $X\\times V$ a diagonal action of $G$ , assume furthermore that there exist a principal $G$ -bundle $\\pi _{X}:X\\times U\\rightarrow \\left(X\\times U\\right)/G$ .",
"We will use $X\\times _{G}U$ to denote $\\left(X\\times U\\right)/G$ .",
"Assume also that $V\\setminus U$ has codimension greater than $d-i$ , then the equivariant Chow group is defined as $A_{i}^{G}(X):=A_{i+l-g}(X\\times _{G}U).$ The definition is independent up to isomorphism of the choice of a representation as long as $V\\setminus U$ is of codimension greater than $d-i$ .",
"For a $G$ -equivariant map $f:X\\rightarrow Y$ with property $P$ where $P$ is either proper, flat, smooth, or regular embedding the $G$ -equivariant map $f\\times 1:X\\times U\\rightarrow Y\\times U$ has the property $P$ since all of these properties are preserved by a flat base change.",
"Moreover, the corresponding morphism $f_{G}:X\\times _{G}U\\rightarrow Y\\times _{G}U$ also has property $P$ .",
"In fact, these properties are local on the target in the Zariski topology and for any trivialization $(V_{i},\\bar{\\varphi }_{i})_{i\\in \\Lambda }$ of $\\pi :U\\rightarrow U_{G}$ the restriction of $f_{G}$ on $\\pi _{X}(X\\times \\pi ^{-1}(V_{i}))$ is isomorphic to $f\\times \\text{id}_{V_{i}}$ .",
"So from the definition, for a flat $G$ -map $f:X\\rightarrow Y$ of codimension $l$ we can define pullback map $f^{*}:A_{i}^{G}(Y)\\rightarrow A_{i+l}^{G}(X)$ for equivariant Chow groups.",
"Similarly, for regular embedding $f:X\\rightarrow Y$ of codimension $d$ we have a Gysin homomorphism $f^{*}:A_{i}^{G}(Y)\\rightarrow A_{i-d}^{G}(X)$ and for proper $G$ -map $f:X\\rightarrow Y$ we can define pushforward $f_{*}:A_{i}^{G}(X)\\rightarrow A_{i}^{G}(Y)$ for equivariant Chow groups.",
"For $G=T_{1}$ and an $l+1$ -dimensional weight space $V_{\\chi }$ we have a principal $G$ -bundle $\\pi _{U}:=V_{\\chi }\\setminus \\lbrace 0\\rbrace \\rightarrow \\mathbb {P}(V_{\\chi })$ .",
"There exist a principal $G$ -bundle $\\pi _{X}:X\\times U\\rightarrow X\\times _{G}U$ .",
"And since $\\text{codim}\\,V_{\\chi }\\setminus U$ is $l+1$ , for each $i\\in \\text{$\\mathbb {Z}\\ $}$ we can take $A_{i+l}\\left(X\\times _{G}U)\\right)$ to represent $A_{i}^{G}(X)$ if $l+i\\ge d$ .",
"We can also fix $\\chi $ to be $-1$ to cover all $i$ .",
"Thus we fix the following notation.",
"For each positive integer $l$ let $V_{l}$ be a $T_{1}$ -space of weight $-1$ with coordinate $x_{0},\\ldots ,x_{l}$ .",
"Thus $V_{l-1}$ is the zero locus of the last coordinate of $V_{l}$ .",
"We use $U_{l}$ to denote $V_{l}\\setminus \\lbrace 0\\rbrace $ and $X_{l}$ to denote $X\\times _{G}U_{l}$ and $\\pi _{X,l}:X\\times U_{l}\\rightarrow X_{l}$ the corresponding principal bundle.",
"Thus we have the following direct system ${\\ldots [r] & X_{l-1}[r]^{j_{X,l-1}} & X_{l}[r]^{j_{X,l}} & X_{l+1}[r]^{j_{X,l+1}} & \\ldots }$ americanThere is a projection from $\\xi :V_{l+1}\\rightarrow V_{l}$ by forgetting the last coordinate such that $j_{l}:V_{l}\\rightarrow V_{l+1}$ is the zero section of $\\xi .$ By removing the fiber of $p:=(0:0:\\ldots :0:1)\\in \\mathbb {P}(V_{l+1})$ , the corresponding projection $\\xi :X_{l+1}\\setminus \\pi _{X}^{-1}(p)\\rightarrow X_{l}$ is a line bundle over $X_{l}$ such that $j_{X,l}:X_{l}\\rightarrow X_{l+1}\\setminus \\pi _{X,l+1}^{-1}(p)$ is the zero section.",
"Note that $\\dim \\text{$\\pi _{X,l+1}^{-1}(x)$=$\\dim \\,X=d$}$ .",
"Thus for $i\\ge d-l$ the restriction map $A_{i+l+1}(X_{l+1})\\rightarrow A_{i+l+1}(X_{l+1}\\setminus \\pi _{X,l+1}^{-1}(p))$ is an isomorphism.",
"In general this restriction is a surjection.",
"Since $\\hat{j}_{X,l}:X_{l}\\rightarrow X_{l+1}\\setminus \\pi _{X,l+1}^{-1}(p)$ is the zero section of $\\xi $ , the Gysin homomorphism $\\hat{j}_{X,n}^{!",
"}:A_{k+1}(X_{l+1}\\setminus \\pi _{X,l+1}^{-1}(p))\\rightarrow A_{k}(X_{l})$ is an isomorphism.",
"Since $j$ is a regular embedding we have a Gysin homomorphism $j^{!",
"}:A_{k+1}\\left(X_{l+1}\\right)\\rightarrow A_{k}(X_{l})$ which is the composition of the above homomorphisms.",
"The direct system REF induces an inverse system ${\\ldots & A_{*}(X_{l-1})[l] & A_{*}(X_{l})[l]_{j_{X,l-1}^{!}}",
"& A_{*}(X_{l+1})[l]_{j_{X,l}^{!}}",
"& \\ldots }$ of abelian groups.",
"Let $({\\displaystyle \\lim _{\\leftarrow }}A(X_{l}),\\lambda _{l})$ be the inverse limit of the above inverse system.",
"englishFrom the definition of equivariant Chow groups, $A_{i}^{G}(X)=A_{i+n}(X_{n})$ for $i\\ge d-n$ so that we can identify $\\prod _{i=d-n}^{d}A_{i}^{G}(X)$ with the group $\\prod _{i=d}^{d+n}A_{i}(X_{n})$ .",
"Recall that $\\left({\\displaystyle \\prod _{i=-\\infty }^{d}}A_{i}^{G}(X),\\nu _{i}\\right)$ where $\\nu _{n}:$${\\displaystyle \\prod _{i=-\\infty }^{d}}A_{i}^{G}(X)\\rightarrow \\prod _{i=d-n}^{d}A^{G}(X)$ is defined by $(a_{d},a_{d-1}\\ldots )\\mapsto \\left(a_{d},\\ldots ,a_{d-n}\\right)$ is the inverse limit of the inverse system defined by the projection $p_{X,n}:$$\\prod _{i=d-n-1}^{d}A^{G}(X)\\rightarrow \\prod _{i=d-n}^{d}A^{G}(X)$ , $(a_{d},\\ldots ,a_{d-n},a_{d-n-1})\\mapsto (a_{d},\\ldots ,a_{d-n})$ .",
"After indentifying $\\prod _{i=d-n}^{d}A_{i}^{G}(X)$ with english$\\prod _{i=d}^{d+n}A_{i}(X_{n})$ , $p_{X,n}$ and $j_{X,n}^{!",
"}$ are the same homomorphism.",
"The compostion of the projections $\\hat{\\xi }_{n}:$$A_{*}(X_{n})\\rightarrow \\prod _{i=d}^{d+n}A(X_{n})$ with $\\lambda _{n}:$${\\displaystyle \\lim _{\\leftarrow }}A_{*}(X_{l})\\rightarrow A_{*}(X_{n})$ are homorphisms $\\xi _{i}:$${\\displaystyle \\lim _{\\leftarrow }}A_{*}(X)\\rightarrow \\prod _{i=d-n}^{d}A_{i}^{G}(X)$ satisfying $p_{X,n+1}\\circ \\xi _{n}=p_{X,n}$ so that by the universal property of inverse limit we have a group homomorphism $\\xi :$${\\displaystyle \\lim _{\\leftarrow }}A_{*}(X_{n})\\rightarrow {\\displaystyle \\prod _{i=-\\infty }^{d}}A_{i}^{G}(X)$ satisfying $p_{X,n}\\circ \\xi =\\xi _{n}$ .",
"We will use the following proposition later.",
"Proposition 1 $\\xi :{\\displaystyle \\lim _{\\leftarrow }}A_{*}(X_{l})\\rightarrow {\\displaystyle \\prod _{i=-\\infty }^{d}}A_{i}^{G}(X)$ is an isomorphism .",
"english"
],
[
"Equivariant $K$ -theory",
"Let $G$ acts on a scheme $X$ .",
"Let $Vec_{G}(X)$ (resp.",
"$Coh_{G}(X)$ ) be the category of equivariant vector bundle (resp.",
"equivariant coherent sheaves) on scheme $X$ .",
"We will use $K^{G}(X)$ (resp.",
"$G^{G}(X)$ ) to denote the grup generated by equivariant vector bundle (resp.",
"coherent sheaves) modulo short exact sequence.",
"More generally for any exact category $\\mathcal {N}$ we can construct the group $K_{0}\\mathcal {N}$ generated by objects of $\\mathcal {N}$ modulo short exact sequence.",
"americanWe will skecth the construction of pushforward map $f_{*}:K^{G}(X)\\rightarrow K^{G}(Y)$ in some special cases induced by direct image functor.",
"For more details, readers should consult chapter 2 of [24] or section 7 and 8 of [20].",
"Since taking pullback is exact on vector bundles, the definition of $f^{*}:K^{G}(Y)\\rightarrow K^{G}(X)$ is straightforward.",
"First we need the following Lemma.",
"Lemma 2 Let $\\mathcal {N}_{X}$ be a full subcategory of $Coh_{G}(X)$ staisfying the following conditions: 1.",
"$\\mathcal {N}_{X}$ contains $Vec_{G}(X)$ 2.",
"$\\mathcal {N}_{X}$ is closed under extension 3.",
"Each objects of $\\mathcal {N}_{X}$ has a resolution by a bounded complex of elements in $Vec_{G}(X)$ 4.",
"$\\mathcal {N}_{X}$ is closed under kernels of surjections.",
"Then 1.",
"$\\mathcal {N}_{X}$ is exact and the inclusion $Vec_{G}(X)\\subset \\mathcal {N}_{X}$ induce the group homomorphism $i:K^{G}(X)\\rightarrow K_{0}\\left(\\mathcal {N}_{X}\\right)$ by mapping the class $\\left[\\mathcal {P}\\right]_{Vec_{G}(X)}$ of any locally free sheaf $\\mathcal {P}$ to its class $\\left[\\mathcal {P}\\right]_{\\mathcal {N}_{X}}$ in $K_{0}(\\mathcal {N}_{x})$ 2. all resolutions of $\\mathcal {\\mathcal {F}}$ by equivariant locally free sheaves ${0[r] & \\mathcal {P}_{n}[r] & \\mathcal {P}_{n-1}[r] & \\ldots [r] & \\mathcal {P}_{1}[r] & \\mathcal {P}_{0}[r] & \\mathcal {\\mathcal {F}}[r] & 0}$ define the same element $\\chi (\\mathcal {\\mathcal {F}}):=\\sum _{i=0}^{n}\\left(-1\\right)^{-i}\\left[\\mathcal {P}_{i}\\right]$ in $K^{G}(X)$ .",
"Furthermore, $\\chi $ define a group homomorphism $\\chi :K_{0}(\\mathcal {N}_{X})\\rightarrow K^{G}(X)$ which is the inverse of $i:K^{G}(X)\\rightarrow K_{0}(\\mathcal {N}_{X})$ .",
"Corollary 3 Let $f:X\\rightarrow Y$ be a finite $G$ -morphism such that $f_{*}:Vec_{G}(X)\\rightarrow Coh_{G}(X)$ factors through a subcatcategory $\\mathcal {N}_{Y}\\subset Coh_{G}(Y)$ satisfying all 4 conditions of lemma REF above.",
"Then there exist a group homomorphism $f_{*}:K^{G}(X)\\rightarrow K^{G}(Y)$ such that $f_{*}[\\mathcal {E}]=\\chi (f_{*}\\mathcal {E})$ for any locally free sheaf $\\mathcal {E}$ on $X$ .",
"Proposition 4 (Projection Formula) Let $f:X\\rightarrow Y$ be a morphism satisfying the condition in corollary REF or the projection $\\varphi :\\mathbb {P}_{Y}(V)\\rightarrow Y$ where $V$ is a $G$ -equivariant vector bundle.",
"Then for any $x\\in K^{G}(X)$ and $y\\in K^{G}(Y)$ we have $f_{*}\\left(x.f^{*}y\\right)=\\left(f_{*}x\\right).y\\in K^{G}(Y).$ Proposition 5 (Base change formula) english Consider the following cartesian diagram ${\\bar{X}[r]^{\\bar{g}}[d]_{\\bar{f}} & X[d]^{f}\\\\\\bar{Y}[r]_{g} & X}$ such that $f$ and $f^{\\prime }$ are $G$ -regular embeddings of codimension $r$ .",
"Then $g^{*}\\circ f_{*}=\\bar{f}_{*}\\circ \\bar{g}^{*}:K^{G}(X)\\rightarrow K^{G}(\\bar{Y})$ american Let $A$ be a smooth projective variety and let $p:A\\times Y\\rightarrow Y$ be the projection to the second factor.",
"Let $g:\\bar{Y}\\rightarrow Y$ be any morphism and consider the following cartesian diagram ${A\\times \\bar{Y}[d]_{\\bar{p}}[r]^{\\bar{g}} & A\\times Y[d]^{p}\\\\\\bar{Y}[r]_{g} & Y}.$ Then the pushforward maps $p_{*}:K^{G}(A\\times Y)\\rightarrow K^{G}\\left(Y\\right)$ and $\\bar{p}_{*}:K^{G}(A\\times \\bar{Y})\\rightarrow K^{G}\\left(\\bar{Y}\\right)$ are well defined and $\\bar{p}_{*}\\circ \\bar{g}^{*}=g^{*}\\circ p_{*}:K^{G}(A\\times Y)\\rightarrow K^{G}\\left(\\bar{Y}\\right)$ .",
"Let $d:D\\rightarrow A\\times Y$ be a $G$ -closed embedding such that $D$ is flat over $Y$ and let $d^{\\prime }:D^{\\prime }\\rightarrow A\\times Y^{\\prime }$ be the corresponding pullback so that we have the following cartesian diagram ${\\bar{D}[r]^{\\hat{g}}[d]_{\\bar{d}} & D[d]^{d}\\\\A\\times \\bar{Y}[r]_{\\bar{g}} & A\\times Y.",
"}$ Then $\\bar{g}^{*}\\left[\\mathcal {O}_{D}\\right]=\\left[\\mathcal {O}_{\\bar{D}}\\right]\\in K^{G}(A\\times \\bar{Y})$ .",
"americanLet $i:X\\rightarrow Y$ be a $G$ -equivariant closed embdedding and let $U=Y\\setminus X$ with open embedding $j:U\\rightarrow Y$ .",
"Then there exist group homomorphism $i_{*}:G^{G}(X)\\rightarrow G^{G}(Y)$ and $j^{*}:G^{G}(Y)\\rightarrow G^{G}(U)$ .",
"These two homomorphism is related as follows: Lemma 6 The following complex of abelian groups is exact ${G^{G}(X)[r]^{i_{*}} & G^{G}(Y)[r]^{j^{*}} & G^{G}(U)[r] & 0}.$ This is Theorem 2.7 of [23].",
"We call a class $\\beta \\in G^{G}(Y)$ is supported on $X$ if $\\beta $ is in the image of $i_{*}$ .",
"Equivalently $\\beta $ is supported on $X$ if $j^{*}\\beta =0$ .",
"Let $Coh_{G}^{X}(Y)$ be the abelian group of coherent sheaves supported on $X$ .",
"Note that $\\mathcal {\\mathcal {F}}\\in Coh_{G}^{X}(Y)$ is not necessarily an $\\mathcal {O}_{X}$ -module.",
"Let $G_{X}^{G}(Y)$ be the corresponding Grothendieck group.",
"The pushforward functor $i_{*}:Coh_{G}(X)\\rightarrow Coh_{G}(Y)$ factors through $Coh_{G}^{X}(Y)$ so that there exist a group homomorphism $\\bar{i}:G^{G}(X)\\rightarrow G_{X}^{G}(Y)$ , $[\\mathcal {\\mathcal {F}}]\\mapsto [i_{*}\\mathcal {\\mathcal {F}}]$ .",
"There exist an inverse of $\\bar{i}$ described as follows.",
"Let $\\mathcal {\\mathcal {F}}\\in Coh_{G}^{X}(Y)$ and let $\\mathcal {I}$ be the ideal of $X.$ Then there exist positive integer $n$ such that $\\mathcal {I}^{n}\\mathcal {\\mathcal {F}}=0$ so that we have a filtration $\\mathcal {\\mathcal {F}}\\supseteq \\mathcal {I}\\mathcal {\\mathcal {F}}\\supseteq \\mathcal {I}^{2}\\mathcal {\\mathcal {F}}\\supseteq \\ldots \\supseteq \\mathcal {I}^{n-1}\\mathcal {\\mathcal {F}}\\supseteq \\mathcal {I}^{n}\\mathcal {\\mathcal {F}}=0.$ Note that each $\\mathcal {I}^{r}\\mathcal {\\mathcal {F}}/\\mathcal {I}^{r+1}\\mathcal {\\mathcal {F}}$ is an $\\mathcal {O}_{X}$ -module.",
"One can show that $[\\mathcal {\\mathcal {F}}]\\mapsto \\sum _{r=0}^{n-1}[\\mathcal {I}^{r}\\mathcal {\\mathcal {F}}/\\mathcal {I}^{r+1}\\mathcal {\\mathcal {F}}]$ defines a group homomorphism $\\bar{i}^{-1}:G_{X}^{G}(Y)\\rightarrow G^{G}(X)$ .",
"Lemma 7 $\\bar{i}:G^{G}(X)\\rightarrow G_{X}^{G}(Y)$ is an isomorphism.",
"englishGiven a cartesian diagram ${\\bar{X}[r]^{\\bar{f}}[d]_{\\bar{i}} & \\bar{Y}[d]^{i}\\\\X[r]_{f} & Y}$ with $i$ , $f$ are closed embeddings and a coherent sheaf $\\mathcal {E}$ on $X$ such that $f_{*}\\mathcal {E}$ has a finite resolution by a complex of locally free sheaves.",
"Then we can define a group homomorphism $f^{[\\mathcal {E}]}:G^{G}(\\bar{Y})\\rightarrow G^{G}(\\bar{X}),$ described as follows.",
"Let $\\mathcal {\\mathcal {F}}$ be a coherent sheaf on $Y$ supported on $\\bar{Y}$ .",
"For each $y\\in Y$ , the stalk of $\\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}})$ on $y$ is $\\mathcal {T}or_{\\mathcal {O}_{Y,y}}^{i}\\left(\\left(f_{*}\\mathcal {E}\\right)_{y},\\mathcal {\\mathcal {F}}_{y}\\right)$ so that $\\mathcal {T}or_{y}^{i}\\left(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}}\\right)$ is supported on $\\bar{X}$ .",
"For any exact sequence $@C=8pt{0[r] & \\mathcal {\\mathcal {F}}^{\\prime }[r] & \\mathcal {\\mathcal {F}}[r] & \\mathcal {\\mathcal {F}}\"[r] & 0}$ of coherent sheaves on $\\bar{Y}$ we have a long exact sequence $@C=8pt{\\mathcal {T}or_{Y}^{i+1}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}}\")[r] & \\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}}^{\\prime })[r] & \\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}})[r] & \\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}}^{\\prime })[r] & \\,}$ so that $\\sum _{i\\ge 0}(-1)^{i}[\\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}})]=\\sum _{i\\ge 0}(-1)^{i}[\\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}}^{\\prime })+\\sum _{i\\ge 0}(-1)^{i}[\\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}}\")]\\in G_{\\bar{X}}^{G}(Y).$ Thus there exist a group homomorphism $\\bar{f}^{[\\mathcal {E}]}:G^{G}(\\bar{Y})\\rightarrow G_{\\bar{X}}^{G}(Y)$ .",
"By Lemma REF , we can define $f^{[\\mathcal {E}]}$ as the composition $\\bar{i}^{-1}\\circ \\bar{f}^{[\\mathcal {E}]}$ .",
"Lemma 8 Let $f:X\\rightarrow Y$ be a closed embedding and a coherent sheaf $\\mathcal {E}$ on $X$ such that $f_{*}\\mathcal {E}$ has a finite resolution by locally free sheaves.",
"For any closed embedding $i:\\bar{Y}\\rightarrow Y$ , there exist a group homomorphism $f^{[\\mathcal {E}]}:G^{G}(\\bar{Y})\\rightarrow G^{G}(\\bar{Y}\\cap X)$ that maps $[\\mathcal {\\mathcal {F}}]$ to $\\sum _{i=0}(-1)^{-1}\\left[\\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}})\\right]_{\\bar{Y}\\cap X}$ .",
"Furthermore, $i_{*}f^{[\\mathcal {E}]}(\\left[\\mathcal {\\mathcal {F}}\\right])=\\sum _{i=0}(-1)^{-1}\\left[\\mathcal {T}or_{Y}^{i}(f_{*}\\mathcal {E},\\mathcal {\\mathcal {F}})\\right]_{Y}$ .",
"Let $G$ be the torus $T_{1}$ and let $X$ be a $G$ -scheme.",
"Recall that by Proposition REF there exist an isomorphism $\\xi :{\\displaystyle \\lim _{\\leftarrow }}A_{*}(X_{n})\\rightarrow \\prod _{i=-\\infty }^{d}A_{i}^{G}(X)$ .",
"In this section we want to recall some results of the corresponding ${\\displaystyle \\lim _{\\leftarrow }}K(X_{n}).$ americanFrom the direct system REF , we have the inverse system ${\\ldots [r] & K(X_{l-1})[l] & K(X_{l})[l]_{j_{X,l-1}^{*}} & K(X_{l+1})[l]_{j_{X,l}^{*}} & \\ldots }$ We denote the inverse limit of the above inverse system as ${\\displaystyle \\lim _{\\leftarrow }}\\,K(X_{l})$ and use $\\rho _{X,l}$ to denote the canonical morphism ${\\displaystyle \\lim _{\\leftarrow }}K(X_{l})\\rightarrow K(X_{l})$ .",
"The pullback functor induced from the projection map $pr_{X}:X\\times U_{l}\\rightarrow X$ and the equivalence between $Vec_{G}(X\\times U_{l})$ and $Vec(X_{l})$ induces group homomorphims $\\kappa _{X,l}:K^{G}(X)\\rightarrow K(X_{l})$ .",
"It's easy to show that $\\kappa _{X,l}=j_{X,l}^{*}\\circ \\kappa _{X,l+1}$ so that we have a uniqe group homomorphism $\\kappa _{X}:K^{G}(X)\\rightarrow {\\displaystyle \\lim _{\\leftarrow }}K(X_{l})$ such that $\\kappa _{X,l}=\\rho _{X,l}\\circ \\kappa _{X}$ .",
"In this section, to distinguish bertween the ordinary and the equivariant version of pullback and pushforward map, we will use superscript $^{G}$ to denote the equivariant version, for example we will use $f^{G,*}$ to denote the pullback in the equivariant setting.",
"There is a canonical way to define pullback map $\\overleftarrow{f^{*}}:{\\displaystyle \\lim _{\\leftarrow }}K(Y)\\rightarrow {\\displaystyle \\lim _{\\leftarrow }}K(X)$ and pushforward map $\\overleftarrow{f_{*}}:{\\displaystyle \\lim _{\\leftarrow }}K(X_{l})\\rightarrow {\\displaystyle \\lim _{\\leftarrow }}K(Y_{l})$ for ${\\displaystyle \\lim _{\\leftarrow }}\\,K(X_{l})$ .",
"english Lemma 9 Let $f:X\\rightarrow X$ be a $G$ morphism.",
"1.",
"If $f:X\\rightarrow Y$ is a finite $G$ -morphism satisfying the condition in corrolary REF .",
"Assume also that for all $l$ , $\\left(f\\times \\text{id}_{U_{l}}\\right)$ also satisfies the condition in corrolary REF .",
"Then there exist a group homomorphism $\\overleftarrow{f_{*}}:{\\displaystyle \\lim _{\\leftarrow }}K(X_{l})\\rightarrow {\\displaystyle \\lim _{\\leftarrow }}K(Y_{l})$ satisfying the identity $\\kappa _{Y}\\circ f_{*}^{G}=\\overleftarrow{f_{*}}\\circ \\kappa _{X}$ .",
"2.",
"If $f:X\\rightarrow Y$ is the structure morphism $\\mathbb {P}_{Y}(V)\\rightarrow Y$ where $V$ is a $G$ -equivariant vector bundle.",
"Then there exist a group homomorphism $\\overleftarrow{f}_{*}:{\\displaystyle \\lim _{\\leftarrow }}K(X_{l})\\rightarrow {\\displaystyle \\lim _{\\leftarrow }}K(Y_{l})$ satisfying the identity $\\kappa _{Y}\\circ f_{*}^{G}=\\overleftarrow{f_{*}}\\circ \\kappa _{X}$ .",
"3.",
"If $f:X\\rightarrow Y$ is a $G$ -morphism that can be factorized into $p\\circ i$ where $i:X\\rightarrow Z$ is a finite morphism satisfying the condition 1. and $p$ satisfies condition 2. then the group homomorphsim $\\overleftarrow{f_{*}}:=\\overleftarrow{p_{*}}\\circ \\overleftarrow{i_{*}}:$${\\displaystyle \\lim _{\\leftarrow }}K(X)\\rightarrow {\\displaystyle \\lim _{\\leftarrow }}K(Y)$ is independent of the factorization.",
"For each equivariant vector bundle $\\mathcal {E}$ on $X$ , its pullback $\\tilde{\\mathcal {E}}$ to $X\\times U_{n}$ correspond to a vector bundle $\\mathcal {E}_{n}$ on $X_{n}$ such that $\\pi ^{*}\\mathcal {E}_{n}=\\tilde{\\mathcal {E}}$ .",
"By the identification $A_{j}^{G}(X)=A_{j+n}(X_{n})$ , $c_{G}^{i}(\\mathcal {E}):A_{j}^{G}(X)\\rightarrow A_{j-i}^{G}(X)$ is given by $c^{i}(\\mathcal {E}_{n}):A_{j+n}(X_{n})\\rightarrow A_{j-i+n}(X_{n})$ .",
"Since Chern class commutes with pullback this definition is well defined.",
"Furthermore, $c_{G}^{j}(\\mathcal {E})$ is an element of $A_{G}^{i}(X)$ .",
"In the non equivariant case, each vector bundle $\\mathcal {E}$ of rank $r$ has Chern roots $x_{1},\\ldots ,x_{r}$ such that $c^{i}(\\mathcal {E})=e_{i}(x_{1},\\ldots ,x_{r})$ where $e_{i}$ is the $i^{\\text{th}}$ symmetric polynomial.",
"Furthermore, its Chern character is defined as $ch(\\mathcal {E})=\\sum _{i=1}^{r}e^{x_{i}}$ .",
"From this definition, we have the following formula of Chern chararacter in terms of Chern classes $ch(\\mathcal {E}) & =r+c^{1}(\\mathcal {E})+\\frac{1}{2}\\left(c^{1}(\\mathcal {E})^{2}-2c^{2}(\\mathcal {E})\\right)+..\\\\& =\\sum _{i=0}^{\\infty }P_{j}(c^{1}(\\mathcal {E}),\\ldots ,c^{i}(\\mathcal {E}))$ where $P_{j}\\left(c^{1}(\\mathcal {E}),\\ldots ,c^{j}(\\mathcal {E})\\right)$ is a polynomial of order $j$ with $c^{i}(\\mathcal {E})$ has weight $i$ .",
"In [4], Edidin and Graham define an equivariant Chern character map $ch^{G}:K^{G}(X)\\rightarrow \\prod _{i=0}^{\\infty }A_{G}^{i}(X)$ by the following formula $ch^{G}\\left(\\mathcal {E}\\right)=\\sum _{i=0}^{\\infty }P_{i}(c_{G}^{1}(\\mathcal {E}),\\ldots ,c_{G}^{i}(\\mathcal {E})).$ One can show that $ch^{G}$ is a ring homomorphism.",
"Let $\\overleftarrow{ch}:K^{G}(X)\\rightarrow {\\displaystyle \\lim _{\\leftarrow }}A^{*}(X_{n})$ denote the composition $\\alpha \\circ ch^{G}$ .",
"For each $n$ there is a Chern character map $ch_{n}:K(X_{n})\\rightarrow A^{*}(X_{n})$ which commutes with refined Gysin homomorphisms.",
"By the universal property of inverse limits we have a ring homomorphism $\\widehat{ch}:{\\displaystyle \\lim _{\\leftarrow }}K(X_{n})\\rightarrow {\\displaystyle \\lim _{\\leftarrow }}A^{*}(X_{n})$ .",
"Since each $ch_{n}$ is a ring homomorphis, $\\widehat{ch}$ is also a ring homomorphism.",
"Furthermore the following diagram commutes $\\begin{tikzcd}K^G(X)[r,\"ch^G\"][d,\"\\kappa \"^{\\prime }]&\\displaystyle {\\prod _{i=0}^{\\infty }}A^i _G(X)[d,\"\\alpha \"^{\\prime }]\\\\{\\displaystyle \\lim _{\\leftarrow }}K(X_n)[r,\"\\widehat{ch}\"^{\\prime }]&{\\displaystyle \\lim _{\\leftarrow }}A^*(X_n)\\end{tikzcd}$ Lemma 10 For all $x\\in {\\displaystyle \\lim _{\\leftarrow }}A_{*}(X_{n})$ and for any $\\beta \\in K^{G}(X)$ we have $\\xi \\left(\\overleftarrow{ch}(\\beta )(x)\\right)$$=ch^{G}(\\beta )(\\xi x)$ ."
],
[
"Kool-Thomas invariants ",
"In this section we will review the definition of Kool-Thomas invariants and its relation to curve counting.",
"Let $X$ be a projective smooth varietiy of dimension 3.",
"A pair $\\left(\\mathcal {\\mathcal {F}},s\\right)$ where $\\mathcal {\\mathcal {F}}$ is a coherent sheaf of dimension 1 and $s$ is a section of $\\mathcal {\\mathcal {F}}$ is called stable if the following two conditions holds: $\\mathcal {\\mathcal {F}}$ is pure The cokernel $Q$ of $s$ is of dimension 0.",
"Let $X$ be a smooth projective 3-fold and let $\\chi $ be an interger and $\\beta $ be a class in $H_{2}(X,\\mathbb {Z})$ , there exists a projective scheme $\\mathcal {P}_{\\chi }\\left(X,\\beta \\right)$ parametrizing pairs $\\left(\\mathcal {\\mathcal {F}},s\\right)$ satisfying the above conditions with scheme theoretic support of $\\mathcal {\\mathcal {F}}$ is of class $\\beta $ and holomorphic Euler characteristic $\\chi (\\mathcal {\\mathcal {F}})$ equals to $\\chi $[18].",
"We wiil use $\\mathcal {P}$ to denote $\\mathcal {P}_{\\chi }\\left(X,\\beta \\right)$ whenever the context is clear.",
"There exist a universal sheaf $\\mathbb {F}$ and universal section $\\mathbb {S}:\\mathcal {O}_{\\mathcal {P}\\times X}\\rightarrow \\mathbb {F}$ on the product space $X\\times \\mathcal {P}.$ We denote by $p$ and $q$ the projection from $\\mathcal {P}_{\\chi }\\left(X,\\beta \\right)\\times X$ to the factor $\\mathcal {P}_{\\chi }\\left(X,\\beta \\right)$ and $X$ respectively.",
"Note that in general $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ is singular.",
"If $G$ acts on $X$ there is a natural $G$ action on $\\mathcal {P}$ .",
"Moreover, if $G$ acts diagonally on $\\mathcal {P}\\times X$ i.e.",
"$\\sigma _{\\mathcal {P}\\times X}:G\\times \\mathcal {P}\\times X\\rightarrow \\mathcal {P}\\times X$ , $(g,p,x)\\mapsto (g.p,g.x)$ , then the universal sheaf $\\mathbb {F}$ is an equivariant sheaf and $\\mathbb {S}:\\mathcal {O}_{\\mathcal {P}\\times X}\\rightarrow \\mathbb {F}$ is an equivariant morphism of sheaves.",
"Given a map $\\phi :E^{\\bullet }\\rightarrow \\mathbb {L}_{Y}$ , where $E^{\\bullet }=\\lbrace E^{-1}\\rightarrow E^{0}\\rbrace $ is a two term complex of vector bundles and $\\mathbb {L}_{Y}$ is the cotangent complex of $Y$ , such that the induced map $\\phi ^{0}$ , $\\phi ^{-1}$ is isomorphism and epimorphism respectively, Behrend and Fantechi construct a class $[Y]^{vir}\\in A_{\\text{rk}E^{0}-\\text{rk}E^{-1}}(Y)$ called virtual fundamental class[1].",
"The map $\\phi $ is called a perfect obstruction theory of $Y$ and $vd:=\\text{rk}E^{0}-\\text{rk}E^{-1}$ is the virtual dimension of $Y$ .",
"The perfect obstruction theory gives a virtual class even though the space $Y$ is badly singular.",
"Let $\\mathbb {I}^{\\bullet }=\\lbrace \\mathcal {O}_{\\mathcal {P}\\times X}\\rightarrow \\mathbb {F}\\rbrace $ be the universal complex on $\\mathcal {P}\\times X$ .",
"Pandharipande and Thomas have shown that $Rp_{*}\\left(R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet },\\mathbb {I}^{\\bullet }\\right)_{0}\\otimes \\omega _{X}\\right)[2]$ is a two term complex of locally free sheaves and there exist a map $\\phi :Rp_{*}\\left(R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet },\\mathbb {I}^{\\bullet }\\right)_{0}\\otimes \\omega _{X}\\right)[2]\\rightarrow \\mathbb {L}_{\\mathcal {P}}$ satisfying the above conditions.",
"We will use $\\mathbb {E}^{\\bullet }=\\lbrace E^{-1}\\rightarrow E^{0}\\rbrace $ to denote the complex $Rp_{*}\\left(R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet },\\mathbb {I}^{\\bullet }\\right)_{0}\\otimes \\omega _{p}\\right)[2]$ on $\\mathcal {P}$ .",
"The virtual dimension of $\\mathcal {P}_{\\chi }\\left(X,\\beta \\right)$ is then $-\\chi (R\\mathcal {H}om\\left(I^{\\bullet },I^{\\bullet }\\right)_{0})=\\int _{\\beta }c_{1}(X)$ .",
"If $X$ is Calabi-Yau the dualizing sheaf $\\omega _{X}\\simeq \\mathcal {O}_{X}$ so that by Serre duality $vd=\\text{rk}E^{0}-\\text{rk}E^{-1}=0$ .",
"If $vd=0$ then $P_{X,\\beta ,\\chi }:=\\int _{\\left[\\mathcal {P}\\right]^{vir}}1\\in \\mathbb {Z}$ and is invariant along a deformation of $X$ .",
"$P_{X,\\beta ,\\chi }$ is called Pandharipande-Thomas invariant or PT-invariant.",
"One technique to compute PT-invariants is using the virtual localization formula by Graber and Pandharipande.",
"If $G=\\mathbb {C}^{\\times }$ acts on $\\mathcal {P}_{\\chi }\\left(X,\\beta \\right)$ then $\\mathbb {L}_{\\mathcal {P}_{\\chi }\\left(X,\\beta \\right)}$ has a natural equivariant structure.",
"Let $\\mathcal {P}^{G}$ be the fixed locus of $\\mathcal {P}$ , then $\\mathbb {E}^{\\bullet }$ has a sub-bundle $\\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}^{G}}\\right)^{fix}$ which has weight 0 and a sub-bundle $\\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}^{G}}\\right)^{mov}$ with non zero weight such that $\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}^{G}}=\\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}^{G}}\\right)^{fix}\\oplus \\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}^{G}}\\right)^{mov}$ .",
"Graber and Pandharipande showed that there exists a canonical morphism $\\hat{\\phi }:\\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}^{G}}\\right)^{fix}\\rightarrow \\mathbb {L}_{\\mathcal {P}^{G}}$ that induces a perfect obstruction theory for $\\mathcal {P}^{G}$ .",
"So that we have the virtual fundamental class $\\left[\\text{$\\mathcal {P}$}^{G}\\right]^{vir}$ of $\\mathcal {P}^{G}$ .",
"Graber and Pandaripandhe gives a formula that relates $[\\mathcal {P}^{G}]^{vir}$ with $\\left[\\mathcal {P}\\right]^{vir}$ as follows : $\\left[\\mathcal {P}\\right]^{vir}=i_{*}\\left(\\frac{[\\mathcal {P}^{G}]^{vir}}{e(N^{vir})}\\right)\\in A_{*}^{G}\\otimes _{\\mathbb {Z}}\\mathbb {Q}[t,t^{-1}]$ where $e\\left(N^{vir}\\right)$ is the top Chern class of the vector bundle $N^{vir}=\\left(\\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}^{G}}\\right)^{mov}\\right)^{\\vee }$ and $t$ is the first Chern class of the equivariant line bundle with weight 1.",
"Let $S$ be a nonsingular projective surface with canonical bundle $\\omega _{S}$ and let $X$ be the total space of $\\omega _{S}$ i.e.",
"$X=Spec\\left(\\text{Sym}^ {}(\\omega _{S}^{\\vee })\\right)$ .",
"Then there is a closed embedding $i$ of $S$ into $X$ as the zero section.",
"Let $\\pi :X\\rightarrow S$ be the structure morphism.",
"Since $\\omega _{X}\\simeq \\pi ^{*}\\omega _{S}\\otimes \\pi ^{*}\\omega _{S}^{\\vee }\\simeq \\mathcal {O}_{X}$ , $X$ is Calabi-Yau.",
"Let $\\bar{X}=\\mathbb {P}(X\\oplus \\mathbb {A}_{S}^{1})$ , then $X$ is an open subscheme of $\\bar{X}$ and let $j:X\\rightarrow \\bar{X}$ be the inclusion and $\\bar{\\pi }:\\bar{X}\\rightarrow S$ be the structure morphism of $\\bar{X}$ as a projective bundle over $S$ .",
"Since $S$ is projective, $\\bar{i}:=j\\circ i:S\\rightarrow \\bar{X}$ is a closed embedding.",
"Let $\\beta \\in H_{2}(S,\\mathbb {Z})$ be an effective class and $\\chi \\in \\mathbb {Z}$ .",
"By [16] there is a projective scheme $\\mathcal {P}_{\\chi }\\left(\\bar{X},\\bar{i}_{*}\\beta \\right)$ parametrizing stable pairs $(\\mathcal {\\mathcal {F}},s)$ with $\\chi (\\mathcal {\\mathcal {F}})=\\chi $ and the cycle $[C_{\\mathcal {\\mathcal {F}}}]$ of the supporting curve is in class $\\beta $ .",
"By removing the pairs $(\\mathcal {\\mathcal {F}},s)$ with supporting curve $C_{\\mathcal {\\mathcal {F}}}$ which intersect the closed subschem $\\bar{X}\\setminus X$ , we have an open subscheme $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ that parametrize stable pairs $(\\mathcal {\\mathcal {F}},s)$ with $\\mathcal {\\mathcal {F}}$ supported on $X$ and let $\\hat{j}:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\rightarrow \\mathcal {P}_{\\chi }\\left(\\bar{X},\\bar{i}_{*}\\beta \\right)$ be the inclusion.",
"Let $\\bar{\\mathbb {F}}$ be the universal sheaf on $\\mathcal {P}_{\\chi }\\left(\\bar{X},\\bar{i}_{*}\\beta \\right)\\times \\bar{X}$ and $\\bar{\\mathbb {S}}:\\mathcal {O}_{\\mathcal {P}_{\\chi }\\left(\\bar{X},\\bar{i}_{*}\\beta \\right)\\times \\bar{X}}\\rightarrow \\bar{\\mathbb {F}}$ be the universal section, then their restriction $\\mathbb {F}$ , $\\mathbb {S}$ to $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X$ is the universal sheaf and the universal section corresponding to the moduli space $\\mathcal {P}_{\\chi }(X,i_{*}\\beta ).$ Notice that $\\left(\\text{id}_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta )}\\times j\\right)_{*}\\mathbb {F}=\\left(\\hat{j}\\times \\text{id}_{\\bar{X}}\\right)^{*}\\bar{\\mathbb {F}}$ on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}$ .",
"We also use $\\mathbb {F}$ to denote $\\left(\\text{id}_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta )}\\times j\\right)_{*}\\mathbb {F}$ on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}$ .",
"There exists an action of $G=\\mathbb {C}^{\\times }$ on $\\bar{X}$ by scaling the fiber such that $X$ is an invariant open subscheme.",
"It follows that there exist a canonical action of $G$ on $\\mathcal {P}_{\\chi }\\left(\\bar{X},\\bar{i}_{*}\\beta \\right)$ and on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ .",
"Since $X$ is an invariant open subscheme, $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ is also invariant in $\\mathcal {P}_{\\chi }\\left(\\bar{X},\\bar{i}_{*}\\beta \\right)$ .",
"Thus $\\bar{\\mathbb {F}}$ and $\\mathbb {F}$ are equivariant sheaves and $\\bar{\\mathbb {S}}$ and $\\mathbb {S}$ are equivariant morphism of sheaves.",
"Consider the following diagrams $\\begin{tikzcd}[column sep=tiny]&\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X[ld,\"p\"^{\\prime }][rd,\"q\"]&\\\\\\mathcal {P}_{\\chi }(X,i_{*}\\beta )&&X.\\end{tikzcd}$ Let $\\mathbb {I}^{\\bullet }$ be the complex $[{\\mathcal {O}_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X}[r]^{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\mathbb {S}} & \\mathbb {F}]}$ in $D(\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X)$ .",
"Let $\\mathbb {E}^{\\bullet }$ be the complex $Rp_{*}\\left(R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet },\\mathbb {I}^{\\bullet }\\right)_{0}\\otimes \\omega _{X}\\right)[2].$ Maulik, Pandharipande and Thomas has shown that the above complex define a perfect obstruction theory on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ [14].",
"Notice that $\\omega _{X}\\simeq \\mathcal {O}_{X}\\otimes \\mathfrak {t}^{*}$ so that by Serre's duality we have an isomorphism $\\left(\\mathbb {E}^{\\bullet }\\right)^{\\vee }\\rightarrow \\mathbb {E}^{\\bullet }[-1]\\otimes \\mathfrak {t}$ and $\\mathbb {E}$ is a symmetric equivariant obstruction theory.",
"Let $\\mathcal {P}_{\\chi }(S,\\beta )$ be the scheme parameterizing stable pairs $\\left(\\mathcal {\\mathcal {F}},s\\right)$ on $S$ such that the support $C_{\\mathcal {\\mathcal {F}}}$ of $\\mathcal {\\mathcal {F}}$ is in class $\\beta $ and $\\mathcal {\\mathcal {F}}$ has Euler characteristic $\\chi (\\mathcal {\\mathcal {F}})=\\chi $ .",
"On $\\mathcal {P}_{\\chi }(S,\\beta )\\times S$ there exists a universal sheaves $\\mathbb {F}$ and universal section $\\mathbb {S}$ .",
"With the closed embedding $\\hat{i}:=\\text{id}_{\\mathcal {P}_{\\chi }(S,\\beta )}\\times i$$:\\mathcal {P}_{\\chi }(S,\\beta )\\times S\\rightarrow \\mathcal {P}_{\\chi }(S,\\beta )\\times X$ , ${\\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )\\times X}[r] & \\,}{\\hat{i}_{*}\\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )\\times S}[r]^{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\hat{i}_{*}\\mathbb {S}} & \\hat{i}_{*}\\mathbb {F}}$ is a family of pairs over $\\mathcal {P}_{\\chi }(S,\\beta )$ .",
"This family induces a closed embedding $\\mathcal {P}_{\\chi }(S,\\beta )\\rightarrow \\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ .",
"Indeed, $\\mathcal {P}_{\\chi }(S,\\beta )$ is a connected component of $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{G}$ .",
"Let $\\mathbb {I}^{\\bullet }_{S}$ denote the complex $[\\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )\\times S}\\rightarrow \\mathbb {F}]$ and $\\mathbb {I}^{\\bullet }$ denotes the complex $[\\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )\\times X}\\rightarrow \\hat{i}_{*}\\mathbb {F}]$ .",
"Proposition 3.4 of [12] gives us the decomposition of $\\left.\\mathbb {E}\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}$ into its fixed and moving part as follows: $\\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}\\right)^{fix}\\simeq R\\hat{p}_{*}R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet }_{S},\\mathbb {F}\\right)^{\\vee }\\qquad \\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}\\right)^{mov}\\simeq R\\hat{p}_{*}R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet }_{S},\\mathbb {F}\\right)[1]\\otimes \\mathfrak {t}^{*}$ We will use $\\mathcal {E}^{\\bullet }$ to denote $\\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}\\right)^{fix}$ .",
"$\\left(\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}\\right)^{fix}$ defines a perfect obstruction theory on $\\mathcal {P}_{\\chi }(S,\\beta )$ with virtual dimension $v=\\beta ^{2}+n$ .",
"If there is a deformation of $S$ such that the class $\\beta $ is no longer algebraic, then the virtual fundamental class will be zero because the the virtual class is deformation invariant.",
"If we restrict the deformation inside the locus when $\\beta $ is always algebraic we get the reduced obstruction theory.",
"In [12], Kool and Thomas also construct a reduded obstruction theory on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ of virtual dimension $h^{0,2}(S)$ .",
"Proposition 3.4 of [12] gives us the decomposition of $\\left.\\mathbb {E}_{red}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}$ into fixed part and moving part as follows: $\\left(\\left(\\left.\\mathbb {E}_{red}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}\\right)^{fix}\\right)^{\\vee } & =\\text{Cone}\\left({R\\hat{p}_{*}R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet }_{S},\\mathbb {F}\\right)[r]^{\\psi \\,\\,\\,\\,\\,\\,\\,\\,} & H^{2}(\\mathcal {O}_{S})\\otimes \\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )}[-1]}\\right)\\\\\\left(\\left.\\mathbb {E}_{red}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}\\right)^{mov} & =R\\hat{p}_{*}R\\mathcal {H}om(\\mathbb {I}^{\\bullet }_{S},\\mathbb {F})[1]\\otimes \\mathfrak {t}$ where $\\psi $ is the composition ${R\\hat{p}_{*}R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet }_{S},\\mathbb {F}\\right)[r] & R\\hat{p}_{*}R\\mathcal {H}om(\\mathbb {F},\\mathbb {F})[1][r]^{\\text{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,tr}} & R\\hat{p}_{*}\\mathcal {O}[1][r] & R^{2}\\hat{p}_{*}\\mathcal {O}[-1]}$ We will use $\\mathcal {E}_{red}^{\\bullet }$ to denote $\\left(\\left.\\mathbb {E}_{red}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}\\right)^{fix}$ .",
"$\\mathcal {E}_{red}^{\\bullet }$ defines a perfect obstruction theory on $\\mathcal {P}_{\\chi }(S,\\beta )$ of virtual dimension $v_{red}=\\beta ^{2}+n+h^{0,2}(S)$ .",
"For a cohomology classes $\\sigma _{i}\\in H^{*}(X,\\mathbb {Z})$ , $i=1,\\dots ,m$ Kool and Thomas assign a class $\\tau (\\sigma _{i}):=p_{*}\\left(ch_{2}\\left(\\mathbb {F}\\right)q^{*}\\sigma \\right)\\in H^{*}\\left(\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\right)$ where $ch_{2}\\left(\\mathbb {F}\\right)$ is the second Chern character of $\\mathbb {F}$ and define the reduced invariants as $\\mathcal {P}_{\\beta ,\\chi }^{red}(X,\\sigma _{1},\\ldots ,\\sigma _{m}):=\\int _{[\\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{G}]^{vir}}\\frac{1}{e\\left(N^{vir}\\right)}\\prod _{i=1}^{m}\\tau (\\sigma _{i}).$ americanAssume that $b_{1}(S)=0$ so that $\\text{Hilb}_{\\beta }=|\\mathcal {L}|$ .",
"It was shown that if for all $i$ , $\\sigma _{i}$ is the pullback of the Poincaré dual of the $[pt]\\in H^{4}(S,\\mathbb {Z})$ represented by a closed point then $\\mathcal {P}_{\\beta ,\\chi }^{red}\\left(X,[pt]^{m}\\right)=\\int _{j^{!",
"}[\\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{G}]^{vir}}\\frac{1}{e\\left(N^{vir}\\right)}$ where $j^{!",
"}$ is the refined Gysin homomorphim corresponding to the following cartesian diagram ${\\mathbb {P}^{\\epsilon }\\times _{|\\mathcal {L}|}\\mathcal {P}_{\\chi }(X,i_{*}\\beta )[d][r] & \\mathcal {P}_{\\chi }(X,i_{*}\\beta )[d]^{\\text{div}}\\\\\\mathbb {P}^{\\epsilon }[r]^{j} & |\\mathcal {L}|}$ where $j$ is a regular embedding $\\mathbb {P}^{\\epsilon }\\subset |\\mathcal {L}|$ of a sublinear system and $\\epsilon =\\dim |\\mathcal {L}|-m$ .",
"american"
],
[
"$\\delta $ -nodal Curve Counting via Kool-Thomas invariants",
"englishRecall that a line bundle $\\mathcal {L}$ on a surface $S$ is $n$ -very ample if for any subscheme $Z$ with length $\\le n+1$ the natural morphsim $H^{0}(X,\\mathcal {L})\\rightarrow H^{0}(Z,\\left.\\mathcal {L}\\right|_{Z})$ is surjective.",
"americanWe assume that $b_{1}(S)=0$ and let $\\mathcal {L}$ be $\\left(2\\delta +1\\right)$ -very ample line bundle on $S$ with $H^{1}(\\mathcal {L})=0$ .",
"We also assume that the first Chern class $c_{1}(\\mathcal {L})=\\beta \\in H^{2}(S,\\mathbb {Z})$ of $\\mathcal {L}$ satisfies the condition that the the morphism $\\cup \\beta :H^{1}(T_{S})\\rightarrow H^{2}(\\mathcal {O}_{S})$ is surjective; in particular then $H^{2}(\\mathcal {L})=0$ also.",
"Given a curve $C$ not necessarily reduced and connected, we let $g(C)$ to denote its arithmetic genus, defined by $1-g(C):=\\chi (\\mathcal {O}_{C})$ .",
"If $C$ is reduced its geometric genus $\\bar{g}(C)$ is defined to be the $g(\\bar{C})$ the genus of its normalisation.",
"And let $h$ denote the arithmetic genus of curves in $\\left|\\mathcal {L}\\right|$ , so that $2h-2=\\beta ^{2}-c_{1}(S)\\beta $ .",
"englishProposition 2.1 of [11] and Proposition 5.1 of [12] tells us that the general $\\delta $ -dimensional linear system $\\mathbb {P}^{\\delta }\\subset |\\mathcal {L}|$ only contains reduced and irreducible curves.",
"Moreover $\\mathbb {P}^{\\delta }$ contains finitely many $\\delta $ -nodal curves with geometric genus $h-\\delta $ and other curves has geometric genus $>h-\\delta .$ Kool and Thomas also define $\\mathcal {P}_{\\chi ,\\beta }^{red}(S,[pt]^{m}):=\\int _{\\left[\\mathcal {P}_{\\chi }(S,\\beta )\\right]^{red}}\\frac{1}{e\\left(N^{vir}\\right)}\\tau \\left(\\left[pt\\right]\\right)^{m}.$ They compute $P_{\\chi ,\\beta }^{red}(S,\\left[pt\\right]^{m})$ in [13] and $P_{\\chi ,\\beta }^{red}(S,\\left[pt\\right]^{m})$ is given by the following expression $t^{n+\\chi (\\mathcal {L})-\\chi (\\mathcal {O}_{S})}\\left(-\\frac{1}{t}\\right)^{n+\\chi (\\mathcal {L})-1-m}\\int \\limits _{S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1-m}}c_{n}(\\mathcal {L}^{[n]}(1))\\frac{c_{\\bullet }(T_{S^{[n]}})c_{\\bullet }\\left(\\mathcal {O}(1)^{\\oplus \\chi (\\mathcal {L})}\\right)}{c_{\\bullet }\\left(\\mathcal {L}^{[n]}(1)\\right),}$ where $\\mathcal {L}^{[n]}$ is the vector bundle of rank $n$ on $S^{[n]}$ with fiber $H^{0}(\\mathcal {L}|_{Z})$ for a point $Z\\in S^{[n]}$ and $\\mathcal {L}^{[n]}(1)=\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)$ .",
"Under the above assumption, only the contribution from $\\mathcal {P}_{\\chi }(S,\\beta )$ counts for $\\mathcal {P}_{\\beta ,\\chi }^{red}\\left(X,[pt]^{m}\\right)$ so $\\mathcal {P}_{\\beta ,\\chi }^{red}\\left(X,[pt]^{m}\\right)=\\mathcal {P}_{\\chi ,\\beta }^{red}(S,[pt]^{m})$ .",
"Define the generating function for $\\mathcal {P}_{\\beta ,\\chi }^{red}(X,[pt]^{m})$ as $\\sum _{\\chi \\in \\mathbb {Z}}\\mathcal {P}_{\\beta ,\\chi }^{red}(X,[pt]^{m})q^{\\chi }$ then define $\\bar{q}=q^{1-i}(1+q)^{2i-2}$ then the coefficient of $\\bar{q}^{h-\\delta }$ is $n_{\\delta }(\\mathcal {L})t^{h-\\delta -1+\\int _{\\beta }c_{1}(S)}$ where $n_{\\delta }(\\mathcal {L})$ is the number of $\\delta $ -nodal curves in $\\mathbb {P}^{\\delta }.$ $n_{\\delta }(\\mathcal {L})$ has been studied for example in [7] and [11].",
"In [11], it is shown that after the same change of variable $n_{\\delta }(\\mathcal {L})$ can be computed as the coefficient of $\\bar{q}^{h-\\delta }$ of the generating function $\\sum _{i=0}^{\\infty }e(\\text{Hilb}^{n}(\\mathcal {C}/\\mathbb {P}^{\\delta }))q^{i+1-h}$ where $e(\\text{Hilb}^{i}(\\mathcal {C}/\\mathbb {P}^{\\delta })$ is the Euler characteristic of the relative Hilbert scheme of points.",
"Moreover $e(\\text{Hilb}^{n}(\\mathcal {C}/\\mathbb {P}^{\\delta }))$ can be computed as $\\int _{S^{[n]}\\times \\mathbb {P}^{\\delta }}c_{i}(\\mathcal {L}^{[n]}(1))\\frac{c_{\\bullet }\\left(T_{S^{[n]}}\\right)c_{\\bullet }\\left(\\mathcal {O}(1)^{\\oplus \\delta +1}\\right)}{c_{\\bullet }\\left(\\mathcal {L}^{[n]}(1)\\right)}.$ In [11], we have to assume that $\\mathcal {L}$ is sufficiently ample and $H^{i}(\\mathcal {L})=0$ for $i>0$ so that $\\text{Hilb}^{n}(\\mathcal {C}/\\mathbb {P}^{\\delta })$ are smooth.",
"While in [12], $\\mathcal {P}_{\\chi ,\\beta }^{red}(S,[pt]^{m})$ can be defined under the assumption that $H^{2}(\\mathcal {L})=0$ for all $\\mathcal {L}$ with $c_{1}(\\mathcal {L})=0$ .",
"We can think $n_{\\delta }(\\mathcal {L})$ as a generalization of the one studied in [11].",
"In particular, we can think $n_{\\delta }(\\mathcal {L})$ as a virtual count of $\\delta $ -nodal curves for not necessarily ample line bundle $\\mathcal {L}$ ."
],
[
"Equivariant $K$ -theoretic PT invariants of local surfaces",
"In this section we will recall the $K$ -theoretic invariants proposed by Neklrasof and Okounkov in [15] and introduce a class that will account for the incidence of the supporting curve of a stable pairs and a point.",
"The definition of this class is motivated by the definition of points insertions in [12].",
"Given a perfect obstruction theory $\\phi :E^{\\bullet }\\rightarrow \\mathbb {L}_{Y}$ the $K$ -theoretic version of the virtual class is given in [5] as follows: $\\mathcal {O}_{Y}^{vir}:=\\sum _{i}^{\\infty }(-1)^{i}[\\mathcal {T}or_{\\mathcal {O}_{E_{1}}}^{i}(\\mathcal {O}_{Y},\\mathcal {O}_{D})]_{Y}\\in G(Y)$ where $D$ is the coneamerican $D\\subset E_{1}$ that gives the virtual class $[Y]^{vir}.$ We call $\\mathcal {O}_{Y}^{vir}$ the virtual structure sheaf of $Y$ .",
"Note that $\\mathcal {O}_{Y}^{vir}$ is not a sheaf but a class in the Grothendieck group of coherent sheaves on $Y$ .",
"If $\\phi $ is an equivariant perfect deformation theory, $D$ is an invariant subscheme of $E_{1}$ and we can construct $\\mathcal {O}_{Y}^{vir}\\in G^{G}(Y)$ .",
"If $Y$ is proper over $\\mathbb {C}$ , the virtual fundamental class and virtual structure sheaf are related by the following virtual Riemann-Roch formula by Fantechi and Göttsche in [5] american $\\chi (\\mathcal {O}_{Y}^{vir})=\\int _{[Y]^{vir}}\\text{td}(T^{vir})$ where $T_{Y}^{vir}:=\\left[E_{0}\\right]-\\left[E_{1}\\right]\\in K(Y)$ .",
"We call $T_{Y}^{vir}$ the virtual tangent bundle and the dual of it's determinant $K_{Y,vir}:=\\left(\\det E_{0}\\right)^{-1}\\otimes \\det E_{1}$ $=\\det E^{0}\\otimes \\left(\\det E^{-1}\\right)^{-1}\\in \\text{Pic}(Y)$ the virtual canonical bundle.",
"If $vd=0$ , by equation (REF ) we have $\\chi (\\mathcal {O}_{Y}^{vir})=\\int _{[Y]^{vir}}1\\in \\mathbb {Z}$ so that we can use either virtual structure sheaf or virtual fundamental class to define a numerical invariant.",
"If there exist an isomorphism $\\theta :E^{\\bullet }\\rightarrow \\left(E^{\\bullet }\\right)^{\\vee }[1]$ then $\\text{rk}E^{\\bullet }=\\text{rk}\\left(\\left(E^{\\bullet }\\right)^{\\vee }[1]\\right)=-\\text{rk}E^{\\bullet }$ so that $vd=0$ .",
"One advantage of working equivariantly is that to compute $\\chi \\left(\\hat{\\mathcal {O}}_{Y}^{vir}\\right)$ , we can use the virtual localization formula for the Grothendieck group of coherent sheaves from [19] by Qu .",
"Let $G=\\mathbb {C}^{\\times }$ act on $Y$ and $\\phi :\\mathbb {E}^{\\bullet }\\rightarrow \\mathbb {L}_{Y}$ be an equivariant perfect obstruction theory.",
"Similar to the virtual localization formula by Graber and Pandaripandhe, it states that, the virtual structure sheaf equals a class coming from the fixed locus.",
"On $Y^{G}$ we can decompose $\\mathbb {E}^{\\bullet }$ into $\\left(\\mathbb {E}^{\\bullet }\\right)^{fix}\\oplus \\left(\\mathbb {E}^{\\bullet }\\right)^{mov}$ where $\\left(\\mathbb {E}^{\\bullet }\\right)^{fix}$ is a two term complex with zero weight and $\\left(\\mathbb {E}^{\\bullet }\\right)^{mov}$ is a two term complex with non zero weight.",
"Let $i:Y^{G}\\rightarrow Y$ be the closed embedding and let $N^{vir}=((\\mathbb {E}^{\\bullet })^{mov})^{\\vee }.$ Then the virtual localization formula can be stated as $i_{*}\\left(\\frac{\\mathcal {O}_{Y^{G}}^{vir}}{\\bigwedge ^{\\bullet }\\left(N^{vir}\\right)^{\\vee }}\\right)=\\mathcal {O}_{Y}^{vir}\\qquad \\in G^{G}(Y)\\otimes _{\\mathbb {Z}[\\mathfrak {t},\\mathfrak {t}^{-1}]}\\mathbb {Q}(\\mathfrak {t})$ where for a two term complex $F^{\\bullet }=[F^{-1}\\rightarrow F^{0}]$ , $\\bigwedge ^{\\bullet }F^{\\bullet }=\\frac{\\sum _{i=0}^{r_{0}}(-1)^{i}\\bigwedge ^{i}F^{0}}{\\sum _{j=0}^{r_{1}}(-1)^{j}\\bigwedge ^{j}F^{-1}}$ with $r_{i}=\\text{rk}F^{-i}$ .",
"On the fixed locus, the Grothendieck group of coherent sheaves is isomorphic to the tensor product $G(Y^{G})\\otimes _{\\mathbb {Z}}K^{G}(pt)$ which is easier to work with.",
"In [15], Nekrasov and Okounkov propose that we should choose a square root of $K^{vir}$ and work with the twisted virtual structure sheaf [20] $\\hat{\\mathcal {O}}_{Y}^{vir}:=K_{Y,vir}^{\\frac{1}{2}}\\otimes \\mathcal {O}_{Y}^{vir}.$ To get a refinement of (REF ), we have to consider the action of the symmetry group of $Y$ so that $\\chi \\left(\\hat{\\mathcal {O}}_{Y}^{vir}\\right)$ is a function with the equivariant parameter as variables.",
"For example let $Y$ be the moduli space of stable pairs on a toric 3-folds $X$ and $\\left(\\mathbb {C}^{\\times }\\right)^{3}$ acts on $Y$ .",
"Choi, Katz and Klemm have calculated $\\chi (\\hat{\\mathcal {O}}_{Y}^{vir})$ where $X$ is the total space of the canonical bundle $K_{S}$ for $S=\\mathbb {P}^{2}$ and $S=\\mathbb {P}^{1}\\times \\mathbb {P}^{1}$ in [2].",
"They have shown that the generating function with coefficients $\\chi (\\hat{\\mathcal {O}}_{Y}^{vir})$ calculates a refinement of BPS invariants.",
"To incorporate $K_{Y,vir}^{\\frac{1}{2}}$ in our computation we will consider a double cover $G^{\\prime }$ of $G$ so that $\\mathfrak {t}^{\\frac{1}{2}}$ is a representation of $G^{\\prime }$ .",
"Explicitly let $\\zeta :G^{\\prime }:=\\mathbb {C}^{\\times }\\rightarrow \\mathbb {C}^{\\times }=G$ , $z\\mapsto z^{2}$ be the double cover.",
"Then $G^{\\prime }$ acts on $Y$ via $\\zeta $ by defining $\\sigma ^{\\prime }_{Y}:G^{\\prime }\\times Y\\rightarrow Y,$$(g^{\\prime },y)\\mapsto \\sigma _{Y}(\\zeta (g^{\\prime }),y)$ where $\\sigma :G\\times Y\\rightarrow Y$ is the morphism defining the action of $G$ on $Y$ .",
"Also via $\\zeta $ any $G$ -equivariant sheaf $\\mathcal {\\mathcal {F}}$ on $Y$ is a $G^{\\prime }$ -equivariant sheaf by pulling back the equivariant structure via $\\zeta $ .",
"This gives an exact functor $Coh^{G}(Y)\\rightarrow Coh^{G^{\\prime }}(Y)$ and a group homomorphism $\\hat{\\zeta }:G^{G}(Y)\\rightarrow G^{G^{\\prime }}(Y)$ .",
"Moreover $\\hat{\\zeta }$ is a morphism of $K^{G}(\\text{pt})$ -modules.",
"For example, the primitive representation $\\mathfrak {t}$ of $G$ has weight 2 at $G^{\\prime }$ module.",
"We can take the primitive representation of $G^{\\prime }$ as the canonical square root of $\\mathfrak {t}$ and denote it by $\\mathfrak {t}^{\\frac{1}{2}}$ .",
"Next we have to compute the restriction of $K_{Y,vir}^{\\frac{1}{2}}$ on the fixed locus.",
"Notice that $Y^{G^{\\prime }}=Y^{G}$ .",
"Assume that there exist an isomorphism $\\theta :E^{\\bullet }\\rightarrow \\left(E^{\\bullet }\\right)^{\\vee }[1]\\otimes \\mathfrak {t}$ .",
"By the argument of Richard Thomas in [22], it shows that on $Y^{G}$ , $K_{Y,vir}^{\\frac{1}{2}}$ has a canonical equivariant structure.",
"We decompose $\\left.E^{\\bullet }\\right|_{Y^{G}}$ into its weight spaces so that $\\left.E^{\\bullet }\\right|_{Y^{\\mathbb {C}^{\\times }}}=\\bigoplus _{i\\in \\mathbb {Z}}F^{i}\\mathfrak {t}^{i}$ where $F^{i}$ are two-term complex of non-equivariant vector bundle which only finitely many of them are nonzero and $\\mathfrak {t}$ is a representation of $G$ of weight 1.",
"$\\det E^{\\bullet }$ can be computed as the determinant of its class in $K^{G}(Y).$ The isomorphism $\\theta $ implies that $[(F^{i})^{\\vee }]=[F^{-i-1}[-1]]$ in $K^{G}(Y)$ .",
"Thus $K_{Y,vir}$ is a squre twisted by a power of $\\mathfrak {t}$ , explicitly $K_{Y,vir}=\\left(\\bigotimes _{i\\ge 0}\\det \\left(F^{i}\\mathfrak {t}^{i}\\right)\\right)^{\\otimes 2}\\mathfrak {t}^{r_{0}+r_{1}+\\ldots }$ where $r_{i}=\\text{rk}F^{i}$ .",
"Thus the canonical choice for $\\left.K_{Y,vir}^{\\frac{1}{2}}\\right|_{Y^{G}}$ is $\\bigotimes _{i\\ge 0}\\det \\left(F^{i}\\mathfrak {t}^{i}\\right)\\otimes \\mathfrak {t}^{\\frac{1}{2}(r_{0}+r_{1}+\\ldots )}\\in K^{G}(Y^{G})\\otimes _{\\mathbb {Z}[\\mathfrak {t},\\mathfrak {t}^{-1}]}\\mathbb {Z}[\\mathfrak {t}^{\\frac{1}{2}},\\mathfrak {t}^{-\\frac{1}{2}}].$ Recall that $N^{vir}$ is the moving part of the dual of $\\left.E^{\\bullet }\\right|_{Y^{G}}$ so that in our case $\\left(N^{vir}\\right)^{\\vee }=\\bigoplus _{i\\ne 0}F^{i}\\mathfrak {t}^{i}$ .",
"After choosing a square root of $K_{Y,vir}$ , and that the square root has an equivariant structure, by equation (REF ) we then have $i_{*}\\left(\\frac{\\mathcal {O}_{Y^{G}}^{vir}\\otimes \\left.K_{Y,vir}^{\\frac{1}{2}}\\right|_{Y^{G}}}{\\bigwedge ^{\\bullet }\\left(N^{vir}\\right)^{\\vee }}\\right)=\\hat{\\mathcal {O}}_{Y}^{vir}\\qquad \\in K^{G}(Y)\\otimes _{\\mathbb {Z}[\\mathfrak {t},\\mathfrak {t}^{-1}]}\\mathbb {Q}(\\mathfrak {t}^{\\frac{1}{2}})$ If $Y$ is compact we can apply the right derived functor $R\\Gamma $ to both sides of the above equation and we have $R\\Gamma \\left(Y^{G},\\frac{\\mathcal {O}_{Y^{G}}^{vir}\\otimes \\left.K_{Y,vir}^{\\frac{1}{2}}\\right|_{Y^{G}}}{\\bigwedge ^{\\bullet }\\left(N^{vir}\\right)^{\\vee }}\\right)=R\\Gamma \\left(Y,\\hat{\\mathcal {O}}_{Y}^{vir}\\right)\\in \\mathbb {Q}(\\mathfrak {t}^{\\frac{1}{2}}).$ Thomas has proved the above identity in without using equation (REF ).",
"Furthermore Thomas has shown that $\\left.R\\Gamma \\left(Y^{G},\\frac{\\mathcal {O}_{Y^{G}}^{vir}\\otimes \\left.K_{Y,vir}^{\\frac{1}{2}}\\right|_{Y^{G}}}{\\bigwedge ^{\\bullet }\\left(N^{vir}\\right)^{\\vee }}\\right)\\right|_{\\mathfrak {t}=1}=\\int _{[Y]^{vir}}\\frac{1}{e\\left(N^{vir}\\right)}\\in \\mathbb {Q}$ In the case that we are interested on, the moduli space $Y$ is not compact.",
"Thus we will use the left hand side of equation (REF ) to define our invariants."
],
[
"Equivarinat $K$ -theoretic invariants",
"Let $Y$ be the moduli space of stable pairs on the canonical bundle $X:=Spec\\left(\\text{$\\text{Sym}$}\\,\\omega _{S}^{\\vee }\\right)$ of a smooth projective surface i.e.",
"$Y=\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ for some $\\chi \\in \\mathbb {Z}$ and $\\beta \\in H_{2}(S,\\mathbb {Z})$ where $i:S\\rightarrow X$ is the zero section.",
"We will use $\\pi $ to denote the structure map $X\\rightarrow S$ of $X$ as a vector bundle over $S$ .",
"Note that $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ is a quasiprojective scheme over $\\mathbb {C}$ .",
"In particular, $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ is separated and of finite type.",
"Let $G=\\mathbb {C}^{\\times }$ act on $X$ by scaling the fiber of $\\pi $ .",
"Consider the following diagram: $\\begin{tikzcd}[column sep=small]& \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X[ld,\"p\"^{\\prime }][rd,\"q\"]&\\\\\\mathcal {P}_{\\chi }(X,i_{*}\\beta )&& X\\end{tikzcd}$ Since $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ has an equivariant perfect obstructrion theory $\\phi :\\mathbb {E}^{\\bullet }\\rightarrow \\mathbb {L}_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta )}$ where $\\mathbb {E}^{\\bullet }$ is the complex $Rp_{*}\\left(R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet },\\mathbb {I}^{\\bullet }\\right)_{0}\\otimes \\omega _{p}\\right)[2]$ with $\\omega _{P}=q^{*}\\omega _{X}$ and since $X$ is Calabi-Yau $\\omega _{X}\\simeq \\mathcal {O}\\otimes \\mathfrak {t}^{*}$ Serre duality gives us the isomorphism $\\left(\\mathbb {E}^{\\bullet }\\right)^{\\vee }\\simeq \\mathbb {E}^{\\bullet }[-1]\\otimes \\mathfrak {t}.$ So that by Proposition 2.6 of [22] we have an equivariant line bundle $\\left.K_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta ),vir}^{\\frac{1}{2}}\\right|_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{G}}$ on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{G}$ .",
"We want to study how to define a class that contains the information about the incidence between a $K$ -theory class in $K^{T}(X)$ and the class of the universal sheaf $\\mathbb {F}$ .",
"From another direction we also want to give a refinement for the Kool-Thomas invariants.",
"In [12], Kool and Thomas take the cup product of the second Chern character of the universal sheaf $\\mathbb {F}$ with the cohomology class of points coming from $X$ .",
"Informally we could think that as taking the intersection between the universal supporting curve and the points of $X$ .",
"In this article we are exploring two approaches.",
"In the first approach we are trying to immitate the definition of descendent used in the article [12].",
"In [12] the authors are cupping the cohomology class coming from $X$ with the second Chern class of $\\mathbb {F}$ .",
"Since we are unfamiliar on how to define Chern classes as a $K$ -theory class, we are considering to take the class of the structure sheaf of the supporting scheme $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ and take the tensor product of $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ with the the class coming from $X$ through the projection $q:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X\\rightarrow X$ .",
"In the second approach we use the $K$ -theory class on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S$ of the structure sheaf of the divisor $\\text{div}\\,\\pi _{*}\\mathbb {F}$ and take the tensor product of $\\mathcal {O}_{\\text{div}\\pi _{*}\\mathbb {F}}$ with the class coming from $S$ through the projection $q_{S}:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S\\rightarrow S$ .",
"The following proposition is an equivariant version of Proposition 2.1.0 in [10] which we will use to define the $K$ -theory class.",
"Proposition 11 Let $f:Y\\rightarrow T$ be a smooth projective $G$ -map of relative dimension $n$ with $G$ -equivariant $f$ -very ample line bundle $\\mathcal {O}_{Y}(1)$ .",
"Let $\\mathcal {\\mathcal {F}}$ be a $G$ -equivariant sheaf flat over $T$ .",
"Then there is a resolution of $\\mathcal {\\mathcal {F}}$ by a bounded complex of $G$ -equivariant locally free sheaves : ${0[r] & \\mathcal {\\mathcal {F}}_{n}[r] & \\mathcal {\\mathcal {F}}_{n-1}[r] & \\ldots [r] & \\mathcal {\\mathcal {F}}_{0}[r] & \\mathcal {\\mathcal {F}}}$ where all morphisms are $G$ -equivariant such that $R^{n}f_{*}F_{\\nu }$ is locally free for $\\nu =0,\\ldots ,n$ and $R^{i}f_{*}F_{\\nu }=0$ for $i\\ne n$ and $\\nu =0,\\ldots ,n$ .",
"The equivariant structure of all sheaves constructed in the proof of Proposition 2.1.10 in [10] can be defined canonically.",
"If $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ is flat over $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ then $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ define a $K$ -theory class in $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X$ .",
"To push the tensor product down to a $K$ -theory class in $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ , we push forward $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ to $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}$ where $\\bar{X}$ is $\\mathbb {P}(K_{S}\\oplus \\mathcal {O}_{S})$ the projective completion of $X$ .",
"Since $\\mathcal {C}_{\\mathbb {F}}$ is proper relative to $\\mathcal {P}_{\\chi }(S,\\beta )$ the push forward $i_{*}\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ by the open embedding $i:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X\\rightarrow \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}$ is a coherent sheaf on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}$ .",
"Then Proposition REF implies that $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ has a resolution by a finite complex of locally free sheaf $F^{\\bullet }$ on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}$ so that we can take $[\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}]:=\\sum _{i}(-1)^{i}[F^{i}]$ .",
"The class $[\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}]$ is independent of the resolution.",
"In section we have described the ring homomorphism $f^{*}:K^{G}(\\bar{Y})\\rightarrow K^{G}(Y)$ for any morphism of sheaves $f:Y\\rightarrow \\bar{Y}$ .",
"We also described the group homomorphism $f_{*}:K^{G}(Y)\\rightarrow K^{G}(\\bar{Y})$ when $f$ is the structure morphism of a projective bundle or when $f$ is finite and $f_{*}\\mathcal {\\mathcal {F}}$ has a resolution by locally free sheaves.",
"Consider the following diagram $\\begin{tikzcd}[column sep=small]& \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}[ld,\"\\bar{p}\"^{\\prime }][rd,\"\\bar{q}\"]&\\\\\\mathcal {P}_{\\chi }(X,i_{*}\\beta )&& \\bar{X}\\end{tikzcd}$ Let $\\bar{\\pi }:\\bar{X}\\rightarrow S$ be the structure morphism of $\\bar{X}$ as a projective bundle over $S$ .",
"We assign for each class $\\alpha \\in K^{T}(X)$ a class $\\gamma \\left(\\alpha \\right)$ in $K^{T}(\\mathcal {P}_{\\chi }(X,i_{*}\\beta ))$ as follows.",
"The pullback map $\\pi ^{*}:K^{T}(S)\\rightarrow K^{T}(X)$ is an isomorphism.",
"Thus there exist a unique class $\\hat{\\alpha }\\in K^{T}(S)$ such that $\\pi ^{*}\\hat{\\alpha }=\\alpha $ .",
"We define $\\gamma \\left(\\alpha \\right):=\\bar{p}_{*}\\left(\\left[\\mathcal {O}_{\\mathcal {C}_{\\bar{\\mathbb {F}}}}\\right].\\left[\\bar{q}^{*}\\circ \\bar{\\pi }^{*}\\hat{\\alpha }\\right]\\right)$ .",
"By Proposition REF , $\\left[\\mathcal {O}_{\\mathcal {C}_{\\bar{\\mathbb {F}}}}\\right]\\in K^{T}(\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X})$ and since $\\bar{X}$ is smooth and projective over $\\mathbb {C}$ , $\\bar{p}_{*}$ can be defined as the composition of $i_{*}$ and $r_{*}$ where $i$ is a regular embedding and $r$ is the structure morphism $\\mathbb {P}_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta )}^{N}$$\\rightarrow \\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ .",
"Thus the class $\\gamma (\\alpha )$ is well defined.",
"In particular for every subscheme $Z\\subset X$ , $\\gamma (\\mathcal {O}_{Z})$ is an element in $K^{T}(\\mathcal {P}_{\\chi }(X,i_{*}\\beta ))$ .",
"For the second approach, $\\text{div}\\,\\pi _{*}\\mathbb {F}$ is a Cartier divisor on $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S$ so that we have a line bundle $\\mathcal {O}(\\text{div}\\,\\pi _{*}\\mathbb {F})$ and exact sequence ${0[r] & \\mathcal {O}(-\\text{div}\\,\\pi _{*}\\mathbb {F})[r] & \\mathcal {O}[r] & \\mathcal {O}_{\\text{div}\\,\\pi _{*}\\mathbb {F}}[r] & 0}.$ Thus the $K$ -theory class of $\\mathcal {O}_{\\text{div}\\,\\pi _{*}\\mathbb {F}}$ is $1-[\\mathcal {O}(-\\text{div}\\,\\pi _{*}\\mathbb {F})]$ .",
"Consider the following diagram $\\begin{tikzcd}[column sep=small]&\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S[dl,\"\\hat{p}\"^{\\prime }][dr,\"q_S\"]&\\\\\\mathcal {P}_{\\chi }(X,i_{*}\\beta )&&S.\\\\\\end{tikzcd}$ Similar to the first approach we assign for each $\\alpha \\in K^{T}(X)$ the class $\\bar{\\gamma }(\\alpha ):=\\hat{p}_{*}\\left([\\mathcal {O}_{\\text{div}\\pi _{*}\\mathbb {F}}].q_{S}^{*}\\hat{\\alpha }\\right).$ In this article we only working for the case when $\\alpha $ is represented by the class of the pullback of a closed point $s\\in S$ .",
"Instead of $\\gamma \\left(\\pi ^{*}\\left[\\mathcal {O}_{s}\\right]\\right)$ we will use $\\gamma \\left(\\left[\\mathcal {O}_{s}\\right]\\right)$ to denote this class.",
"We also assume that $b_{1}(S)=0$ so that $\\text{Hilb}_{\\beta }$ is simply $|\\mathcal {L}|$ for a line bundle $\\mathcal {L}$ on $S$ with $c_{1}(\\mathcal {L})=\\beta $ .",
"In this article, we want to study the following invariants $R\\Gamma \\left(\\mathcal {P}^{G},\\frac{\\mathcal {O}_{\\mathcal {P}^{G}}^{vir}}{\\bigwedge ^{\\bullet }\\left(N^{vir}\\right)^{\\vee }}\\otimes K_{\\mathcal {P},vir}^{\\frac{1}{2}}\\vert _{\\mathcal {P}^{G}}\\otimes \\left.\\prod _{i=1}^{m}\\beta _{i}\\right|_{\\mathcal {P}^{G}}\\right)\\in \\mathbb {Q}\\left(\\mathfrak {t}^{\\frac{1}{2}}\\right)$ where $\\beta _{i}$ is either $\\gamma \\left(\\mathcal {O}_{s_{i}}\\right)$ or $\\bar{\\gamma }(\\mathcal {O}_{s_{i}})$ with $\\mathcal {O}_{s_{i}}$ are the classes of the structure sheaves of closed points $s_{i}\\in S$ .",
"In a special case that we have worked out in this article in order to make the invariant coincide with Kool-Thomas invariant when we evaluate it at $\\mathfrak {t}=1$ we have to replace $\\gamma (\\mathcal {O}_{s_{i}})$ by $\\frac{\\gamma (\\mathcal {O}_{s_{i}})}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}$ and $\\bar{\\gamma }(\\mathcal {O}_{s_{i}})$ with $\\frac{\\bar{\\gamma }(\\mathcal {O}_{s_{i}})}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}$ .",
"Thus we define the following invariants $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m}):=R\\Gamma \\left(\\mathcal {P}^{G},\\frac{\\mathcal {O}_{\\mathcal {P}^{G}}^{vir}}{\\bigwedge ^{\\bullet }\\left(N^{vir}\\right)^{\\vee }}\\otimes K_{\\mathcal {P},vir}^{\\frac{1}{2}}\\vert _{\\mathcal {P}^{G}}\\otimes \\left.\\prod _{i=1}^{m}\\frac{\\gamma (\\mathcal {O}_{s_{i}})}{\\mathfrak {t}^{-\\frac{1}{2}}-\\mathfrak {t}^{\\frac{1}{2}}}\\right|_{\\mathcal {P}^{G}}\\right)$ when $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ is flat and $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right):=R\\Gamma \\left(\\mathcal {P}^{G},\\frac{\\mathcal {O}_{\\mathcal {P}^{G}}^{vir}}{\\bigwedge ^{\\bullet }\\left(N^{vir}\\right)^{\\vee }}\\otimes K_{\\mathcal {P},vir}^{\\frac{1}{2}}\\vert _{\\mathcal {P}^{G}}\\otimes \\left.\\prod _{i=1}^{m}\\frac{\\bar{\\gamma }(\\mathcal {O}_{s_{i}})}{\\mathfrak {t}^{-\\frac{1}{2}}-\\mathfrak {t}^{\\frac{1}{2}}}\\right|_{\\mathcal {P}^{G}}\\right)$"
],
[
"Vanisihing of the contribution of pairs supported on thickening of\n$S$ in {{formula:aca9c345-5ad1-4f65-8a64-05bcdbc3aea5}}",
"In this subsection we will prove that under the assumption that all curve that pass through all the $m$ points are reduced and irreducible, the contribution to the invariants $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ and $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)$ of curves not supported on $S$ is zero.",
"Proposition 2.1 of [11] tells us that if $\\mathcal {L}$ is a $2\\delta +1$ -very ample line bundle on $S$ then the $\\delta $ -dimensional general sublinear system $\\mathbb {P}^{\\delta }\\subset |\\mathcal {L}|$ only contain reduced curves.",
"Proposition 5.1 of [12] also implies that these curves are also irreducible.",
"Thus our assumption that all curves passing through all $m$ points are reduced and irreducible is more likely to happen.",
"If for all $s_{i}$ , $\\mathcal {O}_{s_{i}}$ are in the same class, our assumption does not depend on a particular set of $s_{i}$ but only on the number of points.",
"First we work for $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ .",
"Let $\\bar{\\pi }^{\\mathcal {P}}:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}\\rightarrow \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S$ be the pullback of $\\bar{\\pi }$ and let $i:\\mathcal {C}\\rightarrow \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}$ be the closed embedding of the universal curve.",
"As the composition of projective morphisms is projective then the composition $\\bar{\\pi }^{\\mathcal {P}}\\circ i$ is also projective.",
"Notice the above composition equals to the composition $\\mathcal {C}\\rightarrow \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times X\\rightarrow \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S$ which is affine.",
"Thus we can conclude that $\\bar{\\pi }^{\\mathcal {P}}\\circ i$ is a finite morphism.",
"We denote this morphism by $\\rho $ .",
"Recall the morphism $\\text{div}:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\rightarrow |\\mathcal {L}|$ from section that maps the stable pairs $\\left(\\mathcal {\\mathcal {F}},s\\right)$ to the supporting curve $C_{\\mathcal {\\mathcal {F}}}\\in |\\mathcal {L}|$ of $\\mathcal {\\mathcal {F}}$ .",
"Let $\\mathcal {D}\\subset |\\mathcal {L}|\\times S$ be the universal divisor and let $\\mathcal {D}_{\\mathcal {P}}\\subset \\mathcal {P}\\times S$ be the family of divisors that correspond to the morphism $\\text{div}:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\rightarrow |\\mathcal {L}|$ and let $j:\\mathcal {D}_{\\mathcal {P}}\\rightarrow \\mathcal {P}\\times S$ be the closed embedding.",
"Equivalently $\\mathcal {D}_{\\mathcal {P}}=\\text{div}^{-1}\\mathcal {D}$ .",
"english Lemma 12 $\\rho $ factors through $j$ .",
"american The ideal $I$ in $\\mathcal {O}_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S}$ corresponding to the divisor $\\mathcal {D}_{\\mathcal {P}}$ is flat over $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ and $\\rho $ factorize through $j$ if the composition $I\\rightarrow \\mathcal {O}_{\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S}\\rightarrow \\rho _{*}\\mathcal {O}_{\\mathcal {C}}$ is zero.",
"By Nakayama's Lemma it is sufficient to check whether the composition is zero for each $p\\in \\mathcal {P}_{\\chi }(X,i_{*}\\beta ).$ Or equivalently, we can check whether $\\rho $ factorize through $j$ at each point $p\\in \\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ .",
"Let $\\rho ^{p}:\\mathcal {C}_{p}\\rightarrow \\lbrace p\\rbrace \\times S=S$ be the restriction of $\\rho $ to the point $p\\in \\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ and let $W\\subset S$ be the scheme theoretic support of $\\rho _{*}^{p}\\mathcal {O}_{\\mathcal {C}_{p}}$ .",
"Notice that $|W|=\\text{Supp}(\\rho _{*}\\mathcal {O}_{\\mathcal {C}_{p}})$ is a curve.",
"We claim that $W$ is a Cartier Divisor.",
"We will show that $W$ is a subscheme of $\\text{div}\\,\\mathcal {\\mathcal {F}}=\\text{div}\\,\\rho _{*}\\mathcal {O}_{\\mathcal {C}_{p}}$ so that $\\rho ^{p}$ factorize through $j^{p}$ .",
"Let $\\sigma :\\mathcal {O}_{S}\\rightarrow \\rho _{*}^{p}\\mathcal {O}_{\\mathcal {C}_{p}}$ be the morphism of sheaves corresponding to the morphism $\\rho ^{p}:\\mathcal {C}_{p}\\rightarrow S$ .",
"Then $\\mathcal {O}_{W}$ is the image of $\\sigma $ so that we have an injection $\\mathcal {O}_{W}\\rightarrow \\rho _{*}^{p}\\mathcal {O}_{\\mathcal {C}_{p}}\\rightarrow \\rho _{*}^{p}\\mathcal {\\mathcal {F}}_{p}.$ By Proposition 1 of [6] we have $\\text{div}\\,\\rho _{*}^{p}\\mathcal {\\mathcal {F}}_{p}=\\text{div}\\,\\mathcal {O}_{W}+D$ where $D$ is some effective divisor.",
"Since $W$ is a Cartier divisor then $\\text{div}\\,\\mathcal {O}_{W}=W$ .",
"So that we can conclude that $W$ is a subscheme of $\\text{div}\\,\\mathcal {\\mathcal {F}}$ .",
"It remains to show that $W$ is a Cartier divisor.",
"Let $I\\subset \\mathcal {O}_{S}$ be the ideal sheaf of $W$ .",
"It is sufficient to show that $I_{x}$ is a free $\\mathcal {O}_{S,x}$ -module of rank 1 for every $x\\in X$ .",
"For $U=S\\setminus W$ , the inclusion $I\\subset \\mathcal {O}_{S}$ is an isomorphism so that if $x\\notin W,$ $I_{x}$ is isomorphic to $\\mathcal {O}_{S,x}$ .",
"Since $S$ is nonsingular $\\mathcal {O}_{S,x}$ is a domain so that it is sufficient to show that $I_{x}$ is generated by one element $f\\in \\mathcal {O}_{S,x}$ .",
"Note that the morphism $\\rho :\\mathcal {C}_{p}\\rightarrow S$ is a finite morphism so that $\\left(\\rho _{*}^{p}\\mathcal {O}_{\\mathcal {C}_{p}}\\right)_{x}$ is a finitely generated $\\mathcal {O}_{S,x}$ -module.",
"In particular, $\\left(\\rho _{*}^{p}\\mathcal {O}_{\\mathcal {C}_{p}}\\right)_{x}$ is a Cohen-Macaulay $\\mathcal {O}_{S,x}$ -module.",
"By Proposition IV.13 of [21], any prime $\\textbf {\\mathfrak {p}}\\subset \\mathcal {O}_{S,x}$ such that $\\mathcal {O}_{S,x}/\\mathbf {\\mathfrak {p}}$ is isomorphic to a submodule of $\\left(\\rho _{*}^{p}\\mathcal {O}_{\\mathcal {C}_{p}}\\right)_{x}$ must be generated by a single irreducible element $g\\in \\mathcal {O}_{S,x}.$ There are finitely many of such $\\mathfrak {p}$ and we denote them by $\\mathfrak {p}_{1},\\ldots ,\\text{$\\mathfrak {p}_{k}$}$ .",
"Let $g_{i}$ generate $\\mathfrak {p}_{i}$ .",
"By Proposition IV.11 of [21], $I_{x}$ is the intersection $\\bigcap _{i=1}^{k}\\mathbf {\\mathfrak {q}}_{i}$ where $\\mathfrak {q}_{i}$ is an ideal of $\\mathcal {O}_{S,x}$ such that $\\mathfrak {p}_{i}^{n_{i}}\\subset \\mathfrak {\\mathfrak {q}}_{i}\\subset \\mathbf {\\mathfrak {p}}_{i}$ for some positive integer $n_{i}$ .",
"Since $\\mathcal {O}_{S,x}$ is a domain, $\\mathfrak {q}_{i}$ must be generated by a single element $g_{i}^{m_{i}}$ for some positive integer $m_{i}$ .",
"Thus we conclude that $I_{x}$ is generated by a single element $\\prod _{i=1}^{k}g_{i}^{m_{i}}$ .",
"Let $R\\subset \\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{G}$ be a connected component different from $\\mathcal {P}_{\\chi }(S,\\beta )$ .",
"We denote the inclusion $R\\subset \\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ by $\\iota $ .",
"For every $(F,s)\\in R$ the supporting curve $C\\subset X$ is not supported by $S$ but $F$ is supported on an infinitesimal thickening of $S$ in $X$ .",
"So we have the following diagram where all square are Cartesian $\\begin{tikzcd}\\mathcal {C}_{R}[d,\"i^R\"^{\\prime }][r] & \\mathcal {C}[d,\"i\"^{\\prime }]&\\\\R\\times \\bar{X}[r,\"\\iota _{\\bar{X}}\"][d,\"\\bar{\\pi }^R\"^{\\prime }] & \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times \\bar{X}[r,\"\\bar{q}\"][d,\"\\bar{\\pi }^{\\mathcal {P}}\"^{\\prime }] & \\bar{X}[d,\"\\bar{\\pi }\"]\\\\R\\times S[d,\"\\hat{p}^R\"^{\\prime }][r,\"\\iota _S\"] & \\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\times S[d,\"\\hat{p}\"^{\\prime }][r,\"q_S\"] & S\\\\R[r,\"\\iota \"]&\\mathcal {P}_{\\chi }(X,i_{*}\\beta )&.\\end{tikzcd}$ By base change formula REF and projection formula REF we have $\\iota ^{*}\\gamma \\left(\\mathcal {O}_{s}\\right)& =\\left(\\hat{p}^{R}\\circ \\bar{\\pi }^{R}\\right)_{*}\\left(\\iota _{\\bar{X}}^{*}[\\mathcal {O}_{\\mathcal {C}}].\\iota _{\\bar{X}}^{*}\\bar{q}^{*}\\bar{\\pi }^{*}\\left[\\mathcal {O}_{s}\\right]\\right)\\nonumber \\\\& =\\hat{p}_{x*}\\left(\\bar{\\pi }_{*}^{R}\\left[\\mathcal {O}_{\\mathcal {C}_{R}}\\right].\\iota _{S}^{*}q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right).\\nonumber \\\\$ Now we restrict $\\rho $ from REF to $R\\subset \\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ .",
"By the Lemma REF we can write $\\rho ^{R}$ as the composition $j^{R}\\circ \\lambda ^{R}$ .",
"So now we have the following diagram ${\\mathcal {C}_{R}[r]^{\\lambda ^{R}} & \\mathcal {D}_{R}[r]^{j^{R}} & R\\times S[d]_{\\hat{p}^{R}}[r]^{\\,\\,\\,\\,\\,\\,q_{S}\\circ \\iota _{S}} & S\\\\& & R}$ By Proposition REF the subcategory of flat coherent sheaves on $\\mathcal {D}_{R}$ satisfies all conditions in Lemma REF so that by Corollary REF we have a group homomorphism $\\lambda _{*}^{R}:K^{G}(\\mathcal {C}_{R})\\rightarrow K^{G}\\left(\\mathcal {D}_{R}\\right)$ that maps $\\left[\\mathcal {\\mathcal {F}}\\right]$ to $\\chi \\left(\\lambda _{*}^{R}\\mathcal {\\mathcal {F}}\\right)$ .",
"By the same argument we can conclude the existence of the group homomorphism $j_{*}^{R}:K^{G}\\left(\\mathcal {D}_{R}\\right)\\rightarrow K^{G}(R\\times S)$ .",
"Recall the definition of the ring homomorphism $\\kappa :K^{G}(Y)\\rightarrow \\lim K(Y_{l})$ from Section .",
"Although we have not proved that $\\pi _{*}^{R}\\circ i_{*}^{R}[\\mathcal {O}_{\\mathcal {C}}]=j_{*}^{R}\\circ \\lambda _{*}^{R}[\\mathcal {O}_{\\mathcal {C}}]$ , by Lemma REF we still have $\\kappa _{R\\times S}\\circ \\pi _{*}^{R}\\circ i_{*}^{R}=\\kappa _{R\\times S}\\circ j_{*}^{R}\\circ \\lambda _{*}^{R}$ .",
"english Lemma 13 $\\kappa _{R}\\left(\\left.\\gamma \\left(\\mathcal {O}_{s}\\right)\\right|_{R}\\right) & :=\\kappa _{R}\\left(\\hat{p}_{*}^{R}\\left(\\bar{\\pi }_{*}^{R}\\circ i_{*}^{R}[\\mathcal {O}_{\\mathcal {C}_{R}}]\\otimes \\iota _{S}^{*}q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right)\\right)\\\\& \\,=\\kappa _{R}\\left(\\hat{p}_{*}^{R}\\left(\\left(j_{*}^{R}\\circ \\lambda _{*}^{R}[\\mathcal {O}_{\\mathcal {C}}]\\right)\\otimes \\iota _{S}^{*}q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right)\\right)$ We will use $\\left.\\hat{\\gamma }\\left(\\mathcal {O}_{s}\\right)\\right|_{R}$ to denote $\\hat{p}_{*}^{R}\\left(\\left(j_{*}^{R}\\circ \\lambda _{*}^{R}[\\mathcal {O}_{\\mathcal {C}_{R}}]\\right)\\otimes \\iota _{S}^{*}q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right)$ and $\\left[\\mathcal {O}_{\\mathcal {C}_{R}}\\right]$ to denote $\\lambda _{*}[\\mathcal {O}_{\\mathcal {C}_{R}}]$ .",
"Lemma 14 $R\\Gamma \\left(R,\\frac{\\mathcal {O}_{R}^{vir}\\otimes K_{vir}^{\\frac{1}{2}}\\vert _{R}}{\\bigwedge ^{\\bullet }\\left(N_{vir}^{\\bullet }\\right)^{\\vee }}\\left.\\prod _{i=1}^{m}\\gamma \\left(\\mathcal {O}_{s_{i}}\\right)\\right|_{R}\\right) & =R\\Gamma \\left(R,\\frac{\\mathcal {O}_{R}^{vir}\\otimes \\left.K_{vir}^{\\frac{1}{2}}\\right|_{R}}{\\bigwedge ^{\\bullet }\\left(N_{vir}^{\\bullet }\\right)^{\\vee }}\\left.\\prod _{i=1}^{m}\\hat{\\gamma }\\left(\\mathcal {O}_{s_{i}}\\right)\\right|_{R}\\right)$ american The Chern character map $ch^{G}:\\mathbb {Q}(\\mathfrak {t}^{\\frac{1}{2}})\\rightarrow \\mathbb {Q}((t))$ , $\\mathfrak {t}^{\\frac{1}{2}}\\mapsto e^{\\frac{1}{2}t}$ where $t$ is the equivariant first Chern class of $\\mathfrak {t}$ is an injection since $e^{\\frac{1}{2}t}$ is invertible in $\\mathbb {Q}((t))$ .english By virtual Riemann-Roch theorem of [5], Lemma REF and Lemma REF we have $ch^{G}R\\Gamma \\left(R,\\frac{\\mathcal {O}_{R}^{vir}\\otimes K_{vir}^{\\frac{1}{2}}\\vert _{R}}{\\wedge \\left(N_{vir}^{\\bullet }\\right)^{\\vee }}\\left.\\prod _{i=1}^{m}\\gamma \\left(\\mathcal {O}_{s_{i}}\\right)\\right|_{R}\\right) & =ch^{G}R\\Gamma \\left(R,\\frac{\\mathcal {O}_{R}^{vir}\\otimes \\left.K_{vir}^{\\frac{1}{2}}\\right|_{R}}{\\bigwedge ^{\\bullet }\\left(N_{vir}^{\\bullet }\\right)^{\\vee }}\\left.\\prod _{i=1}^{m}\\hat{\\gamma }\\left(\\mathcal {O}_{s_{i}}\\right)\\right|_{R}\\right).$ The injectivity of $ch^{G}:\\mathbb {Q}(\\mathfrak {t}^{\\frac{1}{2}})\\rightarrow \\mathbb {Q}((t))$ implies the lemma.",
"englishamericanThe above lemma also holds if we replace $\\left.K_{vir}^{\\frac{1}{2}}\\right|_{R}$ by any class $\\alpha \\in K^{G}(R)$ .",
"By the above lemma we can replace $\\gamma \\left(\\mathcal {O}_{s}\\right)$ with $\\hat{\\gamma }\\left(\\mathcal {O}_{s}\\right)=\\hat{p}_{*}\\left(\\rho _{*}\\left[\\mathcal {O}_{\\mathcal {C}}\\right].q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right)$ .",
"The advantage of using $\\hat{\\gamma }\\left(\\mathcal {O}_{s}\\right)$ will become clear later.",
"Lemma 15 Let $\\mathcal {L}$ be a globally generated line bundle on $S$ .",
"Let $\\dim \\,|\\mathcal {L}|=n$ and $\\mathcal {D}\\subset |\\mathcal {L}|\\times S$ be the universal divisor.",
"Then for any point $s\\in S$ the fiber product $\\mathcal {D}\\times _{|\\mathcal {L}|\\times S}\\left(|\\mathcal {L}|\\times \\lbrace s\\rbrace \\right)$ is a hyperplane $\\mathbb {P}^{n-1}\\subset |\\mathcal {L}|\\times \\lbrace s\\rbrace $ .",
"Let $\\mathcal {L}$ be globally generated line bundle on $S$ and let $f:S\\rightarrow Spec\\,\\mathbb {C}$ be the structure morphism.",
"Then $S\\times |\\mathcal {L}|=Proj\\,\\left(\\text{Sym}f^{*}\\left(f_{*}\\mathcal {L}\\right)^{\\vee }\\right)$ and the canonical morphism $\\xi :f^{*}f_{*}\\mathcal {L}\\rightarrow \\mathcal {L}$ is surjective.",
"Let $\\xi ^{\\vee }:\\mathcal {L}^{\\vee }\\rightarrow f^{*}\\left(f_{*}\\mathcal {L}\\right)^{\\vee }$ be the dual of $\\xi $ .",
"Let $e_{i}$ be the basis of $f_{*}\\mathcal {L}$ and let $e_{i}^{\\vee }\\in \\left(f_{*}\\mathcal {L}\\right)^{\\vee }$ defined as $e_{i}^{\\vee }(e_{j})=1$ if $i=j$ and 0 if $i\\ne j$ .",
"Then $\\xi ^{\\vee }$ sends a local section $\\psi $ of $\\mathcal {L}^{\\vee }$ to $\\xi ^{\\vee }(\\psi ):\\sum _{i}a_{i}e_{i}\\mapsto a_{i}\\psi \\left(e_{i}\\right)e_{i}^{\\vee }$ .",
"Sections of $f^{*}\\left(f_{*}\\mathcal {L}\\right)^{\\vee }$ are linear combinations $v$ of $\\lbrace e_{i}^{\\vee }\\rbrace $ with coefficient in $\\mathcal {O}_{S}$ and sections of $\\text{Sym}f^{*}(f_{*}\\mathcal {L})^{\\vee }$ are polynomials $P$ in $\\lbrace e_{i}^{\\vee }\\rbrace $ with coefficient in $\\mathcal {O}_{S}$ .",
"There is a canonical graded morphism $\\phi :f^{*}(f_{*}\\mathcal {L})^{\\vee }\\otimes \\text{Sym}f^{*}(f_{*}\\mathcal {L})^{\\vee }(-1)\\rightarrow \\text{Sym}f^{*}(f_{*}\\mathcal {L})^{\\vee }$ , that sends $v\\otimes P$ to the products of the polynomials $v.P$ .",
"The composition of $\\xi ^{\\vee }\\otimes \\text{id}_{\\text{Sym}f^{*}\\left(f_{*}\\mathcal {L}\\right)^{\\vee }(-1)}$ with $\\phi $ sends $\\psi \\otimes P$ to $\\xi ^{\\vee }(\\psi ).P$ .",
"Let $\\theta $ be this composition.",
"This composition is injective since $\\xi ^{\\vee }$ is injective.",
"This composition correspond to the morphism $\\sigma :\\mathcal {L}^{\\vee }\\boxtimes \\mathcal {O}(-1)\\rightarrow \\mathcal {O}$ on $S\\times |\\mathcal {L}|$ which is injective because $\\theta $ is injective and $Proj$ construction preserve injective morphism.",
"The cokernel $\\sigma $ is the structure sheaf of the universal divisor $\\mathcal {D}\\subset S\\times |\\mathcal {L}|$ .",
"For any closed point $s\\in S$ , we want to show that the restriction of $\\sigma $ to $|\\mathcal {L}|$ is still injective.",
"In this case $\\mathcal {D}\\times _{|\\mathcal {L}|\\times S}\\left(|\\mathcal {L}|\\times \\lbrace s\\rbrace \\right)$ is an effective divisor with ideal $\\mathcal {O}(-1)$ so that $\\mathcal {D}\\times _{|\\mathcal {L}|\\times S}\\left(|\\mathcal {L}|\\times \\lbrace s\\rbrace \\right)$ is a hyperplane $\\mathbb {P}^{n-1}$ .",
"Since $\\xi $ is surjective, its restriction to $s$ is also surjective.",
"Any element $\\alpha \\in \\mathcal {L}^{\\vee }|_{s}$ is the restriction of a local section $\\psi \\in \\mathcal {L}^{\\vee }$ .",
"Thus if $\\alpha $ is not zero there exist $\\psi \\in \\mathcal {L}^{\\vee }$ such that its restriction to $s$ is $\\alpha $ and $e_{i}$ such that the $\\left.\\psi (e_{i})\\right|_{s}=\\left.\\psi \\right|_{s}\\left(\\left.e_{i}\\right|_{s}\\right)$ is not zero.",
"We can conclude that $\\left.\\xi ^{\\vee }\\right|_{s}$ is injective.",
"Because $\\left.\\sigma \\right|_{s}:\\left.\\psi \\right|_{s}\\otimes \\left.P\\right|_{s}\\mapsto \\left.\\xi ^{\\vee }(\\psi )\\right|_{s}\\left.P\\right|_{s}$ we can conclude that $\\left.\\sigma \\right|_{s}$ is injective.",
"We will use $\\mathbb {P}_{s_{i}}^{n-1}$ to denote $\\mathcal {D}\\times _{|\\mathcal {L}|\\times S}\\left(|\\mathcal {L}|\\times \\lbrace s\\rbrace \\right)$ .",
"Lemma 16 Let $c_{1}\\left(\\mathcal {L}\\right)=\\beta $ and let $\\mathcal {P}=\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ .",
"Then all squares in the following diagram are Cartesian.",
"$\\begin{tikzcd}[row sep={40,between origins}, column sep={40,between origins}]&[-] \\mathcal {D}_{\\mathcal {P}}\\times _{\\mathcal {D}}\\mathbb {P}^{n-1}{rr}{dd}[near end]{\\bar{h}}{dl}[near start]{\\bar{j}} &[] &[-] \\mathbb {P}^{n-1}\\vphantom{\\times _{S_1}} {dd}{dl} \\\\[-] \\mathcal {P}\\times \\lbrace s\\rbrace [crossing over]{rr} {dd}{h} & & \\left|\\mathcal {L}\\right|\\times \\lbrace s\\rbrace \\\\[] & \\mathcal {D}_{\\mathcal {P}} {rr} {dl}[near start]{j} & & \\mathcal {D}\\vphantom{\\times _{S_1}} {dl} \\\\[-] \\mathcal {P}\\times S {rr} && \\left|\\mathcal {L}\\right|\\times S [from=uu,crossing over]\\end{tikzcd}$ Lemma 17 If $\\beta \\in G^{T}(\\mathcal {P})$ is supported on $V\\subset \\mathcal {P}$ then $\\beta .\\hat{\\gamma }\\left(\\mathcal {O}_{s}\\right)$ is supported on $V\\times _{\\mathcal {P}}W_{s}$ where $W_{s}:=\\mathcal {D}_{\\mathcal {P}}\\times _{\\mathcal {P}\\times S}\\left(\\mathcal {P}\\times \\lbrace s\\rbrace \\right)$ .",
"Recall the morphism $\\hat{p}$ from diagram (REF ) and $h,\\bar{h}$ from (REF ).",
"Since $\\hat{p}\\circ h=\\text{id}_{\\mathcal {P}}$ we can conclude that $\\text{$\\hat{\\gamma }\\left(\\mathcal {O}_{s}\\right)$}=h^{*}j_{*}\\left[\\mathcal {O}_{\\mathcal {C}}\\right]=h^{*}\\left[j_{*}\\mathcal {O}_{\\mathcal {C}}\\right]$ .",
"Let $E^{\\bullet }$ be a finite resolution of $j_{*}\\mathcal {O}_{\\mathcal {C}}$ by locally free sheaves.",
"It's sufficient prove the statement for the case when $\\beta $ is the class of a coherent sheaf $\\mathcal {\\mathcal {F}}$ on $V$ .",
"By Lemma REF , we have $[\\mathcal {\\mathcal {F}}].\\hat{\\gamma }\\left(\\mathcal {O}_{s}\\right)& =\\bar{j}_{*}k_{*}j^{[\\mathcal {O}_{\\mathcal {C}}]}(\\mathcal {\\mathcal {F}})$ where $j^{[\\mathcal {O}_{\\mathcal {C}}]}$ is the refined Gysin homomorphism and $k$ is the closed embedding $V\\times _{\\mathcal {P}\\times \\lbrace s\\rbrace }W_{s}\\rightarrow W_{s}$ where $W_{s}=\\mathcal {D}_{\\mathcal {P}}\\times _{\\mathcal {D}}\\mathbb {P}_{s}^{n-1}$ .",
"Lemma 18 Given $m$ points $s_{1},\\ldots ,s_{m}\\in S$ in general position such that all curves in $|\\mathcal {L}|$ that passes through all $m$ points are reduced and irreducible, then for any component $R\\subset \\mathcal {P}^{G}$ different from $\\mathcal {P}_{\\chi }(S,\\beta )$ we have $\\iota _{*}\\mathcal {O}_{R}^{vir}.\\prod _{i=1}^{m}\\hat{\\gamma }\\left(\\mathcal {O}_{s_{i}}\\right)=0$ .",
"Let $\\beta _{l}=\\iota _{*}\\mathcal {O}_{R}^{vir}.\\prod _{i=1}^{l}\\hat{\\gamma }\\left(\\mathcal {O}_{s_{i}}\\right)$ .",
"By Lemma REF , $\\beta _{1}$ is supported on $R\\times _{\\mathcal {P}}W_{s}=R\\times _{|\\mathcal {L}|}\\mathbb {P}_{s_{1}}^{n-1}$ .",
"Our assumptions implies that for any $1\\le l\\le m$ , $\\bigcap _{i=1}^{l-1}\\mathbb {P}_{s_{1}}^{n-1}$ is not contained in $\\mathbb {P}_{s_{l}}^{n-1}$ .",
"In particular, $\\bigcap _{i=l}^{l}\\mathbb {P}_{s_{l}}^{n-1}=\\mathbb {P}^{n-m}$ and by induction we can conclude that $\\beta _{m}$ is supported on $R\\times _{|\\mathcal {L}|}\\mathbb {P}^{n-m}$ .",
"Note that all curves in $\\mathbb {P}^{n-m}$ is reduced and irreducible.",
"We will show that for any $\\left(\\mathcal {\\mathcal {F}},s\\right)\\in R$ , $\\text{div}\\,\\left(\\mathcal {\\mathcal {F}},s\\right)$ is not in $\\mathbb {P}^{n-m}$ .",
"Let $C_{\\mathcal {\\mathcal {F}}}$ be the curve on $X$ supporting an element $(\\mathcal {\\mathcal {F}},s)\\in R$ .",
"Note that the reduced subscheme $C_{\\mathcal {\\mathcal {F}}}^{red}$ of $C_{\\mathcal {\\mathcal {F}}}$ is a curve on $S$ so that if $C_{\\mathcal {\\mathcal {F}}}$ is reduced and irreducible then $C_{\\mathcal {\\mathcal {F}}}=C_{\\mathcal {\\mathcal {F}}}^{red}$ is a curve on $S$ and $\\left(\\mathcal {\\mathcal {F}},s\\right)$ can't be in $R$ .",
"If $C_{\\mathcal {\\mathcal {F}}}$ is not irreducible, then the support of $\\pi _{*}\\mathcal {O}_{C_{\\mathcal {\\mathcal {F}}}}$ is not irreduble so that $\\text{div}\\,(\\mathcal {\\mathcal {F}},s)$ is not in $\\mathbb {P}^{n-m}$ .",
"So we are left with the case when $C_{\\mathcal {\\mathcal {F}}}$ is irreducible.",
"Let $C$ be the reduced subscheme of $C_{\\mathcal {\\mathcal {F}}}$ .",
"Let $Spec\\,A\\subset S$ be an open subset such that $K_{S}$ is a free line bundle over $Spec\\,A$ .",
"We can write $C=Spec\\,A/(f)$ for an irreducible element $f\\in A$ and $X|_{Spec\\,A}=Spec\\,A[x]$ .",
"Then $\\mathcal {O}_{C_{\\mathcal {\\mathcal {F}}}}$ can be written as $M:=\\oplus _{i=0}^{r}A/(f^{n_{i}})x^{i}$ for some positive integers $r,n_{i}$ and $\\text{div}\\,M$ is described by the ideal $(f^{\\sum _{i}n_{i}})$ .",
"Since $C_{\\mathcal {\\mathcal {F}}}$ is not supported on $S$ , then $\\sum _{i}n_{i}\\ge 2$ and $\\text{div}\\,M$ is not reduced.",
"Thus in this case $\\text{div}\\,(\\mathcal {\\mathcal {F}},s)$ is not in $\\mathbb {P}^{n-m}$ .",
"Since $\\text{div}\\,(R)$ is disjoint from $\\mathbb {P}^{n-m}$ , we can conclude that $R\\times _{|\\mathcal {L}|}\\mathbb {P}^{n-m}$ is empty.",
"By lemma REF , $\\beta _{m}$ is zero.",
"Following the proof of Lemma REF and Lemma REF and by replacing $[\\mathcal {O}_{\\mathcal {C}}]$ with $[\\mathcal {O}_{\\mathcal {D}}]$ we can prove that the contribution to $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)$ of the component $R\\subset \\mathcal {P}^{G}$ where $R\\ne \\mathcal {P}_{\\chi }(S,\\beta )$ is zero when $s_{1},\\ldots ,s_{m}$ is in general position and all curves on $S$ that passthrough all $m$ points are reduced and irreducible.",
"Actually we have a stronger result for $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)$ .",
"By Proposition REF for any point $s\\in S$ , $\\bar{\\gamma }(\\mathcal {O}_{s})$ is $1-[\\text{div}^{*}\\mathcal {O}(-1)]$ .",
"In particular it's independent from the choosen point.",
"english Proposition 19 Given a positive integer $\\delta $ , let $S$ be a smooth projective surface with $b_{1}(S)=0$ .",
"Let $\\mathcal {L}$ be a $2\\delta +1$ -very ample line bundle on $S$ with $c_{1}(\\mathcal {L})=\\beta $ and $H^{i}(\\mathcal {L})=0$ for $i>0$ .",
"Let $X=K_{S}$ be the canonical line bundle over $S$ .",
"Then for any connected component $R$ of $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{\\mathbb {C}^{\\times }}$ different from $\\mathcal {P}_{\\chi }(S,\\beta )$ and for $m\\ge H^{0}(\\mathcal {L})-1-\\delta $ , we have $R\\Gamma \\left(R,\\frac{\\mathcal {O}_{R}^{vir}}{\\bigwedge ^{\\bullet }\\left(N_{vir}^{\\bullet }\\right)^{\\vee }}K_{vir}^{\\frac{1}{2}}\\vert _{R}\\otimes \\prod _{i=1}^{m}\\frac{\\bar{\\gamma }(\\mathcal {O}_{s_{i}})}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)=0$ where $s_{1},\\ldots s_{m}$ are closed points of $S$ which can be identical.",
"We then can conclude that $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)=R\\Gamma \\left(\\mathcal {P}_{\\chi }(S,\\beta ),\\frac{\\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )}^{vir}}{\\bigwedge ^{\\bullet }\\left(N_{vir}^{\\bullet }\\right)^{\\vee }}K_{vir}^{\\frac{1}{2}}\\vert _{\\mathcal {P}_{\\chi }(S,\\beta )}\\otimes \\prod _{i=1}^{m}\\frac{\\bar{\\gamma }(\\mathcal {O}_{s_{i}})}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right).$ The same result also holds for $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ under additional assumption that the structure sheaf $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ of the universal supporting curve $\\mathcal {C}_{\\mathbb {F}}$ is flat over $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ and $s_{1},\\ldots ,s_{m}$ are closed points in $S$ in general position such that all curves in $|\\mathcal {L}|$ passing through all the given $m$ points are irreducible.",
"american"
],
[
"The contribution of $\\protect \\mathcal {P}_{\\chi }(S,\\beta )$",
"englishThe component $\\mathcal {P}_{\\chi }(S,\\beta )$ of $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{G}$ parametrize stable pairs $(F,s)$ supported on $S\\subset X$ where $S$ is the zero section.",
"The restriction of $\\mathbb {I}^{\\bullet }$ to $\\mathcal {P}_{\\chi }(S,\\beta )\\times X$ is $\\mathbb {I}^{\\bullet }_{X}:=\\lbrace \\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )\\times X}\\rightarrow \\mathcal {\\mathcal {F}}\\rbrace $ , where $\\mathcal {\\mathcal {F}}$ is the universal sheaf restricted to $\\mathcal {P}_{\\chi }(S,\\beta )\\times X$ , so that the restriction of $\\mathbb {E}^{\\bullet }$ to $\\mathcal {P}_{\\chi }(S,\\beta )$ is $Rp_{*}R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet }_{X},\\mathbb {I}^{\\bullet }_{X}\\otimes \\mathfrak {t}^{*}\\right)_{0}[2]$ .",
"Thomas and Kool showed that on $\\mathcal {P}_{\\chi }(S,\\beta )$ , the decomposition of $\\left.\\mathbb {E}^{\\bullet }\\right|_{\\mathcal {P}_{\\chi }(S,\\beta )}$ into fixed and moving part is $\\left(\\mathbb {E}^{\\bullet }\\right)^{mov}\\simeq Rp_{*}R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet }_{S},\\mathcal {\\mathcal {F}}\\right)[1]\\otimes \\mathfrak {t}^{*}\\qquad \\left(\\mathbb {E}^{\\bullet }\\right)^{fix}\\simeq \\left(Rp_{*}R\\mathcal {H}om\\left(\\mathbb {I}^{\\bullet }_{S},\\mathcal {\\mathcal {F}}\\right)\\right)^{\\vee }$ where $\\mathbb {I}^{\\bullet }_{S}=\\lbrace \\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )\\times S}\\rightarrow \\mathcal {\\mathcal {F}}\\rbrace $ .",
"$\\left(\\mathbb {E}^{\\bullet }\\right)^{fix}$ gives $\\mathcal {P}_{\\chi }(S,\\beta )$ a perfect obstruction theory.",
"We will use $\\mathcal {E}^{\\bullet }$ to denote $\\left(\\mathbb {E}^{\\bullet }\\right)^{fix}$ .",
"From equation (REF ) and (REF ) we have $\\left(\\mathbb {E}^{\\bullet }\\right)^{mov}\\simeq \\left(\\mathcal {E}^{\\bullet }\\right)^{\\vee }[1]\\otimes \\mathfrak {t}^{*}$ .",
"Proposition 20 On $\\mathcal {P}_{\\chi }(S,\\beta )$ we have $\\frac{K_{vir}^{\\frac{1}{2}}\\vert _{\\mathcal {P}_{\\chi }(S,\\beta )}}{\\bigwedge ^{\\bullet }\\left(N_{vir}^{\\bullet }\\right)^{\\vee }}=\\left(-\\mathfrak {t}^{-\\frac{1}{2}}\\right)^{v}{\\textstyle \\bigwedge _{-\\mathfrak {t}}}\\mathcal {E}^{\\bullet }$ where $vd=\\text{rk}\\mathcal {E}^{\\bullet }$ and $\\bigwedge _{-\\mathfrak {t}}\\mathcal {E}^{\\bullet }=\\frac{\\sum _{i=0}^{\\text{rk}E^{0}}\\left(-\\mathfrak {t}\\right)^{i}\\bigwedge ^{i}\\mathcal {E}^{0}}{\\sum _{j=0}^{\\text{rk}E^{-1}}\\left(-\\mathfrak {t}\\right)^{j}\\bigwedge ^{j}\\mathcal {E}^{-1}}$ for $\\mathcal {E}^{\\bullet }=[\\mathcal {E}^{-1}\\rightarrow \\mathcal {E}^{0}]$ .",
"By equation (REF ) and (REF ) we have $K_{vir}\\vert _{\\mathcal {P}_{\\chi }(S,\\beta )} & =\\det \\mathcal {E}^{\\bullet }\\det \\left(\\left(\\mathcal {E}^{\\bullet }\\right)^{\\vee }\\otimes \\mathfrak {t}^{*}\\right)^{\\vee }=\\det \\mathcal {E}^{\\bullet }\\det \\mathcal {E}^{\\bullet }\\mathfrak {t}^{v}$ where $v=rk$$\\mathcal {E}^{\\bullet }.$ Thus we can take $K_{vir}^{\\frac{1}{2}}\\vert _{\\mathcal {P}_{\\chi }(S,\\beta )}=\\det \\mathcal {E}^{\\bullet }\\mathfrak {t}^{\\frac{1}{2}v}.$ Let $\\mathcal {E}^{\\bullet }=[\\mathcal {E}^{-1}\\rightarrow \\mathcal {E}^{0}]$ so that $\\left(\\mathcal {E}^{\\bullet }\\right)^{\\vee }[1]\\otimes \\mathfrak {t}^{*}=[\\left(\\mathcal {E}^{0}\\right)^{\\vee }\\otimes \\mathfrak {t}^{*}\\rightarrow \\left(\\mathcal {E}^{-1}\\right)^{\\vee }\\otimes \\mathfrak {t}^{*}]$ in the place of $-1$ and 0.",
"Let $r_{i}=rk\\mathcal {E}^{i}$ for $i=-1$ and $i=0$ .",
"Thus in $K^{G}(\\mathcal {P}_{\\chi }(S,\\beta ))$ we have $\\frac{K_{vir}^{\\frac{1}{2}}\\vert _{\\mathcal {P}_{\\chi }(S,\\beta )}}{\\bigwedge ^{\\bullet }\\left(N_{vir}^{\\bullet }\\right)^{\\vee }} & =\\left(-\\mathfrak {t}^{-\\frac{1}{2}}\\right)^{vd}{\\textstyle \\bigwedge _{-\\mathfrak {t}}}\\mathcal {E}^{\\bullet }$ The calculation of the contribution from this component is given in the next section.",
"We recall Corollary REF here.",
"Under the assumption of Proposition REF the formula for $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)$ is $\\left(-1\\right)^{vd}\\int \\limits _{\\left[\\mathcal {P}_{\\chi }(S,\\beta )\\right]^{red}}\\frac{X_{-\\mathfrak {t}}\\left(TS^{[n]}\\right)X_{-\\mathfrak {t}}\\left(\\mathcal {O}(1)\\right)^{\\delta +1}}{X_{-\\mathfrak {t}}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)}\\left(\\frac{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}e^{-H\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}H^{m}$ where $vd$ is the virtual dimension of $\\mathcal {P}_{\\chi }(S,\\beta )$ and $\\mathcal {O}(1)$ is the dual of the pullback by the morphism $\\text{div}:\\mathcal {P}_{\\chi }(X,i_{*}\\beta )\\rightarrow |\\mathcal {L}|$ of the tautological line bundle and $H=c_{1}(\\mathcal {O}(1))$ and for any vector bundle $E$ of rank $r$ with Chern roots $x_{1},\\ldots ,x_{r}$ , $X_{-\\mathfrak {t}}(E)=\\prod _{i=1}^{r}\\frac{x_{i}\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}e^{-x_{i}\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}\\right)}{1-e^{-x_{i}\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}}.$ We have the same formula for $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ whenever $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ can be defined.",
"This is because the restriction of $\\gamma \\left(\\mathcal {O}_{s_{i}}\\right)$ and $\\bar{\\gamma }(\\mathcal {O}_{s_{i}})$ to $\\mathcal {P}_{\\chi }(S,\\beta )$ are identical.",
"americanWe can observe from the above formula that $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ is independent from the choosen points.",
"It's natural to ask if without assuming that $s_{1},\\ldots ,s_{m}$ are in general positions such that all curves passing through all these points are reduced and irreducible the above proposition still holds.",
"english"
],
[
"Refinemnet of Kool-Thomas Invariants",
"american"
],
[
"Reduced obstruction theory of moduli space of stable pairs on surface",
"In Section 4 we have reviewed the construction of reduced obstruction theory by Kool and Thomas in [12].",
"In this section we will review the description of it's restriction to $\\mathcal {P}_{\\chi }(S,\\beta )$ as a two term complex of locally free sheaves following Appendix $A$ of [12].",
"The Appendix is written by Martijn Kool, Richard P. Thomas and Dmitri Panov.",
"englishPandharipande and Thomas showed that $\\mathcal {P}_{\\chi }(S,\\beta )$ is isomorphic to the relative Hilbert scheme of points $\\text{Hilb}^{n}(\\mathcal {C}/\\text{Hilb}_{\\beta }(S))$ where $\\mathcal {C}\\rightarrow \\text{Hilb}_{\\beta }(S)$ is the universal family of curves $C$ in $S$ in class $\\beta \\in H_{2}(S,\\mathbb {Z})$ and $\\chi =n+1-h$ where $h$ is the arithmetic genus of $C$ .",
"Notice that for $n=1$ , $\\mathcal {P}_{\\chi }(S,\\beta )=\\text{Hilb}^{1}(\\mathcal {C}/\\text{Hilb}_{\\beta }(S))=\\text{Hilb}_{\\beta }(S)$ .",
"We will review first the description of $\\mathcal {P}_{\\chi }(S,\\beta )$ as the zero locus of a vector bundle on a smooth scheme.",
"We assume that $b_{1}(S)=0$ for simplicity and also because we are only working for this case in this article.",
"The following construction does not need this assumption.",
"americanFor $n=0$ , pick a sufficiently ample line divisor $A$ on $S$ such that $\\mathcal {L}(A)=\\mathcal {L}\\otimes \\mathcal {O}(A)$ satisfies $H^{i}(\\mathcal {L}(A))=0$ for $i>0$ .",
"Let $\\gamma =\\beta +[A]$ .",
"Then $\\text{Hilb}_{\\gamma }(S)=|\\mathcal {L}(A)|=\\mathbb {P}^{\\chi (\\mathcal {L}(A)-1}$ has the right dimension.",
"The map that send $C\\in |\\mathcal {L}|$ to $C+A\\in |\\mathcal {L}(A)|$ defines a closed embedding $\\text{Hilb}_{\\beta }(S)\\rightarrow \\text{Hilb}_{\\gamma }(S)$ .",
"englishLet $\\mathcal {D}\\subset H_{\\gamma }\\left(S\\right)\\times S$ be the universal divisor and let $\\hat{p}$ and $q_{S}$ be the projections $H_{\\gamma }(S)\\times S\\rightarrow H_{\\gamma }(S)$ and $H_{\\gamma }(S)\\times S\\rightarrow S$ respectively.",
"Let $s_{\\mathcal {D}}\\in H^{0}\\left(\\mathcal {O}(\\mathcal {D})\\right)$ be the section defining $\\mathcal {D}$ and restrict it to $H_{\\gamma }\\left(S\\right)\\times A$ and consider the section $\\zeta & :=s_{\\mathcal {D}}|_{\\pi _{S}^{-1}A}\\in H^{0}(H_{\\gamma }(S)\\times A,\\mathcal {O}(\\mathcal {D})|_{\\pi _{S}^{-1}A})=H^{0}(H_{\\gamma }(S),\\pi _{H*}(\\mathcal {O}(\\mathcal {D})|_{\\pi _{S}^{-1}A}))$ where for a point $D\\in H_{\\gamma }\\left(S\\right)$ we have $\\zeta |_{D}=s_{D}|_{A}\\in H^{0}(A,\\mathcal {L}(A))$ where $s_{D}$ is the section of $\\mathcal {L}(A)$ defining $D$ .",
"$s_{D}|_{A}=0$ if and only if $A\\subset D$ i.e $D=A+C$ for some effective divisor $C$ with $\\mathcal {O}(C)\\otimes \\mathcal {O}(A)=\\mathcal {L}(A)$ .",
"Thus the zero locus of $\\zeta $ is the image of the closed embedding $\\text{Hilb}_{\\beta }(S)\\rightarrow \\text{Hilb}_{\\gamma }(S).$ If $H^{2}(\\mathcal {L})=0$ then $F=\\pi _{H*}(\\mathcal {O}(\\mathcal {D})|_{\\pi _{S}^{-1}A})$ is a vector bundle of rank $\\chi (\\mathcal {L}(A))-\\chi (\\mathcal {L})=h^{0}(\\mathcal {L}(A))-h^{0}(\\mathcal {L})+h^{1}(\\mathcal {L})$ on $\\text{Hilb}_{\\gamma }(S)$ since $R^{i}\\pi _{H_{*}}\\left(\\mathcal {O}(\\mathcal {D}\\right)|_{\\pi _{S}^{-1}A})=0$ for $i>0$ .",
"Consider the following diagram ${F_{red}^{\\bullet }= & \\lbrace F^{*}\\,\\,\\,\\,\\,\\,[r]^{d\\circ \\zeta ^{*}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}[d]_{\\zeta ^{*}} & \\Omega _{H_{\\gamma }(S)}|_{H_{\\beta }(S)}\\rbrace [d]^{\\text{id}}\\\\\\mathbb {L}_{H_{\\beta }(S)}= & \\lbrace I/I^{2}|_{H_{\\beta }(S)}[r]^{d\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,} & \\Omega _{H_{\\gamma }(S)}|_{H_{\\beta }(S)}\\rbrace .",
"}$ The above morphism is a perfect obstruction theory for $\\text{Hilb}_{\\beta }(S)$ .",
"Next, we embed $\\text{Hilb}^{n}(\\mathcal {C}/\\text{Hilb}_{\\beta }(S))$ into $S^{[n]}\\times \\text{Hilb}_{\\beta }(S)$ .",
"Let $\\mathcal {Z}\\subset S^{[n]}\\times \\text{Hilb}_{\\beta }(S)\\times S$ be the pullback of the universal length $n$ subscheme of $S^{[n]}\\times S$ .",
"Let $\\mathcal {C}\\subset S^{[n]}\\times \\text{Hilb}_{\\beta }(S)\\times S$ be the pullback of the universal divisor of $\\text{Hilb}_{\\beta }\\times S$ and let $\\pi :S^{[n]}\\times \\text{Hilb}_{\\beta }(S)\\times S\\rightarrow S^{[n]}\\times \\text{Hilb}_{\\beta }(S)$ be the projection.",
"Then $\\mathcal {C}$ correspond to a section $s_{\\mathcal {C}}$ of the line bundle $\\mathcal {O}(\\mathcal {C})$ on $S^{[n]}\\times \\text{Hilb}_{\\beta }(S)\\times S.$ A point $(Z,C)\\in S^{[n]}\\times \\text{Hilb}_{\\beta }(S)$ is in the image of $\\text{Hilb}^{n}(\\mathcal {C}/\\text{Hilb}_{\\beta }(S))$ if $Z\\subset C.$ We denote by $\\mathcal {O}(\\mathcal {C})^{[n]}$ the vector bundle $\\pi _{*}\\left(\\mathcal {O}(\\mathcal {C})|_{\\mathcal {Z}}\\right)$ of rank $n$ .",
"Let $\\sigma _{\\mathcal {C}}$ be the pushforward of $s_{\\mathcal {C}}$ so that $\\sigma _{\\mathcal {C}}|_{(Z,C)}=s_{C}|_{Z}\\in H^{0}(\\mathcal {L}|_{Z})$ .",
"Thus a point $(Z,C)\\in S^{[n]}\\times \\text{Hilb}_{\\beta }(S)$ is in the image of $\\text{Hilb}^{n}(\\mathcal {C}/\\text{Hilb}_{\\beta }(S))$ if and only if $\\sigma _{\\mathcal {C}}|_{(Z,C)}=s_{C}|_{Z}=0$ .",
"Thus we get a perfect relative obstruction theory : ${\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,E^{\\bullet }= & \\lbrace \\left(\\mathcal {O}(\\mathcal {C})^{[n]}\\right){}^{*}[d]_{s^{*}}[r]^{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,d\\circ s^{*}} & \\Omega _{S^{[n]}}[d]^{\\text{id}}\\rbrace \\\\\\mathbb {L}_{\\text{Hilb}^{n}\\left(\\mathcal {C}/\\text{Hilb}_{\\beta }(S)\\right)/\\text{Hilb}_{\\beta }(S)}= & \\lbrace J/J^{2}[r]^{d} & \\Omega _{S^{[n]}}\\rbrace }$ where $J$ is the ideal describing $\\text{Hilb}^{n}\\left(\\mathcal {C}/\\text{Hilb}_{\\beta }(S)\\right)$ as a subscheme of $S^{[n]}\\times \\text{Hilb}_{\\beta }(S).$ Notice that in general $|\\mathcal {L}|$ is not of the right dimension.",
"americanAppendix A of [12] shows how to combine the above obstruction theories to define an absolute perfect obstruction theory for $\\text{Hilb}^{n}\\left(\\mathcal {C}/\\text{Hilb}_{\\beta }(S)\\right)$ .",
"To do it we have to consider the embedding of $\\text{Hilb}^{n}\\left(\\mathcal {C}/\\text{Hilb}_{\\beta }(S)\\right)$ into $S^{[n]}\\times \\text{Hilb}_{\\gamma }(S)$ .",
"$E^{\\bullet }$ is the restriction of $[(\\mathcal {O}(\\mathcal {D}-A)^{[n]})^{*}\\rightarrow \\Omega _{S^{[n]}}]$ to $\\text{Hilb}^{n}\\left(\\mathcal {C}/\\text{Hilb}_{\\beta }(S)\\right)$ .",
"It was shown that the complex $E_{red}^{\\bullet }$ that correspond to the combined obstruction theory sits in the following exact triangle ${F_{red}^{\\bullet }[r] & E_{red}^{\\bullet }[r] & E^{\\bullet }}.$ Also in Appendix A of [12], it was shown that the combination of the above obstruction theory have the same $K$ -theory class with the reduced obstruction theory $\\mathcal {E}_{red}^{\\bullet }$ .",
"Thus we can conclude that the $K$ -theory class of $\\mathcal {E}_{red}^{\\bullet }$ is $[\\Omega _{S^{[n]}\\times \\text{Hilb}_{\\gamma }(S)}]-[(\\mathcal {O}(\\mathcal {D}-A)^{[n]})^{*}]-\\left[F^{*}\\right]$ Moreover, Theorem A.7 of [12] gives the virtual class corresponding to the reduced obstruction theory $\\left[\\mathcal {P}_{\\chi }(S,\\beta )\\right]^{red}$ as the class $c_{n}\\left(\\mathcal {O}(\\mathcal {D}-A^{[n]}\\right).c_{top}\\left(F\\right)\\cap [S^{[n]}\\times \\text{Hilb}_{\\gamma }(S)]$ ."
],
[
"Point insertion and linear subsystem",
"englishIn this section we assume that $h^{0,1}(S)=0$ i.e.",
"$\\text{Pic}_{\\beta }=\\left\\lbrace \\mathcal {L}\\right\\rbrace $ and $\\text{Hilb}_{\\beta }(S)=|\\mathcal {L}|$ .",
"Let $\\mathcal {D}\\subset S\\times |\\mathcal {L}|$ be the universal curve.",
"Pandharipande and Thomas showed in [17] that $\\mathcal {P}_{\\chi }(S,\\beta )$ is isomorphic to the relative Hilbert scheme of points $\\text{Hilb}^{n}(\\mathcal {D}\\rightarrow |\\mathcal {L}|)$ .",
"There is an embedding of $\\text{Hilb}^{n}(\\mathcal {D}\\rightarrow |\\mathcal {L}|)$ into $S^{[n]}\\times |\\mathcal {L}|$ and the projection $\\text{Hilb}^{n}(\\mathcal {D}\\rightarrow |\\mathcal {L}|)\\rightarrow |\\mathcal {L}|$ gives a morphism $\\text{div}:\\mathcal {P}_{\\chi }(S,\\beta )\\rightarrow |\\mathcal {L}|$ that maps $(\\mathcal {\\mathcal {F}},s)\\in \\mathcal {P}_{\\chi }(S,\\beta )$ to the supporting curve $C_{\\mathcal {\\mathcal {F}}}\\in |\\mathcal {L}|$ of $\\mathcal {\\mathcal {F}}$ .",
"Fix $\\chi \\in \\mathbb {Z}$ and let $\\mathcal {C}$ be the universal curve supporting the universal sheaf $\\mathcal {\\mathcal {F}}$ on $S\\times \\mathcal {P}_{\\chi }(S,\\beta ).$ Consider the following diagram ${\\mathcal {P}_{\\chi }(S,\\beta )\\times S[r]^{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,q_{S}}[d]_{\\hat{p}} & S\\\\\\mathcal {P}_{\\chi }(S,\\beta )}$ Of course when $n=1$ , $\\mathcal {P}_{\\chi }(S,\\beta )$ is $|\\mathcal {L}|$ and $\\mathcal {C}=\\mathcal {D}$ .",
"Here we will compute explicitly the class $\\gamma \\left(\\mathcal {O}_{s}\\right)$ restricted to $\\mathcal {P}_{\\chi }(S,\\beta )\\rightarrow \\mathcal {P}_{\\chi }(X,i_{*}\\beta )^{G}$ .",
"Note that $G$ acts trivially on $S$ and on $\\mathcal {P}_{\\chi }(S,\\beta )$ .",
"Let $\\mathcal {C}\\subset \\mathcal {P}_{\\chi }(S,\\beta )\\times \\bar{X}$ be the support of the universal sheaf.",
"Note that $\\mathcal {C}$ is supported on $\\mathcal {P}_{\\chi }(S,\\beta )\\times S$ where $S$ is the zero section of the bundle $X\\rightarrow S$ .",
"Thus $\\bar{\\pi \\circ i:\\mathcal {C}\\rightarrow \\mathcal {P}_{\\chi }(S,\\beta )\\times S}$ is a closed embedding.",
"By equation (REF ), $\\gamma \\left(\\mathcal {O}_{s}\\right)=\\hat{p}_{*}\\left([\\mathcal {O}_{\\mathcal {C}}]\\otimes q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right).$ Notice that $G$ acts on $\\mathcal {O}_{s}$ and $\\mathcal {O}_{\\mathcal {C}}$ trivially.",
"Proposition 21 Let $s\\in S$ be a point with structure sheaf $\\mathcal {O}_{s}$ .",
"Let $[\\mathcal {O}_{s}]$ be its class in $K(S).$ Then $\\hat{p}_{*}\\left(\\left[\\mathcal {O}_{\\mathcal {C}}\\right].q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right)=1-[\\text{div}^{*}\\mathcal {O}(-1)].$ where $\\mathcal {O}(-1)$ is the tautological line bundle on $|\\mathcal {L}|$ .",
"First consider the following diagram ${|\\mathcal {L}|\\times S[r]^{\\,\\,\\,\\,\\,\\,\\,q_{S}}[d]^{\\hat{p}^{|\\mathcal {L}|}} & S\\\\|\\mathcal {L}|}.$ We will show that $\\hat{p}_{*}\\left(q_{S}^{*}[\\mathcal {O}_{z}].",
"[\\mathcal {O}_{\\mathcal {D}}]\\right)=1-\\left[\\mathcal {O}(-1)\\right]$ .",
"Since $q_{S}$ is a flat morphism $q_{S}^{*}[\\mathcal {O}_{z}]=[q_{S}^{*}\\mathcal {O}_{z}]=k_{*}\\left[\\mathcal {O}_{|\\mathcal {L}|\\times \\left\\lbrace z\\right\\rbrace }\\right]$ where $k$ is the inclusion $k:|\\mathcal {L}|\\times \\lbrace z\\rbrace \\rightarrow |\\mathcal {L}|\\times S$ .",
"$\\mathcal {C}$ is the universal divisor with $\\mathcal {L}^{*}\\boxtimes \\mathcal {O}(-1)$ as the defining ideal.",
"By the projection formula $q_{S}^{*}[\\mathcal {O}_{s}].",
"[\\mathcal {O}_{\\mathcal {D}}]$ is equal to $k_{*}\\left[\\mathcal {O}_{|\\mathcal {L}|\\times \\left\\lbrace s\\right\\rbrace }\\right].\\left(1-\\left[\\mathcal {L}^{*}\\boxtimes \\mathcal {O}(-1)\\right]\\right)=k_{*}\\left([k^{*}\\mathcal {O}_{|\\mathcal {L}|\\times S}]-\\left[k^{*}q_{S}^{*}\\mathcal {L}^{*}\\otimes k^{*}\\hat{p}^{*}\\mathcal {O}(-1)\\right]\\right).$ $k^{*}q_{S}^{*}\\mathcal {L}^{*}=q_{s}^{*}\\mathcal {L}^{*}|_{s}=\\mathcal {O}_{|\\mathcal {L}|\\times \\lbrace s\\rbrace }$ where $q_{s}=q_{S}|_{|\\mathcal {L}|\\times \\lbrace s\\rbrace }$ and $k^{*}\\hat{p}^{*}\\mathcal {O}(-1)=\\mathcal {O}(-1)$ since $\\hat{p}\\circ k$ is the identity morphism.",
"Thus we conclude that $\\hat{p}_{*}\\left(q_{S}^{*}[\\mathcal {O}_{s}].",
"[\\mathcal {O}_{\\mathcal {D}}]\\right)=\\hat{p}_{*}k_{*}\\left(\\left[\\mathcal {O}_{|\\mathcal {L}|\\times \\lbrace s\\rbrace }\\right]-\\left[\\mathcal {O}(-1)\\right]\\right)=1-[\\mathcal {O}(-1)]$ Now we are working on $\\mathcal {P}_{\\chi }(S,\\beta ).$ Consider the following Cartesian diagram ${\\text{div}^{-1}\\mathcal {D}[r][d] & \\mathcal {D}[d]\\\\\\mathcal {P}_{\\chi }(S,\\beta )\\times S[r]^{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{div}}[d]_{\\hat{p}^{\\mathcal {P}_{\\chi }(S,\\beta )}} & |\\mathcal {L}|\\times S[d]_{\\hat{p}^{|\\mathcal {L}|}}\\\\\\mathcal {P}_{\\chi }(S,\\beta )[r]^{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{div}} & |\\mathcal {L}|.",
"}$ $\\text{div}^{-1}\\mathcal {D}$ is the family of effective Cartier divisor corresponding to the morphism $\\text{div}:\\mathcal {P}_{\\chi }(S,\\beta )\\rightarrow |\\mathcal {L}|,$ For each point $p\\in \\mathcal {P}_{\\chi }(S,\\beta ),$ $\\text{div}^{-1}\\mathcal {D}|_{p}$ is the corresponding curve $\\mathcal {C}_{\\mathcal {\\mathcal {F}}_{p}}$ supporting the sheaf $\\mathcal {\\mathcal {F}}_{p}$ .",
"We conclude that $\\mathcal {C}$ and $\\text{div}^{-1}\\mathcal {C}$ are the same families of divisors on $S$ so that we have a short exact sequence ${0[r] & \\text{div}^{*}(\\mathcal {L}^{*}\\boxtimes \\mathcal {O}(-1))[r] & \\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )\\times S}[r] & \\mathcal {O}_{\\mathcal {C}}[r] & 0}$ and $[\\mathcal {O}_{\\mathcal {C}}]=\\text{div}^{*}[\\mathcal {O}_{\\mathcal {D}}]$ .",
"Thus we have $\\hat{p}_{*}^{\\mathcal {P}_{\\chi }(S,\\beta )}\\left(\\left[\\mathcal {O}_{\\mathcal {C}}\\right]q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right) & =\\hat{p}_{*}^{\\mathcal {P}_{\\chi }(S,\\beta )}\\left(\\text{div}^{*}\\left[\\mathcal {O}_{\\mathcal {D}}\\right].\\text{div}^{*}q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right)\\\\& =\\text{div}^{*}\\hat{p}_{*}^{|\\mathcal {L}|}\\left(\\left[\\mathcal {O}_{\\mathcal {C}}\\right].q_{S}^{*}\\left[\\mathcal {O}_{s}\\right]\\right)\\\\& =\\text{div}^{*}\\left(1-\\left[\\mathcal {O}(-1)\\right]\\right)$ americanWe also have similar result for $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ if we replace $\\mathcal {O}_{\\mathcal {C}}$ with $\\mathcal {O}_{\\text{div}\\,\\pi _{*}\\mathcal {\\mathcal {F}}}$ .",
"Proposition 22 Let $\\mathcal {O}_{s}$ be the structure sheaf of the points $s\\in S$ .",
"Then $\\hat{p}([\\mathcal {O}_{\\text{div}\\pi _{*}\\mathcal {\\mathcal {F}}}].q_{S}^{*}[\\mathcal {O}_{s}])=1-\\text{div}^{*}(\\mathcal {O}(-1))$ where $\\mathcal {O}(-1)$ is the tautological bundle of $|\\mathcal {L}|$ and $\\hat{p}$ , $q_{S}$ are morphism from diagram REF .",
"From the definition of the morphism $\\text{div}$ , $\\text{div}\\,\\pi _{*}\\mathcal {\\mathcal {F}}$ is exactly $\\text{div}^{-1}\\mathcal {D}$ .",
"Thus we can use exactly the same proof as the previous Proposition.",
"englishLater we will drop $\\text{div}^{*}$ from $\\text{div}^{*}\\mathcal {O}(-1)$ for simplicity.",
"american"
],
[
"Refinement of Kool-Thomas invariants",
"Assume that $b_{1}(S)=0$ .",
"From Proposition REF and Proposition REF , the contribution of $\\mathcal {P}_{\\chi }(S,\\beta )$ to $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ and to $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)$ are equal.",
"Consider the contribution of $\\mathcal {P}_{\\chi }(S,\\beta )$ to $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)$ invariants, i.e.",
"$\\Xi =R\\Gamma \\left(\\mathcal {P}_{\\chi }(S,\\beta ),\\frac{\\mathcal {O}_{\\mathcal {P}_{\\chi }(S,\\beta )}^{vir}\\otimes K_{vir}^{\\frac{1}{2}}}{\\bigwedge ^{\\bullet }(N^{vir})^{\\vee }}\\prod _{i=1}^{m}\\frac{\\bar{\\gamma }(\\mathcal {O}_{s_{i}})}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right).$ englishOn $\\text{Hilb}_{\\beta }(S)\\times S$ we have the following exact sequence ${0[r] & \\mathcal {O}[r]^{s_{\\mathcal {C}}} & \\mathcal {O}(\\mathcal {C})[r] & \\mathcal {O}_{\\mathcal {C}}(\\mathcal {C})[r] & 0}$ which induces the exact sequence ${H^{1}(\\mathcal {O}_{\\mathcal {C}}(\\mathcal {C}))[r]^{\\hat{\\phi }} & H^{2}(\\mathcal {O}_{S})[r] & H^{2}(\\mathcal {L})}.$ If $H^{2}(\\mathcal {L})=0$ then $\\hat{\\phi }$ is surjective.",
"Observe that $R\\pi _{H*}\\mathcal {O}_{\\mathcal {C}}(\\mathcal {C})$ is the complex $\\mathcal {E}^{\\bullet }$ from Subsection 2.2.1 when $\\chi =2-h$ or equivalently when $n=1$ .",
"For $n>1$ , it was shown in Appendix A of [12] that $\\mathcal {E}^{\\bullet }$ sits in the exact triangle ${R\\pi _{H*}\\mathcal {O}_{\\mathcal {C}}(\\mathcal {C})[r] & \\mathcal {E}^{\\bullet }[r] & E^{\\bullet }}.$ Thus if $h^{2}(\\mathcal {O}_{S})>0$ then $\\mathcal {E}^{\\bullet }$ contain a trivial bundle so that $[\\mathcal {P}_{\\chi }(S,\\beta )]^{vir}$ vanish.",
"In particular, by virtual Riemann-Roch the contribution of $\\mathcal {P}_{\\chi }(S,\\beta )$ is zero.",
"If $H^{2}(\\mathcal {O}_{S})=0$ , $\\mathcal {E}_{red}^{\\bullet }$ and $\\mathcal {E}^{\\bullet }$ are quasi isomorphic.",
"Let $P$ english be the moduli space $\\mathcal {P}_{\\chi }(S,\\beta ).$ By the virtual Riemann-Roch theorem and by Lemma REF we then have english $ch^{G}\\left(\\Xi \\right) & =\\left(-\\mathfrak {t}^{-\\frac{1}{2}}\\right)^{vd}\\int _{[P]^{red}}\\text{ch}\\left({\\textstyle \\bigwedge _{-\\mathfrak {t}}}\\mathcal {E}_{red}^{\\bullet }\\left(\\frac{{\\textstyle \\bigwedge _{-1}}\\mathcal {O}(-1)}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}\\right).\\text{td}\\left(T_{P}^{red}\\right)$ where $T_{P}^{red}$ is the derived dual of $\\mathcal {E}_{red}^{\\bullet }$ and $\\left(-\\mathfrak {t}^{-\\frac{1}{2}}\\right)^{vd}$ should be understood as $\\left(-e^{-\\frac{1}{2}t}\\right)^{vd}$ where $t$ is the equivariant first Chern class of $\\mathfrak {t}$ .",
"Observe that $ch^{G}\\left(\\Xi \\right)$ can be computed whenever $H^{2}(\\mathcal {L})=0$ without assuming $h^{2}(\\mathcal {O}_{S})=0$ .",
"Thus for $S$ with $b_{1}(S)=0$ and a line bundle $\\mathcal {L}$ with $H^{2}(\\mathcal {L})=0$ , we define $P_{S,\\mathcal {L},m,\\chi }=ch^{G}(\\Xi )$ .",
"The $K$ -theory class of $\\mathcal {E}_{red}^{\\bullet }$ is given by equation (REF ).",
"Since $\\mathcal {O}(\\mathcal {C})=\\mathcal {L}\\boxtimes \\mathcal {O}(1)$ , by the projection formula we have $F=H^{0}(\\mathcal {L}(A)\\vert _{A})\\otimes \\mathcal {O}(1)$ .",
"From the exact sequence ${0[r] & \\mathcal {O}(\\mathcal {C})[r] & \\mathcal {O}(\\mathcal {C}+A)[r] & \\mathcal {O}_{\\pi _{S}^{-1}A}(\\mathcal {C}+\\pi _{S}^{-1}A)[r] & 0}$ on $P$ , and since $H^{i>0}(\\mathcal {L}(A)\\vert _{A})=0$ , we conclude that $F & =\\mathcal {O}(1)^{\\oplus \\chi (\\mathcal {L}(A))-\\chi (\\mathcal {L})}$ And again by projection formula we have $\\mathcal {O}(\\mathcal {C})^{[n]}=\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)$ .",
"By Theorem A.7 of [12] we then can compute $P_{S,\\mathcal {L},m,\\chi }$ as $\\left(-\\mathfrak {t}^{-\\frac{1}{2}}\\right)^{v}\\int \\limits _{S^{[n]}\\times |\\mathcal {L}(A)|}H^{\\chi (\\mathcal {L}(A))-\\chi (\\mathcal {L})}c_{n}(\\mathcal {O}(\\mathcal {D}-A)^{[n]})\\\\\\text{ch}\\left(\\frac{{\\textstyle \\bigwedge _{-\\mathfrak {t}}}\\mathcal {E}_{red}^{\\bullet }\\left(\\bigwedge _{-1}\\mathcal {O}(-1)\\right)^{m}}{\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)^{m}}\\right)\\text{td}\\left(T_{P}^{red}\\right)$ where $H=c_{1}(\\mathcal {O}(1)$ and $n=\\chi +h-1$ .",
"Theorem 23 $P_{S,\\mathcal {L},m,\\chi }\\vert _{\\mathfrak {t}=1}=(-1)^{vd}\\int _{S^{[n]}\\times \\mathbb {P}^{\\varepsilon }}c_{n}(\\mathcal {L}^{[n]}\\otimes \\mathcal {O}(1))\\frac{c_{\\bullet }\\left(TS^{[n]}\\right)c_{\\bullet }(\\mathcal {O}(1))^{\\chi (\\mathcal {L})}}{c_{\\bullet }\\left(\\mathcal {L}^{\\lbrace n]}\\boxtimes \\mathcal {O}(1)\\right)}$ where $\\varepsilon =\\chi (\\mathcal {L})-1-m$ .",
"Thus we can relate Kool-Thomas invariants with our invariants as follows: $\\mathcal {P}_{\\chi ,\\beta }^{red}\\left(S,[pt]^{m}\\right) & = & \\left(-1\\right)^{m}t^{m+1-\\chi (\\mathcal {O}_{S})}\\left.P_{S,\\mathcal {L},m,\\chi }\\right|_{\\mathfrak {t}=1}.$ Let $\\mathcal {X}_{-\\mathfrak {t}}(T_{P}^{red}):=\\text{ch}\\left(\\bigwedge _{-\\mathfrak {t}}\\mathcal {E}_{red}^{\\bullet }\\right)\\text{td}\\left(T_{P}^{red}\\right)$ and let $d:=\\text{rk}\\mathcal {E}_{red}^{\\bullet }=n+\\chi (\\mathcal {L})-1$ be the virtual dimension of $P$ so that we can rewrite (REF ) as $\\left(-1\\right)^{m}\\left(-\\mathfrak {t}^{-\\frac{1}{2}}\\right)^{d-m}\\int _{S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}}c_{n}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)\\mathcal {X}_{-\\mathfrak {t}}\\left(T_{P}^{red}\\right)\\text{ch}\\left(\\frac{{\\textstyle \\bigwedge _{-1}}\\left(\\mathcal {O}(-1)\\right)}{1-\\mathfrak {t}}\\right)^{m}$ By Proposition 5.3 of [5] we can write $\\mathcal {X}_{-\\mathfrak {t}}(T_{P}^{red})=\\sum _{l=0}^{d}\\left(1-\\mathfrak {t}\\right)^{d-l}\\mathcal {X}^{l}$ where $\\mathcal {X}^{l}=c_{l}(T_{P}^{red})+b_{l}$ where $b_{l}\\in A^{>l}(P)$ .",
"Then we can write $P_{S,\\mathcal {L},m,\\chi }$ as $\\left(-1\\right)^{m}\\left(-\\mathfrak {t}^{-\\frac{1}{2}}\\right)^{d-m}\\int _{S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}}c_{n}(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1))\\sum _{l=0}^{d}(1-\\mathfrak {t})^{d-m-l}\\mathcal {X}^{l}\\text{ch}\\left({\\textstyle \\bigwedge _{-1}}\\left(\\mathcal {O}(-1)\\right)\\right)^{m}.$ Note that $\\text{ch}\\left({\\textstyle \\bigwedge _{-1}}\\left(\\mathcal {O}(-1)\\right)\\right)^{m}=H^{m}+O\\left(H^{m+1}\\right)$ so that $\\int _{S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}}c_{n}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)\\mathcal {X}^{l}\\text{ch}\\left({\\textstyle \\bigwedge _{-1}}\\left(\\mathcal {O}(-1)\\right)\\right)^{m}=0$ for $l>d-m$ .",
"Thus the summation ranges from $l=0$ to $l=d-m$ .",
"In this range the power of $(1-\\mathfrak {t})$ is positive except when $l=d-m$ in which the power of $(1-\\mathfrak {t})$ is zero.",
"Thus we can conclude that $P_{S,\\mathcal {L},m,\\chi }\\vert _{\\mathfrak {t}=1}$ equals to $\\left(-1\\right)^{m}\\left(-\\mathfrak {t}^{-\\frac{1}{2}}\\right)^{d-m}\\int _{S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}}c_{n}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)\\mathcal {X}^{d-m}\\text{ch}\\left({\\textstyle \\bigwedge _{-1}}\\left(\\mathcal {O}(-1)\\right)\\right)^{m}.$ Since $b_{d-m}\\in A^{>d-m}(P)$ and $c_{d-m}(T_{P}^{red})\\in A^{d-m}(P)$ we have $\\int _{S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}}c_{n}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)b_{d-m}\\text{ch}\\left({\\textstyle \\bigwedge _{-1}}\\left(\\mathcal {O}(-1)\\right)\\right)^{m}=0$ and $\\int _{S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}}c_{n}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)c_{d-m}(T_{P}^{red})H^{k}=0$ for $k>m$ and we can conclude that $P_{S,\\mathcal {L},m,\\chi }\\vert _{\\mathfrak {t}=1} & =\\left(-1\\right)^{\\frac{1}{2}d}\\int _{S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}}c_{n}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right).H^{m}.c_{d-m}(T_{P}^{red})$ From (REF ) and (REF ) we have $T_{P}^{red}=T\\left(S^{[n]}\\right)+\\mathcal {O}(1)^{\\chi (\\mathcal {L}(A))}-\\mathcal {O}-\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)-\\mathcal {O}(1)^{\\chi (\\mathcal {L}(A))-\\chi (\\mathcal {L})}$ and $c_{d-m}(T_{P}^{red})=\\text{Coeff}_{t^{d-m}}\\left[\\frac{c_{t}\\left(TS^{[n]}\\right)c_{t}\\left(\\mathcal {O}(1)\\right)^{\\chi (\\mathcal {L})}}{c_{t}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)}\\right].$ Finally we conclude that $P_{S,\\mathcal {L},m,\\chi }\\vert _{\\mathfrak {t}=1}=\\left(-1\\right)^{-\\frac{1}{2}d}\\int _{S^{[n]}\\times \\mathbb {P}^{\\delta }}c_{n}(\\mathcal {L}^{[n]}\\otimes \\mathcal {O}(1))\\frac{c_{\\bullet }\\left(TS^{[n]}\\right)c_{\\bullet }(\\mathcal {O}(1))^{\\chi (\\mathcal {L})}}{c_{\\bullet }\\left(\\mathcal {L}^{\\lbrace n]}\\boxtimes \\mathcal {O}(1)\\right)}$ Let $X_{-y}(x)\\in \\mathbb {Q}[[x,y]]$ defined by $X_{-y}(x):=\\frac{x\\left(y^{-\\frac{1}{2}}-y^{\\frac{1}{2}}e^{-x(y^{-\\frac{1}{2}}-y^{\\frac{1}{2}})}\\right)}{1-e^{-x\\left(y^{-\\frac{1}{2}}-y^{\\frac{1}{2}}\\right)}}.$ For a vector bundle $E$ on a scheme $Y$ of rank $r$ with Chern roots $x_{1},\\ldots ,x_{r}$ we will use $X_{-y}(E)$ to denote $\\prod _{i=1}^{r}\\frac{x_{i}\\left(y^{-\\frac{1}{2}}-y^{\\frac{1}{2}}e^{-x_{i}(y^{-\\frac{1}{2}}-y^{\\frac{1}{2}})}\\right)}{1-e^{-x_{i}\\left(y^{-\\frac{1}{2}}-y^{\\frac{1}{2}}\\right)}}.$ Observe that $X_{-y}$ is additive on an exact sequence of vector bundle.",
"Thus we can extend $X_{y}$ to $K(Y)$ .",
"For a class $\\beta \\in K(Y)$ we can write $\\beta =\\sum _{i}[E_{i}^{+}]-\\sum _{j}[E_{j}^{-}]$ for vector bundles $E_{i}^{+},E_{j}^{-}$ and we can define $X_{-y}(\\beta )=\\frac{\\prod _{i}X_{-y}(E_{i}^{+})}{\\prod _{j}X_{-y}(E_{j}^{-})}.$ For a proper nonsingular scheme $Y$ with tangent bundle $T_{Y}$ $\\int _{Y}X_{-y}\\left(T_{Y}\\right)=\\left(\\frac{1}{y}\\right)^{\\frac{1}{2}d}\\sum _{i}(-1)^{p+q}y^{q}h^{p,q}(Y)$ where $h^{p,q}(Y)$ are the Hodge number of $Y$ i.e.",
"$\\int _{Y}X_{-y}\\left(T_{Y}\\right)$ is the normalized $\\chi _{-y}$ genus.",
"Theorem 24 $P_{S,\\mathcal {L},m,\\chi }=\\\\(-1)^{vd}\\int \\limits _{\\left[P\\right]^{red}}\\frac{X_{-\\mathfrak {t}}\\left(TS^{[n]}\\right)}{X_{-\\mathfrak {t}}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)}X_{-\\mathfrak {t}}\\left(\\mathcal {O}(1)\\right)^{\\delta +1}\\left(\\frac{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}e^{-H\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}H^{m}$ where $[P]^{red}$ is $c_{n}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)\\cap [S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}]$ .",
"american $P_{S,\\mathcal {L},m,\\chi }$ equals to (REF ), and we can rewrite it as english $P_{S,\\mathcal {L},m,\\chi }=\\left(-1\\right)^{vd}\\int _{\\left[P\\right]^{red}}\\frac{\\prod _{i=1}^{2n+\\chi (\\mathcal {L})-1}\\frac{\\phi _{-\\mathfrak {t}}(\\alpha _{i})}{\\mathfrak {t}^{1/2}}}{\\prod _{i=1}^{n}\\frac{\\phi _{-\\mathfrak {t}}(\\beta _{i})}{\\mathfrak {t}^{1/2}}}\\left(\\frac{1-e^{-H}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}$ where $\\phi _{-\\mathfrak {t}}(x)=\\frac{x\\left(1-\\mathfrak {t}e^{-x}\\right)}{1-e^{-x}}$ and $\\alpha _{i}$ are the Chern roots of $T(S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1})$ and $\\beta _{i}$ are the Chern roots of $\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)$ .",
"Let's define $\\bar{\\phi }_{-\\mathfrak {t}}(x):=\\frac{\\phi _{-\\mathfrak {t}}(x)}{\\mathfrak {t}^{1/2}}=\\frac{x\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}e^{-x}\\right)}{1-e^{-x}}=\\sum _{i\\ge 0}\\bar{\\phi }_{i}x^{i}.$ Note that this power series starts with $\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}$ .",
"By substituting $x$ with $x\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)$ and dividing it by $\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)$ we have the power series $X_{-\\mathfrak {t}}(x)=\\frac{x\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}e^{-x\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}\\right)}{1-e^{-x\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}}=\\sum _{i\\ge 0}\\xi _{i}x^{i}$ such that $\\xi _{0}=1$ and $\\xi _{i}=\\bar{\\phi }_{i}\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)^{i-1}$ .",
"Thus by substituting $x$ in $\\frac{\\prod _{i=1}^{2n+\\chi (\\mathcal {L})-1}\\frac{\\phi _{-y}(\\alpha _{i})}{\\mathfrak {t}^{1/2}}}{\\prod _{i=1}^{n}\\frac{\\phi _{-y}(\\beta _{i})}{\\mathfrak {t}^{1/2}}}\\left(\\frac{1-e^{-H}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}$ with $x\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)$ whenever $x=\\alpha _{i},\\beta _{i},H$ and dividing it by $\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)^{n+\\chi (\\mathcal {L})-1}$ so that the coefficients of $q^{n+\\chi (\\mathcal {L})-1}$ in $\\frac{\\prod _{i=1}^{2n+\\chi (\\mathcal {L})-1}X_{-\\mathfrak {t}}(\\alpha _{i}q)}{\\prod _{i=1}^{n}X_{-\\mathfrak {t}}(\\beta _{i}q)}\\left(\\frac{1-e^{-Hq\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}$ and $\\frac{\\prod _{i=1}^{2n+\\chi (\\mathcal {L})-1}\\bar{\\phi }_{-\\mathfrak {t}}(\\alpha _{i}q)}{\\prod _{i=1}^{n}\\bar{\\phi }_{-\\mathfrak {t}}(\\beta _{i}q)}\\left(\\frac{1-e^{-Hq}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}$ are the same.",
"Since $[T\\mathbb {P}^{\\chi (\\mathcal {L})-1}]=[\\oplus _{i=1}^{\\chi (\\mathcal {L})}\\mathcal {O}(1)]-[\\mathcal {O}_{\\mathbb {P}^{\\chi (\\mathcal {L})-1}}],$ $P_{S,\\mathcal {L},m,\\chi }$ equals $\\left(-1\\right)^{vd}\\int _{[P]^{red}}\\frac{X_{-\\mathfrak {t}}\\left(TS^{[n]}\\right)X_{-\\mathfrak {t}}\\left(\\mathcal {O}(1)\\right)^{\\chi (\\mathcal {L})}}{X_{-\\mathfrak {t}}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)}\\left(\\frac{1-e^{-H\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}=\\\\\\left(-1\\right)^{vd}\\int _{[P]^{red}}\\frac{X_{-\\mathfrak {t}}\\left(TS^{[n]}\\right)X_{-\\mathfrak {t}}\\left(\\mathcal {O}(1)\\right)^{\\delta +1}}{X_{-\\mathfrak {t}}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)}\\left(\\frac{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}e^{-H\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}H^{m}$ englishIn the following Corollary we want to complete the computation of $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ .",
"Corollary 25 Given a positive integer $\\delta $ , let $S$ be a smooth projective surface with $b_{1}(S)=0$ .",
"Let $\\mathcal {L}$ be a $2\\delta +1$ -very ample line bundle on $S$ with $c_{1}(\\mathcal {L})=\\beta $ and $H^{i}(\\mathcal {L})=0$ for $i>0$ .",
"Let $X=K_{S}$ be the canonical line bundle over $S$ .",
"Then for $m=\\chi (\\mathcal {L})-1-\\delta $ points $s_{1},\\ldots ,s_{m}$ which is not necessarily different $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)=\\\\\\left(-1\\right)^{vd}\\int _{\\left[P\\right]^{red}}\\frac{X_{-\\mathfrak {t}}\\left(TS^{[n]}\\right)X_{-\\mathfrak {t}}\\left(\\mathcal {O}(1)\\right)^{\\delta +1}}{X_{-\\mathfrak {t}}\\left(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1)\\right)}\\left(\\frac{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}e^{-H\\left(\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}\\right)}}{\\mathfrak {t}^{-1/2}-\\mathfrak {t}^{1/2}}\\right)^{m}H^{m}$ where $[P]^{red}=c_{n}(\\mathcal {L}^{[n]}\\boxtimes \\mathcal {O}(1))\\cap [S^{[n]}\\times \\mathbb {P}^{\\chi (\\mathcal {L})-1}]$ for $m\\ge H^{0}(\\mathcal {L})-1-\\delta $ .",
"americanIf additionally $\\mathcal {O}_{\\mathcal {C}_{\\mathbb {F}}}$ is flat over $\\mathcal {P}_{\\chi }(X,i_{*}\\beta )$ and $s_{1},\\dots ,s_{m}$ are closed points of $S$ in general position such that all curves on $S$ that pass through all $m$ points are reduced and irredcible then $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ is given by the same formula.",
"american By Proposition REF $\\bar{P}_{X,\\beta ,\\chi }\\left(s_{1},\\ldots ,s_{m}\\right)=P_{S,\\mathcal {L},m,\\chi }$ .",
"Similarly for $P_{X,\\beta ,\\chi }(s_{1},\\ldots ,s_{m})$ .",
"In [8], [9], for every smooth projective surface $S$ and line bundle $\\mathcal {L}$ on $S$ , Göttsche and Shende defined the following power series $D^{S,\\mathcal {L}}(x,y,w):=\\sum _{n\\ge 0}w^{n}\\int _{S^{[n]}}X_{-y}\\left(TS^{[n]}\\right)\\frac{c_{n}\\left(\\mathcal {L}^{[n]}\\otimes e^{x}\\right)}{X_{-y}\\left(\\mathcal {L}^{[n]}\\otimes e^{x}\\right)}\\in \\mathbb {Q}\\llbracket x,y,w\\rrbracket $ where $e^{x}$ denotes a trivial line bundle with nontrivial $\\mathbb {C}^{\\times }$ action with equivariant first Chern class $x$ .",
"Motivated by this power series we define a generating function $P_{S,\\mathcal {L},m}:=\\sum _{n\\ge 0}\\left(-w\\right)^{n}P_{S,\\mathcal {L},m,n+1-h}.$ where $h$ is the arithmetic genus of the curve $C$ in $S$ with $\\mathcal {O}(C)\\simeq \\mathcal {L}$ so that for the pair $(\\mathcal {\\mathcal {F}},s)\\in \\mathcal {P}_{\\chi }(S,\\beta )$ , $n=\\chi -1+h$ .",
"By Theorem REF , after substituting $\\mathfrak {t}$ by $y$ we can rewrite $P_{S,\\mathcal {L},m}$ as $\\text{Coeff}_{x^{\\delta }}\\left[D^{S,\\mathcal {L}}(x,y,w)X_{-y}(x)^{\\delta +1}\\left(\\frac{y^{-1/2}-y^{1/2}e^{-x\\left(y^{-1/2}-y^{1/2}\\right)}}{y^{-1/2}-y^{1/2}}\\right)^{m}\\right]$ Note that $Q_{S,\\mathcal {L},m}:=\\text{Coeff}_{x^{\\delta }}\\left[D^{S,\\mathcal {L}}(x,y,w)X_{-y}(x)^{\\delta +1}\\right]$ is equation (2.1) of [9] and $\\left(\\frac{y^{-1/2}-y^{1/2}e^{-x\\left(y^{-1/2}-y^{1/2}\\right)}}{y^{-1/2}-y^{1/2}}\\right)^{m}$ is a power series starting with 1. englishIn [9], Gottsche and Shende defined the power series $N_{\\chi (\\mathcal {L})-1-k,[S,\\mathcal {L}]}^{i}(y)$ by the following equation: $\\sum _{i\\in \\mathbb {Z}}N_{\\chi (\\mathcal {L})-1-k,[S,\\mathcal {L}]}^{i}(y)\\left(\\frac{w}{(1-y^{-1/2}w)(1-y^{1/2}w)}\\right)^{i+1-g}=Q_{S,\\mathcal {L},m}$ Motivated by this we also define $M_{\\chi (\\mathcal {L})-1-m,[S,\\mathcal {L}]}^{i}(y)$ as $\\sum _{i\\in \\mathbb {Z}}M_{\\chi (\\mathcal {L})-1-m,[S,\\mathcal {L}]}^{i}(y)\\left(\\frac{w}{(1-y^{-1/2}w)(1-y^{1/2}w)}\\right)^{i+1-g}=P_{S,\\mathcal {L},m}$ Let's define $\\frac{1}{Q}=\\frac{(1-y^{-1/2}w)(1-y^{1/2}w)}{w}=w+w^{-1}-y^{-1/2}-y^{1/2}$ and recall a conjecture from [8].",
"Conjecture 26 (Conjecture 55 of [8]) $\\left(\\frac{w(Q)}{Q}\\right)^{1-g(\\mathcal {L})}D^{S,\\mathcal {L}}(x,y,w(Q))\\in \\mathbb {Q}[y^{-1/2},y^{1/2}]\\llbracket x,xQ\\rrbracket $ Motivated by the conjecture above we define another power series $\\tilde{D}^{S,\\mathcal {L}}(x,y,Q):=\\left(\\frac{w(Q)}{Q}\\right)^{1-g(\\mathcal {L})}D^{S,\\mathcal {L}}(x,y,w(Q)).$ Proposition 27 Assume Conjecture REF .",
"For $\\chi (\\mathcal {L})-1\\ge k\\ge 0$ we have 1.",
"$M_{\\chi (\\mathcal {L})-1-k,[S,\\mathcal {L}]}^{i}(y)=0$ and $N_{\\chi (\\mathcal {L})-1-k}^{i}(y)=0$ for $i>\\chi (\\mathcal {L})-1-k$ and for $i\\le 0$ .",
"2.",
"$M_{\\chi (\\mathcal {L})-1-k,[S,\\mathcal {L}]}^{i}(y)$ and $N_{\\chi (\\mathcal {L})-1-k}^{i}(y)$ are Laurent polynomials in $y^{1/2}$ .",
"3.",
"Furthermore $M_{\\chi (\\mathcal {L})-1-k,[S,\\mathcal {L}]}^{\\chi (\\mathcal {L})-1-k}(y)=N_{\\chi (\\mathcal {L})-1-k,[S,\\mathcal {L}]}^{\\chi (\\mathcal {L})-1-k}(y)$ .",
"Moreover $\\sum _{i\\ge 0}M_{\\delta ,[S,\\mathcal {L}]}^{\\delta }(y)\\left(s\\right)^{\\delta }=\\tilde{D}^{S,\\mathcal {L}}(x,y,\\frac{s}{x})|_{x=0}=\\sum _{\\delta \\ge 0}N_{\\delta ,[S,\\mathcal {L}]}^{\\delta }(y)s^{\\delta }$ After substituting $w$ by $w(Q)$ we rewrite equation (REF ) and (REF ) $\\sum _{i\\in \\mathbb {Z}}N_{\\delta ,[S,\\mathcal {L}]}^{i}(y)x^{\\delta -i}\\left(xQ\\right)^{i}=\\left[\\tilde{D}^{S,\\mathcal {L}}(x,y,Q)X_{-y}(x)^{\\delta +1}\\right]_{x^{\\delta }}$ $\\sum _{i\\in \\mathbb {Z}}M_{\\delta ,[S,\\mathcal {L}]}^{i}(y)x^{\\delta -i}\\left(xQ\\right)^{i}=\\\\\\left[\\tilde{D}^{S,\\mathcal {L}}(x,y,Q)X_{-y}(x)^{\\delta +1}\\left(\\frac{y^{1-/2}-y^{1/2}e^{-x\\left(y^{-1/2}-y^{1/2}\\right)}}{y^{-1/2}-y^{1/2}}\\right)^{m}\\right]_{x^{\\delta }}.$ By Conjecture REF $\\sum _{i\\in \\mathbb {Z}}N_{\\delta ,[S,\\mathcal {L}]}^{i}(y)x^{\\delta -i}\\left(xQ\\right)^{i},\\sum _{i\\in \\mathbb {Z}}M_{\\delta ,[S,\\mathcal {L}]}^{i}(y)x^{\\delta -i}\\left(xQ\\right)^{i}\\in \\mathbb {Q}[y^{-1/2},y^{1/2}]\\llbracket x,xQ\\rrbracket $ so that the only possible power of $Q$ that could appear is $i=0,\\ldots ,\\delta $ .",
"We can directly conclude that $N_{\\delta ,[S,\\mathcal {L}]}^{i},M_{\\delta ,[S,\\mathcal {L}]}^{i}$ are Laurent polynomial in $y^{1/2}.$ Set $s=xQ,$ so that by Conjecture REF we can write $\\tilde{D}^{S,\\mathcal {L}}(x,y,Q)$ as power series of $x$ and $s$ i.e $\\tilde{D}^{S,\\mathcal {L}}(x,y,\\frac{s}{x})\\in \\mathbb {Q}[y^{-1/2},y^{1/2}]\\llbracket x,s\\rrbracket .$ And since $X_{-y}(x=0) & =1\\\\\\left(\\frac{y^{1-/2}-y^{1/2}e^{-x\\left(y^{-1/2}-y^{1/2}\\right)}|_{x=0}}{y^{-1/2}-y^{1/2}}\\right)^{m} & =1$ we can conclude that $\\sum _{i\\ge 0}M_{\\delta ,[S,\\mathcal {L}]}^{\\delta }(y)\\left(s\\right)^{\\delta } & =\\tilde{D}^{S,\\mathcal {L}}(x,y,\\frac{s}{x})|_{x=0}\\\\& =\\sum _{\\delta \\ge 0}N_{\\delta ,[S,\\mathcal {L}]}^{\\delta }(y)s^{\\delta }$ americanIf $H^{i}(\\mathcal {L})=0$ for $i>0$ and $\\mathcal {L}$ is $\\delta $ -very ample, then $N_{\\delta ,[S,\\mathcal {L}]}^{\\delta }(y)$ is the refinement defined by Goettsche and Shende in [8] of $n_{\\delta }(\\mathcal {L})$ that computes the number of $\\delta $ -nodal curves in $|\\mathcal {L}|$ .",
"Theorem REF and Theorem REF gives geometric argument for the equality $M_{\\delta ,[S,\\mathcal {L}]}^{\\delta }(y)|_{y=1}=N_{\\delta ,[S,\\mathcal {L}]}^{\\delta }(y)|_{y=1}$ .",
"Without assuming the conjecture above we would like to know if Proposition REF still true.",
"english"
],
[
"Acknowledgements",
"This project start as my PhD thesis under the supervision of Lothar Göttsche under ICTP/SISSA joint PhD program.",
"The finisihing of this project is supported by P3MI ITB research funds.",
"american tocsectionBibliographyenglish"
]
] | 2012.05278 |
[
[
"Dynamical System Segmentation for Information Measures in Motion"
],
[
"Abstract Motions carry information about the underlying task being executed.",
"Previous work in human motion analysis suggests that complex motions may result from the composition of fundamental submovements called movemes.",
"The existence of finite structure in motion motivates information-theoretic approaches to motion analysis and robotic assistance.",
"We define task embodiment as the amount of task information encoded in an agent's motions.",
"By decoding task-specific information embedded in motion, we can use task embodiment to create detailed performance assessments.",
"We extract an alphabet of behaviors comprising a motion without \\textit{a priori} knowledge using a novel algorithm, which we call dynamical system segmentation.",
"For a given task, we specify an optimal agent, and compute an alphabet of behaviors representative of the task.",
"We identify these behaviors in data from agent executions, and compare their relative frequencies against that of the optimal agent using the Kullback-Leibler divergence.",
"We validate this approach using a dataset of human subjects (n=53) performing a dynamic task, and under this measure find that individuals receiving assistance better embody the task.",
"Moreover, we find that task embodiment is a better predictor of assistance than integrated mean-squared-error."
],
[
"Introduction",
"Motion signals encode information about the underlying task being executed, yet the form this information takes may vary.",
"Typically, we represent motion using continuous real-valued signals.",
"While this representation can provide detailed descriptions of an agent's motions, it can be cumbersome.",
"However, based on our choice of representation, we can compress motion signals while preserving information about the task [1].",
"In [2], the authors propose that human motions are the result of the composition of a finite set of premotor signals emanating from the spinal cord.",
"As a consequence, the neurological feasibility of motion decomposition forms the basis for action in movement primitives, also known as movemes [3].",
"Movemes are fundamental units of motion, and derive their name from their linguistic analogue: phonemes.",
"Thus, all smooth human motions may be comprised of symbolic sequences drawn from an alphabet of movemes.",
"Movemes motivate the application of information measures in human motion analysis, because they provide evidence of finite structure in otherwise continuous motion signals.",
"Moreover, the existence of movemes indicates that under some choice of representation human motion can be discretized without loss of information.",
"In the human motion analysis literature, movemes are often characterized using causal dynamical systems [3], [4], or hybrid system identification methods, such as autoregressive models [5].",
"Most motor signal segmentation methods demand prior specification of the moveme alphabet either through direct template matching or manual labeling of training data, which limits their use in exploratory analyses where the structure of the alphabet may not be known a priori.",
"Techniques in symbolic dynamic filtering can generate symbolic alphabets by creating partitions of the state-space using methods such as maximum entropy partitioning [6].",
"Additionally, state-space partition techniques can be applied to nonlinear transformations of the space via methods such as wavelet transforms [7].",
"However, the symbols synthesized by these spatial techniques are quasi-static, and are not designed to describe the dynamic nature of movemes.",
"Automatic segmentation methods based on Hidden Markov Models (HMMs) describe complex motion as the stochastic evolution of discrete hidden states.",
"Pure HMM-based approaches have been used to directly classify movemes from sequential observations.",
"States in the HMM learn an observation distribution that describes each moveme.",
"Movement primitive HMMs (MP-HMMs) model the temporal phases of an individual primitive and have been used to assess movement performance and generate detailed models of motion [8].",
"Since the model's hidden states are represented by observation distributions, typical HMMs and MP-HMMs—as well as their higher-order and hierarchical variants—all use static symbols, which limits their use in characterizing movemes.",
"Alternatively, Switched Linear Dynamical Systems (SLDS) partition nonlinear systems into a set of piecewise-continuous linear dynamical systems, whose transitions are modeled by the stochastic dynamics of a hidden switching variable described by an HMM [9].",
"Movemes in an SLDS model are represented by linear dynamical systems, which ensures the symbolic alphabet is comprised of dynamic elements.",
"In [10], the authors apply SLDS to model discrete movements, and generate automatic segmentations after training on a manually labeled dataset.",
"While these methods give a trained user tools to analyze motor performance, they rely on the user's ability to identify movemes by themselves or with the aid of a template.",
"This is particularly an issue for analyses of atypical or impaired motion where asymmetries arise, and signals may not match a given template with standard features.",
"Thus, supervised learning approaches are generally not as effective when the structure of the symbolic alphabet is not known ahead of time.",
"In order to address cases with an unknown number of symbols in the alphabet, the authors of [11] introduced a recursive identification framework using switched autoregressive systems without presupposing the number of models required, while work by [12] established a similar algorithm based on Bayesian inference over SLDS model parameters given some prior belief.",
"Additionally, unsupervised learning techniques have been employed for decades in the fields of image processing and computer vision to find and label key features in dense sources of data such as video [13], [14], [15].",
"Within human motion analysis, motion capture setups are a common means of data acquisition.",
"In this setting, unsupervised segmentations of human motion capture data based on cascading linear dynamical systems may correspond to the identification of movemes from video sequences [16].",
"However, by modeling the switches between movemes with exclusively temporal dependencies (i.e.",
"with Markovian dynamics), any state-space dependence on the switching conditions between movemes is unmodeled, failing to provide transition guard conditions.",
"Our approach synthesizes finite sets of dynamic symbols from agents' motions without any prior system knowledge, while modeling state-space dependencies between symbols.",
"Motivated by movemes and movement primitives, we define behaviors as moveme analogues for general systems, and specify them using finite-dimensional nonlinear causal dynamical systems.",
"Making a choice of representation for behaviors is very important.",
"While there exist many data-driven function approximation methods, we choose the Koopman operator to represent behaviors because they are capable of capturing nonlinearities within a linear systems framework [17].",
"By identifying features from motion signals and constructing an alphabet of behaviors, we directly encode task-specific information into the symbolic representation.",
"Information is an ordered sequence of symbols drawn from an alphabet generated by a source [1].",
"We define task information with respect to a source, such that the source generates sequences of symbols comprising realizations of the task.",
"Any physical system or agent attempting a given task is a task-specific information source.",
"Throughout this study, we represent task information using the distribution of relative frequencies of symbols within a given set of realizations of the underlying system.",
"Additionally, we define task embodiment as a measure of task information encoded in an agent's trajectories by calculating their relative entropy with respect to a reference symbol distribution.",
"The task embodiment formalism is important because it is agnostic to specification of the system, task, or symbolic representation, which allows us to analyze motion signals generally.",
"Task embodiment provides a framework for making motor performance assessments in tasks where measures such as error are unsuitable.",
"Additionally, the proposed unsupervised segmentation technique is capable of generating symbolic partitions of complex, continuous movements that are not amenable to supervised or template-based approaches—even when the movements are atypical and asymmetrical.",
"Consider human walking as an example.",
"Gait phase partitioning is a canonical problem in the motor segmentation literature [18], [19].",
"Supervised segmentation methods work well for analyzing healthy gaits due to an abundance of data and clinically verified motion templates.",
"Since most impairments are unique there are no equivalent databases for atypical gaits, demanding unsupervised techniques to model gaits such as those proposed in this work.",
"Detailed models of atypical motion can facilitate the development of sophisticated robotic assistance, including methods for exoskeleton-assisted gait.",
"Additionally, performance assessments of gaits typically use heuristic-based approaches.",
"Typical notions of error are ill-suited to comparing gaits since they depend on analyzing joint trajectories which do not easily generalize from one individual to another.",
"However, gait cycles could be systematically compared by using measures of task embodiment.",
"The primary contributions of this paper are the following.",
"First, we develop a methodology for data-driven partitioning of dynamical systems.",
"These partitions are projections onto the state-space that can be used to extract an alphabet of system behaviors, and can be represented by a graph.",
"Second, we demonstrate that by tracking relative frequencies of behaviors we can discern relationships in human motion, such as whether an individual is receiving task assistance.",
"We apply our information-theoretic approach to a dataset of human subjects $(n=53)$ performing a dynamic task where assistance is sometimes provided, and extract an alphabet of optimal behaviors based on a synthesized exemplar agent.",
"By tracking the relative frequencies of finite behaviors in human subjects and comparing to those of the optimal agent, we are able to quantify the degree of task embodiment, and determine whether a subject received assistance.",
"We validate the performance of task embodiment by using integrated mean-squared-error (MSE) as a baseline, and find that task embodiment is a good predictor of assistance."
],
[
"Methods",
"An agent's state trajectories simultaneously encode information about the system dynamics and the task it executes.",
"By examining system trajectories, one can uncover patterns in how it traverses the underlying state-space manifold.",
"We propose Dynamical System Segmentation (DSS): a nonparametric, unsupervised, data-driven algorithm for creating low-dimensional, graphical representations of system behaviors by generating partitions of the state-space manifold sensitive to the underlying distribution of task information."
],
[
"Koopman Operators",
"We use Koopman operators as our choice of representation for system behaviors because they are capable of representing nonlinear systems within a compact, linear form amenable to control hierarchies.",
"Consider dynamical systems described by $x_{k+1} = F(x_k) = x_k + \\int _{t_k}^{t_k+\\Delta t} f(x(\\tau ))d\\tau ,$ where $x \\in \\pazocal {M}$ is an $n$ -dimensional state evolving on a smooth manifold according to the flow map $F: \\pazocal {M} \\rightarrow \\pazocal {M}$ , which can be related to an analogous continuous-time system $f(x)$ by the discretization shown above.",
"The Koopman operator is an infinite-dimensional linear operator capable of describing the evolution of any measure-preserving dynamical system, through its action on system observables [17].",
"The operator describes the evolution of observables $g: \\pazocal {M} \\rightarrow \\mathbb {R}$ , which are elements of an infinite-dimensional Hilbert space.",
"Although typically observables are taken from the space of Lebesgue square-integrable functions, other $L^p$ measure spaces are valid as well [20].",
"The action of the infinite-dimensional Koopman operator, $\\pazocal {K}$ , on an observable $g$ is given by $g(x_{k+1}) = g(F(x_k)) = \\pazocal {K}g(x_k).$ Despite their infinite-dimensionality, Koopman operators can be approximated in finite dimensions, and used to describe nonlinear dynamical systems as a result of the development of methods such as those in [21], [22].",
"Given a dataset $X = [x_0,...,x_M]$ consisting of a single time-series of observations from a realization of a dynamical system, one must first choose a set of basis functions with which to span some subspace of the underlying function space, $\\lbrace z_1(x),...,z_N(x)\\rbrace , \\ s.t.",
"\\ z_i: \\pazocal {M} \\rightarrow \\mathbb {R}, \\ \\forall i \\in \\lbrace 1,...,N\\rbrace $ , where we can define $\\psi (x) = [z_1(x),...,z_N(x)]^T, \\ s.t.",
"\\ \\psi : \\pazocal {M} \\rightarrow \\mathbb {R}^N$ as a vector-valued function encompassing the action of all basis functions on a given system state.",
"We want to develop a mapping between current states and their evolution from the time-series of observations $X$ .",
"By defining a transformed dataset $\\Psi _{X} = [\\psi (x_0),...,\\psi (x_{M-1})]^T$ , and its evolution $\\Psi _{X^{\\prime }} = [\\psi (x_1),...,\\psi (x_{M})]^T$ , we can describe such a mapping as $\\Psi _{X^{\\prime }} = \\Psi _XK+r(X)$ , where $K$ is a finite-dimensional approximation of the infinite operator $\\pazocal {K}$ , and $r(X)$ is a residual error due to the approximation.",
"We can minimize the residual $r(X)$ over the squared-error loss functional $\\underset{K}{\\text{min}} \\ \\frac{1}{2}\\sum _{k=1}^{M-1}||\\psi (x_{k+1})-K\\psi (x_k)||^2,$ with closed-form solution $K = G^\\dagger A,$ where $\\dagger $ denotes the Moore-Penrose pseudoinverse and $G = \\frac{1}{M}\\sum _{k=0}^{M-1} \\psi (x_k)\\psi (x_k)^T, \\ A = \\frac{1}{M}\\sum _{k=0}^{M-1} \\psi (x_k)\\psi (x_{k+1})^T.$ We obtain a matrix $K \\in \\mathbb {R}^{N\\times N}$ that is an estimate of the system dynamics over the observed domain of the data [23].",
"We will use Koopman operators to describe individual behaviors expressed in data.",
"Figure: Segmentation of the (θ,θ ˙)(\\theta ,\\dot{\\theta }) phase portrait of an optimal control solution to the cart-pendulum inversion problem."
],
[
"Dynamical System Segmentation",
"DSS characterizes all system behaviors over the state-space by synthesizing a non-redundant set of local estimates of the true system dynamics using a collection of Koopman operators.",
"Given a dataset $X = [x_0,...,x_M]$ consisting of a single realization of a dynamical system of the same form as in Eq.",
"REF , and a set of basis functions described by the vector-valued function $\\psi (x) \\ s.t \\ \\psi : \\pazocal {M} \\rightarrow \\mathbb {R}^N$ , we can apply the basis functions onto the dataset $X$ in order to generate a transformed dataset $\\Psi _{X} = [\\psi (x_0),...,\\psi (x_{M})]^T$ .",
"We then split the transformed dataset $\\Psi _{X}$ into a set of $W+1$ overlapping rectangular windows, and calculate a Koopman operator for each, thereby generating a set of symbols $\\mathbb {K} = \\lbrace K_0,...,K_W\\rbrace $ .",
"However, depending on the system under study, the size of the dataset, and choice of window size and overlap percentage, some of these symbols may be redundant.",
"We are interested in creating a minimal alphabet of Koopman operators with which to span all system behaviors.",
"Unsupervised learning methods such as clustering algorithms that specialize in the identification of classes within datasets are well-suited for this task.",
"By considering each $\\mathbb {R}^{N\\times N}$ Koopman operator as a point in $\\mathbb {R}^{N^2}$ space, we can divide the set $\\mathbb {K}$ into subsets using a clustering algorithm.",
"In particular, we use Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN), which is a nonparametric clustering algorithm that performs well in large databases subject to noise [24].",
"The algorithm groups the operators into $B+1$ classes $\\lbrace C_0,...,C_B\\rbrace $ using only the minimum of number of points required to make up a cluster as a parameter.",
"We compose a set $\\overline{\\mathbb {K}}=\\lbrace \\overline{K}_0,...,\\overline{K}_B\\rbrace $ of class exemplars by taking a weighted-average of all $K_i \\in C_j, \\ \\forall j \\in \\lbrace 0,...,B\\rbrace $ , according to the class-membership probability $p(K_i|K_i \\in C_j)$ .",
"The class-membership probability function is provided by the HDBSCAN software package [25].",
"Although we have created a minimal alphabet $\\overline{\\mathbb {K}}$ of system behaviors, it is of interest to project these behaviors onto the state-space manifold from this abstract operator space.",
"We label all points in the transformed dataset $\\Psi _X$ with a label $l\\in \\lbrace 0,...,B\\rbrace $ according to the class label of the Koopman operator each point was used to generate.",
"Then, we train a support vector machine (SVM) classifier, $\\Phi (\\psi (x))$ to project the class labels onto the state-space manifold, thereby generating partitions of the state-space [26].",
"Figures REF & REF depict a cart-pendulum system used for an example application of DSS, where we segment an optimal control solution to the pendulum inversion task.",
"Often the partitions generated by dynamical system segmentation may be intuitively related to the examined task.",
"In Fig.",
"REF , one can see that modes 0 and 1 represent trajectories with negative and positive velocities respectively, while mode 2 represents lower velocity motion and stabilization.",
"Since the state-space trajectories used to train the model encode task-specific information, the behavioral modes do as well.",
"Once a dynamical system has been segmented, the SVM's partitions of the state-space are set, and new data points will be classified according to which partition they fall into.",
"Figure REF shows a cross-section of the partitioned state-space manifold of the optimal controller solution to the cart-pendulum inversion shown in Fig.",
"REF ."
],
[
"Graphical Representation",
"The product of DSS is best represented by a graph.",
"We can define a graph $\\mathbb {G} = (\\overline{\\mathbb {K}},\\ \\mathbb {E})$ where the node set $\\overline{\\mathbb {K}}$ contains the exemplar Koopman operators synthesized from the clustering procedure.",
"The set of edges $\\mathbb {E}$ is determined by directly observing the sequences of class labels in the dataset, and tracking all unique transitions.",
"Figure REF illustrates how DSS relates to the resulting graph.",
"Each node in the graph represents a distinct dynamical system over its respective partition of the state-space manifold.",
"By traversing the graph symbolically from one node to another, traversal of the state manifold is implied.",
"The DSS algorithm is summarized in Algorithm 1.",
"While initialization of a DSS model requires that the dataset $X$ be from a single realization of a system, additional realizations can be used to account for variability in system roll-outs.",
"The graph itself encodes task-specific information embedded in the state trajectories of the training dataset.",
"In particular, the graph's state distribution is an information-rich object that can be used for data analysis purposes.",
"Given an optimal agent's graph $\\mathbb {G}_{opt}$ constructed with DSS, we can use the trained SVM classifier $\\Phi _{opt}(\\psi (x))$ to identify behaviors from the optimal agent in data from other agents.",
"By tracking the relative frequencies of behaviors $\\overline{\\mathbb {K}}$ from the optimal agent in another agent's trajectories, we can calculate a distribution $q(\\overline{\\mathbb {K}})$ , and directly compare it to $\\mathbb {G}_{opt}$ 's optimal state distribution $p(\\overline{\\mathbb {K}})$ using the Kullback-Leibler divergence ($D_{KL}$ ) [27] [!hpb] Dynamical System Segmentation (DSS) [1] Dataset $X = [x_0,...,x_M]$ from a single realization of a dynamical system, basis functions $\\lbrace \\psi (x)| \\psi : \\pazocal {M} \\rightarrow \\mathbb {R}^N\\rbrace $ , window size $S_{w}$ , overlap percentage $P_{ov}$ , minimum number of points required to form a cluster $N_c$ Transform the $X$ dataset into $\\Psi _{X} = [\\psi (x_0),...,\\psi (x_{M})]^T$ Split $\\Psi _X$ into $W+1$ windows of size $S_{w}$ overlapping by $P_{ov}$ Calculate a Koopman operator $K_i$ for each window to construct the set $\\mathbb {K} = \\lbrace K_0,...,K_W\\rbrace $ Construct a feature array $\\mathbb {K}_{flat}$ by flattening all $K_i \\in \\mathbb {R}^{N\\times N}$ in $\\mathbb {K}$ into points in $\\mathbb {R}^{N^2}$ and appending them Cluster using HDBSCAN($\\mathbb {K}_{flat}$ , $N_c$ ), and label all $K_i$ 's from one of $B+1$ discerned classes $\\lbrace C_0,...,C_B\\rbrace $ Construct a set $\\overline{\\mathbb {K}}=\\lbrace \\overline{K}_0,...,\\overline{K}_B\\rbrace $ of class exemplars by taking a weighted-average of all $K_i \\in C_j, \\ \\forall j \\in \\lbrace 0,...,B\\rbrace $ , according to the membership probability $p(K_i|K_i \\in C_j)$ Label all points in $\\Psi _X$ with the label $l \\in \\lbrace 0,...,B\\rbrace $ of the Koopman operator they were used to generate Train an SVM $\\Phi (\\psi (x))$ on the labeled points Construct a set of unique transitions $\\mathbb {E}$ by tracking all sequential labels in the dataset Construct a graph $\\mathbb {G} = (\\overline{\\mathbb {K}},\\ \\mathbb {E})$ Graphical model $\\mathbb {G}$ , and trained SVM $\\Phi (\\psi (x))$ Figure: State-space partitions generated by an SVM trained on an optimal controller's cart-pendulum inversion.",
"DSS identified 3 modes, and the SVM partitions are shown at the (x c ,x ˙ c )=(1,-1)(x_c,\\dot{x}_c) = (1,-1) cross-section of the manifold.$D_{KL}(p(\\overline{\\mathbb {K}})||q(\\overline{\\mathbb {K}})) = -\\sum _{i=0}^{B}p(\\overline{K}_i)log\\big (\\frac{q(\\overline{K}_i)}{p(\\overline{K}_i)}\\big ).$ The state distributions encode coarse-grained information about the task, and their comparison can be used for performance assessment.",
"Since an optimal agent's distribution is the most representative state distribution of a task, $D_{KL}(p(\\overline{\\mathbb {K}})||q(\\overline{\\mathbb {K}}))$ represents the amount of task information embedded in an agent's motions, which we refer to as task embodiment.",
"The proposed assessment of task embodiment was applied to data collected from human subjects performing a cart-pendulum inversion taskThe authors utilized de-identified data from a study approved by the Northwestern Institutional Review Board.. Data was collected using the NACT-3D—an admittance-controlled haptic robot, similar to that described in [28] and [29].",
"We synthesize a dataset representative of an optimal user using an optimal controller.",
"Data from the expert is segmented by applying the DSS algorithm proposed in Section II in order to generate a graphical model $\\mathbb {G}_{opt}$ , and a set of optimal behaviors to track.",
"$\\mathbb {G}_{opt}$ 's state distribution is then used as a reference to compare against the human subjects, and assess their task embodiment."
],
[
"Human Subjects Dataset",
"A filter-based assistance algorithm proposed in [30] for pure noise inputs, and adapted for user input in [31] and [32] was applied to a virtual cart-pendulum inversion task on the NACT-3D.",
"The assistance physically filters the user's inputs—accelerations in this case—such that their actions are always in the direction of an optimal control policy calculated in real time.",
"All subjects were instructed to attempt to invert a virtual cart-pendulum with the goal of spending as much time as possible in the unstable equilibrium during a thirty second trial, where the cart-pendulum states were sampled at $60Hz$ .",
"Subjects repeated this task for 30 trials in each of two sessions.",
"Forty subjects completed this task with assistance in one session and without assistance in the other session.",
"The order in which the subjects received assistance was counterbalanced to account for learning effects.",
"An additional thirteen subjects were placed in a control group which completed both sessions without assistance.",
"Figure REF depicts the effect of assistance on the state trajectories of a representative subject, and includes an optimal trajectory for comparison.",
"For the assisted trial, the subject reaches the goal state of $\\theta =0$ and is able to balance the pendulum starting around $t=5s$ .",
"At this point, the assistance restricts the user's input motion $x_c(t)$ such that the inversion is maintained until $t=13s$ .",
"However, the same subject is unable to maintain the inverted configuration without assistance.",
"Figure: Output of the Dynamical System Segmentation algorithm: each node in the graph is a distinct dynamical system that governs its partition of the state-space manifold generated by the SVM Φ(ψ(x))\\Phi (\\psi (x)).Figure: Sample trajectories from subject 16's trials depicting the effect of assistance in time-domain, as well as a trial from an optimal controller.Figure: Graph 𝔾 opt \\mathbb {G}_{opt} resulting from the segmentation of a dataset of 30 optimal control solutions to the cart-pendulum inversion task.",
"The set of segmented behaviors are shown projected onto the system's phase portrait over the domain {(θ,θ ˙):(-π,π)×(-2π,2π)}\\lbrace (\\theta ,\\dot{\\theta }):(-\\pi ,\\pi ) \\times (-2\\pi ,2\\pi )\\rbrace ."
],
[
"Training Dataset",
"To assess task embodiment using our dynamical system segmentation technique, we synthesized an optimal baseline to compare subjects against.",
"We generated optimal control solutions to the pendulum inversion problem using Sequential Action Control [33], a receding-horizon model predictive optimal controller for nonlinear and nonsmooth systems, over a randomized set of initial conditions.",
"The controller's objective was $(\\theta ,x_c,\\dot{\\theta },\\dot{x}_c)=(0,1,0,0)$ , with linear quadratic cost parameters of $Q=diag([200,80,0.01,0.2])$ .",
"Thirty optimal control trials of thirty seconds each were generated so as to mirror the amount of data collected from human subjects."
],
[
"Optimal Graph",
"We apply the DSS algorithm to the synthesized trials to generate an optimal graphical model.",
"The choice of basis functions has the greatest effect on the algorithm's performance because of how they reshape the state-space boundaries.",
"The set of basis functions, $\\psi (x)$ , selected for this task were $\\psi (x) = [\\theta ,\\ x_c,\\ \\dot{\\theta },\\ \\dot{x}_c,\\ u,\\ u\\ cos(\\theta ),\\ u\\ cos(\\dot{\\theta }),\\\\|u_{sat}|cos^2\\big (\\frac{u \\pi }{|u_{sat}|}\\big ),\\ \\dot{x}_c^2,\\ 1],$ where $|u_{sat}|$ is the optimal controller's saturation limit on the control effort.",
"The basis functions were selected from the set of linear combinations of second order polynomial and sinusoidal functions.",
"Since clustering occurs in $\\mathbb {R}^{N^2}$ space, where $N$ is the number of basis functions, we chose a low-dimensional set ($N=10$ ) of representative basis functions in Eq.",
"REF from the larger set of linear combinations of polynomial and sinusoidal functions.",
"This dimensionality reduction can be achieved via multiple methods, such as principal component analysis [27].",
"Figure REF depicts the behaviors identified from the exemplar trial.",
"The identified modes 0, 1 and 2 correspond to energy pumping and swing-up, energy removal and slow-down, and stabilization, respectively.",
"These modes represent a set of behaviors that an expert user should exhibit in succeeding at the task.",
"We synthesize the optimal graph $\\mathbb {G}_{opt}$ using the identified behaviors, and then use the graph's state distribution, $p(\\overline{\\mathbb {K}})=[0.2437,\\ 0.1275,\\ 0.6288]$ , as the reference baseline with which to assess the subjects' task embodiment.",
"The graph $\\mathbb {G}_{opt}$ and the segmented behaviors projected onto the $(\\theta ,\\dot{\\theta })$ phase portrait by the trained SVM $\\Phi _{opt}(\\psi (x))$ is shown in Fig.",
"REF .",
"Figure: Summary of experimental results: subjects in the experimental group who received assistance (blue) were compared to their own unassisted trials.",
"The control group subjects (red) were compared from their initial session to their final session.",
"The pair of plots to the left show the difference in task embodiment between the sessions of the experimental and control groups.",
"The plots to the right show the difference between the same groups using the integrated MSE instead.",
"Both task embodiment and MSE are good predictors of assistance, validating task embodiment as a performance measure.The human data is analyzed by using the trained SVM $\\Phi _{opt}(\\psi (x))$ to detect the identified behaviors in each subject's trials with and without the presence of assistance.",
"By tracking the relative frequencies of behaviors $\\overline{\\mathbb {K}}$ we can generate a distribution $q(\\overline{\\mathbb {K}})$ with which to compare to $\\mathbb {G}_{opt}$ 's state distribution $p(\\overline{\\mathbb {K}})$ .",
"We compare the distributions using task embodiment quantified by $D_{KL}(p(\\overline{\\mathbb {K}})||q(\\overline{\\mathbb {K}}))$ , where a lower $D_{KL}$ indicates greater embodiment of the task.",
"This same procedure is applied to the two sets of data from the control group subjects."
],
[
"Results",
"We analyzed the human subjects dataset, and found that task embodiment is a reliable predictor of physical assistance.",
"All subjects better embodied the task in their assisted trials, whereas there was no observed difference in the control group.",
"In addition to comparing the groups using task embodiment, we also evaluated a standard metric for assessing task performance, the integrated MSE.",
"Specifically, we calculated the integrated MSE with respect to a goal state of $(\\theta ,\\dot{\\theta })=(0,0)$ .",
"Integrated MSE is a reasonable performance metric for this task since success is defined as the ability to reach a single system configuration.",
"However, we find that it predicts assistance at a lower significance level, and lower effect size than task embodiment.",
"A paired two-sample t-test on the task embodiment of each subject with and without assistance showed that the subjects' sessions with assistance $(\\mu =0.0756,\\ \\sigma =0.0436)$ significantly outperformed the sessions without assistance $(\\mu =0.2084,\\ \\sigma =0.0560)$ , with $p=\\text{2.8633e-16}, \\ t(39)=13.4876$ , and an effect size of $d = 2.1326$ .",
"In contrast, there was no significant difference between the first session $(\\mu =0.2039,\\ \\sigma =0.0406)$ and the second session $(\\mu =0.1943,\\ \\sigma =0.0400)$ of the control group when a paired two-sample t-test was performed $p=0.5546, \\ t(12)=-0.6051$ .",
"These results indicate that task embodiment reliably captures assistance and lack thereof.",
"We also performed a paired two-sample t-test on the MSE of each subject with and without assistance, and found that the session with assistance $(\\mu =124.66,\\ \\sigma =119.96)$ significantly outperformed the session without assistance $(\\mu =428.88,\\ \\sigma =307.46)$ , but with a lower significance and effect size than task embodiment, with $p=\\text{1.2353e-7}, \\ t(39)=6.4526$ and an effect size of $d = 1.0202$ .",
"Again, we applied the paired two-sample t-test to the control group and found that the first session $(\\mu =352.83,\\ \\sigma =217.67)$ did not significantly outperform the second $(\\mu =546.31,\\ \\sigma =446.10)$ , had $p=0.0651, \\ t(12)=2.0320$ .",
"These results indicate that MSE can also predict the presence of assistance, but not as reliably as task embodiment.",
"The task embodiment measure has both a significance level several orders of magnitude greater than that of integrated MSE, and showed an effect size that was twice as large as integrated MSE.",
"This demonstrates that task embodiment captures the large difference between the assisted and unassisted trials.",
"These results are summarized in Fig.",
"REF , where we see that the change in task embodiment ($\\Delta TE$ ) from assisted to unassisted trials is always positive.",
"When we limit our alphabet to linear symbols using the same DSS hyperparameters as those in the outlined results, $\\Delta TE$ was positive only in $5\\%$ of subjects.",
"When we tune the DSS parameters to generate an alphabet of the same size as in the primary results, $\\Delta TE$ was positive only in $30\\%$ of subjects.",
"Therefore, for this system, linear symbols do not capture sufficient task information to reveal the presence of assistance.",
"The experimental methodology presented in this study analyzed subject data by means of comparison to an optimal baseline.",
"While the methodology is informative, it cannot detail subject performance without comparison to the optimal agent.",
"Given the same choice of basis functions and algorithm parameters, we can use DSS to generate graphs of each subject with and without assistance, and analyze the identified behaviors in each graph directly.",
"This alternative methodology allows us to take human motion data and represent it graphically, which creates the opportunity for analyzing human motion using graph-theoretic principles.",
"Figures REF & REF illustrate the graphical models constructed from the assisted and unassisted trials of a representative subject.",
"We note that the extracted behaviors from the unassisted trials in Fig.",
"REF lack structure, and more closely resemble noise-driven behaviors.",
"In contrast, by inspecting the graph from the subject's assisted trials in Fig.",
"REF , we observe the emergence of finite structure in the identified behaviors.",
"Figure: Resulting graph and state-space projections from the segmentation of subject 16's unassisted trials of the pendulum inversion task."
],
[
"Conclusions",
"In this study, we proposed an information-theoretic approach to human motion analysis.",
"The DSS algorithm formulated in Section II produces graphical models that encode task-specific information.",
"By tracking the degree of task embodiment, we are able to decode complex relationships in human motion.",
"We applied DSS to a dataset of human subjects performing a virtual cart-pendulum inversion task with and without assistance.",
"We determined that task embodiment is a good predictor of assistance, and validated the results by comparing task embodiment to integrated MSE.",
"Moreover, task embodiment identified the presence of task assistance at a higher significance level and with a larger effect size than integrated MSE.",
"Thus, the experimental results provide strong support for the use of information measures in human motion analysis."
]
] | 2012.05183 |
[
[
"A repository of vanilla long term integrations of the Solar System"
],
[
"Abstract We share the source code and a 121 GB dataset of 96 long-term N-body simulations of the Solar System.",
"This dataset can be analyzed by itself to study the dynamics of the Solar System.",
"In addition, our simulations can be the starting point for future studies wishing to explore different initial conditions or additional physical effects on the Solar System.",
"Our simulations can also be used as a comparison and benchmark for new numerical algorithms."
],
[
"Motivation",
"The long term dynamical evolution of the Solar System has been investigated analytically for centuries.",
"However, direct numerical integrations of the Solar System over billions of years have only become possible in the last few decades thanks to fast computers and better numerical algorithms.",
"Since then, many authors have run such simulations, including [1], [2], and [8].",
"These simulations differ in the initial conditions and numerical methods used, as well as in the physical effects that are being modelled.",
"For example, some simulations only include gravitational forces, while others also include stellar mass loss or tidal forces.",
"The Solar System is chaotic and even small changes in the initial conditions lead to rapidly diverging trajectories.",
"For this reason, one typically has to integrate many simulations with slightly different initial conditions and make statistical conclusions.",
"Each such simulation is computationally expensive and takes of the order of a month of wall-time to complete a 5 Gyr integration even on modern computers.",
"Because $N$ -body simulations are inherently sequential for small $N$ , it is practically impossible to parallelize them.",
"The best one can do is run one simulation on one core, but this still requires a minimum wall-time of one month, even if a large number of cores is available.",
"We share a set of 96 long-term integrations of the Solar System for which the average relative energy error is $\\Delta E/E\\sim 2.5\\cdot 10^{-9}$ .",
"However, our aim is not to generate the most accurate ephermeris possible, but to provide a repository of vanilla simulations that capture the most important dynamical properties of the system.",
"We hope that these vanilla simulations will allow other studies to 1) analyze this dataset by itself, and 2) run their own simulations and use this dataset as a comparison.",
"To facilitate this, we not only share the data, but also all the code used to generate it.",
"Our simulations can therefore be used as a benchmark and reference in future studies exploring additional physical effects, additional bodies, different initial conditions, or different numerical algorithms."
],
[
"Numerical setup",
"We share all scripts to exactly reproduce our simulations together with our dataset.",
"In this section, we only summarize the important aspects of the numerical scheme used.",
"We use the $N$ -body code REBOUND [10] and the symplectic integrator WHFast [17], [11] in Jacobi coordinates [13].",
"To achieve a high accuracy we employ 17th order symplectic correctors [16] and a modified kick step [18], [14].",
"Specifically, we use the lazy implementer's kernel method which is compatible with general relativistic corrections.",
"This configuration corresponds to using the shorthand WHCKL for the integrator in REBOUND, having a generalized order (18,4,3), and leading error terms $O(\\epsilon dt^{18}+\\epsilon ^2 dt^4 + \\epsilon ^3 dt^3)$ .",
"For more details on various high order symplectic schemes in REBOUND see [14].",
"We use a fixed timestep of $dt\\approx 8.062$ days.",
"This implies that our simulations can be trusted as long as the eccentricity of any planet remains moderate, $e\\lesssim 0.4$ .",
"We query NASA's Horizon systemhttps://ssd.jpl.nasa.gov/?horizons to retrieve initial conditions of the Sun and all 8 major planets on 1st of January 2000, 12:00 UTC.",
"The effects of general relativistic precession are important for the long-term evolution of the Solar System [7].",
"We model these effects by including an additional non-Newtonian $1/r^3$ term in the potential using the gr_potential module of REBOUNDx [15].",
"Our simulations use a system of units where the unit of length is one astronomical unit, the unit of mass is one Solar mass, and the gravitational constant $G$ is 1.",
"One Earth year then corresponds to $2\\pi $ in code units.",
"All 96 simulations are identical except for a small perturbation in the $x$ coordinate of Mercury.",
"The file names indicate by how much Mercury has been shifted.",
"For example, in the simulation labelled p750, Mercury has been shifted by $750\\cdot 0.38$ mm in the positive $x$ direction.",
"Note that as long as the remain small the details of these initial perturbations are not important because the system is chaotic.",
"Figure: Mercury's eccentricity (top) and the relative energy error (bottom) as a function of time.",
"Showing all 96 simulations.The simulations are integrated for 5 Gyr into the future.",
"We make use of the SimulationArchive [12] to store 500,000 snapshots for each simulation, resulting in a file size of 1.3 GB.",
"These snapshots can be used to sample a simulation at 10,000 year intervals.",
"Furthermore, the simulations can be restarted and resampled at any snapshot.",
"The trajectories will be bit-by-bit identical to the original integration."
],
[
"Data access",
"The 96 SimulationArchive files as well as all our scripts are made available on the general-purpose open-access repository Zenodo at https://zenodo.org/record/4299102 [3].",
"The full dataset is over 121 GB in size but each SimulationArchive can also be downloaded as an individual 1.3 GB file.",
"There are two different ways to think about a SimulationArchive.",
"It can simply be used to access the data stored at each of the 500,000 snapshots.",
"However, one can also think of each snapshot as a point where one can restart the simulation.",
"Because two consecutive snapshots are only 3s apart in wall-time, one can access any arbitrary time in the 5 Gyr integration within at most 3s (1.5s on average).",
"And because the SimulationArchive allows the simulation to be reconstructed bit-by-bit, this is not an interpolation but exact repetition of the original simulation.",
"Fig.",
"REF shows the eccentricity of Mercury for all 96 integrations in the top panel and the relative energy error in the bottom panel.",
"Even though we need to sift through 121 GB of data, generating this plot only takes about one minute on a desktop computer with a solid state drive.",
"We can see that the eccentricity of Mercury is correlated across all simulations for the first $\\sim 100$ Myrs.",
"All simulations remain stable for the full 5 Gyr which is consistent with previous results showing that only about 1% of simulations should go unstable [8].",
"The bottom panel shows that the relative energy error remains approximately constant throughout the integration.",
"The average relative energy error of our simulations is about $2.5\\cdot 10^{-9}$ and is shown as a thick black line in the bottom panel.",
"The Zenodo repository includes example scripts to query the SimulationArchive files and reproduce the above figure as well as a few others.",
"The dataset we share took about 6 core years of computation time on a 2.4GHz Intel Xeon CPU, resulting in 200kg of $\\text{CO}_2$ emissionsAssuming 0.5kg $\\text{CO}_2$ per kWh, https://www.eia.gov/tools/faqs/faq.php?id=74..",
"This research was made possible by the open-source projects Jupyter [6], iPython [9], and matplotlib [5], [4]."
]
] | 2012.05177 |
[
[
"Synthesis to Deployment: Cyber-Physical Control Architectures"
],
[
"Abstract We consider the problem of how to deploy a controller to a (networked) cyber-physical system (CPS).",
"Controlling a CPS is an involved task, and synthesizing a controller to respect sensing, actuation, and communication constraints is only part of the challenge.",
"In addition to controller synthesis, one should also consider how the controller will be incorporated within the CPS.",
"Put another way, the cyber layer and its interaction with the physical layer need to be taken into account.",
"In this work, we aim to bridge the gap between theoretical controller synthesis and practical CPS deployment.",
"We adopt the system level synthesis (SLS) framework to synthesize a controller, either state-feedback or output-feedback, and provide deployment architectures for the standard SLS controllers.",
"Furthermore, we derive new controller realizations for open-loop stable systems and introduce different state-feedback and output-feedback architectures for deployment, ranging from fully centralized to fully distributed.",
"Finally, we compare the trade-offs among them in terms of robustness, memory, computation, and communication overhead."
],
[
"Introduction",
"We consider a linear time-invariant (LTI) system with a set of sensors $s_i, i = 1, \\dots , N_y$ and a set of actuators $a_k, k = 1, \\dots , N_u$ , that eveolves according to the dynamics $x[t+1]=&\\ A x[t]+ B u[t]+ d_x[t] \\\\y[t]=&\\ C x[t]+ D u[t]+ d_y[t]$ where $x[t] \\in \\mathbb {R}^{N_x}$ is the state at time $t$ , $y[t] \\in \\mathbb {R}^{N_y}$ is the measured output at time $t$ , $u[t] \\in \\mathbb {R}^{N_u}$ is the control action at time $t$ , and $d_x[t] \\in \\mathbb {R}^{N_x}$ , $d_y[t] \\in \\mathbb {R}^{N_y}$ are the disturbances.",
"Notice that the model in () reduces to a state-feedback model when $C = I$ , $D = 0$ , and ${\\bf d}_{\\bf y}= 0$ .",
"Suppose the system is open-loop stable, i.e., $(zI-A)^{-1} \\in \\mathcal {R}\\mathcal {H}_{\\infty }$ .",
"The goal of this paper is to address how a feedback controller (we will discuss state and output feedback) can be deployed to this system and what the corresponding cyber-physical structures and trade-offs are.",
"Figure: A model-based system control scheme consists of two phases–synthesis and deployment.",
"Here we employ SLS in the synthesis phase to obtain a controller.",
"The focus of this work is how to deploy 𝐊{\\bf K} to the underlying CPS.A model-based approach to control design involves two phases: the synthesis phase and the deployment phase, as illustrated in Fig.",
"REF .",
"In the synthesis phase, the control engineer derives the desired controller model by some synthesis algorithm (e.g.",
"$\\mu $ , $\\mathcal {H}_2$ , $\\mathcal {H}_{\\infty }$ , $L_1$ , etc.)",
"based on a model of the system dynamics (), a suitable objective function, and operating constraints on sensing, actuation, and communication capabilities.",
"The optimal controller in the model-based sense is thus the controller model achieving the best objective value.",
"In some cases the controller model is associated with a realization, for example, the block diagram representation of the “central controller” in $\\mathcal {H}_2$ and $\\mathcal {H}_{\\infty }$ control [1].",
"Synthesizing an optimal controller for a cyber-physical system is, in general, a daunting task.",
"In contrast to centralized systems, cyber-physical systems comprise sensors and actuators distributed over the network, and it is expensive or intractable to maintain a central controller with global information.",
"Recently, the system level synthesis (SLS) framework was proposed to facilitate distributed controller synthesis for large-scale (networked) systems [2], [3], [4].",
"Instead of designing the controller itself, SLS directly synthesizes desired closed-loop system responses subject to system level constraints, such as localization constraints [5] and state and input constraints [6].",
"SLS allows for the optimal linear controller model to be “backed out” of the system response in such a manner that the closed-loop structure is preserved by the controller.",
"The controller admits multiple (mathematically equivalent) control block diagrams/state space realizations (or simply realizations) [3], [7], [8].",
"On the other hand, the deployment phase (often referred to as implementation) is concerned with how to map the derived controller model/realization to the target system.",
"It is usually possible to implement one controller realization by multiple architectures.",
"Although all architectures lead to the “optimal” controller, they can differ in aspects other than the objective, for example; memory requirements, robustness to failure, scalability, and financial cost.",
"Therefore, it is important to consider the trade-offs among those architectures, in order to deploy the most suitable architecture.",
"Typically, control engineers pay less attention to deployment than they do synthesis.",
"Our goal is to develop a theory for deployment, i.e., provide a systematic theory for controller implementation.",
"Approaches to deployment vary greatly in the literature.",
"In the control literature, most work gives the controller design at the realization level and implicitly relies on some interface provided by the underlying CPS for deployment [9], [10], [11].",
"Providing the appropriate abstraction and programming interface is itself a design challenge [12], [13], [14], [15].",
"Some papers also examine the deployment down to the circuit level [16], [17].",
"For instance, a long-standing research area in control has looked at controller implementation using passive components [18], [19], [20], [21].",
"The networking/system community, on the other hand, mostly adopts a bottom-up instead of a top-down approach to system control.",
"It usually involves some carefully designed gadgets/protocols and a coordination algorithm [22], [23], [24].",
"In this work we take an alternative approach to deployment, which lies between the realization and the circuit level.",
"Rather than binding the design to some specific hardware, we specify a set of basic components, i.e.",
"components with well defined functionality that systems can be built from, and use these to implement the derived controller realization.",
"As such, we can easily map our architectures to the real CPS.",
"The paper is organized as follows.",
"In Section , we briefly review SLS, propose new, simpler state-feedback and output-feedback control realizations, and show they are internally stabilizing.",
"With those realizations, we propose different partitions and their corresponding deployment architectures in Section , namely, the centralized (Section REF ), global state (Section REF ), and four different distributed (Section REF ) architectures.",
"Then, in Section Section we discuss the trade-offs made by the architectures on robustness, memory, computation, and communication.",
"Finally, we conclude the paper with possible future research directions in Section ."
],
[
"Synthesis Phase",
"We briefly review distributed control using system level synthesis, describing the “standard” SLS controller realizations and draw attention to their respective architectures.",
"Finally, we derive new controller realizations for open-loop stable systems which in certain settings have favorable properties in the deployment stage."
],
[
"System Level Synthesis (SLS) – An Overview",
"To synthesize a closed-loop controller for the system described in (), SLS introduces the system response transfer matrices: $\\lbrace \\mathbf {\\Phi }_{\\bf x}, \\mathbf {\\Phi }_{\\bf u}\\rbrace $ for state-feedback and $\\lbrace \\mathbf {\\Phi }_{{\\bf x}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf x}{\\bf y}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}\\rbrace $ for output-feedback systems.",
"The system response is the closed-loop mapping from the disturbances to the state ${\\bf x}$ and control action ${\\bf u}$ , under the feedback policy ${\\bf u}= {\\bf K}{\\bf y}$ .",
"Full derivations and proofs of material in this section can be found in [3], [4].",
"Consider the LTI system defined by () , the system response matrices are defined by $\\begin{bmatrix}{\\bf x}\\\\ {\\bf u}\\end{bmatrix}=\\begin{bmatrix}\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} & \\mathbf {\\Phi }_{{\\bf x}{\\bf y}} \\\\\\mathbf {\\Phi }_{{\\bf u}{\\bf x}} & \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}\\end{bmatrix}\\begin{bmatrix}{\\bf d}_{\\bf x}\\\\ {\\bf d}_{\\bf y}\\end{bmatrix}$ where ${\\bf d}_{\\bf x}$ and ${\\bf d}_{\\bf y}$ denote the disturbance to the state and output respectively.",
"Let ${\\bf L}= ({\\bf K}^{-1} - D)^{-1}$ , the system response transfer functions take the following form: $\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} =&\\ (zI-A-B {\\bf L}C)^{-1}, &\\quad \\mathbf {\\Phi }_{{\\bf x}{\\bf y}} =&\\ \\mathbf {\\Phi }_{{\\bf x}{\\bf x}} B {\\bf L} \\\\\\mathbf {\\Phi }_{{\\bf u}{\\bf x}} =&\\ {\\bf L}C \\mathbf {\\Phi }_{{\\bf x}{\\bf x}} ,&\\quad \\mathbf {\\Phi }_{{\\bf u}{\\bf y}} =&\\ {\\bf L}+ {\\bf L}C \\mathbf {\\Phi }_{{\\bf x}{\\bf x}} B {\\bf L}.",
"\\nonumber $ An SLS problem is a convex program defined by the system response transfer functions.",
"In it's most general setting, an SLS problem takes the form: $\\text{minimize}\\ &\\ g(\\mathbf {\\Phi }_{{\\bf x}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf x}{\\bf y}},\\mathbf {\\Phi }_{{\\bf u}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}) \\nonumber \\\\{\\rm s.t.",
"}\\ &\\ \\begin{bmatrix}zI-A & -B\\end{bmatrix}\\begin{bmatrix}\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} & \\mathbf {\\Phi }_{{\\bf x}{\\bf y}} \\\\\\mathbf {\\Phi }_{{\\bf u}{\\bf x}} & \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}\\end{bmatrix}=\\begin{bmatrix}I & 0\\end{bmatrix} \\\\&\\ \\begin{bmatrix}\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} & \\mathbf {\\Phi }_{{\\bf x}{\\bf y}} \\\\\\mathbf {\\Phi }_{{\\bf u}{\\bf x}} & \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}\\end{bmatrix}\\begin{bmatrix}zI-A \\\\ -C\\end{bmatrix}=\\begin{bmatrix}I \\\\ 0\\end{bmatrix} \\\\&\\ \\mathbf {\\Phi }_{{\\bf x}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf x}{\\bf y}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf x}} \\in \\frac{1}{z}\\mathcal {R}\\mathcal {H}_{\\infty },\\ \\ \\mathbf {\\Phi }_{{\\bf u}{\\bf y}} \\in \\mathcal {R}\\mathcal {H}_{\\infty } \\nonumber \\\\&\\ \\begin{bmatrix}\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} & \\mathbf {\\Phi }_{{\\bf x}{\\bf y}} \\\\\\mathbf {\\Phi }_{{\\bf u}{\\bf x}} & \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}\\end{bmatrix} \\in \\mathcal {S}.",
"\\nonumber $ Observe that as as long as $g$ is convex, this is convex problem because i) (REF ) and () are affine in the decision variables, ii) the stability and strictly proper constraints are convex, and iii) $\\mathcal {S}$ encodes localization constraints which are shown to be convex in [5].For the sake of this paper, the constraint set $\\mathcal {S}$ does nothing other than impose sparsity constraints on the spectral components of the system response matrices.",
"Later it will be used to impose a finite impulse response on the system response.",
"The core SLS result states that the affine space parameterized by $\\lbrace \\mathbf {\\Phi }_{{\\bf x}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf x}{\\bf y}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}\\rbrace $ in (REF )–() parameterizes all system responses (REF ) achievable by an internally stabilizing controller [3], [4].",
"Moreover, any feasible system response can realize an internally stabilizing controller via ${\\bf K}=&\\ \\left(\\left( \\mathbf {\\Phi }_{{\\bf u}{\\bf y}} - \\mathbf {\\Phi }_{{\\bf u}{\\bf x}} \\mathbf {\\Phi }_{{\\bf x}{\\bf x}}^{-1} \\mathbf {\\Phi }_{{\\bf x}{\\bf y}} \\right)^{-1}+ D\\right)^{-1}.$ Note that the constraint set $\\mathcal {S}$ imposes sparsity constraints (amongst other things) on the system response transfer matrices.",
"The controller defined in (REF ) will not inherit this sparsity and will thus be unstructured – from an implementation perspective, this is problematic.",
"However, the realization $\\beta [t+1]=&\\ - \\sum _{\\tau \\ge 2}\\Phi _{xx}[\\tau ]\\beta [t + 2 - \\tau ] -\\sum _{\\tau \\ge 1}\\Phi _{xy}[\\tau ]\\overline{y}[t + 1 - \\tau ], \\nonumber \\\\u[t]=&\\ \\sum _{\\tau \\ge 1}\\Phi _{ux}[\\tau ]\\beta [t + 1 - \\tau ] +\\sum _{\\tau \\ge 0}\\Phi _{uy}[\\tau ]\\overline{y}[t - \\tau ], \\nonumber \\\\\\overline{y}[t]=&\\ y[t]- D u[t],$ corresponding to the block diagram shown in Fig.",
"REF maintains the localized structure of the system response.",
"Figure: Output-feedback SLS control block diagrams derived in .",
"The top realization is applicable to general systems.",
"When the plant is open-loop stable and D=0D = 0, a simplified realization (bottom) is possible."
],
[
"State-Feedback",
"When the state information is available to the controller, the control action is given by ${\\bf u}= {\\bf K}{\\bf x}$ , and the system response is characterized by just two transfer matrices $\\lbrace \\mathbf {\\Phi }_{\\bf x}, \\mathbf {\\Phi }_{\\bf u}\\rbrace $ , where $\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} &\\rightarrow \\mathbf {\\Phi }_{\\bf x}:= (zI-A-B {\\bf K})^{-1},\\\\\\mathbf {\\Phi }_{{\\bf u}{\\bf x}} &\\rightarrow \\mathbf {\\Phi }_{\\bf u}:= {\\bf K}\\mathbf {\\Phi }_{\\bf x}.$ The resulting SLS problem greatly simplifies; constraint () disappears, and the $2\\times 2$ block transfer matrices in the remaining constraints reduce to $2\\times 1$ block transfer matrices.",
"Any feasible system response $\\lbrace \\mathbf {\\Phi }_{\\bf x}, \\mathbf {\\Phi }_{\\bf u}\\rbrace $ can be used to construct an internally stabilizing controller via ${\\bf K}= \\mathbf {\\Phi }_{{\\bf u}}\\mathbf {\\Phi }_{{\\bf x}}^{-1}$ .",
"As with the output-feedback case, this controller is not ideal for implementation.",
"Instead, the controller depicted in Fig.",
"REF inherits the structure imposed on $\\lbrace \\mathbf {\\Phi }_{\\bf x}, \\mathbf {\\Phi }_{\\bf u}\\rbrace $ .",
"Analogously to (REF ), the controller dynamics are given by $\\delta [t]= x[t]- \\hat{x}[t], \\quad u[t]= \\sum _{\\tau \\ge 1}^{T-1}\\Phi _u[\\tau ]\\delta [t + 1 - \\tau ],\\\\\\hat{x}[t+1]= \\sum _{\\tau \\ge 2}^{T-2}\\Phi _x[\\tau ]\\delta [t + 2 - \\tau ].$ This paper focuses on how to actually implement and deploy such controllers into cyber-physical systems.",
"We will characterize the necessary hardware resources, and dig further into the architectures of the controllers.",
"Figure: State-feedback SLS control block diagram derived in corresponding to ().Our first result is to show that when the system to be controlled is open-loop stable, there exist alternative controller realizations that simplify the control law computation with fewer convolutions.",
"In some circumstances, the simplified realizations may be more desirable than those described previously.",
"The controller realizations given by the pairs (REF ) and Fig.",
"REF , and (REF ) and Fig.",
"REF are preferrable to (REF ) and ${\\bf K}= {\\mathbf {\\Phi }_{\\bf u}}{\\mathbf {\\Phi }_{\\bf x}^{-1}} $ .",
"In the output-feedback setting, four convolution operations involving $\\lbrace \\mathbf {\\Phi }_{{\\bf x}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf x}{\\bf y}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}\\rbrace $ are required, and in the state-feedback setting, two convolutions involving $\\lbrace \\mathbf {\\Phi }_{\\bf x}, \\mathbf {\\Phi }_{\\bf u}\\rbrace $ are used.",
"We will now show that the number of convolutions can be reduced when the plant is open-loop stable.",
"In the sequel, we will explicitly consider the savings/trade-offs in terms of memory and computation.",
"Theorem 1 Assume that $A \\in \\mathbb {R}^{N_x \\times N_x}$ in () is Schur stable.",
"The following are true For the state-feedback problem with the plant (REF ), the dynamic state-feedback controller ${\\bf u}= {\\bf K}{\\bf x}$ realized via $\\delta [t]=&\\ x[t]- A x[t-1]- B u[t-1],\\\\u[t]=&\\ \\sum \\limits _{\\tau \\ge 1} {\\Phi _u}[\\tau ]\\delta [t+1-\\tau ],$ is internally stabilizing and is realized by the block diagram in Fig.",
"REF .",
"For the output-feedback problem with the plant (), the dynamic output-feedback controller ${\\bf u}= {\\bf K}{\\bf y}$ realized via $\\hat{x}[t+1]=&\\ A \\hat{x}[t]+ B u[t],\\\\\\delta [t]=&\\ y[t]- C \\hat{x}[t]- D u[t]\\\\u[t]=&\\ \\sum \\limits _{\\tau \\ge 0} {\\Phi _{uy}}[\\tau ]\\delta [t-\\tau ],$ is internally stabilizing and is described by the block diagram in Fig.",
"REF .",
"Figure: SLS controller realizations corresponding to Theorem .See Appendix.",
"Remark 1 The controller realizations given by (REF ) and (REF ) require only one convolution operation.",
"In contrast, the standard SLS realizations require at least two convolutions.",
"Both realizations in Theorem REF preserve the structure of the system response matrices as can be seen in Fig.",
"REF .",
"Remark 2 In both controllers in Theorem REF the summation has infinite support over the non-negative reals, i.e.",
"they have an infinite impulse response.",
"However, from an optimization perspective, and a performance perspective, it is beneficial to enforce a finite impulse response constraint which we encode in $\\mathcal {S}$ .",
"The theorem above holds for both IIR and FIR controllers, however, for the remainder of the paper, we will assume all controllers are FIR with horizon length $T$ .",
"See [25] for work that removes the FIR constraint and produces a tractable convex optimization problem.",
"Remark 3 The realization (REF ) can mix FIR and IIR blocks.",
"When $\\mathbf {\\Phi }_{\\bf u}$ has finite impulse response (FIR) with horizon $T$ , we have $u[t]=&\\ \\sum \\limits _{\\tau =1}^{T} {\\Phi _u}[\\tau ]\\delta [t+1-\\tau ]$ regardless of whether ${\\mathbf {\\Phi }_{\\bf x}}$ is FIR or not.",
"In the state-feedback setting, it is possible to internally stabilize some open-loop unstable systems by the controller (REF ).",
"Consider a decomposition of the system matrix such that $A = A_u + A_s$ where $A_u$ is unstable and $A_s$ is Schur stable.",
"Then using the robustness results from [26], the controller designed for $(A_s, B)$ will stabilize $(A,B)$ if $\\Vert A_u \\mathbf {\\Phi }_{\\bf x}\\Vert <1$ for any induced norm.",
"Of course, the “standard” SLS controllers do not have such an open-loop stability requirement.",
"In the remainder of the paper we consider how a controller can be implemented by a set of basic components, and compare several architectures in terms of their memory and compute requirements."
],
[
"Deployment Architectures",
"We now explore the controller architectures for the deployment phase.",
"The crux of our designs centers on the partitions of the controller realization described in Theorem REF and illustrated in Fig.",
"REF .",
"We first introduce the basic components.",
"We then propose the centralized, global state, and distributed architectures accordingly.",
"As long as the real system is capable of providing the basic components, mapping the architectures into the system is straightforward."
],
[
"Basic Components",
"We assume that the nodes in the target system are capable of performing the functions described below and depicted in Fig.",
"REF .",
"These functions cover variable storage, arithmetic computation, and communication.",
"A buffer is where the system keeps the value of a variable.",
"It can be a memory device such as a register or merely a collection of some buses (wires).",
"In contrast, a delay buffer is the physical implementation of $z^{-n}$ in a block diagram.",
"It keeps the variable received at time $t$ and releases it at time $t+n$ .",
"For computation, a multiplier takes a vector from a buffer as an input and computes a matrix-vector multiply operation.",
"An adder performs entry-wise addition of two vectors of compatible dimensions.",
"Of course, we can merge cascaded adders into a multiple-input adder.",
"A node can also communicate with other nodes through disseminator-collector pairs.",
"A disseminator sends some parts of a variable (i.e., a subset of a vector) to designated nodes.",
"At the receiving side, a collector assembles the received parts appropriately to reconstruct the desired variable."
],
[
"Architectures for Original SLS Block Diagrams",
"Internally stabilizing controller realizations are provided in the original SLS papers [4], [3].",
"Using the basic components defined earlier, we demonstrate the architectures corresponding to the original realizations."
],
[
"State-Feedback",
"The original realization of a state-feedback controller is shown in Fig.",
"REF .",
"To derive the corresponding architecture, we consider its state space expression (REF ) and translate the computation into basic components.",
"The resulting architecture is in Fig.",
"REF .",
"We note that this controller is applicable to both stable and unstable plants.",
"Figure: The architecture of the original controller model in Fig.",
"."
],
[
"Output-Feedback",
"Similarly, we can derive the architecture of the original output-feedback realization in Fig.",
"REF .",
"To derive the architecture, it is easier to start with a mapping from $y$ to $u$ .",
"Therefore, we separate $u[t]$ into two terms by defining $u^{\\prime }[t]=&\\ \\sum _{\\tau \\ge 1}\\Phi _{ux}[\\tau ]\\beta [t + 1 - \\tau ] +\\sum _{\\tau \\ge 1}\\Phi _{uy}[\\tau ]\\overline{y}[t - \\tau ].$ As a result, we have $u[t]=\\ \\Phi _{uy}[0]\\overline{y}[t] + u^{\\prime }[t].$ Substituting (REF ) into the above equation, we obtain $(I + \\Phi _{uy}[0]D)u[t]= \\Phi _{uy}[0]y[t]+ u^{\\prime }[t].$ Due to lack of space, we omit the full block diagram of this controller.",
"However, it can be intuitively constructed by following the state-feedback case and additionally implementing the left and right hand sides of (REF ) as a multiplier and buffer respectively."
],
[
"Centralized Architecture",
"The most straightforward deployment architecture for the new realizations derived in Theorem REF is the centralized architecture, which packs all the control functions into one node, the centralized controller.",
"The partitions of the block diagrams are shown in Fig.",
"REF .",
"The centralized controller maintains communications with all sensors and actuators to collect state/measurement information and dispatch the control signal.",
"Remark 4 In the complexity analysis that follows, we consider worst case problem instances, i.e., we ignore sparsity.",
"In reality, $A$ and $B$ will have some sparsity structure and the SLS program will try to produce sparse/structured spectral components $\\Phi _{xx}[\\cdot ], \\Phi _{xy}[\\cdot ], \\Phi _{ux}[\\cdot ]$ , and $\\Phi _{uy}[\\cdot ]$ – although not all systems can be localized, and localization comes with a performance cost.",
"For systems that are easily localizable, the original SLS controller design (Fig.",
"REF and Fig.",
"REF ) may be more efficient than our analysis shows.",
"But likewise, we do not take sparsity in $(A,B)$ into account which would benefit the new realizations.",
"Our goal is to characterize different implementations for difficult to control systems and arm engineers with the appropriate trade-off information.",
"Figure: The partitions of the output-feedback block diagrams for the centralized architecture.",
"State feedback follows in exactly the same manner – with fewer components."
],
[
"State-Feedback",
"Fig.",
"REF shows the architecture of the centralized state-feedback controller.",
"For each time step $t$ , the controller first collects the state information $x_i[t]$ from each sensor $s_i$ for all $i = 1,\\dots ,N_x$ .",
"Along with the stored control signal $u$ , the centralized controller computes $\\delta [t]$ (the disturbance estimate) as in (REF ).",
"$\\delta [t]$ is then fed into an array of delay buffers and multipliers to perform the convolution () and generate the control signals.",
"The control signals $u_k[t]$ are then sent to each actuator $a_k$ for all $k = 1,\\dots , N_u$ .",
"Figure: The architecture of the centralized state-feedback controller.Deploying a synthesized solution to the centralized architecture is simple: We take the spectral components ${\\Phi _u}[\\tau ]$ of ${\\mathbf {\\Phi }_{\\bf u}}$ from $\\tau = 1,2,\\dots $ and insert them into the array of the multipliers.",
"We can compare the centralized architecture of Fig.",
"REF with the architecture of the original SLS controller (c.f.",
"Fig.",
"REF ).",
"As mentioned in Section , the original architecture is expensive both computationally and storage-wise.",
"Specifically, when both ${\\mathbf {\\Phi }_{\\bf x}}$ and ${\\mathbf {\\Phi }_{\\bf u}}$ are FIR with horizon $T$ and not localizable, the original architecture depicted in Fig.",
"REF performs $\\underbrace{\\strut (T-1) N_x (2 N_x - 1)}_{\\Phi _x[\\cdot ]\\delta [\\cdot ]} +\\underbrace{\\strut T N_u (2N_x - 1)}_{\\Phi _u[\\cdot ]\\delta [\\cdot ]} +\\underbrace{\\strut T - 1}_{\\text{additions}}$ floating point operations per time step (flops)Our notion of a flop is limited to scalar additions, subtraction, division, and multiplication.",
"and needs $\\underbrace{\\strut (T-1)N_x^2}_{\\Phi _x[\\cdot ]} +\\underbrace{\\strut T N_x N_u}_{\\Phi _u[\\cdot ]} +\\underbrace{\\strut (T + 2) N_x + N_u}_{\\delta [\\cdot ],\\ x[t],\\ -\\hat{x}[t+1], \\text{~and~} u[t]}\\nonumber $ memory locations to store all variables and multipliers.",
"In contrast, the architecture corresponding to Theorem REF and depicted in Fig.",
"REF requires $\\underbrace{\\strut (N_x + N_u)(2 N_x - 1)}_{-Ax[t]\\text{~and~}-Bu[t]} +\\underbrace{\\strut T N_u (2 N_x - 1)}_{\\Phi _u[\\cdot ]\\delta [\\cdot ]} +\\underbrace{\\strut T + 1}_{\\text{additions}}$ flops, and needs $\\hspace{-4.25pt}\\underbrace{\\strut N_x^2 + N_x N_u}_{A\\text{~and~}B} +\\underbrace{\\strut 2 N_x }_{Ax[t]\\text{,}Bu[t]} +\\underbrace{\\strut TN_x N_u}_{\\Phi _u[\\cdot ]\\delta [\\cdot ]} +\\underbrace{\\strut (T + 1) N_x + N_u}_{\\delta [\\cdot ],\\ x[t], \\text{~and~} u[t]}$ memory locations.",
"The above analysis tells us that if the $\\Phi _x[\\cdot ]$ and $\\Phi _u[\\cdot ]$ are dense matrices (i.e., for systems with no localization), the controller architecture corresponding to Theorem REF is more economic in terms of computation and storage requirements when $N_x \\ge N_u$ , $N_x \\ge 2$ , and $T > 3$ .",
"We note that having fewer inputs than states is the default setting for distributed control problems.",
"For systems that can be strongly localized (which roughly corresponds to the spectral components having a banded structure with small bandwidth), the original SLS architecture will likely be more economical.",
"A detailed analysis for specific localization regimes is beyond the scope of the paper, here we aim to approximately capture the scaling behavior.",
"It should also be pointed out that we have not taken into account sparsity in $A$ or $B$ for either architectures."
],
[
"Output-Feedback",
"The centralized output-feedback controller from Theorem REF is shown in Fig.",
"REF (due to space limitations, we omit the architecture diagram corresponding to the standard SLS controller ).",
"The controller collects the measurements $y_i[t]$ from each sensor $s_i$ for all $i = 1, \\dots , N_y$ , maintains internal states $\\hat{x}[t]$ and $\\delta [t]$ , and generates control signals according to (REF ).",
"Figure: The architecture of the centralized output-feedback controller.Comparing to the architecture formed by the original SLS controller in Fig.",
"REF , the new centralized architecture in Fig.",
"REF is much cheaper in both computation and storage.",
"For FIR $\\mathbf {\\Phi }_{{\\bf x}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf x}{\\bf y}}$ and $\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}$ with horizon $T$ , the original architecture requires $(N_u + N_y) (2 N_u - 1) + (T N_u + (T-1) N_x)(2N_x-1) \\\\+ (N_u + T N_x)(2N_y - 1) + 4T - 1$ flops and uses $N_u(N_u + 2N_y) + (T-1) (N_u + N_x) (N_x + N_y) \\\\+ (T + 1) N_y + T N_x + 2 N_u$ scalar memory locations, which are more than $2(N_x + N_y)(N_x + N_u - 1) + T N_u(2 N_x - 1) + T + 2$ flops and $T N_u N_y + (N_u + N_x) (N_x + N_y) \\\\+ (T + 1) N_y + 2 N_x + N_u$ scalar memory locations used in Fig.",
"REF .",
"Again, the analysis is based on the most general case.",
"With sparsity or structure, one could potentially further reduce the resource requirement."
],
[
"Architecture Trade-Offs",
"Despite the intuitive design, the centralized architecture raises several operational concerns.",
"First, the centralized controller becomes the single point of failure in the CPS.",
"Also, the scalability of the centralized scheme is poor: The centralized controller has to ensure communication with all the sensors/actuators and deal with the burden of high computational load.",
"In the following subsections, we explore various distributed system architectures."
],
[
"Global State Architecture",
"The first attempt to reduce the burden on the controller is to offload the computation to the nodes in the system.",
"A first step in this direction is the global state architecture.",
"With this architecture, the sensing and actuation parts of the controller are pushed to the nodes, and the appropriate information is communicated to a centralized global state keeper (GSK) which keeps track of the full system state instead of the input/output signals at each time $t$ .",
"This partitioning is illustrated in Fig.",
"REF .",
"Figure: The partitions of the state-feedback block diagram for the global state architecture."
],
[
"State-Feedback",
"The GSK for a state-feedback controller keeps track of the global state $\\delta [t]$ instead of the raw state $x[t]$ .",
"Rather than directly dispatching the control signals $u[t]$ to the actuators, GSK supplies $\\delta [t]$ to the actuators and relies on the actuators to compute $u[t]$ .",
"We illustrate the details of each node in Fig.",
"REF (and in algorithmic form in Algorithm REF where we describe the the computation required to implement a control policy at the sensors, actuators, and GSK).",
"GSK collects $\\delta _i[t]$ from each sensor $s_i$ .",
"To compute $\\delta _i[t]$ , each $s_i$ stores a column vector $-A^{\\star i}$ .",
"Using the sensed state $x_i[t]$ , $s_i$ computes $-A^{ji}x_i[t]$ and sends it to $s_j$ .",
"Meanwhile, $s_i$ collects $-A^{ij}x_j[t]$ from each $x_j$ and $-B^{ik}u_k[t]$ from each $a_k$ .",
"Together, $s_i$ can compute $\\delta _i[t]$ by $\\delta _i[t]=&\\ x_i[t]-A^{i\\star }x[t-1]- B^{i\\star } u[t-1]\\\\=&\\ x_i[t]- \\sum \\limits _{j} A^{ij}x_j[t-1]- \\sum \\limits _{k} B^{ik}u_k[t-1].$ The $\\delta [t]$ term is then forwarded to each actuator by GSK.",
"The actuator $a_k$ can compute the control signal using the multiplier array similar to the structure in the centralized architecture.",
"The difference is that each actuator only needs to store the rows of the spectral components ${\\Phi _u^{k\\star }}[\\tau ]$ of ${\\mathbf {\\Phi }_{\\bf u}}$ .",
"After getting the control signal $u_k[t]$ , $a_k$ computes $-B^{ik}u_k[t]$ for each sensor $s_i$ .",
"Figure: The state-feedback global state architecture.One outcome of this communication pattern is that $s_i$ only needs to receive $-A^{ij}x_j[t]$ from $s_j$ if $A^{ij} \\ne 0$ .",
"Similarly, only when $B^{ik} \\ne 0$ does $s_i$ need to receive information from $a_k$ .",
"This property tells us that the nodes only exchange information with their neighbors when a non-zero entry in $A$ and $B$ implies the adjacency of the corresponding nodes (as shown in Fig.",
"REF (subfig:cyber-physical-comparison-global-state)).",
"Notice that this property holds for any feasible $\\lbrace \\mathbf {\\Phi }_{\\bf x}, \\mathbf {\\Phi }_{\\bf u}\\rbrace $ , regardless of the constraint set $\\mathcal {S}$ .",
"[!t] [1] $\\bullet $ $\\delta [t] = $ global_state_keeper $\\left(\\left\\lbrace \\delta _i[t] \\right\\rbrace _i\\right)$ : Stack received $\\left\\lbrace \\delta _i[t] \\right\\rbrace _i$ from all sensors $s_i$ to form $\\delta [t]$ .",
"[1] $\\bullet $ $\\left\\lbrace -A^{ji} x_i[t]\\right\\rbrace _j, \\delta _i[t] = $ sensor_i$\\left(x_i[t],\\left\\lbrace -A^{ij} x_j[t]\\right\\rbrace _j,\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _k\\right)$ : Extract $\\left\\lbrace -A^{ji} x_i[t]\\right\\rbrace _j$ from $-A^{\\star i} x_i[t]$ .",
"Stack received $\\left\\lbrace -A^{ij} x_j[t]\\right\\rbrace _j$ and $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _k$ to form $-A^{i\\star }x[t]$ and $-B^{i\\star }u[t]$ .",
"$\\delta _i[t] = x_i[t] - A^{i\\star }x[t-1] - B^{i\\star }u[t-1]$ .",
"[1] $\\bullet $ $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _i = $ actuator_k$(\\delta [t])$ : $u_k[t] = \\sum \\limits _{\\tau \\ge 1} \\Phi _{u}^{k\\star }[\\tau ] \\delta [t+1-\\tau ]$ .",
"Extract $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _i$ from $-B^{\\star k} u_k[t]$ .",
"The state-feedback global state architecture.",
"Architecture Trade-Offs The global state architecture mitigates the computation workload of a CPS with a centralized architecture an.",
"However, such an architecture is subject to a single point of failure – the GSK.",
"We now shift our focus to distributed architectures that are not as vulnerable.",
"Distributed Architectures To avoid the single point of failure problem, we can either reinforce the centralized unit through redundancy, or, decompose it into multiple sub-units, each responsible for a smaller region.",
"The former option trades resource efficiency for robustness, while the latter prevents a total blackout by localizing the failures.",
"Here we take the latter option to an extreme: We remove GSK from the partitions of the control diagram (Fig.",
"REF ) and distribute all control functions to the nodes in the network.",
"As such, the impact of a failed node is contained within a small part of the network.",
"State-Feedback A naive way to remove the GSK from the state-feedback partitions is to send the state $\\delta [t]$ directly from the sensors to the actuators, as shown in Fig.",
"REF .",
"More specifically, each sensor $s_i$ would send $\\delta _i[t]$ to all the actuators.",
"Each actuator $a_k$ then assembles $\\delta [t]$ from the received $\\delta _j[t]$ for all $j$ and computes $u_k[t]$ accordingly.",
"We summarize the function of each sensor/actuator in Algorithm REF .",
"Figure: A naive way to partition the block diagram in a distributed manner: The sensors directly send δ[t]\\delta [t] to each actuator.",
"Those duplicated copies of δ[t]\\delta [t] waste memory resources.",
"[!t] [1] $\\bullet $ $\\delta _i[t], \\left\\lbrace -A^{ji} x_i[t]\\right\\rbrace _j = $ sensor_i$\\left(x_i[t],\\left\\lbrace -A^{ij} x_j[t]\\right\\rbrace _j,\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _k\\right)$ : Extract $\\left\\lbrace -A^{ji} x_i[t]\\right\\rbrace _j$ from $-A^{\\star i} x_i[t]$ .",
"Stack received $\\left\\lbrace -A^{ij} x_j[t]\\right\\rbrace _j$ and $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _k$ to form $-A^{i\\star }x[t]$ and $-B^{i\\star }u[t]$ .",
"$\\delta _i[t] = x_i[t] - A^{i\\star }x[t-1] - B^{i\\star }u[t-1]$ .",
"[1] $\\bullet $ $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _i = $ actuator_k$\\left(\\left\\lbrace \\delta _i[t] \\right\\rbrace _i\\right)$ : Stack received $\\left\\lbrace \\delta _i[t] \\right\\rbrace _i$ to form $\\delta [t]$ .",
"$u_k[t] = \\sum \\limits _{\\tau \\ge 1} \\Phi _{u}^{k\\star }[\\tau ] \\delta [t+1-\\tau ]$ .",
"Extract $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _i$ from $-B^{\\star k} u_k[t]$ .",
"The state-feedback naive distributed architecture.",
"This approach avoids the single point of failure problem.",
"However, it stores duplicated copies of $\\delta _i[t]$ at each actuator, which wastes memory resources.",
"To conserve memory usage, we propose to send processed information to the actuators instead of the raw state $\\delta [t]$ .",
"We depict such a memory-conservative distributed scheme in Fig.",
"REF .",
"Figure: The partition of the block diagram for the distributed architecture that conserves memory usage.The difference between Fig.",
"REF and Fig.",
"REF is that we move the convolution $z{\\mathbf {\\Phi }_{\\bf u}}$ from the actuator side to the sensor side.",
"Implementation-wise, this change leads to the architectures in Fig.",
"REF (and in algorithmic form in Algorithm REF ).",
"Figure: The memory conservative distributed architecture.In Fig.",
"REF , each sensor $s_i$ not only computes $\\delta _i[t]$ , but $s_i$ also feeds $\\delta _i[t]$ into a multiplier array for convolution.",
"Each multiplier in the array stores the $i^{\\rm th}$ column of a spectral component of ${\\mathbf {\\Phi }_{\\bf u}}$ .",
"The convolution result is then disseminated to each actuator.",
"At the actuator $a_k$ , the control signal $u_k[t]$ is given by the sum of the convolution results from each sensor: $u_k[t]=&\\ \\sum \\limits _{\\tau \\ge 1} {\\Phi _u^{k\\star }}[\\tau ]\\delta [t+1-\\tau ] \\nonumber \\\\=&\\ \\sum \\limits _{i = 1}^{N_x} \\sum \\limits _{\\tau \\ge 1} {\\Phi _u^{ki}}[\\tau ]\\delta _i[t+1-\\tau ].",
"\\nonumber $ To confirm that Fig.",
"REF is more memory efficient than Fig.",
"REF , we can count the number of scalar memory locations in the corresponding architectures.",
"Since both architectures store the matrices $A$ , $B$ , and $\\Phi _u[\\cdot ]$ in a distributed manner, the total number of scalar memory locations needed for the multipliers is $N_x^2 + N_x N_u + T N_x N_u.$ We now consider the memory requirements for the buffers.",
"Suppose ${\\mathbf {\\Phi }_{\\bf u}}$ is FIR with horizon $T$ (or, there are $T$ multipliers in the $z{\\mathbf {\\Phi }_{\\bf u}}$ convolution).",
"In this case Fig.",
"REF has the same architecture as Fig.",
"REF without the GSK.",
"Therefore, the total number of stored scalars (buffers) is as follows: $\\text{At each $s_i$:} &\\ N_x + 4, \\nonumber \\\\\\text{At each $a_k$:} &\\ (T + 1) N_x + 1, \\nonumber \\\\\\text{Total:} &\\ (T + 1) N_x N_u + N_x^2 + 4 N_x + N_u.$ On the other hand, the memory conservative distributed architecture in Fig.",
"REF uses the following numbers of scalars: $\\text{At each $s_i$:} &\\ N_x + N_u + T + 3, \\nonumber \\\\\\text{At each $a_k$:} &\\ N_x + 1, \\nonumber \\\\\\text{Total:} &\\ 2 N_x N_u + N_x^2 + (T + 3) N_x + N_u.$ In sum, the memory conservative distributed architecture requires fewer scalars in the system, and the difference is $(T - 1) N_x (N_u - 1)$ .",
"[!t] [1] $\\bullet $ $\\left\\lbrace \\sum \\limits _{\\tau \\ge 1} \\Phi _{u}^{ki}[\\tau ] \\delta _i[t+1-\\tau ] \\right\\rbrace _k,\\left\\lbrace -A^{ji} x_i[t]\\right\\rbrace _j = $ sensor_i$\\left(x_i[t],\\left\\lbrace -A^{ij} x_j[t]\\right\\rbrace _j,\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _k\\right)$ : Extract $\\left\\lbrace -A^{ji} x_i[t]\\right\\rbrace _j$ from $-A^{\\star i} x_i[t]$ .",
"Stack received $\\left\\lbrace -A^{ij} x_j[t]\\right\\rbrace _j$ and $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _k$ to form $-A^{i\\star }x[t]$ and $-B^{i\\star }u[t]$ .",
"$\\delta _i[t] = x_i[t] - A^{i\\star }x[t-1] - B^{i\\star }u[t-1]$ .",
"Extract $\\left\\lbrace \\sum \\limits _{\\tau \\ge 1} \\Phi _{u}^{ki}[\\tau ] \\delta _i[t+1-\\tau ] \\right\\rbrace _k$ from $\\sum \\limits _{\\tau \\ge 1} \\Phi _{u}^{\\star i}[\\tau ] \\delta _i[t+1-\\tau ]$ .",
"[1] $\\bullet $ $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _i = $ actuator_k$\\left(\\left\\lbrace \\sum \\limits _{\\tau \\ge 1} \\Phi _{u}^{ki}[\\tau ] \\delta _i[t+1-\\tau ] \\right\\rbrace _i\\right)$ : $u_k[t] = \\sum \\limits _{i=1}^{N_x} \\left( \\sum \\limits _{\\tau \\ge 1} \\Phi _{u}^{ki}[\\tau ] \\delta _i[t+1-\\tau ] \\right)$ .",
"Extract $\\left\\lbrace -B^{ik} u_k[t]\\right\\rbrace _i$ from $-B^{\\star k} u_k[t]$ .",
"The (state-feedback) memory conservative distributed architecture.",
"Output-Feedback For output-feedback systems, there are multiple ways to remove the role of a GSK and distribute the computation of the controller.",
"Two possibilities are depicted in Fig.",
"REF .",
"In one, the global state information is pushed to the sensor, and in the other, to the actuator side.",
"Figure: Partition choices for the output feedback block diagram for the distributed architecture.When the sensors take over the global state management, the actuators transmit information to each sensor, the sensors then build the global state $\\hat{x}$ locally as in Fig.",
"REF (and in algorithmic form in Algorithm REF ).",
"This approach prevents the single point of failure problem, but it imposes heavy communication load as each sensor needs to receive two messages; $-D^{ik}u_k[t]$ , $B^{\\star k} u_k[t]$ from each actuator $a_k$ .",
"Figure: The architecture of sensor ii (s i s_i) in the sensor-side global-state distributed architecture.",
"[!t] [1] $\\bullet $ $\\left\\lbrace \\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{ki}[\\tau ]\\delta _i[t-\\tau ] \\right\\rbrace _k = $ sensor_i$\\left(y_i[t],\\left\\lbrace B^{\\star k} u_k[t]\\right\\rbrace _k,\\left\\lbrace -D^{ik} u_k[t]\\right\\rbrace _k\\right)$ : Stack received $\\left\\lbrace B^{\\star k} u_k[t]\\right\\rbrace _k$ to form $Bu[t]$ .",
"Update $\\hat{x}[t]$ by (REF ).",
"Stack received $\\left\\lbrace -D^{ik} u_k[t]\\right\\rbrace _k$ to form $-D^{i\\star }u[t]$ .",
"$\\delta _i[t] = y_i[t] - C^{i\\star } \\hat{x}[t] - D^{i\\star }u[t]$ .",
"Extract $\\left\\lbrace \\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{ki}[\\tau ]\\delta _i[t-\\tau ] \\right\\rbrace _k$ from $\\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{\\star i}[\\tau ]\\delta _i[t-\\tau ]$ .",
"[1] $\\bullet $ $\\left\\lbrace -D^{ik} u_k[t]\\right\\rbrace _i, B^{\\star k} u_k[t] = $ actuator_k$\\left(\\left\\lbrace \\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{ki}[\\tau ]\\delta _i[t-\\tau ] \\right\\rbrace _i\\right)$ : $u_k[t] = \\sum \\limits _{i = 1}^{N_y} \\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{ki}[\\tau ]\\delta _i[t-\\tau ]$ .",
"Extract $\\left\\lbrace -D^{ik} u_k[t]\\right\\rbrace _i$ from $-D^{\\star k} u_k[t]$ .",
"Compute $B^{\\star k} u_k[t]$ .",
"The output-feedback sensor-side global-state distributed architecture.",
"One way to alleviate the communication load is to have the actuators send one summarized message instead of two messages to the sensors.",
"However, one copy of $B^{\\star k} u_k[t]$ is necessary if each sensor needs to compute $\\hat{x}[t]$ , which restrains the actuator from summarizing the information.",
"Therefore, we can merge GSK to the actuator side, which leads to the distributed architecture in Fig.",
"REF (and in algorithmic form in Algorithm REF ).",
"To compute the summarized information, we notice that if we can express $\\hat{x}[t]$ as the sum of some vectors $\\hat{x}_{(k)}[t]$ for $k = 1, \\dots , N_u$ , we have $\\hat{x}[t+1]=&\\ A\\hat{x}[t]+ Bu[t]= A\\sum \\limits _{k=1}^{N_u} \\hat{x}_{(k)}[t]+ \\sum \\limits _{k=1}^{N_u} B^{\\star k}u_k[t]\\\\=&\\ \\sum \\limits _{k=1}^{N_u} \\left( A\\hat{x}_{(k)}[t]+ B^{\\star k}u_k[t]\\right)$ As such, we can enforce $\\hat{x}[t]= \\sum \\limits _{k=1}^{N_u} \\hat{x}_{(k)}[t]$ for all future $t$ by $\\hat{x}_{(k)}[t+1]= A\\hat{x}_{(k)}[t]+ B^{\\star k}u_k[t],$ which defines the internal state $\\hat{x}_{(k)}[t]$ of each $a_k$ .",
"With $\\hat{x}_{(k)}[t]$ , we derive the summarized message at each $a_k$ by $\\delta _i[t]=&\\ y_i[t]- C^{i\\star }\\hat{x}[t]- \\sum \\limits _{k=1}^{N_u} D^{ik} u_k[t]\\nonumber \\\\=&\\ y_i[t]+ \\sum \\limits _{k=1}^{N_u} \\left(- C^{i\\star }\\hat{x}_{(k)}[t]- D^{ik} u_k[t]\\right).$ Figure: The actuator-side global-state distributed architecture.",
"[!t] [1] $\\bullet $ $\\left\\lbrace \\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{ki}[\\tau ]\\delta _i[t-\\tau ] \\right\\rbrace _k = $ sensor_$i$$\\left(y_i[t],\\left\\lbrace -C^{i\\star }\\hat{x}_{(k)}[t] -D^{ik} u_k[t] \\right\\rbrace _k\\right)$ : Compute $\\delta _i[t]$ by (REF ).",
"Extract $\\left\\lbrace \\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{ki}[\\tau ]\\delta _i[t-\\tau ] \\right\\rbrace _k$ from $\\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{\\star i}[\\tau ]\\delta _i[t-\\tau ]$ .",
"[1] $\\bullet $ $\\left\\lbrace -C^{i\\star }\\hat{x}_{(k)}[t] -D^{ik} u_k[t] \\right\\rbrace _i = $ actuator_k$\\left(\\left\\lbrace \\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{ki}[\\tau ]\\delta _i[t-\\tau ] \\right\\rbrace _i\\right)$ : $u_k[t] = \\sum \\limits _{i = 1}^{N_y} \\sum \\limits _{\\tau \\ge 0} \\Phi _{uy}^{ki}[\\tau ]\\delta _i[t-\\tau ]$ .",
"Update $\\hat{x}_{(k)}[t]$ by (REF ).",
"Extract $\\left\\lbrace -C^{i\\star }\\hat{x}_{(k)}[t] -D^{ik} u_k[t] \\right\\rbrace _i$ from $-C^{i\\star }\\hat{x}_{(k)}[t] -D^{ik} u_k[t]$ .",
"The output-feedback actuator-side global-state distributed architecture.",
"More Distributed Options There exist more distributed architectures under different partitions of the block diagram.",
"For example, one can also assign the output-feedback convolution $\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}$ to the actuators instead of the sensors.",
"And different partition choices accompany different computation/memory costs.",
"Future work will look at specializing the above computation/memory costs to specific localization constraints.",
"The above expressions serve as upper bounds that are tight for systems that are difficult to localize in space (in the sense of $(d,T)$ -localization defined in [5]).",
"For systems localizable to smaller regions, the actuators only collect local information, and hence we don't need every sensor to report to every actuator.",
"Architecture Comparison Table: Comparison Amongst the Proposed ArchitecturesFigure: The cyber-physical structures of the proposed architectures.",
"The horizontal dashed line separates the cyber and the physical layers of the system.",
"Each node (the boxes) is equipped with a sensor, which collects the state information in the state-feedback scenarios and the measurements in the output-feedback counterparts.Solid bullets represents a computation unit, and the arrow links are the communication channels.",
"The cyber structure of the centralized architecture is ignorant about the underlying physical system, while the other architectures manifest some correlations between the cyber and the physical structures.",
"The distributed architectures replace GSK by direct communications, which can be trimmed by imposing appropriate localization constraints on Φ 𝐮 {\\mathbf {\\Phi }_{\\bf u}} or Φ 𝐮𝐲 {\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}} during the synthesis phase.We have now seen how different architectures implementing the same controller model allows the engineer to consider different trade-offs when it comes to implementation and deployment.",
"Here we compare the proposed architectures and discuss their differences.",
"Our findings are summarized in Table REF .",
"In terms of robustness, the centralized and the global state architectures suffer from a single point of failure, i.e., the loss of the centralized controller or the GSK paralyzes the whole system.",
"This also makes the system vulnerable from a cyber-security perspective.",
"On the contrary, the distributed architectures can still function with some nodes knocked out of the network.The system operator might need to update ${\\mathbf {\\Phi }_{\\bf u}}$ or ${\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}}$ to maintain performance – such a re-design is well within the scope of the SLS framework.",
"For information storage, the centralized controller uses the fewest buffers, and the state-feedback global state architecture (because its GSK has to relay $\\delta [t]$ ) and the output-feedback sensor-side GS distributed architecture (because the sensors maintain duplicated $\\hat{x}[t]$ ) store the most variables.",
"We remark that although the centralized scheme achieves the minimum storage usage at the system level, the single node memory requirement is high for the centralized controller.",
"Conversely, the other architectures store information in a distributed manner, and a small memory is sufficient for each node.",
"We evaluate the computational load at each node by counting the number of performed multiplication operations.",
"The centralized architecture aggregates all the computation at the centralized controller, while the other architectures perform distributed computing.",
"For distributed settings, the computation overhead is slightly different at each node.",
"For the state-feedback scenario, the global state and naive distributed architectures let the actuators compute the convolution.",
"Instead, the memory conservative distributed architecture puts the multiplier arrays at the sensors.",
"For the output-feedback case, convolutions are performed on the sensor side, and the management of global state $\\hat{x}[t]$ imposes additional computational load on the components.",
"Comparing to the sensor-side GS distributed architecture, it is more computationally balanced for the actuator to handle $\\hat{x}[t]$ .",
"Finally, we discuss the communication loading of the architectures.",
"In Fig.",
"REF , we sketch the resulting cyber-physical structures of each scheme.",
"In the state-feedback scenarios, each node has its own sensor to measure the state, but some nodes can have no sensors in the output-feedback cases.",
"The centralized architecture ignores the underlying system interconnection and installs a centralized controller to collect measurements and dispatch control actions.",
"Under this framework, the sensors and the actuators only need to recognize the centralized controller, but the centralized controller must keep track of all the sensors in the system, which limits the scalability of the scheme.",
"The global state architecture also introduces an additional node into the system, the GSK, with which all nodes should be contact with.",
"Meanwhile, the sensors and actuators also communicate with each other.",
"State-feedback sensors and actuators communicate according to the matrices $A$ and $B$ .",
"In other words, if two nodes are not directly interacting with each other in the system dynamics, they don't need to establish a direct connection.",
"Output-feedback actuators also communicate with the sensors according to matrix $D$ , but sensors compute and distributed control signals to actuators.",
"Although the output-feedback sensors communicate more with the actuators, it is possible to localize the communications by regulating the structure of ${\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}}$ .",
"Similarly, in the distributed architectures, the sensors and the actuators maintain connections according to $A$ and $B$ for state-feedback or $C$ and $D$ for output-feedback.",
"Additionally, direct communications, which are governed by the structure of ${\\mathbf {\\Phi }_{\\bf u}}$ or ${\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}}$ , are added to replace the role of GSK.",
"Although it would be slightly more complicated than having a GSK as the relay, we can localize ${\\mathbf {\\Phi }_{\\bf u}}$ or ${\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}}$ at the synthesis phase to have a sparse communication pattern.",
"Besides the centralized architecture, the physical structure has a direct influence on the cyber structure in all other architectures.",
"As such, we believe further research on the deployment architectures of SLS would lead to better cyber-physical control systems.",
"Conclusion and Future Directions New internally stabilizing state-feedback and output-feedback controllers were derived for systems that are open-loop stable.",
"The controllers were shown to have block diagram realizations that are in some ways simpler than the “standard” SLS controllers.",
"We considered various architectures to deploy this controller to a real CPS.",
"We illustrated and compared the memory and computation trade-offs among different deployment architectures: centralized, global state, and four different distributed architectures.",
"There are still many decentralized architecture options left to explore.",
"We are also investigating how robustness and virtual localizability [26] can be integrated into this framework.",
"Proof of Theorem REF The full proof of the state feedback case can be found in [27].",
"To provide some insight: observe that ${\\bf K}= {\\mathbf {\\Phi }_{\\bf u}}{\\mathbf {\\Phi }_{\\bf x}^{-1}} = (z{\\mathbf {\\Phi }_{\\bf u}})(z{\\mathbf {\\Phi }_{\\bf x}})^{-1}.$ The realization in Fig.",
"REF then follows by putting $z{\\mathbf {\\Phi }_{\\bf u}}$ in the forward path and realizing $(z{\\mathbf {\\Phi }_{\\bf x}})^{-1}$ as the feedback path through the $I-z{\\mathbf {\\Phi }_{\\bf x}}$ block.",
"For the output-feedback case, using the Woodbury matrix identity, we know that $\\mathbf {\\Phi }_{{\\bf u}{\\bf y}} =&\\ {\\bf L}+ {\\bf L}C ((zI - A) - B {\\bf L}C )^{-1} B {\\bf L}\\\\=&\\ \\left({\\bf L}^{-1} - C(zI - A)^{-1}B\\right)^{-1}= \\left( {\\bf K}^{-1} - G \\right)^{-1}$ where $G = C(zI - A)^{-1}B + D$ is the open-loop transfer matrix.",
"As a result, we know ${\\bf K}=\\ \\left( \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}^{-1} + G \\right)^{-1}.$ The block diagram then follows immediately.",
"To show internal stability, we consider additional closed-loop perturbations as in Fig.",
"REF .",
"It is sufficient to examine how the internal states ${\\bf x}$ , ${\\bf u}$ , ${\\bf y}$ , and $\\hat{{\\bf x}}$ are affected by external signals.",
"The relations among the signals are described by $z{\\bf x}=&\\ A {\\bf x}+ B{\\bf u}+ {\\bf d}_{\\bf x},\\\\{\\bf u}=&\\ \\mathbf {\\Phi }_{{\\bf u}{\\bf y}} \\left({\\bf y}- C\\hat{{\\bf x}} - D ({\\bf u}- {\\bf d}_{\\bf u})\\right) + {\\bf d}_{\\bf u},\\\\{\\bf y}=&\\ C {\\bf x}+ D {\\bf u}+ {\\bf d}_{\\bf y},\\\\z\\hat{{\\bf x}} =&\\ A \\hat{{\\bf x}} + B ({\\bf u}- {\\bf d}_{\\bf u}) + {\\bf d}_{\\hat{{\\bf x}}}.$ Rearranging the equations above yields the matrix representation $\\begin{bmatrix}{\\bf x}\\\\{\\bf u}\\\\{\\bf y}\\\\\\hat{{\\bf x}}\\end{bmatrix} =\\begin{bmatrix}\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} & \\Delta B \\Gamma &\\mathbf {\\Phi }_{{\\bf x}{\\bf y}} & \\Delta - \\mathbf {\\Phi }_{{\\bf x}{\\bf x}} \\\\\\mathbf {\\Phi }_{{\\bf u}{\\bf x}} & \\Gamma &\\mathbf {\\Phi }_{{\\bf u}{\\bf y}} & -\\mathbf {\\Phi }_{{\\bf u}{\\bf x}}\\\\\\Lambda C\\Delta & \\Lambda G &\\Lambda & -G\\mathbf {\\Phi }_{{\\bf u}{\\bf x}}\\\\\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} - \\Delta & \\mathbf {\\Phi }_{{\\bf x}{\\bf y}}G &\\mathbf {\\Phi }_{{\\bf x}{\\bf y}} & 2\\Delta - \\mathbf {\\Phi }_{{\\bf x}{\\bf x}} \\\\\\end{bmatrix}\\begin{bmatrix}{\\bf d}_{\\bf x}\\\\{\\bf d}_{\\bf u}\\\\{\\bf d}_{\\bf y}\\\\{\\bf d}_{\\hat{{\\bf x}}}\\end{bmatrix},$ where $\\Delta = (zI-A)^{-1},\\quad \\Gamma = \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}G + I,\\quad \\Lambda = G\\mathbf {\\Phi }_{{\\bf u}{\\bf y}} + I.$ By assumption, $\\Delta $ is stable, and $\\mathbf {\\Phi }_{{\\bf x}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf x}},\\mathbf {\\Phi }_{{\\bf x}{\\bf y}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}$ are constrained to be stable by the SLS optimization problem.",
"It can then be verifies that all 16 entries in the matrix are stable, which concludes the proof.",
"Figure: Perturbations required for internal stability analysis."
],
[
"Architecture Comparison",
"We have now seen how different architectures implementing the same controller model allows the engineer to consider different trade-offs when it comes to implementation and deployment.",
"Here we compare the proposed architectures and discuss their differences.",
"Our findings are summarized in Table REF .",
"In terms of robustness, the centralized and the global state architectures suffer from a single point of failure, i.e., the loss of the centralized controller or the GSK paralyzes the whole system.",
"This also makes the system vulnerable from a cyber-security perspective.",
"On the contrary, the distributed architectures can still function with some nodes knocked out of the network.The system operator might need to update ${\\mathbf {\\Phi }_{\\bf u}}$ or ${\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}}$ to maintain performance – such a re-design is well within the scope of the SLS framework.",
"For information storage, the centralized controller uses the fewest buffers, and the state-feedback global state architecture (because its GSK has to relay $\\delta [t]$ ) and the output-feedback sensor-side GS distributed architecture (because the sensors maintain duplicated $\\hat{x}[t]$ ) store the most variables.",
"We remark that although the centralized scheme achieves the minimum storage usage at the system level, the single node memory requirement is high for the centralized controller.",
"Conversely, the other architectures store information in a distributed manner, and a small memory is sufficient for each node.",
"We evaluate the computational load at each node by counting the number of performed multiplication operations.",
"The centralized architecture aggregates all the computation at the centralized controller, while the other architectures perform distributed computing.",
"For distributed settings, the computation overhead is slightly different at each node.",
"For the state-feedback scenario, the global state and naive distributed architectures let the actuators compute the convolution.",
"Instead, the memory conservative distributed architecture puts the multiplier arrays at the sensors.",
"For the output-feedback case, convolutions are performed on the sensor side, and the management of global state $\\hat{x}[t]$ imposes additional computational load on the components.",
"Comparing to the sensor-side GS distributed architecture, it is more computationally balanced for the actuator to handle $\\hat{x}[t]$ .",
"Finally, we discuss the communication loading of the architectures.",
"In Fig.",
"REF , we sketch the resulting cyber-physical structures of each scheme.",
"In the state-feedback scenarios, each node has its own sensor to measure the state, but some nodes can have no sensors in the output-feedback cases.",
"The centralized architecture ignores the underlying system interconnection and installs a centralized controller to collect measurements and dispatch control actions.",
"Under this framework, the sensors and the actuators only need to recognize the centralized controller, but the centralized controller must keep track of all the sensors in the system, which limits the scalability of the scheme.",
"The global state architecture also introduces an additional node into the system, the GSK, with which all nodes should be contact with.",
"Meanwhile, the sensors and actuators also communicate with each other.",
"State-feedback sensors and actuators communicate according to the matrices $A$ and $B$ .",
"In other words, if two nodes are not directly interacting with each other in the system dynamics, they don't need to establish a direct connection.",
"Output-feedback actuators also communicate with the sensors according to matrix $D$ , but sensors compute and distributed control signals to actuators.",
"Although the output-feedback sensors communicate more with the actuators, it is possible to localize the communications by regulating the structure of ${\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}}$ .",
"Similarly, in the distributed architectures, the sensors and the actuators maintain connections according to $A$ and $B$ for state-feedback or $C$ and $D$ for output-feedback.",
"Additionally, direct communications, which are governed by the structure of ${\\mathbf {\\Phi }_{\\bf u}}$ or ${\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}}$ , are added to replace the role of GSK.",
"Although it would be slightly more complicated than having a GSK as the relay, we can localize ${\\mathbf {\\Phi }_{\\bf u}}$ or ${\\mathbf {\\Phi }_{{\\bf u}{\\bf y}}}$ at the synthesis phase to have a sparse communication pattern.",
"Besides the centralized architecture, the physical structure has a direct influence on the cyber structure in all other architectures.",
"As such, we believe further research on the deployment architectures of SLS would lead to better cyber-physical control systems."
],
[
"Conclusion and Future Directions",
"New internally stabilizing state-feedback and output-feedback controllers were derived for systems that are open-loop stable.",
"The controllers were shown to have block diagram realizations that are in some ways simpler than the “standard” SLS controllers.",
"We considered various architectures to deploy this controller to a real CPS.",
"We illustrated and compared the memory and computation trade-offs among different deployment architectures: centralized, global state, and four different distributed architectures.",
"There are still many decentralized architecture options left to explore.",
"We are also investigating how robustness and virtual localizability [26] can be integrated into this framework."
],
[
"Proof of Theorem ",
"The full proof of the state feedback case can be found in [27].",
"To provide some insight: observe that ${\\bf K}= {\\mathbf {\\Phi }_{\\bf u}}{\\mathbf {\\Phi }_{\\bf x}^{-1}} = (z{\\mathbf {\\Phi }_{\\bf u}})(z{\\mathbf {\\Phi }_{\\bf x}})^{-1}.$ The realization in Fig.",
"REF then follows by putting $z{\\mathbf {\\Phi }_{\\bf u}}$ in the forward path and realizing $(z{\\mathbf {\\Phi }_{\\bf x}})^{-1}$ as the feedback path through the $I-z{\\mathbf {\\Phi }_{\\bf x}}$ block.",
"For the output-feedback case, using the Woodbury matrix identity, we know that $\\mathbf {\\Phi }_{{\\bf u}{\\bf y}} =&\\ {\\bf L}+ {\\bf L}C ((zI - A) - B {\\bf L}C )^{-1} B {\\bf L}\\\\=&\\ \\left({\\bf L}^{-1} - C(zI - A)^{-1}B\\right)^{-1}= \\left( {\\bf K}^{-1} - G \\right)^{-1}$ where $G = C(zI - A)^{-1}B + D$ is the open-loop transfer matrix.",
"As a result, we know ${\\bf K}=\\ \\left( \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}^{-1} + G \\right)^{-1}.$ The block diagram then follows immediately.",
"To show internal stability, we consider additional closed-loop perturbations as in Fig.",
"REF .",
"It is sufficient to examine how the internal states ${\\bf x}$ , ${\\bf u}$ , ${\\bf y}$ , and $\\hat{{\\bf x}}$ are affected by external signals.",
"The relations among the signals are described by $z{\\bf x}=&\\ A {\\bf x}+ B{\\bf u}+ {\\bf d}_{\\bf x},\\\\{\\bf u}=&\\ \\mathbf {\\Phi }_{{\\bf u}{\\bf y}} \\left({\\bf y}- C\\hat{{\\bf x}} - D ({\\bf u}- {\\bf d}_{\\bf u})\\right) + {\\bf d}_{\\bf u},\\\\{\\bf y}=&\\ C {\\bf x}+ D {\\bf u}+ {\\bf d}_{\\bf y},\\\\z\\hat{{\\bf x}} =&\\ A \\hat{{\\bf x}} + B ({\\bf u}- {\\bf d}_{\\bf u}) + {\\bf d}_{\\hat{{\\bf x}}}.$ Rearranging the equations above yields the matrix representation $\\begin{bmatrix}{\\bf x}\\\\{\\bf u}\\\\{\\bf y}\\\\\\hat{{\\bf x}}\\end{bmatrix} =\\begin{bmatrix}\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} & \\Delta B \\Gamma &\\mathbf {\\Phi }_{{\\bf x}{\\bf y}} & \\Delta - \\mathbf {\\Phi }_{{\\bf x}{\\bf x}} \\\\\\mathbf {\\Phi }_{{\\bf u}{\\bf x}} & \\Gamma &\\mathbf {\\Phi }_{{\\bf u}{\\bf y}} & -\\mathbf {\\Phi }_{{\\bf u}{\\bf x}}\\\\\\Lambda C\\Delta & \\Lambda G &\\Lambda & -G\\mathbf {\\Phi }_{{\\bf u}{\\bf x}}\\\\\\mathbf {\\Phi }_{{\\bf x}{\\bf x}} - \\Delta & \\mathbf {\\Phi }_{{\\bf x}{\\bf y}}G &\\mathbf {\\Phi }_{{\\bf x}{\\bf y}} & 2\\Delta - \\mathbf {\\Phi }_{{\\bf x}{\\bf x}} \\\\\\end{bmatrix}\\begin{bmatrix}{\\bf d}_{\\bf x}\\\\{\\bf d}_{\\bf u}\\\\{\\bf d}_{\\bf y}\\\\{\\bf d}_{\\hat{{\\bf x}}}\\end{bmatrix},$ where $\\Delta = (zI-A)^{-1},\\quad \\Gamma = \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}G + I,\\quad \\Lambda = G\\mathbf {\\Phi }_{{\\bf u}{\\bf y}} + I.$ By assumption, $\\Delta $ is stable, and $\\mathbf {\\Phi }_{{\\bf x}{\\bf x}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf x}},\\mathbf {\\Phi }_{{\\bf x}{\\bf y}}, \\mathbf {\\Phi }_{{\\bf u}{\\bf y}}$ are constrained to be stable by the SLS optimization problem.",
"It can then be verifies that all 16 entries in the matrix are stable, which concludes the proof.",
"Figure: Perturbations required for internal stability analysis."
]
] | 2012.05211 |
[
[
"Conformity: A Path-Aware Homophily Measure for Node-Attributed Networks"
],
[
"Abstract Unveil the homophilic/heterophilic behaviors that characterize the wiring patterns of complex networks is an important task in social network analysis, often approached studying the assortative mixing of node attributes.",
"Recent works underlined that a global measure to quantify node homophily necessarily provides a partial, often deceiving, picture of the reality.",
"Moving from such literature, in this work, we propose a novel measure, namely Conformity, designed to overcome such limitation by providing a node-centric quantification of assortative mixing patterns.",
"Differently from the measures proposed so far, Conformity is designed to be path-aware, thus allowing for a more detailed evaluation of the impact that nodes at different degrees of separations have on the homophilic embeddedness of a target.",
"Experimental analysis on synthetic and real data allowed us to observe that Conformity can unveil valuable insights from node-attributed graphs."
],
[
"Introduction",
"During the last decades, network science has become one of the fastest growing multidisciplinary research fields.",
"Every year, countless researchers, from heterogeneous backgrounds, leverage network theory to analyze complex data describing alternative facets of real world phenomena.",
"From sociology to biology, more and more domains study entities composed of several components - each having its internal complexity and peculiar functionalities - all of them strictly tied in functional relationships.",
"Such complex organizations can naturally be modeled as networks, and as such, analyzed.",
"While reasoning on networks built on top of contextual data, topology is only one of the aspects to take into account: nodes and edges often carry additional semantic information that are of uttermost importance to properly understand the phenomena expressed by the underlying topological structure.",
"Often, such augmented structures are referred to as Feature-rich networks [1].",
"That general term acts as an umbrella for several, more specific, class of network extensions including temporal as well as probabilistic and attributed (or labeled) networks.",
"In this work, we are particularly interested in labeled or node-attributed networks, where reliable external information is added to the nodes as categorical or numerical attributes.",
"Node-attributed graphs are a quite expressive model of social network environments since several salient dimensions (age, gender, nationality...) can be meaningfully studied by leveraging such a framework.",
"Indeed, one of the salient aspects that makes network science a widespread research methodology is its ability to unveil emergent behaviors of complex systems.",
"Network topology is, perhaps, the clearest example of how the overall complexity of a whole system is more than the sum of the coupled interactions among its components.",
"Several modeling works have shown how some universal network properties are the results of emergent behaviors: classic examples are the long-tail degree distribution [2] and the meso-scale modular organization [3] that describe complex systems as sparsely connected dense components.",
"Another relevant emerging behavior is homophily.",
"It has been observed that individuals are more likely to group in social circles if they share common features and stay apart when some specificity diverges.",
"Social network analysis has deeply investigated such a phenomenon, trying to measure its impact and propose a mechanistic explanation to its existence.",
"A proxy often used to estimate for homophilic behaviors fall under the name of Newman's assortativity [4].",
"Such a measure aims to classify a whole network in a range that goes among two extremes: disassortative mixing, where nodes are likely to be connected if they are anti-correlated w.r.t.",
"a given property, and assortative mixing, where, conversely, nodes are likely to be connected if they share a given property.",
"Assortativity has been widely studied and applied to characterize several phenomena such as degree correlation and node-attribute correlations.",
"One of the major drawbacks of such a measure, and similar ones, lies in its definition scale: a complex behavior is summarized in a single, average, score.",
"Recently, a few works [5] tried to overcome such limitation by proposing a multiscale extension of Newman's assortativity, thus allowing to analyze multimodal behaviors that the original score makes impossible to observe (e.g., identifying different, even conflicting, homophilic/heterophilic behaviors within the same complex system).",
"In this work, we move from such a line of research, proposing an alternative proxy for measuring multiscale node homophilic couplings: Conformity, a node-centric path-aware measure, able to unveil heterogeneous mixing patterns in node-attributed networks, designed to cope with categorical (single and multi)-attributes.",
"Inspired by a higher-order assortativity definition, namely the clumpiness score [6], Conformity takes into consideration the evidence that nodes with similar characteristics are not divided by long chains.",
"Experimental results carried out on real world node-attributed networks underline that Conformity allows to study homophilic patterns from a novel point of view and make valuable inference on the social contexts it is applied to.",
"The work is organized as follows.",
"Section introduces the relevant literature to frame the proposed contribution; Section formally introduces Conformity; Section discusses experimental results obtained applying Conformity to synthetic as well as real-world data.",
"Finally, Section concludes the paper."
],
[
"Related",
"Literature defines social homophily as the tendency of people to interact with similar others in respect of dimensions such as age, gender, education, as well as values, attitudes, and political beliefs, sourced by geographical distances, households, workplaces, and universal human cognitive processes [7].",
"Hidden social dynamics can be unveiled studying homophily as well as heterophily among people: in the presence of segregation, interracial friendships are less probable when social class is correlated with race [8]; in the early school grades, boys tend to form larger and more heterogeneous cliques compared to the smaller and more homogeneous cliques of girls [9]; intergroup mixing is also a key factor in academic success when interdisciplinary research is involved [10].",
"Such a brief set of examples let us know how both homophily, and its counterpart, act as fundamental principles in the choice of people's social circles.",
"In the language of network science, they act as a discriminant factor for node neighborhood selection.",
"Network homophily can refer either to explicit topology (e.g., nodes with a similar degree preferably connect) or to the interactions between nodes sharing similar labels.",
"Newman's assortativity coefficient [4] is the most known and used measure for quantifying homophily in complex networks.",
"Based on modularity, the coefficient is calculated as the sum of the differences between the observed and the expected fraction of edges between nodes sharing similar values of an attribute.",
"Some recent extensions or alternative approaches, like ProNe [11] or the VA-Index [12], are also able to cope with pairs of attributes or vector of features, shedding light, more than Newman's coefficient, on the phenomenon of similarity between two or more attributes based on network structure.",
"Such global and aggregated measures flatten and simplify a heterogeneous context in one only score, and avoid the presence of outliers or different mixing interactions characterizing different zones of networks and perhaps also single nodes.",
"In such scenarios, local or node-centric approaches (able to assign a score to each graph node) should help for quantifying a more reliable and exploitable network description.",
"Since the only direct neighborhood (or ego-network) of nodes can not be taken into consideration due to its limited expressive power (inherited in the scale-free-like degree distribution of complex networks), the issue is to define connectivity boundednesses able to circumscribe those nodes whose importance is fundamental in the assortative attitude measurement of a target one.",
"While some lines of research focused on degree assortativity [13] (extended to cope with higher-order notions of node neighborhood such as a two-walks degree correlation [14] or transsortativity [15]), the node-attributed counterpart of the problem has not received much of attention.",
"Only a few studies address such a task in this latter scenario.",
"Recent works aimed to study the existence of possible relations among network structure and label distribution among nodes (e.g., how structure and minority size generate perception biases [16]) as well as shed light on the individual differences in mixing (e.g., in the analysis of monophily, a concept aiming to identify those individuals with extreme preferences for different labels [17]).",
"Accordingly, inferring and quantifying individual differences as well as different local mixing comes as a hard task in complex networks studies.",
"A model able to characterize the within-group mean and variation of mixing patterns was recently proposed in the framework of Bayesian inference [18]: when variation is consistently present, the group mean only is not able to fully describe individual node preferences.",
"In some work, locality is exploited through a definition of assortativity based on the correlation between two consecutive nodes visited by a random walker.",
"For instance, this rationale is used in [19], and applied in the graph classification task; a multi-hop assortativity is defined, here, as the probability that a randomly selected node and a randomly selected $t$ -hop neighbor belong to the same category, where $t$ indicates the time of the visit of the random walker.",
"Closer to the current work, a node-centric and Newman's-normalized measure, namely Peel's assortativity [5], was recently proposed in the context of local-aware homophily, modeling similarities between nodes as an autocorrelation of a time-series defined as a sequence of node labels visited by a random walker with restart."
],
[
"We aim to design a local proxy to measure the degree of homophilic embeddedness of network nodes w.r.t.",
"the attributes they carry.",
"Such a task has been recently approached by Peel et al.",
"[5] to overcome the limitation of classical approaches that usually propose a single aggregate score to characterize the overall assortativity of network nodes.",
"A multiscale strategy to estimate the presence of homophilic patterns within a complex system enables the discovery of emergent behaviors that classical indexes often are not capable of unveiling.",
"The score proposed in [5] moves from the classical Newman's assortativity [4] that, in turn, poses its ground on a reinterpretation of the modularity score - a measure often used to quantify the quality of network clustering partitions.",
"Modularity, $Q$ , computes the difference between the observed and the expected fraction of edges between nodes sharing similar attribute values: in the assortativity coefficient, $r_{global}$ , such quantity is normalized in the range $-1\\le r_{global} \\le 1$ .",
"Thus: $r_{global}=1$ implies that all edges only connect nodes labeled with the same value; $r_{global}=0$ that all edges are randomly connected, and; hypothetically, $r_{global}=-1$ that all edges only connect nodes with a different value.",
"Formally, $r_{global} = \\frac{Q}{Q_{max}} = \\frac{\\sum _{g}{e_{gg}} - \\sum _{g}{a^{2}_g}}{1- \\sum _{g}{a^{2}_g}}$ where $e_{gg}$ is the proportion of edges connecting nodes of the same type $g$ , and $a_{g}=\\sum _{i\\in g}{k_i/2m}$ is the sum of degrees ($k_i$ ) of nodes with type $g$ .",
"Indeed, the approach in [5] yields valuable results; however, it misses a fundamental high-order property of networks: the length of paths connecting nodes.",
"To address such an issue, we define a novel measure, namely ConformityPython code available at https://github.com/GiulioRossetti/conformity .",
"Given an undirected attributed network $G=(V,E,A)$ , where $V=\\lbrace v_1, v_2, \\dots , v_n\\rbrace $ is the set of nodes, $E=\\lbrace (v_i,v_j)|v_i,v_j \\in V\\rbrace $ the set of edges among them, and $A=\\lbrace l_1,l_2,\\dots , l_n\\rbrace $ the set of node attributes, Conformity computes the similarity between the attributes of the node $u\\in V$ with the ones of the other nodes of the network, weighing it with the distance among them.",
"Here, we will focus only on networks with nodes carrying categorical attributes.",
"Figure: Toy Examples.",
"(a-c) Node colors map categorical attribute values, while node sizes encode the respective Conformity scores (the smaller the size, the lower the score).",
"(a) A scenario in which all nodes minimize the Conformity score: all nodes have the same size, ψ(u,α)=-1\\psi (u,\\alpha )=-1, since no connection exists among pairs sharing the same color.",
"(b) The effect of distance on the ψ(u,α)\\psi (u,\\alpha ) value: the central node's score approaches 1, while moving toward the graph periphery (composed by nodes having different color) nodes' ψ(u,α) \\psi (u,\\alpha ) decreases – reaching negative values for the extreme periphery.",
"(c) Karate Club.",
"Node colors encode the two factions of the Karate Club dispute, node sizes are proportional to Conformity scores for α=2.5\\alpha =2.5To facilitate the introduction of Conformity we need to define a few support functions.",
"Considering a node $u\\in V$ we define the set $N_{u,d}$ as the set of $u$ neighboring nodes at a distance $d$ : $N_{u,d} = \\lbrace v| dist(u,v)=d\\rbrace .$ Moreover, lets call $I(u,v)$ the indicator function that compares the attribute values of two nodes $u,v \\in V$ $I_{u,v} =\\left\\lbrace \\begin{array}{ll}1 & \\mbox{if } l_u=l_v \\\\-1 & \\mbox{otherwise}\\end{array}\\right.$ and $f_{u,l_u}$ the function that, if among the neighboring nodes of $u$ there is at least one node sharing the same attribute value $l_u$ , computes the ratio of $u$ 's neighbors sharing it $f_{u,l_u} = \\frac{|\\lbrace v|v\\in \\Gamma (u) \\wedge l_u=l_v\\rbrace |}{|\\Gamma (u)|},$ where $\\Gamma (u)$ is the first order neighborhood of node $u$ , i.e., the set of nodes adjacent to it.",
"Moreover, to assure a consistent interpretation of Conformity, we force $f_{u,l_u}$ to assume values in $(0, 1]$ by setting its value to 1 when its numerator nullifies.",
"Finally, we define the Conformity score for a node $u\\in V$ and a given real number $\\alpha $ in $[0, +\\infty )$ as: $\\psi (u,\\alpha ) = \\frac{\\sum _{d\\in D} \\frac{\\sum _{v \\in N_{u,d}} I_{u,v} f_{v,l_v}}{|N_{u,d}| d^\\alpha }}{\\sum _{d\\in D} d^{-\\alpha }},$ where $D$ is $max(\\lbrace dist(i,j)|i,j \\in V\\rbrace )$ , and the parameter $\\alpha $ controls the level of interaction between nodes, which exponentially decreases while the distances among nodes increase; thus, imposing $\\alpha =1$ , we force a linear decrease w.r.t.",
"the distance, while $\\alpha >1$ imposes a sublinear decrease which reduces the level of interaction between relatively distant nodes.",
"Conformity can be algorithmically interpreted as follows.",
"For each node pair $u,v\\in V$ , with $v\\in N_{u,d}$ with $1\\le d \\le max(\\lbrace dist(i,j)| i,j\\in V\\rbrace )$ the nodes attribute concordance – given by $I(u,v)$ – is weighted by $f_{v,l_v}$ , namely the degree of homophily of the node $v$ toward its first order neighborhood; The average of such score aggregated over all the nodes in $N_{u,d}$ is then damped by a factor $d^\\alpha $ , to account for the distance that separates the nodes considered by the source $u$ .",
"Note that we used an inverse power-law distance decay – that recalls well-known physical measures such as the Coulombic and gravitational ones – since such an approach has already proven its consistency in the definition of the clumpiness measure [6], a widely used degree dispersion index.",
"Finally, the computed score is normalized to ensure that Conformity lies in the range $[-1, 1]$ .",
"Intuitively, the value of $\\psi (u, \\alpha )$ is maximized when a node $u$ is surrounded by neighbors having the same attribute value, minimized in the opposite scenario.",
"Fig.",
"REF (a) shows a network whose nodes (colored by their attribute value) always minimize their Conformity value independently from the chosen decay exponent.",
"Such a limit case example perfectly captures the essence of anti-conformity: edges always connect nodes with a different attribute value, resulting in the absence of homophilic islands.",
"Conversely, Fig.",
"REF (b) shows a simple scenario where the length of the paths among nodes sharing different labels plays a crucial role in the Conformity values.",
"We can easily observe how Conformity (coded with the relative node size) tends to decrease moving from the inner layer to the outer ones – e.g., moving from the more homophilic embedded nodes to the more heterophilic ones.",
"As discussed, Conformity is a node-related measure: we can define the overall degree of Conformity of a network as: $\\Psi (\\alpha ) = \\frac{1}{|V|}\\sum _{u\\in V} \\psi (u, \\alpha ).$ Indeed, such average score is only able to capture a general trend, not to provide a clear picture of the emergent homophilic behaviors at a local level.",
"To better understand the information that the proposed measure can unveil, let us consider the classic example offered by Karate Club dataset [20], representing the small social network of a club after a conflict arose between the administrator, “John A.",
"\", and an instructor, `Mr. Hi\".",
"The graph is classically used as a toy example for characterizing community discovery algorithms since it is neatly divided into two factions and very suitable for explaining a clustering methodology.",
"Moreover, since each node is labeled with the club it belongs (“John A.\"",
"or “Mr.",
"Hi\"), this external information is commonly exploited as a ground truth to test the goodness of the algorithm outputs, even if it has been shown not to be a proper approach [21].",
"In Fig.",
"REF (c), different colors encode the two categorical node attribute values characterizing the network while, as in the previous example, the node sizes are proportional to the node Conformity score ($\\alpha =2.5$ ).",
"As we expected, the highest Conformity values are assigned to those nodes that prevalently connect to same attributed peers while, on the other hand, the lowest ones characterize bridge-nodes.",
"Particular attention must be paid to node 8, which registers the lowest Conformity score ($\\simeq $ -0.18).",
"Indeed, the data paper that discusses the origin of the Karate Club network dataset [20] help us in providing a neat justification for such Conformity value: node 8 identifies a weak supporter of “Mr.",
"Hi\", that joined with the “John A.",
"\"'s faction, after the split, for personal advantage, so he represents a bridge between the two opposite sides of the Karate Club dispute indeed.",
"Figure: Peel's quintet toy example.",
"KDE's distributions of several local mixing patterns according to Conformity, for different values of α\\alpha : the higher the value, the less the contribution of distant nodes to the target final score – as shown by the progressive amplification of the distributions toward close-to-bound values."
],
[
"Experimental analysis",
"Studying the homophilic patterns of actors embedded in a network is a way to unveil emergent behaviors that are otherwise hard to identify.",
"In this section, we propose a characterization of both synthetic and real-world networks using the proposed Conformity score.",
"Figure: Copenhagen Network analysis.",
"(a) Conformity analysis (α=2.5\\alpha =2.5) in the SMS layer analysis; (b-c) Conformity analysis (α=2.5\\alpha =2.5) in the proximity graphs of Monday and Saturday."
],
[
"Synthetic data",
"Inspired by the Peel's quintet [5], in Fig.",
"REF we replicate the building of a set of five small synthetic graphs with the same number of nodes and edges (40 nodes – 20 red, 20 green – and 160 edges), but involving a rewiring of edges leading to the emergence of different local mixing patterns that Newman's assortativity coefficient, $r$ , is not able to detect (i.e., $r=0$ ).",
"Indeed, Newman's score is a valid indicator only for the leftmost graph of the figure, the only one where all edges are randomly rewired across all nodes.",
"This is showed by the unimodal distribution in the Conformity plot for (a).",
"In contrast, other plots reflect and capture the heterogeneous patterns obtained by planting homophilic relations among nodes: in such scenarios, the unimodal distribution breaks down into bimodal ones – e.g., the twin peaks observed for the rightmost graph describe the most extreme case where exactly half the nodes is perfectly homogeneous; in contrast, the other half is entirely heterogeneous.",
"An aspect worth noticing is the effect played by the $\\alpha $ parameter on the $\\psi $ values.",
"As discussed, the $\\alpha $ exponent allows tuning Conformity sensitivity w.r.t.",
"the distance among node pairs.",
"For $\\alpha =0$ , all nodes are perceived at the same distance from the source node, thus contributing equally to its final score; for $\\alpha >0$ , the contribution of nodes is weighted w.r.t.",
"their distance, and progressively dumped while increasing such value.",
"The effect of increasing $\\alpha $ , as shown by the KDEs distributions in Fig.",
"REF , is to concentrate the actual contribution to low-distance neighborhoods, thus favoring a polarization of the scores to the extreme values of the domain.",
"Indeed, there is no one-fits-all value for such parameter: it needs to be fitted to the analytical needs and the underlying network topology."
],
[
"Real data",
"Copenhagen Network Study.",
"We firstly consider a small real-world network, namely the interaction data from Copenhagen Network Study [22].",
"It is composed of different layers connecting a sample of 700 among male and female university students for four weeks: we consider, here, the SMS layer and the proximity estimated via Bluetooth signal strength.",
"Since information about node gender is available, we mainly aim to relate a characterization of the network based on Conformity to some of the analysis already shown in the original data paper, e.g., more frequent male-male interaction than male-female and female-female ones [22].",
"Since the underlying network reflects these frequencies, we describe homophily by gender leveraging Conformity, trying to give more insights than the only number of exchanged messages.",
"Fig.",
"REF shows that several male nodes are perfectly homophilic w.r.t.",
"gender, but also that there exist a few highly heterophilic ones among them.",
"The same (i.e., the same mixing pattern) is not true observing female node Conformity distribution, even taking into account the fact that the two populations are unbalanced.",
"Considering the proximity layer, we show the graph analysis of two days, namely Monday and Saturday.",
"Fig.",
"REF (b-c) underlines how different mixing patterns arise considering different days of the week.",
"Facebook100.",
"Facebook100 [23] is a collection of 100 Facebook friendships networks among 100 U.S. colleges, built during the early history of the social network.",
"Nodes are labeled with several categorical attributes, profiling people by gender, college year, dormitory...In the following, we will focus on the first 50 networks ordered by size, considering two single-attributes analyses - namely gender and college year - and a multi-attribute overview.",
"Be aware that the gender attribute yields three values, referring to male, female and missing information; quoting the original data paper, we use a “missing\" label for situations in which individuals did not volunteer a particular characteristic [23], namely that the individual itself does not specify his gender.",
"Figure: Gender analysis (α=2.5\\alpha =2.5).The box-plot above compares male (blue diamonds) and female (red diamonds) distributions of the analyzed colleges, while the box-plot below compares male-and-female (purple diamonds) and missing values (green diamonds) distributions.",
"Three binned networks show heterogeneity of distributions along the colleges.Figure: A comparison between Conformity (α=2.5\\alpha =2.5) and Peel's assortativity in the three female colleges.Gender.",
"Fig.",
"REF shows gender assortativity of the 50 selected networks.",
"In general, we can not state a male/female tendency to homophily/heterophily as a common behavior across all networks; even if it seems that females' average behavior is more assortative than males, this should be examined on a case-by-case basis.",
"Nevertheless, for the work, it is more interesting to focus on the scoring of three specific networks, whose male and female homophilic behaviors are different w.r.t.",
"the other colleges.",
"They are Simmons, Smith and Wellesley, whose distributions are also highlighted in Fig.",
"REF , in view of a comparison with Peel's assortativity [5] This other measure has also a parameter $\\alpha \\in [0,1]$ , which is integrated over all its possible values in the paper where it is defined and presented [5]: we replicated the same approach for the comparison in the current study.. First of all, referring to the analysis present in the original data paper [23], they are three predominantly female colleges whose Newman's assortativity coefficient tends to 0.",
"Leveraging Conformity, we can observe (Fig.",
"REF ) how i) the few male nodes connect disassortatively by gender (i.e., form ties only with females), inducing the emergence of two extreme and distinct mixing, and meanwhile ii) we observe some differences with Peel's assortativity, where the same overall strong assortative behavior of the networks is not maintained (Fig.",
"REF ).",
"Apparently, the extreme disassortative behavior of few nodes should not so strongly affect the entire network mixing.",
"Since a real comparison between the two measures is not possible i) due to the absence of ground truth, but mostly because ii) they capture different aspects of mixing, our interpretation is that the local assortativity variant we face suffers from the same limits about network constraints impacting on the reaching of the whole measure range, as already studied in [24].",
"Also, the presence of missing values has a non trivial effect on the resulting Conformity distribution.",
"The ability to discriminate noisy information from sensible one is important while analyzing a complex system.",
"Since nodes with missing information are homogeneously distributed within the network tissue, Conformity can correctly classify them as noise, as shown in the labeled vs. missing box-plot of Fig.",
"REF .",
"This observation simply implies that these nodes can not induce to homophilic behaviors since missing information is not a real social dimension implying assortative attitudes.",
"Figure: Year analysis (α=2.5\\alpha =2.5).The box-plot compares the first year (red diamonds) and other years (green diamonds) distributions of the selected colleges.",
"Three networks are selected, where also a distinction between first and second year students is highlighted.Year.",
"Fig.",
"REF shows year assortativity of the 50 selected networks.",
"As already shown in [5], first year students highly contribute to the homophilic behavior of the attribute, even when the network attitude does not tend to be globally assortative (see Haverford in Fig.",
"REF ).",
"According to the original data paper [23], the year attribute is the most assortative in terms of Newman's coefficient.",
"Also, in this case, a node-centric measure tends to discover different mixing pattern and allows to differentiate the values that show high homogeneity from the ones that prefer a heterogeneous neighborhood; the three binned networks in Fig.",
"REF also suggest that homophilic behavior tends to decrease with the increase of enrolment years in a coherent way with the ordinal nature of the attribute.",
"Multi-attribute.",
"In a multi-attribute scenario, we want to measure homophily among complex node profile composed by multiple independent fields.",
"Fig.",
"REF focuses on dorm-gender and dorm-year assortativity of two selected networks.",
"Smith college, as Wellesley, shows a consistent difference between male-female distributions when the only gender attribute is considered, while no substantial differences are highlighted when the only dormitory attribute is analyzed.",
"However, male nodes tend to be more assortative than female ones when the two attributes are measured together, allowing us to provide a more reliable description of the social media friendships mirroring college interactions.",
"Like all other colleges, first year students are highly assortative w.r.t.",
"the other years, while the same pattern does not emerge considering the dormitory attribute.",
"However, such a pattern emerges anew when dormitory and years are analyzed in a multi-attribute scenario.",
"Figure: Multi-attribute (α=2.5\\alpha =2.5).",
"Dorm-gender and dorm-year analysis of Smith and Bucknell colleges: respectively, male-female and first year vs. other years differences are highlighted in the distributions."
],
[
"Discussion and future work",
"This work introduced Conformity, a novel strategy to measure the homophilic mixing of network nodes w.r.t.",
"their categorical attributes.",
"The proposed measure aims to address some limitations of the well-known assortativity coefficient, in its classic definition given by Newman's work [4].",
"The main reason behind Conformity is the need to take into account (the often neglected) impact of node distance on the homophilic/heterophilic behaviors that, in social contexts, favor the creation of social ties.",
"As shown, the proposed measure can unveil interesting nodes' behaviors and can, in practice, be fruitfully adapted to support several tasks (e.g., the identification/measuring of echo-chambers or polarized islands among users living in a social media ecosystem).",
"In particular, the multi-attribute analysis it enables can support fine grained analysis of complex homophilic patterns to uncover, for instance, homogeneous nuclei among individuals w.r.t.",
"their age and political views, thus supporting tasks such as attributed community discovery [25].",
"Moreover, Conformity ability to characterize different extreme behavior of even handfuls of nodes (as seen both in homophily by gender analysis of colleges as Smith and Wellesley and in noise isolation when in the presence of missing values) is a promising feature that can support a wide set of network related task as, for instance, graph-based anomaly detection.",
"As future works, since in the current study we focused only on networks encoding categorical attributes, we plan to extend Conformity to handle scalar attributes.",
"We also plan to propose an approximate version of Conformity to lower its computational complexity and to study its effectiveness as support for network analysis tasks in heterogeneous applicative scenarios."
],
[
"Acknowledgments",
"This work is supported by the scheme 'INFRAIA-01-2018-2019: Research and Innovation action', Grant Agreement n. 871042 'SoBigData++: European Integrated Infrastructure for Social Mining and Big Data Analytics'.",
"Giulio Rossetti is a permanent researcher at the Information Science and Technology Institute of the Italian National Research Council (ISTI-CNR).",
"Giulio holds a PhD in Computer Science from the University of Pisa (2015).",
"Since 2011, he has been performing research in the fields of complex network analysis and data science as a member of the Knowledge Discovery and Data Mining laboratory.",
"His recent work focuses on the modeling and study of dynamics of and on networks and on the definition of data-driven models for the forecast of rare events.",
"He is the main contributor of several open-source software applications (https://github.com/GiulioRossetti), some of which have been developed to support big data analysis carried out in a series of EU projects.",
"Salvatore Citraro is a PhD student in Computer Science at the University of Pisa and a member of the Knowledge Discovery and Data Mining Laboratory (KDD-Lab), a joint research group with the Information Science and Technology Institute of the Italian National Research Council in Pisa.",
"Currently, his main research interests focus on complex networks, including classic and feature-rich network data analysis, with a focus on attribute-aware community discovery tasks.",
"His PhD line of research will also include the study of linguistic networks and how their hidden structures and properties can be meaningfully exploited and applied in cognitive contexts.",
"Letizia Milli received in 2018 the PhD in Computer Science from the University of Pisa with a thesis on the study of spreading phenomena over complex networks.",
"She is a post doc researcher in Computer Science at the University of Pisa and a member of the Knowledge Discovery and Data Mining Laboratory (KDD-Lab), a joint research group with the Information Science and Technology Institute of the Italian National Research Council in Pisa.",
"Her research interests include data mining, quantification, diffusion of phenomena, and innovation in complex networks and the Science of Success."
]
] | 2012.05195 |
[
[
"Local and global interpolations along the adiabatic connection of DFT: A\n study at different correlation regimes"
],
[
"Abstract Interpolating the exchange-correlation energy along the density-fixed adiabatic connection of density functional theory is a promising way to build approximations that are not biased towards the weakly correlated regime.",
"These interpolations can be done at the global (integrated over all spaces) or at the local level, using energy densities.",
"Many features of the relevant energy densities as well as several different ways to construct these interpolations, including comparisons between global and local variants, are investigated here for the analytically solvable Hooke's atom series, which allows for an exploration of different correlation regimes.",
"We also analyze different ways to define the correlation kinetic energy density, focusing on the peak in the kinetic correlation potential."
],
[
"Introduction",
"The density-fixed adiabatic connection [1] of Kohn-Sham (KS) density functional theory (DFT) is a powerful theoretical tool for the construction of approximate exchange-correlation (XC) functionals: for example, hybrid [2] and double-hybrid functionals [3] can be constructed from simple models of the adiabatic connection integrand [4], [5], [6].",
"These approximations, however, use exact ingredients only for the limit of small coupling strength, and are thus biased towards the weakly-correlated regime.",
"A class of approximations that removes this bias is based on the idea of Seidl and coworkers [7], [8], [9] to interpolate the adiabatic connection integrand between its weak and strong interaction limits.",
"This way, information from both extreme correlation regimes is taken into account on a similar footing.",
"These interpolations can be done on the global [7], [8], [9], [10], [11], [12] (i.e., integrated over all space) ingredients, or in each point of space, using energy densities [13], [14], [15].",
"As well known, energy densities are not uniquely defined and one should be sure, when doing an interpolation between weak and strong coupling in each point of space, that all the input local quantities are defined in the same way [13], [14], [15], [16], which makes the use of semilocal approximations very difficult, a problem shared with local hybrids [17], [18], [19], [20].",
"Non-local functionals for the strong-interaction limit [21], [22] or the physical regime [23] are needed in this context, as full compatibility with the exact exchange energy density is required.",
"Interpolations constructed from the global ingredients are in general computationally cheaper than their local counterpart, not only because they can use semilocal approximations for the strong-interaction functionals, but also because they do not need energy densities from exact exchange and from second-order perturbation theory, but only their global values.",
"These global interpolations are in principle not size consistent, but it has been recently shown that their size-consistency error can be fully corrected at no additional computational cost [12], allowing for the calculation of meaningful interaction energies [12].",
"On the other hand, in all the tests performed so far on small chemical systems [14], [15], the local interpolations have always been found to be more accurate than the corresponding global ones for systems with more than two electrons.",
"In the Helium isoelectronic series, the global and local interpolation perform similarly [14].",
"The purpose of the present work is to further compare and analyze local and global interpolations when the physical system is in different correlation regimes.",
"In order to disentangle the errors coming from the interpolation itself from those on the input ingredients, we use a model system, two Coulombically interacting electrons in the harmonic potential (“Hooke's atoms”) [24], [25], [26], which allows us to explore the whole range from weak to strong correlation always using exact input ingredients.",
"We also analyze the kinetic correlation energy density, and particularly how its peak in the origin, which in systems with Coulomb confinement plays an important role for strong correlation [27], [28], [29], varies as the system becomes more and more correlated.",
"By defining the $\\lambda $ -dependent density functional $F_{\\lambda }[\\rho ]$ in the Levy constrained-search formalism [30], $ F_{\\lambda }[\\rho ]\\equiv \\min _{\\Psi \\rightarrow \\rho }\\langle \\Psi |\\hat{T}+\\lambda \\hat{W}|\\Psi \\rangle ,$ with $\\hat{T}$ the electronic kinetic energy operator, $\\hat{W}$ the Coulomb electron-electron interaction operator, and “$\\Psi \\rightarrow \\rho $ ” indicating all fermionic wavefunctions yielding the one-electron density $\\rho (\\mathbf {r})$ , one obtains an exact formula [1] for the XC energy functional of KS DFT, $E_{xc}[\\rho ]=\\int _0^1 W_{\\lambda }[\\rho ]\\,d\\lambda .$ In Eq.",
"(REF ) $W_{\\lambda }[\\rho ]$ is the global adiabatic connection integrand, $W_{\\lambda }[\\rho ]\\equiv \\langle \\Psi _{\\lambda }[\\rho ]|\\hat{W}|\\Psi _{\\lambda }[\\rho ]\\rangle -U[\\rho ],$ where $\\Psi _{\\lambda }[\\rho ]$ is the minimizing wavefunction in Eq.",
"(REF ) and $U[\\rho ]$ is the Hartree repulsion energy.",
"The real parameter $\\lambda $ is a knob that controls the interaction strength, defining an infinite set of systems all with the same one-electron density $\\rho (\\mathbf {r})=\\rho _{\\lambda =1}(\\mathbf {r})$ , but with different correlation.",
"The global adiabatic connection integrand has the known expansions at small and large $\\lambda $ , $W_{\\lambda \\rightarrow 0}[\\rho ] & = & W_0[\\rho ]+\\lambda \\,W_0^{\\prime }[\\rho ]+...,\\\\W_{\\lambda \\rightarrow \\infty }[\\rho ] & = & W_\\infty [\\rho ]+\\frac{W^{\\prime }_\\infty [\\rho ]}{\\sqrt{\\lambda }}+... ,$ where $W_0[\\rho ]=E_x[\\rho ]$ is the exact exchange energy (the same expression as the Hartree-Fock exchange, but with KS orbitals), $W_0^{\\prime }[\\rho ]=2 E_c^{\\rm GL2}[\\rho ]$ is twice the Görling-Levy [31] second-order correlation energy (GL2), $W_\\infty [\\rho ]$ is the indirect part of the minimum possible expectation value of the electron-electron repulsion in a given density [32], and $W^{\\prime }_\\infty [\\rho ]$ is the potential energy of coupled zero-point oscillations around the manifold that determines $W_\\infty [\\rho ]$ [33]."
],
[
"Energy densities",
"Equation (REF ) can also be written in terms of real-space energy densities $w_\\lambda (\\mathbf {r};[\\rho ])$ , $E_{xc}[\\rho ] = \\int d \\mathbf {r} \\, \\rho (\\mathbf {r})\\int _0^1 \\text{d} \\lambda \\, w_\\lambda (\\mathbf {r}; [\\rho ]),$ which are, of course, not uniquely defined.",
"For the purpose of building $\\lambda $ -interpolation models on energy densities, the choice of the gauge of the electrostatic potential of the exchange-correlation hole $h^\\lambda _{xc}(\\mathbf {r}_1, \\mathbf {r}_2)$ seems so far to be the most suitable [16], $w_\\lambda (\\mathbf {r}) = \\frac{1}{2} \\int \\frac{h^\\lambda _{xc}(\\mathbf {r}, \\mathbf {r}_2)}{|\\mathbf {r}-\\mathbf {r}_2|} d \\mathbf {r}_2,$ where $h^\\lambda _{xc}(\\mathbf {r}_1, \\mathbf {r}_2)$ is defined in terms of the pair-density $P_2^\\lambda (\\mathbf {r}_1, \\mathbf {r}_2)$ and the density $\\rho $ (see also [34]), $h^\\lambda _{xc}(\\mathbf {r}_1, \\mathbf {r}_2) = \\frac{P_2^\\lambda (\\mathbf {r}_1, \\mathbf {r}_2)}{\\rho (\\mathbf {r}_1)} - \\rho (\\mathbf {r}_2),$ with $P_2^\\lambda $ obtained from $\\Psi _\\lambda [\\rho ]$ , $P_2^\\lambda (\\mathbf {r}, \\mathbf {r}^{\\prime }) = N(N-1) \\sum _{\\sigma , \\sigma ^{\\prime }, \\sigma _3 \\dots \\sigma _N} \\int | \\Psi _\\lambda (\\mathbf {r}\\sigma , \\mathbf {r}^{\\prime }\\sigma ^{\\prime }, \\mathbf {r}_3\\sigma _3 \\dots r_N \\sigma _N)|^2 d \\mathbf {r}_3 \\dots d \\mathbf {r}_N.$"
],
[
"Energy density at $\\lambda = 0$ .",
"At $\\lambda = 0$ we have the Kohn-Sham or exchange hole, which yields, in the case of a closed-shell singlet considered in this work (with real orbitals) $ w_0(\\mathbf {r}) = - \\frac{1}{2 \\rho (\\mathbf {r})} \\sum _{i, j}^{N/2} \\phi _i(\\mathbf {r}) \\phi _j(\\mathbf {r}) \\int \\mathrm {d} \\mathbf {r}^{\\prime } \\frac{\\phi _j(\\mathbf {r}^{\\prime }) \\phi _i(\\mathbf {r}^{\\prime })}{|\\mathbf {r}-\\mathbf {r}^{\\prime }|},$ where $\\phi _i(\\mathbf {r})$ are the occupied KS spatial orbitals."
],
[
"Slope of the energy density at $\\lambda = 0$ .",
"The slope $w^{\\prime }_0(\\mathbf {r})$ of the energy density at $\\lambda = 0$ in the gauge of Eq.",
"(REF ) is given, again for a closed shell singlet with real orbitals, by [14] $ w^{\\prime }_0(\\mathbf {r}) = - \\frac{1}{\\rho (\\mathbf {r})} \\sum _{abij} \\frac{4 \\langle ij | ab \\rangle -2 \\langle ij | ba \\rangle }{\\epsilon _a + \\epsilon _b - \\epsilon _i - \\epsilon _j} \\phi _i(\\mathbf {r}) \\phi _a(\\mathbf {r}) \\int d \\mathbf {r}^{\\prime } \\frac{\\phi _j(\\mathbf {r}^{\\prime }) \\phi _b(\\mathbf {r}^{\\prime }) }{|\\mathbf {r} - \\mathbf {r}^{\\prime }|},$ where $\\phi _a$ and $\\phi _b$ are unoccupied and $\\phi _i$ and $\\phi _j$ are occupied Kohn-Sham orbitals, $\\langle ij | ab \\rangle $ denotes the Coulomb integral over the spatial orbitals, and the $\\epsilon _i$ are the Kohn-Sham orbital energies.",
"For systems with $N>2$ , there should be also a term with single excitations [31], which we do not consider here as we focus on $N=2$ ."
],
[
"Energy density at $\\lambda = \\infty $ .",
"In the $\\lambda \\rightarrow \\infty $ limit we obtain a system of strictly correlated electrons (SCE), for which it has been shown [13] that $ w_\\infty (\\mathbf {r}) = \\frac{1}{2} \\sum _{i=2}^N \\frac{1}{|\\mathbf {r}-\\mathbf {f}_i(\\mathbf {r})|} - \\frac{1}{2} v_H(\\mathbf {r}),$ where $ v_H(\\mathbf {r})$ is the Hartree potential and $\\mathbf {f}_i(\\mathbf {r})$ are co-motion functions that determine the position of the $i^{\\rm th}$ electron given the position $\\mathbf {r}$ of a chosen reference electron (as the $\\mathbf {f}_i(\\mathbf {r})$ satisfy cyclic group properties it does not matter which electron is chosen as reference), and are non-local functionals of the density $\\rho (\\mathbf {r})$ [32], [35].",
"There is at present no local expression in the gauge of Eq.",
"(REF ) for the next leading term $W_\\infty ^{\\prime }[\\rho ]$ in the $\\lambda \\rightarrow \\infty $ asymptotic expansion.",
"In fact, the functional $W_\\infty ^{\\prime }[\\rho ]$ can be computed from an integral on position-dependent zero-point energies [33], which, however, do not provide an energy density within the definition of Eq.",
"(REF ).",
"The original idea of Seidl and coworkers [7], [8], [9] was to build an approximate adiabatic connection integrand $W_\\lambda ^{\\rm ISI}[\\rho ]$ by interpolating between the two limits of Eqs.",
"(REF ) and ().",
"These interaction-strength interpolation (ISI) functionals typically use as input the four ingredients (or a subset thereof) appearing in Eqs.",
"(REF ) and (): $\\lbrace W_0[\\rho ],W_0^{\\prime }[\\rho ],W_\\infty [\\rho ], W_\\infty ^{\\prime }[\\rho ]\\rbrace $ , denoted ${\\bf W}$ in short.",
"The XC energy functional $E_{xc}^{\\rm ISI}[\\rho ]$ is then obtained from Eq.",
"(REF ), by integrating $W_\\lambda ^{\\rm ISI}[\\rho ]$ over $\\lambda $ , which will result in a non-linear function of the input ingredients ${\\bf W}$ .",
"Because of this non linear dependence, the ISI-type functionals are not size consistent when a system dissociates into unequal fragments, even when the input ingredients are size-consistent themselves.",
"However, in this latter case, size-consistency can be easily restored with a very simple correction [12].",
"The ISI-type functionals are, instead, automatically size extensive [12].",
"Several formulas for interpolating between the two limits of Eqs.",
"(REF ) and () have been proposed in the literature, and are reported in Appendix .",
"More recently, these same interpolation formulas have been used to build, in each point of space, a model energy density $w_\\lambda ^{\\rm ISI}(\\mathbf {r};[\\rho ])$ , with Eqs.",
"(REF )-(REF ) as input ingredients [14], [15].",
"This way, by integrating $w_\\lambda ^{\\rm ISI}(\\mathbf {r};[\\rho ])$ over $\\lambda $ between 0 and 1, one obtains an exchange-correlation energy density in the gauge of the coupling-constant averaged exchange-correlation hole.",
"Such interpolations done in each point of space are size consistent in the usual DFT sense [36], [37]."
],
[
"Hooke's atom series",
"The Hooke's atom series consists of two electrons bound by an harmonic external potential, with hamiltonian $\\hat{H}= - \\frac{1}{2}\\left(\\nabla _1^2 + \\nabla _2^2\\right) + \\frac{\\omega ^2}{2} \\left(r_1^2 + r_2^2\\right) + \\frac{1}{r_{12}},$ with $r_i=|\\mathbf {r}_i|$ and $r_{12}=|\\mathbf {r}_1-\\mathbf {r}_2|$ .",
"At large $\\omega $ the system has high-density and is in the weakly correlated regime, which can be fully described by using the scaled coordinates $\\mathbf {s}_i\\equiv \\sqrt{\\omega }\\,\\mathbf {r}_i$ , while as $\\omega \\rightarrow 0$ the system becomes more and more correlated [25], and the relevant scaled variables are $\\tilde{\\mathbf {s}}_i\\equiv \\omega ^{2/3}\\,\\mathbf {r}_i$ .",
"As well known, there is an infinite set of special values of $\\omega $ for which the hamiltonian (REF ) is analytically solvable [24] once rewritten in terms of center of mass and relative coordinates.",
"These analytic solutions have the center of mass in the ground-state of an harmonic oscillator with mass $m=2$ and frequency $\\sqrt{2}\\,\\omega $ , and the relative coordinate in an $s$ -wave with the radial part described by a gaussian times a polynomial [24].",
"We denote here the various analytic solutions with the degree $n-1$ of the polynomial in $r_{12}$ .",
"At $n=1$ we have the non-interacting system, and as $n$ increases the system becomes more and more correlated, with $\\omega $ smaller and smaller [24].",
"The values of $\\omega $ corresponding to the different values of $n$ considered here are reported in Table REF .",
"Table: Values of ω\\omega for the various analytic solutions of the hamiltonian of Eq.",
"() considered here, corresponding to different degrees n-1n-1 of the polynomial in the solution for the relative coordinate r 12 r_{12} ."
],
[
"Computation of exact energy densities",
"Given the analytic solutions [24] $\\Psi (r_1,r_2,r_{12})$ of the hamiltonian (REF ) for $n=2,\\dots ,6$ , we have computed the corresponding densities $\\rho (r)$ , which are also analytic.",
"Although leading to cumbersome expressions, these densities allowed us to obtain analytic Kohn-Sham potentials $v_s(r)=\\frac{\\nabla ^2\\sqrt{\\rho (r)}}{2\\sqrt{\\rho (r)}}+\\epsilon $ , with $\\epsilon =E_2-E_1$ , the energy difference between the physical state with two and one electrons."
],
[
"Energy densities at $\\lambda =0$",
"For a singlet $N=2$ state Eq.",
"(REF ) reduces to $w_0(r)=-\\frac{1}{4}v_H(r)$ , with $v_H(r)$ the Hartree potential, leading to the simple expression $w_0(r)=-\\pi \\int _r^\\infty r^{\\prime }\\rho (r^{\\prime })\\,dr^{\\prime }-\\frac{N_e(r)}{4\\, r},$ with the cumulant $N_e(r)$ defined as $ N_e(r)=4 \\pi \\int _0^r r^{\\prime 2}\\rho (r^{\\prime })\\,dr^{\\prime }.$ We have obtained these energy densities analytically from the exact densities.",
"They are shown in Fig.",
"REF for the different analytic solutions considered here.",
"Figure: Energy densities at λ=0\\lambda =0 for the Hooke's atoms series with n=2,⋯,6n=2,\\dots ,6, corresponding to the ω\\omega values of Table .",
"In the second panel the energy density has been multiplied by the density and by the volume element.",
"The high-density scaling has been used."
],
[
"Energy densities for the slope at $\\lambda =0$",
"The analytic exact Kohn-Sham potentials were used to obtain the virtual Kohn-Sham orbitals needed for the evaluation of Eq.",
"(REF ).",
"We used an isotropic spherical Gaussian basis with $\\omega $ as the width parameter.",
"Angular momentum values were included from $l=0$ to $l=9$ , with 5 to 30 basis states for every value of $l$ .",
"All matrix elements were obtained analytically in this basis, including the Coulomb integrals.",
"We first analyze the convergence of the global slope of the coupling constant integrand, $W_0^{\\prime }=2\\,E_c^{\\rm GL2}$ , with increasing basis set size $n_{\\rm basis}$ in the first panel of Fig.",
"REF .",
"The number of basis states is that per angular momentum quantum number, with all $l$ up to $l=9$ included.",
"As $\\omega $ decreases (the quantum number $n$ increases), the $l=0$ contribution becomes less important, with the $l>0$ contributions gaining more weight, as shown in the second panel of Fig.",
"REF , where the result from each channel $l$ with $n_{\\rm basis}=30$ is reported.",
"Figure: Convergence of W 0 ' =2E c GL 2 W_0^{\\prime }=2\\,E_c^{\\rm GL2} with the size n basis n_{\\rm basis} of the gaussian basis set used to expand the KS orbitals, relative to n basis =30n_{\\text{basis}}=30, (first panel) and contribution of the different angular momentum ll (second panel)For the local slope $w_0^{\\prime }(\\mathbf {r})$ only 10 basis states are used.",
"In the present case of a two-electron system, $w_0^{\\prime }(\\mathbf {r})$ can also be simplified, as there is only one occupied Kohn-Sham spatial orbital.",
"Additional utilization of the spherical symmetry then yields the following expression, by using the spherical harmonic expansion of the Coulomb potential, $\\begin{split}w^{\\prime }_0(r) = - \\frac{2}{\\rho (r)} \\sum _{n_an_b l} \\frac{1}{\\epsilon _a + \\epsilon _b - 2\\epsilon _{occ}} \\langle (occ)(occ) | ab \\rangle \\\\ R_{occ}^{0}(r) R_{n_a}^{l}(r) (r^{-l-1}\\int _0^r dr^{\\prime } r^{\\prime l +2} R_{occ}^{0}(r^{\\prime }) R_{n_b}^{l}(r^{\\prime }) \\\\+ r^l \\int _r^\\infty dr^{\\prime } r^{\\prime -l+1} R_{n_j}^{0}(r^{\\prime }) R_{n_b}^{l}(r^{\\prime })),\\end{split}$ where the functions $R_n^l(r)$ are the radial functions of the spatial orbitals and $occ$ is the occupied Kohn-Sham orbital.",
"The full local slope is shown in the first panel of Fig.",
"REF .",
"Numerical issues appear at around the scaled variable values $s \\gtrsim 4.5$ , but this is of no relevance to the integrated energy as it is clear upon multiplication by the volume element and the density (second panel of Fig.",
"REF ).",
"Figure: The local slope (first panel) and the local slope multiplied by the volume element and density (second panel)."
],
[
"Energy densities at $\\lambda =\\infty $",
"The energy density $w_\\infty (\\mathbf {r})$ of Eq.",
"(REF ) in the case of $N=2$ electrons in a spherical density is known to be determined by the radial co-motion function $f(r)$ , which gives the full $\\mathbf {f}(\\mathbf {r})$ via $\\mathbf {f}(\\mathbf {r})=-\\frac{f(r)}{r}\\,\\mathbf {r}$ [38], [32], [39], [13], yielding $w_\\infty (r) = \\frac{1}{2(r+f(r))} - \\frac{1}{2} v_H(r).$ In turn, $f(r)$ is a fully non-local functional of the density $\\rho (r)$ , given in terms of the cumulant $N_e(r)$ of Eq.",
"(REF ) and its inverse $N_e^{-1}$ , $f(r) = N_e^{-1}(2-N_e(r)).$ In Fig.",
"REF , we report the energy densities $w_\\infty (r)$ for the analytical solutions corresponding to the $\\omega $ values of Table REF .",
"Figure: Energy densities corresponding to λ=∞\\lambda =\\infty (first panel), and energy densities corresponding to λ=∞\\lambda =\\infty multiplied by the density and the volume element (second panel).",
"The coordinates and energy densities are scaled according to the large ω\\omega limit."
],
[
"Energy densities at $\\lambda =1$",
"Since we have exact analytic wavefunctions we can also compute the exact energy densities at physical coupling strength $\\lambda =1$ , which can be used to test the accuracy of local interpolations between $\\lambda =0$ and $\\lambda =\\infty $ , as well to study features of the energy densities as the interaction strength is changed.",
"The exact $w_1(r)$ are reported in Fig.",
"REF .",
"We see that the physical energy densities $w_1(r)$ for the Hooke's atom series differ more among each other at large $r$ , unlike $w_0(r)$ and $w_\\infty (r)$ .",
"This is clearer if we look at the correlation energy density $w_c(r) = w_1(r)-w_0(r)$ , which is reported in Fig.",
"REF .",
"The correlation energy density $w_c(r)$ decays $\\propto -\\frac{1}{r^3}$ , but with different coefficients for different values of $\\omega $ .",
"Figure: Energy densities corresponding to λ=1\\lambda =1 (first panel), and energy densities corresponding to λ=1\\lambda =1 multiplied by the density and the volume element (second panel).",
"The coordinates and energy densities are scaled according to the large ω\\omega limit.Figure: Correlation energy densities (first panel) and correlation energy densities multiplied by the density and the volume element (second panel).",
"The coordinate and energy density are scaled according to the large ω\\omega limitA comparison of the three energy densities $w_0$ , $w_1$ and $w_\\infty $ is given in Fig.",
"REF for the Hooke's atom with $n=6$ .",
"An interesting feature of these energy densities, already observed in Ref.",
"[13], is that for large $r$ it can be seen that $w_1(r) < w_\\infty (r)$ , while for the corresponding global quantities we have the strict inequality $W_1[\\rho ] > W_\\infty [\\rho ]$ .",
"However taking $w_1(r) \\approx w_\\infty (r)$ for large $r$ only has a small effect on the energy even for the most strongly correlated Hooke's atom considered here ($n=6$ ), as it becomes clear once the energy densities are multiplied by the density and the volume element (second panel of Fig REF ), which is what ultimately determines the correlation energy.",
"This crossing of energy densities has never been observed, so far, in systems with the Coulomb external potential.",
"Figure: Energy densities for the most strongly correlated Hooke's atom considered here (n=6n=6), at different values of λ\\lambda (first panel).",
"In the second panel the energy densities have been multiplied by the density and the volume element."
],
[
"Interpolations using global ingredients",
"The global ingredients $W_0[\\rho ]$ , $W_0^{\\prime }[\\rho ]$ have been obtained as described in Sections REF and REF , while $W_\\infty [\\rho ]$ has been obtained by integrating the energy density of Eq.",
"(REF ).",
"Additionally, we have also obtained $W_\\infty ^{\\prime }[\\rho ]$ of Eq.",
"(), which in this case is given by [33] $W_\\infty ^{\\prime }[\\rho ]=\\frac{1}{2}\\int _0^\\infty 4\\pi \\, r^2 \\frac{\\rho (r)}{2} \\left(\\omega _1(r)^2+\\frac{\\omega _2(r)^2}{2}\\right)\\,dr,$ with $\\omega _1(r)^2 & = & \\frac{r^2+f(r)^2}{r f(r)(r+f(r))^3} \\\\\\omega _2(r)^2 & = & -\\frac{2(1+f^{\\prime }(r)^2)}{f^{\\prime }(r)(r+f(r))^3},$ and with $f(r)$ given by Eq.",
"(REF ).",
"Notice that $f^{\\prime }(r)<0$ , so that $\\omega _2(r)^2>0$ .",
"We have used the interpolation formulas reported in Appendix , namely SPL [7], LB [40], ISI [9] and revISI [33].",
"The first two, SPL and LB, use only three ingredients (they do not include $W_\\infty ^{\\prime }[\\rho ]$ ), while ISI and revISI use all the four ingredients of Eqs.",
"(REF )-().",
"Additionally, we have also used a Padé approximant (see Appendix ) which uses $W_0[\\rho ], W_0^{\\prime }[\\rho ]$ and the exact $W_1[\\rho ]$ , to generate plausible reference adiabatic connection curves, which are shown in Fig.",
"REF .",
"As expected, as the Hooke's atoms get more correlated, the AC integrand displays a stronger curvature.",
"Figure: The scaled adiabatic connection integrand as a function of λ\\lambda obtained from a Padé interpolation that includes the exact W 1 [ρ]W_1[\\rho ] (see Appendix ).The error resulting in the correlation energy $E_c[\\rho ]$ with the different global interpolations is shown in Fig.",
"REF .",
"We consider only the correlation energy, since all the methods utilize 100% exact exchange.",
"The Padé method performs best as expected, since it uses the exact $W_1$ , which in practical situations is unavailable.",
"The LB interpolation formula performs second best, while SPL, containing the same ingredients, performs much worse.",
"The ISI and revISI methods improve slightly the SPL formula, but are still outperformed by LB, despite containing more exact information in the form of $W_\\infty ^{\\prime }[\\rho ]$ .",
"Figure: Errors in the correlation energy resulting from the application of several global interpolations (see Appendix ).For comparison with traditional Density Functional Approximations (DFAs), such as the local density approximation (LDA) [41] and the PBE GGA [42], we show the error in the exchange-correlation energy $E_{xc}[\\rho ]$ in the first panel of Fig.",
"REF .",
"It is clear that the adiabatic connection interpolation methods outperform the PBE method, however at the increased computational cost of a double hybrid.",
"In the second panel of Fig.",
"REF we compare the performance of LDA (PW92 [41]) with GL2 alone and with the $\\lambda \\rightarrow \\infty $ expansion of Eq.",
"() alone, which yields $E_{xc}[\\rho ]=W_\\infty [\\rho ]$ if we retain only the first term, and $E_{xc}[\\rho ]=W_\\infty [\\rho ]+2 W_\\infty ^{\\prime }[\\rho ]$ , if we include also the second term.",
"The LDA performs poorly already for the first Hooke's atom and its performance worsens as correlation increases.",
"The GL2 method works well for the first Hooke's atom, which is expected since its adiabatic connection integrand resembles a straight line in Fig.",
"REF , but it is way too negative for the exchange-correlation energy in the more correlated Hooke's atoms.",
"The $\\lambda \\rightarrow \\infty $ expansion alone performs better as the Hooke's atoms become more correlated, but with the first term only is still too negative by about 15% in the strongest correlated Hooke's atom.",
"Adding the second term contribution reduces the error for $n>3$ , and the resulting XC energy becomes now less negative than the exact one.",
"Figure: Errors in the exchange-correlation energy resulting from the application of several global interpolations and approximations (see text)."
],
[
"Interpolations on energy densities",
"As already mentioned at the end of Sec.",
", an expression for the energy density corresponding to $W^{\\prime }_\\infty [\\rho ]$ in the gauge of Eq.",
"(REF ) is not available.",
"For this reason, we can only test local interpolations using the LB and SPL interpolation formulas, which do not use the information from $W^{\\prime }_\\infty [\\rho ]$ .",
"We first compare the resulting $w_c(r)=w_1(r)-w_0(r)$ from the two interpolation formulas in the first panels of Figs.",
"REF (LB) and REF (SPL) with the exact result obtained from the analytic wavefunctions.",
"The errors are small on an absolute scale, so we show in both figures $\\delta w_c(\\mathbf {r}) = w_{c, exact}(\\mathbf {r})-w_{c, model}(\\mathbf {r})$ and include the volume element and density.",
"Notice that $\\delta w_c(\\mathbf {r}) = \\delta w_1(\\mathbf {r})$ since we use the exact $w_0(\\mathbf {r})$ in the construction of both the LB and SPL approximations.",
"In order to assess the coupling constant integrated energy density $\\bar{w}_c$ , which is not known exactly for any of the Hooke's atoms, we compare it with the one obtained from the Padé interpolation, which includes the exact $w_0(r)$ , $w_0^{\\prime }(r)$ and $w_1(r)$ .",
"We see that in the case of LB there is an over-estimation of the coupling-constant averaged energy density at small $r$ , which cancels quite well with an underestimation at large $r$ , achieving almost perfect error cancellation.",
"In the case of SPL, there is a smaller overestimation of the correlation at small $r$ , coupled with a stronger underestimation of the correlation energy at large $r$ , which worsens its performance.",
"Figure: Error δw c (𝐫)=w c,exact (𝐫)-w c,model (𝐫)\\delta w_c(\\mathbf {r}) = w_{c, exact}(\\mathbf {r})-w_{c, model}(\\mathbf {r}) multiplied by the volume element and density obtained with the LB approximation (first panel) and error in w ¯ c (r)\\bar{w}_c(r) obtained with the same LB approximation (second panel).",
"The high density scaling is applied.",
"For the LB interpolation formula, see Appendix Figure: Error δw c (𝐫)=w c,exact (𝐫)-w c,model (𝐫)\\delta w_c(\\mathbf {r}) = w_{c, exact}(\\mathbf {r})-w_{c, model}(\\mathbf {r}) multiplied by the volume element and density obtained with the SPL approximation (first panel) and error in w ¯ c (r)\\bar{w}_c(r) obtained with the same SPL approximation (second panel).",
"The high density scaling is applied.",
"For the SPL interpolation formula, see Appendix"
],
[
"Comparison between global and local interpolations",
"Of interest is then comparing the performance of the global and local variants of the Padé, LB and SPL interpolations.",
"In Fig.",
"REF the relative error on the correlation energy obtained from the local and global interpolation is shown, where in this case we use for both 10 basis states per angular momentum quantum number for the slope.",
"In the case of the Padé interpolation the performance worsens only slightly going from the global to the local interpolation, while for the SPL interpolation there is a dramatic worsening.",
"In the case of the LB interpolation the error switches sign for $n \\ge 3$ and in general worsens.",
"This is somehow surprising as, instead, for small chemical systems the local interpolations have been found to outperform their global counterparts [15], [14].",
"Figure: Comparison of the local and global adiabatic interpolations in terms of the relative error in the correlation energy E c E_c."
],
[
"Kinetic correlation energy densities",
"The coupling-constant integration is one possible way to recover the correlation part due to the difference between the true, interacting, kinetic energy $T[\\rho ]$ and the Kohn-Sham kinetic energy $T_s[\\rho ]$ , $T_c[\\rho ]=T[\\rho ]-T_s[\\rho ]$ .",
"We have $T_c[\\rho ]=\\int \\rho (\\mathbf {r}) (\\overline{w}(\\mathbf {r})-w_1(\\mathbf {r}))\\,d\\mathbf {r},$ where $\\overline{w}(\\mathbf {r})$ is obtained by integrating $w_\\lambda (\\mathbf {r})$ over $\\lambda $ between 0 and 1.",
"Equation (REF ) defines a possible kinetic correlation energy density equal to $\\overline{w}(\\mathbf {r})-w_1(\\mathbf {r}) $ .",
"Another correlation kinetic energy density that has been defined [27] and studied [43], [44], [45] in the literature, and that has been found to display very interesting features for strongly correlated systems [46], [28], [29], [47], arises from the work of Baerends and coworkers [27], [43], [44], [45], $v_{\\rm c,kin}(\\mathbf {r})=\\frac{1}{2} \\int \\left( |\\nabla _\\mathbf {r}\\Phi (2,...,N|\\mathbf {r})|^2 - |\\nabla _\\mathbf {r}\\Phi _s(2,...,N|\\mathbf {r})|^2\\right) \\, d2..d N,$ where $\\Phi (2,...,N|\\mathbf {r})$ is a conditional amplitude defined in terms of a wavefunction $\\Psi $ and its density $\\rho $ , $\\Phi (2,...,N|1)=\\sqrt{\\frac{N}{\\rho (1)}}\\Psi (1,...,N),$ $1,...N$ denote the spatial and spin coordinates of the $N$ electrons, and in Eq.",
"(REF ) we consider the conditional amplitude from the exact wavefunction (denoted with $\\Phi $ ) and for the KS determinant (denoted with $\\Phi _s$ ).",
"Equation (REF ) can also be rewritten in several different interesting and more practical forms, for example in terms of first order density matrices, or in terms of natural orbitals, or with Dyson orbitals (see, e.g., [43], [44], [45], [48], [49], [50], [51], [52], [53]).",
"In the present case of $N=2$ electrons, Eq.",
"(REF ) takes the simple form $v_{\\rm c,kin}(r)=\\frac{1}{2 \\rho (r)} \\int |\\nabla _\\mathbf {r}\\Psi (\\mathbf {r},\\mathbf {r}^{\\prime })|^2 d\\mathbf {r}^{\\prime } - \\frac{|\\nabla \\rho (r)|^2}{8 \\rho (r)^2},$ where $\\Psi (\\mathbf {r}_1,\\mathbf {r}_2)$ is the exact ground-state wavefunction of the interacting system.",
"Both $\\overline{w}(\\mathbf {r})-w_1(\\mathbf {r})$ and $v_{\\rm c,kin}(\\mathbf {r})$ integrate to $T_c[\\rho ]$ when multiplied by the density $\\rho (\\mathbf {r})$ , but they describe the kinetic correlation energy locally in a different way.",
"Here we compare the features of these two definitions, as the correlation kinetic energy is important to capture strong correlation.",
"Also, very recently, it has been proposed to use the correlated kinetic energy density as an additional variable in an extended KS DFT theory for lattice hamiltonians [54], and it is thus important to understand which definition is the most suitable to generalize this theory to the continuum.",
"In Fig.",
"REF we show the two different kinetic correlation energy densities, where for $\\overline{w}(r)$ we have used the integration over $\\lambda $ of the Padé model, which uses the exact $w_0$ , $w_0^{\\prime }$ and $w_1$ as input.",
"We see that the two are rather different: $v_{\\rm c,kin}(r)$ displays a peak in the center of the harmonic trap, reminiscent of the one appearing in a stretched bond [27], [28], [29], [47], while $\\overline{w}(\\mathbf {r})-w_1(\\mathbf {r})$ displays a weaker peak, which is not located at the center.",
"Figure: The two kinetic correlation energy densities of Eqs.",
"() and () for the different Hooke's atoms considered here.",
"The high density scaling is applied."
],
[
"Analysis of the peak of $v_{\\rm c,kin}(\\mathbf {r})$",
"In the case of a stretched bond, it has been shown that the height of the peak of $v_{\\rm c,kin}(\\mathbf {r})$ at the midbond saturates as the bond is stretched [28], displaying an anomalous scaling [29], which is the way in which exact KS DFT can describe Mott-insulator physics [29], and which is not captured by any approximate XC functional.",
"In the low-density (small $\\omega $ or large $n$ ) Hooke's atom, the system forms a “Wigner molecule”, with the maximum of the density located away from the center of the harmonic trap.",
"It is interesting to analyze how the height $v_{\\rm c,kin}(0)$ of the peak scales when the system becomes very correlated ($\\omega \\rightarrow 0$ ), as in Fig.",
"REF it seems to saturate when one uses the high-density scaling.",
"For any 2-electron wavefunction of the form $\\Psi (r_1,r_2,r_{12})=e^{-\\frac{\\omega }{2}(r_1^2+r_2^2)}p(r_{12})$ , the peak's height is given by the simple expression $v_{\\rm c,kin}(0)=\\frac{\\int _0^\\infty e^{-\\omega x^2}\\, x^2\\, p^{\\prime }(x)^2 \\,d x}{2 \\int _0^\\infty e^{-\\omega x^2}\\, x^2\\, p(x)^2 \\,d x}.$ We have used up to the second-order of the small-$\\omega $ (strong correlation) expansion of the exact wavefunction [25], finding that in the scaling used in Fig.",
"REF the peak does not saturate, but eventually will decrease and then go to zero very slowly, as $\\omega ^{1/6}$ .",
"In Fig.",
"REF , we show the peak's height as a function of $\\omega $ for the analytic solutions, compared to the first three orders in the small-$\\omega $ (strong correlation) expansion (Eq.",
"(32) of [25]), and with the large-$\\omega $ (weak correlation) expansion (Eq.",
"(22) of [25]).",
"We see that the strong-correlation expansion for the peak is much more accurate than ordinary perturbation theory from the weak correlation limit even for very moderate correlation (the Hooke's atom with $\\omega =1/2$ resembles the He atom as far as the degree of correlation is concerned).",
"Figure: The peak v c, kin (0)v_{\\rm c,kin}(0) as a function of ω\\omega .",
"The first three orders in the small-ω\\omega (strong correlation) expansion are compared with the values (dots) from the exact wavefunctions of Taut , and with the large-ω\\omega (weak correlation) expansion."
],
[
"Conclusions",
"We have analyzed the performances of exchange-correlation functionals built from global and local interpolations between the weak- and the strong-interaction limits of DFT for the Hooke's atom series.",
"This case study allows for the use of exact analytical input ingredients, thus disentangling the errors coming from the interpolation itself from those on the input quantities.",
"Surprisingly, we have found that for these systems the global interpolations always outperform their local counterparts, in striking contrast with what had been observed so far for small chemical systems [14], [15].",
"We have also compared two different definitions of the kinetic correlation energy density, which plays a crucial role for strongly correlated systems [28], [29], and that can help in understanding how to extend to the continuum a KS theory that recovers the exact kinetic energy density recently proposed for lattice models [54].",
"Financial support from European Research Council under H2020/ERC Consolidator Grant corr-DFT (Grant Number 648932) is acknowledged.",
"We thank S. Giarusso and S. Vuckovic for insightful discussions."
],
[
"Interpolation Formulas",
"In the following we report the interpolation formulas in terms of the global ingredients $W_0$ , $W_0^{\\prime }$ , $W_\\infty $ and $W_\\infty ^{\\prime }$ .",
"For the interpolation on energy densities, we have used the same SPL, LB and Padé[1/1] formulas below in each point of space, replacing the global quantities $W_i$ with their local counterparts $w_i(\\mathbf {r})$ .",
"Interaction Strength Interpolation (ISI) formula [9], [8] $W_\\lambda ^\\mathrm {ISI}= W_\\infty + \\frac{X }{\\sqrt{1+\\lambda Y }+Z }\\ ,$ with $&&X=\\frac{xy^2}{z^2}\\; ,\\; Y=\\frac{x^2y^2}{z^4}\\; , \\; Z=\\frac{xy^2}{z^3}-1\\ ;\\\\&& x=-2 W_0^{\\prime } ,\\; y=W_\\infty ^{\\prime }\\; , \\; z=W_0-W_\\infty \\ .$ After integration in Eq.",
"(REF ), we have $E_{xc}^\\mathrm {ISI}= W_\\infty + \\frac{2X}{Y}\\left[\\sqrt{1+Y}-1-Z\\ln \\left(\\frac{\\sqrt{1+Y}+Z}{1+Z}\\right)\\right]\\ .$ Revised ISI (revISI) formula [33] $W_\\lambda ^\\mathrm {revISI}= W_\\infty + \\frac{b \\left( 2 + c \\lambda + 2 d \\sqrt{1 + c \\lambda }\\right)}{2 \\sqrt{1 + c \\lambda } \\left( d + \\sqrt{1 + c \\lambda }\\right) ^2} ,$ where $\\nonumber b & = &-\\frac{4 W_0^{\\prime } (W_\\infty ^{\\prime }) ^{2}}{\\left(W_0-W_\\infty \\right)^2}\\; ,\\; c=\\frac{4 (W_0^{\\prime } W_\\infty ^{\\prime })^2}{\\left(W_0-W_\\infty \\right)^4}\\; ,\\; \\\\d & = & -1-\\frac{4 W_0^{\\prime } (W_\\infty ^{\\prime }) ^{2}}{\\left(W_0-W_\\infty \\right)^3} \\ .$ The corresponding XC functional is $E_{xc}^\\mathrm {revISI}= W_\\infty + \\frac{b}{\\sqrt{1+c}+d}\\ .$ Seidl-Perdew-Levy (SPL) formula [7] $W_\\lambda ^\\mathrm {SPL}= W_\\infty +\\frac{W_0 -W_\\infty }{\\sqrt{1+2\\lambda \\chi }}\\ ,$ with $\\chi = \\frac{W_0^{\\prime }}{W_\\infty -W_0}\\ .$ The SPL XC functional reads $E_{xc}^\\mathrm {SPL}= \\left(W_0-W_\\infty \\right)\\left[\\frac{\\sqrt{1+2\\chi }-1-\\chi }{\\chi }\\right] + W_0\\ .$ Notice that this functional does not make use of the information from $W_\\infty ^{\\prime }$ .",
"Liu-Burke (LB) formula [40] $W_\\lambda ^\\mathrm {LB}= W_\\infty + \\beta (y + y^4 )\\ ,$ where $y = \\frac{1}{\\sqrt{1+\\gamma \\lambda }}\\; , \\; \\beta =\\frac{W_0-W_\\infty }{2}\\; , \\; \\gamma =\\frac{4 W_0^{\\prime }}{5(W_\\infty -W_0)}\\ .$ Using Eq.",
"(REF ), the LB XC functional is found to be $E_{xc}^\\mathrm {LB}= W_0 +2\\beta \\left[\\frac{1}{\\gamma }\\left(\\sqrt{1+\\gamma }-\\frac{1+\\gamma /2}{1+\\gamma }\\right)-1\\right]\\ .$ Also the LB functional does not use the information from $W_\\infty ^{\\prime }$ .",
"Padé[1/1] formula with the exact $W_1$ [55] $W_\\lambda ^{\\rm Pade} = a + \\frac{b\\, \\lambda }{1+ c\\, \\lambda },$ with $a & = & W_0 \\\\b & = & W_0^{\\prime } \\\\c & = & \\frac{W_1-W_0 - W_0^{\\prime }}{W_0-W_1},$ yielding $E_{xc}^{\\rm Pade}=a+b\\,\\left(\\frac{c-\\log (1+c)}{c^2}\\right)$"
]
] | 2012.05167 |
[
[
"Estimating distances from parallaxes. V: Geometric and photogeometric\n distances to 1.47 billion stars in Gaia Early Data Release 3"
],
[
"Abstract Stellar distances constitute a foundational pillar of astrophysics.",
"The publication of 1.47 billion stellar parallaxes from Gaia is a major contribution to this.",
"Yet despite Gaia's precision, the majority of these stars are so distant or faint that their fractional parallax uncertainties are large, thereby precluding a simple inversion of parallax to provide a distance.",
"Here we take a probabilistic approach to estimating stellar distances that uses a prior constructed from a three-dimensional model of our Galaxy.",
"This model includes interstellar extinction and Gaia's variable magnitude limit.",
"We infer two types of distance.",
"The first, geometric, uses the parallax together with a direction-dependent prior on distance.",
"The second, photogeometric, additionally uses the colour and apparent magnitude of a star, by exploiting the fact that stars of a given colour have a restricted range of probable absolute magnitudes (plus extinction).",
"Tests on simulated data and external validations show that the photogeometric estimates generally have higher accuracy and precision for stars with poor parallaxes.",
"We provide a catalogue of 1.47 billion geometric and 1.35 billion photogeometric distances together with asymmetric uncertainty measures.",
"Our estimates are quantiles of a posterior probability distribution, so they transform invariably and can therefore also be used directly in the distance modulus (5log10(r)-5).",
"The catalogue may be downloaded or queried using ADQL at various sites (see http://www.mpia.de/homes/calj/gedr3_distances.html) where it can also be cross-matched with the Gaia catalogue."
],
[
"Introduction",
"There are various ways to determine astrophysical distances.",
"Near the base of the distance ladder on which almost all other distance measures are built are geometric parallaxes of stars.",
"In recognition of this, the European Space Agency (ESA) implemented the Gaia mission to obtain parallaxes for over one billion stars in our Galaxy down to $G\\simeq 20$ mag, with accuracies to tens of microarcseconds [12].",
"The first two data releases [13], [14] presented a significant leap forward in both the number and accuracy of stellar parallaxes.",
"The recently published early third release [15] (hereafter EDR3) reduces the random and systematic errors in the parallaxes by another 30%.",
"While parallaxes ($\\varpi $ ) are the basis for a distance determination, they are not themselves distances ($r$ ).",
"This is due to the nonlinear transformation between them ($\\varpi \\sim 1/r$ ) and the presence of significant noise for more distant stars.",
"Small absolute uncertainties in parallax can translate into large uncertainties in distance, and while parallaxes can be negative, distances cannot be.",
"Thus for anything but the most precise parallaxes, the inverse parallax is a poor distance estimate.",
"An explicit probabilistic approach to inferring distances may instead be taken.",
"This has been discussed and applied to parallax data in various publications in recent years; a recent overview is given by [24].",
"The simplest approach uses just the parallax and parallax uncertainty together with a one-dimensional prior over distance.",
"This yields a posterior probability distribution over distance to an individual star [5].",
"A suitable prior ensures that the posterior converges to something sensible as the precision of the parallax degrades.",
"This important when working with Gaia data, because its truly revolutionary nature notwithstanding, in EDR3 43% of the sources have parallax uncertainties greater than 50% (63% greater than 20%), and a further 24% have negative parallaxes.",
"The shape and scale of the prior distribution should reflect the expected distribution of stars in the sample, including observational selection effects such as magnitude limits.",
"The prior's characteristic length scale will typically need to vary with direction in the Galaxy [6].",
"More sophisticated approaches use other types of data, such as the star's magnitude and colour [3], [26], [1], [21], velocity [35], [38], or spectroscopic [34], [29] or asteroseismic [18] parameters.",
"In order to exploit such additional data, these methods must make deeper astrophysical assumptions than parallax-only approaches, and may also have more complex priors.",
"The benefit is that the inferred distances will usually be more precise (lower random errors), and hopefully also more accurate (lower systematic errors) if the extra assumptions are correct.",
"In the present paper, the fifth in a series, we determine distances for sources in EDR3 using data exclusively from EDR3.",
"The resulting catalogue should be more accurate and more useful than our earlier work, on account of both the more accurate parallaxes in EDR3 and improvements in our method.",
"We determine two types of distance.",
"The first, which we call “geometric\", uses only the parallaxes and their uncertainties.",
"We explored this approach in detail in the first two papers in this series [5], [3] (hereafter papers I and II), and applied it to estimate distances for 2 million stars in the first Gaia data release [4] (paper III) and 1.33 billion stars in the second Gaia data release [6] (paper IV).",
"Both papers used a (different) direction-dependent distance prior that reflected the Galaxy's stellar populations and Gaia's selection thereof.",
"Our second type of distance estimate uses, in addition to the parallax, the colour and magnitude of the star.",
"We call such distances “photogeometric\".",
"As well as the distance prior, this uses a model of the direction-dependent distribution of (extincted) stellar absolute magnitudes.",
"We construct our priors from the GeDR3 mock catalogue of [33].",
"This lists, among other things, the (noise-free) positions, distances, magnitudes, colours, and extinctions of 1.5 billion individual stars in the Galaxy as a mock-up of what was expected to appear in EDR3.",
"GeDR3mock is based on the Besançon Galactic model and PARSEC stellar evolutionary tracks.",
"We exclude stars from GeDR3mock that simulate the Magellanic Clouds (popid=10) and stellar open clusters (popid=11).",
"We divide the sky into the 12288 equal-area (3.36 sq. deg.)",
"regions defined by the HEALpixel schemehttps://healpix.sourceforge.io at level 5, and fit our prior models separately to each.",
"In doing this we only retain from GeDR3mock those stars that are brighter than the 90th percentile of the EDR3 magnitude distribution in that HEALpixel [32], [16].",
"This is done to mimic the variable magnitude limit of Gaia over the sky, and varies from 19.2 mag around the Galactic centre to 20.7 mag over much of the rest of the sky (the median over HEALpixels is 20.5 mag).",
"We apply our inference to all sources in EDR3 that have parallaxes.",
"As our prior only reflects single stars in the Galaxy, our distances will be incorrect for the small fraction of extragalactic source in the Gaia catalogue, and may also be wrong for some unresolved binaries, depending on their luminosity ratios.",
"As some readers may be familiar with our previous catalogue using GDR2 data (paper IV), here is a summary of the main changes in the new method (which we describe fully in section ).",
"We update the source of our prior from a mock catalogue of GDR2 [31] to one of EDR3 [33].",
"We replace the one-parameter exponential decreasing space density (EDSD) distance prior with a more more flexible three-parameter distance prior (section REF ).",
"We again fit the distance prior to a mock catalogue, but we no longer use spherical harmonics to smooth the length scale of the prior over the sky.",
"We instead adopt a common distance prior for all stars within a small area (level 5 HEALpixels).",
"We introduce photogeometric distances (section REF ) using a model for the (extincted) colour-absolute magnitude diagram, also defined per HEALpixel (section REF ).",
"In paper IV we summarized each posterior with the mode and the highest density interval (HDI).",
"The mode has the disadvantage that it is not invariant under nonlinear transformations.",
"This means that if we inferred $r_{\\rm mode}$ as the mode of the posterior in distance, then $5\\log _{\\rm 10}r_{\\rm mode}-5$ would not, in general, be the mode of the posterior in distance modulus.",
"This is also the case for the mean.",
"The quantiles of a distribution, in contrast, are invariant under (monotonic) nonlinear transformations.",
"We therefore provide the median (the 50th percentile) of the posterior as our distance estimate.",
"To characterize the uncertainty in this we quote the 14th and 86th percentiles (an equal-tailed interval, ETI).",
"These are therefore also the quantiles on the absolute magnitude inferred from the distance.",
"In the next section we describe our method and the construction of the priors.",
"In section we apply our method to the GeDR3mock catalogue, giving some insights into how it performs.",
"We present the results on EDR3 in section , and describe the resulting distance catalogue in section along with its use and limitations.",
"We summarize in section .",
"Auxiliary information, including additional plots for all HEALpixels, for both the prior and the results, can be found onlinehttp://www.mpia.de/~calj/gedr3_distances.html."
],
[
"Method",
"For each source we compute the following two posterior probability density functions (PDFs) over the distance $r$ $\\textrm {Geometric:} \\ \\ && P_{\\rm g}^*(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}\\varpi , \\sigma _{\\varpi }, p) && \\nonumber \\\\\\textrm {Photogeometric:} \\ \\ && P_{\\rm pg}^*(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}\\varpi , \\sigma _{\\varpi }, p, \\, && G, c) \\nonumber $ where $\\varpi $ is the parallax, $\\sigma _{\\varpi }$ is the uncertainty in the parallax, $p$ is the HEALpixel number (which depends on Galactic latitude and longitude), $G$ is the apparent magnitude, and $c$ is the ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ colour.",
"The parallax and apparent magnitude will be adjusted to accommodate known issues with the EDR3 data, as detailed below.",
"The star $^*$ symbol indicates that we infer unnormalized posteriors.",
"The geometric posterior uses just a distance prior.",
"The photogeometric posterior uses this distance prior as well as a colour–magnitude prior that we explain below.",
"The posteriors are summarized using quantiles computed by Markov Chain Monte Carlo (MCMC) sampling."
],
[
"Geometric distance",
"The unnormalized posterior PDF is the product of the likelihood and prior: $P_{\\rm g}^*(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}\\varpi , \\sigma _{\\varpi }, p) \\,=\\ P(\\varpi \\hspace{0.50003pt}\\mid \\hspace{0.50003pt}r, \\sigma _{\\varpi }) \\, P(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}p) \\ .$ The likelihood is conditionally independent of $p$ .",
"We chose to make the second term, which we define in section REF , independent of $\\sigma _{\\varpi }$ ."
],
[
"Likelihood",
"Under the assumption of Gaussian parallax uncertainties the likelihood is $P(\\varpi \\hspace{0.50003pt}\\mid \\hspace{0.50003pt}r, \\sigma _{\\varpi }) \\,=\\, \\frac{1}{\\sqrt{2 \\pi }\\sigma _{\\varpi }} \\exp { \\left[ -\\frac{1}{2\\sigma _{\\varpi }^2}\\left(\\varpi -\\varpi _{\\rm zp}-\\frac{1}{r}\\right)^2 \\right] }$ where $\\varpi _{\\rm zp}$ is the parallax zeropoint.",
"In paper IV we adopted a constant value of $-0.029$ mas for this zeropoint, as recommended in the GDR2 release.",
"For EDR3 the Gaia team has published a more sophisticated parallax zeropoint based on analyses of quasars, binary stars, and the Large Magellanic Cloud (LMC) [22].",
"This is a function of $G$ , the ecliptic latitude, and the effective wavenumber used in the astrometric solution.",
"Ideally this last term was derived from the ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ colour, and this is the case for the standard 5-parameter (5p) astrometric solutions used for 585 million sources [15].",
"But where ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ was unavailable or deemed of insufficient quality, the effective wavenumber was derived as a sixth parameter in the astrometric solution (6p solutions) [23], which is the case for 882 million sources.",
"Overall the zeropoint ranges between about $-0.150$ and $+0.130$ mas (it is narrower for the 5p solutions), although the RMS range is only 0.020 mas.",
"We use this zeropoint correction in equation REF .",
"Our geometric distances are therefore weakly conditioned also on $G$ and $c$ , but we omit this in the mathematical notation for brevity.",
"For the 2.5 million sources that have parallaxes but no $G$ (strictly, no phot_g_mean_mag), we use the EDR3 global zeropoint of $-0.017$ mas [23]."
],
[
"Distance prior",
"In paper IV we used the one-parameter EDSD distance prior, which models the space density of stars as dropping exponentially away from the Sun according to a (direction-dependent) length scale.",
"Here we adopt the more flexible, three-parameter Generalized Gamma Distribution (GGD), which can be written as $P(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}p) \\ = \\ \\begin{dcases}\\ \\frac{1}{\\Gamma (\\frac{\\beta +1}{\\alpha })}\\frac{\\alpha }{L^{\\beta +1}}\\,r^\\beta \\,e^{-(r/L)^\\alpha } & \\:{\\rm if}~~ r\\ge 0 \\\\\\ 0 & \\:{\\rm otherwise}\\end{dcases}$ for $\\alpha > 0$ , $\\beta > -1$ , and $L> 0$ .",
"$\\Gamma ()$ is the gamma function.",
"This PDF is unimodal with an exponentially decreasing tail to larger distances.",
"The mode is $L(\\beta /\\alpha )^{1/\\alpha }$ for $\\beta >0$ , and zero otherwise.",
"The EDSD is a special case of the GGD with $\\alpha =1, \\beta =2$ .",
"We fit the GGD prior for each HEALpixel separately via maximum likelihood using stars from the mock catalogue.",
"The HEALpixel ($p$ ) dependency on the left side of equation REF is equivalent to a dependency on $\\alpha , \\beta , L$ .",
"Figure: The variation of the median of the distance prior over the sky shown in Galactic coordinates on a Mollweide equal-area projection.",
"The LMC/SMC are excluded from our prior.Example fits for two HEALpixels, one at low Galactic latitude and one at high Galactic latitude, are shown in Figure REF .",
"Although the GGD prior provides a better fit than the EDSD prior – which is why we use it – the parameter $L$ may no longer be interpreted as a meaningful length scale, because it varies from 3e-7 to 1e4 pc over all HEALpixels.",
"The appropriate characteristic scale of the GGD prior in this work is its median, for which which there is no closed-form expression.",
"The median varies between 745 and 7185 pc depending on HEALpixel (Figure REF ).",
"Fits for each HEALpixel can be found in the auxiliary information online.",
"In the limit of uninformative parallaxes, the geometric posterior converges on the GGD prior, and so the median distance converges on the median of this prior.",
"In paper IV this convergence was on the mode of the EDSD prior.",
"For the prior fits used in the present paper, the ratio of the GGD median to the EDSD mode ranges from 1.17 to 1.57.",
"There are potential improvements one could make to the prior to give a better convergence in the limit of poor data.",
"Some considerations are in appendix ."
],
[
"Photogeometric distance",
"We define the quantity $Q_G$ as $Q_G\\equiv M_G+ A_G\\,=\\, G- 5\\log _{\\rm 10}r+ 5 \\ .$ The equality ($=$ ), which is a statement of flux conservation, holds only when all the quantities are noise-free.",
"If we knew $Q_G$ for a star, then a measurement of $G$ gives us an estimate of $r$ .",
"Given that the uncertainties on $G$ in EDR3 are generally less than a few millimagnitudes (0.3 to 6 mmag for $G< 20$ mag; [15]), this would be a reasonably precise estimate.",
"We do not know $Q_G$ , but we can take advantage of the fact that the two-dimensional colour–$Q_G$ space for stars is not uniformly populated.",
"This space (e.g.",
"Figure REF ) which we call the CQD – in analogy to the CMD (colour–magnitude diagram) – would be identical to the colour-absolute-magnitude diagram if there were no interstellar extinction.",
"Thus if we know the ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ colour of the star, this diagram places limits on possible values of $Q_G$ , and therefore on the distance to the star.",
"We will use the mock catalogue to model the CQD (per HEALpixel) and from this compute a prior over $Q_G$ given the magnitude and colour of the star.",
"The formal procedure is as follows.",
"Using Bayes' theorem, the unnormalized posterior we want to estimate can be decomposed into a product of two terms $P_{\\rm pg}^*(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}\\varpi , \\sigma _{\\varpi }, G, c, p) \\,=\\ P(\\varpi \\hspace{0.50003pt}\\mid \\hspace{0.50003pt}r, \\sigma _{\\varpi }) \\, P(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}G, c, p) \\ .$ The first term on the right side is the parallax likelihood (section REF ).",
"It is independent of $G$ , $c$ , and $p$ once it is conditioned on $\\sigma _{\\varpi }$ , which is estimated in the Gaia astrometric solution using quantities that depend on the magnitude, colour, scanning law, etc.",
"[23].",
"The second term is independent of the parallax measurement process and thus of $\\varpi $ and $\\sigma _{\\varpi }$ .",
"We may write this second term as a marginalization over $Q_G$ and then apply Bayes' theorem as follows $& P(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}G, c, p) \\\\\\,&=\\ \\int P(r, Q_G\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}G, c, p) \\ \\mathrm {d}Q_G\\nonumber \\\\\\,&=\\ \\int \\frac{1}{P(G\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}c, p)} \\, P(G\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}r, Q_G) \\, P(r, Q_G\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}c, p) \\ \\mathrm {d}Q_G\\nonumber \\\\\\,&=\\ \\frac{P(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}c, p)}{P(G\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}c, p)} \\int P(G\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}r, Q_G) \\, P(Q_G\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}r, c, p) \\ \\mathrm {d}Q_G\\ .",
"\\nonumber $ The first term under the integral is formally the likelihood for $G$ (and is conditionally independent of $c$ and $p$ due to equation REF ).",
"However, as $G$ is measured much more precisely than the intrinsic spread in $Q_G$ – that is, the second term under the integral is a much broader function – we can consider $G$ to be noise-free to a good approximation.",
"This makes the first term a delta function and so the integral is non-zero only when equation REF is satisfied.",
"Hence the (unnormalized) posterior can be written as $& P_{\\rm pg}^*(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}\\varpi , \\sigma _{\\varpi }, G, c, p) \\, \\simeq \\ P(\\varpi \\hspace{0.50003pt}\\mid \\hspace{0.50003pt}r, \\sigma _{\\varpi }) \\, P(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}p) \\, \\times \\nonumber \\\\& \\hspace*{70.0pt} P(Q_G= G- 5\\log _{\\rm 10}r+ 5 \\hspace{0.50003pt}\\mid \\hspace{0.50003pt}c, p) \\ .",
"$ The missing normalization constant, $1/P(\\varpi , G\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}c, p)$ , is not required.",
"The second term on the right side of equation REF is the distance prior.",
"We have chosen to make this colour-independent and so have set $P(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}c, p) \\rightarrow P(r\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}p)$ .",
"The posterior in equation REF is simply the geometric posterior (equation REF ) multiplied by an additional prior over $Q_G$ ."
],
[
"$Q_G$ prior",
"We construct the prior P($Q_G$$\\hspace{0.50003pt}\\mid \\hspace{0.50003pt}$$c$ , $p$ ) from the mock catalogue.",
"Given the complexity of the CQD and its variation over the sky, we do not attempt to fit the prior as a continuous 3D (position and colour) parametric function.",
"We instead compute a CQD for each HEALpixel, two examples of which are shown in Figure REF .",
"Within each we compute a series of one-dimensional functions at a series of colours in the following way.",
"We divide the full colour range of a given HEALpixel into strips of 0.1 mag width in colour, then for each strip fit a model to the stellar number density as a function of $Q_G$ (now ignoring the colour variation in each strip).",
"If there are more than 40 stars in a strip, we bin the data into bins of 0.1 mag and fit a smoothing spline with $\\min (\\lfloor {N/4}\\rfloor , 50)$ degrees of freedom (df), where $N$ is the number of stars in the strip (which can be many thousands).",
"If there are fewer than 40 stars we cannot fit a good spline.",
"This generally occurs at the bluest and reddest ends of the CQD.",
"Here the $Q_G$ distribution is often characterized by two widely-separated components, either the main sequence (MS) and white dwarf (WD) branches, or the MS and giant star branches (see Figure REF ).",
"Thus when $N<40$ we instead fit a two-component Gaussian mixture model, with the constraint that the minimum and maximum standard deviation of each component be $\\sigma _{\\rm min}=0.08$ mag and $\\sigma _{\\rm max}=1.0$ mag respectively.",
"A full fit requires at least five stars, so if there are as few as two stars we constrain the solution to first have equal standard deviations and then to have standard deviations of $\\sigma _{\\rm min}$ .",
"If $N=1$ our model is a one-component Gaussian with mean equal to the $Q_G$ of the star and standard deviation equal to $\\sigma _{\\rm min}$ .",
"If there are no stars the model is null.",
"Examples of the fits are shown in Figures REF and REF .",
"As a smoothing spline can give a negative fit, and both these and the Gaussian models can yield very small values for the density, we impose that the minimum density is never less than $10^{-3}$ of the integrated density (computed prior to fitting the model).",
"Thus our prior is nowhere zero, meaning that even if the data indicate a $Q_G$ in the regions where the mock catalogue is empty, the posterior will not be zero.",
"This allows sources to achieve distances that place them outside the occupied regions of the mock CQD.",
"For a given HEALpixel, each prior model refers to a specific colour, namely the centre of a 0.1 mag-wide strip.",
"When evaluating the prior during the inference process, we compute $Q_G$ from equation REF , evaluate the densities of the two priors that bracket its colour, then linearly interpolate.",
"This ensures that our prior is continuous in colour.",
"If one of the models is null we use the other model as is.",
"If both models are null, or if the source is outside of the colour range of the mock CQD, we do not infer a photogeometric distance.",
"The flag field in our catalogue indicates what kind of $Q_G$ models were used (see section ).",
"The computation of $Q_G$ in equation REF requires the G-band magnitude of the source.",
"For this we use the phot_g_mean_mag field in EDR3 corrected for the processing error described in section 8.3 of [30].",
"This correction, which is a function of magnitude and colour, can be as large as 25 mmag."
],
[
"Posterior sampling and summary",
"We have now defined the posteriors for the geometric and photogeometric distances.",
"These posteriors are formally the answer to our inference process.",
"The geometric posterior has a simple parametric form which may be computed by the reader using the data in the EDR3 catalogue and the parameters of our prior (available with the auxiliary information online).",
"The photogeometric posterior is generally non-parametric.",
"Both posteriors are asymmetric and not necessarily unimodal (section REF ).",
"There are a variety of statistics one could use to summarize these PDFs, such as the mean, median, or mode.",
"There is no theoretically correct measure, and all have their drawbacks.",
"We use quantiles, primarily because they are invariant under nonlinear transformations, and so are simultaneously the quantiles of the posterior in distance modulus, $5\\log _{\\rm 10}r-5$ .",
"We use the three quantiles at 0.159, 0.5, and 0.841, which we label $r_{\\rm lo}$ , $r_{\\rm med}$ , and $r_{\\rm hi}$ respectively.",
"The central quantile is the median.",
"The outer two quantiles give a 68% confidence interval around the median.",
"The difference between each quantile and the median is a Gaussian 1$\\sigma $ -like estimate of the uncertainty.",
"Due to the intrinsic asymmetry of the posteriors we report the lower and upper values separately."
],
[
"Markov Chain Monte Carlo",
"Neither the geometric nor photogeometric posteriors have closed-form expressions for their quantiles so we must compute these numerically.",
"Conventional numerical integration schemes, whether fixed or adaptive, are not very efficient in our application due to the large range of possible shapes and scales of the posteriors.",
"We therefore sample the posteriors using Markov Chain Monte Carlo (MCMC), specifically the Metropolis algorithm.",
"We adopt the following scheme for the MCMC initialization and step size.",
"We first compute the geometric distance posterior using the EDSD prior from paper IV.",
"The length scale of this prior is set to 0.374$r_{\\rm med}$ , where $r_{\\rm med}$ is the median distance of the stars in the mock catalogue for that HEALpixel.In paper IV we used $(1/3)r_{\\rm med}$ on the basis that the maximum likelihood fit of the length scale is a third of the mean.",
"For the EDSD, however, the median is a slightly biased estimator of the mean.",
"We found empirically that for the typical length scales involved, the mean is about 12% (0.374/0.333) larger than the median, which is why we use the factor of 0.374 here.",
"We compute the mode of this posterior, $r^{\\rm EDSD}_{\\rm mode}$ , which has a closed-form solution (paper I), and use this as the initialization for the geometric posterior sampling.",
"The initialization scheme for the photogeometric posterior is more complicated, in accordance with its more complicated shape, and depends on $r^{\\rm EDSD}_{\\rm mode}$ , fractional parallax uncertainty (fpu, $\\sigma _{\\varpi }/\\varpi $ ), and the characteristic length scale of the $Q_G$ prior model(s).",
"For both types of posterior the step size needs to be adapted to the characteristic width of the posterior, which is generally wider the larger the fpu.",
"We found a suitable step size to be $(3/4)r_{\\rm init}\\times \\min (|\\sigma _{\\varpi }/\\varpi |, 1/3)$ , where $r_{\\rm init}$ is the initialization value.",
"The main advantage of this scheme is that relatively short burn-ins are sufficient.",
"We use just 50.",
"We experimented with chains of various lengths, employing various tests of convergence.",
"Longer chains are always better, but as we need to sample around three billion posteriors, some parsimony is called for.",
"We settled on 500 samples (post burn-in).",
"The chains are not always settled, but even in those cases they are generally good enough to compute the required quantiles with reasonable precision.",
"To quantify this we obtained 20 different MCMC chains (using different random number seeds) and computed the standard deviation of the median distance estimates and half the mean of the confidence intervals.",
"The ratio of these is a measure of the convergence noise.",
"Doing this for thousands of stars we find this to be between 0.1 and 0.2 in general.",
"For the geometric posteriors in particular it can get larger for fractional parallax uncertainties larger than about 0.3.",
"We also tested the parallel chain MCMC method of [11], but found that it required more iterations to achieve similar accuracy than did Metropolis."
],
[
"Multimodality",
"As both posteriors are the products of density functions they are not guaranteed to be unimodal.",
"This is more likely to be the case for the photometric posterior at large fpu, as its prior can be multimodal.",
"Multimodality is very rare for the geometric posterior.",
"Multimodality is a challenge for MCMC sampling methods, although we find that even widely-separated modes can be sampled in our scheme.",
"Moreover, our 68% confidence interval often encompasses the span of the multimodality.",
"This can be a blessing and a curse: the distance precision in a single mode may be quite good, yet a large confidence interval is obtained due to the presence of a second mode.",
"Properly characterizing the large range of shapes of posterior and multimodality that could occur with over one billion sources would be complicated and non-robust.",
"We avoid this problem and simply report the three quantiles mentioned.",
"To assist in identifying possible multimodality, we compute, for every posterior, the Hartigan dip test [19].",
"This is a classical statistical test in which the null hypothesis is a unimodal posterior.",
"A small p-value therefore suggests the distribution may not be unimodal.",
"We select a threshold of $10^{-3}$ and set a flag to 1 if the p-value is lower than this, thereby suggesting possible multimodality.",
"If the p-value is above this threshold or the test does not work for any reason, the flag is 0.",
"The test is not particularly accurate, however, so should not be over-interpreted.",
"Furthermore, the test is done on the MCMC samples, not on the true posterior, so tends to be raised more often than we would expect because of the intrinsic noise in MCMC sampling."
],
[
"Performance on the mock catalogue",
"Before looking at the results on EDR3, we evaluate the performance of our method using the mock catalogue, as here we know the true distances.",
"In doing this we add Gaussian random noise to the parallaxes using the parallax_error field in GeDR3mock, which is a model of the expected uncertainties in the EDR3 parallaxes.",
"As the data are drawn from the same distance distribution and CQD from which the prior was constructed, this is a somewhat optimistic test, despite the noise.",
"Unless noted otherwise, throughout this section the term “fpu\" refers to the true fractional parallax uncertainty, i.e.",
"that computed using the true parallax"
],
[
"Example posteriors",
"Figure REF shows examples of both types of posterior compared to their priors.",
"At small fpu, e.g.",
"panels (a) to (c), the two posteriors are very similar, with a median (and mode) near to the true distance, shown as the vertical line.",
"As long as the fpu is not too large, the prior plays little role and the posterior can be quite different, e.g.",
"panel (d), although this can also occur at larger fpu, e.g.",
"panels (i) and (l).",
"Panel (f) shows a multimodal photogeometric prior and posterior.",
"The two types of prior sometimes disagree, as can the posteriors.",
"In panel (h), which is for a 30% parallax uncertainty, the geometric posterior is more consistent with the true distance.",
"Note that the parallax that the algorithm sees does not correspond to the vertical line, so for large fpu we cannot expect either posterior to peak near this.",
"Panel (k) shows a multimodal posterior in which the true distance is close to a smaller mode.",
"This happens here because the parallax has 50% noise, so the measured parallax corresponds to a smaller distance (where both geometric and photogeometric posteriors peak).",
"At larger fpu – the bottom row is all for more than 1.0 – the photogeometric prior is often more consistent with the true distance than the geometric one.",
"Distance inference results for two HEALpixels are shown in Figures REF and REF .",
"We see a good correlation between the inferred and true distances out to several kpc (left columns).",
"The degradation at larger distances is mostly due to stars with larger fpu, as can be seen in the middle columns of these figures.",
"The fractional residual is defined as the estimated minus true distance, divided by the true distance.",
"Note that these middle columns show the true fpu, i.e.",
"as computed from the noise-free parallax, which is not the same as the measured (noisy) fpu that the inference algorithm encounters.",
"(See section for a consequence of this difference.)",
"At large fpu the photogeometric distances perform better than the geometric ones, because even when the parallax is of limited use there is still distance information from the colour and magnitude via the $Q_G$ model.",
"For geometric distances, in contrast, as the measured fpu increases, the distance prior dominates the likelihood, so the median of the posterior is pushed towards the median of the prior.",
"Hence at large fpu, the geometric distances to stars that are truly more distant than the median of the prior will generally be underestimated.",
"Faraway stars tend to have larger fpu than nearby stars, because they have both smaller parallaxes and larger parallax uncertainties (as they are fainter).",
"Thus as a whole, any underestimation of geometric distances to stars that are beyond the median of the prior will tend to be larger than the overestimation of the geometric distances to stars that are closer than the median of the prior.",
"This explains why the distribution in the top-left panels of Figures REF and REF flatten at larger distances.",
"This feature is suppressed in the photogeometric distances (bottom-left panels) because for large fpu, the $Q_G$ prior can overrule the geometric prior.",
"We also see more flattening for the low latitude HEALpixel in Figure REF than the high latitude HEALpixel in Figure REF because the low latitude HEALpixel has larger fpus on average.",
"The right columns of Figures REF and REF assess how well the estimated distance uncertainties explain the residuals, by plotting the distribution of residual$/$ uncertainty.",
"This is shown using three different representations of the uncertainty.",
"The upper uncertainty, $r_{\\rm hi}-r_{\\rm med}$ , and symmetrized uncertainty$(r_{\\rm hi}-r_{\\rm lo})/2$ , shown in blue and black respectively, yield almost identical distributions.",
"For the high latitude HEALpixel 6200 (Figure REF ) they are quite close to a unit Gaussian, in particular for the photogeometric estimates.",
"The lower uncertainty, $r_{\\rm med}-r_{\\rm lo}$ , shown in orange, is negatively skewed (larger tail to negative values), suggesting that the lower uncertainty measure, $r_{\\rm lo}$ , is slightly underestimated.",
"This is more noticeable in the low latitude HEALpixel 7593 (Figure REF ), where we also see that the photogeometric estimates are slightly more skewed than the geometric ones."
],
[
"Quantitative analysis",
"To quantify the accuracy of our results we use the median of the fractional distance residual, which we call the bias, and the median absolute of the fractional distance residual, which we call the scatter.",
"These are robust versions of the mean and standard deviation, respectively.",
"For normally-distributed residuals the mean equals the median, and the standard deviation is 1.48 times the median absolute deviation.",
"For HEALpixel 6200 the bias and scatter for the geometric distances over all stars are +0.29e-3 and 0.10 respectively.",
"If we limit the computation of these metrics to the 50% of stars in this HEALpixel with $0 < \\sigma _{\\varpi }/\\varpi < 0.20$ , the bias is +5.3e-3 and the scatter is 0.037.",
"The scatter in this subsample is smaller, as expected.",
"The bias is larger because stars with small fpu tend to be nearer stars, whereas the distance prior is characteristic of all the stars, which are more distant on average.",
"Hence the prior pulls up the distances for the small fpu subsample, leading to a more positive bias.",
"For the photogeometric distances, the bias and scatter over all stars are +5.7e-3 and 0.059 respectively, and for the $0 < \\sigma _{\\varpi }/\\varpi < 0.20$ subsample are +2.5e-3 and 0.032 respectively.",
"The scatter over the full sample is smaller for the photogeometric estimates than for the geometric ones, because the former benefit from the additional information in the stars' colours and magnitudes.",
"The situation is particularly fortuitous here because of the near-perfect match between the $Q_G$ models and the actual distribution of $Q_G$ in the data.",
"For the full sample the bias is larger for the photogeometric distances than for the geometric ones, although still small on an absolute scale.",
"For the small fpu subsample the photometric distances are not much more accurate than the geometric ones, because the parallax dominates the distance estimate.",
"Turning now to the low latitude HEALpixel 7593 (Figure REF ), the bias and scatter in the geometric distances over all stars are $-0.16$ e-3 and 0.27 respectively.",
"There are two reasons for the larger scatter in this HEALpixel.",
"The first is that the parallax uncertainties are larger: the median parallax uncertainty is 0.32 mas, as opposed to 0.15 mas in HEALpixel 6200.",
"This in turn is because the stars are on average 0.9 magnitude fainter in HEALpixel 7593 (one reason for which is the larger extinction, as is apparent from Figure REF ).",
"The second reason is that the median true distance to stars is larger in this low latitude HEALpixel than in the high latitude one (4.0 kpc vs 1.2 kpc; see Figure REF ).",
"This may seem counter-intuitive, but is a consequence of distant disk (and bulge) stars at low latitudes that remain visible to larger distances despite the higher average extinction.",
"At higher latitudes, in contrast, there are no distant disk stars, and hardly any halo stars (which are scarce in Gaia anyway).",
"Both of these facts contribute to the larger fpu in the low latitude pixel – median of 1.18, central 90% range of 0.21–3.57 – than in the high latitude HEALpixel – median of 0.20, central 90% range of 0.03–1.08.",
"Even if we look at just the 9% of stars in the low latitude HEALpixel with $0 < \\sigma _{\\varpi }/\\varpi < 0.20$ , we get a bias and scatter of +25e-3 and 0.069 respectively, which are still significantly worse than the higher latitude HEALpixel for the same fpu range.",
"Concerning the photogeometric distances in HEALpixel 7593, the bias and scatter for all stars are $-3.8$ e-3 and 0.17 respectively, and for the $0 < \\sigma _{\\varpi }/\\varpi < 0.20$ subsample are +20e-3 and 0.062 respectively.",
"For the full sample we again see a significant decrease in the scatter compared to the geometric distances.",
"In a real application we may get less benefit from the $Q_G$ prior at low latitudes because our model CQD may differ from the true (unknown) CQD more than at high latitudes, on account of the increased complexity of the stellar populations and interstellar extinction near the Galactic plane."
],
[
"Inferred CQDs",
"We can also assess the quality of our distance estimates by computing $Q_G\\,=\\, G- 5\\log _{\\rm 10}r_{\\rm med}+ 5$ and plotting the resulting CQD.",
"We do this for both the geometric and photogeometric distances, for three ranges of fpu, for HEALpixel 6200 in Figure REF and HEALpixel 7593 in Figure REF .",
"These can be compared to the CQD for the same HEALpixels constructed using the true distances shown in Figure REF .",
"Imperfect distance estimates can only move sources vertically in this diagram as the ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ colours are not changed.",
"We see how the inferred main sequence is wider for the larger fpu samples for the geometric distances (left two columns in both plots), but much less so for the photogeometric distances.",
"This is again due to the stablizing influence of the $Q_G$ prior.",
"Both distance estimates are able to recover the primary structures: the main sequence, white dwarf sequence, giant branch, and horizontal branch.",
"These plots will be useful when it comes to analysing the results on the real EDR3 data, because they do not involve the truth as a reference."
],
[
"Analysis of distance results in EDR3",
"We applied our inference code (written in R) to the 1.47 billion sources in Gaia EDR3 that have parallaxes.",
"This required $1.6\\times 10^{12}$ evaluations of the posteriors and took 57 000 CPU-core-hours.",
"Throughout this section the term “fpu\" of course refers to the measured fractional parallax uncertainty, as we do not know the true parallax."
],
[
"Distance distributions and uncertainties",
"Results for our two example HEALpixels are shown in Figures REF and REF .",
"The two panels in the left column compare the two types of distance estimates.",
"As expected, the photogeometric estimates extend to larger distances (see section REF for an explanation).",
"The middle columns plot the ratio of the inferred distance to the inverse parallax distance (corrected for the zeropoint).",
"The latter is of course generally a poor measure of distance because it is not the true parallax, and this is the whole point of using an appropriate prior (see section and references therein).",
"We see that both of our distance estimates converge to $1/\\varpi $ in the limit of small fpu.",
"Although the apparent lack of sources at large fpu in the lower middle panels is primarily a plotting artefact, the two samples in the upper and lower panels are not identical, because not all sources have photogeometric distances.",
"For HEALpixel 6200 there are 24 007 sources with geometric distances and 23 829 with photogeometric distances.",
"For HEALpixel 7592 these numbers are 385 902 and 369 608 respectively.",
"The panels in the right columns of Figures REF and REF show how the fractional symmetrized distance uncertainty varies with fpu.",
"At small fpu they are nearly equal for both geometric and photogeometric distances, because here the likelihood dominates the posterior.",
"At larger fpu the geometric distances become more uncertain, which is commensurate with their lower expected accuracy.",
"For very large fpu ($\\gg 1$ ) the geometric distances and their uncertainties will be dominated by the prior, which for HEALpixel 7593 has a median of 3.98 kpc and lower (16th) and upper (84th) quantiles of 2.06 kpc and 6.74 kpc respectively (corresponding to a fractional distance uncertainty of 0.59).",
"The photogeometric fractional distance uncertainties tend to be smaller than the geometric ones.",
"This is because the $Q_G$ prior (section REF ) is usually more informative than the distance prior."
],
[
"Colour–$Q_G$ diagrams",
"From the inferred median distances we can compute the median $Q_G$ via equation REF and then plot the CQD.",
"This is shown in Figure REF for HEALpixel 6200 for the geometric distance (top row) and photogeometric distance (bottom row) for three different ranges of the fpu.",
"As interstellar extinction should be low towards this high latitude field (around 0.15 mag in GeDR3mock), $Q_G\\simeq M_G$ so this CQD is similar to the colour-absolute magnitude diagram.",
"In all of the panels we see a well-defined main sequence and giant branch, as well as a white dwarf sequence in some of the panels.",
"Comparing the upper and lower panels we see how the photogeometric distances constrain the $Q_G$ distribution more than the geometric distance do.",
"The puffing-up of the geometric CQD is due to sources with large fpu: their distances tend to be underestimated (see section REF ) so $Q_G$ becomes larger – intrinsically fainter – for a given $G$ (see equation REF ).",
"This puffing-up diminishes as we successively reduce the range of fpu, as shown in the middle and right columns of Figure REF .",
"The photogeometric CQD for the full fpu range (bottom left panel of Figure REF ) shows a conspicuous blob of sources at ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}\\simeq 0.5$ mag between the MS and WD sequences.",
"These are sources with spuriously large parallaxes, well known from GDR2 [2] and still present, if less so, in EDR3 [10], [16].",
"They are usually close pairs of sources that receive incorrect astrometric solutions, as the EDR3 astrometric model is only suitable for single stars [23].",
"Figure REF shows that spurious parallaxes are less common among the smaller fpu subsample.",
"The $Q_G$ prior will often help to constrain the distance of these spurious solutions and thus place them on the correct part of the CQD.",
"This is only partially successful at around ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}\\simeq 0.5$ mag in this HEALpixel, however, because the distance prior may still be pulling truly very distant sources with larger fpu towards us.",
"Sources with spurious parallaxes are preferentially faint.",
"To quote from [15]: “For faint sources ($G> 17$ for 6-p astrometric solutions and $G> 19$ for 5-p solutions) and in crowded regions the fractions of spurious solutions can reach 10 percent or more.\"",
"This can be seen in Figure REF where we replot the CQD only for sources with $G< 19.0$ mag.",
"This also reduces the puffing-up of the geometric CQD, although some of this reduction is simply because magnitude is correlated with fpu, so a magnitude cut also lowers the fpu.",
"Figure: As Figure but now for HEALpixel 7593.",
"All sources are shown (no magnitude cut).In total there are 385 902 sources with geometric distances and 369 608 with photogeometric distances.Figure: As Figure but now excluding the 70% of sources in this HEALpixel with G>19.0G> 19.0 mag to remove spurious sources.These effects can be seen more prominently in the low latitude HEALpixel 7593, shown in Figures REF and REF .",
"Due to the larger mean distance of stars at low latitudes (see section REF ), as well as the more complex stellar populations and larger mean extinction (up to 3.5 mag), the CQD is more complex.",
"For the full fpu range, the geometric CQD in Figure REF is quite washed out, due in part to large fpus and spurious parallaxes, although an extincted red clump is visible.",
"The photogeometric CQDs are cleaner, with a better defined main sequence.",
"The CQD for the $G< 19.0$ mag subsample (Figure REF ) again shows the removal of spurious sources.",
"Section 3.2 of [10] analyses spurious astrometric solutions and offers more sophisticated ways of identifying them than a simple magnitude cut.",
"We now look at a representative sample of the entire catalogue.",
"All plots and analyses in this section use a random selection of 0.5% of all sources from each HEALpixel.",
"This has 7 344 896 geometric and 6 739 764 photogeometric distances.",
"Figure: Distribution of inferred geometric and photogeometric median distances, r med r_{\\rm med}, in EDR3.",
"This plot uses a random sample of 0.5% of all sources in each HEALpixel.Figure: Fractional symmetrized distance uncertainty, (r hi -r lo )/2r med (r_{\\rm hi}-r_{\\rm lo})/2r_{\\rm med}, vs distance for the geometric distance estimates (top) and photogeometric distance estimates (bottom) for the three different GG ranges.The colour scale is a logarithmic density (base 10) scale relative to the highest density cell in each panel.This plot uses a random sample of 0.5% of all sources in each HEALpixel.Figure REF shows the distribution of distances.",
"As expected, the photogeometric distances extend to larger distances that the geometric one.",
"The fractional symmetrized distance uncertainties as a function of distance are shown in Figure REF for three different magnitude ranges.",
"As noted earlier, the photogeometric distance uncertainties are generally smaller than the geometric ones, at least for fainter sources.",
"This plot also shows again that photogeometric estimates extend to larger distances."
],
[
"Colour–$Q_G$ diagrams",
"Figure REF shows the CQD over the whole sky.",
"Because the sample is a constant random fraction per HEALpixel it is numerically dominated by sources at low latitude Galactic latitudes where there can be significant interstellar extinction.",
"This is apparent from the upper diagonal feature – especially clear in the photogeometric panel – which is the red clump stretched by extinction/reddening.",
"The white dwarf sequence appears clearly in the photogeometric CQD.",
"Although some white dwarfs are correctly placed in the CQD by the geometric distances, they are not visible here due to the finite dynamic range of the plotted density scale.",
"Furthermore, for reasons explained in section REF , faint nearby sources with large fpu tend of have their geometric distances overestimated and therefore their $Q_G$ underestimated, thereby pushing them up from the true white dwarf sequence.",
"These plots have not filtered out spurious sources, some of which are clearly visible in the photogeometric CQD as the blob between the upper MS and the white dwarf sequence.",
"Other broad differences between the geometric and photogeometric CQDs were explained in section REF ."
],
[
"Distribution on the sky",
"Figure REF shows the mean distance of sources (i.e.",
"mean of $r_{\\rm med}$ ) in each HEALpixel in our catalogue, as well as the ratio of these in log base 2.",
"Over all HEALpixels the 5th, 50th, and 95th percentiles of the mean of the geometric distances are 1.3, 2.1, and 4.4 kpc respectively.",
"The percentiles for the mean of the photogeometric distances are 2.2, 3.3, and 5.0 kpc.",
"These translate into low ratios of geometric to photogeometric distances in general.",
"Only in the Galactic plane and the bulge are the two mean distances comparable.",
"At high Galactic latitudes the photogeometric average is easily twice as large as the geometric average."
],
[
"Galactic spatial distribution",
"Figure REF shows the projected distribution of stars in EDR3 in the Galaxy using our distance estimates.",
"The Sun is at the origin, and we see the expected larger density of sources in the first and fourth Galactic quadrants.",
"Finer asymmetries in the distribution projected onto the Galactic plane (upper panels) are presumably due to both a genuine asymmetry in the Galactic population and Gaia's scanning law.",
"These, as well as nearby dust clouds, also explain the various radial lines pointing out from the origin.",
"The lack of sources in the fan around the positive x-axis in the lower panels is due to extinction in the Galactic plane.",
"The overdensity in the same direction in the upper panels is the projection of the bulge.",
"The lower panels demonstrate the point made earlier (section REF ) about being able to see sources to larger mean distances at lower Galactic latitudes.",
"The high density rays extending below the Galactic plane (lower panels of Figure REF ) are in the directions of the Magellanic Clouds.",
"Many stars in these satellite galaxies are in EDR3 – they are some of the densest HEALpixels – yet they are so far away (50–60 kpc) that most have poor (and often negative) parallaxes, such that the inferred geometric distances are dominated by the prior (see appendix for further discussion).",
"Our photogeometric distances are similarly poor, because we excluded the Magellanic Clouds from the mock CQD out of which our $Q_G$ priors are built.",
"This was intentional: anyone interested in estimating distances to sources in the Magellanic clouds can do better than just use Gaia parallaxes and photometry.",
"Figure REF shows the fractional distance uncertainties also in Galactic projection.",
"As expected, the uncertainties generally increase with distance from the Sun, but there are exceptions due to bright distant stars having more precise distances than faint nearby ones.",
"The rays towards the Magellanic clouds also stand out as having larger uncertainties on the whole.",
"Figures REF and REF show our geometric and photogeometric distances and their uncertainties for members of various star clusters.",
"The membership lists have been drawn from paper IV.",
"NCG6254 (= M10) and NGC6626 (= M28) are globular clusters; the rest are open clusters.",
"Recall that our prior does not include star clusters.",
"The horizontal dashed line in each panel shows the inverse of the variance-weighted mean parallax of the members, i.e.",
"a pure parallax distance for the cluster.",
"Both of our distance estimates congregate around this for small, positive fpu, but deviate for large or negative fpu, as one would expect.",
"We generally see a larger deviation and/or scatter for the geometric distance: compare in particular the panels for NGC2437 (=M46) and NGC6254.",
"Despite this, the weighted mean of our distances is often quite close to the pure parallax distance, even for clusters up to several kpc away.",
"We nevertheless emphasise that the variance-weighted mean parallax distance will usually be a better estimate for the distance to a cluster than the mean of the distances.",
"This is because any combination of our individual distances will re-use the same prior many times.",
"If stars have large fpus, this product of priors will dominate and introduce a strong bias into the combined distance.",
"This would particularly affect clusters beyond a few kpc."
],
[
"Comparison to other distance estimates",
"Figure REF compares our distance estimates for 36 858 red clump (RC) stars with those estimated by [7] using high-resolution APOGEE [25] DR16 spectra.",
"This method selects sources using colour, effective temperature, metallicity, and surface gravity, and is calibrated via stellar evolution models and high-quality asteroseismology data.",
"Given the narrowness of the red clump locus in the parameter space, their distances are expected to be precise to 5% with a bias of no more than 2%.",
"The 5th, 50th, and 95th percentiles of fpu for this sample are 0.01, 0.05, and 0.27 respectively, and of $G$ are 10.4, 13.4, and 16.2 mag respectively.",
"The fractional bias and rms of the deviations of our estimates relative to those of Bovy et al.",
"are $+0.05$ and 0.31 respectively for the geometric distances, and $+0.03$ and 0.29 respectively for the photogeometric distances.",
"For reference, the fractional bias and rms of the deviations of the APOGEE red clump estimates relative to the StarHorse [29] estimates (see next paragraph) for the same sample are $+0.05$ and 0.21 respectively.",
"The parallaxes for this sample are mostly of such high quality that the prior does not strongly effect our posteriors, although we still see a slight improvement in the photogeometric distances over the geometric ones.",
"When counting the percentage of sources where the Bovy et al.",
"estimate is within our upper and lower bounds (+ 7% error margin from Bovy et al.)",
"we find that 65% are compatible with the geometric distances and 69% with photogeometric (we expect 68% to be within 1$\\sigma $ ).",
"If we do the same for the StarHorse estimates (which also have upper and lower percentiles) for the red clump sample we see that 84% of the StarHorse estimates are within 1$\\sigma $ pf the Bovy et al.",
"estimates.",
"Figure: Comparison of StarHorse distance estimates fromto our geometric estimates (top panel) and to our photogeometric estimates (bottom panel) for a common sample of 307 105 sources.Figure REF compares our distance estimates for 307 105 stars with those estimated by [29] using their StarHorse method, which uses APOGEE DR16 spectra, multiband photometry, and GDR2 parallaxes.",
"This sample comprises around $1/3$ main sequence stars; the rest are turnoff star and giants, excluding the red clump stars used in the previous comparison.",
"StarHorse estimates a posterior probability distribution which the authors likewise summarize with a median, so our distance estimates are directly comparable.",
"They report achieving typical distance uncertainties of 11% for giants and 5% for dwarfs.",
"The 5th, 50th, and 95th percentiles of fpu for this sample are 0.002, 0.02, and 0.46 respectively, and of $G$ are 10.2, 13.3, and 16.6 mag respectively.",
"The fractional bias and rms of the deviations of our distance estimates relative to the StarHorse estimates are $0.00$ and 0.30 respectively for the geometric distances, and $-0.01$ and 0.23 respectively for the photogeometric distances.",
"As this sample extends to larger distances (and larger fpu) than the sample in Figure REF , we begin to see that our geometric distances (and to a lesser extent our photogeometric distances) are smaller than the Starhorse distances beyond about 6 kpc, which is where some of the large fpu sources will have true distances beyond the median of the distance prior."
],
[
"Content",
"The distance catalogue includes an entry for all 1 467 744 818 sources in EDR3 that have a parallax.",
"All of these have geometric distances and 92% have photogeometric distances.",
"In comparison there are 1 347 293 721 sources in EDR3 that have defined G-band magnitudesBy this we mean the phot_g_mean_mag field is defined.",
"We do not make use of the other estimates of $G$ from the Gaia catalogue if this field is null., ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ colours, and parallaxes, and so could in principle have received a photogeometric distance estimate, but did not due to missing $Q_G$ prior models.",
"The fields in our catalogue are defined in Table REF .",
"3% of the sources have changed their source_id identifier from GDR2 to EDR3 [10], so the source_id cross-match table dr2_neighbourhood provided with EDR3 should be used to find the best match before doing source-by-source comparisons between the two releases.",
"r_med_geo in Table REF is the median ($r_{\\rm med}$ ) of the geometric distance posterior and should be taken as the geometric distance estimate.",
"r_lo_geo ($r_{\\rm lo}$ ) and r_hi_geo ($r_{\\rm hi}$ ) are the 16th and 84th percentiles of the posterior and so together form a 68% confidence interval around the median.",
"$r_{\\rm hi}- r_{\\rm med}$ and $r_{\\rm med}- r_{\\rm lo}$ are therefore both 1$\\sigma $ -like uncertainties on the distance estimate, and are generally unequal due to asymmetry of the posterior.",
"The fields r_med_photogeo, r_lo_photogeo, and r_hi_photogeo are defined in the same way for the photogeometric distance posterior.",
"We cannot overstate the importance of the uncertainties provided.",
"They reflect the genuine uncertainty in the distance estimate provided by the median.",
"As $r_{\\rm hi}-r_{\\rm lo}$ is a 68% confidence interval, we expect the true distance to lie outside of this range for a third of the sources.",
"This is the nature of statistical uncertainty and should never be ignored.",
"The field flag is a string of five decimal digits defined in Table REF .",
"Flag A is set to 2 if the source is fainter than the faintest mock source used to make the prior for that HEALpixel.",
"The estimated distances can still be used.",
"Faint stars tend to have poor parallaxes so the distance uncertainties will generally be larger in these cases.",
"The two digits of flag B refer to the Hartigan dip test, as explained in section REF .",
"We find that 2% of geometric posteriors and 3% of photogeometric posteriors may not be unimodal according to this test, although this test is not particularly accurate, so this is only a rough guide.",
"Even when the sampled posterior shows a true, significant bimodality (or even multimodality), the 68% confidence interval sometimes spans all modes.",
"The two digits of flag C indicate the nature of the two $Q_G$ models that were used to construct the $Q_G$ prior.",
"If both numbers are between 1 and 3 then two models bracket the source's colour and were combined by linear interpolation, as explained in section REF .",
"If only one of them is 0 then only a single model was used.",
"If both flags are 0 then there is no non-null model within 0.1 mag colour of the source, so the photogeometric posterior is not computed.",
"There are is one special value of this flag: 99 means the star lacked the necessary data to compute the photogeometric distance.",
"We provide additional information on the prior for each HEALpixel in the auxiliary information online, including plots like Figures REF , REF , and REF , and a table with the three parameters of the geometric prior (equation REF )."
],
[
"Filtering",
"We have not filtered out any results from our catalogue.",
"Parallaxes with spurious parallaxes remain, as do sources with negative parallaxes (the latter is no barrier to inferring a sensible distance; [5]).",
"Any filtering should be done with care, as it often introduces sample biases.",
"The flag field we provide is for information purposes; we do not recommend to use it for filtering.",
"Lower quality distances will arise from lower quality input data.",
"These can be identified using the various quality fields in the main Gaia catalogue of EDR3, which is easily cross-matched to our catalogue using the source_id field, as shown in the example in section REF .",
"Useful quality metrics may be ruwe, parallax_over_error, and astrometric_excess_noise, as defined in the EDR3 documentation, where users will also find advice on their use.",
"See in particular section 3.2 of [10] for suggestions for filtering spurious parallaxes.",
"Parallaxes from the 6p astrometric solutions (identified by astrometric_params_solved = 95) are not as accurate as those from the 5p solutions [23], because they were normally used in more problematic situations, such as crowded fields, and are also fainter on average than the 5p solutions.",
"Sources with 6p solutions should not be automatically removed, however.",
"Their larger parallax uncertainties reflect their lower quality.",
"In some applications users may want to filter out sources with large absolute or relative distance uncertainties.",
"One must exercise caution here, however, because uncertainty generally correlates with distance and/or magnitude (among other things), so filtering on these quantities will introduce sample biases."
],
[
"Use cases",
"For stars with positive parallaxes and $\\sigma _{\\varpi }/\\varpi < 0.1$ , the inverse parallax is often a reasonably good distance estimate for many purposes (when using a suitable parallax zeropoint).",
"This applies to 98 million sources in EDR3.",
"For sources with negative parallaxes or $\\sigma _{\\varpi }/\\varpi > 1$ (704 million sources), our distances will generally be prior dominated, and while the photogeometric distances could still be useful, the geometric ones are probably less so.",
"The sweet spot where our catalogue adds most value is for the remaining 665 million sources with $0.1 < \\sigma _{\\varpi }/\\varpi < 1$ .",
"The choice of whether to use our geometric or photometric distance depends on the specific situation and what assumptions you are willing to accept.",
"In the limit of negligible parallax uncertainties they will agree.",
"At large fractional parallax uncertainties our photogeometric distances will generally be more precise than geometric ones, because they use more information and have a stronger prior (see Figures REF and REF ).",
"Whether they are also more accurate depends on how well the $Q_G$ prior matches to the true (but unknown) $Q_G$ distribution.",
"The $Q_G$ model reflects the stellar population and interstellar extinction in a small patch of sky (HEALpixel of area 3.36 sq. deg).",
"The GeDR3mock catalogue and our prior should model these reasonably well at higher Galactic latitudes, but may be less accurate at lower latitudes where extinction is higher and the stellar populations along the line-of-sight are more complicated.",
"If you do not want to rely on colour and magnitude information in the distance inference, use the geometric distance, as the distance prior is less sensitive to the exact stellar population in GeDR3mock.",
"Some example use cases are as follows.",
"Look-up of distance (or distance modulus) for particular sources of interest using their source_id or other identifier matched to this.",
"EDR3 includes a crossmatch to many existing catalogues.",
"Positional crossmatches can also be done on the EDR3 data site or using TAP uploads, and at other sites that host our catalogue.",
"Identification of sources within a given distance (or distance modulus) range.",
"The confidence intervals should be used to find all sources with a distance $r$ satisfying $k(r_{\\rm med}- r_{\\rm lo}) < r< k(r_{\\rm hi}- r_{\\rm med})$ , where the size of $k$ will depend on the desired balance between completeness and purity of the resulting sample.",
"A better approach would be to use the actual posterior to get a probability-weighted sample.",
"For the geometric distances our posterior can be reconstructed using the geometric distance prior provided for each HEALpixel in the auxiliary information online.",
"Readers interested in using our photogeometric priors should contact the authors.",
"Construction of absolute-colour-magnitude diagrams.",
"One of the reasons that we provide quantiles for our distance estimates is that $5\\log _{\\rm 10}(r_{\\rm med})-5$ is the median of the distance modulus posterior.",
"(This would not be the case if we provided the mean or mode, for example.)",
"Using $G$ from EDR3 one can then compute $Q_G$ , and from this the absolute magnitude $M_G$ , if the extinction is zero or otherwise known.",
"The same can be done for any photometric band from any other catalogue.",
"When computing $Q_G$ in this way with equation REF , the user should remember to apply the correction to the EDR3 G-band magnitude as described in section 8.3 of [30].",
"For constructing the three-dimensional spatial distribution of stars in some region of space.",
"This may also assist selection of candidates in targeted follow-up surveys.",
"As a baseline for comparison of distance or absolute magnitude estimates obtained by other means.",
"Our distances could be used for another layer of inference, such as computing transverse velocities using also the EDR3 proper motions, although users will need to consider the appropriate error propagation.",
"In particular, if the error budget is not dominated by a single source (e.g.",
"not just the distance), users are advised to infer their desired quantities directly from the original parallaxes, perhaps using the priors provided here.",
"Users should realise that uncertainties in the parallaxes in EDR3 are correlated between different sources to a greater or less degree depending on their angular separations [23], [10].",
"Caution must therefore be exercised when combining either the parallaxes or our distances, e.g.",
"averaging them to determine the distance to a star cluster.",
"In such a case the simple “standard error in the mean\" may underestimate the true uncertainty, and the same prior would be used multiple times.",
"One should instead set up a joint likelihood for the sources that accommodates the between-source correlations and solve for the cluster distance directly."
],
[
"Access",
"Our distance catalogue is available from the German Astrophysical Virtual Observatory at http://dc.g-vo.org/tableinfo/gedr3dist.main where it can be queried via TAP and ADQL.",
"This server also hosts a reduced version of the main Gaia EDR3 catalogue (and GeDR3mock).",
"Typical queries are likely to involve a join of the two catalogues.",
"By way of example, the following query returns coordinates, our distances, ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ , and the two $Q_G$ values using the median distances, for all stars with a low ruwe in a one-degree cone in the center of the Pleiades.",
"This should run in about one second and return 22 959 sources.",
"SELECT source_id, ra, dec, r_med_geo, r_lo_geo, r_hi_geo, r_med_photogeo, r_lo_photogeo, r_hi_photogeo, phot_bp_mean_mag-phot_rp_mean_mag AS bp_rp, phot_g_mean_mag-5*LOG10(r_med_geo)+5 AS qg_geo, phot_g_mean_mag-5*LOG10(r_med_photogeo)+5 AS gq_photogeo FROM gedr3dist.main JOIN gaia.edr3lite USING (source_id) WHERE ruwe<1.4 AND DISTANCE(ra, dec, 56.75, 24.12)<1 A bulk download for the catalogue is also available at the URL given above.",
"Our catalogue will also become available soon together with the full EDR3 catalogue hosted at https://gea.esac.esa.int/archive/ and its partner data centers.",
"At these sites the table names gedr3dist.main and gaia.edr3lite may well be different."
],
[
"Limitations",
"When using our catalogue users should be aware of its assumptions and limitations.",
"We summarize the posteriors using only three numbers (quantiles), which cannot capture the full complexity of these distributions.",
"This is more of a limitation for the photogeometric posteriors.",
"The confidence intervals should not be ignored.",
"Most sources in EDR3 have large fractional parallax uncertainties and our distances correspondingly have large fractional uncertainties, especially for the geometric distances.",
"The poorer the data, the more our prior dominates the distance estimates.",
"Our prior is built using a sophisticated model of the Galaxy that includes 3D extinction, but it will not be perfect.",
"If the true stellar population, extinction, or reddening law are very different in reality, our distances will be affected.",
"In section REF we explained, using results on simulated data, what biases can occur and why.",
"Sources with very large parallax uncertainties will have a posterior dominated by the prior.",
"The median of this varies between 745 and 7185 pc depending on HEALpixel (Figure REF ).",
"Stars with large fpus that truly lie well beyond the prior's median will have their geometric distances underestimated; stars with large fpus that lie closer than the prior's median will have their geometric distances overestimated.",
"As distant stars generally have larger fpu than nearby stars, and distant stars are more numerous, the former characteristic will dominate among poor quality data.",
"This leads to a bias in distance estimates, one that is probably unavoidable (see appendix ).",
"Poor data remain poor data.",
"Our prior is spatially discretized at HEALpixel level 5, i.e.",
"in patches of 3.36 sq. deg.",
"on the sky.",
"The distance prior and CQD change discontinuously between HEALpixels, and this may be visible in sky maps of posterior distances.",
"The $Q_G$ priors (constructed from the CQD) are formed by a linear interpolation over colour whenever possible, so in these cases there should be no discontinuity of distance with colour within a HEALpixel.",
"Our inferred distances retain all of the issues affecting the parallaxes, some of which have been explored in the EDR3 release papers [23], [10].",
"We applied the parallax zeropoint correction derived by [22], which is better than no correction or a single global correction, but is not perfect.",
"Any error in this will propagate into our distance estimates.",
"The published parallax uncertainties are also probably also underestimated to some degree [10].",
"[15] and [30] report some issues with the EDR3 photometry, such as biased BP photometry and therefore ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ colours for very faint sources, which could affect our photogeometric distances.",
"These distance estimates additionally suffer from any mismatch between the published EDR3 photometry and the modelling of this – in particular the passbands – used in the GeDR3mock catalogue, which forms the basis for our $Q_G$ priors.We compared simulations of the G-band magnitude and the BP-RP colour between the GeDR3mock passbands and those published for EDR3, using isochrones at 4 Myr and 1 Gyr.",
"The differences in the G magnitudes are below 6 mmag, except for sources bluer than $-0.15$ , where it can be as high as 700 mmag.",
"For ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ using the BP bright band (in GeDR3mock), the difference is around 10 mmag, but up to 25 mmag for sources with ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}> 1.2$ mag and up to 100 mmag for sources with ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}< -0.2$ mag.",
"For BP faint, the ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ difference is around 20 mmag, but up to 60 mmag for sources with ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}> 0.5$ mag and up to 100 mmag for sources with ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}< -0.15$ mag.",
"Note that we applied the G-band magnitude correction to the EDR3 photometry as described by [30].",
"We implicitly assume that all sources are single stars in the Galaxy.",
"Our distances will be incorrect for extragalactic sources.",
"The geometric distances will be wrong for unresolved binaries if the parallax for the composite source is affected by the orbital motion.",
"Even when this is not the case the photogeometric distance may still be wrong, because the G-band magnitude will be brighter than the $Q_G$ prior expects (binaries were not included in the prior).",
"By design we infer distances for each source independently.",
"If a set of stars is known to be in cluster, and thus have a similar distance, this could be exploited to infer the distances to the individual stars more accurately than we have done here.",
"In its most general form this involves a joint inference.",
"Various methods exist in the literature for doing this, such as [28], [9], and [27]."
],
[
"Summary",
"We have produced a catalogue of geometric distances for 1.47 billion stars and photogeometric distances for 92% of these.",
"These estimates, and their uncertainties, can also be used as estimates of the distance modulus.",
"Geometric distances use only the EDR3 parallaxes.",
"Photogeometric distances additionally use the $G$ magnitude and ${\\rm BP} \\!",
"- \\!",
"{\\rm RP}$ colour from EDR3.",
"Both types of estimate involve direction-dependent priors constructed from a sophisticated model of the 3D distribution, colours, and magnitudes of stars in the Galaxy as seen by Gaia, i.e.",
"accommodating both interstellar extinction and a Gaia selection function.",
"Tests on mock data, but moreover validation against independent estimates and open clusters, suggest our estimates are reliable out to several kpc.",
"For faint or more distant stars the prior will often dominate the estimates.",
"We have identified various use cases and limitations of our catalogue.",
"Our goal has been one of inclusion: to provide distances to as many stars in the EDR3 catalogue as possible.",
"This has required us to make broad, general assumptions.",
"If one focuses on a restricted set of stars with some approximately known properties, it will be possible to construct more specific priors, and to use these to infer more precise and more accurate distances.",
"Better distances may also be achievable by using additional data, such as spectroscopy or additional photometry.",
"We thank the IT departments at MPIA and ARI for computing support.",
"This work was funded in part by the DLR (German space agency) via grant 50 QG 1403.",
"It has made use of data from the European Space Agency (ESA) mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium).",
"Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.",
"This research made use of: TOPCAT, an interactive graphical viewer and editor for tabular data [36]; Vaex, a tool to visualize and explore big tabular data [8]; matplotlib, a Python graphics library [20]; HEALpix [17] and healpy [37]; the NASA Astrophysics Data System; the VizieR catalogue access tool, CDS, Strasbourg.",
"Gaia"
],
[
"Thoughts on a better distance prior",
"The strong dependence of the geometric posterior on the distance prior in the limit of large parallax uncertainties is an unavoidable consequence of inference with noisy data.",
"We saw something similar in paper IV.",
"This leads to a distance bias mostly for distant stars with large fpu.",
"Could this be avoided?",
"Conceptually one would like a distance prior that depends on the true fpu, but this is impossible because the true parallax is not known.",
"One may be tempted to use the measured fpu instead, but this is not what we want: a star with a large true fpu could have a small measured fpu due to noise, and thereby be treated incorrectly.",
"Its use is also be theoretically dubious because it places the parallax – a measurable – in the prior, as well as in the likelihood.",
"We experimented with using a prior conditioned on $\\sigma _{\\varpi }$ , but found that this did not help (see the technical note GAIA-C8-TN-MPIA-CBJ-089 with the auxiliary information online).",
"One may achieve something close to what is desired by simply shifting the distance prior to greater distances, so that it better represents stars with a larger true fpu, which is where the prior is needed more.",
"Yet this would detrimentally affect the distance estimates for nearby stars.",
"It seems a poor trade-off to sacrifice accuracy on high-quality data for a better prior on low-quality data.",
"Conditioning the prior on the star's magnitude may help, and this is what our photogeometric distances do (section REF )."
],
[
"The limit of poor parallaxes",
"We tend to think that a large fpu means that the likelihood is uninformative and that the posterior converges towards the prior.",
"Consider a red clump star in the LMC with a true parallax of 0.02 mas and a typical parallax uncertainty of 0.2 mas for a star with $G= 19$ mag.",
"The true fpu is $0.2/0.02=10$ .",
"Let's assume initially that we actually measure a parallax of 0.02 mas, i.e.",
"we have an measured fpu of 10.",
"(Of course in this lucky case the inverse parallax would be the correct distance, but it's very rare in practice.)",
"In the LMC HEALpixel 8275 our distance prior has a median of 1.2 kpc because we exclude the LMC from our prior, so we might expect to see many sources with this inferred distance.",
"In fact we see many sources with larger inferred distances (see the plot with the auxiliary online information).",
"The reason is that the likelihood of a measurement of 1 mas (corresponding to a distance of 1 kpc) is still at 4.9 $\\sigma $ and therefore quite unlikely.",
"This shows that even when the fpu is large the parallax can be quite informative.",
"One should remember, however, that our inference never sees the true parallax but only the measured parallax, which is normally distributed around the unknown true value (with a standard deviation which is also only estimated).",
"So it is quite likely that our measurement of the above red clump star gives us a parallax measurement of, say, 0.4 mas.",
"In that case the measured fpu is 0.5 and the likelihood of 1 mas, i.e.",
"a 1 kpc distance, is only 3$\\,\\sigma $ away from this measurement.",
"Taking the parallax measurement into account essentially redistributes probability mass into the wings of the likelihood and therefore to higher and lower (also negative) parallax values.",
"Given the truncation of negative parallaxes when calculating the posterior, this implies that the median distance estimate is lower for the true measurements, compared to the idealised inference using the true parallax.",
"Similarly, one should be careful not to interpret plots involving the measured fpu as though it were the true fpu."
]
] | 2012.05220 |
[
[
"Tactile Object Pose Estimation from the First Touch with Geometric\n Contact Rendering"
],
[
"Abstract In this paper, we present an approach to tactile pose estimation from the first touch for known objects.",
"First, we create an object-agnostic map from real tactile observations to contact shapes.",
"Next, for a new object with known geometry, we learn a tailored perception model completely in simulation.",
"To do so, we simulate the contact shapes that a dense set of object poses would produce on the sensor.",
"Then, given a new contact shape obtained from the sensor output, we match it against the pre-computed set using the object-specific embedding learned purely in simulation using contrastive learning.",
"This results in a perception model that can localize objects from a single tactile observation.",
"It also allows reasoning over pose distributions and including additional pose constraints coming from other perception systems or multiple contacts.",
"We provide quantitative results for four objects.",
"Our approach provides high accuracy pose estimations from distinctive tactile observations while regressing pose distributions to account for those contact shapes that could result from different object poses.",
"We further extend and test our approach in multi-contact scenarios where several tactile sensors are simultaneously in contact with the object.",
"Website: http://mcube.mit.edu/research/tactile_loc_first_touch.html"
],
[
"Introduction",
"Robotics history sends a clear lesson: accurate and reliable perception is an enabler of progress in robotics.",
"From depth cameras to convolutional neural networks, we have seen how advances in perception foster the development of new techniques and applications.",
"For instance, the invention of high-resolution LIDAR fueled self-driving cars, and the generalization capacity of deep neural networks has dominated progress in perception and grasp planning in warehouse automation [1], [2], [3].",
"The long term goal of our research is to understand the key role that tactile sensing plays in that progress.",
"In particular, we are interested in robotic manipulation applications, where occlusions difficult accurate object pose estimation, and where behavior is dominated by contact interactions.",
"Figure: Tactile pose estimation.",
"(Bottom row) In simulation, we render geometric contact shapes of the object from a dense set of possible contacts between object and tactile sensor.",
"(Top row) The real sensor generates a tactile image from which we estimate its geometric contact shape.",
"We then match it against the simulated set of contact shapes to find the distribution of contact poses that are more likely to have generated it.",
"For efficiency and robustness, we do the contact shape matching in an embedding learned for that particular object.In this paper, we propose a framework to estimate the pose of a touched object, as illustrated in Figure REF .",
"Given a 3D model of the object, our method learns an object-specific perception model in simulation, tailored at estimating the pose of the object from one–or possibly multiple–tactile images.",
"As a result, the approach localizes the object from the first touch, i.e., without requiring any previous interaction.",
"The perception model is based on merging two key ideas: Geometric contact rendering: we use the object model to render the contact shapes that the tactile sensor would observe for a dense set of contact poses.",
"Contact shape matching: given the estimated contact shape from a tactile observation, we match it against the precomputed dense set of simulated contact shapes.",
"The comparison happens in an object-specific embedding for contact shapes learned in simulation using contrastive learning tools [4].",
"This provides robustness and speed compared to other methods based on direct pixel comparisons.",
"Accounting by the discriminative power of tactile sensing, the proposed approach is motivated by scenarios where the main requirement is estimation accuracy and where object models will be available beforehand.",
"Many industrial scenarios fit this category.",
"Most previous solutions to tactile pose estimation require prior exploration of the object [5], [6].",
"Acquiring this tactile experience can be expensive, and in many cases, unrealistic.",
"In this paper, instead, we learn the perception model directly from the object geometry.",
"The results in Sec.",
"show that the model learned in simulation directly transfers to the real world.",
"We attribute this both to the object-specific nature of the learned model and to the high-resolution nature of the tactile sensors used.",
"Also, key to the approach is that, by simulating a dense set of tactile imprints, the algorithm can reason over pose distributions, not only the best estimate.",
"The learned embedding allows us to efficiently compute the likelihood of each contact shape in the simulated dense set to match with the predicted contact shape from the tactile sensor.",
"This results in a probability distribution over object poses rather than just a single pose estimate.",
"Predicting distributions is key given that tactile sensing provides local observations, which sometimes might not be sufficiently discriminative.",
"Finally, by maintaining probability distributions in pose space, we can incorporate extra constraints over the likelihood of each pose.",
"We illustrate it in the case of multi-contact, where information from multiple tactile observations must be combined simultaneously.",
"By operating in a discretization of the pose space, the framework can potentially handle other pose constraints including those coming from other perception systems (e.g., vision), previous observations, or kinematics.",
"In summary, the main contribution of this work is a framework for tactile pose estimation for objects with known geometry, with the following primary strengths: 1.",
"Provides accurate pose estimation from the first touch, without requiring any previous interactions with the object.",
"2.",
"Reasons over pose distributions by efficiently computing probabilities between a real contact shape and a dense set of simulated contact shapes.",
"3.",
"Integrates pose constraints, such as those arising from multi-contact scenarios where multiple observations and sensor poses must be considered."
],
[
"Related Work",
" Tactile perception has been extensively explored in the robotics community.",
"Relevant to this paper, this has resulted in the development of high-resolution tactile sensors and their use in a wide range of robotic manipulation applications.",
"In this section, we review works that study tactile pose estimation and refer the reader to [7] for a more in-depth review of tactile applications.",
"While we propose to use high-resolution tactile sensors that are discriminative and rich in contact information, most initial works in tactile localization were meant for low-resolution tactile sensors [8], [9], [10], [11], [12], [13], [14], [15].",
"Moreover, in many cases, the sensors are bulky, or the objects considered are planar or consist of simple geometries.",
"Finally, some works explore how to combine multiple tactile readings and reason in the space of contact manifolds [16], [17].",
"However, these are still based on low-resolution tactile feedback, often a binary contact/no-contact signal, and require many tactile readings to narrow pose estimates.",
"Given the challenges from the locality of tactile sensing, recent works have gravitated towards two different approaches.",
"Combining tactile and vision to obtain better global estimates of the object pose or using higher-resolution tactile sensors that can better discriminate different contacts.",
"Among the solutions that combine vision and tactile, most rely on tactile sensors as binary contact detectors whose main purpose is to refine the predictions from vision [18], [19], [20], [21], [22].",
"Other works, more in line with our approach, have focused on using high-resolution tactile sensors as the main sensing source for object localization.",
"Initial works in this direction used image-based tactile sensors to recover the contact shape of an object and then use it to filter the object pose [23], [24], [25].",
"However, these approaches only provide results on planar objects and require previous tactile exploration.",
"There has also been some recent work on highly deformable tactile sensors for object localization [26].",
"These sensors are large enough to fully cover the touched objects, which eases localization but limits any complex manipulation.",
"In this work, we use the image-based tactile sensor GelSlim [27].",
"The sensing capabilities of these high-resolution sensors have already proven useful in multiple robotic applications, including assessing grasp quality [28], improving 3D shape perception [29] or directly learning from tactile images how to do contour following [30] or tactile servoing [31].",
"For the task of tactile object localization, [5] proposed to extract local contact shapes from objects to build a map of the object and then use it to localize new contacts.",
"The approach is meant to deal with small parts with discriminative features.",
"Later [32] proposed to compute pointclouds from the sensor and use them to complement a vision-based tracker.",
"Their tracker is unimodal and cannot deal with the uncertainty that arises from the locality of tactile sensing.",
"Finally, in previous work [6] we proposed to extract local contact shapes from the sensors and match them to the tactile map of the objects to do object pose estimation.",
"This approach requires the estimation of a tactile map for each object by extensively exploring them with the sensor.",
"In comparison, our approach moves all object-specific computations to simulation and only requires an object-agnostic calibration step of the tactile sensor to predict contact shapes.",
"As a result, we can render in simulation contact shapes and learn object-specific models for pose estimation that translate well to the real world and achieve good accuracy from the first contact.",
"Finally, our approach to tactile pose estimation is related to methods recently explored in the computer vision community where they render realistic images of objects and learn how to estimate the orientation of an object given a new image of it [33], [34].",
"While in vision, the most likely estimate is often sufficient, in tactile sensing, different object poses are more likely to produce the same observation.",
"To address this problem, we explicitly reason over pose distributions when assessing the performance of our perception models."
],
[
"Method",
"We present an approach to object pose estimation based on tactile sensing and known object models; illustrated in Fig.",
"REF .",
"In an object-specific embedding, we match a dense set of simulated contact shapes against the estimated contact shape from a real tactile observation.",
"This results in a probability distribution over contact poses that can be later refined using other pose constraints or registration techniques.",
"The algorithm starts from a geometric model of the object, and a description of the local geometry of a region of the object, a.k.a.",
"contact shape, captured by a tactile sensor.",
"In the case of this paper, we predict real contact shapes directly from the raw tactile images that the sensor outputs (Sec.",
"REF ).",
"The next steps of our approach exploit the object model to estimate the object pose and are learned in simulation without using any real tactile observations.",
"First, we develop geometric contact rendering, an approach to simulate/render contact shapes in the form of images using the object model (Sec.",
"REF ).",
"Next, we generate a dense set of contact poses and their respective contact shapes, and use contrastive learning to match contact shapes depending on the closeness of their contact poses (Sec.",
"REF ).",
"As a result, given the estimated contact shape from a real tactile observation, we can match it against this pre-computed dense set to obtain a probability distribution over contact poses.",
"To predict contact poses beyond the resolution of the pre-computed set, we combine our approach with registration techniques on the contact shapes (Sec.",
"REF ).",
"Finally, we show our perception model is not restricted to single tactile observations and can handle multi-contact scenarios, and additional pose constraints (Sec.",
"REF )."
],
[
"Contact shape prediction from tactile observations",
"Given a tactile observation, our goal is to extract the contact shape that produces it.",
"To that aim, we train a neural network (NN) that maps tactile observations to contact shapes following the approach we proposed in [6].",
"The input to the NN is a normalized rescaled RGB tactile image of size 200x200.",
"The output corresponds to a normalized one-channel depth image of size 200x200 that represents the contact shape.",
"The training data is collected autonomously in a controlled 4-axis stage that generates controlled touches on known 3D-printed shapes.",
"Note that, for each tactile sensor, we only need to gather calibration data once because the map between tactile observations and contact shapes is object-independent.",
"We provide further implementation details in Sec.",
"of the appendix."
],
[
"Contact shape rendering in simulation",
"Given the geometric model of an object and its pose w.r.t.",
"the sensor, we use rendering techniques to simulate local contact shapes from object poses.",
"We refer to this process as geometric contact rendering.",
"Below, we describe how we compute object poses w.r.t.",
"the sensor that would result in contact without penetration, and their associated contact shapes: We create a rendering environment using the open-source library Pyrender [35].",
"In this environment, we place a virtual depth camera at the origin looking in the positive z-axis.",
"The sensor can be then imagined as a flat surface (a rectangle in our case) orthogonal to the z-axis and at a positive distance $d$ from the camera.",
"We place the object in any configuration (6D pose) such that all points in the object are at least at a distance $d$ in the z-axis from the origin.",
"Next, we compute the smallest translation in the z-axis that would make the object contact the surface that represents the sensor.",
"Finally, we move the object accordingly and render a depth image.",
"Its smallest pixel value corresponds to a depth $d$ , and we consider that only pixels between distances $d$ and $d + \\Delta d$ are in contact with the sensor.",
"The rest are marked as non-contact.",
"The resulting depth image corresponds to the simulated contact shape of the object at that particular pose.",
"As a result, given a contact pose, we can easily compute its corresponding simulated contact shape by rendering a depth image from the object mesh.",
"For our sensor, depth images have a width and height of 470x470 (later rescaled to 200x200 for faster to compute), the distance to the origin is $d=25$ mm, and the contact threshold is $\\Delta d = 2$ mm.",
"The intrinsic parameters of the virtual camera are $f_x =291.5$ , $f_y = 289$ , $c_x = 235$ and $c_y = 235$ .",
"Figure: Similarity function.",
"We build a similarity function that learns to encode contact shapes into a low dimensional space and predicts, given a new contact shape, the likelihood of being the closest match of each contact shape in the pre-computed set.",
"By learning an encoder for the contact shapes, we can compare them very efficiently."
],
[
"Global tactile pose estimation",
"Once we know how to compute contact shapes both in simulation and from real tactile imprints, we reduce the problem of object pose estimation to finding what poses are more likely to produce a given contact shape.",
"We solve this problem by first discretizing the space of possible contact poses as a parametrized grid, and then learning a similarity function that compares contact shapes.",
"Object-dependent grids.",
"Using the 3D model of an object, we discretize the space of object poses in a multidimensional grid.",
"Building a grid in the space of poses is a well-studied problem [36], [37] that makes finding nearby neighbors trivial.",
"It also allows each point on the grid to be seen as the representative of a volumetric part of the space which helps to reason over distributions.",
"We prune the grid by only keeping the poses that result in contact, and pair each of them with their respective contact shapes.",
"Since we only consider poses that result in contact, the z-dimension is fully determined by this condition, which results in 5-dimensional grids.",
"Using a discrete structured set of poses allows us to easily account for object symmetries which can significantly reduce the grid size.",
"Similarity metric for contact shapes.",
"Given a new contact shape, we want to compare it to all pre-computed contact shapes in the grid to find what poses are more likely to produce it.",
"To that aim, we modify MoCo [4], a state-of-the-art algorithm in contrastive learning, to encode contact shapes into a low dimensional embedding based on their pose distance.",
"Compared to the original MoCo algorithm, the elements in the queue are fixed and assigned to each of the poses in the object's grid.",
"Given a new contact shape, our model predicts the likelihood that each pose in the grid has produced the given shape.",
"To implement the encoder, we also use a ResNet-50 [38], but cropped before the average-pooling layer to preserve spatial information, making it a fully-convolutional architecture.",
"The loss function is the Cross-Entropy loss which allows us to predict probabilities.",
"The training data comes from selecting a random contact pose and finding its closest element in the dense grid.",
"Then, we use as desired probabilities a vector of all zeros except for the closest element which gets assigned to probability one (see Fig.",
"REF ).",
"Sec.",
"in the appendix contains further details on the learning method.",
"Once we have created a dense grid and trained a similarity encoder for an object, given a new contact shape, we can estimate which poses from the grid are more likely to generate it.",
"To run our method in real-time, we first encode the given contact shape (0.0075s, 133Hz) and then compare it to all pre-computed encodings from the grid, which requires a single matrix-vector multiplication.",
"Finally, we perform a softmax over the resulting vector to obtain a probability distribution over contact poses"
],
[
"Contact shape refinement",
"To avoid limiting pose estimations by the resolution of the grid, we refine the most likely poses and their contact shapes to better match the contact shape we are interested in.",
"Because contact shapes come from depth images, we can convert them to pointclouds and use registration techniques to find the best transformation between them.",
"We evaluated several pointcloud registration algorithms in simulation and concluded that FilterReg [39], a state of the art registration algorithm twice as fast as ICP, performed the best even with just one iteration.",
"Applying FilterReg between two contact shapes takes less than 0.01s (100Hz)."
],
[
"Multi-contact pose estimation",
"In this section, we show how to extend our approach to multi-contact settings where we simultaneously need to reason over several tactile observations.",
"In Sec.",
"in the appendix we show that the likelihood of an object pose from the grid, $x$ , given the estimated contact shapes, $CS_{1,..., N}$ , from N sensors is proportional to $P(x | CS_1, ... , CS_N) \\propto P(x | CS_1) \\cdot ... \\cdot P(x | CS_N) $ when there are no priors on object pose, i.e., all elements in the grid are equally likely, and the embedding network has been trained using uniformly-sampled poses.",
"$P(x | CS_i)$ is the likelihood that pose $x$ produces the contact shape $CS_i$ on sensor $i$ .",
"When a prior over poses is available, we can add a prior term, $P_{task}(x)$ , to compute $P(x | CS_1, ... , CS_N)$ (see appendix Sec.",
"REF ).",
"This allows combining our approach with additional pose constraints such as the ones coming from kinematics, previous tactile observations, or other perceptions systems.",
"The terms $P(x | CS_i)$ come directly from computing the similarity function between $CS_i$ and the contact shape from the grid of sensor $i$ that is closest to the contact pose $x$ .",
"As a result, the computational cost of considering multiple contacts scales linearly with the number of sensors."
],
[
"Real data collection",
"While most computations of the algorithm are done in simulation, the end goal of our approach is to provide accurate pose estimation in the real world.",
"To that aim, we designed a system that collects tactile observations on accurately-controlled poses.",
"Below we describe the tactile sensor, the robot platform, and the objects used to perform the experiments.",
"Tactile sensor.",
"We consider the tactile sensor GelSlim [27] which provides high-resolution tactile readings.",
"The sensor consists of a membrane that deforms when contacted and a camera that records the deformation.",
"The sensor publishes tactile observations through ROS as 470x470 compressed images at a frequency of 90Hz.",
"Some regions of the image are masked because they do not record the deformed membrane.",
"Robot platform.",
"To get calibrated real pairs of contact poses and tactile observations, we fix the sensor to the environment and use a 4-axis robotic stage with translation and rotation in the horizontal plane and vertical motion.",
"The platform is composed of 3 Newmark linear stages, two horizontal (x, y-axis), and one vertical (z-axis), and a rotational stage in the vertical direction.",
"We use ROS to control the setup, which moves with submillimeter precision and high repeatability.",
"To calibrate the sensor w.r.t.",
"the platform, we use a set of marks at known poses to make sure the tactile images from touching them match the simulated contact shapes at these poses.",
"We provide further details on the collection of labeled real data in Sec.",
"REF of the appendix.",
"Figure: Normalized pose errors, i.e., pose errors w.r.t.",
"to the average random error, for the damping pin.",
"The first case corresponds to the average closest distance in its grid, the 3rd to the median results from Best-1, and the last one to the average random error.",
"Finally, the example with 0.84 normalized error depicts a non-unique contact shape, i.e., the two object poses result in very similar contact shapes that are not possible to distinguish without additional information.Objects.",
"We test our algorithm on 4 objects from the McMaster dataset [40] (Figure REF ).",
"We scaled up pin and damping pin to make them bigger and thus more challenging.",
"In the video linked to this paper, we also included the objects cotter and hook, which are also from McMaster, to show additional qualitative results on the generalization of the algorithm.",
"For each object, we build a dense grid that contains the set of poses that would result in contact with the sensor.",
"The distance between closest neighbors is no larger than 2mm on average.",
"For the quantitative results, we restrict these grids to one face of the object which results in grids with 5k to 20k elements depending on the object dimensions.",
"Figure: Pose estimation results.",
"We show in blue the error distributions for the best match and the best out of 10.",
"Green distributions show the pose error after also applying the pointcloud registration.",
"Most of the distributions are far below random error (black line) and close to the error obtained when selecting the closest element from the grid (red line).",
"For some objects like damping pin, we see multimodality in the error distributions due to different contact poses resulting in similar contact shapes."
],
[
" Real pose estimation results",
"First, we test the accuracy of our approach at estimating object poses from single tactile imprints.",
"For each object, we collected at least 150 pairs of tactile images and object poses.",
"Given two poses, we measure their distance by sampling 10K points on the object 3D model and averaging the distance between these points when the object is at either of the two poses.",
"This distance is sometimes called ADD (average 3D distance) but, for simplicity, we just refer to it as the pose error.",
"To be able to compare errors across shapes and object sizes, we also compute the normalized pose error which divides the original pose error by the average error obtained from predicting a random contact pose.",
"Figure REF shows examples of different normalized pose errors.",
"Figure REF shows the accuracy results for tactile pose estimation for each of the four objects, in the form of error distributions.",
"The black line in each plot represents the random error, i.e., the average error when predicting a random contact pose.",
"The red line measures the average pose error between the ground truth pose and its closest pose in the grid.",
"This sets a lower bound on the average performance when only considering poses from the grid, without point-cloud registration.",
"For each object, we include the error distributions for: Best-1: only considers the most likely pose of the grid.",
"Reg-1: refines the most likely pose using FilterReg.",
"Best-10: considers the 10 most likely poses of the grid and selects the one that leads to the lowest pose error.",
"This approach requires knowledge of the true pose and it is not applicable in practice.",
"However, it helps to understand the quality of the predicted pose distributions, which is especially relevant when different object poses can lead to very similar contact shapes.",
"Reg-10: takes the 10 most likely poses of the grid, refines them using FilterReg, and selects the one with the lowest error.",
"Again, this is not a viable approach but a measure of the quality of the pose distributions.",
"For all objects, we observe that most of the error distributions are below the expected random error.",
"Moreover, a large fraction of the errors are small and gathered around the closest error from the grid (red line).",
"Fig.",
"REF from the appendix compares these error distributions against the one obtained from randomly selecting contact poses.",
"In some cases, the error distributions are multimodal.",
"This happens especially when there is non-uniqueness, i.e, different poses of the object lead to very similar contact shapes.",
"These poses exist for all 4 objects, but are most important in the damping pin and the elbow pipe.",
"Figure REF shows an example for the damping pin of non-uniqueness in the contact shape in the example with a normalized error of 0.84.",
"This error matches the second mode in the error distributions for this object.",
"Selecting the best error out of the 10 best poses results in considerably lower errors and suggests that our approach can provide useful pose distributions.",
"Multimodality also disappears or gets reduced in these distributions, meaning that we can capture non-unique cases.",
"The case of the elbow pipe is especially interesting because it has multiple non-unique poses.",
"Therefore, from a single tactile observation, it is often not possible to predict its pose.",
"Because our method provides distributions, we can capture the non-unique cases when considering several pose predictions instead of a single estimation.",
"If we observe the best out of 10, the error distribution becomes much closer to zero, meaning that with 10 samples we can already capture several poses that result in very similar contact shapes.",
"Another important observation coming from these plots is that the median errors are low even when only considering the most likely pose.",
"For Best-1, we get median normalized errors of 0.11 for pin (4.8mm), 0.16 for damping pin (4.6mm), and 0.18 for head (7.3mm).",
"The elbow pipe is more challenging, and we get median normalized errors of 0.87 (54.4mm), however, if we consider the Best10 we get 0.26 (16.3mm), and 0.08 (5mm) for Best50 which represents selecting only 50 poses out of more than the 20k in its grids.",
"Finally, adding FilterReg further improves the results.",
"This is because it can refine the pose estimates when the initial distance between contact shapes is small enough and locally transforming them is possible.",
"Figure: Multi-contact results.",
"We compare the median and mean errors for Best-1 and kinematic.",
"Kinematic does not deliver good results even after many contacts, suggesting that using contact shape information is key for Best-1 to achieve good results.The predictions from Best-1 improve with the number of contacts.",
"The mean and median become closer while the standard deviation also decreases, confirming that our approach can successfully integrate multi-contact information."
],
[
" Multi-contact pose estimation results",
"Next, we evaluate our solution to pose estimation using multiple contacts.",
"For each object, we evaluate our approach on 100 examples.",
"We synthesize multi-contact examples by combining data from single contacts.",
"For each example we add, one by one, pairs of tactile images and sensor-object poses to increase the number of contacts from 1 to 7.",
"We assess the performance of our pose estimations by reporting both Best-1, the best match from the grid, and kinematic, which only considers the kinematic constraints imposed by the sensors poses and whether the sensor is in contact or not.",
"Figure REF compares both approaches using the normalized error and the original pose error.",
"We observe that only using kinematic information is not enough to substantially reduce the uncertainty in object pose.",
"This is because only detecting if there is contact does not inform on what part of the sensor is being touched which results in many poses remaining possible even after multiple contacts.",
"In contrast, for our approach Best-1, adding more contacts results in lower mean and median errors, and the standard deviation decreases.",
"In the case of elbow pipe, just adding one additional contact already reduces dramatically the pose error.",
"This reinforces the idea that adding sensed contacts helps to reduce the ambiguities coming from having different poses that result in similar contact shapes."
],
[
"Discussion",
" This paper presents an approach to tactile pose estimation for objects with known geometry.",
"For tactile sensors that provide local contact shape information, we learn in simulation how to match real estimated contact shapes to a precomputed dense set of poses by rendering their contact shapes.",
"The approach allows to reason over pose distributions and to handle additional pose constraints.",
"Our approach relies on learning an embedding that facilitates comparing simulated and real contact shapes.",
"However, real contact shapes come from a distribution that differs from the training data.",
"To bridge this gap, we rely on improving the methods that extract contact shapes and explore data augmentation techniques to learn embeddings that better accommodate the discrepancies between real and simulated contact shapes.",
"Similarly, we assume we have access to accurate geometric models of objects.",
"This makes our approach applicable to many real scenarios like assembly automation but also limits its use-cases.",
"Our solution could effectively be combined with reconstructed object models, which would require addressing the combined challenges of pose and shape uncertainty.",
"This might demand to learn contact shape embeddings that adapt to both.",
"We believe the idea of matching estimated contact shapes to a dense precomputed set opens the door to moving many computations into simulation and improving how robots learn to perceive and manipulate their environment.",
"We thank Ferran Alet for providing detailed insights and carefully reviewing the paper.",
"We also thank Siyuan Song and Ian Taylor for helping with the real setup, and Yen-Chen Lin for helpful discussions.",
"This work was supported by the Toyota Research Institute (TRI).",
"This article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity.",
"Maria Bauza is the recipient of a Facebook Fellowship."
],
[
"Contact shape prediction from tactile observations",
"Given a tactile observation, our goal is to extract the contact shape that produces it.",
"To that aim, we train a neural network (NN) that maps tactile observations to contact shapes following the approach proposed in [6] and described below for completeness.",
"As shown in Fig.",
"REF , the input to the NN is a normalized rescaled RGB tactile image of size 200x200.",
"The output corresponds to a normalized one-channel depth image of size 200x200 that represents the contact shape.",
"The training data is collected autonomously in a controlled 4-axis stage that generates controlled touches on known 3D-printed shapes.",
"In our case, we collect the calibration data from two 3D-printed boards with simple geometric shapes on them (see the top of Fig.",
"REF ).",
"From each controlled touch, we obtain a tactile observation and the pose of the board w.r.t.",
"to the sensor.",
"From this pose, we can later simulate the corresponding contact shape to the tactile observation using geometric contact rendering and the 3D model of the board.",
"Note that, for each tactile sensor, to do tactile localization on any object, we only need to gather calibration data once because the map between tactile observations and contact shapes is object-independent.",
"Given the training data collected from our sensor, we normalize tactile images (input) and contact shapes (output) using the mean and standard deviation of the training set.",
"Before normalization, the contact shape values range between 0 and 2 mm, where zero means maximum penetration into the sensor, and 2 mm indicates no contact.",
"The architecture of the NN consists of 6 convolutional layers with ReLU as the activation function and a kernel size of 3 except for the last layer which is 1.",
"Between layers 5 and 6, we add a Dropout layer with a fraction rate of 0.5.",
"The optimizer is Adam with a learning rate of 0.0001, and the loss function is the RMSE between pixels.",
"In practice, we collected 1000 pairs of tactile observations and contact shapes for each board, and trained the NN for 30 epochs.",
"Each forward pass on the NN takes less than 0.005s (200Hz).",
"Figure: Training data set-up to estimate real contact shapes from real tactile observations.",
"We use the boards at the top to collect real tactile observations from them as well as simulated contact shapes.",
"By placing the boards on a robotic platform, we can move them w.r.t.",
"the sensor to perform calibrated touches on them.",
"From each touch, we recover both a tactile image and a simulated contact shape that we compute using the 3D model of the boards and its pose w.r.t the sensor when the touch happened."
],
[
"Collecting ground-truth data for real objects",
"To evaluate our approach to tactile localization, we collected ground-truth data from novel objects using the same set-up.",
"The resulting data from the set-up consists of tactile observations paired with the calibrated pose of the object.",
"This allows us to compute pose errors between our method predictions from just the tactile observations, and the true calibrated object poses.",
"Figure REF shows two of the boards used to collect ground-truth data on the real objects.",
"The results displayed in the paper's video show examples of tactile images and the predicted contact shapes obtained using the NN that goes from tactile observations to contact shapes.",
"We note that this NN has only been trained once using only calibration data from the calibration boards in Fig.",
"REF .",
"Figure: Boards used for ground-truth data collection.",
"We sliced the models of the objects and create boards with them.",
"As a result, we can 3D-print these boards and use them to collect ground-truth data for test objects."
],
[
"Similarity metric for contact shapes",
"Given a new contact shape, we want to compare it to all pre-computed contact shapes in the grid to find what poses are more likely to produce it.",
"To that aim, we use MOCO [4], a state-of-the-art tool in contrastive learning, to train a NN that encodes contact shapes into a low dimensional embedding allowing to compare them using the distance between their encodings.",
"The NN encoder is based on a pre-trained Resnet-50 [38] cropped before the average-pooling layer to preserve spatial information.",
"It encodes contact shapes into vectors of dimension 3200.",
"To train, we use the same optimizer as [4]: stochastic gradient descent with a learning rate that starts at 0.03 and decays over time, momentum of 0.07, and weight decay of 0.0001.",
"Each training datapoint comes from selecting a random contact pose, rendering its contact shape, and finding its closest pose in the pre-computed grid.",
"Then our input is the contact shape.",
"The output or label is a vector with all entries zeros except the one that corresponds to the closest element that gets a one.",
"This vector represents the probability of each poses in the grid to be the closest to the target pose.",
"The loss function is the cross-entropy loss between the label and the softmax between the target encoder and the ones from the grid (see Fig.",
"REF ).",
"As the NN gets trained, we recompute the encoding of the queue following [4].",
"We train the models for 30 epochs, as performance does not significantly change after that.",
"To make the training data resemble more the real distribution of contact shapes, we train using as target, contact shapes with different depth thresholds, $\\Delta d$ , selected randomly between 1 and 2 mm.",
"This accounts for changes in the contact force applied to the sensor that can make contacts more or less deep (see Fig.",
"REF ).",
"Finally, before encoding the contact shapes, we converted them into binary masks to facilitate the learning process and prevent that numerical mismatches between real and simulated contact shapes affect the results."
],
[
"Multi-contact for N sensors",
"To derive Equation 1 in Section 3.5, we will prove a more general equation for the object pose $x$ when it is in contact with N sensors: $P(x | CS_1, ..., CS_N) = \\frac{P(x | CS_1) \\cdot ... \\cdot P(x | CS_N) \\cdot P_{task}(x)/ P_{train}(x)^{N}}{\\sum _{x^{\\prime } }P(x^{\\prime } | CS_1) \\cdot ... \\cdot P(x^{\\prime } | CS_N) \\cdot P_{task}(x^{\\prime })/ P_{train}(x^{\\prime })^{N}} $ where $P(x | CS_i)$ is the likelihood that pose $x$ has produced a contact shape $CS_i$ on sensor $i$ .",
"$P_{task}(x)$ is a prior over all possible contact poses that would result in contact with the N sensors in the current task.",
"$P_{train}(x)$ is the distribution over the contact poses used to train the similarity function in Sec.",
"REF .",
"Next, we prove Equation (REF ).",
"Assuming that we have access to the poses of the N sensors, given an object pose $x$ , the contact shapes $\\lbrace CS_i\\rbrace $ become determined.",
"This allows us to write the joint distribution of pose and contact shapes as: $P(x, CS_1, ..., CS_N) = P_{task}(x) \\cdot P(CS_1 | x) \\cdot ... \\cdot P(CS_N | x)$ Now we can use Bayes theorem on the $P (CS_i | x)$ terms: $P(CS_i| x ) = P(x | CS_i) \\cdot P( CS_i)/P_{train}(x)$ and obtain: $P(x, CS_1, ..., CS_N) = P(x| CS_1) \\cdot P(CS_1) \\cdot ... \\cdot P(x| CS_N) \\cdot P(CS_N) \\cdot P_{task}(x)/ P_{train}(x)^N$ where we take into account that the $P(x|CS_i)$ have been trained using a predefined distribution of poses $P_{train}(x)$ that does not necessarily match $P_{task}(x)$ .",
"Now, we can compute $P(x| CS_1, ... , CS_N)$ using again the Bayes theorem: $P(x| CS_1, ... , CS_N) = \\frac{P(x, CS_1, ... , CS_N)}{P(CS_1, ... , CS_N)} = \\frac{P(x, CS_1, ... , CS_N)}{\\sum _{x^{\\prime }}P(x^{\\prime },CS_1, ... , CS_N)}$ and thus $P(x| CS_1, ... , CS_N) = \\frac{\\left(\\Pi _i^N P(x| CS_i) \\cdot P(CS_i) \\right) \\cdot P_{task}(x)/ P_{train}(x)^{N} }{\\sum _{x^{\\prime }} \\left( \\Pi _i^N P(x^{\\prime }| CS_i) \\cdot P(CS_i) \\right) \\cdot P_{task}(x^{\\prime })/ P_{train}(x^{\\prime })^{N}}$ We can now cancel the terms $P(CS_i)$ that appear both in the numerator and denominator, getting: $P(x| CS_1, ... , CS_N) = \\frac{\\left(\\Pi _i^N P(x| CS_i) \\right) \\cdot P_{task}(x)/ P_{train}(x)^{N} }{\\sum _{x^{\\prime }} \\left( \\Pi _i^N P(x^{\\prime }| CS_i) \\right) \\cdot P_{task}(x^{\\prime })/ P_{train}(x^{\\prime })^{N}}$ This concludes the proof of Equation (REF ).",
"In practice, we discretize the set of contact poses $\\lbrace x \\rbrace $ that result in contact with the N sensors using the grid over of sensor 1 (this is an arbitrary choice).",
"Then, we use the transformation between sensor 1 and sensor $i$ to compute each $P(x| CS_i)$ using the closest pose to $x$ in the grid of sensor $i$ .",
"Note that often a pose that results in contact with sensor 1 will not contact sensor $i$ .",
"In that case, we do not consider that pose as we are only interested in poses that contact the N sensors.",
"Finally, because the grids are dense and structured, finding the closest pose to $x$ in a grid has a minor effect on performance and is fast to compute.",
"We can use Equation (REF ) to derive Equation (REF ) in Sec.",
"REF under two extra assumptions.",
"First, we observe that the denominator is constant because all contact shapes $CS_i$ are given.",
"Next, if we have no prior over the contact poses both during the task and training, then $P_{task}(x)$ and $P_{train}(x)$ are constant, and that leads to Equation REF repeated here for completeness: $P(x | CS_1, ... , CS_N) \\propto P(x | CS_1) \\cdot ... \\cdot P(x | CS_N)$ ."
]
] | 2012.05205 |
[
[
"Determining the Variations of Ca-K index and Features using a Century\n Long Equal Contrast Images from Kodaikanal Observatory"
],
[
"Abstract In the earlier analysis of Ca-K spectroheliograms obtained at Kodaikanal Observatory, the \"Good\" images were used to investigate variations in the chromosphere.",
"Still, the contrast of the images varied on a day-to-day basis.",
"We developed a new methodology to generate images to form a uniform time series.",
"We adjusted each image's contrast until the FWHM of the normalized intensity distribution attained a value between 0.10 and 0.11.",
"This methodology of the \"Equal Contrast technique\" is expected to compensate for the change of emulsion, development, contrast of the images due to centering of Ca-K line on the exit slit and sky transparency.",
"Besides, this procedure will correct variations in density-to-intensity conversion for different images.",
"We find that the correlation between sunspot and Ca-K line data improves by a large amount.",
"For example, correlation coefficient (CC) between monthly averaged sunspots and Ca-K plage areas for the equal contrast data improves to 0.9 compared to 0.75 for the \"Good\" data with unequal contrast.",
"The CC for equal contrast images improves to $\\sim$0.78 from $\\sim$0.46 for the \"Okay\" data with unequal contrast.",
"Even the CC between the plage area and the daily sunspot number is 0.85 for 100 years of data.",
"This methodology also permits us to study the variations in Enhanced, Active, and Quiet networks with time along with good accuracy for about a century, for the first time.",
"Further, this procedure can be used to combine data from different observatories to make a long time series."
],
[
"Introduction",
"The long term periodicity or quasi-periodicities of the sun's large and small scale magnetic fields can be investigated using the long term spectroscopic and imaging data.",
"The sun's images in the Ca-K line can be used as a proxy to study the variations in magnetic fields as there is a strong spatial correlation between Ca-K and magnetic features such as plages and networks [1], [10], [13].",
"At the Mount Wilson observatory (MWO) and the Kodaikanal observatory (KO) spectroheliograms were obtained in Ca-K and H-alpha wavelengths daily for about 100 years during the 20th century.",
"These data sets provide a time series of sun’s images to study the long term variations of the solar magnetic field.",
"The earlier analysis of the digitized data of Ca-K images used density-to-intensity conversion using the theoretical equation or mean curve as step-wedge calibration was not recorded on all the images obtained during the long period [9], [22], [8], [21], [4], [17], [16], [5], [6].",
"Several observatories have adapted their procedures to correct the images for the limb darkening and instrumental effects.",
"After completing the task of making the quiet chromosphere of uniform nature, [18] displayed each image along with the intensity distribution curve of the same on the computer screen.",
"By visual inspection of these images and examining the corresponding intensity distribution curves, the images are classified as “Good” and “Okay”, to generate two separate time series.",
"Those two-time series analysis showed very different results [18].",
"The “Good” time series with uniform images indicated smooth variations of Ca-K plage areas and Ca-K index with solar cycle phase.",
"In contrast, the time series of remaining “Okay” images showed a large amount of scatter in the Ca-K parameters.",
"Further, an inspection of the “Good” images indicated that there are variations in the contrast of images on a day-to-day basis as well as and on long term basis.",
"The effect of contrast variations in Ca-K images still needs to be studied on the small and large scale activities on long-term.",
"[14] have found that the Ca-K line intensity at the centre of the sun does not vary with the solar cycle phase.",
"This result implies that quiet chromosphere does not show the solar cycle or long term variations.",
"But [16] found that the intensity of the quiet network varies with the solar cycle but with a very small amplitude.",
"Therefore, it is reasonable to assume that the quiet chromosphere's intensity distribution does not vary with time.",
"[22] showed that the intensity contrast of active regions is beyond the intensity contrast of quiet chromosphere.",
"[18] found that in the normalized intensity distribution of the image, intensity contrast greater than 1.30 represents plages and enhanced network (EN), intensity contrast between 1.20 to 1.30 represents active network (AN), and intensity contrast between 1.10 to 1.20 represents quiet network (QN).",
"The remaining pixels, forming a large area of the sun, with intensity contrast lying between about 0.9 to 1.10 values, approximately follows the Gaussian distribution representing a quiet chromosphere.",
"The extended tail of the intensity distribution of the image indicate the active area of the chromosphere.",
"The Gaussian part of the intensity distribution representing the background chromosphere is expected to remain the same irrespective of the solar cycle phase [14].",
"The area under the extended tail varies significantly with time and is expected to show solar cycle variations.",
"Therefore, all the images obtained on the day-to-day basis are expected to show similar intensity distribution for the quiet background chromosphere.",
"This implies that the contrast of the images, thereby, full width at half maximum (FWHM) of the intensity distribution of all the images obtained during long periods should remain approximately the same.",
"There might be a small change in the total area occupied by the quiet chromosphere over the solar cycle.",
"Still, the FWHM of the intensity distribution is likely to remain the same with minor variations, if any.",
"But FWHM of the intensity distribution was found to be related to the contrast of the images and much larger for high contrast images than for images with very low contrast [18].",
"Therefore, by making the FWHM of the intensity distribution the same for all the images, it is possible to make the contrast of the long-time series data the same.",
"It may be noted that images need to be corrected for the limb darkening and instrumental effects effectively before correcting for the contrast of the obtained images so that the FWHM of the intensity distribution becomes uniform within the specified limits.",
"In this paper, we describe the methodology adopted to make the contrast of all the images uniform and compare the results of the present analysis with those of earlier analysis obtained using data without correcting for the intensity contrast of images.",
"The adapted methodology is somewhat similar approach used in studies of the Mt.",
"Wilson archive of Ca-K observations [15], [3]."
],
[
"Data Preparation and Analysis",
"In our earlier paper [18], we have divided the data into two groups depending on the image quality and intensity distribution.",
"Out of these two, one forming a time series of uniform images termed as “Good” and the other of remaining images termed as “Okay” series.",
"Some bad quality images were discarded.",
"Even though we have selected normal contrast data to make uniform time series termed as “Good” but still the contrast of the images varied daily due to sky conditions, development of photographic film, and variations in visually setting the centre of the Ca-K line on the exit slit of the spectrograph [17].",
"The contrast of the images also varied due to the emulsion change on a long-term basis.",
"In the earlier analysis, we found that FWHM of the intensity distribution varied systematically for the “Good” (hereafter called “P-Good”) images.",
"The average decrease in FWHM of the intensity distribution from 1907 to 1950 and then increase indicates that the contrast of the images varied with time on a long-time basis, probably due to change in emulsion and on a day-to-day basis due to sky conditions.",
"Whereas in the “Okay” (hereafter called “P-Okay”) series, the FWHM of the intensity distribution showed two branches due to the inclusion of very high and very low contrast images in this data set.",
"These two series were analyzed separately.",
"Now also, we have decided to analyse the two series of data separately to compare the earlier results with those of the new methodology being adopted.",
"Sometimes we may refer “P-Good” and “P-Okay” images as “P-image” in general.",
"Figure: The upper panel of the figure shows a histogram of the number of images per year with FWHM of the intensity distribution between 0.10 and 0.11, >> 0.11 and << 0.1 in blue, red, and yellow lines, respectively for the images termed as “P-Good” images.",
"The Lower panel indicates the same for “P-Okay” time series.We have determined the FWHM of the Gaussian fit to the intensity distribution of each image of the “P-Good” and “P-Okay” time series.",
"The upper panel of Figure REF shows the histogram of a number of “P-Good” images per year whose FWHM of intensity distribution lies between 0.10 and 0.11 (shown in blue colour).",
"The number of images per year with FWHM $>$ 0.11 and with FWHM $<$ 0.10 is also shown in the same plot in red and yellow color bars.",
"The lower panel of the same figure shows the histogram of the “P-Okay” time series.",
"The histogram of “P-Good” images indicates that the number of images with high contrast is more than low contrast images at the beginning of observations until 1930.",
"But after 1930, the number of images with low contrast is more than the high contrast images.",
"The histogram for “P-Okay” images shows a similar trend in general.",
"Besides, it indicates that the number of high contrast images are more during the period 1975 – 1990.",
"This comparison is relative.",
"It may be noted that the number of images in the “P-Okay” data with FWHM in the range of 0.10 – 0.11 is less compared to that for in the “P-Good” data.",
"In the present analysis, we studied the data of FWHM for a large number of images selected at a random spread over several years.",
"We noted that images with FWHM in the range of 0.10 to 0.11 show chromospheric features very well.",
"Therefore, it was decided to keep the FWHM of the intensity distribution between 0.10 – 0.11 for all the images by changing their contrast.",
"We compute the intensities of the modified image using the relation, $I_{ECI} = I_{p}^{\\gamma }$ Where, $I_{ECI}$ is the intensity of a pixel of new image (Equal Contrast Image), $I_{p}$ is the intensity of the pixel of the corrected image [18], and $\\gamma $ is the contrast of the image.",
"First, we assume gamma = 1.0 and then compute the FWHM of the intensity distribution of the image.",
"The images with FWHM of the intensity distribution less than 0.10 are low contrast images.",
"The contrast of these images ($\\gamma $ ) was increased in steps of 0.025 until the value of the FWHM lie between 0.10 and 0.11.",
"The step of 0.025 was chosen after doing several experiments with different step sizes on many images.",
"For the images with FWHM of the intensity distribution larger than 0.11 (high contrast images), the gamma value was decreased in steps 0.025 until the value of FWHM of the distribution becomes between 0.10 and 0.11.",
"We term this methodology as “equal contrast technique” and can be used anywhere provided the images have been properly corrected for limb darkening and instrumental effects.",
"The new methodology has been applied to both the “P-Good” and the “P-Okay” time series.",
"After changing their contrast, the new images are called hereafter as “ECI-Good” (Equal Contrast Image) and “ECI-Okay” or “ECI-image” in general."
],
[
"Comparison of “P-images” and “ECI-images” during the Quiet Phase",
"The left panel in the top row of Figure REF shows a typical image belongs to the “P-Good” data set taken during the quiet phase of the solar cycle.",
"The details of the image seen and FWHM of intensity distribution greater than 0.11 shown in the right-side panel imply high contrast.",
"The FWHM of the intensity distribution is large, and the peak value of the frequency distribution is about 1.6%.",
"The extended tail beyond intensity contrast of 1.2 of the intensity distribution in the “P-image” indicates a significant existence of EN and AN even during the quiet phase of the sun, which is contrary to expectation.",
"The left panel in the bottom row of this figure shows the “ECI-image” after adjusting the contrast of the image so that the FWHM of intensity distribution curve becomes between 0.10 and 0.11 (right-side panel).",
"After changing the intensity contrast of the image, the tail of the intensity distribution curve reduces, indicating the less number of AN and almost absence of EN as expected during the quiet phase of the sun.",
"Because of this procedure, the peak value of the frequency distribution increases to $\\sim $ 2.5%.",
"On the contrary, the top row of Figure REF shows a low contrast image (also selected from the “P-Good” time series) taken during the quiet period of the solar cycle.",
"This image exhibits a small FWHM and a high peak value ($\\sim $ 3.8%) in the frequency distribution.",
"In the plot, due to the low contrast of the image, the intensity distribution $>$ 1.10 indicates an insignificant existence of QN.",
"The bottom row of this figure shows the image after increasing the contrast such that FWHM of the intensity distribution lies between 0.10 – 0.11.",
"After increasing the contrast of the image, QN's contribution becomes significant approximately 2 – 3 times.",
"After adjusting the contrast of the images, both the high and low contrast (in “P-Good” time series) images acquire similar intensity distribution, as seen in the bottom row panels of Figures REF and REF .",
"A similar intensity distribution of the two types of images from the “ECI-Good” time series with a peak value of $\\sim $ 2.5% obtained during the quiet phase of the sun confirms the methodology adopted works well in both the cases.",
"Hence, these two types of data can be used reliably for the study of short and long period variations in solar activity."
],
[
"Comparison of “P-images” and “ECI-images” during the Active Phase",
"Four panels of Figure REF shows the images of P-Good (top) and ECI-Good (bottom) and their intensity distributions for a typical high contrast image obtained during the active phase of the solar cycle.",
"The intensity contrast for part of the plage region exceeds 2, but we have plotted the intensity contrast up to 1.6 to clearly show intensity distribution.",
"The intensity distribution of the image after adjusting the contrast appears realistic, indicating the area of the quiet chromosphere and the plage and EN regions’ existence.",
"The peak value of the intensity distribution during the active phase appears to be less by a very small amount than that during the quiet phase to account for plage and EN regions’ existence.",
"The area under the extended tail with an intensity contrast of more than 1.2 indicates the area occupied by the plages, EN, and AN networks.",
"Four panels of Figure REF shows the images of P-image and ECI-image after adjusting the contrast of the image and their intensity distributions representing low contrast image obtained during the active phase of the solar cycle.",
"The tail of the distribution curve for ECI-image beyond 1.2 indicates the area of plages, EN, and AN.",
"A Comparison of P-images in figures REF , REF , REF , and REF indicates that FWHM of intensity distribution for the low contrast image is considerably less than that of high contrast and also the peak value of the frequency distribution is about 2 to 2.5 times more.",
"In addition to this, the intensity distribution of the P-images of low contrast in Figures REF and REF shows a significantly very less area under the extended tail representing plages, EN and AN.",
"However, a large number of plages are visible in the P-image.",
"After adjusting the contrast of the P-image, the area under the extended tail increases for ECI-image; thereby, the detected area of plages, EN, and AN increases.",
"Similarly, after tuning the contrasts of high contrast P-images, intensity distributions of the ECI-images showed the area of detected plages, EN, and AN decreased.",
"Thus, the values of detected plages, EN, and AN become realistic in both cases.",
"A Comparison of all the intensity distributions of images after the fine-tune of contrast indicates that the peak values ( 2.5%) and FWHM become similar.",
"This suggests that the area occupied by a quiet chromosphere may vary by a small percentage only in all the images.",
"In contrast, the area under the extended tail varies significantly (percentage), showing the variations in active part of the sun.",
"The results of the analysis of the images in the “P-Okay” time series agrees well with the “P-Good” time series.",
"Therefore, all the data can be combined and analysed together to reduce the gaps in the long time series to study solar variations with time."
],
[
"Results",
"To study the chromosphere's long-term variations, we generally determine the Ca-K plage area, Ca-K index, and other such parameters with time.",
"We also define the sum of plage, EN, AN and QN area as the “total active area” of the image.",
"In other words, the “total active area” is the percentage area of an image occupied with intensity greater than 1.10 after normalizing the intensity of the image.",
"It is also necessary to establish the correctness of the procedure adopted by comparing these derived parameters with the reliable other solar indices such as sunspots.",
"First, we compare the recent measurements and similar data determined earlier with the sunspot data and then study the variations in other parameters of Ca-K line images such as networks."
],
[
"Comparison of daily Ca-K parameters with sunspot data for equal and non-equal contrast images",
"To assess the improvement in the results due to equal contrast technique on the historical data, we first compare the results of this analysis, with some reliable works of earlier solar activity indices.",
"Data on sunspot numbers and areas are available on a daily basis for the extended period for the comparison.",
"The sunspot data obtained from various observatories were rescaled, combined them and formed a long daily time-series with negligible gap, if any.",
"Thus, to compare the solar activity related indices, the sunspot data is most reliable.",
"The left-side top panel of Figure REF shows a scatter plots between the SILSO Sunspot numbers (WDC-SILSO, Royal Observatory of Belgium, Brussels) versus identified plage areas using ECI-Good time series, whereas the right panel shows for the total active area on a daily basis.",
"The middle two panels show the same for the total data i.e., combined data of the ECI-Good and ECI-Okay images, whereas two bottom panels show the same for combined P-images.",
"Table: Values of correlation coefficients between sunspots and Ca-K plage areas on daily basis.",
"The ECI-Good, ECI-Combined and P-Combined represents the “equal contrast images” with good, good and okay (combined) and good and okay combined images.In Table REF , we list the values of correlation coefficients between sunspot data and Ca-K parameters derived by using the daily ECI images and earlier analysed daily images of [18].",
"Table shows the values of correlation coefficients are large for the ECI-Good time series than others.",
"The correlation coefficient for the sunspot number and plage area of ECI-Good images is 0.85 and 0.80 for the total active area.",
"An excellent values of correlation coefficient for the data on a daily basis spread over about a 100-year period.",
"The correlation coefficients for the combined ECI-images are marginally lower than those for ECI-Good data but still have a confidence level of greater than 99%.",
"The values of correlation coefficients for the combined P-images are significantly lower than those for the ECI-data.",
"Also, there is a large scatter around the linear fit in scatter plots of P-images as compared to ECI-images.",
"Further, in the case of the total active area, correlation coefficient of P-images is much less, 0.32 only as compared to 0.80 for the ECI-images.",
"This indicates that the plage area can be determined to show the solar cycle variations with averages of the data over a long time, even with the non-uniform images of the time series.",
"The maximum value of the total active region reaches up to 50% in the case of P-images but decreases to about 25% after tuning the contrast of images.",
"The large value of the total active region may be due to high contrast images.",
"This implies that it is almost impossible to study the periodic variations in the networks representing the small scale magnetic field with data of non-uniform time series."
],
[
"Comparison of monthly averaged Ca-K parameters with Sunspot Data",
"The monthly averaged plage areas determined using the MWO, KO and some other data are available [9], [21] for comparison for this period.",
"Therefore, we compare sunspot numbers and plage areas on a monthly average basis to determine the most accurate methodology to analyze the historical data.",
"In the top row of Figure REF , we plot the monthly averaged plage areas determined using ECI-Good data versus WDC-SILSO sunspot number and RGO sunspot area for the period 1905 – 2004 (http://www.sidc.be/silso/datafiles).",
"The middle row of this figure shows the plage areas determined from the P-images before applying the equal contrast technique versus sunspot number (left) and area (right).",
"The number of data points in all four plots is the same.",
"In the bottom row of Figure REF , we show the scatter plot of sunspot number versus plage area identified by Foukal (1996) using MWO data for the period 1915 – 1985 only, (ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SOLAR_CALCIUM/DATA/Mt_Wilson/) in the left-side panel and by [21] using KO data in the right-side panel for the period 1907 – 1999.",
"The values of correlation coefficients are indicated in each panel.",
"Similarly, two panels in the top row of Figure REF show the scatter plots of the plage area determined from ECI-Okay images versus monthly average sunspot numbers (left) and sunspot area (right).",
"Two panels in the bottom row show the same for P-Okay images.",
"The values of the correlation coefficient are indicated in the panels.",
"Table: Correlation coefficients for the monthly averaged data for comparison.",
"In the Table ssn and ssa represent sunspot number and sunspots area respectively.The plots in the middle row (Figure REF ) of monthly averaged data (P-Good) show a large scatter with correlation coefficients around 0.75.",
"But, the plage area obtained from the ECI-images shows that the correlation coefficient improved significantly to a value of $\\sim $ 0.9.",
"Similarly, the plots in Figure REF for the “ECI-Okay” images indicate a decrease in the scatter and large improvement in the correlation between plage areas and sunspot parameter.",
"The correlation coefficient improves from a value of $\\sim $ 0.45 (P-Okay) to $\\sim $ 0.8 for ECI-Okay images.",
"In Table REF , we show the values of correlation coefficients for MWO and KO data sets derived by different authors for easy comparison.",
"The obtained values of correlation coefficients are significantly larger for ECI-Good and ECI-combined data sets compared to others.",
"It may be noted that the correlation coefficients are better for the “ECI-Good” data as compared to that for the “ECI-Okay” data, even in the case of equal contrast images.",
"The scatter plots between sunspot data and the plage areas indicate a much better correlation after making contrast equal, of all the images for the “P-Good” data and the “P-Okay” data.",
"From the scatter plots, we learn that it is possible to combine the “ECI-Good” and “ECI-Okay” after applying the equal contrast technique.",
"This will reduce the data gaps in the time series.",
"The good values of correlation coefficients between sunspot data and plage area obtained using ECI-images compared to the other procedures adopted earlier to analyze Ca-K images of the historical data [18], [9], [21] indicate that the difference in the correlation coefficients is likely due to a combination of the different techniques, spectral bandwidth, and spatial resolution between the two databases.",
"Figure: Two panels of the top row of the figure show the scatter plots between monthly averaged intensity (Ca-K index) determined for “ECI-Good” images and sunspot numbers (left) and areas (right).",
"Two panels in the bottom row show the scatter plot between averaged intensity (Ca-K index) determined earlier using P-Good and sunspot data.Figure: Two panels of the top row of the figure show the scatter plots between monthly averaged intensity (Ca-K index) determined for “ECI-Okay” images and sunspot numbers (left) and areas (right).",
"Two panels in the bottom row show the scatter plot between averaged intensity (Ca-K index) determined earlier for P-Okay images and sunspot data.Further, we have computed the average intensity (Ca-K index) over the whole of the disk image as done in our earlier paper [18].",
"In the top-row of Figure REF , we show the scatter plot of the full-disk intensity of the “ECI-Good” images versus sunspot numbers (left-panel) and sunspot area (right) for monthly averaged data .",
"The bottom-row of this figure shows a similar scatter plot for the P-Good images.",
"The scatter plots between the monthly averaged sunspot data and the averaged intensity of the “ECI-Good” images indicate an excellent correlation with a value of 0.85 and confidence level $>$ 99% compared to the correlation coefficient of 0.65 for the plots in the bottom row of P-Good images.",
"The plots pertaining to the recent study indicate less scatter in the data than earlier work [18].",
"We show the linear least square fits for all the plots.",
"There appears a polynomial relation between sunspot numbers and average Ca-K intensity for the ECI-images.",
"But linear fit to the data points is more satisfactory as compared to 2 and 3-degree polynomial fits.",
"The plots between sunspot data and the average intensity of the full-disk image for the ECI-Okay and P-Okay images, indicate a similar behavior, as shown in Figure REF .",
"The correlation coefficients of about 0.7 for ECI-Okay and 0.35 for P-Okay indicate more improvement in the correlation between monthly averaged Ca-K index and sunspot parameters in the present methodology.",
"This is because a many low and high contrast images in this data set have been converted to images of the same contrast.",
"It may also be noted that the correlation between the monthly averaged Ca-K index and sunspot data is better for the “Good” data as compared to the “Okay” data in both the cases with and without, conversion of images to equal contrast."
],
[
"Variation of Ca-K Area Index with Time using ECI-images",
"In [18], the plage areas determined from the “P-Good”, and “P-Okay” data exhibit solar cycle variations, and their amplitude is in general agreement with the sunspot data used with a 12-months running average.",
"That time the procedure adopted to analyze the historic data provided reliable results till 1984 only, and the amplitude of variation for a couple of solar cycles differed with those of sunspot cycles.",
"Now, we have developed the methodology of the Equal contrast technique to analyze that data further.",
"We have shown that ECI-Okay data become comparable with the ECI-Good data.",
"But to make a detailed comparison, we analyze these two series separately before combining the whole data.",
"Figure: The black dots in the figure show the percentage of daily Ca-K area index as a function of time for “ECI-Good” images.",
"The red curve indicates the monthly average of the Ca-K area index, and the green curve shows the monthly average of sunspot area for the period 1905 – 2004.",
"The cycle numbers are also indicated in the figure.Figure: The black dots in the figure show the percentage of daily Ca-K area index as a function of time for “ECI-Okay” images.",
"The red curve indicates the monthly average of the Ca-K area index, and green the monthly average of sunspot area for the period 1906 to 2007.",
"The cycle number is also indicated in the figure.After making all the images of equal contrast, we again examined the intensity threshold values to identify the plages and network features.",
"After several experiments, we found the earlier values of intensity threshold determined by [18] still holds good.",
"This is because average values of FWHM of intensity distributions for the P-Good and ECI-Good data are similar.",
"We have defined plages with intensity contrast $>$ 1.30 and consecutive area $>$ 1000 pixel$^{2}$ , equivalent to about 0.2 arcmin$^{2}$ .",
"In addition, we define intensity $>$ 1.30 and consecutive area $>$ 4 pixels as enhanced network (EN) and intensity $>$ 1.20 and $<$ 1.30 with consecutive area $>$ 4 pixels as active network (AN).",
"The regions with intensity $>$ 1.1 and $<$ 1.2 with consecutive area $>$ 4 pixels are treated as quiet network (QN).",
"To compare the Ca-K and sunspot data, we computed the sum of the percentage of plage, EN, and AN (hereafter called Ca-K area index) from daily data.",
"The difference between the total active area and Ca-K area index is that the total active area includes the QN area, also.",
"We compare both the sum of plages, EN, and AN area variation (Ca-K area index mostly related with activity) and total active area (including QN) with the sunspot activity.",
"The QN may be related with the global characteristic of sun rather than sunspot activity .",
"In Figure REF , we plot the Ca-K area index represented by black dots, monthly averages of Ca-K area index indicated by the red curve, and monthly averages of sunspot areas shown in green for the “ECI-Good” time series for the period of 1905 – 2004.",
"The daily Ca-K area index shows that the scatter in the derived Ca-K area index is very less as compared to similar plots by [18] and others.",
"In our earlier paper [18], we plotted the data for the period of 1907 – 1984 as there was a large scatter in the determined values of plage areas because of varying sky conditions and change of photographic emulsion.",
"But after equal contrast technique applied to the images, it has become possible to determine the plage areas reliably for 1985 – 2007 data, also.",
"The respective solar cycle numbers are also written in the figure.",
"The plot indicates an excellent agreement between the maximum amplitudes of solar cycles in the Ca-K area index with sunspot areas except for solar cycle number 21 (around the year of 1980).",
"The difference may be due to the availability of a fewer number of images per year after the year 1980.",
"Generally, the red curve amplitudes appear more than those of green curves due to the selected scales of the Ca-K area index and sunspot areas.",
"Figure REF shows the daily Ca-K area index, monthly averages of Ca-K area index, and the monthly average of sunspot areas from 1905 to 2007 by black dots, red and green curves, respectively, for the “ECI-Okay” data.",
"The period of “ECI-Good” and “ECI-Okay” data differs because there are no “P-Good” images available from 2005 to 2007.",
"The derived values of the Ca-K area index appear more reliable as these show much less scatter as compared to earlier studies by [18] and others.",
"Generally, the monthly average Ca-K area index shows good correlations with sunspot area values.",
"But, the amplitude of monthly averages of the Ca-K area index for the “ECI-Okay” data differs from those for the “ECI-Good” data, especially for cycle numbers 18 and 19."
],
[
"Variation of Ca-K intensity index with time using ECI-images",
"Black dots in the upper panel of Figure REF show average intensity over the whole solar image on a daily basis and monthly average intensity (red curve) for the period of 1905 – 2004 for the “ECI-Good” time series.",
"We have also computed the average intensity of the active region with pixels having intensity contrast $>$ 1.1 and is shown in the bottom panel of the same figure.",
"The full-disk intensity and active region intensity for the “ECI-Good” data show that the amplitude variation increases from solar cycle 14 to 19, similar to that of sunspots data shown in Figures REF and REF .",
"The determined intensities vary smoothly with the phase of the solar cycle.",
"This confirms that by fine-tuning the contrast of the images, the data has become uniform in quality.",
"We show the same parameters for the “ECI-Okay” images for the period of 1905 – 2007 in two panels of Figure REF that indicates results similar to those of “ECI-Good” data but with some scatter.",
"This is probably due to the quality of images but not due to the contrast of images or could be because of some other reason."
],
[
"Variation of Ca-K networks area and total active area with time using ECI-images",
"The top panel of Figure REF shows the time-variation of enhanced network area (EN), representing the decaying plage regions as a function of time on a daily basis shown by black dots and monthly averaged data displayed by the red curve for the “ECI-Good” time series.",
"The monthly averages of sunspot numbers are over-plotted for the comparison.",
"The middle panel of the figure shows the variation of active network area (AN) on a daily basis (black dots), monthly averages (red curve), and the monthly averages of sunspot area (green curve).",
"The bottom panel indicates the variation of quiet network area (QN) on a daily basis (black dots), month averages (red curve).",
"The plots of EN and AN show that amplitudes of variations agree well with those of sunspot data, whereas the amplitude of variations for QN remains more or less the same throughout all the solar cycles, 14 – 23.",
"Three panels of Figure REF show variations of EN, AN, and QN on daily and monthly average basis for the data termed as “ECI-Okay” as a function of time.",
"The plots indicate variations in “ECI-Okay” data similar to those for the “ECI-Good” data.",
"It may be noted that there appears some scatter in “ECI-Okay” data in addition to temporal variations.",
"Figure: The Figures's upper panel shows the percentage of the total active area of images on a daily basis shown as black dots and monthly averages are shown in the red for “ECI-Good” data as a function of time.",
"We have over-plotted monthly averages of sunspot numbers and areas for comparison.",
"The bottom panel shows the same for “ECI-Okay” data.We have also computed the total active area in the images with intensity contrast of pixels $>$ 1.1 and consecutive area $>$ 4 pixels (3 arcsec$^{2}$ ).",
"The upper panel of Figure REF shows the percentage of the total active area on daily basis (black dots), monthly averages (red curve), monthly averages of sunspot numbers (blue), and monthly averages of sunspot area in green as a function of time for the “ECI-Good” data.",
"The amplitude of variations in all the parameters agrees well with each other for all the solar cycles.",
"This figure’s bottom panel indicates the same parameters for “ECI-Okay” data for the period of 1905 –2007.",
"The plots show the results similar to those for ECI-Good data but with some scatter due to the quality of some of the images in the ECI-Okay time series."
],
[
"Variations with combined “ECI-Good” and “ECI-Okay” data",
"Further, we have combined both the “ECI-Good” and “ECI-Okay” times series to study the variations due to all the Ca-K line images obtained at Kodaikanal observatory.",
"In Figure REF , we show the Ca-K area index on a daily basis by black dots, its monthly averages by the red curve, and monthly averages of sunspot areas by a green curve as a function of time for the period 1905 – 2007.",
"The plots indicate a good correlation between the amplitudes of the Ca-K area index and sunspots areas for all the solar cycles.",
"It may be noted that the inclusion of “ECI-Okay” images has decreased the amplitude of the Ca-K area index for the solar cycle numbers 18 and 19 as compared to those for the “ECI-Good” data.",
"Figure: Top two panels of the figure show the scatter plots of sunspot numbers versus % of Ca-K plage area (left) and % of EN (right) considering all the data of “ECI-Good” and “E-Okay” time series on a monthly average basis.",
"The middle two panels show the scatter plot of sunspot number versus % of AN (left) and % of total active area (right).",
"The bottom two panels show average intensity (left) and total active intensity (right).",
"The value of the correlation coefficient is indicated in each panel.Figure: The top left panel of the figure shows relative variations in the monthly plage areas of the combined ECI-images for the period of 1905 – 2007 and the top-right panel shows the power spectrum.",
"The bottom-left panel shows the wavelet power spectrum, and the right-side panel shows the global power spectrum of the data.",
"The colour bar at the bottom indicates the relative power in the power spectrum.Six panels of Figure REF show the scatter plots between the monthly average of sunspot numbers and monthly averages of Ca-K plages, EN, AN, total active area, the average intensity of image, and active region intensity (average intensity of all areas with normalized intensity greater than 1.1).",
"The plots for AN and total active area appear to indicate polynomial relation but the linear fits were found to be better than polynomial fits.",
"Generally, it can be stated that there is a linear relation between numbers and areas of sunspots and Ca-K line parameters.",
"An excellent correlation between the monthly averaged sunspot numbers and various features visible in Ca-K line images indicates that this methodology permits us to study the long term systematic variations in small scale activity apart from the large scale activity."
],
[
"Periodic behaviour in Ca-K feature",
"We have performed the wavelet analysis of monthly averaged data of plage, EN, AN, and QN areas for the combined data of “ECI-Good” and “ECI-Okay” times series to investigate the periodicities in variations of the large and small scale activity on the sun.",
"The top row two panels of Figure REF show the relative variations in the plage areas as a function of time (left) and its power spectra (right).",
"The bottom two panels show the wavelet power spectrum (left) and the global power spectrum (right).",
"The color bar indicates the relative power in the wavelet power spectrum.",
"The power spectrum and the global power spectrum indicates a strong periodicity around 10.8-years period and negligible power at periods $>$ 16 and $<$ 5 years.",
"The existence of some quasi-periods $<$ 4 years and the 11-years period was observed in the earlier analysis.",
"The wavelet analysis of the EN and AN indicates similar results as seen in Figures REF , REF (Annexure 1).",
"But the wavelet analysis QN area in Figure REF (Annexure 1) shows some power at periods $>$ 16 and $<$ 4 years along with large power around 11-years.",
"This data set is more uniform than earlier ones and will be subjected to more rigorous analysis to study the existence of quasi-periods."
],
[
"Discussions",
"We have developed a new methodology to analyze the Ca-K line images to investigate long-term variations in the chromosphere's large and small scale activity.",
"First, we made all the images of equal contrast.",
"This is to compensate for the different photographic emulsions used for recording the images, different sky transparency and seeing conditions affecting the contrast of the images, changes in contrast due to centering of the Ca-K line on the exit slit of spectro-heliograph, different developing conditions, and chemicals used for the developing photographs over long periods.",
"This methodology has also overcome the difficulty of intensity calibration of the time series as a significant part of the data did not have step wedge calibration.",
"Even the step wedge calibration has some uncertainty due to the non-uniformity of the light source used at edges of photographic plates [17].",
"The variations in the parameters of Ca-K line features obtained from the analysis of each image represent variations on the sun, free from the effects of above mentioned observational parameters and sky conditions.",
"Most of the papers ([18], [7] and the references therein) have done the 12-months averages or running averages of the data to show the correlation between sunspots and Ca-K plage areas.",
"We have shown the correlation between plage areas and sunspot data on daily and monthly mean basis.",
"The percentage of plage area vary with the solar cycle phase smoothly and their amplitude agrees well with the amplitude of sunspot data till 1975 for the data termed as “ECI-Good”.",
"After 1975 the amplitudes of plage areas differs with those of sunspots by a small amount, probably due to available data for a fewer number of days per year.",
"Apart from the study of variation of large scale activity represented by plage areas, we have successfully investigated the variations in small scale activity represented by EN, AN, and QN for the first time for about 100 years.",
"Only [22] has defined the threshold values of intensity for EN and AN using some selected data from 1980 to 1996 to study their relationship with solar cycle phase.",
"The amplitudes of solar cycle variations for EN and AN agree well with those of sunspot data.",
"The amplitude of solar cycle variations for QN remains almost the same irrespective of the amplitude of sunspot data.",
"It may be noted that the values of average monthly intensity include the intensity of a quiet background chromosphere.",
"This data will be very useful for making the realistic models of solar cycle variations and will be made available to the interested scientist.",
"The Ca-K line images obtained at various observatories with different spatial resolution and passband lead to different contrast.",
"For combining the data of different observatories and types, the derived parameters have been compared with each other with large temporal averages and then up or downscaled one of the data [2].",
"But in this process, data on day to day basis remains uncorrected.",
"In the present methodology, we first adjust each image's contrast and then identify the features, such as plages, EN, AN, and QN.",
"Thus, we make the correction on a day to day basis.",
"Hence, this method is likely to work for different spatial resolution and with different passband images within certain limits and will help in combining the data to yield better results.",
"Most of the solar cycle variation models are based on the observed changes in large scale magnetic activity.",
"Some small scale magnetic field measurements have been done and analysed to study solar cycle variations [20].",
"Using the full-disk observations of solar magnetograms from the Michelson Doppler Imager [19] instrument, [12] have found two components of varying small scale fields.",
"One of the components whose magnetic flux is smaller than 32$\\times $ 10$^{18}$ Mx exhibits cyclic variation in anti-phase with the sunspot cycle.",
"Another one with the flux between (4.3 – 38)$\\times $ 10$^{19}$ Mx correlated with the solar cycle.",
"It is possible to observe the magnetic fields at 0.1$^{\\prime \\prime }$ or better, but with a smaller field-of-view [11].",
"The measurement of the magnetic fields was also made with an accuracy better than 5 G. With the results obtained from the accurate analysis of 100 years of Ca-K images of the sun, it will become possible to study and understand the small scale network and internetwork magnetic fields and its variations over the solar cycle better."
],
[
"Conclusion",
"We have developed a new methodology of the “Equal Contrast Technique” to analyze the Ca-K line's photographic images for long periods under different environmental conditions, uniformly.",
"This procedure will help to analyze similar data sets obtained at several observatories with different instruments, uniformly.",
"Then combine all the data to form a long time series with fewer gaps in the data for further studies.",
"This technique helps to determine the variations in large scale and small scale activity with very high accuracy and reliability for the good quality data.",
"It also helps to minimize the errors in the low-quality data.",
"It will be possible to extend the study of large and small scale magnetic activity on the sun back in time as the activity observed in the Ca-K line is related to the magnetic field of the sun."
],
[
"Acknowledgments",
"We thank the referee, Alexei Pevtsov and Kiran Jain for providing valuable comments on the paper.",
"Jagdev Singh designed and developed the digitizer units with the help of P. U. Kamath and F. Gabriel.",
"He trained the team to digitize the data and supervised the process to digitize the Ca-K line images.",
"We thank all the observers at Kodaikanal Observatory who made the observations since 1904, kept the data in suitable environmental conditions, and digitized all the images.",
"The daily sunspot data used here was from the SILSO sunspot numbers (WDC-SILSO, Royal Observatory of Belgium, Brussels).",
"The monthly sunspot data was taken from http://www.sidc.be/silso/datafiles.",
"The monthly Ca-K index was dowloaded from ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SOLAR CALCIUM/DATA/Mt Wilson/."
],
[
"Annexure 1",
"The Figures REF , REF and REF shows the wavelet analysis of the EN, AN and QN areas as explained in the section REF"
]
] | 2012.05238 |
[
[
"Participatory Budgeting with Project Groups"
],
[
"Abstract We study a generalization of the standard approval-based model of participatory budgeting (PB), in which voters are providing approval ballots over a set of predefined projects and -- in addition to a global budget limit, there are several groupings of the projects, each group with its own budget limit.",
"We study the computational complexity of identifying project bundles that maximize voter satisfaction while respecting all budget limits.",
"We show that the problem is generally intractable and describe efficient exact algorithms for several special cases, including instances with only few groups and instances where the group structure is close to be hierarchical, as well as efficient approximation algorithms.",
"Our results could allow, e.g., municipalities to hold richer PB processes that are thematically and geographically inclusive."
],
[
"Introduction",
"In the standard approval-based model of participatory budgeting (PB), specifically, the model of Combinatorial PB [2], we are given $n$ approval votes over a set of $m$ projects, each with its cost, and a budget limit.",
"The task is to aggregate the votes to select a bundle (i.e., a subset) of projects that respects the budget limit.",
"PB has caught quite considerable attention lately [1], [33], [2] as it is being used around the world to decide upon the spending of public (mostly municipal) money.",
"Here we consider a setting of participatory budgeting in which the projects are classified into groups that might be intersecting, and each group comes with its own budget constraint, so that the result of the PB—i.e., the aggregated bundle—shall respect not only the global budget limit, but also the limits of each of the groups.",
"To make things concrete and formal, below is the decision version of our problem (in the optimization version of Group-PB, denoted Max-Group-PB, the goal is to maximize the utility): Table: NO_CAPTIONTable: NO_CAPTION W.l.o.g., we assume $b(F) \\le B$ for every $F \\in \\mathcal {F}$ , and that no two sets in $\\mathcal {F}$ are identical, i.e., $\\forall _{S_1,S_2 \\in \\mathcal {F}} \\quad S_1 \\ne S_2$ .",
"Also, without loss of generality, we assume that every project in $P$ is approved by at least one voter.",
"Example 1 Let $P=\\lbrace p_1,p_2,p_3,p_4\\rbrace $ .",
"Let the cost of projects be as follows: $c(p_1)=2,c(p_2)=1, c(p_3)=3, c(p_4)=1$ .",
"Suppose that we have only 2 voters, say $v,v^{\\prime }$ and $P_v=\\lbrace p_1,p_2,p_3\\rbrace , P_{v^{\\prime }}=\\lbrace p_3,p_4\\rbrace $ .",
"Let $\\mathcal {F}= \\lbrace F_1=\\lbrace p_1,p_3\\rbrace , F_2=\\lbrace p_2,p_4\\rbrace \\rbrace $ .",
"Let $b(F_1)=3, b(F_2)=2$ .",
"Note that we are allowed to take only one project from $F_1$ .",
"Let $B=5$ and $u=3$ .",
"Let $X=\\lbrace p_3,p_4\\rbrace $ .",
"Note that $|P_v \\cap X|=1$ and $|P_{v^{\\prime }}\\cap X|=2$ .",
"Thus, $\\sum _{v \\in \\mathcal {V}}|P_v \\cap X|= 3 =u$ .",
"Further, $c(p_3)+c(p_4)=4\\le B$ , and the cost of projects from sets $F_1$ and $F_2$ are 3 and 1, respectively, that is, $\\sum _{p\\in F_1}c(p) =3= b(F_1)$ and $\\sum _{p\\in F_2}c(p)=1 \\le b(F_2)$ .",
"Our work is motivated mainly by the following use-cases: Geographical budgeting: Consider a city (say, Paris), consisting of several districts.",
"To not spend all public funds on projects from, say, only one district, Group-PB is useful: group projects according by districts and select appropriate budget limits (making sure that, e.g., none of Paris's 20 districts would use more than $10\\%$ of the total budget).",
"(A more fine-grained solution, incorporating neighborhoods, streets, etc., is also possible.",
"Currently, such geographic inclusiveness is usually achieved ad hoc by holding separate per-district elections).",
"Thematic budgeting: Projects usually can be naturally grouped into types, e.g., education projects, recreational projects, and so on.",
"Group-PB is useful here: group projects accordingly, making sure that not all the budget is being spent on, say, projects of only one type.",
"Non-budgeting use-cases: Group-PB is useful in contexts other than PB: e.g., to decide which processes to run on a time-limited computing server, where available processes can be naturally grouped into types and it is not desired to use all computing power for, say, processes of only one type."
],
[
"Related Work",
"The literature on PB is quite rich [1]; formally, we generalize the framework of Talmon and Faliszewski [33] by adding group structures to approval-based PB.",
"Jain et al.",
"[23] and Patel et al.",
"[31] also consider—albeit significantly simpler—group structures (with layerwidth 1; see Definition REF ).",
"Fairness constraints are studied, e.g., in the contexts of influence maximization [35], clustering [11], and allocation problems [3].",
"Our focus is on fairness in PB (e.g., not spending all funds on one district).",
"Recently, Hershkowitz et al.",
"[18] introduced a district-fairness notion by allowing projects have different utility for different districts.",
"There are papers on fairness and group structures for the special model of multiwinner elections [22], [10], [36], [20], [17].",
"Technically, Group-PB is a special case of the $d$ -Dimensional Knapsack problem ($d$ -DK; also called Multidimensional Knapsack) [25]: given a set of items, each having a $d$ -dimensional size-vector and its utility, a $d$ -dimensional knapsack capacity vector $\\beta $ with an entry for each dimension, and required integer utility—with all input numbers being non-negative integers—the goal is to choose a subset of the items with at least the required total utility and such that the sum of the chosen items' sizes is bounded by the knapsack capacity, in each dimension.",
"$d$ -DK generalizes Group-PB: items in $d$ -DK correspond to projects; fix an order on $\\mathcal {F}$ , i.e., $(F_1,F_2,\\dots ,F_g)$ , resulting in $d = g+1$ many dimensions, a $(g+1)$ -dimensional size vector $\\gamma $ for an item $p \\in P$ , defined by $\\gamma _p(i) = c(p)$ if $p \\in F_i$ and $\\gamma _p(i) = 0$ otherwise, $\\gamma _p(g+1) = c(p)$ for every $p\\in P$ , corresponding to a global budget, utility of an item $p \\in P$ equals its approval score, required utility in $d$ -DK equals $u$ , and the $(g+1)$ -dimensional bin $\\beta $ is defined via $\\beta (i) = b(F_i)$ for $i \\in \\lbrace 1,2,\\dots ,g\\rbrace $ , with $\\beta (g+1) = B$ .",
"So, Group-PB is an instance of $(g+1)$ -DK where each item $p \\in P$ has only two possible sizes over dimensions, i.e., 0 and $c(p)$ .",
"Crucially, as our model is a special case we can use our special instance structure; hence, we treat results for $d$ -DK as a good benchmark, in particular, the (in)approximability results for $d$ -DK [25].",
"Recently, many papers introducing fairness components to well known computational problems have appeared.",
"One of them models fairness among voters by introducing a more general utility function for the voters.",
"In essence, instead of a linear function one may apply any function.",
"For example, a harmonic utility function $f(q) = \\sum _{q=1}^i 1/q$ (or any concave) models the law of diminishing marginal utility known in economics.",
"Having unit-cost projects and harmonic utility function we get the Proportional Approval Voting problem [34], [24], [7], [4], [14].",
"When additionally allowing projects to have different costs, we fall into Fair Knapsack [15].",
"Note that these problems model fairness by specifying what is the total utility of an outcome.",
"In essence, a voting rule with properly chosen utility functions disallows a large group of voters to decide a final outcome being their most preferred projects.",
"Hence preferences of a small group impact a solution as well.",
"A more general framework was proposed by Jain et al.",
"[23] in which projects are partitioned into groups and a utility function can be a general non-decreasing function.",
"For each voter a utility function is applied for each group of projects separately.",
"This models interaction between projects.",
"The most interesting utility functions are concave (and convex) ones that model substitution (and complementarity respectively) effects for groups of projects.",
"For such functions Jain et al.",
"[23] achieved more positive algorithmic results (polynomial-time or fixed-parameter tractable (FPT) algorithms) in contrast to general utility functions that are not possible to approximate up to a factor better than $n^{o(1)}$ (assuming Gap-ETH).",
"For concave utility functions also utility-lost minimization variants where considered [32], [7], motivated by some fairness notion in facility location context.",
"In contrast to the works described in the previous paragraph, in this paper we focus on the most natural and the simplest linear utility function, which has practical advantages, but to achieve fairness we group projects and add budget-limits for each group, i.e., every feasible outcome does not exceed a budget-limit among the given groups of projects.",
"A project may be contained in a few or none of the groups.",
"Therefore, a large group of voters cannot force choosing only one type of projects (educational, environmental etc.)",
"that is supported by the large group.",
"There is a recent paper [31] which introduce categories for the projects, and consider additional fairness notions among the categories.",
"First of all, they allow to have both a lower-bound and an upper-bound on the total cost of the projects taken from each category.",
"As a separate model they propose to have a lower-bound and an upper-bound on the cardinality of the set of projects taken from each category.",
"The third model makes use of lower-bounds and upper-bounds on the total utility achieved by the projects taken from each category.",
"Further, instead of approval ballots they consider general values for utilities (they call it values) and general costs of projects, i.e., they can be any non-negative real numbers.",
"Therefore it may seem that Patel et al.",
"[31] considered a much more general model than we do, but there is a very crucial difference: their categories of projects make a partition of projects, i.e., each project belongs to exactly one category (similarly to a project interaction model introduced by Jain et al. [23]).",
"Our model allows to have a family of groups of projects $\\mathcal {F}$ that is any subset of $2^P$ .",
"It means we could have even $2^{|P|}$ many groups in an instance in contrast to the model of Patel et al.",
"[31] that can have at most $|P|$ groups.",
"It shows that the model of Patel et al.",
"[31] and our model have a non-empty intersection and they generalize the intersection in different directions.",
"In particular, the model proposed by Patel et al.",
"[31] with only cost upper-bounds on categories is a special case of our model that has layerwidth (see Definition REF ) equal to 1 and which is a special case of hierarchical instances that we show how to solve in polynomial time.",
"In contrast to our results, the results presented by Patel et al.",
"[31] for cost upper-bounds provide the required total utility (an exact solution) but violate group cost bounds and a global budget by a fixed factor $\\epsilon $ .",
"They rely on a standard bucketing technique extensively, which was used to construct a fully polynomial-time approximation scheme (FPTAS) for the classical knapsack problem [21], [28].",
"This limits the number of different costs and also causes possible bound violations.",
"Our results do not violate any of the constraints.",
"This is possible because the utilities of the projects come from approval ballots hence they are integral and (we may think) they are decoded in unary.",
"In contrast to the work of Patel et al.",
"[31], which considers approximation algorithms, we consider also parameterized complexity (hence exact feasible solutions)."
],
[
"Our Contributions",
"We introduce and study Group-PB, first demonstrating its computational intractability even for some very restricted cases (Theorem REF ); positively, we show that Group-PB can be solved efficiently for constant number of groups (Theorem REF ), for instances with hierarchical group structure (i.e., any pair of groups must be either non-intersecting or in containment relation; Lemma REF ) or close to being such (Theorems REF and REF ).",
"Efficient approximation algorithms exist for some cases (Theorem REF ).",
"Fixed parameter tractability wrt.",
"the number of groups is open (Open Question REF ), however we have an approximation scheme that is ${\\mathsf {FPT}}$ wrt.",
"the number of groups (Theorem REF ) as well as ${\\mathsf {W}}[1]$ -hardness for a slightly more general problem (Theorem REF ).",
"Furthermore, Group-PB can be solved in polynomial time if a project belongs to at most one group as such instances are hierarchical, but becomes ${\\mathsf {NP}}$ -hard as soon as a project can belong to two groups (Theorem REF ).",
"Table REF lists most of our complexity results.",
"Tables REF and REF summarize our (in)approximability results.",
"Table: Parameterized complexity of Group-PB wrt.",
"the layerwidth ℓ\\ell (see Definition ), b max =max{b(F):F∈ℱ}b_{\\max }=\\max \\lbrace b(F): F \\in \\cal {F}\\rbrace , s=max{|F|:F∈ℱ}s=\\max \\lbrace |F|: F \\in \\mathcal {F}\\rbrace , the maximum number 𝚊𝚙𝚙\\mathtt {app} of projects a voter approves, the number nn of voters, the required utility uu, the number D g D_g (D p D_p) of groups (resp., projects) to delete to get a hierarchical structure, the number of groups g=|ℱ|g=|\\mathcal {F}|, and the total budget BB."
],
[
"Preliminaries",
"We use the notation $[n] = \\lbrace 1,2,\\dots ,n\\rbrace $ for $n \\in \\mathbb {N}$ .",
"In parameterized complexity, problem instances are associated with a parameter, so that a parameterized problem $\\rm \\Pi $ is a subset of $\\Sigma ^{*} \\times \\mathbb {N}$ , where $\\Sigma $ is a finite alphabet.",
"A corresponding instance is a tuple $(x,k)$ , where $x$ is a classical problem instance and $k$ is the parameter.",
"A central notion in parameterized complexity is fixed-parameter tractability ($\\mathsf {FPT}$ , in short), which means, for a given instance $(x,k)$ , decidability in $f(k) \\cdot {\\sf poly}(|x|)$ time, where $f(\\cdot )$ is an arbitrary computable function and ${\\sf poly}(\\cdot )$ is a polynomial function.",
"The Exponential Time Hypothesis (ETH) is a conjecture stating that 3-SAT cannot be solved in time that is subexponential in the number of variables.",
"For more about parameterized complexity see, e.g., the book of Cygan et al.",
"[9]."
],
[
"Layer Decompositions",
"The following is a useful structural property.",
"Definition 2 (Layer Decomposition) A layer decomposition of a family of sets $\\mathcal {F}$ is a partition of the sets in $\\mathcal {F}$ such that every two sets in a part are disjoint.",
"Each part is a layer.",
"Definition 3 (Layerwidth $\\ell $ ) The width of a layer decomposition is the number of layers in it.",
"The layerwidth of a family of sets $\\mathcal {F}$ , denoted by $\\ell (\\mathcal {F})$ (or simply $\\ell $ if $\\mathcal {F}$ is clear from the context), is the minimum width among all possible layer decompositions of $\\mathcal {F}$ .",
"We consider the problem of finding a layer decomposition, and refer to the problem of finding one of minimum width as Min Layer Decomposition, with its decision version called Layer Decomposition, which, given a family $\\mathcal {F}$ of sets and an integer $\\ell $ , asks for the existence of a layer decomposition with width $\\ell $ .",
"Unfortunately, Layer Decomposition is intractable via a reduction from Edge Coloring.",
"Theorem 4 Layer Decomposition is ${\\mathsf {NP}}$ -hard even when $\\ell =3$ and $s=2$ , where $s$ is the maximum size of a set in the given family $\\mathcal {F}$ .",
"We provide a polynomial time reduction from the Edge Coloring problem, in which we are given a graph $G$ and an integer $k$ ; we shall decide the existence of a mapping $\\Psi \\colon E(G)\\rightarrow [k]$ such that if two edges $e,e^{\\prime } \\in E(G)$ share an endpoint, then $\\Psi (e)\\ne \\Psi (e^{\\prime })$ .",
"This problem, even when $k=3$ , is known to be ${\\mathsf {NP}}$ -hard [19].",
"We call such a function a proper coloring function.",
"We create a family of sets, $\\mathcal {F}$ , as follows.",
"For every edge $uv \\in E(G)$ , we have a set $\\lbrace u,v\\rbrace $ in the family $\\mathcal {F}$ .",
"We set $\\ell =k$ .",
"Next, we prove the equivalence between the instances $(G,k)$ of the Edge Coloring problem and $(\\mathcal {F},\\ell )$ of the Layer Decomposition problem.",
"In the forward direction, let $\\Psi \\colon E(G)\\rightarrow [k]$ be a proper coloring function.",
"We create a partition, $L_1,\\ldots ,L_\\ell $ , of the sets in $\\cal F$ as follows: $L_i = \\lbrace \\lbrace u,v\\rbrace \\colon \\Psi (uv)\\in i\\rbrace $ .",
"Due to the definition of a proper coloring function, every two sets in a part $L_i$ , $i\\in [\\ell ]$ , are disjoint.",
"In the backward direction, let $L_1,\\ldots ,L_\\ell $ be a layer decomposition of $\\mathcal {F}$ .",
"We define proper coloring function $\\Psi $ as follows: if the set $\\lbrace u,v\\rbrace \\in L_i$ , $i\\in [k]$ , then $\\Psi (uv) = i$ .",
"We claim that $\\Psi $ is a proper coloring function.",
"Suppose not, then there are two edges $e,e^{\\prime }$ that share endpoints and $\\Psi (e)=\\Psi (e^{\\prime })=i$ .",
"Without loss of generality, let $e=uv$ and $e^{\\prime }=uw$ .",
"Since $\\Psi (e)=\\Psi (e^{\\prime })=i$ , $\\lbrace u,v\\rbrace ,\\lbrace u,w\\rbrace \\in L_i$ , contradicting that $L_i$ is a part in a layer decomposition of $\\mathcal {F}$ .",
"A reduction to 2-Graph Coloring gives a polynomial-time algorithm for layerwidth 2.",
"Theorem 5 There exists a polynomial-time algorithm that finds a layer decomposition of layerwidth two, if it exists.",
"We begin with giving a polynomial time reduction from Layer Decomposition to the Vertex Coloring problem, in which given a graph $G$ and an integer $k$ , we need to decide the existence of a mapping $\\Psi \\colon V(G)\\rightarrow [k]$ such that if $uv\\in E(G)$ , then $\\Psi (u)\\ne \\Psi (v)$ .",
"We call this mapping a proper vertex coloring function.",
"Given an instance $({\\cal F},\\ell )$ of Layer Decomposition, we create an instance $(G,k)$ of Vertex Coloring as follows.",
"For every set $F \\in {\\cal F}$ , we add a vertex $u_F$ in the graph $G$ .",
"Next, we define the edge set of $G$ .",
"Let $F$ and $F^{\\prime }$ be two sets in ${\\cal F}$ such that $F \\cap F^{\\prime } \\ne \\emptyset $ , then we add an edge $u_Fu_{F^{\\prime }}$ to $G$ .",
"We set $k=\\ell $ .",
"Next, we prove the equivalence between the instance $({\\cal F},\\ell )$ of Layer Decomposition and $(G,k)$ of Vertex Coloring.",
"In the forward direction, let $L_1,\\ldots ,L_\\ell $ be a layer decomposition of $\\mathcal {F}$ .",
"We construct a proper vertex coloring function $\\Psi $ as follows.",
"If a set $F$ belongs to the set $L_i$ , where $i\\in [\\ell ]$ , then $\\Psi (u_F)=i$ .",
"We claim that if $u_Fu_{F^{\\prime }} \\in E(G)$ , then $\\Psi (u_F)\\ne \\Psi (u_{F^{\\prime }})$ .",
"Consider an edge $u_Fu_{F^{\\prime }} \\in E(G)$ .",
"Note that by the construction of the graph $G$ , $F\\cap F^{\\prime }\\ne \\emptyset $ .",
"Therefore, sets $F$ and $F^{\\prime }$ belong to different sets in $\\lbrace L_1,\\ldots ,L_\\ell \\rbrace $ .",
"Hence, $\\Psi (u_F)\\ne \\Psi (u_{F^{\\prime }})$ .",
"In the backward direction, let $\\Psi $ be a proper vertex coloring function.",
"We create a layer decomposition $L_1,\\ldots ,L_\\ell $ as follows.",
"If for a vertex $u_F$ , $\\Psi (u_F)=i$ , then we add $F$ to layer $L_i$ .",
"Next, we prove that every two sets in every layer $L_i$ , $i\\in [\\ell ]$ , are disjoint.",
"Towards the contradiction, suppose that there exist two sets $F,F^{\\prime }$ in a layer $L_i$ , $i\\in [\\ell ]$ such that $F\\cap F^{\\prime } \\ne \\emptyset $ .",
"Since $F,F^{\\prime }$ are in layer $L_i$ , by the construction of the layer decomposition, we have that $\\Psi (u_F)=\\Psi (u_{F^{\\prime }})=i$ .",
"Since $F\\cap F^{\\prime } \\ne \\emptyset $ , by the construction of $G$ , $u_Fu_{F^{\\prime }}\\in E(G)$ .",
"This contradicts the fact that $\\Psi $ is a proper vertex coloring function.",
"Since we can find a proper coloring function $\\Psi \\colon V(G)\\rightarrow [2]$ in polynomial time, if it exists, as it is equivalent to checking if the given graph is a bipartite graph (see, e.g., [12]), the proof is complete.",
"An important special case is when the family of sets is hierarchical.",
"Definition 6 (Hierarchical Family) A family of sets $\\cal {F}$ is called hierarchical, if every two sets $F_1$ and $F_2$ in the family $\\cal {F}$ are either disjoint or $F_1\\subset F_2$ or $F_2\\subset F_1$ .",
"Theorem 7 There exists a polynomial-time algorithm that solves a given instance $({\\mathcal {F}},\\ell )$ of Layer Decomposition when $\\mathcal {F}$ is a hierarchical family.",
"Let $\\mathcal {F}^{\\prime }=\\mathcal {F}\\cup \\lbrace S\\rbrace $ , where $S$ is the set of all the elements which are in the sets in the family $\\cal {F}$ , and $\\ell ^{\\prime }=\\ell +1$ .",
"Clearly, $(\\cal {F},\\ell )$ is a yes-instance of Layer Decomposition if and only if $(\\cal {F}^{\\prime },\\ell ^{\\prime })$ is a yes-instance of Layer Decomposition.",
"We first create a directed graph, $G$ , called a subset relationship graph, as follows.",
"For every set $F \\in {\\mathcal {F}}^{\\prime }$ , we add a vertex $v_F$ in $G$ .",
"Now, if $F\\subset F^{\\prime }$ , where $F,F^{\\prime } \\in {\\mathcal {F}}$ , then we add an arc $(v_{F^{\\prime }},v_{F})$ to the graph $G$ .",
"Note that $G$ is a transitive acyclic graph.",
"Let $\\sigma $ be a topological ordering (a topological ordering is an ordering, $\\sigma $ , of the vertices of a digraph such that if $(u,v)$ is an arc in the digraph, then $\\sigma (u)<\\sigma (v)$ ; and it can be found for an acyclic graph in polynomial time [27]) of $G$ .",
"Next, we create an out-tree rooted at $\\sigma (1)$ , say $T$ , using DFS in which we first traverse the neighbor of a vertex which appears first in the ordering $\\sigma $ .",
"That is, if the currently visited vertex is $v$ , and $z_1,z_2$ are out-neighbors of $v$ , then we first visit $z_1$ if $\\sigma (z_1)<\\sigma (z_2)$ , and we first visit $z_2$ otherwise.",
"We create a partition, say $\\cal {L}$ , of $\\mathcal {F}$ as follows: Sets corresponding to the vertices at a level of out-tree $T$ forms a layer in $\\cal {L}$ .",
"We first prove that $\\cal {L}$ is a layer decomposition.",
"Suppose not, then there exist two sets $F_1,F_2$ in a layer in $\\cal {L}$ which are not disjoint.",
"Without loss of generality, let $F_1 \\subset F_2$ .",
"Then, due the construction of $G$ , $(v_{F_2},v_{F_1})$ is an arc in $G$ .",
"This implies that $\\sigma (F_2)<\\sigma (F_1)$ .",
"Since $\\mathcal {F}$ is a hierarchical family, if $F_2 \\subset F$ , then $F_1\\subset F$ .",
"Thus, an in-neighbor of $v_{F_1}$ is also an in-neighbor of $v_{F_2}$ .",
"Therefore, by our construction of out-tree, we first visit $v_{F_2}$ than $v_{F_1}$ .",
"Since $(v_{F_2},v_{F_1})$ is an arc in $G$ , it contradicts that $v_{F_2}$ and $v_{F_1}$ are at the same level in the out-tree $T$ which is constructed using DFS.",
"Next, we argue that the width of $\\cal {L}$ is $\\ell ^{\\prime }$ if and only if $(\\mathcal {F}^{\\prime }, \\ell ^{\\prime })$ is a yes-instance of Layer Decomposition.",
"The forward direction follows trivially.",
"Next, we show that if the width of $\\cal {L}$ is more than $\\ell ^{\\prime }$ , then $(\\mathcal {F}^{\\prime }, \\ell ^{\\prime })$ is a no-instance of Layer Decomposition.",
"Suppose that the width of $\\cal {L}$ is more than $\\ell ^{\\prime }$ , then due to the construction of the layer decomposition, the number of levels in the out-tree $T$ is more than $\\ell ^{\\prime }$ .",
"This implies that there exists a directed path from root whose length is at least $\\ell ^{\\prime }+1$ .",
"Due to the transitivity of graph $G$ , directed graph induced on the vertices in this path forms a tournament.",
"Thus, all the sets corresponding to vertices in this path belong to distinct layers.",
"Thus, the layerwidth of $\\mathcal {F}^{\\prime }$ is at least $\\ell ^{\\prime }+1$ .",
"Remark 8 The general idea of the algorithm described in the proof of Theorem REF is to build a graph with one vertex for each group and edges corresponding to group intersections, followed by traversing the graph in topological order and constructing the corresponding hierarchical tree.",
"Note that, conveniently, the algorithm can be modified to construct an ordered layer decomposition such that every set in the $i$ -th layer is a subset of a set in the $(i-1)$ -th layer."
],
[
"Parameters for Group-PB",
"Next we discuss the parameters we consider for Group-PB: the number of projects ($m$ ), the number of voters ($n$ ), the maximum size of an approval set ($\\mathtt {app}$ ), the budget ($B$ ), the maximum size of a group in the family $\\mathcal {F}$ ($s$ ), the utility ($u$ ), the layerwidth ($\\ell $ ), the size of the family $\\mathcal {F}$ ($g$ ), and $b_{\\max }=\\max _{F\\in \\mathcal {F}}b(F)$ .",
"The motivation for considering these is the following: the number of voters, $n$ , can be small in cases when we do PB by the council or in a small community; $m$ is sometimes quite small (e.g., the PB instances of Stanford Participatory Budgeting Platformhttps://pbstanford.org/ usually consist of only 10-20 projects); $\\ell $ and $b_{\\max }$ can be set by designer, but they are generally not small, yet we add them for completeness; the budget, $B$ and $u$ , are also generally not small, but added for completeness; $\\mathtt {app},s$ and $g$ can be set by the designer, and they are usually rather small, e.g.",
"5 to 10.",
"In particular, the main focus of our paper is on adding a group structure on top of a standard PB instance; from this point of view, a PB election designer can choose how complex she wants the group structure to be.",
"Thus, studying the complexity of Group-PB wrt.",
"$g$ sheds light on the effect of the group structure on the complexity of the problem (which is polynomial-time solvable when there are no groups); similarly, studying distance parameters to a hierarchical instance also serves the same point.",
"So, following our parameterized tractability results, a PB election designer can practically use our group structures, albeit perhaps not with arbitrary structural complexity."
],
[
"Initial Observations",
"For completeness, we mention that Group-PB is trivially ${\\mathsf {FPT}}$ wrt.",
"$m$ , by a brute-force algorithm in $\\mathcal {O}^*(2^m)$ time$\\mathcal {O}^*$ hides factors that are polynomial in the input size.",
"(and, as the Exponential Time Hypothesis implies a lower bound of $2^{o(|V|)}$ for Independent Set, we conclude a lower bound of $2^{o(m)}$ following the reduction in the proof of Theorem REF ).",
"Furthermore, Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$\\mathtt {app}+n$ as every project is approved by at least one voter, implying $m \\le \\mathtt {app}\\cdot n$ ."
],
[
"Intractability of General Instances",
"Next we prove intractability, showing that Group-PB is ${\\mathsf {NP}}$ -hard even when some of the input parameters are constant.",
"Note, importantly, that we can solve the standard PB problem—without project groups—in polynomial time (as it can be solved using dynamic programming via equivalence to Unary Knapsack [33]).",
"The following result is obtained via reductions from the Independent Set (IS) problem on 3-regular 3-edge colorable graphs.",
"Theorem 9 Group-PB is ${\\mathsf {NP}}$ -complete even when $b_{\\max }=1$ , $s=2$ , $\\mathtt {app}=1$ , and $\\ell =3$ ; and even when $b_{\\max }=1$ , $s=2$ , $n=1$ , and $\\ell =3$ .",
"We first show the proof for $b_{\\max }=1$ , $s=2$ , $\\mathtt {app}=1$ , and $\\ell =3$ ; afterwards, we show how to modify it to get the other claim.",
"We describe a polynomial-time reduction from the Independent Set (IS) problem on 3-regular 3-edge colorable graphs, in which given a graph $G=(V,E)$ , where each vertex has degree 3 and the edges of graph can be properly colored using 3 colors (no two edges that share an end-point are colored using the same color), and an integer $k$ , we need to decide the existence of a $k$ -sized independent set (there is no edge between any pair of vertices in the set).",
"IS is known to be ${\\mathsf {NP}}$ -complete on 3-regular 3-edge colorable graphs [8].",
"For every vertex $x\\in V(G)$ , we have a project $p_x$ in $P$ ; and for every edge $e \\in E(G)$ , we have two voters $w_e$ and $w_{\\hat{e}}$ in $\\mathcal {V}$ .",
"If $e=xy$ , then the voter $w_e$ approves projects $p_x$ and $w_{\\hat{e}}$ approves $p_y$ , that is, $P_{w_e}=\\lbrace p_x\\rbrace $ and $P_{w_{\\hat{e}}}=\\lbrace p_y\\rbrace $ .",
"For every edge $e=xy$ , we have a set $\\lbrace p_x,p_y\\rbrace $ in the family $\\mathcal {F}$ .",
"For every project $p\\in P$ , $c(p)=1$ .",
"For every set $F\\in \\mathcal {F}$ , $b(F)=1$ .",
"We set $B=k$ and $u=3k$ .",
"Since $G$ is 3-edge colorable, we can partition the sets in $\\mathcal {F}$ in 3 groups such that every two sets in a group are disjoint, and hence $\\ell =3$ .",
"Next, we prove the equivalence between the instance $(G,k)$ of IS and $(\\mathcal {V},P,\\mathcal {E},\\mathcal {F},c,B,b,u)$ of Group-PB.",
"In the forward direction, let $S$ be a solution to $(G,k)$ .",
"We claim that $P_S=\\lbrace p_x\\in P \\colon x\\in S\\rbrace $ is a solution to $(\\mathcal {V},P,\\mathcal {E},\\mathcal {F},c,B,b,u)$ .",
"Since $c(p)=1$ , for every project $p$ in $P$ , we have that $\\sum _{p\\in P_S}c(p)=k$ .",
"Since $S$ is an independent set in $G$ , for every $F \\in \\mathcal {F}$ , $\\sum _{p\\in P_S\\cap F}c(p)=1$ .",
"Since $G$ is a 3-regular graph, every project in $P_S$ is approved by three voters.",
"Moreover, since every voter approves only one project, $\\sum _{v\\in V}|P_v\\cap P_S|=3k$ .",
"In the backward direction, let $X$ be a solution to $(\\mathcal {V},P,\\mathcal {E},\\mathcal {F},c,B,b,u)$ .",
"We claim that $S=\\lbrace x\\in V(G) \\colon p_x\\in X\\rbrace $ is a solution to $(G,k)$ .",
"Since $b(F)=1$ , for every $F\\in \\mathcal {F}$ , clearly, $S$ is an independent set in $G$ .",
"We claim that $|S|=k$ .",
"Clearly, $|S|\\le k$ , otherwise $\\sum _{p\\in X}c(p) > k$ , a contradiction.",
"If $|S|<k$ , then as argued above, $\\sum _{v\\in V}|P_v\\cap X|<3k$ , a contradiction.",
"To get the other claim, in the above reduction, instead of adding two voters for every edge in the graph, we add only one voter who approves all the projects and set $u=k$ .",
"The rest of construction remains the same.",
"Then, using the same arguments as above, we have the second claim.",
"As Independent Set on general graphs is ${\\mathsf {W}}[1]$ -hard wrt.",
"the solution size, the next result follows using similar arguments as used in the proof of Theorem REF .",
"Theorem 10 Group-PB is $\\mathsf {W}$$[1]$ -hard wrt.",
"$u$ even when $s=2$ , $\\mathtt {app}=1$ , and $b_{\\max }=1$ ; and even when $s=2$ , $n=1$ , and $b_{\\max }=1$ .",
"Here, we give a polynomial-time reduction from the Independent Set problem, which is known to be ${\\mathsf {W}}[1]$ -hard wrt.",
"$k$ , where $k$ is a solution size [13].",
"For the first claim, the set of projects $P$ , the cost function $c$ , the family $\\mathcal {F}$ , the function $b$ , and the total budget $B$ is the same as in Theorem REF .",
"Next, for every vertex $x \\in V(G)$ , we have a voter $w_x$ which approves only the project $p_x$ .",
"Thus, $\\mathtt {app}=1$ .",
"We set $u=k$ .",
"The proof of correctness is similar to the proof of Theorem REF .",
"To get the other claim, we use exactly the same reduction as in the second part of the proof of Theorem REF .",
"Note that we have also $u=k$ , hence ${\\mathsf {W}}[1]$ -hardness wrt.",
"$u$ follows."
],
[
"Tractability of Hierarchical Instances",
"We start our quest for tractability by considering Group-PB instances whose group structure is hierarchical; when $\\mathcal {F}$ is hierarchical, we refer to the Group-PB problem as Hierarchical-PB.",
"Fortunately, such instances can be solved in polynomial time.",
"Lemma 11 Hierarchical-PB can be solved in polynomial time.",
"Let $(\\mathcal {V},P,\\mathcal {E},\\mathcal {F},c,B,b,u)$ be a given instance of Hierarchical-PB.",
"W.l.o.g., assume that $P$ is a set in the family ${\\cal F}$ (otherwise, add it to $\\mathcal {F}$ and set $b(P)=B$ ).",
"Using Remark REF , let $\\cal {L}$ be an ordered layer decomposition of ${\\cal F}$ such that every set is a subset of some set in the preceding layer, and note that the first layer is $\\lbrace P\\rbrace $ .",
"Let $S$ be a set in some layer, say $L_i$ , such that $|S|>1$ .",
"Suppose that there exists a project $p\\in S$ such that $p$ is not in any set of $L_{i+1}$ .",
"We add the set $\\lbrace p\\rbrace $ to $\\mathcal {F}$ and $L_{i+1}$ , and set $b(\\lbrace p\\rbrace )=c(p)$ (if the layer $L_{i+1}$ does not exist, then we add this new layer).",
"Note that we might increase the number of layers by 1.",
"Let $\\ell $ be the number of layers.",
"Now, we solve the problem using dynamic programming.",
"For a set $S \\in \\mathcal {F}$ in the $i$ -th layer, where $i\\in [\\ell -1]$ , such that $|S|>1$ , let $\\mathtt {Part}_S$ denote the partition of a set $S$ such that every part in $\\mathtt {Part}_S$ is a set in the $(i+1)$ -th layer.",
"For a singleton set $S$ , $\\mathtt {Part}_S=\\lbrace S\\rbrace $ .",
"For every set $S \\in {\\cal F}$ , we order the parts in $\\mathtt {Part}_S$ arbitrarily.",
"Let $|\\mathtt {Part}_S|$ denote the number of parts in $\\mathtt {Part}_S$ and let $\\mathtt {Part}_S(i)$ denote the $i$ -th part in $\\mathtt {Part}_S$ .",
"Our DP table entries are defined as follows: For every set $S \\in \\mathcal {F}$ , $j\\in \\vert \\mathtt {Part}_S \\vert $ , and utility $z\\in [u]$ $\\begin{split}T[S,j,z] = &\\text{ minimum cost of a bundle that has utility } z \\text{ and that is a subset of projects in first } j \\text{ parts in } \\mathtt {Part}_S.\\end{split}$ For a project $p$ , let $a(p)$ denote the number of voters who approve the project $p$ (approval score).",
"For a set $S$ , let $a(S) = \\sum _{p\\in S} a(p)$ .",
"We compute the table entries level-wise in bottom-up order, that is, we first compute the value corresponding to sets at lower levels.",
"Base case: For every set $S$ , where $S=\\emptyset $ or $S\\in {\\mathcal {F}}$ and $0\\le z \\le u$ $\\begin{split}T[S,0,z] = {\\left\\lbrace \\begin{array}{ll}0 & \\text{if } z=0 \\\\\\infty & \\text{otherwise}\\end{array}\\right.",
"}\\end{split}$ Recursive Step: For every set $S\\in {\\mathcal {F}}$ , $j \\in [|\\mathtt {Part}_S|]$ , and $0\\le z \\le u$ , we compute as follows: $\\begin{split}T[S,j,z] =\\min \\limits _{0\\le z^{\\prime } \\le z} \\lbrace T[S,j-1,z-z^{\\prime }] +T[\\mathtt {Part}_S(j),\\vert \\mathtt {Part}_S(j) \\vert ,z^{\\prime }]\\rbrace \\end{split}$ Next, we prove that (REF ) and (REF ) correctly compute $T[S,j,z]$ for all sets $S\\in {\\mathcal {F}}$ , $0\\le j \\le \\vert \\mathtt {Part}_S \\vert $ , and $0\\le z \\le u$ .",
"Note first that, for $j=0$ , the subset of $S$ that contains projects only from the first $j$ parts is $\\emptyset $ , with cost 0 and $a(\\emptyset ) = 0$ , implying the correctness of the base case.",
"For (REF ), we provide inequalities in both directions.",
"In one direction, let $\\hat{S}\\subseteq S$ be a bundle of minimum cost that is a subset of projects in first $j$ parts in $\\mathtt {Part}_S$ and has utility $z$ .",
"Note that cost of bundle $\\hat{S}\\cap \\mathtt {Part}_S(j)$ is a valid entry for $T[\\mathtt {Part}_S(j),\\vert \\mathtt {Part}_S(j) \\vert , a(\\hat{S}\\cap \\mathtt {Part}_S(j))]$ and cost of the bundle $\\hat{S}\\setminus \\mathtt {Part}_S(j)$ is a valid entry for $T[S,j-1,z-a(\\hat{S}\\cap \\mathtt {Part}_S(j))]$ .",
"Thus, $T[S,j,z] \\ge \\min \\limits _{0\\le z^{\\prime } \\le z} \\lbrace T[S,j-1,z-z^{\\prime }]+ T[\\mathtt {Part}_S(j),\\vert \\mathtt {Part}_S(j) \\vert ,z^{\\prime }]\\rbrace .$ For the other direction, for any $0\\le z^{\\prime } \\le z$ , let $\\hat{S} \\subseteq S$ be a bundle of minimum cost which is a subset of projects in first $j-1$ parts in $\\mathtt {Part}_S$ and has utility $z-z^{\\prime }$ and $\\tilde{S} \\subseteq \\mathtt {Part}_S(j)$ be a bundle of minimum cost with utility $z^{\\prime }$ .",
"Note that $\\hat{S}\\cap \\tilde{S}=\\emptyset $ .",
"Thus, $\\hat{S}\\cup \\tilde{S}$ is a valid entry for $T[S,j,z]$ ; so $\\begin{split}T[S,j,z] \\le \\min \\limits _{0\\le z^{\\prime } \\le z} \\lbrace T[S,j-1,z-z^{\\prime }]+T[\\mathtt {Part}_S(j),\\vert \\mathtt {Part}_S(j) \\vert ,z^{\\prime }]\\rbrace \\end{split}$ This proves the correctness of (REF ).",
"Using (REF ) and (REF ), we compute $T[S,j,z]$ , for all $S\\in {\\mathcal {F}}$ , $0\\le j \\le \\vert \\mathtt {Part}_S \\vert $ , and $0\\le z\\le u$ level-wise in bottom-up order.",
"Note that $T[P,|\\mathtt {Part}_P|,u]$ has the answer.",
"The running time of the algorithm is $\\mathcal {O}(|\\mathcal {F}|su)$ as the size of any set in ${\\mathcal {F}}$ is at most $s$ .",
"Some instances might not be hierarchical but only close to being such, thus we study two distance parameters, namely, the minimum number $D_g$ of groups and the minimum number $D_p$ of projects, respectively, whose deletion results in a hierarchical instance.",
"We have the following lemmas—used later in the proofs for the parameterized complexity of Group-PB wrt.",
"$D_g+s$ and $D_p$ —which are concerned with computing the set of groups/projects whose deletion leads to hierarchical instance; their proofs follow branching arguments, as, for $D_g$ , at least one set from each pair of conflicting groups shall be deleted, and, for $D_p$ , for a pair of conflicting groups $G_1, G_2$ , either $G_1 \\setminus G_2$ or $G_2 \\setminus G_1$ or $G_1 \\cap G_2$ shall be removed.",
"Lemma 12 There exists an algorithm, running in $\\mathcal {O}^*(2^{D_g})$ time, that finds a minimum-sized set of groups whose deletion results in a hierarchical instance.",
"First, observe that, if there are two non-disjoint groups $G_1,G_2$ such that neither $G_1 \\subseteq G_2$ nor $G_2 \\subseteq G_1$ , then we have to delete one of this group.",
"So, we branch as follows: delete either $G_1$ or $G_2$ , and decrease $D_g$ by 1 in each branch.",
"We only branch further if $D_g >0$ .",
"If there is a leaf in this branching tree for which the reduced family is hierarchical, then this leaf gives us required deletion set.",
"The algorithm runs in $\\mathcal {O}^*(2^{D_g})$ -time.",
"Lemma 13 There exists an algorithm, running in $\\mathcal {O}^*(3^{D_p})$ time, that finds a minimum-sized set of projects whose deletion results in a hierarchical instance.",
"First, observe that, if there are two non-disjoint groups $G_1,G_2$ such that neither $G_1 \\subseteq G_2$ nor $G_2 \\subseteq G_1$ , then we have to delete either intersecting part so that both the groups are disjoint or we delete $G_1\\setminus G_2$ or $G_2 \\setminus G_1$ so that one of them is subset of other.",
"So, we branch as follows: Delete $G_1\\cap G_2$ and decrease $D_p$ by $|G_1 \\cap G_2|$ or delete $G_1\\setminus G_2$ and decrease $D_p$ by $|G_1\\setminus G_2|$ or delete $G_2 \\setminus G_1$ and decrease $D_p$ by $|G_2\\setminus G_1|$ .",
"Clearly, in each branch, we decrease $D_p$ by at least one.",
"If there is a leaf in this branching tree for which the reduced family is hierarchical, then this leaf gives us required deletion set.",
"The algorithm runs in $\\mathcal {O}^*(3^{D_p})$ -time.",
"Unfortunately, we have the following intractability result.",
"Theorem 14 Group-PB is ${\\mathsf {NP}}$ -hard even when $D_g=2$ and $\\ell =2$ .",
"We provide a polynomial-time reduction from the Partition problem, in which, given a set of natural numbers $X=\\lbrace x_1,\\ldots ,x_{\\tilde{n}}\\rbrace $ , the goal is to partition $X$ into two parts, $X_1$ and $X_2$ , such that the sum of the numbers in $X_1$ is same as the sum of the numbers in $X_2$ .",
"The Partition problem is known to be ${\\mathsf {NP}}$ -hard [16].",
"We construct an instance $(\\mathcal {V},P,\\mathcal {E},\\mathcal {F},c,B,b,u)$ of Group-PB as follows: for every integer $x_i\\in X$ , we add two projects in $P$ , say $x_i^1$ and $x_i^2$ , and set $c(x_i^1)=c(x_i^2)=x_i$ .",
"We define only one voter $v$ approving all the projects.",
"For every $x_i \\in X$ , we add $\\lbrace x_i^1,x_i^2\\rbrace $ to $\\mathcal {F}$ and define $b(\\lbrace x_i^1,x_i^2\\rbrace )=x_i$ .",
"Furthermore, we add sets $\\lbrace x_i^1\\colon x_i\\in X\\rbrace $ and $\\lbrace x_i^2\\colon x_i\\in X\\rbrace $ to $\\mathcal {F}$ ; and set $b(\\lbrace x_i^1\\colon x_i\\in X\\rbrace )= b(\\lbrace x_i^2\\colon x_i\\in X\\rbrace )=\\sum _{x_i\\in X} {x_i}{2}$ .",
"Let $B=\\sum _{x_i\\in X}x_i$ and $u=\\tilde{n}$ .",
"Note that the layerwidth of $\\mathcal {F}$ is 2 and by removing the sets $\\lbrace x_i^1\\colon x_i\\in X\\rbrace $ and $\\lbrace x_i^2\\colon x_i\\in X\\rbrace $ from $\\mathcal {F}$ leads to a hierarchical family.",
"The main idea of the correctness is that the decision whether to fund project $x_i^1$ or project $x_i^2$ (as both cannot be funded) corresponds to deciding whether to take $x_i$ to the first part or the second part of the solution to the instance of Partition.",
"We next prove formally the equivalence between the instance $X$ of the Partition problem and the instance $(\\mathcal {V},P,\\mathcal {E},\\mathcal {F},c,B,b,u)$ of Group-PB.",
"In the forward direction, let $(X_1,X_2)$ be a solution to $X$ .",
"We create a subset of projects, $S\\subseteq X$ , as follows.",
"If $x_i\\in X_1$ , where $i\\in [\\tilde{n}]$ , then we add $x_i^1$ to $S$ , otherwise we add $x_i^2$ to $S$ .",
"Since $\\vert S \\vert = \\tilde{n}$ , clearly, the utility of $S$ is $\\tilde{n}$ .",
"Since we pick either $x_i^1$ or $x_i^2$ in $S$ , for every $i\\in [\\tilde{n}]$ , for every set $F=\\lbrace x_i^1,x_i^2\\rbrace \\in \\mathcal {F}$ , $\\sum _{p\\in F\\cap S} = x_i$ .",
"Since the sum of all the numbers in $X_1$ is ${\\sum _{x_i\\in X}x_i}{2}$ , for the set $F=\\lbrace x_i^1\\colon x_i\\in X\\rbrace \\in \\mathcal {F}$ , $\\sum _{p\\in F\\cap S}= {\\sum _{x_i\\in X}x_i}{2}$ .",
"Similarly, for the set $F=\\lbrace x_i^2\\colon x_i\\in X\\rbrace \\in \\mathcal {F}$ , $\\sum _{p\\in F\\cap S}= {\\sum _{x_i\\in X}x_i}{2}$ .",
"In the backward direction, let $S$ be a solution to $(\\mathcal {V},P,\\mathcal {E},\\mathcal {F},c,B,b,u)$ .",
"Since $u\\ge \\tilde{n}$ , clearly, at least $\\tilde{n}$ many projects belong to $S$ .",
"Note that for any $i\\in [\\tilde{n}]$ , $x_i^1$ and $x_i^2$ both cannot belong to $S$ as $c(x_i^1)=c(x_i^2)=x_i$ and $b(\\lbrace x_i^1,x_i^2\\rbrace )=x_i$ .",
"Thus, as there are only $2\\tilde{n}$ many projects, either $x_i^1$ or $x_i^2$ belong to $S$ , but not both.",
"We construct a partition of $X=(X_1,X_2)$ as follows: if $x_i^1 \\in S$ , where $i\\in [\\tilde{n}]$ , then add $x_i$ to $X_1$ , otherwise add $x_i$ to $X_2$ .",
"Clearly, the sum of the numbers in $X_1$ is at most ${\\sum _{x_i\\in X}x_i}{2}$ as $b(\\lbrace x_i^1\\colon x_i\\in X\\rbrace )={\\sum _{x_i\\in X}x_i}{2}$ .",
"Similarly, the sum of the numbers in $X_2$ is at most ${\\sum _{x_i\\in X}x_i}{2}$ .",
"Now, suppose the sum of the numbers in at least one of these sets is less than ${\\sum _{x_i\\in X}x_i}{2}$ .",
"This implies that the total sum of the numbers in $X_1$ or $X_2$ is less than $\\sum _{x_i\\in X}x_i$ , this contradicts that for every $i\\in [\\tilde{n}]$ , either $x_i^1$ or $x_i^2$ is in $S$ .",
"This completes the proof.",
"Combining with $s$ helps.",
"Theorem 15 Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$D_g+s$ .",
"We can try all possible subsets of the deletion set; then, for each deletion set, we can try all possibilities of which projects to fund (i.e., going over all subsets of projects in the union of the groups in the deletion set); then, for the remaining budget, we solve using the polynomial-time algorithm for Hierarchical-PB, as described in the proof of Lemma REF .",
"The running time of the algorithm is $\\mathcal {O}^*(2^{D_g \\cdot s})$ .",
"Parametrization by the delete-project-distance of an instance to be hierarchical is tractable.",
"Theorem 16 Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$D_p$ .",
"We can use Lemma REF to find the deletion set.",
"Then we can go over all possibilities of which projects to fund from the deletion set and solve the remaining hierarchical instance, after updating the budgets, using the polynomial-time algorithm of Lemma REF ."
],
[
"Tractability with Few Groups",
"Next we concentrate on the number $g$ of groups as a parameter as, indeed, the groups are the new ingredient we bring to the standard model of PB.",
"First, we have the following result for the parameter $g$ .",
"Theorem 17 Group-PB is ${\\mathsf {XP}}$ wrt.",
"$g$ .",
"The algorithm follows from an enumeration over project types $t_R$ , $R \\subseteq \\mathcal {F}$ , where $t_R$ is defined as follows: a project of type $t_R$ belongs to all groups in $R$ and to none of the groups in $\\mathcal {F}\\setminus R$ .",
"We override the notation and write $p \\in t_R$ to say that project $p \\in P$ has type $t_R$ .",
"Note that each project has exactly one type out of the total of $2^g$ types of projects.",
"For every project type $t_R$ , and for every value $\\mu \\in \\lbrace 0, 1, 2, \\ldots , u\\rbrace $ , we use standard dynamic programming to find the cheapest set of projects of type $t_R$ that achieves at least utility $\\mu $ .",
"(For each of the $2^g$ types, this takes polynomial-time.)",
"Then, we go over all possibilities of achieving some utility from each of the types.",
"As we have $u+1$ possibilities for each project type, it follows that, in total, we consider at most $(u+1)^{2^g}$ cases.",
"If some solution is feasible then the instance is a yes-instance, otherwise it is a no-instance.",
"For correctness we focus only on yes-instances (for no-instances the correctness is straightforward).",
"Fix some feasible solution $X^*$ to the instance.",
"It achieves the desired total utility $u$ by getting some utility $u^*(R) \\in \\mathbb {N}$ from projects of type $t_R$ .",
"Note that our algorithm considers a solution $X$ constructed in a case when it receives utility at least $\\min \\lbrace u^*(R),u\\rbrace $ from projects of type $t_R$ .",
"We will argue that this solution is feasible.",
"$X$ achieves utility of at least $\\sum _{R \\subseteq \\mathcal {F}} \\min \\lbrace u^*(R),u\\rbrace \\ge u$ , where the inequality holds because of the following: (1) if there exists some $R \\subseteq \\mathcal {F}$ such that $u^*(R) \\ge u$ , then this is straightforward; (2) if for all $R \\subseteq \\mathcal {F}$ we have that $u^*(R) < u$ , then this follows from the feasibility of $X^*$ , i.e., $\\sum _{R \\subseteq \\mathcal {F}} u^*(R) \\ge u$ .",
"For group budget limits we note that $X$ is constructed from min-cost bundles of projects of type $t_R$ that achieve utility of at least $\\min \\lbrace u^*(R),u\\rbrace $ .",
"It means that the cost of projects of type $t_R$ in $X$ cannot be more expensive than the cost of projects of type $t_R$ in $X^*$ .",
"By aggregating this cost for each group $F \\in \\mathcal {F}$ separately we get that $\\sum _{p \\in F \\cap X} c(p) = \\sum _{R: F \\in R \\subseteq \\mathcal {F}}\\sum _{p \\in t_R \\cap X} c(p) \\le \\sum _{R: F \\in R \\subseteq \\mathcal {F}}\\sum _{p \\in t_R \\cap X^*} c(p) = \\sum _{p \\in F \\cap X^*} c(p) \\le b(F)$ .",
"Analogously we show feasibility regarding a global budget.",
"Since the running time of the algorithm above is $\\mathcal {O}^*((u+1)^{2^g})$ , we finish the proof.",
"Unfortunately, we do not know whether Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$g$ ; indeed, this is the main question left open.",
"Open Question 18 Is Group-PB ${\\mathsf {FPT}}$ wrt.",
"$g$ ?",
"Note, however, that in Subsection REF we provide a ${\\mathsf {W}}$ -hardness wrt.",
"$g$ proof, albeit for a slightly more general problem, in which we are also given utility requirements for each group.",
"Next we consider combined parameters.",
"Theorem 19 Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$g+u$ .",
"Follows from the proof of Theorem REF as the running time of the algorithm is $\\mathcal {O}^*((u+1)^{2^g})$ .",
"Careful MILP formulation implies the following.",
"Theorem 20 Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$g+n$ .",
"Recall that w.l.o.g.",
"we assume that every project is approved by at least one voter.",
"We construct a Mixed Integer Linear Program (MILP) in the following way.",
"We define a type of a project by a pair $(R,w)$ , where $R \\subseteq \\mathcal {F}$ and $w \\in [n]$ .",
"A project of type $(R,w)$ belongs to all the groups in $R$ (and to none of the groups in $\\mathcal {F}\\setminus R$ ) and it is approved by exactly $w$ voters.",
"Note that we have $n \\cdot 2^g$ types of projects.",
"We define an integer variable $x_{R,w}$ for all $R \\subseteq \\mathcal {F}$ and $ w \\in [n]$ , meaning how many projects of type $(R,w)$ are in a solution.",
"Let $|(R,w)|$ be the number of projects of type $(R,w)$ .",
"Next, we split $x_{R,w}$ into a sum of $|(R,w)|$ real variables: $x_{R,w} = \\sum _{i \\in [|(R,w)|]} y_{R,w,i},$ where $y_{R,w,i} \\in [0,1]$ is a continuous extension of a binary variable that indicates whether we take the $i$ -th cheapest projects of type $(R,w)$ to a solution.",
"From equation (REF ) we get also $x_{R,w} \\in \\lbrace 0,1,\\dots , |(R,w)|\\rbrace $ .",
"Note that we have $\\sum _{R \\subseteq \\mathcal {F}} \\sum _{w \\in [n]} |(R,w)| = m$ real variables $y_{R,w,i}$ because each project has exactly one type.",
"We need to implement the budget function.",
"We write a constraint for each group $F \\in \\mathcal {F}$ : $\\sum _{R: F \\in R \\subseteq \\mathcal {F}} \\sum _{w \\in [n]} \\sum _{i \\in [|(R,w)|]} y_{R,w,i} \\cdot c(R,w,i) \\le b(F),$ where $c(R,w,i)$ is the cost of the $i$ -th cheapest project of type $(R,w)$ .",
"Similarly, we add a global budget limit constraint as follows: $\\sum _{R \\subseteq \\mathcal {F}} \\sum _{w \\in [n]} \\sum _{i \\in [|(R,w)|]} y_{R,w,i} \\cdot c(R,w,i) \\le B.$ The remaining ingredient of MILP is the objective function: $\\max \\sum _{R \\subseteq \\mathcal {F}} \\sum _{w \\in [n]} w \\cdot x_{R,w}.$ We can transform any optimal solution $(x^*,y^*)$ of the MILP into an optimal solution $(x^*,y^{\\operatornamewithlimits{int}})$ consisting of integer variables only.",
"In particular, we define $y^{\\operatornamewithlimits{int}}_{R,w,i} = 1$ for $i \\in \\lbrace 1,\\dots , x^*_{R,w}\\rbrace $ and $y^{\\operatornamewithlimits{int}}_{R,w,i} = 0$ for $i \\in \\lbrace x^*_{R,w}+1, \\dots , |(R,w)|\\rbrace $ ."
],
[
"Optimality.",
"The objective value of such a new solution $(x^*,y^{\\operatornamewithlimits{int}})$ is the same as for $(x^*,y^*)$ (hence optimal) because the objective function depends only on $x_{R,w}$ and both the solutions have the same value for variables $x_{R,w}$ ."
],
[
"Feasibility.",
"We have $\\sum _{i \\in [|(R,w)|]} y^{\\operatornamewithlimits{int}}_{R,w,i}= \\sum _{i = 1}^{x^*_{R,w}} y^{\\operatornamewithlimits{int}}_{R,w,i} + \\sum _{i = x^*_{R,w}+1}^{|(R,w)|} y^{\\operatornamewithlimits{int}}_{R,w,i}= \\sum _{i = 1}^{x^*_{R,w}} 1 + \\sum _{i = x^*_{R,w}+1}^{|(R,w)|} 0= x^*_{R,w}.$ hence the integer solution $(x^*,y^{\\operatornamewithlimits{int}})$ is feasible for equation (REF ).",
"We have $c(R,w,i) \\le c(R,w,i+1)$ and $\\sum _{i \\in [|(R,w)|]} y^{\\operatornamewithlimits{int}}_{R,w,i} = x^*_{R,w} = \\sum _{i \\in [|(R,w)|]} y^*_{R,w,i}$ , hence we get $\\sum _{i \\in [|(R,w)|]} y^{\\operatornamewithlimits{int}}_{R,w,i} \\cdot c(R,w,i) \\le \\sum _{i \\in [|(R,w)|]} y^*_{R,w,i} \\cdot c(R,w,i).$ From this and feasibility of $(x^*,y^*)$ we get that the integer solution $(x^*,y^{\\operatornamewithlimits{int}})$ is also feasible for equations (REF ) and (REF )."
],
[
"Running time.",
"The MILP has at most $n \\cdot 2^g$ integer variables, $m$ real variables and $n \\cdot 2^g +2m+g+1$ constraints ($n \\cdot 2^g$ many constraints of type (REF ), $2m$ many constraints for variables $y_{R,w,i}$ , $g$ many group budget constraints (REF ) and 1 global budget constraint (REF )).",
"We can solve MILP using $\\mathcal {O}(p^{2.5p + o(p)} \\cdot |I|)$ arithmetic operations, where $|I|$ is the input size and $p$ is the number of integer variables [29], [5].",
"In particular, our MILP can be solved in $\\mathcal {O}^*((n \\cdot 2^g)^{\\mathcal {O}(n \\cdot 2^g)}) \\le \\mathcal {O}^*(2^{n \\log (n) \\cdot 2^{\\mathcal {O}(g)}})$ -time.",
"Remark 21 Note that $n \\cdot 2^g$ is only an upper-bound for the number of integer variables in the MILP.",
"Indeed, we can write a more strict upper-bound.",
"Let $t$ be the number of non-empty types $(R,w)$ , i.e., $t = |\\lbrace (R,w): R \\subseteq \\mathcal {F}, w \\in [n], |(R,w)|>0 \\rbrace |$ .",
"Let $A$ be the maximum approval score over projects, i.e., $A = \\max _{p \\in P} |\\lbrace v \\in \\mathcal {V}: p \\in P_v \\rbrace |$ .",
"Constraint (REF ) sets $x_{R,w}$ to be 0 for empty types $(R,w)$ .",
"It means that we have at most $A \\cdot t$ free integer variables, hence Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$A \\cdot t$ .",
"Also combining $g$ with the budget $B$ helps.",
"Theorem 22 Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$g+B$ .",
"Recall that w.l.o.g.",
"we assumed that $b(F) \\le B$ hence $b_{\\max } \\le B$ .",
"We can apply the DP for $(g+1)$ -DK, which runs in time upper-bounded by $\\mathcal {O}^*{(n \\cdot (b_{\\max }+1)^g (B+1))} \\le \\mathcal {O}^*(n \\cdot (B+1)^{(g+1)})$ [25].",
"Remark 23 We mention that the running time claimed by Kellerer et al.",
"is $\\mathcal {O}^*{(n \\cdot b_{\\max }^g \\cdot B)}$ .",
"This would give a polynomial time algorithm for hard instances after the reduction in Theorem REF because there we have $b_{\\max }=1$ .",
"It would imply ${\\mathsf {P}}={\\mathsf {NP}}$ .",
"The logical flaw of Kellerer et al.",
"[25] comes from counting the size of the DP table, in particular, in every dimension we should count the number of sub-problems by considering all possible values for a budget, i.e., all integer numbers between 0 and the budget limit in a dimension.",
"Indeed, asymptotically this $+1$ difference usually does not matter; for us, however, it is crucial."
],
[
"FPT Approximation Scheme for g",
"Recall that Group-PB is ${\\mathsf {XP}}$ wrt.",
"$g$ (Theorem REF ) and recall our open question regarding whether Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$g$ (Open Question REF ).",
"Next we show an approximation scheme for Max-Group-PB that is ${\\mathsf {FPT}}$ wrt.",
"$g$ (compare this result also to that described later, in Theorem REF , showing that there does not exist a constant-factor approximation algorithm unless ${\\mathsf {P}}={\\mathsf {NP}}$ , even if $g$ is as small as $m^2$ ).",
"In particular, our approximation notion is the following: an algorithm has an approximation factor $\\alpha \\ge 1$ if it always outputs a solution that has at most $\\alpha $ factor less utility than the optimal solution.",
"Theorem 24 There exists an algorithm that for any fixed $\\epsilon >0$ finds an $(1+\\epsilon )$ -approximate solution to Max-Group-PB in ${\\mathsf {FPT}}$ time wrt.",
"$g$ .",
"The idea of the algorithm is as follows.",
"First, we reduce the given instance of Group-PB to an instance of Group-PB with an additional feasibility restriction, in particular, such that a feasible solution has to contain exactly one project from each project type, where a type of a project is uniquely defined by the family of groups to which the project belongs.",
"The reduction, shown below, takes ${\\mathsf {FPT}}$ time wrt.",
"$g$ .",
"In the second step we will round down the approval score of each project to the closest multiplicity of $(1+\\epsilon )$ , in effect bounding the number of different approval scores of a project to the logarithmic function of the input size.",
"Then we will apply a brute-force enumeration that runs in ${\\mathsf {FPT}}$ time wrt.",
"$g$ .",
"Details follow.",
"Let us fix $\\epsilon >0$ and an instance $\\mathcal {I}= (\\mathcal {V},P,\\mathcal {E},\\mathcal {F},c,B,b)$ of Max-Group-PB.",
"Recall that w.l.o.g.",
"we assumed that each project is approved by at least one voter.",
"Let $a\\colon P \\rightarrow \\lbrace 1,2,\\dots ,|\\mathcal {V}|\\rbrace $ be an approval score function, i.e., $a(p) = |\\lbrace v \\in \\mathcal {V}: p \\in P_v \\rbrace |$ .",
"Notice that the approval score function $a(\\cdot )$ can be encoded in unary (instead of having voters explicitly).",
"Let $A$ be the total approval score of all the projects, e.g., $A = \\sum _{p \\in P} a(p)$ .",
"Let $u^*(\\mathcal {I})$ be the value (total utility) of an optimal solution to $\\mathcal {I}$ .",
"To avoid triviality, w.l.o.g.",
"we assume $u^*(\\mathcal {I})>0$ .",
"Now, given a subfamily $R \\subseteq \\mathcal {F}$ , we say that a project $p$ is of type $R$ if it belongs to all the groups in $R$ and to none of the groups in $\\mathcal {F}\\setminus R$ (so every project has an unique type).",
"We have at most $2^g$ types of projects.",
"First, we fix an optimal solution $X^*$ and we do the following preprocessing on the instance $\\mathcal {I}$ .",
"For every project type, we guess whether at least one project of the type is contained in $X^*$ , and if none, we delete all projects of that type.",
"We can do this preprocessing in $\\mathcal {O}^*(2^{2^g})$ time.",
"Note that, after this step, the number of project types cannot increase because the number of projects cannot increase.",
"(As this is just a preprocessing, in our next steps we override the notation and use $P$ for the set of projects after the preprocessing.)",
"We define the number of project types after the preprocessing as $t$ , with $t \\le 2^g$ .",
"For every project type $R \\subseteq \\mathcal {F}$ we run a dynamic programming procedure that outputs the following: For every value $v \\in \\lbrace 1,2,\\dots ,A\\rbrace $ , we compute $\\operatornamewithlimits{cost}(R,v)$ , which is the minimum cost of a subset of projects of type $R$ whose total value is exactly $v$ (it is equal to $\\infty $ if there is no such subset).",
"Also we store a bundle of projects, $\\operatornamewithlimits{bundle}(R,v)$ , that realizes the minimum cost $\\operatornamewithlimits{cost}(R,v)$ .",
"We can compute $\\operatornamewithlimits{cost}(R,v)$ together with $\\operatornamewithlimits{bundle}(R,v)$ in time upper-bounded by $\\mathcal {O}^*(t \\cdot A \\cdot |P|) = \\mathcal {O}^*(2^g)$ .",
"Now, we create a new instance $\\mathcal {I}^{\\prime } = (\\mathcal {V}^{\\prime },P^{\\prime },\\mathcal {E}^{\\prime },\\mathcal {F}^{\\prime },c^{\\prime },B^{\\prime },b^{\\prime })$ of Group-PB as follows.",
"For every project type $R \\subseteq \\mathcal {F}$ and every value $v \\in \\lbrace 1,2,\\dots ,A\\rbrace $ such that $\\operatornamewithlimits{cost}(R,v)$ is not $\\infty $ , we define a project $\\operatornamewithlimits{proj}(R,v) \\in P^{\\prime }$ of cost $c^{\\prime }(\\operatornamewithlimits{proj}(R,v)) = \\operatornamewithlimits{cost}(R,v)$ and approval score $a^{\\prime }(\\operatornamewithlimits{proj}(R,v))$ equals to $v$ (equivalently we can define $A$ many voters in $\\mathcal {V}^{\\prime }$ , where the $i$ -th voter approves all the projects $\\operatornamewithlimits{proj}(R,v)$ such that $v \\ge i$ ).",
"We define the type of all the projects $\\operatornamewithlimits{proj}(R,v)$ , $v \\in \\lbrace 1,2,\\dots ,A\\rbrace $ , as $R$ .",
"Note that a project $\\operatornamewithlimits{proj}(R,v)$ corresponds to a bundle of projects of type $R$ from the original instance.",
"We keep the same global budget limit, i.e., $B^{\\prime } = B$ .",
"For every group $F \\in \\mathcal {F}$ , we define a group $T_F \\in \\mathcal {F}^{\\prime }$ that contains all projects $\\operatornamewithlimits{proj}(R,v)$ whose type $R$ contains the group $F$ , i.e., $T_F = \\lbrace \\operatornamewithlimits{proj}(R,v): F \\in R\\rbrace $ .",
"We define $b^{\\prime }(T_F) = b(F)$ .",
"We show correspondence of feasible solutions in both instances.",
"Let $\\mathcal {I}^{\\prime }_1$ be the instance $\\mathcal {I}^{\\prime }$ of Group-PB restricted to solutions containing exactly one project of each type.",
"Lemma 25 Every feasible solution to $\\mathcal {I}^{\\prime }_1$ can be transformed into a feasible solution to $\\mathcal {I}$ with the same utility.",
"Let $X^{\\prime }$ be a feasible solution to $\\mathcal {I}^{\\prime }_1$ .",
"We construct a solution $X$ to $\\mathcal {I}$ as follows.",
"For every $\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }$ we add to $X$ projects stored in $\\operatornamewithlimits{bundle}(R,v)$ .",
"First, we show that $X$ is feasible to $\\mathcal {I}$ .",
"For warm-up we show that $X$ keeps a global budget $B$ .",
"We have the following chain of (in)equalities: $\\sum _{p \\in X} c(p) = \\sum _{\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }} \\sum _{p \\in \\operatornamewithlimits{bundle}(R,v)} c(p)= \\sum _{\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }} \\operatornamewithlimits{cost}(R,v)= \\sum _{\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }} c^{\\prime }(\\operatornamewithlimits{proj}(R,v)) \\le B^{\\prime } = B,$ where the first equality follows from definition of $X$ , the second equality follows from definition of $\\operatornamewithlimits{cost}(R,v)$ , the third equality follows from definition of $c^{\\prime }$ , and the inequality follows from feasibility of $X^{\\prime }$ to $\\mathcal {I}^{\\prime }_1$ .",
"Next, we show that $X$ keeps budgets $b(\\mathcal {F})$ for every $F \\subseteq \\mathcal {F}$ .",
"The proof of this fact is analogous to the proof of feasibility for a global budget $B$ , but we limit the first sum to projects of type $R$ that contains group $F$ .",
"We have $\\sum _{p \\in F \\cap X} c(p) = \\sum _{R \\subseteq \\mathcal {F}: F \\in R}\\sum _{\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }} \\sum _{p \\in \\operatornamewithlimits{bundle}(R,v)} c(p)= &\\sum _{R \\subseteq \\mathcal {F}: F \\in R}\\sum _{\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }} \\operatornamewithlimits{cost}(R,v)\\\\= &\\sum _{R \\subseteq \\mathcal {F}: F \\in R}\\sum _{\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }} c^{\\prime }(\\operatornamewithlimits{proj}(R,v))\\le b^{\\prime }(T_F) = b(F),$ where the inequality follows from feasibility of $X^{\\prime }$ to $\\mathcal {I}^{\\prime }$ .",
"It finishes the proof of feasibility of $X$ to $\\mathcal {I}$ .",
"The last one step is to show that the utility achieved by $X$ is equal to the utility achieved by $X^{\\prime }$ .",
"Indeed we have $\\sum _{p \\in X} a(p)= \\sum _{\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }} \\sum _{p \\in \\operatornamewithlimits{bundle}(R,v)}\\hspace{-10.0pt}a(p)= \\sum _{\\operatornamewithlimits{proj}(R,v) \\in X^{\\prime }} \\hspace{-10.0pt} v= \\sum _{p^{\\prime } \\in X^{\\prime }} a^{\\prime }(p^{\\prime }),$ where the second equality follows from definition of $X$ , the third equality follows from definition of $\\operatornamewithlimits{bundle}(R,v)$ , the forth equality follows from definition of $a^{\\prime }$ .",
"Lemma 26 We have $u^*(\\mathcal {I}) \\le u^*(\\mathcal {I}^{\\prime }_1)$ .",
"Recall that $X^*$ is a fixed optimal solution to $\\mathcal {I}$ .",
"We will construct a feasible solution $X^{\\prime }$ to $\\mathcal {I}^{\\prime }_1$ .",
"Intuitively, for every type $R \\subseteq \\mathcal {F}$ , we take projects of type $R$ from $X^*$ , we count their total utility and we put to $X^{\\prime }$ a min-cost bundle of type $R$ that achieve the same utility.",
"Formally, for every project type $R \\subseteq \\mathcal {F}$ , a solution $X^{\\prime }$ contains project $\\operatornamewithlimits{proj}(R,\\sum _{p \\in X^* \\cap R} a(p))$ , where $p \\in R$ denotes that $p$ has type $R$ .",
"Obviously, $X^{\\prime }$ contains exactly one project of each type.",
"Next we show that $X^{\\prime }$ keeps a global and all group budgets.",
"Cost of solution $X^{\\prime }$ is equal to $\\sum _{p^{\\prime } \\in X^{\\prime }} c^{\\prime }(p^{\\prime })= \\sum _{R \\subseteq \\mathcal {F}} c^{\\prime }(\\operatornamewithlimits{proj}(R,\\sum _{p \\in X^* \\cap R} a(p)))= \\sum _{R \\subseteq \\mathcal {F}} \\operatornamewithlimits{cost}(R,\\sum _{p \\in X^* \\cap R} a(p))\\le \\sum _{R \\subseteq \\mathcal {F}} \\sum _{p \\in X^* \\cap R} c(p)= \\sum _{p \\in X^*} c(p)\\le B = B^{\\prime },$ where the first equality follows from definition of $X^{\\prime }$ , the second equality follows from definition of $c^{\\prime }$ , the first inequality follows from definition of $\\operatornamewithlimits{cost}(R,v)$ being a proper min-cost bundle, and the second inequality follows from feasibility of $X^*$ to $\\mathcal {I}$ .",
"Next, we show that $X^{\\prime }$ keeps group budgets $b^{\\prime }(T_F)$ for every $F \\subseteq \\mathcal {F}$ .",
"The proof of this fact is analogous to the proof of feasibility for a global budget $B$ .",
"The cost of projects taken from $T_F$ is equal to $\\sum _{p^{\\prime } \\in T_F \\cap X^{\\prime }} c^{\\prime }(p^{\\prime })= \\sum _{R \\subseteq \\mathcal {F}: F \\in R} c^{\\prime }(\\operatornamewithlimits{proj}(R,\\sum _{p \\in X^* \\cap R} a(p)))= &\\sum _{R \\subseteq \\mathcal {F}: F \\in R} \\hspace{-10.0pt} \\operatornamewithlimits{cost}(R,\\sum _{p \\in X^* \\cap R} a(p))\\\\\\le &\\sum _{R \\subseteq \\mathcal {F}: F \\in R} \\sum _{p \\in X^* \\cap R} \\hspace{-7.0pt} c(p)= \\sum _{p \\in F \\cap X^*} \\hspace{-5.0pt} c(p)\\le b(F) = b^{\\prime }(T_F).$ It finishes the proof of feasibility of $X^{\\prime }$ to $\\mathcal {I}^{\\prime }_1$ .",
"The last one step is to show that the utility achieved by $X^{\\prime }$ is equal to $\\sum _{p^{\\prime } \\in X^{\\prime }} a^{\\prime }(p^{\\prime })= \\sum _{R \\subseteq \\mathcal {F}} a^{\\prime }(\\operatornamewithlimits{proj}(R,\\sum _{p \\in X^* \\cap R} a(p)))= \\sum _{R \\subseteq \\mathcal {F}} \\sum _{p \\in X^* \\cap R} a(p)= \\sum _{p \\in X^*} a(p)= u^*(\\mathcal {I}),$ where the first equality follows from definition of $X^{\\prime }$ , the second equality follows from definition of $a^{\\prime }$ , and the last equality follows from optimality of $X^*$ .",
"As $X^{\\prime }$ is a feasible solution to $\\mathcal {I}^{\\prime }_1$ hence $u^*(\\mathcal {I}^{\\prime }_1) \\ge u^*(\\mathcal {I})$ as required.",
"In the second step (called bucketing) we round down the approval score of each project to the closest multiple of $(1+\\epsilon )$ .",
"Let $\\mathcal {I}^{\\prime \\prime }_1$ be an instance after the bucketing procedure (note that we do not change costs and budget limits when bucketing the approval scores).",
"Lemma 27 We have $u^*(\\mathcal {I}^{\\prime \\prime }_1) \\ge \\frac{u^*(\\mathcal {I}^{\\prime }_1)}{1+\\epsilon }$ .",
"Let $a^{\\prime \\prime }$ be an approval score function in instance $\\mathcal {I}^{\\prime \\prime }_1$ and let $X^{\\prime }$ be an optimal solution to $\\mathcal {I}^{\\prime }_1$ .",
"A set $X^{\\prime }$ is a feasible solution to $\\mathcal {I}^{\\prime \\prime }_1$ as well.",
"Therefore we have $u^*(\\mathcal {I}^{\\prime \\prime }_1)\\ge \\sum _{p^{\\prime } \\in X^{\\prime }} a^{\\prime \\prime }(p^{\\prime })\\ge \\sum _{p^{\\prime } \\in X^{\\prime }}\\frac{a^{\\prime }(p^{\\prime })}{1+\\epsilon }= \\frac{u^*(\\mathcal {I}^{\\prime }_1)}{1+\\epsilon },$ where the first inequality follows from feasibility of $X^{\\prime }$ to instance $\\mathcal {I}^{\\prime \\prime }_1$ , the second inequality follows from definition of bucketing, and the equality follows from optimality of $X^{\\prime }$ to instance $\\mathcal {I}^{\\prime }_1$ .",
"Because of bucketing, it is possible that two projects of the same type but with different approval scores are rounded down to the same value.",
"Hence we keep the project of minimum cost.",
"So, overall, after bucketing we have at most $\\log _{1+\\epsilon }(A)$ projects of each type.",
"For each project type, we branch on which project of that type is selected and we store the best solution $X^{\\prime \\prime }$ (it has utility $u^*(\\mathcal {I}^{\\prime \\prime }_1)$ ).",
"This takes $\\mathcal {O}^*((\\log _{1+\\epsilon }(A))^{t})$ time, which can be bounded as follows.",
"Lemma 28 For a fixed $\\epsilon >0$ we have $\\mathcal {O}^*((\\log _{1+\\epsilon }(A))^{t}) \\le \\mathcal {O}^*(2^{\\frac{1}{2} \\cdot 4^g}).$ The left-hand side is equal to $\\mathcal {O}^*(2^{2^g \\cdot \\log \\log _{1+\\epsilon }(A)})$ and, using $xy \\le (x^2+y^2)/2$ , it is not bigger than $\\mathcal {O}^*(2^{\\frac{1}{2} \\cdot 4^g + \\frac{1}{2} \\cdot \\log ^2\\log _{1+\\epsilon }(A)}) \\le \\mathcal {O}^*(2^{\\frac{1}{2} \\cdot 4^g + \\delta (\\epsilon ) \\cdot \\log (A)})$ , where $\\delta (\\epsilon )$ is a constant dependent on fixed $\\epsilon $ .",
"Hence, the expression is bounded by $\\mathcal {O}^*(2^{\\frac{1}{2} \\cdot 4^g}) \\cdot \\mathcal {O}^*(A^{\\delta (\\epsilon )}) = \\mathcal {O}^*(2^{\\frac{1}{2} \\cdot 4^g})$ because $A \\le |\\mathcal {V}||P|$ .",
"It shows the algorithm runs in ${\\mathsf {FPT}}$ time wrt.",
"$g$ .",
"Note that $\\epsilon $ (which is a fixed constant) is present only in the polynomial on the input size term.",
"The solution $X^{\\prime \\prime }$ is feasible to $\\mathcal {I}^{\\prime }_1$ , hence using Lemma REF we can construct a feasible solution to $\\mathcal {I}$ with utility equal to $u^*(\\mathcal {I}^{\\prime \\prime }_1)\\ge \\frac{u^*(\\mathcal {I}^{\\prime }_1)}{1+\\epsilon } \\ge \\frac{u^*(\\mathcal {I})}{1+\\epsilon },$ where the first inequality follows from Lemma REF and the second inequality follows from Lemma REF ."
],
[
"W-hardness for g with Utility Requirements",
"While we do not know whether Group-PB is ${\\mathsf {FPT}}$ wrt.",
"$g$ , here we partially resolve this question by providing a proof of ${\\mathsf {W}}[1]$ -hardness wrt.",
"$g$ for a slightly more general problem that we call Utility-Group-PB.",
"Utility-Group-PB has the same input and output as Group-PB with only one difference: additionally we define the utility requirement function $u \\colon \\mathcal {F}\\rightarrow \\mathbb {N}$ which means how much utility we have to get from projects belonging to each project-group.",
"Let $a \\colon P \\rightarrow \\lbrace 1,\\dots ,|\\mathcal {V}|\\rbrace $ be an approval score function, i.e., $a(p) = |\\lbrace v \\in \\mathcal {V}: p \\in P_v \\rbrace |$ .",
"Formally Utility-Group-PB is defined as follows.",
"Table: NO_CAPTIONTable: NO_CAPTION Note that we override the notation for $u$ , i.e., $u(\\cdot )$ is a function, $u$ is a desired utility value and hence we have $u(P) = u$ .",
"We will construct an FPT-reduction from the Arc Supply problem that is ${\\mathsf {W}}[1]$ -hard wrt.",
"the number of vertices in the input graph (even on planar graphs) [6].",
"In Arc Supply we are given a simple directed graph (no self-loops, no parallel arcs).",
"For each vertex we are given a positive integer (demand) and for every directed edge $e$ we have a list $L_e$ of supply pairs, i.e., $L_e = \\lbrace (x_1^e,y_1^e), \\dots , (x_{|L_e|}^e,y_{|L_e|}^e) \\rbrace $ , where $x_i^e, y_i^e \\in \\mathbb {N}_{>0}$ .",
"The task is to choose exactly one supply pair from each of the lists $L_e$ in such a way that each vertex gets at least as much supply as its demand.",
"A supply pair $(x_i^e,y_i^e)$ for $e=(\\alpha ,\\beta )$ gives supply $x_i^e$ to vertex $\\alpha $ and $y_i^e$ to vertex $\\beta $ .",
"Formally the problem is defined as follows.",
"Table: NO_CAPTIONTable: NO_CAPTION Without loss of generality we can assume that for every $e \\in E$ we have $x_1^e > x_2^e > \\cdots > x_{|L_e|}^e$ and $y_1^e < y_2^e < \\cdots < y_{|L_e|}^e$ .",
"We can satisfy this after a preprocessing step by removing dominated pairs and sorting the remaining ones, where a pair $(x_i,y_i)$ dominates $(x_j,y_j)$ if $x_i \\ge x_j$ and $y_i \\ge y_j$ .",
"The preprocessing step takes polynomial-time: $\\mathcal {O}(|E|\\cdot \\max _{e \\in E}|L_e|^2)$ .",
"Theorem 29 Utility-Group-PB is ${\\mathsf {W}}[1]$ -hard wrt.",
"$g$ .",
"We construct a reduction from Arc Supply which is ${\\mathsf {W}}[1]$ -hard parameterized by the number of vertices in the input graph [6].",
"Let $(G=(V,E), d, (L_e)_{e \\in E})$ be an instance of Arc Supply with $x_1^e > x_2^e > \\cdots > x_{|L_e|}^e$ and $y_1^e < y_2^e < \\cdots < y_{|L_e|}^e$ for every edge $e \\in E$ .",
"We define a large number $N = 2 \\cdot \\sum _{e \\in E}\\sum _{i=1}^{|L_e|} (x_i^e+y_i^e)$ , and let $L$ be the input size of all lists $(L_e)_{e \\in E}$ .",
"For every edge $e=(\\alpha ,\\beta ) \\in E$ and every supply pair $(x_i^e,y_i^e) \\in L_e$ we create two projects in $P$ : project $p(e,\\alpha ,i)$ of cost $N-x_i^e$ and approval score $i$ , and project $p(e,\\beta ,i)$ of cost $N-y_i^e$ and approval score $|L_e|-i+1$ .",
"In order to maintain the desired approval scores, we define voters $v_1, v_2, \\dots , v_{\\max _{e \\in E}|L_e|}$ , where a voter $v_i$ approves all projects with approval score equal-or-greater than $i$ .",
"Then indeed, a project $p$ is approved by exactly $v_1, v_2, \\dots , v_{a(p)}$ .",
"Next we define groups of projects belonging to $\\mathcal {F}$ and their budget limits.",
"For every vertex $\\alpha \\in V$ we create a group of projects $F_\\alpha = \\lbrace p(e,\\alpha ,i) \\in P \\rbrace $ with its budget limit $b(F_\\alpha ) = N \\cdot \\deg (\\alpha ) - d(\\alpha )$ , where $\\deg (\\alpha )$ is a degree of $\\alpha $ in $G$ ($\\deg (\\cdot )$ counts both in-edges and out-edges).",
"Budget limit $b(F_\\alpha )$ corresponds to supply demand for vertex $\\alpha $ .",
"For every edge $e=(\\alpha ,\\beta ) \\in E$ we create three groups of projects: $F_{e,\\alpha } = \\lbrace p(e,\\alpha ,i) \\in P \\rbrace $ , $F_{e,\\beta } = \\lbrace p(e,\\beta ,i ) \\in P \\rbrace $ and $F_e = F_{e,\\alpha } \\cup F_{e,\\beta }$ .",
"We define budget limits for them as follows: $b(F_{e,\\alpha }) = b(F_{e,\\beta }) = N$ and $b(F_e) = 2N$ .",
"Budget limit $b(F_{e,\\alpha })$ corresponds to choosing at most one $x_i^e$ from $L_e$ and $b(F_{e,\\beta })$ corresponds to choosing at most one $y_i^e$ from $L_e$ .",
"We define the global budget as $B=\\sum _{p \\in P} c(p)$ .",
"The last ingredient is to specify utility requirements for groups of projects.",
"We only restrict utility requirement on $F_e$ for every $e \\in E$ by setting $u(F_e) = |L_e|+1$ .",
"Utility requirement $u(F_e)$ together with dominating pairs property of $(x_i^e,y_i^e)$ give us possibility that even if in a solution we have $x_i^e$ and $y_j^e$ with $i \\ne j$ we can replace them with $x_i^e$ and $y_i^e$ (which corresponds to a supply pair in the original instance).",
"For the remaining groups utility requirement is 0, also $u=u(P)=0$ .",
"The reduction takes time at most $\\mathcal {O}(|E| \\cdot L^2)$ for creating the projects and voters and $\\mathcal {O}(|V||E|^2 \\cdot L)$ to create groups together with budget limits and utility requirements.",
"Hence, in total, the running time is bounded by $\\mathcal {O}(|V||E|^2 \\cdot L^2)$ that is polynomial in the input size.",
"We have $|V|$ many groups of type $F_\\alpha $ , $2|E|$ many groups of type $F_{e,\\alpha }$ and $|E|$ many groups of type $F_e$ .",
"Hence we get $g = |\\mathcal {F}| = |V|+3|E| \\le 4|V|^2 = \\mathcal {O}(|V|^2)$ .",
"$g$ is bounded by a function of the parameter $|V|$ for which Arc Supply is ${\\mathsf {W}}[1]$ -hard [6], hence this reduction is an FPT-reduction.",
"Next we show correctness of the reduction.",
"Correctness: Let $S$ be a solution to an Arc Supply instance.",
"We construct a solution to Utility-Group-PB by simply choosing the corresponding projects, i.e., for every edge $e = (\\alpha ,\\beta ) \\in E$ , we take to the solution $X$ projects $p(e,\\alpha ,S(e))$ and $p(e,\\beta ,S(e))$ .",
"Recall that desired total utility value is 0 and global budget $B$ is large enough to allow any subset of projects be a solution.",
"Therefore, we need to check only budget and utility requirements in the groups from $\\mathcal {F}$ .",
"For every $\\alpha \\in V$ we check budget limit for $F_\\alpha $ .",
"Cost of the projects taken from $F_\\alpha $ is equal to $\\sum _{p \\in F_\\alpha \\cap X} c(p)&= \\sum _{e=(\\alpha ,\\beta ) \\in E} \\sum _{i: p(e,\\alpha ,i) \\in X} (N-x_i^e)+\\sum _{e=(\\beta ,\\alpha ) \\in E} \\sum _{i: p(e,\\alpha ,i) \\in X} (N-y_i^e)\\\\&= \\sum _{e=(\\alpha ,\\beta ) \\in E} (N-x_{S(e)}^e)+\\sum _{e=(\\beta ,\\alpha ) \\in E} (N-y_{S(e)}^e)= \\deg (\\alpha ) \\cdot N-\\sum _{e=(\\alpha ,\\beta ) \\in E} x_{S(e)}^e-\\sum _{e=(\\beta ,\\alpha ) \\in E} y_{S(e)}^e,$ and this is upper-bounded by $\\deg (\\alpha ) \\cdot N -d(\\alpha ) = b(F_\\alpha )$ because $S$ is feasible.",
"Therefore budget limit for $F_\\alpha $ is kept.",
"Utility requirement for $F_\\alpha $ is 0 hence always feasible.",
"Next, for every edge $e=(\\alpha ,\\beta ) \\in E$ we consider groups $F_{e,\\alpha }, F_{e,\\beta }$ and $F_e$ .",
"Cost of projects taken from $F_{e,\\alpha }$ is equal to $\\sum _{p \\in F_{e,\\alpha } \\cap X} c(p) = \\sum _{i: p(e,\\alpha ,i) \\in X} (N-x_i^e) = N-x_{S(e)}^e\\le N = b(F_{e,\\alpha }).$ Similarly we can show that the cost of projects taken from $F_{e,\\beta }$ is at most $N = b(F_{e,\\beta })$ .",
"It follows from the above and $F_e = F_{e,\\alpha } \\cup F_{e,\\beta }$ that cost of projects taken from $F_e$ is at most $2N = b(F_e)$ .",
"For $F_{e,\\alpha }$ and $F_{e,\\beta }$ the utility requirement is 0, hence always feasible.",
"The utility requirement for $F_e$ is kept because $\\sum _{p \\in F_e \\cap X} a(p) &= \\sum _{p \\in F_{e,\\alpha } \\cap X} a(p) + \\sum _{p \\in F_{e,\\beta } \\cap X} a(p)= \\sum _{i: p(e,\\alpha ,i) \\in X} i + \\sum _{i: p(e,\\beta ,i) \\in X} (|L_e|-i+1)= S(e) + |L_e|-S(e)+1 = u(F_e).$ Soundness: Let $X$ be a solution to the Utility-Group-PB instance.",
"First we show that for every edge $e=(\\alpha ,\\beta ) \\in E$ there are exactly one project from $F_{e,\\alpha }$ and exactly one project from $F_{e,\\beta }$ in $X$ .",
"Lemma 30 For every edge $e=(\\alpha ,\\beta ) \\in E$ we have $|X \\cap F_{e,\\alpha }|=|X \\cap F_{e,\\beta }|=1.$ Let $e=(\\alpha ,\\beta )$ be any edge from $E$ .",
"We have $|X \\cap F_{e,\\alpha }| \\le 1$ because otherwise, if $|X \\cap F_{e,\\alpha }| > 1$ then $ \\sum _{p \\in F_{e,\\alpha } \\cap X} c(p) \\ge (N-x_1^e)+(N-x_2^e) > N = b(F_{e,\\alpha })$ that is a contradiction with feasibility of $X$ .",
"Analogously we show that $|X \\cap F_{e,\\beta }| \\le 1$ .",
"If $|X \\cap F_{e,\\alpha }| = 0$ then utility we get in $F_e$ is equal to $ \\sum _{p \\in F_e \\cap X} a(p) = \\sum _{p \\in F_{e,\\beta } \\cap X} a(p)$ that is at most $ |F_{e,\\beta } \\cap X| \\cdot |L_e| \\le |L_e| < u(F_e)$ and this would be a contradiction with feasibility of $X$ .",
"Therefore $|X \\cap F_{e,\\alpha }| > 0$ and analogously we show that $|X \\cap F_{e,\\beta }| > 0$ what finishes the proof.",
"Due to Lemma REF , for every edge $e=(\\alpha ,\\beta ) \\in E$ we define $i(e)$ being such that $\\lbrace p(e,\\alpha ,i(e)) \\rbrace = X \\cap F_{e,\\alpha }$ and $j(e)$ being such that $\\lbrace p(e,\\beta ,j(e)) \\rbrace = X \\cap F_{e,\\beta }$ .",
"In every feasible solution to Utility-Group-PB there is a dependence between $i(e)$ and $j(e)$ .",
"Lemma 31 For every edge $e\\in E$ we have $y_{j(e)}^e \\le y_{i(e)}^e$ .",
"Let $e=(\\alpha ,\\beta )$ be an edge from $E$ .",
"The utility obtained in group $F_e$ is equal to $\\sum _{p \\in F_e \\cap X} a(p)= \\sum _{p \\in F_{e,\\alpha } \\cap X} a(p) + \\sum _{p \\in F_{e,\\beta } \\cap X} a(p)= a(p(e,\\alpha ,i(e))) + a(p(e,\\beta ,j(e)))= i(e) + (|L_e|-j(e)+1).$ As $X$ is a feasible solution to Utility-Group-PB we have $|L_e|+1 = u(F_e) \\le i(e) + (|L_e|-j(e)+1)$ , hence $j(e) \\le i(e)$ .",
"Together with $y_1^e < y_2^e < \\cdots < y_{|L_e|}^e$ we get the required inequality.",
"We construct a solution $S_X$ to Arc Supply setting $S_X(e) := i(e)$ for every edge $e \\in E$ .",
"We will show that $S_X$ is a feasible solution by arguing that every $\\alpha \\in V$ is supplied by at least its demand.",
"Let us fix $\\alpha \\in V$ .",
"We know that the budget limit of $F_\\alpha $ is kept, so we have $N \\cdot \\deg (\\alpha ) - d(\\alpha )= b(F_\\alpha )\\ge \\sum _{p \\in F_\\alpha \\cap X} c(p)&= \\sum _{e \\in E} \\sum _{p(e,\\alpha ,i) \\in X} c(p(e,\\alpha ,i))\\\\&= \\sum _{e=(\\alpha ,\\beta ) \\in E} \\sum _{p(e,\\alpha ,i) \\in F_{e,\\alpha } \\cap X} (N-x_i^e)+ \\sum _{f=(\\beta ,\\alpha ) \\in E} \\sum _{p(f,\\alpha ,j) \\in F_{f,\\alpha } \\cap X} (N-y_j^f)\\\\&= \\sum _{e=(\\alpha ,\\beta ) \\in E} (N-x_{i(e)}^e)+ \\sum _{f=(\\beta ,\\alpha ) \\in E} (N-y_{j(f)}^f)\\\\&= N \\cdot \\deg (\\alpha )- \\sum _{e=(\\alpha ,\\beta ) \\in E} x_{i(e)}^e- \\sum _{f=(\\beta ,\\alpha ) \\in E} y_{j(f)}^f\\\\&\\ge N \\cdot \\deg (\\alpha )- \\sum _{e=(\\alpha ,\\beta ) \\in E} x_{i(e)}^e- \\sum _{f=(\\beta ,\\alpha ) \\in E} y_{i(f)}^f,$ where the last one inequality follows from Lemma REF .",
"From this and definition of $S_X$ we get $ d(\\alpha )\\le \\sum _{e=(\\alpha ,\\beta ) \\in E} x_{S_X(e)}^e+\\sum _{f=(\\beta ,\\alpha ) \\in E} y_{S_X(f)}^f.$ so the demand requirement for $\\alpha $ is kept in a solution $S_X$ ."
],
[
"Approximation Algorithms",
"Recall that an algorithm for Max-Group-PB has an approximation factor $\\alpha $ , $\\alpha \\ge 1$ , if it always outputs a solution with at least an $\\alpha $ fraction of the utility achieved by an optimal solution.",
"In the Related Work subsection we showed that an instance of Max-Group-PB is an instance of $(g+1)$ -DK where each project $p \\in P$ has only two possible costs $\\lbrace 0,c(p)\\rbrace $ over $g+1$ dimensions.",
"There is a polynomial-time approximation scheme (PTAS) for $d$ -DK when $d$ is a constant, running in $\\mathcal {O}(m^{\\lceil d/\\epsilon \\rceil -d})$ -time [25]; hence we have a PTAS for Group-PB with $g=\\mathcal {O}(1)$ , running in $\\mathcal {O}^*(m^{\\mathcal {O}(1/\\epsilon )})$ -time.",
"We can improve this result using our ${\\mathsf {XP}}$ wrt.",
"$g$ algorithm from Theorem REF , as for $g=\\mathcal {O}(1)$ , the algorithm provides an exact solution in polynomial-time.",
"Corollary 32 For $g=\\mathcal {O}(1)$ , Group-PB can be solved in polynomial-time.",
"In contrast, there is no FPTAS for 2-DK unless ${\\mathsf {P}}={\\mathsf {NP}}$ [25].",
"In that proof the authors use an ${\\mathsf {NP}}$ -hardness reduction from the Partition problem to 2-DK with two different costs of a project $p$ in the dimensions, i.e., $c(p)$ and $(\\max _{r \\in P} c(r))-c(p)$ .",
"Furthermore, unless ${\\mathsf {FPT}}={\\mathsf {W}}[1]$ , there is no efficient polynomial-time approximation scheme (EPTAS) for 2-DK [26].",
"Next we focus on non-constant values of $g$ .",
"$d$ -DK can be approximated in polynomial time up to $d+1$ factor [25], thus we have the following.",
"Corollary 33 There exists a polynomial-time $(g+2)$ -approximation algorithm for Max-Group-PB.",
"While seemingly very weak, this approximation guarantee is almost tight.",
"Theorem 34 For any $\\epsilon >0$ , there does not exist a polynomial-time $(g^{1/2-\\epsilon })$ -approximation algorithm for Max-Group-PB unless ${\\mathsf {P}}={\\mathsf {NP}}$ even if $g \\le m^2$ .",
"We rely on the inapproximability of Maximum Independent Set (MaxIS): for any $\\epsilon ^{\\prime }>0$ , MaxIS is ${\\mathsf {NP}}$ -hard to approximate to within $|V|^{1-\\epsilon ^{\\prime }}$ factor [37].",
"Fix $\\epsilon >0$ , and use the reduction from Theorem REF with setting the global budget to be $B=m$ .",
"This reduction is preserving the approximation factor, i.e., an $\\alpha $ -approximate solution to the resulting Max-Group-PB instance is also an $\\alpha $ -approximate solution to Maximum Independent Set; this is so as any feasible solution to Max-Group-PB of total utility $u$ contains exactly $u$ projects that correspond to an $u$ -element subset of $V$ , which is an independent set (from every edge we choose at most one vertex, as otherwise, when choosing both endpoints, we would violate a group budget constraint).",
"Assume there exists a $(g^{1/2-\\epsilon })$ -approximation algorithm for Max-Group-PB.",
"Then, we also have $(g^{1/2-\\epsilon })$ -approximation algorithm for MaxIS.",
"We have $g=|E| \\le |V|^2$ , hence $g^{1/2-\\epsilon } = g^{(1-2\\epsilon )/2} \\le |V|^{1-2\\epsilon }$ , which means that we achieved $|V|^{1-\\epsilon ^{\\prime }}$ -approximation algorithm for MaxIS for $\\epsilon ^{\\prime } = 2\\epsilon >0$ , in contradiction to the hardness of approximation of MaxIS [37].",
"As we have $m = |V|$ in the proof above, we can also derive $(m^{1-\\epsilon })$ -hardness of approximation (unless ${\\mathsf {P}}={\\mathsf {NP}}$ ).",
"One may consider that Max-Group-PB is easier if $g = o(m^2)$ , but even if $g$ is linear on $m$ we can exclude a PTAS: Theorem 35 Assuming ${\\mathsf {P}}\\ne {\\mathsf {NP}}$ , there does not exist a PTAS for Max-Group-PB, even if $g \\le \\frac{3}{2}m$ .",
"We will use ${\\mathsf {APX}}$ -completeness of MaxIS on 3-regular 3-edge-colorable graphs [8] and the reduction from Theorem REF in which we are changing a global budget into $B=m$ .",
"Analogously to the proof of Theorem REF this reduction is approximation preserving.",
"It means that Max-Group-PB is ${\\mathsf {APX}}$ -hard, which excludes a PTAS assuming ${\\mathsf {P}}\\ne {\\mathsf {NP}}$ .",
"Additionally, in the reduction we have $g=|E|=\\frac{3}{2}|V|=\\frac{3}{2}m$ because the input graph is 3-regular.",
"What happens for $g = o(m)$ ?",
"Here is a partial answer that uses our FPT approximation scheme wrt.",
"$g$ (Theorem REF ).",
"Theorem 36 Let $|\\mathcal {I}|$ be the size of an instance $\\mathcal {I}$ of Max-Group-PB.",
"There exists a PTAS for Max-Group-PB if $g \\le \\log _4\\log (|\\mathcal {I}|^{\\mathcal {O}(1)})$ .",
"For a fixed $\\epsilon >0$ the running time of the FPT approximation scheme is bounded by $\\mathcal {O}^*(2^{\\frac{1}{2} \\cdot 4^g})$ (see Lemma REF ).",
"Hence, putting there $g \\le \\log _4\\log (|\\mathcal {I}|^{\\mathcal {O}(1)})$ we get a following bound on the running time: $\\mathcal {O}^*(2^{\\frac{1}{2} \\cdot 4^{\\log _4\\log (|\\mathcal {I}|^{\\mathcal {O}(1)})}}) = \\mathcal {O}^*(2^{\\frac{1}{2} \\cdot \\log (|\\mathcal {I}|^{\\mathcal {O}(1)})}) =\\mathcal {O}^*(|\\mathcal {I}|^{\\mathcal {O}(1)}) = \\mathcal {O}^*(1),$ that is polynomial for a fixed $\\epsilon $ hence we get a PTAS.",
"As $m$ is bounded by the input size we get the following.",
"Corollary 37 There exists a PTAS for Max-Group-PB if $g \\le \\log _4\\log (m^{\\mathcal {O}(1)})$ .",
"Our approximability and inapproximability results are summarized in Tables REF and REF , respectively.",
"Table: Achieved approximation ratios depending on the number gg of groups.The smaller gg compared to mm, the better approximation for Max-Group-PB we can achieve.Table: Achieved inapproximability results depending on the number gg of groups.The larger gg compared to mm, the higher is the approximation ratio excluded."
],
[
"Outlook",
"Motivated by PB scenarios in which it is useful to consider geographic constraints and thematic constraints, we enriched the standard approval-based model of PB (in particular, the model of Combinatorial PB [2]), by introducing a group structure over the projects and requiring group-specific budget limits for each group.",
"We have showed that the corresponding combinatorial problem is generally computationally intractable, however, fortunately, enriching PB instances with such group structure and its corresponding budget constraints comes at essentially no computational cost if there are not so many such groups or if the structure of these groups is hierarchical or close to being such; we complemented our analysis with lower bounds and approximation algorithms.",
"Some future research directions are the following: Complexity wrt.",
"$g$ : We have one main open question remaining, namely, the parameterized complexity of Group-PB wrt.",
"$g$ , that is worth settling.",
"Further parameters: Considering more parameters, including parameter combinations, would be natural for future research.",
"Furthermore, our reduction from Independent Set in Theorem REF can give also time lower-bounds under the Exponential Time Hypothesis (see, e.g., a survey by Lokshtanov et al. [30]).",
"Generally speaking, it would be interesting to evaluate what are limitations of parameterized algorithms for Group-PB (in particular with respect to parameter $g$ ) and how our current results are close to such lower bounds.",
"Different aggregation methods: Another direction would be to consider other functions of aggregating utilities of voters.",
"For example, instead of maximizing the sum over voter utilities, one might be egalitarian and aim at maximizing the minimum over voter utilities.",
"Cardinality constraints: Another direction would be to consider additional cardinality constraints.",
"While all hardness results hold for the more general variant where we consider both knapsack and cardinality constraints, a careful analysis shall be taken to explore which of the tractability results hold for this setting as well.",
"Project interactions: Another direction would be to consider interactions among projects, such as substitution and complementarities; these can be modeled via general functions, corresponding to the utility of voters from a certain number of projects funded and approved from each group, in the spirit of the work of Jain et al.",
"[23]."
],
[
"Acknowledgements",
"Krzysztof Sornat was partially supported by the Foundation for Polish Science (FNP) within the START programme, the National Science Centre, Poland (NCN; Grant No.",
"2018/28/T/ST6/00366) and the Israel Science Foundation (ISF; Grant No.",
"630/19).",
"Nimrod Talmon was supported by the Israel Science Foundation (ISF; Grant No.",
"630/19).",
"Meirav Zehavi was supported by the Israel Science Foundation (ISF; Grant No.",
"1176/18) and the United States-Israel Binational Science Foundation (BSF; Grant No.",
"2018302)."
]
] | 2012.05213 |
[
[
"Kaluza-Klein fermion mass matrices from Exceptional Field Theory and\n ${\\cal N} =1$ spectra"
],
[
"Abstract Using Exceptional Field Theory, we determine the infinite-dimensional mass matrices for the gravitino and spin-$1/2$ Kaluza-Klein perturbations above a class of anti-de Sitter solutions of M-theory and massive type IIA string theory with topologically-spherical internal spaces.",
"We then use these mass matrices to compute the spectrum of Kaluza-Klein fermions about some solutions in this class with internal symmetry groups containing SU(3).",
"Combining these results with previously known bosonic sectors of the spectra, we give the complete spectrum about some ${\\cal N}=1$ and some non-supersymmetric solutions in this class.",
"The complete spectra are shown to enjoy certain generic features."
],
[
"Introduction",
"By construction, the duality-covariant reformulations , , , , , , (see for a review) of $D=11$ and type II , , supergravities are particularly helpful to investigate the imprint that these higher-dimensional theories instill on their lower-dimensional counterparts.",
"In particular, Exceptional Field Theory (ExFT) , , has been recently shown to provide a powerful framework to compute the spectrum of Kaluza-Klein (KK) perturbations above a certain class of anti-de Sitter (AdS) backgrounds of string and M-theory , .",
"The relevant class of solutions involves the product, possibly warped, of AdS and a topologically-spherical manifold equipped with a possibly inhomogeneous metric, supported in general by fluxes.",
"These solutions typically lie beyond the range of applicability of coset-space techniques , , , for the calculation of KK spectra.",
"Direct calculation methods for these solutions become hard to the point of essentially unsuitable either.",
"Using these novel ExFT techniques, the complete KK spectrum of some supersymmetric AdS solutions of the higher-dimensional supergravities in the relevant class has now been computed , , , .",
"The KK scalar , and vector spectrum of some non-supersymmetric AdS solutions in the same class has also been determined.",
"Specifically, the ExFT methods of , apply to $D=10, 11$ solutions with lower-dimensional AdS factors, which consistently uplift on spheres from AdS vacua of gauged supergravities in lower dimensions with certain gaugings.",
"Under these conditions, infinite-dimensional mass matrices for each species of fields (scalars, vectors, etc.)",
"on AdS exist which are block-diagonal KK level by KK level.",
"The consistent truncation requirement is critical for this block-diagonal structure, as the latter is absent for AdS solutions which do not uplift from a maximally supersymmetric lower-dimensional gauged supergravity , .",
"The infinite-dimensional mass matrices for the KK bosonic perturbations above the relevant class of AdS solutions have been determined from ExFT in , .",
"See also for an early derivation of a covariant mass matrix for the KK gravitons, and for an alternative rederivation of the KK vector mass matrix from ExFT.",
"In this paper, we complete this programme by providing the mass matrices for the fermionic, gravitino and spin-$1/2$ , KK mass matrices.",
"For definiteness, we will focus on E$_{7(7)}$ ExFT , , and extract the KK fermion mass matrices from the fermionic completion of this theory .",
"We thus focus on fermionic KK spectra above AdS$_4$ solutions, but our methods are readily extensible to other instances of ExFT with different duality groups.",
"Also for concreteness, we will restrict our attention to the AdS$_4$ solutions of $D=11$ and massive type IIA supergravity that uplift consistently on $S^7$ and $S^6$ , from AdS vacua of $D=4$ ${\\cal N}=8$ supergravity with concrete gaugings.",
"We take these to be, respectively, the SO(8) gauging and the dyonic, in the sense of , , , ISO(7) gauging .",
"The latest classification results for this type of AdS$_4$ backgrounds of M-theory and type IIA string theory can be found in the recent references , .",
"Section below presents the KK fermionic mass matrices for this class of AdS$_4$ solutions.",
"Table: OSp(4|1)(4|1) supermultiplets that appear in the 𝒩=1{\\cal N}=1 KK spectra.",
"For each type of supermultiplet, the spin and energy (s 0 ,E 0 )(s_0, E_0) of the superconformal primary is given, and the energies of the constituent states, all of them with spins between 0 and 2, are listed.We have then used our mass matrices to compute the KK gravitino and spin-$1/2$ spectra above concrete AdS$_4$ solutions in this class: those that preserve at least SU(3) internal symmetry.",
"Prior to the general scans of , , these particular solutions were classified in and for the SO(8) and the ISO(7) gaugings, respectively.",
"Their corresponding $D=11$ , , , , , , and type IIA uplifts , , are all known.",
"For some of these AdS$_4$ solutions, the complete KK spectrum is also known , , , , , , and we reproduce the corresponding fermionic sectors.",
"More interestingly, there are three ${\\cal N}=1$ solutions in this class whose KK spectrum was only known partially , , , , , until now.",
"One of these AdS$_4$ solutions has internal symmetry G$_2$ in the $D=4$ ${\\cal N}=8$ SO(8) and $D=11$ supergravities .",
"The other two are solutions of $D=4$ ${\\cal N}=8$ ISO(7) supergravity and type IIA.",
"The first one of these also has residual symmetry G$_2$ in $D=4$ and type IIA , .",
"The second one has SU(3) symmetry in $D=4$ , and type IIA .",
"Combining our new fermionic spectra and previous partial results on the bosonic spectra , , , , , , we are able to give the complete supersymmetric spectra for all these three ${\\cal N}=1$ solutions.",
"The details can be found in sections and .",
"Interestingly, a pattern emerges.",
"The KK spectra for these three ${\\cal N}=1$ AdS$_4$ solutions are organised in representations of OSp$(4|1) \\times G$ , with $G= \\textrm {G}_2$ or $G= \\textrm {SU}(3)$ .",
"The supermultiplets of OSp$(4|1)$ have been reviewed for convenience in table REF above, and $G$ , being of rank 2, has its representations labelled by two non-negative integer Dynkin labels $[p,q]$ .",
"At Kaluza-Klein level $n=0 , 1, 2 , \\ldots $ , the dimension $E_0$ of a given OSp$(4|1)$ supermultiplet with superconformal primary spin $s_0$ , arising in the $[p,q]$ representation of $G$ , is found to be given by $ E_0 = 1 + \\sqrt{ 6-s_0(s_0+1) + \\alpha \\, n (n + d-1) - \\beta \\, {\\cal C}_2 (p,q) } \\; .$ Here, $n( n + d-1)$ is the eigenvalue of the scalar Laplacian on the $S^{d}$ sphere, with $d=7$ in M-theory and $d=6$ in type IIA; $\\alpha $ is a positive constant that takes on the value $\\alpha = \\tfrac{5}{8}$ in M-theory and $\\alpha = \\tfrac{5}{6}$ in type IIA for the specific ${\\cal N}=1$ solutions with $G= \\textrm {G}_2$ or $G= \\textrm {SU}(3)$ symmetry; $\\beta $ is a positive constant that depends on the symmetry preserved by the solution, $\\beta = \\tfrac{5}{4}$ for $G= \\textrm {G}_2$ and $\\beta = \\tfrac{5}{3}$ for $G= \\textrm {SU}(3)$ , regardless of whether it lives in $D=11$ or in type IIA; and, finally, ${\\cal C}_2 (p,q)$ is the eigenvalue of the quadratic Casimir operator of $G$ in the $[p,q]$ representation.",
"Although we mainly focus on complete ${\\cal N}=1$ spectra, in section we turn to give the fermionic spectra of the non-supersymmetric solutions in the same class.",
"Of course, these ${\\cal N}=0$ solutions are of limited significance, since they are either manifestly unstable at the perturbative level , or expected to be so in the full string theory .",
"These are either $D=11$ or type IIA solutions preserving $G = \\textrm {SO}(7)$ , , , , , $G = \\textrm {SO}(6) \\sim \\textrm {SU}(4)$ , , , or $G = \\textrm {G}_2$ , , .",
"A formula (see (REF )), similar to (REF ) but now for the individual squared masses, exists in terms of the eigenvalues of the scalar Laplacian on $S^d$ and the quadratic Casimir operator of $G$ .",
"For the non-supersymmetric type IIA solution with G$_2$ invariance , , , our fermionic results combined with the previously known bosonic KK sector , , , allow us to give its complete KK spectrum.",
"For the other non-supersymmetric solutions, the only sector of the KK spectrum that remains to be explicitly computed after our analysis is the scalar sector.",
"However, the pattern displayed by the generic mass formula (REF ) is sufficiently strong to allow us to confidently conjecture the form of the KK scalar spectra for these solutions.",
"Section concludes with further discussion.",
"Our conventions are summarised in appendix , where some explicit results for the eigenvalues of our fermionic mass matrices on selected solutions are also included."
],
[
"KK fermion mass matrices from ExFT",
"We will now determine the mass matrices for the KK fermion perturbations above the AdS$_4$ class of solutions of string and M-theory that uplift from four-dimensional gauged supergravity.",
"We will extract these mass matrices from the fermionic completion of E$_{7(7)}$ ExFT , , by setting the ExFT bosonic fields to the Scherk-Schwarz configuration that gives rise to $D=4$ ${\\cal N}=8$ supergravity upon consistent truncation, while retaining the full tower of KK fermion perturbations."
],
[
"Generalised Scherk-Schwarz–Kaluza-Klein reduction",
"The fermionic content of ExFT includes a gravitino $\\mathbf {\\psi }_\\mu ^i$ and a spin-$1/2$ fermion $\\mathbf {\\chi }^{ijk} = \\mathbf {\\chi }^{[ijk]}$ , neutral under local E$_{7(7)}$ but transforming in the $\\mathbf {8}$ and the $\\mathbf {56}$ of global SU(8), respectively.",
"The bosonic sector of ExFT contains external, $\\mathbf {e}_\\mu {}^\\alpha $ , and internal, $\\mathbf { \\mathcal {V}}_M{}^{\\underline{A}} = \\big ( \\mathbf { \\mathcal {V}}_M{}^{ij} , \\mathbf { \\mathcal {V}}_{M \\,ij} \\big )$ , vielbeine which give rise to metrics $\\mathbf {g}_{\\mu \\nu } = \\eta _{\\alpha \\beta } \\, \\mathbf {e}_\\mu {}^\\alpha \\mathbf {e}_\\nu {}^\\beta $ and $\\mathbf {\\mathcal {M}}_{MN} = 2 \\, \\mathbf { \\mathcal {V}}_{(M | \\,ij} \\mathbf { \\mathcal {V}}_{|N)}{}^{ij} $ .",
"The indices $\\mu = 0, \\ldots , 3$ and $M=1 , \\ldots , 56$ are local fundamental indices of SO$(1,3)$ and E$_{7(7)}$ , while $\\underline{A} = \\big ( \\phantom{}^{ij} , \\phantom{}_{ij} \\big ) $ and $i$ are global indices in the $\\mathbf {28} + \\overline{\\mathbf {28}}$ and the fundamental of SU(8), respectively, so that $\\mathbf { \\mathcal {V}}_{M \\,ij} = \\big ( \\mathbf { \\mathcal {V}}_M{}^{ij} \\big )^*$ and $\\mathbf { \\mathcal {V}}_M{}^{ij} = \\mathbf { \\mathcal {V}}_M{}^{[ij]} $ .",
"The bosonic sector of E$_{7(7)}$ ExFT further includes vectors and two-forms which will not play a role in the present analysis.",
"All these fields depend on both the external and internal coordinates $(x^\\mu ,Y^M)$ , and are subject to the appropriate section constraints.",
"Our starting point is the fermionic action of ExFT .",
"The terms that contribute to the kinetic, mass, and quadratic interaction terms for the $D=4$ KK fermions are, in our conventions $\\mathcal {L}_{\\text{ExFT fermi}} &=& - i \\epsilon ^{\\mu \\nu \\rho \\sigma }\\bar{\\mathbf {\\psi }}_\\mu ^i \\gamma _\\nu \\mathcal {D}_\\rho \\mathbf {\\psi }_{\\sigma i}- \\tfrac{1}{6}\\, \\mathbf {e} \\,\\bar{\\mathbf {\\chi }}^{ijk} \\gamma ^\\mu \\mathcal {D}_\\mu \\mathbf {\\chi }_{ijk} \\nonumber \\\\[4pt]& & -4 i \\, \\epsilon ^{\\mu \\nu \\rho \\sigma } \\big ( \\mathbf { {\\cal V}}^{-1} \\big )_{ij}{}^M \\, \\bar{\\mathbf {\\psi }}_\\mu ^i \\gamma _\\nu \\nabla _M\\left(\\gamma _\\rho \\mathbf {\\psi }_\\sigma ^j\\right)- 4 \\sqrt{2} \\, \\mathbf {e} \\,\\big ( \\mathbf { {\\cal V}}^{-1}\\big )^{ij \\, M} \\, \\bar{\\mathbf {\\psi }}_\\mu ^k \\, \\nabla _M \\left(\\gamma ^\\mu \\mathbf {\\chi }_{ijk}\\right) \\nonumber \\\\[4pt]&& + \\tfrac{1}{9}\\, \\mathbf {e} \\, \\epsilon _{ijklmnpq } \\big ( \\mathbf { {\\cal V}}^{-1}\\big )^{ij \\, M} \\bar{\\mathbf {\\chi }}^{klm} \\nabla _M \\mathbf {\\chi }^{npq} \\, +\\, \\text{c.c.",
"}$ Here, $\\mathbf {e} \\equiv \\sqrt{\\vert \\text{det} \\, \\mathbf {g}_{\\mu \\nu } \\vert }$ , $\\gamma _\\alpha $ are the SO$(1,3)$ Dirac matrices subject to the Clifford algebra $\\lbrace \\gamma _\\alpha , \\gamma _\\beta \\rbrace = 2 \\eta _{\\alpha \\beta }$ , and $\\gamma _\\mu \\equiv \\mathbf {e}_\\mu {}^\\alpha \\, \\gamma _\\alpha $ .",
"The external covariant derivatives featuring in the kinetic terms (the first two terms in (REF )) take on the schematic form $\\mathcal {D}_\\mu \\equiv D_\\mu + \\tfrac{1}{4} \\omega _{\\mu \\, \\;\\beta }^{\\;\\;\\alpha }+ \\tfrac{1}{2} \\mathcal {Q}_{\\mu \\,\\;j}^{\\;\\;i}$ , in terms of SO(1,3) and SU(8) connections, $\\omega _{\\mu \\, \\;\\beta }^{\\;\\;\\alpha }$ and $\\mathcal {Q}_{\\mu \\,\\;j}^{\\;\\;i}$ .",
"The portion $D_\\mu \\equiv \\partial _\\mu - \\mathbb {L}_{\\mathcal {A}_\\mu }$ , covariant under generalised diffeomorphisms generated by the ExFT gauge fields, will not be significant.",
"The second and third lines in (REF ) contain the internal covariant derivative $\\nabla _M$ .",
"On an SU(8) vector $\\mathbf {\\xi }_i$ , with suppressed SO(1,3) spinor indices and weight $\\lambda $ under generalised diffeomorphisms, this derivative acts as $ \\nabla _M \\mathbf {\\xi }_i = \\partial _M \\mathbf {\\xi }_i -\\tfrac{1}{4} \\omega _M{}^{\\alpha \\beta } \\gamma _{\\alpha \\beta } \\, \\mathbf {\\xi }_i + \\tfrac{1}{2} {\\cal Q}_{M \\, i}{}^j \\mathbf {\\xi }_j -\\tfrac{2}{3} \\, \\lambda \\, \\Gamma _{KM}{}^K \\mathbf {\\xi }_i \\; ,$ in terms of internal SO$(1,3)$ , SU(8) and Christoffel connections $\\omega _M{}^{\\alpha \\beta } \\equiv \\mathbf {e}^{\\mu [\\alpha }\\partial _M \\mathbf {e}_\\mu {}^{\\beta ]}$ , ${\\cal Q}_{M \\, i}{}^j $ and $\\Gamma _{MN}{}^P$ .",
"There are other terms in the quadratic fermionic action that we have not retained in (REF ), since they do not contribute to the mass matrices.",
"For example, there is a coupling of the gravitino and the spin-$1/2$ fermion to the spacetime Maurer-Cartan form of the scalars.",
"For similar reasons, a contribution to the SO$(1,3)$ connection $\\omega _M{}^{\\alpha \\beta }$ containing $\\mathbf {\\mathcal {M}}_{MN}$ together with the ExFT gauge field strengths has been disregarded.",
"See for full details.",
"We would now like to fix the external and internal vielbeine to the configurations that give rise to $D=4$ ${\\cal N}=8$ gauged supergravity upon consistent truncation.",
"We therefore write for them the generalised Scherk-Schwarz expressions , , , $ \\mathbf {e}_\\mu {}^\\alpha (x,Y) = \\rho (Y)^{-1} \\, e_\\mu {}^\\alpha (x) \\; , \\qquad \\mathbf { \\mathcal {V}}_M{}^{\\underline{A}} (x, Y) = U_{M}{}^{\\underline{M}} (Y) \\, \\mathcal {V}_{\\underline{M}}{}^{\\underline{A}} (x) \\; ,$ in terms of the $D=4$ vielbein $e_\\mu {}^\\alpha (x)$ and $\\textrm {E}_{7(7)}/\\textrm {SU}(8)$ coset representative $\\mathcal {V}_{\\underline{M}}{}^{\\underline{A}} (x)$ .",
"The $Y$ -dependent function $\\rho $ and twist matrix $U_{M}{}^{\\underline{M}} (Y) $ here obey the consistency conditions , , , $ && \\partial _N \\, (U^{-1})_{\\underline{M}}{}^N - 3 \\rho ^{-1} (U^{-1})_{\\underline{M}}{}^N\\partial _N \\, \\rho =0 \\; , \\nonumber \\\\[5pt]&& 7 \\rho ^{-1} \\, \\left( (U^{-1})_{\\underline{M}}{}^P \\, (U^{-1})_{\\underline{N}}{}^Q \\, \\partial _P \\, U_Q{}^{\\underline{K} }\\right)_{\\mathbf {912}} + F_{\\underline{M} \\underline{N}}{}^{\\underline{K}} = X_{\\underline{M} \\underline{N}}{}^{\\underline{K}} \\; ,$ where $X_{\\underline{M} \\underline{N}}{}^{\\underline{P}} \\equiv \\Theta _{\\underline{M}}{}^\\alpha \\, (t_\\alpha )_{\\underline{N}}{}^{\\underline{P}} $ is the usual contraction of the $D=4$ ${\\cal N}=8$ embedding tensor and the E$_{7(7)}$ generators, the subindex $\\mathbf {912}$ denotes projection to that representation of E$_{7(7)}$ , and $F_{\\underline{M} \\underline{N}}{}^{\\underline{P}}$ is a deformation , of E$_{7(7)}$ ExFT .",
"The latter either vanishes or codifies the Romans mass for $D=11$ and type IIA configurations, respectively.",
"In (REF ) and elsewhere, $\\underline{M} = 1, \\ldots , 56$ is a flat, fundamental $\\textrm {E}_{7(7)}$ index.",
"We would also like to keep the full tower of KK gravitini and spin-$1/2$ fermion perturbations over every AdS vacuum of the $D=4$ ${\\cal N}=8$ supergravities under consideration.",
"Identifying, in our conventions, the $\\mathbf {8}$ of SU(8) with the $\\mathbf {8}_s$ of SO(8), the $D=4$ ${\\cal N}=8$ gravitino $\\psi _\\mu ^i$ and spin-$1/2$ fermion $\\chi ^{ijk}$ respectively lie in the $\\mathbf {8}_s \\equiv [0,0,0,1]$ and $\\mathbf {56}_s \\equiv [1,0,1,0]$ of SO(8) for AdS vacua of the SO(8) gauging (or branchings thereof for vacua with reduced symmetry $G \\subset \\textrm {SO}(8)$ ), and in the $\\mathbf {8} \\equiv [0,0,1]$ and $\\mathbf {48} + \\mathbf {8} \\equiv [1,0,1] + [0,0,1]$ of SO(7) (or branchings thereof) for vacua of the ISO(7) gauging.",
"These $D=4$ ${\\cal N}=8$ states are identified with the KK level $n=0$ states.",
"The $n \\ge 1$ states up the KK tower lie in the infinite-dimensional, reducible representation obtained by tensoring the representations above with the symmetric-traceless representations , , $ \\oplus _{n=0}^\\infty [n,0,0,0] \\; \\textrm {of SO(8)} \\quad \\textrm {or} \\quad \\oplus _{n=0}^\\infty [n,0,0] \\; \\textrm {of SO(7)} \\; ,$ at least, for the spin-$1/2$ fermions, before super-Higgsing takes place: see the discussion around (REF ) and (REF ) below.",
"It is thus convenient to denote the KK gravitino and spin-$1/2$ fermions with a double set of indices carrying this tensor product structure, as $\\psi _\\mu ^{i \\Lambda } (x)$ and $\\chi ^{ijk \\Lambda } (x) $ .",
"For the ExFT fermions we thus write $\\mathbf {\\psi } _\\mu ^i\\left(x,Y\\right) = \\rho (Y )^{-\\frac{1}{2}} \\,\\psi _\\mu ^{i \\Lambda }(x)\\, \\mathcal {Y}_\\Lambda (Y ) \\, , \\qquad \\mathbf {\\chi }^{ijk}\\left(x,Y\\right) = \\rho (Y )^{\\frac{1}{2}} \\,\\chi ^{ijk \\Lambda }(x)\\,\\mathcal {Y}_\\Lambda (Y ) \\, ,$ building on , .",
"Here, ${\\cal Y}_\\Lambda $ denotes the infinite tower of scalar spherical harmonics on the round $S^7$ or $S^6$ spheres.",
"These lie in the representations (REF ) of SO(8) or SO(7).",
"Of the ${\\cal Y}_\\Lambda $ we will only need to note that they are subject to the relation , $ \\rho ^{-1} \\, (U^{-1})_{\\underline{N}}{}^M \\partial _M \\, {\\cal Y}_\\Lambda = -({\\cal T}_{\\underline{N}})_\\Lambda {}^\\Sigma \\, {\\cal Y}_\\Sigma \\; ,$ where the constant, real matrices $({\\cal T}_{\\underline{N}})_\\Lambda {}^\\Sigma $ are the generators of SO(8) or SO(7) in the infinite-dimensional, reducible representations (REF ), normalised as , $ [ {\\cal T}_{\\underline{M}} , {\\cal T}_{\\underline{N}} ] = -X_{\\underline{M} \\underline{N}}{}^{\\underline{P}} \\, {\\cal T}_{\\underline{P}} \\; .$ These are of course traceless, $({\\cal T}_{\\underline{N}})_\\Lambda {}^\\Lambda =0$ .",
"Indices $\\Lambda $ , $\\Sigma $ are raised and lowered with $\\delta _{\\Sigma \\Lambda }$ , and the generators with same-level indices are antisymmetric, $ ({\\cal T}_{\\underline{M}})_{\\Lambda \\Sigma } \\equiv ({\\cal T}_{\\underline{M}})_\\Lambda {}^\\Xi \\, \\delta _{\\Xi \\Sigma } = - ({\\cal T}_{\\underline{M}})_{\\Sigma \\Lambda } \\; ,\\qquad ({\\cal T}_{\\underline{M}})^{\\Lambda \\Sigma } \\equiv \\delta ^{\\Lambda \\Xi } ({\\cal T}_{\\underline{M}})_\\Xi {}^\\Sigma = - ({\\cal T}_{\\underline{M}})^{\\Sigma \\Lambda } \\; .$ At KK level $n=0$ , ${\\cal Y}_0 = 1$ and (REF ) reduces to the expressions given in for the consistent truncation of the ExFT fermions down to their $D=4$ ${\\cal N}=8$ counterparts.",
"For higher KK levels $n \\ge 1$ , (REF ) is a straightforward extension to the fermion sector of the expressions given in , for the embedding of the bosonic KK modes into the ExFT bosonic fields.",
"Finally, we still need to give expressions for the ExFT connections in terms of $D=4$ quantities.",
"With the ExFT scalars fixed via the rightmost equation in (REF ) to their ${\\cal N}=8$ four-dimensional counterparts (and the latter eventually frozen to their vacuum expectation values at an AdS critical point of the ${\\cal N}=8$ scalar potential), the ExFT connections simply take on their expressions for consistent truncation configurations .",
"In particular, the internal SO$(1,3)$ connection $\\omega _M{}^{\\alpha \\beta }$ and the relevant components, ${\\cal Q}^{ik}{}_k{}^j$ and ${\\cal Q}^{[ij}{}_k{}^{l]}$ , of the flattened SU(8) connection ${\\cal Q}^{ij}{}_k{}^{l} \\equiv \\big ( \\mathbf { {\\cal V}}^{-1}\\big )^{ij \\, M } \\, {\\cal Q}_{M \\, k}{}^l$ are set, in our conventions, to $ \\omega _M{}^{\\alpha \\beta } (x, Y) = 0 \\; , \\qquad \\mathcal {Q}^{ik}{}_k{}^j =- \\tfrac{1}{4} \\, \\rho (Y) \\, A_1^{ij} (x) \\; , \\qquad {\\cal Q}^{[ij}{}_k{}^{l]} =- \\tfrac{1}{12} \\, \\rho (Y) \\, A_{2 \\, k}{}^{ijl} (x) \\; .$ Here, $A_1^{ij}$ and $A_{2 \\, k}{}^{ijl}$ are the fermion shifts of ${\\cal N}=8$ gauged supergravity.",
"Recall that these arise as contractions $A_1^{ij} =\\tfrac{4}{21} \\, T^{ikjl}{}_{kl} \\quad , \\quad A_{2h}{}^{ijk} = 2 \\, T_{mh}{}^{mijk}\\, ,$ of the $D=4$ ${\\cal N}=8$ $T$ -tensor $T_{\\underline{A}\\underline{B}}{}^{\\underline{C}}=\\left(\\mathcal {V}^{-1}\\right)_{\\underline{A}}{}^{\\underline{M}} \\left(\\mathcal {V}^{-1}\\right)_{\\underline{B}}{}^{\\underline{N}} \\, X_{ \\underline{M} \\underline{N}}{}^{\\underline{P}} \\, \\mathcal {V}_{\\underline{P}}{}^{\\underline{C}} \\; .$ More specifically, $A_1^{ij}$ and $A_{2 \\, k}{}^{ijl}$ are the $\\mathbf {36}$ and $\\mathbf {420}$ components of the $T$ -tensor (REF ), so that $ A_{1}^{ ij} = A_{ 1}^{ (ij)} \\; , \\quad A_{2 h}{}^{ijk} = A_{2 h}{}^{[ijk]} \\; , \\quad A_{2 k}{}^{ijk} = 0 \\; .$ These are related to the $\\overline{\\mathbf {36}}$ and $\\overline{\\mathbf {420}}$ components $A_{1 ij}$ , $A_2{}^h{}_{ijk}$ of (REF ) by complex conjugation: $( A_{1 ij})^* = A_1^{ij}$ , $(A_2{}^h{}_{ijk})^* = A_{2h}{}^{ijk}$ .",
"The ${\\cal N}=8$ fermion shift $A_{1 ij}$ serves also as the gravitino mass matrix, while the ${\\cal N}=8$ spin-$1/2$ fermion mass matrix $A_{3 \\, ijk , lmn}$ is symmetric, $ A_{3 \\, ijk , lmn} =A_{3 \\, lmn, ijk} \\; ,$ and related to $A_{2 h}{}^{ijk}$ viaThe relation between $A_2$ and $A_3$ is usually given as $ A_{3 \\, ijk , lmn} = \\tfrac{\\sqrt{2}}{144} \\, \\epsilon _{ijkpqr[lm} \\, A_{2 \\, n]}{}^{pqr} \\; ,$ which, unlike (REF ), makes manifest the symmetry (REF ) of $A_3$ .",
"Using the tracelessness condition of $A_2$ in (REF ), the expression (REF ) becomes equivalent to (REF ).",
"The KK analogue of $A_2$ satisfies a similar trace condition only if the KK indices are contracted as well.",
"For this reason, it is (REF ) and not (REF ) that naturally extrapolates to higher KK levels: see (REF ) $ A_{3 \\, ijk , lmn} = \\tfrac{\\sqrt{2}}{108} \\, \\epsilon _{ijkpq[rlm} \\, A_{2 n]}{}^{pqr} \\; .$ Armed with all these definitions, our task now is to obtain from (REF ) the $D=4$ quadratic action for the KK fermion pertubations $\\psi ^{i \\Lambda } (x)$ and $\\chi ^{ijk \\Lambda } (x) $ .",
"The calculation follows closely that described in to obtain the supersymmetry variations of the $D=4$ ${\\cal N}=8$ fermions from ExFT.",
"One should now take into account the dependence (REF ) of the ExFT fermions on the spherical harmonics ${\\cal Y}_\\Lambda $ .",
"On top of the ${\\cal N}=8$ contributions , these amount to new terms in the generators $({\\cal T}_{\\underline{N}})_\\Lambda {}^\\Sigma $ .",
"As will be shown in sections REF and REF , the end result is $ {\\cal L}_{\\text{KK fermi}} &=& -i \\epsilon ^{\\mu \\nu \\rho \\sigma }\\bar{\\psi }_{\\mu }^{i \\Lambda }\\, \\gamma _\\nu \\,\\mathcal {D}_\\rho \\psi _{\\sigma i \\Lambda }-\\tfrac{1}{6}\\,e\\,\\bar{\\chi }^{ijk \\Lambda } \\, \\gamma ^\\mu \\, \\mathcal {D}_\\mu \\chi _{ijk \\Lambda } \\nonumber \\\\[4pt]&& + e \\, A_{1 \\, i \\Lambda , j \\Sigma } \\; \\bar{\\psi }_\\mu ^{i \\Lambda }\\,\\gamma ^{\\mu \\nu } \\, \\psi _{\\nu }^{j \\Sigma }+ \\tfrac{\\sqrt{2}}{6} \\,e \\, A_{2 \\, i \\Lambda }{}^{jkl \\Sigma } \\; \\bar{\\psi }_{\\mu }^{i \\Lambda }\\, \\gamma ^\\mu \\,\\chi _{jkl \\Sigma }\\nonumber \\\\[4pt]&& + \\sqrt{2} \\, e \\, A_3^{ijk \\Lambda , lmn \\Sigma } \\, \\bar{\\chi }_{ijk \\Lambda }\\, \\chi _{lmn \\Sigma } \\; + \\, \\text{c.c.",
"}$ External indices are now raised and lowered with the $D=4$ metric $g_{\\mu \\nu } = \\eta _{\\alpha \\beta } \\, e_\\mu {}^\\alpha e_\\nu {}^\\beta $ , and $e \\equiv \\sqrt{\\vert \\text{det} \\, g_{\\mu \\nu } \\vert }$ .",
"The external covariant derivatives in the kinetic terms of the $D=4$ action (REF ) reduce as in , now also including new couplings to the $D=4$ ${\\cal N}=8$ gauge fields $A_\\mu ^{\\underline{M}} (x)$ , had we retained them, of the form $A_\\mu ^{\\underline{M}} \\, ({\\cal T}_{\\underline{M}})_\\Lambda {}^\\Sigma \\, \\psi _{\\nu i \\Sigma }$ .",
"More importantly for our purposes, the gravitino and spin-$1/2$ mass terms and their quadratic interactions in (REF ) are codified by new tensors $A_{1 \\, i \\Lambda , j \\Sigma }$ , $A_{2 \\, i \\Lambda }{}^{jkl \\Sigma }$ , $A_3{}^{ijk \\Lambda , lmn \\Sigma }$ .",
"These reduce to their gauged supergravity counterparts $A_{1 i j}$ , $A_{2i }{}^{jkl }$ , $A_3^{ijk , lmn }$ at KK level $n=0$ and extrapolate them at higher KK levels.",
"Like their $n=0$ versions, the new tensors $A_{1 \\, i \\Lambda , j \\Sigma }$ , $A_{2 \\, i \\Lambda }{}^{jkl \\Sigma }$ , $A_3{}^{ijk \\Lambda , lmn \\Sigma }$ depend on the $D=4$ ${\\cal N}=8$ scalars.",
"Further, they obey $ & A_{1 \\, i \\Lambda , j \\Sigma } = A_{1 \\, j \\Sigma , i \\Lambda } \\; , \\qquad A_{2 \\, h \\Lambda }{}^{ijk \\Sigma } = A_{2 \\, h \\Lambda }{}^{[ijk] \\Sigma } \\; , \\qquad A_{2 \\, k \\Lambda }{}^{ijk \\Lambda } = 0 \\; , \\nonumber \\\\[5pt]& A_{3 \\, ijk\\Lambda , lmn \\Sigma } = A_{3 \\, lmn \\Sigma , ijk\\Lambda } \\; , \\qquad A_{3 \\, ijk\\Lambda , lmn \\Sigma } = \\tfrac{\\sqrt{2}}{108} \\, \\epsilon _{ijkpq[rlm} \\, A_{2 \\, n ] \\Sigma }{}^{pqr \\Omega } \\, \\delta _{\\Omega \\Lambda } \\; ,$ in direct analogy with the relations (REF ), (REF ) and (REF ) satisfied by their $D=4$ ${\\cal N}=8$ counterparts."
],
[
"The KK gravitino mass matrix",
"The KK gravitino mass term in the $D=4$ action (REF ) derives from the following piece in the ExFT action (REF ) $ \\mathcal {L}_{ \\mathbf {\\psi } \\mathbf {\\psi }} = - 4 i \\epsilon ^{\\mu \\nu \\rho \\sigma } \\big ( \\mathbf { {\\cal V}}^{-1} \\big )_{ij}{}^M \\, \\bar{\\mathbf {\\psi }}_\\mu ^i \\gamma _\\nu \\nabla _M\\left(\\gamma _\\rho \\mathbf {\\psi }_\\sigma ^j\\right) \\; .$ Taking into account the form of the covariant derivative (REF ) with weight $\\lambda \\big (\\gamma _\\rho \\mathbf {\\psi }_\\sigma ^j \\big ) = \\tfrac{3}{4}$ , using the generalised Scherk-Schwarz expressions (REF ), (REF ), (REF ) and getting rid of the Levi-Civita tensor (through $\\gamma ^{\\mu \\nu } = \\frac{i}{2} \\, e^{-1} \\, \\epsilon ^{\\mu \\nu \\rho \\sigma } \\gamma _{\\rho \\sigma } \\gamma _5$ and the chirality of the KK gravitini, $\\gamma _5 \\, \\psi _\\mu ^{i\\Lambda } = \\psi _\\mu ^{i\\Lambda }$ ), (REF ) becomes $ \\mathcal {L}_{ \\mathbf {\\psi } \\mathbf {\\psi }} = e\\left( \\rho ^{-2} A_{1 ij} \\, \\mathcal {Y}_\\Lambda \\mathcal {Y}_\\Sigma - 8 \\, \\rho ^{-3} \\,( {\\cal V}^{-1} )_{ij}{}^{\\underline{N}} (U^{-1})_{\\underline{N}}{}^M {\\cal Y}_\\Lambda \\partial _M \\, {\\cal Y}_\\Sigma \\right) \\, \\bar{\\psi }_\\mu ^{i\\Lambda } \\, \\gamma ^{\\mu \\nu } \\psi _\\nu ^{j\\Sigma } \\; .$ The second term in the parenthesis can be further simplified using (REF ).",
"After this substitution, (REF ) scales with an overall factor of $\\rho ^{-2}$ , as in fact does the entire ExFT action upon generalised Scherk-Schwarz reduction.",
"This factor thus drops at the level of the equations of motion.",
"Also, (REF ) acquires a quadratic dependence on $\\mathcal {Y}_\\Lambda \\mathcal {Y}_\\Sigma $ after (REF ) is used.",
"This dependence reduces to $\\delta _{\\Lambda \\Sigma }$ at the level of the action, by virtue of the orthogonality of the spherical harmonics.",
"Thus, (REF ) gives rise to the KK gravitino mass term in the $D=4$ action (REF ), with mass matrix $ A_{1 \\, i \\Lambda , j \\Sigma } \\equiv A_{1 ij} \\, \\delta _{\\Lambda \\Sigma } - 8 \\, ( {\\cal V}^{-1} )_{ij}{}^{ \\underline{M} } ({\\cal T}_{\\underline{M}})_{\\Lambda \\Sigma } \\; .$ Due to the symmetry of $A_{1 ij}$ and the antisymmetry of $( {\\cal V}^{-1} )_{ij}{}^{ \\underline{M} }$ and $({\\cal T}_{\\underline{M}})_{\\Lambda \\Sigma }$ (in the indices $ij$ and $\\Lambda \\Sigma $ , respectively), the tensor $ A_{1 \\, i \\Lambda , j \\Sigma }$ defined in (REF ) is symmetric in the product index $i\\Lambda $ , as it must from the action (REF ) and asserted in the first relation of (REF ).",
"Likewise, the complex conjugate (c.c.)",
"contribution of (REF ) to the ExFT action (REF ) reduces to a c.c.",
"contribution to the $D=4$ action (REF ) with $ A_1{}^{i \\Lambda , j \\Sigma } \\equiv A_{1}^{ ij} \\, \\delta ^{\\Lambda \\Sigma } - 8 \\, ( {\\cal V}^{-1} )^{ ij \\underline{M} } ({\\cal T}_{\\underline{M}})^{\\Lambda \\Sigma } \\; ,$ so that $ \\big ( A_{1 \\, i \\Lambda , j \\Sigma } \\big )^* = A_1{}^{i \\Lambda , j \\Sigma }$ manifestly.",
"By (REF ), the KK gravitino mass matrix (REF ) is an infinite-dimensional, block-diagonal square matrix.",
"With the conventions specified above (REF ), for $D=11$ AdS$_4$ vacua that uplift from the SO(8) gauging, the square block at level $n$ comes in the SO(8) representations $ [0,0,0,1] \\times [n,0,0,0] \\, \\longrightarrow \\, [n,0,0,1] + [n-1,0,1,0] \\; ,$ or their branchings thereof under $G \\subset \\textrm {SO}(8)$ for AdS$_4$ solutions with residual symmetry group $G$ .",
"For AdS$_4$ solutions of type IIA that uplift from the ISO(7) gauging, the block at KK level $n$ comes instead in the SO(7) representations $ [0,0,1] \\times [n,0,0] \\, \\longrightarrow \\, [n,0,1]+[n-1,0,1] \\; ,$ or their branchings under $G \\subset \\textrm {SO}(7)$ for AdS$_4$ solutions with symmetry group $G$ .",
"In (REF ) and (REF ) only representations with positive Dynkin labels must be kept.",
"In both cases, all eigenvalues of the mass matrix (REF ) are physical and contribute to the spectrum of physical KK gravitini."
],
[
"The KK spin-$1/2$ fermion mass matrix",
"Following similar steps, we can determine the contributions of the gravitino–spin-$1/2$ fermion and fermion-fermion terms from the ExFT quadratic action (REF ) ${\\cal L}_{\\mathbf {\\psi } \\mathbf {\\chi }} = -4\\sqrt{2} \\, \\mathbf {e} \\,\\big ( \\mathbf { {\\cal V}}^{-1}\\big )^{ij \\, M} \\, \\bar{\\mathbf {\\psi }}_\\mu ^k \\, \\nabla _M \\left(\\gamma ^\\mu \\mathbf {\\chi }_{ijk}\\right) , \\quad {\\cal L}_{\\mathbf {\\chi } \\mathbf {\\chi }} = \\tfrac{1}{9}\\, \\mathbf {e} \\, \\epsilon _{ijklmnpq } \\big ( \\mathbf { {\\cal V}}^{-1}\\big )^{ij \\, M} \\bar{\\mathbf {\\chi }}^{klm} \\nabla _M \\mathbf {\\chi }^{npq} .$ The ${\\cal L}_{\\mathbf {\\psi } \\mathbf {\\chi }}$ term and its complex conjugate give rise to the corresponding KK gravitino-fermion terms in the $D=4$ KK action (REF ), with a KK tensor $A_2$ defined as $ A_{2 \\, i \\Lambda }{}^{jkl \\Sigma } \\equiv A_{2i }{}^{jkl } \\, \\delta _\\Lambda ^\\Sigma - 24 \\, \\delta _i^{[j } \\, ( {\\cal V}^{-1} )^{kl ] \\underline{N} } ({\\cal T}_{\\underline{N}})_{\\Lambda }{}^\\Sigma \\; ,$ and $ A_2{}^{i \\Lambda }{}_{jkl \\Sigma } \\equiv A_2{}^{i }{}_{jkl } \\, \\delta ^\\Lambda _\\Sigma - 24 \\, \\delta ^i_{[j } \\, ( {\\cal V}^{-1} )_{kl ]}{}^{\\underline{N} } ({\\cal T}_{\\underline{N}})^{\\Lambda }{}_\\Sigma \\; ,$ so that $(A_{2i \\Lambda }{}^{jkl \\Sigma } )^* = A_2{}^{i \\Lambda }{}_{jkl \\Sigma }$ .",
"The tensor $A_2$ given in (REF ) manifestly satisfies the antisymmmetry and tracelessness relations stated in (REF ).",
"Finally, the term ${\\cal L}_{\\mathbf {\\chi } \\mathbf {\\chi }}$ in the ExFT action gives rise to the mass terms for the spin-$1/2$ fields in the $D=4$ KK action (REF ) with mass matrix proportional to $ A_{3 \\, ijk \\Lambda , lmn \\Sigma } \\equiv A_{3 \\, ijk , lmn} \\, \\delta _{\\Lambda \\Sigma } + \\tfrac{\\sqrt{2}}{18} \\, \\epsilon _{ijklmnpq} \\, {\\cal } ( {\\cal V}^{-1} )^{pq \\underline{N} } ({\\cal T}_{\\underline{N}})_{\\Lambda \\Sigma } \\; ,$ along with its complex conjugate, $ A_{3}{}^{ ijk \\Lambda , lmn \\Sigma } \\equiv A_{3}{}^{ ijk , lmn} \\, \\delta ^{\\Lambda \\Sigma } + \\tfrac{\\sqrt{2}}{18} \\, \\epsilon ^{ijklmnpq} \\, {\\cal } ( {\\cal V}^{-1} )_{pq}{}^{ \\underline{N} } ({\\cal T}_{\\underline{N}})^{\\Lambda \\Sigma } \\; ,$ so that $(A_{3 \\, ijk \\Lambda , lmn \\Sigma })^* = A_{3}{}^{ ijk \\Lambda , lmn \\Sigma }$ .",
"The tensor (REF ) is manifestly symmetric in its product indices, as required by the $D=4$ action (REF ) and anticipated in (REF ).",
"Further, some algebra allows one to verify that the KK tensors $A_3$ and $A_2$ in (REF ) and (REF ) are indeed related as in (REF ), in analogy with the ${\\cal N}=8$ relation (REF ).",
"The KK spin-$1/2$ fermion mass matrix (REF ) is infinite-dimensional and block-diagonal.",
"The block at KK level $n$ is in the representations of SO(8) (or branchings thereof) $ [1,0,1,0] \\times [n,0,0,0] & \\longrightarrow & \\underline{[n,0,0,1]}+ \\underline{[n-1,0,1,0]} + [n+1,0,1,0] + [n-1,1,1,0] \\nonumber \\\\&& + [n-2,1,0,1]+ [n-2,0,0,1] \\; ,$ for solutions that uplift from the SO(8) gauging.",
"For solutions that uplift from the ISO(7) gauging, the blocks lie in the following SO(7) representations (or their branchings): $ ([1,0,1] + [0,0,1]) \\times [n,0,0] & \\longrightarrow & \\underline{[n,0,1]}+ \\underline{[n-1,0,1]} +[n+1,0,1] + [n-1,1,1] \\hspace{30.0pt} \\\\&& + [n-2,1,1] +[n-2,0,1] + [n,0,1] + [n-1,0,1] \\; ,\\nonumber $ for $n=0, 1, 2, \\ldots $ Again, only representations with non-negative Dynkin labels are actually present in both (REF ) and (REF ).",
"Unlike its counterpart (REF ) for the KK gravitini, the spin-$1/2$ KK fermion mass matrix (REF ) contains unphysical states that must be removed from the spectrum.",
"These correspond to the underlined representations in (REF ) and (REF ), which contain the Goldstini eaten by the massive gravitini at the same KK level $n$ .",
"Only the eigenvalues of (REF ) that belong to representations not underlined in (REF ) and (REF ) constitute the physical KK fermion states of spin one-half."
],
[
"KK fermion shifts",
"Like their $D=4$ ${\\cal N}=8$ counterparts $A_{1 i j}$ , $A_{2i }{}^{jkl }$ , the tensors $A_{1 \\, i \\Lambda , j \\Sigma }$ , $A_{2i \\Lambda }{}^{jkl \\Sigma }$ defined in (REF ) and (REF ) are also `fermion shifts', in the sense that they analogously appear in the supersymmetry variations of the KK fermions.",
"Indeed, the ExFT supersymmetry parameter can be expanded as $\\mathbf {\\epsilon }^i\\left(x,Y\\right) = \\rho (Y )^{-\\frac{1}{2}} \\,\\epsilon ^{i \\Lambda }(x)\\, \\mathcal {Y}_\\Lambda (Y ) \\; ,$ building again on , .",
"Inserting (REF ), (REF ) into the supersymmetry variations of the ExFT fermions a calculation analogous to using (REF ) allows us to compute $ \\delta \\psi _\\mu ^{i\\Lambda } = 2 \\, A_1{}^{i\\Lambda , j \\Sigma } \\, \\gamma _{\\mu } \\epsilon _{j \\Sigma } + \\ldots \\; , \\qquad \\delta \\chi ^{ijk \\Lambda } = -2\\sqrt{2} \\, A_{2 \\, h \\Sigma }{}^{ijk \\Lambda } \\, \\epsilon ^{ h \\Sigma } + \\ldots $ with $A_1{}^{i\\Lambda , j \\Sigma }$ and $A_{2 \\, h \\Sigma }{}^{ijk \\Lambda } $ respectively reproducing (REF ) and (REF ).",
"The terms shown here contain all the ${\\cal N}=8$ scalar contributions with no derivatives, while the ellipses hide contributions from the ${\\cal N}=8$ gauge fields, from derivatives of the scalars, and from derivatives of the supersymmetry parameter $\\epsilon ^{ i \\Lambda }$ .",
"All the dependence on the internal coordinates drops from the ExFT supersymmetry variations when the coefficients of the harmonics ${\\cal Y}_\\Lambda $ is equated KK level by KK level on both sides of the equal sign, leaving the supersymmetry transformations (REF ) for the KK fermions under which the full $D=4$ KK action is invariant."
],
[
"Complete KK spectra of $D=11$ {{formula:7155a84a-cd00-4d0b-8bc7-f2cdb9a88ca2}} AdS{{formula:52664aca-19a2-48ab-a06b-35f1e66f5a5b}} solutions",
"We can now use our fermionic mass matrices to compute the KK fermion spectra of specific AdS$_4$ solutions.",
"In this section, we will focus on AdS$_4$ solutions of $D=11$ supergravity that uplift from the $D=4$ ${\\cal N}=8$ SO(8) gauging .",
"For concreteness, we will restrict ourselves to the solutions that preserve at least the SU(3) subgroup of SO(8).",
"These were classified in $D=4$ in and uplifted to $D=11$ in , , , , , , .",
"The bosonic and fermionic KK level $n=0$ spectrum for these solutions is known (see , ).",
"Our results reproduce the fermionic spectra and extend them to higher KK levels $n \\ge 1$ .",
"See table REF in appendix for a summary of the spectrum of KK gravitini for these solutions up to KK level $n=2$ .",
"There are three supersymmetric solutions in this sector, with (super)symmetry ${\\cal N}=8$ SO(8) , ${\\cal N}=2$ $\\textrm {SU}(3) \\times \\textrm {U}(1)$ , and ${\\cal N}=1$ G$_2$ , , .",
"The complete supersymmetric KK spectrum for the former two is known , , , , , , and we reproduce the corresponding fermionic sectors.",
"For the ${\\cal N}=1$ G$_2$ solution, the fermionic spectrum is new.",
"Combining this with previously known sectors of the bosonic spectrum , , we can determine the complete supersymmmetric spectrum for this ${\\cal N}=1$ solution."
],
[
"Spectrum of the ${\\cal N}=1$ G{{formula:716a44ee-bff9-4bde-b976-2c10fb34f1c4}} -invariant solution ",
"The ${\\cal N}=1$ G$_2$ -invariant AdS$_4$ solution was first found as a critical point of $D=4$ ${\\cal N}=8$ SO(8)-gauged supergravity in .",
"The $S^7$ uplift of this solution to $D=11$ was determined in and .",
"The first of these references provided the $D=11$ metric, while the second completed the uplift to include the three- and four-form fluxes.",
"Geometrically, the ${\\cal N}=1$ G$_2$ solution in $D=11$ corresponds to a warped product of AdS$_4$ with a topological $S^7$ .",
"The metric on the latter can be written as a cohomogeneity-one, SO(7)-invariant deformation of the sine-cone metric foliated with round $S^6$ leaves.",
"The latter is naturally equipped with its canonical, homogeneous nearly-Kähler structure.",
"The three- and four-form fluxes can be written in terms of the nearly-Kähler forms, and break the SO(7) isometry down to a G$_2$ symmetry for the full solution.",
"See for further details.",
"The KK spectrum of this solution is partially known.",
"The spectrum at KK level $n=0$ may be determined by linearisation of $D=4$ ${\\cal N}=8$ SO(8) supergravity around its ${\\cal N}=1$ G$_2$ critical point .",
"The $n=0$ scalar , spin-$1/2$ (see ), vector and gravitino (see ) spectra are thus known.",
"The bosonic KK spectra at higher KK levels is also known partially.",
"The KK graviton spectrum was calculated following the standard spin-2 methods of .",
"The KK vector spectrum was computed using ExFT techniques , .",
"Now, we can use the fermionic mass matrices derived in section to compute the spectrum of KK gravitini and spin-$1/2$ fermions.",
"The results are summarised for the gravitini up to KK level $n=2$ in table REF of appendix .",
"Further, we may use all these previous and new results to determine the complete ${\\cal N}=1$ KK spectrum about this AdS$_4$ solution, to which we now turn.",
"The complete KK spectrum arranges itself in representations of the residual supersymmetry and bosonic symmetry groups, $\\textrm {OSp}(4|1) \\times \\textrm {G}_2$ .",
"These descend KK level by KK level from the OSp$(4|8)$ supermultiplets present in the spectrum at the ${\\cal N}=8$ SO(8) point (see also e.g.",
"table 2 of for a convenient summary).",
"At fixed KK level $n$ , the fields of all spins between 0 and 2 come in the (real) representations $[p,q]$ of G$_2$ determined by the branching $\\textrm {G}_2 \\subset \\textrm {SO}(8)$ .",
"Fields in the same G$_2$ representations but different spin must then be allocated into OSp$(4|1)$ supermultiplets, starting from higher to lower spins.",
"Table REF in the introduction comes in handy to carry out this exercise.",
"Only the MGRAV and MVEC (please refer to the table for the acronyms used throughout) OSp$(4|1)$ supermultiplets have their conformal dimensions $E_0$ fixed in terms of the spin $s_0$ of the superconformal primary.",
"For all other multiplets present in the spectrum, $E_0$ cannot be determined from group theory alone.",
"Satisfactorily enough, the dimensions and their multiplicities computed from the previously known bosonic , and from our new fermionic mass spectra match this $\\textrm {OSp}(4|1) \\times \\textrm {G}_2$ structure and bring in the dimensions $E_0$ .",
"We find that the complete supersymmetric KK spectrum of the ${\\cal N}=1$ G$_2$ invariant solution , , has the following structure.",
"At KK level $n=0$ , there are, as expected, a MGRAV and a MVEC, which respectively lie in the trivial and the adjoint representations of G$_2$ .",
"The former corresponds to the ${\\cal N}=1$ supergravity multiplet, which includes the massless graviton and gravitino.",
"The latter contains the vectors that gauge the residual symmetry G$_2$ , along with their spin-$1/2$ superpartners.",
"KK level $n=0$ is completed with a GINO multiplet containing the $\\mathbf {7}$ massive gravitini, along with $\\mathbf {1}$ and $\\mathbf {27}$ CHIRAL multiplets.",
"Higher KK levels $n\\ge 1$ contain all four generic supermultiplets, GRAV, GINO, VEC and CHIRAL, in suitable representations of G$_2$ .",
"The supersymmetric KK spectrum for the first four levels, $n=0,1,2,3$ , is summarised in tables REF –REF below.",
"For each OSp$(4|1)$ supermultiplet with given G$_2$ quantum numbers $[p,q]$ , the dimension $E_0$ is shown next to the corresponding acronym.",
"An entry of the form $m \\times (E_0)$ indicates that there are $m$ such supermultiplets.",
"Whenever there is only one multiplet, $m=1$ , we simply write $(E_0)$ .",
"Table: Supermultiplets at KK level n=0n=0 for the D=11D=11 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution.",
"Table: Supermultiplets at KK level n=1n=1 for the D=11D=11 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution.Table: Supermultiplets at KK level n=2n=2 for the D=11D=11 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution.Table: Supermultiplets at KK level n=3n=3 for the D=11D=11 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution.Table: All KK scalars with dimension Δ≤3\\Delta \\le 3 around the D=11D=11 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution.",
"From tables REF –REF it is possible to infer that the conformal dimension $E_0$ for each of the OSp$(4|1)$ supermultiplets present in the spectrum at KK level $n=0 , 1 , 2 \\ldots $ , with G$_2$ quantum numbers $[p,q]$ , is $ \\textrm {(M)GRAV} & : & E_0 = 1 + \\sqrt{ \\tfrac{9}{4} + \\tfrac{5}{8} n (n+6)-\\tfrac{5}{4} \\, {\\cal C}_2 ( p,q ) } , \\hspace{10.0pt} \\\\[5pt]\\textrm {GINO} & : & E_0 = 1 + \\sqrt{ 4 + \\tfrac{5}{8} n (n+6) - \\tfrac{5}{4} \\, {\\cal C}_2 ( p,q ) } \\; , \\hspace{10.0pt} \\\\[5pt]\\textrm {(M)VEC} & : & E_0 = 1 + \\sqrt{ \\tfrac{21}{4} + \\tfrac{5}{8} n (n+6) - \\tfrac{5}{4} \\, {\\cal C}_2 ( p,q ) } \\; , \\hspace{10.0pt} \\\\[5pt]\\textrm {CHIRAL} & : & E_0 = 1 + \\sqrt{ 6 + \\tfrac{5}{8} n (n+6) - \\tfrac{5}{4} \\, {\\cal C}_2 ( p,q ) } \\; .$ In these expressions, $n(n+6)$ are the eigenvalues of the scalar Laplacian on $S^7$ , andRecall that the overall normalisation of the Casimir operator is arbitrary.",
"A popular normalisation, which we use for the eigenvalue ${\\cal C}_2$ of the G$_2$ (REF ), SU(3) (REF ), SO(7) (REF ) and SO(6) () quadratic Casimir operator in the representation $R$ , is ${\\cal C}_2 = d_G/( 2 \\, d_R ) \\, {\\cal I}_R$ , where $d_R$ , $d_G$ and ${\\cal I}_R$ are respectively the dimension of $R$ , the dimension of the adjoint, and the Dynkin index of $R$ , see e.g.",
".",
"$ {\\cal C}_2( p,q ) \\equiv \\tfrac{1}{3} \\, p(p+5) + q(q+3) + \\, pq \\; ,$ is the eigenvalue of the quadratic Casimir operator of G$_2$ in the $[p,q]$ representation.",
"The dimension (REF ) of the (M)GRAV supermultiplets is in agreement with the masses of the individual graviton states given in table 2 of , with $n_\\textrm {here} = p_\\textrm {here} = k_\\textrm {there}$ , $q_\\textrm {here} = 0$ .",
"Likewise, the individual vector states contained in the supermultiplets with dimensions (REF )–() match the masses reported in table 14 of up to second KK level.",
"More generally, recalling from table REF the value of the conformal primary spin $s_0$ for each supermultiplet, the formulae (REF )–() can be collectively written as in equation (REF ) with $d=7$ , $\\alpha = \\tfrac{5}{8}$ and $\\beta = \\tfrac{5}{4}$ therein, as already advertised in the introduction.",
"The spectrum of individual KK scalar states can be inferred from the complete supersymmetric KK spectrum.",
"Table REF lists all the scalars with conformal dimensions $\\Delta \\le 3$ .",
"The table includes the analytical value of $\\Delta $ together with a convenient numerical approximation.",
"Also shown in the table is the KK level $n$ at which each scalar appears, as well as its $\\textrm {G}_2$ charges $[p,q]$ .",
"The OSp$(4|1)$ supermultiplet with dimension $E_0$ , at the same KK level $n$ and with the same G$_2$ charges $[p,q]$ , that contains each scalar is also shown.",
"The dimension $\\Delta $ will only match $E_0$ if the scalar in question is the superconformal primary of its multiplet.",
"The scalars in table REF are dual to relevant ($\\Delta < 3$ ) or classically marginal ($\\Delta = 3$ ) operators in the dual field theory.",
"All scalars with $\\Delta \\le 3$ turn out to arise at KK levels $n=0,1,2,3$ .",
"Each of these KK levels contain scalars dual to irrelevant ($\\Delta >3$ ) operators as well.",
"At KK levels $n \\ge 4$ , all scalars are dual to irrelevant operators."
],
[
"Complete KK spectra of ${\\cal N}=1$ AdS{{formula:a180120e-6397-44c1-a579-8ba01dad6daa}} solutions of type IIA",
"Next we turn to compute the KK fermionic spectra of the AdS$_4$ solutions of massive type IIA supergravity that are obtained from vacua of $D=4$ ${\\cal N}=8$ dyonic ISO(7) supergravity upon consistent uplift on $S^6$ , .",
"Again for definiteness, we will focus on those solutions that preserve at least the SU(3) subgroup of SO(7).",
"From a four-dimensional perspective, these were classified in .",
"Their type IIA uplifts were given in , , .",
"Using our mass matrices from section , we have computed the spectrum of KK gravitini and spin-$1/2$ fermions for the first few KK levels.",
"A summary of the gravitino spectra of these solutions up to KK level $n=2$ is provided in table REF of appendix .",
"This sector contains three supersymmetric solutions with residual (super)symmetry ${\\cal N}=2$ $\\textrm {SU}(3) \\times \\textrm {U}(1)$ , ${\\cal N}=1$ G$_2$ , , , and ${\\cal N}=1$ SU(3) , , .",
"The complete supersymmetric spectrum of the ${\\cal N}=2$ solution was recently obtained in , and our results match the fermionic spectrum that can be inferred from the results of that reference.",
"Here, we will give the complete supersymmetric spectrum of the ${\\cal N}=1$ solutions.",
"We now move on to discuss them in turn."
],
[
"Spectrum of the ${\\cal N}=1$ G{{formula:247844f2-5754-46d4-8a59-ec9920c6a9fa}} -invariant solution ",
"The AdS$_4$ solution with ${\\cal N}=1$ supersymmetry and G$_2$ symmetry was first reported as a critical point of $D=4$ ${\\cal N}=8$ dyonic ISO(7) supergravity in .",
"The $S^6$ uplift to massive IIA was performed in , (see (4.6) of the latter reference), and shown to coincide with a previously known solution first written in .",
"The ten-dimensional solution is a direct product of AdS$_4$ and the round $S^6$ sphere, endowed with its SO(7)-invariant homogeneous Einstein metric.",
"The latter is inherited from the canonical homogeneous nearly-Kähler structure on $S^6$ .",
"All type IIA forms are active and take values along the nearly-Kähler forms, thereby reducing the symmetry of the full solution to G$_2$ .",
"Some details of the KK spectrum about this solution are already known.",
"As always, the spectrum at KK level $n=0$ is found by linearising $D=4$ ${\\cal N}=8$ ISO(7) supergravity around the ${\\cal N}=1$ G$_2$ critical point .",
"The $n=0$ scalar and vector spectrum was given in that reference.",
"The $n=0$ fermion spectrum, which can be deduced by supersymmetry from the $n=0$ bosonic spectrum , has been explicitly given in the recent .",
"At higher KK levels $n \\ge 1$ , the spectrum of KK gravitons and vectors is also known.",
"These were respectively computed using standard spin-2 techniques and ExFT methods , .",
"Here, we have computed the spectrum of KK gravitini and spin-$1/2$ fermions using the mass matrices (REF ) and (REF ).",
"Combining these new results with the previously known ones, we can further obtain the complete ${\\cal N}=1$ KK spectrum of this solution.",
"Like in the $D=11$ case of section REF , the complete KK spectrum combines itself in representations of $\\textrm {OSp}(4|1) \\times \\textrm {G}_2$ .",
"The process to find the multiplet structure of the spectrum is very similar to that explained in detail in section REF and, for that reason, we shall be brief.",
"The most important difference with respect to the $D=11$ case is that the G$_2$ representations at fixed KK level $n$ branch from the putative SO(7) representations summarised in table 1 of .",
"Proceeding as explained in section REF , we find the $\\textrm {OSp}(4|1) \\times \\textrm {G}_2$ structure of the spectrum.",
"Except for the massless multiplets, the dimensions $E_0$ of the multiplets are again left undetermined by group theory.",
"Fortunately, the dimensions and their multiplicities of the known bosonic KK fields , and the new fermionic KK fields are compatible with this $\\textrm {OSp}(4|1) \\times \\textrm {G}_2$ structure and allow us to give explicitly the supermultiplet dimensions.",
"At KK level $n=0$ , the spectrum of the G$_2$ ${\\cal N}=1$ solution , , of type IIA coincides with the spectrum of its $D=11$ counterpart , , , as already noted in .",
"This spectrum is summarised in table REF below, which is included for completeness although this table is identical to table REF for the $D=11$ case.",
"The spectra in the IIA and $D=11$ cases do differ at higher KK levels: see tables REF –REF below for the supersymmetric spectrum at levels $n=1,2,3$ in the type IIA case.",
"The dimension $(E_0)$ is shown next to each supermultiplet.",
"An entry of the form $m \\times (E_0)$ indicates that there are $m$ such supermultiplets, with the label $m$ omitted when $m=1$ .",
"Table: Supermultiplets at KK level n=0n=0 for the 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution of type IIA.Table: Supermultiplets at KK level n=1n=1 for the 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution of type IIA.",
"Table: Supermultiplets at KK level n=2n=2 for the 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution of type IIA.Table: Supermultiplets at KK level n=3n=3 for the 𝒩=1{\\cal N}=1 G 2 _2-invariant AdS 4 _4 solution of type IIA.Inspection of tables REF –REF allows us to deduce generic expressions for the conformal dimensions of the OSp$(4|1)$ supermultiplets contained in the KK spectrum.",
"Each type of supermultiplet in the $[p,q]$ representation of G$_2$ at KK level $n=0,1,2, \\ldots $ has the following scaling dimension: $ \\textrm {(M)GRAV} & : & E_0 = 1 + \\sqrt{ \\tfrac{9}{4} + \\tfrac{5}{12} n (n+5) } , \\hspace{10.0pt} \\\\[5pt]\\textrm {GINO} & : & E_0 = 1 + \\sqrt{ 4 + \\tfrac{5}{6} n (n+5) - \\tfrac{5}{4} \\, {\\cal C}_2 (p,q) } \\; , \\hspace{10.0pt} \\\\[5pt]\\textrm {(M)VEC} & : & E_0 = 1 + \\sqrt{ \\tfrac{21}{4} + \\tfrac{5}{6} n (n+5) - \\tfrac{5}{4} \\, {\\cal C}_2 (p,q) } \\; , \\hspace{10.0pt} \\\\[5pt]\\textrm {CHIRAL} & : & E_0 = 1 + \\sqrt{ 6 + \\tfrac{5}{6} n (n+5) - \\tfrac{5}{4} \\, {\\cal C}_2 (p,q) } \\; .$ Here, $n(n+5)$ are the eigenvalues of the scalar Laplacian on $S^6$ and ${\\cal C}_2( p,q )$ are the eigenvalues (REF ) of the quadratic Casimir operator of G$_2$ in the $[p,q]$ representation.",
"The (M)GRAV supermultiplets have dimensions (REF ) that agree with the individual graviton masses given in (3.1) of with $n_\\textrm {here} = k_\\textrm {there}$ .",
"In addition, the vector states contained in the supermultiplets with dimensions (REF )–() reproduce the masses given in table 15 of up to KK level $n=2$ .",
"Like their counterparts (REF )–() for the ${\\cal N}=1$ G$_2$ spectrum in $D=11$ , all the dimensions (REF )–() for the type IIA spectrum also conform to the generic expression (REF ) brought to the introduction, now with $d=6$ , $\\alpha = \\tfrac{5}{6}$ , $\\beta = \\tfrac{5}{4}$ therein.",
"This is straightforward to see for the GINO, (M)VEC and CHIRAL dimensions ()–(), by using the relevant values of $s_0$ from table REF .",
"To see that the (M)GRAV dimension (REF ) can be also rewritten as in (REF ), $E_0 = 1 + \\sqrt{ \\tfrac{9}{4} + \\tfrac{5}{6} n (n+5) - \\tfrac{5}{4} \\, {\\cal C}_2 (p,q) } \\; ,$ note from tables REF –REF that, at KK level $n$ , there is a unique (M)GRAV supermultiplet that occurs with G$_2$ quantum numbers $p=n$ , $q=0$ , and that ${\\cal C}_2( n,0) = \\tfrac{1}{3} n (n+5)$ by (REF ).",
"Finally, like for the ${\\cal N}=1$ G$_2$ solution of $D=11$ supergravity, the spectrum of individual KK scalar states for the ${\\cal N}=1$ G$_2$ solution of type IIA can be deduced from the complete supersymmetric spectrum that we have presented in this section.",
"Table REF compiles the result for all scalars in the spectrum with conformal dimension $\\Delta \\le 3$ , following the same layout as table REF .",
"In this case, all scalars with $\\Delta \\le 3$ arise at KK levels $n=0,1,2$ .",
"Each of these KK levels also contain scalars with $\\Delta >3$ .",
"At KK levels $n \\ge 3$ , all scalars have dimensions $\\Delta >3$ .",
"Table: All KK scalars with dimension Δ≤3\\Delta \\le 3 around the 𝒩=1{\\cal N}=1 SU(3)-invariant AdS 4 _4 solution of type IIA."
],
[
"Spectrum of the ${\\cal N}=1$ SU(3)-invariant solution ",
"We finally turn to the ${\\cal N}=1$ AdS$_4$ solution of type IIA with SU(3) residual symmetry.",
"A critical point of maximal supergravity with dyonic gaugings , , with this residual (super)symmetry was first reported in .",
"This vacuum was more precisely identified as a critical point of the dyonic ISO(7) gauging in .",
"The resulting ten-dimensional AdS$_4$ solution was constructed in using the uplifting formulae of , .",
"The massive type IIA solution, (4.4), (4.5) of , features a warped product of AdS$_4$ with a topological $S^6$ .",
"The latter is equipped with a cohomogeneity-one metric.",
"This may be seen as a deformation of the usual sine-cone metric over $S^5$ , where the U(1) Hopf fibre of the latter is inhomogeneously squashed against the $\\mathbb {CP}_2$ base, so that the isometry is $\\textrm {SU}(3) \\times \\textrm {U}(1)$ .",
"The $S^5$ is endowed with its canonical Sasaki-Einstein structure, along whose forms take values the type IIA fluxes.",
"The symmetry of the full solution is thus reduced to SU(3).",
"The ${\\cal N}=1$ supersymmetry is captured by a type of $G$ -structure discussed in .",
"Like in the previous cases, the KK spectrum of this solution is known partially.",
"The bosonic spectrum at KK level $n=0$ was given in .",
"The $n=0$ fermion spectrum follows by supersymmetry from its bosonic counterpart, and has been explicitly given in .",
"At higher KK levels, only the spectra of KK gravitons and KK vectors are known.",
"In the present paper, we have determined the spectrum of KK gravitini and spin-$1/2$ fermions diagonalising the mass matrices of section .",
"These new and previous results allow us to determine the complete supersymmetric KK spectrum above this ${\\cal N}=1$ AdS$_4$ solution.",
"The complete KK spectrum in this case comes in representations of $\\textrm {OSp}(4|1) \\times \\textrm {SU}(3)$ .",
"Other than this, the allocation of the spectra into supermultiplets proceeds as in section REF .",
"The conformal dimensions are again left undertermined by the group theory, but these can be brought in from , and from the explicit calculation of the fermionic sector described above.",
"The result up to KK level $n=3$ is summarised in tables REF –REF below.",
"Again, the dimension $(E_0)$ is shown next to each supermultiplet.",
"An entry of the form $m \\times (E_0)$ indicates that there are $m$ such supermultiplets, with the label $m$ omitted when $m=1$ .",
"Recall that the representation $[p,q]$ of SU(3) with $p \\ne q$ is complex, and its conjugate is $[q,p]$ .",
"In order to avoid repetition, the supermultiplets with SU(3) quantum numbers $[p,q]$ with $q>p$ are simply indicated as the complex conjugates of those with quantum numbers $[q,p]$ .",
"Supermultiplets in conjugate representations have the same conformal dimension $E_0$ .",
"For example, from table REF , the KK spectrum includes $\\overline{\\mathbf {6}}$ CHIRAL multiplets with dimension $E_0 = \\frac{10}{3}$ at KK level $n=1$ .",
"Table: Supermultiplets at KK level n=0n=0 for the 𝒩=1{\\cal N}=1 SU(3)-invariant AdS 4 _4 solution of type IIA.",
"Table: Supermultiplets at KK level n=1n=1 for the 𝒩=1{\\cal N}=1 SU(3)-invariant AdS 4 _4 solution of type IIA.",
"Table: Supermultiplets at KK level n=2n=2 for the 𝒩=1{\\cal N}=1 SU(3)-invariant AdS 4 _4 solution of type IIA.Table: Supermultiplets at KK level n=3n=3 for the 𝒩=1{\\cal N}=1 SU(3)-invariant AdS 4 _4 solution of type IIA.The $\\textrm {OSp}(4|1)$ representations at level $n=0$ have recently appeared in , and table REF matches their results.",
"As expected, the $n=0$ spectrum contains a singlet MGRAV and $\\mathbf {8}$ MVECs: the former is the ${\\cal N}=1$ supergravity multiplet and the latter contains the vectors that gauge the residual SU(3) symmetry.",
"More generally, like in the previous ${\\cal N}=1$ cases discussed in sections REF and REF , closed form expressions may be given for the conformal dimensions at all levels $n \\ge 0$ .",
"From tables REF –REF , the conformal dimension $E_0$ of each type of OSp$(4|1)$ supermultiplet at KK level $n$ with SU(3) Dynkin labels $[p,q]$ turns out to be given by $ \\textrm {(M)GRAV } & : & E_0 = 1 + \\sqrt{ \\tfrac{9}{4} + \\tfrac{5}{6} n (n+5) - \\tfrac{5}{3} \\, {\\cal C}_2 (p,q) } \\; , \\hspace{10.0pt} \\\\[5pt]\\textrm {GINO} & : & E_0 = 1 + \\sqrt{ 4 + \\tfrac{5}{6} n (n+5) - \\tfrac{5}{3} \\, {\\cal C}_2 (p,q) } \\; , \\hspace{10.0pt} \\\\[5pt]\\textrm {(M)VEC} & : & E_0 = 1 + \\sqrt{ \\tfrac{21}{4} + \\tfrac{5}{6} n (n+5) - \\tfrac{5}{3} \\, {\\cal C}_2 (p,q) } \\; , \\hspace{10.0pt} \\\\[5pt]\\textrm {CHIRAL} & : & E_0 = 1 + \\sqrt{ 6 + \\tfrac{5}{6} n (n+5) - \\tfrac{5}{3} \\, {\\cal C}_2 (p,q) } \\; .$ Here, like in (REF )–() for the type IIA ${\\cal N}=1$ G$_2$ solution, $n(n+5)$ are the eigenvalues of the scalar Laplacian on $S^6$ , but now $ {\\cal C}_2( p,q) \\equiv \\tfrac{1}{3} \\left[ p (p+3) + q (q+3) + pq \\right]$ are the eigenvalues of the quadratic Casimir operator of SU(3) in the $[p,q]$ representation, normalised as indicated in footnote REF .",
"The (M)GRAV dimensions (REF ) match the masses for the graviton states, given in (3.1) of , with $k_\\textrm {there} = n_\\textrm {here}$ , $\\ell _\\textrm {there} = p_\\textrm {here} + q_\\textrm {here}$ , $p_\\textrm {there} = p_\\textrm {here}$ .",
"The individual vector masses that follow from (REF )–() match table 15 of up to KK level 2.",
"As for the previous cases, the dimensions (REF )–() can also be written compactly as (REF ) of the introduction with $d=6$ , $\\alpha = \\tfrac{5}{6}$ and $\\beta = \\tfrac{5}{3}$ .",
"Table: All KK scalars with dimension Δ≤3\\Delta \\le 3 around the 𝒩=1{\\cal N}=1 SU(3)-invariant AdS 4 _4 solution of type IIA.",
"Again like in the previous cases, the KK scalar spectrum for the ${\\cal N}=1$ SU(3) type IIA solution follows from the complete supersymmetric spectrum.",
"All KK scalars with dimensions $\\Delta \\le 3$ are summarised in table REF , following the same notation and conventions as tables REF and REF .",
"As already noted in , massless ($\\Delta = 3$ ) scalars, in the adjoint of SU(3), already appear at KK level $n=0$ .",
"This is an unusual feature for this type of AdS$_4$ solutions, for which massless scalars tend to appear at higher KK levels.",
"The massless scalar spectrum is completed at level $n=1$ with $\\mathbf {10} + \\overline{\\mathbf {10}}$ more scalars.",
"All scalars with dimension $\\Delta <3$ appear at KK levels $n = 0, 1, 2 , 3$ .",
"These levels also contain scalars with $\\Delta >3$ .",
"For KK levels $n \\ge 4$ , all scalars have $\\Delta >3$ ."
],
[
"Complete non-supersymmetric KK spectra",
"Within the class of $D=11$ and type IIA AdS$_4$ solutions with at least SU(3) symmetry that we are considering, there are non-supersymmetric solutions besides the ${\\cal N}=1$ cases discussed in sections and .",
"These include solutions with symmetry SO(7)$_v$ , SO(7)$_c$ and SU$(4)_c$ in $D=11$ , and SO(7) , , , SO(6) , and G$_2$ , , in type IIA.",
"In addition, there are two type IIA solutions with SU(3) symmetry which are only known numerically , and will be excluded from our discussion.",
"See respectively and for these eleven- and ten-dimensional AdS$_4$ solutions in our conventions.",
"In this section we turn to address the KK spectrum for these solutions.",
"The state-of-the-art for their KK spectra is the following.",
"For all of these, the bosonic , , and fermionic , spectra at KK level $n=0$ are known.",
"At higher KK levels only the bosonic spectra are known, either partially or completely.",
"The spectra of KK gravitons , and vectors are known and, for the G$_2$ solution of type IIA, also the KK scalar spectrum is known .",
"Thus, the bosonic KK spectrum of the G$_2$ solution is completely known.",
"For the other solutions, the bosonic spectrum is known short of the KK scalars.",
"In this section, we will give the fermionic spectra for all these solutions, thereby completing the KK spectrum for the type IIA G$_2$ solution.",
"For all other solutions, we will conjecture a formula for the KK scalar masses based on strong plausibility arguments.",
"This will effectively complete their KK spectra as well.",
"Using our fermionic KK mass matrices, we have computed the KK gravitino and spin-$1/2$ fermion spectra for all these solutions.",
"We have recorded the gravitino mass eigenvalues up to KK level $n=2$ in tables REF and REF of appendix .",
"For the G$_2$ solution, also the spin-$1/2$ fermion spectrum is tabulated in table REF of the appendix, again up to second KK level.",
"The fields of different spin present in these spectra are organised KK level by KK level in representations of the residual symmetry group $G$ , with $G= \\textrm {SO}(7)$ or $G= \\textrm {SO}(6) \\sim \\textrm {SU} (4) $ , with Dynkin labels $[p,q,r]$ , or $G= \\textrm {G}_2$ , with Dynkin labels $[p,q]$ .",
"For the $D=11$ solutions, these representations branch from the SO(8) representations of the spectrum at the ${\\cal N}=8$ SO(8) point (see also e.g.",
"table 2 of for a summary).",
"For the type IIA solutions, the representations split instead from the SO(7) representations given in table 1 of .",
"For the G$_2$ solution, we have recorded the G$_2$ representation content of the spectrum up to KK level $n=3$ in tables REF –REF below.",
"Together with this group theory analysis and the previously known results for the bosonic sector , , , of the KK spectrum, our new fermionic results finally complete the spectrum of the non-supersymmetric G$_2$ -invariant solution of type IIA.",
"Further, closed-form formulae can be given for the masses at all KK levels, for this and the other solutions, as we will see momentarily.",
"Table: The complete KK spectra for the analytic non-supersymmetric AdS 4 _4 solutions of D=11D=11 supergravity (left) and massive type IIA supergravity (right) that respectively uplift from critical points of D=4D=4 𝒩=8{\\cal N}=8 SO(8) and ISO(7) supergravities, with residual symmetry groups larger than SU(3).",
"For each KK field of spin ss, its squared mass M 2 L 2 M^2L^2 is given at all KK level nn.",
"The spectra also depend on the quadratic Casimir operators 𝒞 2 {\\cal C}_2 specified in the text.",
"The scalar spectra marked with (*) ^{(*)} are conjectured.The spectra of the ${\\cal N}=1$ solutions reported in sections and exhibit significant degeneracy, in the sense that all OSp$(4|1)$ supermultiplets of the same type, at the same KK level $n$ , and with same SU(3) or G$_2$ quantum numbers $[p,q]$ , all have the same conformal dimension $E_0$ .",
"However, individual states with the same spin within the same OSp$(4|1)$ supermultiplet necessarily have different masses, as their conformal dimensions must differ by one (see table REF ).",
"Obviously, this restriction does not affect the non-supersymmetric solutions, as the states do not fill out OSp$(4|1)$ supermultiplets in the first place.",
"In fact, for these ${\\cal N}=0$ cases the spectra show an even larger degeneracy: states at the same KK level, with the same spin and the same $G$ quantum numbers, all have the same mass.",
"This high degeneracy leads to the existence of closed-form formulae for the mass spectra of these solutions, as already announced above.",
"Closed formulae were given in , for the KK graviton spectra of all these solutions.",
"A mass formula has been similarly given for the KK scalar spectrum of the G$_2$ solution .",
"For this solution, the same scalar mass formula has been shown to also fit the KK vector spectrum given in up to KK level $n=2$ .",
"Now, we have derived mass formulae for the vector spectra of all other solutions using the data in tables 14 and 15 of .",
"For our new fermionic results, we have also been able to deduce closed-form mass formulae.",
"Table REF summarises all of these.",
"The $D=11$ and IIA graviton spectra listed on the table reproduce the corresponding entries in table 2 of and equation (3.1) of , respectively, with the following dictionary of quantum numbers.",
"For the $D=11$ SO$(7)_v$ solution, $n_\\textrm {here} = n_\\textrm {there}$ , $ p_\\textrm {here} = k_\\textrm {there}$ , $q_\\textrm {here} =r_\\textrm {here} = 0$ ; for the SO$(7)_c$ solution, $n_\\textrm {here} = r_\\textrm {here} = n_\\textrm {there}$ , $ p_\\textrm {here} = q_\\textrm {here}=0$ ; and for the SU$(4)_c$ solution, $n_\\textrm {here} = n_\\textrm {there} $ , $p_\\textrm {here} = r_\\textrm {there} $ , $q_\\textrm {here}=0$ , $r_\\textrm {here} = n_\\textrm {there} - r_\\textrm {there} $ .",
"For the type IIA gravitons we have, for the SO(7) solution, $n_\\textrm {here} =n_\\textrm {there}$ , $q_\\textrm {here} = \\ell _\\textrm {there} $ , $p_\\textrm {here} =r_\\textrm {here} = 0$ ; for the SO(6) solution, $n_\\textrm {here} =n_\\textrm {there}$ , $q_\\textrm {here} = \\ell _\\textrm {there} $ , $p_\\textrm {here} =r_\\textrm {here} = 0$ ; and for the G$_2$ solution, $n_\\textrm {here} = p_\\textrm {here} = n_\\textrm {there}$ , $q_\\textrm {here} = 0$ .",
"The KK scalar and vector formulae for the IIA G$_2$ solution have been imported from (18) of with $n_\\textrm {here} =\\ell _\\textrm {there}$ , $p_\\textrm {here} =n_{1 \\, \\textrm {there}}$ , $q_\\textrm {here} =n_{2 \\, \\textrm {there}}$ .",
"All other mass formulae in table REF are new.",
"As is apparent from the table, a pattern emerges.",
"The squared mass of a state of spin $s$ at KK level $n$ , with Dynkin labels $[p,q,r]$ under SO(7) or $\\textrm {SU}(4)$ ($[q,p,r]$ for $\\textrm {SO} (6) $ ), or $[p,q]$ under G$_2$ , is given by $ M^2 L^2 = \\gamma _s + \\alpha \\, n (n+ d-1) - \\beta \\, {\\cal C}_2 \\; .$ Here, $d=7$ in M-theory and $d=6$ in type IIA, as usual.",
"Also, $\\gamma _s$ is a constant, the same for all solutions, that only depends on the spin $s$ of the field in question: $\\gamma _2 = 0$ , $\\gamma _{3/2} = \\tfrac{9}{2}$ , $\\gamma _1 = 6$ , $\\gamma _{1/2} = \\tfrac{15}{2}$ and, for the G$_2$ solution, $\\gamma _0 = 6$ as follows from .",
"The positive constant $\\alpha $ only depends on whether the solution is a solution in M-theory ($\\alpha = \\tfrac{3}{4}$ ) or type IIA ($\\alpha = 1$ ).",
"The positive constant $\\beta $ is the same for all solutions with the same symmetry in both M-theory and type IIA, with $\\beta = \\tfrac{6}{5}$ for the SO(7) solutions and $\\beta = \\tfrac{3}{2}$ for the $\\textrm {SO}(6) \\sim \\textrm {SU}(4)$ solutions.",
"The IIA G$_2$ solution happens to have $\\beta = \\tfrac{3}{2}$ as well.",
"Finally, ${\\cal C}_2$ is the eigenvalue, normalised as indicated in footnote REF , of the quadratic Casimir operator in the $[p,q,r]$ representation for SO(7) or $\\textrm {SU}(4)$ , $ \\textrm {SO}(7) & : & {\\cal C}_2 (p,q,r) = \\tfrac{1}{8} \\left[ 4 p (p+5) + 8 q (q+4) + 3 r (r+6) + 8 pq + 4 pr + 8qr \\right] \\; , \\\\[5pt]\\textrm {SU}(4) & : & {\\cal C}_2 (p,q,r) = \\tfrac{1}{8} \\left[ 3 p (p+4) + 4 q (q+4) + 3 r (r+4) + 4 pq +2 pr + 4 qr \\right] \\; , \\hspace{10.0pt}$ (or in the representation $[q,p,r]$ for SO(6), with $ {\\cal C}^{\\textrm {SU}(4)}_2 (p,q,r) = {\\cal C}^{\\textrm {SO}(6)}_2 (q,p,r)$ ) for the solutions with those residual symmetry groups.",
"For the the G$_2$ solution of type IIA, ${\\cal C}_2$ is the $[p,q]$ quadratic Casimir eigenvalue (REF ).",
"Although we have not computed the KK scalar spectra for the SO(7) and $\\textrm {SO}(6) \\sim \\textrm {SU}(4)$ solutions, and to do so is beyond the scope of this paper, it is natural to assume that these will follow the rigid pattern shown by table REF and equation (REF ), as the KK scalar spectrum of the G$_2$ solution does.",
"Assuming that the KK scalar masses for the other solutions also take on the form (REF ), only by choosing $\\gamma _0 = 6$ and letting $\\beta $ follow the pattern above, are the known spectra at KK level $n=0$ , reproduced.",
"Level $n=0$ does not fix the coefficient $\\alpha $ , but it is natural to assume that this coefficient will follow the same pattern as fields of all other spins.",
"Following this logic, one arrives at the KK scalar mass formulae marked with $^{(*)}$ in table REF .",
"For the type IIA SO(7) solution, the proposed mass formula reproduces the G$_2$ -singlet masses at KK level $n=2$ given in table REF of section REF below.",
"Except for this minor caveat on the KK scalar sector, table REF thus gives the masses in the complete KK spectrum for all the non-supersymmetric AdS$_4$ solutions under consideration in this section.",
"Together with the table, the complete spectra are characterised by the representation content under the relevant residual symmetry group $G$ , obtained as described above.",
"For example, the mass formulae given in table REF for the $D=11$ SO$(7)_v$ and SO$(7)_c$ solutions are identical.",
"But the KK spectra of these two solutions are not the same: they differ in their SO(7)-representation content.",
"For the ${\\cal N}=0$ G$_2$ solution, the G$_2$ content of the KK spectrum has been summarised up to KK level $n=3$ in tables REF –REF below.",
"In these tables, each cell lists the states with a certain spin $s_0$ and the number $m$ of them (in the format $( s = s_0 ) \\times m)$ , in an allowed representation $[p,q]$ of G$_2$ .",
"The corresponding masses follow from table REF .",
"For example, table REF indicates the existence of one vector, two spin-$1/2$ fermions and one scalar in the $\\mathbf {189}$ of G$_2$ , with masses 14, $\\frac{35}{2}$ and 14, respectively, at KK level 2.",
"Table: States at KK level n=0n=0 for the 𝒩=0{\\cal N}=0 G 2 _2-invariant AdS 4 _4 solution of type IIA.",
"Table: States at KK level n=1n=1 for the 𝒩=0{\\cal N}=0 G 2 _2-invariant AdS 4 _4 solution of type IIA.Table: States at KK level n=2n=2 for the 𝒩=0{\\cal N}=0 G 2 _2-invariant AdS 4 _4 solution of type IIA.Table: States at KK level n=3n=3 for the 𝒩=0{\\cal N}=0 G 2 _2-invariant AdS 4 _4 solution of type IIA."
],
[
"Discussion",
"We have derived from ExFT the KK fermionic mass matrices for a class of AdS solutions of string and M-theory that uplift on spheres from maximal gauged supergravity.",
"We have focused on E$_{7(7)}$ ExFT, but similar mass matrices can be derived for other instances of ExFT with other duality groups.",
"We have also used these mass matrices to obtain the spectrum of KK fermions about some concrete AdS$_4$ solutions of M-theory and massive type IIA supergravity.",
"These results, together with previously known sectors of the bosonic spectra, have allowed us to give the complete spectrum for some ${\\cal N}=1$ and some non-supersymmetric solutions in this class."
],
[
"A more general pattern for the conformal dimensions",
"A generic formula, (REF ), exists for the conformal dimensions of the OSp$(4|1)$ supermultiplets present in the KK spectra of the ${\\cal N}=1$ AdS$_4$ solutions of M-theory and type IIA that we have covered in this work.",
"The expression (REF ) can be further generalised to account for all the spectra known so far of supersymmetric $D=11$ and type IIA AdS$_4$ solutions that uplift from the SO(8) or ISO(7) maximal supergravities.",
"Consider an AdS$_4$ solution in this class preserving ${\\cal N}$ supersymmetries, invariant under a residual symmetry group $G \\subset \\textrm {SO}(8)$ in $D=11$ or $G \\subset \\textrm {SO}(7)$ in type IIA.",
"We find that the dimension $E_0$ of an OSp$(4|{\\cal N})$ supermultipletThe dimension of a supermultiplet is defined to be the dimension of its superconformal primary.",
"in the KK spectrum of this solution, with superconformal primary of spin $s_0$ and arising at KK level $n$ , turns out to be given generically by $ \\quad E_0 = s_0^{{\\scriptscriptstyle (2)}} -\\tfrac{1}{2} + \\sqrt{ \\tfrac{9}{4} + s_0^{{\\scriptscriptstyle (2)}} (s_0^{{\\scriptscriptstyle (2)}} +1) -s_0(s_0+1) + \\alpha \\, n (n + d-1) + {\\cal Q}_2} \\; .$ Here, $n (n + d-1)$ is the eigenvalue of the scalar Laplacian on $S^d$ , with $d=7$ for $D=11$ and $d=6$ for type IIA; $\\alpha $ is a solution-dependent constant; ${\\cal Q}_2$ is a solution-dependent homogeneous quadratic polynomial in the integer Dynkin labels of $G$ ; and $s_0^{{\\scriptscriptstyle (2)}} =\\left\\lbrace \\begin{array}{lll}\\tfrac{1}{2} \\, (4- {\\cal N}) & , & \\quad \\textrm { if 1 \\le {\\cal N}\\le 4 } \\\\[4pt]0 & , & \\quad \\textrm { if 4 \\le {\\cal N}\\le 8 }\\end{array}\\right.$ is the spin of the superconformal primary of any of the Osp$(4|{\\cal N})$ supermultiplets containing a graviton as its highest spin state.",
"For ${\\cal N}=1$ supersymmetry, the massless (MGRAV, in the notation of table REF ) or generic (GRAV) graviton supermultiplets have $s_0^{{\\scriptscriptstyle (2)}} = \\tfrac{3}{2}$ , and (REF ) is indeed of the form (REF ) with ${\\cal Q}_2 \\equiv - \\beta \\, {\\cal C}_2 (p,q) $ , for the particular values of the constant $\\beta $ specified in the text and the relevant quadratic Casimir eigenvalues ${\\cal C}_2 (p,q) $ in (REF ) or (REF ).",
"Formula (REF ) also describes the spectrum for all the ${\\cal N}\\ge 2$ solutions in the class we are considering.",
"Specifically, a generic formula that agrees with (REF ) can be written for the dimensions of the OSp$(4|2)$ supermultiplets present in the KK spectrum of the ${\\cal N}=2$ $\\textrm {SU}(3) \\times \\textrm {U}(1)$ -invariant AdS$_4$ solutions of M-theory and type IIA .",
"From the spectral results for these solutions , , it follows that the dimension $E_0$ of an OSp$(4|2)$ supermultiplet with conformal primary spin $s_0$ , present in the spectrum at KK level $n$ with $\\textrm {SU}(3) \\times \\textrm {U}(1)$ charges $[p,q]_{y_0}$ is $ {\\cal N}=2 \\; : \\quad E_0 = \\tfrac{1}{2} + \\sqrt{ \\tfrac{17}{4} -s_0(s_0+1) + \\alpha \\, n (n + d-1) - \\tfrac{4}{3} \\, {\\cal C}_2 (p,q) + \\tfrac{1}{2} \\, y_0^2} \\; .$ Here, $ {\\cal C}_2 (p,q)$ is again the SU(3) Casimir eigenvalue (REF ), and now $d=7$ , $\\alpha = \\tfrac{1}{2}$ for the $D=11$ solution and $d=6$ , $\\alpha =\\tfrac{2}{3}$ for the type IIA one .",
"With these definitions, (REF ) agrees with the expressions provided in , for the various OSp$(4|2)$ supermultiplets, including the hypermultiplets.",
"In order to compare (REF ) with the expressions provided in those references recall that ${\\cal N}=2$ (massless, short and long) graviton, (short and long) gravitino, (massless, short and long) vector multiplets, and hypermultiplets respectively have $s_0 \\equiv s_0^{{\\scriptscriptstyle (2)}} = 1$ , $s_0 =\\tfrac{1}{2}$ , $s_0=0$ and $s_0=0$ .",
"Since (massless, short or long) ${\\cal N}=2$ graviton multiplets have $s_0^{{\\scriptscriptstyle (2)}} = 1$ , (REF ) is also of the form (REF ) with ${\\cal Q}_2 \\equiv - \\tfrac{4}{3} \\, {\\cal C}_2 (p,q) + \\tfrac{1}{2} \\, y_0^2$ .",
"A similar observation holds for the ${\\cal N}=3$ AdS$_4$ solution of type IIA , , with $ \\textrm {SO}(3)_{\\cal R} \\times \\textrm {SO}(3)_{\\cal F}$ invariance.",
"It follows from that an OSp$(4|3)$ supermultiplet with conformal primary spin $s_0$ , present at KK level $n$ with $ \\textrm {SO}(3)_{\\cal R} \\times \\textrm {SO}(3)_{\\cal F}$ quantum numbers $(j,h)$ has conformal dimension $ {\\cal N}=3 \\; : \\quad E_0 = \\sqrt{ 3 -s_0(s_0+1) + \\tfrac{1}{2} \\, n (n + d-1) + \\tfrac{1}{2} \\, j ( j +1) - \\tfrac{3}{2} \\, h ( h +1) } \\; .$ With $d=6$ , this formula indeed reproduces (4.6) and (4.7) of for the (massless, short and long) graviton and the (short and long) gravitino multiplets.",
"These have respectively $s_0 \\equiv s_0^{{\\scriptscriptstyle (2)}} = \\tfrac{1}{2}$ and $s_0 = 0$ , see e.g.",
"appendix B of .",
"Equation (REF ) also reproduces the dimension for the (necessarily short for ${\\cal N}=3$ ) vector multiplets, which have $s_0 = 0$ .",
"This was given in table 8 of .",
"Indeed, (REF ) reduces to $E_0 = \\frac{1}{2} (n+2)$ as given in that table upon using that (the unique) vector multiplet at KK level $n$ has quantum numbers $ j = h = \\frac{1}{2} (n+2) $ .",
"Further, (REF ) is also of the generic form (REF ) with $\\alpha = \\tfrac{1}{2}$ , given that the (massless, short, or long) ${\\cal N}=3$ graviton multiplet has $s_0^{{\\scriptscriptstyle (2)}} = \\tfrac{1}{2}$ .",
"In this case, ${\\cal Q}_2 \\equiv \\tfrac{1}{2} \\, j ( j +1) - \\tfrac{3}{2} \\, h ( h +1)$ .",
"Finally, the dimension of the (unique) Osp$(4|8)$ supermultiplet present at level $n$ in the KK spectrum , , of the ${\\cal N}=8$ Freund-Rubin solution of $D=11$ supergravity is $E_0 = \\tfrac{1}{2} (n+2)$ (see e.g.",
"table 9 of ).",
"This may be straightforwardly rewritten as $ {\\cal N}=8 \\; : \\quad E_0 = -\\tfrac{1}{2} + \\sqrt{ \\tfrac{9}{4} + \\tfrac{1}{4} \\, n (n + d-1) }$ with $d=7$ .",
"Since these Osp$(4|8)$ multiplets all have scalar, $s_0 = s_0^{{\\scriptscriptstyle (2)}} = 0 $ , superconformal primaries, (REF ) also conforms to the generic expression (REF ) with ${\\cal Q}_2 \\equiv 0 $ .",
"Of course, the formulae (REF ), (REF ) may not necessarily extrapolate to other supersymmetric AdS$_4$ solutions of M-theory and type IIA that still uplift from the SO(8) or ISO(7) gaugings, but preserve other symmetry groups."
],
[
"G$_2$ -singlet spectra in type IIA and consistent truncations",
"On a different note, table REF of section REF shows that there are no G$_2$ -singlet supermultiplets at KK level $n=3$ for the ${\\cal N}=1$ G$_2$ -invariant solution of type IIA.",
"The claim is in fact stronger: the number of G$_2$ singlets in the KK spectrum of this solution is finite, there are no singlets for $n \\ge 3$ , and all of them appear at levels $n=0,1,2$ .",
"This can be seen by branching the SO(7) representations in table 1 of under $\\textrm {G}_2 \\subset \\textrm {SO}(7)$ for all $n$ .",
"The same holds for the non-supersymmetric G$_2$ -invariant solution of type IIA (see section ), relative to the individual G$_2$ -singlet KK states as there is obviously no supermultiplet structure in that case.",
"For the non-supersymmetric SO(7)-invariant solution of type IIA, something similar happens: its complete KK spectrum comes in an infinite number of SO(7) representations, but the number of singlets under the branching $\\textrm {SO}(7) \\supset \\textrm {G}_2$ is also finite.",
"The complete spectrum of G$_2$ -singlet states for each of these three AdS$_4$ solutions of type IIA is summarised in table REF .",
"For the ${\\cal N}=1$ G$_2$ solution, the states combine into the OSp$(4|1)$ supermultiplets indicated in the table, as follows from tables REF –REF .",
"This feature of the KK spectra for these three AdS$_4$ solutions was expected on the following grounds.",
"Massive type IIA supergravity admits a fully non-linear consistent truncation on $S^6$ down to a certain $D=4$ ${\\cal N}=2$ gauged supergravity coupled to a vector multiplet and a hypermultiplet .",
"This truncation is obtained by expanding the type IIA fluxes along the forms that define the canonical, homogeneous nearly-Kähler structure on $S^6$ with $D=4$ field coefficients, and is in fact valid for any nearly-Kähler six-dimensional manifold .",
"This $D=4$ ${\\cal N}=2$ theory is not contained in $D=4$ ${\\cal N}=8$ ISO(7) supergravity.",
"Rather, these two theories overlap in the G$_2$ -invariant sector of the latter.",
"The ${\\cal N}=8$ supergravity captures the modes at KK level $n=0$ in the compactification of massive IIA on $S^6$ and reconstructs their full non-linear interactions.",
"It was argued in that the ${\\cal N}=2$ theory should do likewise for the G$_2$ -singlet states up the KK towers around any of its three vacua (thereby identified with the three vacua of the ${\\cal N}=8$ ISO(7) theory that appear in table REF ).",
"Table: The KK gravitino spectra up to KK level n=2n=2about the massive type IIA AdS 4 _4 solutions that uplift on S 6 S^6 from vacua of D=4D=4 𝒩=8{\\cal N}=8 ISO(7) dyonically-gauged supergravity with at least SU(3) symmetry."
]
] | 2012.05249 |
[
[
"Quantization and soliton-like solutions for the $\\Phi\\Psi$-model in a\n optic fiber"
],
[
"Abstract In the framework of a mesoscopical model for dielectric media we provide an analytical description for the electromagnetic field confined in a cylindrical cavity containing a finite dielectric sample.",
"This system is apted to simulate the electromagnetic field in a optic fiber, in which two different regions, a vacuum region and a dielectric one, appear.",
"A complete description for the scattering basis is introduced, together with field quantization and the two-point function.",
"Furthermore, we also determine soliton-like solutions in the dielectric, propagating in the sample of nonlinear dielectric medium."
],
[
"Introduction",
"Dielectric media in the framework of analogue gravity are an active subject of investigation, with particular reference to the Hawking effect in nonlinear dielectrics.",
"See e.g.",
"[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].",
"As to experiments with dielectric media and their debate one may refer to [14], [4], [15], [16], [7], [5] and also to the (uncontroversial) experiment in a optic fiber reported in [17].",
"In general, the problem is quite difficult, because of dispersive effects associated with condensed matter systems.",
"Notwithstanding, a framework can be provided where, in the limit of weak dispersive effects, in a precise mathematical sense, one is able to find how the Hawking effect manifests itself when the system is affected by the presence of horizon(s) (mathematically, turning point(s)) [13].",
"Mostly, calculations are carried out for a dielectric medium filling all the space.",
"Furthermore, in order to avoid technical difficulties arising mainly because of the gauge field nature of the electromagnetic field, which arise naturally in the Hopfield model (see [8], [19], [20]), we have introduced a simplified model, called the $\\Phi \\Psi $ -model, where the original fields of the Hopfield model are replaced by two scalar fields: $\\Phi $ in place of the electromagnetic field, and $\\Psi $ in place of the polarization [8].",
"An exact quantization for the fully relativistically covariant version of the model have been provided in [18].",
"We have also taken into account the case of dielectric medium filling only an half-space [21].",
"We have verified that, in the latter case, spectral boundary conditions are required, because of the peculiar role played by the polarization field.",
"Herein, we extend our analysis by taking into account a cylindrical geometry, where the dielectric field fills only a finite cylindrical region of length $2L$ and radius $R$ .",
"The remaining region of radius $R$ is filled by vacuum.",
"This simplified setting can be still interesting because the vacuum regions could be also replaced by regions containing dielectrics with different refractive index.",
"In the first part of our analysis, we discuss in details the problem of the boundary conditions to be imposed on the fields, and a complete scattering basis for the problem is introduced through separation of variables allowed by our peculiar geometrical setting.",
"We also provide the full propagator for the model at hand.",
"In the second part, we take into account a fully nonlinear dielectric, where the nonlinearity is simulated by introducing a term proportional to the fourth power of the polarization field $\\Psi $ .",
"Our aim is to show that solitonic solutions exist, representing a dielectric perturbation travelling with constant velocity in the direction of the cylindrical fiber axis.",
"We can show that, by taking into account `homogeneous' solutions which do not vary in the radial direction and also in the azimuthal one, soliton-like solutions exist, with different characteristics depending on suitable parameters.",
"A linearization around the solitonic solution is the natural set-up for studying the analog Hawking effect also in the present case.",
"The present analysis is in preparation of the analysis of the Hawking effect in the geometrical setting described above."
],
[
"The relativistic Kerr-$\\Phi \\Psi $ model in a cylindrical fibre",
"Let us consider the electromagnetic field in a cylindrical cavity, along the $z$ direction, where an Hopfield dielectric is at rest in the lab, filling the region $C_{\\chi }=\\lbrace (t,x,y,z)| -L\\le z\\le L,\\ x^2+y^2\\le R^2 \\rbrace $ .",
"In general, we consider inertial frames which are boosted in the $z$ direction with respect to the Lab frame.",
"See also figure REF .",
"Figure: The geometry for the problem at hand is displayed.",
"The region C χ C_\\chi contains the dielectric medium, the other two cylindricalregions C ± C_\\pm with the same radius RR are void.",
"The inner boundaries indicated in the text as Σ ±L \\Sigma _{\\pm L} are for simplicityboth indicated with Σ\\Sigma .",
"Again, for simplicity, we have not indicated the other boundaries we take into consideration in the main text.If $n$ is the four-vector with covariant components $\\underline{n}=(0,0,0,1)$ in the Lab frame, then, in an arbitrary inertial frame centered in 0 (the origin of the Lab frame) the confining cylindrical region is $C=\\lbrace x \\in M^{1,3}_{0}| -(x- (x \\cdot v)v+ (x \\cdot n)n)^2\\le R^2 \\rbrace $ , where $\\underline{v}=(1,0,0,0)$ in the Lab frame, and the dielectric region is $C_\\chi =\\lbrace x \\in C_\\chi | -L\\le x\\cdot n \\le L\\rbrace $ .",
"The $\\phi -\\psi $ model is thus described by the action principle as follows: $S[\\phi ,\\psi ]=\\frac{1}{2} \\int _C\\partial _\\mu \\phi \\partial ^\\mu \\phi \\ d^4x +\\int _{C_\\chi } \\biggl (\\frac{1}{2} v^\\mu \\partial _\\mu \\psi v^\\nu \\partial _\\nu \\psi -\\frac{\\omega _0^2}{2} \\psi ^2 -g\\phi v^\\mu \\partial _\\mu \\psi \\biggr ) d^4x.$ This is because we require for $\\psi $ to vanish outside $C_\\chi $ by definition."
],
[
"Equations of motion",
"There are several interesting discussions regarding the deduction of the equations of motion.",
"Nevertheless we will follow a simple deduction, by using local variations and then by choosing boundary conditions.",
"With “local variations” we mean the following.",
"Fix a point $p$ internal to $C_\\chi $ (in the topological sense) or internal to $C\\backslash C_\\chi $ .",
"Thus, there is at least an open set $U(p)$ such that is completely contained in $C_\\chi $ (or in $C\\backslash C_\\chi $ ).",
"A local variation is a variation of the fields with support in such a $U(p)$ .",
"Using local variations we get for the equations of motion $\\Box \\phi &=0, \\\\\\psi &=0 ,$ outside $C_\\chi $ , and $\\Box \\phi +gv^\\mu \\partial _\\mu \\psi &=0, \\\\(v^\\mu \\partial _\\mu )^2 \\psi +\\omega _0^2\\psi -gv^\\mu \\partial _\\mu \\phi &=0,$ inside $C_\\chi $ .",
"Now, we are left with the choice of the boundary conditions in order to completely define the theory.",
"The boundary consists in $\\partial C$ , which includes the conditions at infinity and $C_\\chi \\cap \\partial C$ , and $\\partial C_\\chi $ that adds $\\partial C_\\chi -C_\\chi \\cap \\partial C=: \\Sigma _{-L}\\cup \\Sigma _L$ in obvious notations.",
"In order to choose such conditions, let us start by considering (global) variations in $\\psi $ .",
"The support of $\\psi $ is compact, so we choose to work with variations which are $\\mathcal {C}^\\infty (C_\\chi )$ with support in $C_\\chi $ .",
"In particular, we do not require for them to be continuous on $\\Sigma _{-L}$ and $\\Sigma _L$ .",
"Nevertheless, since $v$ is orthogonal to $N$ , where the latter is the suitably oriented normal field to $\\partial C$ and $\\partial C_\\chi $ , there are no boundary terms in $\\delta S$ under variations of $\\psi $ , and then we are not required to choose any particular condition on $\\psi $ (apart from requiring that it must be at least $\\mathcal {C}^2(C_\\chi )$ ).",
"A little bit more involved is the (global) variation in $\\phi $ .",
"In this case we have to tackle the variation $\\delta \\frac{1}{2} \\int _C\\partial _\\mu \\phi \\partial ^\\mu \\phi \\ d^4x= \\int _C\\partial _\\mu \\delta \\phi \\partial ^\\mu \\phi \\ d^4x= \\int _C\\partial _\\mu (\\delta \\phi \\partial ^\\mu \\phi )\\ d^4x- \\int _C \\delta \\phi \\partial _\\mu \\partial ^\\mu \\phi \\ d^4x.$ Requiring for $\\phi $ to be continuous in $C$ implies that $\\delta \\phi $ must be continuous.",
"However, requiring also the continuity of $\\partial _\\mu \\phi $ looks too much restrictive in general.",
"In order to manipulate the divergence in the last expression, let us notice that if we do not require for $\\partial _\\mu \\phi $ to be continuous on $\\Sigma _0$ and $\\Sigma _L$ , then we cannot apply the divergence theorem directly but we need to separate $C$ into three regions as $C=C_\\chi \\cup C_- \\cup C_+$ , where $C_-\\cup C_+:=C-C_\\chi $ (with obvious notation).",
"This way $\\int _C\\partial _\\mu (\\delta \\phi \\partial ^\\mu \\phi )\\ d^4x=\\int _{C_-}\\partial _\\mu (\\delta \\phi \\partial ^\\mu \\phi )\\ d^4x+\\int _{C_+}\\partial _\\mu (\\delta \\phi \\partial ^\\mu \\phi )\\ d^4x+\\int _{C_\\chi }\\partial _\\mu (\\delta \\phi \\partial ^\\mu \\phi )\\ d^4x,$ and $\\delta \\phi \\partial ^\\mu \\phi $ is continuous and, indeed, smooth in each of the three regions.",
"So we can apply the divergence theorem to each of the three regions.",
"The result is that, if we assume that $n^\\mu \\partial _\\mu \\phi $ is continuous in an open neighbourhood of $\\Sigma _A$ , $A=-L,L$ , then $\\int _C\\partial _\\mu (\\delta \\phi \\partial ^\\mu \\phi )\\ d^4x=\\int _{\\partial C} N^\\mu \\partial _\\mu \\phi \\ \\delta \\phi \\ d^3\\sigma .$ Since we do not mean to fix the value of $\\phi $ on the boundary, we can get rid of the boundary term by imposing the Neuman condition $N^\\mu \\partial _\\mu \\phi |_{\\partial C}=0$ .",
"More precisely this condition is clear on the cylindrical boundary but it should also include a condition at $z\\rightarrow \\pm \\infty $ .",
"There, the above Neuman condition looks not suitable if we want to allow for sources or, say, fluxes.",
"In this sense it seems that at infinity some other condition could be better, but it is not a case of interest here.",
"On the boundaries $\\Sigma _A$ of the dielectric we are left with the condition of continuity of the normal derivative of $\\phi $ .",
"Let us investigate a little bit more at this condition by looking at the equations (REF ) and (REF ).",
"They show that $\\Box \\phi $ is not continuous on $\\Sigma _A$ .",
"Now, since $\\psi $ vanishes outside $C_\\chi $ , we can write both these equations as $\\partial ^\\mu (\\partial _\\mu \\phi +gv_\\mu \\psi )=0.$ Since $\\psi $ is discontinuous, $\\partial ^\\mu \\psi $ is expected to produce $\\delta $ -function contributions supported on $\\Sigma _A$ .",
"However, $v\\cdot n =0$ , so that $v^\\mu \\partial _\\mu \\psi $ is discontinuous but does not contains $\\delta $ contributions.",
"Thus, the same happens for $\\Box \\phi $ .",
"Now, since $C$ is contractible we can add to $n$ and $v$ two other vectors on $C$ , $e_i$ ($i=1,2$ ) in order to get a complete constant orthonormal frame.",
"Thus, we can write $0=\\partial _\\mu (v^\\mu \\partial _{v}\\phi -n^\\mu \\partial _{n}\\phi -e_1^\\mu \\partial _{e_1}\\phi -e_2^\\mu \\partial _{e_2}\\phi )=\\partial _{v} \\partial _{v}\\phi -\\partial _{n} \\partial _{n}\\phi -\\partial _{e_1} \\partial _{e_1}\\phi -\\partial _{e_2} \\partial _{e_2}\\phi .$ Now, $\\partial _{e_1}\\phi $ , $\\partial _{e_2}\\phi $ and $\\partial _{v}\\phi $ are discontinuous, but no $\\delta $ contribution arises in further deriving, since $\\partial _{e_1} $ , $\\partial _{e_2} $ and $\\partial _{v}$ derive in directions orthogonal to $n$ (and, so, tangent to the separating hypersurface).",
"The remaining term $\\partial _{n}\\phi $ is continuous and the further derivative $\\partial _{n}$ introduce at most new discontinuities.",
"This shows that no further conditions are necessary for having consistent equations: all the condition we have to require inside $C$ are the continuity of $\\phi $ and $\\partial _{n}\\phi $ everywhere, with the last vanishing on $\\partial C$ ."
],
[
"General solution.",
"Let us work in a frame with four-velocity $\\underline{v}=\\gamma (\\nu )(1,0,0,\\nu ),$ so that $\\underline{n}=\\gamma (\\nu )(\\nu ,0,0,1).$"
],
[
"Outside the dielectric",
"The dielectric region is defined by $-\\nu t -L/\\gamma (\\nu ) \\le z \\le -\\nu t+L/\\gamma (\\nu ).$ If we choose cylindrical coordinates, outside the dielectric the equations of motion are simply $\\psi =0$ and $\\partial ^2_t \\phi -\\partial ^2_z \\phi -\\partial ^2_\\rho \\phi -\\frac{1}{\\rho }\\partial _\\rho \\phi -\\frac{1}{\\rho ^2} \\partial ^2_\\theta \\phi =0.$ Separating the variables as $\\phi (t,\\rho ,z,\\theta )=K \\phi _T(t) \\phi _R(\\rho ) \\phi _Z(z) \\phi _\\Theta (\\theta ),$ with $K$ a constant, we find that $\\phi _T(t)=e^{-ik_0 t}, \\quad \\phi _Z(z)=e^{ik_z z}, \\quad \\phi _\\Theta (\\theta )=e^{im\\theta },$ with $m\\in \\mathbb {Z}$ , and $\\phi ^{\\prime \\prime }_R+\\frac{1}{\\rho }\\phi ^{\\prime }_R+\\left( k_\\rho ^2 -\\frac{m^2}{\\rho ^2} \\right) \\phi =0,$ where $k_\\rho $ must satisfy $k_0^2-k_z^2-k_\\rho ^2=0$ .",
"The only solutions continuous in $\\rho =0$ are $\\phi _R(\\rho )=J_m(k_\\rho \\rho )$ , where $J_m$ are the usual Bessel functions.",
"The boundary condition on $\\partial C$ reduces to $J^{\\prime }_m(k_\\rho R)=0,$ so that at the end we have $\\phi _R(\\rho )=J_{|m|,s}(\\rho )=J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ), \\quad |m|,s\\in \\mathbb {N},$ where $z_{ms}$ is the $s$ -th positive zero of $J^{\\prime }_m$ .",
"The corresponding dispersion relations are $k_0^2=k_z^2+z_{ms}^2/R^2$ , which can be codified in $K=c_{m,s}(k_0,k_z) \\delta \\biggl (k_0^2-k_z^2 -\\frac{z_{ms}^2}{R^2}\\biggr ).$ In conclusion, we can write the general solution outside the dielectric in the form $&\\phi =\\sum _{s\\in \\mathbb {N}}\\sum _{m\\in \\mathbb {Z}}\\int _{-\\infty }^\\infty \\frac{dk_z}{4\\pi k^0}\\left(c_{m,s}(k_z) e^{-ik^0t+ik_zz+im\\theta } +c^*_{m,s}(k_z) e^{ik^0t-ik_zz-im\\theta }\\right) J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ),\\\\&\\psi =0,$ where $k^0(k_z,m,s)=\\sqrt{k_z^2+\\frac{z_{ms}^2}{R^2}}.$"
],
[
"Inside the dielectric",
"Inside the dielectric, the equations take the form ($v^0=\\gamma (\\nu )$ and $v=\\nu \\gamma (\\nu )$ ) $\\partial ^2_t \\phi -\\partial ^2_z \\phi -\\partial ^2_\\rho \\phi -\\frac{1}{\\rho }\\partial _\\rho \\phi -\\frac{1}{\\rho ^2} \\partial ^2_\\theta \\phi +gv^0\\partial _t \\psi +gv\\partial _z \\psi &=0,\\\\(v^0\\partial _t+v\\partial _z)^2\\psi +\\omega _0^2\\psi -gv^0\\partial _t\\phi -gv\\partial _z\\phi &=0.$ By means of the separation ansatz $\\phi (t,\\rho ,z,\\theta )&=\\tilde{\\phi }\\phi _T(t) \\phi _R(\\rho ) \\phi _Z(z) \\phi _\\Theta (\\theta ),\\\\\\psi (t,\\rho ,z,\\theta )&=\\tilde{\\psi }\\psi _T(t) \\psi _R(\\rho ) \\psi _Z(z) \\psi _\\Theta (\\theta ),$ we find that $&\\psi _T(t)=\\phi _T(t)=e^{-ik_0 t}, \\quad \\psi _Z(z)=\\phi _Z(z)=e^{ik_z z}, \\quad \\psi _\\Theta (\\theta )=\\phi _\\Theta (\\theta )=e^{im\\theta },\\\\&\\psi _R(\\rho )=\\phi _R(\\rho )=J_{|m|,s}(\\rho )=J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ), \\quad s\\in \\mathbb {N},\\ m\\in \\mathbb {Z},$ and $\\tilde{\\phi }$ , $\\tilde{\\psi }$ must satisfy the algebraic system $\\begin{pmatrix}-k_0^2+k_z^2+k_\\rho ^2 & ig\\omega \\\\-ig\\omega & \\omega _0^2-\\omega ^2\\end{pmatrix}\\begin{pmatrix}\\tilde{\\phi }\\\\ \\tilde{\\psi }\\end{pmatrix}=\\begin{pmatrix}0\\\\0\\end{pmatrix},$ where we have put $k_\\rho =z_{ms}/R$ and $\\omega =v^\\mu k_\\mu =k^0 v^0-vk_z.$ This has nontrivial solutions if the determinant of the matrix vanishes, which means $DR:=k_0^2-k_z^2-k_\\rho ^2+\\frac{g^2\\omega ^2}{\\omega _0^2-\\omega ^2}=0.$ Notice that this is an implicit equation in $k_0=k^0$ , since $\\omega $ is a function of $k_0$ .",
"With this condition, the solution of the algebraic system takes the form $\\tilde{\\phi }&=b(k_0,k_z,m,s)\\delta (DR),\\\\\\tilde{\\psi }&=\\frac{k_0^2-k_z^2-k_\\rho ^2}{ig\\omega } b(k_0,k_z,m,s)\\delta (DR).$ If we set $DR^{\\prime }:=\\partial _{k_0} DR=2k^0 \\left(1+ \\frac{g^2\\omega _0^2 v^0}{(\\omega _0^2-\\omega ^2)^2}\\right) ,$ then we can write the general solution inside the dielectric as $&\\phi =\\sum _{a=1}^2\\sum _{s\\in \\mathbb {N}}\\sum _{m\\in \\mathbb {Z}}\\int _{-\\infty }^\\infty \\frac{dk_z}{2\\pi DR^{\\prime }_{(a)}}\\left(b_{(a)ms}(k_z) e^{-ik_{(a)}^0t+ik_zz+im\\theta } +c.c.\\right) J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ),\\cr &\\psi =\\sum _{a=1}^2\\sum _{s\\in \\mathbb {N}}\\sum _{m\\in \\mathbb {Z}}\\int _{-\\infty }^\\infty \\frac{dk_z}{2\\pi DR^{\\prime }_{(a)}}\\left(b_{(a)ms}(k_z) e^{-ik_{(a)}^0t+ik_zz+im\\theta } -c.c.\\right)\\frac{-ig\\omega _{(a)}}{\\omega ^2_{(a)}-\\omega ^2_0}J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ),$ where $(a)$ indicates the branch like in [18], [20]."
],
[
"Gluing conditions",
"At this point we must impose the continuity of $\\phi $ and $\\partial _{n}\\phi $ .",
"From now on, we will work in the lab frame where $\\underline{v}\\equiv (1,0,0,0)$ and $\\omega =k^0$ .",
"In this case, the normal direction is $z$ and the gluing conditions are $\\phi (-L^-) = \\phi (-L^+), \\quad \\phi (L^-) = \\phi (L^+), \\\\\\partial _z\\phi (-L^-) = \\partial _z\\phi (-L^+), \\quad \\partial _z\\phi (L^-) = \\partial _z\\phi (L^+),$ where we defined for short $\\phi (z^\\pm ):=\\lim _{\\epsilon \\rightarrow 0} \\phi (t,\\rho ,z\\pm |\\epsilon |,\\theta ).$ It is convenient to define a scattering basis, defined by replacing the Fourier modes with solutions of the form (omitting the angular and radial parts, at time $t=0$ ) $\\begin{split}\\phi _k(z) =&\\left(e^{ik z} +R_{ksm} e^{-ikz})\\chi _{(-\\infty ,-L)}(z)\\right) + \\\\& + \\left(M_{ksm} e^{iq_{s,m}(k)z}+N_{ksm} e^{-iq_{sm}(k)z}\\right)\\chi _{[-L,L]}(z)+ T_{ksm} e^{ikz} \\chi _{(L,\\infty )},\\end{split}$ going from the left to the right, and the analogous left moving modes.",
"Here $R_{ksm}$ , $T_{ksm}$ , $M_{ksm}$ , and $N_{ksm}$ are reflection and transmission coefficients.",
"Moreover, we choose the measure to be $dk_z/(2\\pi ) 2k^0 $ everywhere so that inside the dielectric $b_{(a)ms}$ and $c_{ms}$ reabsorb the normalization factor $1+ \\frac{g^2\\omega _0^2 v^0}{(\\omega _0^2-\\omega ^2)^2}$ from $DR^{\\prime }$ .",
"Writing the conditions (REF ) explicitly, we obtain the following algebraic system ${\\left\\lbrace \\begin{array}{ll}e^{-ikL} + R_{ksm} e^{ikL} = M_{ksm} e^{-iqL} + N_{ksm} e^{iqL} \\\\k e^{-ikL} - R_{ksm} k e^{ikL} = M_{ksm} q e^{-iqL} - N_{ksm} q e^{iqL} \\\\T_{ksm} e^{ikL} = M_{ksm} e^{iqL} + N_{ksm} e^{-iqL} \\\\T_{ksm} k e^{ikL} = M_{ksm} q e^{iqL} - N_{ksm} q e^{-iqL}\\end{array}\\right.",
"}$ that has solution $R_{ksm}=&\\frac{2i(k^2-q^2_{sm}(k)) \\sin (2q_{sm}(k)L) e^{2i(q_{sm}(k)-k)L}}{(k-q_{sm}(k))^2e^{4iq_{sm}(k)}-(k+q_{sm}(k))^2},\\\\M_{ksm}=& -\\frac{2k (k+q_{sm}(k)) e^{i(q_{sm}(k)-k)L}}{(k-q_{sm}(k))^2e^{4iq_{sm}(k)}-(k+q_{sm}(k))^2}, \\\\N_{ksm}=& \\frac{2k (k-q_{sm}(k)) e^{i(3q_{ms}(k)-k)L}}{(k-q_{sm}(k))^2e^{4iq_{sm}(k)}-(k+q_{sm}(k))^2}, \\\\T_{ksm}=&-\\frac{4kq_{sm}(k) e^{2iq_{sm}(k)L}e^{-2ikL}}{(k-q_{sm}(k))^2e^{4iq_{sm}(k)}-(k+q_{sm}(k))^2}.$"
],
[
"The scattering basis",
"The positive energy scattering basis consists in the dielectric modes and the gap modes.",
"In the lab frame, the dielectric modes are (we write the $\\phi $ -component only): $\\begin{split}\\phi ^R_{D,ksm}(t,\\rho ,z,\\theta )&=\\kappa _{ksm} e^{-i\\omega _{ksm}t} e^{im\\theta } J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ) \\Bigl [ \\Bigl (e^{ik z} +R_{ksm} e^{-ikz}\\Bigr )\\chi _{(-\\infty ,-L)}(z) \\\\& + \\Bigl (M_{ksm} e^{iq_{s,m}(k)z}+N_{ksm} e^{-iq_{sm}(k)z}\\Bigr )\\chi _{[-L,L]}(z)+ T_{ksm} e^{ikz} \\chi _{(L,\\infty )} \\Bigr ],\\end{split}\\\\\\phi ^L_{D,ksm}(t,\\rho ,z,\\theta )& = \\phi ^R_{D,ksm}(t,\\rho ,-z,\\theta ),$ where $\\omega _{ksm}&=\\sqrt{k^2+k_\\rho ^2},\\\\q_{sm}(k)^2+k_\\rho ^2&=(k^2+k_\\rho ^2)\\frac{\\omega _0^2+g^2-k^2-k_\\rho ^2}{\\omega _0^2-k^2-k_\\rho ^2},\\\\m\\in \\mathbb {Z}, & \\qquad \\ s\\in \\mathbb {N}, \\qquad \\ k>0,$ and the coefficients are $R_{ksm}=&\\frac{2i(k^2-q^2_{sm}(k)) \\sin (2q_{sm}(k)L) e^{2i(q_{sm}(k)-k)L}}{(k-q_{sm}(k))^2e^{4iq_{sm}(k)L}-(k+q_{sm}(k))^2},\\\\M_{ksm}=& -\\frac{2k (k+q_{sm}(k)) e^{i(q_{sm}(k)-k)L}}{(k-q_{sm}(k))^2e^{4iq_{sm}(k)L}-(k+q_{sm}(k))^2}, \\\\N_{ksm}=& \\frac{2k (k-q_{sm}(k)) e^{i(3q_{ms}(k)-k)L}}{(k-q_{sm}(k))^2e^{4iq_{sm}(k)L}-(k+q_{sm}(k))^2}, \\\\T_{ksm}=&-\\frac{4kq_{sm}(k) e^{2iq_{sm}(k)L}e^{-2ikL}}{(k-q_{sm}(k))^2e^{4iq_{sm}(k)L}-(k+q_{sm}(k))^2}.$ Notice that $|T_{ksm}|^2+|R_{ksm}|^2=1,$ so that there is no trapping in the dielectric.",
"The dielectric modes are defined in the range for $\\omega _{k,s,m}$ not in the gap $[\\omega _0,\\bar{\\omega }]$ , and do not form a complete basis for all possible initial conditions.",
"In order to get a complete basis we have to add the gap modes $\\phi ^R_{G,ksm}(t,\\rho ,z,\\theta )&=\\tilde{\\kappa }_{ksm} e^{-i\\omega _{ksm}t} e^{im\\theta } J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr )\\sin ((k+L)z) \\chi _{(-\\infty ,-L]}(z), \\\\\\phi ^L_{G,ksm}(t,\\rho ,z,\\theta )&=\\tilde{\\kappa }_{ksm} e^{-i\\omega _{ksm}t} e^{im\\theta } J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr )\\sin ((k-L)z) \\chi _{[L,\\infty )},$ defined for $\\omega _0^2 \\le k^2+k_\\rho ^2 \\le \\bar{\\omega }^2.$ These represent modes that are totally reflected by the dielectric.",
"The normalisation constants $\\kappa _{ksm}$ and $\\tilde{\\kappa }_{ksm}$ can be computed by using the results in appendix .",
"If we choose $\\kappa _{ksm}=\\frac{ \\tilde{\\kappa }_{ksm}}{2}=\\frac{1}{ \\left( 1-\\frac{m^2}{z^2_{ms}} \\right)^{\\frac{1}{2}} \\sqrt{\\pi }R J_{|m|}(z_{ms})},$ then the scattering solution are orthonormalised (with measure $dk/{2\\pi (2k^0)}$ )."
],
[
"Quantization",
"Here we determine the scalar product and invert the expressions to compute the amplitudes fields $c$ , $b$ in terms of the fields and their conjugate momenta, and impose the equal time canonical commutation relations (ETCCR).",
"We write the full field as a superposition of the component fields $\\phi ^R_D$ etc, so: $\\begin{split}\\phi (t,z,\\rho ,\\theta ) & = \\sum _{s,m} \\int _{R_{m,s}} \\frac{dk}{4\\pi k^0} \\bigg \\lbrace a^R_{D,ksm} \\phi ^R_{D,ksm}(t,z,\\rho ,\\theta ) + a^L_{D,ksm} \\phi ^L_{D,ksm}(t,z,\\rho ,\\theta ) + \\\\&\\quad + d^R_{G,ksm} \\phi ^R_{G,ksm}(t,z,\\rho ,\\theta ) + d^L_{G,ksm} \\phi ^L_{G,ksm}(t,z,\\rho ,\\theta ) + \\text{ h.c.} \\biggl \\rbrace ,\\end{split}$ where $R_{m,s}$ is the range of $k$ satisfying the right spectral conditions for any given $s,m$ .",
"In the particular case when $s=m=0$ one has that the integration is on $[0,\\omega _0] \\cup [\\bar{\\omega },\\infty )$ for $D$ modes and $[\\omega _0,\\bar{\\omega }]$ for the $G$ modes (see [18]).",
"Now, our purpose is to determine the commutation relations between the operators $a$ 's and $d$ 's: to do this, we note that the scalar product defined in appendix can also be written as $(f|\\tilde{f}) = \\left\\langle {\\Psi , \\tilde{\\Psi }}\\right\\rangle := \\frac{i}{2} \\int d^3 \\, \\Psi \\Omega \\tilde{\\Psi } ,$ where $\\Psi =\\begin{pmatrix}\\phi \\\\ \\psi \\\\ \\pi _\\phi \\\\ \\pi _\\psi \\end{pmatrix}$ and $\\Omega =\\begin{pmatrix}0_{2x2} & 1_{2x2} \\\\-1_{2x2} & 0_{2x2}\\end{pmatrix} .$ Given the orthogonality relations, we can write the coefficients as the scalar product $a^R_{D,ksm} = \\left\\langle {\\Psi ^R_{D,ksm},\\Psi _{ksm}}\\right\\rangle ,$ and similarly fo the other coefficients.",
"With some algebra, we can evaluate the products $a^R_{D,ksm} {a^R_{D,k^{\\prime }s^{\\prime }m^{\\prime }}}^\\dagger $ and ${a^R_{D,k^{\\prime }s^{\\prime }m^{\\prime }}}^\\dagger a^R_{D,ksm}$ , and imposing the ETCCR $[\\phi (x),\\pi _\\phi (y)] = i\\delta ^{(3)}(x-y)$ , we get that the commutator is equal to the scalar product $[a^R_{D,ksm}, {a^R_{D,k^{\\prime }s^{\\prime }m^{\\prime }}}^\\dagger ] = \\left\\langle {\\Psi ^R_{D,ksm},\\Psi ^R_{D,k^{\\prime }s^{\\prime }m^{\\prime }}}\\right\\rangle .$ Therefore we can easily evaluate the commutation relations, using the orthogonality relations calculated in appendix .",
"The relevant commutators are $\\Bigl [a^R_{D,ksm}, {a^R_{D,k^{\\prime }s^{\\prime }m^{\\prime }}}^\\dagger \\Bigr ] & = 4\\pi \\omega _{ksm} \\delta _{mm^{\\prime }} \\delta _{ss^{\\prime }} \\delta (k^{\\prime }-k) , \\\\\\Bigl [a^L_{D,ksm}, {a^L_{D,k^{\\prime }s^{\\prime }m^{\\prime }}}^\\dagger \\Bigr ] & = 4\\pi \\omega _{ksm} \\delta _{mm^{\\prime }} \\delta _{ss^{\\prime }} \\delta (k^{\\prime }-k) , \\\\\\Bigl [d^R_{G,ksm}, {d^R_{G,k^{\\prime }s^{\\prime }m^{\\prime }}}^\\dagger \\Bigr ] & = 4\\pi \\omega _{ksm} \\delta _{mm^{\\prime }} \\delta _{ss^{\\prime }} \\delta (k^{\\prime }-k) , \\\\\\Bigl [d^L_{G,ksm}, {d^L_{G,k^{\\prime }s^{\\prime }m^{\\prime }}}^\\dagger \\Bigr ] & = 4\\pi \\omega _{ksm} \\delta _{mm^{\\prime }} \\delta _{ss^{\\prime }} \\delta (k^{\\prime }-k) ,$ and all other vanish."
],
[
"The two-point function of the Kerr-$\\Phi \\Psi $ model",
"The full quantum theory, in absence of the Kerr nonlinear term, is fully defined by the free propagator.",
"Let us compute the two-point function of the free theory $iG_{\\psi \\psi }^0(x,x^{\\prime }) = \\left.",
"\\mathinner {\\langle {\\psi (x)\\psi (x^{\\prime })}\\rangle }\\right|_{\\lambda =0} \\,.$ In general, its explicit expression will depend on where we choose the points $x$ and $x^{\\prime }$ .",
"We will not consider the gap modes.",
"In fact, the right and left gap modes can be included by noting that they are equivalent to the $\\phi ^R$ and $\\phi ^L$ modes respectively, with reflection coefficient $R=1$ and all other coefficients vanishing, so that, if necessary, the corresponding contributions can be deduced by taking the limit $R\\rightarrow 1$ .",
"We start by discussing the case $x,x^{\\prime } \\in C_\\chi $ , which is the most important one for computing Feynman diagrams in perturbation theory, when the nonlinearity is included.",
"For simplicity, we restrict to the case $\\rho =\\rho ^{\\prime }$ , $\\theta =\\theta ^{\\prime }$ in the following computation, but at the end we will give the general result: $\\begin{split}iG_{\\psi \\psi }^0(x,x^{\\prime })= \\theta (t-t^{\\prime }) \\sum _{s,m} \\int _0^{\\infty } \\frac{dk}{4\\pi \\omega _{ksm}} \\left[ \\psi _{ksm}^R(x){\\psi _{ksm}^R(x^{\\prime })}^* + \\psi _{ksm}^L(x){\\psi _{ksm}^L(x^{\\prime })}^* \\right] + (x \\leftrightarrow x^{\\prime }) \\\\= \\frac{i}{2\\pi } \\int d\\omega \\sum _{s,m} \\int _0^{\\infty } \\frac{dk}{4\\pi \\omega _{ksm}}\\frac{e^{-i(\\omega +\\omega _{ksm})(t-t^{\\prime })}}{\\omega + i\\epsilon } \\Big [ \\left( M_{ksm}e^{iq(k)z} + N_{ksm}e^{-iq(k)z} \\right)\\left( M_{ksm}^*e^{-iq(k)z^{\\prime }} + N_{ksm}^*e^{iq(k)z^{\\prime }} \\right) \\\\+ \\left( M_{ksm}e^{-iq(k)z} + N_{ksm}e^{iq(k)z} \\right)\\left( M_{ksm}^*e^{iq(k)z^{\\prime }} + N_{ksm}^*e^{-iq(k)z^{\\prime }} \\right)\\Big ] \\, \\frac{g^2\\omega _{ksm}^2\\kappa _{sm}^2|J_m(\\rho )|^2}{(\\omega _{ksm}^2 - \\omega _0^2)^2} \\\\-\\frac{i}{2\\pi } \\int d\\omega \\sum _{s,m} \\int _0^{\\infty } \\frac{dk}{4\\pi \\omega _{ksm}}\\frac{e^{i(\\omega +\\omega _{ksm})(t-t^{\\prime })}}{\\omega + i\\epsilon } \\Big [ \\left( M_{ksm}e^{iq(k)z^{\\prime }} + N_{ksm}e^{-iq(k)z^{\\prime }} \\right)\\left( M_{ksm}^*e^{-iq(k)z} + N_{ksm}^*e^{iq(k)z} \\right) \\\\+ \\left( M_{ksm}e^{-iq(k)z^{\\prime }} + N_{ksm}e^{iq(k)z^{\\prime }} \\right)\\left( M_{ksm}^*e^{iq(k)z} + N_{ksm}^*e^{-iq(k)z} \\right)\\Big ] \\, \\frac{g^2\\omega _{ksm}^2\\kappa _{sm}^2|J_m(\\rho )|^2}{(\\omega _{ksm}^2 - \\omega _0^2)^2} \\\\= \\frac{i}{2\\pi } \\int d\\tilde{\\omega }\\sum _{s,m} \\int _0^{\\infty } \\frac{dk}{4\\pi \\omega _{ksm}} \\Big [ 2\\left(|M_{ksm}|^2 + |N_{ksm}|^2\\right)\\cos (q(k)(z-z^{\\prime })) + 2\\text{Re}(M_{ksm}N_{ksm}^*)\\cos (q(k)(z+z^{\\prime })) \\Big ] \\\\\\times \\frac{g^2\\omega _{ksm}^2\\kappa _{sm}^2|J_m(\\rho )|^2}{(\\omega _{kms}^2 - \\omega _0^2)^2} e^{-i\\tilde{\\omega }(t-t^{\\prime })} \\left( \\frac{1}{\\tilde{\\omega }-\\omega _{kms} + i\\epsilon } - \\frac{1}{\\tilde{\\omega }+\\omega _{kms} - i\\epsilon } \\right)\\,.\\end{split}$ In the second step we used the integral representation $\\theta (\\tau ) = \\frac{i}{2\\pi } \\int d\\omega \\frac{e^{-i\\omega \\tau }}{\\omega + i\\epsilon }$ for the Heaviside $\\theta $ -function, and in the third step we have performed the change of variables $\\tilde{\\omega }= \\omega + \\omega _{ksm}$ in the first integral, and $\\tilde{\\omega }=- \\omega - \\omega _{ksm}$ in the second one.",
"Notice that the two-point function (REF ) can be written as the sum of two functions $G_{\\psi \\psi }^0(x,x^{\\prime }) = G_1(t-t^{\\prime },\\rho ,\\theta ,z-z^{\\prime }) + G_2(t-t^{\\prime },\\rho ,\\theta ,z+z^{\\prime }) \\,.$ This makes evident the breaking of translation invariance along the $z$ axis.",
"While $G_1$ is translationally invariant, $G_2$ can be interpreted as depending on the reflections at the boundaries of the dielectric region.",
"Let us focus first on the $G_1$ part.",
"By performing a change of variable $k = k_a(q)$ , where the subscript $a=\\pm $ denotes the two branches of the dispersion relation, we find $\\begin{split}G_1(t-t^{\\prime },\\rho ,\\theta ,z-z^{\\prime }) &= \\sum _{s,m,a}\\int \\frac{d\\omega }{2\\pi } \\int _{0}^{\\infty } \\frac{dq}{2\\pi DR^{\\prime }_a(q)} e^{-i\\tilde{\\omega }(t-t^{\\prime })} \\frac{g^2\\omega _{asm}^2(q)\\kappa _{sm}^2|J_m(\\rho )|^2}{(\\omega _{asm}^2(q)- \\omega _0^2)^2} \\frac{2\\omega _{asm}(q)}{\\omega ^2 - \\omega _{asm}^2(q) +i\\epsilon } \\\\&\\quad \\times \\left(|M_{asm}(q)|^2 + |N_{asm}(q)|^2\\right)\\left( e^{iq(z-z^{\\prime })} + e^{-iq(z-z^{\\prime })} \\right) \\,,\\end{split}$ where $\\omega _{asm}(q) &:= \\omega _{k_a(q)sm}, \\\\M_{asm}(q) &:= M_{k_a(q)sm}, \\\\DR_a^{\\prime }(q) &= 2\\omega _{asm}(q)\\left(1 + \\frac{g^2\\omega _0^2}{(\\omega _{asm}^2(q) - \\omega _0^2)^2}\\right).$ Notice that $\\omega _{asm}(q)\\equiv \\omega _a \\left(\\sqrt{q^2 + k_\\rho ^2}\\right)$ , where $\\omega _a(\\cdot )$ denotes the two solutions of the dispersion relation as in [18].",
"After defining $k_a(-q):=-k_a(q)$ , we can rewrite the integral in $q$ over the whole $(-\\infty ,+\\infty )$ range.",
"Performing the sum over $a$ explicitly and extending and noting that $M_{asm}(-q)=M_{asm}^*(q)$ , we obtain the final expression $\\begin{split}&G_1(t-t^{\\prime },\\rho ,\\theta ,z-z^{\\prime }) = \\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dq}{2\\pi } \\kappa _{sm}^2|J_m(\\rho )|^2 e^{-i\\omega (t-t^{\\prime }) + iq(z-z^{\\prime })} \\frac{\\omega ^2 - q^2 - k_\\rho ^2}{(\\omega ^2 - q^2 - k_\\rho ^2)(\\omega ^2 - \\omega _0^2) - g^2\\omega ^2} \\\\ &\\times \\frac{1}{(\\omega ^2 - q^2 - k_\\rho ^2)\\,(\\omega _{+sm}^2(q) - \\omega _{-sm}^2(q))} [(\\omega ^2 - \\omega _{-sm}^2)(\\omega _{+sm}^2 - q^2 - k_\\rho ^2)(|M_{+sm}(q)|^2+|N_{+sm}(q)|^2) \\\\&- (\\omega ^2 - \\omega _{+sm}^2)(\\omega _{-sm}^2 - q^2 - k_\\rho ^2)(|M_{-sm}(q)|^2+|N_{-sm}(q)|^2)] \\\\&=: \\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dq}{2\\pi } \\,\\kappa _{sm}^2|J_m(\\rho )|^2 \\,e^{-i\\omega (t-t^{\\prime }) + iq(z-z^{\\prime })} D(\\omega ,q,s,m)\\, \\ell _1(\\omega ,q,s,m) \\,,\\end{split}$ where $D(\\omega ,q,s,m)$ is the factor in the first line, which equals the free propagator $G_{\\psi \\psi }^0$ computed in [18] for the case of the infinite dielectric medium.",
"Also we defined $\\ell _1(\\omega ,q,s,m)$ , which includes the corrections due to the reflections and transmissions appearing in the finite dielectric.",
"Notice that in the case $|M_q|^2+|N_q|^2=1$ we have $\\ell _1(\\omega ,q) = 1$ .",
"From the expression (REF ) we can read the Fourier transform of the $G_1$ part of the propagator.",
"It has the same poles as the ones in the infinite dielectric case, corresponding to the Sellmeier dispersion relation.",
"It can be of interest to understand the asymptotic behaviour of the factor $\\ell _1$ for large momenta.",
"For the $|M_{asm}(q)|^2+|N_{asm}(q)|^2$ factor, we have $|M_{asm}(q)|^2+|N_{asm}(q)|^2 = \\frac{k_a^2(q)\\left(k_a^2(q) + q^2\\right)}{\\left(k_a^2(q) + q^2\\right)^2 \\sin ^2(2qL) + 4q^2 k_a^2(q) \\cos ^2(2qL)} \\,.$ In the case $a=+$ , we have $k_a(q) \\sim q$ for $q \\gg \\omega _0$ .",
"Therefore, in the limit $q\\rightarrow \\infty $ the whole factor (REF ) tends to $1/2$ .",
"In the case $a=-$ , instead, $\\lim _{q\\rightarrow \\infty } k_a(q) = \\omega _0$ , and equation (REF ) has a point dependent limit.",
"Indeed, we can take a succession ${q_n}$ such that $\\sin (2q_n L) = 0$ , in such a way that for $n\\rightarrow \\infty $ equation (REF ) tends to $1/4$ .",
"For all other successions such that $\\sin (2q_n L) = C \\ne 0$ , we have $ |M_{-sm}(q)|^2+|N_{-sm}(q)|^2 \\sim \\frac{\\omega _0^2}{q^2 \\sin ^2C} \\rightarrow 0 \\,.$ Therefore, we see that the Fourier transform of the propagator (REF ) is $\\sim q^{-2}$ , with the exception of small neighbourhoods of the points $q_n=n\\pi /(2L)$ , where sharp peaks of height $1/4$ appear.",
"For the $G_2$ part of the propagator, with similar manipulations, we obtain $\\begin{split}&G_2(t-t^{\\prime },\\rho ,\\theta ,z+z^{\\prime }) = \\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dq}{2\\pi } \\kappa _{sm}^2|J_m(\\rho )|^2 e^{-i\\omega (t-t^{\\prime }) + iq(z+z^{\\prime })} \\frac{\\omega ^2 - q^2 - k_\\rho ^2}{(\\omega ^2 - q^2 - k_\\rho ^2)(\\omega ^2 - \\omega _0^2) - g^2\\omega ^2} \\\\& \\times \\frac{(\\omega ^2 - \\omega _{-sm}^2)(\\omega _{+sm}^2 - q^2 - k_\\rho ^2)\\text{Re}(M_{+sm}(q)N_{+sm}(q)^*) - (\\omega ^2 - \\omega _{+sm}^2)(\\omega _{-sm}^2 - q^2 - k_\\rho ^2)\\text{Re}(M_{-sm}(q)N_{-sm}(q)^*)}{(\\omega ^2 - q^2 - k_\\rho ^2)\\,(\\omega _{+sm}^2(q) - \\omega _{-sm}^2(q))} \\\\&=: \\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dq}{2\\pi } \\,\\kappa _{ms}^2|J_m(\\rho )|^2 \\, e^{-i\\omega (t-t^{\\prime }) + iq(z+z^{\\prime })} D(\\omega ,q,s,m)\\, \\ell _2(\\omega ,q,s,m)\\end{split}$ The factors $M_{asm}(q)N_{asm}(q)^*$ have a very similar behaviour as the factors $|M_{asm}(q)|^2+|N_{asm}(q)|^2$ studied before, so also the function $G_2$ is vanishing for large $q$ .",
"The final expression of the propagator, in the general case $\\rho ^{\\prime }\\ne \\rho $ , $\\theta ^{\\prime }\\ne \\theta $ , is therefore: $\\begin{split}G_{\\psi \\psi }^0(t-t^{\\prime },\\rho ,\\rho ^{\\prime },\\theta -\\theta ^{\\prime },z,z^{\\prime })\\,\\chi _{[-L,L]}(z)\\,\\chi _{[-L,L]}(z^{\\prime })= \\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dk}{2\\pi } e^{-i\\omega (t-t^{\\prime })} J_{|m|,s}(\\rho )J_{|m|,s}(\\rho ^{\\prime })e^{im(\\theta -\\theta ^{\\prime })} \\\\\\times \\kappa _{sm}^2D(\\omega ,q,s,m) \\left[ \\ell _1(\\omega ,q,s,m)e^{ iq(z-z^{\\prime })} + \\ell _2(\\omega ,q,s,m) e^{ iq(z+z^{\\prime })} \\right] .\\end{split}$ The remaining components of the propagator, $G_{\\phi \\phi }$ and $G_{\\phi \\psi }$ , can be derived in a very similar way.",
"We express the result as a matrix propagator $\\begin{pmatrix}G_{\\phi \\phi }^0 & G_{\\phi \\psi }^0 \\\\ G_{\\psi \\phi }^0 & G_{\\psi \\psi }^0\\end{pmatrix}\\,\\chi _{[-L,L]}(z)\\,\\chi _{[-L,L]}(z^{\\prime }) = \\mathcal {G}_1(\\tau ,\\rho ,\\rho ^{\\prime },\\Theta ,\\xi _-) + \\mathcal {G}_2(\\tau ,\\rho ,\\rho ^{\\prime },\\Theta ,\\xi _+) \\,,$ where $\\tau =t-t^{\\prime }$ , $\\Theta = \\theta -\\theta ^{\\prime }$ , $\\xi _\\mp =z\\mp z^{\\prime }$ .",
"The matrices $\\mathcal {G}_{a}$ , $a=1,2$ , result as the Fourier transform in $\\tau $ and $\\xi _\\mp $ of $\\tilde{\\mathcal {G}}_{a}(\\omega ,\\rho ,\\rho ^{\\prime },\\Theta ,k) = \\sum _{s,m} \\kappa _{sm}^2J_{|m|,s}(\\rho )J_{|m|,s}(\\rho ^{\\prime })e^{im\\Theta } \\frac{1}{(\\omega ^2 - k^2 - k_\\rho ^2)(\\omega ^2 - \\omega _0^2) - g^2\\omega ^2} \\mathcal {M}_{a}\\,,$ The matrix $\\mathcal {M}_1$ has the following components (here we omit the obvious dependences on $m$ , $s$ and $q$ ): $\\mathcal {M}_1^{\\phi \\phi } &= \\frac{(\\omega ^2 - \\omega _{-}^2)(\\omega _{+}^2 - \\omega _0^2)(|M_{+}|^2+|N_{+}|^2) - (\\omega ^2 - \\omega _{+}^2)(\\omega _{-}^2 - \\omega _0^2)(|M_{-}|^2+ |N_{-}|^2)}{(\\omega _{+}^2 - \\omega _{-}^2)} ,\\\\\\mathcal {M}_1^{\\phi \\psi } &= \\left(\\mathcal {M}_1^{\\psi \\phi } \\right)^* = -ig\\omega \\, \\frac{(\\omega ^2 - \\omega _{-}^2)(|M_{+}|^2+|N_{+}|^2) - (\\omega ^2 - \\omega _{+}^2)(|M_{-}|^2+ |N_{-}|^2)}{(\\omega _{+}^2 - \\omega _{-}^2)} , \\\\\\mathcal {M}_1^{\\psi \\psi } &= \\frac{(\\omega ^2 - \\omega _{-}^2)(\\omega _{+}^2 - q^2 - k_\\rho ^2)(|M_{+}|^2+|N_{+}|^2) - (\\omega ^2 - \\omega _{+}^2)(\\omega _{-}^2 - q^2 - k_\\rho ^2)(|M_{-}|^2+ |N_{-}|^2)}{(\\omega _{+}^2 - \\omega _{-}^2)} .$ The matrix $\\mathcal {M}_2$ is obtained from $\\mathcal {M}_1$ by replacing the factors $(|M_a|^2+|N_a|^2)$ by $\\text{Re}(M_aN_a^*)$ .",
"The calculations for the case $z<-L$ , $z^{\\prime }>L$ and $z,z^{\\prime }<-L$ are very similar, and actually simpler, to the previous case.",
"The $\\phi $ –$\\phi $ propagator in this case is the only non vanishing one, and it is given by $\\begin{split}G_{\\phi \\phi }^0&(t-t^{\\prime },\\rho ,\\theta ,z,z^{\\prime }) \\,\\chi _{[-\\infty ,-L]}(z)\\,\\chi _{[L,+\\infty ]}(z^{\\prime }) = \\\\ &\\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dk}{2\\pi } e^{-i\\omega (t-t^{\\prime })} \\kappa _{sm}^2 J_{|m|,s}(\\rho )J_{|m|,s}(\\rho ^{\\prime })e^{im(\\theta -\\theta ^{\\prime })} \\frac{1}{\\omega ^2 - \\omega _{ksm}^2} T_{ksm}^*e^{ik(z-z^{\\prime })} .\\end{split} \\\\\\begin{split}G_{\\phi \\phi }^0&(t-t^{\\prime },\\rho ,\\theta ,z,z^{\\prime }) \\,\\chi _{[-\\infty ,-L]}(z)\\,\\chi _{[-\\infty ,-L]}(z^{\\prime }) = \\\\ &\\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dk}{2\\pi } e^{-i\\omega (t-t^{\\prime })} \\kappa _{sm}^2 J_{|m|,s}(\\rho )J_{|m|,s}(\\rho ^{\\prime })e^{im(\\theta -\\theta ^{\\prime })}\\frac{1}{\\omega ^2 - \\omega _{ksm}^2}\\Big [ e^{ik(z-z^{\\prime })} + R_{ksm}^*e^{ik(z+z^{\\prime })} \\Big ] .\\end{split}$ In the second case again we find a dependence on $z+z^{\\prime }$ which is due to reflections; in the first case this does not happen due to the fact that $R_{kms} T_{kms}^*$ is purely imaginary, so that $R_{kms} T_{kms}^*+ R_{kms}^* T_{kms}=0$ .",
"We also notice that for $T=1$ and $R=0$ we find the usual Feynman propagator for a scalar field.",
"The propagators in the cases $z<-L$ , $-L<z^{\\prime }<L$ and $-L<z<L$ , $z^{\\prime }>L$ have less trivial dependence on $z$ and $z^{\\prime }$ .",
"The non vanishing components in this cases are $\\begin{split}G_{\\phi \\phi }^0&(t-t^{\\prime },\\rho ,\\theta ,z,z^{\\prime }) \\,\\chi _{[-\\infty ,-L]}(z)\\,\\chi _{[-L,L]}(z^{\\prime }) = \\\\ &\\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dk}{2\\pi } e^{-i\\omega (t-t^{\\prime })}\\frac{\\kappa _{sm}^2J_{|m|,s}(\\rho )J_{|m|,s}(\\rho ^{\\prime })e^{im(\\theta -\\theta ^{\\prime })}}{\\omega ^2 - \\omega _{ksm}^2} \\Big [ M_{ksm}^* e^{ikz-iq(k)z^{\\prime }} + N_{ksm}^* e^{ikz+iq(k)z^{\\prime }} \\Big ] ,\\end{split}\\\\\\begin{split}G_{\\phi \\psi }^0&(t-t^{\\prime },\\rho ,\\theta ,z,z^{\\prime }) \\,\\chi _{[-\\infty ,-L]}(z)\\,\\chi _{[-L,L]}(z^{\\prime }) = \\\\ &\\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dk}{2\\pi } e^{-i\\omega (t-t^{\\prime })}\\frac{\\kappa _{sm}^2J_{|m|,s}(\\rho )J_{|m|,s}(\\rho ^{\\prime })e^{im(\\theta -\\theta ^{\\prime })}}{\\omega ^2 - \\omega _{ksm}^2} \\,\\frac{-ig\\omega }{\\omega _{ksm} ^2 - \\omega _0^2} \\Big [ M_{ksm}^* e^{ikz-iq(k)z^{\\prime }} + N_{ksm}^* e^{ikz+iq(k)z^{\\prime }} \\Big ] ,\\end{split}\\\\\\begin{split}G_{\\psi \\phi }^0&(t-t^{\\prime },\\rho ,\\theta ,z,z^{\\prime }) \\,\\chi _{[-\\infty ,-L]}(z)\\,\\chi _{[-L,L]}(z^{\\prime }) = \\\\ &\\sum _{s,m} \\int \\frac{d\\omega }{2\\pi } \\frac{dk}{2\\pi } e^{-i\\omega (t-t^{\\prime })}\\frac{\\kappa _{sm}^2J_{|m|,s}(\\rho )J_{|m|,s}(\\rho ^{\\prime })e^{im(\\theta -\\theta ^{\\prime })}}{\\omega ^2 - \\omega _{ksm}^2} \\, \\frac{ig\\omega }{\\omega _{ksm}^2 - \\omega _0^2}\\Big [ M_{ksm}^* e^{ikz-iq(k)z^{\\prime }} + N_{ksm}^* e^{ikz+iq(k)z^{\\prime }} \\Big ] ,\\end{split}$ where the factor in square brackets is the same for all three components.",
"These expressions are obtained by making use of the following equalities, that are easily checked: $M_{kms} = R_{kms} N_{kms}^* + T_{kms} M_{kms}^* \\,, \\\\N_{kms} = R_{kms} M_{kms}^* + T_{kms} N_{kms}^* \\,.$"
],
[
"Solitonic solutions",
"In this section we introduce a nonlinearity in the model, with the aim of describing the perturbation of refractive index propagating in the nonlinear dielectric medium when a strong laser pulse is shot into the dielectric and the Kerr effect is stimulated.",
"The propagating perturbation breaks the homogeneity of the dielectric sample described in the previous section.",
"Still, the solutions for the homogeneous case represent a good asymptotic scattering basis for the full nonlinear problem in the linearisation of the theory around the dielectric perturbation represented by the solitonic solutions we are going to describe.",
"With this aim, we add a fourth order term in the polarization field $\\psi $ , as in [21].",
"A fourth order term in nonlinear optical media appears also e.g.",
"in [22], where a fourth order term in the displacement field can be introduced in the case of an optical fiber.",
"We have discussed a fourth order term in the polarization field for the Hopfield model in [11], where we have shown that our solitonic solutions can be associated with the Kerr effect in a proper way.",
"It is also to be remarked that our approach does not represent the standard way to approach the Kerr effect (see also [24], [23]).",
"and that our solitonic solutions are more constrained that the usual solutions of the nonlinear Schrödinger equation studied in [24].",
"See also the discussion in [25].",
"Our non-linear theory has the following action $S[\\phi ,\\psi ] = \\int _C \\frac{1}{2} \\partial _\\mu \\phi \\partial ^\\mu \\phi \\, d^4x +\\int _{C_\\chi } \\biggl [ \\frac{1}{2} (v^\\mu \\partial _\\mu \\psi )^2 - \\frac{{\\omega _0}^2}{2} \\psi ^2 -g \\phi v^\\mu \\partial _\\mu \\psi - \\frac{\\lambda }{4!}",
"\\psi ^4 \\biggr ] \\, d^4x ;$ the equations of motion are $\\square \\phi + gv^\\mu \\partial _\\mu \\psi &= 0 , \\\\(v^\\mu \\partial _\\mu )^2\\psi + \\omega _0^2\\psi - gv^\\mu \\partial _\\mu \\phi + \\frac{\\lambda }{3!}",
"\\psi ^3 &= 0 .$ In lab $v^\\mu =(1,0,0,0)$ , so the equations become $\\square \\phi + g\\dot{\\psi } &= 0 , \\\\\\ddot{\\psi } + \\omega _0^2\\psi - g\\dot{\\phi } + \\frac{\\lambda }{3!}",
"\\psi ^3 &= 0 .$ We are not looking for the general solution of the system above, still we are interested in finding out analytical solutions.",
"We attempt the ansatz $\\phi (t,z,\\rho ,\\theta ) = f(z-Vt) Y(\\rho ,\\theta ) , \\\\\\psi (t,z,\\rho ,\\theta ) = h(z-Vt) Y(\\rho ,\\theta ) ;$ the radial and angular parts can be separated if $Y=const$ This would be equivalent to a calculation involving only $s$ -waves in presence of spherical symmetry.",
"to obtain the equations $(1-V^2)f^{\\prime \\prime } + gVh^{\\prime } &= 0 , \\\\V^2 h^{\\prime \\prime } + gVf^{\\prime } + \\omega _0^2h + \\frac{Y^2\\lambda }{3!",
"}h^3 &= 0 ,$ where the prime stands for the derivative with respect to the argument $z-Vt$ .",
"Integrating the first equation and inserting in the second, after a new integration we obtain $\\frac{Vh^{\\prime }}{i\\sqrt{\\frac{Y^2\\lambda }{12}h^4 + (\\omega _0^2-g^2V^2\\gamma ^2)h^2 + 2g\\kappa Vh - 2\\chi }} = 1 ,$ where $\\kappa $ and $\\chi $ are constants of integration, and $\\gamma =(1-V^2)^{-\\frac{1}{2}}$ .",
"We will also use the notation $v:=\\gamma V.$ It is worth to mention here that we accept solution having finite energy.",
"It is immediate to see that the energy density inside the dielectric is $\\mathcal {E}=\\frac{1}{2} \\dot{\\phi }^2+\\frac{1}{2} \\nabla \\phi \\cdot \\nabla \\phi +\\frac{1}{2} \\dot{\\psi }^2+\\frac{1}{2} \\omega _0^2\\psi ^2 +\\frac{\\lambda }{4!",
"}\\psi ^4=Y^2(\\chi + (gV\\gamma ^2 h(z-Vt)-\\kappa )^2-(1-V^2)\\kappa ).$ Thus, the energy is finite if $h$ has no poles.",
"Notice that, in particular, $h$ has to be limited, which implies that the quartic radicand in (REF ) must have real roots.",
"So, the constants must be constrained in order to ensure this condition."
],
[
"$\\kappa =\\chi =0$",
"In this case, the solution inside the dielectric is the solitonic solution obtained in [21].",
"With $a=\\sqrt{ \\frac{12}{\\lambda Y^2} (g^2v^2 - \\omega _0^2) }$ and $b=\\frac{1}{v} \\sqrt{g^2v^2 - \\omega _0^2 }$ , we find (for $-L \\le z \\le L$ ) $h & = \\frac{a}{\\cosh (b\\gamma (z-Vt))} , \\\\f & = \\frac{2agv}{b} \\arctan \\biggl [ \\tanh \\biggl (\\frac{b}{2}\\gamma (z-Vt)\\biggr ) \\biggr ] ,$ while the solution in vacuum is a superposition of a progressive and a regressive wave, whose form is determined by the continuity of $\\phi $ and $\\partial _z\\phi $ , as discussed in section REF : $\\partial _z f={\\left\\lbrace \\begin{array}{ll}\\frac{1+V}{2} \\frac{ag\\gamma v}{\\cosh [b\\gamma (L(1-V)-V(z-t))]} + \\frac{1-V}{2} \\frac{ag\\gamma v}{\\cosh [b\\gamma (L(1+V)+V(z+t))]}, & z\\le -L\\\\\\frac{1+V}{2} \\frac{ag\\gamma }{\\cosh [b\\gamma (L(1-V)+V(z-t))]} + \\frac{1-V}{2} \\frac{ag\\gamma }{\\cosh [b\\gamma (L(1+V)-V(z+t))]}, & z\\ge L\\end{array}\\right.}",
".$ Note that the solution exists only if $v>\\omega _0/g$ , that is $V^2>\\frac{\\omega _0^2}{g^2+\\omega _0^2}.$"
],
[
"$\\kappa \\ne 0$ , {{formula:47bac87f-c13a-438b-bd42-dee2f24903e1}}",
"We look for a Möbius transformation $h=\\frac{as+b}{cs+d} ,$ which maps (REF ) into the form $\\frac{i}{V}=\\frac{s^{\\prime }}{\\sqrt{4s^3-g_2s-g_3}}.",
"$ The assumption that the original quartic equation has at least one real root ensures that such a transformation exists with real coefficients $a,b,c,d$ that can be chosen to satisfy $ad-bc=\\pm 1$ , see Appendix .",
"Equation (REF ) has general solution $s(x)=\\wp (g_2,g_3;i(x-x_0)/V)=-\\wp (g_2,-g_3;(x-x_0)/V),$ where $g_2$ , $g_3$ are defined in Appendix and $\\wp $ is the Weierstrass elliptic function defined by $\\wp (z)=\\frac{1}{z^2} +\\sum _{(n,m)\\in \\mathbb {Z}^2-\\lbrace 0,0\\rbrace } \\left( \\frac{1}{(z+n\\omega _1+m\\omega _2)^2}-\\frac{1}{(n\\omega _1+m\\omega _2)^2} \\right)$ with $\\omega _1$ , $\\omega _2$ the two periods satisfying $\\tau :=\\omega _2/\\omega _1\\notin \\mathbb {R}$ , and $x_0$ is an integration constant.",
"Indeed, we can be more precise and notice that there are two distinct situations, when all three roots of $4z^3-g_2z-g_3$ are distinct.",
"In our case, $g_2$ and $g_3$ are real and so we may have three real roots $e_3<e_2<e_1$ , or one real root $e_2$ and two complex roots $e_1, e_3$ , with $e_1=\\bar{e}_3$ .",
"Let us shortly discuss the two cases."
],
[
"3 REAL ROOTS",
"$e_3<e_2<e_1$ .",
"The periods of $\\wp (g_2,-g_3,z)$ are $\\omega _1&= 2\\int _{e_1}^\\infty \\frac{dz}{\\sqrt{4z^3-g_2z+g_3}}\\ \\in \\mathbb {R}_{>}, \\\\\\omega _2&= 2i\\int _{-\\infty }^{e_3} \\frac{dz}{\\sqrt{-4z^3+g_2z-g_3}}\\ \\in i\\mathbb {R}_{>},$ and we get two distinct solutions for $h$ .",
"The first one is $h(z-Vt)=\\frac{a \\wp (g_2,-g_3;t-(z-z_0)/V)-b}{c\\wp (g_2,-g_3;t-(z-z_0)/V)-d}.$ In this case $\\wp $ assumes all values in $[e_1,\\infty )$ .",
"Since we are interested in solutions with $h$ of class $C^2$ everywhere inside the dielectric, we must discard solutions such that the denominator above vanishes somewhere.",
"This happens only if the condition $ce_1 -d>0 $ is satisfied.",
"Analysing this condition in general depends on several details.",
"A partial analysis can be found in appendix .",
"This solutions represent trains of pulses having period $\\omega _1$ , and moving with constant velocity as we can see from figure REF .",
"Figure: Plot of the solution () corresponding to the three real root case.",
"The parameters have the following values:Y=ω 0 =g=1Y=\\omega _0=g=1, λ=V=0.5\\lambda =V=0.5, χ=2\\chi =2, κ=3\\kappa =3, z 0 =0z_0=0.The second solution is $h(z-Vt)=\\frac{a \\wp (g_2,-g_3;t-(z-z_0)/V+\\frac{\\omega _2}{2})-b}{c\\wp (g_2,-g_3;t-(z-z_0)/V+\\frac{\\omega _2}{2})-d}.$ In this case $\\wp $ oscillates in the interval $[e_3,e_2]$ , which is acceptable if the condition $\\frac{d}{c} \\notin [e_3,e_2]$ is satisfied.",
"Again, we get a train of pulses which is shifted along the horizontal axis, as shown in figure REF ."
],
[
"1 REAL ROOT",
"$e_2\\in \\mathbb {R}$ .",
"In this case we have two complex conjugate periods $\\omega _1=\\omega $ and $\\omega _2=\\bar{\\omega }$ with $\\omega =\\int _{e_2}^\\infty \\frac{dz}{\\sqrt{4z^3-g_2z+g_3}}+i \\int _{-\\infty }^{e_2} \\frac{dz}{\\sqrt{-4z^3+g_2z-g_3}}.$ In this case there is only one kind of solutions, having the form (REF ) with condition (REF ).",
"This represents a train of pulses having period $\\omega +\\bar{\\omega }$ , like for example in figure REF .",
"Figure: Plot of the solution () corresponding to the three real root case.",
"The parameters have the following values:Y=ω 0 =g=1Y=\\omega _0=g=1, λ=V=0.5\\lambda =V=0.5, χ=2\\chi =2, κ=3\\kappa =3, z 0 =0z_0=0."
],
[
"ELEMENTARY SOLUTIONS",
"These solutions correspond to the cases of degenerate roots and could be directly deduced as particular cases of the Weierstrass cases.",
"However, since classifying all possible Weierstrass configurations is quite cumbersome, as shown in Appendix , it is easier to construct them directly.",
"In order to obtain a solution expressible in an algebraic form, we set equal to zero the discriminant of the fourth degree polynomial at the denominator of (REF ), and resolve it for the constant $\\lambda $ (here $\\Omega :=\\omega _0^2-g^2v^2$ ): $Y^2\\lambda _\\pm =-\\frac{3}{64\\chi ^3}\\Bigl ( 27g^4\\kappa ^4V^4+72g^2\\kappa ^2V^2\\chi \\Omega +32\\chi ^2\\Omega ^2 \\pm \\sqrt{g^2\\kappa ^2V^2(9g^2\\kappa ^2V^2+16\\chi \\Omega )^3} \\Bigr ) ;$ $\\lambda _\\pm $ is real if and only if $\\Omega \\chi \\ge -9g^2\\kappa ^2V^2/16$ .",
"Recall that at least one between $\\lambda _+$ and $\\lambda _-$ must be positive in order to be consistent with our physical assumptions.",
"On the other hand, with the aid of the general theory of quartic equations [27], we discover the nature of the roots of our polynomial, depending of how the parameters change.",
"It holds (one must choose the positive one between $\\lambda _+$ and $\\lambda _-$ ): if $\\Omega <0$ and $-\\frac{3\\Omega ^2}{2\\lambda _\\pm } < \\chi < \\frac{\\Omega ^2}{2\\lambda _\\pm }$ , then we have four real roots, of which two are double; if $\\Omega <0$ and $\\chi <-\\frac{3\\Omega ^2}{2\\lambda _\\pm }$ , there are a double real root and two complex conjugate roots; if $\\Omega >0$ and $\\chi <0$ , then there are again a double real root and two complex conjugate roots.",
"This is what the theory tells us.",
"We can say something more: from the equation $\\frac{\\lambda }{12}h^4+\\Omega h^2+2g\\kappa V h-2\\chi =0$ , we see that if $\\chi >0$ , it can be chosen $h$ such that the polynomial is zero.",
"So, we conclude that for every $\\chi >0$ the equation has two double real roots.",
"Figure: Plot of the solution () corresponding to the one real root case.",
"The parameters have the following values:Y=10 3 Y=10^3, ω 0 =g=1\\omega _0=g=1, λ=V=0.5\\lambda =V=0.5, χ=2\\chi =2, κ=10 3 \\kappa =10^3, z 0 =0z_0=0.Note that not all of these conditions are compatible with our choice of $\\lambda _\\pm $ .",
"Consider, for example, $\\lambda _+$ : if we want to guarantee the positivity of $\\lambda _+$ , then the combination $\\Omega >0$ , $\\chi >0$ is not acceptable, while the other combinations allow $\\lambda _+>0$ .",
"Moreover, the reality condition $\\chi \\Omega \\ge -9g^2\\kappa ^2V^2/16$ provide further constraints.",
"Independently by the value of the parameters, the general solution of (REF ) can be easily written: if $\\alpha $ is the double real root, and $\\beta $ , $\\delta $ are real (or complex), then, recalling that in our case $\\delta +\\beta =-2\\alpha $ , $h =\\Biggl [\\frac{\\beta -\\delta }{2(\\alpha -\\beta )(\\alpha -\\delta )}\\cos \\biggl (\\sqrt{\\frac{\\lambda (\\alpha -\\beta )(\\alpha -\\delta )}{12}} \\frac{z-Vt}{V}\\biggr )- \\frac{2\\alpha }{(\\alpha -\\beta )(\\alpha -\\delta )} \\Biggr ]^{-1} + \\alpha .$ The term $(\\beta -\\delta )$ shows that if $\\beta $ and $\\delta $ are not real, then the final solution is not real too and must be excluded.",
"So, the parameters must be such that the roots are all real.",
"In this case $(\\alpha -\\beta )(\\alpha -\\delta )=4\\alpha ^2-(\\alpha +\\beta )^2=4\\alpha ^2-(\\alpha +\\delta )^2$ shows that we may have both signs for $(\\alpha -\\beta )(\\alpha -\\delta )$ , so that we have $h =\\Biggl [\\frac{\\beta -\\delta }{2(\\alpha -\\beta )(\\alpha -\\delta )}\\cos \\biggl (\\sqrt{\\frac{\\lambda (\\alpha -\\beta )(\\alpha -\\delta )}{12}} \\frac{z-Vt}{V}\\biggr )- \\frac{2\\alpha }{(\\alpha -\\beta )(\\alpha -\\delta )} \\Biggr ]^{-1} + \\alpha , &\\ {\\rm if}\\ (\\alpha -\\beta )(\\alpha -\\delta )>0,\\\\h =\\Biggl [\\frac{\\beta -\\delta }{2(\\alpha -\\beta )(\\alpha -\\delta )}\\cosh \\biggl (\\sqrt{\\frac{\\lambda (\\alpha -\\beta )(\\alpha -\\delta )}{12}} \\frac{z-Vt}{V}\\biggr )- \\frac{2\\alpha }{(\\beta -\\alpha )(\\alpha -\\delta )} \\Biggr ]^{-1} + \\alpha , &\\ {\\rm if}\\ (\\alpha -\\beta )(\\alpha -\\delta )<0,$ The last case includes the simplest situation $k=0$ , $\\chi =0$ studied above.",
"The first case, instead, includes the simple case studied below."
],
[
"$\\kappa \\ne 0$ , {{formula:0d52707b-7e3b-47b9-900b-c4d8673d5945}}",
"As a particular subcase, we can study equation (REF ) with $\\chi =0$ ; here the analysis is simplified, because we have to study a cubic equation.",
"Requiring the discriminant of the cubic to be zero, we find $\\lambda = -\\frac{4\\Omega ^3}{9g^2\\kappa ^2V^2} ,$ so the positivity of $\\lambda $ requires $\\Omega <0$ , that is $v>\\omega _0/g$ as usual.",
"With this value for $\\lambda $ there are three real roots, of which two are double.",
"The double root is $-3g\\kappa V/\\Omega $ , and the simple root is $6g\\kappa V/\\Omega $ .",
"The solution of (REF ) is $h =-\\frac{3g\\kappa V}{\\omega _0^2-g^2v^2} \\Biggl [\\frac{3}{\\cos \\Bigl ( \\frac{3g\\kappa \\sqrt{\\lambda }}{2(\\omega _0^2-g^2v^2)}(z-Vt) \\Bigr ) - 2}+ 1 \\Biggr ] .$"
],
[
"Conclusions",
"We have studied, in the simplified framework of the $\\phi \\psi $ -model, the propagation of the electromagnetic field in a spatially finite sample of dielectric medium.",
"This situation is physically relevant in the Analogue Gravity picture for the Hawking effect, as experiments involve necessarily finite samples of dielectrics.",
"We have chosen to work in a cylindrical geometry, where the dielectric field fills only a finite cylindrical region of length $2L$ and radius $R$ .",
"The remaining region of radius $R$ is filled by vacuum.",
"This may be considered as a model for a optic fiber, which are an active benchmark for experiments in Analogue Gravity [17].",
"Our present study concerns analytical properties of the solutions for the equations of motion of the involved fields.",
"As a preliminary analysis, we have considered the boundary conditions to be imposed on the fields, together with a complete scattering basis and the quantization of the fields in the case of a still homogeneous dielectric sample.",
"We have also described the propagator for the fields in the given setting.",
"Then we have introduced a nonlinearity in the model, as the dielectric media we are interested in must be associated with the Kerr effect.",
"Indeed, as pointed out firstly in [1], a possibility to obtain analogous black hole in dielectrics consists in generating strong laser pulses which propagate inside a nonlinear dielectric medium.",
"The Kerr effect gives rise to a propagating perturbation of the refractive index which plays the role of the analogous black hole, and is indeed involved with a horizon.",
"Our interest has been to find out solitonic solutions describing the aforementioned perturbation, i.e.",
"the background solutions around which a linearization is performed, and the perturbations are quantized.",
"We have shown that solitonic solutions exist, representing a dielectric perturbation travelling with constant velocity in the direction of the cylindrical fiber axis.",
"Further developments can involve different aspects, all with a noticeable physical interest.",
"One may study perturbation theory for the model, looking for quantum effects induced by surface effects, e.g.",
"transition radiation [26].",
"Also, one can study absorption in the model, associated with the fourth order perturbation.",
"Our main research focus, which is represented by the analogous Hawking effect, requires the analysis of the linearization of the model around the solitonic solution and its quantization.",
"One may also limit to consider, in the comoving frame of the dielectric perturbation, the dependence of the dielectric susceptibility and of the resonance frequency on space (induced by the Kerr effect), and analyze the Hawking effect with simpler background profiles.",
"Future works will be devoted to the aforementioned goals."
],
[
"Acknowledgements",
"A.V.",
"was partially supported by Ministero dell'Università e della Ricerca MIUR-PRIN contract 2017CC72MK_003."
],
[
"Orthogonality relations",
"For $f=\\begin{pmatrix}\\phi \\\\ \\psi \\end{pmatrix},\\qquad \\tilde{f}=\\begin{pmatrix}\\tilde{\\phi }\\\\ \\tilde{\\psi }\\end{pmatrix},$ we define $j^\\mu _{f,\\tilde{f}}(x):= i [\\phi ^*(x) \\partial ^\\mu \\tilde{\\phi }(x)-\\tilde{\\phi }(x) \\partial ^\\mu \\phi ^*(x)+v^\\mu (\\psi ^*(x)\\dot{\\tilde{\\psi }}(x)-\\tilde{\\psi }(x)\\dot{\\psi }^*(x))+gv^\\mu (\\psi ^*(x) \\tilde{\\phi }(x)-\\phi ^*(x)\\tilde{\\psi }(x))].$ It is a conserved current $\\partial _\\mu j^\\mu _{f,\\tilde{f}}(x)=0,$ and the scalar product is $(f|\\tilde{f})=\\int _{C_t} j^0_{f,\\tilde{f}}(x),$ where $C_t$ is the slice of $C$ obtained by fixing $t$ .",
"The conservation law is particularly helpful for computing the scalar product among plane wave solutions or scattering wave solutions.",
"These solutions have the form $\\phi (x)=e^{-i\\omega _{ksm}t} \\varphi _{_{ksm}}(\\vec{x}) ,$ so if we take $f(x)=e^{-i\\omega _{ksm}t}\\begin{pmatrix}\\varphi _{ksm}(\\vec{x}) \\\\\\varrho _{ksm}(\\vec{x})\\end{pmatrix},\\qquad \\tilde{f}(x)=e^{-i\\omega _{k^{\\prime }s^{\\prime }m^{\\prime }}t}\\begin{pmatrix}\\varphi _{k^{\\prime }s^{\\prime }m^{\\prime }}(\\vec{x}) \\\\\\varrho _{k^{\\prime }s^{\\prime }m^{\\prime }}(\\vec{x})\\end{pmatrix},$ then $\\partial _0 j^0_{f,\\tilde{f}}(x)=i(\\omega _{ksm}-\\omega _{k^{\\prime }s^{\\prime }m^{\\prime }}) j^0_{f,\\tilde{f}}(x),$ and integrating over the spatial slice and using the continuity equation we get $(f|\\tilde{f})=-\\frac{1}{i(\\omega _{ksm}-\\omega _{k^{\\prime }s^{\\prime }m^{\\prime }})} \\int _{\\partial C_t} \\vec{j}_{f,\\tilde{f}}(x) \\cdot \\vec{n}(x) \\ d^2\\sigma (x),$ where we have used the continuity of $\\vec{j}_{f,\\tilde{f}}(x) \\cdot \\vec{n}(x)$ , as a consequence of the boundary conditions.",
"In order to compute this integral let us restrict it on the compact cylinder $C_t^Z=\\lbrace (\\rho ,\\theta ,z)\\in C_t| -Z\\le z\\le Z\\rbrace ,$ so that $(f|\\tilde{f})=-\\frac{1}{i(\\omega _{ksm}-\\omega _{k^{\\prime }s^{\\prime }m^{\\prime }})} \\lim _{Z\\rightarrow +\\infty }\\int _{\\partial C^Z_t} \\vec{j}_{f,\\tilde{f}}(x) \\cdot \\vec{n}(x) \\ d^2\\sigma (x).$ Finally, by taking into account the boundary condition for the fields we get $(f|\\tilde{f})=-\\frac{1}{i(\\omega _{ksm}-\\omega _{k^{\\prime }s^{\\prime }m^{\\prime }})} \\lim _{Z\\rightarrow +\\infty } \\left( \\int _{D_Z} j^z_{f,\\tilde{f}}(x) \\rho ^2 d\\rho d\\theta -\\int _{D_{-Z}} j^z_{f,\\tilde{f}}(x) \\rho ^2 d\\rho d\\theta \\right),$ where $D_{\\pm Z}=\\lbrace (\\rho ,\\theta ,z)\\in C_t| z=\\pm Z \\rbrace .$ Using $\\int _0^{2\\pi } e^{i(m^{\\prime }-m)\\theta } d\\theta =2\\pi \\delta _{mm^{\\prime }} ,$ for two right dielectric scattering functions we get $\\begin{split}\\frac{(f|\\tilde{f})}{\\kappa ^*_{ksm}\\kappa _{ksm}}&=-\\frac{e^{i(\\omega _{ksm}-\\omega _{k^{\\prime }s^{\\prime }m})t}}{i(\\omega _{ksm}-\\omega _{k^{\\prime }s^{\\prime }m})} 2\\pi \\delta _{mm^{\\prime }}\\int _0^R J_{|m|} \\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ) J_{|m|} \\biggl (z_{ms^{\\prime }}\\frac{\\rho }{R}\\biggr ) \\rho d\\rho \\ \\\\&\\quad \\times \\lim _{Z\\rightarrow +\\infty }\\Bigl [ (k-k^{\\prime }) R^*_{ksm} e^{-i(k+k^{\\prime })Z} -(k-k^{\\prime }) R_{k^{\\prime }s^{\\prime }m} e^{i(k+k^{\\prime })Z} +(k+k^{\\prime }) \\\\&\\quad \\times \\Bigl ( (T^*_{ksm} T_{k^{\\prime }s^{\\prime }m}+R^*_{ksm} R_{k^{\\prime }s^{\\prime }m})e^{-(k-k^{\\prime })Z} -e^{i(k-k^{\\prime })Z}\\Bigl ) \\Bigr ].\\end{split}$ We first show that Lemma 1.1 It holds $\\int _0^R J_{|m|} \\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ) J_{|m|} \\biggl (z_{ms^{\\prime }}\\frac{\\rho }{R}\\biggr ) \\rho d\\rho =\\delta _{ss^{\\prime }} \\frac{R^2}{2z^2_{ms}}(z_{ms}^2-m^2)J^2_m(z_{ms}).$ The Bessel equation can be written in the form $\\frac{d}{d\\rho } \\biggl ( \\rho \\frac{d}{d\\rho } J_{|m|} \\biggl (z_{ms}\\frac{\\rho }{R}\\biggl ) \\biggr ) +\\biggl (\\rho \\frac{z^2_{ms}}{R}^2-\\frac{m^2}{\\rho }\\biggr )J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr )=0 ,$ from which we get $\\begin{split}& \\frac{d}{d\\rho } \\biggl [ \\rho J_{|m|}\\biggl (z_{ms^{\\prime }}\\frac{\\rho }{R}\\biggr ) \\frac{d}{d\\rho } J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr )-\\rho J_{|m|}\\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ) \\frac{d}{d\\rho } J_{|m|}\\biggl (z_{ms^{\\prime }}\\frac{\\rho }{R}\\biggr ) \\biggr ] \\\\&\\quad +\\frac{z^2_{ms}-z^2_{ms^{\\prime }}}{R^2} \\rho J_{|m|} \\biggl (z_{ms}\\frac{\\rho }{R}\\biggl ) J_{|m|} \\biggl (z_{ms^{\\prime }}\\frac{\\rho }{R}\\biggr )=0 ,\\end{split}$ that integrated from 0 to $R$ in $d\\rho $ gives $\\int _0^R J_{|m|} \\biggl (z_{ms}\\frac{\\rho }{R}\\biggl ) J_{|m|} \\biggl (z_{ms^{\\prime }}\\frac{\\rho }{R}\\biggr ) \\rho d\\rho =0 ,$ if $s\\ne s^{\\prime }$ .",
"Moreover, from (REF ) we get $\\begin{split}0&=\\frac{d}{d\\rho } \\left( \\rho J_{|m|} (z_{ms}\\frac{\\rho }{R}) \\frac{d}{d\\rho } \\left( \\rho \\frac{d}{d\\rho } J_{|m|} (z_{ms}\\frac{\\rho }{R}) \\right) \\right)+\\frac{z^2_{ms}}{R^2} 2\\rho J^2_{m}(z_{ms}\\frac{\\rho }{R}) \\\\&\\quad +(\\rho ^2\\frac{z^2_{ms}}{R^2}-m^2) 2J_{|m|}(z_{ms}\\frac{\\rho }{R}) \\frac{d}{d\\rho }J_{|m|}(z_{ms}\\frac{\\rho }{R})\\\\&=\\frac{d}{d\\rho } \\left( \\rho J_{|m|} (z_{ms}\\frac{\\rho }{R}) \\frac{d}{d\\rho } \\left( \\rho \\frac{d}{d\\rho } J_{|m|} (z_{ms^{\\prime }}\\frac{\\rho }{R}) \\right) \\right)+\\frac{z^2_{ms}}{R^2} 2\\rho J^2_{m}(z_{ms}\\frac{\\rho }{R}) \\\\&\\quad -\\frac{d}{d\\rho } \\left( \\rho \\frac{d}{d\\rho }J_{|m|}(z_{ms}\\frac{\\rho }{R}) \\right)^2 .\\end{split}$ After integration in $d\\rho $ from 0 to $R$ , and using the definition of $z_{ms}$ , we get $2\\frac{z^2_{ms}}{R^2} \\int _0^R J^2_{|m|} \\biggl (z_{ms}\\frac{\\rho }{R}\\biggr ) \\rho d\\rho =-R^2 J^2_{|m|} (z_{ms}) \\frac{d^2}{d\\rho ^2} J^2_{|m|} \\biggl (z_{ms}\\frac{\\rho }{R}\\biggr )\\bigg |_{\\rho =R}.$ Using again the Bessel equation we finally get the assert.",
"Thus $\\begin{split}\\frac{(f|\\tilde{f})}{\\kappa ^*_{ksm}\\kappa _{ksm}}=&-\\frac{e^{i(\\omega _{ksm}-\\omega _{k^{\\prime }sm})t}}{i(\\omega _{ksm}-\\omega _{k^{\\prime }sm})} 2\\pi \\delta _{mm^{\\prime }} \\delta _{ss^{\\prime }} \\frac{R^2}{2z^2_{ms}}(z_{ms}^2-m^2)J^2_m(z_{ms})\\ \\lim _{Z\\rightarrow +\\infty }\\left[ (k-k^{\\prime }) R^*_{ksm} e^{-i(k+k^{\\prime })Z} \\right.",
"\\cr & \\left.",
"-(k-k^{\\prime }) R_{k^{\\prime }sm} e^{i(k+k^{\\prime })Z} +(k+k^{\\prime }) \\left( (T^*_{ksm} T_{k^{\\prime }sm}+R^*_{ksm} R_{k^{\\prime }sm})e^{-(k-k^{\\prime })Z} -e^{i(k-k^{\\prime })Z}\\right) \\right].\\end{split}$ In order to compute this limit, we rewrite it in the form $\\begin{split}\\lim _{Z\\rightarrow \\infty } \\frac{k-k^{\\prime }}{\\omega _{ksm}-\\omega _{k^{\\prime }sm}} &\\left[ (k+k^{\\prime }) \\frac{e^{-(k-k^{\\prime })Z} -e^{i(k-k^{\\prime })Z}}{k^{\\prime }-k} +\\left( R^*_{ksm} e^{-i(k+k^{\\prime })Z} -R_{k^{\\prime }sm} e^{i(k+k^{\\prime })Z} \\right) \\right.",
"\\cr &\\left.",
"+(k+k^{\\prime }) \\frac{T^*_{ksm} T_{k^{\\prime }sm}+R^*_{ksm} R_{k^{\\prime }sm}-1}{k^{\\prime }-k}e^{-(k-k^{\\prime })Z} \\right].\\end{split}$ Since $k$ and $k^{\\prime }$ are positive, the second term in the square brackets vanishes in the limit because of the Riemann–Lebesgue theorem.",
"The second term vanishes for the same reason unless $k=k^{\\prime }$ .",
"Since $|T_{ksm}|^2+|R_{ksm}|^2=1$ , it stays finite for $k=k^{\\prime }$ (if we take the continuation by the limit $k^{\\prime }\\rightarrow k$ ).",
"Thus, in the distributional sense, it vanishes.",
"So, the surviving limit is $\\lim _{Z\\rightarrow \\infty } \\frac{k-k^{\\prime }}{\\omega _{ksm}-\\omega _{k^{\\prime }sm}} &\\left[ (k+k^{\\prime }) \\frac{2i \\sin ((k-k^{\\prime })Z)}{k^{\\prime }-k} \\right]= 2\\pi k \\frac{dk}{d\\omega } 2i \\delta (k^{\\prime }-k) , $ and we finally get $(f|\\tilde{f})=|\\kappa _{ksm}|^2 4\\omega _{ksm}\\pi ^2 R^2 \\left( 1-\\frac{m^2}{z^2_{ms}} \\right) J^2_m(z_{ms}) \\delta _{mm^{\\prime }} \\delta _{ss^{\\prime }} \\delta (k^{\\prime }-k).$ The same result is true for two left dielectric wave functions.",
"The same procedure can be used to compute the scalar product between two right (or left) gap wave functions $g$ , $\\tilde{g}$ $(g|\\tilde{g})=|\\tilde{\\kappa }_{ksm}|^2 \\omega _{ksm} \\pi ^2 R^2 \\left( 1-\\frac{m^2}{z^2_{ms}} \\right) J^2_m(z_{ms}) \\delta _{mm^{\\prime }} \\delta _{ss^{\\prime }} \\delta (k^{\\prime }-k).$ All other combinations vanish.",
"It is worth to mention that for the particular case $m=0$ there is also the zero $z_{0,0}=0$ , for which $(f|\\tilde{f})&=|\\kappa _{k00}|^2 4 k\\pi ^2 R^2 \\delta (k^{\\prime }-k), \\\\(g|\\tilde{g})&=|\\tilde{\\kappa }_{k00}|^2 k \\pi ^2 R^2 \\delta (k^{\\prime }-k).$"
],
[
"Study of equation (",
"Let us write the quartic as $\\frac{Y^2\\lambda }{12}h^4 + (\\omega _0^2-g^2V^2\\gamma ^2)h^2 + 2g\\kappa Vh - 2\\chi =p(h) ,$ with $p(x)=\\alpha _0 x^4+\\alpha _2 x^2+\\alpha _3 x+\\alpha _4 \\equiv \\alpha _0 (x-E_0)(x-E_1)(x-E_2)(x-E_3),$ where $E_j$ are the polynomial roots, satisfying $E_0+E_1+E_2+E_3=0$ .",
"We are assuming that there is at least one real root and so define $E_0$ to be the largest real root.",
"Let us consider the change of variables $h=\\frac{as+b}{cs+d},$ with $a$ , $b$ , $c$ , $d$ all real.",
"These parameters are defined up to a global real rescaling, which can be fixed so that $ad-bc=\\varepsilon $ , with $\\varepsilon =\\pm 1$ .",
"We get $h^{\\prime }=\\frac{\\varepsilon s^{\\prime }}{(cs+d)^2},$ so that $\\frac{h^{\\prime }}{\\sqrt{p(h)}}=\\frac{\\varepsilon s^{\\prime }}{q(s)},$ where $q(s)=a_0 s^4+a_1 s^3+a_2 s^2+a_3 s+ a_4,$ with $a_0 = c^4 p\\biggl (\\frac{a}{c}\\biggr ),$ which we easily set to zero by imposing $a=c E_0.$ With this position, the remaining coefficients are $a_1&=c^3 (b-E_0 d)\\ p^{\\prime }(E_0), \\\\a_2&=(b-E_0 d)c^2 d\\ p^{\\prime }(E_0) -(b-E_0 d) \\varepsilon \\frac{c}{2}\\ p^{\\prime \\prime }(E_0), \\\\a_3&=4cd^3\\ p\\biggl (\\frac{b}{d}\\biggr )+\\varepsilon d^2\\ p^{\\prime }\\biggl (\\frac{b}{d}\\biggr ),\\\\a_4&=d^4\\ p\\biggl (\\frac{b}{d}\\biggr ).$ We have to impose the condition $a_2=0$ and $a_1=4$ .",
"Notice that $b-E_0 d=-\\varepsilon /c$ .",
"Moreover, since we are assuming the roots are generic, therefore all distinct, $p^{\\prime }(E_0)\\ne 0$ .",
"We finally get: $a=E_0 c, \\qquad b = \\frac{c}{4} \\left(p^{\\prime }(E_0)-\\frac{1}{2} E_0 p^{\\prime \\prime }(E_0)\\right), \\qquad c= \\frac{2}{\\sqrt{|p^{\\prime }(E_0)|}} \\qquad d=-\\frac{p^{\\prime \\prime }(E_0)}{8} c .$ and $\\varepsilon = -{\\rm sign}(p^{\\prime }(E_0))$ .",
"With the assumption $E_0$ real, these coefficients are all real and lead us to equation (REF ), with $g_2=4cd^3\\ p\\biggl (\\frac{b}{d}\\biggr )+\\varepsilon d^2\\ p^{\\prime }\\biggl (\\frac{b}{d}\\biggr ), \\qquad \\ g_3=d^4\\ p\\biggl (\\frac{b}{d}\\biggr ).$ The conditions leading to this solution are essentially the ones guaranteeing the existence of at least one real solution of $p(x)=0$ .",
"The discriminant of our quartic equation is $\\Delta =256 \\alpha _0^3\\alpha _4^3-128 \\alpha _0^2 \\alpha _2^2 \\alpha _4^2+144 \\alpha _0^2\\alpha _2 \\alpha _3^2 \\alpha _4 -27 \\alpha _0^2 \\alpha _3^4+16 \\alpha _0 \\alpha _2^4\\alpha _4-4\\alpha _0 \\alpha _2^3 \\alpha _3^2.$ From [27], Theorem 7, we see that if $\\Delta <0$ there are always 2 real roots and 2 complex conjugate roots.",
"For $\\Delta =0$ we boil down to the case of degenerate solutions, studied apart in the main text.",
"Finally, if $\\Delta >0$ the only case with real roots is when the conditions $M\\equiv \\alpha _0 \\alpha _2<0$ and $N\\equiv 4\\alpha _0 \\alpha _4-\\alpha _2^2<0$ are satisfied.",
"If the case, then there are four real solutions.",
"More explicitly, since $\\alpha _0=\\frac{\\lambda }{12} Y^2>0$ , we have to consider the sign of $\\tilde{\\Delta }\\equiv \\frac{\\Delta }{16\\alpha _0}=-128 \\alpha _0^2 \\chi ^3-32 \\alpha _0\\Omega ^2 \\chi ^2-72\\alpha _0 \\Omega g^2\\kappa ^2 V^2 \\chi -27\\alpha _0 g^4 \\kappa ^4 V^4-2\\Omega ^4 \\chi -g^2\\kappa ^2V^2 \\Omega ^3,$ where $\\Omega =\\omega _0^2-g^2 V^2\\gamma ^2.$ Moreover, $\\frac{M}{\\alpha _0}&\\equiv \\alpha _2= \\Omega , \\\\N&\\equiv 4\\alpha _0\\alpha _4-\\alpha _2^2=-8\\alpha _0 \\chi -\\Omega ^2.$ Therefore, if $\\chi \\ge 0$ and $\\Omega >0$ , $\\tilde{\\Delta }<0$ we have always two real roots and two complex conjugate roots.",
"If $\\chi \\ge 0$ and $\\omega <0$ then $\\tilde{\\Delta }$ may have any sign but $M$ and $N$ are both negative, so we always have two or four real roots.",
"This was also evident from the fact that $p(0)=-2\\chi $ so, if $\\chi >0$ , we always have at least one real root (two if $\\chi >0$ or if $\\kappa \\ne 0$ when $\\chi =0$ .",
"For $\\chi =\\kappa =0$ , $x=0$ is a double root of $p$ and the discriminant vanishes).",
"When $\\chi $ is negative things are little bit more complicate.",
"In this case one has to study more carefully the sign of $\\tilde{\\Delta }$ .",
"When it is negative then we are done, while when it is positive then $\\Omega $ must be negative, providing the same condition $v>\\omega _0/g$ as for the case $\\chi =\\kappa =0$ .",
"In this case, we have also to impose the condition $N<0$ , which gives $g^2 v^2>\\omega _0^2+\\sqrt{8\\alpha _0 |\\chi |}$ , that is $V^2> \\frac{\\omega _0^2+|Y|\\sqrt{\\frac{2}{3} \\lambda |\\chi |}}{g^2+\\omega _0^2+|Y|\\sqrt{\\frac{2}{3} \\lambda |\\chi |}},$ which generalizes condition (REF ).",
"So, one is left with the study of the general conditions for which $\\tilde{\\Delta }$ have a specific sign when $\\chi $ is negative.",
"We will not pursue this here, but we limit ourselves to the following considerations.",
"We can look at $\\tilde{\\Delta }>0$ as a second order inequality in $\\alpha _0$ , recalling the physical constraint $\\alpha _0>0$ .",
"Since $\\chi $ is negative, this is always true if the discriminant is negative, while if it is positive we are led to the condition $\\alpha _0> \\max (0, \\alpha _+)$ , $\\alpha _+$ being the higher root of the quadric.",
"A complete classification of all possibilities is not difficult but quite cumbersome and out of the task of the present work."
]
] | 2012.05203 |
[
[
"Uncertainty Intervals for Graph-based Spatio-Temporal Traffic Prediction"
],
[
"Abstract Many traffic prediction applications rely on uncertainty estimates instead of the mean prediction.",
"Statistical traffic prediction literature has a complete subfield devoted to uncertainty modelling, but recent deep learning traffic prediction models either lack this feature or make specific assumptions that restrict its practicality.",
"We propose Quantile Graph Wavenet, a Spatio-Temporal neural network that is trained to estimate a density given the measurements of previous timesteps, conditioned on a quantile.",
"Our method of density estimation is fully parameterised by our neural network and does not use a likelihood approximation internally.",
"The quantile loss function is asymmetric and this makes it possible to model skewed densities.",
"This approach produces uncertainty estimates without the need to sample during inference, such as in Monte Carlo Dropout, which makes our method also efficient."
],
[
"Introduction",
"Currently, most traffic prediction models predict the average traffic conditions from minutes 5 up to an hour ahead.",
"While impressive, this problem setting largely ignores the uncertainty of the generated predictions, making the results more difficult to interpret.",
"Many real-world applications rely on confidence intervals or certainty bounds for these predictions, instead of the mean predicted value.",
"For example, when scheduling road maintenance or when planning a route with minimal delays.",
"Using Monte Carlo (MC) Dropout [1], it is possible to model the trips within a city, and utilize vehicle trajectories to predict future traffic speeds [2], [3].",
"This technique requires B stochastic forward passes to compute the sample variance.",
"Increasing the number of stochastic inference samples B improves the quality variance.",
"Monte Carlo (MC) Dropout makes the assumption that uncertainty in traffic speed and volume can be modelled as multivariate Gaussian.",
"The reality however, is that these distributions are skewed and asymmetric and thus do not satisfy the assumption made here.",
"For instance, when the traffic speed is measured to be maximum, it is far more likely to decrease than increase.",
"In this extended abstract, we introduce a novel traffic prediction model based on the spatio-temporal neural network Graph Wavenet [4].",
"However, instead of optimizing our mean prediction, we train using a Quantile loss function, similar to Autoregressive Implicit Quantile Networks [5].",
"This makes it possible to estimate the density at specific points, conditioned on a quantile.",
"Our method allows for non-gaussian uncertainty modelling, which remains greatly unexplored in the deep learning traffic prediction literature.",
"The contribution of this work can be summarized as follows: We introduce a method to approximate asymmetric and skewed density functions to model the uncertainty estimate.",
"Instead of approximating the variance using multiple forward passes, our technique requires only one pass for each requested density quantile."
],
[
"Quantile Regression",
"Quantile Regression is a method to estimate the quantile function of a distribution at chosen points, which is equal to the inverse cumulative distribution function (cdf).",
"It has been shown that when minimized using stochastic approximation, quantile regression converges to the true quantile function value [6].",
"This allows us to approximate a distribution using a neural network approximation of its quantile funtion, acting as reparameterization of a random sample from the uniform distribution.",
"Let us define the quantile regression loss $\\rho _\\tau (u) = (\\tau - \\mathbb {I}[u \\le 0]) u$ [6] for the error $u$ and the quantile $\\tau \\in \\mathcal {U}[0,1]$ .",
"When $u$ is positive $F$ underestimates $Z$ i.e.",
"the estimate falls short of the true value.",
"Now for a given scalar distribution $Z$ with cdf $F_Z$ and a quantile $\\tau $ we obtain the inverse cdf $F^{-1}_Z(\\tau ) = q$ , which minimizes the expected quantile regression loss $\\mathbb {E}_{z\\sim Z}[\\rho _\\tau (z - q)]$ .",
"Figure: 1-Wasserstein minimizing projection ∏ W \\prod _W onto N = 4uniformly weighted Diracs.",
"Shaded regions sum to form the1-Wasserstein error.",
"For detailed explanation, we refer the reader to ."
],
[
"Graph WaveNet",
"Graph WaveNet is neural network architecture for spatio-temporal traffic prediction.",
"Different from the AIRAI competition, this model assumes traffic measurements in the form of sensors on a road network graph.",
"The architecture consists of temporal causal convolutions (TCN) [8] with graph diffusion convolutions applied to every layer."
],
[
"Temporal Convolutions",
"Gating mechanisms are critical in recurrent neural networks and they have been shown to be powerful to control information flow through layers for temporal convolution networks as well.",
"The gating mechanism in Graph Wavenet is two parallel TCN layers configured with a gate: $\\mathbf {h}=g\\left(\\Theta _{1} \\star \\mathcal {X}+\\mathbf {b}\\right) \\odot \\sigma \\left(\\Theta _{2} \\star \\mathcal {X}+\\mathbf {c}\\right)$ Specifically, the LSTM-style gating $(g=\\text{tanh})$ also shared with PixelCNN [9] and WaveNet [10] is used.",
"Figure: Dilated casual convolution with kernel size 2 and dilation factor k, it picks inputs every k step and applies the standard1D convolution to the selected inputs.Another advantage is that the temporal receptive field grows exponentially w.r.t.",
"the number of layers and the dilation factor.",
"To achieve this, we artificially design the receptive field size of Graph WaveNet equals to the sequence length of the inputs so that in the last spatial-temporal layer the temporal dimension of the outputs exactly equals to one.",
"After that we set the number of output channels of the last layer as a factor of step length T to get our desired output dimension."
],
[
"Spatial Graph Diffusion Convolution",
"Graph Wavenet uses spatial graph convolution [11] to share information on the graph structure.",
"This type of graph convolution differs from the popularised GCN of Kipf and Welling [12], since it is based on epidemic diffusion [13]: $X_{:, c} \\star _{\\mathcal {G}} f_{\\theta }=\\sum _{k=0}^{K-1}\\left(\\theta _{k, 1}\\left(D_{O}^{-1} W\\right)^{k}+\\theta _{k, 2}\\left(D_{I}^{-1} W^{\\top }\\right)^{k}\\right) X_{:, c} \\quad \\text{ for } c \\in \\lbrace 1, \\cdots , C\\rbrace $ where $\\theta \\in \\mathbb {R}^{N\\times 2}$ are the filter parameters and $D_{O}^{-1} W, D_{I}^{-1} W^T$ represent the transition matrices of the diffusion process and the reverse one, respectively.",
"They note that since $T_{k+1} = (D^{-1}_o W) T_k(x)$ and $D^{-1}_o W$ is sparse, it is possible to use recursive sparse-dense matrix multiplication arriving at a time complexity $O(K|\\mathcal {E}|)$ of their update function, which is similar in complexity to the method proposed by Kipf and Welling.",
"However, unlike the method of Kipf and Welling, the edges between the nodes are non-symmetric.",
"Another advantage is that this method only uses a sparse graph neighbourhood in its update function, rather than the full graph Laplacian, and is therefore more resilient against structural changes."
],
[
"Loss Function",
"The training objective is the Mean Absolute Error (MAE) over Q prediction timesteps, N locations, each with C different measurements.",
"$L\\left(\\hat{\\mathbf {X}}^{(t+1):(t+Q)} ; \\mathbf {\\Theta }\\right)=\\frac{1}{Q N C} \\sum _{i=1}^{i=Q} \\sum _{j=1}^{j=N} \\sum _{k=1}^{k=C}\\left|\\hat{\\mathbf {X}}_{j k}^{(t+i)}-\\mathbf {X}_{j k}^{(t+i)}\\right|$"
],
[
"Quantile Graph WaveNet",
"We modify Graph WaveNet to implicitly predict the cdf, instead of optimizing for the mean prediction error.",
"This involves conditioning the input on a quantile $\\tau $ and training with a quantile loss function.",
"One drawback to the original quantile regression loss is that gradients do not scale with the magnitude of the error, but instead with the sign of the error and the quantile weight $\\tau $ [5].",
"The Huber quantile regression loss introduces a threshold $\\kappa $ , such that if the error is within the threshold $\\kappa $ , scaling is performed w.r.t.",
"the magnitude of the errors.",
"In our experiments we find that $\\kappa =0.05$ works well.",
"$\\rho ^\\kappa _\\tau (u) &= |\\tau - \\mathbb {I}\\lbrace u \\le 0\\rbrace | \\mathcal {L}_\\tau ^\\kappa (u),\\text{ with \\ }\\mathcal {L}_\\tau ^\\kappa (u) ={\\left\\lbrace \\begin{array}{ll}\\frac{1}{2\\kappa } u^2 & \\text{if } |u| < \\kappa \\\\|u| - \\frac{1}{2}\\kappa & \\text{otherwise}\\\\\\end{array}\\right.",
"}$"
],
[
"Data Preparation",
"We evaluate our model on traffic in Los Angelos (METR-LA [14]), Istanbul and Berlin[15].",
"The dataset METR-LA recorded 207 loop detectors in the metropolitan area of Los Angelos from March 1st, 2012 until June 30th, 2012.",
"We partition this timeline into 3 non-overlapped sections: training, validation and test with the respective ratios 7:1:2.",
"The Traffic4cast competition offers more complicated real-world datasets from the cities: Berlin, Istanbul and Moscow.",
"The data is presented as grid with the resolution of 495x436 pixels for each city, and every pixel consitutes 100m$^2$ .",
"The training and validation set contain 181 and 18 days, respectively.",
"In their main challenge it is expected from participants to make 500 predictions of up to 1h into the future (test set), spread over 163 days.",
"The Berlin and Istanbul dataset are an excellent opportunity to test our uncertainty prediction model on a larger scale.",
"However, Graph WaveNet expects a sensors on a graph, which is different from the 495x436x9 image in the competition.",
"In theory, one could consider every pixel as a node on the graph, however it is the case that most pixels report traffic measurements infrequently.",
"In contrast to METR-LA, where the percentage of operational sensors is typically $>90\\%$ , as seen in Table REF .",
"For practicality reasons, we sampled pixels from the 495x436x9 image with a value density of at least $\\frac{1}{16}$ in the outskirts and $\\frac{1}{2}$ towards the city centre.",
"From the remaining pixels we sample equidistant points (d = 1200m) on the traffic graph, in Berlin this yields a graph of approximately 1300 nodes.",
"Measurements of the average traffic speed are used at an interval of 5 minutes.",
"Figure: Extraction of equidistant measurement locations on the traffic network.Table: Traffic dataset statistics (after preprocessing)."
],
[
"Results",
"We evaluate the results of our model on the datasets METR-LA, ISTANBUL and BERLIN.",
"Since we do not have the complete test set of ISTANBUL and BERLIN, we instead have partitioned the 181 days training set into train, validation and test with the ratio: (0.89, 0.01, 0.10).",
"Our main results are the uncertainty estimates and their calibration, we compare this to MC Dropout uncertainty estimates.",
"Additionally, we also provide the mean prediction accuracy and compare this with previous methods.",
"Qualitative results To demonstrate that we can successfully learn uncertainty intervals that are skewed and asymmetric, we plot the 0.9 CI in Figure REF , for both Graph WaveNet with the proposed Quantile estimation and Graph WaveNet with MC Dropout.",
"We also project the mean prediction of the original measurements.",
"The locations of the sensors are visible in the map on the left.",
"Figure: Randomly selected sensors grouped by percentage of given measurements.A benefit of the Quantile uncertainty interval (lightgreen) is that each boundary or mean is generated by a distinct $\\tau $ value, on which the network is conditioned.",
"This is efficient to compute and greatly improves the flexibility of uncertainty intervals, which is expressed as skewed and asymmetric densities.",
"Calibrated uncertainty After training, we need to calibrate our model on the validation set.",
"Figure REF shows the uncertainty calibration of BERLIN: before calibration, after calibration on the validation set and on the test set.",
"Figure: Expected confidence vs observed cdf, before and after calibration on the validation set.Both Monte Carlo Dropout and Quantile Graph WaveNet require an additional calibration step, however the advantage of the latter is that we can re-map every tau value, as suggested by [16].",
"The underestimation that is present in many Bayesian Neural networks [16] is not present in the Quantile Graph WaveNet.",
"However, recalibration by reassigning the Quantiles is not perfect either since exploring the region outside of what is learned may cause the density to decrease rather than increase as wished.",
"We do assume the CDF to start with 0.0 and end with 1.0, but there is no mathematical law that requires this start- and endpoint.",
"In theory we can shrink and expand our space if needed, but retraining on these areas with remapped tau values may help prevent reversal of the density.",
"Mean prediction accuracy Lastly, we compare the mean prediction accuracies for the default Graph Wavenet and our proposed Quantile Graph Wavenet (denoted by Quantile est in table below).",
"Note that this is just to demonstrate the difference and this is not the main contribution of this research.",
"center Table: NO_CAPTION $\\dagger :$ The mean of 3 experiments.",
"$*:$ Historical weekly averages, predictions of missing values are counted as average target prediction error.",
"$\\ddag :$ Results of 1 experiment, the adaptive adjacency matrix is not enabled in the model.",
"We observe a decrease in the mean prediction accuracy if we compare Graph Wavenet with its Quantile estimation counterpart, and this is consistent for all datasets.",
"A possible explanation why Quantile Graph WaveNet performs worse on the Traffic4cast datasets may be due to more frequently missing measurements.",
"The versatility of Quantile Graph WaveNet can also be a weakness, using the same number of parameters to predict for any quantile, almost certainly reduces the complexity of the mean approximation.",
"For instance if the density to be learned is shaped differently from the mean.",
"Quantile Graph Wavenet predicts on a subset of the pixelspace used by the Traffic4cast competition and direct accuracy score comparison with other contestents should be avoided.",
"However, making a fair comparison is possible by masking the pixels not used in the graph and computing the MSE of the active pixels.",
"This table omits many recently proposed models and for a comprehensive study on traffic prediction models, we refer the reader to [17]."
],
[
"Discussion",
"The method presented here is a step towards reliable traffic prediction uncertainty estimates.",
"Initial results appear promising but we also want to highlight the limitations of our method.",
"Crossing quantiles: define two quantiles $\\tau _1$ and $\\tau _2$ , with $\\tau _1 < \\tau _2$ , then we should never have that $f(x, \\tau _1) < f(x, \\tau _2)$ .",
"Uncertainty missing for missing values: estimating a density when there are no values has no solution.",
"Additional calibration step is required after training.",
"Since uncertainty intervals are highly useful in traffic prediction applications, we believe that densities beyond the Gaussian should be considered the topic of future research.",
"We chose to model the density directly, following a technique developed in Reinforcement Learning, thereby omitting the likelihood computation all together.",
"In conjunction, more descriptive confidence metrics such as [18] and [19] could be developed with traffic prediction uncertainty applications in mind."
],
[
"Broader Impact",
"Improvements in traffic prediction have the potential to improve traffic conditions and reduce travel delays by facilitating better utilization of the available capacity.",
"Traffic uncertainty estimation allows better planning for scheduled road maintanance and uncertainty estimates are more informative when routing critical trips.",
"Work done partly during intership at HAL24K."
]
] | 2012.05207 |
[
[
"Applications of Mean Field Games in Financial Engineering and Economic\n Theory"
],
[
"Abstract This is an expanded version of the lecture given at the AMS Short Course on Mean Field Games, on January 13, 2020 in Denver CO.",
"The assignment was to discuss applications of Mean Field Games in finance and economics.",
"I need to admit upfront that several of the examples reviewed in this chapter were already discussed in book form.",
"Still, they are here accompanied with discussions of, and references to, works which appeared over the last three years.",
"Moreover, several completely new sections are added to show how recent developments in financial engineering and economics can benefit from being viewed through the lens of the Mean Field Game paradigm.",
"The new financial engineering applications deal with bitcoin mining and the energy markets, while the new economic applications concern models offering a smooth transition between macro-economics and finance, and contract theory."
],
[
"Introduction The goal of this chapter is to review applications in financial engineering and economics which can be cast and tackled within the framework of the Mean Field Game (often abbreviated as MFG in the sequel) paradigm.",
"Sections , and revisit some of the applications discussed in [21].",
"Here, we try to give some context, and examine developments which occurred since the publication of the book.",
"While we do not delve into proofs, we provide detailed references where the interested reader will be able to find complements on the subject matters.",
"This is clearly the case for Section reviewing applications to financial engineering.",
"Economic models often involve optimization over sets of actions, behaviors and strategies.",
"Engineers and biologists facing similar optimization problems would be likely to restrict themselves to finite action sets, but many economists prefer continuous action sets in order to be able to write first order conditions of optimality in differential forms, in hope of resolving them with explicit solutions.",
"Furthermore, they have no qualms about using models with a continuum of agents.",
"They explain that this guarantees an environment with perfect competition and tractable equilibrium behavior.",
"However, these formulations often raise eyebrows from colleagues, and especially mathematicians.",
"Our discussion of macro-economic models is motivated in part by the desire to reconcile the approach to competitive equilibria with a continuum of players with the paradigm of mean field games.",
"Additionally, while not all economists like to work with continuous time, it has been argued over the last two decades that models in macro-economic, contract theory, finance, $\\ldots $ , greatly benefit from the switch to continuous time from the more traditional discrete time models.",
"Yuli Sannikov is certainly one of the most visible crusaders in this respect, and we shall follow his lead and consider only continuous time models in this review chapter.",
"Like it is often the case in the published literature in economics, economies are modeled as populations with a continuum of players, and to capture the fact that the influence of any individual member on the aggregate should be infinitesimally small, these individuals are modeled as elements of a measure space with a continuous measure, say the unit interval $[0,1]$ with its Lebesgue's measure.",
"This starting point is different from the typical set-up of mean field games which usually starts from $N$ player games with $N$ finite, and considers the limit $N\\rightarrow \\infty $ .",
"We shall use both modeling paradigms, and refere the reader to [21] for a discussion of the links between the two.",
"None of the macro-economic papers which we review in this chapter mention the name Nash, or use the terminology Nash equilibrium.",
"The authors of these papers are concerned with general equilibria, not Nash equilibria.",
"They first assume that all the aggregate quantities in the economy are given, and they let the participants in the economy optimize their utilities independently of each other.",
"Next, they identify the constraints to be satisfied according to economic theory, compile them in a set which they call set of clearing conditions, and check that the results of the participants' optimizations are compatible with the clearing conditions.",
"As long as the clearing conditions are satisfied while the participants in the economy optimize their expected utilities, a general equilibrium is said to take place.",
"This procedure is very similar to the search for a Nash equilibrium.",
"Identifying the best response function amounts to having the participants optimize their expected utilities given the strategies of the other players, and given the constraints of economic theory, the search for a fixed point of the best response function appears as an analog of the check that the clearing conditions are satisfied.",
"The parallel is even more striking when the clearing conditions can be written in terms of aggregate quantities.",
"Indeed these aggregates quantify the interactions between the economy participants, and since these aggregates are most often nothing more than empirical means of state variables, they spell out mean field interactions in the model.",
"It is thus reasonable to cast these general equilibria as Nash equilibria for Mean Field Game models, and take full advantage of the technology already developed for the analysis of these game models to gain a better understanding of these general equilibria.",
"Acknowledgements: I would like to thank Markus Brunnermeier for relentless attempts at educating me on some of the subtleties of the economic models discussed in Section .",
"Also, I would like to acknowledge the useful comments and suggestions from an anonymous referee."
],
[
"Financial Engineering Applications"
],
[
"Systemic Risk The study of systemic risk is concerned with the identification and analysis of events or a sequences of events which could trigger severe instability, or even collapse of the financial system and the entire economy as a result.",
"In finance, systemic risk is approached differently than the risk associated with any one individual institution or portfolio.",
"In the US, it was first brought to the forefront by the Federal Reserve Bank of New York after the September 11, 2001 attack.",
"See [62].",
"The Central Bank spearheaded several collaborative research initiatives involving economists and academic researchers including engineers working on the safety of the electric grid, mathematicians experts in graph theory and network analysis, population biologists, .... .",
"But unfortunately, what brought this line of research in the limelight is the financial crisis of 2008 for which systemic risk was a major contributor.",
"While the mean field game models we discuss below give identical importance to the various institutions involved in the model, it is clear that a realistic model of systemic risk in finance should differentiate the roles of those companies considered to be too big to fail, and which have been identified officially under the name of SIFI for Systemically Important Financial Institutions.",
"They are the banks, insurance companies, or other financial institutions that U.S. federal regulators think would pose a serious risk to the economy if they were to collapse.",
"See [47] for an account of the state of the art after the financial crisis.",
"We shall briefly discuss extensions of the models reviewed below which could possibly account for the presence of such important players in the models.",
"Our first example is borrowed from [26].",
"We view its main merit as being pedagogical.",
"It gives us the opportunity to review some of the main features of Linear Quadratic (LQ) finite player game models, and in so doing, explain how one can solve them.",
"Moreover, despite its very simple structure, it exhibits several very useful characteristics: for each integer $N\\ge 1$ , the $N$ -player version of the game has a unique closed loop equilibrium, a unique open loop equilibrium, this open loop equilibrium is in closed loop form, but it is different from the closed loop equilibrium.",
"However, when $N\\rightarrow \\infty $ , both equilibria converge toward the unique equilibrium of the mean field game.",
"To describe the model, let us assume that the log-monetary reserves of $N$ banks, say $X^i_t, i=1,\\dots , N$ , satisfy $dX^i_t=\\left[a(\\overline{X}_t-X^i_t)+\\alpha ^i_t\\right]dt +\\sigma \\bigg (\\sqrt{1-\\rho ^2} dW^i_t+\\rho dW^0_t\\bigg ), \\quad i=1,\\ldots ,N$ where $W^i_t, i=0,1,\\dots , N$ are independent Brownian motions and $\\sigma >0$ is a constant.",
"The notation $\\bar{X}_t$ stands for the empirical mean of the $X^i_t$ for $i=1,\\ldots ,N$ , $a$ is a constant regulating the mean reversion of $X^i_t$ toward the mean, and $\\rho $ correlates the idiosyncratic shocks $dW^i_t$ and the common shock $dW^0_t$ .",
"So in this model, borrowing and lending is done through the drifts.",
"In fact, If $X^i_t$ is small (relative to the empirical mean $\\overline{X}_t$ ) then bank $i$ wants to borrow ($\\alpha ^i_t>0$ ) If $X^i_t$ is large then bank $i$ will want to lend ($\\alpha ^i_t<0$ ) The adapted stochastic process $\\alpha ^i=(\\alpha ^i_t)_{t\\ge 0}$ is the control strategy of bank $i$ which tries to minimize the quantity: $\\begin{split}&J^i(\\alpha ^1,\\ldots ,\\alpha ^N)\\\\&\\hspace{15.0pt}=\\mathbb {E} \\left\\lbrace \\int _0^T\\left[\\frac{1}{2}(\\alpha ^i_t)^2-q\\alpha ^i_t(\\overline{X}_t-X^i_t)+\\frac{\\epsilon }{2}(\\overline{X}_t-X^i_t)^2\\right]dt+\\frac{\\epsilon }{2}(\\overline{X}_T-X^i_T)^2\\right\\rbrace \\end{split}$ We could imagine that the quantity $q>0$ is chosen by the regulator to control the cost of borrowing and lending.",
"This model is a simple example of an $N$ - player stochastic differential game with Mean Field Interactions since the interactions are through the empirical means of the $N$ states, and a common noise."
],
[
"While it is usually very difficult to identify and compute Nash equilibria for finite player games, especially when the games are stochastic and dynamic, the very special Linear-Quadratic (LQ) nature of the model allows for explicit solutions.",
"Searching for an open loop Nash equilibrium $\\alpha =(\\alpha _t)_{0\\le t\\le T}$ for the game with $\\alpha _t=(\\alpha ^1_t,\\ldots ,\\alpha ^N_t)$ being adapted to the filtration generated by the Brownian motions can be done using the Pontryagin stochastic maximum principle.",
"In this particular model, one finds that the strategy profile $\\alpha $ defined by $\\alpha ^i_t=\\bigl [q+(1-\\frac{1}{N})\\eta _t\\bigr ](\\overline{X}_t-X^i_t)$ where the deterministic function $t\\mapsto \\eta _t$ solves the Riccati equation $\\dot{\\eta }_t=2\\bigl [a+q -\\frac{1}{N}q]\\eta _t +\\bigl (1-\\frac{1}{N}\\bigr )\\eta _t^2+q^2-\\epsilon $ with terminal condition $\\eta _T=c$ .",
"This equation is uniquely solvable if we assume $\\epsilon \\ge q^2$ , is the unique open loop Nash equilibrium.",
"Notice that $\\alpha _t$ is in closed loop / feedback form since it is of the form $\\alpha _t=\\phi _t({\\mathbf {X}}_t)$ for the deterministic function $(t,x)\\mapsto \\phi ^i_t(x)=\\bigl [q+(1-\\frac{1}{N})\\eta _t\\bigr ](\\overline{x}-x^i), \\qquad \\text{with}\\quad \\bar{x}=\\frac{1}{N}\\sum _{i=1}^N x^i,$ for $i=1,\\ldots ,N$ .",
"Note also that in equilibrium, the states of the banks satisfy: $dX^i_t= \\bigl [a+q+(1-\\frac{1}{N})\\eta _t \\bigr ] \\bigl (\\overline{X}_t-X^i_t\\bigr )dt +\\sigma \\rho dW^0_t+\\sigma \\sqrt{1-\\rho ^2}dW^i_t,$ for $i=1,\\ldots ,N$ .",
"So the states evolve according to an Ornstein-Ulhenbeck like process which is Gaussian if the initial conditions are Gaussian (or deterministic).",
"Searching for a closed loop Nash equilibrium $\\beta =(\\beta _t)_{0\\le t\\le T}$ with $\\beta _t=\\psi _t({\\mathbf {X}}_t)$ is usually done by deriving the Hamilton-Jacobi-Bellman equation for the system of value functions of the players, but in this particular instance, it can also be done using the Pontryagin stochastic maximum principle.",
"In any case, one finds that there exists a unique Nash equilibrium, and like in the open loop case, the equilibrium strategy profile is given by feedback functions $\\psi =(\\psi ^1,\\ldots ,\\psi ^N)$ given by the same formula (REF ) except for the fact that the deterministic function $t\\mapsto \\eta _t$ now solves a slightly different Riccati equation, namely: $\\dot{\\eta }_t=2(a+q)\\eta _t +\\bigl (1-\\frac{1}{N^2}\\bigr )\\eta _t^2+q^2-\\epsilon ,$ with the same terminal condition $\\eta _T=c$ .",
"The same remarks apply to the Ornstein-Ulhenbeck nature of the state evolutions in equilibrium.",
"However, what is remarkable is the fact that the open loop Nash equilibrium happens to be in closed loop form, but still, it is not a closed loop Nash equilibrium.",
"This is very different from the situation for plain optimization.",
"In fact, it reinforces the message that looking for a Nash equilibrium is a far cry from solving an optimization problem.",
"The second point we want to emphasize is that both Riccati equations, and hence both solutions, coincide in the limit $N\\rightarrow \\infty $ .",
"More on that below.",
"The interested reader can find complements and detailed proofs in [21].",
"Mean Field Games have been touted as the appropriate limits of $N$ player games when $N\\rightarrow \\infty $ .",
"However, beyond appealing intuitive arguments and rigorous proofs that MFG solutions can be used to construct strategy profiles forming approximate Nash equilibria for $N$ player games, the larger $N$ , the better the approximation, proving actual convergence is a difficult problem.",
"See for example D.Lacker chapter in this volume.",
"In the present situation, because of the explicit nature of the solutions of the finite player games, we can take the limit $N\\rightarrow \\infty $ in the dynamics of the states, in the Nash equilibrium controls (open and closed loop), and in the expected costs optimized by the players, all that despite the presence of the common noise.",
"In fact, we can read off the impact of the common noise in the limit $N\\rightarrow \\infty $ where the open and closed loop models coincide.",
"This limit can formally be identified to the so-called Mean Field Game model, and we can even identify the Master Equation from the large $N$ behavior of the system of HJB equations.",
"Being set up in continuous time, the above model is multi-period in nature.",
"This is in sharp contrast with most of the existing mathematical models of systemic risk which are most often cast as static one-period models.",
"Still, one of the most valuable feature of the model presented above is being explicitly solvable.",
"On the shortcoming side, I need to admit that this is a very naive model of bank lending and borrowing.",
"Among its undesirable features, the model does not have any provision for borrowers to repay their debts !",
"Moreover, despite explicit solutions, it gives only a small jab at the stability properties of the system.",
"Still, the model raises interesting challenges and it seems reasonable that it can be made more realistic and more useful with some mathematically tractable add-ons.",
"We cite three of them for the sake of illustration.",
"The introduction of major and minor players in this model will allow to capture the crucial role played by the SIFIs discussed in the introduction to this section.",
"Game models with major and minor players experienced a renewal of interest recently, and systemic risk seems to be a perfect testbed for their implementation.",
"In order to palliate some of the unrealistic features of the original model, Carmona, Fouque, Moussavi and Sun suggested the introduction of time delays in the controls in [25].",
"While increasing the technicalities of the proofs and precluding explicit solutions, this extension of the model includes provisions for borrowers to repay their debts in a fixed amount of time.",
"As another example of the fact that systemic risk is a fertile ground for the introduction and analysis of new MFG models, we mention the recent paper by Benazzoli, Ciampi and Di Persio [17] in which the authors study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity.",
"Also noteworthy is a recent paper [69] of Nadtochyi and Shkolnikov who study the mean field limit of systems of particles interacting through hitting times.",
"Their model was motivated by the contagion of the times of default of financial institutions in periods of economic stress.",
"It would be interesting to add the optimization component to their model and study the endogenously made decisions of the participants.",
"Finally, the introduction of graphical constraints should be a good way to quantify the various levels of exchanges between the financial institutions.",
"Introducing a weighted graph of interactions between the players of the game changes dramatically the mean field nature of the model, and new solution approaches will have to be worked out for any significant progress to be made in this direction.",
"Still, despite the obvious challenges of this research program, it is under active investigation by many financial engineers."
],
[
"High frequency markets offer another fertile ground for applications of financial engineering.",
"Among other things, they highlighted the importance of price impact and optimal execution.",
"The search for the best possible way to execute a given trade is an old problem and the presence of price impact did not have to wait for the popularity of the high frequency markets."
],
[
"We briefly review an MFG model of price impact introduced by Carmona and Lacker in [27].",
"There, it was solved in the weak formulation, but for the purpose of the present discussion the specific approach used to get to a solution will not really matter.",
"We start with a model for $N$ traders.",
"We denote by $X^i_t$ the inventory (i.e.",
"the number of shares owned) at time $t$ by trader $i$ , and we assume that this inventory evolves as an Itô process according to $dX^i_t=\\alpha ^i_t\\,dt+\\sigma ^i dW^i_t$ where $\\alpha ^i_t$ represents the rate of trading of trader $i$ .",
"This will be their control.",
"${\\mathbf {W}}^i=(W^i_t)_{t\\ge 0}$ are independent Wiener processes for $i=1,\\ldots ,N$ , and $\\sigma ^i$ represents an idiosyncratic volatility.",
"We assume that it is independent of $i$ for simplicity.",
"Note that in essentially all the papers on the subject this volatility is assumed to be 0.",
"In other words, inventories are assumed to be differentiable in time.",
"Our decision to work with $\\sigma ^i=\\sigma >0$ is backed by empirical evidence, at least in the high frequency markets, as demonstrated by Carmona and Webster in [30].",
"Next, we denote by $K^i_t$ the amount of cash held by trader $i$ at time $t$ .",
"We have: $dK^i_t=-[\\alpha ^i_t S_t +c(\\alpha ^i_t)]\\, dt,$ where $S_t$ is the transaction price of one share at time $t$ , and the function $\\alpha \\rightarrow c(\\alpha )\\ge 0$ models the cost for trading at rate $\\alpha $ .",
"As explained in [30], this function $c$ should be thought of as the Legendre transform of the shape of the order book.",
"So for a flat order book we should have: $c(\\alpha )=c\\alpha ^2.$ We model the time evolution of the price $S_t$ using the natural extension to the case of $N$ traders of the price impact formula of Almgren and Chriss [7]: $dS_t=\\frac{1}{N}\\sum _{i=1}^N h(\\alpha ^i_t)\\, dt +\\sigma _0 dW^0_t$ for some non-negative increasing function $\\alpha \\mapsto h(\\alpha )$ and a Wiener process ${\\mathbf {W}}^0=(W^0_t)_{t\\ge }$ independent of the other ones.",
"In this model, the wealth $V_t^i$ of trader $i$ at time $t$ is given by the sum of his holdings in cash and the value of his holdings in the stock, as marked at the current value of the stock: $V^i_t=K^i_t+X^i_tS_t.$ Using the standard self-financing condition and Itô's formula we see that: $\\begin{split}dV_t^i &= dK^i_t +X^i_t\\,dS_t + S_t \\,dX^i_t\\\\&=\\bigg [-c(\\alpha _t^i) + X_t^i\\frac{1}{N} \\sum _{j=1}^N h(\\alpha _t^j)\\bigg ] dt + \\sigma S_tdW_t^i+ \\sigma _0 X_t^i dW^0_t,\\end{split}$ so if player $i$ minimizes their expected trading costs $J^i(\\alpha ^1, ..., \\alpha ^N )= \\mathbb {E} \\bigg [ \\int _0^T c_X(X_t^i) dt + g(X_T^i) - V_T^i\\bigg ]$ where $x\\mapsto c_X(x)$ represents the cost of holding an inventory of sixe $x$ and $g(x)$ a form of terminal inventory cost.",
"Using (REF ), we can rewrite these expected costs as; $J^i(\\alpha ^1, ..., \\alpha ^N )= \\mathbb {E} \\bigg [ \\int _0^T f(t,X^i_t,\\bar{\\nu }^N_t,\\alpha ^i_t)dt + g(X^i_T)\\bigg ]$ if $\\bar{\\nu }^N_t$ denotes the empirical distribution of $\\alpha ^1_t$ , $\\ldots $ , $\\alpha ^N_t$ , and the function $f$ is defined by $f(t,x,\\nu ,\\alpha )=c(\\alpha )+c_X(x)-x\\int h \\,d\\nu $ .",
"Remark 2.1 Several important remarks are in order at this stage.",
"The above model is the epitome of a $N$ -player stochastic differential game.",
"The state equations are given by (REF ).",
"They are driven by the $N$ independent idiosyncratic noise term $dW^i_t$ .",
"The common noise ${\\mathbf {W}}^0$ disappeared from the optimization problem only because of the risk neutrality of the traders, namely the fact that they minimize the expectations of their costs.",
"Should they choose to minimize a nonlinear utility function, the common noise would not disappear !",
"Another remarkable property of this model, and one of the reasons its analysis was of great interest, is that it is one of the earliest MFG models for which the mean field interaction occurs naturally through the controls.",
"An early analysis of these MFGs within the probabilistic approach was given in [21].",
"While the model formulated above is for $N$ traders, the mean field formulation $N\\rightarrow \\infty $ is clear.",
"A complete solution of this limit MFG in the weak formulation can be found in [27] and in [21] for the strong formulation.",
"Later on, it was revisited by Cardaliaguet and Lehalle in [18] where the authors consider agents' possible heterogeneities, and the introduction of fictitious plays which gives a learning twist to the model.",
"It was also extended by Cartea and Jaimungal in [32], to formulate an optimal execution problem for which the authors could still provide solution formulas in quasi explicit forms.",
"Game models for optimal execution in the presence of price impact did not wait for the theory of MFGs to catch the interest of financial economists and financial engineers.",
"The interested reader may want to check the papers [13], [19], and [31] for games models shedding light on predatory trading."
],
[
"Models for Bank Runs & Mean Field Games of Timing I came across the potential application of Mean Field Games to the important problem of bank runs by attending Jean Charles Rochet's lectures at the Vancouver Systemic Risk Summer School in July 2014, and the talk given by Olivier Gossner during the conference following the summer school.",
"Both works [71], [46] are reported in detail in [21], [22].",
"Here, we only review the second one because it fits better in the class of continuous time models on which we concentrate in this chapter.",
"In the spirit of the discussion to follow, it is worth mentioning the works of Morris and Shin [68] and He and Xiong [56].",
"Like most economists, they use a model of the economy with a continuum of players based on an atomless measure space.",
"Our goal is to recover their models starting with finitely many players and then, analyze the mean field limit."
],
[
"Following Gossner's talk mentioned earlier, we consider a group of $N$ depositors with individual initial deposits in the amount $D^i_0=1/N$ for $i=1,\\ldots , N$ .",
"They are promised a rate of return $\\overline{r}>r$ where $r$ is the current prevailing interest rate.",
"We assume that the value $Y_t$ of the assets of the bank at time $t$ follows an Itô process and that $Y_0\\ge 1$ .",
"We also assume the existence of a deterministic function $y\\mapsto L(y)$ giving the liquidation value of the bank assets when $Y_t=y$ .",
"One can imagine that the bank has a rate $\\overline{r}$ credit line of size $L(Y_t)$ at time $t$ , and that the bank uses this credit line each time a depositor runs (withdraws their deposit).",
"Next, we assume that the assets mature at time $T$ , and that no transaction takes place after that.",
"If at that time $Y_T\\ge 1$ , every one is paid in full, but if $Y_T<1$ we treat this case as an exogenous default.",
"We talk about an endogenous default at time $t$ if depositors try to withdraw more than $L(Y_t)$ at that time.",
"As time passes, each depositor has access to a private signal $X^i_t$ satisfying: $dX^i_t = dY_t + \\sigma dW^i_t,\\qquad i=1,\\ldots ,N,$ and at a time $\\tau ^i$ of their choosing, they can attempt to withdraw their deposit, de facto collecting the return $\\overline{r}$ until time $\\tau ^i$ , and trying to maximize: $J^i(\\tau ^{1},\\ldots ,\\tau ^{N})=\\mathbb {E} \\bigg [g(\\tau ^i,Y_{\\tau ^i},\\tau ^{-i})\\bigg ]$ where we use the standard notation $\\tau ^{-i}=(\\tau ^1,\\ldots ,\\tau ^{i-1},\\tau ^{i+1},\\ldots ,\\tau ^N)$ , and for example $g(t,Y_t,\\tau ^1,\\ldots ,\\tau ^N)=e^{(\\overline{r}-r)t\\wedge \\tau } + e^{-rt\\wedge \\tau }(L(Y_t)-N_t/N)^+\\wedge \\frac{1}{N},$ $N_t=\\#\\lbrace i;\\; \\tau ^i\\le t\\rbrace $ is number of withdrawals before $t$ , and $\\tau =\\inf \\lbrace t;\\;L(Y_t)<N_t/N\\rbrace $ is the first time the bank cannot withstand the withdrawal requests.",
"First, let us try to derive some conclusions if the depositors had full information, in which case $Y_t$ would be public knowledge, and $\\sigma $ would be 0, i.e.$\\sigma =0$ .",
"If we also assume that the function $y\\hookrightarrow L(y)$ is known to the depositors, then it is easy to check that in any equilibrium: $\\tau ^i=\\inf \\lbrace t;\\; L(Y_t)\\le 1\\rbrace .$ So all the depositors withdraw at the same time (they all run on the bank simultaneously) and each depositor gets their deposit back: no one gets hurt!.",
"Clearly this scenario is unfortunately, highly unrealistic.",
"We should expect that depositors wait longer before running on the bank, presumably because they only have imperfect information (i.e.",
"noisy private signals) on the health of the bank.",
"Let us consider a population of $N$ players with individual states $X^{N,i}_t$ at time $t$ satisfying equations of the form $dX^{N,i}_t=b(t,X^{N,i}_t,\\overline{\\nu }^N_t)dt+\\sigma (t,X^{N,i}_t) dW^{i}_t,\\qquad i=1,\\ldots ,N$ coupled through their empirical distribution $\\overline{\\nu }^N_t=\\frac{1}{N}\\sum _{i=1}^N\\delta _{X^{N,i}_t}.$ Each player chooses a $\\mathcal {F} ^{X^i}$ - stopping time $\\tau ^i$ and tries to maximize $J^i(\\tau ^{1},\\ldots ,\\tau ^{N})=\\mathbb {E} \\bigg [g(\\tau ^i,X_{\\tau ^i},\\overline{\\mu }^N([0,\\tau ^i])\\bigg ]$ where $\\overline{\\mu }=\\frac{1}{N}\\sum _{i=1}^N\\delta _{\\tau ^i}$ is the empirical distribution of the $\\tau ^i$ 's, $g(t,x,p)$ is the reward to a player for exercising their timing decision at time $t$ when their private signal is $X^i_t=x$ , and the proportion of players who already exercised their right is $p$ .",
"Taking formally the limit $N\\rightarrow \\infty $ in this set-up, we obtain the following MFG formulation of a mean field game of timing.",
"Assuming that the drift is independent of the empirical distribution of the states for the sake of simplicity, i.e.",
"$b(t,x,\\nu )=b(t,x)$ the dynamics of the state of a generic player are given by an Itô equation of the form: $dX_t = b(t,X_t)dt + \\sigma (t,X_t) dW_t.$ We denote by $\\mathbb {F} ^X=(\\mathcal {F} ^X_t)_{0\\le t\\le T}$ the information available to the agent at time $t$ , and by $\\mathcal {S} ^X$ the set of $\\mathbb {F} ^X$ -stopping times.",
"The MFG of timing paradigm can then be formulated as follows: Best Response Optimization: for each fixed environment $\\mu \\in \\mathcal {P} ([0,T])$ solve $\\hat{\\theta }\\in \\text{arg}\\sup _{\\theta \\in \\mathcal {S} ^X, \\theta \\le T}\\mathbb {E} [g(\\theta ,X_\\theta , \\mu ([0,\\theta ]))]$ Fixed Point Step: find $\\mu $ so that $\\mu ([0,t])=\\mathbb {P} [\\hat{\\theta }\\le t]$ Here and throughout, we denote by $\\mathcal {P} (A)$ the space of probability measures on $A$ .",
"In an unpublished PhD thesis, Geoffrey Zhu proposed an existence proof for randomized stopping times, providing an analog of Nash's original existence theorem for the existence of equilibria in mixed strategies.",
"Before we go any further, recall the sobering shortcoming of convergence in distribution which says that even if ${\\left\\lbrace \\begin{array}{ll}&\\lim _{n\\rightarrow \\infty } (X,Y_n) = (X,Y) \\quad \\text{in law}\\\\& Y_n\\;\\text{ is a function of }\\; X\\end{array}\\right.",
"}$ then $Y$ is not necessarily a function of $X$ , in other words, $Y\\in \\sigma \\lbrace X\\rbrace $ may not hold.",
"For the purpose of this existence proof, let us assume that the reward function $g:[0,T]\\times \\mathbb {R} \\times \\mathcal {P} ([0,T]) \\ni (t,x,\\mu )\\mapsto g(t,x,\\mu )\\in \\mathbb {R} $ is bounded, continuous in $(t,x)$ for $\\mu $ fixed, and Lipschitz continuous in $\\mu $ for $(t,x)$ fixed.",
"Note that, unfortunately, this last assumption is not satisfied for functions $t\\hookrightarrow \\mu ([0,t])$ , unless $t\\in \\mathbb {T} \\subset [0,T]$ with $\\mathbb {T} $ finite!",
"This will prevent us from using this existence result in the model of bank run discussed earlier.",
"In any case, under the present assumptions $\\Pi :\\mathcal {P} ([0,T])\\times \\mathcal {P} (C([0,T]\\times [0,T])) \\mapsto \\mathbb {R} $ $(\\mu ,\\xi ) \\hookrightarrow \\Pi (\\nu ,\\xi )=\\int _{C([0,T])\\times [0,T]} g(t,x_t,\\mu )\\xi (dx,dt)$ is continuous, and since the space $\\tilde{\\mathcal {S} }$ of randomized stopping times is compact because of an old result of Baxter and Chacon, Berge's maximum theorem implies that the multivalued function $\\mathcal {P} ([0,T])\\ni \\nu \\hookrightarrow \\text{arg}\\sup _{\\xi \\in \\tilde{\\mathcal {S} }}\\Pi (\\nu ,\\xi )$ is upper hemi-continuous and compact-valued.",
"Followed by the projection on the first marginal, it is still upper hemi-continuous and compact-valued, and Kakutani's fixed point theorem implies the desired existence result.",
"Existence for standard stopping times can be shown to hold under a different set of assumptions, using for example the order structure of the space of stopping times instead of topological properties of this space.",
"For example, if we assume that the time increments of $g$ are monotone in $\\nu $ , then we can use the fact that the space of stopping times is a complete lattice, check that $\\tau \\hookrightarrow \\text{arg}\\sup _{\\tau ^{\\prime }\\in \\mathcal {S} }\\mathbb {E} [g(\\tau ^{\\prime },X_{\\tau ^{\\prime }},F_\\tau (\\tau ^{\\prime }))]$ is monotone, and use Tarski's fixed point theorem.",
"Here $F_\\tau (t)=\\mathbb {P} [\\tau \\le t]$ is the cumulative distribution function of $\\tau $ .",
"Unfortunately, once more, this existence result does not apply to the model of bank run discussed earlier.",
"Beyond a simple example presented by M. Nutz in [70], the solution in the general bank run set-up introduced earlier is much more difficult and technical than originally thought.",
"A complete solution was given by Carmona, Delarue and Lacker in [23].",
"See also [11] by C. Bertucci for an approach relying purely on partial differential equations and quasi-variational inequalities, and [48] by Bouveret, Dumitrescu and Tankov for more on the use of relaxed stopping times."
],
[
"Given the fact that the Bitcoin mania hits the financial markets on a recurrent basis, and because of the competitive nature of the mining process involving a large number or miners, it is no surprise that mean field game models have been proposed to analyze some of the features of the cryptocurrency space.",
"Here we briefly review two very recent papers by Li, Reppen and Sircar [65] and Bertucci, Bertucci, Lasry and Lions [12] which use continuous time mean field games, though in very different ways, to analyze some of the features of cryptocurrency generation.",
"Both papers envision a continuum of miners interacting through the aggregate computational power they allocate to mining the blockchain.",
"Even though it is not the only cryptocurrency, we shall only talk about Bitcoin because it is the one getting the most press.",
"There are many reasons for that, wild price moves being definitely one of them.",
"After briefly exceeding $12,000, it crashed down to the $10,000 range, before quickly moving up again to high levels.",
"The generation of bitcoins is based on blockchain technology.",
"The latter was introduced for the purpose of record keeping in a decentralized ledger.",
"Still, it is at the core of bitcoin generation.",
"In bitcoin production, independent miners compete for the right to record the next transaction block on the blockchain.",
"They follow a Proof-of-Work (PoW) protocol.",
"Their goal is to solve mathematical puzzles designed to be solved by brute force only.",
"Computations to solve puzzles (create a block and earn a reward) are otherwise totally useless as they are not applicable anywhere else.",
"Once a miner obtains a solution, the corresponding block is added on top of the blockchain and the miner obtains their reward.",
"This reward is paid out in cryptocurrency (a fixed number of bitcoins, currently $12.50$ bitcoins for adding a block) while electricity and mining hardware need to be paid with traditional fiat currency (like the US Dollar).",
"The supply of bitcoins is constantly growing.",
"However, it is limited to 21 million, of which more than 17 million are already in circulation.",
"The security of the network is a serious issue.",
"A major fear is a majority attack also called $51\\%$ attack when a group of users controls the majority of mining power.",
"These instances are rare, mostly because they are very difficult to realize due to their enormous costs.",
"They are not considered in [65], [12].",
"The computing power devoted to mining is called the hash rate.",
"It captures the number of trials per second trying to solve the mining puzzle.",
"In order to maintain stability in the blockchain, the mining puzzle difficulty is dynamically adjusted so that, on average, the time between the creation of two consecutive blocks is constant, currently approximately 10 minutes.",
"Therefore as the hash rate increases, the difficulty increases so that it is required to compute more hashes for a given block.",
"Miners control their hash rate to increases their share of the blockchain reward, all other things being equal, while reducing the share of the other miners.",
"On the other hand, intensive computations consume a lot of energy and each miner faces significant electricity costs.",
"In a nutshell, this dilemma is the core of the individual miner optimization problem.",
"The aggregate hash rate in the system represents the total computational power devoted to block creation.",
"In both papers this aggregate hash rate will be the source of the mean field interaction between the miners.",
"In [65], Li, Reppen, and Sircar focus on the risk borne by risk-averse miners and study mining concentration.",
"They use a jump process to represent the acquisition of the reward, the jump intensity being the control of the typical miner.",
"In their model, the jump intensity reflects the computer power, or hash rate, devoted to the effort, and the mean of the controls of the individual miners is what creates the mean field interaction in the model.",
"Given our simplistic description of how bitcoins are generated, it is natural to assume that the miner’s probability of receiving the next mining reward is proportional to the ratio of their hash rate to that of the population.",
"The number of rewards each miner receives is modeled by a counting process $N_t$ with jump intensity $\\lambda _t>0$ .",
"If the number of miners is $M+1$ , this intensity is given by $\\lambda _t=\\frac{\\alpha _t}{D(\\alpha _t+M\\bar{\\alpha }_t)}$ where $M\\bar{\\alpha }_t$ approximates the total hash rate of the other miners.",
"The wealth of the miner is used as state variable.",
"It evolves as an Itô process of the form $dX_t=-c\\alpha _t dt+PdN_t$ where $P$ is the bitcoin price, so the value of each reward is the product of its quantity by $P$ .",
"As we mentioned earlier, $12.50$ bitcoins are granted to a miner for adding a block successfully.",
"The total number of rewards in the system as a whole is a Poisson process with a constant intensity $D > 0$ .",
"This will play the role of a common noise in the model.",
"Given an adapted process $\\bar{\\alpha }=(\\bar{\\alpha }_t)_{t\\ge 0}$ representing the conditional mean of the controls given the common noise, the miner optimization problem is to maximize the expected utility of wealth at a fixed terminal time $T$ : $v^{\\bar{\\alpha }}(t,x) = \\sup _{\\alpha \\in [0,A(x)]}\\mathbb {E} \\bigl [U(X_T)\\,|\\, X_t=x\\bigr ]$ where the controls $\\alpha $ are restricted to the interval $[0,A(x)]$ when the state is $x$ .",
"The authors tackle the optimization problem by solving the HJB equation: $\\partial _tv^{\\bar{\\alpha }} + \\sup _{\\alpha \\in [0,A(x)]}\\Bigl (-c\\alpha \\partial _x v^{\\bar{\\alpha }} +\\frac{\\alpha }{D(\\alpha +M\\bar{\\alpha }_t)}\\Delta v^{\\bar{\\alpha }}\\Bigr )=0$ and the solution is completed by solving the fixed point equation for the average effort rate $\\bar{\\alpha }$ .",
"The authors provide an explicit solution for exponential utility and no liquidity constraint.",
"They go on to the analysis of the effect of liquidity constraints and more general utility functions.",
"They provide robust numerical procedures to compute the equilibrium.",
"In the case of the CRRA power utility function, they study the concentration of wealth among the miners.",
"Their conclusion is that the richer the miner, the wealthier they will get.",
"The authors also introduce a model in which one special miner is singled out for having a significant cost advantage (e.g.",
"benefitting from cheaper electricity prices) over the remaining field of miners.",
"Naturally this special miner is shown to contribute more to the hash rate.",
"Bertucci, Bertucci, Lasry and Lions use a different modeling approach.",
"Very much in the spirit of the work [64] proposing a mean field approach to the dynamics of an order book in the high frequency markets, they directly introduce the master equation, arguing that this is the best way to study mean field games.",
"Notice that this is in sharp contrast with the usual approach starting from the introduction of the agents maximization problems.",
"Using their notation system, $P$ is now the nominal hashrate, (number of hashes per second).",
"They define the real hashrate $K_t=e^{-\\delta t}P_t$ arguing that the rate $\\delta $ quantifies technological progress.",
"The evolution of $K_t$ is modeled in continuous time as miners continuously acquire computing power to compute hashes.",
"As we explained earlier, the blockchain outputs a fixed number of coins per unit of time, so the miners compete against each other to earn a share of this fixed output.",
"From the above description, it is reasonable to assume that the share they get is proportional to their relative share of the total computational power.",
"The authors posit dynamics of the real hash rate in the form: $d K_t=-\\delta K_t dt +\\lambda U_t(K_t) dt,\\qquad K_0=K$ where $U_t$ represents the flow of entry of computing devices, or in other words, the value of a unit of real hash rate.",
"They introduce the relationship: $0=-(r+\\delta )U+(-\\delta K +\\lambda )\\partial _K U +(K+\\epsilon )^{-1}-c.$ This Partial Differential Equation (PDE) should be viewed as the master equation of the competitive mean field game with finitely many states.",
"The players are the miners, and we should think of $K$ as a measure of an aggregate of the population of miners responsible for the mean field interaction.",
"If we now assume that the reward is of the form $g(P_t)/(K_t+\\epsilon )$ where $g$ is a smooth positive function of $P_t$ which evolves according to ${\\left\\lbrace \\begin{array}{ll}dP_t&=\\alpha (P_t)dt +\\sqrt{2\\nu }dW_t, \\qquad P_0=P\\\\d K_t&=-\\delta K_t dt +\\lambda U_t(K_t,P_t) dt,\\qquad K_0=K\\end{array}\\right.",
"}$ where the function $\\alpha $ is Lipschitz, $\\nu >0$ and ${\\mathbf {W}}=(W_t)_{t\\ge 0}$ is a standard Wiener process, then $U(K,P)=\\int _0^\\infty e^{-(r+\\delta )t}\\Bigl (\\frac{g(P_t)}{\\epsilon +K_t}-c\\Bigr ) dt$ is the value function of one unit of real hashrate.",
"In this model, $P_t$ captures the exchange rate between the value of the cryptocurrency and fiat money and ${\\mathbf {W}}=(W_t)_{t\\ge 0}$ is a form of common noise.",
"In this case the master equation on $[0,\\infty )\\times \\mathbb {R} $ reads: $0=-(r+\\delta )U+(-\\delta K +\\lambda U)\\partial _K U +\\alpha \\partial _PU +\\nu \\partial ^2_{PP}U+\\frac{g(P)}{K+\\epsilon }-c$ This is the master equation of a MFG with finite state space and common noise.",
"Again, the monotone structure of the MFG plays a key role in the well-posedness of these models.",
"Indeed, existence and uniqueness follow from monotonicity, and the existence of a stationary state is also proven and analyzed.",
"This is proven in [12] when $g$ is bounded from above and below.",
"[12] also discusses model security against attacks, proposes extensions to several competing populations of miners facing different mining costs, and a market where mining equipment can be bought and sold."
],
[
"Game Models for Energy and the Environment In this section we review some of the MFG models which have been touted and used to revisit and extend earlier economic analyses of energy and environment markets.",
"We first summarize the discussion given in [21] of a first model proposed by Guéant, Lasry and Lions in [54]."
],
[
"If we denote by $x^1_0,\\ldots , x^N_0$ the initial reserves of $N$ oil producers who control their own rates of production, and if we denote by $X^i_t$ the oil reserves of producer $i$ at time $t$ , the changes in reserves should be given by equations of the form $dX^i_t=-\\alpha ^i_tdt+\\sigma X^i_t dW^i_t$ where $\\sigma >0$ is a volatility level common to all the producers, the non-negative adapted and square integrable processes $\\alpha ^i=(\\alpha ^i_t)_{t\\ge 0}$ being the controls exerted by the producers, and the ${\\mathbf {W}}^i=(W^i_t)_{t\\ge 0}$ independent standard Wiener processes.",
"If we denote by $P_t$ the price of one barrel of oil at time $t$ , and if we denote by $C(\\alpha )=\\frac{b}{2} \\alpha ^2 + a\\alpha $ the cost of producing $\\alpha $ barrels of oil, then producer $i$ tries to maximize: $J^i(\\alpha ^1,\\ldots ,\\alpha ^N)=\\sup _{(\\alpha _t)_{t\\ge 0}, \\alpha _t\\ge 0}\\mathbb {E} \\bigg [\\int _0^\\infty e^{-rt}[\\alpha ^i_t P_t - C(\\alpha ^i_t)] dt\\bigg ],$ where $r>0$ is a discount factor.",
"As we are about to explain, the price $P_t$ is the source of coupling between the producers' strategies.",
"The notion of general equilibrium is intended to characterize situations in which all the producers manage to maximize their profits simultaneously, and the market clears in the sense that demand matches supply.",
"Let us denote by $D(t,p)$ the demand at time $t$ when the price is $p$ .",
"The function $D(t,p)=we^{\\rho t}p^{-\\gamma }$ was used in [54].",
"We use the obvious notation $D^{-1}$ for the inverse demand function, i.e.",
"$q=D(t,p)\\Longleftrightarrow p=D^{-1}(t,q)$ ."
],
[
"In the present context, the MFG paradigm can be articulated easily.",
"For each fixed deterministic flow $(\\mu _t)_{t\\ge 0}$ of probability measures, we compute the price $P_t$ from the formula: $P_t=D^{-1}\\bigg (t,-\\frac{d}{dt}\\int x\\mu _t(dx)\\bigg ),$ and the best response to this flow of distributions is given by the solution of the discounted infinite horizon optimal control problem for the instantaneous cost function $f(t,x,\\mu ,\\alpha )=[\\alpha p - C(\\alpha )]e^{-rt}$ under the dynamic constraint (REF ).",
"The MFG will be solved if one can find a measure flow $(\\mu _t)_{t\\ge 0}$ such that the marginal distribution $\\mathcal {L} (X_t)$ of the solution of the control problem matches the flow we started from, namely $\\mu _t=\\mathcal {L} (X_t)$ for all $t\\ge 0$ .",
"The analytic approach to MFGs based on the solution of a system of coupled HJB and Fokker-Planck-Kolmogorov equations is used in [54] to give a solution to this problem.",
"Numerical illustrations provide comparative statics of the solutions are also given.",
"In [36], Chan and Sircar propose to look at dynamics $dX_s=-\\alpha _s ds +\\sigma dW_s,\\qquad X_t=x>0.$ with a Dirichlet boundary condition at $X_t=0$ to guarantee that the reserves of a generic oil producer do not become negative.",
"As before, $\\alpha _t$ represents the rate of production of a generic producer and $X_t$ represents the remaining reserves.",
"As in most models for Cournot games, the price experienced by the producer, call it $P_t$ for the sake of definiteness, is given by a linear inverse demand function of the rates of production, and it is chosen to be of the form $P_t=1-\\alpha _t -\\epsilon \\bar{\\alpha }_t$ where $\\bar{\\alpha }_t$ is the mean production rate for all the exhaustible resources.",
"so that the cost function becomes $f(t,x,\\mu ,\\alpha )=\\alpha [1-\\alpha p - \\epsilon \\overline{\\alpha }]$ where $\\mu $ denotes the distribution of the control $\\alpha $ giving the rate of production, $\\bar{\\alpha }$ the mean of this distribution, and $p$ the price given by the above inverse demand function.",
"This is a typical extended MFG (because the mean field interaction is through the controls) with a boundary condition to guarantee that the remaining reserves do not become negative.",
"In parallel, the authors propose a slightly modified model for producers of renewable energy and analyze the oil market in the presence of both populations of producers.",
"In this paper, they also propose several variations on the above model.",
"Their goal was to include several realistic features like strategic blockading the entrance of renewable producers, and exploration and discovery of new reserves.",
"While not always worrying about all the subtleties of mathematical existence theorems, they provide enlightening numerical illustrations of their conclusions.",
"This prompted more mathematically inclined authors like Bensoussan and Graber to pursue in [52] a complete existence analysis based on partial differential equations techniques of the models proposed by Sircar and Chan.",
"For the sake of completeness, we mention that plain models for a macro perspective on the behavior of a large population of oil producers were proposed by Giraud, Guéant, Lasry, and Lions.",
"See for example [49], [53], [54].",
"More recently, Achdou, Giraud, Lasry and Lions revisited some of these models including the presence of major and minor players.",
"See [2].",
"Also, note that game theoretical approaches, though not involving mean field games per se, were used by Ludkovski and Sircar in [66] to analyze oil production."
],
[
"General equilibrium models have a long history in the engineering literature on electricity pricing.",
"See for example [57] and the references therein.",
"More recently, Djehiche, Barreiro-Gomez and Tembine proposed a mean field game model for pricing electricity in a smart grid.",
"See [41].",
"Still, to model individual decision in a smart grid, Alasseur, Ben Tahar and Matoussi use in [6] a game with mean field interactions through the controls as a framework to manage storage.",
"In [5], Aïd, Dumitrescu, and Tankov use one of the mean field game of timing models reviewed earlier to capture the time at which renewable producers choose to enter a new market, and when conventional producers using fossil fuels should exit the market.",
"In a different context, Aïd, Basei and Pham investigate in [4] a Stackelberg game model where the leader (an electricity producing firm) and the follower (consumer) choose strategies possibly depending upon their distributions.",
"So they solve optimal control problems of the McKean-Vlasov type.",
"The main emphasis of the paper is to show that the Stackelberg equilibrium is not Pareto optimal, and to explain the economic consequences of this disparity.",
"Investigating the valuation of demand response contracts in a model with a continuum of consumers with mean field interactions and the presence of a common noise impacting their consumptions, Elie, Hubert, Mastrolia, and Possamaï formulate in [43] the problem as a contract theory problem with moral hazard as those we shall discuss in more detail in Section below.",
"In their model, the Principal is an electricity producer who observes continuously the consumption of a continuum of risk averse consumers, and designs contracts in order to reduce the production costs.",
"To be more specific, the producer incentivizes the consumers to reduce the average consumptions as well as their volatilities in different regimes, without observing the efforts they potentially make.",
"This is exactly the type of models we shall investigate in Section .",
"The recent paper by Shrivats, Firoozi, and Jaimungal [74], still in the context of the electricity markets, offers a smooth transition with the next discussion of the environment markets.",
"Indeed, it proposes MFG models to derive the optimal behavior of electricity producers and an equilibrium price for Solar Renewable Energy Certificate (SRECs) in market-based systems designed to incentivize solar energy generation."
],
[
"Early general equilibrium models aimed at understanding the effects and the control of externalities and taxes (in the spirit of Tobin taxes) were proposed by Golosov, Hassler, Krusell, and Tsyvinski in [50].",
"General equilibrium models were also used in early works on the emissions markets by Bueler [16] and Haurie [55] and more significantly in the analysis of the European Union Emission Trading System (EU ETS) by Carmona, Fehr, Hinz and Porchet in [24].",
"See also the references therein.",
"We argue later on in Section that many general equilibrium models can be recast as Mean Field Game models.",
"More recently, ideas which first appeared in the treatment of MFGs were used by Bahn, Haurie and Malhamé to model negotiations related to environment policies.",
"See [9].",
"Finally, we mention the recent work of Carmona, Dayanikli and Laurière [20] who use MFG models with major and minor players, very much in the spirit of the contract theory models we review in Section , to derive equilibrium analyses of externalities and regulation on one end, and investments in renewables on the other, when dealing with electricity production."
],
[
"Macro-Economic Growth Models In this section, we review several general equilibrium economic growth models.",
"We borrowed the first one from a paper by Guéant, Lasry and Lions on mean field games [54].",
"We chose to present it here because, by cleverly adapting ideas from a model of Aghion and Howitt, the authors present a model with common noise which can be solved explicitly, all the way to the master equation.",
"Our choice of the second model was driven by a remarkable property: even though the original contribution [63] of Krusell and Smith appeared long before the mean field game paradigm was articulated, the numerical algorithm proposed by the authors to approximate numerically the equilibrium characteristics, reads as if it had been designed for the computation of an MFG equilibrium.",
"Indeed, it is eerie to see how closely the description of their numerical algorithm mimics, step by step, the mean field game strategy based on the alternate iteration of steps to approach the solution of the HJB equation and the Fokker-Planck Kolmogorov equation.",
"We learned of the third example presented in this section, from a private conversation with Benjamin Moll.",
"We include it in this review because it can be solved completely, both analytically and numerically.",
"Unfortunately, those examples are few and far between.",
"The interested reader may also want to consult [3] by Achdou, Han, Lasry, Lions and Moll for another discussion of continuous time macro-economic models recast as MFGs."
],
[
"A First Example based on Calculus with Pareto Distributions We introduce directly the mean field formulation of the game without starting from the definition of the finite player game because the interaction between the players is local in the sense that it is a function of the density of the statistical distribution of the states of the players.",
"In the case of finitely many players, this distribution is the empirical distribution of the finitely many states and as such, it does not have a density per se.",
"So in order to avoid the introduction of smoothing of the empirical measures to define the costs to the players, we jump directly to the mean field game formulation which can be done directly with densities without any need for mollification arguments.",
"In this model, there are no idiosyncratic shocks, just aggregate shocks common to all the players.",
"They are given by the increments of a one dimensional Wiener process ${\\mathbf {W}}^0=(W^0_t)_{ t\\ge 0}$ .",
"We denote by $\\mathbb {F} ^0=(\\mathcal {F} ^0_t)_{t\\ge 0}$ its filtration.",
"We also assume that the volatility of the state of a generic player is linear, that is $\\sigma ^0(x)=\\sigma x$ for some positive constant $\\sigma $ , and that each player controls the drift of their state so that the dynamics of their state read: $dX_t=\\alpha _t dt+\\sigma X_t dW_t^0.$ We shall restrict ourselves to Markovian controls of the form $\\alpha _t=\\alpha (t,X_t)$ for a deterministic function $(t,x)\\mapsto \\alpha (t,x)$ , which will be assumed to be non-negative and Lipschitz in the variable $x$ .",
"Under these conditions, $X_{t} \\ge 0$ at all times $t>0$ if $X_{0} \\ge 0$ .",
"Note that if $X_t$ and $\\tilde{X}_t$ are solutions of (REF ) for the same linear control $\\alpha (t,x)=\\gamma _{t} x$ for some continuous path $[0,T] \\ni t \\mapsto \\gamma _{t} \\in [0,+\\infty )$ , with initial conditions $X_0\\le \\tilde{X}_0$ , then $\\tilde{X}_t=X_t+(\\tilde{X}_0-X_0)e^{\\int _{0}^t \\gamma _{s} ds - (\\sigma ^2/2) t+\\sigma W_t^0}.$ We assume that $k>0$ is a fixed parameter, and we introduce a special notation for the family of one-sided scaled Pareto distributions with decay parameter $k$ .",
"For any real number $q>0$ , we denote by $\\mu ^{(q)}$ the one-sided Pareto distribution on the interval $[q,\\infty )$ : $\\mu ^{(q)}(dx)=k\\frac{q^k}{x^{k+1}}{\\bf 1}_{[q,\\infty )}(x)dx.$ Notice that for any random variable $X$ , $X\\sim \\mu ^{(1)}$ is equivalent to $qX\\sim \\mu ^{(q)}$ .",
"For each $t\\ge 0$ we define $\\mu _t(dx)=\\mathbb {P} [X_t\\in dx|\\mathcal {F} _t^0]$ .",
"The flow $(\\mu _t)_{t\\ge 0}$ of probability measures is adapted to the filtration $\\mathbb {F} ^0$ of the common noise.",
"Recall that the MFG paradigm in the presence of a common noise is to solve, for each fixed $\\mathbb {F} ^0$ -adapted flow of probability measures $(\\mu _t)_{t\\ge 0}$ , the optimization problem of a generic player, and then solve the fixed point problem to guarantee that the flow $(\\mu _t)_{t\\ge 0}$ we started from is in fact the flow of conditional marginal laws of the solution of the optimization problem.",
"For this particular family of distributions, if $\\mu _0=\\mu ^{(1)}$ , then $\\mu _t=\\mu ^{(q_t)}$ where $q_t=e^{\\int _{0}^t \\gamma _{s} ds - (\\sigma ^2/2) t+\\sigma W_t^0}$ .",
"In other words, conditioned on the history of the common noise, the distribution of the states of the players remains Pareto with parameter $k$ if it starts that way, and the left-hand point of the distribution $q_t$ can be understood as a sufficient statistic characterizing the distribution $\\mu _t$ .",
"So if $X_0\\sim \\mu ^{(1)}$ , then $\\mu _t\\sim \\mu ^{(q_t)}$ .",
"This simple remark provides an explicit formula for the time evolution of the (conditional) marginal distributions of the states given the common noise.",
"In general MFGs with a common noise, this time evolution is difficult to determine as it requires the solution of a forward Stochastic Partial Differential Equation (SPDE for short).",
"Using the same notation as in [54], we define the running cost function $f$ by $f(x,\\mu ,\\alpha )= c \\frac{x^a}{[(d\\mu /dx)(x)]^b}-\\frac{E}{p}\\frac{\\alpha ^p}{[\\mu ([x,\\infty ))]^b},$ for positive constants $a$ , $b$ , $c$ , $E$ and $p >1$ .",
"The economic rationale for the form of this cost function and the meanings of the parameters are discussed in [54].",
"By convention, the density appearing in this formula is the density of the absolutely continuous part of the Lebesgue's decomposition of the measure $\\mu $ , and it is set to 0 when the measure is singular.",
"The argument of the optimization of the Hamiltonian is given by $\\hat{\\alpha }(x,\\mu ,y)=\\Bigl (\\frac{y}{E}\\bigl [\\mu ([x,\\infty ))\\bigr ]^b\\Bigr )^{1/(p-1)}.$ This formula can be used to write the master equation which, when restricted to one-sided Pareto distributions, can be reduced to a finite dimensional PDE because of the above remark.",
"Accordingly, Nash equilibria can be identified in this family of Pareto distributions.",
"The details, far too technical for this review, can be found in Section 4.5.2 of [22]."
],
[
"The Krusell - Smith's Growth Model One major difference with the growth model discussed in the previous subsection is the fact that, on the top of the common noise affecting all the states, we also have idiosyncratic random shocks specific to each individual agent in the economy.",
"In [63] the shocks take only finitely many values.",
"We suspect that this restrictive assumption was made for the purpose of numerical implementation.",
"In the next subsection, we change the nature of the random shocks by introducing Wiener processes to recast the model in the framework of stochastic differential games."
],
[
"While economists usually work with models comprising a continuum of players (this is indeed the case in [63]), in order to avoid the discussion of measurability issues related to continuum families of independent random variables, we first discuss the model of an economy comprising $N$ consumers.",
"The random shocks are given by a set of $N$ continuous time Markov chains $(z_t,\\eta ^i_t)_{t\\ge 0}$ for $i=1,\\ldots , N$ .",
"The common component $z$ captures the health of the overall economy, like an aggregate productivity measure, so for some constant $\\Delta _z\\ge 0$ , $z_t=1+\\Delta _z$ in good times, and $z_t=1-\\Delta _z$ in bad times.",
"The idiosyncratic component $\\eta $ is specific to the consumer, $\\eta ^i_t=1$ when consumer $i$ is employed, and $\\eta ^i_t=0$ whenever they are unemployed.",
"$\\Delta _z=0$ corresponds to the absence of common noise.",
"The production technology is modeled by a Cobb - Douglas production function in the sense that the per-capita output is given by $Y_t=z_t K^\\alpha _t(\\overline{\\ell }L_t)^{1-\\alpha }$ where $K_t$ and $L_t$ stand for per-capita capital and employment rates respectively.",
"The constant $\\overline{\\ell }$ can be interpreted as the number of units of labor produced by an employed individual.",
"The power $\\alpha \\in (0,1)$ is a constant of the model.",
"In such a model, two quantities play an important role: the capital rent $r_t$ and the wage rate $w_t$ .",
"Economic theory says that in equilibrium, these marginal rates are defined as the partial derivatives of the per-capita output $Y_t$ with respect to capital and employment rate respectively.",
"So $r_t=r(K_t,L_t,z_t)=\\alpha z_t\\big (\\frac{K_t}{L_t}\\big )^{\\alpha -1}$ and $w_t=w(K_t,L_t,z_t)=(1-\\alpha ) z_t\\big (\\frac{K_t}{L_t}\\big )^\\alpha .$ Consumers control their capital consumption rate $c^i_t$ at time $t$ , and maximize their expected utilities of consumption $\\mathbb {E} \\bigg [\\int _0^\\infty e^{-\\rho t} u(c^i_t)dt\\bigg ],$ for some discount factor $\\rho >0$ .",
"We use the power utility function $u(c)=\\frac{c^{1-\\gamma }-1}{1-\\gamma }$ for some $\\gamma \\in (0,1)$ , also known as CRRA (short for Constant Relative Risk Aversion) utility function.",
"Consumers must choose their consumptions while making sure that their individual capitals $k^i_t$ remain non-negative at all times.",
"The individual capitals evolve according to the equation $d k^i_t=\\big [ (r_t-\\delta )k^i_t + [(1-\\tau _t)\\overline{\\ell }\\eta ^i_t + \\mu (1-\\eta ^i_t)]w_t\\big ] dt - c^i_t dt.$ Here, the constant $\\delta >0$ represents a depreciation rate.",
"The second term in the above right hand side represents the wages earned by the consumer.",
"It is equal to $\\mu w_t$ when the consumer is unemployed, quantity which should be understood as an unemployment benefit rate.",
"On the other hand, it is equal to $(1-\\tau _t)\\overline{\\ell }w_t$ after adjustment for taxes, when they are employed.",
"Here $\\tau _t=\\frac{\\mu u_t}{\\overline{\\ell }L_t}$ where $u_t=1-L_t$ is the unemployment rate.",
"By de Finetti's law of large numbers, we expect that the empirical measures $\\overline{\\mu }^{k,N}_t$ of capital and $\\overline{\\mu }^{\\eta ,N}_t$ of labor: $\\overline{\\mu }^{k,N}_t=\\frac{1}{N}\\sum _{i=1}^N\\delta _{k^i_t},\\qquad \\text{and}\\qquad \\overline{\\mu }^{\\eta ,N}_t=\\frac{1}{N}\\sum _{i=1}^N\\delta _{\\eta ^i_t},$ converge as $N\\rightarrow \\infty $ toward a limit which we denote by $\\mu ^{k,z}_t$ and $\\mu ^{\\eta ,z}_t$ .",
"These limits give the conditional distributions of capital and labor $k_t$ and $\\eta _t$ given the state $z_t=z$ of the economy.",
"Since $z_t$ can only take two values $1-\\Delta _z$ and $1+\\Delta _z$ , we only need the knowledge of deterministic flows of measures, $(\\mu ^{k,d}_t)_{t\\ge 0}$ , $(\\mu ^{k,u}_t)_{t\\ge 0}$ , $(\\mu ^{\\eta ,d}_t)_{t\\ge 0}$ , and $(\\mu ^{\\eta ,u}_t)_{t\\ge )}$ corresponding to the two values of $z$ , say down and up, namely $d=1-\\Delta _z$ and $u=1+\\Delta _z$ .",
"Once the flows of conditional measures are known, the computation of the best response of a representative agent reduces to the solution of the optimal control problem $\\max _c\\mathbb {E} \\bigg [\\int _0^\\infty e^{-\\rho t} u(c_t)dt\\bigg ]$ under the constraints $k_t\\ge 0$ and $d k_t=\\big [ (r(K_t,L_t,z_t)-\\delta )k_t + [(1-\\tau _t)\\overline{\\ell }\\eta _t + \\mu (1-\\eta _t)]w(K_t,L_t,z_t)\\big ] dt - c_t dt.$ Here, $(z_t,\\eta _t)_{t\\ge 0}$ is a continuous time Markov chain with the same law as any of the $(z_t,\\eta ^i_t)_{t\\ge 0}$ introduced earlier, the rental rate function $r$ and the wage level function $w$ are as in (REF ) and (REF ), and $K_t=\\overline{k}_t^{z_t}$ is the mean of the conditional measure $\\mu ^{z_t}_t$ , namely $K_t=\\int _{[0,\\infty )} k\\mu ^{k,u}_t(dk)\\;\\;\\text{if } z_t=1+\\Delta _z,\\quad \\text{and}\\quad K_t=\\int _{[0,\\infty )} k\\mu ^{k,d}_t(dk)\\;\\;\\text{if } z_t=1-\\Delta _z,$ and where the aggregate labor $L_t$ is defined similarly as the conditional mean of $\\mu _t^{\\eta ,z}$ .",
"The aggregates $K_t$ and $L_t$ are the conditional means of the capital and labor given the common noise: they carry the mean field interactions in the model.",
"As we recalled during our discussion of the previous example, the MFG paradigm in the presence of a common noise is to solve, for each fixed flow of probability measures adapted to the filtration of the common noise, the optimization problem of a generic player, and then solve the fixed point problem to guarantee that the flow we started from is in fact the flow of conditional marginal laws of the solution of the optimization problem.",
"Note that in the Krusell-Smith's model, the common noise and the idiosyncratic noise are correlated and that the labor state variable $\\eta _t$ (whose aggregate is $L_t$ ) is nothing but the idiosyncratic noise.",
"So clearly, the MFG paradigm reduces to the solution of the individual optimization problem given the aggregate $K_t$ and then solving for the fixed point.",
"This is exactly what the numerical algorithm proposed in [63] does.",
"Time discretization is not needed in [63] since the model is introduced in discrete time there.",
"Then a form of dynamic programming is used to solve the optimization problem given the aggregate $K_t$ , and then an update of $K_t$ is done by Monte Carlo simulation, before going back to the solution of the optimization problem given the update of $K_t$ , and so on and so forth.",
"While the authors realize that the entire distribution $\\mu _t^{k,z}$ should be updated, they argue that updating the mean is sufficient to get reasonable numerical results for a problem whose complexity should have been prohibitive.",
"As explained in the introduction, even though they never used the term Nash equilibrium, their numerical search for a recursive competitive equilibrium is exactly the algorithm based on the iteration of the numerical approximation of the solution of the HJB equation followed by the Fokker-Planck-Kolmogorov equation, algorithm (re)introduced and used over 15 years later for the numerical solution of Mean Field Games."
],
[
"A Diffusion Form of Aiyagari's Growth Model As explained in the introduction to this section, we learned about the model presented in this subsection from a private conversation with Benjamin Moll.",
"It is one of the models discussed in the review [1] by Achdou, Buera, Lasry, Lions and Moll devoted to partial differential equation models in macroeconomics.",
"As far as we know, the first, and most likely the only, complete mathematical solution as a mean field game of this model can be found in Chapter 3 of [21].",
"We first describe the finite player form of the model.",
"We shall solve it as a mean field game model later on.",
"The $N$ agents $i\\in \\lbrace 1,\\ldots ,N\\rbrace $ are the workers comprising the economy.",
"The private state at time $t$ of agent $i$ is a two-dimensional vector $X^i_t=(Z^i_t,A^i_t)$ .",
"For the purpose of this model, $A^i_t$ is the wealth at time $t$ of worker $i$ , and $Z^i_t$ their labor productivity.",
"The time evolutions of the states are given by stochastic differential equations ${\\left\\lbrace \\begin{array}{ll}dZ^i_t&=\\mu _Z(Z^i_t)dt\\, + \\, \\sigma _Z(Z^i_t) dW^i_t\\\\dA^i_t&=[w^i_tZ^i_t \\, + \\, r_t A^i_t \\,-\\,c^i_t] dt.\\end{array}\\right.",
"}$ The functions $\\mu _{Z},\\sigma _{Z}: \\mathbb {R} \\rightarrow \\mathbb {R} $ are known.",
"We shall specify them later on in the examples we treat theoretically and numerically.",
"The random shocks are given by $N$ independent Wiener processes ${\\mathbf {W}}^i=(W^i_t)_{t\\ge 0}$ , for $i=1,\\ldots , N$ .",
"$r_t$ is the interest rate at time $t$ , $w^i_t$ represents the wages of worker $i$ at time $t$ and the consumption process ${c}^i=(c^i_t)_{t\\ge 0}$ is the control of player $i$ .",
"Remark 4.1 In many economic applications, a borrowing limit is imposed.",
"Mathematically, this means that the wealths must satisfy the constraints $A^i_t\\ge \\underline{a}\\;$ for some nonpositive constant $\\underline{a}\\le 0$ .",
"Moreover, the labor productivity processes ${\\mathbf {Z}}^i=(Z^i_t)_{t\\ge 0}$ are also restricted by requiring that they are ergodic, or even restricted to an interval $[\\underline{z},\\overline{z}]$ for some finite constants $0\\le \\underline{z} <\\overline{z}<\\infty $ .",
"We do not know of an economic rationale for these constraints and we suspect that these assumptions are made for the sole benefits of the technical proofs.",
"In this model, given adapted processes ${r}=(r_t)_{t\\ge 0}$ and ${w}^i=(w^i_t)_{t\\ge 0}$ for $i=1,\\ldots ,N$ , the workers choose their consumptions $c^1,\\ldots ,c^N$ in order to maximize their expected discounted utilities: $J^i({c}^1,\\ldots ,{c}^N)=\\mathbb {E} \\int _0^\\infty e^{-\\rho t}u(c^i_t)dt.$ As usual in economic applications, the model is set up in infinite horizon, and $u$ is an increasing concave utility function, the same for all the workers.",
"So far, it seems like the workers do not interact.",
"Also, we need to explain how the interest rate and the wage processes appear in equilibrium.",
"As in the Krusell-Smith model discussed earlier, we assume that the aggregate production in the economy is given by a production function $Y=F(K,L)$ , the total capital supplied in the economy at time $t$ , say $K_t$ being given by the aggregate wealth $K_t=\\int a \\;d\\overline{\\mu }^N_{X_t}(dz,da)=\\frac{1}{N}\\sum _{i=1}^N A^i_t$ while the total amount of labor $L_t$ supplied in the economy at time $t$ is normalized to 1.",
"Here, we denote by $\\overline{\\mu }^N_{X_t}$ the empirical measure of the sample $X^1_t,\\ldots ,X^N_t$ .",
"Note that only the $A$ -marginal enters the definition of $K_t$ .",
"Remark 4.2 As explained in the introduction, the fact that the economic agents interact through aggregate quantities is the reason why mean field models and mean field game formulations are so natural for these macro-economic models.",
"Economic theory says that in a competitive equilibrium, the interest rate and the wages are given by the partial derivatives of the production function ${\\left\\lbrace \\begin{array}{ll}r_t &= [\\partial _KF](K_t,L_t)|_{L_t=1} -\\delta \\\\w_t &= [\\partial _LF](K_t,L_t)|_{L_t=1}\\end{array}\\right.",
"}$ where $\\delta \\ge 0$ is the rate of capital depreciation.",
"So in equilibrium, the interaction between the agents in the economy is through the mean $K_t=\\int a\\overline{\\mu }^N_X(a,z)$ of the empirical distribution of the workers' wealths $A^i_t$ ."
],
[
"We now specify the model further to solve it as a mean field game.",
"We use the CRRA isoelastic utility function with constant relative risk aversion introduced above in (REF ).",
"Note that $u^{\\prime }(c)=c^{-\\gamma }\\qquad \\text{and}\\qquad (u^{\\prime })^{-1}(y)=y^{-1/\\gamma }.$ Next, we use the Cobb - Douglas production function $F(K,L)=A\\,K^\\alpha \\, L^{1-\\alpha }$ for some constants $A>0$ and $\\alpha \\in (0,1)$ .",
"With this choice, in equilibrium, $r_t=\\alpha A K_t^{\\alpha -1}L_t^{1-\\alpha } - \\delta \\qquad \\text{and}\\qquad w_t=(1-\\alpha ) A K_t^\\alpha L_t^{-\\alpha }$ and since we normalized the aggregate supply of labor to 1, $r_t=\\frac{\\alpha A}{ K_t^{1-\\alpha }} - \\delta \\qquad \\text{and}\\qquad w_t=(1-\\alpha ) A K_t^\\alpha ,$ where $K_t$ is given by (REF ) and provides the mean field interaction.",
"Finally, we use an Ornstein-Uhlenbeck process for the mean reverting labor productivity process ${\\mathbf {Z}}=(Z_t)_{t\\ge 0}$ by choosing $\\mu _{Z}(z) = 1-z$ and $\\sigma _{Z} \\equiv 1$ for the sake of definiteness.",
"Moving to the mean field game formulation of the model, the state $X_t=(A_t,Z_t)$ evolves according to ${\\left\\lbrace \\begin{array}{ll}&dZ_{t} = -(Z_{t}-1)\\, dt + dW_{t},\\\\&dA_{t} = \\bigl [ (1-\\alpha ) \\bar{\\mu }_{t}^\\alpha Z_{t}+ \\bigl ( \\alpha \\bar{\\mu }_{t}^{\\alpha -1}-\\delta \\bigr )A_{t} - c_{t}\\bigr ] dt, \\quad t \\in [0,T],\\end{array}\\right.",
"}$ where $(\\bar{\\mu }_{t})_{0 \\le t \\le T}$ denotes the flow of average wealths in the population in equilibrium.",
"It is assumed to take (strictly) positive values.",
"The set $\\mathbb {A} $ of admissible controls is the set $\\mathbb {H} ^{2,1}_+$ of real valued square-integrable $\\mathbb {F} $ -adapted processes ${c}=(c_{t})_{0 \\le t \\le T}$ with non-negative values, and the cost functional is defined by: $J({c}) = \\mathbb {E} \\biggl [\\int _{0}^T (-u)(c_{t}) dt - \\tilde{u}(A_{T}) \\biggr ],$ for the CRRA utility function $u$ given by (REF ) and $\\tilde{u}(a)=a$ .",
"Notice the additional minus signs due to the fact that we want to treat the optimization problem as a minimization problem.",
"Here we chose to take 0 for the discount rate since we are working on a finite horizon.",
"Throughout the analysis, we shall assume that $A_{0}>0$ and $Z_{0}=1$ , so that $\\mathbb {E} [Z_{t}]=1$ for any $t \\ge 0$ .",
"In order to solve this MFG using the Pontryagin Maximum Principle, we introduce the Hamiltonian: $H(t, z,a,\\mu ,y_z,y_a,c)=(1-z)y_z+\\big (-c +(1-\\alpha )\\overline{\\mu }_t^\\alpha z+ (\\alpha \\overline{\\mu }^{\\alpha -1} -\\delta ) a \\big ) y_a - u(c),$ where $\\overline{\\mu }=\\int a\\;d\\mu (z,a)$ denotes the mean of the second marginal of the measure $\\mu $ .",
"The first adjoint equation reads $dY_{z,t}=-\\partial _z H(t,Z_t,A_t,Y_{z,t},Y_{a,t},c_t)dt +\\tilde{Z}_{z,t} dW_t=Y_{z,t}dt+\\tilde{Z}_{z,t} dW_t.$ Its solution is $Y_{z,t}=0$ because its terminal condition is $Y_{z,T}=0$ .",
"Since the variables $z$ and $y_z$ do not play any role in the minimization of the Hamiltonian with respect to the control variable $c$ , we use the reduced Hamiltonian: $H(t, a,\\mu ,y,c)=\\big (-c + (\\alpha \\overline{\\mu }^{\\alpha -1} -\\delta ) a \\big ) y - u(c),$ which is convex in $(a,c)$ and strictly convex in $c$ .",
"The form (REF ) of the derivative of the utility function implies that the value of the control minimizing the Hamiltonian is $\\hat{c}=(-u^{\\prime })^{-1}(y) = (-y)^{-1/\\gamma }$ .",
"Therefore, the FBSDE derived from the Pontryagin stochastic maximum principle reads ${\\left\\lbrace \\begin{array}{ll}&dA_t= \\bigl [(1-\\alpha ) \\overline{\\mu }^\\alpha _t Z_t+ [\\alpha \\overline{\\mu }_t^{\\alpha -1}-\\delta ] A_t- (-Y_{t})^{-1/\\gamma }\\big ] dt\\\\&dY_t =-Y_t[ \\alpha \\overline{\\mu }_t^{\\alpha -1}-\\delta ] dt + Z_t^{\\prime } dW_{t}, \\quad t \\in [0,T] \\ ; \\quad Y_{T}=-1,\\end{array}\\right.",
"}$ where we used the notation $(Z_{t}^{\\prime })_{0 \\le t \\le T}$ to denote the integrand in the quadratic variation part of the backward equation in order to distinguish it from the process $(Z_{t})_{0 \\le t \\le T}$ used in the model as the first component of the state.",
"Despite the fact that the utility function has a singularity at 0, it is not difficult to check that the proof of the sufficient part of the Pontryagin principle goes through provided that the adjoint process $(Y_{t})_{0 \\le t \\le T}$ lives, with probability 1, in a compact subset of $(-\\infty ,0)$ .",
"We shall refrain from going through the gory details of the rest of the proof.",
"We refer the interested reader to [21].",
"The major insight is to notice that the backward equation may be decoupled from the forward equation and that its solution is deterministic and is obtained by solving the backward ordinary differential equation: $dY_{t} = -Y_t[ \\alpha \\overline{\\mu }_t^{\\alpha -1}-\\delta ] dt, \\quad t \\in [0,T] \\ ; \\quad Y_{T}=-1.$ The remaining of the proof follows easily."
],
[
"From Macro to Finance In this section, we review two recent works of M. Brunnermeier and Y. Sannikov [14], [15] in which the authors compare the historical evolutions of macro-economic and finance models, arguing that properly framed, the analysis of continuous time stochastic models should provide a unifying thread for these sub-fields of economics which so far, developed in parallel.",
"To make their point, the authors introduce models of the economy comprising households maximizing consumption like in classical macro-economic growth models, as well as experts trading in financial markets.",
"As explained in the introduction, those models lead to MFGs with a common noise.",
"The importance of common shocks in macro-economics points to the need of a better mathematical understanding of MFGs with a common noise.",
"The first model reviewed in this section fits in the class of MFGs with one population of individuals facing idiosyncratic noise terms as well as random shocks common to all.",
"It was first introduced in [14] in discrete time.",
"While the second model does not have idiosyncratic noise terms, it involves two populations of agents.",
"This gives us an opportunity to quickly review some of the features of MFGs with several populations, which are not discussed often enough in the mathematical literature on mean field games."
],
[
"Economy with One Type of Agents We consider a one-sector economy with a continuum of households with identical preferences (we shall use the logarithmic utility function $u(x)=\\log x$ ) and different levels of wealth.",
"We denote by $I$ the set of households.",
"We choose $I=[0,1]$ for the sake of definiteness.",
"In this model, because each agent's influence on the economy is infinitesimal, we use a continuous probability measure $\\lambda $ on $I$ to sample households.",
"For practical purposes, we can think of $\\lambda $ as the Lebesgue measure on $[0,1]$ .",
"Remark 5.1 This suggestion to think of the space of households as the unit interval $[0,1]$ equipped with its Lebesgue measure is a flagrant expediency.",
"Indeed, mathematically speaking, it does not pass the smell test, as in order to manipulate a continuum of idiosyncratic shocks without having to face severe measurability issues, we would have to jump through several hoops, for example using rich Fubini extensions instead of the Lebesgue unit interval.",
"See for example [21] for a discussion of such a rigorous approach.",
"In this model, each household operates a firm and holds money.",
"The capital stock of a generic household $h$ evolves according to the equation: $\\frac{dk^h_t}{k^h_t}= (\\phi (\\iota ^h_t) -\\delta ) dt +\\sigma ^0 dW^0_t + \\sigma dW^h_t$ where $\\delta >0$ is a depreciation rate, and the function $\\phi $ reflects adjustment costs in capital stock.",
"This function is assumed to satisfy $\\phi (0)=0$ , $\\phi ^{\\prime }(0)=1$ , $\\phi ^{\\prime }(\\cdot )>0$ and $\\phi ^{\\prime \\prime }(\\cdot )<0$ .",
"Its concavity captures technological illiquidity.",
"$\\iota ^h_t$ represents the investment rate of household $h$ in physical capital at time $t$ .",
"Essentially, it gives how many units of physical capital are used in order to produce new physical capital.",
"${\\mathbf {W}}^0=(W^0_t)_{t\\ge 0}$ and ${\\mathbf {W}}^h=(W^h_t)_{t\\ge 0}$ are independent Wiener processes modeling random shocks.",
"$dW^h_t$ represents an idiosyncratic shock specific to the household $h$ , while $dW^0_t$ represents a shock common to all the households.",
"We shall often call it the common noise.",
"The volatilities $\\sigma ^0$ and $\\sigma $ are positive constants.",
"Households hold money.",
"We denote by $m^h_t$ the amount of money held at time $t$ by household $h$ .",
"They consume in the amount $c^h_t$ .",
"We denote by $\\theta ^h_t$ the fraction of the household wealth in money at time $t$ .",
"So the control of a household is the triple $(\\iota ^h_t,\\theta ^h_t, c^h_t)$ .",
"The goal of a household is to maximize its long-run discounted expected utility of consumption: $J^h(\\iota ,\\theta ,c)=\\mathbb {E} \\Bigl [\\int _0^\\infty e^{-\\rho t}u(c_t)dt\\Bigr ]$ over the control strategies $(\\iota ,\\theta ,c)=(\\iota _t,\\theta _t,c_t)_{t\\ge 0}$ .",
"The constant $\\rho >0$ provides actualization.",
"We now derive the dynamic constraint under which this optimization is performed by each household.",
"It is expressed in terms of the wealth $n^h_t$ of the household at time $t$ .",
"We use capital letters to denote the aggregates (i.e.",
"the empirical means) of each of the variables $k^h_t$ , $m^h_t$ and $n^h_t$ .",
"In other words: $K_t=\\int _{I^h}k^h_t\\;\\lambda (dh),\\qquad M_t=\\int _{I^h}m^h_t\\;\\lambda (dh),\\qquad N_t=\\int _{I^h}n^h_t\\;\\lambda (dh).$ Anticipating on the fact that we shall discover that in equilibrium, $(\\iota ^h_t)_{t\\ge 0}$ is independent of the household and adapted to the filtration of the common noise (which is the case if $\\iota ^h_t$ depends only upon aggregate quantities at time $t$ ), which implies that all the households use the same investment in physical capital strategy, we can integrate (REF ) over $h$ and find that $\\frac{dK_t}{K_t}= (\\phi (\\bar{\\iota }_t) -\\delta ) dt +\\sigma ^0 dW^0_t$ which is a stochastic differential equation only driven by the common noise.",
"The idiosyncratic shocks disappear because of a continuous form of the exact law of large numbers.",
"See for example [21].",
"We used the notation $\\bar{\\iota }_t$ to distinguish this aggregate return on capital from the individual households' $\\iota ^h_t$ .",
"We introduce two more constants: $q$ for the price of one unit of physical capital (so the real value of aggregate physical capital is $qK_t$ ), and $p$ for the real value of money normalized by the size of the economy as measured by $K_t$ (so $pK_t$ is the real value of total money supply).",
"These could be Itô processes, say $(q_t)_{t\\ge 0}$ and $(p_t)_{t\\ge 0}$ driven by the common noise ${\\mathbf {W}}^0$ , but for the sake of simplicity, we shall assume them to be deterministic constants for the purpose of this presentation.",
"Given the definition of the constants $q$ and $p$ , the total wealth in the economy is $N_t=(p+q)K_t,$ $qK_t$ representing the value of the physical capital and $pK_t$ the value of the nominal capital.",
"We denote by $\\vartheta $ the fraction of nominal wealth: $\\vartheta =\\frac{p}{p+q}.$ The quantity of money in the economy is controlled exogenously by a central bank.",
"We assume that money supply follows the following stochastic differential equation $\\frac{dM_t}{M_t} = \\mu ^M dt +\\sigma ^M dW^0_t$ driven by the common noise."
],
[
"We first derive the stochastic differential equation driving the dynamics of the wealth of a generic household, and then tackle the optimization of the expected utility of consumption by the Pontryagin stochastic maximum principle.",
"Changes in the wealth $n^h_t$ of household $h$ at time $t$ are the sums of three contributions.",
"We have: $dn^h_t= \\theta ^h_tn^h_t dr^{M}_t +(1-\\theta ^h_t)n^h_t\\; dr^{h,K}_t(\\iota ^h_t) -c^h_tdt$ where $r^{M}_t$ denotes the rate of return on money, and $r^{h,K}_t(\\iota ^h_t)$ the rate of return on capital.",
"If $ \\theta ^h_tn^h_t $ is the amount the household holds in money, and if we denote by $p^m_t$ the value of one unit of money, namely $p^m_t=\\frac{pK_t}{M_t},$ then the return on this investment is $dr^M_t=\\frac{d p^m_t}{p^m_t}=\\frac{d(K_t/M_t)}{K_t/M_t}$ since we assume that $p$ is a constant.",
"Using Itô's formula with (REF ) and (REF ) we get: $dr^{M}_t= \\Bigl [\\phi (\\bar{\\iota }_t)-\\delta -[\\mu ^M-\\sigma ^M(\\sigma ^M-\\sigma ^0)]\\Bigr ]dt+(\\sigma ^0-\\sigma ^M)dW^0_t.$ We now identify the time evolution of the rate of return on capital $r^{h,K}_t(\\iota ^h_t)$ .",
"It has three components: the return of investment in physical capital, the return on the household capital $qk^h_t$ , and the seigniorage.",
"Seigniorage is the amount of money which is transferred to money holders proportionally to their capital.",
"Given the definition (REF ) of the value of one unit of money, we can easily understand the change in the seigniorage over a period $[t, t+\\Delta t]$ .",
"It is given by: $T^h_{t+\\Delta t}-T^h_t=p^m_{t+\\Delta t}(M_{t+\\Delta t}-M_t)=p^m_t(M_{t+\\Delta t}-M_t)+(p^m_{t+\\Delta t}-p^m_t).",
"(M_{t+\\Delta t}-M_t).$ So in continuous time, i.e.",
"after taking the limit $\\Delta t\\searrow 0$ : $\\begin{split}dT_t&=p^m_t\\; dM_t + d[p^m,M]_t\\\\&= pK_t\\Bigl [ [\\mu ^M + (\\sigma ^0-\\sigma ^M)\\sigma ^M]dt +\\sigma ^M dW^0_t\\Bigr ].\\end{split}$ Consequently: $\\begin{split}dr^{h,K}_t(\\iota ^h_t)&=\\frac{a-\\iota ^h_t}{q}dt + \\frac{d\\bigl (qk^h_t\\bigr )}{qk^h_t}+\\frac{dT_t}{qK_t}\\\\&= \\Bigl [\\frac{a-\\iota ^h_t}{q} + \\phi (\\iota ^h_t)-\\delta + \\frac{p}{q}[\\mu ^M+(\\sigma ^0-\\sigma ^M)\\sigma ^M]\\Bigr ]dt+(\\sigma ^0+\\frac{p}{q}\\sigma ^M)dW^0_t + \\sigma dW^h_t.\\end{split}$ Putting together (REF ), (REF ) and (REF ) we get: $\\begin{split}dn^h_t&= \\theta ^h_tn^h_t dr^{M}_t +(1-\\theta ^h_t)n^h_t\\,dr^{h,K}_t(\\iota ^h_t) -c^h_tdt\\\\&=\\theta ^h_tn^h_t \\Bigl [\\phi (\\bar{\\iota }_t)-\\delta -[\\mu ^M+(\\sigma ^0-\\sigma ^M)\\sigma ^M]\\Bigr ]dt+\\theta ^h_tn^h_t (\\sigma ^0-\\sigma ^M)dW^0_t\\\\&\\hspace{35.0pt}+(1-\\theta ^h_t)n^h_t\\Bigl [\\frac{a-\\iota ^h_t}{q} + \\phi (\\iota ^h_t)-\\delta + \\frac{p}{q}[\\mu ^M+(\\sigma ^0-\\sigma ^M)\\sigma ^M]\\Bigr ]dt\\\\&\\hspace{35.0pt}+(1-\\theta ^h_t)n^h_t(\\sigma ^0+\\frac{p}{q}\\sigma ^M)dW^0_t + (1-\\theta ^h_t)n^h_t\\sigma dW^h_t-c^h_tdt\\\\&=\\Bigl (n^h_t \\Bigl [\\theta ^h_t\\bigl (\\phi (\\bar{\\iota }_t)-\\delta \\bigr )+(1-\\theta ^h_t)\\bigl [\\frac{a-\\iota ^h_t}{q}+ \\phi (\\iota ^h_t)-\\delta \\bigr ]\\\\&\\hspace{35.0pt}+[\\mu ^M+(\\sigma ^0-\\sigma ^M)\\sigma ^M]\\bigl (\\frac{p}{q}-\\theta ^h_t\\frac{p+q}{q}\\bigr )\\Bigr ]-c^h_t\\Bigr )dt\\\\&\\hspace{35.0pt}+n^h_t\\bigl [\\sigma ^0+\\bigl (\\frac{p}{q}-\\theta ^h_t\\frac{p+q}{q}\\bigr )\\sigma ^M\\bigr ]dW^0_t + (1-\\theta ^h_t)n^h_t\\sigma dW^h_t\\end{split}$ The Hamiltonian of the optimization problem of a generic household reads: $\\begin{split}&H(t,n,y,z^0,z,\\iota ,\\theta ,c)\\\\&\\hspace{25.0pt}=\\Bigl (n \\Bigl [\\theta \\bigl (\\phi (\\bar{\\iota }_t)-\\delta \\bigr )+(1-\\theta )\\bigl [\\frac{a-\\iota }{q}+ \\phi (\\iota )-\\delta \\bigr ]\\\\&\\hspace{75.0pt}+[\\mu ^M+(\\sigma ^0-\\sigma ^M)\\sigma ^M]\\bigl (\\frac{p}{q}-\\theta \\frac{p+q}{q}\\bigr )\\Bigr ]-c\\Bigr )y\\\\&\\hspace{75.0pt}+n\\bigl [\\sigma ^0+\\bigl (\\frac{p}{q}-\\theta \\frac{p+q}{q}\\bigr )\\sigma ^M\\bigr ]z^0 + (1-\\theta )n\\sigma z-e^{-\\rho t} u(c)\\end{split}$ if we use the notations $y$ , $z^0$ and $z$ for the adjoint variables (sometimes called co-states).",
"The necessary part of the Pontryagin maximum principle suggests to minimize the Hamiltonian with respect to the control variables $\\iota $ , $\\theta $ and $c$ .",
"Moreover, since $(1-\\theta )\\ge 0$ , we can isolate $\\iota $ and minimize $(a-\\iota )/q+\\phi (\\iota )$ which gives Tobin's $q$ equation: $-\\frac{1}{q}+\\phi ^{\\prime }(\\iota )=0 \\quad \\Leftrightarrow \\quad \\iota =(\\phi ^{\\prime })^{-1}\\bigl (\\frac{1}{q}\\bigr ),$ and in the case of the function $\\phi (\\iota )=(1/\\kappa )\\log (1+\\kappa \\iota )$ used in [15], we get: $\\kappa \\hat{\\iota }_t=q-1.$ Notice that the optimal $\\hat{\\iota }_t$ is a constant independent of $t$ .",
"In general, if $q$ is an Itô process adapted to the filtration of the common noise, so is $\\hat{\\iota }$ .",
"But the fact to remember at this stage is that the optimal $\\hat{\\iota }_t$ is the same for all the households.",
"So from now on $\\bar{\\iota }_t=\\hat{\\iota }_t=\\kappa ^{-1}(q-1)$ .",
"Expecting that $\\theta \\in [0,1]$ , minimizing the Hamiltonian over $\\theta $ could lead to $\\partial _\\theta H=0$ , i.e.",
"$ny\\Bigl [\\frac{a-\\iota }{q} + \\frac{p+q}{q}[\\mu ^M+(\\sigma ^0-\\sigma ^M)\\sigma ^M]\\Bigr ]+nz^0\\frac{p+q}{q}\\sigma ^M+n\\sigma z=0.$ For obvious reasons we write the adjoint variables $z$ and $z^0$ in the form $z=-y\\zeta $ and $z^0=-y\\zeta ^0$ so we can rewrite the first order condition $\\partial _\\theta H=0$ in equilibrium as: $\\frac{a-\\iota }{q} +\\frac{p+q}{q}[\\mu ^M+(\\sigma ^0-\\sigma ^M)\\sigma ^M]=\\frac{p+q}{q}\\sigma ^M\\zeta ^0+\\sigma \\zeta .$ This equation does not determine directly the optimal value of $\\theta _t$ .",
"It is sometimes called the pricing equation because it can also be derived from the HJB equation of the optimization problem offering a pricing interpretation.",
"Finally, since we restrict ourselves to $y<0$ , we can write the third First Order Condition (FOC) as: $\\partial _cH=0\\quad \\Leftrightarrow \\quad -y-e^{-\\rho t}u^{\\prime }(c)=0 \\quad \\Leftrightarrow \\quad c=(u^{\\prime })^{-1}\\bigl (-e^{\\rho t}y\\bigr )$ so that in the case of logarithmic utility, the optimal consumption rate should be the process $(\\hat{c}_t)_{t\\ge 0}$ given by $\\hat{c}^h_t=-e^{-\\rho t}\\frac{1}{y^h_t}$ where the adjoint process $(y_t)_{t\\ge 0}$ is the first component of the solution of the adjoint equation, namely the Backward Stochastic Differential Equation (BSDE) equation: $dy^h_t=-\\partial _nH(t,n^h_t,y^h_t,z^{0,h}_t,z^h_t,\\hat{\\iota }_t,\\hat{\\theta }_t,\\hat{c}_t) dt+z^{0,h}_t dW^0_t+z^h_t dW^h_t.$ Computing $\\partial _nH$ from (REF ) and using (REF ) we get: $\\begin{split}\\partial _nH&=y\\bigl [\\phi (\\iota )-\\delta - [ \\mu ^M+\\sigma ^M(\\sigma ^0-\\sigma ^M)] -(\\sigma ^0-\\sigma ^M)\\zeta ^0\\bigr ] \\\\&=yr^h_t\\end{split}$ if we define the individual household effective interest by: $r^h_t=\\phi (\\iota )-\\delta - [ \\mu ^M+\\sigma ^M(\\sigma ^0-\\sigma ^M)] -(\\sigma ^0-\\sigma ^M)\\zeta ^{0,h}_t.$ So the adjoint Backward Stochastic Differential Equation (BSDE) reads $\\frac{dy^h_t}{y^h_t}=-r^h_tdt -\\zeta ^{0,h} dW^0_t -\\zeta ^h_t dW^h_t,$ hence the interpretation of $r^h_t$ as an individual household short interest rate and $y^h_t$ as an individual stochastic discount factor.",
"Using Itô's formula with (REF ) and (REF ), and the definition (REF ) of the Hamiltonian we get: $\\frac{d (y^h_t n^h_t)}{y^h_t n^h_t}= -\\frac{c^h_t}{n^h_t} dt+\\Bigl (\\bigl [\\sigma ^0+\\bigl (\\frac{p}{q}-\\theta ^h_t\\frac{p+q}{q}\\bigr )\\sigma ^M\\bigr ]-\\zeta ^{0,h}_t\\Bigr ) +\\Bigl ((1-\\theta ^h_t)\\sigma -\\zeta ^h_t\\Bigr ),$ so if we choose: $\\zeta ^{0,h}_t=\\sigma ^0+\\bigl (\\frac{p}{q}-\\theta ^h_t\\frac{p+q}{q}\\bigr )\\sigma ^M\\quad \\text{and}\\quad \\zeta ^h_t=\\sigma (1-\\theta _t)\\;,$ we have that $y^h_t n^h_t=-e^{-\\rho t}/\\rho $ and consequently: $\\hat{c}^h_t=\\rho n^h_t.$ NB: The fact that the optimal rate of consumption is proportional to the wealth is to be expected when using logarithmic utility.",
"Plugging the expressions (REF ) for $\\zeta ^{0,h}_t$ and $\\zeta ^h_t$ into the pricing equation (REF ), we find: $\\frac{a-\\iota }{q} +\\frac{p+q}{q}[\\mu ^M+(\\sigma ^0-\\sigma ^M)\\sigma ^M]=\\frac{p+q}{q}\\sigma ^M\\Bigl (\\sigma ^0+\\bigl (\\frac{p}{q}-\\theta ^h_t\\frac{p+q}{q}\\bigr )\\sigma ^M\\Bigr )+\\sigma ^2 (1-\\theta _t)$ from which we derive $1-\\theta ^h_t=\\frac{\\frac{a-\\iota }{q} + \\frac{p+q}{q}\\mu ^M}{\\bigl (\\frac{p+q}{q}\\bigr )^2(\\sigma ^M)^2+\\sigma ^2}.$ Using the fraction of nominal wealth defined in (REF ) this gives: $1-\\theta ^h_t=\\frac{\\frac{a-\\iota }{q}(1-\\vartheta ) +\\mu ^M}{(\\sigma ^M)^2+\\sigma ^2(1-\\vartheta )^2}(1-\\vartheta ).$ So not only is the optimal portfolio the same for all the households, but we also learn that it is a constant.",
"Moreover: $\\zeta ^{0,h}_t=\\sigma ^0-\\sigma ^M +\\sigma ^M\\frac{1-\\theta ^h_t}{1-\\vartheta }=\\sigma ^0-\\sigma ^M +\\sigma ^M\\frac{\\frac{a-\\iota }{q}(1-\\vartheta ) +\\mu ^M}{(\\sigma ^M)^2+\\sigma ^2(1-\\vartheta )^2},$ and inserting this value of $\\zeta ^{0,h}_t$ into the formula (REF ) we get: $r^h_t=\\phi (\\iota )-\\delta - [ \\mu ^M+\\sigma ^M(\\sigma ^0-\\sigma ^M)] -(\\sigma ^0-\\sigma ^M)\\Bigl (\\sigma ^0-\\sigma ^M +\\sigma ^M\\frac{\\frac{a-\\iota }{q}(1-\\vartheta ) +\\mu ^M}{(\\sigma ^M)^2+\\sigma ^2(1-\\vartheta )^2}\\Bigr ),$ which shows that the individual interest rate is in fact the same constant for all the households.",
"The goods market clears if total output $aK_t$ equals the sum of investment $\\iota _t K_t$ and consumption $C_t$ .",
"So the overall consumption $C_t=\\int c^h_t \\lambda (dh)$ should be equal to $(a-\\iota _t)K_t$ since $aK_t$ represents the overall production and $\\iota _tK_t$ represents the overall reinvestment in capital.",
"If we recall that we are using logarithmic utility, we saw that the optimal consumption was proportional to the wealth so: $C_t=\\int c^h_t \\lambda (dh)=\\rho \\int n^h_t\\lambda (dh)=\\rho N_t=\\rho (p+q)K_t$ so that the clearing condition amounts to $\\rho (p+q)=a-\\iota _t$ which gives $\\frac{a-\\hat{\\iota }}{q}= \\frac{\\rho }{1-\\vartheta }.$ The capital market clears if aggregate capital demand equals capital supply $K_t$ , in other words if: $1-\\theta _t\\frac{N_t}{q}=K_t$ and using the fact that $N_t=(p+q)K_t$ we get: $1-\\hat{\\theta }_t=\\frac{q}{p+q}=1-\\vartheta .$ The money market clears by Walras law.",
"Using the clearing condition (REF ) and the optimal value of $\\hat{\\iota }_t$ (REF ) we get: $q=(1-\\vartheta )\\frac{1+\\kappa a}{1-\\vartheta +\\kappa \\rho },$ from which we derive: $\\hat{\\iota }=\\frac{(1-\\vartheta ) a -\\rho }{1-\\vartheta +\\kappa \\rho },\\qquad \\text{and}\\qquad p=\\vartheta \\frac{1-\\kappa a }{1-\\vartheta +\\kappa \\rho }.$ Finally, injecting (REF ) and (REF ) into the pricing equation (REF ) we get: $1-\\vartheta =\\sqrt{\\frac{\\rho +\\mu ^M-(\\sigma ^M)^2}{\\sigma ^2}}.$ which shows that a stationary (meaning the processes $(p_t)_{t\\ge 0}$ and $(q_t)_{t\\ge 0}$ are deterministic and constant given by the real numbers $p$ and $q$ ) general equilibrium is possible if $\\rho +\\mu ^M-(\\sigma ^M)^2 >0\\qquad \\text{and}\\qquad \\sigma >\\sqrt{\\rho +\\mu ^M-(\\sigma ^M)^2}.$ Since the state variable of an individual household is its wealth $n^h_t$ , the typical interaction one should expect if this general equilibrium can be recast as a mean field game should be the aggregate wealth $N_t$ .",
"So in the presence of the common noise $W^0_t$ one should fix the flow of conditional distributions of the wealth $n^h_t$ given the filtration of the common noise, and search for the best response of this household.",
"In other words, given the knowledge of $(N_t)_{t\\ge 0}$ which is a stochastic process adapted to the filtration $\\mathbb {F} ^0$ of ${\\mathbf {W}}^0$ , the individual household should find optimal investment rate in physical capital $(\\hat{\\iota }_t)_{t\\ge 0}$ , optimal investment portfolio $(\\hat{\\theta }_t)_{t\\ge 0}$ , and optimal consumption rate $(\\hat{c}_t)_{t\\ge 0}$ , to maximize its long-run discounted expected utility of consumption (REF ).",
"This is exactly what was done in the section dealing with the individual household optimization problem.",
"The next step of the MFG paradigm is the fixed point step according to which one tries to identify a flow of conditional distributions which ends up being the flow of conditional distributions of the solution of the optimization problem underpinning the search for the best response.",
"In typical macro-economic general equilibrium problems, individual optimizations are performed assuming that the aggregates are known.",
"If aggregates can be interpreted as means of some state variables, fixing the aggregates amounts to fixing the distributions of these state variables.",
"In this example, assuming the knowledge of $(N_t)_{t\\ge 0}$ is the same thing as assuming the knowledge of $(K_t)_{t\\ge 0}$ since as we saw, $N_t=\\rho (p+q)K_t$ , which in turn, is assuming the knowledge of the process $(\\bar{\\iota }_t)_{t\\ge 0}$ representing the aggregate investment rate in physical capital.",
"This is the mean field interaction appearing explicitly in the dynamics (REF ) of the state of the individual household.",
"Since the individual household optimal investment rate in capital is constant as given by Tobin's $q$ equation (REF ), a necessary condition for the fixed point step is that $\\bar{\\iota }_t=\\hat{\\iota }_t$ .",
"Added to the necessary conditions of optimality (which we derived from the Pontryagin stochastic maximum principle) and the capital market clearing condition, this fixed point step leads to the equilibrium solution under the conditions (REF ).",
"Because we chose to restrict ourselves to the search for a stationary general equilibrium in which the processes $(p_t)_{t\\ge 0}$ and $(q_t)_{t\\ge 0}$ are deterministic and constant given by the real numbers $p$ and $q$ , the deterministic nature of most of the characteristics of the equilibria are rather anti-climatic, and the reformulation of the solution as the search for Nash equilibria in a mean field game is rather contrived.",
"We chose to present this model because of the presence of both idiosyncratic and common shocks.",
"We refer the interested readert to [14], [15] for extensions with deeper financial meaning.",
"The next example will be more illustrative of the deep connection with the paradigm of mean field games.",
"While it does not involve idiosymcratic shocks, it involves two populations and this will give us a chance to highlight the possible benefits of a mean field game reformulation of the model."
],
[
"Economy with Two Types of Agents We present the analysis of the model discussed in [15] mutatis mutandis.",
"We consider an economy with a continuum of households and experts.",
"We denote by $I^h$ (resp.",
"$I^e$ ) the space of households (resp.",
"experts).",
"Typically, we choose $I^h=I^e=[0,1]$ which we assume to be equipped with its Borel $\\sigma $ -field.",
"We shall alo use continuous probability measures $\\lambda ^h$ and $\\lambda ^e$ on $I^h$ and $I^e$ respectively.",
"Again, for practical purposes, and modulo the contents of Remark REF at the beginning of the discussion of the previous model, we can think of them both as equal to the Lebesgue measure on $[0,1]$ .",
"In this economy, households consume and lend money to experts.",
"On the other end, experts borrow money from households, invest in the production of a single good, and consume.",
"The goal of each agent is to maximize their long run expected utility.",
"In this model, all agents use the logarithmic utility function $u(c)=\\log c$ .",
"So if we denote by $c^e_t$ and $c^h_t$ the consumptions at time $t$ of expert $e$ and household $h$ respectively, the optimization problem is: $\\sup _{(c^i_t)_{t\\ge 0}} \\mathbb {E} \\Bigl [\\int _0^\\infty e^{-\\rho t}\\log c^i_t\\;dt\\Bigr ],\\qquad i=e,h,$ where $\\rho >0$ is a discount factor common to the two classes of agents.",
"To be consistent with the computation done throughout this chapter, we shall in fact minimize the negative of the above expected utility of consumption."
],
[
"If we denote by $n^h_t$ the wealth of household $h$ at time $t$ , we have: $dn^h_t= r_tn^h_t dt -c^h_tdt$ Here, the process $(r_t)_{t\\ge 0}$ represents the interest rate common to all agents.",
"It is one of the stochastic processes to be determined endogenously.",
"The Hamiltonian of the optimization problem of a generic household reads: $H(t,n,\\xi ,c)=(r_t n-c)\\xi - e^{-\\rho t}u(c)$ if we use the notation $\\xi $ for the adjoint variable (sometimes called the co-state) which we shall restrict to be negative.",
"The necessary part of the Pontryagin maximum principle suggests to minimize the Hamiltonian with respect to the control variable $c$ .",
"This gives the First Order Condition (FOC): $\\partial _cH=0\\quad \\Leftrightarrow \\quad -\\xi -e^{-\\rho t}u^{\\prime }(c)=0 \\quad \\Leftrightarrow \\quad c=(u^{\\prime })^{-1}\\bigl (-e^{\\rho t}\\xi \\bigr )$ so that in the case of logarithmic utility, the optimal consumption rate is given by $\\hat{c}_t=-e^{-\\rho t}\\frac{1}{\\xi _t}$ where the adjoint function $t\\mapsto \\xi _t$ solves the adjoint equation: $d\\xi _t=-\\partial _nH dt= -r_t \\xi _t dt.$ The differentiation product rule gives: $d (\\xi _t n^h_t)= c^h_t\\xi _t dt=-e^{-\\rho t} dt$ implying that $\\xi _t n^h_t=-e^{-\\rho t}/\\rho $ and consequently: $\\hat{c}^h_t=\\rho n^h_t.$ As noted in the previous example, the fact that the optimal rate of consumption is proportional to the wealth (and is independent of the interest rate) is a well known property of the logarithmic utility function.",
"If at time $t$ we denote by $n^e_t$ the wealth of expert $e$ , by $\\theta ^e_t$ the proportion of self worth invested in bonds (i.e.",
"borrowed from the households, so $\\theta ^e_t\\le 0$ ) and by $\\iota ^e_t$ the investment in physical capital, we have: $dn^e_t= \\theta ^e_tn^e_t\\;r_t dt +(1-\\theta ^e_t)n^e_t \\; dr^k_t(\\iota ^e_t)-c^e_tdt$ where $r^k_t(\\iota ^e_t)$ denotes the return from the investment $\\iota ^e_t$ in physical capital.",
"The capital stock of a generic expert $e$ evolves according to the equation: $\\frac{dk^e_t}{k^e_t}= (\\phi (\\iota ^e_t) -\\delta ) dt +\\sigma dW^0_t$ where $\\delta >0$ is a depreciation rate, and the function $\\phi $ reflects adjustment costs in capital stock.",
"It is assumed to satisfy $\\phi (0)=0$ , $\\phi ^{\\prime }(0)=1$ , $\\phi ^{\\prime }(\\cdot )>0$ and $\\phi ^{\\prime \\prime }(\\cdot )<0$ .",
"Its concavity captures technological illiquidity.",
"The volatility $\\sigma >0$ is a positive constant and ${\\mathbf {W}}^0=(W^0_t)_{t\\ge 0}$ is a Wiener process modeling random shocks.",
"Note that this is the same process for all the experts.",
"This is an instance of what we call a common noise.",
"There is no source of idiosyncratic noise in this model.",
"Let the price $q_t$ at time $t$ of one unit of capital be an Itô process satisfying $\\frac{dq_t}{q_t}= \\mu ^q_tdt +\\sigma ^q_t dW^0_t$ for two processes $(\\mu ^q_t)_{t\\ge 0}$ and $(\\sigma ^q_t)_{t\\ge 0}$ adapted to the filtration $\\mathbb {F} ^0$ of the common noise, which will be specified later on.",
"The return on capital $r^k_t(\\iota ^e_t)$ is defined as: $dr^k_t(\\iota ^e_t)= \\frac{a-\\iota ^e_t}{q_t}dt + \\frac{d(q_tk^e_t)}{q_tk^e_t}.$ The first term in the right hand side represents the dividend yield while the second one gives the capital gain.",
"Using the definitions (REF ) and (REF ) and Itô's formula for the differential of a product we get: $dr^k_t(\\iota ^e_t)=\\Bigl [ \\frac{a-\\iota ^e_t}{q_t} +\\phi (\\iota ^e_t)-\\delta +\\mu ^q_t+\\sigma \\sigma ^q_t\\Bigr ]dt + (\\sigma +\\sigma ^q_t)dW^0_t.$ Plugging this formula in the dynamics (REF ) of the wealth of a generic expert we get: $dn^e_t= \\Bigl [\\theta ^e_tn^e_t\\;r_t +(1-\\theta ^e_t)n^e_t \\; \\Bigl ( \\frac{a-\\iota ^e_t}{q_t} +\\phi (\\iota ^e_t)-\\delta +\\mu ^q_t+\\sigma \\sigma ^q_t \\Bigr )-c^e_t\\Bigr ]dt + (1-\\theta _t)n^e_t(\\sigma +\\sigma ^q_t)dW^0_t.$ This equation should be viewed as giving the dynamics of the state variable $n^e_t$ as controlled by $(c^e_t,\\iota ^e_t,\\theta ^e_t)$ .",
"As before, we use Pontryagin stochastic maximum principle to solve the optimization of the expected utility of consumption.",
"For the sake of simplicity of notation, we skip the superscript ${}^e$ throughout the remaining of this subsection.",
"No confusion is possible since we are only dealing with the expert optimization problem.",
"The Hamiltonian of this optimization problem reads: $\\begin{split}&H(t,n,\\xi ,\\zeta ,c,\\iota ,\\theta )= \\Bigl [\\theta n\\;r_t +(1-\\theta )n \\; \\Bigl ( \\frac{a-\\iota }{q_t} +\\phi (\\iota )-\\delta +\\mu ^q_t+\\sigma \\sigma ^q_t \\Bigr )-c\\Bigr ]\\xi \\\\&\\hspace{125.0pt}- (1-\\theta ) n(\\sigma +\\sigma ^q_t)\\zeta \\xi - e^{-\\rho t}u(c)\\end{split}$ where for reasons which will become clear soon, we used the notations $\\xi $ (which is assumed to be negative) and $-\\xi \\zeta $ for the adjoint variables.",
"We now have three First Order Conditions.",
"Since $\\xi \\le 0$ and $(1-\\theta )\\ge 1$ we can isolate the contribution of the control $\\iota $ .",
"This leads to the maximization of the quantity $(a-\\iota )/q_t+\\phi (\\iota )$ which leads to\" $-\\frac{1}{q_t}+\\phi ^{\\prime }(\\iota )=0 \\quad \\Leftrightarrow \\quad \\iota =(\\phi ^{\\prime })^{-1}\\bigl (-\\frac{1}{q_t}\\bigr ).$ In the case of the function $\\phi (\\iota )=(1/\\kappa )\\log (1+\\kappa \\iota )$ used in [15], we get: $\\hat{\\iota }_t=\\frac{1}{\\kappa }(q_t-1).$ In any case, this value is the same for all the experts $e$ , and as a control process, it is adapted to the filtration of the common noise.",
"As before: $\\partial _cH=0\\quad \\Leftrightarrow \\quad -\\xi -e^{-\\rho t}u^{\\prime }(c)=0 \\quad \\Leftrightarrow \\quad c=(u^{\\prime })^{-1}\\bigl (-e^{\\rho t}\\xi \\bigr )$ so that in the case of logarithmic utility, the optimal consumption rate is given by $\\hat{c}_t=-e^{-\\rho t}/\\xi _t$ where the adjoint process $(\\xi _t)_{t\\ge 0}$ solves the adjoint equation: $d\\xi _t=-\\partial _nH dt-\\xi _t\\zeta _t dW^0_t.$ Notice that the FOC $\\partial _\\theta H=0$ gives: $r_t= \\frac{a-\\iota }{q_t} +\\phi (\\iota )-\\delta +\\mu ^q_t+\\sigma \\sigma ^q_t -\\zeta _t(\\sigma +\\sigma ^q_t)$ and we shall see below that this formula will help us identify the individual expert optimal investment $\\hat{\\theta }^e_t$ in terms of the processes $(r_t)_{t\\ge 0}$ , $(q_t)_{t\\ge 0}$ , $(\\mu ^q_t)_{t\\ge 0}$ , $(\\sigma ^q_t)_{t\\ge 0}$ .",
"Computing $\\partial _nH$ from (REF ) we get $\\partial _nH(t,n,\\xi ,\\zeta ,c,\\iota ,\\theta )= \\Bigl [\\theta \\;r_t +(1-\\theta ) \\; \\Bigl ( \\frac{a-\\iota }{q_t} +\\phi (\\iota )-\\delta +\\mu ^q_t+\\sigma \\sigma ^q_t \\Bigr )\\Bigr ]\\xi -(1-\\theta )(\\sigma +\\sigma ^q_t)\\zeta \\xi $ and using (REF ) we get $\\partial _nH(t,n,\\xi ,\\zeta ,c,\\iota ,\\theta )=\\xi r_t$ and the adjoint equation rewrites: $\\frac{d\\xi _t}{\\xi _t}= -r_t\\;dt -\\zeta _t dW^0_t$ which justifies our choice of the form of the second adjoint variable.",
"Applying Itô formula to (REF ) and (REF ) we get: $\\frac{d (\\xi _t n_t)}{\\xi _t n_t}= -\\frac{c_t}{n_t} dt+ \\bigl [-\\zeta _t+(1-\\theta _t)(\\sigma +\\sigma ^q_t)\\bigr ]dW^0_t$ so that, choosing $\\zeta _t=(1-\\theta _t)(\\sigma +\\sigma ^q_t),$ we find that as in the case of the computation of the optimal consumption rate of the households, $\\xi _t n_t=-e^{-\\rho t}/\\rho $ and consequently: $\\hat{c}^e_t=\\rho n^e_t.$ So the fact that the optimal rate of consumption is proportional to the wealth was not affected by the presence of the random shocks.",
"It is typical in the case of logarithmic utility.",
"Plugging our choice (REF ) for $\\zeta _t$ in (REF ) we find: $r_t= \\frac{a-\\hat{\\iota }_t}{q_t} +\\phi (\\hat{\\iota }_t)-\\delta +\\mu ^q_t+\\sigma \\sigma ^q_t -(1-\\theta ^e_t)(\\sigma +\\sigma ^q_t)^2$ from which we can easily extract $\\hat{\\theta }^e_t$ as desired.",
"The next step in the search for a general equilibrium for this macro-economic model is to articulate the constraints imposed by the need to have all the markets clear, and to show that one can identify processes $(r_t)_{t\\ge 0}$ , $(q_t)_{t\\ge 0}$ , $(\\mu ^q_t)_{t\\ge 0}$ , $(\\sigma ^q_t)_{t\\ge 0}$ satisfying these constraints and allowing the simultaneous optimizations of all the agents.",
"Clearing is best expressed in terms of aggregate quantities.",
"For each $i\\in \\lbrace h,e\\rbrace $ , we denote by $C^i_t$ the aggregate consumption for the agents of type $i$ .",
"Formally we write: $C^h_t=\\int _{I^h}\\hat{c}^h_t \\;\\lambda ^h(dh),\\quad \\text{and}\\quad C^e_t=\\int _{I^e}\\hat{c}^e_t \\;\\lambda ^e(de).$ Given (REF ) and (REF ) we see that $C^h_t=\\rho N^h_t$ and $C^e_t=\\rho N^e_t$ where $N^h_t$ and $N^e_t$ are the aggregate worths of the populations of households and experts respectively, i.e.",
"$N^h_t=\\int _{I^h}\\hat{n}^h_t \\;\\lambda ^h(dh),\\quad \\text{and}\\quad N^e_t=\\int _{I^e}\\hat{n}^e_t \\;\\lambda ^e(de).$ If we denote by $K_t$ the aggregate physical capital in the economy at time $t$ , i.e.",
"$K_t=\\int _{I^e}k^e_t \\;\\lambda ^e(de),$ the aggregate wealth in the economy is equal to $q_tK_t$ Clearing of the loan market requires that at each time $t$ , the aggregate debt of the experts, say $D^e_t$ , be equal to the aggregate loans of the households, so that: $D^e_t=\\int _{I^h}n^h_t\\lambda ^h(dh)=N^h_t.$ So $N^e_t=q_tK_t-D^e_t=q_tK_t-N^h_t$ which implies that $q_tK_t=N^h_t+N^e_t.$ It will be convenient to use the quantity: $\\eta _t=\\frac{N^e_t}{N^h_t+N^e_t}=\\frac{N^e_t}{q_tK_t}$ representing the wealth share of the experts.",
"Notice that all these aggregate quantities are random since they depend upon the common noise ${\\mathbf {W}}^0$ which does not average out in the computation of the aggregates because it is common to all the agents.",
"Clearing of consumption on the market for goods requires $C_t=(a-\\iota ^e_t)K_t$ in other words $\\rho q_tK_t=(a-\\iota ^e_t(q_t))K_t$ which implies $\\rho q_t=a-\\iota ^e_t(q_t)$ which in turn implies that the process $(q_t)_{t\\ge 0}$ is in fact a positive constant, say $q$ .",
"As a consequence, $\\mu ^q_t\\equiv 0$ and $\\sigma ^q_t\\equiv 0$ , and if we use the function $\\phi (\\iota )=(1/\\kappa )\\log (1+\\kappa \\iota )$ proposed in [15] we get $q=\\frac{1+\\kappa a}{1+\\kappa \\rho }\\qquad \\text{and}\\qquad \\iota ^e=\\frac{a-\\rho }{1+\\kappa \\rho }.$ Capital market clearing yields: $1-\\theta ^e_t=\\frac{q_tK_t}{N^e_t}=\\frac{1}{\\eta _t}$ Knowing that $q_t$ has to be a deterministic constant, we can use the facts that $\\begin{split}\\frac{dN^e_t}{N^e_t}&= \\frac{dn^e_t}{n^e_t}\\\\&= \\Bigl [r_t +(1-\\theta ^e_t)^2\\sigma ^2-\\frac{c^e_t}{n^e_t}\\Bigr ]dt + (1-\\theta ^e_t)\\sigma dW^0_t\\end{split}$ and $\\begin{split}\\frac{dK^e_t}{K^e_t}&= \\frac{dk^e_t}{k^e_t}\\\\&= \\Bigl [ r_t +(1-\\theta ^e_t) \\sigma ^2-\\rho \\Bigr ]dt + \\sigma dW^0_t\\end{split}$ to derive from Itô's formula that: $\\begin{split}\\frac{d\\eta _t}{\\eta _t}&=\\frac{d\\bigl (N^e_t/K_t\\bigr )}{N^e_t/K_t}\\\\&=\\Bigl [ -\\frac{c^e_t}{n^e_t}+\\rho +(\\theta ^e_t)^2\\sigma ^2\\Bigr ] dt -\\theta ^e_t\\sigma dW^0_t\\\\&=(\\theta ^e_t)^2\\sigma ^2 dt -\\theta ^e_t\\sigma dW^0_t\\end{split}$ which we can rewrite as $d\\eta _t=\\sigma ^2\\frac{(1-\\eta _t)^2}{\\eta _t} dt +\\sigma (1-\\eta _t)dW^0_t$ if we use the capital market clearing condition (REF ).",
"This is a stochastic differential equation on the open interval $(0,1)$ .",
"According to Feller's theory of one dimensional diffusions, the scale function $p(x)$ and the speed measure $m(dx)$ are given by: $p(x)=\\frac{1}{2}\\bigl (1-\\frac{1}{2x}\\bigr )\\qquad \\text{and}\\qquad m(dx)=\\frac{8}{\\sigma ^2}\\frac{x^2}{(1-x)^2}dx.$ Feller's explosion test can be computed and it says that if started inside the interval $(0,1)$ , the diffusion remains inside the interval for ever and in fact $\\lim _{t\\rightarrow \\infty }\\eta _t=1$ almost surely.",
"Note also that the drift is always positive, and very large when $\\eta _t$ is small, so up to the fluctuations due to the random shocks (whose sizes $\\sigma (1-\\eta _t)$ decrease as $\\eta _t$ get closer to 1), one should expect that $\\eta _t$ would grow quickly toward 1 and become mostly flat when it gets close to 1.",
"This is illustrated in Figure REF .",
"Figure: Typical sample path of η t \\eta _t.From an economic point of view, this means that the proportion of the wealth held by the experts grows quickly toward a high value close to $100\\%$ , and eventually converges to this $100\\%$ level, leaving the households helpless.",
"Finally, revisiting the constraint (REF ), we see that in equilibrium we must have: $\\begin{split}r_t&= \\frac{a-\\iota ^e_t}{q_t} +\\phi (\\iota ^e_t)-\\delta +\\mu ^q_t+\\sigma \\sigma ^q_t -(1-\\theta ^e_t)\\sigma ^2\\\\&= \\frac{a-\\iota ^e_t}{q_t} +\\phi (\\iota ^e_t)-\\delta -(1-\\theta ^e_t)\\sigma ^2\\\\&= \\frac{a-\\iota ^e_t}{q_t} +\\phi (\\iota ^e_t)-\\delta -\\frac{\\sigma ^2}{\\eta _t}\\\\&=\\rho +\\frac{1}{\\kappa }\\log \\Bigl ( \\frac{1+\\kappa a}{1+\\kappa \\rho }\\Bigr )-\\delta -\\frac{\\sigma ^2}{\\eta _t}\\end{split}$ if we use the function $\\phi (\\iota )=(1/\\kappa )\\log (1+\\kappa \\iota )$ proposed in [15].",
"NB : This interest rate is negative for small values of $\\eta _t$ .",
"Given the common random shock process ${\\mathbf {W}}^0$ , we solve the stochastic differential equation (REF ) to find a process $(\\eta _t)_{t\\ge 0}$ which stays in $(0,1)$ .",
"Next we define the short interest rate process $(r_t)_{t\\ge 0}$ by (REF ) and with the constant price of capital $q$ given by (REF ) all the agents can maximize their expected long run discounted utility of consumption simultaneously and all the markets clear.",
"These are the elements of the desired equilibrium.",
"The model described in this section is the epitome of an infinite-horizon, two-population Mean Field Game (MFG) with a common noise and no idiosyncratic noise.",
"We make it explicit directly in the limiting mean field limit without motivating it with the description of the finite player analogue.",
"Because we do not know of examples of this type treated in the existing literature, we formulate a rigorous definition in the spirit and with the notations of [21] and [22], and we accommodate the possibility of idiosyncratic random shocks.",
"The sources of random shocks are three independent $\\mathbb {R} ^d$ -valued Wiener processes ${\\mathbf {W}}^1=(W^1_t)_{t\\ge 0}$ , standing for the idiosyncratic noise for the players of the first population, ${\\mathbf {W}}^2=(W^2_t)_{t\\ge 0}$ standing for the idiosyncratic noise for the players of the second population, and ${\\mathbf {W}}^0=(W^0_t)_{t\\ge 0}$ standing for the noise common to all the players.",
"For $i=0,1,2$ , we denote by $\\mathbb {F} ^i=(\\mathcal {F} ^i_t)_{t\\ge 0}$ the filtration generated by ${\\mathbf {W}}^i$ .",
"The MFG problem can be formulated as the conjunction of the following two bullet points: For any two probability measure $\\mu ^1_0$ and $\\mu ^2_0$ on $\\mathbb {R} ^d$ and two stochastic flows of (random) probability measures $\\mu ^1=(\\mu _{t}^1)_{t>0}$ and $\\mu ^2=(\\mu _{t}^2)_{t>0}$ on $\\mathbb {R} ^d$ , both adapted to the filtration $\\mathbb {F} ^0$ of the common noise, solve the two optimal control problems: $\\sup _{\\alpha ^1}J^{1,\\mu ^1,\\mu ^2}(\\alpha ^1)\\qquad \\textrm {and}\\qquad \\sup _{\\alpha ^2} J^{2,\\mu ^1,\\mu ^2}(\\alpha ^2)$ over $\\mathbb {F} ^1$ -progressively measurable $\\mathbb {R} ^{k_1}$ -valued processes $\\alpha ^1=(\\alpha ^1_t)_{t\\ge 0}$ and $\\mathbb {F} ^2$ -progressively measurable $\\mathbb {R} ^{k_2}$ -valued processes $\\alpha ^2=(\\alpha ^2_t)_{t\\ge 0}$ , where $\\begin{split}&J^{1,\\mu ^1,\\mu ^2}(\\alpha ^1)= \\mathbb {E} \\biggl [ \\int _{0}^\\infty e^{-\\rho t} f_{1}\\bigl (t,X_{t}^1,\\mu _{t}^1,\\mu _{t}^2,\\alpha _{t}^1\\bigr ) dt \\biggr ],\\\\&J^{2,\\mu ^1,\\mu ^2}(\\alpha ^2)= \\mathbb {E} \\biggl [ \\int _{0}^\\infty e^{-\\rho t} f_{2}\\bigl (t,X_{t}^2,\\mu _{t}^1,\\mu _{t}^2,\\alpha _{t}^2\\bigr ) dt \\biggr ],\\end{split}$ with $\\begin{split}dX_{t}^1 &= b_{1}\\bigl (t,X_{t}^1,\\mu _{t}^1,\\mu _{t}^2,\\alpha _{t}^1\\bigr ) dt + \\sigma _{1}\\bigl (t,X_{t}^1,\\mu _{t}^1,\\mu _{t}^2,\\alpha _{t}^1\\bigr ) dW_{t}^1\\\\&\\hspace{125.0pt}+ \\sigma _{1,0}\\bigl (t,X_{t}^1,\\mu _{t}^1,\\mu _{t}^2,\\alpha _{t}^1\\bigr ) dW_{t}^0,\\\\dX_{t}^2 &= b_{2}\\bigl (t,X_{t}^2,\\mu _{t}^1,\\mu _{t}^2,\\alpha _{t}^2\\bigr ) dt + \\sigma _{2}\\bigl (t,X_{t}^2,\\mu _{t}^1,\\mu _{t}^2,\\alpha _{t}^2\\bigr ) dW_{t}^2\\\\&\\hspace{125.0pt}+ \\sigma _{2,0}\\bigl (t,X_{t}^2,\\mu _{t}^1,\\mu _{t}^2,\\alpha _{t}^2\\bigr ) dW_{t}^0,\\end{split}$ for $t >0$ , and $\\mathcal {L} (X_{0}^1)=\\mu _{0}^1$ and $\\mathcal {L} (X_{0}^2)=\\mu _{0}^2$ .",
"Find $\\mathbb {F} ^0$ -adapted random flows $\\mu ^1=(\\mu _{t}^1)_{t>0}$ and $\\mu ^2=(\\mu _{t}^2)_{t>0}$ such that conditioned on the past of the common noise, almost surely, the marginal distributions of the solutions of the above stochastic control problems coincide with the elements of the probability flows we started from.",
"In other words: $\\forall t \\in [0,T], \\quad \\mu _{t}^1=\\mathcal {L} \\bigl (\\hat{X}_{t}^{1,\\mu ^1,\\mu ^2}|\\mathcal {F} ^0_t\\bigr ),\\quad \\mu _{t}^2=\\mathcal {L} \\bigl (\\hat{X}_{t}^{2,\\mu ^1,\\mu ^2}|\\mathcal {F} ^0_t\\bigr ),$ if we denote by $\\hat{{\\mathbf {X}}}^{1,\\mu ^1,\\mu ^2}$ and $\\hat{{\\mathbf {X}}}^{2,\\mu ^1,\\mu ^2}$ the solutions of the above optimal control problems.",
"In practice, specific assumptions are required on the coefficients $b_1$ , $\\sigma _1$ , $\\sigma _{1,0}$ , $b_2$ , $\\sigma _2$ , $\\sigma _{2,0}$ and the running reward functions $f_1$ and $f_2$ for the stochastic differential equations determining the generic states $X^1_t$ and $X^2_t$ of the two populations to have solutions, and the expected costs to make sense.",
"Moreover, much more restrictive assumptions are required for the existence of a couple of measure flows satisfying the fixed point conditions (2).",
"Our goal is to explain how the macro-economic model presented in the previous subsection is an instance of such a Mean Field Game.",
"The individuals in the first population are the households and the individuals of the second population are the experts.",
"In the present situation, the idiosyncratic noises ${\\mathbf {W}}^1$ and ${\\mathbf {W}}^2$ are not present so we can take $\\sigma _1=\\sigma _2=0$ .",
"The generic states $X^1_t$ and $X^2_t$ are the wealths $n^h_t$ and $n^e_t$ .",
"As a result, the interpretation of the fixed point condition in item (2) of the above definition of a solution to the MFG is that in equilibrium, the (random) measures $\\mu ^1_t$ and $\\mu ^2_t$ should be the conditional distributions of the states $n^h_t$ and $n^e_t$ given the past $\\lbrace W^0_s;\\;0\\le s\\le t\\rbrace $ of the common noise.",
"In fact as we are about to see, the forms of the coefficients of the state equations as well as of the reward functions are such that the means (first moments) of the probability measures $\\mu ^1_t$ and $\\mu ^2_t$ are sufficient statistics.",
"So instead of working with the full random measures $\\mu ^1_t$ and $\\mu ^2_t$ , we can restrict ourselves to their means $\\bar{\\mu }^1_t=\\int x\\mu ^1_t(dx)$ and $\\bar{\\mu }^2_t=\\int x\\mu ^2_t(dx)$ which are still functions of the past of the common noise.",
"According to item (1) of the above definition of a solution to the MFG, for each couple of stochastic flows of (random) probability measures $\\mu ^1=(\\mu _{t}^1)_{t>0}$ and $\\mu ^2=(\\mu _{t}^2)_{t>0}$ adapted to the filtration $\\mathbb {F} ^0$ of the common noise ${\\mathbf {W}}^0$ , we need to be able to solve the two optimal control problems before we tackle the fixed point problem stated in item (2) of this definition.",
"Let me argue that this is exactly what we did in the optimization steps of the construction of a general equilibrium for the macro-economic model studied earlier in this section.",
"For the following discussion to be more transparent, we should think of the means $\\bar{\\mu }^1_t$ and $\\bar{\\mu }^2_t$ as the stochastic processes $(N^h_t)_{t\\ge 0}$ and $(N^e_t)_{t\\ge 0}$ of the aggregate wealths in the households and experts populations.",
"At each time $t\\ge 0$ , knowing the averages $\\bar{\\mu }^1_t$ and $\\bar{\\mu }^2_t$ we can compute $\\eta _t=\\bar{\\mu }^2_t/(\\bar{\\mu }^1_t+\\bar{\\mu }^2_t)$ and then the quatity $r_t$ from formula (REF ).",
"Choosing the control $\\alpha ^1_t=c^h_t$ , the drift function $b_1(t,n^h,\\mu ^1_t,\\mu ^2_t,\\alpha ^1_t)=r_t n^h_t-c^h_t$ , the volatility $\\sigma _{1,0}(t,n^h,\\mu ^1_t,\\mu ^2_t,\\alpha ^1_t)=0$ , and the reward function $f_1(t,n^h,\\mu ^1_t,\\mu ^2_t,\\alpha ^1_t)=\\log c^h_t$ , for the utility of the household, we see that the first optimal control problem in item (1) is exactly what we called the optimization problem of the households which we solved using the Pontryagin maximum principle.",
"Notice that in the dynamics of the state, namely in the function $b_1$ , the dependence upon the random probability measures $\\mu ^1_t$ and $\\mu ^2_t$ appears implicitly through the process $r_t$ .",
"Choosing the control $\\alpha ^2_t=(\\iota _t,\\theta _t,c^e_t)$ , the drift function $b_2(t,n^e,\\mu ^1_t,\\mu ^2_t,\\alpha ^2_t)=\\theta _t r_t n^e_t-c^e_t+ (1-\\theta _t)n^e_t\\bigl (\\frac{a-\\iota }{q}+\\phi (\\iota _t)-\\delta \\bigr )$ , the volatility $\\sigma _{2,0}(t,n^e,\\mu ^1_t,\\mu ^2_t,\\alpha ^2_t)=(1-\\theta _t)n^e_t\\sigma $ , and the reward function $f_2(t,n^e,\\mu ^1_t,\\mu ^2_t,\\alpha ^2_t)=\\log c^e_t$ we see that the second optimal control problem in item (1) is exactly what we called the optimization problem of the experts which we solved using the Pontryagin maximum principle.",
"As before, the dependence upon the random probability measures $\\mu ^1_t$ and $\\mu ^2_t$ appears implicitly through the process $r_t$ .",
"The fixed point condition in item (2) of the definition of the MFG guarantees that $\\bar{\\mu }^1_t=N^h_t$ and $\\bar{\\mu }^2_t=N^e_t$ so that the process $(\\eta _t)_{t\\ge 0}$ is indeed the wealth share of the experts, and the process $(r_t)_{t\\ge 0}$ is indeed the short interest rate process and all the clearing conditions are satisfied."
],
[
"So the Nash equilibrium of the two-population Mean Field Game coincides with the general equilibrium constructed in the previous subsections.",
"But what have we gained?",
"Aren't we making matters worse?",
"Johann Wolfgang von Goethe once said \"Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.\"",
"and I would add that it has to be even worse when the mathematician is French !",
"Still, once the macro-economic general equilibrium model is reformulated as a Mean Field Game, the technology developed during the last 15 years to analyze these models can be brought to bear to gain insight in these macro-economic models.",
"Among them, 1) numerical methods to compute solutions and statics, 2) convergence of finite populations models and quantification of the finite size effects, 3) analysis of the uniqueness of the equilibria, or lack thereof, 4) analysis of the centralized optimization counterpart and comparisons of the global welfares (e.g.",
"computation of the price of anarchy), $\\ldots $ ."
],
[
"Moral Hazard & Contract Theory After reviewing some of the major historical developments in contract theory, and explaining the basics of the models, we concentrate on two recent papers using mean field games for the purpose of extending the reach of possible applications of the theory to large populations of agents.",
"Because this topic was not discussed as a possible application of Mean Field Games in the books [21], [22], we spend a significant amount of time reviewing the underpinnings of the economic theory as well as some of the recent applications involving mean field models.",
"Put in layman's terms, the purpose of economic contract theory is to address questions of the type: a) How should a government control a flu outbreak by encouraging citizens to vaccinate?",
"b) How should taxes be levied to influence people's consumption, saving and investment decisions?",
"c) How should an employer incentivize and compensate their employees in order to boost productivity?",
"While these are mundane objectives, they should shed some initial light on the type of problems contract theory can encompass.",
"The economic lingo we shall use and try to elucidate includes terms like Agency Problem, Contract Theory, Moral Hazard, and Information Asymmetry.",
"The agency problem refers to a conflict of interest between two parties when one of them is expected to behave in the other party's best interests.",
"The purpose of contract theory is to study how an economic agent can design and structure a contractual agreement to incentivize another agent to behave in his or her best interest.",
"This problem is most often set up when the informations available to the two agents are not the same.",
"In economics and in finance, moral hazard refers to a situation when an agent has an incentive to take excess risk because they do not bear the full consequences of that risk.",
"In contract theory and economics, information asymmetry deals with the study of decisions in transactions where one party has more or better information than the other.",
"The principal - agent problem occurs when one agent makes decisions on behalf of another person or entity called the principal.",
"Earliest contributions to this important field of economics concentrated on static one period models.",
"They are attached to the names of Mirrlees [67] and Holmström [58], [59].",
"The first dynamic models were introduced by Holmström and Milgrom [60], [61] and it had to wait almost two decades for Sannikov's breakthrough [72], [73].",
"By focusing on Markov diffusion models in continuous time, and using the weak formulation of stochastic control problems to capture moral hazard, Sannilov created a new wave of interest, especially among financial mathematicians.",
"The book of Cvitanic and Zhang focusing on the use of Backward Stochastic Differential Equations (BSDEs) is a case in point, and the more recent work of Cvitanic, Possamaï and Touzi [40] highlighting the real nature of Sannikov's trick, will be instrumental in the developments we present in this section.",
"NB: Bengt Holmström and Oliver Hart were awarded the Nobel Memorial Prize in Economic Sciences in 2016 for their work on Contract Theory."
],
[
"In a classical contract theory model, two parties are present: the principal who devises a contract, according to which incentives are given to, and/or penalties are imposed on, the agent who may accept the contract and work for the principal.",
"The following are the major assumptions which are usually made implicitly or explicitly in a contract theory model.",
"It is assumed that all the agents are rational in the sense that they behave optimally to maximize their own utilities, controlling the tradeoff between the rewards/penalties they received and the efforts they put in.",
"The principal designs the contract After reviewing the terms of the contract, the agent may walk away.",
"We assume that the agent has a threshold level (e.g.",
"minimum reward, ..... ) below which they think the contract is not worth the effort.",
"We call this threshold the reservation utility of the agent.",
"The principal observes the agent actions only partially.",
"This is the source of information asymmetry and moral hazard in the problem.",
"From a mathematical point of view, the fact that the principal de facto does not see all of the agent behavior forces a special formulation of the optimization problem faced by the principal.",
"As far as we know, Sannikov was the first one to realize that this situation was not accommodated properly by the usual mathematical strong formulation of stochastic games and stochastic control.",
"He proposed to use the weak formulation of stochastic control (also called the martingale approach to stochastic control) to set the principal optimization problem.",
"We now explain what we mean by weak formulation to accommodate moral hazard."
],
[
"We denote by $X^0_t$ the state of the system at time $t$ .",
"We shall give detailed explanations of what the state is in each of the applications considered below.",
"We assume that the system is controlled by one single controller who takes actions to implement their control.",
"The actions of the controller affect the state of the system and the level of the cost / reward.",
"This describes the set-up of a classical control problem.",
"Games and stochastic games differ in the sense that several controllers can act on the same system, i.e.",
"several controllers (players, agents) take actions.",
"In this case, each player has a cost / reward to worry about, and the whole system is affected by the individual actions and their interactions.",
"The weak formulation of these optimization problems, also called the martingale approach, is perfectly suited to information asymmetry and moral hazard.",
"In this set-up the trajectories $t\\hookrightarrow X^0_t(\\omega )$ are not affected by the actions taken by the controllers.",
"Only the likelihood of the scenarios given by the trajectories can be changed by the controls.",
"In other words, only the distribution of the process ${\\mathbf {X}}^0=(X^0_t)_{0\\le t\\le T}$ is affected by controls.",
"In words, one can surmise that the choice of a control is equivalent to the choice of a law for the state process.",
"Still, it may not be completely clear why the presence of moral hazard and the lack of symmetry of information suggest the use of the weak formulation.",
"Given the terms of the contract, the Agent chooses controls which influence their own reward, as well as the rewards of the Principal.",
"Let us denote by $\\alpha _t$ the control (effort level) of the agent at time $t$ .",
"The agent sees the state $X_t$ as it is impacted by the control, and the agent optimizes their cost / reward according to the terms of the contract.",
"On the other end, through the terms of the contract, the Principal chooses the remuneration conditions of the agent.",
"But he or she does so without observing directly the effort level $\\alpha _t$ of the agent, observing only partially the state $X_t$ , and seeing the impact of the agent's effort only through the expected returns he or she is getting, de facto through the values of expected quantities Roughly speaking, the principal guesses the expected value of his or her returns through the distribution of the output of the agent.",
"This is exactly what the weak formulation approach is trying to capture.",
"See below for the mathematical details of this approach.",
"After this quick and informal review of moral hazard and classical contract theory, we now introduce the application we have in mind in this chapter."
],
[
"The main features of the new framework can be summarized as follows: ONE principal who devises one single contract, according to which incentives are given to, and/or penalties are imposed on, MULTIPLE agents who see the same contract and may accept it and work for the principal.",
"As before, the major assumptions can be captured in a small number of bulllet points: The principal designs a contract hoping to maximize their own utility; The agents are rational so they will also try to maximize their own utilities; The agents have their reservation utilities and so they may decide to walk away; The agents are statistically identical in that their contracts are the same; They behave selfishly and maximize their utilities; We assume that they reach a Nash equilibrium; The Principal designs the contract anticipating that the agents will reach a Nash equilibrium.",
"In some sense, we can say that we are considering the problem of one principal contracting a field of agents.",
"As before, the principal does not see (and cannot control) the individual actions taken by the agents.",
"The principal only feels the overall expected value of the reward he or she gets from the actions of the agents.",
"This information asymmetry creates the moral hazard which the model captures through the weak formulation of the optimization problem.",
"Below, we give mathematical details when the state space is the Euclidean space $\\mathbb {R} ^d$ (originally treated by Elie, Mastrolia and Possamaï in [45]) and the case of finite states treated by Carmona and Wang in [29], [28]."
],
[
"The weak formulation is best set-up using the canonical representation of the state process by assuming that $\\Omega $ is the space of continuous functions from $[0,T]$ to $E$ (typically $E=\\mathbb {R} $ or $E=\\mathbb {R} ^d$ ), $W_t(\\omega )=\\omega (t)$ for $t\\ge 0$ gives the coordinate process, $\\mathbb {F} := (\\mathcal {F}_t)_{t \\in [0,T]}$ is the natural filtration generated by the process ${\\mathbf {W}}=(W_t)_{t\\ge 0}$ , $\\mu _0$ is a fixed probability measure on $E$ which serves as the initial distribution of the state, i.e.",
"$X_0\\sim \\mu _0$ , $\\mathcal {F} := \\mathcal {F}_T$ if we work on a finite time horizon $[0,T]$ .",
"$\\mathbb {P}$ is the Wiener measure on $(\\Omega , \\mathbb {F}, \\mathcal {F})$ so that ${\\mathbf {W}}$ is a Wiener process, and $X^0_t=\\xi _0+\\int _0^t\\sigma (s,X^0_\\cdot )dW_s$ for some Lipschitz function $(s,x)\\mapsto \\sigma (s,x)$ which is assumed to be bounded from above and below away from 0 uniformly in $s$ and $x$ .",
"In order to be consistent with the existing literature on the subject, we allow the coefficients to depend upon the past history of the states.",
"We use the notation $X_\\cdot $ and $x_\\cdot $ to denote the whole trajectories of the state.",
"Note also that: $dX^0_t=\\sigma (t,X^0_\\cdot )dW_t,\\qquad \\text{ under \\;} \\mathbb {P} $ irrespective of which control is chosen by the agent.",
"Next we introduce $\\mathbb {A} $ , the space of admissible control strategies (representing the agents effort levels).",
"The elements of $\\mathbb {A} $ are adapted processes $\\alpha = (\\alpha _t)_{0\\le t\\le T}$ which may satisfy further properties to be specified later on.",
"Next we introduce the drift function $b$ (the only part of the dynamics of the state controlled by the agent).",
"We assume that the drift $(t,x,\\alpha )\\mapsto b(t,x_\\cdot ,\\alpha )\\in \\mathbb {R} ^d$ is bounded and progressively measurable.",
"For each admissible control strategy $\\alpha $ , we denote by $\\mathbb {P} ^{\\alpha }$ the state distribution when the effort level of the agent is $\\alpha $ .",
"It is defined by its density with respect to the measure $\\mathbb {P} $ given by: $\\frac{d\\mathbb {P} ^{\\alpha }}{d\\mathbb {P} }=\\mathcal {E} \\Bigl [\\int _0^T\\sigma (t,X^0_\\cdot )^{-1}b(t,X^0_\\cdot ,\\alpha _t)dW_t\\Bigr ]$ where $\\mathcal {E} ({\\mathbf {M}})=\\exp [M_t-\\frac{1}{2}<M,M>_t]$ denotes the Doleans exponential of the continuous square integrable martingale ${\\mathbf {M}}=(M_t)_{0\\le t\\le T}$ .",
"Girsanov theorem implies that: $dX^0_t= b(t,X^0_\\cdot ,\\alpha _t)dt +\\sigma (t,X^0_\\cdot )dW^{\\alpha }_t,\\qquad \\text{ under } \\mathbb {P} ^{\\alpha }$ where $W^{\\alpha }_t=W_t-\\int _0^t\\sigma (s,X_\\cdot )b(s,X_\\cdot ,\\alpha _s)ds$ is a Brownian motion under the measure $\\mathbb {P} ^{\\alpha }$ .",
"So the same state process ${\\mathbf {X}}^0$ , constructed in (REF ) independently of the controls $\\alpha $ , now appears as the state of the process controlled by $\\alpha $ if one looks at its evolution under the probability measure $\\mathbb {P} ^{\\alpha }$ .",
"More on that remark below.",
"We now finalize the weak formulation of the problem by describing the behaviors of the agents in this set-up.",
"The principal offers a contract $(r,\\xi )$ where $r=(r_t)_{0\\le t\\le T}$ is an adapted process representing the payment stream; $\\xi $ is a random variable representing a terminal payment.",
"The agent decides whether or not to accept the contract and work for the principal, and if he or she does accept, chooses an effort level $\\alpha =(\\alpha _t)_{0\\le t\\le T}$ to maximize their expected overall reward: $J^{r,\\xi }(\\alpha ,\\mu _0)=\\mathbb {E} ^{\\mathbb {P} ^{\\alpha }}\\Bigl [U_A(\\xi ) +\\int _0^T[u_A(r_t)-c(t,X_\\cdot ,\\alpha _t)]dt\\Bigr ]$ where $u_A$ is the agent running utility, $U_A$ is the agent terminal utility, and $c(t,x_\\cdot ,\\alpha )$ is the cost for applying the effort level $\\alpha $ at time $t$ when the history of the state is $x_{[0,t]}$ .",
"Given this rational expected behavior of the agent, the optimization problem of the principal can be formulated in the following way: For each contract $(r,\\xi )$ , assuming knowledge of the utility and cost functions of the agent and assuming that the agent is rational, the Principal computes an optimal effort level $\\alpha ^*=(\\alpha ^*_t)_{0\\le t\\le T}$ $\\alpha ^*\\in \\arg \\inf _{\\alpha \\in \\mathbb {A} }J^{r,\\xi }(\\alpha ,\\mu _0)$ which the agent should choose, and then, search for an optimal contract $(r^*,\\xi ^*)$ $(r^*,\\xi ^*)\\in \\arg \\inf _{(r,\\xi )}\\mathbb {E} ^{\\mathbb {P} ^{\\alpha ^*}}\\Bigl [U_P\\Bigl (X_T-\\xi -\\int _0^Tr_tdt\\Bigr )\\Bigr ]$ where $U_P$ is the (terminal) utility of the principal.",
"This is a typical instance of a Stackelberg game between the principal going first and the agent."
],
[
"We assume that the agents, while competing with each other, behave similarly (this is the form of symmetry assumption in force in mean field game models), and because of their large number, their individual influences on the aggregate quantities are negligeable.",
"In these conditions, the optimization problem of the Principal can be formulated as before.",
"Knowing the utility and cost functions of the agents, the Principal assumes that for each contract $(r,\\xi )$ , the agents settle in a Mean Field Nash Equilibrium, so for each $(r,\\xi )$ , the Principal solves the MFG of the agents computes the effort level $\\alpha ^*=(\\alpha ^*_t)_{0\\le t\\le T}$ of the Nash Equilibria he or she can compute then search for an optimal contract $(r^*,\\xi ^*)$ $(r^*,\\xi ^*)\\in \\arg \\inf _{(r,\\xi )}\\mathbb {E} ^{\\mathbb {P} ^{\\alpha ^*}}\\Bigl [U_P\\Bigl (X_T-\\xi -\\int _0^Tr_tdt\\Bigr )\\Bigr ]$ where $U_P$ is the (terminal) utility of the principal.",
"As before, this is a form of Stackelberg game between the principal going first, and the field of agents going next.",
"But now, the dynamics of the state and the cost/reward functions depend upon the distribution of the state in the sense that: $b(t,X_t,\\alpha _t,\\mu _t)\\qquad \\text{and}\\qquad c(t,X_t,\\alpha _t,\\mu _t)$ where $\\mu _t$ is the distribution at time $t$ of the state $X_t$ under $\\mathbb {P} ^{\\alpha }$ .",
"Details about the formulation of the problem and an example of solvable model (essentially from the linear-quadratic family) can be found in [45].",
"Remark 6.1 One of the major shortcomings of the approach described above is the fact that the agents can only control the drifts of their states.",
"This is due to the reliance on Girsanov's change of measure.",
"Allowing the volatility to be controlled requires the representation of the value functions of the optimization problems by so-called 2BSDEs instead of regular Backward Stochastic Differential Equations (BSDEs).",
"The analysis becomes significantly more technical.",
"The interested reader may want to look at the recent work [43] of Elie, Hubert, Mastrolia, and Possamaï for an attempt in this direction.",
"Next we consider the same contract theory model when the state space is finite.",
"We go over a numerical application in detail to illustrate with a few statics, the informational content of the equilibrium when we can actually compute it."
],
[
"The framework of the above discussion is based on the theory of diffusion processes in continuous time and its application to problems of stochastic control.",
"The states are living in Euclidean spaces, their dynamics are modeled by stochastic differential equations, and sophisticated tools from stochastic analysis, starting with Girsanov's theory of changes of measure, are brought to bear in order to formalize the asymmetry of information and the weak formulation appropriate for the optimization of the Principal.",
"Stochastic dynamical systems taking values with finite state spaces are often used in applications for which numerical implementations are of crucial importance.",
"Strangely enough, what seems like a simplification at first, after all finite state spaces should be easier to handle than continuous spaces, may not always make the theoretical analysis easier.",
"Here, we review recent works attempting to port the strategy outlined in the previous section to this case.",
"In particular, we explain how to set up the weak formulation for Mean Field Games with finitely many states, and we implement the steps previously outlined in the diffusion case in the framework of finitely valued state processes.",
"Remark 6.2 Mean Field Games with finitely many states have caught the attention of the MFG crowd throughout the past decade.",
"All the works dated before 2017 are reviewed thoroughly and commented from an historical perspective in [21].",
"See also [22] for a discussion of models with major and minor players, the analysis of which bears much resemblence to some of the steps taken to state and solve contract theory problems with a large number of agents.",
"For the sake of completeness, we mention some of the works on finite state space mean field games which appeared since then, and which we know of.",
"The probabilistic approach to finite state mean field games is advocated by Cecchin and Fischer in [33], Bayraktar and Cohen derived the equivalent of the master equation in [10], and the convergence problem is studied in [42] by Doncel, Gast and Gaujal, and in [34] by Cecchin and Pelino.",
"Finally, we note that these models can exhibit all sorts of behavior as shown for example in [35] where Cecchin, Dai Pra, Fischer and Pelino identify a two-state model without uniqueness.",
"The following is a review of results of Carmona and Wang borrowed from [29], [28]."
],
[
"In this section, we assume that the state space is the finite set $E=\\lbrace e_1,\\dots ,e_m\\rbrace $ , where for the sake of mathematical convenience we shall assume that the $e_i$ 's are the unit vectors of the canonical basis of $\\mathbb {R}^m$ .",
"The state process ${\\mathbf {X}}^0 = (X^0_t)_{0\\le t\\le T}$ will be a continuous-time Markov chain with $m$ states whose sample paths $t\\rightarrow X_t$ are càdlàg, i.e.",
"right continuous with left limits, and continuous at $T$ (i.e.",
"$X_{T-}=X_T$ ).",
"In analogy with the Euclidean state space case, we introduce the following canonical representation: $\\Omega $ is the space of càdlàg functions from $[0,T]$ to $E$ , continuous at $T$ ; $X^0_t(\\omega ) := \\omega _t$ is the coordinate process; $\\mathbb {F} := (\\mathcal {F}_t)_{t \\in [0,T]}$ is the natural filtration generated by ${\\mathbf {X}}^0$ ; $\\mathbf {p}^{\\circ }$ is a fixed probability on $E$ ; $\\mathcal {F} := \\mathcal {F}_T$ ; $\\mathbb {P}$ is the unique probability on $(\\Omega , \\mathbb {F}, \\mathcal {F})$ for which ${\\mathbf {X}}^0$ is a continuous-time Markov chain with initial distribution $\\mathbf {p}^{\\circ }$ and transition rates between any two different states equal to 1.",
"So if $i\\ne j$ and $\\Delta t >0$ , $\\mathbb {P}[X_{t+\\Delta t} = e_j | \\mathcal {F}_t ] = \\mathbb {P}[X_{t+\\Delta t} = e_j| X_t]\\quad \\text{and}\\quad \\mathbb {P}[X_{t+\\Delta t} = e_j | X_t = e_i] = \\Delta t + o(\\Delta t)$ Using the result of [38], [37] we see that the process ${\\mathbf {X}}^0$ has the representation: $X^0_t = X^0_0 + \\int _{(0,t]} Q^0\\cdot X^0_{t-} dt + \\mathcal {M}_t,$ where $Q^0$ is the square matrix whose entries are given by: $Q^0_{i,i}=-(m-1), \\qquad i=1,\\ldots ,m$ $Q^0_{i,j}=1$ if $i\\ne j$ and $\\mathcal {M}=(\\mathcal {M}_t)_{t\\ge 0}$ is a $\\mathbb {R}^m$ -valued $\\mathbb {P}$ -martingale.",
"We sometime use the symbol $\\cdot $ to emphasize matrix multiplication.",
"The predictable quadratic variation of the martingale $\\mathcal {M}$ under $\\mathbb {P}$ is given by the formula: $\\langle \\mathcal {M}, \\mathcal {M}\\rangle _t = \\int _0^t \\psi _t dt,$ where $\\psi _t$ is given by: $\\psi _t := diag(Q^0 \\cdot X^0_{t-}) - Q^0 \\cdot diag( X^0_{t-}) - diag( X^0_{t-}) \\cdot Q^0.$ We assume that all the agents can take actions which are elements $\\alpha $ of a closed convex subset $A$ of a Euclidean space $\\mathbb {R} ^k$ .",
"For any agent, the set $\\mathbb {A} $ of admissible (control) strategies is the set of $A$ -valued, $\\mathbb {F} $ -predictable process $\\alpha =(\\alpha _t)_{0\\le t\\le T}$ .",
"The space of probability measures on the state space being the simplex $\\mathcal {P} (E)=\\mathcal {S} := \\lbrace p \\in \\mathbb {R}^m;\\; \\sum _{i=1}^m p_i = 1, p_i \\ge 0\\rbrace ,$ the controlled state processes will have dynamics determined by $Q$ -matrices $Q(t,\\alpha ,p,\\nu )=[q(t, i, j, \\alpha , p,\\nu )]_{1\\le i,j\\le m}$ where $q$ is a function $[0,T] \\times \\lbrace 1,\\dots ,m\\rbrace ^2 \\times A \\times \\mathcal {S}\\times \\mathcal {P}(A) \\rightarrow q(t, i, j, \\alpha , p,\\nu )\\in \\mathbb {R} .$ We shall assume that: (i) $Q(t, \\alpha , p,\\nu )$ is a Q-matrix.",
"(ii) $0 < C_1< q(t,i,j,\\alpha ,p,\\nu ) < C_2$ .",
"(iii) For all $(t,i,j)\\in [0,T]\\times E^2$ , $\\alpha ,\\alpha ^{\\prime } \\in A$ , $p,p^{\\prime } \\in \\mathcal {S}$ and $\\nu ,\\nu ^{\\prime }\\in \\mathcal {P}(A)$ , we have: $|q(t,i,j,\\alpha ,p,\\nu ) - q(t,i,j,\\alpha ^{\\prime }, p^{\\prime }, \\nu ^{\\prime })| \\le C( \\Vert \\alpha - \\alpha ^{\\prime }\\Vert + \\Vert p - p^{\\prime }\\Vert + \\mathcal {W}_1(\\nu ,\\nu ^{\\prime }))$ where $\\mathcal {W}_1$ is the 1-Wasserstein distance on $\\mathcal {P} (A)$ .",
"Assumption (i) is natural given that we start from a canonical process ${\\mathbf {X}}^0$ which is already a continuous time Markov chain.",
"The strictly positive lower bound of assumption (ii) may appear to be restrictive at first, but if we understand that in fact, it is sufficient that it is satisfied for a given power of the matrix, this assumption guarantees that all states are attainable through appropriate actions, and this is a desirable feature for control problems to be solvable.",
"Finally, assumption (iii) is to be expected if one thinks of the mathematical analysis needed to study these models.",
"Now, given $\\alpha =(\\alpha _t)_{0\\le t\\le T}\\in \\mathbb {A} $ $p=(p_t)_{0\\le t\\le T}$ a flow of probability measures on $E$ $\\nu =(\\nu _t)_{0\\le t\\le T}$ a flow of probability measures on $A$ we define the martingale ${\\mathbf {L}}^{(\\alpha ,p,\\nu )}=(L^{(\\alpha ,p,\\nu )}_t)_{0\\le t\\le T}$ by $L^{(\\alpha ,p,\\nu )}_t := \\int _0^t X_{s^-}^* \\cdot (Q(s, \\alpha _s, p_s, \\nu _s) - Q^0)\\cdot \\psi _s^+ \\cdot d\\mathcal {M}_s.$ Simple calculations show that $\\Delta L^{(\\alpha ,p,\\nu )}_t =&\\;\\; X_{t-}^*\\cdot (Q(t, \\alpha _t, p_t, \\nu _t) - Q^0)\\cdot \\psi _t^+\\cdot \\Delta X_t,$ which is either 0 when there is no jump at time $t$ , or $q(t,i,j,\\alpha _t,p_t,\\nu _t) - 1$ if the state jumps from state $i$ to state $j$ at time $t$ .",
"In any case, $\\Delta L^{(\\alpha ,p,\\nu )}_t \\ge -1$ .",
"Also, the Doleans exponential $\\mathcal {E}({\\mathbf {L}}^{(\\alpha ,p,\\nu )})$ is uniformly integrable so we can apply the extension of Girsanov's theorem to processes with jumps, and define the probability measure $\\mathbb {Q}^{(\\alpha ,p,\\nu )}$ by its density with respect to $\\mathbb {P} $ : $\\frac{d\\mathbb {Q}^{(\\alpha ,p,\\nu )}}{d\\mathbb {P}} := \\mathcal {E}({\\mathbf {L}}^{(\\alpha ,p,\\nu )})_T,$ which guarantees that the process $\\mathcal {M}^{(\\alpha ,p,\\nu )}=(\\mathcal {M}^{(\\alpha ,p,\\nu )}_t)_{0\\le t\\le T}$ defined as: $\\mathcal {M}^{(\\alpha ,p,\\nu )}_t := \\mathcal {M}_t - \\int _{0}^t ( Q^*(s, \\alpha _s, p_s, \\nu _s) - Q^0)\\cdot X_{s-} ds,$ is a $\\mathbb {Q}^{(\\alpha ,p,\\nu )}$ -martingale, and the canonical decomposition of ${\\mathbf {X}}^0$ under $\\mathbb {Q}^{(\\alpha ,p,\\nu )}$ reads: $X^0_t = X^0_0 + \\int _0^t Q^*(s, \\alpha _s, p_s, \\nu _s)\\cdot X^0_{s-} dt + \\mathcal {M}^{(\\alpha ,p,\\nu )}_t,$ showing that under $\\mathbb {Q}^{(\\alpha ,p,\\nu )}$ , the stochastic intensity rate of ${\\mathbf {X}}^0$ is $Q(t, \\alpha _t, p_t, \\nu _t)$ .",
"Notice that $X^0_0$ has still distribution $\\mathbf {p}^{\\circ }$ and if $\\alpha _t = \\phi (t, X^0_{t-})$ for some measurable function $\\phi $ , ${\\mathbf {X}}^0$ is a continuous-time Markov chain with jump rate intensity $q(t,i,j,\\phi (t,i),p_t,\\nu _t)$ under the measure $\\mathbb {Q}^{(\\alpha ,p,\\nu )}$ .",
"So as explained earlier in our first mention of the weak formulation, the choice of the control of the agents does not affect the trajectories of the state process, but it does influence the probability distribution, $\\mathbb {Q}^{(\\alpha ,p,\\nu )}$ in the present case, which determines the expected costs and rewards of the principal.",
"The reward of the Principal depends on the distribution of the agents' states and the payments made to the agents.",
"We use the notation $c_0:[0,T]\\times \\mathcal {S} \\rightarrow \\mathbb {R}$ for the running cost function $C_0:\\mathcal {S} \\rightarrow \\mathbb {R}$ for the terminal cost function defining the costs of the Principal.",
"Now, assuming that all the agents choose $\\alpha =(\\alpha _t)_{0\\le t\\le T}$ as their control strategy, that the resulting flow of marginal distribution of the agents' states is $p=(p(t))_{t\\in [0, T]}$ , and the contract offered by the principal is $(r, \\xi )$ , the principal's expected total cost is given by: $J_0^{\\alpha ,p}(r,\\xi ):=\\mathbb {E}^{\\mathbb {Q}^{(\\alpha ,p)}}\\left[\\int _0^T [c_0(t, p(t)) + r_t] dt +C_0(p(T)) + \\xi \\right].$ We assume that, for a given contract $(r,\\xi )$ proposed by the principal, the agents reach a Nash equilibrium as defined rigorously in the following statement.",
"Definition 6.3 The couple $(\\hat{\\alpha },\\hat{p})$ is a Nash equilibrium for the contract $(r,\\xi )$ , $(\\hat{\\alpha },\\hat{p}) \\in \\mathcal {N}(r,\\xi )$ in notation, if: (i) $\\hat{\\alpha }$ is the best response to the behavior of the other agents, i.e.",
"it minimizes the cost when the agent is committed to the contract $(r,\\xi )$ and the flow of marginal distributions of all the agents is given by the flow $\\hat{p}$ : $\\hat{\\alpha }= \\arg \\inf _{\\alpha \\in \\mathbb {A}}\\mathbb {E}^{\\mathbb {Q}^{(\\alpha ,\\hat{p})}}\\left[\\int _0^T [c(t, X_t, \\alpha _t, \\hat{p}(t)) - u(r_t)] dt - U(\\xi )\\right].$ (ii) $(\\hat{\\alpha },\\hat{p})$ satisfies the fixed point condition: $\\forall t \\in [0,T] \\qquad \\hat{p}(t) = \\mathbb {E}^{\\mathbb {Q}^{(\\hat{\\alpha },\\hat{p})}}[X_t].$ Notice that this equation is equivalent to $\\hat{p}_i(t) = \\mathbb {Q}^{(\\hat{\\alpha },\\hat{p})}[X_t = e_i]$ for all $t \\in [0,T]$ and $i\\in \\lbrace 1,\\dots ,m\\rbrace $ .",
"As we already explained, the Principal minimizes his or her total expected cost assuming the agents reach a Nash equilibrium.",
"So we only consider contracts $(r,\\xi )$ that result in at least one Nash equilibrium.",
"We denote by $\\mathcal {C}$ the set of all admissible contracts.",
"To implement the participation constraint, we disregard the equilibria in which the agent's expected total cost is above a given threshold $\\kappa $ , i.e.",
"take-it-or-leave-it behavior of the agents in contract theory: if the agents' expected total costs exceed a certain threshold, they should be able to turn down the contract.",
"In summary, the optimization problem for the principal reads: $V(\\kappa ) := \\inf _{(r,\\xi )\\in \\mathcal {C}}\\inf _{\\begin{array}{c}(\\alpha ,p) \\in \\mathcal {N}(r,\\xi )\\\\ J^{r,\\xi }(\\alpha ,p) \\le \\kappa \\end{array}}\\mathbb {E}^{\\mathbb {Q}^{(\\alpha ,p)}}\\left[\\int _0^T [c_0(t, p(t)) + r_t] dt +C_0(p(T)) +\\xi \\right],$ For the agent's optimization problem, we introduce the Hamiltonian $H: [0,T] \\times E \\times \\mathbb {R}^m\\times A \\times \\mathcal {S} \\times \\mathbb {R} \\rightarrow \\mathbb {R}$ defined by: $H(t,x,z,\\alpha ,p, r) := c(t,x,\\alpha ,p) - r + x^*(Q(t, \\alpha , p) - Q^0)z.$ and $H_i(t,z,\\alpha ,p, r) = H(t,e_i,z,\\alpha ,p, r)$ .",
"We assume that there exists a unique minimizer $\\hat{\\alpha }_i(t,z,p)$ of $\\alpha \\rightarrow H_i(t,z,\\alpha ,p,r)$ and that it is uniformly Lipschitz in $z$ , and we use the notations: $\\hat{H}_i(t,z,p, r) = H_i(t,z,\\hat{\\alpha }_i(t,z,p),p, r)\\;\\text{and}\\;\\hat{H}(t,x,z,p, r) = \\sum _{i=1}^m \\hat{H}_i(t,z,p, r) \\bf {1}_{x=e_i}$ for the maximized Hamiltonians.",
"Following the strategy at the root of the weak formulation of stochastic control problems, we introduce the BSDEs : $Y_t = -U(\\xi ) + \\int _t^T H(s, X_{s-}, Z_s, \\alpha _s, p(s), u(r_t)) ds - \\int _t^T Z_s^*d\\mathcal {M}_s.$ $Y_t = -U(\\xi ) + \\int _t^T \\hat{H}(s, X_{s-}, Z_s, p(s), u(r_t)) ds - \\int _t^T Z_s^*d\\mathcal {M}_s.$ and we prove the following representation theorems by inspection.",
"Notice that in the present situation, the BSDEs are driven by continuous time Markov chains.",
"Lemma 6.4 For each fixed contract $(r,\\xi )$ , $\\alpha \\in \\mathbb {A}$ and measurable mapping $p: [0,T]\\rightarrow \\mathcal {S}$ , (i) the BSDE (REF ) admits a unique solution $({\\mathbf {Y}}, {\\mathbf {Z}})$ and we have $J^{r,\\xi }(\\alpha ,p) = \\mathbb {E}^{\\mathbb {P}}[Y_0].$ (ii) The BSDE (REF ) admits a unique solution $({\\mathbf {Y}},{\\mathbf {Z}})$ and we have $\\inf _{\\alpha \\in \\mathbb {A}}J^{r,\\xi }(\\alpha ,p) = \\mathbb {E}^{\\mathbb {P}}[Y_0].$ In addition, the optimal control of the agent is $\\hat{\\alpha }(t, X_{t-}, Z_t, p(t))$ .",
"Let $({\\mathbf {Y}},{\\mathbf {Z}},\\alpha ,p,\\mathbb {Q})$ be a solution to the McKean-Vlasov BSDE system: $Y_t =&\\;\\; -U(\\xi ) + \\int _t^T \\hat{H}(s, X_{s-}, Z_s, p(s), u(r_s)) ds - \\int _t^T Z_s^* d\\mathcal {M}_s,\\\\\\mathcal {E}_t =&\\;\\; 1 + \\int _0^t \\mathcal {E}_{s-} X_{s-}^* (Q(s, \\alpha _s, p(s)) - Q^0)\\psi _s^+ d\\mathcal {M}_s,\\\\\\alpha _t = &\\;\\; \\hat{\\alpha }(t,X_{t-},Z_t,p(t)),\\\\p(t) = &\\;\\; \\mathbb {E}^{\\mathbb {Q}}[X_t],\\;\\;\\frac{d\\mathbb {Q}}{d\\mathbb {P}} = \\mathcal {E}_T.$ ${\\mathbf {Y}}$ is an adapted càdlàg process such that $\\mathbb {E}^{\\mathbb {P}}[\\int _0^T Y_t^2] < +\\infty $ for all $t \\in [0,T]$ , ${\\mathbf {Z}}$ is an adapted square integrable left-continuous process, $\\alpha \\in \\mathbb {A}$ , $p:[0,T]\\rightarrow \\mathcal {S}$ is measurable, $\\mathbb {Q}$ is a probability on $\\Omega $ The following result links the solution of the McKean-Vlasov BSDE (REF )-() to the Nash equilibria of the agents.",
"Theorem 6.5 If the BSDE (REF )-() admits a solution $({\\mathbf {Y}},{\\mathbf {Z}},\\alpha ,p,\\mathbb {Q})$ , then $(\\alpha ,p)$ is a Nash equilibrium.",
"Conversely if $(\\hat{\\alpha },\\hat{p})$ is a Nash equilibrium, then the BSDE (REF )-() admits a solution $({\\mathbf {Y}},{\\mathbf {Z}},\\alpha ,p,\\mathbb {Q})$ such that $\\alpha = \\hat{\\alpha }$ , $d\\mathbb {P}\\otimes dt$ -a.e.",
"and $p(t) = \\hat{p}(t)$ $dt$ -a.e.",
"Recall the optimization problem for the principal: $V(\\kappa ) := \\inf _{(r,\\xi )\\in \\mathcal {C}}\\inf _{\\begin{array}{c}(\\alpha ,p) \\in \\mathcal {N}(r,\\xi )\\\\ J^{r,\\xi }(\\alpha ,p) \\le \\kappa \\end{array}}\\mathbb {E}^{\\mathbb {Q}^{(\\alpha ,p)}}\\left[\\int _0^T [c_0(t, p(t)) + r_t] dt +C_0(p(T)) +\\xi \\right],$ Unfortunatly, this problem is totally intractable !",
"!!!",
"So we transform it into a more familiar control problem.",
"This is often called the Sannikov trick.",
"Its nature was clearly elucidated by Cvitanic, Possamaï and Touzi in [39].",
"We consider the following system of (forward) McKean-Vlasov SDEs: $Y_t =&\\;\\;Y_0-\\int _0^t\\hat{H}(s, X_{s-}, Z_s, p(s), u(r_s)) ds + \\int _0^t Z_s^* d\\mathcal {M}_s,\\\\\\mathcal {E}_t =&\\;\\;1 + \\int _0^t \\mathcal {E}_{s-} X_{s-}^* (Q(s, \\alpha _s, p(s)) - Q^0)\\psi _s^+ d\\mathcal {M}_s,\\\\\\alpha _t = &\\;\\;\\hat{\\alpha }(t,X_{t-},Z_t,p(t)),\\\\p(t) = &\\;\\;\\mathbb {E}^{\\mathbb {Q}}[X_t],\\;\\;\\frac{d\\mathbb {Q}}{d\\mathbb {P}} = \\mathcal {E}_T.$ This is the same type of equations as before, except that we write the dynamic of ${\\mathbf {Y}}$ in the forward direction of time.",
"That makes the whole difference.",
"Indeed, if we denote its solution by $({\\mathbf {Y}}^{{\\mathbf {Z}},br, Y_0)},{\\mathbf {Z}}^{({\\mathbf {Z}},r,Y_0)},\\alpha ^{({\\mathbf {Z}},r,Y_0)}, p^{({\\mathbf {Z}},r,Y_0)}, \\mathbb {P}^{({\\mathbf {Z}},r,Y_0)})$ , the expectation under $\\mathbb {P}^{({\\mathbf {Z}},r,Y_0)}$ by $\\mathbb {E}^{({\\mathbf {Z}},r,Y_0)}$ , and if we consider the optimal control problem: $&&\\tilde{V}(\\kappa ) := \\inf _{\\mathbb {E}^{\\mathbb {P}}[Y_0] \\le \\kappa } \\inf _{\\begin{array}{c}{\\mathbf {Z}}\\in \\mathcal {H}_{X}^2\\\\ r\\in \\mathcal {R}\\end{array}}\\\\&&\\hspace{15.0pt}\\mathbb {E}^{({\\mathbf {Z}},r,Y_0)}\\bigg [\\int _0^T [c_0(t, p^{({\\mathbf {Z}},r,Y_0)}(t)) + r_t] dt +C_0(p^{({\\mathbf {Z}},r,Y_0)}(T))+ U^{-1}(-Y_T^{({\\mathbf {Z}},r,Y_0)})\\bigg ],$ then, as a direct consequence of the previous theorem, we have $\\tilde{V}(\\kappa ) = V(\\kappa ).$ While informative at the theoretical level, still, the above results remain of little practical value if they cannot be implemented in the solution of practical problems.",
"In this respect, it is rewarding to discover that under a reasonable set of assumptions, computable solutions can be identified.",
"Here is an example.",
"We fix $p^{\\circ } \\in \\mathcal {S}$ , we assume that the space of actions is a bounded interval, say $A := [\\underline{\\alpha }, \\overline{\\alpha }]\\subset \\mathbb {R}^+$ , and that the transition rates are linear in the control in the sense that: $q(t,i,j,\\alpha ,p) &:=& \\bar{q}_{i,j}(t,p) + \\lambda _{i,j}(\\alpha - \\underline{\\alpha }),\\;\\;\\text{for}\\;\\;i\\ne j,\\\\q(t,i,i,\\alpha ,p) &:=& -\\sum _{j\\ne i}q(t,i,j,\\alpha ,p),$ where $\\lambda _{i,j} \\in \\mathbb {R}^+$ for all $i\\ne j$ , and $\\sum _{j\\ne i}\\lambda _{i,j} > 0$ for all $i$ , $\\bar{q}_{i,j}:[0,T]\\times \\mathcal {S}\\rightarrow \\mathbb {R}^+$ are continuous mappings for all $i\\ne j$ .",
"Furthermore, we assume that the agent running costs are of the form: $c(t, e_i, \\alpha , p) := c_1(t, e_i, p) + \\frac{\\gamma _i}{2}\\alpha ^2,$ where $\\gamma _i>0$ , and the mapping $(t,p)\\rightarrow c_1(t, e_i, p)$ is continuous for all $i\\in \\lbrace 1,\\dots ,m\\rbrace $ .",
"Finally, we assume that the utility function of continuous reward $u$ is continuous, concave and increasing, and that the utility of terminal reward is linear, say $U(\\xi ) = \\xi $ .",
"Under these conditions it is possible to show that the minimizer of the Hamiltonian is given by: $\\hat{\\alpha }(t,e_i, z,p) = \\hat{\\alpha }(e_i, z) = b\\left(-\\frac{1}{\\gamma _i}\\sum _{j\\ne i}\\lambda _{i,j}(z_j- z_i)\\right),$ for $i\\in \\lbrace 1,\\dots ,m\\rbrace $ , where $b(z) := \\min \\lbrace \\max \\lbrace z,\\underline{\\alpha }\\rbrace ,\\bar{\\alpha }\\rbrace $ .",
"Under these assumptions, one can reduce the problem to the optimal control of a flow of probability measures, and construct an optimal contract!",
"See [28] for details.",
"We illustrate this result on a concrete example.",
"I feel compelled to offer a disclaimer before presenting the gory details of the model I propose to use as illustration.",
"Peiqi Wang and I concocted this model over three years ago for the purpose of illustrating the inner workings of the theory and the analytic computations presented in [28].",
"In the Spring of 2020, when the paper was accepted for publication in Management Science, the Editor in Chief asked if we could add a discussion to highlight the relevance of this kind of model to the understanding of the COVID-19 pandemic.",
"We obliged, and while doing so, I realized the potential of these new tools to inform policy makers in the control of the spread of epidemics, and the localized re-opening of an economy after shut-down.",
"Given the dire conditions in which we are finding ourselves at this very moment, Aurrell, Dayanikli, Laurière and I embarked in a systematic investigation of what extensions of the model could bring to the understanding of the health and economic consequences of regulations.",
"This effort resulted in [8].",
"The reader interested in applications of similar equilibrium view to epidemic control can also consult the recent work of Elie, Hubert and Turinici [44].",
"Below, we present the model originally introduced in [28], where plenty numerical illustrations are given illustrating the influence of the various parameters of the model, and in particular, how the contract proposed by the regulator can influence the propensity of the agents to move from one city to another.",
"A regulator tries to control the spread of a virus over a time period $[0,T]$ .",
"The jurisdiction of the regulator consists of two cities, say $A$ and $B$ .",
"Each individual is either infected ($I$ ) or healthy ($H$ ), lives in city $A$ or $B$ .",
"So the state space of the model is $E = \\lbrace AI, AH, BI, BH\\rbrace $ and we denote by $\\pi _{AI}, \\pi _{AH}, \\pi _{BI}, \\pi _{BH}$ the proportions of individuals in each of these states.",
"To describe the time evolution of the state of each individual we introduce the following assumptions: (1) the rate of contracting the virus depends on the proportion of infected individuals in the city so the transition rate from state $AH$ to state $AI$ is $\\theta _A^- (\\frac{\\pi _{AI}}{\\pi _{AI} + \\pi _{AH}}) $ transition rate from state $BH$ to state $BI$ is $\\theta _B^- (\\frac{\\pi _{BI}}{\\pi _{BI} + \\pi _{BH}}) $ .",
"(2) the rate of recovery is a function of the proportion of healthy individuals in the city, so the transition rate from state $AI$ to state $AH$ is $\\theta _A^+(\\frac{\\pi _{AH}}{\\pi _{AI} + \\pi _{AH}})$ transition rate from state $BH$ to state $BI$ is $\\theta _B^+(\\frac{\\pi _{BH}}{\\pi _{BI} + \\pi _{BH}})$ .",
"(3) Each individual can try to move to the other city: we denote by $\\nu _I\\alpha $ the transition rates between the states $AI$ and $BI$ , and by $\\nu _H\\alpha $ the transition rates between the states $AH$ and $BH$ .",
"(4) Status of infection does not change when individual moves between cities.",
"The non-negative functions $\\theta _A^-$ , $\\theta _B^-$ , $\\theta _A^+$ and $\\theta _B^+$ are increasing, differentiable on $[0,1]$ .",
"They characterize the quality of health care in the cities $A$ and $B$ .",
"So we can change their parameters to make it more or less attractive to individuals to move from one city to the other, or to stay put.",
"In any case, with these simple prescription, the Q-matrix of the system reads: $\\hspace{45.0pt}AI \\hspace{45.0pt}AH \\hspace{45.0pt}BI \\hspace{45.0pt}BH$ $Q(t,\\alpha ,\\pi ):=\\left[\\begin{array}{cccc}\\ldots & \\theta _A^+(\\frac{\\pi _{AH}}{\\pi _{AI} + \\pi _{AH}}) & \\nu _I \\alpha & 0\\\\\\theta _A^-(\\frac{\\pi _{AI}}{\\pi _{AI} + \\pi _{AH}}) & \\ldots & 0 & \\nu _H\\alpha \\\\\\nu _I\\alpha & 0 & \\ldots & \\theta _B^+(\\frac{\\pi _{BH}}{\\pi _{BI} + \\pi _{BH}}) \\\\0 & \\nu _H\\alpha & \\theta _B^-(\\frac{\\pi _{BI}}{\\pi _{BI} + \\pi _{BH}}) & \\ldots \\end{array}\\right]\\begin{array}{c}AI\\\\AH\\\\BI\\\\BH\\end{array}$ We now introduce the costs, first for the agents: $c_1(t, AI, \\pi ) =&\\;\\; c_1(t, AH, \\pi ) := \\phi _A\\left(\\frac{\\pi _{AI}}{\\pi _{AI} + \\pi _{AH}}\\right),\\\\c_1(t, BI, \\pi ) =&\\;\\; c_1(t, BH, \\pi ) := \\phi _B\\left(\\frac{\\pi _{BI}}{\\pi _{BI} + \\pi _{BH}}\\right),\\\\\\gamma _{AI} =&\\;\\; \\gamma _{BI} := \\gamma _I,\\;\\;\\gamma _{AH}=\\gamma _{BH} := \\gamma _H,$ where $\\phi _A$ and $\\phi _B$ are two increasing functions, and next for the regulator (namely the Principal) for whom the running and terminal costs are given in the form: $c_0(t,\\pi ) =&\\;\\; \\exp (\\sigma _A \\pi _{AI} + \\sigma _B \\pi _{BI}),\\\\C_0(\\pi ) =&\\;\\; \\sigma _P\\cdot (\\pi _{AI} + \\pi _{AH} - \\pi _A^0)^2,$ where $\\pi _A^0$ is the population of city $A$ at time 0.",
"Choosing the values of the parameters $\\sigma _A$ , $\\sigma _B$ and $\\sigma _P$ offer a trade-off between the control of the epidemic and population planning; to try to minimize the infection rate of both cities.",
"In fact, $\\sigma _A$ , $\\sigma _B$ and $\\sigma _P$ weigh the relative importance the regulator attributes to each of these objectives.",
"The analysis of this model reduces to the solution of an explicit forward-backward system of Ordinary Differential Equations (ODEs) which can easily be solved numerically, allowing for the computation of statics of the model.",
"Numerical illustrations are provided in [28]."
],
[
"Comparison with Plain Nash Equilibria",
"It is natural and enlightening to compare the equilibrium computed from the solution of the principal-agent problem to the Nash equilibrium of the mean field game reached by the individuals in the absence of the regulator.",
"In its absence, the states of the individuals are still governed by the same transition rates, but the individuals' rewards or penalties from the authority are not present in the objective functions they optimize.",
"In other words, they minimize selfishly their expected costs $\\mathbb {E}^{\\mathbb {Q}^{(\\alpha ,\\pi )}}\\left[\\int _0^T c(t, X_t, \\alpha _t, \\pi (t)) dt\\right],$ as part of a regular mean field game, and it is plain to compute some of the numerical characteristics of its Nash equilibrium.",
"Note that this formula for the expected costs of the agents does not contain the payment stream $r=(r_t)_{0\\le t\\le T}$ and the terminal payment $\\xi $ which enter the costs to the agents as part of the covenants between them and the principal.",
"This comparison it very much in the spirit of the computation of the so-called Price of Anarchy in classical game theory.",
"Following the analytical approach to finite-state mean field games introduced in [51], it is straightforward to derive the system of forward-backward ODEs characterizing the Nash equilibrium.",
"See the system of ODEs (12)-(13) in [51] or [21].",
"In the particular case of the model discussed in this section, the exact form of this system of ODEs is given in the appendix of [28]."
]
] | 2012.05237 |
[
[
"Pressure control of the magnetic anisotropy of the quasi-two-dimensional\n van der Waals ferromagnet Cr$_2$Ge$_2$Te$_6$"
],
[
"Abstract We report the results of the pressure-dependent measurements of the static magnetization and of the ferromagnetic resonance (FMR) of Cr$_2$Ge$_2$Te$_6$ to address the properties of the ferromagnetic phase of this quasi-two-dimensional van der Waals magnet.",
"The static magnetic data at hydrostatic pressures up to 3.4 GPa reveal a gradual suppression of ferromagnetism in terms of a reduction of the critical transition temperature, a broadening of the transition width and an increase of the field necessary to fully saturate the magnetization $M_{\\rm s}$.",
"The value of $M_{\\rm s} \\simeq 3\\mu_{\\rm B}$/Cr remains constant within the error bars up to a pressure of 2.8 GPa.",
"The anisotropy of the FMR signal continuously diminishes in the studied hydrostatic pressure range up to 2.39 GPa suggesting a reduction of the easy-axis type magnetocrystalline anisotropy energy (MAE).",
"A quantitative analysis of the FMR data gives evidence that up to this pressure the MAE constant $K_{\\rm U}$, although getting significantly smaller, still remains finite and positive, i.e.",
"of the easy-axis type.",
"Therefore, a recently discussed possibility of switching the sign of the magnetocrystalline anisotropy in Cr$_2$Ge$_2$Te$_6$ could only be expected at still higher pressures, if possible at all due to the observed weakening of the ferromagnetism under pressure.",
"This circumstance may be of relevance for the design of strain-engineered functional heterostructures containing layers of Cr$_2$Ge$_2$Te$_6$."
],
[
"Introduction",
"The layered van der Waals magnet Cr$_2$ Ge$_2$ Te$_6$ [1] belongs to a class of materials which are currently the subject of intense research activities as these systems offer, on the one hand, a materials base for experimentally exploring details of two-dimensional magnetism [2], [3], [4].",
"On the other hand, the weak couplings between individual layers render these materials promising as (magnetic) components in so-called van der Waals heterostructures [5].",
"In both respects it is essential to properly characterize the magnetic anisotropies of the investigated materials.",
"At zero external pressure, Cr$_2$ Ge$_2$ Te$_6$ features a uniaxial magnetic anisotropy with the magnetic easy axis being oriented parallel to the crystallographic $c$ axis, i.e., perpendicular to the honeycomb layers extending in the $ab$ plane (see, e.g., Refs.",
"[1], [6], [7], [8]).",
"In a previous study [9] we established the value of the uniaxial magnetocrystalline anisotropy energy density (MAE) $K_{\\rm U} = 0.48\\cdot 10^6$ erg/cm$^3$ by means of ferromagnetic resonance (FMR) measurements under standard conditions at ambient pressure.",
"A detailed discussion of this result and its comparison with the band structure calculations can be found in Ref. [9].",
"One route to modify the magnetic properties of van der Waals magnets is the application of hydrostatic pressure to the crystals, as, in the case of Cr$_2$ Ge$_2$ Te$_6$ , was already attempted in Refs.",
"[10], [11].",
"Z. Lin et al.",
"[11] reported a change of the MAE upon applying external pressures up to 2 GPa, in particular a change of the sign of $K_{\\rm U}$ .",
"This implies that the anisotropy of the system transforms from easy-axis type (easy axis parallel to the crystallographic $c$ axis) to easy-plane type (easy direction perpendicular to the $c$ axis and, thus, in the $ab$ plane).",
"However, details of the MAE as a function of applied pressure have not been investigated so far.",
"This motivates a systematic, combined magnetization and FMR study on Cr$_2$ Ge$_2$ Te$_6$ at various hydrostatic pressures aiming at a quantification of the pressure dependence of $K_{\\rm U}$ .",
"To our knowledge, the FMR experiments on Cr$_2$ Ge$_2$ Te$_6$ under pressure and the magnetization studies at pressures exceeding 1 GPa were not reported so far.",
"In the present work we communicate a comprehensive investigation of the effect of the hydrostatic pressure on the static magnetic and FMR properties of single crystals of Cr$_2$ Ge$_2$ Te$_6$ at pressures up to $P = 2.39$ and 3.4 GPa applied in the FMR and magnetization experiments, respectively.",
"With increasing pressure, the magnetic ordering temperature gradually decreases from $T_{\\rm c} = 66$ K at $P = 0$ down to 45 K at $P = 3.4$ GPa.",
"Concomitantly, the transition width broadens and the saturation field increases.",
"Remarkably, the saturation magnetization stays practically constant up to $P= 2.8$ GPa and decreases at a higher pressure.",
"The FMR measurements performed at $T = 4.2$ K $\\ll T_{\\rm c}$ in a frequency range $50 - 260$ GHz for the two orientations of the applied magnetic field ${\\bf H}\\parallel c$ and ${\\bf H}\\perp c$ evidence that the excitation energy gap characteristic of an easy-axis ferromagnet closes at $P > 2$ GPa.",
"The data analysis in the frame of a phenomenological theory of FMR reveals that the seemingly isotropic magnetism of the studied crystals manifesting in the gap closure is a result of the compensation of the intrinsic easy-axis magnetocrystalline anisotropy of Cr$_2$ Ge$_2$ Te$_6$ and the shape anisotropy of the sample.",
"It follows from our analysis that the MAE constant $K_{\\rm U}$ is reduced by a factor of 2 at the highest applied pressure in the FMR experiment but still remains sizable.",
"This suggests a robustness of the easy-axis-type of ferromagnetism of Cr$_2$ Ge$_2$ Te$_6$ to the application of pressure up to $P = 2.39$ GPa despite a reduction of $T_{\\rm c}$ .",
"Our findings motivate further pressure experiments to address the question if the sign of the MAE could be changed at still high pressures before the ferromagnetic state would be fully suppressed.",
"From the technological perspective, the results of the present work on the pressure-tunability of the magnetic anisotropy of Cr$_2$ Ge$_2$ Te$_6$ may be insightful for accessing the use of this material as a magnetic element of spintronic devices with the strained architecture."
],
[
"Samples and methods",
"Single crystals of Cr$_2$ Ge$_2$ Te$_6$ were grown with the self flux technique and thoroughly characterized as described in detail in our previous work in Ref. [9].",
"As grown single crystals were thoroughly characterized by powder x-ray diffraction and energy dispersive x-ray spectroscopy, both agree well with the crystal structure in the R space group as well as with the expected stoichiometry of Cr$_2$ Ge$_2$ Te$_6$ [12].",
"The bulk magnetization data were measured using a custom-built pressure cell for a commercial Quantum Design superconducting quantum interference device (SQUID) magnetometer (MPMS-XL).",
"Inside the CuBe cell, two opposing cone-shaped ceramic anvils compress a CuBe gasket with a cylindrical hole used as sample chamber [13], [14].",
"The platelet-like shaped single-crystalline sample ($m = 3.3 \\times 10^{-6}$ g) of Cr$_2$ Ge$_2$ Te$_6$ with the $c$ axis normal to the plate was installed into the gasket hole (diameter Ø = 0.6 mm and height $h$ = 0.8 mm) and rested on the flat part of the ceramic anvil.",
"Given the longitudinal magnetic field direction in the SQUID magnetometer the ${\\bf H} \\parallel c$ orientation was easily achieved.",
"The uniaxial force is applied at ambient temperature, and it is converted into hydrostatic pressure in the sample chamber using Daphne oil 7575 as pressure-transmitting medium.",
"We checked the pressure value and its homogeneity at low temperature by measuring the pressure dependent diamagnetic response associated with the superconducting transition of a small Pb manometer inserted in the sample space [15].",
"Such measurements were performed in applied fields of $H = 5$ Oe.",
"In order to ensure a stable pressure throughout all measurements due to the thermal expansion of the cell upon temperature sweeps, a thermal cycling was performed once from ambient temperature to 2 K and back to ambient temperature before the actual data acquisition.",
"Note, that a magnetic background is detected in our DC magnetization measurements arising from the pressure cell itself.",
"Here, we performed a detailed magnetic characterization of the empty pressure cell within the same conditions of the real experiments for the aim of a quantitative and reliable disentanglement of the sample signal from the overall magnetic response.",
"The background correction was done by subtracting the fitted signal of a gasket without a sample measured under the same conditions, given that the raw signal of the gasket in our setup can be described by a magnetic dipole in our SQUID detection coils.",
"While at temperatures above $\\sim $ 20 K a temperature-independent pressure cell background ensures an excellent background subtraction, at low temperatures a strongly temperature dependent magnetization of the CuBe cell limits the resolution of our experiments.",
"Consequently, only background subtracted data for $T >$ 10 K are shown.",
"Small kinks in the magnetization curves around 40 K were identified as instrumental artifacts.",
"More detailed information about the background of our pressure cell can be found in Ref. [16].",
"Still, the combination of the very small magnetization above the ferromagnetic ordering temperature of our samples with a very small mass ($ m \\sim 3.3 \\times 10^{-6}$ g) and the large uncertainty of absolute pressure values for $T > 150$ K using Pb as a manometer at low $T$ in our ceramic anvil pressure cell restrains us from a quantitative Curie-Weiss analysis in the high-temperature regime.",
"Ferromagnetic resonance is a resonance response of the total magnetization of the ferromagnetically ordered material exposed to the microwave radiation.",
"The FMR frequency is determined not only by the strength of the applied external magnetic field but also sensitively depends on the magnetic anisotropy of the studied sample, as will be explained in Sect.",
"REF .",
"The magnetic anisotropy can be accurately quantified by measuring FMR at different excitation frequencies.",
"Therefore, in the present work FMR experiments were carried out using a multi-frequency electron spin resonance setup equipped with a piston-cylinder pressure cell with a maximum pressure of about 2.5 GPa.",
"Its detailed description can be found in Ref. [17].",
"Microwave radiation with frequencies in a range 50 – 260 GHz was provided by a set of Gunn-diode oscillators and detected by a $^4$ He-liquid cooled hot-electron InSb bolometer.",
"Magnetic fields up to 10 T were generated by a cryogen-free superconducting magnet.",
"Platelet-like single-crystalline samples of Cr$_2$ Ge$_2$ Te$_6$ with the typical lateral dimensions of $\\sim 3-4$ mm were put inside the pressure cell in a Teflon capsule filled with a pressure-transmitting fluid (Daphne 7373 oil from Idemitsu Kosan Co., Ltd [18], [19], [20]).",
"The advantage of having a well-defined $c$ axis direction of the samples facilitated their orientation in the FMR pressure cell.",
"For the ${\\bf H} \\parallel {\\bf c}$ field geometry the studied sample was placed on the flat bottom of the Teflon capsule.",
"For ${\\bf H} \\perp {\\bf c}$ it was firmly fixed with a thin Teflon tape to the side of a small rectangular parallelepiped placed on the bottom of the Teflon capsule.",
"The inner pistons of the pressure cell made of ZrO$_2$ -based ceramics ensured a low-loss propagation of the microwaves through the cell.",
"The pressure was calibrated using a superconducting tin-based pressure gauge.",
"All FMR measurements were performed at a temperature $T = 4.2$ K for two orientations of the sample with respect to the applied magnetic field, ${\\bf H}\\parallel c$ axis and ${\\bf H}\\perp c$ axis, respectively."
],
[
"Pressure-dependent magnetization study",
"The field-cooled (FC) magnetization curves $M(T)$ as a function of temperature are shown in Fig.",
"REF for an applied magnetic field $\\mu _0H = 0.1$ T oriented parallel to the $c$ axis under an applied pressure up to 3.4 GPa.",
"Due to the small magnetization values, resulting from the very small mass of the sample, the absolute values at high temperature show an artificial shift in the magnetization axis, coming from a highly sensitive background subtraction.",
"Two main characteristics are observed.",
"First, the onset of the magnetic transition at approximately 80 K is persistent within $\\pm 1$ K despite the pressure change.",
"The onset was determined from the derivative $dM/dT$ curve (Fig.",
"REF , inset) while moving from high to low temperature as the point of its departure from a very small, close to zero value $dM/dT \\sim 0$ caused by just a small polarization of paramagnetic moments to the negative values due the gradual development of the spontaneous ferromagnetic moment.",
"Second, the transition broadens continuously, and the ferromagnetic (FM) transition temperature $T_c$ , defined as the minimum of the $dM(T)/dT$ curve [21], [22], shifts to lower temperature upon increasing pressure, as shown in the inset of Fig.",
"REF .",
"This behavior is in line with an earlier report [10] where magnetization measurements at pressures up to 1 GPa showed the same tendency.",
"Remarkably, the absolute value of the magnetization in the FM state at the given field of the measurement is gradually reduced with increasing the pressure up to 3.4 GPa.",
"This can be explained by a shift of the saturation field $H_{\\rm sat}$ as function of pressure to higher values, which reduces consequently the magnetization at small applied fields such as 0.1 T (see Fig.",
"REF and discussion below).",
"The magnetic field dependence of the magnetization $M(H)$ is depicted in Fig.",
"REF for the same pressure values as shown for the temperature dependences.",
"To minimize the influence of the non-perfect subtraction of the background signal from the pressure cell at low temperatures, the curves were measured at $T = 15$ K, which is still low enough to capture the characteristic behavior of the FM state (cf.",
"Fig.",
"REF ).",
"The $M(H)$ data set in Fig.",
"REF includes a reference measurement performed at ambient pressure on a more massive single crystal ($m \\sim 4$ mg) and without the pressure cell.",
"The ambient pressure data yield a saturation moment $M_{\\rm s}$ of 3.05 $\\mu _{\\rm B}$ /Cr, which is consistent with the value found in our previous study [9].",
"Importantly, the data in Fig.",
"REF show that $M_{\\rm s}$ is pressure independent up to about 2.8 GPa within the experimental error bar, the latter mostly being determined by the comparably strong background signal due to the reduced mass of our sample.",
"At higher pressure, however, $M_{\\rm s}$ reduces to about 2.5 $\\mu _{\\rm B}$ /Cr at 3.4 GPa.",
"Another important observation is that, contrary to the saturation moment, the saturation field $H_{\\rm sat}$ continuously shifts to higher values under applied hydrostatic pressure.",
"While saturation can be achieved at $\\sim 0.1$ T for ambient pressure, $\\mu _0H_{\\rm sat}$ increases to $\\sim 1.5$ T at 3.4 GPa.",
"A possible explanation for the shift of the saturation field has been proposed by Sun et al.",
"[10] through first principles calculations.",
"According to them, the pressure-induced decrease in the Cr-Cr bond length favors antiferromagnetic (AFM) exchange, while a concomitant deviation from the 90$^\\circ $ Cr-Te-Cr bond angle leads to a suppression of the FM superexchange interaction.",
"An increasing competition between the easy-axis MAE and the easy-plane shape anisotropy under pressure (see below) may also contribute to an increase of $H_{\\rm sat}$ .",
"Magnetization measurements at still higher pressures would be enlightening in this regard.",
"Figure: Magnetization MM as a function of applied field 𝐇∥c{\\bf H} \\parallel c at T=15T = 15 K for pressures up to 3.4 GPa.",
"The solid lines are guides to the eye.A demagnetization correction has been applied .The results of the above analysis of the $M(T)$ and $M(H)$ dependences are summarized in Fig.",
"REF , where we plot the $M_{\\rm s}$ and $T_{\\rm c}$ values as a function of the applied pressure with the corresponding experimental error bars.",
"As already noticed above, the ordering temperature value estimated from the first derivative criterion is less accurate for higher pressures due to the broadening of the transition under pressure (Fig.",
"REF , inset).",
"Still, we could clearly validate the long-range ordered ferromagnetic ground state of Cr$_2$ Ge$_2$ Te$_6$ at 4.2 K and up to 3.4 GPa, i.e., for the parameter ranges where FMR experiments were performed (see below).",
"In contrast to $T_{\\rm c}$ , the FM saturation moment is approximately constant for pressures up to 2.8 GPa with a subsequent reduction for $P > 2.8$ GPa.",
"Figure: Saturation magnetization M s M_{\\rm s} and the FM transition temperature T c T_{\\rm c} as a function of the applied pressure (symbols).",
"M s M_{\\rm s} was calculated as the average magnetization value for applied fields μ 0 H≥2\\mu _0H \\ge 2 T, which was determined as the lower limit of the applied field at which dM/dH=0dM/dH = 0 for all the curves within the experimental error bar.",
"The solid lines are guides to the eye."
],
[
"Pressure-dependent FMR study",
"The FMR data sets presented in this section were collected by measuring three different single-crystalline samples of Cr$_2$ Ge$_2$ Te$_6$ from the same batch with similar, but slightly different, lateral dimensions.",
"As will be discussed below, the differences in demagnetization factors associated with different sample dimensions are, however, negligible in the analysis of the FMR data.",
"Selected FMR spectra measured for the two field geometries used in this work are shown in Figs.",
"REF (a) and (b).",
"A shift of the signal to higher fields for ${\\bf H}\\parallel c$ and to smaller fields for ${\\bf H}\\perp c$ , respectively, by applying pressure is clearly visible.",
"The results of frequency-dependent FMR measurements at a temperature of 4.2 K with ${\\bf H} \\parallel c$ and at different applied external pressure between 0 and 2.39 GPa are summarized in Fig.",
"REF (c).",
"At 0 GPa, Cr$_2$ Ge$_2$ Te$_6$ features an easy-axis type MAE with the easy axis oriented parallel to the $c$ axis as has been investigated in detail in our previous study [9].",
"This kind of anisotropy directly manifests in an FMR measurement by a shift of the resonance line towards smaller magnetic fields with respect to the resonance position in the paramagnetic state if the external magnetic field ${\\bf H}$ is applied parallel to the magnetic easy axis ($c$ axis) and towards higher magnetic fields if ${\\bf H}$ is normal to the magnetic easy axis.",
"This behavior is illustrated in Figs.",
"REF (a) and (b) where, depending on the direction of the applied field, the FMR signal at $P = 0$ is shifted either to the left or to the right side of the signal from the paramagnetic reference sample 2,2-diphenyl-1-picrylhydrazyl (DPPH) with a $g$ factor $g\\simeq 2$ which is commonly used as a magnetic field marker.",
"Remarkably, the application of an external pressure counteracts these tendencies by shifting the FMR line towards the paramagnetic position for both field geometries.",
"As a result, with increasing the applied pressure from $P = 0$ to 2.01 GPa the FMR signal from the sample becomes isotropic, i.e., the resonance field $H_{\\rm res}$ does not depend on the direction of the applied field at this value of pressure [Fig.",
"REF (b)].",
"The frequency-field dependence of the paramagnetic reference sample DPPH is shown in Fig.",
"REF as a blue dashed line for comparison with the ferromagnetic resonance fields $H_{\\text{res}}$ measured on the Cr$_2$ Ge$_2$ Te$_6$ samples for ${\\bf H}\\parallel c$ axis.",
"$H_{\\rm res}$ shifts continuously to higher magnetic fields with increasing $P$ and thereby approach, but do not cross, the paramagnetic resonance branch in the frequency-field diagram.",
"This systematic shift is emphasized in the inset of Fig.",
"REF where details of the frequency-field diagram are shown for intermediate field strengths.",
"The change of the FMR line position as a function of external pressure evidences a reduction of the MAE of the studied system.",
"However, the shift of the resonance fields of Cr$_2$ Ge$_2$ Te$_6$ is not only determined by the strength and the sign of the MAE constant $K_{\\rm U}$ but also by demagnetization fields resulting from the specific sample shapes which favor, in the case of the plate-like shaped Cr$_2$ Ge$_2$ Te$_6$ crystals, an in-plane magnetization.",
"As a consequence, the internal demagnetization fields will lead to a shift of the resonance position to higher fields if the external magnetic field is applied perpendicular to the honeycomb layers, i.e., for ${\\bf H}\\parallel c$ axis.",
"Thus, the size of the anisotropy constant $K_{\\rm U}$ derived from shifts of the resonance position would be underestimated, if demagnetization effects are not taken into account (cf.",
"the related discussion in Ref. [9]).",
"These demagnetization fields $H_{\\rm D}$ can be calculated using the elements of the demagnetization tensor $N_i$ ($i = x,y,z$ , with $x,y,z$ being the principal axes of the tensor) [23], [26], [27]: $H_{\\rm D}^{\\rm i} = -4\\pi N_{\\rm i}M \\ \\ \\ ,$ where $M$ denotes the magnetization of the sample.",
"Since the thickness of the plate-like samples used in this study could not be quantified precisely, the real sample shape was approximated using the demagnetization factors for a flat (infinite) plate oriented perpendicular to the $z$ axis (which was chosen to coincide with the crystallographic $c$ axis of Cr$_2$ Ge$_2$ Te$_6$ ), i.e., $N_{\\rm x} = N_{\\rm y} = 0$ , $N_{\\rm z} = 1$ .",
"This assumption is supported by the fact that the lateral dimensions in the $ab$ plane of the used Cr$_2$ Ge$_2$ Te$_6$ crystals were much larger than the thicknesses of the samples parallel to the $c$ axis.",
"Using the value of the saturation magnetization $M_{\\rm s} \\simeq 3$ $\\mu _{\\rm B}$ /Cr (which corresponds to $M_{\\rm s} \\approx 201$ erg/G cm$^3$ for Cr$_2$ Ge$_2$ Te$_6$ ) yields a demagnetization field $\\mu _0 H_{\\rm D}$ in the fully saturated ferromagnetic phase of -0.253 T if the external field is applied parallel to the $c$ axis.",
"The frequency-field dependences measured with this field orientation were then fitted by the following expression $\\nu _{\\text{res}} = \\frac{g\\mu _{\\rm B}\\mu _0}{h}[H_{\\text{res}}+H_{\\rm D}+H_{\\rm A}] \\ \\ \\ ,$ where $\\nu _{\\text{res}}$ denotes the resonance/microwave frequency and $H_{\\rm A}$ is the anisotropy field describing the intrinsic magnetocrystalline anisotropy.",
"Fits of Eq.",
"(REF ) to the measured data are shown in Fig.",
"REF (c) as solid lines.",
"For the fitting, the $g$ factor as well as $H_{\\rm A}$ were treated as free fit parameters.",
"The $g$ -factor varied very little within the error bars around the mean value of 2.03 (see, Sect. )",
"whereas $H_{\\rm A}$ was significantly decreasing with increasing the pressure.",
"Using the latter parameter it is possible to calculate the uniaxial MAE $K_{\\rm U}$ according to (see, for instance, Ref.",
"[28]) $K_{\\rm U}^{\\text{fit}} = \\frac{H_{\\rm A}M_{\\rm S}}{2} \\ \\ \\ .$ The results obtained from this fitting procedure are discussed in the following section together with results from an alternative approach for a quantitative analysis of the FMR data.",
"Figure: Examples for simulations (solid lines) of the measured frequency-field dependences (symbols) at (a) 0 GPa and (b) 2.01 GPa, respectively.",
"In these simulations experimental data sets obtained for 𝐇∥c{\\bf H} \\parallel c (circles) and 𝐇⊥c{\\bf H} \\perp c (squares) were taken into account simultaneously in order to determine the value of K U K_{\\rm U}."
],
[
"Simulations of the frequency dependence and determination of the magnetic anisotropies",
"In order to simultaneously take into account the results of our FMR measurements performed with ${\\bf H} \\parallel c$ and ${\\bf H} \\perp c$ field geometries, the measured frequency-field dependences were numerically simulated for each pressure value.",
"This lead to a refinement of the determination of $K_{\\rm U}$ , in particular at higher pressures.",
"For the simulations we used a well established phenomenological model of FMR (see, e.g., Refs.",
"[29], [28], [30]) where the resonance frequency $\\nu _{\\text{res}}$ is expressed as $\\nu _{res}^2 = \\frac{g^2 \\mu _B^2}{h^2 M_s^2\\sin ^2\\theta } \\bigg (\\frac{\\partial ^2{F}}{\\partial \\theta ^2}\\frac{\\partial ^2{F}}{\\partial \\varphi ^2} - \\Big (\\frac{\\partial ^2{F}}{\\partial \\theta \\partial \\varphi }\\Big )^2\\bigg ) \\ \\ \\ .$ Here, $F$ is the free energy density, and $\\theta $ and $\\varphi $ are the spherical coordinates of the magnetization vector $M(M_s, \\varphi , \\theta )$ .",
"This phenomenological model is applicable for ferromagnets with the fully saturated magnetization at temperatures sufficiently lower than $T_c$ , i.e., in the case of the fully developed ferromagnetic phase.",
"According to the above discussed results of our pressure-dependent magnetization study both criteria are safely fulfilled for the $T$ , $P$ and $H$ parameter ranges of the FMR study.",
"To obtain the resonance position of the FMR signal, $F$ in Eq.",
"(REF ) should be taken at the equilibrium angles $\\varphi _0$ and $\\theta _0$ of $M$ .",
"In the simulations, the minimum of $F$ with respect to $\\theta $ and $\\varphi $ for a given set of experimental parameters, such as the microwave frequency and the direction and the strength of the magnetic field, was found numerically.",
"For Cr$_2$ Ge$_2$ Te$_6$ , an account of both an intrinsic uniaxial magnetic anisotropy and of (extrinsic) shape anisotropy of the particular studied sample enabled an accurate description of all FMR data sets.",
"In this case the free energy density (in cgs units) is defined as $\\begin{split}F &= -H\\cdot M - K_U\\cos ^2(\\theta ) + \\\\ & \\ \\ \\ \\ 2\\pi M^2_s(N_x\\sin ^2(\\theta )\\cos ^2(\\varphi ) + \\\\ & \\ \\ \\ \\ N_y\\sin ^2(\\theta )\\sin ^2(\\varphi ) + N_z\\cos ^2(\\theta )) \\ \\ \\ \\end{split}$ and comprises, besides the Zeeman energy density expressed by the first term, contributions due to the uniaxial and shape anisotropies, second and third terms, respectively.",
"Representative simulations of frequency-dependent measurements at 0 and 2.01 GPa are shown in Figs.",
"REF (a) and (b), respectively.",
"At 0 GPa, the separation of the $\\nu (H_{\\text{res}})$ curves obtained for the two different field orientations clearly evidences the easy-axis type magnetic anisotropy with the ${\\bf H} \\parallel c$ curve being shifted to smaller fields and the ${\\bf H} \\perp c$ curve being shifted to higher fields as compared to the paramagnetic position, respectively.",
"The 0 GPa data could be simulated using the value for $K_{\\rm U}$ of $4.8 \\cdot 10^5$ erg/cm$^3$ obtained in our previous study [9].",
"Keeping the value of $K_{\\rm U}$ fixed, the $g$ factor as well as demagnetization factors were adjusted in order to consistently describe different studied samples.",
"It turned out that all 0 GPa data could be described successfully with an isotropic $g$ factor of 2.03, which is consistent with our previous investigations, and the same set of demagnetization factors $N_x = N_y = 0$ , $N_z = 1$ despite the (small) differences in the shapes of the samples.",
"These parameters were kept fixed in the simulations of the frequency dependences measured at several non-zero pressures.",
"The only free parameter in these simulations was the anisotropy constant $K_{\\rm U}$ which allowed to determine the pressure dependence of the MAE.",
"As can be seen in Fig.",
"REF (b), at a pressure of 2.01 GPa, the measured frequency dependences are nearly identical for both orientations of the external magnetic field (a similar frequency-field diagram was obtained for 2.39 GPa), giving the impression of an isotropic behavior.",
"However, the contributions of the magnetocrystalline anisotropy and the shape anisotropy to the free energy density [Eq.",
"(REF )] have opposite sign within the pressure range of this study.",
"Thus, the apparently isotropic frequency dependence is, in fact, a consequence of the similar strength of these two contributions.",
"This allows to conclude that an application of pressures between 2.01 and 2.39 GPa reduces the uniaxial MAE $K_{\\rm U}$ to a value that compensates the shape anisotropy of the crystals.",
"With the demagnetization field $\\mu _0H_{\\rm D} = -0.253$ T and the saturation magnetization $M_{\\rm s} \\approx 201$ erg/G cm$^3$ estimated above one obtains the shape anisotropy constant $K_{\\rm shape} = (H_{\\rm D}M_{\\rm s})/2 \\approx -2.54 \\cdot 10^5$ erg/cm$^3$ .",
"Its absolute value is indicated by the dashed line in Fig.",
"REF .",
"Moreover, the pressure-dependent FMR studies do not provide any evidence for a pressure-induced sign change of the magnetocrystalline anisotropy or a vanishing of the intrinsic uniaxial anisotropy up to pressures of 2.39 GPa.",
"This is in contrast to the conclusions drawn in Ref.",
"[11] on a spin reorientation transition at pressures between 1.0 and 1.5 GPa based on magneto-transport investigations of Cr$_2$ Ge$_2$ Te$_6$ .",
"The pressure dependence of $K_{\\rm U}$ as obtained from fitting of the ${\\bf H} \\parallel c$ data to Eq.",
"(REF ) and from the simulations of all data available for both field orientations according to Eq.",
"(REF ) is shown in Fig.",
"REF .",
"Qualitatively, both methods reveal a similar behavior, in particular a reduction of $K_{\\rm U}$ with increasing pressure.",
"However, the deviation between the $K_{\\rm U}$ values derived from the fits and the simulation increases at higher pressures.",
"This could be attributed to the fact that the simulations simultaneously take into account the frequency dependences measured for both field orientations.",
"Therefore, the reliability of the $K_{\\rm U}$ values from simulations is higher despite the larger error bars, in particular at higher pressures at which differences between both field orientations become very small.",
"Finally, it is worthwhile mentioning that in Ref.",
"[31] a pressure-induced difference in the unit-cell volume of Cr$_2$ Ge$_2$ Te$_6$ between applied pressures of 0 and 3 GPa of about 4 % was reported.",
"As the unit-cell volume enters into the calculation of the saturation magnetization, such a reduction of volume with increasing pressure could change the absolute value determined for $K_{\\rm U}$ .",
"However, it was found that the pressure-induced contraction of the unit-cell volume by 4 % does not affect the determination of $K_{\\rm U}$ but is within the given error bars.",
"Figure: Pressure dependence of the uniaxial magnetocrystalline anisotropy constant K U K_{\\rm U} of Cr 2 _2Ge 2 _2Te 6 _6.",
"Red circles denote the results obtained from fitting the different ν(H res )\\nu (H_{\\text{res}}) curves according to Eq.",
"() for 𝐇∥c{\\bf H} \\parallel c. Blue squares are the K U K_{\\rm U} values determined from simulations according to Eq.",
"() considering the experimental data sets for 𝐇∥c{\\bf H}\\parallel c and 𝐇⊥c{\\bf H}\\perp c simultaneously.",
"The horizontal dashed line shows the absolute value of the negative shape anisotropy constant K shape ≈-2.54·10 5 K_{\\rm shape} \\approx -2.54 \\cdot 10^5 erg/cm 3 ^3.",
"It compensates the decreasing positive value of K U K_{\\rm U} at p≳2p \\gtrsim 2 GPa yielding a seemingly isotropic behavior."
],
[
"Conclusions and outlook",
"Our combined magnetization and ferromagnetic resonance study on single crystals of the quasi-two-dimensional ferromagnet Cr$_2$ Ge$_2$ Te$_6$ has revealed a significant effect of the application of hydrostatic pressure $P$ on the properties of the ferromagnetic state of this compound.",
"It manifests in a reduction of the critical temperature $T_{\\rm c}$ and a concomitant broadening of the transition to the FM state, as well as in a gradual increase of the saturation field $H_{\\rm sat}$ .",
"In contrast, the saturation magnetization remains practically constant up to $P = 2.8$ GPa and reduces by approaching the highest applied pressure of 3.4 GPa.",
"The frequency dependent FMR measurements performed in a fully developed FM state at $T\\ll T_{\\rm c}$ and $H > H_{\\rm sat}$ at various external pressures up to 2.39 GPa revealed a systematic shift of the resonance fields with increasing $P$ evidencing a continuous reduction of the magnetocrystalline anisotropy constant $K_{\\rm U}$ .",
"From the quantitative analysis of the data, which takes into account demagnetization effects, it follows that $K_{\\rm U}$ , although being significantly reduced at highest pressures applied in this study, neither vanishes nor changes its sign up to pressures of about 2.4 GPa.",
"Therefore, noting that FMR is a very direct method for the quantitative determination of magnetic anisotropy, it can be concluded with confidence that there is no evidence for a switching of the magnetic anisotropy from the easy-axis to the easy-plane type within the pressure range under consideration.",
"Further pressure dependent studies of Cr$_2$ Ge$_2$ Te$_6$ are appealing to understand the apparent discrepancy between our work and the magneto-transport study in Ref.",
"[11] claiming a spin-reorientation transition in Cr$_2$ Ge$_2$ Te$_6$ at pressures 1 GPa $< P < 1.5$ GPa, and to address the question whether a pressure-induced sign-switching of the magnetic anisotropy could be achieved at pressures exceeding 2.39 GPa.",
"Although in terms of the spatial dimensionality we investigated bulk, 3-dimensional crystals of Cr$_2$ Ge$_2$ Te$_6$ , we showed in our previous ESR/FMR study that the low-temperature magnetism of the bulk crystals is essentially of a 2D nature due to a very weak interlayer magnetic coupling in this van der Waals compound [9].",
"Considering this intrinsic magnetic two-deminsionality of the bulk material our pressure dependent study can be relevant for the research on Cr$_2$ Ge$_2$ Te$_6$ in the truly 2D spatial limit.",
"In this respect, the pressure control of the magnetic anisotropy investigated in our work may provide important hints for a targeted design of functional magneto-electrical heterostructures containing layers of Cr$_2$ Ge$_2$ Te$_6$ .",
"In particular, a recent prediction of a remarkable strain and electric field tunability of a single layer Cr$_2$ Ge$_2$ Te$_6$ is encouraging [32].",
"It remains yet an open question regarding the design of a heterostructure where the strain $\\epsilon $ of the order $\\pm (1 - 2)$ % for the Cr$_2$ Ge$_2$ Te$_6$ layer, as proposed in Ref.",
"[32], can be achieved.",
"However, a sizable effect on the magnetic anisotropy which we observed at pressures up to 2.39 GPa corresponds to an even smaller compressive in-plane strain $\\epsilon \\sim 0.7$ % [31].",
"Such straining seems to be plausible to achieve since, in general, 2D van der Waals materials are known to sustain large strain.",
"For example, it was shown that the biaxial compressive and tensile strain of $\\sim 1$ % can be achieved in the single layer molybdenum dichalcogenide deposited on a thermally compressed or expanded polymer substrate [33]."
],
[
"Acknowledgments",
"The work in Kobe was partially supported by Grants-in-Aid for Scientific Research (C) (No.",
"19K03746) from Japan Society for the Promotion of Science.",
"The work in Dresden was supported by the Deutsche Forschungsgemeinschaft (DFG) through Grant KA1694/12-1 and within the Collaborative Research Center SFB 1143 “Correlated Magnetism – From Frustration to Topology” (project-id 247310070) and the Dresden-Würzburg Cluster of Excellence (EXC 2147) “ct.qmat - Complexity and Topology in Quantum Matter\" (project-id 39085490).",
"S.A. acknowledges financial support by the DFG through Grant AS 523/4-1.",
"A.A. acknowledges financial support by the DFG through Grant No.",
"AL 1771/4-1.",
"S.S. acknowledges financial support from GRK-1621 Graduate Academy.",
"V.K.",
"gratefully acknowledges hospitality and financial support during his research visit at Kobe University."
]
] | 2012.05193 |
[
[
"High Fermi velocities and small cyclotron masses in LaAlGe"
],
[
"Abstract We report quantum oscillation measurements of LaAlGe, a Lorentz-violating type-II Weyl semimetal with tilted Weyl cones.",
"Very small quasiparticle masses and very high Fermi velocities were detected at the Fermi surface.",
"Whereas three main frequencies have been observed, angular dependence of two Fermi surface sheets indicates possible two-dimensional (2D) character despite the absence of the 2D structural features such as van der Waals bonds.",
"Such conducting states may offer a good platform for low-dimensional polarized spin current in magnetic RAlGe (R = Ce, Pr) materials."
],
[
"DATA AVAILABILITY",
"The data that support the findings of this study are available from the corresponding author upon reasonable request."
]
] | 2012.05234 |
[
[
"Streaming Algorithms for Stochastic Multi-armed Bandits"
],
[
"Abstract We study the Stochastic Multi-armed Bandit problem under bounded arm-memory.",
"In this setting, the arms arrive in a stream, and the number of arms that can be stored in the memory at any time, is bounded.",
"The decision-maker can only pull arms that are present in the memory.",
"We address the problem from the perspective of two standard objectives: 1) regret minimization, and 2) best-arm identification.",
"For regret minimization, we settle an important open question by showing an almost tight hardness.",
"We show {\\Omega}(T^{2/3}) cumulative regret in expectation for arm-memory size of (n-1), where n is the number of arms.",
"For best-arm identification, we study two algorithms.",
"First, we present an O(r) arm-memory r-round adaptive streaming algorithm to find an {\\epsilon}-best arm.",
"In r-round adaptive streaming algorithm for best-arm identification, the arm pulls in each round are decided based on the observed outcomes in the earlier rounds.",
"The best-arm is the output at the end of r rounds.",
"The upper bound on the sample complexity of our algorithm matches with the lower bound for any r-round adaptive streaming algorithm.",
"Secondly, we present a heuristic to find the {\\epsilon}-best arm with optimal sample complexity, by storing only one extra arm in the memory."
],
[
"Introduction",
"The Stochastic Multi-armed Bandits (MAB) problem is a classical framework used to capture decision-making in uncertain environments.",
"Starting with the seminal work of [17], a significant body of work has been developed to address theoretical as well as practical aspects of the MAB problem.",
"See, e.g.",
"[7] for a textbook treatment of the area.",
"In addition to being theoretically interesting, the MAB problem also finds many practical applications in multiple areas, including on-line advertising [20], crowd-sourcing [21], and clinical trials [8].",
"Hence, the study of MAB and its variants is of central interest in multiple fields, including online learning and reinforcement learning.",
"In the MAB setting, a decision-maker is faced with $n$ choices (called arms) and has to sequentially choose one of the $n$ arms (referred to as pulling an arm).",
"Based on the pulled arm, the decision-maker gets a reward drawn from a corresponding reward distribution that is unknown to the decision-maker.",
"The MAB problem has been extensively studied with one of the following two goals: regret minimization and best-arm identification.",
"In the regret minimization literature, several algorithms such as UCB1 [6], Thompson Sampling [19], [3], and KL-UCB [1] have been proposed whose regret bounds are within a constant factor of the optimal regret [13].",
"Algorithms for the best-arm identification problem such as the Median-Elimination algorithm by [10] have optimal (upto constants) sample complexity for this problem.",
"Each of the algorithms mentioned above needs to store the reward statistics (e. g. the number of pulls and mean reward of an arm observed so far) of all the arms in the memory.",
"In many of the applications of the MAB problem, the number of arms (set of advertisements, crowd-workers, etc.)",
"could be very large and the algorithm may not be able to simultaneously store all the arms in the memory.",
"Additionally, the arms could arrive online, i.e., the algorithm may not have access to the entire set of arms at the beginning.",
"The streaming model, first formalized in the seminal work of [4], has been developed to handle data streams where the data arrives online and an algorithm has access to only a limited amount of memory.",
"In this work, we study a setting where an algorithm can store statistics from only a fixed number of arms $m < n$ , where $m$ is called the space complexity of the algorithm.",
"Now, our revised goal is to study the trade-off between space complexity vs. expected regret and space complexity vs. sample complexity, respectively.",
"We follow a standard model in this setup and address both regret minimization and best-arm identification under streaming constraints.",
"Here, $n$ arms arrive one by one in a stream, and we have bounded arm-memory of $m$ , i.e., at any time step at most $m$ ($<n$ ) number of arms can be stored in the memory.",
"We call $m$ the space complexity of the algorithm.",
"At any time step $t$ , the algorithm can only pull an arm that is currently in memory and then, if needed, the algorithm can choose to discard some of the arms that are currently in memory.",
"If, at some time step $t$ , the number of arms in memory is less than $m$ , then the algorithm can choose to store upto total $m$ arms in the memory.",
"Note that, if the arm that is read into memory had previously been in memory and was subsequently discarded before being read back into memory, then the algorithm does not have access to any of the previous reward statistics of the arm.",
"In streaming terminology, algorithms that are allowed to read back an arm that was previously discarded from memory are called multi-pass algorithms.",
"Otherwise, they are called single-pass algorithms.",
"In one of the earliest works that studies the MAB problem with bounded arm-memory, [11] study the infinite-armed MAB problem where only a single arm is stored in the memory at any time step and an arm that is once discarded from the memory cannot be recalled.",
"The work of [15] studies a different variation of bounded memory in the learning from experts setting, where the constraint is on the number of time steps for which the rewards can be stored by the algorithm and not on the number of arms.",
"In our work, we focus on the finite-armed MAB problem.",
"Recently, the MAB problem where arms arrive in a stream and arm-memory is bounded, has been studied in both regret minimization and best-arm identification frameworks.",
"The work of [14] and [9] studies the MAB problem with bounded arm-memory to minimize the expected cumulative regret over $T$ time steps.",
"The algorithms in both these works are multi-pass algorithms, i.e., they assume that an arm discarded from the memory can be again read back into the memory later.",
"[14] propose an upper-confidence bound based algorithm with $O(1)$ space complexity, which achieves an expected cumulative regret bound of $O\\big (\\sum _{i\\ne i^*} \\log (\\frac{\\Delta _i}{\\Delta })\\frac{\\log T}{\\Delta _i}\\big )$ , which is within $\\log (\\frac{\\Delta _i}{\\Delta })$ factor of the UCB-1 regret bound ([6]).",
"Depending on the instance, the regret of this algorithm can be very high.",
"[9] propose an algorithmic framework, which is given $m$ as input and uses a MAB algorithm as a black-box.",
"When the MAB algorithm used is UCB-1, their algorithm achieves expected regret of $O(nm + n^{3/2}m\\sqrt{T \\log (T/nm)})$ .",
"The recent work of [5] studies the best-arm identification variant of this problem.",
"First, they propose an algorithm for the best-coin identification problem (equivalent to MAB with Bernoulli reward distributions), which keeps exactly one extra coin in the memory at any time and has optimal sample complexity.",
"They further extend their algorithm to the top-$k$ coins identification problem, which stores $k$ coins in the memory and has optimal sample complexity.",
"Crucially though, both algorithms assume that the gap parameter $\\Delta $ , which is the difference in the expected rewards of the best and the second-best coins, is known to the algorithm.",
"Throughout our work, we deal with the case when $\\Delta $ is not known to the algorithm."
],
[
"Our Contribution",
"We study the MAB problem under bounded arm-memory, where $n$ arms arrive in a stream and at most $m < n$ arms can be stored in the memory at any time.",
"In this work, we study the trade-off between space complexity vs. expected regret and space complexity vs. sample complexity.",
"Regret minimization: Our first result settles an open question stated in both ([14] and [9]) pertaining to the lower bound on the expected cumulative regret in this model.",
"Using information-theoretic machinery related to $\\mathtt {KL}$ -divergence, we show that any single-pass algorithm in this model will incur an expected regret of $\\Omega \\big ({T^{2/3}}/{m^{7/3}}\\big )$ .",
"Interestingly, this result holds for any $m < n$ which shows that even if the algorithm is allowed to store $n-1$ arms in memory at any time, we cannot hope to get a better regret guarantee.",
"This almost matches with the $\\tilde{O}(T^{2/3})$ bound on the expected cumulative regret, obtained by the standard uniform-exploration algorithm.",
"Best-arm identification: We propose an $r$ -round ($1\\le r \\le \\log ^*n$ ) adaptive $(\\varepsilon ,\\delta )$ -PAC streaming algorithm.",
"Adaptive algorithms are well-studied in active learning.",
"In $r$ -round adaptive streaming algorithm, the arm pulls in each round is decided based on the observed outcomes in the previous rounds, and the best-arm is then output at the end of $r$ rounds.",
"Our algorithm stores $O(r)$ arms in memory at any time, and its sample complexity asymptotically matches with the lower bound for any $r$ -round adaptive algorithm by [2].",
"In particular, when $r = \\log ^* n$ , our algorithm achieves the optimal worst-case sample complexity $O\\big (\\frac{n}{\\varepsilon ^2}\\log (1/\\delta )\\big )$ for any best-arm identification algorithm ([10]) and has space complexity $O(\\log ^* n)$ .",
"This problem was also studied by [5] and their algorithm was claimed to be an $(\\varepsilon ,\\delta )$ -PAC algorithm with optimal sample complexity and $O(\\log ^* n)$ space complexity.",
"However, we show that due to an oversight in their analysis, the algorithm of [5] is not $(\\varepsilon ,\\delta )$ -PAC.",
"In Appendix , we construct a family of input instances for which the algorithm will output a non-$\\varepsilon $ -best arm with probability significantly larger than $\\delta $ .",
"We note here that for the special case when $r = \\log ^* n$ , our algorithm does provide the guarantees claimed in [5].",
"This leads us to the question of finding $(\\varepsilon ,\\delta )$ -PAC guarantee with optimal sample complexity while using only $O(1)$ arm-memory.",
"Towards this, we propose an algorithm that stores exactly one extra arm in the memory.",
"We then show that under the assumption of random-order arrival of arms, our algorithm outputs an $\\varepsilon $ -best arm with high confidence when the expected rewards of the arms are drawn from some standard distributions.",
"We conclude by experimentally showing that our algorithm performs well on randomly generated input without any assumptions on the arrival order of the arms."
],
[
"Notation",
"Let $[k]$ (where $k \\in \\mathbb {N}$ ) denote the set $\\lbrace 1,2,\\ldots , k\\rbrace $ .",
"Let $\\log $ denote the binary logarithm.",
"For integers $r\\ge 0$ , and $a\\ge 1$ , $\\mathtt {ilog}^{(r)}(\\cdot )$ denotes the iterated logarithm of order $r$ , i.e., $\\mathtt {ilog}^{(r)}(a)=\\max \\lbrace \\log (\\mathtt {ilog}^{(r-1)}(a)),1\\rbrace $ and $\\mathtt {ilog}^{(0)}(a)=a$ .",
"Hence, $\\mathtt {ilog}^{(\\log ^* n)}(n)=1$ .",
"Let $\\mathbb {P}$ , $\\mathbb {E}$ denote probability and expectation, respectively."
],
[
"Model and Problem Definition",
"An instance of the MAB problem is defined as the tuple $\\langle n, (\\mu _i)_{i\\in [n]} \\rangle $ , where $n$ is the number of arms.",
"A pull of $\\mathtt {arm}_i$ gives a reward in $[0,1]$ drawn from a distribution with mean $\\mu _i \\in [0,1]$ that is unknown to the decision-maker beforehand.",
"We study this problem in a bounded arm-memory setting where the arms arrive in a stream, and at any time-step, the algorithm can only store a subset of the arms in memory.",
"Any arm that the algorithm wants to pull, either immediately or in the future, has to be present in the memory.",
"An arm that is not present in the memory cannot be pulled.",
"In the literature, MAB problems have been studied with the following objectives: 1) regret minimization, and 2) best-arm identification.",
"Next, we formalize these two notions and their adaptation to our setting.",
"The regret of a MAB algorithm can be thought of as the loss suffered by it due to not knowing the reward distributions of the arms beforehand.",
"Let $i^* = \\arg \\max _{i\\in [n]}\\mu _i$ .",
"Then $\\mathtt {arm}_{i^*}$ is the best arm and let $\\mu ^* = \\mu _{i^*}$ .",
"The cumulative regret (also called the pseudo-regret) of an algorithm over $T$ time-steps is defined as follows: Definition 1 (Cumulative Regret) Given an instance $\\langle n, (\\mu _i)_{i\\in [n]} \\rangle $ of the MAB problem, the cumulative regret of an algorithm after $T$ rounds is defined as $R(T) = \\mu ^*\\cdot T - \\sum _{t=1}^T \\mu _{i_t}$ , where $\\mathtt {arm}_{i_t}$ is the arm pulled by the algorithm at time $t \\in [T]$ , and $\\mu ^*=\\max _{i\\in [n]}\\mu _i$ .",
"The expected cumulative regret of an algorithm is defined as $\\mathbb {E}[R(T)] = \\mu ^*\\cdot T - \\sum _{t=1}^T\\mathbb {E}[\\mu _{i_t}]$ , where the expectation is over the randomness in the algorithm and the distribution of rewards.",
"In the model with bounded arm-memory, the goal is to minimize expected cumulative regret while storing at most $m$ $(< n)$ arms in memory at any time-step.",
"Note that, popular algorithms such as UCB-1 ([6]) and Thompson sampling ([19]) store all $n$ arms in memory, i.e., they have space complexity $O(n)$ .",
"For best-arm identification, the goal of a decision-maker is to output the best arm $\\mathtt {arm}_{i^*}$ using the minimum number of arm pulls.",
"In practice, a relaxed goal is to find an arm which is close to the best arm in terms of the expected reward.",
"We formalize this notion below.",
"Definition 2 ($\\varepsilon $ -best arm) Given a parameter $\\varepsilon \\in (0,1)$ , $\\mathtt {arm}_i$ with mean reward $\\mu _i$ is said to be an $\\varepsilon $ -best arm if $\\mu _i \\ge \\mu ^* - \\varepsilon $ .",
"Otherwise we call the arm a non-$\\varepsilon $ -best arm.",
"The reward gap of $\\mathtt {arm}_i$ is defined as $\\Delta _i = \\mu ^* - \\mu _i$ .",
"Without loss of generality, we assume that the best arm is unique, i.e., $\\Delta _i > 0$ for all $i \\ne i^*$ .",
"Definition 3 ($(\\varepsilon ,\\delta )$ -PAC Algorithm) Given an approximation parameter $\\varepsilon \\in [0,1)$ and a confidence parameter $\\delta \\in [0,1/2)$ , an algorithm $\\mathcal {A}$ is said to be an $(\\varepsilon ,\\delta )$ -PAC algorithm if it outputs an $\\varepsilon $ -best arm with probability at least $1 - \\delta $ .",
"Traditionally, the goal in the best-arm identification problem is to design an $(\\varepsilon ,\\delta )$ -PAC algorithm that minimizes the total number of arm pulls.",
"Under the streaming setup, given bounded arm-memory $m$ , the goal now is to find an $(\\varepsilon ,\\delta )$ -PAC algorithm that minimizes the total number of arm pulls while storing at most $m$ arms in the memory at any time.",
"Our best-arm identification algorithm in Section is an $r$ -round adaptive streaming algorithm, where in each round $j \\in [r]$ , only a subset of the arms processed in round $j$ is sent to round $j+1$ and the rest of the arms are discarded from the memory.",
"Additionally, once an arm is discarded, it cannot be pulled in any subsequent rounds, i.e., it is a single-pass algorithm.",
"The set of arms to be sent to round $j+1$ is decided based only on the outcomes in rounds 1 to $j$ .",
"Further, once an arm reaches round $j+1$ , the number of times the arm will be sampled in round $j+1$ gets decided before the sampling begins.",
"This number only depends on the round index, i.e., $j+1$ and the outcomes of the pulls of any arm up to the round $j$ .",
"All arms in the stream are pulled in round 1 and the arm output after round $r$ is the best-arm guess of the algorithm.",
"The $r$ -round adaptive algorithm model was discussed in great detail by [2].",
"If $r=1$ , then the algorithm is said to be non-adaptive.",
"The algorithm is said to be fully adaptive if $r$ is unbounded.",
"If the algorithm is fully adaptive then there is a potential to reduce the sample complexity but the downside of full-adaptivity is that such algorithms are highly sequential.",
"This is because the set of arms to be sampled in a given round can only be determined after we observe the outcomes of pulls of the arms up to the previous round.",
"In contrast, algorithms with only a few rounds of adaptivity enable us to enjoy the benefits of parallelism.",
"Some of our results in Section hold for random-order arrival of arms, which we define next.",
"Let $\\langle n, (\\mu _i)_{i\\in [n]} \\rangle $ be an instance of the MAB problem.",
"Let, $\\sigma : [n] \\rightarrow [n]$ be a permutation and let $(\\mathtt {arm}_{\\sigma (i)})_{i\\in [n]}$ be the ordering of $(\\mathtt {arm}_i)_{i\\in [n]}$ under $\\sigma $ .",
"Define $\\mathcal {S}_n = \\lbrace \\sigma : \\sigma \\text{~is a permutation of ~}[n]\\rbrace $ .",
"Under the random-order arrival model, we assume that the arrival order of the arms in the stream is determined by a permutation $\\sigma \\in \\mathcal {S}_n$ , which is drawn uniformly at random from the set $\\mathcal {S}_n$ .",
"The arms arrive in the order in which they appear in the tuple $(\\mathtt {arm}_{\\sigma (i)})_{i\\in [n]}$ , i.e., the first arm to arrive in the stream is $\\mathtt {arm}_{\\sigma (1)}$ , followed by $\\mathtt {arm}_{\\sigma (2)}$ , and so on.",
"Random-order arrival is a well-studied model in optimization under uncertainty due to its connectiond with secretary problem and optimal stopping theory [12].",
"The algorithm of [9] also uses an analogous random shuffling of arms."
],
[
"Regret Minimization",
"In this section, we study limitations of bounded arm-memory for regret minimization.",
"An adaptation of uniform-exploration algorithm (see [18]) achieves expected cumulative regret of $\\tilde{O}(T^{2/3})$ with an arm-memory of two.",
"The algorithm keeps in memory one arm $\\mathtt {arm}^*$ , called the king, with the best empirical mean $\\widehat{\\mu }_{\\mathtt {arm}^*}$ among the arms seen so far.",
"Whenever a new arm $\\mathtt {arm}_i$ arrives, $\\mathtt {arm}_i$ is sampled $(T/n)^{2/3} O(\\log T)^{1/3}$ times to obtain its empirical mean $\\widehat{\\mu }_i$ .",
"Then $\\widehat{\\mu }_i$ is compared with $\\widehat{\\mu }_{\\mathtt {arm}^*}$ .",
"If $\\widehat{\\mu }_{\\mathtt {arm}^*} < \\widehat{\\mu }_i$ , then $\\mathtt {arm}_i$ becomes the new king, replacing $\\mathtt {arm}^*$ .",
"After the algorithm tries out all the arms, it returns the king as the best-arm and continues to sample it for the rest of the time horizon.",
"A question left open in (Chaudhuri et al.",
"[9]) is to provide a lower bound on the expected cumulative regret of an algorithm with bounded arm-memory.",
"We settle the question by showing that any single-pass algorithm for such a setting incurs at least $\\Omega (T^{2/3})$ regret.",
"Our result is based on $\\mathtt {KL}$ -divergence, which we define below.",
"Definition 4 Let $\\mathtt {\\Omega }$ be a finite sample space and $p, q$ be two probability distributions on $\\mathtt {\\Omega }$ .",
"$\\mathtt {KL}$ -divergence is defined as: $ \\mathtt {KL}(p,q)=\\sum _{x \\in \\mathtt {\\Omega }} p(x) \\ln (p(x)/q(x)) =\\mathbb {E}_p[\\ln (p(x)/q(x))].$ Now we state some fundamental properties of $\\mathtt {KL}$ -divergence that will be needed in this section.",
"Theorem 1 ([18]) $\\mathtt {KL}$ -divergence satisfies the following properties: Pinsker's inequality: For any event $A \\subset \\mathtt {\\Omega }$ , we have $2(p(A)-q(A))^2 \\le \\mathtt {KL}(p,q)$ .",
"Chain rule for product distributions: Let the sample space be a product $\\mathtt {\\Omega }= \\mathtt {\\Omega }_1\\times \\mathtt {\\Omega }_2 \\times \\ldots \\times \\mathtt {\\Omega }_t$ .",
"Let $p, q$ be two distributionas on $\\mathtt {\\Omega }$ such that $p = p_1 \\times p_2 \\times \\ldots \\times p_t$ and $q = q_1 \\times q_2 \\times \\ldots \\times q_t$ , where $p_j, q_j$ are distributions on $\\mathtt {\\Omega }_j$ , for each $j \\in [t]$ .",
"Then $\\mathtt {KL}(p,q)=\\sum _{j=1}^t \\mathtt {KL}(p_j, q_j)$ .",
"Random coins: Let $\\mathtt {B}_\\epsilon $ denote a Bernoulli distribution with mean $(1+\\epsilon )/2$ .",
"Then $\\mathtt {KL}(\\mathtt {B}_\\epsilon , \\mathtt {B}_0) \\le 2 \\epsilon ^2$ and $\\mathtt {KL}(\\mathtt {B}_0, \\mathtt {B}_\\epsilon ) \\le \\epsilon ^2$ , for all $\\epsilon \\in (0,1/2)$ .",
"Our main result is the following theorem.",
"Theorem 2 In the MAB setting, fix the number of arms $n$ and the time horizon $T$ .",
"For any online MAB algorithm, if we are allowed to store at most $m < n$ arms, then there exists a problem instance such that $\\mathbb {E}[R(T)] \\ge \\Omega ({T^{2/3}}/{m^{7/3}})$ We consider 0-1 rewards and the following family of problem instances $\\lbrace \\mathcal {I}_j:j\\in \\lbrace 0,\\ldots ,m\\rbrace \\rbrace $ each containing $n$ arms, with parameter $\\epsilon >0$ (where $\\epsilon =\\frac{1}{m^{1/3}T^{1/3}}$ ): $\\mathcal {I}_0= \\left\\lbrace \\begin{array}{rcl}\\mu _i &= 1/2,~~~~~~~ &\\mbox{for~~} i\\ne n;\\\\\\mu _i &= 1,~~~~~~~~~~ & \\mbox{for~~} i=n.\\end{array}\\right.\\\\\\forall j\\in [m],~\\mathcal {I}_j = \\left\\lbrace \\begin{array}{rcl}\\mu _i & =(1+\\epsilon )/2,& \\mbox{for~~} i=j; \\\\\\mu _i & = 1/2,~~~~~~~ & \\mbox{for~~} i \\ne j.\\end{array}\\right.$ In the above instances, $\\mu _i$ denotes the expected reward of $\\mathtt {arm}_i$ , the $i$ -th arm to arrive in the stream.",
"Note that a deterministic algorithm that directly stores the first $m$ arms in the memory and has the least expected regret among all such deterministic algorithms, can not have a worse regret compared to any other algorithm that processes the first $m$ arms in some different manner.",
"This is because the processing of any such algorithm can be replicated by an algorithm which directly stores the first $m$ arms.",
"So we fix a deterministic algorithm $\\mathcal {A}$ which directly stores the the first $m$ arms in the memory.",
"We next set up the sample space.",
"Let $L = {1}/({4m^2\\epsilon ^2})$ .",
"Further, let $(r_s(i): i\\in [m], s \\in [L])$ be a tuple of mutually independent Bernoulli random variables where $r_s(i)$ has expectation $\\mu _i$ .",
"We interpret $r_s(i)$ as the reward obtained when $\\mathtt {arm}_i$ is pulled for the $s$ -th time and the tuple is called the rewards table.",
"The sample space is then expressed as $\\mathtt {\\Omega }= \\lbrace 0,1\\rbrace ^{m\\times L}$ and any $\\omega \\in \\mathtt {\\Omega }$ can be interpreted as a realization of the rewards table.",
"Each instance $\\mathcal {I}_j$ , where $j \\in \\lbrace 0,\\ldots , m\\rbrace $ , defines a distribution $P_j$ on $\\mathtt {\\Omega }$ as follows: $P_j(A)= \\mathbb {P}[A~|~\\mathcal {I}_j], \\text{~~~~~for each $A\\subseteq \\mathtt {\\Omega }$}$ Given an instance $\\mathcal {I}_j$ where $j\\in \\lbrace 0,\\ldots , m\\rbrace $ , let $P_j^{i,s}$ be the distribution of $r_s(i)$ under this instance.",
"Then we have that $P_j = \\prod _{i\\in [m],s\\in [L]} P_{j}^{i,s}$ .",
"Let $S_{t} \\subseteq \\lbrace \\mathtt {arm}_1, \\mathtt {arm}_2, \\dots , \\mathtt {arm}_m\\rbrace $ denote the subset of first $m$ arms which are discarded from memory till (and including) time step $t$ by the algorithm $\\mathcal {A}$ .",
"If $\\mathtt {arm}_i$ is discarded before the algorithm begins pulling arms, call this time step 0, then we include $\\mathtt {arm}_i$ in the set $S_0$ where $i\\in [m]$ .",
"As time horizon $T$ is fixed and we will eventually discard all arms at the end of time horizon, we can assume that $S_T= \\lbrace \\mathtt {arm}_1, \\mathtt {arm}_2, \\dots , \\mathtt {arm}_m\\rbrace $ .",
"For all $\\omega \\in \\mathtt {\\Omega }$ , let $T^{\\prime }_\\omega = \\arg \\min _{0\\le t \\le T}\\lbrace t : S_t \\ne \\emptyset \\rbrace $ , i.e., $T^{\\prime }_{\\omega }$ is the number of time steps since the beginning of the algorithm $\\mathcal {A}$ when some arm in $\\lbrace \\mathtt {arm}_1, \\mathtt {arm}_2, \\dots , \\mathtt {arm}_m\\rbrace $ is discarded from memory for the first time.",
"Let $A_1=\\lbrace \\omega \\in \\mathtt {\\Omega }: ~T^{\\prime }_\\omega \\le L\\rbrace $ be the set of reward realizations for which $T^{\\prime }_{\\omega } \\le L$ .",
"Now fix some arm $i \\in \\lbrace \\mathtt {arm}_1, \\mathtt {arm}_2, \\dots , \\mathtt {arm}_m\\rbrace $ .",
"Define $A_2^{i}=\\lbrace \\omega \\in \\mathtt {\\Omega }: \\mathtt {arm}_i \\in S_{T^{\\prime }_\\omega }\\rbrace $ to be the event that the $\\mathtt {arm}_i$ belongs to $S_{T^{\\prime }_{\\omega }}$ .",
"Now, let $A^{i} = A_1 \\cap A_2^{i}$ be the set of reward realizations such that $\\forall \\omega \\in A^i$ , $T^{\\prime }_\\omega \\le L$ and $\\mathtt {arm}_i$ is discarded from memory at the time step $T^{\\prime }_\\omega $ .",
"Also, for any event $A\\subseteq \\mathtt {\\Omega }$ , let $\\overline{A}=\\mathtt {\\Omega }\\setminus A$ .",
"Now we have the following observation for instance $\\mathcal {I}_0$ .",
"Observation 1 If $\\omega \\in \\overline{A_1}$ , then the algorithm $\\mathcal {A}$ would incur regret of $\\Omega (\\frac{1}{m^2 \\epsilon ^2})$ on the instance $\\mathcal {I}_0$ .",
"Let $i^{\\prime }=\\arg \\max _{i\\in [m]}P_0(A^i)$ .",
"We obtain the next observation due to the fact that $L=o(T)$ .",
"Observation 2 For all $\\omega \\in A^{i^{\\prime }}$ , the regret for instance $\\mathcal {I}_{i^{\\prime }}$ is at least $\\frac{\\epsilon (T-T^{\\prime }_\\omega )}{2}=\\Omega (\\epsilon T)$ .",
"Now we will prove the following inequality which will be useful in our analysis: $m\\cdot P_{i^{\\prime }}(A^{i^{\\prime }})+P_0(\\overline{A_1})\\ge \\frac{1}{4}.$ The above inequality is trivially true if $P_0(\\overline{A_1})\\ge 1/4$ .",
"Therefore, let us assume $P_0(\\overline{A_1})\\le 1/4$ , i.e., $P_0(A_1)\\ge 3/4$ .",
"Then $P_0(A^{i^{\\prime }}) \\ge {3}/{(4m)}$ , by averaging argument.",
"Using Theorem REF for distributions $P_{0}$ and $P_{i^{\\prime }}$ , we obtain: $2(P_{0}(A^{i^{\\prime }})-P_{i^{\\prime }}(A^{i^{\\prime }}))^2 & \\le \\mathtt {KL}(P_{0},P_{i^{\\prime }}) \\\\& = \\sum _{i\\in [m]}\\sum _{t=1}^{L} \\mathtt {KL}(P_0^{i,t},P_{i^{\\prime }}^{i,t}) \\\\& = \\sum _{i\\in [m] \\setminus \\lbrace i^{\\prime }\\rbrace } \\sum _{t=1}^{L} \\mathtt {KL}(P_0^{i,t},P_{i^{\\prime }}^{i,t}) +\\sum _{t=1}^{L} \\mathtt {KL}(P_0^{i^{\\prime },t},P_{i^{\\prime }}^{i^{\\prime },t})\\\\& \\le 0 + L\\cdot 2\\epsilon ^2.$ In the last inequality, the first term of the summation is zero because all arms $\\mathtt {arm}_i$ , where $i \\in [m]\\setminus \\lbrace i^{\\prime }\\rbrace $ , have identical reward distributions under instances $\\mathcal {I}_0$ and $\\mathcal {I}_{i^{\\prime }}$ .",
"To bound the second term in the summation, we use the last property from Theorem REF .",
"Thus we have, $P_0(A^{i^{\\prime }})- P_{i^{\\prime }}(A^{i^{\\prime }}) \\le \\epsilon \\sqrt{L}$ .",
"Hence, $P_{i^{\\prime }}(A^{i^{\\prime }}) \\ge P_0(A^{i^{\\prime }})-\\epsilon \\sqrt{L} \\ge ({3}/{(4m)})-({1}/{(2m)}) = {1}/{(4m)}$ .",
"Here, we use $P_0(A^{i^{\\prime }})\\ge {3}/{(4m)}$ and $L= 1/(4 m^2 \\epsilon ^2)$ .",
"Hence, if $P_0(\\overline{A_1})\\le \\frac{1}{4}$ then $m\\cdot P_{i^{\\prime }}(A^{i^{\\prime }})\\ge \\frac{1}{4}$ .",
"This proves Inequality (REF ).",
"Now suppose that we choose an instance $\\mathcal {I}$ uniformly at random from the family of $m+1$ instances $\\lbrace \\mathcal {I}_j:j\\in \\lbrace 0,\\ldots ,m\\rbrace \\rbrace $ , i.e., we have $\\mathbb {P}\\big [\\mathcal {I} = \\mathcal {I}_{j}\\big ] = {1}/{(m+1)}$ for any $j\\in \\lbrace 0,\\ldots ,m\\rbrace $ .",
"Then, $\\mathbb {E}[R(T)] &\\ge ~\\mathbb {P}\\big [\\mathcal {I} = \\mathcal {I}_{i^{\\prime }}\\big ]\\cdot P_{i^{\\prime }}(A^{i^{\\prime }})\\cdot \\Omega (\\epsilon T) + \\mathbb {P}\\big [\\mathcal {I} = \\mathcal {I}_{0}\\big ]\\cdot P_0(\\overline{A_1}) \\cdot \\Omega \\Big (\\frac{1}{m^2\\cdot \\epsilon ^2}\\Big ) \\\\&\\ge ~\\frac{1}{m+1}\\cdot (m\\cdot P_{i^{\\prime }}(A^{i^{\\prime }}))\\cdot \\Omega \\Big (\\frac{\\epsilon T}{m}\\Big ) + \\frac{1}{m+1}\\cdot P_0(\\overline{A_1}) \\cdot \\Omega \\Big (\\frac{1}{m^2\\cdot \\epsilon ^2}\\Big ) \\\\& = ~\\Big (m\\cdot P_{i^{\\prime }}(A^{i^{\\prime }}) +P_0(\\overline{A_1})\\Big )\\cdot \\Omega \\Big (\\frac{T^{2/3}}{m^{7/3}}\\Big )\\\\&=\\Omega \\Big (\\frac{T^{2/3}}{m^{7/3}}\\Big )$ The first inequality follows from Observation REF and REF .",
"In the last equality, we have used Inequality (REF ).",
"Note that the expectation is taken over that choice of input instance and randomness in reward.",
"Since a randomized algorithm is a distribution over deterministic algorithms, the above result also holds for any randomized algorithm.",
"Also, by slight modification to the above family of instances, we can show the same lower bound on the expected cumulative regret even under the assumption that arms arrive in a random-order (see Appendix for details)."
],
[
"Best-arm Identification",
"In this section, we design an $(\\varepsilon ,\\delta )$ -PAC algorithm which minimizes the total number of arm pulls while storing at most $m < n$ arms in memory.",
"Towards this goal, we propose a general algorithmic framework (Algorithm ), which is an $r$ -round adaptive streaming algorithm for $r \\in [\\log ^*n]$ .",
"This algorithm has the optimal sample complexity for any $r$ -round adaptive streaming algorithm (refer to Appendix for a detailed discussion) and stores $O(r)$ arms in memory at any time step.",
"Intuitively, in each of the $r$ rounds, we keep a running best arm candidate (denoted $\\mathtt {arm}^*_i$ for round $i$ ) in the memory.",
"Once we see sufficient number ($= c_i$ for round $i$ ) of arms in a round, we send $\\mathtt {arm}_i$ to the next round.",
"At each round $i$ , only one out of every $c_i$ arms is sent to round $i+1$ .",
"For higher round indices, the number of arms reaching that round decreases rapidly.",
"Hence, each arm can be sampled more number of times for a more refined comparison without affecting the sample complexity.",
"Our algorithm is related to the recent work by (Assadi et al.",
"[5]).",
"They proposed an $(\\varepsilon ,\\delta )$ -PAC algorithm for this setting which has optimal sample complexity and stores at most $\\log ^* n$ arms in memory at any time step.",
"We remark here that their analysis has an oversight, due to which their algorithm will output a non-$\\varepsilon $ -best arm with probability much greater than $\\delta $ for some input sequences.",
"We refer the reader to Appendix for a detailed discussion.",
"Unlike their algorithm, whenever a new arm arrives we do not again sample the stored best arm for comparison.",
"Instead, we reuse the stored empirical mean of the best arm for the comparison.",
"Due to this subtle difference, for $r = \\log ^* n$ , our algorithm does in fact provide the guarantees claimed in (Assadi et al.",
"[5]).",
"In this section we use the terms round and level interchangeably.",
"For simplicity during the analysis, we ignore the ceil in the expression of $s_\\ell $ and $c_\\ell $ .",
"[ht!]",
"[1] $\\lbrace \\varepsilon _\\ell \\rbrace _{\\ell =1}^{r}: \\varepsilon _\\ell ={\\varepsilon }/{2^{\\ell +1}}.$ //Intermediate gap parameter.",
"$\\lbrace \\beta _\\ell \\rbrace _{\\ell =1}^{r}: \\beta _\\ell ={1}/{\\varepsilon ^2_\\ell }.$ $\\lbrace s_\\ell \\rbrace _{\\ell =1}^{r}: s_\\ell = \\lceil 2\\beta _\\ell \\big (\\mathtt {ilog}^{(r+1-\\ell )}(n)+\\log (\\frac{2^{\\ell +2}}{\\delta })\\big )\\rceil .$ //Samples per arm in level $\\ell $ .",
"$\\lbrace c_\\ell \\rbrace _{\\ell =1}^{r}: c_\\ell =\\lceil \\mathtt {ilog}^{(r-\\ell )}(n)\\rceil .$ //Number of arms in level $\\ell $ .",
"Counters: $C_1,C_2,\\ldots ,C_r$ initialized to 0.",
"Stored arms: $\\mathtt {arm}^*_1, \\mathtt {arm}^*_2, \\ldots , \\mathtt {arm}^*_r$ , where $\\mathtt {arm}^*_\\ell $ is the arm with the highest empirical mean at $\\ell $ -th level.",
"Stored empirical means: $\\mu ^*_1,\\mu ^*_2,\\ldots ,\\mu ^*_r$ , where $\\mu ^*_\\ell $ is the highest empirical mean of $\\ell $ -th level, initialized to 0. a new arm $\\mathtt {arm}_i$ arrives in the stream Read $\\mathtt {arm}_i$ to memory.",
"Modified Selective Promotion: Starting from level $\\ell =1$ : Sample $\\mathtt {arm}_i$ for $s_\\ell $ times and compare its empirical mean with $\\mu ^*_\\ell $ .",
"If $\\widehat{\\mu }_{\\mathtt {arm}_i} < \\mu ^*_\\ell $ , drop $\\mathtt {arm}_i$ .",
"Otherwise, replace $\\mathtt {arm}^*_\\ell $ with $\\mathtt {arm}_i$ and make $\\mu ^*_\\ell $ equal to $\\widehat{\\mu }_{\\mathtt {arm}_i}$ .",
"Increase $C_\\ell $ by 1.",
"If $C_\\ell =c_\\ell $ and $r=1$ , return $\\mathtt {arm}^*_\\ell $ as the selected arm and terminate the Algorithm.",
"If $C_\\ell =c_\\ell $ , make $C_\\ell $ and $\\mu ^*_\\ell $ equal to 0, send $\\mathtt {arm}^*_\\ell $ to the next level by calling Line with $(\\ell =\\ell +1)$ .",
"If there is any arm which is stored in a level below $r$ , then promote it to level $r$ and sample it for $s_r$ times.",
"Let $\\mathtt {arm}^*_{\\ell ^{\\prime }}$ be the arm with highest empirical mean $\\mu ^*_{\\ell ^{\\prime }}$ among the arms which were sampled, where $\\ell ^{\\prime }\\in [r-1]$ .",
"If $\\mu ^*_r > \\mu ^*_{\\ell ^{\\prime }}$ then Return $\\mathtt {arm}^*_r$ Else Return $\\mathtt {arm}^*_{\\ell ^{\\prime }}$ Theorem 3 Algorithm is an $(\\varepsilon ,\\delta )$ -PAC algorithm with sample complexity $O(\\frac{n}{\\varepsilon ^2}\\cdot (\\mathtt {ilog}^{(r)}(n)+\\log (\\frac{1}{\\delta })))$ and space complexity $O(r)$ where $1\\le r \\le \\log ^*(n)$ .",
"We prove this theorem using the following lemmas.",
"Lemma REF shows that the sample complexity of our algorithm is $O(\\frac{n}{\\varepsilon ^2}\\cdot (\\mathtt {ilog}^{(r)}(n)+\\ln (\\frac{1}{\\delta })))$ .",
"Lemma REF gives the proof of correctness.",
"Further, at each of the $r$ levels we store a single arm along with its empirical mean, implying the space complexity of our algorithm.",
"Lemma 4 The sample complexity of the algorithm is $O(\\frac{n}{\\varepsilon ^2}\\cdot (\\mathtt {ilog}^{(r)}(n)+\\ln (\\frac{1}{\\delta })))$ .",
"If $r=1$ , then the total number of samples is $n\\cdot s_1= O(\\frac{n}{\\varepsilon ^2}\\cdot (\\mathtt {ilog}^{(r)}(n)+\\log (\\frac{1}{\\delta })))$ .",
"Let $c_0 :=1$ .",
"So for the rest of the analysis we assume that $r\\ge 2$ and define $c_i:=2^{c_{i-1}},\\forall i>r$ .",
"Note that since $2\\le r\\le \\log ^*(n)$ , $c_2\\ge 2$ and $c_i= 2^{c_{i-1}}$ , $\\forall i\\ge 3$ .",
"For any level $\\ell -1$ , we send one arm from level $\\ell -1$ to level $\\ell $ for every $c_{\\ell -1}$ arms seen (this is excluding the arms sampled in Step ).",
"Hence during the Modified Selective Promotion, the number of arms that can reach any level $\\ell $ is at most ${n}/({\\prod _{i=0}^{\\ell -1}c_i})$ .",
"Each arm arriving at level $\\ell $ is pulled exactly $s_\\ell $ times.",
"Also note that we can sample up to $(r-1)\\cdot s_r$ times in the Step .",
"Since, we have $r$ levels, the total number of samples can be bounded as: $&\\sum _{\\ell =1}^{r} \\frac{n}{\\prod _{i=0}^{\\ell -1}c_i} \\cdot s_\\ell +(r-1)\\cdot s_r \\\\\\le & \\sum _{\\ell =1}^{r} \\frac{2n\\beta _\\ell \\big ({\\mathtt {ilog}}^{(r+1-\\ell )}(n)+\\log (\\frac{2^{\\ell +2}}{\\delta })\\big )}{\\prod _{i=0}^{\\ell -1}c_i}+r\\cdot s_r\\\\\\le & n\\cdot s_1+ r\\cdot s_r+ \\frac{2n}{\\varepsilon ^2} \\cdot \\sum _{\\ell =2}^{r} 4^{\\ell +1} \\cdot \\bigg ( \\frac{{\\mathtt {ilog}}^{(r+1-\\ell )}(n)}{c_{\\ell -1}\\cdot c_{\\ell -2}} + \\frac{2\\ell \\cdot \\log (\\frac{2}{\\delta })}{c_{\\ell -1}\\cdot c_{\\ell -2}}\\bigg )\\\\&\\text{\\big (Since, $\\prod _{i=0}^{\\ell -1}c_i \\ge c_{\\ell -1}\\cdot c_{\\ell -2}$, $\\beta _{\\ell }={4^{\\ell +1}}/{\\varepsilon ^2},$ and }\\text{$\\log ({2^{\\ell +2}}/{\\delta })\\le \\log ({2^{2\\ell }}/{\\delta ^{2\\ell }})\\le 2\\ell \\log ({2}/{\\delta })$\\big )}\\\\\\le & n\\cdot s_1+ r\\cdot s_r+({2\\cdot 4^5\\cdot n}/{\\varepsilon ^2})\\sum _{\\ell =2}^{\\infty }({4^{\\ell - 4}}/{c_{\\ell -2}})+ ({2^2\\cdot 4^5\\cdot n }/{\\varepsilon ^2})\\cdot \\log ({2}/{\\delta })\\sum _{\\ell =2}^{\\infty }({4^{\\ell - 3}}/{c_{\\ell -1}})\\\\&\\text{\\big (Since, $c_{\\ell -1}={\\mathtt {ilog}}^{(r+1-\\ell )}(n), \\text{ and } ({\\ell }/{c_{\\ell -2}})\\le 4$\\big )}\\\\\\le & n\\cdot s_1+ r\\cdot s_r +{O(n/\\varepsilon ^2)} \\big (1 + \\log ({2}/{\\delta })\\big )\\\\&\\text{\\big (Since, $\\sum _{\\ell =0}^{5}\\frac{4^{\\ell -2}}{c_\\ell } = O(1)$, $\\sum _{\\ell =6}^{\\infty }\\frac{4^{\\ell -2}}{c_\\ell } < \\sum _{\\ell =6}^{\\infty }\\frac{4^{\\ell -2}}{8^{\\ell -2}} <1$\\big )}\\\\\\le & O({n}/{\\varepsilon ^2}) \\cdot ({\\mathtt {ilog}}^{(r)}(n)+\\log ({1}/{\\delta }))\\\\& \\text{\\big (Since, $r\\cdot s_r= O(n/\\varepsilon ^2)(1+\\log (1/\\delta ))$ and }\\text{$n\\cdot s_1= O({n}/{\\varepsilon ^2})\\cdot ({\\mathtt {ilog}}^{(r)}(n)+\\log ({1}/{\\delta }))\\big )$.",
"}$ Hence, we have that the sample complexity is $O(\\frac{n}{\\varepsilon ^2}\\cdot (\\mathtt {ilog}^{(r)}(n)+\\log (\\frac{1}{\\delta })))$ .",
"We now use the following two claims to prove the correctness of our algorithm.",
"The proofs of both these claims follow from the application of Hoeffding's inequality followed by taking a union bound over the number of arms that are compared at a given level (see Appendix for the detailed proof).",
"claimRoundAlgoLevelSuboptimality For any level $\\ell <r$ , let $\\mathtt {arm}^{\\prime }_\\ell $ be the best arm to ever reach this level.",
"Then, with probability at least $1 - \\frac{\\delta }{2^{\\ell +1}}$ , an arm with reward gap at most $\\varepsilon _\\ell $ from $\\mathtt {arm}^{\\prime }_\\ell $ is sent to level $\\ell + 1$ or is sampled in Step .",
"claimRoundAlgoLevelSuboptimalityTwo Let $\\mathtt {arm}^{\\prime }_r$ be the best arm among the arms which reached level $r$ including the arms which were sampled in Step of Algorithm .",
"Then, with probability at least $1 - \\frac{\\delta }{2^{r+1}}$ , an arm with reward gap at most $\\varepsilon _r$ from $\\mathtt {arm}^{\\prime }_r$ is returned by the Algorithm.",
"Lemma 5 With probability at least $1-\\delta $ , the arm selected by the algorithm is an $\\varepsilon $ -best arm.",
"Let the best arm be $\\mathtt {arm}^*$ .",
"By union bound and Claim , the probability that an arm with reward gap at most $\\sum _{i=1}^{r-1}\\varepsilon _i$ from $\\mathtt {arm}^*$ does not reach either level $r$ via Modified Selective Promotion or is not sampled in Step is upper bounded by $\\sum _{\\ell = 1}^{r-1} \\frac{\\delta }{2^{\\ell +1}}$ .",
"Given this upper bound and Claim , the probability that an arm with reward gap at most $\\sum _{i=1}^{r}\\varepsilon _i$ from $\\mathtt {arm}^*$ is not returned by the Algorithm is upper bounded by: $ \\sum _{\\ell = 1}^{r} \\frac{\\delta }{2^{\\ell +1}} \\le \\delta \\sum _{\\ell =1}^\\infty \\frac{1}{2^{\\ell +1}} = \\frac{\\delta }{2} < \\delta $ .",
"Now with probability at least $1-\\delta $ , an arm with reward gap at most: $\\sum _{\\ell = 1}^{r} \\varepsilon _\\ell = \\sum _{\\ell = 1}^{r} \\frac{\\varepsilon }{2^{\\ell + 1}} \\le \\frac{\\varepsilon }{2}< \\varepsilon $ from $\\mathtt {arm}^*$ is returned by the algorithm.",
"This concludes the proof of this lemma."
],
[
"Towards Constant Arm-Memory",
"In this section, we take a step towards designing an $(\\varepsilon ,\\delta )$ -PAC algorithm which has optimal sample complexity, while using only $O(1)$ arm-memory.",
"[5] have proposed an algorithm which stores only one extra arm, assuming $\\Delta $ is known.",
"They maintain a candidate best-arm (king) and assign it a certain budget, essentially denoting the number of permissible arm pulls.",
"For each arriving arm, both king and the new arm are sampled for some number of times, if needed in multiple levels, until either king wins against the new arm (by having a higher empirical mean at one of the levels) or the budget of the king is exhausted.",
"If the budget is exhasuted then the king is replaced with the new arm and the process is repeated.",
"The number of samples and budget is proportional to $1/\\Delta ^2$ and this careful choice of budget ensured smaller sample complexity.",
"However, when $\\Delta $ is not known a similar approach will not work.",
"See Appendix for more details.",
"Inspired by their framework, we propose Algorithm for the case when $\\Delta $ is not known to the algorithm.",
"In Step , we go to the next level of challenge subroutine only if there is a good chance that the newly arrived arm is significantly better compared to the king.",
"Intuitively, we ensure two properties: (i) king only lose to an arm that has significantly better mean compared to the king, and (ii) when the true best-arm arrives, it can only lose to a king if their means are quite close.",
"See Appendix for more details.",
"[ht!]",
"[1] $\\lbrace s_\\ell \\rbrace ^{\\infty }_{\\ell =1}:s_\\ell =\\lceil \\frac{2}{(\\frac{\\varepsilon }{200})^2}\\cdot \\ln \\left(\\frac{4}{\\delta }\\right)\\cdot 3^\\ell \\rceil $ .",
"$b:=\\lceil \\frac{2}{(\\frac{\\varepsilon }{200})^2}\\cdot C \\cdot \\ln \\left(\\frac{4}{\\delta }\\right)+s_1\\rceil $ .",
"//Here, $C$ is a large constant that we choose later.",
"Let king be the first available arm and set its budget $\\phi :=\\phi (\\textbf {king})=0$ .",
"A new arm $\\mathtt {arm}_i$ arrives in the stream Increase the budget $\\phi (\\textbf {king})$ by $b$ .",
"Challenge subroutine: For level $\\ell =1$ to $+\\infty $ : If $\\phi (\\textbf {king})<s_\\ell $ : we declare $\\textbf {king}$ defeated, make $\\mathtt {arm}_i$ the king, initialize its budget to 0 and go to Step .",
"Otherwise, we decrease $\\phi (\\textbf {king})$ by $s_\\ell $ and sample both king and $\\mathtt {arm}_i$ for $s_\\ell $ times.",
"Let $\\widehat{\\mu }_{king}$ and $\\widehat{\\mu }_i$ denote the empirical means of king and $\\mathtt {arm}_i$ in this trial.",
"If $\\widehat{\\mu }_{king} > \\widehat{\\mu }_i-0.495\\varepsilon $ , we declare king winner and go to the next arm in the stream; otherwise we go to the next level of challenge (increment $\\ell $ by one).",
"Return $\\textbf {king}$ as the selected best-arm.",
"We experimentally show that this algorithm performs well on randomly generated input and provide a theoretical justification for these experimental results as follows."
],
[
"Random Order Arrival",
"Definition 5 Let the P.D.F.",
"and C.D.F.",
"of a distribution $\\mathcal {D}$ be $g(x)$ and $F(x)$ , respectively.",
"Then the truncated distribution $\\mathcal {D}$ with support $(a,b]$ is a distribution where P.D.F.",
"and C.D.F.",
"are $\\big \\lbrace \\frac{g(x)}{F(b)-F(a)}\\text{ for }x\\in (a,b], 0 \\text{ for } x\\notin (a,b]\\big \\rbrace $ and $\\big \\lbrace \\frac{F(x)-F(a)}{F(b)-F(a)}\\text{ for } x\\in (a,b],1\\text{ for } x>b,0 \\text{ for }x\\le a\\big \\rbrace $ , respectively.",
"In the following theorem we show that if the arms arrive in random order and the means of arms come from various common distributions, then Algorithm is successful with reasonable probability.",
"See Appendix for the proof.",
"Theorem 6 Let the means of the arms come from one of the following distributions with support in (0,1] : uniform, truncated normal, truncated lognormal, truncated exponential, beta, truncated gamma, truncated weibull.",
"Then, under random order arrival, asymptotically (i.e., when $n \\rightarrow \\infty $ ) the probability that Algorithm returns an $\\varepsilon $ -best arm is greater than or equal to $0.9(1-\\delta )$ , $\\forall \\varepsilon \\le {1}/{10}$ ."
],
[
"Randomly generated input stream",
"We now consider finding the $\\varepsilon $ -best arm when the means of $n$ arms are i.i.d.",
"samples from a distribution with C.D.F.",
"$F(x)$ .",
"Let $F_n(x)$ be the empirical distributions of the means of these $n$ arms.",
"Then, we have that $\\sup _{x\\in \\mathbb {R}}|F_n(x)-F(x)| \\rightarrow 0$ as $n \\rightarrow \\infty $ , due to Glivenko-Cantelli theorem.",
"For practical purposes, $F(x) \\approx F_n(x)$ for $n\\ge 10^5$ .",
"Due to Theorem REF , for certain well-known distributions, under random order arrival, Algorithm returns an $\\varepsilon $ -best arm with probability at least $0.9(1-\\delta )$ , $\\forall \\varepsilon \\le \\frac{1}{10}$ .",
"This implies that for any set of such $n$ arms, for at least $\\frac{9}{10}\\cdot n!$ out of a total $n!$ permutations, Algorithm returns an $\\varepsilon $ -best arm with probability at least $(1-\\delta )$ .",
"Hence, the probability that for a randomly generated input stream, Algorithm will output an $\\varepsilon $ -best arm is at least $0.9(1-\\delta )$ .",
"Here, the probability is calculated over all possible input streams."
],
[
"Experimental evaluation",
"We now give experimental evidence that in practice Algorithm returns an $\\varepsilon $ -arm with high confidence, even when we reduce the number of samplings by a factor of 40000.",
"In the experiments, we run the algorithm on $R$ different instances and for each instance means of $n$ arms were sampled from a distribution $\\mathcal {D}$ with support in $(0,1]$ , mean $=\\mu $ and variance $=\\sigma ^2$ .",
"Also we set $C=117, \\varepsilon ={1}/{10}, \\delta ={1}/{10}$ .",
"After we obtain the mean of an arm, the arm has Bernoulli reward distributions with that mean.",
"In Figure (REF ), $R=1000$ , $n=10^6$ , $\\mathcal {D}$ is the truncated normal distribution, $\\mu =1/2$ , $\\sigma ^2=1$ .",
"In Figure (REF ), $R=100$ , $n=10^5$ , $\\mathcal {D}$ is the truncated normal distribution, $\\mu =1/2$ , $\\sigma ^2=1/2$ .",
"In Figure (REF ), $R=100$ , $n=10^6$ , $\\mathcal {D}$ is the uniform distribution, $\\mu =1/2$ , $\\sigma ^2={1}/{12}$ .",
"In Figure (REF ), $R=1000$ , $n=10^5$ , $\\mathcal {D}$ is the uniform distribution, $\\mu =1/2$ , $\\sigma ^2={1}/{12}$ .",
"See Appendix for the experimental evaluation for more distributions.",
"In all these cases, we almost always return an arm with mean within at most 0.05 ($<1/10=\\varepsilon $ ) from the mean of the best-arm.",
"Figure: X-axis: Gap between means of the best arm and the arm returned by Algorithm , Y-axis: Count of such arms."
],
[
"Conclusion",
"We study the MAB problem with bounded arm-memory where the arms arrive in a stream.",
"We study two standard objectives: regret minimization and best-arm identification.",
"For regret minimization, [16] show that when $m=1$ , a single-pass algorithm can have linear regret unless the reward distributions of arms satisfy some additional conditions.",
"[14] conjecture an instance-dependent lower bound on the expected regret.",
"(Chaudhuri et al.",
"[9]) leave it as an open problem to prove a lower bound on the expected regret of any bounded arm-memory algorithm.",
"Our first result shows a lower bound of $\\Omega \\big ({T^{2/3}}/{m^{7/3}}\\big )$ on the expected cumulative regret for single-pass MAB algorithms with bounded arm-memory.",
"The lower bound holds for any $m < n$ .",
"This shows a nice dichotomy for $T >> n$ , as one can obtain expected cumulative regret of $\\tilde{O}\\big ({\\sqrt{nT}})$ by standard UCB1 algorithm, where we are allowed to store $n$ arms.",
"Note that the question of proving a lower bound on the regret of multi-pass algorithms remains open.",
"The best-arm identification problem in a streaming model has been studied by (Assadi et al.",
"[5]).",
"We propose an $r$ -round adaptive $(\\varepsilon ,\\delta )$ -PAC streaming algorithm that uses $O(r)$ arm-memory ($r \\in [\\log ^* n]$ ) and has tight sample complexity.",
"We also propose a best-arm identification algorithm that stores exactly one extra arm in the memory and outputs an $\\varepsilon $ -best arm with high confidence for most standard distributions.",
"We note that Algorithm and the techniques used in its analysis for random order arrival may help in resolving the problem with adversarial order arrival of arms.",
"Another interesting question is to find the top-k arms in the stream.",
"For this problem, under the assumption that $\\Delta $ is known, (Assadi et al.",
"[5]) propose an algorithm that has optimal sample complexity and stores exactly $k$ arms in the memory at any time-step.",
"Note that our algorithm in Section can be directly extended to find the top-k arms having optimal $r$ -round sample complexity and $O(kr)$ space complexity, even when $\\Delta $ is not known beforehand.",
"It would be interesting to see if these ideas can be extended to an $(\\varepsilon ,\\delta )$ -PAC algorithm that finds the top-k arms using the optimal number of samples, while improving the space complexity to say, $O(k + r)$ .",
"Regret Minimization under Random Order Arrival Theorem 7 In the MAB setting, fix the number of arms $n$ and the time horizon $T$ .",
"For any online MAB algorithm, if we are allowed to store at most $m < n$ arms, then in the setting of random order arrival there exists an input instance such that $\\mathbb {E}[R(T)] \\ge \\Omega (\\frac{T^{2/3}}{m^{7/3}})$ We consider 0-1 rewards and the 2 input instances $\\mathcal {I}_1,\\mathcal {I}_2$ each containing $n$ arms, with parameter $\\epsilon >0$ (where $\\epsilon =\\frac{1}{m^{1/3}T^{1/3}}$ ): $\\mathcal {I}_1 = \\left\\lbrace \\begin{array}{rcl}\\mu _i & =(1+\\epsilon )/2& \\mbox{for~~} i=1 \\\\\\mu _i & = 1/2,~~~~~~~ & \\mbox{for~~} i \\ne 1\\end{array}\\right.$ $\\mathcal {I}_2= \\left\\lbrace \\begin{array}{rcl}\\mu _i &= 1/2~~~~~~~ &\\mbox{for~~} i\\ne n\\\\\\mu _i &= 1.~~~~~~~~~~ & \\mbox{for~~} i=n\\end{array}\\right.$ In the above instances, $\\mu _i$ denotes the expected reward of the $i^{th}$ arm in the input instance.",
"Let us fix a deterministic algorithm $\\mathcal {A}$ that directly stores the first $m$ arms in the memory.",
"We choose an input instance uniformly at random.",
"Let this input instance be $\\mathcal {I}^{\\prime }$ .",
"Then under random-order arrival setting one of the $n$ permutations of $\\mathcal {I}^{\\prime }$ is chosen uniformly at random and is sent as the input stream to the Algorithm $\\mathcal {A}$ .",
"Note that this equivalent to choosing a permutation $\\mathcal {P}$ from $2n$ total distinct permutations of $\\mathcal {I}_1$ and $\\mathcal {I}_2$ uniformly at random and sending it to the Algorithm $\\mathcal {A}$ .",
"Now assume that $m\\le n-1$ .",
"Let $\\mathcal {I}_1^{\\prime }$ be the collection of distinct permutations of $\\mathcal {I}_1$ such that the arm with expected reward of $(1+\\epsilon )/2$ is in the first $m$ positions of the permutation.",
"Similarly, let $\\mathcal {I}_2^{\\prime }$ be the collection of distinct permutations of $\\mathcal {I}_2$ such that the arm with expected reward of 1 is not in the first $m$ positions of the permutation.",
"Clearly $|\\mathcal {I}_1^{\\prime }|=m$ and $|\\mathcal {I}_2^{\\prime }|=n-m$ .",
"Using arguments similar to Section , we can show that $\\mathbb {E}[R(T)|\\mathcal {P}\\in \\mathcal {I}_1^{\\prime }\\cup \\mathcal {I}_2^{\\prime }]\\ge \\Omega \\Big (\\frac{T^{2/3}}{m^{7/3}}\\Big )$ .",
"Hence, we have the following: $\\mathbb {E}[R(T)] &\\ge \\mathbb {P}[\\mathcal {P}\\in \\mathcal {I}_1^{\\prime }\\cup \\mathcal {I}_2^{\\prime }]\\cdot \\mathbb {E}[R(T)|\\mathcal {P}\\in \\mathcal {I}_1^{\\prime }\\cup \\mathcal {I}_2^{\\prime }]\\\\&\\ge \\frac{n-m+m}{2n}\\cdot \\Omega \\Big (\\frac{T^{2/3}}{m^{7/3}}\\Big )\\\\&\\ge \\Omega \\Big (\\frac{T^{2/3}}{m^{7/3}}\\Big )$ The above result should hold for any randomized algorithm too as randomized algorithm are a distribution over deterministic algorithms.",
"Important Inequalities Lemma 8 (Hoeffding's inequality).",
"Let $Z_1,\\ldots ,Z_n$ be independent bounded variables with $Z_i\\in [0,1]$ for all $i\\in [n]$ .",
"Then $\\mathbb {P}\\Big (\\frac{1}{n}\\sum _{i=1}^{n}(Z_i-\\mathbb {E}[Z_i])\\ge t)\\le e^{-2nt^2}\\Big ), \\text{ and }$ $\\mathbb {P}\\Big (\\frac{1}{n}\\sum _{i=1}^{n}(Z_i-\\mathbb {E}[Z_i])\\le -t)\\le e^{-2nt^2}\\Big ),\\text{ for all } t\\ge 0.$ Lemma 9 Let $\\mathtt {arm}_1$ and $\\mathtt {arm}_2$ be two different arms with means $\\mu _1$ and $\\mu _2$ .",
"Suppose $\\mu _1 - \\mu _2 \\ge \\theta $ and we sample each arm $\\frac{K}{\\theta ^2}$ times to obtain empirical biases $\\widehat{\\mu }_1$ and $\\widehat{\\mu }_2$ .",
"Then, $\\mathbb {P}(\\widehat{\\mu }_1 \\le \\widehat{\\mu }_2)\\le 2\\cdot e^{(-K/2)}$ $\\mathbb {P}(\\widehat{\\mu }_1 > \\widehat{\\mu }_2) & \\ge \\mathbb {P}\\Big (\\mu _1 - \\frac{\\theta }{2}< \\widehat{\\mu }_1 \\text{ and } \\widehat{\\mu }_2 < \\mu _2 + \\frac{\\theta }{2}\\Big )\\\\&= \\mathbb {P}\\Big (\\mu _1 - \\frac{\\theta }{2}< \\widehat{\\mu }_1\\Big ) \\cdot \\mathbb {P}\\Big (\\widehat{\\mu }_2 < \\mu _2 + \\frac{\\theta }{2}\\Big ) \\\\&\\ge \\big (1- e^{-2\\cdot \\frac{K}{\\theta ^2}\\cdot (\\frac{\\theta }{2})^2}\\big )\\cdot \\big (1- e^{-2\\cdot \\frac{K}{\\theta ^2}\\cdot (\\frac{\\theta }{2})^2}\\big ) \\\\& \\ge (1-2\\cdot e^{-K/2}).$ Hence, $\\mathbb {P}(\\widehat{\\mu }_1 \\le \\widehat{\\mu }_2)\\le 2\\cdot e^{-K/2}$ .",
"Theorem 10 (Berry-Esseen Theorem) There exists a positive constant $C\\le 1$ such that if $X_1, X_2, \\ldots , X_n$ are i.i.d.",
"random variables with $\\mathbb {E}[X_i] = 0$ , $\\mathbb {E}[X_i^2] = \\sigma ^2 > 0$ , and $\\mathbb {E}[|X_i|^3] = \\rho < \\infty $ , and if we define $Y_n = \\frac{X_1 + X_2 + \\ldots + X_n}{n}$ to be the sample mean, with $F_n$ being the cumulative distribution function of $\\frac{Y_n\\sqrt{n}}{\\sigma }$ , and $\\Phi $ be the cumulative distribution function of the standard normal distribution $\\mathcal {N}(0,1)$ , then for all $x$ and $n$ , $|F_n(x) - \\Phi (x)| \\le \\frac{C\\rho }{\\sigma ^3\\sqrt{n}}.$ Omitted Proofs from Section * Let us consider some level $\\ell $ and a set of arms $\\mathtt {arm}_{\\ell _1},\\mathtt {arm}_{\\ell _2},\\ldots ,\\mathtt {arm}_{\\ell c_\\ell }$ , which increases the counter $C_\\ell $ from 0 to $c_\\ell $ .",
"Denote the best arm among them as $\\mathtt {arm}^{\\prime }$ and let $\\mathtt {arm}^*_\\ell $ be the empirically best arm seen so far since the arrival of $\\mathtt {arm}_{\\ell _1}$ .",
"Our algorithm works as follows: once an arm arrives at level $\\ell $ , it is pulled $s_\\ell $ number of times.",
"Next, we compare its empirical mean with that of $\\mathtt {arm}^*_\\ell $ , which was computed when the arm corresponding to $\\mathtt {arm}^*_\\ell $ arrived at level $\\ell $ .",
"The arm with the greater empirical mean is maintained as $\\mathtt {arm}^*_\\ell $ and its empirical mean is stored for future comparisons.",
"Once $C_\\ell = c_\\ell $ , $\\mathtt {arm}^*_\\ell $ is sent to level $\\ell + 1$ .",
"Note that if $0<C_\\ell < c_\\ell $ , and the condition in Step is not satisfied then we will sample $\\mathtt {arm}^*_\\ell $ in the Step .",
"Note that this is equivalent to simultaneously pulling each of the $c_\\ell $ arms $s_\\ell $ number of times, and then sending the empirically best arm to level $\\ell +1$ .",
"Using Lemma REF , we can show that at any level $\\ell $ , if two arms have reward gap $\\mu _i - \\mu _j \\ge \\varepsilon _\\ell $ then $\\mathbb {P}[\\widehat{\\mu }_i < \\widehat{\\mu }_j] \\le \\frac{\\delta }{2^{\\ell +1}\\cdot c_\\ell }$ .",
"Since at most $c_\\ell $ arms arrive at level $\\ell $ before the counter $C_\\ell $ reaches $c_\\ell $ or the condition in Step is not satisfied, taking union bound we have that the probability that an arm with reward gap $\\ge \\varepsilon _\\ell $ with respect to $\\mathtt {arm}^{\\prime }_\\ell $ is either sent to level $\\ell +1$ or is sampled in the Step is at most $\\frac{\\delta }{2^{\\ell +1}\\cdot c_\\ell }\\cdot c_\\ell =\\frac{\\delta }{2^{\\ell +1}}$ .",
"* Using Lemma REF , we can show that among the arms that are being considered here, if two arms have reward gap $\\mu _i - \\mu _j \\ge \\varepsilon _r$ then $\\mathbb {P}[\\widehat{\\mu }_i < \\widehat{\\mu }_j] \\le \\frac{\\delta }{2^{r+1}\\cdot c_r}$ .",
"As $c_r=n$ , by taking union bound we get that with probability at least $1 - \\frac{\\delta }{2^{r+1}}$ , an arm with reward gap at most $\\varepsilon _r$ from $\\mathtt {arm}^{\\prime }_r$ is returned by the Algorithm.",
"Adversarial Example for [5] In this section, we show an adversarial example to show that the algorithm of [5] is not $(\\varepsilon , \\delta )$ -PAC.",
"First, we state the algorithm of [5] (Algorithm ) here for completeness.",
"The algorithm takes as input $n \\in \\mathbb {N}$ arms arriving in a stream in an arbitrary order, an approximation parameter $\\varepsilon \\in [0,1/2)$ , and the confidence parameter $\\delta \\in (0,1)$ .",
"We first define some notation that is used throughout this work, which is consistent with the notation used in [5].",
"$&\\lbrace r_\\ell \\rbrace _{\\ell =1}^\\infty :\\hspace{8.5359pt} r_1 = 4;~~ r_\\ell = 2^{r_\\ell }; \\\\&\\varepsilon _\\ell = \\varepsilon /({10\\cdot 2^{\\ell - 1}}); &\\\\&\\beta _\\ell = {1}/{\\varepsilon _\\ell ^2}; \\\\&\\lbrace s_\\ell \\rbrace _{\\ell =1}^\\infty :\\hspace{8.5359pt} s_\\ell = 4\\beta _\\ell \\big (\\ln (1/\\delta ) + 3r_\\ell \\big ); \\\\&\\lbrace c_\\ell \\rbrace _{\\ell =1}^{\\lceil \\log ^* n\\rceil + 1}:\\hspace{8.5359pt} c_1 = 2^{r_1};~~ c_\\ell = {2^{r_\\ell }}/{2^{\\ell - 1}};\\\\& \\text{(the number of arms processed in level \\qquad \\mathrm {(intermediate parameters used to define s_\\ell below,intermediate estimate of gap parameter,number of samples per arm in level \\ell )}$\\ell $ before}\\\\ &\\text{sending $\\mathtt {arm}^*_\\ell $ to level $\\ell +1$)}$ For Algorithm to be $(\\varepsilon ,\\delta )$ -PAC, it has to find an $\\varepsilon $ -best arm with probability at least $1-\\delta $ .",
"We next provide a formal argument for why Algorithm is not an $(\\varepsilon ,\\delta )$ -PAC algorithm by providing a counterexample.",
"[ht!]",
"[1] $\\lbrace r_\\ell \\rbrace ^{\\infty }_{\\ell =1}:r_1:=4,r_{\\ell +1}=2^{r_\\ell }$ ; $\\varepsilon _\\ell =\\frac{\\varepsilon }{10\\cdot 2^{\\ell -1}}$ ; $\\beta _\\ell =\\frac{1}{\\varepsilon ^2_\\ell }$ ; $s_\\ell =4\\beta _\\ell \\big (\\ln (\\frac{1}{\\delta })+3r_\\ell \\big )$ ; $c_1=2^{r_1},$ $c_\\ell =\\frac{2^{r_\\ell }}{2^{\\ell -1}}(\\ell \\ge 2)$ ; Counters: $C_1,C_2,\\ldots ,C_t$ initialized to 0 where $t=\\lceil \\log ^*(n)\\rceil +1$ .",
"Stored arms: $\\mathtt {arm}^*_1,\\mathtt {arm}^*_2,\\ldots ,\\mathtt {arm}^*_t$ the most biased arm of $\\ell $ -th level.",
"Stored empirical means:$p_1,p_2,\\ldots ,p_t$ the highest empirical mean of $\\ell $ -th level.",
"A new arm $\\mathtt {arm}_i$ arrives in the stream Read $\\mathtt {arm}_i$ to memory Aggressive Selective Promotion: Starting from level $\\ell =1$ : Sample both $\\mathtt {arm}_i$ and $\\mathtt {arm}^*_\\ell $ for $s_\\ell $ times.",
"Drop $\\mathtt {arm}_i$ if $\\hat{p}_{\\mathtt {arm}_i} < \\hat{p}_{\\mathtt {arm}^*_\\ell }$ , otherwise replace $\\mathtt {arm}^*_\\ell $ with $\\mathtt {arm}_i$ .",
"Increase $C_\\ell $ by 1.",
"If $C_\\ell =c_\\ell $ , make $C_\\ell $ equal to 0, send $\\mathtt {arm}^*_\\ell $ to the next level by calling Line with $(\\ell =\\ell +1)$ .",
"Return $\\mathtt {arm}^*_t$ as the selected most bias arm.",
"Let the arms arrive in the stream be $\\mathtt {arm}_1,\\ldots ,\\mathtt {arm}_n$ , where $n>(c_1\\cdot c_2)$ .",
"Let $\\mathtt {arm}_i$ be the $i^{th}$ arm to arrive in the stream and has a mean $p_i$ , where $i\\in [n]$ .",
"For all $i>c_1\\cdot c_2$ , let $p_i=0$ .",
"For all $i\\le c_1\\cdot c_2$ , let $p_i=\\frac{1}{2}-(\\lceil \\frac{i}{c_1}-1\\rceil )\\cdot \\frac{\\varepsilon }{c_2-2}$ .",
"Let $\\mathtt {arm}_{k_1},\\mathtt {arm}_{k_2},\\ldots ,\\mathtt {arm}_{k_{c_2}}$ be the first $c_2$ arms which arrive at level 2 (note that all the arms which arrive at level 2 after $\\mathtt {arm}_{k_{c_2}}$ will have a mean of 0).",
"Let $\\mathtt {arm}^*_2$ be the most biased arm (based on the sampling) at the end of Aggressive Selection Promotion step for level $\\ell =2$ for $\\mathtt {arm}_{k_{c_2}}$ .",
"Now $C_2 = c_2$ after the arrival of $\\mathtt {arm}_{k_{c_2}}$ .",
"Thus $\\mathtt {arm}^*_2$ will be sent to level 3.",
"As all the following remaining arms have lesser means, at the end, the algorithm finally returns an arm with mean less than or equal to the mean of $\\mathtt {arm}^*_2$ .",
"Note that $\\forall i\\in [c_2]$ , all the arms in the set $\\lbrace \\mathtt {arm}_{(i-1)\\cdot c_1+1},\\ldots ,\\mathtt {arm}_{i\\cdot c_1}\\rbrace $ have the same mean and one among them is sent as $\\mathtt {arm}_{k_{i}}$ to level 2.",
"Therefore $p_{k_i} = p_{k_1}-(i-1)\\frac{\\varepsilon }{b}$ , where $b = 2^{15}-2$ and $p_{k_1}=\\frac{1}{2}$ .",
"So for any $i \\in [c_{2}-1]$ , $p_{k_i} - p_{k_{i+1}} = \\frac{\\varepsilon }{b}$ .",
"We will show that with probability $>\\delta $ , we send $\\mathtt {arm}_{k_{c_2}}$ to level 3, and $\\frac{1}{2}-p_{k_{c_2}}>\\varepsilon $ .",
"For $i\\in [c_2]$ , let $Y_i^t$ denote the reward when we sample the arm $\\mathtt {arm}_{k_i}$ for the $t$ -th time.",
"We assume that $Y_i^t \\sim \\text{Bern}(p_{k_i})$ and $\\mathrm {Var}[Y_i^t]=p_{k_i}(1-p_{k_i})$ (Note that this is a reasonable assumption as the Algorithm should work for any distribution).",
"For $i \\in [c_2 - 1]$ , let $Z_i^t = Y_i^t - Y_{i+1}^t$ .",
"Clearly, $\\mu _i:=\\mathbb {E}[Z_i^t] = \\frac{\\varepsilon }{b}$ .",
"Let $\\sigma _i^2:=\\mathrm {Var}[Z_{i}^t]$ .",
"Let us assume that $\\varepsilon <\\frac{1}{5}$ (Later we will choose $\\varepsilon $ in such a way so that this condition is satisfied).",
"In this case, $\\sigma _i^2=\\mathrm {Var}[Y_i^t]+\\mathrm {Var}[Y_{i+1}^t]>2(p_{k_{c_2}})(1-p_{k_{c_2}})>\\frac{2}{5}$ .",
"Let $Z_i = Z_i^1 + Z_i^2 + \\ldots + Z_i^{s_2}$ .",
"Note that, if every arm from the set $\\mathtt {arm}_{k_1},\\mathtt {arm}_{k_2},\\ldots ,\\mathtt {arm}_{k_{c_2}}$ when it arrives in the level 2 beats $\\mathtt {arm}^*_2$ in the challenge, then $\\mathtt {arm}_{k_{c_2}}$ will be sent to level 3.",
"Thus, $\\lbrace Z_i < 0, \\forall i\\in [c_2-1]\\rbrace \\subseteq \\lbrace \\mathtt {arm}_{k_{c_2}} ~\\text{is sent to level}~ 3\\rbrace $ .",
"Assuming that $\\delta $ and $\\varepsilon $ are very small (which we will choose appropriately to bound the error), we approximate (using the central limit theorem) the distribution of $Z_i$ using the normal distribution $\\mathcal {N}(s_2\\mu _i,s_2\\sigma _i^2)$ .",
"$\\mathbb {P}[Z_i<0] &= \\mathbb {P}[Z_i > 2s_2\\mu _i]\\\\& = 1-\\frac{1}{2}\\bigg (1+\\text{erf}\\Big (\\frac{s_2\\mu _i}{\\sqrt{2s_2 \\sigma _i^2}}\\Big )\\bigg )\\\\&= \\frac{\\text{erfc}\\Big (\\frac{s_2\\mu _i}{\\sqrt{2s_2 \\sigma _i^2}}\\Big )}{2}\\\\& \\ge \\frac{\\text{erfc}\\Big (s_2\\mu _i \\cdot \\sqrt{\\frac{5}{4s_2}}\\Big )}{2} \\\\& = \\frac{\\text{erfc}\\Big (\\frac{20\\sqrt{5}}{b}\\sqrt{\\ln \\big (\\frac{e^{3r_2}}{\\delta }\\big )}\\Big )}{2} \\\\&\\ge \\frac{\\sqrt{\\gamma -1}}{2}e^{-\\frac{2000\\cdot \\gamma }{b^2}\\ln (\\frac{e^{3r_2}}{\\delta })}\\\\&= \\frac{\\sqrt{\\gamma -1}}{2}\\Big (\\frac{\\delta }{e^{3r_2}}\\Big )^{\\frac{2000\\cdot \\gamma }{b^2}}.\\\\$ Thus, we can lower bound the probability that $\\mathtt {arm}_{k_{c_2}}$ is sent to level 3 as follows: $\\mathbb {P}[\\mathtt {arm}_{k_{c_2}} ~\\text{is sent to level}~ 3]&\\ge \\mathbb {P}[Z_i < 0, \\forall i\\in [c_2-1]]\\\\& = \\prod _{i\\in [c_2 -1]}\\mathbb {P}[Z_i < 0] \\\\& \\ge \\Big ( \\frac{\\sqrt{\\gamma -1}}{2}\\Big (\\frac{\\delta }{e^{3r_2}}\\Big )^{\\frac{2000\\cdot \\gamma }{b^2}}\\Big )^{c_2-1}\\\\& = \\frac{\\delta ^{\\frac{2000\\cdot \\gamma \\cdot (c_2-1)}{b^2}}}{K}, \\text{ where } K= \\Big (\\frac{2\\cdot e^{\\frac{3r_2\\cdot 2000\\cdot \\gamma }{b^2}}}{\\sqrt{\\gamma -1}}\\Big )^{c_2-1}.$ Consider that function $f(x)=\\frac{e^{x\\left(1-\\frac{2000\\cdot \\gamma \\cdot (c_2-1)}{b^2}\\right)}}{K}$ .",
"Since $f(x)$ is an increasing and convex function, there is a constant $c$ such that $f(c)>2$ .",
"This implies that for $\\delta =e^{-c}$ we have the following: $\\mathbb {P}[\\mathtt {arm}_{k_{c_2}} ~\\text{is sent to level}~ 3] & \\ge \\frac{\\delta ^{\\frac{2000\\cdot \\gamma \\cdot (c_2-1)}{b^2}}}{K}\\\\& = f(c)e^{-c}\\\\& >2\\delta .$ Now, we bound the error in calculation of the above probability.",
"Using the Berry-Esseen theorem, the error $\\epsilon _i$ of calculating $\\mathbb {P}[Z_i<0]$ is upper bounded by $\\frac{C \\rho }{\\sigma _i^3\\sqrt{s_2}}\\le \\frac{ \\varepsilon }{\\sqrt{\\ln (\\frac{1}{\\delta })}}$ , where $C\\le 1$ , $\\rho =\\mathbb {E}[|Z_i^t-\\mu _i|^3]\\le 8 \\text{ (as } |Z_i^t-\\mu _i|\\le 2)$ and $\\sigma _i^2=\\mathrm {Var}[Z_i^t-\\mu _i]=\\mathrm {Var}[Z_i^t]$ .",
"Also we assumed $\\mathrm {Var}[Z_i^t]>\\frac{2}{5}$ (we will choose $\\varepsilon $ in such a way that this is satisfied).",
"If we choose $\\varepsilon $ such that $\\varepsilon < \\frac{\\delta \\sqrt{\\ln (\\frac{1}{\\delta })}}{c_2}$ and $\\varepsilon <1/5$ , then $\\epsilon _i<\\frac{\\delta }{c_2}$ .",
"Hence, we can conclude that $\\mathbb {P}[\\mathtt {arm}_{k_{c_2}} ~\\text{is sent to level}~ 3]>2\\delta -\\sum _{i=1}^{c_2-1}\\epsilon _i > 2\\delta -\\delta =\\delta $ .",
"As $\\frac{1}{2}-p_{k_{c_2}}=\\Big (\\frac{c_2-1}{c_2-2}\\Big )\\cdot \\varepsilon $ , we can conclude that with probability $>\\delta $ , the Algorithm returns an arm with reward gap $> \\varepsilon $ .",
"Lower bound for r-round adaptive streaming algorithm In this section, we use the following lower bound for $r$ -round adaptive offline algorithm model defined in [2] to provide a lower bound on the sample complexity for any $r$ -round adaptive streaming algorithm.",
"Lemma 11 ([2]).",
"For any parameter $\\Delta \\in (0,1/2)$ and any integer $n,k\\ge 1$ , there exists a distribution $\\mathcal {D}$ on input instances of the $k$ most biased coins problem with $n$ coins and gap parameter $\\Delta _k=\\Delta $ such that for any integer $r\\le 1$ , any $r$ -round algorithm that finds the $k$ most biased coins in the instances sampled from $\\mathcal {D}$ with probability at least $3/4$ has a sample complexity $\\Omega (\\frac{n}{\\varepsilon ^2\\cdot r^4}\\cdot \\mathtt {ilog}^{(r)}(n/k))$ Any $r$ -round adaptive offline algorithm is same as $r$ -round adaptive streaming algorithm except for the following two points: In an $r$ -round adaptive offline algorithm all the arms can be simultaneously stored in the memory, whereas arm-memory is usually bounded in an $r$ -round adaptive streaming algorithm.",
"In an $r$ -round adaptive offline algorithm, in any round $j$ , all the arms are sampled simultaneously.",
"On the other hand, in an $r$ -round adaptive streaming algorithm all the arms in the round $j$ need not be sampled simultaneously, and they can also be sampled one after the other.",
"Any $r$ -round adaptive streaming algorithm can be replicated by an $r$ -round adaptive offline algorithm such that the worst-case sample complexity is the same in both the algorithms.",
"Next, we present the following lemma.",
"Lemma 12 For any approximation parameter $\\varepsilon \\in (0,1/2)$ and any integer $n\\ge 1$ , there exists a distribution $\\mathcal {D}$ on input instances of the best-arm identification problem with $n$ arms such that for any integer $r\\ge 1$ , any $r$ -round adaptive streaming algorithm that finds the $\\varepsilon $ -best arm in the instances sampled from $\\mathcal {D}$ with probability at least $3/4$ has a sample complexity $\\Omega (\\frac{n}{\\varepsilon ^2\\cdot r^4}\\cdot \\mathtt {ilog}^{(r)}(n))$ We present the proof idea.",
"Let $\\mathcal {A}$ be an $r$ -round adaptive streaming algorithm with the lowest worst-case sample complexity which finds the $\\varepsilon $ -best arm with probability at least $3/4$ .",
"Now we replicate the algorithm $\\mathcal {A}$ using an $r$ -round offline algorithm $\\mathcal {B}$ for best-arm identification such that its worst-case sample complexity is equal to that of $\\mathcal {A}$ .",
"Due to Lemma REF , the worst-case sample complexity of $\\mathcal {B}$ is $\\Omega (\\frac{n}{\\varepsilon ^2\\cdot r^4}\\cdot \\mathtt {ilog}^{(r)}(n))$ .",
"Hence, the worst-case sample complexity of $\\mathcal {A}$ is $\\Omega (\\frac{n}{\\varepsilon ^2\\cdot r^4}\\cdot \\mathtt {ilog}^{(r)}(n))$ .",
"Adversarial Example for Constant Arm-Memory Algorithm based on [5] In this section, we show an adversarial example for Constant Arm-memory Algorithm based on [5] when the parameter $\\Delta $ is unknown where $\\Delta $ is the gap between the best arm and the second best arm.",
"For completeness, we present this algorithm below (Algorithm ).",
"[ht!]",
"[1] $\\lbrace r_\\ell \\rbrace ^{\\infty }_{\\ell =1}:r_\\ell =3^\\ell $ ; $\\lbrace s_\\ell \\rbrace ^{\\infty }_{\\ell =1}:s_\\ell =\\frac{2}{\\varepsilon ^2}\\cdot \\ln \\left(\\frac{1}{\\delta }\\right)\\cdot r_\\ell $ ; $b:=\\frac{2}{\\epsilon ^2}\\cdot C \\cdot \\ln \\left(\\frac{1}{\\delta }\\right)+s_1$ ; Let king be the first available arm and set its budget $\\phi :=\\phi (king)=0$ .",
"A new arm $\\mathtt {arm}_i$ arrives in the stream Increase the budget $\\phi (\\textbf {king})$ by $b$ .",
"Challenge subroutine: For level $\\ell =1$ to $+\\infty $ : If $\\phi (king)<s_\\ell $ : we declare $\\textbf {king}$ defeated , make $\\mathtt {arm}_i$ the king, initialize its budget to 0 and go to Step .",
"Otherwise, we decrease $\\phi (\\textbf {king})$ by $s_\\ell $ and sample both king and $\\mathtt {arm}_i$ for $s_\\ell $ times.",
"Let $\\widehat{\\mu }_{king}$ and $\\widehat{\\mu }_i$ denote the empirical biases of king and $\\mathtt {arm}_i$ in this trial.",
"If $\\widehat{\\mu }_{king} > \\widehat{\\mu }_i$ , we declare $king$ winner and go to the next arm in the stream; otherwise we go to the next level of challenge (increment $\\ell $ by one).",
"Return $\\textbf {king}$ as the selected most bias arm.",
"Let $\\mathtt {arm}_{1},\\mathtt {arm}_{2},\\ldots ,\\mathtt {arm}_{n}$ be the stream of arms which arrive.",
"For each $i \\in [n]$ , $\\mu _{i}$ is the expected reward of arm $\\mathtt {arm}_{i}$ .",
"Define the input stream such that $\\mu _{i} = \\mu _{1}-(i-1)\\frac{\\varepsilon }{n-2}$ .",
"Note that for any $i \\in [n-1]$ , $\\mu _{i} - \\mu _{i+1} = \\frac{\\varepsilon }{n-2}$ .",
"Let $\\mu _1=\\frac{1}{2}$ .",
"We now show that there exist $\\delta ,\\varepsilon >0$ such that with probability $>\\delta $ , an arm with reward gap $> \\varepsilon $ is returned by the Algorithm.",
"Let $k$ be the maximum value of $\\ell $ such that $\\sum _{i=2}^{\\ell }3^i\\le C$ .",
"Let $\\forall i\\in [n]$ , $Y_{i,t}^\\ell $ denote the reward when we sample the arm $\\mathtt {arm}_i$ for the $t^{th}$ time at level $\\ell $ .",
"Then, $\\mathrm {Var}[Y_{i,t}^\\ell ]= \\mu _i(1 - \\mu _i)$ .",
"Let $Z_{i,t}^\\ell =Y_{i,t}^\\ell -Y_{i+1,t}^\\ell $ .",
"Clearly $p_i:=\\mathbb {E}[Z_{i,t}^\\ell ]=\\frac{\\varepsilon }{n-2}$ .",
"Let $\\sigma _i^2:=\\mathrm {Var}[Z_{i,t}^\\ell ]$ .",
"Let us assume that $\\varepsilon <\\frac{1}{5}$ and $n>>100$ (We will later choose $\\varepsilon $ and $n$ in a way so that this condition is satisfied).",
"In this case, $\\sigma _i^2=\\mathrm {Var}[Y_{i,t}^\\ell ]+\\mathrm {Var}[Y_{i+1,t}^\\ell ]>2(\\mu _n)(1-\\mu _n)>\\frac{2}{5}$ .",
"Let $Z_i^\\ell =Z_{i,1}^\\ell +Z_{i,2}^\\ell +\\ldots +Z_{i,{s_\\ell }}^\\ell $ .",
"Assuming that $\\delta ,\\varepsilon $ are very small (which we will choose appropriately to bound the error) we approximate (using Central Limit Theorem) the distribution of $Z_i^\\ell $ using the normal distribution $\\mathcal {N}(s_\\ell p_i,s_\\ell \\sigma _i^2)$ .",
"$\\mathbb {P}[Z_i^\\ell <0]&=\\mathbb {P}[Z_i^\\ell > 2s_\\ell p_i]\\\\&=1-\\frac{1}{2}\\bigg [1+\\text{erf}\\bigg (\\frac{s_\\ell p_i}{\\sqrt{2s_\\ell \\sigma _i^2}}\\bigg )\\bigg ]\\\\&=\\frac{\\text{erfc}\\Big (\\frac{s_\\ell p_i}{\\sqrt{2s_\\ell \\sigma _i^2}}\\Big )}{2}\\\\& \\ge \\frac{\\text{erfc}(\\frac{s_\\ell p_i\\sqrt{5}}{2\\sqrt{s_\\ell }})}{2}\\\\& \\ge \\frac{\\text{erfc}\\Big (\\frac{\\sqrt{5}\\cdot 3^{\\ell /2}}{n-2}\\sqrt{\\ln (\\frac{1}{\\delta })}\\Big )}{2}\\\\&\\ge \\frac{\\sqrt{\\gamma -1}}{2}e^{-\\frac{5\\cdot 3^{\\ell }\\cdot \\gamma }{(n-2)^2}\\ln (\\frac{1}{\\delta })} \\\\&= \\frac{\\sqrt{\\gamma -1}}{2}\\delta ^{\\frac{5\\cdot 3^{\\ell }\\cdot \\gamma }{(n-2)^2}}.$ Hence, we now have $\\mathbb {P}[\\mathtt {arm}_{i+1}\\text{ becomes king by defeating }\\mathtt {arm}_{i}] &\\ge \\mathbb {P}[\\forall \\ell \\in [k], Z_i^\\ell <0]\\\\& \\ge \\left(\\frac{\\sqrt{\\gamma -1}}{2}\\delta ^{\\frac{5\\cdot 3^{k}\\cdot \\gamma }{(n-2)^2}}\\right)^{k}\\\\& = \\frac{\\delta ^{\\frac{5\\cdot 3^{k}\\cdot k \\cdot \\gamma }{(n-2)^2}}}{K} $ Now we choose $n$ such that $\\frac{(n-2)^2}{(n-1)}>>5\\cdot 3^{k}\\cdot k \\cdot \\gamma $ .",
"Consider the function $f(x)=\\frac{e^{x\\left(1-\\frac{5\\cdot 3^{k}\\cdot k \\cdot \\gamma \\cdot (n-1)}{(n-2)^2}\\right)}}{K^{n-1}}$ .",
"Since $f(x)$ is an increasing and convex function, there is a constant $c$ such that $f(c)>2$ .",
"This implies that for $\\delta =e^{-c}$ we have the following: $\\mathbb {P}[\\mathtt {arm}_{n}\\text{ is returned as king by the algorithm}]& \\ge \\mathbb {P}[\\forall i\\in [n-1], \\mathtt {arm}_{i+1}\\text{ becomes king by defeating }\\mathtt {arm}_{i}] \\\\& \\ge \\left(\\frac{\\delta ^{\\frac{5\\cdot 3^{k}\\cdot k \\cdot \\gamma }{(n-2)^2}}}{K}\\right)^{n-1}\\\\& = \\frac{e^{c\\left(1-\\frac{5\\cdot 3^{k}\\cdot k \\cdot \\gamma \\cdot (n-1)}{(n-2)^2}\\right)}}{K^{n-1}}\\cdot \\delta \\\\& = f(c)\\cdot \\delta \\\\& >2\\delta .$ Now we bound the error in calculation of the above probability.",
"By Berry-Esseen theorem, the error $\\epsilon _i^\\ell $ of calculating $\\mathbb {P}[Z_i^\\ell <0]$ is upper bounded by $\\frac{C \\rho }{\\sigma _i^3\\sqrt{s_1}}\\le \\frac{16 \\varepsilon }{\\sqrt{\\ln (\\frac{1}{\\delta })}}$ .",
"Here $C\\le 1$ , $\\rho =\\mathbb {E}[|Z_{i,t}^\\ell -p_i|^3]\\le 8( \\text{as } |Z_{i,t}^\\ell -p_i|\\le 2)$ and $\\sigma _i^2=\\mathrm {Var}[Z_{i,t}^\\ell -p_i]=\\mathrm {Var}[Z_{i,t}^\\ell ]$ .",
"Also we assumed $\\mathrm {Var}[Z_{i,t}^\\ell ]>\\frac{2}{5}$ (we will choose $\\varepsilon $ in such a way that this is satisfied).",
"If we choose $\\varepsilon $ such that it is less than $\\frac{\\delta \\sqrt{\\ln (\\frac{1}{\\delta })}}{16\\cdot k\\cdot n}$ and it is also less than $\\frac{1}{5}$ , then $\\epsilon _i^\\ell <\\frac{\\delta }{k\\cdot n}$ .",
"Therefore $\\mathbb {P}[\\mathtt {arm}_{n}\\text{ is returned as king by the algorithm}]>2\\delta -\\sum _{i=1}^{n-1}\\sum _{\\ell =1}^{k}\\epsilon _i^\\ell >2\\delta -\\delta =\\delta $ .",
"As $\\mu _{1} - \\mu _n>\\varepsilon $ , we can conclude that there exists $\\delta ,\\varepsilon >0$ such that with probability $>\\delta $ , a king with reward gap $> \\varepsilon $ is returned by the algorithm.",
"We also have experimental evidence for this result.",
"We ran the experiment on problem instances of this adversial type with number of arms $= 5500001$ , mean of the best arm $=1/2$ , $\\varepsilon =1/10$ .",
"We ran ten independent experiments with different realizations due to different values from sampling.",
"Each time the algorithm didn't return $\\varepsilon $ -best arm.",
"Algorithm with constant arm-memory We now tweak the Algorithm in Section and show that our new proposed Algorithm deals with the following: works well both theoretically and experimentally on the counter example for the algorithm mentioned in Section .",
"works well theoretically on random order arrival for well known distributions works well experimentally on any randomly generated input from some well known distributions.",
"Lemma 13 Let $\\mathtt {arm}_1$ and $\\mathtt {arm}_2$ be two different arms with biases $\\mu _1$ and $\\mu _2$ .",
"Suppose $\\mu _1 - \\mu _2 \\ge 0.5\\varepsilon $ and we sample each arm $s_\\ell $ times to obtain empirical biases $\\widehat{\\mu }_1$ and $\\widehat{\\mu }_2$ .",
"Then, $\\mathbb {P}(\\widehat{\\mu }_1 \\le \\widehat{\\mu }_2 + 0.495\\varepsilon )\\le 2\\cdot e^{(-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell )}.$ $\\mathbb {P}(\\widehat{\\mu }_1 > \\widehat{\\mu }_2 + 0.495\\varepsilon )& \\ge \\mathbb {P}\\Big (\\mu _1 - \\frac{\\varepsilon }{400} < \\widehat{\\mu }_1 \\text{ and } \\widehat{\\mu }_2 < \\mu _2 + \\frac{\\varepsilon }{400} \\Big )\\\\&= \\mathbb {P}\\Big (\\mu _1-\\frac{\\varepsilon }{400}< \\widehat{\\mu }_1\\Big ) \\cdot \\mathbb {P}\\Big (\\widehat{\\mu }_2 < \\mu _2 + \\frac{\\varepsilon }{400}\\Big ) \\\\&\\ge (1- e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell })\\cdot (1- e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }) \\\\& \\ge (1- 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }).$ Hence, we have $\\mathbb {P}(\\widehat{\\mu }_1 \\le \\widehat{\\mu }_2 + 0.495\\varepsilon ) \\le 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }$ .",
"Lemma 14 Let $\\mathtt {arm}_1$ and $\\mathtt {arm}_2$ be two different arms with biases $\\mu _1$ and $\\mu _2$ .",
"Suppose $\\mu _1 - \\mu _2 \\le 0.49\\varepsilon $ and we sample each arm $s_\\ell $ times to obtain empirical biases $\\widehat{\\mu }_1$ and $\\widehat{\\mu }_2$ .",
"Then, $\\mathbb {P}(\\widehat{\\mu }_1\\ge \\widehat{\\mu }_2+0.495\\varepsilon )\\le 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }.$ $\\mathbb {P}(\\widehat{\\mu }_1 < \\widehat{\\mu }_2 + 0.495\\varepsilon )& \\ge \\mathbb {P}\\Big (\\widehat{\\mu }_1 < \\mu _1 + \\frac{\\varepsilon }{400} \\text{ and } \\mu _2 - \\frac{\\varepsilon }{400}< \\widehat{\\mu }_2\\Big )\\\\&= \\mathbb {P}\\Big ( \\widehat{\\mu }_1 < \\mu _1 + \\frac{\\varepsilon }{400}\\Big )\\cdot \\mathbb {P}\\Big (\\mu _2 - \\frac{\\varepsilon }{400}< \\widehat{\\mu }_2\\Big ) \\\\&\\ge (1- e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell })\\cdot (1- e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }) \\\\& \\ge (1- 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }).$ Hence, we have $\\mathbb {P}(\\widehat{\\mu }_1 \\ge \\widehat{\\mu }_2 + 0.495\\varepsilon )\\le 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }$ .",
"Lemma 15 In a challenge subroutine, if $\\mu _i - \\mu _{king} \\ge 0.5\\varepsilon $ , then the probability that $\\mathtt {arm}_i$ does not become the king is at most $\\frac{\\delta }{8}$ .",
"$\\mathbb {P}(\\mathtt {arm}_i\\text{ loses to king })&\\le \\sum _{\\ell =1}^\\infty \\mathbb {P}(\\mathtt {arm}_i\\text{ loses to king }\\text{at level $\\ell |\\mathtt {arm}_i$ has not lost until }\\ell -1)\\\\&\\le \\sum _{\\ell =1}^\\infty 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell } \\\\& < (\\delta /2) \\cdot \\sum _{\\ell =1}^\\infty e^{-3^\\ell }\\\\& < \\delta /8.$ Since the budget is finite, king will lose to $\\mathtt {arm}_i$ with probability at least $(1-\\delta /8)$ in finite time.",
"The next lemma is an adaptation of Lemma 3.3 in (Assadi et al.",
"[5]).",
"Lemma 16 In Algorithm , if any incoming arm does not lose to the king (denoted $\\mathtt {arm}_{king}$ ) at a level $\\ell $ with probability at most $2\\cdot e^{-\\ln (\\frac{1}{\\delta ^{\\prime }})\\cdot 3^\\ell }$ , then the probability that $\\mathtt {arm}_{king}$ loses to any incoming arm is at most $\\delta ^{\\prime }/2$ .",
"Lemma 17 Let $\\mu _{king}$ be the bias of the current king ($\\mathtt {arm}_{king}$ ).",
"If the future arms in the stream don't have a bias in the range $(\\mu _{king} + 0.49\\epsilon , \\mu ^*]$ where $\\mu ^*$ is the bias of the most-biased arm, then the probability that the king is ever defeated is at most $\\delta /8$ .",
"The lemma follow from Lemma REF and Lemma REF by substituting $\\delta ^{\\prime }=\\delta /4$ .",
"Lemma 18 Let the $\\mu ^*$ be the bias of the most-biased arm.",
"If a $\\mathtt {arm}_i$ with bias $\\mu _i \\in [\\mu ^* - 0.49\\varepsilon ,\\mu ^*]$ becomes the king, then the probability that $arm_i$ is ever defeated as a king is at most $\\delta /8$ .",
"This follows directly from Lemma REF .",
"Corollary 19 If the input stream is the counter example for the algorithm mentioned in the Section , then the probability that Algorithm returns a non-$\\varepsilon $ -best arm is at most $\\delta /8$ .",
"Random Order Arrival Let the number of arms $n$ in the input set of arms be very large such that the distribution of means of the input set of arms becomes sort of continuous.",
"Let this distribution have a P.D.F $f(x)$ .",
"Let $\\mu ^*$ be the mean of the best arm.",
"We choose a random permutation of our input and send it as an input stream to Algorithm .",
"Lemma 20 Under random order arrival, probability that Algorithm returns an $\\varepsilon $ -best arm is at least $\\inf _{p^{\\prime }\\in [\\mu ^*-0.99\\varepsilon ,\\mu ^*-0.5\\varepsilon ]}\\Bigg \\lbrace (1-\\delta )\\cdot \\bigg (\\frac{\\int _{\\min \\lbrace p^{\\prime }+0.01\\varepsilon ,\\mu ^*-0.5\\varepsilon \\rbrace }^{\\mu ^*-0.5\\varepsilon } f(x) \\frac{\\int _{x+0.5\\varepsilon }^{\\mu ^*} f(x)dx}{\\int _{x+0.49\\varepsilon }^{\\mu ^*} f(x) dx}dx}{\\int _{p^{\\prime }}^{\\mu ^*} f(x)dx}+\\frac{\\int _{\\mu ^*-0.49\\varepsilon }^{\\mu ^*} f(x)dx}{\\int _{p^{\\prime }}^{\\mu ^*} f(x)dx}\\bigg )\\Bigg \\rbrace .$ Let the number of arms in the input stream having their means in the range $[0,\\mu ^*-0.99\\varepsilon )$ be $k$ .",
"For all $1\\le \\ell \\le k$ , let $S_\\ell $ be the set of all possible input streams containing exactly $\\ell $ out of the $k$ arms above.",
"Let $t_\\ell $ denote the sequence of first $\\ell $ arms to arrive in the stream and let $t_\\ell =\\lbrace c_1,c_2,\\ldots ,c_\\ell \\rbrace \\in S_\\ell $ .",
"Let $king^\\ell $ denote the arm that is the king after the $\\ell $ -th arm has been processed by the algorithm.",
"Let $king^\\ell = \\mathtt {arm}_i$ with probability $q_{i}$ , $\\forall i\\in [\\ell ]$ , where $q_{i}\\ge 0$ for all $i \\in [\\ell ]$ , $\\sum _{i=1}^{\\ell }q_{i} = 1$ .",
"Let $X$ be a random variable such that $X=i$ if and only if the first arm in the stream which has mean in the range $[\\mu ^*-0.99\\varepsilon ,\\mu ^*]$ is the $i$ -th arm in the stream.",
"Let $X=\\ell +1$ .",
"Let us assume that the king is $\\mathtt {arm}_i$ (with mean $\\mu _i$ ) just before the $(\\ell +1)$ -th arm arrives where $i\\in [\\ell ]$ .",
"If the arms arriving in the stream at position $\\ell +1$ and later have means in the range $[0,\\mu _i+0.49\\varepsilon ]$ , then from Lemma REF we know that $\\mathtt {arm}_i$ continues to be the king with probability at least $(1-\\delta /8)$ .",
"Let $p^{\\prime }$ =max$\\lbrace \\mu ^*-0.99\\varepsilon ,\\mu _i+0.49\\varepsilon \\rbrace $ .",
"Clearly $\\mu ^*-0.99\\varepsilon \\le p^{\\prime } < \\mu ^* - 0.5\\varepsilon $ .",
"Consider the first time an arm with mean in the range $[p^{\\prime },\\mu ^*]$ arrives in the stream.",
"Let $T_1$ be the set of arms whose biases lie in the range $[p^{\\prime },\\mu ^*]$ and $T_2$ be the set of arms whose biases lie in the range $[\\mu ^*-0.49\\varepsilon ,\\mu ^*]$ .",
"Let $\\mathcal {A}_1$ be the event that the first arm $a_1$ from $T_1$ which arrives in the stream belongs to $T_2$ .",
"Let $\\mathcal {A}_2$ be the event that the first arm $a_1$ from $T_1$ which arrives in the stream does not belong to $T_2$ .",
"If $king^\\ell = \\mathtt {arm}_i$ and the event $\\mathcal {A}_1$ occurs then $a_1$ is returned as the king by our Algorithm at the end with a probability of at least $(1-\\delta /8)^3$ .",
"This happens because when the arm $a_1$ arrives, the king at that time has mean less than $\\mu ^*-0.99\\varepsilon $ with probability at least $(1-\\delta /8)$ , and due to Lemma REF , $a_1$ becomes the new king with probability at least $(1-\\delta /8)$ .",
"Due to Lemma REF , $a_1$ continues to remain as king with probability at least $(1-\\delta /8)$ .",
"Hence if the event $\\mathcal {A}_1$ occurs then $a_1$ is returned as the king by our Algorithm at the end with a probability of at least $(1-\\delta /8)^3$ .",
"Now assume that instead the event $\\mathcal {A}_2$ has occurred.",
"Let $\\mathcal {B}_{a_2}$ be the event that the first arm $a_2$ from the set $T_1$ to arrive in the stream has a mean $\\mu _{a_2}$ .",
"Let us assume that $\\mathcal {B}_{a_2}$ has occurred and $\\mu _{a_2}$ belongs to the range $[p^{\\prime }+0.01\\varepsilon ,\\mu ^*-0.5\\varepsilon )$ .",
"Due to Lemma REF , $a_2$ becomes the king with probability at least $(1-\\delta /8)$ .",
"If the means of the arms coming to stream after $a_2$ belongs to the range $[0,\\mu _{a_2}+0.49\\varepsilon ]$ , then $a_2$ continues to be the king with probability at least $(1-\\delta /8)$ .",
"If the first arm $a_3$ arriving in the stream with mean in the range $[\\mu _{a_2}+0.5\\varepsilon ,\\mu ^*]$ comes before the first arm $a_4$ arriving in the stream with bias in the range $(\\mu _{a_2}+0.49\\varepsilon ,\\mu _{a_2}+0.5\\varepsilon ]$ , then $a_3$ becomes the king with probability at least $(1-\\delta /8)$ and continues to remain as the king with probability at least $(1-\\delta /8)$ .",
"Let us denote this event of $a_3$ coming before $a_4$ by $\\mathcal {B}_1$ .",
"Note that if $X=1$ , then we can repeat the above analysis by considering $p^{\\prime } = \\mu ^*-0.99\\varepsilon $ .",
"Let $\\mathcal {C}_1$ be the event that $\\varepsilon $ -best arm is returned by the Algorithm and let $\\mathcal {C}_2$ be the event that an arm with mean in the range $[\\mu ^*-0.49\\varepsilon ,\\mu ^*]$ is returned by the algorithm.",
"$&\\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{28.45274pt}\\ge \\mathbb {P}[\\mathcal {C}_2|\\mathcal {A}_1,X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\cdot \\mathbb {P}[\\mathcal {A}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{42.67912pt}+ \\sum _{a_2:p_{a_2}\\in [p^{\\prime }+0.01\\varepsilon ,p-0.5\\varepsilon )}\\Big (\\mathbb {P}[\\mathcal {C}_2|\\mathcal {B}_1,\\mathcal {B}_{a_2},X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{42.67912pt}\\cdot \\mathbb {P}[\\mathcal {B}_1|\\mathcal {B}_{a_2},X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ] \\cdot \\mathbb {P}[\\mathcal {B}_{a_2}|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\Big )\\\\&\\hspace{28.45274pt} \\gtrapprox (1-\\delta /8)^3\\cdot \\frac{\\int _{p-0.49\\varepsilon }^{p} f(x)dx}{\\int _{p^{\\prime }}^{p} f(x)dx}+\\int _{\\min \\lbrace p^{\\prime }+0.01\\varepsilon ,p-0.5\\varepsilon \\rbrace }^{p-0.5\\varepsilon } (1-\\delta /8)^5 \\cdot \\Bigg (\\frac{\\int _{x+0.5\\varepsilon }^{p} f(x)dx}{\\int _{x+0.49\\varepsilon }^{p} f(x) dx}\\cdot \\frac{f(x)dx}{\\int _{p^{\\prime }}^{p} f(x)dx}\\Bigg )\\\\&\\hspace{28.45274pt}= (1-\\delta )\\cdot \\Bigg (\\frac{\\int _{\\min \\lbrace p^{\\prime }+0.01\\varepsilon ,p-0.5\\varepsilon \\rbrace }^{p-0.5\\varepsilon } f(x) \\frac{\\int _{x+0.5\\varepsilon }^{p} f(x)dx}{\\int _{x+0.49\\varepsilon }^{p} f(x) dx}dx}{\\int _{p^{\\prime }}^{p} f(x)dx}+\\frac{\\int _{p-0.49\\varepsilon }^{p} f(x)dx}{\\int _{p^{\\prime }}^{p} f(x)dx}\\Bigg ).\\\\$ Similarly, if $X=1$ then we have the following: $&\\mathbb {P}[\\mathcal {C}_1|X=1]\\gtrapprox (1-\\delta )\\cdot \\Bigg (\\frac{\\int _{p-0.98\\varepsilon }^{p-0.5\\varepsilon } f(x) \\frac{\\int _{x+0.5\\varepsilon }^{p} f(x)dx}{\\int _{x+0.49\\varepsilon }^{p} f(x) dx}dx}{\\int _{p-0.99\\varepsilon }^{p} f(x)dx}+\\frac{\\int _{p-0.49\\varepsilon }^{p} f(x)dx}{\\int _{p-0.99\\varepsilon }^{p} f(x)dx}\\Bigg ).\\\\$ $&\\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{28.45274pt}= \\sum _{i=1}^{\\ell }\\mathbb {P}[king^\\ell =c_i|X=\\ell +1,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\cdot \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{28.45274pt}=\\sum _{i=1}^{\\ell }q_{c_i}\\cdot \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{28.45274pt}\\ge \\min _{i\\in [\\ell ]} \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ].\\\\$ We have, $\\mathbb {P}[\\mathcal {C}_1|X=\\ell +1] &= \\sum _{(c_i)_{i\\in [\\ell ]} \\sim \\mathcal {S}_\\ell }\\mathbb {P}[t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace |X=\\ell +1]\\cdot \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\ge \\min _{(c_i)_{i\\in [\\ell ]}\\sim \\mathcal {S}_\\ell }\\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ].\\\\$ $\\mathbb {P}[\\mathcal {C}_1] &= \\sum _{i\\in [k+1]}\\mathbb {P}[X=i]\\cdot \\mathbb {P}[\\mathcal {C}_1|X=i]\\\\& \\ge \\min _{i\\in [k+1]}\\mathbb {P}[\\mathcal {C}_1|X=i]\\\\&\\ge \\min \\Big \\lbrace \\mathbb {P}[\\mathcal {C}_1|X=1],\\min _{(c_i)_{i\\in [\\ell ]}\\sim \\mathcal {S}_\\ell }\\min _{i\\in [\\ell ]} \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\Big \\rbrace \\\\&\\gtrapprox \\inf _{p^{\\prime }\\in [p-0.99\\varepsilon ,p-0.5\\varepsilon ]}\\Bigg \\lbrace (1-\\delta )\\cdot \\Bigg (\\frac{\\int _{\\min \\lbrace p^{\\prime }+0.01\\varepsilon ,p-0.5\\varepsilon \\rbrace }^{p-0.5\\varepsilon } f(x) \\frac{\\int _{x+0.5\\varepsilon }^{p} f(x)dx}{\\int _{x+0.49\\varepsilon }^{p} f(x) dx}dx}{\\int _{p^{\\prime }}^{p} f(x)dx}+\\frac{\\int _{p-0.49\\varepsilon }^{p} f(x)dx}{\\int _{p^{\\prime }}^{p} f(x)dx}\\Bigg )\\Bigg \\rbrace .$ Performance under various distributions The following distributions are truncated distributions and the support is (0,1].",
"Note that the following calculations are made assuming $\\varepsilon =\\frac{1}{10}$ .",
"Lower bound on the probability that Algorithm returns an $\\varepsilon $ -best arm, for various truncated distributions like Normal, lognormal, exponential, beta, gamma, Weibull, and uniform distribution is at least $0.9(1-\\delta )$ .",
"Note that as $\\varepsilon $ tends to 0, the distributions mentioned earlier behave similar to uniform distribution on the range $[1-\\varepsilon ,1]$ .",
"So the lower bound of $\\mathbb {P}$ [Algorithm returns an $\\varepsilon $ -best arm] tends to $0.927(1-\\delta )$ which is the lower bound on this probability for the uniform distribution and it does not change with $\\varepsilon $ .",
"Table: NO_CAPTION Experiments We now provide detailed experimental evaluation of Algorithm and show that it returns an $\\varepsilon $ -best arm with high confidence even when we reduce the number of samples $(s_\\ell )$ per arm at each level $\\ell $ by a factor of 40000.",
"We ran the algorithm on $R=100$ different instances.",
"For each instance, the means of each of the $n=10^5$ arms were sampled from a distribution $\\mathcal {D}$ with support $(0,1]$ , mean $=\\mu $ , and variance $=\\sigma ^2$ .",
"Note that if $\\mathcal {D}$ is a truncated distribution then $\\mu $ and $\\sigma ^2$ denote the mean and variance of the non-truncated version of $\\mathcal {D}$ , denoted $\\mathcal {D}^\\prime $ .",
"Also we set $C=117, \\varepsilon ={1}/{10}, \\delta ={1}/{10}$ .",
"For $i \\in [n]$ , let $\\mu _i$ be the mean obtained for $\\mathtt {arm}_i$ .",
"The reward distribution of $\\mathtt {arm}_i$ is then Bernoulli$(\\mu _i)$ .",
"We next provide the details of the distributions corresponding to the figures in Figure REF .",
"Note that we consider the truncated version, $\\mathcal {D}$ , of the following distributions, $\\mathcal {D}^\\prime $ , supported on $(0,1]$ .",
"Figure (REF ): $\\mathcal {D}^\\prime = \\text{Beta}(\\alpha = 10, \\beta = 1)$ Figure (REF ): $\\mathcal {D}^\\prime = \\text{Exp}(\\lambda = 2)$ Figure (REF ): $\\mathcal {D}^\\prime = \\text{Gamma}(k = 0.5,\\theta = 1)$ Figure (REF ): $\\mathcal {D}^\\prime = \\mathcal {N}(\\mu =1/2,\\sigma ^2=1/10)$ Figure (REF ): $\\mathcal {D}^\\prime = \\mathcal {N}(\\mu =1/2,\\sigma ^2=1/5)$ Figure (REF ): $\\mathcal {D}^\\prime = \\mathcal {N}(\\mu =1/2,\\sigma ^2=1/100)$ Figure (REF ): $\\mathcal {D}^\\prime = \\mathcal {N}(\\mu =1/2,\\sigma ^2=1/20)$ Figure (REF ): $\\mathcal {D}^\\prime = \\text{lognormal}(\\mu =0,\\sigma ^2=3/2)$ Figure (REF ): $\\mathcal {D}^\\prime = \\text{Weibull}(\\lambda = 1, k = 2)$ In all these cases, we almost always return an arm with mean within at most 0.05 ($<1/10=\\varepsilon $ ) from the mean of the best-arm.",
"Figure: X-axis: Gap between the means of the best arm and arm returned by Algorithm , Y-axis: Count of such arms."
],
[
"Regret Minimization under Random Order Arrival",
"Theorem 7 In the MAB setting, fix the number of arms $n$ and the time horizon $T$ .",
"For any online MAB algorithm, if we are allowed to store at most $m < n$ arms, then in the setting of random order arrival there exists an input instance such that $\\mathbb {E}[R(T)] \\ge \\Omega (\\frac{T^{2/3}}{m^{7/3}})$ We consider 0-1 rewards and the 2 input instances $\\mathcal {I}_1,\\mathcal {I}_2$ each containing $n$ arms, with parameter $\\epsilon >0$ (where $\\epsilon =\\frac{1}{m^{1/3}T^{1/3}}$ ): $\\mathcal {I}_1 = \\left\\lbrace \\begin{array}{rcl}\\mu _i & =(1+\\epsilon )/2& \\mbox{for~~} i=1 \\\\\\mu _i & = 1/2,~~~~~~~ & \\mbox{for~~} i \\ne 1\\end{array}\\right.$ $\\mathcal {I}_2= \\left\\lbrace \\begin{array}{rcl}\\mu _i &= 1/2~~~~~~~ &\\mbox{for~~} i\\ne n\\\\\\mu _i &= 1.~~~~~~~~~~ & \\mbox{for~~} i=n\\end{array}\\right.$ In the above instances, $\\mu _i$ denotes the expected reward of the $i^{th}$ arm in the input instance.",
"Let us fix a deterministic algorithm $\\mathcal {A}$ that directly stores the first $m$ arms in the memory.",
"We choose an input instance uniformly at random.",
"Let this input instance be $\\mathcal {I}^{\\prime }$ .",
"Then under random-order arrival setting one of the $n$ permutations of $\\mathcal {I}^{\\prime }$ is chosen uniformly at random and is sent as the input stream to the Algorithm $\\mathcal {A}$ .",
"Note that this equivalent to choosing a permutation $\\mathcal {P}$ from $2n$ total distinct permutations of $\\mathcal {I}_1$ and $\\mathcal {I}_2$ uniformly at random and sending it to the Algorithm $\\mathcal {A}$ .",
"Now assume that $m\\le n-1$ .",
"Let $\\mathcal {I}_1^{\\prime }$ be the collection of distinct permutations of $\\mathcal {I}_1$ such that the arm with expected reward of $(1+\\epsilon )/2$ is in the first $m$ positions of the permutation.",
"Similarly, let $\\mathcal {I}_2^{\\prime }$ be the collection of distinct permutations of $\\mathcal {I}_2$ such that the arm with expected reward of 1 is not in the first $m$ positions of the permutation.",
"Clearly $|\\mathcal {I}_1^{\\prime }|=m$ and $|\\mathcal {I}_2^{\\prime }|=n-m$ .",
"Using arguments similar to Section , we can show that $\\mathbb {E}[R(T)|\\mathcal {P}\\in \\mathcal {I}_1^{\\prime }\\cup \\mathcal {I}_2^{\\prime }]\\ge \\Omega \\Big (\\frac{T^{2/3}}{m^{7/3}}\\Big )$ .",
"Hence, we have the following: $\\mathbb {E}[R(T)] &\\ge \\mathbb {P}[\\mathcal {P}\\in \\mathcal {I}_1^{\\prime }\\cup \\mathcal {I}_2^{\\prime }]\\cdot \\mathbb {E}[R(T)|\\mathcal {P}\\in \\mathcal {I}_1^{\\prime }\\cup \\mathcal {I}_2^{\\prime }]\\\\&\\ge \\frac{n-m+m}{2n}\\cdot \\Omega \\Big (\\frac{T^{2/3}}{m^{7/3}}\\Big )\\\\&\\ge \\Omega \\Big (\\frac{T^{2/3}}{m^{7/3}}\\Big )$ The above result should hold for any randomized algorithm too as randomized algorithm are a distribution over deterministic algorithms."
],
[
"Important Inequalities",
"Lemma 8 (Hoeffding's inequality).",
"Let $Z_1,\\ldots ,Z_n$ be independent bounded variables with $Z_i\\in [0,1]$ for all $i\\in [n]$ .",
"Then $\\mathbb {P}\\Big (\\frac{1}{n}\\sum _{i=1}^{n}(Z_i-\\mathbb {E}[Z_i])\\ge t)\\le e^{-2nt^2}\\Big ), \\text{ and }$ $\\mathbb {P}\\Big (\\frac{1}{n}\\sum _{i=1}^{n}(Z_i-\\mathbb {E}[Z_i])\\le -t)\\le e^{-2nt^2}\\Big ),\\text{ for all } t\\ge 0.$ Lemma 9 Let $\\mathtt {arm}_1$ and $\\mathtt {arm}_2$ be two different arms with means $\\mu _1$ and $\\mu _2$ .",
"Suppose $\\mu _1 - \\mu _2 \\ge \\theta $ and we sample each arm $\\frac{K}{\\theta ^2}$ times to obtain empirical biases $\\widehat{\\mu }_1$ and $\\widehat{\\mu }_2$ .",
"Then, $\\mathbb {P}(\\widehat{\\mu }_1 \\le \\widehat{\\mu }_2)\\le 2\\cdot e^{(-K/2)}$ $\\mathbb {P}(\\widehat{\\mu }_1 > \\widehat{\\mu }_2) & \\ge \\mathbb {P}\\Big (\\mu _1 - \\frac{\\theta }{2}< \\widehat{\\mu }_1 \\text{ and } \\widehat{\\mu }_2 < \\mu _2 + \\frac{\\theta }{2}\\Big )\\\\&= \\mathbb {P}\\Big (\\mu _1 - \\frac{\\theta }{2}< \\widehat{\\mu }_1\\Big ) \\cdot \\mathbb {P}\\Big (\\widehat{\\mu }_2 < \\mu _2 + \\frac{\\theta }{2}\\Big ) \\\\&\\ge \\big (1- e^{-2\\cdot \\frac{K}{\\theta ^2}\\cdot (\\frac{\\theta }{2})^2}\\big )\\cdot \\big (1- e^{-2\\cdot \\frac{K}{\\theta ^2}\\cdot (\\frac{\\theta }{2})^2}\\big ) \\\\& \\ge (1-2\\cdot e^{-K/2}).$ Hence, $\\mathbb {P}(\\widehat{\\mu }_1 \\le \\widehat{\\mu }_2)\\le 2\\cdot e^{-K/2}$ .",
"Theorem 10 (Berry-Esseen Theorem) There exists a positive constant $C\\le 1$ such that if $X_1, X_2, \\ldots , X_n$ are i.i.d.",
"random variables with $\\mathbb {E}[X_i] = 0$ , $\\mathbb {E}[X_i^2] = \\sigma ^2 > 0$ , and $\\mathbb {E}[|X_i|^3] = \\rho < \\infty $ , and if we define $Y_n = \\frac{X_1 + X_2 + \\ldots + X_n}{n}$ to be the sample mean, with $F_n$ being the cumulative distribution function of $\\frac{Y_n\\sqrt{n}}{\\sigma }$ , and $\\Phi $ be the cumulative distribution function of the standard normal distribution $\\mathcal {N}(0,1)$ , then for all $x$ and $n$ , $|F_n(x) - \\Phi (x)| \\le \\frac{C\\rho }{\\sigma ^3\\sqrt{n}}.$"
],
[
"Omitted Proofs from Section ",
"* Let us consider some level $\\ell $ and a set of arms $\\mathtt {arm}_{\\ell _1},\\mathtt {arm}_{\\ell _2},\\ldots ,\\mathtt {arm}_{\\ell c_\\ell }$ , which increases the counter $C_\\ell $ from 0 to $c_\\ell $ .",
"Denote the best arm among them as $\\mathtt {arm}^{\\prime }$ and let $\\mathtt {arm}^*_\\ell $ be the empirically best arm seen so far since the arrival of $\\mathtt {arm}_{\\ell _1}$ .",
"Our algorithm works as follows: once an arm arrives at level $\\ell $ , it is pulled $s_\\ell $ number of times.",
"Next, we compare its empirical mean with that of $\\mathtt {arm}^*_\\ell $ , which was computed when the arm corresponding to $\\mathtt {arm}^*_\\ell $ arrived at level $\\ell $ .",
"The arm with the greater empirical mean is maintained as $\\mathtt {arm}^*_\\ell $ and its empirical mean is stored for future comparisons.",
"Once $C_\\ell = c_\\ell $ , $\\mathtt {arm}^*_\\ell $ is sent to level $\\ell + 1$ .",
"Note that if $0<C_\\ell < c_\\ell $ , and the condition in Step is not satisfied then we will sample $\\mathtt {arm}^*_\\ell $ in the Step .",
"Note that this is equivalent to simultaneously pulling each of the $c_\\ell $ arms $s_\\ell $ number of times, and then sending the empirically best arm to level $\\ell +1$ .",
"Using Lemma REF , we can show that at any level $\\ell $ , if two arms have reward gap $\\mu _i - \\mu _j \\ge \\varepsilon _\\ell $ then $\\mathbb {P}[\\widehat{\\mu }_i < \\widehat{\\mu }_j] \\le \\frac{\\delta }{2^{\\ell +1}\\cdot c_\\ell }$ .",
"Since at most $c_\\ell $ arms arrive at level $\\ell $ before the counter $C_\\ell $ reaches $c_\\ell $ or the condition in Step is not satisfied, taking union bound we have that the probability that an arm with reward gap $\\ge \\varepsilon _\\ell $ with respect to $\\mathtt {arm}^{\\prime }_\\ell $ is either sent to level $\\ell +1$ or is sampled in the Step is at most $\\frac{\\delta }{2^{\\ell +1}\\cdot c_\\ell }\\cdot c_\\ell =\\frac{\\delta }{2^{\\ell +1}}$ .",
"* Using Lemma REF , we can show that among the arms that are being considered here, if two arms have reward gap $\\mu _i - \\mu _j \\ge \\varepsilon _r$ then $\\mathbb {P}[\\widehat{\\mu }_i < \\widehat{\\mu }_j] \\le \\frac{\\delta }{2^{r+1}\\cdot c_r}$ .",
"As $c_r=n$ , by taking union bound we get that with probability at least $1 - \\frac{\\delta }{2^{r+1}}$ , an arm with reward gap at most $\\varepsilon _r$ from $\\mathtt {arm}^{\\prime }_r$ is returned by the Algorithm."
],
[
"Adversarial Example for {{cite:3a80cfbc12f550517576043701a54bccb5bec1e3}}",
"In this section, we show an adversarial example to show that the algorithm of [5] is not $(\\varepsilon , \\delta )$ -PAC.",
"First, we state the algorithm of [5] (Algorithm ) here for completeness.",
"The algorithm takes as input $n \\in \\mathbb {N}$ arms arriving in a stream in an arbitrary order, an approximation parameter $\\varepsilon \\in [0,1/2)$ , and the confidence parameter $\\delta \\in (0,1)$ .",
"We first define some notation that is used throughout this work, which is consistent with the notation used in [5].",
"$&\\lbrace r_\\ell \\rbrace _{\\ell =1}^\\infty :\\hspace{8.5359pt} r_1 = 4;~~ r_\\ell = 2^{r_\\ell }; \\\\&\\varepsilon _\\ell = \\varepsilon /({10\\cdot 2^{\\ell - 1}}); &\\\\&\\beta _\\ell = {1}/{\\varepsilon _\\ell ^2}; \\\\&\\lbrace s_\\ell \\rbrace _{\\ell =1}^\\infty :\\hspace{8.5359pt} s_\\ell = 4\\beta _\\ell \\big (\\ln (1/\\delta ) + 3r_\\ell \\big ); \\\\&\\lbrace c_\\ell \\rbrace _{\\ell =1}^{\\lceil \\log ^* n\\rceil + 1}:\\hspace{8.5359pt} c_1 = 2^{r_1};~~ c_\\ell = {2^{r_\\ell }}/{2^{\\ell - 1}};\\\\& \\text{(the number of arms processed in level \\qquad \\mathrm {(intermediate parameters used to define s_\\ell below,intermediate estimate of gap parameter,number of samples per arm in level \\ell )}$\\ell $ before}\\\\ &\\text{sending $\\mathtt {arm}^*_\\ell $ to level $\\ell +1$)}$ For Algorithm to be $(\\varepsilon ,\\delta )$ -PAC, it has to find an $\\varepsilon $ -best arm with probability at least $1-\\delta $ .",
"We next provide a formal argument for why Algorithm is not an $(\\varepsilon ,\\delta )$ -PAC algorithm by providing a counterexample.",
"[ht!]",
"[1] $\\lbrace r_\\ell \\rbrace ^{\\infty }_{\\ell =1}:r_1:=4,r_{\\ell +1}=2^{r_\\ell }$ ; $\\varepsilon _\\ell =\\frac{\\varepsilon }{10\\cdot 2^{\\ell -1}}$ ; $\\beta _\\ell =\\frac{1}{\\varepsilon ^2_\\ell }$ ; $s_\\ell =4\\beta _\\ell \\big (\\ln (\\frac{1}{\\delta })+3r_\\ell \\big )$ ; $c_1=2^{r_1},$ $c_\\ell =\\frac{2^{r_\\ell }}{2^{\\ell -1}}(\\ell \\ge 2)$ ; Counters: $C_1,C_2,\\ldots ,C_t$ initialized to 0 where $t=\\lceil \\log ^*(n)\\rceil +1$ .",
"Stored arms: $\\mathtt {arm}^*_1,\\mathtt {arm}^*_2,\\ldots ,\\mathtt {arm}^*_t$ the most biased arm of $\\ell $ -th level.",
"Stored empirical means:$p_1,p_2,\\ldots ,p_t$ the highest empirical mean of $\\ell $ -th level.",
"A new arm $\\mathtt {arm}_i$ arrives in the stream Read $\\mathtt {arm}_i$ to memory Aggressive Selective Promotion: Starting from level $\\ell =1$ : Sample both $\\mathtt {arm}_i$ and $\\mathtt {arm}^*_\\ell $ for $s_\\ell $ times.",
"Drop $\\mathtt {arm}_i$ if $\\hat{p}_{\\mathtt {arm}_i} < \\hat{p}_{\\mathtt {arm}^*_\\ell }$ , otherwise replace $\\mathtt {arm}^*_\\ell $ with $\\mathtt {arm}_i$ .",
"Increase $C_\\ell $ by 1.",
"If $C_\\ell =c_\\ell $ , make $C_\\ell $ equal to 0, send $\\mathtt {arm}^*_\\ell $ to the next level by calling Line with $(\\ell =\\ell +1)$ .",
"Return $\\mathtt {arm}^*_t$ as the selected most bias arm.",
"Let the arms arrive in the stream be $\\mathtt {arm}_1,\\ldots ,\\mathtt {arm}_n$ , where $n>(c_1\\cdot c_2)$ .",
"Let $\\mathtt {arm}_i$ be the $i^{th}$ arm to arrive in the stream and has a mean $p_i$ , where $i\\in [n]$ .",
"For all $i>c_1\\cdot c_2$ , let $p_i=0$ .",
"For all $i\\le c_1\\cdot c_2$ , let $p_i=\\frac{1}{2}-(\\lceil \\frac{i}{c_1}-1\\rceil )\\cdot \\frac{\\varepsilon }{c_2-2}$ .",
"Let $\\mathtt {arm}_{k_1},\\mathtt {arm}_{k_2},\\ldots ,\\mathtt {arm}_{k_{c_2}}$ be the first $c_2$ arms which arrive at level 2 (note that all the arms which arrive at level 2 after $\\mathtt {arm}_{k_{c_2}}$ will have a mean of 0).",
"Let $\\mathtt {arm}^*_2$ be the most biased arm (based on the sampling) at the end of Aggressive Selection Promotion step for level $\\ell =2$ for $\\mathtt {arm}_{k_{c_2}}$ .",
"Now $C_2 = c_2$ after the arrival of $\\mathtt {arm}_{k_{c_2}}$ .",
"Thus $\\mathtt {arm}^*_2$ will be sent to level 3.",
"As all the following remaining arms have lesser means, at the end, the algorithm finally returns an arm with mean less than or equal to the mean of $\\mathtt {arm}^*_2$ .",
"Note that $\\forall i\\in [c_2]$ , all the arms in the set $\\lbrace \\mathtt {arm}_{(i-1)\\cdot c_1+1},\\ldots ,\\mathtt {arm}_{i\\cdot c_1}\\rbrace $ have the same mean and one among them is sent as $\\mathtt {arm}_{k_{i}}$ to level 2.",
"Therefore $p_{k_i} = p_{k_1}-(i-1)\\frac{\\varepsilon }{b}$ , where $b = 2^{15}-2$ and $p_{k_1}=\\frac{1}{2}$ .",
"So for any $i \\in [c_{2}-1]$ , $p_{k_i} - p_{k_{i+1}} = \\frac{\\varepsilon }{b}$ .",
"We will show that with probability $>\\delta $ , we send $\\mathtt {arm}_{k_{c_2}}$ to level 3, and $\\frac{1}{2}-p_{k_{c_2}}>\\varepsilon $ .",
"For $i\\in [c_2]$ , let $Y_i^t$ denote the reward when we sample the arm $\\mathtt {arm}_{k_i}$ for the $t$ -th time.",
"We assume that $Y_i^t \\sim \\text{Bern}(p_{k_i})$ and $\\mathrm {Var}[Y_i^t]=p_{k_i}(1-p_{k_i})$ (Note that this is a reasonable assumption as the Algorithm should work for any distribution).",
"For $i \\in [c_2 - 1]$ , let $Z_i^t = Y_i^t - Y_{i+1}^t$ .",
"Clearly, $\\mu _i:=\\mathbb {E}[Z_i^t] = \\frac{\\varepsilon }{b}$ .",
"Let $\\sigma _i^2:=\\mathrm {Var}[Z_{i}^t]$ .",
"Let us assume that $\\varepsilon <\\frac{1}{5}$ (Later we will choose $\\varepsilon $ in such a way so that this condition is satisfied).",
"In this case, $\\sigma _i^2=\\mathrm {Var}[Y_i^t]+\\mathrm {Var}[Y_{i+1}^t]>2(p_{k_{c_2}})(1-p_{k_{c_2}})>\\frac{2}{5}$ .",
"Let $Z_i = Z_i^1 + Z_i^2 + \\ldots + Z_i^{s_2}$ .",
"Note that, if every arm from the set $\\mathtt {arm}_{k_1},\\mathtt {arm}_{k_2},\\ldots ,\\mathtt {arm}_{k_{c_2}}$ when it arrives in the level 2 beats $\\mathtt {arm}^*_2$ in the challenge, then $\\mathtt {arm}_{k_{c_2}}$ will be sent to level 3.",
"Thus, $\\lbrace Z_i < 0, \\forall i\\in [c_2-1]\\rbrace \\subseteq \\lbrace \\mathtt {arm}_{k_{c_2}} ~\\text{is sent to level}~ 3\\rbrace $ .",
"Assuming that $\\delta $ and $\\varepsilon $ are very small (which we will choose appropriately to bound the error), we approximate (using the central limit theorem) the distribution of $Z_i$ using the normal distribution $\\mathcal {N}(s_2\\mu _i,s_2\\sigma _i^2)$ .",
"$\\mathbb {P}[Z_i<0] &= \\mathbb {P}[Z_i > 2s_2\\mu _i]\\\\& = 1-\\frac{1}{2}\\bigg (1+\\text{erf}\\Big (\\frac{s_2\\mu _i}{\\sqrt{2s_2 \\sigma _i^2}}\\Big )\\bigg )\\\\&= \\frac{\\text{erfc}\\Big (\\frac{s_2\\mu _i}{\\sqrt{2s_2 \\sigma _i^2}}\\Big )}{2}\\\\& \\ge \\frac{\\text{erfc}\\Big (s_2\\mu _i \\cdot \\sqrt{\\frac{5}{4s_2}}\\Big )}{2} \\\\& = \\frac{\\text{erfc}\\Big (\\frac{20\\sqrt{5}}{b}\\sqrt{\\ln \\big (\\frac{e^{3r_2}}{\\delta }\\big )}\\Big )}{2} \\\\&\\ge \\frac{\\sqrt{\\gamma -1}}{2}e^{-\\frac{2000\\cdot \\gamma }{b^2}\\ln (\\frac{e^{3r_2}}{\\delta })}\\\\&= \\frac{\\sqrt{\\gamma -1}}{2}\\Big (\\frac{\\delta }{e^{3r_2}}\\Big )^{\\frac{2000\\cdot \\gamma }{b^2}}.\\\\$ Thus, we can lower bound the probability that $\\mathtt {arm}_{k_{c_2}}$ is sent to level 3 as follows: $\\mathbb {P}[\\mathtt {arm}_{k_{c_2}} ~\\text{is sent to level}~ 3]&\\ge \\mathbb {P}[Z_i < 0, \\forall i\\in [c_2-1]]\\\\& = \\prod _{i\\in [c_2 -1]}\\mathbb {P}[Z_i < 0] \\\\& \\ge \\Big ( \\frac{\\sqrt{\\gamma -1}}{2}\\Big (\\frac{\\delta }{e^{3r_2}}\\Big )^{\\frac{2000\\cdot \\gamma }{b^2}}\\Big )^{c_2-1}\\\\& = \\frac{\\delta ^{\\frac{2000\\cdot \\gamma \\cdot (c_2-1)}{b^2}}}{K}, \\text{ where } K= \\Big (\\frac{2\\cdot e^{\\frac{3r_2\\cdot 2000\\cdot \\gamma }{b^2}}}{\\sqrt{\\gamma -1}}\\Big )^{c_2-1}.$ Consider that function $f(x)=\\frac{e^{x\\left(1-\\frac{2000\\cdot \\gamma \\cdot (c_2-1)}{b^2}\\right)}}{K}$ .",
"Since $f(x)$ is an increasing and convex function, there is a constant $c$ such that $f(c)>2$ .",
"This implies that for $\\delta =e^{-c}$ we have the following: $\\mathbb {P}[\\mathtt {arm}_{k_{c_2}} ~\\text{is sent to level}~ 3] & \\ge \\frac{\\delta ^{\\frac{2000\\cdot \\gamma \\cdot (c_2-1)}{b^2}}}{K}\\\\& = f(c)e^{-c}\\\\& >2\\delta .$ Now, we bound the error in calculation of the above probability.",
"Using the Berry-Esseen theorem, the error $\\epsilon _i$ of calculating $\\mathbb {P}[Z_i<0]$ is upper bounded by $\\frac{C \\rho }{\\sigma _i^3\\sqrt{s_2}}\\le \\frac{ \\varepsilon }{\\sqrt{\\ln (\\frac{1}{\\delta })}}$ , where $C\\le 1$ , $\\rho =\\mathbb {E}[|Z_i^t-\\mu _i|^3]\\le 8 \\text{ (as } |Z_i^t-\\mu _i|\\le 2)$ and $\\sigma _i^2=\\mathrm {Var}[Z_i^t-\\mu _i]=\\mathrm {Var}[Z_i^t]$ .",
"Also we assumed $\\mathrm {Var}[Z_i^t]>\\frac{2}{5}$ (we will choose $\\varepsilon $ in such a way that this is satisfied).",
"If we choose $\\varepsilon $ such that $\\varepsilon < \\frac{\\delta \\sqrt{\\ln (\\frac{1}{\\delta })}}{c_2}$ and $\\varepsilon <1/5$ , then $\\epsilon _i<\\frac{\\delta }{c_2}$ .",
"Hence, we can conclude that $\\mathbb {P}[\\mathtt {arm}_{k_{c_2}} ~\\text{is sent to level}~ 3]>2\\delta -\\sum _{i=1}^{c_2-1}\\epsilon _i > 2\\delta -\\delta =\\delta $ .",
"As $\\frac{1}{2}-p_{k_{c_2}}=\\Big (\\frac{c_2-1}{c_2-2}\\Big )\\cdot \\varepsilon $ , we can conclude that with probability $>\\delta $ , the Algorithm returns an arm with reward gap $> \\varepsilon $ ."
],
[
"Lower bound for r-round adaptive streaming algorithm",
"In this section, we use the following lower bound for $r$ -round adaptive offline algorithm model defined in [2] to provide a lower bound on the sample complexity for any $r$ -round adaptive streaming algorithm.",
"Lemma 11 ([2]).",
"For any parameter $\\Delta \\in (0,1/2)$ and any integer $n,k\\ge 1$ , there exists a distribution $\\mathcal {D}$ on input instances of the $k$ most biased coins problem with $n$ coins and gap parameter $\\Delta _k=\\Delta $ such that for any integer $r\\le 1$ , any $r$ -round algorithm that finds the $k$ most biased coins in the instances sampled from $\\mathcal {D}$ with probability at least $3/4$ has a sample complexity $\\Omega (\\frac{n}{\\varepsilon ^2\\cdot r^4}\\cdot \\mathtt {ilog}^{(r)}(n/k))$ Any $r$ -round adaptive offline algorithm is same as $r$ -round adaptive streaming algorithm except for the following two points: In an $r$ -round adaptive offline algorithm all the arms can be simultaneously stored in the memory, whereas arm-memory is usually bounded in an $r$ -round adaptive streaming algorithm.",
"In an $r$ -round adaptive offline algorithm, in any round $j$ , all the arms are sampled simultaneously.",
"On the other hand, in an $r$ -round adaptive streaming algorithm all the arms in the round $j$ need not be sampled simultaneously, and they can also be sampled one after the other.",
"Any $r$ -round adaptive streaming algorithm can be replicated by an $r$ -round adaptive offline algorithm such that the worst-case sample complexity is the same in both the algorithms.",
"Next, we present the following lemma.",
"Lemma 12 For any approximation parameter $\\varepsilon \\in (0,1/2)$ and any integer $n\\ge 1$ , there exists a distribution $\\mathcal {D}$ on input instances of the best-arm identification problem with $n$ arms such that for any integer $r\\ge 1$ , any $r$ -round adaptive streaming algorithm that finds the $\\varepsilon $ -best arm in the instances sampled from $\\mathcal {D}$ with probability at least $3/4$ has a sample complexity $\\Omega (\\frac{n}{\\varepsilon ^2\\cdot r^4}\\cdot \\mathtt {ilog}^{(r)}(n))$ We present the proof idea.",
"Let $\\mathcal {A}$ be an $r$ -round adaptive streaming algorithm with the lowest worst-case sample complexity which finds the $\\varepsilon $ -best arm with probability at least $3/4$ .",
"Now we replicate the algorithm $\\mathcal {A}$ using an $r$ -round offline algorithm $\\mathcal {B}$ for best-arm identification such that its worst-case sample complexity is equal to that of $\\mathcal {A}$ .",
"Due to Lemma REF , the worst-case sample complexity of $\\mathcal {B}$ is $\\Omega (\\frac{n}{\\varepsilon ^2\\cdot r^4}\\cdot \\mathtt {ilog}^{(r)}(n))$ .",
"Hence, the worst-case sample complexity of $\\mathcal {A}$ is $\\Omega (\\frac{n}{\\varepsilon ^2\\cdot r^4}\\cdot \\mathtt {ilog}^{(r)}(n))$ ."
],
[
"Adversarial Example for Constant Arm-Memory Algorithm based on {{cite:3a80cfbc12f550517576043701a54bccb5bec1e3}}",
"In this section, we show an adversarial example for Constant Arm-memory Algorithm based on [5] when the parameter $\\Delta $ is unknown where $\\Delta $ is the gap between the best arm and the second best arm.",
"For completeness, we present this algorithm below (Algorithm ).",
"[ht!]",
"[1] $\\lbrace r_\\ell \\rbrace ^{\\infty }_{\\ell =1}:r_\\ell =3^\\ell $ ; $\\lbrace s_\\ell \\rbrace ^{\\infty }_{\\ell =1}:s_\\ell =\\frac{2}{\\varepsilon ^2}\\cdot \\ln \\left(\\frac{1}{\\delta }\\right)\\cdot r_\\ell $ ; $b:=\\frac{2}{\\epsilon ^2}\\cdot C \\cdot \\ln \\left(\\frac{1}{\\delta }\\right)+s_1$ ; Let king be the first available arm and set its budget $\\phi :=\\phi (king)=0$ .",
"A new arm $\\mathtt {arm}_i$ arrives in the stream Increase the budget $\\phi (\\textbf {king})$ by $b$ .",
"Challenge subroutine: For level $\\ell =1$ to $+\\infty $ : If $\\phi (king)<s_\\ell $ : we declare $\\textbf {king}$ defeated , make $\\mathtt {arm}_i$ the king, initialize its budget to 0 and go to Step .",
"Otherwise, we decrease $\\phi (\\textbf {king})$ by $s_\\ell $ and sample both king and $\\mathtt {arm}_i$ for $s_\\ell $ times.",
"Let $\\widehat{\\mu }_{king}$ and $\\widehat{\\mu }_i$ denote the empirical biases of king and $\\mathtt {arm}_i$ in this trial.",
"If $\\widehat{\\mu }_{king} > \\widehat{\\mu }_i$ , we declare $king$ winner and go to the next arm in the stream; otherwise we go to the next level of challenge (increment $\\ell $ by one).",
"Return $\\textbf {king}$ as the selected most bias arm.",
"Let $\\mathtt {arm}_{1},\\mathtt {arm}_{2},\\ldots ,\\mathtt {arm}_{n}$ be the stream of arms which arrive.",
"For each $i \\in [n]$ , $\\mu _{i}$ is the expected reward of arm $\\mathtt {arm}_{i}$ .",
"Define the input stream such that $\\mu _{i} = \\mu _{1}-(i-1)\\frac{\\varepsilon }{n-2}$ .",
"Note that for any $i \\in [n-1]$ , $\\mu _{i} - \\mu _{i+1} = \\frac{\\varepsilon }{n-2}$ .",
"Let $\\mu _1=\\frac{1}{2}$ .",
"We now show that there exist $\\delta ,\\varepsilon >0$ such that with probability $>\\delta $ , an arm with reward gap $> \\varepsilon $ is returned by the Algorithm.",
"Let $k$ be the maximum value of $\\ell $ such that $\\sum _{i=2}^{\\ell }3^i\\le C$ .",
"Let $\\forall i\\in [n]$ , $Y_{i,t}^\\ell $ denote the reward when we sample the arm $\\mathtt {arm}_i$ for the $t^{th}$ time at level $\\ell $ .",
"Then, $\\mathrm {Var}[Y_{i,t}^\\ell ]= \\mu _i(1 - \\mu _i)$ .",
"Let $Z_{i,t}^\\ell =Y_{i,t}^\\ell -Y_{i+1,t}^\\ell $ .",
"Clearly $p_i:=\\mathbb {E}[Z_{i,t}^\\ell ]=\\frac{\\varepsilon }{n-2}$ .",
"Let $\\sigma _i^2:=\\mathrm {Var}[Z_{i,t}^\\ell ]$ .",
"Let us assume that $\\varepsilon <\\frac{1}{5}$ and $n>>100$ (We will later choose $\\varepsilon $ and $n$ in a way so that this condition is satisfied).",
"In this case, $\\sigma _i^2=\\mathrm {Var}[Y_{i,t}^\\ell ]+\\mathrm {Var}[Y_{i+1,t}^\\ell ]>2(\\mu _n)(1-\\mu _n)>\\frac{2}{5}$ .",
"Let $Z_i^\\ell =Z_{i,1}^\\ell +Z_{i,2}^\\ell +\\ldots +Z_{i,{s_\\ell }}^\\ell $ .",
"Assuming that $\\delta ,\\varepsilon $ are very small (which we will choose appropriately to bound the error) we approximate (using Central Limit Theorem) the distribution of $Z_i^\\ell $ using the normal distribution $\\mathcal {N}(s_\\ell p_i,s_\\ell \\sigma _i^2)$ .",
"$\\mathbb {P}[Z_i^\\ell <0]&=\\mathbb {P}[Z_i^\\ell > 2s_\\ell p_i]\\\\&=1-\\frac{1}{2}\\bigg [1+\\text{erf}\\bigg (\\frac{s_\\ell p_i}{\\sqrt{2s_\\ell \\sigma _i^2}}\\bigg )\\bigg ]\\\\&=\\frac{\\text{erfc}\\Big (\\frac{s_\\ell p_i}{\\sqrt{2s_\\ell \\sigma _i^2}}\\Big )}{2}\\\\& \\ge \\frac{\\text{erfc}(\\frac{s_\\ell p_i\\sqrt{5}}{2\\sqrt{s_\\ell }})}{2}\\\\& \\ge \\frac{\\text{erfc}\\Big (\\frac{\\sqrt{5}\\cdot 3^{\\ell /2}}{n-2}\\sqrt{\\ln (\\frac{1}{\\delta })}\\Big )}{2}\\\\&\\ge \\frac{\\sqrt{\\gamma -1}}{2}e^{-\\frac{5\\cdot 3^{\\ell }\\cdot \\gamma }{(n-2)^2}\\ln (\\frac{1}{\\delta })} \\\\&= \\frac{\\sqrt{\\gamma -1}}{2}\\delta ^{\\frac{5\\cdot 3^{\\ell }\\cdot \\gamma }{(n-2)^2}}.$ Hence, we now have $\\mathbb {P}[\\mathtt {arm}_{i+1}\\text{ becomes king by defeating }\\mathtt {arm}_{i}] &\\ge \\mathbb {P}[\\forall \\ell \\in [k], Z_i^\\ell <0]\\\\& \\ge \\left(\\frac{\\sqrt{\\gamma -1}}{2}\\delta ^{\\frac{5\\cdot 3^{k}\\cdot \\gamma }{(n-2)^2}}\\right)^{k}\\\\& = \\frac{\\delta ^{\\frac{5\\cdot 3^{k}\\cdot k \\cdot \\gamma }{(n-2)^2}}}{K} $ Now we choose $n$ such that $\\frac{(n-2)^2}{(n-1)}>>5\\cdot 3^{k}\\cdot k \\cdot \\gamma $ .",
"Consider the function $f(x)=\\frac{e^{x\\left(1-\\frac{5\\cdot 3^{k}\\cdot k \\cdot \\gamma \\cdot (n-1)}{(n-2)^2}\\right)}}{K^{n-1}}$ .",
"Since $f(x)$ is an increasing and convex function, there is a constant $c$ such that $f(c)>2$ .",
"This implies that for $\\delta =e^{-c}$ we have the following: $\\mathbb {P}[\\mathtt {arm}_{n}\\text{ is returned as king by the algorithm}]& \\ge \\mathbb {P}[\\forall i\\in [n-1], \\mathtt {arm}_{i+1}\\text{ becomes king by defeating }\\mathtt {arm}_{i}] \\\\& \\ge \\left(\\frac{\\delta ^{\\frac{5\\cdot 3^{k}\\cdot k \\cdot \\gamma }{(n-2)^2}}}{K}\\right)^{n-1}\\\\& = \\frac{e^{c\\left(1-\\frac{5\\cdot 3^{k}\\cdot k \\cdot \\gamma \\cdot (n-1)}{(n-2)^2}\\right)}}{K^{n-1}}\\cdot \\delta \\\\& = f(c)\\cdot \\delta \\\\& >2\\delta .$ Now we bound the error in calculation of the above probability.",
"By Berry-Esseen theorem, the error $\\epsilon _i^\\ell $ of calculating $\\mathbb {P}[Z_i^\\ell <0]$ is upper bounded by $\\frac{C \\rho }{\\sigma _i^3\\sqrt{s_1}}\\le \\frac{16 \\varepsilon }{\\sqrt{\\ln (\\frac{1}{\\delta })}}$ .",
"Here $C\\le 1$ , $\\rho =\\mathbb {E}[|Z_{i,t}^\\ell -p_i|^3]\\le 8( \\text{as } |Z_{i,t}^\\ell -p_i|\\le 2)$ and $\\sigma _i^2=\\mathrm {Var}[Z_{i,t}^\\ell -p_i]=\\mathrm {Var}[Z_{i,t}^\\ell ]$ .",
"Also we assumed $\\mathrm {Var}[Z_{i,t}^\\ell ]>\\frac{2}{5}$ (we will choose $\\varepsilon $ in such a way that this is satisfied).",
"If we choose $\\varepsilon $ such that it is less than $\\frac{\\delta \\sqrt{\\ln (\\frac{1}{\\delta })}}{16\\cdot k\\cdot n}$ and it is also less than $\\frac{1}{5}$ , then $\\epsilon _i^\\ell <\\frac{\\delta }{k\\cdot n}$ .",
"Therefore $\\mathbb {P}[\\mathtt {arm}_{n}\\text{ is returned as king by the algorithm}]>2\\delta -\\sum _{i=1}^{n-1}\\sum _{\\ell =1}^{k}\\epsilon _i^\\ell >2\\delta -\\delta =\\delta $ .",
"As $\\mu _{1} - \\mu _n>\\varepsilon $ , we can conclude that there exists $\\delta ,\\varepsilon >0$ such that with probability $>\\delta $ , a king with reward gap $> \\varepsilon $ is returned by the algorithm.",
"We also have experimental evidence for this result.",
"We ran the experiment on problem instances of this adversial type with number of arms $= 5500001$ , mean of the best arm $=1/2$ , $\\varepsilon =1/10$ .",
"We ran ten independent experiments with different realizations due to different values from sampling.",
"Each time the algorithm didn't return $\\varepsilon $ -best arm."
],
[
"Algorithm with constant arm-memory",
"We now tweak the Algorithm in Section and show that our new proposed Algorithm deals with the following: works well both theoretically and experimentally on the counter example for the algorithm mentioned in Section .",
"works well theoretically on random order arrival for well known distributions works well experimentally on any randomly generated input from some well known distributions.",
"Lemma 13 Let $\\mathtt {arm}_1$ and $\\mathtt {arm}_2$ be two different arms with biases $\\mu _1$ and $\\mu _2$ .",
"Suppose $\\mu _1 - \\mu _2 \\ge 0.5\\varepsilon $ and we sample each arm $s_\\ell $ times to obtain empirical biases $\\widehat{\\mu }_1$ and $\\widehat{\\mu }_2$ .",
"Then, $\\mathbb {P}(\\widehat{\\mu }_1 \\le \\widehat{\\mu }_2 + 0.495\\varepsilon )\\le 2\\cdot e^{(-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell )}.$ $\\mathbb {P}(\\widehat{\\mu }_1 > \\widehat{\\mu }_2 + 0.495\\varepsilon )& \\ge \\mathbb {P}\\Big (\\mu _1 - \\frac{\\varepsilon }{400} < \\widehat{\\mu }_1 \\text{ and } \\widehat{\\mu }_2 < \\mu _2 + \\frac{\\varepsilon }{400} \\Big )\\\\&= \\mathbb {P}\\Big (\\mu _1-\\frac{\\varepsilon }{400}< \\widehat{\\mu }_1\\Big ) \\cdot \\mathbb {P}\\Big (\\widehat{\\mu }_2 < \\mu _2 + \\frac{\\varepsilon }{400}\\Big ) \\\\&\\ge (1- e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell })\\cdot (1- e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }) \\\\& \\ge (1- 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }).$ Hence, we have $\\mathbb {P}(\\widehat{\\mu }_1 \\le \\widehat{\\mu }_2 + 0.495\\varepsilon ) \\le 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }$ .",
"Lemma 14 Let $\\mathtt {arm}_1$ and $\\mathtt {arm}_2$ be two different arms with biases $\\mu _1$ and $\\mu _2$ .",
"Suppose $\\mu _1 - \\mu _2 \\le 0.49\\varepsilon $ and we sample each arm $s_\\ell $ times to obtain empirical biases $\\widehat{\\mu }_1$ and $\\widehat{\\mu }_2$ .",
"Then, $\\mathbb {P}(\\widehat{\\mu }_1\\ge \\widehat{\\mu }_2+0.495\\varepsilon )\\le 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }.$ $\\mathbb {P}(\\widehat{\\mu }_1 < \\widehat{\\mu }_2 + 0.495\\varepsilon )& \\ge \\mathbb {P}\\Big (\\widehat{\\mu }_1 < \\mu _1 + \\frac{\\varepsilon }{400} \\text{ and } \\mu _2 - \\frac{\\varepsilon }{400}< \\widehat{\\mu }_2\\Big )\\\\&= \\mathbb {P}\\Big ( \\widehat{\\mu }_1 < \\mu _1 + \\frac{\\varepsilon }{400}\\Big )\\cdot \\mathbb {P}\\Big (\\mu _2 - \\frac{\\varepsilon }{400}< \\widehat{\\mu }_2\\Big ) \\\\&\\ge (1- e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell })\\cdot (1- e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }) \\\\& \\ge (1- 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }).$ Hence, we have $\\mathbb {P}(\\widehat{\\mu }_1 \\ge \\widehat{\\mu }_2 + 0.495\\varepsilon )\\le 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell }$ .",
"Lemma 15 In a challenge subroutine, if $\\mu _i - \\mu _{king} \\ge 0.5\\varepsilon $ , then the probability that $\\mathtt {arm}_i$ does not become the king is at most $\\frac{\\delta }{8}$ .",
"$\\mathbb {P}(\\mathtt {arm}_i\\text{ loses to king })&\\le \\sum _{\\ell =1}^\\infty \\mathbb {P}(\\mathtt {arm}_i\\text{ loses to king }\\text{at level $\\ell |\\mathtt {arm}_i$ has not lost until }\\ell -1)\\\\&\\le \\sum _{\\ell =1}^\\infty 2\\cdot e^{-\\ln (\\frac{4}{\\delta })\\cdot r_\\ell } \\\\& < (\\delta /2) \\cdot \\sum _{\\ell =1}^\\infty e^{-3^\\ell }\\\\& < \\delta /8.$ Since the budget is finite, king will lose to $\\mathtt {arm}_i$ with probability at least $(1-\\delta /8)$ in finite time.",
"The next lemma is an adaptation of Lemma 3.3 in (Assadi et al.",
"[5]).",
"Lemma 16 In Algorithm , if any incoming arm does not lose to the king (denoted $\\mathtt {arm}_{king}$ ) at a level $\\ell $ with probability at most $2\\cdot e^{-\\ln (\\frac{1}{\\delta ^{\\prime }})\\cdot 3^\\ell }$ , then the probability that $\\mathtt {arm}_{king}$ loses to any incoming arm is at most $\\delta ^{\\prime }/2$ .",
"Lemma 17 Let $\\mu _{king}$ be the bias of the current king ($\\mathtt {arm}_{king}$ ).",
"If the future arms in the stream don't have a bias in the range $(\\mu _{king} + 0.49\\epsilon , \\mu ^*]$ where $\\mu ^*$ is the bias of the most-biased arm, then the probability that the king is ever defeated is at most $\\delta /8$ .",
"The lemma follow from Lemma REF and Lemma REF by substituting $\\delta ^{\\prime }=\\delta /4$ .",
"Lemma 18 Let the $\\mu ^*$ be the bias of the most-biased arm.",
"If a $\\mathtt {arm}_i$ with bias $\\mu _i \\in [\\mu ^* - 0.49\\varepsilon ,\\mu ^*]$ becomes the king, then the probability that $arm_i$ is ever defeated as a king is at most $\\delta /8$ .",
"This follows directly from Lemma REF .",
"Corollary 19 If the input stream is the counter example for the algorithm mentioned in the Section , then the probability that Algorithm returns a non-$\\varepsilon $ -best arm is at most $\\delta /8$ ."
],
[
"Random Order Arrival",
"Let the number of arms $n$ in the input set of arms be very large such that the distribution of means of the input set of arms becomes sort of continuous.",
"Let this distribution have a P.D.F $f(x)$ .",
"Let $\\mu ^*$ be the mean of the best arm.",
"We choose a random permutation of our input and send it as an input stream to Algorithm .",
"Lemma 20 Under random order arrival, probability that Algorithm returns an $\\varepsilon $ -best arm is at least $\\inf _{p^{\\prime }\\in [\\mu ^*-0.99\\varepsilon ,\\mu ^*-0.5\\varepsilon ]}\\Bigg \\lbrace (1-\\delta )\\cdot \\bigg (\\frac{\\int _{\\min \\lbrace p^{\\prime }+0.01\\varepsilon ,\\mu ^*-0.5\\varepsilon \\rbrace }^{\\mu ^*-0.5\\varepsilon } f(x) \\frac{\\int _{x+0.5\\varepsilon }^{\\mu ^*} f(x)dx}{\\int _{x+0.49\\varepsilon }^{\\mu ^*} f(x) dx}dx}{\\int _{p^{\\prime }}^{\\mu ^*} f(x)dx}+\\frac{\\int _{\\mu ^*-0.49\\varepsilon }^{\\mu ^*} f(x)dx}{\\int _{p^{\\prime }}^{\\mu ^*} f(x)dx}\\bigg )\\Bigg \\rbrace .$ Let the number of arms in the input stream having their means in the range $[0,\\mu ^*-0.99\\varepsilon )$ be $k$ .",
"For all $1\\le \\ell \\le k$ , let $S_\\ell $ be the set of all possible input streams containing exactly $\\ell $ out of the $k$ arms above.",
"Let $t_\\ell $ denote the sequence of first $\\ell $ arms to arrive in the stream and let $t_\\ell =\\lbrace c_1,c_2,\\ldots ,c_\\ell \\rbrace \\in S_\\ell $ .",
"Let $king^\\ell $ denote the arm that is the king after the $\\ell $ -th arm has been processed by the algorithm.",
"Let $king^\\ell = \\mathtt {arm}_i$ with probability $q_{i}$ , $\\forall i\\in [\\ell ]$ , where $q_{i}\\ge 0$ for all $i \\in [\\ell ]$ , $\\sum _{i=1}^{\\ell }q_{i} = 1$ .",
"Let $X$ be a random variable such that $X=i$ if and only if the first arm in the stream which has mean in the range $[\\mu ^*-0.99\\varepsilon ,\\mu ^*]$ is the $i$ -th arm in the stream.",
"Let $X=\\ell +1$ .",
"Let us assume that the king is $\\mathtt {arm}_i$ (with mean $\\mu _i$ ) just before the $(\\ell +1)$ -th arm arrives where $i\\in [\\ell ]$ .",
"If the arms arriving in the stream at position $\\ell +1$ and later have means in the range $[0,\\mu _i+0.49\\varepsilon ]$ , then from Lemma REF we know that $\\mathtt {arm}_i$ continues to be the king with probability at least $(1-\\delta /8)$ .",
"Let $p^{\\prime }$ =max$\\lbrace \\mu ^*-0.99\\varepsilon ,\\mu _i+0.49\\varepsilon \\rbrace $ .",
"Clearly $\\mu ^*-0.99\\varepsilon \\le p^{\\prime } < \\mu ^* - 0.5\\varepsilon $ .",
"Consider the first time an arm with mean in the range $[p^{\\prime },\\mu ^*]$ arrives in the stream.",
"Let $T_1$ be the set of arms whose biases lie in the range $[p^{\\prime },\\mu ^*]$ and $T_2$ be the set of arms whose biases lie in the range $[\\mu ^*-0.49\\varepsilon ,\\mu ^*]$ .",
"Let $\\mathcal {A}_1$ be the event that the first arm $a_1$ from $T_1$ which arrives in the stream belongs to $T_2$ .",
"Let $\\mathcal {A}_2$ be the event that the first arm $a_1$ from $T_1$ which arrives in the stream does not belong to $T_2$ .",
"If $king^\\ell = \\mathtt {arm}_i$ and the event $\\mathcal {A}_1$ occurs then $a_1$ is returned as the king by our Algorithm at the end with a probability of at least $(1-\\delta /8)^3$ .",
"This happens because when the arm $a_1$ arrives, the king at that time has mean less than $\\mu ^*-0.99\\varepsilon $ with probability at least $(1-\\delta /8)$ , and due to Lemma REF , $a_1$ becomes the new king with probability at least $(1-\\delta /8)$ .",
"Due to Lemma REF , $a_1$ continues to remain as king with probability at least $(1-\\delta /8)$ .",
"Hence if the event $\\mathcal {A}_1$ occurs then $a_1$ is returned as the king by our Algorithm at the end with a probability of at least $(1-\\delta /8)^3$ .",
"Now assume that instead the event $\\mathcal {A}_2$ has occurred.",
"Let $\\mathcal {B}_{a_2}$ be the event that the first arm $a_2$ from the set $T_1$ to arrive in the stream has a mean $\\mu _{a_2}$ .",
"Let us assume that $\\mathcal {B}_{a_2}$ has occurred and $\\mu _{a_2}$ belongs to the range $[p^{\\prime }+0.01\\varepsilon ,\\mu ^*-0.5\\varepsilon )$ .",
"Due to Lemma REF , $a_2$ becomes the king with probability at least $(1-\\delta /8)$ .",
"If the means of the arms coming to stream after $a_2$ belongs to the range $[0,\\mu _{a_2}+0.49\\varepsilon ]$ , then $a_2$ continues to be the king with probability at least $(1-\\delta /8)$ .",
"If the first arm $a_3$ arriving in the stream with mean in the range $[\\mu _{a_2}+0.5\\varepsilon ,\\mu ^*]$ comes before the first arm $a_4$ arriving in the stream with bias in the range $(\\mu _{a_2}+0.49\\varepsilon ,\\mu _{a_2}+0.5\\varepsilon ]$ , then $a_3$ becomes the king with probability at least $(1-\\delta /8)$ and continues to remain as the king with probability at least $(1-\\delta /8)$ .",
"Let us denote this event of $a_3$ coming before $a_4$ by $\\mathcal {B}_1$ .",
"Note that if $X=1$ , then we can repeat the above analysis by considering $p^{\\prime } = \\mu ^*-0.99\\varepsilon $ .",
"Let $\\mathcal {C}_1$ be the event that $\\varepsilon $ -best arm is returned by the Algorithm and let $\\mathcal {C}_2$ be the event that an arm with mean in the range $[\\mu ^*-0.49\\varepsilon ,\\mu ^*]$ is returned by the algorithm.",
"$&\\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{28.45274pt}\\ge \\mathbb {P}[\\mathcal {C}_2|\\mathcal {A}_1,X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\cdot \\mathbb {P}[\\mathcal {A}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{42.67912pt}+ \\sum _{a_2:p_{a_2}\\in [p^{\\prime }+0.01\\varepsilon ,p-0.5\\varepsilon )}\\Big (\\mathbb {P}[\\mathcal {C}_2|\\mathcal {B}_1,\\mathcal {B}_{a_2},X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{42.67912pt}\\cdot \\mathbb {P}[\\mathcal {B}_1|\\mathcal {B}_{a_2},X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ] \\cdot \\mathbb {P}[\\mathcal {B}_{a_2}|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\Big )\\\\&\\hspace{28.45274pt} \\gtrapprox (1-\\delta /8)^3\\cdot \\frac{\\int _{p-0.49\\varepsilon }^{p} f(x)dx}{\\int _{p^{\\prime }}^{p} f(x)dx}+\\int _{\\min \\lbrace p^{\\prime }+0.01\\varepsilon ,p-0.5\\varepsilon \\rbrace }^{p-0.5\\varepsilon } (1-\\delta /8)^5 \\cdot \\Bigg (\\frac{\\int _{x+0.5\\varepsilon }^{p} f(x)dx}{\\int _{x+0.49\\varepsilon }^{p} f(x) dx}\\cdot \\frac{f(x)dx}{\\int _{p^{\\prime }}^{p} f(x)dx}\\Bigg )\\\\&\\hspace{28.45274pt}= (1-\\delta )\\cdot \\Bigg (\\frac{\\int _{\\min \\lbrace p^{\\prime }+0.01\\varepsilon ,p-0.5\\varepsilon \\rbrace }^{p-0.5\\varepsilon } f(x) \\frac{\\int _{x+0.5\\varepsilon }^{p} f(x)dx}{\\int _{x+0.49\\varepsilon }^{p} f(x) dx}dx}{\\int _{p^{\\prime }}^{p} f(x)dx}+\\frac{\\int _{p-0.49\\varepsilon }^{p} f(x)dx}{\\int _{p^{\\prime }}^{p} f(x)dx}\\Bigg ).\\\\$ Similarly, if $X=1$ then we have the following: $&\\mathbb {P}[\\mathcal {C}_1|X=1]\\gtrapprox (1-\\delta )\\cdot \\Bigg (\\frac{\\int _{p-0.98\\varepsilon }^{p-0.5\\varepsilon } f(x) \\frac{\\int _{x+0.5\\varepsilon }^{p} f(x)dx}{\\int _{x+0.49\\varepsilon }^{p} f(x) dx}dx}{\\int _{p-0.99\\varepsilon }^{p} f(x)dx}+\\frac{\\int _{p-0.49\\varepsilon }^{p} f(x)dx}{\\int _{p-0.99\\varepsilon }^{p} f(x)dx}\\Bigg ).\\\\$ $&\\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{28.45274pt}= \\sum _{i=1}^{\\ell }\\mathbb {P}[king^\\ell =c_i|X=\\ell +1,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\cdot \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{28.45274pt}=\\sum _{i=1}^{\\ell }q_{c_i}\\cdot \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\hspace{28.45274pt}\\ge \\min _{i\\in [\\ell ]} \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ].\\\\$ We have, $\\mathbb {P}[\\mathcal {C}_1|X=\\ell +1] &= \\sum _{(c_i)_{i\\in [\\ell ]} \\sim \\mathcal {S}_\\ell }\\mathbb {P}[t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace |X=\\ell +1]\\cdot \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\\\&\\ge \\min _{(c_i)_{i\\in [\\ell ]}\\sim \\mathcal {S}_\\ell }\\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ].\\\\$ $\\mathbb {P}[\\mathcal {C}_1] &= \\sum _{i\\in [k+1]}\\mathbb {P}[X=i]\\cdot \\mathbb {P}[\\mathcal {C}_1|X=i]\\\\& \\ge \\min _{i\\in [k+1]}\\mathbb {P}[\\mathcal {C}_1|X=i]\\\\&\\ge \\min \\Big \\lbrace \\mathbb {P}[\\mathcal {C}_1|X=1],\\min _{(c_i)_{i\\in [\\ell ]}\\sim \\mathcal {S}_\\ell }\\min _{i\\in [\\ell ]} \\mathbb {P}[\\mathcal {C}_1|X=\\ell +1,king^\\ell =c_i,t_\\ell =\\lbrace c_1,\\ldots ,c_\\ell \\rbrace ]\\Big \\rbrace \\\\&\\gtrapprox \\inf _{p^{\\prime }\\in [p-0.99\\varepsilon ,p-0.5\\varepsilon ]}\\Bigg \\lbrace (1-\\delta )\\cdot \\Bigg (\\frac{\\int _{\\min \\lbrace p^{\\prime }+0.01\\varepsilon ,p-0.5\\varepsilon \\rbrace }^{p-0.5\\varepsilon } f(x) \\frac{\\int _{x+0.5\\varepsilon }^{p} f(x)dx}{\\int _{x+0.49\\varepsilon }^{p} f(x) dx}dx}{\\int _{p^{\\prime }}^{p} f(x)dx}+\\frac{\\int _{p-0.49\\varepsilon }^{p} f(x)dx}{\\int _{p^{\\prime }}^{p} f(x)dx}\\Bigg )\\Bigg \\rbrace .$"
],
[
"Performance under various distributions",
"The following distributions are truncated distributions and the support is (0,1].",
"Note that the following calculations are made assuming $\\varepsilon =\\frac{1}{10}$ .",
"Lower bound on the probability that Algorithm returns an $\\varepsilon $ -best arm, for various truncated distributions like Normal, lognormal, exponential, beta, gamma, Weibull, and uniform distribution is at least $0.9(1-\\delta )$ .",
"Note that as $\\varepsilon $ tends to 0, the distributions mentioned earlier behave similar to uniform distribution on the range $[1-\\varepsilon ,1]$ .",
"So the lower bound of $\\mathbb {P}$ [Algorithm returns an $\\varepsilon $ -best arm] tends to $0.927(1-\\delta )$ which is the lower bound on this probability for the uniform distribution and it does not change with $\\varepsilon $ .",
"Table: NO_CAPTION"
],
[
"Experiments",
"We now provide detailed experimental evaluation of Algorithm and show that it returns an $\\varepsilon $ -best arm with high confidence even when we reduce the number of samples $(s_\\ell )$ per arm at each level $\\ell $ by a factor of 40000.",
"We ran the algorithm on $R=100$ different instances.",
"For each instance, the means of each of the $n=10^5$ arms were sampled from a distribution $\\mathcal {D}$ with support $(0,1]$ , mean $=\\mu $ , and variance $=\\sigma ^2$ .",
"Note that if $\\mathcal {D}$ is a truncated distribution then $\\mu $ and $\\sigma ^2$ denote the mean and variance of the non-truncated version of $\\mathcal {D}$ , denoted $\\mathcal {D}^\\prime $ .",
"Also we set $C=117, \\varepsilon ={1}/{10}, \\delta ={1}/{10}$ .",
"For $i \\in [n]$ , let $\\mu _i$ be the mean obtained for $\\mathtt {arm}_i$ .",
"The reward distribution of $\\mathtt {arm}_i$ is then Bernoulli$(\\mu _i)$ .",
"We next provide the details of the distributions corresponding to the figures in Figure REF .",
"Note that we consider the truncated version, $\\mathcal {D}$ , of the following distributions, $\\mathcal {D}^\\prime $ , supported on $(0,1]$ .",
"Figure (REF ): $\\mathcal {D}^\\prime = \\text{Beta}(\\alpha = 10, \\beta = 1)$ Figure (REF ): $\\mathcal {D}^\\prime = \\text{Exp}(\\lambda = 2)$ Figure (REF ): $\\mathcal {D}^\\prime = \\text{Gamma}(k = 0.5,\\theta = 1)$ Figure (REF ): $\\mathcal {D}^\\prime = \\mathcal {N}(\\mu =1/2,\\sigma ^2=1/10)$ Figure (REF ): $\\mathcal {D}^\\prime = \\mathcal {N}(\\mu =1/2,\\sigma ^2=1/5)$ Figure (REF ): $\\mathcal {D}^\\prime = \\mathcal {N}(\\mu =1/2,\\sigma ^2=1/100)$ Figure (REF ): $\\mathcal {D}^\\prime = \\mathcal {N}(\\mu =1/2,\\sigma ^2=1/20)$ Figure (REF ): $\\mathcal {D}^\\prime = \\text{lognormal}(\\mu =0,\\sigma ^2=3/2)$ Figure (REF ): $\\mathcal {D}^\\prime = \\text{Weibull}(\\lambda = 1, k = 2)$ In all these cases, we almost always return an arm with mean within at most 0.05 ($<1/10=\\varepsilon $ ) from the mean of the best-arm.",
"Figure: X-axis: Gap between the means of the best arm and arm returned by Algorithm , Y-axis: Count of such arms."
]
] | 2012.05142 |
[
[
"Universal $L^{-3}$ finite-size effects in the viscoelasticity of\n confined amorphous systems"
],
[
"Abstract We present a theory of viscoelasticity of amorphous media, which takes into account the effects of confinement along one of three spatial dimensions.",
"The framework is based on the nonaffine extension of lattice dynamics to amorphous systems, or nonaffine response theory.",
"The size effects due to the confinement are taken into account via the nonaffine part of the shear storage modulus $G'$.",
"The nonaffine contribution is written as a sum over modes in $k$-space.",
"With a rigorous argument based on the analysis of the $k$-space integral over modes, it is shown that the confinement size $L$ in one spatial dimension, e.g.",
"the $z$ axis, leads to a infrared cut-off for the modes contributing to the nonaffine (softening) correction to the modulus that scales as $L^{-3}$.",
"Corrections for finite sample size $D$ in the two perpendicular dimensions scale as $\\sim (L/D)^4$, and are negligible for $L \\ll D$.",
"For liquids it is predicted that $G'\\sim L^{-3}$ in agreement with a previous more approximate analysis, whereas for amorphous materials $G' \\sim G'_{bulk} + \\beta L^{-3}$.",
"For the case of liquids, four different experimental systems are shown to be very well described by the $L^{-3}$ law."
],
[
"Introduction",
"Lattice dynamics can be extended to deal with disordered systems where the positions of atoms or molecules are completely random, to arrive at theoretical expressions for the elastic constants and for the viscoleastic moduli[1], [2], [3], [4].",
"The resulting theoretical framework is sometimes referred to as nonaffine lattice dynamics or NALD[1], [3].",
"The theory has proved effective in quantitatively describing elastic, viscoelastic and plastic response of systems as diverse as jammed random packings and random networks[2], glassy polymers[4], [5], [6], metallic glass[7], colloidal glasses[8], and non-centrosymmetric crystals like quartz [9].",
"Furthermore, NALD intrinsically takes into account long-range correlation phenomena[10], [11] that are present also in liquids and give rise to acoustic wave propagation.",
"Because of its microscopic character, and to its ability to represent contributions to elasticity in terms of eigenmodes of the Hessian or dynamical matrix of the systems, NALD is thus a promising framework to describe size-dependent effects due to confinement.",
"Understanding these effects at the microscopic level is important for a wide variety of systems in condensed matter and materials physics [12], [13], polymers [14], [15], [16], [17], and amorphous and glassy systems [18], [19].",
"In this paper, we present a detailed analysis of size-dependent effects on the viscoelastic shear modulus of amorphous systems confined in one spatial dimension, including liquids and glasses.",
"We evaluate the nonaffine integral over $k$ -space generally by allowing the “infrared” limit to vary with polar angle $\\theta $ and hence evaluate the scaling properties of the nonaffine contribution to the shear storage modulus.",
"An analysis of four published experimental data sets shows that this scaling law is shared by many different systems, and remains valid for arbitrary chemical composition and microscopic or mesoscopic structure of the system.",
"Although we focus on linear viscoelasticity, these results could be useful also for understanding of plasticity of confined systems [20], and also for understanding mechanical fragmentation processes in dispersed, colloidal and biological systems, where mesoscopic aggregates display size-dependent mechanical properties [21]."
],
[
"Nonaffine viscoelastic theory",
"The usual starting point is the equation of motion of a microscopic building block, i.e.",
"an atom or a molecule for atomic liquids or molecular liquids, respectively.",
"In the case of polymers, the building block could be identified with a monomer of the polymer chain[4].",
"Following previous literature [1], [2], we introduce the Hessian matrix of the system $\\underline{\\underline{H}}_{ij}=-\\partial ^2\\mathcal {U}/\\partial \\underline{\\mathring{q}}_i\\partial \\underline{\\mathring{q}}_j$ and the affine force field $\\underline{\\Xi }_{i,\\kappa \\chi }=\\partial \\underline{f}_i/\\partial \\eta _{\\kappa \\chi }$ , where $\\eta _{\\kappa \\chi }$ is the strain tensor.",
"For example, for simple shear deformation the $xy$ entry of tensor $\\eta _{\\kappa \\chi }$ is given by a scalar $\\gamma $ , which coincides with the angle of deformation.",
"As shown in previous works [1], [4], the equation of motion of an atom $i$ in a disordered medium subjected to an external strain, in mass-rescaled coordinates, can be written as: $\\frac{d^2\\underline{x}_i}{dt^2}+\\nu \\frac{d\\underline{x}_i}{dt}+\\underline{\\underline{H}}_{ij}\\underline{x}_j=\\underline{\\Xi }_{i,\\kappa \\chi }\\eta _{\\kappa \\chi } $ where $\\underline{\\underline{\\eta }}$ is the (Green-Saint Venant) strain tensor and $\\nu $ is a microscopic friction coefficient which arises from dynamical couplings mediated by the anharmonicity of the pair potential.",
"The term on the r.h.s.",
"physically represents the effect of the disordered (non-centrosymmetric) environment leading to nonaffine motions: a net force acts on the atom $i$ in the affine position (i.e.",
"the position prescribed by the external strain tensor $\\eta _{\\kappa \\chi }$ ).",
"In a disordered or non-centrosymmetric bonding environment, in order to keep mechanical equilibrium on all atoms throughout the deformation, an additional nonaffine displacement is required in order to relax the force $f_{i}$ acting in the affine position.",
"This displacement brings each atom $i$ to a new (nonaffine) position.",
"The equation of motion Eq.",
"(REF ) can also be derived from first principles, from a model particle-bath Hamiltonian as shown in previous work [4].",
"Using standard manipulations (Fourier transformation and eigenmode decomposition from time to eigenfrequency [1]), and applying the definition of mechanical stress as derivative of the energy, one obtains the following expression for the viscoelastic (complex) elastic constants[1], [4]: $C_{\\alpha \\beta \\kappa \\chi }(\\omega )=C_{\\alpha \\beta \\kappa \\chi }^{\\textit {Born}}-\\frac{1}{V}\\sum _n\\frac{\\hat{\\Xi }_{n,\\alpha \\beta }\\hat{\\Xi }_{n,\\kappa \\chi }}{\\omega _{p,n}^2-\\omega ^2+i\\omega \\nu } $ where $C_{\\alpha \\beta \\kappa \\chi }^{\\textit {Born}}$ is the Born or affine part of the elastic constant, which is what survives in the infinite-frequency limit.",
"Here, $\\omega $ represents the oscillation frequency of the external strain field, whereas $\\omega _p$ denotes the internal eigenfrequency of the liquid (which results, e.g., from diagonalization of the Hessian matrix [4]).",
"We use the notation $\\omega _{p}$ to differentiate the eigenfrequency from the external oscillation frequency $\\omega $ .",
"An atomistic expression for $G_{\\infty } \\equiv C_{xyxy}^{\\textit {Born}}$ is provided by the well known Zwanzig-Mountain (ZM) formula [22], in terms of the pair potential $V(r)$ and the radial distribution function $g(r)$ .",
"The sum over $n$ in Eq.",
"(REF ) runs over all $3N$ degrees of freedom (given by the atomic or molecular building blocks with central-force interactions).",
"Also, we recognize the typical form of a Green's function, with an imaginary part given by damping and poles $\\omega _{p,n}$ that correspond to the eigenfrequencies of the excitations.",
"At this point, we consider the dynamics of elastic waves in liquids.",
"The propagation of longitudinal acoustic waves in liquids is of course a well known fact, with firmly established both experimental and theoretical evidence of longitudinal acoustic dispersion relations[23], [24], [25].",
"For transverse or shear acoustic waves in liquids, instead, there is no propagation below a characteristic wavenumber.",
"Indeed, there is an onset value of $k$ , that we shall denote $k_{g}$ , above which these modes can propagate in liquids.",
"This represents a gapped momentum state seen in a number of different systems, including liquids, supercritical fluids, plasma, Keldysh-Schwinger theory, relativistic hydrodynamics, holographic and other models such as the sine-Gordon model[26].",
"The gap increases with temperature and the inverse of liquid relaxation time (see, e.g., Refs.",
"[27], [28]).",
"Following the analytical steps presented in Ref.",
"PNAS2020, we arrive at the following expression for the frequency-dependent storage modulus $G^{\\prime }$ , $G^{*}(\\omega )=&G_{\\infty }-B \\int _{k_{\\text{min}}}^{k_{D}}\\frac{\\omega _{p,L}^{2}(k)}{\\omega _{p,L}^2(k)-\\omega ^2+i\\omega \\,\\nu }k^{2}dk\\nonumber \\\\& - B\\int _{k_{\\text{min}}}^{k_{D}}\\frac{\\omega _{p,T}^{2}(k)}{\\omega _{p,T}^2(k)-\\omega ^2+i\\omega \\, \\nu }k^{2}dk\\,, $ where the first integral represents the nonaffine (negative or softening) contribution due to longitudinal (L) acoustic modes, while the second integral represents the nonaffine (also softening) contribution due to the transverse (T) acoustic modes.",
"In the above expression, $k_{\\text{min}}$ is an “infrared” cutoff, which is $k_{\\text{min}}=0$ for a standard bulk material, which can be considered as large in all spatial dimensions ($L=\\infty $ ).",
"$B$ is an arbitrary prefactor.",
"For liquids, $k_{\\text{min}}=\\max \\left(k_g,k_{\\text{conf}}\\right)$ , for the transverse modes, with $k_g$ the onset wavenumber for transverse phonons in liquids (the $k$ -gap), and $k_{\\text{conf}}$ is the wavenumber set by the confinement length (see below).",
"Upon taking the real part of $G^{*}$ , which gives the storage modulus $G^{\\prime }$ , and focusing on low external oscillation frequencies $\\omega \\ll \\omega _{p}$ , in both integrals numerator and denominator cancel out, so that both integrals reduce to the same expression, a volume in $k$ -space.",
"Therefore, as anticipated above, the final low-frequency result does not depend on the actual form of $\\omega _{p,L}(k)$ , nor of $\\omega _{p,T}(k)$ , although the latter, in liquids, due to the $k$ -gap, plays an important role.",
"In the experiments where the size effect of confinement is seen [30], $k_g\\ll \\frac{1}{L}$ .",
"Assuming that $k_{min}\\approx \\frac{1}{L}$ , an approximation that we further expand on in the following section, we have $G^{\\prime }= G_{\\infty } - \\alpha \\int _{1/L}^{k_{D}}k^{2}dk =G_{\\infty } - \\frac{\\alpha }{3} k_{D}^{3} + \\frac{\\beta }{3} L^{-3}.",
"$ For bulk (unconfined) liquids in thermodynamic equilibrium, it can be shown [31] that $G_{\\infty } - \\frac{\\alpha }{3} k_{D}^{3}=0$ , thus leaving: $G^{\\prime } = \\beta ^{\\prime } L^{-3}.$ For amorphous solids, instead, $G_{\\infty } - \\frac{\\alpha }{3} k_{D}^{3} > 0$ , and one has the final scaling on $L$ given by $G^{\\prime } = G^{\\prime }_\\text{bulk}+\\beta ^{\\prime } L^{-3},$ where $G^{\\prime }_\\text{bulk}$ is the value of shear modulus for unconfined, bulk samples.",
"In equations (REF –REF ), $\\alpha $ , $\\beta $ , and $\\beta ^{\\prime }$ are arbitrary prefactors.",
"Evidently, the scaling $L^{-3}$ is easier to observe in liquids, as in amorphous solids it may be overshadowed by noise.",
"Nonetheless, it is important to present the theoretical prediction also for amorphous solids, as it may be verified experimentally or in simulations in future work.",
"Figure: (a) Schematic section in real space of the confined cylindrical sample.",
"(b) Geometry of the different regions over which the kk-space integral () can be taken; see full explanation in text.",
"This is not to scale; in fact k D ≫2π/Lk_D\\gg 2\\pi /L.",
"Both parts of this diagram have full rotational symmetry about the zz axis.Figure: 3D rendering of the geometry of integration in kk-space for the confined system of Fig.",
"."
],
[
"General proof of the $L^{-3}$ law",
"We consider a cylindrical system confined to length $L$ in the $z$ direction; for now we allow its extent in the perpendicular directions (that is, the cylinder's diameter) to be infinite.",
"We use spherical polar coordinates, measuring the polar angle $\\theta $ from the $z$ axis (Fig.",
"REF (a)).",
"Since our system has cylindrical symmetry, no quantities depend on the azimuthal angle $\\phi $ and its origin is therefore arbitrary.",
"In this notation the volume element in $k$ -space is $\\mathrm {d}V_k = k^2\\mathrm {d}k\\,\\sin \\theta \\mathrm {d}\\theta \\,\\mathrm {d}\\phi $ .",
"If an integrand does not depend on $\\theta $ or $\\phi $ , then $\\mathrm {d}V_k = 4\\pi k^2\\mathrm {d}k$ , demonstrating that the integrals in (REF ) represent, to within a constant factor that can be absorbed into the prefactor, a volume in $k$ -space.",
"If the system were unconfined, the lower limit on $k$ would simply be zero.",
"Thus the region of allowable states would be a sphere in $k$ -space, with radius equal to the Debye wavevector $k_D$ and hence volume $\\tfrac{4}{3}\\pi k_D^3$ .",
"[32], [33], [34], [35] In our confined system, however, the maximum possible wavelength in the $z$ direction is approximately $\\lambda _\\text{max}\\approx L$ , giving a minimum wavevector of $k_\\text{min}\\approx 2\\pi /L$ .",
"In our previous analysis,[29], we made the simplifying approximation that the lower (“infrared”) limit of the $k$ -space integral (REF ) is $k_\\text{min}$ regardless of the direction of propagation of the wave.",
"In this approximation, the lower limit is a spherical surface in $k$ -space with radius $2\\pi /L$ , so that the integral should be taken over the pink narrow hatched volume in Fig.",
"REF (b).",
"Here we relax that assumption, showing that the $L^{-3}$ scaling holds even if we allow the lower limit to vary with $\\theta $ .",
"If measured at an angle $\\theta $ from the $z$ confinement axis, the extent of the confined medium is $L/\\cos \\theta $ (Fig.",
"REF (a)).",
"Taking this value, as before, to be the maximum allowed wavelength in that direction, we now have $k_\\text{max} = 2\\pi \\cos \\theta /L$ .",
"In the range $0\\le \\theta \\le \\pi $ , this equation describes two spheres with radius $\\pi /L$ , centred at $(0,0,\\pm \\pi /L)$ in $k$ -space.",
"The integral (REF ) must now be taken over the wide blue hatched volume in Fig.",
"REF (b).",
"A 3D rendered version of the same geometry is presented in Fig.",
"REF .",
"The volume of the two small spheres is $V_{k,\\text{min}} = 2\\times \\tfrac{4}{3}\\pi \\left(\\frac{\\pi }{L}\\right)^3 = \\frac{8\\,\\pi ^4}{3\\,L^3}.$ The allowable volume in $k$ -space is therefore $V_k = \\tfrac{4}{3}\\,\\pi \\, k_D^3 - \\tfrac{8}{3}\\,\\pi ^4\\,L^{-3}\\,,$ displaying the same $L^{-3}$ scaling as derived previously.",
"Figure: The same construction as Fig.",
", but allowing for the finite diameter DD of the cylinder.",
"Here D/L=4D/L=4.",
"Compared to Fig.",
", (b) has been enlarged for clarity.We can extend this analysis still further by considering the effects of the finite cylinder diameter $D\\gg L$ (Fig.",
"REF ).",
"The analysis proceeds as before except that the extent is now limited by $D$ rather than $L$ near $\\theta =\\pi /2$ : $k_\\text{min} ={\\left\\lbrace \\begin{array}{ll}2\\pi \\cos \\theta /L & \\vert \\tan \\theta \\vert \\le D/L \\\\2\\pi \\sin \\theta /D & \\vert \\tan \\theta \\vert \\ge D/L\\end{array}\\right.",
"}.$ In this case, the infrared limit surface is the intersection of the two spheres and a toroidal shape.",
"The internal volume is $\\begin{split}V_{k,\\text{min}} &= 2\\times 2\\pi \\int _0^{\\tan ^{-1}(D/L)}\\int _0^{2\\pi \\cos \\theta /L}k^2\\mathrm {d}k\\,\\sin \\theta \\mathrm {d}\\theta \\\\&\\qquad + 2\\times 2\\pi \\int _{\\tan ^{-1}(D/L)}^{\\pi /2}\\int _0^{2\\pi \\sin \\theta /D}k^2\\mathrm {d}k\\,\\sin \\theta \\mathrm {d}\\theta \\\\&= \\frac{8\\pi ^4}{3L^3}\\bigg (1 - \\frac{1 + L^2/D^2}{2(1 + D^2/L^2)^2} + \\frac{2L^2/D^2}{1 + D^2/L^2}\\\\&\\qquad + \\frac{3L^2}{2D^2}\\left[\\frac{\\pi }{2} - \\tan ^{-1}\\left(\\frac{D}{L}\\right)\\right]\\bigg )\\\\&= \\frac{8\\pi ^4}{3L^3}\\Big (1 + 3(L/D)^4 + \\mathcal {O}\\big ((L/D)^6\\big )\\Big ).\\end{split}$ As expected, this recovers the previous result (REF ) in the limit as $D$ tends to infinity.",
"Furthermore, it approaches this limit rather quickly, with the difference term being fourth-order in the aspect ratio $L/D$ .",
"In typical experiments, $L\\approx 0.1D$ , so that the difference from (REF ) is negligible within experimental error.",
"We conclude that the $G^{\\prime } \\sim L^{-3}$ scaling presented above in (REF ) is robust in two senses.",
"First, it does not depend on the simplifying assumption previously made in Ref. PNAS2020.",
"Second, the correction term to allow for finite system size in the non-confined direction scales as the fourth power of the aspect ratio, making this correction negligible for typical experimental conditions where confinement is along the $z$ axis only."
],
[
"Discussion and comparison to experiments",
"The above theory clarifies that the confinement between two plates is able to “remove” certain low-frequency normal mode collective oscillations of molecules, associated with the nonaffine motions (i.e.",
"negative contributions to the elasticity), which are otherwise responsible for the fluid response of liquids under standard macroscopic (“unconfined”) conditions.",
"These nonaffine motions are directly responsible for reducing the shear modulus, basically to zero in macroscopic liquids and to $G^{\\prime }_\\text{bulk}$ in amorphous solids.",
"Under confinement, instead, the shear modulus becomes non-zero for liquids, because these collective oscillations modes are suppressed, and the theory we (A.Z.",
"and K.T.)",
"have recently reported[29] provides the law by which the shear modulus grows upon reducing the confinement size $L$ .",
"In particular, the static shear modulus grows with the inverse cubic power of the confinement size $L$ .",
"For amorphous solids, the bulk shear modulus acquires an additional positive contribution $\\sim L^{-3}$ , due to confinement.",
"Figure: Experimental data of low-frequency shear modulus G ' G^{\\prime } versus confinement length LL for different systems: (a) PAOCH 3 _3, an isotropic liquid crystal; (b) an ionic liquid; (c) short-chain (non-entangled) polystyrene melts; (d) nanoconfined water.",
"Circles represent experimental data while the solid line is the law G ' ∼L -3 G^{\\prime } \\sim L^{-3}, with a prefactor determined by fitting to the data.In Ref.",
"PNAS2020, the law $G^{\\prime } \\sim L^{-3}$ was found to provide a good description of experimental data of the isotropic liquid crystal PAOCH$_3$ upon varying the confinement $L$ , using a conventional rheometer.",
"Given our conclusion in the previous section that this result should be robust across a wide range of media and experimental conditions, we now extend our comparison to more experimental systems.",
"In Fig.",
"REF , we show the fits of this scaling law to three more experimental data sets.",
"We observe that the scaling law agrees well with experiments performed within short chain polymers, isotropic liquid crystals, ionic liquids as well as nano-confined water."
],
[
"Conclusions",
"In summary, we presented a microscopic theoretical framework for the size-dependent viscoelasticity of confined amorphous systems, both liquids and solids.",
"For the case of liquids, a previous approximate treatment [29] has unveiled the surprising solid-like response under confinement, where the confinement effectively cuts off some nonaffine softening modes, leading to the scaling $G^{\\prime } \\sim L^{-3}$ for the low-frequency shear modulus.",
"In that earlier description, the integral over $k$ -space, which provides the negative nonaffine correction, was evaluated approximately assuming that waves in any direction have the same maximal wavelength.",
"Here, we presented a rigorous and general proof of the same result that takes the full $k$ -space geometry of the problem into account, allowing the maximum wavelength to vary with the polar angle $\\theta $ .",
"Our analysis shows that the $G^{\\prime } \\sim L^{-3}$ law still holds when the initial approximation is relaxed.",
"Furthermore, it is extremely robust with respect to finite sample size in the two perpendicular directions.",
"These results are supported by an analysis of experimental data from the literature on four different liquids and complex fluids, all of which obey the $G^{\\prime } \\sim L^{-3}$ law.",
"We also derived a similar law for amorphous solids, with a predicted confinement-induced enhancement term in the low-frequency shear modulus that also scales with $L^{-3}$ .",
"This correction for amorphous solids is probably more challenging to verify, either experimentally or in simulations, but it may inspire further investigations.",
"Finally, it would be interesting in future work to study the interplay between confinement or boundary effects like those presented here and other low-$k$ phenomena in condensed matter such as hyperuniformity [40] and its ramifications [41].",
"Also, our $\\sim L^{-3}$ law for the shear modulus, derived from the viewpoint of nonaffinity, may connect to the $1/N$ (where $N \\sim L^{3}$ is the number of particles) finite-size correction to the shear modulus observed near the jamming transition of random jammed packings [42], [43], [44], a connection that should be explored in future work.",
"M.B.",
"acknowledges the support of the Shanghai Municipal Science and Technology Major Project (Grant No.2019SHZDZX01) and of the Spanish MINECO ”Centro de Excelencia Severo Ochoa” Programme under grant SEV-2012-0249.",
"C.S.",
"is supported by the U.S. DOE grant number DE-FG02-05ER46236.",
"A.Z.",
"acknowledges financial support from US Army Research Laboratory and US Army Research Office through contract nr.",
"W911NF-19-2-0055."
]
] | 2012.05149 |
[
[
"Optimising cost vs accuracy of decentralised analytics in fog computing\n environments"
],
[
"Abstract The exponential growth of devices and data at the edges of the Internet is rising scalability and privacy concerns on approaches based exclusively on remote cloud platforms.",
"Data gravity, a fundamental concept in Fog Computing, points towards decentralisation of computation for data analysis, as a viable alternative to address those concerns.",
"Decentralising AI tasks on several cooperative devices means identifying the optimal set of locations or Collection Points (CP for short) to use, in the continuum between full centralisation (i.e., all data on a single device) and full decentralisation (i.e., data on source locations).",
"We propose an analytical framework able to find the optimal operating point in this continuum, linking the accuracy of the learning task with the corresponding network and computational cost for moving data and running the distributed training at the CPs.",
"We show through simulations that the model accurately predicts the optimal trade-off, quite often an intermediate point between full centralisation and full decentralisation, showing also a significant cost saving w.r.t.",
"both of them.",
"Finally, the analytical model admits closed-form or numeric solutions, making it not only a performance evaluation instrument but also a design tool to configure a given distributed learning task optimally before its deployment."
],
[
"Introduction",
"We are facing exponential growth in the number of personal mobile and IoT devices at the edge of the Internet.",
"According to Cisco [1] the number of connected devices will exceed by three times the global population by 2023.",
"Most importantly, this is a trend that, apparently, not only will not slow down in the near future, but it also will be the cause of the so-called “data tsunami”, i.e., the explosion of the amount of data generated by these devices at the edge of the Internet.",
"Ericsson [2] foresees that global mobile data traffic will grow by almost a factor of 5 to reach 164EB per month in 2025.",
"Most of the value of data consists in the possibility of being processed and analyzed to extract useful knowledge out of them.",
"For example, this applies to many relevant cases of Industry 4.0 and Smart Cities where, in many applications, raw data are of little use, while the real value comes from the knowledge extracted through AI and Big Data Analytics.",
"The current approach for extracting knowledge from data is to centralize them to remote data centres, as many IoT architectures demonstrate [3].",
"This is the case, among others, of the ETSI M2M architecture [4] where data are transferred from the physical locations where they are generated to some global cloud platform to be processed.",
"Such an approach might not be sustainable in the long run because, despite the evolution of the mobile networks, their capacity is growing only linearly [1], which makes it impractical or simply impossible to transfer all the data to a remote cloud platform at reasonable costs.",
"Furthermore, data might also have privacy and confidentiality constraints, which might make it impossible to transfer them to third parties such as global cloud platform operators.",
"There are several scenarios where these constraints are relevant.",
"One of the most important are applications in the Industry 4.0 (or Industrial Internet) area, where data analytics is one of the cornerstones [5].",
"However, companies might have severe concerns in moving their data to some external cloud provider infrastructure due to confidentiality reasons.",
"On the other hand, they might not have the competences and resources to build and manage a private cloud platform.",
"Moreover, real-time delay constraints might require that data elaboration or storage is performed at the edge, i.e., close to where data is needed, rather than in remote data centres.",
"These trends and needs push towards a decentralization of data analytics approaches towards the edge of the network, where the paradigms of edge computing, such as Fog Computing [6], Multi-access Edge Computing [7], Cloudlets [8], can address the aforementioned problemsWhile there are architectural differences between these paradigms and in particularly between edge and fog computing, in the following we use the terms interchangeably, as those differences do not impact on the focus of the paper.",
"Luckily, many machine learning (ML) algorithms at the basis of fundamental data analytics tasks such as classification, regression, and clustering admit distributed formulations, which can be used to implement decentralised data analytics.",
"The main idea of distributed ML algorithms is to derive local models from partial datasets at several locations, and then exchange information across locations to refine the partial models.Clearly, distributed ML assumes a certain degree of collaboration between nodes.",
"This might be guaranteed either because nodes may be under the same controlling entity (e.g., the nodes inside a factory), or more in general through appropriate incentive schemes (e.g., for crowdsensing applications [9], [10], [11]).",
"This is an orthogonal problem concerning the focus of this paper.",
"In this paper, we tackle the issue of optimal configuration of such distributed ML algorithms.",
"Specifically, given a set of nodes generating data, it is possible to identify a whole range of configurations, from fully centralised to fully decentralised, for a given ML algorithm.",
"Each configuration is characterized by the set of collection points where data are collected (possibly from more nodes generating them), and partial model computed (and refined via collaboration).",
"As we will discuss in detail in the paper, given a target accuracy for the learning task, each configuration is characterized by (i) a network cost, required to move data to the collection points and exchange information for collaborative learning, and (ii) a computational cost at the collection points.",
"The key contribution of this paper is to provide an analytical model that identifies the optimal operating point, i.e., the optimal set of collection points to be used.",
"More in detail, the contribution of the paper is as follows.",
"As explained in Section , it is known that, by exchanging partial models updates between collection points and through multiple rounds of training on the local (partial) datasets, decentralised ML algorithms can achieve any target accuracy.",
"However, this comes at a cost in terms of network traffic and computation.",
"We define appropriate cost functions for both classes of costs in Section .",
"Thus, we develop an analytical model that, for any given number of collection points, provides the total cost of achieving a target accuracy.",
"The model takes into consideration (i) the number of collection points, (ii) the amount of data at each collection point, (iii) the cost for transferring data on the network (iv) the cost for processing data.",
"Based on the model, we are finally able to obtain the optimal operating point for a target accuracy, i.e., the optimal number of collection points such that the target accuracy is achieved at the minimal total cost.",
"We extensively validate the model through simulations.",
"Specifically, we compare the predictions of our model with the optimal operational point (obtained from exhaustive search) for different types of computational cost functions.",
"We show that the model is in general very accurate in predicting cost of training a decentralised ML algorithm, and identifying the optimal operating point.",
"The model shows that the optimal operating point is almost always at an intermediate aggregation level, between full centralisation and full decentralisation.",
"Moreover, it also shows that a significant additional cost is paid when the ML algorithm works in the simplest fully-decentralised and fully-centralised configurations.",
"This justifies the use of analytical tools like the one proposed in the paper to optimally configure decentralised ML operations.",
"In fact, our model admits numerical and, in some cases, closed-form expressions for the optimal operating point, which are quite efficient to compute.",
"Therefore, it can be used as a design tool to configure a system based on decentralised ML.",
"The key take-home messages of this paper are: “decentralisation helps”, since the optimal operating point is, in almost all cases, an intermediate point between full centralisation and decentralisation; “modelling tools help\", since operating a decentralised ML at optimal operating point reduces quite significantly the cost with respect to more naive solutions; the shape of the compute cost function plays a significant role in determining the optimal operating point.",
"This means it can be tuned by edge service providers to drive the usage of their infrastructure according to specific policies; the relative costs of communication vs. computing may push the optimal operating point either towards more or less decentralisation.",
"The rest of the paper is organised as follows.",
"In Section we review the related literature.",
"In Section , we present the reference scenario, and we define the general structure of the cost model.",
"In Section we present the distributed learning algorithm considered in the paper, while in Section we provide the mathematical formulation of its cost, for a given accuracy.",
"In Section , we validate the model through simulations, and present a comparative analysis of the performance at the optimal operating point, as well as a sensitivity analysis of the optimal operating point against key parameters.",
"Finally, Section concludes the paper."
],
[
"Related work",
"The execution of machine learning tasks at the edge, in the literature, is considered from several perspectives.",
"A very important body of work deals with distributed learning algorithms that are suitable for being executed in fog scenarios.",
"This is the case of the Federated Learning Framework initially proposed by [12], [13].",
"According to this framework, several devices coordinated by a central entity (i.e., a parameter server) and holding some local data, collaboratively train a global model (e.g., an artificial neural network) on the local data.",
"During the process, the information exchanged between the device are only models' updates and other related information, but never raw data.",
"Federated Learning is an iterative procedure spanning over several communication rounds until convergence is reached.",
"Based on this paradigm, several modifications have been proposed concerning (i) new distributed optimisation algorithms [14], [15], [16], [17] and (ii) privacy-preserving methods for federated learning [18], [19].",
"Alternatively, other approaches do not rely on a centralised coordinating server.",
"In [10], [11] authors propose a distributed and decentralised learning approach based on Hypothesis Transfer Learning (HTL).",
"Similarly to the Federated Learning framework, authors assume that several devices hold a portion of a dataset to be analysed by some distributed machine learning algorithm.",
"The aim of [10], [11] is to provide a learning procedure able to train, in a decentralised way, an accurate model while drastically limiting the network traffic generated by the learning process.",
"According to this approach, each device trains a local model (using the local data), which is then shared with the other devices.",
"Each device exploits an HTL algorithm to combine the models received by the other devices in order to improve its local model.",
"After a number of communication rounds, all the devices have an accurate global model which includes the knowledge contained in all the local datasets held by the devices.",
"Other decentralised learning mechanisms propose methods based on gossiping [20], [21], [22], where the devices exchange the models updates only with their neighbourhood.",
"The main focus of these papers is on the convergence properties of these distributed algorithms, taking into account different kinds connectivity, i.e., fixed vs. random, between the devices involved.",
"Another body of work focuses on the coordination/orchestration of the learning process at the edge.",
"Specifically, authors of [23], [24] address the problem of offloading the computation needed to train a learning model from mobile devices to some edge/fog/cloud server.",
"Authors of [25] presents an edge solution in which the edge server coordinates the collaboration of several mobile devices that have to train models for object recognition.",
"The closest work to the one proposed in this paper is presented in [9] where authors studied if there exists a trade-off between the accuracy and the network performance connected to the execution of a decentralised analytics task in IoT scenarios.",
"Although the methodology followed in [9] is mostly empirical, the paper shows that there could exist such type of trade-off and that, once found, it is possible to optimise the configuration of the distributed learning process to reduce the related network cost significantly.",
"Another interesting solution with some similarity with this paper (i.e., the idea of using the convergence bounds of a learning algorithm) is presented in [26].",
"Authors propose an adaptive federated learning mechanism suitable for resource-constrained edge systems.",
"The solution exploits the convergence bounds of the learning algorithm to design a control algorithm that adapts the number of local model's updates during training and the number of communication rounds in order to be efficient.",
"A similar solution is presented in [27] where authors propose a control algorithm for optimising the trade-off between execution time and energy spent during the execution of a federated learning algorithm.",
"In this paper, which extends our prior work in [28], we neither propose a decentralised learning procedure suitable for fog scenarios, nor a control algorithm for optimising the execution of such a procedure.",
"Conversely, we address the problem of identifying, through an analytical tool, the optimal configuration of a broad class of decentralised learning algorithms, trough which a number of fog devices can (i) aggregate a certain amount of data and (ii) run a specific decentralised learning algorithm on them.",
"Differently from [28], in our model we take into account, at the same time, both the network traffic generated by the data collection, the communications triggered by the distributed learning algorithm and the computational cost associated to the process.",
"We solve the model both numerically and analytically (when possible), showing that it also admits closed-form expressions of the optimal aggregation level in certain cases.",
"We then analyse the accuracy of the model predictions compared to the optimal points found via simulations with exhaustive search over all possible configurations.",
"We compare the cost penalty paid when configuring the ML task, instead of at the optimal point identified by our model, at the two simple extreme configurations, fully centralised and fully decentralised, Finally, we study, via the model, the sensitiveness of the optimal operating point with respect to the values of its key parameters.",
"To the best of our knowledge, this is the first time that such an analytical framework is proposed in the literature.",
"A key distinguishing feature of our work is that our methodology can be applied to any distributed learning algorithm for which it is possible to find an analytical expression for the communication and the computation needed to converge to a fixed accuracy."
],
[
"Problem definition",
"We consider a scenario (represented in Fig.",
"REF ) in which there are $m_0$ devices (e.g., high-end sensors of workers' devices in a smart factory, handheld devices in a smart city, but also more powerful devices such as edge gateways) collecting data in their local storage.",
"In our analysis, we assume that each device collects, on average, a certain amount of data $n_0$ , each made up of d features of equal sizeThis feature model is grounded in the fact that, for example, smart devices are equipped with a number of sensors, whose readings can be considered together to enrich the extractable knowledge..",
"Therefore, the average total amount of data collected by all the devices is $N=m_0n_0d$ .",
"Although this is a simplifying assumption taken for pure modelling reasons, it does not affect the generality of our results, as shown by the validation results presented in Section .",
"Table: Main notation used in the paper.Moreover, we assume that all the devices in the system involved in the learning process are computationally capable of processing the amount of data they hold.",
"We expect that even devices such as Raspberry PIs would belong to this class.",
"Low-end devices (such as low-end sensors) involved in the data collection process are assumed to relay their data to more capable devices, where the distributed learning process can be performed.",
"Figure: Reference scenario.In our model, the data analytics process can be executed on a number $m_1$ of collection points (CPs) in the range $[1,m_0]$ , where $m_1=1$ represents the fully centralised case and $m_1=m_0$ the fully decentralised one.",
"The average number $n_1$ of data points collected on the $m_1$ CPs is calculated as $n_1=\\frac{n_0m_0}{m_1}$ .",
"Our model provides, for a given fixed target accuracy of the learning task, the optimal number of collection points, i.e., the number of collection points that minimises the associated cost.",
"The starting point for our model is the definition of the costs of training the distributed ML model.",
"Specifically, we define the cost as $C &=& C_{N} + C_{P}$ where $C_{N}$ represents the traffic generated on the network during the data collection/aggregation and the learning process and $C_{P}$ is the term that represents the computational burden that CPs have to face in order to execute the distributed learning process (we discuss later on how to make the two cost terms homogeneous, and how to flexibly account for the relative importance of networking and computation costs in the total balance).",
"Therefore, the cost function can be interpreted in many ways depending on the context.",
"For example, it could be seen as the energy cost imposed on the devices, but also as the monetary cost imposed by network and computing operators to a third party requiring the data analytics service.",
"Without loss of generality, to have a single control parameter in the model, in the following we define $\\gamma $ as $\\gamma =\\frac{m_0}{m_1}$ representing the level at which we group data at collection points (we call it grouping parameter).",
"Clearly, it holds that $\\gamma \\in [1,m_0]$ , ranging from completely decentralised ($\\gamma =1$ ) to completely centralised ($\\gamma =m_0$ ).",
"The goal of this paper is to find the optimal value of $\\gamma $ for a given target accuracy $\\varepsilon $ , i.e.",
"the value of $\\gamma $ that minimises the total cost.",
"Mathematically, we want to solve the following problem: $\\operatornamewithlimits{argmin}_{\\gamma } & C_N(\\gamma ) + C_P(\\gamma )\\\\\\mathrm {s.t.}",
"& \\nonumber \\\\& \\varepsilon _{\\gamma } \\le \\varepsilon \\nonumber \\\\& 1\\le \\gamma \\le m_0 \\nonumber $ Where $\\varepsilon _{\\gamma }$ is the estimated accuracy obtained by the distributed learning algorithm for a given $\\gamma $ .",
"In Section we provide the formal definition of $\\varepsilon _{\\gamma }$ used in the paper.",
"The key challenge to solve Equation REF is to obtain an analytical formula for the two components of the cost, for a general enough class of learning algorithms.",
"Before presenting all the mathematical details of the solution of our model, in the next section we describe the class of learning algorithms that we consider in the rest of the paper."
],
[
"Distributed learning algorithms",
"In this section we briefly present the details about “Distributed Stochastic Variance Reduced Gradient\" (DSVRG) [29] method we consider as a specific learning task.",
"DSVRG is a distributed stochastic optimisation algorithm based on the very well known Stochastic Gradient Descent Algorithm.",
"As explained in the following, this has been selected as it is a reference method for machine learning both in centralised and in distributed settings.",
"Specifically, SGD can be used to solve a very significant class of supervised learning problems, including classification and regression.",
"This means that our framework is applicable to all of them.",
"Moreover, the analytical form to describe (i) the cost in terms of generated network traffic and (ii) the cost in terms of computation, can be derived from the literature.",
"As far as the generality of our approach is concerned, although in this paper we focus on DSVRG, any learning method for which is possible to express analytically (i) and (ii) can be analysed as we do for DSVRG.",
"In the following, we provide a concise mathematical formulation of general supervised learning problems that con be solved via DSVRGInterested readers are referred to [30] for a detailed presentation..",
"The goal of supervised learning is to learn the association between data patterns and corresponding labels.",
"More formally, let us suppose that there exists a set of i.i.d.",
"data points $\\lbrace s_1, s_2, \\dots , s_N\\rbrace $ belonging to an unknown distribution $\\mathcal {D}$ .",
"Each data point $s$ is a pair $(x,y)$ where $x\\in \\mathbb {R}^d$ is a vector of features (a pattern) and $y\\in \\mathbb {R}$ is the label associated to $x$ .",
"$y$ can be either a continuous value (for regression problems) or a discrete value (for classification problems).",
"Moreover, let us suppose that the points of the dataset $D$ are grouped in $K$ separate (and possibly non-overlapping) subsets $ S_k$ physically stored on $k$ different devices.",
"Therefore, the complete dataset is defined as $D=\\bigcup _{k=1}^K S_k$ .",
"In distributed supervised leaning each device can only operate on local data.",
"During training, it computes a loss function defined as follows.",
"$\\mathcal {L}_k(S_k,w_k)=\\frac{1}{|S_k|}\\sum _{i=1}^{|S_k|}\\ell (q(x_{i,k},w_k),y_{i,k}) + \\lambda \\mathcal {R}(w_k)$ In Equation (REF ), the first term is the average approximation error on the training set, while the second term is a regularisation factor that we describe in the following.",
"The function $\\ell (q(x_{i,k},w_k),y_{i,k})$ is a loss function computed on the device $k$ , i.e.",
"it measures the error of the model $q(x_{i,k},w_k)$ predicting the real values of $y_{i,k}$ for the data points in $S_k$ .",
"$w_k$ is the parameters' vector of the model $q$ .",
"The function $\\mathcal {R}(w_k)$ is a regularisation term whose purpose is to ease the search for the solution of the optimisation problem, producing a model which is less complex and, thus, less prone to over-fitting.",
"Finally, the scalar $\\lambda $ is a tuning parameter used to regulate the balance between the error term and the regularisation term.",
"In the following we denote with $w$ the union of all model parameters $w_k$ (note that, the model parameters will be also referred to as weights, without loss of generality).",
"The goal of DSVRG is to find the values of parameters $w$ that minimise the average loss over all devices.",
"Formally, the objective is to find the parameters $w^*$ such that $w^*=\\operatornamewithlimits{argmin}_{w} \\frac{1}{K}\\sum _{k=1}^K \\mathcal {L}_k(S_k,w_k)$ DSVRG is a distributed algorithm to solve Equation REF .",
"It is known to have superior convergence properties with respect to the alternative algorithms [29].",
"Precisely, DSVRG assumes the presence of a central unit whose task is to coordinate the communication between the devices and monitor the overall learning process.",
"Therefore, DSVRG belongs to the class of algorithms suitable for being used in the Federated Learning Framework [13].",
"A typical assumption is that the central unit (also called centre) has complete knowledge of the system, i.e.",
"information about devices, about the data they hold, etc.",
"We will discuss in the following how to relax this assumption, thus making the algorithm decentralized.",
"DSVRG is the distributed implementation of the Stochastic Variance Reduced Gradient (SVRG) algorithm [31].",
"The SVRG algorithm represents an improvement of the well known Stochastic Gradient Descent method (SGD) [32].",
"All these algorithms work \"in turns\", i.e., they progressively update the weights based on the loss computed on subsets of the dataset.",
"SVRG improves the convergence speed of SGD by reducing the variance of each weights update, mainly due to the randomness of the selection of the subset of the dataset used during training.",
"The algorithmic description of DSVRG can be split into two logically separate procedures, one executed by the “centre\" and one executed by each of the other devices.",
"For ease of explanation, in the following we describe DSVRG as if a centre would actually exist.",
"However, in our scenario we consider that collection points assume the role of centre in a round robin fashion, thus making DSVRG totally decentralized and, as we show, also reducing the overall network cost.",
"Algorithms and describe the behaviour of DSVRG.",
"Specifically, Algorithm describes the actions taken by the centre, while Algorithm describes the behaviour of each individual collection point.",
"In the beginning, the centre broadcasts an initial estimate of the solution $\\tilde{w}_{t}$ to all the devices (Alg.",
"line 3).",
"Each of them computes in parallel the new gradient over all data points contained in their local datasets and sends it back to the centre (Alg.",
"lines 2-4).",
"Gradients are essentially functions of the losses on the local datasets, which are used (as shown in the following) to update the weights.",
"Specifically, in general the higher the loss, the higher the values of gradients, the higher the “adjustments\" to be made to the weights.",
"The centre collects all gradients and computes their average, $\\tilde{h}_{t}$ (Alg.",
"lines 4-5).",
"At this point, the centre selects a single device $k$ and sends to it the average gradient $\\tilde{h}_t$ (Alg.",
"line 6).",
"Device $k$ updates its local estimate of the weights using equation (REF ) (Alg.",
"line 6), whose meaning we explain next.",
"Then, it computes the new global current estimate of the weights $\\tilde{w_t}$ (using Eq.",
"REF ) and sends it to the centre (Alg.",
"lines 7-8).",
"Finally the centre is rotated (line 8 of Alg.",
").",
"$w_{k,t+1}=w_{k,t}-\\eta (g(w_{k,t},s_{k,t}) - g(\\tilde{w_t},s_{k,t}) + \\tilde{h}_t)$ $\\tilde{w}_{t+1}=\\frac{w_{k,t+1}+t \\tilde{w}_{t}}{t+1}$ The rationale of Eq.",
"REF is to update the estimate of the parameters with a quantity that is proportional to (i) the local gradient ($g(w_{k,t},s_{k,t})$ ), (ii) the gradient of the global current solution ($g(\\tilde{w}_t,s_{k,t})$ ) and (iii) the average global gradient ($\\tilde{h}_t$ ).",
"Specifically, the local gradient in (i) is computed by using only the current (at time step $t$ ) values of the local weights $w_k$ on the local dataset.",
"The gradient in (ii) is computed using the current (at time step $t$ ) estimate of the overall parameters ($\\tilde{w}$ ) on the local dataset.",
"Each individual collection point $k$ has all the input parameters to compute both gradients.",
"Therefore, the update of the local weights through Equation REF takes into account both local and global models parameters.",
"Finally, the rationale of Equation REF is to update the global weights as a weighted average of the old value in the previous round and the updated local weights computed through Equation REF .",
"Both components weight proportionally to the number of rounds over which they have been computed (i.e., 1 and $t$ , respectively).",
"This procedure is repeated for a number $R$ of rounds.",
"Note that the Algorithms presented here are only a short summary of the complete ones.",
"Their purpose is to provide to the reader an intuitive though precise idea of how DSVRG works.",
"For more details on the internals of the algorithm, the interested reader can refer to the original paper [29], [33].",
"[tb!]",
"DSVRG [1] Center-DSVRG $t = 0,1,2,\\dots R-1$ sendToAll$(\\tilde{w}_t)$ $H\\leftarrow $ receiveFromAll$(h_{k=1,\\ldots ,K})$ $\\tilde{h}_t\\leftarrow $ computeAvgGrad$(H)$ sendToDevice$(\\tilde{h}_t,k)$ $\\tilde{w}_{t+1}\\leftarrow $ receiveGlobalWeightsFromDev$(k)$ updateCenter$(k)$ [tb!]",
"On device part of DSVRG [1] Device-DSVRGk $\\tilde{w}_t\\leftarrow $ receiveGlobalWeightsFromCentre() $g_t\\leftarrow $ computeNewGradient$(\\tilde{w}_t)$ sendToCenter$(h_{k,t})$ $h_t\\leftarrow $ receiveFromCentre$(\\tilde{h}_t)$ $w_{t+1}\\leftarrow $ updateLocalWeigths$(\\tilde{w}_t,\\tilde{h}_t)$ as in Eq (REF ) $\\tilde{w}_{t+1}\\leftarrow $ updateGlobalWeights$(w_{t+1})$ as in Eq.",
"(REF ) sendToCenter$(\\tilde{w}_{t+1})$"
],
[
"Network Cost of DSVRG",
"To analyse the network cost of DSVRG it is necessary to first define the $\\varepsilon $ -accuracy of the algorithm.",
"Remember that the goal of our analyses is to identify the optimal operating point given a target accuracy.",
"The $\\varepsilon $ -accuracy is defined as $|\\mathcal {L}(\\tilde{w},s)- \\mathcal {L}(w^*,s)|\\le \\varepsilon $ and means that for the model with parameters $\\tilde{w}$ and any point $s$ the value of the objective function is far from the optimal solution $w^*$ at most by $\\varepsilon $ .",
"It is important to note that the value of $\\varepsilon $ must not be confused with the value of the generalisation performance of the learning algorithm.",
"Clearly, for convex objective functions, these quantities are related to each other because a more accurate solution translates into a better prediction performance of the model.",
"However, quantifying in advance the exact relation between the two is very difficult (if not impossible) and strongly problem-dependent.",
"The optimal solution $w^*$ is in general unknown.",
"Therefore, the accuracy is estimated as the change of parameter values across rounds, measured via gradients.",
"The intuition is that when the updates do not change significantly from one round to the next, the solution is close to the optimal one (assuming a convex loss function as in DSVRG).",
"Specifically, since DSRVG is a gradient-based algorithm, the accuracy is estimated as the squared norm of the gradients, i.e., $\\Vert \\nabla \\mathcal {L}(\\tilde{w},x)\\Vert $ where $\\nabla \\mathcal {L}(\\bar{w},s)$ is the gradient of the model evaluated on point $s$ .",
"Therefore, the value of $\\varepsilon $ is used as stopping condition, i.e., the algorithm stops when the estimate of the accuracy drops below this value.",
"As we will show in the following, $\\varepsilon $ directly affects the number of communication rounds and therefore both the communication and compute costs.",
"Precisely, as stated in [29], the relationship between the $\\varepsilon $ -accuracy and the number of communication rounds between the centre and the devices is expressed by $R = \\left(1+\\frac{\\kappa }{n_1}\\right)\\log _2\\left(\\frac{1}{\\varepsilon }\\right)$ where $n_1$ is the size of the local dataset held by each Collection Point and $\\kappa $ is the condition number of the problem.",
"The condition number is defined as the ratio between the first and the last eigenvalues of $S_k$ .",
"Precisely, $\\kappa $ measures how much the output value of the function can change for a small change in the input argument.",
"Computing the exact value of $\\kappa $ is computationally expensive, but in the machine learning literature, it is typically considered to be proportional to $O(\\sqrt{N})$ , where $N$ is the cardinality of the entire dataset.",
"Note that, the number of rounds depends on the aggregation level, as $n_1$ (the number of data points available at a collection point) is equal to $\\gamma n_0$ .",
"In DSRVG, the amount of communication generated in a single round follows directly from Algorithm and can be computed as in Equation (REF ).",
"Note that we consider as unitary cost of communication the cost of sending a single model (i.e., all its parameters).",
"$C_R = 2(m_1-1)\\omega $ Specifically, for each round the centre sends to all the other $(m_1-1)$ Collection Points a parameters' vector $w$ (line 3) and receives from them $(m_1-1)$ gradient vectors.",
"The size of the parameters' vector and of each gradient is expressed by $\\omega $ (line 4).",
"The reason why we consider the communication cost only due to sending the parameters' vector and the gradient, is explained as follows.",
"Without loss of generality, we assume that the node acting as centre also executes the local updates according to Algorithm .",
"This means that lines 6 and 7 of Algorithm are not executed and do not generate traffic.",
"Finally, we assume that, at a given round, the current centre indicates the center for the next round while sending the new model (line 3 of Algorithm ), thus with negligible network cost.",
"Again without loss generality, we consider that the centres are selected according to a round robin policy.",
"The specific policy does not impact on the network cost as long as the current centre has locally the information required to identify the next one.",
"The total amount of traffic generated to obtain an $\\varepsilon $ -accurate solution is therefore $C_A = C_R * R = 2\\omega (m_1-1)(1+\\frac{\\kappa }{n_1})\\log _2\\frac{1}{\\varepsilon }$ In addition to $C_A$ , which is the amount of traffic generated by the learning algorithm, we also have to account for the network traffic to collect raw data at the $m_1$ collection points, defined as $C_D$ .",
"This is easily computed as each node that is not a collection point ($m_0-m_1$ nodes in total) sends the local data ($d n_0$ on average) to one collection point.",
"The final cost in terms of network traffic is thus provided in Equation (REF ), where $\\theta $ is the cost for transmitting a single feature between two nodesMore precisely, it is the cost of sending a message of average size, computed over the sizes of messages required to transfer raw data (each data point generates $d$ messages) and of the messages required by Algorithms and (each model generates $\\omega $ messages).",
"Note that $\\theta $ highly depends on the specific communication technology used.",
"$C_{N} &=& \\theta (C_A + C_D)\\\\&=& \\theta C_A + \\theta (m_0-m_1)d n_0$"
],
[
"Computational Cost of DSVRG",
"First of all, clearly only collection points incur computation costs.",
"We model the computation required by all collection points with Equation REF below: $P(\\gamma )= G(\\gamma )R(\\gamma ).$ In Equation (REF ), $R(\\gamma )$ denotes the number of rounds to achieve $\\varepsilon $ accuracy, and is again provided by Equation (REF ) (in this case we only make it explicit the dependence on $\\gamma $ ).",
"Moreover, $G(\\gamma )$ is the number of floating point operations needed to compute the gradients by all collection points during a round.",
"Specifically, this is equal to the number of data points over which the gradients need to be computed multiplied by the cost for computing a single gradient, that we refer to as $\\tau $ .",
"Specifically, $\\tau $ is the number of floating-point operations (FLOPS) needed to compute a single gradient vector, given an input pattern, and it is typically proportional to the size of the model, $\\omega $ in our case.",
"According to Alg.",
", the number of gradients to compute at each collection point is equal to the number of local data points, i.e., $n_1 m_1$ .",
"In addition, the centre needs to compute one additional gradient per local point, needed by the term $g(\\tilde{w}_t,s_{k,t})$ in Equation REF .",
"Thus, the total number of gradients to compute is $n_1 (m_1+1)$ .",
"Therefore, $G(\\gamma )$ becomes as follows: $G(\\gamma ) = \\tau n_1 (m_1+1)$ From Equation (REF ) we obtain the final expression of the cost for computing the data $C_P(\\gamma )$ at a certain aggregation level $\\gamma $ : $C_P(\\gamma )=\\psi (\\gamma )P(\\gamma ).$ $\\psi (\\gamma )$ represents the cost of a single floating-point operation.",
"We assume that $\\psi (\\gamma )$ depends on the total number of operations needed for the computations at each collection point, or, better, on the aggregation level $\\gamma $ .",
"This allows us to model the cost of using computation on resource-limited collection points, as well as the economy of scale available at big collection points run by infrastructure operators, as well as their strategies to attract compute services.",
"For example, in cases where collection points are resource-constrained devices (e.g., smartphones or Raspberry PIs), computations might easily saturate local resources.",
"Therefore, it is reasonable to use a cost function that scales super-linearly with the amount of data handled by each collection point, and thus with the aggregation level.",
"On the other hand, when CPs are nodes of an operator's infrastructure, computation might be very cheap (assume, e.g., cases where CPs are virtual machines running on a set of edge gateways, typically over-provisioned with respect to the specific needs of the learning task).",
"In such cases, it might be reasonable to model the cost as a linear or even sub-linear function of the amount of data each CP handles.",
"To capture all these possibilities at once, we define $\\psi (\\gamma )$ as in Eq.",
"( REF ): $\\psi (\\gamma ) &=& \\beta f(\\gamma )$ where $\\beta $ is the cost of a single floating point operation required to compute gradients and $f(\\gamma )$ captures the sub-/super-linear dependence with the number of data handled by a CP.",
"It is thus defined as in Equation REF .",
"$f(\\gamma ) = \\gamma ^{\\alpha }$ where $\\alpha $ is a tuning parameter that we use to control the shape of $f(\\gamma )$ .",
"Precisely, by tuning appropriately $\\alpha $ we can obtain three different regimes of cost, as shown in Table REF .",
"In the paper, we analyse and solve our model with each one of them.",
"Table: Possible parametrisation of f(γ)f(\\gamma )Before presenting, in the next section, the analysis of the optimal operating point that can be obtained through our model, let us highlight a few important features.",
"First, in our model, the balance between the communication and compute costs are defined by the two parameters $\\theta $ and $\\beta $ .",
"The former defines the cost of sending a single paramenter (e.g.",
"a float), while the latter defines the cost of performing a floating point operation in the gradient computation.",
"Moreover, the dependency of the model with the specific algorithm considered is limited to the formula of the number of rounds $R(\\gamma )$ to achieve $\\varepsilon $ -accuracy (Equation REF ) and the number of floating point operations to compute all gradients, $G(\\gamma )$ (Equation REF ).",
"The rest of the model does not depend on the specific definition of the DSRVG algorithm, and can thus be used also with other ML algorithms.",
"Also remember that DRSVG already represents a broad class of ML algorithms itself."
],
[
"Cost model analysis",
"In this section we present the analytical solution of the optimization problem defined in Section for the case of the learning algorithm presented in Section .",
"First, let us rewrite the cost model expanding $C_A(\\gamma )$ and $C_P(\\gamma )$ : $C(\\gamma ) &=& \\theta \\left((C_A(\\gamma ) + C_D(\\gamma )\\right)+C_P(\\gamma ) \\nonumber \\\\&= & 2 \\omega \\theta \\left(\\frac{m_0}{\\gamma }-1\\right)R(\\gamma ) +\\nonumber \\\\& & + \\theta \\left(m_0 - \\frac{m_0}{\\gamma }\\right)n_0 d +\\nonumber \\\\& & + \\beta f(\\gamma ) \\tau (n_0m_0+n_0\\gamma )R(\\gamma )$ Now we analyse the analytical behaviour of the function $C(\\gamma )$ in order to find an optimal value of $\\gamma $ in the domain $[1,m_0]\\in \\mathbb {R}$ , i.e., the value for which the total cost is minimised.",
"We anticipate that we have been able to find the closed-form solution of the optimal value only for the case when: i) we consider the network cost alone, i.e.. $C(\\gamma )=\\theta (C_A(\\gamma )+C_D(\\gamma ))$ , as already presented in [28] and reported here for completeness and, ii) when $f(\\gamma )$ is linear, i.e.",
"with $\\alpha =1$ .",
"Conversely, for the general case, i.e, when $\\alpha $ is left symbolic, we have not been able to find a closed-form solution but, as we will show in Section , it is possible to find the numerical solution of our model using a standard solver.",
"We present now the analytical result considering the network cost only, which represents a situation in which the computational costs are so negligible that can be considered as null.",
"In our model this is achieved setting $\\beta =0$ .",
"Therefore, Eq.",
"(REF ) reads: $C(\\gamma ) &=& C_N(\\gamma ) \\nonumber \\\\& =& \\theta (C_A(\\gamma ) + C_D(\\gamma )) \\nonumber \\\\& = & 2 \\omega \\theta \\left(\\frac{m_0}{\\gamma }-1\\right)R(\\gamma ) +\\nonumber \\\\& & + \\theta \\left(m_0 - \\frac{m_0}{\\gamma }\\right)n_0 d$ The solution of Eq.",
"(REF ) is enclosed in the following theorem.",
"Theorem 1 When $\\beta =0$ , $C(\\gamma )$ admits only one solution in $\\mathbb {R}$ .",
"Its expression is given in Equation (REF ).",
"$\\tilde{\\gamma } = \\frac{4 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m \\omega }{d m n^{2} + 2 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\omega - 2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)m n \\omega }$ The proof is provided in [28] where it is shown that the sign of first derivative of $C_N(\\gamma )$ before $\\tilde{\\gamma }$ is negative and after $\\tilde{\\gamma }$ is positive, identifying $\\tilde{\\gamma }$ as a minimum.",
"Let us now present the analytical solution for the case in which $\\beta >0$ and the computational cost is linear, in the following Theorem.",
"Theorem 2 When $f(\\gamma )=\\gamma $ , $C(\\gamma )$ admits only one minimum in the range $[0,m_0]$ .",
"Its expression is given in Equation (REF ), where $\\gamma _1$ is a function whose definition is provided in Appendix .",
"$\\tilde{\\gamma } = \\gamma _1$ See Appendix .",
"We can exploit the result of Theorems REF and REF to identify the optimal operating point of our problem in the range $[1,m_0]$ as in Equation (REF ) $\\widehat{\\gamma } =\\left\\lbrace \\begin{array}{lcl}1 & \\mathrm {,for } &\\tilde{\\gamma }< 1 \\\\\\tilde{\\gamma } & \\mathrm {,for } &1\\le \\tilde{\\gamma }\\le m_0 \\\\m_0 & \\mathrm {,for } &\\tilde{\\gamma }>m_0\\end{array} \\right.$ The interpretation of Equation (REF ) is as follows and holds for the both Theorems.",
"If $\\tilde{\\gamma }$ is below 1, the function $C_N$ crossing the domain $[1,m_0]$ is increasing for $\\gamma >\\tilde{\\gamma }$ .",
"Therefore the only viable option for minimising the costs is to adopt a fully distributed configuration, leaving data on the source locations.",
"Conversely, if $\\tilde{\\gamma }$ is beyond $m_0$ then $C_N$ crossing the domain $[1,m_0]$ is strictly decreasing when $\\gamma <\\tilde{\\gamma }$ .",
"This means that in this case the minimum cost is equivalent to centralised all the data on a single device.",
"For all the case in which $\\tilde{\\gamma }$ gets values in the range $[1,m_0]$ the best option is represented by an intermediate solution.",
"In the simplest case when $\\beta =0$ , it is possible to provide very intuitive explanations of the form of $\\tilde{\\gamma }$ as in Eq.",
"REF , see [28]."
],
[
"Performance analysis",
"In this section, we provide an extensive performance analysis to compare the cost of a system configured to operate at the optimal point estimated by our model, as compared to full centralisation and full decentralisation.",
"Before that, in Section REF we present the reference dataset used in our evaluation and the evaluation settings, while in Section REF we assess the accuracy of the model in estimating the optimal operating point across the entire range of possible aggregation levels.",
"Without loss of generality, in the following we consider a binary classification task, and, as customary in the literature (e.g., [28]), we use the logistic function as error function, i.e, the function $\\ell $ of Eq.",
"(REF ), and the squared norm as regularization term (i.e., $\\mathcal {R}(w_k)=\\Vert w_k\\Vert _2^2$ )The squared norm is a common regularizer in ML.",
"Its physical meaning is to penalise the model's parameters with large values, thus preventing over-fitting issues.",
": $\\mathcal {L}_k(S_k,w){=}\\frac{1}{|S_k|}\\sum _{i=1}^{|S_k|}\\log (1+\\exp (-y_i(x_i^Tw_k))) + \\frac{\\lambda }{|S_k|}\\Vert w_k\\Vert ^2_2 \\\\=\\frac{1}{|S_k|}\\sum _{i=1}^{|S_k|}\\log (1+\\exp (-y_i(x_i^Tw_k))) + \\frac{\\lambda }{|S_k|}\\sum _{j=1}^\\omega w_{k,j}^2 \\\\$"
],
[
"Dataset description and methodology",
"We use the Covtype dataset that contains real-world observations related to an environmental monitoring task.",
"It is a publicly available datasetDataset available at https://archive.ics.uci.edu/ml/datasets/Covertype that contains 581012 observations.",
"Each observation is a vector of 54 features containing both cartographic and soil information, corresponding to a $30m\\times 30m$ area of forest.",
"The learning task is to use this information to predict what is the main kind of trees covering the area.",
"Note that the Covtype dataset has been selected because it is a reference dataset in the ML literature, and provides sufficient data to test our model in various configurations, as clarified in the following.",
"It is not necessarily representative of the scenarios described in Section (such as Industry 4.0).",
"However, this does not impact on the generality of our results, as the main purpose of the paper is to assess the accuracy of our analysis and quantify the advantages of decentralised learning, which does not depend on the specific environment used for evaluation.",
"The dataset contains seven classes.",
"Since in this paper we test our model by means of a binary classification problem, we evaluate our model separately on the classes 3,7.",
"We selected these classes because the sizes of the associated data have the same order of magnitude, i.e.",
"this allows us to evenly distribute the data on devices, letting each of them have a balanced representation of each class.",
"The dataset size in our simulations is $\\sim 45000$ points.",
"In order to validate and analyse our model, we used the following procedure.",
"The dataset is split into training and test sets, with a proportion $(80\\%,20\\%)$ .",
"We set a number $m_0$ of devices, each one holding, on average, $n_0$ observations of the training set.",
"In order to simulate the data collection process, each device draws, from a Poisson distribution with average $n_0$ , the number of data generated locally, and stores in the local cache this amount of data picking them from the original dataset (we make sure to avoid replicating the same data on multiple devices).",
"Note that, before assigning the data to devices, the training set has been randomly shuffled (using a uniform distribution) in order to maintain in each sub-sample the same class distribution of the entire dataset.",
"We select a target accuracy $\\varepsilon $ , and we impose, as it is common practice in the machine learning field, that $\\kappa $ is equal to $\\sqrt{N\\cdot d}$ , where $N=m_0\\cdot n_0$ is the total amount of data held by the devices in the system.",
"In order to make communication and computation costs comparable we assume that $\\theta , \\beta $ , which in our model are respectively the generic cost for sending a feature and the generic cost of an operation for computing gradients, are expressed in energy units (Eu).",
"We also assume that the two quantities have the following relationship: $\\beta = \\theta *\\mu $ , where $\\mu \\ge 0$ .",
"In this way, with a single parameter, we can control the contribution of the two terms of the cost function.",
"Note that the optimal point identified by our model is insensitive to the specific values of $\\beta $ and the $\\theta $ , but only depends on their ratio, i.e.",
"$\\mu $ .",
"In the following analysis, we evaluate the optimal operating point, derived by solving our analytical model as explained in Section .",
"We refer to this value as $\\hat{\\gamma }$ .",
"To validate our model, we proceed as follows.",
"We run the distributed learning algorithm for all values of $\\gamma $ in the range $[1,m_0]$ .",
"For each level of aggregation, we collect the number of communication rounds and the number of FLOPS executed to reach the target $\\varepsilon $ -accuracy.",
"All the simulations have been repeated ten times in order to reduce the variance of the results.",
"Therefore, the results reported in this paper are average values for which we report also the confidence intervals at 95% level of confidence.",
"These values are use as input to Equation (REF ) to calculate the empirical costs $C_A(\\gamma ),C_D(\\gamma ),C_P(\\gamma )$ at the various aggregation levels.",
"We define as empirical optimal operating point the value of $\\gamma $ for which we obtain minimum cost, hereafter referred to as $\\gamma ^*$ .",
"Notice that, for a given aggregation level $\\gamma $ , the analytical expression of the total cost and the one obtained via simulation differ, as in the former case we are estimating both the number of rounds needed for convergence and the number of computations required.",
"In turns, this means that the optimal operating point estimated by our analytical model is an approximation of the real optimal operating point found in simulations (which requires an exhaustive search over the $\\gamma $ parameter range).",
"Comparing the values of $\\gamma ^*$ and $\\hat{\\gamma }$ , as well as the costs incurred at those aggregation levels, we are able to validate the accuracy of our solution.",
"Specifically, we define overhead the percentage additional cost incurred when using an aggregation level $\\hat{\\gamma }$ instead of $\\gamma ^*$ .",
"After validating the model, in Section REF we compare the cost achieved at the optimal operating point with respect to a fully centralised $\\gamma =m_0$ and a fully decentralised $\\gamma =1$ configurations.",
"This allows us to highlight the advantage of configuring the distributed ML system through our model, as opposed to use two straightforward configurations.",
"Finally, in Section REF we present a sensitiveness analysis of the optimal configuration point and related cost, by varying key parameters.",
"Table REF shows the default values for the main parameters used in the following analysis.",
"The size of the dataset $N$ is derived from the CovType dataset classes that we selected.",
"The number of devices generating data $m_0$ is used to represent a densely populated fog environment.",
"The values of $\\varepsilon $ are typical accuracy levels considered in the literature.",
"As for the relative costs of computation vs. networking (i.e., $\\mu $ ) we consider, by default, a case where computation is much cheaper ($10^{-4}$ ) than communications, under the assumption that a wireless transmission of data is more power consuming than processing the same amount of data, e.g., computing a gradient.",
"Note that in Section REF we analyse the impact of varying all of these parameters, considering also cases where computation is basically for free.",
"Table: Default parameters values."
],
[
"Model Validation",
"In this section, we validate our model for three configurations of computational cost.",
"We begin analysing the linear computational cost, corresponding to a baseline case where the computation cost is proportional to the number of data points handled by each collection point.",
"Fig.",
"REF shows, for each value of $\\varepsilon $ in the range $[10^{-2},10^{-7}]$ , the curves of the empirical cost function $C(\\gamma )$ (solid blue line) and the one calculated by our model (dashed purple line), the optimal empirical aggregation point (red circle), the optimal operating point found by solving numerically our model (green triangle) and the one found though the analytical solution (black star), i.e., only for the linear computational cost.",
"Fig.",
"REF reveals that for high requested $\\varepsilon $ accuracies, the model's cost function (dashed line) approximates the empirical one (blue line) with increasing accuracy.",
"This is reflected by the fact the optimal operating points identified by our model are the same or very close to the empirical ones.",
"An interesting case is that of $\\varepsilon =10^{-4}$ in which our model suggests a slightly more aggressive aggregation, while the empirical optimal solution would be to keep the training fully distributed (i.e., $\\gamma =1$ ).",
"The inaccuracy of our model in this case is due to the fact that the empirical cost function in the first trait has two distinct regimes: for lower accuracies ($\\varepsilon \\ge 10^{-4}$ ) the curve is concave-down while for higher accuracies ($\\varepsilon \\le 10^{-5}$ ) it is concave-up.",
"The model, although being able to catch this behaviour, switches from one regime to the other too soon, thus causing the imprecise approximation.",
"To get a quantitative understanding of the effect of the approximation induced by the model's solution, we refer to Table REF .",
"In the table we report, for each target accuracy, the empirical optimal aggregation point ($\\gamma ^*$ ) the one found by our model ($\\hat{\\gamma }$ ), the number of rounds actually needed to reach the target accuracy ($R(\\gamma ^*)$ ) and those induced by configuring the system at the estimated optimal operating point ($R(\\hat{\\gamma })$ ).",
"For each solution, we also report the optimal empirical cost ($C(\\gamma ^*)$ ) and the one obtained by using the estimated optimal aggregation ($C(\\hat{\\gamma })$ ).",
"Finally, we provide the percentage of additional cost due to the approximation of the model, i.e., the overhead (OH).",
"For $\\varepsilon =10^{-6},10^{-7}$ our model overestimates the number of rounds (twice with respect to the optimal solution) needed to converge, while for $\\varepsilon =10^{-4}$ it underestimates them.",
"In either case, this results in an overhead greater than 0, i.e., an increment of the the cost with respect to the optimal case.",
"However, we observe that for higher accuracies ($\\varepsilon =10^{-6},10^{-7}$ ) the overhead is lower than for lower ones, specifically for $\\varepsilon =10^{-4}$ .",
"This initial set of results (Fig.",
"REF and Table REF ) already shows that our model is quite accurate in estimating the optimal operating point.",
"Interestingly, unless in the specific case of $\\varepsilon =10^{-4}$ , the error in estimating the optimal operating point leads to marginal additional costs (below 10%).",
"Moreover, the accuracy of the model in estimating the actual cost diminishes as higher and higher $\\varepsilon $ -accuracy is targeted, which is a very positive feature.",
"Table: Validation of the model with linear f(γ)f(\\gamma ) varying ε\\varepsilon .Figure: Linear computational cost (α=1\\alpha =1).",
"Empirical curves with optimal operational point (red dot) and the one identified by our model analytically (black star) and numerically (green triangle).We present now the simulation results related to a computational cost that grows non-linearly with the amount of data to be processed on a device.",
"Remember that sub-linear cases represent configurations where “it is cheep to aggregate\", such as edge infrastructure operators implementing cost policies to push moving data on their devices.",
"Super-linear cases represent configurations where “it is costly to aggregate\", such as when collection points are resource constrained devices which risk saturation.",
"In our model, super-linear and sub-linear cases correspond to values of $\\alpha $ greater and lower than 1, respectively.",
"In our simulations we set $\\alpha =2$ and $\\alpha =0.5$ for the super-linear and sub-linear cases.",
"Results are shown in Fig.",
"REF and Table REF (sub-linear case), and Fig.",
"REF and Table REF (super-linear case).",
"Figure: Sub-linear computational cost (α=0.5\\alpha =0.5).",
"Empirical curves with optimal operational point (red dot) and the one identified by our model (green triangle).Figure: Super-linear computational cost (α=2)\\alpha =2).",
"Empirical curves with optimal operational point (red dot) and the one identified by our model (green triangle).Table: Validation of the model with sub-linear f(γ)f(\\gamma ) varying ε\\varepsilon .We focus first on the case when the computational cost is sub-linear.",
"Results show that the general behaviour for the sub-linear case is similar to the linear one.",
"Also here the model solutions push for lower aggregation levels at higher accuracies, i.e., for $\\varepsilon =10^{-6},10^{-7}$ the optimal number of collection points are $m_1=80$ while our model suggests $m_1=133$ and $m_1=100$ , respectively.",
"However, the overhead remains quite limited, as the additional cost is no more than $2.58\\%$ with respect to the optimal empirical solution.",
"For $\\varepsilon =10^{-4}$ , we notice the same approximation issue observed in the linear case, leading to a similar overhead ($41.70\\%$ ).",
"Table: Validation of the model with super-linear f(γ)f(\\gamma ) varying ε\\varepsilon .For the super-linear case, at each $\\varepsilon $ , the optimal operating point is at lower aggregation levels with respect to the linear and sub-linear cases.",
"This is expected, because the exponential growth of the computational cost pushes the optimal aggregation point towards more decentralised solutions (lower values of $\\gamma $ ).",
"From a systems point of view, this means that the distributed learning process performs more communication rounds.",
"This is confirmed by the number of rounds performed at, for example, $\\varepsilon =10^{-6},10^{-7}$ (see columns $R(\\gamma ^*),R(\\hat{\\gamma })$ of Table REF compared to the corresponding ones in Table REF ).",
"In terms of accuracy of the model, the optimal operating point is exactly predicted at lower accuracies ($\\varepsilon \\in [10^{-5},10^{-2}]$ ), while for higher accuracies ($\\varepsilon =10^{-6},10^{-7}$ ) it is approximated.",
"Also in this case, though, the overhead is quite limited ($1.30\\%$ and $19.16\\%$ , respectively).",
"Summarising this set of results, we can observe some common features across the three cases, which highlights a coherent behaviour of our model.",
"First, the model tends to be more accurate in estimating the costs as the required accuracy increases.",
"Moreover, in many cases the model is able to estimate the exact optimal operating point, i.e., the best level of aggregation that drives the system to achieve minimal cost.",
"Finally, even when the optimal operating point is only approximated, the incurred overhead is most of the time quite limited, with only a very few exceptions.",
"We can thus conclude that the model is a very accurate tool to both (i) analyse the cost incurred at the different aggregation levels, and (ii) identify optimal operation points of the decentralised learning task."
],
[
"Comparison of optimal, fully centralised and fully decentralised operating points",
"An important feature to be noted is that, in most of the analysed cases, the optimal operating point is an intermediate aggregation level, i.e., the optimal configuration is neither to fully centralise ($\\gamma =400$ ), neither to fully decentralise ($\\gamma =1)$ the learning task.",
"It is thus interesting to quantify the advantage of using our model and configure the system at the predicted optimal operating point, with respect to applying simpler, but less accurate, policies corresponding to the two extreme aggregation configurations.",
"For this analysis we refer to Table REF , where we report the results for the linear ($\\alpha =1$ ), sub-linear ($\\alpha =0.5$ ) and super-linear ($\\alpha =2$ ) cases, respectively.",
"Moreover, we report in Figure REF the gain/loss (%) deriving by the usage of the model's intermediate solution in place of the fully distributed and fully centralised ones.",
"Table: Costs at the predicted optimal operating point compared to those of full decentralisation and centralisation.Figure: Gain/loss (%) comparison, for all the computational costs, of the model's solution with respect to centralised and fully decentralised solutionsTable REF sheds light on the reasons why moving from less to more expensive computation pushes the optimal aggregation point towards higher decentralisation.",
"Precisely, looking at column ($C(400)$ ) of the table, the costs connected to the full centralisation increase up to 4 order of magnitude (e.g.",
"$\\varepsilon =10^{-6},10^{-7}$ ) when we move form the sub-linear to the super-linear regimes.",
"Consequently, the optimal operating point reduces accordingly, doubling the optimal number of collection point in the super-linear regime for the same level of accuracy (i.e., $\\varepsilon =10^{-7}$ ).",
"The advantage of using the intermediate solution instead of centralising all the data on one device becomes explicit looking at the gain curve shown in Figure REF .",
"In the sub-linear computational cost case, we can save form $37\\%$ to $95\\%$ Eu, according to the target accuracy of interest.",
"Clearly, the highest gain corresponds to the lowest accuracy ($\\varepsilon =10^{-2}$ ), since it means that the DSVRG can stop the computation earlier, thus limiting the number of communication rounds and local computation.",
"Interestingly, the gain does not constantly decrease as we request higher target accuracies.",
"Conversely, the gain reduction touches its minimum at $\\varepsilon =10^{-4},10^{-5}$ after which it starts increasing up to $72\\%$ .",
"The reason of this behaviour is the contrasting effect of the network and the computation components of the cost function.",
"Precisely, since the number of collection points is higher for lower accuracies the network traffic generated by the data collection and the communication rounds of the algorithm dominates the computation costs.",
"As we aggregate on less collection points, the number of data points per CP increases along with the local computation which adds up significantly over the network.",
"The process ends up with the centralised solution for which the cost includes not only transfer of almost all the data on one device, but also the execution on the learning algorithm on the whole data for several iterations.",
"This explains the overall increase of the costs for centralising the data on a single collection point which makes the intermediate solution more convenient.",
"Using more aggressive cost functions (linear and super-linear) magnifies such a behaviour.",
"The costs for the full centralisation explode and, thus, the costs connected to intermediate solutions results in greater gains ranging from $73\\%$ to $97\\%$ (see Fig.",
"REF ) in the linear case and even better ones for the super linear case where they achieve a constant $99\\%$ for all the requested accuracies, as shown in Fig.",
"REF .",
"In comparison with full decentralisation the gain obtainable using an intermediate solution, like the one identified by the model, ranges from $\\%0$ to $52\\%$ (apart from the solution connected to the bad model approximation already discussed).",
"Overall, looking at Fig.",
"REF we see that in this case the different regimes of computational cost do not affect significantly the gain obtainable at the different target accuracies.",
"Precisely, for $\\varepsilon \\ge 10^{-5}$ in the sub-linear case, working at the optimal operating point saves from $15\\%$ to $52\\%$ Eu while in the linear case the gains are between $15\\%$ and $48\\%$ .",
"In the case of super-linear computational costs, looking at the same accuracies, the gains are slightly less than in the sub-linear and linear cases (up to 39%), although with the very same trend.",
"The reason of such invariance is that all the computational cost functions (sub-super-linear) are very similar to each others for small values of $\\gamma $ , which is the regime where both full decentralisation and our approach work.",
"Moreover, the plots provide also the intuition behind the convenience of using intermediate solutions in place of full decentralisation.",
"The increasing trend of saving for higher accuracies is due the fact that the computational costs increases with the accuracy required, contributing more to the overall cost.",
"Since full decentralisation is the configuration in which devices have the least quantity of local data, this configuration forces the algorithm to perform more communication rounds (and therefore also more local computation) which might be avoided with an intermediate level of data aggregation."
],
[
"Sensitivity analysis",
"We finally present a sensitivity analysis of the model with respect to key parameters.",
"Note that, for this analysis we adopt the same configuration used during the validation, with the aim of making the results easier to interpret.",
"By default, we set the parameters as already indicated in Table REF .",
"We fix the target accuracy $\\varepsilon =10^{-5}$ and we consider that the input data have dimensionality $d=54$ as in the Covtype dataset.",
"The unitary cost for networking is $\\theta =1 Eu$ , and we set the $\\mu $ parameter to $10^{-4}$ .",
"Therefore, the computational unitary cost is $\\beta =10^{-4}$ Eu.",
"We show the results for all the considered computational cost regimes: sub-linear, linear and super-linear which correspond to $\\alpha =0.5, \\alpha =1, \\alpha =2$ , respectively.",
"Unless differently specified, we assume to have $m_0=400$ devices, each one holding $n_0=112$ data points.",
"First, we analyse the sensitivity of our model to increasingly higher amount of data generated at each device ($n_0$ ).",
"Results in Table REF show that, for all the regimes of $\\alpha $ , our model tends to aggregate the data on fewer collection points when at each device the amount of data is smaller.",
"Therefore, the optimal number of collection points is directly proportional to the initial quantity of data per device.",
"The reason behind this behaviour is that, when $n_0$ increases, the local models can be trained with more data, and thus are more accurate.",
"Therefore, fewer rounds are required, and thus the networking costs at a given aggregation level decreases.",
"This pushes towards more decentralised optimal operating points.",
"As expected, for each value of $n_0$ the optimal operating point moves towards more collection points as the computational cost grows faster with $\\gamma $ .",
"Table: Optimal operational points varying the amount of data generated per device.We then analyse the effect of the number of devices, $m_0$ , by keeping the data they individually generate ($n_0$ ) fixed.",
"As it was the case when increasing $n_0$ , also by increasing $m_0$ we are increasing the total amount of data in the system.",
"However, looking at results in Table REF we notice that in the latter case, the optimal behaviour is to aggregate (slightly) more (rather than less) as the total amount of data increases.",
"The reason is that, when increasing the number of nodes at a given aggregation level, we are keeping constant the number of nodes that contribute their data to an individual Collection Point (remember that $\\gamma $ has exactly this physical meaning), and we are increasing the number of Collection Points.",
"Therefore, the amount of data over which Collection Points compute local models are the same, which means the accuracy of the local models are similar.",
"However, with more Collection Points the network traffic per round of model update increases.",
"Therefore, it is more efficient to aggregate data on fewer Collection Point, paying the initial cost of moving more raw data, but saving on the network traffic during collective training.",
"This is an interesting result as it shows that variations in the total amount of data in the system may generate opposite variations in terms of optimal operating point, depending on whether this is a side effect of a higher amount of data generated at each individual device, or a higher number of devices collaborating in the system.",
"Table: Optimal operational points varying the initial amount of data per device.Finally we analyse the impact of the relationship between the communication and computing costs.",
"To this end we vary the $\\mu $ parameter, which controls the ratio between $\\theta $ and $\\beta $ , in the range $[10^{-5},10]$ .",
"In this way we are considering both scenarios in which the computational cost per device is negligible (low values of $\\mu $ ) and others where they exceed the communication costs (values of $\\mu $ higher than 1).",
"We jointly analyse the cases where the amount of data generated at each device increases.",
"We perform five blocks of simulations.",
"For each block we keep fixed the number of devices ($m_0=400$ ) and we increase the amount of data they individually generate ($n_0=50,75,100,125,200$ ).",
"For each pair $(m_0,n_0)$ we vary the parameter $\\mu $ .",
"Note that, considering cases where the computation cost is negligible allows us to also include in the analysis cases where computation is offered “for free\".",
"Indeed, in such cases we found that the computation component of the cost function is orders of magnitude lower than the network component.",
"Even in such cases, our results show that aggregating all data in a unique centralisation point is not the optimal choice.",
"Results are reported in Table REF .",
"Table: Optimal operational points varying the amount of data per device and the ratio between the communication and the computation cost.In most cases, when the computational cost has a negligible impact with respect to the communication cost, the optimal solution is to increase the level of aggregation.",
"This is quite interesting, and somewhat counter-intuitive, as increasing aggregation means spending more on moving data initially collected to fewer collection points, which requires using a “costly\" resource, i.e., communication.",
"However, aggregating data then produces more precise local models at a small increase of cost, due to the low cost of computation, also reducing the number of rounds to achieve the target accuracy, thus eventually saving on the “costly\" resource.",
"For example, note the case in the last row of the first block the optimal point reduces by a factor between 13 and 9 the number of collection points with respect to the set of nodes generating data.",
"However, it is also interesting to note that neither in this case full aggregation (i.e., $\\gamma =m_0$ ) is the best choice.",
"Conversely, as the computation and communication costs become similar, it is more efficient to move less the data, “pay\" less at collection points in terms of computation (remember that, due to our computation cost function model, computing on the same amount of data is more costly on a single node than separately on two nodes), even though this requires additional communication cost due to a higher number of rounds to achieve the required accuracy.",
"An interesting behaviour is shown at the last three blocks of the sub-linear regime ($\\hat{\\gamma }_{\\alpha =0.5}$ ) where within the block the optimal aggregation decreases as the cost for computing becomes more and more negligible, which is the opposite of what we have just observed.",
"This is actually a joint effect of the interplay between the two components of the cost, and the high numerosity of data available at each node ($n_0$ ).",
"On the one hand, as observed before, when the computation cost becomes negligible compared to the communication cost, aggregating more becomes convenient, as the fewer collection points can work with a higher number of data, thus local models are more precise and lower communication rounds are needed to achieve a target accuracy.",
"This is indeed what happens at low values of $n_0$ .",
"However, aggregating more means moving more data from original devices to the (fewer) collection points.",
"This increases the networking cost.",
"At high values of $n_0$ , i.e., when a large amount of data is already available at individual devices, this additional cost is not balanced by a reduced networking cost due to fewer rounds of communication required to converge.",
"In other words, low levels of aggregation result in precise enough local models not to require a large number of communication rounds, which pushes towards lower aggregation levels.",
"However, increased computation costs might change the picture.",
"When computation becomes costly with respect to communication (high values of $\\mu $ ) the overall computation cost becomes predominant over communication costs.",
"Therefore, it pays of to aggregate more, as this allows for fewer rounds to refine the models, and thus to a lower total computation cost.",
"Summarising from the obtained results, we can draw a few general remarks.",
"First, “decentralisation helps\".",
"In all examined cases, the optimal operating point is far from centralising all data in a single collection point.",
"This tells that decentralised machine learning, in addition to advantages in terms of privacy, is also beneficial in terms of efficiency, without compromising on model accuracy.",
"Second, we have also shown that the shape of the computation cost function with respect to the aggregation level has a key role in moving the optimal point between higher and lower aggregation.",
"This might give interesting indications (and a useful tool) to configure “computational offers\" by distributed infrastructure providers, which might tune, e.g., the cost of their VM resources also based on whether they prefer to work with higher or lower levels of aggregation.",
"Third, we have shown that the relative cost of communication vs computation might also play a big role in setting the optimal operating point.",
"This might lead to apparently counter-intuitive results: high communication costs might result in few collection points aggregating more data (which have to be moved from the originating nodes), if this allows to obtain significantly more precise local models, or more collection points if the amount of data generated at individual devices is large enough to obtain sufficiently accurate local models.",
"High computation costs might result in few collection points (where computation is more expensive) if the amount of data at individual nodes it too little to sustain precise enough local models."
],
[
"Conclusion",
"In this paper we consider a scenario in which edge devices (IoT, personal mobile devices) accomplish a data analytics process on some generated or collected data.",
"This is an increasingly relevant scenario in many use cases, as it is becoming more and more clear that distributed data analytics (machine learning, in our specific case) may bring significant advantages with respect to conventional centralised approaches.",
"For example, in Industry 4.0 contexts this addresses data confidentiality and real-time operational constraints.",
"However, in principle centralising data analytics is considered more effective, as machine learning models can be trained with more data.",
"In this paper we addressed the research question whether this is actually the case, when resource consumption is also considered.",
"Specifically, decentralised machine learning algorithms allow to collaboratively train models without sharing data, via successive rounds of communications through which local models are exchanged and fine tuned.",
"Any target accuracy can be obtained, at the cost of increasing the number of refinement rounds.",
"As each round has an associated cost in terms of communication traffic and computation, in this paper we investigate what is the optimal level at which to aggregate raw data (i.e., the optimal number of nodes where to collect subsets of the raw data) such that the data analytics process achieves a target accuracy, while minimising the overall communication and computation resources.",
"To this end we propose an analytical framework able to cope with this problem.",
"We exemplify the complete derivation of the optimal operating point in the case of one of the reference distributed learning algorithms, named DSVRG, which represents a large number of analytics tasks (e.g., classification and regression).",
"Exploiting analytical expressions of the number of data exchanged between the nodes to run the decentralised algorithm, we derive a closed form expression for the computational and network resources required to achieve a target accuracy.",
"We thus solve the model to obtain the optimal operating point, and study the properties of the cost function in the range of the possible levels of aggregation, ranging from complete decentralisation to complete centralisation.",
"The analysis highlights that in most of the cases the optimal operating point is neither of these two extremes aggregation points.",
"Rather, the typical optimal solution consists in aggregating raw data in an intermediate number of collection points, which run the decentralised machine learning algorithm in a collaborative fashion.",
"The paper presents closed form expressions for the optimal operating point in significant cases, validates the accuracy of the analytical model, quantifies the advantage of configuring the system at the optimal operating point as opposed to the two extreme cases, and finally presents a sensitivity analysis showing the impact of various parameters on the optimal configuration.",
"Moreover, we highlight that our analytical model not only provides a performance evaluation tool, but can also be used as a design too, to find (analytically or numerically) the optimal configuration of a decentralised machine learning system."
],
[
"Acknowledgments",
"This work is partially supported by two projects: HumanE AI Network (EU H2020 HumanAI-Net, GA #952026) and Operational Knowledge from Insights and Analytics on Industrial Data (MIUR PON OK-INSAID, ARS01_00917) Proof of Theorem REF We want to prove that the Equation REF (reported here for convenience) has only one minimum in the range $[1,m_0]$ .",
"To this end we prove that, i) there is only one solution in $[1,m_0]$ and ii) that it is a minimum for the function $C(\\gamma )$ .",
"Note we conducted this analysis by means of Sympy (http://www.sympy.org), a standard and open-source mathematical manipulation software.",
"Therefore, our analysis is completely reproducible.",
"Considering the equation $C(\\gamma ) &=& C_A(\\gamma ) + C_D(\\gamma )+C_P(\\gamma ) \\nonumber \\\\&= & 2 w \\theta \\left(\\frac{m_0}{\\gamma }-1\\right)R(\\gamma ) +\\nonumber \\\\& & + \\theta \\left(m_0 - \\frac{m_0}{\\gamma }\\right)n_0 d +\\nonumber \\\\& & + \\beta f(\\gamma ) \\tau (n_0m_0+n_0\\gamma )R(\\gamma )$ and its first derivative $C^{\\prime }(\\gamma )=\\frac{\\partial C(\\gamma )}{\\partial \\gamma }$ : $C^{\\prime }(\\gamma ) = \\\\ \\frac{1}{\\gamma ^{3} n_0}\\Bigg (\\beta \\gamma ^{3} \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^{2} \\tau \\left(\\gamma + m_0\\right) + \\beta \\gamma ^{3} \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0 \\tau \\left(\\gamma n_0 + k\\right) + d \\gamma m_0 n_0^{2} \\theta + 2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\omega \\theta \\left(k \\left(\\gamma - m_0\\right) - m_0 \\left(\\gamma n_0 + k\\right)\\right)\\Bigg )$ it is possible to find, after some algebraic manipulations, the four roots of $C^{\\prime }(\\gamma )=0$ w.r.t.",
"$\\gamma $ , all in closed-form and denoted as $\\gamma _1,\\gamma _2,\\gamma _3,\\gamma _4$ .",
"Moreover, $\\gamma _1 \\in \\mathbb {R}^+$ , $\\gamma _2 \\in \\mathbb {R}^-$ and $\\gamma _3,\\gamma _4 \\in \\mathbb {C}$ .",
"The definition of $\\gamma _1$ provided in Appendix .",
"Since in our problem $\\gamma \\in [1,m_0]$ , the only solution of interest is $\\gamma _1$ .",
"Therefore, we conclude the first part of the proof stating that there exists one and only one solution in the domain of our problem.",
"Let us now prove that $\\gamma _1$ is a minimum for $C(\\gamma )$ .",
"In order to prove it we consider the following limits: $\\lim _{\\gamma \\rightarrow 0^+} C^{\\prime }(\\gamma ) = -\\infty $ $\\lim _{\\gamma \\rightarrow m_0} C^{\\prime }(\\gamma ) = \\\\ \\frac{1}{m^{2} n}(\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m^{2} n \\tau + 3 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m^{3} n^{2} \\tau + d m n^{2} \\theta - 2 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\omega \\theta - 2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)m n \\omega \\theta ) > 0$ $\\lim _{\\gamma \\rightarrow +\\infty } C^{\\prime }(\\gamma ) = +\\infty $ From Equations (REF )-(REF ) we have that $C^{\\prime }(\\gamma ) < 0$ for $0<\\gamma <\\gamma _1$ and $C^{\\prime }(\\gamma ) > 0$ for $\\gamma >\\gamma _1$ , therefore we conclude that $\\gamma _1$ is a minimum of $C(\\gamma )$ .$\\Box $ Definitions of functions The function $\\gamma _1$ of Theorem REF is defined as: $\\gamma _1 = A(m,n,d,k,\\tau ,\\mu ,\\varepsilon ,\\theta ,\\beta ,\\omega )$ where $A(m,n,d,k,\\tau ,\\mu ,\\varepsilon ,\\theta ,\\beta ,\\omega )$ , is a function defined as follows: $-\\frac{1}{24 n_{0\\omega }}\\Bigg ( n_0m_0(3 k+3 m_0 n_0+\\sqrt{3} n_0 \\Bigg (\\frac{1}{n_0^2}[3 (k+m_0 n_0)^2+\\frac{1}{\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau }\\Bigg [8 \\@root 3 \\of {3} (-18 k\\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega \\Big )^2+\\sqrt{3} \\\\\\Big (\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+ 27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0 \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2 \\omega \\Big )^2\\Big )\\Big )^{1/2})^{1/3}\\Bigg ]\\\\-\\Big [8\\ 3^{2/3} n_0 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )\\Big ]\\Big [\\Big (-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0)\\omega \\Big )^2+\\sqrt{3} \\\\\\sqrt{\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2\\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0 \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2\\omega \\Big )^2\\Big )}\\Big )^{1/3}\\Big ]^{-1}]\\Bigg )^{1/2} \\\\ -\\sqrt{6} n_0 \\Bigg \\lbrace \\frac{1}{n_0^3}\\Bigg [\\Big (4\\ 3^{2/3} \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big ) n_0^2\\Big ) \\Big (\\Big (-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0)\\omega \\Big )^2+\\sqrt{3} \\\\\\sqrt{\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2\\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0 \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2 \\omega \\Big )^2\\Big )}\\Big )^{1/3}\\Big )^{-1}+ \\\\3 (k+m_0 n_0)^2n_0-\\frac{1}{\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau }\\Big [4 \\@root 3 \\of {3} \\Big (-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega \\Big )^2+\\sqrt{3} \\\\\\sqrt{\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0\\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2 \\omega \\Big )^2\\Big )})^{1/3} n_0\\Big ]+\\\\\\Big (3 \\sqrt{3} \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau (k+m_0 n_0)^3+32 n_0 \\theta \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0)\\omega \\Big )\\Big )\\Big )\\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau \\Big (\\frac{1}{n_0^2}\\Bigg [3 (k+m_0 n_0)^2+\\frac{1}{\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau }\\\\(8 \\@root 3 \\of {3} (-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega \\Big )^2+\\sqrt{3} \\\\\\sqrt{\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+27 \\Big (\\beta \\tau \\theta \\Big (d m_0n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0 \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2\\omega \\Big )^2\\Big )})^{1/3})\\\\- (8\\ 3^{2/3} n_0 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big ))((-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega \\Big )^2+\\sqrt{3}\\\\ (\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+\\\\27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0\\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2 \\omega \\Big )^2\\Big ))^{1/2})^{1/3})^{-1}\\Bigg ]\\Big )^{1/2}\\Bigg \\rbrace ^{-1}\\Bigg ]\\Bigg )^{1/2}\\Bigg )$"
],
[
"Proof of Theorem ",
"We want to prove that the Equation REF (reported here for convenience) has only one minimum in the range $[1,m_0]$ .",
"To this end we prove that, i) there is only one solution in $[1,m_0]$ and ii) that it is a minimum for the function $C(\\gamma )$ .",
"Note we conducted this analysis by means of Sympy (http://www.sympy.org), a standard and open-source mathematical manipulation software.",
"Therefore, our analysis is completely reproducible.",
"Considering the equation $C(\\gamma ) &=& C_A(\\gamma ) + C_D(\\gamma )+C_P(\\gamma ) \\nonumber \\\\&= & 2 w \\theta \\left(\\frac{m_0}{\\gamma }-1\\right)R(\\gamma ) +\\nonumber \\\\& & + \\theta \\left(m_0 - \\frac{m_0}{\\gamma }\\right)n_0 d +\\nonumber \\\\& & + \\beta f(\\gamma ) \\tau (n_0m_0+n_0\\gamma )R(\\gamma )$ and its first derivative $C^{\\prime }(\\gamma )=\\frac{\\partial C(\\gamma )}{\\partial \\gamma }$ : $C^{\\prime }(\\gamma ) = \\\\ \\frac{1}{\\gamma ^{3} n_0}\\Bigg (\\beta \\gamma ^{3} \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^{2} \\tau \\left(\\gamma + m_0\\right) + \\beta \\gamma ^{3} \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0 \\tau \\left(\\gamma n_0 + k\\right) + d \\gamma m_0 n_0^{2} \\theta + 2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\omega \\theta \\left(k \\left(\\gamma - m_0\\right) - m_0 \\left(\\gamma n_0 + k\\right)\\right)\\Bigg )$ it is possible to find, after some algebraic manipulations, the four roots of $C^{\\prime }(\\gamma )=0$ w.r.t.",
"$\\gamma $ , all in closed-form and denoted as $\\gamma _1,\\gamma _2,\\gamma _3,\\gamma _4$ .",
"Moreover, $\\gamma _1 \\in \\mathbb {R}^+$ , $\\gamma _2 \\in \\mathbb {R}^-$ and $\\gamma _3,\\gamma _4 \\in \\mathbb {C}$ .",
"The definition of $\\gamma _1$ provided in Appendix .",
"Since in our problem $\\gamma \\in [1,m_0]$ , the only solution of interest is $\\gamma _1$ .",
"Therefore, we conclude the first part of the proof stating that there exists one and only one solution in the domain of our problem.",
"Let us now prove that $\\gamma _1$ is a minimum for $C(\\gamma )$ .",
"In order to prove it we consider the following limits: $\\lim _{\\gamma \\rightarrow 0^+} C^{\\prime }(\\gamma ) = -\\infty $ $\\lim _{\\gamma \\rightarrow m_0} C^{\\prime }(\\gamma ) = \\\\ \\frac{1}{m^{2} n}(\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m^{2} n \\tau + 3 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m^{3} n^{2} \\tau + d m n^{2} \\theta - 2 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\omega \\theta - 2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)m n \\omega \\theta ) > 0$ $\\lim _{\\gamma \\rightarrow +\\infty } C^{\\prime }(\\gamma ) = +\\infty $ From Equations (REF )-(REF ) we have that $C^{\\prime }(\\gamma ) < 0$ for $0<\\gamma <\\gamma _1$ and $C^{\\prime }(\\gamma ) > 0$ for $\\gamma >\\gamma _1$ , therefore we conclude that $\\gamma _1$ is a minimum of $C(\\gamma )$ .$\\Box $"
],
[
"Definitions of functions",
"The function $\\gamma _1$ of Theorem REF is defined as: $\\gamma _1 = A(m,n,d,k,\\tau ,\\mu ,\\varepsilon ,\\theta ,\\beta ,\\omega )$ where $A(m,n,d,k,\\tau ,\\mu ,\\varepsilon ,\\theta ,\\beta ,\\omega )$ , is a function defined as follows: $-\\frac{1}{24 n_{0\\omega }}\\Bigg ( n_0m_0(3 k+3 m_0 n_0+\\sqrt{3} n_0 \\Bigg (\\frac{1}{n_0^2}[3 (k+m_0 n_0)^2+\\frac{1}{\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau }\\Bigg [8 \\@root 3 \\of {3} (-18 k\\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega \\Big )^2+\\sqrt{3} \\\\\\Big (\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+ 27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0 \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2 \\omega \\Big )^2\\Big )\\Big )^{1/2})^{1/3}\\Bigg ]\\\\-\\Big [8\\ 3^{2/3} n_0 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )\\Big ]\\Big [\\Big (-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0)\\omega \\Big )^2+\\sqrt{3} \\\\\\sqrt{\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2\\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0 \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2\\omega \\Big )^2\\Big )}\\Big )^{1/3}\\Big ]^{-1}]\\Bigg )^{1/2} \\\\ -\\sqrt{6} n_0 \\Bigg \\lbrace \\frac{1}{n_0^3}\\Bigg [\\Big (4\\ 3^{2/3} \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big ) n_0^2\\Big ) \\Big (\\Big (-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0)\\omega \\Big )^2+\\sqrt{3} \\\\\\sqrt{\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2\\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0 \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2 \\omega \\Big )^2\\Big )}\\Big )^{1/3}\\Big )^{-1}+ \\\\3 (k+m_0 n_0)^2n_0-\\frac{1}{\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau }\\Big [4 \\@root 3 \\of {3} \\Big (-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega \\Big )^2+\\sqrt{3} \\\\\\sqrt{\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0\\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2 \\omega \\Big )^2\\Big )})^{1/3} n_0\\Big ]+\\\\\\Big (3 \\sqrt{3} \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau (k+m_0 n_0)^3+32 n_0 \\theta \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0)\\omega \\Big )\\Big )\\Big )\\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau \\Big (\\frac{1}{n_0^2}\\Bigg [3 (k+m_0 n_0)^2+\\frac{1}{\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau }\\\\(8 \\@root 3 \\of {3} (-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega \\Big )^2+\\sqrt{3} \\\\\\sqrt{\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+27 \\Big (\\beta \\tau \\theta \\Big (d m_0n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0 \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2\\omega \\Big )^2\\Big )})^{1/3})\\\\- (8\\ 3^{2/3} n_0 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16 m_0n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big ))((-18 k \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\theta \\omega \\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2+9 \\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^2 \\tau \\theta ^2 \\Big (d m_0 n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega \\Big )^2+\\sqrt{3}\\\\ (\\log _2\\left(\\frac{1}{\\epsilon }\\right)^2 \\theta ^2 \\Big (\\beta ^3 \\log _2\\left(\\frac{1}{\\epsilon }\\right)n_0^3 \\tau ^3 \\theta \\Big (d m_0 (k+m_0 n_0) n_0^2+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\Big (k^2+16m_0 n_0 k-m_0^2 n_0^2\\Big ) \\omega \\Big )^3+\\\\27 \\Big (\\beta \\tau \\theta \\Big (d m_0 n_0^3+2 \\log _2\\left(\\frac{1}{\\epsilon }\\right)(k-m_0 n_0) \\omega n_0\\Big )^2-2 k m_0\\Big (\\beta \\log _2\\left(\\frac{1}{\\epsilon }\\right)m_0 \\tau n_0^2+\\beta k \\log _2\\left(\\frac{1}{\\epsilon }\\right)\\tau n_0\\Big )^2 \\omega \\Big )^2\\Big ))^{1/2})^{1/3})^{-1}\\Bigg ]\\Big )^{1/2}\\Bigg \\rbrace ^{-1}\\Bigg ]\\Bigg )^{1/2}\\Bigg )$"
]
] | 2012.05266 |
[
[
"Germs in a poset"
],
[
"Abstract Motivated by the theory of correspondence functors, we introduce the notion of {\\em germ} in a finite poset, and the notion of {\\em germ extension} of a poset.",
"We show that any finite poset admits a largest germ extension called its {\\em germ closure}.",
"We say that a subset $U$ of a finite lattice $T$ is {\\em germ extensible} in $T$ if the germ closure of $U$ naturally embeds in $T$.",
"We show that any for any subset $S$ of a finite lattice $T$, there is a unique germ extensible subset $U$ of $T$ such that $U\\subseteq S\\subseteq \\overline{G}(U)$, where $\\overline{G}(U)\\subseteq T$ is the embedding of the germ closure of $U$."
],
[
"Germs in a poset Serge Bouc 1.05 Abstract: Motivated by the theory of correspondence functors, we introduce the notion of germ in a finite poset, and the notion of germ extension of a poset.",
"We show that any finite poset admits a largest germ extension called its germ closure.",
"We say that a subset $U$ of a finite lattice $T$ is germ extensible in $T$ if the germ closure of $U$ naturally embeds in $T$ .",
"We show that any for any subset $S$ of a finite lattice $T$ , there is a unique germ extensible subset $U$ of $T$ such that $U\\subseteq S\\subseteq \\,\\overline{\\!",
"G}(U)$ , where $\\,\\overline{\\!",
"G}(U)\\subseteq T$ is the embedding of the germ closure of $U$ .",
"MSC2020: 06A07, 06A11, 06A12, 18B05.",
"Keywords: Germ, poset, lattice, correspondence functor.",
"1.05 Abstract: Motivated by the theory of correspondence functors, we introduce the notion of germ in a finite poset, and the notion of germ extension of a poset.",
"We show that any finite poset admits a largest germ extension called its germ closure.",
"We say that a subset $U$ of a finite lattice $T$ is germ extensible in $T$ if the germ closure of $U$ naturally embeds in $T$ .",
"We show that any for any subset $S$ of a finite lattice $T$ , there is a unique germ extensible subset $U$ of $T$ such that $U\\subseteq S\\subseteq \\,\\overline{\\!",
"G}(U)$ , where $\\,\\overline{\\!",
"G}(U)\\subseteq T$ is the embedding of the germ closure of $U$ .",
"MSC2020: 06A07, 06A11, 06A12, 18B05.",
"Keywords: Germ, poset, lattice, correspondence functor.",
"section1-3.5ex plus-1ex minus -.2ex2.3ex plus.2exIntroduction In a series of papers with Jacques Thévenaz (, , , , ), we develop the theory of correspondence functors over a commutative ring $k$ , i.e.",
"linear representations over $k$ of the category of finite sets, where morphisms are correspondences instead of maps.",
"In this theory, finite lattices and finite posets play a crucial role, at various places.",
"In particular, we show (, Theorem 4.7) that the simple correspondence functors are parametrized by triples $(E,R,V)$ , where $E$ is a finite set, $R$ is a partial order relation on $E$ - that is, $(E,R)$ is a finite poset - and $V$ is a simple $k{\\rm Aut}(E,R)$ -module.",
"Moreover, the evaluation at a finite set $X$ of the simple functor $S_{E,R,V}$ parametrized by the triple $(E,R,V)$ can be completely described (, Theorem 6.6 and Theorem 7.9).",
"It follows (, Theorem 8.2) that when $k$ is a field, the dimension of $S_{E,R,V}(X)$ is given by $\\dim _k S_{E,R,V}(X)=\\frac{\\dim _kV}{|{\\rm Aut}(E,R)|}\\sum _{i=0}^{|E|}(-1)^i\\binom{|E|}{i}\\big (|G|-i\\big )^{|X|}\\;\\;.\\qquad \\mathrm {{(0.1)}}$ The main consequence of these results is a complete description of the simple modules for the algebra over $k$ of the monoid of all relations on $X$ (, Section 8).",
"Formula REF is obtained by first choosing a finite lattice $T$ such that the poset ${\\rm Irr}(T)$ of join-irreducible elements of $T$ is isomorphic to the opposite poset $(E,R^{op})$ , and then constructing a specific subset $G=G_T$ of $T$ (see () for a precise definition of $G_T$ ), which appears in the right hand side.",
"Now for a given $X$ , the left hand side of (REF ) only depends on the poset $(E,R)$ and the simple $k{\\rm Aut}(E,R)$ -module $V$ , whereas the right hand side depends in addition on the cardinality of the set $G$ , which a priori depends on $T$ , and not just on $(E,R)$ .",
"We have checked ( Corollary 6.7) that $|G|$ indeed only depends on $(E,R)$ .",
"A natural question is then to ask if the subposet $G$ of $T$ only depends on $(E,R)$ , up to isomorphism, and not really on $T$ itself.",
"One of the aims of the present paper is to answer this question.",
"In fact, the main aim is to introduce various structural results on posets and lattices, which appear to be new.",
"The first notion we introduce is the notion of germ of a finite poset.",
"A germ of a finite poset $S$ is an element of $S$ with specific properties (Definition REF ).",
"A related notion is the following: When $U$ is a subset of $S$ , the poset $S$ is called a germ extension of $U$ if any element of $S-U$ is a germ of $S$ .",
"The main result of the paper (Theorem ) is that conversely, being given a finite poset $U$ , there is a (explicitly defined) finite poset $G(U)$ , containing $U$ as a full subposet, which is the largest germ extension of $U$ , in the following sense: First $G(U)$ is a germ extension of $U$ , and moreover, if $S$ is a finite poset containing $U$ as a full subposet, and such that $S$ is a germ extension of $U$ , then there is a unique full poset embedding $S\\rightarrow G(U)$ which restricts to the identity map of $U$ .",
"For this reason, the poset $G(U)$ will be called the germ closure of $U$ .",
"This poset $G(U)$ can be viewed as a structural answer to the above question: In the case of a finite lattice $T$ with poset $(E,R)$ of join-irreducible elements, we show (Theorem ) that the poset set $G$ identifies canonically with $G(E,R)$ , and in particular, it only depends on the poset $(E,R)$ .",
"In Section 3, we consider germ extensible subsets of a finite lattice.",
"For any full subposet $U$ of a finite lattice $T$ , the inclusion map $U\\hookrightarrow T$ extends to a canonical map of posets $\\nu :G(U)\\rightarrow T$ .",
"We say that $U$ is germ extensible in $T$ if this map $\\nu $ is injective, and in this case, we denote by $\\,\\overline{\\!",
"G}(U)\\subseteq T$ its image.",
"We give a characterization of germ extensible subsets of a lattice (Theorem ), and then show (Theorem ) that for any subset $S$ of $T$ , there exists a unique germ extensible subset $U$ of $T$ such that $U\\subseteq S\\subseteq \\,\\overline{\\!",
"G}(U)$ .",
"In other words, the poset of subsets of $T$ is partitioned by the intervals $[U,\\,\\overline{\\!",
"G}(U)]$ , where $U$ is a germ extensible subset of $T$ .",
"In a forthcoming paper, we will show how this rather surprising result yields a natural filtration of the correspondence functor $F_T$ associated to $T$ (, Definition 4.1), by fundamental functors indexed by germ extensible subsets of $T$ .",
"The last section of the paper (Section 5) lists some examples of germs, germ closures, and germ extensible subsets of lattices.section1-3.5ex plus-1ex minus -.2ex2.3ex plus.2exGerms in a poset Throughout the paper, we use the symbol $\\subseteq $ for inclusion of sets, and the symbol $\\subset $ for proper inclusion.",
"We denote by $\\sqcup $ the disjoint union of sets.",
"If $(U,\\le )$ is a poset, and $u,v$ are elements of $U$ , we set $[u,v]_U&=\\lbrace w\\in U\\mid u\\le w\\le v\\rbrace ,\\; &[u,v[_U&=\\lbrace w\\in U\\mid u\\le w< v\\rbrace ,\\\\]u,v]_U&=\\lbrace w\\in U\\mid u< w\\le v\\rbrace ,\\; &]u,v[_U&=\\lbrace w\\in U\\mid u< w< v\\rbrace ,\\\\\\,]\\,.\\,,v]_U&=\\lbrace w\\in U\\mid w\\le v\\rbrace ,\\; &\\,]\\,.\\,,v[_U&=\\lbrace w\\in U\\mid w< v\\rbrace ,\\\\[v,\\,.\\,[_U&=\\lbrace w\\in U\\mid w\\ge v\\rbrace ,\\; &]v,\\,.\\,[_U&=\\lbrace w\\in U\\mid w> v\\rbrace .$ When $V$ is a subset of a poset $U$ , we denote by ${\\rm Sup}_UV$ the least upper bound of $V$ in $U$ , when it exists.",
"Similarly, we denote by ${\\rm Inf}_UV$ the greatest lower bound of $V$ in $U$ , when it exists.",
"For $u\\in U$ , when we write $u={\\rm Sup}_UV$ (resp.",
"$u={\\rm Inf}_UV$ ), we mean that ${\\rm Sup}_UV$ exists (resp.",
"that ${\\rm Inf}_UV$ exists) and is equal to $u$ ."
]
] | 2012.05171 |
[
[
"Isomorphic Bisections of Cubic Graphs"
],
[
"Abstract Graph partitioning, or the dividing of a graph into two or more parts based on certain conditions, arises naturally throughout discrete mathematics, and problems of this kind have been studied extensively.",
"In the 1990s, Ando conjectured that the vertices of every cubic graph can be partitioned into two parts that induce isomorphic subgraphs.",
"Using probabilistic methods together with delicate recolouring arguments, we prove Ando's conjecture for large connected graphs."
],
[
"Introduction",
"Graph theory enjoys applications to a wide range of disciplines because graphs are incredibly flexible mathematical structures, capable of modelling very complex systems.",
"The study of complicated networks motivates the following important question: to what extent can a graph be decomposed into simpler subgraphs?",
"Spearheading this line of research is the classic problem of graph colouring, one of the oldest branches of graph theory.",
"When colouring a graph, one seeks to partition the vertices into as few independent (edgeless) sets as possible.",
"Indeed, subgraphs cannot get much simpler than independent sets, but such ambitious goals come at a cost.",
"Not only is the determination of a graph's chromatic number a notoriously difficult problem, but even when dealing with sparse graph classes, one can require many colours.",
"For instance, many $d$ -regular graphs cannot be partitioned into much fewer than $d$ independent sets.",
"Often one does not want to have such a large number of parts, and so it is natural to ask what can be achieved with fewer colours.",
"An early result along these lines was provided by Lovász [13], who proved that, given a graph $G$ of maximum degree $d$ , some number of colours $t$ , and a sequence $d_1, d_2, \\hdots , d_t$ with $\\sum _i d_i = d - t + 1$ , one can $t$ -colour the vertices of $G$ such that the $i$ th colour class induces a subgraph with maximum degree $d_i$ .",
"In particular, we can partition any graph into two subgraphs, each with half the maximum degree.",
"While reducing the maximum degree certainly simplifies graphs, it still allows for large connected subgraphs within the colour classes.",
"A different objective, therefore, is to find clustered colourings, which are colourings where each monochromatic component is of bounded size.",
"Alon, Ding, Oporowski and Vertigan [6] proved, among other results, that any graph with maximum degree four can be two-coloured such that the largest monochromatic components are of order at most 57.",
"However, they also constructed six-regular graphs with arbitrarily large monochromatic components in every two-colouring.",
"Answering one of their questions, Haxell, Szabó and Tardos [10] proved that clustered two-colourings of graphs of maximum degree five always exist.",
"Aside from improving the bounds on the monochromatic component sizes, subsequent research has sought to explore which graph classes admit clustered colourings, and what is possible with more colours.",
"A more general setting, in which the size of monochromatic components can grow with the size of graph, has also been studied.",
"For example, it was proved in [12] that any planar $n$ -vertex graph has a two-colouring in which all monochromatic components have size at most $O(n^{2/3})$ , and this is tight.",
"For more results on vertex colourings with small monochromatic components, we refer the interested reader to the survey of Wood [16].",
"The above results show that when we restrict our attention to graphs of small maximum degree, there is less room for complexity, and therefore we can prove strong partitioning results.",
"One might further try to achieve more than simply bounding the size of monochromatic components.",
"Indeed, it appears that in the case of cubic graphs is also possible to control the structure of such components.",
"An early result along these lines is due to Akiyama, Exoo and Harary [4], who proved that every cubic graph admits a two-edge-colouring in which every monochromatic component is a path, with a short proof of this result provided by Akiyama and Chvátal [3] soon after.",
"Although these results allowed for paths of unbounded length, Bermond, Fouquet, Habib and Péroche [8] conjectured that paths of length at most five suffice, a result that would be best possible.",
"Partial results, with larger but finite bounds on the path lengths, were obtained by Jackson and Wormald [11] and by Aldred and Wormald [5], before the conjecture was proven by Thomassen [15] in 1999.",
"Theorem 1.1 The edges of any cubic graph can be two-coloured such that each monochromatic component is a path of length at most five.",
"One can also try to find different structure in the partition than just finding path forests with small components.",
"One attractive conjecture of Wormald [17] from 1987 asks whether we can colour the edges of every cubic graph with an even number of edges so that the red and blue subgraphs form isomorphic linear forests.",
"This is known to hold for particular classes of graphs — it was proved for Jaeger graphs in [8], [17], and for some further classes of cubic graphs in [9].",
"Given Wormald's Conjecture, it is then natural to ask for analogues when colouring vertices rather than edges.",
"In this direction, the following striking conjecture was made by Ando in 1990's.",
"Conjecture 1.2 The vertices of any cubic graph can be two-coloured such that the two colour classes induce isomorphic subgraphs.",
"This conjecture was first mentioned in print by Abreu, Goedgebeur, Labbate and Mazzuoccolo [2], who drew connections between Conjecture REF , Wormald's conjecture and a conjecture of Ban and Linial [7] (see Section for discussion of the latter).",
"They further obtained computational results verifying the conjecture for graphs on at most 32 vertices.",
"Hence any counterexample to Conjecture REF must have at least 34 vertices.",
"Observe that any minimal counterexample must be connected, as the components of a graph can be coloured independently.",
"In this paper we essentially resolve this problem and prove that large connected cubic graphs satisfy Ando's conjecture.",
"This shows that there can be at most finitely many minimal counterexamples.",
"Theorem 1.3 Every sufficiently large connected cubic graph admits a two-vertex-colouring $\\varphi $ whose colour classes induce isomorphic subgraphs.",
"The remainder of this paper is organized as follows.",
"In Section , we present the proof of Theorem REF .",
"The proof uses probabilistic methods together with delicate recolouring arguments.",
"In the process of recolouring we use some gadgets which we call $P_t$ -reducers.",
"The existence of these structures is shown in Section .",
"Finally, we make some concluding remarks and discuss open problems in Section .",
"Notation.",
"We shall call the colours used in our two-colourings red and blue.",
"To show that the red and blue subgraphs are isomorphic, we will need to keep track of the monochromatic components.",
"Given a fixed graph $H$ and a red-/blue-colouring $\\varphi $ of the vertices of a graph $G$ , we denote by $r_H(G, \\varphi )$ the number of red components of $G$ under $\\varphi $ that are isomorphic to $H$ , and define $b_H(G,\\varphi )$ similarly for the blue components.",
"Paths will play a significant role in our argument, and we write $P_t$ for the path of length $t-1$ on $t$ vertices.",
"Given a colouring of a graph, we call the colouring with colours reversed the opposite colouring.",
"We will also often explore the neighbourhood of a vertex or a subset in a graph, and write $B_d(v)$ for the radius-$d$ ball around a vertex $v$ and write $B_d(X)$ for the set of all vertices of $G$ within distance at most $d$ from a subset $X$ .",
"We use $N^d(X)$ for the set of vertices at distance exactly $d$ from $X$ , and abbreviate $N^1(X)$ as $N(X)$ (so $N(X)$ is the set of external neighbours of $X$ in $G$ ).",
"Finally, all logarithms are to the base $e$ ."
],
[
"Proving the theorem",
"Given a large connected cubic graph, our goal is to colour the vertices such that the colour classes induce isomorphic subgraphs, and we shall find this colouring in two stages.",
"In the first, we take a semi-random vertex colouring, and show that this is very close to having the desired properties.",
"Then, in the second stage, we make some deterministic local recolourings to balance the two subgraphs and ensure they are truly isomorphic."
],
[
"A random colouring",
"We begin our search for the isomorphic bisection by showing that one may leverage Thomassen's result to define a random colouring that will produce monochromatic subgraphs that are nearly isomorphic.",
"The following proposition collects the properties of this initial colouring.",
"Proposition 2.1 For any $d \\in \\mathbb {N}$ and any sufficiently large cubic graph $G$ , there is a red-/blue-colouring $\\varphi $ of the vertices for which the following hold.",
"(a) Each monochromatic component is a path of length at most 5.",
"(b) For each $2 \\le t \\le 6$ , $\\left| r_{P_t}(G,\\varphi ) - b_{P_t}(G,\\varphi ) \\right| \\le 3 \\sqrt{n \\log n}$ .",
"(c) There are sequences of vertices $(u_i)_{i \\in [s]}$ and $(w_i)_{i \\in [s]}$ , for some $s \\ge 2^{-2d-5}n$ , such that all the balls $B_d(u_i)$ and $B_d(w_i)$ are pairwise disjoint, and for each $i \\in [s]$ , $B_d(u_i)$ and $B_d(w_i)$ induce isomorphic subgraphs with opposite colourings.",
"(d) The colouring $\\varphi $ is a bisection; that is, there are an equal number of red and blue vertices.",
"In our proof, we will need to show that several random variables, each a function of the random colouring, lie close to their expected values.",
"For this we employ McDiarmid's Inequality [14], which bounds large deviations in random variables defined on product probability spaces, provided they do not vary too much in response to changes in individual coordinates.",
"Theorem 2.2 (McDiarmid's Inequality, 1989) Let $X = (X_1, X_2, \\hdots , X_n)$ be a family of independent random variables, with $X_k$ taking values in a set $A_k$ for each $k \\in [n]$ .",
"Suppose further that there is some real-valued function $f$ defined on $\\prod _{k \\in [n]} A_k$ and some $c > 0$ such that $\\left| f(x)-f(x^{\\prime }) \\right| \\le c$ whenever $x$ and $x^{\\prime }$ differ only in a single coordinate.",
"Then, for any $m \\ge 0$ , $ \\mathbb {P} \\left( \\left| f(X) - \\mathbb {E}[f(X)] \\right| \\ge m \\right) \\le 2 \\exp \\left( \\frac{-2m^2}{c^2n} \\right).",
"$ Armed with this tool, we can now proceed with our proof.",
"[Proof of Proposition REF ] To prove the proposition, we shall first define a random colouring $\\varphi ^{\\prime }$ that satisfies properties (a), (b) and (c) while being very close to a bisection.",
"We shall then make a few small changes to obtain a bona fide bisection $\\varphi $ , maintaining the other properties in the process.",
"To define $\\varphi ^{\\prime }$ , we first apply Theorem REF to our cubic graph $G$ .",
"This results in a partition of the edges of $G$ into two spanning linear forests, $F_1$ and $F_2$ , neither of which contains a path of length 6 or more.",
"We then take $\\varphi ^{\\prime }$ to be a uniformly random proper colouring of $F_1$ .",
"Observe that this is equivalent to selecting, independently for each path component of $F_1$ , one of its two proper colourings uniformly.",
"Thus, our probability space is a product space, with each coordinate corresponding to a colouring of a path component of $F_1$ , of which there are at most $n$ .",
"We now verify in turn that the first three properties hold.",
"For (a), observe that since $\\varphi ^{\\prime }$ is a proper colouring of $F_1$ , the only monochromatic components are subgraphs of $F_2$ .",
"It thus follows immediately that each such component is a path of length at most 5.",
"To establish (b), we seek to apply Theorem REF to the random variable $r_{P_t}(G,\\varphi ^{\\prime })$ .",
"To do so, we must understand how changing the colouring of an individual path component of $F_1$ can affect $r_{P_t}(G, \\varphi ^{\\prime })$ .",
"Since the component in $F_2$ of each vertex is simply a path, changing the colouring of an individual vertex can affect at most two monochromatic copies of $P_t$ .",
"As each path component in $F_1$ is of length at most 5, the colouring of each such path therefore affects at most 12 copies of $P_t$ .",
"Hence, we can take $c = 12$ in Theorem REF .",
"It then follows that, for each $m > 0$ , $\\mathbb {P}\\left( \\left| r_{P_t}(G,\\varphi ^{\\prime }) - \\mathbb {E}[ r_{P_t}(G,\\varphi ^{\\prime }) ] \\right| \\ge m \\right) \\le 2 \\exp \\left( - m^2 / (72n) \\right)$ .",
"In particular, the probability that $r_{P_t}(G, \\varphi ^{\\prime })$ deviates from its expectation by more than $\\sqrt{n \\log n}$ is at most $2n^{-1/72} = o(1)$ .",
"The same argument holds for $b_{P_t}(G,\\varphi ^{\\prime })$ , and, by symmetry, we have $\\mathbb {E}[r_{P_t}(G,\\varphi ^{\\prime })] = \\mathbb {E}[b_{P_t}(G,\\varphi ^{\\prime })]$ .",
"Hence, the probability that $\\left| r_{P_t}(G,\\varphi ^{\\prime }) - b_{P_t}(G,\\varphi ^{\\prime }) \\right| \\ge 2 \\sqrt{n \\log n}$ is $o(1)$ .",
"Taking a union bound over all choices of $2 \\le t \\le 6$ , we find that with high probability we have $\\left| r_{P_t}(G, \\varphi ^{\\prime }) - b_{P_t}(G, \\varphi ^{\\prime }) \\right| \\le 2 \\sqrt{n \\log n}$ for all $2 \\le t \\le 6$ .",
"We next turn our attention to (c).",
"To start, we wish to select a sequence of vertices whose pairwise distances are all at least $2d+1$ .",
"Observe that, since $G$ is cubic, for each vertex $v$ we have $\\left| B_{2d}(v) \\right| \\le 1 + \\sum _{i=1}^{2d} 3 \\cdot 2^i < 3 \\cdot 2^{2d+1}$ .",
"We can then greedily select our desired vertices.",
"Indeed, after we add a vertex $v_i$ to our sequence, we eliminate the vertices in $B_{2d}(v_i)$ from our consideration.",
"This guarantees that we are now free to select any of the remaining vertices.",
"This results in a sequence of vertices $v_1, v_2, \\hdots , v_{s^{\\prime }}$ for some $s^{\\prime } \\ge \\tfrac{1}{3} 2^{-2d-1} n$ , with each pair at distance $2d+1$ or more.",
"In particular, the balls $B_d(v_i)$ are pairwise disjoint.",
"Now consider the subgraphs induced by each ball $B_d(v_i)$ .",
"These are subcubic graphs on at most $3 \\cdot 2^{d+1}$ vertices, and hence there are a finite number of possible isomorphism types.",
"Furthermore, the subgraphs inherit a vertex-colouring from $\\varphi ^{\\prime }$ .",
"Since the number of vertices in the ball is bounded, there can be at most $2^{3 \\cdot 2^{d+1}}$ different colourings.",
"Therefore, considering the isomorphism type and colouring of each ball, there is some finite number $\\kappa = \\kappa (d)$ of different classes the balls can fall into.",
"For each $j \\in [\\kappa ]$ , we denote by $Y_j$ the random variable counting the number of balls $B_d(v_i)$ , $i \\in [s^{\\prime }]$ , that belong to the $j$ th class.",
"Note that for each $i$ , the distribution of the colouring of $B_d(v_i)$ depends on how the paths of $F_1$ appear in the ball.",
"However, by symmetry, opposite colourings of $B_d(v_i)$ appear with equal probabilities.",
"It follows that if $j$ and $\\bar{j}$ represent opposite colourings of isomorphic balls, then $\\mathbb {E}[Y_j] = \\mathbb {E}[Y_{\\bar{j}}]$ .",
"Since the balls are pairwise disjoint, changing the colour of a single vertex can affect at most 1 ball $B_d(v_i)$ .",
"Hence, changing the colouring of a path component of $F_1$ can affect at most 6 balls.",
"Applying Theorem REF to the random variables $Y_j$ , we can therefore take $c = 6$ .",
"Setting $m = \\sqrt{n \\log n}$ , we find that with probability $1 - o(1)$ we have $\\left| Y_j - \\mathbb {E}[Y_j] \\right| \\le \\sqrt{n \\log n}$ for each $j \\in [\\kappa ]$ .",
"It then follows that for opposite colourings $j$ and $\\bar{j}$ we have $\\left| Y_j - Y_{\\bar{j}} \\right| \\le 2 \\sqrt{n \\log n}$ .",
"We can therefore match the balls $B_d(v_i)$ into isomorphic pairs with opposite colourings, with at most $2\\sqrt{n \\log n}$ unmatched balls for each of the $\\kappa $ isomorphism types.",
"Provided $n$ is suitably large, this is a total of at most $\\tfrac{1}{2} s^{\\prime }$ unmatched balls, leaving us with at least $\\tfrac{1}{4} s^{\\prime }$ matched pairs of balls.",
"This shows that the random colouring $\\varphi ^{\\prime }$ is very likely to satisfy (a), (b) and (c).",
"To finish the proof, we shall show that it is typically also close to being a bisection, and we can make it one without destroying the other properties.",
"Consider the random variable that is the difference between the numbers of red and blue vertices.",
"Since every path component in $F_1$ of odd length contributes an equal number of red and blue vertices to the colouring, any discrepancy in the colour class sizes must come from the paths of even length.",
"Furthermore, each such path contributes a difference in the colour class sizes of exactly 1, with equal probability in either direction.",
"Thus, the expected difference is 0, and recolouring a path component can affect the difference by at most 2.",
"We thus make a final appeal to Theorem REF , finding that with probability $1 - o(1)$ , the difference in the two colour classes is of size at most $\\tfrac{1}{10} \\sqrt{n \\log n}$ .",
"In summary, with positive probability it holds that the colouring $\\varphi ^{\\prime }$ satisfies properties (a), (b) and (c), and that the difference $\\Delta $ between the number of red and blue vertices is at most $\\tfrac{1}{10} \\sqrt{n \\log n}$ .",
"We can therefore obtain a bisection $\\varphi $ by taking $\\varphi ^{\\prime }$ and replacing the colouring of $\\Delta $ even paths in $F_1$ with their opposite colourings, thereby satisfying (d).",
"Note that $\\varphi $ remains a proper colouring of $F_1$ , and therefore property (a) is preserved.",
"Moreover, we are changing the colours of at most $5 \\Delta $ vertices.",
"As previously discussed, changing the colour of a vertex can affect at most two monochromatic copies of $P_t$ , and hence, for each $2 \\le t \\le 6$ , the difference $\\left| r_{P_t}(G,\\varphi ) - b_{P_t}(G,\\varphi ) \\right|$ deviates from $\\left| r_{P_t}(G,\\varphi ^{\\prime }) - b_{P_t}(G, \\varphi ^{\\prime }) \\right|$ by at most $\\sqrt{n \\log n}$ .",
"In particular, we have $\\left| r_{P_t}(G,\\varphi ) - b_{P_t}(G, \\varphi ) \\right| \\le 3 \\sqrt{n \\log n}$ for each $2 \\le t \\le 6$ , establishing (b).",
"Similarly, under $\\varphi ^{\\prime }$ we had at least $\\tfrac{1}{4} s^{\\prime } \\ge \\tfrac{1}{3} 2^{-2d-3} n$ pairs of isomorphic balls with opposite colourings.",
"Since these balls are pairwise disjoint, each recoloured vertex can affect at most one such pair.",
"Hence, under $\\varphi $ we still have at least $\\tfrac{1}{4} s^{\\prime } - 5 \\Delta $ such pairs which, if $n$ is large enough, is at least $2^{-2d - 5} n$ pairs.",
"Thus, (c) holds for $\\varphi $ as well, completing the proof."
],
[
"Correcting the bisection",
"The colouring $\\varphi $ from Proposition REF is close to being the bisection we need, but the number of short paths in the red and blue subgraphs can be a little off.",
"In the second stage of our argument, we will make local changes to the colouring to correct these discrepancies.",
"We do this via gadgets we call $P_t$ -reducers.",
"These are flexible subgraphs, in the sense that they admit two different colourings which have the same monochromatic subgraph counts, except for short paths.",
"By choosing an appropriate colouring, then, we can adjust the values of $r_{P_t}(G,\\varphi )$ and $b_{P_t}(G, \\varphi )$ , making them equal.",
"When doing so, though, we do need to take care that our changes do not leak out and affect subgraph counts elsewhere.",
"We thus insulate the $P_t$ -reducer by colouring its boundary with alternating colours, thus preventing any monochromatic components from extending outwards.",
"Definition 2.3 ($P_t$ -reducer) Given some $t \\ge 3$ , an induced subgraph $R \\subseteq G$ is a $P_t$ -reducer if there are two vertex colourings $\\psi _1, \\psi _2$ of $B_2(R)=R\\cup N(R)\\cup N^2(R)$ such that: (i) the two colourings have the same number of red (and therefore also blue) vertices, (ii) in both $\\psi _1$ and $\\psi _2$ , all vertices of $N(R)$ are blue and all vertices of $N^2(R)$ are red, (iii) $r_H(B_2(R), \\psi _1) = r_H(B_2(R), \\psi _2)$ and $b_H(B_2(R), \\psi _1) = b_H(B_2(R), \\psi _2)$ , unless $H = P_{\\ell }$ for some $2 \\le \\ell \\le t$ , and (iv) $r_{P_t}(B_2(R), \\psi _2) = r_{P_t}(B_2(R), \\psi _1) -1 $ and $b_{P_t}(B_2(R),\\psi _1) = b_{P_t}(B_2(R), \\psi _2)$ .",
"Of course, this definition is only useful if we can actually find $P_t$ -reducers in our graph.",
"Fortunately, they happen to be ubiquitous in cubic graphs.",
"Proposition 2.4 Let $G$ be a connected cubic graph on more than $3 \\cdot 2^{50}$ vertices, and let $v \\in V(G)$ be arbitrary.",
"Then, for every $3 \\le t \\le 6$ , there is a $P_t$ -reducer in $B_{50}(v)$ .",
"We shall prove Proposition REF in Section , but first we show how one can use $P_t$ -reducers to obtain the desired isomorphic bisection.",
"[Proof of Theorem REF ] Given a sufficiently large connected cubic graph $G$ , set $d = 57$ and let $\\varphi _0$ be the bisection given by Proposition REF with this $d$ .",
"We then have $r_H(G,\\varphi _0) = b_H(G,\\varphi _0)$ for all $H$ except $H = P_t$ , $2 \\le t \\le 6$ .",
"For these paths, we have $\\left| r_{P_t}(G,\\varphi _0) - b_{P_t}(G,\\varphi _0) \\right| \\le 3\\sqrt{n \\log n}$ , and we shall correct these imbalances one at a time, in decreasing order of path length.",
"We start with $t = 6$ .",
"Suppose, without loss of generality, that we have $r_{P_6}(G,\\varphi _0) > b_{P_6}(G,\\varphi _0)$ .",
"Take the first pair of isomorphic and oppositely-coloured balls, $B_{57}(u_1)$ and $B_{57}(w_1)$ .",
"By Proposition REF , we can find some $P_6$ -reducer $R \\subseteq G[B_{50}(u_1)]$ , and therefore we find a corresponding oppositely-coloured copy $\\bar{R} \\subseteq G[B_{50}(w_1)]$ as well.",
"We then recolour $B_2(R)$ with $\\psi _2$ , and colour $B_2(\\bar{R})$ with $\\bar{\\psi }_1$ , the opposite colouring of $\\psi _1$ .",
"Let $\\varphi _1$ be the resulting colouring.",
"Note that this is still a bisection, since $\\psi _1$ and $\\psi _2$ have the same number of red vertices, and we have made symmetric changes in $B_2(R)$ and $B_2(\\bar{R})$ .",
"We next claim that $r_H(G,\\varphi _1) = b_H(G,\\varphi _1)$ for all graphs except $P_t$ , $2 \\le t \\le 6$ , and that $r_{P_6}(G,\\varphi _1) - b_{P_6}(G,\\varphi _1) = r_{P_6}(G,\\varphi _0) - b_{P_6}(G, \\varphi _0) - 1$ ; that is, the difference in monochromatic copies of $P_6$ is reduced by 1.",
"Observe that we only need to concern ourselves with monochromatic components contained within $B_{57}(u_1)$ and $B_{57}(w_1)$ .",
"Indeed, in $\\varphi _0$ , all monochromatic components were paths of length at most 5.",
"As we only recoloured vertices in $B_2(R)$ and $B_2(\\bar{R})$ , which are contained in $B_{50}(u_1)$ and $B_{50}(w_1)$ respectively, any affected components do not reach outside the original balls $B_{57}(u_1)$ and $B_{57}(w_1)$ .",
"Also note that, since these balls had the opposite colorings, before the recoloring there was one to one correspondence between red components in the first ball and blue components in the second ball and vice versa.",
"First we consider components not fully contained within $B_2(R)$ or $B_2(\\bar{R})$ .",
"Recall that in $\\psi _2$ , the vertices in $N(R)$ and $N^2(R)$ receive opposite colours, with the same being true of the colouring $\\bar{\\psi }_1$ of $B_2(\\bar{R})$ .",
"Thus, any such component in $B_{57}(u_1)$ can only contain (red) vertices from $N^2(R)$ , together with some vertices in $B_{57}(u_1) \\setminus B_2(R)$ .",
"However, since $B_{57}(u_1)$ and $B_{57}(w_1)$ are isomorphic and oppositely-coloured, these components are in bijection with isomorphic blue components in $B_{57}(w_1)$ , and hence no additional discrepancy is introduced through these components.",
"This leaves us with components fully within $B_2(R)$ and $B_2(\\bar{R})$ , where the properties of the $P_6$ -reducer come into play.",
"For any component $H$ that is not a path of length at most 6, we have $r_H(B_2(R),\\psi _2) = r_H(B_2(R),\\psi _1) = b_H(B_2(\\bar{R}), \\bar{\\psi }_1)$ , and similarly $b_H(B_2(R),\\psi _2) = r_H(B_2(\\bar{R}),\\bar{\\psi }_1)$ .",
"Thus the monochromatic copies of these components remain balanced.",
"As for paths of length 6, we have $r_{P_6}(B_2(R),\\psi _2) = r_{P_6}(B_2(R),\\psi _1) - 1 = b_{P_6}(B_2(\\bar{R}),\\bar{\\psi }_1) - 1,$ while $b_{P_6}(B_2(R), \\psi _2) = r_{P_6}(B_2(\\bar{R}),\\bar{\\psi }_1)$ , and so the difference between red and blue copies of $P_6$ is indeed reduced by 1 in $\\varphi _1$ .",
"Although this recolouring could create monochromatic components that are not paths of length at most 6, these must be fully contained within the balls $B_{57}(u_1)$ and $B_{57}(w_1)$ , which are disjoint from all the other balls.",
"We can thus repeat this process a further $r_{P_6}(G, \\varphi _1) - b_{P_6}(G, \\varphi _1)$ times, using the next pairs of balls $B_{57}(u_i)$ and $B_{57}(w_i)$ in the sequence.",
"This gives us a sequence of colourings, $\\varphi _1, \\varphi _2, \\hdots , \\varphi _k$ , the last of which will satisfy $r_{P_6}(G, \\varphi _k) = b_{P_6}(G, \\varphi _k)$ .",
"Furthermore, since each $P_6$ -reducer is of constant size, every step of the process can only have created a constant number of monochromatic $P_5$ -components.",
"Thus, we still have $\\left| r_{P_5}(G,\\varphi _k) - r_{P_5}(G, \\varphi _k) \\right| = O \\left( \\sqrt{n \\log n} \\right)$ .",
"We can therefore use Proposition REF to find $P_5$ -reducers in the next $O \\left( \\sqrt{n \\log n} \\right)$ pairs of balls and balance the monochromatic $P_5$ counts.",
"Once those are handled, we proceed to fixing the $P_4$ counts, and then finally the $P_3$ counts.",
"Observe that we require a total of $O \\left( \\sqrt{n \\log n} \\right)$ steps, and since Proposition REF guarantees us $\\Omega (n)$ pairs of isomorphic balls, we can see this process through to completion.",
"Let $\\varphi $ be the final colouring obtained.",
"Following our corrections with the $P_t$ -reducers, we know that for every $H \\ne P_2$ , we have $r_H(G,\\varphi ) = b_H(G, \\varphi )$ .",
"However, since $G$ is cubic and $\\varphi $ is a bisection, double-counting the edges between the colour classes shows that the total number of monochromatic red edges must equal the number of blue edges.",
"It therefore follows that $r_{P_2}(G,\\varphi ) = b_{P_2}(G, \\varphi )$ as well, and hence the subgraphs induced by the red and the blue vertices are isomorphic."
],
[
"Constructing $P_t$ -reducers",
"To complete the proof of Theorem REF , we need to prove Proposition REF , showing that we can find $P_t$ -reducers in the local neighbourhoods of every vertex.",
"We will first show a very simple construction that works in graphs of girth at least 7, thereby providing a short proof of Conjecture REF for large cubic graphs without short cycles.",
"The proof in the general case is a little more involved, requiring analysis of a few different cases, and can be found in Section REF ."
],
[
"Graphs of girth at least 7",
"Our construction is based around geodesics — that is, shortest paths in the graph between a pair of vertices.",
"We start with a lemma about neighbourhoods of geodesics.",
"Lemma 3.1 Let $G$ be a cubic graph of girth at least 7, and let $P$ be a geodesic in $G$ .",
"Then $N(P)$ , the set of external neighbours of vertices on $P$ , is an independent set.",
"Let $u, w \\in N(P)$ be arbitrary vertices in the neighbourhood of $P$ , and suppose for contradiction we have an edge $\\lbrace u,w\\rbrace $ .",
"Let $u^{\\prime }, w^{\\prime } \\in P$ be neighbours of $u$ and $w$ respectively.",
"If $u^{\\prime }$ and $w^{\\prime }$ are at distance at most 3 along the path $P$ , then, together with the edges $\\lbrace u^{\\prime }, u\\rbrace , \\lbrace u,w\\rbrace $ and $\\lbrace w,w^{\\prime }\\rbrace $ , we would obtain a cycle of length at most 6, contradicting our assumption on the girth of $G$ .",
"On the other hand, if $u^{\\prime }$ and $w^{\\prime }$ are at distance at least 4 on the path $P$ , then we could shorten the path by rerouting it between $u^{\\prime }$ and $w^{\\prime }$ through $u$ and $w$ instead.",
"This contradicts $P$ being a geodesic.",
"Hence $u$ and $w$ cannot be adjacent, and thus it follows that $N(P)$ is an independent set.",
"This shows that the structure around a geodesic is particularly simple.",
"As a result, we can easily find $P_t$ -reducers, as we now show.",
"[Proof of Proposition REF (girth $\\ge 7$ )] Since there are at most $3 \\cdot 2^{t+1}<3 \\cdot 2^{50}$ vertices at distance at most $t$ from the vertex $v$ , there must be a vertex $w$ at distance exactly $t+1$ from $v$ .",
"Let $P$ be a geodesic from $v$ to $w$ , with vertices $v = v_0, v_1, v_2, \\hdots , v_t, v_{t+1} = w$ .",
"We'll show that $P$ is a $P_t$ -reducer.",
"We now define the colourings $\\psi _1$ and $\\psi _2$ of $B_2(P)=P\\cup N(P)\\cup N^2(P)$ .",
"In both colourings, we colour all vertices in $N^2(P)$ red and all vertices in $N(P)$ blue.",
"When colouring the path $P$ , we make all vertices red, except in $\\psi _1$ the vertex $v_t$ is coloured blue, while in $\\psi _2$ the vertex $v_{t-1}$ is coloured blue instead.",
"Figure: NO_CAPTIONClearly both $\\psi _1$ and $\\psi _2$ have the same number of red vertices, and thus property (i) is satisfied.",
"Property (ii) is satisfied by the definition of the colouring on $N(P)$ and $N^2(P)$ .",
"Finally, we inspect the monochromatic components of $B_2(P)$ under $\\psi _1$ and $\\psi _2$ to verify properties (iii) and (iv).",
"Let us start with the blue components.",
"In the two colourings, the blue vertices are those in $N(P)$ , together with one of $v_{t-1}$ or $v_t$ .",
"By Lemma REF , the vertices in $N(P)$ form an independent set.",
"As $v_{t-1}$ and $v_t$ are internal vertices on the path $P$ , they have exactly one neighbour in $N(P)$ .",
"Thus in both $\\psi _1$ and $\\psi _2$ , the blue vertices induce one isolated edge and $\\left| N(P) \\right| - 1$ isolated vertices, and thus $b_H(B_2(P),\\psi _1) = b_H(B_2(P), \\psi _2)$ for every $H$ .",
"Since $N(P)$ is entirely blue, the red components of $B_2(P)$ are either wholly contained in $N^2(P)$ or in $P$ .",
"In the former case, since all vertices in $N^2(P)$ are red, both colourings have the same components.",
"In the latter case, the path $P$ is broken into two red paths by the sole blue vertex, which is $v_t$ in $\\psi _1$ and $v_{t-1}$ in $\\psi _2$ .",
"Thus $\\psi _1$ has a red $P_t$ and a red $P_1$ , while $\\psi _2$ has a red $P_{t-1}$ and a red $P_2$ .",
"This shows that (iii) and (iv) are indeed satisfied, and so the colourings $\\psi _1$ and $\\psi _2$ bear witness to $P$ being a $P_t$ -reducer."
],
[
"The general case",
"In general, we cannot expect such simple structure around geodesic paths.",
"However, induced paths will still play a key role in our construction of $P_t$ -reducers, as shown by the following lemma.",
"Lemma 3.2 In a cubic graph $G$ , let $Q$ be an induced path of length $t+1$ or $t+2$ with vertices labelled as $Q=(u, x,z, q_{1}, \\dots , q_{t-1})$ or $Q=(u, x,y, z, q_{1}, \\dots , q_{t-1})$ respectively.",
"Let $v$ a vertex outside $Q$ with the edges $vx$ and $vz$ present.",
"Then $B_{3}(Q)$ contains a $P_t$ -reducer.",
"Depending on the length of $Q$ , define $\\psi _1$ and $\\psi _2$ on $R:=Q\\cup \\lbrace v\\rbrace $ as follows: Figure: NO_CAPTIONDefine both $\\psi _1$ and $\\psi _2$ to be blue on $N(R)$ and red on $N^2(R)$ .",
"We claim that this makes $R$ a $P_t$ -reducer.",
"Properties (i) and (ii) are immediate by construction.",
"Notice that $b_{H}(B_2(R),\\psi _1)=b_{H}(B_2(R),\\psi _2)$ for all $H$ .",
"Indeed the only blue component that changes between the two colourings is the one that contains $v$ .",
"However, since the graph is cubic, neither $x$ nor $z$ has any neighbours in $N(R)$ , and therefore it is easy to see that this component is isomorphic between the two colourings.",
"Finally notice that the only red components that change between the two colourings are the two red subpaths of $Q$ .",
"The colouring $\\psi _1$ has a red $P_t$ and a red $P_1$ , while $\\psi _2$ has a red $P_{t-1}$ and a red $P_2$ .",
"We also have $r_{H}(B_2(R),\\psi _1)=r_{H}(B_2(R),\\psi _2)$ for all other $H$ .",
"This completes the proof of (iii) and (iv) Recall that a geodesic path $Q$ is a shortest length path between its endpoints, and note that a length 14 geodesic path exists in $B_{14}(v)$ for every vertex $v$ in a large connected cubic graph.",
"An important consequence of the above lemma is that it reduces us to the case when every vertex outside a geodesic path $Q$ has at most one neighbour on $Q$ .",
"Using this we can prove the following lemma.",
"Lemma 3.3 Let $Q$ be a geodesic path of length 14 in a cubic graph $G$ .",
"Then $B_3(Q)$ either contains $P_t$ -reducers for all $3\\le t\\le 6$ or it contains, for each $3 \\le t \\le 6$ , an induced subgraph $R$ which has two colourings $\\psi _1$ and $\\psi _2$ of $B_2(R)$ that satisfy (ii), (iii), and (iv) of the definition of $P_{t}$ -reducer and also: (i') $\\psi _1$ has one more red vertex than $\\psi _2$ .",
"Let $Q=(q_1, \\dots , q_{15})$ .",
"For each $2 \\le i \\le 14$ , let $r_i$ be the neighbour of $q_i$ outside $Q$ .",
"These are unique because the graph is cubic.",
"If $r_i=r_j$ for some $2 \\le i< j \\le 14$ , the fact that $Q$ is a geodesic path forces $j=i+ 1$ or $j=i+ 2$ .",
"In either case, we find $P_t$ -reducers for all $3\\le t\\le 6$ using Lemma REF .",
"Thus we can assume that $r_2, \\dots , r_{14}$ are all distinct.",
"Next, suppose that $r_ir_{i+1}$ is an edge for all $3 \\le i\\le 10$ .",
"Since $G$ is cubic, this determines all the neighbours of $r_i$ for $4 \\le i \\le 10$ .",
"It follows that for $t\\in \\lbrace 3,4,5,6\\rbrace $ , the path $Q^{\\prime }=(q_2, q_3, q_4, r_4, r_5, \\dots , r_{t+4})$ is induced, has length $t+3$ , and the vertex $v:=r_3$ has neighbours $q_3$ and $r_4$ .",
"Thus Lemma REF applies to give us $P_t$ -reducers for all $3 \\le t \\le 6$ .",
"Hence, we may assume that $r_ir_{i+1}$ is a non-edge for some $3\\le i\\le 10$ .",
"Fix $t\\in \\lbrace 3,4,5,6\\rbrace $ .",
"Define the subpath $Q^{\\prime }=(q_{i-1}, \\dots , q_{i+t-1})$ Define $\\psi _1$ and $\\psi _2$ on $R:=Q^{\\prime }\\cup \\lbrace r_i, r_{i+1}\\rbrace $ as follows: Figure: NO_CAPTIONExtend both $\\psi _1$ , and $\\psi _2$ to be blue on $N(R)$ and red on $N^2(R)$ .",
"We claim that then these colourings of $B_2(R)$ satisfy (i'), (ii), (iii), and (iv).",
"Properties (i') and (ii) are immediate by construction.",
"For (iii) and (iv), notice that the only components that change between the two colourings are the blue component containing $q_i$ (which, by virtue of the fact that all neighbours of $q_i$ and $q_{i+1}$ lie within $R$ , is a $P_1$ in $\\psi _1$ and a $P_2$ in $\\psi _2$ ) and the red component containing $q_{i+2}$ (which is a $P_t$ in $\\psi _1$ and splits into a $P_1$ and a $P_{t-2}$ in $\\psi _2$ ).",
"The above lemma implies Proposition REF for $t\\ge 4$ .",
"[Proof of Proposition REF for $t\\ge 4$ ] Note that within $B_{45}(v)$ , we can find two geodesic paths of length 14, which we call $Q$ and $R$ , at distance 10 from each other.",
"Thus the balls $B_3(Q)$ and $B_3(R)$ are disjoint.",
"If either of these balls contains a $P_t$ -reducer, we are done.",
"Otherwise, by Lemma REF , we can assume the existence of some $S_1 \\subseteq B_3(Q)$ with colourings $\\psi _1^{S_1,t}$ and $\\psi _2^{S_1,t}$ of $B_2(S_1)$ satisfying (ii), (iii), (iv) of the definition of a $P_t$ -reducer and also (i') from Lemma REF .",
"Similarly, we find $S_2 \\subseteq B_3(R)$ with colourings $\\psi _1^{S_2,t-1}, \\psi _2^{S_2,t-1}$ of $B_2(S_2)$ satisfying (ii), (iii), (iv) of the definition of a $P_{t-1}$ -reducer and also (i') above.",
"We now construct colourings $\\psi _1$ and $\\psi _2$ of $B_{50}(v)$ satisfying the definition of a $P_t$ -reducer.",
"On $B_2(S_1)$ , $\\psi _1$ and $\\psi _2$ agree with $\\psi _1^{S_1,t}$ and $\\psi _2^{S_1,t}$ respectively.",
"On $B_2(S_2)$ , $\\psi _1$ and $\\psi _2$ have the opposite colourings of $\\psi _1^{S_2,t-1}$ and $\\psi _2^{S_2,t-1}$ .",
"Outside $B_2(S_1)$ and $B_2(S_2)$ , $\\psi _1$ and $\\psi _2$ are entirely blue, except on $N^{50}(v)$ , where they are red.",
"We check that these colourings do indeed result in a $P_t$ -reducer $B_{48}(v)$ .",
"Using property (i') of the pairs $\\psi _1^{S_1,t}, \\psi _2^{S_1,t}$ and $\\psi _1^{S_2,t-1}, \\psi _2^{S_2,t-1}$ , and the fact that we use the opposite colourings on $S_2$ , we see that $\\psi _1$ and $\\psi _2$ have an equal number of red vertices in total, satisfying property (i) of a $P_t$ -reducer.",
"Property (ii) holds by the third bullet point of the construction.",
"Property (iii) is immediate from it holding for the colourings $\\psi _1^{S_1,t}$ , $\\psi _2^{S_1,t}$ , $\\psi _1^{S_2,t-1}$ and $\\psi _2^{S_2,t-1}$ , since the property (ii) for these colourings ensures that the monochromatic components involving vertices whose colours change are fully contained within $B_2(S_1)$ and $B_2(S_2)$ respectively.",
"For property (iv), note we lose exactly one red copy of $P_t$ when going from $\\psi _1^{S_1,t}$ to $\\psi _2^{S_1,t}$ on $B_2(S_1)$ , while the number of blue copies of $P_t$ stays the same.",
"On the other hand, going from $\\psi _1^{S_2,t-1}$ to $\\psi _2^{S_2,t-1}$ on $B_2(S_2)$ does not affect the number of the monochromatic copies of $P_t$ (by property (iii)).",
"Thus in total exactly one red $P_t$ is lost, as required.",
"The above proof doesn't work for $t=3$ because our proof of Lemma REF doesn't work for $t=2$ .",
"Thus for $t=3$ , we need a different proof of the proposition.",
"[Proof of Proposition REF for $t=3$ ] Let $Q=(q_0, \\dots , q_{20})$ be a length 20 geodesic within distance 20 of $v$ .",
"For each $1 \\le i \\le 19$ , let $r_i$ be the (unique) neighbor of $q_i$ .",
"As in the proof of Lemma REF , if $r_i=r_j$ for some $1 \\le i< j\\le 19$ , we must have $j=i+ 1$ or $j=i+ 2$ , and then we find a $P_3$ -reducer using Lemma REF .",
"Thus we may assume that $r_1, \\dots , r_{19}$ are all distinct.",
"Claim 3.4 We have at least one of the following: For some $3 \\le i \\le 9$ , none of the edges $r_ir_{i+1}, r_ir_{i+2}$ or $r_{i+1}r_{i+2}$ are present.",
"For some $3 \\le i \\le 10$ , the edge $r_{i}r_{i+1}$ is present and either the edges $r_ir_{i+2}$ and $r_ir_{i+3}$ are absent, or the edges $r_{i-1}r_{i+1}$ and $r_{i-2}r_{i+1}$ are absent.",
"For some $3 \\le i \\le 8$ , the edge $r_{i}r_{i+1}$ is present, the edges $r_{i+1}r_{i+2}, r_{i+1}r_{i+3}, r_{i-1}r_i$ and $r_{i-2}r_i$ are absent, and either $r_ir_{i+3}$ or $r_{i-2}r_{i+1}$ is present.",
"First observe that the three edges $r_3 r_5, r_5 r_7$ and $r_7 r_9$ cannot all be present, as that would contradict $Q$ being a geodesic.",
"Hence, if (a) does not hold, we must have the edge $r_i r_{i+1}$ for some $3 \\le i \\le 8$ .",
"Next, suppose that $r_{i+1}r_{i+2}$ is also present.",
"In this case, $N(r_{i+1})=\\lbrace q_{i+1}, r_i, r_{i+2}\\rbrace $ , which implies that $r_{i-1}r_{i+1}$ and $r_{i-2}r_{i+1}$ are both absent (since $r_{i-1}, r_{i-2}$ are distinct from $q_{i+1}, r_i, r_{i+1}$ ).",
"This leaves us in case (b).",
"The same argument shows that we are in case (b) if any of the edges $r_{i+1}r_{i+3}, r_{i-1}r_i$ or $r_{i-2}r_i$ were present.",
"Thus we can assume that $r_{i+1}r_{i+2}, r_{i+1}r_{i+3}, r_{i-1}r_i$ and $r_{i-2}r_i$ are absent.",
"If (c) doesn't hold, then both $r_ir_{i+3}$ and $r_{i-2}r_{i+1}$ must be absent.",
"Then, to avoid case (b), both $r_ir_{i+2}$ and $r_{i-1}r_{i+1}$ must be present.",
"Hence $N(r_i)=\\lbrace q_i, r_{i+1}, r_{i+2}\\rbrace $ and $N(r_{i+1})=\\lbrace q_{i+1}, r_{i-1}, r_i\\rbrace $ .",
"Recalling that $r_{i+1}r_{i+2}$ and $r_{i+1}r_{i+3}$ are absent, note that $r_{i+2}r_{i+3}$ must be present, as otherwise we would be in case (a) for $i^{\\prime } = i+1$ .",
"However, as $r_ir_{i+3}$ and $r_{i+1}r_{i+3}$ are both absent, we are then in case (b) with $i^{\\prime }=i+2$ .",
"We now find $P_3$ -reducers in each of the above three cases.",
"Set $R=\\lbrace q_{i-1}, q_i, q_{i+1}, q_{i+2}, q_{i+3}, r_i, r_{i+1}, r_{i+2} \\rbrace $ and define colourings $\\psi _1, \\psi _2$ on $R$ as follows: Figure: NO_CAPTIONDefine both $\\psi _1$ , and $\\psi _2$ to be blue on $N(R)$ and red on $N^2(R)$ .",
"It is immediate that this colouring satisfies the definition of $P_3$ -reducer.",
"Without loss of generality, we may suppose that $r_ir_{i+1}$ is present and the edges $r_ir_{i+2}, r_ir_{i+3}$ are absent (the other case is symmetric to this by reversing the order of the path $Q$ ).",
"Let $u$ be the third neighbour of $r_i$ , i.e.",
"$N(r_{i})=\\lbrace q_i, r_{i+1}, u\\rbrace .$ Consider the path $Q^{\\prime }=(u, r_i, q_i, q_{i+1}, q_{i+2}, q_{i+3})$ .",
"We claim that this is an induced path.",
"Indeed the edges $r_iq_{j}$ are all absent for $j\\ne i$ , so the only way this could be non-induced is if $u=r_{i+1}, r_{i+2}$ or $r_{i+3}$ .",
"But $u$ was the third neighbour of $r_i$ so $u \\ne r_{i+1}$ by definition, and $r_{i+2}, r_{i+3}\\notin N(r_i)$ by the definition of case (b).",
"Thus $Q^{\\prime }$ is indeed induced, and by Lemma REF (applied with $v=r_{i+1}$ ) there is a $P_3$ -reducer in $B_3(Q^{\\prime })$ .",
"Without loss of generality, we may suppose that $r_ir_{i+1}$ and $r_{i}r_{i+3}$ are present and $r_{i+1}r_{i+2},$ $r_{i+1}r_{i+3},$ $r_{i-1}r_i,$ $r_{i-2}r_i$ are absent (the other case is symmetric).",
"Set $R=\\lbrace q_{i-1}, q_i, q_{i+1}, q_{i+2}, q_{i+3}, r_i, r_{i+1}, r_{i+2}, r_{i+3} \\rbrace $ and define colourings $\\psi _1, \\psi _2$ on $R$ as shown below.",
"Extend both $\\psi _1$ and $\\psi _2$ to be blue on $N(R)$ and red on $N^2(R)$ .",
"Since $q_i, r_i, q_{i+1}$ have no neighbours in $N(R)$ , it is easy to check that this colouring satisfies the definition of $P_3$ -reducer.",
"Figure: NO_CAPTION"
],
[
"Concluding remarks",
"In this paper we have proven Ando's conjecture for all large connected cubic graphs.",
"While this only leaves a finite number of graphs to be checked, and there is room to optimise the constants in our proof, there will still be too many cases to be handled computationally.",
"Hence, a complete resolution of Conjecture REF will likely require some additional ideas.",
"In this section we indicate some ways in which our proof could be modified, which might help make progress towards the full conjecture, and close with some related open problems."
],
[
"A simpler starting block",
"While our proof is relatively short, one could argue that it is not fully self-contained, as we use Thomassen's theorem, which is already a very significant result.",
"However, it is not crucial in our proof that, in the partition of the edges of $G$ into linear forests $F_1$ and $F_2$ , the paths have length at most five.",
"We could therefore replace Theorem REF with one of its predecessors [5], [11], which have simpler proofs, but allow for longer paths.",
"Although this comes at the cost of requiring $P_t$ -reducers for larger values of $t$ , our constructions readily generalise to longer paths.",
"This is especially easy when we assume $G$ has large girth, resulting in a truly short proof of this special case."
],
[
"Fewer reducers",
"Alternatively, one might seek to reduce the amount of work done in Section , when constructing the $P_t$ -reducers.",
"A potential route to simplification lies in the observation that, when using Theorem REF , it was not very important that the paths in $F_1$ were so strongly bounded in length.",
"Indeed, we only used the lengths of the paths in $F_1$ to bound the Lipschitz constant $c$ in our application of Theorem REF , and we can afford for this to be as large as $n^{o(1)}$ .",
"On the other hand, if we can limit the lengths of the paths in $F_2$ to some $\\ell \\le 4$ , then we would only need to construct $P_t$ -reducers for $2 \\le t \\le \\ell +1$ .",
"Question 4.1 What is the smallest $\\ell $ for which the edges of any connected cubic graph on $n$ vertices can be partitioned into two spanning linear forests $F_1$ and $F_2$ , such that the paths in $F_1$ are of length $n^{o(1)}$ , and the paths in $F_2$ are of length at most $\\ell $ ?",
"It is worth noting that, while the five in Theorem REF is best possible, the examples of tightness given in [15] are the two cubic graphs on six vertices.",
"It would be interesting to know if there are arbitrarily large tight constructions, or if, when dealing with large connected graphs, one can achieve $\\ell = 4$ even in the symmetric setting.",
"Furthermore, we can weaken Question REF , as in our application $F_1$ does not have to be a linear forest, but rather a bipartite graph with bounded components."
],
[
"Stronger conjectures",
"While Theorem REF sheds light on the structure of large cubic graphs, showing that they can be partitioned into isomorphic induced subgraphs, it does not directly address the motivating question raised in Section , as there are no guarantees that these subgraphs are simple.",
"However, by analysing our proof, one can obtain some further information about the subgraphs obtained.",
"As stated in Proposition REF , the only monochromatic components in the initial random colouring are paths of length at most five.",
"When we then use the $P_t$ -reducers to make the subgraphs isomorphic, we can introduce more complicated monochromatic components.",
"However, since the $P_t$ -reducers are all isolated within balls of bounded radius, it follows that the components in the isomorphic subgraphs are of bounded size.",
"In particular, if we assume that our connected graph has large girth,By carefully considering the $P_t$ -reducers from Section REF , it suffices to assume girth at least 15. it follows that the isomorphic subgraphs are forests.",
"Moreover, we need never have vertices of degree three in the isomorphic subgraphs, as these can be recoloured (in pairs) to become isolated vertices of the opposite colour.",
"Thus, we in fact partition large connected cubic graphs of large girth into isomorphic linear forests.",
"It was conjectured in [2] that every cubic graph should admit such a partition; the challenge lies in removing the girth condition.",
"Ban and Linial [7] went even further, conjecturing that much more should be true when we restrict our attention to two-edge-connected cubic graphs.",
"Conjecture 4.2 The vertices of every bridgeless cubic graph, with the exception of the Petersen graph, can be two-coloured such that the two colour classes induce isomorphic matchings.",
"The conjecture has been proven for three-edge-colourable graphs [7] and for claw-free graphs [1], but is otherwise open.",
"It would be very interesting to see to what extent our methods can be applied to this conjecture, as well as to Wormald's conjecture on partitioning the edges of cubic graphs into isomorphic linear forests."
]
] | 2012.05222 |
[
[
"On the Lattice of Conceptual Measurements"
],
[
"Abstract We present a novel approach for data set scaling based on scale-measures from formal concept analysis, i.e., continuous maps between closure systems, and derive a canonical representation.",
"Moreover, we prove said scale-measures are lattice ordered with respect to the closure systems.",
"This enables exploring the set of scale-measures through by the use of meet and join operations.",
"Furthermore we show that the lattice of scale-measures is isomorphic to the lattice of sub-closure systems that arises from the original data.",
"Finally, we provide another representation of scale-measures using propositional logic in terms of data set features.",
"Our theoretical findings are discussed by means of examples."
],
[
"Introduction",
"The discovery and analysis of patterns and dependencies in the realm of data science does strongly depend on the measurement of the data.",
"Each data set is subject to one or more scales of measure [1], i.e., maps from the data into variable of some (mathematical) space, e.g., the real line, an ordered set, etc.",
"Beyond that, almost every data set is further scaled prior to (data)processing to meet the requirements of the employed data analysis method, such as the introduction of artificial metrics, the numerical representation of nominal features, etc.",
"This scaling is usually accompanied by a grade of detail, which in turn is becoming more and more of a problem for data science tasks as the availability of features increases and their human explainability decreases.",
"Often used methods to deal with this problem from the field of machine learning, such as principal component analysis, do enforce particular, possible inapt, levels of measurement, e.g., food tastes represented by real numbers, and amplify the problem for explainability.",
"Therefore, understanding the set of possible scaling maps, identifying its (algebraic) properties, and deriving to some extent human explainable control over it, is a pressing problem.",
"This is especially important since found patterns and dependencies may be artifacts of some scaling map and may therefore corrupt any subsequent task,e.g., classification tasks.",
"In the case Boolean data sets the field of formal concept analysis provides a well-formalized, yet insufficiently studied, approach for mathematically grasping the process of data scaling, called scale-measure maps.",
"These maps are continuous with respect to the closures systems that emerge from the original Boolean data set and scale, which resembles also a Boolean data set, i.e., the preimage of a closed set is closed.",
"Equipped with this notion for data scaling we discover and characterize consistent scale-refinements and derive a theory that is able to provide new insights to data sets by comparing different scale-measures.",
"Building up on this we prove that the set of all scale-measures bears a lattice structure and we show how to transform scale-measures using lattice operations.",
"Moreover, we introduce an equivalent representation of scale-measures using propositional logic expressions and how they emerge naturally while scaling data.",
"Altogether, we present methods that are able to generate different conceptual measurements of a data set by computing meaningful features such that they are consistent with the conceptual knowledge of the original data set.",
"Formalizing and understanding the process of measurement is, in particular in data science, an ongoing discussion.",
"Representational Theory of Measurement (RTM) [2], [3] reflects the most recent and widely acknowledged current standpoint on this.",
"RTM relies on homomorphisms from an (empirical) relational structure $\\mathbf {E}=(E,(R_i)_{i\\in I})$ to a numerical relational structure $\\mathbf {B}=(B,(S_i)_{i\\in I})$ , very well explained by J. Pfanzagl [4], where $B$ is often chosen to be the real line $\\mathbb {R}$ or a $n$ dimensional vector space on it.",
"However, it might be beneficial to allow for other, more algebraic (measurement) structures [5].",
"This is particularly true in cases where the empirical data does not allow for a meaningful measurement into the ratio level (cf [1]), e.g., taxonomic ranks in biology or types of faults in software engineering.",
"Both examples are instances of categorical data, which is classified to the nominal level with respect to S. S. Stevens [1].",
"If such data is also naturally equipped with an rank order relation, e.g., the Likert scale or school grades, it is situated on the ordinal level.",
"A mathematical framework well equipped for the nominal as well as the ordinal level is formal concept analysis (FCA) [6], [7].",
"In FCA we represent data in the form of formal contexts as see fig:bj1 (top).",
"A formal context is a triple $(G,M,I)$ with $G$ beeing a finite set of object, $M$ beeing a finite set of attributes and $I \\subseteq G \\times M$ an incidence relation between them.",
"With $(g,m) \\in I$ means that object $g$ has attribute $m$ .",
"We visualize formal context using cross tables, as depicted for the running example Ben and Jerry's in bjice (top).",
"A cross in the table indicates that an object (ice cream flavor) has an attribute (ice cream ingredient).",
"A context $\\mathbb {S}=(H,N,J)$ is called an induced sub-context of $$ , if $H\\subseteq G, N\\subseteq M$ and $I_\\mathbb {S}=I\\cap (H_\\mathbb {S}\\times N)$ , denoted $\\mathbb {S}\\le $ .",
"The incidence relation gives rise to two derivation operators.",
"The first is the derivation of an attribute $A \\subseteq M$ where $A^{\\prime }=\\lbrace g\\in G \\mid \\forall m \\in A: (g,m)\\in I\\rbrace $ .",
"The object derivation $B^{\\prime }$ for $B\\subseteq G$ is defined analogously.",
"The consecutive application of the two derivation operators on an attribute set (object set) constitutes a closure operators, i.e., a idempotent, monotone, and extensive, map.",
"Therefore, the pairs $(G,^{\\prime \\prime })$ and $(M,^{\\prime \\prime })$ are closure spaces with $\\cdot ^{\\prime \\prime }: \\mathcal {P}(G)\\rightarrow \\mathcal {P}(G)$ and $\\cdot ^{\\prime \\prime }: \\mathcal {P}(M)\\rightarrow \\mathcal {P}(M)$ .",
"For example, $\\lbrace \\text{Dough, Vanilla}\\rbrace ^{\\prime \\prime }=\\lbrace \\text{Choco, Dough, Vanilla}\\rbrace $ in bjice.",
"A formal concept is a pair $(A,B) \\in \\mathcal {P}(G)\\times \\mathcal {P}(M)$ with $A^{\\prime }=B$ and $A = B^{\\prime }$ , where $A$ is called extent and $B$ intent.",
"We denote with $\\operatorname{Ext}()$ and $\\operatorname{Int}()$ the sets of all extents and intents, respectively.",
"Each of these sets forms a closure system associated to the closure operator on the respective base set, i.e., the object set or the attribute set.",
"Both closure systems are represented in the (concept) lattice $()=(\\mathcal {B}(),\\subseteq )$ , where $\\mathcal {B}()$ denotes the set of all concepts in $$ and for $(A,B), (C,D)\\in \\mathcal {B}()$ we have $(A,B)\\le (C,D)\\Leftrightarrow A\\subseteq C$ ."
],
[
"Scales",
"A fundamental problem for the analysis, the computational treatment, and the visualization of data is the high dimensionality and complex structure of modern data sets.",
"Hence, the tasks for scaling data sets to a lower number of dimensions and decreasing their complexity has growing importance.",
"Many unsupervised (machine learning) procedures were developed and are applied, for example, multidimensional scaling [8], [9] or principal component analysis.",
"These scaling methods use non-linear projections of data objects (points) into a lower dimensional space.",
"While preserving the notion of object they loose the interpretability of features as well as the original algebraic object-feature relation.",
"Therefore, the advantage of explainability when analyzing nominal or ordinal data cannot be preserved.",
"Furthermore, most scaling approaches require the representation of the data points in a real coordinate space of some dimension, which is in turn, already a scaling for many data sets.",
"A more fundamental approach to scaling, in particular for nominal and ordinal data, that preserves the interpretable features can be found in FCA.",
"Definition 1 (Scale-Measure (cf.",
"Definition 91, [7])) Let $= (G,M,I)$ and $\\mathbb {S}=(G_{\\mathbb {S}},M_{\\mathbb {S}},I_{\\mathbb {S}})$ be a formal contexts.",
"The map $\\sigma :G \\rightarrow G_{\\mathbb {S}}$ is called an $\\mathbb {S}$ -measure of $$ into the scale $\\mathbb {S}$ iff the preimage $\\sigma ^{-1}(A)\\lbrace g\\in G\\mid \\sigma (g)\\in A\\rbrace $ of every extent $A\\in \\operatorname{Ext}(\\mathbb {S})$ is an extent of $$ .",
"This definition corresponds the notion for continuity between closure spaces $(G_1,c_1)$ and $(G_2,c_2)$ , i.e., a map $f:G_1\\rightarrow G_2$ is continuous iff $\\text{for all}\\ A\\in \\mathcal {P}(G_2) we havec_1(f^{-1}(A))\\subseteq f^{-1}(c_2(A)).$ This property is equivalent to the requirement in def:sm that the preimage of closed sets is closed, more formally, $\\text{for all}\\ A\\in \\mathcal {P}(G_2)\\ \\text{with}\\ c_2(A)=A\\ \\text{we have}\\ f^{-1}(A)=c_1(f^{-1}(A)).$ Conditions in (REF ) and (REF ) are known to be equivalent, since $(\\ref {eq:cont})\\Rightarrow (\\ref {eq:scales})$ follows from $x\\in c_1(f^{-1}(A))\\Rightarrow x\\in f^{-1}(c_2(A)){c_2(A)=A} x\\in f^{-1}(A)$ .",
"Also, from $x\\in c_1(f^{-1}(A))\\Rightarrow x\\in c_1(f^{-1}(c_2(A))){(\\ref {eq:scales})} x\\in f^{-1}(c_2(A))$ results (REF )$\\Rightarrow $ (REF ).",
"In the following we may address by $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))$ the set of all extents of $$ that are reflected by the scale context, i.e., $\\bigcup _{A\\in \\operatorname{Ext}(\\mathbb {S})}\\sigma ^{-1}(A)$ .",
"Furthermore, we want to nourish the understanding of scale-measures as consistent measurements (or views) of the objects in some scale context.",
"In this sense we understand the map $\\sigma $ as an interpretation of the objects from $$ in $\\mathbb {S}$ .",
"The following corollary can be deduced from the continuity property above and will be used frequently throughout our work.",
"Corollary 2 (Composition Scale-Measures) Let $$ be a formal context, $\\sigma $ a $\\mathbb {S}$ -measure of $$ and $\\psi $ a $\\mathbb {T}$ -measure of $\\mathbb {S}$ .",
"Then is $\\psi \\circ \\sigma $ a $\\mathbb {T}$ -measure of $$ .",
"Figure: A scale context (top), its concept lattice (bottom right)for which id G \\operatorname{id}_G is a scale-measure of the context inbjice and the reflected extentsσ -1 (Ext())\\sigma ^{-1}(\\operatorname{Ext}()) (bottem left) indicated asnon-transparent.In bjicemeasure we depict a scale-measure and its concept lattice for our running example context Ben and Jerry's $_{\\text{BJ}}$ , cf. bjice.",
"This scale-measure uses the same object set as the original context and maps every object to itself.",
"The attribute set is comprised of six elements, which may reflect the taste, instead of the original nine attributes that indicated the used ingredients.",
"The specified scale-measure map allows for a human comprehensible interpretation of $\\sigma ^{-1}$ , as indicated by the grey colored concepts in bjicemeasure (bottom).",
"In this figure we observe that the concept lattice of the scale-measure reflects ten out of the sixteen concepts in $\\mathfrak {B}(_{\\text{BJ}})$ .",
"The empirical observations about the afore presented example scale-measure for some context $$ lead to the question whether scale-measures are always at least as comprehensible as the context $$ itself.",
"A typical (objective) measure for the complexity of lattices is given by the following quantity.",
"Definition 3 (Order Dimension (cf.",
"Definition 82, [7])) An ordered set $(P,\\le )$ has order dimension $\\dim (P,\\le )=n$ iff it can be embedded in a direct product of $n$ chains and $n$ is the smallest number for which this is possible.",
"The order dimension of $(_{\\text{BJ}})$ is three whereas the concept lattice of the given scale-measure is two.",
"Finding low dimensional scale-measures for large and complex data sets is a natural approach towards comprehensible data analysis, as demonstrated in prop:dim.",
"In particular, we will answer the question if the order dimension of scale-measures is bound by the order dimension of $()$ .",
"Another notion for comparing scale-measures is provided by a natural order relation amongst scales [7]).",
"We may present in the following a more general definition within the scope of scale-measures.",
"Definition 4 (Scale-Measure Refinement) Let the set of all scale-measures of a context be denoted by $\\mathfrak {S}()\\lbrace (\\sigma , \\mathbb {S})\\mid \\sigma $ is a $\\mathbb {S}-$ measure of $\\rbrace $ .",
"For $(\\sigma ,\\mathbb {S}),(\\psi ,\\mathbb {T})\\in \\mathfrak {S}()$ we say $(\\sigma ,\\mathbb {S})$ is a coarser scale-measure of $$ than $(\\psi ,\\mathbb {T})$ , iff $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})) {\\subseteq }\\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T})$ .",
"Analogously we then say $(\\psi ,\\mathbb {T})$ is finer than $(\\sigma ,\\mathbb {S})$ .",
"If $(\\sigma ,\\mathbb {S})$ is finer and coarser than $(\\psi ,\\mathbb {T})$ we call them equivalent scale-measures.",
"We remark that the finer relation as well as coarser relation constitute (partial) order relations on the set of all scale-measure for context $$ , since they are obviously reflexive, anti-symmetric, and the transitivity follow from the continuity of the composition of scale maps.",
"Hence, we may refer to the refinement (order) using the symbol $\\le $ .",
"By computing scale-measures with coarser scale contexts with respect to the refinement order we can provide a more general conceptual view on a data set.",
"The study of such views, e.g.",
"the ice cream tastes in our running example presented in bjicemeasure, is in a similar fashion to the Online Analytical Processing tools for multidimensional databases.",
"Moreover, the set of all scale-measure for some formal context enables an abstract analytical structure to navigate and explore a data set with.",
"Yet, despite the supposed usefulness of the scale-measures, there are up until now no existing methods, to the best of our knowledge, for the generation and evaluation of scale-measures, in particular with respect to data science applications.",
"Both tasks, the generation and the evaluation of scale-measures, will be tackled in the next section using a novel navigation approach among them."
],
[
"Navigation though Conceptual Measurement",
"Based on the just introduced refinement order of scale-measures we provide in this section the means for efficiently browsing this structure.",
"Given a data set, the presented methods are able to compute arbitrary scale abstractions and the structure operations that connect them, which resembles a navigation through conceptual measurements.",
"To lay the foundation for the navigation methods we start with analyzing the structure of all scale-measures.",
"Thereafter we will present a thorough description of the navigation problem and its solution.",
"Lemma 5 The scale-measure equivalence is an equivalence relation on the set of scale-measures.",
"Let $(\\sigma ,\\mathbb {S}),(\\psi ,\\mathbb {T}),(\\phi ,\\mathbb {O})\\in \\mathfrak {S}()$ be scale-measures of context $$ .",
"Using def:sm-refine we know from $(\\sigma ,\\mathbb {S})\\sim (\\psi ,\\mathbb {T})$ that $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})) = \\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))$ , from which the reflexivity and the symmetry of $\\sim $ can be inferred.",
"Analogously we can infer for $(\\sigma ,\\mathbb {S})\\sim (\\psi ,\\mathbb {T})$ and $(\\psi ,\\mathbb {T})\\sim (\\phi ,\\mathbb {O})$ that $(\\sigma ,\\mathbb {S})\\sim (\\phi ,\\mathbb {O})$ .$\\Box $ Note that for two given equivalent scale-measures that their scale-measure equivalence does not imply the existence of an bijective scale-measure between them.",
"Yet, a minor requirement to the scale-measure map leads to a useful link.",
"Lemma 6 Let $(\\sigma ,\\mathbb {S}),(\\psi , \\mathbb {T})\\in \\mathfrak {S}()$ with $(\\sigma ,\\mathbb {S})\\sim (\\psi , \\mathbb {T})$ and $\\sigma ,\\psi $ are surjective maps.",
"Then $\\sigma ^{-1}\\circ \\psi $ is an order isomorphism from $(\\operatorname{Ext}(\\mathbb {S}),\\subseteq )$ to $(\\operatorname{Ext}(\\mathbb {T}),\\subseteq )$ .",
"From [7] we have that $\\sigma ^{-1}$ is a injective $\\wedge $ -preserving order embedding of $(\\operatorname{Ext}(\\mathbb {S}),\\subseteq )$ into $(\\operatorname{Ext}(),\\subseteq )$ and thereby a bijective $\\wedge $ -preserving order embedding into $(\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})),\\subseteq )$ .",
"The analogue holds for $\\psi ^{-1}$ from $\\operatorname{Ext}(\\mathbb {T})$ into $\\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))$ .",
"Due to $(\\sigma ,\\mathbb {S})\\sim (\\psi , \\mathbb {T})$ we know that $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))=\\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))$ , which results in $\\sigma ^{-1}$ being a bijective $\\wedge $ -preserving order embedding into $\\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))$ .",
"Hence, when restricting $\\sigma ^{-1}\\circ \\psi :\\mathcal {P}(G_\\mathbb {S})\\rightarrow \\mathcal {P}(G_{\\mathbb {T}})$ to the respective extent set we obtain a bijective map.",
"The fact that all formal contexts are finite (throughout this work) and the monotonicity of the lifts of $\\sigma ^{-1}$ and $\\psi $ to their respective power sets imply the required order preserving property follow.$\\Box $ We may stress that the required surjectivity is not constraining the application of scale-measures, since any object $g$ of a scale-context having an empty preimage may just be removed from the scale-context without consequences to the analysis.",
"The just discussed equivalence relation together with the refinement order allows to cope with the set of all scale-measures $\\mathfrak {S}()$ in a meaningful way.",
"Definition 7 (Scale-Hierarchy) Given a formal context $$ and its set of all scale-measures $\\mathfrak {S}()$ , we call $\\underline{\\mathfrak {S}}()({\\mathfrak {S}()}{\\sim },\\le )$ the scale-hierarchy of $$ .",
"The order structure thus given represents all possible means of scaling a (contextual) data set.",
"Yet, it seems hardly comprehensible or even applicable in that form.",
"Therefore the goal for the rest of this section is to achieve a characterization of said structure in terms of closure systems.",
"Lemma 8 Let $G$ be a set and $\\mathcal {A}\\subseteq \\mathcal {P}(G)$ be a closure system.",
"Furthermore, let $_{\\mathcal {A}}=(G,\\mathcal {A},\\in )$ be a formal context using the element relation as incidence.",
"Then the set of extents $\\operatorname{Ext}(_{\\mathcal {A}})$ is equal to the closure system $\\mathcal {A}$ .",
"For any set $D\\subseteq G$ and $A\\in \\mathcal {A}$ we find ($\\ast $ ) $D\\subseteq A\\Rightarrow A\\in D^{\\prime }$ .",
"Since $\\mathcal {A}$ is a closure system and $D^{\\prime \\prime }=\\bigcap D^{\\prime }$ we see that $D^{\\prime \\prime }\\in \\mathcal {A}$ , hence, $\\operatorname{Ext}(_{\\mathcal {A}})\\subseteq \\mathcal {A}$ .",
"Conversely, for $A\\in \\mathcal {A}$ we can draw from ($\\ast $ ) that $A^{\\prime \\prime }=A$ , thus $A\\in \\operatorname{Ext}(_{\\mathcal {A}})$ .$\\Box $ We want to further motivate the constructed formal context $_{\\mathcal {A}}$ and its particular utility with respect to scale-measures for some context $$ .",
"Since both contexts have the same set of objects, we may study the use of the identity map $\\operatorname{id}:G\\rightarrow G, g\\mapsto g$ as scale-measure map.",
"Lemma 9 (Canonical Construction) For a context $$ and any $\\mathbb {S}$ -measure $\\sigma $ is $\\operatorname{id}$ a $_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))}$ -measure of $$ , i.e., $(\\operatorname{id},_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))})\\in \\mathfrak {S}()$ .",
"lem:csctx gives that $\\operatorname{Ext}(_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))})$ is equal to $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))$ .",
"Since $(\\sigma ,\\mathbb {S})\\in \\mathfrak {S}()$ , i.e., $(\\sigma ,\\mathbb {S})$ is a scale-measure of $$ , we see that the preimage $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))\\subseteq \\operatorname{Ext}()$ , and thus $\\operatorname{id}^{-1}(\\operatorname{Ext}(_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))}))\\subseteq \\operatorname{Ext}()$ .",
"$\\Box $ Using the canonical construction of a scale-measure, as given above, we can facilitate the understanding of the scale-hierarchy $\\underline{\\mathfrak {S}}()$ .",
"Proposition 10 (Canonical Representation) Let $= (G,M,I)$ be a formal context with scale-measure $(\\mathbb {S},\\sigma )\\in \\mathfrak {S}()$ , then $(\\sigma ,\\mathbb {S})\\sim (\\operatorname{id}, _{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))})$ .",
"lem:cssm states that $\\operatorname{id}$ is a $_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))}$ -measure of $$ .",
"Furthermore, from lem:csctx we know that the extent set of $_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))}$ is $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))$ , as required by def:sm-refine.$\\Box $ Equipped with this proposition we are now able to compare sets of scale-measures for a given formal context $$ solely based on their respective attribute sets in the canonical representation.",
"Furthermore, since these representation sets are sub-closure systems of $\\operatorname{Ext}()$ , by def:sm, we may reformulate the problem for navigating scale-measures using sub-closure systems and their relations.",
"For this we want to nourish the understanding of the correspondence of scale-measures and sub-closure systems in the following.",
"Proposition 11 For a formal context $$ and the set of all sub-closure systems $\\mathfrak {C}()\\subseteq \\mathcal {P}(\\operatorname{Ext}())$ together with the inclusion order, the following map is an order isomorphism: ${i}:\\mathfrak {C}()\\rightarrow \\mathfrak {S}()_{/\\sim },\\ \\mathcal {A}\\mapsto i(\\mathcal {A})(\\operatorname{id},_{\\mathcal {A}})$ Let $\\mathcal {A},\\mathcal {B}\\subseteq \\operatorname{Ext}()$ be two closure systems on $G$ .",
"Then the images of $\\mathcal {A}$ respectively $\\mathcal {B}$ under $i$ are a scale-measures of $$ , according to lem:cssm, with extents $\\mathcal {A}$ and $\\mathcal {B}$ , respectively.",
"Since $\\mathcal {A}\\ne \\mathcal {B}\\iff \\operatorname{Ext}(_{\\mathcal {A}})\\ne \\operatorname{Ext}(_{\\mathcal {B}})$ are different and therefore $(\\operatorname{id},_{\\mathcal {B}})\\lnot \\sim (\\operatorname{id},_{\\mathcal {B}})$ , thus, $i$ is an injective map.",
"For the surjectivity of $i$ let $[(\\sigma ,\\mathbb {S})]\\in \\mathfrak {S}()_{/\\sim }$ , then $(\\operatorname{id},_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))})\\sim (\\sigma ,\\mathbb {S})$ , i.e., an equivalent representation having extents $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))\\subseteq \\operatorname{Ext}()$ and $i(\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})))=(\\operatorname{id},_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))})$ .",
"Finally, for $\\mathcal {A}\\subseteq \\mathcal {B}$ we find that $i(\\mathcal {A})\\subseteq i(\\mathcal {B})$ , since $\\operatorname{Ext}(_{\\mathcal {A}}) \\subseteq \\operatorname{Ext}(_{\\mathcal {B}})$ , as required.$\\Box $ Figure: Scale-Hierarchy of (right) and embedded in boolean𝔹 G \\mathbb {B}_GThis order isomorphism allows us to analyze the structure of the scale-hierarchy by studying the related closure systems.",
"For instance, the problem of computing $|\\underline{\\mathfrak {S}}()|$ , i.e., the size of the scale-hierarchy.",
"In the case of the boolean context $_{\\mathcal {P}(G)}$ this problem equivalent to the question for the number of Moore families, i.e., the number of closure systems on $G$ .",
"This number grows tremendously in $|G|$ and is known up to $|G|=7$ , for which it is known [10], [11], [12] to be $14\\,087\\,648\\,235\\,707\\,352\\,472$ .",
"In the general case the size of the scale-hierarchy is equal to the size of the order ideal $\\downarrow \\operatorname{Ext}()$ in $\\mathfrak {C}(_{\\mathcal {P}(G)})$ .",
"The fact that the set of all closure systems on $G$ is again a closure system [13], which is lattice ordered by set inclusion, allows for the following statement.",
"Corollary 12 (Scale-hierarchy Order) For a formal context $$ , the scale-hierarchy $\\underline{\\mathfrak {S}}()$ is lattice ordered.",
"We depicted this lattice order relation in the form of abstract visualizations in fig:SmAsCl.",
"In the bottom (right) we see the most simple scale which has only one attribute, $G$ .",
"The top (right) element in this figure is then the scale which has all extents of $$ .",
"On the left we see the lattice ordered set of all closure systems on a set $G$ , in which we find the embedding of the hierarchy of scales.",
"Proposition 13 Let $(\\sigma ,\\mathbb {S}),(\\psi ,\\mathbb {T})\\in \\underline{\\mathfrak {S}}()$ and let $\\wedge ,\\vee $ be the natural lattice operations in $\\underline{\\mathfrak {S}}()$ , (induced by the lattice order relation).",
"We then find that: Meet : $(\\sigma ,\\mathbb {S})\\wedge (\\psi ,\\mathbb {T})=(\\operatorname{id},_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))\\cap \\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))})$ , Join : $(\\sigma ,\\mathbb {S})\\vee (\\psi ,\\mathbb {T})=(\\operatorname{id},_{\\lbrace A \\cap B \\mid A \\in \\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})), B\\in \\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))\\rbrace })$ .",
"For the preimages $i^{-1}(\\sigma ,\\mathbb {S})$ , $i^{-1}(\\psi ,\\mathbb {T})$ (cor:size) we can compute their meet [13], which yields $i^{-1}(\\sigma ,\\mathbb {S})\\wedge i^{-1}(\\psi ,\\mathbb {T})=\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))\\cap \\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T})).$ The join [13] of the scale-measure preimages under $i$ (cor:size) is equal to $\\lbrace A \\cap B \\mid A \\in \\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})), B \\in \\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))\\rbrace $ , which results in the required expression by applying the order isomorphism $i$ .",
"$\\Box $ Propositional Navigation through Scale-Measures Although the canonical representation of scale-measures is complete up to equivalence prop:eqi-scale, this representation eludes human explanation to some degree.",
"In particular the use of the extentional structure of $$ as attributes provides insight to the scale-hierarchy itself, however, not to the data, i.e., the objects, attributes, and their relation.",
"A formulation of scales using attributes from $$ , and their combinations, seems more natural and more comprehensible.",
"For this, we employ an approach as used in [14].",
"In their work the authors used a logic on the context's attributes to introduce new attributes.",
"The advantage is that the so newly introduced attributes have a real-world semantic in terms of the measured properties.",
"In this work we use propositional logic, which leads to the following problem description.",
"Problem 14 (Navigation Problem) For a formal context $$ , a scale-measure $(\\sigma ,\\mathbb {S})\\in \\mathfrak {S}()$ and $M_\\mathbb {T}\\subseteq \\mathcal {L}(M,\\lbrace \\wedge ,\\vee ,\\lnot \\rbrace )$ , compute an equivalent scale-measure $(\\psi ,\\mathbb {T})\\in \\mathfrak {S}()$ , i.e., $(\\sigma ,\\mathbb {S})\\sim (\\psi ,\\mathbb {T})$ , where $(h,m)\\in I_\\mathbb {T}\\Leftrightarrow \\psi ^{-1}(h)^{I}\\models m$ .",
"The attributes of $\\mathbb {T}$ are logical expression build from the attributes of $$ , and are thus interpretable in terms of the measurements by the attributes $M$ from $$ .",
"For example, we can express the Choco taste attribute of our running example (bjicemeasure) as the disjunction of the ingredients Choco Ice or Choco Pieces, i.e.",
"Choco$$Choco Ice$\\vee $Choco Pieces.",
"For any scale-measure $(\\sigma ,\\mathbb {S})$ , such an equivalent scale-measure, as searched for in problem:navi, is not necessarily unique, and the problem statement does not favor any of the possible solutions.",
"To understand the semantics of the logical operations, we first investigate their contextual derivations.",
"For $\\phi \\in \\mathcal {L}(M,\\lbrace \\wedge ,\\vee ,\\lnot \\rbrace )$ we let $\\operatorname{Var}(\\phi )$ be the set of all propositional variables in the expression $\\phi $ .",
"We require from $\\phi \\in \\mathcal {L}(M,\\lbrace \\wedge ,\\vee ,\\lnot \\rbrace )$ that $|\\operatorname{Var}(\\phi )|>0$ .",
"Lemma 15 (Logical Derivations) Let $=(G,M,I)$ be a formal context, $\\phi _\\wedge \\in \\mathcal {L}(M,\\lbrace \\wedge \\rbrace )$ , $\\phi _\\vee \\in \\mathcal {L}(M,\\lbrace \\vee \\rbrace )$ , $\\phi _\\lnot \\in \\mathcal {L}(M,\\lbrace \\lnot \\rbrace )$ , with scale contexts $(G,\\lbrace \\phi \\rbrace ,I_{\\phi })$ having the incidence $(g,\\phi )\\in I_{\\phi }\\iff g^{I}\\models \\phi $ for $\\phi \\in \\lbrace \\phi _\\vee ,\\phi _\\wedge ,\\phi _\\lnot \\rbrace $ .",
"Then we find $\\lbrace \\phi _\\wedge \\rbrace ^{I_{\\phi _{\\wedge }}} = \\operatorname{Var}(\\phi _{\\wedge })^{I}$ , $\\lbrace \\phi _\\vee \\rbrace ^{I_{\\phi _{\\vee }}}=\\bigcup _{m\\in \\operatorname{Var}(\\phi _{\\vee })}\\lbrace m\\rbrace ^{I}$ , $\\lbrace \\phi _\\lnot \\rbrace ^{I_{\\phi _{\\lnot }}} = G\\setminus \\lbrace n\\rbrace ^{I}$ with $\\phi _\\lnot = \\lnot n$ for $n\\in M$ .",
"[i)] For $g\\in G$ if $gI_{\\phi _\\wedge }\\phi _{\\wedge }$ , then $\\lbrace g\\rbrace ^I\\models \\phi _\\wedge $ and thereby $\\operatorname{Var}(\\phi _{\\wedge })\\subseteq \\lbrace g\\rbrace ^{I}$ .",
"Hence $g\\in \\operatorname{Var}(\\phi _{\\wedge })^{I}$ .",
"In case $(g,\\phi _{\\wedge })\\notin I_{\\phi _\\wedge }$ it holds that $\\operatorname{Var}(\\phi _{\\wedge })\\lnot \\subseteq \\lbrace g\\rbrace ^{I}$ and thereby $g\\notin \\operatorname{Var}(\\phi _{\\wedge })^{I}$ .",
"For $g\\in G$ if $gI_{\\phi _\\vee }\\phi _\\vee $ we have $\\lbrace g\\rbrace ^I\\models \\phi _{\\vee }$ .",
"Hence, $\\exists m\\in \\operatorname{Var}(\\phi _{\\vee })$ with $g\\in m^{I}$ and therefore $g$ is in the union.",
"If $(g,\\phi _{\\vee })\\notin I_{\\phi _\\vee }$ there does not exists such a $m\\in \\operatorname{Var}(\\phi _{\\wedge })$ and $g\\notin \\bigcup _{m\\in \\operatorname{Var}(\\phi )}m^{I}$ .",
"For any $n\\in M$ we have $\\phi _\\lnot = \\lnot n$ .",
"Hence, for $g\\in G$ if $gI_{\\phi _\\lnot }\\phi _\\lnot $ we find $g\\notin \\lbrace n\\rbrace ^{I}$ .",
"Conversely, if $(g,\\phi _\\lnot )\\notin I_{\\phi _\\lnot }$ it follows that $g\\in \\lbrace n\\rbrace ^{I}$ .",
"$\\Box $ Naturally, the results from the lemma above generalize to scale contexts with more than one logical expression in the set of attributes.",
"How this is done is demonstrated in sec:apos.",
"Moreover, more complex formulas, i.e., $\\phi \\in \\mathcal {L}(M,\\lbrace \\wedge ,\\vee ,\\lnot \\rbrace )$ , can be recursively deconstructed and then treated with lem:deri.",
"In particular, with respect to unsupervised machine learning, we may mention the connection to the task of clustering attributes, as studied by Kwuida et al. [15].",
"Proposition 16 (Logical Scale-Measure) Let $$ be a formal context and let $\\phi \\in \\mathcal {L}(M,\\lbrace \\wedge , \\vee , \\lnot \\rbrace )$ , then $\\operatorname{id}_G$ is a $(G,\\lbrace \\phi \\rbrace ,I_\\phi )$ -measure of $$ iff $\\lbrace \\phi \\rbrace ^{I_{\\phi }}\\in \\operatorname{Ext}()$ .",
"Since $|\\lbrace \\phi \\rbrace |=1$ we find that $(G, \\lbrace \\phi \\rbrace ,I_\\phi )$ has at least one and most two possible extents, $\\lbrace \\lbrace \\phi \\rbrace ^{I_{\\phi }},G\\rbrace $ .",
"If the map $\\operatorname{id}_{G}$ is a scale-measure of $$ , then $\\operatorname{id}_G^{-1}(\\lbrace \\phi \\rbrace ^{I_{\\phi }}) = \\lbrace \\phi \\rbrace ^{I_{\\phi }} \\in \\operatorname{Ext}()$ .",
"Conversely, if $ \\lbrace \\phi \\rbrace ^{I_{\\phi }} \\in \\operatorname{Ext}()$ so is $\\operatorname{id}_G^{-1}(\\lbrace \\phi \\rbrace ^{I_{\\phi }})$ , hence, $\\operatorname{id}_{G}$ is $(G, \\lbrace \\phi \\rbrace ,I_\\phi )$ -measure of $$ .",
"$\\Box $ Figure: Counter examples for which id G \\operatorname{id}_G is not a(G,{φ ∨ },I φ ∨ )(G, \\lbrace \\phi _\\vee \\rbrace ,I_{\\phi _\\vee })- or(G,{φ ¬ },I φ ¬ )(G, \\lbrace \\phi _\\lnot \\rbrace ,I_{\\phi _\\lnot })-measure of a .",
"Theconflicting extents are marked in red.This result raises the question for which formulas $\\phi $ is $\\operatorname{id}_{G}$ a $(G, \\lbrace \\phi \\rbrace ,I_\\phi )$ -measure of $$ .",
"Counter examples for which $\\operatorname{id}_G$ is not a $(G, \\lbrace \\phi _\\vee \\rbrace ,I_{\\phi _\\vee })$ - or $(G,\\lbrace \\phi _\\lnot \\rbrace ,I_{\\phi _\\lnot })$ -measure of a $$ are depicted in fig:counter.",
"Corollary 17 (Conjunctive Logical Scale-Measures) Let $=(G,M,I)$ be a formal context and $\\phi _\\wedge \\in \\mathcal {L}(M,\\lbrace \\wedge \\rbrace )$ , then $(\\operatorname{id}_G,(G,\\lbrace \\phi _\\wedge \\rbrace ,I_{\\phi _{\\wedge }}))\\in \\mathfrak {S}()$ .",
"According to lem:deri $(\\phi _\\wedge )^{I_{\\phi _{\\wedge }}} =\\operatorname{Var}(\\phi )^{I}$ , hence, by prop:logiattr $(\\operatorname{id}_G,(G,\\lbrace \\phi _\\wedge \\rbrace ,I_{\\phi _{\\wedge }}))\\in \\mathfrak {S}()$ .$\\Box $ Context Apposition for Scale Construction To build more complex scale-measures we employ the apposition operator of contexts and transfer it to the realm of scale-measures.",
"We remind the reader that the apposition of two contexts $_1,_1$ with $G_1=G_2$ and $M_1\\cap M_2 = \\emptyset $ is defined as $_1\\operatorname{\\mid \\,}_2(G,M_1\\cup M_2,I_1\\cup I_2)$ .",
"The set of extents of $_1 \\operatorname{\\mid \\,}_2$ is known to be the set of all pairwise extents of $_1$ and $_2$ .",
"In the case of $M_{1}\\cap M_{2}\\ne \\emptyset $ the apposition is defined alike by coloring the attribute sets.",
"Definition 18 (Apposition of Scale-Measures) Let $(\\sigma ,\\mathbb {S}),(\\psi ,\\mathbb {T})$ be scale-measures of $$ .",
"Then the apposition of scale-measures $(\\sigma ,\\mathbb {S})\\operatorname{\\mid \\,}(\\psi ,\\mathbb {T})$ is: $(\\sigma ,\\mathbb {S})\\operatorname{\\mid \\,}(\\psi ,\\mathbb {T})\\begin{dcases}(\\sigma , \\mathbb {S}\\,\\operatorname{\\mid \\,}\\mathbb {T})&\\text{if}\\ G_\\mathbb {S}=G_\\mathbb {T}, \\sigma =\\psi \\\\(\\sigma ,\\mathbb {S})\\vee (\\psi ,\\mathbb {T})&\\text{else}\\end{dcases}$ Note that also in the case of $G_\\mathbb {S}=G_\\mathbb {T}, \\sigma =\\psi $ is the scale-measure apposition is as well a join up to equivalence in the scale-hierarchy, cf. prop:lattice.",
"Proposition 19 (Apposition Scale-Measure) Let $(\\sigma ,\\mathbb {S}),(\\psi ,\\mathbb {T})$ be two scale-measures of $$ .",
"Then $(\\sigma ,\\mathbb {S})\\operatorname{\\mid \\,}(\\psi ,\\mathbb {T})\\in \\mathfrak {S}()$ .",
"In the first case we know that set of extents $\\operatorname{Ext}(\\mathbb {S}\\mid \\mathbb {T})$ contains all intersections $A\\cap B$ for $A\\in \\operatorname{Ext}(\\mathbb {S})$ and $B\\in \\operatorname{Ext}(\\mathbb {T})$ [7].",
"Furthermore, we know that we can represent $\\sigma ^{-1}(A\\cap B)=\\sigma ^{-1}(A)\\cap \\sigma ^{-1}(B)=\\sigma ^{-1}(A)\\cap \\psi ^{-1}(B)$ .",
"Since $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})),\\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))\\subseteq \\operatorname{Ext}()$ , we can infer that the intersection $\\sigma ^{-1}(A)\\cap \\psi ^{-1}(B)\\in \\operatorname{Ext}()$ .",
"The second case follows from prop:lattice.",
"$\\Box $ The apposition operator combines two scale-measures, and therefore two views, on a data context to a new single one.",
"We may note that the special case of $(\\sigma ,\\mathbb {S})=(\\operatorname{id}_G,)$ was already discussed by Ganter and Wille [7].",
"Proposition 20 Let $= (G,M,I)$ and $\\mathbb {S}=(G_{\\mathbb {S}},M_{\\mathbb {S}},I_{\\mathbb {S}})$ be two formal contexts and $\\sigma :G \\rightarrow G_{\\mathbb {S}}$ , then TFAE: $\\sigma \\text{ is a } \\mathbb {S}\\text{-measure of }$ $\\sigma \\text{ is a } (G_{\\mathbb {S}},\\lbrace n\\rbrace ,I_{\\mathbb {S}}\\cap (G_{\\mathbb {S}}\\times \\lbrace n\\rbrace ))\\text{-measure of }\\ \\text{forall}\\ n\\in M_{\\mathbb {S}} $ $(i)\\Rightarrow (ii):$ Assume $\\hat{n}\\in M_{\\mathbb {S}}$ s.t.",
"$\\sigma $ is not a $(G_{\\mathbb {S}},\\lbrace \\hat{n}\\rbrace ,\\overbrace{I_{\\mathbb {S}}\\cap (G_{\\mathbb {S}}\\times \\lbrace \\hat{n}\\rbrace )}^{J})$ -measure of $$ .",
"Then the only non-trivial extent $\\lbrace \\hat{n}\\rbrace ^{J}$ has a preimage $\\sigma ^{-1}(\\lbrace \\hat{n}\\rbrace ^{J})\\notin \\operatorname{Ext}()$ .",
"Since $\\lbrace \\hat{n}\\rbrace ^{J}\\in \\operatorname{Ext}(\\mathbb {S})$ we can conclude that $\\sigma $ is not a $\\mathbb {S}$ -measure of $$ .",
"$(ii)\\Rightarrow (i):$ From prop:app follows $\\operatorname{\\mid \\,}_{n\\in M_{\\mathbb {S}}}(\\sigma ,(G_{\\mathbb {S}},\\lbrace n\\rbrace ,I_{\\mathbb {S}}\\cap (G_{\\mathbb {S}}\\times \\lbrace n\\rbrace )))$ is again a scale-measure.",
"Furthermore, by def:app we know that $\\mathbb {S}=\\operatorname{\\mid \\,}_{n\\in M_{\\mathbb {S}}}(G_{\\mathbb {S}},\\lbrace n\\rbrace ,I_{\\mathbb {S}}\\cap (G_{\\mathbb {S}}\\times \\lbrace n\\rbrace ))$ .$\\Box $ Corollary 21 (Deciding the Scale-measure Problem) Given a formal context $(G,M,I)$ and scale-context $\\mathbb {S}(G_{\\mathbb {S}},M_{\\mathbb {S}},I_{\\mathbb {S}})$ and a map $\\sigma :G\\rightarrow G_{\\mathbb {S}}$ , deciding if $(\\sigma ,\\mathbb {S})$ is a scale-measure of $$ is in $P$ .",
"More specifically, to answer this question does require $O(|M_{\\mathbb {S}}|\\cdot |G_{\\mathbb {S}}|\\cdot |G|\\cdot |M|)$ .",
"We may not that this result is favorable since the naive solution would be to compute $\\operatorname{Ext}(\\mathbb {S})$ , which is potentially exponential in the size of $\\mathbb {S}$ , and checking all its elements in $$ for their closure, which consumes $O(|G|\\cdot |M|)$ for all $A\\in \\operatorname{Ext}(\\mathbb {S})$ .",
"Moreover, if the formal context $$ is fixed as well as $G_{\\mathbb {S}}$ , the computational cost for deciding the scale-measure problem grows linearly in $|M_{\\mathbb {S}}|$ .",
"Altogether, this enables a feasible navigation in the scale-hierarchy.",
"Corollary 22 (Attribute Projection) Let $=(G,M,I)$ be a formal context, $M_{\\mathbb {S}}\\subseteq M$ , and $I_{\\mathbb {S}}I\\cap (G\\times M_{\\mathbb {S}})$ , then $\\sigma =\\operatorname{id}_G$ is a $(G,M_{\\mathbb {S}},I_{\\mathbb {S}})$ -measure of $$ .",
"The map $id_G$ is a $$ -measure of $$ , hence $id_G$ is a $(G,\\lbrace n\\rbrace ,I\\cap (G\\times \\lbrace n\\rbrace ))$ -measure of $$ for every $n\\in M$ , and in particular $n\\in M_{\\mathbb {S}}$ , by prop:attr, leading to $(\\operatorname{id}_{G},(G,M_{\\mathbb {S}},I_{\\mathbb {S}}))$ being a scale-measure of $$ , cf.",
"prop:app.$\\Box $ Due to duality one may also investigate an object projection based on the just presented attribute projection.",
"However, an investigation of dualities in the realm of scale-measures is deemed future work.",
"Combining our results on scale-measure apposition (prop:app) with the logical attributes (prop:logiattr) we are now tackle the navigation problem as stated in problem:navi.",
"When we look at this problem again, we find that in its generality it does not always permit a solution.",
"For example, consider the well-known Boolean formal context $\\mathbb {B}_{n}([n],[n],\\ne )$ , a standard scale context, where $[n]\\lbrace 1,\\cdots ,n\\rbrace $ and $n>2$ .",
"This context allows a scale-measure into the standard nominal scale $\\mathbb {N}_{n}([n],[n],=)$ , the map $\\operatorname{id}_{[n]}$ .",
"Restricted to any disjunctive combination of attributes, i.e., $M_{\\mathbb {T}}\\subseteq \\mathcal {L}(M,\\lbrace \\vee \\rbrace )$ , the afore mentioned scale-measure does not have an equivalent logical scale-measure $(\\psi ,\\mathbb {T}([n],M_{\\mathbb {T}},I_{\\mathbb {T}}))$ .",
"This is due to the fact that in nominal contexts there is for every object $g$ there is an attribute $m$ , such that ${m}^{\\prime }={g}$ , also $|{m}^{\\prime }|=1$ , all attribute derivations in Boolean context $\\mathbb {B}_{n}$ are of cardinality $n-1$ , the derivation of a disjunctive formula (over $[n]$ ) is the union of the elemental attribute derivations (lem:deri).",
"Hence, the derivation of an disjunctive formula is at least of cardinality $n-1$ in $\\mathbb {T}$ and therefore there must not exist an $m\\in M_{\\mathbb {T}}$ such that $|\\lbrace m\\rbrace ^{I_{\\mathbb {T}}}|=1$ , and therefore $\\operatorname{Ext}(\\mathbb {N})\\ne \\operatorname{Ext}(\\mathbb {T})$ .",
"Despite this result, we may also report positive answers for particular instances of problem:navi that use conjunctive formulas for $M_{\\mathbb {T}}$ .",
"Proposition 23 (Conjunctive Normalform of Scale-measures) Let $$ be a context, $(\\sigma ,\\mathbb {S})\\in \\mathfrak {S}()$ .",
"Then the scale-measure $(\\psi ,\\mathbb {T})\\in \\mathfrak {S}()$ given by $\\psi = \\operatorname{id}_G\\quad \\text{ and }\\quad \\mathbb {T}= \\operatorname{\\mid \\,}\\limits _{A\\in \\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))} (G,\\lbrace \\phi =\\wedge \\ A^{I}\\rbrace ,I_{\\phi }) $ is equivalent to $(\\sigma ,\\mathbb {S})$ and is called conjunctive normalform of $(\\sigma ,\\mathbb {S})$ .",
"We know that every formal context $(G,\\lbrace \\phi =\\wedge A^{I}\\rbrace ,I_{\\phi })$ together with $\\operatorname{id}_{G}$ is a scale-measure (prop:clattr).",
"Moreover, every apposition of scale-measures (for some formal context $$ ) is again a scale-measure (prop:app).",
"Hence, the resulting $(\\psi ,\\mathbb {T})$ is a scale-measure of $$ .",
"It remains to be shown that $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})) =\\operatorname{id}_G(\\operatorname{Ext}(\\mathbb {T}))$ .",
"Scale-measure equivalence holds if $(\\psi ,\\mathbb {T})$ reflects the same set of extents in $\\operatorname{Ext}()$ as $(\\sigma ,\\mathbb {S})$ , thus if Each $(G,\\lbrace \\phi =\\wedge A^{I}\\rbrace ,I_{\\phi })$ has the extent set $\\lbrace G,(\\wedge A^{I})^{I_\\phi }\\rbrace $ .",
"In this set we find that $(\\wedge A^{I})^{I_\\phi }=A$ by lem:deri.",
"Due to the apposition property the resulting context has the intersections of all subsets of $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))$ as extents.",
"This set is closed under intersection.",
"Therefor, $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S})) =\\operatorname{id}_G(\\operatorname{Ext}(\\mathbb {T}))$ .",
"$\\Box $ The conjunctive normalform $(\\psi ,\\mathbb {T})$ of a scale-measure $(\\sigma ,\\mathbb {S})$ may constitute a more human-accessible representation of the same scaling information.",
"To demonstrate this in a more practical manner we applied our method to the well-known Zoo data set by R. S. Forsyth, which we obtained from the UCI repository [16].",
"For this we computed a canonical scale-measure (lem:cssm), for which we computed an equivalent scale-measure (fig:zoo) according to lem:appconst.",
"In the presented example we see that the intent of animal taxons emerge naturally, which are indicated using red colored names in fig:zoo, (instead of extents as used by the canonical representation).",
"Order Dimension of Scale-measures An important property of formal contexts, and therefore of scale-measures, is the order dimension (def:orddim).",
"We already motivated their investigation with respect to our running example, specifically the decrease of dimension (bjicemeasure).",
"The substantiate formally our experimental finding we investigate the correspondence between order dimension and scale-hierarchies.",
"For this we employ the Ferrers dimension of contexts, which is equal to their order dimension [7].",
"A Ferrers relation is a binary relation $F\\subseteq G\\times M$ such that for $(g,m),(h,n)\\in F$ it holds that $(g,n)\\notin F \\Rightarrow (h,m)\\in F$ .",
"The Ferrers dimension of the formal context $$ is equal to the minimum number of ferrers relations $F_t\\subseteq G\\times M, t\\in T$ such that $I=\\bigcap _{t\\in T} F_t$ .",
"Proposition 24 For a context $$ and scale-measures $(\\sigma ,\\mathbb {S}),(\\psi ,\\mathbb {T})\\in \\underline{\\mathfrak {S}}()$ with $(\\sigma ,\\mathbb {S})\\le (\\psi ,\\mathbb {T})$ , where $\\sigma $ and $\\psi $ are surjective, it holds that $\\dim (\\mathbb {S})\\le \\dim (\\mathbb {T})$ .",
"We know that $(\\sigma ,\\mathbb {S})$ has the canonical representation $(\\operatorname{id}_G,_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))})$ , cf.",
"prop:eqi-scale, and the same is true for $(\\psi ,\\mathbb {T})$ .",
"Since $(\\sigma ,\\mathbb {S})\\le (\\psi ,\\mathbb {T})$ it holds that $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))\\subseteq \\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))$ and the scale $_{\\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))}$ restricted to the set $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))$ as attributes is equal to $_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))}$ .",
"Hence, a Ferrers set $F_{T}$ such that $\\bigcap _{t\\in T}F_{t}$ is equal to the incidence of $_{\\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))}$ , can be restricted to the attribute set $\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))$ and is then equal to the incidence of $_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))}$ .",
"Thus, as required, $\\dim (_{\\sigma ^{-1}(\\operatorname{Ext}(\\mathbb {S}))}) \\le \\dim (_{\\psi ^{-1}(\\operatorname{Ext}(\\mathbb {T}))})$ .$\\Box $ Building up on this result we can provide an upper bound for the dimension of apposition of scale-measures for some formal context $$ .",
"Proposition 25 For a context $$ and scale-measures $(\\sigma ,\\mathbb {S}),(\\psi ,\\mathbb {T})\\in \\underline{\\mathfrak {S}}()$ with $(\\sigma ,\\mathbb {S})\\operatorname{\\mid \\,}(\\psi , \\mathbb {T}){=}(\\delta , \\mathbb {O})$ .",
"Then order dim.",
"of $\\mathbb {O}$ is bound by $dim(\\mathbb {O}){\\le }\\dim (\\mathbb {S}){+}\\dim (\\mathbb {T})$ .",
"Without loss of generality we consider for all scale-measures their canonical representation, only.",
"Let $F_T$ be a Ferrers set of the formal context $\\mathbb {T}$ such that $\\bigcap _{t\\in T}F_{t}=I_{\\mathbb {T}}$ and similarly $\\bigcap _{s\\in S} F_s=I_{\\mathbb {S}}$ .",
"For any Ferrers relation $F$ of $\\mathbb {S}$ it follows that $F\\cup (G\\times M_{\\mathbb {T}})$ is a Ferrers relation of $\\mathbb {S}\\,\\operatorname{\\mid \\,}\\mathbb {T}$ .",
"Hence, the intersection of $\\bigcap _{s\\in S} F_s\\cup (G\\times M_\\mathbb {T})$ and $\\bigcap _{t\\in T} F_t\\cup (G\\times M_\\mathbb {S})$ is a Ferrers set and is equal to $I_{\\, \\mathbb {S}\\,\\operatorname{\\mid \\,}\\mathbb {T}}$ .",
"Since this construction does neither change the cardinality of index set $T$ nor the index set $S$ , the required inequality follows.",
"$\\Box $ Implications for Data Set Scaling We revisit the running example $_{\\text{BJ}}$ (bjice) and want to outline a semi-automatically procedure to obtain a human-meaningful scale-measure from it, as depicted in bjicemeasure, based on the insights from sec:methods.",
"In this example, we derive new attributes $M_{\\mathbb {T}}\\subseteq \\mathcal {L}(M,\\lbrace \\wedge ,\\vee ,\\lnot \\rbrace )$ from the original attribute set $M$ of $_{\\text{BJ}}$ using background knowledge.",
"This process results in $M_{\\mathbb {T}}=\\lbrace &\\textbf {Choco}=\\texttt {Choco Ice}\\vee \\texttt {ChocoPieces},\\\\&\\textbf {Caramel}=\\texttt {CaramelIce}\\vee \\texttt {Caramel},\\\\&\\textbf {Peanut}=\\texttt {PeanutIce}\\vee \\texttt {Peanut Butter},\\\\&\\textbf {Brownie, Dough, Vanilla}\\rbrace $ Such propositional features can be bear various meanings, in our example we interpret $M_{\\mathbb {T}}$ as taste attributes (as opposed to ingredients).",
"Another possible set $M_{\\mathbb {T}}$ could represent ingredient mixtures ($\\wedge $ ) to generate a recipe view on the presented ice creams.",
"From $M_{\\mathbb {T}}$ we can now derive semi-automatically a scale-measure (prop:app,prop:attr) if it exists (cor:decide).",
"Scaling of Larger Data Set To demonstrate the benefits of the scale-measure navigation on a larger data set, we evaluate our method on a data set that related spices to dishes [17], [18].",
"We decided for another food related data set, since we assume that this knowledge domain is easily to grasp.",
"Specifically, the data set is comprised of 56 dishes as objects and 37 spices as their attributes, and the resulting context is in the following denoted by $_{\\text{Spices}}$ .",
"The dishes in the data set are picked from multiple categories, such as vegetables, meat, or fish dishes.",
"The incidence $I_{_{\\text{Spices}}}$ indicates that a spice $m$ is necessary to cook a dish $g$ .",
"The concept lattice of $_{\\text{Spices}}$ has 421 concepts and is therefore too large for a meaningful human comprehension.",
"Thus, using scale-measures through our methods, we are able to generate two small-scaled views of readable size.",
"Both scales, as depicted in fig:gewscale, measure the dishes in terms of spice mixtures $M_{\\mathbb {T}}\\subseteq \\mathcal {L}(M,\\lbrace \\wedge \\rbrace )$ .",
"For the conjunction of spices we transformed intent sets $B\\in \\operatorname{Int}(_{\\text{Spices}})$ to propositional formulas $\\bigwedge _{m\\in B} m$ .",
"However, in order to retrieve a small scale context we decided for using intents with high support, only, i.e., $B^{\\prime }/G$ is high with respect to some selection criterion.",
"We employed two different selection criteria: A) high support in all dishes; B) high support in meat dishes.",
"Afterwards we derive semi-automatically two scale-measures (prop:app,prop:attr).",
"Both scale-measures include five spices mixtures.",
"The concept lattice for the scale context of A) is depicted in fig:gewscale (bottom), and for B) in fig:gewscale (top).",
"We named all selected intent sets to make them more easily addressable.",
"Both scales can be used to identify similar flavored dishes, e.g., a menu such as deer in combination with red cabbage, which share the bay leaf mix.",
"Based on the scale-measures one might be interested to further navigate in the scale-hierarchy by adding additional spice mixtures (prop:app), or employing other selection criterion, which result in different views on the data set $_{\\text{Spices}}$ , e.g., vegetarian.",
"Finally, we may point out that in contrast to feature compression techniques, such as LSA (which use linear combinations of attributes), the scale-measure attributes are directly interpretable by the semantics of propositional logics on the original data set attributes.",
"Figure: In this figure, we display the concept lattices of two scalecontexts for which the identity map is a scale-measures of thespices context.",
"The attributes of the scales are spice mixturesgenerated by propositional logic.",
"By Other we identify allobjects in the top concept for better readability.",
"Related Work Measurement is an important field of study in many (scientific) disciplines that involve the collection and analysis of data.",
"According to [1] [1] there are four feature categories that can be measured, i.e.",
"nominal, ordinal, interval and ratio features.",
"Although there are multiple extensions and re-categorizations of the original four categories, e.g., most recently [19] introduced ten [19], for the purpose of our work the original four suffice.",
"Each of these categories describe which operations are supported per feature category.",
"In the realm of formal concept analysis we work often with nominal and ordinal features, supporting value comparisons by $=$ and $<,>$ .",
"Hence grades of detail/membership cannot be expressed.",
"A framework to describe and analyze the measurement for Boolean data sets has been introduced in [20] and [21], called scale-measures.",
"It characterizes the measurement based on object clusters that are formed according to common feature (attribute) value combinations.",
"An accompanied notion of dependency has been studied [22], which led to attribute selection based measurements of boolean data.",
"The formalism includes a notion of consistency enabling the determination of different views and abstractions, called scales, to the data set.",
"This approach is comparable to OLAP [23] for databases, but on a conceptual level.",
"Similar to the feature dependency study is an approach for selecting relevant attributes in contexts based on a mix of lattice structural features and entropy maximization [24].",
"All discussed abstractions reduce the complexity of the data, making it easier to understand by humans.",
"Despite the in this work demonstrated expressiveness of the scale-measure framework, it is so far insufficiently studied in the literature.",
"In particular algorithmical and practical calculation approaches are missing.",
"Comparable and popular machine learning approaches, such as feature compressed techniques, e.g., Latent Semantic Analysis [25], [26], have the disadvantage that the newly compressed features are not interpretable by means of the original data and are not guaranteed to be consistent with said original data.",
"The methods presented in this paper do not have these disadvantages, as they are based on meaningful and interpretable features with respect to the original features using propositional expressions.",
"In particular preserving consistency, as we did, is not a given, which was explicitly investigated in the realm scaling many-valued formal contexts [14] and implicitly studied for generalized attributes [15].",
"Earlier approaches to use scale contexts for complexity reduction in data used constructs such as $(G_N\\subseteq \\mathcal {P}(N),N,\\ni )$ for a formal context $=(G,M,I)$ with $N\\subseteq M$ and the restriction that at least all intents of $$ restricted to $N$ are also intent in the scale [27].",
"Hence, the size of the scale context concept lattice depends directly on the size of the concept lattice of $$ .",
"This is particularly infeasible if the number of intents is exponential, leading to incomprehensible scale lattices.",
"This is in contrast to the notion of scale-measures, which cover at most the extents of the original context, and can thereby display selected and interesting object dependencies of scalable size.",
"Conclusion Our work has broadened the understanding of the data scaling process and has paved the way for the development of novel scaling algorithms, in particular for Boolean data, which we summarize under the term Exploring Conceptual Measurements.",
"We build our framework on the notion of scale-measures, which themselves are interpretations of formal contexts.",
"By studying and extending the theory on scale-measures, we found that the set of all possible measurements for a formal context is lattice ordered, up to equivalence.",
"Thus, this set is navigable using the lattice's meet and join operations.",
"Furthermore, we found that the problem of deciding whether for a given formal context $$ and a tuple $(\\sigma ,\\mathbb {S})$ the latter represents a scale-measure for the former is PTIME with respect to the respective object and attribute set sizes.",
"All this and the following is based on our main result that for a given formal context $$ the set of all scale-measures and the set of all sub-closure systems of $()$ are isomorphic.",
"To ensure our goal for human comprehensible scaling we derived a propositional logic scaling of formal contexts by transferring and extending results from conceptual scaling [14].",
"With this approach, we are able to introduce new features that lead to interpretable scale features in terms of a logical formula and with respect to the original data set attributes.",
"Moreover, these features are suitable to create any possible scale measurement of the data.",
"Finally, we found that the order dimension decreases monotonously when scale-measures are coarsened, hinting the principal improved readability of scale-measures in contrast to the original data set.",
"We have substantiated our theoretical results with three exemplary data analyses.",
"In particular we demonstrated that employing propositional logic on the attribute set enables us to express and apply meaningful scale features, which improved the human readability in a natural manner.",
"All methods used throughout this work are published with the open source software conexp-clj[28], a research tool for Formal Concept Analysis.",
"We identified three different research directions for future work, which together may lead to an efficient and comprehensible data scaling framework.",
"First of all, the development of meaningful criteria for ranking or valuing scale-measures is necessary.",
"Although our results enable an efficient navigation in the lattice of scale-measures, it cannot provide a promising direction, except from decreasing the order dimension.",
"Secondly, efficient algorithms for computing an initial, well ranked/rated scale-measure and the subsequent navigation are required.",
"Even though we showed a bound for the computational run time complexity, we assume that this can still be improved.",
"Thirdly, a natural approach for decreasing the computational cost of navigating conceptual measurements would be to employ a set of minimal closure generators instead of the closure system.",
"We speculate that our results hold in this case.",
"Yet, it is an open questions if procedures, such as TITANIC [29], can be adapted to efficiently navigate the scale-hierarchy of a formal context.",
"Example Figure: Concept lattice of a scale-measure of the zoo data set withtwenty-seven of the original 4579 concepts.",
"Contained objects areanimals and attributes are characteristics.",
"Newly introducedlogical attributes are a characterization of animal taxons.",
"Theobjects girl,frogB were omitted.",
"We groupedOtherFishes={seahorse, sole, herring, piranha, pike,chub, haddock, stingray, carp, bass, dogfish, catfish,tuna},OtherMammals={reindeer, aardvark, polecat,wolf, mole, vole, hare, boar, cavy, antelope, goat, puma,mongoose, pony, bear, pussycat, lynx, elephant, calf, mink,opossum, leopard, buffalo, lion, giraffe, cheetah, oryx, deer,hamster, raccoon},OtherBirds={gull, parakeet, crow,skua, swan, hawk, sparrow, lark, wren, dove, vulture, penguin,duck, flamingo, pheasant, rhea, ostrich, skimmer, chicken,kiwi}"
],
[
"Implications for Data Set Scaling",
"We revisit the running example $_{\\text{BJ}}$ (bjice) and want to outline a semi-automatically procedure to obtain a human-meaningful scale-measure from it, as depicted in bjicemeasure, based on the insights from sec:methods.",
"In this example, we derive new attributes $M_{\\mathbb {T}}\\subseteq \\mathcal {L}(M,\\lbrace \\wedge ,\\vee ,\\lnot \\rbrace )$ from the original attribute set $M$ of $_{\\text{BJ}}$ using background knowledge.",
"This process results in $M_{\\mathbb {T}}=\\lbrace &\\textbf {Choco}=\\texttt {Choco Ice}\\vee \\texttt {ChocoPieces},\\\\&\\textbf {Caramel}=\\texttt {CaramelIce}\\vee \\texttt {Caramel},\\\\&\\textbf {Peanut}=\\texttt {PeanutIce}\\vee \\texttt {Peanut Butter},\\\\&\\textbf {Brownie, Dough, Vanilla}\\rbrace $ Such propositional features can be bear various meanings, in our example we interpret $M_{\\mathbb {T}}$ as taste attributes (as opposed to ingredients).",
"Another possible set $M_{\\mathbb {T}}$ could represent ingredient mixtures ($\\wedge $ ) to generate a recipe view on the presented ice creams.",
"From $M_{\\mathbb {T}}$ we can now derive semi-automatically a scale-measure (prop:app,prop:attr) if it exists (cor:decide)."
],
[
"Scaling of Larger Data Set",
"To demonstrate the benefits of the scale-measure navigation on a larger data set, we evaluate our method on a data set that related spices to dishes [17], [18].",
"We decided for another food related data set, since we assume that this knowledge domain is easily to grasp.",
"Specifically, the data set is comprised of 56 dishes as objects and 37 spices as their attributes, and the resulting context is in the following denoted by $_{\\text{Spices}}$ .",
"The dishes in the data set are picked from multiple categories, such as vegetables, meat, or fish dishes.",
"The incidence $I_{_{\\text{Spices}}}$ indicates that a spice $m$ is necessary to cook a dish $g$ .",
"The concept lattice of $_{\\text{Spices}}$ has 421 concepts and is therefore too large for a meaningful human comprehension.",
"Thus, using scale-measures through our methods, we are able to generate two small-scaled views of readable size.",
"Both scales, as depicted in fig:gewscale, measure the dishes in terms of spice mixtures $M_{\\mathbb {T}}\\subseteq \\mathcal {L}(M,\\lbrace \\wedge \\rbrace )$ .",
"For the conjunction of spices we transformed intent sets $B\\in \\operatorname{Int}(_{\\text{Spices}})$ to propositional formulas $\\bigwedge _{m\\in B} m$ .",
"However, in order to retrieve a small scale context we decided for using intents with high support, only, i.e., $B^{\\prime }/G$ is high with respect to some selection criterion.",
"We employed two different selection criteria: A) high support in all dishes; B) high support in meat dishes.",
"Afterwards we derive semi-automatically two scale-measures (prop:app,prop:attr).",
"Both scale-measures include five spices mixtures.",
"The concept lattice for the scale context of A) is depicted in fig:gewscale (bottom), and for B) in fig:gewscale (top).",
"We named all selected intent sets to make them more easily addressable.",
"Both scales can be used to identify similar flavored dishes, e.g., a menu such as deer in combination with red cabbage, which share the bay leaf mix.",
"Based on the scale-measures one might be interested to further navigate in the scale-hierarchy by adding additional spice mixtures (prop:app), or employing other selection criterion, which result in different views on the data set $_{\\text{Spices}}$ , e.g., vegetarian.",
"Finally, we may point out that in contrast to feature compression techniques, such as LSA (which use linear combinations of attributes), the scale-measure attributes are directly interpretable by the semantics of propositional logics on the original data set attributes.",
"Figure: In this figure, we display the concept lattices of two scalecontexts for which the identity map is a scale-measures of thespices context.",
"The attributes of the scales are spice mixturesgenerated by propositional logic.",
"By Other we identify allobjects in the top concept for better readability."
],
[
"Related Work",
"Measurement is an important field of study in many (scientific) disciplines that involve the collection and analysis of data.",
"According to [1] [1] there are four feature categories that can be measured, i.e.",
"nominal, ordinal, interval and ratio features.",
"Although there are multiple extensions and re-categorizations of the original four categories, e.g., most recently [19] introduced ten [19], for the purpose of our work the original four suffice.",
"Each of these categories describe which operations are supported per feature category.",
"In the realm of formal concept analysis we work often with nominal and ordinal features, supporting value comparisons by $=$ and $<,>$ .",
"Hence grades of detail/membership cannot be expressed.",
"A framework to describe and analyze the measurement for Boolean data sets has been introduced in [20] and [21], called scale-measures.",
"It characterizes the measurement based on object clusters that are formed according to common feature (attribute) value combinations.",
"An accompanied notion of dependency has been studied [22], which led to attribute selection based measurements of boolean data.",
"The formalism includes a notion of consistency enabling the determination of different views and abstractions, called scales, to the data set.",
"This approach is comparable to OLAP [23] for databases, but on a conceptual level.",
"Similar to the feature dependency study is an approach for selecting relevant attributes in contexts based on a mix of lattice structural features and entropy maximization [24].",
"All discussed abstractions reduce the complexity of the data, making it easier to understand by humans.",
"Despite the in this work demonstrated expressiveness of the scale-measure framework, it is so far insufficiently studied in the literature.",
"In particular algorithmical and practical calculation approaches are missing.",
"Comparable and popular machine learning approaches, such as feature compressed techniques, e.g., Latent Semantic Analysis [25], [26], have the disadvantage that the newly compressed features are not interpretable by means of the original data and are not guaranteed to be consistent with said original data.",
"The methods presented in this paper do not have these disadvantages, as they are based on meaningful and interpretable features with respect to the original features using propositional expressions.",
"In particular preserving consistency, as we did, is not a given, which was explicitly investigated in the realm scaling many-valued formal contexts [14] and implicitly studied for generalized attributes [15].",
"Earlier approaches to use scale contexts for complexity reduction in data used constructs such as $(G_N\\subseteq \\mathcal {P}(N),N,\\ni )$ for a formal context $=(G,M,I)$ with $N\\subseteq M$ and the restriction that at least all intents of $$ restricted to $N$ are also intent in the scale [27].",
"Hence, the size of the scale context concept lattice depends directly on the size of the concept lattice of $$ .",
"This is particularly infeasible if the number of intents is exponential, leading to incomprehensible scale lattices.",
"This is in contrast to the notion of scale-measures, which cover at most the extents of the original context, and can thereby display selected and interesting object dependencies of scalable size."
],
[
"Conclusion",
"Our work has broadened the understanding of the data scaling process and has paved the way for the development of novel scaling algorithms, in particular for Boolean data, which we summarize under the term Exploring Conceptual Measurements.",
"We build our framework on the notion of scale-measures, which themselves are interpretations of formal contexts.",
"By studying and extending the theory on scale-measures, we found that the set of all possible measurements for a formal context is lattice ordered, up to equivalence.",
"Thus, this set is navigable using the lattice's meet and join operations.",
"Furthermore, we found that the problem of deciding whether for a given formal context $$ and a tuple $(\\sigma ,\\mathbb {S})$ the latter represents a scale-measure for the former is PTIME with respect to the respective object and attribute set sizes.",
"All this and the following is based on our main result that for a given formal context $$ the set of all scale-measures and the set of all sub-closure systems of $()$ are isomorphic.",
"To ensure our goal for human comprehensible scaling we derived a propositional logic scaling of formal contexts by transferring and extending results from conceptual scaling [14].",
"With this approach, we are able to introduce new features that lead to interpretable scale features in terms of a logical formula and with respect to the original data set attributes.",
"Moreover, these features are suitable to create any possible scale measurement of the data.",
"Finally, we found that the order dimension decreases monotonously when scale-measures are coarsened, hinting the principal improved readability of scale-measures in contrast to the original data set.",
"We have substantiated our theoretical results with three exemplary data analyses.",
"In particular we demonstrated that employing propositional logic on the attribute set enables us to express and apply meaningful scale features, which improved the human readability in a natural manner.",
"All methods used throughout this work are published with the open source software conexp-clj[28], a research tool for Formal Concept Analysis.",
"We identified three different research directions for future work, which together may lead to an efficient and comprehensible data scaling framework.",
"First of all, the development of meaningful criteria for ranking or valuing scale-measures is necessary.",
"Although our results enable an efficient navigation in the lattice of scale-measures, it cannot provide a promising direction, except from decreasing the order dimension.",
"Secondly, efficient algorithms for computing an initial, well ranked/rated scale-measure and the subsequent navigation are required.",
"Even though we showed a bound for the computational run time complexity, we assume that this can still be improved.",
"Thirdly, a natural approach for decreasing the computational cost of navigating conceptual measurements would be to employ a set of minimal closure generators instead of the closure system.",
"We speculate that our results hold in this case.",
"Yet, it is an open questions if procedures, such as TITANIC [29], can be adapted to efficiently navigate the scale-hierarchy of a formal context."
]
] | 2012.05267 |
[
[
"On-sky performance and recent results from the Subaru coronagraphic\n extreme adaptive optics system"
],
[
"Abstract We describe the current on-sky performance of the Subaru Coronagraphic Extreme Adaptive Optics (SCExAO) instrument on the Subaru telescope on Maunakea, Hawaii.",
"SCExAO is continuing to advance its AO performance, delivering H band Strehl ratios in excess of 0.9 for bright stars.",
"We describe new advances with SCExAO's wavefront control that lead to a more stable corrected wavefront and diffraction-limited imaging in the optical, modifications to code that better handle read noise suppression within CHARIS, and tests of the spectrophotometric precision and accuracy within CHARIS.",
"We outline steps in the CHARIS Data Processing Pipeline that output publication-grade data products.",
"Finally, we note recent and upcoming science results, including the discovery of new directly-imaged systems and multiwavelength, deeper characterization of planet-forming disks, and upcoming technical advances that will improve SCExAO's sciencec capabilities."
],
[
"Introduction",
"In the past twelve years, facility adaptive optics (AO) systems on 8-10m class telescopes and now successor extreme AO platforms have provided the first images of (super-)jovian extrasolar planets orbiting nearby, young stars[1], [2], [3], [4], [5], [6], [7].",
"Depending on the exact adopted definition for a planet vs. a brown dwarf, about 12–20 directly-imaged planets have been discovered.",
"Nearly all imaged exoplanets have masses exceeding 5 jovian masses ($M_{\\rm J}$ ) and orbit beyond 10–30 au from their host star.",
"Typical planet-to-star contrasts in the near-infrared range from 10$^{-4}$ to 10$^{-6}$ ; on the sky, projected separations range from 0$.\\!\\!^{\\prime \\prime }$ 1 to 1$^{\\prime \\prime }$.",
"Recent extreme AO direct imaging surveys from the Gemini Planet Imager (GPI)[8] on Gemini-South and Spectro-Polarimetric High-contrast Exoplanet REsearch instrument (SPHERE)[9] on the Very Large Telescope (VLT) show that superjovian planets at 10–100 au are rare, but the frequency of these planets may increase closer to 5 au, consistent with a turnover in the jovian planet population suggested from radial-velocity surveys[10], [11].",
"Photometric and spectroscopic follow-up observations of directly-imaged planets reveal key features in young jovian exoplanet/substellar atmospheres, including the presence of copious dust and thicker clouds from near-IR colors, non-equilibrium carbon chemistry from near-IR spectra and mid-IR photometry, low surface gravities from near-IR spectra, and molecular abundances from medium-resolution spectra [12], [13], [14], [15], [16].",
"Longer-term, ground based extreme AO systems on the next generation of 30-m class telescopes are designed with the goal of directly imaging planets in reflected light around mature stars, including Earth-like habitable zone worlds[17].",
"However, current ground-based capabilities only probe the extremes of planet formation: moderate to wide-separation superjovian planets.",
"The best near-IR post-processed contrasts achieved thus far – $\\sim $ 10$^{-6}$ at 0$.\\!\\!^{\\prime \\prime }$ 2 and 5$\\times $ 10$^{-7}$ at 0$.\\!\\!^{\\prime \\prime }$ 5[18] – are still 10–1000 times too bright to detect reflected-light jovian-sized planets around the nearest mature stars or self-luminous Saturn to Jupiter-mass planets around more distant young Suns on Jupiter-to-Neptune-like orbits.",
"Detecting and then characterizing solar system-like planets from the ground requires significant advances over capabilities demonstrated with the first generation of extreme AO systems.",
"In this paper, we describe the status, recent hardware/software advances, and recent science results for the Subaru Coronagraphic Extreme Adaptive Optics project (SCExAO) at the Subaru Telescope on Maunakea [19] coupled with its main near-IR planet-imaging science instrument, the CHARIS integral field spectrograph [20].",
"Section §2 motivates and describes wavefront control within SCExAO, noting recent changes that have improved its performance stability and correction for low-order modes.",
"Section §3 summarizes CHARIS, notes current its read noise level and changes to the data reduction pipeline to work around these problems, notes spectrophotometric calibration and astrometric calibration, and performance.",
"Section §4 provides a walkthrough of CHARIS data reduction steps currently adopted within the CHARIS data processing pipeline (DPP) [21].",
"Section §5 gives a summary of recent science results, focusing on newly discovered companions to nearby stars and detailed characterization of planet-forming disks.",
"Section §6 outlines updated future directions for SCExAO and for CHARIS, describing upgrades that will improve the instrument combination's performance, enhance its planet detection capabilities, and broaden its science capabilities.",
"Figure: Wavefront error budget for a spatially-filtered Shack-Hartmann wavefront sensor running at 1 kHz and correcting turbulence for an II = 8 star on an 8m telescope.",
"See , for a more in-depth discussion.Figure REF sketches an example wavefront error budget that illustrates the technical challenges imaging planets below the 10$^{-6}$ level from the ground with 8m class telescopes.",
"At small angles, contrasts are nominally most impacted by the temporal bandwidth error (“servo lag\") and photon noise on the wavefront sensor (WFS).",
"These two terms are also coupled: a faster-sampling WFS loop reduces temporal errors, as the WFS loop runs faster relative to the atmospheric coherence time, but suffers increased photon noise error.",
"Chromatic wavefront errors and scintillation further impede raw contrasts below 10$^{-5}$ .",
"Demonstrated contrast gains using advanced post-processing techniques range between a factor of 10 and 100[23].",
"Thus, reaching post-processed contrasts approaching 10$^{-7}$ requires raw contrasts below 10$^{-5}$ and, in turn, significant reduction of dominant coupled error terms and then some suppression of chromatic wavefront errors.",
"SCExAO's wavefront control architecture is designed to significantly mitigate coupled temporal bandwidth and photon noise error terms, while providing a path forward to reducing chromatic errors.",
"Figure: The current schematic of SCExAO.",
"Note that the coronagraphic low-order loop is not in normal operation; the MKIDs camera (MEC) is undergoing commissioning."
],
[
"SCExAO Architecture",
"Figure REF displays a schematic of high-contrast imaging with SCExAO.",
"Incoming starlight is partially corrected for atmospheric turbulence using AO-188, Subaru's facility adaptive optics system, which typically achieves $\\sim $ 30–40% Strehl at 1.6 $\\mu m$ under median to good conditions.",
"Then it is further sharpened by the SCExAO wavefront loop, reaching typical $H$ band Strehl ratios of 80–90% under median to good conditions, with higher values for the brightest stars under the best conditions[24].",
"The PSF core from sharpened starlight is then blocked and Airy rings suppressed by a suite of coronagraphs, where the standard Lyot coronagraph is typically used for observations requiring data over multiple bandpasses.",
"Finally, this light is fed into a suite of science instruments, where the CHARIS integral field spectrograph is the workhorse instrument operating over the $JHK$ bandpass.",
"Common observing modes include sending $Y$ band light to a high speed broadband camera or MKIDS-based detector (MEC) and $JHK$ to CHARIS.",
"Alternatively, the broadband camera/MEC can take a broader bandpass and CHARIS focus on a single filter (e.g.",
"$H$ or $K$ ).",
"For science cases focused on circumstellar material (e.g.",
"protoplanetary disks) targeting optically bright stars, the bluer ($<$ 800 nm) optical wavelengths normally sent to the WFS (see below) are redirected to the VAMPIRES instrument[25]."
],
[
"Wavefront Control with SCExAO: Recent Advances",
"SCExAO's wavefront control (WFC) loop consists of a 2000-element MEMS deformable mirror from Boston MicroMachines driven by a modulated Pyramid wavefront sensor (PyWFS) using a double roof pyramid prism optic and an OCAM$^{2}$ K EMCCD camera from First-Light Imaging that operates over a 600–900 nm bandpass (smaller if used in combination with VAMPIRES).",
"The SCExAO WFC loop can run at speeds up to 3.5 kHz.",
"However, for most science observations it operates at a slower 2 kHz for bright stars coupled with predictive wavefront control[26] (see below) to decrease temporal bandwidth error and wavefront sensor noise, limited by an overall latency of just under 1 ms. For very faint stars($I$ $>$ 9), we typically run the loop at 1 kHz.",
"The loop can correct for up to 1200 modes of dynamic aberrations.",
"To operate SCExAO's wavefront control loop, we use the Command and Control for Adaptive Optics (CACAO) open-use software framework[27]: see Paper 11448-145 for details.",
"CACAO uses a unified shared memory structure to read in images from multiple cameras and then issue DM commands and apply various offsets to the PyWFS.",
"SCExAO leverages the very high bandwidth of its MEMS DM, coupled to a precise calibration of the hardware latency, to acquire AO response matrices (RM) – influence functions that map actuator offsets to changes in the focal plane – at a significant speed.",
"A complete sequence of Hadamard modes can be probed in just 1.25 s at the usual framerate of 2 kHz.",
"Stabilized RMs with optimized signal-to noise ratio (SNR) are usually completed within $\\sim $ 10 minutes.",
"The bandwidth of the DM enables us to probe the WFS response faster than the typical turbulence fluctuations on sky, and SCExAO has had the capability to acquire or refine RMs on sky (e.g.",
"\"RM bootstrapping\") for many several years.",
"This is ideal, as nonlinear effects from the PyWFSs[28] can be measured in-situ and compensated for.",
"Under the best seeing conditions (e.g.",
"$\\theta _{\\rm V}$ $\\sim $ 0$.\\!\\!^{\\prime \\prime }$ 15–0$.\\!\\!^{\\prime \\prime }$ 4), on-sky RMs result in extremely well-corrected point-spread functions, with estimated $H$ -band Strehl ratios reaching 94% for bright stars and extreme AO corrections even for stars as faint as R = 11.6 [29], [24].",
"However, with PyWFSs nonlinearities being quite volatile, such a strategy is only paying for much better-than-median seeing and temporally stable conditions.",
"Otherwise, the measured RM is mostly filled with randomized nonlinear components from the PyWFS, and the control capability is reduced to $\\sim $ 500 modes, with reduced stability.",
"The reliability of on-sky RMs to compute low-order modes is further compromised by the tendency of AO-188 to habitually split PSF cores or first Airy rings and the “low-wind effect\"[30].",
"Figure: Laboratory-based response matrix calculations for SCExAO blend a) Hadamard pokes sampling high spatial frequencies and b) Zernike/Fourier pokes sampling low spatial frequencies.",
"This approach, when combined with predictive control and in lieu of acquiring an RM on sky, yields far superior control of low-order modes and diffraction-limited imaging in the red optical (panel c).",
"Note that the rotation angle for the DM and PyWFS are offset by 45 o ^{o}, explaining the tilt of the poke vs. PyWFS response patterns.Thus, we now prefer using command laws calibrated in the lab, coupled to frequency-dependent modal gain optimization.",
"As shown in panels a) and b) of Figure REF , our lab RM calculation blends a high spatial frequency Hadamard poke sequence sensitive to high-order aberrations (12,000 pokes) with low spatial frequency poke patterns consisting of an explicit (non-orthogonal) mix of the first few Zernike modes and Fourier mode pokes up to three cycles per aperture (120).",
"While Hadamard pokes sequences provide a high SNR estimate of the WFS response at high spatial frequencies, they perform much more poorly at sampling low spatial frequencies.",
"Explicit Fourier/Zernike modes compensate for these shortcomings.",
"This strategy has considerably improved AO stability, and enables a significant fraction of “salvageable time” during nights with poor seeing and high wind speed.",
"Noiseless, daytime RMs – which lack non-linearities – also provide a more stable baseline for on-sky training of predictive control[31].",
"The raw broadband contrast for moderately bright stars ($I$ $\\sim $ 5-7) under “good\" conditions (e.g.",
"$\\theta _{\\rm V}$ $\\sim $ 0$.\\!\\!^{\\prime \\prime }$ 5) using the lab-calibrated RM approaches and sometimes exceeds the best performance SCExAO has achieved on 1st and 2nd magnitude stars using an on-sky RM taken under exceptional conditions ($\\theta _{\\rm V}$ $\\sim $ 0$.\\!\\!^{\\prime \\prime }$ 18–0$.\\!\\!^{\\prime \\prime }$ 45).",
"While formally an on-sky RM could provide better sensing and control of the highest order modes, this advantage appears to be mitigated by the lab RM's superior low-order control, which allows the loop to be run at a higher gain, combined with improved efficacy of our coronagraphs, whose performance is particularly sensitive to low-order aberrations like tip-tilt (see [32]).",
"The visible PSF demonstrates that the lowest order modes are in fact exceptionally well controlled (Figure REF , right panel).",
"In good conditions, the visible PSF is diffraction limited with a well defined PSF core and first Airy ring.",
"Finally, predictive control using empirical orthogonal functions[33] is now part of SCExAO's normal operation.",
"Predictive control allows us to reduce the servo lag error in the wavefront sensing error budget, improving contrast at small angles, while running at a slightly slower 2 kHz loop speed.",
"As shown in Figure REF (panel c), predictive control improves PSF stability and contributes to diffraction limited imaging over a wide wavelength range."
],
[
"Overview",
"CHARIS is a lenslet-based cryogenic integral field spectrograph capable of operating from 1.1 to 2.4 $\\mu m$[20].",
"After receiving light that is well corrected from SCExAO and partially suppressed by a coronagraph, a sparse image is formed on the lenslet array.",
"After a pinhole array mitigates lenslet diffraction, light from the lenslet array is dispersed from one of two prisms onto a 2048x2048-pixel Hawaii 2RG detector into 135x135 30 pixel-long microspectra.",
"With the low-resolution prism in position, CHARIS spectra have a resolution of $\\mathcal {R}$ $\\sim $ 20 and cover 22 channels with central wavelengths of range 1.15–2.37 $\\mu m$ .",
"In its high-resolution mode ($\\mathcal {R}$ $\\sim $ 70), CHARIS spectra cover the J, H, or K passbands.",
"The CHARIS Data Reduction Pipeline (DRP) converts raw CHARIS detector data into data cubes ($x$ by $y$ by $\\lambda $ ).",
"Common among Hawaii 2RG detectors, CHARIS's detector suffers from $1/f$ read noise correlated among readout channels.",
"CHARIS's correlated read noise on the Nasmyth platform at Subaru is significantly higher than it was during pre-shipment laboratory testing at Princeton, enough to compromise the instrument's science capabilities[34].",
"To suppress this read noise, the $\\chi ^{2}$ cube extraction method constructs and then subtracts from the detector data a two-dimensional model of the entire detector readout.",
"These residuals are then used to model and thus remove the correlated component of the read noise to levels approaching that of laboratory values.",
"As shown in Figure REF , the typical CHARIS read noise level has fluctuated wildly for a subset of channels.",
"No one single cause for this change has been pinpointed, although the movement of CHARIS and other instruments on the Nasmyth platform may be connected to some fluctuations.",
"Our October 2018 run immediately followed an observatory power outage and warm-up of CHARIS.",
"The DRP's cube extraction program's read noise suppression failed, leaving swaths of the CHARIS image plane with exceptionally noisy (unusable) pixels along one axis and NaN stripes along a perpendicular axis (middle panel) for over 90% of our exposures.",
"While elevated noise levels dropped to slightly lower levels during the next run, they reemerged in Summer 2020.",
"To compensate, we relaxed the outlier rejection thresholds in fitramps.pyx to 30-$\\sigma $ (line 603) and 20-$\\sigma $ (line 609).",
"As shown in Figure REF (right panel), this modification recovered CHARIS data.",
"Comparing results using these new threshholds and previous ones for exposures with lower read noise levels showed negligible differences in data quality.",
"In addition to modifying the source code to adjust relax read noise suppression, we pointed the DRP to a different URL for populating fits header metadata responsible for determining the parallactic angle.",
"The previous one is down for maintainence indefinitely.",
"Specifically, in calc$\\_$ metadata.py, we added the following two lines in succession at the end of “imports\": 1) from astropy.utils import iers 2) iers.Conf.iers$\\_$ auto$\\_$ url.set('ftp://cddis.gsfc.nasa.gov/pub/products/iers/finals2000A.all') CHARIS users are strongly encouraged to verify their installation's settings for read noise suppression and the URL for fits header metadata and then modify them as described above, if needed."
],
[
"Spectrophotometric Precision and Accuracy",
"To perform spectrophotometric and astrometric calibration, CHARIS uses artificial satellite speckles.",
"These speckles are generated by applying sine-waves to the DM and are unocculted copies of the central PSF.",
"Their intensity scales as $\\lambda ^{-2}$ .",
"Using tests with SCExAO's internal source over a narrow bandpass centered on 1.55 $\\mu m$ in Summer 2017, T. Currie measured the contrast of the speckles to be 6.412 $\\pm $ 0.05 mags (2.72$\\times $ 10$^{-3}$ $\\pm $ 1.3$\\times $ 10$^{-4}$ ) for a 25nm modulation.",
"The predicted contrast for a 50nm modulation should be exactly a factor of 4 higher (4.91 mags), which is completely consistent with our measurements: 4.92 $\\pm $ 0.05 mag (1.08$\\times $ 10$^{-2}$ ).",
"Separate tests performed by J. Lozi in Fall 2018 used the superK laser in combination with CHARIS with the Lyot focal plane mask in place, simulating a CHARIS observation (Figure REF , top-left panel).",
"At 1.55 $\\mu m$ , the estimated contrast is $\\sim $ 2.94$\\times $ 10$^{-3}$ , or $\\sim $ 8% higher.",
"The median-average of spot contrasts nearly perfectly matches the predicted $\\lambda ^{-2}$ trend, with a deviation of $\\sim $ 1% or less for each channel.",
"The contrasts for two of the four satellite spots show extremely small deviations from the median ($\\sigma (spot/median)$ $\\sim $ 2.2–2.4%).",
"While the two other spots show slightly larger deviations when the full wavelength range is considered (5–6.4%), over a smaller range ($\\lambda $ $>$ 1.25 $\\mu m$ ) their residuals are nearly the same as for the other two spots (2.6–3.2%).",
"Both tests were performed using the internal source where we differenced exposures with and without satellite spots to remove background speckles.",
"On sky under normal operation, the satellite speckles are always on.",
"Background speckles therefore limit the SNR of the satellite speckles and thus the precision of our spectrophotometric calibration.",
"The speckles lie at $\\sim $ 15.9 $\\lambda $ /D in each channel, at the edge of the region well corrected by the DM.",
"In median to good seeing conditions when observing a bright star, the satellite speckles with a 25nm modulation typically have SNR $\\sim $ 25–40 for most channels but lower for fainter stars, in channels covering telluric features, or at the reddest $K$ -band channel where the thermal background is highest.",
"The righthand panel of Figure REF compares the CHARIS spectrum for for the HD 1160 B brown dwarf [36] in broadband mode and in higher-resolution $J$ , $H$ , and $K$ bands to the published spectrum from [35].",
"No PSF subtraction techniques were applied to the CHARIS data, the total sequences spanned just a few minutes each, and the satellite spots were used only for the first few CHARIS data cubes for each band.",
"We adopted the 2017 spot absolute calibration.",
"The CHARIS broadband and higher-resolution spectra agree extremely well with one another modulo a slight (2.5%) absolute calibration offset, which could be explained by slight changes in the AO performance or transmission between the satellite-on exposures and those lacking the satellite spots.",
"While the shape of our spectra agree with that for SPHERE/IFS outside of telluric-dominated passbands, our spectra are $\\sim $ 20% brighter.",
"However, our photometry integrated over the $JH$ passbands agrees far better with published values using unsaturated SCExAO/HiCIAO images[37] and also with subsequent SPHERE long-slit spectra[38].",
"Photometry adopting the 2018 superK-derived absolute calibration agrees marginally less with SCExAO/HiCIAO photometry but is still consistent within errors.",
"In summary, CHARIS can offer extremely good spectrophotometric precision and accuracy.",
"The median value of the satellite spots per channel follows the predicted $\\lambda ^{-2}$ trend to within 1%; the contrast scaling vs. modulation amplitude is also consistent with expectations.",
"However, we note a slight offset in absolute spectrophotometric calibration on order of $\\sim $ 8% between separate tests.",
"This difference is plausibly due to slight changes in the placement of SCExAO's focal-plane masks between 2017 and 2018, a parameter that will change on sky if the star is reacquired or observing suffers stretches of significant tip-tilt (e.g.",
"before and after a transit near zenith).",
"Thus, achieving absolute calibration necessary to study variability at the $<$ 5% level is not yet feasible with SCExAO/CHARIS.",
"Finally, the intrinsic SNRs of the satellite speckles used to flux-calibrate data cubes limit the precision of extracted spectra to $\\sim $ 3–5% precision per channel.",
"Figure: SCExAO/CHARIS 5-σ\\sigma contrast curve scaled to one hour integration time (magenta) under good conditions with excellent (Δ\\Delta PA >> 50 o ^{o}) parallactic angle motion compared to performance reported in Ref.",
"Currie2017 for SCExAO/HiCIAO in Fall 2016, expected performance given recent improvements in PSF subtraction methods, and projected performance in 2022.Horizontal dotted lines note the contrasts expected for jovian planets around a 50 Myr-old Sun-like star at a distance of 25 pcpc."
],
[
"SCExAO/CHARIS High-Contrast Performance",
"Figure REF displays a 5-$\\sigma $ , 1 hour sequence contrast curve with CHARIS for a 5th magnitude star under good observing conditions and excellent parallactic angle rotation, showing contrasts of 10$^{-5}$ , 2$\\times $ 10$^{-6}$ , and 10$^{-6}$ 0$.\\!\\!^{\\prime \\prime }$ 25, 0$.\\!\\!^{\\prime \\prime }$ 4, and 0$.\\!\\!^{\\prime \\prime }$ 8, respectively.",
"Performance degrades for poor field rotation ($\\Delta $ PA $\\lesssim $ 30$^{o}$ ) or poorer AO corrections.",
"The data were processed with A-LOCI[41] first applied to ADI data and then with SDI on the post-ADI residuals.",
"The SDI PSF subtraction step typically gains a factor of 1.5–2 at mid spatial frequencies; A-LOCI typically outperforms KLIP[42] by 20–80%.",
"Recent updates to our PSF subtraction algorithm, in particularly more optimal optimization/subtraction zone geometries[43] in the SDI step suggest an additional gain of 20–40% at most separations (dashed line).",
"Simultaneous ADI+SDI PSF subtraction as has been well-demonstrated with SPHERE likewise would provide an additional performance gain[18] and is in development.",
"SCExAO/CHARIS performance is equal to or slightly exceeds that of GPI under good conditions (e.g.",
"2$\\times $ 10$^{-5}$ at 0$.\\!\\!^{\\prime \\prime }$ 25, 5$\\times $ 10$^{-6}$ at 0$.\\!\\!^{\\prime \\prime }$ 4[23]), while it slightly trails SPHERE's performance under good conditions at small angles (0$.\\!\\!^{\\prime \\prime }$ 1–0$.\\!\\!^{\\prime \\prime }$ 4)[18].",
"The near future will see the replacement of AO-188 with a higher order DM, which simulations show should gain a factor of $\\sim $ 5 in raw contrast.",
"This hardware advance coupled with focal-plane wavefront sensing improvements to be commissioned by the end of 2021 should improve performance to $\\sim $ 10$^{-6}$ contrast at 0$.\\!\\!^{\\prime \\prime }$ 25 and slightly deeper at wider separations (dark green line).",
"Figure: Selected reduction steps in the CHARIS Data Post-Processing Pipeline.",
"(top-left) A CHARIS data cube after sky-subtraction and precise image registration have been performed.",
"(top-right) Kurucz model atmosphere for an F8V star binned to CHARIS's spectral resolution and resampled along the CHARIS broadband mode wavelength grid.",
"(bottom-left) PSF-subtracted image of HR 8799 showing extremely high SNR detections of HR 8799 cde.",
"(bottom-right) Grid of synthetic L-type planets forward-modeled through our data using the approach of Ref.",
"Currie2018a to simulate signal loss due to A-LOCI in ADI+SDI mode."
],
[
"The CHARIS Data Processing Pipeline: A Walkthrough",
"The CHARIS DPP package has been used in nearly all SCExAO/CHARIS science papers thus far and functions as an end-to-end pipeline, ingesting rectified CHARIS data cubes and producing publication-grade science products: PSF-subtracted images, planet and disk forward-modeling, extracted and calibrated spectra, contrast curves, and basic empirical comparisons.",
"It is currently written in IDL with plans to translate into Python 3.7 starting in mid/late 2021.",
"In efforts led by K. Lawson, we are currently beta-testing code to support polarimetric differential imaging with CHARIS's integral field polarimetry mode[44].",
"While the package is currently proprietary, we expect a public version release later in 2021.",
"Below we describe key reduction steps, a subset of which are shown in Figure REF .",
"Data cube preparation – The program charis$\\_$ newobs creates a directory structure and a parameter file with syntax of [targetname]+$\\_$ +[filter(broadband,J,H,K)]+'.info', populating it with critical information about the target from SIMBAD and VizieR and default parameters for PSF algorithms.",
"Next, charis$\\_$ imprep populates the fits headers with standardized keywords, checks data quality, classifies cubes into science frames and sky frames, and edits the string variables for file number in the .info file accordingly.",
"Sky subtraction and Image Registration – The program charis$\\_$ subtract$\\_$ sky performs sky subtraction of each cube element.",
"As the sky background level may fluctuate, the user can opt for a simple median sky subtraction or one scaled to the match the background intensity of the reddest $K$ -band channels at wide separations.",
"Each data cube element is registered to coordinate [100,100] (indexed from zero) with charis$\\_$ register$\\_$ cube (Figure REF a).",
"By default precise registration relies on measuring the centroids of satellite spots in each cube slice, using centroid starting guesses from the middle (usually highest SNR) channel for the bandpass, and refining this estimate using a quadratic functional fit across all high SNR channels.",
"Registration for unsaturated, unocculated data may use the star itself.",
"The program also can register a sequence of cubes lacking satellite spots when at least one cube with spots are acquired before/after the sequence by cross-correlating the halo.",
"Spectrophotometric Calibration – In charis$\\_$ specphot$\\_$ cal, each cube is spectrophotometrically calibrated using the satellite spots and a spectrum of the star (Figure REF b)As a general rule the satellite spots should remain on AT ALL TIMES.",
"Some satellite spots observations are usually required for proper spectrophotometric calibration.",
"The spots incur negligible loss in speckle suppression.",
"While SCExAO/CHARIS can take exposures without satellite spots, removing them usually degrades image registration precision and causes significant additional uncertainties in absolute calibration due to transmission and AO performance variations.",
"Exceptions to this above guideline include programs where science goals require and CHARIS enables unsaturated images of the star.",
"Likewise, for most programs sky frames before or after science exposures should be taken to remove thermal background at K band.",
"They also appear to yield small reductions in the background rms at shorter wavelengths..",
"The spectrum may draw from the Kurucz atmosphere library, the Pickles library, or be an empirical spectrum.",
"By default, flux densities are normalized to one PSF core, although this choice can be overridden.",
"Spatial Filtering (Optional) and PSF subtraction – Each data cube slice can be spatially filtered using either a moving-box median or a radial profile (charis$\\_$ imrsub) before PSF subtraction.",
"The spatial filtering function provides a quick-look sequence-combined cube and wavelength-collapsed image with the halo suppressed by classical PSF subtraction.",
"Current available publication-grade PSF subtraction approaches include (for reference star differential imaging, RDI) KLIP, (for ADI) A-LOCI and KLIP, and (for SDI, SDI on the post ADI residuals) A-LOCI: e.g.",
"charis$\\_$ adiklip (Figure REF c).",
"The pipeline nominally adopts default PSF subtraction parameters in the .info file, all of which can be overridden at command line.",
"Throughtput Corrections, Forward-Modeling, Spectral Extraction,and Contrast Curves – Spectra for a detected source are extracted (charis$\\_$ extract$\\_$ 1d$\\_$ spectrum) but must be corrected for signal loss.",
"The pipeline uses forward-modeling[21], [45] to determine signal loss due to processing.",
"In charis$\\_$ aloci(klip)$\\_$ fwdmod$\\_$ planet, forward-modeling is applied at a specific location; similar programs forward-model a grid of synthetic disks.",
"charis$\\_$ aloci(klip)$\\_$ attenmap$\\_$ planet injects a grid of point sources to yield an attenuation map needed for calculating contrast curves in charis$\\_$ calc$\\_$ final$\\_$ contrast, which produces 1D, 5-$\\sigma $ contrast curves in each channel, and in each bandpass.",
"Basic Empirical Analysis – The pipeline calculates the spectral covariance at a predefined location[46] (charis$\\_$ calc$\\_$ spec$\\_$ covar) and compares an extracted spectrum to spectral libraries (charis$\\_$ empbdplanspec).",
"Auto-Reduce – To enable quick-look reductions potentially suitable for real-time on-sky analysis, we also incorporate an automatic reduction script (charis$\\_$ autoreduce) which performs most of these steps with a single command and includes blind source detection and extraction from ADI-reduced data."
],
[
"Recent SCExAO/CHARIS Science Results",
"The second (COVID-19 affected) year of full science operations for SCExAO/CHARIS has yielded four peer-reviewed publications from our collaboration [47], [48], [49], [50], a complementary publication on SCExAO/VAMPIRES focused on $H_{\\rm \\alpha }$ imaging searches for protoplanets[51], and several more CHARIS studies in preparation.",
"Most results focus on new companions imaged with SCExAO/CHARIS or multi-wavelength characterization of previously known planet-forming disks.",
"We summarize these published results and several studies in preparation below."
],
[
"New Companion Detections",
"HD 33632 Ab[50] – In SCExAO/CHARIS's first discovery paper, we presented the detection of HD 33632 Ab, a substellar companion to a nearby mature Sun-like star, HD 33632 Aa imaged at a projected separation of $\\sim $ 20 au (Figure REF ).",
"The companion's colors and near-IR spectrum are best-matched by field objects at the L/T dwarf transition, where clouds/dust in substellar atmospheres dissipate/sink below the photosphere.",
"The companion may be a particularly useful reference point for understanding the first directly imaged exoplanets, as its colors overlap with and its temperature is just slightly exceeds those of the young exoplanets HR 8799 cde.",
"Unlike nearly all other “benchmark” L/T transition substellar objects, HD 33632 Ab has both a high-quality near-IR spectrum AND a direct dynamical mass measurement because of the astrometric acceleration it induces on its host star, identified from the Hipparcos and Gaia satellite data.",
"Assuming it is solely responsible for the primary's acceleration, HD 33632 Ab's inferred mass is $\\sim $ 46 $M_{\\rm J}$ $\\pm $ 8 $M_{\\rm J}$ , which is comfortably above the deuterium-burning limit nominally separating planets from brown dwarfs.",
"However, its mass, mass ratio, and separation are comparable to multiple companions near the nominal planet to brown dwarf boundary.",
"Its eccentricity must be low and may be more similar values for bona fide directly imaged planets than companions identified as brown dwarfs based on their masses.",
"Future follow-up CHARIS data at higher resolution will refine its spectral properties and more precisely determine its mass, assuming that it is the only massive object accelerating its host star.",
"Deep follow-up CHARIS data could identify any hitherto unseen companions at small separations; however, due to the system's age, JWST/NIRCam thermal IR imaging likely will provide a more sensitive search for cool companions lower in mass than HD 33632 Ab.",
"Multi-band thermal IR imaging could further probe filters carbon chemistry and absorption from other species (e.g.",
"CO$_{\\rm 2}$ ).",
"While the ground could provide some of these data (e.g.",
"at 3.1 and 3.3 $\\mu m$ ), high thermal backgrounds likely make a high SNR detection and precise photometry at $M_{\\rm s}$ (i.e.",
"$\\sigma $ $\\lesssim $ 0.1 mag) implausible with current facilities (see [13]) .",
"Here again, JWST is natural complement, as NIRCam imaging – e.g.",
"3.6 $\\mu m$ and 4.3–4.8 $\\mu m$ – will be essential for better characterizing this system and providing a context for HR 8799 bcde and other L/T transition planet-mass objects.",
"Other Systems – In addition to HD 33632 Ab, SCExAO/CHARIS has yielded over a half-dozen newly-detected (likely) companions, three of which are shown in Figure REF .",
"The most unique of these shown in a new candidate infant planet detected in over half a dozen CHARIS data sets (right panel) (Currie et al.",
"in prep).",
"While distinguishing between bona fide protoplanets and disk features is notoriously challenging, the signal's spectrum, astrometry, and other features provide good evidence for a protoplanet interpretation.",
"In addition to these objects and several other more massive stellar companions, SCExAO/CHARIS also has a multiple data set detection of a candidate fully-formed directly imaged planet with properties plausibly similar to HR 8799 bcde (T. Currie in prog.).",
"This detection is not shown in this paper.",
"Figure: Three additional new discoveries with SCExAO/CHARIS."
],
[
"Characterization of Planet-Forming Disks",
"HD 15115[49] – The first discovery with SCExAO – then coupled with the HiCIAO infrared camera – was a resolved $H$ -band image of the bright debris disk around the 5–10 $Myr$ old star, HD 36546 [39].",
"SCExAO/CHARIS imaging of another debris disk – the cold debris disk around HD 15115, also known as the “Blue Needle\"[52] – provided a multi-wavelength look at Kuiper belt-like structures to constrain dust composition and scattering properties(Figure REF a).",
"We detected the HD 15115 disk down to separations of $\\sim $ 0$.\\!\\!^{\\prime \\prime }$ 2, a factor of 3–5 smaller than previous studies.",
"We recover an east-west disk brightness asymmetry previously seen at wider separations and at other wavelengthsKalas2007.",
"However, the intrinsic disk colors appear red at small separations, in contrast to the disk's blue colors at wide separations.",
"While a SPHERE study suggested a misaligned two-ring geometry[53], our more sensitive data showed that a single ring with a Hong-like scattering phase function fit the data well.",
"HD 34700 A[47] – We resolve the bright broken ring around HD 34700 A and recover multiple spirals arms previously seen in polarimetric data[54](middle panel).",
"Geometric albedos derived from the ring's surface brightness profile point to a large scale height or copious submicron-sized dust at position angles between $\\sim $ 45$^{o}$ and 90$^{o}$ .",
"A stellar flyby or envelope infall may explain the spirals' very large pitch angles.",
"MWC 758 – In an unpublished work, we resolved multiple spiral arms in the MWC 758 protoplanetary disk (right panel).",
"The presence of spiral density waves has been attributed to massive, hidden planets in this system[55].",
"Two studies have identified candidate protoplanets that may be connected to these spirals: a bright inner companion at $\\rho $ $\\sim $ 0$.\\!\\!^{\\prime \\prime }$ 11[56] and a much fainter outer one at $\\rho $ $\\sim $ 0$.\\!\\!^{\\prime \\prime }$ 62[57], both detected at $L_{\\rm p}$ .",
"CHARIS data well resolve MWC 758's spiral arms, both of which appear to have two components, and provide constraints on other disk material and bright protoplanets down to 0$.\\!\\!^{\\prime \\prime }$ 05.",
"Our data might not be sensitive enough to assess the nature of the faint, wide-separation candidate protoplanet.",
"Analysis of these data provide evidence against the protoplanet reported in Ref.",
"Reggiani2018: no astrophysical is seen at its predicted location (dashed circle).",
"To further assess its nature, we downloaded and examined the Ref.",
"Reggiani2018 images provided by VizieR.",
"While Ref.",
"Reggiani2018 report an SNR of $\\sim $ 5 and 6 for the two epochs, using our adopted computation for SNR, we instead find that the detections are not statistically significant compared to other signal at the same angular separations: SNRs $\\sim $ 1.8 and 2.0, respectively This result reaffirms the significant challenge of distinguishing between bona fide protoplanets and disk features[29].",
"It also a case study of complications with interpreting the nature of signals near the diffraction limit that emerge after PSF subtraction techniques are used to (imperfectly) whiten highly non-Gaussian noise.",
"Figure: Planet-forming disks resolved with SCExAO/CHARIS.",
"(left) The HD 15115 (“Blue Needle\") debris disk.",
"(middle) The broken ring resolved around HD 34700 A.",
"(right) Unpublished image of the MWC 758 protoplanetary disk.",
"The previously claimed protoplanet candidate (dashed circle) is not detected."
],
[
"Technical Advances",
"A detailed overview of recent and upcoming technical improvements to SCExAO is discussed in various complementary SPIE submissions.",
"A subset of these upgrades are discussed below, particularly focusing on wavefront sensing/control advances.",
"Replacement and Upgrade of AO-188 - We plan to replace Subaru's venerable facility AO system, which uses a curvature wavefront sensor driving a DM with only 188 actuators.",
"AO-188's low actuator density across the telescope pupil allows a modest correction of atmospheric turbulence and exhibits strong vibration modes precluding full-speed operation[58].",
"Its successor will operate with a much faster (2 kHz) and far higher-order DM (3200 actuators).",
"Our plan is to upgrade its wavefront sensor from the current 188-element avalanche photodiode array to an EMCCD.",
"The camera upgrade will allow the “AO-3000\" to substantially reduce both the fitting error and temporal bandwidth error, yielding $H$ band Strehl ratios approaching 85% and a reduction in the PSF halo by nearly a factor of 5.",
"Self-adjusting modal AO control - Work is in progress to implement an automatic modal gain algorithm[59], which continuously tracks the adequate gain for minimum variance control of all AO modes, while compensating for the varying sensititivity of the PyWFS (a nonlinear effect depending on seeing, source brightness, etc.).",
"Daytime tests have shown adequate control with varying turbulence conditions for up to 400 DM modes.",
"On-sky testing is ongoing to improve the robustness of the algorithm.",
"AO optimization through reinforcement learning - We are developing an algorithm - Dr WHO (Direct Reinforcement Wavefront Heuristic Optimization) aiming to correct on-sky the static, quasi-static and dynamic non-common path aberrations by updating the reference of the pyramid wavefront sensor.",
"The new references are chosen based on the lucky imaging of the focal plane camera.",
"See SPIE paper 11448-255 for more details.",
"Focal-Plane Wavefront Sensing with MEC - The MKID Exoplanet Camera (MEC) is a noiseless, ultra-cooled photon-counting detector able to measure the energy and wavelength of every photon.",
"and can function both as a science instrument and a focal-plane wavefront sensing sensor integrated with SCExAO[60].",
"MEC operates from $z$ to $J$ band (0.8–1.4 $\\mu m$ ) and often runs in parallel with CHARIS observations, taking $Y$ band light.",
"Operating at the focal plane, MEC is now demonstrating the ability to perform slow speckle nulling capable of cancelling bright, “slow\" quasi-static speckles that evolve on tens of minutes timescales.",
"Given its exceptional sensitivity and fast readout ($>$ 2 kHz), MEC can also cancel fast ($\\tau _{\\rm o}$ $\\sim $ 10 ms) atmospheric speckles and generate an extremely deep-contrast dark hole (DH).",
"In both cases, MEC will enable significantly deeper raw contrasts beyond those provided by AO-3000 and SCExAO alone.",
"Linear Dark Field Control – After generating a dark hole using a method like speckle nulling with MEC or another focal-plane camera, Linear Dark Field Control(LDFC)[61] could “freeze\" the residual DH state without the need for probing.",
"Laboratory tests of LDFC at raw contrasts ($\\sim $ 5$\\times $ 10$^{-7}$ ) relevant for imaging reflected-light planets from ground-based telescopes demonstrated its ability to correct for a range of phase perturbations with improved efficiency relative to classical speckle nulling[62].",
"Tests of LDFC at slightly milder contrasts using SCExAO's internal source and simulated turbulence as well as preliminary on-sky tests were extremely promising (K. Miller et al.",
"2020 submitted; S. Bos 2021 in prep.).",
"Alternating Speckle Grid for High-Precision Spectroscopy – To achieve higher-precision spectroscopy without the need for very bright satellite spots, we have developed an alternating speckle grid strategy (Figure REF ).",
"The underlying incoherent background is subtracted by taking two exposures with alternating speckle patterns [63].",
"Current on-sky photometric and astrometric precision (obtained by measuring the flux ratio and relative position between calibration spots) for a 30 minute observation time with is 0.3$\\%$ and 1.7mas respectively.",
"We plan to begin support of this mode within the CHARIS DPP in 2021.",
"Figure: On-sky images of two consecutive reduced CHARIS data slices of θ\\theta Hydrae with satellite spots (a) Incoherent speckles (without background subtraction) and (b) Alternating technique; after background subtraction obtained from CHARIS at 1.63 μm\\mu m with the two speckle patterns at ρ\\rho ∼\\sim 11λ/D\\lambda /D."
],
[
"SCExAO/CHARIS Direct Imaging Survey",
"We have recently initiated an exoplanet direct imaging survey focused on targeting stars showing indirect evidence for a planet from astrometry.",
"The low yield of blind direct imaging surveys show that exoplanets directly detectable with current instruments are rare at 5–100 au[10].",
"Companions detected by these surveys are typically more than 2–5 $M_{\\rm J}$ and orbit beyond $\\sim $ 3 au, where the jovian planet frequency peaks.",
"Limited sample sizes and sparse coverage of ages, temperatures, and surface gravities impedes our understanding of the atmospheric evolution of gas giant planets.",
"However, targeted searches focused on stars showing evidence of gravitational pulls from massive planets could improve survey yields.",
"The Hipparcos-Gaia Catalog of Acclerations (HGCA) provides absolute astrometry for 115,000 nearby stars, including those with clear dynamical evidence for unseen massive companions[64].",
"Accelerations derived from the HGCA can provide dynamical masses of known imaged exoplanets and low-mass brown dwarfs independent of luminosity evolution models and irrespective of uncertainties in stellar ages[65].",
"So far, our survey has been a resounding success.",
"This approach has already yielded the discovery of HD 33632 Ab and and numerous other low-mass companions and other planet/brown dwarf candidates.",
"Thus far, our detection rate of (candidate) companions is $\\sim $ 30%.",
"For target selection, we have explored identifying accelerating stars 1) in young moving groups and 2) among the field.",
"Thus far, we favor selecting field objects, as they are typically nearer and have higher quality astrometry than stars in many moving groups or associations: moving group members with statistically signficant accelerations in HGCA are often stars with K or M dwarf companions (see also Ref.",
"DeRosa2019).",
"Other objects – e.g.",
"white dwarfs – may also be responsible for accelerations identified from astrometric data[67].",
"Age estimates for field stars are more uncertain than for moving group members.",
"However, this has negligible impact on our derived masses since Hipparcos and Gaia absolute astrometry and SCExAO/CHARIS relative astrometry provide dynamical masses.",
"The Gaia Early Data Release 3 and full Data Release 3 (scheduled for 2022) will provide additional precise astrometry for nearby stars and will reveal more systems showing evidence for hitherto unidentified but imageable planets.",
"Combining these new measurements with previous HGCA absolute astrometry for SCExAO/CHARIS-discovered planets and brown dwarfs will yield an even more robust mapping between a substellar object's atmospheric properties and its bulk properties (mass).",
"The authors acknowledge the very significant cultural role and reverence that the summit of Mauna Kea holds within the Hawaiian community.",
"We are most fortunate to have the opportunity to conduct observations from this mountain.",
"We support and endeavor to contribute to respectful, effective stewardship of cultural, natural, and scientific resources that properly honors these lands.",
"We acknowledge the critical importance of the current and recent Subaru Telescope daycrew, technicians, support astronomers, telescope operators, computer support, and office staff employees, especially during the challenging times presented by the COVID-19 pandemic.",
"Their expertise, ingenuity, and dedication is indispensable to the continued successful operation of these observatories.",
"We thank the Subaru Time Allocation Committee for their generous support of this program.",
"TC was supported by a NASA Senior Postdoctoral Fellowship and NASA/Keck grant LK-2663-948181.",
"TB gratefully acknowledges support from the Heising-Simons foundation and from NASA under grant #80NSSC18K0439.",
"The development of SCExAO was supported by JSPS (Grant-in-Aid for Research #23340051, #26220704 & #23103002), Astrobiology Center of NINS, Japan, the Mt Cuba Foundation, and the director's contingency fund at Subaru Telescope.",
"CHARIS was developed under the support by the Grant-in-Aid for Scientific Research on Innovative Areas #2302."
]
] | 2012.05241 |
[
[
"A Geometric View of Closure Phases in Interferometry"
],
[
"Abstract Closure phase is the phase of a closed-loop product of correlations in a $\\ge 3$-element interferometer array.",
"Its invariance to element-based phase corruption makes it invaluable for interferometric applications that otherwise require high-accuracy phase calibration.",
"However, its understanding has remained mainly mathematical and limited to the aperture plane (Fourier dual of image plane).",
"Here, we lay the foundations for a geometrical insight.",
"we show that closure phase and its invariance to element-based corruption and to translation are intricately related to the conserved properties (shape, orientation, and size, or SOS) of the principal triangle enclosed by the three fringes formed by a closed triad of array elements, which is referred herein as the \"SOS conservation principle\".",
"When element-based amplitude calibration is not needed, as is typical in optical interferometry, the 3-element interference image formed from phase-uncalibrated correlations is a true and uncorrupted representation of the source object's morphology, except for a possible shift.",
"Based on this SOS conservation principle, we present two geometric methods to measure the closure phase directly from a 3-element interference image (without requiring an aperture-plane view): (i) the closure phase is directly measurable from any one of the triangle's heights, and (ii) the squared closure phase is proportional to the product of the areas enclosed by the triad of array elements and the principal triangle in the aperture and image planes, respectively.",
"We validate this geometric understanding across a wide range range of interferometric conditions using data from the Very Large Array and the Event Horizon Telescope.",
"This geometric insight can be potentially valuable to other interferometric applications such as optical interferometry.",
"These geometric relationships are generalised for an $N$-element interferometer."
],
[
"Introduction",
"The concept of closure phase in radio interferometry can be traced back to Jennison in 1958 [1].",
"Closure phase provides information on the phase encoded in the spatial coherences due only to the intensity distribution of sources of electromagnetic (EM) radiation in the sky, without the need for calibration to correct for corruption of the phases of the EM waves due to propagation effects and the array receiver elements themselves.",
"The invariance of the closure phase to phase corruptions of the incident EM wave that can be factorized into element-based phase terms, has been extensively tested and applied in interferometry.",
"This property has played a significant role in the development of a popular calibration scheme called “self-calibration” [2], [3], [4].",
"Moreover, closure phase is known for its measure of the centrosymmetry or point-symmetry (morphological symmetry around a point) as well as for its invariance to translation of the spatial intensity distribution of the EM radiation [5].",
"These properties have made it a valuable tool in experiments that face challenges due to the requirement of high-accuracy phase calibration of the instrument to correct for the phase corruptions introduced by array elements, as well as by the propagation medium.",
"Closure phases have thus been used in optical interferometry to deduce the structures of stars [6], [7], [8], [9], Very Long Baseline Interferometry (VLBI) experiments, such as the Event Horizon Telescope (EHT) imaging of the shadow of the supermassive black hole in M87 [10], and seismic imaging [11].",
"Recently, closure phase has provided a useful avenue towards detecting the neutral Hydrogen structures during the cosmic reionization (at redshifts, $z\\gtrsim 6$ ) using its characteristic 21 cm spectral line redshifted to low radio frequencies with interferometer arrays [12], [13], [14], [15], [16].",
"Despite extensive use and successful applications spanning several decades, a physical insight into the interferometric closure phase has remained elusive.",
"The complex, higher-order dependence on the moments of the spatial intensity and spatial coherence [17], [15] makes it very challenging to gain a geometric intuition of this special quantity.",
"However, the interferometric closure phase has close parallels with the “structure invariants” (such as the triplet and quartet phases) in crystallography [18], [19], as well as with the phase of the $N$ -vertex Bargmann invariants [20], also known as the geometric phase or the Pancharatnam phase or the Berry phase [21], [22], [23], [24] in optics and quantum mechanics, the understanding of which has made significant advances [25].",
"The primary aim of this paper is to provide foundational steps towards building a geometric insight for closure phases measured in an interferometer array.",
"It is anticipated that this insight will result in the aiding of, and the benefiting from, parallels in other fields such as those mentioned above as well as in widening the spectrum of similar synthesis interferometry applications, including optical interferometry.",
"The paper is organized as follows.",
"Section sets up the interferometry context.",
"Section introduces the closure phase of an $N$ -polygon interferometer array.",
"In section , we present the geometrical characteristics, and direct geometrical methods for the estimation of closure phase in the image plane using a 3-element interferometer, through a derivation of the shape-orientation-size conserving property of closure phase.",
"A validation via an application to real radio interferometric data from observations of bright cosmic objects at radio wavelengths using the Jansky Very Large Array (JVLA) radio telescope is provided in Section .",
"We also identify and discuss analogs of closure phase in other fields such as optical interferometry, crystallography, and quantum mechanics in Section .",
"In Section , we provide a generalization to a closed $N$ -polygon array of interferometer elements accompanied by details of the derivation in the appendices.",
"The findings are summarized in Section ."
],
[
"The Interferometry Context",
"Consider measurements of a single polarization state of a complex-valued, quasi-monochromatic electric field, $E_a(\\lambda )$ , integrated over a narrow band around the wavelength, $\\lambda $ , of the incident EM radiation by $N_\\textrm {A} $ array elements at locations ${x}_a$ , with $a=1, 2, \\ldots N_\\textrm {A} $ in the aperture plane.",
"The spacing between any pair of array elements (commonly referred as to as the baseline vector in radio interferometry) is denoted by ${x}_{ab} \\equiv {x}_b - {x}_a$ .",
"The spatial distribution of the intensity of the EM radiation in the image plane, $I(\\widehat{{s}},\\lambda )$ , and the corresponding spatial coherence of the electric fields (also known as visibilities in radio interferometry) in the aperture plane, $V_{ab}(\\lambda )$ , exhibit a Fourier-transform relationship with each other [26], [27], [28], [29], Vab() Ea()Eb() = (s,) I(s,) e-i 2uabs d, where, the angular brackets, $\\langle \\cdot \\rangle $ , represent a true ensemble average, $\\hat{{s}}$ denotes a unit vector in the direction of any location in the image, ${u}_{ab}\\equiv {x}_{ab}/\\lambda $ defined on the aperture plane denotes the array element spacings projected on the plane perpendicular to the direction of the phase center, $\\hat{{s}}_0$ , in the image.",
"In the Fourier relationship, ${u}_{ab}$ , by definition, represents the spatial frequencies of the structures in $I(\\hat{{s}},\\lambda )$ .",
"The array element's power response in a given direction, $\\hat{{s}}$ , is denoted by $\\Theta (\\hat{{s}},\\lambda )$ , and $\\mathrm {d}\\Omega $ denotes the differential solid angle in the image plane perpendicular to $\\hat{{s}}$ .",
"The vectors $\\hat{{s}}$ and ${u}_{ab}$ can be represented on a Cartesian coordinate frame with orthogonal basis vectors, $\\hat{{e}}_x$ , $\\hat{{e}}_y$ , and $\\hat{{e}}_z$ .",
"In this frame, $\\hat{{s}}\\equiv \\ell \\,\\hat{{e}}_x + m\\,\\hat{{e}}_y + n\\hat{{e}}_z$ with $\\ell ^2 + m^2 + n^2 = 1$ , where $l$ , $m$ , and $n$ denote the direction-cosines of $\\hat{{s}}$ .",
"And, ${u}_{ab}\\equiv u_{ab}\\,\\hat{{e}}_x + v_{ab}\\,\\hat{{e}}_y + w_{ab}\\hat{{e}}_z$ .",
"Figure REF depicts the modeled locations of three array elements in units of wavelengths (chosen at $\\lambda =21$ cm) that will be used in the initial examples that follow.",
"The cyclic ordering of the element indices is indicated by the arrowed circle.",
"The three encircled elements can be considered as three antennas in a radio interferometer, or optical mirrors or aperture mask openings in an optical interferometer.",
"Figure: A triad of aperture elements with positions, x a /λ{x}_a/\\lambda , and spacings, u ab {u}_{ab}, both in units of wavelengths, with a,b=1,2,3a,b=1, 2, 3, and b≠ab\\ne a. u ab {u}_{ab} represents the spatial frequencies of the image-plane intensity distribution, I(s ^,λ)I(\\hat{{s}},\\lambda ), in the aperture plane.",
"V ab (λ)V_{ab}(\\lambda ) denotes the complex-valued spatial coherence of I(s ^,λ)I(\\hat{{s}},\\lambda ) measured at u ab {u}_{ab} in the aperture plane.",
"The cyclic ordering of the element spacings is indicated by the arrowed (anti-clockwise) circle.",
"The three spatial frequencies, u ab {u}_{ab}, are shown by dashed, dash-dotted, and dotted lines, which will be used to denote the corresponding fringes in the image plane in subsequent figures.In practice, the EM voltage measurements at the array elements are inevitably corrupted by complex-valued “gain” factors introduced by the intervening medium as well as the array element response.",
"The corrupted measurements are denoted by $E_a^\\prime (\\lambda ) = g_a(\\lambda ) E_a(\\lambda )$ , where $g_a(\\lambda )$ denotes the net corruption factors introduced in the measurement process factorizable in such a way that it is attributable to the individual elements.",
"Thus, a calibration process, which estimates $g_a(\\lambda )$ [denoted by $\\widehat{g}_a(\\lambda )$ ], is usually required to correct for these gains to estimate the true electric fields, Ea() = ga-1()ga() Ea() = Ga() Ea(), where, $G_a(\\lambda ) \\equiv \\widehat{g}_a^{-1}(\\lambda )g_a(\\lambda )$ is the net residual gain after calibration.",
"The calibrated estimate of the spatial coherence is Vab() Ga() Gb() Ea()Eb() = Ga() Gb() Vab() = |Ga() Gb() Vab()| ei[b()-a() + ab()] where, $\\xi _a(\\lambda )\\equiv \\arg G_a(\\lambda )$ , $\\phi _{ab}(\\lambda )\\equiv \\arg V_{ab}(\\lambda )$ , and $\\widehat{\\phi }_{ab}(\\lambda )\\equiv \\arg \\widehat{V}_{ab}(\\lambda ) = \\phi _{ab}(\\lambda ) + \\xi _b(\\lambda )-\\xi _a(\\lambda )$ .",
"In an ideal calibration process, $G_a(\\lambda )\\equiv 1$ , and thus $\\xi _a(\\lambda )\\equiv 0,\\, \\forall \\, a$ .",
"However, it is often difficult to realize in practice.",
"A basic image of the apparent intensity, $\\Theta (\\hat{{s}},\\lambda )\\,\\widehat{I}(\\hat{{s}},\\lambda )$ (also known as “dirty” image in synthesis imaging in radio interferometry), can be reconstructed by superposition of the image plane responses from the calibrated spatial coherence measured on each of the array element spacings in the aperture plane, (s,) I(s,) a,b=1NA Fab(s,) = a,b=1NA Vab() ei 2uabs, where, $\\widehat{F}_{ab}(\\hat{{s}},\\lambda )$ is the response of a single interferometer (visibility measured on one baseline), called the “fringe” on the image plane corresponding to $\\widehat{V}_{ab}(\\lambda )$ , and is defined as Fab(s,) |Vab()| ei [2uabs + ab() + b()-a()] with $\\arg \\widehat{F}_{ab}(\\hat{{s}},\\lambda ) = 2\\pi {u}_{ab}\\cdot \\hat{{s}} + \\phi _{ab}(\\lambda ) + \\xi _b(\\lambda )-\\xi _a(\\lambda )$ .",
"Ideally, $\\xi _a(\\lambda )=\\xi _b(\\lambda )=0$ , in which case $\\arg F_{ab}(\\hat{{s}},\\lambda )\\equiv 2\\pi {u}_{ab}\\cdot \\hat{{s}} + \\phi _{ab}(\\lambda )$ .",
"The null-valued (or zero-valued) isophase contours (and their equivalents offset by multiples of $2\\pi $ ), hereafter referred to as the null phase curves (NPC) of the fringes, are given by 2uabs + ab() = 0, a,b=1,...N, with ab, where, $\\psi _{ab}(\\lambda ) = \\phi _{ab}(\\lambda ) + 2\\pi n_{ab}$ , and $n_{ab}$ (an integer) accounts for the NPC offset from the principal NPC ($n_{ab}=0$ ) by integer multiples ($n_{ab}$ ) of $2\\pi $ .",
"When traversing anywhere on these NPC, no change in phase is acquired, and hence this set of NPCs can be considered as isophase contours in the image plane.",
"Using coordinate geometry, it can be shown that the signed positional offset, $\\delta s_{ab}(\\lambda )$ , of the fringe NPC from the phase center (origin) along a perpendicular and the corresponding phase offset, $\\psi _{ab}(\\lambda )$ , are related by ab() = 2|uab| sab().",
"Because ${u}_{ab}$ is the spatial frequency of a fringe, $1/|{u}_{ab}|$ represents the spatial period of the periodic fringe (or the fringe spacing) in the image plane and corresponds to a phase change of $2\\pi $ , as verified by setting $\\delta s_{ab}(\\lambda )=1/|{u}_{ab}|$ in Equation ().",
"Figure REF shows the ideal fringes, $F_{ab}(\\hat{{s}},\\lambda )$ , in the image plane in direction-cosine coordinates, ($\\ell , m$ ), given by Equation () for the three modeled array element spacings and the corresponding visibilities shown in Figure REF .",
"The $+$ symbol marks the phase center (origin).",
"The fringe NPCs, described by Equation (), are shown in line styles corresponding to those in Figure REF .",
"The black line in each panel denotes the principal NPC ($n_{ab}=0$ ) of the corresponding fringe.",
"The various gray lines denote the secondary NPCs ($|n_{ab}|>0$ ) of the fringes.",
"The positional offset, $\\delta s_{ab}(\\lambda )$ , of the principal fringe NPC from the phase center is shown by the magenta segments and corresponds to $\\phi _{ab}(\\lambda )$ (the principal visibility phase) according to Equation ().",
"In the case of uncalibrated visibilities, these phase offsets also include the corruption due to the complex voltage gains, $\\xi _a(\\lambda )$ , of the array elements.",
"Figure: (Color) Ideal fringes [in color map and annotated by F ab (s ^,λ)F_{ab}(\\hat{{s}},\\lambda )] and the respective NPCs (lines) in the image plane in direction-cosine (ℓ,m\\ell , m) coordinates, with the line style in each panel corresponding to that of the array element spacings, u ab {u}_{ab}, in Figure .",
"Equation () yields the fringe NPCs.",
"The black lines in each line style corresponds to the principal fringe NPC (n ab =0n_{ab}=0), while the varying shades of gray correspond to secondary (|n ab |>0|n_{ab}|>0) fringe NPCs.",
"The phase center (origin) is marked (with a ++ symbol).",
"The positional offset from the phase center to each of the principal fringe NPCs is shown in magenta and is related to the visibility phase, φ ab (λ)\\phi _{ab}(\\lambda ), by Equation ()."
],
[
"Closure Phase in Interferometry",
"Hereafter, we will assume that the apparent intensity as “seen” by the array elements, $\\Theta (\\hat{{s}},\\lambda )\\, I(\\hat{{s}},\\lambda )$ , remains identical between them.",
"Consider $N$ elements forming an $N$ -vertex polygon in the aperture plane.",
"The element spacings in the adjacent sides in the polygon are given by ${x}_{a\\lceil a+1\\rfloor _N} \\equiv {x}_{\\lceil a+1\\rfloor _N} - {x}_a$ , where $\\lceil a\\rfloor _N \\equiv a \\mod {N}$ .",
"Thus, a closed loop in the aperture plane is expressed as a=1N uaa+1N 0.",
"The interferometric closure phase measured post-calibration on the closed $N$ -polygon is N() a=1N Vaa+1N() = a=1N Vaa+1N().",
"Because $\\sum _{a=1}^N \\arg G_a^\\star (\\lambda ) G_{\\lceil a+1\\rfloor _N}(\\lambda ) \\equiv 0$ , N() = N() = a=1N Vaa+1N(), where, $\\phi _N(\\lambda )$ is the true closure phase on the $N$ -polygon.",
"Therefore, the closure phase is invariant to element-based voltage gains, $g_a(\\lambda )$ , the corrections from calibration, $\\widehat{g}_a(\\lambda )$ and $G_a(\\lambda )$ , as well as the errors therein, making it a true observable physical property of the structures in the image-plane intensity distribution.",
"This property is a form of gauge-invariance with respect to any element-based phases acquired during the measurement process.",
"One of the consequences of this gauge-invariance in interferometry is that the closure phase is also invariant to translation in the image plane.",
"This can be shown by replacing $\\hat{{s}}$ with $\\hat{{s}}^\\prime = \\hat{{s}} - \\hat{{s}}_0$ , where $\\hat{{s}}_0$ is an arbitrary choice for the origin of the image plane, usually referred to as the phase center in interferometry.",
"From Equation (), such a translation modifies the spatial coherence as Vab() = ei 2uabs0 Vab(), which simply introduces an additional phase factor, $e^{i 2\\pi {u}_{ab}\\cdot \\hat{{s}}_0}$ , that is factorizable into element-based phase factors as $e^{i 2\\pi {x}_b\\cdot \\hat{{s}}_0 /\\lambda }\\, e^{-i 2\\pi {x}_a\\cdot \\hat{{s}}_0 /\\lambda }$ .",
"Due to the gauge-invariance discussed above, the closure phase is therefore independent of the phase factors introduced by translation in the image plane.",
"Conversely, the translation invariance of the closure phase is simply a special case of the gauge-invariance to the phase factors attributable locally to the array elements.",
"The gauge-invariance property of the closure phase has proved to be invaluable in astronomy applications involving detection of structures using interferometry, particularly in situations where the phase calibration of the array elements to a very high level of accuracy is challenging.",
"A few of the diverse applications include the goal of detecting of neutral hydrogen structures in the cosmic web from the early universe [14], [15], [16], [12], [13], the EHT imaging of the supermassive black hole in M87 [10], and characterizing the complex structures on stellar surfaces and their surroundings [6], [7], [8], [9]."
],
[
"Shape-Orientation-Size (SOS) Conserving Geometry of Closure Phase",
"In the following, we mathematically derive, and then demonstrate with model and real data, the underlying geometric nature of closure phase using the image-plane fringes of a closed triad of array elements, in the case when visibility phase corruption can be assigned to individual elements, and is not idiosyncratic to a given baseline.",
"The geometric behavior manifests itself as a `shape-orientation-size' (SOS) conservation, in which the relative locations and orientations of the three NPCs from a closed triad of array elements are conserved in the presence of large phase errors, except possibly an overall translation in the image plane.",
"The derivation relies on two key points.",
"First is the well known fact that the position of an interferometric fringe in the image plane relative to some chosen reference point (such as the phase tracking center in radio interferometry), is directly related to the visibility phase relative to that phase reference point.",
"The phase reference point itself can be adjusted to fit the situation, with no loss of generality.",
"And second, if the phase of one element of the closed triad is corrupted, this corruption affects the phases of the two fringes that contain this array element in an opposite sense, such that the location of the three-fringe pattern in the image plane for a closed loop of elements just shifts, while the pattern itself is preserved exactly.",
"The element-based phase corruption for a triad can be visualized as a tilt of the aperture plane of the triad, which then shifts, but does not alter the geometry of the three-fringe interference pattern in the image plane (discussed in detail later using Figure REF ).",
"Below, we provide a mathematical basis for the geometric characteristics of closure phase in the image plane, beginning with a triad (3-polygon).",
"We derive mathematical methods to measure the closure phase using the three-fringe image pattern, without resort to an aperture plane transform.",
"Hereafter, we assume that the locations of the array elements are coplanar, and thus without loss of generality, we choose a plane where $w_{ab}\\equiv 0$ .",
"We further assume a flat image plane without significant curvature effects, usually referred to as narrow field of view or “flat sky” approximation ($\\ell ,m\\ll 1$ ) in radio interferometry.",
"The effects of non-coplanarity and curvature of the image plane will be the subject of future study.",
"A triad ($N=3$ ) is the simplest closed shape for studying the closure phase or the bispectrum phase and will form the basis later for characterizing the behavior on $N$ -polygons with $N>3$ .",
"Consider the three fringes $F_{a\\lceil a+1\\rfloor _N}(\\hat{{s}},\\lambda )$ corresponding to $V_{ab}(\\lambda )$ with $N=3$ and $a,b=1,2,3$ .",
"The set of NPCs for each fringe, $F_{a\\lceil a+1\\rfloor _3}(\\hat{{s}},\\lambda )$ , in the image plane is simply obtained from Equation () as 2uaa+13s + aa+13() = 0, a=1,2,3."
],
[
"Relation to the Phase Center",
"In Equation (), $\\psi _{a\\lceil a+1\\rfloor _3}(\\lambda )$ represents the phase offset from the phase center, which has been implicitly assumed to be at $\\hat{{s}}_0\\equiv (0,0,1)$ .",
"Thus, the closure phase on the 3-polygon is 3() a=13 aa+13(), which is the sum of the phase offsets of the individual fringe NPC from the phase center.",
"Geometrically, the phase offsets are obtained from Equation () by measuring the respective positional offsets along the perpendiculars dropped from the phase center to each of these fringe NPC [Equation ()] normalized by the respective fringe spacing along the perpendiculars.",
"For a calibrated interferometer, these measured phase offsets from the phase center for the fringes relate directly to the object's position and structure on the sky, modulo $2\\pi n_{a\\lceil a+1\\rfloor _3}$ .",
"If the phase center is shifted to some arbitrary $\\hat{{s}}_0$ , then by defining $\\hat{{s}}^\\prime = \\hat{{s}} - \\hat{{s}}_0$ , Equation () can be written as 2uaa+13s+ aa+13() = 0, a=1,2,3.",
"Then, the closure phase with the shifted phase center is 3() a=13 aa+13() = a=13 aa+13() + 2s0a=13 uaa+13 = 3(), where, we have used Equations () and (REF ).",
"This reiterates, using a fringe-based geometric viewpoint in the image plane, that the closure phase is invariant under translation.",
"Moreover, the phase center, $\\hat{{s}}_0$ , can be conveniently chosen to be at the point of intersection of any of the two fringe NPCs, for instance, $F_{12}(\\hat{{s}},\\lambda )$ and $F_{23}(\\hat{{s}},\\lambda )$ .",
"Because $\\hat{{s}}_0$ lies on the NPCs of both $F_{12}(\\hat{{s}},\\lambda )$ and $F_{23}(\\hat{{s}},\\lambda )$ , by definition, $\\delta s_{12}^\\prime (\\lambda )= \\delta s_{23}^\\prime (\\lambda ) = 0$ , and therefore, $\\psi _{12}^\\prime (\\lambda ) = \\psi _{23}^\\prime (\\lambda ) = 0$ from Equation ().",
"Hence, 3() = 31() = 31() + 2u31s0 = 31() - 2(u12+u23)s0 = a=13 aa+13() = 3().",
"Thus, when the phase center is chosen to be at the intersection of any of the two fringe NPCs, the closure phase has a simple relation 3() = aa+13() = 2|uaa+13| saa+13() from Equations () and (REF ), where, $\\delta s_{a\\lceil a+1\\rfloor _3}^\\prime (\\lambda )$ is the positional separation of the intersection vertex, which is now the chosen phase center, from the opposite fringe NPC corresponding to $F_{a\\lceil a+1\\rfloor _3}(\\hat{{s}},\\lambda )$ .",
"Figure REF illustrates these relations geometrically.",
"The color scale shows the net superposition of the three ideal interfering fringes, $F_{a\\lceil a+1\\rfloor _3}(\\hat{{s}},\\lambda )$ , in the image plane shown individually earlier in Figure REF .",
"The black and gray lines denote the principal and secondary NPCs of the fringes, respectively, with line styles corresponding to those in Figures REF and REF .",
"The $+$ symbol marks the phase center (or the origin) and is denoted by $\\mathcal {O}$ in magenta.",
"The positional offsets, $\\delta s_{a\\lceil a+1\\rfloor _3}(\\lambda )$ , of the principal fringe NPCs from the phase center are shown as magenta lines annotated by the corresponding principal visibility phases, $\\phi _{a\\lceil a+1\\rfloor _3}(\\lambda )$ , obtained using Equation ().",
"When the phase center is conveniently chosen to be the intersection of any two of the three principal fringe NPCs (denoted by $\\mathcal {O}^\\prime $ in red, blue, and brown), the modified visibility phases, $\\phi _{a\\lceil a+1\\rfloor _3}^\\prime (\\lambda )$ , are proportional to the positional offsets, $\\delta s_{a\\lceil a+1\\rfloor _3}^\\prime (\\lambda )$ , of the principal fringe NPCs from the so-chosen phase center shown by the corresponding colored lines, according to Equation (REF )We note that, for a given vertex, there can be flipped or complementary triangles in the image plane from which the closure phase can be derived.",
"Two of these can be seen to the left and right of the brown $\\mathcal {O}^\\prime $ vertex in Figure REF .",
"The sum of the two closure phases from the complementary triangles sharing a vertex must be, by definition, $2\\pi $ , thereby demonstrating the $2\\pi $ ambiguity of phase, encapsulated by $n_{ab}$ following Equation ()..",
"The same equation also implies that each of these modified principal visibility phases, $\\phi _{a\\lceil a+1\\rfloor _3}^\\prime (\\lambda )$ , is equal to the principal closure phase, $\\phi _3(\\lambda )$ , or in general, $\\psi _3(\\lambda ) = \\psi _{a\\lceil a+1\\rfloor _3}^\\prime (\\lambda ), \\forall a$ when the $2\\pi $ phase ambiguity (represented by $n_{ab}$ ) is accounted for.",
"Figure REF illustrates geometrically the gauge-invariance of the 3-polygon closure phase for uncalibrated and translated (in the image plane) fringes in the left and the right panels, respectively, but the discussion applies to both scenarios equally.",
"Both scenarios cause a displacement of the fringes and the NPCs relative to the ideal case in Figure REF .",
"As a result, the individual principal visibility phases, $\\phi _{a\\lceil a+1\\rfloor _3}(\\lambda )$ , relative to the default phase center, $\\mathcal {O}$ , are differently offset relative to the ideal case.",
"However, the closure phase, which is the sum of these three phases remains unchanged.",
"This is also clear when the phase center is shifted to one of the three vertices of interesection between any pair of the fringe NPCs (denoted by $\\mathcal {O}^\\prime $ in red, blue, and brown), the modified phase offset, $\\phi _{a\\lceil a+1\\rfloor _3}^\\prime (\\lambda )$ , corresponding to the positional offset of an intersection vertex from its corresponding opposite fringe NPC, $\\delta s_{a\\lceil a+1\\rfloor _3}^\\prime (\\lambda )$ given by Equation (REF ), remains unchanged compared to the ideal case.",
"It is important to note that the displacement of the fringes in either case is constrained to be parallel to themselves such that the triangle enclosed by the three vertices of intersection (the gray shaded region), while translated, conserves its shape, orientation, and size (SOS), and thus the area too, independent of the choice of the phase center.",
"Through a geometric picture, we can clearly confirm that the closure phase is gauge-invariant and closely related to the properties of the triangle enclosed by the fringe NPCs in the image plane, and not to the phase center, calibration, or image-plane translations.",
"Figure: (Color) Illustration of the gauge-invariant and shape-orientation-size (SOS) conserving nature of closure phase.",
"(a) Visibility fringes and phases, and closure phase on ideal (or perfectly calibrated) fringes, F ab (s ^,λ)F_{ab}(\\hat{{s}},\\lambda ) for a=1,2,3a = 1, 2, 3, b=⌈a+1⌋ 3 b = \\lceil a+1\\rfloor _3.",
"The three principal fringe NPCs are annotated and shown in black lines with the line style corresponding to that in Figures and .",
"Gray lines denote secondary fringe NPCs.",
"The three principal visibility phases, φ ab (λ)\\phi _{ab}(\\lambda ), are proportional to the positional offsets [see Equation ()] shown in magenta from the phase center (origin) marked by ++ and annotated by 𝒪\\mathcal {O}.",
"The closure phase from the principal fringes, φ 3 (λ)\\phi _3(\\lambda ), is the sum of the three visibility phases.",
"The phase center can be conveniently shifted to any one of the points of intersection between the three principal fringe NPCs, 𝒪 ' \\mathcal {O}^\\prime , marked in brown, blue, or red, in which case the closure phase reduces simply to φ ab ' (λ)\\phi _{ab}^\\prime (\\lambda ), which are shown corresponding to positional offsets from the intersection vertex to the opposite principal fringe NPC in brown (dashed), blue (dash-dotted), or red (dotted), respectively, according to Equation ().",
"The area enclosed by the three principal fringe NPCs (gray shaded region) is proportional to the closure phase squared (see Section ).",
"(b) Same as the ideal case in panel (a) but when considering uncalibrated (middle) and translated fringes (right).",
"As a result, all the fringe NPCs are displaced parallel to themselves relative to the phase center compared to the ideal case.",
"The closure phase, which is still the sum of the three uncalibrated or translated visibility phases (corresponding to the positional offsets in magenta), remains unchanged.",
"Equivalently, the closure phase which is proportional to the positional offsets (brown dashed, blue dash-dotted, or red dotted lines) of the intersection vertices of any of the two principal fringe NPCs from the opposite fringe NPC are independent of these shifts as well as of the phase center.",
"Though the fringes and the triangle enclosed by them are displaced, their displacements are constrained to be parallel to themselves with the only degree of freedom being translation, thereby conserving SOS (hence, the area too).",
"The SOS conservation despite electromagnetic phase corruption attributable to individual array elements, and overall translation in the image plane, demonstrates the gauge-invariance of the closure phase.Note that the option to choose the phase center to be a point of intersection of the fringe NPCs applies only when the fringe NPCs are not parallel to each other.",
"A special case arises when the array elements lie on a collinear arrangement on the aperture plane.",
"The resulting fringes are all parallel to each other yielding no definite intersections between the fringe NPCs that could serve as the preferred phase centers.",
"In fact, due to the presence of the element-based phase terms, the uncalibrated fringes will be offset from each other differently relative to the calibrated case.",
"However, the closure phase is well-defined even in this scenario.",
"An arbitrary phase center can be still chosen, including anywhere on one of the fringe NPCs, and the closure phase is given by Equation (REF ), which is still valid and cancels the element-based phase terms as in the general case."
],
[
"Relation between Areas in Aperture and Image planes",
"In the preceding section, we have seen clear evidence that the closure phase is intricately linked to the the geometric characteristics of the triangle determined by the fringe NPCs, encapsulated by SOS conservation.",
"This motivates further investigation of geometric properties, specifically the relation between the areas of the triangles enclosed by the fringes and the array elements in the image and aperture planes, and the closure phase.",
"Indeed, it can be shown that 32() = 162 AA3() AI3(), where, $A_{\\mathcal {I}3}(\\lambda )$ is the area of the triangle enclosed by the three fringe NPCs in the image plane, $A_{\\mathcal {A}3}(\\lambda )$ denotes the area of the triangle formed by the three vertices denoting the array elements in the aperture plane in units of wavelengths squared.",
"The subscripts $\\mathcal {I}$ and $\\mathcal {A}$ in $A_{\\mathcal {A}3}(\\lambda )$ $A_{\\mathcal {I}3}(\\lambda )$ and $A_{\\mathcal {A}3}(\\lambda )$ denote the image and the aperture plane, respectively, while the subscript 3 denotes a 3-polygon.",
"$A_{\\mathcal {I}3}(\\lambda )$ is dimensionless as it is obtained using direction-cosine coordinates.",
"See Appendix for a detailed derivation of this result and associated caveats.",
"Thus, the product of the areas enclosed by the triad of array elements and the 3-fringe NPC in the aperture and image planes, respectively, is proportional to the closure phase squared, and is thus gauge-invariant too.",
"Figure REF illustrates the quantities in this relationship.",
"$A_{\\mathcal {I}3}(\\lambda )$ is denoted by the gray shaded area, while $A_{\\mathcal {A}3}(\\lambda )$ is the area enclosed by the array elements in Figure REF in wavelength squared units.",
"In this example, $A_{\\mathcal {I}3}(\\lambda )\\approx 1.78\\times 10^{-6}$ , $A_{\\mathcal {A}3}(\\lambda )\\approx 34410.43$ , and $\\psi _3(\\lambda )\\approx -3.11$ radians, thereby confirming the validity of Equation (REF ).",
"Thus, using coordinate geometry in the image plane in angular coordinates, this work provides a detailed derivation of the equivalent findings using a quantum mechanical formalism in [30]."
],
[
"Application to Real Data",
"We present two examples of closure phase visualization in the image plane, and the image-plane methods to estimate closure phase, using data from the Jansky Very Large Array (JVLA).",
"The JVLA is a radio interferometer in New Mexico, comprised of 27 antennas of 25 m diameter each, arranged in a Y-pattern [31].",
"The first example is that of the compact radio quasar 3C 286, including both calibrated data and uncalibrated data.",
"The second involves the powerful extended radio galaxy with a complex morphology, Cygnus A, using calibrated data, and then purposefully phase-corrupted data.",
"There are two main objectives with these two examples that span simple to complex morphologies in the image plane.",
"First, these extreme scenarios will be used to demonstrate geometrically the well known fact that the closure phase is robust to phase errors that are interferometric element-based, but not baseline-based.",
"We will characterize the geometric behavior of the fringe patterns in the image plane from calibrated and uncalibrated, or corrupted data from a closed triad of element spacings, thereby demonstrating the SOS conservation property of closure phase while remaining completely invariant to clearly noticeable translations in the image plane resulting from element-based phase corruptions.",
"Second, we will show that the closure phases can be estimated geometrically from the image plane and that they agree remarkably well with those derived from the aperture plane data (visibilities) to within the estimated uncertainties."
],
[
"Radio Quasar 3C 286",
"The first example employs observations of the radio quasar, 3C 286, which is a bright and highly compact object, often used for flux density and complex bandpass calibration at radio wavelengths.",
"We employ the JVLA in its largest (`A') configuration, and selected three antennas from the array, corresponding to a triangle with spacings (baselines) of 12.4 km, 7.5 km, and 15.0 km projected on a plane perpendicular to the direction of the phase center (coincident with the position of 3C 286).",
"The flux density of the target object, 3C 286, at the observing wavelength of $\\lambda =3.2$ cm ($\\nu =9.4$ GHz), measured on these antenna spacings is $\\simeq 4.4$ Jy ($1\\,\\textrm {Jy} = 10^{-26} \\,\\textrm {W}\\,\\textrm {m}^{-2}\\,\\textrm {Hz}^{-1}$ ).",
"3C 286 is the dominant source of emission in the field of view.",
"It has a compact core-jet structure, which on the spatial frequencies being considered herein essentially appears as an unresolved, point-like object [32].",
"The nearly point-like structure of 3C 286 implies a closure phase very close to zero, which further implies that the three fringe NPCs will intersect nearly at a point.",
"Equation (REF ) then implies that the positional offset of the fringe NPC along its perpendicular from the opposite intersection vertex will be $\\delta s_{ab}^\\prime (\\lambda )\\approx 0$ .",
"We use a short 20 s observation made at $\\lambda =3.2$ cm with a narrow bandwidth of 20 MHz.",
"At this wavelength, the spatial frequencies (in units of number of wavelengths) are $|{u}_{ab}|\\approx 3.912\\times 10^5$ , $\\approx 2.371\\times 10^5$ , and $\\approx 4.749\\times 10^5$ , respectively.",
"The root-mean-square (RMS) level of thermal noise in the calibrated visibilities is $\\approx 33$ mJy, estimated using the JVLA exposure calculatorhttps://obs.vla.nrao.edu/ect/ using a 2 MHz spectral channel and a 20 s averaging time interval.",
"We consider both calibrated and uncalibrated data.",
"With the former, the visibilities are expected to add coherently for a sky image, since instrumental and tropospheric phase terms at each element have been determined via a strong celestial calibrator (in this case, 3C 286 itself).",
"The uncalibrated data includes electronics- and troposphere-induced phase offsets for each aperture element in the interferometer, which need to be corrected via calibration before a coherent image of the target object can be synthesized.",
"From the superposed interference pattern from a triad of elements, there are numerous ways in which the fringe NPCs can be geometrically and directly determined from the image plane without recourse to the visibility data in the aperture plane.",
"Here, we employed the following method.",
"In the first step, we determine the intersecting vertices, $(\\ell _{a\\underline{b}c}, m_{a\\underline{b}c})$ , from the interference pattern of any pair of fringes in the image plane, typically using a peak-fitting algorithm.",
"Of the many possible possible vertices, we preferentially choose the ones closest to the peak of the antenna power pattern which will yield the best signal-noise-ratio ($S/N$ ).",
"Next, given this vertex and the slopes of the two fringe NPCs from the accurately pre-determined projected element spacings (or spatial frequencies), ${u}_{ab}$ , the individual fringe NPCs that contain this intersecting vertex can be determined.",
"Finally, with the three vertices and the fringe NPCs determined, the closure phase can be measured geometrically using either the positional offset of an intersecting vertex from the opposite fringe NPC [see Equation (REF )] or the area enclosed by the fringe NPCs [see Equation (REF )].",
"The thermal noise in the measurements and other systematics will lead to uncertainties in the determined positions of the intersecting vertices of the fringe NPCs which will result in an uncertainty on the measured closure phase.",
"The phase deviations on the measured visibility phases, $\\psi _{ab}(\\lambda )$ , from thermal noise and random systematics in a high-$S/N$ regime ($S/N\\gg 1$ ) follow a Gaussian distribution with a standard deviation that is inversely proportional to the $S/N$ [28], [29].",
"The corresponding position error in the fringe NPCs is given by standard error propagation between the pertinent quantities, $\\psi _{ab}(\\lambda )$ and $\\delta s_{ab}(\\lambda )$ , in Equation () as [Var(sab())]1/2 = [Var(ab())]1/22|uab| (S/N)-12|uab|, which implies that the fractional phase error (relative to $2\\pi $ ), $\\bigl [\\textrm {Var}\\left(\\psi _{ab}(\\lambda )\\right)\\bigr ]^{1/2}/(2\\pi )$ , is equal to the fractional position error (relative to the fringe spacing, $1/|{u}_{ab}|$ ), $\\bigl [\\textrm {Var}\\left(\\delta s_{ab}(\\lambda )\\right)\\bigr ]^{1/2}/(1/|{u}_{ab}|)$ , in the image plane.",
"This is also typically the case with astrometric errors in VLBI applications [28].",
"This error will also propagate into the estimate of closure phase.",
"In the 3C 286 data analyzed here (2 MHz spectral channel, 20 s integration), the signal strength from 3C 286 and the thermal noise RMS in the visibilities are roughly uniform across the different aperture element spacings giving a $S/N\\approx 133$ on each visibility.",
"Figure REF shows the individual visibility fringes (color maps) before any amplitude or phase calibration for the projected array element spacings of $\\approx 12.4$ km, $\\approx 7.5$ km, and $\\approx 15.0$ km in the left, middle, and right panels, respectively.",
"The fringe NPCs shown by black lines (dashed, dot-dashed, and dotted for 12.4 km, 7.5 km, and 15.0 km element spacings, respectively) and the uncertainties therein were geometrically determined using the simple peak-fitting procedure described above.",
"Note that the fitted fringe NPCs align remarkably well with the fringes due to the relatively high $S/N$ in the 3C 286 data.",
"It must be emphasized that in the determination of the fringe NPCs, we did not use any aperture-plane measurements involving the visibilities, except the mathematically pre-determined aperture element spacings, ${u}_{ab}$ .",
"The J2000 Right Ascension (R.A.) and the Declination (Dec.) coordinates used are equivalent to the direction-cosine coordinates introduced earlier [28], [29].",
"Figure: (Color) The three individual fringe patterns (color maps) annotated by F ab (s ^,λ)F_{ab}(\\hat{{s}},\\lambda ), a=1,2,3a=1,2,3 and b=⌈a+1⌋ 3 b=\\lceil a+1\\rfloor _3, in the image plane for the uncalibrated 3C 286 data at λ=3.2\\lambda =3.2 cm with the JVLA, as explained in Section .",
"The three panels correspond to the element spacings of 7.5 km, 12.4 km, and 15 km.",
"The principal fringe NPCs (black lines) were determined entirely from the image plane using the peak location from the intersections and the slopes of the intersecting fringes [determined by the element spacings u ab {u}_{ab}, see Equation ()].",
"The RMS errors in the determined positions of these fringe NPCs are illustrated in Figure .",
"The phase center is at the center of the images.",
"The image coordinates are in Right Ascension (R.A.) and Declination (Dec.) at the J2000 epoch, which are equivalent to the direction-cosine coordinates used earlier , .Figure REF shows the images made by superposing the three fringes from calibrated (left panel) and uncalibrated (right panel) data.",
"The principal fringe NPCs are shown as black lines using the same line styles as in Figure REF .",
"For the calibrated data, the fringe NPCs nearly intersect at a point, indeed, on a grid of points, including the position of 3C 286.",
"Importantly, the uncalibrated fringes also result in a similar grid of points.",
"The only change is that the grid shifts by about 0$.\\!\\!^{\\prime \\prime }$ 2 relative to the pattern seen in the calibrated data.",
"Figure: (Color) (a) Images made with the superposition of the three fringe patterns shown in Figure from calibrated (left) and uncalibrated (right) data using the same line styles as before for the principal fringe NPCs.",
"The lack of calibration results in a net displacement of the interference pattern by ≈0. ''",
"2\\approx 0\\hbox{$.\\!\\!^{\\prime \\prime }$}2 relative to the calibrated fringes, which indicates the magnitude of the required phase calibration terms.",
"Independent of calibration, the principal fringe NPCs in both cases are nearly coincident with each other which geometrically confirm that 3C 286 has a highly compact structure and the closure phase, φ 3 (λ)≈0\\phi _3(\\lambda )\\approx 0 as expected, remains invariant even when the element-based instrumental and tropospheric phase corruption terms remain undetermined.",
"(b) A zoomed-in view of the upper panel.",
"The gray-shaded regions indicate twice the RMS uncertainties in the determined positions of the fringe NPCs, which depend on the RMS phase errors [≈(S/N) -1 \\approx (S/N)^{-1} when S/N≫1S/N\\gg 1] in the measured visibilities, as in Equation ().",
"In this case, S/N≈133S/N\\approx 133.",
"The level of these uncertainties imply that the differences in the measured closure phases [based on both the positional offset from any vertex to the opposite fringe NPC given by Equation (), and the areas enclosed by the three fringes given by Equation ()] using the calibrated and the uncalibrated cases are statistically consistent with each other (only ≲1σ\\lesssim 1\\sigma significance in difference) and are also consistent with zero (only ≲2σ\\lesssim 2\\sigma significance of being non-zero).Figure REF shows a highly zoomed-in view of Figure REF around the intersection points of the fringe NPCs.",
"The gray-shaded regions denote twice the best-case RMS error expected in the geometrical determination of the principal fringe NPCs due to thermal noise as given by Equation (REF ).",
"The three fringe NPCs are not perfectly coincident thereby implying that the closure phase does not vanish completely.",
"Specifically, the principal closure phases were measured to be $\\phi _3(\\lambda )\\approx 1.7^\\circ $ and $\\phi _3(\\lambda )\\approx 2^\\circ $ from calibrated and uncalibrated data, respectively.",
"However, the errors derived from the fitting process are $\\sim 1.3^\\circ $ , implying that both results are consistent with zero closure phase statistically, as expected for a point-like structure.",
"For comparison, it is also possible to calculate the closure phase using the visibilities (in the Fourier- or aperture domain), as is typical in radio interferometry.",
"From the individual visibility phases for each baseline in the triad, we calculate, using Equation (REF ), a closure phase of $2.6^\\circ \\pm 0.74^\\circ $ and $2.0^\\circ \\pm 0.74^\\circ $ for the calibrated and uncalibrated data, respectively.",
"The RMS uncertainty in the visibility phases was again calculated as a reciprocal of the $S/N$ , in radians.",
"The closure phase is the sum of three visibility phases.",
"Hence, the phase noise, which is uncorrelated between the three visibilities, increases by a factor of $\\simeq \\sqrt{3}$ , to $\\simeq 0.74^\\circ $ in the closure phase.",
"These aperture-plane estimates of closure phase are statistically consistent with that from the image plane discussed earlier.",
"The quoted uncertainties represent that expected from thermal noise alone ignoring any systematic errors, and are thus optimistic.",
"This result demonstrates a few important principles.",
"First, the fact that the fringes intersect at a point even for the uncalibrated data confirms the invariance of closure phase (zero, for a point-like morphology), for an instrument in which the instrumental and tropospheric phase contributions can be factored into element-based terms as in Equation ().",
"Second, the shift in the grid pattern in Figure REF is a measure of the magnitude of antenna-based phase corruptions due to the instrument and troposphere.",
"And third, the fact that the fringes intersect at close to a point implies that, for the VLA, the atmospheric and electronic phase corruptions to the data are predominantly factorizable into antenna-based gains, and are not dominated by corruptions that may be idiosyncratic to a given interferometric baseline.",
"Note that adding fringes from other sets of triads to the uncalibrated image in Figure REF , will result in different sets of grids whose peaks will not coincide with each other, and hence a coherent sum of the peaks will not be seen.",
"Thus, from an imaging viewpoint, only after calibration of the instrumental and tropospheric phases in a standard synthesis imaging approach, or alternatively, by imposing a priori image constraints in a forward-modeling approach using closure quantities [33], [34], will the various triad grids align to yield a coherent source of emission at the target object position."
],
[
"Radio Galaxy Cygnus A (3C 405)",
"As a second example, we employ JVLA observations at $\\lambda =3.75$ cm ($\\nu =8.0$ GHz) of the bright, extended radio galaxy, Cygnus A [35].",
"Cygnus A has a total flux density of 170 Jy at this wavelength, distributed in two extended lobes with a full extent of $120^{\\prime \\prime }$ .",
"The observations were made in the `D' configuration of the JVLA, which has a longest baseline of 1 km, corresponding to a spatial resolution of $\\approx 8^{\\prime \\prime }$ .",
"Figure REF shows an image synthesized from 4 min and 128 MHz of these data.",
"Cygnus A is noted to have complex spatial structure typical of an FR II morphology (edge-brightened with bright hotspots at the outer edges of their lobes) [36].",
"Figure: (Color) Image of Cygnus A, a bright radio galaxy, synthesized from 4 min and 128 MHz of JVLA data at λ=3.75\\lambda =3.75 cm .",
"Cygnus A has a complex structure at these wavelengths: a bright core centered on the active galactic nucleus (AGN) and two bright and non-symmetric lobes, classified as an FR II morphology.",
"The angular resolution of the image (“beam size”) is ≈8 '' \\approx 8^{\\prime \\prime }.",
"The contours correspond to -2.5σ-2.5 \\sigma (dashed), 2.5σ2.5 \\sigma , 5σ5 \\sigma , 20σ20 \\sigma , 80σ80 \\sigma , and 320σ320 \\sigma , where, σ≈0.1\\sigma \\approx 0.1 Jy/beam is the RMS of noise in the image.",
"The color bar uses a “symmetric” logarithmic scale to represent both negative and positive values of brightness.We choose three baselines in a rough equilateral triangle for estimating the closure phase, with baseline lengths and correlated flux densities = (797.1 m, 22.7 Jy), (773.7 m, 26.4 Jy), and (819.7 m, 38.3 Jy).",
"We employ a single record with an integration time of 8 s and spectral channel width of 8 MHz, giving a thermal noise of $\\simeq 82$ mJy in a single polarization.",
"We employ calibrated data, and then corrupt the phase of one of the array elements in a closed triad by $80^\\circ $ , as would occur if, for instance, there was a significant mis-calibration.",
"From the aperture-plane visibilities, we calculate a closure phase for both the calibrated and corrupted visibilities of $112.7^\\circ \\pm 0.3^\\circ $ , where the uncertainty is set by the quadrature sum of the individual phase errors based on the respective visibility $S/N$ ($\\gtrsim 275$ ) using Equation (REF ).",
"The images from summing all three fringes for calibrated and corrupted data are shown in Figure REF .",
"In this case, the closure phase is clearly non-zero, and hence the three fringe NPCs do not intersect in a grid of points, as for 3C 286.",
"However, a grid-like pattern remains visible in the three-fringe images, and this pattern repeats exactly, with a simple shift between the calibrated and corrupted data in the left and right panels, respectively.",
"The phase corruption of a single antenna in the triad leads to a corresponding phase (or position) shift of the two corrupted fringes containing this antenna, and no change in the third fringe.",
"The shifting of the pattern will then occur parallel to the uncorrupted fringe, as seen in Figure REF .",
"From these, we calculate the closure phase in the image plane using the same process as employed for 3C 286 above, and find it to be $112.9^\\circ \\pm 1.5^\\circ $ , where the uncertainties were determined using the uncertainties in the determined points of intersection using the peak-fitting procedure.",
"Figure: (Color) Same as Figure but for Cygnus A data.",
"The phase corruption of one antenna measurement results in the parallel displacement of the interference pattern relative to the calibrated fringes, which indicates the magnitude of the required phase calibration terms.",
"Independent of calibration, the principal fringe NPCs in both cases are clearly non-coincident with each other which geometrically confirms that Cygnus A has a complex structure (see Figure ) in contrast to 3C 286.",
"Gray-shaded regions indicate twice the RMS uncertainties in the determined positions of the fringe NPCs as determined from Equation (), but they are barely visible due to the high S/NS/N (≳275\\gtrsim 275) in the visibilities.",
"The closure phase calculated from the positional offsets between the vertices and the opposite fringe NPCs is φ 3 (λ)≈112.9 ∘ \\phi _3(\\lambda )\\approx 112.9^\\circ (see Section ) with an RMS uncertainty of ≈1.5 ∘ \\approx 1.5^\\circ , and remains invariant even after the element-based phase of one antenna measurement was corrupted by 80 ∘ 80^\\circ .",
"φ 3 (λ)\\phi _3(\\lambda ) estimated from the area relations in Section are ≈112.5 ∘ \\approx 112.5^\\circ and ≈113.7 ∘ \\approx 113.7^\\circ from the calibrated and uncalibrated fringe NPCs, respectively.In terms of visualizing closure phase, these images show clearly the SOS conservation theorem, meaning that, for a closed triad of array elements, the resulting images are a true representation of the sky brightness distribution, independent of element-based phase corruption, besides an overall translation of the pattern.",
"If the phase error was dependent on the baseline vector instead of an antenna, only one of the NPCs that corresponds to the affected baseline will be displaced while the other two will remain unchanged and unconstrained by this phase perturbation, thereby changing the size of the resulting triangle in the image plane.",
"Thus, in the presence of a baseline-dependent phase error, the SOS conservation theorem will not apply.The closure phases were also estimated using the relations between the areas in the aperture and image planes.",
"For the chosen triad, $A_{\\mathcal {A}3}(\\lambda )\\approx 1.976\\times 10^8$ (in units of wavelengths squared).",
"The corresponding image-plane areas enclosed by the NPCs, $A_{\\mathcal {I}3}(\\lambda )$ , are found to be $\\approx 1.236\\times 10^{-10}$ and $\\approx 1.263\\times 10^{-10}$ for the calibrated and corrupted fringes, respectively.",
"Hence, the respective closure phases estimated are $\\approx 112.5^\\circ $ and $\\approx 113.7^\\circ $ , which are consistent with the estimates above and confirm the relations derived in Section .",
"Although the image-plane estimate appears to have a higher uncertainty, it must be noted that our aperture-plane uncertainty calculation represents a best-case scenario assuming ideal thermal noise, ignoring imaging systematics around a bright, complex object such as Cygnus A.",
"The value of closure phase inferred from the image plane is not only consistent with that estimated from the corrupted visibilities in the aperture plane, but also geometrically confirms that it is indeed independent of element-based calibration.",
"We also consider a counter-example in which the phase corruption occurs in a baseline instead of an individual antenna.",
"In this case, two of the NPCs whose fringes were not corrupted will remain unchanged as they are unaffected by the corruption.",
"Only the fringe NPC of the phase-corrupted baseline will be shifted.",
"This will not result in a change of shape or orientation (which are set by the geometry of the baseline vectors) but will change the size of the triangle enclosed by the three NPCs which effectively modifies the closure phase.",
"Therefore, the closure phase and the SOS characteristic will no longer be conserved.",
"This clearly demonstrates that strict closure phase and SOS conservation only occurs if the phase error can be attributed to individual array elements (thereby affecting the visibilities in a constrained manner with opposite signs), not an individual baseline.",
"In summary, our two real data examples show clearly the closure phase and SOS conservation of three fringe images with respect to element-based phase corruption, for a closed triad of elements.",
"The implication is that both calibrated and corrupted images in Figure REF and REF , are true representations of the sky brightness (assuming the amplitudes have been calibrated) as seen by the closed triad, without correction of the element-based phase errors.",
"The only effect of an element-based phase corruption is a translation in the image pattern."
],
[
"Parallels to Other Applications",
"Here, we present a few examples of close parallels to the interferometric closure phase in other areas of physics."
],
[
"Optical Interferometry and Aperture Masking",
"We have approached this problem from the perspective of radio interferometric imaging, but the insight is applicable to optical inteferometry, with particular relevance to aperture masking interferometry [37], [38], [7].",
"Indeed, consideration of simple aperture masking provides further physical insight into the interpretation of closure phase in the image domain [39], [40], [41], [42].",
"In radio astronomy, the visibility phases are measured as the argument of the complex cross-correlation products of voltages between the antennas, as per Equation (), where the voltages are generated via coherent amplification of the radio signals at each antenna in the aperture plane.",
"These visibility phases can then be summed in closed triangles to produce closure phases.",
"In optical interferometry, voltages in the aperture plane cannot be captured and coherently amplified, and thus the antenna-pair visibilities are generated via mirrors (e.g., siderostats or unmasked regions of a larger aperture) and lenses, beam splitters, and/or beam combiners, then coherently reflect, focus, and interfere the light from different aperture elements onto a photon detector, typically a charge-coupled device (CCD), resulting in interference fringes.",
"The phase and amplitude of the visibilities can then be extracted through a Fourier analysis of the image (using knowledge of the beam combination and reimaging optics), and closure phases are generated by summing these visibility phases [43], [44], [37].",
"In the context of aperture masking, the three circles (indexed by 1, 2, and 3) in the aperture plane (Figure REF ), correspond to the small unmasked regions of a larger parabolic mirrorThe mask, of course, is usually implemented in the pupil plane..",
"If we assume beam combination of the type used in most aperture masking experiments, i.e., image-plane combination where pupil rescaling is the only type of pupil remapping performed, the fringe patterns in Figure REF then correspond to the imaged fringes at the CCD in the focal plane of the telescope.",
"In this case, the geometric delays are set by the shape and accuracy of the parabolic surface, and sidereal tracking of the fringes is performed by moving the full telescope.",
"In radio interferometers, the array elements in the aperture plane coherently amplify the voltages, and geometric delays and sidereal fringe tracking are performed electronically, followed by cross-correlation of voltages from different array elements [see Equation ()].",
"In this picture, a distortion of the wavefront's phase at one of the unmasked apertures caused by turbulence in the propagation medium along its path, effectively translates to a simple displacement of the aperture element toward or away from the prime focus, resulting in a net path length or phase difference to the focus.",
"We have shown that such a disturbance will shift the closed three-fringe pattern on the image plane, but will obey SOS conservation.",
"It is easy to see why the three angles of the fringe triangle, and its orientation, are preserved, since these are set strictly by the geometry of the projected baselines which are predetermined and thus the fringes can only shift perpendicular to the fringe length, as seen earlier and described by Equations () or ().",
"While less obvious, it remains physically intuitive that the lengths of the triangle's sides are also preserved, since a phase distortion that can be associated with a single aperture affects the visibilities on the two baseline vectors that include this aperture with equal but opposite values, so that the two fringes involved shift relative to each other in such a way that the lengths between the intersecting vertices are preserved.",
"Figure REF shows a schematic representation of what occurs when the electronic phase of one element in a closed triad is corrupted.",
"The three dark circles indicate the elements in the aperture plane (in dark shade of gray), assumed to be at $Z=0$ whose normal vector is indicated by the thick, solid upward arrow.",
"These apertures can be considered unmasked regions in an aperture mask of an optical telescope, or radio antennas in a radio interferometer.",
"The radiation is then directed from the elements to the focal (image) plane, wherein a three-fringe image is synthesized by the interference of the EM waves.",
"Consider a phase corruption of one array element (indexed by $a$ ) by an amount $\\delta \\xi _a(\\lambda )$ .",
"Such a phase corruption is equivalent to a change in path length, $\\Delta D_a$ related by $\\delta \\xi _a(\\lambda )=2\\pi \\Delta D_a/\\lambda $ , of that aperture element to the focal plane.",
"Since three points determine a plane, one can visualize this phase corruption, or the extra path length, at one of the aperture elements as a tilting of the aperture plane relative to the original.",
"The tilted aperture plane and its normal are shown by the light gray-shaded region and the dashed arrow, respectively.",
"Such a tilt then directs the light in a slightly different direction, leading to a shift of the interference pattern in the image plane.",
"Each of the fringes from baselines that contain the phase-corrupted aperture element will be subject to a position offset in the image plane given by Equation (), $\\Delta s_{ab}(\\lambda )=\\delta \\xi _a(\\lambda ) / (2\\pi |{u}_{ab}|)$ .",
"Except for the shift, the three-fringe pattern, including the SOS characteristic, is otherwise conserved.",
"This argument can be generalized to a scenario when an arbitrary number of aperture elements are subject to phase corruption.",
"Figure: A schematic diagram of the effect of a phase error attributable to a single element in a close triad of elements (denoted by dark circles) in an interferometer array.",
"The original aperture plane (in dark gray shade) is at Z=0Z=0 with normal vector shown by the thick, solid upward arrow, with the focal (image) plane, in the e ^ z \\hat{{e}}_z direction.",
"The phase error, δξ a (λ)\\delta \\xi _a(\\lambda ), at one array element (indexed by aa) can be effectively characterized as a change in path length, ΔD a \\Delta D_a, from that array element to the focal plane (sometimes referred to as `the piston effect' ) given by δξ a (λ)=2πΔD a /λ\\delta \\xi _a(\\lambda ) = 2\\pi \\Delta D_a/\\lambda .",
"This change in effective path length leads to a tilt of the aperture plane (in light gray shade) as indicated by the new normal vector (tilted, dashed arrow), and hence a corresponding shift of the image plane.",
"Thus, the image appears displaced relative to the original image plane.",
"The fringes of all baseline vectors that contain the array element with the phase error will each be subject to a position offset as governed by Equation (), Δs ab (λ)=δξ a (λ)/(2π|u ab |)\\Delta s_{ab}(\\lambda )=\\delta \\xi _a(\\lambda ) / (2\\pi |{u}_{ab}|).",
"Regardless of the shift, SOS conservation will apply to the three-fringe interference image.SOS conservation for an image synthesized from a closed triad of baselines is an implicit criterion in the theory of optical speckle imaging with a non-redundant mask, sometimes called triple correlation (or triple product or bispectrum) imaging [42], [46], [47], [48].",
"In a speckle imaging process, which employs exposures shorter than the atmospheric coherence time and a non-redundant aperture maskA non-redundant mask ensures that only one aperture pair, or baseline, contributes to a given spatial frequency in the image plane.",
"Without the mask, the many redundant spatial frequencies that would normally occur using the full mirror, will incoherently add in the image plane (incoherence arising from turbulent phase structure over the telescope), leading to decoherence of the measured visibility.",
"The exception is in the high Strehl ratio regime, meaning close to diffraction-limited optics, where the element-based phase errors, or `piston phases', are small, and hence decoherence of redundant fringes is small.",
"Such is the case for space telescopes [45]., a Fourier transform of a given speckle image contains a set of spatial frequencies that are unique to a given aperture pair, or baseline, such that the visibilities derived can be traced directly and uniquely back to specific aperture pairs.",
"In radio astronomy parlance, the resulting data set corresponds to an uncalibrated set of aperture-plane visibilities.",
"From these, meaningful closure phases can be derived from the visibilities, and a standard hybrid imaging and element-based self-calibration process can be performed, in which closure phase is inherently preserved [42], [4].",
"In aperture masking optical interferometry, and in some other applications of interferometric structure determination, the magnitude of amplitude errors in the aperture element-based complex gains, and of non-closing (i.e., baseline-based) phase errors, is negligible.",
"In this case, the conservation of the relative positions of the NPCs on a closed triad of apertures implies a stricter conservation of the true image of the sky itself for that closed triad, except possibly an overall shift of the image."
],
[
"Crystallography",
"Here, we provide a short summary of the key similarities between closure phase in interferometry and structure invariants in crystal lattice structure analysis using X-ray, electron, or neutron crystallography.",
"Rigorous details are provided in the references cited herein.",
"When a beam of radiation (X-rays, for example) is incident on a crystal, the radiation is scattered in discrete directions by the electron density distribution, $\\rho ({r})$ , in the crystal lattice.",
"${r}$ denotes the position vector in three-dimensional (3-D) space.",
"In such a scattering experiment, the scattered wave or the “reflection”, $S_{{h}}$ , is represented in reciprocal lattice space (or the Fourier space denoted by ${h}$ ) by the 3-D Fourier transformWe use a different sign convention in the Fourier transform compared to most references on the subject in order to remain internally consistent with our previous definitions.",
"of $\\rho ({r})$ .",
"At discrete locations, ${h}_j$ , Shj = V (r) e-i 2hjr dV, where, the integration is performed over the unit cell volume, $V$ .",
"In crystallography literature, $S_{{h}}$ is commonly referred to as the “structure factor”We use $S_{{h}}$ instead of $F_{{h}}$ (which is frequently used in standard crystallography literature) to avoid conflicts with symbols used in this paper., which is the Fourier-space dual of $\\rho ({r})$ .",
"${h}_a$ is often expressed in terms of Miller indices along the basis vectors in the reciprocal lattice space.",
"$S_{{h}_j}$ is a complex number with an amplitude, $|S_{{h}_j}|$ and a phase, $\\psi _{{h}_j}$ .",
"If the complex-valued structure factor is known, then the electron density distribution and the underlying crystal structure can be inferred via a Fourier series summationSee footnote above on the Fourier transform convention.",
": (r) = 1Vj Shj ei 2hjr.",
"Similarities between this formulation and the interferometry context introduced in Section are conspicuous when we identify the correspondence between ${r}$ and $\\hat{{s}}$ , ${h}_j$ and ${u}_{ab}$ , $S_{{h}_j}$ and $V_{ab}(\\lambda )$ , and $\\rho ({r})$ and $\\Theta (\\hat{{s}},\\lambda ) I(\\hat{{s}},\\lambda )$ .",
"One of the differences is that typically the Fourier transforms are three-dimensional and two-dimensional in crystallography and radio interferometric synthesis imaging, respectively, and the implications are discussed further below.",
"In crystal structure analysis, only the amplitudes of the structure factor are measured in the form of intensities, $|S_{{h}_j}|^2$ , but not their phases.",
"And any guessed phases will have ambiguities due to the arbitrariness in the choice of the origin (or the phase center).",
"The lack of information about the phases constitutes the classic “phase problem” or the “origin problem” in crystallography [18], [19], [49], [50].",
"A number of methods have been developed to estimate the phases and ultimately the lattice structure, some of which are referred to as the “direct” methods in contrast to others such as the Patterson [51], molecular replacement [52], isomorphous replacement [53], [54], and anomalous dispersion techniques [55], [56].",
"The direct methods typically use a priori structural and symmetry constraints [18], [19], [57], [49], [50] to estimate the phases and the structure.",
"They rely heavily on the use of structure invariants [58], [18], [19], [49], [50], defined as N = j=1N Shj = j=1N hj, which are invariant to origin translation if j=1N hj = 0.",
"Notice the close correspondence of the definition and the properties of these structure invariants to those of the interferometric closure phase developed in Section .",
"The most important structure variants in crystallography are the triplet ($N=3$ ) and quartet ($N=4$ ) phases.",
"They have played a key role in crystallography in making the determination of many molecular structures possible [59].",
"Another difference to note here is that unlike in radio interferometry, the structure invariants in crystallography are not measured, but are estimated from the intensities by the use of a priori information, such as the fact that the electron density distribution function, $\\rho ({r})$ , consists of well separated peaks.",
"Now we extend the geometric insight developed for the interferometric closure phase to the the structure invariants in crystallography.",
"Because the latter is described in 3-D, the NPCs derived in 2-D now become null phase surfaces (NPS) in 3-D real space.",
"Let us consider $N=3$ .",
"The fringe NPS in real space is given by 2hjr + hj = 0, j=1,2,3.",
"The phases, $\\psi _{{h}_j}$ , only change along the direction of the vector, ${h}_j$ , but remain constant in the plane perpendicular to ${h}_j$ .",
"Thus, Equation (REF ) is the 3-D equivalent of Equation (), where the NPC straight lines are replaced with NPS planes.",
"Translating the origin to ${r}_0$ simply modifies the phases to hj= hj + 2 hjr0 as seen earlier in Section REF .",
"Assuming that ${h}_j$ are not collinear, the intersection of each of the three fringe NPS planes with the other will be a straight line (instead of a point in the 2-D interferometric closure phase case).",
"Since the three locations, ${h}_j$ , determine a plane in reciprocal lattice space, each of the three fringe NPS planes in real space will be perpendicular to this plane.",
"Thus, a situation where the three planes could intersect at a single point will not arise and does not require further consideration.",
"We consider two plausible scenarios: (1) each fringe NPS plane intersects the other two along a line resulting in three parallel lines each perpendicular to the plane determined by the three ${h}_j$ , and (2) all three planes intersect along a single line which is also perpendicular to the plane determined by the three ${h}_j$ .",
"Since the structure invariant is invariant to origin translation, using the same reasoning as in Section REF , the phase center (origin), ${r}_0$ , can be chosen to lie anywhere along one of the intersecting lines.",
"Then the triplet invariant phase is simply the phase corresponding to the positional offset, $\\delta r_j^\\prime $ , of the phase center (which is now chosen to lie on one of the intersecting lines) from the opposite fringe NPS plane (which is not participating in the intersection and is parallel to the intersecting line), and is given by 3 = hj= 2|hj| rj When the intersection of three planes results in a single line, then $r_j^\\prime =0$ and $\\psi _3=0$ , and this would indicate centrosymmetry in the crystal [60].",
"By slicing the three fringe NPS planes using a plane that is parallel to the one determined by ${h}_j$ in the reciprocal lattice space, which will yield three intersecting lines just as in Section REF , the SOS conservation and area relations in Section REF can be readily extended to the triplet phase invariant.",
"Equation (REF ) will continue to apply, where the area in real space corresponding to $A_{\\mathcal {I}3}(\\lambda )$ will be the area of this sliced cross-section that is parallel to the triangle in the reciprocal lattice (Fourier) space and $A_{\\mathcal {A}3}(\\lambda )$ will be half the area enclosed by ${h}_j$ in the reciprocal lattice space.",
"When ${h}_j$ are collinear, the three planes are parallel to each other wherein two or all of them may be coincident and a distinct line of intersection between the fringe NPS may not be available.",
"Nevertheless, the structure invariant is still well-defined.",
"Though we focussed here on a specific application, namely, crystallography, the geometric insight and the formalism presented here are generic.",
"Therefore, they can be readily extended to other interferometry applications used in the determination of structures."
],
[
"Quantum Mechanics and Polarized Light",
"A close analogy found in quantum mechanics is noteworthy here.",
"Using the Dirac bra-ket notation, by identifying the signals, $E_a(\\lambda )$ , with the states, ${E_a(\\lambda )}$ , as vectors in complex Hilbert space, the true and the calibrated spatial coherence, $V_{ab}(\\lambda )$ and $\\widehat{V}_{ab}(\\lambda )$ , can be expressed as ${E_a(\\lambda )}{E_b(\\lambda )}$ and $G_a^\\star (\\lambda ) G_b(\\lambda ) {E_a(\\lambda )}{E_b(\\lambda )}$ , respectively, which are inner products of the signal vectors in Hilbert space.",
"Then, $\\phi _{ab}(\\lambda )\\equiv \\arg {E_a(\\lambda )}{E_b(\\lambda )}$ and $\\widehat{\\phi }_{ab}(\\lambda ) = \\phi _{ab}(\\lambda ) + \\xi _b(\\lambda )-\\xi _a(\\lambda )$ .",
"Such a representation of signals in Hilbert space has also been used in radio astronomy [61].",
"In this section, we will assume implicit dependence on $\\lambda $ , if any, and drop it from our expressions for convenience.",
"The closure phase on an $N$ -polygon has a close correspondence to the phase of the $N$ -vertex Bargmann invariant [20] in quantum mechanics, which is defined as N = N(E1, E2, ...EN) a=1N EaEa+1N = a=1N EaEa+1N = a=1N aa+1N It must be emphasized that the Bargmann phase is independent of the choice of local phase factors such as $\\xi _a$ that each state could be subject to in the same way that the closure phase is independent of aperture element-based phase corruptions because of the high degree of similarity in their underlying mathematical formulation [see Equation ()].",
"A close connection between the phase of the Bargmann invariant and the geometric phase [25] has been known to exist [62].",
"When the quantum states refer to the states of polarization of light, the geometric phase is also familiarly recognized as the Pancharatnam phase [21], [22], [63].",
"When the variation of states is cyclic, the Bargmann phase or the geometric phase is gauge-invariant to the local phases acquired by the states and depends only on the cyclic path taken, and is thus a true observable property of the system.",
"The geometric phase acquired during this cyclical state change is equal to half the solid angle the cyclical path, $E_1 \\rightarrow E_2 \\rightarrow \\ldots \\rightarrow E_N \\rightarrow E_1$ , subtends on the Bloch sphere, or the Poincaré sphere in case of the Pancharatnam phase arising from the polarized states of light.",
"The correspondence between the geometric phase and the spurious interferometric closure phase introduced by the polarization leakage in individual antennas of a radio interferometer even for a point-like object was pointed out in [64].",
"The contribution to the intrinsic closure phase from the polarization leakage is indeed equal to half the solid angle subtended by the points represented by the antenna measurements, $E_a$ , on the Poincaré sphere, which is zero for perfectly co-polar visibilities when the polarizations states of the antennas are identical, and is non-zero otherwise even for a compact, point-like object."
],
[
"Closure Phases on $N$ -Polygons",
"The relations established for closure phases on a triad of array elements can be extended to generic closed $N$ -polygons in the aperture plane.",
"A closed $N$ -polygon can be decomposed into $N-2$ adjacent triads with each adjacent pair sharing a side and all such triads sharing a common vertex.",
"The net closure phase on the $N$ -polygon is simply the sum of the closure phases on the adjacent elemental triads defined here.",
"This is because the visibility phase measured by the element spacing on the shared side between adjacent triads appears as the negative of each other and thus vanishes perfectly in the net sum (see also [42]).",
"As before, assuming non-parallel fringes, the intersection between the NPCs of fringes, $F_{12}(\\hat{{s}},\\lambda )$ and $F_{N1}(\\hat{{s}},\\lambda )$ , can be chosen, for example, as the phase center, $\\hat{{s}}_0$ .",
"Then, the visibility phases on these two fringes vanish because the NPCs of fringes $F_{12}(\\hat{{s}},\\lambda )$ and $F_{N1}(\\hat{{s}},\\lambda )$ pass through $\\hat{{s}}_0$ .",
"The closure phase on the $N$ -polygon is then determined by the rest of the $N-2$ fringe NPCs.",
"From Equation (), the visibility phases of the fringe NPCs for the chosen phase center are ab() = {ll 0, a=1,N, 2uabs0 + ab(), otherwise, .",
"with, b = a+1N, and, aa+1N() = -2uaa+1Ns, a=1,2,...N. Here, $\\psi _{ab}^\\prime (\\lambda )$ is simply the phase offset proportional to the positional offset between $\\hat{{s}}_0$ and each of the fringe NPCs given by Equation ().",
"Using Equation (), the closure phase is obtained by summing the closure phases of each of the adjacent triads, which are effectively identical to the phase offsets, $\\psi _{a\\lceil a+1\\rfloor _N}^\\prime (\\lambda )$ , corresponding to these position offsets.",
"Thus, similar to Equation (REF ), we get N() = q=1N-2 3(q)() = a=1N aa+1N(), where, the subscript $q$ indexes the $N-2$ adjacent triads constituting the closed $N$ -polygon, and $\\psi _{3(q)}(\\lambda )$ denotes the closure phase on triad $q$ .",
"Note that, by choice of the phase center adopted here, $\\psi _{12}^\\prime (\\lambda ) = \\psi _{N1}^\\prime (\\lambda ) = 0$ from Equation ().",
"Equation () is a generalization of Equation (REF ) for the $N$ -polygon.",
"Figure REF illustrates the above relation.",
"Figure: An aperture-plane view of an NN-polygon interferometric array, indexed by a=1,2,...,Na=1,2,\\ldots ,N. The aperture element spacing in wavelength units (or spatial frequencies) and the corresponding spatial coherence are indicated by u a⌈a+1⌋ N {u}_{a\\lceil a+1\\rfloor _N} and V a⌈a+1⌋ N (λ)V_{a\\lceil a+1\\rfloor _N}(\\lambda ), respectively, on the adjacent sides.",
"By choosing a vertex (indexed by 1 in this case), adjacent triads sharing this common vertex and having one overlapping side (shown by dashed lines) with the next triad can be defined, each with its own closure phase, ψ 3(q) (λ),q=1,2,...N-2\\psi _{3(q)}(\\lambda ),\\, q=1,2,\\ldots N-2.",
"The closure phase on the NN-polygon is the sum of the closure phases on these adjacent triads with a consistent cyclic rotation of the vertices as indicated by the arrowed circles, ψ N (λ)=∑ q=1 N-2 ψ 3(q) (λ)\\psi _N(\\lambda ) = \\sum _{q=1}^{N-2} \\psi _{3(q)}(\\lambda ).Note that all the relations throughout the paper hold for any arbitrary closed polygon in any configuration between the available vertices, including self-intersecting polygons, and not limited to only the convex or concave configurations.",
"Each polygon configuration will have a unique closure phase, in general, of course.",
"We now examine whether the SOS conservation property applies directly to an $N$ -polygon of fringe NPCs in the image plane when $N\\ge 4$ .",
"This can be understood by perturbing the phase of one of the aperture array elements.",
"This phase perturbation will affect two fringes whose baseline vectors contain this aperture element with opposite displacements of their respective fringe NPCs.",
"However, the rest of the fringe NPCs will remain unchanged and are unconstrained by this change.",
"Therefore, the superposed fringe interference pattern from all the $N$ array elements ($N\\ge 4$ ) will not be conserved on the whole.",
"However, when the $N$ -polygon is decomposed into adjacent, elemental triads, as described above, then the individual triad patterns will obey the SOS conservation theorem as discussed in earlier sections.",
"This is further explained using a 4-polygon example in Appendix .",
"The relationship established in Section REF between the closure phase and the areas in the aperture and the image planes can be extended to an $N$ -polygon by expressing it in terms of adjacent and elemental 3-polygon units as above, each of which obey gauge-invariance and SOS conservation.",
"Consider the elemental triads (indexed by $q=1,2,\\ldots N-2$ ) all sharing a common vertex (denoted by index $a=1$ ) in the aperture plane.",
"In a simple example with a 4-polygon ($N=4$ , $q=1,2$ ), it can be shown that 42() = 162 q=12 AA3(q)() AI3(q)() + 2 q=11r=q+12 3(q)() 3(r)().",
"It can be easily generalized to an $N$ -polygon as N2() = 162 q=1N-2 AA3(q)() AI3(q)() + 2 q=1N-3r=q+1N-2 3(q)() 3(r)().",
"Alternatively, we can also express the relation between the area of the $N$ -polygon in the aperture plane and the closure phases in the adjacent elemental triads as AAN() = q=1N-2 AA3(q)() = 1162 q=1N-2 3(q)2()AI3(q)().",
"Both Equations () and () are gauge-invariant.",
"The former expresses the closure phase on the $N$ -polygon in terms of its adjacent elemental triads.",
"The latter expresses the area of the $N$ -polygon in the aperture plane as a weighted sum of closure phases on the adjacent elemental triads where the weights are inversely proportional to the areas enclosed by the fringe NPCs of the elemental triads.",
"See Appendix for details of the derivation and related caveats.",
"Although a detailed discussion of the propagation of measurement noise into the measured closure phases is beyond the scope of this paper and discussed in detail elsewhere [65], [66], the general trends of the noise properties of closure phases on $N$ -polygons can be easily inferred.",
"The phase noise in the individual fringe is, in general, analytically involved but is well approximated by a Gaussian distribution in a high $S/N$ regime [28], [29].",
"The same applies to closure phases as well [65], [66].",
"Since the closure phase of an $N$ -polygon interferometric array is the sum of the $N$ individual fringe phases from Equation (), the net uncertainty increases if the individual phase noises of the fringes are uncorrelated.",
"As $N$ increases, the net uncertainty in the closure phase will tend to follow a Gaussian distribution as governed by the Central Limit Theorem.",
"In a high $S/N$ regime, the net uncertainty will follow closely a Gaussian distribution and grow as $\\sim N^{1/2}$ ."
],
[
"Summary",
"Although the closure phase has been critically useful in interferometric applications, especially in astronomy, its inherently higher-order nature has made a detailed geometric intuition of this extremely valuable quantity elusive.",
"This paper provides a basis for such an insight.",
"We show how the closure phase can be visualized in the image plane, and we derive and demonstrate the shape-orientation-size (SOS) conservation theorem, in which the relative location and orientation of the three NPCs of a closed triad of array elements are preserved, even in the presence of large element-based phase errors, besides possibly an overall translation of the fringe pattern.",
"We measure closure phase directly from interferometric images using two geometric methods, as opposed to using conventional visibility-based phase measurements in the Fourier domain (or the aperture plane).",
"The closure phase from a triad (3-polygon) of aperture elements is geometrically derived in the image plane to be the sum of the phase offsets of the fringe null phase curves (NPC), which are related to the positional offsets of the fringe NPCs from the phase center and the fringe spacings (inverse of the spatial frequencies of the image-plane intensity distribution or the element spacings, in units of number of wavelengths, projected onto a plane perpendicular to the direction of the phase center) through Equation ().",
"In most cases, the phase center can be conveniently chosen to be the vertex of intersection between any pair of fringe NPCs, and the closure phase is then the phase offset of the third fringe NPC relative to this vertex, which is obtained from the corresponding positional offset between this vertex and the third fringe NPC using Equation (REF ).",
"Additionally, a gauge-invariant relationship is found to exist between the squared closure phase and the product of the areas enclosed by the triad of array elements and the triad of fringe NPCs in the aperture and image planes, respectively.",
"We have now geometrically demonstrated in the image plane, via the SOS conservation theorem, the gauge-invariant nature of closure phase, namely, its invariance to phase corruptions introduced by the propagation medium and the measurement elements, as well as any translations of the intensity distribution in the image, which has only been understood mathematically from the viewpoint of the aperture plane so far.",
"By analyzing real interferometric observations of the bright radio quasar 3C 286 and the radio galaxy Cygnus A using the Jansky Very Large Array (JVLA) radio telescope, we have independently estimated the closure phase using the conventional aperture-plane method and the direct geometric method in the image plane.",
"The resulting closure phase values derived from the image and aperture plane are consistent with each other to within the expected levels of uncertainties.",
"We also verify that the closure phase in real data is robust to antenna-based phase errors and calibration.",
"In future work, we will investigate different geometric methods of estimating closure phases directly in the image plane, compare their relative merits and performance against existing aperture-plane methods.",
"Although this geometric understanding of closure phase in the image plane (namely, SOS conservation), was motivated by radio interferometry for astronomy applications, we identify the existence of close parallels in, and potential extensions of, these geometric methods to optical interferometry, structure analysis in crystallography, and other similar interferometric applications.",
"The closure phase on an $N$ -polygon is identified as a close and an interferometric analog of the structure invariants (for example, the triplet and quartet phases) in crystallography, as well as the phase of the $N$ -vertex Bargmann invariant in quantum mechanics, which is also closely identified with the geometric phase or the Pancharatnam phase or the Berry phase.",
"The invariance of the interferometric closure phase to phase corruptions locally acquired during the propagation and the measurement processes that are locally attributable to the array elements is a form of gauge invariance, and thus, the closure phase is a true observable physical property of the spatial intensity distribution on the image plane, specifically the degree of centrosymmetry.",
"Its invariance to translation is simply a manifestation of this gauge invariance.",
"We have generalized these gauge invariant relationships derived for closure phases on a triad, to an $N$ -polygon interferometric array.",
"Such higher-order closure phases are expected to have higher levels of uncertainty that grow as $\\sim N^{1/2}$ due to intrinsic noise in real-world measurements.",
"We acknowledge valuable inputs from Rajaram Nityananda, Arul Lakshminarayanan, David Buscher, Rick Perley, Craig Walker, Michael Carilli, and James Moran.",
"We thank Kumar Golap for help in using the Common Astronomical Software Applications [67].",
"We thank L. Sebokolodi and R. Perley for permitting use of the Cygnus A data.",
"We acknowledge the use of software packages including AstroUtilsAstroUtils is publicly available for use under the MIT license at https://github.com/nithyanandan/AstroUtils [68], Precision Radio Interferometry Simulator https://github.com/nithyanandan/PRISim;PRISimsoftware, Astropy [70], [71], NumPy [72], [73], SciPy [74], Matplotlib [75], Pyuvdata [76], and Python.",
"Nithyanandan Thyagarajan is a Jansky Fellow of the National Radio Astronomy Observatory.",
"This work makes use of the following JVLA data: VLA/19A-024, VLA/14B-336.",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc."
],
[
"Derivation of Closure Phase Relation to Areas in the Aperture and the Image Planes for a Closed Triad of Aperture elements",
"There are multiple ways in which the area of a triangle formed by three intersecting coplanar lines can be determined.",
"Here we present two methods of deriving the relationship between the closure phase and the areas in the image and the aperture planes given in Section REF ."
],
[
"Method 1",
"This method relies on determining the area of the triangle when the three vertices are known.",
"As shown in Figure REF , Equation () representing the three fringe NPCs reduces to three straight lines when expressed in the $(\\hat{{e}}_x, \\hat{{e}}_y, \\hat{{e}}_z)$ -basis, Ks + () = 0, where, K= 2[u12 v12 u23 v23 u31 v31], s = [ m], () = [12() 23() 31()], where the subscript $\\Delta $ denotes that these equations apply to a closed 3-polygon consisting of three visibility phases.",
"When the fringe NPCs are not parallel to each other (the special case when they are parallel is discussed later), there are 3 points of intersection – one for each pair of adjacent rows permuted cyclically.",
"Using the Cramer's rule, the intersection between the fringe NPCs, $F_{ab}(\\hat{{s}},\\lambda )$ and $F_{bc}(\\hat{{s}},\\lambda )$ , is given by (abc, mabc) = (2-ab() vab -bc() vbcCabc, 2uac -ab() ubc -bc()Cabc), with, Cabc = (2)2 ([uab vab ubc vbc]), b = a+13, c=a+23, and a = 1,2,3.",
"The underline used in index $b$ denotes that this index is shared by the two intersecting fringes, $F_{ab}(\\hat{{s}},\\lambda )$ and $F_{bc}(\\hat{{s}},\\lambda )$ .",
"In Equation (), $|{M}| \\equiv \\det ({M})$ for an arbitrary matrix ${M}$ .",
"The three points of intersection can be obtained by permuting the indices $a$ , $b$ , and $c$ cyclically as $1\\rightarrow 2$ , $2\\rightarrow 3$ , and $3\\rightarrow 1$ .",
"If ${K}_\\Delta ^\\prime (\\lambda )$ , an “augmented” version of ${K}_\\Delta $ , is defined as K() = [2u12 2v12 12() 2u23 2v23 23() 2u31 2v31 31()], then the area of the triangle enclosed by the three fringe NPCs in the image plane is obtained using standard coordinate geometry as AI3() = [(K())]22 a=1 b=a+13 c=a+233 Cabc = [a=1 b=a+13 c=a+233 ab() Cbca]22a=1 b=a+13 c=a+233 Cabc = [12() C231 + 23() C312 + 31() C123]22 C123C231C312, where, the subscripts $\\mathcal {I}$ and 3 in $A_{\\mathcal {I}3}(\\lambda )$ denote the image plane and a 3-polygon (triad), respectively.",
"$A_{\\mathcal {I}3}(\\lambda )$ is dimensionless as it is obtained using direction-cosine coordinates.",
"Noting that $k = 2\\pi /\\lambda $ denotes the wavenumber of the incident EM radiation, it can be shown using Equation () that for a 3-polygon (triad) formed by vertices 1, 2, and 3, C123 = C231 = C312 = 2 k2 AA3 = 82 AA3(), with, AA3() AA3/2 where, $\\widetilde{A}_{\\mathcal {A}3}$ denotes the area of the triangle formed by the vertices $a$ , $b$ , and $c$ in the aperture plane in units of physical distance squared, and $A_{\\mathcal {A}3}(\\lambda )$ denotes the same area normalized to have units of wavelengths squared.",
"The subscripts $\\mathcal {A}$ and 3 in $A_{\\mathcal {A}3}(\\lambda )$ and $\\widetilde{A}_{\\mathcal {A}3}$ denote the aperture plane and a 3-polygon, respectively.",
"Therefore, 32() = 162 AA3() AI3().",
"Note that both the areas, $A_{\\mathcal {A}3}(\\lambda )$ and $A_{\\mathcal {I}3}(\\lambda )$ , individually are signed quantities depending on the sense of the cyclic order adopted while traversing the triad vertices, but they will have the same signs thereby ensuring that $\\psi _3^2(\\lambda ) \\ge 0$ .",
"This method is simpler and relies on the standard algebraic expression for the area of a triangle, namely, half the product of the base, $b(\\lambda )$ , and the height, $h(\\lambda )$ .",
"In Figure REF , consider the segment bounded by the vertices of the intersection of the NPC of fringe $F_{12}(\\hat{{s}},\\lambda )$ with the other two fringe NPCs as the base of the triangle.",
"Let $\\theta _{1\\underline{2}3}(\\lambda )$ be the angle between the NPCs of the fringes $F_{12}(\\hat{{s}},\\lambda )$ and $F_{23}(\\hat{{s}},\\lambda )$ .",
"Then $\\theta _{1\\underline{2}3}(\\lambda )$ is also the angle between ${u}_{12}$ and ${u}_{23}$ in the aperture plane.",
"Thus, $b(\\lambda ) = \\delta s_{23}^\\prime (\\lambda )/\\sin _{1\\underline{2}3}\\theta (\\lambda )$ , where $\\delta s_{23}^\\prime (\\lambda )$ is the perpendicular positional offset of the NPC of the fringe $F_{23}(\\hat{{s}},\\lambda )$ from its opposite vertex.",
"The height is simply given by $h(\\lambda )=\\delta s_{12}^\\prime (\\lambda )$ .",
"Then, using Equation (REF ), the area enclosed by the fringe NPCs in the image plane is AI3() = 12 s12() s23()123() = 12() 23()82 |u12| |u23| 123() = 32()162 AA3(), where, $A_{\\mathcal {A}3}(\\lambda ) = (1/2)\\,|{u}_{12}|\\, |{u}_{23}|\\, \\sin \\theta _{1\\underline{2}3}(\\lambda )$ .",
"This result is identical to that derived using the first method above.",
"Note that the equations in this section are directly applicable only to non-parallel fringes (or equivalently, the array elements are non-collinear in the aperture plane) for which $C_{a\\underline{b}c}\\ne 0$ , or equivalently, $\\theta _{1\\underline{2}3}(\\lambda )\\ne 0$ and $A_{\\mathcal {A}3}(\\lambda )\\ne 0$ .",
"In the limiting case when the triad of array elements are collinear in the aperture plane, $\\theta _{1\\underline{2}3}(\\lambda )=0$ , and hence, $C_{a\\underline{b}c}=0$ and $A_{\\mathcal {A}3}(\\lambda )=0$ .",
"Because the fringe NPCs are parallel to each other and do not have a distinct point of intersection between them, the area enclosed by the fringe NPCs on the tangent-plane of the image is infinite or indeterminate from Equation ().",
"However, the product of these two areas is still a well-defined, finite value proportional to the closure phase squared, given by Equation ()."
],
[
"Generalization of Closure Phase Relation to Areas in the Aperture and the Image Planes in a Closed $N$ -polygon",
"Following Section , consider adjacent triangles all sharing a common vertex (denoted by index $a=1$ ) in the aperture plane.",
"As a simple example, consider a 4-polygon in the aperture plane with four vertices indexed by $a=1,\\ldots N$ , with $N=4$ .",
"The two adjacent triangles with a common vertex at $a=1$ are denoted by $\\Delta _{123}$ and $\\Delta _{134}$ with areas $A_{\\mathcal {A}3(q)}(\\lambda )$ with $q=1$ and $q=2$ , respectively, in the aperture plane.",
"The area of the 4-polygon is $A_{\\mathcal {A}4}(\\lambda ) = \\sum _{q=1}^2 A_{\\mathcal {A}3(q)}(\\lambda )$ .",
"Note that the segment joining the vertices 1 and 3 in these elemental triangle units is a diagonal and not a side in the chosen 4-polygon configuration.",
"However, this construction is only intermediate and eventually the visibility phase on the baseline between vertices 1 and 3 will be immaterial as we will express the results using only gauge-invariant quantities from the individual elemental triads.",
"The closure phase relations apply to each of the $N-2$ adjacent elemental triads (indexed by $q$ ) constituting the $N$ -polygon.",
"For the 4-polygon, $q=1,2$ .",
"Thus, from Equation () or (), 3(q)2() = 162 AA3(q)() AI3(q)(), q=1,2.",
"Because $\\psi _4(\\lambda ) = \\sum _{q=1}^2 \\psi _{3(q)}(\\lambda )$ from Equation (), 42() = 162 q=12 AA3(q)() AI3(q)() + 2 q=11r=q+12 3(q)() 3(r)(), which can be generalized to an $N$ -polygon as N2() = 162 q=1N-2 AA3(q)() AI3(q)() + 2 q=1N-3r=q+1N-2 3(q)() 3(r)().",
"Alternatively, we can also express the relation between the area of the $N$ -polygon in the aperture plane and the closure phases in the adjacent elemental triads as AAN() = q=1N-2 AA3(q)() = 1162 q=1N-2 3(q)2()AI3(q)().",
"It is noted that both Equations () and () are gauge invariant.",
"In either case, the gauge invariant closure phase relations on each of the elemental triads, and hence on the $N$ -polygon, can be measured geometrically as shown using the 4-polygon example in Figure REF .",
"Figure: (Color) Left: An aperture-plane view of a 4-polygon interferometric array decomposed as two adjacent triads sharing a side (dashed lines).",
"The element spacing of the shared side in one triad is negative of that in the adjacent triad as indicated.",
"Thus the corresponding spatial coherences are conjugates of each other.",
"The area of the 4-polygon is A 𝒜4 (λ)=∑ q=1 2 A 𝒜3(q) (λ)A_{\\mathcal {A}4}(\\lambda ) = \\sum _{q=1}^2 A_{\\mathcal {A}3(q)}(\\lambda ).",
"Right: An image-plane view of the visibility phases on the 4-polygon and the adjacent triads using the principal NPCs of the corresponding fringes, F ab (s ^,λ),a,b=1,2,...N,b≠aF_{ab}(\\hat{{s}},\\lambda ),\\, a,b=1,2,\\ldots N,\\, b\\ne a.",
"The principal fringe NPCs from adjacent spacings in the 4-polygon are shown by the thick, solid black lines, while that of the spacing shared by the adjacent triads is shown by the two dashed lines where one phase is negative of the other [φ 13 (λ)=-φ 31 (λ)\\phi _{13}(\\lambda ) = -\\phi _{31}(\\lambda )] due to the conjugate relationship between their spatial coherences.",
"The closure phases of the two triads are φ 3(1) (λ)=φ 12 (λ)+φ 23 (λ)+φ 31 (λ)\\phi _{3(1)}(\\lambda )=\\phi _{12}(\\lambda )+\\phi _{23}(\\lambda )+\\phi _{31}(\\lambda ) and φ 3(2) (λ)=φ 13 (λ)+φ 34 (λ)+φ 41 (λ)\\phi _{3(2)}(\\lambda )=\\phi _{13}(\\lambda )+\\phi _{34}(\\lambda )+\\phi _{41}(\\lambda ), where the visibility phases, φ ab (λ)\\phi _{ab}(\\lambda ) are the phase offsets associated with the positional offsets of the phase center (origin) from the respective fringe NPCs according to Equation ().",
"The closure phase of the 4-polygon is the sum of closure phases of the two adjacent triads, φ 4 (λ)=∑ q=1 2 φ 3(q) (λ)=∑ a=1 4 φ a⌈a+a⌋ 4 (λ)\\phi _4(\\lambda )=\\sum _{q=1}^2 \\phi _{3(q)}(\\lambda ) = \\sum _{a=1}^4 \\phi _{a\\lceil a+a\\rfloor _4}(\\lambda ).",
"However, the area enclosed by the fringe NPCs of the 4-polygon (area enclosed between the four thick, solid black lines), A ℐ4 (λ)A_{\\mathcal {I}4}(\\lambda ), is not equal to the sum of the areas enclosed by the two sets of triad fringe NPCs (the two yellow-shaded regions).",
"Thus, A ℐ4 (λ)≠∑ q=1 2 A ℐ3(q) (λ)A_{\\mathcal {I}4}(\\lambda )\\ne \\sum _{q=1}^2 A_{\\mathcal {I}3(q)}(\\lambda ).",
"The SOS conservation does not apply directly to the 4-fringe pattern (denoted by their NPCs in solid black lines) as a whole.",
"However, the SOS conservation holds individually for the elemental triad fringe patterns denoted by the yellow shaded regions.Note that in either of the equations above, the area under the fringe NPCs is expressed only in terms of the elemental triangle NPCs and not the $N$ -polygon in the image plane.",
"This is because the area enclosed by the fringe NPCs of the $N$ -polygon is not the sum of the elemental triad fringe NPCs in the image plane, as illustrated in Figure REF .",
"Therefore, $A_{\\mathcal {I}4}(\\lambda )\\ne \\sum _{q=1}^2 A_{\\mathcal {I}3(q)}(\\lambda )$ .",
"This inequality results from the fact that the SOS conservation is not expected to directly apply for the 4-fringe pattern.",
"For example, perturbing the phase of array element “2” will only displace the NPCs of fringes $F_{12}(\\lambda )$ and $F_{23}(\\lambda )$ leaving the NPCs of fringes $F_{34}(\\lambda )$ and $F_{41}(\\lambda )$ unchanged.",
"The resulting change in NPCs of fringes $F_{12}(\\lambda )$ and $F_{23}(\\lambda )$ and the lack of constraint on NPCs of fringes $F_{34}(\\lambda )$ and $F_{41}(\\lambda )$ will result in a distortion or shearing of the 4-fringe interference pattern (solid black lines) in the image plane shown on the right panel of Figure REF .",
"Hence, the SOS conservation does not apply to the 4-fringe interference pattern as a whole.",
"This explains the area inequality expressed above.",
"However, the yellow regions denoting the 3-fringe interference patterns from the two adjacent, elemental triads will individually obey the SOS conservation property despite the non-conservation of the net 4-fringe interference pattern."
]
] | 2012.05254 |
[
[
"Physics-Guided Spoof Trace Disentanglement for Generic Face\n Anti-Spoofing"
],
[
"Abstract Prior studies show that the key to face anti-spoofing lies in the subtle image pattern, termed \"spoof trace\", e.g., color distortion, 3D mask edge, Moire pattern, and many others.",
"Designing a generic face anti-spoofing model to estimate those spoof traces can improve not only the generalization of the spoof detection, but also the interpretability of the model's decision.",
"Yet, this is a challenging task due to the diversity of spoof types and the lack of ground truth in spoof traces.",
"In this work, we design a novel adversarial learning framework to disentangle spoof faces into the spoof traces and the live counterparts.",
"Guided by physical properties, the spoof generation is represented as a combination of additive process and inpainting process.",
"Additive process describes spoofing as spoof material introducing extra patterns (e.g., moire pattern), where the live counterpart can be recovered by removing those patterns.",
"Inpainting process describes spoofing as spoof material fully covering certain regions, where the live counterpart of those regions has to be \"guessed\".",
"We use 3 additive components and 1 inpainting component to represent traces at different frequency bands.",
"The disentangled spoof traces can be utilized to synthesize realistic new spoof faces after proper geometric correction, and the synthesized spoof can be used for training and improve the generalization of spoof detection.",
"Our approach demonstrates superior spoof detection performance on 3 testing scenarios: known attacks, unknown attacks, and open-set attacks.",
"Meanwhile, it provides a visually-convincing estimation of the spoof traces.",
"Source code and pre-trained models will be publicly available upon publication."
],
[
"Introduction",
" In recent years, the vulnerability of face biometric systems has been widely recognized and increasingly brought attention to the computer vision community.",
"The attacks to the face biometric systems attempt to deceive the systems to make wrong identity recognition: either recognize the attackers as a target person (i.e., impersonation), or cover up the original identity (i.e., obfuscation).",
"Figure: The proposed approach can detect spoof faces, disentangle the spoof traces, and reconstruct the live counterparts.",
"It can be applied to diverse spoof types and estimate distinct traces (e.g., Moiré pattern in replay attack, artificial eyebrow and wax in makeup attack, color distortion in print attack, and specular highlights in 3D mask attack).",
"Zoom in for details.There are various types of digital and physical attacks, including face morphing [1], [2], [3], face adversarial attacks [4], [5], [6], face manipulation attacks (e.g., deepfake, face swap) [7], [8], and face spoofing [9], [10], [11].",
"Among the above-mentioned attacks, face spoofing is the only physical attack to deceive the systems, where attackers present faces from spoof mediums, such as photograph, screen, mask and makeup, instead of a live human.",
"These spoof mediums can be easily manufactured by ordinary people, and hence they pose huge threats to face biometric applications such as mobile face unlock, building access control, and transportation security.",
"Therefore, face biometric systems need to be secured with face anti-spoofing (FAS) techniques to distinguish the source of the face before performing the face recognition task.",
"Figure: The comparison of different deep-learning based face anti-spoofing.",
"(a) direct FAS only provides a binary decision of spoofness; (b) auxiliary FAS can provide simple interpretation of spoofness.",
"𝐌\\mathbf {M} denotes the auxiliary task, such as depth map estimation; (c) generative FAS can provide more intuitive interpretation of spoofness, but only for a limited number of spoof attacks; (d) the proposed method can provide spoof trace estimation for generic face spoof attacks.As most face recognition systems are based on a monocular RGB camera, monocular RGB based face anti-spoofing has been studied for over a decade, and one of the most common approaches is based on texture analysis [12], [13], [14].",
"Researchers noticed that presenting faces from spoof mediums introduces special texture differences, such as color distortions, unnatural specular highlights, Moiré patterns, etc.",
"Those texture differences are inherent within spoof mediums and thus hard to remove or camouflage.",
"Conventional approaches build a feature extractor plus classifier pipeline, such as LBP+SVM and HOG+SVM [15], [16], and show good performance on several small databases with constraint environments.",
"In recent years, many works leverage deep learning techniques and show great progress in face anti-spoofing performance [17], [18], [19], [20], [21].",
"Deep learning based methods can be generally grouped into 3 categories: direct FAS, auxiliary FAS, and generative FAS, as illustrated in Fig.",
"REF .",
"Early works [22], [23] build vanilla CNN with binary output to directly predict the spoofness of an input face (Fig.REF a).",
"Methods [18], [21] propose to learn an intermediate representation, e.g., depth, rPPG, reflection, instead of binary classes, which can lead to better generalization and performance (Fig.REF b).",
"[24], [25], [26] additionally attempt to generate the visual patterns existing in the spoof samples (Fig.REF c), providing a more intuitive interpretation of the sample's spoofness.",
"Despite the success, there are still at least three unsolved problems in the topic of deep learning-based face anti-spoofing.",
"First, most prior works are designed to tackle limited spoof types, either print/replay or 3D mask solely, while a real-world anti-spoofing system may encounter a wide variety of spoof types including print, replay, various 3D masks, facial makeup, and even unseen attack types.",
"Therefore, to better reflect real-world performance, we need a benchmark to evaluate face anti-spoofing under known attacks, unknown attacks, and their combination (termed open-set setting).",
"Second, many approaches formulate face anti-spoofing as a classification/regression problem, with a single score as the output.",
"Although auxiliary FAS and generative FAS attempt to offer some extent of interpretation by fixation, saliency, or noise analysis, there is little understanding on what the exact differences are between live and spoof, and what patterns the classifier's decision is based upon.",
"A better interpretation can be estimating the exact patterns differentiating a spoof face and its live counterpart, termed spoof trace.",
"Thirdly, compared with other face analysis tasks such as recognition or alignment, the data for face anti-spoofing has several limitations.",
"Most FAS databases are captured in the constraint indoor environment, which has limited intra-subject variation and environment variation.",
"For some special spoof types such as makeup and customized silicone mask, they require highly skilled experts to apply or create, with high cost, which results in very limited samples (i.e., long-tail data).",
"Thus, how to learn from data with limited variations or samples is a challenge for FAS.",
"In this work, we aim to design a face anti-spoofing model that is applicable to a wide variety of spoof types, termed generic face anti-spoofing.",
"We equip this model with the ability to explicitly disentangle the spoof traces from the input faces.",
"Some examples of spoof trace disentanglement are shown in Fig.",
"REF .",
"This is a challenging objective due to the diversity of spoof traces and the lack of ground truth during model learning.",
"However, we believe that fulfilling this objective can bring several benefits: Binary classification for face anti-spoofing would harvest any cue that helps classification, which might include spoof-irrelevant cues such as lighting, and thus hinder generalization.",
"In contrast, spoof trace disentanglement explicitly tackles the most fundamental cue in spoofing, upon which the classification can be more grounded and witness better generalization.",
"With the trend of pursuing explainable AI [27], [28], it is desirable for the face anti-spoofing model to generate the spoof patterns that support its binary decision, since spoof trace serves as a good visual explanation of the model's decision.",
"Certain properties (e.g., severity, methodology) of spoof attacks might potentially be revealed from the traces.",
"Disentangled spoof traces can enable the synthesis of realistic spoof samples, which addresses the issue of limited training data for the minority spoof types, such as special 3D masks and makeup.",
"As shown in Fig.",
"REF d, we propose a Physics-guided Spoof Trace Disentanglement (PhySTD) to explore the spoof traces for generic face anti-spoofing.",
"To model all types of spoofs, we formulate the spoof trace disentanglement as a combination of additive process and inpainting process.",
"Additive process describes spoofing as spoof material introducing extra patterns (e.g., moire pattern), where the live counterpart can be recovered by removing those patterns.",
"Inpainting process describes spoofing as spoof material fully covering certain regions of the original face, where the live counterpart of those regions has to be “guessed\" [29], [30].",
"We further decompose the spoof traces into frequency-dependent components, so that traces with different frequency properties can be equally handled.",
"For the network architecture, we extend a backbone network for auxiliary FAS with a decoder to perform the disentanglement.",
"With no ground truth of spoof traces, we adopt an overall GAN-based training strategy.",
"The generator takes an input face, estimates its spoofness, and disentangles the spoof trace.",
"After obtaining the spoof trace, we can reconstruct the live counterpart from the spoof and synthesize new spoof from the live.",
"The synthesized samples are then sent to multiple discriminators with real samples for adversarial training.",
"The synthesized spoof samples are further utilized to train the generator in a fully supervised fashion, thanks to disentangled spoof traces as ground truth for the synthesized samples.",
"To correct possible geometric discrepancy during spoof synthesis, we propose a novel 3D warping layer to deform spoof traces toward the target live face.",
"A preliminary version of this work was published in the Proceedings European Conference on Computer Vision (ECCV) 2020 [31].",
"We extend the work from three aspects.",
"1) Guided by the physics of how a spoof is generated, we introduce a spoof generation function (SGF) to model the spoof trace disentanglement as a combination of additive and inpainting processes.",
"SGF has a better and more natural modeling of generic spoof attacks, such as paper glass.",
"2) Previous trace components $\\lbrace \\mathbf {S},\\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ are not supervised hierarchically so that there exists semantic ambiguity.",
"In this work, we introduce several hierarchical designs in the GAN framework to remedy such ambiguity.",
"3) We propose an open-set testing scenario to further evaluate the real-world performance for face anti-spoofing models.",
"Both known and unknown attacks are included in the open-set testing.",
"We perform a side-by-side comparison between the proposed approach and the state-of-the-art (SOTA) face anti-spoofing solutions on multiple datasets and protocols.",
"In summary, the main contributions of this work are as follows: $\\bullet $ We for the first time study spoof trace for generic face anti-spoofing, where a wide variety of spoof types are tackled with one unified framework; $\\bullet $ We propose a novel physics-guided model to disentangle spoof traces, and utilize the spoof traces to synthesize new data samples for enhanced training; $\\bullet $ We propose novel protocols for a generic open-set face anti-spoofing; $\\bullet $ We achieve SOTA anti-spoofing performance and provide convincing visualization for a wide variety of spoof types."
],
[
"Related Work",
"Face Anti-Spoofing Face anti-spoofing has been studied for more than a decade and its development can be roughly divided into three stages.",
"In the early years, researchers leverage spontaneous human movement, such as eye blinking and head motion, to detect simple print photograph or static replay attacks [32], [33].",
"However, when facing counter attacks, such as print face with eye region cut, and replaying a face video, those methods would fail.",
"In the second stage, researchers pay more attention to texture differences between live and spoof, which are inherent to spoof mediums.",
"Researchers mainly extract hand-crafted features from the faces, e.g., LBP [12], [15], [34], [35], HoG [16], [36], SIFT [14] and SURF [13], and train a classifier to split the live vs. spoof, e.g., SVM and LDA.",
"Recently, face anti-spoofing solutions equipped with deep learning techniques have demonstrated significant improvements over the conventional methods.",
"Methods in [37], [38], [39], [22] train a deep neural network to learn a binary classification between live and spoof.",
"In [17], [18], [19], [20], [21], additional supervisions, such as face depth map and rPPG signal, are utilized to help the network to learn more generalizable features.",
"As the latest approaches achieving saturated performance on several benchmarks, researchers start to explore more challenging cases, such as few-shot/zero-shot face anti-spoofing [19], [40], [41] and domain adaptation in face anti-spoofing [20], [42].",
"In this work, we aim to solve an interesting yet very challenging problem: disentangling and visualizing the spoof traces from an input face.",
"A related work [24] also adopts GAN seeking to estimate the spoof traces.",
"However, they formulate the traces as low-intensity noises, which is limited to print and replay attacks only and cannot provide convincing visual results.",
"In contrast, we explore spoof traces for a much wider range of spoof attacks, visualize them with novel disentanglement, and also evaluate the proposed method on the challenging cases, e.g., zero-shot face anti-spoofing.",
"Disentanglement Learning Disentanglement learning is often adopted to better represent complex data and features.",
"DR-GAN [43] disentangles a face into identity and pose vectors for pose-invariant face recognition and view synthesis.",
"Similarly in gait recognition, [44] disentangles the representations of appearance, canonical, and pose features from an input gait video.",
"3D reconstruction works [45], [46] also disentangle the representation of a 3D face into identity, expressions, poses, albedo, and illuminations.",
"For image synthesis, [47] disentangles an image into appearance and shape with U-Net and Variational Auto Encoder (VAE).",
"Different from [45], [43], [44], we intend to disentangle features that have different scales and contain geometric information.",
"We leverage the multiple outputs to represent features at different scales, and adopt multiple-scale discriminators to properly learn them.",
"Moreover, we propose a novel warping layer to tackle the geometric discrepancy during the disentanglement and reconstruction.",
"Image Trace Modeling Image traces are certain signals existing in the image that can reveal information about the capturing camera, imaging setting, environment, and so on.",
"Those signals often have much lower energy compared to the image content, which needs proper modeling to explore them.",
"[48], [49], [50] observe the difference of image noises, and use them to recognize the capture cameras.",
"From the frequency domain, [25] shows the image noises from different cameras obey different noise distributions.",
"Such techniques are applied to the field of image forensics, and later [51], [52] propose methods to remove such traces for image anti-forensics.",
"Recently, image trace modeling is widely used in image forgery detection and image adversarial attack detection [53], [54].",
"In this work, we attempt to explore the traces of spoof face presentation.",
"Due to different spoof mediums, spoof traces show large variations in content, intensity, and frequency distribution.",
"We propose to disentangle the traces as additive traces and inpainting trace.",
"And for additive traces, we further decompose them based on different frequency bands.",
"Figure: Overview of the proposed Physics-guided Spoof Trace Disentanglement (PhySTD)."
],
[
"Problem Formulation",
" Let the domain of live faces be denoted as $\\mathcal {L}\\!",
"\\subset \\!",
"\\mathbb {R}^{N\\!\\times \\!N \\!\\times \\!3}$ and spoof faces as $\\mathcal {S}\\!",
"\\subset \\!\\mathbb {R}^{N\\!\\times \\!N\\!",
"\\times \\!3}$ , where $N$ is the image size.",
"We intend to obtain not only the correct prediction (live vs. spoof) of the input face, but also a convincing estimation of the spoof trace and live face reconstruction.",
"To represent the spoof trace, our preliminary version assumes an additive relation between live and spoof, and uses 4 trace components $\\lbrace \\mathbf {S},\\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ at different frequency bands as: $\\mathbf {I}_{\\textit {spoof}} = (1+\\lfloor \\mathbf {S}\\rfloor _{n_1})\\mathbf {I}_{\\textit {live}} + \\lfloor \\mathbf {B}\\rfloor _{n_1} + \\lfloor \\mathbf {C}\\rfloor _{n_2} + \\mathbf {T},$ where $\\mathbf {S},\\mathbf {B}$ represent low-frequency traces, $\\mathbf {C}$ represents mid-frequency ones, and $\\mathbf {T}$ represents high-frequency ones.",
"$\\lfloor \\cdot \\rfloor $ is the low bandpass filtering operation, and in practice, we achieve this by downsampling the original image and upsampling it back.",
"In the previous setting, $n_1\\!=\\!1$ and $n_2\\!=\\!64$ .",
"Compared to the simple representation with only a single component [24], this multi-scale representation of $\\lbrace \\mathbf {S},\\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ can largely improve disentanglement quality and suppress undesired artifacts due to its coarse-to-fine process.",
"The model is designed to provide a valid estimation of spoof traces $\\lbrace \\mathbf {S},\\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ without respective ground truth.",
"Our preliminary version [31] aims to find a minimum intensity change that transfers an input face to the live domain: $\\operatornamewithlimits{arg\\,min}_{\\hat{\\mathbf {I}}} \\Vert \\mathbf {I} - \\hat{\\mathbf {I}}\\Vert _F \\; s.t.",
"\\; \\mathbf {I} \\in (\\mathcal {S}\\cup \\mathcal {L}) \\; \\text{and} \\; \\hat{\\mathbf {I}} \\in \\mathcal {L},$ where $\\mathbf {I}$ is the source face, $\\hat{\\mathbf {I}}$ is the target face to be optimized, and $\\mathbf {I}-\\hat{\\mathbf {I}}$ is defined as the spoof trace.",
"When the source face is live $\\mathbf {I}_{\\text{live}}$ , $\\mathbf {I}-\\hat{\\mathbf {I}}$ should be 0 as $\\mathbf {I}$ is already in $\\mathcal {L}$ .",
"When the source face is spoof $\\mathbf {I}_{\\text{spoof}}$ , $\\mathbf {I}-\\hat{\\mathbf {I}}$ should be regularized to prevent unnecessary changes such as identity shift.",
"Despite the effectiveness of this representation, there are still two drawbacks: First, the spoof trace disentanglement is mainly formulated as an additive processing.",
"The optimization of Eqn.",
"REF limits the trace intensity, and the reconstruction for spoof regions with large appearance divergence might be sub-optimal, such as spoof glasses or mask.",
"For those spoof regions, the physical relationship between the live and the spoof is better described as replacement rather than addition; Second, while our preliminary version representing the traces with hierarchical components, these components are learned with losses on their summation.",
"Without careful supervision, the learned components can be ambiguous in their semantic meanings, e.g., the high-frequency component may include low-frequency information.",
"To address the first drawback, we introduce a spoof generation function (SGF) as an additive process followed by an inpainting process: $\\mathbf {I}_{\\textit {spoof}} = (1-\\mathbf {P})(\\mathbf {I}_{\\textit {live}} +\\mathbf {T}_A) + \\mathbf {P}\\cdot \\mathbf {T}_P,$ where $\\mathbf {T}_A\\!\\in \\mathbb {R}^{N\\!\\times \\!N\\!\\times \\!3}$ indicates the traces from additive process, $\\mathbf {T}_P$ indicates the traces from inpainting process, and $\\mathbf {P} \\in \\mathbb {R}^{N\\!\\times \\!N\\!\\times \\!1}$ denotes the inpainting region.",
"Given a spoof face, one may reconstruct the live counterpart by inversing Eqn.",
"REF : $ \\hat{\\mathbf {I}}_{\\textit {live}} =(1-\\mathbf {P})(\\mathbf {I}_{\\textit {spoof}} - \\mathbf {T}_\\text{A}) + \\mathbf {P}\\cdot (\\hat{\\mathbf {I}}_{\\textit {live}} + \\mathbf {I}_{\\textit {spoof}} - \\mathbf {T}_P),$ As the inpainting physically replaces content, the spoof trace $\\mathbf {T}_P$ in the inpainting region $\\mathbf {P}$ is identical to the spoof image $\\mathbf {I}_{\\textit {spoof}}$ in the same region, and thus both cancel out in the second term of Eqn.",
"REF .",
"We further rename the $\\hat{\\mathbf {I}}_{\\textit {live}}$ in the second term as $\\mathbf {I}_P$ to indicate the inpainting content within the inpainting region that should be estimated from the model.",
"Therefore, the reconstruction of the live image becomes: $ \\hat{\\mathbf {I}}_{\\textit {live}} =(1-\\mathbf {P})(\\mathbf {I}_{\\textit {spoof}} - \\mathbf {T}_\\text{A}) + \\mathbf {P}\\cdot \\mathbf {I}_P,$ where $\\mathbf {T}_A \\!=\\!",
"\\lfloor \\mathbf {B}\\rfloor _{n_1} \\!+\\!",
"\\lfloor \\mathbf {C}\\rfloor _{n_2}\\!+\\!",
"\\mathbf {T}$ denotes the additive trace represented by three hierarchical components.",
"$n_1$ and $n_2$ are set to be 32 and 128 respectively.",
"With a larger $n_1$ , the effect of component $\\mathbf {S}$ in the preliminary version can be incorporated into $\\mathbf {B}$ , and hence we remove $\\mathbf {S}$ for simplicity.",
"Besides the additive traces, the model is further required to estimate the inpainting region $\\mathbf {P}$ and inpainting live content $\\mathbf {I}_P$ .",
"$\\mathbf {I}_P$ is estimated based on the rest of the live facial region without intensity constraint.",
"We use a function $G(\\cdot )$ to represent the reconstruction process of Eqn.",
"REF .",
"Accordingly, the optimization of Eqn.",
"REF is re-formulated by replacing $\\hat{\\mathbf {I}}$ with Eqn.",
"REF as: ${\\begin{array}{c}\\operatornamewithlimits{arg\\,min}_{\\mathbf {T}_A,\\mathbf {P},\\mathbf {I}_P} \\Vert \\mathbf {I}-(1-\\mathbf {P})(\\mathbf {I} -\\mathbf {T}_A) - \\mathbf {P}\\cdot \\mathbf {I}_P\\Vert _F \\; \\\\\\rightarrow \\operatornamewithlimits{arg\\,min}_{\\mathbf {T}_A,\\mathbf {P},\\mathbf {I}_P} \\Vert (1-\\mathbf {P})\\mathbf {T}_A\\Vert _F + \\Vert \\mathbf {P}\\cdot (\\mathbf {I}-\\mathbf {I}_P)\\Vert _F.",
"\\; \\\\\\end{array}}$ As we do not wish to impose any intensity constraint on $\\mathbf {I}_P$ , the final objective is formulated as: ${\\begin{array}{c}\\operatornamewithlimits{arg\\,min}_{\\mathbf {T}_A,\\mathbf {P}} \\Vert (1\\!-\\!\\mathbf {P})\\mathbf {T}_A\\Vert _F+\\lambda \\Vert \\mathbf {P}\\Vert _F \\;s.t.",
"\\; \\mathbf {I} \\!",
"\\in \\!",
"\\mathcal {S}\\cup \\mathcal {L}, \\hat{\\mathbf {I}} \\in \\mathcal {L},\\end{array}}$ where $\\lambda $ is a weight to balance two terms.",
"In addition, based on Eqn.",
"REF , we can define another function $G^-(\\cdot )$ to synthesize new spoof faces, by transferring the spoof traces from $\\mathbf {I}^i$ to $\\mathbf {I}^j$ : $\\hat{\\mathbf {I}}_{\\textit {spoof}}^{i\\rightarrow j} = G^-(\\mathbf {I}^j|\\mathbf {I}^i) = (1-\\mathbf {P}^i)(\\mathbf {I}^j+ \\mathbf {T}_\\text{A}^i) + \\mathbf {P}^i\\cdot \\mathbf {I}^i.$ Note that $\\mathbf {T}_P$ in Eqn.",
"REF has been replaced with $\\mathbf {I}^i$ since the spoof image $\\mathbf {I}^i$ contains the spoof trace for the inpainting region.",
"Figure: The proposed PhySTD network architecture.",
"Except the last layer, each conv and transposed conv is concatenated with a batch normalizstion layer and a leaky ReLU layer.",
"‘k3c64s2’ indicates the kernel size of 3×33\\times 3, the convolution channel of 64 and the stride of 2.Estimating $\\lbrace \\mathbf {T}_\\text{A},\\mathbf {P},\\mathbf {I}_P\\rbrace $ from an input face $\\mathbf {I}$ is termed as spoof trace disentanglement.",
"Given that no ground truth of traces is available, this disentanglement can be achieved via generative adversarial based training.",
"As shown in Fig.",
"REF , the proposed Physics-guided Spoof Trace Disentanglement (PhySTD) consists of a generator and discriminator.",
"Given an input image, the generator is designed to predict the spoofness (represented by the pseudo depth map) as well as estimate the additive traces $\\lbrace \\mathbf {B}, \\mathbf {C}, \\mathbf {T}\\rbrace $ and the inpainting components $\\lbrace \\mathbf {P},\\mathbf {I}_P\\rbrace $ .",
"With the traces, we can apply function $G(\\cdot )$ to reconstruct the live counterpart and function $G^-(\\cdot )$ to synthesize new spoof faces.",
"We adopt a set of discriminators at multiple image resolutions to distinguish the real faces $\\lbrace \\mathbf {I}_{\\textit {live}},\\mathbf {I}_{\\textit {spoof}}\\rbrace $ with the synthetic faces $\\lbrace \\hat{\\mathbf {I}}_{\\textit {live}},\\hat{\\mathbf {I}}_{\\textit {spoof}}\\rbrace $ .",
"To remedy the semantic ambiguity during $\\lbrace \\mathbf {B}, \\mathbf {C}, \\mathbf {T}\\rbrace $ learning, three trace component combinations, $\\lbrace \\mathbf {B}\\rbrace $ , $\\lbrace \\mathbf {B}, \\mathbf {C}\\rbrace $ , and $\\lbrace \\mathbf {B}, \\mathbf {C}, \\mathbf {T}\\rbrace $ , will contribute to the synthesis of live reconstruction at one particular resolution, which is then supervised by a respective discriminator (details in Sec.REF ).",
"To learn a proper inpainting region $\\mathbf {P}$ , we leverage both the prior knowledge and the information from the additive traces.",
"In the rest of this section, we present the details of the generator, the discriminators, the details of face reconstruction and synthesis, and the losses and training steps used in PhySTD."
],
[
"Disentanglement Generator",
" As shown in Fig.",
"REF , the disentanglement generator consists of a backbone encoder, a spoof trace decoder and a depth estimation network.",
"The backbone encoder aims to extract multi-scale features, the depth estimation network leverages the features to estimate the facial depth map, and a spoof trace decoder to estimate the additive trace components $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ and the inpainting components $\\lbrace \\mathbf {P},\\mathbf {I}_P\\rbrace $ .",
"The depth map and the spoof traces will be used to compute the final spoofness score.",
"Figure: The visualization of image decomposition for different input faces: (a) live face (b) 3D mask attack (c) replay attack (d) print attack.Backbone encoder Backbone encoder extracts features from the input images for both depth map estimation and spoof trace disentanglement.",
"As shown in our preliminary work [31], the spoof traces consists of components from different frequency bands: low-frequency traces includes color distortion, mid-frequency traces includes makeup strikes, and high-frequency traces includes Moiré patterns and mask edges.",
"However, a vanilla CNN model might overlook high-frequency traces since the energy of high-frequency traces is often much weaker than that of low-frequency traces.",
"In order to encourage the network to equally regard traces with different physical properties, we explicitly decompose the image into three elements $\\lbrace \\mathbf {I}_\\mathbf {B},\\mathbf {I}_\\mathbf {C},\\mathbf {I}_\\mathbf {T}\\rbrace $ as: $\\begin{split}\\mathbf {I}_\\mathbf {B}=&\\lfloor \\mathbf {I}\\rfloor _{n_1},\\\\\\mathbf {I}_\\mathbf {C}=&\\lfloor \\mathbf {I}\\rfloor _{n_2}-\\lfloor \\mathbf {I}\\rfloor _{n_1},\\\\\\mathbf {I}_\\mathbf {T}=& \\mathbf {I} - \\lfloor \\mathbf {I}\\rfloor _{n2}, \\\\\\end{split}$ where $n_1\\!=\\!32$ , $n_2\\!=\\!128$ and the image size $N\\!=\\!256$ .",
"In addition, we amplify the value in $\\mathbf {I}_\\mathbf {C}, \\mathbf {I}_\\mathbf {T}$ by two constants 15 and 25, and then feed the concatenation of three elements to the backbone network.",
"Fig.",
"REF provides the visualization of image decomposition.",
"We observe that the traces that are less distinct in the original images become more highlighted in the $\\mathbf {I}_\\mathbf {T}$ component: 3D mask and replay attack bring unique patterns different with the live face pattern, while print attack is lacking of necessary high frequency details.",
"Semantically, $\\mathbf {I}_\\mathbf {B},\\mathbf {I}_\\mathbf {C},\\mathbf {I}_\\mathbf {T}$ share the same frequency domains with $\\mathbf {B},\\mathbf {C},\\mathbf {T}$ respectively, and thus the decomposition potentially eases the learning of $\\mathbf {B},\\mathbf {C},\\mathbf {T}$ .",
"After that, the encoder progressively downsamples the decomposed image components 3 times to obtain features $\\mathbf {F}_1\\!",
"\\in \\!\\mathbb {R}^{128\\!\\times \\!128\\!\\times \\!64}$ , $\\mathbf {F}_2\\!",
"\\in \\!\\mathbb {R}^{64\\!\\times \\!64\\!\\times \\!96}$ , $\\mathbf {F}_3\\!",
"\\in \\!\\mathbb {R}^{32\\!\\times \\!32\\!\\times \\!128}$ via conv layers.",
"Spoof trace decoder The decoder upsamples the feature $\\mathbf {F}_3$ with transpose conv layers back to the input face size 256.",
"The last layer outputs both additive traces $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ and inpainting components $\\lbrace \\mathbf {P},\\mathbf {I}_P\\rbrace $ .",
"Similar to U-Net [55], we apply the short-cut connection between the backbone encoder and decoder to bypass the multiple scale details for a high-quality trace estimation.",
"Depth estimation network We still recognize the importance of the discriminative supervision used in auxiliary FAS, and thus introduce a depth estimation network to perform the pseudo-depth estimation for face anti-spoofing, as proposed in [18].",
"The depth estimation network takes the concatenated features of $\\mathbf {F}_1$ , $\\mathbf {F}_2$ , $\\mathbf {F}_3$ from the backbone encoder and $\\mathbf {U}_3$ from the decoder as input.",
"The features are put through a spatial attention mechanism from [56] and resize to the same size of $K=32$ .",
"It outputs a face depth map $\\mathbf {M}\\!\\in \\!",
"\\mathbb {R}^{32\\!\\times \\!32}$ , where the depth values are normalized within $[0,1]$ .",
"Regarding the number of parameters, both spoof trace decoder and depth estimation network are light weighed, while the backbone network is much heavier.",
"With more network layers being shared to tackle both depth estimation and spoof trace disentanglement, the knowledge learnt from spoof trace disentanglement can be better shared with depth estimation task, which can lead to a better anti-spoofing performance.",
"Final scoring In the testing phase, we use the norm of the depth map and the intensity of spoof traces for real vs. spoof classification: $\\text{score} = \\frac{1}{2K^2}\\Vert \\mathbf {M}\\Vert _1 \\!+ \\!\\frac{\\alpha _0}{2N^2}(\\Vert \\mathbf {B}\\Vert _1\\!+\\!\\Vert \\mathbf {C}\\Vert _1\\!+\\!\\Vert \\mathbf {T}\\Vert _1\\!+\\!\\Vert \\mathbf {P}\\Vert _1),$ where $\\alpha _0$ is the weight for the spoof trace."
],
[
"Reconstruction and Synthesis",
" There are multiple options to use the disentangled spoof traces: 1) live reconstruction, 2) spoof synthesis, and 3) “harder\" sample synthesis, which will be described below respectively.",
"Live reconstruction: Based on Eqn.",
"REF , we propose a hierarchical reconstruction of the live face counterpart from the input images.",
"To reconstruct faces at a certain resolution, each additive trace is included only if its frequency domain is lower than the target resolution.",
"We apply $\\lbrace \\textit {hi},\\textit {mid},\\textit {low}\\rbrace $ three resolution settings as: $\\begin{split}\\hat{\\mathbf {I}}_{\\textit {hi}}=&(1-\\mathbf {P})(\\mathbf {I} - \\lfloor \\mathbf {B}\\rfloor _{n_1} - \\lfloor \\mathbf {C}\\rfloor _{n_2} - \\mathbf {T}) + \\mathbf {P}\\cdot \\mathbf {I}_P,\\\\\\hat{\\mathbf {I}}_{\\textit {mid}}=&(1-\\mathbf {P})(\\lfloor \\mathbf {I}\\rfloor _{n_2} - \\lfloor \\mathbf {B}\\rfloor _{n_1} - \\lfloor \\mathbf {C}\\rfloor _{n_2}) + \\mathbf {P}\\cdot \\mathbf {I}_P,\\\\\\hat{\\mathbf {I}}_{\\textit {low}}=&(1-\\mathbf {P})(\\lfloor \\mathbf {I}\\rfloor _{n_1} - \\lfloor \\mathbf {B}\\rfloor _{n_1}) + \\mathbf {P}\\cdot \\mathbf {I}_P.\\\\\\end{split}$ Spoof synthesis: Based on Eqn.",
"REF , we can obtain a new spoof face via applying the spoof traces disentangled from a spoof face $\\mathbf {I}_i$ to a live face $\\mathbf {I}_j$ .",
"However, spoof traces may contain face-dependent content associated with the original spoof subject.",
"Directly applying them to a new face with different shapes or poses may result in mis-alignment and strong visual implausibility.",
"Therefore, the spoof trace should go through a geometry correction before performing this synthesis.",
"We propose an online 3D warping layer and will introduce it in the following subsection.",
"“Harder\" sample synthesis: The disentangled spoof traces can not only reconstruct live and synthesize new spoof, but also synthesize “harder\" spoof samples by removing or amplifying part of the spoof traces.",
"We can tune one or some of the trace elements $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T},\\mathbf {P}\\rbrace $ to make the spoof sample to become “less spoofed\", which is thus closer to a live face since the spoof traces are weakened.",
"Such spoof data can be regarded as harder samples and may benefit the generalization of the disentanglement generator.",
"For instance, while removing the low frequency element $\\mathbf {B}$ from a replay spoof trace, the generator may be forced to rely on other elements such as high-level texture patterns.",
"To synthesize the “harder\" sample $\\hat{\\mathbf {I}}_{\\textit {hard}}$ , we follow Eqn.",
"REF with two minor changes: 1) generate 3 random weights between $[0,1]$ and multiple each with one component of $\\lbrace \\mathbf {B}, \\mathbf {C}, \\mathbf {T}\\rbrace $ ; 2) randomly remove the inpainting process (i.e., set $\\mathbf {P}=0$ ) with a probability of $0.5$ .",
"Compared with other methods, such as brightness and contrast change [57], reflection and blurriness effect [21], or 3D distortion [58], our approach can introduce more realistic and effective data samples, as shown in Sec.",
"."
],
[
"Online 3D Warping Layer",
" We propose an online 3D warping layer to correct the shape discrepancy.",
"To obtain the warping, previous methods in [59], [18] use offline face swapping and pre-computed dense offset respectively, where both methods are non-differentiable as well as memory intensive.",
"In contrast, our warping layer is designed to be both differentiable and computationally efficient, which is necessary for online synthesis during the training.",
"Figure: The online 3D warping layer.",
"(a) Given the corresponding dense offset, we warp the spoof trace and add them to the target live face to create a new spoof.",
"E.g.",
"pixel (x,y)(x,y) with offset (3,5)(3,5) is warped to pixel(x+3,y+5)(x+3,y+5) in the new image.",
"(b) To obtain a dense offsets from the spare offsets of the selected face shape vertices, Delaunay triangulation interpolation is adopted.First, the live reconstruction of a spoof face $\\mathbf {I}^i$ can be expressed as: $G^i = G(\\mathbf {I}^i)[\\mathbf {p}_0],$ where $\\mathbf {p}^0=\\lbrace (0,0),(0,1),...,(255,255)\\rbrace \\in \\mathbb {R}^{256\\times 256\\times 2}$ enumerates pixel locations in $\\mathbf {I}^i$ .",
"To align the spoof traces while synthesizing a new spoof face, a dense offset $\\Delta \\mathbf {p}^{i\\rightarrow j}\\in \\mathbb {R}^{256\\times 256\\times 2}$ is required to indicate the deformation between face $\\mathbf {I}^i$ and face $\\mathbf {I}^j$ .",
"A discrete deformation can be acquired from the distances of the corresponding facial landmarks between two faces.",
"During the data preparation, we use [60] to fit a 3DMM model and extract the 2D locations of $Q$ facial vertices for each face: $\\mathbf {s}=\\lbrace (x_0,y_0),(x_1,y_1),...,(x_N,y_N)\\rbrace \\in \\mathbb {R}^{Q\\times 2}.$ A sparse offset on the corresponding vertices can then be computed two faces as $\\Delta \\mathbf {s}^{i\\rightarrow j} = \\mathbf {s}^j - \\mathbf {s}^i$ .",
"To convert the sparse offset $\\Delta \\mathbf {s}^{i\\rightarrow j}$ to the dense offset $\\Delta \\mathbf {p}^{i\\rightarrow j}$ , we apply a triangulation interpolation: $\\Delta \\mathbf {p}^{i\\rightarrow j}=\\text{Tri}(\\mathbf {p}^0, \\mathbf {s}^i,\\Delta \\mathbf {s}^{i\\rightarrow j}),$ where $\\text{Tri}(\\cdot )$ is the interpolation, $\\mathbf {s}^i$ denotes the vertex locations, $\\Delta \\mathbf {s}^{i\\rightarrow j}$ are the vertex values, and we adopt Delaunay triangulation.",
"The warping operation can be denoted as: $G^{-i\\rightarrow j} = G^-(\\mathbf {I}^j|\\mathbf {I}^i)[\\mathbf {p}^0+\\Delta \\mathbf {p}^{i\\rightarrow j}],$ where the offset $\\Delta \\mathbf {p}^{i\\rightarrow j}$ applies to all subject $i$ related elements $\\lbrace \\mathbf {T}_A^i, \\mathbf {I}^i, \\mathbf {P}^i\\rbrace $ .",
"Since the offset $\\Delta \\mathbf {p}^{i\\rightarrow j}$ is typically composed of fractional numbers, we implement the bilinear interpolation to sample the fractional pixel locations.",
"We select $Q=140$ vertices to cover the face region so that they can represent non-rigid deformation, due to pose and expression.",
"As the pixel values in the warped face are a linear combination of pixel values of the triangulation vertices, this entire process is differentiable.",
"This process is illustrated in Fig.",
"REF .",
"[t] Input: live faces $\\mathbf {I}_{\\textit {live}}$ and facial landmarks $\\mathbf {s}_{\\textit {live}}$ , spoof faces $\\mathbf {I}_{\\textit {spoof}}$ and facial landmarks $\\mathbf {s}_{\\textit {spoof}}$ , ground truth depth map $\\mathbf {M}_0$ , preliminary mask $\\mathbf {P}_0$ ; Output: reconstructed live $\\hat{\\mathbf {I}}_{\\textit {live}}$ , synthesized spoof $\\hat{\\mathbf {I}}_{\\textit {spoof}}$ , spoof traces $\\lbrace \\mathbf {T}_A^{\\textit {l}}$ , $\\mathbf {P}^{\\textit {l}}$ , $\\mathbf {I}_P^{\\textit {l}}, \\mathbf {T}_A^{\\textit {s}}$ , $\\mathbf {P}^{\\textit {s}}$ , $\\mathbf {I}_P^{\\textit {s}}\\rbrace $ , depth maps $\\lbrace \\mathbf {M}^{\\textit {l}}$ , $\\mathbf {M}^{\\textit {s}}\\rbrace $ ; iteration $<$ max_iteration training step 1 $\\textbf {1}$ : compute $\\mathbf {T}_A^{\\textit {l}}$ , $\\mathbf {P}^{\\textit {l}}$ , $\\mathbf {I}_P^{\\textit {l}}$ $\\leftarrow $ $G(\\mathbf {I}_{\\textit {live}})$ and compute $\\mathbf {T}_A^{\\textit {s}}$ , $\\mathbf {P}^{\\textit {s}}$ , $\\mathbf {I}_P^{\\textit {s}}$ $\\leftarrow $ $G(\\mathbf {I}_{\\textit {spoof}})$ $\\textbf {2}$ : estimate the depth map $\\mathbf {M}^{\\textit {l}}$ , $\\mathbf {M}^{\\textit {s}}$ $\\textbf {3}$ : compute losses $L_{\\textit {depth}}$ , $L_P$ , $L_R$ training step 2 $\\textbf {4}$ : compute $\\hat{\\mathbf {I}}_{\\textit {low}}$ , $\\hat{\\mathbf {I}}_{\\textit {mid}}$ , $\\hat{\\mathbf {I}}_{\\textit {hi}}$ from $\\mathbf {T}_A^s$ , $\\mathbf {P}^s$ , $\\mathbf {I}_P^s$ and $\\mathbf {I}_{\\textit {spoof}}$ (Eqn.",
"REF ) $\\textbf {5}$ : compute warping offset $\\Delta \\mathbf {p}^{s\\rightarrow l}$ from $\\mathbf {s}_{\\textit {live}}$ , $\\mathbf {s}_{\\textit {spoof}}$ (Eqn.",
"REF ) $\\textbf {6}$ : compute $\\hat{\\mathbf {I}}_{\\textit {spoof}}$ from warped $\\mathbf {T}_A^{s\\rightarrow l}$ , $\\mathbf {P}^{s\\rightarrow l}$ and $\\mathbf {I}_{\\textit {live}}$ (Eqn.",
"REF ) $\\textbf {7}$ : send $\\mathbf {I}_{\\textit {live}}$ , $\\mathbf {I}_{\\textit {spoof}}$ , $\\hat{\\mathbf {I}}_{\\textit {low}}$ , $\\hat{\\mathbf {I}}_{\\textit {mid}}$ , $\\hat{\\mathbf {I}}_{\\textit {hi}}$ , $\\hat{\\mathbf {I}}_{\\textit {spoof}}$ to discriminators $\\textbf {8}$ : compute the adversarial loss for generator $L_{\\textit {G}}$ and for discriminators $L_{\\textit {D}}$ training step 3 $\\textbf {9}$ : create harder samples $\\mathbf {I}_{\\textit {hard}}$ from $\\mathbf {T}_A^{s\\rightarrow l}$ , $\\mathbf {P}^{s\\rightarrow l}$ and $\\mathbf {I}_{\\textit {live}}$ with random perturbation on traces $\\textbf {10}$ : compute $\\mathbf {T}_A^h$ , $\\mathbf {P}^h$ , $\\mathbf {I}_P^h$ $\\leftarrow $ $G(\\mathbf {I}_{\\textit {hard}})$ $\\textbf {11}$ : compute depth map $\\mathbf {M}^h$ for $\\mathbf {I}_{\\textit {hard}}$ $\\textbf {12}$ : compute losses $L_S$ , $L_H$ back propagation $\\textbf {13}$ : back-propagate the losses from step $3,8,12$ to corresponding parts and update the network; PhySTD Training Iteration."
],
[
"Multi-scale Discriminators",
" Motivated by [61], we adopt multiple discriminators at different resolutions (e.g., 32, 96, and 256) in our GAN architecture.",
"We follow the design of PatchGAN [62], which essentially is a fully convolutional network.",
"Fully convolutional networks are shown to be effective to not only synthesize high-quality images [62], [61], but also tackle face anti-spoofing problems [18].",
"For each discriminator, we adopt the same structure but do not share the weights.",
"As shown in Fig.",
"REF , we use in total 4 discriminators in our work: $D_1$ , working in the lowest resolution of 32, focuses on low frequency elements since the higher-frequency traces are erased by downsampling.",
"$D_2$ , working at the resolution of 96, focuses on the middle level content pattern.",
"$D_3$ and $D_4$ , working on the highest resolution of 256, focus on the fine texture details.",
"Our preliminary version resizes real and synthetic samples $\\lbrace \\mathbf {I}, \\hat{\\mathbf {I}}\\rbrace $ to different resolutions and assign to each discriminator.",
"To remove semantic ambiguity and provide correspondence to the trace components, we instead assign the hierarchical reconstruction from Eqn.",
"REF to the discriminators: we send low frequency pairs $\\lbrace \\mathbf {I}_{\\textit {live}}, \\hat{\\mathbf {I}}_{\\textit {low}}\\rbrace $ to $D_1$ , middle frequency pairs $\\lbrace \\mathbf {I}_{\\textit {live}}, \\hat{\\mathbf {I}}_{\\textit {mid}}\\rbrace $ to $D_2$ , high frequency pairs $\\lbrace \\mathbf {I}_{\\textit {live}}, \\hat{\\mathbf {I}}_{\\textit {hi}}\\rbrace $ to $D_3$ , and real/synthetic spoof $\\lbrace \\mathbf {I}_{\\textit {spoof}}, \\hat{\\mathbf {I}}_{\\textit {spoof}}\\rbrace $ to $D_4$ .",
"Each discriminator outputs a 1-channel map in the range of $[0,1]$ , where 0 denotes fake and 1 denotes real."
],
[
"Loss Functions and Training Steps",
" We utilize multiple loss functions to supervise the learning of depth maps and spoof traces.",
"Each training iteration consists of three training steps.",
"We first introduce the loss function, followed by how they are used in the training steps.",
"Depth map loss: We follow the auxiliary FAS [18] to estimate an auxiliary depth map $\\mathbf {M}$ , where the depth ground truth $\\mathbf {M}_0$ for a live face contains face-like shape and the depth for spoof should be zero.",
"We apply the $\\mathcal {L}$ -1 norm on this loss as: $L_{\\textit {depth}} = \\frac{1}{K^2}\\mathbb {E}_{i\\sim \\mathcal {L}\\cup \\mathcal {S}}{\\Vert }\\mathbf {M}^i-\\mathbf {M}^i_0{\\Vert }_F, $ where $K\\!=\\!32$ is the size of $\\mathbf {M}$ .",
"We apply the dense face alignment [60] to estimate the 3D shape and render the depth ground truth $\\mathbf {M}_0$ .",
"Figure: Preliminary mask 𝐏 0 \\mathbf {P}_0 for the negative term in inpainting mask loss.",
"White pixels denote 1 and black pixels denote 0.",
"White indicates the area should not be inpainted.",
"𝐏 0 \\mathbf {P}_0 for: (a) print, replay; (b) 3D mask and makeup; (c) partial attacks that cover the eye portion; (d) partial attacks that cover the mouth portion.Adversarial loss for $G$ : We employ the LSGANs [63] on reconstructed live faces and synthesized spoof faces.",
"It encourages the reconstructed live to look similar to real live from domain $\\mathcal {L}$ , and the synthesized spoof faces to look similar to faces from domain $\\mathcal {S}$ : $\\begin{aligned}L_{G} = \\mathbb {E}_{i\\sim \\mathcal {L}, j\\sim \\mathcal {S}}\\Big [{\\Vert }D_1(\\hat{\\mathbf {I}}_{\\textit {low}}^j)\\!- \\!\\mathbf {1}{\\Vert }_F^2 + {\\Vert }D_2(\\hat{\\mathbf {I}}_{\\textit {mid}}^j)\\!- \\!\\mathbf {1}{\\Vert }_F^2 + \\\\{\\Vert }D_3(\\hat{\\mathbf {I}}_{\\textit {hi}}^j)\\!- \\!\\mathbf {1}{\\Vert }_F^2 +{\\Vert }D_4(\\hat{\\mathbf {I}}_{\\textit {spoof}}^{j\\rightarrow i})\\!",
"-\\mathbf {1}{\\Vert }_F^2\\Big ].\\end{aligned}$ Adversarial loss for $D$ : The adversarial loss for discriminators encourages $D(\\cdot )$ to distinguish between real live vs. reconstructed live, and real spoof vs. synthesized spoof: $\\begin{aligned}L_{D} = \\mathbb {E}_{i\\sim \\mathcal {L}, j\\sim \\mathcal {S}}\\Big [{\\Vert }D_1(\\mathbf {I}^i)\\!- \\!\\mathbf {1}{\\Vert }_F^2 +{\\Vert }D_2(\\mathbf {I}^i)\\!- \\!\\mathbf {1}{\\Vert }_F^2 + \\\\{\\Vert }D_3(\\mathbf {I}^i)\\!- \\!\\mathbf {1}{\\Vert }_F^2 +{\\Vert }D_4(\\mathbf {I}^j)\\!",
"-\\mathbf {1}{\\Vert }_F^2 +{\\Vert }D_1(\\hat{\\mathbf {I}}_{\\textit {low}}^j){\\Vert }_F^2 + \\\\{\\Vert }D_2(\\hat{\\mathbf {I}}_{\\textit {mid}}^j){\\Vert }_F^2 +{\\Vert }D_3(\\hat{\\mathbf {I}}_{\\textit {hi}}^j){\\Vert }_F^2 + {\\Vert }D_4(\\hat{\\mathbf {I}}_{\\textit {spoof}}^{j\\rightarrow i}){\\Vert }_F^2\\Big ].\\end{aligned}$ Inpainting mask loss: The ground truth inpainting region for all spoof attacks is barely possible to obtain, hence a fully supervised training [64] for inpainting mask is out of the question.",
"However, we may still leverage the prior knowledge of spoof attacks to facilitate the estimation of inpainting masks.",
"The inpainting mask loss consists of a positive term and a negative term.",
"First, the positive term encourages certain region to be inpainted.",
"As the goal of inpainting process is to allow certain region to change without intensity constraint, the region with larger magnitude of additive traces would have a higher probability to be inpainted.",
"Hence, the positive term adopts a $\\mathcal {L}$ -2 norm between the inpainting region $\\mathbf {P}$ and the region where the additive trace is larger than a threshold $\\beta $ .",
"Second, the negative term discourages certain region to be inpainted.",
"While the ground truth inpainting mask is unknown, it's straightforward to mark a large portion of region that should not be inpainted.",
"For instance, the inpainting region for funny eye glasses should not appear in the lower part of a face.",
"Hence, we provide a preliminary mask $\\mathbf {P}_0$ to indicate the not-to-be-inpainted region, and adopt a normalized $\\mathcal {L}$ -2 norm on the masked inpainting region $\\mathbf {P}\\cdot \\mathbf {P}_0$ as the negative term.",
"The preliminary mask $\\mathbf {P}_0$ is illustrated in Fig.",
"REF .",
"Overall, the inpainting mask loss is formed as: $L_{P} = \\mathbb {E}_{\\mathbf {i}\\sim \\mathcal {S}}\\Big [{\\Vert }\\mathbf {P}^i-(\\mathbf {T}_A^i>\\beta ){\\Vert }_F^2 +\\frac{{\\Vert }\\mathbf {P}^i\\cdot \\mathbf {P}_0^i{\\Vert }_F^2 }{{\\Vert }\\mathbf {P}_0^i{\\Vert }_F^2}\\Big ].$ Trace regularization: Based on Eqn.",
"REF with $\\lambda =1$ , we regularize the intensity of additive traces $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ and inpainting region $\\mathbf {P}$ .",
"The regularizer loss is denoted as: $L_{R} = \\mathbb {E}_{\\mathbf {i}\\sim \\mathcal {L}\\cup \\mathcal {S}}\\Big [{\\Vert }\\mathbf {B}{\\Vert }_F^2 +{\\Vert }\\mathbf {C}{\\Vert }_F^2 +{\\Vert }\\mathbf {T}{\\Vert }_F^2 +{\\Vert }\\mathbf {P}{\\Vert }_F^2\\Big ].$ Synthesized spoof loss: Synthesized spoof data come with ground truth spoof traces.",
"As a result, we are able to define a supervised pixel loss for the generator to disentangle the exact spoof traces that were added: $ L_{S} = \\mathbb {E}_{\\mathbf {i}\\sim \\mathcal {L}, \\mathbf {j}\\sim \\mathcal {S}}\\Big [{\\Vert }G(\\lceil G^{-j\\rightarrow i} \\rceil ) -\\lceil G^{j\\rightarrow i}\\rceil {\\Vert }_F^1\\Big ],$ where $G^{j\\rightarrow i}$ is the overall effect of $\\lbrace \\mathbf {P}^j,\\mathbf {I}_P^j,\\mathbf {B}^j,\\mathbf {C}^j,\\mathbf {T}^j\\rbrace $ after warping to subject $i$ , and $\\lceil \\cdot \\rceil $ is the stop_gradient operation.",
"Without stopping the gradient, $G^{j\\rightarrow i}$ may collapse to 0.",
"Depth map loss for “harder” samples: We send the “harder” synthesized spoof data to depth estimation network to improve the data diversity, and hope to increase the FAS model's generalization: $L_{\\textit {H}} = \\frac{1}{K^2}\\mathbb {E}_{i\\sim \\hat{\\mathcal {S}}}\\Big [{\\Vert }\\mathbf {M}^i-\\mathbf {M}_0^i{\\Vert }_F\\Big ],$ where $\\hat{\\mathcal {S}}$ denotes the domain of synthesized spoof faces.",
"Training steps and total loss: Each training iteration has 3 training steps.",
"In the training step 1, live faces $\\mathbf {I}_{\\textit {live}}$ and spoof faces $\\mathbf {I}_{\\textit {spoof}}$ are fed into generator $G(\\cdot )$ to disentangle the spoof traces.",
"The spoof traces are used to reconstruct the live counterpart $\\hat{\\mathbf {I}}_{\\textit {live}}$ and synthesize new spoof $\\hat{\\mathbf {I}}_{\\textit {spoof}}$ .",
"The generator is updated with respect to the depth map loss $L_{\\textit {depth}}$ , adversarial loss $L_G$ , inpainting mask loss $L_P$ , and regularizer loss $L_R$ : $L = \\alpha _1L_{\\textit {depth}} + \\alpha _2 L_{G} + \\alpha _3 L_P + \\alpha _4 L_R.$ In the training step 2, the discriminators are supervised with the adversarial loss $L_D$ to compete with the generator.",
"In the training step 3, $\\mathbf {I}_{\\textit {live}}$ and $\\hat{\\mathbf {I}}_{\\textit {hard}}$ are fed into the generator with the ground truth label and trace to minimize the synthesized spoof loss $L_S$ and depth map loss $L_H$ : $L = \\alpha _5 L_{S} + \\alpha _6L_H,$ where $\\alpha _1$ -$\\alpha _6$ are the weights to balance the multitask training.",
"To note that, we send the original live faces $\\mathbf {I}_{\\textit {live}}$ with $\\hat{\\mathbf {I}}_{\\textit {hard}}$ for a balanced mini-batch, which is important when computing the moving average in the batch normalization layer.",
"We execute all 3 steps in each minibatch iteration, but reduce the learning rate for discriminator step by half.",
"The whole training process is depicted in Alg.",
"REF ."
],
[
"Experiments",
" In this section, we first introduce the experimental setup, and then present the results in the known, unknown, and open-set spoof scenarios, with comparisons to respective baselines.",
"Next, we quantitatively evaluate the spoof traces by performing a spoof medium classification, and conduct an ablation study on each design in the proposed method.",
"Finally, we provide visualization results on the spoof trace disentanglement, new spoof synthesis and t-SNE visualization.",
"Table: The evaluation on four protocols in OULU-NPU.",
"Bold indicates the best score in each protocol."
],
[
"Experimental Setup",
"Databases We conduct experiments on three major databases: Oulu-NPU [67], SiW [18], and SiW-M [19].",
"Oulu-NPU and SiW include print/replay attacks, while SiW-M includes 13 spoof types.",
"We follow all the existing testing protocols and compare with SOTA methods.",
"Similar to most prior works, we only use the face region for training and testing.",
"Evaluation metrics Two common metrics are used in this work for comparison: EER and APCER/BPCER/ACER.",
"EER describes the theoretical performance and predetermines the threshold for making decisions.",
"APCER/BPCER/ACER[68] describe the practical performance given a predetermined threshold.",
"For both evaluation metrics, lower value means better performance.",
"The threshold for APCER/BPCER/ACER is computed from either training set or validation set.",
"In addition, we also report the True Detection Rate (TDR) at a given False Detection Rate (FDR).",
"This metric describes the spoof detection rate at a strict tolerance to live errors, which is widely used to evaluate real-world systems [69].",
"In this work, we report TDR at FDR$=0.5\\%$ .",
"For TDR, the higher the better.",
"Parameter setting PhySTD is implemented in Tensorflow with an initial learning rate of $5e$ -5.",
"We train in total $150,000$ iterations with a batch size of 8, and decrease the learning rate by a ratio of 10 every $45,000$ iterations.",
"We initialize the weights with $[0,0.02]$ normal distribution.",
"$\\lbrace \\alpha _1,\\alpha _2,\\alpha _3,\\alpha _4,\\alpha _5,\\alpha _6\\rbrace $ are set to be $\\lbrace 100,5,1,1e$ -$4,10,1\\rbrace $ , and $\\beta =0.1$ .",
"$\\alpha _0$ is empirically determined from the training or validation set.",
"We use the open-source face alignment [70] and 3DMM fitting [60] to crop the face and provide 140 landmarks.",
"Table: The evaluation on three protocols in SiW Dataset.",
"We compare with the top 7 performances."
],
[
"Anti-Spoofing for Known Spoof Types",
"Oulu-NPU Oulu-NPU[67] is a commonly used face anti-spoofing benchmark due to its high-quality data and challenging testing protocols.",
"Tab.",
"REF shows our anti-spoofing performance on Oulu-NPU, compared with SOTA algorithms.",
"Our method achieves the best overall performance on this database.",
"Compared with our preliminary version [31], we demonstrate improvements in all 4 protocols, with significant improvement on protocol 1 and protocol 3, i.e., reducing the ACER by $63.6\\%$ and $32.1\\%$ respectively.",
"Compared with the SOTA, our approach achieves similar best performances on the first three protocols and outperforms the SOTA on the fourth protocol, which is the most challenging one.",
"To note that, in protocol 3 and protocol 4, the performances of testing camera 6 are much lower than those of cameras 1-5: the ACER for camera 6 are $6.4\\%$ and $10.2\\%$ , while the average ACER for the other cameras are $1.0\\%$ and $2.0\\%$ respectively.",
"Compared with other cameras, we notice that camera 6 has stronger sensor noises and our model recognizes them as unknown spoof traces, which leads to an increased false negative rate (i.e., BPCER).",
"How to separate sensor noises from spoof traces can be an important future research topic.",
"Table: The evaluation and ablation study on SiW-M Protocol I: known spoof detection.Table: The evaluation on SiW-M Protocol II: unknown spoof detection.SiW SiW[18] is another recent high-quality database.",
"It includes fewer capture cameras but more spoof mediums and environment variations, such as pose, illumination, and expression.",
"The comparison on three protocols is shown in Tab.",
"REF .",
"We outperform the previous works on the first two protocols and rank in the middle on protocol 3.",
"Protocol 3 aims to test the performance of unknown spoof detection, where the model is trained on one spoof attack (print or replay) and tested on the other.",
"As we can see from Fig.REF -REF , the traces of print and replay are significantly different, where the replay traces are more on the high-frequency part (i.e., trace component $\\textbf {T}$ ) and the print traces are more on the low-frequency part (i.e., trace component $\\textbf {S}$ ).",
"These pattern divergence leads to the adaption gap of our method while training on one attack and testing on the other.",
"SiW-M SiW-M[19] contains a large diversity of spoof types, including print, replay, 3D mask, makeup, and partial attacks.",
"This allows us to have a comprehensive evaluation of the proposed approach with different spoof attacks.",
"To use SiW-M for known spoof detection, we randomly split the data of all types into train/test set with a ratio of $60\\%$ vs. $40\\%$ , and the results are shown in Tab.",
"REF .",
"Compared to the preliminary version [31], our method outperforms on most spoof types as well as the overall EER performance by $47.9\\%$ relatively, which demonstrates the superiority of our anti-spoofing on known spoof attacks.",
"For experiments on SiW-M (protocol I, II, and III), we additionally report the TPR at FNR equal to $0.5\\%$ .",
"While EER and ACER provide the theoretical evaluation, the users in real-world applications care more about the true spoof detection rate under a given live detection error rate, and hence TPR can better reflect how well the model can detect one or a few spoof attacks in practices.",
"As shown in Tab.",
"REF , we improve the overall TDR of our preliminary version [31] by $29.5\\%$ .",
"Table: The evaluation on SiW-M Protocol III: openset spoof detection.Figure: Examples of each spoof trace components.",
"(a) the input sample faces.",
"(b) 𝐁\\textbf {B}.",
"(c) 𝐂\\textbf {C}.",
"(d) 𝐓\\textbf {T}.",
"(e) 𝐏\\textbf {P}.",
"(f) the final live counterpart reconstruction and zoom-in details.",
"(g) results from .",
"(h) results from Step1+Step2 with a single trace representation."
],
[
"Anti-Spoofing for Unknown and Open-set Spoofs",
" Another important aspect is to test the anti-spoofing performance on unknown spoof.",
"To use SiW-M for unknown spoof detection, The work [19] defines the leave-one-out testing protocols, termed as SiW-M Protocol II.",
"In this protocol, each model (i.e., one column in Tab.",
"REF ) is trained with 12 types of spoof attacks (as known attacks) plus the $80\\%$ of the live faces, and tested on the remaining 1 attack (as unknown attack) plus the $20\\%$ of live faces.",
"As shown in Tab.",
"REF , our PhySTD achieves significant improvement over our preliminary version, with relatively $11.7\\%$ on the overall EER, $19.0\\%$ on the overall ACER, $50.4\\%$ on the overall TPR.",
"Specifically, we reduce the EERs of half mask, paper glasses, transparent mask, replay attack, and partial paper relatively by $47.6\\%$ , $40.4\\%$ , $37.7\\%$ , $31.6\\%$ , $56.3\\%$ , respectively.",
"Overall, compared with the top 7 performances, we outperform the SOTA performance of EER/TPR and achieve comparable ACER.",
"Among all, the detection of silicone mask, paper-crafted mask, mannequin head, impersonation makeup, and partial paper attacks are relatively good, with the detection accuracy (i.e., TPR@FNR=$0.5\\%$ ) above $65\\%$ .",
"Obfuscation makeup is the most challenging one with TPR of $0\\%$ , where we predict all the spoof samples as live.",
"This is due to the fact that the makeup looks very similar to the live faces, while being dissimilar to any other spoof types.",
"However, once we obtain a few samples, our model can quickly recognize the spoof traces on the eyebrow and cheek, synthesize new spoof samples, and successfully detect the attack (TPR=$41.1 \\%$ in Tab.",
"REF ).",
"Figure: Examples of spoof trace disentanglement on SiW (a-h) and SiW-M (i-x).",
"(a)-(d) items are print attacks and (e)-(h) items are replay attacks.",
"(i)-(x) items are live, print, replay, half mask, silicone mask, paper mask, transparent mask, obfuscation makeup, impersonation makeup, cosmetic makeup, paper glasses, partial paper, funny eye glasses, and mannequin head.",
"The first column is the input face, the second column is the overall spoof trace (𝐈-𝐈 ^\\textbf {I}-\\hat{\\textbf {I}}), the third column is the reconstructed live.Moreover, in the real-world scenario, the testing samples can be either a known spoof attack or an unknown one.",
"Thus, we propose SiW-M Protocol III to evaluate this open-set testing situation.",
"In Protocol III, we first follow the train/test split from protocol I, and then further remove one spoof type as the unknown attack.",
"During the testing, we test on the entire unknown spoof samples as well the test split set of the know spoof samples.",
"The results are reported in Tab.",
"REF .",
"Compared to the SOTA face anti-spoofing method [18], our approach substantially outperforms it in all three metrics.",
"Table: Confusion matrices of spoof mediums classification based on spoof traces.",
"The results are compared with the previous method .",
"Green represents improvement over .",
"Red represents performance drop.Table: Confusion matrices of 6-class spoof traces classification on SiW-M database.Figure: Examples of the spoof data synthesis.",
"The first row are the source spoof faces, the first column are the target live faces, and the remaining are the synthesized spoof faces from the live face with the corresponding spoof traces."
],
[
"Spoof Traces Classification",
" To quantitatively evaluate the spoof trace disentanglement, we perform a spoof medium classification on the disentangled spoof traces and report the classification accuracy.",
"The spoof traces should contain spoof medium-specific information, so that they can be used for clustering without seeing the face.",
"To make a fair comparison with [24], we remove the additional spoof type information from the preliminary mask $\\mathbf {P}_0$ .",
"That is, for this specific experiment, we only use the additive traces $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ to learn the trace classification.",
"After $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ finish training with only binary labels, we fix PhySTD and apply a simple CNN (i.e., AlexNet) on the estimated additive traces to do a supervised spoof medium classification.",
"We follow the same 5-class testing protocol in [24] in Oulu-NPU Protocol 1.",
"We report the classification accuracy as the ratio between correctly predicted samples from all classes and all testing samples.",
"Shown in Tab.",
"REF .",
"Our model can achieve a 5-class classification accuracy of $83.3\\%$ .",
"If we treat two print attacks as the same class and two replay as the same class, our model can achieve a 3-class classification accuracy of $92.0\\%$ .",
"Compared with the prior method [24], we show an improvement of $29\\%$ on the 5-class model.",
"In addition, we train the same CNN on the original images instead of the estimated spoof traces for the same spoof medium classification task, and the classification accuracy can only reach $80.6\\%$ .",
"This further demonstrates that the estimated traces do contain significant information to distinguish different spoof mediums.",
"We also execute the spoof traces classification task on more spoof types in SiW-M database.",
"We leverage the train/test split on SiW-M Protocol 1.",
"We first train the PhySTD till convergence, and use the estimated traces from the training set to train the trace classification network.",
"We explore the 6-class scenario, shown in Tab.",
"REF .",
"Our 6-class model can achieve the classification accuracy of $92.0\\%$ .",
"Since the traces are more distinct among different spoof types, this performance is even better than 5-class classification on print/replay scenario in Oulu-NPU Protocol 1.",
"This further demonstrates that PhySTD can estimate spoof traces that contain significant information of spoof mediums and can be applied to multiple spoof types."
],
[
"Ablation Study",
" In this section, we show the importance of each design of our proposed approach on the SiW-M Protocol I, in Tab.REF .",
"Our baseline is the auxiliary FAS [18], without the temporal module.",
"It consists of the backbone encoder and depth estimation network.",
"When including the image decomposition, the baseline becomes the training step 1 in Alg.",
"REF , as the traces are not activated without the training step 2.",
"To validate the effectiveness of GAN training, we report the results from the baseline model with our GAN design, denoted as Step1+Step2.",
"We also provide the control experiment where the traces are represented by a single component to demonstrate the effectiveness of the proposed 5-element trace representation.",
"This model is denoted as Step1+Step2 with single trace.",
"In addition, we evaluate the effect of training with more synthesized data via enabling the training step 3 as Step1+Step2+Step3, which is our final approach.",
"As shown in Tab.",
"REF , the baseline model (Auxiliary) can achieve a decent performance of EER $6.7\\%$ .",
"Adding image decomposition to the baseline (Step 1) can improve the EER from $6.7\\%$ to $4.3\\%$ , but more live samples are predicted with higher scores, causing a worse ACER.",
"Adding simple GAN design (Step1+Step2 with single trace) may lead to a similar EER performance of $5.8\\%$ , but based on the TPR ($59.3\\%\\rightarrow 74.8\\%$ ) its practical performance may be improved.",
"With the proper physics-guided trace disentanglement, we can improve the EER to $2.8\\%$ and TPR to $89.7\\%$ .",
"And our final design can achieve the performance of HTER $2.8\\%$ , EER $2.5\\%$ , and TPR $91.2\\%$ .",
"Compared with our preliminary version, the EER is improved by $47.9\\%$ , HTER is improved by $31.7\\%$ and TPR is improved by $29.5\\%$ .",
"Figure: NO_CAPTIONFigure: NO_CAPTIONSpoof trace components In Fig.REF , we provide illustration of each spoof trace component.",
"Strong color distortion (low-frequency trace) shows up in the print attacks.",
"Moiré patterns in the replay attack are well detected in the high-frequency trace.",
"The local specular highlights in transparent mask are well presented in the low- and mid-frequency components, and the inpainting process further fine-tunes the most highlighted area.",
"For the two glasses attacks, the color discrepancy is corrected in the low-frequency trace, and the sharp edges are corrected in the mid- and high-frequency traces.",
"Each component shows a consistent semantic meaning on different spoof samples, and this successful trace disentanglement can lead to better final visual results.",
"As shown on the right side of Fig.",
"REF , we compare with our preliminary version [31] and the ablated GAN design with a single trace representation.",
"The result of single trace representation shows strong artifacts on most of the live reconstruction.",
"The multi-scale from our preliminary version has already shown a large visual quality improvement, but still have some spoof traces (e.g., glass edges) remained in the live reconstruction.",
"In contrast, our approach can further handle the missing traces and achieve better visualization.",
"Live reconstruction In Fig.",
"REF , we show more examples from different spoof types in SiW and SiW-M databases.",
"The overall trace is the exact difference between the input face and its live reconstruction.",
"For the live faces, the trace is zero, and for the spoof faces, our method removes spoof traces without unnecessary changes, such as identity shift, and make them look like live faces.",
"For example, strong color distortion shows up in print/replay attacks (Fig.",
"REF a-h) and some 3D mask attacks (Fig.",
"REF l-o).",
"For makeup attacks (Fig.",
"REF q-s), the fake eyebrows, lipstick, artificial wax, and cheek shade are clearly detected.",
"The folds and edges (Fig.",
"REF t-w) are well detected and removed in paper-crafted masks, paper glasses, and partial paper attacks.",
"Spoof synthesis Additionally, we show examples of new spoof synthesis using the disentangled spoof traces, which is an important contribution of this work.",
"As shown in Fig.",
"REF , the spoof traces can be precisely transferred to a new face without changing the identity of the target face.",
"Due to the additional inpainting process, spoof attacks such as transparent mask and partial attacks can be better attached to the new live face.",
"Thanks to the proposed 3D warping layer, the geometric discrepancy between the source spoof trace and the target face can be corrected during the synthesis.",
"Especially on the second source spoof, the right part of the traces is successfully transferred to the new live face while the left side remains to be still live.",
"It demonstrates that our trace regularization can suppress unnecessary artifacts generated by the network.",
"Both the live reconstruction results in Fig.",
"REF and the spoof synthesis results in Fig.",
"REF demonstrate that our approach disentangles visually convincing spoof traces that help face anti-spoofing.",
"Spoof trace removing process As shown in Fig.",
"REF , we illustrate the effects of trace components by progressively removing them one by one.",
"For the replay attack, the spoof sample comes with strong over-exposure as well as clear Moiré pattern.",
"Removing the low-frequency trace can effectively correct the over-exposure and color distortion caused by the digital screen.",
"And removing the texture pattern in the high-frequency trace can peel off the high-frequency grid effect and reconstruct the live counterpart.",
"For the makeup attack, since there is no strong color range bias, removing estimated low-frequency trace would mainly remove the lip-stick color and fake eyebrow, but in the meantime bring a few artifacts at the edges.",
"Next, while removing the content pattern, the shadow on the cheek and the fake eyebrows are adequately lightened.",
"Finally, removing the texture pattern would significantly correct the spoof traces from artificial wax, eyeliner, and shadow on the cheek.",
"Similarly, in mask and partial attacks, the reconstruction will be gradually refined as we removing components one by one.",
"t-SNE visualization We use t-SNE [71] to visualize the encoder features $\\mathbf {F}_1$ ,$\\mathbf {F}_2$ ,$\\mathbf {F}_3$ , and the features that produce $\\lbrace \\mathbf {B},\\mathbf {C},\\mathbf {T}\\rbrace $ and $\\lbrace \\mathbf {P},\\mathbf {I}_P\\rbrace $ .",
"The t-SNE is able to project the output of features from different scales and layers to 2D by preserving the KL divergence distance.",
"As shown in Fig.",
"REF , among the three feature scales in the encoder, $F_3$ is the most separable feature space, the next is $F_1$ , and the worst is $F_2$ .",
"The features for additive traces $\\lbrace \\textbf {B},\\textbf {C},\\textbf {T}\\rbrace $ are well-clustered as semantic sub-groups of live, makeup, mask, and partial attacks.",
"As we know the inpainting masks for live samples are close to zero, the feature for inpainting traces $\\lbrace \\textbf {P},\\textbf {I}_P\\rbrace $ shows the inpainting process mostly update the partial attacks, and then some makeup attacks and mask attacks, i.e., the green dots being further away from the black dots means they have greater magnitude.",
"This validates our prior knowledge of the inpainting process.",
"This work proposes a physics-guided spoof traces disentanglement network (PhySTD) to tackle the challenging problem of disentangling spoof traces from the input faces.",
"With the spoof traces, we reconstruct the live faces as well as synthesize new spoofs.",
"To correct the geometric discrepancy in synthesis, we propose a 3D warping layer to deform the traces.",
"The disentanglement not only improves the SOTA of face anti-spoofing in known, unknown, and open-set spoof settings, but also provides visual evidence to support the model's decision."
],
[
"Acknowledgment",
"This research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R$\\&$ D Contract No.",
"2017-17020200004.",
"The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government.",
"The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION"
]
] | 2012.05185 |
[
[
"Direct measurement of ferroelectric polarization in a tunable semimetal"
],
[
"Abstract Ferroelectricity, the electrostatic counterpart to ferromagnetism, has long been thought to be incompatible with metallicity due to screening of electric dipoles and external electric fields by itinerant charges.",
"Recent measurements, however, demonstrated signatures of ferroelectric switching in the electrical conductance of bilayers and trilayers of WTe$_2$, a semimetallic transition metal dichalcogenide with broken inversion symmetry.",
"An especially promising aspect of this system is that the density of electrons and holes can be continuously tuned by an external gate voltage.",
"This degree of freedom enables investigation of the interplay between ferroelectricity and free carriers, a previously unexplored regime.",
"Here, we employ capacitive sensing in dual-gated mesoscopic devices of bilayer WTe$_2$ to directly measure the spontaneous polarization in the metallic state and quantify the effect of free carriers on the polarization in the conduction and valence bands, separately.",
"We compare our results to a low-energy model for the electronic bands and identify the layer-polarized states that contribute to transport and polarization simultaneously.",
"Bilayer WTe$_2$ is thus shown to be a canonical example of a ferroelectric metal and an ideal platform for exploring polar ordering, ferroelectric transitions, and applications in the presence of free carriers."
],
[
"Direct measurement of ferroelectric polarization in a tunable semimetal Sergio C. de la Barrera These authors contributed equally.",
"Qingrui Cao These authors contributed equally.",
"Yang Gao Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Yuan Gao Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China Vineetha S. Bheemarasetty Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Jiaqiang Yan Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA David G. Mandrus Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Wenguang Zhu International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China Di Xiao Benjamin M. Hunt [email protected] Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 Ferroelectricity, the electrostatic counterpart to ferromagnetism, has long been thought to be incompatible with metallicity due to screening of electric dipoles and external electric fields by itinerant charges.",
"Recent measurements, however, demonstrated signatures of ferroelectric switching in the electrical conductance of bilayers and trilayers of WTe2, a semimetallic transition metal dichalcogenide with broken inversion symmetry .",
"An especially promising aspect of this system is that the density of electrons and holes can be continuously tuned by an external gate voltage.",
"This degree of freedom enables investigation of the interplay between ferroelectricity and free carriers, a previously unexplored regime.",
"Here, we employ capacitive sensing in dual-gated mesoscopic devices of bilayer WTe2 to directly measure the spontaneous polarization in the metallic state and quantify the effect of free carriers on the polarization in the conduction and valence bands, separately.",
"We compare our results to a low-energy model for the electronic bands and identify the layer-polarized states that contribute to transport and polarization simultaneously.",
"Bilayer WTe2 is thus shown to be a canonical example of a ferroelectric metal and an ideal platform for exploring polar ordering, ferroelectric transitions, and applications in the presence of free carriers.",
"Polar materials exhibit charge separation in the absence of an applied electric field, an effect of broken inversion symmetry and a unique polar axis in the crystal , .",
"In certain polar systems, the charge polarization can be switched by an external electric field, an effect known as ferroelectricity.",
"In principle, the presence or absence of ferroelectric effects depends only on the crystal class and not the details of the electronic structure.",
"Despite this, nearly all known conventional ferroelectrics are electrically insulating.",
"Since the first theoretical proposals for ferroelectric metals in 1965 , only a handful of experimental claims of ferroelectric-like phases in metallic systems have been reported , , , , , and no clear case for a canonical ferroelectric metal has emerged.",
"Many such claims fail to demonstrate two key signatures of ferroelectric behavior, direct evidence of the polarization and ferroelectric switching, due to bulk screening effects.",
"Here, we focus on the polar, semimetallic van der Waals crystal, T$_\\text{d}$ -WTe2, in the limit of two atomic layers, thin enough to admit an external electric field (Fig.",
"REFd-e).",
"Few-layer crystals of WTe2 have drawn recent interest for exhibiting a wide variety of low-temperature phases , , , , .",
"Recent transport measurements showed that bilayer (2L) and trilayer (3L) WTe2 exhibit intrinsic, switchable electrical polarization in the conducting state , and separately, surfaces of bulk WTe2 crystals display hysteresis in piezoresponse force microscopy .",
"Subsequent first-principles calculations indicated that the net polarization points only in the out-of-plane direction, and that the underlying mechanism results from a subtle interlayer sliding between the layers in two stable configurations , .",
"These findings are exciting given the semimetallic and tunable nature of bilayer WTe2, which enables reaching both electron and hole bands by electrostatic gating in the ferroelectric state.",
"While the hysteretic behavior observed in previous experiments is promising, a direct measurement of the metallic polarization is still missing.",
"Due to methodological limitations it was previously not possible to measure the polarization while varying a pure electric field.",
"More importantly, these limitations also prevented observing the effect of free carriers on the polarization and its dependence on carrier density, a fundamental open question for ferroelectric metals.",
"In this work, we directly measure the charge polarization and electronic compressibility as a function of density for electrons and holes with independent control of the electric field.",
"We study the simplest polar WTe2 system, a bilayer, via capacitive sensing in a dual-gated, planar capacitance device (Fig.",
"REFb).",
"Capacitance measures the electronic compressibility (and thus metallicity) of a 2D system.",
"In a bilayer 2D system the top-gate and bottom-gate capacitances provide a direct measurement of the layer-specific charge distribution , and thus the out-of-plane polarization.",
"Furthermore, the parallel-plate geometry enables this charge sensing with simultaneous and independent control of the vertical electric field and the carrier density in the bilayer by electrostatic gating.",
"Our devices each consist of a bilayer WTe2 crystal encapsulated by two hexagonal boron nitride (hBN) dielectric layers, with metallic top and bottom gates, and contacts integrated into the top hBN layer (Fig REFa).",
"Figure: Parameters employed in 𝐤·𝐩\\mathbf {k \\cdot p} calculation, following Eqs.",
"–.These parameters are obtained by fitting the polarization from the low-energy model to Fig.",
"a, and hence are different from those in Ref.",
"du2018band."
]
] | 2012.05261 |
[
[
"Decomposition of $(2k+1)$-regular graphs containing special spanning\n $2k$-regular Cayley graphs into paths of length $2k+1$"
],
[
"Abstract A $P_\\ell$-decomposition of a graph $G$ is a set of paths with $\\ell$ edges in $G$ that cover the edge set of $G$.",
"Favaron, Genest, and Kouider (2010) conjectured that every $(2k+1)$-regular graph that contains a perfect matching admits a $P_{2k+1}$-decomposition.",
"They also verified this conjecture for $5$-regular graphs without cycles of length $4$.",
"In 2015, Botler, Mota, and Wakabayashi verified this conjecture for $5$-regular graphs without triangles.",
"In this paper, we verify it for $(2k+1)$-regular graphs that contain the $k$th power of a spanning cycle; and for $5$-regular graphs that contain special spanning $4$-regular Cayley graphs."
],
[
"Introduction",
"All graphs in this paper are simple, i.e., have no loops nor multiple edges.",
"A decomposition of a graph $G$ is a set $of edge-disjoint subgraphs of $ G$ that cover its edge set.If every element of $ is isomorphic to a fixed graph $H$ , then we say that $ is an \\emph {$ H$-decomposition}.In this paper, we focus on the case $ H$is the simple path with $ 2k+1$ edges, which we denote by $ P2k+1$.Note that this notation is not standard.In 1957, Kotzig~\\cite {Ko57} (see also~\\cite {BoFo83})proved that a $ 3$-regular graph $ G$admits a $ P3$-decomposition if and only if $ G$ contains a perfect matching.In 2010, Favaron, Genest, and Kouider~\\cite {FaGeKo10} extended this result by provingthat every $ 5$-regular graph thatcontains a perfect matching and no cycles of length $ 4$admits a $ P5$-decomposition;and proposed the following conjecture toextend Kotzig^{\\prime }s result.\\begin{conjecture}[Favaron--Genest--Kouider, 2010]If G is a (2k+1)-regular graph that contains a perfect matching,then G admits a P_{2k+1}-decomposition.\\end{conjecture}$ In 2015, Botler, Mota, and Wakabayashi [3] verified Conjecture for triangle-free 5-regular, and, more recently, Botler, Mota, Oshiro, and Wakabayashi [2] generalized this result for $(2k+1)$ -regular graphs with girth at least $2k$ .",
"It is clear that a 5-regular graph contains a perfect matching if and only if it contains a spanning 4-regular graph.",
"In fact, by using a theorem of Petersen [10], one can prove that a $(2k+1)$ -regular graph contains a perfect matching if and only if it contains a spanning $2k^{\\prime }$ -regular graph for every $k^{\\prime }\\le k$ .",
"Theorem 1 (Petersen, 1891) If $G$ is a $2k$ -regular graph, then $G$ admits a decomposition into spanning 2-regular graphs.",
"In this paper, we explore Conjecture for $(2k+1)$ -regular graphs that contain special spanning $2k$ -regular graphs as follows.",
"Throughout the text, $\\Gamma $ denotes a finite group of order $n$ ; $+$ denotes the group operation of $\\Gamma $ ; and 0 denotes the identity of $\\Gamma $ .",
"As usual, for each $x\\in \\Gamma $ , we denote by $-x$ the inverse of $x$ , i.e., the element $y\\in \\Gamma $ for which $x+y=0$ , and the operation $-$ denotes the default binary operation $(x,y)\\mapsto x+(-y)$ .",
"Let $S\\subseteq \\Gamma $ be a set not containing 0, and such that $-x\\in S$ for every $x\\in S$ (i.e., $S$ is closed under taking inverses).",
"The Cayley graph $X(\\Gamma ,S)$ is the graph $H$ with $V(H)=\\Gamma $ , and $E(H) = \\big \\lbrace xy\\colon y-x\\in S\\big \\rbrace $ (see [7]).",
"In this paper, we allow $S$ to be a set not generating $\\Gamma $ , and hence $X(\\Gamma ,S)$ is not necessarily connected.",
"We say that $H$ is simply commutative if (i) $x + y = y + x$ for every $x,y\\in S$ , and (ii) $-x\\ne x$ for every $x\\in S$ .",
"Condition (ii) implies that $H$ has no multiple edges, and, since $0\\notin S$ , $H$ is simple.",
"It is not hard to check that, in such a graph, the neighborhood of a vertex $v\\in \\Gamma $ is $N(v) = \\lbrace v+x\\colon x\\in S\\rbrace $ .",
"Although the definition of Cayley graphs can be extended to multigraphs and directed graphs, Conjecture considers only simple graphs.",
"In fact, we explore some structure of the colored directed Cayley graph (see [5]) in which the edge set consists of the pairs $(x,x+s)$ with color $s\\in S$ .",
"We present two results regarding Conjecture .",
"We verify it for $(2k+1)$ -regular graphs that contain the $k$ th power of a spanning cycle (see Section ); and for 5-regular graphs that contain spanning simply commutative 4-regular Cayley graphs (see Section ).",
"Since the graphs in these classes may contain cycles of lengths 3 and 4, these results extend the family of graphs for which Conjecture is known to hold.",
"We believe that, due to the underlying group structure, the techniques developed here can be extended for dealing with $(2k+1)$ -regular graphs that contain more general spanning Cayley graphs, and also $(2k+1)$ -regular graphs that contain special spanning Schreier graphs, which could give us significant insight with respect to the general case of Conjecture (see Section ).",
"Notation.",
"A graph $T$ is a trail if there is a sequence $x_0,\\ldots , x_{\\ell }$ of its vertices for which $E(T)=\\lbrace x_{i}x_{i+1}\\colon 0\\le i \\le \\ell -1\\rbrace $ and $x_{i}x_{i+1}\\ne x_{j}x_{j+1}$ , for every $i\\ne j$ .",
"Moreover, if $x_i\\ne x_j$ for every $i \\ne j$ , we say that $T$ is a path.",
"A subgraph $F$ of a graph $G$ is a factor of $G$ or a spanning subgraph of $G$ , if $V(F)=V(G)$ .",
"If, additionally, $F$ is $r$ -regular, then we say that $F$ is an $r$ -factor.",
"In particular, a perfect matching is the edge set of a 1-factor.",
"Moreover, we say that $F$ is an $H$ -factor if $F$ is a factor that consists of vertex-disjoints copies of $H$ .",
"The reader may refer to [1] for standard definitions of graph theory."
],
[
"Regular graphs that contain powers of cycles",
"Given a perfect matching $M$ in a graph $G$ , we say that a $P_{\\ell }$ -decomposition $ of a graph $ G$ is$ M$-\\emph {centered} if for every$ P=a0a1a-1a , we have $a_{i}a_{i+1} \\in M$ for $i=(\\ell -1)/2$ , i.e., if the middle edges of the paths in $ are precisely the edges of $ M$.The next results are examples of $ M$-centered decompositionthat are used in the proof of Theorems~\\ref {theorem:no-g2r2} and~\\ref {theorem:2g2r!=0}.$ Proposition 2 If $G$ is a 5-regular graph that contains a spanning copy $K$ of $K_{4,4}$ , and ${M = E(G)\\setminus E(K)}$ , then $G$ admits an $M$ -centered $P_5$ -decomposition.",
"Let $G$ , $K$ , and $M$ be as in the statement.",
"Let $(R,L)$ be the bipartition of $K$ , where $R=\\lbrace r_1,r_2,r_3,r_4\\rbrace $ and $L=\\lbrace l_1,l_2,l_3,l_4\\rbrace $ .",
"Since $K$ is a complete bipartite graph, if $xy\\in M$ , then either $x,y \\in R$ or $x,y \\in L$ .",
"Thus, we may assume, without loss of generality, that $M=\\lbrace r_1r_2,r_3r_4,l_1l_2,l_3l_4\\rbrace $ , and hence, ${\\lbrace l_1r_1l_3l_4r_2l_2,~l_3r_3l_1l_2r_4l_4,~r_1l_2r_3r_4l_1r_2,~r_3l_4r_1r_2l_3r_4\\rbrace }$ is an $M$ -centered decomposition of $G$ as desired (see Figure REF ).",
"Figure: P 5 P_5-decomposition of a 5-regular graph that contains a spanning copy of a K 4,4 K_{4,4}.Given positive integers $k$ and $n$ , the $k$ th power of the cycle on $n$ vertices, denoted by $C_n^k$ , is the graph on the vertex set $\\lbrace 0,\\ldots ,n-1\\rbrace $ and such that, for every vertex $v$ , we have $x\\in N(v)$ if and only if $x = v+r\\pmod {n}$ , where $r\\in \\lbrace -k,\\ldots ,-1\\rbrace \\cup \\lbrace 1,\\ldots ,k\\rbrace $ .",
"Proposition 3 Let $n$ and $k$ be positive integers for which $k < n/2$ .",
"If $G$ is a simple $(2k+1)$ -regular graph on $n$ vertices that contains a copy $C$ of $C_n^k$ , and $M = E(G) \\setminus E(C)$ , then $G$ admits an $M$ -centered $P_{2k+1}$ -decomposition.",
"Let $G$ , $C$ , and $M$ be as in the statement, and let $V(C) = \\lbrace 0,\\ldots , n-1\\rbrace $ as above.",
"The operations on the vertices of $C$ are taken modulo $n$ .",
"Since $C$ is a $2k$ -regular graph, $M$ is a perfect matching of $G$ .",
"Given $i\\in V(C)$ , let $Q_i$ be the path $v_0v_1\\cdots v_k$ in which $v_0=i$ ; and, for $j=1,\\ldots ,k$ , we have $v_j = v_{j-1}+j$ if $j$ is odd; and $v_j = v_{j-1}-j$ if $j$ is even (see Figure REF ).",
"Note that for every $j=1,\\ldots ,k$ , the path $Q_i$ contains an edge $xy$ such that $|x-y|=j$ .",
"Also, we have $V(Q_i) = \\big \\lbrace i+r\\colon r\\in \\lbrace -\\lfloor k/2\\rfloor ,-\\lfloor k/2\\rfloor +1,\\ldots , \\lceil k/2\\rceil \\rbrace \\big \\rbrace $ .",
"It is not hard to check that the set $\\mathcal {Q}=\\big \\lbrace Q_i\\colon i\\in V(C)\\big \\rbrace $ is a $P_k$ -decomposition of $C$ .",
"Given an edge $e=ij\\in M$ , let $P_e = Q_i \\cup \\lbrace ij\\rbrace \\cup Q_j$ .",
"Since $Q_i$ and $Q_j$ have, respectively, $i$ and $j$ as end vertices, and $E(Q_i)\\cap E(Q_j)=\\emptyset $ , the graph $P_e$ is a trail of length $2k+1$ .",
"Thus, since $\\mathcal {Q}$ is a $P_k$ -decomposition of $C$ , and $M$ is a perfect matching of $G$ , the set $\\lbrace P_e\\colon e\\in M\\rbrace $ is a decomposition of $G$ into trails of length $2k+1$ .",
"We claim that, in fact, $is a $ P2k+1$-decomposition of $ G$.For that, we prove that if $ ijM$,then $ V(Qi)V(Qj)=$.Indeed, note that for every $ e=ijM$,we have $ |i-j|>k$, otherwise we have $ ijE(C)$.Now, suppose that there is a vertex $ v$ in $ V(Qi)V(Qj)$.Then, there are $ r1,r2$ with$ -k/2r1,r2k/2$,and such that $ i+r1 = v = j+r2$.Suppose, without loss of generality, that $ i>j$.Then, we have $ r2-r1 = i-j > k$,but $ r2-r1 k/2+k/2= k$,a contradiction.$ Figure: The path P uv P_{uv}, with u=2u=2 and v=8,v=8, in the proof of Proposition for a 7-regular graph that contains a spanning copy of a C 10 3 C^3_{10}.Note that, from the proof of Proposition REF we also obtain a construction for the Hamilton path decomposition of complete graphs of even order.",
"Corollary 3.1 Let $\\ell $ be odd.",
"The complete graph $K_{\\ell +1}$ admits a $P_\\ell $ -decomposition.",
"A slight variation of the proof of Proposition REF also provides the following result.",
"Proposition 4 Let $\\ell $ be odd, and let $G$ be an $\\ell $ -regular graph with a perfect matching $M$ .",
"If each component of $G\\setminus M$ is the $\\frac{\\ell -1}{2}$ -th power of a cycle, then $G$ admits an $M$ -centered $P_\\ell $ -decomposition.",
"Let $G_1$ and $G_2$ be disjoint graphs with perfect matchings $M_1$ and $M_2$ , respectively.",
"Let $a_1b_1,\\ldots ,a_kb_k \\in M_1$ and $x_1y_1,\\ldots , x_ky_k\\in M_2$ be distinct edges, and let $G$ be the graph obtained from the disjoint union $G_1\\cup G_2$ by removing $a_1b_1,\\ldots ,a_kb_k,x_1y_1,\\ldots , x_ky_k$ and adding the edges $a_1x_1,b_1y_1,\\ldots ,a_kx_k,b_ky_k$ .",
"We say that $G$ is a collage of $G_1$ and $G_2$ over edges of $M_1$ and $M_2$ , and denote by $M_G$ the perfect matching ${\\big (M_1\\cup M_2 \\cup \\lbrace a_1x_1,b_1y_1,\\ldots ,a_kx_k,b_ky_k\\rbrace \\big ) \\setminus \\lbrace a_1b_1,\\ldots ,a_kb_k,x_1y_1,\\ldots , x_ky_k\\rbrace }$ .",
"When $M_1$ and $M_2$ are clear from the context, we say simply that $G$ is a collage of $G_1$ and $G_2$ .",
"Note that $G$ is $\\ell $ -regular if and only if $G_1$ and $G_2$ are $\\ell $ -regular.",
"Let $G$ be an $\\ell $ -regular graph, where $\\ell $ is an odd positive integer, and let $M$ be a perfect matching of $G$ .",
"We say that $G$ is $M$ -constructable if either $G$ admits an $M$ -centered $P_\\ell $ -decomposition, or $G$ is the collage of an $M_1$ -constructable graph and an $M_2$ -constructable graph over edges of $M_1$ and $M_2$ and $M=M_G$ .",
"The next straightforward result is a useful tool in the proof of Theorem REF .",
"Lemma 5 Let $\\ell $ be odd, and $G$ be an $\\ell $ -regular graph.",
"If $G$ is $M$ -constructable, then $G$ admits an $M$ -centered $P_\\ell $ -decomposition.",
"The proof follows by induction on $|V(G)|$ .",
"By the definition of $M$ -constructable, we may assume that $G$ is the collage of an $M_1$ -constructable graph $G_1$ and an $M_2$ -constructable graph $G_2$ over edges of $M_1$ and $M_2$ .",
"By the induction hypothesis, $G_i$ admits an $M_i$ -centered $P_\\ell $ -decomposition $i$ , for $i=1,2$ .",
"Let $a_i,b_i,x_i,y_i$ , for $i=1,\\ldots ,k$ be such that $G$ is the graph obtained from $G_1\\cup G_2$ by removing $a_1b_1,\\ldots ,a_kb_k,x_1y_1,\\ldots , x_ky_k$ and adding $a_1x_1,b_1y_1,\\ldots ,a_kx_k,b_ky_k$ as above.",
"For ${i=1,\\ldots ,k}$ , let $P_i\\in 1$ and $Q_i\\in 2$ be the paths containing the edges $a_ib_i$ and $x_iy_i$ , respectively.",
"By the definition of $M_1$ - and $M_2$ -centered $P_\\ell $ -decomposition, for $i=1,\\ldots ,k$ , we may write $P_i = P_{i,1}a_ib_i P_{i,2}$ and $Q_i = Q_{i,1}x_iy_iQ_{i,2}$ , where $P_{i,1},P_{i,2},Q_{i,1}$ and $Q_{i,2}$ are paths of length $(\\ell -1)/2$ .",
"Since $G_1$ and $G_2$ are disjoint, $V(P_{i,j})\\cap V(Q_{i,j}) = \\emptyset $ for $i=1,\\ldots ,k$ and $j=1,2$ .",
"Let $R_{i,1} = P_{i,1}\\cup \\lbrace a_ix_i\\rbrace \\cup Q_{i,1}$ and $R_{i,2} = P_{i,2}\\cup \\lbrace b_iy_i\\rbrace \\cup Q_{i,2}$ , and note that $ \\big (1\\setminus \\lbrace P_1,\\ldots ,P_k\\rbrace \\big )\\cup \\big (2\\setminus \\lbrace Q_1,\\ldots ,Q_k\\rbrace \\big )\\cup \\lbrace R_{i,j}\\colon i=1,\\ldots ,k\\text{ and }j=1,2\\rbrace $ is an $M_G$ -centered $P_\\ell $ -decomposition of $G$ as desired.",
"By Proposition REF , if $G$ contains a spanning copy $K$ of $K_{4,4}$ and $M=E(G)\\setminus E(K)$ , then $G$ is $M$ -constructable.",
"Therefore, Lemma REF yields the following result.",
"Corollary 5.1 If $G$ is a 5-regular graph that contains a $K_{4,4}$ -factor $K$ and ${M=E(G)\\setminus E(K)}$ , then $G$ admits an $M$ -centered $P_5$ -decomposition."
],
[
"5-regular graphs that contain Cayley graphs",
"In this section, we explore 5-regular graphs that contain spanning simply commutative 4-regular Cayley graphs.",
"Botler, Mota, and Wakabayashi [3] showed that every triangle-free 5-regular graph $G$ that has a perfect matching admits a $P_5$ -decomposition.",
"For that, they applied the following strategy: i) to find an initial decomposition of $G$ into paths and trails; and ii) to perform exchanges of edges between the elements of $, preserving a special invariant, while minimizing the number of trails that are not paths.$ The proof of our main result (Theorem REF ) consists of four steps.",
"First, we deal with a somehow degenerate case (Theorem REF ).",
"After that, we follow the framework used by Botler, Mota, Wakabayashi [3], i.e, from the structure of Cayley graphs, we find an initial decomposition into trails, not necessarily paths (Proposition REF ), and then we exchange edges between the elements of the decomposition in order to reduce the number of bad elements (the trails that are not paths).",
"For that, we first show that the bad elements are distributed in a circular fashion (Lemma REF ), and then we show how to deal with these “cycles of bad elements” (Theorem REF ).",
"The invariants preserved by the operations in the proofs of Lemma REF and Theorem REF are presented, respectively, in Definitions REF and REF .",
"The following lemma is used often throughout the text.",
"Lemma 6 Let $\\ell $ be odd, and $G$ be an $\\ell $ -regular graph.",
"If $ is a decomposition of~$ G$ into trails of length $$,then each vertex of $ G$ is the end vertex of precisely one element of~$ .",
"Let $k$ , $G$ and $ be as in the statement.Let $ n=|V(G)|$.Given an element $ T, we denote by $o(T)$ the number of vertices $v$ in $T$ for which $d_T(v)$ is odd, and given a vertex $v\\in V(G)$ , we denote by $v)$ the number of trails in $ for which $ dT(v)$ is odd.Clearly, $ T o(T) = vV(G)v)$.Moreover, for every trail $ T$, we have $ o(T)2$.Also, since every element of $ has $\\ell $ edges, we have $| = \\frac{1}{\\ell }|E(G)|= \\frac{1}{\\ell }\\frac{1}{2}\\ell n = \\frac{1}{2}n$ .",
"Thus, we have $\\sum _{T\\in o(T) \\le 2| = n.Now, since v\\in V(G) has odd degree (in G),v must have odd degree in at least one element of ,and hence v) \\ge 1.Thus, we have \\sum _{v\\in V(G)}v)\\ge n,and hence n \\le \\sum _{v\\in V(G)}D(v) = \\sum _{T\\in o(T) \\le n.This implies that v) = 1 for every v\\in V(G), as desired.",
"}}Recall that $$ is a finite group of order $ n$ and operation $ +$.Fix two elements $ g,r$ of~$$,we say that $ {g,r}$ is a \\emph {simple commutative generator} (SCG)if (a) $ 0{g, r, 2g, 2r}$;(b) $ g{r,-r}$;and (c) $ g+r = r+g$.Let $ {g,r}$ be an SCG, put $ S = {g, -g, r, -r}$, and consider the Cayley graph $ X=X(,S)$.By construction, $ X$ is a simply commutative Cayley graph (see Section~\\ref {sec:introduction}).Conditions (a) and (b) guarantee that $ X$ is a simple graph,while condition (c) introduces the main restriction explored in this paper.In this case, we say that~$ X$ is the graph \\emph {generated} by $ {g,r}$,and that $ {g,r}$ is the \\emph {generator} of $ X$.Finally, we say that a simple $ 5$-regular graph $ G$ with vertex set $$is a \\emph {$ {g,r}$-graph}if $ G$ contains a spanning Cayley graph $ X$ generated by $ {g,r}$.We say that $ G$ is a \\emph {simply commutative generated graph} or, for short, \\emph {SCG-graph} if~$ G$is a $ {g,r}$-graph for some SCG $ {g,r}$.In this section, we verify Conjecture~\\ref {conj:favaron} for SCG-graphs.",
"In particular,Proposition~\\ref {lemma:cycle-factorization} implies that every $ {g, r}$-graph for which $ 2g = r$ admits an $ Mg,r$-centered decomposition;and as a consequence of Corollary~\\ref {corollary:K44}, we obtain the following result,which is also a special case of our main result.$ Theorem 7 Every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r=0$ and $2g-2r=0$ admits an $M_{g,r}$ -centered decomposition.",
"Let $G$ be a $\\lbrace g,r\\rbrace $ -graph for which $2g+2r=0$ and $2g-2r=0$ and put ${M=M_{g,r}}$ .",
"Note that we also have $4g=4r=0$ .",
"Let $u$ be a vertex of $G$ , and let $H$ be the component of $G\\setminus E(M)$ that contains $u$ .",
"In what follows, we prove that $H$ is a copy of $K_{4,4}$ .",
"Since $g$ and $r$ commute, if $v\\in V(H)$ , we have $v = u + ig + jr$ , where $i,j\\in \\mathbb {N}$ .",
"Since $4g=4r=0$ , we may assume $i,j\\in \\lbrace 0,1,2,3\\rbrace $ .",
"Moreover, since $2g - 2r = 0$ (and hence $2g=2r$ ), we may assume $j\\in \\lbrace 0,1\\rbrace $ .",
"Therefore, there are at most eight vertices in $H$ , namely, $V(H) = \\lbrace u, u+g,u+2g,u+3g, u+r,u+2g+r,u+3g+r\\rbrace $ .",
"We claim that $H$ is bipartite.",
"Indeed, suppose that there is an odd cycle $C$ in $H$ .",
"Then, there is an element $x \\in V(C)$ such that $x+ig+jr=x$ , where $i,j \\in \\mathbb {N}$ .",
"Note that $i+j$ can be obtained from the length of $C$ by ignoring pairs of edges with the same color and different directions.",
"Since $C$ is odd, precisely one between $i$ and $j$ is odd.",
"Suppose, without loss of generality, that $i$ is odd, and hence $j$ is even.",
"Note that, since $2g=2r$ , we have $jr = jg$ .",
"Therefore, $(i+j)g = ig + jr = 0$ .",
"Let $s\\in \\lbrace 1,3\\rbrace $ be such that $i+j = 4q + s$ for some $q\\in \\mathbb {N}$ .",
"Then we have $0 = (i+j)g = 4qg + sg$ , which implies $sg = 0$ .",
"Thus, if $s = 1$ , then $g=0$ ; and if $s = 3$ , then $g = 4g-sg = 0$ , a contradiction to the definition of SCG.",
"Thus, since $H$ is 4-regular, $H$ is a copy of $K_{4,4}$ .",
"Now, since every component of $G\\setminus E(M)$ is isomorphic to $K_{4,4}$ .",
"Therefore, $G$ is a 5-regular graph that contains a $K_{4,4}$ -factor, and hence by Corollary REF , $G$ admits an $M$ -centered decomposition as desired.",
"If $X$ is the graph generated by an SCG $\\lbrace g,r\\rbrace $ , and $x\\in \\lbrace g,r\\rbrace $ , then we denote by $F_x$ the 2-factor of $X$ with edge set $E(F_x) = \\lbrace v+x\\colon v\\in \\Gamma \\rbrace $ .",
"If $G$ is a $\\lbrace g,r\\rbrace $ -graph, then we denote by $M_{g,r}$ the perfect matching $G\\setminus E(F_g \\cup F_r)$ , and the triple $\\lbrace M_{g,r},F_g,F_r\\rbrace $ is called the base factorization of $G$ .",
"Although $G$ is a simple graph, for ease of notation, we refer to an edge $uv\\in F_x$ , with $x\\in \\lbrace g,r\\rbrace $ , as a green (resp.",
"red) out edge of $u$ and in edge of $v$ if $v = u+x$ and $x=g$ (resp.",
"$x=r$ ).",
"In the figures throughout the text, the edges in $F_g$ , $F_r$ , $M_{g,r}$ are illustrated, respectively, in dotted green, dashed red, and double black patterns, while edges without specific affiliation are illustrated in straight gray pattern.",
"Moreover, if such an edge has a specific direction (i.e., in edge or out edge), it is illustrated accordingly.",
"Note that each vertex $u$ has precisely one edge of each type (green in edge, green out edge, red in edge, red out edge), and is incident to precisely one edge of $M_{g,r}$ .",
"In particular, the group structure overcomes Theorem REF by giving a decomposition of $X$ into 2-factors in terms of the elements $g$ and $r$ .",
"In the rest of the paper we deal with the case $2g+2r\\ne 0$ .",
"For that, we characterize the elements of the forthcoming decompositions.",
"Definition 1 We say that a trail $T$ in a $\\lbrace g,r\\rbrace $ -graph is of type A, B, C, or D if $T$ can be written as $a_0a_1a_2a_3a_4a_5$ , where $a_0,a_1,a_2,a_3,a_4$ are distinct vertices, as follows.",
"$a_2 = a_5$ , $a_2a_3\\in M_{g,r}$ , $a_2a_1,a_3a_4\\in F_g$ , $a_4a_5 \\in F_r$ , and $a_1a_0\\in F_g\\cup F_r\\cup M_{g,r}$ , i.e., $a_1a_0$ is an out edge of $a_1$ , or $a_1a_0\\in M_{g,r}$ (see Figure REF fig:typeA).",
"In this case, we say that $a_3$ is the primary connection vertex of $T$ , $a_2$ is the secondary connection vertex of $T$ ; $a_1$ is the auxiliary vertex of $T$ ; and $a_4$ is the tricky vertex of $T$ .",
"We denote these vertices, respectively, by ${\\rm {cv}}_1(T)$ , ${\\rm {cv}}_2(T)$ , ${\\rm {aux}}(T)$ , and ${\\rm {tr}}(T)$ ; $a_5\\notin \\lbrace a_0,a_1,a_2,a_3,a_4\\rbrace $ , $a_2a_3\\in M_{g,r}$ $a_2a_1,a_3a_4\\in F_g$ , $a_1a_0,a_4a_5\\in F_g\\cup F_r\\cup M_{g,r}$ (see Figure REF fig:typeB); $a_5\\notin \\lbrace a_0,a_1,a_2,a_3,a_4\\rbrace $ , $a_2a_1,a_4a_3\\in F_g$ , $a_3a_2,a_4a_5\\in F_r$ , $a_1a_0\\in F_g\\cup F_r\\cup M_{g,r}$ , and, moreover, we have $a_2a_4\\in E(G)$ and $a_2a_4 \\in M_{g,r}$ (see Figure REF fig:typeC); $a_5\\notin \\lbrace a_0,a_1,a_2,a_3,a_4\\rbrace $ , $a_1a_0,a_4a_5\\in F_r$ , $a_1a_2,a_3a_4\\in M_{g,r}$ , and $a_3a_2\\in F_g$ (see Figure REF fig:typeD).",
"Figure: The main types of trails.We remark that elements of type A are not paths, while elements of type B, C, and D are paths.",
"Moreover, the connection vertices are defined only for elements of type A, and the connection vertices of an element $T$ are always incident to an edge of $M_{g,r}$ in $T$ , and hence, no vertex of a $\\lbrace g,r\\rbrace $ -graph is a connection vertex of two edge-disjoint elements of type A in a graph.",
"Given a trail (not necessarily a path) $T= a_0a_1a_2a_3a_4a_5$ in a decomposition $ of a $ {g,r}$-graph $ G$,we say that the edge $ a1a0$ (resp.",
"$ a4a5$) is a \\emph {hanging edge}at $ a1$ (resp.",
"$ a4$) if $ a1a0Mg,rFgFr$ (resp.",
"$ a4a5Mg,rFgFr$),i.e., the hanging edges of $ T$ are the end edges of $ T$ that are in $ Mg,r$ or that are in edges of its end vertices.By Definition~\\ref {def:types}, all end edges of elements of type~A, B, C, or D are hanging edges.Note that if $ T$ is an element of type~A where $ a5 = a2$, then $ a1a0$, $ a2a3$ and $ a4a2$ are hanging edgesof $ T$ at, respectively, $ a1$, $ a3$, and $ a4$.Given a trail decomposition $ of a graph $G$ and a vertex $u\\in V(G)$ , we denote by ${\\rm {hang}}_u)$ the number of edges of $G$ that are hanging edges at $u$ .",
"The next lemma presents a consequence of the exchange of hanging edges at primary connection vertices.",
"Lemma 8 If $T= a_0a_1a_2a_3a_4a_5$ is an element of type A in a decomposition of a $\\lbrace g,r\\rbrace $ -graph $G$ into trails of length 5, where $a_5=a_2$ and $a_2a_3\\in M_{g,r}$ , and $u\\in V(G)$ is such that $a_3u$ is a hanging edge at $a_3 = {\\rm {cv}}_1(T)$ , then $T^{\\prime }= a_0a_1a_2a_4a_3u$ is of type C. Let $T$ , $u$ , and $T^{\\prime }$ be as in the statement.",
"Since $a_3a_4$ is a green out edge of $a_3$ and $a_2a_3$ is an edge of $M_{g,r}$ incident to $a_3$ , we conclude that $a_3u$ is a red out edge of $a_3$ , and hence $u = a_3 + r$ .",
"Now, since $G$ is simple, we have $u\\notin \\lbrace a_2,a_3,a_4\\rbrace $ ; if $u=a_1$ , then we have $a_3 + r= u = a_1 = a_3 + g + r + g$ , which implies $2g=0$ , a contradiction to the definition of SCG.",
"Finally, by Lemma REF we have $u\\ne a_0$ .",
"Thus, $T^{\\prime }$ is a path.",
"Since $a_3u\\in F_r$ , $T^{\\prime }$ is of type C. Figure: Exchange of edges performed in the proof of Lemma .The following lemma shows how two elements of type A may be connected.",
"Lemma 9 If $T_1$ and $T_2$ are two edge-disjoint elements of type A in a $\\lbrace g,r\\rbrace $ -graph $G$ for which ${\\rm {tr}}(T_2) = {\\rm {cv}}_1(T_1)$ , then ${\\rm {aux}}(T_2) = {\\rm {cv}}_2(T_1)$ .",
"Let $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ , where $a_5 = a_2$ and $b_5 = b_2$ and $a_2a_3,b_2b_3\\in M_{g,r}$ .",
"If ${\\rm {cv}}_1(T_1)={\\rm {tr}}(T_2)$ , then $a_3=b_4$ .",
"Since $b_1 = b_4 + r+g$ and $a_2 = a_3 + g + r$ .",
"Thus, ${\\rm {aux}}(T_2) = b_1 = b_4 + r + g = a_3 + r + g = a_2 = {\\rm {cv}}_2(T_1)$ , as desired (see Figure REF ).",
"Figure: Identities given by Lemma when b 3 =0b_3 = 0."
],
[
"Complete decompositions",
"The following definition consists of two properties that are invariant under a series of operations performed throughout the proof of Lemma REF .",
"Definition 2 A decomposition $ of a $ {g,r}$-graph $ G$into trails of length $ 5$is \\emph {complete} if the following hold for every $ T. $T$ is of type A, B, C or D; If $T$ is of type A, then ${\\rm {hang}}_({\\rm {cv}}_1(T)\\big )\\ge 2$ and ${\\rm {hang}}_({\\rm {cv}}_2(T)\\big )\\ge 1$ .",
"The first step of our proof is given by the next proposition, which presents an initial decomposition for the graphs studied.",
"Proposition 10 If $G$ is a $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ , then $G$ admits a complete decomposition.",
"Let $\\lbrace M_{g,r},F_g,F_r\\rbrace $ be the base factorization of $G$ .",
"For each $e=xy\\in M_{g,r}$ , let $P_e=a_0a_1a_2a_3a_4a_5$ , where $a_1a_0,a_4a_5\\in F_r$ , $a_2a_1,a_3a_4\\in F_g$ , $a_2=x$ , and $a_3 = y$ .",
"We claim that $ \\lbrace P_e\\colon e\\in M_{g,r}\\rbrace $ is complete.",
"Clearly, $P_e$ is an element of type A or B, for every $e\\in M_{g,r}$ , and hence $ satisfiesDefinition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-types}).Moreover, note that $ a0a1$ (resp.",
"$ a4a5$) is a hanging edgeat $ a1$ (resp.",
"$ a4$).Thus, given $ zV(G)$, let $ e' = xyMg,r$ be such that $ x= z-g$,then $ Pe'$ contains a hanging edge at $ z$,namely, the red out edge of $ z$,and hence there is a hanging edge at every vertex of $ G$.Moreover, if $ z=cv1(T)$ for some element $ T of type A, and $e\\in M_{g,r}\\cap E(T)$ , then $e$ is a second hanging edge at $z$ .",
"This proves Definition REF (REF ).",
"We say that an element $T$ of type A in a decomposition $ is \\emph {free}if $ tr(T)cvi(T')$ for every element $ T' of type A and $i\\in \\lbrace 1,2\\rbrace $ .",
"An A-chain is a sequence $T_0, T_1, \\ldots , T_{s-1}$ of elements of type A such that for each $j\\in \\lbrace 0,\\ldots ,s-1\\rbrace $ , we have ${\\rm {tr}}(T_j) = {\\rm {cv}}_i(T_{j-1})$ , for some $i \\in \\lbrace 1,2\\rbrace $ (subtraction on the indexes are taken modulo $s$ ).",
"Note that A-chains do not consider the auxiliary vertex when allowing two elements to be consecutive.",
"Thus, elements, say $T$ and $T^{\\prime }$ , of type A that are not consecutive in an A-chain, or that are in different A-chains, may still share a vertex $u$ for which ${\\rm {cv}}_i(T) = u = {\\rm {aux}}(T^{\\prime })$ .",
"Given a decomposition $ of a graph $ G$ into trails of length $ 5$, denote by$ ($ the number of elements that are not paths.By exchanging edges between the elements of a decompositiongiven by Proposition~\\ref {proposition:initial-decomposition},we can show that a complete decomposition that minimizes $ ($has no free element, and hence its elements of type~A are partitionedinto A-chains.$ Lemma 11 Every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ admits a complete decomposition in which the elements of type A are partitioned into A-chains.",
"Let $g$ and $r$ be as in the statement, let $G$ be a $\\lbrace g,r\\rbrace $ -graph, and put ${M=M_{g,r}}$ .",
"By Proposition REF , $G$ admits a complete decomposition.",
"Let $ be a complete decomposition of~$ G$ that minimizes $ ($.In what follows, we prove that $ contains no free element of type A.",
"For that, we prove three claims regarding the relation between some elements of $.In the proof of each such claim, we exchange edges between some elements of $ and obtain a complete decomposition $$ into trails of length 5 such that $\\tau () <\\tau ($ , which is a contradiction to the minimality of $.To check that $$ is a complete decomposition, we observe the two following items:(i) the vertices $ u$ for which $ hangD(u) >hang(u)$ are verticesthat are not connection vertices of $$,for example, tricky vertices of free elements of type~A,or connection vertices of elements of type~A in $ that become paths in $$ .",
"Hence, Definition REF (REF ) holds for $$ ; and (ii) every element of $$ that is not an element of $, i.e.,the elements involved in the exchange of edges, are of type~A, B, C, or~D,and hence \\ref {def:complete-commutative}(\\ref {def:complete-commutative-types})holds for $$.$ Claim 11.1 No element of type B or C has a hanging edge at the primary connection vertex of a free element of type A.",
"Let $T_1 = a_0a_1a_2a_3a_4a_5\\in , where $ a5=a2$ and $ a2a3M$, be a free element of type~A,and let $ T2= b0b1b2b3b4b5 be an element of type B or C that contains a hanging edge at ${\\rm {cv}}_1(T_1)$ .",
"We divide the proof depending on whether $T_2$ is of type B or C. $\\mathbf {T_2}$ is of type B.",
"Suppose, for a contradiction, that $b_4={\\rm {cv}}_1(T_1)=a_3$ .",
"Put ${T_1^{\\prime } = a_0a_1a_2a_4a_3b_5}$ , $T_2^{\\prime } = b_0b_1b_2b_3b_4a_2$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime }\\rbrace $ .",
"Note that $$ is a decomposition of $G$ into trails of length 5.",
"By Lemma REF , $T_1^{\\prime }$ is an element of Type C. In what follows, we prove that $T_2^{\\prime }$ is of type B, i.e., $a_2\\notin \\lbrace b_0,b_1,b_2,b_3,b_4\\rbrace $ .",
"Indeed, since $G$ has no loops or multiple edges, $a_2\\notin \\lbrace b_3,b_4\\rbrace $ .",
"Since $M$ is a matching, $a_2\\ne b_2$ .",
"If $a_2 = b_1$ , then $b_2 = b_1-g = a_2 -g = a_3 + g +r -g = b_5$ , and hence $T_2$ is of type A, a contradiction.",
"Finally, by Lemma REF , $a_2 \\ne b_0$ .",
"Thus, $T^{\\prime }_2$ is an element of type B, and hence Definition REF (REF ) holds for $$ .",
"Note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_4\\rbrace .Since a_4 is not a connection vertex of ,and by Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}),we have {\\rm {hang}}_{\\big ({\\rm {cv}}_1(T)\\big )\\ge 2 and {\\rm {hang}}_{\\big ({\\rm {cv}}_2(T)\\big )\\ge 1for every T\\in ,we have{\\rm {hang}}_{}\\big ({\\rm {cv}}_1(T)\\big )\\ge 2 and {\\rm {hang}}_{}\\big ({\\rm {cv}}_2(T)\\big )\\ge 1for every T\\in ,Thus Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}) holds for .Therefore, is a complete decomposition such that\\tau ()=\\tau (-1<\\tau (, a contradictionto the minimality of .",
"}\\begin{figure}[H]\\centering \\begin{subfigure}{.45}\\centering \\scalebox {.8}{\\begin{tikzpicture}[scale=1.5]\\end{tikzpicture}[fatpath,backcolor1] (120:1)-- ((-1,0)+(120:1)) -- (120:1) -- (0,0) -- (0:1) -- (60:1) -- (0,0) -- (60:1);}\\node (0) [black vertex] at ((-1,0)+(120:1)) {};\\node (1) [black vertex] at (120:1) {};\\node (2) [black vertex] at (0,0) {};\\node (3) [black vertex] at (0:1) {};\\node (4) [black vertex] at (60:1) {};\\node () [] at ((2,0)+(60:1)) {};\\end{subfigure}\\end{figure}\\node (4^{\\prime }) [black vertex] at (1,0) {};\\node (3^{\\prime }) [black vertex] at ((4^{\\prime })-(60:1)) {}; \\node (2^{\\prime }) [black vertex] at ((3^{\\prime })-(1,0)) {};\\node (1^{\\prime }) [black vertex] at ((2^{\\prime })+(120:1)) {};\\node (0^{\\prime }) [black vertex] at ((2^{\\prime })+(-1,0)+(120:1)) {};\\node (5^{\\prime }) [black vertex] at ((4^{\\prime })+(1,0)) {};}}\\node () [] at ($ (0)+(90:.2)$) {a_0};\\node () [] at ($ (1)+(90:.2)$) {a_1};\\node () [] at ($ (2)+(-90:.2)$) {a_2};\\node () [] at ($ (3)+(45:.3)$) {a_3};\\node () [] at ($ (4)+(90:.2)$) {a_4};$ ) [] at ($(0^{\\prime })+(90:.2)$ ) $b_0$ ; ) [] at ($(1^{\\prime })+(45:.3)$ ) $b_1$ ; ) [] at ($(2^{\\prime })+(-90:.2)$ ) $b_2$ ; ) [] at ($(3^{\\prime })+(-90:.2)$ ) $b_3$ ; ) [] at ($(4^{\\prime })+(-45:.3)$ ) $b_4$ ; ) [] at ($(5^{\\prime })+(90:.2)$ ) $b_5$ ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.3pt,color=gray,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (5'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); .45 [scale=1.5] [fatpath,backcolor1] (120:1) – ($(-1,0)+(120:1)$ ) – (120:1) – (0,0) (60:1) – (0,0) – (60:1) – (0:1) – ($(2,0)$ ) – (0:1); 0) [black vertex] at ($(-1,0)+(120:1)$ ) ; 1) [black vertex] at (120:1) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (0:1) ; 4) [black vertex] at (60:1) ; ) [] at ($(2,0)+(60:1)$ ) ; 4') [black vertex] at (1,0) ; 3') [black vertex] at ($(4^{\\prime })-(60:1)$ ) ; 2') [black vertex] at ($(3^{\\prime })-(1,0)$ ) ; 1') [black vertex] at ($(2^{\\prime })+(120:1)$ ) ; 0') [black vertex] at ($(2^{\\prime })+(-1,0)+(120:1)$ ) ; 5') [black vertex] at ($(4^{\\prime })+(1,0)$ ) ; ) [] at ($(0)+(90:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(-90:.2)$ ) $a_2$ ; ) [] at ($(3)+(45:.3)$ ) $a_3$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(90:.2)$ ) $b_0$ ; ) [] at ($(1^{\\prime })+(45:.3)$ ) $b_1$ ; ) [] at ($(2^{\\prime })+(-90:.2)$ ) $b_2$ ; ) [] at ($(3^{\\prime })+(-90:.2)$ ) $b_3$ ; ) [] at ($(4^{\\prime })+(-45:.3)$ ) $b_4$ ; ) [] at ($(5^{\\prime })+(90:.2)$ ) $b_5$ ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.3pt,color=gray,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (5'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); Exchange of edges between elements of type A and B in the proof of Claim REF .",
"$\\mathbf {T_2}$ is of type C. We may assume $b_3b_2\\in F_r$ .",
"In this case we have $b_4b_3\\in F_g$ .",
"Since $T_2$ contains a hanging edge at ${\\rm {cv}}_1(T_1)$ , we have $a_3={\\rm {cv}}_1(T_1)\\in \\lbrace b_1,b_4\\rbrace $ .",
"If $b_4=a_3$ , then there are two green out edges at $a_3$ , namely $a_3a_4, b_4a_3$ , a contradiction.",
"Thus, we may assume that $a_3 = b_1$ .",
"Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_0$ , $T_2^{\\prime } = a_2b_1b_2b_4b_3b_5$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime }\\rbrace $ .",
"Note that $$ is a decomposition of $G$ into trails of length 5.",
"By Lemma REF , $T_1^{\\prime }$ is an element of Type C. In what follows we prove that $T_2^{\\prime }$ is a path.",
"For that, we prove that $a_2\\notin \\lbrace b_0,b_1,b_2,b_3,b_4\\rbrace $ .",
"Indeed, since $G$ is simple, ${a_2\\notin \\lbrace b_1,b_2\\rbrace }$ .",
"If $a_2 = b_4$ , then $a_2a_1$ and $b_4b_3$ are two green out edges at $a_2$ , a contradiction.",
"By Lemma REF , $a_2\\ne b_5$ .",
"Finally, $a_3 + g + r = a_2$ and ${b_1 = b_3 + r + g}$ , if $a_2 = b_3$ , then we have $a_3 + g + r = a_2 = b_3 = b_1 - g - r = a_3 - g - r$ , which implies $2g+2r = 0$ , a contradiction.",
"Thus, $T^{\\prime }_2$ is an element of type C, and hence Definition REF (REF ) holds for $$ .",
"Analogously to the case above $$ is a complete decomposition of $G$ such that $\\tau ()=\\tau (-1<\\tau ($ , a contradiction to the minimality of $.",
"$ Figure: Exchange of edges between elements of type Aand C in the proof of Claim .Claim 11.2 Let $T_1$ and $T_2$ be two elements of type A in $.If $ T1$ is free and $ T2$ contains a hangingedge on $ cv1(T1)$, then no element of type~A, B, or C in $ {T1,T2}$contains a hanging edge at $ cv2(T2)$.",
"$ Let $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ be two elements of $,where $ a5=a2$ and $ b5=b2$ and $ a2a3,b2b3M$.First, we prove that $ cv1(T1)=tr(T2)$,and hence, by Lemma~\\ref {lemma:type12}, we have $ cv2(T1) = aux(T2)$.Suppose, for contradiction, that $ cv1(T1)tr(T2)$.Since $ b2b3M$, we must have $ b1=cv1(T1)=a3$.Now, put $ T1' = a0a1a2a4a3b0$,$ T2' = a2b1b2b3b4b2$ (see Figure~\\ref {fig:case3}) and let$ = ({T1,T2}){T1',T2'}$.By Lemma~\\ref {lemma:T^{\\prime }1-is-path}, $ T1'$ is an element of type~C.We claim that $ T2'$ is an element of type~A.For that we prove that $ a2 {b1,b2,b3,b4}$.Again, since $ G$ is a simple graph, we have $ a2{b1,b2}$.Since every vertex is incident to precisely one edge of $ M$, we have $ a2b3$,and if $ a2 = b4$, then we have $ a3 + g + r = a2 = b4 = a3 - g - r$, which implies $ 2g+2r=0$, a contradiction.Thus, Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-types}) holds for $$.Analogously to the cases above,$$ is a complete decomposition such that$ ()=(-1<($, a contradictionto the minimality of $ .",
"Finally, by Lemma REF , we have ${\\rm {cv}}_2(T_1) = {\\rm {aux}}(T_2)$ .",
"Figure: Exchange of edges between two elements of type Ain the proof of Claim .Now, let $T_3\\in \\lbrace T_1,T_2\\rbrace $ be an element of type A, B, or C, and suppose, for a contradiction, that $T_3$ contains a hanging edge at ${\\rm {cv}}_2(T_2)$ .",
"Since ${\\rm {cv}}_1(T_1)={\\rm {tr}}(T_2)$ and ${\\rm {cv}}_2(T_1) = {\\rm {aux}}(T_2)$ , we have $a_3=b_4$ , $a_5=a_2=b_1$ and $b_5=b_2$ .",
"In what follows, we divide the proof according to the type of $T_3$ .",
"$\\mathbf {T_3}$ is of type A.",
"Let $T_3 = c_0c_1c_2c_3c_4c_5$ , where $c_2=c_5$ and $c_2c_3\\in M$ .",
"Since each vertex is incident to precisely one edge of $M$ we have $c_3\\ne b_2 = {\\rm {cv}}_2(T_2)$ .",
"Therefore, we have ${\\rm {cv}}_2(T_2)\\in \\lbrace c_1,c_4\\rbrace $ .",
"Suppose that ${\\rm {cv}}_2(T_2)=c_4$ .",
"Thus, we have $b_5 = b_2 = c_4$ .",
"Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_5$ , $T_2^{\\prime } = b_0b_1b_4b_3b_2c_2$ , $T_3^{\\prime } = b_1c_4c_3c_2c_1c_0$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .",
"In what follows, we prove that $T_1^{\\prime }$ , $T_2^{\\prime }$ and $T_3^{\\prime }$ are paths.",
"By Lemma REF , $T_1^{\\prime }$ is an element of type C. Since $G$ is simple, we have $c_2\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_2,c_3,c_4\\rbrace $ .",
"By Lemma REF , $c_2 \\ne b_0,~b_1 \\ne c_0$ .",
"Therefore, $T_2^{\\prime }$ is an element of type D. If $b_1 = c_1$ , then $b_2b_1$ and $c_2c_1$ are two green in edges at $c_1$ , a contradiction.",
"Thus, $T^{\\prime }_3$ is an element of type B, and hence Definition REF (REF ) holds for $$ .",
"Analogously to the cases above, we have ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)\\ge 0for every v\\in V(G)\\setminus \\lbrace a_3,a_4,b_3,c_3\\rbrace .Since a_3,a_4,b_3,c_3 are not connection vertices in ,Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}) holds for .Thus, is a complete decomposition such that\\tau ()=\\tau (-3<\\tau (, a contradiction to the minimality of~.",
"}\\begin{figure}\\centering \\begin{subfigure}{.45}\\centering \\scalebox {.8}{\\begin{tikzpicture}[scale=1.5]\\end{tikzpicture}[fatpath,backcolor1] (-0.5,0.8) -- (-1.5,0.8) -- (-0.5,0.8) -- (0,0) -- (0,0) -- (1,0) -- (0.5,0.8) -- (0,0);[fatpath,backcolor2] (-0.5,-0.8) -- (-1.5,-0.8)-- (-0.5,-0.8) --(0,-1.6) -- (1,-1.6) -- (0.5,-0.8) -- (0,-1.6);}\\node (0) [black vertex] at (-1.5,0.8) {};\\node (1) [black vertex] at (-0.5,0.8) {};\\node (2) [black vertex] at (0,0) {};\\node (3) [black vertex] at (1,0) {};\\node (4) [black vertex] at (0.5,0.8) {};\\node () [] at ((0,0)+(60:1)) {};\\end{subfigure}\\end{figure}\\node (4^{\\prime }) [black vertex] at ($ (1,0)$) {};\\node (3^{\\prime }) [black vertex] at ($ (3)+(0.5,-0.8)$) {}; \\node (2^{\\prime }) [black vertex] at ($ (2)+(0.5,-0.8)$) {};\\node (1^{\\prime }) [black vertex] at ($ (0,0)$) {};\\node (0^{\\prime }) [black vertex] at ($ (2')-(1,0)$) {};\\node (5^{\\prime }) [] at ($ (2,0)$) {};$ 4”) [black vertex] at ($(2^{\\prime })$ ) ; 3”) [black vertex] at ($(3)-(0,1.6)$ ) ; 2”) [black vertex] at ($(2)-(0,1.6)$ ) ; 1”) [black vertex] at ($(4^{\\prime \\prime })-(1,0)$ ) ; 0”) [black vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(2,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.6)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(-90:.3)-(0.35,0)$ ) $b_0=c_1$ ; ) [] at ($(1^{\\prime })+(180:.4)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\prime })+(-90:.25)+(0.7,0)$ ) $b_2=c_4$ ; ) [] at ($(3^{\\prime })+(0:.3)$ ) $b_3$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.4)$ ) ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.3)$ ) $c_3$ ; ) [] at ($(4^{\\prime \\prime })+(90:.3)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.3pt,color=gray,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (0.5,0.8) – (1,0) – (0.5,-0.8); [fatpath,backcolor2] (-0.5,-0.8) – (-1.5,-0.8)– (-0.5,-0.8) –(0,-1.6) – (1,-1.6) – (0.5,-0.8) – (0,0); 0) [black vertex] at (-1.5,0.8) ; 1) [black vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [black vertex] at (0.5,0.8) ; ) [] at ($(0,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(2^{\\prime })-(1,0)$ ) ; 5') [] at ($(2,0)$ ) ; 4”) [black vertex] at ($(2^{\\prime })$ ) ; 3”) [black vertex] at ($(3)-(0,1.6)$ ) ; 2”) [black vertex] at ($(2)-(0,1.6)$ ) ; 1”) [black vertex] at ($(4^{\\prime \\prime })-(1,0)$ ) ; 0”) [black vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(2,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.6)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(-90:.3)-(0.35,0)$ ) $b_0=c_1$ ; ) [] at ($(1^{\\prime })+(180:.4)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\prime })+(-90:.25)+(0.7,0)$ ) $b_2=c_4$ ; ) [] at ($(3^{\\prime })+(0:.3)$ ) $b_3$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.4)$ ) ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.3)$ ) $c_3$ ; ) [] at ($(4^{\\prime \\prime })+(90:.3)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.3pt,color=gray,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange of edges between three elements of type A in the proof of Claim REF .",
"Thus, we may assume ${\\rm {cv}}_2(T_2) = c_1$ .",
"This implies that $b_5=b_2=c_1$ , and hence we have $b_3 = b_5 - r - g = c_1 - g-r = c_4$ .",
"Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\prime } = b_1c_1c_2c_3c_4c_2$ (see Figure REF ) and $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .",
"Again, by Lemma REF , $T_1^{\\prime }$ is an element of type C. We claim that $T_2^{\\prime }$ , $T_3^{\\prime }$ are, respectively, of type D and A.",
"Since $G$ is simple, $c_0\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_1,c_2,c_4\\rbrace $ .",
"By Lemma REF we have $c_0 \\ne b_0$ .",
"Therefore, $T_2^{\\prime }$ is of type D. Finally, if $b_1 = c_3$ , then $d(b_1)\\ge 7>5$ , a contradiction.",
"Thus, $T^{\\prime }_3$ is an element of type A, and hence Definition REF (REF ) holds for $$ .",
"Analogously to the case above, we have ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)\\ge 0 for every v \\in V(G)\\setminus \\lbrace a_4,a_3,b_3\\rbrace .Since a_4,a_3,b_3 are not connection vertices in ,Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}) holds for .Thus, is a complete decomposition such that {\\tau () = \\tau ( -2<\\tau (}, a contradiction to the minimality of~.",
"}$ Figure: Exchange of edges between three elements of type Ain the proof of Claim .$\\mathbf {T_3}$ is of type B.",
"Let $T_3 = c_0c_1c_2c_3c_4c_5$ be an element of type B.",
"Since $T_3$ contains a hanging edge on ${\\rm {cv}}_2(T_2)=b_2$ , we have $b_2\\in \\lbrace c_1,c_4\\rbrace $ .",
"By symmetry we may assume $b_2=c_1$ .",
"Thus, put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\prime } = b_1c_1c_2c_3c_4c_5$ (see Figure REF ) and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .",
"Again, by Lemma REF , $T_1^{\\prime }$ is an element of type C. We prove that $T_2^{\\prime }$ and $T_3^{\\prime }$ are, respectively, of type D and B.",
"Since $G$ is simple, we have $c_0\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_1,c_2\\rbrace $ .",
"By Lemma REF , we have $c_0 \\ne b_0$ and $b_1 \\ne c_5$ .",
"Therefore, $T_2^{\\prime }$ is an element of type D. Since $c_4=c_3+g$ and $b_1=c_1+g$ , if $c_4=b_1$ , then $c_3=c_1$ , a contradiction.",
"If $b_1 \\in \\lbrace c_3,c_4\\rbrace $ , then $d(b_1)\\ge 7>5$ , a contradiction.",
"Therefore, $T_3^{\\prime }$ is an element of type B. Analogously to the case above, $$ is a complete decomposition such that $\\tau () = \\tau ( -2<\\tau ($ , a contradiction to the minimality of $.$ Figure: Exchange of edges between two elements of type Aand an element of type Bin the proof of Claim .$\\mathbf {T_3}$ is of type C. Let $T_3 = c_0c_1c_2c_3c_4c_5$ be an element of type C, where $c_3c_2\\in F_r$ .",
"This implies that $c_4c_3\\in F_g$ .",
"Since $T_3$ contains a hanging edge on ${\\rm {cv}}_2(T_2)=b_2$ , we have $b_2\\in \\lbrace c_1,c_4\\rbrace $ .",
"If $b_2 = c_4$ , then $c_4c_3$ and $b_2b_1$ are two green out edges of $b_2$ , a contradiction.",
"Thus, we may assume $b_2 = c_1$ .",
"Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_2$ , $T_2^{\\prime } = b_0b_1b_4b_3b_2c_0$ , $T_3^{\\prime } = b_1c_1c_2c_3c_4c_5$ (see Figure REF ) and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .",
"Again, by Lemma REF , $T_1^{\\prime }$ is an element of type C. We prove that $T_2^{\\prime }$ and $T_3^{\\prime }$ are, respectively, of type D and C. Since $G$ is simple, $c_0\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_1,c_2\\rbrace $ .",
"By Lemma REF , we have $c_0 \\ne b_0$ and $b_1 \\ne c_5$ .",
"Therefore, $T_2^{\\prime }$ is an element of type D. Analogously to the case above, If $b_1 \\in \\lbrace c_3,c_4\\rbrace $ , then $d(b_1)\\ge 7>5$ , a contradiction.",
"Therefore, $T_3^{\\prime }$ is an element of type C. Once more, analogously to the cases above, $$ is a complete decomposition such that $\\tau () = \\tau ( -2<\\tau ($ , a contradiction to the minimality of $.$ Figure: Exchange of edges between two elements of type Aand an element of type Cin the proof of Claim .Claim 11.3 There is no free element of type A.",
"Suppose, for a contradiction, that $contains a free element, say $ T1$, of type~A.By Definition~\\ref {def:complete-commutative}(\\ref {def:complete-commutative-hanging-edge}),there are two hanging edges $ e2$ and $ e2'$ at $ cv1(T1)$.We may assume $ e2 E(T1)$.Let $ T2$ be the element of $ that contains $e_2$ .",
"By Claim REF , $T_2$ is not of type B or C, and since $M$ is a matching, $T_2$ is not of type D. Thus, $T_2$ is of type A.",
"By Definition REF (REF ), there is a hanging edge $e_3$ on ${\\rm {cv}}_2(T_2)$ .",
"Note that $e_3\\notin E(T_2)$ .",
"Let $T_3$ be the element of $ that contains $ e3$.By Claim~\\ref {claim:AAB,AAC},$ T3$ is of type~D,which implies that there are two edges of $ M$incident to $ cv2(T2)$, a contradiction.$ Now, consider the auxiliary directed graph $D_ in which $ V(D = and $(T_1,T_2)$ is an arc of $D_ if and only if $ tr(T2) = cvi(T1)$ for some $ i{1,2}$.It is clear that the elements of type~A in $ are partitioned into A-chains if and only if $D_ consists of vertex-disjoint directed cycles and isolated vertices.",
"Since every vertex of $ G$ is a connection vertex of at most one element of $ , by Claim REF , every vertex of $D_ has in degree precisely $ 1$.$ Note also that given two elements $T_1$ and $T_2$ we have ${\\rm {tr}}(T_1) \\ne {\\rm {tr}}(T_2)$ , otherwise there would be a vertex with two green in edges.",
"This implies that every vertex of $D_ has out degree at most $ 2$.Now, if $ T1$ and $ T2$ are two elements of type~A in $ such that ${\\rm {cv}}_1(T_1) = {\\rm {tr}}(T_2) = u_1$ , by Lemma REF , we have ${\\rm {aux}}(T_2) = {\\rm {cv}}_2(T_1) = u_2$ , which means that $E(T_1)\\cup E(T_2)$ contains the four edges in $E(G)$ incident to $u_1$ and five edges incident to $u_2$ , and hence, no other element of $ contains $ u2$,and no other element of $ has $u_1$ as its tricky vertex.",
"This implies that every vertex of $D_ has out degree at most $ 1$,and hence $ D consists of vertex-disjoint directed cycles and isolated vertices as desired."
],
[
"Admissible decompositions",
"In this section, we present a new decomposition invariant, which we call admissibility, and conclude our proof.",
"For that, we introduce an important object, the exceptional pair.",
"Let $G$ be a $\\lbrace g,r\\rbrace $ -graph, and let $ be a decomposition of $ G$ into trails of length $ 5$.We say that a pair $ (T1,T2)$ of elements of $ is an exceptional pair if $T_1$ and $T_2$ are elements of type A and C, respectively, and can be written as $T_1 = a_0a_1a_2a_3a_4a_5$ and $T_2 = b_0b_1b_2b_3b_4b_5$ such that $a_2a_3\\in M_{g,r}$ , $a_2=a_5 = b_3$ , and $a_2a_1,a_3a_4,b_2b_1,b_4b_3\\in F_g$ , $a_4a_5,b_3b_2,b_4b_5\\in F_r$ , $a_1a_0,b_1b_0\\in M_{g,r}\\cup F_g\\cup F_r$ (see Figure REF ).",
"Note that since $G$ is a simple graph, we have $b_4\\ne a_3$ .",
"Also, if $2g + 2r \\ne 0$ , then we have $b_1 \\ne a_3$ .",
"This yields the following remark.",
"Remark 1 If $G$ is a $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ and $(T_1,T_2)$ is an exceptional pair, then $T_2$ does not contain a hanging edge at ${\\rm {cv}}_1(T_1)$ .",
"Figure: An exceptional pair.An open chain is a sequence $T_0, T_1,\\ldots , T_{s-1}$ of $s\\ge 2$ elements of $with the following properties.",
"(i) $ T0$ is a free element of type~A;(ii) $ Tj$ is an element of type~A and $ tr(Tj) = cvi(Tj-1)$,for every $ j {0,...,s-2}$ and some $ i{1,2}$; and(iii) $ Ts-1$ is an element of type~C for which $ (Ts-2,Ts-1)$ is an exceptional pair.We remark that open chains are not A-chains.The next definition describes the invariant studied in this section.$ Definition 3 We say that a decomposition $ of a $ {g,r}$-graph $ G$into trails of length $ 5$ is \\emph {admissible} if the following hold.\\begin{enumerate}[(a)]\\item Every element in is either a path or an element of type~A;\\item For every element T\\in of type~A, we have {\\rm {hang}}\\big ({\\rm {cv}}_1(T)\\big )\\ge 2,and there is at most one element T\\in of type~A for which{\\rm {hang}}\\big ({\\rm {cv}}_2(T)\\big ) = 0,and, in this case, there is an open chain {S = T_0,\\ldots , T_{s-2},T_{s-1}} in ,for which T_{s-2} = T;\\item The elements of type~A in are partitioned into A-chains and at most one open chain.\\end{enumerate}$ It is not hard to check that the decomposition given by Lemma REF is an admissible decomposition.",
"Therefore, every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ admits an admissible decomposition.",
"By performing a few more exchanges of edges between the elements of a same A-chain of an admissible decomposition, we can show that an admissible decomposition that minimizes its number of elements of type A is in fact a $P_5$ -decomposition.",
"Theorem 12 Every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r\\ne 0$ admits a $P_5$ -decomposition.",
"Let $g$ and $r$ be as in the statement, let $G$ be a $\\lbrace g,r\\rbrace $ -graph, and put ${M=M_{g,r}}$ .",
"By Lemma REF , $G$ admits an admissible decomposition.",
"Let $ be an admissible decomposition of $ G$ that minimizes $ ($.In what follows, we prove that $ ( = 0$.Suppose, for a contradiction, that $ (>0$.We divide A-chains into three types, according to the connections between its elements.Given $ i{1,2}$,we say that an A-chain $ S=T0, T1,..., Ts-1$ is of type~$ i$ if $ tr(Tj)=cvi(Tj-1)$for every $ j {0,...,s-1}$;and we say that $ S$ is a \\emph {mixed} A-chain if $ S$ is not of type~$ 1$ or $ 2$.$ Similarly to the proof of Lemma REF , in each step, we exchange edges between some elements of $and obtain an admissible decomposition $$ into trails of length~5such that $ () <($, which is a contradiction to the minimality of $ .",
"To check that $$ is an admissible decomposition, we observe the three following items: (i) The only connection vertex that has fewer hanging edges in $$ than in $is the secondary connection vertex of an element $ T1$ of type~A,and in this case there is an element $ T2$ of type~Csuch that $ (T1,T2)$ is an exceptional pair,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$;(ii) every element of $$ that is not an element of $ , i.e., the elements involved in the exchange of edges, is a path or an element of type A, and hence REF () holds for $$ ; (iii) either an open chain is shortened by at least one element, an A-chain is converted into an open chain, or all the elements of an A-chain are replaced by paths of length 5, and hence REF () holds for $$ .",
"Claim 12.1 Every A-chain in $ is mixed.",
"$ Suppose, for a contradiction, that there is an A-chain $S = T_0,T_1,\\ldots , T_{s-1}$ of type 1 or 2.",
"Let $T_j=a_{0,j}a_{1,j}a_{2,j}a_{3,j}a_{4,j}a_{5,j}$ , where $a_{5,j} = a_{2,j}$ , $a_{2,j}a_{3,j}\\in M$ , ${a_{2,j}a_{1,j}, a_{3,j}a_{4,j}\\in F_g}$ , $a_{4,j}a_{2,j}\\in F_r$ and $a_{1,j}a_{0,j}\\in M\\cup F_g \\cup F_r$ .",
"For $i\\in \\lbrace 1,2,3,4,5\\rbrace $ , the edge $a_{i-1,j}a_{i,j}$ is called the $i$ th edge of $T_j$.",
"In what follows, we divide the proof according to the type of S. $\\mathbf {S}$ is of type 1.",
"In this case, we have ${a_{3,j} = {\\rm {cv}}_1(T_{j}) = {\\rm {tr}}(T_{j+1}) = a_{4,j+1}}$ for each ${j\\in \\lbrace 0,\\ldots ,s-1}\\rbrace $ , and hence, by Lemma REF , we have $a_{2,j} = {\\rm {cv}}_2(T_{j}) = {\\rm {aux}}(T_{j+1})=a_{1,j+1}$ .",
"Now, for each $j=0,\\ldots ,s-1$ , let $T^{\\prime }_j = a_{2,j+1}a_{3,j}a_{4,j}a_{1,j}a_{2,j}a_{0,j+1}$ (see Figure REF ).",
"Note that $T^{\\prime }_j = T_j- a_{1,j}a_{0,j} + a_{1,j+1}a_{0,j+1}- a_{2,j}a_{3,j} + a_{2,j-1}a_{3,j-1}- a_{4,j}a_{2,j} + a_{4,j+1}a_{2,j+1}$ .",
"More specifically, $a_{2,j+1}a_{3,j} = a_{4,j+1}a_{5,j+1}$ is the 5th edge of $T_{j+1}$ ; $a_{3,j}a_{4,j}$ is the 4th edge of $T_{j}$ ; $a_{4,j}a_{1,j} = a_{2,j-1}a_{3,j-1}$ is the 3rd edge of $T_{j-1}$ ; $a_{1,j}a_{2,j}$ is the 2nd edge of $T_{j}$ ; $a_{2,j}a_{0,j+1} = a_{1,j+1}a_{0,j+1}$ is the 1st edge of $T_{j+1}$ .",
"Clearly, $T^{\\prime }_j$ is a trail of length 5.",
"Moreover, since, for each $i\\in \\lbrace 1,2,3,4,5\\rbrace $ , the element $T^{\\prime }_j$ contains the $i$ th edge of an element of $S$ , and, if $j\\ne j^{\\prime }$ , the elements $T^{\\prime }_j$ and $T^{\\prime }_{j^{\\prime }}$ contain the $i$ th edge of different elements of $S$ , the set $=\\big (\\lbrace T_j\\colon j=0,\\ldots ,s-1\\rbrace \\big )\\cup \\lbrace T^{\\prime }_j\\colon j=0,\\ldots ,s-1\\rbrace $ is a decomposition of $G$ into trails of length 5.",
"We may regard $$ as the decomposition obtained by reversing the direction of two components of $F_g$ , namely, the green edges in $S$ , and applying the same strategy used in Proposition REF .",
"Figure: Exchange of edges between the elements of an A-chain of type 1 withfive elements in the proof of Claim .In order to prove that $T^{\\prime }_j$ is a path, we show that $a_{2,j+1},a_{0,j+1}\\notin \\lbrace a_{3,j},a_{4,j},a_{1,j},a_{2,j}\\rbrace $ .",
"Note that, since for each $j\\in \\lbrace 0,\\ldots ,s-1\\rbrace $ , $T_j$ is a path, we have $a_{i,j} \\ne a_{i^{\\prime },j}$ for every $i\\ne i^{\\prime }$ .",
"Since $G$ is a simple graph, we have $a_{2,j+1}\\notin \\lbrace a_{3,j},a_{4,j},a_{2,j}\\rbrace $ and $a_{0,j+1}\\notin \\lbrace a_{3,j},a_{4,j},a_{1,j},a_{2,j}\\rbrace $ ; and if $a_{2,j+1} = a_{1,j}$ , then $a_{4,j+1}a_{2,j+1}$ and $a_{4,j-1,}a_{2,j-1}$ are two distinct red in edges of $a_{1,j}$ , a contradiction.",
"We claim that $$ is an admissible decomposition.",
"Indeed, the only vertices of the elements of $S$ that can be connection vertices of elements in $$ are the vertices $a_{0,j}$ , for $j=0,\\ldots ,s-1$ .",
"But a hanging edge at a vertex $a_{0,j}$ is in $T_{j^{\\prime }}\\in if and only if $ a0,j = a3,j'$ for some $ j'j$,and, in this case $ a3,j'$ is not a connection vertex in $$ becauseall edges incident to it are in elements of $ {T'jj=0,...,s-1}$.Therefore, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$.Moreover, since $ T'j$ is a path, for $ j=0,...,s-1$, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:complete-commutative-types}) holds for $$.Finally, $ and $$ have the same number of open chains, and hence Definition REF () holds for $$ .",
"Therefore, $$ is an admissible decomposition of $G$ such that ${\\tau () = \\tau ( - s}$ , a contradiction to the minimality of $.$ $\\mathbf {S}$ is of type 2.",
"In this case, for each $j=0,\\ldots ,s-1$ , we have $a_{2,j} = {\\rm {cv}}_2(T_{j}) = {\\rm {tr}}(T_{j+1}) = a_{4,j+1}$ .",
"Now, for each $j=0,\\ldots ,s-1$ , let $T^{\\prime }_j = a_{0,j}a_{1,j}a_{2,j}a_{3,j}a_{4,j}a_{4,j-1}$ (see Figure REF ).",
"Clearly, $T^{\\prime }_j$ is a trail of length 5.",
"Note that $T^{\\prime }_j = T_j - a_{4,j}a_{2,j} + a_{4,j-1}a_{2,j-1}$ , i.e., $T^{\\prime }_j$ is the element obtained from $T_j$ by exchanging its 5th edge by the 5th edge of $T_{j-1}$ .",
"Thus, the set $=\\big (\\lbrace T_j\\colon j=0,\\ldots ,s-1\\rbrace \\big )\\cup \\lbrace T^{\\prime }_j\\colon j=0,\\ldots ,s-1\\rbrace $ is a decomposition of $G$ into trails of length 5.",
"We may regard $$ as the decomposition obtained by reversing the direction of one component of $F_r$ and applying the same strategy used in Proposition REF .",
"In order to prove that $T^{\\prime }_j$ is a path, we show that $a_{4,j-1}\\notin \\lbrace a_{0,j},a_{1,j},a_{2,j},a_{3,j},a_{4,j}\\rbrace $ .",
"Note that, since for each $j\\in \\lbrace 0,\\ldots ,s-1\\rbrace $ , $T_j$ is a path, we have $a_{i,j} \\ne a_{i^{\\prime },j}$ for every $i\\ne i^{\\prime }$ .",
"Since $G$ is a simple graph, we have $a_{4,j-1}\\notin \\lbrace a_{2,j},a_{3,j},a_{4,j}\\rbrace $ ; also, by Lemma REF , we have $a_{4,j-1}\\ne a_{0,j}$ ; and if $a_{4,j-1} = a_{1,j}$ , then $a_{2,j}a_{1,j}$ and $a_{3,j-1}a_{4,j-1}$ are two distinct green in edges of $a_{4,j-1}$ , a contradiction.",
"Figure: Exchange of edges between the elements of an A-chain of type 2 withfive elements in the proof of Claim .We claim that $$ is an admissible decomposition.",
"Indeed, the only vertices that have hanging edges in $ and may not have hanging edges in $$are the vertices $ a3,j$ and $ a4,j = a2,j-1$, for $ j=0,...,s-1$,but these vertices are connection vertices of the elements in $ S$,and hence can^{\\prime }t be connection vertices of elements in $$.Therefore, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) holds for $$.Moreover, since $ T'j$ is a path, for $ j=0,...,s-1$, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:complete-commutative-types}) holds for $$.Finally, $ and $$ have the same number of open chains, and hence Definition REF () holds for $$ .",
"Therefore, $$ is an admissible decomposition of $G$ such that $\\tau () = \\tau ( - s$ , a contradiction to the minimality of $.$ Claim 12.2 Every A-chain contains at least four elements First, note that if an A-chain consists of two elements, then $G$ contains a parallel edge, which is a contradiction.",
"Thus, let $S$ be an A-chain in $ with precisely three elements, say $ T1$, $ T2$, and $ T3$.By Claim~\\ref {claim:mixed-A-chain}, we may assume $ tr(T1)=cv2(T3)$,$ tr(T2)=cv1(T1)$ and $ tr(T3)=cvi(T2)$, for $ i {1,2}$.",
"In what follows, we divide the proof depending on whether $ i=1$ or $ i=2$.$ Let $T_1=a_0a_1a_2a_3a_4a_5$ , $T_2=b_0b_1b_2b_3b_4b_5$ and $T_3 = c_0c_1c_2c_3c_4c_5$ be the elements of $S$ where $a_4={\\rm {tr}}(T_1)={\\rm {cv}}_2(T_3)=c_2$ , $b_4={\\rm {tr}}(T_2)={\\rm {cv}}_1(T_1)=a_3$ and ${c_4={\\rm {tr}}(T_3)={\\rm {cv}}_i(T_2)}$ .",
"Case $\\mathbf {i=1}$ .",
"In this case, $c_4={\\rm {tr}}(T_3)={\\rm {cv}}_1(T_2)=b_3$ .",
"Since $b_3=c_4$ , we have $b_2=c_1$ and $c_0=a_1$ .",
"Put $T_1^{\\prime } = a_0a_1a_2a_3a_4c_1$ , $T_2^{\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_3^{\\prime } = c_0c_1c_4c_3c_2a_2$ (see Figure REF ) and ${ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace }$ .",
"We claim that $T_1^{\\prime }$ , $T_2^{\\prime }$ and $T_3^{\\prime }$ are paths.",
"By Lemma REF , $T_2^{\\prime }$ is a path.",
"Since $G$ is simple, $c_1\\notin \\lbrace a_2,a_3,a_4\\rbrace $ and $a_2\\notin \\lbrace c_1,c_2,c_3,c_4\\rbrace $ .",
"By Lemma REF , we have $c_1 \\ne a_0$ and $a_2\\ne c_0$ .",
"Therefore, $T_3^{\\prime }$ is a path.",
"Finally, if $c_1 = a_1$ , then $c_2c_1$ and $a_2a_1$ are two green out edges at $a_1$ , a contradiction.",
"Therefore, $T_1^{\\prime }$ is a path, and hence definition REF () holds.",
"Also, ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_3,b_3,c_3\\rbrace .Thus, definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) holds for .Since is admissible and the new elements are paths, the elements of type~Aare still partitioned into A-chains and at most one open chain,and hence \\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free})holds for~.Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 3, a contradiction to the minimality of~.",
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[line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (1,0) – (0.5,0.8); [fatpath,backcolor1] (1,-1.6) – (0.5,-0.8) –(1,-1.6) – (0.5,-0.8); [fatpath,backcolor2] (0.5,-0.8) – (-0.5,-0.8) – (0.5,-0.8) – (-0.5,-0.8); [fatpath,backcolor2] (2,-1.6) – (1,-1.6) – (2,-1.6) – (1.5,-0.8) – (0.5,-0.8) – (1.5,-0.8); [fatpath,backcolor2] (0,0) – (0.5,0.8) – (0,0) – (0.5,0.8); 0) [black vertex] at (-1.5,0.8) ; 1) [blue vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [red vertex] at (0.5,0.8) ; ) [] at ($(1,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [red vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\prime })$ ) ; 0”) [blue vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $a_0$ ; ) [] at ($(1)+(90:.2)$ ) $a_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $a_3=b_4$ ; ) [] at ($(4)+(90:.2)$ ) $a_4$ ; ) [] at ($(0^{\\prime })+(180:.2)$ ) $b_0$ ; ) [] at ($(1^{\\prime })+(-90:.3)-(0.3,0)$ ) $b_1=a_2$ ; ) [] at ($(2^{\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\prime })+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $c_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.25)$ ) $c_3$ ; ) [] at ($(4^{\\prime \\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange performed in the proof of Claim REF in the case ${\\rm {tr}}(T_3) = {\\rm {cv}}_1(T_2)$ .",
"The red (resp.",
"blue) circles illustrate the same vertex, i.e., $a_4=c_2$ (resp.",
"$a_1=c_0$ ).",
"Case $\\mathbf {i=2}$ .",
"In this case, $c_4={\\rm {tr}}(T_3)={\\rm {cv}}_2(T_2)=b_2$ .",
"Put $T_1^{\\prime } = a_0a_1a_2a_4a_3b_5$ , ${T_2^{\\prime } = b_0b_1b_4b_3b_2c_2}$ , $T_3^{\\prime } = c_0c_1c_2c_3c_4b_1$ (see Figure REF ) and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .",
"By Lemma REF , $T_1^{\\prime }$ is a path.",
"Since $G$ is simple, we have $c_2\\notin \\lbrace b_1,b_2,b_3,b_4\\rbrace $ and $b_1\\notin \\lbrace c_2,c_3,c_4\\rbrace $ .",
"By Lemma REF , $b_2 \\ne a_0, ~c_2 \\ne b_0,~b_1 \\ne c_0$ .",
"Therefore, $T_2^{\\prime }$ is a path.",
"If $b_1 = c_1$ , then $b_2b_1$ and $c_2c_1$ are two green in edges at $c_1$ , a contradiction.",
"Therefore, $T_3^{\\prime }$ is a path.",
"Analogously to the case above, $$ is an admissible decomposition of $G$ such that $\\tau () = \\tau ( - 3$ , a contradiction to the minimality of $.$ Figure: Exchange performed in the proof of Claim in the case tr (T 3 )= cv 2 (T 2 ){\\rm {tr}}(T_3) = {\\rm {cv}}_2(T_2).",
"The red circles illustrate the same vertex.Claim 12.3 $ contains an open chain.$ Suppose, for a contradiction, that there is no open chain in $.Since $ (>0$, {~contains} an A-chain $ S = T0,T1,..., Ts-1$.By Claim~\\ref {claim:mixed-A-chain}, $ S$ is a mixed A-chain.Then we can find three consecutive elements in $ S$, say $ Tj, Tj+1,Tj+2$,such that $ cv2(Tj)= tr(Tj+1)$ and $ cv1(Tj+1) = tr(Tj+2)$.By the cyclic structure of $ S$, we may assume, without loss of generality, that $ j=0$.By Claim~\\ref {claim:no-short-A-chain}, we have $ s4$,and hence there is an element $ T3 such that ${\\rm {tr}}(T_3) = {\\rm {cv}}_i(T_2)$ , for some $i\\in \\lbrace 1,2\\rbrace $ .",
"In what follows, the proof is divided according to $i$ .",
"Let $T_0 = a_0a_1a_2a_3a_4a_5$ , $T_1 = b_0b_1b_2b_3b_4b_5$ , $T_2 = c_0c_1c_2c_3c_4c_5$ , and $T_3 = d_0d_1d_2d_3d_4d_5$ , where $a_5 = a_2$ , $b_5 = b_2$ , $c_5 = c_2$ , $d_5 = d_2$ , and $a_2a_3, b_2b_3, c_2c_3, d_2d_3 \\in M$ .",
"By the choice of $T_0$ , $T_1$ , and $T_2$ , we have ${b_4 = {\\rm {tr}}(T_1) = {\\rm {cv}}_2(T_0) = a_2}$ , ${c_4 = {\\rm {tr}}(T_2) = {\\rm {cv}}_1(T_1) = b_3}$ .",
"The exchanges of edges performed here are analogous to the exchanges performed on the proof of Claim REF of Lemma REF for elements of type A.",
"Case $\\mathbf {{\\mathbf {\\rm }{tr}}(T_3) = {\\mathbf {\\rm }{cv}}_1(T_2)}$.",
"In this case, we have $d_4 = c_3$ and, by Lemma REF , $c_2 = d_1$ .",
"Put $T_1^{\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_2^{\\prime } = c_0c_1c_4c_3c_2d_0$ , $T_3^{\\prime } = c_1d_1d_2d_3d_4d_2$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .",
"By Lemma REF , $T^{\\prime }_1$ is an element of type C. In what follows, we prove that $T_2^{\\prime }$ is a path and $T_3^{\\prime }$ is an element of type A.",
"Since $G$ is simple, we have $d_0\\notin \\lbrace c_1,c_2,c_3,c_4\\rbrace $ , and $c_1\\notin \\lbrace d_1,d_2,d_4\\rbrace $ .",
"By Lemma REF , $d_0 \\ne c_0$ , and hence, $T_2^{\\prime }$ is a path.",
"If $c_1 = d_3$ , then $b_2b_3$ and $d_2d_3$ are two edges of $M$ incident to $c_1$ , a contradiction.",
"Therefore, $T_3^{\\prime }$ is an element of type A, and hence definition REF () holds.",
"Also, ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_2=b_4,b_3,c_3\\rbrace .Note also that b_4b_2 is a hanging edge at a_2 = {\\rm {cv}}_2(T_0) in ,but not in .However, (T_0,T_1^{\\prime }) is an exceptional pair.Also, b_3 and c_3 are not connection vertices of .Since c_3 is not a connection vertex in ,the element T^{\\prime }_3 is free.Therefore, S^{\\prime }=T^{\\prime }_3,\\ldots , T_{s-1},T_1,T^{\\prime }_2 is an open chain,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for .Finally, note that an element T of type~A in \\lbrace T_1,T_2,T_3\\rbrace is either in an A-chain of different from S,which implies that T is in an A-chain of ,or is in S, which implies that T is in S^{\\prime }.Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 2, a contradiction to the minimality of .",
"}\\begin{figure}[h]\\centering \\begin{subfigure}{.45}\\centering \\scalebox {.8}{\\begin{tikzpicture}[scale=1.5]\\end{tikzpicture}[fatpath,backcolor1] (-0.5,0.8) -- (-1.5,0.8) -- (-0.5,0.8) -- (0,0) -- (0,0) -- (1,0) -- (0.5,0.8) -- (0,0);[fatpath,backcolor2] (0.5,-0.8) -- (-0.5,-0.8)-- (0.5,-0.8) -- (1,-1.6) -- (2,-1.6) -- (1.5,-0.8) -- (1,-1.6);}\\end{subfigure}\\node (0) [black vertex] at (-1.5,0.8) {};\\node (1) [black vertex] at (-0.5,0.8) {};\\node (2) [black vertex] at (0,0) {};\\node (3) [black vertex] at (1,0) {};\\node (4) [black vertex] at (0.5,0.8) {};\\node () [] at ((1,0)+(60:1)) {};\\end{figure}$4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [black vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\prime })$ ) ; 0”) [black vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $b_0$ ; ) [] at ($(1)+(90:.2)$ ) $b_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4)+(90:.2)$ ) $b_4=a_2$ ; ) [] at ($(0^{\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime })+(-90:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\prime })+(0:.55)$ ) $c_3=d_4$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $d_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.3)-(0.3,0)$ ) $d_1=c_2$ ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $d_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.25)$ ) $d_3$ ; ) [] at ($(4^{\\prime \\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); .45 [scale=1.5] [fatpath,backcolor1] (-0.5,0.8) – (-1.5,0.8) – (-0.5,0.8) – (0,0) – (0.5,0.8) – (1,0) – (0.5,-0.8); [fatpath,backcolor2] (0,0)– (0.5,-0.8) –(1,-1.6) – (2,-1.6) – (1.5,-0.8) – (1,-1.6); 0) [black vertex] at (-1.5,0.8) ; 1) [black vertex] at (-0.5,0.8) ; 2) [black vertex] at (0,0) ; 3) [black vertex] at (1,0) ; 4) [black vertex] at (0.5,0.8) ; ) [] at ($(1,0)+(60:1)$ ) ; 4') [black vertex] at ($(1,0)$ ) ; 3') [black vertex] at ($(3)+(0.5,-0.8)$ ) ; 2') [black vertex] at ($(2)+(0.5,-0.8)$ ) ; 1') [black vertex] at ($(0,0)$ ) ; 0') [black vertex] at ($(0,0)+(-1,0)$ ) ; 5') [] at ($(3,0)$ ) ; 4”) [black vertex] at ($(3^{\\prime })$ ) ; 3”) [black vertex] at ($(3)+(1,-1.6)$ ) ; 2”) [black vertex] at ($(2)+(1,-1.6)$ ) ; 1”) [black vertex] at ($(2^{\\prime })$ ) ; 0”) [black vertex] at ($(1^{\\prime \\prime })+(-1,0)$ ) ; 5”) [] at ($(3,0)$ ) ; ) [] at ($(0)+(180:.2)$ ) $b_0$ ; ) [] at ($(1)+(90:.2)$ ) $b_1$ ; ) [] at ($(2)+(90:.3)$ ) ; ) [] at ($(3)+(0:.55)$ ) $b_3=c_4$ ; ) [] at ($(4)+(90:.2)$ ) $b_4=a_2$ ; ) [] at ($(0^{\\prime })+(180:.2)$ ) $c_0$ ; ) [] at ($(1^{\\prime })+(-90:.3)-(0.3,0)$ ) $c_1=b_2$ ; ) [] at ($(2^{\\prime })+(90:.35)$ ) ; ) [] at ($(3^{\\prime })+(0:.55)$ ) $c_3=d_4$ ; ) [] at ($(4^{\\prime })+(-90:.3)$ ) ; ) [] at ($(0^{\\prime \\prime })+(180:.2)$ ) $d_0$ ; ) [] at ($(1^{\\prime \\prime })+(225:.3)-(0.3,0)$ ) $d_1=c_2$ ; ) [] at ($(2^{\\prime \\prime })+(180:.2)$ ) $d_2$ ; ) [] at ($(3^{\\prime \\prime })+(0:.25)$ ) $d_3$ ; ) [] at ($(4^{\\prime \\prime })+(270:.35)$ ) ; [line width=1.3pt,color=gray,<-] (0) – (1); [line width=1.5pt,dotted,color=green,->] (2) – (1); [line width=1.5pt,dotted,color=green,->] (3) – (4); [line width=1.5pt,dashed,color=red,->] (4) – (2); [line width=1.3pt,M edge] (2) – (3); [line width=1.5pt,dashed,color=red,<-] (0') – (1'); [line width=1.5pt,dashed,color=red,->] (4') – (2'); [line width=1.5pt,dotted,color=green,->] (2') – (1'); [line width=1.5pt,dotted,color=green,->] (3') – (4'); [line width=1.3pt,M edge] (2') – (3'); [line width=1.5pt,dashed,color=red,<-] (0”) – (1”); [line width=1.5pt,dashed,color=red,->] (4”) – (2”); [line width=1.5pt,dotted,color=green,->] (2”) – (1”); [line width=1.5pt,dotted,color=green,->] (3”) – (4”); [line width=1.3pt,M edge] (2”) – (3”); Exchange performed in the proof of Claim REF in the case ${\\rm {tr}}(T_3) = {\\rm {cv}}_1(T_2)$ .",
"Case $\\mathbf {{\\mathbf {\\rm }{tr}}(T_3) = {\\mathbf {\\rm }{cv}}_2(T_2)}$.",
"Put $T_1^{\\prime } = b_0b_1b_2b_4b_3c_2$ , $T_2^{\\prime } = c_0c_1c_4c_3c_2d_2$ , $T_3^{\\prime } = d_0d_1d_2d_3d_4c_1$ (see Figure REF ), and let $ = \\big (\\lbrace T_1,T_2,T_3\\rbrace \\big )\\cup \\lbrace T_1^{\\prime },T_2^{\\prime },T_3^{\\prime }\\rbrace $ .",
"By Lemma REF , $T^{\\prime }_1$ is an element of type C. In what follows, we prove that $T_2^{\\prime }$ and $T_3^{\\prime }$ are paths.",
"Since $G$ is simple, we have $d_2\\notin \\lbrace c_1,c_2,c_3,c_4\\rbrace $ , and $c_1\\notin \\lbrace d_2,d_3,d_4\\rbrace $ .",
"By Lemma REF , $d_2 \\ne c_0$ , and $c_1 \\ne d_0$ .",
"Therefore, $T_2^{\\prime }$ is a path.",
"If $c_1 = d_1$ , then $d_2d_1$ and $c_2c_1$ are two green in edges of $c_1$ .",
"Therefore, $T_3^{\\prime }$ is a path, and hence definition REF () holds.",
"Also, ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_2=b_4,b_3,c_3,d_3\\rbrace .Note also that b_4b_2 is a hanging edge at a_2 in but not in .However, (T_0,T_1^{\\prime }) is an exceptional pair.Also, b_3, c_3 and d_3 are not connection vertices of .Thus, since d_2 and d_3 are not connection vertices in ,the element T_4 (or T_1, if s=4) is free.Therefore, S^{\\prime }=T_4,\\ldots , T_{s-1},T_1,T^{\\prime }_2 is an open chain,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for .",
"Finally, note that an element T of type~A in \\lbrace T_1,T_2,T_3\\rbrace is either in an A-chain of different from S,which implies that T is in an A-chain of ,or is in S, which implies that T is in S^{\\prime }.Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 3, a contradiction to the minimality of .",
"}$ Figure: Exchange performed in the proof of Claim in the case tr (T 3 )= cv 2 (T 2 ){\\rm {tr}}(T_3) = {\\rm {cv}}_2(T_2).Now, let $S = T_0, T_1,\\ldots , T_{s-1}$ be an open chain in $.Let $ Tj = a0,ja1,ja2,ja3,ja4,ja5,j$, for $ j{0,...,s-1}$, where $ a2,ja3,jM$ and $ a5,j=a2,j$ for $ j{0,...,s-2}$.$ Claim 12.4 $T_1$ is an element of type A and ${\\rm {tr}}(T_1) = {\\rm {cv}}_1(T_0)$ .",
"Suppose, for a contradiction, that $T_1$ is not an element of type A or ${{\\rm {tr}}(T_1) = {\\rm {cv}}_2(T_0)}$ .",
"We claim that $T_1$ does not contain a hanging edge at ${\\rm {cv}}_1(T_0)$ .",
"Indeed, if $T_1$ is not an element of type A, then, by the definition of open chain, $T_1$ is an element of type C, and hence, by Remark REF , $T_1$ does not have a hanging edge at ${\\rm {cv}}_1(T_0)$ ; and if $T_1$ is an element of type A for which ${\\rm {tr}}(T_1) = {\\rm {cv}}_2(T_0)$ , then we have $a_{4,1} = {\\rm {tr}}(T_1) = {\\rm {cv}}_2(T_0) = a_{2,0}$ , and hence, if $a_{1,1} = a_{3,0}$ , then we have $a_{4,1} + r + g = a_{1,1} = a_{3,0} = a_{2,0} - r - g$ , which implies that $2g + 2r = 0$ , a contradiction.",
"Therefore, $T_1$ does not contain a hanging edge at ${\\rm {cv}}_1(T_0)$ .",
"By Definition REF (), there are two hanging edges at ${\\rm {cv}}_1(T_0)$ .",
"Thus, there is an element $T = u_0u_1u_2u_3u_4u_5$ in $\\lbrace T_0,T_1\\rbrace $ that contains a hanging edge, say $u_1u_0$ , at ${\\rm {cv}}_1(T_0)$ .",
"Note that all the edges incident to $a_{2,0}$ are in $E(T_0)\\cup E(T_1)$ .",
"Let $T_0^{\\prime } = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}u_0$ and $T^{\\prime } = a_{2,0}u_1u_2u_3u_4u_5$ and put $ = \\big (\\lbrace T_0,T\\rbrace \\big )\\cup \\lbrace T_0^{\\prime },T^{\\prime }\\rbrace $ .",
"By Lemma REF , $T_0^{\\prime }$ is a path; and since all the edges incident to $a_{2,0}$ are in $E(T_0)\\cup E(T_1)$ , we have $a_{2,0}\\notin \\lbrace u_1,u_2,u_3,u_4,u_5\\rbrace $ , and hence if $T$ is a path (resp.",
"an element of type A), then $T^{\\prime }$ is a path (resp.",
"an element of type A).",
"Thus Definition REF (REF ) holds for $$ .",
"To check that $$ is an admissible decomposition first note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_{4,0}\\rbrace .Thus, since T_0 is a free element, a_{4,0} is not a connection vertex in ,and hence a_{4,0} is not a connection vertex in .Note also that T_1 is either an element of type~C or a free element of type~A,and hence S^{\\prime } = T_1,\\ldots , T_{s-1} is an open chain.Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for .Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 1, a contradiction to the minimality of .",
"}$ By Claim REF , we have $s\\ge 3$ , and hence, there is an element $T_2$ in $S$ .",
"Note that, by Lemma REF , since ${\\rm {tr}}(T_1) = {\\rm {cv}}_1(T_0)$ , we have ${\\rm {aux}}(T_1) = {\\rm {cv}}_2(T_0)$ .",
"This implies that $a_{1,1}a_{0,1}\\in F_r$ because all the edges incident to $a_{1,1} = {\\rm {cv}}_2(T_0)$ are in $E(T_0)\\cup E(T_1)$ .",
"Claim 12.5 $T_2$ is of type A.",
"Suppose, for a contradiction, that $T_2$ is not of type A, then $T_2$ is an element of type C and $(T_1,T_2)$ is an exceptional pair.",
"Thus, we can write $T_2 = a_{0,2}a_{1,2}a_{2,2}a_{3,2}a_{4,2}a_{5,2}$ such that $a_{2,1}=a_{5,1} = a_{3,2}$ , and $a_{2,2}a_{1,2},a_{4,2}a_{3,2}\\in F_g$ , $a_{3,2}a_{2,2},a_{4,2}a_{5,2}\\in F_r$ , $a_{1,2}a_{0,2}\\in M_{g,r}\\cup F_g\\cup F_r$ .",
"We claim that $a_{0,1} = a_{1,2}$ .",
"Indeed, since ${a_{1,1}a_{0,1}\\in F_r}$ , we have ${a_{0,1} = a_{1,1} + r = a_{2,1} + g + r}$ , but by the definition of type C, we have ${a_{1,2} = a_{2,2} + g = a_{3,2} + r + g}$ .",
"Thus, since $a_{3,2} = a_{2,1}$ , we obtain $a_{0,1} = a_{1,2}$ .",
"Now, put $T^{\\prime }_0 = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}a_{2,1}$ , $T^{\\prime }_1 = a_{1,1}a_{4,1}a_{3,1}a_{2,1}a_{2,2}a_{0,1}$ , and $T^{\\prime }_2 = a_{0,2}a_{1,2}a_{1,1}a_{3,2}a_{4,2}a_{5,2}$ (see Figure REF ), and put $ = \\big (\\lbrace T_0,T_1,T_2\\rbrace \\big )\\cup \\lbrace T^{\\prime }_0,T^{\\prime }_1,T^{\\prime }_2\\rbrace $ .",
"By Lemma REF , $T_0^{\\prime }$ is a path; since $G$ is a simple graph, $a_{2,2}\\notin \\lbrace a_{1,1},a_{4,1},a_{3,1},a_{2,1},a_{0,1}\\rbrace $ , and hence $T^{\\prime }_1$ is a path; and since all edges incident to $a_{1,1}$ are in $E(T_0)\\cup E(T_1)$ , we have $a_{1,1} \\notin V(T_2)$ , which implies that $T^{\\prime }_2$ is a path.",
"Thus Definition REF (REF ) holds for $$ .",
"Figure: Exchange of edges between two elements of type Aand the elements of an exceptional pairin the proof of Claim .To check that $$ is an admissible decomposition first note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_{4,0},a_{1,1},a_{3,1}\\rbrace .Thus, since T_0 is a free element, a_{4,0} is not a connection vertex in ,and hence a_{4,0} is not a connection vertex in ;and since the edges of M incident to a_{1,1} and a_{3,1} are in T^{\\prime }_1,the vertices a_{1,1} and a_{3,1} are not connection vertices in .Note also that no element of S is in , and hence there are no open chains in .Thus, Definitions~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge}) and~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free})hold for~.Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 2, a contradiction to the minimality of~.",
"}$ Now, by Claim REF , we have $s\\ge 4$ .",
"In what follows, we divide the proof depending on whether ${\\rm {tr}}(T_2) = {\\rm {cv}}_1(T_1)$ or ${\\rm {tr}}(T_2) = {\\rm {cv}}_2(T_1)$ .",
"Case $\\mathbf {{\\mathbf {\\rm }{tr}}(T_2) = {\\mathbf {\\rm }{cv}}_1(T_1)}$.",
"By Lemma REF , we have $a_{1,2}={\\rm {aux}}(T_2) = {\\rm {cv}}_2(T_1)=a_{2,1}$ .",
"Put $T_0^{\\prime } = a_{0,0}a_{1,0}a_{2,0}a_{4,0}a_{3,0}a_{2,1}$ , $T_1^{\\prime } = a_{0,1}a_{1,1}a_{4,1}a_{3,1}a_{2,1}a_{0,2}$ , $T_2^{\\prime } = a_{1,1}a_{1,2}a_{2,2}a_{3,2}a_{4,2}a_{2,2}$ (see Figure REF ), let $ = \\big (\\lbrace T_0,T_1,T_2\\rbrace \\big )\\cup \\lbrace T_0^{\\prime },T_1^{\\prime },T_2^{\\prime }\\rbrace $ , and let $S^{\\prime } = T^{\\prime }_2, T_3, \\dots , T_{s-1}$ .",
"By Lemma REF , $T^{\\prime }_0$ is an element of type C. In what follows, we prove that $T_1^{\\prime }$ is a path and $T_2^{\\prime }$ is an element of type A.",
"Since $G$ is simple, we have $a_{0,2}\\notin \\lbrace a_{1,1},a_{2,1},a_{3,1},a_{4,1}\\rbrace $ , and $a_{1,1}\\notin \\lbrace a_{1,2},a_{2,2},a_{4,2}\\rbrace $ .",
"By Lemma REF , $a_{0,2} \\ne a_{0,1}$ , and hence, $T_1^{\\prime }$ is a path.",
"If $a_{1,1} = a_{3,2}$ , then $a_{2,0}a_{3,0}$ and $a_{2,2}a_{3,2}$ are two edges of $M$ incident to $a_{1,1}$ , a contradiction.",
"Therefore, $T_2^{\\prime }$ is an element of type A.",
"Figure: Exchange of edges between the three first elements in an open A-chain with at least four elements, in the case tr (T 3 )= cv 1 (T 2 ){\\rm {tr}}(T_3) = {\\rm {cv}}_1(T_2).To check that $$ is an admissible decomposition first note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_{3,0}, a_{3,1}, a_{4,0}\\rbrace ,but since T_0 is free, a_{4,0} is not a connection vertex in ,and hence a_{3,0}, a_{3,1},a_{4,0} are not connection vertices in .Thus, since a_{3,1} is not a connection vertex in ,the element T^{\\prime }_2 is free.Therefore, S^{\\prime } is an open chain,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for .Since, T^{\\prime }_0 and T^{\\prime }_1 are paths and T^{\\prime }_2 is an element of type~A,Definition~\\ref {def:semi-complete-commutative}(\\ref {def:complete-commutative-types}) holds for .",
"Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 2, a contradiction to the minimality of .",
"}\\smallskip \\textbf {Case} $ tr(T2) = cv2(T1)$\\textbf {.",
"}Put $ T0' = a0,0a1,0a2,0a4,0a3,0a2,1$,$ T1' = a0,1a1,1a4,1a3,1a2,1a2,2$,$ T2' = a0,2a1,2a2,2a3,2a4,2a1,1$ (see Figure~\\ref {fig:case5-similar2}),let $ = ({T0,T1,T2}){T0',T1',T2'}$,and let $ S' = T3, ..., Ts-1$.By Lemma~\\ref {lemma:T^{\\prime }1-is-path}, $ T'1$ is an element of type~C.",
"In what follows, we prove that $ T1'$ and $ T2'$ are paths.Since $ G$ is simple, we have $ a2,2{a1,1,a2,1,a3,1,a4,1}$,and $ a1,1{a2,2,a3,2,a4,2}$.By Lemma~\\ref {lemma:no-cycles}, $ a2,2 a0,1$, and $ a1,1 a0,2$.Therefore, $ T2'$ is a path.If $ a1,1 = a1,2$,then $ a2,2a1,2$ and $ a2,1a1,1$ are two green in edges of $ a1,1$.Therefore, $ T2'$ is a path.$ Figure: Exchange of edges between the three first elements in an open A-chain with at least four elements, in the case tr (T 3 )= cv 2 (T 2 ){\\rm {tr}}(T_3) = {\\rm {cv}}_2(T_2).To check that $$ is an admissible decomposition first note that ${\\rm {hang}}_{}(v) \\ge {\\rm {hang}}_{(v)for every v\\in V(G)\\setminus \\lbrace a_{3,0}, a_{3,1}, a_{3,2},a_{4,0}\\rbrace ,but since T_0 is free, a_{3,0}, a_{3,1}, a_{3,2},a_{4,0} are not connection vertices in ,and hence a_{4,0} is not a connection vertex in .Thus, since a_{2,2} and a_{3,2} are not connection vertices in~,the element T_3 is free.Therefore, S^{\\prime } is either an open chain or contains only one element,which is of type~C,and hence Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-commutative-hanging-edge})holds for~.Since, T^{\\prime }_0, T^{\\prime }_1 and T^{\\prime }_2 are paths,Definition~\\ref {def:semi-complete-commutative}(\\ref {def:complete-commutative-types}) holds for .",
"Analogously to the cases above, every element of type~A in is in an A-chain of .Thus, Definition~\\ref {def:semi-complete-commutative}(\\ref {def:semi-complete-free}) holds for .Therefore, is an admissible decomposition of Gsuch that \\tau () = \\tau ( - 3, a contradiction to the minimality of .This concludes the proof.",
"}$ Recall that a $\\lbrace g,r\\rbrace $ -graph $G$ is a 5-regular graph that contains the Cayley graph $X(\\Gamma ,S)$ , where $S = \\lbrace g,-g,r,-r\\rbrace $ .",
"Thus, since $S$ is closed under taking inverses, $G$ is also a $\\lbrace g,-r\\rbrace $ -, $\\lbrace -g,r\\rbrace $ -, $\\lbrace -g,-r\\rbrace $ -graph, which yields the following corollary of Theorem REF .",
"Corollary 12.1 Every $\\lbrace g,r\\rbrace $ -graph for which $2g+2r \\ne 0$ or $2g-2r\\ne 0$ admits a $P_5$ -decomposition.",
"The main result of this paper is a straightforward consequence of Corollary REF and Theorem REF .",
"Theorem 13 Every $\\lbrace g,r\\rbrace $ -graph admits a $P_5$ -decomposition."
],
[
"Conclusion and future works",
"In this paper, we verified Conjecture for (i) $(2k +1)$ -regular graphs containing a spanning $2k$ -regular power of a cycle, and (ii) 5-regular graphs containing special spanning 4-regular Cayley graphs.",
"We believe that the techniques developed here can be extended for a more general class of graphs, such as Schreier Coset Graphs (see [8]).",
"Let $G$ be a group and let $H$ be a subgroup of $G$ .",
"For $s \\in G$ , the right coset of $H$ corresponding to $s$ is the set $Hs=\\lbrace hs\\colon h \\in H\\rbrace $ .",
"Left cosets can be defined analogously.",
"Let $g_1,\\dots ,g_r$ be a sequence in $G$ whose members generate $G$ , the Schreier Right Coset Graph (SRCG) is defined as follows.",
"Its vertex set is the set of right cosets of $H$ in $G$ , for each coset $H^{\\prime }$ and each generator $g_i$ there is an edge from $H^{\\prime }$ to the right coset $H^{\\prime }g_i$ .",
"In particular, a Cayley graph is an SRCG where $H=\\lbrace 0\\rbrace $ .",
"Schreier coset graphs are generalization of Cayley “color” graphs using cosets of some specified subgroup as vertices instead of group elements.",
"In 1977, Gross [8] showed that every connected regular graph of even degree is an SRCG.",
"This implies that, if we extend our result for 5-regular graphs that contain any spanning 4-regular SRCG, then we verify the conjecture for $k=2$ .",
"Finally, we can also explore others graphs containing special spanning Cayley graphs.",
"For instance, a natural step is to examine 7-regular graphs containing a spanning 4- or 6-regular Cayley graph.",
"Also, note that the definitions of simple commutative generator and $\\lbrace g,r\\rbrace $ -graph are equivalent to Cayley graphs under the restriction of the equation $g+r = r+g$ for every pair of generators.",
"Therefore, we plan to explore other restrictions, such as $g+r \\ne r+g$ , which would extend our result for 5-regular graphs containing every spanning 4-regular Cayley graph."
],
[
"Acknowlegments",
"This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.",
"F. Botler is partially supported by CNPq (Grant 423395/2018-1) and by FAPERJ (Grant 211.305/2019)."
],
[
"Conclusion and future works",
"In this paper, we verified Conjecture for (i) $(2k +1)$ -regular graphs containing a spanning $2k$ -regular power of a cycle, and (ii) 5-regular graphs containing special spanning 4-regular Cayley graphs.",
"We believe that the techniques developed here can be extended for a more general class of graphs, such as Schreier Coset Graphs (see [8]).",
"Let $G$ be a group and let $H$ be a subgroup of $G$ .",
"For $s \\in G$ , the right coset of $H$ corresponding to $s$ is the set $Hs=\\lbrace hs\\colon h \\in H\\rbrace $ .",
"Left cosets can be defined analogously.",
"Let $g_1,\\dots ,g_r$ be a sequence in $G$ whose members generate $G$ , the Schreier Right Coset Graph (SRCG) is defined as follows.",
"Its vertex set is the set of right cosets of $H$ in $G$ , for each coset $H^{\\prime }$ and each generator $g_i$ there is an edge from $H^{\\prime }$ to the right coset $H^{\\prime }g_i$ .",
"In particular, a Cayley graph is an SRCG where $H=\\lbrace 0\\rbrace $ .",
"Schreier coset graphs are generalization of Cayley “color” graphs using cosets of some specified subgroup as vertices instead of group elements.",
"In 1977, Gross [8] showed that every connected regular graph of even degree is an SRCG.",
"This implies that, if we extend our result for 5-regular graphs that contain any spanning 4-regular SRCG, then we verify the conjecture for $k=2$ .",
"Finally, we can also explore others graphs containing special spanning Cayley graphs.",
"For instance, a natural step is to examine 7-regular graphs containing a spanning 4- or 6-regular Cayley graph.",
"Also, note that the definitions of simple commutative generator and $\\lbrace g,r\\rbrace $ -graph are equivalent to Cayley graphs under the restriction of the equation $g+r = r+g$ for every pair of generators.",
"Therefore, we plan to explore other restrictions, such as $g+r \\ne r+g$ , which would extend our result for 5-regular graphs containing every spanning 4-regular Cayley graph."
],
[
"Acknowlegments",
"This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.",
"F. Botler is partially supported by CNPq (Grant 423395/2018-1) and by FAPERJ (Grant 211.305/2019)."
]
] | 2012.05145 |
[
[
"The ACTIONFINDER: An unsupervised deep learning algorithm for\n calculating actions and the acceleration field from a set of orbit segments"
],
[
"Abstract We introduce the \"ACTIONFINDER\", a deep learning algorithm designed to transform a sample of phase-space measurements along orbits in a static potential into action and angle coordinates.",
"The algorithm finds the mapping from positions and velocities to actions and angles in an unsupervised way, by using the fact that points along the same orbit have identical actions.",
"Here we present the workings of the method, and test it on simple axisymmetric models, comparing the derived actions to those generated with the Torus Mapping technique.",
"We show that it recovers the Torus actions for halo-type orbits in a realistic model of the Milky Way to $\\sim 0.6$% accuracy with as few as 1024 input phase-space measurements.",
"These actions are much better conserved along orbits than those estimated with the St\\\"ackel fudge.",
"In our case, the reciprocal mapping from actions and angles to positions and velocities can also be learned.",
"One of the advantages of the ACTIONFINDER is that it does not require the underlying potential to be known in advance, indeed it is designed to return the acceleration field.",
"We expect the algorithm to be useful for analysing the properties of dynamical systems in numerical simulations.",
"However, our ultimate goal with this effort will be to apply it to real stellar streams to recover the Galactic acceleration field in a way that is relatively agnostic about the underlying dark matter properties or the behavior of gravity."
],
[
"Introduction",
"Actions and angles are the natural phase-space coordinates to represent the state of the constituents of an integrable, or close to integrable dynamical system, such as typical Galactic potentials [3].",
"In these coordinates, motion along an orbit is very simple, as the actions ${J}$ are preserved, while their canonically conjugate angles ${\\theta }$ advance through a cycle at a uniform rate in time (with corresponding fixed frequencies ${\\Omega }$ ).",
"These are also the natural coordinates for perturbation theory [15].",
"Furthermore, the actions are adiabatic invariants, so slow changes in the evolution of the parent system will preserve these quantities.",
"With the superb astrometry being gathered by the Gaia mission [10], it has now become possible to study the real six-dimensional phase space structure of our Galaxy in unprecedented detail.",
"Action-angle variables are being used extensively in this exploration [5], [27], [22], [29], [6], [23], notably because actions can be used to construct equilibrium distribution functions from the Jeans theorem, because they provide (in principle) the best “archaeological” information on the dynamics of the Galaxy, and because they are convenient for exploring perturbations from, for example, the Galactic bar [19], [20], [17], [26].",
"While actions and angles are thus undoubtedly useful, they are not always easy to calculate.",
"Indeed, the only potentials in which the actions are expressed analytically are those of from the family of isochrone models of which the Kepler and spherical harmonic potentials are special cases.",
"Although this model is very useful, it does not provide a good approximation to interesting stellar systems (other than those dominated by a central object).",
"Substantial theoretical efforts were therefore undertaken over the past decades to calculate approximations for the transformation from position $x$ and velocity $v$ to $J$ and $\\theta $ , and conversely, for more general and realistic galactic potentials.",
"[18] developed a method known as the Torus Mapping [2], starting from action-angle coordinates in an isochrone potential, close enough but different from the true potential.",
"Their insight was to search for a generating function in order to transform from the true actions and angles to those of the isochrone.",
"This generating function is expressed as a Fourier series expansion on the isochrone angles, the fitted Fourier coefficients being such that the true Hamiltonian remains, after the associated canonical transformation, constant for a set of true actions.",
"Once this generating function is found, the transformation from actions and angles to positions and velocities is known.",
"For the reciprocal transformation, one usually relies on separable potentials, denoted Stäckel potentials [7], [9], for which three exact integrals of the motion exist.",
"These potentials are best expressed in spheroidal coordinates, associated to a focal distance directly related to the first and second derivatives of the Stäckel potential.",
"This focal distance can be computed for the true potential at any configuration space point as if the potential was a Stäckel one, and the corresponding integrals of the motion and respective actions can be evaluated.",
"This method introduced by [1] is known as the “Stäckel fudge\".",
"Unfortunately, this transformation is not the exact inverse of the one going from actions and angles to positions and velocities as obtained from the Torus Mapping.",
"All these methods are reviewed in [25], and have all been implemented in the AGAMA dynamics package [28], [30].",
"Our aim in the present work is to provide an innovative calculation of action-angle coordinates, a method which would be able to jointly determine, from a sample of segments of orbits in positions and velocities, (i) the corresponding actions and angles, as well as (ii) the true acceleration field in which these orbits reside.",
"Moreover, (iii) the inverse transformation from actions and angles to positions and velocities should be determined easily.",
"To this end we will build a deep neural network that will try to learn on its own, and in an unsupervised way, the coordinate transformation from observables into $J$ and $\\theta $ for the particular dataset under study.",
"Figure: Sketch of the method.",
"We consider a dynamical system for which we have measured the kinematics of NN particles or “stars” along each of a sample of SS different orbits which we loosely refer to as “streams”.",
"In panel (a) we have sketched two such “streams”, each with four “stars”.",
"The observable sky positions, parallaxes (or distances) as well as the radial velocities and proper motions are first converted by the algorithm into the Cartesian system ( i n x, i n y)({}_{i}^{n}{x}, {}_{i}^{n}{y}), where the subscript denotes the stream number and the superscript denotes the star number in that stream.",
"The Cartesian coordinates are converted into action-angle coordinates of an analytic isochrone model (b), which serves as a “toy” first approximation to the real actions.",
"The central ingredient of the algorithm is the network that proposes trial generating functions G(θ,J ' )G({\\theta }, {J^{\\prime }}) of the canonical transformation between the toy coordinates (θ,J)({\\theta }, {J}) and the refined coordinates (θ ' ,J ' )({\\theta ^{\\prime }}, {J^{\\prime }}).",
"Note that because J ' {J^{\\prime }} is not known in advance, we need to iterate at this stage (c) to find the J ' {J^{\\prime }} values that are consistent with the isochrone's values of J{J}.",
"Finally (d), we also allow for a simpler canonical point-transformation with the generating function P(θ ' )P({\\theta ^{\\prime }}) to calculate the target coordinates (θ '' ,J '' )({\\theta ^{\\prime \\prime }}, {J^{\\prime \\prime }}).",
"The two networks (green arrows) are refined over the course of the training procedure by attempting to find the network weights that minimize the spread in the J '' {J^{\\prime \\prime }} values in each stream."
],
[
"The Algorithm",
"We designed the algorithm around the fact that different phase space points along an orbit have identical $J$ (and frequencies $\\Omega $ ).",
"For this to be useful, we imagine having access to groups of such phase space points along the same orbit.",
"This could be the case if we had the output of a numerical N-body simulation at a series of different time steps, or alternatively we might have such information from observations of a real dynamical system.",
"In the code we refer to a group of such phase space points along the same orbit as a “stream”, while each individual phase space point is referred to as a “star”.",
"Our use of the terms “streams” and “stars” in the software obviously betrays our ultimate motive for building it, but we note here that star streams do not precisely follow orbits, and that a significant amount of further work is required to adapt the present algorithm to properly model such structures.",
"It is relatively straightforward to devise an algorithm to implement the coordinate mapping from Cartesian $\\eta \\equiv (x, y, z, v_x, v_y, v_z)$ to $\\xi \\equiv (\\theta _1, \\theta _2, \\theta _3, J_1, J_2, J_3)$ with a deep neural network for a system for which we have a set of known $\\eta $ and $\\xi $ values.",
"(This is what we construct for the inverse transformations in Section ).",
"In this case of so-called “supervised learning”, the network calculates the derivatives of the loss function with respect to the system parameters (the millions of weights in the neural net), and adapts these parameters using a stochastic gradient descent algorithm (in our work we use the “Adam” optimization algorithm, [16]) so as to mimimize the difference between the predicted actions and angles and the corresponding “ground truth” values.",
"The learned map can subsequently be used to attribute action and angle values to data not previously seen by the network.",
"However, for the forward transformation in the present work we do not wish to rely on external software to provide us with a training dataset of action-angle coordinates in galaxy models of interest.",
"The fact that nature does not provide us with ground-truth action and angle coordinate labels provides the incentive to be able to calculate these quantities without reference to a training set.",
"The potentials of galaxies probably also do not follow simple mathematical forms, so we have a further motivation to be able to calculate the canonical transformation in a model-independent way.",
"How then can the stochastic gradient algorithm be steered towards the right solution automatically without labeled data?",
"As stated above, we know that points along the same orbit should have identical actions and frequencies.",
"In addition, the map from $\\eta $ to $\\xi $ obeys the symplectic condition for the Jacobian matrix: $M^T \\mathbb {J} M = \\mathbb {J} ,$ where $M$ is the symplectic Jacobian matrix $M_{ij} = {{\\partial \\eta _i}/{\\partial \\xi _j}}$ , and $\\mathbb {J}$ is the antisymmetric block matrix $\\mathbb {J} =\\begin{pmatrix}0 & \\mathbb {I}_3\\\\-\\mathbb {I}_3 & 0\\end{pmatrix} \\, ,$ with $\\mathbb {I}_3$ being the $3\\times 3$ identity matrix.",
"This makes it tempting to simply build a deep network that directly maps $\\eta $ to $\\xi $ , while including the above conditions as terms in the loss function (together with additional conditions that will be presented in Section ).",
"We were disappointed to find that this did not work very well.",
"In order to approximate the coordinate transformation to $\\sim 1$ % accuracy or better, one requires a deep neural net with tens of millions of parameters.",
"Yet the symplectic condition creates a very complex loss function landscape in this parameter space, which the stochastic gradient descent optimizer explores in a very inefficient way.",
"To overcome this problem, we decided to make use of a generating function for the canonical transformation, to guarantee that the coordinate transformation will be symplectic.",
"As pioneered by [18], the transformation can be made much simpler by using the action and angle coordinates of a “toy” model as a stepping stone.",
"Our procedure is sketched in Fig.",
"REF .",
"We begin by converting the observed astrometric data into Cartesian coordinates $({x},{v})$ .",
"It is worth noting that thanks to the automatic differentiation module in the pytorch [21] machine learning library, the gradients of the output quantities with respect to the astrometric observables can be calculated easily; we will make use of this feature in future work to account for observational uncertainties.",
"For the toy model, we use an isochrone potential: $\\Phi (r) = - {{G M}\\over {b + \\sqrt{b^2+r^2}}} \\, .$ Here $r$ is the (spherical) radius coordinate, $G$ is the universal gravitational constant, and the two model parameters are the mass $M$ and scale radius $b$ .",
"One may then choose the three actions in the isochrone model to be: $\\begin{split}J_{1, \\rm iso} &= L_z\\\\J_{2, \\rm iso} &= L - |L_z|\\\\J_{3, \\rm iso} &= {{G M}\\over {\\sqrt{-2 E}}} - {{1}\\over {2}} \\Bigg ( L + {{1}\\over {2}} \\sqrt{L^2 - 4 G M b} \\Bigg ) \\, ,\\end{split}$ where $L$ is the angular momentum of the particle, $L_z$ is the $z$ -component of angular momentum, and $E$ is the particle's energy.",
"The procedure to convert from $({x},{v})$ to the isochrone model's $({\\theta },{J})$ is detailed in [18], and our pytorch version is heavily inspired by the galpy isochrone implementation [4].",
"[18] found that their algorithm converged irrespective of the initial chosen values of the two isochrone parameters $M$ and $b$ , although values closer to the real system produced faster convergence.",
"Our algorithm does not appear to be very sensitive to this initial choice either, as long as $M$ is set high enough that all orbits are bound.",
"In our algorithm both the $M$ and $b$ parameters can be fitted by pytorch in the stochastic gradient descent procedure, or if desired, they can be held fixed at their initial values.",
"We now aim to find new coordinates $({\\theta ^{\\prime }},{J^{\\prime }})$ that are closer to those of the real system.",
"To this end we define an indirect type 2 generating function $G=G({\\theta },{J^{\\prime }})$ , whose derivatives give the implicit transformation: ${J} = \\, {J^{\\prime }} + {{\\partial G}\\over {\\partial {\\theta }}} \\\\{\\theta ^{\\prime }} = \\, {\\theta } + {{\\partial G}\\over {\\partial {J^{\\prime }}}} \\, .",
"$ A deep learning network will be used to propose trial variations on $G$ , and again thanks to the automatic differentiation in pytorch it is straightforward to find the ${J}$ and ${\\theta ^{\\prime }}$ values generated by $G$ .",
"Since we are actually interested in ${J^{\\prime }}$ , we iteratively find the ${J^{\\prime }}$ value that yields values of ${J}$ from Eqn.",
"REF that are the same as those of the toy isochrone.",
"We now finally update the $({\\theta ^{\\prime }},{J^{\\prime }})$ coordinates using a further transformation ${\\theta ^{\\prime \\prime }}=P({\\theta ^{\\prime }})$ that is only dependent on the updated angles.",
"This point-transformation is much simpler than $G$ , as we can obtain the updated angles and actions directly with no need for an iterative procedure, ${\\theta ^{\\prime \\prime }} = \\, P({\\theta ^{\\prime }}) \\\\{J^{\\prime \\prime }} = \\, {{\\partial {\\theta ^{\\prime }}}\\over {\\partial {\\theta ^{\\prime \\prime }}}} {J^{\\prime }} \\, ,$ and is therefore easy for the algorithm to fit.",
"As demonstrated by [14], such a transformation allows for deformations of the toy model that may be better adapted to the geometry of the real system's orbits.",
"In our experiments we found that this additional network improved the quality of the predicted actions by up to $\\sim 50$ %.",
"Python style pseudocode for the central function of the algorithm is shown in the Appendix in Listing .",
"Figure: Sketch of the (shorter) net P {\\rm net}_P neural network.",
"The ResNet-like unit blocks (left, and Eqn.",
"), are incorporated as a series of layers (right) that progressively increase in complexity up to a chosen maximum width, and then decrease symmetrically up to the layer immediately before the output.",
"The number of features nf {\\rm nf} in each unit block is indicated."
],
[
"The Action-Angle Network",
"All of the neural networks used here have the same basic architecture.",
"Thus, the generating function $G$ is given by: $G = {\\rm net}_G(\\cos ({\\theta }),\\sin ({\\theta }), {J^{\\prime }}) \\, , $ while the canonical point-transformation $P$ has the form: $P = {\\rm net}_P(\\cos ({\\theta }),\\sin ({\\theta })) \\, .",
"$ After extensive experimentation, we found that the angle coordinates were best learned by first introducing the pair of auxiliary variables $t_{x}\\equiv \\cos (\\theta )$ and $t_{y}\\equiv \\sin (\\theta )$ , as used in Eqns.",
"REF and REF .",
"This naturally takes care of the cyclic property of the angle variables.",
"The ${\\rm net}_G$ network consists of a series of 15 blocks similar to a Residual neural network (ResNet, [12]), adapted for the case of Dense (Linear) layersA linear layer $W$ with bias $b$ simply transforms an input vector $x$ to $y=x W^T+b$ .",
"The parameters $W$ and $b$ are learned by the algorithm.. An initial fully-connected linear layer takes the input features (9 in the case of ${\\rm net}_G$ and 6 in the case of ${\\rm net}_P$ ) and passes them onto a layer of 64 features.",
"Deeper layers increase in width, becoming a factor of 2 wider in the number of features per layer, up to a maximum of 1024 features per layer.",
"After the chosen maximum depth is reached (at layer 6), the layers decrease in size in a symmetric way.",
"A final fully-connected linear layer is applied to give the chosen output features, which in the case of the generating functions $G$ and $P$ is a scalar value.",
"This architecture is sketched in Fig.",
"REF .",
"The ${\\rm net}_P$ network is identical to ${\\rm net}_G$ , but consists of only 11 blocks.",
"The unit blocks were constructed as follows: $\\begin{split}{\\rm out} = W_2(& D_2( {\\rm ReLU}( \\\\W_1(& D_1( {\\rm ReLU}( {\\rm input} ))) \\,\\, ))) + {\\rm input}\\end{split}$ where $D_1$ and $D_2$ are fully connected linear layers, ${\\rm ReLU}(x)=\\max (0,x)$ is the Rectified Linear Unit activation function, and $W_1$ and $W_2$ are Weight Normalisation layers [24].",
"This setup is very similar to the canonical ResNet, designed so that the blocks learn successive corrections to the input information.",
"We show the pseudo-code of this module in Listing in the Appendix.",
"Each unit block is connected to the following block with a Dense layer, without applying an activation function.",
"Our use of Weight Normalisation layers in Eqn.",
"REF is noteworthy.",
"Most modern networks in computer vision make use of the “Batch Normalization” (BN, [13]) procedure to decouple as much as possible the fitting of the parameters in the different layers of a network.",
"This mitigates against the co-variance between layers, allowing the parameters of a deep layer to be refined even though the parameters of the higher layers are also being adjusted at the same time by the optimization algorithm.",
"However, after much experimentation, we found that the BN layers we initially used were limiting the accuracy of our algorithm.",
"This is due to the fact that this procedure operates (during training) on the data presented to it in each “batch” (i.e.",
"in small sub-samples which are chosen so that the data may fill the graphics card memory), from which it normalizes the data using the mean and standard deviation of the sub-sample.",
"Since the data at different positions within the batch belong to different streams, the unavoidable shot noise then creates substantial variation between batches, resulting in unacceptably large errors for a study such as ours.",
"We devised a work-around by training in the normal way with the BN procedure until it reached an equilibrium state, and then restarting with the BN layers frozen using parameter values calculated from the whole dataset.",
"We later realised that the “Weight Normalization” scheme gives similar accuracy to our BN “hack”, and have adopted it for the present work as it is an accepted machine learning method.",
"Between them, with the chosen depth of 15 and 11 layers, the two generating function networks have a total of $\\sim 44$ M free parameters.",
"The chosen loss function is very simple: $L = L_{J^{\\prime \\prime }, \\, \\rm spread} + L_{J_{2,3}^{\\prime \\prime }>0} + \\alpha _1 L_{{\\theta _0}^{\\prime \\prime }} \\, .$ $L_{J^{\\prime \\prime }, \\, \\rm spread}$ is the mean absolute deviation of the difference between the predicted action $J^{\\prime \\prime }$ of the stars in a stream and the mean action of that stream $\\langle J^{\\prime \\prime }\\rangle $ .",
"The term $L_{J_{2,3}^{\\prime \\prime }>0}$ penalises unphysical negative values of ${J_2}^{\\prime \\prime }$ and ${J_3}^{\\prime \\prime }$ , as follows: $L_{J_{2,3}^{\\prime \\prime }>0} = \\langle | J_2-|J_2| | \\rangle + \\langle | J_3-|J_3| | \\rangle \\, .$ Finally, the loss term $\\alpha _1 L_{{\\theta _0}^{\\prime \\prime }}$ is the mean absolute deviation of the target angles ${\\theta ^{\\prime \\prime }}$ when $\\theta =0$ is fed into the network (i.e.",
"it encourages the zero-point of the target angles to coincide with those of the isochrone toy model).",
"Following common machine learning practise, we normalised the position and velocity variables (to $20{\\rm \\,kpc}$ and $200{\\rm \\,km\\,s^{-1}}$ , respectively); thus the actions are normalized to $4000 {\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ .",
"However, for convenience, we left the angle variables in radians.",
"We chose to set the hyper-parameter to $\\alpha _1=0.1$ in Eqn.",
"REF , so as compensate for this difference in the range of the angle and action variables.",
"The algorithm proceeds by iterating over discrete “epochs” when the network derivatives are calculated and the parameters are consequently updated to obtain improved estimates for $J^{\\prime \\prime }$ and $\\theta ^{\\prime \\prime }$ ."
],
[
"The acceleration network",
"After the algorithm has converged on values of ${J^{\\prime \\prime }}$ and ${\\theta ^{\\prime \\prime }}$ that minimize the spread of ${J^{\\prime \\prime }}$ in the streams, we can use these transformed coordinates to study the system they inhabit.",
"To this end, we re-run the network once more with an updated loss function, and with an additional network (the acceleration network) turned on.",
"The structure of the acceleration network is set up as an additive correction to the acceleration of the toy isochrone model ${a_{\\rm iso}}$ : $a = {\\rm net}_{a}(x, y, z, {a_{\\rm iso}}) + a_{\\rm iso} \\, .$ Note that here we only feed in the position part of the $\\eta $ phase space vector.",
"We use a depth of 15 blocks for this network, which contains $\\sim 17.5$ M free parameters.",
"The structure of ${\\rm net}_{a}$ is essentially identical to that of ${\\rm net}_{G}$ , except that it outputs a vector.",
"If ${J^{\\prime \\prime }}$ and ${\\theta ^{\\prime \\prime }}$ are only functions of position ${x}$ and velocity ${v}$ , the chain rule implies that the mapping is constrained by the following relations: ${{d {J_i}^{\\prime \\prime }}\\over {d t}} &= {{\\partial {J_i}^{\\prime \\prime }}\\over {\\partial x_j}} \\dot{x}_j + {{\\partial {J_i}^{\\prime \\prime }}\\over {\\partial \\dot{x}_j}} \\ddot{x}_j = 0\\\\{{d {\\theta _i}^{\\prime \\prime }}\\over {d t}} &= {{\\partial {\\theta _i}^{\\prime \\prime }}\\over {\\partial x_j}} \\dot{x}_j + {{\\partial {\\theta _i}^{\\prime \\prime }}\\over {\\partial \\dot{x}_j}} \\ddot{x}_j = {\\Omega _i}^{\\prime \\prime } \\, ,$ where the $i$ index here is associated to each of the (usually three) integrals of motion $J_i$ , and $j$ is a dummy index denoting Einstein summation.",
"A brief inspection of Eqn.",
"REF reveals that once we have access to a network that delivers predictions for the position and velocity derivatives of $J^{\\prime \\prime }$ , we can solve the simultaneous equations to calculate the acceleration terms $\\ddot{x}$ .",
"While this suggests that the acceleration network (Eqn.",
"REF ) is superfluous, we nevertheless opted to employ a network to calculate $a$ for two main reasons.",
"First, it allows us to train a network that can be subsequently applied to make predictions for the acceleration on new unseen data (for instance, for “stars” without full 6-dimensional information).",
"Second, we found that solving Eqn.",
"REF for $\\ddot{x}$ using linear algebra can occasionally be problematic, due to ill-conditioned matrices.",
"Creating a separate acceleration network eliminates these problems.",
"In this second run through the algorithm, we choose the following loss function: $\\begin{split}L =& L_{dJ^{\\prime \\prime }/dt} + \\alpha _2 L_{\\rm symmetry} + \\alpha _3 L_{\\Omega ^{\\prime \\prime }, \\, \\rm spread} \\, ,\\end{split}$ where $L_{dJ^{\\prime \\prime }/dt}$ is the mean absolute value of $dJ^{\\prime \\prime }_i/dt$ , as calculated from Eqn.",
"REF .",
"$L_{\\rm symmetry}$ is an optional term to enforce a desired symmetry on the solution of the acceleration network.",
"For instance for axisymmetric potential models, we experimented with a cosine anti-similarity criterion $L_{\\rm symmetry} = 1 + \\vec{a}_R \\cdot \\vec{R}/(||\\vec{a}_R|| \\, ||\\vec{R}||)$ between the cylindrical-$R$ component of the acceleration field $\\vec{a}_R$ and the same component of the (Galactocentric) position vector $\\vec{R}$ .",
"Although this loss term helps to constrain the acceleration, for the purpose of the tests presented in Section we decided to suppress this symmetry constraint on $a$ by setting $\\alpha _2=0$ .",
"Finally, $L_{\\Omega ^{\\prime \\prime }, \\, \\rm spread}$ is another optional loss term to ensure that the frequencies calculated from Eqn.",
"for each “stream” have minimum spread.",
"This is done by calculating the mean absolute difference of the frequency of each “star” from the mean of the frequencies in the “stream”.",
"However, we found in our tests that including this term introduced slight additional scatter in the solutions (most likely because angles are much less accurately constrained than actions in our algorithm), so it too was suppressed by setting $\\alpha _3=0$ in the present work."
],
[
"Inverse Transformation",
"Deep learning can also be used to calculate the inverse transformation from actions and angles back to positions and velocities.",
"However, there are many different situations that one can envisage being confronted with.",
"For instance, one may or may not have frequencies in addition to the actions and angles, or one may already know the potential.",
"Alternatively, it is possible that the potential and frequencies are not known, but one may have groups of particles along orbits (i.e.",
"our “stars??",
"in “streams??).",
"Each of these cases would require a different network to be constructed in order to learn the inverse transformation in an unsupervised way.",
"The approach we take here is slightly different, but perhaps more realistic given the forward transformation we have presented previously.",
"We imagine having derived the forward transformation from $\\eta $ to $\\xi $ for a number of orbits.",
"This information can now be used to train a supervised network to predict $\\eta $ from $\\xi $ .",
"Since this is a supervised learning task, compared to the previous problem, it is a much easier for the network to find the correct direction for the stochastic gradient descent to improve the loss function.",
"We therefore choose a simple network architecture: $(t_x,t_y,J) = {\\rm net}_{\\rm inverse}(t_x^{\\prime \\prime },t_y^{\\prime \\prime },J^{\\prime \\prime }) \\, ,$ from which the toy model angles are calculated with $\\theta =\\operatorname{atan2}(t_y,t_x) + \\pi /2$ .",
"Just like the forward network, the inverse transformation thus employs the isochrone model as a stepping stone, with the final step being the analytic inversion of the toy model's $(\\theta ,J)$ coordinates to the 6-dimensional vector $\\eta $ .",
"Since this inverse network is not constrained by differential equations (and hence does not require any Jacobians to be calculated) it is feasible to implement it as a deeper network of 23 (or more if necessary) layers of blocks (Eqn.",
"REF ), using a total of $\\sim 34$ M free parameters."
],
[
"Results",
"The algorithm was constructed to accept as input data a set of $S$ “streams”, each with $N$ “stars”.",
"The method works with $N\\ge 2$ “stars” per “stream”, but for these initial tests we have chosen a more conservative $N=8$ , simply to make it easier for the algorithm to verify that the actions and frequencies calculated for each “stream” are constant.",
"Pytorch achieves its speed by processing the data in parallel, and so it is much more efficient to pass pytorch “tensors” of the same size onto the graphics processing unit (GPU).",
"Because of this, we expect that when the method is applied to real data, it will be convenient to break up streams with a large number of known members into smaller sub-groups (of the same size $N$ ).",
"In all the experiments below, we split the input sample into two sets, a training set with 50% and a test set with 50% of the sample.",
"Clearly, with real data we would be much more sparing with the fraction allocated to the test set!",
"Typically, we run the training process with a learning rate of $10^{-4}$ .",
"In all the tests reported below we iterate for 1024 “epochs” even though the training losses often stabilize much earlier.",
"We found that re-running the network with smaller learning rates (which is the standard procedure in machine learning to improve accuracy), resulted in only very marginal improvements to the loss function values.",
"Test samples are used in machine learning primarily to verify that the training procedure is not overfitting the data.",
"When this occurs, the loss of the training set continues to improve, while the loss values in the test set (which the algorithm does not see during training) become worse.",
"We simply ignore all further epochs once the algorithm begins to overfit.",
"Both the data and network are expressed internally as double precision floating-point numbers, as we found that the Jacobian matrices were not always calculated to sufficient precision with single precision numbers, resulting in a network that would not update correctly due to vanishing gradients.",
"Because of the fact that we find the actions iteratively starting from an initial guess provided by the toy model, the speed of the algorithm depends on how close the toy model is to the target system.",
"But in typical cases using $S=1024$ and $N=8$ , the algorithm takes $\\sim 3$ hours to complete a training run of 1024 epochs on an NVIDIA GV100 GPU with 32 GB of card memory.",
"Running times on larger datasets should scale approximately linearly with the number of data."
],
[
"Fitting isochrone models",
"Although we have built the ACTIONFINDER as a series of transformations from the analytic isochrone, it is still worth checking whether the software can fit a sample of orbital points drawn from different models of this family.",
"To this end we generated orbits within an isochrone potential with $M=3.334\\times 10^{11}{\\rm \\,M_\\odot }$ and $b=5{\\rm \\,kpc}$ (i.e.",
"$M=1.5\\times 10^6$ in N-body units where $G=1$ and distances are in ${\\rm \\,kpc}$ and time is in ${\\rm \\,Gyr}$ ), which gives a circular velocity at the Solar neighborhood ($R_\\odot =8.122{\\rm \\,kpc}$ , [11]) of $v_c(R_\\odot )=217{\\rm \\,km\\,s^{-1}}$ .",
"One of the advantages of the ACTIONFINDER is that we do not need to provide it with a fair sample of orbits within the system, almost any sample that covers the region of interest will do.",
"In the following, we imagine having access to the orbits of some objects in the “Galaxy” halo.",
"To model this situation we select a random 3-dimensional initial radius drawn uniformly in Galactocentric distance between $r=[6$ –$16]{\\rm \\,kpc}$ , together with a velocity vector that is oriented randomly (i.e.",
"isotropic) with magnitude drawn from a Gaussian of dispersion $150{\\rm \\,km\\,s^{-1}}$ .",
"Using a symplectic Leapfrog scheme, we integrate from these initial phase space locations for $0.1{\\rm \\,Gyr}$ , ensuring energy conservation to 1 part in $10^7$ , and randomly select $N=8$ phase space points along each path.",
"Since our aim is to eventually work with real astrometric data, we convert the positions and velocities of the set of orbit locations to the observable quantities: $d, \\ell , b, v_h, \\mu _\\ell , \\mu _b$ , as they would be measured from our vantage point in the Galaxy, where $d$ is Heliocentric distance, ($\\ell , b$ ) are Galactic coordinates, $v_h$ is the Heliocentric radial velocity, and $\\mu _\\ell $ and $\\mu _b$ are the proper motions along the Galactic coordinate directions.",
"With the correct input values of $M$ and $b$ , the network quickly finds the correct $J^{\\prime \\prime }$ and $\\theta ^{\\prime \\prime }$ to better than 0.01% (with a training set size of $S=1024$ ), which is reassuring given that the algorithm effectively just has to learn (in an unsupervised way) the identity operation.",
"A more interesting case occurs when we try to fit the isochrone “stream” sample above using fixed and incorrect reference $M$ and $b$ values.",
"With $M$ and $b$ both 10% lower (higher) in the toy model compared to the simulated data, using $S=1024$ we obtain action errors of $\\delta J=1.2 (2.0) {\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ and angle errors of $0.7^\\circ (1.2^\\circ )$ .",
"When $M$ and $b$ are 20% higher, the action error is $\\delta J=3.0 {\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ and the angle error is $1.4^\\circ $ .",
"The algorithm fails with a fixed 20% lower value of $M$ , because some orbits are unbound in the toy model, and thus do not have valid $(\\theta ,J)$ coordinate values.",
"Using $S=1024$ with fixed 20% higher values of $M$ and $b$ in the toy model, the inverse transformation from $(\\theta ^{\\prime },J^{\\prime })$ to $(x,v)$ is recovered to an accuracy of $\\delta x=0.3{\\rm \\,kpc}$ and $\\delta v=2.6{\\rm \\,km\\,s^{-1}}$ (again, mean absolute deviation errors)."
],
[
"Fitting a realistic Galaxy model",
"[8] fitted plausible axisymmetric density models of the main Galactic components to kinematic observations of the Milky Way to derive the potential of the Galaxy.",
"These useful models have been incorporated into the AGAMA package [30], which now allows one to transform the actions and angles of test particles in these potentials to positions and velocities with the Torus Mapping.",
"We created a set of random orbits (each with fixed triplet of actions) in the [8] model `1' in a similar way to that described in Section REF , and as before we selected $N=8$ random points along the orbits, recording their input actions and angles values, their $(x, v)$ values generated with the Torus Mapping, as well as their output $(\\theta ^{\\prime \\prime }, J^{\\prime \\prime })$ values found by ACTIONFINDER.",
"For these tests the toy isochrone $M$ and $b$ parameters are left free.",
"Using $S=1024$ , the ACTIONFINDER recovers the Torus Mapping input action-angle coordinates with an error of $\\delta J^{\\prime \\prime }=8.0{\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ in action (i.e.",
"$\\sim 0.4$ % of the Sun's action) and $\\delta \\theta ^{\\prime \\prime }=7.4^\\circ $ in angle.",
"The model accelerations are also recovered to $\\delta a=2.6$ %.",
"These errors are calculated as the mean absolute deviation between the Torus Mapping input values and the ACTIONFINDER predictions.",
"With these noise-less data (possessing only shot-noise), the derived $(\\theta ^{\\prime \\prime },J^{\\prime \\prime })$ are not very sensitive to the sample size.",
"If instead we set $S=128 (16)$ , we obtain an action error of $\\delta J^{\\prime \\prime }=11.2 (18.1){\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ and the angle error is $\\delta \\theta ^{\\prime \\prime }=8.0^\\circ $ ($9.1^\\circ $ ).",
"In Fig.",
"REF we show the evolution of the loss function in these three tests.",
"As a comparison, we also transformed back the positions and velocities of each particle into action-space with the Stäckel-fudge in AGAMA.",
"In this case, the mean absolute deviation between the Torus Mapping input actions and the Stäckel-fudge estimate is $44 {\\rm \\,km\\,s^{-1}}{\\rm \\,kpc}$ , 5 times less accurate than with the ACTIONFINDER.",
"Figure: Convergence of the loss function for different sample sizes.",
"All three tests use simulated orbits in the potential model `1', each with N=8N=8 phase-space points.",
"The sample sizes are S=1024S=1024, S=128S=128 and S=16S=16 for the top, middle and bottom panels, respectively.",
"In each case, the blue line shows the training loss while the orange line shows the validation loss.",
"The red dot shows the position of the best validation loss, which defines the epoch at which the (θ '' ,J '' )(\\theta ^{\\prime \\prime },J^{\\prime \\prime }) coordinates are extracted.With $S=1024$ the inverse transformation algorithm (which is basically the same transformation as the one done with the Torus Mapping to generate positions and velocities) recovers the $(x,v)$ originally generated with the Torus Mapping to an accuracy of $\\delta x=0.24{\\rm \\,kpc}$ and $\\delta v=4.4{\\rm \\,km\\,s^{-1}}$ ."
],
[
"Discussion and Conclusions",
"Using modern deep learning tools we have constructed a method to transform observable astrometric coordinates into action and angle coordinates with a scheme that is effectively a back-to-front version of that developed by [18], together with the canonical point-transformation improvement by [14].",
"The deep neural nets used here are able to generate non-linear functions that are more flexible than a linear decomposition into Fourier coefficients, and this flexibility may make it easier to fit more general dynamical systems.",
"A further advantage of our method is that it simultaneously uses all the data at its disposal to fit the generating functions of the canonical transformations.",
"This contrasts to earlier methods where each orbit is fitted independently to derive the best set of Fourier coefficients for that orbit, bearing in mind that each such orbit requires many constraining data points due to the large number of Fourier coefficients that have to be fit.",
"Moreover, once the transformation from positions and velocities is found, it is easy to inverse it in a supervised way.",
"The main advantage of the method, however, is that we do not need to know in advance the Hamiltonian or potential of the system under investigation.",
"The algorithm finds it for us.",
"The data used in the tests presented here are noise free (apart from shot noise), and as such they do not portray a realistic picture of the limitations of the method as applied to real data.",
"However, it is beyond the scope of the present contribution to attempt an exploration of the consequences of the observational limitations.",
"With this caveat in mind, we are able to train the network to predict the actions from positions and velocities in a realistic axisymmetric Milky Way model, with an uncertainty of $\\sim 0.6$ % of the action value of the Sun (compared to the input actions transformed into positions and velocities with the Torus Mapping), by training on only $N=8$ phase-space points along $S=128$ orbit segments, i.e.",
"with only 1024 phase-space points.",
"With $S=1024$ such orbit segments, an action uncertainty of $\\sim 0.4$ % of the action value of the Sun can be attained.",
"This is 5 times more accurate than the estimate of actions from positions and velocities with the Stäckel fudge in AGAMA.",
"In principle, the present algorithm is not limited to axisymmetric systems, and given that the canonical coordinate transformation encoded in the generating function networks can be made arbitrarily complex by increasing the depth of the networks, one may be able to learn and model complex mass distributions given sufficient data and computational resources.",
"However, the presence of chaotic dynamics in complex potentials gives cause for concern, since our method relies on the actions being conserved, which is not the case for chaotic orbits.",
"It is thus likely that in such systems the method will only work for a subset of the orbits, while returning nonsensical results for the rest.",
"We suspect that this could be turned from a bug into a feature of the method, as it may allow the present algorithm to be adapted into an automatic orbit classifier.",
"We have attempted some very preliminary tests of the method on a non-axisymmetric system by applying the algorithm to orbits integrated in a triaxial logarithmic halo model.",
"We found that the present algorithm is not able to adapt automatically to fit all orbital families in a triaxial model simultaneously.",
"The reason for this may be both because of the presence of chaotic orbits, and because the isochrone model is not a good starting point for the canonical transformation in a triaxial potential.",
"Nevertheless, we were encouraged to find that we were able to fit a generating function that gave small scatter in $J^{\\prime \\prime }$ between points on the same orbit, by selecting the input sample of “streams” from high angular momentum tube orbits.",
"In future work we will attempt to replace the isochrone model with a triaxial Stäckel model [7] as the “toy” starting point; it is plausible that this may provide the key to unlock unsupervised fitting in triaxial and more general systems.",
"But we will also need to be able to identify chaotic orbits in the system.",
"The method currently relies on the fact that the potential is static.",
"However, with the same caveats as above regarding chaos, it may be possible to generalize the algorithm so that motion is analysed in a rotating frame in which, for instance, a barred structure would appear static.",
"This might allow the method to be used for analysing non-axisymmetric rotating systems as well.",
"The algorithm was deliberately built to accept astrometric data as inputs.",
"Because pytorch processes data in parallel, this architecture makes it very simple to supply the network with multiple inputs for the same star, where the different instances could, for example, sample over the uncertainties in the astrometry, or scan over missing information in some input dimensions.",
"Thus, if a star has a missing radial velocity measurement, one may attempt to find the radial velocity value that makes the derived actions for the star agree with those of the group.",
"It is thus plausible that the present software can be adapted into a new method for detecting stellar streams, especially structures that do not possess any obvious spatial correlation.",
"The unsupervised learning technique developed here of building a network based on potentially complex corrections to simple analytic models and applying the physical constraints as loss function terms may have substantially wider applications.",
"The pytorch tensor structure is particularly powerful for this purpose, with its ability to implement automatically the derivatives of the analytic model and the correction function.",
"As we have seen, this provides an easy means to incorporate differential equations into a network.",
"A simplified demonstration version of the code, along with a sample of test data from the [8] potential model `1' will be made available on github upon publication of this manuscript.",
"RI, BF, and GM acknowledge funding from the Agence Nationale de la Recherche (ANR project ANR-18-CE31-0006, ANR-18-CE31-0017 and ANR-19-CE31-0017), from CNRS/INSU through the Programme National Galaxies et Cosmologie, and from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No.",
"834148)."
],
[
"Appendix",
"[emphstyle=, caption=Pytorch pseudocode for the ActionFinder module., emph=forward, ActionFinder,label=list:DECODE] class ActionFinder(torch.nn.Module): \"\"\" Inputs: d,l,b,vh,mul,mub astrometric phase space coordinates Output: Jddvals - the J” values Tddvals - the theta” values lossJddspread - loss due to the spread of the J” values \"\"\" def init(self,verbose=False): super().init() self.convinput2xv = convInput2xv(verbose=verbose) self.isochroneanalytic = IsochroneAnalytic(verbose=verbose) self.encoderGFG = EncoderGFG(verbose=verbose) self.encoderGFP = EncoderGFP(verbose=verbose) def forward(self,inputs): inputsxv = self.convinput2xv(inputs) convert d,l,b,vh,mul,mub to xv (J,Theta) in isochrone toy model Jiso, Tiso = self.isochroneanalytic(inputsxv, M, b) We now iterate to find J'.",
"Start with J' = mean(Jiso for each stream) Jdmean = torch.mean( Jiso, dim=1, keepdim=True) for iter in range(itermax): Jdtrial = Jdmean.clone() trial J' Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) expand mean value to fill NStars dimension TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) (T,J') GFG = self.encoderGFG(TJd) generating function G J-J' = d(G)/d(T): JmJd = jacobian(Tiso,GFG)[:,:,0,:].data remove from computational graph GFG.detach() remove from computational graph update Jdmean = torch.mean( Jiso - JmJd, dim=1, keepdim=True) if ( torch.max( torch.abs(Jdtrial - Jdmean)) < 5.e-5 and test for acceptable convergence torch.mean(torch.abs(Jdtrial - Jdmean)) < 1.e-6): break Find Jdmean once more, but now retain the computational graph Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) GFG = self.encoderGFG(TJd) JmJd = jacobian(Tiso,GFG)[:,:,0,:] Jdvals = Jiso - JmJd J' given by generating function G Jdmean = torch.mean( Jdvals, dim=1, keepdim=True) Calculate T' from generating function derivative Jdmeanfill = torch.cat([Jdmean]*NStars,dim=1) TJd = torch.cat( [Tiso,Jdmeanfill],dim=-1 ) GFG = self.encoderGFG(TJd) T' = T + d(G)/d(J'): Tdvals = (Tiso + jacobian(Jdmeanfill,GFG)[:,:,0,:]) Add in the extra freedom of a point-transformation Tddvals = self.encoderGFP( Tdvals ) final T” from generating function P dTdddTd = jacobian(Tdvals,Tddvals) d(T”)/d(T') dTddTdd = torch.inverse( dTdddTd ) d(T')/d(T”) Ji” = dT'/dTi” .",
"J' Jddvals = torch.einsum('bsji,bsj->bsi',dTddTdd,Jdvals) final J” values Jddmean = torch.mean( Jddvals, dim=1, keepdim=True) Jddspread = Jddmean - Jddvals lossJddspread = torch.mean( torch.abs( Jddspread ) ) return Jddvals, Tddvals, lossJddspread [emphstyle=, caption=Pytorch pseudocode for the unit ResDense block., emph=forward, ResDenseblock,label=list:UNITBLOCK] class ResDenseblock(torch.nn.Module): \"\"\" The ResDense unit block.",
"This layer does not change the number of features. \"\"\"",
"def init(self, nunits): super().init() \"\"\" Declaration of the layers that will be used: nunits is the number of features. \"\"\"",
"self.nunits = nunits self.dense1 = torch.nn.Linear(infeatures=self.nunits, outfeatures= self.nunits, bias=False) self.WN1 = torch.nn.utils.weightnorm(self.dense1) self.dense2 = torch.nn.Linear(infeatures=self.nunits, outfeatures= self.nunits) self.WN2 = torch.nn.utils.weightnorm(self.dense2) def forward(self, input): \"\"\" The computational graph (how the above layers are used): \"\"\" xx = torch.relu(input) xx = self.dense1(xx) xx = self.WN1(xx) xx = torch.relu(xx) xx = self.dense2(xx) xx = self.WN2(xx) output = xx + input return output"
]
] | 2012.05250 |
[
[
"A Deep Learning Approach to Anomaly Sequence Detection for\n High-Resolution Monitoring of Power Systems"
],
[
"Abstract A deep learning approach is proposed to detect data and system anomalies using high-resolution continuous point-on-wave (CPOW) or phasor measurements.",
"Both the anomaly and anomaly-free measurement models are assumed to have unknown temporal dependencies and probability distributions.",
"Historical training samples are assumed for the anomaly-free model, while no training samples are available for the anomaly measurements.",
"By transforming the anomaly-free observations into uniform independent and identically distributed sequences via a generative adversarial network, the proposed approach deploys a uniformity test for anomaly detection at the sensor level.",
"A distributed detection scheme that combines sensor level detections at the control center is also proposed that combines local detections to form more reliable detections.",
"Numerical results demonstrate significant improvement over the state-of-the-art solutions for various bad-data cases using real and synthetic CPOW and PMU data sets."
],
[
"Introduction",
"Bad-data detection is an essential and challenging part of a power grid monitoring system.",
"Bad data arise in many ways, from natural data anomaly caused by malfunctioning sensors to data attacks that are carefully designed to evade detection.",
"Whereas anomaly-free data share common characteristics, bad data may have vastly different types; in practice, rarely there are adequate historical anomaly data that can be used as training samples for bad-data detection and classification.",
"A less discussed and more challenging problem, especially for power system monitoring, is bad-sequence detection (BSD).",
"By BSD we mean, in particular, the testing of a hypothesis on whether a part of the measurement time series or a block of measurements is corrupted.",
"Instead of detecting a single anomaly data sample that may be an isolated incidence of little significance, BSD aims to capture bad sequences that are indicative of persistent anomalies.",
"A direct application of BSD is anomaly detection in the phasor-measurement unit (PMU) data stream where there are strong dependencies among data samples, and such data anomalies tend to happen not in isolation but in blocks of samples.",
"More significantly, BSD is essential in the detection of dynamic data attacks that aim to affect dispatch sequences from the control center [2], [3], [4].",
"Because such attacks are strategically designed to alter measurement sequences in specific patterns, a sample-by-sample bad-data detection is not effective.",
"Bad-data detection, identification, and removal in power system monitoring have been studied extensively [5], although the topic of BSD has not been treated.",
"Here we highlight aspects of bad-data detection most relevant to BSD and the use of machine learning techniques.",
"Some methods reviewed here are not designed specifically for BSD but the underlying ideas apply.",
"Originally proposed by Schweppe and Wildes [6], [7], a classic technique of bad-data detection is the residue error test (a.k.a $J(x)$ -test) in which the system state is first estimated using the weighted least-squares method as if there were no bad data.",
"Data anomaly is declared when the residue error (computed using the estimated state) is greater than a certain threshold.",
"The conventional residue error test and its many variations belong to the category of post-estimation bad-data detections where state estimation must be performed first.",
"As a result, inaccuracy of state estimation caused by bad data circulates back to affect the detection of bad data.",
"An alternative to post-estimation detection is the class of pre-estimation bad-data detection techniques that detect data anomaly prior to state estimation, thus breaking the path of estimation error propagation.",
"A key idea is to replace the estimated state used in the post-estimation scheme by the predicted state using the past measurements and apply residue test on the predicted measurement [8], [9].",
"A more direct pre-estimation approach is to detect anomalies directly based on features of anomaly-free data.",
"One of the earliest such techniques is the use of a neural network classifier trained by anomaly-free data [10].",
"A separate line of approaches is to extract features from the anomaly-free data.",
"Examples include the use of principal component analysis to characterize the signal subspace of the anomaly-free data [11] and the formulation of the problem as the detection of a change in measurement probability distribution [12].",
"The pre-estimation techniques are particularly relevant here because many of these techniques involve time-series data, and the idea of extracting innovation features of data for anomaly detection is particularly powerful and amenable to machine learning techniques.",
"A significant line of existing work on bad-data detection focuses on detecting the so-called false data attack by an adversary who can inject, remove, and substitute data to affect system and market operations [13], [14], [2], [15], [16], [17], [18].",
"When such an attacker is able to manipulate data from a sufficiently large number of sensors, the bad data introduced by the attacker become unobservable.",
"Specifically, the attacker can create a fake sequence of system states such that the manipulated measurements and the fake state sequence satisfy the underlying power flow equation.",
"Therefore, such a type of unobservable attack cannot be detected by techniques that rely only on the power flow model.",
"Consequently, post-estimation bad-data detection techniques tend to be ineffective.",
"There is a growing literature on the use of machine learning for bad-data detection in power systems since the mid-1990s [10].",
"A natural classification of these techniques can be based on how data are used in learning.",
"Supervised learning requires the labeled training data in the anomaly-free and anomaly cases [19], [20], [21], semi-supervised learning requires training samples for the anomaly-free data [4], [1], [22], and unsupervised learning requires no training data [19], [12].",
"An ensemble learning techniques is proposed in [23] that combines a collection of bad-data detectors.",
"Because it is difficult to obtain labeled anomaly data for training, the semi-supervised and unsupervised learning paradigms are of special significance.",
"Although not designed for power system state estimation, two types of semi-supervised anomaly detectors that use only training samples under the anomaly-free model can be applied for bad-data detection in power systems.",
"One is the one-class support vector machine (OC-SVM) [24] that separates anomaly and anomaly-free data deterministically.",
"The other is based on the idea of auto-encoder in deep neural network [25].",
"In both cases, there is an implicit assumption that anomaly data share non-overlapping domains with that of the anomaly-free data.",
"Such assumptions rarely hold in power system measurement models.",
"Statistical learning approaches to anomaly detection start from the premise that anomaly and anomaly-free data come from different probability distributions.",
"To this end, a recent work of particular relevance is [4] that focuses on dynamic data attacks of power system state estimation.",
"Although the attack models in [4] clearly suggests a BSD problem, the proposed mitigation strategy is a sample-by-sample detection scheme based on anomaly-free probability distributions from historical samples.",
"The idea of universal bad-data detection methods developed in [1], [22] is a semi-supervised learning technique that learns the inverse generative model of the anomaly-free data using a Wasserstein GAN approach [26], followed by a coincidence test.",
"The approach developed in [1], [22] is perhaps the only one that offers a detection consistency guarantee.",
"Specifically, under the assumption that measurement data samples are independent and identically distributed (i.i.d.",
"), the detection by an ideally trained deep learning network has both the false-positive and false-negative probabilities go to zero as the number of samples used in detection increases.",
"The assumption on i.i.d.",
"observations, however, does not hold for sequence detection.",
"For PMU measurements, in particular, measurements are strongly correlated, and simple decorrelation techniques do not produce independent samples essential to guarantee detection consistency."
],
[
"Summary of approach and contributions",
"We consider the problem of detecting BSD as part of the power system state estimation when neither the distribution of anomaly-free nor that of the anomaly measurements is known.",
"Also assumed unknown are the temporal dependencies of the measurements.",
"The main contribution of this work is a deep learning technique based on training samples from anomaly-free data.",
"The proposed approach builds upon and extends two classic ideas in the literature.",
"One is on the classical problem of uniformity test based on occupancy statistics [27], [28], [29] It was shown in [28] that, when the anomaly-free data are uniformly distributed i.i.d.",
"samples, the coincidence test achieves diminishing false positive and false negative error probabilities.",
"Such consistency is lost when the samples are correlated.",
"The power system measurements, however, are neither uniformly distributed nor i.i.d.",
"Combining generative adversarial network (GAN) learning and independent component analysis (ICA), we propose an ICA-GAN neural network trained to transform highly dependent time series measurements with unknown distributions to (approximately) uniformly distributed and i.i.d samples.",
"The output sequence of the ICA-GAN generator is used in a uniformity test that combines coincidence statistics to form an asymptotic most powerful test.",
"The proposed technique is applied to the measurements from the EPFL PMU network [30] and a larger synthetic North Texas network with PMU measurements [31]."
],
[
"System and Measurement model",
"The power system state $s_t$ at time $t$ is a column vector consisting of real and imaginary parts of the voltage phasors.",
"The measurement vector $z_{t}$ may include standard types of measurements such as branch power flows, power injections, and current magnitudes.",
"The measurement vector $z_{t}$ and the system state $s_{t}$ satisfy $ z_{t}=h(s_{t})+w_{t},$ where $h(\\cdot )$ is defined by the network topology, locations of measurement devices, and network parameters such as the network admittance matrix.",
"Measurement noise is denoted by $w_{t}$ .",
"Without loss of generality, we can treat the measurement variables as real by taking either the rectangular or the polar form of the complex variables and modifying and $h(\\cdot )$ in (REF ) accordingly."
],
[
"Models of data anomaly",
"We consider two types of data anomalies: natural data anomaly and adversarial data anomaly.",
"In both cases, the measurement under anomaly is given by $z_t=h(s_t) + w_t + a_t,$ where $a_t$ is the additional measurement error beyond the standard ambient noise.",
"Under natural anomaly, $a_t$ is likely non-Gaussian and independent across sensor locations.",
"Under the adversarial data anomaly, very little can be assumed about the statistical properties of $a_t$ , because an attacker with access to a substantial number of measurements and detailed system-level information can design $a_t$ to evade detection.",
"An extreme caseNote that the BSD model considered here do not make assumptions on the underlying network model and do not require information on the network topology and network parameters.",
"is when the attacker injects bad data $a_t=h(s_t+\\Delta _t)-h(s_t)$ into the measurements such that the system control center observes $z_{t}=h(s_{t}) + w_{t} + a_{t}=h(s_{t} + \\Delta _{t}) + w_{t},$ which means that the control center is deceived to believe that the actual state is $s^{\\prime }_{t}=s_{t}+ \\Delta _{t}$ using only $z_{t}$ for state estimation, .",
"Despite that the extreme adversarial attack is unobservable by any algebraic means, the attack vector $a_{t}$ does change the underlying distribution of $z$ , which is where the proposed BSD scheme can be effective in detecting such an unobservable attack."
],
[
"BSD as non-parametric hypothesis testing",
"We modeled the bad data in the most general form.",
"For simplicity, consider a stationary random sequence $(z_{t})$ .",
"Let the null hypothesis ${\\cal H}_0$ model the anomaly-free data and the alternative ${\\cal H}_1$ for the anomaly data.",
"We assume a simple hypothesis ${\\cal H}_0$ with (anomaly-free) distribution $f_0$ and a composite hypothesis ${\\cal H}_1$ consisting of all distributions some distance away from $f_0$ .",
"Specifically, $\\begin{array}{ll}{\\cal H}_0: z_{t} \\sim f_0~~{\\rm vs.}~~{\\cal H}_1: z_{t}\\sim f_1 \\in {F}_\\epsilon \\\\{F}_\\epsilon :=\\lbrace f, ||f-f_0|| \\ge \\epsilon \\rbrace \\end{array}$ where ${F}_\\epsilon $ is the set of probability distributions that are $\\epsilon $ distance (divergence) away from $f_0$ .",
"We assume that neither $f_0$ nor $f_1$ is known.",
"Under ${\\cal H}_0$ , however, we have some training samples distributed in $f_0$ and none under ${\\cal H}_1$ .",
"In the context of modern statistical machine learning, the model assumed here is a semi-supervised non-parametric inference.",
"Fig.",
"REF shows a schematic of the proposed technique.",
"At time $t$ , a measurement vector $Z_{t}=(z_{t},z_{t-1}, \\cdots ,z_{t-(M-1)})$ is passed through the ICA-GAN, which is a deep neural network trained to produce a block of uniformly-distributed independent components $V_{t}=(v_{t,1},v_{t,2},\\cdots , v_{t,N})$ .",
"The training of ICA-GAN is discussed in Sec REF , and it requires some historical measurements under the anomaly-free model.",
"A stratification (binning) layer, discussed in Sec.",
"REF , maps independent component vector $v$ to a $K$ -dimensional occupancy vector $x_t=(x_{t,1}, \\cdots , x_{t,K})$ for detection.",
"The output layer is linear classifier that produces a binary classification label $y_{t} \\in \\lbrace 0, 1\\rbrace $ , “1” for anomaly and “0” for anomaly-free.",
"The uniformity test is implemented as a coincidence test that passes a linear combination of occupancy statistics through a threshold designed to control the false positive rate of the test.",
"The threshold depends on the size of the test, the quantization level $K$ , and the number of samples for the coincidence test $N$ .",
"Figure: A schematics of ICA-GAN for BSD."
],
[
"Uniformity Test",
"The key idea that allows us to distinguish the null hypothesis under $f_0$ from the alternative distributions in ${F}_\\epsilon $ is rooted in the classical birthday problem [32]: given $N$ people in a room, what is the coincidence probability $P_c$ that there are at least two people having the same birthday?",
"It turns out that this probability is the lowest when the underlying birthday distribution is uniform [32].",
"This suggests that a test on some measure of coincidence can serve as a way to distinguish the uniform distribution from all other distributions.",
"Such a test was proposed earlier by David in [27] and more recently by Paninski [28].",
"The coincidence test by Paninski [28] considers the following binary hypotheses using conditionally i.i.d.",
"samples $(X_i \\in \\lbrace 1,\\cdots , K\\rbrace )$ from $K$ -alphabet discrete distributions $&{\\cal H}_0^{\\prime }:& X_i \\sim P_0=(\\frac{1}{K},\\cdots ,\\frac{1}{K}),\\nonumber \\\\&{\\cal H}_1^{\\prime }:& X_i \\sim P_1 \\in \\lbrace p=(p_1,\\cdots , p_K)|~ ||p-P_0|| \\ge \\epsilon \\rbrace .\\nonumber $ The intuition of the coincidence test is that, when $X_i$ are from the uniform distribution, the probability of coincidence is the lowest.",
"Specifically, let $K_1(x)$ be the number of alphabets that are realized by a single sample.",
"When $(x_i)$ s are realizations of i.i.d.",
"uniformly distributed random variables, $K_1(x)$ is statistically the highest.",
"$K_1(x)\\begin{array}{c}{\\small {\\cal H}_0^{\\prime }}\\\\\\gtrless \\\\{\\small {\\cal H}_1^{\\prime }}\\\\\\end{array}T_\\alpha $ where the threshold $T_\\alpha $ is a function of false positive level $\\alpha $ as well as the alphabet size (quantization level) $K$ , and the sample size $N$ .",
"Paninski showed that the coincidence test is consistent so long as $N$ grows faster than $\\sqrt{K}$ as $N = o(\\frac{1}{\\epsilon ^4} \\sqrt{K})$ .",
"Remarkably, the sample complexity can be significantly less than the size of the alphabet.",
"Huan and Meyn [33] later showed that the coincidence-based test also attains the best achievable probability of error decay rate with $P_e = exp(-(N^2/K)J(1 + o(1)))$ for some $J>0.2$ ."
],
[
"Stratification",
"The coincidence test is efficient to test uniformity on discrete distributions.",
"However, testing the uniformity of continuously distributed random samples without any assumptions on the density function is nontrivial [34].",
"Here we apply the $K$ -level uniform quantization to $v_t$ , which gives us $K$ -ary discrete random samples $x_t$ .",
"The quantization level $K$ and the threshold $T_\\alpha $ are the hyperparameters of the test.",
"Selecting a number of bins $K$ smaller than the number of samples would make the $K_{1}$ number not the highest under the uniform distribution and it would perform poorly.",
"However, a very big $K$ can also underperform as the alternative distribution also would have a large $K_{1}$ .",
"Guided by the asymptotic properties of $K_{1}(x)$ shown in [28], [33], we choose $K$ using a simple rule of the square of the block size of the test vector.",
"The threshold of the test statistics affects the true and false-positive probabilities of the detection.",
"The threshold $T_\\alpha $ of the $K_1$ coincidence test with the constraint on the false-positive probability to no greater than $\\alpha $ is given by $T_\\alpha = \\min \\lbrace k: \\Pr (K_1\\le k; {\\cal H}_0) \\le \\alpha \\rbrace .$ The computation of $T_\\alpha $ amounts to evaluating $P_k := \\Pr (K_1 = k; {\\cal H}_0)$ , which was given by Von Mises in [32]: $P_k=\\sum _{j=k}^{K}(-1)^{j+k} \\binom{j}{k} \\binom{K}{j}\\frac{ N!}{(N-j)!",
"}\\frac{(K-j)^{N-j}}{K^{N}}.$"
],
[
"Asymptotically most powerful test.",
"Extending the definition of coincidence we can define $K_{i}$ -coincidence as the number of bins which exactly have $i$ samples.",
"The original test proposed by David used $K_{0}$ coincidence to test.",
"Paninski used the $K_{1}$ coincidence.",
"The higher coincidence statistics $K_{i}, i\\ge 2$ , can also be used as each of these statistics provides additional information.",
"The importance of these statistics depend on the number of bins $K$ , and the number of samples $N$ .",
"Viktorova proposed to use a linear combination of statistics $(K_{i})$ with the optimal coefficients as a function of $K$ and $N$ to achieve the asymptotically most powerful test, $\\sum \\limits _{i=0}^{r}c_{i}K_{i}(x)\\begin{array}{c}{\\tiny {\\cal H}_0}\\\\\\gtrless \\\\{\\tiny {\\cal H}_1}\\end{array} T_{\\alpha ^{\\prime }}$ where the coefficient $c_{i}$ given in [29].",
"In this paper, in addition to using only $K_0(x)$ , only $K_1(x)$ and the linear combination of all higher coincidence $(K_i(x))$ , we also considered the use of a nonlinear combination of $(K_i(x))$ via an OC-SVM.",
"We trained OC-SVM with the $K_{0}, K_{1}, \\cdots K_{r}$ as input.",
"We trained the model with the anomaly-free samples and analyzed the results in the simulation section."
],
[
"ICA-GAN for Time Series",
"The coincidence test is developed to test anomalies on independent samples from distributions.",
"However, in power systems the measurements are highly correlated over time.",
"To this end, we propose ICA-GAN, a neural network that produces, approximately, i.i.d.",
"uniformly distributed samples under ${\\cal H}_0$ .",
"Independent Component Analysis (ICA) is a statistical technique originally proposed in [35] to find a linear transformation to represent a set of random variables as statistically independent random variables.",
"Nonlinear ICA was later proposed in [35] where a measurement vector $Z_t$ is represented as a nonlinear transform of latent vector $V_t$ with statistical independent components: $Z_{t} = f(V_{t})$ The goal is then to recover the inverse function $f^{-1}$ as well as the independent components $V_{t}$ based on observations of $Z_{t}$ alone.",
"ICA can be achieved by various machine learning models [36], [37].",
"ICA-GAN transforms temporally dependent data into independent components.",
"Fig.",
"REF shows a learning architecture of ICA-GAN that consists of two simultaneously trained neural networks: (i) an ICA generator and (ii) an a discriminator.",
"The training data passes through the ICA generator and the output is tested against synthetic uniformly distributed data by a discriminator.",
"Ideally, the ICA generator converges to a function that transforms the distribution of the data to the uniform distribution.",
"Figure: Learning of an ICA generator.",
"The measurement vector Z t Z_{t} is used as input to the generator.",
"The discriminator is fed by the output of the generator (converges to V t V_t) and i.i.d.",
"uniformly distributed 𝒰 t {\\cal U}_t.ICA-GAN.",
"The experiments in the paper used the values $\\alpha = 0.0001$ , $\\lambda = 0.1$ , $b = 100$ , $c = 10$ , $M = 80$ , $N = 50$ .",
"[1] : $\\alpha $ , the learning rate.",
"$\\lambda $ , the gradient penalty coefficient.",
"$b$ , the batch size.",
"$c$ , the number of iterations of the discriminator per generator iteration.",
"$M$ , the block size for the data sequence.",
"$N$ , the number of the independent components for ICA.",
"Number of training iterations $k = 0,1,\\cdots ,c$ $l = 1,\\cdots ,b$ Sample the uniform distribution samples $U_t = (u_{t,1},u_{t,2}\\cdots , u_{t,N}) $ , $u_{t,i} \\stackrel{{\\normalfont \\mbox{\\tiny \\rm i.i.d}}}{\\sim }{\\cal U}(0,1)$ .",
"Choose a random time $t$ for the time sequence.",
"Get $Z_{t}=(z_{t},z_{t-1}, \\cdots ,z_{t-(M-1)})$ measurements sequence from real data sequence.",
"$\\tilde{U} \\leftarrow g_{\\theta }(Z_t)$ $\\hat{U} \\leftarrow \\epsilon U+ (1-\\epsilon ) \\tilde{U}$ $L_{l} \\leftarrow {f_{\\omega }(\\tilde{U})}-{f_{\\omega }(U)}+ \\lambda (\\Vert \\nabla _{\\hat{U}} f_{\\omega (\\hat{U})}\\Vert _{2}-1)^{2}$ Update the discriminator parameters $\\omega $ by descending its stochastic gradient: $\\nabla _{\\omega } \\big [ \\frac{1}{b}\\sum \\limits _{l=1}^{b}{L_{l}}\\big ]$ $l = 1,\\cdots ,b$ Choose a random time $t$ for the time sequence.",
"Get $Z^{l}_{t}=(z_{t},z_{t-1}, \\cdots ,z_{t-(M-1)})$ measurements sequence from real data sequence.",
"Update the ICA-GAN generator parameters $\\theta $ by descending its stochastic gradient: $\\nabla _{\\theta } \\big [ \\frac{1}{b}\\sum \\limits _{i=1}^{b}{-f_{\\omega }(g_{\\theta }(Z^{l}_{\\lbrace t\\rbrace })} \\big ]$"
],
[
"Training ICA-GAN",
"The pseudo-code for the training of ICA-GAN is given in Algorithm REF .",
"The weights in both networks $\\theta $ , and $\\omega $ are initialized randomly and updated with the learning rate of $\\alpha $ .",
"Then ICA is trained to transform the observed measurement sequence $Z$ to $N$ independent components.",
"We use the 1-Wasserstein distance to measure the similarity between probability distributions.",
"As shown in [26], the Wasserstein distance measure tends to have improved stability of the training process.",
"To enforce the Lipschitz constraint of the 1-Wasserstein distance, we used gradient penalty on the discriminator's loss function as it is proposed in [38].",
"The discriminator $f_{\\omega }$ is updated more frequently than the ICA generator $g_{\\theta }$ .",
"We used Adam algorithm [39] for the weight updates."
],
[
"Systems simulated",
"Two sets of simulations were performed.",
"One was a four-bus system used in the EPFL Smart Grid Project[30].",
"We used the 50 Hz PMU measurements collected on April 1st, 2016 from 5 PM to 6 PM.",
"The second was based on the 133-bus synthetic North Texas transmission system [31] where one hour of 30 Hz PMU measurements were used.",
"We simulated non-interacting bad-sequences and unobservable attack sequences for each system.",
"For both systems, we used the one-phase equivalent of the three-phase systems."
],
[
"Performance measure",
"For BSD, the test was based on a block of N samples.",
"The detection performance of the algorithms was evaluated using receiver operating characteristic (ROC) curves.",
"We varied the thresholds of the algorithms to obtain a set of different true positive rates (TPR) and false positive rates (FPR) the tests.",
"We plotted the ROC curves with these values.",
"In power system applications, low false positive rates are preferred.",
"To this end, we paid special attention to the standard setting of 0.05 FPR in our performance evaluation.",
"We presented the true positive rate for the 0.05 false positive rate for each of the algorithms."
],
[
"ICA-GAN Implementation. ",
"We trained the generator with three hidden layers and 100 neurons at each hidden layer.",
"Rectified Linear Units (ReLU) at hidden layers and a linear activation function at the final layer were used as the activation functions.",
"For the discriminative network, we also used three hidden layers with 100 neurons.",
"ReLU at hidden layers and a linear activation function at the final layer were used as the activation functions.",
"Adam optimization [39] algorithm with mini-batches of 100 was used as the optimizer.",
"We separated the available data into training, validation and test sets.",
"Using EPFL data sequence we created a training set that has 600 batches of 80 consecutive anomaly-free samples, a validation set that has 400 batches of 80 consecutive anomaly-free samples and a test set that has 500 batches of 80 anomaly sequences and 500 batches of 80 anomaly-free sequences for each measurement.",
"Each test sequence consisted of 1.6 seconds of PMU measurement.",
"Using the North Texas data similarly we created a training, validation and test sets with 600, 300, 225 and another 225 batches of 80 consecutive anomaly-free samples and anomaly samples.",
"Each test sequence consisted of 2.66 seconds of PMU measurement.",
"The Wasserstein ICA-GAN was trained to obtain the transformation function from the measurements to the independent components.",
"$b = 80$ consecutive measurements were used as inputs for the generator and $N = 50$ -dimensional output of the generator was transferred to the discriminator.",
"We fed another 50-dimensional i.i.d.",
"uniform samples to the discriminator.",
"As a preprocessing step before applying ICA-GAN, we used a linear least-squares prediction to decorrelate the measurement samples.",
"The input layer of the ICA-GAN neural network was a linear least-squares predictor that whitens the input sequence.",
"After the ICA-GAN generator was used to convert the samples to i.i.d.",
"sequence samples, we used an additional step to convert the distribution of the ICA sequence to uniform distribution.",
"We used the empirical CDF of anomaly-free training samples to achieve this transformation.",
"After these steps, with the trained ICA-GAN we constructed the BSD algorithm.",
"We used the samples to apply the $K_{1}$ -coincidence test as defined in (REF ).",
"We used this approach for each measurement sequence individually.",
"If at least one anomaly measurement sequence was detected we assumed it was a successful detection.",
"The histogram of anomaly-free and anomaly test samples for one measurement are shown in Fig.",
"REF anomaly-free test data followed a uniform distribution, where the anomaly data distribution was far from that.",
"Figure: Histograms of anomaly-free test samples and anomaly test samples after ICA-GAN for simulation Case 1."
],
[
"Benchmark Techniques and Implementations",
"We compared the performance of ICA-GAN with $J(x)$ test [5], OC-SVM [24] and F-AnoGAN [25].",
"The $J(x)$ -test is the post-estimation detection method that requires no historical data for training.",
"When dealing with bad data sequence, we used the standard recursive techniques that apply the J(x)-test to isolate the measurement with the largest total residue error that was calculated over the sequence.",
"If it failed the $J(x)$ -test, the data will be declared bad and removed from the system until either the measurement data pass the test or the system becomes unobservable.",
"OC-SVM is a machine learning method that uses only anomaly-free historical data.",
"To deal with multiple bad data we implemented F-AnoGAN and OC-SVM for each individual measurement sequence individually.",
"Then, if at least one anomaly measurement sequence was detected we assumed it was a successful detection.",
"We used models that takes 80 consecutive measurement samples from a single measurement unit and gives an anomaly score as output.",
"We used the radial basis function as a nonlinear kernel.",
"We evaluated the results on the test sequences using the anomaly score function we achieved.",
"We varied the parameter $\\nu $ of SVM to get different points on the ROC curve.",
"F-AnoGAN is an auto-encoder based technique trained based on anomaly-free data.",
"We used a generator and a discriminator with 3 hidden layers in Wasserstein GAN and a deep neural network with 2 hidden layers and 100 neurons in each layer in the auto-encoder.",
"We used the 80 consecutive measurement samples as input and reduced the dimension to 50.",
"We evaluated the results considering the reconstruction error of the auto-encoder.",
"We thresholded the reconstruction error with different values to get the different points on the ROC curve."
],
[
"Case 1: Natural Anomalies",
"We considered natural anomalies involving multiple non-interacting noises.",
"We used a Gaussian Mixture noise on these multiple measurements.",
"We used bad data sequence on 4 of the measurements out of 10 measurements on EPFL system data, and on 6 measurements out of 266 measurements on North Texas synthetic system data.",
"Figure: ROC curves for anomaly case 1.",
"Left: The EPFL System.",
"Right:The Synthetic North Texas system.In Fig.",
"REF we presented the ROC curve for case 1.",
"It was seen that the ICA-GAN achieved a higher TPR than the compared methods for each level of FPR.",
"We also presented the TPR for $\\alpha = 0.05$ significance level in the in the legend of Fig.",
"REF .",
"ICA-GAN achieved the highest true positive rate on both systems.",
"The $J(x)$ -test did not perform well on the EPFL system simulation compared to Texas system simulation possibly because bad sequences are used on a larger ratio of measurements.",
"OC-SVM had close to ICA-GAN's performance on EPFL system, however it was less successful on the Texas system simulation.",
"F-AnoGAN had a worse performance than ICA-GAN and OC-SVM in both cases.",
"ICA-GAN had the best performance on both systems and had higher than $90\\%$ TPR even for small FPRs."
],
[
"Case 2: Adversarial Anomalies",
"We simulated the unobservable attacks to test the performance of the algorithms.",
"An unobservable data attack is constructed on the EPFL system and the North Texas system.",
"We used the method in [13] to obtain an attack vector $a_{t}$ .",
"We assumed the measurements in the system makes the system barely observable.",
"Then it was possible to design an unobservable data attack by manipulating some of the measurements.",
"The attack vector can be added to the measurements and it was physically not possible to detect.",
"$z_{t}^{\\prime }=z_{t} + w_{t} a_{t}.$ where we constructed $w_{t}$ from independent samples from a Gaussian Mixture Model.",
"In Fig.",
"REF we presented the ROC curve for case 2.",
"It was also seen in this case ICA-GAN had better performance than all compared methods with each FPR.It also had the highest TPR for 0.05 FPR level.",
"The $J(x)$ -test performed as if it was a random selection without using measurements in both cases.",
"OC-SVM performance was the closest to ICA-GAN, but there was still a significant difference.",
"F-AnoGAN had a worse performance than ICA-GAN and OC-SVM on both systems.",
"ICA-GAN had the best performance in both cases because; i) the independent component analysis approach transformed the correlated measurements to an independent sequence of samples that is easier to work on.",
"ii) it detected the changes in the probability distribution rather than trying to find outlier samples.",
"Figure: ROC curves for anomaly case 2.",
"Left: The EPFL System.",
"Right:The Synthetic North Texas system."
],
[
"Variants of the coincidence test",
"We compared four variants of the coincidence test: David's test [27] based on $K_0$ , Paniski's test [28] based on $K_1$ , Victorova-Chistyakov (VC) test [29] based on a linear combination of $\\lbrace K_i(x), i=1,\\cdots 15\\rbrace $ , and OC-SVM that linearly combines $\\lbrace K_i(x)\\rbrace $ using labeled samples under anomaly-free historical data.",
"Table REF shows the TPR under FPR=0.05.",
"We observed that under this setting VC test was the most powerful, i.e.",
"that minimizes the type 2 error.",
"$K_{0}$ and One-Class SVM was close to each other and $K_{1}$ test was the worst.",
"The asymptotically most powerful coincidence test also was the most powerful among the tests we tried in our simulation.",
"Table: TPR of different coincidence tests for 0.05 FPR constraint.",
"Case 1.a and Case 1.b are for natural anomalies for the EPFL and the North Texas systems, respectively.",
"Case 2.a and Case 2.b are for adversarial anomalies for the EPFL and the North Texas systems, respectively."
],
[
"Conclusion",
"We developed a deep learning architecture for BSD and cleansing for state estimation.",
"A major contribution of this work is to extend the capacities of the universal bad-data detection to time-sequential data using independent component analysis approaches.",
"We improved the universal anomaly detection algorithm by proposing to use a combination of coincidences instead of only $K_{1}$ .",
"We simulated bad-data scenarios including the challenging unobservable data attack.",
"Numerical tests show a considerable gain for the detection performance and state estimation performances over the state-of-the-art benchmark solutions."
]
] | 2012.05163 |
[
[
"Iterative Collision Resolution for Slotted ALOHA with NOMA for\n Heterogeneous Devices"
],
[
"Abstract In this paper, the problem of using uncoordinated multiple access (UMA) to serve a massive amount of heterogeneous users is investigated.",
"Leveraging the heterogeneity, we propose a novel UMA protocol, called iterative collision resolution for slotted ALOHA (IRSA) with non-orthogonal multiple access (NOMA), to improve the conventional IRSA.",
"In addition to the inter-slot successive interference cancellation (SIC) technique used in existing IRSA-based schemes, the proposed protocol further employs the intra-slot SIC technique that enables collision resolution for certain configurations of collided packets.",
"A novel multi-dimensional density evolution is then proposed to analyze and to optimize the proposed protocol.",
"Simulation results show that the proposed IRSA with NOMA protocol can efficiently exploit the heterogeneity among users and the multi-dimensional density evolution can accurately predict the throughput performance.",
"Last, an extension of the proposed IRSA with NOMA protocol to the frame-asynchronous setting is investigated, where a boundary effect similar to that in spatially-coupled low-density parity check codes can be observed to bootstrap the decoding process."
],
[
"Introduction",
"To realize the overarching ambition of internet of things (IoT) [1], [2], the current 3GPP Specification of the fifth generation cellular network technology has defined a use scenario, termed massive machine-type communication (mMTC) [3], [4].",
"This use scenario is expected to accommodate a massive number of users with sporadic activities.",
"Moreover, in such applications, the payload is assumed to be rather small.",
"In this circumstance, traditional philosophy of first coordinating then communicating becomes extremely inefficient, as there are so many users to coordinate while so small amount of data for each user to transmit.",
"Hence, uncoordinated multiple access (UMA) techniques that can communicate without first establishing coordination are highly desirable.",
"Among many UMA techniques, slotted ALOHA (SA) [5] has been adopted in many practical communication systems.",
"In an SA protocol, each user transmits a packet at the time slot next to that in which the data packet arrived.",
"When multiple users send packets at the same slot, i.e., the packets collide, these collided packets are discarded black (i.e., collision channel model) and retransmissions are scheduled according to some back-off mechanism.",
"Recently, this concept has even been adopted in modern cellular systems such as SigFoX and LoRaWAN [6].",
"Notwithstanding the success and popularity, it is well known that the conventional SA achieves an efficiency (to be defined in Section ) of at most $1/e\\approx 36.8 \\%$ [5].",
"To improve the efficiency, Casini et al.",
"in [7] proposed contention resolution diversity slotted ALOHA (CRDSA) that improves the efficiency to be at most $55\\%$ blackasymptotically as the frame size goes to infinity.",
"In CRDSA, each active user (those having data to send) sends replicated packets on twoblackHigher efficiencies can be achieved by sending more replicas.",
"randomly selected slots .",
"The collided packets are not discarded in CRDSA; instead, they are stored with the hope that inter-slotHere, we particularly highlight the term “inter-slot\" for distinguishing it from the intra-slot SIC discussed in Section .",
"successive interference cancellation (SIC) can resolve collisions by their replicas.",
"Due to its improved efficiency, a version of CRDSA has been included into the digital video broadcasting (DVB) standardization [8].",
"After the paradigm-shifting idea [7], much effort has been devoted to improve CRDSA by making connections to error-correction codes [9].",
"To the best of our knowledge, the connection was first made in [9] where a CRDSA system was shown to have a left-regular bipartite graph representation and inter-slot SIC is interpreted as belief propagation (BP) decoding [10].",
"After this connection, Liva in [9] then proposed irregular repetition slotted ALOHA (IRSA), in which each active user randomly chooses the number of repetitions and randomly selects time slots to send those replicas.",
"It can be observed that an IRSA scheme corresponds to an irregular bipartite graph.",
"It was shown in [9] that by optimizing the left degree distribution with the help of density evolution [10], one can achieve a significantly improved efficiency of $96.5\\%$ blackasymptotically.",
"It was then recognized in [11] that the coding problem induced by IRSA is similar to that of LT codes [12] with the peeling decoder [13].",
"Remarkably, with this observation, Narayanan and Pfister showed that applying the robust Soliton distribution as the left degree distribution of IRSA achieves the efficiency of $100\\%$ asymptotically [11], which is optimal among all IRSA schemes.",
"blackAlso, in [14], Liva et al.",
"proposed grouping many frames together to form a super frame and employing spatially-coupled IRSA on the super frame to achieve the efficiency of $100\\%$ asymptotically.",
"The performance of these UMA schemes were then assessed in a more realistic model in [15].",
"Several variants of IRSA have also been proposed and analyzed.",
"In [16], the frame-asynchronous version of IRSA was investigated and a boundary effect similar to that exhibited in spatially-coupled low-density parity-check (LDPC) codes [17] is observed.",
"Thanks to this boundary effect, it was shown that regular left degree distributions suffice to attain the optimal efficiency for frame-asynchronous IRSA.",
"In [18], a coded slotted ALOHA protocol was proposed where maximum distance separable (MDS) codes replace repetition codes when sending replicas.",
"It was shown that this class of schemes can achieve rates higher than that obtained by IRSA.",
"In [19], a combination of IRSA and physical-layer network coding [20] (a.k.a.",
"compute-and-forward [21]) was proposed, where the receiver decodes the received signal at a slot to an integer linear combination of collided packets.",
"These combinations are later used to recover individual packets that would not be decodable in conventional IRSA.",
"In contrast to the aforementioned works focusing on the asymptotic analysis, another series of work in the literature investigated non-asymptotic analysis of IRSA or coded slotted ALOHA.",
"For example, using packet loss rate and frame error rate as the metrics, the performance of IRSA in the waterfall region was analyzed in [22].",
"Moreover, the performance of coded slotted ALOHA in the error floor region and that in the waterfall region were separately investigated in [23] and [24], respectively.",
"Another highly related line of research named unsourced multiple access was initiated by Polyanskiy in [25], where a large number of homogeneous users wish to communicate with a receiver who is only interested in recovering as many users' data as possible without concern for their identities.",
"Using random Gaussian codebooks together with maximum likelihood decoding, [25] presented finite block-length achievability bounds.",
"A low-complexity coding scheme which employs the concatenation of an inner compute-and-forward code [21] and a outer binary adder channel code was then introduced and analyzed for unsourced multiple access in [26].",
"The current state of the art is [27] where a coded compress sensing algorithm followed by a tree-like code was proposed and analyzed.",
"Most of the works in the UMA literature, including those discussed above, considered homogeneous users.",
"However, in many applications in the mMTC use scenarios, several heterogeneous types of users coexist [4], [28], [29].",
"Therefore, it is imperative to understand how the heterogeneity would influence the system performance and it is crucial to design UMA schemes that can exploit such heterogeneity among users.",
"To address the heterogeneity among users in UMA, in this work, we incorporate the power-domain non-orthogonal multiple access (NOMA) [30], [31], [32], [33] into IRSA and proposed the IRSA with NOMA protocol.",
"In the proposed scheme, leveraging the power-domain NOMA in the physical layer, when packets coming from heterogenous types of users collide at a slot, it is possible that all the packets can be resolved by intra-slot SIC.",
"These decoded packets are then used to resolve collisions in other slots by inter-slot SIC.",
"We must emphasize that this is in sharp contrast to the idea of multi-packet reception [34].",
"In SA with multi-packet reception [35], [36], as long as the number of collided packets at a slot is smaller than some predefined parameter, all the packets can be decoded.",
"Evidently, those users are still homogeneous and the benefit of having heterogeneous users remains unexploited.",
"However, in our proposed IRSA with NOMA, not only the number of collided packets but also the types of users who transmit the packets will affect the decodabilityA more detailed comparison between SA with multi-packet reception and that with NOMA is relegated to Section ..",
"The main contributions of this paper are provided as follows.",
"A novel IRSA protocol that incorporates the power-domain NOMA, namely the IRSA with NOMA scheme, is proposed.",
"In such a protocol, depending on the types of users, multiple collided packets at a time slot may be decodable by intra-slot SIC.",
"The decoded packets are then used to resolve collisions in other slots by inter-slot SIC.",
"The connection between the proposed IRSA with NOMA and a bipartite graph with heterogeneous variable nodes and special check nodes is drawn.",
"This connection is then leveraged to propose a novel multi-dimensional density evolution, which assists us in analyzing the proposed IRSA with NOMA protocol.",
"According to the proposed multi-dimensional density evolution, a constrained optimization problem for maximizing the efficiency is formulated.",
"Through solving this constrained optimization problem, the best degree distribution of each type of users for the proposed IRSA with NOMA protocol can be found.",
"Extensive simulations are provided.",
"For the considered simulation environment with a practical frame size, it is shown that when there are two and three heterogeneous types, the proposed scheme achieves the sum (over types) efficiencies of up to $132\\%$ ($143\\%$ asymptotically) and $170\\%$ ($185\\%$ asymptotically), respectively.",
"Moreover, it also indicates that the analysis based on the proposed multi-dimensional density evolution can accurately predict the actual performance of the proposed scheme.",
"The extension of the proposed protocol to the frame-asynchronous IRSA is presented and the corresponding multi-dimensional density evolution is analyzed.",
"It is observed that under the frame-asynchronous circumstance, the proposed IRSA with NOMA also exhibits a boundary effect similar to that in [16].",
"Hence, left-regular degree distributions again suffice to achieve performance that is as good as that achieved by an optimal degree distribution.",
"blackDuring the revision of this paper, we became aware of a pioneering work [37], in which CRDSA with power unbalance is investigated from a joint physical and MAC layers perspective.",
"By allowing each user randomly selects its power according to a log-normal distribution, Mengali et al.",
"show that significant gains can be obtained by intra-slot SIC.",
"However, how to exploit heterogeneity inherent in the network, how to analyze the asymptotic performance, and how to obtain optimal degree distributions when there are multiple types in the network are left undiscussed.",
"In Section , we formally state the problem of UMA with heterogeneous types of users and propose the IRSA with NOMA protocol.",
"In Section , we present a graph representation of the proposed protocol and translate the decoding process as a modified peeling decoder for the corresponding graph.",
"In Section , to analyze the asymptotic performance of the proposed protocol, we propose a multi-dimensional density evolution technique, with which we formulate an optimization problem for obtaining optimal policies for our proposed protocol.",
"Optimal policies are then obtained by the differential evolution technique.",
"Simulation results are presented in Section , followed by blackthe extension of the proposed IRSA with NOMA scheme to the frame asynchronous setting in Sections .",
"Finally, some conclusion remarks are given in Section ."
],
[
"Notational Convention",
"We use $\\mathcal {F}[x]$ to denote a polynomial with variable $x$ and we denote by $\\mathcal {F}^{\\prime }[x]=\\mathrm {d}\\mathcal {F}[x]/\\mathrm {d}x$ and $\\mathcal {F}^{\\lbrace k\\rbrace }[x]=\\mathrm {d^k\\mathcal {F}[x]}/\\mathrm {d} x^k$ the first and $k$ -th derivative of $\\mathcal {F}[x]$ , respectively.",
"We denote by $\\mathbb {N}$ the set of natural numbers.",
"For two natural numbers $N_1, N_2\\in \\mathbb {N}$ with $N_1 < N_2$ , we denote by $[N_1]=\\lbrace 1, 2, \\ldots , N_1\\rbrace $ and $[N_1:N_2]=\\lbrace N_1, N_1+1, \\ldots , N_2\\rbrace $ .",
"We consider an UMA setting where a massive number of users coming from $T\\in \\mathbb {N}$ different types of applications wish to communicate with the base station as shown in Fig.",
"REF .",
"Suppose time is slotted and synchronized.",
"Furthermore, every $N\\in \\mathbb {N}$ successive slots are organized into a frame; frames are also perfectly synchronous among users, as in other conventional IRSA schemes [7], [9], [11].",
"A frame-asynchronous version of this problem is relegated to Section .",
"A user who has data to send within a frame is said to be active in this frame.",
"Potentially, there could be an arbitrary number of users for each type; Suppose $K^{(t)}\\in \\mathbb {N}$ of type $t\\in [T]$ users are active.",
"Each active user of type $t\\in [T]$ constructs $L$ replicas of its packet, where the value $L$ is sampled from a degree distribution $\\mathcal {L}^{(t)}$ ; then, the user sends the $L$ replicas in $L$ slots out of $N$ slots chosen uniformly at random in the frame.",
"The degree distribution is our design issue for maximizing efficiency (that will be discussed soon).",
"Figure: UMA with T=3T=3 heterogeneous types of users.blackLeveraging the natural heterogeneity and the maturity of SIC technique [38], we propose the decoder that employs power-domain NOMA [30], [31], [33], [32] at each slot.",
"Specifically, without loss of generality, we assume the decoding order of power-domain NOMA in the physical layer is given by $1\\rightarrow 2\\rightarrow \\ldots \\rightarrow T$ and we define the intra-SIC decodable patterns as follows.",
"Definition 1 (Type $t$ decodable pattern) For a slot $n\\in [N]$ , we define the transmission pattern $\\mathbf {c}_n=(c_1,\\ldots ,c_T)$ to be a $T$ -dimensional vector where the entry $c_t$ represents the total number of unresolved type $t$ packets at this slot.",
"Such a vector $\\mathbf {c}_n$ is type $t$ decodable if and only if the following conditions hold: $&\\text{(i) $c_t=1$},\\quad \\quad \\quad \\text{(ii) $c_{t^{\\prime }}\\le 1$ for $t^{\\prime }\\in [t-1]$}\\\\&\\text{(iii) $c_{\\tilde{t}}\\le \\tilde{t}-t$ for $\\tilde{t}\\in [t+1:T]$,} \\quad \\text{(iv) $\\sum _{\\tilde{t}=t+1}^Tc_{\\tilde{t}}\\le T-t$.",
"}$ The type $t$ decodable pattern is defined based on the principle of power-domain NOMA.",
"In power-domain NOMA, it is assumed that a packet can be successfully decoded by intra-slot SIC as long as each interfered type has at most one packet collided at the same slot.",
"This justifies the above conditions (i) and (ii).",
"Moreover, types having later decoding order means that they will cause smaller interference.",
"Therefore, trading a type $\\tilde{t}\\in [t+1:T]$ packet for a type $\\hat{t}>\\tilde{t}$ packet would not affect the decodability, justifying the conditions (iii) and (iv).",
"Definition 2 (Type $t$ decodable set) The type $t$ decodable set $\\mathcal {D}^{(t)}$ consists of all the type $t$ decodable patterns.",
"i.e., $\\mathcal {D}^{(t)}=\\lbrace \\mathbf {c}:\\mathbf {c}~\\text{is type $t$ decodable}\\rbrace $ .",
"Example 3 When $T=2$ , it is easy to check that $\\mathcal {D}^{(1)}=\\lbrace (1,0),(1,1)\\rbrace $ and $\\mathcal {D}^{(2)}=\\lbrace (0,1),(1,1)\\rbrace .$ When $T=3$ , we have $\\mathcal {D}^{(1)}&=\\lbrace (1,0,0), (1,0,1), (1,0,2),(1,1,0), (1,1,1)\\rbrace , \\\\\\mathcal {D}^{(2)}&=\\lbrace (0,1,0), (0,1,1), (1,1,0), (1,1,1)\\rbrace , \\\\\\mathcal {D}^{(3)}&=\\lbrace (0,0,1), (0,1,1), (1,0,1), (1,1,1)\\rbrace .$ The decoding process performed at the end of a frame then proceeds as follows.",
"For every slot $n\\in [N]$ during a frame, the corresponding transmission pattern $\\mathbf {c}_n$ is computedThe transmission patterns $\\mathbf {c}_n$ can evolve with iteration; however, we ignore the index for iterations to avoid heavy notation..",
"The intra-slot SIC is then employed to resolve a type $t$ packet if there is a $\\mathbf {c}_n$ for $n\\in [N]$ belonging to the type $t$ decodable set $\\mathcal {D}^{(t)}$ .",
"We assume that in the burst payload of each packet, there is information about the (other) slots containing copies of this packet.",
"Hence, the decoded packets are then used to cancel other replicas via inter-slot SIC.",
"This completes an iteration and the decoding process continues iteratively until no more packets are decodable for all $t\\in [T]$ and all $n\\in [N]$ or all the packets in this frame are decoded.",
"This process is done, independently, for every frame.",
"An example of the proposed IRSA with NOMA for $T=2$ can be found in Fig.",
"REF .",
"In this figure, the replicas of packets transmitted from the type 1 users, namely users 1 and 2, are colored in blue and that from the type 2 users, namely users 3 and 4, are colored in red.",
"In this example, user 1's packet can be easily decoded from slot 1.",
"Subtracting user 1's packet at slot 3 from the decoded packet, there's no more degree-1 slot and traditional IRSA would stop the decoding process.",
"For the proposed IRSA with NOMA, at slot 2, we receive the collision of a packet from type 1 (user 2) and another one from type 2 (user 4), both packets can be decoded via intra-slot SIC.",
"Hence, our proposed IRSA with NOMA would continue the decoding process by subtracting users 2 and 4's packets in slot 4 and decode user 4's packet.",
"Figure: An example of SA with heterogeneous users transmitting in a frame of 5 slots.",
"Here, we have T=2T=2 types where the first type consists of users 1 and 2 and the second type consists of users 3 and 4.At the end of the decoding process, suppose the base station can successfully decode $\\hat{K}^{(t)}$ packets of type $t\\in [T]$ , we define the (actual) efficiency of type $t\\in [T]$ achieved by the scheme as $\\hat{\\eta }^{(t)} = \\frac{\\hat{K}^{(t)}}{N},\\quad t\\in [T],$ and the (actual) sum efficiency of the scheme as $ \\hat{\\eta } = \\sum _{i=1}^T \\hat{\\eta }^{(t)}.$ It is clear that choosing different $\\mathcal {L}^{(t)}$ would result in different efficiency.",
"Throughout the paper, we call a collection of $T$ degree distributions $\\lbrace \\mathcal {L}^{(1)}, \\mathcal {L}^{(2)}, \\ldots , \\mathcal {L}^{(T)}\\rbrace $ , one for each type $t\\in [T]$ , a policy.",
"The goal of this paper is then to design the policy, such that the achieved efficiency is maximized."
],
[
"Comparison with existing models",
"In this subsection, we discuss differences between the model considered in this paper and existing models.",
"Remark 4 It is not difficult to see that when $T=1$ , i.e., there is only one type of homogeneous users, the proposed protocol reduces to the IRSA protocol in [9].",
"In light of this, the proposed protocol can be regarded as a generalization of the IRSA protocol to accommodate heterogeneous types of users.",
"Remark 5 We note that both multi-packet reception [34] and power-domain NOMA [30], [31], [32], [33] are based on the same physical-layer model, namely the multiple access channel [39].",
"However, the multi-packet reception technique treats all the users equally and assumes collided packets are decodable as long as the number of users collided at a slot is smaller than a predefined number.",
"Hence, in this homogeneous setting, the problem in the physical layer becomes to establish reliable multiple access with homogeneous users that have the same codebook and code rate.",
"To the best of our knowledge, this is not an easy task that is tackled with some success in [40] and [41], which involves either extremely large blocklength due to spatial coupling [40] or very complex operations of coding over prime fieldsThis is due to the adoption of Construction A lattices [42].. On the contrary, the power-domain NOMA lets the users transmit with different power levels and/or rates so that low-complex SIC can be employed to achieve a corner point of the capacity regionblackHere, the standard information-theoretic sense of achievability [39] is considered.. An illustration can be found in Fig.",
"REF .",
"In this paper, the considered problem exhibits natural heterogeneity among users; hence, it is more natural and suitable to work with power-domain NOMA so that the heterogeneity can be exploited by intra-SIC.",
"The main theme of this paper then centers around analyzing policies for IRSA with power-domain NOMA and designing policies that optimally exploit the heterogeneity inherent in the considered problem.",
"Figure: The capacity region of a two-user multiple access channel.Remark 6 The combination of NOMA and SA has been investigated before in [43], [44] by Choi.",
"However, in these works, it is assumed that there is only one type of homogeneous users.",
"The focus of [43] is mainly to analyze the performance of SA when users randomly choose their power levels and the receiver employs intra-slot SIC.",
"Moreover, in both works [43], [44], the inter-slot SIC technique is not considered.",
"Remark 7 We also note that our model is fundamentally different from IRSA with capture effect discussed in [9] and [45].",
"With capture effect, it is assumed that the receiver may be able to decode more than one packet with some probability.",
"This opportunity usually comes from the effect of random channel fading.",
"On the other hand, in the considered model, the receiver is able to decode more than one packet deterministically for certain configurations of collided users, due to the natural heterogeneity inherent in the problem."
],
[
"Implementation Issues",
"We now address potential implementation issues of the proposed IRSA with NOMA.",
"We would like to note that many of these issues have been encountered and addressed by the conventional IRSA schemes [9].",
"However, we still present them here for the sake of self-containedness.",
"Slot synchronization: This issue is inherent in all slotted multiple access schemes.",
"One easy solution to provide slot synchronization is to rely on stable clocks and a small amount of guard time between packets [5].",
"blackMoreover, for applications with cheap user devices, stable clocks may be unaffordable.",
"For such scenarios, [46] provides a technique to achieve slot synchronization, where each node constantly re-synchronizes its clock with the base station by exploiting the ACK signals fed back from the base station.",
"Another source of slot synchronization errors which is less seen in traditional communication systems is from significantly different travel distances experienced by different devices' signals.",
"This type of source of slot synchronization errors has become important in non-terrestrial networks (NTN) [47].",
"Fortunately, in NTN, each device is required to be GNSSGNSS stands for Global Navigation Satellite System.-capable for acquiring its own position and together with satellite ephemeris.",
"Each user can calculate and then pre-compensate its timing advance (TA) (see Section 6.3 in [47]).",
"To reduce devices' computation loading, another candidate technique currently under discussion in 3GPP is that the base station estimates and broadcasts TA to a group of devices who have similar TA (see Section 6.3 in [47]).",
"Frame synchronization: This issue is also not new and has to be cope with in every frame-synchronous systems.",
"This issue can be addressed by letting the base station periodically broadcasting a beacon signal at the beginning of each frame.",
"In Section , extension to frame asynchronous setting will be discussed.",
"Inter-slot SIC: Similar to CRDSA [7] and IRSA schemes [9], to enable inter-slot SIC, the proposed protocol requires the knowledge of the locations of other replicas once a packet is successfully decoded.",
"blackThis can be achieved by including pointers to the locations of other replicas into the burst payload or by some pseudo-random mechanism known by both the transmit and receive ends.",
"For example, the DVB system [8] adopts the latter one, where each active user reports its MAC address and a pseudo-random number to the base station.",
"Both the base station and the user can then compute the slots used for sending replicas by a common deterministic function of the MAC address and the pseudo-random number.",
"Intra-slot SIC: blackFor the proposed scheme, in order to enable intra-slot SIC at a slot, it appears that the receiver has to know which types of packets are collided in this slot.",
"This issue is new in our proposed protocol.",
"blackWe provide two ways to address this issue.",
"We note that in DVB-RCS2 [8], each active user selects the number of transmissions and a random seed for uniquely determining a pseudo random pattern.",
"This information is then stored in the burst payload and sent to the base station.",
"Given the existing data structure, to enable intra-slot SIC, each user can include additional $\\log (T)$ bits to the payload for revealing its type.",
"The second method described in the sequel eliminates the need for sending extra $\\log (T)$ bits by increasing the decoding burden.",
"Let $R^{(t)}$ be the rate of the physical-layer channel code adopted by users of type $t\\in [T]$ .",
"Let us assume the We assume that the rate tuple $(R^{(1)}, R^{(2)}, \\ldots , R^{(T)})$ can be achieved by intra-slot SIC with ascending decoding order $1\\rightarrow 2\\rightarrow \\ldots \\rightarrow T$ , without loss of generality.",
"Following the NOMA principle, a means to address this issue is as follows: Starting from $t=1$ , the base station tries to decode a packet of type $t$ ; if it succeedsThe success of decoding can be checked by a cyclic redundancy check mechanism., the decoded packet is subtracted from the received signal and we set $t=t+1$ ; otherwise, directly set $t=t+1$ and look for decoding opportunity of the next type.",
"This procedure would continue until $t=T$ ."
],
[
"Graph Representation",
"In this section, we introduce a graph representation of the proposed IRSA with NOMA protocol for determining an optimal policy in Section ."
],
[
"Bipartite Graph and Degree Distributions",
"We first note that in UMA with $T$ types of heterogeneous users, each type has its own transmission policy and number of users that may be different from other types.",
"We construct a bipartite graph representation of the proposed IRSA with NOMA protocol.",
"The graph consists of $\\sum _{t=1}^T K^{(t)}$ variable nodes and $N$ super check nodes, as shown in Fig.",
"REF .",
"Each variable node $\\mathsf {v}^{(t)}_j$ represents a user $j\\in [K^{(t)}]$ of type $t\\in [T]$ .",
"Each super check node consists of $T$ check nodes $\\mathsf {c}^{(t)}_n$ , one for each type $t\\in [T]$ .",
"A user $j$ of type $t\\in [T]$ transmits a packet in slot $n\\in [N]$ if and only if there is an edge connecting $\\mathsf {v}^{(t)}_j$ with $\\mathsf {c}^{(t)}_n$ .",
"For simplicity, we use $\\mathsf {v}^{(t)}$ and $\\mathsf {c}^{(t)}$ to denote a generic variable and check nodes of type $t\\in [T]$ .",
"Figure: A bipartite graph representation of the proposed IRSA with NOMA.With this bipartite graph representation, a user of type $t$ who transmits $d$ replicas corresponds to a degree $d$ variable node $\\mathsf {v}^{(t)}$ .",
"Moreover, a time slot in which $d$ type $t$ variable nodes transmit corresponds to a degree $d$ check node $\\mathsf {c}^{(t)}$ .",
"We define the node perspective left and right degree distributions as $\\mathcal {L}^{(t)}[x] = \\sum _{d} L^{(t)}_d x^d \\quad \\text{and}\\quad \\mathcal {R}^{(t)}[x] = \\sum _d R^{(t)}_d x^d,$ respectively, where $L^{(t)}_d$ is the probability that a variable node $\\mathsf {v}^{(t)}$ has degree $d$ and $R^{(t)}_d$ is the probability that a check node $\\mathsf {c}^{(t)}$ has degree $d$ .",
"Through the node perspective degree distributions, the edge perspective left and right degree distributions can be given by $\\lambda ^{(t)}[x] = \\sum _d \\lambda ^{(t)}_d x^{d-1}=\\frac{\\mathcal {L}^{^{\\prime }(t)}[x]}{\\mathcal {L}^{^{\\prime }(t)}[1]},$ and $\\rho ^{(t)}[x] &= \\sum _d\\rho ^{(t)}_d x^{d-1} =\\frac{\\mathcal {R}^{^{\\prime }(t)}[x]}{\\mathcal {R}^{^{\\prime }(t)}[1]},$ respectively, where $\\lambda ^{(t)}_d$ stands for the probability that an edge connects to a type $t$ variable node of degree $d$ and $\\rho ^{(t)}_d$ is the probability that an edge connects to a type $t$ check node of degree $d$ .",
"The target efficiency of type $t\\in [T]$ is then defined as $\\eta ^{(t)}= \\frac{K^{(t)}}{N} = \\frac{\\mathcal {R}^{^{\\prime }(t)}[1]}{\\mathcal {L}^{^{\\prime }(t)}[1]},$ and the target sum efficiency of the scheme is defined as $ \\eta = \\sum _{t=1}^T \\eta ^{(t)}.$ Note that the target efficiency $\\eta ^{(t)}$ is different from the actual efficiency $\\hat{\\eta }^{(t)}$ in (REF ).",
"In general, $\\hat{\\eta }^{(t)}\\le \\eta ^{(t)}$ with equality if and only if all packets are successfully decoded.",
"Example 8 Consider the scheme described in Fig.",
"REF where there is one type 1 variable node of degree 2 and one type 1 variable node of degree 3.",
"Hence, $\\mathcal {L}^{(1)}[x]=0.5x^2 +0.5x^3$ .",
"One can obtain $\\mathcal {L}^{(2)}[x]=x^3$ in a similar fashion.",
"Also, there are three type 1 check nodes of degree 1 and one type 1 check node of degree 2.",
"Therefore, $\\mathcal {R}^{(1)}[x]=0.75x+0.25x^2$ .",
"Similarly, $\\mathcal {R}^{(2)}[x]=0.5x+0.5x^2.$ Moreover, one can easily verify that the edge perspective degree distributions are given by $\\lambda ^{(1)}[x]= \\frac{2}{5}x + \\frac{3}{5}x^2, \\quad \\lambda ^{(2)}[x]=x^2, \\quad \\rho ^{(1)}[x]=\\frac{3}{5}+\\frac{2}{5}x,\\quad \\rho ^{(2)}[x]=\\frac{1}{3}+\\frac{2}{3}x.$ The graph representation of this example can be found in Fig.",
"REF -(a).",
"Figure: The graph representation of the scheme in Fig.",
"and its decoding output at each iteration of the modified peeling decoder."
],
[
"Modified Peeling Decoder",
"Here, we propose a modified peeling decoder for the bipartite graph that corresponds to the decoding procedure of the proposed IRSA with NOMA.",
"The decoder operates in an iterative fashion.",
"For each iteration, the decoder looks at every super check node $n\\in [N]$ and computes the remaining degrees of check nodes $\\mathsf {c}^{(t)}_1, \\mathsf {c}^{(t)}_2, \\ldots , \\mathsf {c}^{(t)}_T$ , which are stored in the transmission pattern $\\mathbf {c}_n$ .",
"We then look for an $n$ such that $\\mathbf {c}_n$ belongs to the type $t$ decodable set $\\mathcal {D}^{(t)}$ for some $t\\in [T]$ .",
"If such an $n$ exists, we use intra-slot SIC to decode all the type $t$ packets satisfying $\\mathbf {c}_n\\in \\mathcal {D}^{(t)}$ in this super check node $n$ .",
"For a variable node $\\mathsf {v}^{(t)}$ that is decoded, all the edges connected with it are removed from the graph.",
"This concludes an iteration.",
"The modified peeling decoder operates iteratively until no more decoding opportunity is found.",
"Example 8 (continued) In this example, we illustrate how to use the modified peeling decoder to decode the scheme in Example REF .",
"As shown in Fig.",
"REF -(b), in iteration 1, $\\mathsf {c}^{(1)}_1$ has degree 1 and $\\mathsf {c}^{(2)}_1$ has degree 0, i.e., $\\mathbf {c}_1=(1,0)\\in \\mathcal {D}^{(1)}$ ; hence, $\\mathsf {v}^{(1)}_1$ is decoded and all its edges are removed.",
"Then, in iteration 2 shown in Fig.",
"REF -(c), $\\mathsf {c}^{(1)}_2$ has degree 1 and $\\mathsf {c}^{(2)}_2$ has degree 1, i.e., $\\mathbf {c}_2=(1,1)$ belonging to both $\\mathcal {D}^{(1)}$ and $\\mathcal {D}^{(2)}$ ; thus, both $\\mathsf {v}^{(1)}_2$ and $\\mathsf {v}^{(2)}_2$ can be decoded by intra-slot SIC.",
"All the edges connected to $\\mathsf {v}^{(1)}_2$ and $\\mathsf {v}^{(2)}_2$ are subsequently removed.",
"Last, in iteration 3 shown in Fig.",
"REF -(d), one can easily see that $\\mathsf {v}^{(2)}_1$ can be decoded."
],
[
"Multi-Dimensional Density Evolution and Convergence Analysis",
"In this section, to analyze the IRSA with NOMA, we propose a novel multi-dimensional density evolution based on the graph representation described in Section .",
"With this multi-dimensional density evolution, we then provide the convergence analysis of the IRSA with NOMA under modified peeling decoding.",
"With an assist from the convergence analysis, we formulate and numerically solve an optimization problem that finds best left degree distributions, leading to the highest asymptotic efficiency.",
"For clearly explaining the ideas behind our scheme and analysis, we focus on $T=2$ .",
"The general $T$ case can be similarly analyzed."
],
[
"Proposed Multi-Dimensional Density Evolution",
"It is well known that for LDPC codes over a binary erasure channel, BP and peeling decoders have the same performanceIt essentially means that the order of the limit of average residue erasure probability as the number of iterations tends to $\\infty $ and the limit of that as the blocklength tends to $\\infty $ is exchangeable.",
"The interested reader is referred to [10]The reason that we propose using the peeling decoder instead of BP is because the peeling decoder potentially leads to a smaller decoding latency in practice as it does not have to wait until the end of the frame in order to start decoding; it can start decoding right away once a decodable slot shows up..",
"Moreover, density evolution is an outstanding tool for analyzing the performance of BP decoding over a sparse graph [10].",
"Thus, this section proposes a novel multi-dimensional density evolution to analyze the performance of the proposed scheme under BP decoding.",
"blackSimilar to that introduced in [10], the BP decoding is an iterative algorithm where in each iteration, each variable node computes the belief of its own message and passes an extrinsic version of it to each connected check node, while each check node computes its belief of the message of each connected variable node and passes an extrinsic version to that variable node.",
"After a predefined number of iterations, the decoding algorithm halts and outputs hard decisions according to the last beliefs held by the variable nodes.",
"blackConsider BP decoding described above.",
"For $t\\in [2]$ , we denote by $x_\\ell ^{(t)}$ and $y_\\ell ^{(t)}$ the average erasure probability of the message passed along an edge from $\\mathsf {v}^{(t)}$ to $\\mathsf {c}^{(t)}$ and that of the message passed along an edge from $\\mathsf {c}^{(t)}$ to $\\mathsf {v}^{(t)}$ in iteration $\\ell $ , respectively.",
"At iteration $\\ell =0$ , according to the decodable set $\\mathcal {D}^{(t)}$ specified in Example REF , for an edge connected to a type $t$ check node $\\mathsf {c}^{(t)}$ , the outgoing message is not in erasure if and only if a) $\\mathsf {c}^{(t)}$ has degree 1; b) $\\mathsf {c}^{(\\bar{t})}$ has a degree either 0 or 1 for every $\\bar{t}\\in [2],~\\bar{t}\\ne t$ .",
"Hence, we initialize $y_0^{(t)}$ for initial iteration $\\ell =0$ to be $y_0^{(t)}= 1-\\rho ^{(t)}[0]\\cdot \\left( \\mathcal {R}^{(\\bar{t})}[0] + \\mathcal {R}^{^{\\prime }(\\bar{t})}[0] \\right),$ where $\\mathcal {R}^{(\\bar{t})}[0]=R^{(\\bar{t})}_0$ and $\\mathcal {R}^{^{\\prime }(\\bar{t})}[0]=R^{(\\bar{t})}_1$ are the fractions of type $\\bar{t}$ check nodes having degree 0 and degree 1, respectively.",
"Moreover, the term $\\rho ^{(t)}[0]=\\rho _1^{(t)}$ is the probability that an edge connects to a type $t$ check node of degree 1.",
"Suppose we have obtained $x_\\ell ^{(t)}$ and $y_\\ell ^{(t)}$ for all $t\\in [T]$ from iteration $\\ell $ .",
"In iteration $\\ell +1$ , for an edge incident to a variable node $\\mathsf {v}^{(t)}$ with degree $d$ , the only possibility that the message along this edge to a $\\mathsf {c}^{(t)}$ is in erasure is that all the other $d-1$ edges are in erasure.",
"Therefore, the probability that the message passed along this edge is in erasure is $(y_\\ell ^{(t)})^{d-1}$ .",
"Now, averaging over all the edges results in the average erasure probability $x_\\ell ^{(t)}= \\sum _d \\lambda ^{(t)}_d (y_\\ell ^{(t)})^{d-1} = \\lambda ^{(t)}[y_\\ell ^{(t)}].$ For an edge incident to a check node $\\mathsf {c}^{(t)}$ with degree $d$ , the message passed along this edge to a $\\mathsf {v}^{(t)}$ is not in erasure if and only if a) all the other $d-1$ edges incident to $\\mathsf {c}^{(t)}$ are not in erasure in the previous iteration; and b) all but at most one of the edges incident to $\\mathsf {c}^{(\\bar{t})}$ are not in erasure in the previous iteration for every $\\bar{t}\\in [2],~\\bar{t}\\ne t$ .",
"Suppose in the same super check node $n$ , $\\mathsf {c}^{(\\bar{t})}_n$ has degree $d_{\\bar{t}}$ .",
"Then the above event has probability $(1-x_\\ell ^{(t)})^{d-1}\\cdot \\left( (1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}} + d_{\\bar{t}} x_\\ell ^{(\\bar{t})}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-1}\\right),$ where $(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}}$ is the probability that all $d_{\\bar{t}}$ edges are not erased and $d_{\\bar{t}} x_\\ell ^{(\\bar{t})}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-1}$ is the probability that all but one edges are not erased.",
"Now, averaging over all the edges and over all the type $\\bar{t}$ check nodes for $\\bar{t}\\in [2],~\\bar{t}\\ne t$ shows that the average probability of correct decoding is given by $&\\sum _d \\lambda ^{(t)}_d(1-x_\\ell ^{(t)})^{d-1}\\cdot \\nonumber \\sum _{d_{\\bar{t}}}R^{(\\bar{t})}_{d_{\\bar{t}}}\\left( (1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}} + d_{\\bar{t}} x_\\ell ^{(\\bar{t})}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-1}\\right) \\nonumber \\\\&=\\rho ^{(t)}[1-x_\\ell ^{(t)}] \\left( \\mathcal {R}^{(\\bar{t})}[1-x_\\ell ^{(\\bar{t})}] + x_\\ell ^{(\\bar{t})}\\mathcal {R}^{^{\\prime }(\\bar{t})}[1-x_\\ell ^{(\\bar{t})}]\\right).$ Therefore, the average erasure probability becomes $y_{\\ell +1}^{(t)}= 1-\\rho ^{(t)}[1-x_\\ell ^{(t)}]\\cdot \\left( \\mathcal {R}^{(\\bar{t})}[1-x_\\ell ^{(\\bar{t})}] + x_\\ell ^{(\\bar{t})}\\mathcal {R}^{^{\\prime }(\\bar{t})}[1-x_\\ell ^{(\\bar{t})}]\\right).$ Plugging (REF ) into (REF ) leads to the evolution of average erasure probability of a type $t$ check node as shown in (REF ) in the bottom of next page.",
"Figure: NO_CAPTION"
],
[
"Convergence and Stability Condition",
"After obtaining the density evolution in (REF ), for any given degree distributions $\\mathcal {L}^{(t)}$ (or $\\lambda ^{(t)}$ ) and $\\mathcal {R}^{(t)}$ (or $\\rho ^{(t)}$ ), one can now analyze whether the average erasure probability converges to 0 by checking whether $y_\\ell ^{(t)}> y_{\\ell +1}^{(t)}$ for every $\\ell $ , starting from $y_0^{(t)}=1$ for $t\\in [T]$ .",
"Moreover, to make sure that the average erasure probability indeed vanishes instead of hovering around 0, we derive the following stability conditionblackNote that the term “stability\" used here is referred to the stability of fixed points rather than that in the stability analysis of slotted ALOHA [48], [5].. We enforce $\\lambda _1^{(t)}=0$ for all $t\\in [2]$ because we certainly do not want degree 1 variable nodes.",
"Then, assuming $y^{(t)}$ is very small for all $t\\in [2]$ , we expand the degree distributions and approximate them by keeping only the linear terms as follows, $\\lambda ^{(t)}[y^{(t)}] &\\approx \\lambda _2^{(t)}y^{(t)}, \\\\\\rho ^{(t)}[1-\\lambda ^{(t)}[y^{(t)}]] &\\approx 1-\\rho ^{^{\\prime }{(t)}}[1]\\lambda _2^{(t)}y^{(t)}, \\\\R^{(\\bar{t})}[1-\\lambda ^{(\\bar{t})}[y^{(\\bar{t})}]] &\\approx 1-R^{^{\\prime }(\\bar{t})}[1] \\lambda _2^{(\\bar{t})} y^{(\\bar{t})}, \\\\R^{^{\\prime }(\\bar{t})}[1-\\lambda ^{(\\bar{t})}[y^{(\\bar{t})}]] &\\approx R^{^{\\prime }(\\bar{t})}[1] - R^{^{\\prime \\prime }(\\bar{t})}[1]\\lambda _2^{(\\bar{t})}y^{(\\bar{t})}.$ We can now linearize the recursion around 0 by plugging (REF )-() into (REF ) to get $y^{(t)}> 1- (1-\\rho ^{^{\\prime }{(t)}}[1]\\lambda _2^{(t)}y^{(t)}) \\left(1-R^{^{\\prime \\prime }(\\bar{t})}[1](\\lambda _2^{(t)}y^{(\\bar{t})})^2\\right) \\approx \\rho ^{^{\\prime }{(t)}}[1]\\lambda _2^{(t)}y^{(t)},$ which leads to the following stability condition $\\lambda _2^{(t)}< \\frac{1}{\\rho ^{^{\\prime }{(t)}}[1]}\\quad \\text{for $t\\in [2]$}.$"
],
[
"Optimization Problem",
"Before we formulate the optimization problem, we note that there are two sets of degree distributions $\\lbrace \\mathcal {L}^{(t)}[x]\\rbrace $ and $\\lbrace \\mathcal {R}^{(t)}[x]\\rbrace $ in the proposed problem, but we only have control over $\\lbrace \\mathcal {L}^{(t)}[x]\\rbrace $ .",
"The behavior of $\\lbrace \\mathcal {R}^{(t)}[x]\\rbrace $ ; however, are completely determined by how the users behave.",
"Specifically, according to the protocol, a type $t$ user $\\mathsf {v}^{(t)}$ having degree $L$ will send a replica in slot $\\mathsf {c}^{(t)}$ with probability $L/N$ .",
"Hence the average probability that a user $\\mathsf {v}^{(t)}$ sending a replica in slot $\\mathsf {c}^{(t)}$ is given by $ \\frac{\\mathcal {L}^{^{\\prime }(t)}[1]}{N} = \\frac{\\mathcal {R}^{^{\\prime }(t)}[1]}{K^{(t)}}.$ Thus, the degrees of $\\mathsf {c}^{(t)}$ follows the Binomial distribution with parameter $\\frac{\\mathcal {L}^{^{\\prime }(t)}[1]}{N}$ .",
"Similar to [9], by Poisson approximation [49], we have $\\mathcal {R}^{(t)}[x] \\approx \\exp \\left( -\\mathcal {R}^{^{\\prime }(t)}[1](1-x) \\right) = \\exp \\left( -\\eta ^{(t)}\\mathcal {L}^{^{\\prime }(t)}[1](1-x) \\right).$ Moreover, from (REF ), we have $\\rho ^{(t)}[x]= \\exp \\left( -\\eta ^{(t)}\\mathcal {L}^{^{\\prime }(t)}[1](1-x) \\right).$ blackNow, the convergence condition becomes (REF ) in the bottom of the this page by plugging (REF ) into (REF ) and noting that it is sufficient that $y^{(t)}_{\\ell +1}<y^{(t)}_{\\ell }$ for every $\\ell $ with $y^{(t)}_{\\ell }\\ne 0$ .",
"Figure: NO_CAPTIONNow, we are ready to formulate the optimization problem that maximizes the target efficiency subject to conditions derived thus far: $&\\!\\max _{\\lbrace \\mathcal {L}^{(t)}[x]\\rbrace } &\\qquad & \\eta =\\sum _{t=1}^2 \\eta ^{(t)}\\\\&\\text{subject to} & & \\text{convergence condition~}(\\ref {eqn:DE_full_poisson}),\\\\& & & \\lambda _1^{(t)}=0~\\text{for $t\\in [2]$},\\\\& & & L_d^{(t)}\\ge 0, ~\\text{for $d\\in [d_{\\mathrm {max}}^{(t)}]$ and $t\\in [2]$}, \\\\& & & \\mathcal {L}^{(t)}[1]=1~\\text{for $t\\in [2]$},$ where $d_{\\mathrm {max}}^{(t)}$ is the maximum degree of $\\mathcal {L}^{(t)}[x]$ that has to be imposed in practice.",
"blackWe solve this problem and provide some optimized degree distributions in the next subsection.",
"We note that the stability condition (REF ) is not strictly required for maximizing the target efficiency as above.",
"However, for applications that require very low packet loss rates, we do need to include (REF ) into our optimization problem in order to make sure that the average erasure probability does vanish."
],
[
"Solving the Optimization Problem",
"In what follows, we again focus on $T=2$ solely but the discussion and intuitions apply to any $T\\ge 2$ .",
"For $T=2$ , let $y^{(t)}$ and $y^{(t)^{\\prime }}$ be blackthe LHS and RHS of (REF ), representing the average erasure probability before and that after an iteration, respectively.",
"Similar to the conventional IRSA in [9], we can have a sufficient condition for decodability that $y^{(t)^{\\prime }}<y^{(t)}$ for $t\\in [2]$ and every $(y^{(1)},y^{(2)})\\in (0,1]\\times (0,1]$ .",
"This admits a graphical interpretation as follows.",
"Note that it is a $T=2$ dimensional problem and we need a curve of $(y^{(1)^{\\prime }},y^{(2)^{\\prime }})$ against $(y^{(1)},y^{(2)})$ , which is in 4 dimensional space.",
"We instead plot $y^{(1)^{\\prime }}$ against $(y^{(1)},y^{(2)})$ and $y^{(2)^{\\prime }}$ against $(y^{(1)},y^{(2)})$ separately in Figs REF and REF , respectively.",
"blackIn Fig.",
"REF , the identity plane is the hyperplane consisting of $(y^{(1)},y^{(2)},y^{(1)})$ and the evolution plane for the degree distribution pair $\\mathsf {P}_2$ shown in Table REF consists of $(y^{(1)},y^{(2)},y^{(1)^{\\prime }})$ .",
"Similarly, in Fig.",
"REF , we show the identity plane consisting of $(y^{(1)},y^{(2)},y^{(2)})$ and also plot the evolution plane for $\\mathsf {P}_2$ that consists of $(y^{(1)},y^{(2)},y^{(2)^{\\prime }})$ .",
"The sufficient condition mentioned above is then to ask the entire evolution plane lie beneath the identity plane in both figures.",
"However, this is by no means necessary and is way too strong as after each iteration, both $y^{(1)}$ and $y^{(2)}$ drop and some pairs like $(0,1)$ and $(1,0)$ will never be visited.",
"Also shown in Figs.",
"REF and REF is the evolution path of the considered distribution pair that is generated by using the output $(y^{(1)^{\\prime }},y^{(2)^{\\prime }})$ of the previous iteration as input.",
"This evolution path depicts the trajectory of the pair of erasure probabilities evolving with the iterative decoding algorithm.",
"Although the evolution plane does not lie entirely beneath the identity plane, since the evolution path in this example lies entirely beneath the identity plane and has the unique fixed point at the origin, this degree distribution pair is decodable.",
"Figure: Identity plane, evolution plane, and evolution pathWith the above observation, we propose a new sufficient condition for a degree distribution pair to be decodable: There exists an evolution path that lies entirely beneath the identity plane.",
"Hence, the optimization problem becomes to find a degree distribution pair with the evolution path all but touches the identity plane.",
"We then adopt the differential evolution technique [50] to solve this optimization problem.",
"Specifically, in the mutation step of differential evolution, we adopt the DEEP algorithm in [51] and use the “bounce-back\" method [50] to handle the boundary conditions.",
"Some optimized degree distributions $\\mathcal {L}^{(t)}[x]$ with $d_{\\mathrm {max}}^{(t)}=8$ for $t\\in [T]$ are shown in Table REF .",
"In addition, the corresponding thresholds $\\eta ^*$ , the analytic result of $\\eta $ in the limit as $N\\rightarrow \\infty $ , are also shown.",
"In this table, the policies $\\mathsf {P}_1$ , $\\mathsf {P}_2$ , and $\\mathsf {P}_3$ are produced by solving the above optimization problem for the cases when $K^{(1)}=K^{(2)}$ , $3K^{(1)}=K^{(2)}$ , and $7K^{(1)}=K^{(2)}$ , respectively.",
"We also provide an example with $T=3$ whose density evolution and the corresponding optimization problem can be found in Appendix .",
"The policy $\\mathsf {P}_4$ provides a set of optimized degree distributions for $T=3$ with $K^{(1)}=K^{(2)}=K^{(3)}$ .",
"Finally, the policy $\\mathsf {P}_5$ is that for conventional IRSA with $T=1$ and is directly borrowed from [9].",
"These policies will be tested with simulations in the next section.",
"Remark 9 From Table REF , one can observe that every proposed policy has an threshold larger than $100\\%$ .",
"At first glance, this seems violating the fundamental limit of UMA that at most 1 packet can be successfully delivered in 1 slot, even under perfect coordination.",
"However, we would like to stress that this is not the fundamental limit for our setting, as in our protocol, leveraging heterogeneity among users, it is entirely possible that one can use intra-SIC to decode at most $T$ packets in a slot.",
"Remark 10 It is worth mentioning that for $T=2$ , when the numbers of active users of type $t\\in [2]$ are the same, the optimized degree distributions $\\mathcal {L}^{(t)}[x]$ become the same for $t\\in [2]$ .",
"See for e.g.",
"$\\mathsf {P}_1$ in Table REF .",
"This makes perfect sense as when $K^{(1)}= K^{(2)}$ , the decodable sets $\\mathcal {D}^{(1)}$ and $\\mathcal {D}^{(2)}$ become symmetric as shown in Example REF .",
"However, when $T=3$ , the degree distributions become different.",
"This can also be explained by the different decodable sets in Example REF .",
"Table: Optimized degree distributions and thresholds"
],
[
"Simulation Results",
"blackIn this section, we validate the proposed IRSA with NOMA via extensive simulations under practical frame sizes.",
"In particular, to demonstrate the effectiveness of the proposed method, we will use the efficiency $\\hat{\\eta }$ versus the total active users per slot, $\\sum _{t=1}^N K^{(t)}/N$ (that is $\\eta $ ), as the metric.",
"In Fig.",
"REF , we compare the achieved efficiency of the proposed IRSA with NOMA for different $T$ .",
"Specifically, in Fig.",
"REF , for both $N=150$ and 1500, we compare the efficiencies of $\\mathsf {P}_5$ , $\\mathsf {P}_1$ , and $\\mathsf {P}_4$ , that are optimized for $T=1$ , $T=2$ , and $T=3$ , respectively.",
"From these figures, we can observe that the proposed multi-dimensional density evolution indeed accurately predicts the efficiency of the proposed IRSA with NOMA as $N$ increases.",
"Moreover, we observe that the more types, the higher efficiency the proposed algorithm can achieve.",
"The proposed multi-dimensional density evolution predicts an efficiency of $143\\%$ for $\\mathsf {P}_1$ and $185\\%$ for $\\mathsf {P}_4$ , which are both significantly higher than the $93.8\\%$ achieved by the conventional IRSA with $\\mathsf {P}_5$ .",
"This indicates that when the application at hand presents natural heterogeneity, the proposed IRSA with NOMA can effectively exploit it.",
"Figure: Efficiency versus target efficiency (active users per slot) at N=150N=150 and 1500 for 𝖯 5 \\mathsf {P}_5 (T=1T=1), 𝖯 1 \\mathsf {P}_1 (T=2T=2), and 𝖯 4 \\mathsf {P}_4 (T=3T=3).In Fig.",
"REF , we show the efficiency of the proposed IRSA with NOMA for $T=2$ where the two groups have different numbers of users.",
"Again, for both $N=150$ and 1500, we plot the efficiencies of $\\mathsf {P}_1$ , $\\mathsf {P}_2$ , and $\\mathsf {P}_3$ , that are optimized for the cases $K^{(1)}=K^{(2)}$ , $3K^{(1)}=K^{(2)}$ , and $7K^{(1)}=K^{(2)}$ , respectively.",
"One can see from this figure that the larger the difference between the numbers of users, the smaller efficiency.",
"This is because the small difference means that the number of users in a group is roughly the same from group to group; thereby, it means the heterogeneity is large and more decoding opportunity may be introduced by intra-slot SIC.",
"On the contrary, if the difference is large, it means that one of the group contains most of the users in the network; hence, heterogeneity is small.",
"An extreme example can be seen by considering $M\\cdot K^{(1)}= K^{(2)}$ and let $M\\rightarrow \\infty $ , under which the problem would become the homogeneous setting and no heterogeneity can be exploited.",
"Figure: Efficiency versus target efficiency (active users per slot) at N=150N=150 and 1500 for 𝖯 1 \\mathsf {P}_1 (K (1) =K (2) K^{(1)}=K^{(2)}), 𝖯 2 \\mathsf {P}_2 (K (1) =3K (2) K^{(1)}=3K^{(2)}), and 𝖯 3 \\mathsf {P}_3 (K (1) =7K (2) K^{(1)}=7K^{(2)}).Figs.",
"REF and REF have demonstrated that thanks to NOMA, the proposed protocol can efficiently exploit the heterogeneity inherent in the network.",
"One natural question arising at this point is whether this gain comes solely from the intrinsic benefit of NOMA or the analysis and optimization proposed in Sections REF and REF indeed play a role.",
"In other words, do the optimized policies in Table REF indeed provide non-negligible gains over $\\mathsf {P}_5$ when $T=2$ and $T=3$ ?",
"We note that with a larger decodable set enabled by intra-slot SIC, each node should transmit less in order to maintain the same chance to be successfully decoded while reducing the probability of causing unresolvable collisions.",
"However, $\\mathsf {P}_5$ is optimized for $T=1$ [9], which do not take the larger decodable set into account.",
"Hence, nodes adopting this distribution would tend to over-transmit.",
"To confirm this, we look into the degree distributions and observe that the average degrees are 3.6, 3.25, and 3.068 for $\\mathsf {P}_5$ , $\\mathsf {P}_1$ , and $\\mathsf {P}_4$ , respectively.",
"Moreover, for the optimized degree distributions, as $T$ increases, the fraction of degree 2 nodes increases while that of degrees 3 and 8 nodes decreases.",
"In Fig.",
"REF , we compare the performance of $\\mathsf {P}_5$ optimized for $T=1$ and the respective optimized degree distributions at $N=150$ .",
"The analytic results obtained by the multi-dimensional density evolution are also plotted, which indicate that the optimized degree distributions provide efficiency gains of 0.083 and 0.174 over $\\mathsf {P}_5$ when $T=2$ and $T=3$ , respectively.",
"Simulation results also show similar gains, which corroborate our analysis.",
"This gain of the optimized degree distribution over $\\mathsf {P}_5$ is expected to increase as $T$ increases due to the larger and larger decodable set.",
"In contrast, when the numbers of users of different types are drastically different, the gain of the optimized degree distribution over $\\mathsf {P}_5$ becomes negligible.",
"For such a scenario, we end up focusing more and more on the type with the largest size and the optimized degree distribution becomes more and more like $\\mathsf {P}_5$ .",
"This is evident by observing that $\\mathsf {P}_5$ and $\\Lambda ^{(2)}$ (for the type that is 7 times larger than the other) in $\\mathsf {P}_3$ are very similar to each other and that the density evolution shows almost identical efficiency.",
"That said, the optimized degree distribution still enjoys a significantly less average degree of 2.89 as opposed to 3.6 in $\\mathsf {P}_5$ .",
"Figure: Comparison of the suboptimal policy (𝖯 5 \\mathsf {P}_5) and the optimized policy (𝖯 1 \\mathsf {P}_1 when T=2T=2 and 𝖯 4 \\mathsf {P}_4 when T=3T=3) at N=150N=150."
],
[
"Extension to Frame Asynchronous Case",
"In this section, we extend the proposed IRSA with NOMA and the corresponding multi-dimensional density evolution to the frame asynchronous setting.",
"Both the analysis and the simulation results indicate that similar to [16], such asynchrony results in the boundary effect [17] that facilitates our system design blackby enabling easily implemented regular left degrees to be asymptotically optimal.",
"Moreover, since there is no concept of frame, the users do not have to wait until the next frame and can immediately start the transmission upon the arrival of a packet."
],
[
"Problem and Protocol",
"In the frame asynchronous setting, the users are allowed to join the network without frame synchronization; but slots are still synchronous.",
"We assume that the arrival of type $t$ users' data follows a Poisson distribution with arrival rate $g^{(t)}$ .",
"That is, the probability that $m^{(t)}$ type $t$ users join the system in a given slot is given by $ \\frac{\\exp (-g^{(t)})(g^{(t)})^{m^{(t)}}}{m^{(t)}!",
"}.$ We note here that $g^{(t)}$ has the unit “users per slot\" and plays a similar role with the target efficiency $\\eta ^{(t)}$ in the frame synchronous setting.",
"Also, the sum arrival rate $g=\\sum _{t=1}^T g^{(t)}$ plays a similar role with $\\eta $ in the frame synchronous setting.",
"Under this frame asynchronous setting, each user has its local view about “frame\" of size $N$ slots starting from the arrival of its packet.",
"For example, for a user joining the network at slot $i$ , its local view of frame comprises the slots $[i: i+N-1]$ as this user can transmit its packet in one or multiple of slots within $[i: i+N-1]$ .",
"An illustration of this model can be found in Fig.",
"REF .",
"Figure: Frame asynchronous ALOHA with two types of heterogeneous users, where users 1 and 2 belong to the type 1 and users 3 and 4 belong to the type 2.",
"Each user has its own view about frame of size 5 slots, starting from the arrival of its packet.Under the Poisson arrival process, by the superposition of independent Poisson distributions, the number of active users of type $t$ , namely $K^{(t)}$ , at slot $i$ again follows a Poisson distribution of rate $\\mu ^{(t)}_i$ , whereblackThis corresponds to the model with a boundary effect in [16].",
"As discussed therein, this boundary effect is possible if the receiver is turned on and monitors users' activities before their transmissions or if we artificially create a guard interval occasionally.",
"$\\mu ^{(t)}_i = {\\left\\lbrace \\begin{array}{ll}ig^{(t)}, & \\mbox{\\text{for $1\\le i < N$}} \\\\Ng^{(t)}, & \\mbox{\\text{for $i\\ge N$}}.\\end{array}\\right.",
"}$ The total number of active users at slot $i$ then follows a Poisson distribution of rate $\\mu _i=\\sum _{t=1}^T \\mu ^{(t)}_i$ .",
"In the presence of frame asynchrony, the proposed IRSA with NOMA protocol is modified as follows.",
"Upon the arrival of its packet at slot $i$ , a type $t$ user samples from a degree distribution $L\\sim \\mathcal {L}^{(t)}$ to determine the number of replicas it sends within its local view of frame.",
"It then immediately sends one packet in slot $i$ and uniformly selects $L-1$ slots from $\\mathsf {I}_i=[i+1:i+N-1]$ for sending the remaining $L-1$ replicas.",
"In the decoding process, we again allow both intra-slot and inter-slot SIC, but with a sliding window fashion [52], [16].",
"Specifically, to decode the packets of the users joining at slot $i$ , the decoder considers a sliding window of size $5N$ , namely $[i:i+5N-1]$ .",
"The problem can then be treated as a realization of the frame synchronous IRSA with NOMA with frame size $5N$ and can therefore be decoded in a similar fashion."
],
[
"Proposed Multi-Dimensional Density Evolution",
"To analyze the asymptotic performance of the proposed IRSA with NOMA in the presence of frame asynchrony, we extend the proposed multi-dimensional density evolution to the frame asynchronous setting.",
"Here, for clearly delivering our innovation, we focus on $T=2$ again.",
"Note that the extension to the general $T$ can be completed in the similar way to the synchronous case in Appendix .",
"We recall that the definitions of the node perspective degree distributions $\\mathcal {L}^{(t)}[x], \\mathcal {R}^{(t)}[x]$ and edge perspective degree distributions $\\lambda ^{(t)}[x], \\rho ^{(t)}[x]$ are in (REF )-(REF ).",
"Moreover, for a type $t$ user active at slot $i$ , referred to as a class $i$ type $t$ variable node, its behavior at slot $i$ and that in slots $\\mathsf {I}_i$ are different.",
"We thus need to define the node perspective left degree distributions $\\mathcal {L}^{(t)}_{i\\rightarrow i}[x] = x \\quad \\text{and}\\quad \\mathcal {L}^{(t)}_{i\\rightarrow \\mathsf {I}_i}[x] = \\sum _d L^{(t)}_{i\\rightarrow \\mathsf {I}_i, d} x^d =\\sum _d L^{(t)}_d x^{d-1},$ where $L^{(t)}_{d,i\\rightarrow \\mathsf {I}_i}$ is the probability of a type $t$ node joining at slot $i$ that would connect with $d$ of the check nodes in $\\mathsf {I}_i$ , which is equal to $L^{(t)}_{d+1}$ .",
"As for the check nodes corresponding to slot $i$ , all the edges must be incident to variable nodes with class $i$ or those with class $j\\in \\mathsf {K}_i$ where $\\mathsf {K}_i= {\\left\\lbrace \\begin{array}{ll}\\emptyset , & \\mbox{\\text{for $i=1$}} \\\\[1:i-1], & \\mbox{\\text{for $2\\le i <N$}} \\\\[i-N+1:i-1], & \\mbox{\\text{for $i \\ge N$}}.\\end{array}\\right.",
"}$ We then define the right degree distributions of a class $i$ type $t$ check node that has $d_1$ edges incident to class $i$ type $t$ variable nodes and that of a class $i$ type $t$ check node that has $d_2$ edges incident to type $t$ variable nodes in $\\mathsf {K}_i$ as $\\mathcal {R}^{(t)}_{i\\rightarrow i}[x] = \\sum _{d_1} R^{(t)}_{i\\rightarrow i, d_1} x^{d_1} \\quad \\text{and}\\quad \\mathcal {R}^{(t)}_{i\\rightarrow \\mathsf {K}_i}[x] = \\sum _{d_2} R^{(t)}_{i\\rightarrow \\mathsf {K}_i, d_2} x^{d_2},$ respectively.",
"The edge perspective degree distributions can be similarly derived as $\\lambda ^{(t)}_{i\\rightarrow i}[x] = 1 \\quad \\text{and} \\quad \\lambda ^{(t)}_{i\\rightarrow \\mathsf {I}_i}[x] = \\frac{\\mathcal {L}^{^{\\prime }(t)}_{i\\rightarrow \\mathsf {I}_i}[x]}{\\mathcal {L}^{^{\\prime }(t)}_{i\\rightarrow \\mathsf {I}_i}[1]} = \\sum _d \\lambda ^{(t)}_{i\\rightarrow \\mathsf {I}_i, d} x^{d-2} ,$ and $\\rho ^{(t)}_{i\\rightarrow i}[x] = \\frac{\\mathcal {R}^{^{\\prime }(t)}_{i\\rightarrow i}[x]}{\\mathcal {R}^{^{\\prime }(t)}_{i\\rightarrow i}[1]}= \\sum _{d_1} \\rho ^{(t)}_{i\\rightarrow i, d_1} x^{d_1-1} \\quad \\text{and}\\quad \\rho ^{(t)}_{i\\rightarrow \\mathsf {K}_i}[x] = \\frac{\\mathcal {R}^{^{\\prime }(t)}_{i\\rightarrow \\mathsf {K}_i}[x]}{\\mathcal {R}^{^{\\prime }(t)}_{i\\rightarrow \\mathsf {K}_i}[1]}= \\sum _{d_2} \\rho ^{(t)}_{i\\rightarrow \\mathsf {K}_i, d_2} x^{d_2-1}.$ We note that with the proposed protocol, similar to [16], we have Proposition 11 $\\mathcal {R}^{(t)}_{i\\rightarrow i}[x]=\\rho ^{(t)}_{i\\rightarrow i}[x]=\\exp (-g^{(t)}(1-x)),$ and $\\mathcal {R}^{(t)}_{i\\rightarrow \\mathsf {K}_i}[x]=\\rho ^{(t)}_{i\\rightarrow \\mathsf {K}_i}[x]=\\exp \\left(-\\frac{\\delta ^{(t)}_i(\\mathcal {L}^{^{\\prime }(t)}[1]-1)}{N-1}(1-x)\\right),$ where $\\delta ^{(t)}_i=\\min (i-1,N-1)g^{(t)}$ .",
"For $t\\in [T]$ , let $x_{{i\\rightarrow i},\\ell }^{(t)}$ and $x_{{i\\rightarrow j},\\ell }^{(t)}$ be the average erasure probability of the message passed along an edge from a class $i$ type $t$ variable node to a class $i$ type $t$ check node and that of the message passed along an edge from a class $i$ type $t$ variable node to a class $j$ type $t$ check node in iteration $\\ell $ , respectively.",
"Also, we denote by $y_{{i\\rightarrow i},\\ell }^{(t)}$ and $y_{{i\\rightarrow j},\\ell }^{(t)}$ the average erasure probability of the message passed along an edge from a class $i$ type $t$ check node to a class $i$ type $t$ variable node and that of the message passed along an edge from a class $i$ type $t$ check node to a class $j$ type $t$ variable node in iteration $\\ell $ , respectively.",
"In the sequel, we study how these average erasure probabilities evolve with iterations.",
"The average erasure probability of a message from a type $t$ check node in $\\mathsf {I}_i$ to a class $i$ type $t$ variable node in iteration $\\ell $ is given by $ \\tilde{y}_{i,\\ell }^{(t)}= \\frac{1}{N-1}\\sum _{j\\in \\mathsf {I}_i} y_{{j\\rightarrow i},\\ell }^{(t)}.$ Then, similar to [16], we have $x_{{i\\rightarrow i},\\ell }^{(t)}= \\mathcal {L}^{(t)}[\\tilde{y}_{i,\\ell }^{(t)}]~\\quad \\text{and}\\quad x_{{i\\rightarrow j},\\ell }^{(t)}= y_{{i\\rightarrow i},\\ell }^{(t)}\\lambda ^{(t)}_{i\\rightarrow \\mathsf {I}_i}[\\tilde{y}_{i,\\ell }^{(t)}].$ Also, the average erasure probability of a message from a type $t$ variable node in $\\mathsf {K}_i$ to a class $i$ type $t$ check node in iteration $\\ell $ is given by $\\tilde{x}_{i,\\ell }^{(t)}= {\\left\\lbrace \\begin{array}{ll}0, & \\mbox{\\text{for $i=1$}} \\\\\\frac{1}{i-1}\\sum _{k\\in \\mathsf {K}_i} x_{{k\\rightarrow i},\\ell }^{(t)}, & \\mbox{\\text{for $2\\le i <N$}} \\\\\\frac{1}{N-1}\\sum _{k\\in \\mathsf {K}_i} x_{{k\\rightarrow i},\\ell }^{(t)}, & \\mbox{\\text{for $i \\ge N$}}.\\end{array}\\right.",
"}$ With the above results and Proposition REF , we are now ready to present the multi-dimensional density evolution for the proposed IRSA with NOMA under the frame-asynchronous setting.",
"Proposition 12 For $t\\in [2]$ , define $f_1(t) &= \\exp (-g^{(t)}x_{{i\\rightarrow i},\\ell }^{(t)}), \\\\ f_2(t) &= \\exp \\left(-\\frac{\\delta _i^{(t)}(\\mathcal {L}^{^{\\prime }(t)}[1]-1)}{N-1}\\tilde{x}_{i,\\ell }^{(t)}\\right), \\\\ f_3(t) &= x_{{i\\rightarrow i},\\ell }^{(t)}g^{(t)}\\exp (-g^{(t)}x_{{i\\rightarrow i},\\ell }^{(t)}), \\\\ f_4(t) &= \\tilde{x}_{i,\\ell }^{(t)}\\frac{\\delta _i^{(t)}(\\mathcal {L}^{^{\\prime }(t)}[1]-1)}{N-1}\\exp \\left(-\\frac{\\delta _i^{(t)}(\\mathcal {L}^{^{\\prime }(t)}[1]-1)}{N-1}\\tilde{x}_{i,\\ell }^{(t)}\\right).",
"$ The multi-dimensional density evolution is as follows.",
"For $t\\in [2]$ , $i\\in \\mathbb {N}$ , and $j\\in \\mathsf {K}_i$ , we have $y_{{i\\rightarrow i},\\ell +1}^{(t)}=y_{{i\\rightarrow j},\\ell +1}^{(t)}= 1-f_1(t)f_2(t) \\prod _{\\bar{t}\\in [2],\\bar{t}\\ne t}\\left(f_1(\\bar{t})f_2(\\bar{t})+f_3(\\bar{t})f_2(\\bar{t})+ f_1(\\bar{t})f_4(\\bar{t})\\right).$ The proof of this result is omitted for the sake of brevity.",
"But we point out that in (REF ), for $\\bar{t}\\ne t$ , $(f_1(\\bar{t})f_2(\\bar{t})+f_3(\\bar{t})f_2(\\bar{t})+ f_1(\\bar{t})f_4(\\bar{t}))$ corresponds to the average probability that there is at most one erasure among edges connecting to a type $\\bar{t}$ check node."
],
[
"Simulation Results",
"blackFirst, the threshold of the proposed IRSA with NOMA for the left-regular $\\Lambda ^{(t)}=x^3$ is evaluated under the frame asynchronous setting as $g=1.42$ .",
"Note that for the same left-regular degree distribution, the density evolution shows a $g=1.24$ threshold for the frame synchronous setting.",
"blackThis shows that, as predicted, the boundary effect helps bootstrapping the decoding process and allows the left-regular degree distribution $x^3$ to achieve a threshold that is close to $g=1.433$ achieved by the optimized degree distribution $\\mathsf {P}_1$ in Table REF in the frame synchronous setting.",
"Simulation results are then provided in Fig.",
"REF , where the packet loss rate $P_e$ versus sum arrival rate $g$ is plotted for $N=200$ , 500, and 1600.",
"Figure: Packet loss rate versus sum arrival rate gg in the frame asynchronous setting.",
"The thresholds obtained by the proposed density evolution are also plotted."
],
[
"Conclusion",
"In this paper, we have investigated UMA in the presence of heterogeneous users.",
"A novel protocol, IRSA with NOMA, has been proposed to leverage the heterogeneity inherent in the problem.",
"To analyze the proposed protocol, a novel multi-dimensional density evolution has been proposed, which has been shown to be able to accurately predict the asymptotic performance of IRSA with NOMA under the modified peeling decoding.",
"An optimization problem has then been formulated and solved for finding optimal degree distributions for our IRSA with NOMA.",
"Simulation results have demonstrated that the proposed IRSA with NOMA can exploit the natural heterogeneity and obtained efficiency higher than that achieved by conventional IRSA.",
"blackFinally, an extension of the proposed protocol to the frame-asynchronous setting has been investigated, where a boundary effect that bootstraps decoding process has been discovered via both analysis and simulation.",
"blackThroughout the paper, we have taken the MAC layer perspective.",
"One potential future work is to extend our idea to a more practical model as in [37] and to see how randomly selecting power can further improve the performance in the presence of heterogeneity inherent in the network.",
"Another interesting direction is to analyze how much more we can gain from having more types.",
"From Fig.",
"REF , we have already observed diminishing returns going from $T=2$ to $T=3$ .",
"We would expect that the $\\eta ^*$ converges to a constant as $T\\rightarrow \\infty $ .",
"The analysis of this convergence is left for future work."
],
[
"Multi-Dimensional Density Evolution for General $T$",
"In this appendix, we discuss the multi-dimensional density evolution for general $T$ ."
],
[
"Density Evolution for General $T$",
"For $t\\in [T]$ , we recall that $x_\\ell ^{(t)}$ and $y_\\ell ^{(t)}$ are the average erasure probability of the message passed along an edge from $\\mathsf {v}^{(t)}$ to $\\mathsf {c}^{(t)}$ and that of the message passed along an edge from $\\mathsf {c}^{(t)}$ to $\\mathsf {v}^{(t)}$ in iteration $\\ell $ , respectively.",
"In iteration $\\ell $ , for an edge incident to a variable node $\\mathsf {v}^{(t)}$ with degree $d$ , the only possibility that the message along this edge to a $\\mathsf {c}^{(t)}$ is in erasure is that all the other $d-1$ edges are in erasure.",
"Therefore, the probability that the message passed along this edge is in erasure is $(y_\\ell ^{(t)})^{d-1}$ .",
"Now, averaging over all the edges results in the average erasure probability $x_\\ell ^{(t)}= \\sum _d \\lambda ^{(t)}_d (y_\\ell ^{(t)})^{d-1} = \\lambda ^{(t)}[y_\\ell ^{(t)}].$ blackWe denote by $\\mathbf {c}$ the vector whose $t$ -th entry stores the number of unresolved (erased) packets.",
"For an edge incident to a check node $\\mathsf {c}^{(t)}$ with degree $d_t$ , the message passed along this edge to a $\\mathsf {v}^{(t)}$ is revealed if and only if the corresponding $\\mathbf {c}$ belongs to the decodable set $\\mathcal {D}^{(t)}$ .",
"Suppose in the same super check node $n$ , $\\mathsf {c}^{(\\bar{t})}_n$ has degree $d_{\\bar{t}}$ .",
"Then the above event has probability $&\\sum _{\\mathbf {c}\\in \\mathcal {D}^{(t)}} (1-x_\\ell ^{(t)})^{d_t-c_t} \\prod _{\\bar{t}=1, \\bar{t}\\ne t}^T \\binom{d_{\\bar{t}}}{c_{\\bar{t}}} (x_\\ell ^{(\\bar{t})})^{c_{\\bar{t}}}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-c_{\\bar{t}}} \\nonumber \\\\&=(1-x_\\ell ^{(t)})^{d_t-1} \\sum _{\\mathbf {c}\\in \\mathcal {D}^{(t)}} \\prod _{\\bar{t}=1, \\bar{t}\\ne t}^T \\binom{d_{\\bar{t}}}{c_{\\bar{t}}} (x_\\ell ^{(\\bar{t})})^{c_{\\bar{t}}}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-c_{\\bar{t}}},$ where the equality is due to the fact that every vector in $\\mathcal {D}^{(t)}$ has $c_t=1$ .",
"Averaging over all the edges and over all the type $\\bar{t}$ check nodes for $\\bar{t}\\in [T]\\setminus \\lbrace t\\rbrace $ shows that the average probability of correct decoding is given by $&\\sum _{d_t} \\lambda ^{(t)}_{d_t}(1-x_\\ell ^{(t)})^{d_t-1}\\cdot \\nonumber \\\\&\\sum _{\\mathbf {c}\\in \\mathcal {D}^{(t)}}\\prod _{\\bar{t}=1, \\bar{t}\\ne t}^T \\sum _{d_{\\bar{t}}}R^{(\\bar{t})}_{d_{\\bar{t}}}\\binom{d_{\\bar{t}}}{c_{\\bar{t}}} (x_\\ell ^{(\\bar{t})})^{c_{\\bar{t}}}(1-x_\\ell ^{(\\bar{t})})^{d_{\\bar{t}}-c_{\\bar{t}}} \\nonumber \\\\&=\\rho ^{(t)}[1-x_\\ell ^{(t)}]\\sum _{\\mathbf {c}\\in \\mathcal {D}^{(t)}}\\prod _{\\bar{t}=1, \\bar{t}\\ne t}^T (x_\\ell ^{(\\bar{t})})^{c_{\\bar{t}}}(\\mathcal {R}^{(\\bar{t})})^{\\lbrace c_{\\bar{t}}\\rbrace }[1-x_\\ell ^{(\\bar{t})}]/c_{\\bar{t}}!.$ Therefore, the average erasure probability becomes black $&y_{\\ell +1}^{(t)}= 1-\\rho ^{(t)}[1-x_\\ell ^{(t)}]\\cdot \\nonumber \\\\&\\sum _{\\mathbf {c}\\in \\mathcal {D}^{(t)}}\\prod _{\\bar{t}=1, \\bar{t}\\ne t}^T (x_\\ell ^{(\\bar{t})})^{c_{\\bar{t}}}(\\mathcal {R}^{(\\bar{t})})^{\\lbrace c_{\\bar{t}}\\rbrace }[1-x_\\ell ^{(\\bar{t})}]/c_{\\bar{t}}!.$ Plugging (REF ) into (REF ) leads to the evolution of average erasure probability of a type $t$ check node as shown in (REF ) in the bottom of this page.",
"Figure: NO_CAPTION"
],
[
"Convergence and Stability Condition",
"After obtaining the density evolution in (REF ), for any given degree distributions $\\mathcal {L}^{(t)}$ (or $\\lambda ^{(t)}$ ) and $\\mathcal {R}^{(t)}$ (or $\\rho ^{(t)}$ ), we again can analyze whether the average erasure probability converges to 0 by checking whether $y_\\ell ^{(t)}> y_{\\ell +1}^{(t)}$ for every $\\ell $ and every $y_\\ell ^{(t)}>0$ .",
"In what follows, we again derive the stability condition.",
"Similar to the $T=2$ case, we enforce $\\lambda _1^{(t)}=0$ for all $t\\in [T]$ .",
"Assuming $y^{(t)}$ is very small for all $t\\in [T]$ , we expand the degree distributions and approximate them by keeping only the linear terms as follows, black $\\lambda ^{(t)}[y^{(t)}] &\\approx \\lambda _2^{(t)}y^{(t)}, \\\\\\rho ^{(t)}[1-\\lambda ^{(t)}[y^{(t)}]] &\\approx 1-\\rho ^{^{\\prime }{(t)}}[1]\\lambda _2^{(t)}y^{(t)}, \\\\R^{(\\bar{t})}[1-\\lambda ^{(\\bar{t})}[y^{(\\bar{t})}]] &\\approx 1-R^{^{\\prime }(\\bar{t})}[1] \\lambda _2^{(\\bar{t})} y^{(\\bar{t})}, \\\\(R^{(\\bar{t})})^{\\lbrace k\\rbrace }[1-\\lambda ^{(\\bar{t})}[y^{(\\bar{t})}]] &\\approx (R^{(\\bar{t})})^{\\lbrace k\\rbrace }[1] \\nonumber \\\\&- (R^{(\\bar{t})}[1])^{\\lbrace k+1\\rbrace }\\lambda _2^{(\\bar{t})}y^{(\\bar{t})}.$ We can now linearize the recursion around 0 by plugging (REF )-() into (REF ) to get the same stability condition $\\lambda _2^{(t)}< \\frac{1}{\\rho ^{^{\\prime }{(t)}}[1]}\\quad \\text{for $t\\in [T]$}.$"
],
[
"Optimization Problem and Optimized Degree Distributions",
"Similar to the $T=2$ case, we again apply the Poisson approximation and rewrite the convergence condition in (REF ) as (REF ) in the bottom of the next page.",
"Figure: NO_CAPTIONFinally, we are able to formulate the optimization problem that maximizes the target efficiency subject to derived conditions: $&\\!\\max _{\\lbrace \\mathcal {L}^{(t)}[x]\\rbrace } &\\qquad & \\eta =\\sum _{t=1}^T \\eta ^{(t)}\\\\&\\text{subject to} & & \\text{convergence condition~}(\\ref {eqn:DE_full_poisson_T}),\\\\& & & \\lambda _1^{(t)}=0~\\text{for all $t\\in [T]$},\\\\& & & L_d^{(t)}\\ge 0, ~\\text{for all $d\\in [d_{\\mathrm {max}}^{(t)}]$ and $t\\in [T]$}, \\\\& & & \\mathcal {L}^{(t)}[1]=1~\\text{for all $t\\in [T]$}.$ Similar to $T=2$ , one may include the stability condition (REF ) into the above optimization problem to make sure that the packet loss rate indeed vanishes as $N\\rightarrow \\infty $ ."
]
] | 2012.05159 |
[
[
"Smoothed Quantile Regression with Large-Scale Inference"
],
[
"Abstract Quantile regression is a powerful tool for learning the relationship between a response variable and a multivariate predictor while exploring heterogeneous effects.",
"In this paper, we consider statistical inference for quantile regression with large-scale data in the \"increasing dimension\" regime.",
"We provide a comprehensive and in-depth analysis of a convolution-type smoothing approach that achieves adequate approximation to computation and inference for quantile regression.",
"This method, which we refer to as {\\it{conquer}}, turns the non-differentiable quantile loss function into a twice-differentiable, convex and locally strongly convex surrogate, which admits a fast and scalable Barzilai-Borwein gradient-based algorithm to perform optimization, and multiplier bootstrap for statistical inference.",
"Theoretically, we establish explicit non-asymptotic bounds on both estimation and Bahadur-Kiefer linearization errors, from which we show that the asymptotic normality of the conquer estimator holds under a weaker requirement on the number of the regressors than needed for conventional quantile regression.",
"Moreover, we prove the validity of multiplier bootstrap confidence constructions.",
"Our numerical studies confirm the conquer estimator as a practical and reliable approach to large-scale inference for quantile regression.",
"Software implementing the methodology is available in the \\texttt{R} package \\texttt{conquer}."
],
[
"Introduction",
"Quantile regression (QR) is a useful statistical tool for modeling and inferring the relationship between a scalar response $y$ and a $p$ -dimensional predictor $$ [35].",
"Compared to the least squares regression that focuses on modeling the conditional mean of $y$ given $$ , QR allows modeling of the entire conditional distribution of $y$ given $$ , and thus provides valuable insights into heterogeneity in the relationship between $$ and $y$ .",
"Moreover, quantile regression is robust against outliers and can be performed for skewed or heavy-tailed response distributions without a correct specification of the likelihood.",
"These advantages make QR an appealing method to explore data features that are invisible to the least squares regression.",
"We refer the reader to [33] and [36] for an extensive overview of QR in terms of methods, theory, computation, and various extensions under complex data structures.",
"Quantile regression involves a convex optimization problem with a piecewise linear loss function, also known as the check function and pinball loss.",
"One can formulate QR as a linear programming problem, solvable by the interior point method with a computational complexity of $(n^{1+\\alpha } p^3 \\log n)$ with $\\alpha \\in (0, 1/2)$ , where $n$ is the sample size and $p$ is the parametric dimension.",
"When $n$ is large relative to $p$ , an efficient algorithm based on pre-processing has an improved complexity of $\\lbrace (np)^{2(1+\\alpha )/3} p^3 \\log n + np\\rbrace $ , and thus can be more efficient than solving a least squares problem when $p$ is small [48].",
"However, primarily due to the non-differentiability and lack of strong convexity of the loss function, QR remains computationally expensive for large-scale data when both $n$ and $p$ are large.",
"We refer to Chapter 5 of [36] for an overview of prevailing computational methods for quantile regression, such as simplex-based algorithms [4], [37], interior point methods [48], and alternating direction method of multipliers among other first-order proximal methods [44].",
"We consider conducting large-scale inference for quantile regression under the setting in which $p$ is large and $n$ is even larger.",
"Two general principles have been widely used to suit this purpose.",
"The first uses a nonparametric estimate of the asymptotic variance [23] that involves the conditional density of the response given the covariates, yet such an estimate can be fairly unstable.",
"Even if the asymptotic variance is well estimated, its approximation accuracy to the finite-sample variance depends on the design matrix and the quantile level.",
"Resampling methods, on the other hand, provide a more reliable approach to inference for QR under a wide variety of settings [45], [25], [31], [19].",
"Inevitably, the resampling approach requires repeatedly computing QR estimates up to thousands of times, and therefore is unduly expensive for large-scale data.",
"Theoretically, valid statistical inference is often justified by asymptotic normal approximations to QR estimators.",
"The Bahadur-Kiefer representation of the nonlinear QR estimators are essential to this end, as shown in [2] and [24].",
"In large-$p$ (non)asymptotic settings in which the parametric dimension $p$ may tend to infinity with the sample size, we refer to [54], [26], and [43] for normal approximation results of the QR estimators under fixed and random designs.",
"The question of how large $p$ can be relative to $n$ to ensure asymptotic normality has been addressed by those authors.",
"It is now recognized that we may have to pay a price here as compared to $M$ -estimators with smooth loss functions that are at least continuously twice differentiable.",
"To circumvent the non-differentiability of the QR loss function, [27] proposed to smooth the indicator part of the check function via the survival function of a kernel.",
"This smoothing method, which we refer to as Horowitz's smoothing throughout, has been widely used for various QR-related problems [55], [30], [21], [52], [56], [14], [10].",
"However, Horowitz's smoothing gains smoothness at the cost of convexity, which inevitably raises optimization issues.",
"In general, computing a global minimum of a non-convex function is intractable: finding an $\\epsilon $ -suboptimal point for a $k$ -times continuously differentiable function $f:\\mathbb {R}^p \\rightarrow $ requires at least as many as $(1/\\epsilon )^{p/k}$ evaluations of the function and its first $k$ derivatives [42].",
"As we shall see from the numerical studies in Section , the convergence of gradient-based algorithms can be relatively slow for high and low quantile levels.",
"To address the aforementioned issue, [20] proposed a convolution-type smoothing method that yields a convex and twice differentiable loss function, and studied the asymptotic properties of the smoothed estimator when $p$ is fixed.",
"To distinguish this approach from Horowitz's smoothing, we adopt the term conquer for convolution-type smoothed quantile regression.",
"In this paper, we first provide an in-depth statistical analysis of conquer under various array (non)asymptotic settings in which $p$ increases with $n$ .",
"Our results reveal a key feature of the smoothing parameter, often referred to as the bandwidth: the bandwidth adapts to both the sample size $n$ and dimensionality $p$ , so as to achieve a tradeoff between statistical accuracy and computational stability.",
"Since the convolution smoothed loss function is globally convex and locally strongly convex, we propose an efficient gradient descent algorithm with the Barzilai-Borwein stepsize and a Huber-type initialization.",
"The proposed algorithm is implemented via RcppArmadillo [17] in the R package conquer.",
"We next focus on large-scale statistical inference (hypothesis testing and confidence estimation) with large $p$ and large $n$ .",
"We propose a bootstrapped conquer method that has reduced computational complexity when the conquer estimator is used as initialization.",
"Under appropriate restrictions on dimension, we establish the consistency (or concentration), Bahadur representation, asymptotic normality of the conquer estimator as well as the validity of the bootstrap approximation.",
"In the following, we provide more details on the computational and statistical contributions of this paper.",
"Theoretically, by allowing $p$ to grow with $n$ , the `complexity' of the function classes that we come across in the analysis also increases with $n$ .",
"Conventional asymptotic tools for proving the bootstrap validity are based on weak convergence arguments [50], which are not directly applicable in the finite-sample setting.",
"In this paper we turn to a more refined and self-contained analysis, and prove a new local restricted strong convexity (RSC) property for the empirical smoothed quantile loss.",
"This validates the key merit of convolution-type smoothing, i.e., local strong convexity.",
"The smoothing method involves a bandwidth, denoted by $h$ .",
"Theoretically, we show that with sub-exponential random covariates (relaxing the bounded covariates assumption in [20]), conquer exhibits an $\\ell _2$ -error in the order of $\\sqrt{(p+t)/n}+h^2$ with probability at least $1-2e^{-t}$ .",
"When $h$ is of order $\\lbrace (p+t)/n \\rbrace ^{\\gamma }$ for any $\\gamma \\in [1/4, 1/2]$ , the conquer estimation is first-order equivalent to QR.",
"Under slightly more stringent sub-Gaussian condition on the covariates, we show that the Bahadur-Kiefer linearization error of conquer is of order $(p+t)/(nh^{1/2})+ h \\sqrt{(p+t)/n}+h^3$ with probability at least $1-3e^{-t}$ .",
"Based on such a representation, we establish a Berry-Esseen bound for linear functionals of conquer, which lays the theoretical foundation for testing general linear hypotheses, encompassing covariate-effect analysis, analysis of variance, and model comparisons, to name a few.",
"It is worth noting that with a properly chosen $h$ , the linear functional of conquer is asymptotically normal as long as $p^{8/3}/n\\rightarrow 0$ , which improves the best known growth condition on $p$ for standard QR [54], [26], [43].",
"We attribute this gain to the effect of smoothing.",
"Under similar conditions, we further establish upper bounds on both estimation and Bahadur-Kiefer linearization errors for the bootstrapped conquer estimator.",
"In the context of nonparametric density or regression estimation, it is known that when higher-order kernels are used (and if the density or regression function has enough derivatives), the bias is proportional to $h^\\nu $ for some $\\nu \\ge 4$ which is of better order than $h^2$ .",
"While a higher-order kernel has negative parts, the resulting smoothed loss is non-convex and thus brings the computational issue once again.",
"Motivated by the two-stage procedure proposed by [7] whose original idea is to improve an initial estimator that is already consistent but not efficient, we further propose a one-step conquer estimator using higher-order kernels but without the need for solving a large-scale non-convex optimization.",
"With increasing degrees of smoothness, the one-step conquer is asymptotically normal under a milder dimension constraint of roughly $p^2/n \\rightarrow 0$ .",
"To better appreciate the computational feasibility of conquer for large-scale problems, we compare it with standard QR on large synthetic datasets, where the latter is implemented by the R package quantreg [34] using the Frisch-Newton approach after preprocessing “pfn”.",
"We generate independent data vectors $\\lbrace y_i, _i\\rbrace _{i = 1}^n$ from a linear model $y_i = \\beta ^*_0 + \\langle _i , \\beta ^* \\rangle + \\varepsilon _i$ , where $(\\beta ^*_0, {\\beta ^*}^^ (1, \\ldots , 1)^^{p+1}$ , $_i \\sim \\mathcal {N}_p(0, )$ , and the independent errors $\\varepsilon _i \\sim t_2$ , for $i = 1, 2, \\dots , n$ .",
"We report the estimation error and elapsed time for increasing sample sizes $n\\in \\lbrace 1000,5000,10000,\\ldots ,100000\\rbrace $ and dimension $p = \\lfloor n^{1/2} \\rfloor $ , the largest integer that is less than or equal to $n^{1/2}$ .",
"Figure REF displays the average estimation error and average elapsed time based on 20 Monte Carlo samples.",
"This experiment shows promise of conquer as a practically useful tool for large-scale quantile regression analysis.",
"More empirical evidence will be given in the latter section.",
"Figure: RuntimeThe rest of the paper is organized as follows.",
"We start with a brief review of linear quantile regression and the convolution-type smoothing method in Section .",
"Explicit forms of the smoothed check functions are provided for several representative kernel functions in nonparametric statistics.",
"We introduce the multiplier bootstrap for statistical inference in Section REF .",
"In Section , we provide a comprehensive theoretical study of conquer from a nonasymptotic viewpoint, which directly leads to array asymptotic results.",
"Specifically, the bias incurred by smoothing the quantile loss is characterized in Section REF .",
"In Section REF , we establish the rate of convergence, Bahadur-Kiefer representation, and Berry-Esseen bound for conquer in a large-$p$ and larger-$n$ regime.",
"Results for its bootstrap counterpart are provided in Section REF .",
"A Barzilai-Borwein gradient-based algorithm with a Huber-type warm start is detailed in Section .",
"We conclude the paper with an extensive numerical study in Section to illustrate the finite-sample performance of conquer in large-scale quantile regression analysis.",
"We defer the proofs of all theoretical results as well as the full details of the one-step conquer to online supplementary materials.",
"Notation: For every integer $k\\ge 1$ , we use $^k$ to denote the the $k$ -dimensional Euclidean space.",
"The inner product of any two vectors $=(u_1, \\ldots , u_k)^ =(v_1, \\ldots ,v_k)^^k$ is defined by $^= \\langle , \\rangle = \\sum _{i=1}^k u_i v_i$ .",
"We use $\\Vert \\cdot \\Vert _p$ $(1\\le p \\le \\infty )$ to denote the $\\ell _p$ -norm in $^k$ : $\\Vert \\Vert _p = ( \\sum _{i=1}^k | u_i |^p )^{1/p}$ and $\\Vert \\Vert _\\infty = \\max _{1\\le i\\le k} |u_i|$ .",
"Throughout this paper, we use bold capital letters to represent matrices.",
"For $k\\ge 2$ , $_k$ represents the identity matrix of size $k$ .",
"For any $k\\times k$ symmetric matrix $\\in ^{k\\times k}$ , $\\Vert \\Vert _2$ denotes the operator norm of $$ .",
"If $$ is positive semidefinite, we use $\\Vert \\cdot \\Vert _{}$ to denote the vector norm linked to $$ given by $\\Vert \\Vert _{} = \\Vert ^{1/2} \\Vert _2$ , $\\in ^k$ .",
"For $r \\ge 0$ , define the Euclidean ball and sphere in $^k$ as $\\mathbb {B}^k(r) = \\lbrace \\in ^k : \\Vert \\Vert _2 \\le r\\rbrace $ and $\\mathbb {S}^{k-1}(r) = \\partial \\mathbb {B}^k(r) = \\lbrace \\in ^k: \\Vert \\Vert _2 =r \\rbrace $ , respectively.",
"For two sequences of non-negative numbers $\\lbrace a_n \\rbrace _{n\\ge 1}$ and $\\lbrace b_n \\rbrace _{n\\ge 1}$ , $a_n \\lesssim b_n$ indicates that there exists a constant $C>0$ independent of $n$ such that $a_n \\le Cb_n$ ; $a_n \\gtrsim b_n$ is equivalent to $b_n \\lesssim a_n$ ; $a_n \\asymp b_n$ is equivalent to $a_n \\lesssim b_n$ and $b_n \\lesssim a_n$ ."
],
[
"The linear quantile regression model",
"Given a univariate response variable $y\\in $ and a $p$ -dimensional covariate vector $= (x_1, \\ldots , x_p)^^p$ with $x_1 \\equiv 1$ , the primary goal here is to learn the effect of $$ on the distribution of $y$ .",
"Let $F_{y|}(\\cdot )$ be the conditional distribution function of $y$ given $$ .",
"The dependence between $y$ and $$ is then fully characterized by the conditional quantile functions of $y$ given $$ , denoted as $F^{-1}_{y|}(\\tau )$ , for $0<\\tau <1$ .",
"We consider a linear quantile regression model at a given $\\tau \\in (0,1)$ , that is, the $\\tau $ -th conditional quantile function is $F^{-1}_{y|} (\\tau ) = \\langle , \\beta ^*(\\tau ) \\rangle ,$ where $\\beta ^*(\\tau ) = (\\beta ^*_{1}(\\tau ) , \\ldots , \\beta ^*_{ p}(\\tau ) )^^p$ is the true quantile regression coefficient.",
"Let $\\lbrace (y_i, _i) \\rbrace _{i=1}^n$ be a random sample from $(y,)$ .",
"The standard quantile regression estimator [35] is then given as $ \\hat{\\beta }(\\tau ) \\in \\min _{\\beta \\in ^p } \\hat{Q} (\\beta ) = \\min _{\\beta \\in ^p } \\frac{1}{n} \\sum _{i=1}^n\\rho _\\tau ( y_i - \\langle _i , \\beta \\rangle ) ,$ where $ \\rho _\\tau (u) = u \\lbrace \\tau - \\mathbb {1}(u<0)\\rbrace $ is the $\\tau $ -quantile loss function, also referred to as the pinball loss and check function.",
"Statistical properties of $\\hat{\\beta }(\\tau )$ have been extensively studied.",
"We refer the reader to [33] and [36] for more details."
],
[
"Convolution-type smoothing",
"We consider the convolution-type smoothing approach proposed by [20].",
"For every $\\beta \\in ^p$ , let $\\hat{F}(\\cdot ;\\beta )$ be the empirical distribution function of the residuals $\\lbrace r_i(\\beta ) := y_i - \\langle _i,\\beta \\rangle \\rbrace _{i=1}^n$ , i.e., $\\hat{F}(t ;\\beta ) = (1/n) \\sum _{i=1}^n\\mathbb {1}\\lbrace r_i(\\beta ) \\le t \\rbrace $ .",
"Given a fixed quantile level $\\tau $ , the empirical quantile loss $\\hat{Q}(\\cdot )$ in (REF ) can be represented as $\\hat{Q}(\\beta ) = \\int _{-\\infty }^\\infty \\rho _\\tau (u) \\,{\\rm d} \\hat{F}(u;\\beta ).",
"$ Given a bandwidth $h=h_n >0$ , let $\\hat{F}_h(\\cdot ;\\beta )$ be the distribution function of the classical Rosenblatt–Parzen kernel density estimator: $\\hat{F}_h(u;\\beta ) = \\int _{-\\infty }^u \\hat{f}_h(t;\\beta ) \\,{\\rm d} t~~\\mbox{ with }~~ \\hat{f}_h(t;\\beta ) = \\frac{1}{nh} \\sum _{i=1}^nK\\bigg ( \\frac{t-r_i(\\beta )}{h} \\bigg ) , $ where $K:\\mathbb {R} \\rightarrow ^+ := \\mathbb {[}0,\\infty )$ is a symmetric kernel that integrates to one.",
"Replacing the empirical distribution function in (REF ) with $\\hat{F}_h $ yields the following smoothed empirical quantile loss $\\hat{Q}_h(\\beta ) := \\int _{-\\infty }^\\infty \\rho _\\tau (u) \\,{\\rm d} \\hat{F}_h(u;\\beta ) = \\frac{1}{nh} \\sum _{i=1}^n\\int _{-\\infty }^\\infty \\rho _\\tau (u) K \\bigg ( \\frac{u + \\langle _i, \\beta \\rangle - y_i }{h} \\bigg ) \\,{\\rm d} u .",
"$ Accordingly, the smoothed QR estimator is given by $ \\hat{\\beta }_h = \\hat{\\beta }_h(\\tau ) \\in _{\\beta \\in ^p} \\hat{Q}_h(\\beta ) .$ As we shall see later, the ideal choice of bandwidth should adapt to the sample size $n$ and dimension $p$ , while the quantile level $\\tau $ is prespecified and fixed.",
"Thus, the dependence of $\\hat{\\beta }_h$ and $\\hat{Q}_h(\\cdot )$ on $\\tau $ will be assumed without display.",
"Given a kernel function $K(\\cdot )$ and bandwidth $h>0$ , the smoothed quantile loss $\\hat{Q}_h(\\cdot )$ defined in (REF ) can be equivalently written as $\\hat{Q}_h(\\beta ) = (1/n) \\sum _{i=1}^n\\ell _h(y_i - \\langle _i, \\beta \\rangle )$ , where $\\ell _h(u) = (\\rho _\\tau * K_h )(u) = \\int _{-\\infty }^{\\infty } \\rho _\\tau (v) K_h(v- u ) \\, {\\rm d} v $ with $*$ denoting the convolution operator and $K_h(u)=(1/h)K(u/h)$ .",
"Therefore, as stated in the Introduction, we refer to the aforementioned smoothing method as conquer throughout the paper.",
"Commonly used kernel functions include: (a) uniform kernel $K(u) = (1/2) \\mathbb {1} (|u|\\le 1)$ , (b) Gaussian kernel $K(u) = \\phi (u):= (2\\pi )^{-1/2} e^{-u^2/2}$ , (c) logistic kernel $K(u) = e^{-u}/(1+ e^{-u})^2 $ , (d) Epanechnikov kernel $K(u) = (3 / 4) (1 - u^2) \\mathbb {1}(|u|\\le 1)$ , and (e) triangular kernel $K(u) = (1 - |u|) \\mathbb {1}(|u|\\le 1)$ .",
"Explicit expressions of the corresponding smoothed loss function $\\rho _\\tau * K_h$ will be given in Section .",
"The convolution-type kernel smoothing yields an objective function $\\beta \\mapsto \\hat{Q}_{h}(\\beta )$ that is twice continuously differentiable with gradient and hessian matrix $\\nabla \\hat{Q}_{h}(\\beta ) = \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace {K}\\bigl ( -r_i(\\beta )/h \\bigr ) - \\tau \\bigr \\rbrace _i ~~\\mbox{ and }~~\\nabla ^2 \\hat{Q}_{h}(\\beta ) = \\frac{1}{n} \\sum _{i=1}^nK_h( r_i(\\beta )) _i _i^ $ respectively, where $K(u) := \\int _{-\\infty }^u K(t) \\,{\\rm d} t$ .",
"Provided that $K$ is non-negative, $\\hat{Q}_{h}(\\cdot )$ is a convex function for any $h>0$ , and $\\hat{\\beta }_h = \\hat{\\beta }_h(\\tau ) $ satisfies the first-order condition $\\nabla \\hat{Q}_{ h}(\\hat{\\beta }_h ) = \\frac{1}{n} \\sum _{i=1}^n\\Biggl \\lbrace {K}\\Biggl ( \\frac{\\langle _i , \\hat{\\beta }_h \\rangle - y_i }{h} \\Biggr ) - \\tau \\Biggr \\rbrace _i = \\textbf {0} .\\nonumber $ The smoothness and convexity of $\\hat{Q}_{h}$ warrant the superior computation efficiency of first-order gradient based algorithms for solving large-scale smoothed quantile regressions.",
"Theoretically, with $p$ fixed and $n \\rightarrow \\infty $ , [20] showed that the convolution-type smoothed QR estimator has a lower asymptotic mean squared error than Horowitz's smoothed estimator, and also has a smaller Bahadur linearization error than the standard QR in the almost sure sense.",
"The optimal order of the bandwidth based on the asymptotic mean squared error is unveiled as a function of $n$ .",
"Our theoretical results complement the results in [20] by providing a comprehensive analysis of conquer in the regime in which $p$ is allowed to increase with $n$ ."
],
[
"Connection to related existing work",
"The idea of smoothing the piecewise linear loss, to the best of our knowledge, dates back to [1] in the context of median regression.",
"Specifically, [1] proposed a smoothed approximation of the absolute value function that has the form $u \\mapsto u + 2 h \\log (1 + e^{-u / h}) , \\ \\ u\\in ,$ where $h= n^{-\\gamma }$ with $ \\gamma \\in (1/3, 1/2)$ .",
"This smoothing device, however, is mainly used therein to simplify the analysis of the asymptotic behavior of a two-stage median regression estimator.",
"[27] proposed a kernel smoothing method of similar type for bootstrapping the median regression estimator.",
"The idea is to replace the indicator function in $\\rho _\\tau (u) = u \\lbrace \\tau - \\mathbb {1}(u<0) \\rbrace $ with a smoothed counterpart, leading to $ \\ell ^{\\mathrm {Horo}}_{ h}(u) = u \\bigl \\lbrace \\tau - {K}(-u / h) \\bigr \\rbrace ,$ where $K(u) = \\int _{-\\infty }^u K(t) \\,{\\rm d} t$ , and $K(\\cdot )$ is a symmetric kernel function.",
"The key difference between the conquer loss (REF ) and Horowitz's loss (REF ) is that the former is globally convex, while Horowitz's loss is not.",
"This is illustrated in Figure REF .",
"We refer to [20] for an in-depth comparison between convolution smoothing and Horowitz's smoothing in terms of asymptotic mean squared errors of the smoothed estimators.",
"A closer inspection reveals that the smoothed function considered by [1] is a convolution-type smoothed loss with a logistic kernel.",
"More recently, [57] considered a smoothing approximation for the quantile loss, which they refer to as the Huber approximation.",
"Their goal was to compute the regularized QR estimator but via a smoothed optimization method.",
"The corresponding loss function also falls into the general framework of convolution-type smoothing, with $K$ taken as the uniform kernel.",
"The general statistical theory developed in this paper applies to those special cases.",
"Figure: Uniform kernel under τ=0.7\\tau = 0.7."
],
[
"Multiplier bootstrap inference",
"In this section, we consider a multiplier bootstrap procedure to construct confidence intervals for conquer.",
"Independent of the observed sample $\\mathcal {X}_n = \\lbrace (y_i,_i) \\rbrace _{i=1}^n$ , let $\\lbrace w_i\\rbrace _{i = 1}^n$ be independent and identically distributed random variables with $(w_i)=1$ and $\\textnormal {var}(w_i)=1$ .",
"Recall that $\\hat{\\beta }_h = \\hat{\\beta }_h(\\tau ) = \\min _{\\beta \\in ^p} \\hat{Q}_{h}(\\beta )$ is the conquer estimator.",
"If the minimizer is not unique, we take any of the minima as $\\hat{\\beta }_h = (\\hat{\\beta }_{h,1},\\ldots , \\hat{\\beta }_{h,p})^.$ The proposed bootstrap method, which dates back to [16] and [3], is based on reweighting the summands of $\\hat{Q}_{h}(\\cdot )$ with random weights $w_i$ .",
"The resulting weighted quantile loss $\\hat{Q}^{\\flat }_h :^p \\rightarrow $ is $\\hat{Q}_{ h}^\\flat (\\beta ) = \\frac{1}{n} \\sum _{i=1}^nw_i \\ell _{h}(y_i - \\langle _i, \\beta \\rangle ) , $ where $\\ell _{ h}(u) = (\\rho _\\tau * K_h)(u)$ is as in (REF ).",
"We refer to [9] for a general asymptotic theory for weighted bootstrap for estimating equations, where a class of bootstrap weights is considered.",
"Extensions to semiparametric $M$ -estimation can be found in [41] and [12].",
"Let $\\mathbb {E}^*$ and $^*$ be the conditional expectation and probability given the observed data $_n$ , respectively.",
"Observe that $\\mathbb {E}^*\\lbrace \\hat{Q}_{h}^{\\flat }(\\beta ) \\rbrace = \\hat{Q}_{h}(\\beta )$ for any $\\beta \\in ^{p}$ .",
"Consequently, we have $_{\\beta \\in ^p} ^* \\lbrace \\hat{Q}_{h}^{\\flat }(\\beta ) \\rbrace = _{\\beta \\in ^p} \\hat{Q}_{h}(\\beta ) = \\hat{\\beta }_h .$ This simple and yet important observation motivates the following multiplier bootstrap statistic: $\\hat{\\beta }_h^\\flat = \\hat{\\beta }_h^\\flat (\\tau ) \\in _{\\beta \\in ^p } \\hat{Q}_{h}^{\\flat }(\\beta ) .",
"$ To retain convexity of the loss function, non-negative random weights are preferred, such as $w_i \\sim {\\rm Exp}(1)$ , i.e., exponential distribution with rate 1, and $w_i =1 + e_i$ , where $e_i$ are independent Rademacher random variables.",
"We can construct confidence intervals based on the bootstrap estimates using one of the three classical methods, the percentile method, the pivotal method, and the normal-based method.",
"To be specific, for each $q\\in (0,1)$ and $1\\le j\\le p$ , define the (conditional) $q$ -quantile of $\\hat{\\beta }^\\flat _{h,j}$ —the $j^{{\\rm th}}$ coordinate of $\\hat{\\beta }^\\flat _h \\in ^p$ —given the observed data as ${\\rm c}^\\flat _j(q) = \\inf \\lbrace t \\in : ^* ( \\hat{\\beta }^\\flat _{h,j} \\le t ) \\ge q \\rbrace $ .",
"Then, for a prespecified nominal level $\\alpha \\in (0,1)$ , the corresponding $1-\\alpha $ bootstrap percentile and pivotal confidence intervals (CIs) for $\\beta ^*_j$ ($j=1,\\ldots ,p$ ) are, respectively, $\\left[ {\\rm c}^\\flat _j(\\alpha /2 ) , \\, {\\rm c}^\\flat _j( 1 -\\alpha /2) \\right] ~\\mbox{ and }~ \\left[ 2\\hat{\\beta }_{h,j} - {\\rm c}^\\flat _j( 1 -\\alpha /2) , \\, 2 \\hat{\\beta }_{h,j} - {\\rm c}^\\flat _j(\\alpha /2 ) \\right] .$ Numerically, ${\\rm c}^\\flat _j(q)$ ($q\\in \\lbrace \\alpha , 1-\\alpha /2\\rbrace $ ) can be calculated with any specified precision by the simulation.",
"In the R package conquer, the default number of bootstrap replications is set as $B=1000$ .",
"In the next section, we will present a finite-sample theoretical framework for convolution-type smoothed quantile regression, including the concentration inequality and nonasymptotic Bahadur representation for both the conquer estimator (REF ) and its bootstrap counterpart (REF ) using Rademacher multipliers.",
"As a by-product, a Berry-Esseen-type inequality (see Theorem REF ) states that, under certain constraints on the (growing) dimensionality and bandwidth, the distribution of any linear projection of $\\hat{\\beta }_h$ converges to a normal distribution as the sample size increases to infinity.",
"Informally, for any given deterministic vector $\\in ^p$ , the scaled statistic $n^{1/2} \\langle , \\hat{\\beta }^*_h - \\beta ^* \\rangle $ is asymptotically normally distributed with asymptotic variance $\\sigma _\\tau ^2 := \\tau (1-\\tau ) \\,^^{-1} ^{-1} $ .",
"To make inference based on such asymptotic results, we need to consistently estimate the asymptotic variance.",
"[20] suggested the following estimators $\\hat{}_h := \\nabla ^2 \\hat{}_h(\\hat{\\beta }_h) = \\frac{1}{n} \\sum _{i=1}^nK(\\hat{\\varepsilon }_i /h) \\cdot _i^~\\mbox{ and }~~ \\hat{}_h := \\frac{1}{n h} \\sum _{i=1}^n\\bigl \\lbrace K(-\\hat{\\varepsilon }_i/ h) - \\tau \\bigr \\rbrace ^2 ^{align*}of and \\tau (1-\\tau ) \\, , respectively, where \\hat{\\varepsilon }_i = y_i - \\langle _i, \\hat{\\beta }_h\\rangle are fitted residuals.",
"The ensuing 1-\\alpha normal-based CIs are given by \\hat{\\beta }_{h,j} \\pm \\Phi ^{-1}(1-\\alpha /2) \\cdot n^{-1/2} (\\hat{}_h^{-1} \\hat{}_h \\hat{}_h^{-1} )_{jj}, j=1,\\ldots ,p.However, the normal approximations to the CI suffer from the sensitivity to the smoothing needed to estimated the conditional densities, namely, the matrix = \\lbrace f_{\\varepsilon | }(0) ^.",
"This is especially true when \\tau is in the upper or lower tail.",
"See Section~\\ref {sec:normal-boot} for a numerical comparison between normal approximation and bootstrap calibration for confidence construction at high and low quantile levels.",
"As we shall see, the normal-based CIs can be exceedingly wide and thus inaccurate under these situations.$"
],
[
"Statistical Analysis",
"Under the linear quantile regression model in (REF ), we write, for convenience, the generic data vector $(y,)$ in a linear model form: $y = \\langle , \\beta ^*(\\tau ) \\rangle + \\varepsilon (\\tau ), $ where the random variable $\\varepsilon (\\tau )$ satisfies $\\lbrace \\varepsilon (\\tau ) \\le 0 | \\rbrace = \\tau $ .",
"Let $ f_{\\varepsilon |}(\\cdot )$ be the conditional density function of the regression error $\\varepsilon = \\varepsilon (\\tau )$ given $$ .",
"We first derive upper bounds for the smoothing bias under mild regularity conditions on the conditional density $ f_{\\varepsilon |}$ and the kernel function."
],
[
"Smoothing bias",
"[Kernel function] Let $K(\\cdot )$ be a symmetric and non-negative function that integrates to one, that is, $K(u)= K(-u)$ , $K(u) \\ge 0$ for all $u \\in $ and $\\int _{-\\infty }^\\infty K(u) \\, {\\rm d}u = 1$ .",
"Moreover, $\\int _{-\\infty }^\\infty u^2 K(u) \\, {\\rm d} u <\\infty $ .",
"We will use the notation $\\kappa _k = \\int _{-\\infty }^\\infty |u |^k K(u) \\,{\\rm d} u$ for $k \\ge 1$ .",
"Furthermore, we define the population smoothed loss function $Q_{ h} (\\beta ) = \\lbrace \\hat{Q}_{ h}(\\beta )\\rbrace $ , $\\beta \\in ^p$ and the pseudo parameter $\\beta ^*_h (\\tau ) \\in _{\\beta \\in ^p } Q_{ h}(\\beta ), $ which is the population minimizer under the smoothed quantile loss.",
"For simplicity, we write $\\beta ^* = \\beta ^*(\\tau )$ and $\\beta ^*_h = \\beta _h^*(\\tau )$ hereinafter.",
"In general, $\\beta ^*_h$ differs from $\\beta ^*$ , and we refer to $\\Vert \\beta ^*_h - \\beta ^* \\Vert _2$ as the approximation error or smoothing bias.",
"[Conditional density] There exist $\\bar{f} \\ge {f} >0$ such that $ {f} \\le f_{\\varepsilon |}(0 ) \\le \\bar{f}$ almost surely (for all $$ ).",
"Moreover, there exists a constant $l_0 >0$ such that $|f_{\\varepsilon |}(u)- f_{\\varepsilon |}(0)| \\le l_0 |u |$ for all $u \\in $ almost surely.",
"[Random design: moments] The (random) vector $\\in \\mathbb {R}^p$ of covariates satisfies $\\mu _3 := \\sup _{\\in \\mathbb {S}^{p-1}} \\bigl ( | \\langle ^{-1/2} , \\rangle |^3 \\bigr )<\\infty $ , where $= (^$ is positive definite.",
"If $$ is Gaussian, then Condition REF holds trivially.",
"Heavier-tailed distributions of $$ is excluded here so that we can expect standard rates of convergence for the quantile regression estimates.",
"The following result characterizes the smoothing bias from a nonasymptotic viewpoint.",
"Assume Conditions REF –REF hold.",
"There exist constants $c_1, c_2>0$ depending only on $({f} , l_0, \\mu _3, \\kappa _1, \\kappa _2)$ such that for any $0<h \\le c_1$ , $\\beta ^*_h$ is the unique minimizer of $\\beta \\mapsto Q_{ h}(\\beta )$ and satisfies $\\Vert \\beta ^*_h - \\beta ^* \\Vert _{} \\le c_2 h^2 , $ with $= \\bigl \\lbrace f_{\\varepsilon |}( 0 ) ^\\rbrace $ .",
"In addition, assume $\\kappa _3 <\\infty $ , $f_{\\varepsilon |}(\\cdot )$ is continuously differentiable and satisfies almost surely that $ | f^{\\prime }_{\\varepsilon |} (u) - f^{\\prime }_{ \\varepsilon |} (0) | \\le l_1 | u |$ for some constant $l_1 >0$ .",
"Then $ \\biggl \\Vert \\beta ^*_h - \\beta ^* + \\frac{1}{2} \\kappa _2 h^2 \\cdot ^{-1} \\bigl \\lbrace f_{\\varepsilon |}^{\\prime }( 0 )\\bigr \\rbrace \\biggr \\Vert _{}\\le c_3 h^3,$ where $c_3 >0$ is a constant depending only on $({f} , l_0 , l_1 , \\mu _3 )$ and the kernel $K$ .",
"To better understand inequality (REF ), consider for simplicity a special case where $ f_{\\varepsilon |} ( 0 ) $ is a constant (independent of $$ ).",
"Hence, $= f_{\\varepsilon |} ( 0 ) $ and (REF ) becomes $( \\langle , \\beta ^*_h -\\beta ^* \\rangle ^2 )^{1/2} = \\Vert \\beta ^*_h -\\beta ^* \\Vert _{} \\le f^{-1/2}_{\\varepsilon |} ( 0 ) c_2 h^2$ .",
"On the left-hand side, $\\Vert \\beta ^*_h -\\beta ^* \\Vert _{}$ is the average prediction smoothing error.",
"Interestingly, the upper bound on the right-hand side could be dimension-free given $h$ .",
"Another interesting implication is that, when both $f_{\\varepsilon |}(0)$ and $f^{\\prime }_{\\varepsilon |}(0)$ are independent of $$ , i.e., $f_{\\varepsilon |}(0)=f_{\\varepsilon }(0)$ and $f^{\\prime }_{\\varepsilon |}(0)=f^{\\prime }_{\\varepsilon }(0)$ , the leading term in the bias simplifies to $\\frac{1}{2} \\kappa _2 h^2 \\cdot ^{-1} \\bigl \\lbrace f_{\\varepsilon |}^{\\prime }(0)\\bigr \\rbrace = \\frac{f^{\\prime }_{\\varepsilon }(0)}{2f_{ \\varepsilon }(0)} \\kappa _2 h^2 \\cdot ^{-1} () = \\frac{f^{\\prime }_{\\varepsilon }(0)}{2f_{\\varepsilon }(0)} \\kappa _2 h^2 \\cdot \\begin{bmatrix}1 \\\\\\textbf {0}_{p-1}\\end{bmatrix} .",
"\\nonumber $ In other words, the smoothing bias is concentrated primarily on the intercept.",
"In the asymptotic setting where $p$ is fixed, and $h=o(1)$ as $n\\rightarrow \\infty $ , we refer to Theorem 1 in [20] for the expression of asymptotic bias."
],
[
"Finite sample theory",
"In this section, we provide two nonasymptotic results, the concentration inequality and the Bahadur-Kiefer representation, for the conquer estimator under random design.",
"[Random design: sub-exponential case] The predictor $\\in ^p$ is sub-exponential: there exists $ \\upsilon _0>0$ such that $\\lbrace |\\langle , \\rangle | \\ge \\upsilon _0 \\Vert \\Vert _2 t \\rbrace \\le 2e^{-t }$ for all $\\in ^p$ and $t \\ge 0$ , where $= ^{-1/2} $ with $= (^$ positive-definite.",
"Condition REF assumes a sub-exponential condition on the random covariates, which encompasses the bounded case considered by [20].",
"For the standardized predictor $= ^{-1/2}$ , we define the uniform moment parameters $\\mu _k = \\sup _{\\in \\mathbb {S}^{p-1} } | \\langle , \\rangle |^k , \\ \\ k = 1,2 ,\\ldots , $ with $\\mu _2=1$ .",
"Under Condition REF , a straightforward calculation shows that $\\mu _k \\le 2 \\upsilon _0^k k!$ , valid for all $k\\ge 1$ , where $\\upsilon _0$ is defined in Condition REF .",
"In general, $\\upsilon _0$ may increase with $p$ , but in this paper we are primarily interested in the cases where $\\upsilon _0$ is a dimension-free constant.",
"This is indeed the case if $$ is multivariate Gaussian, multivariate Bernoulli, uniform on $[-1,1]^p$ , or uniform on the sphere/ball with radius $p^{1/2}$ ; see Section 3.4 in [51] for more examples.",
"Assume that Conditions REF –REF hold with $\\kappa _l = \\min _{|u|\\le 1} K(u) >0$ .",
"For any $t > 0$ , the smoothed quantile regression estimator $\\hat{\\beta }_{h}$ with $\\sqrt{(p+t)/n} \\lesssim h \\lesssim 1$ satisfies $\\Vert \\hat{\\beta }_h - \\beta ^* \\Vert _{} \\le C (\\kappa _l {f} )^{-1} \\Biggl ( \\sqrt{\\frac{p+t}{n}} + h^2 \\Biggr ), $ with probability at least $1-2e^{-t}$ , where $C>0$ is a constant depending only on $(\\upsilon _0, \\kappa _2,l_0)$ .",
"The estimation error in (REF ) is upper bounded by two terms, $h^2$ and $\\sqrt{(p+t)/n}$ , and can be interpreted as the upper bounds for bias and variance, respectively.",
"The condition $\\min _{|u|\\le 1} K(u) >0$ can be relaxed to $ \\min _{|u|\\le c} K(u) >0$ for some $c\\in (0,1)$ , which will only change the constants encountered in the proof.",
"In particular, for kernels that are compactly supported on $[-1,1]$ , we may choose $c=1/2$ in Theorem REF .",
"Next, we establish a Bahadur representation for conquer.",
"To this end, we impose a slightly more stringent sub-Gaussian condition on the covariates.",
"[Random design: sub-Gaussian case] The predictor $\\in ^p$ is sub-Gaussian: there exists $ \\upsilon _1 >0$ such that $\\lbrace |\\langle , \\rangle | \\ge \\upsilon _1 \\Vert \\Vert _2 t \\rbrace \\le 2 e^{-t^2/2}$ for all $\\in ^p$ and $t\\ge 0$ , where $= ^{-1/2} $ .",
"We are primarily concerned with the cases where $\\upsilon _1$ is a dimension-free constant that does not depend on $p$ ; see the remarks made earlier about $\\upsilon _0$ .",
"Assume that the data $\\lbrace (y_i, _i) \\rbrace _{i=1}^n$ are generated from the conditional quantile model (REF ), and write $\\varepsilon _i = y_i - \\langle _i, \\beta ^* \\rangle $ which satisfy $(\\varepsilon _i\\le 0 | _i) =\\tau $ .",
"Assume Conditions REF , REF , and REF hold with $\\kappa _l = \\min _{|u|\\le 1} K(u) >0$ and $\\kappa _u = \\sup _{u \\in } K(u)$ .",
"Moreover, assume $\\sup _{u \\in } f_{\\varepsilon | } (u) \\le \\bar{f} $ almost surely.",
"For any $t > 0$ , let the bandwidth $h$ satisfy $\\sqrt{(p+t)/n} \\lesssim h \\lesssim 1$ .",
"Then, $\\Biggl \\Vert \\hat{\\beta }_h - \\beta ^* - ^{-1} \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace \\tau - K(- \\varepsilon _i /h) \\bigr \\rbrace _i \\Biggr \\Vert _2 \\le C \\Biggl ( \\frac{p+ t}{ h^{1/2} n } + h\\sqrt{\\frac{p+t}{n}} + h^3\\Biggr ) , $ with probability at least $1-3e^{-t}$ , where $= \\bigl \\lbrace f_{\\varepsilon |}( 0 ) ^\\rbrace $ , and $C>0$ is a constant depending only on $(\\upsilon _1, \\kappa _2, \\kappa _u, \\kappa _l, l_0,\\bar{f}, {f})$ .",
"The Bahadur representation can be used to establish the limiting distribution of the estimator or its functionals.",
"Here we consider a fundamental statistical inference problem for testing the linear hypothesis $H_0: \\langle , \\beta ^* \\rangle = 0$ , where $\\in ^p$ is a deterministic vector that defines a linear functional of interest.",
"It is then natural to consider a test statistic that depends on $n^{1/2} \\langle , \\hat{\\beta }_h \\rangle $ .",
"Based on the nonasymptotic result in Theorem REF , we establish a Berry-Esseen bound for the linear projection of conquer.",
"Assume that the conditions in Theorem REF hold, and $\\sqrt{(p+\\log n)/n}\\lesssim h \\lesssim 1$ .",
"Then, $\\Delta _{n,p}( h) :=\\sup _{x\\in , \\, \\in ^p } \\left| \\left\\lbrace \\frac{n^{1/2} \\langle , \\hat{\\beta }_h - \\beta ^* \\rangle }{\\sqrt{\\tau (1-\\tau ) \\, ^^{-1} ^{-1} }} \\le x \\right\\rbrace - \\Phi (x ) \\right| \\lesssim \\frac{p + \\log n}{(n h)^{1/2}} + n^{1/2} h^2 , $ where $\\Phi (\\cdot )$ denotes the standard normal distribution function.",
"In particular, with the choice of bandwidth $h \\asymp \\lbrace (p+\\log n)/n \\rbrace ^{2/5}$ , the normal approximation error $ \\Delta _{n,p}( h) $ is of order $(p+\\log n)^{4/5} n^{-3/10}$ .",
"It is worth noticing that although the bound in (REF ) holds uniformly over all $$ , the asymptotic variance $\\sigma _\\tau ^2 := \\tau (1-\\tau ) \\, ^^{-1} ^{-1} $ depends on $$ .",
"[Large-$p$ asymptotics] A broader view of classical asymptotics recognizes that the parametric dimension of appropriate model sequences may tend to infinity with the sample size; that is $p=p_n \\rightarrow \\infty $ as $n\\rightarrow \\infty $ .",
"Results with increasing $p$ are available in the context of regularized quantile regression [6], [53], [36].",
"In the large-$p$ and larger-$n$ setting with quantile regression estimation without regularization, [54] shows that $p^3(\\log n)^2 / n \\rightarrow 0$ suffices for a normal approximation, which provides some support to the viability of observed rates of parametric growth in the applied literature.",
"[32] considers a sample of 733 wage models appeared in the econometric literature, and finds that $p_n = (n^{1/4})$ is roughly consistent with empirical practice.",
"For the (convolution-type) smoothed quantile regression, the Berry-Esseen bound (REF ) in Theorem REF immediately yields a large-$p$ asymptotic result.",
"Taking $h=h_n = \\lbrace (p+\\log n) /n\\rbrace ^{2/5}$ therein, we find that $n^{1/2} \\langle , \\hat{\\beta }_h - \\beta ^* \\rangle $ , for any given $\\in ^p$ , is asymptotically normally distributed as long as $p^{8/3} /n \\rightarrow 0$ , which improves the best known growth condition on $p$ for quantile regression [54].",
"For smooth robust regression estimators, asymptotic normality can be proven under less restrictive conditions on $p$ .",
"[28] showed that if the loss is twice differentiable, the asymptotic normality for $\\langle , \\hat{\\beta }\\rangle $ , where $\\in ^p$ , holds if $p^3/n \\rightarrow 0$ as $n$ increases.",
"[46] weakened the condition to $(p \\log n)^{3/2} /n \\rightarrow 0$ if the loss function is four times differentiable and the error distribution is symmetric.",
"For Huber loss that has a Lipschitz continuous derivative, [26] obtained the scaling $p^2 \\log p = o(n)$ that ensures the asymptotic normality of arbitrary linear combinations of $\\hat{\\beta }$ .",
"Table 1 summarizes our discussion here and show that the smoothing for conquer helps ensure asymptotic normality of the estimator under weaker conditions on $p$ than what we need for the usual quantile regression estimator.",
"Table: Summary of scaling conditions required for normal approximation under various loss functions.In this paper, we show that the accuracy of conquer-based inference via the Bahadur representation (and normal approximations) has an error of rate faster than $n^{-1/4}$ yet slower than $n^{-1/2}$ ; see Theorems REF and REF .",
"For standard regression quantiles, [47] proposed an alternative expansion for the quantile process using the “Hungarian\" construction of Komlós, Major and Tusnády.",
"This stochastic approximation yields an error of order $n^{-1/2}$ (up to a factor of $\\log n$ ), and hence provides a theoretical justification for accurate approximations for inference in regression quantile models."
],
[
"Theoretical guarantees for bootstrap inference",
"We next investigate the statistical properties of the bootstrapped estimator defined in (REF ), with a particular focus on the Rademacher multiplier bootstrap.",
"To be specific, we use, in this section and the rest of the paper, the random weights $w_i=1+e_i$ for $i=1,\\ldots , n$ , where $e_1,\\ldots , e_n$ are independent Rademacher random variables, that is, $(e_i=1) = (e_i=-1)=1/2$ .",
"As before, we consider array (non)asymptotics, so that the bootstrap approximation errors depend explicitly on $(n,p)$ .",
"Assume Conditions REF –REF , and REF hold with $\\kappa _l = \\min _{|u|\\le 1}K(u)>0$ .",
"For any $t\\ge 0$ , there exists some event $(t)$ with $\\lbrace (t) \\rbrace \\ge 1-5e^{-t}$ such that, with $^*$ -probability at least $1-2e^{-t}$ conditioned on $(t)$ , the bootstrap estimator $\\hat{\\beta }^\\flat _h$ given in (REF ) satisfies $\\Vert \\hat{\\beta }^\\flat _h - \\beta ^* \\Vert _{} \\le C \\Biggl ( \\sqrt{\\frac{p+t}{n} } + h^2 \\Biggr )$ as long as $\\max \\lbrace (p+\\log n + t )/n, \\sqrt{(p+t)/n } \\rbrace \\lesssim h \\lesssim 1$ , where $C>0$ is a constant depending only on $(\\upsilon _1,\\kappa _2, \\kappa _l, l_0, {f})$ .",
"Analogously to Theorem REF , we further provide a Bahadur representation result for the bootstrap counterpart $\\hat{\\beta }^\\flat _h $ , which lays the theoretical foundation for the validity of Rademacher multiplier bootstrap for conquer.",
"Assume Conditions REF –REF and REF hold with $\\kappa _l = \\min _{|u|\\le 1}K(u)>0$ and $\\kappa _u = \\sup _{u\\in } K(u)$ .",
"Moreover, assume $\\sup _{u\\in } f_{\\varepsilon | } (u) \\le \\bar{f}$ almost surely in $$ .",
"Let the bandwidth satisfy $h \\asymp \\lbrace (p+\\log n)/n\\rbrace ^{\\delta }$ for some $\\delta \\in [1/3, 1/2)$ .",
"Provided that $n\\gtrsim p + \\log n$ , there exists a sequence of events $\\lbrace _n\\rbrace $ with $(_n) \\ge 1-6n^{-1}$ such that, with $^*$ -probability at least $1-2n^{-1}$ conditioned on $_n$ , $\\hat{\\beta }^\\flat _h - \\hat{\\beta }_h = ^{-1 } \\frac{1}{n} \\sum _{i=1}^ne_i \\cdot _i \\bigl \\lbrace \\tau - K(- \\varepsilon _i /h) \\bigr \\rbrace + ^\\flat _n, $ where $^{\\flat }_n$ depends on $\\lbrace (e_i, y_i, _i) \\rbrace _1^n$ and satisfies $\\Vert ^\\flat _n \\Vert _2 = _{^*}(\\chi _n)$ , where $\\chi _n = \\chi _n(\\lbrace y_i, _i \\rbrace _{i=1}^n) = _{}\\lbrace \\sqrt{(p+\\log n) p \\log n} / (h^{1/2} n) + (p+\\log n)^{1/2} p \\log (n) /n \\rbrace $ .",
"As suggested by Theorem REF and Remark REF , if we take $h$ to be $\\lbrace (p+\\log n)/n \\rbrace ^{2/5}$ , the normal approximation to the conquer estimator is asymptotically accurate provided that $p^{8/3} = o(n)$ as $n\\rightarrow \\infty $ .",
"For the same $h$ , the remainder $^\\flat _n$ in (REF ) satisfies $\\Vert ^\\flat _n \\Vert _2 = o_{^*}(n^{-1/2})$ as long as $p^3 (\\log n)^2 = o(n)$ .",
"Taken these two parts together, we have the following asymptotic bootstrap approximation result.",
"If the dimension $p=p_n$ satisfies $p^3 (\\log n)^2 = o(n)$ , then, as $n\\rightarrow \\infty $ , $\\sup _{x\\in } \\, \\bigl | \\bigl ( n^{1/2} \\langle , \\hat{\\beta }_h - \\beta ^* \\rangle \\le x \\bigr ) - ^*\\bigl ( n^{1/2} \\langle , \\hat{\\beta }^\\flat _h - \\hat{\\beta }_h \\rangle \\le x \\bigr ) \\bigr | \\stackrel{}{\\rightarrow } 0 .",
"$ The proof of (REF ) follows the same argument as that in the proof of Theorem REF , and therefore is omitted.",
"The additional logarithmic factor in the scaling may be an artifact of the proof technique.",
"For standard quantile regression, [19] established a fixed-$p$ asymptotic bootstrap approximation result for wild bootstrap under fixed design."
],
[
"Gradient Descent Methods for Conquer",
"To solve optimization problems (REF ) and (REF ) with non-negative weights, arguably the simplest algorithm is a vanilla gradient descent algorithm (GD).",
"For a prespecified $\\tau \\in (0,1)$ and bandwidth $h>0$ , recall that $ \\hat{Q}_h (\\beta ) = (1/n) \\sum _{i=1}^n\\ell _h(y_i - \\langle _i, \\beta \\rangle )$ .",
"Starting with an initial value $\\hat{\\beta }^0 \\in ^p$ , at iteration $t=0, 1,2,\\ldots $ , GD computes $\\hat{\\beta }^{t+1} = \\hat{\\beta }^t - \\eta _t \\cdot \\nabla \\hat{Q}_h (\\hat{\\beta }^t) ,$ where $\\eta _t>0$ is the stepsize and $\\nabla \\hat{Q}_h (\\beta ) = \\frac{1}{n} \\sum _{i=1}^n\\biggl \\lbrace {K}\\biggl ( - \\frac{ y_i - \\langle _i, \\beta \\rangle }{h} \\biggr ) - \\tau \\biggr \\rbrace _i ~~\\mbox{ with }~~{K}(u ) = \\int _{-\\infty }^u K(t) \\,{\\rm d} t .",
"\\nonumber $ In the classical GD method, the stepsize is usually obtained by employing line search techniques.",
"However, line search is computationally intensive for large-scale settings.",
"One of the most important issues in GD is to determine a proper update step $\\eta _t$ decay schedule.",
"A common practice in the literature is to use a diminishing stepsize or a best-tuned fixed stepsize.",
"Neither of these two approaches can be efficient, at least compared to the Newton-Frisch algorithm with preprocessing [48].",
"Recall that the smoothed loss $\\hat{Q}_h(\\cdot )$ is twice differentiable with Hessian $\\nabla ^2 \\hat{Q}_h (\\beta ) = \\frac{1}{n h } \\sum _{i=1}^nK \\biggl ( \\frac{y_i - \\langle _i, \\beta \\rangle }{h} \\biggr ) _i _i^ \\nonumber $ It is therefore intriguing to employ the Newton-Raphson method, which at iteration $t$ would read $\\hat{\\beta }^{t+1} = \\hat{\\beta }^t - \\bigl \\lbrace \\nabla ^2 \\hat{Q}_h (\\hat{\\beta }^t) \\bigr \\rbrace ^{-1} \\nabla \\hat{Q}_h (\\hat{\\beta }^t) .$ Computing the inverse of the Hessian for large problems can be an expensive operation.",
"Moreover, when $h$ is small, $\\nabla ^2 \\hat{Q}_h (\\cdot )$ can have a very large condition number, which leads to slow convergence.",
"Below we list the explicit expressions of the convolution smoothed check functions using several commonly used kernels.",
"Recall that the check function can be written as $\\rho _\\tau (u) =|u|/2+ (\\tau -1/2)u$ , which, after convolution smoothing, becomes $\\ell _h (u) = \\int _{-\\infty }^\\infty \\frac{1}{2}|u + hv | K (v) \\, {\\rm d} v + (\\tau -1/2)u .$ (Uniform kernel $K(u) = (1/2) \\mathbb {1} (|u|\\le 1)$ ): $\\ell _h(u) = (h/2) \\ell ^{{\\rm U}}(u/h) + (\\tau - 1/2) u$ , where $ \\ell ^{{\\rm U}}(u) := (u^2/2 + 1/2)\\mathbb {1}( |u|\\le 1) + |u| \\mathbb {1}(|u|>1)$ is a Huber-type loss [28].",
"(Gaussian kernel $K(u)=(2\\pi )^{-1/2} e^{-u^2/2}$ ): $\\ell _{h}(u) = (h/2)\\ell ^{{\\rm G}}(u/h) + (\\tau - 1/2 ) u$ , where $ \\ell ^{{\\rm G}}(u) := (2/\\pi )^{1/2} e^{-u^2/2 } + u \\lbrace 1- 2\\Phi (-u) \\rbrace $ .",
"(Logistic kernel $K(u) = e^{-u}/(1+ e^{-u})^2 $ ): $\\ell _h (u) = (h/2) \\ell ^{{\\rm L}}(u/h) + (\\tau -1/2) u$ , where $ \\ell ^{{\\rm L}}(u) := u + 2\\log (1+e^{-u})$ .",
"(Epanechnikov kernel $K(u) = (3 / 4) (1 - u^2) \\mathbb {1}(|u|\\le 1)$ ): $\\ell _{h}(u) = (h/2) \\ell ^{{\\rm E}}(u/h) +(\\tau - 1/2)u$ , where $ \\ell ^{{\\rm E}}(u) := (3u^2/4 - u^4/8+ 3/8) \\mathbb {1}(|u|\\le 1) + |u|\\mathbb {1}(|u|>1)$ .",
"(Triangular kernel $K(u) = (1 - |u|) \\mathbb {1}(|u|\\le 1)$ ): $ \\ell _{h}(u) = (h/2) \\ell ^{{\\rm T}}(u/h) + ( \\tau - 1/2) u$ , where $ \\ell ^{{\\rm T}}(u ) := ( u^2 - |u|^3/3 + 1/3 ) \\mathbb {1}(|u|\\le 1) + |u|\\mathbb {1}(|u| >1) $ ."
],
[
"The Barzilai-Borwein stepsize rule",
"In this paper, we propose to solve conquer by means of the gradient descent with a Barzilai-Borwein update step [5], which we refer to as the GD-BB algorithm.",
"The BB method, which is motivated by quasi-Newton methods, has been proven to be very successful in solving nonlinear optimization problems.",
"Recall the Newton-Raphson method (REF ), which can be computationally expensive and needs modifications if the Hessian is ill-conditioned especially when $h$ is too small.",
"For this reason, many quasi-Newton methods seek a simple approximation of the inverse Hessian matrix, say $(^t)^{-1}$ , which satisfies the secant equation $^t \\delta ^t = ^t$ , where for $t=1, 2, \\ldots $ , $\\delta ^t = \\hat{\\beta }^t - \\hat{\\beta }^{t-1} ~~\\mbox{ and }~~ ^t = \\nabla \\hat{Q}_h (\\hat{\\beta }^t) - \\nabla \\hat{Q}_h (\\hat{\\beta }^{t-1}).",
"$ To relieve the computational bottleneck of inverting a large matrix, the BB method chooses $\\eta $ so that $\\eta \\nabla \\hat{Q}_h (\\hat{\\beta }^t) = (\\eta ^{-1} _p)^{-1}\\nabla \\hat{Q}_h (\\hat{\\beta }^t)$ “approximates\" $(^t)^{-1} \\nabla \\hat{Q}_h (\\hat{\\beta }^t)$ .",
"Since $^t$ satisfies $^t \\delta ^t = ^t$ , it is more practical to choose $\\eta $ such that $ (1/\\eta ) \\delta ^t \\approx ^t$ or $ \\delta ^t \\approx \\eta ^t$ .",
"Via least squares approximations, one may use $\\eta _{1,t}^{-1} = _{ \\alpha } \\Vert \\alpha \\delta ^t - ^t \\Vert _2^2$ or $\\eta _{2,t} = _{ \\eta } \\Vert \\delta ^t - \\eta ^t \\Vert _2^2$ .",
"The BB stepsizes are then defined as $\\eta _{1,t} = \\frac{\\langle \\delta ^t , \\delta ^t\\rangle }{\\langle \\delta ^t, ^t \\rangle } ~~\\mbox{ and }~~ \\eta _{2,t} = \\frac{\\langle \\delta ^t , ^t \\rangle }{ \\langle ^t, ^t \\rangle } .",
"$ Consequently, the BB iteration takes the form $\\hat{\\beta }^{t+1} = \\hat{\\beta }^t - \\eta _{\\ell ,t} \\nabla \\hat{Q}_h (\\hat{\\beta }^t) , \\quad \\ell = 1 {\\rm ~or~} 2 .",
"$ Note that the BB step starts at iteration 1, while at iteration 0, we compute $\\hat{\\beta }^1$ using the standard gradient descent with an initial estimate $\\hat{\\beta }^0$ .",
"The procedure is summarized in Algorithm REF .",
"Based on extensive numerical studies, we find that at a fixed $\\tau $ , the number of iterations is insensitive to varying $(n,p)$ combinations.",
"Moreover, as $h$ increases, the number of iterations declines because the loss function is “more convex\" for larger $h$ .",
"In Algorithm REF , the quantity $\\delta >0$ is called the gradient tolerance, ensuring that the obtained estimate, say $\\hat{\\beta }$ , satisfies $\\Vert \\nabla \\hat{Q}_h(\\hat{\\beta }) \\Vert _2 \\le \\delta $ .",
"Provided that $\\delta \\lesssim \\sqrt{p/n}$ , the statistical theory developed in Section prevails.",
"In our R package conquer, we set $\\delta = 10^{-4}$ as the default value; this value can also be specified by the user.",
"As $\\tau $ approaches 0 or 1, the Hessian matrix becomes ill-conditioned.",
"As a result, the stepsizes computed in GD-BB may sometimes vibrate drastically, causing instability of the algorithm.",
"Therefore, in practice, we set a lower bound for the stepsizes by taking $\\eta _t = \\min \\lbrace \\eta _{1,t}, \\eta _{2,t} , 100\\rbrace $ , for $t=1, 2,\\ldots $ .",
"Another cause of an ill-conditioned Hessian arises when we have covariates with very different scales.",
"In this case, the stepsize should be different for each covariate, and a constant stepsize will be either too small or too large for one or more covariates, which correspond to slow convergence.",
"To address this issue, we scale the covariate inputs to have zero mean and unit variance before applying gradient descent.",
"[!t] Gradient descent with Barzilai-Borwin stepsize (GD-BB) for solving conquer.",
"Input: data vectors $\\lbrace (y_i, _i)\\rbrace _{i=1}^n$ , $\\tau \\in (0,1)$ , bandwidth $h\\in (0,1)$ , initialization $\\hat{\\beta }^{(0)}$ , and gradient tolerance $\\delta $ .",
"[1] Compute $\\hat{\\beta }^1 \\leftarrow \\hat{\\beta }^0 - \\nabla \\hat{Q}_h( \\hat{\\beta }^0)$ $t=1,2 \\ldots $ $\\delta ^t \\leftarrow \\hat{\\beta }^t - \\hat{\\beta }^{t-1}$ , $^t \\leftarrow \\nabla \\hat{Q}_h (\\hat{\\beta }^{t})-\\nabla \\hat{Q}_h (\\hat{\\beta }^{t-1})$ $\\eta _{1,t} \\leftarrow \\langle \\delta ^t, \\delta ^t \\rangle / \\langle \\delta ^t, ^t \\rangle $ , $\\eta _{2,t} \\leftarrow \\langle \\delta ^t, ^t\\rangle / \\langle ^t, ^t \\rangle $ $\\eta _t \\leftarrow \\min \\lbrace \\eta _{1,t},\\eta _{2,t}, 100\\rbrace $ if $\\eta _{1,t} >0$ and $\\eta _t \\leftarrow 1$ otherwise $ \\hat{\\beta }^{t+1} \\leftarrow \\hat{\\beta }^t - \\eta _t \\nabla \\hat{Q}_h (\\hat{\\beta }^{t})$ when $\\Vert \\nabla \\hat{Q}_h (\\hat{\\beta }^{t}) \\Vert _2\\le \\delta $"
],
[
"Warm start via Huber regression",
"A good initialization helps reduce the number of iterations for GD, and hence facilitates fast convergence.",
"Recall from Remark that with a uniform kernel, the smoothed check function is proximal to a Huber loss [28].",
"Motivated by this subtle proximity, we propose using the Huber $M$ -estimator as an initial estimate, and then proceed by iteratively applying gradient descent with BB update step.",
"Let $H_\\gamma (u) = (u^2/2) \\mathbb {1}(|u|\\le \\gamma ) + \\gamma (|u| - \\gamma /2) \\mathbb {1}(|u|> \\gamma )$ be the Huber loss parametrized by $ \\gamma >0$ .",
"The Huber $M$ -estimator is then defined as $\\widetilde{\\beta }_\\gamma \\in _{\\beta \\in ^p} \\hat{}_\\gamma (\\beta ), ~~\\mbox{where}~~ \\hat{}_\\gamma (\\beta ) = \\frac{1}{n} \\sum _{i=1}^nH_\\gamma (y_i - \\langle _i, \\beta \\rangle ).",
"$ Note that $H_1(u) = \\ell ^{{\\rm U}}(u)-1/2$ for $\\ell ^{{\\rm U}}(\\cdot )$ defined in Remark , we have $ \\hat{Q}_{1/2, \\gamma }(\\beta ) = \\hat{}_ \\gamma (\\beta )$ .",
"The quantity $ \\gamma $ is a shape parameter that controls the amount of robustness.",
"The main reason for choosing a fixed (neither diminishing nor diverging) tuning parameter $ \\gamma $ in [29] is to guarantee robustness towards arbitrary contamination in a neighborhood of the model.",
"This is at the core of the robust statistics idiosyncrasy.",
"In particular, [29] proposed $ \\gamma =1.35 \\sigma $ to gain as much robustness as possible while retaining 95% asymptotic efficiency for normally distributed data, where $\\sigma >0$ is the standard deviation of the random noise.",
"We estimate $\\sigma $ using the median absolute deviation on the residuals at each iteration, i.e., ${\\rm MAD}(\\lbrace r_i^t \\rbrace _{i=1}^n)= \\text{median}(|r_i^t - r^t_{\\textrm {median}}|)$ , where $r^t_{\\textrm {median}}$ is the median of $r_1^t,\\ldots ,r_n^t$ .",
"Noting that the Huber loss is continuously differentiable, convex, and locally strongly convex, we use the GD-BB method described in the previous section to solve the optimization problem (REF ).",
"Starting at iteration 0 with $\\widetilde{\\beta }^0 = \\textbf {0}$ , at iteration $t=0, 1,2,\\ldots $ , we compute $\\widetilde{\\beta }^{t+1} = \\widetilde{\\beta }^t - \\eta _t \\nabla \\hat{}_\\gamma (\\widetilde{\\beta }^t) = \\widetilde{\\beta }^t + \\frac{\\eta _t}{n} \\sum _{i=1}^n\\psi _\\gamma (y_i-\\langle _i, \\widetilde{\\beta }^t \\rangle ) _i$ with $\\eta _t>0$ automatically obtained by the BB method, where $\\psi _\\gamma (u) = H_\\gamma ^{\\prime }(u) =\\min \\lbrace \\max (- \\gamma , u), \\gamma \\rbrace $ .",
"We summarize the details in Algorithm REF .",
"[!t] GD-BB method for solving (REF ).",
"Input: $\\lbrace (y_i, _i)\\rbrace _{i=1}^n$ and convergence criterion $\\delta $ .",
"[1] Initialize $\\tilde{\\beta }^{(0)} = \\textbf {0}$ Compute $ \\gamma ^0 = 1.35 \\cdot {\\rm MAD}(\\lbrace r_i^0 \\rbrace _{i=1}^n)$ , where $r^0_i \\leftarrow y_i - \\langle _i, \\widetilde{\\beta }^{0} \\rangle $ , $i=1,\\ldots , n$ , where $\\mathrm {MAD}(\\cdot )$ is the median absolute deviation $\\tilde{\\beta }^1 \\leftarrow \\tilde{\\beta }^0 - \\nabla \\hat{L}_{\\gamma ^0}( \\tilde{\\beta }^0)$ $t=1,2 \\ldots $ $\\gamma ^t = 1.35 \\cdot {\\rm MAD}(\\lbrace r^t_i\\rbrace _{i=1}^n)$ , where $r^t_i \\leftarrow y_i - \\langle _i, \\widetilde{\\beta }^{t} \\rangle $ , $i=1,\\ldots , n$ $\\delta ^t \\leftarrow \\tilde{\\beta }^t - \\tilde{\\beta }^{t-1}$ , $^t \\leftarrow \\nabla \\hat{}_{\\gamma ^t}(\\tilde{\\beta }^{t})-\\nabla \\hat{}_{\\gamma ^t}(\\tilde{\\beta }^{t-1})$ $\\eta _{1,t} \\leftarrow \\langle \\delta ^t, \\delta ^t \\rangle / \\langle \\delta ^t, ^t \\rangle $ , $\\eta _{2,t} \\leftarrow \\langle \\delta ^t, ^t\\rangle / \\langle ^t, ^t \\rangle $ .",
"$\\eta _t \\leftarrow \\min \\lbrace \\eta _{1,t},\\eta _{2,t}, 100\\rbrace $ if $\\eta _{1,t} >0$ and $\\eta _t \\leftarrow 1$ otherwise $ \\widetilde{\\beta }^{t+1} \\leftarrow \\tilde{\\beta }^t - \\eta _t \\nabla \\hat{}_{\\gamma ^t}(\\tilde{\\beta }^{t})$ when $\\Vert \\nabla \\hat{}_{\\gamma ^t}(\\tilde{\\beta }^{t}) \\Vert _2\\le \\delta $ The Huber loss $H_\\gamma (\\cdot )$ approximates the quantile loss function with $\\tau =1/2$ as $\\gamma \\rightarrow 0$ .",
"Therefore, an alternative method for QR computing is to solve the Huber regression via gradient descent with a shrinking gamma.",
"To evaluate its performance, we implement the above idea by setting $\\gamma ^{t} = c \\cdot \\gamma ^{t-1}$ for some $c\\in (0,1)$ at the $t$ -th iteration.",
"We found that the aforementioned idea is not stable numerically across several simulated data sets, unless one is very careful in controlling the minimal magnitude of $\\gamma $ .",
"In addition, the solution obtained has a higher estimation error than that of standard QR and conquer."
],
[
"Numerical Studies",
"In this section, we assess the finite-sample performance of conquer via extensive numerical studies.",
"We compare conquer to standard QR [35] and Horowitz's smoothed QR [27].",
"Both the convolution-type and Horowitz's smoothed methods involve a smoothing parameter $h$ .",
"In view of Theorem REF , we take $h = \\lbrace (p+\\log n) /n\\rbrace ^{2/5}$ in all of the numerical experiments.",
"As we will see from Figure REF , the proposed method is insensitive to the choice of $h$ .",
"Therefore, we leave the fine tuning of $h$ as an optional rather than imperative choice.",
"In all the numerical experiments, the convergence criterion in Algorithms REF and REF is taken as $\\delta =10^{-4}$ .",
"We first generate the covariates $_i = (x_{i , 1}, \\dots , x_{i , p})^ from a multivariate uniform distribution on the cube $ 31/2 [-1, 1]p$ with covariance matrix $ =(0.7|j - k|)1j,kp$ using the \\texttt {R} package \\texttt {MultiRNG} \\cite {F1999}.The random noise $ i$ are generated from two different distributions: (i) Gaussian distribution, $ N(0, 4)$; and (ii) $ t$ distribution with degrees of freedom 2, $ t2$.Let $ i- = (xi, 1, ..., xi, p - 1), $\\beta ^* = (1, \\dots , 1)^{p - 1}$ , and $\\beta ^*_0 = 1$ .",
"Given $\\tau \\in (0, 1)$ , we then generate the response $y_i$ from the following homogeneous and heterogeneous models, all of which satisfy the Assumption REF : Homogeneous model: $y_i = \\beta ^*_0 + \\langle _i^- , \\beta ^* \\rangle + \\lbrace \\varepsilon _i - F^{-1}_{\\varepsilon _i}(\\tau ) \\rbrace , ~~~~i = 1, \\dots , n; $ Linear heterogeneous model: $y_i = \\beta ^*_0 + \\langle _i^- , \\beta ^* \\rangle + (0.5 x_{i, p} + 1) \\lbrace \\varepsilon _i - F^{-1}_{\\varepsilon _i}(\\tau ) \\rbrace , ~~~~i = 1, \\dots , n; $ Quadratic heterogeneous model: $y_i = \\beta ^*_0 + \\langle _i^- , \\beta ^* \\rangle + 0.5\\lbrace 1+ ( x_{i, p}-1)^2 \\rbrace \\lbrace \\varepsilon _i - F^{-1}_{\\varepsilon _i}(\\tau )\\rbrace , ~~~~i = 1, \\dots , n. $ To evaluate the performance of different methods, we calculate the estimation error under the $\\ell _2$ -norm, i.e., $\\Vert \\hat{\\beta }-\\beta ^*\\Vert _2$ , and record the elapsed time.",
"The details are in Section REF .",
"In Section REF , we examine the finite-sample performance of the multiplier bootstrap method for constructing confidence intervals in terms of coverage probability, width of the interval, and computing time."
],
[
"Estimation",
"For all the numerical studies in this section, we consider a wide range of the sample size $n$ , with the size-dimension ratio fixed at $n / p = 20$ .",
"That is, we allow the dimension $p$ to increase as a function of $n$ .",
"We implement conquer with four different kernel functions as described in Remark : (i) Gaussian; (ii) uniform; (iii) Epanechnikov; and (iv) triangular.",
"The classical quantile regression is implemented via a modified version of the Barrodale and Roberts algorithm [37], [38] by setting method= “br\" in the R package quantreg, which is recommended for problems up to several thousands of observations in [34].",
"For very large problems, the Frisch-Newton approach after preprocessing “pfn\" is preferred.",
"Since the same size taken to be at most 5000 throughout this section, the two methods, “br” and “pfn\", have nearly identical runtime behaviors.",
"In some applications where there are a lot of discrete covariates, it is advantageous to use method “sfn\", a sparse version of Frisch-Newton algorithm that exploits sparse algebra to compute iterates [39].",
"Moreover, we implement Horowitz's smoothed quantile regression using the Gaussian kernel, and solve the resulting non-convex optimization via gradient descent with random initialization and stepsize calibrated by backtracking line search (Section 9.3 of [8]).",
"The results, averaged over 100 replications, are reported.",
"Figure REF depicts estimation error of the different methods under the simulation settings described in Section with $\\tau = 0.9$ .",
"We see that conquer has a lower estimation error than the classical QR across all scenarios, indicating that smoothing can improve estimation accuracy under the finite-sample setting.",
"Moreover, compared to Horowitz's smoothing, conquer has a lower estimation error in most settings.",
"Estimation error under various quantile levels $\\tau \\in \\lbrace 0.1, 0.3, 0.5, 0.7\\rbrace $ under the $t_2$ random noise are also examined.",
"The results are reported in Figure REF in Appendix , from which we observe evident advantages of conquer, especially at low and high quantile levels.",
"Figure: Model () with t 2 t_2 error.To assess the computational efficiency, we compute the elapsed time for fitting the different methods.",
"Figures REF and REF in Appendix report the runtime for the different methods with growing sample size and dimension under the same settings as in Figures REF and REF , respectively.",
"We observe that conquer is computationally efficient and stable across all scenarios, and the runtime is insensitive to the choice of kernel functions.",
"In contrast, the runtime for classical quantile regression grows rapidly as the sample size and dimension increase.",
"Figures REF and REF in Appendix show that the runtime of Horowitz's smoothing method increases significantly at extreme quantile levels $\\tau \\in \\lbrace 0.1,0.9\\rbrace $ , possibly due to the combination of its non-convex nature and flatter gradient.",
"In summary, we conclude that conquer significantly improves computational efficiency while retaining high statistical accuracy for fitting large-scale linear quantile regression models.",
"Next, we conduct a sensitivity analysis for conquer regarding the smoothing bandwidth $h$ .",
"We set $(n, p) = (2000, 100)$ and consider the simulation settings (REF )–(REF ) with $\\mathcal {N}(0, 4)$ and $t_2$ noise.",
"We perform conquer with $h \\in \\lbrace 0.2, 0.22,\\dots , 0.5\\rbrace $ , including the default value $h_{{\\rm de}} = \\lbrace (p+\\log n) /n\\rbrace ^{2/5} = 0.3107$ , and compare the estimation error with that of QR in Figure REF .",
"We see that the estimation error of conquer is uniformly lower than that of QR over a range of $h$ , suggesting that conquer is insensitive to the choice of bandwidth $h$ .",
"Figure: Model () with t 2 t_2 error.Figure: Model () with t 2 t_2 error."
],
[
"Inference",
"In this section, we assess the performance of the multiplier bootstrap procedure for constructing confidence interval for each of the regression coefficients obtained from conquer.",
"We implement conquer using the Gaussian kernel, and construct three types of confidence intervals: (i) the percentile mb-per; (ii) pivotal mb-piv; (iii) and regular mb-norm confidence intervals, as described in Section REF .",
"We also refer to the proposed multiplier bootstrap procedure as mb-conquer for simplicity.",
"We compare the proposed method to several widely used inference methods for QR.",
"In particular, we consider confidence intervals by inverting a rank score test, rank ([23]; Section 3.5 of [33]); a bootstrap variant based on pivotal estimating functions, pwy [45]; and wild bootstrap with Rademacher weights, wild [19].",
"The three methods rank, pwy, and wild are implemented using the R package quantreg.",
"Note that rank is a non-resampling based procedure that relies on prior knowledge on the random noise, i.e., a user needs to specify whether the random noise are independent and identically distributed.",
"In our simulation studies, we provide rank an unfair advantage by specifying the correct random noise structure.",
"We set $(n, p) = (800, 20)$ , $\\tau \\in \\lbrace 0.5, 0.9\\rbrace $ , and significance level $\\alpha = 0.05$ .",
"All of the resampling methods are implemented with $B = 1000$ bootstrap samples.",
"To measure the reliability, accuracy, and computational efficiency of different methods for constructing confidence intervals, we calculate the average empirical coverage probability, average width of confidence interval, and the average runtime.",
"The average is taken over all regression coefficients without the intercept.",
"Results based on 200 replications are reported in Figure REF , and Figures REF –REF in Appendix .",
"Figure: Runtime under model ().In Figure REF , and Figures REF –REF in Appendix , we use the rank-inversion method, rank, as a benchmark since we implement rank using information about the true underlying random noise, which is practically infeasible.",
"In the case of $\\tau =0.9$ , pwy is most conservative as it produces the widest confidence intervals with slightly inflated coverage probability, and wild gives the narrowest confidence intervals but at the cost of coverage probability.",
"The proposed methods mb-per, mb-piv, and mb-norm achieve a good balance between reliability (high coverage probability) and accuracy (narrow CI width), and moreover, has the lowest runtime.",
"To further highlight the computational gain of the proposed method, we now perform numerical studies with larger $n$ and $p$ .",
"In this case, the rank inversion method rank is computationally infeasible.",
"For example, when $(n,p)= (5000,250)$ , rank inversion takes approximately 80 minutes while conquer with multiplier bootstrap takes 41 seconds for constructing confidence intervals.",
"We therefore omit rank from the following comparison.",
"We consider the quadratic heterogeneous model (REF ) with $(n, p) = (4000, 100)$ and $t_2$ noise.",
"The results are reported in Figure REF .",
"We see that pwy and wild take approximately 300 seconds while mb-conquer takes approximately 15 seconds.",
"In summary, mb-conquer leads to a huge computational gain without sacrificing statistical efficiency.",
"Figure: Runtime"
],
[
"Comparison between normal approximation and bootstrap calibration",
"Finally, we complement the above studies with a comparison between the normal approximation and bootstrap calibration methods for confidence estimation.",
"We consider model (REF ) with $(n,p) = (2000,10)$ .",
"We use the same $\\beta ^*\\in ^{p-1}$ and $\\beta ^*_0$ as before, and generate random covariates and noise from a multivariate uniform distribution and $t_{1.5}$ -distribution, respectively.",
"For each of the $p-1$ regression coefficients, we apply the proposed bootstrap percentile method and the normal-based method [20] to construct pointwise confidence intervals at quantile indices close to 0 and 1, that is, $\\tau \\in \\lbrace 0.05, 0.1, 0.9, 0.95\\rbrace $ .",
"Boxplots of the empirical coverage and CI width for the two methods are reported in Figure REF .",
"Considering that extreme quantile regressions are notoriously hard to estimate, the bootstrap method can produce much more reliable (high coverage) and accurate (narrow width) confidence intervals than the normal-based counterpart.",
"Therefore, for applications in which extreme quantiles are of particular interest, such as the problem of forecasting the conditional value-at-risk of a financial institution [13], the bootstrap provides a more reliable approach for quantifying the uncertainty of the estimates.",
"Figure: Boxplots of CI width"
],
[
"Discussion",
"In this paper, we provide a comprehensive study on the statistical properties of conquer—namely, convolution-type smoothed quantile regression, under the array nonasymptotic setting in which $p$ is allowed to increase as a function of $n$ while $p/n$ being small.",
"An efficient gradient-based algorithm is proposed to compute conquer, which proves to be scalable to large dimensions and even larger sample sizes.",
"Recently, there has been a growing interest in studying the asymptotic behavior of regression estimates under the regime in which $p$ grows proportionally with the sample size $n$ [15], [40].",
"The current studies are mainly focused on $M$ -estimation with strongly convex loss functions.",
"New theoretical tools are needed to establish results for conquer under such a regime, and we leave it for future work.",
"In the high-dimensional setting in which $p \\gg n$ , various authors have studied the regularized quantile regression under the sparsity assumption that most of the regression coefficients are zero [6], [53].",
"Using $\\ell _1$ penalties, the computation of regularized QR is based on either a linear programming reformulation or alternating direction method of multiplier algorithms [22].",
"Since the conquer loss is convex and twice differentiable, we expect that gradient-based algorithms, such as coordinate gradient descent or proximal gradient descent, will enjoy superior computational efficiency for solving regularized conquer without sacrificing statistical accuracy.",
"One future work is to establish the statistical theory and computational complexity of the regularized conquer estimator."
],
[
"One-step Conquer with Higher-order Kernels",
"As noted in Section REF , the smoothing bias is of order $h^2$ when a non-negative kernel is used.",
"The ensuing empirical loss $\\beta \\mapsto (1/n) \\sum _{i=1}^n(\\rho _\\tau * K_h)(y_i - \\langle _i , \\beta \\rangle )$ is not only twice-differentiable and convex, but also (provably) strongly convex in a local vicinity of $\\beta ^*$ with high probability.",
"Kernel smoothing is ubiquitous in nonparametric statistics.",
"The order of a kernel, $\\nu $ , is defined as the order of the first non-zero moment.",
"The order of a symmetric kernel is always even.",
"A kernel is called high-order if $\\nu >2$ , which inevitably has negative parts and thus is no longer a probability density.",
"Thus far we have focused on conquer with second-order kernels, and the resulting estimator achieves an $\\ell _2$ -error of the order $\\sqrt{p/n} + h^2$ .",
"Let $G(\\cdot )$ be a higher-order symmetric kernel with order $\\nu \\ge 4$ , and $b >0$ be a bandwidth.",
"Again, via convolution smoothing, we may consider a bias-reduced estimator that minimizes the empirical loss $\\beta \\mapsto \\hat{Q}_b^G(\\beta ):= (1/n) \\sum _{i=1}^n(\\rho _\\tau * G_b )(y_i - \\langle _i , \\beta \\rangle )$ .",
"This, however, leads to a non-convex optimization.",
"Without further assumptions, finding a global minimum is computationally intractable: finding an $\\epsilon $ -suboptimal point for a $k$ -times continuously differentiable loss function requires at least $\\Omega \\lbrace (1/\\epsilon )^{p/k}\\rbrace $ evaluations of the function and its first $k$ derivatives, ignoring problem-dependent constants; see Section 1.6 in [42].",
"Instead, various gradient-based methods have been developed for computing stationary points, which are points $\\beta $ with sufficiently small gradient $\\Vert \\nabla \\hat{Q}_b^G(\\beta ) \\Vert _2 \\le \\epsilon $ , where $\\epsilon \\ge 0$ is optimization error.",
"However, the equation $\\nabla \\hat{Q}_b^G(\\beta ) = \\textbf {0}$ does not necessarily have a unique solution, whose statistical guarantees remain unknown.",
"Motivated by the classical one-step estimator [7], we further propose a one-step conquer estimator using high-order kernels, which bypasses solving a large-scale non-convex optimization.",
"To begin with, we choose two symmetric kernel functions, $K: \\rightarrow [0,\\infty )$ with order two and $G(\\cdot )$ with order $\\nu \\ge 4$ , and let $h, b >0$ be two bandwidths.",
"First, compute an initial conquer estimator $\\beta \\in _{\\beta \\in ^p} \\hat{Q}_{h}^K(\\beta )$ , where $\\hat{Q}_{h}^K(\\beta ) = (1/n) \\sum _{i=1}^n(\\rho _\\tau * K_h) (y_i - \\langle _i , \\beta \\rangle )$ .",
"Denote by $\\bar{r}_i = y_i - \\langle _i, \\beta \\rangle $ for $i=1,\\ldots , n$ the fitted residuals.",
"Next, with slight abuse of notation, we define the one-step conquer estimator $\\hat{\\beta }$ as a solution to the equation $ \\nabla ^2 \\hat{Q}_b^G (\\beta ) ( \\hat{\\beta }- \\beta ) = - \\nabla \\hat{Q}_b^G(\\beta )$ , or equivalently, $\\Biggl \\lbrace \\frac{1}{n } \\sum _{i=1}^nG_b ( \\bar{r}_i ) _i _i^\\rbrace ( \\hat{\\beta }- \\beta ) = \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace G ( \\bar{r}_i / b ) + \\tau - 1 \\bigr \\rbrace _i .",
"$ where $\\hat{Q}_b^G(\\beta ) = (1/n) \\sum _{i=1}^n(\\rho _\\tau * G_b ) (y_i - \\langle _i , \\beta \\rangle )$ .",
"Provided that $\\nabla ^2 \\hat{Q}_b^G (\\beta )$ is positive definite, the one-step conquer estimate $\\hat{\\beta }$ essentially performs a Newton-type step based on $\\beta $ : $\\hat{\\beta }= \\beta - \\bigl \\lbrace \\nabla ^2 \\hat{Q}_b^G (\\beta ) \\bigr \\rbrace ^{-1} \\nabla \\hat{Q}_b^G(\\beta ).",
"$ In this case, $\\hat{\\beta }$ can be computed by the conjugate gradient method [62].",
"Theoretical properties of the one-step estimator $\\hat{\\beta }$ defined in (REF ), including the Bahadur representation and asymptotic normality with explicit Berry-Esseen bound, will be provided in on-line supplementary materials.",
"For practical implementation, we consider higher-order Gaussian-based kernels.",
"For $r=1,2,\\ldots ,$ the $(2r)$ -th order Gaussian kernels are $G_{2r}( u) = \\frac{(-1)^r \\phi ^{(2r-1)}(u)}{2^{r-1} (r-1)!",
"u} = \\sum _{\\ell =0}^{r-1} \\frac{(-1)^\\ell }{2^\\ell \\ell !}",
"\\phi ^{(2\\ell )} (u) ;$ see Section 2 of [70].",
"Integrating $G_{2r}(\\cdot )$ yields ${G}_{2r}(v) = \\int _{-\\infty }^v G_{2r}( u) \\, {\\rm d}u = \\sum _{\\ell =0}^{r-1} \\frac{(-1)^\\ell }{2^\\ell \\ell !}",
"\\phi ^{(2\\ell -1)}(v).$ In fact, both $G_{2r}$ and ${G}_{2r}$ have simpler forms $G_{2r}(u) = p_r(u) \\phi (u)$ and ${G}_{2r}(u) = \\Phi (u)+ P_r(u) \\phi (u)$ , where $p_r(\\cdot )$ and $P_r(\\cdot )$ are polynomials in $u$ .",
"For example, $p_1(u) = 1$ , $P_1(u)=0$ , $p_2(u) =(-u^2+3)/2$ , $P_2(u)=u/2$ , $p_3(u)= (u^4 - 10 u^2 + 15)/8$ , and $P_3(u)=(-u^3 +7u)/8$ .",
"We refer to [65] for more details when $r$ is large."
],
[
"Proof of Proposition ",
"For every $r>0$ , define the ellipse $\\Theta (r) = \\lbrace \\in ^p: \\Vert \\Vert _{} \\le r \\rbrace $ and the local vicinity $\\Theta ^* = \\lbrace \\beta \\in ^p: \\beta - \\beta ^* \\in \\Theta ( \\kappa _2^{1/2} h ) \\rbrace $ .",
"Let $\\eta = \\sup \\lbrace u \\in [0,1] : u (\\beta ^*_h -\\beta ^*) \\in \\Theta (\\kappa _2^{1/2} h) \\rbrace $ and $\\widetilde{\\beta }^* = \\beta ^* + \\eta (\\beta ^*_h -\\beta ^*)$ .",
"By definition, $\\eta = 1$ if $\\beta ^*_h \\in \\Theta ^*$ and $\\eta <1$ if $\\beta ^*_h \\notin \\Theta ^*$ .",
"In the latter case, $\\widetilde{\\beta }^* \\in \\partial \\Theta ^*$ .",
"By the convexity of $\\beta \\mapsto Q_h(\\beta )$ and Lemma C.1 in the supplementary material of [68], $0 & \\le \\langle \\nabla Q_h( \\widetilde{\\beta }^* ) - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle \\nonumber \\\\& \\le \\eta \\cdot \\langle \\nabla Q_h(\\beta ^*_h ) - \\nabla Q_h(\\beta ^*) , \\beta ^*_h - \\beta ^* \\rangle = \\langle - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle .",
"$ It follows from the mean value theorem for vector-valued functions that $\\nabla Q_h( \\widetilde{\\beta }^* ) - \\nabla Q_h(\\beta ^*) = \\int _0^1 \\nabla ^2 Q_h( (1-t) \\beta ^* + t \\widetilde{\\beta }^* ) \\, {\\rm d} t \\, \\big ( \\widetilde{\\beta }^* - \\beta ^* \\bigr ) , $ where $\\nabla ^2 Q_h( \\beta ) = \\bigl \\lbrace K_h( y - \\langle , \\beta \\rangle ) ^\\rbrace $ for $\\beta \\in ^p$ .",
"With $\\delta = \\beta - \\beta ^*$ , note that $\\bigl \\lbrace K_h( y - \\langle , \\beta \\rangle ) | \\bigr \\rbrace = \\frac{1}{h} \\int _{-\\infty }^\\infty K\\bigg ( \\frac{u- \\langle , \\delta \\rangle }{h} \\bigg ) f_{\\varepsilon |} (u) \\, {\\rm d} u = \\int _{-\\infty }^\\infty K (v ) f_{\\varepsilon |} ( \\langle , \\delta \\rangle + h v ) \\, {\\rm d} v .",
"\\nonumber $ By the Lipschitz continuity of $f_{ \\varepsilon |}(\\cdot )$ , $\\bigl \\lbrace K_h( y - \\langle , \\beta \\rangle ) | \\bigr \\rbrace = f_{ \\varepsilon |}( 0 ) + R_h(\\delta ) $ with $R_h(\\delta )$ satisfying $|R_h(\\delta )| \\le l_0 \\bigl ( |\\langle , \\delta \\rangle | + \\kappa _1 h \\bigr )$ .",
"Substituting (REF ) into (REF ) and (REF ) yields $& \\langle \\nabla Q_h( \\widetilde{\\beta }^* ) - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle \\nonumber \\\\& \\ge \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 - \\frac{l_0}{2} | \\langle , \\widetilde{\\beta }^* - \\beta ^* \\rangle |^3 - l_0 \\kappa _1 h \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 \\nonumber \\\\& \\ge \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 - \\frac{l_0}{2} \\mu _3 \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert ^3_{} - l_0 \\kappa _1 h \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 .",
"$ On the other hand, under model (REF ) we have $& \\langle - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle \\le \\Vert ^{-1/2} \\nabla Q_h(\\beta ^*) \\Vert _2 \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} , \\nonumber $ where $\\nabla Q_h(\\beta ^*) = \\lbrace K(-\\varepsilon /h) - \\tau \\rbrace $ .",
"By integration by parts and a Taylor series expansion, $\\bigl \\lbrace K(-\\varepsilon /h) | \\bigr \\rbrace & = \\int _{-\\infty }^\\infty K(-t/h) \\, {\\rm d} F_{ \\varepsilon |}( t) \\nonumber \\\\& = -\\frac{1}{h}\\int _{-\\infty }^\\infty K(-t/h) F_{ \\varepsilon |}( t) \\, {\\rm d}t = \\int _{-\\infty }^\\infty K(u) F_{\\varepsilon |}( -h u) \\, {\\rm d}u \\nonumber \\\\& = \\tau + \\int _{-\\infty }^\\infty K(u) \\int _{0}^{-hu} \\bigl \\lbrace f_{ \\varepsilon |} ( t) - f_{\\varepsilon |} (0) \\bigr \\rbrace \\, {\\rm d} t \\, {\\rm d}u ,\\nonumber $ from which it follows that $| \\lbrace K(-\\varepsilon /h) | \\rbrace - \\tau | \\le \\frac{l_0}{2} \\kappa _2 h^2$ .",
"Consequently, $& \\Vert ^{-1/2} \\nabla Q_h(\\beta ^*) \\Vert _2 = \\sup _{\\in \\mathbb {S}^{p-1}} \\bigl \\lbrace K(-\\varepsilon /h) -\\tau \\bigr \\rbrace \\langle ^{-1/2} , \\rangle \\le \\frac{l_0}{2} \\kappa _2 h^2 .",
"$ Putting together the pieces, we conclude that $\\langle - \\nabla Q_h(\\beta ^*) , \\widetilde{\\beta }^* - \\beta ^* \\rangle \\le \\frac{l_0}{2} \\kappa _2 h^2 \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} .",
"$ Recall that $f_{\\varepsilon | }(0) \\ge {f} >0$ almost surely.",
"Combining (REF ) and (REF ) with (REF ) yields ${f} \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}^2 & \\le {f} ^{1/2} \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} \\nonumber \\\\& \\le \\bigg (\\frac{\\mu _3 +1 }{2} \\kappa _2 + \\kappa _1 \\kappa _2^{1/2} \\bigg ) l_0 h^2 \\cdot \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} .",
"\\nonumber $ Canceling $\\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}$ on both sides we obtain $\\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{} \\le \\underbrace{ \\bigg (\\frac{ \\mu _3 +1 }{2} \\kappa _2^{1/2} + \\kappa _1 \\bigg ) }_{= c_K} \\frac{l_0 h }{{f} } \\kappa _2^{1/2} h = \\frac{c_K l_0 h}{{f}} \\kappa _2^{1/2} h .",
"\\nonumber $ Provided that $h < {f}/(c_K l_0)$ , $\\widetilde{\\beta }^*$ falls in the interior of $\\Theta ^*$ , i.e., $ \\Vert \\widetilde{\\beta }^* - \\beta ^* \\Vert _{}< \\kappa _2^{1/2} h$ , enforcing $\\eta =1$ (otherwise, by construction $\\widetilde{\\beta }^*$ must lie on the boundary which leads to contradiction) and hence $\\widetilde{\\beta }^* = \\beta ^*_h$ .",
"In addition, by (REF ), $Q_h(\\cdot )$ is strictly convex in a neighborhood of $\\beta ^*_h$ so that $\\beta ^*_h$ is the unique minimizer and satisfies the stated bound (REF ).",
"Next, to investigate the leading term in the bias, define $\\Delta = ^{-1/2} \\bigl \\lbrace \\nabla Q_h(\\beta ^*_h) - \\nabla Q_h(\\beta ^*) \\bigr \\rbrace - ^{1/2} ( \\beta ^*_h -\\beta ^* ).",
"\\nonumber $ Again, by the mean value theorem for vector-valued functions, $\\Delta & = \\Bigg \\lbrace ^{-1/2} \\int _0^1 \\nabla ^2 Q_h((1-t) \\beta ^* + t \\beta ^*_h) \\, {\\rm d}t \\, ^{-1/2} - _{p} \\Bigg \\rbrace ^{1/2} ( \\beta ^*_h - \\beta ^* ) .",
"$ Write $= ^{-1/2}$ , we have $& \\Bigg \\Vert ^{-1/2} \\int _0^1 \\nabla ^2 Q_h((1-t) \\beta ^* + t \\beta ^*_h) \\, {\\rm d}t \\, ^{-1/2} - _p \\Bigg \\Vert _2 \\nonumber \\\\&=\\Bigg \\Vert \\int _0^1 \\int _{-\\infty }^\\infty K(u) \\bigl \\lbrace f_{\\varepsilon |}( t \\langle , \\beta ^*_h -\\beta ^* \\rangle - hu ) - f_{ \\varepsilon |}( 0) \\bigr \\rbrace \\, {\\rm d} u \\, {\\rm d} t \\, ^\\Vert _2 \\nonumber \\\\& \\le l_0 \\sup _{\\in \\mathbb {S}^{p-1}} \\int _0^1 \\int _{-\\infty }^\\infty K(u) \\bigl ( | t \\langle , \\beta ^*_h - \\beta ^* \\rangle | + h |u | \\bigr ) \\, {\\rm d}u \\, {\\rm d} t \\, \\langle , \\rangle ^2 \\nonumber \\\\& \\le \\frac{l_0}{2 } \\sup _{\\in \\mathbb {S}^{p-1}} \\bigl ( |\\langle , \\beta ^*_h - \\beta ^* \\rangle | \\langle , \\rangle ^2 \\bigr ) + \\frac{l_0}{{f}} \\kappa _1 h \\nonumber \\\\& \\le \\frac{l_0}{2{f}} \\mu _3 \\Vert \\beta _h^* - \\beta ^* \\Vert _{} + \\frac{l_0}{{f}} \\kappa _1 h. \\nonumber $ This bound, together with (REF ), implies $\\Vert \\Delta \\Vert _2 \\le \\frac{l_0 }{{f}} \\bigl ( 0.5 \\mu _3 \\Vert \\beta _h^* - \\beta ^* \\Vert _{} + \\kappa _1 h \\bigr ) \\Vert \\beta _h^* - \\beta ^* \\Vert _{} .",
"$ Moreover, applying a second-order Taylor series expansion to $f_{\\varepsilon |}$ yields $& \\bigl \\lbrace K(-\\varepsilon /h) | \\bigr \\rbrace - \\tau \\nonumber \\\\& = \\int _{-\\infty }^\\infty K(u) \\int _{0}^{-h u} \\bigl \\lbrace f_{ \\varepsilon |} ( t) - f_{\\varepsilon |} ( 0) \\bigr \\rbrace \\, {\\rm d} t \\, {\\rm d}u \\nonumber \\\\& = \\frac{1 }{2} \\kappa _2 h^2 \\cdot f_{ \\varepsilon |}^{\\prime }(0 ) + \\int _{-\\infty }^\\infty \\int _{0}^{-hu} \\int _0^t K(u) \\bigl \\lbrace f_{ \\varepsilon |}^{\\prime }(v) - f_{ \\varepsilon |}^{\\prime }( 0 ) \\bigr \\rbrace \\, {\\rm d} v \\, {\\rm d} t \\, {\\rm d}u .",
"\\nonumber $ Recalling that $\\nabla Q_h(\\beta ^*) = \\lbrace K(-\\varepsilon /h) - \\tau \\rbrace $ , we get $\\bigg \\Vert ^{-1} \\nabla Q_h(\\beta ^*) - \\frac{1}{2} \\kappa _2 h^2 \\cdot ^{-1} \\bigl \\lbrace f_{\\varepsilon |}^{\\prime }( 0)\\bigr \\rbrace \\bigg \\Vert _{} \\le \\frac{l_1}{6 {f}^{1/2}} \\kappa _3 h^3 .",
"$ Finally, combining (REF ) and (REF ) proves (REF ).",
"$\\Box $"
],
[
"Proof of Theorem ",
"Recall from (REF ) that $\\hat{\\beta }_h $ is the smoothed quantile regression estimator obtained by minimizing $\\hat{Q}_{ h} (\\cdot )$ .",
"For any given $\\tau \\in (0,1)$ and $h>0$ , it follows from the optimality of $\\hat{\\beta }_h $ that $\\nabla \\hat{Q}_h ( \\hat{\\beta }_h ) = \\mathbf {0} $ and by convexity, $ \\langle \\nabla \\hat{Q}_h ( \\hat{\\beta }_h ) - \\nabla \\hat{Q}_h ( \\beta ^*) , \\hat{\\beta }_h - \\beta ^* \\rangle \\ge 0$ .",
"Recall from the proof of Proposition REF that $\\Theta (t) = \\lbrace \\in ^p : \\Vert \\Vert _{} \\le t \\rbrace $ for $t\\ge 0$ .",
"For some $r>0$ to be determined, let $\\eta = \\sup \\lbrace u \\in [0,1] : u (\\hat{\\beta }_h -\\beta ^*) \\in \\Theta (r) \\rbrace $ and $\\widetilde{\\beta }= \\beta ^* + \\eta (\\hat{\\beta }_h -\\beta ^*)$ .",
"Thus, by definition, $\\eta =1$ if $\\hat{\\beta }_h \\in \\beta ^* + \\Theta (r)$ , and $\\eta <1$ if $\\hat{\\beta }_h \\notin \\beta ^* + \\Theta (r)$ .",
"In the latter case, $\\widetilde{\\beta }\\in \\beta ^* + \\partial \\Theta (r)$ , where $\\partial \\Theta (r)$ is the boundary of $\\Theta (r)$ .",
"The symmetrized Bregman divergence associated with $\\hat{Q}_h(\\cdot )$ for points $\\beta _1, \\beta _2$ is given by $ D (\\beta _1, \\beta _2) = \\langle \\nabla \\hat{Q}_h (\\beta _1)- \\nabla \\hat{Q} _h (\\beta _2), \\beta _1 - \\beta _2 \\rangle .$ By Lemma C.1 in [68], the three points $\\hat{\\beta }_h$ , $\\widetilde{\\beta }$ , and $\\beta ^*$ satisfy $D( \\widetilde{\\beta }, \\beta ^* ) \\le \\eta D ( \\hat{\\beta }, \\beta ^*)$ .",
"Together with the properties that $ \\widetilde{\\beta }- \\beta ^* = \\eta (\\hat{\\beta }_h - \\beta ^* ) \\in \\Theta (r)$ and $\\nabla \\hat{Q}_h (\\hat{\\beta }_h) = \\textbf {0}$ , we obtain $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{}^2 \\cdot \\frac{ D( \\widetilde{\\beta }, \\beta ^* ) }{\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{}^2} \\le - \\eta \\langle \\nabla \\hat{Q}_h (\\beta ^* ), \\hat{\\beta }_h - \\beta ^* \\rangle \\le \\Vert ^{-1/2} \\nabla \\hat{Q}_h (\\beta ^* ) \\Vert _2 \\cdot \\Vert \\widetilde{\\beta }- \\beta ^* \\Vert _{} .",
"\\nonumber $ Canceling $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{}$ on both sides, we obtain $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{} \\le \\frac{ \\Vert ^{-1/2} \\nabla \\hat{Q}_h (\\beta ^* ) \\Vert _2 }{\\inf _{\\beta \\in \\beta ^* + \\Theta (r)} \\lbrace D( \\beta , \\beta ^* ) / \\Vert \\beta -\\beta ^* \\Vert _{}^2 \\rbrace } .",
"$ The following two lemmas provide, respectively, upper and lower bounds on $\\Vert ^{-1/2} \\nabla \\hat{Q}_h (\\beta ^* ) \\Vert _2$ and $\\inf _{\\beta \\in \\beta ^* + \\Theta (r)} \\lbrace D( \\beta , \\beta ^* ) / \\Vert \\beta -\\beta ^* \\Vert _{}^2\\rbrace $ .",
"Assume that Conditions REF –REF hold.",
"For any $t\\ge 0$ , $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2 \\le 1.46 \\upsilon _0 \\Biggl \\lbrace C_\\tau ^{1/2 }\\sqrt{ \\frac{ 4p +2 t }{n}} + 2\\max (1-\\tau , \\tau ) \\frac{ 2p+t }{n} \\Biggr \\rbrace + \\frac{1}{2} l_0\\kappa _2 h^2 $ with probability at least $1-e^{-t}$ , where $C_\\tau = \\tau (1-\\tau ) + (1+\\tau ) l_0 \\kappa _2 h^2$ .",
"Under Condition REF , there exist constants $\\bar{f}_{ h} \\ge {f}_{ h}$ such that ${f}_{ h} \\le \\inf _{|u|\\le h/2} f_{\\varepsilon | } (u) \\le \\sup _{|u|\\le h/2} f_{\\varepsilon | } (u) \\le \\bar{f}_{ h} $ almost surely.",
"In fact, by the Lipschitz continuity, we can take $\\bar{f}_{ h} = \\bar{f} + l_0 h/2$ and ${f}_{ h} = {f} - l_0 h/2$ .",
"Throughout the following, we assume (REF ) holds.",
"For every $\\delta \\in (0,1]$ , we define $\\eta _\\delta \\ge 0$ as $\\eta _\\delta = \\inf \\bigl \\lbrace \\eta >0 : \\bigl \\lbrace \\langle \\delta , \\rangle ^2 \\mathbb {1}\\big ( |\\langle \\delta , \\rangle | > \\eta _\\delta \\big ) \\bigr \\rbrace \\le \\delta ~\\mbox{ for all } \\delta \\in \\mathbb {S}^{p-1} \\bigr \\rbrace , $ where $= ^{-1/2} $ is the standardized predictor that satisfies $(^ = _p$ .",
"It can be shown that $\\eta _\\delta $ depends only on $\\delta $ and $\\upsilon _0$ in Condition REF , and the map $\\delta \\mapsto \\eta _\\delta $ is non-increasing with $\\eta _\\delta \\downarrow 0$ as $\\delta \\uparrow 1$ .",
"For any $t>0$ , $0 < h < 2 {f} / l_0$ and $0< r \\le h/(4 \\eta _{1/4})$ with $\\eta _{1/4}$ defined in (REF ), we have $\\inf _{\\beta \\in \\beta ^* + \\Theta (r) } \\frac{ D( \\beta , \\beta ^* ) }{ \\kappa _l \\Vert \\beta - \\beta ^* \\Vert _{}^2 } \\ge \\frac{3}{4} {f}_{h } - \\bar{f}_{ h}^{1/2} \\left( \\frac{5}{4}\\sqrt{\\frac{h p}{r^2 n}} + \\sqrt{\\frac{h t}{8r^2 n}} \\, \\right) - \\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .",
"In view of (REF ), (REF ), and (REF ), we take $r = h/(4 \\eta _{1/4})$ so that as long as $(p+t)/n \\lesssim h\\lesssim 1$ , $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{} < \\frac{3 \\upsilon _0 }{ \\kappa _l {f} } \\Biggl \\lbrace C_\\tau ^{1/2} \\sqrt{ \\frac{ 4p + 2t }{n}} + 2\\max (1-\\tau , \\tau ) \\frac{2 p+t }{n} \\Biggr \\rbrace + \\frac{l_0 \\kappa _2 }{ \\kappa _l {f} } h^2 $ with probability at least $1-2e^{-t}$ .",
"With this choice of $r$ , we see that under the constraint $\\sqrt{(p+t)/n} \\lesssim h \\lesssim 1$ , $\\Vert \\widetilde{\\beta }-\\beta ^* \\Vert _{} <r$ with probability at least $1-2e^{-t}$ .",
"In other words, on an event that occurs with high probability, $\\widetilde{\\beta }$ falls in the interior of $\\beta ^* + \\Theta (r)$ , enforcing $\\eta =1$ and $\\hat{\\beta }_h = \\widetilde{\\beta }$ .",
"The claimed bound for $\\hat{\\beta }_h$ then follows immediately.",
"$\\Box $"
],
[
"Proof of Lemma ",
"Write $Q_{ h } (\\beta ) = \\lbrace \\hat{Q}_h(\\beta )\\rbrace $ , and define $\\xi _i = K(-\\varepsilon _i /h) - \\tau $ for $i=1,\\ldots , n$ .",
"By the triangle inequality and (REF ), we have $\\begin{split}\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h(\\beta ^*) \\bigr \\Vert _2 &\\le \\bigl \\Vert ^{-1/2} \\lbrace \\nabla \\hat{Q}_h(\\beta ^*) -\\nabla {Q}_h(\\beta ^*) \\rbrace \\bigr \\Vert _2 + \\bigl \\Vert ^{-1/2} \\nabla Q_h(\\beta ^*) \\bigr \\Vert _2 \\\\&\\le \\bigl \\Vert ^{-1/2} \\lbrace \\nabla \\hat{Q}_h(\\beta ^*) -\\nabla {Q}_h(\\beta ^*) \\rbrace \\bigr \\Vert _2 +l_0 \\kappa _2 h^2/2.\\end{split}$ It suffices to obtain an upper bound for the centered score $ \\nabla \\hat{Q}_h (\\beta ^*) - \\nabla Q_h (\\beta ^*) = (1/n) \\sum _{i=1}^n\\lbrace \\xi _i _i - (\\xi _i _i)\\rbrace \\in ^p$ .",
"Using a covering argument, for any $\\epsilon \\in (0,1)$ , there exists an $\\epsilon $ -net $_\\epsilon $ of the unit sphere with cardinality $|_{\\epsilon }|\\le (1+2/\\epsilon )^{p}$ such that $\\bigl \\Vert ^{-1/2} \\lbrace \\nabla \\hat{Q}_h (\\beta ^*) - \\nabla Q_h (\\beta ^*) \\rbrace \\bigr \\Vert _2 \\le (1-\\epsilon )^{-1} \\max _{\\in _{\\epsilon }} \\bigl \\langle , ^{-1/2} \\lbrace \\nabla \\hat{Q}_h (\\beta ^*) - \\nabla Q_h (\\beta ^*) \\rbrace \\bigr \\rangle .$ For each unit vector $\\in _{\\epsilon }$ , define centered random variables $\\gamma _{,i} = \\langle , ^{-1/2} \\lbrace \\xi _i _i -(\\xi _i _i )\\rbrace \\rangle $ .",
"We first show that $(\\xi _i^2|_i)$ is bounded.",
"By a change of variable and integration by parts, it can be shown that $\\lbrace {K}^2(-\\varepsilon /h) |\\rbrace &= 2\\int _{-\\infty }^{\\infty } K(v){K}(v) F_{\\varepsilon |}(-vh)\\mathrm {d}v\\nonumber \\\\&=2 \\tau \\int _{-\\infty }^{\\infty } K(v){K}(v) \\mathrm {d}v -2h f_{\\varepsilon |}(0) \\int _{-\\infty }^{\\infty }v K(v) {K}(v) \\mathrm {d}v \\nonumber \\\\&\\quad + 2\\int _{-\\infty }^{\\infty }\\int _{0}^{-vh} \\lbrace f_{\\varepsilon | }(t) -f_{\\varepsilon | }(0) \\rbrace K(v) {K}(v) \\mathrm {d}t\\mathrm {d}v\\nonumber \\\\&\\le \\tau + l_0 \\kappa _2 h^2\\nonumber ,$ where $\\kappa _2$ and $l_0$ are constants that appear in Conditions REF and REF , respectively.",
"It then follows that $(\\xi _i^2 | _i ) \\le C_\\tau := \\tau (1-\\tau ) + (1+\\tau ) l_0 \\kappa _2 h^2$ .",
"Moreover, $|\\xi _i| \\le \\max (1-\\tau ,\\tau )$ .",
"Hence, for $k=2,3,\\ldots $ , $\\bigl ( | \\langle , ^{-1/2} \\xi _i _i \\rangle |^k\\bigr ) &\\le \\max ( 1-\\tau , \\tau )^{k-2}\\, \\bigl \\lbrace | \\langle ,^{-1/2} _i \\rangle |^k \\cdot (\\xi _i^2 | _i ) \\bigr \\rbrace \\nonumber \\\\&\\le C_\\tau \\max ( 1-\\tau , \\tau )^{k-2} \\upsilon ^k_0 \\int _0^{\\infty }(| \\langle ,^{-1/2} _i \\rangle |\\ge \\upsilon _0 t) kt^{k-1}\\,\\mathrm {d}t \\nonumber \\\\&\\le C_\\tau \\max ( 1-\\tau , \\tau )^{k-2} \\upsilon ^k_0 k \\int _0^{\\infty } t^{k-1}e^{-t}\\, \\mathrm {d}t\\nonumber \\\\&= C_\\tau k!",
"\\max ( 1-\\tau , \\tau )^{k-2} \\upsilon ^k_0 \\nonumber \\\\&\\le \\frac{k!",
"}{2} \\cdot C_\\tau \\upsilon _0^2 \\cdot \\left\\lbrace 2 \\max ( 1-\\tau , \\tau ) \\upsilon _0\\right\\rbrace ^{k-2}.",
"\\nonumber $ Consequently, it follows from Bernstein's inequality that for every $u\\ge 0$ , $\\frac{1}{n} \\sum _{i=1}^n\\gamma _{,i} \\le \\upsilon _0 \\Biggl \\lbrace C_\\tau ^{1/2} \\sqrt{ \\frac{2 u}{n}} + \\max (1-\\tau , \\tau ) \\frac{2 u}{n} \\Biggr \\rbrace \\nonumber $ with probability at least $1-e^{-u}$ .",
"Finally, applying a union bound over all vectors $\\in _{\\epsilon }$ yields $\\bigl \\Vert ^{-1/2} \\lbrace \\nabla \\hat{Q}_h (\\beta ^*) - \\nabla Q_h (\\beta ^*) \\rbrace \\bigr \\Vert _2 \\le \\frac{\\upsilon _0}{1-\\epsilon }\\Biggl \\lbrace C_\\tau ^{1/2} \\sqrt{ \\frac{2 u}{n}} + \\max (1-\\tau , \\tau ) \\frac{2 u}{n} \\Biggr \\rbrace \\nonumber $ with probability at least $1- e^{\\log (1+2/\\epsilon ) p -u}$ .",
"Taking $\\epsilon = 2/(e^2-1)$ and $u= 2p+ t$ ($t\\ge 0$ ) implies the claimed result.",
"$\\Box $"
],
[
"Proof of Lemma ",
"Recall that the empirical loss $ \\hat{Q}_h(\\cdot )$ in (REF ) is convex and twice continuously differentiable with $\\nabla \\hat{Q}_h(\\beta ) = (1/n) \\sum _{i=1}^n[ K\\lbrace ( \\langle _i, \\beta \\rangle - y_i ) /h\\rbrace - \\tau ] _i$ and $\\nabla ^2 \\hat{Q}_h(\\beta ) = (1/n) \\sum _{i=1}^nK_h(\\langle _i, \\beta \\rangle - y_i) _i _i^.",
"For the symmetrized Bregman divergence $ D: p p +$ defined in (\\ref {Bregman.div}), we have{\\begin{@align}{1}{-1}D(\\beta , \\beta ^* ) = \\frac{1}{n} \\sum _{i=1}^n\\left\\lbrace {K} \\left( \\frac{\\langle _i, \\beta \\rangle - y_i}{h}\\right) - {K}\\left( \\frac{ -\\varepsilon _i}{h} \\right) \\right\\rbrace \\langle _i, \\beta - \\beta ^* \\rangle .",
"\\end{@align}}Define the events $ i = { | i | h/2 } { |i, $\\beta $ - $\\beta $ * | $\\beta $ - $\\beta $ * h/(2r ) }$ for $ i=1,..., n$.",
"For any $$\\beta $$\\beta $ * + (r)$, note that $ |yi - i, $\\beta $ | h$ on $ i$, implying{\\begin{@align}{1}{-1}D(\\beta , \\beta ^*) \\ge \\frac{\\kappa _l}{n h} \\sum _{i=1}^n\\langle _i , \\beta - \\beta ^* \\rangle ^2 \\mathbb {1}_{ _i } , \\end{@align}}where $ 1i $ is the indicator function of $ i $ and $ l = |u|1 K(u)$.",
"It then suffices to bound the right-hand side of the above inequality from below uniformly over $$\\beta $$\\beta $ * + (r)$.$ For $R>0$ , define the function $\\varphi _R(u)=u^2 \\mathbb {1}(|u|\\le R/2) + \\lbrace u(u) -R \\rbrace ^2 \\mathbb {1}( R/2 < |u| \\le R)$ , which is $R$ -Lipschitz continuous and satisfies $u^2 \\mathbb {1}(|u| \\le R / 2) \\le \\varphi _R(u) \\le u^2 \\mathbb {1}(|u| \\le R ) .",
"$ Moreover, note that $\\varphi _{cR}(cu) = c^2 \\varphi _{R}(u)$ for any $c > 0$ and $\\varphi _0(u) = 0$ .",
"For $\\beta \\in \\beta ^* + \\Theta (r)$ , consider a change of variable $\\delta = ^{1/2} (\\beta -\\beta ^*)/\\Vert \\beta -\\beta ^* \\Vert _{} $ so that $\\delta \\in \\mathbb {S}^{p-1}$ .",
"Together, () and (REF ) imply $\\frac{ D(\\beta , \\beta ^* ) }{\\Vert \\beta -\\beta ^* \\Vert _{}^2} \\ge \\kappa _l \\cdot \\underbrace{ \\frac{1}{n h } \\sum _{i=1}^n\\omega _i \\cdot \\varphi _{ h/(2r) } (\\langle _i, \\delta \\rangle ) }_{=: D_0( \\delta )} , $ where $\\omega _i := \\mathbb {1}( | \\varepsilon _i | \\le h/2 )$ with $\\varepsilon _i = y_i - \\langle _i, \\beta ^* \\rangle $ , and $_i = ^{-1/2} _i$ .",
"Next, we bound the expectation $\\lbrace D_0(\\delta ) \\rbrace $ and the random fluctuation $ D_0(\\delta ) - \\lbrace D_0(\\delta )\\rbrace $ , separately, starting with the former.",
"By (REF ), ${f}_{ h} h \\le ( \\omega _i | _i ) = \\int _{-h/2}^{h/2} f_{\\varepsilon _i | _i} (u) \\, {\\rm d}u \\le \\bar{f}_{ h} h. $ Moreover, define $\\xi _{\\delta } = \\langle , \\delta \\rangle $ such that $(\\xi _{\\delta }^2) =1$ .",
"By (REF ) and (REF ), $& \\bigl \\lbrace \\omega _i \\cdot \\varphi _{ h/(2r)} (\\langle _i , \\delta \\rangle ) \\bigr \\rbrace \\ge {f}_{ h} h \\cdot \\varphi _{ h/(2r) } (\\langle _i , \\delta \\rangle )\\ge {f}_{ h} h \\cdot \\bigl [ 1 - \\bigl \\lbrace \\xi _{\\delta }^2 \\mathbb {1}_{ | \\xi _{\\delta } | > h/(4r) } \\bigr \\rbrace \\bigr ] , \\nonumber $ from which it follows that $\\inf _{ \\delta \\in \\mathbb {S}^{p-1} } \\lbrace D_0(\\delta ) \\rbrace \\ge {f}_h \\cdot \\left( 1 - \\sup _{\\in \\mathbb {S}^{p-1}} \\bigl \\lbrace \\langle , \\rangle ^2 \\mathbb {1}_{ |\\langle , \\rangle | >h/(4r)} \\bigr \\rbrace \\right) .",
"$ By the definition of $\\eta _\\delta $ in (REF ), we see that as long as $0<r \\le h/(4 \\eta _{1/4})$ , $\\inf _{ \\delta \\in \\mathbb {S}^{p-1} } \\lbrace D_0(\\delta )\\rbrace \\ge \\frac{3}{4} {f}_h .",
"$ Turning to the random fluctuation, we will use Theorem 7.3 in [60] (a refined Talagrand's inequality) to bound $\\Delta = \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\bigl [ D^-_0(\\delta ) - \\lbrace D^-_0(\\delta ) \\rbrace \\bigr ], $ where $D^-_0(\\delta ) := -D_0(\\delta ) $ .",
"Note that $0\\le \\varphi _R(u) \\le \\min \\lbrace (R/2)^2, (R/2)|u| \\rbrace $ for all $u\\in $ and $\\omega _i \\in \\lbrace 0, 1\\rbrace $ .",
"Therefore, $0\\le \\chi _i := \\frac{\\omega _i}{h } \\varphi _{ h/(2r) } (\\langle _i, \\delta \\rangle ) \\le \\omega _i \\cdot \\frac{h}{(4r)^2} \\bigwedge \\frac{ |\\langle _i, \\delta \\rangle | }{4r} .",
"\\nonumber $ This, combined with (REF ), yields $(\\chi _i^2) \\le \\frac{(\\omega _i) }{(4r)^2} \\le \\frac{ \\bar{f}_h h }{(4r)^2} .",
"\\nonumber $ With the above preparations, it follows from Theorem 7.3 in [60] that for any $t>0$ , $\\Delta & \\le (\\Delta ) + \\lbrace ( \\Delta )\\rbrace ^{1/2} \\sqrt{\\frac{h t}{4 r^2 n}} + \\sqrt{ \\bar{f}_h h \\frac{ t }{ 8 r^2 n }} + \\frac{h t}{48 r^2 n} \\le \\frac{5}{4} (\\Delta ) + \\sqrt{ \\bar{f}_h h \\frac{ t }{ 8 r^2 n }} + \\frac{h t}{3 r^2 n} $ with probability at least $1-e^{-t}$ , where the second step follows from the inequality that $ab\\le a^2/4 + b^2$ for all $a,b\\in $ .",
"It then remains to bound the expectation $(\\Delta ) $ .",
"Define $(\\delta ; _i) &= \\frac{1}{h } \\omega _i\\varphi _{ h/(2r) } (\\langle _i, \\delta \\rangle )= \\frac{1}{h} \\varphi _{ \\omega _i h/(2r) } (\\langle \\omega _i _i , \\delta \\rangle ) , \\ \\ \\delta \\in \\mathbb {S}^{p-1} ,\\nonumber $ where $_i = (_i , \\varepsilon _i )$ and $\\omega _i = \\mathbb {1} ( | \\varepsilon _i | \\le h/2 ) \\in \\lbrace 0, 1\\rbrace $ .",
"By Rademacher symmetrization, $(\\Delta ) \\le 2 \\Bigg \\lbrace \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^ne_i (\\delta ; _i) \\Bigg \\rbrace , \\nonumber $ where $e_1,\\ldots , e_n$ are independent Rademacher random variables.",
"Since $\\varphi _R(\\cdot )$ is $R$ -Lipschitz, $(\\delta ; _i)$ is a $(1/2r)$ -Lipschitz function in $\\langle \\omega _i _i , \\delta \\rangle $ , i.e., for any sample $_i = (_i, \\varepsilon _i)$ and parameters $\\delta , \\delta ^{\\prime }\\in \\mathbb {S}^{p-1}$ , $\\bigl | (\\delta ; _i) - (\\delta ^{\\prime }; _i) \\bigr | \\le \\frac{1}{2r} \\bigl | \\langle \\omega _i _i , \\delta \\rangle - \\langle \\omega _i _i , \\delta ^{\\prime } \\rangle \\bigr | .$ Moreover, observe that $(\\delta ; _i) = 0$ for any $\\delta $ such that $ \\langle \\omega _i _i , \\delta \\rangle = 0$ .",
"With the above preparations, we are ready to use Talagrand's contraction principle to bound $( \\Delta )$ .",
"Define the subset $T\\subseteq ^n$ as $T = \\bigl \\lbrace = (t_1, \\ldots , t_n)^ t_i = \\langle \\omega _i_i , \\delta \\rangle , i=1,\\ldots , n, \\, \\delta \\in \\mathbb {S}^{p-1} \\bigr \\rbrace ,$ and contractions $\\phi _i:\\rightarrow $ as $\\phi _i(t) = (2r/h) \\cdot \\varphi _{ h \\omega _i/(2r)} (t)$ .",
"By (REF ), $|\\phi (t)-\\phi (s)|\\le |t-s|$ for all $t, s \\in $ .",
"Applying Talagrand's contraction principle (see, e.g., Theorem 4.12 and (4.20) in [64]), we have $(\\Delta ) & \\le 2 \\Bigg \\lbrace \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^ne_i (\\delta ; _i) \\Bigg \\rbrace = \\frac{1}{r} \\Bigg \\lbrace \\sup _{\\in T} \\frac{1}{n} \\sum _{i=1}^ne_i \\phi _i(t_i) \\Bigg \\rbrace \\le \\frac{1}{r} \\Bigg ( \\sup _{\\in T} \\frac{1}{n} \\sum _{i=1}^ne_i t_i \\Bigg ) \\nonumber \\\\&= \\frac{1}{r}\\Bigg \\lbrace \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^ne_i \\langle \\omega _i _i , \\delta \\rangle \\Bigg \\rbrace \\le \\frac{1}{r} \\bigg \\Vert \\frac{1}{n} \\sum _{i=1}^ne_i \\omega _i _i \\bigg \\Vert _2 \\le \\frac{1}{r} \\sqrt{ \\bar{f}_h h \\frac{p}{n} }.",
"\\nonumber $ This, together with (REF ) and (REF ), yields $\\Delta \\le \\bar{f}_h^{1/2} \\Biggl ( \\frac{5}{4}\\sqrt{\\frac{h p}{r^2 n}} + \\sqrt{\\frac{h t}{8r^2 n}} \\Biggr ) + \\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .",
"Finally, combining (), (REF ), (REF ), and (REF ) proves (REF ).",
"$\\Box $"
],
[
"Proof of Theorem ",
"We keep the notation used in the proof of Theorem REF , and for any $t\\ge 0$ , let $r=r(n,p,t) \\asymp \\sqrt{(p+t)/n} + h^2 >0$ be such that $\\lbrace \\hat{\\beta }_h \\in \\beta ^* + \\Theta (r) \\rbrace \\ge 1-2e^{-t}$ , provided $\\sqrt{(p+t)/n} \\lesssim h \\lesssim 1$ .",
"Define the vector-valued random process $\\Delta (\\beta ) = ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}_h(\\beta ) - \\nabla \\hat{Q}_h(\\beta ^*) - ( \\beta - \\beta ^* ) \\bigr \\rbrace , $ where $= \\lbrace f_{\\varepsilon | } (0) ^$ .",
"Since $\\hat{\\beta }_h$ falls in a local neighborhood of $\\beta ^*$ with high probability, it suffices to bound the local fluctuation $\\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta (\\beta ) \\Vert _2$ .",
"By the triangle inequality, $\\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta (\\beta ) \\Vert _2 \\le \\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta (\\beta ) \\Vert _2+ \\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta (\\beta ) - \\Delta (\\beta ) \\Vert _2 :=I_1+I_2.$ We now provide upper bounds for $I_1$ and $I_2$ , respectively.",
"Upper bound for $I_1$ : By the mean value theorem for vector-valued functions, $\\Delta (\\beta ) & = ^{-1/2} \\biggl \\langle \\int _0^1 \\nabla ^2 Q_h\\bigl ( (1-t) \\beta ^* + t \\beta \\bigr ) {\\rm d}t , \\beta - \\beta ^* \\biggr \\rangle - ^{-1/2} (\\beta - \\beta ^* ) \\nonumber \\\\& =\\biggl \\langle ^{-1/2} \\int _0^1 \\nabla ^2 Q_h\\bigl ( (1-t) \\beta ^* + t \\beta \\bigr ) {\\rm d}t \\, ^{-1/2} - _0 , ^{1/2} ( \\beta - \\beta ^* ) \\biggr \\rangle , \\nonumber $ where $_0 := ^{-1/2} ^{-1/2} = \\lbrace f_{\\varepsilon | }(0) ^$ .",
"By law of iterative expectation and by a change of variable, $^{-1/2} \\nabla ^2 Q_h(\\beta ) ^{-1/2} & = \\bigl \\lbrace K_h( \\langle , \\beta \\rangle - y ) ^\\rbrace = \\Biggl \\lbrace \\int _{-\\infty }^\\infty K(u) f_{\\varepsilon | }(\\langle , \\beta - \\beta ^* \\rangle - h u ) \\, {\\rm d} u \\cdot ^\\rbrace .",
"\\nonumber $ For notational convenience, let $= ^{-1/2} ( \\beta -\\beta ^*)$ with $\\beta \\in \\beta ^* + \\Theta (r)$ , so that $\\Vert \\Vert _2 \\le r$ and $\\nabla ^2 Q_h \\bigl ( (1-t) \\beta ^* + t \\beta \\bigr ) = \\Biggl \\lbrace \\int _{-\\infty }^\\infty K(u) f_{\\varepsilon | }( t \\langle ,\\rangle - h u ) \\, {\\rm d} u \\cdot ^\\rbrace .$ By the Lipschitz continuity of $f_{\\varepsilon |}(\\cdot )$ , i.e.",
"$|f_{\\varepsilon |}(u)- f_{\\varepsilon |}(0)| \\le l_0 |u |$ for all $u \\in $ almost surely for $$ , we have $& \\bigl \\Vert ^{-1/2} \\nabla ^2 Q_h\\bigl ( (1-t) \\beta ^* + t \\beta \\bigr ) ^{-1/2} - _0 \\bigr \\Vert _2 \\nonumber \\\\& = \\Biggl \\Vert \\int K(u) \\bigl \\lbrace f_{\\varepsilon |} ( t \\langle , \\rangle - h u) - f_{\\varepsilon |} ( 0 ) \\bigr \\rbrace \\, {\\rm d} u \\, \\cdot ^\\Vert _2 \\nonumber \\\\& \\le l_0 t \\sup _{\\Vert \\Vert _2 =1} \\bigl ( \\langle , \\rangle ^2 |\\langle , \\rangle | \\bigr )+ l_0 \\kappa _1 h \\sup _{\\Vert \\Vert _2 =1} \\langle , \\rangle ^2 \\nonumber \\\\& \\le l_0 t \\,\\Biggl ( \\sup _{\\Vert \\Vert _2 =1} | \\langle , \\rangle |^3 \\Biggr )^{2/3} \\bigl ( |\\langle , \\rangle |^3 \\bigr )^{1/3} + l_0 \\kappa _1 h \\le l_0 \\big ( \\mu _3 r t + \\kappa _1 h \\big ),$ where the third inequality holds by the Cauchy-Schwarz inequality.",
"Consequently, $\\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta ( \\beta ) \\Vert _2 \\le l_0 \\bigl (\\mu _3 r /2 + \\kappa _1 h \\bigr ) \\cdot r .",
"$ Upper bound for $I_2$ : Next, we provide an upper bound for $\\Delta (\\beta ) - \\Delta (\\beta )$ .",
"Define the centered gradient process $G(\\beta ) = ^{-1/2} \\lbrace \\nabla \\hat{Q}_h(\\beta ) - \\nabla Q_h(\\beta ) \\rbrace $ , so that $\\Delta (\\beta ) - \\Delta (\\beta ) = G(\\beta ) - G(\\beta ^*)$ .",
"Again, by a change of variable $= ^{ 1/2} ( \\beta - \\beta ^*)$ , we have $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta (\\beta ) - \\Delta (\\beta ) \\Vert _2& \\le \\sup _{ \\beta \\in \\beta ^* + \\Theta (r) } \\Vert G(\\beta ) - G(\\beta ^*) \\Vert _2 \\nonumber \\\\& = \\sup _{ \\Vert \\Vert _2 \\le r } \\Vert \\underbrace{ G(\\beta ^* + ^{-1/2} ) - G(\\beta ^*) }_{ =: \\Delta _0() } \\Vert _2 .\\nonumber $ We will employ Theorem A.3 in [67] to bound the supremum $\\sup _{ \\Vert \\Vert _2 \\le r } \\Vert \\Delta _0() \\Vert _2$ , where $\\Delta _0(\\cdot )$ defined above satisfies $\\Delta _0(\\textbf {0}) = \\textbf {0}$ and $\\lbrace \\Delta _0()\\rbrace =\\textbf {0}$ .",
"Taking the gradient with respect to $$ yields $\\nabla \\Delta _0 () = \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace K_{i,} _i _i^ \\bigl ( K_{i,} _i _i^) \\bigr \\rbrace , \\nonumber $ where $K_{i,} : = K_h(\\langle _i, \\rangle - \\varepsilon _i )$ satisfies $0\\le K_{i,} \\le \\kappa _u h^{-1}$ .",
"For any $, \\in \\mathbb {S}^{p-1}$ and $\\lambda \\in $ , using the elementary inequality $|e^u- 1 - u| \\le u^2 e^{|u|}/2$ , we obtain $& \\exp \\bigl \\lbrace \\lambda n^{1/2} \\langle , \\nabla \\Delta _0 () \\rangle \\bigr \\rbrace \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl \\lbrace 1 + \\frac{\\lambda ^2 }{2n} e^{ \\frac{ \\bar{f} |\\lambda | }{ \\sqrt{n}} | \\langle _i, \\rangle \\langle _i, \\rangle |}\\bigl \\lbrace K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle - ( K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle ) \\bigr \\rbrace ^2 e^{\\frac{ \\kappa _u|\\lambda | }{ h \\sqrt{n}} | \\langle _i, \\rangle \\langle _i, \\rangle | } \\Biggr \\rbrace \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl \\lbrace 1 + \\frac{\\lambda ^2 }{2n} e^{ \\frac{ \\bar{f} |\\lambda | }{ \\sqrt{n}} }\\bigl \\lbrace K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle - ( K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle ) \\bigr \\rbrace ^2 e^{\\frac{ \\kappa _u|\\lambda | }{ h \\sqrt{n}} | \\langle _i, \\rangle \\langle _i, \\rangle | } \\Biggr \\rbrace , $ where we use the bound $| \\langle _i, \\rangle \\langle _i, \\rangle | \\le (\\langle _i, \\rangle ^2 )^{1/2} (\\langle _i, \\rangle ^2 )^{1/2} = 1$ in the second inequality.",
"Moreover, the first and second conditional moments of $K_{i,}$ can be rewritten as follows: $ \\,\\bigl ( K_{i,} | _i \\bigr ) = \\frac{1}{h} \\int _{-\\infty }^\\infty K \\biggl ( \\frac{\\langle _i, \\rangle - t}{h} \\biggr ) f_{\\varepsilon _i(\\tau ) | _i } (t) \\, {\\rm d} t = \\int _{-\\infty }^\\infty K(u) f_{\\varepsilon _i (\\tau ) | _i } (\\langle _i, \\rangle - hu ) \\, {\\rm d} u; \\nonumber \\\\\\bigl ( K^2_{i,} | _i \\bigr ) = \\frac{1}{h^2} \\int _{-\\infty }^\\infty K^2 \\biggl ( \\frac{\\langle _i, \\rangle - t}{h} \\biggr ) f_{\\varepsilon _i(\\tau ) | _i } (t) \\, {\\rm d} t = \\frac{1}{h} \\int _{-\\infty }^\\infty K^2(u) f_{\\varepsilon _i(\\tau ) | _i } (\\langle _i, \\rangle - hu ) \\, {\\rm d} u, \\nonumber $ from which it follows that $|( K_{i,} | _i ) | \\le \\bar{f} $ and $( K_{i,}^2 | _i )\\le \\kappa _u \\bar{f} h^{-1}$ almost surely.",
"By the Cauchy-Schwarz inequality and the inequality $ab\\le a^2/2+b^2/2$ , $a,b\\in $ , we have $& ( \\langle _i, \\rangle \\langle _i, \\rangle )^2 e^{t | \\langle _i, \\rangle \\langle _i, \\rangle | } \\nonumber \\\\& \\le ( \\langle _i, \\rangle \\langle _i, \\rangle )^2 e^{ \\frac{ t }{ 2 } \\langle _i, \\rangle ^2 +\\frac{ t }{ 2 } \\langle _i, \\rangle ^2 } \\nonumber \\\\& \\le \\Bigl ( \\langle _i, \\rangle ^4 e^{ t \\langle _i, \\rangle ^2 } \\Bigr )^{1/2} \\Bigl ( \\langle _i, \\rangle ^4 e^{ t \\langle _i, \\rangle ^2 } \\Bigr )^{1/2} , ~\\mbox{ valid for any } t>0 .\\nonumber $ Given a unit vector $$ , let $\\chi = \\langle , \\rangle ^2/(2\\upsilon _1)^2$ so that under Condition REF , $( \\chi \\ge u) \\le 2 e^{-2u}$ for any $u\\ge 0$ .",
"It follows that $(e^\\chi ) = 1 + \\int _0^\\infty e^u (\\chi \\ge u) {\\rm d}u \\le 1 + 2 \\int _0^\\infty e^{-u} {\\rm d}u = 3$ , and $(\\chi ^2 e^{\\chi } ) = \\int _0^\\infty (u^2 + 2u) e^{u} (\\chi \\ge u) {\\rm d} u \\le 2 \\int _0^\\infty (u^2 + 2u) e^{-u} {\\rm d} u = 8.",
"\\nonumber $ Taking the supremum over $\\in \\mathbb {S}^{p-1}$ , we have $\\sup _{\\in \\mathbb {S}^{p-1} } e^{\\langle , \\rangle ^2 / (2\\upsilon _1)^2 } \\le 3~\\mbox{ and } ~ \\sup _{\\in \\mathbb {S}^{p-1} } \\langle , \\rangle ^4 e^{\\langle , \\rangle ^2 / (2\\upsilon _1)^2 } \\le 8(2\\upsilon _1)^4 .$ Substituting the above bounds into (REF ) yields that, for any $|\\lambda | \\le \\min \\lbrace n^{1/2} h/( 4 \\kappa _u \\upsilon _1^2 ), n^{1/2} / \\bar{f} \\rbrace $ , $& \\exp \\bigl \\lbrace \\lambda n^{1/2} \\langle , \\nabla \\Delta _0 () \\rangle \\bigr \\rbrace \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl [ 1 + \\frac{ e\\lambda ^2 }{2n}\\bigl \\lbrace K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle - (K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle ) \\bigr \\rbrace ^2 e^{ | \\langle _i, \\rangle \\langle _i, \\rangle | /(4\\upsilon _1^2)} \\Biggr ] \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl [ 1 + \\frac{ e\\lambda ^2 }{ n}\\bigl ( K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle \\bigr )^2 e^{ | \\langle _i, \\rangle \\langle _i, \\rangle | / (4\\upsilon _1^2) } \\nonumber \\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + \\frac{e \\lambda ^2}{n} \\bigl \\lbrace \\bigl ( K_{i,} \\langle _i, \\rangle \\langle _i, \\rangle \\bigr )\\bigr \\rbrace ^2 e^{ | \\langle _i, \\rangle \\langle _i, \\rangle |/ (4\\upsilon _1^2) } \\Biggr ] \\nonumber \\\\& \\le \\prod _{i=1}^n \\Biggl ( 1 + C_0^2 \\frac{\\lambda ^2}{2 n h} \\Biggr ) \\le \\exp \\bigl \\lbrace C_0^2 \\lambda ^2 /(2h) \\bigr \\rbrace ,\\nonumber $ where $C_0>0$ depends only on $(\\upsilon _1, \\kappa _u, \\bar{f} )$ .",
"We have thus verified condition (A.4) in [67] with ${\\rm g}= \\min \\lbrace h /( 4\\kappa _u \\upsilon _1^2 ), 1 / \\bar{f} \\rbrace (n/2)^{1/2}$ and $\\nu _0= C_0h^{-1/2}$ .",
"Applying Theorem A.3 therein, we obtain that with probability at least $1-e^{-t}$ , $\\sup _{ \\Vert \\Vert _2 \\le r } \\Vert \\Delta _0() \\Vert _2 \\le 6 C_0 r \\sqrt{\\frac{4 p + 2 t}{n h }} \\nonumber $ as long as $h \\ge 8 \\kappa _u \\upsilon _1^2 \\sqrt{(2p+t)/n}$ and $n\\ge 4 \\bar{f}^2 (2p+t)$ .",
"Together with (REF ) and (REF ), this implies that with probability at least $1-e^{-t}$ , $\\sup _{\\beta \\in \\beta ^* + \\Theta (r) } \\Vert \\Delta ( \\beta ) \\Vert _2 \\le 6 C_0 r \\sqrt{\\frac{4 p + 2 t}{n h }} + l_0 \\bigl ( \\mu _3 r /2 + \\kappa _1 h \\bigr ) r .$ Recall from the beginning of the proof that $\\hat{\\beta }_h \\in \\beta ^* + \\Theta (r)$ with probability at least $1-2e^{-t}$ with $r=r(n,p,t) \\asymp \\sqrt{(p+t)/n} +h^2$ .",
"Combined with (REF ), we conclude that with probability at least $1-3e^{-t}$ , $\\Vert \\Delta (\\hat{\\beta }_h ) \\Vert _2 \\lesssim (p+t)/(h^{1/2}n) + h \\sqrt{(p+t)/n} +h^3$ , as claimed.",
"$\\Box $"
],
[
"Proof of Theorem ",
"Let $\\in ^p$ be an arbitrary vector defining a linear functional of interest.",
"Given $h=h_n>0$ , define $S_n = n^{-1/2} \\sum _{i=1}^n\\gamma _i \\xi _i$ and its centered version $S_n^0= S_n - (S_n)$ , where $\\xi _i = \\tau - K(-\\varepsilon _i /h)$ and $\\gamma _i = \\langle ^{-1} , _i \\rangle $ .",
"Moreover, write $\\delta _n = (p+\\log n)/n$ .",
"By the Lipschitz continuity of $f_{\\varepsilon | }$ around 0 and the fundamental theorem of calculus, it can be shown that $| (\\xi _i | _i) |\\le 0.5 l_0 \\kappa _2 h^2$ , from which it follows by the law of iterated expectation that $|(\\gamma _i \\xi _i) | \\le 0.5 l_0 \\kappa _2 \\Vert ^{-1} \\Vert _{} \\cdot h^2$ .",
"Applying (REF ) and (REF ) with $t=\\log n$ and the triangle inequality, we obtain that under the constraint $\\delta _n^{1/2} \\lesssim h \\lesssim 1$ , $& \\bigl | n^{1/2} \\langle , \\hat{\\beta }_h - \\beta ^* \\rangle - S_n^0 \\bigr | \\nonumber \\\\&= n^{1/2} \\biggl | \\bigg \\langle ^{ 1/2} ^{-1} , ^{-1/2} (\\hat{\\beta }_h - \\beta ^* ) - ^{-1/2} \\frac{1}{n} \\sum _{i=1}^n\\bigl \\lbrace \\tau - K(-\\varepsilon _i/h) \\bigr \\rbrace _i \\bigg \\rangle \\biggr | + | (S_n) | \\nonumber \\\\& \\le c_1 \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr ) $ with probability at least $1 - 3 n^{-1}$ for some constant $c_1>0$ .",
"For the centered partial sum $S_n^0 = S_n - (S_n) = n^{-1/2} \\sum _{i=1}^n(1-) \\gamma _i \\xi _i$ , we have $\\textnormal {var}(S_n^0) = \\textnormal {var}(S_n) = (\\gamma \\xi )^2 - \\lbrace (\\gamma \\xi )\\rbrace ^2$ , where $\\gamma = \\langle ^{-1} , \\rangle $ and $\\xi = \\tau - K(-\\varepsilon /h)$ .",
"By the Berry-Esseen inequality (see, e.g., [69]), $\\sup _{x\\in } \\bigl | \\bigl \\lbrace S_n^0 \\le \\textnormal {var}(S_n)^{1/2} x \\bigr \\rbrace - \\Phi (x) \\bigr | \\le \\frac{\\lbrace |\\gamma \\xi - (\\gamma \\xi ) |^3\\rbrace }{2 [( \\gamma \\xi )^2 - \\lbrace ( \\gamma \\xi )\\rbrace ^2 ]^{3/2} \\sqrt{n} } .",
"$ We have shown that $|(\\gamma \\xi ) | \\lesssim \\Vert ^{-1} \\Vert _{} \\cdot h^2$ .",
"Following a similar argument in the proof of Lemma REF , we have $(\\xi ^2| )\\lesssim \\tau (1-\\tau )+h^2$ .",
"For $h$ sufficiently small, $\\textnormal {var}(S_n)= \\lbrace \\tau (1-\\tau ) + O(h)\\rbrace \\Vert ^{-1} \\Vert _{}^2$ and $(|\\gamma \\xi |^3) \\le \\max (\\tau , 1-\\tau ) (\\xi ^2 |\\gamma |^3) \\le \\mu _3 \\lbrace \\tau (1-\\tau ) + O(h^2) \\rbrace \\Vert ^{-1} \\Vert _{}^3$ .",
"Substituting these bounds into (REF ) yields $\\sup _{x\\in } \\bigl | \\bigl \\lbrace S_n^0 \\le \\textnormal {var}(S_n)^{1/2} x \\bigr \\rbrace - \\Phi (x) \\bigr | \\le c_2 n^{-1/2} $ for some constant $c_2>0$ .",
"Set $\\sigma _\\tau ^2 = \\tau (1-\\tau )\\, \\Vert ^{-1} \\Vert _{}^2$ .",
"By an application of Lemma A.7 in the supplement of [66], for sufficiently small $h$ , we have $\\sup _{x\\in } \\bigl | \\Phi ( x/\\textnormal {var}(S_n)^{1/2} ) -\\Phi (x/\\sigma _\\tau ) \\bigr | \\le c_3 h. $ Before proceeding, we note that the constants $c_1$ –$c_3$ appeared above are all independent of $$ .",
"Let $G\\sim N(0,1)$ .",
"Putting together the above derivations, for any $x\\in $ and $\\in ^p$ , we obtain $& \\bigl ( n^{1/2} \\langle , \\hat{\\beta }_h -\\beta ^* \\rangle \\le x \\bigr ) \\nonumber \\\\& \\le \\bigl \\lbrace S_n^0 \\le x + c_1 \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr ) \\bigr \\rbrace + 3 n^{-1} \\nonumber \\\\& \\le \\bigl \\lbrace \\textnormal {var}(S_n)^{1/2} G \\le x+ c_1 \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr ) \\bigr \\rbrace +c_2n^{-1/2} + 3 n^{-1} \\nonumber \\\\&\\le \\bigl \\lbrace \\sigma _\\tau G \\le x+ c_1 \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr ) \\bigr \\rbrace + c_2 n^{-1/2} + c_3 h + 3 n^{-1} \\nonumber \\\\& \\le \\bigl ( \\sigma _\\tau G \\le x \\bigr ) + c_1(2\\pi )^{-1/2} \\Vert ^{-1} \\Vert _{} \\cdot n^{1/2} \\bigl ( h^{-1/2} \\delta _n + h^2 \\bigr )/\\sigma _{\\tau } + c_2 n^{-1/2} + c_3 h^2 + 3 n^{-1} , \\nonumber $ where the first, second, and third inequalities holds by (REF ), (REF ), and (REF ), respectively, and the last inequality holds by the fact that for any $a\\le b$ , $\\Phi (b/\\sigma _\\tau ) - \\Phi (a/\\sigma _\\tau ) \\le (2\\pi )^{-1/2}(b-a)/\\sigma _\\tau $ .",
"A similar argument leads to a series of reverse inequalities.",
"Note the above bounds are independent of $x$ and $$ , and therefore hold uniformly over all $x$ and $$ .",
"Putting together the pieces, we conclude that under the bandwidth requirement $\\delta _n^{1/2} \\lesssim h \\lesssim 1$ , $\\sup _{x\\in , \\, \\in ^p } \\bigl | \\bigl ( n^{1/2} \\langle , \\hat{\\beta }_h -\\beta ^* \\rangle \\le \\sigma _\\tau x \\bigr ) - \\Phi (x ) \\bigr | \\lesssim \\frac{p+ \\log n}{ (n h)^{1/2}} + n^{1/2} h^2 , \\nonumber $ as claimed.",
"$\\Box $"
],
[
"Proof of Theorem ",
"Keep the notation used in the proof of Theorem REF .",
"With nonnegative weights $w_i$ , the corresponding weighted loss $\\hat{Q}_h^\\flat $ in (REF ) is convex, and thus the first-order condition $\\nabla \\hat{Q}_h^\\flat (\\hat{\\beta }_h^\\flat ) = \\textbf {0}$ holds.",
"We use the same localized argument as in the proof of Theorem REF .",
"For some $r>0$ to be determined, define $\\widetilde{\\beta }_h^\\flat = \\beta ^* + \\eta (\\hat{\\beta }_h^\\flat - \\beta ^*)$ , where $\\eta = \\sup \\lbrace u\\in [0,1] : u (\\hat{\\beta }_h^\\flat - \\beta ^*) \\in \\Theta (r) \\rbrace $ .",
"Similar to (REF ), we have $\\Vert \\widetilde{\\beta }_h^\\flat - \\beta ^* \\Vert _{} \\le \\frac{\\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^*) \\Vert _2 }{\\inf _{\\beta \\in \\beta ^* + \\Theta (r)} D^\\flat (\\beta , \\beta ^*) / \\Vert \\beta - \\beta ^* \\Vert _{}^2 } , $ where $D^\\flat : ^p\\times ^p \\rightarrow ^+$ denotes the symmetrized Bregman divergence associated with $\\hat{Q}_h^\\flat $ .",
"Next, we present two lemmas on the upper and lower bounds of the numerator and denominator of (REF ), respectively.",
"Assume that Conditions REF , REF , and REF hold.",
"For any $t\\ge 0$ , there exists some event $_1(t)$ with $\\lbrace _1(t) \\rbrace \\ge 1- 2e^{-t}$ such that, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_1(t)$ , $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) \\bigr \\Vert _2 \\le C_1 \\Biggl ( \\sqrt{\\frac{p+t}{n}} + h^2 \\Biggr )$ as long as $n\\gtrsim p+t$ , where $C_1>0$ is a constant depending only on $(\\upsilon _1, \\kappa _2, l_0)$ and $\\tau $ .",
"Let $r= h/(4\\eta _{1/4})$ with $\\eta _{1/4}$ defined in (REF ).",
"For any $t\\ge 0$ , there exists some event $_2(t)$ with $\\lbrace _2(t) \\rbrace \\ge 1- 3e^{-t}$ such that, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_2(t)$ , $^* \\Biggl \\lbrace \\inf _{\\beta \\in \\beta ^* + \\Theta (r)}\\frac{D^\\flat (\\beta , \\beta ^* )}{\\kappa _l \\Vert \\beta -\\beta ^* \\Vert _{}^2} \\ge \\frac{1}{2} {f} \\Biggr \\rbrace \\ge 1-e^{-t} .",
"$ as long as $ (p+\\log n + t )/n \\lesssim h \\lesssim 1$ .",
"Recall from the proof of Theorem REF that the upper bound on $\\Vert \\hat{\\beta }_h - \\beta ^* \\Vert _{}$ depends crucially on $\\Vert ^{-1/2} \\nabla \\hat{Q}_h(\\beta ^* ) \\Vert _2$ and $\\inf _{\\beta \\in \\beta ^* + \\Theta (r)} \\lbrace D(\\beta , \\beta ^*)/\\Vert \\beta - \\beta ^* \\Vert _{}^2\\rbrace $ .",
"In view of Lemmas REF and REF , we take $r=h/( 4 \\eta _{1/4})$ and for $t\\ge 0$ , set $(t) = _1(t) \\cup _2(t)$ so that $\\lbrace (t) \\rbrace \\ge 1-5 e^{-t}$ and $\\Vert \\hat{\\beta }_h - \\beta ^* \\Vert _{} \\le C_0 \\Biggl ( \\sqrt{\\frac{p+t}{n}} + h^2 \\Biggr ) ~\\mbox{ on $(t)$ provided that } \\sqrt{\\frac{p+t}{n}} \\lesssim h \\lesssim 1,$ where $C_0>0$ depends only on $(\\upsilon _1, \\kappa _2, \\kappa _l, l_0, {f} )$ and $\\tau $ .",
"Moreover, by (REF ), with $^*$ -probability at least $1-2e^{-t}$ conditioned on $(t)$ , $\\Vert \\widetilde{\\beta }_h^\\flat - \\beta ^* \\Vert _{} \\le \\frac{2C_1}{\\kappa _l {f} }\\Biggl ( \\sqrt{\\frac{p+t}{n}} + h^2 \\Biggr ) < r \\nonumber $ as long as $\\max \\lbrace (p+\\log n + t )/n, \\sqrt{(p+t)/n } \\rbrace \\lesssim h \\lesssim 1$ .",
"Consequently, $\\widetilde{\\beta }_h^\\flat $ falls in the interior of $\\beta ^* + \\Theta (r)$ , thereby implying $\\eta =1$ and $\\hat{\\beta }_h^\\flat = \\widetilde{\\beta }_h^\\flat $ .",
"This completes the proof.",
"$\\Box $"
],
[
"Proof of Lemma ",
"By the triangle inequality, we have $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) \\bigr \\Vert _2 \\le \\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) - ^{-1/2}\\nabla \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2 + \\bigl \\Vert ^{-1/2}\\nabla \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2:= I_1+I_2.$ The term $I_2$ can be upper bounded by Lemma REF , and thus it suffices to bound $I_1$ .",
"To this end, we will employ Hoeffding's and Bernstein's inequalities.",
"Recall from Section REF that $w_i$ is such that $(w_i)=1$ and $\\textnormal {var}(w_i)=1$ .",
"Thus, $^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) = (1/n) \\sum _{i=1}^nw_i \\xi _i _i$ and $^* \\lbrace ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) \\rbrace = ^{-1/2} \\hat{Q}_h (\\beta ^* ) $ , where $_i = ^{-1/2} _i$ and $\\xi _i = K(-\\varepsilon _i /h) - \\tau $ with $\\varepsilon _i = y_i - \\langle _i, \\beta ^* \\rangle $ .",
"Using a similar covering argument as in the proof of Lemma REF , for any $\\epsilon \\in ( 0, 1)$ , there exists an $\\epsilon $ -net $_\\epsilon $ of the unit sphere $\\mathbb {S}^{p-1}$ with $|_\\epsilon |\\le ( 1+2/\\epsilon )^p$ such that $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) - ^{-1/2}\\nabla \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2 \\le \\frac{1}{1-\\epsilon } \\max _{\\in _\\epsilon } \\frac{1}{n} \\sum _{i=1}^ne_i \\xi _i \\langle , _i \\rangle , \\nonumber $ where $e_i = w_i-1$ are i.i.d.",
"Rademacher random variables.",
"By Hoeffding's inequality, for any $u\\ge 0$ , $^* \\left\\lbrace \\frac{1}{n} \\sum _{i=1}^ne_i \\xi _i \\langle , _i \\rangle \\ge \\Biggl ( \\frac{1}{n} \\sum _{i=1}^n\\xi _i^2 \\langle , _i \\rangle ^2 \\Biggr )^{1/2} \\sqrt{\\frac{2 u }{n}} \\right\\rbrace \\le e^{-u}.$ For the data-dependent quantity $(1/n) \\sum _{i=1}^n\\xi _i^2 \\langle , _i \\rangle ^2 $ , as in the proof of Lemma REF we have $| \\xi _i | \\le B_\\tau := \\max ( \\tau , 1-\\tau )$ , $(\\xi _i^2 | _i ) \\le C_\\tau := \\tau (1-\\tau ) + (1+\\tau ) l_0 \\kappa _2 h^2$ almost surely and note that $\\langle , _i \\rangle ^2 =1$ .",
"Moreover, for $k=2,3,\\ldots $ , $\\bigl ( \\xi _i^2 \\langle , _i \\rangle ^2 \\bigr )^k & \\le C_\\tau B_\\tau ^{2(k-1)} \\upsilon _1^{2k} \\cdot 2k \\int _0^\\infty \\bigl ( | \\langle , _i \\rangle | \\ge \\upsilon _1 u \\bigr ) u^{2k-1} \\, {\\rm d} u \\nonumber \\\\& \\le C_\\tau B_\\tau ^{2(k-1)} \\upsilon _1^{2k} \\cdot 4k \\int _0^\\infty u^{2k-1} e^{-u^2/2 } \\, {\\rm d} u \\nonumber \\\\& = C_\\tau B_\\tau ^{2(k-1)} 2^k \\upsilon _1^{2k} \\cdot 2k \\int _0^\\infty v^{ k-1} e^{-v } \\, {\\rm d}v = C_\\tau B_\\tau ^{2(k-1)} 2^{k+1} \\upsilon _1^{2k} k!",
".",
"\\nonumber $ In particular, $(\\xi _i^4 \\langle , _i \\rangle ^4)\\le B_\\tau ^2 C_\\tau 16 \\upsilon _1^4$ and $( \\xi _i^2 \\langle , _i \\rangle ^2 )^k \\le \\frac{k!",
"}{2} \\cdot B_\\tau ^2 C_\\tau 16 \\upsilon _1^4 \\cdot (2 B_\\tau ^2 \\upsilon _1^2 )^{k-2}$ for $k\\ge 3$ .",
"Since $(\\xi _i^2 | _i ) \\le C_\\tau $ , it then follows from Bernstein's inequality that, for any $v\\ge 0$ , $\\Biggl ( \\frac{1}{n} \\sum _{i=1}^n\\xi _i^2 \\langle , _i \\rangle ^2 \\ge C_\\tau + 4 B_\\tau C_\\tau ^{1/2} \\upsilon _1^2 \\sqrt{\\frac{2 v}{n}} + B_\\tau ^2 \\upsilon _1^2 \\frac{2v}{n} \\Biggr ) \\le e^{-v} .$ In the above analysis, we set $\\epsilon = 2/(e^2-1)$ so that $(1+2/\\epsilon )^p = e^{2p}$ , and take $u=v=2p +t$ .",
"Applying the union bound, we conclude that $\\max _{\\in _\\epsilon } \\frac{1}{n} \\sum _{i=1}^n\\xi _i^2 \\langle , _i \\rangle ^2 \\le C_\\tau + 4 B_\\tau C_\\tau ^{1/2} \\upsilon _1^2 \\sqrt{\\frac{4p+2t}{n}} + 2 B_\\tau ^2 \\upsilon _1^2 \\frac{2p+t}{n} $ with probability at least $1-e^{-t}$ .",
"Let $_1(t)$ be the event that (REF ) holds.",
"Then, with $^*$ -probability at least $1-e^{-t}$ conditioned on $_1(t)$ , and by an application of union bound, we obtain $\\bigl \\Vert ^{-1/2} \\nabla \\hat{Q}_h^\\flat (\\beta ^* ) - ^{-1/2} \\hat{Q}_h (\\beta ^* ) \\bigr \\Vert _2 \\le C \\sqrt{\\frac{p+t}{n}}$ as long as $n\\gtrsim p+t$ , where $C>0$ depends only on $(\\upsilon _1,\\kappa _2, l_0)$ and $\\tau $ .",
"This, together with (REF ) and Lemma REF , proves the claimed bound.",
"$\\Box $"
],
[
"Proof of Lemma ",
"Using arguments similar to (), (), and (REF ) in the proof of Lemma REF , we obtain $D^\\flat (\\beta , \\beta ^*) &= \\frac{1}{n} \\sum _{i=1}^n\\left\\lbrace {K} \\left( \\frac{\\langle _i, \\beta \\rangle - y_i}{h}\\right) - {K}\\left( \\frac{ -\\varepsilon _i}{h} \\right) \\right\\rbrace w_i \\langle _i, \\beta - \\beta ^* \\rangle \\nonumber \\\\&\\ge \\kappa _l \\Vert \\beta - \\beta ^* \\Vert _{}^2 \\cdot \\underbrace{ \\frac{1}{n h } \\sum _{i=1}^nw_i \\omega _i \\varphi _{h/(2r)} ( \\langle _i , \\delta \\rangle ) }_{ =: D_0^\\flat (\\delta ) } , \\nonumber $ where $\\delta = ^{1/2}( \\beta - \\beta ^*) /\\Vert \\beta - \\beta ^* \\Vert _{}$ , $\\omega _i = \\mathbb {1}( |\\varepsilon _i | \\le h/2)$ , and $\\varphi (\\cdot )$ is as defined in (REF ).",
"Recall from (REF ) the definition of $D_0 (\\delta )$ .",
"We have $\\inf _{\\beta \\in \\beta ^* + \\Theta (r)}\\frac{D^\\flat (\\beta , \\beta ^* )}{\\kappa _l \\Vert \\beta -\\beta ^* \\Vert _{}^2} \\ge \\inf _{\\delta \\in \\mathbb {S}^{p-1} } D_0(\\delta ) - \\sup _{\\delta \\in \\mathbb {S}^{p-1} } \\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\rbrace .$ We now obtain lower and upper bounds for $ D_0(\\delta ) $ and $\\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\rbrace $ , respectively.",
"Recall that $w_i$ is a random variable that is independent of $_i$ with $(w_i)=1$ .",
"Let $e_i = w_i -1 \\in \\lbrace -1,1\\rbrace $ .",
"Then, we have $D_0^\\flat (\\delta ) - D_0(\\delta ) = \\frac{1}{n h } \\sum _{i=1}^ne_i \\omega _i \\varphi _{h/(2r)} ( \\langle _i , \\delta \\rangle ) .$ Define $\\Gamma _n = \\gamma (e_1,\\ldots ,e_n) = \\sup _{ \\delta \\in \\mathbb {S}^{p-1}} \\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\rbrace $ .",
"Recall that $\\varphi _{R}(u) \\le (R/2)^2 $ , we have $^* \\lbrace e_i \\omega _i \\varphi _{h/(2r)} ( \\langle _i , \\delta \\rangle ) \\rbrace ^2 \\le (h/4r)^4 \\omega _i $ and $| e_i \\omega _i \\varphi _{h/(2r)} ( \\langle _i , \\delta \\rangle ) |\\le (h/4r)^2$ .",
"Then, by the Talagrand's inequality (see Theorem 7.3 in [60]), for every $t\\ge 0$ , $\\Gamma _n & \\le ^*( \\Gamma _n ) + \\sqrt{ \\frac{h^2}{(4r)^4}\\frac{ \\sum _{i=1}^n\\omega _i}{ n} \\frac{2t}{n} + 4^*( \\Gamma _n ) \\frac{h}{(4r)^2}\\frac{t}{n} } + \\frac{h}{(4r)^2} \\frac{t}{3 n} \\nonumber \\\\& \\le 2^*( \\Gamma _n ) + \\frac{h}{(4r)^2} \\Biggl \\lbrace \\Biggl ( \\frac{1}{n}\\sum _{i=1}^n\\omega _i \\Biggr )^{1/2} \\sqrt{ \\frac{2t}{n} } + \\frac{4t}{3 n} \\Biggr \\rbrace \\nonumber $ with probability at least $1-e^{-t}$ .",
"Further, by the Lipschitz continuity of $u\\rightarrow \\varphi _R(u)$ and Talagrand's contraction principle, $^*( \\Gamma _n ) & \\le \\frac{1}{2 r} ^* \\Biggl \\lbrace \\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^ne_i \\langle \\omega _i _i, \\delta \\rangle \\Biggr \\rbrace \\nonumber \\\\& \\le \\frac{1}{2 r} ^* \\Biggl \\Vert \\frac{1}{n} \\sum _{i=1}^ne_i \\omega _i _i \\Biggr \\Vert _2 \\nonumber \\\\& \\le \\frac{1}{2 r n } \\Biggl ( \\sum _{i=1}^n\\omega _i \\Vert _i \\Vert _2^2 \\Biggr )^{1/2} \\nonumber \\\\& \\le \\frac{\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 }{2 r \\sqrt{n}} \\Biggl ( \\frac{1}{n}\\sum _{i=1}^n\\omega _i \\Biggr )^{1/2} .",
"\\nonumber $ Together, the last two displays imply $\\sup _{ \\delta \\in \\mathbb {S}^{p-1} } \\bigl \\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\bigr \\rbrace \\le \\Biggl ( \\frac{1}{n}\\sum _{i=1}^n\\omega _i \\Biggr )^{1/2} \\Biggl \\lbrace \\frac{\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 }{r \\sqrt{n}} +\\frac{h}{(4r)^2} \\sqrt{\\frac{2t}{n} } \\Biggr \\rbrace + \\frac{h}{(4r)^2} \\frac{4t}{3 n} $ with $^*$ -probability at least $1-e^{-t}$ .",
"Next we provide upper bounds for the data-dependent terms $\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 $ and $(1/n) \\sum _{i=1}^n\\omega _i$ .",
"As in the proof of Lemma REF , for any $\\epsilon \\in (0,1)$ , there exits $_\\epsilon \\subseteq \\mathbb {S}^{p-1}$ with $|_\\epsilon | \\le (1+2/\\epsilon )^p$ such that $\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 \\le (1-\\epsilon )^{-1} \\max _{1\\le i\\le n} \\max _{\\in _\\epsilon } \\langle , _i \\rangle $ .",
"Given $1\\le i\\le n$ and $\\in _\\epsilon $ , Condition REF indicates $( |\\langle , _i \\rangle | \\ge \\upsilon _1 u) \\le 2 e^{-u^2/2}$ for any $u \\in $ .",
"Taking the union bound over $i$ and $$ , and setting $u=\\sqrt{2t + 2\\log (2n)+2p\\log (1+2/\\epsilon )}$ ($t>0$ ), we obtain that with probability at least $1-2 n (1+2/\\epsilon )^p e^{-u^2/2}= 1-e^{-t}$ , $\\max _{1\\le i\\le n} \\Vert _i \\Vert _2 \\le (1-\\epsilon )^{-1} \\upsilon _1 \\sqrt{2t + 2\\log (2n)+2p\\log (1+2/\\epsilon )}$ .",
"Minimizing this upper bound with respect to $\\epsilon \\in (0,1)$ , we conclude that for any $t>0$ , $\\biggl [ \\max _{1\\le i\\le n} \\Vert _i \\Vert _2^2 \\ge 2 \\upsilon _1^2 \\bigl \\lbrace 3.7p + \\log (2n) + t \\bigr \\rbrace \\biggr ] \\le e^{-t}.",
"$ Moreover, applying Bernstein's inequality to $(1/n) \\sum _{i=1}^n\\omega _i = (1/n) \\sum _{i=1}^n\\mathbb {1}( |\\varepsilon _i | \\le h/2)$ yields $\\frac{1}{n} \\sum _{i=1}^n\\omega _i \\le (\\omega _i) + \\sqrt{ 2(\\omega _i) \\frac{ t}{n}} + \\frac{t}{3n} \\le \\Biggl ( \\sqrt{(\\omega _i)} + \\sqrt{\\frac{t}{2n}} \\Biggr )^2$ with probability greater than $1-e^{-t}$ .",
"Note that $(\\omega _i) \\le \\bar{f}_{ h} h$ .",
"Putting together the pieces, we conclude that conditioned on some event that occurs with probability at least $1-2e^{-t}$ , $& \\sup _{\\delta \\in \\mathbb {S}^{p-1} } \\bigl \\lbrace D_0(\\delta ) -D_0^\\flat (\\delta ) \\bigr \\rbrace \\nonumber \\\\& \\le \\bigl \\lbrace ( \\bar{f}_h h )^{1/2} + (0.5 t/ n)^{1/2} \\bigr \\rbrace \\left\\lbrace \\frac{\\upsilon _1 \\bigl ( 7.4 p + 2\\log (2n) + 2 t \\bigr )^{1/2} }{r \\sqrt{n}} + \\frac{h}{(4r)^2} \\sqrt{\\frac{2t}{n} } \\right\\rbrace + \\frac{h}{(4r)^2}\\sqrt{\\frac{4 t}{3 n}}$ with $^*$ -probability greater than $1-e^{-t}$ .",
"Turning to $D_0(\\delta )$ , applying Lemma REF yields that as long as $0< r\\le h/(4\\eta _{1/4} )$ , $\\inf _{\\delta \\in \\mathbb {S}^{p-1}} D_0(\\delta ) \\ge \\frac{3}{4} {f}_h - \\bar{f}_h^{1/2} \\left( \\frac{5}{4}\\sqrt{\\frac{h p}{r^2 n}} + \\sqrt{\\frac{h t}{8r^2 n}} \\right) - \\frac{h t}{3 r^2 n } $ with probability at least $1-e^{-t}$ .",
"Substituting (REF ) and (REF ) into (REF ), and taking $r=h/(4\\eta _{1/4} )$ with $h\\gtrsim t/n$ , we conclude that conditioned on some event that occurs with probability at least $1-3 e^{-t}$ , $\\inf _{\\beta \\in \\beta ^* + \\Theta (r)}\\frac{D^\\flat (\\beta , \\beta ^* )}{\\kappa _l \\Vert \\beta -\\beta ^* \\Vert _{}^2} \\ge \\frac{3}{4} {f}_h - C \\sqrt{\\frac{p + \\log n + t}{ n h }} \\nonumber $ with $^*$ -probability greater than $1-e^{-t}$ , where $C>0$ depends only on $(\\upsilon _1, l_0 ,\\bar{f} )$ .",
"This completes the proof of (REF ).",
"$\\Box $"
],
[
"Proof of Theorem ",
"The proof is based on an argument similar to that used in the proof of Theorem REF .",
"To begin with, define the random process $\\Delta ^\\flat (\\beta ) = ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}_h^\\flat (\\beta ) - \\nabla \\hat{Q}_h^\\flat (\\beta ^*) - (\\beta -\\beta ^* )\\bigr \\rbrace , ~~ \\beta \\in ^p.$ For a prespecified $r>0$ , a key step is to bound the local fluctuation $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta ^\\flat (\\beta ) \\Vert _2$ .",
"Since $(w_i)=1$ , we have $^* \\lbrace \\nabla \\hat{Q}_h^\\flat (\\beta ) \\rbrace = \\nabla \\hat{Q}_h (\\beta )$ .",
"Define the (conditionally) centered process $G^\\flat (\\beta ) = ^{-1/2} \\big \\lbrace \\nabla \\hat{Q}_h^\\flat (\\beta ) - \\nabla \\hat{Q}_h (\\beta ) \\big \\rbrace = \\frac{1}{n} \\sum _{i=1}^ne_i \\lbrace K(-(y_i - \\langle _i, \\beta \\rangle )/h) - \\tau \\rbrace _i,$ so that $\\Delta ^\\flat (\\beta )$ be be written as $\\Delta ^\\flat (\\beta ) = \\bigl \\lbrace G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\bigr \\rbrace + \\Delta (\\beta ) , \\nonumber $ where $\\Delta (\\beta )$ is defined in (REF ).",
"By the triangle inequality, $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta ^\\flat (\\beta ) \\Vert _2 \\le \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\Vert _2 + \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta (\\beta ) \\Vert _2 .",
"$ It suffices to bound the first term on the right-hand side of (REF ).",
"By a change of variable $= ^{1/2} ( \\beta -\\beta ^*)$ , we have $y_i -\\langle _i, \\beta \\rangle = \\varepsilon _i - \\langle _i, \\rangle $ and $& \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\Vert _2 \\nonumber \\\\ & \\le \\sup _{ \\in \\mathbb {B}^p(r), \\in \\mathbb {B}^p(1) } \\langle G^\\flat ( \\beta ^* + ^{-1/2} ) - G^\\flat ( \\beta ^* ) , \\rangle \\nonumber \\\\& = r^{-1} n^{-1/2} \\sup _{ , \\in \\mathbb {B}^p(r) } \\underbrace{ n^{1/2} \\bigl \\langle G^\\flat ( \\beta ^* + ^{-1/2} ) - G^\\flat ( \\beta ^* ) , \\bigr \\rangle }_{ =: \\Psi ( , ) } , $ where $\\Psi (, ) = n^{-1/2} \\sum _{i=1}^ne_i\\langle _i, \\rangle \\lbrace K( \\langle _i, \\rangle / h - \\varepsilon _i /h ) - K(-\\varepsilon _i/h) \\rbrace $ .",
"To characterize the magnitude of $\\sup _{, \\in \\mathbb {B}^p(r)} \\Psi (, )$ , define the following classes of measurable functions $_1 =\\bigl \\lbrace ( , \\varepsilon ) \\mapsto \\langle , \\rangle : \\in \\mathbb {B}^p(r) \\bigr \\rbrace ~\\mbox{ and }~ _2 = \\bigl \\lbrace ( , \\varepsilon ) \\mapsto \\langle , \\rangle - \\varepsilon : \\in \\mathbb {B}^p(r) \\bigr \\rbrace .$ Also, define functions $f_0 : ( , \\varepsilon ) \\rightarrow K(-\\varepsilon /h)$ and $\\phi : u \\rightarrow K(u/h)$ .",
"With this notation, we can write $\\sup _{, \\in \\mathbb {B}^p(r)}\\Psi (, ) = \\sup _{g\\in } \\frac{1}{\\sqrt{n}} \\sum _{i=1}^ne_i g(_i), $ where $_i = (_i , \\varepsilon _i ) \\in ^p\\times $ and $= _1 \\cdot (\\phi \\circ _2 - f_0)$ denotes the pointwise product between $_1$ and $\\phi \\circ _2 - f_0$ .",
"Let $Z = ^* \\lbrace \\sup _{g\\in } (1/n)\\sum _{i=1}^ne_i g(_i) \\rbrace $ be the conditional Rademacher average.",
"By Theorem 13 in [59] and the bound $\\sup _{1\\le i\\le n, g\\in } g(_i )\\le r \\max _{1\\le i\\le n} \\Vert _i \\Vert _2$ , we have $\\bigl [ \\lbrace Z - (Z)\\rbrace ^{2k}_+ \\bigr ]^{1/(2k)} \\le 2 \\sqrt{(Z) \\cdot k \\kappa r \\frac{M_{n,k}}{n}} + 2 k \\kappa r \\frac{M_{n,k}}{n} , ~\\mbox{ valid for any } k\\ge 1, $ where $\\kappa = \\sqrt{e} / (2\\sqrt{e}-2)$ and $M_{n,k} = \\lbrace (\\max _{1\\le i\\le n} \\Vert _i \\Vert _2^{2k})\\rbrace ^{1/(2k)}$ .",
"Next we bound $(Z)$ via a maximal inequality specialized to VC type classes.",
"Note that the function class $_1$ is a $p$ -dimensional vector space, and $_2$ is included in a $(p + 1)$ -dimensional vector space.",
"By Lemma 2.6.15 of [50], both $_1$ and $_2$ have VC-subgraph of index smaller than or equal to $p+2$ .",
"Moreover, since $\\phi (\\cdot ) = K(\\cdot /h)$ is an non-decreasing function, it follows from Lemma 9.9 of [63] that $\\phi \\circ _2$ is VC with index less than or equal $ p+2$ , and so is $\\phi \\circ _2 - f_0$ .",
"Let $F_1(, \\varepsilon ) = r \\Vert \\Vert _2$ and $F_2 (, \\varepsilon ) \\equiv 1$ be the envelops of $_1$ and $\\phi \\circ _2 - f_0$ , respectively.",
"Hence, by Theorem 9.3 in [63] (or Theorem 2.6.7 in [50]), we obtain the following bounds on the covering numbers of function classes $_1$ and $\\phi \\circ _2 - f_0$ : there exists a universal constant $A_0>0$ such that, for any probability measure $Q$ with $\\Vert F_1 \\Vert _{Q,2} >0$ and any $\\epsilon \\in (0,1)$ , $N( _1, L_2(Q) , \\epsilon \\Vert F_1 \\Vert _{Q,2} ) \\le A_0 (p+2)(4e)^{p+2} \\biggl ( \\frac{2}{\\epsilon } \\biggr )^{2(p+1)}; \\nonumber \\\\N( \\phi \\circ _2 - f_0 , L_2(Q) , \\epsilon \\Vert F_2 \\Vert _{Q,2} ) \\le A_0 (p+2)(4e)^{p+2} \\biggl ( \\frac{2}{\\epsilon } \\biggr )^{2(p+1)} .\\nonumber $ For the product class $$ with envelop $G(, \\varepsilon )= F_1(, \\varepsilon ) F_2(, \\varepsilon ) = r \\Vert \\Vert _2$ , applying Corollary A.1 in the supplement of [61] yields that, for any $\\epsilon \\in (0 ,1)$ , $& N( , L_2(Q) , \\epsilon \\Vert G \\Vert _{Q,2} ) \\nonumber \\\\& \\le N( _1, L_2(Q) , 2^{-1/2} \\epsilon \\Vert F_1 \\Vert _{Q,2} ) N( \\phi \\circ _2 - f_0 , L_2(Q) , 2^{-1/2} \\epsilon \\Vert F_2 \\Vert _{Q,2} ) \\nonumber \\\\& \\le A_0^2 (p+2)^2 (4 e)^{2(p+2)} \\biggl ( \\frac{2^{3/2}}{\\epsilon } \\biggr )^{4(p+1)} \\le \\biggl ( \\frac{A_1}{\\epsilon } \\biggr )^{4(p+2)} \\nonumber $ for some $A_1 > e$ .",
"Another key quantity is the variance proxy $\\sup _{g\\in } \\lbrace g^2(, \\varepsilon )\\rbrace $ .",
"Given some $g = g_{, } \\in $ with $, \\in \\mathbb {B}^p(r)$ , we have $\\lbrace g^2(, \\varepsilon )\\rbrace & = ( \\langle , \\rangle ^2 [K\\lbrace ( \\langle , \\rangle - \\varepsilon ) /h\\rbrace - K(-\\varepsilon /h) ]^2) \\nonumber \\\\& = \\biggl (\\langle , \\rangle ^2 \\int _{-\\infty }^\\infty \\bigl \\lbrace K\\bigl ( ( \\langle , \\rangle -u ) /h \\bigr ) - K(-u/h) \\bigr \\rbrace ^2 f_{\\varepsilon | } (u) {\\rm d} u\\biggr ) \\nonumber \\\\& = h \\, \\biggl (\\langle , \\rangle ^2 \\int _{-\\infty }^\\infty \\bigl \\lbrace K \\bigl ( \\langle , \\rangle /h + v \\bigr ) - K(v) \\bigr \\rbrace ^2 f_{\\varepsilon | } (-vh) {\\rm d} v\\nonumber \\biggr ) \\\\& \\le \\bar{f} h^{-1} \\biggl (\\langle , \\rangle ^2 \\langle , \\rangle ^2 \\int _{-\\infty }^\\infty \\biggl \\lbrace \\int _0^1 K(v+ s \\langle ,\\rangle /h ) {\\rm d} s \\biggr \\rbrace ^2 {\\rm d} v\\biggr ) \\nonumber \\\\& \\le \\bar{f} h^{-1} \\biggl (\\langle , \\rangle ^2 \\langle , \\rangle ^2\\Biggl [ \\int _0^1 \\biggl \\lbrace \\int _{-\\infty }^\\infty K^2(v+ s \\langle ,\\rangle /h ) {\\rm d} v \\biggr \\rbrace ^{1/2} {\\rm d}s \\Biggr ]^2 \\nonumber \\biggr ) \\\\& \\le \\kappa _u \\bar{f} h^{-1} (\\langle , \\rangle ^2 \\langle , \\rangle ^2 ), \\nonumber $ where the second inequality follows from an application of generalized Minkowski's integral inequality.",
"Hence, $\\sup _{g\\in } \\lbrace g^2(, \\varepsilon )\\rbrace \\le \\kappa _u \\bar{f} \\mu _4 r^4h^{-1}$ .",
"For the envelop $G$ for $$ , it can be shown that $\\lbrace G^2(, \\varepsilon )\\rbrace = r^2 p$ , $\\max _{1\\le i\\le n}\\sup _{g\\in } | g(_i) | \\le r\\max _{1\\le i\\le n} \\Vert _i \\Vert _2$ , and $\\lbrace \\max _{1\\le i\\le n}G^2(_i, \\varepsilon _i)\\rbrace = r^2 M_{n,1}^2$ with $M_{n,1}$ given in (REF ).",
"Equipped with the above bounds, we apply Corollary 5.1 in [61] to obtain $(Z) \\lesssim r^2 \\sqrt{\\frac{p}{nh} \\log ( A_2 p h /r^2 ) } + r M_{n,1} \\log ( A_2 p h /r^2 ) \\frac{ p }{n} , $ where $A_2>0$ depends on $(\\bar{f} , \\kappa _u, \\mu _4)$ and $A_1$ .",
"It remains to bound $M_{n,1}$ , which appears in both (REF ) and (REF ).",
"In fact, the exponential tail probability (REF ) can be directly used to bound the expectation.",
"Given a non-negative random variable $X$ and constants $A, a>0$ , we have $(X) & = \\int _0^\\infty (X\\ge t) {\\rm d} t \\le A + \\int _A^\\infty (X\\ge t) {\\rm d} t \\nonumber \\\\& = A + \\int _0^\\infty (X\\ge A+ t) {\\rm d} t = A + a \\int _0^\\infty (X\\ge A + a s ) {\\rm d} s .",
"\\nonumber $ Taking $X= \\max _{1\\le i\\le n} \\Vert _i \\Vert _2^2$ , $A= 2 \\upsilon _1^2 \\lbrace 3.7 p + \\log (2n) \\rbrace $ and $a=2 \\upsilon _1^2 $ , we obtain $M_{n,1}^2 = \\Biggl ( \\max _{1\\le i\\le n} \\Vert _i \\Vert _2^2 \\Biggr ) \\le 2 \\upsilon _1^2 \\bigl \\lbrace 3.7 p + \\log (2e n) \\bigr \\rbrace .",
"$ Back to the supremum $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\Vert _2$ , first by (REF ), (REF ), and the definition of $Z$ below (REF ), we obtain $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert G^\\flat (\\beta ) - G^\\flat (\\beta ^*) \\Vert _2 = O_{^*} (r^{-1} Z ) .",
"$ For the conditional Rademacher average $Z$ , it follows from Markov's inequality that $\\lbrace Z \\ge (Z) + u \\rbrace \\le u^{-2} \\lbrace Z- (Z)\\rbrace _+^2$ , valid for any $u >0$ .",
"Applying the bound (REF ) with $k=1$ , (REF ) and (REF ) yields $& \\bigl [ \\lbrace Z- (Z)\\rbrace _+^2 \\bigr ]^{1/2} \\lesssim \\sqrt{ (Z) \\cdot (p +\\log n)^{1/2} \\frac{r}{n} } + (p+\\log n)^{1/2} \\frac{r}{n}; \\nonumber \\\\&(Z) \\lesssim r^2 \\sqrt{ \\frac{p}{n h} \\log (A_2ph / r^2 ) } + r \\log (A_2ph / r^2 ) (p+\\log n)^{1/2} \\frac{p}{n} , \\nonumber $ from which we conclude that $r^{-1 }Z = O_{} \\Biggl ( r \\sqrt{ \\frac{p}{n h} \\log (A_2ph / r^2 ) } + \\log (A_2ph / r^2 ) (p+\\log n)^{1/2} \\frac{p}{n} \\Biggr ).$ Turning to the second term on the right-hand side of (REF ), the earlier bound (REF ) implies that there exists some event $_1(t)$ with $\\lbrace _1(t) \\rbrace \\ge 1-e^{-t}$ such that $\\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta (\\beta ) \\Vert _2 \\le C_1 \\Biggl ( r \\sqrt{\\frac{p+t}{n h}} + h r + r^2 \\Biggr ) ~\\mbox{ on } _1 (t), $ where $C_1>0$ depends on $( \\upsilon _1, \\kappa _1, \\kappa _u, l_0, \\bar{f} , {f})$ .",
"With the above preparations, we are ready to prove the claim.",
"Assume that the triplet $(n,p,h)$ satisfies the scaling $ \\sqrt{(p+\\log n)/n} \\lesssim h \\lesssim 1$ .",
"By Theorems REF and REF , there exist some event $_n$ with $(_n ) \\ge 1-5n^{-1}$ and some $r=r(n,p,h)\\asymp \\sqrt{(p+\\log n)/n} + h^2>0$ such that, $\\Vert \\hat{\\beta }_h - \\beta ^* \\Vert _{} \\le r$ on the event $_n$ , $ \\Vert \\hat{\\beta }_h^\\flat - \\beta ^* \\Vert _{} \\le r$ with $^*$ -probability at least $1-2 n^{-1}$ conditioned on $_n$ .",
"Taking $t=\\log n$ and $_{1n} = _1(\\log n)$ in (REF ), we have on the event $_{1n} \\cap _n$ that $\\chi _{1n} := \\Vert \\Delta (\\hat{\\beta }_h ) \\Vert _2 \\lesssim \\frac{p + \\log n}{ h^{1/2} n } + h \\sqrt{\\frac{p + \\log n}{n}} + h^3.",
"\\nonumber $ Moreover, with $^*$ -probability at least $1-2n^{-1}$ conditioned on $_n$ , $\\Biggl \\Vert ^{-1/2} \\Biggl [ ( \\hat{\\beta }_h^\\flat - \\beta ^* ) - \\frac{1}{n} \\sum _{i=1}^nw_i \\bigl \\lbrace \\tau - K(- \\varepsilon _i /h) \\bigr \\rbrace _i \\Biggr ] \\Biggr \\Vert _2 \\le \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta ^\\flat (\\beta ) \\Vert _2 .",
"\\nonumber $ Let $\\chi _{2n} = ^* \\lbrace \\sup _{\\beta \\in \\beta ^* + \\Theta (r)} \\Vert \\Delta ^\\flat (\\beta ) \\Vert _2 \\rbrace $ , for which it follows from (REF ), (REF ), and (REF ) with $t=\\log n$ that $\\chi _{2n} = O_{} \\Biggl \\lbrace \\frac{\\sqrt{(p+\\log n)p \\log n}}{h^{1/2} n} + h^{3/2} \\sqrt{\\frac{p \\log n }{n}} + h \\sqrt{\\frac{p + \\log n}{n}} + h^3 + (p+\\log n)^{1/2} \\frac{p \\log n}{n} \\Biggr \\rbrace .$ With $h=h_n\\asymp \\lbrace (p+\\log n)/n \\rbrace ^{\\delta }$ for some $\\delta \\in (1/3 , 1/2]$ , the first term on the right-hand side dominates the rest except the last one.",
"Then, putting together the pieces establishes the claim.",
"$\\Box $"
],
[
"Theoretical Properties of One-step Conquer",
"In this section, we provide theoretical properties of the one-step conquer estimator $\\hat{\\beta }$ , defined in Section REF .",
"The key message is that, when higher-order kernels are used (and if the conditional density $f_{\\varepsilon |}(\\cdot )$ has enough derivatives), the asymptotic normality of the one-step estimator holds under weaker growth conditions on $p$ .",
"For example, the scaling condition $p=o(n^{3/8})$ that is required for the conquer estimator can be reduced to roughly $p=o(n^{7/16})$ for the one-step conquer estimator using a kernel of order 4.",
"Let $G(\\cdot )$ be a symmetric kernel of order $\\nu >2$ , that is, $\\int _{-\\infty }^\\infty u^k G(u) \\, {\\rm d}u=0$ for $k=1,\\ldots , \\nu -1$ and $\\int _{-\\infty }^\\infty u^\\nu G(u) \\, {\\rm d} u \\ne 0$ .",
"Moreover, $\\kappa ^G_k = \\int _{-\\infty }^\\infty |u^k G(u) | \\, {\\rm d}u <\\infty $ for $1\\le k\\le \\nu $ , $G$ is uniformly bounded with $\\kappa ^G_u = \\sup _{u\\in } |G(u) | <\\infty $ and is $L^G$ -Lipschitz continuous for some $L^G>0$ .",
"The use of a higher-order kernel does not necessarily reduce bias unless the conditional density $ f_{\\varepsilon | }(\\cdot )$ of $\\varepsilon $ given $$ is sufficiently smooth.",
"Therefore, we further impose the following smoothness conditions on $f_{\\varepsilon | }(\\cdot )$ .",
"Let $\\nu \\ge 4$ be the integer in Condition .",
"The conditional density $ f_{\\varepsilon | }(\\cdot )$ is $(\\nu -1)$ -times differentiable, and satisfies $| f_{\\varepsilon | }^{(\\nu -2)} (u) - f_{\\varepsilon | }^{(\\nu -2)} (0) | \\le L_{\\nu -2} |u|$ for all $u\\in $ almost surely (over the random vector $$ ), where $L_{\\nu -2}>0$ is a constant.",
"Also, there exists some constant $C_G>0$ such that $ \\int _{-\\infty }^\\infty | u^{\\nu -1} G(u)| \\cdot \\sup _{|t| \\le |u|} | f_{\\varepsilon | }^{(\\nu -1)} (t)- f_{\\varepsilon | }^{(\\nu -1)} (0) | \\, {\\rm d}u \\le C_G$ almost surely.",
"Notably, we have $\\nabla Q^G_b( \\beta ) = \\bigl \\lbrace G\\big ( ( \\langle , \\beta \\rangle - y) /b \\big ) - \\tau \\bigr \\rbrace ~~\\mbox{ and }~~\\nabla ^2 Q^G_b (\\beta ) = \\big \\lbrace G_b( y- \\langle , \\beta \\rangle ) ^\\rbrace ,$ representing the population score and Hessian of $Q^G_b (\\cdot ) = \\hat{Q}^G_b(\\cdot )$ .",
"As $b \\rightarrow 0$ , we expect $\\nabla Q^G_b( \\beta ^*)$ and $\\nabla ^2 Q^G_b (\\beta ^*)$ to converge to $\\textbf {0}$ (zero vector in $^p$ ) and $= \\lbrace f_{\\varepsilon | } (0) ^$ , respectively.",
"The following proposition validates this claim by providing explicit error bounds.",
"Let $b\\in (0, 1)$ be a bandwidth.",
"Under Conditions and , we have $\\big \\Vert ^{-1/2} \\nabla Q^G_b(\\beta ^*) \\big \\Vert _2 & \\le L_{\\nu -2} \\kappa ^G_\\nu \\, b^\\nu / \\nu !",
"\\nonumber \\\\~~\\mbox{ and }~~\\big \\Vert ^{-1/2} \\nabla ^2 Q^G_b(\\beta ^*) ^{-1/2} - _0 \\big \\Vert _2 & \\le C_G \\, b^{\\nu -1} / (\\nu -1)!",
", \\nonumber $ where $_0 = ^{-1/2} ^{-1/2} = \\lbrace f_{\\varepsilon |} (0) ^$ with $= ^{-1/2} $ .",
"Proposition shows that when a higher-order kernel is used, the bias is significantly reduced in the sense that $\\Vert \\nabla Q^G_b(\\beta ^*) \\Vert _2 = ( b^\\nu )$ and $\\Vert \\nabla ^2 Q^G_b(\\beta ^*) - \\Vert _2 = (b^{\\nu -1})$ , where $\\nu \\ge 4$ is an even integer.",
"Notably, even if the kernel $H$ has negative parts, the population Hessian $\\nabla ^2 Q^G_b(\\beta ^*) $ preserves the positive definiteness of $$ as long as the bandwidth $g$ is sufficient small, To construct the one-step conquer estimator, two key quantities are the sample Hessian $\\nabla ^2 \\hat{Q}^G_b( \\cdot ) $ and sample gradient $\\nabla \\hat{Q}^G_b(\\cdot )$ , both evaluated at $\\beta $ , a consistent initial estimate.",
"In the next two propositions, we establish uniform convergence results of the Hessian and gradient of the empirical smoothed loss to their population counterparts.",
"As a direct consequence, $\\nabla ^2 \\hat{Q}^G_b( \\beta )$ is positive definite with high probability, provided that $\\beta $ is consistent (i.e., in a local vicinity of $\\beta ^*$ ).",
"To be more specific, for $r>0$ , we define the local neighborhood $\\Theta ^*(r) = \\bigl \\lbrace \\beta \\in ^p : \\Vert \\beta - \\beta ^* \\Vert _{} \\le r \\bigr \\rbrace .",
"$ Conditions , and REF ensure that, with probability at least $1-e^{-t}$ , $& \\sup _{\\beta \\in \\Theta ^*(r)}\\big \\Vert ^{-1/2} \\big \\lbrace \\nabla ^2 \\hat{Q}^G_b(\\beta ) - \\nabla ^2 Q^G_b(\\beta ) \\big \\rbrace \\nonumber ^{-1/2} \\big \\Vert _2 \\lesssim \\sqrt{\\frac{p \\log n + t}{n b }} + \\frac{p\\log n+t}{n b } + \\frac{(p + t)^{1/2} r }{n b^2} \\nonumber $ as long as $n\\gtrsim p+t$ .",
"Conditions , and REF ensure that, with probability at least $1- e^{-t}$ , $& \\sup _{\\beta \\in \\Theta ^*(r)} \\big \\Vert ^{-1/2} \\big \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) - ( \\beta - \\beta ^*) \\big \\rbrace \\big \\Vert _2 \\lesssim r \\,\\Biggl ( \\sqrt{\\frac{p+ t}{n b }} + r +b^{\\nu -1} \\Bigg ) $ as long as $\\sqrt{(p+t) /n } \\lesssim b \\lesssim 1$ .",
"With the above preparations, we are ready to present the Bahadur representation for the one-step conquer estimator $\\hat{\\beta }$ .",
"Assume Conditions REF , REF and REF in the main text and Conditions and hold.",
"For any $t>0$ , let the sample size $n$ , dimension $p$ and the bandwidths $h , b>0$ satisfy $n\\gtrsim p(\\log n)^2 +t$ , $\\sqrt{(p+t)/n} \\lesssim h \\lesssim \\lbrace (p+t)/n \\rbrace ^{1/4}$ and $\\sqrt{(p+t)/n } \\lesssim b \\lesssim \\lbrace (p+t)/n \\rbrace ^{1/(2\\nu )}$ .",
"Then, the one-step conquer estimator $\\hat{\\beta }$ satisfies the bound $ \\Bigg \\Vert \\, \\hat{\\beta }- \\beta ^* - ^{-1} \\frac{ 1}{n} \\sum _{i=1}^n\\bigl \\lbrace \\tau - G(-\\varepsilon _i/b ) \\bigr \\rbrace _i \\Bigg \\Vert _2 \\lesssim \\Biggl \\lbrace \\underbrace{ \\sqrt{ (p\\log n + t)/(n b ) } }_{{\\rm variance~ term}}+ \\underbrace{ b^{\\nu -1} }_{{\\rm bias~term}} \\Biggr \\rbrace \\sqrt{\\frac{p+t}{n}}$ with probability at least $1-5e^{-t}$ , where $G(u) = \\int _{-\\infty }^u G(v)\\,{\\rm d}v$ .",
"Theorem shows that using a higher-order kernel ($\\nu \\ge 4$ ) allows one to choose larger bandwidth, thereby reducing the “variance\" and the total Bahadur linearization error.",
"Similarly to Theorem REF in the main text, the following asymptotic normal approximation result for linear projections of one-step conquer is a direct consequence of Theorem .",
"Assume Conditions REF , REF and REF in the main text and Conditions and hold.",
"Let the bandwidths satisfy $( \\frac{p+\\log n}{n} )^{1/2} \\lesssim h \\lesssim ( \\frac{p+\\log n}{n} )^{1/4}$ and $( \\frac{p+\\log n}{n} )^{1/2} \\lesssim b \\lesssim (\\frac{p+\\log n}{n})^{1/(2\\nu )}$ .",
"Then, $\\sup _{x\\in , \\, \\in ^p } \\bigl | \\bigl ( n^{1/2} \\langle , \\hat{\\beta }- \\beta ^* \\rangle \\le \\sigma _\\tau x \\bigr ) - \\Phi (x ) \\bigr | \\lesssim \\sqrt{\\frac{ (p+\\log n)p \\log n}{n b}} + n^{1/2} b^{\\nu } , $ where $\\sigma _\\tau ^2 = \\tau (1-\\tau ) \\Vert ^{-1} \\Vert _{ }^2$ .",
"In particular, with a choice of bandwidth $b \\asymp (\\frac{p+\\log n}{n})^{2/(2\\nu +1)}$ , $\\sup _{x\\in , \\, \\in ^p } \\bigl | \\bigl ( n^{1/2} \\langle , \\hat{\\beta }- \\beta ^* \\rangle \\le \\sigma _\\tau x \\bigr ) - \\Phi (x) \\bigr | \\rightarrow 0 \\nonumber $ as $n, p \\rightarrow \\infty $ under the scaling $p^{4\\nu /(2\\nu -1)} (\\log n)^{(2\\nu +1 )/(2\\nu -1)} =o(n)$ .",
"Let $G(\\cdot )$ be a kernel of order $\\nu =4$ .",
"In view of Theorem , we take $h \\asymp \\lbrace (p+\\log n)/n\\rbrace ^{2/5}$ as in the main text and $b = \\lbrace (p+\\log n)/n\\rbrace ^{2/9}$ , thereby obtaining that $n^{1/2}\\langle , \\hat{\\beta }-\\beta \\rangle $ , for an arbitrary $\\in ^p$ , is asymptotically normally distributed as long as $p^{16/7} (\\log n)^{9/7} = o(n)$ as $n\\rightarrow \\infty $ ."
],
[
"Proof of Proposition ",
"We start from the gradient $^{-1/2}\\nabla Q^G_b(\\beta ^*)=\\lbrace G ( - \\varepsilon /b ) - \\tau \\rbrace $ with $=^{-1/2} $ .",
"Let $_{}$ be the conditional expectation given $$ .",
"By integration by parts, $_{}G ( -\\varepsilon /b ) & = \\int _{-\\infty }^\\infty G(-t/b ) \\, {\\rm d} F_{\\varepsilon | } (t) = \\int _{-\\infty }^\\infty G(u) F_{\\varepsilon | } (- b u ) \\, {\\rm d} u.",
"$ Applying a Taylor expansion with integral remainder on $F_{\\varepsilon | } (- b u )$ yields $F_{\\varepsilon | } (- b u ) &= F_{\\varepsilon | }(0) + \\sum _{\\ell =1}^{\\nu -1} F_{\\varepsilon | }^{(\\ell )} (0) \\frac{(- b u)^\\ell }{\\ell !}",
"+ \\frac{(-b u)^{\\nu -1}}{(\\nu -2)!}",
"\\int _0^1 (1-w)^{\\nu -2} \\big \\lbrace F_{\\varepsilon | }^{(\\nu -1)} (- b u w) - F_{\\varepsilon | }^{(\\nu -1)} (0) \\bigr \\rbrace {\\rm d} w \\\\&= \\tau + \\sum _{\\ell =0}^{\\nu -2} f_{\\varepsilon | }^{(\\ell )} (0) \\frac{(- b u)^{\\ell +1}}{(\\ell +1)!}",
"+ \\frac{(- b u)^{\\nu -1}}{(\\nu -2)!}",
"\\int _0^1 (1-w)^{\\nu -2} \\big \\lbrace f_{\\varepsilon | }^{(\\nu -2)} (- b u w) - f_{\\varepsilon | }^{(\\nu -2)} (0) \\bigr \\rbrace {\\rm d} w .$ Recall that $G$ is a kernel of order $\\nu \\ge 4$ (an even integer) and $\\kappa ^G_\\nu = \\int _{-\\infty }^\\infty |u^\\nu G (u) |\\, {\\rm d}u <\\infty $ .",
"Substituting the above expansion into (REF ), we obtain $_{}G ( -\\varepsilon /b ) = \\tau - \\frac{b^{\\nu -1}}{(\\nu -2)!}",
"\\int _{-\\infty }^\\infty \\int _0^1 u^{\\nu -1} G(u) (1-w)^{\\nu -2} \\big \\lbrace f_{\\varepsilon | }^{(\\nu -2)} (- b u w) - f_{\\varepsilon | }^{(\\nu -2)} (0) \\bigr \\rbrace {\\rm d} w {\\rm d} u .\\nonumber $ Furthermore, by the Lipschitz continuity of $f_{\\varepsilon | }^{(\\nu -2)}(\\cdot )$ around 0, $| _{}G ( -\\varepsilon /b ) - \\tau | & \\le \\frac{L_{\\nu -2} b^{\\nu } }{(\\nu -2)!}",
"\\int _{-\\infty }^\\infty \\int _0^1 | u^{\\nu } G(u)| (1-w)^{\\nu -2} w \\,{\\rm d} w {\\rm d} u \\nonumber \\\\& = B(2,\\nu -1) L_{\\nu -2} \\kappa ^G_\\nu \\, b^{\\nu } / (\\nu -2)!",
", \\nonumber $ where $B(x,y) := \\int _0^1 t^{x-1} (1-t)^{y-1} {\\rm d}t$ denotes the beta function.",
"In particular, $B(2,\\nu -1)= \\Gamma (2)\\Gamma (\\nu -1)/\\Gamma (\\nu +1)= (\\nu -2)!/\\nu !$ .",
"Putting together the pieces yields $\\Vert ^{-1/2} \\nabla Q^G_b(\\beta ^*) \\Vert _2 & = \\sup _{\\in \\mathbb {S}^{p-1}} \\, _{} \\big \\lbrace G(-\\varepsilon /b ) - \\tau \\big \\rbrace ^\\le L_{\\nu -2} \\kappa ^G_\\nu \\, b^\\nu / \\nu !",
".",
"\\nonumber $ Turning to the Hessian, note that $& \\big \\Vert ^{-1/2} \\nabla ^2 Q^G_b(\\beta ^*) ^{-1/2} - _0 \\big \\Vert _2 = \\Bigg \\Vert \\int _{-\\infty }^\\infty G(u) \\big \\lbrace f_{\\varepsilon | } (- b u) - f_{\\varepsilon | }(0) \\big \\rbrace {\\rm d}u \\, ^\\Vert _2 .\\nonumber $ Applying a similar Taylor expansion as above, we have $f_{\\varepsilon | } (t) &= f_{\\varepsilon | }(0) + \\sum _{\\ell =1}^{\\nu -1} f_{\\varepsilon | }^{(\\ell )} (0) \\frac{t^\\ell }{\\ell !}",
"+ \\frac{t^{\\nu -1}}{(\\nu -2)!}",
"\\int _0^1 (1-w)^{\\nu -2} \\big \\lbrace f_{\\varepsilon | }^{(\\nu -1)} (t w ) - f_{\\varepsilon | }^{(\\nu -1)} (0) \\bigr \\rbrace {\\rm d} w .",
"$ Under Conditions and , it follows that $& \\big \\Vert ^{-1/2} \\nabla ^2 Q^G_b(\\beta ^*) ^{-1/2} - _0 \\big \\Vert _2 \\\\& \\le \\frac{b^{\\nu -1}}{(\\nu -2)!}",
"\\sup _{, \\delta \\in \\mathbb {S}^{p-1}} \\int _{-\\infty }^\\infty \\int _0^1 u^{\\nu -1} G(u) (1-w)^{\\nu -2}\\big \\lbrace f_{\\varepsilon | }^{(\\nu -1)} (- b u w) - f_{\\varepsilon | }^{(\\nu -1)} (0) \\bigr \\rbrace {\\rm d} w\\, {\\rm d}u \\, \\big \\langle , \\big \\rangle \\big \\langle , \\delta \\big \\rangle \\\\& \\le \\frac{b^{\\nu -1}}{(\\nu -1)!}",
"\\sup _{, \\delta \\in \\mathbb {S}^{p-1}} \\int _{-\\infty }^\\infty | u^{\\nu -1} G(u) | \\sup _{|t| \\le b |u|} \\big | f_{\\varepsilon | }^{(\\nu -1)} (t) - f_{\\varepsilon | }^{(\\nu -1)} (0) \\bigr | \\, {\\rm d}u \\cdot | \\langle , \\rangle \\langle , \\delta \\big \\rangle | \\\\& \\le \\frac{C_G b^{\\nu -1 }}{(\\nu -1)!}",
"\\sup _{ \\in \\mathbb {S}^{p-1}} \\big \\langle , \\big \\rangle ^2 = \\frac{C_G}{(\\nu -1)!}",
"b^{\\nu -1 } .$ This completes the proof.",
"$\\Box $"
],
[
"Proof of Proposition ",
"Consider the change of variable $\\delta = ^{1/2} (\\beta - \\beta ^*)$ , so that $\\beta \\in \\Theta ^*(r)$ is equivalent to $\\delta \\in \\mathbb {B}^p(r)$ .",
"Write $_i = ^{-1/2} _i \\in ^p$ , which are isotropic random vectors, and define $_n(\\delta ) = ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} = \\frac{1}{n} \\sum _{i=1}^nG_b(\\varepsilon _i - _i^) _i _i^ \\quad (\\delta ) = \\bigl \\lbrace _n(\\delta ) \\bigr \\rbrace .$ For any $\\epsilon \\in (0,r)$ , there exists an $\\epsilon $ -net $\\lbrace \\delta _1, \\ldots , \\delta _{d_\\epsilon }\\rbrace $ with $d_\\epsilon \\le (1+ 2r/\\epsilon )^p$ satisfying that, for each $\\delta \\in \\mathbb {B}^p(r)$ , there exists some $1\\le j\\le d_\\epsilon $ such that $\\Vert \\delta - \\delta _j \\Vert _2 \\le \\epsilon $ .",
"Hence, $& \\Vert _n(\\delta ) - (\\delta ) \\Vert _2 \\nonumber \\\\& \\le \\Vert _n(\\delta ) - _n(\\delta _j) \\Vert _2 + \\Vert _n(\\delta _j) - (\\delta _j) \\Vert _2 + \\Vert (\\delta _j) - (\\delta ) \\Vert _2 \\nonumber \\\\& =: I_1(\\delta ) + I_2 (\\delta _j)+ I_3(\\delta ).",
"\\nonumber $ For $I_1(\\delta )$ , note that $G_b(u) = (1/b)G(u/b)$ is Lipschitz continuous, i.e.",
"$|G_b(u) -G_b(v)| \\le L^G b^{-2} |u-v|$ for all $u,v \\in $ .",
"It follows that $I_1(\\delta ) & \\le \\sup _{, \\in \\mathbb {S}^{p-1}} \\frac{1}{n} \\sum _{i=1}^n| G_b(\\varepsilon _i - _i^) - G_b(\\varepsilon _i -_i^_j ) | \\cdot | _i^\\cdot _i^| \\nonumber \\\\& \\le L^G b^{-2}\\sup _{, \\in \\mathbb {S}^{p-1}} \\frac{1}{n} \\sum _{i=1}^n| _i^ \\delta -\\delta _j ) \\cdot _i^\\cdot _i^| \\nonumber \\\\& \\le L^G b^{-2} \\epsilon \\underbrace{ \\sup _{\\in \\mathbb {S}^{p-1} } \\frac{1}{n} \\sum _{i=1}^n| _i^|^3 }_{=: M_{n,3}} .",
"$ Next, we use the standard covering argument to bound $M_{n,3}$ .",
"Given $\\epsilon _1 \\in (0,1)$ , let $_1$ be an $\\epsilon _1$ -net of the unit sphere $\\mathbb {S}^{p-1}$ with $d_{\\epsilon _1} := | _1 | \\le (1+2/\\epsilon _1)^p$ such that for every $\\in \\mathbb {S}^{p-1}$ , there exists some $\\in _1$ satisfying $\\Vert - \\Vert _2 \\le \\epsilon _1$ .",
"Define the (standardized) design matrix $_n = n^{-1/3}(_1,\\ldots , _n)^^{n\\times p}$ , so that $M_{n,3} = \\sup _{\\in \\mathbb {S}^{p-1}} \\Vert _n \\Vert _3^3$ .",
"By the triangle inequality, $\\Vert _n \\Vert _3 & \\le \\Vert _n \\Vert _3 + \\Vert _n ( - ) \\Vert _3 \\nonumber \\\\& = \\Vert _n \\delta \\Vert _3 +\\bigg ( \\frac{1}{n} \\sum _{i=1}^n| _i^ - ) |^3 \\bigg )^{1/3} \\le \\Vert _n \\Vert _3 + \\epsilon _1 M_{n,3}^{1/3} .",
"\\nonumber $ Taking the maximum over $\\in _1$ , and then taking the supremum over $\\in \\mathbb {S}^{p-1}$ , we arrive at $M_{n,3} \\le (1-\\epsilon _1)^{-3} \\widetilde{M}_{n, 3} , $ where $ \\widetilde{M}_{n, 3} = \\max _{\\in _1 } (1/n) \\sum _{i=1}^n| _i^|^3$ .",
"For every $\\in _1$ , note that $( |_i^|^3 \\ge y\\rbrace \\le 2e^{-y^{2/3}/(2\\upsilon _1^2)}$ for any $y>0$ .",
"Hence, by inequality (3.6) in [58] with $s=2/3$ , we obtain that for any $z\\ge 3$ , $\\frac{1}{n} \\sum _{i=1}^n|_i^|^3 \\le |^|^3 + C_1 \\Biggl ( \\sqrt{\\frac{z }{n}} + \\frac{z^{3/2}}{n} \\Bigg )$ with probability at least $1-e^{3-z}$ , where $C_1>0$ is a constant depending only on $\\upsilon _1$ .",
"Taking the union bound over all vectors $$ in $_1$ yields that, with probability at least $1-d_{\\epsilon _1} e^{3-z} \\ge 1- e^{3+p\\log (1+2/\\epsilon _1) - z}$ , $\\widetilde{M}_{n,3} \\le \\mu _3 + C_1\\Biggl ( \\sqrt{\\frac{z }{n}} + \\frac{z^{3/2}}{n} \\Bigg )$ where $\\mu _3 = \\sup _{\\in \\mathbb {S}^{p-1} } |^|^3$ .",
"Reorganizing the terms, we have $\\widetilde{M}_{n,3} \\le \\mu _3 + C_1 \\Bigg [ \\sqrt{\\frac{p\\log (1+2/\\epsilon _1) +4+t }{n}} + \\frac{\\lbrace p\\log (1+2/\\epsilon _1) + 4 + t \\rbrace ^{3/2}}{n} \\Bigg ] $ with probability at least $1- e^{-t}/2$ .",
"Hence, taking $\\epsilon =1/8$ in (REF ) and (REF ) implies $M_{n,3} \\le 1.5\\mu _3 + 1.5 C_1 \\Bigg \\lbrace \\sqrt{\\frac{3p+ 4 +t }{n}} + \\frac{ ( 3p + 4+ t )^{3/2}}{n} \\Bigg \\rbrace .",
"\\nonumber $ Plugging the above bound into (REF ) yields that as long as $n \\gtrsim p+t$ , $\\sup _{\\delta \\in \\mathbb {B}^p(r)} I_1(\\delta ) \\lesssim (p + t)^{1/2} b^{-2} \\epsilon $ with probability at least $1- e^{-t}/2$ .",
"For $I_3(\\delta )$ , it can be similarly obtained that $I_3(\\delta ) \\le L^G b^{-2} \\sup _{, \\in \\mathbb {S}^{p-1}} | ^ \\delta - \\delta _j ) \\cdot ^\\cdot ^| \\le L^G \\mu _3 \\, b^{-2}\\epsilon $ uniformly over all $\\delta \\in \\mathbb {B}^p(r)$ .",
"Turning to $I_2(\\delta _j)$ , note that $_n(\\delta _j) - (\\delta _j ) = (1/n) \\sum _{i=1}^n(1-) \\phi _{ij} _i _i^, where $ ij = Gb(i - ij )$ satisfy $ |ij| Gu b-1$ and{\\begin{@align}{1}{-1}\\bigl ( \\phi _{ij}^2 | _i \\bigr ) = \\frac{1}{b^2} \\int _{-\\infty }^\\infty G^2 \\biggl ( \\frac{ \\langle _i , \\delta \\rangle - t}{b} \\biggr ) f_{\\varepsilon _i | _i } (t) \\, {\\rm d} t = \\frac{1}{b} \\int _{-\\infty }^\\infty G^2(u) f_{\\varepsilon _i | _i } (_i^- b u ) \\, {\\rm d} u \\le \\frac{ \\bar{f} m_{G^2 } }{b} \\nonumber \\end{@align}}almost surely, where $ mG2 := -G2(u) du <$.Given $ 2(0,1/2)$, there exits an $ 2$-net $$ of the sphere $ Sp-1$ with $ ||(1+2/2)p$ such that $ n($\\delta $ j) - ($\\delta $ j )2 (1-22)-1 | n($\\delta $ j) - ($\\delta $ j ) }|$.",
"Given $$ and $ k=2,3,...$, we bound the higher order moments of $ ij (i)2$ by{\\begin{@align}{1}{-1}|\\phi _{ij} (_i^)^2 |^k & \\le \\bar{f} m_{G^2 } b^{-1} \\cdot (\\kappa ^G_u b^{-1})^{k-2} \\upsilon _1^{2k} \\cdot 2k \\int _0^\\infty \\bigl ( |_i^| \\ge \\upsilon _1 u \\bigr ) u^{2k-1} {\\rm d} u \\nonumber \\\\& \\le \\bar{f} m_{G^2 } b^{-1} \\cdot ( \\kappa ^G_u b^{-1})^{k-2} \\upsilon _1^{2k} \\cdot 4k \\int _0^\\infty u^{2k-1} e^{-u^2/2} {\\rm d} u \\nonumber \\\\& \\le \\bar{f} m_{G^2} b^{-1} \\cdot ( \\kappa ^G_u b^{-1})^{k-2} \\upsilon _1^{2k} \\cdot 2^{k+1} k!",
".\\nonumber \\end{@align}}In particular, $ ij2 (i)4 16 14 f mG2 b-1 $, and $ |ij (i)2 |kk!2 16 14 f mG2 b-1 (212Gu b-1 )k-2$ for $ k3$.",
"Applying Bernstein^{\\prime }s inequality and the union bound, we find that for any $ u0$,{\\begin{@align}{1}{-1}& \\Vert _n(\\delta _j) - (\\delta _j ) \\Vert _2 \\nonumber \\\\& \\le \\frac{1}{1-2\\epsilon _2} \\max _{\\in } \\Biggl | \\frac{1}{n} \\sum _{i=1}^n(1-) \\phi _{ij} (_i^)^2 \\Biggr | \\le \\frac{ 2\\upsilon _1^2}{1- 2\\epsilon _2} \\Biggl ( 2 \\sqrt{2 \\bar{f} m_{G^2 } \\frac{u}{n b }} +\\kappa ^G_u \\frac{u}{n b} \\Biggr ) \\nonumber \\end{@align}}with probability at least $ 1-2 (1+2/2)p e-u = 1 - (1/2) e(4) + p(1+2/2) - u$.",
"Setting $ 2=2/(e3-1)$ and $ u= (4) + 3p + v$, it follows that with probability at least $ 1 -e-v/2$,{\\begin{@align}{1}{-1}I_2(\\delta _j) \\lesssim \\sqrt{ \\frac{ p + v }{n b } } + \\frac{ p + v }{n b} .",
"\\nonumber \\end{@align}}Once again, taking the union bound over $ j=1,..., d$ and setting $ v= p (1 + 2r/)+ t$, we obtain that with probability at least $ 1-de-v 1-e-t/2$,{\\begin{@align}{1}{-1}\\max _{1\\le j\\le N_{\\epsilon }} I_2(\\delta _j) \\lesssim \\sqrt{\\frac{p\\log (3 e r/\\epsilon )+t}{n b }} + \\frac{p\\log (3e r/\\epsilon )+t}{n b} .",
"\\end{@align}}$ Finally, combining (REF ), (REF ) and (), and taking $\\epsilon = r/n \\in (0, r)$ in the beginning of the proof, we conclude that with probability at least $1- e^{-t}$ , $& \\sup _{\\beta \\in \\Theta ^*(r)}\\big \\Vert ^{-1/2} \\big \\lbrace \\nabla ^2 \\hat{Q}^G_b(\\beta ) - \\nabla ^2 Q^G_b(\\beta ) \\big \\rbrace \\nonumber ^{-1/2} \\big \\Vert _2 \\lesssim \\sqrt{\\frac{p \\log n + t}{n b }} + \\frac{p\\log n+t}{n b } + \\frac{(p + t)^{1/2} r }{n b^2} \\nonumber $ as long as $n\\gtrsim p+t$ .",
"This completes the proof.",
"$\\Box $"
],
[
"Proof of Proposition ",
"Define the stochastic process $\\Delta _b(\\beta ) = ^{-1/2} [ \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\rbrace - ( \\beta - \\beta ^*) ]$ .",
"By the triangle inequality, $\\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) \\Vert _2 \\le \\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) \\Vert _2 + \\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) - \\Delta _b(\\beta ) \\Vert _2$ Recall that $_0 = ^{-1/2} ^{-1/2} = \\lbrace f_{\\varepsilon |}(0) ^$ .",
"For the first term on the right-hand side, using the mean value theorem for vector-valued functions yields $\\Delta _b(\\beta ) = \\Biggl \\lbrace ^{-1/2} \\int _0^1 \\nabla ^2 Q^G_b\\bigl ( (1-s) \\beta ^* + s \\beta \\bigr ) \\, {\\rm d}s \\, ^{-1/2} - _0 \\Biggr \\rbrace ^{1/2} ( \\beta -\\beta ^* ) .$ By a change of variable $\\delta = ^{1/2}(\\beta - \\beta ^*)$ , $\\nabla ^2 Q^G_b\\bigl ( (1-s) \\beta ^* + s \\beta \\bigr ) = \\int _{-\\infty }^\\infty G(u) f_{\\varepsilon | } (s ^- b u ) \\, {\\rm d}u \\cdot ^$ For every $s \\in [0,1]$ and $u\\in $ , it ensures from $f_{\\varepsilon | }(\\cdot )$ being Lipschitz that $| f_{\\varepsilon | } (s ^- b u ) - f_{\\varepsilon | } ( - b u ) | \\le l_0 s \\cdot | ^| .$ Moreover, by the Taylor expansion (REF ), $f_{\\varepsilon | } ( - b u ) & = f_{\\varepsilon | }(0) + \\sum _{\\ell =1}^{\\nu -1} f_{\\varepsilon | }^{(\\ell )}(0) \\frac{( - b u )^\\ell }{\\ell !}",
"+ \\frac{( - b u)^{\\nu -1}}{(\\nu -2)!}",
"\\int _0^1 (1-w)^{\\nu -2} \\bigl \\lbrace f_{\\varepsilon | }^{(\\nu -1)}( - b u w ) -f_{\\varepsilon | }^{(\\nu -1)}(0)\\bigr \\rbrace \\, {\\rm d}w.$ Consequently, $& \\Bigg \\Vert ^{-1/2} \\int _0^1 \\nabla ^2 Q^G_b\\bigl ( (1-s) \\beta ^* + s\\beta \\bigr )\\, {\\rm d}s \\, ^{-1/2} - _0 \\Bigg \\Vert _2 \\\\&\\le \\Bigg \\Vert \\frac{b^{\\nu -1}}{(\\nu -2)!}",
"\\int _{-\\infty }^\\infty \\int _0^1 u^{\\nu -1} G(u) (1-w)^{\\nu -2} \\bigl \\lbrace f_{\\varepsilon | }^{(\\nu -1)}( - b u w ) -f_{\\varepsilon | }^{(\\nu -1)}(0)\\bigr \\rbrace \\, {\\rm d}w {\\rm d}u \\cdot ^\\Vert _2 \\\\&~~~~ + \\frac{l_0}{2} \\cdot \\big \\Vert \\, |^| \\cdot ^\\Vert _2 \\\\& \\le \\frac{C_G b^{\\nu -1}}{(\\nu -1)!}",
"\\sup _{\\in \\mathbb {S}^{p-1}} \\big \\langle , \\big \\rangle ^2 + \\frac{l_0}{2} \\sup _{\\in \\mathbb {S}^{p-1}} \\big ( | ^| \\langle , \\rangle ^2 \\big ) \\\\& \\le \\Bigg ( \\frac{ C_G }{(\\nu -1)! }",
"b^{\\nu -1} + 0.5 l_0 \\mu _3\\Vert \\delta \\Vert _2 \\Bigg ) .$ Taking the supremum over $\\beta \\in \\Theta ^*(r)$ , or equivalently $\\delta \\in \\mathbb {B}^p(r)$ , yields $\\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) \\Vert \\le \\Bigg ( \\frac{ C_G }{(\\nu -1)! }",
"b^{\\nu -1} + 0.5 l_0 \\mu _3 r \\Bigg ) r \\lesssim (b^{\\nu -1} + r ) r .",
"\\nonumber $ For the stochastic term $ \\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) - \\Delta _b(\\beta ) \\Vert _2$ , following the proof of Theorem REF , it can be similarly shown that with probability at least $1-e^{-t}$ , $\\sup _{\\beta \\in \\Theta ^*(r)} \\Vert \\Delta _b(\\beta ) - \\Delta _b(\\beta ) \\Vert _2 \\lesssim r \\sqrt{\\frac{p+t}{n b}} \\nonumber $ as long as $\\sqrt{(p+t)/n} \\lesssim b\\lesssim 1$ .",
"Combining the last two displays completes the proof of the claim (REF ).",
"$\\Box $"
],
[
"Proof of Theorem ",
"Step 1 (Consistency of the initial estimate).",
"First, note that the consistency of the initial estimator $\\beta $ —namely, $\\beta $ lies in a local neighborhood of $\\beta ^*$ with high probability, is a direct consequence of Theorem REF .",
"Given a non-negative kernel $K(\\cdot )$ and for any $t>0$ , the ensuing estimator $\\beta $ satisfies $\\Vert \\beta - \\beta ^* \\Vert _{} \\le r_n \\asymp \\sqrt{\\frac{p+t}{n}} $ with probability at least $1-2e^{-t}$ as long as $ (\\frac{p+t}{n})^{1/2} \\lesssim h \\lesssim (\\frac{p+t}{n})^{1/4}$ .",
"Provided that the sample Hessian $\\nabla ^2 \\hat{Q}^G_b(\\beta )$ is invertible, by the definition of $\\hat{\\beta }$ we obtain $^{1/2} ( \\hat{\\beta }- \\beta ) & = - \\bigl \\lbrace ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} \\bigr \\rbrace ^{-1} ^{-1/2}\\nabla \\hat{Q}^G_b(\\beta ) \\nonumber \\\\& = - \\bigl \\lbrace ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} \\bigr \\rbrace ^{-1} \\bigl [ ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\bigr \\rbrace - ^{-1/2} (\\beta - \\beta ^* ) \\bigr ] \\nonumber \\\\& ~~~~~ - \\bigl \\lbrace ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} \\bigr \\rbrace ^{-1} \\bigl \\lbrace ^{-1/2} (\\beta - \\beta ^* ) + ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* )\\bigr \\rbrace ,\\nonumber $ or equivalently, $^{1/2} (\\hat{\\beta }- \\beta ^*) & = ^{1/2} (\\beta -\\beta ^* ) - _0^{-1} _0 ^{1/2} (\\beta - \\beta ^* ) - _0^{-1} ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* ) \\nonumber \\\\& ~~~~ - _0^{-1} ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) - (\\beta - \\beta ^* ) \\bigr \\rbrace , \\nonumber $ where $_0 = \\lbrace f_{\\varepsilon |}(0) ^ = ^{-1/2} ^{-1/2}$ and $_0 := ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} $ .",
"It follows that $& \\bigl \\Vert ^{1/2} ( \\hat{\\beta }- \\beta ^* ) + ^{1/2} ^{-1} \\nabla \\hat{}_b^G(\\beta ^* ) \\bigr \\Vert _2= \\bigl \\Vert ^{1/2} (\\hat{\\beta }- \\beta ^*) + _0^{-1} ^{-1/2}\\nabla \\hat{}_b^G(\\beta ^* ) \\bigr \\Vert _2 \\nonumber \\\\& \\le \\underbrace{ \\Vert _p - _0^{-1} _0 \\Vert _2 }_{ {\\rm see~Proposition~\\ref {prop:Hessian.uniform}} }\\cdot \\bigl \\lbrace \\Vert \\beta -\\beta ^* \\Vert _{} + \\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* ) \\Vert _2 \\bigr \\rbrace \\\\&~~~~~ + \\Vert _0^{-1} \\Vert _2 \\cdot \\underbrace{ \\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) - (\\beta - \\beta ^* ) \\bigr \\rbrace \\big \\Vert _2 }_{ {\\rm see~Proposition~\\ref {prop:gradient.uniform}}}.",
"\\nonumber $ According to the above bound, we need to further control the following three terms: $\\Vert _p - _0^{-1} _0 \\Vert _2 , ~~\\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* ) \\Vert _2~\\mbox{ and }~\\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) -(\\beta - \\beta ^* ) \\bigr \\rbrace \\big \\Vert _2.$ Step 2 (Invertibility of the sample Hessian $\\nabla ^2 \\hat{Q}^G_b(\\beta )$ ).",
"Recall that $_0 = ^{-1/2} \\nabla ^2 \\hat{Q}^G_b(\\beta ) ^{-1/2} $ and $_0 = ^{-1/2} ^{-1/2}$ .",
"By the triangle inequality, $& \\Vert _0 - _0 \\Vert _2 \\le \\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla ^2 \\hat{Q}^G_b(\\beta ) - \\nabla ^2 Q^G_b( \\beta ^*) \\bigr \\rbrace ^{-1/2} \\big \\Vert _2 + \\big \\Vert ^{-1/2} \\nabla ^2 Q^G_b( \\beta ^*) ^{-1/2} - _0 \\big \\Vert _2 .",
"\\nonumber $ Let the bandwidth $b$ satisfy $ \\max \\lbrace \\frac{p\\log n + t}{n} , n^{-1/2}\\rbrace \\lesssim b \\lesssim 1$ .",
"Then, applying Propositions and with $r=r_n$ yields that with probability $1-3e^{-t}$ , $& \\Vert _0 - _0 \\Vert _2 \\le \\delta _n \\asymp \\sqrt{\\frac{p\\log n + t}{ nb }} + b^{\\nu -1} .",
"\\nonumber $ Note that, under Condition REF , $0< {f} \\le \\lambda _{\\min }(_0) \\le \\lambda _{\\max }(_0) \\le \\bar{f}$ .",
"For sufficiently large $n$ , this implies $\\Vert _0^{-1} _0 - _p \\Vert _2 \\le {f}^{-1} \\delta _n < 1$ , and hence $\\Vert _0^{-1} _0 - _p \\Vert _2 \\le \\frac{\\delta _n}{ {f} - \\delta _n } ~~\\mbox{ and }~~ \\Vert _0^{-1} \\Vert _2 \\le \\frac{1}{{f} - \\delta _n} .$ Step 3 (Controlling the score).",
"In view of (REF ), it remains to control $\\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* )\\Vert _2$ and $\\Vert ^{-1/2} \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\rbrace - ^{-1/2} (\\beta - \\beta ^* ) \\Vert _2$ .",
"For the latter, applying the concentration bound (REF ) and Proposition yields that, with probability at least $1-3e^{-t}$ , $\\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\bigr \\rbrace - ^{-1/2} (\\beta - \\beta ^* ) \\big \\Vert _2 \\lesssim \\frac{p+ t}{n b^{1/2}} + b^{\\nu -1} \\sqrt{\\frac{p+t}{n}} $ as long as $ (\\frac{p+t}{n})^{1/2} \\lesssim b \\lesssim 1$ .",
"Turning to $\\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* )\\Vert _2$ , it follows from Lemma REF and Proposition that with probability at least $1-e^{-t}$ , $\\Vert ^{-1/2} \\nabla \\hat{Q}^G_b(\\beta ^* ) \\Vert _2 \\lesssim \\sqrt{\\frac{p+ t}{n}} + b^\\nu .",
"$ Finally, combining the bounds (REF )–(REF ), we conclude that with probability at least $1-5e^{-t}$ , $\\big \\Vert ^{-1/2} \\bigl \\lbrace \\nabla \\hat{Q}^G_b(\\beta ) - \\nabla \\hat{Q}^G_b(\\beta ^* ) \\bigr \\rbrace - ^{-1/2} (\\beta - \\beta ^* ) \\big \\Vert _2\\lesssim \\Biggl ( \\sqrt{\\frac{p\\log n+ t}{n b}}+ b^{\\nu -1} \\Biggr ) \\Bigg ( \\sqrt{\\frac{p+t}{n}} + b^\\nu \\Bigg ) , \\nonumber $ provided that $ \\max \\bigl \\lbrace \\frac{p\\log n + t}{n} , (\\frac{p + t}{n})^{1/2} \\bigr \\rbrace \\lesssim b \\lesssim 1$ .",
"Under the sample size requirement $n\\gtrsim p \\,(\\log n)^2+t$ and additional constraint $b \\lesssim (\\frac{p+t}{n})^{1/(2\\nu )}$ , this leads to the claimed bound (REF ).",
"$\\Box $"
],
[
"Additional Simulation Results",
"In this section, we present additional results for the numerical studies as described in Section in Figures REF –REF ."
]
] | 2012.05187 |
[
[
"First observation of the decay $B_s^0 \\to K^-\\mu^+\\nu_\\mu$ and a\n measurement of $|V_{ub}|/|V_{cb}|$"
],
[
"Abstract The first observation of the suppressed semileptonic $B_s^0 \\to K^-\\mu^+\\nu_\\mu$ decay is reported.",
"Using a data sample recorded in {\\it pp} collisions in 2012 with the LHCb detector, corresponding to an integrated luminosity of 2 $\\mathrm{fb}^{-1}$, the branching fraction \\mbox{$\\mathcal{B}(B_s^0 \\to K^-\\mu^+\\nu_\\mu)$} is measured to be $(1.06\\pm0.05~(\\mathrm{stat})\\pm0.08~(\\mathrm{syst}))\\times 10^{-4}$, where the first uncertainty is statistical and the second one represents the combined systematic uncertainties.",
"The decay $B_s^0 \\to D_s^-\\mu^+\\nu_\\mu$, where $D_s^-$ is reconstructed in the final state $K^+K^-\\pi^-$, is used as a normalization channel to minimize the experimental systematic uncertainty.",
"Theoretical calculations on the form factors of the $B_s^0 \\to K^-$ and $B_s^0 \\to D_s^-$ transitions are employed to determine the ratio of the CKM matrix elements ${|V_{ub}|}/{|V_{cb}|}$ at low and high $B_s^0 \\to K^-$ momentum transfer."
],
[
"0.5em"
]
] | 2012.05143 |
[
[
"Long-time behaviour of a model for p62-ubiquitin aggregation in cellular\n autophagy"
],
[
"Abstract The qualitative behavior of a recently formulated ODE model for the dynamics of heterogenous aggregates is analyzed.",
"Aggregates contain two types of particles, oligomers and cross-linkers.",
"The motivation is a preparatory step of cellular autophagy, the aggregation of oligomers of the protein p62 in the presence of ubiquitin cross-linkers.",
"A combination of explicit computations, formal asymptotics, and numerical simulations has led to conjectures on the bifurcation behavior, certain aspects of which are proven rigorously in this work.",
"In particular, the stability of the zero state, where the model has a smoothness deficit is analyzed by a combination of regularizing transformations and blow-up techniques.",
"On the other hand, in a different parameter regime, the existence of polynomially growing solutions is shown by Poincar\\'e compactification, combined with a singular perturbation analysis ."
],
[
"Introduction",
"A preparatory step of cellular autophagy is the aggregation of cellular waste material before inclusion in an autophagosome and, later, a lysosome, which are vesicular structures, where the waste is eventually decomposed.",
"In vitro studies of the evolution of heterogeneous aggregates of the proteins p62 and ubiquitin [8] have motivated the formulation of a mathematical model of this process [1].",
"The model has the form of an ODE system, which shows different qualitative behaviour in three different regions of parameter space.",
"This statement is based on formal asymptotics and numerical simulations carried out in [1].",
"Since some of these observations are not accessible to standard dynamical systems methods, it is the purpose of this work to provide a rigorous analysis.",
"The model is based on the assumption that ubiquitinated waste material serves as a cross-linker between p62 oligomers of a fixed size $n\\ge 3$ , where each monomer can serve as a binding site for cross-linking.",
"It describes the evolution of the size of aggregates through the evolution of three parameters $p$ , $q$ and $r$ , where $p$ represents the number of one-hand bound ubiquitin links in the aggregate (in green in Figs.",
"REF and REF ), $q$ represents the number of both-hand bound cross-links (in red in Figs.",
"REF and REF ), and $r$ represents the number of p62$_n$ oligomers in the aggregate (in black in Figs.",
"REF and REF with $n=5$ ).",
"Since we are interested in large aggregates, the variables $(p,q,r)$ are considered as continuous after an appropriate scaling.",
"The state space is a subset of the positive octant of $\\mathbb {R}^3$ determined by two constraints: For a connected aggregate the number of two-hand bound cross-links has to be at least the number of p62 oligomers minus one.",
"In the continuous description this becomes the constraint $q\\ge r$ .",
"Since the total number of binding sites on the p62 oligomers in an aggregate is $nr$ , the number of free binding sites is equal to $nr-p-2q$ , which has to be nonnegative.",
"The dynamics of an aggregate is governed by the basic binding and unbinding reactions between cross-linkers and p62 oligomers.",
"Since the reaction rates depend on the state of the reaction partners and of the aggregate, six different reactions have been considered in [1] (see Figs.",
"REF and REF ).",
"The models for the reaction rates are based on the law of mass action.",
"However, since the shape of an aggregate is not described unambiguously by the parameters $(p,q,r)$ , some additional empirical modeling assumptions are required.",
"Figure: Left: Reaction 1: addition of a free cross-linker to the aggregate.",
"Right: Reaction 2: addition of a p62 5 _5 oligomer to the aggregate.Figure: Left: Reaction 3 is a rearrangement making the aggregate more cohesive.",
"Right: the breaking of a cross-link bound can lead either to the reverse of Reaction 2 of of Reaction 3.Reaction 1 is the addition of a free cross-linker to the aggregate.",
"This is a second order reaction with a rate proportional to the concentration of free cross-linkers and to the number of free binding sites on oligomers.",
"Since the supply of free cross-linkers and oligomers has been modeled as not limiting in [1], the reaction rate is written as $\\kappa _1(nr-p-2q)$ with rate constant $\\kappa _1$ , which can be seen as proportional to the cross-linker concentration, modeled as constant.",
"Similarly Reaction 2, the addition of a free oligomer to the aggregate, is modeled as a first order reaction with rate $\\kappa _2 p$ proportional to the number of one-hand bound cross-linkers.",
"Reaction 3 is consolidating the aggregate by building an additional cross-link using a so far only one-hand bound cross-linker.",
"Its rate is $\\kappa _3 p(nr-p-2q)$ .",
"For the reverse of Reaction 1, the rate $\\kappa _{-1}p$ should not be a surprise.",
"The reverses of Reactions 2 and 3 are actually the same reaction with rate $\\kappa _- q$ , but with possibly different outcomes.",
"Therefore we write their rate constants as $\\kappa _{-2}:= \\kappa _- \\alpha $ and $\\kappa _{-3}:= \\kappa _- (1-\\alpha )$ , with $\\alpha =\\alpha (q,r)\\in [0,1]$ .",
"The reverse of Reaction 2, i.e.",
"loss of an oligomer, never happens in a fully connected aggregate with $nr=2q$ .",
"It always happens in a minimally connected aggregate with $q=r$ .",
"This motivates the choice $\\alpha = \\frac{nr-2q}{(n-2)r}$ .",
"Concerning the outcome of this reverse reaction, it has to be taken into account that the loss of an oligomer might also mean a loss of one-hand bound cross-links attached to it.",
"This produces the loss term $\\kappa _- q \\frac{(n-1)p}{(n-2)r}$ (see [1] for details).",
"It is now straightforward to write down the ODE problem governing the evolution of the state variables: $ \\begin{aligned}\\dot{p} &= (\\kappa _1 - \\kappa _3 p) (nr-p-2q) + \\kappa _{-}q \\left( 1 - \\frac{(n-1)p}{(n-2)r}\\right) - (\\kappa _2 + \\kappa _{-1})p \\,, &\\qquad p(0)=p_0\\,,\\\\\\dot{q} &= \\kappa _2 p + \\kappa _3 p(nr-p-2q) - \\kappa _{-}q\\,, &\\qquad q(0)=q_0\\,, \\\\\\dot{r} &= \\kappa _2 p - \\kappa _{-} q \\alpha (q,r), \\qquad \\alpha (q,r) = \\frac{nr -2q}{(n-2) r}\\,, &\\qquad r(0)=r_0\\,,\\end{aligned}$ with the inequalities $ nr - p - 2q \\ge 0 \\,, \\qquad q\\ge r \\,,$ implying $0 \\le \\alpha (q,r) \\le 1 \\,.$ We recall from [1] that for initial data $p_0,q_0,r_0>0$ satisfying (REF ), which we assume in the following, the initial value problem (REF ) has a unique, global solution propagating (REF ), the nonnegativity of the components, and in particular $r(t),q(t) > 0 \\,,\\qquad t\\ge 0 \\,.$ Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5,κ 1 =κ 2 =κ 3 =κ -1 =1\\kappa _1= \\kappa _2=\\kappa _3=\\kappa _{-1}=1, and κ - =0.6\\kappa _{-}=0.6, giving 0<α ¯<10<\\bar{\\alpha }< 1 and convergence to the nontrivial equilibrium.Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5, κ 1 =κ 2 =κ 3 =κ -1 =1\\kappa _1= \\kappa _2=\\kappa _3=\\kappa _{-1} = 1 ,and κ - =0.93\\kappa _{-}=0.93, giving α ¯>1\\bar{\\alpha }>1 and convergence to the zero steady state.Figure: Evolution of a state (p,q,r)(p,q,r) of initial size (2,4,3)(2,4,3) with parameters n=5n=5,κ 1 =κ 2 =κ 3 =κ -1 =1\\kappa _1= \\kappa _2=\\kappa _3=\\kappa _{-1} = 1, and κ - =0.2\\kappa _{-}=0.2, giving α ¯<0\\bar{\\alpha }<0 and a polynomially growing aggregate.The search for steady states [1] has suggested a splitting of the parameter space into three regions.",
"Besides the trivial steady state $(p,q,r)=(0,0,0)$ , only one other equilibrium may exist, which can be computed explicitly: $& \\bar{p} =\\frac{\\kappa _- A}{\\kappa _2} \\frac{1-\\bar{\\alpha }}{\\bar{\\alpha }} \\,,\\qquad \\bar{q} = A \\frac{1-\\bar{\\alpha }}{\\bar{\\alpha }^2} \\,, \\qquad \\bar{r} = \\frac{2A}{n-(n-2)\\bar{\\alpha }} \\frac{1-\\bar{\\alpha }}{\\bar{\\alpha }^2}\\,, \\\\&\\mbox{with}\\quad \\bar{\\alpha }= \\frac{n}{n-2} + \\frac{\\kappa _{-1} + \\kappa _1 - \\sqrt{(\\kappa _1 + \\kappa _{-1})^2 + 4 \\kappa _1 \\kappa _2 (n-1)}}{\\kappa _{-} (n-1)} \\,, \\quad A = \\frac{2\\kappa _1 \\kappa _2^2 (n-2)}{\\kappa _3 \\kappa _- (\\kappa _-(n-1)(n - (n-2)\\bar{\\alpha }) + 2\\kappa _{-1}(n-2))} \\,.\\nonumber $ Since $\\bar{\\alpha }= \\alpha (\\bar{q},\\bar{r})$ is the equilibrium value of $\\alpha $ , the nontrivial steady state is relevant only in the parameter region defined by $0<\\bar{\\alpha }<1$ .",
"It has been conjectured in [1] that in this parameter region $(\\bar{p},\\bar{q}, \\bar{r})$ is globally attracting, which has been supported by numerical simulations (see also Fig.",
"REF ).",
"Local stability could in principle be examined by linearization.",
"However, the complexity of the resulting formulas has been prohibitive.",
"Since $(\\bar{p},\\bar{q},\\bar{r})\\rightarrow (0,0,0)$ as $\\bar{\\alpha }\\rightarrow 1-$ , it seems natural to expect a transcritical bifurcation at $\\bar{\\alpha }=1$ with stability of the trivial steady state for $\\bar{\\alpha }>1$ .",
"Again the conjecture of global asymptotic stability of $(0,0,0)$ for $\\bar{\\alpha }>1$ has been supported by simulations (see for example Fig.",
"REF ).",
"The right hand sides of (REF ) are continuous up to the origin (when considered as an element of the set of admissible states), since $0\\le \\alpha (q,r)\\le 1$ and $p/r\\le n$ .",
"However, their nonsmoothness prohibits a standard local stability or bifurcation analysis.",
"The expected local stability behaviour (asymptotic stability for $\\bar{\\alpha }>1$ , instability for $\\bar{\\alpha }<1$ ) is proven in Section .",
"The analysis is based on a regularizing transformation, which makes the steady state very degenerate, combined with a blow-up analysis [2].",
"The fact that the components of the nontrivial equilibrium tend to infinity when $\\bar{\\alpha }\\rightarrow 0+$ suggests that solutions might be unbounded for $\\bar{\\alpha }<0$ .",
"In this parameter region approximate solutions with polynomial growth of the form $p(t) = p_1 t+ \\textit {o}(t) \\,,\\qquad q(t) = q_2 t^2 + \\textit {o}(t^2) \\,,\\qquad r(t) = \\frac{2q_2}{n} t^2+ \\textit {o}(t^2) \\,,\\qquad \\mbox{as } t\\rightarrow \\infty \\,,$ have been constructed in [1] by formal asymptotic methods.",
"It has also been shown that no other growth behaviour (polynomial with other powers or exponential) should be expected, and the conjecture that all solutions have the constructed asymptotic behaviour is again verified by simulations (see Fig.",
"for example REF ).",
"We justify the formal asymptotics in Section .",
"A variant of Poincaré compactification [4] produces a problem with bounded solutions and with three different time scales, which is analyzed by singular perturbation methods [3].",
"The final result is existence and semi-local stability of the polynomially growing solutions, where 'semi-local' means that initial data have to be large with relative sizes as in (REF ).",
"The article is concluded by a discussion section about biological interpretation of our results as well as perspectives."
],
[
"Local stability of the zero steady state",
"In this section, we study under which conditions small aggregates tend to disaggregate.",
"This is equivalent to studying the stability of the zero-steady-state $(p,q,r)=(0,0,0)$ of the system (REF ).",
"Because of the appearance of the ratios $\\frac{p}{r}$ and $\\frac{q}{r}$ , the Jacobian of the right hand side of (REF ) is not defined there.",
"As a consequence of (REF ) the regularizing transformation $\\tau := \\int _0^t r(s)^{-1} \\mathrm {d}s$ is well defined and leads to $ \\begin{aligned}\\frac{dp}{d\\tau } &= r(\\kappa _1 - \\kappa _3 p) (nr-p-2q) + \\kappa _{-}q \\left( r - \\frac{(n-1)p}{n-2}\\right) - (\\kappa _2 + \\kappa _{-1})pr \\,, \\\\\\frac{dq}{d\\tau } &= \\kappa _2 pr + \\kappa _3 pr(nr-p-2q) - \\kappa _{-}qr\\,, \\\\\\frac{dr}{d\\tau } &= \\kappa _2 pr - \\kappa _{-} q \\frac{nr -2q}{n-2}\\,.\\end{aligned}$ The regularization came at the expense that the zero steady state is degenerate in (REF ), since the right hand side is of second order in terms of the densities.",
"A classical approach to study such non-hyperbolic points is blow-up [2].",
"The standard blow-up transformation would be the introduction of spherical coordinates, blowing up the origin to the part of $\\mathbb {S}^2$ in the positive octant.",
"It is also common to work with charts instead.",
"In our case this preserves the polynomial form of the right hand side.",
"Although the charts in the different coordinate directions are equivalent, since the state space is a subset of the positive octant, it has turned out to be convenient to use the $q$ -chart, whence the blow-up transformation $(p,q,r) \\rightarrow (p_1,q_1,r_1)$ is given by $p= p_1q_1\\,, \\qquad q=q_1\\,,\\qquad r=r_1 q_1 \\,,$ and we also introduce another change of time scale: $T := \\int _0^\\tau q_1(\\sigma )d\\sigma $ , again justified by (REF ), leading to $\\frac{dq_1}{dT} &=& q_1 r_1\\left( \\kappa _2 p_1 - \\kappa _- \\right) + \\kappa _3 p_1 r_1 q_1^2(nr_1-p_1-2)\\,,\\nonumber \\\\\\frac{dp_1}{dT} &=& r_1(\\kappa _1 - \\kappa _3 p_1 q_1)(nr_1-p_1-2) + \\kappa _- \\left(r_1 - \\frac{n-1}{n-2}p_1\\right) - (\\kappa _2 + \\kappa _{-1})p_1r_1- p_1 r_1(\\kappa _2 p_1 - \\kappa _-) \\nonumber \\\\&& -\\kappa _3 p_1^2 r_1 q_1 (nr_1 - p_1 - 2) \\,,\\\\\\frac{dr_1}{dT} &=& (1-r_1)\\left( \\kappa _2 p_1 r_1 + \\kappa _- \\left( \\frac{2}{n-2} - r_1\\right) \\right) - \\kappa _3 p_1 r_1^2 q_1 (nr_1 - p_1 - 2)\\,.\\nonumber $ The invariant manifold $q_1=0$ of this system corresponds to the zero steady state of (REF ).",
"The inequalities (REF ) become $r_1 \\le 1 \\,,\\qquad 0\\le p_1 \\le nr_1-2 \\,,$ in terms of the new variables, i.e.",
"the dynamics of $(p_1,r_1)$ remains in the triangle depicted in Fig.",
"REF .",
"Since $r_1\\ge 2/n$ , we conclude from the equation for $q_1$ that the invariant manifold is locally exponentially attracting in the region to the left of the line $p_1 = \\kappa _-/\\kappa _2$ .",
"Since $p_1\\le n-2$ , the inequality $\\kappa _- > (n-2)\\kappa _2$ already implies local asymptotic stability of the invariant manifold $q_1=0$ of (REF ) and therefore of the zero steady state of (REF ).",
"Note that $\\kappa _- > (n-2)\\kappa _2$ also implies $\\bar{\\alpha }> 1$ for $\\bar{\\alpha }$ defined by (REF ).",
"Figure: The dynamics in the (p 1 ,r 1 )(p_1,r_1)-plane is limited to the shaded triangle because of the inequalities ().In the following we therefore consider the case $\\kappa _- \\le (n-2)\\kappa _2$ (see Fig.",
"REF ) and $\\bar{\\alpha }> 1$ , where the latter is equivalent to $\\kappa _1\\kappa _2(n-2)^2 < \\kappa _-(\\kappa _1 + \\kappa _{-1})(n-2) + \\kappa _-^2(n-1) \\,,$ see also [1].",
"The flow on the invariant manifold $q_1=0$ of (REF ) is governed by the system $\\frac{dp_1}{dT} &=& r_1\\kappa _1 (nr_1-p_1-2) + \\kappa _- \\left(r_1 - \\frac{n-1}{n-2}p_1\\right) - (\\kappa _2 + \\kappa _{-1})p_1r_1- p_1 r_1(\\kappa _2 p_1 - \\kappa _-) \\nonumber \\\\\\frac{dr_1}{dT} &=& (1-r_1)\\left( \\kappa _2 p_1 r_1 + \\kappa _- \\left( \\frac{2}{n-2} - r_1\\right) \\right) \\,.$ In the right part of the triangle, i.e.",
"for $r_1 \\le 1 \\,,\\qquad \\frac{\\kappa _-}{\\kappa _2}\\le p_1 \\le nr_1-2 \\,,$ we have $\\frac{dp_1}{dT} &\\le & r_1\\kappa _1 \\left( n - \\frac{\\kappa _-}{\\kappa _2} - 2\\right) + \\kappa _- \\left( r_1 - \\frac{n-1}{n-2}\\,\\frac{\\kappa _-}{\\kappa _2}\\right)- (\\kappa _2 + \\kappa _{-1})\\frac{\\kappa _-}{\\kappa _2}r_1 \\\\&=& \\frac{r_1\\bigl (\\kappa _1\\kappa _2(n-2)^2 - \\kappa _-(\\kappa _1 + \\kappa _{-1})(n-2)\\bigr ) - \\kappa _-^2(n-1)}{\\kappa _2(n-2)}< \\frac{(r_1-1)\\kappa _-^2(n-1)}{\\kappa _2(n-2)} \\le 0 \\,,$ where the strict inequality is due to (REF ).",
"This implies that all trajectories reach the left part of the triangle, i.e.",
"$p_1 < \\kappa _-/\\kappa _2$ in finite time.",
"By standard regular perturbation theory the dynamics for the full system (REF ), when started close to the invariant manifold $q_1=0$ , remains close to the dynamics on the invariant manifold for finite time, until the region $p_1 < \\kappa _-/\\kappa _2$ is reached, where the invariant manifold is attracting.",
"Thus $q=q_1$ tends to zero and, by the inequalities (REF ), the same is true for $p$ and $r$ .",
"Now we consider the case $\\bar{\\alpha }<1$ , i.e.",
"the opposite of inequality (REF ), and look for a steady state on the invariant manifold $r_1=1$ of the system (REF ).",
"Since $\\frac{dp_1}{dT}\\Bigm |_{r_1=1,p_1=\\kappa _-/\\kappa _2} &=& \\frac{\\kappa _1\\kappa _2(n-2)^2 - \\kappa _-(\\kappa _1 + \\kappa _{-1})(n-2) - \\kappa _-^2(n-1)}{\\kappa _2(n-2)} >0 \\,,\\\\\\frac{dp_1}{dT}\\Bigm |_{r_1=1,p_1=n-2} &=& -(n-2)(n\\kappa _2 + \\kappa _{-1}) < 0 \\,,$ there exists a steady state $(p_1,r_1) = (p_1^*,1)$ with $\\kappa _-/\\kappa _2 < p_1^* < n-2$ , which is stable under the flow along $r_1=1$ .",
"On the other hand $\\frac{1}{1-r_1}\\, \\frac{dr_1}{dT}\\Bigm |_{r_1=1,p_1=p_1^*} = \\left( \\kappa _2 p_1^* + \\kappa _- \\left( \\frac{2}{n-2} - 1\\right) \\right) > \\frac{2\\kappa _-}{n-2} >0 \\,,$ which implies stability of the manifold $r_1=1$ close to the steady state, and therefore stability of the steady state.",
"The existence of a stable steady state on the invariant manifold $q_1=0$ of (REF ) in the region, where the manifold is repulsive, implies instability of the manifold and therefore also of the zero steady state of (REF ).",
"This completes the proof of the main result of this section.",
"Theorem 1 Let $\\bar{\\alpha }$ be defined by (REF ).",
"Then the steady state $(0,0,0)$ of the system (REF ) is locally asymptotically stable for $\\bar{\\alpha }>1$ and unstable for $\\bar{\\alpha }<1$ ."
],
[
"Polynomially growing regime",
"The goal of this section is a rigorous justification of the formal asymptotics (REF ) (see [1]) under the assumption $\\bar{\\alpha }<0$ with $\\bar{\\alpha }$ defined in (REF ), i.e.",
"$4\\kappa _1\\kappa _2(n-2)^2 > n\\kappa _-\\bigl (2(\\kappa _1 + \\kappa _{-1})(n-2) + \\kappa _-n(n-1) \\bigr )\\,,$ see also [1].",
"Considering (REF ), it would be natural to write an equation for $p(t)/t$ .",
"It is easily seen from (REF ) that its derivative contains terms of the order of $t^2$ .",
"Similarly the derivative of $q(t)/t^2$ has contributions up to the order of $t$ , whereas the derivative of $r(t)/t^2$ is a combination of terms bounded as $t\\rightarrow \\infty $ .",
"This shows that we are confronted with a problem with different time scales, which will put us into the realm of singular perturbation theory (see, e.g.",
"[3], [6]).",
"The leading order term in the fastest equation, i.e.",
"the $p$ -equation, is $-\\kappa _3 p (nr-2q)$ , from which it has been concluded in [1] that $nr(t)\\approx 2q(t)$ as $t\\rightarrow \\infty $ .",
"In a standard singular perturbation setting, it should be possible to express $p(t)$ from this relation.",
"Since this is not the case, our problem belongs to the family of singular singularly perturbed problems (see e.g.",
"[5]) which, however, can be transformed to the standard regular form in many cases.",
"The introduction of $p(t)/t$ , $q(t)/t^2$ , $r(t)/t^2$ , as new variables would lead to a study of bounded solutions, but to a non-autonomous system.",
"We shall use a variant of the Poincaré compactification method [4] instead.",
"The previous observations led us to the introduction of the new variables $u= \\frac{p}{\\sqrt{p+q}} \\,,\\qquad v=\\frac{2p+2q -nr}{\\sqrt{p+q}} \\,,\\qquad w=\\frac{1}{\\sqrt{p+q}} \\,,$ where we expect that $w(t)$ tends to zero as $t^{-1}$ , and that $u(t)$ and $v(t)$ converge to nontrivial limits.",
"Since this coordinate change produces a singularity at $w=0$ , we also change the time variable by $\\tau = \\int _0^t ds/w(s)$ .",
"In terms of the new variables system (REF ) becomes $\\frac{du}{d\\tau } &=& (\\kappa _1 w - \\kappa _3 u)(u-v) + \\kappa _-(1-uw)\\left( 1 - \\frac{n(n-1)uw}{(n-2)(2-vw)}\\right) - (\\kappa _2+\\kappa _{-1})uw \\nonumber \\\\&& - uw^2A(u,v,w) \\,,\\nonumber \\\\\\frac{dv}{d\\tau } &=& w\\left(2\\kappa _1(u-v) - (2\\kappa _{-1} + n\\kappa _2)u + \\kappa _- (1-uw)n \\frac{2u-nv}{(n-2)(2-vw)}\\right) - vw^2A(u,v,w) \\,,\\\\\\frac{dw}{d\\tau } &=& - w^3 A(u,v,w) \\,,\\quad A(u,v,w) := \\frac{1}{2}\\left(\\kappa _1(u-v) - \\kappa _{-1}u - \\kappa _-(1-uw)\\frac{n(n-1)u}{(n-2)(2-vw)}\\right) \\,.\\nonumber $ Our goal is to prove that solutions converge to a steady state $(u^*,v^*,w^*)$ with $w^*=0$ , which obviously has to satisfy $- \\kappa _3 u^*(u^*-v^*) + \\kappa _-=0$ , implying $u^* = U(v^*) := \\frac{1}{2} \\left( v^* +\\sqrt{(v^*)^2 + 4\\kappa _-/\\kappa _3}\\right) \\,,$ since we need $u^*>0$ .",
"We intend to show that $v^*$ is determined from the requirement that the large parenthesis in the $v$ -equation vanishes.",
"The argument is essentially that for small values of $w$ , the variable $v$ evolves much faster than $w$ .",
"In order to make the slow-fast structure of this system more apparent and to allow the application of basic results from singular perturbation theory, we assume that the initial value for $w$ is small and define $\\varepsilon := (p_0+q_0)^{-1/2}\\ll 1$ and the rescaled variable $W= w/\\varepsilon $ , leading to $\\frac{du}{d\\tau } &=& - \\kappa _3 u(u-v) + \\kappa _- + O(\\varepsilon ) \\,,\\nonumber \\\\\\frac{dv}{d\\tau } &=& \\varepsilon W\\left(2\\kappa _1(u-v) - (2\\kappa _{-1} + n\\kappa _2)u + \\kappa _- n \\frac{2u-nv}{2(n-2)}\\right) +O(\\varepsilon ^2) \\,,\\\\\\frac{dW}{d\\tau } &=& - \\varepsilon ^2 W^3 A(u,v,0) + O(\\varepsilon ^3) \\,.\\nonumber $ The initial data are denoted by $u(0)=u_0 := \\frac{p_0}{\\sqrt{p_0+q_0}} >0\\,,\\qquad v(0)=v_0 := \\frac{2p_0+2q_0 -nr_0}{\\sqrt{p_0+q_0}}\\,,\\qquad W(0)=1\\,,$ where in the following we consider $u_0$ and $v_0$ as fixed when $\\varepsilon \\rightarrow 0$ .",
"This is a singular perturbation problem in standard form, where $\\tau $ plays the role of an initial layer variable.",
"We pass to the limit $\\varepsilon \\rightarrow 0$ to obtain the initial layer problem $\\frac{d\\hat{u}}{d\\tau } &=& - \\kappa _3 \\hat{u}(\\hat{u}-\\hat{v}) + \\kappa _- \\,,\\\\\\frac{d\\hat{v}}{d\\tau } &=& \\frac{d\\hat{W}}{d\\tau } = 0 \\,,\\nonumber $ subject to the initial conditions.",
"By the qualitative behaviour of the right hand side of the first equation, the solution satisfies $\\hat{v}(\\tau )=v_0$ , $\\hat{W}(\\tau )=1$ , and $\\lim _{\\tau \\rightarrow \\infty } \\hat{u}(\\tau ) = U(v_0) \\,,$ with exponential convergence, where $U$ has been defined in (REF ).",
"The equation $u=U(v)$ defines the so called reduced manifold.",
"Since it is exponentially attracting, the Tikhonov theorem [7] (or rather its extension [3]) implies that, after the initial layer, i.e.",
"when written in terms of the slow variable $\\sigma = \\varepsilon \\tau $ , the solution trajectory remains exponentially close to the slow manifold, which is approximated by the reduced manifold, and the flow on the slow manifold satisfies $\\frac{dv}{d\\sigma } &=& W\\left(2\\kappa _1(U(v)-v) - (2\\kappa _{-1} + n\\kappa _2)U(v) + \\kappa _- n \\frac{2U(v)-nv}{2(n-2)}\\right) +O(\\varepsilon ) \\,,\\nonumber \\\\\\frac{dW}{d\\sigma } &=& - \\varepsilon W^3 A(U(v),v,0) + O(\\varepsilon ^2) \\,,$ with $v(0)=v_0$ , $W(0)=1$ .",
"This is again a singular perturbation problem in standard form, where now $\\sigma $ is the initial layer variable.",
"We repeat the above procedure and consider the limiting layer problem $\\frac{d\\tilde{v}}{d\\sigma } &=& \\tilde{W}\\left(2\\kappa _1(U(y\\tilde{v})-\\tilde{v}) - (2\\kappa _{-1} + n\\kappa _2)U(\\tilde{v})+ \\kappa _- n \\frac{2U(\\tilde{v})-n\\tilde{v}}{2(n-2)}\\right) \\,,\\\\\\frac{d\\tilde{W}}{d\\sigma } &=& 0 \\,.\\nonumber $ The observations $U(-\\infty ) = 0 \\,,\\qquad U(\\infty )=\\infty \\,,\\qquad 0< U^{\\prime }(v) < 1 \\,,$ suffice to show that the right hand side of the first equation is a strictly decreasing function of $v$ with a unique zero $v^*$ , which can actually be computed explicitly: $v^* = B\\left(\\kappa _1 - \\kappa _{-1} - \\frac{n}{2}\\kappa _2 + \\frac{n}{2(n-2)}\\kappa _-\\right)$ $\\mbox{with}\\quad B = 2\\sqrt{\\frac{\\kappa _-}{\\kappa _3}} \\left( \\frac{n^3}{4(n-2)}\\kappa _-^2 + 4\\kappa _1 \\kappa _{-1} + 2n \\kappa _1\\kappa _2+ n\\kappa _1 \\kappa _- + \\frac{n^2}{n-2} \\kappa _{-1}\\kappa _- + \\frac{n^3}{2(n-2)\\kappa _2 \\kappa _-}\\right)^{-1/2}$ The solution of (REF ) with $\\tilde{v}(0)=v_0$ satisfies $\\lim _{\\sigma \\rightarrow \\infty } \\tilde{v}(\\sigma ) = v^*$ with exponential convergence.",
"Another application of the Tikhonov theorem shows that the slowest part of the dynamics with $t=O(\\varepsilon ^{-1})$ can be approximated by $\\frac{dW}{d\\sigma } = - \\varepsilon W^3 A^* \\,,\\qquad W(0)=1 \\,,$ with $A^* := A(U(v^*),v^*,0) = \\frac{nB}{16(n-2)^2}(4(n-2)^2\\kappa _1\\kappa _2 - 2n(n-2)\\kappa _-(\\kappa _1+\\kappa _{-1}) - n^2(n-1)\\kappa _-^2) >0 \\,,$ by (REF ).",
"This gives the approximation $W(\\sigma ) = (1+2 A^*\\varepsilon \\sigma )^{-1/2} \\,.$ The results of [3] imply that the approximations are accurate with errors of order $\\varepsilon $ uniformly with respect to time.",
"Theorem 2 Let (REF ) hold.",
"Then, for $\\varepsilon >0$ small enough, the solution of (REF ) with initial conditions $u(0) = u_0>0 \\,,\\qquad v(0) = v_0 \\in \\mathbb {R} \\,, \\qquad w(0)=\\varepsilon \\,,$ satisfies $u(\\tau ) &=& \\hat{u}(\\tau ) - U(v_0) + U(\\tilde{v}(\\varepsilon \\tau )) + O(\\varepsilon ) \\,,\\\\v(\\tau ) &=& \\tilde{v}(\\varepsilon \\tau ) + O(\\varepsilon ) \\,,\\\\w(\\tau ) &=& \\varepsilon (1 + 2A^* \\varepsilon ^2\\tau )^{-1/2} + O(\\varepsilon ^2) \\,,$ uniformly in $\\tau \\ge 0$ , where $U$ is given in (REF ), $\\hat{u}$ solves (REF ), $\\tilde{v}$ solves (REF ), and $A^*$ is given in (REF ).",
"Actually more can be deduced.",
"In terms of the original time variable $t$ , the equation for $w$ in (REF ) becomes $\\dot{w} = -w^2 A(u,v,w) \\,.$ Under the assumptions of Theorem REF , $A(u,v,w)$ is uniformly close to the positive constant $A^*$ and therefore uniformly positive for large enough $t$ .",
"This implies that $w$ tends to zero as $t\\rightarrow \\infty $ .",
"The slow manifold of the system (REF ) reduces to the steady state $(v,W)=(v^*,0)$ for $W=0$ .",
"Therefore $v$ tends to $v^*$ as $t\\rightarrow \\infty $ .",
"Analogously, the slow manifold of (REF ) reduces to the steady state $(u,v,W)=(u^*=U(v^*),v^*,0)$ at $W=0$ , implying convergence of $u$ to $u^*$ .",
"This in turn implies convergence of $A(u,v,w)$ to $A^*$ , which can be used in (REF ).",
"Corollary 1 Let the assumptions of Theorem REF hold.",
"Then $\\lim _{t\\rightarrow \\infty } u(t) = u^* \\,,\\qquad \\lim _{t\\rightarrow \\infty } v(t) = v^* \\,,\\qquad w(t) = \\frac{1}{A^* t} + O\\left( \\frac{1}{t^2}\\right) \\quad \\mbox{as } t\\rightarrow \\infty \\,.$ Finally, we reformulate these results in terms of the original variables, verifying the formal asymptotics of [1] for initial data, which are in a sense already 'close enough' to the polynomially growing solutions.",
"Theorem 3 Let (REF ) hold, let $c_2\\ge c_1>0$ , and let $\\delta >0$ be small enough.",
"Let the initial data satisfy $p_0 = \\frac{c_1}{\\delta } \\,,\\qquad q_0 = \\frac{1}{\\delta ^2} \\,,\\qquad r_0 = \\frac{2}{n\\delta ^2} + \\frac{c_2}{n\\delta }$ Then the solution of (REF ) with $(p(0),q(0),r(0)) = (p_0,q_0,r_0)$ satisfies $p(t) = u^*A^* t + o(t)\\,,\\qquad q(t) = (A^*)^2 t^2 + o(t^2) \\,,\\qquad r(t) = \\frac{2}{n}(A^*)^2 t^2 + o(t^2) \\,,\\qquad \\mbox{as } t\\rightarrow \\infty \\,.$ We just need to verify that the assumptions of this theorem imply the assumptions of Theorem REF .",
"The result is then a direct consequence of Corollary REF .",
"Actually the assumptions of Theorem REF hold with $\\varepsilon \\approx \\delta $ , since $u_0 = \\frac{c_1}{\\sqrt{1+c_1\\delta }} \\,,\\qquad v_0 = \\frac{2c_1-c_2}{\\sqrt{1+c_1\\delta }} \\,,\\qquad w_0 = \\frac{\\delta }{\\sqrt{1+c_1\\delta }} \\,.$"
],
[
"Discussion",
"In this work a mathematical model for aggregation via cross-linking has been analyzed.",
"Besides the basic assumption that aggregating particles (here p62 oligomers) need to have at least $n=3$ binding sites for cross-linkers (here ubiqutinated cargo), the rate constants for binding reactions need to be large enough compared to those for the unbinding reactions (the opposite of inequality (REF )) for stable aggregates to exist.",
"Under a stronger condition (inequality (REF )) aggregates grow indefinitely in the presence of an unlimited supply of free particles and cross-linkers.",
"These conjectures from [1], where the model has been formulated, have been partially proven in this work.",
"It has been shown in Section that small aggregates get completely degraded under the condition (REF ) and that they grow under the opposite condition.",
"In the latter case, but when (REF ) does not hold, there exists an equilibrium configuration with positive aggregate size.",
"Finally, it has been shown in Section that under the condition (REF ) aggregate size grows polynomially with time (actually like $t^2$ ) for appropriate initial states.",
"Figure: Bifurcation diagram obtained for κ -1 =κ - =1\\kappa _{-1}=\\kappa _{-}=1The constants $\\kappa _1$ , $\\kappa _2$ in the model have to be interpreted as the products of rate constants with the concentrations of free cross-linkers and, respectively, of free particles.",
"This means that the conditions (REF ) and (REF ) are actually conditions for these concentrations.",
"Fig.",
"REF shows a bifurcation diagram in terms of $\\kappa _1$ and $\\kappa _2$ with the curves $\\bar{\\alpha }=1$ , corresponding to equality in (REF ), and $\\bar{\\alpha }=0$ , corresponding to equality in (REF ).",
"The qualitative behaviour is no surprise: Close to the origin, i.e.",
"for small concentrations of free particles and cross-linkers, aggregates are unstable.",
"Moving to the right and/or up we pass through two bifurcations to stable finite aggregate size and, subsequently, to polynomial growth of aggregates.",
"Less obvious is the fact that the picture is rather unsymmetric with respect to the two parameters.",
"The condition $(n-2)\\kappa _2> \\kappa _-$ is necessary for the existence of stable aggregates, regardless of the value of $\\kappa _1$ , whereas arbitrarily small values of $\\kappa _1$ can be compensated by large enough $\\kappa _2$ .",
"This means that, if the concentration of free particles is below a threshold, even a large concentration of cross-linkers does not lead to aggregation, whereas arbitrarily small numbers of cross-linkers are used for aggregation if the particle concentration is high.",
"For the application in cellular autophagy this means that aggregation will only happen for large enough concentrations of p62 oligomers.",
"However, arbitrarily small amounts of ubiquitinated cargo can be aggregated in the presence of a large enough supply of oligomers.",
"This work has been motivated by the experimental results of [8], where aggregates have been detected by light microscopy.",
"If the evolution of single aggregates can be followed, the growth like $t^2$ might be observed as a fluorescence signal of tagged oligomers, which goes like $t^2$ , or cross section areas going like $t^{4/3}$ , if a roughly spherical shape of aggregates is assumed.",
"For quantitative predictions of such experiments, the model should be extended in various ways.",
"First, the limited supply of free p62 oligomers and of free cross-linkers should be taken into account.",
"This is straightforward for the modeling of a single aggregate, but if many aggregates develop simultaneously, they will compete for the free particles.",
"Apart from that the number of aggregates has to be predicted, which requires modeling of the nucleation process.",
"Finally, it is very likely that the coagulation of aggregates plays an important role.",
"A growth-coagulation model for distributions of aggregates, based on the growth model (REF ) would be prohibitively complex.",
"It is therefore the subject of ongoing work to formulate, analyze, and simulate a growth-coagulation model based on the multiscale analysis of Section , where aggregates are only described by the size parameter $r$ (number of p62 oligomers in the aggregate), whose evolution is determined by the slow dynamics (REF ), which translates to an equation of the form $\\dot{r} = C\\sqrt{r}$ for $r$ .",
"This approach raises several challenging issues such as the development of an efficient simulation algorithm or the existence and stability of equilibrium aggregate distributions."
]
] | 2012.05201 |
[
[
"1000 days of lowest frequency emission from the low-luminosity GRB\n 171205A"
],
[
"Abstract We report the lowest frequency measurements of gamma-ray burst (GRB) 171205A with the upgraded Giant Metrewave Radio Telescope (uGMRT) covering a frequency range from 250--1450 MHz and a period of $4-937$ days.",
"It is the first GRB afterglow detected at 250--500 MHz frequency range and the second brightest GRB detected with the uGMRT.",
"Even though the GRB is observed for nearly 1000 days, there is no evidence of transition to non-relativistic regime.",
"We also analyse the archival ${\\it Chandra}$ X-ray data on day $\\sim 70$ and day $\\sim 200$.",
"We also find no evidence of a jet break from the analysis of combined data.",
"We fit synchrotron afterglow emission arising from a relativistic, isotropic, self-similar deceleration as well as from a shock-breakout of wide-angle cocoon.",
"Our data also allow us to discern the nature and the density of the circumburst medium.",
"We find that the density profile deviates from a standard constant density medium and suggests that the GRB exploded in a stratified wind like medium.",
"Our analysis shows that the lowest frequency measurements covering the absorbed part of the light curves are critical to unravel the GRB environment.",
"Our data combined with other published measurements indicate that the radio afterglow has contribution from two components: a weak, possibly slightly off-axis jet and a surrounding wider cocoon, consistent with the results of Izzo et al.",
"(2019).",
"The cocoon emission likely dominates at early epochs, whereas the jet starts to dominate at later epochs, resulting in flatter radio lightcurves."
],
[
"Gamma Ray Bursts (GRBs) are the most energetic flashes of gamma rays, with $T_{90}$ duration (time interval between which 5% to 95% of fluence is collected by the detector) ranging between a few milli-seconds to thousands of seconds [87].",
"The GRBs can be classified in two broad classes i.e short/hard GRBs with duration less than 2 secs, and long/soft GRBs of duration greater than 2 secs [42].",
"According to well accepted theories, most of the long soft GRBs originate from gravitational collapse of massive stars (collapsar model); and short hard GRBs result from explosive binary compact object mergers [86].",
"The GRBs from both channels of formation power relativistic collimated jets which give Doppler-boosted high luminosities in gamma-rays.",
"GRBs are cosmological events with an average redshift $z\\approx 2.2$ [29], and have isotropic-equivalent gamma ray luminosities ($L_{\\rm iso}$ ) of the order of $10^{51}$ to $10^{53}$ erg s$^{-1}$ .",
"However, a handful of long/soft GRBs with spectroscopically identified supernovae have been discovered with luminosities that are 3–5 orders of magnitude lower than the average, i.e.",
"[69], [10].",
"Their low luminosities allow them to be detected only at low redshifts, though they may be 10–100 times more abundant than regular GRBs [68].",
"The prompt light-curves of typical low-luminosity GRBs are smooth and spectra have a single peak with the peak energy generally below $\\sim $ 50 keV, which softens further with time [56], [10].",
"The radio afterglow of these GRBs tend to indicate similar energy content in mildly relativistic ejecta [43].",
"Many of these GRBs are associated with broad-line Type Ic supernovae.",
"The list of such GRBs-supernovae include some of the well studied cases such as, GRB 980425/SN 1998bw [31], GRB 030329/SN 2003dh [36], GRB 031203/SN 2003lw [48], GRB 060218/SN 2006aj [9], GRB 100316D/SN 2010bh [74], GRB 111209A/SN 2011kl [32], GRB 120422A/SN 2012bz [50], GRB 130427A/SN 2013cq [51], GRB 130702A/SN 2013dx [12], GRB 161219B/SN 2016jca [11], GRB 171010A/SN 2017htp [52], GRB 190829A/ SN 2019oyu [80], although only the five are nearby, $z\\lesssim 0.1$ [10].",
"A significant amount of work has gone towards understanding whether the low-luminosity GRBs are simply the low-energy counterparts of the cosmological GRBs, or have a different emission mechanism.",
"Many low-luminosity GRBs do not follow the $E_{\\rm iso}-E_p$ Amati relation [1], indicating that their emission mechanism should be different from that of canonical, more distant GRBs.",
"Technically an off-axis jet can also explain low-luminosity emission from GRB, but predicts an achromatic steepening of the light curve, absent in many low-luminosity GRBs.",
"There have been suggestions that in contrast to the emission from an ultra-relativistic jet driven by a central engine, these low-luminosity GRBs are powered by shock breakouts [43], [56], [54], [5], [76].",
"In some cases observations have suggested a mildly relativistic blast wave being responsible for producing the radio afterglow.",
"This has supported the relativistic shock breakout model, e.g.",
"GRB 980425 [43].",
"The shock-break out model got further support in case of GRB 060218, in which a thermal component was also seen which cooled and shifted to optical/UV band with time.",
"This was interpreted to be arising from the break out of a shock driven by a mildly relativistic supernova shell in the progenitor wind [9], although late time photospheric emission from a jet [27], or thermal emission from a cocoon [78] can also explain it.",
"[8] have investigated whether the low-luminosity GRBs launch relativistic jets like their high energy counterparts, but incur resistance by the stellar envelopes surrounding their progenitor stars.",
"They found that some low-luminosity GRBs have much shorter durations compared to the jet breakout time, This is inconsistent with the collapsar model which is largely successful in explaining the cosmological GRBs [87].",
"While [5] and [56] have developed spherical relativistic shock break out models in context of low-luminosity GRBs, [54] addressed some of the problems of spherical shock-break out model by introducing a low mass optically thick stellar envelope surrounding the progenitor star.",
"In this model, the explosion powering the low-luminosity GRBs was not the spherical breakout of the supernova shock, but by a jet that gets choked in the envelope and powers quasi-spherical explosion.",
"To extend this idea further, [55] considered a cocoon breakout model.",
"In this model, as the GRB jet pushes through the stellar material, it heats the surrounding gas and produces a high-pressure sub-relativistic cocoon, which at the time of breakout, produces a relatively faint flare of $\\gamma $ -rays.",
"This break out will not be as spherical as a supernova break-out, but will be wider than a jet.",
"In this model, interaction of the cocoon with the surrounding medium can give rise to a late time radio and X-ray afterglow.",
"However, [37] have provided an alternative mechanism, where the composite emission of GRB 060218/SN 2006aj could be explained by a weak jet, along with a quasi-spherical supernova ejecta.",
"llGBRs also have a radio afterglow, which indicates a comparable energy in mildly relativistic ejecta (Kulkarni et al.",
"1998; Soderberg et al.",
"2004, 2006; Margutti et al.",
"2013).",
"The lack of bright, late-time, radio emission from ll-GRBs strongly constrain the total energy of any relativistic outflow involved in these events (Waxman 2004; Soderberg et al.",
"2004, 2006b).",
"Additionally, statistical arguments rule out the possibility that ll-GRBs are regular LGRBs viewed at a large angle (e.g., Daigne & Mochkovitch 2007).",
"Thus, if ll-GRBs are generated by relativistic jets these jets must be weak and have a large opening angle.",
"The bursts with $L_\\gamma > 2 \\times 10^{48}$ erg s$^{-1}$ are considered as regular GRBs and are separated into LGRBs and SGRBs according to the standard criterion of whether T90 in the observer frame is above or below 2 s (Bromberg, Nakar, Piran 2011).",
"GRB 171205A is a nearby low-luminosity GRB with a $T_{90}$ duration of almost 189.4 secs [24].",
"It was first discovered by the Burst Alert Telescope (BAT) onboard Swift on 5th December, 2017 [23].",
"It has a redshift of 0.0368 [23], [38].",
"The isotropic energy release in gamma ray band at GRB rest frame was $2.18_{-0.50}^{+0.63}\\times 10^{49}$ ergs [24].",
"The host of this event was a bright spiral galaxy named 2MASX J11093966-1235116 [38], with a mass of the order of $10^{10}M_{\\odot }$ and a star formation rate of $3\\pm 1M_{\\odot }/\\rm {yr}$ [63].",
"An emergent supernova event (SN 2017iuk) was seen three days after the burst [24].",
"There are a handful of observations for GRB 171205A from mm to radio bands.",
"It was detected by [22] using Northern Extended Millimeter Array (NOEMA) with a flux density of $\\sim 35$ mJy at 150 GHz after 20.2 hours of the burst.",
"Atacama Large Millimeter/Submillimeter Array (ALMA) detected a bright afterglow of significance more than 100$\\sigma $ at 92 GHz and 340 GHz on 10-11 December, 2017 [62].",
"RATAN-600 radio telescope detected it at 4.7 and 8.2 GHz bands during 9th December to 16th December, 2017 [81].",
"Karl G. Jansky Very Large Array (VLA) also observed the afterglow in the frequency range 4.5 to 16.5 GHz [44].",
"This GRB also had a Very Long Baseline Array (VLBA) detection at different frequencies [61].",
"These observations showed a steeply rising spectrum($\\propto \\nu ^{2}$ ) at low frequency which indicates a synchrotron self-absorbed spectrum.",
"This is the first GRB for which [82] claimed to have detected polarization at the ALMA 90GHz frequency, though this claim has been disputed by [45].",
"The upgraded Giant Metrewave Radio Telescope (uGMRT) first detected it on 20th December, 2017 at 1400 MHz [15] after a non detection on 10th and 11th December, 2017 [16].",
"The observed flux density at that epoch was $782\\pm 57 \\rm {\\mu Jy}$ .",
"Radio afterglow emission from GRBs evolve slowly which gives us the opportunity to observe it for a long time and obtain the distribution of the kinetic energy in the velocity space.",
"Since this distribution is different for various models, especially central engine driven versus shock break out, radio observations provide unique opportunity to distinguish between various emission models [43].",
"Additionally, the early radio emission is likely to be absorbed via synchrotron self-absorption (SSA), and thus constrain the circumburst medium (CBM) density [17].",
"The late time radio observations in the Newtonian limit, when the jet becomes sub-relativistic, are nearly independent of jet geometry and measure the kinetic energy of the afterglow accurately [26].",
"In this paper we present low frequency observations of GRB 171205A taken with the uGMRT for around 1000 days.",
"We summarize our observations and data analysis in §.",
"We discuss our model in §REF and results in §REF .",
"In §, we discuss the properties of GRB 171205A in conjunction with published measurements at higher frequencies and present our main conclusions.",
"Unless otherwise stated, we assume a cosmology with $H_0=67.3$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _m=0.315$ , $\\Omega _\\Lambda =0.685$ [64]."
],
[
"The GMRT observed GRB 171205A starting 2017 December 10 and continued observing until 2020 June 26.",
"The observations were taken in band 5 (1000–1450 MHz), band 4 (550–900 MHz) and band 3 (250–500 MHz).",
"The bandwidth for band 4 and band 5 was 400 MHz while for band 3 it was 200 MHz.",
"The duration of each observation was around 2-3 hours including overheads (on source time 1.5 hours).",
"We observed flux density calibrators 3C286 and 3C48; and a phase calibrator J1130-148.",
"Flux calibrators were also used as bandpass calibrators.",
"We use a package Common Astronomy Software Applications( CASA) for data analysis.",
"The data were analyzed in three major steps, i.e flagging, calibration and imaging.",
"The CASA task `flagdata' was used to remove dead antennas and bad data.",
"In addition, the tasks `tfcrop'(http://www.aoc.nrao.edu/~rurvashi/TFCrop/TFCropV1/node2.html) and `rflag'(https://casa.nrao.edu/Release4.2.2/docs/userman/UserMansu167.html) were used to flag the radio frequency interference (RFI).",
"The calibration [30] was performed to remove the instrumental and atmospheric effects from the measurement.",
"The final part of processing was imaging.",
"The continuum imaging of the target source was done using CASA task `tclean'.",
"Finally, a few rounds of `phase only' mode and two rounds of `amplitude-phase' self-calibrations were run.",
"We fit a Gaussian to determine the GRB flux density at the GRB position.",
"The flux densities are shown in Table .",
"Sample radio images of GRB171205A at bands 5, 4 and 3 are shown in Figure REF .",
"The errors in flux densities in Table show only the statistical errors.",
"We also add 15% of flux densities in quadrature to account the uncertainties due to calibration and other systematics for GMRT bands during our model fit.",
"We closely follow the procedure shown in [14].",
"The table list the details of observations and flux densities at various epochs.",
"Figure: GMRT images of GRB171205A.",
"Panel a) Band 5 detection on 2019 May 13.",
"Panel b) Band 4 detection on 2019September 10.Panel c) Band 3 detection on 2019 May 13.",
"The contours in black lines show detection significance and are at 20σ\\sigma , 40σ\\sigma and 60σ\\sigma for bands 4 and 5; and at 6σ\\sigma , 8σ\\sigma , and 10σ\\sigma for band 3, where σ\\sigma is the map rms of corresponding images.",
"For the images displayed, they are following: Band 5, 17 μ\\mu Jy, Band 4, 30 μ\\mu Jy and Band 5, 80 μ\\mu Jy.We also extract Swift-XRT 0.3–10 keV flux light curve from the the Swift online repositoryhttps://www.swift.ac.uk/xrt_products/00794972.",
"The light curve post day 1 indicate a photon index $\\Gamma =1.94^{+0.23}_{-0.22}$ and column density $N_H=(1.2^{+0.8}_{-0.7} \\times 10^{21}$ cm$^{-2}$https://www.swift.ac.uk/xrt_live_cat/00794972/.",
"This is in addition to Galactic column density of $5.89\\times 10^{20}$ cm$^{-2}$ .",
"We converted X-ray flux into into 1 keV spectral flux density using this photon index.",
"Swift data covered observations until 2020 May 27.",
"In addition, we analysed two archival data from Chandra ACIS-S on 2018 Feb 14 and 2018 June 29 (PI: Margutti).",
"We used Chandra Interactive Analysis of Observations software [28] task specextractor to extract the spectra, response and ancillary matrices.",
"We used CIAO version 4.6 along with CALDB version 4.5.9.",
"The HEAsofthttp://heasarc.gsfc.nasa.gov/docs/software/lheasoft/ package Xspec version 12.1 [3] was used to carry out the analysis of the Chandra spectra.",
"The GRB was detected in the first obserations at 71 days with 0.3–10 keV unabsorbed flux of $(11.25\\pm 3.38)\\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ .",
"The second Chandra observation on second epoch, i.e 206 days, resulted in a 3-$\\sigma $ upper limit $< 3.98\\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ .",
"lcccrc 1 Flux Densities of GRB171205A 0pt Date of Observation Band Frequency Days since explosion Flux Densitya Map RMS MHz (mJy) $\\mu $ Jy/beam 2017 Dec 10.10 5 1255 4.79 $<0.07$ 24 2017 Dec 11.02 4 648 5.71 $<$ 0.06 20 2017 Dec 19.91 5 1265 14.60 $0.64\\pm 0.05$ 17 2017 Dec 26.90 5 1265 21.59 1.00$\\pm $ 0.07 17 2017 Dec 28.91 4 607 23.60 $<$ 0.39 130 2018 Jan 16.94 5 1265 42.63 1.75$\\pm $ 0.05 17 2018 Feb 12.85 5 1370 68.54 3.04$\\pm $ 0.09 41 2018 Feb 17.85 4 607 73.54 1.37$\\pm $ 0.15 115 2018 Mar 20.68 5 1352 105.37 5.79$\\pm $ 0.08 34 2018 Jun 08.44 3 402 185.13 $2.94\\pm 0.41$ 106 2018 Jun 10.68 5 1255 187.37 3.07$\\pm $ 0.11 26 2018 Jun 11.42 4 745 188.11 $2.46\\pm 0.49$ 141 2018 Jul 13.47 5 1250 220.16 3.55$\\pm $ 0.12 17 2018 Jul 15.35 4 610 222.043.13$\\pm $ 0.18 66 2018 Jul 23.35 3 402 230.04 $2.30\\pm 0.22$ 64 2018 Jul 26.57 5 1265 233.26 $2.60\\pm 0.08$ 26 2018 Jul 28.35 4 750 235 2.92$\\pm $ 0.31 62 2018 Aug 24.32 5 1265 262.01 $3.32\\pm 0.06$ 33 2018 Aug 25.22 4 607 262.91 $1.06\\pm 0.16$ 81 2018 Sep 21.35 5 1265 290.04 $3.03\\pm 0.10$ 59 2018 Sep 23.16 3 402 291.85 $2.01\\pm 0.15$ 63 2018 Oct 20.28 5 1250 319 3.18 $\\pm $ 0.13 52 2018 Oct 26.09 3 400 324.78 1.77 $ \\pm $ 0.27 131 2018 Oct 26.33 4 607 325.02 $2.61\\pm 0.26$ 143 2018 Dec 21.88 3 402 381.57 $1.61\\pm 0.31$ 49 2018 Dec 22.02 5 1255381.71 $2.83\\pm 0.06$ 21 2018 Dec 22.14 4 610 381.83 $1.52\\pm 0.17$ 52 2019 Feb 26.98 4 610 448.67 1.52$\\pm $ 0.21 79 2019 Feb 26.70 3 402 448.39 1.05$\\pm $ 0.16 40 2019 Feb 26.87 5 1250 448.56 1.87$\\pm $ 0.06 17 2019 May 13.49 5 1250 524.18 1.74$\\pm $ 0.04 17 2019 May 13.62 3 402 524.31 1.43 $\\pm $ 0.26 79 2019 May 13.77 4 607 524.46 2.09 $\\pm $ 0.18 81 2019 Sep 10.16 5 1250 643.85 1.74 $\\pm $ 0.03 19 2019 Sep 10.31 3 402 644.00 1.67 $\\pm $ 0.21 50 2019 Sep 10.43 4 750 644.12 1.93 $\\pm $ 0.10 30 2019 Dec 09.91 4 647 734.60 1.38 $\\pm $ 0.12 37 2019 Dec 10.02 5 1265 734.71 1.71 $\\pm $ 0.04 20 2019 Dec 10.19 3 402 734.88 1.30 $\\pm $ 0.17 59 2020 June 26.37 4 648 934.06 $1.29\\pm 0.11$ 20 2020 June 26.49 5 1255 934.18 $1.23\\pm 0.04$ 21 2020 June 29.44 3 402 937.13 $1.12\\pm 0.16$ 55 aThe uncertainties reflect the statistical errors."
],
[
"GRB afterglow Model",
"We use external synchrotron model for the GRB afterglow emission, which arises due to the interaction between the GRB outflow and the surrounding CBM [34].",
"As the outflow moves into the CBM, a `forward shock' or a `blast wave' shock moving into the CBM, and a `reverse shock' moving into the ejected outflow are created.",
"These shocks have the ability to accelerate charged particles to relativistic speeds via Fermi acceleration [47].",
"Radio afterglow emission is expected to be synchrotron emission arising due to these relativistic charged particles in the shocks in the presence of magnetic fields.",
"The evolution of the blast wave is `self-similar' [7], and the dynamics depends only on the density of the CBM and the blast wave energy.",
"The CBM is usually modelled to be one of the two forms, a constant density medium and a wind like density medium [18].",
"The number density profile of the ambient medium is usually modelled as a powerlaw $n \\propto r^{-k}$ .",
"For constant density case, the parameter $k=0$ and $n=n_0$ .",
"For wind like case, the mass flows radially outwards at uniform speed and rate from the of GRB progenitor giving $k=2$ ; hence for a mass-loss rate from the progenitor $\\dot{M}_W$ and the progenitor wind velocity $V_W$ , the density can be defined as [33], [19]: $n=\\frac{\\dot{M}_W}{4\\pi r^{2} m_p V_W}=3\\times 10^{35} A_* r^{-2}$ Where $m_p$ is the mass of the proton, $A_*$ is in units of $3\\times 10^{35}$ cm$^{-1}$ (or $5\\times 10^{11}$ g cm$^{-1}$ for mass-density), corresponding to $\\dot{M}_{W,-5}/V_{W,3}$ .",
"Here $\\dot{M}_{W,-5}$ is mass loss rate in $10^{-5}M_{\\odot }/\\rm {yr}$ and $V_{W,3}=10^3\\,\\rm {km/s}$ .",
"The GRB wideband afterglow spectrum has several breaks characterized by various characteristic frequencies, namely $\\nu _a$ (the transition from optically thick to thin region, i.e.",
"synchrotron self-absorption (SSA) peak), $\\nu _c$ (synchrotron cooling frequency) and $\\nu _m$ (frequency corresponding to minimum injected Lorentz factor).",
"In the fast cooling regime the frequency ordering is $\\nu _m > \\nu _c$ , while in the slow cooling regime the ordering is opposite.",
"Generally afterglow modelling is done in the slow cooling regime, where the most relevant ordering in radio frequencies in first few days are $\\nu _a \\le \\nu _m \\le \\nu _c$ ; and then $\\nu _m \\le \\nu _a \\le \\nu _c$ at later times [35].",
"However, there has been evidence for fast cooling in some GRBs with high density environments [17].",
"The afterglow spectra evolves as $F_{\\nu } \\propto \\nu ^2$ ($\\nu <\\nu _a$ ) and $F_{\\nu } \\propto \\nu ^{1/3}$ ($\\nu _a < \\nu < \\min (\\nu _m,\\nu _c)$ ) for both fast as well as the slow cooling.",
"In the regime $\\min (\\nu _m,\\nu _c) < \\nu < \\max (\\nu _m,\\nu _c) $ , the evolution changes to $F_{\\nu } \\propto \\nu ^{-1/2}$ and $F_{\\nu } \\propto \\nu ^{-(p-1)/2}$ for fast and slow cooling regimes, respectively, and then evolves as $F_{\\nu } \\propto \\nu ^{-p/2}$ for $ \\nu > \\max (\\nu _m,\\nu _c) $ , where $p$ is the usual power law index showing particle number distribution with energy in non-thermal emission case.",
"In addition to standard afterglow models, shock breakout model too has been favoured for low-luminosity GRBs, where the breakout of a shock travelling through the stellar envelope may be responsible for gamma-ray emission.",
"Due to decreasing density of the stellar matter outwards, the shock breakout velocity increases and may become relativistic.",
"[56] have defined a ‘relativistic breakout closure relation’ between the breakout energy $E_{\\rm bo}$ , temperature $T_{\\rm bo}$ and duration $t_{\\rm bo}^{\\rm obs}$ , i.e.",
"$( t_{\\rm bo}^{\\rm obs}/20 \\,\\rm s) \\sim (E_{\\rm bo}/10^{46}\\, \\rm erg)^{1/2} (T_{bo}/50 \\, \\rm keV)^{-2.68}$ .",
"This relation has been found to be followed by several low luminosity GRBs.",
"For GRB 171205A, the relation gives $\\sim 80$ s, which is roughly 1/3rd of the observed duration.",
"However, the large uncertainties in the $E_p$ and $E_{\\rm iso}$ [24] do not rule out this model."
],
[
"Inputs from high frequency data",
"The Swift-XRT light curve covers epoch until day $\\sim 902$ .",
"In addition, Chandra observations are on day 71 and day 205 (Fig.",
"REF ).",
"We fit a power law to the X-ray data post day 1.",
"This is to avoid possible energy injection due to central engine activities at $\\le 1$ d. The light curve is fit with power law index $\\alpha =1.06\\pm 0.06$ .",
"[24] also find a power law fit with index $1.08$ to the light curve post $\\sim $ 1 day, consistent with our value.",
"This is typical value for the standard X-ray light curve decay before jet break [59], suggesting that no jet-break was seen until the last Swift-XRT epoch.",
"The jet break time is thus constrained to $t_{\\rm jet}\\ge 71\\,\\rm d$ [66], i.e.",
"the last detected Chandra epoch.",
"The photon index of the Swift-XRT data is a photon index $\\Gamma =1.94^{+0.23}_{-0.22}$ https://www.swift.ac.uk/xrt_live_cat/00794972/.",
"This suggests $\\beta =0.94^{+0.23}_{-0.22}$ .",
"The values of $\\alpha $ and $\\beta $ are consistent with X-ray frequency ($\\nu _X$ ) being above the cooling frequency ($\\nu _X>\\nu _c$ ), for both wind as well as ISM density profiles.",
"The X-ray temporal index is consistent with $\\alpha =(3p-2)/4$ and spectral index $\\beta =p/2$ .",
"These values gives $p$ to be $2.08\\pm 0.08$ and $1.88\\pm 0.46$ , respectively, which, within the errorbars, are consistent with each other.",
"Within the errorbars, this is also consistent with $\\nu _X<\\nu _c$ for a ISM like medium where $\\alpha =3(p-1)/4$ and $\\beta =(p-1)/2$ .",
"We also plot early time 230 and 345 GHz light curves GRB 171205A, taken from [82].",
"We estimate the realistic errorbars in the flux density values by adding 5% of the flux density in quadrature to the map rms to account for the systematic errors.",
"The mm highest flux density is significantly higher than the peak flux densities at uGMRT bands, indicating most likely presence of wind like medium.",
"The mm light curves are jointly fit with a broken powerlaw with common post peak index.",
"The data are best fit with post break index $-1.37\\pm 0.07$ (bottom panel of Fig.",
"REF ).",
"For above values of $p$ , this is consistent with an evolution of $(3p-1)/4$ , for $\\nu _m<\\nu _{\\rm mm}<\\nu _c$ for wind density profile.",
"The pre-break indices $-0.17\\pm 0.14$ and $-0.54 \\pm 0.31$ for 230 and 345 GHz bands, respectively.",
"The breaks in 230 and 345 GHz are at $3.14\\pm 0.28$ day and $6.12\\pm 1.17$ day, respectively.",
"At the epochs of the breaks, the flux density of the 230 and 345 GHz light curves are $43.83\\pm 3.29$ and $14.85\\pm 2.86$ mJy, respectively.",
"We also fit the 345 GHz data with a single powerlaw model to determine the significance of the break.",
"The single powerlaw model fits with an index of $-1.05\\pm 0.14$ , however results in a much larger reduced-$\\chi ^2$ value of 2.47 as compared to 0.81 in the broken power law case.",
"For the broken powerlaw model, the characteristic frequency evolves with an index of $+0.71\\pm 0.12$ ., hence the breaks can not be due to passage of $\\nu _m$ , which decreases with time.",
"The only possibility is the breaks are being due to $\\nu _c$ in the wind medium, where $\\nu _c \\propto t^{1/2}$ .",
"Thus the data indicate.",
"$\\nu _c\\approx 230$ GHz on day 3.14 and $\\nu _c\\approx 345$ GHz on day 6.12.",
"However, there is one concern.",
"The pre-break evolution is rather flat.",
"This could be reconciled if mm bands are close to passage of $\\nu _m$ .",
"Another possibility is that if there is a reverse shock component which is contributing to the mm band at the early epoch.",
"The thick shell reverse shock model during the reverse shock crossing phase will evolve as $-(p-2)/2$ , for $\\nu _{\\rm mm}>\\nu _c$ in the slow cooling phase and $\\nu _{\\rm mm}>\\nu _m$ in fast cooling phase.",
"The characteristic frequency evolves with an index of $+0.71\\pm 0.12$ , this is in between evolution of the $\\nu _c\\, (\\propto t^{1/2}) $ in the forward shock and the evolution of $\\nu _c\\, (\\propto t^{1}) $ in the reverse shock.",
"This also suggests the possibility of reverse shock contributing to the afterglow model.",
"If so, then the breaks at day 3.14 and 6.12 in 230 and 345 GHz are artificial breaks and may not reflect the cooling break.",
"Figure: Top panel: Near-simultaneous spectra on day ∼4\\sim 4, day ∼11\\sim 11 and day ∼909\\sim 909.",
"Here we use only the cm and mmdata.",
"Bottom panel: The same as top panel but including the X-ray data as well.",
"The spectra are fit with smoothed single broken powerlaw fits.",
"For day 4.2, we also fit the data witha powerlaw with two breaks.",
"The double broken powerlaw fit is indicated with a black continuous line and the single one with dashed blue line.We also combine the early epoch published cm and mm data from [82], late epoch Australian Telescope Compact Array (ATCA) data from [46] with the uGMRT and the X-ray data, to obtain near simultaneous spectra on around day 4, day 11 and day 909 (Fig.",
"REF ).",
"For the early epoch spectrum, the VLA data are on day 4.3 and the ALMA data are on day 5.2.",
"We use the nearest epoch temporal evolution to derive the ALMA values on day 4.3.",
"In the figure, we plot the values on day 5.2 as well as their derived values on day 4.3 along with the VLA data.",
"We do not use optical data as supernova signatures appeared by day 3 and hence the optical data are likely to be heavily contaminated by the underlying supernova.",
"We fit a smoothed single broken powerlaw (SBPL) and smoothed double broken powerlaw (DBPL) fits to day 4.3 spectrum.",
"We used formalism of [34] for the treatment of smoothening of powerlaws at the break frequencies.",
"We first fit only the cm and mm data.",
"For day 4.3, the best fit SBPL model peaks at $30.41\\pm 4.08$ GHz and afterwards evolves as $-0.41\\pm 0.25$ .",
"Here we have fixed the pre-break spectral index to 2.",
"However, even when we use this as a free parameter, the best-fit index is consistent with 2 within errorbars.",
"The DBPL model fits the data well and give the breaks at $7.25\\pm 2.11$ GHz and $44.41\\pm 3.62$ GHz with post break indices $1.26\\pm 0.12$ , and $-0.41\\pm 0.02$ , respectively.",
"The peak flux density is $43.80\\pm 1.12$ mJy.",
"We also fit SBPL for the spectrum on day 11 and 909.",
"The data indicates a fit $-0.68\\pm 0.02$ with peak $<90$ GHz, with the peak flux density $>16.2$ mJy.",
"The pre-break index is $0.43\\pm 0.46$ .",
"The spectrum on day 909 are fit with pre-break and post-break indices of $0.15\\pm 0.13$ and $-0.90\\pm 0.04$ , respectively, and a peak at $1.55\\pm 0.48$ GHz.",
"The peak flux at day 909 is $1.15\\pm 0.16$ mJy.",
"Now we carry out the above fits including the X-ray data as well.",
"The data to mm to X-ray data are fit by indices close to $-1$ .",
"These values are $-1.01\\pm 0.18$ and $-1.01\\pm 0.02$ for the first two spectra, respectively.",
"For the day 909, the X-ray upper limit do not constrain the model.",
"The mm to X-ray indices consistent with the X-ray spectral indices between 0.3–10 keV.",
"This indicates that the cooling frequency ($\\nu _c$ ) is probably close to mm values if $ \\nu _X > \\nu _c$ .",
"Due to lack of optical data, we cannot constrain the cooling frequencies more precisely.",
"The peak flux density and the frequency of the peak in the three cases are $37.79\\pm 3.95$ mJy at $160.93\\pm 35.31$ GHz, $>16.2$ mJy at $<90$ GHz and $1.15\\pm 0.16$ mJy at $1.55\\pm 0.48$ GHz., respectively.",
"Our analysis indicate that the peak of the spectra on day 4 and day 909 are due to $\\nu _m$ or $\\nu _a$ .",
"Between day 4 and 909, the peak flux density evolves as $-0.63\\pm 0.02$ .",
"This clearly rules out ISM model and supports the wind model.",
"[88] has estimated kinetic energy in the synchrotron afterglow, $E_K$ , from the X-ray data at the time of shallow to normal decay, which for GRB 171205A is 1.05 day.",
"For $\\nu _X>\\nu _c$ , $E_K$ is independent of density and is only weakly depends on $B$ and $p$ , and therefore an ideal regime to measure $E_K$ .",
"One can then derive $E_K$ from the X-ray band using Eq 9 of [88], which in this case is $E_K\\approx 1.4\\times 10^{50} (\\epsilon _B/0.01)^{2-p/2+p} (\\epsilon _e/0.1)^{4(1-p)/2+p}$ erg."
],
[
"Initial inferences from uGMRT data",
"We plot uGMRT radio light curves and fit them jointly with a smoothed broken powerlaw (SBPL) model.",
"We allow the normalization to vary but fix the indices before and after the peak.",
"We show the light curves in Fig.",
"REF .",
"In the figure, the band 4 and 5 values are scaled by factors of 10 and 100, respectively, for clarity.",
"The indicies before and after the peak are $1.37\\pm 0.20$ and $-0.72\\pm 0.06$ .",
"The evolution before peak at uGMRT frequencies ($\\nu _{\\rm radio}$ ) is in rough agreement with wind slow cooling case for $\\nu _{\\rm radio}<\\nu _a$ for $\\nu _a< \\rm min(\\nu _m,\\nu _c)$ ; the wind fast cooling phase for $\\nu _a<\\nu _c<\\nu _m$ if the observing frequency is in the transition zone between $\\nu _{\\rm radio}<\\nu _a$ to $\\nu _a<\\nu _{\\rm radio}<\\nu _c$ , as well as wind slow cooling phase for $\\nu _m<\\nu _a<\\nu _c$ for transition between $\\nu _{\\rm radio}<\\nu _m$ to $\\nu _m<\\nu _{\\rm radio}<\\nu _a$ .",
"The post peak index is rather shallow and is consistent only with the wind fast cooling in the regime $\\nu _a<\\nu _{\\rm radio}<\\nu _c$ .",
"It is rather shallow for the post jet-break or non-relativistic evolution.",
"However, as we discuss in the next section, the shallow decline of radio light curve in seen in other GRBs as well, and other emission components may contribute to it.",
"From our fits, the epochs of the peak flux densities are $5.19\\pm 0.05$ and $3.23\\pm 0.15$ mJy, respectively, in bands 5 and 4 on day $101.08\\pm 9.58$ and day $153.71\\pm 40.31$ , respectively.",
"In band 3, the data are optically thin, which constraints the peak to be $<185.13$ day and flux $>2.81$ mJy.",
"The peak flux density evolves as $-1.13 \\pm 0.76$ .",
"While this value has a large error, it is consistent with the evolution in the stratified wind within 2-$\\sigma $ and most likely rule out all the models involving ISM, where $F_{\\rm max} \\propto t^0$ .",
"This also rules out $1<p<2$ case for which $F_{\\nu , \\rm max}$ is expected to remain constant [33].",
"Figure: The uGMRT bands 5, 4 and 3 radio light curves (the band 4 and 5 values are scaled by factors of 10 and 100).",
"The data are best fit with pre- and post peak spectral indices of 1.37±0.201.37\\pm 0.20 and -0.72±0.06-0.72\\pm 0.06."
],
[
"Model fits",
"In this section, we carry out detailed model fits to the uGMRT data.",
"The simple closure relations seem to suggest that the GRB is in a slow cooling regime with wind density medium.",
"We fit the data with all three scenarios, i.e.",
"wind density slow cooling regime $\\nu _a<\\nu _m<\\nu _c$ , $\\nu _m<\\nu _a<\\nu _c$ and wind density fast cooling regime $\\nu _a<\\nu _c<\\nu _m$ .",
"We also account for the non-standard wind profile, i.e.",
"$k \\ne 2$ .",
"For this we keep $k$ as a free parameter and adopt the expressions shown in [83].",
"Models are proposed for different stages of blastwave expansion.",
"The precise coefficients associated with specified model parameters can be computed by numerical simulations.",
"For decelerating blastwave and adiabatic wind like case the parameter dependencies of peak flux density and characteristic frequencies are given by [33] $F_{\\nu , \\rm max} & = 17.0\\,\\rm {mJy} \\left(\\frac{1+z}{2}\\right)^{3/2}\\left(\\frac{E_{\\rm KE}}{10^{49}\\, {\\rm erg}}\\right)^{1/2} \\left(\\frac{\\epsilon _{B}}{0.01}\\right)^{1/2}\\nonumber \\\\& \\left(\\frac{A_*}{0.1 \\dot{M}_{W,-5}/V_{W,3} }\\right) \\left(\\frac{D}{100\\,\\rm Mpc}\\right)^{-2} \\left(\\frac{t}{10\\,\\rm d}\\right)^{-1/2}$ $\\nu _c & =1.7\\times 10^{17}\\,\\rm {Hz}\\left(\\frac{1+z}{2}\\right)^{-3/2}\\left(\\frac{E_{\\rm KE}}{10^{49}\\, {\\rm erg}}\\right)^{1/2} \\nonumber \\\\& \\left(\\frac{A_*}{0.1 \\dot{M}_{W,-5}/V_{W,3} }\\right) ^{-2}\\left(\\frac{\\epsilon _{B}}{0.01}\\right)^{-3/2}\\left(\\frac{t}{10\\,\\rm d}\\right)^{1/2}$ $\\nu _m & =2.7\\times 10^{7}\\,\\rm {Hz} \\,G^{\\prime }(p) \\left(\\frac{1+z}{2}\\right)^{1/2}\\left(\\frac{E_{\\rm KE}}{10^{49}\\, {\\rm erg}}\\right)^{1/2} \\nonumber \\\\& \\left(\\frac{\\epsilon _{e}}{0.1}\\right)^{2}\\left(\\frac{\\epsilon _{B}}{0.01}\\right)^{1/2}\\left(\\frac{t}{10\\,\\rm d}\\right)^{-3/2}$ here the parameters $\\epsilon _B$ and $\\epsilon _e$ are microscopic parameters indicating fraction of energy into magnetic field and relativistic electrons, respectively and $E_{\\rm KE}$ is the afterglow kinetic energy.",
"$G^{\\prime }(p)=0.053 ((p-2)/(p-1))^2$ .",
"The expression $\\nu _a$ for the three regimes are: $\\nu _a{\\left\\lbrace \\begin{array}{ll}=4.3\\times 10^{9}\\,{ \\rm Hz}\\, g^{\\prime }(p) \\left(\\frac{1+z}{2}\\right)^{-2/5} \\left(\\frac{E_{\\rm KE}}{10^{49}\\, {\\rm erg}}\\right)^{-2/5} \\\\\\left(\\frac{A_*}{0.1 \\dot{M}_{W,-5}/V_{W,3} }\\right) ^{6/5}\\left(\\frac{\\epsilon _{B}}{0.01}\\right)^{1/5} \\left(\\frac{\\epsilon _{e}}{0.1}\\right)^{-1} \\left(\\frac{t}{10\\,\\rm d}\\right)^{-3/5}, & \\\\ \\text{for} \\ \\nu _a<\\nu _m<\\nu _c.",
"\\\\=4.9\\times 10^{8}\\,{ \\rm Hz}\\, g^{\\prime \\prime }(p) \\left(\\frac{1+z}{2}\\right)^{\\frac{p-2}{2(p+4)}} \\left(\\frac{E_{\\rm KE}}{10^{49}\\, {\\rm erg}}\\right)^{\\frac{p-2}{2(p+4)}}\\\\\\left(\\frac{A_*}{0.1 \\dot{M}_{W,-5}/V_{W,3} }\\right) ^{\\frac{4}{p+4}}\\left(\\frac{\\epsilon _{B}}{0.01}\\right)^{\\frac{p+2}{2(p+4)}} \\left(\\frac{\\epsilon _{e}}{0.1}\\right)^{\\frac{2(p-1)}{p+4}} \\left(\\frac{t}{10\\,\\rm d}\\right)^{-\\frac{3(p+2)}{2(p+4)}}, & \\\\ \\text{for} \\ \\nu _m<\\nu _a<\\nu _c.\\\\=6.0\\times 10^{4}\\,{ \\rm Hz} \\, g^{\\prime \\prime \\prime }(p) \\left(\\frac{1+z}{2}\\right)^{3/5}\\left(\\frac{E_{\\rm KE}}{10^{49}\\, {\\rm erg}}\\right)^{-2/5} \\\\\\left(\\frac{A_*}{0.1 \\dot{M}_{W,-5}/V_{W,3} }\\right) ^{11/5}\\left(\\frac{\\epsilon _{B}}{0.01}\\right)^{6/5}\\left(\\frac{t}{10\\,\\rm d}\\right)^{-8/5}, & \\\\ \\text{for} \\ \\nu _a<\\nu _c<\\nu _m.",
"\\\\\\end{array}\\right.",
"}$ Here are expressions for $g^{\\prime }(p)$ , $g^{\\prime \\prime }(p)$ and $g^{\\prime \\prime \\prime }(p)$ are derived in [33] and for $p=2.1$ .",
"Using these expressions, the temporal and spectral evolution in different transitions regimes can be derived and are mentioned in [33].",
"We also carry out modeling for the shock breakout cases, for which we adopt methodology of [5].",
"In this model, due to decreasing outer ejecta density, the outer parts of the shock envelope are faster and less energetic, and inner parts are slower and more energetic.",
"As slower material catches up with the decelerating ejecta it re-energizes the forward shock and the blastwave energy continuously changes with time.",
"Thus this model can be treated as a series of successive shells which accelerate and catch up to the boundary and hence explain the increasing afterglow energy via continuous injection [5].",
"If $\\eta $ is the ratio of the prompt to afterglow energy ($\\eta \\equiv E_{\\rm \\gamma .iso}/ E_{\\rm k,iso})$ , then for shock breakout case, we parameterize the model as $\\eta _{\\rm eff} E_{\\rm k,iso} = E_{\\rm \\gamma .iso}(\\frac{tA_*}{1+z})^s$ , where $s$ is free parameter characterising energy injection [5].",
"Using this expression along with the scalings for $\\nu _a$ , $\\nu _m$ , $\\nu _c$ and $F_{\\rm max}$ for various regimes, provided by [5], leads to the following closure relations for a wind like medium: Case 1 ($\\nu _a<\\nu _m<\\nu _c$ ): $F_{\\nu }(t) \\propto {\\left\\lbrace \\begin{array}{ll} \\nu ^2 t^{s+1}, & \\text{for} \\ \\nu <\\nu _a \\\\\\nu ^{1/3} t^{\\frac{s}{3}}, & \\text{for} \\ \\nu _a<\\nu <\\nu _m \\ \\\\\\nu ^{-(p-1)/2} t^{-\\frac{3p-1 -s(p+1)}{4}} , & \\text{for} \\ \\nu _m<\\nu <\\nu _c \\\\\\nu ^{-p/2} t^{-\\frac{3p-2 -s(p+2)}{4}}, & \\text{for}\\ \\nu >\\nu _c\\end{array}\\right.",
"}$ Case 2 ($\\nu _m<\\nu _a<\\nu _c$ ): $F_{\\nu }(t) \\propto {\\left\\lbrace \\begin{array}{ll} \\nu ^2 t^{s+1}, & \\text{for} \\ \\nu <\\nu _m \\\\\\nu ^{5/2} t^{\\frac{7+3s}{4}}, & \\text{for} \\ \\nu _m<\\nu <\\nu _a \\ \\\\\\nu ^{-(p-1)/2} t^{-\\frac{3p-1 -s(p+1)}{4}} , & \\text{for} \\ \\nu _a<\\nu <\\nu _c \\\\\\nu ^{-p/2} t^{-\\frac{3p-2 -s(p+2)}{4}}, & \\text{for}\\ \\nu >\\nu _c\\end{array}\\right.",
"}$ Case 3 ($\\nu _a<\\nu _c<\\nu _m$ ): $F_{\\nu }(t) \\propto {\\left\\lbrace \\begin{array}{ll} \\nu ^2 t^{2+s}, & \\text{for} \\ \\nu <\\nu _a \\\\\\nu ^{1/3} t^{-\\frac{2(s+1)}{3}}, & \\text{for} \\ \\nu _a<\\nu <\\nu _c \\ \\\\\\nu ^{-1/2} t^{-\\frac{s+1}{4}} , & \\text{for} \\ \\nu _c<\\nu <\\nu _m \\\\\\nu ^{-p/2} t^{-\\frac{3p-2 +(p+2)s}{4}}, & \\text{for}\\ \\nu >\\nu _m\\end{array}\\right.",
"}$ We now fit the uGMRT data with both standard isotropic afterglow and shock break-out afterglow models.",
"We use smoothed broken powerlaw models for various regimes following the recipe of [34].",
"We use $z=0.036$ , $D=163$ Mpc.",
"We define $E_{\\rm KE}=E_{\\rm \\gamma .iso}/\\eta $ , and keep $\\eta $ ($\\eta _{\\rm eff}$ for SBO) as the free parameter.",
"The parameters $p$ , $A^*$ , $\\epsilon _B$ and $\\epsilon _e$ are also free parameters.",
"With the inputs above, we carry out the detailed modelling using Markov chain Monte Carlo (MCMC) fitting using the Python package emcee [25].",
"We choose 150 walkers, 2000 steps.",
"Even though the analytical modelling suggests wind like medium, we still start with fits to a constant density medium.",
"The fit results in a high values of reduced-$\\chi ^2$ further ruling out the constant density model.",
"We fit the standard wind model with $k=2$ for all the cases.",
"In addition, we also account for non-standard wind density medium keeping $k$ as a free parameter.",
"Table REF shows the fit statistics for different parameters using the above mentioned models.",
"For $\\nu _a<\\nu _m<\\nu _c$ case, we do not list general $k$ model as this model performed quite poorly for both afterglow as well as shock breakout.",
"$\\nu _a<\\nu _m<\\nu _c$ generally performs very poorly, with shock breakout model performing slightly better than the isotropic afterglow model.",
"In addition, the parameters obtained in this model are rather unphysical.",
"While the fast cooling model gives best reduced $\\chi _{\\nu }^{2}$ , this case is unlikely to be true.",
"The analysis of mm and X-ray light published data have already revealed that $\\nu _c$ lies between the mm and X-ray frequencies.",
"Since $\\nu _c \\propto t^{1/2}$ in the wind model, uGMRT radio frequencies cannot be in the fast cooling regime.",
"The most viable model fits are obtained for $\\nu _m<\\nu _a<\\nu _c$ case.",
"This is quite viable since in wind density provide, $\\nu _m$ evolves faster than $\\nu _a$ and may reach $\\nu _m<\\nu _a$ regime at late epochs [35].",
"Here keeping $k$ as free parameter also results in $k \\sim 2$ .",
"Our model fits are equally good for the standard afterglow model and the shock breakout model, and uGMRT data alone cannot differentiate between the two.",
"In Fig.",
"REF , we show the light curve for a standard wind $k=2$ model for $\\nu _m<\\nu _a<\\nu _c$ case for both isotropic forward shock afterglow as well as the shock breakout afterglow model.",
"|l|ll|llll|llll| Best fit parameters for GRB 171205A uGMRT data 9 0pt Param.",
"2|c|$\\nu _a<\\nu _m<\\nu _c$ 4|c|$\\nu _m<\\nu _a<\\nu _c$ 4|c|$\\nu _a<\\nu _c<\\nu _m$ AG SBO 2|c|AG 2|c|SBO 2|c|AG 2|c|SBO $k=2$ $k=2$ $k=2$ general $k$ $k=2$ general $k$ $k=2$ general $k$ $k=2$ general $k$ $A_*$ $7.37^{+0.95}_{-0.80}$ $2.82^{+0.46}_{-0.39}$ $1.58^{+0.11}_{-0.75}$ $1.11^{+1.03}_{-0.54}$ $2.89^{+1.95}_{-1.26}$ $3.54^{+2.46}_{-1.55}$ $1.69^{+1.15}_{-0.51}$ $0.17^{+0.19}_{-0.09}$ $2.15^{+1.53}_{-0.78}$ $0.22^{+0.16}_{-0.11}$ $\\eta (\\eta _{\\rm eff}$ for SBO) $0.005^{+0.0004}_{-0.0004}$ $0.30^{+0.13}_{-0.12}$ $0.02^{+0.02}_{-0.01}$ $0.02^{+0.03}_{-0.02}$ $0.07^{+0.09}_{-0.05}$ $0.13^{+0.12}_{-0.07}$ $0.014^{+0.002}_{-0.002}$ $0.03^{+0.003}_{-0.003}$ $0.06^{+0.05}_{-0.03}$ $0.03^{+0.02}_{-0.01}$ $\\epsilon _B$ $ 0.94^{+0.05}_{-0.09}$ $0.70^{+0.20}_{-0.20}$ $0.21^{+0.24}_{-0.13}$ $0.11^{+0.17}_{-0.06}$ $0.03^{+0.03}_{-0.02}$ $0.01^{+0.01}_{-0.01}$ $0.17^{+0.14}_{-0.10}$ $0.02^{+0.01}_{-0.01}$ $0.08^{+0.09}_{-0.05}$ $0.01^{+0.01}_{-0.01}$ $\\epsilon _e$ $0.99^{+0.01}_{-0.02}$ $0.78^{+0.16}_{-0.23}$ $0.12^{+0.09}_{-0.06}$ $0.13^{+0.11}_{-0.07}$ $0.24^{+0.17}_{-0.11}$ $0.12^{+0.07}_{-0.06}$ $\\cdots $ $\\cdots $ $\\cdots $ $\\cdots $ $p$ $2.55^{+0.08}_{-0.06}$ $3.87^{+0.10}_{-0.19}$ $2.22^{+0.04}_{-0.04}$ $2.18^{+0.14}_{-0.09}$ $2.23^{+0.04}_{-0.05}$ $2.22^{+0.16}_{-0.13}$ $\\cdots $ $\\cdots $ $\\cdots $ $\\cdots $ $k$ $\\cdots $ $\\cdots $ $\\cdots $ $1.99^{+0.01}_{-0.02} $ $\\cdots $ $1.99^{+0.01}_{-0.01} $ $\\cdots $ $1.91^{+0.02}_{-0.01} $ $\\cdots $ $1.92^{+0.02}_{-0.02} $ $s$ $\\cdots $ $0.55^{+0.04}_{-0.04}$ $\\cdots $ $\\cdots $ $0.13^{+0.08}_{-0.07}$ $0.15^{+0.08}_{-0.08}$ $\\cdots $ $\\cdots $ $0.35^{+0.13}_{-0.16}$ $0.08^{+0.16}_{-0.18}$ $\\chi _{\\nu }^{2}= 7.55$ $\\chi _{\\nu }^{2}=2.58$ $\\chi _{\\nu }^{2}=1.74$ $\\chi _{\\nu }^{2}=1.80$ $\\chi _{\\nu }^{2}=1.72$ $\\chi _{\\nu }^{2}=1.78$ $\\chi _{\\nu }^{2}=1.60$ $\\chi _{\\nu }^{2}=1.52$ $\\chi _{\\nu }^{2}=1.58$ $\\chi _{\\nu }^{2}=1.57$ Here AG is the standard isotropic afterglow model and SBO is the shock breakout model.",
"In case of fast cooling, the data are only in the regime $\\nu _a$ to $\\nu _c$ ( 2 to 1/3 transition of spectra), which we don't have $p$ dependencies in temporal or spectral slopes.",
"The only $p$ dependency is in the expression of $\\nu _a$ via that ratio of $G(p)$ parameter which we have taken to be of order unity for $p\\sim 2.1$ .",
"The $\\epsilon _e$ dependency is also not there as $\\nu _m$ is unconstrained.",
"Figure: Upper left panel: Light curves of GRB171205A in radio regime using the slow cooling model (ν m <ν a <ν c \\nu _m<\\nu _a<\\nu _c) and standard wind (k=2k=2).",
"Upper right panel: Posterior Distributions of parameters for this model.",
"Lower left panel: Light curves of GRB171205A in radio regime using the slow cooling model (ν m <ν a <ν c \\nu _m<\\nu _a<\\nu _c) with standard wind (k=2k=2) and shock breakout scenario.",
"Lower right panel: Posterior Distributions of parameters for this model.In the left panels, the blue, red and green points are observed data in band 5 , band 4 and band 3 respectively which are included in the fit.",
"The points with arrow have only the upper limit of flux.",
"The lines are the best fits.",
"In the right panels, the 2D plots show the joint probability distribution of any two parameters.",
"The contours are at 0.5σ\\sigma , 1σ\\sigma , 2σ\\sigma , 3σ\\sigma .",
"The middle dotted lines in the 1D parameter distribution is the median value of posterior followed by 1σ\\sigma and 2σ\\sigma lines on both sides.",
"σ\\sigma is standard deviation of the corresponding distribution."
],
[
"Properties of GRB 171205A from radio modelling",
"The peak radio flux density of GRB 171205A at 1.3 GHz is $\\sim 10^{29}$ erg s$^{-1}$ Hz$^{-1}$ .",
"This is two orders of magnitude fainter than cosmological GRBs at this frequency (Fig.",
"REF ).",
"However, these values are comparable to other low-luminosity GRBs, e.g.",
"GRB 031203 [70], 980425 [43].",
"Figure: Plot of 1.4 GHz luminosities of canonical GRBs taken from .",
"Here we overlay the uGMRT 1.3 GHz measurementsfor GRB 171205A.",
"Our values are at least two orders of magnitude smaller than canonical GRBs.The value of $A_*$ in the standard afterglow and the SBO models are $1.58^{+0.11}_{-0.75}$ and $2.89^{+1.95}_{-1.26}$ , respectively, which, assuming a wind velocity of 1000 km s$^{-1}$ , translate to mass-loss rates of $1.58^{+0.11}_{-0.75}\\times 10^{-6} $ M$_\\odot \\,\\rm yr^{-1}$ and $2.89^{+1.95}_{-1.26}\\times 10^{-6} $ M$_\\odot \\,\\rm yr^{-1}$ , respectively, for the two models.",
"The nature of the surrounding ambient medium reflects on the progenitor nature of GRBs.",
"It is expected that the progenitors of long GRBs are massive stars (Wolf-Rayet) and in most of the cases a long GRB is associated with a supernova [43], [86].",
"Another evidence for massive star progenitors is that the long GRBs generally have star-forming host galaxies [87].",
"In such a case, one expects the association with a wind like circumburst medium.",
"However, several GRBs from massive stars collapse have shown homogeneous density [58], [59].",
"A constant density medium can be produced around a massive star if the wind faces a shock termination [19].",
"The low frequency observations present here provide a unique opportunity to determine the nature of the circumburst medium of GRB 171205A and establish that GRB 171205A exploded in a wind like environment.",
"At uGMRT frequencies, the optically thick to thin transition peak arises at a long time ($t > 100$ d) after the burst, indicating a relatively high density medium.",
"This may be created due to a large stellar mass-loss rate or a low wind velocity.",
"Some previous works [20] have shown that the large mass-loss rate for Wolf-Rayet stars are associated with large metallicity of the medium.",
"Thus GRBs in wind medium can be potential tools for studying metallicity variation at different redshifts.",
"The uGMRT light curve declines as $\\sim t^{-0.7}$ .",
"This indicates that there is no jet break until 3 years.",
"There are several explanations for the lack of jet breaks in some GRBs.",
"In the cases of GRB 980326 and GRB 980519, [35] have argued that a wind medium can dilute the jet break even for highly collimated bursts.",
"The jet break will be absent if the radio emission indeed arises from a quasi-spherical afterglow, such as that due to shock breakout [56] or cocoon [54].",
"In case of GRB 030329, [6] have argued that radio emission may be arising from two components, a narrow jet, surrounded by a wider component (e.g.",
"cocoon) and the radio emission is being dominated by the wider component.",
"However, the requirement of this model is that the contribution to the radio afterglow from the narrow jet may be negligible.",
"X-ray observations cover the period of $\\sim 200$ days and show no indication of a jet break at least until the last detection on day $\\sim 70$ .",
"Using $t_j >71$ d, gives a limit $\\theta _j > 1.2$ radians for the AG model and $\\theta _j > 1.9$ for SBO model [57], [84].",
"[49] have shown that ejecta kinetic energy profiles in stripped enveloped supernovae vary based on different explosion mechanisms.",
"While stripped-envelope supernovae have a steep dependence $E_K \\propto (\\Gamma \\beta )^{-5.2}$ indicating no central engine activity, relativistic supernovae, sub-energetic GRBs with SBO mechanisms are flatter with $E_K \\propto (\\Gamma \\beta )^{-2.4}$ showing weak activity from central engine.",
"Canonical GRBs, on the other hand, follow $E_K \\propto (\\Gamma \\beta )^{-0.4}$ typical of jet-driven explosions with long-lasting central engines.",
"Our radio modeling and the relativistic treatment of [4], results in $E_K \\approx 1.1 \\times 10^{51}$ ergs and $\\Gamma \\beta \\sim 1 $ .",
"For the non-relativistic supernova component, we use values from [39], i.e.",
"supernova kinetic energy $E_K=2.4 \\times 10^{52}$ ergs and ejecta velocity 55000 km s$^{-1}$ .",
"Using these values, $E_K \\sim (\\Gamma \\beta )^{-1.9}$ .",
"While GRB 171205A is a sub-energetic GRB, it follows the energy-velocity profile somewhere between canonical GRBs and the SBOs.",
"These arguments suggest that the jet and shock-breakout both may play an important role in the late time afterglow emission in GRB 171205A.",
"Since we have radio light curve peaks at two uGMRT frequencies, we could also estimate some more parameter evolutions.",
"The relativistic energy under equipartition assumption at the two epochs are $E_{\\rm Eq}=3.6\\times 10^{48}$ ergs and $4.9\\times 10^{48}$ ergs [4].",
"Thus there is an indication of enhancement of energy $E \\propto t^{0.48}$ .",
"This along with flatter light curve decays may also be explained if there is an energy injection from the central engine to the shock.",
"For an injection luminosity $L(t)=L_0(t/t_0)^{-q}$ , $E \\propto t^{1-q}$ .",
"This implies $q=0.52$ .",
"However, our best fits result in much larger value of $q$ ($s=0.13$ , $q=1-s\\equiv 0.87$ .",
"This may imply that if energy injection, it is not continuous and probably lasted for a small amount of time.",
"The equipartition size obtained from the above formulation [4] at the epochs of two peaks in band 5 and band 4 follow R(t) $\\propto $ $t^{0.49}$ .",
"We note that for ISM and Wind, density profiles, $R$ follows as $R\\propto t^{1/4}$ and $R\\propto t^{1/2}$ , so it also points towards a wind like medium surrounding GRB171205A.",
"Our uGMRT observations cover the period of around 1000 days.",
"However, our data does not suggest the GRB to be in the Newtonian regime yet (Fig.",
"REF ).",
"This is not uncommon for low-luminosity GRBs [37].",
"We note that the value of $\\Gamma \\beta $ indicates mildly relativistic outflow.",
"Hence it is likely that the GRB is making a transition into Newtonian regime soon."
],
[
"Shallow decay of radio afterglow",
"We note that the decay of the radio afterglow is much shallower than that of the X-ray afterglow.",
"The shallowness of radio lightcurves was first pointed out by [60].",
"For a reasonable afterglow parameters, they estimated that the afterglow is supposed to cross $\\nu _m$ at around 10 days for 10 GHz and follow a decay slope of $(3p-1)/4$ for a wind medium, which was not the case for some GRBs, e.g.",
"GRB 991216 and GRB 000926.",
"They explored the difference between the radio and the optical decay indices could be caused by the fact that the injection frequency remains above the radio domain ($\\sim $ 10 GHz), or a different population of electrons, or a variability of microparameters.",
"Finally, they concluded that a long-lived reverse shock in the radio regime could cause this flattening.",
"However, this is unlikely in GRB 171205A as a strong persistent reverse shock requires a low wind density [65].",
"Here the late rising of the radio afterglow suggests a comparatively high density medium which is against the previous statement.",
"So, this scenario can be excluded.",
"[40] noted that radio afterglows of some GRBs deviate at late times and low frequencies from the standard model, and attempted to explain it with the two-component jet model, a narrow jet core and a wider cocoon surrounding the jet.",
"[34] have explained this flattening due to counter-jet which becomes visible when turning sub-relativistic.",
"While such two components should result in a bump, for stratified wind medium, the revelation of counter-jet is more gradual, causing mild flattening.",
"An energy injection event or a different component dominating the radio emission can also produce this flattening.",
"In case of slightly off-axis jet, the early radio emission could possibly be from the cocoon, the accelerated polar ejecta and at a late phase the contribution from the off-axis jet coming to the line of sight can increase the total radio flux [21].",
"The lack of X-ray data at such late times prevents us from directly distinguishing between these scenarios.",
"A population of quasi-thermal electrons has also been argued as one of the reasons [85], which would mainly dominate at radio frequencies, as it would result in increased $\\nu _a$ and suppressing radio emission below this.",
"In GRB 171205A, this has been tentatively supported from the polarization measurements.",
"As per [82], the mm data revealed 0.27% level linear-polarisation which is a factor of 4 smaller than optical polarization measurements.",
"This has been explained as Faraday depolarization by non-accelerated, cool electrons in the shocked region.",
"However, one cannot rely on these results due to dispute of the detection claimed by [45]."
],
[
"Origin of radio emission",
"There have been suggestions that sub-energetic bursts are simply canonical GRBs viewed off-axis [53].",
"However, such bursts will have two distinguishing characters, a) low $E_p$ b) a rise in the afterglow energy while the shocked ejecta gradually comes into our line of sight.",
"In GRB 171205A, the afterglow energy increases slightly from $3.6\\times 10^{48}$ erg to $4.9\\times 10^{48}$ erg between $\\sim 100$ and $\\sim 200$ days.",
"However, the $E_p$ is comparable to that of canonical GRBs.",
"Additionally, off-axis jet is a geometric effect, which would result in a frequency independent break in the light curve, which has not been seen for GRB 171205A, ruling out the off-axis model [24].",
"Though a jet somewhat off-axis is not ruled out.",
"[24] found that GRB 171205A is an outlier of the Amati relation, as are some other low redshift GRBs, and its emission mechanism should be different from that of canonical, more distant GRBs.",
"There are two models to explain the electromagnetic emission in low-luminosity GRBs, central engine driven [49], [37] and shock-breakout driven [43], [56], [4], [75].",
"An issue with a purely shock breakout model is the requirement of high $\\gamma $ -ray efficiency, for a quasi-spherical outflow.",
"Another issue is generation of such relativistic quasi-spherical outflow.",
"One can envisage a situation where some fraction of the supernova ejecta is accelerated to relativistic speeds to provide this quasi-spherical relativistic outflow.",
"However, [79] have shown that only a fraction ($\\sim 10^{-4}$ ) of supernova energy goes in relativistic ejecta.",
"[54] has suggested an alternative scenario where a choked jet in a low-mass envelope can put a significant energy into a quasi-spherical, relativistic flow.",
"Our uGMRT model fits are incapable of distinguishing between canonical afterglow versus SBO afterglow models.",
"We check the applicability of this model for GRB 171205A.",
"In this model, the expanding outflow is considered to harbour a series of successive shocks, which accelerate and catch up to the boundary and hence explain the increasing afterglow energy via continuous injection.",
"In this model, eventually the total energy should reach around $2.4 \\times 10^{52}$ erg [39], the kinetic energy of the associated supernova.",
"Generally one does not see such a large amount of energy as radio observations do not cover epochs late enough.",
"However, our radio observations cover a period of nearly 1000 days.",
"The SBO model fit gives $E_K\\approx 3.4\\times 10^{50}$ erg, two orders of magnitude smaller than the one predicted in pure SBO model.",
"[77] carried out multiwavelength hydrodynamical modelling of GRB 171205A in the framework of post shock break-out relativistic SN ejecta-CBM interaction scenario.",
"While they claimed that this model worked well for GRB 171205A and favour the wind model, we note some problems with their model.",
"They had to introduce a centrally concentrated CBM with a sudden density drop to explain the available radio and X-ray data.",
"however, our work includes radio measurements upto 3 years and do not show a sudden density drop.",
"The pure shock breakout model also has some other problems.",
"A shock breakout model predicts a $\\gamma $ -ray emission lasting $\\ge 1000 s$ , lower $E_p$ (not exceeding 50 keV), a large absorption column density, a late time soft X-ray emission, and comparable energy in the X-ray emission and the prompt $\\gamma $ -ray flare.",
"We note that the X-ray spectrum shows an intrinsic hydrogen column density $N_H=7.4^{+4.1}_{-3.6} \\times 10^{20}$ cm$^{-2}$ [24].",
"This intrinsic column density is at the low-end of low-luminosity GRB distribution, even among low redshift Swift-XRT GRBs where the mean is $N_H=2.4 \\times 10^{21}$ cm$^{-2}$ at $z < 0.2$ [2].",
"For the observations post day 1 onwards, there may be indication of a slightly higher column density $N_H=1.2^{+0.8}_{-1.7} \\times 10^{21}$ cm$^{-2}$ https://www.swift.ac.uk/xrt_live_cat/00794972/.",
"But there is no particular indication of significant spectral softening, except of a slight indication of $\\Gamma =1.63\\pm 0.30$ to $\\Gamma =1.94\\pm 0.23$ .",
"The total X-ray energy from the first observation onwards until the last detection is at least an order of magnitude smaller than the prompt energy.",
"[24] have claimed the presence of a thermal component.",
"While shock breakout from supernovae is the most favourable model for a thermal component [56], [75], late time photospheric emission from jet [27], or thermal emission from cocoon [78] can also explain this component.",
"If we assume the blackbody component to be significant, it comprises of 20% flux and has a temperature of 89 eV [24].",
"This corresponds to a radius $R =(E_{\\rm iso} /aT_{BB}^4)^{1/3} \\approx 1.4 \\times 10^{13}$ cm ($a$ is the radiation density constant).",
"This is much larger than the typical Wolf-Rayet star radius, but can be explained if the shock expands in a non-spherical manner.",
"Alternatively, [54] suggested the presence of an optically thick stellar envelope further away from the star, from where the breakout happens.",
"We have seen that our observations are inconsistent with a pure shock breakout due to the short duration, higher $E_p$ , shallow $E$ vs $\\Gamma \\beta $ relation, low column density and a much larger break-out radius predicted by the thermal component.",
"It can also not be explained as merely a canonical GRB seen off-axis.",
"We show below that both these components contribute towards the radio afterglow.",
"GRB 171205A is the first GRB in which direct signatures of a cocoon has been seen [39].",
"This is rare because generally a line-of-sight jet is much brighter than the associated cocoon, hiding the cocoon signatures.",
"An off-axis jet can reveal itself by the associated supernova, but the cocoon signatures are long gone by the time supernova may be discovered.",
"Ideally cocoon signatures are visible only in slightly off-axis GRBs.",
"In GRB 171205A, the cocoon was identified by the broad absorption features overlapped on the supernova spectrum [39] They also estimated, from the energy deposited in cocoon, that the jet was quite energetic [39] .",
"This may imply that we may be seeing a slightly off-axis jet, which enabled us to reveal the cocoon, not overshadowed by the bright jet.",
"With this knowledge, it is likely that radio afterglow has from both components, the sub-relativistic wider cocoon and a slightly off-axis jet.",
"The cocoon radio emission dominates the GRB emission at early times when the GRB jet is off-axis.",
"Later the additional flux contribution comes when the jet spreads sideways and comes in our line of sight.",
"In such a case, the total radio flux can also be large compared to on-axis GRBs since the cocoon and the jet carry comparable energy [21].",
"[39] could not distinguish between the cocoon along with slightly off-axis jet versus only cocoon emission, where the faint $\\gamma $ -rays are the predicted signal of the cocoon breaking out of the stellar envelope.",
"However, our radio modelling rules out the pure shock break-out model in favour of cocoon along with a slightly off-axis jet.",
"As per theoretical models, the indicated speed of the ejecta is also consistent with the sub-relativistic speeds expected in this model."
],
[
"Comparison with other low-$z$ low-luminosity GRBs",
"With $E_{\\rm iso}=2.3\\times 10^{49}$ erg and $z=0.0368$ , GRB 171205A is one of the few low-$z$ ($z\\lesssim 0.1$ ), low-luminosity GRBs.",
"Other GRBs in this category are GRB 980425, [31], GRB 031203, [67], GRB 060218 [9] and GRB 100316D [74].",
"Like other low-luminosity GRBs, the spectrum of GRB 171205A can be fit by a simple power-law model, however, the photon index $\\Gamma \\sim 1.94$ is harder than these GRBs, except that of GRB 031203 [67].",
"Even amongst the intrinsically sub-energetic bursts, GRB 171205A resembles more closely with GRBs 980425 and 031203 and not GRBs 060218 and 100316D.",
"GRBs 980425 and 031203 were difficult to realise in the shock break out models due to their relatively hard spectra, shorter durations and larger $E_p$ , though lack of thermal equilibrium may accommodate it [41].",
"GRBs 060218 and 100316D stand out due to their large durations of a few thousand seconds and lower peak energies $E_p < 50$ keV.",
"The shorter $T_{90}$ and higher $E_{\\rm p}$ for GRB 171205A are are comparable to the respective values for GRBs 980425 and 031203.",
"A major difference between GRB 171205A and other low-luminosity GRBs is the absence of intrinsic absorption column density.",
"All other GRBs in this class show significant neutral hydrogen column density $N_H \\sim (6-7)\\times 10^{21}$ cm$^{-2}$ , as opposed to an order of magnitude lower $N_H$ for GRB 171205A [24].",
"The higher column density is considered an essential feature for supernova shock break out models in low-luminosity GRBs.",
"The peak radio flux density of GRB 171205A at 1.3 GHz is $\\sim 10^{29}$ erg s$^{-1}$ Hz$^{-1}$ .",
"This value is comparable to other low-luminosity GRBs, e.g.",
"GRB 031203 [70].",
"However, the X-ray luminosity at 10 hr is $L_x\\approx 2\\times 10^{42}$ erg s$^{-1}$ , which is 4 times smaller than that of GRB 031203 [70].",
"This discrepancy is even more significant for a wind-like medium, since the measured peak radio luminosities are at 1.3 GHz and 8.5 GHz bands for GRB 171205A and GRB 031203, respectively.",
"It has been found that the X-ray afterglow decays slowly in low-luminosity GRBs.",
"For example, the X-ray emission in GRB 031203 followed $F_{\\rm xray}(t) \\propto \\nu ^{-0.8} t^{-0.4}$ [70], and for GRB 980425 $F_{\\rm xray}(t) \\propto \\nu ^{-1} t^{-0.2}$ [43].",
"This is opposed to the canonical GRBs with $F_{\\rm xray}(t) \\propto \\nu ^{-1.3} t^{-1}$ .",
"[5] have shown that a flat temporal decay of the X-ray light curve can be explained in the shock-break out model, although dominance of an underlying supernova has also been suggested as a possible reason for this flatness [70].",
"With $F_{\\rm xray}(t) \\propto \\nu ^{-0.94} t^{-1.1}$ , GRB 171205A is more like canonical GRBs.",
"On the contrary, the X-ray spectral index was very soft for GRB 060218A with $F_{\\rm xray}(t) \\propto \\nu ^{-2.2} t^{-1.1}$ , which softened even further at later epochs [71].",
"GRB 171205A did not so that much spectral softening.",
"In addition, contrary to smooth light curves of low-luminosity GRBs, the X-ray light curve of GRB 171205A revealed three temporal breaks [24].",
"The lack of jet break seems to be a common feature of low-luminosity GRBs.",
"This may either indicate much wider angle ejecta responsible for afterglow emission, or stratified medium which can dilute the jet break in GRBs and hide its signatures.",
"Another defining feature of these GRBs is the presence of a thermal component, which has also been seen in GRB 171205A [24].",
"Though this feature is quite common in GRBs associated with SNe [9], [74], their origin is still a matter of debate.",
"Shock breakout from the relativistic supernova shell is the most favourable model [75], however, late time photospheric emission from a jet [27], or a thermal emission from a cocoon [78] can also explain this component.",
"However, it appears that the presence of a thermal component is not a unique feature of low-luminosity GRBs.",
"[73], [72] analysed a sample of canonical GRBs and found the evidence of thermal component in a significant number of GRBs, though large radii associated with the blackbody emission argue against the shock breakout model.",
"Finally while low-luminosity GRBs are considered to be a different class, [54] has provided a unified picture of low-luminosity GRBs and cosmological GRBs, in terms of a key difference, namely, the existence of an extended low-mass envelope.",
"In the unified model, the envelope is present in the low-luminosity GRBs, but absent in cosmological GRBs.",
"The lack of envelope allows a jet to launch without any resistance in cosmological GRBs, but the extended envelope in low-luminosity GRBs smothers the jet and deposits a large amount of energy in the stellar envelope, driving a mildly relativistic shock producing low-luminosity GRBs via shock breakout.",
"Our model does not support this picture."
],
[
"Summary and Conclusions",
"In this paper, we have presented the sub-GHz observations of a low-luminosity GRB 171205A upto around 1000 days.",
"These are the best sampled low frequency light curves of any GRB.",
"For the first time we report a lowest frequency (250–500 MHz) detection of a GRB.",
"Our light curves cover a period of two years.",
"While we are able to see the light curve peak transitions in bands 5 and 4, we missed the peak in band 3 due to the lack of early data.",
"The radio data suggest that the GRB exploded in a wind medium.",
"At sub-GHz frequencies at late time, the afterglow is in $\\nu _m<\\nu _a<\\nu _c$ regime, a common phenomenon seen in late time radio afterglows.",
"The late time Chandra X-ray measurements constrain the jet break to be $t_{\\rm jet}>71$ days.",
"Even though GRB 171205A has significant similarities with other low-luminosity GRBs, it deviates from this class in many respects.",
"We suggest that the radio emission arises from both a cocoon and a jet, where the jet is slightly off-axis [39].",
"The early epoch radio emission is dominated by the cocoon surrounding the jet, while the late time radio emission has contribution from the jet.",
"The flatter radio light curves, harder GRB X-ray spectrum, large $E_p$ , shorter $T_{90}$ , kinetic energy - ejecta velocity relation and time dependence of various parameters are consistent with this picture.",
"Our work emphasises the importance of the nature of CBM, which is critical in understanding the evolution of GRB afterglows, and are best revealed by low-frequency radio measurements.",
"We thank the referee for very insightful comments which helped improve the manuscript significantly.",
"We thank Dipankar Bhattacharya for carefully reading the manuscript and providing critical suggestions.",
"B.M.",
"and P.C.",
"acknowledge support of the Department of Atomic Energy, Government of India, under the project no.",
"12-R&D-TFR-5.02-0700.",
"P.C.",
"acknowledges support from the Department of Science and Technology via SwaranaJayanti Fellowship award (file no.DST/SJF/PSA-01/2014-15).",
"We thank the staff of the GMRT that made these observations possible.",
"GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research."
]
] | 2012.05166 |
[
[
"MetaInfoNet: Learning Task-Guided Information for Sample Reweighting"
],
[
"Abstract Deep neural networks have been shown to easily overfit to biased training data with label noise or class imbalance.",
"Meta-learning algorithms are commonly designed to alleviate this issue in the form of sample reweighting, by learning a meta weighting network that takes training losses as inputs to generate sample weights.",
"In this paper, we advocate that choosing proper inputs for the meta weighting network is crucial for desired sample weights in a specific task, while training loss is not always the correct answer.",
"In view of this, we propose a novel meta-learning algorithm, MetaInfoNet, which automatically learns effective representations as inputs for the meta weighting network by emphasizing task-related information with an information bottleneck strategy.",
"Extensive experimental results on benchmark datasets with label noise or class imbalance validate that MetaInfoNet is superior to many state-of-the-art methods."
],
[
"Introduction",
"Deep Neural Networks (DNNs) have achieved remarkable success on various computer vision tasks due to their powerful capacity for modeling complex input patterns.",
"Despite their success, the vulnerability of DNNs has been extensively illustrated in many previous studies [2], [5], [10], [17], [28], [30].",
"One important drawback of DNNs is that DNNs could easily overfit to biased training data, where the distribution of training data is inconsistent with that of the evaluation data.",
"There are many different forms of distribution mismatch, leading to poor performance in generalization.",
"A typical example is class imbalance [7], [15], where the distribution of data across the classes is not equal in the training set.",
"This issue will sometimes lead to biased training models that does not perform well in practice [3], [31], [42].",
"Another popular type of distribution mismatch is label noise, which usually happens when the training set is collected from a crowdsourcing system [49] or search engines [4].",
"It has been shown that a standard CNN can fit any ratio of label noise in the training set and eventually leads to poor generalization performance [2], [53].",
"Therefore, robust learning from these biased data has become an important and challenging problem in machine learning and computer vision.",
"Figure: Sample weight distributions on trainingdata of different instatiations (different inputs) of meta weighting network under the 40% Flip-1 label noise case.",
"It is ideally expected that higher weights are assigned to clean data (green) while lower weights are assigned to noisy data (red).Sample reweighting algorithms [20], [23], [26] are widely studied for robust learning from biased data.",
"The main idea is to generate different weights for training losses to different samples.",
"Some existing algorithms design specific weighting function with training loss by simple rules, such as monotonically increasing [23], [26] or monotonically decreasing [8], [20], [53], which means taking samples with larger or smaller loss values as more important ones.",
"However, these methods need to manually design a specific form of weighting function based on certain assumptions on training data.",
"To make the learning more automatic and reliable across various biased settings, a popular research line is meta sample weighting, which learns weights for each sample implicitly [35] or automatically learns an explicit weighting function [37].",
"Specifically, the weighting function could be parameterized as a multilayer perceptron (MLP) network that maps the training loss to sample weight [37], and the training of the parameters could by guided by a small unbiased validation set.",
"For easy reference, we call this instantiation of meta weighting network (MW-Net) as LossNet [37].",
"In this paper, we advocate that choosing proper inputs for MW-Net is crucial for desired sample weights in a specific task, while training loss is not always the correct answer.",
"As shown in Figure REF , we illustrate the sample weight distributions on training data of different instantiations (different inputs) of MW-Net.",
"We can observe that LossNet tends to produce polarized results, and high weights could be assigned to some noisy data.",
"This is because the commonly used cross-entropy loss is known to be highly overconfident [27], [40].",
"Besides, the training loss may not contain rich and flexible information of the original sample for capturing meaningful sample weights, due to the restriction on the fixed loss function.",
"To alleviate this issue, an intuitive method is to use the logits and labels as inputs (instead of training loss with a specified loss function) to build MW-Net (we call such a instantiation of MW-Net as LogitNet).",
"As shown in Figure REF , LogitNet would not produce polarized results, while it could still assign high sample weights to some noisy data.",
"This is because LogitNet may contain too much redundant information with the unprocessed logits.",
"To address this problem, we propose a novel meta-learning algorithm, MetaInfoNet, which automatically learns effective representations as inputs for MW-Net by emphasizing task-related information with an information bottleneck strategy.",
"As shown in Figure REF , our proposed MetaInfoNet can learn smooth and meaningful sample weights, and the trend that higher weights are assigned to clean data while lower weights are assigned to noisy data is clearly demonstrated.",
"Finally, we conduct extensive experiments on simulated and real-world datasets with class imbalance or label noise, to show that MetaInfoNet significantly improves the robustness of deep learning on training data under various biased settings.",
"Empirical results demonstrate that the robustness of deep models trained by our proposed MetaInfoNet is superior to many state-of-the-art methods."
],
[
"Related Work",
"Sample Reweighting Methods.",
"The idea of sample reweighting has been commonly used in the machine learning literature.",
"For example, hard example mining downsamples the majority class and exploits the most challenging examples [26].",
"Similarly, Focal loss emphasizes harder examples by soft weighting [23].",
"On the contrary, self-paced learning (SPL) takes samples with smaller loss values as more important ones firstly [20].",
"Despite their success under some specific settings, these methods need to manually design a particular form of weighting function based on certain assumptions on training data, which might be impractical.",
"Rather than predefined by human experts, MentorNet uses a bidirectional LSTM network to learn a curriculum from data with label noises [16].",
"However, the weighting function of MentorNet is too complicated and would overfit to the biased data.",
"Meta Learning Methods for Robustness.",
"Meta-learning algorithms are introduced to improve robustness of deep learning in the form of sample reweighting.",
"The first work is L2RW [35], which implicitly learns the weights without a pre-defined weighting function, then uses a small unbiased validation set to guide the training of its parameters.",
"In contrast, LossNet [37] parameterizes the weighting function as an MLP network explicitly, mapping from training loss to sample weight.",
"However, the capacity of LossNet is limited by its input with the fixed loss function.",
"Learning with Class Imbalance.",
"In addition to sample reweighting, there are other methods to handle the class imbalance issue in deep learning.",
"For example, some methods try to transfer the knowledge learned from major classes to minor classes [7], [46].",
"The metric learning based methods have also been developed to effectively exploit the tailed data to improve the generalization ability, e.g., triple-header loss [15] and range loss [52].",
"Learning with Label Noise.",
"For handling noisy label issues [9], [13], [21], [24], [29], [39], [44], [45], some other algorithms focus on estimating the label transition matrix.",
"For example, F-correction [32] proposed a loss correction approach by heuristically estimating the noise transition matrix.",
"In these approaches, the quality of noise rate estimation is a critical factor for improving robustness.",
"However, noise rate estimation is challenging, especially on datasets with a large number of classes.",
"Another popular research line of handling noisy labels is to train models on small-loss instances, which can be viewed as a hard version of sample weighting [12], [16], [47].",
"Information Bottleneck.",
"Information Bottleneck (IB) method was initially proposed in [41], where the idea can be formulated as a variational principle of minimizing the mutual information between the input and the learned representation, while preserving the information about the learning task.",
"The IB method has been successfully applied to supervised learning [1], generative modeling [33] and reinforcement learning [33], [43].",
"In this work, we apply the IB algorithm to enforce the input of MW-Net to focus on the relevant information implicitly defined by the learning task."
],
[
"Preliminaries",
"In this section, we introduce the problem setting of learning from biased training data with sample reweighting and the formulation of LossNet [37], which is a representative related work."
],
[
"Problem Setting",
"For multi-class classification with $K$ classes, we suppose the biased training dataset with $N$ samples is given as $\\mathcal {D}^{\\mathrm {train}} = \\lbrace x_i, y_i\\rbrace ^N_{i=1}$ , where $x_i$ is the $i$ -th instance with its observed label as $y_i \\in \\lbrace 1, \\dots , K\\rbrace $ .",
"Similar to L2RW and LossNet, we assume that there is a small unbiased and clean meta dataset with $M$ samples $\\mathcal {D}^{\\mathrm {val}} = \\lbrace x^{\\mathrm {val}}_i, y^{\\mathrm {val}}_i\\rbrace ^M_{i=1}$ and $M \\ll N$ , representing the meta-knowledge of ground-truth sample-label association.",
"For the classifier to be trained, we denote it as $f_{\\Theta }(x)$ , where $\\Theta $ is the parameters of the classifier.",
"Generally, we optimize the classifier network by minimizing the training loss: $\\frac{1}{N} \\sum ^N_{i=1} L^{\\mathrm {train}}_i(\\Theta ) = \\frac{1}{N} \\sum ^N_{i=1} \\ell (f_{\\Theta }(x_i), y_i)$ , where $\\ell $ denotes the employed loss function (e.g., cross entropy loss), and each input example is weighted equally.",
"For enhancing the robustness of training on the biased training data, we aim to assign weight $w_i$ on the loss of the $i$ -th sample.",
"Without loss of generality, the weighting function can be formulated as $\\mathcal {V}_{\\Phi }((x_i, y_i), f_{\\Theta })$ where $\\Phi $ denotes the parameters of the weighting function.",
"Then the optimal parameter $\\Theta $ of classifier is calculated by minimizing the following weighted loss: $\\begin{split}\\Theta ^{*}(\\Phi ) &= \\underset{\\Theta }{\\arg \\min } \\ \\mathcal {L}^{\\mathrm {train}}(\\Theta ;\\Phi )\\\\&\\triangleq \\underset{\\Theta }{\\arg \\min } \\frac{1}{N} \\sum _{i=1}^{N} w_i L^{\\mathrm {train}}_i(\\Theta ),\\end{split}$ where $w_i = \\mathcal {V}((x_i, y_i), f_{\\Theta }; \\Phi )$ denotes the example weight and $L^{\\mathrm {train}}_i(\\Theta ) = \\ell (f_{\\Theta }(x_i), y_i)$ denotes the loss on the training example $(x_i,y_i)$ ."
],
[
"Formulation of LossNet",
"LossNet [37] formulates the weighting function $\\mathcal {V}$ as an MLP network, and automatically trains it in a meta-learning manner.",
"Specifically, the MLP network learns a mapping from training loss to sample weight, $w_i = \\mathcal {V}_{\\Phi }(L_i^{\\mathrm {train}}(f_{\\Theta }(x_i), y_i))$ , where $\\Phi $ denotes the parameters of the MLP net.",
"In the training process, the optimal parameter $\\Phi ^{*}$ of LossNet is calculated by minimizing the following meta loss: $\\begin{split}\\Phi ^{*} &= \\underset{\\Phi }{\\arg \\min } \\ \\mathcal {L}^{\\mathrm {meta}}(\\Theta ^*(\\Phi ))\\\\&\\triangleq \\underset{\\Phi }{\\arg \\min } \\frac{1}{M} \\sum _{i=1}^{M} L_{i}^{\\mathrm {meta}}(\\Theta ^*(\\Phi )),\\end{split}$ where $L_{i}^{\\mathrm {meta}}(\\Theta ^*) = \\ell (f_{\\Theta ^*}(x^{\\mathrm {val}}_i), y_i^{\\mathrm {val}}))$ denotes the meta loss on the validation example $(x^{\\mathrm {val}}_i,y^{\\mathrm {val}}_i)$ .",
"As we analyzed before, the training loss may not contain rich and flexible information of the original sample for capturing meaningful sample weights, due to the restriction on the fixed loss function.",
"As the commonly used cross-entropy loss is known to be highly overconfident [27], [40], LossNet tends to produce polarized results, and high weights could be assigned to some noisy data."
],
[
"The Proposed Approach",
"In this section, we propose to automatically learn proper inputs for MW-Net, thereby adapting to various biased settings.",
"[t] The Meta Weighting Algorithm 1.1 Training dataset $\\mathcal {D}$ , meta dataset $\\mathcal {D}_m$ , batch size $n,m$ , max iterations $T$ .",
"Parameter $\\Theta $ of classifier $f$ .",
"Initialize parameters of Classifier and MW-Net $t=0$ $\\mathrm {to}$ $T$ $\\lbrace x_i,y_i\\rbrace _{i=1}^n \\leftarrow \\text{SampleMiniBatch}(\\mathcal {D},n)$ $\\lbrace x^{\\mathrm {val}}_i,y^{\\mathrm {val}}_i\\rbrace _{i=1}^m \\leftarrow \\text{SampleMiniBatch}(\\mathcal {D}^{\\mathrm {val}},m)$ $ z_i = f_{\\Theta }(x_i), \\tilde{w} = \\mathcal {V}_{\\Phi ^{(t)}}(z,y)$ $w_i \\leftarrow \\frac{\\tilde{w}_i}{\\sum _{j} \\tilde{w}_j +\\delta \\left(\\sum _{j} \\tilde{w}_j\\right)}$ $\\mathcal {L}^{\\mathrm {train}} = \\frac{1}{n} \\sum _{i=1}^{n} w_i \\cdot \\ell (z_i, y_i)$ $\\nabla \\Theta \\leftarrow \\text{ BackwardAD }\\left(\\mathcal {L}^{\\mathrm {train}}, \\Theta \\right)$ $\\widehat{\\Theta } \\leftarrow \\Theta -\\alpha \\nabla \\Theta $ $\\mathcal {L}^{\\mathrm {meta}} = \\frac{1}{m} \\sum _{i=1}^{m} \\ell (f_{\\widehat{\\Theta }}(x^{\\mathrm {val}}_i), y^{\\mathrm {val}}_i)$ Update MW-Net $\\mathcal {V}$ to $\\Phi ^{(t+1)}$ by $\\mathcal {L}^{\\mathrm {meta}}$ $\\tilde{w} = \\mathcal {V}_{\\Phi ^{(t+1)}}(z,y); w_i \\leftarrow \\frac{\\tilde{w}_i}{\\sum _{j} \\tilde{w}_j+\\delta \\left(\\sum _{j} \\tilde{w}_j\\right)}$ $\\mathcal {L}^{\\mathrm {train}} = \\frac{1}{n} \\sum _{i=1}^{n} w_i L^{\\mathrm {train}}_i(z_i, y_i)$ Update Classifier $f$ to $\\Theta ^{(t+1)}$ by $\\mathcal {L}^{\\mathrm {train}}$"
],
[
"A General Framework for Meta Weighting",
"As shown in Figure REF , we define a general framework for meta weighting, which contains a MW-Net that takes information from the samples $x_i$ and its given labels $y_i$ (instead of training loss) as inputs.",
"To abstract information from the sample $x_i$ , we adopt the outputs of the classifier $f$ , (i.e., $f_{\\Theta }(x_i)$ ) for MW-Net.",
"In this way, LossNet can be interpreted as an instantiation of MW-Net.",
"Intuitively, if we calculate the cross-entropy loss with the logit and its corresponding label, we can exactly recover the LossNet model.",
"The training loss can be seen as a representation of information abstracted from samples $x_i$ and its given labels $y_i$ .",
"The left of Figure REF shows that LossNet is an instantiation of MW-Net.",
"Under the general framework in Figure REF , LossNet could be replaced by a more powerful instantiation of MW-Net, thereby achieving better performance.",
"Similar to L2RW [35] and LossNet [37], calculating the optimal parameters $\\Theta ^*$ and $\\Phi ^*$ requires two nested loops of optimization.",
"To improve the optimization efficiency, we adopt an online strategy to alternatively update the classifier $f$ with $\\Theta $ and the MW-Net $\\mathcal {V}$ with $\\Phi $ through a single optimization loop.",
"We describe the details of the training process in Algorithm .",
"In general, the training process can be separated into three parts: Virtually updating parameters of classifier.",
"In the first part, we randomly sample a mini-batch of training samples ${(x_i, y_i), 1 \\le i \\le n}$ from the training dataset, where $n$ is the mini-batch size.",
"Then the virtual updating of the classifier network parameter can be formulated by moving the current $\\Theta ^{(t)}$ along the descent direction of the objective loss in Eq.",
"(REF ) on a mini-batch training data (lines 5-9) : $\\nonumber \\widehat{\\Theta }^{(t)}(\\Phi ) = \\Theta ^{(t)} - \\alpha \\frac{1}{n} \\left.",
"\\sum _{i=1}^{n} \\mathcal {V}_{\\Phi }((f_{\\Theta }(x_i), y_i)) \\nabla _{\\Theta } L^{\\mathrm {train}}_{i}\\right|_{\\Theta ^{(t)}},$ where $\\alpha $ is the step size.",
"Updating parameters of MW-Net.",
"After receiving the feedback of the virtually updated classifier, i.e., $\\widehat{\\Theta }^{(t)}(\\Phi )$ , the parameter $\\Phi $ of the MW-Net can then be readily updated with Eq.",
"(REF ) calculated on the meta dataset (lines 10-11): $\\Phi ^{(t+1)}=\\Phi ^{(t)}-\\beta \\frac{1}{m} \\left.\\sum _{i=1}^{m} \\nabla _{\\Phi } L_{i}^{\\mathrm {meta}}(\\widehat{\\Theta }^{(t)}(\\Phi ))\\right|_{\\Phi ^{(t)}},$ where $\\beta $ is the step size.",
"Updating parameters of classifier.",
"Then, the updated $\\Phi ^{(t+1)}$ is employed to ameliorate the parameter $\\Theta $ of the classifier (lines 12-14), i.e., $\\nonumber &\\Theta ^{(t+1)}=\\Theta ^{(t)}\\\\\\nonumber &\\quad \\quad \\quad \\quad -\\alpha \\frac{1}{n}\\left.\\sum _{i=1}^{n} \\mathcal {V}_{\\Phi ^{(t+1)}}\\left((f_{\\Theta }(x_i), y_i)\\right) \\nabla _{\\Theta } L_{i}^{\\mathrm {train}}\\right|_{\\Theta ^{(t)}},$ Figure: Three different instantiations of MW-Net, including LossNet, LogitNet, and MetaInfoNet."
],
[
"LogitNet: An Instatiation of MW-Net",
"Now the question becomes how to design MW-Net, for mapping information from the sample and its given label to weight.",
"An intuitive idea is to simply take the model output of $x_i$ , i.e., $f_{\\Theta }(x)$ .",
"For the given label $y_i$ , we employ a label embedding layer to produce an embedding vector $e(y_i)\\in \\mathbb {R}^K$ , which has the same dimension as $f_{\\Theta }(x)$ .",
"It is a fully connected layer that projects the given label to a dense vector.",
"The obtained embedding vector can be seen as a latent vector for describing the properties of a specific class, in the context of MW-Net.",
"Since MW-Net adopts two pathways to extract information from $x_i$ and $y_i$ , it is intuitive to combine the vectors from the two pathways (i.e., $f_{\\Theta }(x)$ and $e(y_i)$ ) by concatenating them.",
"However, a vector concatenation cannot account for any interactions between samples and its given labels.",
"Therefore, we apply an element-wise product to combine them as the input for an MLP in MW-Net, that is, $w_i=\\text{MLP}(f_{\\Theta }(x_i) \\odot e(y_i))$ , where $\\odot $ denotes the element-wise product.",
"In this way, the combined vector can capture the interaction information between the sample and its given label before we feed it into an MLP.",
"For easy reference, we call this instantiation of MW-Net as LogitNet.",
"The structure of LogitNet is shown on the center of Figure REF ."
],
[
"MetaInfoNet: An Instatiation of MW-Net with Information Bottleneck",
"Although LogitNet introduces more information to the weighting function, it also leads to a new problem: the unprocessed logits may contain too much redundant information for the weighting task.",
"To “squeeze out\" the redundant information of the inputs for LogitNet, we propose a novel algorithm by information bottleneck (IB) principle [1], [36], [41], called MetaInfoNet.",
"Instead of directly using the output of classifier $z = f_{\\Theta }(x)$ in LogitNet, we aim to learn a representation $\\psi $ that captures the most relevant information from the logit $z$ and removes redundant formation, with respect to the weighting task.",
"In this way, the weight of sample $x_i$ can be generated by $w_i=\\text{MLP}(\\psi _i \\odot e(y_i))$ .",
"Figure: The detailed structure of IB layer.Concretely, we propose an information-theoretic regularization by limiting the mutual information between the logit and the learned representation: $I(Z;\\Psi ) \\le I_c,$ , where $Z$ is a random variable (for the logit vector) with a probability density function $p_{Z}(z)$ , $\\Psi $ is a random variable (for the learned representation vector) with a probability density function $p_{\\Psi }(\\psi )$ and $I_c$ is a constant that represents threshold of the mutual information.",
"The constraint on mutual information can be viewed as a penalty term restricting the information from the logit vector to the learned representation vector.",
"Consequently, for learning an effective representation $\\psi $ , we add the mutual information constraint for optimizing MW-Net: $\\mathcal {L}^{\\mathrm {meta}}(\\Theta (\\Phi )) = \\sum \\nolimits _{i=1}^M L^{\\mathrm {meta}}_i(\\Theta (\\Phi )), \\text{s.t.}",
"\\ I(Z;\\Psi ) \\le I_c,$ In practice, we propose to minimize the following objective using the information bottleneck Lagrangian: $\\sum \\nolimits _{i=1}^M L^{\\mathrm {meta}}_i(\\Theta (\\Phi )) + \\lambda I(Z;\\Psi ),$ where $\\lambda $ is the Lagrange multiplier.",
"As $\\lambda $ increases, the information from $Z$ to $\\Psi $ becomes denser while keeping relevant to the meta learning task.",
"In the above equation, $I(Z;\\Psi )$ is defined as: $I(Z;\\Psi ) &=\\iint p\\left(z, \\psi \\right) \\log \\frac{p\\left(z, \\psi \\right)}{p\\left(z\\right) p\\left(\\psi \\right)} \\mathrm {d} z \\mathrm {d} \\psi \\\\&=\\iint p\\left(z\\right) p_{ib}\\left(\\psi \\mid z\\right) \\log \\frac{p_{ib}\\left(\\psi \\mid z\\right)}{p\\left(\\psi \\right)} \\mathrm {d} z \\mathrm {d} \\psi \\nonumber $ where $p(z, \\psi )$ is the joint probability of $z$ and $\\psi $ , and $p_{ib}$ denotes the information bottleneck (IB) layer.",
"Unfortunately, computing the marginal distribution $p(\\psi ) = \\int p\\left(\\psi \\mid z\\right) p\\left(z\\right) d z$ is quite challenging as we do not know the prior distribution of $p(z)$ .",
"With the help of variational information bottleneck [1], we use a Gaussian approximation $h(\\psi )$ of the marginal distribution $p(\\psi )$ and view $p_{ib}$ as multivariate variational encoders (see Figure REF ).",
"Since $D_{\\mathrm {KL}}\\left[p\\left(\\psi \\right) \\Vert h\\left(\\psi \\right)\\right] \\ge 0$ , where the $D_{\\mathrm {KL}}$ is the Kullback-Leibler divergence, we expand the KL term and get $\\int p\\left(\\psi \\right) \\log p\\left(\\psi \\right) d \\psi \\ge \\int p\\left(\\psi \\right) \\log h\\left(\\psi \\right) d \\psi $ , an upper bound on the mutual information $I(Z;\\Psi )$ can be obtained via the KL divergence: $\\nonumber I\\left(Z ; \\Psi \\right) & \\le \\int p\\left(z\\right) p_{ib}\\left(\\psi \\mid z\\right) \\log \\frac{p_{ib}\\left(\\psi \\mid z\\right)}{h\\left(\\psi \\right)} d z d \\psi \\\\\\nonumber &=\\mathbb {E}_{z \\sim p\\left(z\\right)}\\left[D_{\\mathrm {KL}}\\left[p_{ib}\\left(\\psi \\mid z\\right) \\Vert h\\left(\\psi \\right)\\right]\\right],$ which further provides an upper bound $\\widetilde{\\mathcal {L}}^{\\mathrm {meta}}(\\Theta (\\Phi ))$ on the regularized empirical objective that we minimize: $\\widetilde{\\mathcal {L}}^{\\mathrm {meta}}(\\Theta (\\Phi )) & = \\sum \\nolimits _{i=1}^M L^{\\mathrm {meta}}_i(\\Theta (\\Phi )) \\\\\\nonumber & + \\lambda \\sum \\nolimits _{j=1}^N\\left[D_{\\mathrm {KL}}\\left[p_{ib}\\left(\\psi _j \\mid z_j\\right) \\Vert h\\left(\\psi _j\\right)\\right]\\right].$ Recall that the learned representation $\\psi $ via IB layer is used to generate the weight $w_i=\\text{MLP}(\\psi _i \\odot e(y_i))$ , then we optimize the MW-Net with the new meta loss in Eq.",
"(REF ) to control the information capacity of the learned representation $\\psi $ , while keeping the task-relevant information.",
"The network structure of the IB layer is shown in Figure REF .",
"Specifically, a linear layer is applied to map logit $z$ to a multivariate normal distribution.",
"Then we use the “re-parameterization\" trick [18] to sample a representation $\\psi $ from the learned distribution, parameterized by a mean vector $\\mu $ and a variance vector $\\sigma $ .",
"Another alternative trick is to implement skip connection by concatenating the representation $\\phi $ and the origin logit $z$ .",
"In such a manner, we emphasize the most relevant information in $\\psi $ , with respect to the learning task, while keeping intact information from the logit $z$ .",
"To keep aligned with the dimension of label embedding for the element-wise product, we add a linear layer to scale the dimension of the learned representation $\\psi $ ."
],
[
"Convergence Analysis",
"Here, we further provide the convergence analysis of MetaInfoNet.",
"We show that MetaInfoNet converges to the critical points of both the meta and training loss function under some mild conditions in the following two theorems.",
"Theorem 1 Suppose the loss function $L$ is Lipshchitz smooth with constant $\\rho $ , and $\\mathcal {V}(\\cdot )$ is differential with a $\\delta $ -bounded gradients and twice differential with its Hessian bounded by $\\mathfrak {B}$ , and the loss function $L$ has $\\gamma $ -bounded gradients with respect to training/meta data.",
"Let the learning rate $\\alpha _t$ (for updating classifier) satisfies $\\alpha _t=\\min \\lbrace 1,\\frac{k}{T}\\rbrace $ , for some constant $k>0$ , such that $\\frac{k}{T}<1$ , and the learning rate $\\beta _t$ (for updating meta weight net) satisfies $\\beta _t=\\min \\lbrace \\frac{1}{L},\\frac{c}{\\sigma \\sqrt{T}}\\rbrace $ for some constant $c>0$ where $\\sigma $ is the variance of drawing uniformly mini-batch sample at random, such that $\\frac{\\sigma \\sqrt{T}}{c}>\\rho $ and $\\sum _{t=1}^{\\infty }\\beta _t\\le \\infty ,\\sum _{t=1}^{\\infty }\\beta _t^2\\le \\infty $ .",
"Then we have $\\nonumber \\lim \\limits _{t\\rightarrow \\infty }\\mathbb {E}\\Big [\\left\\Vert \\nabla _{\\Theta }\\mathcal {L}^{\\mathrm {train}}(\\Theta ^{(t)};\\Phi ^{(t)})\\right\\Vert ^2_2\\Big ] = 0.$ Theorem 2 When the conditions in Theorem REF holds, our proposed algorithm can achieve $\\mathbb {E}[\\left\\Vert \\nabla \\mathcal {L}^{\\mathrm {meta}}(\\Theta ^{(t)})\\right\\Vert _2^2]\\le \\epsilon $ in $\\mathcal {O}(1/\\epsilon ^2)$ steps, more specifically, $\\nonumber \\min _{0\\le t\\le T}\\mathbb {E}[\\left\\Vert \\nabla _{\\Theta } \\mathcal {L}^{\\mathrm {meta}}(\\Theta ^{(t)})\\right\\Vert _2^2]\\le \\mathcal {O}(\\frac{O}{\\sqrt{T}}),$ where $C$ is some constant independent of the convergence process.",
"It is worth noting that we have shown that our proposed method MetaInfoNet and LossNet [37] are different instantiations of MW-Net.",
"Therefore, MetaInfoNet shares the same convergence properties as LossNet if we adopt the same assumptions for analyzing the two methods.",
"We omit the proofs of the two theorems, as they are very similar to those proofs in [37]."
],
[
"Experiments",
"In this section, we first implement experiments on simulated datasets with class imbalance and noisy labels.",
"Then two real-world datasets with biased training data are also used to verify the effectiveness of our proposed algorithms."
],
[
"Experiment setup",
"Datasets.",
"For the class imbalance setting, we verify the effectiveness of our proposed algorithm on the Long-Tailed version of CIFAR10 and CIFAR100 datasets [6].",
"Specifically, we reduce the number of training examples per class according to an exponential function $n = n_i\\mu ^i$ , where $i$ is the class index, $n_i$ is the original number of training samples in the largest class divided by the smallest.",
"Following LossNet [37], we randomly select ten images per class in the validation set as the meta dataset.",
"For noisy label settings, we conduct experiments on simulated noisy datasets, including CIFAR10 and CIFAR100 [19].",
"We study two types of label noise on the training set: 1) Flip noise.",
"Following LossNet [37], we mainly consider the case that the label of each example is flipped to two similar classes with equal probability $\\frac{p}{2}$ (Flip-2).",
"2) Uniform noise.",
"The label of each example is changed to a random class with probability $p$ independently.",
"To verify the effectiveness of our algorithms in various settings, we also compare our algorithms with existing meta reweighting algorithms in a harder case: One class is randomly selected as the similar class with probability $p$ (Flip-1).",
"In these cases, 1000 images with clean labels in the validation set are randomly selected as the meta dataset.",
"To verify the effectiveness of the proposed method on real-world data, we conduct experiments on the Clothing1M dataset [48], containing 1 million images of clothing obtained from online shopping websites with 14 categories, e.g., T-shirt, Shirt, Knitwear.",
"The labels are generated by using surrounding texts of the images provided by the sellers, and therefore contain many errors.",
"We use the 7k clean data as the meta dataset.",
"For preprocessing, we resize the image to $256 \\times 256$ , crop the middle $224 \\times 224$ as input, and perform normalization.",
"Moreover, we also compare our proposed algorithms with existing meta reweighting algorithms on ANIMAL10N dataset [38], which contains 5 pairs of confusing animals with a total of 55,000 images.",
"In our experiments, the training dataset contains 50,000 pictures with noisy labels, and 1000 pictures with clean labels are randomly selected as the meta dataset.",
"The rest 4000 images are set as the test dataset.",
"Network Structure and Optimizer.",
"For experiments on CIFAR10 and CIFAR100, we train ResNet-32 [14] for the class imbalance settings.",
"For the settings of label noise, we adopt WRN-28-10 [51] for uniform noise and ResNet-32 for flip noise.",
"Specifically, SGD optimizer is applied with a momentum 0.9, a weight decay $5 \\times 10^{-4}$ , an initial learning rate 0.1, and the batch size is set to 100.",
"The learning rate of the classifier network is divided by 10 after 80 epochs and 100 epochs (for a total of 120 epochs), and after 30 epochs and 40 epochs (for a total of 50 epochs).",
"For experiments on Clothing1M, we use ResNet-50 pre-trained on ImageNet, following the previous works [32], [37], [39].",
"About the optimizer, we use SGD with a momentum 0.9, a weight decay $10^{-3}$ , and an initial learning rate 0.01, and batch size 32.",
"The learning rate of ResNet-50 is divided by 10 after 5 epochs (for a total 10 epochs).",
"For experiments on ANIMAL10N, we also use ResNet-32.",
"The setting of the SGD optimizer is the same as that on CIFAR10.",
"We run 100 epochs in total, and the learning rate of the classifier network is divided by 10 after 50 epochs and 75 epochs.",
"For the training of meta weighting network, we set the learning rate to $10^{-3}$ and set the weight decay to $5 \\times 10^{-4}$ .",
"About the $\\lambda $ in Eq.",
"(REF ), we search it in $[0, 0.01, 0.03, 0.1, 0.3, 1]$ .",
"Additionally, we can set an interval parameter to control the updating of MW-Net, e.g., when the interval is 10, we update the MW-net every 10 iterations.",
"We implement all methods with default parameters by PyTorch, and conduct all the experiments on NVIDIA Tesla V100 GPUs.",
"We repeated the experiments 5 times with different random seeds for network initialization and label noise generation.",
"Compared Methods.",
"For class imbalance setting, we compare our proposed algorithm with: 1) Standard, which simply uses CE loss to train the DNNs with equal weights; 2) Focal loss [23], which emphasizes harder examples by sample reweighting; 3) Class-balanced [6], which represents one of the state-of-the-arts of the predefined sample reweighting methods; 4) Fine-tuning, which finetunes the standard model on the meta dataset to further enhance its performance; 5) L2RW [35], which leverages an additional meta dataset to adaptively assign weights on training samples; 6) LossNet [37], which automatically learns an explicit loss-weight function in a meta-learning manner.",
"For noisy label setting, the compared methods include: 1) Standard; 2) Bootstrap [34]; 3) S-Model [11]; 4) SPL [20]; 5) Focal Loss [23]; 6) Co-teaching [12]; 7) D2L [25]; 6) Fine-tuning; 7) MentorNet [16]; 8) L2RW [35]; 9) LossNet [37].",
"Table: Test accuracy (%) on long-tailed CIFAR-10 and CIFAR-100.",
"The best results are highlighted in bold.Table: Average test accuracy (%) with standard deviation on synthesized datasets under Flip-2 noise (over 5 trials).",
"The best results are highlighted in bold."
],
[
"Experiments with Class Imbalance",
"Table REF shows the test accuracy of different algorithms on long-tailed CIFAR10 and long-tailed CIFAR100.",
"As we can see, our proposed algorithms work better than all the baselines in most of the long-tailed cases, which demonstrates their robustness against class imbalance.",
"When imbalance factor is 1, e.g., there are the same number of samples in all classes, the fine-tuning method obtains the best performance on both CIFAR10 and CIFAR100, and our algorithms also get comparable performance.",
"As the imbalance factor rises from 10 to 200, the datasets become more and more unbalanced and the accuracy of all algorithms decreases gradually.",
"We observe that LogitNet achieves better or comparable results compared to LossNet in most cases, which indicates introducing more information can improve the effectiveness of MW-Net.",
"Moreover, MetaInfoNet consistently performs better than LogitNet in most cases, verifying the effectiveness of the information bottleneck method while learning the representation.",
"Table: Average test accuracy (%) with standard deviation on CIFAR10 with WRN-28-10 under Unif noise (over 5 trials).",
"The best results are highlighted in bold."
],
[
"Experiments with Noisy Labels",
"Table REF shows the test accuracy of ResNet-32 averaged over 5 repetitions on both CIFAR10 and CIFAR100 under Flip-2 noise.",
"We can observe that MetaInfoNet achieves the best performance across both datasets and all noise rates, indicating the effectiveness of our proposed architecture.",
"When the noise rate is 0%, MetaInfoNet also gets comparable results compared to the Fine-tune baseline while performs better than other meta-reweighting algorithms.",
"Additionally, the results of LogitNet are similar to those of LossNet, which shows that simply introducing more information cannot improve the capacity of meta sample weighting in this case.",
"Table REF presents the test accuracy of WRN-28-10 averaged over 5 repetitions on CIFAR10 under Unif noise.",
"It can be observed that all the meta weighting algorithms, including MetaInforNet, LogitNet, LossNet and L2RW, perform the best among all algorithms, which demonstrates their robustness.",
"The results also indicate that uniform noise is a relatively simple biased setting for meta weighting algorithms.",
"Now we turn to a more difficult noisy setting, Flip-1 noise, that each example's label is flipped to one similar class with total probability $p$ .",
"As shown in Table REF , Best denotes the scores of the epoch where the validation accuracy is optimal, and Last denotes the average accuracy over the last 10 epochs at the end of the training stage.",
"As we can see, LossNet performs poorly even gets lower test accuracy than L2RW, while MetaInfoNet attains more than 8% improvement over LossNet.",
"Moreover, MetaInfoNet keeps the best performance in both CIFAR10 and CIFAR100 and all noise rates.",
"Specifically, in the 20% label noise cases, all four meta weighting algorithms could maintain almost the $best$ accuracy in the last 10 epochs.",
"When it improves to the 40% label noise cases, MetaInfoNet keeps the advantage while the other three algorithms' performance decrease dramatically after reaching the top.",
"The result shows that MetaInfoNet is more robust to label noise compared to the other meta-weighting algorithms and has a better capacity to adapt to different biased settings in training data."
],
[
"Experiments on Real-world Datasets",
"Table REF shows the classification accuracy on the Clothing1M test set.",
"Specifically, MetaInfoNet achieves the best accuracy, while LogitNet performs worse than LossNet.",
"The results show that the IB layer is an efficient way to improving the meta-weighting framework with logits and labels as inputs.",
"Table: Classification accuracy (%) on Clothing1M.We also compare our proposed algorithms to the Finetune baseline and the other meta-weighting algorithms on the ANIMAL-10N dataset, as shown in Table REF .",
"Note that the noise rate is about 8% but it is difficult for even people to distinguish between categories, the performance gains of meta weighting baselines are not huge compared to the Finetune method.",
"In this case, MetaInfoNet still obtains nearly 2% improvement over the other methods on best.",
"After all epochs, MetaInfoNet maintains this advantage over the best baseline method.",
"Table: Classification accuracy (%) on ANIMAL-10N."
],
[
"Conclusion",
"In this paper, we propose an effective approach called MetaInfoNet to improve the robustness of deep neural networks under various biased settings in training data.",
"Compared with current meta reweighting algorithms that directly map loss to weight for each sample, MetaInfoNet could automatically learn effective representations as inputs for the meta weighting network by emphasizing task-related information with an information bottleneck strategy.",
"The Empirical results on simulated and real-world datasets demonstrate that the robustness of deep models trained by our proposed approach is superior to many state-of-the-art approaches in general biased settings, such as class imbalance, label noise, and more complicated real cases."
]
] | 2012.05273 |
[
[
"Quantum Discrimination of Two Noisy Displaced Number States"
],
[
"Abstract The quantum discrimination of two non-coherent states draws much attention recently.",
"In this letter, we first consider the quantum discrimination of two noiseless displaced number states.",
"Then we derive the Fock representation of noisy displaced number states and address the problem of discriminating between two noisy displaced number states.",
"We further prove that the optimal quantum discrimination of two noisy displaced number states can be achieved by the Kennedy receiver with threshold detection.",
"Simulation results verify the theoretical derivations and show that the error probability of on-off keying modulation using a displaced number state is significantly less than that of on-off keying modulation using a coherent state with the same average energy."
],
[
"Introduction",
"Quantum discrimination of two optical quantum states plays a crucial rule in quantum information processing tasks, e.g., continuous-variable quantum key distributions and optical quantum communications [1], [2], [3], [4].",
"Due to the good compatibility with classical optical infrastructures, coherent states [5] generated by lasers are usually employed as the information carriers in quantum communication systems.",
"However, the minimum discrimination error probability (MDEP) of discriminating two coherent states cannot be zero because of the non-orthogonal property of two coherent states [5].",
"To improve the performance of the quantum discrimination, new information carriers using non-coherent states draw much attention recently [6], [7].",
"For example, the problem of discriminating between two noisy photon-added coherent states (PACSs) was addressed in [7].",
"The PACS is generated by sequentially applying the displacement operator and the creation operator on a vacuum state.",
"It was demonstrated that the error probability can be significantly reduced when PACSs instead of coherent states are employed in pulse position modulations [6].",
"Inspired by [7], we focus on the quantum discrimination between two noisy displaced number states (DNSs).",
"The DNS is generated by sequentially applying the creation operator and the displacement operator on a vacuum state.",
"The properties of noiseless DNS were discussed in [8], [9], [10].",
"However, the thermal noise is inevitable in preparing a DNS and the property of a noisy DNS has not been studied yet.",
"Besides, to the best of the authors' knowledge, the problem of discriminating between two noisy DNSs has not been addressed.",
"In this letter, we first address the problem of discriminating between two noiseless DNSs.",
"Then we derive the Fock representation of noisy DNSs and address the problem of discriminating between two noisy DNSs.",
"Using the Fock representation of noisy DNSs, we further prove that the optimal quantum discrimination of two noisy DNSs can be achieved using a Kennedy receiver with a threshold detection.",
"The simulation results verify our theoretical derivations.",
"We also explore the possibility of employing DNSs instead of coherent states in on-off keying (OOK) modulations; and find that the error probability of OOK modulation using a DNS can be significantly reduced compared with the error probability of OOK modulation using a coherent state with the same average energy.",
"The DNS is generated by sequentially applying the creation operator and the displacement operator on a vacuum state; and it can be written as [8], [9] $\\mathinner {|{\\mu ,k}\\rangle }= \\hat{D}(\\mu )\\mathinner {|{k}\\rangle }$ where $\\hat{D}(\\mu )$ is the displacement operator; and $\\mathinner {|{k}\\rangle }$ is the number state containing $k$ photons.",
"The number state decomposition of a DNS can be obtained as [9] $\\mathinner {|{\\mu ,k}\\rangle }=\\sum _{n=0}^{\\infty }b_n\\mathinner {|{n}\\rangle }$ where the coefficient $b_n$ is given by [9] $b_n=\\left\\lbrace \\begin{array}{ll}\\sqrt{\\frac{n!}{k!",
"}}(-\\mu ^*)^{k-n}e^{-\\frac{|\\mu |^2}{2}}L_n^{(k-n)}(|\\mu |^2), \\text{ for } n<k\\\\\\sqrt{\\frac{k!}{n!",
"}}\\mu ^{n-k}e^{-\\frac{|\\mu |^2}{2}}L_n^{(n-k)}(|\\mu |^2), \\text{ for } n\\ge k\\\\\\end{array}\\right.$ and where $L_n^{(a)}(x)$ is the generalized Laguerre polynomial of order $n$ with parameter $a$ .",
"Using the number state decomposition, we can obtain the inner product of two DNSs $\\mathinner {|{\\mu ,k}\\rangle }$ and $\\mathinner {|{\\xi ,h}\\rangle }$ , where we let $h\\ge k$ without loss of generality, as $\\begin{aligned}\\langle \\xi ,h|\\mu ,k \\rangle =&\\mathinner {\\langle {h}|}\\hat{D}(\\mu -\\xi )\\mathinner {|{k}\\rangle }\\\\=&\\sum _{n=0}^{\\infty }\\mathinner {\\langle {h}|}b_n(\\mu -\\xi ,k)\\mathinner {|{n}\\rangle }\\\\=&\\sqrt{\\frac{k!}{h!",
"}}(\\mu -\\xi )^{h-k}e^{-\\frac{|\\mu -\\xi |^2}{2}}L_k^{(h-k)}(|\\mu -\\xi |^2)\\end{aligned}$ where in the first step we have used the properties of displacement operator: $\\hat{D}(\\alpha )=\\hat{D}^{\\dagger }(-\\alpha )$ and $\\hat{D}(\\alpha )\\hat{D}(\\beta )=\\hat{D}(\\alpha +\\beta )$ ."
],
[
"Discriminate Two Noiseless DNSs",
"The key of discriminating between any two quantum states $\\lbrace \\hat{\\rho }_0, \\hat{\\rho }_1\\rbrace $ with prior probabilities $\\lbrace p_0, p_1\\rbrace $ is to find two positive operator-valued measure (POVM) operators $\\lbrace \\hat{\\Pi }_0,\\hat{\\Pi }_1\\rbrace $ that can minimize the discrimination error probability.",
"According to the Helstrom's theory [11], the optimal POVM operators can be obtained as $\\hat{\\Pi }_0=\\sum _{\\lambda _n<0}\\mathinner {|{\\lambda _n}\\rangle }\\mathinner {\\langle {\\lambda _n}|}$ and $\\hat{\\Pi }_1=\\hat{\\mathbb {I}}-\\hat{\\Pi }_0$ , where $\\lambda _n$ and $\\mathinner {|{\\lambda _n}\\rangle }$ are the eigenvalue and the eigenvector of the decision operator $\\hat{\\Delta }=p_1\\hat{\\rho }_1-p_0\\hat{\\rho }_0$ ; $\\hat{\\mathbb {I}}$ is the identity operator.",
"The MDEP of discriminating $\\lbrace \\hat{\\rho }_0, \\hat{\\rho }_1\\rbrace $ is obtained as the Helstrom bound [11] $\\begin{aligned}P_e&=\\frac{1}{2}(1-\\Vert \\hat{\\Delta }\\Vert _1)\\\\&=p_1-\\sum _{\\lambda _n>0}\\lambda _n\\end{aligned}$ where $\\Vert \\hat{A}\\Vert _1=\\text{tr}\\lbrace \\sqrt{\\hat{A}^{\\dagger }\\hat{A}}\\rbrace $ denotes the trace norm of the operator $\\hat{A}$ .",
"For discriminating between two pure states $\\hat{\\rho }_0=\\mathinner {|{\\psi _0}\\rangle }\\mathinner {\\langle {\\psi _0}|}$ and $\\hat{\\rho }_1=\\mathinner {|{\\psi _1}\\rangle }\\mathinner {\\langle {\\psi _1}|}$ , the Helstrom bound (REF ) can be rewritten as $P_e=\\frac{1}{2}-\\frac{1}{2}\\sqrt{1-4p_0p_1|\\left\\langle \\psi _0|\\psi _1 \\right\\rangle |^2}.$ Therefore, the MDEP for discriminating between two noiseless DNSs $\\mathinner {|{\\xi ,h}\\rangle }$ and $\\mathinner {|{\\mu ,k}\\rangle }$ is determined by the inner product $\\langle \\xi ,h|\\mu ,k \\rangle $ .",
"Substituting (REF ) into (REF ), we can obtain the MDEP as $\\begin{aligned}P_e=\\frac{1}{2}-\\frac{1}{2}&\\left\\lbrace 1-4p_0p_1\\frac{k!}{h!",
"}|\\mu -\\xi |^{2(h-k)}e^{-|\\mu -\\xi |^2}\\right.\\\\&\\quad \\times \\left.\\left[L_k^{(h-k)}(|\\mu -\\xi |^2)\\right]^2\\right\\rbrace ^{\\frac{1}{2}}.\\end{aligned}$ From (REF ), we can observe that the perfect discrimination with zero error probability happens in the following two situations: (i) $\\mu =\\xi $ and $h\\ne k$ ; (ii) $L_k(|\\mu -\\xi |^2)=0$ and $h=k\\ne 0$ .",
"Notice that when $h=k=0$ , the two DNSs becomes two non-orthogonal coherent states $\\mathinner {|{\\xi }\\rangle }$ and $\\mathinner {|{\\mu }\\rangle }$ .",
"Then the MDEP approaches zero when $|\\mu -\\xi |^2$ approaches $\\infty $ ."
],
[
"Noisy DNSs",
"A noisy number state $\\hat{\\rho }(k)$ is obtained by applying creation operators on a thermal state $\\hat{\\rho }_{th}$ , which results in $\\hat{\\rho }(k)=\\frac{(\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k}{\\text{tr}\\lbrace (\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k\\rbrace }$ where $\\text{tr}\\lbrace \\cdot \\rbrace $ denotes the trace operation.",
"Then the noisy DNS $\\hat{\\rho }(\\mu ,k)$ is defined as $\\hat{\\rho }(\\mu ,k)\\triangleq \\frac{\\hat{D}(\\mu )(\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k\\hat{D}^{\\dagger }(\\mu )}{N_k}$ where $N_k$ can be obtained as $\\begin{aligned}N_k=\\text{tr}\\lbrace \\hat{D}(\\mu )(\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k\\hat{D}^{\\dagger }(\\mu )\\rbrace =k!",
"(n_t+1)^k\\end{aligned}$ and where $n_t$ is the average number of thermal photons due to the presence of thermal noise.",
"The following theorem presents the Fock representation for a noisy DNS.",
"Theorem 1 (Fock representation) The Fock representation of a noisy DNS $\\hat{\\rho }(\\mu ,k)$ is found to be $\\begin{aligned}\\mathinner {\\langle {n}|}&\\hat{\\rho }(\\mu ,k)\\mathinner {|{m}\\rangle }\\\\&=\\sum _{i=0}^{k}\\sum _{j=0}^{k}I(n\\ge i;m\\ge j)\\frac{(-1)^{i+j}\\binom{m}{j}\\binom{k}{i}}{(k-j)!}\\sqrt{\\frac{n!}{m!",
"}}e^{-\\frac{|\\mu |^2}{n_t+1}}\\\\&\\quad \\times \\frac{|\\mu |^{2(k-j)}(\\mu ^*)^{m-n} n_t^{n-i}}{(n_t+1)^{m+k-j+1}}L_{n-i}^{(m-n+i-j)}\\left(-\\frac{|\\mu |^2}{n_t(n_t+1)}\\right)\\end{aligned}$ where $I(n\\ge i;m\\ge j)$ is an indicator function defined as $\\begin{aligned}I(n\\ge i;m\\ge j)\\triangleq \\left\\lbrace \\begin{array}{ll}1,\\text{ for } n\\ge i \\text{ and } m\\ge j \\\\0,\\text{ otherwise.",
"}\\end{array}\\right.\\end{aligned}$ See Appendix .",
"Using the Fock representation in (REF ), we can obtain the photon statistics of a noisy DNS as $\\begin{aligned}p(n)&= \\mathinner {\\langle {n}|}\\hat{\\rho }(\\mu ,k)\\mathinner {|{n}\\rangle }\\\\&=\\sum _{i=0}^{k}\\sum _{j=0}^{k}I(n\\ge i;n\\ge j)\\frac{(-1)^{i+j}\\binom{n}{j}\\binom{k}{i}}{(k-j)!",
"}e^{-\\frac{|\\mu |^2}{n_t+1}}\\\\&\\quad \\times \\frac{|\\mu |^{2(k-j)} n_t^{n-i}}{(n_t+1)^{n+k-j+1}}L_{n-i}^{(i-j)}\\left(-\\frac{|\\mu |^2}{n_t(n_t+1)}\\right).\\end{aligned}$ Then the average number of photons $n_p$ contained in a noisy DNS can be obtained in the following lemma.",
"Lemma 1 The average number of photons $n_p(\\mu ,k)$ contained in a noisy DNS $\\hat{\\rho }(\\mu ,k)$ is found to be $n_p(\\mu ,k)=|\\mu |^2+k(n_t+1)+n_t.$ See Appendix ."
],
[
"Discriminate Two Noisy DNSs",
"Now we consider the discrimination of two noisy DNSs $\\hat{\\rho }_0=\\hat{\\rho }(\\xi ,h)$ and $\\hat{\\rho }_1=\\hat{\\rho }(\\mu ,k)$ with prior probabilities $p_0$ and $p_1$ , respectively.",
"Here we let $h\\ge k$ without loss of generality.",
"According to the Helstrom bound (REF ), the MDEP is determined by all the positive eigenvalues $\\lambda _n$ of the decision operator $\\hat{\\Delta }=p_1\\hat{\\rho }(\\mu ,k)-p_0\\hat{\\rho }(\\xi ,h)$ .",
"Because the displacement operator is an unitary operator, $\\hat{\\Delta }$ and $\\hat{D}(-\\xi )\\hat{\\Delta }\\hat{D}^{\\dagger }(-\\xi )$ share the same eigenvalues.",
"Then it is readable to show that $\\Vert p_1\\hat{\\rho }(\\mu ,k)-p_0\\hat{\\rho }(\\xi ,h)\\Vert _1=\\Vert p_1\\hat{\\rho }(\\mu -\\xi ,k)-p_0\\hat{\\rho }(0,h)\\Vert _1.$ It is challenging to obtain an analytical form of the eigenvalues $\\lambda _n$ for an arbitrary $\\hat{\\Delta }$ .",
"However, eq.",
"(REF ) indicates that when $\\mu =\\xi $ , we only need to obtain the eigenvalues of $p_1\\hat{\\rho }(0,k)-p_0\\hat{\\rho }(0,h)$ .",
"Using the Fock representation (REF ), we can obtain the eigenvalues of $p_1\\hat{\\rho }(0,k)-p_0\\hat{\\rho }(0,h)$ as $\\begin{aligned}\\lambda _n&=p_1\\mathinner {\\langle {n}|}\\hat{\\rho }(0,k)\\mathinner {|{n}\\rangle }-p_0\\mathinner {\\langle {n}|}\\hat{\\rho }(0,h)\\mathinner {|{n}\\rangle }\\\\&=\\left\\lbrace \\begin{array}{ll}0, \\text{ for } n<k\\\\p_1\\binom{n}{k}\\frac{n_t^{n-k}}{(n_t+1)^{n+1}}, \\text{ for } k\\le n<h\\\\p_1\\binom{n}{k}\\frac{n_t^{n-k}}{(n_t+1)^{n+1}}-p_0\\binom{n}{h}\\frac{n_t^{n-h}}{(n_t+1)^{n+1}}, \\text{ for } n\\ge h.\\end{array}\\right.\\end{aligned}$ Now the key of obtaining a tractable MDEP is to find all the positive eigenvalues.",
"To achieve this, in the following we first introduce the Kennedy receiver with threshold detection [12], and then we prove that the Kennedy receiver with threshold detection can achieve the optimal discrimination and provide a tractable MDEP."
],
[
"Kennedy Receiver with Threshold Detection",
"A Kennedy receiver with threshold detection [12] consists of a displacement operator $\\hat{D}(\\beta )$ and a photon counting process followed by a threshold detection based on the counted photons.",
"The threshold detection is characterized by two POVM operators $\\hat{M}_0=\\hat{\\mathbb {I}}-\\sum _{n=0}^{n_{th}}\\mathinner {|{n}\\rangle }\\mathinner {\\langle {n}|}; \\quad \\quad \\hat{M}_1=\\sum _{n=0}^{n_{th}}\\mathinner {|{n}\\rangle }\\mathinner {\\langle {n}|}$ where $n_{th}$ is the detection threshold of the counting photons.",
"These two POVM operators correspond to the following threshold detection rule $n \\mathop {\\lesseqgtr } \\limits _{\\hat{\\rho }_0}^{\\hat{\\rho }_1} n_{th}.$ If we set the displacement operator as $\\hat{D}(\\beta )=\\hat{D}(-\\mu )$ , then the input states $\\hat{\\rho }(\\mu ,h)$ and $\\hat{\\rho }(\\mu ,k)$ are displaced as $\\hat{\\rho }(0,h)$ and $\\hat{\\rho }(0,k)$ , respectively.",
"Then the error probability of the receiver can be calculated by $\\begin{aligned}P_e&=p_0\\text{tr}\\lbrace \\hat{M}_1\\hat{\\rho }(0,h)\\rbrace +p_1\\text{tr}\\lbrace \\hat{M}_0\\hat{\\rho }(0,k)\\rbrace .\\end{aligned}$ Substituting (REF ) into (REF ), we can obtain $\\begin{aligned}P_e&=p_1-\\sum _{n=0}^{n_{th}}\\left(p_1\\mathinner {\\langle {n}|}\\hat{\\rho }(0,k)\\mathinner {|{n}\\rangle }-p_0\\mathinner {\\langle {n}|}\\hat{\\rho }(0,h)\\mathinner {|{n}\\rangle }\\right)\\\\&=p_1-\\sum _{n=0}^{n_{th}}\\lambda _n.\\end{aligned}$ The optimal threshold $n_{th}$ is obtained by minimizing the error probability in (REF ).",
"The following optimal discrimination theorem guarantees that the Kennedy receiver with optimal threshold $n_{th}$ can always achieve the MDEP.",
"Theorem 2 (Optimal discrimination) The optimal discrimination of two noisy DNSs $\\hat{\\rho }(\\mu ,h)$ and $\\hat{\\rho }(\\mu ,k)$ can be achieved by the Kennedy receiver with threshold detection, where the displacement operator is $\\hat{D}(-\\mu )$ ; and the MDEP can be obtained as $P_e=p_1-\\sum _{n=0}^{n_{th}}\\lambda _n$ where the optimal threshold $n_{th}$ is the maximum $n$ satisfying $\\begin{aligned}\\binom{n}{k}n_t^{h-k}\\ge \\binom{n}{h}\\frac{p_0}{p_1}.\\end{aligned}$ See Appendix .",
"Although the POVM operators of optimal quantum discrimination for two quantum states can be obtained from the Helstrom's theory, the realization of the optical quantum discrimination is usually intractable.",
"However, Theorem REF indicates that the optimal quantum discrimination of two noisy DNSs $\\hat{\\rho }(\\mu ,h)$ and $\\hat{\\rho }(\\mu ,k)$ is realizable by the Kennedy receiver with threshold detection Note that the Kennedy receiver with threshold detection is a near-optimum receiver for discriminating between two coherent states.."
],
[
"Numerical Results",
"The prior probabilities are set as $p_0=p_1=0.5$ in this section.",
"Figs.",
"REF and REF present the MDEP for discriminating two noiseless DNSs and two noisy DNSs, respectively.",
"From Fig.",
"REF , we can observe that when $k=h$ , the MDEP decreases as $|\\mu -\\xi |$ increases; and when $k\\ne h$ , the MDEP achieves zero when $\\mu =\\xi $ .",
"Besides, when $k\\ne h$ , the MDEP decreases as the gap $h-k$ increases.",
"Comparing Fig.",
"REF with Fig.",
"REF , we can see that the MDEP for discriminating two noisy DNSs demonstrates similar properties to that for discriminating two noiseless DNSs.",
"Besides, the MDEP for discriminating two noisy DNSs is always larger than that for discriminating two noiseless DNSs with the same parameters.",
"Figure: Error probabilities under different gap (μ=ξ=1,n t =0.2\\mu =\\xi =1, n_t=0.2)Next we check the error probability for discriminating between two noisy DNSs obtained by the Kennedy receiver with threshold detection with $\\mu =\\xi =1$ under different gaps, shown in Fig.",
"REF .",
"We also plot the MDEP results of Helstrom bound obtained by the optimal quantum discrimination.",
"We can see that the error probabilities obtained by the Kennedy receiver with threshold detection coincide with the MDEP results of Helstrom bound obtained by the optimal quantum discrimination.",
"This verifies the result of Theorem REF .",
"The MDEP decreases as the gap increases for a given $k$ , which is as expected.",
"Besides, when the gap is fixed, the MDEP decreases as $k$ decreases.",
"For a given gap, a smaller $k$ implies a smaller energy requirement.",
"This indicates that if we use DNSs as the information carriers in intensity modulations, a smaller $k$ can achieve a better performance in not only error probability but also energy efficiency.",
"Therefore, the OOK modulation with $k=0$ for a given energy gap is the optimal intensity modulation in terms of both the error probability and the energy efficiency.",
"Figure: Error probabilities of OOK modulation under different thermal noisesAt last, we consider a special case of discriminating two noisy DNSs with $\\mu =\\xi =0$ and $k=0$ under different gaps $h$ , which corresponds to an OOK modulation in communication systems.",
"The error probabilities under different $h$ with different thermal noises are shown in Fig.",
"REF .",
"We also plot the error probabilities of the OOK modulation employing a coherent state with the same average energy per information bit.",
"We can see that the error probability decreases as $h$ increases, which is as expected.",
"Besides, we can also see that the error probability of OOK modulations employing a DNS can be significantly reduced compared with that of employing a coherent state with the same average energy."
],
[
"Conclusion",
"We addressed the problem of discriminating between two noisy DNSs.",
"We first considered the quantum discrimination of two noiseless DNSs.",
"Then we derived the Fock representation of a noisy DNS, and then used the Fock representation to derive the MDEP of discriminating two noisy DNSs.",
"We further proved that the optimal quantum discrimination of two noisy DNSs can be achieved by the Kennedy receiver with threshold detection.",
"The simulation results verified our theoretical derivations.",
"Besides, we found that the error probability of OOK modulation employing a DNS is significantly less than that of OOK modulation employing a coherent state with the same average energy."
],
[
"Proof of Theorem 1",
"We use the coherent-state representation of noisy DNS $\\hat{\\rho }(\\mu ,k)$ to obtain its Fock representation.",
"The coherent-state representation of a noisy DNS can be obtained by [5] $\\begin{aligned}R(\\alpha ^*,\\beta )&=e^{\\frac{1}{2}|\\alpha |^2+\\frac{1}{2}|\\beta |^2}\\mathinner {\\langle {\\alpha }|}\\hat{\\rho }(\\mu ,k)\\mathinner {|{\\beta }\\rangle }.\\end{aligned}$ Substituting (REF ) into (REF ), we obtain $\\begin{aligned}R(\\alpha ^*,\\beta )&=\\frac{e^{\\frac{1}{2}|\\alpha |^2+\\frac{1}{2}|\\beta |^2}}{k!",
"(n_t+1)^k}\\mathinner {\\langle {\\alpha -\\mu }|}(\\hat{A}^{\\dagger })^k\\hat{\\rho }_{th}\\hat{A}^k\\mathinner {|{\\beta -\\mu }\\rangle }\\\\&=\\frac{e^{\\frac{1}{2}|\\alpha |^2+\\frac{1}{2}|\\beta |^2}(\\alpha ^*-\\mu ^*)^k(\\beta -\\mu )^k}{k!",
"(n_t+1)^k}\\mathinner {\\langle {\\alpha }|}\\hat{\\rho }_{th}(\\mu )\\mathinner {|{\\beta }\\rangle }\\end{aligned}$ where $\\hat{\\rho }_{th}(\\mu )$ is the displaced thermal state.",
"Using the coherent-state representation $R_{th}(\\alpha ^*,\\beta )$ of displaced thermal state [7], we can obtain $\\begin{aligned}R(\\alpha ^*&,\\beta )\\\\&=\\frac{(\\alpha ^*-\\mu ^*)^k(\\beta -\\mu )^k}{k!",
"(n_t+1)^k}R_{th}(\\alpha ^*,\\beta )\\\\&=\\sum _{n=0}^{\\infty }\\sum _{m=0}^{\\infty }\\frac{(\\alpha ^*)^n\\beta ^m}{\\sqrt{n!m!",
"}}\\sum _{i=0}^{k}\\sum _{j=0}^{k}I(n\\ge i;m\\ge j)\\frac{(-1)^{i+j}}{(k-j)!",
"}\\\\&\\quad \\times \\binom{m}{j}\\binom{k}{i}\\sqrt{\\frac{n!}{m!",
"}}e^{-\\frac{|\\mu |^2}{n_t+1}}|\\mu |^{2(k-j)}(\\mu ^*)^{m-n}\\\\&\\quad \\times \\frac{n_t^{n-i}}{(n_t+1)^{m+k-j+1}}L_{n-i}^{(m-n+i-j)}\\left(-\\frac{|\\mu |^2}{n_t(n_t+1)}\\right).\\end{aligned}$ Using the relation between the coherent-state representation and the Fock representation [5], we obtain the Fock representation of noisy DNS as (REF )."
],
[
"Proof of Lemma 1",
"The displacement operator is defined as $\\begin{aligned}\\hat{D}(\\alpha )\\triangleq e^{\\alpha \\hat{A}^{\\dagger }-\\alpha ^*\\hat{A}}=e^{-\\frac{1}{2}|\\alpha |^2}e^{\\alpha \\hat{A}^{\\dagger }}e^{-\\alpha ^*\\hat{A}}.\\end{aligned}$ Using the Taylor series of matrix exponential $e^{\\alpha \\hat{A}^{\\dagger }}=\\sum _{k=0}^{\\infty }\\frac{\\alpha ^k}{k!",
"}(\\hat{A}^{\\dagger })^k$ and the commutator $[\\hat{A}^{\\dagger },\\hat{A}]=\\hat{\\mathbb {I}}$ , we can obtain the following commutators $[\\hat{A},\\hat{D}(\\alpha )]=\\alpha \\hat{D}(\\alpha );\\quad [\\hat{A}^{\\dagger },\\hat{D}(\\alpha )]=\\alpha ^*\\hat{D}(\\alpha ).$ The average number of photons $n_p(\\mu ,k)$ of a noisy DNS $\\hat{\\rho }(\\mu ,k)$ is defined as $\\begin{aligned}n_p(\\mu ,k)&\\triangleq \\text{tr}\\lbrace \\hat{\\rho }(\\mu ,k)\\hat{A}^{\\dagger }\\hat{A}\\rbrace \\\\&=\\text{tr}\\lbrace \\hat{\\rho }(0,k)\\hat{D}^{\\dagger }(\\mu )\\hat{A}^{\\dagger }\\hat{A}\\hat{D}(\\mu )\\rbrace .\\end{aligned}$ Using the commutators in (REF ), we can obtain $\\hat{D}^{\\dagger }(\\mu )\\hat{A}^{\\dagger }\\hat{A}\\hat{D}(\\mu )=(\\hat{A}^{\\dagger }\\hat{A}+\\mu \\hat{A}^{\\dagger }+\\mu ^*\\hat{A}+|\\mu |^2\\hat{\\mathbb {I}}).$ Substituting (REF ) into (REF ), we can obtain $\\begin{aligned}n_p(\\mu ,k)&=|\\mu |^2+\\text{tr}\\lbrace \\hat{\\rho }(0,k)\\hat{A}^{\\dagger }\\hat{A}\\rbrace +\\mu \\text{tr}\\lbrace \\hat{\\rho }(0,k)\\hat{A}^{\\dagger }\\rbrace \\\\&\\quad +\\mu ^*\\text{tr}\\lbrace \\hat{\\rho }(0,k)\\hat{A}\\rbrace \\end{aligned}$ where $\\text{tr}\\lbrace \\hat{\\rho }(0,k)\\hat{A}^{\\dagger }\\hat{A}\\rbrace =k(n_t+1)+n_t$ is the average number of photons of a photon-added thermal state [7].",
"Note that $\\text{tr}\\lbrace \\hat{\\rho }(0,k)\\hat{A}^{\\dagger }\\rbrace =\\text{tr}\\lbrace \\hat{\\rho }(0,k)\\hat{A}\\rbrace =0$ where we have used the property that $P(\\alpha )|_{\\mu =0}$ is an even function of $\\alpha $ .",
"Therefore, we have $n_p(\\mu ,k)=|\\mu |^2+k(n_t+1)+n_t$ ."
],
[
"Proof of Theorem 2",
"From (REF ), we can observe that $\\lambda _n\\ge 0$ for any $n<h$ .",
"Then the key is to find all positive $\\lambda _n$ when $n\\ge h$ .",
"Note that if $\\lambda _n<0$ when $n\\ge h$ , then we have $p_1\\binom{n}{k}n_t^{h-k}<p_0\\binom{n}{h}.$ Then for $\\lambda _{n+1}$ , we have $\\begin{aligned}\\lambda _{n+1}&=\\frac{n_t^{n+1-h}}{(n_t+1)^{n+2}}\\left[\\frac{n+1}{n+1-k}p_1\\binom{n}{k}n_t^{h-k}\\right.\\\\&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\left.-\\frac{n+1}{n+1-h}p_0\\binom{n}{h}\\right]\\\\&\\le \\frac{n_t^{n+1-h}}{(n_t+1)^{n+2}}\\frac{n+1}{n+1-h}\\left[p_1\\binom{n}{k}n_t^{h-k}-p_0\\binom{n}{h}\\right]\\end{aligned}$ where we have used the inequality $\\frac{n+1}{n+1-k}\\le \\frac{n+1}{n+1-h}$ for $h\\ge k$ .",
"According to (REF ), we have $\\lambda _{n+1}<0$ .",
"In other words, $\\lambda _{n}<0$ guarantees $\\lambda _{n+1}<0$ .",
"This indicates that there exists a threshold $n_{th}$ such that $\\left\\lbrace \\begin{array}{ll}\\lambda _n\\ge 0, \\text{ } n\\le n_{th}\\\\\\lambda _n < 0, \\text{ } n>n_{th}.\\end{array}\\right.$ Therefore, by optimizing the threshold in (REF ), we can always achieve the MDEP in (REF ), i.e., we have $\\sum _{\\lambda _n>0}\\lambda _n=\\sum _{n=0}^{n_{th}}\\lambda _n.$ Accordingly, the optimal threshold $n_{th}$ is the maximum $n$ satisfying (REF )."
]
] | 2012.05165 |
[
[
"Sneak Preview: PowerDynamics.jl -= An Open-Source library for analyzing\n dynamic stability in power grids with high shares of renewable energy"
],
[
"Abstract PowerDynamics.jl is an Open-Source library for dynamic power grid modeling built in the latest scientific programming language, Julia.",
"It provides all the tools necessary to analyze the dynamical stability of power grids with high share of renewable energy.",
"In contrast to conventional tools, it makes full use of the simplicity and generality that Julia combines with highly-optimized just-in-time compiled Code.",
"Additionally, its ecosystem provides DifferentialEquations.jl, a high-performance library for solving differential equations with built-in solvers and interfaces to industrial grade solvers like Sundials.",
"PowerDynamics.jl provides a multitude of dynamics for different node/bus-types, e.g.",
"rotating masses, droop-control in inverters, and is able to explicitly model time delays of inverters.",
"Furthermore, it includes realistic models of fluctuations from renewable energy sources.",
"In this paper, we demonstrate how to use PowerDynamics.jl for the IEEE 14-bus distribution grid feeder."
],
[
"Introduction",
"Dynamic stability analysis of power grids concerns the investigation of transient stability and power quality.",
"Especially with more intermittent renewable generation sudden changes in power followed by the transient reaction of frequency and voltage dynamics may present a challenge to power system stability.",
"At the same time, fluctuations also challenge power quality, which refers to whether voltage and frequency stay within safety bounds and whether the waveform has undesired harmonic distortions.",
"Furthermore, the wide deployment of power electronics, witch RES are connected to the grid, introduces new features such as delayed reaction into future power grids.",
"These challenges have been the topic of recent research on the stability of renewable power grids [1], [2], [3].",
"However, the usage of state-of-the-art Open-Source software repeatedly posed limitations to the possible modeling scope.",
"One reason is the lack of solvers for the different types of dynamics which need to be described a stochastic differential equations, delay differential equations and differential algebraic equations.",
"At the same time to network size was constrained by computational performance.",
"This means a realistic model implementation and hence reliable stability analysis has been constrained by the available OS software framework.",
"In this conference paper, we give a sneak preview to PowerDynamics.jl, an new Open-Source approach to dynamic power grid modelingThis paper is meant as a sneak preview, as the planned publication date is Oct 15, 2018, i.e.",
"before the Wind Integration Workshop 2018 will take place..",
"While there are multiple modeling suites available they are either proprietary (e.g.",
"PSS® [4]) or need proprietary tools to be used (e.g.",
"PSAT [5], [6]).",
"With PowerDynamics.jl, we aim to provide a modeling framework that tackles all the needs a modeler for power grids with high shares of renewables to execute dynamic analyses and the first publication will provide tools for time-domain modeling.",
"At the moment, PowerDynamics.jl is under high development, with the basic software architecture already being developed and implemented.",
"We are currently adding new types of dynamics and controls to represent different buses, e.g.",
"droop control, multiple inverter types, higher-order representations of synchronous machines, and in particular intermittency representations of renewable energy sources.",
"A second focus of effort goes into usability, in order to improve the experience of modelers and decrease their time they have to concern with software-related problems so they can focus on their modeling instead.",
"Our vision is to provide a modular, fully flexible library where modelers can decide whether they (a) want to simply use repdefined bus models to analyze a power grid quickly, (b) want to into all the detail of modeling each bus type, or (c) want to find their right position in-between (a) and (b).",
"All of that is done in the context of an Open-Source context in order to make it available to and invitre contributions from research and industry.",
"PowerDynamics.jl is written in Julia [7], a new, fast developing programming language targeting the scientific computing community.",
"With its recent release to Julia 1.0, it has matured enough to become a powerful player competing with more traditional programming languages like Python and C in the modeling world.",
"In the wider context, there is a current movement within the Open Energy Modeling community [8] to switch to Julia and unite modeling efforts.",
"With PowerDynamics.jl we are aiming to support these efforts by contributing to dynamic power system analysis.",
"The rest of the paper is structured as follows: in sec:julia “sec:julia” we give a detailed reasoning why we find Julia to be the correct choice as the programming language for our modeling effort; in sec:powerdynamics “sec:powerdynamics” we introduce the basic mathematical notions we use in PowerDynamics.jl; in sec:ieee14 “sec:ieee14” we provide the details of the example system used in this paper; in sec:implementation “sec:implementation” we show how to implement the IEEE 14-bus feeder in PowerDynamics.jl; in sec:modeling “sec:modeling” we present the results for our example model; and in sec:conclusion “sec:conclusion” we conclude the paper."
],
[
"Convention & Notation",
"Within this paper, we take up the convention of PowerDynamics.jl where minor letters are used as variables for dynamic variables and capital letters denote (constant) parameters and all variables are in defined in the co-rotating frame.",
"Further, we take $u_a$ to to be the complex voltage, $v_a = |u_a|$ to be the voltage magnitude, $\\varphi _a = \\arg (u_a)$ to be the voltage angle, $i_a = \\sum _b Y^L_{ab}u_b$ to be the complex current, $s_a = u_a \\cdot i_a^*$ to be the complex power, $p_a = \\text{Re}(s_a)$ the active power and $q_a = \\text{Im}(s_a)$ to be the reactive power of the $$ ath bus.",
"$Y^L_{}$ is the admittance Laplacian.",
"The imaginary element is denoted as $j = \\sqrt{-1}$ ."
],
[
"Julia for Scientific Modeling",
"The advantages of Julia for scientific modeling can be summarized as (adapted from [9]): Julia is fast!",
"It was designed from the beginning as a high-performance language.",
"Using the just-in-time (JIT) compilation method, it produces efficient native code for multiple platforms.",
"Julia allows for rapid development!",
"It uses dynamic typing, so it is easy to use, feels like a scripting language and has good interactive use without losing any performance.",
"That way, it saves the developer a lot of time.",
"Julia is technical!",
"It is developed for numerical computing, the syntax is focusing on precise mathematics and many datatypes and even parallelism is available out of the box.",
"Julia is general!",
"Using multiple dispatch as the fundamental paradigm, it is easy to express object-oriented and functional programming patterns at the same time.",
"Julia is composable!",
"Julia has been designed such that independent packages work well together.",
"Hence, we can use matrices of unit quantities or differentiation with sparse matrices without having these types ever defined explicitly before.",
"Julia has a growing ecosystem!",
"The aforementioned advantages attract a growing communityThe community growth recently reached the point where the organizers had to cut off ticket sales for this year's JuliaCon at some point.",
"and with that a growing ecosystem of Julia packages.",
"Particularly important for modeling is the DifferentialEquations.jl library which provides high-performance solvers and interfaces to industrial grade solvers like SUNDIALS [10] for solving multiple kinds of differential equations (ordinary, algebraic, delayed, stochastic).",
"Julia allows for metaprogramming!",
"This means, at execution time, the source code is available as data and one can easily modify it.",
"Furthermore, one can even add own syntax to Julia.",
"Within PowerDynamics.jl, we make heavy use of metaprogramming to make it as simple as possible for a user to implement her/his own kind of bus model (see sec:implementation-nodes).",
"Also, it allows to write very simple code that can automatically be modified to highly optimized code.",
"Of course, this is just a simply overview and we would strongly recommend the reader to try out Julia and convince herself/himselfhttps://julialang.org/learning/."
],
[
"Mathematics of PowerDynamics.jl",
"Within this paper, we take the approach to describe the dynamics of a power grid by a (set of) semi-explicit differential algebraic equation(s).",
"We explicitly distinguish between the complex voltages $u_a$ (of the $$ ath bus), whose information is transmitted between buses directly through the complex current $i$ , and internal variables $x_{ab}$ (the $$ bth internal variable of the $$ ath bus), that exist locally at a node.",
"An example for an internal variable would be the frequency of a synchronous machine.",
"While the phase angle $\\varphi $ is transmitted directly via the current, its derivative is the angle velocity whose dynamics is defined locally by the the rotating mass.",
"We are aware that a multitude of approaches are possible to write down the dynamics and we have chosen this particular one as it can be easily translated into source code.",
"Writing this idea in formulas yields $i_a = \\sum _c Y^L_{ac} \\cdot u_c , \\\\m^u_a \\dot{u}_a = f_a(u_a, x_a, i_a) , \\\\m^x_{ab} \\dot{x}_{ab} = g_{ab}(u_a, x_a, i_a) , $ where $m^u$ / $m^x$ are the masses for the voltages / internal variables respectivelyWithin PowerDynamics.jl we allow masses to be 1 or 0.. Also, $f_a$ encodes the voltage dynamics of the $$ ath bus and $g_{ab}$ the dynamics of the $$ bth internal variable of the $$ ath bus.",
"Note the convention of using minor letters for dynamical variables.",
"$Y^L_{}$ is the admittance Laplacian.",
"So having the admittances encoded as $Y_{ab}$ where $Y_{ab} = 0$ if there is no line, or $Y_{ab}$ with a complex value (with $\\text{Re}(Y_{ab})\\ge 0$ ) for existing lines, then the admittance Laplacian is defined as $Y^L_{ab} = \\delta _{ab}\\sum _c Y_{ac} - Y_{ab},$ where $\\delta _{ab}$ is Kronecker-$\\delta $ .",
"Particular examples would be an algebraic slack bus as the $$ ath bus with a fixed complex voltage $U_a$ .",
"There are no internal variables and the voltage mass is $m_a^u = 0$ (as its an algebraic equation), so it can simply be written as $0\\cdot \\dot{u}_a = f_a(u_a, x_a, i_a) = u_a - U_a , $ where we have explicitly not removed $\\dot{u}$ in order to keep a strong analogy the the implementation shown later.",
"Similarly, we can treat an algebraic load at the $$ ath bus with a constant complex power $-S_a$ flowing outThe $-$ is due to the definition of all variables such that the power flows always from the respective node into the power grid.. Again, there are no internal variables and the voltage mass is 0 so it reduces to $0\\cdot \\dot{u}_a = f_a(u_a, x_a, i_a) = s_a - S_a = u_a\\cdot i_a^* - S_a .$ A synchronous machine as the $$ ath bus can be represented by the Swing equation (or 2nd-order synchronous machine model) [11], [12] with a produced active power $P_a$ , the damping constant $D_a$ , the rated frequency $\\Omega $ and the inertia $H_a$ .",
"In this case, the voltage mass is 1, there is one internal variable, the frequency, $x_{a1} = \\omega _a$ with a mass of 1 because it is dynamic and not an algebraic constraintIn vector notation this gives $x_{a} = (\\omega _a)$ for the Swing equation..",
"Hence, the equations reduce to $&\\dot{u}_a = f_a(u_a, (\\omega _a), i_a) = u_aj\\omega _a , \\\\&\\begin{aligned}\\dot{x}_{a1} = \\dot{\\omega }_a &= g_{a1}(u_a, (\\omega _a), i_a)\\\\&= \\frac{2\\pi \\Omega }{H_a}\\left( P_a - D_a\\omega _a - \\text{Re}(u_a\\cdot i_a^*) \\right)\\end{aligned} $ where we would like to remind the reader of $j$ being used as the imaginary element in this paper.",
"Note that with $u_a = e^{j\\varphi _a}\\ \\Rightarrow \\ du_a = ju_ad\\varphi _a$ , eq:swing-u reduces to $\\dot{\\varphi }_a = \\omega _a$ and one recovers the usual version in terms of the voltage angle.",
"In the following section, we will introduce the example system in order to demonstrate afterwards, how these kind of dynamics are then described in PowerDynamics.jl."
],
[
"The IEEE 14-bus distribution grid feeder",
"The IEEE 14-bus system is a representation of a medium-voltage distribution grid.",
"A graphical representation is in fig:ieee14grid, which is extracted from [13].",
"There are two generators, where the one at bus 2 is the slack bus, three synchronous compensators, and eight (complex) loads.",
"Bus 7 is for the representation of the transformer only.",
"The generator at bus 1 and the synchronous compensators will be modeled with the swing equation, the slack bus 2 with the algebraic constraint introduced in sec:powerdynamics and the loads with the algebraic constraints for complex loads (see sec:powerdynamics also).",
"The bus parameters haven been extracted from [13] and are presented in tab:ieee14-bus-parameters.",
"The resistance $R$ and reactance $X$ for the lines (taken from [13] as well) are summarized in tab:ieee14-line-parameters.",
"Having a description of the power grid in question enables us to have a look at the implementation in PowerDynamics.jl now.",
"Table: Bus parameters of the IEEE 14-bus test system.",
"INERTIA UNIT IS?Table: Line parameters of the IEEE 14-bus test system."
],
[
"Implementation in PowerDynamics.jl",
"In this section, we will show first how to implement the dynamics for the different buses presented in sec:powerdynamics in PowerDynamics.jl and then how create a power grid model from these bus dynamics definitions, in this case the IEEE 14-bus system."
],
[
"Defining the dynamics",
"The implementation of such a power grid in PowerDynamics.jl is strongly based on the $\\mathtt {@DynamicNode}$ macro.",
"A macro is a metaprogramming function that takes part of the source codes and modifies it at parsing time before giving it to the compiler, that creates the actual bitcode.",
"In case of $\\mathtt {@DynamicNode}$ , this is used to hide all the complicated internals of PowerDynamics.jl and make the definition of a type of dynamics as easy as possible.",
"The following examples are already provided by default in PowerDynamics.jl but using them as an example, any kind of dynamics can be implemented.",
"Using $\\mathtt {@DynamicNode}$ , the slack bus constraint from eq:slack-bus is implemented as [linewidth=] @DynamicNode SlackAlgebraic(U) <: OrdinaryNodeDynamicsWithMass(mu=false) begin end [] begin du = u - U end where line breaks and line numbers have been added for easier orientation.",
"The left part on the first line states the name of the new type $\\mathtt {SlackAlgebraic}$ and the parameter name $\\mathtt {U}$ .",
"Separated by the subtyping operator ($\\mathtt {<:}$ ) comes the statement that it can be represented by an ordinary differential equation with a mass term, where the mass for the voltage m_u is set to $\\mathtt {false}$ , meaning $m^u = 0$ from eq:u-general as it is a constraint.",
"The [] in line 2 states that there are no internal variables.",
"And line 3 simply provides the formula for the dynamics and assigns it to $\\mathtt {du}$ .",
"As m_u = false is given, this simply translates it to a constraint.",
"In a similar manner, the load is defined as an algebraic PQ-constraint [linewidth=] @DynamicNode PQAlgebraic(S) <: OrdinaryNodeDynamicsWithMass(mu=false) begin end [] begin s = u*conj(i) du = S - s end In this case, an additional line was added to calculate the outflowing complex power $\\mathtt {s}$ from the voltage $\\mathtt {u}$ and the current $\\mathtt {i}$ and then the definition of the constraint is written in line 4.",
"Concerning the generator and the synchronous compensators, we implemented the swing equation as [linewidth=] @DynamicNode SwingEq(H, P, D, SOmega) <: OrdinaryNodeDynamics() begin @assert D > 0 \"damping (D) should be >0\" @assert H > 0 \"inertia (H) should be >0\" SOmegaH = SOmega * 2pi / H end [[Somega, dSomega]] begin p = real(u * conj(i)) du = u * im * Somega dSomega = (P - D*Somega - p)*SOmegaH end Again, the left part of the first line provides the name of the new type and the name of the parameters.",
"Lines 2 to 4 are code that should be run only once.",
"In this case these are consistency checks, that the damping and inertia are positive, and the reduction of the rated frequency $\\mathtt {\\Omega }$ and the inertia $\\mathtt {H}$ to a single variable $\\mathtt {\\Omega \\_H}$ .",
"In line 5, the variable name of the internal variable $\\mathtt {\\omega }$ and the name for its derivative $\\mathtt {d\\omega }$ are given.",
"Finally, lines 6 to 8 implement eq:swing-u,eq:swing-omega by simply writing down the mathematical terms."
],
[
"Instantiating the power grid model",
"In order to create the grid model, we first have to instantiate the bus models simply by calling them with the corresponding parameter values from tab:ieee14-bus-parameters, e.g.",
": [linewidth=] SwingEq(H=5.148, P=2.32, D=2, SOmega=50) for bus 1 PQAlgebraic(S=-0.295-0.166im) for bus 9 Within the actual code we simply loaded the data from a $\\mathtt {.csv}$ file into and automatized this instantiationThe full source code is not part of this paper as it would be too long, but it will be published along with PowerDynamics.jl..",
"The instantiated bus models should then be saved in an array called e.g.",
"$\\mathtt {nodes}$ .",
"Similarly, the admittance Laplacian should be generated from the line data in tab:ieee14-line-parameters and saved in a matrix called e.g.",
"$\\mathtt {LY}$ .",
"The actual grid model instatiation is then simply one line where the model is saved in the variable $\\mathtt {g}$ : [linewidth=] g = GridDynamics(nodelist, LY) Now, $\\mathtt {g}$ contains all the information of the power grid and in the following section we will show how to solve it."
],
[
"Modeling Results",
"Within this section, we will analyze two simple cases: (a) a frequency perturbation at the largest generator at bus 1 and (b) a line tripping event at of line 2 (between bus 1 and 5) (cmp.",
"fig:ieee14grid).",
"Before we find the normal point of operation for the power grid, i.e.",
"the fixed point of eq:i-definition,eq:u-general,eq:x-general or synchronous state for the IEEE 14-bus system.",
"For that, we use the grid model $\\mathtt {g}$ generated in the previous sec:implementation and use the function provided by PowerDynamics.jl: [linewidth=] fp = operationpoint(g, ones(SystemSize(g))) where $\\mathtt {ones(SystemSize(g))}$ is a vector of the correct length for the initial condition of the fixed point search.",
"$\\mathtt {fp}$ is now a $\\mathtt {State}$ object that we can use as initial condition for the solving the differential equations corresponding to the power grid."
],
[
"Frequency perturbation",
"In order to model a frequency perturbation, one can simply take a copy of the fixed point as found before and adjust the initial frequency value: [linewidth=] x0 = copy(fp) x0[1, :int, 1] += 0.2 The second line can be read as “add 0.2 to the 1$^\\text{st}$ internal variable of the 1$^\\text{st}$ node” which is the frequency $\\omega $ as there is only one internal variable.",
"Note that the first $\\mathtt {1}$ refers to the node and then second $\\mathtt {1}$ to the internal variable counter.",
"The power grid with the initial condition $\\mathtt {x0}$ can then be solved for a time span of 0.5 seconds by calling: [linewidth=] sol = solve(g, x0, (0.0,.5)) The solution is shown in fig:ieee14-frequency-perturbation.",
"It shows that the system is very stable against frequency perturbations.",
"The actual dynamics is not so exciting as the system is very stable.",
"Please note that this system was taken as an example to present on how easily one can model a power grid using PowerDynamics.jl, not to find new exciting dynamics."
],
[
"Line tripping",
"To show some more dynamic behavior we simulated a line tripping as well.",
"We model this effect by taking the operation point of the full power grid ($\\mathtt {fp}$ ) as initial condition but defining a new admittance Laplacian where line 2 (between bus 1 and 5, see tab:ieee14-line-parameters,fig:ieee14grid) has been taken out (i.e.",
"the admittance is set to 0).",
"Running the model with this new Laplacian yields fig:ieee14-line-tripping.",
"In the frequency plot we identify how the frequency of bus 1, where the line tripping happened, compensates for the momentarily excess power at the bus.",
"The lacking power in the rest of the grid is matched by the synchronous compensators whose frequency decreases in turn.",
"Note that the angular frequency $\\omega $ is shown, so with a division by $2\\pi $ the maximal frequency deviation $f$ is $\\approx 0.05\\,$ Hz.",
"After about one second, the system recovers to the normal state of operation."
],
[
"Conclusion & Outlook",
"Within this paper, we have seen how one one can use the Open-Source library PowerDynamics.jl in order to model the dynamics of a power grid with just a few lines of code.",
"We have seen how the fundamental mathematical equations (given in sec:powerdynamics) translate to source code that reads exactly the same.",
"We employed PowerDynamics.jl for the IEEE 14-bus distribution grid feeder in order to demonstrate how one can easily simulate faults and analyze the transient reaction of the power grid dynamics.",
"As example scenarios we used a frequency perturbation and a line tripping.",
"Finally, this paper is really just a sneak preview with a simple example the publication of PowerDynamics.jl is planned for October 15, 2018.",
"By then, we will have added more inverter control schemes and stochastic descriptions of intermittency due to renewable energy sources."
],
[
"Acknowledgment",
"This paper was presented at the 19th Wind Integration Workshop and published in the workshop’s proceedings.",
"We would like to thank the German Academic Exchange Service for the opportunity to participate at the Wind Integration Workshop 2018 in Stockholm via the funding program “Kongressreisen 2018”.",
"This paper is based on work developed within the Climate-KIC Pathfinder project “elena – electricity network analysis” funded by the European Institute of Innovation & Technology.",
"This work was conducted in the framework of the Complex Energy Networks research group at the Potsdam Institute for Climate Impact Research.",
"We would like to thank Frank Hellmann and Paul Schultz for the discussions on structuring an Open-Source library for dynamic power grid modeling."
]
] | 2012.05175 |
[
[
"Symmetry and Quantum Query-to-Communication Simulation"
],
[
"Abstract Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) < 2 R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively.",
"Chakraborty et al.",
"(CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required.",
"We improve upon their result in several ways.",
"We show that the log n overhead is not required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05).",
"This upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication.",
"To prove this, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function.",
"In view of our first result, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive.",
"We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2.",
"We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free."
],
[
"Motivation and main results",
"The classical model of communication complexity was introduced by Yao [33], who also subsequently introduced its quantum analogue [34].",
"Communication complexity has important applications in several disciplines, in particular for lower bounds on circuits, data structures, streaming algorithms, and many other complexity measures (see, for example, [22] and the references therein).",
"A natural way to derive a communication problem from a Boolean function $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is via composition.",
"Let $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function and let $G: \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a “two-party function”.",
"Then $F = f \\circ G : \\left\\lbrace -1,1 \\right\\rbrace ^{nj} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{nk} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ denotes the function corresponding to the communication problem in which Alice is given input $X = (X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nj}$ , Bob is given $Y = (Y_1, \\dots , Y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nk}$ , and their task is to compute $F(X, Y) = f(G(X_1, Y_1), \\dots , G(X_n, Y_n))$ .",
"Many well-known functions in communication complexity are derived in this way, such as Set-Disjointness ($\\mathsf {DISJ}_n := \\mathsf {NOR}_n \\circ \\mathsf {AND}_2$ ), Inner Product ($\\mathsf {IP}_{n} : = \\mathsf {PARITY}_n \\circ \\mathsf {AND}_2$ ) and Equality ($\\mathsf {EQ}_{n} : = \\mathsf {NOR}_n \\circ \\mathsf {XOR}_2$ ).",
"A natural approach to obtain efficient quantum communication protocols for $f \\circ G$ is to “simulate” a quantum query algorithm for $f$ , where a query to the $i$ th input bit of $f$ is simulated by a communication protocol that computes $G(X_i,Y_i)$ .",
"Buhrman, Cleve and Wigderson [6] observed that such a simulation is indeed possible if $G$ is bitwise AND or XOR.",
"Theorem 1.1 ([6]) For every Boolean function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ , we have $\\mathsf {Q}^{cc}\\left(f \\circ \\square \\right) = O\\left(\\mathsf {Q}(f) \\log n\\right).$ Here $\\mathsf {Q}(f)$ denotes the bounded-error quantum query complexity of $f$ , and $\\mathsf {Q}^{cc}(f\\circ \\square )$ denotes the bounded-error quantum communication complexity for computing $f\\circ \\square $ .",
"Throughout this paper, we refer to Theorem REF as the BCW simulation.",
"[6] used this, for instance, to show that the bounded-error quantum communication complexity of the Set-Disjointness function is $O(\\sqrt{n}\\log n)$ , using Grover's $O(\\sqrt{n})$ -query search algorithm [16] for the $\\mathsf {NOR}_n$ function.",
"It is folklore in the classical world that the analogous simulation does not incur a $\\log n$ factor overhead.",
"That is, $\\mathsf {R}^{cc}\\left(f \\circ \\square \\right) \\le 2\\mathsf {R}(f),$ where $\\mathsf {R}(f)$ denotes the bounded-error randomized query complexity of $f$ and $\\mathsf {R}^{cc}(f\\circ \\square )$ denotes the bounded-error randomized communication complexity for computing $f\\circ \\square $ .",
"Thus, a natural question is whether the multiplicative $\\log n$ blow-up in the communication cost in the BCW simulation is necessary.",
"Chakraborty et al.",
"[12] answered this question and exhibited a total function for which the $\\log n$ blow-up is indeed necessary when $\\mathsf {XOR}_2$ is the inner function.",
"Theorem 1.2 ([12]) There exists a function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ such that $\\mathsf {Q}^{cc,*}(f \\circ \\mathsf {XOR}_2) = \\Omega (\\mathsf {Q}(f) \\log n).$ Here $\\mathsf {Q}^{cc,*}(F)$ denotes the bounded-error quantum communication complexity of two-party function $F$ when Alice and Bob shared an entangled state at the start of the protocol for free.",
"This gives rise to the following basic question: is there a natural class of functions for which the $\\log n$ overhead in the BCW simulation is not required?",
"Improving upon Høyer and de Wolf [17], Aaronson and Ambainis [1] showed that for the canonical problem of Set-Disjointness, the $\\log n$ overhead in the BCW simulation can be avoided.",
"Since the outer function $\\mathsf {NOR}_n$ is symmetric (i.e., it only depends on the Hamming weight of its input, its number of $-1$ s), a natural question is whether the $\\log n$ overhead can be avoided whenever the outer function is symmetric.",
"Our first result gives a positive answer to this question.",
"Theorem 1.3 For every symmetric Boolean function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and two-party function $G:\\left\\lbrace -1,1 \\right\\rbrace ^j\\times \\left\\lbrace -1,1 \\right\\rbrace ^k\\rightarrow \\lbrace 0,1\\rbrace $ , we have $\\mathsf {Q}^{cc,*}(f \\circ G)=O(\\mathsf {Q}(f)\\mathsf {Q}_E^{cc}(G)).$ Here $\\mathsf {Q}_E^{cc}(G)$ denotes the exact quantum communication complexity of $G$ , where the error probability is 0.",
"In particular, if $G\\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ then $\\mathsf {Q}_E^{cc}(G)=1$ and hence $\\mathsf {Q}^{cc,*}(f \\circ G)=O(\\mathsf {Q}(f))$ .",
"Remark 1.4 If $\\mathsf {Q}(f)=\\Theta (\\sqrt{tn})$ , then our protocol in the proof of Theorem REF starts from a shared entangled state of $O(t\\log n)$ EPR-pairs.",
"Note that if $t\\le n\\mathsf {Q}_E^{cc}(G)^2/(\\log n)^2$ (this condition holds for instance if $\\mathsf {Q}_E^{cc}(G)\\ge \\log n$ ) then this number of EPR-pairs is no more than the amount of communication and hence might as well be established in the first message, giving asymptotically the same upper bound $\\mathsf {Q}^{cc}(f \\circ G)=O(\\mathsf {Q}(f)\\mathsf {Q}_E^{cc}(G))$ for the model without prior entanglement.",
"The next question one might ask is whether one can weaken the notion of symmetry required in Theorem REF .",
"A natural generalization of the class of symmetric functions is the class of transitive-symmetric functions.",
"A function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is said to be transitive-symmetric if for all $i, j \\in [n]$ , there exists $\\sigma \\in S_n$ such that $\\sigma (i) = j$ , and $f(x) = f(\\sigma (x))$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"Henceforth we refer to transitive-symmetric functions as simply transitive functions.",
"Can the $\\log n$ overhead in the BCW simulation be avoided whenever the outer function is transitive?",
"We give a negative answer to this question in a strong sense: the $\\log n$ overhead is still necessary even when we allow the quantum communication protocol an error probability that can be arbitrarily close to $1/2$ .",
"Theorem 1.5 There exists a transitive and total function $f : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , such that $\\mathsf {UPP}^{cc}(f \\circ \\square ) & = \\Omega (\\mathsf {Q}(f) \\log n)$ for every $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .",
"Here $\\mathsf {UPP}^{cc}(f \\circ \\square )$ denotes the unbounded-error quantum communication complexity of $f \\circ \\square $ (adding “quantum” here only changes the communication complexity by a constant factor).",
"The unbounded-error model of communication was introduced by Paturi and Simon [30] and is the strongest communication complexity model against which we know how to prove explicit lower bounds.",
"This model is known to be strictly stronger than the bounded-error quantum model.",
"For instance, the Set-Disjointness function on $n$ inputs requires $\\Omega (n)$ bits or $\\Omega (\\sqrt{n})$ qubits of communication in the bounded-error model, but only requires $O(\\log n)$ bits of communication in the unbounded-error model.",
"In fact, it follows from a recent result of Hatami, Hosseini and Lovett [18] that there exists a function $F : \\left\\lbrace -1,1 \\right\\rbrace ^n\\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ with $\\mathsf {Q}^{cc,*}(F) = \\Omega (n)$ while $\\mathsf {UPP}^{cc}(F) = O(1)$ .",
"Theorem REF and Theorem REF clearly demonstrate the role of symmetry in determining the presence of the $\\log n$ overhead in the BCW query-to-communication simulation: this overhead is absent for symmetric functions (Theorem REF ), but present for a transitive function even when the model of communication under consideration is as strong as the unbounded-error model (Theorem REF ).",
"We also give a general recipe to construct functions for which the $\\log n$ overhead is required in the BCW simulation in the bounded-error communication model (see Theorem REF )."
],
[
"Overview of our approach and techniques",
"In this section we discuss the ideas that go into the proofs of Theorem REF and Theorem REF ."
],
[
"Communication complexity upper bound for symmetric functions",
"To prove Theorem REF we use the well-known fact that every symmetric function $f$ has an interval around Hamming weight $n/2$ where the function is constant; for $\\mathsf {NOR}_n$ the length of this interval would be essentially $n$ , while for $\\mathsf {PARITY}_n$ it would be 1.",
"To compute $f$ , it suffices to either determine that the Hamming weight of the input lies in that interval (because the function value is the same throughout that interval) or to count the Hamming weight exactly.",
"For two-party functions of the form $f\\circ G$ , we want to do this type of counting on the $n$ -bit string $z=(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"We show how this can be done with $O(\\mathsf {Q}(f)\\,\\mathsf {Q}^{cc}_E(G))$ qubits of communication if we had a quantum protocol that can find $-1$ s in the string $z$ at a cost of $O(\\sqrt{n}\\,\\mathsf {Q}^{cc}_E(G))$ qubits.",
"Such a protocol was already given by Aaronson and Ambainis for the special case where $G=\\mathsf {AND}_2$ for their optimal quantum protocol for Set-Disjointness, as a corollary of their quantum walk algorithm for search on a grid [1].",
"In this paper we give an alternative $O(\\sqrt{n}\\,\\mathsf {Q}^{cc}_E(G))$ -qubit protocol.",
"This implies the result of Aaronson and Ambainis as a special case, but it is arguably simpler and may be of independent interest.",
"Our protocol can be viewed as an efficient distributed implementation of amplitude amplification with faulty components.",
"In particular, we replace the usual reflection about the uniform superposition by an imperfect reflection about the $n$ -dimensional maximally entangled state ($=\\log n$ EPR-pairs if $n$ is a power of 2).",
"Such a reflection would require $O(\\log n)$ qubits of communication to implement perfectly, but can be implemented with small error using only $O(1)$ qubits of communication, by invoking the efficient protocol of Aharonov et al.",
"[4] that tests whether a given bipartite state equals the $n$ -dimensional maximally entangled state.",
"Still, at the start of this protocol we need to assume (or establish by means of quantum communication) a shared state of $\\log n$ EPR-pairs.",
"If $\\mathsf {Q}(f)=\\Theta (\\sqrt{tn})$ then our protocol for $f\\circ G$ will run the $-1$ -finding protocol $O(t)$ times, which accounts for our assumption that we share $O(t\\log n)$ EPR-pairs at the start of the protocol."
],
[
"Communication complexity lower bound for transitive functions",
"For proving Theorem REF , we exhibit a transitive function $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ whose bounded-error quantum query complexity is $O(n)$ and the unbounded-error communication complexity of $f \\circ \\square $ is $\\Omega (n \\log n)$ for $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .",
"Function construction and transitivity: For the construction of $f$ we first require the definition of Hadamard codewords.",
"The Hadamard codeword of $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , denoted by $H(s) \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , is a list of all parities of $s$ .",
"See Figure REF for a graphical visualization of $f$ .",
"Figure: If the inputs to the jj-th h 𝖨𝖯 logn h_{\\mathsf {IP}_{\\log n}} are the Hadamard codewords H(s j )H(s_j) and H(t j )H(t_j) for all j∈[n]j \\in [n] and some s j ,t j ∈-1,1 logn s_j, t_j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}, then f=𝖯𝖠𝖱𝖨𝖳𝖸(𝖨𝖯 logn (s 1 ,t 1 ),⋯,𝖨𝖯 logn (s n ,t n ))f = \\mathsf {PARITY}(\\mathsf {IP}_{\\log n}(s_1,t_1), \\dots , \\mathsf {IP}_{\\log n}(s_n, t_n)).",
"If there exists at least one j∈[n]j \\in [n] for which either x j1 ,⋯,x jn x_{j1}, \\dots , x_{jn} or y j1 ,⋯,y jn y_{j1}, \\dots , y_{jn} is not a Hadamard codeword, then ff outputs -1-1.Using properties of $\\mathsf {IP}$ and Hadamard codewords, and the symmetry of $\\mathsf {PARITY}_n$ , we are able to show that $f$ is transitive (see Claim REF ).",
"Query upper bound: The query upper bound of $O(n)$ follows along the lines of [12], using the Bernstein-Vazirani algorithm to decode the Hadamard codewords, and Grover's algorithm to check that they actually are Hadamard codewords.",
"This approach was in turn inspired by a query upper bound due to Ambainis and de Wolf [3].",
"See the proof of Theorem REF for the query algorithm and its analysis.",
"Communication lower bound: Towards the unbounded-error communication lower bound, we first recall that each input block of $f$ equals $\\mathsf {IP}_{\\log n}$ if the inputs to each block are promised to be Hadamard codewords.",
"Hence $f$ equals $\\mathsf {IP}_{n \\log n}$ under this promise, since $\\mathsf {PARITY}_n \\circ \\mathsf {IP}_{\\log n} = \\mathsf {IP}_{n \\log n}$ .",
"Thus by setting certain inputs to Alice and Bob suitably, $f \\circ \\square $ is at least as hard as $\\mathsf {IP}_{n \\log n}$ for $\\square \\in \\lbrace \\mathsf {AND}_2, \\mathsf {XOR}_2\\rbrace $ (for a formal statement, see Lemma REF with $r = \\mathsf {PARITY}_n$ and $g = \\mathsf {IP}_{\\log n}$ ).",
"It is known from a seminal result of Forster [15] that the unbounded-error communication complexity of $\\mathsf {IP}_{n \\log n}$ equals $\\Omega (n \\log n)$ , completing the proof of the lower bound.",
"This proof is more general than and arguably simpler than the proof of the lower bound for bounded-error quantum communication complexity in [12]."
],
[
"Other results",
" We give a general recipe for constructing a class of functions that witness tightness of the BCW simulation where the inner gadget is either $\\mathsf {AND}_2$ or $\\mathsf {XOR}_2$ .",
"However, the communication lower bound we obtain here is in the bounded-error model in contrast to Theorem REF , where the communication lower bound is proven in the unbounded-error model.",
"The functions $f$ constructed for this purpose are composed functions similar to the construction in Figure REF , except that we are able to use a more general class of functions in place of the outer $\\mathsf {PARITY}$ function, and also a more general class of functions in place of the inner $\\mathsf {IP}_{\\log n}$ functions.",
"See Figure REF and its caption for an illustration and a more precise definition.",
"Figure: In this figure, G:-1,1 log ×-1,1 logn →-1,1G:\\left\\lbrace -1,1 \\right\\rbrace ^{\\log } \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace .",
"If the inputs to the jj-th h G h_{G} are Hadamard codewords in ±H(s j )\\pm H(s_j) and ±H(t j )\\pm H(t_j) for all j∈[n]j \\in [n] and some s j ,t j ∈-1,1 logn s_j, t_j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}, then f=r(G(s 1 ,t 1 ),⋯,G(s n ,t n ))f = r(G(s_1,t_1), \\dots , G(s_n, t_n)).",
"If there exists at least one j∈[n]j \\in [n] for which either x j1 ,⋯,x jn x_{j1}, \\dots , x_{jn} or y j1 ,⋯,y jn y_{j1}, \\dots , y_{jn} is not a Hadamard codeword, then ff outputs -1-1.We require some additional constraints on the outer and inner functions.",
"First, the approximate degree of $r$ should be $\\Omega (n)$ .",
"Second, the discrepancy of $G$ should be small with respect to some “balanced” probability distribution (see Definition REF and Definition REF for formal definitions of these notions).",
"Theorem 1.6 (Informal version of Theorem REF ) Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be such that $\\widetilde{\\textnormal {deg}}(r) = \\Omega (n)$ and let $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function.",
"Define $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^{2}} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ as in Figure REF .",
"If there exists $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\mathbb {R}$ that is a balanced probability distribution with respect to $G$ and $\\textnormal {disc}_{\\mu }(G) = n^{-\\Omega (1)}$ , then $\\mathsf {Q}(f) & = O(n),\\\\\\mathsf {Q}^{cc, *}(f \\circ \\square ) & = \\Omega (n \\log n),$ for every $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .",
"The query upper bound follows along similar lines as that of Theorem REF .",
"For the lower bound, we first show via a reduction that for $f$ as described in Figure REF and $\\square \\in \\lbrace \\mathsf {AND}_2, \\mathsf {XOR}_2\\rbrace $ , the communication problem $f \\circ \\square $ is at least as hard as $r \\circ G$ (see Lemma REF ).",
"This part of the lower bound proof is the same as in the proof of Theorem REF .",
"For the hardness of $r \\circ G$ (which in the case of Theorem REF turned out to be $\\mathsf {IP}_{n \\log n}$ , for which Forster's theorem yields an unbounded-error communication lower bound), we are able to use a theorem implicit in a work of Lee and Zhang [25].",
"This theorem gives a lower bound on the bounded-error communication complexity of $r \\circ G$ in terms of the approximate degree of $r$ and the discrepancy of $G$ under a balanced distribution.",
"For completeness, we provide an explicit proof in Appendix .",
"We recover the result of Chakraborty et al.",
"(Theorem REF ) using a more general technique, and additionally show that $\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {AND}_2) = \\Omega (\\mathsf {Q}(f) \\log n)$ , where $f$ is as in Theorem REF .",
"This is discussed in Appendix .",
"For a Boolean function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , let $\\log \\Vert \\widehat{f}\\Vert _{1, 1/3}$ denote the log-approximate-spectral norm of $f$ , and $\\widetilde{\\textnormal {deg}}_{1/3}(f)$ denote its approximate degree.",
"As an approach towards proving the Fourier entropy-influence conjecture, Arunachalam et al.",
"[2] asked whether it is true for all total functions $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ that $\\log \\Vert \\widehat{f}\\Vert _{1, 1/3} = O(\\widetilde{\\textnormal {deg}}_{1/3}(f))$ .",
"Chakraborty et al.",
"gave a negative answer to this question by showing that for the function $f$ in Theorem REF , $\\log \\Vert \\widehat{f}\\Vert _{1, 1/3} = \\Omega (\\widetilde{\\textnormal {deg}}_{1/3}(f) \\log n)$ .",
"They also showed that the $\\log n$ overhead is not required for the class of symmetric functions [12].",
"In this paper we complement this result by showing that the $\\log n$ overhead is required for the class of transitive functions.",
"This is discussed in Appendix ."
],
[
"Organization",
"Section introduces some notation and preliminaries.",
"In Section we construct our new one-sided error protocol for finding solutions in the string $z=(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , as a corollary of our distributed version of amplitude amplification.",
"In Section we prove Theorem REF , which shows that the $\\log n$ overhead in the BCW simulation can be avoided when the outer function is symmetric; this relies on the protocol from Section .",
"In Section we prove Theorem REF and Theorem REF , which are our results regarding necessity of the $\\log n$ overhead in the BCW simulation in the unbounded-error and the bounded-error models of communication, respectively.",
"In Appendix we show that the function constructed in [12] also requires a $\\log n$ overhead in the BCW simulation when the inner function is $\\mathsf {AND}_2$ .",
"In Appendix we exhibit a transitive function $f$ on $n$ bits for which $\\log (\\Vert \\widehat{f}\\Vert _{1, 1/3}) = \\Omega (\\widetilde{\\textnormal {deg}}(f) \\log n)$ .",
"Finally in Appendix we prove Theorem REF , which shows a lower bound on the bounded-error quantum communication complexity of $f \\circ G$ in terms of the approximate degree of $f$ and the discrepancy of $G$ under balanced distributions."
],
[
"Notation and preliminaries",
"Without loss of generality, we assume $n$ to be a power of 2 in this paper, unless explicitly stated otherwise.",
"All logarithms in this paper are base 2.",
"Let $S_n$ denote the symmetric group over the set $[n]=\\lbrace 1,\\ldots ,n\\rbrace $ .",
"For a string $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ and $\\sigma \\in S_n$ , let $\\sigma (x)$ denote the string $x_{\\sigma (1)}, \\dots , x_{\\sigma (n)} \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"Consider an arbitrary but fixed bijection between subsets of $[\\log n]$ and elements of $[n]$ .",
"For a string $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , we abuse notation and also use $s$ to denote the equivalent element of $[n]$ .",
"The view we take will be clear from context.",
"For a string $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ and set $S \\subseteq [n]$ , define the string $x_S \\in \\left\\lbrace -1,1 \\right\\rbrace ^{S}$ to be the restriction of $x$ to the coordinates in $S$ .",
"Let $1^n$ and $(-1)^n$ denote the $n$ -bit string $(1, 1, \\dots , 1)$ and $(-1, -1, \\dots , -1)$ , respectively."
],
[
"Boolean functions",
"For strings $x, y \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , let $\\langle x, y \\rangle $ denote the inner product (mod 2) of $x$ and $y$ .",
"That is, $\\langle x, y \\rangle = \\prod _{i = 1}^n (x_i \\wedge y_i).$ For every positive integer $n$ , let $\\mathsf {PARITY}_n : \\lbrace -1,1\\rbrace ^n \\rightarrow \\lbrace -1,1\\rbrace $ be defined as: $\\mathsf {PARITY}_n(x_1, \\dots , x_n) = \\prod _{i \\in [n]} x_i.$ Definition 2.1 (Symmetric functions) A function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is symmetric if for all $\\sigma \\in S_n$ and for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ we have $f(x) = f(\\sigma (x))$ .",
"Definition 2.2 (Transitive functions) A function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive if for all $i,j \\in [n]$ there exists a permutation $\\sigma \\in S_n$ such that $\\sigma (i) = j$ , and $f(x) = f(\\sigma (x))$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"We next discuss function composition.",
"For total functions $f, g$ , let $f \\circ g$ denote the standard composition of the functions $f$ and $g$ .",
"We also require the following notion of composition of a total function $f$ with a partial function $g$ .",
"Definition 2.3 (Composition with partial functions) Let $f : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function and let $g : \\left\\lbrace -1,1 \\right\\rbrace ^{m} \\rightarrow \\left\\lbrace -1, 1, \\star \\right\\rbrace $ be a partial function.",
"Let $f ~\\widetilde{\\circ }~g : \\left\\lbrace -1,1 \\right\\rbrace ^{nm} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ denote the total function that is defined as follows on input $(X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nm}$ , where $X_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^m$ for all $i \\in [n]$ .",
"$f ~\\widetilde{\\circ }~g (X_1, \\dots , X_n) = {\\left\\lbrace \\begin{array}{ll}f(g(X_1), \\dots , g(X_n)) & \\text{if}~g(X_i) \\in \\left\\lbrace -1,1 \\right\\rbrace ~\\text{for all}~i \\in [n],\\\\-1 & \\text{otherwise}.\\end{array}\\right.",
"}$ That is, we use $f ~\\widetilde{\\circ }~g$ to denote the total function that equals $f \\circ g$ on inputs when each copy of $g$ outputs a value in $\\left\\lbrace -1,1 \\right\\rbrace $ , and equals $-1$ otherwise.",
"Definition 2.4 (Approximate degree) For every $\\varepsilon \\ge 0$ , the $\\varepsilon $ -approximate degree of a function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is defined to be the minimum degree of a real polynomial $p : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ that uniformly approximates $f$ to error $\\varepsilon $ .",
"That is, $\\widetilde{\\textnormal {deg}}_{\\varepsilon }(f) = \\min \\left\\lbrace \\textnormal {deg}(p) : |p(x) - f(x)| \\le \\varepsilon ~\\text{for all}~x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n \\right\\rbrace .$ Unless specified otherwise, we drop $\\varepsilon $ from the subscript and assume $\\varepsilon = 1/3$ .",
"We assume familiarity with quantum computing [26], and use $\\mathsf {Q}_{\\varepsilon }(f)$ to denote the $\\varepsilon $ -error query complexity of $f$ .",
"Unless specified otherwise, we drop $\\varepsilon $ from the subscript and assume $\\varepsilon = 1/3$ .",
"Theorem 2.5 ([5]) Let $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function.",
"Then $\\mathsf {Q}(f) \\ge \\widetilde{\\textnormal {deg}}(f)/2$ ."
],
[
"Communication complexity",
"We assume familiarity with communication complexity [22].",
"Definition 2.6 (Two-party function) We call a function $G: \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ a two-party function to indicate that it corresponds to a communication problem in which Alice is given input $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^j$ , Bob is given input $y \\in \\left\\lbrace -1,1 \\right\\rbrace ^k$ , and their task is to compute $G(x, y)$ .",
"Remark 2.7 Throughout this paper, we use uppercase letters to denote two-party functions, and lowercase letters to denote functions which are not two-party functions.",
"Definition 2.8 (Composition with two-party functions) Let $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function and let $G: \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a two-party function.",
"Then $F = f \\circ G : \\left\\lbrace -1,1 \\right\\rbrace ^{nj} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{nk} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ denotes the two-party function corresponding to the communication problem in which Alice is given input $X = (X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nj}$ , Bob is given $Y = (Y_1, \\dots , Y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nk}$ , and their task is compute $F(X, Y) = f(G(X_1, Y_1), \\dots , G(X_n, Y_n))$ .",
"Definition 2.9 (Inner Product function) For every positive integer $n$ , define the function $\\mathsf {IP}_n : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ by $\\mathsf {IP}_n(x,y) = \\langle x,y \\rangle .$ In other words, $\\mathsf {IP}_n = \\mathsf {PARITY}_n \\circ \\mathsf {AND}_2$ .",
"Observation 2.10 For all positive integers $k, t$ , $\\mathsf {PARITY}_k \\circ \\mathsf {IP}_t = \\mathsf {IP}_{kt}$ .",
"We also assume familiarity with quantum communication complexity [32].",
"We use $\\mathsf {Q}_{\\varepsilon }^{cc}(G)$ and $\\mathsf {Q}_{\\varepsilon }^{cc,*}(G)$ to represent the $\\varepsilon $ -error quantum communication complexity of a two-party function $G$ in the models without and with unlimited shared entanglement, respectively.",
"Unless specified otherwise, we drop $\\varepsilon $ from the subscript and assume $\\varepsilon = 1/3$ .",
"Definition 2.11 (Balanced probability distribution) We call a probability distribution $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ balanced with respect to a function $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ if $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} f(x) \\mu (x) = 0$ .",
"Definition 2.12 (Discrepancy) Let $G : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function and $\\lambda $ be a distribution on $\\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k$ .",
"For every $S \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^j$ and $T \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^k$ , define $\\textnormal {disc}_\\lambda (S \\times T, G) = \\left|\\sum _{x, y \\in S \\times T}G(x, y)\\lambda (x, y) \\right|.$ The discrepancy of $G$ under the distribution $\\lambda $ is defined to be $\\textnormal {disc}_{\\lambda }(G) = \\max _{S \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^j, T \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^k} \\textnormal {disc}_{\\lambda }(S \\times T, G),$ and the discrepancy of $f$ is defined to be $\\textnormal {disc}(G) = \\min _{\\lambda }\\textnormal {disc}_{\\lambda }(G).$ Definition 2.13 (Balanced-discrepancy) Let $G : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function and $\\Lambda $ be the set of all balanced distributions on $\\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k$ with respect to $G$ .",
"The balanced-discrepancy of $G$ is defined to be $\\textnormal {bdisc}(G) = \\min _{\\lambda \\in \\Lambda }\\textnormal {disc}_{\\lambda }(G).$ The following theorem is implicit in [25].",
"We prove it in Appendix REF for completeness.",
"Theorem 2.14 Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be functions.",
"Let $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\mathbb {R}$ be a balanced distribution with respect to $G$ and $\\textnormal {disc}_{\\mu }(G) = o(1)$ .",
"If $\\frac{8en}{\\widetilde{\\textnormal {deg}}(r)}\\le \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)}\\right)^{1-\\beta }$ for some constant $\\beta \\in (0,1)$ , then $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)}\\right)\\right).$ In particular, $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)\\right).$"
],
[
"Hadamard encoding",
"Recall that we index coordinates of $n$ -bit strings by integers in $[n]$ , and also interchangeably by strings in $\\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ via the natural correspondence.",
"For $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , let $-x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ be defined as $(-x)_i = -x_i$ for all $i \\in [n]$ .",
"We use the notation $\\pm x$ to denote the set $\\lbrace x, -x\\rbrace $ .",
"Definition 2.15 (Hadamard Codewords) For every positive integer $n$ and $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , let $H(s) \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ be defined as $(H(s))_t = \\prod _{i: s_i = -1} t_i~\\text{for all}~t \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}.$ If $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ is such that $x = H(s)$ for some $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , we say $x$ is a Hadamard codeword corresponding to $s$ .",
"That is, for every $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , there is an $n$ -bit Hadamard codeword corresponding to $s$ .",
"This represents the enumeration of all parities of $s$ .",
"We now define how to encode a two-party total function $G$ on $(\\log j + \\log k)$ input bits to a partial function $h_G$ on $(j+k)$ input bits, using Hadamard encoding.",
"Definition 2.16 (Hadamardization of functions) Let $j, k \\ge 1$ be powers of 2, and let $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a function.",
"Define a partial function $h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{j + k} \\rightarrow \\left\\lbrace -1, 1, \\star \\right\\rbrace $ by $h_G(x, y) = {\\left\\lbrace \\begin{array}{ll}G(s, t) & \\text{if}~x \\in \\pm H(s), y \\in \\pm H(t)~\\text{for some}~s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}, t \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}\\\\\\star & \\text{otherwise}.\\end{array}\\right.",
"}$"
],
[
"Additional concepts from quantum computing",
"The Bernstein-Vazirani algorithm [11] is a quantum query algorithm that takes an $n$ -bit string as input and outputs a $(\\log n)$ -bit string.",
"The algorithm has the following properties: the algorithm makes one quantum query to the input and if the input $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ satisfies $x\\in \\pm H(s)$ for some $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , then the algorithm returns $s$ with probability 1.",
"Consider a symmetric Boolean function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"Define the quantity $\\Gamma (f)=\\min \\lbrace |2k-n+1| : f(x)\\ne f(y)\\text{ if }|x|=k\\text{ and } |y|=k+1\\rbrace $ from [29].",
"One can think of $\\Gamma (f)$ as essentially the length of the interval of Hamming weights around $n/2$ where $f$ is constant (for example, for the majority and parity functions this would be 1, and for $\\mathsf {OR}_n$ this would be $n-1$ ).",
"Theorem 2.17 ([5]) For every symmetric function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , we have $\\mathsf {Q}(f) = \\Theta (\\sqrt{(n-\\Gamma (f))n}).$ The upper bound follows from a quantum algorithm that exactly counts the Hamming weight $|x|$ of the input if $|x|\\le t$ or $|x|\\ge n-t$ for $t=\\left\\lceil (n-\\Gamma (f))/2 \\right\\rceil $ , and that otherwise learns $|x|$ is in the interval $[t+1,n-t-1]$ (which is an interval around $n/2$ where $f(x)$ is constant).",
"By the definition of $\\Gamma (f)$ , this information about $|x|$ suffices to compute $f(x)$ .",
"In Section we use this observation to give an efficient quantum communication protocol for a two-party function $f\\circ G$ .",
"We will need a unitary protocol that allows Alice and Bob to implement an approximate reflection about the $n$ -dimensional maximally entangled state $|\\psi \\rangle =\\frac{1}{\\sqrt{n}}\\sum _{i\\in \\lbrace 0,1\\rbrace ^{\\log n}}|i\\rangle |i\\rangle .$ Ideally, such a reflection would map $|\\psi \\rangle $ to itself, and put a minus sign in front of all states orthogonal to $|\\psi \\rangle $ .",
"Doing this perfectly would requires $O(\\log n)$ qubits of communication.",
"Fortunately we can derive a cheaper protocol from a test that Aharonov et al.",
"[4] designed, which uses $O(\\log (1/\\varepsilon ))$ qubits of communication and checks whether a given bipartite state equals $|\\psi \\rangle $ , with one-sided error probability $\\varepsilon $ .",
"By the usual trick of running this protocol, applying a $Z$ -gate to the answer qubit, and then reversing the protocol, we can implement the desired reflection approximately.Possibly with some auxiliary qubits on Alice and Bob's side which start in $|0\\rangle $ and end in $|0\\rangle $ , except in a part of the final state that has norm at most $\\varepsilon $ .",
"A bit more precisely: Theorem 2.18 Let $R_{\\psi }=2|\\psi \\rangle \\!",
"\\langle \\psi |-I$ be the reflection about the maximally entangled state shared between Alice and Bob.",
"There exists a protocol that uses $O(\\log (1/\\varepsilon ))$ qubits of communication and that implements a unitary $R^\\varepsilon _{\\psi }$ such that $\\left\\Vert R^\\varepsilon _{\\psi } - R_{\\psi } \\right\\Vert \\le \\varepsilon $ and $R^\\varepsilon _{\\psi }|\\psi \\rangle =|\\psi \\rangle $ .",
"We use $\\mathsf {UPP}^{cc}(F)$ to denote unbounded-error quantum communication complexity of two-party function $F$ .",
"It is folklore (see for example [20]) that the unbounded-error quantum communication complexityThe unbounded-error model does not allow shared randomness or prior shared entanglement (which yields shared randomness by measuring) between Alice and Bob, since any two-party function $F$ would have constant communication complexity in that setting.",
"of $F$ equals its classical counterpart up to a factor of at most 2 so it does not really matter much whether we use $\\mathsf {UPP}^{cc}$ for classical unbounded-error communication complexity (as it is commonly used) or for quantum unbounded-error complexity.",
"Crucially, for both the complexity of $\\mathsf {IP}_n$ is linear in $n$ : Theorem 2.19 ([15]) Let $n$ be a positive integer.",
"Then, $\\mathsf {UPP}^{cc}(\\mathsf {IP}_n) = \\Omega (n).$"
],
[
"Noisy amplitude amplification and a new distributed-search protocol",
"In this section we present a version of quantum amplitude amplification that still works if the reflections involved are not perfectly implemented.",
"In particular, the usual reflection about the uniform superposition will be replaced in the communication setting by an imperfect reflection about the $n$ -dimensional maximally entangled state, based on the communication-efficient protocol of Aharonov et al.",
"[4] for testing whether Alice and Bob share that state.",
"This allows us to avoid the $\\log n$ factor that would be incurred if we instead used a BCW-style distributed implementation of standard amplitude amplification, with $O(\\log n)$ qubits of communication to implement each query.",
"Our main result in this section is the following general theorem, which allows us to search among a sequence of two-party instances $(X_1,Y_1),\\ldots ,(X_n,Y_n)$ for an index $i \\in [n]$ where $G(X_i,Y_i)=-1$ , for any two-party function $G$ .",
"Theorem 3.1 Let $G: \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a two-party function, $X = (X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nj}$ and $Y = (Y_1, \\dots , Y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{nk}$ .",
"Define $z=(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"Assume Alice and Bob start with $\\left\\lceil \\log n \\right\\rceil $ shared EPR-pairs.",
"There exists a quantum protocol using $O(\\sqrt{n}\\,\\mathsf {Q}_E^{cc}(G))$ qubits of communication that finds (with success probability $\\ge 0.99$ ) an $i\\in [n]$ such that $z_i=-1$ if such an $i$ exists, and says “no” with probability 1 if no such $i$ exists.",
"If the number of $-1$ s in $z$ is within a factor of 2 from a known integer $t$ , then the communication can be reduced to $O(\\sqrt{n/t}\\,\\mathsf {Q}_E^{cc}(G))$ qubits.",
"Remark 3.2 The $\\log n$ shared EPR-pairs that we assume Alice and Bob share at the start could also be established by means of $\\log n$ qubits of communication at the start of the protocol.",
"For the result in the first bullet, this additional communication does not change the asymptotic bound.",
"For the result of the second bullet, if $t\\le n\\mathsf {Q}_E^{cc}(G)^2/(\\log n)^2$ , then this additional communication does not change the asymptotic bound either.",
"However, if $t=\\omega (n/(\\log n)^2)$ and $\\mathsf {Q}_E^{cc}(G)=O(1)$ then the quantum communication $O(\\sqrt{n/t}\\,\\mathsf {Q}_E^{cc}(G))$ is $o(\\log n)$ and establishing the $\\log n$ EPR-pairs by means of a first message makes a difference.",
"As a corollary, we obtain a new $O(\\sqrt{n})$ -qubit protocol for the distributed search problem composed with $G=\\mathsf {AND}_2$ (whose decision version is the Set-Disjointness problem)."
],
[
"Amplitude amplification with perfect reflections",
"We first describe basic amplitude amplification in a slightly unusual recursive manner, similar to [19].",
"We are dealing with a search problem where some set $\\cal G$ of basis states are deemed “good” and the other basis states are deemed “bad.” Let $P_{\\cal G}=\\sum _{g\\in {\\cal G}}|g\\rangle \\!",
"\\langle g|$ be the projector onto the span of the good basis states, and $O_{\\cal G}=I-2P_{\\cal G}$ be the reflection that puts a `$-$ ' in front of the good basis states: $O_{\\cal G}|g\\rangle =-|g\\rangle $ for all basis states $g\\in {\\cal G}$ , and $O_{\\cal G}|b\\rangle =|b\\rangle $ for all basis states $b\\notin {\\cal G}$ .",
"Suppose we have an initial state $|\\psi \\rangle $ which is a superposition of a good state and a bad state: $|\\psi \\rangle =\\sin (\\theta )|G\\rangle +\\cos (\\theta )|B\\rangle ,$ where $|G\\rangle =P_{\\cal G}|\\psi \\rangle /\\left\\Vert P_{\\cal G}|\\psi \\rangle \\right\\Vert $ and $|B\\rangle =(I-P_{\\cal G})|\\psi \\rangle /\\left\\Vert (I-P_{\\cal G})|\\psi \\rangle \\right\\Vert $ .",
"For example in Grover's algorithm, with a search space of size $n$ containing $t$ solutions, the initial state $|\\psi \\rangle $ would be the uniform superposition, and its overlap (inner product) with the good subspace spanned by the $t$ “good” (sometimes called “marked”) basis states would be $\\sin (\\theta )=\\sqrt{t/n}$ .",
"We'd like to increase the weight of the good state, i.e., move the angle $\\theta $ closer to $\\pi /2$ .",
"Let $R_\\psi $ denote the reflection about the state $|\\psi \\rangle $ , i.e., $R_\\psi |\\psi \\rangle =|\\psi \\rangle $ and $R_\\psi {|\\phi \\rangle }=-|\\phi \\rangle $ for every $|\\phi \\rangle $ that is orthogonal to $|\\psi \\rangle $ .",
"Then the algorithm $A_1=R_\\psi \\cdot O_{\\cal G}$ is the product of two reflections, which (in the 2-dimensional space spanned by $|G\\rangle $ and $|B\\rangle $ ) corresponds to a rotation by an angle $2\\theta $ , thus increasing our angle from $\\theta $ to $3\\theta $ .",
"This is the basic amplitude amplification step.",
"It maps $|\\psi \\rangle \\mapsto A_1|\\psi \\rangle =\\sin (3\\theta )|G\\rangle +\\cos (3\\theta )|B\\rangle .$ We can now repeat this step recursively, defining $A_2=A_1R_\\psi A^*_1\\cdot O_{\\cal G}\\cdot A_1.$ Note that $A_1R_\\psi A^*_1$ is a reflection about the state $A_1|\\psi \\rangle $ .",
"Thus $A_2$ triples the angle between $A_1|\\psi \\rangle $ and $|B\\rangle $ , mapping $|\\psi \\rangle \\mapsto A_2|\\psi \\rangle =\\sin (9\\theta )|G\\rangle +\\cos (9\\theta )|B\\rangle .$ Continuing recursively in this fashion, define the algorithm $A_{j+1}=A_j R_\\psi A_j^*\\cdot O_{\\cal G}\\cdot A_j.$ The last algorithm $A_k$ will map $|\\psi \\rangle \\mapsto A_k|\\psi \\rangle =\\sin (3^k\\theta )|G\\rangle +\\cos (3^k\\theta )|B\\rangle .$ Hence after $k$ recursive amplitude amplification steps, we have angle $3^k\\theta $ .",
"Since we want to end up with angle $\\approx \\pi /2$ , if we know $\\theta $ then we can choose $k=\\left\\lfloor \\log _3(\\pi /(2\\theta )) \\right\\rfloor .$ This gives us an angle $3^k\\theta \\in (\\pi /6,\\pi /2]$ , so the final state $A_k|\\psi \\rangle $ has overlap $\\sin (\\theta _k)>1/2$ with the good state $|G\\rangle $ .",
"Let $C_k$ denote the “cost” (in whatever measure, for example query complexity, or communication complexity, or circuit size) of algorithm $A_k$ .",
"Looking at its recursive definition (Equation (REF )), $C_k$ is 3 times $C_{k-1}$ , plus the cost of $R_\\psi $ plus the cost of $O_{\\cal G}$ .",
"If we just count applications of $O_{\\cal G}$ (“queries”), considering $R_\\psi $ to be free, then $C_{k+1}=3C_k+1$ .",
"This recursion has the closed form $C_k=\\sum _{i=0}^{k-1} 3^i< 3^k$ .",
"With the above choice of $k$ we get $C_k=O(1/\\theta )$ .",
"In the case of Grover's algorithm, where $\\theta =\\arcsin (\\sqrt{t/n})\\approx \\sqrt{t/n}$ , the cost is $C_k=O(\\sqrt{n/t})$ ."
],
[
"Amplitude amplification with imperfect reflections",
"Now we consider the situation where we do not implement the reflections $R_\\psi $ perfectly, but instead implement another unitary $R^\\varepsilon _\\psi $ at operator-norm distance $\\left\\Vert R^\\varepsilon _\\psi -R_\\psi \\right\\Vert \\le \\varepsilon $ from $R_\\psi $ , with the additional property that $R^\\varepsilon _\\psi |\\psi \\rangle =|\\psi \\rangle $ (this one-sided error property will be important for the proof).",
"We can control this error $\\varepsilon $ , but smaller $\\varepsilon $ will typically correspond to higher cost of $R^\\varepsilon _\\psi $ .",
"The reflection $O_{\\cal G}$ will still be implemented perfectly below.",
"We again start with the initial state $|\\psi \\rangle =\\sin (\\theta )|G\\rangle +\\cos (\\theta )|B\\rangle .$ For errors $\\varepsilon _1,\\ldots ,\\varepsilon _k$ that we will specify later, recursively define the following algorithms.",
"$A_1=R^{\\varepsilon _1}_\\psi \\cdot O_{\\cal G}\\mbox{~~and~~}A_{j+1}=A_jR^{\\varepsilon _{j+1}}_\\psi A_j^*\\cdot O_{\\cal G}\\cdot A_j.$ These algorithms will map the initial state as follows: $|\\psi \\rangle \\mapsto |\\psi _j\\rangle =A_j|\\psi \\rangle =\\sin (3^j\\theta )|G\\rangle +\\cos (3^j\\theta )|B\\rangle +|E_j\\rangle ,$ where $|E_j\\rangle $ is some unnormalized error state defined by the above equation; its norm $\\eta _j$ quantifies the extent to which we deviate from perfect amplitude amplification.",
"Our goal here is to upper bound this $\\eta _j$ .",
"In order to see how $\\eta _j$ can grow, let us see how $A_jR^{\\varepsilon _{j+1}}_\\psi A_j^*\\cdot O_{\\cal G}$ acts on $\\sin (3^j\\theta )|G\\rangle +\\cos (3^j\\theta )|B\\rangle $ (we'll take into account the effects of the error term $|E_j\\rangle $ later).",
"If $R^{\\varepsilon _{j+1}}_\\psi $ were equal to $R_\\psi $ , then we would have one perfect round of amplitude amplification and obtain $\\sin (3^{j+1}\\theta )|G\\rangle +\\cos (3^{j+1}\\theta )|B\\rangle $ ; but since $R^{\\varepsilon _{j+1}}_\\psi $ is only $\\varepsilon _{j+1}$ -close to $R_\\psi $ , additional errors can appear.",
"First we apply $O_{\\cal G}$ , which negates $|G\\rangle $ and hence changes the state to $-\\sin (3^j\\theta )|G\\rangle +\\cos (3^j\\theta )|B\\rangle =|\\psi _j\\rangle -|E_j\\rangle -2\\sin (3^j\\theta )|G\\rangle .$ Second we apply $V=A_jR^{\\varepsilon _{j+1}}_\\psi A_j^*$ .",
"Let $V^{\\prime }=A_jR_\\psi A_j^*$ , and note that $V|\\psi _j\\rangle =V^{\\prime }|\\psi _j\\rangle =|\\psi _j\\rangle $ and $\\left\\Vert V^{\\prime }-V \\right\\Vert =\\left\\Vert R_\\psi -R^{\\varepsilon _{j+1}}_\\psi \\right\\Vert \\le \\varepsilon _{j+1}$ .",
"The new state is $V(|\\psi _j\\rangle -|E_j\\rangle -2\\sin (3^j\\theta )|G\\rangle )& = V^{\\prime }(|\\psi _j\\rangle -|E_j\\rangle -2\\sin (3^j\\theta )|G\\rangle ) +(V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle )\\\\& = V^{\\prime }(-\\sin (3^j\\theta )|G\\rangle +\\cos (3^j\\theta )|B\\rangle )+(V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle )\\\\& = \\sin (3^{j+1}\\theta )|G\\rangle +\\cos (3^{j+1}\\theta )|B\\rangle +(V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle ).$ Putting back also the earlier error term $|E_j\\rangle $ from Equation (REF ) (to which the unitary $VO_{\\cal G}$ is applied as well), it follows that the new error state is $|E_{j+1}\\rangle =|\\psi _{j+1}\\rangle -(\\sin (3^{j+1}\\theta )|G\\rangle +\\cos (3^{j+1}\\theta )|B\\rangle )=VO_{\\cal G}|E_j\\rangle +(V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle ).$ Its norm is $\\eta _{j+1} & \\le \\left\\Vert VO_{\\cal G}|E_j\\rangle \\right\\Vert +\\left\\Vert (V^{\\prime }-V)(|E_j\\rangle +2\\sin (3^j\\theta )|G\\rangle ) \\right\\Vert \\\\& \\le \\eta _j + \\varepsilon _{j+1}(\\eta _j+2\\sin (3^j\\theta ))= (1+\\varepsilon _{j+1})\\eta _j + 2\\varepsilon _{j+1}\\sin (3^j\\theta ).$ Since $\\eta _0=0$ , we can “unfold” the above recursive upper bound to the following, which is easy to verify by induction on $k$ : $\\eta _k \\le \\sum _{j=1}^k\\prod _{\\ell =j+1}^k(1+\\varepsilon _\\ell )2\\varepsilon _j\\sin (3^{j-1}\\theta ).$ For each $1 \\le j \\le k$ , choose $\\varepsilon _j=\\frac{1}{100\\cdot 4^j}.$ Note that $\\sigma =\\sum _{j=1}^k\\varepsilon _k\\le 1/300$ .",
"With this choice of $\\varepsilon _j$ 's, and the inequalities $1+x\\le e^x$ , $e^\\sigma \\le 1.5$ and $\\sin (x)\\le x$ for $x \\le \\pi /2$ (which is the case here), we can upper bound the norm of the error term $|E_k\\rangle $ after $k$ iterations (see Equation (REF )) as $\\eta _k \\le \\sum _{j=1}^k e^\\sigma 2\\varepsilon _j 3^{j-1}\\theta \\le \\frac{3\\theta }{400}\\sum _{j=1}^k (3/4)^{j-1}\\le \\frac{3\\theta }{100}.$ Accordingly, up to very small error we have done perfect amplitude amplification."
],
[
"Distributed amplitude amplification with imperfect reflection",
"We will now instantiate the above scheme to the case of distributed search, where our measure of cost is communication, that is, the number of qubits sent between Alice and Bob.",
"Specifically, consider the intersection problem where Alice and Bob have inputs $x\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ and $y\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , respectively.",
"Assume for simplicity that $n$ is a power of 2, so $\\log n$ is an integer.",
"Alice and Bob want to find an $i\\in \\lbrace 0,\\ldots ,n-1\\rbrace =\\lbrace 0,1\\rbrace ^{\\log n}$ such that $x_i=y_i=-1$ , if such an $i$ exists.",
"The basis states in this distributed problem are $|i\\rangle |j\\rangle $ , and we define the set of “good” basis states as ${\\cal G}=\\lbrace |i\\rangle |j\\rangle \\mid x_i=y_j=-1\\rbrace ,$ even though we are only looking for $i,j$ where $i=j$ (it's easier to implement $O_{\\cal G}$ with this more liberal definition of $\\cal G$ ).",
"Our protocol will start with the maximally entangled initial state $|\\psi \\rangle $ in $n$ dimensions, which corresponds to $\\log n$ EPR-pairs: $|\\psi \\rangle =\\frac{1}{\\sqrt{n}}\\sum _{i\\in \\lbrace 0,1\\rbrace ^{\\log n}}|i\\rangle |i\\rangle =\\sin (\\theta )|G\\rangle +\\cos (\\theta )|B\\rangle ,$ where we assume there are $t$ $i$ 's where $x_i=y_i=-1$ , i.e., $t$ solutions to the intersection problem, so $\\theta =\\arcsin (\\sqrt{t/n}).$ and $|G\\rangle =\\frac{1}{\\sqrt{t}}\\sum _{(i,i)\\in {\\cal G}}|i\\rangle |i\\rangle .$ It costs $\\left\\lceil \\log n \\right\\rceil $ qubits of communication between Alice and Bob to establish this initial shared state, or it costs nothing if we assume pre-shared entanglement.",
"Our goal is to end up with a state that has large inner product with $|G\\rangle $ .",
"In order to be able to use amplitude amplification, we would like to be able to reflect about the above state $|\\psi \\rangle $ .",
"However, in general this perfect reflection $R_{\\psi }$ costs a lot of communication: Alice would send her $\\log n$ qubits to Bob, who would unitarily put a $-1$ in front of all states orthogonal to $|\\psi \\rangle $ , and then sends back Alice's qubits.",
"This has a communication cost of $O(\\log n)$ qubits, which is too much for our purposes.",
"Fortunately, Theorem REF gives us a way to implement a one-sided $\\varepsilon $ -error reflection protocol $R^\\varepsilon _{\\psi }$ that only costs $O(\\log (1/\\varepsilon ))$ qubits of communication.",
"The reflection $O_{\\cal G}$ puts a `$-$ ' in front of the basis states $|i\\rangle |j\\rangle $ in $\\cal G$ .",
"This can be implemented perfectly using only 2 qubits of communication, as follows.",
"To implement this reflection on her basis state $|i\\rangle $ , Alice XORs $|x_i\\rangle $ into a fresh auxiliary $|0\\rangle $ -qubit and sends this qubit to Bob.",
"Bob receives this qubit and applies the following unitary map: $|b\\rangle |j\\rangle \\mapsto y_j^{b}|b\\rangle |j\\rangle ,~~~b\\in \\lbrace 0,1\\rbrace ,j\\in [n].$ He sends back the auxiliary qubit.",
"Alice sets the auxiliary qubit back to $|0\\rangle $ by XOR-ing $x_i$ into it.",
"Ignoring the auxiliary qubit (which starts and ends in state $|0\\rangle $ ), this maps $|i\\rangle |j\\rangle \\mapsto (-1)^{[x_i=y_j=-1]}|i\\rangle |j\\rangle $ .",
"Hence we have implemented $O_{\\cal G}$ correctly: a minus sign is applied exactly for the good basis states, the ones where $x_i=y_j=-1$ .",
"Now consider the algorithms (more precisely, communication protocols): $A_1=R^{\\varepsilon _1}_\\psi \\cdot O_{\\cal G}\\mbox{~~and~~}A_{j+1}=A_jR^{\\varepsilon _{j+1}}_\\psi A_j^*\\cdot O_{\\cal G}\\cdot A_j$ with the choice of $\\varepsilon _j$ 's from Equation (REF ).",
"If we pick $k=\\left\\lfloor \\log _3(\\pi /(2\\theta )) \\right\\rfloor $ , like in Equation (REF ), then $3^k\\theta \\in (\\pi /6,\\pi /2]$ .",
"Hence by Equation (REF ) and Equation (REF ), the inner product of our final state with $|G\\rangle $ will be between $\\sin (3^k\\theta )-3\\theta /100\\ge 0.4$ and 1.",
"At this point Alice and Bob can measure, and with probability $\\ge 0.4^2$ they will each see the same $i$ , with the property that $x_i=y_i=-1$ .",
"From Equation (REF ) and Theorem REF , the recursion for the communication costs of these algorithms is $C_{j+1}=3C_j+O(\\log (1/\\varepsilon _{j+1}))+2.$ Solving this recurrence with our $\\varepsilon _j$ 's from Equation (REF ) and the value of $\\theta $ from Equation (REF ) we obtain $C_k=\\sum _{j=1}^k 3^{k-j} (O(\\log (1/\\varepsilon _j))+2)=\\sum _{j=1}^k 3^{k-j}O(j)= O(3^k)=O(\\sqrt{n/t}).$ Thus, using $O(\\sqrt{n/t})$ qubits of communication we can find (with constant success probability) an intersection point $i$ .",
"This also allows us to solve the Set-Disjointness problem (the decision problem whose output is 1 if there is no intersection between $x$ and $y$ ).",
"Note that if the $t$ we used equals the actual number of solutions only up to a factor of 2, the above protocol still has $\\Omega (1)$ probability to find a solution, and $O(1)$ repetitions will boost this success probability to $0.99$ .",
"In case we do not even know $t$ approximately, we can use the standard technique of trying exponentially decreasing guesses for $t$ to find an intersection point with communication $O(\\sqrt{n})$ .",
"Note that there is no log-factor in the communication complexity, in contrast to the original $O(\\sqrt{n}\\log n)$ -qubit Grover-based quantum protocol for the intersection problem of Buhrman et al. [6].",
"Aaronson and Ambainis [1] earlier already managed to remove the log-factor, giving an $O(\\sqrt{n})$ -qubit protocol for Set-Disjointness as a consequence of their local version of quantum search on a grid graph (which is optimal [31]).",
"We have just reproved this result of [1] in a different and arguably simpler way.",
"The above description is geared towards the intersection problem, where the “inner” function is $G=\\mathsf {AND}_2$ : we called a basis state $|i\\rangle |j\\rangle $ “good” if $x_i=y_j=-1$ .",
"However, this can easily be generalized to the situation where Alice and Bob's respective inputs are $X=(X_1,\\ldots ,X_n)$ and $Y=(Y_1,\\ldots ,Y_n)$ and we want to find an $i\\in [n]$ where $G(X_i,Y_i)=-1$ for some two-party function $G$ , and define the set of “good” basis states as ${\\cal G}=\\lbrace |i\\rangle |j\\rangle \\mid G(X_i,Y_j)=-1\\rbrace $ .We intentionally use the letter `$G$ ' to mean “good” in $\\cal G$ and and to refer to the two-party function $G$ , since $G$ determines which basis states $|i\\rangle |j\\rangle $ are “good.” The only thing that changes in the above is the implementation of the reflection $O_{\\cal G}$ , which would now be computed by means of an exact quantum communication protocol for $G(X_i,Y_j)$ , at a cost of $2\\mathsf {Q}_E^{cc}(G)$ qubits of communication.The factor of 2 is to reverse the protocol after the phase $G(X_i,Y_j)$ has been added to basis state $|i\\rangle |j\\rangle $ , in order to set any workspace qubits back to $|0\\rangle $ .",
"Note that because we can check (at the expense of another $\\mathsf {Q}_E^{cc}(G)$ qubits of communication) whether the output index $i$ actually satisfies $G(X_i,Y_i)=-1$ , we may assume the protocol has one-sided error: it always outputs “no” if there is no such $i$ .",
"This concludes the proof of Theorem REF ."
],
[
"No log-factor needed for symmetric functions",
"In this section we prove Theorem REF from the introduction.",
"Consider a symmetric Boolean function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"As explained in Section REF , there is an integer $t=\\left\\lceil (n-\\Gamma (f))/2 \\right\\rceil $ such that we can compute $f$ if we learn the Hamming weight $|z|$ of the input $z\\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ or learn that $|z|\\in [t+1,n-t-1]$ .",
"The bounded-error quantum query complexity is $\\mathsf {Q}(f)=\\Theta (\\sqrt{tn})$ (Theorem REF ).",
"For a given two-party function $G:\\left\\lbrace -1,1 \\right\\rbrace ^j\\times \\left\\lbrace -1,1 \\right\\rbrace ^k\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , we have an induced two-party function $F:\\left\\lbrace -1,1 \\right\\rbrace ^{nj}\\times \\left\\lbrace -1,1 \\right\\rbrace ^{nk}\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ defined as $F(X_1,\\ldots ,X_n,Y_1,\\ldots ,Y_n)=f(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))$ .",
"Define $z=(G(X_1,Y_1),\\ldots ,G(X_n,Y_n))\\in \\left\\lbrace -1,1 \\right\\rbrace ^n.$ Then $F(X,Y)=f(z)$ only depends on the number of $-1$ s in $z$ .",
"The following theorem allows us to count this number using $O(\\mathsf {Q}(f)\\,\\mathsf {Q}_E^{cc}(G))$ qubits of communication.",
"Theorem 4.1 For every $t$ between 1 and $n/2$ , there exists a quantum protocol that starts from $O(t\\log n)$ EPR-pairs, communicates $O(\\sqrt{tn}\\,\\mathsf {Q}_E^{cc}(G))$ qubits, and tells us $|z|$ or tells us that $|z|>t$ , with error probability $\\le 1/8$ .",
"Abbreviate $q=\\mathsf {Q}_E^{cc}(G)$ .",
"Our protocol has two parts: the first filters out the case $|z|\\ge 2t$ , while the second finds all solutions if $|z|<2t$ ."
],
[
"Part 1.",
"First Alice and Bob decide between the case (1) $|z|\\ge 2t$ and (2) $|z|\\le t$ (even though $|z|$ might also lie in $\\lbrace t+1,\\ldots ,2t-1\\rbrace $ ) using $O(\\sqrt{n}q)$ qubits of communication, as follows.",
"They use shared randomness to choose a uniformly random subset $S\\subseteq [n]$ of $\\left\\lceil n/(2t) \\right\\rceil $ elements.",
"Let $E$ be the event that $z_i=-1$ for at least one $i\\in S$ .",
"By standard calculations there exist $p_1,p_2\\in [0,1]$ with $p_1=p_2+\\Omega (1)$ such that $\\Pr [E]\\ge p_1$ in case (1) and $\\Pr [E]\\le p_2$ in case (2).",
"Alice and Bob use the distributed-search protocol from the first bullet of Theorem REF to decide $E$ , with $O(\\sqrt{|S|}\\,q)=O(\\sqrt{n}\\,q)$ qubits of communication (plus a negligible $O(\\log n)$ EPR-pairs) and error probability much smaller than $p_1-p_2$ .",
"By repeating this a sufficiently large constant number of times and seeing whether the fraction of successes was larger or smaller than $(p_1+p_2)/2$ , they can distinguish between cases (1) and (2) with success probability $\\ge 15/16$ .",
"If they conclude they're in case (1) then they output “$|z|>t$ ” and otherwise they proceed to the second part of the protocol.",
"Note that if $|z|\\in \\lbrace t+1,\\ldots ,2t-1\\rbrace $ (the “grey zone” in between cases (1) and (2)), then we can't give high-probability guarantees for one output or the other, but concluding (1) leads to the correct output “$|z|>t$ ” in this case, while concluding (2) means the protocol proceeds to Part 2.",
"So either course of action is fine if $|z|\\in \\lbrace t+1,\\ldots ,2t-1\\rbrace $ .",
"By Newman's theorem [27] the shared randomness used for choosing $S$ can be replaced by $O(\\log n)$ bits of private randomness on Alice's part, which she can send to Bob in her first message, so Part 1 communicates $O(\\sqrt{n}\\,q)$ qubits in total."
],
[
"Part 2.",
"We condition on Part 1 successfully filtering out case (1), so from now on assume $|z|<2t$ .",
"Our goal in this second part of the protocol is to find all indices $i$ such that $z_i=-1$ (we call such $i$ “solutions”), with probability $\\ge 15/16$ , using $O(\\sqrt{tn}\\,q)$ qubits of communication.",
"This will imply that the overall protocol is correct with probability $1-1/16-1/16=7/8$ , and uses $O(\\sqrt{tn}\\,q)$ qubits of communication in total.",
"For an integer $k\\ge 1$ , consider the following protocol $P_k$ .",
"[H] Input: An integer $k \\ge 1$ $200\\sqrt{2^k n}\\,q$ qubits have been sent Run the protocol from the last bullet of Theorem REF with $t=2^{k-1}$ .",
"(suppressing some constant factors, assume for simplicity that this uses $\\sqrt{n/2^k}\\,q$ qubits of communication, $\\log n$ shared EPR-pairs at the start, and has probability $\\ge 1/100$ to find a solution if the actual number of solutions is in $[t/2,2t]$ ).",
"Alice measures and gets outcome $i\\in [n]$ and Bob measures and gets outcome $j\\in [n]$ , respectively.",
"Alice sends $i$ to Bob, Bob sends $j$ to Alice.",
"If $i=j$ then they verify that $G(X_i,Y_i)=-1$ by one run of the protocol for $G$ , and if so then they replace $X_i,Y_i$ by some pre-agreed inputs $X^{\\prime }_i,Y^{\\prime }_i$ , respectively, such that $G(X^{\\prime }_i,Y^{\\prime }_i)=1$ (this reduces the number of $-1$ s in $z$ by 1) Protocol $P_k$ Claim 4.2 Suppose $|z|\\in [2^{k-1},2^k)$ .",
"Then protocol $P_k$ uses $O(\\sqrt{2^k n}\\,q)$ qubits of communication, assumes $O(2^k\\log n)$ EPR-pairs at the start of the protocol, and finds at least $|z|-2^{k-1}+1$ solutions, except with probability $\\le 1/2$ .",
"The upper bound on the communication is obvious from the stopping criterion of $P_k$ .",
"As long as the remaining number of solutions is $\\ge 2^{k-1}$ , each run of the protocol has probability $\\ge 1/100$ to find another solution.",
"Hence the expected number of runs of the protocol of Theorem REF to find at least $|z|-2^{k-1}+1$ solutions, is $\\le 100(|z|-2^{k-1}+1)$ .",
"By Markov's inequality, the probability that we haven't yet found $|z|-2^{k-1}+1$ solutions after $\\le 200(|z|-2^{k-1}+1)\\le 100\\cdot 2^k$ runs, is $\\le 1/2$ .",
"The communication cost of so many runs is $100\\cdot 2^k (\\sqrt{n/2^k}\\,q + \\log n)\\le 200 \\sqrt{2^k n}\\,q$ qubits.",
"Hence by the time that the number of qubits of the stopping criterion have been communicated, we have probability $\\ge 1/2$ of having found at least $|z|-2^{k-1}+1$ solutions.",
"The assumed number of EPR-pairs at the start is $\\log n$ per run, so $O(2^k\\log n)$ in total.",
"Note that if we start with a number of solutions $|z|\\in [2^{k-1},2^k)$ , and $P_k$ succeeds in finding at least $|z|-2^{k-1}+1$ new solutions, then afterwards we have $< 2^{k-1}$ solutions left.",
"The following protocol runs these $P_k$ in sequence, pushing down the remaining number of solutions to 0.",
"[H] $k=\\left\\lceil \\log _2 (2t) \\right\\rceil $ downto 1 Run $P_k$ a total of $r_k=\\left\\lceil \\log _2 (2t) \\right\\rceil -k+5$ times (replacing all $-1$ s found by $+1s$ in $z$ ).",
"Output the total number of solutions found.",
"Protocol $P$ Claim 4.3 If $|z|<2t$ then protocol $\\cal P$ uses $O(\\sqrt{tn}\\,q)$ qubits of communication, assumes $O(t\\log n)$ EPR-pairs at the start of the protocol, and outputs $|z|$ , except with probability $\\le 1/16$ .",
"First, by Claim REF , the total number of qubits communicated is $\\sum _{k=1}^{\\left\\lceil \\log _2 (2t) \\right\\rceil } r_k\\cdot O(\\sqrt{2^k n}\\,q)=O(\\sqrt{tn}\\,q)\\cdot \\sum _{\\ell =0}^{\\left\\lceil \\log _2 (2t) \\right\\rceil -1} (\\ell +5) /\\sqrt{2^\\ell }=O(\\sqrt{tn}\\,q),$ where we used a variable substitution $k=\\left\\lceil \\log _2 (2t) \\right\\rceil -\\ell $ .",
"Second, the number of EPR-pairs we're starting from is $\\sum _{k=1}^{\\left\\lceil \\log _2 (2t) \\right\\rceil }r_k\\cdot O(2^k\\log n)=O(t\\log n)\\cdot \\sum _{\\ell =0}^{\\left\\lceil \\log _2 (2t) \\right\\rceil -1}(\\ell +5) /2^\\ell =O(t\\log n).$ Third, by Claim REF and the fact that we are performing $r_k$ repetitions of $P_k$ , if the $k$ th round of $\\cal P$ starts with a remaining number of solutions that is in the interval $[2^{k-1},2^k)$ then that round ends with $<2^{k-1}$ remaining solutions, except with probability at most $1/2^{r_k}$ .",
"By the union bound, the probability that any one of the $\\left\\lceil \\log _2 (2t) \\right\\rceil $ rounds does not succeed at this, is at most $\\sum _{k=1}^{\\left\\lceil \\log _2 (2t) \\right\\rceil } \\frac{1}{2^{r_k}}=\\sum _{\\ell =0}^{\\left\\lceil \\log _2 (2t) \\right\\rceil -1}\\frac{1}{2^{\\ell +5}}\\le \\frac{1}{16}.$ Since $2^{\\left\\lceil \\log _2 (2t) \\right\\rceil }\\ge 2t$ and we start with $|z|<2t$ , if each round succeeds, then by the end of $\\cal P$ there are no remaining solutions left.",
"Thus, the protocol $\\cal P$ finds all solutions and learns $|z|$ with probability at least $15/16$ .",
"Part 1 and Part 2 each have error probability $\\le 1/16$ , so by the union bound the protocol succeeds except with probability $1/8$ .",
"If $|z|\\ge 2t$ then Part 1 outputs the correct answer “$|z|>t$ ”; if $|z|\\le t$ then all solutions (and hence $|z|$ ) are found by Part 2; and if $|z|\\in \\lbrace t+1,\\ldots ,2t-1\\rbrace $ then either Part 1 already outputs the correct answer “$|z|>t$ ” or the protocol proceeds to Part 2 which then finds all solutions.",
"We can use the above theorem twice: once to count the number of $-1$ s in $z$ (up to $t$ ) and once to count the number of 1s in $z$ (up to $t$ ).",
"This uses $O(\\sqrt{tn}\\,\\mathsf {Q}^{cc}_E(G))=O(\\mathsf {Q}(f)\\,\\mathsf {Q}_E^{cc}(G))$ qubits of communication, assumes $O(t\\log n)$ shared EPR-pairs at the start of the protocol, and gives us enough information about $|z|$ to compute $f(z)=F(X,Y)$ .",
"This concludes the proof of Theorem REF from the introduction, restated below.",
"Theorem 4.4 (Restatement of Theorem REF ) For every symmetric Boolean function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n\\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and two-party function $G:\\left\\lbrace -1,1 \\right\\rbrace ^j\\times \\left\\lbrace -1,1 \\right\\rbrace ^k\\rightarrow \\lbrace 0,1\\rbrace $ , we have $\\mathsf {Q}^{cc,*}(f \\circ G)=O(\\mathsf {Q}(f)\\mathsf {Q}_E^{cc}(G)).$ If $\\mathsf {Q}(f)=\\Theta (\\sqrt{tn})$ , then our protocol assumes a shared state of $O(t\\log n)$ EPR-pairs at the start.",
"We remark that for the special case where $G=\\mathsf {AND}_2$ , our upper bound matches the lower bound proved by Razborov [31], except for symmetric functions $f$ where the first switch of function value happens at Hamming weights very close to $n$ .",
"In particular, if $f=\\mathsf {AND}_n$ and $G=\\mathsf {AND}_2$ , then $\\mathsf {Q}^{cc}(f \\circ G)=1$ but $\\mathsf {Q}(f)=\\Theta (\\sqrt{n})$ ."
],
[
"Necessity of the log-factor overhead in the BCW simulation",
"In this section we prove Theorem REF and Theorem REF .",
"For Theorem REF we exhibit a function $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ for which $\\mathsf {Q}(f)= O(n)$ and $\\mathsf {UPP}(f \\circ \\square ) = \\Omega (n \\log n)$ for $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .",
"In Theorem REF we show a general recipe for constructing total functions $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ such that $\\mathsf {Q}(f) = O(n)$ and $\\mathsf {Q}^{cc, *}(f \\circ \\square ) = \\Omega (n \\log n)$ for $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .",
"We first give a formal statement of Theorem REF .",
"Theorem 5.1 (Restatement of Theorem REF ) Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function with $\\widetilde{\\textnormal {deg}}(r) = \\Omega (n)$ , $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function and $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .",
"Define $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"If there exists $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\mathbb {R}$ that is a balanced probability distribution with respect to $G$ and $\\textnormal {disc}_{\\mu }(G) = n^{-\\Omega (1)}$ , then $\\mathsf {Q}(f) & = O(n),\\\\\\mathsf {Q}^{cc, *}(f \\circ \\square ) & = \\Omega (n \\log n).$ The proofs of Theorem REF and Theorem REF each involve proving a query complexity upper bound and a communication complexity lower bound.",
"The proofs of the query complexity upper bounds are along similar lines and follow from Theorem REF and Corollary REF (see Section REF ).",
"The proofs of the communication complexity lower bounds each involve a reduction from a problem whose communication complexity is easier to analyze (see Lemma REF in Section REF ).",
"Finally, we complete the proofs of Theorem REF and Theorem REF in Section REF ."
],
[
"Quantum query complexity upper bound",
"We start by stating the main theorem in this section.",
"Theorem 5.2 Let $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $r : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"Then the quantum query complexity of the function $r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{n(j+k)} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is given by $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) = O(n + \\sqrt{n(j+k)}).$ As a corollary we obtain the following on instantiating $j = k = n$ and $r$ as a Boolean function with quantum query complexity $\\Theta (n)$ in Theorem REF .",
"Corollary 5.3 Let $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a non-constant function and let $r : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a total function with $\\mathsf {Q}(r) = \\Theta (n)$ .",
"Then the quantum query complexity of the total function $r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) = \\Theta (n).$ The upper bound $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) = O(n)$ follows by plugging in parameters in Theorem REF .",
"For the lower bound, we show that $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) \\ge \\mathsf {Q}(r)$ .",
"Since $G$ is non-constant, there exist $x_1, y_1, x_2, y_2 \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ such that $G(x_1, y_1) = -1$ and $G(x_2, y_2) = 1$ .",
"Let $X_1 = H(x_1), Y_1 = H(y_1)$ , $X_2 = H(x_2)$ and $X_2 = H(y_2)$ .",
"Consider $r ~\\widetilde{\\circ }~h_G$ only restricted to inputs where the inputs to each copy of $h_G$ are either $(X_1, Y_1)$ or $(X_2, Y_2)$ .",
"Under this restriction, $r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is the same as $r : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"Thus $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) \\ge \\mathsf {Q}(r) = \\Omega (n)$ .",
"We now prove Theorem REF .",
"[Proof of Theorem REF ] Recall from Definition REF that the function $h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{j + k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is defined as $h_G(x, y) = G(s, t)$ if $x \\in \\pm H(s)$ and $y \\in \\pm H(t)$ for some $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}$ and $t \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}$ , and $h_G(x, y) = \\star $ otherwise.",
"Also recall from Definition REF that the function $r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{n(j+k)} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is defined as $r ~\\widetilde{\\circ }~h_G((X_1, Y_1), \\dots , (X_n, Y_n)) = r \\circ h_G((X_1, Y_1), \\dots , (X_n, Y_n))$ if $h_G((X_i, Y_i)) \\in \\left\\lbrace -1,1 \\right\\rbrace $ for all $i \\in [n]$ , and $-1$ otherwise."
],
[
"Quantum query algorithm:",
"View inputs to $r ~\\widetilde{\\circ }~h_G$ as $(X_1, Y_1, \\dots , X_n, Y_n)$ , where $X_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^j$ for all $i \\in [n]$ and $Y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^k$ for all $i \\in [n]$ .",
"We give a quantum algorithm and its analysis below.",
"Run $2n$ instances of the Bernstein-Vazirani algorithm: 1 instance on each $X_i$ and 1 instance on each $Y_i$ , to obtain $2n$ strings $x_1, \\dots , x_n, y_1, \\dots , y_n$ , where each $x_i$ is a $(\\log j)$ -bit string and each $y_i$ is a $(\\log k)$ -bit string.",
"For each $X_i$ and $Y_i$ , query $(X_i)_{1^{\\log j}}$ and $(Y_i)_{1^{\\log k}}$ to obtain bits $b_i, c_i \\in \\left\\lbrace -1,1 \\right\\rbrace $ for all $i \\in [n]$ .",
"Run Grover's search [16], [7] to check equality of the following two $(nj + nk)$ -bit strings: $(b_1 H(x_1), \\dots , b_n H(x_n), c_1 H(y_1), \\dots , c_n H(y_n))$ and $(X_1, \\dots , X_n, Y_1, \\dots , Y_n)$ .",
"If the step above outputs that the strings are equal, then output $r(G(x_1, y_1), \\dots , G(x_n, y_n))$ .",
"Else, output $-1$ ."
],
[
"Analysis of the algorithm:",
" If the input is indeed of the form $(X_1, Y_1), \\dots , (X_n, Y_n)$ where each $X_i \\in \\pm H(x_i)$ and $Y_i \\in \\pm H(y_i)$ for some $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}$ and $y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}$ , then Step REF outputs the correct strings $x_{1}, \\dots , x_{n}, y_1, \\dots , y_n$ with probability 1 by the properties of the Bernstein-Vazirani algorithm.",
"Step REF then implies that $X_i = b_i H(x_i)$ and $Y_i = c_i H(y_i)$ for all $i \\in [n]$ .",
"Next, Step REF outputs that the strings are equal with probability 1 (since the strings whose equality are to be checked are equal).",
"Hence the algorithm is correct with probability 1 in this case, since $(r ~\\widetilde{\\circ }~h_G)(X_1, Y_1, \\dots , X_n, Y_n) = r(G(x_1, y_1), \\dots , G(x_n, y_n))$ .",
"If the input is such that there exists an index $i \\in [n]$ for which $X_i \\notin \\pm H(x_i)$ for every $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}$ or $Y_i \\notin \\pm H(y_i)$ for every $y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}$ , then the two strings for which equality is to be checked in the Step REF are not equal.",
"Grover's search catches a discrepancy with probability at least $2/3$ .",
"Hence, the algorithm outputs $-1$ (as does $r ~\\widetilde{\\circ }~h_G$ ), and is correct with probability at least $2/3$ in this case."
],
[
"Cost of the algorithm:",
"Step REF accounts for $2n$ quantum queries.",
"Step REF accounts for $2n$ quantum queries.",
"Step REF accounts for $O(\\sqrt{n(j+k)})$ quantum queries.",
"Thus, $\\mathsf {Q}(r ~\\widetilde{\\circ }~h_G) = O(n + \\sqrt{n(j+k)}).$"
],
[
"Quantum communication complexity lower bound",
"In this section we first show a communication lower bound (under some model) on $(r ~\\widetilde{\\circ }~h_G) \\circ \\square $ in terms of the communication complexity of $r \\circ G$ (in the same model of communication) using a simple reduction.",
"We state the lemma below (Lemma REF ) for the case where the models under consideration are the bounded-error and unbounded-error quantum models, since these are the models of interest to us.",
"Lemma 5.4 Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ and $CC \\in \\lbrace \\mathsf {Q}^{cc, *}, \\mathsf {UPP}^{cc}\\rbrace $ .",
"Then, $CC((r ~\\widetilde{\\circ }~h_G) \\circ \\square ) \\ge CC(r \\circ G).$ We first consider the case $\\square = \\mathsf {AND}_2$ .",
"Consider a protocol $\\Pi $ of cost $\\ell $ that solves $(r ~\\widetilde{\\circ }~h_G) \\circ \\square $ in the $CC$ -model.",
"We exhibit below a protocol of cost $\\ell $ that solves $r \\circ G$ in the same model.",
"Suppose Alice is given input $x = (x_1, \\dots , x_{n}) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n \\log j}$ and Bob is given input $y = (y_1, \\dots , y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n \\log k}$ , where $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j}, y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k}$ for each $i \\in [n]$ ."
],
[
"Preprocessing step:",
"Alice constructs the $(n(j+k))$ -bit string $X = ((H(x_1), (-1)^{k}), \\dots , (H(x_n), (-1)^k)) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n(j+k)},$ and Bob constructs the $(n(j+k))$ -bit string $Y = (((-1)^j, H(y_1)), \\dots , ((-1)^j, H(y_n))) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n(j+k)}.$"
],
[
"Protocol:",
"Alice and Bob run the protocol $\\Pi $ with input $(X, Y)$ and output $\\Pi (X, Y)$ ."
],
[
"Cost:",
"The preprocessing of the inputs to obtain $X$ from $x$ and $Y$ from $y$ takes no communication.",
"Hence the total amount of communication is at most the cost of $\\Pi $ ."
],
[
"Correctness:",
"For $X$ and $Y$ constructed in Equation (REF ) and Equation (REF ), respectively, we now argue that $(r \\circ G)(x, y) = ((r ~\\widetilde{\\circ }~h_G) \\circ \\mathsf {AND}_2) (X, Y)$ , which would conclude the proof for $\\square = \\mathsf {AND}_2$ .",
"$((r ~\\widetilde{\\circ }~h_G) \\circ \\mathsf {AND}_2)(X, Y) & = (r ~\\widetilde{\\circ }~h_G)((H(x_1), H(y_1)), \\dots , (H(x_n), H(y_n)))\\\\& = r(G(x_1, y_1), \\dots , G(x_n, y_n)) \\\\& = (r \\circ G)(x, y).$ Thus, $CC((r ~\\widetilde{\\circ }~h_G) \\circ \\mathsf {AND}_2) \\ge CC(r \\circ G).$ The argument for $\\square = \\mathsf {XOR}_2$ follows along the same lines, with the strings $(-1)^j$ and $(-1)^k$ replaced by $1^j$ and $1^k$ , respectively, in the preprocessing step."
],
[
"On the tightness of the BCW simulation",
"We prove Theorem REF in Section REF and Theorem REF in Section REF ."
],
[
"Proof of Theorem ",
"Towards proving Theorem REF , we first observe how to obtain bounded-error quantum communication complexity lower bounds using Theorem REF and Lemma REF .",
"Lemma 5.5 Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log j} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log k} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be functions such that $\\textnormal {bdisc}(G) = o(1)$ and $\\frac{8en}{\\widetilde{\\textnormal {deg}}(r)} \\le \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)^{1-\\beta }$ for some constant $\\beta \\in (0,1)$ .",
"Let $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .",
"Then, $\\mathsf {Q}^{cc, *} ((r ~\\widetilde{\\circ }~h_G) \\circ \\square ) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)\\right).$ By Lemma REF we have $\\mathsf {Q}^{cc, *} ((r ~\\widetilde{\\circ }~h_G) \\circ \\square ) \\ge \\mathsf {Q}^{cc, *}(r \\circ G)$ .",
"By Theorem REF , $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)\\right)$ .",
"We now prove Theorem REF .",
"[Proof of Theorem REF ] Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $G : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be as in the statement of the theorem.",
"We have $\\mathsf {Q}(r) \\ge \\widetilde{\\textnormal {deg}}(r)/2 = \\Omega (n)$ , where the first inequality follows by Theorem REF and the second equality follows from the assumption that $\\widetilde{\\textnormal {deg}}(r) = \\Omega (n)$ .",
"Moreover, $\\mathsf {Q}(r) \\le n$ since $r$ is a function on $n$ input variables.",
"Hence $\\mathsf {Q}(r) = \\Theta (n)$ .",
"Thus, Corollary REF is applicable, and we have $\\mathsf {Q}(f) = \\Theta (n).$ For the lower bound, $\\widetilde{\\textnormal {deg}}(r) = \\Omega (n)$ by assumption.",
"Thus $\\frac{2en}{\\widetilde{\\textnormal {deg}}(r)} = O(1).$ Also, since by assumption $\\frac{1}{\\textnormal {bdisc}(G)} = n^{\\Omega (1)} = \\omega (1)$ , we have $\\frac{2en}{\\widetilde{\\textnormal {deg}}(r)} \\le \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)^{1-\\beta }$ for every constant $\\beta \\in (0,1)$ .",
"Lemma REF implies $\\mathsf {Q}^{cc, *}(f \\circ \\square ) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right) \\right) = \\Omega (n \\log n).$"
],
[
"Proof of Theorem ",
"The total function $f: \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ that we use to prove Theorem REF is $f = r ~\\widetilde{\\circ }~h_G$ , where $r = \\mathsf {PARITY}_n$ and $G = \\mathsf {IP}_{\\log n}$ .",
"Note that Theorem REF implies $\\mathsf {Q}(f) = O(n)$ .",
"For the quantum communication complexity lower bound in Theorem REF , we are able to show not only $\\mathsf {Q}^{cc, *}(f \\circ \\square ) = \\Omega (n \\log n)$ , but $\\mathsf {UPP}^{cc}(f \\circ \\square ) = \\Omega (n \\log n)$ for $\\square \\in \\lbrace \\mathsf {AND}_2,\\mathsf {XOR}_2\\rbrace $ .",
"The following claim shows that $f$ is transitive.",
"Claim 5.6 Let $n > 0$ be a power of 2.",
"Let $r = \\mathsf {PARITY}_n : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G = \\mathsf {IP}_{\\log n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"The function $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive.",
"We first show that $h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive.",
"We next observe that $s ~\\widetilde{\\circ }~t$ is transitive whenever $s$ is symmetric and $t$ is transitive ($s$ can be assumed to just be transitive rather than symmetric, as noted in Remark REF ).",
"The theorem then follows since $\\mathsf {PARITY}_n$ is symmetric.",
"Towards showing transitivity of $h_G$ , let $\\pi \\in S_{2n}$ , and $(\\sigma _{\\ell }, \\sigma _\\ell ) \\in S_{2n}$ for $\\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ be defined as follows.",
"(Here $\\sigma _\\ell \\in S_n$ ; the first copy acts on the first $n$ coordinates, and the second copy acts on the next $n$ coordinates.)",
"$\\pi (k) = {\\left\\lbrace \\begin{array}{ll}k + n & k \\le n\\\\k - n & k > n.\\end{array}\\right.",
"}$ That is, on every string $(x, y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ , the permutation $\\pi $ maps $(x, y)$ to $(y, x)$ .",
"For every $\\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , the permutation $\\sigma _\\ell \\in S_n$ is defined as $\\sigma _{\\ell }(i) = i \\oplus \\ell ,$ where $i \\oplus \\ell $ denotes the bitwise XOR of the strings $i$ and $\\ell $ .",
"That is, for every input $(x, y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ and every $k \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , the input bit $x_{k}$ is mapped to $x_{k \\oplus \\ell }$ and $y_k$ is mapped to $y_{k\\oplus \\ell }$ .",
"For every $(x,y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ and $i,j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , the permutation $\\sigma _{i \\oplus j}(x,y)$ swaps $x_i$ and $x_j$ , and also swaps $y_i$ and $y_j$ .",
"If for $i, j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , our task was to swap the $i$ 'th index of the first $n$ variables with the $j$ 'th index of the second $n$ variables, then the permutation $\\sigma _{i\\oplus j} \\circ \\pi $ does the job.",
"That is, for every $(x,y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ and $i,j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , the permutation $\\sigma _{i\\oplus j} \\circ \\pi $ maps $x_i$ to $y_j$ .",
"Thus the set of permutations $\\lbrace \\pi , \\lbrace \\sigma _\\ell : \\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}\\rbrace \\rbrace $ acts transitively on $S_{2n}$ .",
"Now we show that for all $x,y \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ and all $\\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , we have $h_G(\\sigma _{\\ell }(x), \\sigma _\\ell (y)) = h_G(x,y)$ .",
"Fix $\\ell \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ .",
"If $x \\in \\pm H(s)$ and $y \\in \\pm H(t)$ are Hadamard codewords, then $x_k = \\langle k, s \\rangle $ and $y_k = \\langle k, t \\rangle $ for all $k \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , and $G(x,y) = \\langle s, t \\rangle $ .",
"Thus, for every $k \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ we have $\\sigma _{\\ell }(x_k) = x_{k \\oplus \\ell } = \\langle k \\oplus \\ell , s \\rangle = \\langle \\ell , s \\rangle \\cdot \\langle k , s \\rangle $ .",
"Hence $\\sigma _{\\ell }(x) \\in \\pm H(s)$ (since $\\langle \\ell , s \\rangle $ does not depend on $k$ , and takes value either 1 or $-1$ ).",
"Similarly, $\\sigma _{\\ell }(y) \\in \\pm H(t)$ .",
"Thus $h_G(\\sigma _{\\ell }(x,y)) = h_G(x,y)$ .",
"If $x$ ($y$ , respectively) is not a Hadamard codeword, then a similar argument shows that for all $\\ell \\in [n]$ , $\\sigma _{\\ell }(x)$ ($\\sigma _{\\ell }(y)$ , respectively) is also not a Hadamard codeword.",
"Using the fact that $\\langle s, t \\rangle = \\langle t, s \\rangle $ for every $s, t \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ , one may verify that $h_G(\\pi (x,y)) = h_G(x,y)$ for all $x,y \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n}$ .",
"Along with the observation that $\\mathsf {PARITY}_n$ is a symmetric function, we have that $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive under the following permutations: $S_n$ acting on the inputs of $\\mathsf {PARITY}_n$ , and The group generated by $\\left\\lbrace \\pi \\right\\rbrace \\cup \\left\\lbrace (\\sigma _\\ell , \\sigma _\\ell ) : \\ell \\in [n] \\right\\rbrace $ acting independently on the inputs of each copy of $h_G$ , where $\\sigma _{\\ell }$ is as in Equation (REF ).",
"We now prove Theorem REF .",
"[Proof of Theorem REF ] Let $n > 0$ be a power of 2.",
"Let $r = \\mathsf {PARITY}_n : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G = \\mathsf {IP}_{\\log n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"Let $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"By Claim REF , $f$ is transitive.",
"By Corollary REF we have $\\mathsf {Q}(f) = \\Theta (n).$ For the communication lower bound we have $\\mathsf {UPP}^{cc}(f \\circ \\square ) & = \\mathsf {UPP}^{cc}((r ~\\widetilde{\\circ }~h_G) \\circ \\square ) \\\\& \\ge \\mathsf {UPP}^{cc}(\\mathsf {PARITY}_n \\circ \\mathsf {IP}_{\\log n}) \\\\& = \\mathsf {UPP}^{cc}(\\mathsf {IP}_{n \\log n}) \\\\& = \\Omega (n \\log n) .$ Remark 5.7 The proof of transitivity of $f$ in Theorem REF can also be used to prove that if $r : \\left\\lbrace -1,1 \\right\\rbrace ^n$ is transitive and $G = \\mathsf {IP}_{\\log n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , then $r ~\\widetilde{\\circ }~h_G$ is transitive as well.",
"By instantiating $r$ to a transitive function with approximate degree $\\Omega (n)$ (e.g.",
"Majority), Theorem REF implies that the BCW simulation is tight w.r.t.",
"the bounded-error communication model for a wide class of transitive functions."
],
[
"Hardness of composing the Chakraborty et al. function with $\\mathsf {AND}_2$",
"Chakraborty et al.",
"[12] exhibited a total function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ for which $\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {XOR}_2) = \\Omega (\\mathsf {Q}(f) \\log n)$ , that is, the log-factor overhead in the BCW simulation is necessary for $f$ when the inner function is $\\mathsf {XOR}_2$ .",
"In this section we show that the log-factor overhead is necessary for $f$ even when the inner function is $\\mathsf {AND}_2$ .",
"Definition A.1 (Addressing function) For an integer $n>0$ that is a power of 2, let the Addressing function, denoted $\\mathsf {ADDR}_{n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , be a two-party function defined as follows.",
"$\\mathsf {ADDR}_n(x_1, \\dots , x_{\\log n}, y_1, \\dots , y_n) = y_{\\textnormal {bin}(i)},$ where $\\textnormal {bin}(i)$ represents the integer in $[n]$ whose binary representation is $i$ .",
"In the corresponding communication problem, Alice holds the inputs $x_1, \\dots , x_{\\log n}$ , and Bob holds the inputs $y_1, \\dots , y_n$ .",
"In [12] a subset of the authors defined the following function $f: \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"Let $X_i, Y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ for all $i \\in [n]$ .",
"$f(X_1, Y_1, \\dots , X_n, Y_n) ={\\left\\lbrace \\begin{array}{ll}\\mathsf {PARITY}(\\mathsf {ADDR}_n(x_1, Y_1), \\dots , \\mathsf {ADDR}_n(x_n, Y_n)) \\textnormal { if } \\forall i \\in [n], X_i \\in \\pm H(x_i)\\\\-1 \\textnormal { otherwise},\\end{array}\\right.",
"}$ That is, if there exist $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ for all $i \\in [n]$ such that $X_i \\in \\pm H(x_i)$ , then $f(X_1, Y_1, \\dots , X_n, Y_n)$ equals the parity of $\\mathsf {ADDR}_n(x_1, Y_1), \\dots , \\mathsf {ADDR}_n(x_n, Y_n)$ , and $f$ equals $-1$ otherwise.",
"Chakraborty et al.",
"proved the following quantum query upper bound on $f$ (set $\\delta = 1/2$ in [12]).",
"Theorem A.2 ([12]) For $f: \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ as defined in Equation (REF ), $\\mathsf {Q}(f) = O(n)$ .",
"Remark A.3 The function considered in [12] is not exactly as in Equation (REF ) because the former one did not consider negations of Hadamard codewords as done in Equation (REF ).",
"However, Theorem REF can still be seen to hold by incorporating a step similar to Step REF in the algorithm given in the proof of [12].",
"Chakraborty et al.",
"showed the BCW simulation theorem is tight for $f$ when the inner function is $\\mathsf {XOR}_2$ , that is $\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {XOR}_2) = \\Omega (n \\log n)$ .",
"They left open the question whether the BCW simulation theorem is tight for $f$ even when the inner function is $\\mathsf {AND}_2$ .",
"We show that this is indeed the case.",
"The quantum query upper bound holds by Theorem REF .",
"For the communication complexity lower bound we use techniques similar to those in the proof of Lemma REF .",
"Lemma A.4 Let $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be as in Equation (REF ), and $CC \\in \\lbrace \\mathsf {Q}^{cc, *}, \\mathsf {UPP}^{cc} \\rbrace $ .",
"Then, $CC(r \\circ \\mathsf {AND}_2) \\ge CC(\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}).$ We require the following lemma.",
"Lemma A.5 Let $n > 0$ be a power of 2, and let $U$ be the uniform distribution on $\\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{n}$ .",
"Then, $\\textnormal {disc}_U(\\mathsf {ADDR}_{n}) \\le \\frac{1}{\\sqrt{n}}.$ We provide a proof for completeness.",
"Consider the uniform distribution $U: \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ defined as $U(x,y) = \\frac{1}{n2^n}$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ and $y \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"Let $A_{n \\times 2^n}$ be the communication matrix of $\\mathsf {ADDR}_n: \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"Observe that for every $i,j \\in [n]$ , $(AA^T)_{ij}= \\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} x_i x_j={\\left\\lbrace \\begin{array}{ll}2^n \\text{ if } i = j \\\\0 \\text{ otherwise}.\\end{array}\\right.",
"}$ Hence $AA^T = 2^n I_n$ , where $I_n$ denotes the $n \\times n$ identity matrix.",
"Thus, $\\Vert A\\Vert = \\sqrt{2^n}$ .",
"For $S \\subseteq [n]$ and $T \\subseteq [2^n]$ , let $1_S: [n] \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ and $1_T: [2^n] \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ be the indicator function of corresponding rows and columns of $A$ , respectively.",
"We now upper bound the discrepancy with respect to the uniform distribution: $\\textnormal {disc}_U(\\mathsf {ADDR}_n)&= \\max _{S \\subseteq [n], T\\subseteq [2^n]} \\left|(1_S)^T \\cdot (U \\circ A) \\cdot 1_T \\right| \\\\&\\le \\sqrt{n} \\cdot \\Vert U \\circ A \\Vert \\cdot \\sqrt{2^n} \\\\&= \\frac{1}{\\sqrt{n2^n}} \\Vert A \\Vert \\\\&= \\frac{1}{\\sqrt{n}}.$ We now prove Lemma REF using Lemma REF .",
"[Proof of Lemma REF ] Consider a protocol $\\Pi $ of cost $\\ell $ that solves $f \\circ \\mathsf {AND}_2$ in the $CC$ -model.",
"We exhibit below a protocol of cost $\\ell $ that solves $\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}$ in the same model.",
"Recall from Definitions REF and REF that in the communication problem $\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}$ , Alice is given the address bits and Bob is given the target bits for each copy of the inner gadget $\\mathsf {ADDR}_{n}$ .",
"Suppose Alice is given input $x = (x_1, \\dots , x_{n}) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n \\log n}$ and Bob is given input $y = (y_1, \\dots , y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n^2}$ , where $x_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}, y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n}$ for each $i \\in [n]$ ."
],
[
"Preprocessing step:",
"Alice constructs the following $(2n^2)$ -bit string $X = (H(x_1), (-1)^{n}), \\dots , (H(x_n), (-1)^n)) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2},$ and Bob constructs the $(2n^2)$ -bit string $Y = (((-1)^n, y_1), \\dots , ((-1)^n, y_n)) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2}.$"
],
[
"Protocol:",
"Alice and Bob run the protocol $\\Pi $ with input $(X, Y)$ , and output $\\Pi (X, Y)$ ."
],
[
"Cost:",
"The preprocessing of the inputs to obtain $X$ from $x$ and $Y$ from $y$ takes no communication.",
"Hence the total amount of communication is at most the cost of $\\Pi $ ."
],
[
"Correctness:",
"We now argue that $\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}(x, y) = (f \\circ \\mathsf {AND}_2) (X, Y)$ , which would conclude the proof.",
"We have from Equation (REF ) and Equation (REF ) that $(f \\circ \\mathsf {AND}_2)(X, Y) & = f((H(x_1), y_1), \\dots , (H(x_n), y_n))\\\\& = \\mathsf {PARITY}_n(\\mathsf {ADDR}_n(x_1, y_1), \\dots , \\mathsf {ADDR}_n(x_n, y_n)) \\\\& = \\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}(x, y).$ Thus, $CC(f \\circ \\mathsf {AND}_2) \\ge CC(\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_{n}),$ which proves the lemma.",
"Theorem A.6 Let $n > 0$ be a sufficiently large power of 2.",
"Let $f : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be as in Equation (REF ).",
"Then, $\\mathsf {Q}(f) & = O(n),\\\\\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {AND}_2) & = \\Omega (n \\log n).$ The upper bound follows from Theorem REF .",
"For the lower bound, we first note that for sufficiently large $n$ and every constant $\\beta \\in (0, 1)$ , we have $\\frac{2en}{\\widetilde{\\textnormal {deg}}(\\mathsf {PARITY}_n)} = 2e < (\\sqrt{n})^{1 - \\beta } = \\left(\\frac{1}{\\textnormal {disc}_{U}(\\mathsf {ADDR}_n)}\\right)^{1-\\beta },$ where $U$ denotes the uniform distribution over $\\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"The first equality above holds since $\\widetilde{\\textnormal {deg}}(\\mathsf {PARITY}_n) = n$ , and the last equality holds by Lemma REF .",
"It is easy to verify that $U$ is a balanced distribution w.r.t.",
"$\\mathsf {ADDR}_n$ .",
"Hence, $\\mathsf {Q}^{cc, *}(f \\circ \\mathsf {AND}_2) & \\ge \\mathsf {Q}^{cc, *}(\\mathsf {PARITY}_n \\circ \\mathsf {ADDR}_n) \\\\& = \\Omega \\left(\\widetilde{\\textnormal {deg}}(\\mathsf {PARITY}_n) \\log \\left(\\frac{1}{\\textnormal {disc}_U(\\mathsf {ADDR}_{n})}\\right)\\right) \\\\& = \\Omega \\left(n \\log \\left(\\frac{1}{\\textnormal {disc}_U(\\mathsf {ADDR}_{n})}\\right)\\right) \\\\& = \\Omega (n \\log n).",
"$"
],
[
"A separation between log-approximate-spectral norm and approximate degree for a transitive function",
"In this section, we exhibit a transitive function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ for which the logarithm of its approximate spectral norm ($\\log (\\Vert \\widehat{f}\\Vert _{1, 1/3})$ ) is at least $\\Omega (\\widetilde{\\textnormal {deg}}(f) \\log n)$ .",
"We refer the reader to [28] for preliminaries on Fourier analysis of Boolean functions.",
"Definition B.1 (Spectral norm) Let $p : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be a function, and let $p = \\sum _{S \\subseteq [n]}\\widehat{p}(S)\\chi _S$ denote its Fourier expansion.",
"The spectral norm of $p$ is define by $\\Vert \\widehat{p}\\Vert _1 := \\sum _{S \\subseteq [n]} \\left|\\widehat{p}(S) \\right|.$ Definition B.2 (Approximate Spectral Norm) The approximate spectral norm of a function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\lbrace -1,1\\rbrace $ , denoted by $\\Vert \\widehat{f}\\Vert _{1,\\varepsilon }$ is defined to be the minimum spectral norm of a real polynomial $p : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ that satisfies $\\left|p(x) - f(x) \\right| \\le \\varepsilon $ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"That is, $\\Vert \\widehat{f}\\Vert _{1,\\varepsilon } := \\min \\left\\lbrace \\Vert \\widehat{p}\\Vert _1 : \\left|p(x) - f(x) \\right| \\le \\varepsilon ~\\text{for all~} x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n \\right\\rbrace .$ In [12] the following question raised in [2] was shown to have a negative answer, using the function $f$ of Equation (REF ) to witness this.",
"Question B.3 ([2]) Is it true that for all Boolean functions $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $\\log (\\Vert \\widehat{f}\\Vert _{1,\\varepsilon }) = O(\\widetilde{\\textnormal {deg}}(f))?$ In this section we show that Question REF is false even for the special class of transitive Boolean functions.",
"This is in stark contrast with the class of symmetric functions, for which Question REF is true (see [12]).",
"Claim B.4 There exists a transitive function $f: \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ such that $\\log \\left(\\Vert \\widehat{f}\\Vert _{1, 1/3} \\right) = \\Omega (\\widetilde{\\textnormal {deg}}(f) \\log n)$ .",
"We first state some required preliminaries.",
"Definition B.5 (Monomial projection) We call a function $g:\\left\\lbrace -1,1 \\right\\rbrace ^m \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ a monomial projection of a function $f:\\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ if $g$ can be expressed as $g(x_1,\\dots , x_m) = f(M_1,\\dots , M_n)$ , where each $M_i$ is a monomial in the variables $x_1,\\dots , x_m$ .",
"It is known that the approximate spectral norm of a function can only decrease upon monomial projections (see, for example, [14]).",
"Observation B.6 ([14]) For $f : \\left\\lbrace -1,1 \\right\\rbrace ^{n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $g : \\left\\lbrace -1,1 \\right\\rbrace ^{m} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ such that $g$ is a monomial projection of $f$ , $\\Vert \\widehat{g}\\Vert _{1, 1/3} \\le \\Vert \\widehat{f}\\Vert _{1, 1/3}.$ Fact B.7 (Fourier coefficients of $\\mathsf {IP}_{n}$ ) Let $\\mathsf {IP}_n: \\left\\lbrace -1,1 \\right\\rbrace ^{2n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be as in Definition REF .",
"Then for all $S \\subseteq [2n]$ we have, $\\left|\\widehat{\\mathsf {IP}_n}(S) \\right| = \\frac{1}{2^n}.$ Fact B.8 (Plancherel's Theorem) Let $f,g: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be functions.",
"Then, $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n}f(x)g(x) = \\sum _{S \\subseteq [n]} \\widehat{f}(S) \\widehat{g}(S).$ [Proof of Claim REF ] Let $n > 0$ be a power of 2.",
"Let $r = \\mathsf {PARITY}_n : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G = \\mathsf {IP}_{\\log n} : \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"From Claim REF , the function $f = r ~\\widetilde{\\circ }~h_G : \\left\\lbrace -1,1 \\right\\rbrace ^{2n^2} \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ is transitive.",
"By Corollary REF we have $\\mathsf {Q}(f) = \\Theta (n).",
"$ By Theorem REF we have $\\mathsf {Q}(f) \\ge \\widetilde{\\textnormal {deg}}(f)/2$ and together with Equation (REF ), this implies $\\widetilde{\\textnormal {deg}}(f) = O(n)$ .",
"Thus to complete the proof of the claim, it suffices to show $\\log \\left(\\Vert \\widehat{f}\\Vert _{1, 1/3} \\right) = \\Omega (n \\log n)$ .",
"We first note that $\\mathsf {IP}_{n \\log n}$ is a monomial projection of $f$ .",
"Consider the function $f$ acting on the input variables $x^{(1)}, \\dots , x^{(n)}, y^{(1)}, \\dots , y^{(n)}$ , where $x^{(i)}, y^{(i)} \\in \\left\\lbrace -1,1 \\right\\rbrace ^{n}$ for all $i \\in [n]$ .",
"For $i \\in [n]$ , set $x^{(i)}_{1^{\\log n}} = y^{(i)}_{1^{\\log n}} = 1$ .",
"For $i \\in [n]$ and string $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n} \\setminus \\left\\lbrace 1^{\\log n} \\right\\rbrace $ , set $x^{(i)}_s = \\prod _{j: s_j = -1}x^{(i)}_j $ and $y^{(i)}_s = \\prod _{j: s_j = -1}y^{(i)}_j$ .",
"That is, in each block of inputs $x^{(i)}$ and $y^{(i)}$ , the coordinate corresponding to $1^{\\log n}$ equals 1, the coordinates corresponding to $j \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ with $|j| = 1$ are free variables, and all other variables are replaced by monomials in these variables.",
"Under the above monomial projection there are $2n \\log n$ free variables, namely $\\left\\lbrace x^{(i)}_{s}, y^{(i)}_{s} : i \\in [n], s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}, |s| = 1\\right\\rbrace $ .",
"Also note that under this projection and every setting of the free variables, the blocks $x^{(i)}$ and $y^{(i)}$ , for $i \\in [n]$ , are always Hadamard codewords.",
"Let $f^{\\prime }$ be the monomial projection (see Definition REF ) of $f$ under the projection defined in this paragraph.",
"For the purpose of the next equality we abbreviate strings $s \\in \\left\\lbrace -1,1 \\right\\rbrace ^{\\log n}$ of Hamming weight 1 by the set $\\left\\lbrace i \\right\\rbrace $ , where $i \\in [\\log n]$ is such that $s_i = -1$ .",
"On the free variables, the projected function $f^{\\prime }$ equals $\\mathsf {PARITY}_n \\left(\\langle (x^{(1)}_{\\lbrace 1\\rbrace }, \\dots , x^{(1)}_{\\lbrace \\log n\\rbrace }), (y^{(1)}_{\\lbrace 1\\rbrace }, \\dots , y^{(1)}_{\\lbrace \\log n\\rbrace })\\rangle , \\dots , \\langle (x^{(n)}_{\\lbrace 1\\rbrace }, \\dots , x^{(n)}_{\\lbrace \\log n\\rbrace }), (y^{(n)}_{\\lbrace 1\\rbrace }, \\dots , y^{(n)}_{\\lbrace \\log n\\rbrace })\\rangle \\right).$ Thus, $f^{\\prime } = \\mathsf {IP}_{n\\log n}.$ It follows from earlier works [8], [9] that every polynomial that approximates $\\mathsf {IP}_{n\\log n}$ to error $1/3$ must have spectral norm $2^{\\Omega (n \\log n)}$ .",
"We include a short proof below for completeness.",
"Let $P$ be a polynomial that approximates $\\mathsf {IP}_{n\\log n}$ to error $1/3$ .",
"Since we have $P(x, y)\\mathsf {IP}_{n\\log n}(x, y) \\ge 2/3$ for all $(x, y) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n\\log n}$ , $2/3 & \\le \\mathbb {E}_{x, y \\in \\left\\lbrace -1,1 \\right\\rbrace ^{2n\\log n}} [P(x, y)\\mathsf {IP}_{n\\log n}(x, y)]\\\\& = \\sum _{S \\subseteq [2n\\log n]}\\widehat{P}(S)\\widehat{\\mathsf {IP}_{n\\log n}}(S) \\\\& \\le \\sum _{S \\subseteq [2n\\log n]}\\frac{|\\widehat{P}(S)|}{2^{n \\log n}} \\\\& = \\frac{\\Vert \\widehat{P}\\Vert _1}{2^{n \\log n}}\\\\& \\Rightarrow \\log (\\Vert \\widehat{P}\\Vert _1) = \\Omega (n \\log n)\\\\& \\Rightarrow \\log (\\Vert \\widehat{\\mathsf {IP}_{n \\log n}}\\Vert _{1, 1/3}) = \\Omega (n \\log n).$ This yields the desired contradiction by Observation REF ."
],
[
"Quantum communication lower bound via the generalized discrepancy method",
"In this section we prove Theorem REF , which gives a lower bound on the quantum communication complexity of a composed function in terms of the approximate degree of the outer function and the discrepancy of the inner function.",
"This result is implicit in [25].",
"Their result is stated in the more general setting of matrix norms, and the $\\log $ factor we require on the right-hand side of Theorem REF is not included in their statement.",
"Theorem REF follows implicitly from the proof of [25] along with the fact that $\\gamma _2^*$ -norm characterizes discrepancy [23].",
"For completeness and clarity, we prove Theorem REF below from first principles.",
"Definition C.1 For functions $f,g : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ and a probability distribution $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ the correlation between $f$ and $g$ with respect to $\\mu $ is defined to be $\\textnormal {corr}_{\\mu }(f, g) = \\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} f(x)g(x) \\mu (x).$ For a Boolean function $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , its approximate degree can be captured by a certain linear program.",
"Writing out the dual of this program and analyzing its optimum yields the following theorem (see, for example, [10]).",
"Theorem C.2 (Dual witness for $\\varepsilon $ -approximate degree) For every $\\varepsilon \\ge 0$ and $f : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ , $\\widetilde{\\textnormal {deg}}_{\\varepsilon }(f) \\ge d$ if and only if there exists a polynomial $\\psi :\\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ such that $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} f(x) \\psi (x) > \\varepsilon $ , $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} |\\psi (x)| = 1$ and $\\widehat{\\psi }(S) = 0$ for all $|S| < d$ .",
"We require the following fact, which follows immediately from the fact that $\\widehat{f}(S)$ is a uniform average of different signed values of $f(x)$ .",
"Fact C.3 (Folklore) For every function $f: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ , $2^n \\max _{S \\subseteq [n]} \\left|\\widehat{f}(S) \\right| \\le \\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\left|f(x) \\right|.$ The following theorem shows that if a two-party function $F$ correlates well with a two-party function $G$ under some distribution $\\lambda $ , and the discrepancy of $G$ under $\\lambda $ is small, then the bounded-error quantum communication complexity of $F$ must be large.",
"This is referred to as the generalized discrepancy method, and was first proposed by Klauck [21].",
"The following version can be found in [13], for example, stated as a lower bound on randomized communication complexity.",
"However, the generalized discrepancy method is also known to give lower bounds on bounded-error quantum communication complexity, even in the model where the parties can share an arbitrary prior entangled state for free.",
"Theorem C.4 (Generalized Discrepancy Bound) Consider functions $E, F : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ .",
"If there exists a distribution $\\lambda : \\left\\lbrace -1,1 \\right\\rbrace ^{m} \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ such that $\\textnormal {corr}_{\\lambda }(E, F) \\ge \\delta $ , then $\\mathsf {Q}^{cc, *}_{\\frac{1}{2} - \\varepsilon }(E)=\\Omega \\left( \\log \\left(\\frac{\\delta + 2\\varepsilon - 1}{\\textnormal {disc}_{\\lambda }(F)} \\right)\\right).$ Definition C.5 For a distribution $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ and integer $k>0$ , define the distribution $\\mu ^{\\otimes k} : \\left\\lbrace -1,1 \\right\\rbrace ^{mk} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{nk} \\rightarrow \\mathbb {R}$ by $\\mu ^{\\otimes k}((X_1, Y_1) \\dots , (X_k, Y_k)) = \\prod _{i \\in [k]}\\mu (X_i, Y_i),$ where $X_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^{m}$ and $Y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ for all $i \\in [k]$ .",
"We require the following XOR lemma for discrepancy due to Lee, Shraibman and Špalek [24].",
"Theorem C.6 ([24]) Let $P : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be a two-party function and $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be a distribution.",
"For every integer $k > 0$ , $\\textnormal {disc}_{\\mu ^{\\otimes k}}(\\mathsf {PARITY}_k \\circ P) \\le (8 \\textnormal {disc}_{\\mu }(P))^{k}.$ We recall Theorem REF below.",
"Theorem C.7 (Restatement of Theorem REF ) Let $r : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ and $G : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ be functions.",
"Let $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^j \\times \\left\\lbrace -1,1 \\right\\rbrace ^k \\rightarrow \\mathbb {R}$ be a balanced distribution with respect to $G$ and $\\textnormal {disc}_{\\mu }(G) = o(1)$ .",
"If $\\frac{8en}{\\widetilde{\\textnormal {deg}}(r)} \\le \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)}\\right)^{1-\\beta }$ for some constant $\\beta \\in (0,1)$ , then $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)}\\right)\\right).$ In particular, $\\mathsf {Q}^{cc, *}(r \\circ G) = \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left(\\frac{1}{\\textnormal {bdisc}(G)}\\right)\\right).$ For simplicity we assume $j = k = m$ .",
"The general proof follows along similar lines.",
"Let $\\widetilde{\\textnormal {deg}}(r) = d$ , and let $\\psi : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be a dual witness for this as given by Theorem REF .",
"Let $\\nu : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ be defined as $\\nu (x) = |\\psi (x)|$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ and also define $h: \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\left\\lbrace -1,1 \\right\\rbrace $ as $h(x) = \\text{sign}(\\psi (x))$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"First note that $\\nu $ is a distribution since $\\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\nu (x) = \\sum _{x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} |\\psi (x)| = 1$ by Theorem REF .",
"From Theorem REF we also have $\\textnormal {corr}_{\\nu }(r, h) & > 1/3 \\\\\\widehat{h\\nu }(S) & = 0 \\quad \\text{for all~} |S| < d,$ where $h\\nu (x) := h(x)\\nu (x) = \\psi (x)$ for all $x \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"We will construct a probability distribution $\\lambda : \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\rightarrow \\mathbb {R}$ using $\\nu : \\left\\lbrace -1,1 \\right\\rbrace ^n \\rightarrow \\mathbb {R}$ and $\\mu : \\left\\lbrace -1,1 \\right\\rbrace ^m \\times \\left\\lbrace -1,1 \\right\\rbrace ^m \\rightarrow \\mathbb {R}$ such that $r \\circ G$ and $h \\circ G$ have large correlation under $\\lambda $ .",
"We will also show that $\\textnormal {disc}_{\\lambda }(h \\circ G)$ is small.",
"The proof of the theorem would then follow from Theorem REF .",
"Let $X,Y \\in \\left\\lbrace -1,1 \\right\\rbrace ^{mn}$ be such that $X = (X_1, \\dots , X_n)$ , $Y = (Y_1, \\dots , Y_n)$ and $X_i, Y_i \\in \\left\\lbrace -1,1 \\right\\rbrace ^m$ for all $i \\in [n]$ .",
"Also, let $G(X,Y) = (G(X_1, Y_1), \\dots , G(X_n, Y_n)) \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ .",
"Define $\\lambda : \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\rightarrow \\mathbb {R}$ as follows.",
"$\\lambda (X,Y)&= 2^n \\cdot \\nu (G(X, Y)) \\cdot \\prod _{i \\in [n]} \\mu (X_i,Y_i).$ Observe that for any $z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n$ , $\\sum _{X,Y : G(X, Y) = z} \\lambda (X,Y)&= \\sum _{X,Y: G(X,Y) = z} 2^n \\cdot \\nu (G(X, Y)) \\cdot \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\nonumber \\\\&= 2^n \\nu (z) \\prod _{i \\in [n]}\\left(\\sum _{X_i, Y_i : G(X_i, Y_i) = z_i} \\mu (X_i, Y_i)\\right)\\nonumber \\\\&= \\nu (z), $ where the last equality follows since $\\mu $ is balanced w.r.t.",
"$G$ by assumption.",
"Thus $\\sum _{X,Y} \\lambda (X,Y) = \\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\nu (z) = 1.$ We next observe that $\\textnormal {corr}_{\\lambda }(r \\circ G, h \\circ G)$ is large.",
"$\\textnormal {corr}_{\\lambda }(r \\circ G, h \\circ G)&= \\sum _{X,Y} {r \\circ G}(X,Y) \\cdot {h \\circ G}(X,Y) \\cdot \\lambda (X,Y) \\nonumber \\\\&= \\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\sum _{X,Y: G(X,Y) = z} r(z)h(z) \\cdot \\lambda (X,Y) \\nonumber \\\\&= \\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} r(z)h(z) \\nu (z) \\nonumber \\\\&= \\textnormal {corr}_{\\nu }(r, h) > 1/3.",
"$ where the last equality follows from Equation (REF ).",
"We now upper bound the discrepancy of $h \\circ G$ with respect to a rectangle $R$ , under the distribution $\\lambda $ .",
"Let $R \\subseteq \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\times \\left\\lbrace -1,1 \\right\\rbrace ^{mn}$ be any rectangle of the form $R(X, Y) = A(X)B(Y)$ for $A : \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ and $B : \\left\\lbrace -1,1 \\right\\rbrace ^{mn} \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ .",
"$\\textnormal {\\textnormal {disc}}_{\\lambda }(h \\circ G, R)&= \\left|\\sum _{X,Y} {h \\circ G}(X,Y) \\cdot R(X,Y) \\cdot \\lambda (X,Y) \\right| \\\\&= \\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\sum _{X,Y: G(X,Y) = z} {h \\circ G}(X,Y) \\cdot R(X,Y) \\cdot \\lambda (X,Y) \\right|\\\\&= 2^n \\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} h(z) \\nu (z) \\sum _{X,Y: G(X,Y) = z} R(X,Y) \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\right| \\\\&= 2^n \\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\left(\\sum _{S \\subseteq [n]} \\widehat{h\\nu }(S)\\chi _S(z)\\right) \\sum _{X,Y: G(X,Y) = z} R(X,Y) \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\right|\\\\&= 2^n\\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\sum _{S \\subseteq [n]} \\widehat{h\\nu }(S) \\sum _{X,Y: G(X,Y) = z} R(X,Y) \\cdot \\chi _S(z) \\cdot \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\right| \\\\&= 2^n\\left|\\sum _{z \\in \\left\\lbrace -1,1 \\right\\rbrace ^n} \\sum _{S \\subseteq [n]} \\widehat{h\\nu }(S) \\sum _{X,Y: G(X,Y) = z} R(X,Y) \\cdot \\prod _{i \\in S}G(X_i,Y_i) \\cdot \\prod _{i \\in [n]} \\mu (X_i,Y_i) \\right| \\\\& \\le 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\cdot \\left|\\sum _{X,Y} R(X,Y) \\cdot \\prod _{i \\in S}G(X_i,Y_i)\\mu (X_i, Y_i) \\cdot \\prod _{j \\notin S}\\mu (X_j,Y_j) \\right| \\\\& \\le 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\cdot \\left|\\sum _{X_j,Y_j : j \\notin S} \\prod _{j \\notin S}\\mu (X_j,Y_j) \\left(\\sum _{X_i,Y_i : i \\in S} A(X) B(Y) \\cdot \\prod _{i \\in S}G\\mu (X_i, Y_i)\\right) \\right|.$ For any $X = (X_1, \\dots , X_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{mn}$ (respectively, $Y = (Y_1, \\dots , Y_n) \\in \\left\\lbrace -1,1 \\right\\rbrace ^{mn}$ ) and set $S \\subseteq [n]$ define the string $X_S \\in \\left\\lbrace -1,1 \\right\\rbrace ^{m|S|}$ (respectively, $Y_S \\in \\left\\lbrace -1,1 \\right\\rbrace ^{m|S|}$ ) by $X_S = (\\dots , X_i, \\dots )$ (respectively, $Y_S = (\\dots , Y_i, \\dots )$ ), where $i$ ranges over all elements of $S$ .",
"For any $S$ and fixed $\\lbrace X_j : j \\notin S\\rbrace $ (respectively, $\\lbrace Y_j : j \\notin S\\rbrace $ ) note that $A(X) = A^{\\prime }(X_S)$ (respectively, $B(X) = B^{\\prime }(X_S)$ ) for some $A^{\\prime }: \\left\\lbrace -1,1 \\right\\rbrace ^{m|S|} \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ (respectively, $B^{\\prime }: \\left\\lbrace -1,1 \\right\\rbrace ^{m|S|} \\rightarrow \\left\\lbrace 0,1 \\right\\rbrace $ ) which is a function of the fixed values $\\lbrace X_j : j \\notin S\\rbrace $ (respectively, $\\lbrace Y_j : j \\notin S\\rbrace $ ).",
"Let $R^{\\prime }(X_S, Y_S) = A^{\\prime }(X_S) B^{\\prime }(Y_S)$ .",
"Continuing from the above we obtain, $\\textnormal {\\textnormal {disc}}_{\\lambda }(h \\circ G, R)&\\le 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\left|\\sum _{X_j,Y_j : j \\notin S} \\prod _{j \\notin S}\\mu (X_j,Y_j) \\left(\\sum _{X_i,Y_i : i \\in S} R^{\\prime }(X_S, Y_S) \\prod _{i \\in S}G\\mu (X_i, Y_i)\\right) \\right|\\\\& \\le 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\left| \\sum _{X_j,Y_j : j \\notin S} \\prod _{j \\notin S}\\mu _j(X_j,Y_j) \\left(\\textnormal {disc}_{\\mu ^{\\otimes |S|}}(\\mathsf {XOR}_{|S|} \\circ G)\\right) \\right| \\\\&= 2^n \\sum _{S \\subseteq [n], |S| \\ge d}\\left|\\widehat{h\\nu }(S) \\right| \\cdot \\textnormal {disc}_{\\mu ^{\\otimes |S|}}(\\mathsf {XOR}_{|S|} \\circ G)\\\\&\\le \\sum _{S \\subseteq [n], |S| \\ge d} \\left(8 \\textnormal {disc}_{\\mu }(G)\\right)^{|S|}\\\\&= \\sum _{k = d}^n \\binom{n}{k} \\left(8 \\textnormal {disc}_{\\mu }(G)\\right)^{k} \\le \\sum _{k = d}^n \\left( \\frac{8en}{k} \\textnormal {disc}_{\\mu }(G)\\right)^{k}\\\\&\\le \\sum _{k = d}^n \\left(\\textnormal {disc}_{\\mu }(G)^{\\beta }\\right)^{k}\\\\&\\le \\frac{\\textnormal {disc}_{\\mu }(G)^{d \\cdot \\beta }}{1 - \\textnormal {disc}_{\\mu }(G)^{\\beta }}$ Since $\\textnormal {disc}_{\\mu }(G) = o(1)$ by assumption ($\\textnormal {disc}_{\\mu }(G) = 1 - \\Omega (1)$ suffices, but we assume $\\textnormal {disc}_{\\mu }(G) = o(1)$ in the statement for readability) and $\\beta \\in (0, 1)$ is a constant, we have $1 - \\textnormal {disc}_\\mu (G)^\\beta = \\Omega (1)$ , and hence $\\textnormal {disc}_{\\lambda }(h \\circ G) = O(\\textnormal {disc}_{\\mu }(G)^{d \\beta }).$ Hence, $\\mathsf {Q}^{cc}_{1/2 - 2/5}(r \\circ G)&\\ge \\log \\left(\\frac{1/3 + 4/5 - 1}{\\textnormal {disc}_{\\lambda }(h \\circ G)}\\right) \\\\&= \\Omega \\left( \\log \\left(\\frac{1}{\\textnormal {disc}_{\\mu }(G)^{d\\beta }}\\right) \\right) \\\\&= \\Omega \\left(\\widetilde{\\textnormal {deg}}(r) \\log \\left( \\frac{1}{\\textnormal {disc}_{\\mu }(g)}\\right)\\right) .$ The theorem follows since $\\mathsf {Q}^{cc}(F) = \\Theta (\\mathsf {Q}^{cc}_{\\varepsilon }(F))$ for all constants $\\varepsilon \\in (0, 1/2)$ ."
]
] | 2012.05233 |
[
[
"Jets and elliptic flow correlations at low and high transverse momenta\n in ultrarelativistic A+A collisions"
],
[
"Abstract Data from the Large Hadron Collider on elliptic flow correlations at low and high $p_T$ from Pb+Pb collisions at $\\sqrt{s_{NN}} = 5.02$~TeV are analyzed and interpreted in the framework of the HYDJET++ model.",
"This model allows us to describe simultaneously the region of both low and high transverse momenta and, therefore, to reproduce the experimentally observed nontrivial centrality dependence of elliptic flow correlations.",
"The origin of the correlations between low and high-$p_T$ flow components in peripheral lead-lead collisions is traced to correlations of particles in jets."
],
[
"Introduction",
"A number of exquisite and intriguing phenomena, which have never been systematically studied at the accelerators of previous generation, has been observed after the start of Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) heavy-ion program.",
"Azimuthal anisotropy of multi-particle production is among them as a powerful probe of collective properties of a new state of matter, quark-gluon plasma (QGP), see, e.g., [1].",
"It is commonly described by the Fourier decomposition of the invariant cross section in a form [2], [3]: $\\displaystyle E\\frac{d^3N}{dp^3} &=&\\frac{d^2N}{2\\pi p_{T}dp_{T}d\\eta }\\nonumber \\\\&\\times & \\left\\lbrace 1+2\\sum \\limits _{n = 1}^\\infty v_{n}(p_{T},\\eta )\\cos { \\left[ n(\\varphi -\\Psi _{n}^{PP}) \\right] }\\right\\rbrace ,$ where $p_{T}$ is the transverse momentum, $\\eta $ is the pseudorapidity, $\\varphi $ is the azimuthal angle with respect to the participant plane $\\Psi _{n}^{PP}$ of $n$ -th order, and $v_{n}$ are the Fourier coefficients: $v_n = \\langle \\langle \\cos { \\left[ n(\\varphi -\\Psi _{n}^{PP}) \\right]}\\rangle \\rangle .$ The averaging in the last equation is performed over all particles in a single event and over all events.",
"The second harmonic, $v_2$ , typically referred to as elliptic flow, is the most thoroughly investigated one, for review see [4] and references therein.",
"The reason is obvious: it directly relates the anisotropic shape of the overlap region of the colliding nuclei to the corresponding anisotropy of the outgoing momentum distribution.",
"At relatively low transverse momenta, $p_{T} < 3\\div 4$ ${\\rm GeV}/c$ , the azimuthal anisotropy results from a pressure-driven anisotropic expansion of the created matter, with more particles emitted in the direction of the largest pressure gradients [5].",
"At higher transverse momenta, this anisotropy is understood to arise from the path-length dependent energy loss of partonic jets as they traverse the matter, with more jet particles emitted in the direction of shortest path-length [6].",
"The correlations between soft and hard contributions to anisotropic flow has attracted a lot of attention, see, e.g., [7], [8], [9] and references therein.",
"Recently, the interesting correlations between $v_2$ values at high and low transverse momenta for different centralities has been reported by the CMS Collaboration [10].",
"These correlations can also be seen in the ATLAS data [11].",
"In the present paper we analyze and interpret this intriguing experimental observation within the hydjet++ model, which allows us to perform such analysis due to its remarkable feature, namely the presence of soft and hard physics simultaneously.",
"The paper is organized as follows.",
"Basic features of the model are sketched in Sec. .",
"Here the origin of elliptic flow in the model, and interplay between the soft and hard physics are discussed.",
"Section presents the results of model calculations of hadronic elliptic flow in Pb+Pb collisions at $\\sqrt{s_{\\mbox{\\tiny {\\it {NN}}}}}= 5.02$ TeV.",
"The calculations are in fair agreement with the experimental data, and the role of jets is clarified.",
"Finally, conclusions are drawn in Sec.",
"."
],
[
"The model hydjet++ [12] is the popular and known event generator, which describes successfully the large number of physical observables measured in heavy-ion collisions during the RHIC and LHC operation.",
"Among them are centrality and pseudorapidity dependence of inclusive charged particle multiplicity, transverse momentum spectra and $\\pi ^\\pm \\pi ^\\pm $ correlation radii in central Pb+Pb collisions [13], momentum and centrality dependence of elliptic and higher-order harmonic coefficients [14], [15], [16], [17], [18], flow fluctuations [19], angular dihadron correlations [20], forward-backward multiplicity correlations [21], jet quenching effects [22], [23] and heavy meson production [24], [25], [26].",
"Details of the model can be found in the hydjet++ manual [12].",
"The event generator includes two independent components: the soft, hydro-type state and the hard state resulting from in-medium multi-parton fragmentation.",
"The soft component is the thermal hadronic state generated on the chemical and thermal freeze-out hypersurfaces prescribed by the parametrization of relativistic hydrodynamics with preset freeze-out conditions.",
"It represents the adapted version of the event generator fastmc [27], [28].",
"Particle multiplicities are calculated using the effective thermal volume approach and Poisson multiplicity distribution around its mean value, which is supposed to be proportional to the number of participating nucleons for a given impact parameter in a $A\\mbox{+}A$ collision.",
"To simulate the elliptic flow effect, the hydro-inspired parametrization for the momentum and spatial anisotropy of soft hadron emission source is implemented [12], [29].",
"Note that there are two parameters which govern the strength and the direction of the elliptic flow in original hydjet++ version [12].",
"The first one is the spatial anisotropy $\\epsilon (b)$ .",
"It is responsible for the elliptic modulation of the final freeze-out hyper-surface at a given impact parameter $b$ .",
"The second one is the momentum anisotropy $\\delta (b)$ , dealing with the modulation of the flow velocity profile.",
"One can treat these two parameters as independent ones and, therefore, adjust it separately for each centrality by comparing to data.",
"Although providing better agreement with the data, this procedure leads to significant increase of the parameters to be tuned.",
"Therefore, for the sake of simplicity we opted for another scenario, which is implemented in basic version of the model.",
"According to it, both parameters are correlated through the dependence of the elliptic flow coefficient $v_2$ on both $\\epsilon (b)$ and $\\delta (b)$ , obtained in the hydrodynamic approach [29]: $v_2(\\epsilon , \\delta ) \\propto \\frac{2(\\delta -\\epsilon )}{(1-\\delta ^2)(1-\\epsilon ^{2})}~.$ Because $v_2(b)$ is proportional to the initial ellipticity $\\epsilon _0 (b)=b/2R_A$ , where $R_A$ is the radius of colliding nucleus, the relation between $\\epsilon (b)$ and $\\delta (b)$ reads [12]: $\\delta = \\frac{\\sqrt{1+4B(\\epsilon +B)}-1}{2B}~, \\quad B=C(1-\\epsilon ^2)\\epsilon ~, \\quad \\epsilon =k \\epsilon _0~.$ Two new parameters $C$ and $k$ , entering the last expression, are independent on centrality and, therefore, should be obtained from the fit to the experimental data.",
"In hard sector the model propagates the hard partons through the expanding quark-gluon plasma and takes into account both collisional loss and gluon radiation due to parton rescattering.",
"It is based on the pyquen partonic energy loss model [30].",
"The number of jets is generated according to the binomial distribution.",
"Their mean number in an $A\\mbox{+}A$ event is calculated as a product of the number of binary nucleon-nucleon (NN) sub-collisions at a given impact parameter and the integral cross section of the hard process in NN collisions with the minimum transverse momentum transfer $p_T^{\\rm min}$ .",
"The latter is the input parameter of the model.",
"In the hydjet++ framework, partons produced in (semi)hard processes with the momentum transfer lower than $p_T^{\\rm min}$ are considered as being “thermalized”, and their hadronization products are automatically included in the soft component of the event.",
"Recall that there are many competing event generators successfully describing the soft [31], [32], [33], [34], [35], [36], [37], [38] and hard [39], [40], [41] momentum components of particle production in heavy-ion collisions separately.",
"The hydjet++ is among the few ones [42], [43] which allows us to study the soft and hard physics simultaneously."
],
[
"Results",
"To measure azimuthal correlations and to extract the Fourier coefficients the CMS Collaboration employs the cumulant and the scalar product (SP) methods.",
"The 2- and 4-particle correlations are defined as: $\\langle \\langle 2\\rangle \\rangle = \\langle \\langle e^{in(\\varphi _1-\\varphi _2)}\\rangle \\rangle , \\quad \\quad \\langle \\langle 4\\rangle \\rangle = \\langle \\langle e^{in(\\varphi _1+\\varphi _2-\\varphi _3-\\varphi _4)}\\rangle \\rangle .$ Here the double averaging is performed over all particle combinations and over all events.",
"The multi-particle cumulant method is applied to measure $v_2$ from 4-particle correlations.",
"The 2-nd order and 4-th order cumulants, $c_n\\lbrace 2\\rbrace $ and $c_n\\lbrace 4\\rbrace $ , respectively, are given as [44] $c_n\\lbrace 2\\rbrace = \\langle \\langle 2\\rangle \\rangle , \\quad \\quad c_n\\lbrace 4\\rbrace = \\langle \\langle 4\\rangle \\rangle -2 \\cdot \\langle \\langle 2\\rangle \\rangle ^{2}.$ The cumulants are expressed in terms of the corresponding $Q_n$ vectors [45].",
"For differential flow calculations the restricted 2- and 4-particle correlations, $\\langle \\langle 2^\\prime \\rangle \\rangle $ and $\\langle \\langle 4^\\prime \\rangle \\rangle $ , are defined, where transverse momentum of one of the particles is limited to being within a certain $p_T$ bin.",
"The differential cumulant $d_n\\lbrace 4\\rbrace $ reads $d_n\\lbrace 4\\rbrace = \\langle \\langle 4^\\prime \\rangle \\rangle -2 \\cdot \\langle \\langle 2^\\prime \\rangle \\rangle \\cdot \\langle \\langle 2\\rangle \\rangle .$ Finally, the differential $v_n\\lbrace 4\\rbrace (p_T)$ coefficients are derived as $v_n\\lbrace 4\\rbrace (p_T) = -d_n\\lbrace 4\\rbrace \\cdot (-c_n\\lbrace 4\\rbrace )^{-3/4}.$ This technique is used by the CMS Collaboration [10].",
"The same technique is applied in the present paper to calculate the corresponding differential $v_2\\lbrace 4\\rbrace (p_T)$ coefficients in the hydjet++ model.",
"Figure: (Color online).The comparison of CMS data for v 2 {4}(p T )v_2\\lbrace 4\\rbrace (p_T) (triangles) and hydjet++ calculations for v 2 {4}(p T )v_2\\lbrace 4\\rbrace (p_T) (squares) andv 2 RP (p T )v_2^{RP}(p_T) (circles) in Pb+Pb collisions at s NN =5.02\\sqrt{s_{\\mbox{\\tiny {\\it {NN}}}}}= 5.02 TeVfor the centrality 30–40%.",
"Lines are drawn to guide the eye.To compare model calculations with the data we generated Pb+Pb collisions at $\\sqrt{s_{\\mbox{\\tiny {\\it {NN}}}}}= 5.02$ TeV in seven centrality bins: $\\sigma /\\sigma _{geo}=$ 5–10,10–15,15–20,20–30,30–40,40–50 and 50–60%.",
"Statistics of generated events varies from 2 million for semi-central to 7 million for very periperal collisions, respectively.",
"Figure REF shows the elliptic flow restored by 4-cumulant method from the hydjet++ calculated events in comparison with the CMS data [10].",
"The model generated “true” value of the elliptic flow coefficient $v_2^{RP}(p_T)$ is also presented.",
"It is calculated with respect to the reaction plane ($RP$ ), which is exactly known and is directed along the impact parameter ${\\bf b}$ in the model.",
"At relatively low transverse momenta, $p_{T}<3\\div 4$ GeV/$c$ , the measured elliptic flow, $v_2\\lbrace 4\\rbrace (p_T)$ (CMS), and the restored one, $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), are practically equal to the generated original elliptic coefficient $v_2^{RP}(p_T)$ .",
"It is not surprising, because in this momentum region the bulk of produced particles are mainly correlated with the reaction plane only, whereas nonflow correlations are relatively small.",
"At higher transverse momenta, $p_{T}>10$ GeV/$c$ , which is the region of the special interest here, the values of the measured elliptic flow and the flow restored from the model calculations are close to each other, but noticeably larger than the generated elliptic flow $v_2^{RP}(p_T)$ .",
"In the intermediate region, $4< p_{T}<10$ GeV/$c$ , the both model calculated coefficients, $v_2^{RP}(p_T)$ and $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), lie below the CMS data.",
"Figure: (Color online).The elliptic flow soft (triangles) and hard (squares) components andresulting total flow v 2 RP (p T )v_2^{RP}(p_T) (circles) in hydjet++ for Pb+Pb collisions at s NN =5.02\\sqrt{s_{NN}}=5.02 TeV with the centrality 30–40%.Figure REF illustrates the structure of elliptic flow in hydjet++ .",
"Here the coefficients $v_2^{RP}(p_T)$ are calculated with respect to the reaction plane separately for each component, soft and hard, together with the resulting total value.",
"This illustration shows the slight model drawback in the intermediate momentum region.",
"Originally, the model includes the adequate description of soft and hard physics.",
"In the intermediate $p_T$ region the result is obtained by a simple superposition of two independent contributions.",
"The crosslinking is regulated by one parameter, $p_T^{\\rm min}$ , which is the minimum transverse momentum transfer in a hard parton subprocess.",
"For the transverse momentum spectra this procedure is painless since the crosslinking takes place for a continuous function, and its fracture is smoothed out by the overlapping of particles from both contributions.",
"As a result, we describe effectively the transverse momentum spectra even in the intermediate region also [13] without any additional mechanism.",
"For elliptic flow coefficients the crosslinking takes place for a “discontinuous” function, see the crosslinking region of $p_T \\simeq 10$ GeV/$c$ , where $v_2^{RP}(p_T)$ (soft) and $v_2^{RP}(p_T)$ (hard) have the quite different values.",
"Thus, the simple smoothing is not enough to describe this region by a simple superposition.",
"It means that some improvements of the model are required to describe successfully the whole $p_T$ -region.",
"For instance, it can be a minijet production or some other mechanism.",
"Fortunately, this “problematic” region of the model is not used in the present flow correlation analysis.",
"Now we focus on the high transverse momentum region, to study which the large number of events should be generated.",
"In this region the hydjet++ flow, restored by cumulant method, is close to the measured one, but it is visibly larger than the original elliptic flow generated in the model.",
"This observation requires an explanation.",
"The azimuthal anisotropy arises in the model as a result of jet quenching, see [22], [23].",
"Due to the path-length dependent energy loss of partonic jets as they traverse the matter, the jet particles become to be correlated with the reaction plane.",
"However, the particle correlations relative to the jet axis remain also.",
"Unlike the region of small transverse momentum, there are at least two singled out directions, the reaction plane and the jet axis, respectively, relative to which particles are correlated.",
"In this case we can decompose the azimuthal distribution in a form $\\displaystyle \\frac{dN}{d\\varphi } &=& N_0 \\big (1+ 2 v_2^{RP} \\cos { \\left[2(\\varphi -\\Psi _{2}^{RP}) \\right] }\\nonumber \\\\&+& 2 v_2^{jet} \\cos { \\left[ 2(\\varphi -\\Psi _{2}^{jet}) \\right] } \\big ),$ where the direction of jet axis $\\Psi _{2}^{jet}$ is randomly oriented with respect to the reaction plane $\\Psi _{2}^{RP}$ , which is set to zero in the model calculations.",
"$N_0$ is the normalization factor.",
"The 2- and 4-particle correlations are estimated as $\\langle \\langle 2\\rangle \\rangle \\simeq (v_2^{RP})^2 + (v_2^{jet})^2 ,\\quad \\quad \\langle \\langle 4\\rangle \\rangle \\simeq [(v_2^{RP})^2 + (v_2^{jet})^2]^2$ and they are sensitive to both types of correlations discussed above.",
"Herewith $\\langle \\langle 2\\rangle \\rangle ^{1/2}$ and $\\langle \\langle 4\\rangle \\rangle ^{1/4}$ are always larger than the “true” elliptic coefficient $v_2^{RP}(p_T)$ which we are interested in.",
"Figure: (Color online).The correlation between v 2 v_2 values at low and highp T p_T in Pb+Pb collisions at s NN =5.02\\sqrt{s_{\\mbox{\\tiny {\\it {NN}}}}}= 5.02 TeVas function of centrality.The points represent the centrality bins 5–10,10–15,15–20,20–30,30–40,40–50, and 50–60%.The model results are presented for v 2 RP v_2^{RP} (circles) andv 2 {4}v_2\\lbrace 4\\rbrace (squares).",
"Data (triangles) are taken from .Model calculations are performed (a) with and(b) without the jet quenching.The correlations between the low-$p_T$ and high-$p_T$ elliptic flow which are the main issue of our study are displayed in Fig.",
"REF (a).",
"The picked up intervals are $14< p_{T}<20$ GeV/$c$ for high and $1.0< p_{T}<1.25$ GeV/$c$ for low transverse momenta, respectively.",
"One can see that the model calculations, $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), reproduce the experimentally observed centrality dependence of the flow correlations fairly well except the last centrality bin, while the generated original elliptic flow $v_2^{RP}(p_T)$ at high $p_T$ goes always lower than the restored flow in accordance with the explanation above.",
"The deviations become significant for the centralities larger than 30–40%.",
"In this centrality interval the anisotropy caused by the jet quenching becomes to die out, but particles remain correlated with the axes of jets.",
"Note that in the low-$p_T$ region the integrated values of $v_2$ in the model are lower than the data.",
"This is due to simplification, discussed in Sec.",
", aiming to reduce the number of free parameters in the model.",
"To reach better quantitative agreement with the data one has to treat both anisotropy parameters, $\\varepsilon (b)$ and $\\delta (b)$ , as independent ones, see [19].",
"This circumstance, however, does not affect the main results associated with identifying the role of jets on the behavior of cumulants.",
"Note also, that in the high-$p_T$ sector the agreement between the model results and the data is good.",
"Figure REF (b) demonstrates the comparison of model calculations without jet quenching effects with the data.",
"In this case $v_2^{RP} =0$ and the correlations relative to the jet axis contributes to the 4-th order cumulant only, but the magnitude of jet correlations is not strong enough to reproduce data.",
"Thus, in the collisions with centrality up to 30–40% the azimuthal anisotropy due to jet quenching reveals itself, whereas in more peripheral collisions the jet correlations contribute significantly to the 4-th order cumulant.",
"The reason is as follows.",
"Since in peripheral heavy-ion collisions there are simply quantitatively fewer nucleon-nucleon collisions, then, fewer jets are produced.",
"In the limiting case there is one back-to-back pair, and the method sees this axis and anisotropy.",
"In more central collisions there are many jet pairs.",
"They are all distributed randomly in azimuth, therefore, the anisotropy caused by jets tends to zero.",
"It is worth noting that jets were the main source of violation of the number-of-constituent quark (NCQ) scaling in hydjet++ calculations of differential elliptic [14], [15] and triangular [18] flow.",
"The linear fit [10], performed by CMS Collaboration to data on elliptic flow correlations at low and high transverse momenta, also indicates some sort of scaling behavior.",
"In contrast to situation with the NCQ scaling, here jets work toward the scaling fulfillment."
],
[
"Conclusions",
"The phenomenological analysis of elliptic flow correlations at low and high $p_T$ in Pb+Pb collisions at center-of-mass energy 5.02 TeV per nucleon pair has been performed within the two-component hydjet++ model.",
"These correlations are stipulated by the fact that the magnitudes of anisotropy at low and high $p_T$ are mainly determined by the value of initial ellipticity of the nuclei overlapping.",
"At relatively low transverse momenta, $p_{T}<3\\div 4$ GeV/$c$ , the model generated elliptic flow $v_2^{RP}(p_T)$ and its value restored by 4-cumulant method, $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), are very close to the differential elliptic flow $v_2\\lbrace 4\\rbrace (p_T)$ measured by the CMS Collaboration.",
"At high transverse momenta $p_{T}> 10$ GeV/$c$ the cumulants are sensitive to both the anisotropy due to jet quenching, $v_2^{RP}$ , and the particle correlations with the jet axis, $v_2^{jet}$ .",
"In the collisions with centrality up to 30–40% the 4-cumulant method “measures” mainly the azimuthal anisotropy due to jet quenching, whereas in more peripheral collisions it is affected by the jet correlations primarily.",
"The model calculations restored by this method, $v_2\\lbrace 4\\rbrace (p_T)$ (hydjet++), reproduce the experimentally observed centrality dependence of elliptic flow correlations rather well without any additional tunes of model parameters.",
"Fruitful discussions with A.I.",
"Demyanov and L.V.",
"Malinina are gratefully acknowledged.",
"This work was supported in parts by Russian Foundation for Basic Research (RFBR) under Grant No.",
"18-02-00155, Grant No.",
"18-02-40084 and Grant No.",
"18-02-40085.",
"L.B.",
"and E.Z.",
"acknowledge support of the Norwegian Research Council (NFR) under grant No.",
"255253/F50 - “CERN Heavy Ion Theory\"."
]
] | 2012.05139 |
[
[
"Extracting the signed backbone of intrinsically dense weighted networks"
],
[
"Abstract Networks provide useful tools for analyzing diverse complex systems from natural, social, and technological domains.",
"Growing size and variety of data such as more nodes and links and associated weights, directions, and signs can provide accessory information.",
"Link and weight abundance, on the other hand, results in denser networks with noisy, insignificant, or otherwise redundant data.",
"Moreover, typical network analysis and visualization techniques presuppose sparsity and are not appropriate or scalable for dense and weighted networks.",
"As a remedy, network backbone extraction methods aim to retain only the important links while preserving the useful and elucidative structure of the original networks for further analyses.",
"Here, we provide the first methods for extracting signed network backbones from intrinsically dense unsigned unipartite weighted networks.",
"Utilizing a null model based on statistical techniques, the proposed significance filter and vigor filter allow inferring edge signs.",
"Empirical analysis on migration, voting, temporal interaction, and species similarity networks reveals that the proposed filters extract meaningful and sparse signed backbones while preserving the multiscale nature of the network.",
"The resulting backbones exhibit characteristics typically associated with signed networks such as reciprocity, structural balance, and community structure.",
"The developed tool is provided as a free, open-source software package."
],
[
"Introduction",
"Networks are increasingly useful in modeling and studying many problems in seemingly unrelated domains from social sciences [9], natural sciences and engineering [74], and arts and humanities [68].",
"A very simple network consists of nodes (vertices, actors) and links (edges, ties) connecting them.",
"The network science tools utilize the information from the interdependence between entities given by the network structure.",
"The data that can be modeled in the network form is not limited only to the node and edge structure.",
"More complex networks have attributes associated with edges and nodes, especially with the advent of advanced data collection methods.",
"In weighted networks, edges have numeric weights indicating its intensity (e.g., coupling strength, amount, similarity, etc.).",
"In directed networks, edges have distinguishable source and target nodes, indicating the direction of the relation (e.g., flow, following, etc.).",
"For instance, the air traffic can be modeled as a very simple network such that a node represents an airfield and an edge between two nodes indicates a flight between them.",
"In this case, the number of connections an airfield has is given by the degree of the corresponding node.",
"The same system can also be modeled as a directed weighted network where an edge carries the take-off and landing airfield data as its direction and passenger capacity data as its weight.",
"In this way, the outgoing and incoming capacity of an airfield is given by the out- and in-strength of the node, that is the total weight of edges leaving and entering the node, respectively.",
"This very simplified example demonstrates the view that as networks get more complex, richer analyses can be made.",
"Another type of networks, which is explored relatively less in the literature, is signed networks where edges have negative or positive signs respectively indicating antagonism (e.g., dislike, distrust, foes, dissimilarity, voting against, inhibition, etc.)",
"or rapport (e.g., like, trust, friendship, similarity, voting for, activation, etc.).",
"Negative edges are not simply negation of positive edges [81], they show distinct behavior in networks [76], and their inclusion enrich the typical tasks on networks such as link prediction [50], [36], recommender systems [79], [89], node classification [60], [78], node centrality and ranking [31], [85], representation learning [46], [21], information diffusion and influence maximization [39], [44], finding cliques [51], community detection, graph partitioning and blockmodels [27], [55], [54], and polarization [8], [87].",
"In addition to the typical and prevalent applications in social media [80], signed networks also find application areas in politics [3], international relations [24], finance [37], biology [42], [63], [40], and ecology [40].",
"The density of a network is usually defined by the ratio of the number of observed links to the number of possible links.",
"Formally, sparse networks are those where density asymptotically goes to zero in the limit of large number of nodes.",
"However, in most empirical networks, it is impossible to evaluate this [61].",
"As a result, in general, an empirical network is called sparse if it has a low density.",
"The majority of real-world networks are sparse [62].",
"For instance, the number of stable social relations of a human is limited by cognitive constraints [34], stations are usually only connected to nearest stations in power grid networks, a webpage points out only so many others in web networks, and so on.",
"On the other hand, some networks might be very dense or even almost completely connected by its own nature.",
"Most countries trade with most other countries, humans move/migrate from many locations to many other locations, many species have many predators and preys [25], most people interact with most others in certain social settings, and so on.",
"The earlier view that additional data allowing richer analyses is restricted by the amount and structure of the data itself, particularly in the case where the additional data is provided by more links between the nodes.",
"In addition to the fact that sparsity is desirable due to the computational complexity of many network algorithms, most of the typical network analysis and visualization methods assume the networks to be sufficiently sparse [61].",
"Apart from computational and methodological concerns, dense networks might have noisy, uninformative, insignificant, or otherwise redundant links.",
"This further worsens the application and interpretation capacity in many network tasks e.g., those relating to node centrality, cliques and communities, and diffusion.",
"Therefore, the relevant and significant information should be extracted from such dense networks such that the original rich data is reduced into a network that is sparse and simpler but maintains adequate structural information for efficient and effective analyses.",
"There is a body of literature for extracting backbones of usually weighted and relatively dense networks, which we review in the next section.",
"Such information filtering task is usually referred to as backbone extraction or network sparsification and aims to remove statistically insignificant or otherwise redundant links while maintaining the informative structural properties for further analysis.",
"To the best of our knowledge, this is the first study in the literature that provides methods for extracting signed network backbones from weighted dense networks.",
"Unlike the earlier studies, our method relies on and requires the input network to be intrinsically dense; which in turn enables inferring the link signs.",
"We use the term intrinsically dense for characterizing networks where all nodes, in a sense, are aware of all other nodes and can interact with them without obvious natural limits such as those mentioned earlier for sparse networks.",
"This definition follows that an edge does not necessarily represent the existence of a positive relationship but might be an artifact of the randomness or even a negative relationship depending on its weight.",
"Such definition of intrinsically dense networks provides a distinction not only from sparse networks but also from certain dense networks.",
"For instance, human interaction network in a workshop with two parallel sessions with a single mutual short break is not intrinsically dense because the participants in different sessions are not given sufficient time to meet with others, thus, absence of a link cannot reasonably indicate if two participants from the different sessions are avoiding each other.",
"As a demonstrating example of intrinsically dense networks, consider a network where link weights denote similarity of respective nodes.",
"Lower similarity does not only mean a lack of similarity but often indicates dissimilarity, a negative underlying link.",
"In the same vein, voting for a set of candidates indicates support for those candidates while also suggesting opposition for the other candidates especially those who are otherwise popular.",
"Finally, we should highlight that whether a network is intrinsically dense or not is a matter of extent and not a strictly binary decision.",
"We suggest that positive and negative links are those with weights significantly and substantially deviating from the random expectation under a suitable null model.",
"We develop an appropriate null model based on hypergeometric distribution and iterative proportional fitting procedure and offer significance filter and vigor filter for extracting signed backbones of intrinsically dense networks.",
"The proposed methodology is capable of handling directed or undirected networks as input and producing weighted or unweighted signed backbones.",
"Our contribution can be summarized as follows.",
"[itemsep=0pt] The first extensive literature review on network backbone extraction methods is provided (Section ).",
"The first methods to extract signed backbones of intrinsically dense weighted networks are proposed (Section ).",
"The methodology is empirically evaluated on real-world networks of different characteristics and its feasibility and usefulness is shown (Section )."
],
[
"Backbone extraction methods for unsigned networks",
"The simplest yet a popular approach is to apply a global threshold where only those edges with weights satisfying the predetermined threshold are retained.",
"There are two major problems with this approach.",
"First, the choice of the threshold is rather arbitrary and non-impartial unless the physical meaning of the weights in the domain of the particular network allows an explanation.",
"The second and more important problem is the case of multiscale networks where the weights are distributed over a broad range of scales.",
"Such characteristic is definitely not an exception but an observed phenomenon in many real-world networks [5].",
"For instance, in many networks where rich club effects [72] are prevalent, nodes with higher degrees/strengths tend to form links with nodes of the same or higher degree/strength which results in a hierarchical and multiscale network structure.",
"Rich club effects and such structures are present in many weighted networks including the global airline traffic network [1], worldwide maritime transportation network [41], world trade network [7], [67], international credit-debt networks [18], population flow network during a national holiday [86], patient referral networks [77], and communication networks in the brains of humans [1] and rats [52].",
"Therefore, weights with small magnitudes are not necessarily noise and weights with large magnitudes are not necessarily important but might be the result of such multiscale weight characteristics.",
"In multiscale networks, the application of a global threshold tends to eliminate the edges of low-strength nodes regardless of the local significance of those edges.",
"As a result, the structure of high-strength nodes can be preserved but local regions characterized by relatively low-strength nodes are underestimated, might become disconnected, and might even be completely eliminated.",
"In summary, increasing the value of a global threshold loses local structures at the bottom of the hierarchy whereas decreasing it results in retained noise in the upper levels.",
"Consequently, global thresholding would only work under the often-unrealistic assumption that edge weights are independent and identically distributed random variables.",
"As a remedy, the global threshold method can be improved to work on weights represented not in universal units but as fractions of the node degrees/strengths.",
"Yet, an edge is associated with two nodes and it again becomes relatively arbitrary to choose the normalization factor.",
"Such normalization might not be robust to nodes with very low and high degrees as well since it potentially overestimates the edges of low-degree nodes and underestimates the edges of high-degree nodes.",
"Overall, it could serve as a practical remedy for certain problems but the need for more statistically-sound methods is clear.",
"Another approach is to extract the spanning tree with maximal weight by appropriately transforming the weights and finding the minimum spanning tree.",
"This method does not require any extrinsic input and ensures that the extracted backbone is connected.",
"However, the backbone will be acyclic and local clustering and community structure cannot be preserved.",
"Using a similar spanning tree approach, Tumminello et al.",
"[82] propose a technique for controlling the genus of the backbone which, in turn, enables preservation of some local structure.",
"Grady et al.",
"[35] extracts the high-salience skeleton of the network which consists of only the salient edges.",
"For each node, they construct the shortest path tree by merging all shortest paths from the node to all other nodes.",
"Then, the saliency of an edge is defined as the fraction of the all shortest path trees where the edge is a member.",
"They show that edge saliency shows a bimodal distribution near the boundaries.",
"Only the edges with saliency near 1 are retained; effectively eliminating the need for choosing an arbitrary threshold.",
"By design, it ensures the connectivity of the extracted backbone.",
"Edge saliency is different from edge betweenness where the former tends to award the edges in the periphery (e.g., low-degree nodes) whereas the latter tends to award the edges in the core (e.g., high-degree nodes).",
"The methods described so far do not assume any underlying null model and do not compare observed weights to expected weights for statistical evaluation.",
"Disparity filter [70] assumes that the normalized weights of the links of a node follow a uniform distribution.",
"Comparing the observed weights to this null model at a desired significance level, the network backbone including only the statistically significant links can be obtained.",
"As mentioned earlier, a weight can be normalized and evaluated separately for the two nodes it connects.",
"Thus, a link can be significant from the viewpoint of one node and not the other.",
"This problem is tackled by retaining a link if it satisfies the significance condition for at least one of the two nodes.",
"Adopting a loosely similar weight normalization scheme, bistochastic filter [73] first scales the weight matrix such that its marginals (i.e., the row and column totals, in- and out-strength sequence) are equal to 1 resulting in a doubly stochastic matrix.",
"Then, the links with highest weights are retained until the network is strongly connected; or until any other stopping criteria which can result in sparser backbones with disconnected components or even denser backbones.",
"In the same line of research, LANS [30] does not assume an underlying distribution for the weights of a node and employs the empirical cumulative density function instead.",
"It is stated that it is more robust to the highly heterogeneous local weight distributions than disparity filter and bistochastic filter are.",
"Rather than adopting a local approach in developing null models, GloSS filter [64] utilizes a global null model.",
"The model considers both the strengths and degrees and aim to preserve the global weight distribution and the topology.",
"GLANB [90] evaluates the statistical significance of link saliency to retain satisfactory links; loosely combining disparity filter and high-salience skeleton.",
"Dianati et al.",
"[22] introduce two interrelated filters.",
"Treating an integer-weighted network as a multiedge network, they assume that each unit edge chooses two nodes in random respecting the degree sequences which results in a binomial distribution for weights.",
"The null model is indicated to blend the approaches of disparity filter and GloSS.",
"Marginal likelihood filter operates by evaluating each edge separately against a chosen significance level whereas global likelihood filter incorporates exponential random graph model with a Monte Carlo simulation scheme to consider all links at once.",
"ECM filter [33] enhances the null model of Dianati et al.",
"[22] with the purpose of retaining the relation between strengths and degrees.",
"Polya filter [58] assumes an underlying null with a self-reinforcing mechanism in which the generation of link weight increases the probability of further weight generations for that link.",
"Tumminello et al.",
"[83] introduce hypergeometric filtering originally for bipartite weighted networks.",
"It is employed for unipartite weighted networks [65], [71] as well.",
"Assuming integer weights and treating weighted links as multiple links, every time a node increases its degree by 1 by adding a new link, the node at the other end of the link is selected randomly among all possible nodes.",
"Then, the edge weight generation process can be described by a hypergeometric distribution which becomes the null model for the filter.",
"Comparing observed values with the null model, statistically significant links can be extracted at desired levels of confidence.",
"Noise-Corrected Bayesian filter [19] assumes edge weight generation follows a binomial distribution where a null model based on the hypergeometric distribution is used in determining Bayesian priors.",
"Lift value for each link (i.e., the ratio of the observed value to the expected value) is calculated, transformed to the $[-1, 1]$ range, and the associated variance is estimated with a Bayesian inference schema.",
"Using the appropriate posterior variances, the links which satisfy a desired significance level are retained.",
"Although more specific approaches exist with different purposes (e.g., to retain the most consistent links in brain networks based on diffusion images [66], to preserve functional backbones based on network motifs [13]), the state-of-the-art concentrates on developing appropriate null models to evaluate link weights against and retain only those satisfying a desired level of statistical significance."
],
[
"Formal Problem Definition",
"We denote an intrinsically dense, undirected or directed, and non-negative weighted network without self-loops by $G := (V, W)$ .",
"$V$ is the set of nodes with $i, j \\in V$ as its general elements and its cardinality is $n = |V|$ .",
"$W$ is the corresponding weight matrix.",
"When $G$ is undirected $W_{ij}$ denotes the weight of the link between $i$ and $j$ and $W_{ij} = W_{ji}$ .",
"When $G$ is directed, $W_{ij}$ denotes the weight of the link from $i$ to $j$ .",
"Self-loops are not allowed hence $W_{ii} = 0 \\; \\forall i$ .",
"Total outflow of $i$ is denoted with $W_{i.}",
"= \\sum _{j} W_{ij}$ .",
"Total inflow to $j$ is denoted with $W_{.j} = \\sum _{i} W_{ij}$ .",
"The sum of all weights is denoted with $W_{..} = \\sum _{i} \\sum _{j} W_{ij}$ .",
"We employ the same definitions of $W_{i.",
"}$ , $W_{.j}$ , and $W_{..}$ for directed and undirected networks, i.e., treating an undirected link as two reciprocal directed links with equal weights.",
"Given $G$ , the aim is to extract a meaningful signed network $\\hat{G} := (\\hat{V}, \\hat{A}, \\hat{W})$ where $\\hat{A}$ is a sparse adjacency matrix in general such that $\\hat{A}_{ij} \\in \\lbrace -1, 0, 1\\rbrace $ where $-1$ denotes a negative link, 1 denotes a positive link, and 0 indicates the absence of a link.",
"$\\hat{W}$ is the corresponding optional weight matrix.",
"If $\\hat{A}_{ij} = 0$ , then $\\hat{W}_{ij} = 0$ ; otherwise $\\hat{W}_{ij}$ has the same sign as $\\hat{A}_{ij}$ ."
],
[
"Proposed Solution",
"The null model.",
"To extract the signed edges, it is necessary to have a null model which empirical edge weights are compared against to distinguish edge weights that are due to chance and edge weights that significantly deviate from the expected values in either positive or negative direction.",
"We make the assumption that every time a node increases its out-strength by one, it randomly chooses the other node with probabilities proportional to their in-strengths.",
"In this way, the process is reduced to sampling without replacement problem, of which the urn problem is a famous example.",
"We describe the process with a simple analogy.",
"For each node $i$ , there are $W_{..} - W_{.i}$ marbles in the urn (not allowing self-loops), $W_{i.",
"}$ marbles are chosen from the urn without replacement, and there are $W_{.j}$ marbles in the urn for each $j$ .",
"Such process is well-characterized by hypergeometric distribution and allows us to calculate statistical quantities for links between all $i$ and $j$ .",
"The mean of the hypergeometric distribution associated with the link from $i$ to $j$ is given by Eq.",
"REF .",
"However, this formulation does not preserve the in-strength and out-strength sequences in the system, thus, is inadequate for our purposes.",
"Given that $N$ is the weight matrix under null model with $N_{ij}$ denoting the expected weight from $i$ to $j$ , the following should be satisfied: $N_{i.}",
"= W_{i.",
"}$ , $N_{.j} = W_{.j}$ , and $N_{..} = W_{..}$ .",
"For this purpose, we utilize the iterative proportional fitting procedure (IPFP) described in [4], [59] which is shown to converge to an optimal solution [10].",
"Given the column and row marginals (e.g., in- and out-strength sequences) and an appropriate prior matrix with certain elements as zero (e.g., diagonal elements to disallow self-loops) and other elements as akin to Eq.",
"REF , IPFP estimates the values for non-zero elements of the matrix through a series of row-scaling and column-scaling operations until achieving a desired precision.",
"Given out- and in-strength ($W_{i.",
"}$ and $W_{.j}$ ) sequences as the marginals and the values obtained from Eq.",
"REF as the elements prior matrix with its diagonal elements set to 0, IPFP finds the suitable matrix $N$ representing the expected weights under our proposed null model.",
"$\\frac{W_{i.}",
"W_{.j}}{W_{..} - W_{.i}}$ In evaluating the statistical significance of an edge weight, only considering the difference of observed and expected values without referencing to a dispersion measure is not viable.",
"For each edge, dispersion of its expected weight must be known to produce confidence intervals.",
"Accordingly, we estimate the variance of $N_{ij}$ based on the associated hypergeometric distribution and obtain its standard deviation as given by Eq.",
"REF .",
"$\\sigma _{ij} = \\left( W_{i.}",
"\\; \\frac{W_{.j}}{W_{..} - W_{.i}} \\; \\frac{(W_{..} - W_{.i}-W_{.j})}{W_{..} - W_{.i}} \\; \\frac{W_{..} - W_{.i}-W_{i.}}{W_{..}",
"- W_{.i}-1} \\right) ^ {\\frac{1}{2}}$ Significance filter.",
"The first filter we propose functions to eliminate links whose weights do not significantly deviate from their expected values.",
"Significance filter retains only those links satisfying the condition $\\sigma _{ij} \\: \\alpha ^- \\: \\le W_{ij} - N_{ij} \\le \\: \\sigma _{ij} \\: \\alpha ^+$ where $\\alpha ^-$ and $\\alpha ^+$ respectively take non-positive and non-negative values and are user-defined hyperparameters specifying the desired significance thresholds for negative and positive signed edges.",
"It should be highlighted that the hypergeometric distribution is a distribution for discrete events.",
"In general, we can slightly abuse it by allowing continuous weights or rounding weights to the nearest integers without much impact on the resulting backbone.",
"However, a larger issue that is not adequately discoursed in the hypergeometric filtering literature is the impact of the magnitude of weights on the statistical significance evaluation.",
"As the magnitude of weights increases, the confidence intervals relative to the weights become narrower, as a consequence of the assumption that the weights are discrete events.",
"For instance, the same monetary network represented in dollars or cents would result in different network backbones for the same values of $\\alpha $ .",
"This issue is less pressing when the link weights essentially correspond to discrete events such as human migration networks or voting networks.",
"Taking these into account, it can be concluded that the values chosen for $\\alpha $ might not accurately translate into traditional p-values in many networks due to their non-discrete nature.",
"Hence, the choice of $\\alpha $ should not necessarily be limited to traditionally employed values especially when the weights do not correspond to discrete events.",
"Vigor filter.",
"The significance filter might retain statistically significant but otherwise very weak links.",
"Yet, an edge can be evaluated also based on whether it is sufficiently strong in terms of intensity.",
"This is useful for multiple reasons: (i) the employed significance filter might be too permissive in certain circumstances, (ii) the signed network backbone is needed to reflect only binary links where opinions are rather strong, or (iii) higher sparsity is desired.",
"We define vigor of an edge from $i$ to $j$ as $\\beta _{ij}$ (pronounced as víta in Modern Greek) in Eq.",
"REF .",
"It takes values in the range $[-1, 1]$ , its magnitude indicates the intensity of the link and $\\beta _{ij} = 0$ when $W_{ij} = N_{ij}$ .",
"Vigor filter retains only those links satisfying the condition $\\beta ^- \\le \\beta _{ij} \\le \\beta ^+$ where $\\beta ^+$ and $\\beta ^-$ respectively take non-positive and non-negative values and are user-defined hyperparameters specifying the desired vigor thresholds for positive and negative signed edges.",
"$\\beta _{ij} = \\frac{W_{ij}/N_{ij} \\; -1}{W_{ij}/N_{ij} \\; +1}$ Signed backbone extraction.",
"The signed backbone of a network can be extracted using a combination of significance filter and vigor filter.",
"We suggest always employing the former since $\\beta $ values might be unreliable in the small-magnitude weight regime characterized by small $N_{ij}$ , relatively large $\\sigma _{ij}$ , and small $W_{ij}$ .",
"Significance filter would function to eliminate such links, therefore, the retained links would have reliable vigor.",
"Vigor values of the remaining edges can be utilized as the signed edge weights in the extracted backbone.",
"In the case of weighted backbone, the use of vigor filter is rather optional since the vigor information is directly carried onto the network.",
"Alternatively, the weights can be ignored to produce a network with binary opinions.",
"In this case, we recommend utilizing the vigor filter to retain only the edges that exhibit appropriate levels of intensity.",
"Overall, for setting appropriate values to the hyperparameters, one should consider (i) the physical meaning of hyperparameters in the context of the specific problem, (ii) the desired level of sparsity for the extracted backbone.",
"Undirected networks.",
"The described method is defined well on the directed networks.",
"The generalization to the undirected networks is ensured in the following way.",
"An undirected link is replaced with two reciprocal directed links of the same weight; effectively transforming the network into a directed network.",
"The null model and the filters produce the same null expectation and nearly the same varianceFor sufficiently large networks, $W_{..} - W_{.j} \\approx W_{..} - W_{.i}$ in In Eq.",
"REF for such reciprocal links.",
"Finally, the directed backbone can be transformed back into an undirected backbone by treating reciprocal directed links as undirected links and removing the redundancy by keeping the link with the highest absolute vigor."
],
[
"Empirical Analysis",
"In this section, relevance and diverse characteristics of the employed network datasets are described, and proposed filters are empirically analyzed in terms of resulting backbone sizes, robustness to multiscale networks, and structures of the extracted signed network backbones.The datasets and code for producing the analyses in this section can be made available upon request."
],
[
"Datasets.",
"The proposed method for signed network backbone extraction is experimented on four real-world networks from different domains and of varying statistical properties.",
"For each network; the number of nodes $n$ , number of nonzero links $m$ , density, five-point summary of weights (minimum, first quantile, median, third quantile, and maximum values), and three-point summary (minimum, median, and maximum values) of node strengths (out- and in-strengths if networks are directed) are presented in Table REF .",
"Table: Employed NetworksMigration network [11] reports interstate migration flows (in terms of people) among the US statesThe dataset contains District of Columbia in addition to the 50 states.",
"in 2018.",
"Such internal migration data is explored for its correlation with economic productivity differences, geographical proximity, political preferences, and other cultural and historical factors [15], [53], [17].",
"Eurovision network [84] represents the votes between participant countries of the song contest in 20032003 was selected because it was the last year that all countries competed in a single round.. Each country, via public votesIreland, Russia, and Bosnia and Herzegovina exercised jury voting instead of public voting., awards the set of points {1,2,3,4,5,6,7,8,10,12} to 10 other countries.",
"Studies employing a network analysis perspective [28], [16], [32], [56], [57], [26], [75] show that voting behavior is not determined only by the music/performance quality but affected by political factors, geographical and cultural similarity, diasporas, and others.",
"Contact network data [43] is gathered via wearable sensors at an academic conferenceThe temporal face-to-face interaction dataset was collected at Hypertext conference in 2009 as part of SocioPatterns research collaboration.",
"with 113 attendees over 2.5 days.",
"The edge weights are generated by aggregating the number of 20-second intervals the respective two participants spent face-to-face over the course of the conference.",
"The networks of such temporal ties are utilized for temporal backbone extraction, investigation of spreading processes, and other behavioral and structural analysis [47], [88], [49], [6].",
"Species network [48] is generated based on the cohabitation patterns of 62 marine species in South Florida.",
"Edge weights represent the similarity of species based on the habitats they co-occupy during the same life stagesGiven the species v. habitat-life stage bipartite matrix $B$ , $W_{ij} := \\sum _{k} B_{ik} B_{jk}$ where $i, j$ are the species and $k$ is the habitat-life stage pair, and $B_{ik}$ is the respective species-habitat-life stage score.",
"More information on the original dataset can be found at https://atlanticfishhabitat.org/species-habitat-matrix/.",
"Information derived from such cohabitation networks is useful in various ecological studies [45], [12], [29].",
"Moreover, none of the employed networks have natural self-loops; which make them particularly suitable for our study.",
"Our proposed method considers the propensity of nodes for incoming and outgoing edge weights by explicitly including the in-strengths and out-strengths in the null model.",
"For instance, high economic productivity for a state would result in more immigration into it.",
"Similarly, low productivity would cause emigration from that state.",
"The effect of such productivity is homogeneous in the sense that all other nodes perceive it in the same way.",
"Such homogeneous effects are eliminated for the large part by our null model.",
"However, for instance, the effect of political affiliation or geographic location is perceived differently by other nodes.",
"Hence, the extracted backbone is expected to reflect non-economic factors in Migration network.",
"With a simple analogy, the quality of the songs and performances in Eurovision song contest is equivalent to productivity.",
"Hence, the elimination of such effect by the null model would result in an extracted signed network representing the non-quality factors in voting behavior.",
"Similarly, the null model largely eliminates the effects of participants' tendency to engage in conversations (e.g., popularity, extroversion) and the extent of species' ability to live in many different habitats."
],
[
"Backbone Size",
"In general, a signed network backbone should be sufficiently sparse such that only the important edges are preserved.",
"At the same time, unlike edges, most nodes should be retained since they are often the subject of the analysis and are largely indispensable in understanding the global network structure.",
"Here, we extract the signed backbones of the four networks under various hyperparameter regimes and observe the size of the resulting backbone in terms of signed edges and nodes.",
"Effects of significance threshold.",
"Figure REF visualizes the fraction of links and nodes retained for meaningful continuous ranges of the significance threshold under different vigor filters.",
"The suitable range of $\\alpha $ differs between networks mostly due to the varying overall magnitude of edge weights.",
"The significance threshold is represented with $\\alpha = \\alpha ^+ = - \\alpha ^-$ on x-axes.",
"Three different vigor threshold settings are represented in columns.",
"For edges, percentage values on y-axes are with respect to all possible edges, that is $n^2 - n$ edges for directed networks and half of it for undirected networks.",
"Figure: Effects of significance filter on the backbone size.Figure: Effects of vigor filter on the backbone size.Except for Species network, all nodes are retained in the backbones even under very strict settings where 80% to 85% of all possible edges are eliminated.",
"When edges are inspected, apart from the Species network where edges represent similarity, the negative edges are more frequent than the positive edges when $\\alpha = 0, (\\beta ^-, \\beta ^+) = (0, 0)$ .",
"This is due to the case that edges with zero weights (i.e., nonexistent edges) or trivially small values are treated as negative links regardless of their statistical significance or vigor.",
"As the values of $\\alpha $ increase, the number of negative links in the extracted backbone decreases faster than the positive links since many of the zero or trivially small weights are rather statistically insignificant.",
"On the other hand, utilizing vigor filter without the significance filter does not eliminate such zero-weighted links since those edges have vigor values of $-1$ (since 0 divided by any null expectation is 0, which maps to vigor of $-1$ ).",
"Hence, the decrease in negative edges in second and third columns in comparison to the first column of Figure REF when $\\alpha = 0$ is not provided by eliminating zero-weighted edges but other weak edges.",
"Therefore, significance filter is always necessary when insignificant zero or trivially small edge weights exist.",
"Employing very large $\\alpha $ values eliminates negative links almost completely.",
"This is largely because the edges with relatively small expected values under the null model have relatively large variances.",
"Increasing $\\alpha $ too much, thus, tends to eliminate negative edges first as well as other edges in local regions characterized by edge weights of small magnitude.",
"In the case of high-magnitude regions, an edge between two high-strength nodes is expected to have a large expectation under the null model.",
"It is much harder for such edges to empirically observe much larger weights than the null expectation since the total weight in the network is fixed.",
"This implies an upper boundary on vigor values for edges connecting to high-strength nodes.",
"Therefore, increasing $\\beta $ too much, especially on its positive range, tends to eliminate edges between central, high-strength nodes.",
"Effects of vigor threshold.",
"In a similar fashion, we have also explored the whole continuous range of vigor threshold under different significance threshold filter in Figure REF .",
"The significance threshold is represented with $\\beta = \\beta ^+ = -\\beta ^-$ on x-axes.",
"Three different significance threshold settings are represented in columns.",
"Except for Species network and except under strict vigor thresholds, all or almost all nodes remain in the backbone.",
"When edges are inspected, in line with the relevant conclusions derived from Figure REF , increasing $\\beta $ does not sufficiently eliminate the negative edges which would be otherwise statistically insignificant.",
"In very large values of $\\beta $ , positive links are largely eliminated.",
"When $\\beta = 1$ , the edges remaining in the backbone are the zero-weighted (i.e., nonexistent) edges.",
"Accordingly, employing very large vigor thresholds should be avoided.",
"Conclusion.",
"Overall, we have shown that the proposed method is able to reduce an intrinsically dense network to a signed backbone of the relative size of $10\\%$ to $20\\%$ such that the resulting backbone contains comparable portions of negative and positive edges.",
"Yet, the hyperparameter selection largely lies with the user and can be changed according to the purpose of analysis and the nature of the network.",
"For instance, Species network is constructed with similarity values as edge weights and the edge weights do not follow a skewed distribution unlike other networks.",
"Therefore, its behavior under the proposed filters partially deviates from the behavior of the other employed networks.",
"Principally, we can conclude that (i) significance filter should almost always be used with a fair threshold, (ii) very large threshold values for both filters should be avoided, and (iii) a sufficiently balanced, statistically meaningful, and sparse backbone can be extracted with a balanced utilization of the two proposed filters."
],
[
"Heterogeneity",
"As discussed in Section , an enviable backbone extraction method should respect the weight and strength heterogeneity in the original network (i.e., its multiscale, hierarchical nature).",
"That is to say, the retained edges should not be only those originally with large weights or those connecting high-strength nodes.",
"Heterogeneity of edge weights.",
"Figure REF presents the original weight distributionSmall random noise is added for Eurovision network for visualization purposes.",
"of the retained edges in backbones of different sizes The backbones are extracted with the following ($\\alpha ^-, \\alpha ^+$ ), ($\\beta ^-, \\beta ^+$ ) settings for the respective backbone sizes: Migration $\\rightarrow $ 25%: (-33, 33), (-0.33, 0.33); 10%: (-40, 40), (-0.57, 0.57); 5%: (-40, 40), (-0.72, 0.72).",
"Eurovision $\\rightarrow $ 25%: (-1.5, 2.25), (0, 0); 10%: (-2, 4.5), (0, 0); 5%: (-2.5, 5), (0, 0).",
"Contact $\\rightarrow $ 25%: (-2, 2), (0, 0); 10%: (-3, 3), (-0.66, 0.66); 5%: (-5, 5), (-0.55, 0.55).",
"Species $\\rightarrow $ 25%: (-3.5, 3.5), (0.15, 0.15); 10%: (-5, 5), (-0.2, 0.2); 5%: (-5.5, 5.5), (-0.33, 0.33).",
".",
"The figure shows that even when the extracted backbone retains only $5\\%$ of all possible edges, the heterogeneity of edge weights are respected.",
"It also visually depicts that there is no global cutoff value for determining the sign of edges as those cutoff values are established individually for each edge by the null model.",
"Figure: Original weights of the retained edgesHeterogeneity of node strengths.",
"Figure REF show the strength distributionSmall random noise is added for Eurovision network for visualization purposes.",
"of the dyads for the original networks in transparent color and the extracted backbones with relative size of $\\approx 20\\%$The backbones in Figure REF are extracted with the following ($\\alpha ^-, \\alpha ^+$ ), ($\\beta ^-, \\beta ^+$ ) settings.",
"Migration $\\rightarrow $ (-40, 40), (-0.25, 0.25).",
"Eurovision $\\rightarrow $ (-1.8, 2.4), (-0.5, 0.3).",
"Contact $\\rightarrow $ (-2.25, 2.25), (-0.25, 0.25).",
"Species $\\rightarrow $ (-3.6, 3.6), (-0.25, 0.25).",
"in opaque color.",
"Specifically, the edge between nodes $i$ and $j$ is represented with a point colored based on its sign.",
"X-axis denotes the (out-)strength of $i$ and y-axis denotes the (in-)strength of $j$For an undirected network, $i$ and $j$ might switch arbitrarily.",
"As demonstrated by the figure, the retained edges are rather evenly distributed in the plotting space and the original heterogeneity is respected.",
"The retained edges between very low-strength nodes are usually positive since null expectations are usually low in that regime and it is difficult to conclude whether small-weighted edges in low-strength regimes are statistically significant.",
"Likewise, the edges between very high-strength nodes are usually negative due to the upper bound on vigor implied by the large null expectation and the fixed amount of total weights.",
"On the other hand, in the majority of strength-strength regions, the edge signs exhibit a mixed distribution.",
"Therefore, this is generally a desired property since the proposed method, by itself, does not allow deducing strong conclusions when the data is limited.",
"Figure: Original strength distribution of the retained dyadsConclusion.",
"We can conclude that the heterogeneity given by (i) the multiscale nature of edge weights and (ii) heterogeneous nature of node strengths are respected, and (iii) the proposed filters do not draw strong conclusions when the data is limited."
],
[
"Structure of the backbones",
"Generally speaking, we expect a real-world signed network to show some extent of reciprocity and have its reciprocal edges be of the same sign, and demonstrate sufficient levels of structural balance.",
"Moreover, most real-world networks exhibit a community structure regardless of whether they are signed or not.",
"Therefore, an informative signed backbone should have these characteristics in general and up to a certain extent.",
"We analyze the extracted backbonesThe backbones in Figure REF are extracted with the following ($\\alpha ^-, \\alpha ^+$ ), ($\\beta ^-, \\beta ^+$ ) settings.",
"Migration $\\rightarrow $ (-40, 40), (-0.33, 0.33).",
"Eurovision $\\rightarrow $ (-1, 0.5), (-0.33, 0.1).",
"Contact $\\rightarrow $ (-3, 3), (-0.33, 0.33).",
"Species $\\rightarrow $ (-5,3), (-0.5, 0.1).",
"of the employed networks in terms of reciprocity, structural balance, and community structure.",
".",
"Reciprocity.",
"Reciprocity in unsigned networks is generally defined as the ratio of the number of reciprocal edges to the number of all edges.",
"In signed networks, however, the edge signs should also be considered in the analyses.",
"Table REF presents the number of nodes, edges, nonreciprocal edges ({$\\times $ , $\\cdot $ }), reciprocated positive edges ({+, +}), reciprocated negative edges ({-, -}), and reciprocated edges with sign conflict ({+, -}) for the directed backbones extracted from directed networks.",
"In Migration backbone, there is considerable reciprocity with no conflicting edge signs between any node pair.",
"In contrast, Eurovision backbone discloses a substantial amount of conflicting edge pairs which might be of interest for further investigation into its voting dynamics (e.g., the influence of diasporas).",
"Table: Reciprocity in extracted directed backbonesFor the rest of the analysis, the directed backbones are transformed into undirected backbones in the following way.",
"For Migration backbone, directed edges are transformed into undirected edges of the same sign with positive signs having a priority over negative signs when in conflict.",
"For Eurovision backbone, a positive (or negative) edge is created between two nodes when there are exactly two positive (or negative) directed edges between them (i.e., reciprocal edges of the same sign).",
"Structural Balance.",
"As put forward by Heider [38] and formalized for signed networks by Cartwright and Harary [14], an undirected triple is said to be balanced if its edges have the signs {+, +, +} or {+, -, -} and unbalanced if its edges have the signs {+, +, -} or {-, -, -}.",
"The balanced triples can be simply described with the following expressions: friend of my friend is my friend and enemy of my friend is my enemy.",
"Davis [20] defines a weaker notion and proposes that enemy of my friend is my enemy is not necessarily required for balance and the only unbalanced triple among the four possible settings is the one with edge signs {+, +, -}.",
"Accordingly, structural balance (SB) and weak structural balance (WSB) of a network can be defined as the ratio of the number of balanced triples to the number of all triples.",
"Frustration index (FI) [2] is a measure of the minimum number of required edges such that upon their deletion the network is partitioned into two communities with no positive edges between the communities and with no negative edges within communities.",
"The number of such edges is scaled so that a fully balanced network of two communities results in a FI of 1.",
"Table REF shows the count of nodes, edges, and four possible triple settings; and three different structural balance measures for the undirected backbones.The number of triples and frustration index is calculated using signnet [69] package in R. The backbones for Migration and Species shows a strong balance for all three measures.",
"The other two backbones are also highly balanced in terms of weak structural balance which is shown to be more appropriate in real-world networks.",
"Moreover, it should be highlighted that in the calculation of the frustration index the network is partitioned into only two communities whereas there are multiple underlying communities for three of the networks as discussed next.",
"Table: Structural characteristics of extracted backbonesCommunity Structure.",
"Based on the partitioning methods for signed networks [23], the extracted backbones are partitioned into communities where the number of negative edges within communities and number of positive edges between communities are minimized via heuristicsCommunities are found and visualized using signnet [69] package in R. Figure REF depicts the community structure via a visual block matrix where the order of nodes is the same for its rows and columns.",
"Diagonal blocks are expected to cover the positive edges and non-diagonal blocks are expected to cover the negative edges.",
"Overall, the extracted backbones manifest community structures.",
"The backbone of Migration has 5 clear densely connected communities and two outlier individual nodes whereas the communities in Eurovision are not as dense.",
"The backbone of Contact consists of one large sparse group and several relatively-small denser groups.",
"The backbone of Species has a very clear structure with two similar-sized communities, visually confirming the very high frustration index ($0.96$ ) obtained for it.",
"Figure: Block models of extracted backbonesTable REF shows members of each group in the order of diagonal blocks where nodes are also listed in the same order as they appear in the rows and columns.",
"We can observe that the states are grouped mostly based on their geographical proximity except for two outlier states which are not part of any groups.",
"Similarly, countries in the Eurovision backbone are grouped mainly based on geographic, cultural, and historical ties.",
"Although the results are seemingly plausible, a sophisticated and conclusive discussion on the found communities and its usefulness is spared for future studies in those domains.",
"Table: Block membershipsConclusion.",
"We showed that the extracted backbones usually (i) reveal interesting reciprocal structures, (ii) are structurally balanced, and (iii) exhibit clear community structure."
],
[
"Conclusion and Future Work",
"In this study, we put forward an initial discussion on intrinsically dense networks and provide significance filter and vigor filter for extracting signed backbones of such networks.",
"Empirical evaluations that utilize the proposed filters on a variety of real-world networks show that sparse backbones can be obtained while maintaining comparable numbers of positive and negative links and respecting the original weight and strength heterogeneity.",
"In general, the obtained backbones exhibits characteristics associated with signed networks such as reciprocity, structural balance, and community structure.",
"The extraction method, by design, does not prematurely arrive at conclusions regarding the existence of signed links between nodes when the data is limited.",
"On the other hand, choosing appropriate hyperparameter values lies with the user and should be guided by empirical analysis and recommendations provided in this work as well as the purpose and the nature of the specific problem.",
"Further studies can improve this work in two major ways.",
"As we presented many examples throughout the manuscript, intrinsically dense networks exist in many different domains.",
"First, a stream of its applications in different fields and utilization of the resulting backbones in different tasks would provide important feedback regarding its usefulness, weaknesses, and strengths.",
"Second, hypergeometric distribution characterizes a discrete process and such discrete distributions are being utilized in the state-of-the-art backbone extraction methods including ours.",
"As we discussed earlier, null models developed based on it can have certain undesirable properties for certain networks.",
"Although such issues are easy to handle for practical purposes, a natural avenue is to develop new null models that are more appropriate for the case of link weights which are continuous or that can be equivalently represented in different units."
]
] | 2012.05216 |
[
[
"Simple is not Easy: A Simple Strong Baseline for TextVQA and TextCaps"
],
[
"Abstract Texts appearing in daily scenes that can be recognized by OCR (Optical Character Recognition) tools contain significant information, such as street name, product brand and prices.",
"Two tasks -- text-based visual question answering and text-based image captioning, with a text extension from existing vision-language applications, are catching on rapidly.",
"To address these problems, many sophisticated multi-modality encoding frameworks (such as heterogeneous graph structure) are being used.",
"In this paper, we argue that a simple attention mechanism can do the same or even better job without any bells and whistles.",
"Under this mechanism, we simply split OCR token features into separate visual- and linguistic-attention branches, and send them to a popular Transformer decoder to generate answers or captions.",
"Surprisingly, we find this simple baseline model is rather strong -- it consistently outperforms state-of-the-art (SOTA) models on two popular benchmarks, TextVQA and all three tasks of ST-VQA, although these SOTA models use far more complex encoding mechanisms.",
"Transferring it to text-based image captioning, we also surpass the TextCaps Challenge 2020 winner.",
"We wish this work to set the new baseline for this two OCR text related applications and to inspire new thinking of multi-modality encoder design.",
"Code is available at https://github.com/ZephyrZhuQi/ssbaseline"
],
[
"Introduction",
"To automatically answer a question or generate a description for images that require scene text understanding and reasoning has broad prospects for commercial applications, such as assisted driving and online shopping.",
"Equipped with these abilities, a model can help drivers decide distance to the next street or help customers get more details about a product.",
"Two kinds of tasks that focus on text in images have recently been introduced, which are text-based visual question answering (TextVQA) [34], [7] and text-based image captioning (TextCaps) [33].",
"For example, in Figure REF , a model is required to answer a question or generate a description by reading and reasoning the texts “tellus mater inc.” in the image.",
"These two tasks pose a challenge to current VQA or image captioning models as they explicitly require understanding of a new modality – Optical Character Recognition (OCR).",
"A model must efficiently utilize text-related features to solve these problems.",
"Figure: An example of TextVQA and TextCaps tasks.",
"The answer and description are generated by our model.",
"Our simple baseline is able to read texts and answer related questions.",
"Besides, it can also observe the image and generate a description with texts embedded in it.Figure: Encoders of different models.",
"(a) Current state-of-the-art model M4C on TextVQA task forwards each feature vector of all modalities indiscriminately into transformer layers, which exhaust tremendous computation.",
"(b) MM-GNN handcrafts three graphs to represent the image and applies three aggregators step by step to pass messages between graphs.",
"(c) SMA introduces a heterogeneous graph and considers object-object, object-text and text-text relationships, upon which a graph attention network is then used to reason over them.",
"(d) Our baseline uses three vanilla attention blocks to highlight most relevant features and combines them into six individually-functioned vectors, which is then sent into transformer-based fusion encoders.",
"The considerably fewer parameters of six vectors save computation.For TextVQA task, the current state-of-the-art model M4C [17] handles all modalities (questions, visual objects and OCR tokens) over a joint embedding space.",
"Although this homogeneous method is easy to implement, fast to train and has made great headway, it considers that texts and visual objects contribute indiscriminately to this problem and uses text features as a whole.",
"For TextCaps problem, the only difference is that it only has two modalities: visual objects and OCR tokens.",
"However, these limitations remain.",
"Some other works proposed even more complex structures to encode and fuse multi-modality features of this task, i.e., questions, OCR tokens and images.",
"For example, SMA [13] uses a heterogeneous graph to encode object-object, object-text and text-text relationships in the image, and then designs a graph attention network to reason over it.",
"MM-GNN [14] represents an image as three graphs and introduces three aggregators to guide message passing from one graph to another.",
"In this paper, we use the vanilla attention mechanism to fuse pairwise modalities.",
"Under this mechanism, we further use a more reasonable method to utilize text features which leads to a higher performance, that is by splitting text features into two functionally different parts, i.e., linguistic- and visual-part which flows into corresponding attention branch.",
"The encoded features are then sent to a popularly-used Transformer-based decoder to generate answers or captions.",
"As compared to the aforementioned M4C models (shown in Figure REF a) that throw each instance of every modality into transformer layers, our model (in Figure REF d) first uses three attention blocks to filter out irrelevant or redundant features and aggregate them into six individually-functioned vectors.",
"In contrast to hundreds of feature vectors in M4C, the six vectors consume much less computation.",
"Moreover, to group text features into visual- and linguistic- parts is more reasonable.",
"When comparing with the graph-based multi-modal encoders such as MM-GNN (Figure REF b) and SMA (Figure REF c), our baseline is extremely simple in design and reduces much space and time complexity.",
"Figure: A simple baseline model for TextVQA.",
"Given an image and a question, we prepare three features (OCR visual-part, OCR linguistic-part and object features) and three question self-attention outputs.",
"The six sequences are put into attention block and fused into six vectors, upon which we calculate element-wise product two by two to get concatenated embeddings.",
"The encoder outputs predict the first word and the rest of answer is produced by an iterative decoder.In addition, for the first time we ask the question: to what extent OCRs contribute to the final performance of TextVQA, in contrast to the other modality - visual contents such as objects and scenes?",
"An interesting phenomenon is observed that OCRs play an almost key role in this special problem while visual contents only serve as assisting factors.",
"A strong model without the use of visual contents surpasses current state-of-the-art model, demonstrating the power of proposed pairwise fusion mechanism and primary role of texts.",
"To demonstrate the effectiveness of our proposed simple baseline model, we test it on both TextVQA [34] and TextCaps [33] tasks.",
"For the TextVQA, we outperforms the state-of-the-art (SOTA) on TextVQA dataset and all three tasks of ST-VQA, and rank the first on both leaderboardshttps://eval.ai/web/challenges/challenge-page/551/leaderboard/1575https://rrc.cvc.uab.es/?ch=11&com=evaluation&task=1.",
"More importantly, all compared SOTA models use the similar Transformer decoder with ours, but with far more complex encoding mechanisms.",
"For TextCaps, we surpass the TextCaps Challenge 2020 winner and now rank the first place on the leaderboardhttps://eval.ai/web/challenges/challenge-page/573/leaderboard/1617.",
"Overall, the major contribution of this work is to provide a simple but rather strong baseline for the text-based vision-and-language research.",
"This could be the new baseline (backbone) model for both TextVQA and TextCaps.",
"More importantly, we wish this work to inspire a new thinking of multi-modality encoder design – simple is not easy."
],
[
"Related Work",
"Text based visual question answering.",
"VQA [5], [18], [19], [39] has seen rapid development in recent years.",
"A new task – TextVQA goes one step further and aims at the understanding and reasoning of scene texts in images.",
"A model needs to read texts first and then answer related questions in natural everyday situations.",
"Two datasets, TextVQA [34] and ST-VQA [7] are introduced concurrently to benchmark progress in this field.",
"To solve this problem, various methods have also been proposed.",
"LoRRA, the baseline model in TextVQA, uses bottom-up and top-down [3] attention on visual objects and texts to select an answer from either vocabulary or fixed-index OCR.",
"M4C [17] is equipped with a vanilla transformer decoder to iteratively generate answers and a flexible pointer network to point back at most probable OCR token at one decoding step.",
"MM-GNN [14] designs a representation of three graphs and introduces three aggregators to update message passing for question answering.",
"Text based image captioning.",
"Image captioning challenges a model to automatically generate a natural language description based on the contents in an image.",
"Existing datasets, e.g., COCO Captions [8] and Flickr30k [43], focus more on visual objects.",
"To enhance text comprehension in the context of an image, a new dataset called TextCaps [33] is proposed.",
"It requires a model to read and reason about texts and generate coherent descriptions.",
"The baseline model in TextCaps is modified from aforementioned M4C slightly, by removing question input directly.",
"Generative transformer decoder.",
"To address the problem that answers in these two text-based tasks are usually concatenated by more than one word, we use the structure of transformer [11] decoder in answer module.",
"Following previous work, we also use the generative transformer decoder for fair comparison."
],
[
"Proposed Method",
"Given the three modalities (questions, OCR tokens, visual objects), the first step of our model is to prepare the features by projecting them into the same dimension.",
"Then we describe the formulation of Attention Block for feature summarizing.",
"Stacking the blocks together yields encoder for downstream tasks.",
"Using encoder output to produce the first word in answer, we then use an iterative decoder to predict the rest of words.",
"This whole process is shown in Figure REF .",
"When transferring to TextCaps, we only make minimal modifications which will be detailed in Section REF .",
"Notation In the remainder of this paper, all $\\mathbf {W}$ are learned linear transformations, with different symbols to denote independent parameters, e.g., $\\mathbf {W}_{fr}$ .",
"$\\mathrm {LN}$ is Layer Normalization [6].",
"$\\circ $ represents element-wise product."
],
[
"Feature Preparation",
"Question features.",
"For a question with $L$ words, we first use a three-layer BERT [11] model to embed it into $Q = \\lbrace q_{i}\\rbrace _{i=1}^{L}$ .",
"This BERT model is finetuned during training.",
"OCR features.",
"In text-based VQA and image captioning problem, texts are of key importance.",
"Simply gather every feature of text together is not efficient enough.",
"When faced with a bunch of OCRs, human recognition system tends to use two complementary methods to select subsequent words, either by finding similarly-looking and spatially-close words or choosing words coherent in linguistic meaning.",
"For this intuitive purpose, we split features of $N$ OCR tokens into two parts: Visual and Linguistic.",
"1) OCR visual-part.",
"Visual features are combined by appearance feature and spatial feature as they show what eyes catch of, without further processing of natural language system.",
"From this part, a model can get visual information such as word font, color and background.",
"These two are extracted by an off-the-shelf Faster R-CNN [30] detector.",
"${\\begin{split}& \\mathbf {x}_{i}^{ocr,v} = \\mathrm {LN}( \\mathbf {W}_{fr}\\mathbf {x}_{i}^{ocr,fr})+ \\mathrm {LN}(\\mathbf {W}_{bx}\\mathbf {x}_{i}^{ocr,bx}),\\end{split}}$ where $\\mathbf {x}_{i}^{ocr,fr}$ is the appearance feature extracted from the fc6 layer of Faster R-CNN detector.",
"The fc7 weights are finetuned on our task.",
"$\\mathbf {x}_{i}^{ocr,bx}$ is the bounding box feature in the format of $[x_{tl},y_{tl},x_{br},y_{br}]$ , where $tl$ and $br$ denotes top left and bottom right coordinates respectively.",
"2) OCR linguistic-part.",
"Linguistic features are made up of 1) FastText feature $\\mathbf {x}_{i}^{ocr,ft}$ , which is extracted from a pretrained word embedding and 2) character-level Pyramidal Histogram of Characters (PHOC) [1] feature $x_{i}^{ocr,ph}$ as they contain natural language related information.",
"${\\begin{split}& \\mathbf {x}_{i}^{ocr,l} = \\mathrm {LN}( \\mathbf {W}_{ft}\\mathbf {x}_{i}^{ocr,ft} + \\mathbf {W}_{ph}\\mathbf {x}_{i}^{ocr,ph})\\end{split}}$ 3) OCR additional features.",
"In the SBD-Trans [25], [40] that we use to recognize OCR tokens, the holistic representations in a specific text region are themselves visual features, however, are employed for linguistic word classification purpose.",
"Therefore, they cover both visual and linguistic context of the OCR token and we thus introduce the Recog-CNN feature $\\mathbf {x}_{i}^{ocr,rg}$ from this network to enrich text features.",
"Finally, Recog-CNN features are added to OCR visual- and linguistic-part simultaneously.",
"${\\begin{split}& \\mathbf {x}_{i}^{ocr,v} = \\mathrm {LN}( \\mathbf {W}_{fr}\\mathbf {x}_{i}^{ocr,fr} + \\mathbf {W}_{rg}\\mathbf {x}_{i}^{ocr,rg})+ \\mathrm {LN}(\\mathbf {W}_{bx}\\mathbf {x}_{i}^{ocr,bx})\\\\& \\mathbf {x}_{i}^{ocr,s} = \\mathrm {LN}( \\mathbf {W}_{ft}\\mathbf {x}_{i}^{ocr,ft} + \\mathbf {W}_{ph}\\mathbf {x}_{i}^{ocr,ph} + \\mathbf {W}_{rg}\\mathbf {x}_{i}^{ocr,rg})\\end{split}}$ Visual features.",
"In text-based tasks, visual contents in an image can be utilized to assist textual information in the reasoning process.",
"To prove that our simple attention block has the power of using visual features in various forms, we adopt either grid-based global features or region-based object features.",
"1) Global features.",
"We obtain image global features $x_{i}^{glob}$ from a ResNet-152 [15] model pretrained on ImageNet, by average pooling $2048D$ features from the res-5c block, yielding a $14 \\times 14 \\times 2048$ feature for one image.",
"To be consistent with other features, we resize the feature into $196 \\times 2048$ , a total of 196 uniformly-cut grids.",
"${\\begin{split}& \\mathbf {x}_{i}^{glob} = \\mathrm {LN}( \\mathbf {W}_{g}\\mathbf {x}_{i}^{glob})\\end{split}}$ 2) Object features.",
"The region-based object features are extracted from the same Faster F-CNN model as mentioned in OCR features part.",
"${\\begin{split}& \\mathbf {x}_{i}^{obj} = \\mathrm {LN}( \\mathbf {W}_{fr}^{^{\\prime }}\\mathbf {x}_{i}^{obj,fr})+ \\mathrm {LN}(\\mathbf {W}_{bx}^{^{\\prime }}\\mathbf {x}_{i}^{obj,bx}),\\end{split}}$ where $\\mathbf {x}_{i}^{obj,fr}$ is the appearance feature and $\\mathbf {x}_{i}^{obj,bx}$ is the bounding box feature."
],
[
"Attention Block as Feature Summarizing",
"In tasks that cross the fields of computer vision and natural language processing, modality fusion is of superior importance.",
"Treating them as homogeneous entities in a joint embedding space might be easy to implement, however, is not carefully tailored to a specific problem.",
"Moreover, the many parameters of all entities in the large model (for example, a Transformer) consume much computation.",
"To grasp interaction between modalities for maximum benefit and filter out irrelevant or redundant features before the entering into a large fusion layer, we use a simple attention block to input two sequences of entities and output two processed vectors, which is shown as the Attention Block in Figure REF .",
"The two sequences of entities might be any sequence we want.",
"For TextVQA problem, question changes in real-time and plays a dominant role in final answering.",
"The design of question needs careful consideration and its existence should contribute throughout the process.",
"For example, here we use question as one input of query in attention block.",
"The sequence of question words goes through a self-attention process before forwarding into attention block.",
"First we put the question word sequence $Q = \\lbrace q_{i}\\rbrace _{i=1}^{L}$ through a fully connected feed-forward network, which consists of two linear transformations (or two convolutions with 1 as kernel size) and one ReLU activation function between them.",
"$\\begin{split}&q_{i}^{fc} = \\mathrm {conv}\\lbrace \\mathrm {ReLU}[\\mathrm {conv}(q_{i})]\\rbrace , \\,\\, i = 1, \\dots , L ;\\end{split}$ A softmax layer is the used to compute the attention on each word in the question.",
"$\\begin{split}&a_i = \\mathrm {Softmax}({q_{i}^{fc}}), \\,\\, i = 1, \\dots , L ;\\\\\\end{split}$ This is known as self-attention and these weights are multiplied with original question embedding to get weighted sum of word embeddings.",
"$\\begin{split}&Q^s = \\textstyle {\\sum _{i=1}^{L}} a_i {q}_{i}.\\end{split}$ If we have several individual entities to combine with question, the corresponding number of parallel self-attention processes are performed on the same question with independent parameters.",
"For example, we can get $Q^s_{v}$ , $Q^s_{l}$ and $Q^s_{o}$ for OCR visual-part, OCR linguistic-part and object regions respectively.",
"Then the $Q^s$ are used as query for corresponding features.",
"We calculate the attention weights under the guidance of $Q^s$ , which are then put into a softmax layer.",
"Finally the weights are multiplied with original queried features to get a filtered vector.",
"Here we take the pair of $Q^s_{v}$ and ${\\mathbf {x}}_{i}^{ocr,v}$ as an example: ${\\begin{split}&p_i = \\mathbf {W}[\\mathrm {ReLU}(\\mathbf {W}_{s} Q^s_{v})\\circ \\mathrm {ReLU}(\\mathbf {W}_{x}{x}_i^{ocr,v})],\\\\&s_i = \\mathrm {Softmax}(p_i), \\quad i=1,\\dots ,N,\\\\&g^{ocr,v} = \\textstyle {\\sum _{i=1}^{N}} s_i {\\mathbf {x}}_{i}^{ocr,v}\\end{split}}$ where $g^{ocr,v}$ is the output of attention block.",
"Similarly, we can get $g^{ocr,l}$ for the OCR-linguistic summarizing feature, $g^{obj}$ for the object summarizing feature.",
"Different from M4C sending every single question tokens, OCR tokens and objects into the transformer feature fusion layer, here we only have 6 feature vectors ($Q^s_{v}$ , $Q^s_{l}$ , $Q^s_{o}$ , $g^{ocr,v}$ , $g^{ocr,l}$ and $g^{obj}$ ) which are sent to the following process.",
"This largely decreases the computation complexity and burden, considering that the transformer is a parameter-heavy network."
],
[
"Stacked-Block Encoder",
"The attention block in Section REF can be stacked together as an encoder which produces combined embedding for downstream tasks.",
"TextVQA baseline model.",
"As presented in the above module, questions are sent through self-attention to output $Q^s_{v}$ , $Q^s_{l}$ and $Q^s_{o}$ .",
"OCR features in images are splitted into visual- and linguistic part, which are $x^{ocr,v}$ and $x^{ocr,l}$ .",
"We also have object features $x^{obj}$ .",
"The six sequences are put into three attention blocks and we get six $768D$ vectors which are then forwarded into a fusion encoder.",
"The fusion encoder, OCR encoder and generative decoder in Figure REF are in the same transformer model but undertaking different roles.",
"After fusion encoder processing, the six vectors conduct element-wise multiplication in a pairwise way to get corresponding embeddings which are concatenated together.",
"Then we use a fully-connected layer to transform the concatenated embeddings to a context embedding with appropriate dimension, upon which we generate the first answer output.",
"Given the first answer word, a generative decoder is then used to select the rest of answer, which will be detailed in Section REF and Supplementary Material.",
"Figure: TextCaps baseline model.",
"It has a same structure as TextVQA baseline model.Table: We ablate our model on TextVQA dataset by testing number of attention blocks, forms of visual object features and addition of OCR representations.Table: Comparison to previous work on TextVQA dataset.",
"Our model sets new state-of-the-art with an extremely simple design.TextCaps baseline model.",
"As there are no questions in TextCaps, we use objects to guide OCR visual- and linguistic-part and use OCRs to guide object features.",
"Technically we simply replace question word sequence with OCR token sequence or object proposal feature sequence.",
"The other settings are the same with TextVQA.",
"Figure REF illustrates our Textcaps baseline model.",
"To easily transfer to another task demonstrates the generalization ability and simplicity of our method."
],
[
"Answer Generation Module",
"To answer a question or generate a caption, we use a generative decoder based on transformer.",
"It takes as input the `context embedding' from the above encoder and select the first word of the answer.",
"Based on the first output word, we then use the decoder to find the next word token either from a pre-built vocabulary or the candidate OCR tokens extracted from the given image, based on a scoring function.",
"Training Loss.",
"Considering that the answer may come from two sources, we use multi-label binary cross-entropy (bce) loss: ${\\begin{split}pred &= \\frac{1}{1+\\exp {(-y_{pred})}}, \\\\\\mathcal {L}_{bce} &= -y_{gt} \\mathrm {log}(pred)-(1-y_{gt}) \\mathrm {log}(1-pred),\\end{split}}$ where $y_{pred}$ is prediction and $y_{gt}$ is ground-truth target.",
"Additional Training Loss.",
"In some cases, the model reasons correctly, however, picks slightly different words than what we expected due to defective reading (OCR) ability.",
"To take advantage of these predictions, we introduce a new policy gradient loss as an auxilliary task inspired by reinforcement learning.",
"In this task, the greater reward, the better.",
"We take Average Normalized Levenshtein Similarity (ANLS) metric$\\mathrm {ANLS}(s_{1},s_{2})=1-d(s_{1},s_{2})/\\mathrm {max}(\\mathrm {len}(s_{1}),\\mathrm {len}(s_{2}))$ , d(,) is edit distance.",
"as the reward which measures the character similarity between predicted answer and ground-truth answer.",
"$\\begin{split}&r = \\mathrm {ANLS}(\\phi (y_{gt}),\\phi (y_{pred})), \\\\&y = \\mathbb {I}(\\mathrm {softmax}(y_{pred})), \\\\&\\mathcal {L}_{pg} = (0.5 - r) (y_{gt} \\mathrm {log}(y) + (1-y_{gt}) \\mathrm {log}(1-y)),\\\\&\\mathcal {L} = \\mathcal {L}_{bce}+\\alpha \\cdot \\mathcal {L}_{pg},\\end{split}$ where $\\phi $ is a mapping function that returns a sentence given predicting score (e.g., , $y_{pred}$ ), ANLS($\\cdot $ ) is used to calculate similarity between two phrases, $\\mathbb {I}$ is an indicator function to choose the maximum probability element.",
"The additional training loss is a weighted sum of $\\mathcal {L}_{bce}$ and $\\mathcal {L}_{pg}$ , where $\\alpha $ is a hyper-parameter to control the trade-off of $\\mathcal {L}_{pg}$ .",
"After introducing policy gradient loss, our model is able to learn fine-grained character composition alongside linguistic information.",
"We only apply this additional loss on ST-VQA dataset, which brings roughly $1\\%$ improvement."
],
[
"Experiments",
"Extensive experiments are conducted across two categories of tasks: TextVQA and TextCap.",
"For TextVQA we set new state-of-the-art on TextVQA dataset and all three tasks of ST-VQA.",
"For TextCaps we surpass 2020 TextCaps Challenge winner.",
"See more experiments details below."
],
[
"Implementation Details",
"The set of methods are built on top of PyTorch.",
"We use Adam as the optimizer.",
"The learning rate for TextVQA and TexCaps is set to $1e-4$ .",
"For TextVQA we multiply the learning rate with a factor of $0.1$ at the $14,000$ and $15,000$ iterations in a total of $24,000$ iterations.",
"For TextCaps the multiplication is done at the $3,000$ and $4,000$ iterations, with $12,000$ total iterations.",
"We set the maximum length of questions to $L = 20$ .",
"We recognize at most $N = 50$ OCR tokens and detect at most $M = 100$ objects.",
"The maximum number of decoding steps is set to 12.",
"Transformer layer in our model uses 12 attention heads.",
"The other hyper-parameters are the same with BERT-BASE [11].",
"We use the same model on TextVQA and three tasks of ST-VQA, only with different answer vocabulary, both with a fixed size of 5000."
],
[
"Ablation Study on TextVQA Dataset",
"TextVQA [34] is a popular benchmark dataset to test scene text understanding and reasoning ability, which contains $45,336$ questions on $28,408$ images.",
"Diverse questions involving inquires about time, names, brands, authors, etc., and dynamic OCR tokens that might be rotated, casual or partially occluded make it a challenging task.",
"We first conduct an experiment of building only one block, with one question self-attention output guiding the whole set of text features as a comparison.",
"This is a one-block model in TableREF which does not perform as good as the state-of-the-art.",
"To investigate how well text features can perform without the usage of visual grid-based or region-based features, we build a two-block model.",
"Given the two categories of OCR features – visual and linguistic, we find that our simple model is already able to perform promisingly on TextVQA problem.",
"Line 1 and Line 2 tell clearly the validity of sorting text features into two groups ($1.77\\%$ difference).",
"When building our third block on the basis of visual contents, either global features or object-level features are at our disposal.",
"The incorporation of a third block has modest improvements ($0.24\\%$ for global and $0.63\\%$ for object features).",
"From Line 4 to Line 5, a new Recog-CNN feature is added to enrich text representation and brings $0.37\\%$ improvement.",
"We also use more transformer layers (from 4 to 8) and get $0.1\\%$ higher result.",
"Then we use a much better OCR system (especially on recognition part) and obtain large performance boost (from $40.38\\%$ to $43.95\\%$ ).",
"Table: Comparison to previous work on ST-VQA dataset.",
"With TextVQA pretraining, our model outperforms current approaches by a large margin.Figure: Qualitative examples of our baseline model in contrast to M4C.",
"While our model can read texts in accordance with written language system, M4C can only pick tokens in a random and errorneous way.Qualitative examples.",
"We present four examples in Figure REF where our model shows the ability to really read scene texts in a way similar to humans, i.e., left-to-right then top-to-bottom.",
"In contrast, current state-of-the-art model M4C fails to read tokens in a correct order."
],
[
"Comparison with state-of-the-art",
"TextVQA dataset.",
"Even stripped of the usage of visual contents in the image, our two-block model already surpasses current state-of-the-art M4C by $0.98\\%$ on test set (Line 2 in Table REF VS. Line 4 in Table REF ).",
"Using the same OCR system, our baseline model further improves upon M4C by $0.98\\%$ on val and $1.91\\%$ on test (Line 7 VS. Line 4 in Table REF ).",
"Compared to the top entries in TextVQA Challenge 2020, our baseline has a significantly simpler model design, especially on the encoder side.",
"M4C and SA-M4C take all parameters of entities into transformer layers and join large amout of computation.",
"SMA uses a heterogeneous graph to explicitly consider different nodes and compute attention weights on 5-neighbored graph.",
"Our model that surpasses all of them only sends six holistic vectors two-by-two into transformer layers, which tremendously saves computation.",
"ST-VQA dataset.",
"The ST-VQA dataset [7] is another popular dataset with three tasks, which gradually increase in difficulty.",
"Task 1 provides a dynamic candidate dictionary of 100 words per image, while Task 2 provides a fixed answer dictionary of $30,000$ words for the whole dataset.",
"As for Task 3, however, the model are supposed to generate answer without extra information.",
"We also evaluate our model on ST-VQA dataset, using the same model from TextVQA for all three tasks.",
"Without any additional training data, our model achieved the highest on Task 2 and Task 3 (Line 4 in Table REF ).",
"Using TextVQA dataset as additional training data, our model sets new state-of-the-art on all three tasks and outperforms current approaches by a large margin.",
"Table: Results on TextCaps dataset.",
"(B: BLEU-4; M: METEOR; R: ROUGE_L; S: SPICE; C: CIDEr)"
],
[
"TextCaps Dataset",
"TextCaps is a new dataset that requires a model to read texts in images and generate descriptions based on scene text understanding and reasoning.",
"In TextCaps, automatic captioning metrics (BLEU [28], METEOR [10], ROUGE_L [23], SPICE [2] and CIDEr [36]) are compared with human evaluation scores.",
"All automatic metrics show high correlation with human scores, among which CIDEr and METEOR have the highest.",
"M4C-Captioner is the method provided in TextCaps, which is modified from M4C model by simply removing question input.",
"Similarly, simply replacing question in our TextVQA baseline model with object or OCR sequence yields our TextCaps baseline model.",
"Using exactly the same OCR system, OCR representations, our baseline with Rosetta-en OCR (Line 2 on Table REF ) already surpass M4C-Captioner (Line 1 on Table REF ), especially on BLUE-4 and CIDEr metric.",
"By upgrading our OCR system to SBD-Trans and using 6-layer transformer in our encoder-decoder structure, our baseline further exceeds TextCaps Challenge Winner on BLUE-4 and CIDEr metric as shown in Line 5 and Line 6 of Table REF .",
"[h] Answer generation in TextVQA Question features $Q$ , OCR features $x^{ocr}$ , Object features $x^{obj}$ An answer with a length of $T$ words $A=\\lbrace a_{i}\\rbrace _{i=0}^{T-1}$ Obtain three independent question signals: $(Q^s_{v}, Q^s_{l}, Q^s_{o}) = \\mathrm {Self\\_Attention}^{\\lbrace v,l,o\\rbrace }(Q)$ Split OCR features into OCR visual- and linguistic-part ($x^{ocr,v}$ , $x^{ocr,l}$ ) Forward question signals and three features into three Attention Blocks in a pairwise way: $g^{ocr,v} = \\mathrm {Att\\_Blk}(Q^s_{v}, x^{ocr,v});\\ g^{ocr,l} = \\mathrm {Att\\_Blk}(Q^s_{l}, x^{ocr,l});\\ g^{obj} = \\mathrm {Att\\_Blk}(Q^s_{o}, x^{obj})$ Forward six vectors into fusion encoder: $\\bar{Q}^s_{v}, \\bar{Q}^s_{l}, \\bar{Q}^s_{o}, \\bar{g}^{ocr,v}, \\bar{g}^{ocr,l}, \\bar{g}^{obj} = \\mathrm {Fusion\\_Encoder}(Q^s_{v}, Q^s_{l}, Q^s_{o}, g^{ocr,v}, g^{ocr,l}, g^{obj})$ Element-wise multiply the six vectors two by two: $\\mathbf {e}^{ocr,v}=\\bar{Q}^s_{v}\\circ \\bar{g}^{ocr,v};\\ \\mathbf {e}^{ocr,l}=\\bar{Q}^s_{l}\\circ \\bar{g}^{ocr,l};\\ \\mathbf {e}^{obj}=\\bar{Q}^s_{o}\\circ \\bar{g}^{obj}$ Concatenate the three results together and transform it into context embedding $\\mathrm {context\\, embedding}=\\mathrm {Linear}([\\mathbf {e}^{ocr,v}; \\mathbf {e}^{ocr,l}; \\mathbf {e}^{obj}])$ $t = 0$ Based on context embedding, calculate the first word $a_{t}$ in the answer: 1) Map context embedding to a $5000D$ vector through a linear layer as scores for vocabulary 2) Get $50D$ dot product between context embedding and all 50 OCR encoder outputs as scores for OCR tokens 3) Concatenate $5000D$ vector and $50D$ vector together, from which select the highest-score index as $a_{t}$ $t=1;t \\le T-1;t++$ the previous output $a_{t-1}$ is from vocabulary Forward linear layer weights corresponding to the previous output $a_{t-1}$ into generative decoder and obtain decoder embedding the previous output $a_{t-1}$ is from OCR tokens Forward OCR features of the previous output $a_{t-1}$ into generative decoder and obtain decoder embedding Based on decoder embedding, calculated the current word $a_{t}$ in the answer: 1) Map decoder embedding to a $5000D$ vector through a linear layer as scores for vocabulary 2) Get $50D$ dot product between decoder embedding and all 50 OCR encoder outputs as scores for OCR tokens 3) Concatenate $5000D$ vector and $50D$ vector together, from which select the highest-score index as $a_{t}$ return $A=\\lbrace a_{i}\\rbrace _{i=0}^{T-1}$ Table: Computational complexity of two model encoders.",
"L=20L=20 is the length of question; N=50N=50 is the number of OCR tokens; M=100M=100 is the number of detected objects.",
"Here we omit all vector dimensions DD for simplicity.",
"In transformer encoder of our model, apart from the six vectors, we also input 50 OCR tokens to compute dot product value in answer choosing."
],
[
"Conclusion",
"In this paper, we provide a simple but rather strong baseline for the text-based vision-and-language research.",
"Instead of handling all modalities over a joint embedding space or via complicated graph structural encoding, we use the vanilla attention mechanism to fuse pairwise modalities.",
"We further split text features into two functionally different parts, i.e., linguistic- and visual-part which flow into corresponding attention branch.",
"We evaluate our simple baseline model on TextVQA, ST-VQA and TextCaps, all leading to the best performance on the public leaderboards.",
"This sets the new state-of-the-art and our model could be the new backbone model for both TextVQA and TextCaps.",
"What's more, we believe this work inspires a new thinking of the multi-modality encoder design."
],
[
"Algorithm",
"Algorithm REF provides the pseudo-code for the process of answer generation in the TextVQA problem.",
"The description generation process in the TextCaps problem is roughly the same, except that the decoding step $T$ is set to 30 rather than 12 for a longer and more expressive image description.",
"Besides, the question features are also replaced by OCR features and Object features to guide each other."
],
[
"Comparison of Computation Complexity between Encoders",
"As our model and M4C have the same generative decoder, we only compare encoders in Table REF .",
"In one Attention Block, main operations are element-wise multiplication and final matrix multiplication, which yield $O(n \\cdot d)=2n \\cdot d$ operations, where n is number of queried features.",
"In Transformer encoder, the complexity is $O(n^2 \\cdot d)$ , where $n$ is the sequence length and $d$ is the representation dimension."
],
[
"Additional Experiment Details",
"OCRs and Objects self-attention in TextCaps.",
"Different from question self-attention in TextVQA problem that uses a BERT model to capture the sequence information, for OCRs and Objects self-attention in TextCaps we use an LSTM [16] layer to capture the recurrent features of isolate regions before the self-attention calculation.",
"SBD-Trans training data.",
"The SBD model is pretrained on a $60k$ dataset, which consists of $30,000$ images from LSVT [35] training set, $10,000$ images from MLT 2019 [27] training set, $5,603$ images from ArT [9] , and $14,859$ images selected from a bunch of datasets – RCTW-17 [32], ICDAR 2013 [22], ICDAR 2015 [21], MSRA-TD500 [41], COCOText [37], and USTB-SV1K [42].",
"The model was finally finetuned on MLT 2019 training set.",
"The robust transformer based network is trained on the following datasets: IIIT 5K-Words [26], Street View Text [38], ICDAR 2013, ICDAR 2015, Street View Text Perspective [29], CUTE80 [31] and ArT."
],
[
"Additional Qualitative Examples",
"To demonstrate that our model not only learns fixed superficial correlation between questions and prominent OCR tokens, we present several images in the dataset with two different questions to which our model both answers correctly, while other models such as M4C fails, in Figure REF .",
"TextCaps examples are in Figure REF .",
"Figure: Qualitative examples of our model on TextCaps.Figure: Examples that our model fails at on TextVQA."
],
[
"Failure Cases on TextVQA",
"The questions that our model struggles at can be categorized into several groups.",
"Firstly, due to the inappropriate tricky design of the dataset, models are required to read the time, find a specific letter in a word, or read a ruler, e.g., Figure REF a), b) and c).",
"Secondly, even though our model answers reasonably, it does not lie in the region of ground-truth answers as shown in Figure REF d).",
"Thirdly, the mistake might occur because of defective reading ability that OCR tokens do not include answer candidates at all, as shown in Figure REF e).",
"Fourthly, it is difficult to discern the color of tokens as shown in Figure REF f).",
"Finally, when it comes to nuanced relationship understanding, our model is not yet complex enough to reason as shown in Figure REF g) and h)."
]
] | 2012.05153 |
[
[
"Cosmological constraints with the Effective Fluid approach for Modified\n Gravity"
],
[
"Abstract Cosmological constraints of Modified Gravity (MG) models are seldom carried out rigorously.",
"First, even though general MG models evolve differently (i.e., background and perturbations) to the standard cosmological model, it is usual to assume a $\\Lambda$CDM background.",
"This treatment is not correct and in the era of precision cosmology could induce undesired biases in cosmological parameters.",
"Second, neutrino mass is usually held fixed in the analyses which could obscure its relation to MG parameters.",
"In a couple of previous papers we showed that by using the Effective Fluid Approach we can accurately compute observables in fairly general MG models.",
"An appealing advantage of our approach is that it allows a pretty easy implementation of this kinds of models in Boltzmann solvers (i.e., less error--prone) while having a useful analytical description of the effective fluid to understand the underlying physics.",
"This paper illustrates how an effective fluid approach can be used to carry out proper analyses of cosmological constraints in MG models.",
"We investigated three MG models including the sum of neutrino masses as a varying parameter in our Markov Chain Monte Carlo analyses.",
"Two models (i.e., Designer $f(R)$ [DES-fR] and Designer Horndeski [HDES]) have a background matching $\\Lambda$CDM, while in a third model (i.e., Hu $\\&$ Sawicki $f(R)$ model [HS]) the background differs from the standard model.",
"In this way we estimate how relevant the background is when constraining MG parameters along with neutrinos' masses.",
"We implement the models in the popular Boltzmann solver CLASS and use recent, available data (i.e., Planck 2018, CMB lensing, BAO, SNIa Pantheon compilation, $H_0$ from SHOES, and RSD Gold-18 compilation) to compute tight cosmological constraints in the MG parameters that account for deviation from the $\\Lambda$CDM model.",
"[abridged]"
],
[
"Introduction ",
"The growing evidence for the late-time accelerating expansion of the Universe represents a major milestone in cosmology [1], [2], [3], [4] and investigations using Machine Learning algorithms have confirmed this fact through model independent analyses [5], [6], [7].",
"Bayesian analyses indicate that the standard model of cosmology $\\Lambda $ CDM is preferred over its alternatives due to its simplicity and lower number of free parameters [8].",
"The concordance model is relatively simple and having just six free parameters is able to make predictions which agree remarkably well with most observations [3], [4], [9], [10].",
"Nevertheless, the cosmological constant problem, our current ignorance on the nature of Dark Matter (DM) as well as a worrying discordance in a few cosmological parameters represent big disadvantages for the $\\Lambda $ CDM model.",
"Over time the disagreement between the Hubble constant determined via distance ladder and the $H_0$ value obtained through analyses of the CMB has become more interesting [11].",
"Although the discrepancy could be due to unaccounted-for systematic errors, there also exists the more appealing possibility of new physics (e.g.",
"Modified Gravity, early Dark Energy).",
"With the coming of latest data analyses disagreements on the values of $\\Omega _m$ and $\\sigma _8$ also became apparent [12] or for example, a $\\sim 4\\sigma $ deviation of the dark energy equation of state $w(z)$ from the $\\Lambda $ CDM model using quasars at high redshift up to $z\\sim 7.5$ [13].",
"The curvature of the Universe has also given rise to a discussion on recent analyses [14], [3], [15], [16].",
"There are some mild hints ($\\approx 2\\sigma $ ) of Modified Gravity (MG) [3], [17], which challenge assumptions made in the standard model.",
"Here we will investigate some viable MG models in light of latest data releases.",
"Dark Energy (DE) and MG have emerged as alternatives to the cosmological constant model.",
"In the case of DE, some of the simplest models include minimally coupled scalar fields in the form of quintessence, that has a standard kinetic term, or k-essence which has a generalized kinetic-term [18].",
"On the other hand, MG models are covariant modifications to General Relativity (GR) that extend the Einstein-Hilbert action in various ways: either by promoting it to a function as in the $f(R)$ models and by introducing higher order curvature invariants (see for example [19]) or by adding non-minimally coupled scalar fields, as in the case of Horndeski theory [20].",
"MG models have the advantage that they are also inspired from high-energy physics, as covariant modifications to GR of a similar form appear naturally when one tries to renormalize GR at one loop order [21].",
"The advantages of these alternatives, especially of the MG models, are clear: there is no need for a cosmological constant as the accelerating expansion of the Universe can be explained by the presence of the extra terms in the modified Friedmann equations due to the new degrees of freedom.",
"Both DE and MG models are also able to describe well the cosmological observations and give equally good fits to the data as the $\\Lambda $ CDM model.",
"On the other hand, there are also some disadvantages for these models, for example the presence of the additional parameters may penalize some of the models when one calculates the Bayesian evidence and uses the Jeffreys' scale, albeit it has been shown that the latter has to be interpreted with care [22].",
"Furthermore, by using N-body simulations it has been shown that a compelling $f(R)$ model fails to reproduce the observed redshift-space clustering on scales $\\sim 1-10 ~ \\mathrm {Mpc}~ h^{-1}$ [23].",
"Finally, as both the extra scalar field degrees of freedom and the higher order corrections to GR are as yet unobserved in a laboratory or in an astrophysical setting, their motivation is obviously somewhat weakened.",
"Not only that, but recently several MG models of the Horndeski type have been ruled out, via the measurement of the speed of propagation of the gravitational waves by the event GW170817 and its optical counterpart GRB170817A [24], [25], [26].",
"Thus, as the available parameter space has shrank remarkably, there are a few remaining models which deserve attention as well as proper analyses.",
"However, many analyses of the remaining models, especially the ones where the background expansion differs significantly from the $\\Lambda $ CDM model, do not consider the background expansion properly and just fix it to either the $\\Lambda $ CDM or a constant $w$ model, as was observed in Refs.",
"[27], [28].",
"This obviously biases the results as it introduces biases in the cosmological parameters and spurious tensions with the data.",
"However, some recent analyses have also acknowledged this discrepancy and newer versions of the Boltzmann solvers now have support for the correct backgrounds in some cases [29].",
"On the other hand, the so-called Effective Fluid Approach has the advantage that it presents a unified approach to analyse all models under the same umbrella, allows for the correct background expansion in the models, all without sacrificing the accuracy of the results [27], [28], [30].",
"In a nutshell, the Effective Fluid Approach works by rewriting the field equations of the MG model as GR and a DE fluid with an equation of state $w$ , a pressure perturbation $\\delta P$ and an anisotropic stress $\\sigma $ .",
"Especially the latter is crucial as sometimes it is ignored in analyses of MG models [31], something which might bias the results [32].",
"Moreover, through a joint Machine Learning analysis applied to the latest cosmological data hints of dark energy anisotropic stress were found [7].",
"In the Effective Fluid Approach we also assume that in the relevant scales, where linear theory applies, the sub-horizon and quasi-static approximations hold.",
"With these, general analytical expressions for the equation of state, the pressure perturbation, and the anisotropic stress were found in Refs.",
"[27], [28].",
"With the latter, one may then just solve numerically the evolution equations for the perturbations, found for example in Ref. [33].",
"Thus, the main advantage of the Effective fluid approach is that once one has the expressions for the variables $w$ , $\\delta P$ and $\\sigma $ , it is very straightforward to also implement them in standard Boltzmann codes, such as CLASS, with very minimal modifications.",
"In fact, in [27], [28], this was done with the EFCLASS code, which implements the aforementioned approach, where it was found that EFCLASS and hi-CLASS [34], a modification of CLASS that solves numerically the whole set of perturbation equations for Horndeski, agree to better than $0.1\\%$ [28].",
"Recently, a comparison of different approaches to the quasi-static approximation in Horndeski models was made by Ref.",
"[35], by applying this approximation to either the field equations, as done in the Effective fluid approach, or the equations for the two metric potentials $\\Phi $ and $\\Psi $ and finally, the use of the attractor solution derived within the Equation of State approach [36].",
"It was found that all three approaches agree exactly on small scales and that in general, this approach is valuable in future model selections analyses for models beyond the $\\Lambda $ CDM model.",
"In this analysis we use the Effective Fluid approach and our EFCLASS code, for the background and first order perturbations, to obtain cosmological constraints with the latest cosmological data sets: we include the Pantheon SNe compilation [37], the Planck 2018 CMB data [3], the $H_0$ Riess measurement [38], various BAO points [39], [40], [41], and a new redshift space distortions (RSD) likelihood (see Ref.",
"[42] for the “Gold 2018\" compilation of Ref. [43]).",
"The paper is organised as follows.",
"In Sec.",
"we briefly summarize our theoretical framework and present the models we consider, in Sec.",
"we present the results of our MCMC analysis with EFCLASS, in Sec.",
"we conclude and lastly, in Appendix we present some details on our RSD likelihood."
],
[
"Theoretical framework ",
"The standard cosmological model $\\Lambda $ CDM assumes the Einstein-Hilbert action $\\mathcal {L} = \\frac{1}{2\\kappa }f(R) + \\mathcal {L}_m$ with $f(R)=R$ , $R$ the Ricci scalar, $\\mathcal {L}_m$ the Lagrangian for matter fields, and the constant $\\kappa \\equiv 8\\pi G_N$ with $G_N$ being the bare Newton's constant.",
"We can derive the field equations by applying the Principle of Least Action; they read $G_{\\mu \\nu } = \\kappa T^{(m)}_{\\mu \\nu }$ where $G_{\\mu \\nu }\\equiv R_{\\mu \\nu }-\\frac{1}{2}g_{\\mu \\nu }R$ is the Einstein tensor, $R_{\\mu \\nu }$ is the Ricci tensor, $g_{\\mu \\nu }$ is the metric, and $T^{(m)}_{\\mu \\nu }$ is the energy-momentum tensor of matter fields.Throughout this paper our conventions are: $(-+++)$ for the metric signature, the Riemann and Ricci tensors are given respectively by $V_{b;cd}-V_{b;dc}=V_{a}R^{a}_{\\;bcd}$ and $R_{ab}=R^{s}_{\\;asb}$ .",
"It is also possible to consider more general theories (e.g., $f(R)\\ne R$ , Horndeski theories) and derive similar field equations.",
"As we clearly explained in Refs.",
"[27], [28], modifications to GR in these kinds of MG models can be interpreted as an effective fluid yielding field equations which schematically look like $G_{\\mu \\nu }&=&\\kappa \\left(T_{\\mu \\nu }^{(m)}+T_{\\mu \\nu }^{(\\textrm {DE})}\\right).$ Here the tensor $T_{\\mu \\nu }^{(\\textrm {DE})}$ depends on the metric and its derivatives, and in the case of general scalar-tensor theories, also on the scalar field and its derivatives (see [28]).",
"Observational evidence indicates that the Universe is statistically homogeneous and isotropic on large scales [44], [45], [46], [47].",
"Therefore we will as usual assume a flat linearly perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) metric $ds^2=a(\\tau )^2\\left[-(1+2\\Psi (\\vec{x},\\tau ))d\\tau ^2+(1-2\\Phi (\\vec{x},\\tau ))d\\vec{x}^2\\right],$ where $a$ is the scale factor, $\\vec{x}$ represents spatial coordinates, $\\tau $ is the conformal time, and $\\Psi $ and $\\Phi $ are the gravitational potentials in the Newtonian gauge.",
"In order to describe matter fields, we will consider them as ideal fluids having small perturbations.",
"We will take into consideration an Effective Fluid Approach (EFA) when dealing with MG models: modifications to GR will be described by an effective fluid having equation of state, pressure perturbation, and anisotropic stress.",
"In our approach, and assuming a metric (REF ), the background evolution is governed by the Friedmann equations $\\mathcal {H}^2&=&\\frac{\\kappa }{3}a^2 \\left(\\bar{\\rho }_{m}+\\bar{\\rho }_{\\textrm {DE}}\\right), \\\\\\dot{\\mathcal {H}}&=&-\\frac{\\kappa }{6}a^2 \\left(\\left(\\bar{\\rho }_{m}+3\\bar{P}_{m}\\right)+\\left(\\bar{\\rho }_{\\textrm {DE}}+3\\bar{P}_{\\textrm {DE}}\\right)\\right),$ where $\\mathcal {H}\\equiv \\dot{a}/a$ is the conformal Hubble parameter.In our notation a dot over a function denote derivative respect to the conformal time $\\dot{f}=df/d\\tau $ .",
"The Hubble parameter $H$ and the conformal Hubble parameter $H$ are related via $\\mathcal {H}=aH$ .",
"Here $\\bar{\\rho }_{\\textrm {DE}}$ and $\\bar{P}_{\\textrm {DE}}$ respectively denote density and pressure of the DE effective fluid.",
"The effective DE equation of state $w_{\\textrm {DE}} \\equiv \\bar{P}_{\\textrm {DE}}/\\bar{\\rho }_{\\textrm {DE}}$ allows us to describe the background evolution in these kinds of models.",
"Another ingredient that we need is the evolution equations for the perturbations obtained through the energy-momentum conservation $T^{\\mu \\nu }_{;\\nu }=0$ [27]: $\\delta ^{\\prime } &=& 3(1+w) \\Phi ^{\\prime }-\\frac{V}{a^2 H}-\\frac{3}{a}\\left(\\frac{\\delta P}{\\bar{\\rho }}-w\\delta \\right),$ $V^{\\prime } &=& -(1-3w)\\frac{V}{a}+\\frac{k^2}{a^2 H}\\frac{\\delta P}{\\bar{\\rho }} +(1+w)\\frac{k^2}{a^2 H} \\Psi \\nonumber \\\\&-&\\frac{2}{3} \\frac{k^2}{a^2 H} \\pi ,$ where $V$ is the scalar velocity perturbation, $\\delta P$ is the pressure perturbation, $\\pi $ is related to the anisotropic stress as $\\pi \\equiv \\frac{3}{2}(1+w)\\sigma $ and the prime $^{\\prime }$ is the derivative with respect to the scale factor $a$ .",
"In [28] we extensively used the EFA on the remaining Horndeski Lagrangian.",
"By applying both sub-horizon and quasi-static approximations, we managed to find analytical expressions for several quantities describing Horndeski models as an effective DE fluid, namely, DE pressure perturbation $\\frac{\\delta P_{\\textrm {DE}}}{\\overline{\\rho }_{DE}}(a,k) \\nonumber ,$ DE scalar velocity perturbation $V_{\\textrm {DE}}(a,k) \\nonumber ,$ and DE anisotropic stress $\\pi _{\\textrm {DE}}(a,k) \\nonumber .$ Having mapped remaining Horndeski models into an effective DE fluid, one can easily implement them in Boltzmann codes which compute observables such as the CMB angular power spectrum and the matter power spectrum.",
"In the following subsections we will provide a few details on the specific models that we study in this work."
],
[
"DES-fR model",
"The remaining Horndeski Lagrangian includes a very important sort of MG models, namely, $f(R)$ .",
"In [27] we showed there is a correspondence between a given $f(R)$ model and its effective DE equation of state $w_{DE}$ .",
"Then it becomes clear that by specifying a background (i.e., an equation of state), it is in principle possible to obtain a corresponding $f(R)$ .",
"The models obtained in this way are the so-called designer f(R) models [48], [49], [50].",
"An interesting case is the $w_{DE}=-1$ designer $f(R)$ model (DES-fR, henceforth) that mimics the standard $\\Lambda $ CDM model at the background, while exhibiting differences in the evolution of the linear perturbations.",
"The DES-fR model satisfying all viability conditions (see, for instance, [51]) is given by [50] $f(R)&=&R-2\\Lambda +\\alpha ~H_0^2\\left(\\frac{\\Lambda }{R-3 \\Lambda }\\right)^{c_{0}} \\times \\nonumber \\\\& & {}_2F_1\\left(c_{0},\\frac{3}{2}+c_{0},\\frac{13}{6}+2c_{0},\\frac{\\Lambda }{R-3 \\Lambda }\\right)\\;,$ where $c_{0}=\\frac{1}{12} \\left(-7+\\sqrt{73}\\right)$ , $\\alpha $ is a free dimensionless parameter, $H_0$ is the Hubble constant, $\\Lambda $ is a constant, and ${}_2F_1$ is a hypergeometric function.",
"In the literature it is common to define $F\\equiv f^{\\prime }(R)$ and we will follow this convention.",
"Instead of using $\\alpha $ in Eq.",
"(REF ) we can parametrise our expressions in terms of $b_\\pi = f_{R,0} \\equiv F(a=1)-1$ .",
"Furthermore, in Ref.",
"[27] we found that in the range $a\\in [0,1]$ the following approximation around $a\\simeq 0$ is equally accurate $&F(a)&\\simeq 1 + \\nonumber \\\\& & f_{R,0}\\frac{\\Omega _{m0}^{-c_0-1}}{\\, _2F_1\\left(c_0+1,c_0+\\frac{3}{2};2 c_0+\\frac{13}{6};1-\\Omega _{m0}\\right)}a^{3 (1 + c_0)} \\nonumber \\\\&+&\\mathcal {O}(a^{3(2+c_0)}) , $ Then, when we rewrite our expressions for the DE effective fluid in terms of $f_{R,0}$ we find that they depend on $g(x) = {}_2F_1(\\frac{3}{2}+c_0,2+c_0;2c_0+\\frac{13}{6},x),$ where $x \\equiv \\frac{a^3(\\Omega _{m,0}-1)}{a^3(\\Omega _{m,0}-1)-\\Omega _{m,0}}.$ Implementing special functions in EFCLASS is not an easy task.",
"Therefore, we used a Taylor approximation for $g(x)$ in Eq.",
"(REF ) which works really well around $x=0$ while keeping 30 terms in the expansion."
],
[
"HS model",
"The popular Hu $\\&$ Sawicki model (HS, henceforward) [52] $f(R)=R-m^2 \\frac{c_1 (R/m^2)^n}{1+c_2 (R/m^2)^n},$ can actually be rewritten, after some algebraic manipulations, as [53] $f(R)= R- \\frac{2\\Lambda }{1+\\left(\\frac{b_{hs} \\Lambda }{R}\\right)^n},$ where $\\Lambda = \\frac{m^2 c_1}{2c_2}$ and $b_{hs}=\\frac{2 c_2^{1-1/n}}{c_1}$ .",
"In [53] the authors found that when written in the form (REF ), it is clear the reason why the HS model satisfies solar system tests: if $b_{hs} \\rightarrow 0$ $\\Lambda $ CDM is recovered, and if $b_{hs} \\rightarrow \\infty $ a matter dominated universe is obtained i.e., $\\lim _{b_{hs}\\rightarrow 0}f(R)&=&R-2\\Lambda , \\nonumber \\\\\\lim _{b_{hs}\\rightarrow \\infty }f(R)&=&R.$ If the parameter $b_{hs}$ is small enough, the HS model can be regarded as a “perturbation” around $\\Lambda $ CDM .",
"This is key in our work since we do not assume the usual approximation of fixing the background to $\\Lambda $ CDM when investigating the HS model.",
"However, solving numerically the equation for the Hubble parameter is not trivial in the HS model and we worked out a different approach.",
"Instead of approximating the background, we solve the field equations and find an approximate, accurate analytical expression for the Hubble parameter.",
"We followed the treatment in [53], where it was shown that HS Hubble parameter can be written as $H_{HS}(a)^2 = H_{\\Lambda }(a)^2 + b_{hs} \\, \\delta H_1(a)^2 + b_{hs}^2\\, \\delta H_2(a)^2 + \\hdots ,~~$ which is an analytical approximation that works extremely well, for example for $b_{hs}\\le 0.1$ the average error with respect to the numerical solution is $10^{-5}\\%$ for redshifts $z\\le 30$ ."
],
[
"HDES model",
"In what follows, we will present a brief overview of a family of designer Horndeski (HDES) models that have been already studied in Ref. [28].",
"These are models whose background is exactly that of the $\\Lambda $ CDM model but at the perturbation level it is dictated by the Horndeski theory.",
"In its full form, Horndeski theory constitutes as the most general Lorentz-invariant extension of GR in four dimensions and contains a few DE and MG models.",
"Due to the recent discovery of gravitational waves by the LIGO Collaboration the Horndeski Lagrangian has been severely reduced.",
"In particular, it has been found the following constrain on the speed of GWs [25] $-3 \\cdot 10^{-15} \\le c_g/c-1 \\le 7 \\cdot 10^{-16},$ which implies that for Horndeski theories $G_{4X}\\approx 0, \\hspace{5.69054pt} G_5 \\approx \\text{constant}.$ Then, the surviving part of the Horndeski Lagrangian reads, $S[g_{\\mu \\nu }, \\phi ] = \\int d^{4}x\\sqrt{-g}\\left[\\sum ^{4}_{i=2} \\mathcal {L}_i\\left[g_{\\mu \\nu },\\phi \\right] + \\mathcal {L}_m \\right],$ where $\\mathcal {L}_2&=& G_2\\left(\\phi ,X\\right) \\equiv K\\left(\\phi ,X\\right),\\\\\\mathcal {L}_3&=&-G_3\\left(\\phi ,X\\right)\\Box \\phi ,\\\\\\mathcal {L}_4&=&G_4\\left(\\phi \\right) R,$ and $\\phi $ is a scalar field, $X \\equiv -\\frac{1}{2}\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi $ is a kinetic term, and $\\Box \\phi \\equiv g^{\\mu \\nu }\\nabla _{\\mu }\\nabla _{\\nu }\\phi $ ; $K$ , $G_3$ and $G_4$ are free functions of $\\phi $ and $X$ .",
"From the action (REF ) one can find several theories, for example $f(R)$ theories [54], Brans-Dicke theories, [55] and Cubic Galileon [56].",
"The HDES family of models that where constructed in Ref.",
"[28] limit to the Kinetic Gravity Braiding (KGB) which is distinguished by the following functions $K=K(X), \\hspace{14.22636pt} G_3=G_3(X), \\hspace{14.22636pt} G_4=\\frac{1}{2 \\kappa }.$ With Eq.",
"(REF ), we will present how to find a specific family of designer models such that $w_{\\textrm {DE}}=-1$ , i.e., the background is always that of the $\\Lambda $ CDM model but at the perturbation level it follows Horndeski's theory.",
"The usefulness of this designer model comes from allowing one to detect deviations from $\\Lambda $ CDM at the perturbations level and is a natural expansion of our previous work [50], [27].",
"For our HDES model we need two functions, the modified Friedmann equation and the scalar field conservation equation, both of which have been presented and analyzed in detail in [28].",
"The modified Friedmann equation reads, $&-H(a)^2-\\frac{K(X)}{3}+H^2_0\\Omega _m(a)+\\nonumber \\\\&+2\\sqrt{2}X^{3/2}H(a)G_{3X}+\\frac{2}{3}X K_X=0,$ where $\\Omega _m(a)$ is the matter density and $H_0$ is the Hubble parameter.",
"The scalar field conservation equation is $\\frac{J_c}{a^3}-6XH(a)G_{3X}-\\sqrt{2}\\sqrt{X}K_X=0.$ The constant $J_c$ quantifies our deviation from the attractor given the KGB model [57].",
"By looking at Eqs.",
"(REF ) and (REF ) we see that we have three unknown functions $(G_{3X}(X),K(X), H(a))$ hence the system is undetermined.",
"Then, we need to describe one of the three unknown functions $(G_{3X}(X),K(X), H(a))$ and find out the other two using Eqs.",
"(REF ) and (REF ).",
"For convenience, we write the Hubble parameter as a function of the kinetic term $X$ , i.e., $H=H(X)$ and then solve the previous equations to find $(G_{3X}(X),K(X))$ .",
"Then we find $K(X) &=& -3 H_0^2 \\Omega _{\\Lambda ,0}+\\frac{J_c \\sqrt{2X} H(X)^2}{H_0^2 \\Omega _{m,0}}-\\frac{J_c \\sqrt{2X} \\Omega _{\\Lambda ,0}}{\\Omega _{m,0}}, \\nonumber \\\\G_{3X}(X) &=& -\\frac{2 J_c H^{\\prime }(X)}{3 H_0^2 \\Omega _{m,0}},$ where $\\Omega _{m,0}$ is the matter density at redshift $z=0$ and $\\Omega _{\\Lambda ,0}=1-\\Omega _{m,0}$ .",
"With Eqs.",
"(REF ) we can create a whole family of designer models.",
"A specific model found, dubbed HDES [28], that has a smooth limit to $\\Lambda $ CDM and also recovers GR when $J_c\\sim 0$ is the following.",
"First, we demand that the kinetic term behaves as $X= \\frac{c_0}{H(a)^n}$ , where $c_0>0$ and $n>0$ .",
"Then, from Eqs.",
"(REF ) and (REF ) we have $G_{3}(X)&=&-\\frac{2 J_c c_0^{1/n} X^{-1/n}}{3 H_0^2 \\Omega _{m,0}},\\\\K(X)&=&\\frac{\\sqrt{2} J_c c_0^{2/n} X^{\\frac{1}{2}-\\frac{2}{n}}}{H_0^2 \\Omega _{m,0}}-3 H_0^2 \\Omega _{\\Lambda ,0}-\\frac{\\sqrt{2} J_c \\sqrt{X} \\Omega _{\\Lambda ,0}}{\\Omega _{m,0}}.\\nonumber $ A particular HDES model that we have selected for our MCMC runs and we also used it in our comparison with hi-CLASS previously [28] comes by setting $n=1$ in Eq.",
"(REF ).",
"In the effective fluid approach one needs to specify three functions: the equation of state, the pressure perturbation and the anisotropic stress $(w,\\delta P,\\sigma )$ to modify properly the background and the linear perturbations in the CLASS code.",
"With the HDES model we only need to specify $\\delta P$ since $w=-1$ and $\\sigma =0$ for this particular model.",
"As it was shown in [28], computing $\\delta P$ through the effective fluid approach and then introducing this function into the evolution equation for the perturbations Eqs.",
"(REF ) and (REF ) we can specify the DE scalar velocity perturbation $V_{\\textrm {DE}}$ as $V_{\\textrm {DE}}\\simeq \\left(-\\frac{14 \\sqrt{2}}{3} \\Omega _{m,0}^{-3/4} J_c~ a^{1/4}\\right)\\frac{\\bar{\\rho }_m}{\\bar{\\rho }_{\\textrm {DE}}} \\delta _m.$ which is what we implement later in our modified version of CLASS."
],
[
"Methodology",
"In order to compute cosmological constraints we use the following data sets.",
"Firstly, we utilise the 2018 release by the Planck Collaboration including temperature and polarisation anisotropies of the CMB (TTTEEE) as well as CMB lensing (lensing) [3].",
"Secondly, we include measurements of Baryon Acoustic Oscillations (BAO) from Refs.",
"[39], [40], [41].",
"Thirdly, Pantheon supernovae (SNe) from [37] were also incorporated in the analysis.",
"Fourthly, we employed local Hubble measurement (H0) from Ref.",
"[38] as a Gaussian prior.",
"Finally, we coded a new likelihood for a compilation of Redshift-Space-Distortions (RSD) measurements (see Appendix and Ref.",
"[42]).",
"Table: Flat prior bounds used in the MCMC analyses.",
"Prior range for other parameters is set as in Table 1 of Ref.",
"The cosmological models previously discussed in Section were implemented in our Boltzmann solver EFCLASS.",
"For a given cosmological model and a set of cosmological parameters we can compute the solution for both background and linear perturbations, that is, we can predict observables such as the matter power spectrum and the CMB angular power spectrum.",
"Since the parameter space not only includes cosmological parameters, but also several nuissance parameters, it becomes hard to find the best fit model as well as the relevant statistical information.",
"The usual approach is to use Markov Chain Monte Carlo (MCMC) techniques [59], [60] and we will do so.",
"We explore the parameter space of the cosmological models with the code Montepython [61], [62] which works along with EFCLASS: theoretical predictions are computed and compared to observations through likelihood functions $\\sim 10^5$ times.",
"The MCMC procedure allows us not only to find the best fit model parameters, but also to obtain the relevant countours confidence.",
"In our analysis we use the set of flat priors in Table REF .",
"In Fig.",
"REF we show the $68\\%$ and $95\\%$ confidence contours for the DES-fR model.",
"Vertical dashed and horizontal dotted lines indicate the values obtained by the Planck Collaboration in their analyses for the standard cosmological model $\\Lambda $ CDM (last column in Table 2 of Ref.",
"[3]).",
"The relevant statistical information (i.e., mean values and $68\\%$ confidence limits) is shown in Table REF .",
"We see there is good agreement for common parameters in both DES-fR and $\\Lambda $ CDM models.",
"Although error on neutrino masses get significantly reduced as we add more data, we can only set an upper limit when combining all data sets.",
"In the case of the MG parameter $b_\\pi $ we do not observe any degeneracy with other parameters in the model.",
"It is interesting that the constraints on MG, although still prior dominated, present different tendencies according to the combination of data sets: i) RSD push the MG constraints towards GR, while the $H_0$ tension remains unresolved; ii) if we exclude RSD from the data sets, we notice a preference for a MG scenario, but still hitting the prior bound on the right and not solving the problem with $H_0$ ; iii) a similar situation occurs when we exclude supernovae, $H_0$ , and RSD from the data sets, because there is a preference for MG (prior dominated though) while obtaining a $H_0$ value that agrees very well with Planck Collaboration results for $\\Lambda $ CDM .",
"Finally, we note that our derived value for the parameter $\\sigma _8 = 0.815^{+0.009}_{-0.007} \\qquad (68\\%),$ when including the whole data set, agrees very well with the value found by DES Collaboration $\\sigma _8 = 0.807^{+0.062}_{-0.041}$ for the $\\Lambda $ CDM model [63].",
"Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\% and 95%95\\%) for the DES-fR model.",
"The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\Lambda CDM parameters (last column in Table 2 of Ref.",
").",
"TTTEEE \\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\rm {lensing} stands for CMB lensing, BAO \\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\rm {SNe} stands for supernovae, H0\\rm {H0} stands for the Hubble constant, and RSD \\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\% confidence limits on cosmological parameters for the DES-fR model.",
"Here ⋯\\left\\lbrace \\dots \\right\\rbrace stands for the inclusion of data from column on the left."
],
[
"HS model",
"In Fig.",
"REF we depict $68\\%$ and $95\\%$ confidence contours for the HS model using a number of data sets.",
"Dashed-vertical and dotted-horizontal lines are the parameter values that the Planck Collaboration reported for its analysis using $\\Lambda $ CDM model (last column in Table 2 of Ref.",
"[3]).",
"Table REF contains relevant statistical information for our analysis: we show mean values and $68\\%$ limits for the HS model.",
"Again, cosmological parameters which are common to both $\\Lambda $ CDM and HS models are in good agreement with Planck Collaboration's results.",
"As in the case for the DES-fR model we can only find an upper limit for the neutrino masses which is slightly smaller for the HS model.",
"Also in this case MG constraints are prior dominated and we observe a preference for departure from GR in most probe combination, the exception being the case including RSD.",
"The latter again goes towards GR while not solving the $H_0$ discrepancy with the local value.",
"Interestingly, in Ref.",
"[64] the authors analyzed galaxy morphology and placed the following constraint for the HS model $f_{R0}<1.4\\times 10^{-8}$ .",
"By using the whole data set we find $\\sigma _8 = 0.816^{+0.008}_{-0.007} \\qquad (68\\%),$ which perfectly agrees with the value found for the DES-fR model.",
"Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\% and 95%95\\%) for the HS model.",
"The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\Lambda CDM parameters (last column in Table 2 of Ref.",
").",
"TTTEEE \\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\rm {lensing} stands for CMB lensing, BAO \\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\rm {SNe} stands for supernovae, H0\\rm {H0} stands for the Hubble constant, and RSD \\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\% confidence limits on cosmological parameters for the HS model.",
"Here ⋯\\left\\lbrace \\dots \\right\\rbrace stands for the inclusion of data from column on the left."
],
[
"HDES model",
"Fig.",
"REF shows confidence contours for the cosmological parameters in the HDES model.",
"We see good agreement in parameters that also play a part in $\\Lambda $ CDM model; the values found by the Planck Collaboration (last column in Table 2 of Ref.",
"[3]) are depicted as vertical-dashed and horizontal-dotted lines in Fig.",
"REF .",
"As for the DES-fR and HS models, in this case the neutrino masses remain unconstrained in our analysis and we can only set an upper limit.",
"Concerning the MG parameter we observe that results are not decisive since posteriors are mostly affected by the prior distribution.",
"Although there exist preference for departure from GR when including $H_0$ and RSD in the data set, the constraints hit the prior bound on the right.",
"Interesting in this case RSD push the constraints far from GR, whereas in the case of DES-fR and HS models the whole data set prefer the GR limit.",
"Finally we note that our derived $\\sigma _8 = 0.814^{+0.009}_{-0.007} \\qquad (68\\%),$ taking into consideration the full data set agrees well with values found for DES-fR and HS models.",
"In Table REF we show mean values and $68\\%$ confidence bounds for the cosmological parameters in the HDES model.",
"Figure: 1D marginalised likelihoods as well as confidence contours (i.e., 68%68\\% and 95%95\\%) for the HDES model.",
"The dashed vertical and horizontal dotted lines correspond to the results obtained by the Planck Collaboration for the Λ\\Lambda CDM parameters (last column in Table 2 of Ref.",
").",
"TTTEEE \\rm {TTTEEE} stands for CMB temperature and E-mode polarisation anisotropies correlations and cross-correlations, lensing \\rm {lensing} stands for CMB lensing, BAO \\rm {BAO} stands for Baryonic Acoustic Oscillations, SNe \\rm {SNe} stands for supernovae, H0\\rm {H0} stands for the Hubble constant, and RSD \\rm {RSD} stands for redshift space distortions.Table: Mean values and 68%68\\% confidence limits on cosmological parameters for the HDES model.",
"Here ⋯\\left\\lbrace \\dots \\right\\rbrace stands for the inclusion of data from column on the left."
],
[
"Conclusions",
"Over the past decades several cosmological models have emerged as a plausible explanation for the late-time accelerating expansion of the Universe.",
"In this paper we investigated three MG models which satisfy solar system tests and also fulfil constraints on the speed of propagation of GWs, namely: DES-fR, HS, and the HDES models.",
"It is possible to interpret MG models as an effective fluid and we followed this approach in this work.",
"We implemented DES-fR, HS, and HDES models in the Boltzmann solver EFCLASS which uses sub-horizon and quasi-static approximations when solving the perturbation equations.",
"We showed in previous works that the observables are accurately computed (i.e., better than $0.1\\%$ as compared to outputs from codes which do not use any approximation) while having the advantage of analytical expressions describing MG as an effective fluid.",
"When constraining the parameter space for the HS model is usual to assume a $\\Lambda $ CDM background.",
"This is however incorrect as the background for the HS model is in general different from the $\\Lambda $ CDM model.",
"In this paper we dropped this assumption and solved the perturbations equations taking into consideration the background evolution too.",
"We found constraints which are in good agreement with results by the Planck Collaboration when the parameter spaces overlap.",
"We also note that the constraints on the MG parameter are dominated by the prior hence unconstrained by current data sets.",
"As the HS model has an additional parameter than the $\\Lambda $ CDM model, the former will be severely penalized in any Bayesian model comparison.",
"Since data indicate a preference for the standard model it is interesting to study models which exactly match the $\\Lambda $ CDM background.",
"These models might rely on new physics while also behaving differently at the perturbations level with respect to the $\\Lambda $ CDM model.",
"By investigating these kinds of models we can also reveal whether or not current data sets can discriminate alternative models from the concordance model.",
"In this paper we investigated two models, namely, a designer $f(R)$ (DES-fR) and a designer Horndeski model (HDES).",
"When considering common cosmological parameters, constraints for the DES-fR model do not exhibit significant discrepancies with results by the Planck Collaboration for the standard model.",
"Concerning the MG parameter we note the results depend on the probe combination.",
"Most cases are dominated by the prior and hence unconstrained.",
"The full data set however, prefers the GR limit.",
"One reason for this might be the strong constraints from the RSD likelihood.",
"As the surveys that make the RSD measurements assume a $\\Lambda $ CDM model in their analysis, the data themselves maybe a bit biased.",
"While this in general can be corrected, up to a point, with the AP correction as mentioned in the Appendix, some residual bias may remain.",
"While this is an important point, it is however outside the scope of our present analysis, thus we leave it for future work.",
"Regarding the constraints for the HDES model we also find good agreement with parameters also appearing in the $\\Lambda $ CDM model.",
"Here constraints on MG are also inconclusive as the posteriors are prior dominated.",
"Interestingly, in this case the full data set shows a slight preference for a departure from GR.",
"In summary, our results do not conclusively indicate the presence of modifications to GR.",
"Since our MG constraints are prior dominated we conclude $\\Lambda $ CDM is still the preferred model.",
"Where data sets overlap, our results fully agree with the investigation carried out by the Planck Collaboration [17]."
],
[
"Acknowledgements",
"The authors would like to thank M. Martinelli for useful discussions.",
"W.C. acknowledges financial support from the Departamento Administrativo de Ciencia, Tecnología e Innovación (COLCIENCIAS) under the project “Discriminación de modelos de energía oscura y gravedad modificada con futuros datos de cartografiado galáctico\" and from Universidad del Valle under the contract 449-2019.",
"S.N.",
"and R.A. acknowledge support from the Research Projects PGC2018-094773-B-C32 and the Centro de Excelencia Severo Ochoa Program SEV-2016-0597.",
"S.N.",
"also acknowledges support from the Ramón y Cajal program through Grant No.",
"RYC-2014-15843.",
"The calculations for this article were carried out on the Datacenter CIBioFi.",
"The statistical analyses as well as the plots were made with the Python package GetDist https://github.com/cmbant/getdist.",
"The numerical codes used by the authors in the analysis of the paper and our modifications to the CLASS code, which we call EFCLASS, can be found on the websites of the EFCLASS here, here and here.",
"The publicly available RSD Montepython likelihood for the growth-rate $f\\sigma _8$ data set can be found at https://github.com/snesseris/RSD-growth."
],
[
"The RSD likelihood ",
"In this appendix we will present some details on the growth RSD likelihood implemented in MontePython and used to analyse the growth $f\\sigma _8$ data of the “Gold 2018\" compilation, namely, the $N=22$ data points of Ref. [43].",
"This likelihood was first presented in Ref.",
"[42], but we summarize here again some of the key details for completeness.",
"The growth data we use in our analysis are based on measurements of the RSD, which is a direct probe of the LSS.",
"In essence, these data directly measure $f\\sigma _8(a)\\equiv f(a)\\cdot \\sigma (a)$ , where $f(a) \\equiv \\frac{d ln\\delta }{d lna}$ is the growth rate and $\\sigma (a) \\equiv \\sigma _{8,0}\\frac{\\delta (a)}{\\delta (1)}$ are the redshift-dependent rms fluctuations of the linear density field within spheres of radius $R=8$ Mpc $h^{-1}$ .",
"Note that by $\\sigma _{8,0}$ we denote the present value of this parameter.",
"This dataset has been shown to be internally robust and unbiased by the authors of Ref.",
"[43] that used the “robustness” criterion of Ref. [65].",
"The latter uses combinations of subsets in the data to perform a Bayesian analysis and establish the dataset's overall consistency.",
"The value of the parameter $f\\sigma _8(a)$ can be measured directly by using the ratio of the monopole to the quadrupole of the redshift-space power spectrum.",
"Thus, one can show with linear theory that $f\\sigma _8(a)$ does not depend on the bias $b(k,z)$ , which maybe both scale and redshift dependent, as it drops from the equations due to the particular combination of variables [66], [67], [68].",
"Furthermore, $f\\sigma _8(a)$ has been show to be a good discriminator of DE models [67].",
"The RSD data points are given explicitly in Ref.",
"[43] as $ f\\sigma _8^\\textrm {obs,i}=\\Big (f\\sigma _8^\\textrm {obs}(z_1),\\dots , f\\sigma _8^\\textrm {obs}(z_n)\\Big )$ .",
"It should be noted that a few of the growth points are correlated, while most also require a fiducial cosmology that has to be adjusted for the Alcock-Paczynski effect (see Refs.",
"[43], [69], [70] as well as Refs.",
"[71], [72], [73] for earlier analyses).",
"The data points that are correlated are the four points from SDSS [74] and the three WiggleZ points from Ref. [75].",
"Their covariance matrices are given by $\\mathbf {C}_{\\text{WiggleZ}}= 10^{-3}\\left(\\begin{array}{ccc}6.400 & 2.570 & 0.000 \\\\2.570 & 3.969 & 2.540 \\\\0.000 & 2.540 & 5.184 \\\\\\end{array}\\right),$ for the WiggleZ data and for the SDSS points by $\\mathbf {C}_{\\text{SDSS-IV}}= 10^{-2}\\left(\\begin{array}{cccc}3.098 & 0.892 & 0.329 & -0.021\\\\0.892 & 0.980 & 0.436 & 0.076\\\\0.329 & 0.436 & 0.490 & 0.350 \\\\-0.021 & 0.076 & 0.350 & 1.124\\end{array}\\right).$ In order to perform the correction for the Alcock-Paczynski effect, we follow the prescription of Ref.",
"[69], which requires the use of a correction factor given by $\\text{fac}(z^i)= \\frac{H(z^i)\\,d_A(z^i)}{H^{\\text{ref},i}(z^i) \\, d_A^{\\text{ref},i}(z^i)}\\;,$ where the label “$\\text{ref},i$ ” stands for the fiducial cosmology at the redshift $z^i$ .",
"Then, the corrected data will be given by [76] $f\\sigma _8^{\\textrm {th,i}}\\rightarrow \\frac{f\\sigma _8^{\\textrm {th,i}}}{\\text{fac}(z^i)}\\;.$ By defining the vector $\\mathbf {V}$ for the data via: $\\mathbf {V} = \\mathbf {f\\sigma _8^\\textrm {obs,i}} - \\frac{f\\sigma _8^{\\textrm {th,i}}}{\\text{fac}(z^i)},$ the chi-squared is then given by $\\chi ^2= \\mathbf {V}^T \\mathbf {C}^{-1} \\mathbf {V}\\;.$ To conclude, we also need the theoretical prediction for the growth $\\delta (k,z)$ at each redshift, which in CLASS can be estimated from the matter power spectrum as $\\delta (k,z)=\\sqrt{\\frac{P(k,z)}{P(k,0)}}$ , where we can obtain $P(k,z)$ via the function cosmo.pk(k,z).",
"Finally, the exact value of the growth-rate $f\\sigma _8(k,z)$ can be calculated numerically with direct differentiation and cubic interpolations."
]
] | 2012.05282 |
[
[
"Product of Matrix Valued Truncated Toeplitz Operators"
],
[
"Abstract Let $A_\\Phi$ be a matrix valued truncated Toeplitz operator-the compression of multiplication operator to vector-valued model space $H^2(E)\\ominus \\Theta H^2(E)$, where $\\Theta$ is a matrix valued non constant inner function.",
"Under supplementary assumptions, we find necessary and sufficient condition that the product $A_\\Phi A_\\Psi$ is itself a matrix valued truncated Toeplitz operator."
],
[
"Introduction",
"Toeplitz operators are the compressions of multiplication operator to the usual Hardy Hilbert space $H^2$ .",
"In [3], Brown and Halmos describe the algebraic properties of Toeplitz operators.",
"Among other things, they found necessary and sufficient conditions for the product of two Toeplitz operators to itself be a Toeplitz operator, namely that either the first operator’s symbol is antiholomorphic or the second operator’s symbol is holomorphic.",
"In either case, the symbol of the product is the product of the symbols.",
"In the last decade, a large amount of research has concentrated on a generalization of Toeplitz matrices, namely truncated Toeplitz operators.",
"These are the compressions of multiplication operator to subspaces of the Hardy space which are invariant under the backward shift operator.",
"They have been formally introduced in [10]; see [5] for a more recent survey.",
"Sarason [10] found equivalents to several of Brown and Halmos’s results for truncated Toeplitz operators on the model spaces $H^2\\ominus \\theta H^2$ , where $\\theta $ is some non-constant inner function.",
"The model spaces are the backward-shift invariant subspaces of $H^2$ (that they are backward shift invariant follows easily from the fact that $\\theta H^2$ is clearly shift invariant).",
"We refer the reader to [1], [2], [5] (see also [6]) for the general theory of these operators.",
"It is well know that the product of two truncated Toeplitz operators is not a truncated Toeplitz operator.",
"In particular, in [11] Sedlock has investigated when a product of truncated Toeplitz operators is itself a truncated Toeplitz operator.",
"Most recently, the basics of corresponding matrix valued truncated Toeplitz operators (MTTOs), which are compressions to ${\\mathcal {K}}_\\Theta $ of multiplications with matrix valued functions on $H^2(E)$ has been developed in [9].",
"In view of the result of [11], there arises a basic question related to the product of matrix valued truncated Toeplitz operators that is when the product of two MTTOs is still an MTTO?",
"But there is no such simple result in the general case, and we need some supplementary assumptions to obtain the main result Theorem 4.6.",
"The purpose of the present paper is to adapt the approach in [4] to the case of MTTOs on an arbitrary model space.",
"The plan of the paper is following: By means of Section 2, we want to make sure that the reader has become acquainted to model spaces and their operators and other useful facts from this area, needed when we are going to start the main work in section 4.",
"In Section 3 we will introduce truncated Toeplitz operators (TTOs) and matrix valued truncated Toeplitz operators (MTTOs).",
"The last section contains a particular case of MTTOs namely the Block Toeplitz matrices."
],
[
"Model Spaces and Operators",
"Let $\\mathbb {C}$ denote the complex plane, $\\mathbb {D}$ the unit disc in $\\mathbb {C}$ and $\\mathbb {T}$ one dimensional torus in $\\mathbb {C}$ .",
"In the sequel $E$ will denote a fixed Hilbert space of dimension $d$ .",
"We designate the algebra of bounded linear operators on $E$ by $\\mathcal {L}(E)$ and by ${\\mathcal {L}}(E,K)$ the space of all bounded linear operators from Hilbert space $E$ to a Hilbert space $K$ .",
"The space $L^2(E)$ is defined as usual, by $L^2(E):=\\left\\lbrace f:\\mathbb {T}\\longrightarrow E: f(e^{it})=\\sum _{n=-\\infty }^{\\infty } a_ne^{int}: a_n\\in E,\\sum _{n=-\\infty }^{\\infty }\\Vert a_n\\Vert ^2<\\infty \\right\\rbrace $ endowed with the inner product $\\langle f , g \\rangle _{L^2 (E)} =\\int _{\\mathbb {T}}\\langle f(e^{it}), g(e^{it})\\rangle _E dm$ where $dm=\\frac{dt}{2\\pi }$ is the normalized Lebesgue measure on $\\mathbb {T}$ .",
"The norm induced by the inner product is given by $\\Vert f\\Vert _{L^2(E)}=\\int _{\\mathbb {T}}\\Vert f\\Vert _{E}^2dm.$ If $\\dim E=1$ (i.e., $E = \\mathbb {C}$ ) then $L^2 (E)$ consists of scalar-valued functions and is denoted by $L^2$ .",
"The Hardy space $H^2(E)$ is the subspace of $L^2(E)$ formed by the functions with vanishing negative Fourier coefficients; it can be identified with a space of $E$ - valued functions analytic in $\\mathbb {D}$ , from which the boundary values can be recovered almost everywhere through radial limits.",
"One can also view $H^2(E)$ as the direct sum of $d$ standard $H^2$ spaces.",
"We have the orthogonal decomposition $L^2(E)=H^2(E)\\oplus (zH^2(E))^*.$ The spaces $ L^\\infty (E)\\subset L^2(E) $ is formed by the essentially bounded functions with values in $ E $ ; then $ H^\\infty (E)\\subset H^2(E) $ are the functions in $ L^\\infty (E) $ with vanishing negative Fourier coefficients.",
"Taking into account that $\\mathcal {L}(E)$ is a Hilbert space endowed with the Hilbert-Schmidt norm, we may similarly define $ H^2({\\mathcal {L}}(E)) \\subset L^2({\\mathcal {L}}(E))$ and $ H^\\infty ({\\mathcal {L}}(E)) \\subset L^\\infty ({\\mathcal {L}}(E)) $ .",
"Note, however, that we prefer to consider on $ L^\\infty ({\\mathcal {L}}(E)) $ and $ H^\\infty ({\\mathcal {L}}(E)) $ the equivalent norm obtained by considering on $ {\\mathcal {L}}(E) $ the operator norm instead of the Hilbert-Schmidt norm.",
"The space $L^2(\\mathcal {L}(E))$ may be identified with the matrices with all the entries in $L^2$ .",
"We have an orthogonal decomposition $L^2 (\\mathcal {L}(E)) =[zH^2 \\mathcal {L}(E)]^*\\oplus H^2(\\mathcal {L}(E)).$ The space $ L^\\infty ({\\mathcal {L}}(E)) $ acts on $L^2 (E)$ by means of multiplication: to $\\Phi \\in \\ L^\\infty (\\mathcal {L}(E))$ we associate the operator $M_\\Phi $ defined by $M_\\Phi (f)=\\Phi f$ for all $f\\in L^2(E)$ .",
"Let $S$ denote the forward shift operator $(Sf)( z ) = zf (z)$ on $H^2 (E)$ ; it is the restriction of $M_z$ to $H^2(E)$ .",
"Its adjoint (the backward shift) is the operator $(S^*f)(z) =\\frac{f(z )-f(0)}{z}$ One sees easily that $I-SS^*$ is precisely the orthogonal projection onto the space of constant functions.",
"The main object of study is formed by the model spaces and the operators acting on them.",
"These are defined as follows.",
"First, an inner function is an element $ \\Theta \\in H^2(\\mathcal {L}(E))$ whose boundary values are almost everywhere unitary operators in $\\mathcal {L}(E)$ .",
"The inner function is called pure if $ \\Vert \\Theta (0)\\Vert <1 $ ; this is equivalent to requiring that $ \\Theta $ has no constant unitary part.",
"Consider then a pure inner function $\\Theta $ , with values in $\\mathcal {L}(E)$ .",
"The model space associated to a pure inner function $\\Theta $ , denoted by $\\mathcal {K}_{\\Theta }$ and is defined by $\\mathcal {K}_{\\Theta }=H^{2}(E)\\ominus \\Theta H^{2}(E).$ Just like the Beurling-type subspaces $\\Theta H^{2}(E)$ constitute nontrivial invariant subspaces for the unilateral shift $S$ on $H^{2}(E)$ , the subspaces $\\mathcal {K}_{\\Theta }$ play an analogous role for the backward shift $S^{*}$ .",
"The orthogonal projection onto $\\mathcal {K}_{\\Theta }$ will be denoted by $P_{\\Theta }$ .",
"It is also known that $ {\\mathcal {K}}_\\Theta \\cap H^\\infty (E) $ is dense in $ {\\mathcal {K}}_\\Theta $ .",
"The analogous space of matrix-valued functions is denoted by $\\mathcal {M}_\\Theta $ ; it is the orthogonal complement of $\\Theta H^2(\\mathcal {L}(E))$ in $H^2(\\mathcal {L}(E))$ .",
"The model operator $S_\\Theta \\in \\mathcal {L}(\\mathcal {K}_\\Theta )$ is defined by the formula $(S_\\Theta f)(z)=P_\\Theta (zf).$ The adjoint of $S_\\Theta $ is given by $(S_\\Theta ^*f)(z)=\\frac{f(z)-f(0)}{z};$ it is the restriction of the left shift in $H^2(E)$ to the $S^*$ -invariant subspace $\\mathcal {K}_\\Theta $ .",
"Let us assume that $ \\Theta (0)=0 $ , so $ \\Theta =z\\Theta _1 $ , which is the case we will consider in the sequel.",
"Then $ I-S_\\Theta S_\\Theta ^* $ is the projection $ P_0 $ onto the constant functions, which are contained in $ {\\mathcal {K}}_\\Theta $ , while $ I-S_\\Theta ^* S_\\Theta $ is the projection $P_{D_*}$ onto the space $ {\\mathcal {D}}_*=\\lbrace \\Theta _1 x: x\\in E \\rbrace $ (which is also contained in $ {\\mathcal {K}}_\\Theta $ ).",
"The scalar valued model spaces and operators are obtained when $ \\dim E=1 $ ; that is, when $ E=\\mathbb {C}$ .",
"We have then the classical spaces $ H^2\\subset L^2 $ and $L^\\infty $ .",
"The inner function is a scalar inner function $ \\theta $ , and the model space is $ \\mathcal {K}_{\\theta }=H^{2}\\ominus \\theta H^{2} $ .",
"In particular, in case $ \\theta (z)=z^n $ , $ {\\mathcal {K}}_\\theta $ becomes the $ n $ -dimensional space of polynomials of degree at most $ n-1 $ .",
"Definition 2.1 A conjugation on a complex Hilbert space $\\mathcal {H}$ is a function $C:\\mathcal {H}\\longrightarrow \\mathcal {H}$ that is (i) conjugate linear: that is $C(\\alpha x+\\beta y)=\\overline{\\alpha }Cx +\\overline{\\beta }Cy$ for all $x,y\\in \\mathcal {H}$ and $\\alpha ,\\beta \\in \\mathbb {C}$ , (ii) involutive: $C^{2}=I$ , (iii) isometric: $\\Vert Cx\\Vert =\\Vert x\\Vert $ for all $x\\in \\mathcal {H}$ .",
"The following result is an immediate consequence of a theorem in [9].",
"Lemma 2.1 Suppose $ \\Theta (0)=0 $ , so $ \\Theta =z\\Theta _1 $ .",
"Let $\\Gamma $ be a conjugation on $E$ , and suppose that $\\Theta (e^{it})^{*}=\\Gamma \\Theta (e^{it})\\Gamma $ a.e.",
"on $\\mathbb {T}$ .",
"Then the map $C_{\\Gamma }$ defined by $C_\\Gamma (f)=z\\Theta _{1}\\Gamma f$ is a conjugation on $z\\mathcal {K}_{\\Theta _1}$ ."
],
[
"Truncated Toeplitz Operators and Matrix valued Truncated Toeplitz Operators",
"If $ \\phi \\in L^\\infty $ , then the compression of the multiplication operator $ M_\\phi $ to $ H^2 $ is called a Toeplitz operator and is denoted by $ T_\\phi $ .",
"That means that $ T_\\phi =P_{H^2}M_\\phi |H^2 $ , where $P_H^2$ is the orthogonal projection of $L^2$ onto $H^2$ .",
"More than a decade ago, Sarason has introduced in [10] the so-called truncated Toeplitz operators.",
"Remember that $ P_\\theta $ is the orthogonal projection onto the model space $ {\\mathcal {K}}_\\theta $ .",
"If $ \\phi \\in L^\\infty $ , then the truncated Toeplitz operator $ A^\\theta _\\phi $ is defined to be the compression of $ M_\\phi $ to $ {\\mathcal {K}}_\\theta $ .",
"That means that $ A^\\theta _\\phi =P_\\theta M_\\phi |{\\mathcal {K}}_\\theta $ .",
"In particular, we see that with this notation $ S_\\theta = A^\\theta _z $ .",
"Let us now remember that for $ \\theta (z)=z^n $ the space $ {\\mathcal {K}}_\\theta $ is formed by the polynomials of degree not greater than $ n-1 $ .",
"The monomials $ z^k $ , $ k=0,\\dots , n-1 $ form an orthonormal basis of $ {\\mathcal {K}}_{z^n} $ .",
"If we write the matrix of an operator with respect to this basis, then one can see that truncated Toeplitz operators correspond precisely to Toeplitz matrices.",
"Passing now beyond the scalar case, let us suppose that $\\Theta $ is a pure inner function.",
"The analogue of truncated Toeplitz operators have been defined in [9], where they are called matrix valued truncated Toeplitz operators.",
"Suppose then that $\\Phi \\in L^2(\\mathcal {L}(E))$ .",
"Consider the linear map $f\\longrightarrow P_\\Theta (\\Phi f)$ , defined on $\\mathcal {K}_\\Theta \\cap H^\\infty (E)$ .",
"If it is bounded, then it uniquely determines an operator in $\\mathcal {L}(\\mathcal {K}_\\Theta )$ , denoted by $A_\\Phi ^\\Theta $ , and called a matrix valued truncated Toeplitz operator (MTTO).",
"The function $\\Phi $ is then called a symbol of the operator.",
"We will usually drop the subscript $\\Theta $ , as we consider a fixed inner function.",
"We denote by $\\mathcal {T}(\\mathcal {K}_{\\Theta })$ the space of all MTTOs on the model space $\\mathcal {K}_\\Theta $ .",
"Note that if $ \\Phi \\in L^\\infty ({\\mathcal {L}}(E)) $ (that is, it is bounded), then it follows that $f\\longrightarrow P_\\Theta (\\Phi f)$ defines a bounded linear operator on the whole of $ {\\mathcal {K}}_\\Theta $ , and thus $A_\\Phi ^\\Theta \\in {\\mathcal {T}}({\\mathcal {K}}_\\Theta )$ .",
"But we may have bounded MTTOs which have no bounded symbols, which is one of the complications of the theory.",
"However, a result of [9] tells that any operator in $ {\\mathcal {T}}({\\mathcal {K}}_\\Theta ) $ has a symbol in $ {\\mathcal {M}}_\\Theta +({\\mathcal {M}}_\\Theta )^* $ ; this is why we will restrict in the sequel to considering operators $A_\\Phi ^\\Theta $ with $ \\Phi \\in {\\mathcal {M}}_\\Theta +({\\mathcal {M}}_\\Theta )^* $ .",
"The operator $S_\\Theta $ is a simple example of MTTO; it is obtained by taking $\\Phi (z)=zI_E$ .",
"This example is rather special because the symbol is scalar valued.",
"It is immediate that $A_\\Phi ^*=A_{\\Phi ^*};$ so $\\mathcal {T}(\\mathcal {K}_{\\Theta })$ is a self adjoint linear space.",
"In section 5 we will see that if $\\Theta (z)=z^N I_E$ , then the MTTOs obtained are actually familiar objects, namely block Toeplitz matrices of dimension $N$ , in which the entries are the matrices of dimension $d$ .",
"The theory of Block Toeplitz matrices has been an inspiration for research in the domain of matrix valued truncated Toeplitz operators.",
"In particular, it should be mentioned that some of the classes of block Toeplitz matrices which are closed to multiplication are found in [7] and [8].",
"As supposed above, we will consider a fixed inner function $ \\Theta $ and different MTTOs acting on $ {\\mathcal {K}}_\\Theta $ .",
"The symbols of these operators will be $ \\Phi , \\Psi ,\\dots $ .",
"Before the ending of this section we need to quote result from [12].",
"Proposition 3.1 Suppose $\\Theta $ be an inner function, and $\\Phi \\in H^\\infty (\\mathcal {L}(E))$ such that $\\Phi \\Theta =\\Theta \\Phi $ .",
"Then (1) $\\Theta H^2(\\mathcal {L}(E))$ is invariant with respect to $M_\\Phi $ (and consequently $\\mathcal {K}_\\Theta )$ is invariant under $M_{\\Phi }^*$ .",
"(2) $A_\\Phi S_\\Theta =S_\\Theta A_\\Phi $ and consequently $A_\\Phi ^*S_\\Theta ^*=S_\\Theta ^*$ Set $\\Delta =I-S_\\Theta S_\\Theta ^*$ .",
"Note that $P_0$ is the orthogonal projection onto the constant functions contained in $\\mathcal {K}_\\Theta $ .",
"The next result from [9] characterizes MTTOs among all operators on $ {\\mathcal {K}}_\\Theta $ .",
"Proposition 3.2 A bounded operator $A$ on $\\mathcal {K}_\\Theta $ belongs to $\\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only if there exist operators $B,B^\\prime $ on $\\mathcal {K}_\\Theta $ such that $\\Delta (A)=BP_0+P_0{B^\\prime }$ In this case $A=A_{\\Phi +{\\Phi ^\\prime }^*}$ , where $\\Phi ,\\Phi ^\\prime \\in H^2(\\mathcal {L}(E))$ ."
],
[
"Main results",
"In view of the result of Sedlock [11], a natural question is to determine when is the product of two MTTOs still an MTTO.",
"However, there is no such simple result in the general case, and we need some supplementary assumptions to obtain the main result, Theorem REF .",
"The path we take is suggested by [4], but the matrix valued situation is much more complicated.",
"We will consider in the rest of this section a fixed inner function $\\Theta \\in H^\\infty (E)$ subjected to the condition $\\Theta (0)=0$ .",
"Then $\\Theta (z)=z\\Theta _{1}(z)$ , where $\\Theta _{1}\\in H^\\infty (E)$ is also inner.",
"We have the orthogonal decomposition ${\\mathcal {K}}_\\Theta =E\\oplus z{\\mathcal {K}}_{\\Theta _1}.$ Take now $\\Phi \\in {\\mathcal {M}}_\\Theta +({\\mathcal {M}}_\\Theta )^* $ .",
"We can write then $\\Phi =z\\Phi _{+}+\\bar{z}\\Phi _{-}^*+\\Phi _0$ with $\\Phi _\\pm \\in \\mathcal {M}_{\\Theta _1}$ and $\\Phi _0\\in \\mathcal {L}(E)$ .",
"If $ \\Phi (e^{it})=\\displaystyle \\sum _{n=-\\infty }^{\\infty }\\Phi _n e^{int}$ with $\\Phi _n\\in \\mathcal {L}(E)$ , then $\\begin{split}\\Phi _+(z)&=\\sum _{n=1}^{\\infty }\\Phi _n z^n=\\sum _{n=1}^{\\infty } \\left( \\int \\Phi (e^{it})e^{-int}\\, dt \\right) z^n\\\\&=\\int \\Phi (e^{it}) \\left( \\sum _{n=1}^{\\infty } e^{-int}z^n \\right)dt =\\int \\Phi (e^{it}) \\frac{e^{it}z}{1-e^{it}z}dt.\\end{split}$ Remember that two operators $ A,B $ are said to doubly commute if $ AB=BA $ and $ AB^*=B^*A $ (whence it follows that also $ A^*B^*=B^*A^* $ and $ A^*B=BA^* $ ).",
"Lemma 4.1 Suppose that $\\Theta (0)=0$ and $\\Phi ,\\Psi \\in \\mathcal {M}_\\Theta +({\\mathcal {M}}_\\Theta )^*$ such that $\\Phi (e^{it})\\Psi (e^{is})=\\Psi (e^{is})\\Phi (e^{it})$ for any $ t,s $ .",
"(i) For any $ s,t $ we have $\\Phi _+(e^{it})\\Psi (e^{is})=\\Psi (e^{is})\\Phi _+(e^{it})$ and $\\Phi _-(e^{it})\\Psi (e^{is})=\\Psi (e^{is})\\Phi _-(e^{it})$ .",
"(ii) If the values of $\\Phi ,\\Psi $ doubly commute with those of $\\Theta $ , then the same is true for $ \\Phi _{\\pm }, \\Psi _{\\pm } $ .",
"(iii) If $\\Gamma $ is a conjugation on $E$ such that $\\Phi (e^{it})^*=\\Gamma \\Phi (e^{it})\\Gamma $ , then $\\Phi _\\pm (e^{it})^*=\\Gamma \\Phi _\\pm (e^{it})\\Gamma $ .",
"We will give the proof only for one of the equalities in (ii); the rest are similar.",
"Using (REF ), we have $\\begin{split}\\Psi (e^{is}) \\Phi _+(z)&= \\Psi (e^{is})\\int \\Phi (e^{it}) \\frac{e^{it}z}{1-e^{it}z}dt=\\int \\Psi (e^{is}) \\Phi (e^{it}) \\frac{e^{it}z}{1-e^{it}z}dt\\\\&=\\int \\Phi (e^{it})\\Psi (e^{is}) \\frac{e^{it}z}{1-e^{it}z}dt=\\left(\\int \\Phi (e^{it}) \\frac{e^{it}z}{1-e^{it}z}dt\\right)\\Psi (e^{is})=\\Phi _+(z)\\Psi (e^{is}).\\end{split}$ By taking radial limits a.e., one obtains the required commutativity.",
"The next lemma gives an identification of elements in $ {\\mathcal {M}}_\\Theta $ .",
"Lemma 4.2 The map $ \\Phi \\mapsto J_\\Phi $ , defined by $J_\\Phi (x)(z)=\\Phi (z)x.$ is a bijection between $ {\\mathcal {M}}_\\Theta $ and $ {\\mathcal {L}}(E, {\\mathcal {K}}_\\Theta ) $ .",
"Fixing a basis $e_1,\\cdots ,e_d$ in $E$ and defining the transformation $J:E\\longrightarrow \\mathcal {K}_\\Theta $ as follow $J(e_k)=\\phi _k,\\qquad 1\\le k\\le d,$ where $\\phi _k\\in \\mathcal {K}_\\Theta $ for every $1\\le k\\le d$ .",
"If we arrange $\\phi _k$ as a column vectors then we obtain $\\Phi \\in \\mathcal {M}_\\Theta $ .",
"Conversely if we have $\\Phi \\in \\mathcal {M}_\\Theta $ then we obtain the map $J_\\Phi : E\\longrightarrow \\mathcal {K}_\\Theta $ which sends $e_k$ to $k$ th column of $\\Phi $ for every $1\\le k\\le d$ .",
"Note that, if $\\Phi \\in \\mathcal {M}_\\Theta $ , then the relation between $J_\\Phi $ and $\\Phi $ is simply $J_\\Phi (x)(z)=\\Phi (z)x.$ We denote by $J_0$ the embedding of $E$ in $K_\\Theta $ ; that is for every $x\\in E$ , $J_0(x)=x$ .",
"It is easy to see that in a given basis the matrices of functions in $z\\mathcal {M}_{\\Theta _1}$ are characterized by the fact that columns are functions in $z\\mathcal {K}_{\\Theta _1}$ .",
"Finally, we define $\\mathbf {C}_\\Gamma (\\Phi ) $ by giving the action of $J_\\Phi $ on $x\\in E$ as $J_{\\mathbf {C}_\\Gamma (\\Phi )} x=C_\\Gamma (\\Phi \\Gamma x),$ where $ C_\\Gamma $ is defined by (REF ).",
"In the rest of this section we assume that $\\mathfrak {F}$ is a commutative algebra of functions contained in ${\\mathcal {M}}_\\Theta +({\\mathcal {M}}_\\Theta )^*$ , such that all the elements of $\\mathfrak {F}$ doubly commute with those of $\\Theta $ .",
"The next lemma is the main technical result of this section.",
"Lemma 4.3 Suppose that $\\Theta (0)=0$ and $\\Phi ,\\Psi \\in \\mathfrak {F}$ .",
"Then there exist operators $ X, Y\\in \\mathcal {M}_\\Theta $ such that $\\Delta (A_\\Phi A_\\Psi )=J_{z\\Phi _+}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*+XP_0+P_0Y$ For any $\\Phi \\in \\mathcal {M}_{\\Theta }+({\\mathcal {M}}_\\Theta )^*$ we will denote $\\hat{\\Phi }=\\Phi -\\Phi _0$ .",
"It follows easily from Lemma REF that $\\Phi _0\\Psi (e^{is})=\\Psi (e^{is})\\Phi _0$ and $\\Phi _0\\hat{\\Psi }(e^{is})=\\hat{\\Psi }(e^{is})\\Phi _0$ .",
"In the same way one can obtain $\\Psi _0\\Phi (e^{is})=\\Phi (e^{is})\\Psi _0$ and $\\Psi _0\\hat{\\Phi }(e^{is})=\\hat{\\Phi }(e^{is})\\Psi _0$ .",
"A similar argument works for double commutation with $\\Theta $ .",
"Since $\\Phi _0,\\Psi _0\\in H^2(\\mathcal {L}(E))$ commutes with $\\Theta $ then by using Proposition REF $S_\\Theta $ commutes with $A_{\\Phi _0}$ and $A_{\\Psi _0}$ , and therefore $\\Delta (A_{\\Phi } A_{\\Psi })&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+\\Delta (A_{\\hat{\\Phi }}A_{\\Psi _0})+\\Delta (A_{\\Phi _0}A_{\\hat{\\Psi }})+\\Delta (A_{\\Phi _0}A_{\\Psi _0}).\\\\&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+\\Psi _0\\Delta (A_{\\hat{\\Phi }})+\\Phi _0\\Delta (A_{\\hat{\\Psi }})+\\Phi _0\\Psi _0\\Delta (I)\\\\&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}) +\\Psi _0\\Delta (A_{\\hat{\\Phi }})+\\Phi _0\\Delta (A_{\\hat{\\Psi }})+\\Phi _0\\Psi _0P_0.$ By Proposition REF there exist operators $B=z\\Phi _{+}$ and $B^\\prime =z\\Phi _{-}$ such that $\\Delta (A_{\\hat{\\Phi }})=z\\Phi _{+}P_0+P_0\\bar{z}\\Phi _{-}^*$ with $\\Phi _{\\pm }\\in \\mathcal {M}_{\\Theta _{1}}$ .",
"Similarly $\\Delta (A_{\\hat{\\Psi }})=z\\Psi _{+}P_0+P_0\\bar{z}\\Psi _{-}^*$ and $\\Psi _{\\pm }\\in \\mathcal {M}_{\\Theta _{1}}$ .",
"Using Lemma REF , we have $\\Delta (A_\\Phi A_\\Psi )&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+\\Psi _0(z\\Phi _{+}P_0+P_0\\bar{z}\\Phi _{-}^*)+\\Phi _0(z\\Psi _{+}P_0+P_0\\bar{z}\\Psi _{-}^*)+\\Phi _0\\Psi _0P_0\\\\&=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+(\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0 \\Psi _0)P_0+(\\Psi _0P_0 \\bar{z}\\Phi _{-}^*+\\Phi _0P_0\\bar{z}\\Psi _{-}^*)$ Since $P_0$ is the projection onto the constants then it must commute with $\\Phi _0$ and $\\Psi _0$ .",
"Therefore $\\Delta (A_\\Phi A_\\Psi )=\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})+(\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0)P_0+P_0(\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*)$ Now, by using the definition of $\\Delta $ , $\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})&=A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}-S_\\Theta A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}S_\\Theta ^*\\\\&=A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}-A_{\\hat{\\Phi }}S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*+A_{\\hat{\\Phi }}S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-S_\\Theta A_{\\hat{\\Phi }}A_{\\hat{\\Psi }}S_\\Theta ^*\\\\&=A_{\\hat{\\Phi }}\\Delta (A_{\\hat{\\Psi }})+\\Delta (A_{\\hat{\\Phi }})S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*\\\\&=A_{\\hat{\\Psi }}(z\\Phi _{+}P_0+P_0\\bar{z}\\Psi _{-}^*)+(z\\Phi _{+}P_0+P_0\\bar{z}\\Phi _{-}^*)S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*}-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*$ or $\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})= A_{\\hat{\\Phi }}z\\Psi _{+}P_0+A_{\\hat{\\Phi }}P_0\\bar{z}\\Psi _{-}^*+z\\Phi _{+}P_0 S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*} +P_0\\bar{z}\\Phi _{-}^*S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*}-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*.$ Since the constant functions are in $\\mathcal {K}_{\\Theta }$ , we have $ P_0P_\\Theta =P_\\Theta P_0 $ .",
"Also since $ P_0zf=0 $ , then $P_0S_\\Theta f=P_0P_\\Theta zf= P_\\Theta P_0 zf=0,$ So the third term in the left hand side of (REF ) is 0.",
"The second term is $A_{\\hat{\\Phi }}P_0\\bar{z}\\Psi _{-}^*=P_\\Theta (z\\Phi _+ +\\bar{z}\\Phi _-^*)P_0\\bar{z}\\Psi _{-}^*.$ But, since $ \\Phi _+\\in \\mathcal {M}_{\\Theta _{1}} $ , we have, for any constant function $ x $ , $ \\Phi _+ x\\in \\mathcal {K}_{\\Theta _1} $ , $ z\\Phi _+ x\\in \\mathcal {K}_{\\Theta } $ , and therefore $ P_\\Theta z\\Phi _+P_0=z\\Phi _+P_0 $ .",
"Also, for any constant function $ x $ , $ \\bar{z}\\Phi _-^*x\\perp H^2(E) $ , so $ P_\\Theta \\bar{z}\\Phi _-^*P_0=0 $ .",
"So $\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})= A_{\\hat{\\Phi }}z\\Psi _{+}P_0+z\\Phi _+ P_0\\bar{z}\\Psi _{-}^*+P_0\\bar{z}\\Phi _{-}^*S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*}-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*.$ Since $ J_0 $ is the embedding of the constants into $ \\mathcal {K}_\\Theta $ , we have, for $ f\\in \\mathcal {K}_\\Theta $ and $ x\\in E $ , $\\langle J_0^*f, x\\rangle =\\langle f,J_0 x\\rangle =\\langle f(0), x\\rangle $ whence $ J^*_0f=f(0) $ .",
"So $J_\\Phi J_0^*f=J_\\Phi f(0)= \\Phi (z)f(0)=\\Phi P_0 f,\\quad \\hbox{for any}\\quad f\\in \\mathcal {K}_\\Theta .$ By taking adjoints we have $J_0 J_\\Phi ^*=P_0\\Phi ^*$ .",
"Therefore we can write (REF ) as $\\Delta (A_{\\hat{\\Phi }}A_{\\hat{\\Psi }})= A_{\\hat{\\Phi }} J_{z\\Psi _{+}}J_0^*+J_{z\\Phi _{+}}P_0 J_{z\\Psi _{-}}^*+P_0\\bar{z}\\Phi _{-}^*S_\\Theta A_{\\hat{\\Psi }}S_{\\Theta }^{*}-S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*.$ Since $ \\mathcal {D}_* $ is the space spanned by $ \\Theta _1 E $ , we can define an isometry $ V:E\\rightarrow \\mathcal {K}_\\Theta $ by the formula $ Vx=\\Theta _1 x $ , and, moreover, $P_{\\mathcal {D}_*}=VV^* $ .",
"Also, we have $A_{\\hat{\\Phi }}Vx=P_\\Theta \\hat{\\Phi }Vx=P_\\Theta z\\Phi _+ Vx +P_\\Theta \\bar{z}\\Phi _-^* Vx.$ Then, using the commutativity between $ \\Theta $ and $ \\Phi _+ $ , $z\\Phi _+ V x=z\\Phi _+ \\Theta _1 x= z\\Theta _1\\Phi _+ x=\\Theta \\Phi _+ x\\perp \\mathcal {K}_\\Theta ,$ and so the first term in (REF ) is 0.",
"We have also $\\bar{z}\\Phi _-^*V x&=\\bar{z}\\Phi _-^*\\Theta _1 x=\\Theta _1 \\bar{z} \\Phi _-^* x= \\Theta _1 \\bar{z} \\Phi _-^* \\Gamma \\Gamma x\\\\&=\\bar{z}( z\\Theta _1 \\Gamma (z \\Phi _-\\Gamma x))=\\bar{z} C_\\Gamma ( z \\Phi _-\\Gamma x).=\\bar{z}J_{\\mathbf {C}_\\Gamma }(z\\Phi _{-})x.$ Since $ C_\\Gamma $ is a conjugation on $ z\\mathcal {K}_{\\Theta _1} $ , $ \\bar{z} C_\\Gamma (z \\Phi _-\\Gamma x)\\in \\mathcal {K}_{\\Theta }$ , and therefore $P_\\Theta \\bar{z}\\Phi _-^*V x=\\bar{z}\\Phi _-^*V x= \\bar{z} C_\\Gamma z \\Phi _-\\Gamma x=\\bar{z}J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}x.$ So $S_\\Theta A_{\\hat{\\Phi }}Vx= z \\bar{z} C_\\Gamma z \\Phi _-\\Gamma x= C_\\Gamma z \\Phi _-\\Gamma x=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}x.$ Similarly, we obtain $S_\\Theta A_{\\hat{\\Psi }^*}Vx=C_\\Gamma z \\Psi _+\\Gamma x=J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})} x.$ Consequently, $S_\\Theta A_{\\hat{\\Phi }}V=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}, \\qquad S_\\Theta A_{\\hat{\\Psi }^*}V=J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})},$ Finally, the last term in (REF ) is $S_\\Theta A_{\\hat{\\Phi }}P_{{\\mathcal {D}}_*}A_{\\hat{\\Psi }}S_\\Theta ^*=S_\\Theta A_{\\hat{\\Phi }}VV^*A_{\\hat{\\Psi }}S_\\Theta ^*=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*.$ Combining (REF ) and (REF ) we get $\\Delta (A_{\\hat{\\Phi }} A_{\\hat{\\Psi }})=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*+J_{z\\Phi _+}P_0J_{z\\Psi _{-}}^*+J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*$ so we have $\\begin{split}\\Delta (A_\\Phi A_\\Psi )&=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*+J_{z\\Phi _+}P_0J_{z\\Psi _{-}}^*+J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*\\\\&+(\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0)P_0+P_0(\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*)\\end{split}$ Now we have the relations $J_\\phi J_0^*=\\Phi P_0, J_0J_\\Phi =P_0\\Phi ^*$ So the second term in (REF ) becomes $z\\Phi _{+}P_0(z\\Phi )^*=J_{z\\Phi _{+}}J_0^*J_0J_{z\\Phi _{-}}^*=J_{z\\Phi _{+}}J_{z\\Phi _{-}}^*$ Also, since $J_0: E\\longrightarrow \\mathcal {K}_{\\Theta }$ is the embedding of the constants, while $P_0:\\mathcal {K}_\\Theta \\longrightarrow \\mathcal {K}_\\Theta $ is the projection onto the constant functions, it follows immediately that $P_0J_0=J_0, J_0^*P_0=J_0^*$ Therefore we can write the first and third term in (REF ) as $A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*P_0,\\quad J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*=P_0J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*.$ So we have $\\begin{split}\\Delta (A_\\Phi A_\\Psi )&=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*P_0+J_{z\\Phi _+}J_{z\\Psi _{-}}^*+P_0J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*\\\\&+(\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0)P_0+P_0(\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*)\\\\&=J_{z\\Phi _+}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*+[A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*+\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0]P_0\\\\&+P_0[J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*+\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*]\\\\&=J_{z\\Phi _+}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*+XP_0+P_0Y\\end{split}$ where, $X=A_{\\hat{\\Phi }}J_{z\\Psi _+}J_0^*+\\Psi _0 z\\Phi _{+}+\\Phi _0z\\Psi _{+}+\\Phi _0\\Psi _0$ and $Y=J_0J_{z\\Phi _{-}}^*S_\\Theta A_{\\hat{\\Psi }}S_\\Theta ^*+\\Psi _0\\bar{z}\\Phi _{-}^*+\\Phi _0\\bar{z}\\Psi _{-}^*$ .",
"Theorem 4.4 Suppose $\\Theta (0)=0$ and $\\Phi ,\\Psi ,\\chi , \\zeta \\in \\mathfrak {F}$ .",
"Then $A_\\Phi A_\\Psi -A_\\chi A_\\zeta \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only if $ J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _-)}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*=J_{z\\chi _+}J_{z\\zeta _-}^*-J_{\\mathbf {C}_\\Gamma (z\\chi _-)}J_{\\mathbf {C}_\\Gamma (z\\zeta _+)}^*.$ By Lemma REF there exists operators $X, Y\\in \\mathcal {M}_\\Theta $ such that $\\Delta (A_\\Phi A_\\Psi -A_\\chi A_\\zeta )=J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}-J_{z\\chi _+}J_{z\\zeta _-}^*+J_{\\mathbf {C}_\\Gamma (z\\chi _-)}J_{\\mathbf {C}_\\Gamma (z\\zeta _+)}+XP_0+P_0Y.$ By Proposition REF , we have $A_\\Phi A_\\Psi -A_\\chi A_\\zeta \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only there exist operators $B, B^\\prime $ such that $\\Delta (A_\\Phi A_\\Psi -A_\\chi A_\\zeta )=BP_0+P_0B.$ The last two equations say that $A_\\Phi A_\\Psi -A_\\chi A_\\zeta \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only if there exist $ X^{\\prime },Y^{\\prime } $ such that $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*-J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}-J_{z\\chi _+}J_{z\\zeta _-}^*+J_{\\mathbf {C}_\\Gamma (z\\chi _-)}J_{\\mathbf {C}_\\Gamma (z\\zeta _+)}=X^{\\prime }P_0+P_0Y^\\prime .$ Since $\\Theta (0)=0$ then we can write $\\mathcal {K}_\\Theta =E\\oplus \\mathcal {K}_{\\Theta _{1}}$ .",
"With respect to this decomposition, the left hand side of (REF ) has zeros on the first row and column, while the right hand side is the general form of an operator that has zeros in the lower right corner.",
"Now it is clear that for (REF ) to be true both sides have to be zero, which proves the theorem.",
"The following result is the main result of this paper: it gives the answer to the question stated at the beginning of this section, namelu when is the product of two MTTOs also an MTTO.",
"Theorem 4.5 Suppose $\\Theta (0)=0$ , $\\Phi ,\\Psi \\in \\mathfrak {F}$ and $A_\\Phi , A_\\Psi , \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ .",
"Then $A_\\Phi A_\\Psi \\in \\mathcal {T}(\\mathcal {K}_\\Theta )$ if and only if $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*=J_{\\mathbf {C}_\\Gamma (z\\Phi _-)}J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*.$ Applying Theorem REF to the case $\\chi =\\zeta =0$ ."
],
[
"A particular case: Block Toeplitz Matrices",
"In this section let $\\Theta (z)=z^NI_E$ for some fixed positive integer $N$ .",
"Then ${\\mathcal {K}}_\\Theta $ is the Hilbert space of all polynomials in $z$ of degree at most $N-1$ with coefficients from $E$ ,i.e., ${\\mathcal {K}}_\\Theta =\\lbrace a_0+a_1z+\\cdots a_{N-1}z^{N-1};a_0,a_1,\\cdots ,a_{N-1}\\in E\\rbrace .$ One can also identify this space with N-copies of $E$ by mapping $\\displaystyle \\sum _{k=0}^{N-1}a_kz^k$ into $\\displaystyle \\otimes _{k=0}^{N-1}a_k$ .",
"Now if we take $A_\\Phi $ corresponding to function $\\Phi \\in L^{2}({\\mathcal {L}}(E))$ having Fourier expansion $\\Phi (e^{it})=\\displaystyle \\sum _{n=-\\infty }^{\\infty }\\Phi _{n}e^{int}$ , with $\\Phi _n\\in {\\mathcal {L}}(E)$ .",
"Then it can be easily seen that with respect to the direct decomposition given above $A_\\Phi $ has a natural representation as a matrices: Let $\\Phi ({k})$ be the $k$ -th Fourier coefficient of $\\Phi $ .",
"Then ${\\mathcal {T}}({\\mathcal {K}}_\\Theta )=\\Biggl \\lbrace A_\\Phi =\\begin{pmatrix}\\Phi (0) & \\Phi (1)&\\cdots &\\Phi (N-1)\\\\\\Phi (-1)& \\Phi (0)&\\cdots & \\Phi (N-2)\\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\\\Phi (1-N)& \\Phi (2-N)& \\cdots &\\Phi (0)\\end{pmatrix}; \\Phi ({k})\\in {\\mathcal {L}}(E)\\Biggr \\rbrace $ Suppose now that $\\Phi ,\\Psi $ belong to a commutative algebra $\\mathfrak {F}$ .",
"Since $\\Theta $ is a scalar valued inner function the double commutation assumption on $\\Phi ,\\Psi $ is satisfied.",
"As stated in section 4, we can write the symbol $\\Phi \\in \\mathfrak {F}$ of any MTTO as $\\Phi =z\\Phi _{+}+\\bar{z}\\Phi _{-}+\\Phi _{0},$ where $\\Phi _{\\pm }\\in {\\mathcal {M}}_{\\Theta _1}$ and $\\Phi _{0}\\in {\\mathcal {L}}(E)$ .",
"Let $\\Phi _+(k)$ and $\\Phi _{-}(k)$ denote the Fourier coefficients of $\\Phi _{+}$ and $\\Phi _{-}$ respectively then we have $A_{z\\Phi _{+}}=\\begin{pmatrix}0 & 0 &\\cdots & 0\\\\\\Phi _+(0) & 0&\\cdots & 0 \\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\\\Phi _+(N-2) & \\Phi _+(N-3)& \\cdots &0\\end{pmatrix}$ and $A_{\\bar{z}\\Phi _{-}^{*}}=\\begin{pmatrix}0 & \\Phi _{-}^{*}(0) &\\cdots & \\Phi _{-}^{*}(N-2) \\\\0 & 0&\\cdots & \\Phi _{-}^{*}(N-3) \\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\0 & 0 & \\cdots &0\\end{pmatrix}$ We have therefore $A_\\Phi =A_{z\\Phi _{+}+\\bar{z}\\Phi _{-}+\\Phi _{0}}=\\begin{pmatrix}\\Phi _{0} & \\Phi _{-}^{*}(0) &\\cdots & \\Phi _{-}^{*}(N-2) \\\\\\Phi _{+}(0) & \\Phi _{0}&\\cdots & \\Phi _{-}^{*}(N-3) \\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\\\Phi _{+}(N-2) & \\Phi _{+}(N-3) & \\cdots &\\Phi _{0}\\end{pmatrix}$ Similarly, $A_\\Psi =A_{z\\Psi _{+}+\\bar{z}\\Psi _{-}+\\Psi _{0}}=\\begin{pmatrix}\\Psi _{0} & \\Psi _{-}^{*}(0) &\\cdots & \\Psi _{-}^{*}(N-2) \\\\\\Psi _{+}(0) & \\Psi _{0}&\\cdots & \\Psi _{-}^{*}(N-3) \\\\\\vdots & \\vdots & \\ddots &\\vdots \\\\\\Psi _{+}(N-2) & \\Psi _{+}(N-3) & \\cdots &\\Psi _{0}\\end{pmatrix}$ Since $\\Theta (z)=z^NI_E$ , ${\\mathcal {M}}_\\Theta $ is the space of all polynomials in $z$ of degree at most $N-1$ with coefficients from ${\\mathcal {L}}(E)$ ; also, $\\Theta (0)=0$ imply that $\\Theta (z)=z\\Theta _1(z)$ , where $\\Theta _1$ is also inner.",
"Then ${\\mathcal {M}}_{\\Theta _1}$ is also the space of polynomials in $z$ with coefficients from ${\\mathcal {L}}(E)$ but of degree not greater than $N-2$ .",
"For any $x\\in E$ , $J_{z\\Phi _{+}}x=z\\Phi _{+}x$ , where $\\Phi _{+}\\in {\\mathcal {L}}(E)$ , and for any $f\\in {\\mathcal {K}}_\\Theta $ , $J_{z\\Psi _{-}}^*f=\\bar{z}\\Psi _{-}^*f$ , with $\\Psi _{-}\\in {\\mathcal {L}}(E)$ .",
"Now we have $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*f&=z\\Phi _{+}\\bar{z}\\Psi _{-}^*f=\\Phi _{+}\\Psi _{-}^*f=\\left(\\sum _{k=0}^{N-2}\\Phi _{+}(k)z^k\\right)\\left(\\sum _{k=0}^{N-2}\\Psi _{-}^*(k)\\bar{z}^k\\right)f\\\\&=\\left(\\sum _{k=0}^{N-2}\\Phi _{+}(k)\\Psi _{-}^*(k)+\\sum _{k=0}^{N-3}\\Phi _{+}(k)\\Psi _{-}^*(k+1)\\bar{z}+\\cdots \\Phi _{+}(0)\\Psi _{-}^*(N-2)\\bar{z}^{N-2}\\right)f+\\\\&\\left(\\sum _{k=0}^{N-3}\\Phi _{+}(k+1)\\Psi _{-}^*(k)z+\\sum _{k=0}^{N-4}\\Phi _{+}(k+2)\\Psi _{-}^*(k)z^2+\\cdots \\Phi _{+}(N-2)\\Psi _{-}^*(0)z^{N-2}\\right)f\\\\&=\\sum _{m=N-2}^{0}\\sum _{k=0}^{m}\\Phi _{+}(k)\\Psi _{-}^*(k+(N-2)-m)\\bar{z}^{N-2-m}f+\\\\&\\sum _{m=N-3}^{0}\\sum _{k=0}^{m}\\Phi _{+}(k-m+N-2)\\Psi _{-}^*(k)z^{N-2-m}f$ $\\begin{split}J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*f=\\sum _{m=N-2}^{0}\\sum _{k=0}^{m}\\Phi _{+}(k)\\Psi _{-}^*(k+(N-2)-m)\\bar{z}^{N-2-m}f+\\\\+\\sum _{m=N-3}^{0}\\sum _{k=0}^{m}\\Phi _{+}(k+(N-2)-m)\\Psi _{-}^*(k)z^{N-2-m}f\\end{split}$ and for any $f$ in ${\\mathcal {K}}_\\Theta $ $\\begin{split}J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})} J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*f=\\Phi _{-}^*\\Psi _{+}f=\\sum _{m=N-2}^{0}\\sum _{k=0}^{m}\\Phi _{-}^*(k)\\Psi _{+}(k+(N-2)-m)\\bar{z}^{N-2-m}f+\\\\+\\sum _{m=N-3}^{0}\\sum _{k=0}^{m}\\Phi _{-}^*(k-m+N-2)\\Psi _{+}(k)z^{N-2-m}f\\end{split}$ By Theorem REF $A_\\Phi A_\\Psi \\in {\\mathcal {T}}({\\mathcal {K}}_\\Theta )$ if and only if $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})} J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*$ .",
"Comparing corresponding coefficients we get $\\bar{z}^{N-2}:\\Phi _{+}(0)\\Psi _{-}^*(N-2)=\\Phi _{-}(0)^*\\Psi _{+}(N-2)\\Rightarrow \\Phi _{+}^*(0)\\Psi _{-}(N-2)=\\Phi _{-}^*(0)\\Psi _{+}(N-2)\\\\$ $\\bar{z}^{N-3}:\\Phi _{+}(0)\\Psi _{-}^*(N-2)+\\Phi _{+}(1)\\Psi _{-}^*(N-3)=\\Phi _{-}^*(0)\\Psi _{+}(N-2)+\\Phi _{-}^*(1)\\Psi _{+}(N-3)$ Using (REF ) in (REF ) we have $\\Phi _{+}(1)\\Psi _{-}^*(N-3)=\\Phi _{-}^*(1)\\Psi _{+}(N-3)$ .",
"In general $\\Phi _{+}(i)\\Psi _{-}^*(N-2-i)=\\Psi _{+}(N-2-i)\\Phi _{-}^*(0)\\quad \\hbox{for every}\\quad i=0,\\cdots ,N-2.$ In the same way compairing coefficients of $z,z^2,\\cdots z^{N-2}$ we obtain $\\Psi _{+}(i)\\Phi _{-}^*(N-2-i)=\\Phi _{+}(N-2-i)\\Psi _{-}^*(i)\\quad \\hbox{for every}\\quad i=0,1,\\cdots ,N-2$ If we take $A_i=\\Phi _{-}^*(i-1)$ , $A_{i-N}=\\Phi _{+}(i-(N-1))$ and $B_i=\\Psi _{-}^*(i-1)$ , $B_{i-N}=\\Psi _{+}(i-(N-1))$ for every $i=1,2,\\cdots , N-1$ then Lemma 3.1(i) of [8] imply that $A_\\Phi A_{\\Psi }$ is a block Toeplitz matrix.",
"Thus the condition $J_{z\\Phi _{+}}J_{z\\Psi _{-}}^*=J_{\\mathbf {C}_\\Gamma (z\\Phi _{-})} J_{\\mathbf {C}_\\Gamma (z\\Psi _{+})}^*$ is equivalent to the condition Lemma 3.1(i) of [8]."
],
[
"Acknowledgements",
"The author is highly grateful to Dr. Dan Timotin for his valuable suggestions and comments."
]
] | 2012.05279 |
[
[
"Off-shell Higgs Couplings in $H^*\\to ZZ\\to \\ell\\ell\\nu\\nu$"
],
[
"Abstract We explore the new physics reach for the off-shell Higgs boson measurement in the ${pp \\to H^* \\rightarrow Z(\\ell^{+}\\ell^{-})Z(\\nu\\bar{\\nu})}$ channel at the high-luminosity LHC.",
"The new physics sensitivity is parametrized in terms of the Higgs boson width, effective field theory framework, and a non-local Higgs-top coupling form factor.",
"Adopting Machine-learning techniques, we demonstrate that the combination of a large signal rate and a precise phenomenological probe for the process energy scale, due to the transverse $ZZ$ mass, leads to significant sensitivities beyond the existing results in the literature for the new physics scenarios considered."
],
[
"Introduction",
"After the Higgs boson discovery at the Large Hadron Collider (LHC) [1], [2], [3], [4], [5], the study of the Higgs properties has been one of the top priorities in searching for new physics beyond the Standard Model (BSM).",
"Indeed, the Higgs boson is a unique class in the SM particle spectrum and is most mysterious in many aspects.",
"The puzzles associated with the Higgs boson include the mass hierarchy between the unprotected electroweak (EW) scale ($v$ ) and the Planck scale ($M_{PL}$ ), the neutrino mass generation, the possible connection to dark matter, the nature of the electroweak phase transition in the early universe, to name a few.",
"Precision studies of the Higgs boson properties can be sensitive to new physics at a higher scale.",
"Parametrically, new physics at a scale $\\Lambda $ may result in the effects of the order $v^2/\\Lambda ^2$ .",
"So far, the measurements at the LHC based on the Higgs signal strength are in full agreement with the SM predictions.",
"However, these measurements mostly focus on the on-shell Higgs boson production, exploring the Higgs properties at low energy scales of the order $v$ .",
"It has been argued that if we explore the Higgs physics at a higher scale $Q$ , the sensitivity can be enhanced as $Q^2/\\Lambda ^2$ .",
"A particularly interesting option is to examine the Higgs sector across different energy scales, using the sizable off-shell Higgs boson rates at the LHC [6], [7], [8], [9], [10].",
"While the off-shell Higgs new physics sensitivity is typically derived at the LHC with the $H^* \\rightarrow ZZ \\rightarrow 4\\ell $ channel [11], [12], [13], [14], [15], [16], [17], [18], we demonstrate in this work that the extension to the channel $ZZ \\rightarrow \\ell \\ell \\nu \\nu $ can significantly contribute to the potential discoveries.",
"This channel provides two key ingredients to probe the high energy regime with enough statistics despite of the presence of two missing neutrinos in the final state.",
"First, it displays a larger event rate by a factor of six than the four charged lepton channel.",
"Second, the transverse mass for the $ZZ$ system sets the physical scale $Q^2$ and results in a precise phenomenological probe to the underlying physics.",
"In this paper, we extend the existing studies and carry out comprehensive analyses for an off-shell channel in the Higgs decay $pp \\rightarrow H^* \\rightarrow ZZ \\rightarrow \\ell ^+ \\ell ^-\\ \\nu \\bar{\\nu },$ where $\\ell =e,\\mu $ and $\\nu =\\nu _e,\\nu _\\mu ,\\nu _\\tau $ .",
"Because of the rather clean decay modes, we focus on the leading production channel of the Higgs boson via the gluon fusion.",
"First, we phenomenologically explore a theoretical scenario with additional unobserved Higgs decay channels leading to an increase in the Higgs boson width, $\\Gamma _H/\\Gamma _H^{SM}>1$ .",
"The distinctive dependence for the on-shell and off-shell cross-sections with the Higgs boson width foster the conditions for a precise measurement for this key ingredient of the Higgs sector.",
"We adopt the Machine-learning techniques in the form of Boosted Decision Tree (BDT) to enhance the signal sensitivity.",
"This analysis sets the stage for our followup explorations.",
"Second, we study the effective field theory framework, taking advantage of the characteristic energy-dependence from some of the operators.",
"Finally, we address a more general hypothesis that features a non-local momentum-dependent Higgs-top interaction [18], namely, a form factor, that generically represents the composite substructure.",
"Overall, the purpose of this paper is to highlight the complementarity across a multitude of frameworks [14], [13], [19], [15], [16], [17], [18] via the promising process at the LHC $H^*\\rightarrow Z(\\ell \\ell )Z(\\nu \\nu )$ , from models that predict invisible Higgs decays, passing by the effective field theory, and a non-local form-factor scenario.",
"Our results demonstrate significant sensitivities at the High-Luminosity LHC (HL-LHC) to the new physics scenarios considered here beyond the existing literature.",
"The rest of the paper is organized as follows.",
"In Sec.",
", we derive the Higgs width limit at HL-LHC.",
"Next, in Sec.",
", we study the new physics sensitivity within effective field theory framework.",
"In Sec.",
", we scrutinize the effects of a non-local Higgs-top form-factor.",
"Finally, we present a summary in Sec.",
"."
],
[
"Higgs Boson Width",
"The combination of on-shell and off-shell Higgs boson rates addresses one of the major shortcomings of the LHC, namely the Higgs boson width measurement [6], [7].",
"This method breaks the degeneracy present on the on-shell Higgs coupling studies $\\sigma _{i\\rightarrow H \\rightarrow f}^{\\text{on-shell}}\\propto \\frac{g_i^2(m_H)g_f^2(m_H)}{\\Gamma _H}\\,,$ where the total on-shell rate can be kept constant under the transformation $g_{i,f}(m_H)\\rightarrow \\xi g_{i,f}(m_H)$ with $\\Gamma _H\\rightarrow \\xi ^4\\Gamma _H$ .",
"The off-shell Higgs rate, due to a sub-leading dependence on the Higgs boson width $\\Gamma _H$ $\\sigma _{i\\rightarrow H^* \\rightarrow f}^{\\text{off-shell}}\\propto g_i^2(\\sqrt{\\hat{s}})g_f^2(\\sqrt{\\hat{s}})\\,,$ breaks this degeneracy, where $\\sqrt{\\hat{s}}$ is the partonic c.m.",
"energy that characterizes the scale of the off-shell Higgs.",
"In particular, if the new physics effects result in the same coupling modifiers at both kinematical regimes [14], [13], [15], [16], the relative measurement of the on-shell and off-shell signal strengths can uncover the Higgs boson width, $\\mu _{\\text{off-shell}}/\\mu _{\\text{on-shell}}=\\Gamma _H/\\Gamma _H^{SM}$ .",
"In this section, we derive a projection for the Higgs boson width measurement at the ${\\sqrt{s}=14}$ TeV high-luminosity LHC, exploring the $ZZ\\rightarrow 2\\ell 2\\nu $ final state.",
"We consider the signal channel as in Eq.",
"(REF ).",
"The signal is characterized by two same-flavor opposite sign leptons, $\\ell =e$ or $\\mu $ , which reconstruct a $Z$ boson and recoil against a large missing transverse momentum from $Z\\rightarrow \\nu \\bar{\\nu }$ .",
"The major backgrounds for this search are the Drell-Yan (DY) processes $q\\bar{q}\\rightarrow ZZ, ZW$ and gluon fusion (GF) $gg\\rightarrow ZZ$ process, see Fig.",
"REF for a sample of the Feynman diagrams.",
"While the Drell-Yan component displays the largest rate, the gluon fusion box diagrams interfere with the Higgs signal, resulting in important contributions mostly at the off-shell Higgs regime [6].",
"In our calculations, the signal and background samples are generated with MadGraph5_aMC@NLO [20], [21].",
"The Drell-Yan background is generated at the NLO with the MC@NLO algorithm [22].",
"Higher order QCD effects to the loop-induced gluon fusion component are included via a universal $K$ -factor [8], [23].",
"Spin correlation effects for the $Z$ and $W$ bosons decays are obtained in our simulations with the MadSpin package [24].",
"The renormalization and factorization scales are set by the invariant mass of the gauge boson pair $Q= m_{VV}/2$ , using the PDF set nn23nlo [25].",
"Hadronization and underlying event effects are simulated with Pythia8 [26], and detector effects are accounted for with the Delphes3 package [27].",
"We start our analysis with some basic lepton selections.",
"We require two same-flavor and opposite sign leptons with $|\\eta _\\ell |<2.5$ and $p_{T\\ell }>10$ GeV in the invariant mass window $76~\\text{GeV}<m_{\\ell \\ell }<106$ GeV.",
"To suppress the SM backgrounds, it is required large missing energy selection $E_{T}^{\\text{miss}} > 175$ GeV and a minimum transverse mass for the $ZZ$ system ${m_T^{ZZ}>250}$ GeV, defined as $m_T ^{ZZ}= \\sqrt{\\left(\\sqrt{m_Z^2 + p_{T(\\ell \\ell )}^2} + \\sqrt{m_Z^2 + (E_T^{\\mathrm {miss}})^2}\\right)^2 - \\left| \\overrightarrow{p}_{TZ} + \\overrightarrow{E}_T^{\\mathrm {miss}}\\right|^2} \\, .$ The consistency of our event simulation and analysis setup is confirmed through a cross-check with the ATLAS study in Ref. [9].",
"To further control the large Drell-Yan background, a Boosted Decision Tree (BDT) is implemented via the Toolkit for Multivariate Data Analysis with ROOT (TMVA) [28].",
"The BDT is trained to distinguish the full background events from the $s$ -channel Higgs production.",
"The variables used in the BDT are missing transverse energy, the momenta and rapidity for the leading and sub-leading leptons $(p_T^{\\ell 1}, \\eta ^{\\ell 1}, p_T^{\\ell 2},\\eta ^{\\ell 2})$ , the leading jet $(p_T^{j1},\\eta ^{j1})$ , the separation between the two charged leptons $\\Delta R_{\\ell \\ell }$ , the azimuthal angle difference between the di-lepton system and the missing transverse energy $\\Delta \\phi (\\vec{p}_{T}^{~\\ell \\ell },\\vec{E}^{\\mathrm {miss}}_T)$ , and the scalar sum of jets and lepton transverse momenta $H_T$ .",
"Finally, we also include the polar $\\theta $ and azimuthal $\\phi $ angles of the charged lepton $\\ell ^- $ in the $Z$ rest frame [29], [30].",
"We choose the coordinate system for the $Z$ rest frame following Collins and Soper (Collins-Soper frame) [31].",
"The signal and background distributions for these observables are illustrated in Fig.",
"REF .",
"We observe significant differences between the $s$ -channel signal and background in the $(\\theta ,\\phi )$ angle distributions.",
"These kinematic features arise from the different $Z$ boson polarizations for the signal and background components at the large di-boson invariant mass $m_{T}^{ZZ}$ [15], [32].",
"Whereas the $s$ -channel Higgs tends to have $Z_L$ dominance, the DY background is mostly $Z_T$ dominated.",
"Figure: BDT distribution for the ss-channel Higgs signal (red) and background (blue).We would like to illustrate the power of the implemented BDT analysis to separate the $s$ -channel Higgs from the background contributions in Fig.",
"REF .",
"The BDT discriminator is defined in the range $[-1,1]$ .",
"The events with discriminant close to $-1$ are classified as background-like and those close to 1 are signal-like.",
"The optimal BDT score selection has been performed with TMVA.",
"To estimate the effectiveness of the BDT treatment, we note that one can reach $S/\\sqrt{S+B}=5$ at an integrated luminosity of $273~\\text{fb}^{-1}$ with signal efficiency 88% and background rejection of 34%, by requiring ${\\text{BDT}_{\\text{response}}>-0.26}$ .",
"Now that we have tamed the dominant backgrounds ${q\\bar{q}\\rightarrow ZZ,ZW}$ , we move on to the new physics sensitivity study.",
"To maximize the sensitivity of the Higgs width measurement, we explore the most sensitive variable, $m_{T}^{ZZ}$ distribution, and perform a binned log-likelihood ratio analysis.",
"In Fig.",
"REF , we display the 95% CL on the Higgs width $\\Gamma _H/ \\Gamma _H^{SM}$ as a function of the $\\sqrt{s}=14$ TeV LHC luminosity.",
"To infer the relevance of the multivariate analysis, that particularly explore the observables $(E_T^{\\mathrm {miss}},\\theta ,\\phi )$ depicted in Fig.",
"REF , we display the results in two analysis scenarios: in blue we show the cut-based analysis and in red the results accounting for the BDT-based framework.",
"The significant sensitivity enhancement due to the BDT highlights the importance of accounting for the full kinematic dependence, including the $Z$ -boson spin correlation effects.",
"Whereas the Higgs width can be constrained to $\\Gamma _H/ \\Gamma _H^{SM}<1.35$ at 95% CL level following the cut-based analysis, $\\Gamma _H/ \\Gamma _H^{SM}<1.31$ in the BDT-based study assuming $\\mathcal {L}=3~\\text{ab}^{-1}$ of data.",
"Hence, the BDT limits result in an improvement of $\\mathcal {O}(5\\%)$ on the final Higgs width sensitivity.",
"These results are competitive to the HL-LHC estimates for the four charged lepton final state derived by ATLAS and CMS, where the respective limits are $\\Gamma _H/ \\Gamma _H^{SM}<\\mathcal {O}(1.3)$ and $\\mathcal {O}(1.5)$ at 68% CL [33], [34].",
"Figure: 95% CL bound on the Higgs width Γ H /Γ H SM \\Gamma _H/ \\Gamma _H^{SM} as a function of the s=14\\sqrt{s}=14 TeV LHC luminosity.",
"We display the results forthe cut-based study (blue) and BDT-based analysis (red)."
],
[
"Effective Field Theory",
"The Effective Field Theory (EFT) provides a consistent framework to parametrize beyond the SM effects in the presence of a mass gap between the SM and new physics states.",
"In this context, the new physics states can be integrated out and parametrized in terms of higher dimension operators [35].",
"In this section we parametrize the new physics effects in terms of the EFT framework [36], [37].",
"Instead of performing a global coupling fit, we will focus on a relevant subset of higher dimension operators that affect the Higgs production via gluon fusion.",
"This will shed light on the new physics sensitivity for the off-shell $pp\\rightarrow H^*\\rightarrow Z(\\ell \\ell )Z(\\nu \\nu )$ channel.",
"Our effective Lagrangian can be written as $\\mathcal {L}\\supset & c_g \\frac{\\alpha _s}{12\\pi v^2}|\\mathcal {H}|^2G_{\\mu \\nu }G^{\\mu \\nu }+c_t\\frac{y_t}{v^2}|\\mathcal {H}|^2 \\bar{Q}_L\\tilde{\\mathcal {H}}t_R+\\text{h.c.} \\,\\,,$ where $\\mathcal {H}$ is the SM Higgs doublet and $v=246$ GeV is the vacuum expectation value of the SM Higgs field.",
"The couplings are normalized in such a way for future convenience.",
"If we wish to make connection with the new physics scale $\\Lambda $ , we would have the scaling as $c_g, c_t \\sim v^2/\\Lambda ^2$ .",
"After electroweak symmetry breaking, Eq.",
"(REF ) renders into the following interaction terms with a single Higgs boson $\\mathcal {L}\\supset & \\kappa _g \\frac{\\alpha _s}{12\\pi v}H G_{\\mu \\nu }G^{\\mu \\nu }-\\kappa _t \\frac{m_t}{v}H\\left(\\bar{t}_Rt_L+\\text{h.c.}\\right) \\,\\,,$ where the coupling modifiers $\\kappa _{g,t}$ and the Wilson coefficients $c_{g,t}$ are related by $\\kappa _g=c_g$ and $\\kappa _t=1-\\text{Re}(c_t)$ .",
"We depict in Fig.",
"REF the $gg\\rightarrow ZZ$ Feynman diagrams that account for these new physics effects.",
"Whereas Eq.",
"(REF ) represents only a sub-set of high dimensional operators affecting the Higgs interactions [36], [37], we focus on it to highlight the effectiveness for the off-shell Higgs measurements to resolve a notorious degeneracy involving these terms.",
"The gluon fusion Higgs production at low energy regime can be well approximated by the Higgs Low Energy Theorem [38], [39], where the total Higgs production cross-section scales as ${\\sigma _{\\text{GF}}\\propto |\\kappa _t+\\kappa _g|^2}$ .",
"Therefore, low energy measurements, such as on-shell and non-boosted Higgs production [40], [41], [42], [43], [44], [13], [45], [15], [46], are unable to resolve the ${|\\kappa _t+\\kappa _g|=\\text{constant}}$ degeneracy.",
"While the combination between the $t\\bar{t}H$ and gluon fusion Higgs production have the potential to break this blind direction [47], we will illustrate that the Higgs production at the off-shell regime can also result into relevant contributions to resolve this degeneracy.",
"Figure: Feynman diagrams for the GF gg→ZZgg\\rightarrow ZZprocess.",
"The new physics effects from Eq.",
"() display deviations on thecoefficients κ t \\kappa _t and κ g \\kappa _g from the SM point (κ t ,κ g )=(1,0)(\\kappa _t,\\kappa _g)=(1,0).Figure: Transverse mass distributions m T ZZ m_T^{ZZ} for the DY and GF Z(ℓℓ)Z(νν)Z(\\ell \\ell )Z(\\nu \\nu ) processes.",
"The new physics effectsare parametrized by deviations from SM point (κ t ,κ g )=(1,0)(\\kappa _t,\\kappa _g)=(1,0).",
"We follow the benchmark analysis defined in Sec .Since the Higgs boson decays mostly to longitudinal gauge bosons at the high energy regime, it is enlightening to inspect the signal amplitude for the longitudinal components.",
"The amplitudes associated to each contribution presented in Fig.",
"REF can be approximated at $m_{ZZ} \\gg m_t,m_H,m_Z$ by [48], [13], [15] $\\mathcal {M}_t^{++00} &\\approx + \\frac{m_t^2}{2m_Z^2} \\log ^2\\frac{m_{ZZ}^2}{m_t^2}\\,, \\\\\\mathcal {M}_g^{++00} &\\approx - \\frac{m_{ZZ}^2}{2m_Z^2}\\,, \\\\\\mathcal {M}_c^{++00} &\\approx - \\frac{m_t^2}{2m_Z^2} \\log ^2\\frac{m_{ZZ}^2}{m_t^2} \\,.$ Two comments are in order.",
"First, both the $s$ -channel top loop $\\mathcal {M}_t$ and the continuum $\\mathcal {M}_c$ amplitudes display logarithmic dependences on $m_{ZZ}/m_t$ at the far off-shell regime.",
"In the SM scenario the ultraviolet logarithm between these two amplitudes cancel, ensuring a proper high energy behavior when calculating the full amplitude.",
"Second, it is worth noting the difference in sign between the $s$ -channel contributions $\\mathcal {M}_t$ and $\\mathcal {M}_g$ .",
"This results into a destructive interference between $\\mathcal {M}_t$ and $\\mathcal {M}_c$ , contrasting to a constructive interference between $\\mathcal {M}_g$ and $\\mathcal {M}_c$ .",
"In the following, we will explore these phenomenological effects pinning down the new physics sensitivity with a higher precision.",
"Figure: 95% CL bound on the coupling modifiers κ t \\kappa _t and κ g \\kappa _g when accounting for the off-shell Higgs measurementin the Z(ℓℓ)Z(νν)Z(\\ell \\ell )Z(\\nu \\nu ) channel.",
"We assume the 14 TeV LHC with 3ab -1 3~\\text{ab}^{-1} of data.Exploiting the larger rate for $ZZ\\rightarrow \\ell \\ell \\nu \\nu $ than that for $ZZ\\rightarrow 4\\ell $ [13], [14], [15], we explore the off-shell Higgs physics at the HL-LHC.",
"To simulate the full loop-induced effects, we implemented Eq.",
"(REF ) into FeynRules/NLOCT [49], [50] through a new fermion state, and adjusting its parameters to match the low-energy Higgs interaction $HG_{\\mu \\nu }G^{\\mu \\nu }$ [38], [39].",
"Feynman rules are exported to a Universal FeynRules Output (UFO) [51] and the Monte Carlo event generation is performed with MadGraph5aMC@NLO [20].",
"In Fig.",
"REF , we present the Drell-Yan (DY) and the gluon-fusion (GF) $m_{T}^{ZZ}$ distributions for different signal hypotheses.",
"In the bottom panel, we display the ratio between the GF beyond the SM (BSM) scenarios with respect to the GF SM.",
"In agreement with Eq.",
"(REF ), we observe a suppression for the full process when accounting for the $s$ -channel top loop contributions and an enhancement when including the new physics terms associated to $\\mathcal {M}_g$ at high energies.",
"We follow the benchmark analysis defined in Sec. .",
"After the BDT study, the resulting events are used in a binned log-likelihood analysis with the $m_T^{ZZ}$ distribution.",
"This approach explores the characteristic high energy behavior for the new physics terms highlighted in Eq.",
"(REF ) and illustrated in Fig.",
"REF .",
"We present in Fig.",
"REF the resulting 95% CL sensitivity to the $(\\kappa _t,\\kappa _g)$ new physics parameters at the high-luminosity LHC.",
"In particular, we observe that the LHC can bound the top Yukawa within $\\kappa _t\\approx [0.4,1.1]$ at 95% CL, using this single off-shell channel.",
"The observed asymmetry in the limit, in respect to the SM point, arises from the large and negative interference term between the $s$ -channel and the continuum amplitudes.",
"The upper bound on $\\kappa _t$ is complementary to the direct Yukawa measurement via $ttH$ [52] and can be further improved through a combination with the additional relevant off-shell Higgs final states.",
"The results derived in this section are competitive to the CMS HL-LHC prediction that considers the boosted Higgs production combining the $H\\rightarrow 4\\ell $ and $H\\rightarrow \\gamma \\gamma $ channels [34].",
"The CMS projection results into an upper bound on the top Yukawa of $\\kappa _t\\lesssim 1.2$ at 95% CL."
],
[
"Higgs-Top Form Factor",
"The fact that the observed Higgs boson mass is much lighter than the Planck scale implies that there is an unnatural cancellation between the bare mass and the quantum corrections.",
"Since the mass of the Higgs particle is not protected from quantum corrections, it is well-motivated to consider that it may not be fundamental, but composite in nature [53], [54], [55], [56].",
"In such a scenario, the Higgs boson is proposed as a bound state of a strongly interacting sector with a composite scale $\\Lambda $ .",
"In addition, the top quark, which is the heaviest particle in the SM, can also be composite.",
"In this case, the top Yukawa coupling will be modified by a momentum-dependent form factor at a scale $q^2$ close to or above the new physics scale $\\Lambda ^2$ .",
"It is challenging to find a general construction for such form factor without knowing the underlying dynamics.",
"Here, we will adopt a phenomenological ansatz motivated by the nucleon form factor [57].",
"It is defined as $\\Gamma (q^2/\\Lambda ^2) = \\frac{1}{(1+q^2/\\Lambda ^2)^n}\\,,$ where $q^2$ is the virtuality of the Higgs boson.",
"For $n=2$ , it is a dipole-form factor and corresponds to an exponential spacial distribution.",
"Building upon Ref.",
"[18], we study the impact of this form factor on $ g g\\rightarrow H^{*} \\rightarrow Z Z $ process now with the complementary final state $ \\ell ^+ \\ell ^- \\nu \\overline{\\nu }$ .",
"Figure: 95% CL sensitivity on the new physics scale Λ\\Lambda as a function of the LHC luminosity.",
"We assume the form factor in Eq.",
"()with n=2n=2 (dashed line) and n=3n=3 (solid line) at the 14 TeV LHC.In Fig.",
"REF , we illustrate the $m_{T}^{ZZ}$ distribution for the full gluon fusion $ g g ( \\rightarrow H^* ) \\rightarrow Z Z $ process.",
"We show the Standard Model (black) and the form factor scenario (red).",
"We assume $n=2$ or 3 and $\\Lambda =1.5$ TeV for the depicted form factor scenarios.",
"The differences between Standard Model and form factor cases become larger when the energy scales are comparable or above $\\Lambda $ due to the suppression of destructive interference between Higgs signal and continuum background.",
"Thus, we perform the same BDT procedure introduced in Sec.",
"followed by a binned log-likelihood ratio test in the $m_T^{ZZ}$ distribution to fully explore this effect.",
"In Fig.",
"REF , we display the sensitivity reach for the LHC in the Higgs-top form factor.",
"We observe that the LHC can bound these new physics effects up to $\\Lambda = 1.5$ TeV for $n = 2 $ and $\\Lambda = 2.1$ TeV for $n=3$ at 95% CL.",
"The large event rate for the $H^*\\rightarrow ZZ\\rightarrow \\ell \\ell \\nu \\nu $ signal results in a more precise probe to the ultraviolet regime than for the $H^*\\rightarrow ZZ\\rightarrow 4\\ell $ channel, where the limits on the new physics scale are $\\Lambda = 0.8$ TeV for $n = 2 $ and $\\Lambda = 1.1$ TeV for $n=3$ at 95% CL [18]."
],
[
"Summary",
"We have systematically studied the off-shell Higgs production in the $pp\\rightarrow H^{*}\\rightarrow Z(\\ell \\ell )Z(\\nu \\nu )$ channel at the high-luminosity LHC.",
"We showed that this signature is crucial to probe the Higgs couplings across different energy scales potentially shedding light on new physics at the ultraviolet regime.",
"To illustrate its physics potential, we derived the LHC sensitivity to three BSM benchmark scenarios where the new physics effects are parametrized in terms of the Higgs boson width, the effective field theory framework, and a non-local Higgs-top coupling form factor.",
"The combination of a large signal rate and a precise phenomenological probe for the process energy scale, due to the transverse $ZZ$ mass, renders strong limits for all considered BSM scenarios.",
"A summary table and comparison with the existing results in the literature are provided in Table REF .",
"Adopting Machine-learning techniques, we demonstrated in the form of BDT that the HL-LHC, with $\\mathcal {L}=3~\\text{ab}^{-1}$ of data, will display large sensitivity to the Higgs boson width, $\\Gamma _H/ \\Gamma _H^{SM}<1.31$ .",
"In addition, the characteristic high energy behavior for the new physics terms within the EFT framework results in relevant bounds on the $(\\kappa _t,\\kappa _g)$ new physics parameters, resolving the low energy degeneracy in the gluon fusion Higgs production.",
"In particular, we observe that the LHC can bound the top Yukawa within $\\kappa _t\\approx [0.4,1.1]$ at 95% CL.",
"The upper bound on $\\kappa _t$ is complementary to the direct Yukawa measurement via $ttH$ and can be further improved in conjunction with additional relevant off-shell Higgs channels.",
"Finally, when considering a more general hypothesis that features a non-local momentum-dependent Higgs-top interaction, we obtain that the HL-LHC is sensitive to new physics effects at large energies with $\\Lambda = 1.5$ TeV for $n = 2 $ and $\\Lambda = 2.1$ TeV for $n=3$ at 95% CL.",
"We conclude that, utilizing the promising $H^*\\rightarrow Z(\\ell ^+\\ell ^-)Z(\\nu \\bar{\\nu })$ channel at the HL-LHC and adopting the Machine-Learning techniques, the combination of a large signal rate and a precise phenomenological probe for the process energy scale renders improved sensitivities beyond the existing literature, to all the three BSM scenarios considered in this work.",
"This work was supported by the U.S. Department of Energy under grant No.",
"DE-FG02- 95ER40896 and by the PITT PACC.",
"DG was supported by the U.S. Department of Energy under grant number DE-SC 0016013."
]
] | 2012.05272 |
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